Attitude dynamics and control - uniroma2.itlocatell/school1-astronet2/material/...Complete attitude...
Transcript of Attitude dynamics and control - uniroma2.itlocatell/school1-astronet2/material/...Complete attitude...
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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.
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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
Outline
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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
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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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
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
Controller Spacecraft dynamics
qspacecraft
qerror
qrequired Sensors
<q>spacecraft
S- Actuators
disturbance torques (design stage 2:
quantify disturbance
environment)
The Design Process
Attitude determination
Attitude control
Estimation
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The Astrodynamics Network
1 1 3 2 2 3 1
2 2 1 3 1 3 2
3 3 2 1 1 2 3
( )
( )
( )
I I I T
I I I T
I I I T
Attitude Dynamics of a spacecraft (Euler 1765):
3 2 11 1
3 1 22 2
2 1 33 3
1 2 34 4
0
01
02
0
q q
q q
q q
q q
1 2 1 2 1 2 1
2 1 2 1 2 22
3 1 1 3
cos sin sin cos sin1
0 cos cos sin coscos
0 sin cos
q q q q q q
q q q q q q
q q q
Attitude kinematics of a spacecraft: 1
2 3
or
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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)
Sensors
Simulation Model
Reality – require sensors to measure angles 1q
2q
3q
Angular position
1q 2q 3q
Angular velocities
Strong assumption in much theoretical
work that state is perfectly known.
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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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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
Determination
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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
2-axis Sun sensor star camera
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.
Magnetometer used on the THEMIS IRES – Infrared earth sensor
Sensors
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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)
Hubble space telescope ESA Mars Express
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Typical ADCS Actuators
Actuator Typical Performance
Range
Weight (kg) Power (W)
Thrusters 1mN to 9N 1-10 1-100
Reaction &
Momentum Wheels
max torque from to
Nm to 1Nm
0.2 to 20 0.6 to 110
CMG 25 to 500 Nm > 40 90 to 150
Magnetic Torquers At 800km around the
Earth from 4e-7 to 0.18
Nm
0.05 to 50 0.6 to 16
64 10
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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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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
reaction wheel assembly reaction wheel assembly
spin
Actuators
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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
I
B
Simple systems based on the interaction
of an electric current with a magnetic
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
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
(~1o accuracy)
Simple, passive, long-life, provides
scan motion, gyroscopic stability for
large burns
Poor manoeuvrability, low solar cell
efficiency (cover entire drum), no fixed
pointing
3-axis stabilised
(~0.001o accuracy)
High pointing accuracy, rapid attitude
slews possible, generate large power
(Sun-facing flat solar arrays)
Expensive (~2 x spinner), complex,
requires active closed-loop control,
actuators for each body axis
Dual-spin stabilised
(~0.1o accuracy)
Provides both fixed pointing (on de-
spun platform) and scanning motion,
gyroscopic stability for large burns
Require de-spin mechanism, low solar
cell efficiency (cover entire drum), cost
can approach 3-axis if high accuracy
Gravity-gradient
(~5o accuracy)
Simple, low cost totally passive,
long-life, provides simple passive
Earth pointing mode
Low accuracy, almost no
manoeuvrability, poor yaw stability,
require deployment mechanism
Magnetic
(~1o accuracy)
Simple, low cost, can be passive
with use of permanent magnet or
active with use of electromagnets
Poor accuracy (uncertainty in Earth's
magnetic field), magnetic interference
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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Attitude control systems test
Control methods
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Spacecraft natural dynamics:
• Symmetric Spacecraft
• Jacobi elliptic functions
• Asymmetric spacecraft
• Spin-stabilisation.
