AHRS algorithms and calibration solutions to

facilitate new applications using low-cost MEMS

Sebastian O. H. Madgwick

Department of Mechanical Engineering

Supervised by

Andrew Harrison

A dissertation submitted to the University of Bristol in accordance with the

requirements for award of the degree of Doctor of Philosophy in the Faculty of
Engineering

March 2014
Declaration

I declare that the work in this dissertation was carried out in accordance with the
requirements of the University’s Regulations and Code of Practice for Research
Degree Programmes and that it has not been submitted for any other academic
award. Except where indicated by specific reference in the text, the work is the
candidate’s own work. Work done in collaboration with, or with the assistance of,
others, is indicated as such. Any views expressed in the dissertation are those of
the author.

SIGNED: .............................................................

DATE: ..........................

i
Declaration

ii
Acknowledgments

I would like to start by thanking my supervisor Dr. Andrew Harrison. Your

input has been invaluable as has your patience. Secondly, thank you to Dr. Ravi
Vaidyanthan, my initial supervisor before leaving to Imperial Collage London. You
offered me a position at the University of Bristol and this project would not have
been possible without your ongoing support and inspiration. Thirdly, I would like
to thank Prof. William Harwin. Much of the work in this project builds on ideas
first explored during my time at the University of Reading, I hope we can work
together again.
Over the past four years at the University, I have got to know many technicians
whose help and advice has been especially useful. I would like to thank Tony
Griffith and Clive Rendall for their advice concerning all things mechanical, Mark
Fitzgerald for your help debugging awkward electronics, and Neil Pierce for the
countless 3D printed parts.
There are many friends whom have help me out along the way. Thank you to Pete
Bennett for all the laser cutting; to Dave Hygh for our the fascinating conversions
about manufacturing sensors; and to Jenny Lannon for putting up with me for as
long as you did. I should reserve special thank you for Tom Mitchell, our many
projects together over the past 2 years have been defining to this project.
I would like to thank Fabio Varesano, creator of the “FreeIMU”. I am sad that
we will not be able to speak again but I am very happy that you found my work
useful and even more so that you shared it with so many people.
Finally, I would like to thank my parents. I would not be in the position I am in
today without you ongoing support and love.

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Acknowledgments

iv
Abstract

Microelectromechanical System (MEMS) technology is advancing rapidly.

Gyroscopes, accelerometers and magnetometers, also referred to as an Inertial
Measurement Unit (IMU), has traditionally been associated with aerospace and
industrial robotics but is now within every smart phone. The proliferation of these
low-cost devices has facilitated countless new applications with many more still
unrealised. This dissertation presents work towards this end.
A significant contribution of this work was the development of novel Attitude and
Heading Reference System (AHRS) algorithms that fuse together sensor data from
an IMU to provide an absolute measurement of orientation relative to the Earth.
The novel work presented on non-gyro IMUs demonstrated the potential practical
benefits of such kinematically redundant sensor arrays.
Low-cost MEMS can only be fully utilised if they are combined with a
calibration solution to provide precise measurements with a determined accuracy.
This dissertation presents a comprehensive calibration solution to the specific
requirements of these sensors based on extensive characterisations investigations.
The calibration solutions enable sensors costing <10 United States Dollar (USD)
to achieve a static pitch/roll accuracy of <1◦ and a static heading accuracy of <2◦ .
This performance is equivalent to commercial IMUs costing up to 3000 USD.
The AHRS algorithm and sensor calibration works were brought together in the
development of three IMU hardware platforms. To date, >500 have been sold and
the open-source associated algorithm downloaded >10,000 times. Each platform
addressed a specific design need and together these facilitated a wide range of new
applications; demonstrated by the numerous scientific publications that resulted
from collaborative projects and user projects.

v
Abstract

vi
Publications arising from this
research

Journal papers
1. Sebastian O.H. Madgwick; Andrew Harrison; Paul Sharkey; Ravi
Vaidyanathan; William Harwin. “Measuring motion with kinematically
redundant accelerometer arrays: theory, simulation and implementation”.
Mechatronics, Elsevier, vol. 23, no. 5, pp. 518-529, 2013

Conference papers
1. Sebastian O.H. Madgwick, Andrew J.L. Harrison, Ravi Vaidyanathan.
“Estimation of IMU and MARG orientation using a gradient descent
algorithm”. IEEE International Conference on Rehabilitation Robotics
(ICORR), 2011
2. James Carberry, Graham Hinchly, James Buckerfield, Edward Tayler.
Thomas Burton, Sebastian Madgwick, Ravi Vaidyanathan. “Parametric
Design of an Active Ankle Foot Orthosis with Passive Compliance”. IEEE
International Conference on Computer-Based Medical Systems
3. Thomas Mitchell, Sebastian Madgwick, Imogen Heap. “Musical Interaction
with Hand Posture and Orientation: A Toolbox of Gestural Control
Mechanisms”. Proceedings of the International Conference on New Interfaces
for Musical Expression (NIME2012), 2012
4. Sebastian Madgwick, Thomas Mitchell. “x-OSC: A VersatileWireless I/O
Device For Creative/Music Applications”. Proceedings of Sound and Music
Computing Conference (SMC), 2013
5. Thomas Mitchell, Sebastian Madgwick, Simon Rankin, Geoff Hilton, Andrew
Nix, “Making the Most of Wi-Fi: Optimisations for Wireless Robust Live
Music Performance”. Proceedings of the International Conference on New
Interfaces for Musical Expression (NIME2014)

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Publications arising from this research

viii
Table of contents

Declaration i

Acknowledgments iii

Abstract v

Publications arising from this research vii

Table of contents ix

List of tables xv

List of figures xvii

Glossary xxiii

1 Introduction 1
1.1 Low-cost MEMS sensors and AHRS applications . . . . . . . . . . . 1
1.1.1 Measuring motion with IMUs . . . . . . . . . . . . . . . . . 2
1.1.2 IMUs platforms . . . . . . . . . . . . . . . . . . . . . . . . . 4
1.2 Research objectives . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.3 Contribution to knowledge . . . . . . . . . . . . . . . . . . . . . . . 5
1.4 Thesis structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

2 Background 11
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.2 MEMS gyroscopes, accelerometers and magnetometers . . . . . . . 11
2.2.1 Accelerometers . . . . . . . . . . . . . . . . . . . . . . . . . 13
2.2.2 Gyroscopes . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.3 Magnetometers . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.2.4 Monolithic devices . . . . . . . . . . . . . . . . . . . . . . . 15
2.3 Attitude and Heading Reference System algorithms . . . . . . . . . 15
2.3.1 Complementary filter AHRS algorithms . . . . . . . . . . . . 17
2.4 Non-gyro IMUs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.5 Representation and manipulation of spatial quantities . . . . . . . . 20
2.5.1 Rotation matrices . . . . . . . . . . . . . . . . . . . . . . . . 20
2.5.1.1 Matrix transpose . . . . . . . . . . . . . . . . . . . 22

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2.5.1.2 Matrix multiplications . . . . . . . . . . . . . . . . 22

2.5.1.3 Rotation of a vector . . . . . . . . . . . . . . . . . 22
2.5.2 Quaternions . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.5.2.1 Quaternion conjugate . . . . . . . . . . . . . . . . 24
2.5.2.2 Quaternion product . . . . . . . . . . . . . . . . . 25
2.5.2.3 Quaternion rotation of a vector . . . . . . . . . . . 25
2.5.2.4 Converting a quaternion to a rotation matrix . . . 25
2.5.2.5 Converting a rotation matrix to quaternion . . . . 26
2.5.3 Euler angles . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5.3.1 Converting a rotation matrix to Euler angles . . . . 28
2.5.3.2 Converting a quaternion to Euler angles . . . . . . 29

3 Initial Attitude and Heading Reference System algorithm 31

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
3.1.1 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2 Algorithm derivation . . . . . . . . . . . . . . . . . . . . . . . . . . 32
3.2.1 Orientation from angular rate . . . . . . . . . . . . . . . . . 32
3.2.2 Orientation from a homogeneous field . . . . . . . . . . . . . 33
3.2.3 Algorithm fusion process . . . . . . . . . . . . . . . . . . . . 38
3.2.4 Magnetic distortion compensation . . . . . . . . . . . . . . . 40
3.2.5 Algorithm adjustable parameter . . . . . . . . . . . . . . . . 41
3.3 Experimental equipment . . . . . . . . . . . . . . . . . . . . . . . . 42
3.4 Experimental results . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4 Kinematically redundant non-gyro IMUs 49

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49
4.2 Solving inverse kinematics with accelerometers . . . . . . . . . . . . 50
4.2.1 Forward kinematics . . . . . . . . . . . . . . . . . . . . . . . 51
4.2.2 Inverse kinematics solutions . . . . . . . . . . . . . . . . . . 52
4.2.2.1 Combinatorial method . . . . . . . . . . . . . . . . 52
4.2.2.2 Pseudoinverse method . . . . . . . . . . . . . . . . 53
4.2.3 Angular velocity estimation . . . . . . . . . . . . . . . . . . 54
4.2.3.1 Stability and response to initial conditions . . . . . 57
4.3 Orientation estimation and angular velocity drift compensation . . . 58
4.4 2 axis pendulum kinematics . . . . . . . . . . . . . . . . . . . . . . 59
4.5 Simulation studies . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
4.6 Experimental studies . . . . . . . . . . . . . . . . . . . . . . . . . . 64
4.6.1 Hardware . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
4.6.1.1 Data acquisition equipment . . . . . . . . . . . . . 65
4.6.1.2 Accelerometer probes . . . . . . . . . . . . . . . . . 65
4.6.1.3 Test rig . . . . . . . . . . . . . . . . . . . . . . . . 66
4.6.2 Calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 67
4.6.2.1 Axis gain and bias . . . . . . . . . . . . . . . . . . 67
4.6.2.2 Alignment . . . . . . . . . . . . . . . . . . . . . . . 69
4.6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.7 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 73

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4.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5 Characterisation of low-cost MEMS sensors 77

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
5.1.1 Devices selected for characterisation . . . . . . . . . . . . . . 79
5.1.2 Scope of investigations . . . . . . . . . . . . . . . . . . . . . 80
5.2 Gyroscope bias thermal response . . . . . . . . . . . . . . . . . . . 80
5.2.1 Thermal chamber . . . . . . . . . . . . . . . . . . . . . . . . 81
5.2.2 Monotonic thermal response . . . . . . . . . . . . . . . . . . 83
5.2.3 Thermal hysteresis . . . . . . . . . . . . . . . . . . . . . . . 85
5.3 Gyroscope bias acceleration sensitivity . . . . . . . . . . . . . . . . 87
5.4 Gyroscope bias random walk . . . . . . . . . . . . . . . . . . . . . . 88
5.4.1 Allan variance . . . . . . . . . . . . . . . . . . . . . . . . . . 89
5.5 Gyroscope, accelerometer and magnetometer thermal response . . . 90
5.5.1 Three-axis thermal chamber . . . . . . . . . . . . . . . . . . 92
5.5.2 Gyroscope sensitivity and bias thermal response . . . . . . . 93
5.5.3 Accelerometer sensitivity and bias thermal response . . . . . 95
5.5.4 Magnetometer sensitivity and bias thermal response . . . . . 97
5.6 Part variation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99
5.6.1 Gyroscope part variation . . . . . . . . . . . . . . . . . . . . 100
5.6.2 Accelerometer part variation . . . . . . . . . . . . . . . . . . 101
5.6.3 Magnetometer . . . . . . . . . . . . . . . . . . . . . . . . . . 102
5.7 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6 Calibration of low-cost MEMS sensors for AHRS applications 105

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.1.1 Scope of calibration . . . . . . . . . . . . . . . . . . . . . . . 106
6.1.2 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . 106
6.2 Commercial IMUs chosen as benchmarks . . . . . . . . . . . . . . . 107
6.3 Calibration cube . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.3.1 Calibration cube alignment errors . . . . . . . . . . . . . . . 110
6.3.2 Calibration cube protractor . . . . . . . . . . . . . . . . . . 111
6.4 Gyroscope calibration . . . . . . . . . . . . . . . . . . . . . . . . . . 112
6.4.1 Gyroscope bias calibration . . . . . . . . . . . . . . . . . . . 114
6.4.2 Gyroscope calibration dataset . . . . . . . . . . . . . . . . . 114
6.4.3 Gyroscope sensitivity calibration . . . . . . . . . . . . . . . 115
6.4.4 Gyroscope alignment calibration . . . . . . . . . . . . . . . . 116
6.5 Accelerometer calibration . . . . . . . . . . . . . . . . . . . . . . . 116
6.5.1 Accelerometer calibration dataset . . . . . . . . . . . . . . . 118
6.5.2 Accelerometer sensitivity and bias calibration . . . . . . . . 119
6.5.3 Accelerometer alignment calibration . . . . . . . . . . . . . . 119
6.5.4 Calibrated accelerometer performance . . . . . . . . . . . . . 120
6.6 Magnetometer calibration . . . . . . . . . . . . . . . . . . . . . . . 123
6.6.1 Magnetometer calibration datasets . . . . . . . . . . . . . . 124
6.6.2 Calibrated magnetometer performance . . . . . . . . . . . . 126
6.6.3 Offset calibration . . . . . . . . . . . . . . . . . . . . . . . . 128
6.6.3.1 Offset calibration performance . . . . . . . . . . . . 130

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6.6.4 Ellipsoid calibration . . . . . . . . . . . . . . . . . . . . . . 131

6.6.4.1 Ellipsoid calibration performance . . . . . . . . . . 135
6.6.5 Alignment calibration . . . . . . . . . . . . . . . . . . . . . . 137
6.6.5.1 Alignment calibration performance . . . . . . . . . 139
6.6.6 Summary of magnetometer calibration solutions . . . . . . . 142
6.7 Simplifying sensor alignment arithmetic . . . . . . . . . . . . . . . . 143
6.8 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

7 Revised AHRS algorithm 145

7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
7.1.1 Algorithm overview . . . . . . . . . . . . . . . . . . . . . . . 146
7.2 Complementary filter estimator . . . . . . . . . . . . . . . . . . . . 147
7.2.1 Algorithm gain and initialisation . . . . . . . . . . . . . . . 148
7.3 Sensor conditioning . . . . . . . . . . . . . . . . . . . . . . . . . . . 149
7.3.1 Gyroscope bias compensation . . . . . . . . . . . . . . . . . 149
7.3.2 Magnetic distortion rejection . . . . . . . . . . . . . . . . . . 150
7.3.3 Linear acceleration rejection . . . . . . . . . . . . . . . . . . 150
7.4 ‘Zero-g’ and global acceleration calculation . . . . . . . . . . . . . . 151
7.5 Algorithm block diagram . . . . . . . . . . . . . . . . . . . . . . . . 151
7.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.6.1 Algorithm gain . . . . . . . . . . . . . . . . . . . . . . . . . 153
7.6.2 Gyroscope bias compensation . . . . . . . . . . . . . . . . . 154
7.6.3 Magnetic distortion rejection . . . . . . . . . . . . . . . . . . 155
7.6.4 Linear acceleration rejection . . . . . . . . . . . . . . . . . . 156
7.6.5 Algorithm outputs . . . . . . . . . . . . . . . . . . . . . . . 156
7.6.6 Omitting the magnetometer . . . . . . . . . . . . . . . . . . 157
7.7 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157

8 IMU platforms and applications 159

8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
8.2 x-IMU: A versatile data acquisition platform . . . . . . . . . . . . . 160
8.2.1 x-IMU overview . . . . . . . . . . . . . . . . . . . . . . . . . 160
8.2.1.1 x-IMU hardware . . . . . . . . . . . . . . . . . . . 161
8.2.1.2 x-IMU software . . . . . . . . . . . . . . . . . . . . 163
8.2.2 Real-time wired and wireless communication . . . . . . . . . 164
8.2.3 Autonomous data logging . . . . . . . . . . . . . . . . . . . 165
8.2.4 Auxiliary port . . . . . . . . . . . . . . . . . . . . . . . . . . 165
8.3 x-BIMU: A low-power, wireless IMU for real-time applications . . . 166
8.3.1 x-BIMU overview . . . . . . . . . . . . . . . . . . . . . . . . 167
8.3.1.1 x-BIMU hardware . . . . . . . . . . . . . . . . . . 169
8.3.1.2 x-BIMU software . . . . . . . . . . . . . . . . . . . 170
8.3.2 Modular communication interface . . . . . . . . . . . . . . . 171
8.3.3 Low-power optimisation and features . . . . . . . . . . . . . 173
8.3.3.1 802.15.4 low-power optimisations . . . . . . . . . . 173
8.3.3.2 Sleep timer and motion trigger wake up . . . . . . 174
8.3.3.3 Measured current consumption . . . . . . . . . . . 174
8.3.4 Synchronisation of multiple devices . . . . . . . . . . . . . . 175

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8.4 x-OSC: High-performance wireless IMU and I/O interface . . . . . . 176

8.4.1 x-OSC overview . . . . . . . . . . . . . . . . . . . . . . . . . 177
8.4.1.1 x-OSC hardware . . . . . . . . . . . . . . . . . . . 178
8.4.1.2 x-OSC browser interface . . . . . . . . . . . . . . . 179
8.4.2 I/O channels . . . . . . . . . . . . . . . . . . . . . . . . . . 180
8.4.3 Network modes . . . . . . . . . . . . . . . . . . . . . . . . . 181
8.4.4 Practical throughput and latency . . . . . . . . . . . . . . . 182
8.4.4.1 Round-trip Latency . . . . . . . . . . . . . . . . . 182
8.4.4.2 Throughput . . . . . . . . . . . . . . . . . . . . . . 183
8.4.4.3 Throughput-latency trade-off . . . . . . . . . . . . 185
8.5 Collaborative applications . . . . . . . . . . . . . . . . . . . . . . . 186
8.5.1 The gloves . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
8.5.1.1 Prototype gloves . . . . . . . . . . . . . . . . . . . 187
8.5.1.2 The performance gloves . . . . . . . . . . . . . . . 188
8.5.2 Gait analysis . . . . . . . . . . . . . . . . . . . . . . . . . . 190
8.6 User applications . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192
8.6.1 Estimating upper limb kinematics . . . . . . . . . . . . . . . 192
8.6.2 Digits: Wrist-worn gloveless sensor . . . . . . . . . . . . . . 193
8.6.3 MONSUN II: A small and inexpensive AUV for underwater
swarms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194
8.6.4 SMART-E: An omnidirectional AUV . . . . . . . . . . . . . 195
8.6.5 Tidal turbulence spectra from a compliant mooring . . . . . 196
8.6.6 Virtual reality based minimally invasive surgery training
system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
8.6.7 Instrumented soccer equipment to collected ankle joint
kinematics in the field . . . . . . . . . . . . . . . . . . . . . 199
8.6.8 Evaluating horse lameness using standalone ‘low-cost’ IMUs 200
8.6.9 Telepresences through “Flobi”, an anthropomorphic robot
head . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
8.6.10 Haptic communication between dance performers . . . . . . 202
8.6.11 SLAM for an autonomous car . . . . . . . . . . . . . . . . . 203
8.6.12 PERSIA: An autonomous humanoid robot to compete in
RoboCup . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
8.6.13 Low-cost structural usage monitoring of small-fleet military
helicopters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
8.6.14 Danceroom spectroscopy . . . . . . . . . . . . . . . . . . . . 207
8.6.15 Changibles: Shape changing constructive assembly . . . . . 207
8.7 Future work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209

9 Conclusions 211

References 213

A Non-gyro IMU planar simplification 231

A.1 Planar simplification . . . . . . . . . . . . . . . . . . . . . . . . . . 231
A.1.1 Angular velocity estimation . . . . . . . . . . . . . . . . . . 232
A.2 Optimal array configurations . . . . . . . . . . . . . . . . . . . . . . 232

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A.2.1 Array volume and kinematic origin . . . . . . . . . . . . . . 233

B Crystal oscillator thermal response 237

B.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
B.2 Crystal error model . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
B.3 Crystal characterisation . . . . . . . . . . . . . . . . . . . . . . . . 239
B.4 Crystal calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
B.4.1 Crystal calibration assuming typical parameters . . . . . . . 240
B.4.2 Crystal calibration based on error model . . . . . . . . . . . 241
B.4.3 Crystal calibration through third order model . . . . . . . . 241
B.5 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242

C Effect of geographic location on AHRS heading accuracy 243

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List of tables

3.1 Static and dynamic RMS error of Kalman-based algorithm and

proposed algorithm IMU and MARG implementations . . . . . . . 44

5.1 Gyroscope bias error assuming an nth order temperature calibration

model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
5.2 Temperature sensitivity of sensor parameters as specified in device
datasheets. “-” indicates that the information is not provided. . . . 91
5.3 Gyroscope sensitivity temperature coefficients . . . . . . . . . . . . 95
5.4 Accelerometer sensitivity temperature coefficients . . . . . . . . . . 96
5.5 Accelerometer bias temperature coefficients . . . . . . . . . . . . . . 97
5.6 Magnetometer sensitivity and bias temperature coefficients . . . . . 98

6.1 Specified performance of each IMU as specified in the

manufacturer’s documentation. “-” indicates that a parameter is
not specified. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
6.2 Heading error for each magnetometer calibration solution achieved
for ‘typical’ and ‘worst case’ magnetic distortion scenarios . . . . . 142

8.1 Measured current consumption of x-BIMU and XBee 802.15.4 module174

8.2 Summary of measured round-trip latency . . . . . . . . . . . . . . . 183

B.1 Crystal error model parameters as specified in datasheet [215] . . . 238

B.2 Crystal error model parameters for eight devices over temperature
range . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239

C.1 Extreme values of b and γ selected from [30] to demonstrate the

least disparity in θerror for differing geographic locations . . . . . . . 246

xv
List of tables

xvi
List of figures

1.1 Subject coordinate system, S, relative to reference coordinate

system, R . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

2.1 Coordinate system B with an arbitrary orientation relative to

coordinate system A . . . . . . . . . . . . . . . . . . . . . . . . . . 21
2.2 Any orientation of coordinate system B relative to coordinate
system A can be achieved through a rotation of angle θ around
the axis A r̂ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.3 The aerospace sequence of Euler angles describe orientation as the
result of angular rotations φ , θ and ψ around the x, y and z axes
respectively . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.4 The Euler angle sequence describes successive rotations of around
each axis to achieve the ultimate orientation of B relative to A . . . 28

3.1 Block diagram representation of the complete orientation estimation

algorithm for an IMU implementation . . . . . . . . . . . . . . . . . 40
3.2 Block diagram representation of the complete orientation estimation
algorithm for an MARG implementation including magnetic
distortion compensation . . . . . . . . . . . . . . . . . . . . . . . . 42
3.3 Typical results for measured and estimated angle θ (top) and error
(bottom) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.4 The effect of the adjustable parameter, β, on the performance of the
proposed algorithm IMU (left) and MARG (right) implementations 45
3.5 The effect of sampling rate on the performance of the proposed
algorithm IMU (left) and MARG (right) implementations . . . . . . 45

4.1 Schematic describing the i’th accelerometer relative to the ridged

body kinematic frame . . . . . . . . . . . . . . . . . . . . . . . . . . 50
4.2 Complementary filter used for angular velocity estimation . . . . . . 57
4.3 2 axis pendulum schematic . . . . . . . . . . . . . . . . . . . . . . . 60
4.4 Schematic of pendulum with triple-axis accelerometers mounted in
helix pattern. Graphics generated by simulation software . . . . . . 61
4.5 Angular acceleration error vs. sensor redundancy . . . . . . . . . . 63
4.6 Angular velocity error vs. sensor redundancy . . . . . . . . . . . . . 63
4.7 Angular accelerations of pendulum measured by accelerometer array
(top) and residuals (bottom) . . . . . . . . . . . . . . . . . . . . . . 64

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List of figures

4.8 Estimated angular velocity compensated by orientation estimation

algorithm (top) and residuals (bottom) . . . . . . . . . . . . . . . . 64
4.9 Complete accelerometer probe next to a 150 mm ruler . . . . . . . . 66
4.10 2 axis pendulum with attached accelerometer probes and annotated
potentiometer axes θ1 and θ2 . . . . . . . . . . . . . . . . . . . . . . 67
4.11 Validation the calibrated accelerometer gains and biases . . . . . . . 69
4.12 Angular accelerations of pendulum measured by accelerometer array
(top) and residuals (bottom) . . . . . . . . . . . . . . . . . . . . . . 71
4.13 Estimated angular velocity compensated by orientation estimation
algorithm (top) and residuals (bottom) . . . . . . . . . . . . . . . . 72
4.14 Relationship between error and sensor redundancy for linear
acceleration (left), angular acceleration (middle), angular velocity
(right). Whiskers indicate 1.5× interquartile range. . . . . . . . . . 72

5.1 Ideal Aerosmith 2003HP three-axis thermal chamber of achieve

precise velocities and orientations for temperatures between -65◦C
to +85◦C. Image source: [115] . . . . . . . . . . . . . . . . . . . . . 78
5.2 MPU-6050 (top IC) and HMC5883L (bottom IC) mounted on the
x-IMU PCB. The larger IC is 4×4 mm in size. . . . . . . . . . . . . 79
5.3 Thermal chamber internal components including the prototype
x-IMU being tested . . . . . . . . . . . . . . . . . . . . . . . . . . . 81
5.4 Sealed thermal chamber with thermocouple meter and separate
variable power supplies for fan and heater . . . . . . . . . . . . . . 82
5.5 Measured gyroscope temperature during experiments for each of the
eight devices tested . . . . . . . . . . . . . . . . . . . . . . . . . . . 83
5.6 Gyroscope bias thermal response of eight devices over a -20◦C to
70◦C temperature range . . . . . . . . . . . . . . . . . . . . . . . . 83
5.7 Measured gyroscope temperature during an experiment to
demonstrate thermal hysteresis . . . . . . . . . . . . . . . . . . . . 85
5.8 Thermal response of the gyroscope bias indicating thermal hysteresis 86
5.9 Gyroscope bias deviation for six different orientations relative to
gravity. Box plots show the distribution for 27 devices (whiskers
indicate 1.5× interquartile range). . . . . . . . . . . . . . . . . . . . 87
5.10 Gyroscope bias random walk over 10 hours . . . . . . . . . . . . . . 88
5.11 Allan variance plot of the gyroscope bias random walk . . . . . . . 90
5.12 Three-axis thermal chamber components including: polystyrene box
with internal thermal mass and fan, cube structure to mount four
prototype x-IMUs, and SL-1200MK2 turntable . . . . . . . . . . . . 93
5.13 Measured temperature ascent of sensors within the three-axis
thermal chamber over 2.5 hours . . . . . . . . . . . . . . . . . . . . 94
5.14 Thermal response of gyroscope sensitivity and bias from -15◦C to
40◦C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
5.15 Thermal response of accelerometer sensitivity and bias from -15◦C
to 40◦C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96
5.16 Thermal response of magnetometer sensitivity and bias from -15◦C
to 40◦C . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

xviii
 List of figures

5.17 Gyroscope sensitivity and bias normalised distribution . . . . . . . 100

5.18 Accelerometer sensitivity and bias normalised distribution . . . . . 101
5.19 Magnetometer sensitivity and bias normalised distribution . . . . . 102

6.1 Commercial IMUs chosen to benchmark performance. From left to

right: MicroStrain 3DM-GX3-25, Xsens MTw, VectorNav VN-100
and CH Robotics UM6 . . . . . . . . . . . . . . . . . . . . . . . . . 107
6.2 Calibration cube with nine prototype x-BIMUs held in alignment
with the cube axes . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
6.3 A misalignment between the calibration cube and sensor x axes
of angle γ prevents the sensor axis from being exposed to the full
magnitude of the reference signal r. . . . . . . . . . . . . . . . . . . 110
6.4 Relationship between calibration cube alignment error and
attenuation of reference signal . . . . . . . . . . . . . . . . . . . . . 111
6.5 Calibration cube mounted on the protractor (left) and close up of
cog teeth and engraved angular increments (right) . . . . . . . . . . 112
6.6 The gyroscope calibration dataset is represented by the six
orientations of the calibration cube on the turntable to expose each
axis to a +ω and −ω reference . . . . . . . . . . . . . . . . . . . . . 115
6.7 The accelerometer calibration dataset is represented by the six
orientations of the calibration cube on a level surface to expose
each axis to a +1 g and −1 g reference . . . . . . . . . . . . . . . . 118
6.8 Calibrated accelerometer performance was evaluated by using the
calibration cube protractor to rotate through precise inclinations . . 121
6.9 Error of each uncalibrated accelerometer’s measurement of
inclination for rotations around each axis. The bottom three box
plots show the distribution of the error in the top three plots
(whiskers indicate 1.5× interquartile range). . . . . . . . . . . . . . 122
6.10 Error of each calibrated accelerometer’s measurement of inclination
for rotations around each axis. The bottom three box plots show
the distribution of the error in the top three plots (whiskers indicate
1.5× interquartile range). . . . . . . . . . . . . . . . . . . . . . . . 123
6.11 Magnetometer in the ‘typical’ (left) and ‘worst case’ (right)
magnetic distortion arrangements. The magnetometer chip is
indicated by the orange circle. . . . . . . . . . . . . . . . . . . . . . 124
6.12 Example 3D plot of uncalibrated magnetometer measurements for
‘typical’ (left) and ‘worst case’ (right) magnetic distortion datasets . 125
6.13 Calibrated magnetometer performance was evaluated by using the
calibration cube protractor to rotate through precise heading angles 127
6.14 Error of each uncalibrated magnetometer’s measurement of heading
for rotations around each axis for the ‘typical’ magnetic distortion
dataset. The bottom three plots show the distribution of the error
in the top three plots. . . . . . . . . . . . . . . . . . . . . . . . . . 128
6.15 Example 3D plot of magnetometer measurements after offset
calibration for ‘typical’ (left) and ‘worst case’ (right) magnetic
distortion datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . 130

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List of figures

6.16 Heading measurement error provided by the nine magnetometers

for the ‘typical’ magnetic distortion scenario after offset calibration. 131
6.17 Heading measurement error provided by the nine magnetometers for
the ‘worst case’ magnetic distortion scenario after offset calibration. 132
6.18 Example 3D plot of magnetometer measurements after ellipsoid
calibration for ‘typical’ (left) and ‘worst case’ (right) magnetic
distortion datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6.19 Heading measurement error provided by the nine magnetometers for
the ‘typical’ magnetic distortion scenario after ellipsoid calibration. 136
6.20 Heading measurement error provided by the nine magnetometers for
the ‘worst case’ magnetic distortion scenario after ellipsoid calibration.137
6.21 Example 3D plot of magnetometer measurements after alignment
calibration for ‘typical’ (left) and ‘worst case’ (right) magnetic
distortion datasets . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
6.22 Heading measurement error provided by the nine magnetometers for
the ‘typical’ magnetic distortion scenario after alignment calibration. 141
6.23 Heading measurement error provided by the nine magnetometers
for the ‘worst case’ magnetic distortion scenario after alignment
calibration. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141

7.1 Complete AHRS block diagram . . . . . . . . . . . . . . . . . . . . 152

8.1 x-IMU board alone (left) and enclosed in its plastic housing with
1000 mAh battery (right) . . . . . . . . . . . . . . . . . . . . . . . 161
8.2 x-IMU top and bottom view with labelled key hardware components 162
8.3 x-IMU GUI displaying real-time data with 2D and 3D graphics . . . 164
8.4 x-BIMU with an unattached XBee 802.15.4 module . . . . . . . . . 167
8.5 x-BIMU in plastic housing with XBee 802.15.4 module and 320
mAh battery (left) and Velcro strap designed for human motion
applications (right) . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
8.6 x-BIMU top with labelled key hardware components . . . . . . . . 169
8.7 x-BIMU Terminal displaying real-time data with 2D and 3D graphics171
8.8 Example XBee-style communication modules compatible with
the x-BIMU, for: 802.15.4 (a) (b), Bluetooth (c), ZigBee (d),
Proprietary 900 MHz (e), Wi-Fi (f), Bluetooth 4.0 Low Energy (g)
and wired serial (h). . . . . . . . . . . . . . . . . . . . . . . . . . . 172
8.9 Four 802.15.4 receivers (Digi XSticks) in a USB hub provide
synchronised measurements from x-BIMUs . . . . . . . . . . . . . . 175
8.10 x-OSC . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
8.11 x-OSC top and bottom view with labelled key hardware components 178
8.12 x-OSC settings viewed on web browser . . . . . . . . . . . . . . . . 180
8.13 Normalised distribution of measured round-trip latency . . . . . . . 183
8.14 Throughput of 1 to 15 x-OSCs sending to a single AP on one channel184
8.15 Throughput of 1 to 15 x-OSCs sending to three Access Points (APs)
on three non-overlapping channels . . . . . . . . . . . . . . . . . . . 185
8.16 SoundGrasp gloves with wrist-mounted microphone and postures
with associated control function. Image source: [170] . . . . . . . . 187

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 List of figures

8.17 Prototype gloves system diagram (left) and being worn by musician
(right) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 188
8.18 Performance glove parts . . . . . . . . . . . . . . . . . . . . . . . . 189
8.19 Gloves being used to create and manipulate visuals projected on to
a screen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
8.20 Measured foot position during walking (3 steps) obtained using an
x-IMU . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
8.21 Mechanical model of upper limb (a), placement of x-IMUs (b) and
instrumented anthropomorphic upper limb used for evaluation (c).
Image source: [187] . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
8.22 Digits wrist-worn hardware components. Image source: [188] . . . . 194
8.23 MONSUN II AUV is 60 cm long, has a hull diameter of 10 cm and
weighs <3kg. Image source: [190] . . . . . . . . . . . . . . . . . . . 194
8.24 Key internal hardware complements including an x-IMU. Image
source: [190] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
8.25 SMART-E AUV uses three actuated thrusters for omnidirectional
movement and a band of RGB LEDs to provide visual feedback
under water. Image source: [195] . . . . . . . . . . . . . . . . . . . 196
8.26 Internal electronics and communication networks between sensors,
actuators and controllers. Image source: [195] . . . . . . . . . . . . 196
8.27 Dimensional drawings of Tidal Turbulence Moorings deployed in
Admiralty Inlet (left) and Chacao Channel (right). Image source:
[196] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
8.28 Instrumented surgical tool uses x-IMU to provide orientation data.
Image source: [196] . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
8.29 Virtual reality game controlled using the instrumented surgical tool.
Image source: [196] . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
8.30 Instrumented equipment comprised an a soccer boot and shin
guard; each with an x-IMU mounted internally. The equipment
is shown here attached the the synchronisation trigger prior to a
data acquisition session. Image source: [198] . . . . . . . . . . . . . 200
8.31 Combined motion capture and display helmet (left) and robot head
(right). Image source: [202] . . . . . . . . . . . . . . . . . . . . . . 202
8.32 Latest generation autonomous car shown fully assembled (left) and
with shell removed (right) . . . . . . . . . . . . . . . . . . . . . . . 203
8.33 Screenshot showing view from camera mounted on car and the
plotted location of the car as estimated by the RatSLAM algorithm 204
8.34 ‘Adult size’ robot (150 cm) robot walking towards football
(Photograph courtesy of Hamidreza Kasaei) . . . . . . . . . . . . . 205
8.35 The x-IMU is being used to collect data from the Army Historic
Aircraft Flight Scout AH Mk 1 to monitor structural usage . . . . . 206
8.36 An x-BIMU mounted on a dancer’s wrist allows her to control the
real-time visualisations of the molecular simulation. Photographer:
Paul Blakemore . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
8.37 Changibles analysis software and resultant physical construction.
Image source: [209] . . . . . . . . . . . . . . . . . . . . . . . . . . . 208

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List of figures

8.38 An individual block containing x-OSC for Wi-Fi control and 2 servos
with scissor lift mechanism. Image source: [209] . . . . . . . . . . . 209

A.1 Schematic describing the planar sensor array characterised by

parameters p (origin displacement) and l (sensor separation) . . . . 233

B.1 Thermal repose of eight crystal oscillators . . . . . . . . . . . . . . 239

B.2 Crystal oscillator error using the generic calibration model and
typical parameters specified in the datasheet . . . . . . . . . . . . . 240
B.3 Crystal oscillator error using the generic calibration model and
specific parameters evaluated empirically . . . . . . . . . . . . . . . 241
B.4 Crystal oscillator error using a third-order calibration model . . . . 242

C.1 Contour interval: 2 degrees, red contours positive (down); blue

negative (up); green zero line. Image source: [30] . . . . . . . . . . 244
C.2 Heading error achieved for an inclination of 0◦ (on the equator),
30◦ , 70◦ (in the UK) and 80◦ and fixed values of b = 0.5 Gauss and
e = 0.0044 Gauss . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246

xxii
Glossary

2D Two Dimensional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 124

3D Three Dimensional . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

ADC Analogue-to-Digital Converter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

ADV Acoustic Doppler Velocimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

AHRS Attitude and Heading Reference System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

ANN Artificial Neural Network . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

AP Access Point . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xx

API Application Programming Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

ASCII American Standard Code for Information Interchange . . . . . . . . . . . . . . . 168

AUV Autonomous Underwater Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 194

BFGS Broyden-Fletcher-Goldfarb-Shanno . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

CO2 Carbon Dioxide. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .78

CSV Comma-Separated Values . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

DCM Direction Cosine Matrix. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .20

DMC Digital Magnetic Compass . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 148

xxiii
Glossary

DMD Digital Micromirror Device. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .12

DMP Digital Motion Processor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174

DOF Degrees of Freedom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

DSC Digital Signal Controller . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

ENU East, North, Up . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146

FTDI Future Technology Devices International . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

GPS Global Positioning System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .3

GUI Graphical User Interface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163

I/O Input/Output . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

I2 C Inter-Integrated Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

IC Integrated Circuit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

IMU Inertial Measurement Unit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

IP Internet Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181

IR Infra-Red . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

LED Light-Emitting Diode . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162

LN2 Liquid Nitrogen . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78

lsb least-significant bit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

MAC Medium Access Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 171

xxiv
 Glossary

MARG Magnetic, Angular Rate and Gravity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

MDF Medium-Density Fibreboard. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .109

MEMS Microelectromechanical System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1

MIPS Million Instructions Per Second . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

MIS Minimally Invasive Surgery . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

OCXO Oven-Controlled Crystal Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237

OS Operating System. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .183

OSC Open Sound Control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177

PCB Printed Circuit Board. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79

ppm parts per million . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

PWM Pulse-Width Modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165

RAM Random-Access Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161

RF Radio Frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

RGB Red, Green, Blue . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

RMS Root Mean Squared . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

RPM Rotations Per Minute . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

RTC Real-Time Clock . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160

SD Secure Digital. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .79

SLAM Simultaneous Localisation And Mapping . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203

xxv
Glossary

SPP Serial Port Profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 164

SRAM Static Random-Access Memory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

TCP Transmission Control Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

TCXO Temperature Compensated Crystal Oscillator . . . . . . . . . . . . . . . . . . . . . . . 237

TTM Tidal Turbulence Moorings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196

UART Universal Asynchronous Receiver/Transmitter . . . . . . . . . . . . . . . . . . . . . . 181

UAV Unmanned Aerial Vehicle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

UDP User Datagram Protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172

UK United Kingdom . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

USB Universal Serial Bus. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .65

USD United States Dollar . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

VR Virtual Reality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197

XLP eXtreme Low Power . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170

xxvi
Chapter 1

Introduction

1.1 Low-cost MEMS sensors and AHRS

applications

Gyroscopes and accelerometers have traditionally been associated with the flight
control systems of an aircraft or the precision instruments within industrial
robotics. Today, they are ubiquitous; they are in our smart phones, cameras,
hard drives and toys. Advances in Microelectromechanical System (MEMS)
technology have facilitated the miniaturisation of these sensors with ever increasing
performance. In the 90s, performance of inertial MEMS was increasing by a factor
of five every year [1]. Today, performance can be achieved equally that of tactical
grade to meet the demands of the military and aerospace industry [2].

An Inertial Measurement Unit (IMU) consists of both a gyroscope and an

accelerometer. The gyroscope provides a measurement of angular velocity and the
accelerometer provides a measurement of acceleration combined with the direction
of gravity. While these quantities are useful in themselves, it is their combined use
that is special. Using a sensor fusion algorithm, the sensors can be combined to

1
Chapter 1. Introduction

provide an absolute measurement of orientation relative to the Earth’s surface. If

a magnetometer is used in conjunction with the IMU, the absolute measurement
of orientation may also be determined relative to magnetic north to create an
Attitude and Heading Reference System (AHRS).

The proliferations of theses sensors has facilitated countless new application.

Smart phones and tablets typically incorporate these sensors alongside cameras
and touch screens and provide an excellent example of the new applications that
have been achieved, including: navigation, image stabilisation, gestural control,
augmented reality and personal health/biometrics.

This project set out to demonstrate how new applications can be realised with
greater precision than previously using modern low-cost MEMS gyroscopes,
accelerometers and magnetometers. This was achieved through the development
of publicly available software and hardware platforms that have facilitated a wide
range of commercial and academic research projects exploring a diverse range of
applications.

1.1.1 Measuring motion with IMUs

The purpose of an IMU is to quantify a subject coordinate frame relative to a

reference point. This is represented in Figure 1.1 where the coordinate system S,
defined by the axes xs , ys and zs , is the subject; and R, defined by the axes xr ,
yr and zr , is the reference coordinate system. The origin of S is distance d from
R and at an arbitrary orientation. For example, the reference coordinate system
may be that of the Earth and the subject may be the position of a robot or a point
on a person’s body.

