AstroNet-II

The Astrodynamics Network

Attitude dynamics
and control
First AstroNet-II Training School
"Astrodynamics of natural and artificial satellites: from regular to
chaotic motions"

Department of Mathematics,
University of Roma Tor Vergata, Roma, Italy
14-17 January 2013.
james.biggs@strath.ac.uk
Outline
AstroNet-II
The Astrodynamics Network
Outline
 Attitude determination and control overview:
• Definitions
• Hardware: Sensors and Actuators
• Control overview
 Spacecraft dynamics:
• Natural Dynamics
• Spin stabilization
 Kinematic representations
 Perturbed dynamics and active Control
• Perturbations
• Detumbling
• Re-pointing
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 Attitude dynamics and control overview:

• Definitions
• Attitude modelling
• Hardware: Sensors (determination) and Actuators (control)
• Control methods

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Definitions
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Attitude definitions

•Attitude: orientation of spacecraft body axes relative to fixed frame

• Attitude determination: use of sensors to estimate attitude (real-time)
• Attitude control: maintain specified attitude with given precision using actuators.
• Attitude error: difference between true and desired spacecraft attitude

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The Design Process
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Attitude control requirements
• Principal requirement is to point the spacecraft payload (instrument, antenna, solar array)
•Required accuracy depends on payload + long/short-term pointing accuracy
•Achievable accuracy depends on actuators and hardware.

Sun-pointing for RHESSI solar physics mission 5

The Design Process
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Attitude control loop (Active)
•Normally have full closed-loop for spacecraft attitude with on-board control
• Have some required attitude (point payload to target, antenna to Earth etc)
• Sensors estimate true attitude of spacecraft to generate attitude error signal
• Attitude error signal input to controller which drives the control actuators
•Control signal is then some function of the error

Attitude control disturbance torques (design stage 2:

quantify disturbance

qerror environment)

S- Controller Actuators Spacecraft dynamics

qrequired
Estimation Sensors
qspacecraft
<q>spacecraft Attitude determination

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Attitude Dynamics of a spacecraft (Euler 1765):

2 3
I11  ( I 3  I 2 )23  T1
I 22  ( I1  I 3 )13  T2
I 33  ( I 2  I1 )12  T3

Attitude kinematics of a spacecraft: 1

 q1   0 3 2 1   q1 
q    q1   cos q 2 sin q1 sin q 2 cos q1 sin q 2   1 
    3 1 2   q2     
 sin q1 cos q 2  2 
1 0 1
2
or q 
  cos q  0 cos q1 cos q 2
 q3  2  2 1 0 3   q3 
2
 0   
     q  2
sin q1 cos q1
 3   3
 4
q  1 2 3 0   q4 

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Sensors
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Attitude determination
•In order to determine the spacecraft’s current attitude it requires the use of sensors.
•Various definitions of attitude angles (popular Earth-centred group below)
q3 Simulation Model

Strong assumption in much theoretical

work that state is perfectly known.

q1
Reality – require sensors to measure angles

Angular position
q2 q1 q 2 q3
Angular velocities

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Attitude sensors

There are two types of measurement system:

•Reference sensor – measures discretely the attitude with respect to some reference frame
defined by the position of objects in space e.g. the Sun, Earth or stars.
•Inertial sensors – measure continuously changes in attitude relative to a gyroscopic rotor.

•Reference sensors fixes the reference but there can be problems during periods of eclipse for
Sun sensors.

•Inertial sensors the errors progressively increase when making continuous measurements.

They are often used in combination where the reference sensors can be used to calibrate the
inertial sensors at discrete times. The inertial sensors can then measure continuously between
each calibration.
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Determination
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Sensor references
•External references must be used to determine the spacecrafts absolute attitude.
• the Sun
• the stars
•The Earth’s IR horizon
• The Earth’s magnetic field direction

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Sensors
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Attitude sensors
•Simple Sun sensors provides unit vector to Sun using masked photocells
• Star camera searches star catalogue database to determine view direction

star camera 2-axis Sun sensor 11

Sensors
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Attitude sensors
•Earth sensors operating principle is based on electromechanical modulation of the radiation
from the Earth’s horizon.
• A magnetometer is a measuring instrument used to measure the strength or direction of the
Earth’s magnetic field.

IRES – Infrared earth sensor Magnetometer used on the THEMIS 12

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Attitude sensor data
Reference Object Potential accuracy
Stars 1 arc second
Sun 1 arc minute
Earth (horizon) 6 arc minutes
Magnetometer 30 arc minutes
Narstar GPS 6 arc minutes

1 arcminute is 1/60th of a degree

1 arcsecond is 1/360th of a degree

Control algorithm is only as good as the hardware.

