2/12/20

Spacecraft Attitude Control

Space System Design, MAE 342, Princeton University
Robert Stengel

• More on Rotation Matrices

• Direction cosine matrix
• Quaternions
• Yo-yo De-Spin
• Continuously Variable Torque Controllers
• On/Off-Torque Controllers

Copyright 2016 by Robert Stengel. All rights reserved. For educational use only.
http://www.princeton.edu/~stengel/MAE342.html 1

Attitude Control System

Fortescue
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UARS Attitude Control System

Spacecraft Attitude Control Inputs

• On-Board Sensors
– Inertial Measurements
• Accelerometers
• Angle sensors
• Angular-rate sensors
– Optical Sensors
• Star sensors
• Sun sensors
• Horizon sensors
• Off-Board Observations
– Ground-Based Tracking
• Radar
• Navigation beacons (VOR/DME, LORAN, …)
– Spaced-Based Tracking
• GPS, GLONASS, …
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Potential Accuracies of External

Attitude Measurements

Fortescue
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Spacecraft Attitude Control Outputs

• Continuous Control Torques
– Control Moment/Reaction Wheel Gyros
– Magnetic Torquers
– Solar Panels
• Pulsed Control Torques
– Reaction Control Thrusters (RCS)
• One-Shot Devices
– RCS Spin-up
– Yo-Yo De-Spin
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Spacecraft Attitude Disturbances

• External Torques
– Solar radiation pressure
– Gravity gradient
– Magnetic fields
– Aerodynamics
– Can be put to good use if related to
attitude control objectives
• Vehicle-Based Torques
– Mass movement
– Elasticity
– Out-gassing
7

More on Rotation
Matrices and Quaternions

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Direction Cosine Matrix

§ Cosines of angles
between each I axis
and each B axis
§ Projections of vector
components in one
frame on the other

⎡ cos δ11 cos δ 21 cos δ 31 ⎤

⎢ ⎥
H BI = ⎢ cos δ12 cos δ 22 cos δ 32 ⎥
⎢ cos δ cos δ 23 cos δ 33 ⎥
⎣ 13
⎦
rB = H BI rI 9

Euler’s Rotation Formula

Angular orientation of one axis
system, B, with respect to another, I
Vector transformation

rB = H BI rI
= ( aT rI ) a + ⎡⎣ rI − ( aT rI ) a ⎤⎦ cos φ + sin φ ( rI × a )
= cos φ rI + (1− cos φ ) ( aT rI ) a − sin φ ( a × rI )

⎡ a1 ⎤
⎢ ⎥
a ! ⎢ a2 ⎥
⎢ a ⎥
⎣ 3 ⎦ 10

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Euler’s Formula
rB = H BI rI = cos φ rI + (1− cos φ ) ( aT rI ) a − sin φ ( a × rI )
Identity

( a r ) a = ( aa ) r
T
I
T
I

Rotation matrix

H BI = cos φ I 3 + (1− cos φ ) aaT − sin φ a!

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Quaternion Derived from Euler

Rotation Angle and Orientation
Quaternion vector
4 parameters based on Euler’s formula

⎡ ⎡ ⎛ a1 ⎞ ⎤
q1 ⎤ ⎢ ⎥
⎢ ⎥
⎢ q2 ⎥ ⎡ aφ ⎤ ⎢ sin (φ 2 ) ⎜ a2 ⎟ ⎥
⎥=⎢ ⎜ ⎟ ⎥
q=⎢ !⎢
q3 ⎥ ⎢ q4 ⎥⎦ ⎢ ⎜ a3
⎝
⎟
⎠ ⎥
⎢ ⎥ ⎣ ⎢ ⎥
⎢⎣ q4 ⎥
⎦ ⎢⎣ cos (φ 2 ) ⎥⎦
§ Not singular at θ = ±90°
§ 4-parameter representation of 3 parameters;
hence, a constraint must be satisfied

q T q = q12 + q2 2 + q32 + q4 2
= sin 2 (φ 2 ) + cos 2 (φ 2 ) = 1 12

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Rotation Matrix Expressed with

