Chapter 7

Sensors

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 7, Slide 1
 Architecture
Destination, Map
obstacles

Path planner
Waypoints
Status

Path manager
Path Definition
Tracking error

Airspeed, Path following

Altitude,
Heading, Position error
commands
Autopilot
Servo commands
State estimator
Wind
Unmanned Vehicle
On-board sensors

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 7, Slide 2
 Sensors for MAVs
• The following types of sensors are commonly used for
guidance and control of MAVs
– accelerometers
– rate gyros
– pressure sensors
– magnetometers (digital compasses)
– GPS

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 7, Slide 3
 MEMS Accelerometer
Newtons 2nd law gives
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mẍ = k(y x)

Note that the acceleration of the proof mass pro-

portional to deflection of the suspension
Taking the Laplace transform gives
1
X(s) = m 2 Y (s)
ks +1
1
=) s2 X(s) = m 2 s2 Y (s)
ks +1
1
=) AX (s) = m 2 AY (s)
k s + 1

Accelerometers also have bias and zero mean Gaus-

sian noise. Therefore, the sensor model is

0
yaccel = kaccel a + accel + ⌘accel .

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 7, Slide 4
 MEMS Accelerometer

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 7, Slide 5
 Acceleration Measurement
Tricky concept: Measured acceleration is the total acceleration of the accelerometer casing minus the
acceleration of gravity

1
a= (Ftotal Fgravity )
m

Example: Set the accelerometer on a table top.

What does it measure?

Accelerometers measure components of linear, coriolis, and externally applied acceleration. They do not
measure gravity, since both the proof mass and the casing are acted on by gravity in exactly the same way
Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 7, Slide 6
 Acceleration Measurement
Said another way, accelerometers measure specific force, which is defined as the sum of the non-
gravitational forces divided by the mass

1 ⇣X ⌘
ameasured = Fnon-gravitational
m
1 ⇣X ⌘
= F Fgravitational
m

Example: Set the accelerometer on a table top.

What does it measure?

Hint: (Draw FBD of accel housing)

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 7, Slide 7
 Acceleration on Fixed-Wing Aircraft
Flift
Fthrust
Fdrag

Fgravity

1
ameasured = (Ftotal Fgravity )
m
1
= ((Flift + Fdrag + Fthrust + Fgravity ) Fgravity )
m
1
= (Flift + Fdrag + Fthrust )
m

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 7, Slide 8
 Acceleration on Fixed-Wing Aircraft
Recall from Chapter 3, that
✓ ◆
dv
m + ! b/i ⇥ v = Ftotal .
dtb

Using the expression

1
ameasured = (Ftotal Fgravity ) ,
m
the output of the accelerometer can be expressed as
dv 1
ameasured = + ! b/i ⇥ v Fgravity .
dtb m
Expressing this relationship in the body frame gives

ax = u̇ + qw rv + g sin ✓
ay = v̇ + ru pw g cos ✓ sin
az = ẇ + pv qu g cos ✓ cos

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 7, Slide 9
 Accelerometer Models
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yaccel,x = u̇ + qw rv + g sin ✓ + ⌘accel,x

yaccel,y = v̇ + ru pw g cos ✓ sin + ⌘accel,y
yaccel,z = ẇ + pv qu g cos ✓ cos + ⌘accel,z
or

⇢Va2 S c̄q
yaccel,x = CX (↵) + CXq (↵) + CX e (↵) e + Tp ( t , Va ) + ⌘accel,x
2m 2Va
⇢Va2 S h bp br i
yaccel,y = C Y0 + C Y + C Yp + C Yr + CY a a + CY r r + ⌘accel,y
2m 2Va 2Va
2

⇢V S c̄q
yaccel,z = a CZ (↵) + CZq (↵) + CZ e (↵) e + ⌘accel,z .
2m 2Va

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 7, Slide 10
 Accelerometer Models

or

fx
yaccel,x = + g sin ✓ + ⌘accel,x
m
fy
yaccel,y = g cos ✓ sin + ⌘accel,y
m
fz
yaccel,z = g cos ✓ cos + ⌘accel,z
m