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Attitude actuators
There are two classes of actuator system:
•Active controllers –to actively control the system means to use a propulsion system to
re-point the spacecraft:
•Thrusters
•Magnetic Torque
•Momentum Wheels
•Control Moment Gyros
•Passive controllers– use mechanisms that exploit the orbit environment to naturally
correct the pointing direction such as solar radiation pressure, Gravity Gradient, Earth’s
magnetic field, spacecraft design:
•Permanent magnet
•Gravity Gradient
•Spin Stabilisation

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Active Control - Three-axis stabilisation
•More complex and expensive method, but delivers high precision pointing
• Require actuators for each rotational axis of spacecraft + control laws
• Best method for missions with frequent payload slews (space telescopes)

ESA Mars Express

Hubble space telescope 15
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Typical ADCS Actuators

Actuator Typical Performance Weight (kg) Power (W)

Range

Thrusters 1mN to 9N 1-10 1-100

Reaction & max torque from to 4 106 0.2 to 20 0.6 to 110

Momentum Wheels Nm to 1Nm
CMG 25 to 500 Nm > 40 90 to 150

Magnetic Torquers At 800km around the 0.05 to 50 0.6 to 16

Earth from 4e-7 to 0.18
Nm

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Thrusters

The conventional, low-risk solution is to use thrusters (usually monopropellant rockets),

organized in a Reaction control system. However, they use fuel.

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Actuators
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Momentum/reaction wheel
•Wheel with reversible DC motor to spin up/spin down metal disc assembly
• Transfer momentum to/from wheel and spacecraft axes (total conserved)
• Have 3 wheels (1 per axis) + spare canted relative to 3 main wheels

spin

reaction wheel assembly reaction wheel assembly

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Control moment gyro

•Use spinning wheel which can be quickly rotated to change its spin axis
• Rapid change in direction of angular momentum vector results in large torque
• Used for fast attitude slews and for applications with high torque demands

Space station CMGs Mini-satellite CMG

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Magnetic Torques Simple systems based on the interaction
of an electric current with a magnetic
B field.
They are generally good for medium to
small satellite flying at low altitude
around the Earth.
They can be used as passive or active
I devices. The control algorithm needs a
good knowledge of the magnetic field of
the planet.

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Spin stabilisation
• Simple and low cost method of attitude stabilisation (largely passive)
• Generally not suitable for imaging payloads (but can use a scan platform)
• Poor power efficiency since entire spacecraft body covered with solar cells

ESA Giotto spacecraft (de-spun antenna) ESA Cluster spacecraft

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Dual-spin stabilisation
•Simplicity of spin-stabilised spacecraft, but de-spun platform at top
• Mount payload on de-spun platform for better pointing, but passive stability
• Popular for some GEO comsats

Boeing SBS 6
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Attitude control trade-off
Type Advantages Disadvantages

Spin-stabilised Simple, passive, long-life, provides Poor manoeuvrability, low solar cell
scan motion, gyroscopic stability for efficiency (cover entire drum), no fixed
(~1o accuracy) large burns pointing

3-axis stabilised High pointing accuracy, rapid attitude Expensive (~2 x spinner), complex,
slews possible, generate large power requires active closed-loop control,
(~0.001o accuracy) (Sun-facing flat solar arrays) actuators for each body axis

Dual-spin stabilised Provides both fixed pointing (on de- Require de-spin mechanism, low solar
spun platform) and scanning motion, cell efficiency (cover entire drum), cost
(~0.1o accuracy) gyroscopic stability for large burns can approach 3-axis if high accuracy

Gravity-gradient Simple, low cost totally passive, Low accuracy, almost no

long-life, provides simple passive manoeuvrability, poor yaw stability,
(~5o accuracy) Earth pointing mode require deployment mechanism

Magnetic Simple, low cost, can be passive Poor accuracy (uncertainty in Earth's
with use of permanent magnet or magnetic field), magnetic interference
(~1o accuracy) active with use of electromagnets with science payload
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Complete attitude control system
•Require complete, integrated attitude control system for spacecraft
• May have extensive software on-board for control law implementation
• Can test some hardware in the loop prior to launch (sensors/actuators)

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Control methods
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Attitude control systems test

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Outline – Lecture 2
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 Spacecraft natural dynamics:

• Symmetric Spacecraft
• Jacobi elliptic functions
• Asymmetric spacecraft
• Spin-stabilisation.