There are several technologies that may be used for the measurement of motion.
Optical systems typically use Infra-Red (IR) light in conjunction with cameras

2
 Chapter 1. Introduction

S zs

R
zr ys

d xs
yr

xr

Figure 1.1: Subject coordinate system, S, relative to reference coordinate system,

R

to determine a Three Dimensional (3D) position; notable examples include the

Vicon system, Microsoft Kinect and Leap Motion. Radio Frequency (RF) systems
such as the Xsens MVN MotionGrid or Global Positioning System (GPS) rely on
transmitters and receivers fixed to the subject and reference. Magnetic systems
operate in a similar way to RF but use active magnetic fields; a notable example
is the Razer Hydra games controller. IMU technology is exceptional in that it
does not require any components external to the subject; the measurement unit
is completely self-contained. Unlike other measurement systems, this means that
IMU systems are not constrained to a specific operating environment and are not
vulnerable to external interferences.

Whereas other measurement systems determine the 3D position and orientation

of the subject, inertial systems are only able to determine the orientation directly.
A gyroscope is used to measure the angular velocity around the subject axes;
these measurements may be numerically integrated to determine the change in
orientation from a known starting point. An accelerometer is used to measure
the accelerations in the directions of the subject axes including the direction of
gravity. As the direction of gravity is known relative to the reference axes, this
may also be used to determine an orientation. Similarly, a magnetometer may be
used to determine orientation relative to the Earth’s magnetic field. Each one of

3
Chapter 1. Introduction

these sensors provides an indicate of direction but only through their combination
in an AHRS algorithm can an absolute measurement of orientation be determined.

An AHRS algorithm provides a best estimate of orientation given the measured

angular rates and observed directions of the Earth’s gravitational and magnetic
fields relative to the subject. An absolute measurement of orientation is of
immediate use to many applications, such as a flight control system stabilising the
attitude of a Unmanned Aerial Vehicle (UAV). IMUs may be used to determine
relative positions of joints such as a robotic manipulator or the human body.
By fusing data from all three types of sensor, an AHRS is able to determine
the direction of gravity independent of other linear accelerations. This allows an
AHRS to subtract gravity from accelerometer measurement to yield a measurement
of acceleration in the Earth coordinate system. Integrating this would yield a
measurement of velocity; integrating the velocity would yield the displacement of
the IMU. The ability to measure displacement is of benefit to many applications.
In practice, this can only be successful if an exponentially increasing error inherent
to numerical integration can be compensated for. For example, pedestrian
dead-reckoning applications achieve this through zero velocity detection [3].

1.1.2 IMUs platforms

Until recently, commercial IMUs were limited to expensive instruments for use
in robotics and aerospace systems, and human motion capture systems. It is
only with the advent of smart phones and the proliferation of low-cost MEMS
sensors that these devices have been more widely available. Many low-cost
IMU platforms are development boards featuring sensors and a processor, for
example, the systems offered by Sparkfun, Polulu and Seeedstudio. Although
such platforms are cheap, they are often of little use as they lack the essential
sensor calibration and AHRS processing. Furthermore, these platforms do not

4
 Chapter 1. Introduction

have the necessary infrastructure for application; for example, wired or wireless
communication and interfacing software. Platforms that do offer a complete
solution typically target human motion capture applications. There is an absence
of a generic IMU platforms incorporating calibrated sensors and AHRS processing
alongside a versatile communication interface.

1.2 Research objectives

The principal goal of this project is to demonstrate how new applications

can be realised using modern low-cost MEMS gyroscopes, accelerometers and
magnetometers. This can be broken down into the following objectives.

• Development of a AHRS algorithm applicable to modern low-cost MEMS

devices

• Calibration of gyroscopes, accelerometers and magnetometers for use in

AHRS applications

• Design and manufacture of IMU platforms that combine the calibration and
AHRS algorithm solutions

• Demonstration of a broad range of applications using the IMU platforms

through collaborations

1.3 Contribution to knowledge

The contributions made by this project are summarised as the following.

5
Chapter 1. Introduction

AHRS algorithms

The initial AHRS algorithm presented in Chapter 3 was obtained through a

novel derivation and offers advantages compared to existing algorithms [4] in
computational efficiency and robust magnetic performance. The algorithm was
published as a conference paper in 2011 [5] with the source code made available on
Google Code1 . The paper and its associated technical report have since totalled
over 100 citations [6] and the code downloaded over 10,000 times [7].

The revised AHRS algorithm presented in Chapter 7 was developed after the work
on sensor characterisation and calibration presented in Chapters 5 and 6, and
simultaneously with the IMU platforms and applications presented in Chapter 8.
The algorithm offers several improvements over the initial algorithm and other
solutions available within the subject literature.

Kinematically redundant non-gyro IMUs

Non-gyro IMUs comprised of only accelerometers may incorporate a redundant

number of sensors with many potential advantages in application. However,
few empirical studies exist for redundant accelerometer arrays; those that do
are limited to only a small margin of redundancy. The work presented in
Chapter 4 investigates the relationship between accuracy and sensor redundancy
is investigated for arrays of up to 100 triple axis (300 accelerometer axes)
accelerometers in simulation and 10 equivalent sensors (30 accelerometer axes) in
a laboratory test rig. An article covering this work was published in the journal,
Mechatronics, in 2013 [8].

1
http://code.google.com/p/imumargalgorithm30042010sohm/

6
 Chapter 1. Introduction

A complete calibration solution for modern low-cost IMUs

The calibration solutions presented in Chapter 6 build on the characterisation

studies presented in Chapter 5 to address the specific requirements of
modern low-cost gyroscopes, accelerometers and magnetometers. The solutions
demonstrate how sensors costing <10 United States Dollars (USD) can achieve an
accuracy equivalent to commercial IMUs priced up to 3000 USD. This opens the
possibilities for new applications.

Facilitation of novel work through IMU platforms

Three IMU platforms, presented in Chapter 8, were created to facilitate new

applications; the x-IMU, x-BIMU and x-OSC. Each device fulfils a different design
specification. x-OSC addresses shortcomings of existing commercial platforms
and was originally presented in a 2013 conference paper [9]; a forthcoming 2014
conference paper focusses on its specific performance abilities [10]. The three IMU
platforms facilitated a number of collaborative investigations including a novel
gestural control system for music which was presented in a 2012 conference paper
[11]; and a gait tracking algorithm which was demonstrated as an evaluation tool
for an active ankle orthosis in a 2011 conference paper [12]. Chapter 8 also presents
selected example user applications that have utilised the platforms developed. The
research projects were selected to demonstrate the impact of the platforms in
peer-reviewed publications.

1.4 Thesis structure

This section details how this dissertation is organised along with a synopsis of each
chapter.

7
Chapter 1. Introduction

Chapter 2: Background

This chapter summaries the relevant subject backgrounds; including a review of

the subject literature. The chapter includes: a description of modern MEMS
sensor technology; a literature review of AHRS algorithms with a particular focus
on complementary filter solutions; a literature review of non-gyro IMU works; and
an introduction to the representation and manipulation of spatial quantities.

Chapter 3: Initial Attitude and Heading Reference System algorithm

Some of the first work conducted in this research project was the development of
an AHRS algorithm. Although the algorithm would be significantly revised over
the proceeding years (see Chapter 7), this initial algorithm offered a novel solution
that would become widely adopted within the academic and commercial projects.
The algorithm was intended to provide an alternative solution to computationally
expensive Kalman-based solutions and explore an alternative approach to the
complementary filter solutions available. The algorithm is benchmarked against
that of a Kalman-based algorithm and demonstrated equivalent performance.

Chapter 4: Kinematically redundant non-gyro IMUs

This chapter presents two schemes to measure the linear and angular kinematics of
a rigid body using a kinematically redundant array of triple-axis accelerometers.
A novel angular velocity estimation algorithm is proposed and evaluated that
can compensate for angular velocity errors using measurements of the direction
of gravity. Analysis and discussion of optimal sensor array characteristics are
provided. A damped 2 axis pendulum was used to excite all 6 Degrees of Freedom
(DOF) of the a suspended accelerometer array through determined complex motion
and is the basis of both simulation and experimental studies. The relationship

8
 Chapter 1. Introduction

between accuracy and sensor redundancy is investigated for arrays of up to 100

triple axis (300 accelerometer axes) accelerometers in simulation and 10 equivalent
sensors (30 accelerometer axes) in the laboratory test rig. The work also reports
on the sensor calibration techniques and hardware implementation.

Chapter 5: Characterisation of low-cost MEMS sensors

This chapter presents the methods and findings of the sensor characterisation
studies that form the basis of the calibration solutions presented in Chapter 6.
Several different gyroscopes, accelerometers and magnetometers from a range
of manufacturers were used throughput this project. This chapter presents the
characterisation of the two devices that would ultimately be incorporated into the
IMU platforms described in Chapter 8.

Chapter 6: Calibration of low-cost MEMS sensors for AHRS

applications

This chapter builds on the findings in Chapter 5 to develop calibration solutions

applicable to low-cost MEMS gyroscopes, accelerometers and magnetometers. The
calibrated performance was benchmarked against the performance of commercial
IMUs to demonstrate equivalent accuracy.

Chapter 7: Revised AHRS algorithm

The original AHRS algorithm presented in Chapter 3 represents some of the first
work of this project. The subsequent characterisation studies presented in Chapter
5 and calibration solutions in Chapter 6 demonstrated how modern low-cost
MEMS sensors can achieve equivalent performance of such commercial IMUs. The
revised AHRS algorithm presented in this chapter was designed to operate with

9
Chapter 1. Introduction

these sensors based on the findings of Chapters 5 and 6. The revised algorithm
addresses the shortcomings of contemporary complementary filter solutions while
introducing several new features developed in conjunction with the IMU platforms
and applications presented in Chapter 8.

Chapter 8: IMU platforms and applications

The chapter describes three IMU platforms; the x-IMU, the x-BIMU and the
x-OSC. Each platform fulfils a different design specification to facilitate a wide
range of applications. Two collaborative application utilising these platforms are
presented; a novel gestural music control system and a gait tracking algorithm.
This chapter also presents selected example user applications that have utilised the
platforms. The research projects chosen to appear in this chapter were selected to
demonstrate the diversity of user applications and the impact of the platforms in
peer-review publications.

Chapter 9: Conclusions

This final chapter summarises the achievements of the project, with reference to
the original research objectives. A brief description of future work is also provided.

10
Chapter 2

Background

2.1 Introduction

This chapter provides an overview of the background subjects central to the

project including a review of the subject literature. The following sections describe
modern MEMS sensors and AHRS algorithms that are used in conjunctions with
these sensors. A survey of non-gyro IMU work is also presented. The chapter
concludes with an overview of the mathematics associated with the representation
and manipulation of spatial quantities which will be fundamental throughout this
dissertation.

2.2 MEMS gyroscopes, accelerometers and

magnetometers

MEMS is a term used to describe a wide range of miniaturised devices combining

mechanical structures with electrical systems on a micrometer scale (1 μm
= 10−6 m) scale. The devices are typically fabricated from silicon using

11
Chapter 2. Background

photolithography techniques derived from the Integrated Circuit (IC) industry

to create moving structures such as levers, masses and actuators on the substrate
alongside conventional digital and analogue ICs. Arguably the first MEMS sensors
were air pressure sensors developed in 1970s that would sense a difference in
pressure through the deflection of a diaphragm [13]. Perhaps the world’s most
complex [13] MEMS structure is the Digital Micromirror Device (DMD) created
by Texas Instruments which incorporates up to two million actuated 16 μm2
aluminium micro-mirrors [14]. The mirrors are fabricated over an array of Static
Random-Access Memory (SRAM) cells so that each pixel may be addressed as a
bit in digital memory. MEMS technology has facilitated the miniaturisation of a
wide range of technologies but it is inertial sensors, gyroscopes and accelerometers,
that have seen the greatest surge in development.

The inertial MEMS market has traditionally been dominated by the automotive
sector with almost 80% of applications being related to automotive safety features
in 2005 [15]. Today it is consumer electronics that drives the market [2] with diverse
applications ranging from hard-disk protection to complete autopilot systems
within flying toys. The Nintendo Wii games console, released in 2006, is a notable
example which attracted broad public interest in inertial MEMS and provided
researchers with a platform to innovate new applications [16, 17, 18]. The demands
of consumer electronics have focussed development on low-cost and low-power
devices. The average price of an accelerometer used in smart phones was 1.7
USD in 2006 and is predicted to be 0.2 USD in 2017 [2]. In 2012, the industries
lowest power accelerometer could provide measurements at 100 Hz while drawing
only 2 μA [19]. Smart phones and tablets typically incorporate these sensors
alongside cameras and touch screens and provide an excellent example of the new
applications that may be achieved, including: navigation, imagine stabilisation,
gestural control, augmented reality and personal health/biometrics.

12
 Chapter 2. Background

2.2.1 Accelerometers

An accelerometer provides a measurement of acceleration; for example, in units

of g or ms−2 . Accelerometers are also sensitive to gravity. The principle of
operation is that a deflection of a mass is measured through a change in an electrical
characteristic. When accelerating, the inertia of the mass exerts a force according
to Newton’s second law of motion. The required mechanical structure within the
MEMS is recognisable as the mass and spring components with fingers on the
silicon substrate that form a variable capacitor array.

Accelerometers can operate open-loop or closed-loop. Open-loop sensor

performance is limited to the characteristics of the mechanical structure but require
simpler interface circuitry and is inherently stable for a bandwidth below their
resonance [1]. Closed-loop sensors use feedback to maintain the proof mass in
a null position and can achieve an extended dynamic range, increased linearity,
flatten frequency response, and improved cross-axis rejection [20].

A three-axis accelerometer requires that the proof mass is able to translate

vertically. This requires a more complex mechanical structure and as a result,
the z performance on triple-axis accelerometer is often worse than that of x
and y. Six-axis accelerometers have been developed and are able to measure
rotational accelerations [15]. However, such devices are not yet available to
commercial markets. The demands of the consumer electronics industry has
driven developments towards low-power and low-cost. However, there have been
advances in recent years to develop high-performance tactical and navigational
grade MEMS accelerometers [21, 22, 23, 24] which are predicted to replace
conventional technologies in the coming years [2].

13
Chapter 2. Background

2.2.2 Gyroscopes

A gyroscope provides a measurement of angular velocity; for example, in units of

degrees per second or radians per second. The most common MEMS gyroscopes
are Coriolis vibratory gyroscopes as they offer the best performance to price
ratio [2]. These devices work by vibrating a mass, when rotated, the Coriolis
effect that causes a secondary vibration in a direction orthogonal to the original
vibration. This active vibration means that gyroscopes will require more power
than accelerometers. Development of alternative technologies attempt to address
this issue [25]. Gyroscopes are particularly sensitive to temperature variations and
this has been the focus of several works [26, 27, 28]. Advances in recent years have
shown that Tactical grade performance achievable with MEMS [26] though such
devices typically cost 2000 USD to 4000 USD per axis [2].

2.2.3 Magnetometers

Triple-axis magnetometers provide a measurement of angular velocity; for

example, in units of Gauss or μT (micro Tesla). They typically use Anisotropic
Magnetoresistive (AMR) technology [29]. Magnetic-based heading measurement
imposes two fundamental limitations: (1) measurements can be corrupted by local
magnetic interfaces; and (2) variations in the Earth’s magnetic field [30] will
mean measurements of magnetic north may be significantly different from geodetic
north. A gyrocompass provides a measurement of geodetic north based on the
rotation of the Earth and so solves this problem. However, this requires extremely
high-performance and is typically only achieved with ring laser gyroscopes or fibre
optic gyroscopes. Achieving this with MEMS is a hot topic and significant advances
have been made in recent years [31, 32, 33, 34]

14
 Chapter 2. Background

2.2.4 Monolithic devices

Some of the most recent developments in MEMS sensors has seen the appearance
of monolithic devices that incorporate multiple sensors into a single package. What
are commonly refereed as ‘9DOF’ sensors incorporated a gyroscope, accelerometer
and magnetometer. Development is primarily driven by the low-cost demands of
consumer electronics; the budget for 9DOF sensors is less than 2.5 dollars in 2012
[2]. Several companies are also developing ‘10DOF’ devices that also incorporate a
barometer [2] though none have reached the market yet. The latest generation of
monolithic devices incorporate on-board processes to implement data processing
tasks such a calibration and AHRS data fusion.

2.3 Attitude and Heading Reference System

algorithms

An AHRS algorithm is a sensor fusion algorithm that combines the data from
multiple sensors to yield a single measurement of orientation. AHRSs may
incorporate a variety of technologies, for example GPS and air speed of a UAV,
but in the context of this research project an AHRS is considered to be comprised
of a gyroscope, accelerometer and magnetometer. The accurate measurement
of orientation plays a critical role in a range of fields including: aerospace [35],
robotics [36, 37], navigation [38, 39], human motion analysis [40, 41] and machine
interaction [42].

The sensor fusion algorithm operates on the principle that each of the sensors is
able to provide some information about the orientation but not able to provide a
complete picture. The gyroscopes is the most important sensor within an AHRS.
It provides a measurement of angular velocities which may be integrated over

15
Chapter 2. Background

time to yield a measurement of orientation relative to the initial orientation. If

the initial orientation of the IMU is known and the gyroscope free from errors
then this would be sufficient for a complete AHRS and there would be no need
to incorporate other types of sensors. However, in practice, the integration of
gyroscope measurements errors will mean that the measurement orientation will
drift over time and is necessary to compensate for this drift using an alternative
measurement of orientation. An accelerometer is able to provide a measurement
of gravity relative the IMU though the measurement will be corrupted by motion.
Similarly, a magnetometer provides a measurement of magnetic North but may
become unreliable in the presence of magnetic distortions local to the environment.
The goal of the AHRS algorithm is therefore to combine the information from all
three sensors to yield measurement of orientation relative to gravity and magnetic
north.

Most AHRS algorithms are either based on a Kalman filter or complementary

filter though there are exceptions that have demonstrated fuzzy logic [37], Artificial
Neural Network (ANN) [43, 44] and non-linear observer solutions [45]. The Kalman
filter [46, 47] has become the accepted basis for the majority of AHRS algorithms
and commercial IMUs; Xsens [48], MicroStrain [49], Intersense [50], VectorNav
[51], CH Robotics [52] and YEI [53] all produce systems founded on its use. The
widespread use of Kalman-based solutions is a testament to their effectiveness,
however, they have a number of disadvantages. They can be complicated to
implement which is reflected by the numerous solutions available in the subject
literature; see [4] for a survey. The non-linearity and large state matrices inherent
to the application present a challenge in tuning the algorithm and impose a
computational load that may be inappropriate for low-power or high-bandwidth
applications. These shortcomings have been cited as the motivation for the
development of complementary filter alternatives [54, 55, 56].

16
 Chapter 2. Background

2.3.1 Complementary filter AHRS algorithms

A complementary filter fuses a low bandwidth measurement of orientation derived

from the accelerometer and magnetometer with the high bandwidth measurement
of angular rate provide by the gyroscope based on a first-order kinematic model.
Mahony et al. provide a review of classical complementary filtering in the appendix
of [55]. The classical solution limited initial implementations to one dimensional
states [57, 58]. However, subsequent works have demonstrated 3D solutions
utilising Euler angles [35, 56], rotation matrices [55] and quaternions [54, 55].
Quaternion implementations are recognised as having an advantage in simplicity
[59, 54] while avoiding the problematic singularity of an Euler angles representation
[60].

Although the algorithm may be derived as the sum of two filters [57, 61], most
authors express the complementary filter as a feedback loop [54, 35, 58, 55, 62,
56]. In this form, different works may be characterised by their computation of
the feedback error. For example: as the difference between angular quantities
[58, 35, 56], as the cross-product of spatial quantities [62], through rotation matrix
operations [55], or through Gauss-Newton iterations [54]. The feedback gain is
typically quantified as the crossover frequency [57, 54, 55, 35, 56]. This may
provide a convenient means of tuning the gain if the frequency characteristics of
an application are known, as in [57, 61, 56]. Most implementations will use a fixed
gain but a variable gain may enable greater accuracy if unreliable sensor data
can be detected [56]. The incorporation of integral feedback provides a means to
compensate for a gyroscope bias [62, 35]. However, this may be an undesirable
solution as it detracts from the simplicity of the complementary filter and risks
worse performance though poorly tuned second-order dynamics.

Complementary filters that use only a gyroscope and accelerometer [35, 61, 62] are
unable to provide an absolute measurement of heading. Some implementations

17
Chapter 2. Background

incorporate open-loop magnetometer measurements [56] though this fails to take

advantage of sensor fusion. Solutions that incorporate the magnetometer within
within the feedback loop [63, 64] are able to provide an absolute measurement
of heading derived from both the gyroscope and magnetometer. However,
these solutions require a fixed inclination of the magnetic field in the operating
environment to be predefined, this is a significant limitation. Although most
commercial system use Kalman-based algorithms, some modern products also offer
complementary filter modes to achieve greater update speeds [53].

2.4 Non-gyro IMUs

Conventional IMUs consist of an accelerometer to measure accelerations and

a gyroscope to measure angular velocities. However, an accelerometer is also
sensitive to centripetal, Coriolis and tangential accelerations associated with
rotational motion. It is therefore possible for an array of accelerometers be used
in conjunction with an inverse kinematic solution to directly measure rotational
kinematics and potentially eliminate the need for a gyroscope. There are a number
of motivations for the exclusion of gyroscopes. A sensor array consisting only
of accelerometers has the potential to incorporate redundancy which may be
exploited for robustness, increased accuracy and fault detection. Redundancy
of inertial sensor arrays is already common in safety critical applications such
as avionics [65]. The ability to measure angular states using an accelerometer
array is also of benefit to existing applications that already feature such an array
since extra or redundant measurements may be obtained at no extra cost. For
example, automotive applications [66, 67] incorporate more than 15 axes of inertial
measurement [68].

The use of linear accelerometers to measure rotational kinematics has been a

18
 Chapter 2. Background

subject of research since the 1960s [69, 70]. For planar motion, the inverse
kinematic solution is trivial [71] and has few practical applications; for example,
the motion of a human knee [72]. Most practical applications require motion in
all six DOF to be accounted for. The so-called ‘cube configuration’ uses six single
axis accelerometers each aligned along the diagonal of a different face of a cube.
This configuration has been the focus of several studies [73, 74, 75], including
an implementation in hardware [76]. Analytical [77] and empirical [78] studies
have shown the six sensor configuration to be intrinsically unstable and limited in
application of over short time intervals. Arrays of nine accelerometers provide an
alternative, stable solution [79, 80, 81].

The arrays of six and nine accelerometer axis use specific configurations of multiple
single-axis accelerometers. Modern MEMS accelerometers are widely available
in triple-axis packages containing three mutually orthogonal accelerometer axes.
The theoretical minimum number of triple-axis accelerometers required to measure
motion in six DOF is four (incorporating a total of 12 linear accelerometers axis).
Schemes using triple-axis accelerometers [67, 82, 83, 84, 85] do not require specific
geometric configurations, only that the position of a sensor within an array is
known. As each sensor is able to provide a vector measurement of acceleration in
three dimensions, the physical orientation of the sensor is irrelevant provided that
the orientation is known. The theoretical irrelevance of each sensor’s position and
orientation within the sensor array gives triple-axis accelerometer arrays a clear
practical advantage over single-axis accelerometer arrays.

Accelerometer arrays designed to incorporate a number of redundant sensors

exploit the invariance of an inverse kinematic solution to the position of any
individual sensor and such schemes typically use a matrix pseudoinverse [86, 87]
to resolve the inverse kinematics. Few studies have addressed the relationship
between the performance and redundancy of the system; most studies are
numerical and limited to sensor arrays of no more than 24 sensitive axes [88]

19
Chapter 2. Background

[87].

Due to past limitations of accelerometer technology, research into accelerometer

sensors arrays has been largely limited to numerical and theoretical work. It has
only been with the relatively recent developments in MEMS that empirical studies
have became feasible [76, 67, 89, 83]. Few empirical studies exist for redundant
accelerometer arrays and such studies have been limited to a number of sensors
that yield a small margin of redundancy [90] [91]. Work presented in Chapter
4 investigates highly redundant systems through both simulation and empirical
studies.

2.5 Representation and manipulation of spatial

quantities

The representation of orientation and the manipulation of spatial quantities are

fundamental to this dissertation. This section provides a brief introduction
to the notation and operations of rotation matrix, quaternion and Euler angle
representations. An implementation of the methods presented in this section is
available as an open-source1 MATLAB library. The convention used throughout
this dissertation is that matrices and vectors are expressed in bold font with
upper-case indicating a matrix and the hat accent denoting a normalised vector of
unit magnitude. For example, M and v̂ would represent a matrix and normalised
vector respectively.

1
https://github.com/xioTechnologies/Quaternion-MATLAB-Library

20
 Chapter 2. Background

2.5.1 Rotation matrices

A rotation matrix, sometimes called a Direction Cosine Matrix (DCM), is a 3×3

matrix that describes the orientation of a subject in Euclidean space relative to a
reference coordinate system. The three columns of the matrix define the directions
of the mutually orthogonal x, y and z axes of the subject coordinate system as unit
vectors within the reference coordinate system. For example, Figure 2.1 shows an
arbitrary orientation of a subject coordinate system, B, relative to the reference
coordinate system, A. The principle axes of the coordinate systems A and B are
defined by the mutually orthogonal units vectors x̂A , ŷA and ẑA ; and x̂B , ŷB and
ẑB respectively.

ẑB ẑA

ŷB
ŷA

x̂B
x̂A

Figure 2.1: Coordinate system B with an arbitrary orientation relative to

coordinate system A

The rotation matrix describing the orientation of B relative to A is constructed

from the available vector quantities as shown in Equation 2.1. A notation system
adopted from Craig [92] is used to denote the coordinate system that a quantity
is in reference to as a leading superscript, and the coordinate system the quantity
describes with a leading subscript.
 
r11 r12 r13   
 
A
R =  
 21 22 23  = x̂B (2.1)
A A A
B r r r ŷB ẑB
 
r31 r32 r33

If A and B are aligned then the rotation matrix would be an identity matrix as

21
Chapter 2. Background

described by Equation 2.2

 
1 0 0
 
A
R = 0 1 0 , for A = B (2.2)
B  
 
0 0 1

A
In application, BR may be a measurement of orientation provided by an AHRS
where A is a coordinate system fixed to the Earth and B represents the axes of
the IMU.

2.5.1.1 Matrix transpose

A matrix transpose operation switches the reference coordinate system. For

example, the transpose of a rotation matrix describing coordinate system B relative
to A would describe A relative to B as shown in 2.3. As a rotation matrix is by
definition an orthogonal matrix, the transpose is equal to the inverse.

B −1 T
AR =A
BR =A
BR (2.3)

2.5.1.2 Matrix multiplications

Matrix multiplications can be used to compound a chain of relative orientations

into a single quantity. For example, the orientation of frame C relative to frame A
can be computed if the separate rotation matrices describing C relative to B and
B relative to A are known as demonstrated in Equation 2.4.

A
CR =A B
B RC R (2.4)

The rotation matrix can be obtained using Craig’s notation system by cancelling
the diagonal opposite subscript and superscript. The requirement for these

22
 Chapter 2. Background

diagonal matching pairs also provides means of validating a sequence.

2.5.1.3 Rotation of a vector

The multiplication of a rotation matrix by a vector quantity will alter the reference
coordinate system of the vector and thus achieve a rotation. For example, A v is
a vector defined in A, the vector can be equivalently described in B through the
multiplication of Equation 2.5.

A
v=A B
BR v (2.5)

An example application could be for B v to be an accelerometer measurement and

for A
B R to describe the known misalignment of the accelerometer due to a mounting

error. Equation 2.5 would provide A v as the accelerometer measurement corrected

for misalignment.

2.5.2 Quaternions

Rotation matrices provide an intuitive means of manipulating three-dimensional

quantities; however, the necessity for orthogonality is incompatible with processes
such as interpolation and integration. Quaternions are four-dimensional complex
numbers that offer an alternative, more compact representation to overcome these
shortcomings.

Any orientation of coordinate system B relative to coordinate system A can

A
be achieved through a single rotation of angle θ around an axis r̂. This is
demonstrated in Figure 2.2 where the mutually orthogonal vectors x̂A , ŷA and
ẑA ; and x̂B , ŷB and ẑB define the principle axes of A and B respectively.

The quaternion describing the orientation of B relative to A is defined by Equation

23
Chapter 2. Background

ẑB ẑA

A
θ r̂
ŷB
ŷA

x̂B
x̂A

Figure 2.2: Any orientation of coordinate system B relative to coordinate system

A can be achieved through a rotation of angle θ around the axis A r̂

2.6 where rx , ry and rz are the elements of A r̂. The last three elements form the
imaginary parts of the quaternion.
 
A
B q̂ = qw qx qy qz
  (2.6)
= cos 2θ −rx sin 2θ −ry sin 2θ −rz sin 2θ

A quaternion used to represent an orientation must be of unit magnitude. If

the coordinate systems were to be in alignment then the quaternion would be an
identity as described by Equation 2.7.

 
A
B q̂ = 1 0 0 0 , for A = B (2.7)

The convention used throughput this dissertation positions qw as the first element,
this is consistent with Kuipers [93] and MATLAB [94]. Other implementations
may position qw as the last element; notable examples include OpenGL libraries
[95] and the Microsoft XNA Framework [96].

2.5.2.1 Quaternion conjugate

The quaternion conjugate, denoted by ∗ , switches the reference coordinate system.

For example, the conjugate of a quaternion describing coordinate system B relative
to A would describe A relative to B. A conjugate is obtained by negating the

24
 Chapter 2. Background

imaginary parts of the quaternion as shown in 2.8.

 
B A ∗
A q̂ = B q̂ = qw −qx −qy −qz (2.8)

2.5.2.2 Quaternion product

The quaternion product, denoted by ⊗, can be used to compound a chain of

relative orientations into single quantity. For example, a quaternion describing C
relative to frame A can be computed from B A
C q̂ and B q̂ through the multiplication

of Equation 2.9.
A
C q̂ =B A
C q̂ ⊗ B q̂ (2.9)

The multiplication of two quaternions, q̂a and q̂b , can be determined as the
Hamilton product and defined as Equation 2.10 [93]. A quaternion product is
not commutative; that is, q̂a ⊗ q̂b 6= q̂b ⊗ q̂a .
   
q̂a ⊗ q̂b = qaw qax qay qaz ⊗ qbw qbx qby qbz
 T
qaw qbw − qax qbx − qay qby − qaz qbz 
  (2.10)
qaw qbx + qax qbw + qay qbz − qaz qby 
 
= 
q q − q q + q q + q q 
 aw by ax bz ay bw az bx 
 
qaw qbz + qax qby − qay qbx + qaz qbw

2.5.2.3 Quaternion rotation of a vector

A three dimensional vector can be rotated by a quaternion using the relationship

described in Equation 2.11 [93]. In this equation, A v and B v are the same vector
described in frame A and frame B respectively with a relative orientation defined
by A
B q̂.
   
A ∗
0 B
vT = B q̂ ⊗ 0 A T
v ⊗A
B q̂ (2.11)

25
Chapter 2. Background

2.5.2.4 Converting a quaternion to a rotation matrix

An equivalent rotation matrix can be obtained from a quaternion through the

conversion described by Equation 2.12 [93] where qw , qx , qy and qz are the elements
of A
B q̂.  
2 2
2qw − 1 + 2qx 2qx qy + 2qw qz 2qx qz − 2qw qy 
 
A 
B R = 2qx qy − 2qw qz 2qw − 1 + 2qy 2qy qz + 2qw qx 
2 2  (2.12)
 
2 2
2qx qz + 2qw qy 2qy qz − 2qw qx 2qw − 1 + 2qz

2.5.2.5 Converting a rotation matrix to quaternion

A poorly formulated rotation matrix could potentially corrupt a direct conversion

to a quaternion. Bar-Itzhack provides an alternative method [97] to extract the
‘best fit’ quaternion from an imprecise and non-orthogonal rotation matrix. The
method requires the construction of the symmetric 4×4 matrix K, constructed
A
from the elements of the rotation matrix BR as defined by Equation 2.13. The
optimal quaternion, A
B q̂, is found as the normalised Eigenvector corresponding to

the maximum Eigenvalue, λmax , of K. The element order must be altered to

account for the alternative convention, as described by Equation 2.14 where v0 to
v3 are the elements of the normalised Eigenvector.

 
r11 − r22 − r33 r21 + r12 r31 + r13 r23 − r32 
 
1
 r 21 + r12 r 22 − r11 − r33 r32 + r 23 r31 − r 13


K=   (2.13)
3 r +r r32 + r23 r33 − r11 − r22 r12 − r21 
 31 13 
 
r23 − r32 r31 − r13 r12 − r21 r11 + r22 + r33

 
A
B q̂ = v3 v0 v1 v2 (2.14)

Some practical applications require that a ‘best fit’ rotation matrix is obtained
from an approximated rotation matrix derived from measurement data. This can

26
 Chapter 2. Background

be achieved by first obtaining a quaternion using the method above and then
converting the quaternion into a rotation matrix as per Equation 2.12.

2.5.3 Euler angles

Although rotation matrices and quaternions provide effective tools for the
manipulation of spatial quantities, their numerical values and not immediately
meaningful. Euler angles describe an orientation as three angular quantities and
so offer a more intuitive representation. Perhaps the most common use of Euler
angles is within the aerospace sequence where the three angular quantities φ, θ
and ψ correspond to a rotations around the x, y and z axes respectively as shown
in Figure 2.3. φ, θ and ψ are commonly referred to as roll, pitch and yaw (or
heading) respectively.

θ
x φ y

Figure 2.3: The aerospace sequence of Euler angles describe orientation as the
result of angular rotations φ , θ and ψ around the x, y and z axes respectively

Euler angles describe an orientation that is achieved through a specific sequence

of rotations. The aerospace sequence, also referred to as the ZYX sequence, is just
one of twelve distinct possibilities and defines the order as a rotation first around
the z axis, then the y axis and then the x axis. The direction of each rotational axis
relative to the original orientation is therefore dependent on the previous rotation
in the sequence. This is demonstrated in Figure 2.4 where the three diagrams from
left to right show the successive rotations of B around each axis to achieve the
ultimate orientation of B relative to A.

27
Chapter 2. Background

ẑA , ẑB00 ẑA ẑA

ẑB0 ẑB
ψ θ
ŷB00 ŷB0 ŷB
ŷA ŷA ŷA
φ
x̂00B
x̂0B x̂B
x̂A x̂A x̂A

Figure 2.4: The Euler angle sequence describes successive rotations of around each
axis to achieve the ultimate orientation of B relative to A

Although an angular representation may be convenient, the sequence introduces

an inherent hazard that must be accounted for in any application utilising Euler
angles. When θ approaches ±90◦ , the rotational axis of φ and ψ relative to the
original orientation align; this singularity means that there are infinite solutions
to the Euler angle sequence. As a result, calculated values of φ and ψ can be
expected to become erratic and unreliable. This phenomena is commonly referred
to as gimbal lock. In practice, applications should utilise a rotation matrix or
quaternion representation wherever possible with Euler angles used only as an
ultimate output where the singularity hazard does not have an opportunity to
feedback into numerical processes.

2.5.3.1 Converting a rotation matrix to Euler angles

The ZYX Euler angles φ, θ and ψ describing the orientation of B relative to A

can be calculated from the rotation matrix A
B R̂ as Equations 2.15 to 2.17 [64].

φ = atan2 (r32 , r33 ) (2.15)

θ = − arcsin (r31 ) (2.16)

28
 Chapter 2. Background

ψ = atan2 (r21 , r11 ) (2.17)

2.5.3.2 Converting a quaternion to Euler angles

The ZYX Euler angles can be obtained from the quaternion A

B q̂ using the Equations

2.15 to 2.17 by substituting each element of the rotation matrix with function of
the quaternion as per Equation 2.12 to yield Equations 2.18 to 2.20.

φ = atan2 2 (qy qz − qw qx ) , 2qw2 − 1 + 2qz2 (2.18)

θ = − arcsin 2 (qx qz + qw qy ) (2.19)

ψ = atan2 2 (qx qy − qw qz ) , 2qw2 − 1 + 2qx2 (2.20)

29
Chapter 2. Background

30
Chapter 3

Initial Attitude and Heading

Reference System algorithm

3.1 Introduction

Some of the first work conducted in this research project was the development of an
AHRS algorithm. Although the algorithm would be significantly revised over the
proceeding years (see Chapter 7), this initial algorithm offered a novel solution
that would become widely adopted within academic and commercial projects.
The algorithm was intended to provide an alternative solution to computationally
expensive Kalman-based solutions and explore an alternative approach to the
complementary filter solutions available. The principle goals were:

• to minimise the computational load for applicability to low-power embedded

systems.

• address the short comings of magnetometer integration previously discussed

in Chapter 2.

31
Chapter 3. Initial Attitude and Heading Reference System algorithm

The algorithm design was inspired by the novel Gauss-Newton-based solution

developed by Bachmann et al. [63]. Their solution demonstrated excellent
performance. However, the least squares regression associated with the
Gauss-Newton process imposes a significant computational load. The solution
presented here operates on a similar principle but implements an analytically
derived gradient descent algorithm and so avoids the need for a matrix inversion.
A quaternion representation facilitates the derivation of the algorithm and avoids
the problematic singularities associated with an Euler angle representation.

3.1.1 Chapter summary

Section 3.2 delineates the mathematical derivation of the orientation estimation

algorithm, including a description of the parametrisation and compensation for
magnetic distortion. Section 3.3 describes the experimental equipment used to
test and verify the performance of the algorithm. Section 3.4 quantifies the
experimental testing and accuracy of the algorithm and compares it to existing
systems. Section 3.5 summarises conclusions and contributions of this work.

3.2 Algorithm derivation

3.2.1 Orientation from angular rate

A tri-axis gyroscope will measure the angular rate about the x, y and z axes of the
sensor frame, termed ωx , ωy and ωz respectively. If these parameters (in rads−1 )
are arranged into the vector S ω defined by Equation 3.1, the quaternion derivative
describing rate of change of the Earth frame relative to the sensor frame SE q̇ can
be calculated [98] as Equation 3.2. The ⊗ operate denotes a quaternion product

32
 Chapter 3. Initial Attitude and Heading Reference System algorithm

and theˆaccent denotes a normalised vector of unit length.

 
S
ω = 0 ωx ωy ωz (3.1)

S 1S S
E q̇ = E q̂ ⊗ ω (3.2)
2

The orientation of the Earth frame relative to the sensor frame at time t, E
S qω,t ,

S
can be computed by numerically integrating the quaternion derivative E q̇ω,t as
described by Equations 3.3 and 3.4, provided that initial conditions are known. In
these equations, S ωt is the angular rate measured at time t, ∆t is the sampling
S
period and E q̂est,t−1 is the previous estimate of orientation. The subscript ω
indicates that the quaternion is calculated from angular rates.

S 1S S
E q̇ ω,t = E q̂est,t−1 ⊗ ωt (3.3)
2

S
E qω,t = SE q̂est,t−1 + SE q̇ ω,t ∆t (3.4)

3.2.2 Orientation from a homogeneous field

In the context of an orientation estimation algorithm, it will initially be assumed

that an accelerometer will measure only gravity and a magnetometer will measure
only the Earth’s magnetic field. If the direction of the Earth’s field is known in
the Earth frame, a measurement of the field’s direction within the sensor frame
will allow an orientation of the sensor frame relative to the Earth frame to be
calculated. However, for any given measurement there will not be a unique sensor
orientation solution, instead there will infinite solutions represented by all those

33
Chapter 3. Initial Attitude and Heading Reference System algorithm

orientations achieved by the rotation of the true orientation around an axis parallel
with the field. A quaternion representation requires a single solution to be found.
This may be achieved through the formulation of an optimisation problem where
an orientation of the sensor, SE q̂, is found as that which aligns a predefined reference
ˆ with the measured field in the sensor
direction of the field in the Earth frame, E d,
frame, S ŝ; thus solving 3.5 where Equation 3.6 defines the objective function.

ˆ S ŝ)
min f (SE q̂, E d, (3.5)
S q̂∈<4
E

ˆ S ŝ) = S q̂ ∗ ⊗ E dˆ ⊗ S q̂ − S ŝ
f (SE q̂, E d, (3.6)
E E

Many optimisation algorithms exist but the gradient descent algorithm is one of
the simplest to both implement and compute. Equation 3.7 describes the gradient
descent algorithm for n iterations resulting in an orientation estimation of SE q̂n+1
S
based on an ‘initial guess’ orientation E q̂0 and a variable step-size µ. Equation
3.8 computes an error direction on the solution surface defined by the objective
function, f , and its Jacobian, J .

ˆ S ŝ)
∇f T (SE q̂ k , E d,
S
E qk+1 = SE q̂ k − µ , k = 0, 1, 2...n (3.7)
∇f T (S q̂ , E d, ˆ S ŝ)
E k

ˆ S ŝ) = J T (S q̂ , E d)f
∇f (SE q̂ k , E d, ˆ (S q̂ , E d,
ˆ S ŝ) (3.8)
E k E k

Equations 3.7 and 3.8 describe the general form of the algorithm applicable to a
field predefined in any direction. However, if the reference direction of the field is
defined to only have components within 1 or 2 of the principle axes of the Earth
coordinate frame then the equations simplify. An appropriate convention would
be to assume that the direction of gravity defines the vertical, z axis as shown

34
 Chapter 3. Initial Attitude and Heading Reference System algorithm

in Equation 3.10. Substituting E ĝ and normalised accelerometer measurement S â

for E dˆ and S ŝ respectively, yields the simplified objective function and Jacobian
defined by Equations 3.12 and 3.13.