Quaternion
From Euler’s formula

( )
H BI = ⎡⎣ q4 2 − aφ T aφ ⎤⎦ I 3 + 2aφ aφ T − 2q4 a! φ
Rotation matrix from quaternion
H BI =
⎡ q2 − q2 − q2 + q2 2 ( q1q2 + q3q4 ) 2 ( q1q3 − q2 q4 ) ⎤
⎢ 1 2 3 4
⎥
⎢ 2 ( q1q2 − q3q4 ) −q12 + q22 − q32 + q42 2 ( q2 q3 + q1q4 ) ⎥
⎢ ⎥
⎢ 2 ( q1q3 + q2 q4 ) 2 ( q2 q3 − q1q4 ) −q12 − q22 + q32 + q42 ⎥
⎣ ⎦
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Quaternion Expressed from

Elements of Rotation Matrix
1
q4 = 1+ h11 + h22 + h33
2
Assuming that q4 ≠ 0

⎡ q1 ⎤ ⎡ (h − h ) ⎤
1 ⎢ ⎥
23 32
⎢ ⎥
aφ = ⎢ q2 ⎥ = ⎢ ( h31 − h13 ) ⎥
⎢ q ⎥ 4q ⎢ ⎥
⎢ ( h12 − h21 )
4
⎣ 3
⎦ ⎥
⎣ ⎦

Pisacane, 2005 14

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Successive Rotations Expressed

by Products of Quaternions and
Rotation Matrices
Rotation from Frame q BA : Rotation from A to B
A to Frame C through qCB : Rotation from B to C
Intermediate Frame B
qCA : Rotation from A to C
Matrix Multiplication Rule

HCA ( qCA ) = HCB ( qCB ) H BA ( q BA )

Quaternion Multiplication Rule
⎡ ⎤
( ( )
q4 )B aφ BA + ( q4 )A aφ CB − a" φ aφ BA
C C B C
⎡ aφ ⎤ ⎢ ⎥
qCA = ⎢ ⎥ = qCB q BA ! ⎢ B
⎥
⎢⎣ q4 ⎥⎦ ( ( )
q4 )B ( q4 )A − aφ CB aφ BA
C B T
⎢ ⎥
A
⎣ ⎦
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Quaternion Vector Kinematics

ODE is linear in both q and ωB

d ⎡ aφ ⎤ 1 ⎡ q4 ω B − ω" B aφ ⎤
q! = ⎢ ⎥= ⎢ ⎥
dt ⎢ q4
⎣ ⎥⎦ 2 ⎢ −ω BT aφ ⎥
⎣ ⎦

⎡ q!1 ⎤ ⎡ 0 ωz −ω y ω x ⎤ ⎡ q1 ⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢ q!2 ⎥ 1 ⎢ −ω z 0 ωx ωy ⎥ ⎢ q2 ⎥
⎢ = ⎢ ⎥
q! 3 ⎥ 2 ⎢ ω y −ω x 0 ωz ⎥ ⎢ q3 ⎥
⎢ ⎥ ⎢ ⎥
⎢⎣ q! 4 ⎥ ⎢ −ω −ω y −ω z 0 ⎥ ⎢⎣ q4 ⎥
⎦ ⎣ x
⎦B ⎦
Pisacane, 2005 16

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Propagate Quaternion Vector

⎡ q!1 ( t ) ⎤ ⎡ 0 ω z ( t ) −ω y ( t ) ω x ( t ) ⎤ ⎡ q1 ( t ) ⎤
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
dq ( t ) ⎢ q!2 ( t ) ⎥ 1 ⎢ −ω z ( t ) 0 ω x (t ) ω y (t ) ⎥ ⎢ q2 ( t ) ⎥
=⎢ ⎥= ⎢ ⎥ ⎢ ⎥
dt ⎢ q! 3 ( t ) ⎥ 2 ⎢ ω y ( t ) −ω x ( t ) 0 ω z (t ) ⎥ ⎢ q3 ( t ) ⎥
⎢ ⎢ ⎥ ⎢
⎢⎣ q! 4 ( t ) ⎥⎥ ⎢⎣ −ω x ( t ) −ω y ( t ) −ω z ( t ) 0 ⎥⎦ B ⎢⎣ q4 ( t ) ⎥⎥
⎦ ⎦