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Dynamics
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Euler's equations

•Have Euler's equations to describe the attitude dynamics of a rigid body

• Body axes coincide with principal axes of inertia so products of inertia zero
• Real spacecraft have flexible modes (solar arrays, booms, fuel slosh . . .)

z
I11  ( I 3  I 2 )23  0
3 y
I 22  ( I1  I 3 )13  0 2
I 33  ( I 2  I1 )12  0
1 x

Euler's equations spacecraft body axes

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1

E  I112  I 222  I332
2

M 2  I1212  I 2222  I3232

1  12  22  32 
E    

2  I1 I2 I3 

M 2  12   22   32

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Example analytic functions:
1. Any polynomial function
2. The exponential function
3. Trigonometric functions.
4. Hyperbolic functions.
5. Jacobi Elliptic functions (1829).
Exponential function
f ( x)  an x n  an 1 x n 1  ...  a2 x 2  a1 x  a0
e.g. Exponential function
dy
ye
d x iq
e  ex x
y ye
dt dx 29
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Geometric definition of sine
and cosine:
a
 sin q q
c
b
 cos q
c

c 2  a 2  b2
Parameterisation of
the circle

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Dynamical Systems definition of sine and cosine:
The trigonometric functions x  sin t and y  cos t
are defined as the solution of the first order equations:
dx
y
dt
dy
 x
dt
x(0)  0, y(0)  1
d
sin t  cos t
dt
d
cos t   sin t
dt
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Conserved quantity the dynamical systems definition with
the geometric one
dx
y
dt
dy
 x
dt

The function 1  x 2  y 2 is conserved

An integral of motion.

d 2
( x  y 2 )  2 xx  2 yy  2 xy  2 yx  0
dt
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Free dynamics defined by Euler equations:

I11  ( I 3  I 2 )23  0
I 22  ( I1  I 3 )13  0
I 33  ( I 2  I1 )12  0

Conserved quantities:
Kinetic Energy: Magnitude of angular momentum:
1

E  I112  I 222  I332
2
 M 2  I1212  I 2222  I3232

Exercise: Prove that these quantities are conserved for a free

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rigid body.
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Natural Spin Motions
•Assume have perfectly symmetric spacecraft with I1=I2=A and I3=C
• Also assume no external torques acting of spacecraft so that M=0
• See that z-axis spin rate 3 =constant, what about 2 and 3 ?

symmetric spacecraft z (I3=C)

1=W
Aω1  (A  C)ω2 ω3  0 2 y (I2=A)
Aω2  (C  A)ω3ω1  0
Cω3  0
3 x (I1=A)

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Spacecraft nutation
•Combine Euler equations for x- and y- body axes to form a single equation
• Have oscillatory solution for x-axis angular velocity (and also y-axis)
• Solution describes nutation (precession) of the spacecraft body axes

(A  C)
ω1 
A
Wω2 E
1
2

I112  I 222  I332 
(C  A)
ω2  Wω1 2E  C W 2
 12  22
A A
ω3  W
1  r sin q
Exercise: Try   r cos q to find solution in terms of initial
2
conditions and constants of motion. 35
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Nutation geometry
•See from solution to Euler's equations that w executes a coning motion
• The angular momentum vector H is fixed since have no external torque
• Have nutation of spacecraft due to angular momentum about x- and y-axes

z nutation
ωz  Ω
ωy  ωo cost  t o 
y
prograde if A>C (prolate)

retrograde if C>A (oblate)

x
ωx  ωo sint  to 
body frame inertial frame

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Nutation damping

• Nutation undesirable for a spin-stabilised satellite (require pure spin state)

• Can damp angular momentum about x- and y-axes using nutation dampers
• Leads to pure spin about z-axis (+ small spin-up to conserve momentum)

nutation
pure spin
z
ωz  Ω
nutation damping
ωy  0

y
ωx  0

x
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Liquid ring nutation damper

•Can use viscous fluid in partially filled ring to damp nutation of spacecraft
• Nutation causes fluid motion along tube - viscous effects lead to dissipation
• Simple, passive means of damping nutation and achieving pure spin state

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Geometric definition of sinh
and cosh:

x  sinh a
y  cosh a

Parameterise the unit

hyperbola

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Dynamical Systems definition of sinh and
cosh:
The hyperbolic functions x  sinh t , y  cosh t are
defined as the solution of the first order equations:
dx
 y,
dt
dy
 x
dt
x(0)  0, y(0)  1
d
sinh t  cosh t
dt
d
cosh t  sinh t
dt
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Conserved quantity connects the dynamical
systems definition with the geometric one
dx
 y,
dt x(0)  0, y(0)  1
dy
 x
dt

The function 1  x2  y 2 is conserved

d 2
( x  y 2 )  2 xx  2 yy  2 xy  2 yx  0
dt
Parameterise the hyperbola.

