Bounded-Rate Multi-Mode Systems Based Motion Planning

TL;DR

This paper investigates motion planning for bounded-rate multi-mode systems, providing algorithms for trajectory following, analyzing computational complexity, and exploring decidability under various restrictions and system structures.

Contribution

It introduces algorithms for precise trajectory following, analyzes complexity, and examines decidability issues in bounded-rate multi-mode systems.

Findings

The motion planning problem is co-NP complete.

Polynomial-time solutions exist for fixed-dimensional systems.

Decidability is affected by dwell-time constraints and system structure.

Abstract

Bounded-rate multi-mode systems are hybrid systems that can switch among a finite set of modes. Its dynamics is specified by a finite number of real-valued variables with mode-dependent rates that can vary within given bounded sets. Given an arbitrary piecewise linear trajectory, we study the problem of following the trajectory with arbitrary precision, using motion primitives given as bounded-rate multi-mode systems. We give an algorithm to solve the problem and show that the problem is co-NP complete. We further prove that the problem can be solved in polynomial time for multi-mode systems with fixed dimension. We study the problem with dwell-time requirement and show the decidability of the problem under certain positivity restriction on the rate vectors. Finally, we show that introducing structure to the multi-mode systems leads to undecidability, even when using only a single clock variable.