Digital integration to compute q(tk)

dq (τ )
tk

q int ( t k ) = q ( t k−1 ) + ∫t dt dτ
k−1

Normalize q(tk) to enforce constraint

q ( t k ) = q int ( t k ) q int T ( t k ) q int ( t k )

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Quaternion Interface with Euler Angles

• Quaternion and its kinematics unaffected by Euler angle convention
• Definition of HIB makes the connection
• Specify Euler angle convention (e.g., 1-2-3 or 3-1-3) ; for (1-2-3),

B
⎡ h11 h12 h13 ⎤
⎢ ⎥
H I = ⎢ h21 h22
B
h23 ⎥
⎢ h h h33 ⎥
⎣ 31 32 ⎦I
⎡ cosθ cosψ cosθ sinψ − sin θ ⎤
⎢ ⎥
= ⎢ − cos φ sinψ + sin φ sin θ cosψ cos φ cosψ + sin φ sin θ sinψ sin φ cosθ ⎥
⎢ sin φ sinψ + cos φ sin θ cosψ − sin φ cosψ + cos φ sin θ sinψ cos φ cosθ ⎥
⎣ ⎦
• Apply equations on earlier slide to find q(0)
• Perform trigonometric inversions as indicated to generate
[Φ(tk), θ(tk),Ψ(tk)] from q(tk)
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Yo-Yo De-Spin

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Mars Odyssey Launch Phases

Booster Separation Stage 2 Separation Stage 2 Ignition

Heat Shield Separation

Stage 3 Spinup Yo-Yo De-Spin

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Yo-Yo De-spin

Kaplan

• Satellite is initially spinning at ωz rad/s

• Angular momentum and rotational energy of satellite
plus expendable masses are conserved
• Masses are released, moment of inertia increases, and
angular velocity of satellite decreases
• With proper cord length (independent of initial spin
rate), satellite is de-spun to zero angular velocity

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Yo-Yo De-spin
Angular momentum
R = spacecraft radius
hz = I zzω z + mR 2 ⎡⎣ω z + φ 2 (ω z + φ! ) ⎤⎦ l = tether length
mR 2 + I zz
c=
Rotational energy mR 2
2

m = mass of 2 deployable objects

I zzω z 2 + mR 2 ⎡ω z 2 + φ 2 (ω z + φ! ) ⎤
1 1 2
T= I zz = satellite moment of inertia
2 2 ⎣ ⎦ φ = angle between split hinge and tangent point

Simultaneous solution for final angular rate

⎛ cR 2 − l 2 ⎞
ω final = ω initial ⎜ 2 2 ⎟ = 0 if l=R c
⎝ cR + l ⎠
Spaceloft 7 Sounding Rocket De-Spin
https://www.youtube.com/watch?v=5ZqbjQ9ASc8
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Continuously Variable
Torque Controllers

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Overview of Control

Single- or multi-axis interpretation

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Single-Axis “Classical”
Control of Non-Spinning
Spacecraft
Pitching motion (about the y axis) is to be controlled

⎡ p! ( t )
⎢
⎤ ⎡ M x ( t ) / I xx
⎥ ⎢ ⎥ ⎢
( )
⎤ ⎡ I zz − I yy q ( t ) r ( t ) / I xx ⎤
⎥
⎢ q! ( t ) ⎥ = ⎢ M y ( t ) / I yy ⎥ − ⎢ ( I xx − I zz ) p ( t ) r ( t ) / I yy ⎥
⎢ ⎥ ⎢ ⎥ ⎢ ⎥
⎢⎣ r! ( t ) ⎥⎦ ⎢⎣ M z ( t ) / I zz ( )
⎥⎦ ⎢ I yy − I xx p ( t ) q ( t ) / I zz
⎢⎣
⎥
⎥⎦
• For motion about the y axis q! ( t ) = M y ( t ) / I yy
only, this reduces to
• Pitch angle equation θ! ( t ) = q ( t )
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Single-Axis Angular Rate Control of

Non-Spinning Spacecraft
• Small angle and angular rate perturbations
• Linear actuator, e.g.,
– Momentum wheel
• Linear measurement, e.g.,
– Angular rate gyro