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 7, Slide 11
 MEMS Rate Gyro
Point translating on a rotating rigid
body experiences a coriolis acceleration:

aC = 2⌦ ⇥ v

MEMS rate gyro – resonating proof

mass:
|v| = A!n sin(!n t)

Sensor measures deflection of proof mass due to coriolis acceleration

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Vgyro = kC |aC |
= 2kC |⌦ ⇥ v| = 2kC ⌦|v|
= 2kC ⌦|A!n sin(!n t)| = 2kC A!n ⌦
= kgyro ⌦, where kgyro = 2kc A!n

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 7, Slide 12
 MEMS Rate Gyro

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 7, Slide 13
 Rate Gyro Model
The manufacturing process implies that rate gyros will have a
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drift term, as well as zero mean Gaussian noise:

0
ygyro = kgyro ⌦ + gyro + ⌘ gyro ,

where ⌥gyro is in units of voltage. Calibrating to units of rad/s

gives
ygyro,x = p + gyro,x + ⌘gyro,x
ygyro,y = q + gyro,y + ⌘gyro,y
ygyro,z = r + gyro,z + ⌘gyro,z

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 7, Slide 14
 Pressure Measurement

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 7, Slide 15
 Pressure Measurement

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 7, Slide 16
 Altitude Measurement
The basic equation of hydrostatics is
<latexit sha1_base64="NUAbKsbtyYromJlMOchPPzHQn9g=">AAAOJXicdZdbb9s2FMfdZpdOW7Z2e9wLsaRD2yWBnabdUKBDuw69IUG9NEkDRI5BUUcSG+oSkorrKNrX2afZ27BhT/sqO5Qdh6YSPdgS/z8ekofniEdBIbjS3e6/164vfPTxJ5/e+Mz7/IvFL7+6eevrPZWXksEuy0Uu9wOqQPAMdjXXAvYLCTQNBLwLjp4Z/d0JSMXzbEePCxikNM54xBnV2DS8df2fnQQIWuCMwHHZtJI8Isk4lLnS+MwU4Yp4/oHXH66TVdIf9shj4sskJzG5c9q0nQ57dz1/4O0qnsVEo8VY5mUWEqqIhAgkZAxWCG1aVJkaiuUZ2s80oVySEDLF9ZjE/ASU5wcQ86yigsfZvdrrr/aHla/hg64mZuvaI3h9/3j1fBrJauIid4nvO9iMefpis659z4csnI3yC4h8RHq9lW63S9IVwmhGSmV8I/MUtEQHRblMS0EfNc5AJ/SHXeILiPQB8SNJWbUz7Nbm54dNVC5Ge4uj1egyHid6QA4nbEy26mqbIFmvGN+NEnST52c5z9Ab2ltG68uPiPFRSGVIcFvRdRII1UQBJQJOQJg1Lu/MgxrSAiTVl7KbDYsqmF220RAYBg4u2GAxQrGkJ1w3EUHFxXYZeRvlMuMmsFCKcZdnMvqo2VGDbdnTSnNBJUlx/83IVKe5KnDJ6FbEV5A3XUYJZwnR9Agw6jKdE8oYbqcmLKFZDNg2C5UR1wmhAmO+DKEJLWs1a97w5lJ3rdtcpH3Tm94sdaZXf3jLe+CHOStTdD4TOM2DXrfQg4pKzAEBtedjNBSUHdEYDvA2oymoQdWkYE1uY0vYLD7KzXRNq92joqlS4zRAMqU6Ua5mGi/TDkod/TSoeFaUGpNoMlBUCoK+MflMQi6BaTHGG8okx7kaX2GAadycuVHQOwkc1553G6MMRixPU/Ra5Rch1Afrg2oSln5hFkzFUq++uF/HjJvroxHFXr3BNMR9SImW9Z2l3l2HZPlJbUHm0SFOqLQJ8+gQSocwZ2XS4FAhvmiOa0PjgiUIM0uOUYN7Vz124QxTeTJ/k7/+2VJvkp7+mQPSQF1wM8ylJI5R+Wb3gqDartvuMrPPC6c5oSKqp35Hd7e8bJKlnphVUZW68lNGxfmweFs9bY37zCGetYgXDvGiRbx0iJct4rVDvG4Rmw6x2SK2HaLtw7cO8bZF7DrEbovYb4gZsO/qNIimchBV1FUDWw1cldkqc1WwVXDVyFYjV+W2yl31va2+d9UjWz1qBZittuKrsNXCVY9t9biVsraqXLW01bL1QrDVVpqPbHXkqh9s9YOrntrqqas+t9XnrvrKVl+1csBWX7vqlq1uueqOre646p6t7rVfjcn5i1HVQx9LL03dwLxA2BUInsS2lSLhro1izkYbUIWas6DaFtSchRYQgtCT/AtyEZqzMhfTVnfRWI/FbbRpddE3l6JvLkNP4ZLxTWP7oIklwORYqPCcPPQZl8xdL5bPzYo5lgnu/DMYQ3PqyVJA1eumaV3dT1sJqEe54ZozoMRKB6uKMs3wnKjO2Nl57/WHs95u9wSneZmB+3MGNjauMsBN5Czzw+m+4VP9+zIyeNL6hczzCI8wheUFLgHQBPKkb5ofkXpKYYE9B97Ddx0XAs34gcBjWuEnh4RlU2gYFg/sE/PR45eZOuLmuPwVsBaTsIXh/6Yp7nKJwYYVe101v1cQFKtOdNPk7yoGP1YQMb+en4yxOMqaYrcqFJRhvipNxYnlEh7uF05pPDHioU6qB6ajZwrNnltWtm/21td6D9c2fltferIxLTlvdL7tfNe50+l1fuw86bzs9Du7Hbbw80K4kC5ki38s/rn41+LfE/T6tWmfbzpz1+J//wNtOVGW</latexit>

