AAE 666 Final Presentation

Spacecraft Attitude Control

Justin SmithChieh-Min OoiApril 30, 2005

Problem Description

The problem considered is that of designing an attitude regulator for rigid body (spacecraft) attitude regulation

The closed-loop system must have exactly one equilibrium point, namely when the body and inertial coordinate systems coincide

The feedback control law has to be chosen carefully to meet the above requirement of only possessing exactly one equilibrium state for the closed-loop system

Equilibrium point must be asymptotically stable for arbitrary initial conditions, when there are no external disturbing torques acting on the body

Compared results of a linear regulator and a non-linear regulator with a numerical example

Dynamics

Two Cartesian coordinate sets chosen; inertially fixed and body-fixed

Origin taken as mass center These assumptions allow for decoupling of

rotational and translational dynamics Body principal moments of inertia are taken as

the body-fixed axes

Euler’s Equations

.

1 1 2 3 2 3 1

.

2 2 3 1 3 1 2

.

3 3 1 2 1 2 3

( )

( )

( )

I I I u

I I I u

I I I u

Quaternions

Problems of singularity (gimbal lock) do not arise due to absence of trigonometric functions

Any change in orientation can be expressed with a simple rotation

The Euler symmetric parameters may be interpreted in terms of a rotation through an angle Φ about an axis defined by a unit vector e = [e1 e2 e3]T via the relations

q0 = cos(Φ/2), qi = ei sin(Φ/2), i = 1, 2, 3

Quaternions (cont’d)

The quaternion differential equations are

The use of a four-parameter scheme rather than a three-parameter scheme results in redundancy of one of the quarternion parameters. This is evident as every solution of the differential equation above satisfies the constraint:

.

1 1 1 2 2 2 3 3 30

.

1 1 0 3 3 2 2 2 31

.

2 2 0 3 3 1 1 1 32

.

3 3 0 2 2 1 1 1 23

q b H q b H q b H q

q b H q b H q b H q

q b H q b H q b H q

q b H q b H q b H q

2 2 2 20 1 2 3( ) ( ) ( ) ( ) 1q t q t q t q t

Control Inputs

Control torques applied to the three body axes Implemented with throttleable reaction jets or

momentum exchange devices Must include the dynamical equations of the

flywheel, introducing three new state variables 10 first-order, coupled, non-linear differential

equations necessary to describe system State variables include spacecraft angular

momentum components, quaternions, and flywheel angular momentum components

Equations of Motion

Define parameters to simplify equations:

1 11 3 2

1 12 1 3

1 13 2 1

11

1, 1,2,3

2 i

a I I

a I I

a I I

b I i

Equations of Motion

.

1 2 3 3 3 2 2 1

.

2 3 1 1 1 3 3 2

.

3 1 2 2 2 1 1 3

/ /

/ /

/ /

h h H I h H I u

h h H I h H I u

h h H I h H I u

.

1 1 2 3 1

.

2 2 3 1 2

.

3 3 1 2 3

.

1 1 1 2 2 2 3 3 30

.

1 1 0 3 3 2 2 2 31

.

2 2 0 3 3 1 1 1 32

.

3 3 0 2 2 1 1 1 23

H a H H u

H a H H u

H a H H u

q b H q b H q b H q

q b H q b H q b H q

q b H q b H q b H q

q b H q b H q b H q

Controller Design Criterion

Devise a feedback control law relating three control torques to 10 state variables

The closed-loop system must have exactly one equilibrium point

Equilibrium point must be asymptotically stable for arbitrary initial conditions, when there are no external disturbing torques acting on the body

Equilibrium State: h1= h2 = h3 = 0; q0 = 1; q1 = q2 = q3 = 0

Equations of Motion

.

1 2 3 3 3 2 2 1

.

2 3 1 1 1 3 3 2

.

3 1 2 2 2 1 1 3

/ /

/ /

/ /

h h H I h H I u

h h H I h H I u

h h H I h H I u

.

1 1 2 3 1

.

2 2 3 1 2

.

