Variational Technique for Spacecraft Trajectory Planning

Abstract

This paper describes a spacecraft trajectory planning algorithm based on the calculus of variations which can solve 6-degree-of-freedom spacecraft docking and proximity operations problems. The design of a cost functional which trades off fuel use, obstacle clearance distance, and arrival time is discussed. The nonlinear orbital dynamic equations are treated as dynamic constraints. The Euler-Lagrange equations for this functional are derived, as is the Pontryagin criteria for the optimal control input given realistic saturating on-off thrusters. The indirect collocation method is chosen to solve the attendant boundary-value problem for its lack of sensitivity to initial conditions; continuation is used to further improve the algorithm’s robustness. The manipulation of the Euler-Lagrange equations and the transversality condition into a form suitable for use with existing collocation codes is discussed. Results are shown for an end-to-end docking maneuver with a tumbling satellite.