Outline – Lecture 2
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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 . . .)
x
y
z
2
1
3
spacecraft body axes Euler's equations
Dynamics
1 1 3 2 2 3
2 2 1 3 1 3
3 3 2 1 1 2
( ) 0
( ) 0
( ) 0
I I I
I I I
I I I
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2 2 21 1 2 2 3 3
1
2E I I I
2 2 2 2 2 2 21 1 2 2 3 3M I I I
2 2 2 21 2 3M
22 231 2
1 2 3
1
2E
I I I
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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).
e.g. Exponential function
dyy
dxxy e
1 2
1 2 1 0( ) ...n n
n nf x a x a x a x a x a
x xde e
dt iy e q
Exponential function
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cosb
cq
sina
cq
2 2 2c a b
Geometric definition of sine
and cosine:
q
Parameterisation of
the circle
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Dynamical Systems definition of sine and cosine:
The trigonometric functions and
are defined as the solution of the first order equations:
sinx t cosy t
dxy
dt
dyx
dt
(0) 0, (0) 1x y
sin cos
cos sin
dt t
dt
dt t
dt
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dxy
dt
dyx
dt
The function is conserved
2 2( ) 2 2 2 2 0d
x y xx yy xy yxdt
Conserved quantity the dynamical systems definition with
the geometric one
2 21 x y
An integral of motion.
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1 1 3 2 2 3
2 2 1 3 1 3
3 3 2 1 1 2
( ) 0
( ) 0
( ) 0
I I I
I I I
I I I
2 2 21 1 2 2 3 3
1
2E I I I
2 2 2 2 2 2 21 1 2 2 3 3M I I I
Free dynamics defined by Euler equations:
Conserved quantities:
Kinetic Energy: Magnitude of angular momentum:
Exercise: Prove that these quantities are conserved for a free
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 =constant, what about and ?
x (I1=A)
y (I2=A)
z (I3=C)
2
3
1=W
1 2 3
2 3 1
3
0
0
0
Aω (A C)ω ω
Aω (C A)ω ω
Cω
symmetric spacecraft
3 2 3
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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
1 2
2 1
3
(A C)ω ω
A
(C A)ω ω
A
ω
W
W
W
2 2 21 1 2 2 3 3
1
2E I I I
22 21 2
2E C
A
W
Exercise: Try to find solution in terms of initial
conditions and constants of motion.
1
2
sin
cos
r
r
q
q
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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
x
y
z
oox ttωω sin
ooy ttωω cos
Ωωz
body frame inertial frame
nutation
prograde if A>C (prolate)
retrograde if C>A (oblate)
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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
nutation damping Ωωz
0xω
0yω
x
y
z
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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:
sinh
cosh
x a
y a
Parameterise the unit
hyperbola
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Dynamical Systems definition of sinh and
cosh:
The hyperbolic functions , are
defined as the solution of the first order equations:
Where
sinhx t coshy t
,dx
ydt
dyx
dt
(0) 0, (0) 1x y
sinh cosh
cosh sinh
dt t
dt
dt t
dt
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2 21 x y The function is conserved
2 2( ) 2 2 2 2 0d
x y xx yy xy yxdt
Parameterise the hyperbola.