 
S
E q̂ = qw qx q y qz (3.9)

 
E
ĝ = 0 0 0 1 (3.10)

 
S
â = 0 ax ay az (3.11)

 
2(qx qz − qw qy ) − ax 
 
fg (SE q̂, S â) =  
2(qw qx + qy qz ) − ay  (3.12)
 
1 2 2
2( 2 − qx − qy ) − az

 
−2qy 2qz −2qw 2qx 
 
Jg (SE q̂) = 
 2qx 2qw 2qz 2qy 
 (3.13)
 
0 −4qx −4qy 0

The Earth’s magnetic field can be considered to have components in one horizontal
axis and the vertical axis; the vertical component due to the inclination of the
field which is between 65◦ and 70◦ to the horizontal in the United Kingdom (UK)
[99]. This can be represented by Equation 3.14. Substituting E b̂ and normalised
magnetometer measurement S m̂ for E dˆ and S ŝ respectively, yields the simplified
objective function and Jacobian defined by Equations 3.16 and 3.17.

 
E
b̂ = 0 bx 0 bz (3.14)

35
Chapter 3. Initial Attitude and Heading Reference System algorithm

 
S
m̂ = 0 mx my mz (3.15)

 
2 2
2bx (0.5 − qy − qz ) + 2bz (qx qz − qw qy ) − mx 
 
fb (SE q̂, E b̂, S m̂) =  2b
 x x y(q q − q q
w z ) + 2b (q q
z w x + q q
y z ) − m 
y  (3.16)
 
2bx (qw qy + qx qz ) + 2bz (0.5 − qx2 − qy2 ) − mz


 −2bz qy 2bz qz

Jb (SE q̂, E b̂) = 
−2bx qz + 2bz qx 2bx qy + 2bz qw

2bx qy 2bx qz − 4bz qx
 (3.17)
−4bx qy − 2bz qw −4bx qz + 2bz qx 

2bx qx + 2bz qz −2bx qw + 2bz qy 


2bx qw − 4bz qy 2bx qx

As has already been discussed, the measurement of gravity or the Earth’s magnetic
field alone will not provide a unique orientation of the sensor. To do so, the
measurements and reference directions of both fields may be combined as described
by Equations 3.18 and 3.19. Whereas the solution surface created by the objective
functions in Equations 3.12 and 3.16 have a global minimum defined by a line,
the solution surface defined by Equation 3.18 has a minimum defined by a single
point, provided that bx 6= 0.

 
 fg (SE q̂, S â) 
fg,b (SE q̂, S â, E b̂, S m̂) =   (3.18)
fb (SE q̂, E b̂, S m̂)

36
 Chapter 3. Initial Attitude and Heading Reference System algorithm

 
 JgT (SE q̂) 
Jg,b (SE q̂, E b̂) =   (3.19)
JbT (SE q̂, E b̂)

A conventional approach to optimisation would require multiple iterations of

Equation 3.7 to be computed for each new orientation and corresponding sensor
measurements. However, it is acceptable to compute one iteration per time sample
provided that the convergence rate of the estimated orientation governed by µt is
equal or greater than the rate of change of physical orientation. Equation 3.20
calculates the estimated orientation SE q ∇,t computed at time t based on a previous
estimate of orientation SE q̂ est,t−1 and the objective function error ∇f defined by
sensor measurements S ât and S m̂t sampled at time t. The form of ∇f is chosen
according to the sensors in use, as shown in Equation 3.21. The subscript ∇
indicates that the quaternion is calculated using the gradient descent algorithm.

S ∇f T
E q ∇,t = SE q̂ est,t−1 − µt (3.20)
k∇f T k



 JgT (SE q̂est,t−1 )fg (SE q̂est,t−1 , S ât )
∇f = (3.21)

Jg,b
T S
(E q̂est,t−1 , E b̂)fg,b (SE q̂est,t−1 , S ât , E b̂, S m̂t )

S
An appropriate value of µt is that which ensures the convergence rate of E q ∇,t

is limited to the physical orientation rate as this avoids overshooting due an

unnecessarily large step size. Therefore µt can be calculated as Equation 3.22
where ∆t is the sampling period, SE q̇ω,t is the rate of change of orientation measured
by gyroscopes and α is an augmentation of µ to account for noise in accelerometer
and magnetometer measurements.

S
µt = α E q̇ω,t ∆t, α > 1 (3.22)

37
Chapter 3. Initial Attitude and Heading Reference System algorithm

3.2.3 Algorithm fusion process

S
In practice, E qω,t may start from incorrect initial conditions and accumulate
S
errors due to gyroscope measurement noise and E q∇,t will provide an incorrect
estimate when the accelerometer is not stationary or the magnetometer exposed
to interferences. The goal of the fusion algorithm is to provide an orientation
estimate where SE qω,t is used to filter out high frequency errors in SE q∇,t , and SE q∇,t
is used both to compensate for integral drift in SE qω,t and to provide convergence
from initial conditions.

An estimated orientation of the Earth frame relative to the sensor frame, SE qest,t ,
is obtained through the fusion of the two separate orientation calculations, SE qω,t
and SE q∇,t as described by Equation 3.23 where γt and (1 − γt ) are weights applied
to each orientation calculation.

S
E qest,t = γt SE q∇,t + (1 − γt )SE qω,t , 0 ≤ γt ≤ 1 (3.23)

An optimal value of γt is therefore that which ensures the weighted rate of

divergence of SE qω due to integral drift is equal to the weighted rate of convergence
µt
of SE q∇ . This is represented by Equation 3.24 where ∆t
is the convergence rate of
S
E q∇ and β is the divergence rate of SE qω expressed as the magnitude of a quaternion
derivative corresponding to the gyroscope measurement error. Equation 3.24 can
be rearranged to define γt as Equation 3.25.

µt
(1 − γt )β = γt (3.24)
∆t

β
γt = µt (3.25)
∆t
+β

38
 Chapter 3. Initial Attitude and Heading Reference System algorithm

S S
The fusion process ensures the optimal fusion of E qω,t and E q∇,t assuming that
the convergence rate of SE q ∇ governed by α is equal or greater than the physical
rate of change of orientation. Therefore α has no upper bound. If α is assumed to
be very large then µt , defined by Equation 3.22, also becomes very large and the
equations simplify. A large value of µt used in Equation 3.20 means that SE q̂est,t−1
becomes negligible and the equation can be re-written as Equation 3.26.

S ∇f
E q ∇,t ≈ −µt (3.26)
k∇f k

The definition of γt in Equation 3.25 also simplifies if the β term in the denominator
becomes negligible and the equation can be rewritten as Equation 3.27. It is
possible from Equation 3.27 to also assume that γt ≈ 0.

β∆t
γt ≈ (3.27)
µt

Substituting Equations 3.4, 3.26 and 3.27 into Equation 3.23 directly yields
Equation 3.28. It is important to note that in Equation 3.28, γt has been
substituted as both as Equation 3.26 and 0.

   
S β∆t ∇f S
E qest,t = −µt + (1 − 0) E q̂est,t−1 + SE q̇ ω,t ∆t (3.28)
µt k∇f k

S
Equation 3.28 can be simplified to Equation 3.29 where E q̇est,t is the estimated
orientation rate defined by Equation 3.30.

S
E qest,t = SE q̂est,t−1 + SE q̇est,t ∆t (3.29)

S ∇f
E q̇est,t = SE q̇ ω,t − β (3.30)
k∇f k

39
Chapter 3. Initial Attitude and Heading Reference System algorithm

It can be seen from Equations 3.29 and 3.30 that the algorithm calculates
S
the orientation E qest by numerically integrating the estimated rate of change
S S
of orientation E q̇ est . The algorithm computes E q̇est as the rate of change of
orientation measured by the gyroscopes, SE q̇ ω , with the magnitude of the gyroscope
measurement error, β, removed in a direction based on accelerometer and
magnetometer measurements. Figure 3.1 shows a block diagram representation
of the complete orientation estimation algorithm implementation for an IMU.

Accelerometer S ât J Tg (SE q̂ est,t−1)f g (SE q̂ est,t−1, S ât )

∇f
k∇f k
z−1
β

Z
1S q S
Gyroscope S ω t q̂ ⊗ S ωt .dt
kqk E q̂ est,t
2 E est,t−1
S
E q̇ est,t
z−1

Figure 3.1: Block diagram representation of the complete orientation estimation

algorithm for an IMU implementation

3.2.4 Magnetic distortion compensation

Investigations into the effect of magnetic distortions on an orientation sensor’s

performance have shown that substantial errors may be introduced by sources
including electrical appliances, metal furniture and metal structures within
a building’s construction [100, 101]. Soft-iron and hard-iron distortions are
interference fixed in the sensor frame and can be removed through calibration
[102, 29, 103]. Sources of interference in the Earth frame may only be removed
if an additional reference of orientation is available. An accelerometer provides a
reference of attitude and so may be used to compensate for inclination errors in
the measured Earth’s magnetic field.

40
 Chapter 3. Initial Attitude and Heading Reference System algorithm

The measured direction of the Earth’s magnetic field in the Earth frame at time
t, E ĥt , can be computed as Equation 3.31. The effect of an erroneous inclination
E
of the measured direction Earth’s magnetic field, ĥt , can be corrected if the
E
algorithm’s reference direction of the Earth’s magnetic field, b̂t , is of the same
E E
inclination. This is achieved by computing b̂t as ĥt normalised to have only
components in the Earth frame x and z axes; as described by Equation 3.32.

 
E ∗
ĥt = 0 hx hy hz = SE q̂est,t−1 ⊗ S m̂t ⊗ SE q̂est,t−1 (3.31)

 
E p 2
b̂t = 0 hx + h2y 0 hz (3.32)

Compensating for magnetic distortions in this way ensures that magnetic

disturbances are limited to only affect the estimated heading component of
orientation. The approach also eliminates the need for the reference direction
of the Earth’s magnetic field to be predefined; a potential disadvantage of
other orientation estimation algorithms [60, 104]. Figure 3.2 shows a block
diagram representation of the complete algorithm implementation for a Magnetic,
Angular Rate and Gravity (MARG) sensor array, including the magnetic distortion
compensation.

3.2.5 Algorithm adjustable parameter

The orientation estimation algorithm requires one adjustable parameter, β,

representing the gyroscope measurement error expressed as the magnitude of a
quaternion derivative. It is convenient to define β using the angular quantity ω̃max
representing the maximum gyroscope measurement error of each axis. Using the
relationship described by Equation 3.2, β may be defined by Equation 3.33 where

41
Chapter 3. Initial Attitude and Heading Reference System algorithm

S
E q̂ est,t−1 ⊗ S m̂t ⊗ SE q̂ ∗est,t−1
E
ĥt
h q i
0 h2x + h2y 0 hz
E
b̂t
S
Magnetometer m̂t
J Tg,b (SE q̂ est,t−1, E b̂t )f g,b (SE q̂ est,t−1, S â, E b̂t , S m̂)
Accelerometer S ât

∇f
k∇f k
z−1
β

Z
1S q S
S
Gyroscope ω t q̂ ⊗ S ωt .dt
kqk E q̂ est,t
2 E est,t−1
S
E q̇ est,t
z−1

Figure 3.2: Block diagram representation of the complete orientation

estimation algorithm for an MARG implementation including magnetic distortion
compensation

q̂ is any unit quaternion.

  r
1 3
β= q̂ ⊗ 0 ω̃max ω̃max ω̃max = ω̃max (3.33)
2 4

3.3 Experimental equipment

The algorithm was tested using the Xsens MTx orientation sensor [105] containing
16-bit resolution tri-axis gyroscopes, accelerometers and magnetometers. Raw
sensor data was logged to a computer at 512 Hz and imported using the
accompanying software to provide calibrated sensor measurements which were then
processed by the proposed orientation estimation algorithm. The software also
incorporates a propriety Kalman-based orientation estimation algorithm. As both
the Kalman-based algorithm and proposed algorithm’s estimates of orientation
were computed using identical sensor data, the performance of each algorithm
could be evaluated relative to one-another, independent of sensor performance.

42
 Chapter 3. Initial Attitude and Heading Reference System algorithm

The Xsens Kalman-based algorithm was used with the factory default settings.

A Vicon system, consisting of eight MX3+ cameras connected to an MXultranet

server [106] and Nexus [107] software, was used to provide reference measurements
of the orientation sensor’s actual orientation. To do so, the sensor was fixed to
an orientation measurement platform. The positions of optical markers attached
to the platform were logged at 120 Hz and then post-processed to compute the
orientation of the measurement platform and sensor. In order for the measurements
of an orientation in the camera coordinate frame to be comparable to the algorithm
estimate of orientation in the Earth frame, an initial calibration procedure was
required where the direction of the Earth’s magnetic and gravitational fields in the
camera coordinate frame were measured using a magnetic compass and pendulum
with attached optical markers.

3.4 Experimental results

Measured and estimated angle θ

100
Measured
50 Kalman−based algorithm
θ (degrees)

Proposed filter (MARG)

0

−50

−100
42 44 46 48 50 52 54 56 58 60 62
Error
5
Kalman−based algorithm
θε (degrees)

Proposed filter (MARG)

−5
42 44 46 48 50 52 54 56 58 60 62
time (seconds)

Figure 3.3: Typical results for measured and estimated angle θ (top) and error
(bottom)

It is common [104, 105, 108, 109, 110] to quantify orientation sensor performance
as the static and dynamic Root Mean Squared (RMS) errors in the decoupled

43
Chapter 3. Initial Attitude and Heading Reference System algorithm

Euler parameter Kalman-based MARG IMU

algorithm algorithm algorithm
RMS[φ ] static 0.789◦ 0.581◦ 0.594◦
RMS[φ ] dynamic 0.769◦ 0.625◦ 0.623◦
RMS[θ ] static 0.819◦ 0.502◦ 0.497◦
RMS[θ ] dynamic 0.847◦ 0.668◦ 0.668◦
RMS[ψ ] static 1.150◦ 1.073◦ N/A
RMS[ψ ] dynamic 1.344◦ 1.110◦ N/A

Table 3.1: Static and dynamic RMS error of Kalman-based algorithm and proposed
algorithm IMU and MARG implementations

Euler parameters describing the pitch, φ, roll, θ and heading, ψ components of an

orientation, corresponding to rotations around the sensor frame x, y, and z axis
respectively. A total of four sets of Euler parameters were computed, corresponding
to the calibrated optical measurements of orientation, the Kalman-based algorithm
estimated orientation and the proposed algorithm estimates orientation for both
the MARG and IMU implementations. The errors of estimated Euler parameters,
φ , θ and ψ , were computed as the difference between estimated values and the
calibrated optical measurements. Results were obtained for a sequence of rotations
around each axis preformed by hand. The experiment was repeated eight times to
compile a dataset representative of system performance. The proposed algorithm’s
adjustable parameter, β, was set to 0.033 for the MARG implementation and
0.041 for the IMU implementation. Trials summarised in Figure 3.4, found these
values to provide optimal performance. Figure 3.3 shows the Kalman-based
algorithm and proposed algorithm MARG implementation results, typical of the
eight experiments.

The static and dynamic RMS values of φ , θ , and ψ were calculated assuming
a static state when the measured corresponding angular rate was < 5◦ /s, and a
dynamic when ≥ 5◦ /s. This threshold was chosen to be sufficiently greater than
the noise floor of the data. The results are summarised in Table 3.4 where each
value represents the mean of all eight experiments.

44
 Chapter 3. Initial Attitude and Heading Reference System algorithm

β vs. filter performance (IMU) β vs. filter performance (MARG)

mean[ RMS[φε], RMS[θε], RMS[ψε] ] (degrees)

2 2

mean[ RMS[φε], RMS[θε] ] (degrees)

Static performance Static performance
Dynamic performance Dynamic performance

1.5 1.5
β = 0.033 β = 0.041

1 1

0.5 0.5
0 0.2 0.4 0 0.2 0.4
β β

Figure 3.4: The effect of the adjustable parameter, β, on the performance of the
proposed algorithm IMU (left) and MARG (right) implementations

mean[ RMS[φε], RMS[θε], RMS[ψε] ] (degrees)

Sampling rate vs. filter performance (IMU) Sampling rate vs. filter performance (MARG)
30 30
mean[ RMS[φ ], RMS[θ ] ] (degrees)

Static performance Static performance

25 Dynamic performance 25 Dynamic performance

20 20
ε

15 15
ε

10 10

5 5

0 0
0 1 2 0 1 2
10 10 10 10 10 10
Sampling rate (Hz) Sampling rate (Hz)

Figure 3.5: The effect of sampling rate on the performance of the proposed
algorithm IMU (left) and MARG (right) implementations

The results of an investigation into the effect of the adjustable parameter β on

algorithm performance are summarised in Figure 3.4. The experimental data was
processed though the separate proposed algorithm IMU and MARG implantations,
using fixed values of β between 0 to 0.5. There is a clear optimal value of β high
enough to minimise errors due to integral drift but sufficiently low enough that
unnecessary noise is not introduced by large steps of gradient descent iterations.

The results of an investigation into the effect of sampling rate on algorithm

performance is summarised in Figure 3.5. The experimental data was processed
though the separate proposed algorithm IMU and MARG implantations, using
the previously defined, optimal values β. Experimental data was decimated to

45
Chapter 3. Initial Attitude and Heading Reference System algorithm

simulate sampling rates between 1 Hz and 512 Hz. It can be seen from Figure 3.5
that the proposed algorithm achieves similar levels of performance at 50 Hz as at
512 Hz. Both algorithm implementations are able to achieve a static error < 2◦
and dynamic error < 7◦ while sampling at 10 Hz. This level of accuracy may be
sufficient for human motion applications though the sampling rate will limit the
bandwidth of the motion that may be measured.

3.5 Conclusions

Orientation estimation algorithms for inertial/magnetic sensors is a is a mature

field of research. Modern techniques [63, 59, 111] have focused on simpler
algorithms that ameliorate the computational load and parameter tuning burdens
associated with conventional Kalman-based approaches. The algorithm presented
in this chapter employs processes similar to other complementary filters but
through a novel derivation, is able to offer some key advantages:

• Computing an error based on an analytically derived Jacobian results in a

significant reduction in the computation load relative to a Gauss-Newton
method [63]; quantified as 109 and 248 scalar arithmetic operations per
update for C code implementations of the IMU and MARG implementations
respectively.

• Normalisation of the feedback error permits optimal gains to be defined based

on observable system characteristics.

• Magnetic distortion compensation algorithm eliminates the need for a

direction of magnetic field to be predefined by the designer.

The elimination of a predefined direction of magnetic field is an advantage over

all other algorithms cited by this chapter; though this feature may be easily
incorporated to other algorithms. Experimental studies have been presented for an

46
 Chapter 3. Initial Attitude and Heading Reference System algorithm

off-the-shelf, leading commercial unit with reference measurements obtained from

a precision optical measurement system. These studies enabled the algorithm
to be benchmarked and have indicated that the algorithm performs as well as the
proprietary Kalman-based system; and is able to maintain this accuracy even with
a full order of magnitude in reduction of sampling rate.

The algorithm presented in this chapter was published in 2011 [5] with the source
code initially made available on Google Code1 . The paper and its associated
technical report have since totalled over 100 citations [6]. Google Code indicates
over 10,000 downloads [7]. However, as the files have since been published in
multiple locations the total number of downloads is assumed to be higher.

1
http://code.google.com/p/imumargalgorithm30042010sohm/

47
Chapter 3. Initial Attitude and Heading Reference System algorithm

48
Chapter 4

Kinematically redundant
non-gyro IMUs

4.1 Introduction

This chapter presents two schemes of measuring the linear and angular kinematics
of a rigid body using a kinematically redundant array of triple-axis accelerometers
with potential applications in biomechanics. A novel angular velocity estimation
algorithm is proposed and evaluated that can compensate for angular velocity
errors using measurements of the direction of gravity. Analysis and discussion
of optimal sensor array characteristics are provided. A damped 2 axis pendulum
was used to excite all 6 DOF of the a suspended accelerometer array through
determined complex motion and is the basis of both simulation and experimental
studies. The relationship between accuracy and sensor redundancy is investigated
for arrays of up to 100 triple axis (300 accelerometer axes) accelerometers in
simulation and 10 equivalent sensors (30 accelerometer axes) in the laboratory
test rig. The work also reports on the sensor calibration techniques and hardware
implementation.

49
Chapter 4. Kinematically redundant non-gyro IMUs

4.2 Solving inverse kinematics with

accelerometers

Consider n triple-axis accelerometers fixed and aligned to a rigid body such as an

orthosis on a subject’s limb at known positions relative to the body coordinate
B
frame. The position of the i’th sensor is described by the vector di defined in
the body frame and provides the measurement vector i α. The motion of the body
is described by the linear accelerations ẍ, ÿ and z̈ of the body origin and the
rotation velocities p, q and r around the body x, y and z axes respectively. This is
represented as the schematic shown as Figure 4.1 where the mutually orthogonal
unit vectors x̂B , ŷB and ẑB , and x̂i , ŷi and ẑi define the coordinated frames of
the body and i’th sensor respectively. If the i’th sensor is not aligned to the body

ẑ i , αx
ẑ i′

ẑ B , z̈
ŷ i′
B
di
ŷ i , αy
r q x̂i , αx x̂ ′
i

p ŷ B , ÿ
x̂B , ẍ

Figure 4.1: Schematic describing the i’th accelerometer relative to the ridged body
kinematic frame

frame then it can be considered aligned to the arbitrary frame i0 defined by the
mutually orthogonal unit vectors x̂i0 , ŷi0 and ẑi0 (shown in Figure 4.1), providing
0
the measurement vector i α. The orientation of frame i0 relative to frame i is
described by the rotation matrix ii0 R so that the aligned acceleration measurement
i
α may be obtained using Equation 4.1.

i 0
α = ii0 R i α (4.1)

50
 Chapter 4. Kinematically redundant non-gyro IMUs

4.2.1 Forward kinematics

The linear acceleration measured by the i’th sensor, i α, may be defined by the
kinematics of the body origin using Equation 4.2 where B ω and B a describe the
angular velocity and linear acceleration of the rigid body origin respectively. This
represents the sum of linear, tangential and centripetal accelerations. There is no
Coriolis term as the accelerometer position is fixed relative to the body.

i
α = B ω̇ × B di + B ω × (B ω × B di ) + B a (4.2)

 T
i
α = αx αy αz (4.3)

 T
B
ω= p q r (4.4)

 T
B
di = dx dy dz (4.5)

 T
B
a = ẍ ÿ z̈ (4.6)

Equation 4.2 may be rearranged as Equation 4.7 where Gi is a constant matrix

defined by the position of the i’th sensor and s is a state vector describing the
kinematics of the body origin.
i
α = Gi s (4.7)

 
 0 -dx -dx 0 dz -dy dy 0 dz 
 
Gi =  I
 3 -d y 0 -d y -d z 0 d x dx d z 0 
 (4.8)
 
-dz -dz 0 dy -dx 0 0 dy dx

51
Chapter 4. Kinematically redundant non-gyro IMUs

and
 T
s= ẍ ÿ z̈ p2 q 2 r2 ṗ q̇ ṙ pq qr rp (4.9)

4.2.2 Inverse kinematics solutions

4.2.2.1 Combinatorial method

The inverse kinematic solution can be found directly from Equation 4.7 through
an inversion of Gi . If the number of sensors fixed to the body is four
then accelerometer measurements may be arranged as the vector αC and the
corresponding square matrix C constructed as described by Equations 4.10 and
4.11. C may be inverted to provide the inverse kinematic solution shown as
Equation 4.12.
 T
αC = 1
αT 2
αT 3
αT 4
αT (4.10)

 T
C = G1 T
G2 T
G3 T
G4 T (4.11)

s = C -1 αC (4.12)

This solution cannot directly incorporate n triple-axis accelerometer for n 6= 4. If

n > 4 then there exists m unique combinations of sensors and possible C matrices
where m is defined by Equation 4.13. A single value of s may be computed as the
mean of the m possible values, as described by Equation 4.14 where sj a value for
a given combination of four triple-axis accelerometer.

 
n n!
m= = (4.13)
4 24(n − 4)!

52
 Chapter 4. Kinematically redundant non-gyro IMUs

1 X
m
s= sj (4.14)
m j=0

If the positions of any sensors are equal or if all sensors exist on a plane (for a
non-planar system) then Gi is singular and cannot be inverted. For n > 4, a
unique value of C -1 must be computed for each combination. This method may
result in a considerable computational load as an array of n sensors requires m
12×12 matrix inversions and multiplications.

4.2.2.2 Pseudoinverse method

For n accelerometer measurements arranged in the vector αP , the non-square

matrix P may be constructed and the inverse kinematic solution may be computed
+
as Equation 4.17 where denotes the pseudoinverse. This is a similar approach
to that of existing inverse kinematic solutions for redundant accelerometer arrays
[86, 87].
 T
αP = 1
α T 2
α T 3
α T
... n
α T (4.15)

 T
P = G1 T G2 T G3 T ... Gn T (4.16)

s = P + αP (4.17)

The Moore-Penrose pseudoinverse of P may be calculated as Equation 4.18

provided that the rows of P are linearly independent and the matrix represents
> 4 sensors.
P + = (P T P )-1 P T (4.18)

If the positions of any sensors are equal or if all sensors exist on a plane (for a
non-planar system) then the rows of P are not linearly independent and Equation

53
Chapter 4. Kinematically redundant non-gyro IMUs

4.18 may not be used. The computational requirement of this method considerably
is less than that of the combinatorial method as only one n×12 matrix inversion
and multiplication is required for n sensors.

It should also be noted that the pseudoinverse is identical to an inverse computed

using the non-zero singular values from a singular value decomposition of the
matrix. Thus the sensitivity of the solution can be evaluated from the ratio of the
largest to the smallest singular value (equivalent to the condition number of the
matrix) as well as the composition of the two unitary matrices associated with the
decomposition. This analysis confirms the findings presented later in Appendix
A, optimal array configurations. Other matrix methods are available to solve
Equation 4.17 such as QR decomposition with back substitution, or minimising
the l1 norm.

4.2.3 Angular velocity estimation

The state vector s does not directly yield individual angular velocity terms. This
is a problem common to all accelerometer-only inertial measurement methods and
many solutions have proposed. Cardou et al. provide a discussion and analysis of
the different approaches to this problem and present a method more robust than
many existing solutions [112].

Here we propose a novel angular velocity estimation algorithm that employs

the gradient descent approach but combines this with a complementary filter
structure to achieve advantages in computational efficiency and well described
error dynamics. The method is similar to that described in Madgwick et al. [5].
The angular velocity estimation is further improved by the orientation estimation
algorithm presented in Section 4.3 to compensate for bias errors in the estimated
angular velocity.

54
 Chapter 4. Kinematically redundant non-gyro IMUs

The available angular velocity terms may be used to construct an objective function
f (ω̃, s) as shown in Equation 4.19 where ω̃ is the estimated angular velocity. If
f (ω̃, s) = 0 then ω̃ = ±ω. As each angular velocity term is only available as
the product of itself or another, the sign of each quantity is lost and cannot be
recovered directly.  
2 2
 p̃ − p 
 
 q̃ 2 − q 2 
 
 
 2 
 r̃ − r2 
f (ω̃, s) = 


 (4.19)
p̃q̃ − pq 
 
 
q̃r̃ − qr 
 
 
r̃p̃ − rp

 T
ω̃ = p̃ q̃ r̃ (4.20)

The gradient of the solution surface (defined by ηp , ηq and ηr ) created by f (ω̃, s)

may be used to compute the direction of the error in the estimated angular velocity

55
Chapter 4. Kinematically redundant non-gyro IMUs

as Equation 4.23 where Jf is the Jacobian of f (ω̃, s).

 
ηp 
 
η  = JfT (ω̃)f (ω̃, s) (4.21)
 q
 
ηr
 
2 2
 p̃ − p 
 
 T  q̃ 2 − q 2 
 
 
2p̃ 0 0 q̃ 0 r̃   2 
   r̃ − r2 
=
 0 2q̃ 0 p̃ r̃ 0 




 (4.22)
  p̃q̃ − pq 
 
0 0 2r̃ 0 q̃ p̃  
q̃r̃ − qr 
 
 
r̃p̃ − rp
 
2 2
2p̃(p̃ − p ) + q̃(p̃q̃ − pq) + r̃(r̃p̃ − rp)
 
=
 2q̃(q̃ 2
− q 2
) + p̃(p̃q̃ − pq) + r̃(q̃r̃ − qr) 
 (4.23)
 
2r̃(r̃2 − r2 ) + q̃(q̃r̃ − qr) + p̃(r̃p̃ − rp)

The direction of this vector may be combined with a magnitude computed as the
difference between the magnitudes of the estimated angular velocities and actual
angular velocities. The error in the angular velocity is therefore estimated, ẽ, as
the dot product shown in Equation 4.24. Due to the loss of sign in the available
angular velocity terms the error will be computed relative to the closest value of
±ω.    
p
sign(η )
p   |p̃| − p2 
   
ẽ =  sign(η )  ·  |q̃| − pq 2  (4.24)
 q   
   √ 
sign(ηr ) |r̃| − r2

The estimated angular velocity is computed from the angular acceleration and
estimated error in the angular velocity using Equation 4.25. This is represented
as the block diagram shown as Figure 4.2.

Z
ω̃ = (ω̇ − K ẽ).dt (4.25)

56
 Chapter 4. Kinematically redundant non-gyro IMUs

R
ω̇ ω̃

K
2

p  

q2 

2 
r
Calculate ẽ
pq 

qr 


rp

Figure 4.2: Complementary filter used for angular velocity estimation

4.2.3.1 Stability and response to initial conditions

A simplified analysis of the proposed complementary filter is possible if the

estimated error term, ẽ, is assumed to be equal to the true error term, e. The
complementary filter can then be seen to represent a first order system with error
dynamics governed by Equation 4.26. The system is therefore stable for all values
of K > 0 and the error can be expected to decay exponentially at a rate governed
by K.
ė = −Ke (4.26)

As the estimated error may only be computed relative to the closest value of ±ω,
ω̃ will converge to the closest value of either ±ω. A correct sign assertion may be
assured if the dynamics in ω̇ are of a sufficiently large magnitude relative to the
convergence rate of ω̃ governed by K, at the zero-crossing point of ω. A sufficient
magnitude of ω̇ relative to the convergence rate of ω̃ would mean that integral
drift would not dominate the ω̃ dynamics and the value of ω̃ would not drift across
zero and hence cause an incorrect sign.

57
Chapter 4. Kinematically redundant non-gyro IMUs

4.3 Orientation estimation and angular velocity

drift compensation

In many applications it can be assumed that the mean linear acceleration measured
over an extended period of time represents gravity. The direction of gravity
measured relative to the accelerometer array may therefore be used to estimate
the attitude of the array relative to the Earth’s surface. As will be shown in
Section 4.6.3, sensor measurement and alignment errors result in a bias error in
the estimated angular velocity at low velocities. Knowledge of the accelerometer
array orientation may be used to correct for steady state errors in the estimation
angular velocity and so compensate for integral drift.

Mahony et al. [59] have presented an orientation estimation algorithm for

conventional IMUs consisting of gyroscopes and accelerometers. The algorithm is
equally applicable to the accelerometer array presented in this work as the inverse
kinematic solution yields linear accelerations and angular velocities. Equations
4.27 to 4.30 represent a quaternion implementation [62] of this algorithm where
B
E q̂ is the estimated orientation of the Earth relative to the body, represented as
a quaternion of elements qw to qz . The gains KP and KI represent proportional
and integral gains respectively. The ˆ accent denotes the normalised unit vector
and the ⊗ operator denotes a quaternion product.

Z
B B
Eq = E q̇ (4.27)

 
B 1B
E q̇ = E q̂ ⊗ 0 (B ω − δ)T (4.28)
2

Z
δ = KP  + KI  (4.29)

58
 Chapter 4. Kinematically redundant non-gyro IMUs

 
 2qx qz − qw qy 
 
 = Ba × 
 2qw qx + qy qz 
 (4.30)
 
2 2 2 2
qw − qx − qy + qz

The error in the estimated angular velocity is represented by the integral feedback
term. The compensated estimated angular velocity is therefore obtained as
Equation 4.31.
Z
0
ω̃ = ω̃ − Ki  (4.31)

4.4 2 axis pendulum kinematics

A 2 axis pendulum was the chosen subject of simulation and experimental studies
as it approximates a number of human and robotic manipulation joints and the
oscillatory motion is analogous to gait and excites all 6 DOF of the suspended
body. The damped motion results in both low and high magnitude rates and the
complex motion of the 2 axis joint results in rotational kinematics in all three axes
of the suspended body. The pendulum is represented by the schematic shown as
Figure 4.3 where θ1 and θ2 represent the angle in each axis of the joint. The 2
axis pendulum equations of motion have been derived as Equations 4.32 and 4.33
where c1 and c2 are the damping of each axis of the joint.

g sin θ1 c1 θ̇1
θ̈1 = 2θ̇1 θ̇2 tan θ2 − − (4.32)
l cos θ2 m cos2 θ2

g c2 θ̇2
θ̈2 = −θ̇12 cos θ2 sin θ2 − cos θ1 sin θ2 − (4.33)
l m

The body kinematics are computed as the forward kinematics described by

B B B
Equations 4.34 to 4.37 where a, ω and ω̇ describe the translational

59
Chapter 4. Kinematically redundant non-gyro IMUs

{E}
{A} z zE
A
g
ϴ1 yA

yE
xE xA ϴ2

l
m

Figure 4.3: 2 axis pendulum schematic

acceleration, angular velocity and angular acceleration respectively of the body

E
frame in directions relative to the body frame. BR describes the orientation of the
body frame relative to the Earth frame.
 
 - sin θ2 
 
B
a = g
 sin θ1 cos θ2

 (4.34)
 
cos θ1 cos θ2

 
θ̇1 cos θ2 
 
B
ω=
 θ̇ 2

 (4.35)
 
θ̇1 sin θ2
 
θ̈1 cos θ2 − θ̇1 θ̇2 sin θ2 
 
B
ω̇ = 
 θ̈ 2

 (4.36)
 
θ̈1 sin θ2 + θ̇1 θ̇2 cos θ2
 
 cos θ2 0 sin θ2 
 
E 
B R =  sin θ1 sin θ2 cos θ1 - sin θ1 cos θ2  (4.37)

 
- cos θ1 sin θ2 sin θ1 cos θ1 cos θ2

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 Chapter 4. Kinematically redundant non-gyro IMUs

4.5 Simulation studies

Simulation studies were used for the initial testing of the proposed methods
and to investigate the relationship between system accuracy and redundancy.
Simulations used the damped 2 axis pendulum model with an array of n triple-axis
accelerometers positioned evenly around a 360◦ helix concentric with the length of
the pendulum and with a radius of 0.05 m. Figure 4.4 shows the pendulum and
triple-axis accelerometers as drawn by the simulation software. For convenience,
all sensors were aligned with the pendulum body coordinate frame; arbitrary
orientations were used and calibrated for in the experimental studies discussed
in Section 4.6.

Two axis pendulum simulation

Earth frame
Pendulum frame and arm
Pendulum axes
Accelerometer
Accelerometer x axes
0.05
Accelerometer y axes
0 Accelerometer z axes

−0.05
Z (m)

−0.1 Mass = 0.5 kg

Length = 0.3 m
−0.15 Axis 1 damping = 0.1 Ns/m
Axis 2 damping = 0.1 Ns/m
−0.2

−0.25
−0.2
0
−0.2 −0.1 0.2
0 0.1 0.2 0.3
X (m)
Y (m)

Figure 4.4: Schematic of pendulum with triple-axis accelerometers mounted in

helix pattern. Graphics generated by simulation software

The forward kinematics were computed to provide ideal sensor accelerometer

measurements and processed through a sensor model to simulate the frequency
response and noise characteristics. The sensor model was a simplification of a
Simulink based accelerometer behavioural model provided by the manufacturer
(Analog Devices). The noise model consisted of zero mean additive Gaussian noise

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Chapter 4. Kinematically redundant non-gyro IMUs

with a standard deviation of 0.045ms−2 chosen to match the experimental data.

The signal was then processed through a first order low-pass filter with a corner
frequency of 50 Hz. The angular velocity estimation filter used a fixed gain of
K = 0.5. Pendulum model parameters were fixed so that m = 0.5 kg, l = 0.3 m,
g = 9.8 ms−2 , c1 = 0.1 Nsm−1 and c2 = 0.3 Nsm−1 , with initial conditions of
θ1 = θ2 = 60◦ , and θ̇1 = θ̇2 = 0. The simulation was run for 30 seconds with a
discrete-time step of 1 ms.

To investigate the relationship between the redundancy and accuracy of a sensor

array, simulations were conducted for arrays consisting of between 5 and 100
triple-axis sensors. For each array the mean of the RMS errors in the x, y and
z dimensions was computed for the linear acceleration, angular acceleration and
angular velocity. The error in the estimated orientation was not investigated as
integral drift would fail to represent a meaningful relationship. Results were
obtained using both the combinatorial and pseudoinverse methods to enable
comparison.

The integral drift inherent to the angular velocity estimation algorithm would
introduce a bias in the estimated velocity at low rates. This random steady
state error meant that the apparent relationship between the angular velocity
error and redundancy was non-monotonic. To compensate for this, the mean
result of 5 simulations was computed for each value of redundancy. The results
for angular acceleration and velocity are shown as Figure 4.5 and Figure 4.6.
It can be seen that the combinatorial method results in an greater error than
the pseudoinverse method at greater computational expense. Simulations of the
combinatorial method for arrays > 33 sensors were abandoned as the time taken
to process arrays of this size became impractical. The non-monotonic relationship
observed in the estimated angular velocity is due to the ‘random walk’ in this
state that comes about at low velocities and is constrained by the feedback in
the angular velocity estimation filter. This ‘random walk’ is due to the noise

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 Chapter 4. Kinematically redundant non-gyro IMUs

(in simulation) of the Analog Devices accelerometers. As the angular velocity is

reduced (due to damping), noise dominates the signal and the estimated angular
velocity is then largely governed by the integration of the angular acceleration
which again is largely made up of the sensor noise causing a random walk. This
error does not directly relate to the number of sensors.

Oscillatory motion of the pendulum and the frequency response of the sensors
results in a minimum achievable error represented by the oscillatory error
components due to the phase-lag of the sensor outputs (due to a simulated
anti-aliasing filter). This minimum achievable error was computed by running a
simulation using an accelerometer error model comprised of only the anti-aliasing
filter, i.e. zero noise. The resultant error is the same for arrays of all sizes and
represents the minimum achievable error. The respective minimum achieve errors
are indicated in Figure 4.5 and Figure 4.6.

0.26 0.038
mean[RM S[pǫ ], RM S[qǫ ], RM S[rǫ ] (rads−1 )
mean[RM S[ṗǫ ], RM S[q̇ǫ ], RM S[ṙǫ ] (rads−2 )

Pseudoinverse method Pseudoinverse method

0.24 Combinatorial method 0.036 Combinatorial method
Minimum achievable Minimum achievable
0.22 0.034

0.2 0.032

0.18 0.03

0.16 0.028

0.14 0.026

0.12 0.024

0.1 0.022
5 20 40 60 80 100 5 20 40 60 80 100
Redundancy (number of tri-axis sensors) Redundancy (number of tri-axis sensors)

Figure 4.5: Angular acceleration error vs.Figure 4.6: Angular velocity error vs.
sensor redundancy sensor redundancy

The pendulum testbed allows more complex motion to be evaluated. If the initial
conditions use a non-zero velocity vector and dissimilar damping coefficients, all
cross terms in Equation 4.7 can be elicited at a range of frequencies. The spectrum
of any individual accelerometer axis will contain frequencies up to the natural
frequency of the pendulum and hence the results can be demonstrated to work
across this frequency range. The upper graphs in Figures 4.7 and 4.8 demonstrate

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Chapter 4. Kinematically redundant non-gyro IMUs

this result for a simulation of a cluster of 20 accelerometers placed in a 5 cm radius

helix beginning half way down the pendulum and continuing to the end. When
viewed in isolation this is readily observed to be a non obvious complex motion.
The error residuals are given in the lower graphs of Figures 4.7 and 4.8.

ṗ
50 q̇
ṙ
rads−2

−50

−100
3 3.5 4 4.5 5 5.5 6

5 ṗǫ
q̇ǫ
ṙǫ
rads−2

−5
3 3.5 4 4.5 5 5.5 6
Time (s)

Figure 4.7: Angular accelerations of pendulum measured by accelerometer array

(top) and residuals (bottom)

p̃′
5 q̃ ′
r̃ ′
rads−1

−5

2 2.5 3 3.5 4 4.5 5 5.5 6 6.5

0.8
p̃′ǫ
0.4 q̃ǫ′
r̃ǫ′
rads−1

−0.4

−0.8
2 2.5 3 3.5 4 4.5 5 5.5 6 6.5
Time (s)

Figure 4.8: Estimated angular velocity compensated by orientation estimation

algorithm (top) and residuals (bottom)

4.6 Experimental studies

Experimental studies were conducted to verify the above simulation results. The
studies were designed to be as similar to the simulation studies as possible and

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 Chapter 4. Kinematically redundant non-gyro IMUs

used a redundant array of 10 triple-axis accelerometers. Unlike the simulation

studies, the experimental studies were subject to 2 significant sources of error:
sensor calibration errors (of gains and biases) and sensor installation errors (of
position and alignment).