P2 P1 = ⇢g(z2 z1 )

Using the ground as reference, and assuming constant air density gives

P Pground = ⇢g(h hground )

= ⇢ghAGL

Below 11,000 m, can use barometric formula:

 gM
T0 RL0
P = P0 ,
T0 + L0 hASL

where P0 : standard pressure at sea level

T0 : standard temperature at sea level
L0 : rate of temperature decrease
g: gravitational constant
R: universal gas constant for air
M : standard molar mass of atmospheric air,

which takes into account change in density with altitude and temperature.

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 7, Slide 17
 Altitude Measurement
We usually assume density is constant:

yabs pres = (Pground P) + abs pres + ⌘abs pres

= ⇢ghAGL + abs pres + ⌘abs pres

Is this valid?
4 4
x 10 x 10
10.2
10 constant density
ideal gas law
10.1
8
10
pressure (N/m )

pressure (N/m2)
6
2

9.9
4
9.8
2
9.7
0
9.6
2
9.5
0 2000 4000 6000 8000 10000 0 100 200 300 400 500
altitude (m) altitude (m)

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 7, Slide 18
 Airspeed Measurement
From Bernoulli’s equation:

⇢Va2 ⇢Va2
Pt = Ps + or = Pt Ps
2 2
Pitot-static pressure sensor measures dynamic pressure:

⇢Va2
ydi↵ pres = + di↵ pres + ⌘di↵ pres
2

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 7, Slide 19
 Magnetometer
magnetic
north
north
iv mi
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iv1 <latexit sha1_base64="OHcURmLnHg1Q7+NF2SNTPcQ19EY=">AAAB/XicbVBPS8MwHE3nv1n/VXf0EtwET6OdBz0OvHic4DZhKyPN0i0sTUvyq1DK8KN4EhTEqx/Ek9/GrOtBNx8EXt77PfLLCxLBNbjut1XZ2Nza3qnu2nv7B4dHzvFJT8epoqxLYxGrh4BoJrhkXeAg2EOiGIkCwfrB7Gbh9x+Z0jyW95AlzI/IRPKQUwJGGjk1LqnJFjfcGPIYSGPk1N2mWwCvE68kdVSiM3K+huOYphGTQAXReuC5Cfg5UcCpYHN7mGqWEDojEzYwVJKIaT8vlp/jc6OMcRgrcyTgQv2dyEmkdRYFZjIiMNWr3kL8zxukEF77OZdJCkzS5UNhKjDEeNEEHnPFKIjMEEIVN7tiOiWKUDB92bZpwVv98zrptZreZdO9a9XbbtlHFZ2iM3SBPHSF2ugWdVAXUZShZ/SK3qwn68V6tz6WoxWrzNTQH1ifP7O9lHA=</latexit>

inclination ◆

declination m
Let e1 = (1, 0, 0)> be a unit vector pointing north.
<latexit sha1_base64="LNB9m8/drbzSfEuocWbkLykqnGk=">AAANZXicdZdtb9s2EMdd76nTmq3d05u9GLGkQNK1hp1m7TCgQIsMbVq0aJcmaYAoMSjpZLGRRIWknDiK9mH2dvtC+wT7GjvSTkOTiRDY9P1/PN5RJx0TVTmTqt//91r3o48/+fSz658HX9xY+PKrm7e+3pG8FjFsxzznYjeiEnJWwrZiKofdSgAtohzeRYfrWn83BiEZL7fUpIL9go5KlrKYKjQNb3W/ewmKLIUFVVmUNtAOB+QRWR7cJX38WzkIFa+WSASEkrpkiowhVlyQirNSsXJESi5U1gsC42VzGSfdCxlX9C4JE8gVXTGTVQZEoFWvSXApwU5IKnihhUBHBIrFaKEFstzgmI9QjOZTa4+QrQzKINwLzkMt2gOGoW6aEC9ZmNg5BeF+wKRxbKehMqqmuUhc0MgJE6jpOHlqDBjeNDoGeXKXCJA8H0NyzjtxBvrCUPXGFDRnp0heJKhdkAKorMWFh4gnk1nuTDoZRjrDYXTQMJ1lS+zkezopvdTMYQGlQg9kxMZQkmhiu5q0wwbDaNHdnPefSRjxPJGTAr+aEBRtjdvhzcV+r28u4g8Gs8FiZ3a9Gd4KHoYJj2sdRJxTKfcG/UrtNxQ3J86hDcJaQkXjQ9zNPRyWmK7cb0wBt+Q2WhKS4h1JOSZhrPaMhhZSx4ikjl66mjZepu3VKv11v2FlVSso4+lCaZ2bIsOnYXa38wkOaCyYvklxRgWNFT4zc6tAUWVw1AbB7SAs4TjmRUHLpAmrBNq91f0mxDsY409qqmFx0F6MV9t2fo5CFGcNcJaCE4X7jk+CaJcXBysOGfNxa0H6p0OMqbAJ/dMhpEpgzsvU4FAJpJiepjFhAbmOktFyhPeueeTCurin8eeQqvBscRAKNspw5IA0khfcB8ylBK7RTOsyajZbf7t09Fj+8+aM5mk723fcbm+XC6zCdupW6nJ35Ccxzc+XxWHzxFt33SHWPeKZQzzziA2H2PCIFw7xwiNeOsRLj9h0CH8P3zrEW4/Ydohtj9g1xAdg19VplM5kfL1QV41sNXLV2FZjVwVbBVdNbTV1VWarzFXf2+p7Vz201UOvwGzVq6/KVitXPbLVI++RtVXpqrWt1t4LwVa9x/zYVo9d9cRWT1z11FZPXfWprT511ee2+tx7Bmz1hau+stVXrrplq1uuumOrO/6rMTt/Mcp2GGI3xgboFOYFEl+ByGrOS5Ux10c158MHZCXnPEjfg5zz4AHmzGNytfu5sbpJ8wJGPmqsLvr6UvT1ZegpXLK+NvqNZiQApm2hwT55EMZMxG6+gpuXt2J4THDjL2ECpuuJOodm0C+KtrlfeA+gOuaaMz2gzrG187wuSuwTzVl8dj579cGH2e70DMO8zMH9OQdra1c5YLpyltjB7L7hr/bPJWSw04aV4DzFFibxeIEpALpAnrzR5t9IO6OgTObAO/iuY3mObsIoxzYtj2oqYEkfNDSLDXus/2UI61IeMt0ufwc8iwl4heX/ugJB8cyLxcZGZduYzysIqmiJ2zT9uoqRDL2YzyDMJng4Ks3xvqkk1Am/J/DsgMe+AJv7xaaYnThmicqatd4vemqgj5oD92DpD3ZWe4MHvbU/Vhcfr80Ondc7P3R+6ix3Bp2Hncedjc6bznYn7p51/+r+3f3nxn8LCwvfLnw/RbvXZnO+6cxdCz/+D1MFDrU=</latexit>

