Kinodynamic control systems and discontinuities in clearance

TL;DR

This paper analyzes the structure and causes of discontinuities in clearance functions for nonlinear control systems with obstacles, revealing how certain conditions lead to such discontinuities on obstacle surfaces and in free space.

Contribution

It provides new theoretical insights into the interactions between trajectories and clearance discontinuities, including conditions that cause these discontinuities to occur.

Findings

Discontinuities can cause instantaneous increases in clearance.

Clearance discontinuities propagate along optimal trajectories.

Certain velocity conditions lead to surface and free space discontinuities.

Abstract

We investigate the structure of discontinuities in clearance (or minimum time) functions for nonlinear control systems with general, closed obstacles (or targets). We establish general results regarding interactions between admissible trajectories and clearance discontinuities: e.g. instantaneous increases in clearance when passing through a discontinuity, and propagation of discontinuity along optimal trajectories. Then, investigating sufficient conditions for discontinuities, we explore a common directionality condition for velocities at a point, characterized by strict positivity of the minimal Hamiltonian. Elementary consequences of this common directionality assumption are explored before demonstrating how, in concert with corresponding obstacle configurations, it gives rise to clearance discontinuities both on the surface of the obstacle and propagating out into free space. Minimal assumptions are made on the topological structure of obstacle sets.