3 3 1 2 3

.

1 1 1 2 2 2 3 3 30

.

1 1 0 3 3 2 2 2 31

.

2 2 0 3 3 1 1 1 32

.

3 3 0 2 2 1 1 1 23

H a H H u

H a H H u

H a H H u

q b H q b H q b H q

q b H q b H q b H q

q b H q b H q b H q

q b H q b H q b H q

Globally Stable Non-Linear Spacecraft Attitude Regulator

An asymptotically stabilizing feedback regulator is defined by:

( , ) [ ( ) ( )], 1, 2,3i i iu q H v q w H i

1

2

2( 1)1

1

2( 1)

1

( ) || ( ) || ,

( ) || || , 0

j

j

mp

i j i eq i eqj

m

i ij ij

with

v q c I q q q q q

w H k H H H

Design Parameters

1

1

2

2

0, 1,2,...,

0, 1,2,...,

0, 1,2,3, 1,2,...,

0, 1,2,...,

j

j

ij

j

c j m

p j m

k i j m

j m

Globally Stable Linear Spacecraft Attitude Regulator

for i=1,2,3, where ki > 0, c > 0 are constant

control gains

( ) ( ) ( )i i i i iu t k H t cb q t

Lyapunov Proof of Stability

Candidate Lyapunov function for the linear regulator case:

Candidate Lyapunov function for the non-linear regulator case:

Global asymptotic stability (GAS) since V is positive definite and Vdot < 0, for all state variables ≠ 0

1

2

21 2

1

3.2( 1)2

1 1

1( , ) || ( ) || || ||

2

( ) || ( ) || , ( ) 0

j

j

mp

j j eqj

m

ij ii j

V q H c p q q H

V k H t H t H t

2 2 2 2 2 2 21 2 3 0 1 2 32 [( 1) ]V H H H c q q q q

3.2

1i i

i

V k H

Simulation Parameters

2 2 21 2 3

(0) 5

1 1 1; ;2 22

(0) (0) 0

1545 ; 340 ; 1518

o

e

H h

I kgm I kgm I kgm

Simulation Results (Linear Regulator)

Simulation Results (Linear Regulator)

Simulation Results (Non-linear Regulator)

Simulation Results (Non-linear Regulator)

Analysis

An initial disturbance in the quaternion parameters causes a disturbance in the spacecraft’s angular momentum and flywheel angular momenta

Flywheel angular momenta directly opposes the angular momentum of the spacecraft to bring it back to its initial attitude

Attitude error, Φ(.), regulates to zero for both linear and non-linear cases

Equilibrium is achieved much faster with the use of non-linear feedback regulators (approx 350 seconds for the non-linear case as compared to approx 3000 seconds for the linear case)

The desired equilibrium state of h1=h2=h3=0; q0=1; q1=q2=q3=0 was successfully achieved

Analysis (Cont’d)

For both linear and non-linear cases, tweaking the gains affects how the system behaves

From Vdot equation previously, it can be seen that large gains will improve stability of the linear regulator

Larger gains result in spacecraft angular momentum and flywheel angular momenta achieving equilibrium conditions faster

Same results could be observed by increasing the gains for the non-linear regulator

Conclusion

The rotational motion of an arbitrary rigid body (spacecraft) subject to control torques may be described by the EOMs defined earlier

If linear feedback control law with constant coefficients is used, the closed-loop system is globally asymptotically stable (GAS)

Lyapunov techniques were used to prove stability For the non-linear feedback regulator, if either pj > 1 or πj

> 1 for some j, then a ‘higher-order’ feedback term is introduced in the control

If pj Є (1/2,1) or πj Є (1/2,1) for some j, then a ‘lower-order’ feedback term is introduced

Lower-order feedback exhibits efficient regulation characteristics near the equilibrium state

References

S.V. Salehi and E.P. Ryan; A non-linear feedback attitude regulator

Richard E. Mortensen; A globally stable linear attitude regulator

Professor K.C. Howell; AAE 440 notes

Questions???