,dx
ydt
dyx
dt
(0) 0, (0) 1x y
Conserved quantity connects the dynamical
systems definition with the geometric one
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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:
Are defined as solutions of the differential equations:
( , ), ( , ), ( , )x sn t m y cn t m z dn t m
dxyz
dt
dyzx
dt
dzmxy
dt
Where (0, ) (0) 0, (0, ) (0) 1, (0, ) (0) 1sn m x cn m y dn m z
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2 21 x y 2 21 mx z
dxyz
dt
dyzx
dt
dzmxy
dt
It is easily shown that and
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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2 31 2 3
1
3 12 1 3
3
1 23 1 2
3
( )
( )
( )
I I
I
I I
I
I I
I
dxyz
dt
dyzx
dt
dzmxy
dt
Jacobi Elliptic function Euler equations
( , ),
( , ),
( , )
x sn t m
y cn t m
z dn t m
1 1
2 2
3 3
( , ),
( , ),
( , )
sn t m
cn t m
dn t m
(0) 0, (0) 1, (0) 1x y z
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2 31 2 3
1
3 12 1 3
2
1 23 1 2
3
( )
( )
( )
I I
I
I I
I
I I
I
1 1
2 2
3 3
( , ),
( , ),
( , )
sn t m
cn t m
dn t m
2 2 21 1 2 2 3 3
1
2E I I I
2 2 2 2 2 2 21 1 2 2 3 3M I I I
Exercise: Solve Euler equations in terms of initial conditions and
constants of motion (conserved quantities)
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Derivatives and identities:
2 2
2 2
1 ( , ) ( , )
1 ( , ) ( , )
sn t m cn t m
msn t m dn t m
( , )( , ) ( , )
( , )( , ) ( , )
( , )( , ) ( , )
dsn t mcn t m dn t m
dt
dcn t msn t m dn t m
dt
ddn t mmsn t m cn t m
dt
Some interesting special solutions:
0m 1m
( , ) sin
( , ) cos
( , ) 1
sn t m t
cn t m t
dn t m
( , ) tanh
( , ) sech
( , ) secht
sn t m t
cn t m t
dn t m
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Other representations
2 31 2 3
2 3
3 12 2 3
1 3
1 23 1 2
1 2
( )
( )
( )
I I
I I
I I
I I
I I
I I
22 231 2
1 2 3
1
2H
I I I
2 2 2 21 2 3M
2L M
L L H
Co-ordinate form Vector form
[ , ]L L H
Matrix form
[ , ]X Y XY YX
Hamiltonian
systems on Lie
groups
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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.
+
Then we can design feedback control systems:
1 2 1 2 1 2 1
2 1 2 1 2 22
3 1 1 3
cos sin sin cos sin1
0 cos cos sin coscos
0 sin cos
q q q q q q
q q q q q q
q q q
Controller Spacecraft dynamics
qerror
qrequired
Sensors
S- Actuators
1 1 3 2 2 3 1
2 2 1 3 1 3 2
3 3 2 1 1 2 3
( )
( )
( )
I I I T
I I I T
I I I T
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Kinematic representations
ˆ ˆ ˆ( )R t x y z
ˆ ˆ ˆ, ,x y z are orthogonal unit vectors
Perhaps the most intuitive (global) representation of rotation is
the 3 x 3 rotation matrix comprised of three orthogonal vectors:
x
Body-fixed orthonormal frame
z
y
3ˆ ˆ ˆ, ,x y z each column is composed on a unit vector
( ) (3)R t SO
It is a Lie group having the
properties of a group and a
differentiable manifold
Group – set with an operator (e.g. matrix multiplication) that satisfies
four conditions of closure, associativity, identity, inverse.
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Kinematic representations
•The structure of a Lie group means that we can define the kinematics:
3 2
3 1
2 1
0( )
( ) 0
0
dR tR t
dt
ˆ ˆ ˆ( )R t x y z
3 2
3 1
2 1
0
0 (3)
0
so
Is a screw-symmetric
matrix whose
components are the
angular velocities.
Mathematically this is the Lie algebra of SO(3).
It has the structure of an algebra. ˆ ˆ ˆ(0) (0) (0) (0)R x y z
The Lie algebra of a Lie group is considered to be the tangent space to the group at the identity:
0
( )
t
dR tX
dt
cos sin 0
( ) sin cos 0
0 0 1
t t
R t t t
Consider a rotation about the third axis in time:
0
0
sin cos 0 0 1 0( )
cos sin 0 1 0 0
0 0 0 0 0 0t
t
t tdR t
t tdt
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Kinematic representations
Parameterize R(t) using Euler angles e.g.
3 3 2 2
3 3 1 1
2 2 1 1
cos sin 0 cos 0 sin 1 0 0
( ) sin cos 0 0 1 0 0 cos sin
0 0 1 sin 0 cos 0 sin cos
R t
q q q q
q q q q
q q q q
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.
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.
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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.