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Define a Jacobi Elliptic function
Let m be a number (0,1) and let t denote a real variable that
we interpret as time. The Jacobi Elliptic Functions:
x  sn(t , m), y  cn(t , m), z  dn(t, m)
Are defined as solutions of the differential equations:
dx
 yz
dt
dy
  zx
dt
dz
  mxy
dt
Where sn(0, m)  x(0)  0, cn(0, m)  y(0)  1, dn(0, m)  z(0)  1 42
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dx
 yz
dt
dy
  zx
dt
dz
  mxy
dt
It is easily shown that 1  x 2  y 2 and 1  mx2  z 2
are both conserved quantities for these differential equations.

Therefore the Jacobi elliptic functions can be interpreted

geometrically as parameterising the circle in the x-y plane
and an ellipse in the x-z plane.
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dx ( I 2  I3 )
 yz
dt 1  23
I1
dy
  zx ( I 3  I1 )
dt 2  13
dz I3
  mxy
dt ( I1  I 2 )
3  12
x(0)  0, y (0)  1, z (0)  1 I3
Jacobi Elliptic function Euler equations

x  sn(t , m), 1  1sn( t   , m),

y  cn(t , m), 2  2 cn( t   , m),
z  dn(t , m) 3  3 dn( t   , m) 44
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( I 2  I3 )
1  23 1

E  I112  I 222  I332 
I1 2
( I 3  I1 )
2  13
I2 M 2  I1212  I 2222  I3232
( I1  I 2 )
3  12
I3
Exercise: Solve Euler equations in terms of initial conditions and
constants of motion (conserved quantities)    sn( t   , m),
1 1
2  2 cn( t   , m),
3  3 dn( t   , m) 45
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Derivatives and identities:
dsn(t , m)
 cn(t , m)dn(t , m) 1  sn 2 (t , m)  cn 2 (t , m)
dt
dcn(t , m) 1  msn 2 (t , m)  dn 2 (t , m)
  sn(t , m)dn(t , m)
dt
ddn(t , m)
  msn(t , m)cn(t , m)
dt

Some interesting special solutions:

m0 m 1
sn(t , m)  sin t sn(t , m)  tanh t
cn(t , m)  cos t cn(t , m)  secht
dn(t , m)  1 dn(t , m)  secht
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Other representations
Co-ordinate form Vector form Matrix form
( I 2  I3 )
1   23 L  M 2
I 2 I3
( I 3  I1 )
2   23 L  L  H L  [ L, H ]
I1 I 3
( I1  I 2 )
3  1 2 [ X , Y ]  XY  YX
I1 I 2

1  12  22  32 
H     Hamiltonian

2  I1 I2 I3 
systems on Lie
groups
M 2  12  22  32
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 Spacecraft Kinematic representations:

• Kinematics
• Euler angles
• Quaternions
• Lie Groups

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 Active pointing control of spacecraft requires knowledge of the
kinematics.
 Previous lecture focused on the natural dynamics:
 We need to know how the angular velocities relate to the
orientation of the spacecraft e.g.
I11  ( I 3  I 2 )23  T1 q1   cos q 2 sin q1 sin q 2 cos q1 sin q 2   1 
   
 sin q1 cos q 2  2 
1
I 22  ( I1  I 3 )13  T2 + q 2   cos q  0 cos q1 cos q 2
q  2  0 sin q1 cos q1   
I 33  ( I 2  I1 )12  T3  3   3

 Then we can design feedback control systems:

qerror
qrequired S- Controller Actuators Spacecraft dynamics

Sensors

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Kinematic representations
Perhaps the most intuitive (global) representation of rotation is
the 3 x 3 rotation matrix comprised of three orthogonal vectors:
R(t )   xˆ yˆ zˆ  each column is composed on a unit vector xˆ , yˆ , zˆ  3

ẑ xˆ, yˆ , zˆ are orthogonal unit vectors

R (t )  SO (3)
ŷ
It is a Lie group having the
properties of a group and a
x̂ differentiable manifold
Body-fixed orthonormal frame

Group – set with an operator (e.g. matrix multiplication) that satisfies

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Kinematic representations
•The structure of a Lie group means that we can define the kinematics:
 0 3 2   0 3 2  Is a screw-symmetric
dR(t )    
 R(t )  3 1  matrix whose
 3 1   so(3)
0 0
dt   0     components are the
 2 1  2 1 0  angular velocities.
R(t )   xˆ yˆ zˆ 
R (0)   xˆ (0) zˆ (0) 
Mathematically this is the Lie algebra of SO(3).
yˆ (0) It has the structure of an algebra.

