Time Adaptive Discrete Mechanics and Optimal Control

Abstract

Space mission design is often achieved through a combination of dynamical systems theory and optimal control. This work focuses on how to adapt DMOC, a method devised with a constant step size, for the highly nonlinear dynamics involved in space problems including trajectory design and reconfiguration and docking of formation flying cubesats, similar to those proposed for the KISS project’s reconfigurable modular space telescope. A time adaptive form of DMOC is developed that allows for a variable step size that is updated throughout the optimization process. Time adapted DMOC is based on a discretization of Hamilton’s principle applied to the time adapted Lagrangian of the optimal control problem. Variations of the discrete action of the optimal control Lagrangian lead to discrete Euler-Lagrange equations that can be enforced as constraints for a boundary value problem. This new form of DMOC leads to the accurate and efficient solution of optimal control problems with highly nonlinear dynamics. Time adapted DMOC is tested on several space trajectory problems including the elliptical orbit transfer in the 2-body problem and the reconfiguration of a cubesat.