Direct Pseudospectral Optimal Control by Orthogonal Polynomial Integral Collocation

TL;DR

This paper introduces a pseudospectral optimal control method using orthogonal polynomial integral collocation, enabling efficient trajectory generation by approximating derivatives and integrating states, with demonstrated effectiveness on complex aerospace problems.

Contribution

The paper presents a novel pseudospectral optimal control approach that approximates derivatives through polynomial integrals, improving computational efficiency and flexibility over existing methods.

Findings

Method accurately solves benchmark orbit raising problem.

Polynomial basis choice significantly impacts performance.

Successfully applied to complex rocket landing maneuver.

Abstract

This paper details a methodology to transcribe an optimal control problem into a nonlinear program for generation of the trajectories that optimize a given functional by approximating only the highest order derivatives of a given system's dynamics. The underlying method uses orthogonal polynomial integral collocation by which successive integrals are taken to approximate all lower order states. Hence, one set of polynomial coefficients can represent an entire coordinate's degree of freedom. Specifically, Chebyshev polynomials of the first and second kind and Legendre polynomials are used over their associated common interpolating grids derived from the bases' roots and extrema. Simple example problems compare different polynomial bases' performance to analytical solutions. The planar circular orbit raising problem is used to verify the method with solutions obtained by other pseudospectral methods in literature. Finally, a rocket landing flip maneuver problem is solved to demonstrate the ability to solve complex problems with multiple states and control variables with constraints. Simulations establish this method's performance, and reveal that the polynomial/node choice for a given problem notably affects the performance.