Simplified Control Law (C = Control Gain)

e(t) = qc (t) − q(t)

u(t) = C e(t) 26

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Angular Rate Control

t t t
g Cg Cg
q(t) = A ∫ u(t) dt = A ∫ e(t) dt = A ∫ [ qc − q(t)] dt
I yy 0 I yy 0 I yy 0

• Iyy: moment of inertia

• q(t): angular rate
• qc(t): desired angular rate
• gA: actuator gain
• gAu(t): control torque

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Step Response of Angular Rate

Controller
Step input :
⎧⎪ 0, t < 0
qc (t) = ⎨
1, t ≥ 0
⎩⎪

⎡ −⎜ A ⎟ t ⎤
⎛ Cg ⎞

q(t) = qc ⎢1− e ⎝ yy ⎠ ⎥ = qc ⎡⎣1− eλt ⎤⎦ = qc ⎡⎢1− e τ ⎤⎥

I −t

⎢ ⎥ ⎣ ⎦
⎣ ⎦

• where
– λ = –CgA/Iyy = eigenvalue or
root of the system (rad/s)
– τ = Iyy/CgA = time constant of
the response (s)

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Angle Control of the Spacecraft

• Small angle and angular rate perturbations
• Linear actuator, e.g.,
– Momentum wheel
• Linear measurement, e.g.,
– Earth horizon sensor

Angle Control Law (C = Control Gain)

e(t) = θ c (t) − θ (t)

u(t) = C e(t)
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Model of Dynamics and Angle Control

• 2nd-order ordinary differential equation
d 2θ (t) !! Cg
= θ (t) = A [θ c − θ (t)]
dt 2
I yy

• Output angle, θ(t), as a function of time

t t t t t t
g Cg Cg
θ (t) = A ∫ ∫ u(t) dt dt = A ∫ ∫ e(t) dt dt = A ∫ ∫ [θ c − θ (t)] dt dt
I yy 0 0 I yy 0 0 I yy 0 0

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Rewrite 2nd-Order Model as Two

1st-Order Equations
θ!(t) = q(t)
Cg
! = A [θ c − θ (t)]
q(t)
I yy

⎡ θ!(t) ⎤ ⎡ 0 1 ⎤ ⎡ θ (t) ⎤ ⎡ 0 ⎤
⎢ ⎥=⎢ ⎥⎢ ⎥+⎢ ⎥ C [θ c (t) − θ (t)]
!
⎢⎣ q(t) ⎥⎦ ⎣ 0 0 ⎦ ⎢⎣ q(t) ⎥⎦ ⎢⎣ gA / I yy ⎥⎦

⎡ θ!(t) ⎤ ⎡ 0 1 ⎤ ⎡ θ (t) ⎤ ⎡ 0 ⎤
⎢ ⎥=⎢ ⎥⎢ ⎥+⎢ ⎥θ c
!
⎢⎣ q(t) ⎥⎦ ⎢⎣ −CgA / I yy 0 ⎥ ⎢ q(t)
⎦⎣ ⎥⎦ ⎢⎣ CgA / I yy ⎥⎦
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Simulation of Step Response

with Angle Feedback
Objective is to control angle to 1 rad, but
solution oscillates about the target

CgA/Iyy = 1, 0.5, and 0.25 32

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What Went Wrong?

• No damping!
• Solution: Add rate feedback

u(t) = c1 [θ c (t) − θ (t)] − c 2q(t)

• Control law with
rate feedback

Closed-loop dynamic equation

⎡ θ!(t) ⎤ ⎡ 0 1 ⎤ ⎡ θ (t) ⎤ ⎡ 0 ⎤
⎢ ⎥=⎢ ⎥⎢ ⎥+⎢ ⎥θ c
!
⎢⎣ q(t) ⎥⎦ ⎢⎣ −c1gA / I yy −c2 gA / I yy ⎥ ⎢ q(t) ⎥ ⎢ c1gA / I yy
⎦⎣ ⎦ ⎣ ⎥⎦
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Step Response with Angle

and Rate Feedback

c1gA /Iyy = 1
c2gA /Iyy = 0, 1.414, 2.828
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2nd-Order Dynamics
Oscillation and damping are induced by linear
feedback control