The Lie algebra of a Lie group is considered to be the tangent space to the group at the identity:

dR (t )
X 
dt t 0
Consider a rotation about the third axis in time:

 cos t sin t 0    sin t cos t 0  0 1 0

  dR (t )    
R (t )    sin t cos t 0     cos t  sin t 0    1 0 0 
 0 1 
dt t 0 
0   0 0 0
 0  0 0
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Kinematic representations
Euler’s Theorem: Any two independent orthonormal coordinate frames may be related by
a minimum sequence of rotations (less than four) about coordinate axes, where no two
successive rotations may be about the same axis.

It is possible to bring a rigid body into an arbitrary orientation by performing three

successive rotations (12 sets are possible):
1. The first rotation is about any axis
2. The second rotation is about any axis not used for the first
3. The third rotation is either of the two axis not used for the second rotation.

Use Euler angles to represent these simple rotations about each axis.

Parameterize R(t) using Euler angles e.g.

 cos q3 sin q3 0  cos q 2 0  sin q 2  1 0 0 
   
R(t )    sin q3 cos q3 0  0 1 0  0 cos q1 sin q1 
 0 1  0 cos q 2  
 0  sin q 2  0  sin q1 cos q1 
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Kinematic representations
•Matrix representation is highly computation – requires integrating 9 couple ODEs.
• Euler angles representation simplifies the procedure and design of controls.
Parameterize R(t)
R(t )   xˆ yˆ zˆ 
 cos q3 sin q3 0  cos q 2 0  sin q 2  1 0 0 
   
R(t )    sin q3 cos q3 0  0 1 0  0 cos q1 sin q1 
xˆ, yˆ , zˆ are orthogonal vectors  0
 0 1 
 sin q 2 0 cos q 2  
 0  sin q1 cos q1 

Kinematics in matrix form

Euler angle representation of kinematics

 0 3 2 
dR(t )  
 R(t )  3 0 1  q1   cos q 2 sin q1 sin q 2 cos q1 sin q 2   1 
   
 sin q1 cos q 2  2 
dt   0 
1
 2 1 q 
  cos q
2  0 cos q1 cos q 2
q  2  0 sin q1 cos q1   
 3   3

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Kinematic representations

•Euler angles representation is useful for small manoeuvres but has singularities at cos q 2  0

q1   cos q 2 sin q1 sin q 2 cos q1 sin q 2   1 

   
 sin q1 cos q 2  2 
1
q 2   cos q  0 cos q1 cos q 2
q  2  0 sin q1 cos q1   
 3   3

•Quaternions often used to represent kinematics on-board spacecraft (non-intuitive)

 q1   0 3 2 1   q1 
q   
    3
2 1 0 1 2   q2 
 q3  2  2 1 0 3   q3 
    
 4
q  1 2 3 0   q4 

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Quaternion representations (Hamilton 1843)
•Quaternions often used to represent kinematics on-board spacecraft (non-intuitive)

 q1   0 3 2 1   q1 
qˆ  [q1 , q2, q3 , q4 ]
T
q   
2   q2  ii  jj  kk  1
    3
2 1 0 1
 q3  2  2
  
1 0 3   q3 
 
ij  k , jk  i, ki  j
q̂  q1  q2i  q3 j  q4 k 0   q4 
 4
q  1 2 3

These two rotations are equivalent:

R (t ) x  x ' Where x  [ x1, x2 , x3 ] is in its vector form

T

ˆ ˆ 1  x '
qxq Where x  [0, x1, x2 , x3 ] is in its quaternion form
T

 q12  q22  q32  q42 2q2 q3  2q1q4 2q2 q4  2q1q3 

 
qˆ  [q1 , q2, q3 , q4 ]T , R (t )   2q2 q3  2q1q4 q12  q22  q32  q42 2q3q4  2q1q2 
 2q2 q4  2q1q3 2q3q4  2q1q2 q12  q22  q32  q42 

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Eigen-axis rotation
Euler’s (1st) Theorem
 cos q3 sin q3 0   cos q 2 0  sin q 2   1 0 0 
   
R (t )    sin q3 cos q3 0 0 1 0   0 cos q1 sin q1   Rx (q3 ) R y (q 2 ) Rz (q1 )
 0 1   sin q 2 0 cos q 2   0  sin q1 cos q1 
 0

Euler’s (2nd) Theorem: Any two independent orthonormal coordinate frames may be related by a
single rotation about some axis.

