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A Feasible Sequential Linear Programming Algorithm with Application to Time-Optimal Path Planning Problems

David Kiessling, Andrea Zanelli, Armin Nurkanović, Joris Gillis, Moritz Diehl, Melanie Zeilinger, Goele Pipeleers, Jan Swevers

TL;DR

A Feasible Sequential Linear Programming algorithm applied to time-optimal control problems (TOCP) obtained through direct multiple shooting discretization motivated by TOCP with nonlinear constraints which arise in motion planning of mechatronic systems.

Abstract

In this paper, we propose a Feasible Sequential Linear Programming (FSLP) algorithm applied to time-optimal control problems (TOCP) obtained through direct multiple shooting discretization. This method is motivated by TOCP with nonlinear constraints which arise in motion planning of mechatronic systems. The algorithm applies a trust-region globalization strategy ensuring global convergence. For fully determined problems our algorithm provides locally quadratic convergence. Moreover, the algorithm keeps all iterates feasible enabling early termination at suboptimal, feasible solutions. This additional feasibility is achieved by an efficient iterative strategy using evaluations of constraints, i.e., zero-order information. Convergence of the feasibility iterations can be enforced by reduction of the trust-region radius. These feasibility iterations maintain feasibility for general Nonlinear Programs (NLP). Therefore, the algorithm is applicable to general NLPs. We demonstrate our algorithm's efficiency and the feasibility update strategy on a TOCP of an overhead crane motion planning simulation case.

A Feasible Sequential Linear Programming Algorithm with Application to Time-Optimal Path Planning Problems

TL;DR

A Feasible Sequential Linear Programming algorithm applied to time-optimal control problems (TOCP) obtained through direct multiple shooting discretization motivated by TOCP with nonlinear constraints which arise in motion planning of mechatronic systems.

Abstract

In this paper, we propose a Feasible Sequential Linear Programming (FSLP) algorithm applied to time-optimal control problems (TOCP) obtained through direct multiple shooting discretization. This method is motivated by TOCP with nonlinear constraints which arise in motion planning of mechatronic systems. The algorithm applies a trust-region globalization strategy ensuring global convergence. For fully determined problems our algorithm provides locally quadratic convergence. Moreover, the algorithm keeps all iterates feasible enabling early termination at suboptimal, feasible solutions. This additional feasibility is achieved by an efficient iterative strategy using evaluations of constraints, i.e., zero-order information. Convergence of the feasibility iterations can be enforced by reduction of the trust-region radius. These feasibility iterations maintain feasibility for general Nonlinear Programs (NLP). Therefore, the algorithm is applicable to general NLPs. We demonstrate our algorithm's efficiency and the feasibility update strategy on a TOCP of an overhead crane motion planning simulation case.
Paper Structure (19 sections, 5 theorems, 32 equations, 7 figures, 1 table, 2 algorithms)

This paper contains 19 sections, 5 theorems, 32 equations, 7 figures, 1 table, 2 algorithms.

Table of Contents

  1. Introduction
  2. Feasible Sequential Linear Programming
  3. Notation & Preliminaries
  4. Description of the algorithm
  5. Global convergence of FP-SQP and FSLP
  6. Local convergence of FSLP
  7. Inner Feasibility Iterations
  8. Description of the algorithm
  9. Termination heuristic
  10. Limit of feasibility iterations
  11. Local convergence of feasibility iterations
  12. Projection ratio
  13. Numerical Example
  14. Implementation
  15. Illustrative example of quadratic convergence
  16. ...and 4 more sections

Key Result

Theorem 1

Suppose that Assumptions ass:boundedL0, ass:boundedDistance hold then all limit points of Algorithm alg:FPSQP either are KKT points or else fail to satisfy the Mangasarian-Fromowitz constraint qualification.

Figures (7)

  • Figure 1: Visualization of feasibility perturbation.
  • Figure 2: Comparison of the convergence of FSLP on a fully determined NLP and a not fully determined NLP.
  • Figure 3: Schematic illustration of the overhead crane.
  • Figure 4: Payload trajectories of the iterates of FSLP on the overhead crane problem.
  • Figure 5: Comparison of local convergence and projection ratio of feasibility iterations for different trust-region radii. For all trust-region radii below a certain value between $2^{-1}$ and $2^{-2}$ we obtain linear convergence and satisfaction of condition \ref{['eq:asymptoticExactness']}.
  • ...and 2 more figures

Theorems & Definitions (9)

  • Theorem 1: Global convergence of FP-SQP Wright2004
  • Corollary 1: Local quadratic convergence
  • proof
  • Lemma 1: Limit of feasibility improvement
  • proof
  • Theorem 2: Local linear convergence proportional to $\Delta$
  • proof
  • Theorem 3: Projection ratio
  • proof