1. Constraint Based Control Method For Precision Formation Flight of Spacecraft AAS 06-122
Try Lam
Jet Propulsion Laboratory
California Institute of Technology
2006 AAS/AIAA Space Flight Mechanics Meeting
Tampa, Florida, January 2006
Aaron Schutte
Aerospace Corporation
Firdaus E. Udwadia
University of Southern California

2. 2006 AAS/AIAA Space Flight Mechanics Meeting
Agenda
Introduction
Theory
Formation Flying Example
Constraints
Numerical Results
Conclusion
1/23/2006
2006 AAS/AIAA Space Flight Mechanics Meeting

3. 2006 AAS/AIAA Space Flight Mechanics Meeting
Introduction
We introduce a very general constraint based control methodology and apply it to the problem of controlling multiple spacecraft in precise formation
Method is simple to implement and equations are explicit
Examples will be given for 2 different formation types, a 2 spacecraft formation and a 4 spacecraft formation
Objective is to keep the spacecraft in a very precise formation and analyze its dynamics
1/23/2006
2006 AAS/AIAA Space Flight Mechanics Meeting

4. Theory: EOM for Constrained System
In general, the equations of motion of a system that is perturbed from its natural state has the form of
where FC is the perturbing or constraint force (Lagrange Multiplier approach).
The task is to find FC and there are a number of ways to solve for it depending on problem. Analytically, the problem is solved by the various form of the Fundamental Equation. The most famous is probably the concept of Virtual Work or the use of Lagrange multipliers.
We apply here another form of the Fundamental Equation of Lagrangian mechanics, which we will call “the Fundamental Equation”, based on Gauss’s principle of Least Constraint (Udwadia and Kalaba [1991])
1/23/2006
2006 AAS/AIAA Space Flight Mechanics Meeting

5. Theory: EOM for Constrained System II
Udwadia’s and Kalaba’s form of the Fundamental Equation:
F, a = free-response force and acceleration
M = diagonal mass matrix
A = from the constraint equation (will be discussed later)
b = from the constraint equation (will be discussed later)
( )+ = pseudo-inverse
All constraints are solved in a least square sense analytically at every time step
Weighted Feedback Gain
Acceleration Error Signal
1/23/2006
2006 AAS/AIAA Space Flight Mechanics Meeting

6. Formation Flying Example
Circular Orbit
378 km Altitude
80 deg inclination
Mars with
4 x 4
Harmonic
Field
2-SC Formation: SC#1,3
4-SC Formation: SC#1,2,3,4
1/23/2006
2006 AAS/AIAA Space Flight Mechanics Meeting

7. Formation Flying Example: Constraints
Constraints applied between the spacecraft
Relative Distance Constraints
Relative Radial Distance Constraints
For precision we use Baumgarte’s stabilization technique
Formation behaves as a “virtual” rigid body
1/23/2006
2006 AAS/AIAA Space Flight Mechanics Meeting

8. Formation Flying Example: Finding A & b
m constraints
Differentiate twice for position constraints
Differentiate once for velocity constraints
As is for acceleration constraints
Example:
Differentiate twice A b
1/23/2006
2006 AAS/AIAA Space Flight Mechanics Meeting

9. Two-Spacecraft Example: Stabilization
Relative Distance Between the 2 Spacecraft
Without Stabilization
With Stabilization
1/23/2006
2006 AAS/AIAA Space Flight Mechanics Meeting

10. Two-Spacecraft Example: Simulation
Circular Orbit
378 km Altitude
80 deg inclination
Relative Distance Error (< 1E-12) 1 2 4
Thrust
1/23/2006
2006 AAS/AIAA Space Flight Mechanics Meeting

11. Four-Spacecraft Example: Simulation
Circular Orbit
378 km Altitude
80 deg inclination
Thrust: SC 1&3 1 2 3 4
Thrust: SC 2&4
1/23/2006
2006 AAS/AIAA Space Flight Mechanics Meeting

12. Four-Spacecraft Example: Pulsating
Time-varying constraint of the form
1/23/2006
2006 AAS/AIAA Space Flight Mechanics Meeting

13. Four-Spacecraft Example: Pulsating II
Thrust
Circular Orbit
378 km Altitude
80 deg inclination 1 2 3 4
Thrust (zoom)
1/23/2006
2006 AAS/AIAA Space Flight Mechanics Meeting

14. Conclusion and Future Work
Introduced and applied different approach to the control of multiple spacecraft in precision formation flight
Method is based on a new form of the fundamental equations of motion for constrained system (in Lagrangian mechanic) using Gauss’s principle
Solutions are explicit and equations are simple to derive
Future Work:
Improve numerical implementation
Add switching function to the thrusting
1/23/2006
2006 AAS/AIAA Space Flight Mechanics Meeting

15. 2006 AAS/AIAA Space Flight Mechanics Meeting
BACK UP
CHARTS
1/23/2006
2006 AAS/AIAA Space Flight Mechanics Meeting

16. Theory: EOM for Constrained System III
Gauss’s principle of Least Constraint states that of all possible acceleration (including those that are non-physical) for the system, the “actual” acceleration is one which minimize the Gaussian
Handling constraints
Given a set of differentiable kinematic constraints
We can differentiate Eq. (2) to be of the form
holonomic or non-holonomic constraints
Gaussian looks like the weighted least square solution with M as the weighted matrix.
1/23/2006
2006 AAS/AIAA Space Flight Mechanics Meeting

17. Theory: EOM for Constrained System IV
Consider a system of n particles, then the free-response motion is
Given a set of differentialable constraints
We can differentiate (2) to be of the form
The constraint force is found to be (next slide)
Thus,
(1)
(2)
(3)
(4)
(5)
1/23/2006
2006 AAS/AIAA Space Flight Mechanics Meeting

18. Formation Flying Example: Formation
Circular obits around a body
Direction of Motion
Direction of Motion
To Planet
To Planet
Two-Spacecraft Formation
Four-Spacecraft Formation
1/23/2006
2006 AAS/AIAA Space Flight Mechanics Meeting