Let R(0, ◆, ) be the rotation matrix from the magnetic frame to the
inertial frame. Then
mi = R> (0, ◆, )e1
is the unit vector that points in the direction of the magetic field, resolved
local in the inertial frame.
magnetic The normalized magnetic field measured in the body frame is
field

mb = Rbi> mi .

The measurement is given by

ymag = mb + ⌘.

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 7, Slide 20
 Digital Compass
magnetic
north
north
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Heading is sum of magnetic declination angle and magnetic heading

iv mi
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= + m
i v1

Magnetic heading determined from measurements of body-frame

components of magnetic field projected onto horizontal plane
declination m 0 v1 1
mx
mv1 = @mv1 y
A = Rv1 b ( , ✓)m
b

mv1
z

= Rv1 v2
v2 (✓)Rb ( )m
b
0 v1 1 0 1
local mx c✓ s✓ s s✓ c
magnetic @mv1 A=@ 0
field y c s A mb
mv1
z s✓ c✓ s c✓ c

Solving for heading gives

m = atan2(mv1 v1
y , mx ).

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 7, Slide 21
 Magnetic Declination Variation
180° 210° 240° 270° 300° 330° 0° 30° 60° 90° 120° 150° 180°

-30
60° 60°

-10
0
-2 0
-1

30° 30°

-10
10

0° 0°

-1
-20 0
-30° -30°
20 -20

-30

-40
0
30
-1
0 -5

-60° -60°

-60

0
-80 -90

-7
180° 210° 240° 270° 300° 330° 0° 30° 60° 90° 120° 150° 180°

World Magnetic Model, National Geophysical Data Center

Magnetic declination in Provo Utah is 12.5 degrees

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 7, Slide 22
 Magnetic Inclination
From Wikipedia: "Magnetic dip or magnetic inclination is the angle made by a compass needle with the
horizontal at any point on the Earth's surface. Positive values of inclination indicate that the field is pointing
downward, into the Earth, at the point of measurement.
US/UK World Magnetic Model -- Epoch 2010.0
Main Field Inclination (I)
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

g
h
-80

-60
70°S 70°S
180° 135°W 90°W 45°W 0° 45°E 90°E 135°E 180°

Main field inclination (I) Map developed by NOAA/NGDC & CIRES

Contour interval: 2 degrees, red contours positive (down); blue negative (up); green zero line. http://ngdc.noaa.gov/geomag/WMM/
Mercator Projection. Map reviewed by NGA/BGS
g : Position of dip poles Published January 2010

Magnetic inclination in Provo Utah is 66 degrees

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 7, Slide 23
 Global Positioning System
• 24 satellites orbiting the earth
• Altitude 20,180 km
• Any point on Earth’s surface can be seen by at
least 4 satellites at all times
• Time of flight of radio signal from 4 satellites to
receiver used to trilaterate location of receiver in
3 dimensions
• 4 range measurements needed to account for
clock offset error
• 4 nonlinear equations in 4 unknowns results:
– latitude
– longitude
– altitude
– receiver clock time offset