⎡ θ!(t) ⎤ ⎡ 0 1 ⎤ ⎡ θ (t) ⎤ ⎡ 0 ⎤
⎢ ⎥=⎢ ⎥⎢ ⎥+⎢ ⎥θ c
!
⎢⎣ q(t) ⎥⎦ ⎢⎣ −c1gA / I yy −c2 gA / I yy ⎥ ⎢ q(t)
⎦⎣ ⎥⎦ ⎢⎣ c1gA / I yy ⎥⎦
⎡ θ!(t) ⎤ ⎡ 0 1 ⎤ ⎡ θ (t) ⎤ ⎡ 0 ⎤
⎢ ⎥=⎢ ⎥⎢ ⎥+⎢ 2 ⎥θ c
⎥⎦ ⎢⎣ −ω n −2ζω n ⎥ ⎢ q(t) ⎥⎦ ⎢⎣ ω n ⎥⎦
2
!
⎢⎣ q(t) ⎦⎣
Natural frequency and damping ratio

ω n = c1gA / I yy

( )
ζ = c2 gA / I yy / 2ω n = c2 / 2 c 1 gA I yy
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Effect of Damping on Eigenvalues,

Damping Ratio, and Natural Frequency
c1gA /Iyy = 1
c2gA /Iyy = 0, 1.414, 2.828

Eigenvalues Damping Ratio, Natural Frequency

λ1, λ2 = ζ= ωn = (rad/s)
0 + 1.0000i 0 1
0 - 1.0000i

-0.7070 + 0.7072i 0.707 1

-0.7070 - 0.7072i

-0.4143 Overdamped
-2.4137

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Control System Design to

Adjust Roots
Choose control gains to satisfy desirable eigenvalue
range

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Control System Design to Adjust

Transient Response
Choose control gains to satisfy step response criteria

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Control System Design to Adjust

Frequency Response
Choose control gains to satisfy frequency response
criteria

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Laplace Transform of the State Vector

Neglecting the initial condition

Adj ( sI − F )
x(s) = G u(s)
Δ(s)
Applied to the closed-loop system

⎡ c1gA I yy ⎤ ⎡ c1gA I yy ⎤
⎢ ⎥ ⎢ ⎥ Δu(s)
⎡ Δθ (s) ⎤ ⎢ sc1gA / I yy ⎥ ⎢ sc1gA / I yy ⎥
⎣ ⎦ Δu(s) = ⎣ ⎦
⎢ ⎥=
⎢⎣ Δq(s) ⎥
⎦ Δ(s) 2
( )
( s ) + c2 gA I yy ( s ) + c1gA I yy
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Frequency Response of the System

σ = jω
Angle Frequency Response

Δθ ( jω ) ωn 2
=
Δu( jω ) ( jω ) 2 + 2ζω n ( jω ) + ω n 2

Rate Frequency Response

Δq( jω )
=
( jω ) ω n 2

Δu( jω ) ( jω ) + 2ζω n ( jω ) + ω n 2
2

• Bode plot
– 20 log(Amplitude Ratio) [dB] vs. log ω
– Phase angle (deg) vs. log ω

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Proportional-Integral-
Derivative (PID) Controller

e(t) = θ C (t) − θ (t)

PID Control Law
de(t)
(or compensator): u(t) = cI ∫ e(t) dt + cP e(t) + cD
dt 42

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Proportional-Integral-
Derivative (PID) Controller
Control Law Transfer Function:

e(s) = θ C (s) − θ (s)

e(s)
u(s) = cP e(s) + cI + cD se(s)
s
u(s) cI + cP s + cD s 2
=
e(s) s
Differentiator produces rate term for damping
Integrator compensates for persistent (bias) disturbance
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Proportional-Integral-Derivative
(PID) Controller
Forward-Loop Angle θ (s) ⎡ cI + cP s + cD s ⎤ ⎡ gA ⎤
2
= ⎥ ⎢ I s2 ⎥
Transfer Function: e(s) ⎢⎣ s ⎦ ⎣ yy ⎦

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Closed-Loop Spacecraft Control