Call the rotation angle about this eigenaxis q then we can define the quaternions as:

q̂  q1  q2i  q3 j  q4k q q q q
q1  cos , q2  sin cos  x , q3  sin cos  y , q4  sin cos  z
2 2 2 2
q̂  q1  q2i  q3 j  q4 k

cos  x , cos  y , cos  z are the direction cosines locating the single axis of rotation
e.g. if the eigenaxis is the x axis then q  cos q , q  sin q , q  0, q  0
1 2 3 4
2 2
q q
qˆ  cos  sin i 56
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Euler’s (1st) Theorem
 cos q3 sin q3 0   cos q 2 0  sin q 2   1 0 0 
   
R (t )    sin q3 cos q3 0 0 1 0   0 cos q1 sin q1   Rx (q3 ) R y (q 2 ) Rz (q1 ) Euler angle representation
 0 1   sin q 2 0 cos q 2   0  sin q1 cos q1 
 0

 q12  q22  q32  q42 2q2 q3  2q1q4 2q2 q4  2q1q3 

 
R(t )   2q2 q3  2q1q4 q12  q22  q32  q42 2q3q4  2q1q2  Quaternion representation
 2q2 q4  2q1q3 2q3q4  2q1q2 q12  q22  q32  q42 


Equating components we can conveniently compute equations for the Euler angles in
terms of the quaternions.
Using the eigenaxis definition of quaternions.
q̂  q1  q2i  q3 j  q4 k q q q q
q1  cos , q2  sin cos  x , q3  sin cos  y , q4  sin cos  z
2 2 2 2
We can express each rotation matrix about a fixed axes as a quaternion.
q q q q q q
qˆ  Rx (q3 ) R y (q 2 ) Rz (q1 )  [cos 3  sin 3 i ][cos 2  sin 2 j ][cos 1  sin 1 k ]
2 2 2 2 2 2
This provides a convenient expression for the quaternions in terms of the Euler angles.

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Quaternion in matrix form
•It can sometimes be convenient to use the matrix form called SU(2):
H  SU (2)
 q  q i q3  q4i 
g 1 2 
q̂  q1  q2i  q3 j  q4 k   q3  q4i q1  q2i 

q̂1  x1  x2i  x3 j  x4 k  x xi x3  x4i 

g1   1 2
  x3  x4i x1  x2i 
Quaternion multiplication Matrix Multiplication

Then the kinematics can be defined on the SU(2):

dg 1  i1 2  i3 
 g 
dt 2  2  i3 i1 

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Summary

• We have considered four different representations of the kinematics: SO(3), Euler

angles, quaternions, SU(2).
• SO(3) – global coordinate system – requires the integration of 9 coupled ODEs.

• Euler angles – local coordinate system – requires the integration of 3 coupled ODEs.
non-unique – suffers from gimbal lock – intuitive

• Quaternions – global coordinate system – requires the integration of 3 coupled ODEs

unique – no singularities – non-intuitive.

• Quaternions in matrix form SU(2) – as above – useful form for certain computations.

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Nonlinear model
I11  ( I 3  I 2 )23  T1
I 22  ( I1  I 3 )13  T2
I 33  ( I 2  I1 )12  T3

 q1   0 3 2 1   q1 
q   
    3
2 1 0 1 2   q2 
 q3  2  2 1 0 3   q3 
    
 4
q  1 2 3 0   q4 

Re-pointing problem – find T1 , T2 , T3 that drives the spacecraft from

q(0)  [q1 (0), q2 (0), q3 (0).q4 (0)]T , (0)  [1 (0), 2 (0), 3 (0)]T ,

to q(T )  [q1 (T ).q2 (T ), q3 (T ).q4 (T )]T , q(0)  [1 (T ), 2 (T ), 3 (T )]T ,

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dg (t )
Lie Group representation of kinematics:  g (t )W
dt
where g (t )  G is a Lie Group and W is a Lie algebra.