Kinematics in matrix form
3 2
3 1
2 1
0( )
( ) 0
0
dR tR t
dt
ˆ ˆ ˆ( )R t x y z
ˆ ˆ ˆ, ,x y z are orthogonal vectors
Parameterize R(t)
3 3 2 2
3 3 1 1
2 2 1 1
cos sin 0 cos 0 sin 1 0 0
( ) sin cos 0 0 1 0 0 cos sin
0 0 1 sin 0 cos 0 sin cos
R t
q q q q
q q q q
q q q q
1 2 1 2 1 2 1
2 1 2 1 2 22
3 1 1 3
cos sin sin cos sin1
0 cos cos sin coscos
0 sin cos
q q q q q q
q q q q q q
q q q
Euler angle representation of kinematics
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Kinematic representations
1 2 1 2 1 2 1
2 1 2 1 2 22
3 1 1 3
cos sin sin cos sin1
0 cos cos sin coscos
0 sin cos
q q q q q q
q q q q q q
q q q
•Euler angles representation is useful for small manoeuvres but has singularities at
•Quaternions often used to represent kinematics on-board spacecraft (non-intuitive)
3 2 11 1
3 1 22 2
2 1 33 3
1 2 34 4
0
01
02
0
q q
q q
q q
q q
2cos 0q
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Quaternion representations (Hamilton 1843)
•Quaternions often used to represent kinematics on-board spacecraft (non-intuitive)
3 2 11 1
3 1 22 2
2 1 33 3
1 2 34 4
0
01
02
0
q q
q q
q q
q q
1 2, 3 4ˆ [ , , ]Tq q q q q
( ) 'R t x x
These two rotations are equivalent:
1ˆ ˆ 'qxq x
1 2 3 4q q q i q j q k
1, 2 3[0, , ]Tx x x x
1, 2 3[ , ]Tx x x xWhere is in its vector form
Where is in its quaternion form
2 2 2 2
1 2 3 4 2 3 1 4 2 4 1 3
2 2 2 2
1 2, 3 4 2 3 1 4 1 2 3 4 3 4 1 2
2 2 2 2
2 4 1 3 3 4 1 2 1 2 3 4
2 2 2 2
ˆ [ , , ] , ( ) 2 2 2 2
2 2 2 2
T
q q q q q q q q q q q q
q q q q q R t q q q q q q q q q q q q
q q q q q q q q q q q q
1
, ,
ii jj kk
ij k jk i ki j
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Eigen-axis rotation
3 3 2 2
3 3 1 1 3 2 1
2 2 1 1
cos sin 0 cos 0 sin 1 0 0
( ) sin cos 0 0 1 0 0 cos sin ( ) ( ) ( )
0 0 1 sin 0 cos 0 sin cos
x y zR t R R R
q q q q
q q q q q q q
q q q q
Euler’s (1st) Theorem
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 then we can define the quaternions as: q
1 2 3 4cos , sin cos , sin cos , sin cos2 2 2 2
x y zq q q qq q q q
1 2 3 4q q q i q j q k
1 2 3 4q q q i q j q k
cos ,cos ,cosx y z are the direction cosines locating the single axis of rotation
e.g. if the eigenaxis is the x axis then 1 2 3 4cos , sin , 0, 0
2 2q q q q
q q
ˆ cos sin2 2
q iq q
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3 3 2 2
3 3 1 1 3 2 1
2 2 1 1
cos sin 0 cos 0 sin 1 0 0
( ) sin cos 0 0 1 0 0 cos sin ( ) ( ) ( )
0 0 1 sin 0 cos 0 sin cos
x y zR t R R R
q q q q
q q q q q q q
q q q q
Euler’s (1st) Theorem
1 2 3 4cos , sin cos , sin cos , sin cos2 2 2 2
x y zq q q qq q q q
1 2 3 4q q q i q j q k
3 3 2 2 1 13 2 1
ˆ ( ) ( ) ( ) [cos sin ][cos sin ][cos sin ]2 2 2 2 2 2
x y zq R R R i j kq q q q q q
q q q
2 2 2 2
1 2 3 4 2 3 1 4 2 4 1 3
2 2 2 2
2 3 1 4 1 2 3 4 3 4 1 2
2 2 2 2
2 4 1 3 3 4 1 2 1 2 3 4
2 2 2 2
( ) 2 2 2 2
2 2 2 2
q q q q q q q q q q q q
R t q q q q q q q q q q q q
q q q q q q q q q q q q
Euler angle representation
Quaternion representation
Equating components we can conveniently compute equations for the Euler angles in
terms of the quaternions.