4.6.1 Hardware

Experimental studies required data to be acquired from an array of 10 triple-axis

accelerometers at a sample rate of at least 1 kHz (to match simulation
investigations) and with as high a resolution as practicable; sensor resolution
had been identified as a limiting factor in similar studies [71]. To meet these
requirements, custom equipment was developed for this research project, consisting
of a DAQ32 data acquisition board PC Interface software and 10 triple-axis
accelerometers.

4.6.1.1 Data acquisition equipment

The DAQ32 is an open-source1 32 channel Universal Serial Bus (USB) data

acquisition board built around Microchip’s high-end dsPIC33FJ256GP710 Digital
Signal Controller. The DAQ32 features a filtered power supply for analogue sensors
and 32 analogue input channels each sampled at 1.024 kHz. Each channel is
oversampled by a factor of 8 and the truncated mean result used for enhanced
filtering and accuracy.

4.6.1.2 Accelerometer probes

Each accelerometer probe consisted of an Analog Devices’ ADXL335 triple-axis

accelerometer with signal and power supply conditioning circuitry assembled on

1
https://github.com/xioTechnologies/DAQ32

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Chapter 4. Kinematically redundant non-gyro IMUs

the accelerometer package to minimise size and weight. The probe’s 0.4 m
lead consisted of wires twisted together so that the power supply wires would
provide a level of electromagnetic shielding to the accelerometer signals. Fine wire
connections were sunk in a silicon sealant to provide mechanical durability. The
complete probe (shown in Figure 4.9) occupied a volume of a 5 mm cube and
weighed less than 1 g (excluding lead).

Figure 4.9: Complete accelerometer probe next to a 150 mm ruler

4.6.1.3 Test rig

The pendulum, shown in Figure 4.10, was constructed from 2 perpendicular

sheets of aluminium so that the 10 accelerometer probes on the surface of the
sheets formed an approximate helix with a length of 350 mm and radius of 50
mm. Each accelerometer was mounted at arbitrary orientations varying up to
180◦ . The position of each accelerometer was measured. The pendulum joint
was constructed from 2 precision servo potentiometers so that angular position
of each axis could be measured. The 32 analogue channels required by the 10
triple-axis accelerometer probes and 2 potentiometers were all accommodated by
the DAQ32. Figure 4.10 shows the pendulum with attached accelerometer probes
and annotated potentiometer axes θ1 and θ2 .

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 Chapter 4. Kinematically redundant non-gyro IMUs

Simulation was based on a helix, the experimental rig needed to be as stiff as

possible so a cross structure was chosen. This is equivalent to a four-spoke helix.
This is a reasonable compromise to allow the experimental rig to be manufactured.
The cross structure in the pendulum allows for greater stiffness in the pendulum
and higher accuracy when placing the sensors.

Figure 4.10: 2 axis pendulum with attached accelerometer probes and annotated
potentiometer axes θ1 and θ2

4.6.2 Calibration

4.6.2.1 Axis gain and bias

Assuming the accelerometer output to be linear, the calibrated measurement, α,

may be obtained from the un-calibrated measurement, u, using the relationship
described by 4.38. The vector b and diagonal matrix K account for the

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Chapter 4. Kinematically redundant non-gyro IMUs

individual axis bias and sensitivity respectively. The skew matrix S accounts
for non-orthogonality and coupling between the accelerometer axes. The gain smn
defines the proportion of axis n affecting axis m.

α = SKu − b
     
 1 sxy sxz  kx 0 0  ux  bx  (4.38)
     
= s
 yx 1 s   0 k 0  u  − b 
yz   y   y  y
     
szx szy 1 0 0 kz uz bz

A stationary accelerometer will measure the direction and magnitude of the

Earth’s gravitational field due to the reaction force on the mass within the MEMS
structure. For a calibrated sensor, the measured magnitude will be constant at
all orientations of the sensor. That is, Equation 4.39 will be true for all said
measurements where g is the magnitude of gravity and the sensor is stationary.
Therefore, K and b may be found as the solution to Equation 4.40 where uj
represents the un-calibrated sensor measurement at the j’th orientation within a
calibration dataset. Although it is possible to compound S and K in to a single
matrix, it was found this would limit the success of an optimisation algorithm as
it became sensitive to the initial guesses of the gain parameters.

g = kSKu − bk (4.39)

X
min (g − kSKuj − bk)2 (4.40)
S,K,b∈<
j

A calibration dataset was collected using an automated calibration gimbal capable

of rotating independently around the pan, roll and tilt axes. The pendulum (with
attached accelerometer probes) was fixed to the gimbal so that it could be rotated
to any orientation. An automated routine was composed to rotate the gimbal

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 Chapter 4. Kinematically redundant non-gyro IMUs

pitch and roll axes through their full range in 10◦ steps so that every combination
of pitch and roll angle was achieved. At each orientation, the gimbal was allowed
10 seconds to come to rest and then the mean of each accelerometer output was
taken for a 15 second period.

A value of g was obtained as 9.812 ms−2 for the geographical location of the
studies [113]. Equation 4.40 was solved using MATLAB’s Optimisation toolbox’s
fminunc function and the Broyden-Fletcher-Goldfarb-Shanno (BFGS) method.
Initial values for gains and biases were chosen as the typical values detailed in
the AXDL335 datasheet. Calibration was verified by comparing the measured
magnitude of gravity with the known magnitude at each orientation within the
calibration dataset. Figure 4.11 shows typical calibrated measurements of one
accelerometer for the full calibration dataset.
Calibrated accelerometer output
10
αx
αy
αz
5
ms−2

−5

−10
Calibrated accelerometer measured magnitude of gravity
9.85 q
α2x + αy2 + αz2
ms−2

9.8 g

9.75
100 200 300 400 500 600 700
Dataset sample

Figure 4.11: Validation the calibrated accelerometer gains and biases

4.6.2.2 Alignment

Hardware installation practicalities mean that misalignment between the

accelerometers and the pendulum coordinate frame must be accounted for in each
triple-axis accelerometer. Alignment errors as low as 0.1◦ have been shown to
result in significant errors in estimated kinematic states [86]. An aligned triple-axis

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Chapter 4. Kinematically redundant non-gyro IMUs

0
accelerometer measurement, i α, is related to the misaligned measurement i α by
the rotation matrix ii0 R as described by Equation 4.1. The alignment may be
equivalently represented by the quaternion ii0 q. If the array is stationary, Equation
4.41 will be true for sensors at all orientations. Therefore ii0 q may be found as the
0
solution to 4.42 where i αj and B aj represent the measured acceleration of the i’th
accelerometer and the acceleration of the body at the j’th orientation respectively.
As the acceleration of the body is unknown, one fiducial sensor must be chosen
B
to provide a measurement equal to a. An alternative kinematic minimisation
approach for alignment calibration was also implemented but performance was
found to be no better than simply choosing a convenient sensor to be the fiducial.

0
B
a = ii0 q̂ ∗ ⊗ i α ⊗ ii0 q̂ (4.41)

Xh i2
i ∗ i0 i B
min
i q∈<
i0 q̂ ⊗ αj ⊗ i0 q̂ − aj (4.42)
i0 j

Equation 4.42 was solved using the same calibration dataset and method described
in Section 4.6.2.1. The fiducial sensor was chosen as that judged to be best
aligned to the pendulum on visual inspection. This gravitational based attitude
calibration scheme is similar to previously proposed techniques; however, the use of
a quaternion representation avoids problems of non-orthogonality associated with
a rotation matrix based approach [114].

4.6.3 Results

The pendulum was left to swing with motion in both axes remaining in phase
until rest while data from each of the 10 triple-axis accelerometers and 2 joint
potentiometers was recorded. The ‘true’ suspended body kinematics were then
estimated for the first 30 seconds of the dataset using the potentiometer data and

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 Chapter 4. Kinematically redundant non-gyro IMUs

kinematic model within a moving horizon state estimation algorithm.

Figure 4.12 and Figure 4.13 show the pendulum kinematic states as estimated
using the pseudoinverse method and data from all 10 sensors. The angular
velocity estimation algorithm used a gain of K = 2 and the orientation estimation
algorithm used gains of KP = 5 and KI = 1.0. Each figure consists of 2 plots: the
upper plot represents the measured or estimated state and the lower plot represents
the corresponding error. Small errors in alignment calibration and the constant
acceleration of gravity result in an angular velocity bias error and low velocities
(see Figure 4.13). The orientation estimation algorithm described in Section 4.3
compensates for this error in p and q but errors r cannot be compensated for as
this axis of rotation eventually aligns with the direction of gravity.
Angular acceleration
40
ṗ
20 q̇
ṙ
rads−2

−20

−40
0 5 10 15 20 25 30
Angular acceleration error
5 ṗǫ
q̇ǫ
ṙǫ
rads −2

−5
0 5 10 15 20 25 30
Time (s)

Figure 4.12: Angular accelerations of pendulum measured by accelerometer array

(top) and residuals (bottom)

To investigate the relationship between error and redundancy, the suspended body
kinematics were computed using the pseudoinverse method for each combination
of sensors for the available array of ten sensors, thus results from four to ten
sensors were possible. Since the number of combinations for a given number of


sensors is defined by 10
n
, 210 results were possible for 4 sensors, rising to 252 for
5 sensors and then falling to give only the single result when the fully redundant
array of 10 sensors was computed. Performance is quantified as the mean of the

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Chapter 4. Kinematically redundant non-gyro IMUs

Compensated estimate of angular velocity

5
p̃′
q̃ ′
0 r̃ ′
rads−1

−5

−10
0 5 10 15 20 25 30
Compensated estimate of angular velocity error
0.8
p̃′ǫ
0.4 q̃ǫ′
r̃ǫ′
rads −1

−0.4

−0.8
0 5 10 15 20 25 30
Time (s)

Figure 4.13: Estimated angular velocity compensated by orientation estimation

algorithm (top) and residuals (bottom)

RMS error in the x, y and z DOF for linear acceleration, angular acceleration and
angular velocity. These results are shown as box plots in Figure 4.14 The results
compare well with the simulation studies given in Section 4.5. The anomalous
rise apparent in the third graph of Figure 4.14 when all 10 sensors are used is an
artefact of the imperfect alignment of sensors causing the information of one sensor
to dominate the calculation. This would suggest a practical system could make a
confidence estimate of individual sensors and down grade sensors that might be
close to saturation, become mechanically misaligned (or knocked), or operating at
a low signal to noise ratio.

Linear acceleration error Angular acceleration error Angular velocity error

mean[RM S[pǫ ], RM S[qǫ ], RM S[rǫ ] (rads−1 )
mean[RM S[ṗǫ ], RM S[q̇ǫ ], RM S[ṙǫ ] (rads−2 )
mean[RM S[ẍǫ ], RM S[ÿǫ ], RM S[z̈ǫ ] (ms−2 )

3.5

0.9 1.6
3
0.8 1.4

0.7 2.5 1.2

0.6 1
2
0.5 0.8

0.4 1.5 0.6

0.3 0.4
1
0.2 0.2

0.1 0.5 0
4 5 6 7 8 9 10 4 5 6 7 8 9 10 4 5 6 7 8 9 10
Redundancy (number of tri-axis sensors) Redundancy (number of tri-axis sensors) Redundancy (number of tri-axis sensors)

Figure 4.14: Relationship between error and sensor redundancy for linear
acceleration (left), angular acceleration (middle), angular velocity (right).
Whiskers indicate 1.5× interquartile range.

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 Chapter 4. Kinematically redundant non-gyro IMUs

4.7 Discussion

This work has shown a clear advantage of higher levels of redundancy.

Incorporating a large number of sensors requires a practical consideration of the
computational load. The pseudoinverse method has shown not to only provide
greater accuracy but to do so at significantly less computational expense. As the
matrix inversion is required to be computed only once, the computational load of
the pseudoinverse is proportional to the number of sensors in the array.

Due to sensor alignment errors and the constant acceleration of gravity, the
estimate of angular velocity r was subject to a bias error at low velocities
(< 0.5 rads−1 ). Observations of the direction of gravity enabled the orientation
estimation algorithm to compensate for such errors in p and q; however,
compensation is not possible when the axis of r is parallel with the gravity
vector. Such compensation may be achieved if the algorithm is extended to
incorporate information from a magnetometer [59]. An accelerometer array cannot
be used to determine directly the direction of an angular velocity. The angular
velocity estimation algorithm presented can potentially assert an incorrect sign
when angular velocities and accelerations are both of low magnitude. The velocity
estimation method appeared to be robust in dealing with this problem although a
sign switch is possible.

An accelerometer array IMU presents a number of benefits to a biomedical

application. Unlike a conventional IMU it is able to provide direct measurement
of angular acceleration which may provide new insight to movement studies.
Redundancy provides increased levels of accuracy, potentially allowing a system
to be created where a single (more expensive) high quality sensor may be replaced
by many (cheaper) lower quality sensors.

Calibration has been limited to being off-line using a specific calibration dataset

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Chapter 4. Kinematically redundant non-gyro IMUs

so that use of accelerometer arrays is limited to ridged structures where the

orientation and location of each accelerometer is fixed. This is immediately
applicable to robotics and avionics applications where the sensor array can be
installed to an ridged chassis. If future work can achieve an on-line calibration
method capable of estimating changes in individual accelerometer locations and
orientations using the natural motion of the subject, then use of accelerometer
arrays can be extended to non-rigid bodies. This would be of interest to biomedical
applications as it would allow the use of accelerometers woven into a wearable
fabric. In the interim, sensors could be embedded within a rigid orthosis such as
an angle foot orthosis, or onto rigid clothing inserts. In either case, the resulting
sensor will allow continuous, low power and accurate measurements with minimum
obstruction to patient movements and lifestyle.

This work has demonstrated how redundancy in an array of accelerometers can

improve measurement accuracy. Redundancy of such homogeneous arrays opens
avenues for further exploration on object dynamics, in particular considering
relaxing the constraints by considering multiple sensors attached to an elastic
object, such as skin, or constrained to move in a complex relationship (such as
clothing), or subject to vibration modes, (such as an aeroplane wing) or included
a segmentation such as biomechanical joint was included as a constraint.

4.8 Conclusion

This work analyses the benefits of using large arrays of accelerometers. Results
include both simulation studies of up to 100 tri-axial sensors and empirical results
of an array of up to 10 MEMS accelerometers. A comparison is given between
simulation and empirical data based on a 2 axis pendulum simulating a knee.

Two methods are reported for the reconstruction of an acceleration state vector

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 Chapter 4. Kinematically redundant non-gyro IMUs

to represent the limb. Both methods allow the pre-computation of the necessary
matrix inverse. The method based on computing a pseudoinverse (using singular
value decomposition) is shown to achieve better accuracy than that based on
combinatorial averaging of the sensors, and at less computational expense.

A limitation of accelerometer arrays is that it is not possible to recover the sign of

the angular velocities directly so a method is given that tracks velocity transitions
through this zero transition. Finally, the work gives a method for assessing the
optimal placement of sensors in a large planar array, describes a hardware solution
and discusses a method for calibrating large arrays of accelerometer sensors.

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Chapter 4. Kinematically redundant non-gyro IMUs

76
Chapter 5

Characterisation of low-cost
MEMS sensors

5.1 Introduction

The objective of this project is to demonstrate how new applications can be realised
using modern low-cost MEMS gyroscopes, accelerometers and magnetometers. A
key part of achieving this is the development of calibration solutions so that each
sensor is able to provide precise measurements with a determined accuracy. For
example, it is shown in Chapter 6 that calibration of the accelerometer bias and
sensitivity may reduce orientation measurement errors from 8◦ to <0.5◦ .

Low-cost MEMS are typically targeted towards consumer electronics that do not
demand a high level of accuracy. A competitive market means that many modern
devices are supported by the manufacture for only a few years before being
superseded by the next generation. As a result, detailed information about sensor
characteristics and performance is often unavailable. It was therefore necessary for
this project to determine significant sensor characteristics as a first step towards
the development of calibration solutions. This chapter presents the methods and

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Chapter 5. Characterisation of low-cost MEMS sensors

findings of the sensor characterisations studies that form the basis of the calibration
solutions presented in the next chapter.

The investigations presented in this the chapter focus on the sensor performance
over an operating temperature range or at a fixed temperature. For gyroscopes
and accelerometers, this requires precise controlled motion while simultaneously
regulating the device temperature. Ideally, this would be achieved using an
industrial three-axis thermal chamber. For example, the Ideal Aerosmith 2003HP
shown in Figure 5.1 is able to achieve an accuracy in angular velocities of 0.0001%
and orientations of ±1 arcsecond (∼0.00027◦ ) for temperatures between -65◦C to
+85◦C [115]. Temperature is regulated using electric heating and Liquid Nitrogen
(LN2 ) or Carbon Dioxide (CO2 ) cooling.

Figure 5.1: Ideal Aerosmith 2003HP three-axis thermal chamber of achieve precise
velocities and orientations for temperatures between -65◦C to +85◦C. Image source:
[115]

Although such industrial equipment would facilitate a comprehensive

characterisation of gyroscopes and accelerometers, the hire cost made it
inaccessible to this project [116]. Similar characterisation studies presented in
the subject literature take advantage of such equipment [117, 28, 118]. The
investigations presented in this chapter are achieved using improvised solutions.

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 Chapter 5. Characterisation of low-cost MEMS sensors

5.1.1 Devices selected for characterisation

Several different gyroscopes, accelerometers and magnetometers from a range of

manufacturers were used throughput this project. This chapter presents the
characterisation of the two devices that would ultimately be incorporated to
the IMU platforms described in Chapter 8. The devices are the InvenSense
MPU-6050 combined triple-axis gyroscope (±2000◦/s range, 16-bit resolution) and
accelerometer (±16 g range, 16-bit resolution), and the Honeywell HMC5883L
triple-axis magnetometer (±8 Gauss range, 12-bit resolution). Both devices
incorporate an Analogue-to-Digital Converter (ADC) and provide measurements in
units of least-significant bit (lsb) via Inter-Integrated Circuit (I2 C) communication
bus. The MPU-6050 also incorporates an on-board thermometer for accurate
measurement of the sensor temperature.

Figure 5.2: MPU-6050 (top IC) and HMC5883L (bottom IC) mounted on the
x-IMU PCB. The larger IC is 4×4 mm in size.

Investigations were conducted using prototypes of the x-IMU and x-BMIU

platforms presented in Chapter 8. Both platform incorporates the MPU-6050 and
HMC5883L along side wireless communication or Secure Digital (SD) card logging
to facilitate data collection. Figure 5.2 shows the MPU-6050 and HMC5883L
mounted on a Printed Circuit Board (PCB).

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Chapter 5. Characterisation of low-cost MEMS sensors

5.1.2 Scope of investigations

Standards such as IEEE 1431-2004 [119] and 1293-1998 [120] offer an extensive
list of gyroscope and accelerometer characteristics that may be quantified. The
investigations presented in this chapter are limited to the characteristics indicated
as being most significant in the the device datasheets [121, 122] with a particular
focus on the gyroscope bias. The scope of the investigations is summarised by the
following.

• Gyroscope bias temperature sensitivity

• Gyroscope bias acceleration sensitivity

• Gyroscope bias random walk

• Gyroscope sensitivity and bias thermal response

• Accelerometer and magnetometer thermal response

• Sensitivity and bias variation between parts for gyroscope, accelerometer and
magnetometer.

The following sections present the methods and findings of each investigation in
the order above.

5.2 Gyroscope bias thermal response

The MPU-6050 datasheet specifies the gyroscope bias “variation over temperature”
as ±20◦/s over -40◦C to 80◦C [121]. Further information about characteristic, such
as linearity or repeatability, is not specified in the datasheet or any other available
documentation. The absence of such information about this potentially very large
measurement error exemplifies the necessity for these characterisation studies. The
gyroscope bias thermal response was characterised for eight devices by logging

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 Chapter 5. Characterisation of low-cost MEMS sensors

the gyroscope output when stationary within a thermal chamber to expose the
gyroscope to a controlled temperature range.

5.2.1 Thermal chamber

A thermal chamber was designed to provide a stable temperature ascent from

-20◦C to 80◦C using a thermal mass and active heating. The chamber was
constructed from a polystyrene box originally intended to enclose cold goods during
transportation. Figure 5.3 shows the chamber internal components including: a
41 m3 /hour fan, 1 kg block of copper with aluminium fins, 20 W fan heater and
a thermalcouple probe. The prototype x-IMU being tested is positioned in the
centre and interfaced externally via a wired connection.

41 m3 /hour fan 1 kg copper thermal mass (with aluminium fins)

Prototype x-IMU Thermocouple probe 20 W fan heater

Figure 5.3: Thermal chamber internal components including the prototype x-IMU
being tested

During operation, the fan continuously blows on to the thermal mass to circulate
air within the chamber and transfer heat produced by the x-IMU to the thermal
mass. The fan heater blows directly on to the thermal mass and away from the

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Chapter 5. Characterisation of low-cost MEMS sensors

x-IMU to prevent rapid thermal dynamics. Separate variable power supplies were
used to power the fan and heater and a thermalcouple meter used to monitor
internal temperature. Software was written to log sensor from the x-IMU and
display temperature measurements in real-time to guide manual control of the
heater. Figure 5.4 shows the sealed thermal chamber with separate variable power
supplies and thermocouple meter.

Figure 5.4: Sealed thermal chamber with thermocouple meter and separate
variable power supplies for fan and heater

Eight x-IMUs were tested individually. For each test, the thermal chamber and
x-IMU were initially left in an industrial freezer at -20◦C for >12 hours. The
chamber was then sealed within the freezer before being removed to conduct the
experiment. The heat produced by the fans and x-IMU caused the temperature to
rise of ∼5◦C over the first 5 minutes. The heater power supply was then gradually
increased to maintain a fixed ascent of ∼0.5◦C per minute over 3 hours. The
temperature ascent for each of the eight devices (named ‘A’ to ‘H’) is shown in
Figure 5.5.

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 Chapter 5. Characterisation of low-cost MEMS sensors

A B C D E F G H
80

70

60

50

Temperature (◦ C) 40

30

20

10

−10

−20
0 15 30 45 60 75 90 105 120 135 150 165 180 195
Time (minutes)

Figure 5.5: Measured gyroscope temperature during experiments for each of the
eight devices tested

5.2.2 Monotonic thermal response

The measured gyroscope bias for each of the eight devices over a -20◦C to 70◦C
temperature range is shown in Figure 5.6. The three plots indicate the bias for
the x, y and z axes termed bωx , bωy and bωz . The gyroscope output originally in
units of lsb was converted to degrees per second using the typical sensitivity of
16.4 lsb/◦/s specified in the datasheet.
A B C D E F G H

3 3 3

2.5 2.5 2.5

2 2 2

1.5 1.5 1.5

1 1 1

0.5 0.5 0.5

0 0 0
bωx (◦ /s)

bωy (◦ /s)

bωz (◦ /s)

−0.5 −0.5 −0.5

−1 −1 −1

−1.5 −1.5 −1.5

−2 −2 −2

−2.5 −2.5 −2.5

−3 −3 −3

−3.5 −3.5 −3.5

−4 −4 −4
−25 −15 −5 5 15 25 35 45 55 65 75 −25 −15 −5 5 15 25 35 45 55 65 75 −25 −15 −5 5 15 25 35 45 55 65 75
Temperature ( C) ◦
Temperature ( C) ◦
Temperature (◦ C)

Figure 5.6: Gyroscope bias thermal response of eight devices over a -20◦C to 70◦C
temperature range

The bias variation over temperature is within the ±20◦/s range specified in the

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Chapter 5. Characterisation of low-cost MEMS sensors

datasheet though the relationship was revealed to be both non-linear and unique to
each device. A bias calibration solution that accounts for this variation would need
to incorporate a non-linear model evaluated uniquely for each device. Although
some devices may be approximated by a linear model over a limited temperature
range, this would not be sufficient for all; notably, bωy of device ‘E’ approximates
a parabola centred at 25◦C.

A calibration solution may model the thermal response of the bias as an nth
order polynomial function of temperature. To determine the necessary order, a
polynomial fit was performed for each device for first to fifth order polynomials
and the resultant maximum error of all devices summarised in Table 5.1. Each
polynomial was computed using the MATLAB polyfit function with data points
computed as the mean of each consecutive 2◦C bin.

Maximum error (◦/s)

Polynomial order, n
bωx bωy bωz
1 0.328 0.400 0.395
2 0.190 0.146 0.144
3 0.041 0.120 0.109
4 0.038 0.053 0.040
5 0.037 0.047 0.035

Table 5.1: Gyroscope bias error assuming an nth order temperature calibration
model

Table 5.1 suggests at least a fourth order polynomial is necessary to account for
the non-linear response over the investigated temperature range. At least five data
points would be required to approximate such a polynomial. In practice, using the
minimum number of data points will result in a less accurate approximation; the
above accuracy was achieved using thousands of data points obtained over a three
hour temperature ascent. Evaluating calibration parameters in this way for a large
number devices would represent a significant practical challenge.

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 Chapter 5. Characterisation of low-cost MEMS sensors

5.2.3 Thermal hysteresis

The gyroscope bias thermal response presented in Section 5.2.2 was obtained for a
slow, monotonic temperature ascent. These experiments were extended to observe
the bias behaviour when exposed to a range of thermal dynamics that could be
achieved with the chamber, including: slow cooling, rapid cooling, rapid heating
and thermal cycling. These investigations exposed a hysteresis characteristic of
the gyroscope that was not described within any of the available manufacturer’s
documentation. Figure 5.7 shows the measured temperature during one such
experiment that exemplifies the observations. The annotated markers ‘a’, ‘b’,
‘c’ and ‘d’ indicate key events during the experiment.

C
80
(b)
70
(c)
60

50
Temperature (◦ C)

40

30 (d)
20

10

0
(a)
−10

−20
0 15 30 45 60 75 90 105 120 135 150 165 180 195 210 225
Time (minutes)

Figure 5.7: Measured gyroscope temperature during an experiment to demonstrate

thermal hysteresis

The first ∼180 minutes of the experiment, from point ‘a’ to point ‘b’, show the
temperature ascent to ∼70◦C presented in Section 5.2.2. The heater was then
switched off and the temperature slowly descended for ∼15 minutes until point
‘c’. The thermal chamber lid was then removed and the temperature descended
rapidly until stabilising at ∼25◦C at point ‘d’. The bias response is shown as
5.8. The annotated markers correspond to the events previously described within
Figure 5.7. These plots indicate thermal hysteresis in all three axes.

Although this thermal hysteresis is a known characteristic [123, 27], there is little

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Chapter 5. Characterisation of low-cost MEMS sensors

2.2 0.1 3
(a) (a)
0.05 2.9 (b)
2
2.8
0
1.8 2.7 (c)
−0.05
(d) 2.6
1.6 −0.1
(b) 2.5
1.4 −0.15 (c) 2.4
−0.2
bωx (◦ /s)

bωy (◦ /s)

bωz (◦ /s)
2.3
1.2
−0.25 2.2
1
−0.3 2.1

0.8 2 (d)
−0.35
1.9
0.6 −0.4 (d)
1.8
−0.45
0.4 (c) 1.7
−0.5
1.6
0.2
(a)
(b) −0.55 1.5
0 −0.6 1.4
−25 −15 −5 5 15 25 35 45 55 65 75 −25 −15 −5 5 15 25 35 45 55 65 75 −25 −15 −5 5 15 25 35 45 55 65 75
◦ ◦
Temperature ( C) Temperature ( C) Temperature (◦ C)

Figure 5.8: Thermal response of the gyroscope bias indicating thermal hysteresis

information available on the specific behaviour that may be expected. InvenSense

were unable to offer an explanation as they do not thermal cycle their devices,
only test that the devices operate over the specific temperature range [124]. The
thermal response seen in Figure 5.8 suggests that the hysteresis is sensitive to both
the change in direction of temperature and the rate of change of temperature. The
hysteresis of bωy is apparent the moment the device begins to cool from point ‘b’.
However, bωz seems to indicate little or no hysteresis for the ∼15 minutes of slow
cooling but immediately shifts when rate of cooling is increased at point ‘c’.

The observed hysteresis indicates that the polynomial approximation of the

thermal response previously suggest in Section 5.2.2 may be insufficient. For
example, a polynomial fit to the initial monotonic response of bωy would yield
an error of 1.55◦/s after cooling.

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5.3 Gyroscope bias acceleration sensitivity

The MPU-6050 datasheet indicates a typical gyroscope bias sensitivity to static

acceleration of 0.1◦/s/g [121]. A sensitivity to static acceleration suggests that the
error is dependent on the orientation of the device relative to gravity and could may
be compensated for through a calibration model that incorporates accelerometer
measurements. The ‘g-sensitivity’ calibration model proposed by Bancroft et al.
incorporates a 3×3 matrix that suggests each axis bias is sensitive to accelerations
in all directions. To investigate this characteristic, the device was mounted on a
cube structure so that the six orientations of the cube on a flat surface would align
each gyroscope axis with +1 g, -1 g and 0 g. The gyroscope bias was calculated
as the mean of ∼40,000 samples per taken over 5 minutes for each orientation. 27
different devices were tested to demonstrate statistical significance.

50 50 50

40 40 40

30 30 30
∆bωx (0.001◦ /s)

∆bωy (0.001◦ /s)

∆bωz (0.001◦ /s)

20 20 20

10 10 10

0 0 0

−10 −10 −10

−20 −20 −20

−30 −30 −30

+X −X +Y −Y +Z −Z +X −X +Y −Y +Z −Z +X −X +Y −Y +Z −Z
Direction of gravity Direction of gravity Direction of gravity

Figure 5.9: Gyroscope bias deviation for six different orientations relative to
gravity. Box plots show the distribution for 27 devices (whiskers indicate 1.5×
interquartile range).

The deviation of the gyroscope bias for each axis, ∆bωx , ∆bωy and ∆bωz , was
calculated as the difference between the gyroscope bias at a given orientation
from the mean for all orientations. The box plots in Figure 5.9 shows the
distribution of ∆bωx , ∆bωy and ∆bωz for all six orientations of the gyroscope
relative to gravity. The data was scaled to units of degrees per second using

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the typical sensitivity specified in the datasheet. Although the specific nature of
the acceleration sensitivity is unknown, the box plots were expected to indicate a
correlation between the direction of gravity and an offset error of with an order
of magnitude of ∼0.1◦/s. Instead, the box plots indicate a mean deviation of
approximately zero and a variance representative of noise. These results suggest
that, contrary to the manufacturer’s specification, the gyroscope bias does not
have a significant sensitivity to static acceleration.

5.4 Gyroscope bias random walk

The MPU-6050 datasheet does not provide any information about the gyroscope
bias time-variant random walk and so an investigation was conducted to quantify
this characteristic. Four x-IMUs were sealed in the thermal chamber, each set up
to log the gyroscope output and thermometer measurements to the on-board SD
card at a rate of 512 samples per second. The thermal chamber was intended only
to isolate the internal temperature from the ambient environment; the internal
fan and heater were not used. Data was logged over a 48 hour period and then
cropped to select a period with minimal temperature variations.
0.1
∆T (◦ C)

−0.1
0 1 2 3 4 5 6 7 8 9 10

X Y Z
150

100
∆bω (◦ /hour)

50

−50

−100
0 1 2 3 4 5 6 7 8 9 10
Time (hours)

Figure 5.10: Gyroscope bias random walk over 10 hours

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 Chapter 5. Characterisation of low-cost MEMS sensors

Figure 5.10 shows the measured temperature variation, ∆T , and gyroscope bias
variation, ∆bω , for one device over a 10 hour period with a temperature variation of
±0.025◦C. The measurement were average for each 30 second window to attenuate
the high-frequency noise. Bias measurements were scaled to units of degrees per
hour using the typical sensitivity specified in the datasheet. The results shown
above are representative of the observations made for all devices and indicate a
bias variation that does not correlate to temperature. This random walk resulted
in a peak error of 190◦ /hour (∼0.053◦/s) for the four devices over the 10 hour
period.

5.4.1 Allan variance

Plotting the gyroscope bias variation over time clearly demonstrates random walk
but does not provide an effective means for quantifying this characteristic. Allan
variance is the accepted standard for evaluating gyroscope bias stability [125].
Stockwell provides a concise introduction to the implementation of the method in
a Crossbow white paper [126].

The Allan variance, AVAR, is calculated by splitting the dataset into multiple
consecutive windows of time and evaluating the variance in the gyroscope bias
averaged for each window of time. The AVAR is computed for a range of different
window sizes, termed τ . The maximum value of τ must not split the dataset into
more than nine windows to maintain statistical significance [125]. Equation 5.1
describes the calculation where AVR2 (τ ) is the squared Allan variance for period
τ , n is the number of windows created by the given value of τ , and y(τ )i is the
averaged gyroscope bias calculated for the it h window.

1 X
2
AVR2 (τ ) = y(τ )i+1 − y(τ )i (5.1)
2(n − 1) i

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Chapter 5. Characterisation of low-cost MEMS sensors

Figure 5.11 shows the AVAR for each axis calculated using values of τ between 0.01
and 1000 seconds. The linear descent created on a logarithmic plot for small values
of τ indicates a reduction in variance due to the averaging of mean-zero random
noise. The AVAR reaches a minimum at which point the length of τ is sufficiently
long to incorporate the changes due to the random walk and the variance beings
to increase again. The stability of the gyroscope bias is quantified as the minimum
Allan variance and corresponding value of τ [126]. In this case the gyroscope bias
stability is quantified as ∼4◦ /hour for τ = 100 seconds.

X Y Z
3
10

2
10
AVAR (◦ /hour)

1
10

0
10
−2 −1 0 1 2 3
10 10 10 10 10 10
Averaging time, τ (seconds)

Figure 5.11: Allan variance plot of the gyroscope bias random walk

5.5 Gyroscope, accelerometer and

magnetometer thermal response

The MPU-6050 and HMC5883L datasheets indicate that the gyroscope,

accelerometer and magnetometer all have a temperature sensitivity that will cause
each sensor’s sensitivity and bias to vary over the operating temperature range.
Table 5.2 summarises the temperature sensitivities as specified in the device
datasheets [121, 122]. Although units such as %/◦C suggest a linear relationship,

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most of the specified quantities are ambiguous. The gyroscope bias investigations
in Section 5.2 have already revealed the potential significance of such ambiguities.
This provides a motivation to investigate the thermal response of the bias and
sensitivity for all three types of sensor.

Parameter Temperature sensitivity Conditions

Gyroscope sensitivity ±2% -
Gyroscope bias ±20◦/s -40◦C to 85◦C
Accelerometer sensitivity ±0.02%/◦C -40◦C to 85◦C
Accelerometer bias (x and y axis) ±35 mg 0◦C to 70◦C
Accelerometer bias (z axis) ±60 mg 0◦C to 70◦C
Magnetometer bias - -
Magnetometer sensitivity -0.3%/◦C -

Table 5.2: Temperature sensitivity of sensor parameters as specified in device

datasheets. “-” indicates that the information is not provided.

The sensitivity and bias and may be evaluated for each sensor by exposing each
axis to both a positive and negative reference. For example, this could be a ±1
g reference for each accelerometer axis. The axis bias may then be calculated as
Equation 5.2 where b is the axis bias in lsb, u+r is the axis output when exposed
to the positive reference and u−r is the axis output when exposed to the negative
reference.
u+r + u−r
b= (5.2)
2

The axis sensitivity may be calculated as Equation 5.3 where s is the axis bias
in lsb per units of the reference signal, u+r is the axis output when exposed to
the positive reference and u−r is the axis output when exposed to the negative
reference, and r is the magnitude of the reference signal.

|u+r | + |u−r |
s= (5.3)
2r

The bias and sensitivity may be determined at a given temperature using

Equations 5.2 and 5.3. To evaluate variation over temperature, each sensor must

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Chapter 5. Characterisation of low-cost MEMS sensors

be exposed to a corresponding positive and negative reference while simultaneously

regulating the device temperature. The gyroscope and accelerometer both require
physical reference quantities that can only be achieved through controlled motion.
A three-axis thermal chamber was created to facilitate these investigations.

5.5.1 Three-axis thermal chamber

The three-axis thermal chamber was created from a polystyrene box and a cube
structure designed to slot within the internal volume to mount four battery
powered prototype x-IMUs in alignment with the outer dimensions of the chamber.
A 1 kg copper plate with aluminium fins and battery powered fan were mounted
internally to blow air directly on to the x-IMUs. The chamber can be orientated at
six orientations on a flat surface to expose each accelerometer axis to +1 g and -1 g;
and at six orientations relative to the rotational axis of a rate table to expose each
gyroscope axis to a positive and negative reference of angular velocity. A Technics
SL-1200MK2 turntable was used to provide a repeatable reference of 200◦/s (331/3
Rotations Per Minute (RPM)). Details on the evaluation of this turntable are
discussed in Chapter 6. Figure 5.12 shows the components that make up the
three-axis thermal chamber.

Each x-IMUs was set up to log sensor data to on-board SD cards at a rate of
256 samples per second while simultaneously streaming the data via Bluetooth
to be viewed in real-time. The HMC5883L was mounted only a few millimetres
from the MPU-6050 and so assumed to be at the same temperature indicated
by the MPU-6050 internal thermometer. The thermal chamber was stored in
a -30◦C industrial freezer for >12 hours prior to the investigation. The x-IMUs
and fan were switched on and thermal chamber sealed before being removed
from the freezer. The thermal chamber was then continuously cycled through
six static orientations relative to gravity and six orientations on the turntable

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 Chapter 5. Characterisation of low-cost MEMS sensors

Figure 5.12: Three-axis thermal chamber components including: polystyrene box

with internal thermal mass and fan, cube structure to mount four prototype
x-IMUs, and SL-1200MK2 turntable

while the internal temperature increased with the heat generated by the internal
electronics from approximately -25◦C to 40◦C over 2.5 hours. Figure 5.13 shows
this temperature ascent for each of the four devices: ‘A’, ‘B’, ‘C’ and ‘D’. The
necessity for the chamber to be wireless prevented the use of active heating which
both limited the maximum temperature temperature and prevented control of the
rate of ascent. Nevertheless, the temperature range achieved was sufficient to
investigate the thermal response of each of the sensors.

5.5.2 Gyroscope sensitivity and bias thermal response

A MATLAB script was written to process the logged data and extract the periods
at which the each gyroscope axis was detected as being at the reference speeds
of -200◦/s and +200◦/s. The average gyroscope output was computed for discrete

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Chapter 5. Characterisation of low-cost MEMS sensors

A B C D
40
35
30
25
20
Temperature (◦ C)

15
10
5
0
−5
−10
−15
−20
−25
0 15 30 45 60 75 90 105 120 135 150
Time (minutes)

Figure 5.13: Measured temperature ascent of sensors within the three-axis thermal
chamber over 2.5 hours

temperature intervals of 2◦C to yield a value of u+r and u−r for each axis at each
temperature interval. The corresponding sensitivities, sωx , sωy and sωz ; and biases,
bωx , bωy and bωz , were calculated using Equations 5.3 and 5.2 to yield the results
shown in Figure 5.14. The bias was scaled to units of degrees per second using the
typical sensitivity specified in the datasheet.

A B C D

16.5 16.5 16.5

sωx (lsb/◦ /s)

sωy (lsb/◦ /s)

sωz (lsb/◦ /s)

16.4 16.4 16.4

16.3 16.3 16.3

16.2 16.2 16.2

−20 −10 0 10 20 30 40 −20 −10 0 10 20 30 40 −20 −10 0 10 20 30 40
Temperature (◦ C) Temperature (◦ C) Temperature (◦ C)
3 1 3
2.5 0 2
2
−1
bωx (◦ /s)

bωy (◦ /s)

bωz (◦ /s)

1.5 1
−2
1 0
−3
0.5
−4 −1
0
−0.5 −5 −2
−20 −10 0 10 20 30 40 −20 −10 0 10 20 30 40 −20 −10 0 10 20 30 40
Temperature ( C) ◦
Temperature ( C)◦
Temperature (◦ C)

Figure 5.14: Thermal response of gyroscope sensitivity and bias from -15◦C to
40◦C

The gyroscope biases bωx , bωy and bωz for the four devices has already been
investigated in greater detail in Section 5.2. The thermal responses shown in Figure

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 Chapter 5. Characterisation of low-cost MEMS sensors

5.14 match those previously demonstrated for devices ‘A’, ‘B’, ‘C’ and ‘D’. The
gyroscope sensitivity temperature coefficient was calculated for the four devices
as the ratio of the best-fit gradient to the mean sensitivity over the temperature
range. The results are summarised in Table 5.3.

Temperature coefficients (%/◦C)

Device
sωx sωy sωz
A -0.0083 -0.0035 -0.0004
B -0.0091 -0.0035 -0.0027
C -0.0097 -0.0051 -0.0016
D -0.0138 0.0043 -0.0012

Table 5.3: Gyroscope sensitivity temperature coefficients

The datasheet specifies a maximum gyroscope sensitivity variation of ±2% over an

unspecified temperature range. This is confirmed by the worst-case coefficient of
-0.0138%/◦C which corresponds to a 1.73% variation over the operating range of
-40◦C to 80◦C. A calibration model may account for this temperature coefficient as
linear approximation. However, the magnitude may already be sufficiently low for
most applications; particular those operating within a limited temperature range.

