Time-Optimal Switching Surfaces for Triple Integrator under Full Box Constraints

TL;DR

This paper characterizes time-optimal switching surfaces for a triple integrator with full box constraints, introducing an efficient algorithm that handles asymmetric and non-stationary conditions with high speed and success rate.

Contribution

It provides a complete geometric characterization of optimal switching surfaces and active position constraints, along with a fast planning algorithm for complex boundary conditions.

Findings

Achieves 100% success rate in trajectory planning.

Computational time per trajectory is approximately 10 microseconds.

Reduces planning time by five orders of magnitude compared to optimization baselines.

Abstract

Time-optimal control for triple integrator under full box constraints is a fundamental problem in the field of optimal control, which has been widely applied in the industry. However, scenarios involving asymmetric constraints, non-stationary boundary conditions, and active position constraints pose significant challenges. This paper provides a complete characterization of time-optimal switching surfaces for the problem, leading to novel insights into the geometric structure of the optimal control. The active condition of position constraints is derived, which is absent from the literature. An efficient algorithm is proposed, capable of planning time-optimal trajectories under asymmetric full constraints and arbitrary boundary states, with a 100% success rate. Computational time for each trajectory is within approximately 10$\mu$s, achieving a 5-order-of-magnitude reduction compared to optimization-based baselines.