Using the eigenaxis definition of quaternions.
We can express each rotation matrix about a fixed axes as a quaternion.
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):
1 2 3 4q q q i q j q k
Then the kinematics can be defined on the SU(2):
1 2 3
2 3 1
1
2
i idgg
i idt
1 2 3 4
1
3 4 1 2
x x i x x ig
x x i x x i
1 2 3 4
3 4 1 2
q q i q q ig
q q i q q i
1 1 2 3 4q x x i x j x k
(2)H SU
Quaternion multiplication Matrix Multiplication
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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
1 1 3 2 2 3 1
2 2 1 3 1 3 2
3 3 2 1 1 2 3
( )
( )
( )
I I I T
I I I T
I I I T
3 2 11 1
3 1 22 2
2 1 33 3
1 2 34 4
0
01
02
0
q q
q q
q q
q q
Re-pointing problem – find that drives the spacecraft from 1 2 3, ,T T T
1 2 3 4 1 2 3(0) [ (0), (0), (0). (0)] , (0) [ (0), (0), (0)] ,T Tq q q q q
1 2 3 4 1 2 3( ) [ ( ). ( ), ( ). ( )] , (0) [ ( ), ( ), ( )] ,T Tq T q T q T q T q T q T T T to
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Lie Group representation of kinematics:
where is a Lie Group and is a Lie algebra.
( )( )
dg tg t
dt W
( )g t G W
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
• Now perturb state of pure spin such that
•Under what conditions does the spacecraft spin remain stable?
2 3 0ω ω ~ 1 3 0ω ω ~
3 1 2, 0 W
3 3 1 1 2 2, , , W
2 31 2
1
3 12 1
2
3
( )
( )
0
I I
I
I I
I
W
W
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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)
2 31 2
1
( )I I
I
W 22 3 3 1
1 11 2
( )( )I I I I
I I
W
2 23 2 3 1
1 3
( )( )I I I IQ
I I
W
2 0Q > Stable
2 0Q < Unstable
21 1 0Q
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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
minor axis spin major axis 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
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1 1 2 3 2 3 1
2 2 3 1 3 1 2
3 3 1 2 2 3 3
1 1 1 2 3 3 2
2 2 2 3 1 1 3
3 3 3 1 2 2 1
( ) ( )
( ) ( )
( ) ( )
( / )
( / )
( / )
I I I I
I I I I
I I I I
I
I
I
m
m
m
m
m
m
Example of a dissipative system with spherical fuel slug
I
m
1 2 3, ,
Spherical fuel slug moment of inertia
Viscous damping coefficient
Relative rates between the rigid body and the fuel slug
2 2 2 2 2 2
1 1 2 2 3 3 1 1 2 2 3 3
1(( ) ( ) ( ) (( ) ( ) ( ) ))
2H I I I I I I I
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The Astrodynamics Network
•Air drag
•Solar pressure
•Gravity Gradient
•Magnetic field
•Spherical harmonics
Perturbations – disturbance torques
1 1 3 2 2 3 1
2 2 1 3 1 3 2
3 3 2 1 1 2 3
( )
( )
( )
I I I M
I I I M
I I I M
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AstroNet-II
The Astrodynamics Network
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
GOES-J solar pressure boom
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:
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.
ft
t
dxx dtML
0
ft
t
dyy dtML
0
ft
t
dzz dtML
0
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Objectives
1 – Detumbling –
Dynamic equations
2 – Repointing –
Dynamics and kinematics
Tumbling motion must be stabilised
or mission will fail.