5.5.3 Accelerometer sensitivity and bias thermal response

As with the gyroscope analysis, accelerometer data was extracted to yield averaged
measurements of u+r and u−r for each axis at each temperature interval. The
corresponding sensitivities sax , say and saz , and biases bax , bay and baz , were
calculated using Equations 5.3 and 5.2 to yield the results shown in Figure 5.15.
The bias was scaled to units of mg using the typical sensitivity specified in the
datasheet.

The accelerometer sensitivity temperature coefficient was calculated for the four
devices as the ratio of the best-fit gradient to the mean magnitude of the measured
sensitivity over the temperature range. The results are summarised in Table 5.4.

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Chapter 5. Characterisation of low-cost MEMS sensors

A B C D

2070 2070 2080

2065 2065 2075

sax (lsb/g)

say (lsb/g)

saz (lsb/g)
2060 2060 2070

2055 2055 2065

2050 2050 2060

2045 2045 2055

2040 2040 2050

−20 −10 0 10 20 30 40 −20 −10 0 10 20 30 40 −20 −10 0 10 20 30 40
◦ ◦
Temperature ( C) Temperature ( C) Temperature (◦ C)
20 14 200
18 12 150
16 10 100
bax (mg)

bay (mg)

baz (mg)
14 8 50
12 6 0
10 4 −50
8 2 −100
6 0 −150
−20 −10 0 10 20 30 40 −20 −10 0 10 20 30 40 −20 −10 0 10 20 30 40
Temperature (◦ C) Temperature (◦ C) Temperature (◦ C)

Figure 5.15: Thermal response of accelerometer sensitivity and bias from -15◦C to
40◦C

The accelerometer sensitivity is indicated as having a temperature coefficient of

±0.02%/◦C in the datasheet. The worst-case measured temperature coefficient of
-0.0155%/◦C is within this range.

Temperature coefficients (%/◦C)

Device
sax say saz
A 0.0042 0.0032 -0.0097
B 0.0040 0.0040 -0.0060
C 0.0027 0.0035 -0.0068
D 0.0027 0.0034 -0.0155

Table 5.4: Accelerometer sensitivity temperature coefficients

The accelerometer bias temperature sensitivity was calculated as the best-fit

gradient and the results summarised in Table 5.5. The datasheet specifies the
temperature variation of ±35 mg for the x and y axis and ±60 mg in the z axis
for a temperature range of 0◦C to 70◦C. The lower plots of 5.15 confirm a linear
relationship. The The worst-case measured temperature sensitivities in Table 5.5
indicate a temperature variation of 17.6 mg in the x axis but 118.6 mg in the z
axis. While the variation in the x and y axis is within the manufacturer’s specified
tolerances, the variation z in the axis for three of the four devices are not.

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Temperature coefficients (mg/◦C)

Device
bax bay baz
A -0.054 0.037 1.001
B -0.096 0.076 0.168
C -0.043 0.046 1.045
D -0.252 -0.050 1.694

Table 5.5: Accelerometer bias temperature coefficients

The accelerometer temperature sensitivities may be compensated for in calibration

though a linear approximation though the sensitivities may be sufficiently low
that thermal calibration is not necessary for most applications. The z axis bias
represents a possible exception as its relatively large ∼100 mg variation observed
over the 50◦C temperature range is beyond the manufacturer’s specified tolerances
and significantly large relative to the 1 g typical measurement expected by most
applications.

5.5.4 Magnetometer sensitivity and bias thermal response

The magnetometer incorporates an internal field generator that functions within

an automated sequence to sample each axis when the field generator is disabled
and then again when it is enabled. The difference between these two samples
is provided as a measurement of the reference field strength independent of
motion and the ambient magnetic field. The prototype x-IMUs were configured to
continuously log this data for positive and negative fields. Averaged measurements
of u+r and u−r were calculated for each axis at each 2◦C temperature interval. The
corresponding sensitivities smx , smy and smz , and biases bmx , bmy and bmz , were
calculated using Equations 5.3 and 5.2 for each temperature interval to yield the
results shown in Figure 5.16.

The magnetometer sensitivity temperature coefficient was calculated for the four
devices as the ratio of the best-fit gradient to the mean magnitude of the measured

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Chapter 5. Characterisation of low-cost MEMS sensors

A B C D

550 525 550

525 500 525

smx (lsb/Ga)

smy (lsb/Ga)

smz (lsb/Ga)
500 475 500

475 450 475

450 425 450

425 400 425

400 375 400

−20 −10 0 10 20 30 40 −20 −10 0 10 20 30 40 −20 −10 0 10 20 30 40
◦ ◦
Temperature ( C) Temperature ( C) Temperature (◦ C)
10 11 1
9.5 10.5
0.5
9 10
0
bmx (lsb)

bmy (lsb)

bmz (lsb)
8.5 9.5
8 9 −0.5
7.5 8.5
−1
7 8
−1.5
6.5 7.5
6 7 −2
−20 −10 0 10 20 30 40 −20 −10 0 10 20 30 40 −20 −10 0 10 20 30 40
Temperature (◦ C) Temperature (◦ C) Temperature (◦ C)

Figure 5.16: Thermal response of magnetometer sensitivity and bias from -15◦C
to 40◦C

sensitivity over the temperature range. The magnetometer bias temperature

coefficient was calculated in the same way. The results are summarised in Table
5.6.

Temperature coefficients (%/◦C)

Device
smx smy smz bmx bmy bmz
A -0.37 -0.39 -0.34 -0.24 -0.43 1.28
B -0.38 -0.40 -0.36 -0.36 -0.36 0.37
C -0.35 -0.39 -0.35 -0.47 -0.33 1.31
D -0.37 -0.39 -0.35 -0.34 -0.46 -190.26

Table 5.6: Magnetometer sensitivity and bias temperature coefficients

The datasheet specifies a sensitivity temperature coefficient of -0.3%/◦C which is

confirmed by the results in Table 5.6. The datasheet does not document any bias
variation over temperature. Although the lower plots of Figure 5.16 suggest that
the bias does vary over temperature, representing this variation as %/◦C provides
an alternative explanation. The bias temperature coefficients calculated in Table
5.6 approximate the -0.3%/◦C expected in the sensitivity which suggests that the
observed bias variation is the result of scaling by the sensitivity and that the bias
itself is subject to little or no variation over temperature. The low magnitudes of

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bmx , bmy and bmz result in the increased variance of the bias temperature coefficient
relative to the sensitivity temperature coefficients. In the case of bmz , calculations
approach divisions by zero and so cannot be expected to provide meaningful results.

A magnetometer sensitivity error is of particular significance to AHRS applications

as hard-iron distortions may shift a small operating range of a magnetometer
axis away from zero. For example, if such distortions shifted an operating range
from ±100 to a range of 800 to 1000, an offset of 900 can be subtracted to yield
a ‘distortion-free’ measurement. However, if a sensitivity error of -10% is also
in effect then the compounded operating rage would become 720 to 900 and
subtracting the assumed 900 offset would yield a range of -180 to 0 corresponding
to a maximum error of 180%. This numerical example represents a hard-iron
distortion with a magnitude nine times that of the Earth’s magnetic field, such a
distortion would be extreme but is not unrealistic.

The absence of a bias variation means that the magnetometer thermal response
can be accounted for by scaling all measurements by approximately -0.3%/◦C,
though the variation in above results indicate that the specific coefficient would
need to be evaluated for each device. An alternative solution would be to use
the magnetometer internal field generators to calculate the sensitivity at discrete
time or temperature intervals; this technique is suggested in the magnetometer
datasheet.

5.6 Part variation

The manufacture of the IMU platforms described in Chapter 8 provided

an opportunity to investigate the variation of gyroscope, accelerometer and
magnetometer characteristics within a population of over 150 x-IMUs. The
MPU-6050 and HMC5883L were each sourced from a single production reel and

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Chapter 5. Characterisation of low-cost MEMS sensors

so are assumed to represent common batch and silicon revision. It is common

for electronic components within a given batch to share common characteristics
which may not be representative of the part in general. The methods described
in Section 5.5 were repeated without the thermal chamber to measure the bias
and sensitivity of each sensor once the device had reached stable temperature of
∼35◦C. The following sections summarise the distribution of these parameters for
each sensor.

5.6.1 Gyroscope part variation

Figure 5.17 shows the normalised distribution of the gyroscope axis sensitivities
sωx , sωy and sωz and biases bωx , bωy and bωz measured for the 150 MPU-6050 parts.
The biases have been scaled using the typical sensitivity specified in the datasheet.

0.3 0.3 0.3

Normalised distribution
Normalised distribution

Normalised distribution

0.25 0.25 0.25

0.2 0.2 0.2

0.15 0.15 0.15

0.1 0.1 0.1

0.05 0.05 0.05

0 0 0
15.8 16 16.2 16.4 16.6 16.8 15.8 16 16.2 16.4 16.6 16.8 15.8 16 16.2 16.4 16.6 16.8
◦ ◦
sωx (lsb/ /s) sωy (lsb/ /s) sωz (lsb/◦ /s)
0.35 0.35 0.35
Normalised distribution
Normalised distribution

Normalised distribution

0.3 0.3 0.3

0.25 0.25 0.25
0.2 0.2 0.2
0.15 0.15 0.15
0.1 0.1 0.1
0.05 0.05 0.05
0 0 0
−8 −6 −4 −2 0 2 4 6 −8 −6 −4 −2 0 2 4 6 −8 −6 −4 −2 0 2 4 6
◦ ◦
bωx ( /s) bωy ( /s) bωz (◦ /s)

Figure 5.17: Gyroscope sensitivity and bias normalised distribution

The datasheet specifies that the gyroscope sensitivity of 16.4 lsb/◦/s ±3% at 25◦C;
corresponding to a range of 15.9 lsb/◦/s to 16.9 lsb/◦/s. The histograms confirm
a mean of ∼16.4 lsb/◦/s with no outliers exceeding these tolerances. The bias is
specified as being within the range ±20◦/s at 25◦C. The lower histograms of Figure
5.17 reveal that the true distribution is more limited. Whereas the sensitivity

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 Chapter 5. Characterisation of low-cost MEMS sensors

distributions are centred on the specified nominal value, the mean bias for each
axis is not centred on zero. For example, the mean value of bωx is approximately
-3◦/s. Although this mean suggests a more accurate nominal value, this may be
specific to the batch.

5.6.2 Accelerometer part variation

Figure 5.18 shows the normalised distribution of the accelerometer axis sensitivities
sax , say and saz and biases bax , bay and baz measured for the 150 MPU-6050 parts.
The biases have been scaled using the typical sensitivity specified in the datasheet.

0.4 0.4 0.4

Normalised distribution
Normalised distribution

Normalised distribution

0.35 0.35 0.35

0.3 0.3 0.3
0.25 0.25 0.25
0.2 0.2 0.2
0.15 0.15 0.15
0.1 0.1 0.1
0.05 0.05 0.05
0 0 0
1950 2000 2050 2100 2150 1950 2000 2050 2100 2150 1950 2000 2050 2100 2150
sax (lsb/g) say (lsb/g) saz (lsb/g)
0.7 0.7 0.7
Normalised distribution
Normalised distribution

Normalised distribution

0.6 0.6 0.6

0.5 0.5 0.5
0.4 0.4 0.4
0.3 0.3 0.3
0.2 0.2 0.2
0.1 0.1 0.1
0 0 0
−150 −100 −50 0 50 100 150 −150 −100 −50 0 50 100 150 −150 −100 −50 0 50 100 150
bax (mg) bay (mg) baz (mg)

Figure 5.18: Accelerometer sensitivity and bias normalised distribution

The datasheet specifies a sensitivity of 2048 lsb/g ±3% at 25◦C; corresponding to

a range of 1987 lsb/g to 2109 lsb/g.The histograms confirm a mean of ∼2048 lsb/g
with no outliers exceeding these tolerances. The bias is specified as being within
the range ±50 mg for x and y axis and ±80 mg for z axis at 25◦C. As with the
gyroscope, the lower histograms of Figure 5.18 reveal that the true distribution is
more limited for the x and y axis. However, the distribution for the z axis exceeds
the specified range. This is consistent with the thermal response of the z axis
which was also found to be outside of the the manufacturer’s tolerances.

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5.6.3 Magnetometer

Figure 5.18 shows the normalised distribution of the magnetometer axis

sensitivities smx , smy and smz and biases bmx , bmy and bmz measured for the 150
HMC5883L parts.

0.8 0.8 0.8

Normalised distribution
Normalised distribution

Normalised distribution

0.7 0.7 0.7

0.6 0.6 0.6
0.5 0.5 0.5
0.4 0.4 0.4
0.3 0.3 0.3
0.2 0.2 0.2
0.1 0.1 0.1
0 0 0
200 300 400 500 600 700 800 900 200 300 400 500 600 700 800 900 200 300 400 500 600 700 800 900
smx (lsb/Gauss) smy (lsb/Gauss) smz (lsb/Gauss)
0.8 0.8 0.8

Normalised distribution
Normalised distribution

Normalised distribution

0.7 0.7 0.7

0.6 0.6 0.6
0.5 0.5 0.5
0.4 0.4 0.4
0.3 0.3 0.3
0.2 0.2 0.2
0.1 0.1 0.1
0 0 0
−30 −20 −10 0 10 20 30 40 50 −30 −20 −10 0 10 20 30 40 50 −30 −20 −10 0 10 20 30 40 50
bmx (lsb) bmy (lsb) bmz (lsb)

Figure 5.19: Magnetometer sensitivity and bias normalised distribution

The x-IMUs were configured for a magnetometer measurement range of ±4.0

Gauss with a corresponding sensitivity of 440 lsb/Gauss specified in the device

datasheet. However, the mean sensitivity indicated in Figure 5.19 is closer to the
820 lsb/Gauss sensitivity associated with the magnetometer’s default full-scale
range of ±1.9 Gauss. There are also second peaks around 440 lsb/Gauss which
suggest that a malfunction in the HMC5883L or prototype x-IMU firmware has
caused an incorrect full-scale range to be selected for a majority of the devices
tested. The tests conducted during the collection of this data focussed on the
static heading error achieved by the magnetometer and so were insensitive to this
scaling error. It was not possible to repeat the data collection for the 150 devices.
It is therefore not possible to comment on the distributions of 5.19.

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5.7 Conclusions

The purpose of this chapter is to determine the significant sensor characteristics

that must be accounted for within a calibration solution. Investigations revealed
undocumented characteristics as well as discrepancies in the datasheets. The
gyroscope bias was found to exhibit thermal hysteresis. The accelerometer z axis
performance was found to be outside of the specified tolerances in both its thermal
response and the part variation survey of over 150 parts. Although the gyroscope
datasheet indicates a bias sensitivity to static acceleration, a correlation could not
be confirmed.

Thermal response investigations confirmed low temperature coefficients of most

characteristics. The gyroscope bias and magnetometer sensitivity are exceptions
that would need to be accounted for in calibration. Magnetometer temperature
calibration may be achieved using the manufacturer’s recommended solution.
Although the gyroscope incorporates a thermometer to facilitate temperature
calibration, the non-linear response and hysteresis may limit success in a practical
implementation. It may therefore be more effective to avoid gyroscope bias
temperature calibration and instead rely on a bias compensation built-in to the
AHRS algorithm. The findings presented in this chapter were used to determine
the scope of the calibration solutions developed in the next chapter.

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104
Chapter 6

Calibration of low-cost MEMS

sensors for AHRS applications

6.1 Introduction

This chapter describes calibration solutions for gyroscopes, accelerometers and

magnetometers for use in AHRS. The proposed solutions are intended to be
practical in the sense that they can be implemented without the need for specialist
equipment and so be a viable solution for production scenarios. The solutions
and methods for IMU calibration are well documented [119, 120, 127, 118].
The solutions presented in this chapter were developed to meet the specific
requirements of the MPU-6050 and HMC-5883L as presented in the previous
chapter. Furthermore, the calibrated performance for each sensor is evaluated
and benchmarked against selected commercial IMUs.

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Chapter 6. Calibration of low-cost MEMS sensors for AHRS applications

6.1.1 Scope of calibration

The calibration solutions presented in this chapter account for the following
characteristics.

• Individual axis bias and sensitivity

• Sensor misalignment

• Hard-iron and soft-iron distortions

Notable characteristics not accounted for by the calibration models include the
gyroscope bias acceleration sensitivity and the thermal response of the sensors.
Previous investigations indicated could not demonstrate a correlation between the
gyroscope bias and static acceleration. Thermal response investigations reveal a
significant sensitivity in the magnetometer that may be accounted for using the
manufactures recommended technique; and of the gyroscope bias which can be
accounted for by the AHRS algorithm.

6.1.2 Chapter summary

The next section summaries the commercial IMUs chosen as benchmarks. Section
6.4 describes the calibration solution for the gyroscope. Section 6.5 describes
the calibration solution for the accelerometer along side an empirical equation
to contrast uncalibrated an calibrated AHRS performance. Section 6.6 describes
three different magnetometer calibration solutions and demonstrates the AHRS
performance of each solution for both ‘typical’ and ‘extreme’ magnetic distortion
scenarios. Section 6.7 presents an optimisation that can be made to simplify
the sensor alignment arithmetic, and significantly reduce the computation load
in application. The chapter concludes with a summary of proposed calibration
solutions.

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 Chapter 6. Calibration of low-cost MEMS sensors for AHRS applications

6.2 Commercial IMUs chosen as benchmarks

Four commercial IMUs were chosen to benchmark the performance of the proposed
calibration solutions. Each product appeared on the market within the last 5 years
and is priced less than 3000 USD. The chosen devices, shown Figure 6.1, were
the MicroStrain 3DM-GX3-25, Xsens MTw, VectorNav VN-100 and CH Robotics
UM6. MicroStrain and Xsens are recognised as long-established and leading
manufacturers whereas VectorNav and CH Robotics represent new manufacturers
whom have taken advantage of recent advances in MEMS technology to offer
lower-end devices at a significantly lower price.

Figure 6.1: Commercial IMUs chosen to benchmark performance. From left

to right: MicroStrain 3DM-GX3-25, Xsens MTw, VectorNav VN-100 and CH
Robotics UM6

Table 6.1 summarises the calibrated performance of each IMU as specified in the
manufacturer’s documentation [108, 128, 109, 52, 129]. Prices and release dates
were either obtained from the manufacturer’s website or confirmed by contacting
the manufacturer directly. The MPU-6050 [121] and HMC-5883L [122] have
also been included in the table for comparison, these devices function only as
an uncalibrated gyroscope, accelerometer and magnetometer and do not have an
associated AHRS performance.

There is a clear correlation between the AHRS performance and price with the
more expensive devices providing greater accuracy. Both MicroStrain and Xsens
specify a dynamic error. However, the meaning of this quantity is ambiguous and it
cannot be used for the purposes of benchmarking; Xsens specify the dynamic error
along side the disclaimer: “May depend on type of motion” [128]. The proposed

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MPU-6050 /
3DM-GX3-25 MTw VN-100 UM6
HMC5883L
Price $2550 $2990 $500 $199 <$10
Year of release (circa) 2009 2011 2008 2009 2010

AHRS performance
Static error (pitch/roll) ±0.5◦ <0.5◦ 0.5◦ <2◦ N/A
◦ ◦
Static error (heading) ± 0.5 1 2◦ < ◦
5 N/A
Dynamic error (pitch/roll) ±2◦ 2◦ - - N/A
Dynamic error (heading) ±2◦ 2◦ - - N/A
Operating temperature -40 to 70◦C 0 to 55◦C 25◦C - N/A

Gyroscope
Range (◦/s) 300 1200 2000 2000 2000
Linearity (%)
√ 0.03 0.1 0.1 0.2 0.2
Noise (◦/s/ Hz) 0.03 0.05 0.005 0.03 0.005

Accelerometer
Range (g) 5 16 16 2 16
Linearity (%)
√ 0.1 0.2 0.5 - 0.5
Noise (μg/ Hz) 80 0.0003 400 - 400

Magnetometer
Range (Gauss) 2.5 1.5 2.5 2.5 8.1
Linearity (%) √ 0.4 0.2 0.1 - 0.1
Noise (μGauss/ Hz) 100 150 140 - -

Table 6.1: Specified performance of each IMU as specified in the manufacturer’s

documentation. “-” indicates that a parameter is not specified.

calibration solutions were therefore benchmarked against the static errors only.

The specified heading accuracy of each device also represents a potential ambiguity.
It can be shown (see Appendix C) that a calibration offset error corresponding to
a 0.5◦ heading error at the equator would result in a 1.5◦ heading error in the
UK due to a varying inclination of the Earth’s magnetic field. This should be
considered when benchmarking the heading performance.

6.3 Calibration cube

The calibration cube is a precision tool used to calibrate triple-axis gyroscopes,

accelerometers and magnetometers. The structure allows sensors to be rotated
through orthogonal orientations relative to a physical reference signal. For

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example, each accelerometer axis may be exposed to a +1 g or -1 g refrence by

placing the cube on a level surface so that the axis is either pointing directly up
or down. The specific use of the cube varies with each type of sensor. Figure 6.2
shows the calibration cube with nine prototype x-BIMUs held in alignment with
the cube axes.

Figure 6.2: Calibration cube with nine prototype x-BIMUs held in alignment with
the cube axes

The calibration cube was laser cut from 3 mm Medium-Density Fibreboard (MDF)
and assembled through press-fit joints to avoid metal components that may
interfere with magnetic calibration. The structure is comprised of three horizontal
levels separated by vertical spacers. The middle and top levels feature grids of
mounting holes and cut-outs for securing the sensors and additional equipment
such as batteries. The bottom level is an empty frame to provide rigidity to the
structure. Sensors mounted on the middle level are held in alignment by a laser cut
MDF plate with cut-outs matching the dimensions of the x-BIMU. Each corner of
the cube functions as a socket to ensure alignment when the cube is mounted on the

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protractor described in Section 6.3.2. The calibration cube design is open-source1 .

6.3.1 Calibration cube alignment errors

In practise, sensors mounted on the calibration cube will incorporate a small

misalignment and this will cause an attenuation in the measured reference signal.
For example, Figure 6.3 shows a misalignment between the calibration cube axes,
xc , yc and zc ; and the sensor axes, xs , ys and zs . xc has been aligned with the
reference signal r but a misalignment between xs and xc of angle γ prevents the
sensor axis from being exposed to the full magnitude of the reference signal; it is
instead exposed to the attenuated reference signal ra .

xs , ra xc
r
γ
zs
zc

ys
yc

Figure 6.3: A misalignment between the calibration cube and sensor x axes of
angle γ prevents the sensor axis from being exposed to the full magnitude of the
reference signal r.

The relationship between the reference signal, r, and the attenuated reference
signal, ra , is defined by Equation 6.1 where γ is the angular misalignment between
the sensor axis and the direction of the reference signal. ra can be approximated
as a truncated Taylor series.

 
γ2
ra = r cos γ ≈ r 1 − (6.1)
2

The cosine relationship means that the error in the reference signal has greater
tolerance to misalignment errors of a small magnitude. The attenuation of the
1
https://github.com/xioTechnologies/Calibration-Cube

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 Chapter 6. Calibration of low-cost MEMS sensors for AHRS applications

reference signal for misalignments between -10◦ and +10◦ is shown in Figure 6.4.
A sensor calibrated against an attenuated reference would incorporate a sensitivity
error equal to this attenuation if the misalignment is not accounted for within the
calibration model.
Reference signal attention (%)
0
−0.2
−0.4
−0.6
−0.8
−1
−1.2
−1.4
−1.6
−10 −8 −6 −4 −2 0 2 4 6 8 10
Alignment error, γ (degrees)

Figure 6.4: Relationship between calibration cube alignment error and attenuation
of reference signal

The calibration of accelerometer alignment method in Section 6.5.3 indicated a

mean alignment error of <0.7◦ for the nine prototype x-BIMUs mounted on the
calibration cube.

6.3.2 Calibration cube protractor

The calibration cube protractor is used in conjunction with the calibration cube
to evaluate the static pitch/roll and heading error achieved by an accelerometer
and magnetometer. The protractor is not required for the calibration. Figure 6.5
shows the calibration cube mounted on the protractor.

The protractor is comprised of a 290 mm diameter cog with 72 teeth, and a base
plate incorporating a socket for the cog. The cog can be mounted in the socket
at 72 different angular positions corresponding to a 360◦ rotation in 5◦ steps. The
cog has four sockets that mate with any four corners of the calibration cube. The
protractor may therefore be used to rotate the cube around all three axes.

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Figure 6.5: Calibration cube mounted on the protractor (left) and close up of cog
teeth and engraved angular increments (right)

The manufacture process was essential in achieving the accuracy of the protractor.
The cog and socket were laser cut from a single sheet of 5 mm acrylic plastic and
the cog flipped within the socket so that the natural chamfers created by the laser
cutter formed parallel surfaces. Theoretically, the protractor accuracy would be
determined by the backlash between the cog and the socket. In practise, the finite
accuracy of the laser cutter results in an elliptical cog and socket that achieve either
a flush fit or a ‘press fit’ as the cog is rotated. This potentially improves accuracy
by distributing geometric errors between the 72 teeth. An accelerometer was
demonstrated as achieving 0.1◦ static pitch/roll accuracy through a 360◦ rotation
using the protractor in Section 6.5.4; it can therefore be inferred that the protractor
accuracy is <0.1◦ .

6.4 Gyroscope calibration

The gyroscope calibration model is described by Equations 6.2 and 6.3 where ω
is the calibrated gyroscope measurement in degrees per second, C
Ω R is a rotation

matrix describing the alignment of the gyroscope relative to the calibrated frame,
uω is the uncalibrated gyroscope output in lsb, Sω is the gyroscope sensitivity
in lsb per degrees per second, and bω is gyroscope bias in lsb. Sω is a diagonal

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matrix as the calibration model does not account for cross-axis coupling or axis
non-orthogonality.
−1
ω=C
Ω RSω (uω − bω ) (6.2)

    −1    
ωx  rω11 rω12 rω13  sωx 0 0  uωx  bωx 
        
ω  = r      
 y   ω21 rω22 rω23   0 sωy 0  uωy  − bωy  (6.3)
        
ωz rω31 rω32 rω33 0 0 sωz uωz bωz

The above calibration model provides a calibrated measurement of angular velocity,

ω, given the uncalibrated gyroscope output, uω , and known values of the fixed
calibration parameters C
Ω R, Sω and bω . These parameters are evaluated through

a one-off calibration process summarised by the following steps.

1. Calculate the bias, bω , as the the mean gyroscope output, uω , while the
gyroscope is stationary.

2. Collect a calibration dataset representing the mean gyroscope output, uω ,

for the six orientations of the calibration cube relative to the rotational axis
of a precision rate table to expose each axis to a known positive and negative
reference of angular velocity ω.

3. Calculate each axis sensitivity, sωx , sωy and sωz , as the average magnitude
of the axis output when exposed the +ω and −ω reference.

C
4. Calculate the misalignment, Ω R, using the previously determined values
bω and Sω and three orthogonal quantities of uω within the dataset that
represent each axis begin exposed to a +ω reference.

The following sections describe the complete gyroscope calibration process in

detail.

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6.4.1 Gyroscope bias calibration

The gyroscope bias parameters bωx , bωy and bωz are calculated as the mean
uncalibrated gyroscope output over a period of several seconds while the device is
stationary. The gyroscope bias is known to vary over time and with temperature
though this is not accounted for in the calibration model. It is assumed that an
AHRS algorithm will provide a means of updating bω during operation.

6.4.2 Gyroscope calibration dataset

The gyroscope calibration dataset requires a rate table to provide a known

reference of angular velocity. Industrial rate tables such as the the Ideal Aerosmith
1270VS provide an accuracy of 0.1% over a range of angular velocities [130].
However, such equipment is not commonly available and the dataset requires only
a single velocity which can be achieved using widely available record turntables.
The Technics SL-1200MK2 turntable was chosen as it features a direct drive motor
with quartz-phase-locked control to provide an accurate and repeatable angular
velocity [131]. A turntable speed of 331/3 RPM was chosen for a reference angular
velocity of ω = 200◦/s. The accuracy was verified using a magnet and a Hall effect
sensor to generate a square wave with a period equal to the angular velocity; a TTi
TF930 frequency counter was used to measure the period with an accuracy of ±0.2
parts per million (ppm) [132]. The turntable speed was measured as 200.0023◦/s;
corresponding to an error of 12 ppm. This accuracy was maintained with a 5 kg
load which is far greater than the expected load during calibration.

The dataset is collected by orientating the calibration cube on the turntable so that
so that each axis is aligned with the vertical rotational axis and pointing either up
or down; exposing each axis to a reference of +ω and −ω. This is represented by
the six orientations shown in Figure 6.6. The mean uncalibrated gyroscope output,

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uω , is taken over a period of several seconds for each orientation to yield a dataset
of six vector quantities. Each pair of measurements representing +ω and −ω for a
given axis should be made contiguously with the calibration cube rotated around
only a single axis relative to the turntable between the two orientations. This
ensures that unexpected time-variant or temperature deviations are minimised
and that a misalignment between the plater and rotational axis results in an equal
attenuation of the +ω and −ω reference.

Figure 6.6: The gyroscope calibration dataset is represented by the six orientations
of the calibration cube on the turntable to expose each axis to a +ω and −ω
reference
.

6.4.3 Gyroscope sensitivity calibration

The gyroscope axis sensitivities sωx , sωy and sωz are calculated as the average
magnitude of the axis output when exposed to the +ω and −ω reference. This is
described by Equation 6.4 where sω is the axis sensitivity in lsb per degrees per
second, u+ω and u−ω are the axis output in lsb when exposed to the +ω and −ω
reference respectively, and ω is the magnitude of the reference angular velocity

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known to be 200◦/s.
|u+ω | + |u−ω
sω = (6.4)
2ω

6.4.4 Gyroscope alignment calibration

The alignment of the gyroscope relative to the calibrated frame, C

Ω R, is calculated

using the values of the Sω and bω previously determined in Sections 6.4.1 and 6.4.3
and the three orthogonal vector quantities of uω within the calibration dataset that
C
represent each axis being exposed to a positive reference of angular velocity. ΩR is
initially approximated as C
G R̃ as described by Equation 6.5 where uωx , uωy and uωz

are the gyroscope outputs when the positive reference is applied to the x, y and
z axes respectively. The three columns of C
G R̃ are constructed as each measured

direction of rotation for three orthogonal orientations of the calibration cube on

the turntable.  T
−1
kSω (uωx − bω )k
 
C  −1
Ω R̃ = kSω (uωy − bω )k
 (6.5)
 
kSω−1 (uωz − bω )k

C
G R̃ can only be assumed to be an approximation of a rotation matrix because
C
measurement errors may prevent it from being orthogonal. ΩR is obtained from
C
G R̃ by computing the best-fit rotation matrix as described in Chapter 2.

6.5 Accelerometer calibration

The accelerometer calibration model is described by Equations 6.6 and 6.7 where
C
a is the calibrated accelerometer measurement in g, AR is a rotation matrix
describing the alignment of the accelerometer relative to the calibrated frame,
ua is the uncalibrated accelerometer output in lsb, Sa is the accelerometer
sensitivity in lsb per g, and ba is accelerometer bias in lsb. Sa is a diagonal

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matrix as the calibration model does not account for cross-axis coupling or axis
non-orthogonality.
−1
a=C
A RSa (ua − ba ) (6.6)

    −1    
ax  ra11 ra12 ra13  sax 0 0  uax  bax 
        
a  = r      
 y   a21 ra22 ra23   0 say 0  uay  − bay  (6.7)
        
az ra31 ra32 ra33 0 0 saz uaz baz

The above calibration model provides a calibrated measurement of acceleration,

a, given the uncalibrated accelerometer output, u, and known values of the fixed
calibration parameters C
A R, Sa and ba . These parameters are evaluated through a

one-off calibration process summarised by the following steps.

1. Collect a calibration dataset representing the mean accelerometer output, ua ,

for the six orientations of the calibration cube on a level surface to expose
each axis to a +1 g and −1 g reference.

2. Calculate each axis bias, bax , bay and baz , as the average of the axis output
when exposed the +1 g and −1 g reference.

3. Calculate each axis sensitivity, sax , say and saz , as the average magnitude of
the axis output when exposed the +1 g and −1 g reference.

C
4. Calculate the misalignment, A R, using the previously determined values
Sa and ba and the three orthogonal quantities of ua within the dataset
representing each axis begin exposed to a +1 g.

The following sections describe the accelerometer calibration process in detail and
present a practical evaluation of the calibrated accelerometer performance. The
calibrated performance in contrasted with the uncalibrated performance and that
of the commercially available IMUs.

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6.5.1 Accelerometer calibration dataset

The accelerometer calibration dataset is collected by orientating the calibration

cube on a flat, level surface so that so that each axis is aligned with gravity and
pointing either vertically up or down, exposing each axis to a reference of +1 g and
−1 g. This dataset is represented by the six orientations of the calibration cube
shown in Figure 6.7. The mean accelerometer output, ua , is taken over a period of
several seconds for each orientation to yield a dataset of six vector quantities. Each
pair of measurements representing +1 g and −1 g for a given axis should be made
contiguously with the calibration cube rotated around only a single axis relative
to the level surface between the two orientations. This ensures that unexpected
time-variant or temperature deviations are minimised and that a misalignment
between the level surface and direction of gravity results in an equal attenuation
of the +1 g and −1 g reference.

Figure 6.7: The accelerometer calibration dataset is represented by the six

orientations of the calibration cube on a level surface to expose each axis to a
+1 g and −1 g reference
.

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6.5.2 Accelerometer sensitivity and bias calibration

The accelerometer axis sensitivities sa x, sa y and sa z and biases ba x, ba y and ba z are

calibrated using the accelerometer calibration dataset and an industry-standard
6-point tumble test [120]. The bias of each axis is calculated as the average of the
axis output when exposed to the +1 g and −1 g reference. This is described by
Equation 6.8 where ba is the axis bias in lsb, and u+g and u−g are the axis output
in lsb when exposed to the +1 g and −1 g reference respectively.

u+g + u−g
ba = (6.8)
2

The sensitivity of each axis is calculated as the average magnitude of the axis
output when exposed to the +1 g and −1 g reference. This is described by Equation
6.9 where sa is the axis sensitivity in lsb per g; and u+g and u+g are the axis output
in lsb when exposed to the +1 g and −1 g reference respectively.

|u+g | + |u−g |
sa = (6.9)
2g

6.5.3 Accelerometer alignment calibration

The alignment of the calibrated frame relative to the physical accelerometer, C

A R, is

calculated using the values of the Sa and ba previously determined in Section 6.5.2
and the three orthogonal vector quantities of ua within the calibration dataset that
represent each axis begin exposed to a +1 g reference. C
A R is initially approximated

as Equation 6.10 where C C

A R̃ is an approximation of A R; and uax , uay and uaz are

the accelerometer outputs when the +1 g reference is applied to the x, y and z

axes respectively. In this equation, the three columns of C
A R̃ are constructed as

the directions of gravity measured by the accelerometer when gravity is aligned to

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the each orthogonal axis of the calibration cube.

 T
−1
kSa (uax − ba )k
 
C  −1
A R̃ = kSa (uay − ba )k
 (6.10)
 
−1
kSa (uaz − ba )k

C
A R̃ can only be assumed to be an approximation of a rotation matrix because
C
measurement errors may prevent it from being orthogonal. AR is obtained from
C
A R̃ by computing the best-fit rotation matrix as described in Chapter 2.

6.5.4 Calibrated accelerometer performance

The accelerometer performance was evaluated as the error in the angular

measurement of inclination provided by the accelerometer when rotated around an
axis perpendicular to gravity. Both uncalibrated and calibrated performance were
evaluated for comparison. The angle of inclination for a rotation around a given
axis can be calculated using Equations 6.11 to 6.14 where nx , ny and nz represent
either the normalise uncalibrated or calibrated accelerometer measurement; and θx ,
θy and θz represent the angle of inclination for rotations around the accelerometer
x, y and z axis respectively.

 T 
û u a
nx ny nz = , where û = â = (6.11)

â kuk kak

θx = atan2 (ny , nz ) (6.12)

θy = atan2 (nz , nx ) (6.13)

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θz = atan2 (nx , ny ) (6.14)

The calibration cube protractor was used to rotate the cube 360◦ around each
axis in 5◦ steps. At each step, the mean accelerometer output was obtained
for approximately 2000 samples over 30 seconds to yield both uncalibrated and
calibrated measurements for each accelerometer. The corresponding values of θx ,
θy or θz for each step would then be computed and compared with the angle
indicated by the protractor to yield the error in the angular measurement of
inclination provided by an accelerometer. Figure 6.8 shows the calibration cube
protractor being used to collect the accelerometer evaluation dataset.

Figure 6.8: Calibrated accelerometer performance was evaluated by using the

calibration cube protractor to rotate through precise inclinations

A misalignment between the zero position of the protractor and the direction of
gravity introduced an offset to each accelerometer’s measurement of inclination.
This offset was removed by subtracting the mean angular error of all nine

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accelerometers from each. Figure 6.9 shows the performance of the uncalibrated
accelerometers. The top three plots indicate the error in each accelerometer’s
measurement of inclination for a 360◦ rotation around each of accelerometer axes.
The distribution of these errors are shown for each accelerometer as bottom three
box plots. Rotations around the z axis can be seen to result in the lowest
error. This is expected because rotations around the x axis or y axis incorporate
measurements from the accelerometer z axis which is indicated as providing a
greater initial offset error in the MPU-6050 data sheet [121]. This is due to the
physical structure of the MEM sensor. The worst case inclination measurement
errors are ±6.5◦ , 6.9◦ and ±2.6◦ for rotations around the x, y and z axes respectively.
These errors are three times greater than that of the lowest performing commercial
IMU and demonstrate the need for calibration.
A B C D E F G H I

10 10 10
8 8 8
θx error (degrees)

θy error (degrees)

θz error (degrees)

6 6 6
4 4 4
2 2 2
0 0 0
−2 −2 −2
−4 −4 −4
−6 −6 −6
−8 −8 −8
−10 −10 −10
0 90 180 270 360 0 90 180 270 360 0 90 180 270 360
Inclination (degrees) Inclination (degrees) Inclination (degrees)
10 10 10
8 8 8
θx error (degrees)

θx error (degrees)

θx error (degrees)

6 6 6
4 4 4
2 2 2
0 0 0
−2 −2 −2
−4 −4 −4
−6 −6 −6
−8 −8 −8
−10 −10 −10
A B C D E F G H I A B C D E F G H I A B C D E F G H I
Accelerometer Accelerometer Accelerometer

Figure 6.9: Error of each uncalibrated accelerometer’s measurement of inclination

for rotations around each axis. The bottom three box plots show the distribution
of the error in the top three plots (whiskers indicate 1.5× interquartile range).

Figure 6.10 shows the performance of the calibrated accelerometers. The worst case
inclination measurement errors are ±0.35◦ , ±0.44◦ and ±0.15◦ for rotations around
the x, y and z axes respectively. This performance matches that of the highest
performing commercial IMU; the 3DM-GX3-25 which is indicated as achieving an

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accuracy of ±0.5◦ .

A B C D E F G H I

0.5 0.5 0.5

0.4 0.4 0.4
θx error (degrees)

θy error (degrees)

θz error (degrees)
0.3 0.3 0.3
0.2 0.2 0.2
0.1 0.1 0.1
0 0 0
−0.1 −0.1 −0.1
−0.2 −0.2 −0.2
−0.3 −0.3 −0.3
−0.4 −0.4 −0.4
−0.5 −0.5 −0.5
0 90 180 270 360 0 90 180 270 360 0 90 180 270 360
Inclination (degrees) Inclination (degrees) Inclination (degrees)
0.5 0.5 0.5
0.4 0.4 0.4
θx error (degrees)

θx error (degrees)

θx error (degrees)
0.3 0.3 0.3
0.2 0.2 0.2
0.1 0.1 0.1
0 0 0
−0.1 −0.1 −0.1
−0.2 −0.2 −0.2
−0.3 −0.3 −0.3
−0.4 −0.4 −0.4
−0.5 −0.5 −0.5
A B C D E F G H I A B C D E F G H I A B C D E F G H I
Accelerometer Accelerometer Accelerometer

Figure 6.10: Error of each calibrated accelerometer’s measurement of inclination

for rotations around each axis. The bottom three box plots show the distribution
of the error in the top three plots (whiskers indicate 1.5× interquartile range).

6.6 Magnetometer calibration

The magnetometer calibration model is described by Equations 6.15 and 6.16

where m is the calibrated magnetometer measurement in Guass, S is the soft-iron
matrix incorporating the individual axis sensitivities and cross-axis coupling, u
the calibrated magnetometer output in lsb, and h is magnetometer bias in Gauss.

m = Su − h (6.15)

      
mx  sxx sxy sxz  ux  hx 
      
m  = s    
 y   yx syy syz  uy  − hy  (6.16)
      
mz szx szy szz uz hz

Three separate calibration solutions to determine the soft-iron matrix, S, and

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hard-iron biases, h, are proposed in the following sections. Each solution represents
a different level of complexity with associated practical implications.

