Constraint Based Control Method For Precision Formation Flight of Spacecraft AAS

Transcription

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

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

3 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/ AAS/AIAA Space Flight Mechanics Meeting TL-3

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 M x = F + F C where F C is the perturbing or constraint force (Lagrange Multiplier approach). The task is to find F C 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/ AAS/AIAA Space Flight Mechanics Meeting TL-4

5 Theory: EOM for Constrained System II Udwadia s and Kalaba s form of the Fundamental Equation: M x = F + M 1/ 2 ( AM 1/ 2 ) + ( b Aa) Weighted Feedback Gain Acceleration Error Signal 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 1/23/ AAS/AIAA Space Flight Mechanics Meeting TL-5

6 Formation Flying Example Circular Orbit Z 378 km Altitude 80 deg inclination Mars with 4x4 Harmonic Field Direction of Motion Y 2 2-SC Formation: SC#1,3 X 1/23/ SC Formation: SC#1,2,3, AAS/AIAA Space Flight Mechanics Meeting TL-6

7 Formation Flying Example: Constraints Constraints applied between the spacecraft Relative Distance Constraints φ = L 2 ( x i x ) 2 j + ( y i y ) 2 j + ( z i z ) 2 j = 0 Relative Radial Distance Constraints r i 2 r j 2 = c For precision we use Baumgarte s stabilization technique φ +α φ + βφ = 0 Formation behaves as a virtual rigid body 1/23/ AAS/AIAA Space Flight Mechanics Meeting TL-7

8 Formation Flying Example: Finding A & b m constraints ϕ m ( x, x,t) = 0 A( x, x,t) x = b x, x,t ( ) Differentiate twice for position constraints Differentiate once for velocity constraints As is for acceleration constraints Example: r 1 2 r 2 2 = const. Differentiate twice [ x 1 y 1 z 1 x 2 y 2 z 2 ] x = [ x y z 2 1 x 2 2 y z ] 2 A b 1/23/ AAS/AIAA Space Flight Mechanics Meeting TL-8

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

10 Two-Spacecraft Example: Simulation Circular Orbit 378 km Altitude Relative Distance Error (< 1E-12) 80 deg inclination Spacecraft 4 4 Spacecraft 3 2 Spacecraft 2 2 Spacecraft 1 1 Direction of Motion Thrust 1/23/ AAS/AIAA Space Flight Mechanics Meeting TL-10

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

12 Four-Spacecraft Example: Pulsating Time-varying constraint of the form L t ( ) +1 ( ) = L 0 abs sin( kt) [ ] 1/23/ AAS/AIAA Space Flight Mechanics Meeting TL-12

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

14 Conclusion: 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/ AAS/AIAA Space Flight Mechanics Meeting TL-14

15 BACK UP CHARTS 1/23/ AAS/AIAA Space Flight Mechanics Meeting TL-15

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 ( ) T M( x a) G = x a Handling constraints Given a set of differentiable kinematic constraints φ( x, x,t) = 0 We can differentiate Eq. (2) to be of the form holonomic or non-holonomic constraints A( x, x,t) x = b x, x,t ( ) 1/23/ AAS/AIAA Space Flight Mechanics Meeting TL-16

17 Theory: EOM for Constrained System IV Consider a system of n particles, then the free-response motion is ( ) a = M 1 F x, x,t Given a set of differentialable constraints We can differentiate (2) to be of the form The constraint force is found to be (next slide) φ i ( x, x,t) = 0 A( x, x,t) x = b x, x,t ( ) (1) (2) (3) Thus, F C = M 1/ 2 ( AM 1/ 2 ) + ( b Aa) M x = F + F C (4) (5) 1/23/ AAS/AIAA Space Flight Mechanics Meeting TL-17

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/ AAS/AIAA Space Flight Mechanics Meeting TL-18