$\mathsf{QuITO}$ $\textsf{v.2}$: Trajectory Optimization with Uniform Error Guarantees under Path Constraints

TL;DR

This paper presents a new trajectory optimization method with uniform error guarantees for nonlinear constrained optimal control problems, incorporating wavelet-based localization and mesh refinement to handle irregularities effectively.

Contribution

It introduces QuITO v.2, a novel direct multiple shooting algorithm with a wavelet-based localization and targeted mesh refinement scheme, providing theoretical error bounds and practical implementation.

Findings

Effective localization of irregular regions in control trajectories.

Guaranteed uniform approximation errors for certain regularity classes.

Successful numerical demonstrations on discontinuous control profiles.

Abstract

This article introduces a new transcription, change point localization, and mesh refinement scheme for direct optimization-based solutions and for uniform approximation of optimal control trajectories associated with a class of nonlinear constrained optimal control problems (OCPs). The base transcription algorithm for which we establish the refinement algorithm is a direct multiple shooting technique -- $\mathsf{QuITO}$ $\textsf{v.2}$ (Quasi-Interpolation based Trajectory Optimization). The mesh refinement technique consists of two steps -- localization of certain irregular regions in an optimal control trajectory via wavelets, followed by a targeted $h$-refinement approach around such regions of irregularity. Theoretical approximation guarantees on uniform grids are presented for optimal controls with certain regularity properties, along with guarantees of localization of change points by wavelet transform. Numerical illustrations are provided for control profiles involving discontinuities to show the effectiveness of the localization and refinement strategy. We also announce, and make freely available, a new software package based on $\mathsf{QuITO}$ $\textsf{v.2}$ along with all its functionalities for completeness. The package is available at: https://github.com/chatterjee-d/QuITOv2.git.