6.6.1 Magnetometer calibration datasets

Magnetometer calibration was evaluated for two scenarios to demonstrate ‘typical’

and ‘worst case’ magnetic distortions. The ‘typical’ distortions are represented
by the magnetometer mounted on a PCB designed according to guidelines [122];
for example, positioning the magnetometer away from ferro-magnetic elements
and high-current traces. The ‘worst case’ distortions are obtained from the same
devices with an additional metal plate positioned directly above the magnetometer.
The plate is the shielding on a radio module and represents a scenario relevant
to any wireless application incorporating a magnetometer. Figure 6.11 shows
the magnetometer in the both the ‘typical’ and ‘worst case’ magnetic distortion
arrangements.

Figure 6.11: Magnetometer in the ‘typical’ (left) and ‘worst case’ (right) magnetic
distortion arrangements. The magnetometer chip is indicated by the orange circle.

An immediate characterisation of the calibrated magnetometer can be achieved

through a 3D plot of the measurements obtained for a set of Two Dimensional (2D)
and 3D rotations within the ambient uniform magnetic field. Magnetometer
measurements for an arbitrary 3D rotation through many different orientations
would ideally form the surface of a sphere centred on the origin and measurements

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for 2D rotations around each calibrated axis would form three circles on the
surface of the sphere, each concentric with the axis of rotations. However, soft-iron
distortions will distort the sphere into an ellipsoid and hard-iron biases will shift
the centre away from the origin. This visual method of evaluation is demonstrated
in some magnetometer manufacturer application notes [133].

Figure 6.12 shows ellipsoid plots of the uncalibrated measurements provided by

one of the nine magnetometers mounted on the calibration cube for the ‘typical’
and ‘worst case’ distortions. The measurements for the arbitrary 3D rotation are
shown as the grey line. The red, green and blue points are measurements for 2D
rotations around the x, y and z axis respectively. The ∼70◦ inclination of the
Earth’s magnetic field in the UK results in a significant equatorial displacement of
each of the coloured circles. Both plots indicate soft-iron and hard-iron distortions
but the ‘worst case’ magnetic distortions causes significantly greater distortions of
the ellipsoid and offset from the origin. The ‘worst case’ distortions are so great the
origin is outside of the ellipsoid manifold, any calculation of heading as a direction
around the origin would be meaningless.
Z

Y X Y X

Figure 6.12: Example 3D plot of uncalibrated magnetometer measurements for

‘typical’ (left) and ‘worst case’ (right) magnetic distortion datasets

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6.6.2 Calibrated magnetometer performance

The magnetometer performance was obtained as the error in the angular

measurement of heading provided by the magnetometer when rotated around an
axis perpendicular with the Earth’s surface. The heading angle for a rotation
around a given axis can be calculated using Equations 6.18 to 6.20 where vx , vy
and vz describe a vector orthogonal to both the measured direction of the magnetic
field provided by the magnetometer, m, and the measured direction of gravity
provided by the accelerometer, a; and θx , θy and θz are the heading angles for
rotations around the x, y and z axis respectively. It is necessary to incorporate
the calibrated accelerometer measurements to eliminate sensitivity to the tilt of
the magnetometer relative to the horizontal.

 T
a m
vx vy vz = × (6.17)
kak kmk

θx = atan2 (vy , vz ) (6.18)

θy = atan2 (vz , vx ) (6.19)

θz = atan2 (vx , vy ) (6.20)

The calibration cube protractor was used to rotate the cube 360◦ around
each axis in 5◦ steps and the mean accelerometer and magnetometer outputs
obtained for approximately 2000 samples over 30 seconds at each step. The
corresponding values of θx , θy or θz for each step were then be computed and
compared with the angle indicated by the protractor to yield the error in the
angular measurement of heading provided by an magnetometer. A misalignment

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between the protractor and magnetic North introduced an offset error to each
magnetometer’s measurement of heading. This offset was removed by subtracting
the mean error of the nine measurements of heading for a given axis of rotation.
Figure 6.13 shows the calibration cube protractor being used to collect this
magnetometer evaluation data.

Figure 6.13: Calibrated magnetometer performance was evaluated by using the

calibration cube protractor to rotate through precise heading angles

The top three plots of 6.14 show the error in each uncalibrated magnetometer’s
measurement of heading for a 360◦ rotation around each of axes for the ‘typical’
magnetic distortion scenario. The distribution of these errors are shown for each
magnetometer as bottom three box plots. The distribution of the errors between
±180 seen in Figure 6.14 demonstrates that magnetometer calibration is essential
for the calculation of heading. The error in uncalibrated measurements of heading
obtained for the ‘worst case’ dataset were all uniformly distributed between ±180.

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A B C D E F G H I

180 180 180

135 135 135
θx error (degrees)

θy error (degrees)

θz error (degrees)
90 90 90
45 45 45
0 0 0
−45 −45 −45
−90 −90 −90
−135 −135 −135
−180 −180 −180
0 90 180 270 360 0 90 180 270 360 0 90 180 270 360
Heading (degrees) Heading (degrees) Heading (degrees)
180 180 180
135 135 135
θx error (degrees)

θx error (degrees)

θx error (degrees)
90 90 90
45 45 45
0 0 0
−45 −45 −45
−90 −90 −90
−135 −135 −135
−180 −180 −180
A B C D E F G H I A B C D E F G H I A B C D E F G H I
Magnetometer Magnetometer Magnetometer

Figure 6.14: Error of each uncalibrated magnetometer’s measurement of heading

for rotations around each axis for the ‘typical’ magnetic distortion dataset. The
bottom three plots show the distribution of the error in the top three plots.

6.6.3 Offset calibration

Magnetometer offset calibration accounts for only the hard-iron distortions. The
soft-iron matrix, S is assumed to be an identity so that the magnetometer
calibration model described by Equation 6.16 simplifies to Equation 6.21. The
calibrated magnetometer measurement and hard-iron bias are both represented in
units of lsb for this model.
     
mx  ux  hx 
     
m  = u  − h  (6.21)
 y  y  y
     
mz uz hz

The hard-iron bias parameters hx , hy and hz are determined through a

least-squares calculation based on a dataset of the uncalibrated magnetometer
measurements when rotated through many different orientations within a uniform
magnetic field. A calibrated magnetometer would measure a constant magnitude
for all orientations so that all measurements would satisfy Equation 6.22 where v

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is the magnitude of the magnetic field in lsb. This can be expressed in vector form
as shown by Equation 6.23.

(mx − hx )2 + (my − hy )2 + (mz − hz )2 = v 2 (6.22)

 
hx 
 
h2 
 x
 
  
hy 
1 −2mz   2 2 2 2
−2mx 1 −2my  2  = v − mx − my − mz (6.23)
hy 
 
 
h 
 z
 
h2z

The dimensions of Equation 6.23 can be extended to accommodate a dataset of n

measurements as shown by Equation 6.24 which can be equivalently represented
by Equation 6.25.
 
hx 
    
 2 2 2 2 2
 −2mx1 1 −2my1 1 −2mz1  hx   v − mx1 − my1 − mz1 
   
 
 −2mx2 1 −2my2 1 −2mz2   h   v 2
− m 2
− m 2
− m 2
   y  x2 y2 z2 
 . .. .. .. ..    = .  (6.24)
 .. .   .. 
 . . .  h2
 
  
y
 
−2mxn 1 −2myn 1 
−2mzn hz   2 2
v − mxn − myn − mzn 2 2
 
h2z

Mh = v (6.25)

The hard-iron biases can therefore the calculated as the solution to Equation 6.25
for a specified value of v and numerical dataset of n measurements. The value of
h may be obtained using the pseudoinverse of M as described by Equation 6.31
for a calibration dataset of dataset of n > 6. Elements hx , hy and hz of vector h

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are invariant to the value of v; a value of 0 can be chosen for convenience.

h = M +v

−1 (6.26)
T T
= M M M v

6.6.3.1 Offset calibration performance

The nine magnetometers mounted on the calibration cube were calibrated as

described above for both the ‘typical’ and ‘worst case’ scenarios. Figure 6.15
shows the ellipsoid plots for each dataset. The ellipsoid is centred on the origin
which indicates that the hard-iron biases have been removed. The absence of the
soft-iron sensitivity matrix, S, means that the deformation from a sphere cannot
be accounted for with this limited calibration model.
Z

Y X Y X

Figure 6.15: Example 3D plot of magnetometer measurements after offset

calibration for ‘typical’ (left) and ‘worst case’ (right) magnetic distortion datasets

Figure 6.16 summarises the heading measurement error provided by the

nine magnetometers for the ‘typical’ magnetic distortion scenario after offset
calibration. The heading error archived for each axis is approximately ±6◦ . This
is greater than that of the worst performing commercial IMU, the UM6. However,
this accuracy may be sufficient for some applications and offers the benefit of a

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simple calibration model.

A B C D E F G H I

10 10 10
8 8 8
θx error (degrees)

θy error (degrees)

θz error (degrees)
6 6 6
4 4 4
2 2 2
0 0 0
−2 −2 −2
−4 −4 −4
−6 −6 −6
−8 −8 −8
−10 −10 −10
0 90 180 270 360 0 90 180 270 360 0 90 180 270 360
Heading (degrees) Heading (degrees) Heading (degrees)
10 10 10
8 8 8
θx error (degrees)

θx error (degrees)

θx error (degrees)
6 6 6
4 4 4
2 2 2
0 0 0
−2 −2 −2
−4 −4 −4
−6 −6 −6
−8 −8 −8
−10 −10 −10
A B C D E F G H I A B C D E F G H I A B C D E F G H I
Magnetometer Magnetometer Magnetometer

Figure 6.16: Heading measurement error provided by the nine magnetometers for
the ‘typical’ magnetic distortion scenario after offset calibration.

Figure 6.17 summarises the heading measurement error provided by the nine
magnetometers for the ‘worst case’ magnetic distortion scenario after offset
calibration. The simplified offset calibration model is clearly insufficient for this
scenario. The distortion of a sphere into the ellipsoid seen in Figure 6.15 causes
the x axis to fall outside of the circle of data points describing θx ; consequently it
is not possible to resolve a heading and θx is distributed between ±180◦ .

6.6.4 Ellipsoid calibration

Ellipsoid calibration requires the fitting an ellipsoid to the data points provided by
the uncalibrated magnetometer when rotated through many different orientation
in a homogeneous magnetic field. The soft-iron matrix, S, and hard-iron vector,
h, may then be obtained as the geometric solution that equated the ellipsoid with
a sphere. Ellipsoid fitting is the basis for many existing magnetometer calibration
solutions [134]. The magnetometer calibration solution presented here builds on

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A B C D E F G H I

180 180 180

135 135 135
θx error (degrees)

θy error (degrees)

θz error (degrees)
90 90 90
45 45 45
0 0 0
−45 −45 −45
−90 −90 −90
−135 −135 −135
−180 −180 −180
0 90 180 270 360 0 90 180 270 360 0 90 180 270 360
Heading (degrees) Heading (degrees) Heading (degrees)
180 180 180
135 135 135
θx error (degrees)

θx error (degrees)

θx error (degrees)
90 90 90
45 45 45
0 0 0
−45 −45 −45
−90 −90 −90
−135 −135 −135
−180 −180 −180
A B C D E F G H I A B C D E F G H I A B C D E F G H I
Magnetometer Magnetometer Magnetometer

Figure 6.17: Heading measurement error provided by the nine magnetometers for
the ‘worst case’ magnetic distortion scenario after offset calibration.

an ellipsoid fitting algorithm shared by Yury Petrov on the MATLAB file exchange
[135] that evaluates the ellipsoid origin, radii and principle axes for a given dataset.
The calibration parameters S and h are derived from these parameters. The
ellipsoid fitting algorithms describes an ellipsoid using with the polynomial shown
as Equations 6.27 and 6.28.

v1 x2 + v2 y 2 + v3 z 2 + 2v4 xy + 2v5 xz + 2v6 yz + 2v7 x + 2v8 y + 2v9 z = 1 (6.27)

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 Chapter 6. Calibration of low-cost MEMS sensors for AHRS applications

 
v1 
 
v 
 2
 
 
v3 
 
 
 
  v4 
 
x2 y 2 z 2 2xy 2xz 2yz 2x 2y 2z  
v5  = 1 (6.28)
 
 
v6 
 
 
v7 
 
 
v 
 8
 
v9

The dimensions of Equation 6.28 can be extended to accommodate a dataset of n

measurements as shown by Equation 6.29 which can be equivalently represented
by Equation 6.30 where x, y, and z correspond to calibrated magnetometer
measurements mx , my and mz in lsb.

 
A
 
B 
 
 
  
C   
2 2 2
 x1 y1 z1 2x1 y1 2x1 z1 2y1 z1 2x1 2y1 2z1   

 1
  D 
   
 x2 y 2 z 2 2x2 y2 2x2 z2 2y2 z2 2x2 2y2 2z2  1
 2 2 2 
   
. .. .. .. .. .. .. .. ..   E = . (6.29)
 .. .    
 . . . . . . .    .. 
  F   
x2n yn2 zn2 2xn yn 2xn zn 2yn zn 2xn 2yn 2zn   
 1
G
 
 
H 
 
 
I

Mh = v (6.30)

The ellipsoid can therefore the calculated as the solution to Equation 6.30 for a
numerical dataset of n measurements. The value of h may be obtained using the

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Chapter 6. Calibration of low-cost MEMS sensors for AHRS applications

pseudoinverse of M as described by Equation 6.31 for a calibration dataset of

dataset of n > 9.

h = M +v

−1 (6.31)
T T
= M M M v

The polynomial parameters can be arranged in a matrix to describe the algebraic

form of the ellipsoid shown Equation 6.32. The origin of the ellipsoid, c, found as
Equation 6.33.  
v1 v4 v5 v7 
 
v4 v2 v6 v8 
 
T =  (6.32)
v v6 v3 v9 
 5 
 
v7 v8 v9 −1
 −1
  v1 v4 v5 
 
c = v7 v8 v9  v v
 4 2 6v  (6.33)
 
v5 v6 v3

The radii of the ellipsoid can be determined from the Eigenvalues of Rc as described
by Equation 6.35. Tc describes the T having been translated to the origin. λ1 to
λ2 are the first three Eigenvalue of Rc .

    T 
t11 t12 t13 t14  1 0 0  1 0 0 
     
t21 t22 t23 t24   1 0 c 0 1 0 c
  0   
Tc =  =  T  (6.34)
t t32 t33 t34   0 1  0 0 1 
 31  0   
     
t41 t42 t43 t44 0 0 0 1 0 0 0 1

 
t11 t12 t13 
  1
Rc =  
t21 t22 t23  t44 (6.35)
 
t31 t32 t33

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 Chapter 6. Calibration of low-cost MEMS sensors for AHRS applications

A rotation matrix describing the orientation of ellipsoid is created as a matrix

where the Eigenvectors of Rc make up the columns.

 
R = λ1 λ2 λ3 (6.36)

The above ellipsoid fitting algorithm describes the ellipsoid as an origin, c, radii,
r and a rotation R. The soft-iron matrix, S, and hard-iron vector, h, can be
obtained using Equations 6.37 and 6.38. r is the intensity of the Earth’s field in
Gauss within the calibration dataset, known to be approximately 0.5 Gauss in the
UK [30].  
r
 rx 0 0
  T
S = R
0
r
rx
0
R (6.37)
 
r
0 0 rx

h = Sc (6.38)

6.6.4.1 Ellipsoid calibration performance

The nine magnetometers mounted on the calibration cube were calibrated as

described above for both the ‘typical’ and ‘worst case’ scenarios, Figure 6.18 shows
the ellipsoid plots for each dataset. The calibrated measurements for both datasets
now form the surface of a sphere centred on the origin. This confirms successful
calibration of the relative axis sensitivities and the hard-iron bias. The red, green
and blue circles describing the rotations around each axis are now concentric with
each corresponding axis for the ‘typical’ dataset. However, these circles indicate
that the magnetometers calibrated for the ‘worst case’ datasets incorporate a
misalignment approximated by a −15◦ rotation around the y axis.

Figure 6.19 summarises the heading measurement error provided by the nine
magnetometers for the ‘typical’ magnetic distortion scenario after ellipsoid

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Chapter 6. Calibration of low-cost MEMS sensors for AHRS applications

Z
Z

Y X Y X

Figure 6.18: Example 3D plot of magnetometer measurements after ellipsoid

calibration for ‘typical’ (left) and ‘worst case’ (right) magnetic distortion datasets

calibration. The error for rotations around each axis is approximately ±5◦ ,

matching the performance of the worst performing commercial IMU, the UM6.

A B C D E F G H I

10 10 10
8 8 8
θx error (degrees)

θy error (degrees)

θz error (degrees)

6 6 6
4 4 4
2 2 2
0 0 0
−2 −2 −2
−4 −4 −4
−6 −6 −6
−8 −8 −8
−10 −10 −10
0 90 180 270 360 0 90 180 270 360 0 90 180 270 360
Heading (degrees) Heading (degrees) Heading (degrees)
10 10 10
8 8 8
θx error (degrees)

θx error (degrees)

θx error (degrees)

6 6 6
4 4 4
2 2 2
0 0 0
−2 −2 −2
−4 −4 −4
−6 −6 −6
−8 −8 −8
−10 −10 −10
A B C D E F G H I A B C D E F G H I A B C D E F G H I
Magnetometer Magnetometer Magnetometer

Figure 6.19: Heading measurement error provided by the nine magnetometers for
the ‘typical’ magnetic distortion scenario after ellipsoid calibration.

The heading measurement errors for the ‘worst case’ magnetic distortion scenario
are summarised in Figure 6.20. The misalignment around the y axis observed in
Figure 6.18 indicates that the variance of errors in θx and θz will be greater than

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 Chapter 6. Calibration of low-cost MEMS sensors for AHRS applications

that in θy . This is confirmed by Figure 6.20. It is evident that ellipsoid calibration

alone is not sufficient for the the ‘worst case’ magnetic distortion scenario.

A B C D E F G H I

40 40 40
30 30 30
θx error (degrees)

θy error (degrees)

θz error (degrees)
20 20 20
10 10 10
0 0 0
−10 −10 −10
−20 −20 −20
−30 −30 −30
−40 −40 −40
0 90 180 270 360 0 90 180 270 360 0 90 180 270 360
Heading (degrees) Heading (degrees) Heading (degrees)
40 40 40
30 30 30
θx error (degrees)

θx error (degrees)

θx error (degrees)
20 20 20
10 10 10
0 0 0
−10 −10 −10
−20 −20 −20
−30 −30 −30
−40 −40 −40
A B C D E F G H I A B C D E F G H I A B C D E F G H I
Magnetometer Magnetometer Magnetometer

Figure 6.20: Heading measurement error provided by the nine magnetometers for
the ‘worst case’ magnetic distortion scenario after ellipsoid calibration.

Ellipsoid calibration is able to successfully evaluate each axis sensitivity, cross-axis

sensitivities and the hard-iron biases but it is not able to compensate for rotations
around the origin. Consequently, the magnetometers calibrated for the ‘worst case’
magnetic distortions provide heading errors of up to 30◦ . Although performance
for ‘typical’ magnetic distortions is better, it is limited by a failure to incorporate a
misalignment between the magnetometer and accelerometer. The shortcomings of
the ellipsoid calibration method can be overcome if the calibrated magnetometer
axes can be brought into alignment with the calibrated frame.

6.6.5 Alignment calibration

The proposed alignment calibration method requires that the magnetometer has
already been calibrated using the ellipsoid method to yield an initial soft-iron
matrix, Ŝ, and hard-iron vector, ĥ. The calibrated datasets describing rotations

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Chapter 6. Calibration of low-cost MEMS sensors for AHRS applications

around the calibration cube x, y and z axis form the red, green and blue circles seen
Figure 6.18 are used to calibrated for misalignment. The data points of each circle
lie on a plane orthogonal to the axis of rotation. A rotation matrix describing the
calibration cube relative to the ellipsoid-calibrated magnetometer can therefore be
found as the solutions to these planes.

Equations 6.39 and 6.40 describe a plane where α, β, γ and δ are fixed
coefficients and mx , bmy and cmz represents an ellipsoid-calibrated magnetometer
measurement that sits on the plane.

αmx + βmy + γmz + δ = 0 (6.39)

 
α
 
β 

 
mx my mz 1  =0 (6.40)
γ 
 
 
δ

The dimensions of this equation can be extended to incorporate the n

measurements as described by Equations 6.41 and 6.42 where v describes a vector
that is orthogonal to the plane and of arbitrary magnitude.

   
 
 mx1 my1 mz1 
α 1
   −δ   
 mx2 my2 mz2    1
  β   
 . .. ..   −δ  =  ..  (6.41)
 .. . .   
 .
  
  γ  
−δ
mxn myn mzn 1

 T
Mv = 1 1 . . . 1 (6.42)

The axis orthogonal to the plane, v, can be found as the solution to Equation 6.43
for a dataset of n measurements where n > 3 where M + is the pseudoinverse of

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 Chapter 6. Calibration of low-cost MEMS sensors for AHRS applications

M.
 T
+
v=M 1 1 ... 1
 T (6.43)
T

−1 T
= M M M 1 1 ... 1

The vectors orthogonal to each planes are found as vx , vy and vz . These vectors
can be normalised and combined to approximate a rotation matrix as shown by
Equation 6.44. A true rotation matrix, R, can be obtained from R̃ by computing
the best-fit rotation matrix as described in chapter 2.

 T
vy
R̃ = vx
kvx k kvy k
vz
kvz k
(6.44)

The aligned soft-iron matrix, S, and hard-iron vector, h, can be obtained from
the initial ellipsoid-calibrated parameters, Ŝ and ĥ by correcting each using the
calculated misalignment, R, as described by Equations 6.45 and 6.46 .

S = RT Ŝ (6.45)

h = RT ĥ (6.46)

6.6.5.1 Alignment calibration performance

The nine magnetometers mounted on the calibration cube were calibrated as

described above for both the ‘typical’ and ‘worst case’ scenarios. Figure 6.21
shows the ellipsoid plots for each dataset. The −15◦ misalignment around the y
axis previously seen in Figure 6.18 for the ‘worst case’ scenario has been corrected
for and the red, green and blue circles now lie concentric with each axis for both
datasets.

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Chapter 6. Calibration of low-cost MEMS sensors for AHRS applications
Z

Z
Y X Y X

Figure 6.21: Example 3D plot of magnetometer measurements after alignment

calibration for ‘typical’ (left) and ‘worst case’ (right) magnetic distortion datasets

Figure 6.22 summarises the heading measurement error provided by the nine
magnetometers for the ‘typical’ magnetic distortion scenario after alignment
calibration. The error remains appropriately equal for each axis of rotation but is
the peak-to-peak error through a 360◦ is now half that of the ellipsoid-calibrated
performance. The performance is now approximates that of the 2◦ accuracy
achieved by the VN-100 commercial IMU. Geographic location is known to affect
the heading accuracy achieved by a magnetometer; for example, a ±0.5◦ at the
equator would result in a ±1.5◦ heading error in the UK (see Appendix C). This
suggests that the approximate ±2◦ heading accuracy demonstrated here may be
closer to that of the high-end MTw and 3DM-GX3-25 if the test were to be repeated
at the equator. However, it was not possible to verify this empirically, nor was
it possible to confirm the test conditions of the heading accuracy specified by the
commercial IMU.

The heading measurement errors for the ‘worst case’ magnetic distortion scenario
are summarised in Figure 6.23. The error for rotations around each axis are now
appropriately equal though the errors remain greater than those for the ‘typical’
magnetic distortions. Although some devices achieve the 2◦ accuracy achieved by

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 Chapter 6. Calibration of low-cost MEMS sensors for AHRS applications

A B C D E F G H I

5 5 5
θx error (degrees) 4 4 4

θy error (degrees)

θz error (degrees)
3 3 3
2 2 2
1 1 1
0 0 0
−1 −1 −1
−2 −2 −2
−3 −3 −3
−4 −4 −4
−5 −5 −5
0 90 180 270 360 0 90 180 270 360 0 90 180 270 360
Heading (degrees) Heading (degrees) Heading (degrees)
5 5 5
4 4 4
θx error (degrees)

θx error (degrees)

θx error (degrees)
3 3 3
2 2 2
1 1 1
0 0 0
−1 −1 −1
−2 −2 −2
−3 −3 −3
−4 −4 −4
−5 −5 −5
A B C D E F G H I A B C D E F G H I A B C D E F G H I
Magnetometer Magnetometer Magnetometer

Figure 6.22: Heading measurement error provided by the nine magnetometers for
the ‘typical’ magnetic distortion scenario after alignment calibration.

the VN-100, the performance is generally closer to that of the UM6 which may
be sufficient for many applications. The contrast between the ‘typical’ and ‘worst
case’ performance demonstrates the need to minimise magnetic distortions within
the physical design of a device.

A B C D E F G H I

10 10 10
8 8 8
θx error (degrees)

θy error (degrees)

θz error (degrees)

6 6 6
4 4 4
2 2 2
0 0 0
−2 −2 −2
−4 −4 −4
−6 −6 −6
−8 −8 −8
−10 −10 −10
0 90 180 270 360 0 90 180 270 360 0 90 180 270 360
Heading (degrees) Heading (degrees) Heading (degrees)
10 10 10
8 8 8
θx error (degrees)

θx error (degrees)

θx error (degrees)

6 6 6
4 4 4
2 2 2
0 0 0
−2 −2 −2
−4 −4 −4
−6 −6 −6
−8 −8 −8
−10 −10 −10
A B C D E F G H I A B C D E F G H I A B C D E F G H I
Magnetometer Magnetometer Magnetometer

Figure 6.23: Heading measurement error provided by the nine magnetometers for
the ‘worst case’ magnetic distortion scenario after alignment calibration.

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Chapter 6. Calibration of low-cost MEMS sensors for AHRS applications

6.6.6 Summary of magnetometer calibration solutions

Three magnetometer calibration solutions have proposed, each achieving a different

level of accuracy and having different practical implications for the end application.
Table 6.2 summarises the heading error for each of the solutions, calculated as the
mean of the maximum of each device tested.

Heading error
Calibration solution
‘Typical’ ‘Worst case’
Hard-iron ±5.5◦ ±55.5◦
Ellipsoid ±3.3◦ ±21.1◦
Ellipsoid + alignment ±1.9◦ ±3.9◦

Table 6.2: Heading error for each magnetometer calibration solution achieved for
‘typical’ and ‘worst case’ magnetic distortion scenarios

The hard-iron calibration solution uses a simplified calibration model requiring

only three parameters and implemented through three scalar operations. This
may appeal to application with limited processing resources. The ±5.5◦ accuracy
achieved for ‘typical’ distortions is sufficient for many applications but hard-iron
calibration alone is unable to compensate for the ‘worst case’ distortions. The ±3.3◦
accuracy achieved through ellipsoid calibration is greater than that of the the UM6
and approaches the accuracy of the VN-100. However, ellipsoid calibration alone
is also insufficient for ‘worst case’ distortion scenarios.

Where as hard-iron and ellipsoid calibration require only the arbitrary rotation
dataset, alignment calibration requires the additional three single-axis rotation
datasets. This increases the complexity the calibration process but enables
significantly improved performance. In the case of ‘typical’ distortions, the ±1.9◦
accuracy achieved matches that of the VN-100. The ±3.9◦ accuracy achieved for
‘worst case’ distortions is less accurate but still sufficient for many applications.

None of the calibration solutions were able to match the ±0.5◦ heading accuracy
of the 3DM-GX3-25. Although it may be assumed that the greatest accuracy

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 Chapter 6. Calibration of low-cost MEMS sensors for AHRS applications

achieved, ±1.9◦ , is limited by the geographical location of the tests, there was not
an opportunity to demonstrate this empirically or to clarify the test conditions of
the 3DM-GX3-25.

6.7 Simplifying sensor alignment arithmetic

The gyroscope, accelerometer and magnetometer calibration models account for

a misalignment of each sensor to provide measurements aligned to a common
calibrated frame. The associated rotation matrix multiplication more than
triples the number of scalar arithmetic operations required by the gyroscope and
accelerometer calibration models which may represent a significant burden for
low-power or high sample rate applications. If the gyroscope and accelerometer
can be assumed to be physically aligned then it is possible to eliminate the rotation
matrix multiplications while maintaining a calibrated alignment between all three
sensors.

In this simplified calibration model, the common calibrated frame becomes that
of the physical accelerometer and the magnetometer calibration parameters are
adjusted accordingly. The gyroscope is assumed to be physically aligned with the
accelerometer so that the calibration models for the three sensors may be redefined
as Equations 6.47 to 6.49.

ω = Sω−1 (uω − bω ) (6.47)

a = Sa−1 (ua − ba ) (6.48)

m = S̃um − h̃ (6.49)

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Chapter 6. Calibration of low-cost MEMS sensors for AHRS applications

The adjusted soft-iron and hard-iron parameters S̃ and h̃ in Equation 6.49 are
calculated as Equations 6.50 and 6.51 where R is the rotation matrix describing
the calibrated frame relative to the physical accelerometer, previously determined
in Section 6.5.3.
T
S̃ = C
AR S (6.50)

T
h̃ = C
AR h (6.51)

This simplification significantly reduces both the computational load of the

calibration models as well as the number of parameters needed to be stored in
the memory of a calibrated device. Monolithic packages such as the MPU-6050
combine both the gyroscope and accelerometer on the same silicon die [136] to
minimise misalignment between these two sensors and justify the assumption
behind this simplification.

6.8 Conclusion

This chapter has presented a complete calibration solution for IMU incorporating a
gyroscope, accelerometer and magnetometer. Practical performance was evaluated
for the low-cost (<10 USD) MPU-6050 and HM-5883L sensors and benchmarked
against four commercial IMUs price between 199 USD and 3000 USD. The results
indicate a static pitch and roll accuracy matching that to the highest performing
commercial IMU and a static heading accuracy matching that of a mid-range
commercial IMU.

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Chapter 7

Revised AHRS algorithm

7.1 Introduction

The original AHRS algorithm presented in Chapter 3 represents some of the

first work of this project. The algorithm was derived analytically and tested
using a fully-calibrated commercial IMU. The subsequent characterisation studies
presented in Chapter 5 and calibration solutions in Chapter 6 demonstrated
how modern low-cost MEMS sensors can achieve equivalent performance of such
commercial IMUs. The revised AHRS algorithm presented in this chapter was
designed to operate with these sensors based on the findings of Chapters 5 and
6. The algorithm design continued to evolve through the development of the IMU
platforms and applications presented in Chapter 8; building upon the experiences
gained from a diverse range of practical applications.

The revised AHRS algorithm builds on the quaternion implementation of the

complementary filter proposed by Mahony et al. [55, 62]. As with other
complementary filters, Mahony’s algorithm is dependent on a predefined fixed
inclination of the magnetic field in the operating environment [63, 64]. This
limitation is incompatible with many applications that must accommodate

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Chapter 7. Revised AHRS algorithm

magnetic variations [100, 101]. It is common for complementary filters to

compensate for a gyroscope bias through integral feedback [62, 35]. However, this is
arguably an inappropriate solution as integral wind-up dynamics will likely exceed
those of the gyroscope bias and so risk worse performance. The revised algorithm
addresses these shortcomings and introduces several new features, including:

• Fast initialisation behaviour

• Gyroscope bias compensation

• Magnetic distortion rejection

• Linear acceleration rejection

• Calculation of ‘zero-g’ and global accelerations

Although the calculation of ‘zero-g’ and global accelerations are not required for
the computation of attitude or heading, these accelerations can only be obtained
in conjunction with an AHRS algorithm and are often the ultimate quantities
required by an application.

7.1.1 Algorithm overview

The algorithm provides a measurement of the IMU orientation relative to the

Earth. The Earth coordinate system is defined according to the East, North,
Up (ENU) convention with the y axis pointing in the direction of magnetic north
and the z pointing directly up. Sensors must be calibrated to provide gyroscope
measurements in units of radians per second and accelerometer measurements in
units of g. The calibrated magnetometer units may be arbitrary. The algorithm
can also operate without a magnetometer as described in the discussion at the end
of the chapter. The complete algorithm is shown as the block diagram in Section
7.5. The following sections describe the individual processes that make up the
revised AHRS algorithm.

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 Chapter 7. Revised AHRS algorithm

7.2 Complementary filter estimator

The unnormalised quaternion IE q describes the orientation of the Earth, E, relative

to the IMU, I, and is obtained through the integration of the rate of change of the
quaternion, IE q̇, as described by Equation 7.1. I
E q̇ is computed as Equation 7.2 [98]
and is a function of the normalised quaternion, IE q̂; the gyroscope measurement,
ω; and an error term, e, scaled by the algorithm gain K.

Z
I I
Eq = E q̇.dt (7.1)

 
I 1I T
E q̇ = E q̂ ⊗ 0 (ω − Ke) (7.2)
2

The error term e is determined from the accelerometer and magnetometer

measurements with each sensor providing a separate error component. The
accelerometer error component, ea , is calculated as the cross product of the
normalised accelerometer measurement, â, and the direction of gravity assumed by
I
E q̂ as shown by Equation 7.3 where qw , qx , qy and qz are the elements of IE q̂. The
second operand of Equation 7.12 is equal to the third column of the rotation matrix
obtained from IE q̂. Equations 7.1 to 7.3 alone describe an algorithm identical to
[62]; subsistent algorithm processes represent novel development.

 
2qx qz − 2qw qy 
  a
ea = â ×  2q
 y z q + 2q q 
w x  , where â = (7.3)
  kak
2 2
2qw − 1 + 2qz

The magnetometer error component, em , is calculated as the cross product

of the normalised magnetometer measurement, m̂, and the direction of East
(perpendicular to magnetic north) assumed by IE q as shown by Equation 7.4. The
last operand of Equation 7.4 is equal to the negated second column of the rotation

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Chapter 7. Revised AHRS algorithm

matrix obtained from IE q̂. The cross product of â and m̂ yields a vector that is
orthogonal to both gravity as assumed by the accelerometer, and the measured
direction of the magnetic field thus providing a vector measurement on a plane
parallel with the Earth’s surface that is insensitive to the orientation of the IMU
and the inclination of the magnetic field, as in [137]. This is a feature of Digital
Magnetic Compasses (DMCs) sometimes refereed to as tilt compensation.

 
2 2
−2qw + 1 − 2qx 
  m
em = â × m̂ × 
 −2q q
x y + 2q q 
w z  , where m̂ = (7.4)
  kmk
−2qx qz − 2qw qy

In normal operation, e is calculated as the sum of ea and em . However, these

individual components are derived from the normalised measurements and so risk
the numerical process being corrupted by a divided by zero. e is therefore defined
by Equation 7.5 where invalid sensor measurements are omitted from the algorithm
update. If e is set to zero then IE q̂ is determined from the integrated gyroscope
measurements alone. This conditional selection of e is a prerequisite of some of
the sensor conditioning described in Section 7.3.




 ea + em if kak > 0 and kmk > 0


e= ea else if kak > 0 (7.5)

  T



 0 0 0 else

7.2.1 Algorithm gain and initialisation

The initial value of IE q̂, assumed to be an identity, will likely be incorrect upon
initialisation of the algorithm. The low value of K and corresponding slow
convergence of IE q̂ desired for normal operation will cause this error to be sustained
for a period of several seconds. This initialisation time can be reduced by ramping

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 Chapter 7. Revised AHRS algorithm

down K from a high value to the value intended during normal operation. This
process is described by Equation 7.6 where tinit is the initialisation period in
seconds, Kinit is the initial large value of K and Knormal is the value intended
during normal operation where Knormal > Kinit .


 Knormal + tinit −t
tinit
(Kinit − Knormal ) if t < tinit
K= (7.6)

 K else
normal

The algorithm output will be unreliable during initialisation. An application may

determine whether the algorithm initialisation has completed as the boolean result
of the condition t ≥ tinit .

7.3 Sensor conditioning

The sensor conditioning processes described in this section are performed prior to
the main complementary filter described in Section 7.2.

7.3.1 Gyroscope bias compensation

Gyroscope bias compensation is achieved by substituting ω 0 in place of ω in

Equation 7.2 where ω 0 is equal to the original gyroscope measurement with an
estimated bias, ωbias , subtracted as described by Equation 7.7.

ω 0 = ω − ωbias (7.7)

ωbias is computed as the low-pass filtered result of ω as described in Equation 7.8

where fc is the corner frequency in Hz and p is a value dynamically determined as

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either 1 or 0 to enable the integrator only when the IMU is stationary

Z
ωbias = 2πfc pω.dt (7.8)

p is determined as Equation 7.9 where the function fb (ω, ωmin ) that computes the
time that the magnitude of each element of ω has been below ωmin . ωmin is the
minimum angular velocity below which the bias estimator can be enabled. tb is
the minimum stationary period after which the bias estimator will be enabled.


 1 if fb (ω, ωmin ) > tb
p= (7.9)

 0 else

7.3.2 Magnetic distortion rejection

Magnetic measurements that deviate from the expected intensity of the Earth’s
magnetic field are rejected to prevent magnetic distortions from corrupting the
algorithm output. This is achieved by substituting m0 in place of m in Equations
7.4 and 7.5. m0 is defined by Equation 7.10 where mmin and mmax are the specified
minimum and maximum magnitudes of the Earth’s magnetic field in units of
the magnetometer. The selection of m0 as zero works in conjunction with the
conditional selection of e in Equation 7.5 to omit the magnetometer from the
algorithm update.



 m if mmin < kmk < mmax
0
m =  T (7.10)

 0 0 0 else

7.3.3 Linear acceleration rejection

Linear acceleration rejection is achieved in a similar way to [56]. The accelerometer

is only omitted if the measured magnitude of gravity exceeds limits for a specified

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 Chapter 7. Revised AHRS algorithm

period of time. The process is described by 7.11 where the function f (a, gmin , gmax )
calculates the period of time that the condition of −gd < (kak − 1) < gd has been
false.   T

 0 0 0 if f (a, gd ) > ta
a0 = (7.11)

 a else

7.4 ‘Zero-g’ and global acceleration calculation

The ‘zero-g’ acceleration, azero , is the accelerometer measurement with gravity

removed. It is calculated as the accelerometer measurement, a, minus the 1 g
offset dependent on the orientation of the IMU as shown by Equation 7.12 where
qw , qx , qy and qz are the elements of IE q̂. The second operand of Equation 7.12 is
equal to the third column of the rotation matrix obtained from IE q̂.

 
2qx qz − 2qw qy 
 
azero =a− 
2qy qz + 2qw qx  (7.12)
 
2qw2 − 1 + 2qz2

The global acceleration, aglobal , is the acceleration of the IMU in the Earth
coordinate system as defined by the ENU convention. It is calculated as Equation
7.13.
   
0 aTglobal =
I
E q̂ ⊗ 0 aTzero ⊗ IE q̂ ∗ (7.13)

7.5 Algorithm block diagram

Figure 7.1 shows a block diagram of the complete AHRS algorithm and indicates
the connection between the separate processes described in Sections 7.2 to 7.4.

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 Chapter 7. Revised AHRS algorithm
Gyroscope bias Sensor conditionaing Complementary filter estimator ‘Zero-g’/global acceleration calculation
ωbias

Gyroscope
Gyroscope bias compensation
ω0 + ω 0 − Ke Calculate IE q̇
I q̇
E
R I q
E I q
Orientation
ω E I q̂
(Equation 7.7) (Equation 7.2) kI q
E k E
−

Ke

Initialisation behaviour
K (Equation 7.6)

Magnetometer m0 e
152

Magnetic distortion rejection

m (Equation 7.10)

Error calculation
(Equations 7.3 to 7.5)

Accelerometer
Linear acceleration rejection
a (Equation 7.11) a0

Global acceleration
Zero-g acceleration calculation Global acceleration calculation
aglobal
(Equation 7.12) (Equation 7.13)

Zero-g acceleration
azero

Figure 7.1: Complete AHRS block diagram

 Chapter 7. Revised AHRS algorithm

7.6 Discussion

The algorithm computes the error in ω 0 from the accelerometer and magnetometer
through a cross product operation, the reason for this may not be immediately
obvious. In the case of the accelerometer, the error is determined as the difference
between the direction of gravity measured by the accelerometer, â, and the
direction of gravity assumed by IE q̂. The cross product yields ea as a vector with a
magnitude equal to the sine of the angular error in IE q̂ and a direction orthogonal
to both â and the direction of gravity assumed by IE q̂. ea therefore describes the
necessary ‘adjustment’ to IE q̂ applicable to the dimensions of ω 0 . In the case of
the magnetometer, em represents an equivalent adjustment though its direction
is always aligned with gravity to prevent the magnetometer from influencing the
pitch and roll components of orientation.

The sine relationship defining kek suggests a potential flaw in the algorithm.
Angular errors exceeding 90◦ would yield an increasingly small feedback term. An
error of 180◦ would result in no feedback and the algorithm would never converge.
In practice, it is may assumed that angular errors in IE q̂ will always be <90◦ after
initialisation and an appropriate selection of Kinit will avoid the hazard associated
with an initial error of 180◦ .

7.6.1 Algorithm gain

The algorithm gain, K, controls the rate at which IE q̂ converges to an orientation

assumed by the accelerometer and magnetometer. A large value of K will mean an
increased rate of convergence but also an greater sensitivity to linear accelerations
and temporal magnetic distortions. A small value of K will mean that the IE q̂ is
predominately determined by the gyroscope but will risk an accumulating error
if K is not sufficiently large enough to compensate for gyroscope measurement

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errors. A minimum value K is therefore limited by the accuracy of the gyroscope,

if K = 0 then IE q̂ is determined from the integrated gyroscope measurements alone.
An optimal value K is that found to provide sufficient attenuation of the errors
imposed by the accelerometer and magnetometer whilst remaining large enough
that gyroscope drift is compensated for.