When a system has this form in is often sufficient to study the Lie
algebra to obtain information about the entire system.
• Controllability properties.
• The co-ordinate free maximum principle
• Lax Pair Integration

Other systems include robotics e.g. unmanned air vehicle,

and autonomous underwater vehicles, quantum control, fine
needle surgery.
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Attitude Dynamics and Control

• Spin Stabilization

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Spin stabilisation
• Simple and low cost method of attitude stabilisation (largely passive)
• Generally not suitable for imaging payloads (but can use a scan platform)
• Poor power efficiency since entire spacecraft body covered with solar cells

ESA Giotto spacecraft (de-spun antenna)

ESA Cluster spacecraft
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Spin axis stability

• Consider rotation about z-axis with angular velocity 3  W, 1  2  0

• Now perturb state of pure spin such that 3  W  3 , 1  1 , 2  2 ,
•Under what conditions does the spacecraft spin remain stable?
ω2 ω3 ~ 0 ω1ω3 ~ 0

( I 2  I3 )
1  2 W
I1
( I 3  I1 )
2  1W
I2
3  0
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Conditions for stability

•Combine perturbed Euler equations x- and y-axis to form a single equation
• Obtain single 2nd order differential equation with constant coefficient
• Obtain (stable) oscillatory solution in coefficient positive (unstable if -ve)

( I 2  I3 ) ( I 2  I 3 )( I 3  I1 ) 2
1  2 W 1  W 1
I1 I1 I 2

1  Q 2 1  0 Q2 > 0 Stable

( I 3  I 2 )( I 3  I1 ) 2
Q 2
W
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Major axis rule
•See that minor axis spin is stable (Q2>0)
• See that major axis spin is stable (Q2>0)
• But, intermediate axis spin is unstable (I3>I2 but I3<I1) since then Q2<0

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Explorer 1 flat spin

•Explorer 1 (first US satellite, 1958) designed as a minor axis spinner !

• Via energy dissipation spacecraft experienced transition to major axis spin
• Energy dissipation caused by flexing of wire antennae on spacecraft body

flexible antennae

minor axis spin major axis spin

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Example of a dissipative system with spherical fuel slug
( I1  I )1  ( I 2  I 3 )23  m 1
( I 2  I )2  ( I 3  I1 )31  m 2
( I 3  I )3  ( I1  I 2 )23  m 3
 1  1  ( m / I ) 1  2 3  3 2
 2  2  ( m / I ) 2  3 1  1 3
 3  3  ( m / I ) 3  1 2  2 1
I Spherical fuel slug moment of inertia
m Viscous damping coefficient
1, 2 , 3 Relative rates between the rigid body and the fuel slug

1
H (( I1  I )12  ( I 2  I )22  ( I 3  I )32  I ((1   1 ) 2  (2   2 ) 2  (3   3 ) 2 ))
2
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Perturbations – disturbance torques
•Air drag

•Solar pressure
I11  ( I 3  I 2 )23  M1
I 22  ( I1  I 3 )13  M 2
•Gravity Gradient

I 33  ( I 2  I1 )12  M 3 •Magnetic field

•Spherical harmonics

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Disturbance torques

•Disturbance torques lead to reduction in pointing accuracy of spacecraft

• Torque magnitudes dependent on spacecraft orbit type and orbit altitude
• Air drag, gravity gradient torques in LEO, solar pressure torques in GEO

solar array

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ADCS Sizing: Momentum wheels, reaction wheels,
magnetic torques
1. Calculate disturbance torques
2. Compute time integral of disturbance torques for each control axis. The resulting
angular impulse on each axis Lx,Ly,Lz represent the accumulated angular
momentum:
tf tf tf

Lx   M dxdt Ly   M dydt Lz   M dz dt
t0 t0 t0

3. Identify cyclic and secular components of disturbing forces

4. Size torquers: wheels and CMGs are sized for cyclic terms, thrusters/magnet
torquers are sized for secular terms if used with wheels (desaturation)
5. This is often referred to a momentum dumping.

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Objectives
1 – Detumbling – 2 – Repointing –
Dynamic equations Dynamics and kinematics

Reorient spacecraft to target

Tumbling motion must be stabilised specific point (e.g. point antenna to
or mission will fail. ground station, point solar cells
towards sun for maximum power.)

