Birkhoff Pseudospectral Method and Convex Programming for Trajectory Optimization

Abstract

AbstractTrajectory optimization, an optimal control problem (OCP) in essence, is an important issue in many engineering applications including space missions, such as orbit insertion of launchers, orbit rescue, formation flying, etc. There exist two kinds of solving methods for OCP, i.e., indirect and direct methods. For some simple OCPs, using the indirect methods can result in analytic solutions, which are not easy to be obtained for complicated systems. Direct methods transcribe an OCPs into a finite-dimensional nonlinear programming (NLP) problem via discretizing the states and the controls at a set of mesh points, which should be carefully designed via compromising the computational burden and the solution accuracy. In general, the larger number of mesh points, the more accurate solution as well as the larger computational cost including CPU time and memory [1]. There are many numerical methods have been developed for the transcription of OCPs, and the most common method is by using Pseudospectral (PS) collocation scheme [2], which is an optimal choice of mesh points in the reason of well-established rules of approximation theory [3]. Actually, there have several mature optimal control toolkits based PS methods, such as DIDO [4], GPOPS [5]. The resulting NLP problem can be solved by the well-known algorithm packages, such as IPOPT [6] or SNOPT [7]. However, these algorithms cannot obtain a solution in polynomial-time, and the resulting solution is locally optimal. Moreover, a good initial guess solution should be provided for complicated problems.