The algorithm process described by Equations 7.1 to 7.5 is a first-order

1
proportional feedback system where kek ≈ π
radians for small angular errors in
I
E q̂. It is therefore possible to approximate performance in terms of K analytically.
For example, it can be shown that K < π is necessary for stability and that the
4π
algorithm will converge within 2% of the final output in K
seconds [138]. The
dynamics may be expressed in the frequency domain to show that IE q̂ incorporates
the accelerometer and magnetometer data through a low-pass filter with a corner
K
frequency of 2π 2
Hz. This may provide a convenient means of choosing K if
the frequency characteristics of an application are known, as in [57, 61, 56]. In
practice, an appropriate value of K is typically found through manual adjustments.
Knormal = 0.5 was found to provide suitable performance for most applications.
Kinit = 10 and tinit = 3 were found to provide reliable initialisation behaviour.
The large value of Kinit deliberately causes a momentary instability to escape an
initial angular error in IE q̂ of ∼180◦ .

7.6.2 Gyroscope bias compensation

The gyroscope bias is estimated each time the IMU is detected as being stationary.
Some manufactures suggest similar mechanisms where measurements below a
threshold are zeroed [139] or steady-state signals are sampled as the bias [140].
Such solutions would risk corrupting small amplitude signals and result in
undesirable step changes of the gyroscope bias. The bias compensation proposed in
this chapter is not subject to these shortcomings. Some applications may involve no

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natural stationary periods. In such scenarios, bias compensation may still effective
if an initial stationary period can be imposed to allow the system to ‘warm up’
prior to use.

The gyroscope bias compensation parameters include the filter corner frequency,
fc , the minimum angular velocity, ωmin , below which the bias estimator can
enabled and the minimum stationary period, tb , after which the bias estimator
will be enabled. A corner frequency of fc = 0.05 Hz achieves convergence (within
2% of final value) within 13 seconds yet is relatively low to reduce the risk of
attenuating low amplitude motion that may falls below ωmin . A value of ωmin may
be determined as the expected variation in the gyroscope bias during operation.
For example, the thermal response and random walk investigations presented in
Chapter 5 indicate that a threshold of ωmin = 4◦/s would be sufficient. The value
should include a margin to account unexpected noise within the application; for
example, mechanical vibrations. A period threshold of tb = 2 seconds would be
sufficient to prevent the bias compensation of activating during zero-crossing in
most applications.

7.6.3 Magnetic distortion rejection

The magnetic distortion rejection parameters, mmin and mmax define the minimum
and maximum valid intensity for the ambient magnetic field. Default values of
mmin = 0.22 Gauss and mmax = 0.67 Gauss may be assumed as these represent
the natural limits of the Earth’s magnetic field [30]. More robust performance
can be achieved if a reduced range can be specified representative of a specific
operating environment.

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Chapter 7. Revised AHRS algorithm

7.6.4 Linear acceleration rejection

The linear acceleration rejection parameters include the period threshold, ta ,

and gravity deviation, gd . Values of ta = 100ms and gd = 0.1 g were
found to compensate for linear accelerations without being too sensitivity that
accelerometer measurements are rejected unnecessarily.

7.6.5 Algorithm outputs

The algorithm provides IE q̂ describing the orientation of the Earth relative to the
IMU. Many applications will require a measurement of the IMU relative to the
I ∗
Earth, E
I q̂, which is obtained as the conjugate E q̂ . An equivalent Euler angle or

rotation matrix may be obtained using the methods described in Chapter 2.

The algorithm also provides the estimated gyroscope bias, ωbias . This
may be useful to processes external to the AHRS algorithm that require a
bias-compensated gyroscope measurement. The provision of ωbias also enables
the gyroscope bias to be saved to non-volatile memory so that the ‘tracked’ value
can be maintained even when the system is shut down.

The calculation of ‘zero-g’ accelerations, azero , and global accelerations, aglobal ,

can only be achieved in conjunction with an AHRS algorithm. aglobal is the second
derivative of position in the Earth coordinate system. Integrating this would yield
a measurement of velocity; integrating the velocity would yield the displacement of
the IMU. The ability to measure displacement is of benefit to many applications.
In practice, this can only be successful if an exponentially increasing error inherent
to numerical integration process can be compensated for. For example, pedestrian
dead-reckoning applications achieve this through zero velocity detection [3].

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7.6.6 Omitting the magnetometer

In some applications it is desirable to omit the magnetometer; for example, if

the magnetic environment is known to be unreliable. This may be achieved by
substituting zeros in place of the magnetometer measurement, m. Without the
I
magnetometer, the heading component of E q̂ is determined from the gyroscope
alone. The heading will initially be relative to the direction of the IMU
immediately after initialisation but will drift over time. The pitch and roll
components of IE q̂ will be unaffected and still provide an absolute measurement
relative to the Earth’s surface. The calculation of azero and aglobal will also be
unaffected.

7.7 Conclusion

The revised AHRS algorithm presented in this chapter builds on the findings of
the sensor characterisation and calibration work presented in Chapters 5 and 6.
It was developed simultaneously with the IMU platforms and applications and
applications in the next chapter and incorporates features that address practical
requirements, verified through these empirical works.

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Chapter 7. Revised AHRS algorithm

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Chapter 8

IMU platforms and applications

8.1 Introduction

This chapter describes the development of IMU platforms and the broad
applications they have facilitated. These platforms take advantage of modern
MEMS devices, the calibration solutions presented in Chapter 6, and the AHRS
algorithm presented in chapter 7. The following three sections (8.2 to 8.3)
summarise the development of three IMU platforms developed through this
research project; the: x-IMU, x-BIMU and x-OSC. Each platform fulfils a different
design specification and together have facilitated a wide range of applications and
research projects. Section 8.5 summarises projects that have utilised the platforms
for specific applications in collaboration with other academic researchers. Section
8.6 section presents a selection of user applications that have utilised the IMU
platforms within commercial or academic research projects. The chapter concludes
with a description of the future developments scheduled for for the IMU platforms
and applications.

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Chapter 8. IMU platforms and applications

8.2 x-IMU: A versatile data acquisition platform

The x-IMU was developed in 2010 to provide a versatile IMU and data acquisition
platform for use in a wide range of potential applications. In the four years since,
the hardware and firmware has continued to evolve to take advantage of modern
MEMS sensors and introduce new functionality based on user feedback. The
original design was intended to meet the following specification.

x-IMU design specification

• Calibrated gyroscope, accelerometer and magnetometer with on-board

AHRS

• Real-time wired and wireless communication

• Autonomous data logging

• Input/Output (I/O) interface for external sensors and output devices

8.2.1 x-IMU overview

The x-IMU is equally applicable to real-time and autonomous data logging

applications. Real-time communication is supported through USB for wired
applications and Bluetooth for wireless. Autonomous data logging is supported
by an on-board SD card and several features including a Real-Time Clock (RTC)
and sleep timer with motion trigger wake-up. An eight channel auxiliary port
provides a configurable interface for external analogue and digital sensors or output
devices. Figure 8.1 shows the x-IMU board alone (left) and enclosed within its
plastic housing with a 1000 mAh battery with a total size of 57 × 38 × 21 mm and
weight of 49 g.

Each x-IMU is calibrated during manufacture and individual sensor calibration

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Figure 8.1: x-IMU board alone (left) and enclosed in its plastic housing with 1000
mAh battery (right)

parameters saved to on-board memory. During normal operation, the sensors are
continuously sampled and the on-board AHRS algorithm updated at 512 Hz. An
extensive set of internal settings can be managed using the x-IMU software to
configure the device behaviour and the send rate of individual packet types up to
512 Hz.

8.2.1.1 x-IMU hardware

The x-IMU board dimensions are 33×42 mm and it weights 12 g. Figure 8.2 shows
an annotated top and bottom view with key hardware components are described
below.

(a) dsPIC33FJ128GP804: 16-bit, 40 Million Instructions Per Second (MIPS)

general purpose Digital Signal Controller (DSC) with 16 kB of
Random-Access Memory (RAM) and 128 kB of program space [141].

(b) Crystal oscillator: 32.768 kHz crystal provides a time base for the RTC
sample timer.

(c) USB bridge: Future Technology Devices International (FTDI) FT232R USB
bridge. See Section 8.2.2 for more information.

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(a) (b) (c)

(p) (d)

(o) (e)

(n)
(f)

(m)

(g)
(l)

(h)
(k)

(j)
(i)

Figure 8.2: x-IMU top and bottom view with labelled key hardware components

(d) USB connector: Provides interface for real-time wired communication as

well as power for the device and battery charging.

(e) Maxim MAX1811: Fully integrated Lithium-ion charger specifically designed

for charging from a USB power supply [142].

(f) Bluetooth LED: Blue Light-Emitting Diode (LED) that indicates the
connection status of Bluetooth.

(g) Battery connector: Socket for 3.7 V Lithium-ion cell. Battery voltage is
measured on-board.

(h) Power switch: Slide switch for switching the device on and off or selecting

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the auxiliary port power input

(i) SD card socket: Micro SD card socket supports FAT16 and FAT32 formats
for capacities up to 32 GB.

(j) Bluetooth: Microchip RN-41 Bluetooth module. See Section 8.2.2 for more
information.

(k) Auxiliary port: Eight channel configurable auxiliary port also includes power
I/O pins. See Section 8.2.4 for more information.

(l) Status LED: Green LED indicates the status of the x-IMU.

(m) Command button: Configurable command button with a difficult mode to

toggle sleep mode.

(n) SD card LED: Amber LED that indicates the connection status of SD card.

(o) InvenSense MPU-6050: ±2000◦/s, 16-bit resolution gyroscope; ±16 g, 16-bit

resolution accelerometer; and 16-bit thermometer [121].

(p) Honeywell HMC5883L: ±8.1 Gauss, 12-bit resolution magnetometer [122].

8.2.1.2 x-IMU software

The x-IMU Graphical User Interface (GUI) is a Windows application that provides
an interface for configuring internal settings and displaying real-time data. Figure
8.3 shows a screen shot of the GUI displaying internal sensor measurements in 2D
plots and the AHRS data as a 3D representation. The software also provides tools
for synchronisation of real-time clock, firmware updates, inertial and magnetic
calibration tools and data logging to Comma-Separated Values (CSV) files for
exporting data to software such as MATLAB and Microsoft Excel. Figure 8.3
shows the x-IMU GUI with gyroscope and accelerometer measurements being
displayed as 2D plots, and AHRS data being displayed as a 3D representation.

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Figure 8.3: x-IMU GUI displaying real-time data with 2D and 3D graphics

The x-IMU GUI is open-source1 and incorporates a generic C# Application

Programming Interface (API) for the x-IMU that can be reused by other C#
projects. There are also interface libraries available for C++2 and Android3 . A
MATLAB library provides tools for importing and organising logged data acquired
via the SD card.

8.2.2 Real-time wired and wireless communication

The x-IMU enables wireless communication via USB, Bluetooth or serial through
the the axillary port. USB communication is enabled through a FTDI FT232R
chip to maximise compatibility with FTDI drivers are available for Windows, OS
X, Linux [143] and Android [144]. The RN-41 Bluetooth module is a class 1 device
with a range of 100 m [145]. It uses the Serial Port Profile (SPP) Bluetooth profile
eliminates the need for host drivers for compatibly with a wide range of platforms
1
https://github.com/xioTechnologies/x-IMU-GUI
2
https://github.com/xioTechnologies/x-IMU-Arduino-Example
3
https://github.com/xioTechnologies/x-IMU-Android-Example

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 Chapter 8. IMU platforms and applications

such laptops and smart phones. The auxiliary port can be configured as a serial
interface for communication with embedded systems. Settings support hardware
flow control and standard baud rates.

8.2.3 Autonomous data logging

A feature of the x-IMU that has proved to be essential to many application is

its ability to operate as an autonomous data logger. There are several features
that facilitate this operation: SD card data logging, real-time clock and calendar,
battery level monitoring and automated sleep/wake mechanisms. The x-IMU can
be configured to log continuously to create a single file for the entire period,
the x-IMU will automatically shut down when the battery reaches a specified
minimum level. Alternately, the motion trigger wake-up and sleep timer used to
automatically enter sleep mode during period of inactivity. Each time the device
wake up, a new file is created on the SD card, time-stamped with the date and time.
This may significantly extend the battery lfe for some applications. The x-IMU
use manual [146] provides suggestions of how internal settings can be configured
to minimise power consumption.

8.2.4 Auxiliary port

The auxiliary port is a 12 pin socket that provides access to power and eight
general purpose analogue/digital I/O pins. The auxiliary port can be configured
in several different modes including: analogue inputs, digital I/O, Pulse-Width
Modulation (PWM), serial and external sleep/wake mode. Analogue and digital
input modes allows data from external sensors to be sent in real-time or logged
to the SD card. PWM and digital output modes allow real-time control of wide
range of device, for example, LEDs or vibration motors for haptic feedback. The

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external sleep/wake mode allows external controls to be used to enter/exit sleep

mode. This has been of use to under water applications that require the x-IMU
to be semi-permanently sealed within water proof housing.

8.3 x-BIMU: A low-power, wireless IMU for

real-time applications

The x-BIMU was developed in 2012 to provide a more versatile wireless alternative
to the x-IMU. The x-IMU uses Bluetooth which imposes several limitations. The
Bluetooth standard limits the maximum number of active devices to seven [147]
which may be insufficient for many applications. Throughput and latency become
unreliable when using multiple devices with a single Bluetooth receiver which may
comprise real-time applications. Bluetooth modules also consume more power that
some other wireless technologies and so reduce the potential battery life.

The x-BIMU was designed to function as a low-power, wireless IMU to provide

low-latency performance for real-time applications. Use of any specific wireless
technology would represent a constraint to some applications; the x-BIMU instead
employs a modular design for compatibly with a wide range of communication
modules. The x-BIMU design was intended to meet the following specification.

x-BIMU design specification

• Calibrated gyroscope, accelerometer and magnetometer with on-board

AHRS

• Low-latency, real-time wireless communication

• Modular design to support a range communication interface

• Low-power operation

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• Synchronisation of data between multiple devices

8.3.1 x-BIMU overview

The x-BIMU is designed to mate with XBee-style communication modules; hence

the name “x-BIMU” which would be pronounced “XBee IMU”. XBee is the
brand name for a product line of wireless communication modules produced by
Digi. The XBee-style connector has been adopted by many manufactures and
so represents a standardised interface. Although the x-BIMU is designed to
operate with any XBee-style module, it is optimised to provide its best low-power,
low-latency and robust wireless performance with XBee 802.15.4 modules. Using
these modules, up to 12 x-BIMUs can stream simultaneously with a single host
to provide synchronised measurement data with sample rates >200 Hz. Figure 8.4
shows the x-BIMU with an unattached XBee 802.15.4 module.

Figure 8.4: x-BIMU with an unattached XBee 802.15.4 module

Size and weight are often important for wireless applications and so the x-BIMU
was designed for a compact form factor. When enclosed within its plastic housing
with a 320 mA battery and wireless module, the x-BIMU has a total size of 30 ×
38 × 22 mm and weight of 22 g. Figure 8.5 shows the x-BIMU in plastic housing
alone (left) and combined with a Velcro body strap (right) designed to secure the
plastic housing for human motion applications.

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Figure 8.5: x-BIMU in plastic housing with XBee 802.15.4 module and 320 mAh
battery (left) and Velcro strap designed for human motion applications (right)

The principle operation of the x-BIMU is to process measurements from

the on-board gyroscope, accelerometer and magnetometer to yield calibrated
quantities alongside AHRS data and to stream this data via the connected
communication module to the host platform; for example, to a computer or smart
phone. Calibration parameters are evaluated for each x-BIMU during production
and saved to internal memory. During normal operation, the on-board sensors are
sampled and AHRS algorithm updated at 256 Hz. The rate at which individual
sensor measurements or orientation data are then transmitted can be configured
in internal settings.

The communication protocol was designed to maximise compatibility. Data can

be transmitted as either American Standard Code for Information Interchange
(ASCII) or binary packets. ASCII packets represent data as human-readable
strings of CSVs terminated by a new-line character. This format may be convenient
for interpretation by high-level programming languages such as MATLAB. Binary
packets represent data in compact packets of fixed lengths to facilitate a
deterministic latency and higher throughput. Binary packets are more convenient
for interpretation by lower-level languages such as C++, C# or Java. An ASCII
communication protocol is also used to configure all internal settings which allows
the x-BIMU to be configured using any serial terminal.

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8.3.1.1 x-BIMU hardware

The x-BIMU board dimensions are 25 × 28 mm and it weights 5 g. Figure 8.6

shows an annotated top view with key hardware components are described below.

(b) (c)
(a) (d)

(e)
(j)

(f)

(i)
(g)
(h)

Figure 8.6: x-BIMU top with labelled key hardware components

(a) RGB status LED: The colour indicates the selected wireless channel of
the XBee 802.15.4 module of the battery charging status if the x-BIMU
is connected to a charger.

(b) Battery connector: Socket for 3.7 V Lithium-ion cell. Battery voltage is
measured on-board and can be transmitted via the connected communication
module to enable to user to monitor the battery level.

(c) USB connector: A socket charging the connected battery, it does not provide
USB communication.

(d) Power button: Used to turn the device on and off. The button also provides
a fail-safe mechanism of recovery if the device is configured with incorrect

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settings.

(e) Microchip MCP73832T: Fully integrated Lithium-ion charge management

controller specifically designed for charging from a USB power supply [148].

(f) Micrel MIC5319: Ultra-low dropout voltage regulator able to supply up to

500 mA [149] to accommodate high-power communication modules.

(g) InvenSense MPU-6050: ±2000◦/s, 16-bit resolution gyroscope; ±16 g, 16-bit

resolution accelerometer; and 16-bit thermometer [121].

(h) Honeywell HMC5883L: ±8.1 Gauss, 12-bit resolution magnetometer [122].

(i) Microchip PIC24FJ64GA102: 16-bit, 16 MIPS general purpose

microcontroller with eXtreme Low Power (XLP) technology [150].

(j) XBee socket: Connection for any XBee-style module. Provides interface to
serial communication as well as control signals such as sleep and reset.

8.3.1.2 x-BIMU software

The x-BIMU Terminal is a windows application that functions as a basic serial

terminal but also provides tools specific to the x-BIMU, including real-time 2D and
3D visualisations, calibration tools data logging to CSV files for exporting data
to software such as MATLAB and Microsoft Excel. Figure 8.7 shows the x-BIMU
Terminal with gyroscope, accelerometer, magnetometer and battery measurements
each being displayed as 2D plots, and AHRS data being displayed as a 3D
representation. The x-BIMU Terminal is open-source4 and incorporates a generic
C# interface library for the x-BIMU that can be reused by other C# projects.
There are also interface libraries available for C++5 and Android6 .

4
https://github.com/xioTechnologies/x-BIMU-Terminal
5
https://github.com/xioTechnologies/x-BIMU-Arduino-Example
6
https://github.com/xioTechnologies/x-BIMU-Android-Example

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Figure 8.7: x-BIMU Terminal displaying real-time data with 2D and 3D graphics

8.3.2 Modular communication interface

The x-BIMU is optimised for XBee 802.15.4 modules but compatible with any
XBee-style communication module. Upon start up, the x-BIMU will automatically
detect if an XBee 802.15.4 module is present and configure it to operate in API
mode. This gives the x-BIMU explicit control over packetisation at the 802.15.4
Medium Access Control (MAC) layer [151] to enable low-latency and low-power
performance. If an XBee 802.15.4 module is not detected, communication reverts
to a generic 115200 baud serial interface compatible with any XBee-style module
with a matching configuration.

Digi support a number of different communication technologies within the XBee

product line. Support for additional technologies is offered by other manufacturers
who have adopted the XBee-style interface. Figure 8.8 shows a range of modules
chosen to demonstrate the versatility of the x-BIMU when combined with different
off-the-shelf modules. Each modules is described below.

(a) Digi XBee 802.15.4 (1 mW): Standard 802.15.4 module with a maximum
transmit power for 1 mW for a range of 100 m [151]. The module shown
incorporates a low-profile PCB antenna.

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(a) (b) (c) (d)

(e) (f) (g) (h)

Figure 8.8: Example XBee-style communication modules compatible with the

x-BIMU, for: 802.15.4 (a) (b), Bluetooth (c), ZigBee (d), Proprietary 900 MHz
(e), Wi-Fi (f), Bluetooth 4.0 Low Energy (g) and wired serial (h).

(b) Digi XBee 802.15.4 (60 mW): High-power variant of the 802.15.4 module;
operates identically but has a maximum transmit power for 60 mW for a
range of 1 km [151]. The module shown has a wire antennae to provide
better omnidirectional performance.

(c) Microchip RN41XV Bluetooth: Class 1 Bluetooth 2.1 module for 100 m
range [152] Supports SPP to avoid the need for host drivers.

(d) Digi XBee ZB: Supports licensed ZigBee for complex network configurations
such as mesh networks, and compatibility with third-party ZigBee systems.
The module shown is a 2 mW variant for 120 m range, 63 mW variants are
also available for 3.2 km range [153].

(e) Digi XBee-PRO 900: Uses proprietary 900 MHz communication protocol
with range of 15.5 km [154]. The module requires an external antennae.

(f) Digi XBee Wi-Fi: 802.11n for high-throughput User Datagram Protocol
(UDP) or Transmission Control Protocol (TCP) transmission over ad hoc
or existing infrastructure Wi-Fi networks [155].

(g) Seeed Studio BLEbee: Bluetooth 4.0 Low Energy module for direct

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connection to modern consumer platforms such as laptops and smart-phones.

BLEbee is an open-source7 project developed by Dr Michael Kroll.

(h) x-BIMU Serial Breakout: Open-source8 serial adapter for direct wired
interface to microelectronics or USB and RS-232 platforms when combined
with a corresponding adapter.

8.3.3 Low-power optimisation and features

The x-BIMU incorporates several low-power optimisations and features to extend

battery life and facilitate potential new applications. Central to the design is
the Microchip PIC24FJ64GA102 XLP microcontroller. This part was chosen as
its 16 MIPS would meet the demands of the calibration and AHRS processing
requirements with a maximum current consumption of 11 mA during normal
operation and 25 nA in its lowest power mode [150]. Standard hardware and
firmware design techniques were employed to minimise power consumption [156];
including extensive use of interrupts allowing the processor maximise time spent
in idle mode and sourcing peripheral component power from a microcontroller
output to minimise current leakage when powered down.

8.3.3.1 802.15.4 low-power optimisations

The 802.15.4 standard is specially designed for low-power devices to spend large
amount of time asleep [157]. The XBee 802.15.4 idle current is 50 mA but also
includes a <10 μA sleep mode with a typical wake up time of 10 ms [151]. The
x-BIMU has explicit control of packetisation of the 802.15.4 MAC layer and so is
able to cycle this sleep mode between packet transmission to achieve significant
power savings for low data-rate applications.

7
https://github.com/michaelkroll/BLEbee
8
https://github.com/xioTechnologies/x-BIMU-Serial-Breakout

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8.3.3.2 Sleep timer and motion trigger wake up

In some applications it may be desirable for the wireless IMU to power down
during periods of inactivity to extend the battery life. The x-BIMU supports this
through a sleep timer and motion trigger wake up. The sleep timer will trigger
the x-BIMU to enter sleep mode if no motion is detected for a specified period
of time. The motion trigger wake up will cause the x-IMU to immediately wake
up again once motion is detected. An important feature of this mechanism is the
low-latency wake up as any missed motion data could compromise measurements.
To minimise power consumption, motion is detected using the MPU-6050 built-in
Digital Motion Processor (DMP) [121]. This allows the all other electronics
to completely power down until a hardware interrupt pulse is generated by the
MPU-6050.

8.3.3.3 Measured current consumption

The current consumption of an x-BIMU with an XBee 802.15.4 module was

measured for different usage scenarios. Table 8.1 summaries these measurements
and indicates the corresponding battery life calculated for the 320 mAh battery
shown previously in Figure 8.5. The nominal voltage of the battery is 3.7 V.

Test condition Current Battery life

Sleep mode 94 μA 142 days
Sleep mode (motion trigger wake up enabled) 1 mA 13 days
Running (transmitting data at 16 Hz) 22 mA 15 hours
Running (transmitting data at 128 Hz) 64 mA 5 hours

Table 8.1: Measured current consumption of x-BIMU and XBee 802.15.4 module

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8.3.4 Synchronisation of multiple devices

Many applications require measurements received from multiple IMUs to

synchronised in time. The x-BIMU is able to achieve this using the XBee
802.15.4 modules. An x-BIMU can be configured to transmit data on one of
12 channels. This allows up to 12 x-BIMUs to transmit simultaneously without
interface from one another. The firmware was designed for a fixed and determined
latency between the sampling of the sensors and transmission of the corresponding
data packet. Synchronisation may therefore be achieved by time-stamping the
time-of-arrival of each packet.

Figure 8.9: Four 802.15.4 receivers (Digi XSticks) in a USB hub provide
synchronised measurements from x-BIMUs

This time-of-arrival synchronisation technique avoids the need to maintain

independent synchronised clocks on each wireless node. However, it does require
one receiving module per x-BIMU. This is current achieved using a USB hub of
multiple 802.15.4 receivers, as shown in Figure 8.9. An open-source9 software
application communicates with each receiver and time-stamps each received
packet.

9
https://github.com/xioTechnologies/x-BIMU-Logger

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8.4 x-OSC: High-performance wireless IMU and

I/O interface

The x-OSC was developed in 2012 and intended to address the need for a combined
high-performance wireless I/O interface and IMU. The x-OSC was introduced
in a 2013 paper [9] which provides a survey of existing wireless I/O devices and
discussion on their shortcomings. Recent years have seen the appearance of several
commercial wireless IMU systems, for example the Xsens MTw [128] and APDM
opal [158]. However, these systems utilise low-bandwidth wireless technologies
and so are typically limited to an update rate of ∼120 Hz for multiple IMUs.
Furthermore, none of these systems offer an I/O interface for external sensors and
output devices.

The x-OSC takes advantage of modern embedded Wi-Fi technology to achieve

superior wireless performance and take advantage of the wide support and
versatility of Wi-Fi and Ethernet networking. The x-OSC design was intended
to meet the following specification.

x-OSC design specification

• Calibrated gyroscope, accelerometer and magnetometer with on-board

AHRS

• 32 configurable I/O channels

• Versatile Wi-Fi network configurations

• Maximise platform compatibility

• Optimise for throughput and latency

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8.4.1 x-OSC overview

The x-OSC, shown in Figure 8.10, is a wireless I/O board that provides just
about any software with access to 32 high-performance analogue/digital channels
and on-board IMU measurements via Open Sound Control (OSC) messages over
Wi-Fi. There is no user programmable firmware and no software or drivers to
install making x-OSC immediately compatible with any Wi-Fi-enabled platform.

Figure 8.10: x-OSC

OSC is a lightweight network protocol designed specifically for communication

between computers and multimedia devices [159]. It is an ideal solution to achieve
a cross-platform and cross-language interface with over 80 languages/platform
implementations [160]. Although OSC is widely supported, many platforms fail
to incorporate the full specification [161]. To maximise compatibility, x-OSC
messages are limited to the four fundamental data types: int32, float32, OSC-string
and OSC-blob. A full description of x-OSC messages is available in the x-OSC user
manual [162].

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8.4.1.1 x-OSC hardware

The x-OSC board dimensions are 32 × 45 mm and it weights 11 g. Figure 8.11

shows an annotated top view with key hardware components are described below.

(d) (e) (f) (g) (h)

(c)

(b)

(a)

(l)
(k) (j) (i)

Figure 8.11: x-OSC top and bottom view with labelled key hardware components

(a) dsPIC33EP512MC806: 16-bit, 70 MIPS DSC that represents one of

Microchip’s highest performing devices with extensive I/O and peripheral
support [163].

(b) Ping button: Used to trigger network ‘pings’ to facilitate device discover.
Also provides a fail-safe mechanism of recovery if the device is configured
with incorrect settings.

(c) 16 Input channels: See Section 8.4.2 for more information. Includes
regulated 3.3 V output to power user electronics.

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(d) Honeywell HMC5883L: ±8.1 Gauss, 12-bit resolution magnetometer [122].

(e) RGB LED: Indicates the networking mode and connection status. Can also
be controlled by user to provide visual feedback within an application.

(f) InvenSense MPU-6050: ±2000◦/s, 16-bit resolution gyroscope; ±16 g, 16-bit

resolution accelerometer; and 16-bit thermometer [121].

(g) 16 Input channels: See Section 8.4.2 for more information. Includes
unregulated battery voltage output to power user electronics.

(h) Microchip MRF24WG0MA: Microchip’s highest performing Wi-Fi module,

capable of up to 5 Mbit sustained throughput and maximum transmit power
of +18 dBm [164].

(i) Analog devices ADP3338: 1A ultralow dropout regulator capable of meeting

the power demands of both Wi-Fi and external user electronics [165]

(j) 74LVC125A Quad buffers : Output buffers enable each of the 16 outputs to
sink/source up to 50 mA [166].

(k) Battery connector: Socket for 3.7 V Lithium-ion cell. Battery voltage is
measured on-board and available to the user as an OSC message.

(l) Crystal oscillator: 16 MHz, ±20 ppm crystal drives system clock [167].

8.4.1.2 x-OSC browser interface

An embedded web server enables all internal settings to be configured using a web
browser, see Figure 8.12. Settings may be viewed and modified during run-time
without interrupting the acOSC messages. This eliminates the need for any
device-specific software and allows the x-OSC to be configured using any laptop,
smart phone or tablet.

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Figure 8.12: x-OSC settings viewed on web browser

8.4.2 I/O channels

16 dedicated inputs can be independently configured to be either analogue or

digital. Digital inputs can be configured to use internal pull-up/down resistors
and to minimise latency their state is only transmitted on change. All 16 analogue
inputs are sampled with 13-bit resolution and sent simultaneously at a specified
update rate up to 400 Hz. Analogue mode inputs also provide a comparator
function to send a message each time a specified threshold is crossed. This enables
low-latency threshold detection without the need for a high message rate.

16 dedicated outputs can be independently configured to digital, pulse or PWM

modes. In digital mode, an output can be set to high or low enabling simple
control of LEDs, relays, or generation of control logic signals. In pulse mode, an
output can be triggered to generate a pulse with a period of 1 ms to 1 minute at
a resolution of 1 ms. This may be useful for momentary actuators such a solenoid
driving the strike mechanism of a percussive instrument. An output in PWM
mode can generate a PWM waveform from 5 Hz to 250 kHz with a duty cycle

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resolution up to 16-bit. PWM is commonly used as a DAC where fixed frequency

and variable duty cycle approximate an analogue signal. For example, this may
be used to control the brightness of a light or the speed of a motor.

In addition to modes described above, the first four inputs and outputs can be
configured to serial mode with each transmit and receive pair utilising a dedicated
hardware Universal Asynchronous Receiver/Transmitter (UART) module. Each
serial channel supports baud rates in the range 9600 to 1 M baud and incorporates
a 2 kB buffer to ensure high throughput without loss of data. Received serial
data is framed before being sent as OSC-blob messages. Framing boundaries are
determined by a user defined buffer size, timeout and optional framing character.

8.4.3 Network modes

x-OSC can be configured to operate in one of two network modes: ad hoc or

infrastructure. In ad hoc mode, x-OSC creates a network for other devices to join.
Multiple devices can connect to a single x-OSC with simultaneous access to its
I/O. Infrastructure mode allows x-OSC to connect to an existing network. The
device Internet Protocol (IP) address can be configured to be static or use DHCP
to be assigned an appropriate IP address by the network server. The assigned IP
address can be discovered by pressing the ping button, which causes x-OSC to
broadcast a message indicating the IP address over the network. Alternatively,
a ping message can be sent to x-OSC by another network device. Infrastructure
mode enables multiple x-OSCs to operate on the same network and be addressed by
multiple host devices also connected to the network. A connection to a router can
also provide an inherent interface to x-OSC via Ethernet or from remote internet
connections.

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8.4.4 Practical throughput and latency

Investigations were conducted to demonstrate the practical latency and throughput

that could be achieved for up to 15 x-OSCs. The results of the investigations were
presented in a 2014 paper that also covered recommended networking optimisations
for best performance [10]. Experiments were carried out in a university lab in the
presence of other Wi-Fi networks, which were revealed using a Wi-Fi scanning
application. A spectrum analyser was used to confirm that any significant use
of the 2.4 GHz spectrum was limited to the visible Wi-Fi networks. When only
one Wi-Fi channel was required, the scanning application was used to select an
appropriate channel. Each investigation used a Late 2013 13” MacBook Pro
running WireShark v1.10.5, fitted with a Broadcom BCM4360 transceiver and
a Thunderbolt to Ethernet adaptor. All OSC messages were sent as unicast
UDP packets. The infrastructure network was provided by a LevelOne WBR-6805
pocket router/AP configured as open (no encryption) in 802.11g mode (54 Mbps)
with an Ethernet connection to the host computer.

8.4.4.1 Round-trip Latency

The round-trip latency was evaluated by wiring an x-OSC digital input and output
together and then measuring the time between an output toggle message being
sent by the host computer and the input change message being received from
x-OSC. The digital input and digital output OSC messages are 100 and 32 bytes
long respectively, each incorporating an address pattern and either 16 or 1 int32
argument/s. A software application was written to send a message to toggle the
digital output every 50 ms and WireShark was used to log the time of each
packet sent and received. This method of evaluating latency is different from
that previously proposed [9] and has the advantage of eliminating the software
application from the loop, which may impose an unknown latency specific to

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the software, Operating System (OS) or processing load of the computer. The
infrastructure network was created by connecting the computer and x-OSC to a
single router/AP.

Ideal (latency test-packets only) Under load (with additional 200 packets per second)

0.6
Normalised distribution

0.5

0.4

0.3

0.2

0.1

0
0 2 4 6 8 10 12 14 16 18
Round-trip latency (ms)

Figure 8.13: Normalised distribution of measured round-trip latency

Figure 8.13 shows the round-trip latency distribution of over 13000 samples
achieved for either ‘ideal’ conditions, where communication is limited to only
the digital input/output messages; and ‘under load’ conditions, where x-OSC
was configured to simultaneously send analogue input data at 200 messages per
second. Table 8.2 summarises the results. It may be assumed that the latency
for communication in either direction is approximately half that of the observed
round-trip latency, for example, <3 ms for ‘ideal’ conditions.

Test condition Mean 95% less than

Ideal 5.30 ms 6.59 ms
Under load 8.09 ms 9.96 ms

Table 8.2: Summary of measured round-trip latency

8.4.4.2 Throughput

Throughput was evaluated as the total packet rate achieved by up to 15 x-OSCs,

each attempting to send 450 analogue input messages per second. This rate is
intentionally greater than can be achieved by a single x-OSC to demonstrate
saturated throughput. Each message is a UDP packet containing a 104 byte

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OSC message including 16 float32 arguments. WireShark was used to log the
time of arrival of each packet and packets per second was calculated as the
number of packets arriving from each x-OSC within each one second window.
Each experiment starts with a single running x-OSC. At one minute intervals, an
additional x-OSC is activated for a period of 15 minutes, to yield a recording of
throughput for 1 to 15 x-OSCs. Tests were conducted with the 15 x-OSCs sharing
a single channel and evenly distributed between three non-overlapping channels to
investigate the benefit.
x−OSC 1 x−OSC 2 x−OSC 3 x−OSC 4... Sum
4000

3600

3200
Packets per second

2800

2400

2000

1600

1200

800

400

0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Minutes

Figure 8.14: Throughput of 1 to 15 x-OSCs sending to a single AP on one channel

Figure 8.14 shows the throughput for 1 to 15 x-OSCs connected to a single AP

and indicates that up to four x-OSCs can operate on a single channel without
significantly impacting the 400 packets per second ceiling of a single x-OSC.
Beyond this, additional x-OSCs reduce the throughput of each device so that when
all 15 are active, the net throughput is ∼3800 packets per second. An important
observation is that this over-saturated network reduces the throughput of each
device equally (from 400 to ∼250 packets per second).

Figure 8.15 shows the throughput for 1 to 15 x-OSCs connected to three APs, each
operating on a separate non-overlapping channel. Distribution between multiple
channels can be seen to produce an increased net throughput of ∼4800 packets per
second. The first group of five x-OSCs were configured on channel 1, the next on

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x−OSC 1 x−OSC 2 x−OSC 3 x−OSC 4... Sum

5600

4800

4000

Packets per second

3200

2400

1600

800

0
0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15
Minutes

Figure 8.15: Throughput of 1 to 15 x-OSCs sending to three APs on three

non-overlapping channels

channel 11 and the final five on channel 6. This specific order demonstrates that the
channel 1 and 11 groups are able to operate simultaneously without interference.
After 10 minutes, the inclusion of the the final group (channel 6) increases the net
throughput proportionally but with significantly increased variance. Crucially, it
can be seen that the the throughput of an x-OSC in the channel 11 group falls
below that of the rest once the channel 6 group appears. This demonstrates the
potential for channel 6 to interfere with channels 1 or 11 [168].

8.4.4.3 Throughput-latency trade-off

Although the above results show that the x-OSC is able to achieve both
high-throughput and low latency, the large channel access overhead involved in
the transmission of 802.11 packets does produce a trade-off between throughput
and latency [169]. For example, the x-OSC firmware was modified to demonstrate
a maximum throughput of ∼400 packets per second with a 1 byte payload (0.4
kB/s), and ∼295 packets with a 1472 byte payload (∼434 kB/s). This corresponds
to a 1000 times increase in throughput at the cost of a ∼25% reduction in send rate.
This indicates the potential for the x-OSC to achieve significantly higher sample
rates by allowing the user to sacrifice latency in favour of bundling multiple OSC

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messages within large UDP packets.

8.5 Collaborative applications

The development of the IMU platforms has coincided with exploration of numerous
applications over the past four years. However, the largest projects were achieved
through collaboration with other people. This section present two selected
projects.

8.5.1 The gloves

“The gloves” are a pair of data gloves that allow the wearer interact with software
though the natural motion and gesticulation of their hands. Although the gloves
have potential for a wide variety of applications, development has focussed on
live music performance where the gloves provide the wearer with an unlimited
number of virtual musical instruments and access to all the tools of a professional
production desk; all played and controlled live, through precise motion and
gestures.

The project started in 2010 as a partnership between a musician Imogen Heap, and
a computer scientist Dr Tom Mitchell of the University of the West of England.
Together they created ‘SoundGrasp’ [170]; a single glove with a wrist-mounted
mic that would allow the wearer to record and play back audio samples. The
glove was a 5DT 14 Ultra glove which incorporated 14 fibre optic bend sensors to
provide measurements finger flexion and abduction via USB. An ANN was used
to identify predefined postures that would either trigger events or switch modes.
The continuous motion of flexion also provided a means to control variable audio
effects including reverb and filtering. Figure 8.16 shows the glove with the postures
and their associated control audio function.

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Figure 8.16: SoundGrasp gloves with wrist-mounted microphone and postures with
associated control function. Image source: [170]

The fingertip-less gloves did not obstruct the wearer from playing most musical
instruments and the wrist-mounted microphone could be used to sample both voice
and acoustic instruments. SoundGrasp was an exciting proof of concept but is was
clear that much more could be achieved with the adoption of IMU technology. The
author of this dissertation joined the partnership to help achieve this.

8.5.1.1 Prototype gloves

The prototype gloves built upon the SoundGrasp system by using a pair of gloves
and mounting an x-IMU to each wrist. Each x-IMU used a USB connection along
side that of the 5DT gloves. The x-IMU auxiliary ports were configured in PWM
mode to drive an Red, Green, Blue (RGB) LED on each hand. A dedicated
headset microphone was also added. Figure 8.17 shows the prototype gloves system
diagram (left) and being worn by musician (right).

The addition of the x-IMUs enabled numerous more dimensions of measurement on

top of the gesture recognition and flexion offered by the 5DT glove alone; including:
orientation, positional displacement and inertial peak detection. Furthermore,
these separate sources control mechanisms could be combined to begin to resemble

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Figure 8.17: Prototype gloves system diagram (left) and being worn by musician
(right)

‘real-world’ control interfaces. For example, by combing a segmented orientation

measurement with velocity peak detection, the wearer could be surrounded by an
invisible drum kit. A toolbox of of gestural control mechanisms based on these
prototype gloves was presented in a 2012 paper [11].

8.5.1.2 The performance gloves

The performance gloves represented a significant advance to the system. The 5DT
gloves were abandoned in order to achieve a wireless system with custom designed
components to meet technical requirements. The complete performance gloves
system is shown in Figure 8.18 with labelled key components.

The system was comprised of the following components.

• Gloves: Custom made gloves incorporating an x-IMU, eight flex sensors, an

RGB LED, vibration motor for haptic feedback and microphone. The x-IMU
acquired the analogue flex sensor measurements via its auxiliary port. The
RGB LED and vibration motor were driven by the auxiliary port of an x-IMU
mounted within the arm bands.

• Arm bands: Contained an x-IMU with a wired connection to the RGB LED
and vibration motor on the hand.

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Back hub Head piece

RGB LED Microphone x-IMU

Arm bands

Flex sensors Vibration motor

Gloves

Figure 8.18: Performance glove parts

• Harness and back hub: Mounts wireless radio units for the two
wrist-mounted microphones, headset microphone and audio earpiece. Also
mounts a x-IMU an custom designed hub that routes wired serial
communication between the five x-IMUs and the Bluetooth radio within
the head piece.

