AAE 666 Final Presentation

1. AAE 666 Final Presentation Spacecraft Attitude Control Justin Smith Chieh-Min Ooi April 30, 2005

2. 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

3. 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

4. Euler’s Equations

5. 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]Tvia the relations q0 = cos(Φ/2), qi = ei sin(Φ/2), i = 1, 2, 3

6. 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:

7. 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

8. Equations of Motion • Define parameters to simplify equations:

9. Equations of Motion

10. 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

11. Equations of Motion

12. Globally Stable Non-Linear Spacecraft Attitude Regulator • An asymptotically stabilizing feedback regulator is defined by:

13. Design Parameters

14. Globally Stable Linear Spacecraft Attitude Regulator for i=1,2,3, where ki > 0, c > 0 are constant control gains

15. 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

16. Simulation Parameters

17. Simulation Results (Linear Regulator)

18. Simulation Results (Linear Regulator)

19. Simulation Results (Non-linear Regulator)

20. Simulation Results (Non-linear Regulator)

21. 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

22. 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

23. 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 pjЄ(1/2,1) or πjЄ(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

24. 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

25. Questions???