Variational Model for Optimization of Finite-Burn Escape Trajectories Using a Direct Method

Abstract

A variational model is used to obtain the necessary gradient information for a multiple finite-burn trajectory problem. The augmented state vector includes position, velocity, mass, and control variables such as thrust magnitude, and the variables that describe the time evolution of the thrust direction unit vector. The augmented state vector and its associated state equation are used in the variational model to formulate analytical expressions for the gradients of the problem functions with respect to the problem variables. A general finite-burn trajectory problem is formulated for an inertially fixed thrust vector steering model and a fixed-plane linearly varying thrust vector steering model that constrains the thrust vector to be normal to a rotation axis, which itself is part of the control variable set. Both steering models are used in the optimization of a three-finite-burn lunar-escape-trajectory example, using a direct optimization method with explicit numerical integration. The performance of the optimization procedure is compared for both thrust vector steeringmodels, using variational gradients and standard numerical finite difference gradients. The ability to obtain the most accurate gradient information for the direct optimization of simple finite-burn thrust steering models is a motivation of the current study.