• Head piece: Mounts a Bluetooth radio that provides a single wireless link
between the five x-IMUs and the host software.

The performance gloves software utilised the same control mechanisms as outlines
for the prototype gloves but was extended to provide a direct interface to existing
professional music production software. The use of five x-IMUs distributed
between the arms and torsos provided an opportunity to implement more detailed
measurement of the body’s motion. A key use was to use the torso orientation

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measurement as a ’zero reference’ for each glove so that orientation controls would
be relative to the wearer and not the Earth.

The performance gloves have provided a unique tool for the composition and
performance of music. They have also have potential for a use in a range of
applications not yet fully explored. For example, Figure 8.19 shows the gloves
being used to create and manipulate visuals projected on to a screen.

Figure 8.19: Gloves being used to create and manipulate visuals projected on to
a screen

8.5.2 Gait analysis

Human motion tracking for biomedical monitoring applications has received

significant attention in the engineering and medical research communities in recent
years; see [41, 171, 172] for recent surveys. Zhoua [41], cited real time operation,
wireless properties, correctness of data, and portability as major deficiencies
that must be addressed to realize a clinically viable system. Body-worn sensors
such as goniometers [173], pressure tubes [174], gyroscopes and accelerometers
[175, 176, 177, 178, 179] have also been implemented in a range of activities
and clinical conditions [179, 180, 181, 177]. There are strong reasons to consider
inertial-based sensors in biomechanical analysis. They are small, light-weight,

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low-powered, inexpensive, and unobtrusive and so can be arranged so that they

do not interfere with the person’s movements. They can be considered as viable for
use outside a specialist laboratory and could be integrated into existing orthotics
and clothing.

This project explored the potential application of the x-IMU to human gait studies.
The autonomous data logging and motion triggered sleep/wake functionality make
the x-IMU an idea tool ambulatory monitoring with the potential to address
concerns outlined by Zhoua.

An algorithm was developed to obtain the translational position data from an

x-IMU mounted on the foot using methods similar to [182, 183, 184, 185, 186].
Figure 8.20 shows data obtained using a prototype unit for tracking of the right
foot of a test subject as they walked in a straight line.

foot y axis
foot z axis

0.2

0
3 2.8 2.6 2.4 2.2 2 1.8 1.6 1.4 1.2 0.2
1 0.8 0
0.6 0.4 0.2
0.2 0

Figure 8.20: Measured foot position during walking (3 steps) obtained using an
x-IMU

This provides numerous sources of measurement in addition to the IMU data,

including: linear accelerations, velocities and position of the foot relative to the
Earth, and a quantification of the stride characteristic such as cadence and stride
length. Furthermore, these individual source of measurement can be combined to
infer new information. For example, to correlate variations in patents gait when
walking on different gradients or at different times of the days.

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The system was used in collaboration with a team of students at Bristol

University to evaluate the effectiveness of an active ankle foot orthosis with passive
compliance. The x-IMU was used to measure the foot kinematics of a subject with
and without the orthosis and so provided an objective qualification of how much
the orthosis would impact natural gait. The project was presented in a 2011 paper
[12]

8.6 User applications

The x-IMU has been on public sale since 2010 and the x-BIMU and x-OSC since
2013. More than 500 of these platforms have been sold to customers world-wide;
including many commercial and academic research institutions. This section
presents 15 selected example user applications that have utilised the platforms
developed. The research projects chosen to appear in this section were selected
to demonstrate the diversity of user applications and impact of the platforms in
peer-reviewed publications.

8.6.1 Estimating upper limb kinematics

Multi IMUs mounted on an upper limb can be used to determine the limb joint
kinematics and provide a valuable tool for rehabilitation of stroke patients. Daniel
Galinski and Bruno Dehez used four x-IMUs to explore this in work conducted at
Center for Research in Mechatronics, Université catholique de Louvain, Belgium.
In a 2012 paper [187] they present an evaluation of initialisation procedures for
this application of IMUs. The evaluation was achieved through a comparison of
the measurement provided by four x-IMU fixed to an anthropomorphic upper limb
and its built in absolute encoders. Figure 8.21 shows the corresponding mechanical
model of upper limb, placement of x-IMUs and instrumented anthropomorphic

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upper limb. Their investigation validated the tested initialisation procedures and
suggested a potential advantage in one procedure for physically weak patients.

(a) (b) (c)

Figure 8.21: Mechanical model of upper limb (a), placement of x-IMUs (b) and
instrumented anthropomorphic upper limb used for evaluation (c). Image source:
[187]

8.6.2 Digits: Wrist-worn gloveless sensor

Digits is a wrist-worn sensor that recovers the full 3D pose of the user’s hand
without requiring any external sensing infrastructure or covering the hand itself.
It is a Microsoft Research project in collaboration with the Culture Lab, Newcastle
University; and FORTH, University of Crete. The hardware, shown in Figure is
comprised on an x-IMU, an IR laser line projector, diffused IR LEDs, an and
IR camera. The x-IMU was connected via USB and used to measure the wrist
orientation while the IR LEDs line projector and LEDs project on to the hand to
be seen by the camera. A wide range of signal processing techniques are used with
a kinematic model of the hand to determine a full 3D posture from the data. The
techniques and an evaluation of performance are discussed in detail in a 2012 paper
[188] published by the team. A range of applications have been demonstrated on
the Microsoft Research website10 with digits providing an eyes-free interaction for
10
http://research.microsoft.com/en-us/projects/digits/

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everyday devices such as a computer or phone.

Figure 8.22: Digits wrist-worn hardware components. Image source: [188]

8.6.3 MONSUN II: A small and inexpensive AUV for

underwater swarms

MONSUN II is a small Autonomous Underwater Vehicle (AUV) developed as a

platform for research into under water swarm. It is an ongoing project at the
Institute of Computer Engineering in the University of Lüebeck, Germany. The
robot, shown in Figure 8.23, is 60 cm long with a hull diameter of 10 cm and
weighs <3 kg. Figure 8.24 summarises the key internal electronics of MONSUN
II. The x-IMU is used to provide real-time orientation data to the microcontroller
via a serial connection. The microcontroller uses this data within a closed-loop
control algorithm to stabilise the craft [189].

Figure 8.23: MONSUN II AUV is 60 cm long, has a hull diameter of 10 cm and

weighs <3kg. Image source: [190]

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Figure 8.24: Key internal hardware complements including an x-IMU. Image

source: [190]

The MONSUN project started in 2009 and has been a source of numerous
publications covering the hardware and software design process [191] including
novel components such as an acoustic modem for underwater communication
[192]; simulation studies exploring practical applications such as environmental
monitoring [190]; and significant towards fault-tolerant and energy-efficient swarms
of underwater robots [193]. The most recent publication presented a preview of
the forthcoming third generation MONSUN in 2013 [194] which may incorporate
the x-BIMU.

8.6.4 SMART-E: An omnidirectional AUV

SMART-E is an omnidirectional AUV at the Institute of Computer Engineering,

University of Lüebeck, Germany. The unique design of SMART-E enable high
manoeuvrability and agility in 6 DOF to meet the challenges of surveying water
bodies and underwater installations. The hardware design is detailed in a
forthcoming paper [195]. The platform, shown in Figure 8.25, utilises a minimal
configuration of three actuated thrusters arranged at 120◦ to each other, an
actuated camera on the underside for use by a remote viewer, and a band of
RGB LEDs to give clear visual feedback under water. Figure 8.26 describes the

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on-board electronics. The x-IMU is used along side the Tritech Micron Scanning
Sonar for localisation and navigation of the craft. SMART-E won the innovation
award at the European SAUC-E competition 2012.

Figure 8.25: SMART-E AUV uses three actuated thrusters for omnidirectional
movement and a band of RGB LEDs to provide visual feedback under water.
Image source: [195]

Figure 8.26: Internal electronics and communication networks between sensors,

actuators and controllers. Image source: [195]

8.6.5 Tidal turbulence spectra from a compliant mooring

Detailed analysis of tidal turbulence is essential for the development of efficient

tidal energy sites. Tidal Turbulence Moorings (TTM) incorporating Acoustic
Doppler Velocimeters (ADVs) to provide an effective tool for acquiring tidal data
but but are subject to motion contamination in strong currents. In a recent paper
[196], a team from the Northwest National Marine Renewable Energy Center at
the University of Washington present a passively stabilised solution that utilises
x-IMUs mounted on the ADVs to achieve motion correction in post-processing.

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 Chapter 8. IMU platforms and applications

The team tested two versions of the TTM, each comprised of a steel float tethered
to an anchor with several instruments mounted along the tether as shown in Figure
8.27. The mounted instruments include: ADVs, current meters, x-IMUs, and an
Acoustic Wave And Current Profiler. The x-IMU logged data to its on-board SD
card to provide orientation data necessary for motion correction. The TTMs were
deployed at different locations through 2012 and 2013. Results showed although
raw data saw sufficient to evaluate turbulence intensities and reasonable turbulence
spectra, Motion correction would more detailed aspects of the turbulence.

8.6.6 Virtual reality based minimally invasive surgery

training system

Minimally Invasive Surgery (MIS) allow invasive surgery to be performed with

only a small surgical wound. This presents many advantages to the patient
including, less pain, reduced risk of haemorrhaging, and a faster recovery. However,
manipulation of the surgical tools demands excellent hand-eye coordination of
the surgeon which must be gained through training. A virtual reality based
MIS training system was presented in a recent paper [197], developed through a
collaboration between the Department of Electronic and Computer Engineering at
the National Taiwan University of Science and Technology, as the Departments of
Surgery at the Tri-Service General Hospital and National Defense Medical Center,
Taiwan. The paper presents a low-cost MIS simulation training system composed
of a surgical training box, surgical instrument with x-IMU, a stereo vision camera,
a DE2-115 FPGA board and a Virtual Reality (VR) game. The x-IMU is mounted
on the surgical instrument as shown in Figure 8.28 and provides orientation data
along side location data provided by the camera.

A virtual reality 3D game developed in C# was used to demonstrate the system.

In this game, the player must use the instrumented surgical tool to pick up balls

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Chapter 8. IMU platforms and applications

Figure 8.27: Dimensional drawings of Tidal Turbulence Moorings deployed in

Admiralty Inlet (left) and Chacao Channel (right). Image source: [196]

and place them on small cylinders within a virtual environment. Figure 8.29 shows
a screen shot of this software. The authors suggest that future development could
improve the system through the creation of virtual human organs.

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 Chapter 8. IMU platforms and applications

Figure 8.28: Instrumented surgical tool uses x-IMU to provide orientation data.
Image source: [196]

Figure 8.29: Virtual reality game controlled using the instrumented surgical tool.
Image source: [196]

8.6.7 Instrumented soccer equipment to collected ankle

joint kinematics in the field

In 2013, Jonathan Akins submitted a Ph.D Thesis [198] at the University of

Pittsburgh documenting the development of instrumented soccer equipment to
collect ankle joint kinematics in the field. The equipment was intended to establish
the reliability and validity of a kinematic assessment during athletic manoeuvres
and identify laboratory manoeuvres that elicited game-like demands from athletes.

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Chapter 8. IMU platforms and applications

The instrumented equipment included a soccer boot with an x-IMU embedded

in the sole and a shin guard with the x-IMU mounted internally. Data from
the two x-IMUs was synchronised by measuring a synchronisation pulse on the
auxiliary ports of both x-IMUs. Figure 8.30 shows the instrumented soccer booth
and shin guard connected to the synchronisation trigger. Akins concluded that
instrumented equipment collected reliable and valid ankle joint kinematics and
represents a viable technology for in-game data collection and injury prevention.

Figure 8.30: Instrumented equipment comprised an a soccer boot and shin guard;
each with an x-IMU mounted internally. The equipment is shown here attached
the the synchronisation trigger prior to a data acquisition session. Image source:
[198]

8.6.8 Evaluating horse lameness using standalone

‘low-cost’ IMUs

Standalone ‘low-cost’ IMUs, such as the x-IMU with its on-board data logging
to an SD card, could facilitate large-scale studies into the orthopaedic deficits
(lameness) of horses. In a 2013 paper [199] submitted to the journal of Computer
Methods in Biomechanics and Biomedical Engineering , Brighton et al. prevent
their investigation to the accuracy and limits of agreement of the x-IMU compared
with an established IMU-based gait analysis system, the Xsens MTx [48]. The

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study involved six horses, both IMUs were mounted on two anatomical landmarks,
the sacrum and sternum. The vertical velocity and position of each location were
estimated by combining the orientation and accelerometer measurements and the
symmetry of the horse gait evaluated. The x-IMU was found to provide sufficient
precision when assessing symmetry but was not able to provide reliable data for
an alternative analytical method that as tested. The authors suggest that further
investigations into specific calibration and processing algorithms could address
these shortcomings.

8.6.9 Telepresences through “Flobi”, an anthropomorphic

robot head

“Flobi” is the anthropomorphic robot head developed at the Bielefeld University

and serves as a long-term research platform to explore the emotional interaction
of humans with robots. Flobi, shown in Figure 8.31, was designed to appear
cartoon-like and babyfaced to avoid the ‘uncanny valley’ effect [200]. Emotions
are expressed through 18 actuators controlling the neck, lips, eyes, eye lids and
eye brows; and two LEDs for blushing cheeks. Studies including 259 participants
demonstrated that Flobi is able to effectively communicate a wide range of
emotions [201].

In a recent publication [202], the research team presented a system that allows
a human operator to control Flobi by wearing a combined motion capture and
display helmet. The system allows the the direct and live transfer of human
facial expressions, gaze and head movements while providing the operator with
a real-time display of the scene as perceived by the robot’s vision sensors.
The operators helmet, shown in Figure 8.31, incorporates a video projector,
face camera, x-IMU and earphone. The x-IMU is used to provide real-time
measurement of the head orientation to control Flobi’s neck actuators. The system

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also records human motion datasets for analysis or playback. Other applications
of Flobi have included its role as a fitness instructor [203] and an investigation into
the effects of loneliness on psychological anthropomorphism [204].

Figure 8.31: Combined motion capture and display helmet (left) and robot head
(right). Image source: [202]

8.6.10 Haptic communication between dance performers

Tychonas Michailidis, a researcher based at the Birmingham Conservatoire, has

been using the x-IMU and x-BIMU in collaborations with musical and dance
performers since 2011. In a recent paper [205], Michailidis at al demonstrate
an attempt to create a “corporeal” link between physical performers and to give
them the ability to become aware of each other’s actions in the space through
haptic feedback. The project utilised x-IMUs as wearable devices to provide
real-time measurements of the performers motion and gestures; as well as control a
vibration motor via a PWM signal generated by the x-IMU’s auxiliary port. Each
x-IMU communicated with a computer wirelessly using Bluetooth and interfaced to
software via an open-source11 x-IMU-OSC bridge, developed by one of the authors.
The investigation explores three scenarios: ‘co-exist’, where performers would
experience the haptic feedback of virtual objects as they moved around the space;
‘collaboration/play’, where the motion of one performer could result in feedback
11
http://birminghamconservatoire.github.io/IMU2OSC/

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 Chapter 8. IMU platforms and applications

for another; and ‘intimacy’, where physical contact between performers is felt by
others and also represented through visualisations to an audience. The authors
conclude that the paper aims to invite further discussion on if such technology can
enhance the experience for both performers and audience.

8.6.11 SLAM for an autonomous car

A team at the Department of Computer Science at the Hamburg University

adopted the x-BIMU for use in their latest generation robot platform for entry
in the annual Carolo-Cup competition. The competition tests the ability of a
robot platform to navigate an urban road environment while identifying obstacles,
crossings and missing road markings. The latest platform, shown in Figure 8.32,
is build around a 45 cm electrics car with several on-board processors and sensors;
including, IR proximity sensors, wheel encoders, an x-BIMU and a video camera.

Figure 8.32: Latest generation autonomous car shown fully assembled (left) and
with shell removed (right)

The x-BIMU is used with a serial breakout board to connect to the main processor
via serial. The angular rate data provided by the x-BIMU is used along side
images provided by the camera to estimate the location of the vehicle on a map
using a RatSLAM algorithm. Simultaneous Localisation And Mapping (SLAM)
encompasses all the processes required for autonomous navigation of a robot in a

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Chapter 8. IMU platforms and applications

new environment. RatSLAM is a SLAM algorithm that attempts to recreate the

neurological mechanisms observed in rats navigating new environments. Figure
8.32 is a screen shot that shows the image provided by the platforms camera along
side the map of the road being built up by the RatSLAM algorithm. The new
platform competed at the 2012 and 2013 competition with limited success but the
team are hopeful for the 2014 competition having now refined software components
and increasing the car speed.

Figure 8.33: Screenshot showing view from camera mounted on car and the plotted
location of the car as estimated by the RatSLAM algorithm

8.6.12 PERSIA: An autonomous humanoid robot to

compete in RoboCup

RoboCup is and international humanoid robot football competition. The

development of competing robots presents challenges in many disciplines including
mechanical design, bipedal gait and balance, robot autonomous navigation,
cooperative robotics, computer vision and artificial intelligence. The PERSIA
team from the Islamic Azad University in Iran formed in 2007 and have ranked first
place in national competitions on several occasions. In 2012 the team adopted the
x-IMU for use on all their competing robots. The hardware and software design of

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the PERSIA platform is described in a paper currently under review for the Elsiver
journal of Mechatronics [206]. The robot, shown in Figure 8.34, is actuated by 20
servo motors and senses the environment though a camera and the x-IMU. The
main processor implements an ANN to maintain balance and gait control. An
x-IMU installed on the hip is used to provide feedback to facilitate the training of
the ANN. Other software mechanisms for path planing and decision making are
also described in the paper. In the recent RobotCup IranOpen 2013 competition,
the team were awarded first place for the ‘adult size’ robot (150 cm) category.

Figure 8.34: ‘Adult size’ robot (150 cm) robot walking towards football
(Photograph courtesy of Hamidreza Kasaei)

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8.6.13 Low-cost structural usage monitoring of small-fleet

military helicopters

Monitoring the in-service usage of military aircraft and comparing this usage with
design assumptions is essential for assuring structural integrity throughout the life
of a fleet. Traditional approach to these validation programmes can be expensive
and so there is a demand for alternative low-cost solutions. A team at the Defence
Science and Technology Laboratory, a part of the UK Ministry of Defence, has
been evaluating the potential of the x-IMU to provide a cost-effective solution for
monitoring the usage of historic and small-fleet rotary-wing aircraft. The x-IMU
is used as a self-contained data logger to record gyroscope and accelerometer data
to the on-aboard SD card. The sleep timer and motion trigger wake up allow the
x-IMU to remain in situ for extended periods of time for minimal maintenance.
The x-IMU has been used to capture flight data from the Army Historic Aircraft
Flight Scout AH Mk 1 aircraft shown in Figure 8.35. Currently, over 20 flying
hours of data have been captured and is being analysed and compared with the
design usage assumptions for the aircraft.

Figure 8.35: The x-IMU is being used to collect data from the Army Historic
Aircraft Flight Scout AH Mk 1 to monitor structural usage

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8.6.14 Danceroom spectroscopy

Danceroom spectroscopy is an interactive installation and dance performance

project that allows people to be become part of real-time quantum partial
simulations projected a large visuals and sonified through generative audio
software. The project is collaboration between artists and scientists lead by
theoretical chemist David Glowacki. The project is described as the chapter of
the book Molecular Aesthetics [207]. The system uses multiple Multiple Microsoft
Kinect to generate a real-time 3D model of people within a space and translate
them to energy fields within a molecular simulation incorporating hundreds of
particles. The simulation implements both quantum and classical mechanics but
allows manipulation of the laws to design the aesthetics of the visualisations and
audio generated by the particle interactions.

A recent development is the project is the use of an x-BIMU worn on a dancer

wrist. The x-BIMU uses the 802.15.4 wireless modules to provide the dancer
low-latency control of selected simulation parameters through their movements.
Figure 8.36 shows a dancer performing with the x-BIMU mounted on her wrist
as she manipulates the visualisations behind her. The project has won multiple
awards and demonstrates the potential of novel approaches for accelerated particle
simulations on GPU architectures [208].

8.6.15 Changibles: Shape changing constructive assembly

Changibles are interactive, shape-changing assembly blocks that connect together

to create animated structures. The system was developed by a team at the
University of Bristol and presented in a forthcoming paper [209]. A key
competent of the project was the development of the algorithms that determine
the construction of a shape. Figure 8.37 shows how the analysis software and
11
http://paulblakemore.co.uk/

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Chapter 8. IMU platforms and applications

Figure 8.36: An x-BIMU mounted on a dancer’s wrist allows her to control

the real-time visualisations of the molecular simulation. Photographer: Paul
Blakemore

resultant physical construction.

Figure 8.37: Changibles analysis software and resultant physical construction.

Image source: [209]

Prototype hardware was developed using x-OSC, which provided the team with
a versatile wireless interface board with the potential to explore a wide range of

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sensor and actuator concepts. The current block incorporate two servo motors
to actuate the one face of the block about two degrees of freedom. Each block
connects to a router to achieve a single Wi-Fi connection to the computer running
the software... x-OSC/Wi-Fi offers the potential for large number of nodes.
The prototype block design, shown in Figure 8.38 incorporates two servo motors
to acute the one face of the block about two degrees of freedom. The paper
demonstrates these six blocks being assembled together to form pulsing heart.

Figure 8.38: An individual block containing x-OSC for Wi-Fi control and 2 servos
with scissor lift mechanism. Image source: [209]

8.7 Future work

Future work will see the IMU platforms continue to evolve based on user feedback
and to take advantage of increasingly high-performance MEMS technology.
Specific tasks have already be outlined for 2014. The modular communication
of the x-BIMU as proved to be a versatile design paradigm but the device is
limited by the processor; A new hardware revision will utilise the latest 32-bit
processors to achieve equivalent low-power performance with superior same rates
and signal processing capabilities. Planned x-OSC firmware developments will
take full advantage of Wi-Fi bandwidth to enable sample rates in the order kHz
with the potential to facilitate new applications.

The gloves project presented in Section 8.5.1 is an ongoing collaboration and will
see the development of next generation hardware and software in 2014 using the
x-OSC.

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Chapter 9

Conclusions

This project set out to demonstrate how new applications can be realised using
modern low-cost MEMS gyroscopes, accelerometers and magnetometers. This was
achieved through the development of publicly available software and hardware
platforms that have facilitated a wide range of commercial and academic research
projects exploring a diverse range of applications. In achieving this, the project
has met the following objectives:

• Development of a AHRS algorithm applicable to modern low-cost MEMS

devices

• Calibration of gyroscopes, accelerometers and magnetometers for use in

AHRS applications

• Design and manufacture of IMU platforms that combine the calibration and
AHRS algorithm solutions

• Demonstration of a broad range of applications using the IMU platforms

through collaborations

In achieving its objectives, this project has made several contributions. Chapters
3, 4 and 7 present sensor fusion algorithms applicable to conventional IMUs

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Chapter 9. Conclusions

and non-gyro IMUs. The conventional AHRS algorithms address the specific
requirements of modern low-cost MEMS sensors and have been proven in the IMU
platforms and applications presented in Chapter 8. The impact of these algorithms
is also demonstrated by the numerous citations and code downloads. The non-gyro
IMU was not demonstrated in application but represents an interesting alternative.
The novel work presented in Chapter 4 demonstrated the potential practical
benefits of such kinematically redundant sensor arrays.

Another contribution of this project is the development of a comprehensive

calibration solution for gyroscopes, accelerometers and magnetometers. The
calibration solutions build on the characterisation studies to address the specific
needs of modern low-cost MEMS sensors. These characterisation studies
revealed significant characteristics of the sensors that are not documented by the
manufacture.

The AHRS algorithm and sensor calibration works were brought together in
the development of the Chapter IMU platforms presented in Chapter 8. Each
of these platforms each addressed a specific design specification and together
facilitated a wide range of new applications; demonstrated by the numerous
scientific publications that resulted from collaborative projects and user projects.

This project achieved its original objectives but MEMS technology is continuing
to advance and there are undoubtedly many more new and exciting applications
that can be facilitated by the work conducted within this project. Future work
will see the ongoing development of the IMU platforms to incorporate improved
MEMS sensors and new features to meet the needs of an expanding user base.
Work will also continue on the AHRS algorithm and sensor calibration solutions
to meet the specific needs of improved MEMS sensors and new applications.

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230
Appendix A

Non-gyro IMU planar

simplification

A.1 Planar simplification

If motion is constrained to a planar coordinate system so that translational motion

occurs only in the x and y dimensions and rotational motion only around the z
axis, each sensor is reduced to a dual-axis arrangement and Equations 4.3, 4.8 and
4.9 simplify to Equations A.1, A.2 and A.3 respectively. For planar systems the
rotational velocity is denoted by ω. The planar simplification gives a useful insight
into the arrangement of redundant sensor constellations (see A.2).

 T
i
α = αx αy (A.1)

 
 1 0 -dx -dy 
Gi =   (A.2)
0 1 -dy dx

 T
s= ẍ ÿ ω 2
ω̇ (A.3)

231
Appendix A. Non-gyro IMU planar simplification

A.1.1 Angular velocity estimation

For the simplified planar system, the objective function is simply the difference
between the squared estimated angular velocity term and the squared angular
velocity term yielded by the state vector s. The error originally defined in Equation
4.24 in the estimated angular velocity may be redefined as the scalar quantity of
Equation A.4.
√
ẽ = sign(ω̃(ω̃ 2 − ω 2 )) |ω̃| − ω2 (A.4)

A.2 Optimal array configurations

For ideal accelerometers, all valid array configurations perform equally. In practice,
errors due to noise and signal saturation in accelerometer measurements will mean
that some array configurations perform better than others. We wish to evaluate
the effect of individual array characteristics to understand what determines an
optimal array. An accelerometer array may be characterised by the following:

1. Position of the kinematic origin relative to a given sensor constellation;

2. Orientation of the kinematic origin relative to a given sensor constellation;

3. Sensor constellation, i.e. a set of specific sensor locations; and

4. Volume occupied by a given the sensor constellation.

Qin et al. [87] use inspection of matrix singular values to determine the suitability
of a sensor configuration and determined that an optimal design is represented by
an isotropic matrix; that is, a matrix with a condition number of one where all
singular values are identical and nonzero. This is achieved by a sensor constellation
represented by the vertices of a Platonic solid. The analysis of other authors exist
for planar [210, 211] and 3D [212] [213] accelerometer arrays.

232
 Appendix A. Non-gyro IMU planar simplification

A.2.1 Array volume and kinematic origin

To analyse the effect of the kinematic origin location and array volume we will
consider a simplified planar array of 2 dual-axis accelerometers represented by the
measurements α10 to α40 , each incorporating an error (δ1 to δ4 ). The kinematic origin
of the array is equidistant from each sensor and may be displaced by a distance p
along the origin y axis (remaining equidistant from each sensor). The volume of
the array is defined by the distance of separation, l. The system is described by
the schematic shown in Figure A.1. Equation A.5 describes the forward kinematic
solution.

ÿ

ω
α4 α2
ẍ
p
α3 α1

1 1
2l 2l

Figure A.1: Schematic describing the planar sensor array characterised by

parameters p (origin displacement) and l (sensor separation)

α0 = Hs + δ (A.5)

      
0
α1  1 0 − 2l p   ẍ  δ1 
      
α0  0 1 p l 
ÿ   
 2  2   δ2 
 =    + (A.6)
α0  1 l
p     
 3  0 2  ω 2  δ3 
      
α40 0 1 p −2l
ω̇ δ4

The inverse kinematic solution may be arranged to provide the measured kinematic
states of the body origin in terms of the true kinematic states and measurement
0
errors as shown by Equations A.7 and A.8. In these equations, ẍ0 , ÿ 0 , ω 2 and ω̇ 0

233
Appendix A. Non-gyro IMU planar simplification

represent the measured kinematic states of the body origin.

s0 = H −1 α0

= H −1 (Hs + δ) (A.7)

= s + H −1 δ

      
0
 ẍ   ẍ  
1
2
− pl 1
2
p
l  δ1 
     p  
 ÿ 0   ÿ   1
− pl 1  
δ
     l 2 2   2
 = +   (A.8)
ω 2 0  ω 2  − 1 0 1
0   
     l l  δ3 
      
ω̇ 0 ω̇ 0 + 1l 0 − 1l δ4

For this simplified equidistant configuration it can be seen from Equation A.8 how
the distance from the origin (p) and sensor separation (l) may be manipulated to
minimise the error in the measured kinematics states. The error in the measured
0
angular states (ω 2 and ω̇ 0 ) is independent of the origin displacement and inversely
proportional to the sensor separation. Error components in the measured linear
states (ẍ0 , ÿ 0 ) are proportional to the origin displacement and inversely proportional
to the sensor separation. This method extends to assessing redundant planar
sensor arrays, and leads to a weighted average (linear accelerations) or a weighted
sum of difference (angular acceleration). Thus it can be demonstrated that an
optimal array would maximise the sensor separation. The kinematics origin should
be located at the point where the expected mean acceleration is minimised. Where
more specific information is not available this could be considered as the centre of
mass of the limb. Other authors have arbitrarily chosen the centre of volume as
the kinematic origin of a sensor array [86, 87], so assuming a homogeneous body
the methodology we describe gives a justification for their decision.

Analysis of the equidistant planar structure supports Qin’s assertion [87] that a
reasonable sensor array would be to distribute the sensors on the vertices of a
Platonic solid. This only applies if the body is assumed to be rotating around the

234
 Appendix A. Non-gyro IMU planar simplification

centre of the sensor constellation. Where more sensors than vertices are used in
the IMU, a consideration of the singular values of the reconstruction matrix is an
appropriate method to optimise sensor placement.

235
Appendix A. Non-gyro IMU planar simplification

236
Appendix B

Crystal oscillator thermal

response

B.1 Introduction

An AHRS determines orientation through numerical integration of gyroscope

measurements. The accuracy of the orientation is therefore dependent on both
the calibrated performance of the gyroscope and the accuracy of the system
clock. The system clock accuracy is also reverent to systems intended to achieve
synchronisation of multiple devices. This appendix describes a characterisation of
the crystal oscillator to determine its accuracy over an operating temperature range
and considers how this error may be compensated for using existing hardware.

These investigations focusses specially on the 32.678 kHz crystal used by the
x-IMU. The is the clock source for both the RTC and sample clock. Crystal
oscillators typically have an error of around ±20 ppm though this will increase with
changes in temperature. Temperature Compensated Crystal Oscillator (TCXO)
and Oven-Controlled Crystal Oscillator (OCXO) provide far greater accuracy over
a temperature range but are large, expensive and require allot of power. IMU

237
Appendix B. Crystal oscillator thermal response

platforms typically already incorporate a thermometer to measure the temperature

of the on-board electronics and so have the potential to achieve thermal calibration
using only the existing hardware.

B.2 Crystal error model

Equation B.1 [214] describes the crystal error model where e is the period error in
ppm, e0 is the error at the turnover frequency, c is the temperature coefficient, T
is the temperature of the crystal and T0 is the turnover temperature. Table B.1
summarises the values of each of these parameters are stated in the datasheet of
the crystal used by the x-IMU [215].

e = e0 + c(T − T0 )2 (B.1)

Parameter Typical value Tolerance

e0 0 ppm ±20 ppm
c 0.035 ppm/◦C 2 ±10%
T0 25◦C ±5◦C

Table B.1: Crystal error model parameters as specified in datasheet [215]

A typical error of ±20 ppm at 25◦C can only be achieved if the design of the
crystal oscillator circuit accounts for electrical characteristics of the PCB layout.
For example, a poorly matched load capacitance can result in an error of almost
400 ppm [216]. Component values were initially selected according to manufacturer
application notes [217] but were found to provide errors of the order of 150 ppm
at 25◦C Instead, component values were selected as those found to provide a error
centred around 0 ppm 25◦C for the eight tested devices.

238
 Appendix B. Crystal oscillator thermal response

B.3 Crystal characterisation

The thermal response of eight crystals were characterised using the thermal chapter
described in Chapter 5. Each device was exposed to a monotonic ascent from -20◦C
to 70◦C over 3 hours. A TTi TF930 frequency counter was used to measure the
crystal frequency with an accuracy of ±0.2 ppm [132]. Software was written to log
the device temperature along side the crystal error. Figure B.1 shows measured
period error for each device over the temperature range.

A B C D E F G H
100
90

80
Crystal period error (ppm)

70
60

50

40

30

20

10

−10
−20 −15 −10 −5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70
Temperature (◦ C)

Figure B.1: Thermal repose of eight crystal oscillators

A second-order polynomial was fit was performed for each device to determine the
model parameters. The results are summarised in Table B.2.

Device e0 (ppm) c (ppm/◦C 2 ) T0 (◦C)

A -0.6 0.0360 27.07
B 17.8 0.0358 23.87
C 9.6 0.0357 26.20
D 10.0 0.0360 25.18
E -3.7 0.0361 27.31
F 8.3 0.0357 29.11
G 3.7 0.0353 25.82
H 10.1 0.0359 24.75

Table B.2: Crystal error model parameters for eight devices over temperature
range

239
Appendix B. Crystal oscillator thermal response

B.4 Crystal calibration

The thermal response of the crystal may be compensated for in calibration by

determining the error for a given temperature and adjusting time calculations
accordantly. The following sections present three different calibration solutions;
each solution is able to offer a different accuracy over the operating temperature
with different piratical implications.

B.4.1 Crystal calibration assuming typical parameters

The simplest calibration solution is to use the generic error model defined by
Equation B.1 and the typical parameters specified in Table B.1. The resultant
error of the eight devices is shown as Figure B.2.

A B C D E F G H
30

25
Crystal period error (ppm)

20

15

10

−5

−10
−20 −15 −10 −5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70
Temperature (◦ C)

Figure B.2: Crystal oscillator error using the generic calibration model and typical
parameters specified in the datasheet

This simple calibration solution is able to reduce the crystal error from 90 ppm to
25 ppm. This solution can be implemented in software and does not require any
specific values t be evaluated through testing.

240
 Appendix B. Crystal oscillator thermal response

B.4.2 Crystal calibration based on error model

A significant reduction in error can be achieved if the generic error model is used
in conjunction with the specific specific parameters evaluated for each device and
summarised in Table B.2. The resultant error of the eight devices is shown as
Figure B.3. This calibration solution reduces the errors to within ppm.

A B C D E F G H
1.5

1
Crystal period error (ppm)

0.5

−0.5

−1

−1.5

−2

−2.5
−20 −15 −10 −5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70
Temperature (◦ C)

Figure B.3: Crystal oscillator error using the generic calibration model and specific
parameters evaluated empirically

B.4.3 Crystal calibration through third order model

The curve of the error shown in Figure B.3 indicates that the second-order generic
error model is too low an order. This may due to an inaccuracy in the model form
or may be due to a non-linearity in the thermometer. The source of the error is
irreverent provide that a calibration solution accounts for the characteristics of
both the thermometer and crystal. Figure B.4 shows the crystal oscillator error
using a third-order calibration model.

This third-order calibration model is able to achieve an error of 0.2 ppm. Initial
large error is thought to be because of an initial temperature difference between
the thermometer device and the crystal oscillator.

241
Appendix B. Crystal oscillator thermal response

A B C D E F G H
0.5

0.4

0.3
Crystal period error (ppm)

0.2

0.1

−0.1

−0.2

−0.3

−0.4

−0.5
−20 −15 −10 −5 0 5 10 15 20 25 30 35 40 45 50 55 60 65 70
Temperature (◦ C)

Figure B.4: Crystal oscillator error using a third-order calibration model

B.5 Conclusion

The crystal characterisation indicates that without calibration crystal variation

over temperature is equivalent to a ±0.009% gyroscope sensitivity error over -20◦C
to 70◦C. This error would not be significant relative to other gyroscope error sources
though it can be reduced then this will be of benefit. Reducing the crystal error
may be of greater interest to achieving synchronisation between multiple devices.
Three crystal calibration solutions were proposed; the first reduced the error to
25 ppm and can be implemented without any imposing any additional tasks on
device production. Both the second and third calibration solution require a precise
dataset to evaluate calibration parameters but yield an accuracy of up to 0.2 ppm.

242
Appendix C

Effect of geographic location on

AHRS heading accuracy

The Earth’s magnetic field is not constant around the world and it is of interest
to consider how this will affect the accuracy of a heading measurement obtained
from a magnetometer-based AHRS. The field varies in declination (deviation from
geodetic north), inclination (deviation from the horizontal plane) and in intensity.
Given that a magnetometer-based AHRS is intended to measure heading relative
to magnetic north and not geodetic north, declination variations are irreverent.
Maus et al. provide a comprehensive explanation and quantification of the Earth’s
magnetic field in The US/UK World Magnetic Model for 2010-2015 [30]. It is
reported that the intensity can be expected to between 0.22 Gauss and 0.67 Gauss
around the world. The varying inclination is presented as the Mercator projection
overlay shown as Figure C.1. The contours indicate that non-polar regions can
expect a maximum inclination of ∼80◦ , with the 0◦ inclination contour remaining
with ±15◦ latitude of the equator. The inclination in the UK is indicted as being
between 66◦ and 70◦ .

A mathematical model can be derived to investigate the effect of varying inclination

243
Appendix C. Effect of geographic location on AHRS heading accuracy

180° 135°W 90°W 45°W 0° 45°E 90°E 135°E 180°

70°N 70°N

80 80
60°N 60°N

60
45°N 45°N
60 60

30°N 30°N
40 40 40

20 20
15°N 15°N
20
0
0
0 -20
0° -20 0°
-20 -40
-40
15°S 15°S
-40 -60

-60

30°S 30°S
-60

45°S 45°S
-80
-60

60°S 60°S

kj
-80

-60
70°S 70°S
180° 135°W 90°W 45°W 0° 45°E 90°E 135°E 180°

Figure C.1: Contour interval: 2 degrees, red contours positive (down); blue
negative (up); green zero line. Image source: [30]

and field intensity on the measurement of heading obtained from a magnetometer.

For this we will assume a two-axis magnetometer aligned with the Earth’s surface
so that the axis measurements, x and y, are equal to the horizontal components of
the Earth’s magnetic field. The magnetometer measurement, m, can be defined
as Equation C.1 where b is the magnetic field intensity, γ is the inclination and θ
is the heading of the device relative to magnetic north.
   
x  sin θ 
m =   = b cos γ   (C.1)
y cos θ

The heading as determined by the magnetometer measurements would be

calculated as Equation C.2.

 
x y
θ = atan2 , (C.2)
||m|| ||m||

For an ideal sensor, the measurement of heading would be insensitive to field

intensity and inclination provided b > 0 and γ < 90. In practice, the magnetometer

244
 Appendix C. Effect of geographic location on AHRS heading accuracy

will be subject to measurement errors from a variety of sources. For the purposes of
this investigation we will consider an error of e in the magnetometer magnetometer
x axis to yield the measurement m̃ as described by Equation C.3.
   
x̃ e
m̃ =   = m +   (C.3)
ỹ 0

The error in the corresponding measurement of heading may be calculated as

Equation C.4.
 
x̃ ỹ
θerror = atan2 , −θ (C.4)
||m̃|| ||m̃||

Using Equations C.1, C.3 and C.4, it is possible to determine θerror for selected
values of b and γ. The relationship of Equation C.1 suggests that θerror will have
a greater sensitivity to variations in γ and so we will first consider a numerical
example for only variations in γ. Fixed values of b = 0.5 Gauss and e = 0.0044
Gauss were chosen as this ratio was found to provided a maximum heading error
±0.5◦ for an inclination angle of 0◦ ; this is heading accuracy specified by high-end
commercial IMUs such as the MicroStrain 3DM-GX3-25 [108]. Figure C.2 shows
the corresponding heading error achieved for an inclination of 0◦ (on the equator),
30◦ , 70◦ (in the UK) and 80◦ .

The above results suggests a significant disparity in the accuracy of heading

measurements dependent on the geographic location. However, this numerical
example assumes a fixed magnetic field intensity. The actual intensity varies
around the world and is typically greater away from the equator which can be
expected to result in lower heading errors. To verify the accuracy of this simplified
model, θerror was calculated for specific locations corresponding to the above values
of inclination. The locations were selected to demonstrate the least disparity in
the values of θerror . The results are summarised in Table C.1.

245
Appendix C. Effect of geographic location on AHRS heading accuracy

γ = 0◦ γ = 30◦ γ = 70◦ γ = 80◦

3

2
θerror (degrees)

−1

−2

−3
0 60 120 180 240 300 360
θ (degrees)

Figure C.2: Heading error achieved for an inclination of 0◦ (on the equator), 30◦ ,
70◦ (in the UK) and 80◦ and fixed values of b = 0.5 Gauss and e = 0.0044 Gauss

Location b γ θerror
Phuket, Thailand 0.41 Gauss 0◦ ±0.6◦
Mina, Peru 0.26 Gauss 0◦ ±1.0◦
Edinburgh, UK 0.5 Gauss 70◦ ±1.5◦
Trua, Russia 0.6 Gauss 80◦ ±2.5◦

Table C.1: Extreme values of b and γ selected from [30] to demonstrate the least
disparity in θerror for differing geographic locations

The results in Table C.1 confirm that the variation in heading performance for
differing geographic locations may be less than that assumed by an inclination-only
model but that the specific geographic location will still have a significant effect on
the heading performance. A heading error performance observed at the equator
may correspond to an error almost times greater in the UK and equally a heading
accuracy observed in the UK may be expected to improve by a factor of three
when operating on the equator.

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