Reorient spacecraft to target
specific point (e.g. point antenna to
ground station, point solar cells
towards sun for maximum power.)
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12
3
1 1 3 2 2 3 1
2 2 1 3 1 3 2
3 3 2 1 1 2 3
( )
( )
( )
I I I T
I I I T
I I I T
Use proportional control
•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)
Detumbling with thrusters
1 1 1 2 2 2 3 3 3, , .T k T k T k
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With magnetorquers controllable parameter is magnetic dipole N:-
Convention is to use BDOT, which uses rate of change of magnetic field as control
(requires a magnetometer and Kalman filter):-
BDOT used on several magnetically actuated spacecraft (e.g. Compass-1, UKube-1.).
T N B
(K ) N B
Detumbling with magnetic torquers
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AstroNet-II
The Astrodynamics Network
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
classical PID controller
0( ) ( ) ( ) ( )
t
p i d
du t K e t K e d K e t
dt
1 1 2 2 3 3( ) [ , , ]T
d d de t q q q q q q
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0( ) ( ) ( ) ( )
t
p i d
du t K e t K e d K e t
dt
Parameter Overshoot Settling time Steady-state error
-
Eliminates
-
p
i
d
K
K
K
Estimated to be used in 95% of industrial processes.
1 1 3 2 2 3 1
2 2 1 3 1 3 2
3 3 2 1 1 2 3
( )
( )
( )
I I I T
I I I T
I I I T
1 2 3( ) [ , , ]Tu t T T T
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Full nonlinear model – quaternion representation
1 1 3 2 2 3 1 1
2 2 1 3 1 3 2 2
3 3 2 1 1 2 3 3
( )
( )
( )
I I I M T
I I I M T
I I I M T
3 2 11 1
3 1 22 2
2 1 33 3
1 2 34 4
0
01
02
0
q q
q q
q q
q q
Re-pointing problem – find that drives the spacecraft from 1 2 3, ,T T T
1 2 3 4 1 2 3(0) [ (0), (0), (0). (0)] , (0) [ (0), (0), (0)] ,T Tq q q q q
1 2 3 4 1 2 3( ) [ ( ). ( ), ( ). ( )] , (0) [ ( ), ( ), ( )] ,T Tq T q T q T q T q T q T T T to
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A simple quaternion feedback controller (rest-to-rest)
1 1 1 1 1
2 2 2 2 2
3 3 3 3 3
T k e c
T k e c
T k e c
where 4 3 2 11 1
3 4 1 22 2
2 1 4 33 3
1 2 3 44 4
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
( ) ( ) ( ) ( )
q T q T q T q Te q
q T q T q T q Te q
q T q T q T q Te q
q T q T q T q Te q
Algorithm is flight tested and stability can be verified. Requires tuning of the
parameters:
Can use adaptive tuning methods e.g fuzzy logic
1 1 2 2 3 3, , , , ,k c k c k c
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The Astrodynamics Network
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
Controller Spacecraft dynamics
qspacecraft
Sensors
<q>spacecraft
Actuators
disturbance torques (design stage 2:
quantify disturbance
environment)
Attitude determination
Attitude control
Estimator
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AstroNet-II
The Astrodynamics Network
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
Controller Spacecraft dynamics
qspacecraft
Sensors
<q>spacecraft
Actuators
disturbance torques (design stage 2:
quantify disturbance
environment)
Attitude determination
Attitude control
Estimator
Motion
Planner
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The Astrodynamics Network
“Open” telescope (noexternal baffling) OTAallows passivecooling to ~50K
DeployablesecondaryMirror (SM)
Beryllium
Primary mirror (PM)
Spacecraft support module SSM (attitude control,communications, power,data handling)
warm side
Science
Instruments
(ISIM)
Large (200m2) deployablesunshield protects from sun,earth and moon IR radiation(ISS)
Isolation truss
cold side
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.
next generation space telescope (NGST) UKube
Rapideye
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The Astrodynamics Network