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 7, Slide 24
 GPS Error Sources
• Time of flight of radio signal from satellite to receiver used to calculate
pseudorange
– Called pseudorange to distinguish it from true range
• Numerous sources of error in time-of-flight measurement:
– Ephemeris Data – errors in satellite location
– Satellite Clock – due to clock drift
– Ionosphere – upper atmosphere, free electrons slow transmission of GPS signal
– Troposphere – lower atmosphere, weather (temperature and density) affect speed of
light, GPS signal transmission
– Multipath Reception – signals not following direct path
– Receiver Measurement – limitations in accuracy of receiver timing
• Small timing errors can result in large position errors
– 10 ns timing error à 3 m pseudorange error

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 7, Slide 25
 GPS Error Characterization
• Cumulative effect of GPS pseudorange errors is described by user
equivalent range error (UERE)
• UERE has two components
138 SENSORS FOR MAVS
– Bias SENSORS FOR MAVS
– Random Table 7.1: Standard pseudorange error model (1- , in meters) [34].
Table 7.1: Standard pseudorange error model (1- , in meters) [34].
Error source Bias Random Total
Error source BiasdataRandom 2.1
Ephemeris Total 0.0 2.1
Ephemeris data Satellite2.1
clock 0.0 2.0
2.1 0.7 2.1
Ionosphere 4.0
Satellite clock 2.0 0.7 2.1 0.5 4.0
Troposphere monitoring 0.5 0.5 0.7
Ionosphere Multipath
4.0 0.5 4.0 1.0
1.0 1.4
Troposphere monitoring Receiver 0.5 0.5
measurement 0.7 0.2
0.5 0.5
Multipath UERE, 1.0rms 1.0 1.4 1.4
5.1 5.3
Receiver measurement Filtered0.5 0.2
UERE, rms 0.5 0.4
5.1 5.1
UERE, rms 5.1 1.4 5.3
Filtered
Beard UERE,
& McLain, rms Aircraft,”
“Small Unmanned 5.1Princeton University
0.4 5.12012,
Press, Chapter 7, Slide 26
 GPS Error Characterization
• Effect of satellite geometry on position calculation is expressed
by Dilution of Precision (DOP)
• Satellites close together à high DOP
• Satellites far apart à low DOP
• DOP varies with time
• Horizontal DOP is smaller than vertical DOP
• Nominal HDOP = 1.3
• Nominal VDOP = 1.8

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 7, Slide 27
 Total GPS Error (RMS)
Standard deviation of RMS error in the north-east plane:

En-e,rms = HDOP ⇥ UERErms

= (1.3)(5.1 m)
= 6.6 m

Standard deviation of RMS altitude error:

Eh,rms = VDOP ⇥ UERErms

= (1.8)(5.1 m)
= 9.2 m
Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 7, Slide 28
 GPS Error Model
• Interested in transient behavior of errors – how does GPS error change
with time
• We use Gauss-Markov error model proposed by Rankin
kGPS Ts
⌫[n + 1] = e ⌫[n] + ⌘GPS [n]

Nominal 1- error (m) Model Parameters

Direction Bias Random Std. Dev. ⌘GPS (m) 1/kGPS (s) Ts (s)
North 4.7 0.4 0.21 1100 1.0
East 4.7 0.4 0.21 1100 1.0
Altitude 9.2 0.7 0.40 1100 1.0

yGPS,n [n] = pn [n] + ⌫n [n]

yGPS,e [n] = pe [n] + ⌫e [n]
yGPS,h [n] = pd [n] + ⌫h [n]
Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 7, Slide 29
 GPS Gauss Markov Process Error Model
20

10

altitude error (m)

0

10

20

30
0 2 4 6 8 10 12
time (hours)

1
altitude error (m)

2
100 110 120 130 140 150 160 170 180 190
time (s)

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 7, Slide 30
 Project
• Add sensor models to the simulation

Beard & McLain, “Small Unmanned Aircraft,” Princeton University Press, 2012, Chapter 7, Slide 31