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Detumbling with thrusters
•Spacecraft have a tendency to tumble post-seperation.
• Can use continuos torque to de-tumble the spacecraft (three reaction wheels are required four
for redundancy)
• Micro and nano spacecraft often use magnetic-torquers (underactuated)

2 I11  ( I3  I 2 )23  T1
1
I 22  ( I1  I 3 )13  T2
I 33  ( I 2  I1 )12  T3
Use proportional control

3 T1  k11 , T2  k22 , T3  k33 .

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Detumbling with magnetic torquers
 With magnetorquers controllable parameter is magnetic dipole N:-

T  N B
 Convention is to use BDOT, which uses rate of change of magnetic field as control
(requires a magnetometer and Kalman filter):-

N  K (B )
 BDOT used on several magnetically actuated spacecraft (e.g. Compass-1, UKube-1.).

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Attitude control laws

• Can use classical linear control laws for fine pointing (PID controller)
• Also, need to ensure robustness of control laws due to disturbance torques

t d
u (t )  K p e(t )  K i  e( )d  K d e(t )
0 dt

e(t )  [q1  q1d , q 2  q 2 d , q3  q3d ]T

classical PID controller

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I11  ( I3  I 2 )23  T1
I 22  ( I1  I 3 )13  T2
u (t )  [T1 , T2 , T3 ]T
I 33  ( I 2  I1 )12  T3

t d
u (t )  K p e(t )  K i  e( )d  K d e(t )
0 dt

Parameter Overshoot Settling time Steady-state error

Kp  - 
Ki   Eliminates
Kd   -

Estimated to be used in 95% of industrial processes.

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Full nonlinear model – quaternion representation
I11  ( I 3  I 2 )23  M1  T1
I 22  ( I1  I3 )13  M 2  T2
I33  ( I 2  I1 )12  M 3  T3

 q1   0 3 2 1   q1 
q   
 2    3
1 0 1 2   q2 
 q3  2  2 1 0 3   q3 
    
 4
q  1 2 3 0   q4 

Re-pointing problem – find T1 , T2 , T3 that drives the spacecraft from

q(0)  [q1 (0), q2 (0), q3 (0).q4 (0)]T , (0)  [1 (0), 2 (0), 3 (0)]T ,

to
q(T )  [q1 (T ).q2 (T ), q3 (T ).q4 (T )]T , q(0)  [1 (T ), 2 (T ), 3 (T )]T ,
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A simple quaternion feedback controller (rest-to-rest)
T1  k1e1  c11
T2  k2 e2  c22
T3  k3e3  c33

where  e1   q4 (T ) q3 (T ) q2 (T ) q1 (T )   q1 

e   q (T ) q (T ) q (T )  q (T )
 
 
2 3 4 1 2   q2 
 e3   q2 (T ) q1 (T ) q4 (T ) q3 (T )   q3 
     
 4  1
e q ( T ) q2 (T ) q3 ( T ) q4 (T )   q4 

Algorithm is flight tested and stability can be verified. Requires tuning of the
parameters:
k1 , c1 , k2 , c2 , k3 , c3
Can use adaptive tuning methods e.g fuzzy logic
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Attitude control loop (precision spacecraft)
•Actuators: Reaction wheels and jet torques.
• Sensors: Star and Sun sensors.
•Estimator – Kalman filter, low pass filter, nonlinear filter.
• Appropriate feedback law e.g. quaternion feedback law

Attitude control disturbance torques (design stage 2:

quantify disturbance
environment)

Controller Actuators Spacecraft dynamics

Estimator Sensors
qspacecraft
<q>spacecraft Attitude determination

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Attitude control loop (precision spacecraft)
•Actuators: Reaction wheels and jet torques.
• Sensors: Star and Sun sensors.
•Estimator – Kalman filter, low pass filter, nonlinear filter.
• Appropriate feedback law e.g. quaternion feedback law

Attitude control disturbance torques (design stage 2:

quantify disturbance
environment)

Motion Controller Actuators Spacecraft dynamics

Planner Estimator Sensors

qspacecraft
<q>spacecraft Attitude determination

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Future developments
• Future spacecraft will pose exciting challenges for attitude control design
• Large deployable, highly flexible membrane structures (telescope Sun-shade)
• Fast slews for agile science/Earth observation (monitor transient phenomena)
•Attitude control of nano and micro spacecraft – low-onboard computational power and small
torque available.
Large (200m 2) deployable
sunshield protects from sun,
earth and moon IR radiation(ISS)
Deployable Spacecraft support module
secondary SSM (attitude control,
Mirror (SM) communications, power,
data handling)

cold side warm side

“Open” telescope (no

external baffling) OTA
allows passive Science
cooling to ~50K Instruments
Beryllium (ISIM) Isolation truss
Primary mirror (PM)
Rapideye
UKube
next generation space telescope (NGST)
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AstroNet-II
The Astrodynamics Network

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