Transfer Function w/PID Control

θ (s) ⎡ c I + cP s + c D s 2 ⎤
⎢ gA ⎥
θ (s) e(s) ⎣ I yy s 3 ⎦
= =
Closed-Loop Angle θ c (s) 1+ θ (s) ⎡ c I + cP s + c D s 2 ⎤
Transfer Function: e(s) 1+ ⎢ I yy s 3
gA ⎥
⎣ ⎦
c I + cP s + c D s 2
=
cI + cP s + cD s 2 + gA / I yy s 3 45

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Closed-Loop Frequency
Response w/PID Control
θ (s) c I + cP s + c D s 2
=
θ c (s) cI + cP s + cD s 2 + gA / I yy s 3

Let s = jω. As ω -> 0

θ ( jω ) c
→ I =1 Steady-state output =
θ c ( jω ) cI desired steady-state input

As ω -> ∞
θ ( jω ) −cDω 2 c jc
→ g = D gA = − D gA High-frequency
θ c ( jω ) − j I yyω 3 A
j I yyω I yyω response “rolls off”
cD and lags input
AR → g ; φ → −90 deg
I yyω A 46

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State (“Phase”)-Plane Plots

Cross-plot of angle (or displacement) against rate

Time not shown explicitly in phase-plane plot

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Effect of Damping Ratio

on State-Plane Plots
Damping ratio = 0.1 Damping ratio = 0.3 Damping ratio = –0.1
(Unstable)

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On/Off-Torque
Controllers

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Single-Axis State History

with Constant Thrust
What if the control torque can only be turned ON or OFF?

⎡ θ!(t) ⎤ ⎡ 0 1 ⎤ ⎡ θ (t) ⎤ ⎡ 0 ⎤ u (t ) =
⎢ ⎥=⎢ ⎥ ⎢ q(t) ⎥ + ⎢ g / I ⎥ u(t) +1, 0, or − 1
!
⎢⎣ q(t) ⎥⎦ ⎣ 0 0 ⎦ ⎢⎣ ⎥⎦ ⎣⎢ A yy ⎥⎦
What is the time evolution of the state while a thruster is
on [u(t) = 1]?

( )
q(t) = gA / I yy t + q(0)

( )
θ (t) = gA / I yy t 2 / 2 + q(0)t + θ (0)
Neglecting initial conditions, what does
the phase-plane plot look like? 50

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Constant-Thrust
(Acceleration) Trajectories
For u = 1, For u = –1,
Acceleration = gA/Iyy Acceleration = –gA/Iyy

Thrusting away from the origin Thrusting to the origin

With zero thrust, what does the phase-plane plot look like?
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Phase Plane Plot with Zero

Thrust

How can you use this information to design

an on-off control law?
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Switching-Curve Control
Law for On-Off Thrusters

• Origin (i.e., zero rate

and attitude error)
can be reached
from any point in
the state space
• Control logic:
– Thrust in one
direction until
switching curve is
reached
– Then reverse
thrust
– Switch thrust off
when errors are
zero
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Switching-Curve Control with

Coasting Zone

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Apollo Lunar Module Control

• 16 reaction control thrusters
– Control about 3 axes
– Redundancy of thrusters
• LM Digital Autopilot

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Apollo Lunar Module

Phase-Plane Control Logic

• Coast zones conserve RCS propellant by limiting angular rate

• With no coast zone, thrusters would chatter on and off at
origin, wasting propellant
• State limit cycles about target attitude
• Switching curve shapes modified to provide robustness
against modeling errors
– RCS thrust level
– Moment of inertia 56

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Apollo Lunar Module Phase-

Plane Control Law

Switching logic implemented in the Apollo

Guidance & Control Computer
More efficient than a linear control law for on-off
actuators

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Typical Phase-Plane Trajectory

• With angle error, RCS turned on until reaching OFF

switching curve
• Phase point drifts until reaching ON switching curve
• RCS turned off when rate is 0-
• Limit cycle maintained with minimum-impulse RCS firings
– Amplitude = ±1 deg (coarse), ±0.1 deg (fine)
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Multi-Axis Spacecraft Control

Asymmetry Introduces Dynamic Coupling, Complicating Control

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Next Time:
Sensors and Actuators

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Supplemental Material

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GOES Attitude Control Sub-System

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