Costate Convergence with Legendre-Lobatto Collocation for Trajectory Optimization

TL;DR

This paper presents a novel Legendre-Lobatto collocation method with an exceptional sample that ensures complete costate convergence in trajectory optimization, demonstrating exponential convergence and robustness against interpolation issues.

Contribution

It introduces an exceptional sample to the Legendre-Lobatto polynomial roots, ensuring full-rank differentiation matrices and complete costate convergence in collocation methods.

Findings

Achieves exponential convergence with discretization points.

Ensures full-rank differentiation matrices for costate variables.

Reduces Runge phenomenon in polynomial interpolation.

Abstract

This paper introduces a new method of discretization that collocates both endpoints of the domain and enables the complete convergence of the costate variables associated with the Hamilton boundary-value problem. This is achieved through the inclusion of an \emph{exceptional sample} to the roots of the Legendre-Lobatto polynomial, thus promoting the associated differentiation matrix to be full-rank. We study the location of the new sample such that the differentiation matrix is the most robust to perturbations and we prove that this location is also the choice that mitigates the Runge phenomenon associated with polynomial interpolation. Two benchmark problems are successfully implemented in support of our theoretical findings. The new method is observed to converge exponentially with the number of discretization points used.