Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Siddhartha Ganguly, Rihan Aaron D'Silva, Debasish Chatterjee

TL;DR

This work introduces \\textsf{QuITO}\\textsf{ v.2}, a direct transcription framework for constrained optimal control that delivers uniform error guarantees on the control trajectory by coupling direct multiple shooting with a quasi-interpolation representation on a piecewise uniform grid. It combines a wavelet-based change-point localization method with an adaptive \$h\$-refinement strategy to effectively capture bang-bang and singular features while maintaining computational efficiency. Theoretical results guarantee that the approximate control \\widehat{u}_{\\mathsf{G},\\mathcal{D}}(\\cdot) can be made arbitrarily close to the optimal \\u^*(\\cdot) in the uniform norm by suitable choices of \(h,\\mathcal{D},\\rho\), and the localization/refinement pipeline is validated across diverse benchmarks including Bang Bang, Bressan, Catalyst Mixing, SIRI, and multi-agent planning. The paper also provides an open-source software package with a GUI, illustrating practical applicability and enabling researchers and practitioners to solve challenging constrained OCPs with improved accuracy and interpretable mesh refinement.

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.

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

TL;DR

This work introduces \\textsf{QuITO}\\textsf{ v.2}, a direct transcription framework for constrained optimal control that delivers uniform error guarantees on the control trajectory by coupling direct multiple shooting with a quasi-interpolation representation on a piecewise uniform grid. It combines a wavelet-based change-point localization method with an adaptive \-refinement strategy to effectively capture bang-bang and singular features while maintaining computational efficiency. Theoretical results guarantee that the approximate control \\widehat{u}_{\\mathsf{G},\\mathcal{D}}(\\cdot) can be made arbitrarily close to the optimal \\u^*(\\cdot) in the uniform norm by suitable choices of , and the localization/refinement pipeline is validated across diverse benchmarks including Bang Bang, Bressan, Catalyst Mixing, SIRI, and multi-agent planning. The paper also provides an open-source software package with a GUI, illustrating practical applicability and enabling researchers and practitioners to solve challenging constrained OCPs with improved accuracy and interpretable mesh refinement.

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 -- (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 -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 along with all its functionalities for completeness. The package is available at: https://github.com/chatterjee-d/QuITOv2.git.
Paper Structure (18 sections, 7 theorems, 78 equations, 21 figures, 7 tables)

This paper contains 18 sections, 7 theorems, 78 equations, 21 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Problem Statement
  3. Mathematical background on approximate approximations
  4. Main results
  5. $\textsf{QuITO}$ $\textsf{v.2}$: The direct multiple shooting method.
  6. A couple of illustrations on how to employ Theorem \ref{['thrm:optimal estimate']}
  7. Change point localization algorithm
  8. Mesh Refinement Scheme.
  9. Numerical experiments and discussion
  10. Bang Bang problem
  11. Bressan problem
  12. Catalyst Mixing Problem
  13. SIRI epidemics
  14. Multi agent robot motion planning
  15. Discussion on CPU times
  16. ...and 3 more sections

Key Result

Theorem 3.1

ref:mazyabook Consider a Lipschitz continuous function $q:\mathbb{R} \to \mathbb{R}$ of Lipschitz rank $L_q$. Let $(h,\mathcal{D}) \in \loro{0}{+\infty}^2$ and let $\mathsf{G} \coloneqq \aset[]{mh \suchthat m \in \mathbb{Z}} \subset \mathbb{R}$ be a uniform grid of mesh size $h$. Suppose that $\psi^ where $\Delta_0(\psi,\mathcal{D}) \coloneqq \mathcal{E}_0(\psi,\mathcal{D})\left\lVert q(\cdot) \ri

Figures (21)

  • Figure 1: A schematic diagram of $\textsf{QuITO}$$\textsf{v.2}$. The grids $\mathsf{G}$ and $\mathcal{G}$ denotes a uniform and a piecewise uniform grid respectively.
  • Figure 2: Error trajectory for different thresholds $\varepsilon$ for the Aly-Chan problem.
  • Figure 3: The left-hand subfigure shows numerical control trajectories obtained via ICLOCS2 by employing LGR collocation using Legendre polynomials for state and control representation for the Aly-Chan problem. The right-hand subfigure depicts Euler, Trapezoidal, and Hermite-Simpson collocation using piecewise cubic hermite interpolating polynomial for state and control representation for the same problem.
  • Figure 4: Error trajectory for different thresholds $\varepsilon$ for the Bryson-Denham problem.
  • Figure 5: The analytical solution $u^\ast(\cdot)$ and the numerical control trajectory obtained by $\textsf{QuITO}$.
  • ...and 16 more figures

Theorems & Definitions (31)

  • Remark 2.1
  • Theorem 3.1
  • Remark 3.2
  • Remark 3.3
  • Theorem 3.4
  • Remark 3.5
  • Remark 4.1
  • Proposition 4.1
  • Remark 4.2
  • Proposition 4.2
  • ...and 21 more