Proofs of Technical Results Justifying and Illustrating an Algorithm of Navigation for Monitoring Unsteady Environmental Boundaries

TL;DR

This paper introduces a sliding mode control algorithm for a non-holonomic robot to detect, locate, and track unsteady environmental boundaries without gradient estimation, with rigorous proofs of convergence and performance illustrations.

Contribution

It provides a novel control method for boundary tracking in dynamic environments, including theoretical proofs and illustrations, without relying on spatial gradient estimation.

Findings

The control algorithm successfully localizes and tracks unsteady boundaries.

The method does not require gradient estimation, reducing computational demands.

Theoretical proofs confirm non-local convergence of the control law.

Abstract

We consider a single non-holonomic Dubins-like robot traveling with a constant longitudinal speed in an a priori unknown and unsteady planar environment. The robot should detect, locate, and track the boundary of a dynamic environmental scalar field. The field is measured by an on-board sensor in a point-wise fashion at the robot's location. The focus is on unsteady boundaries that evolve over time in an arbitrary fashion, including deformations, i.e., changes of shapes and sizes. We present a sliding mode control method for localizing and tracking such boundaries: the robot is steered to the boundary and circulates in its close proximity afterwards. The proposed control algorithm does not require estimation of the spatial gradient of the field and is non-demanding with respect to both computation and motion. The paper offers the proofs of technical facts required for rigorous justification of non-local convergence of the proposed control law, as well as theoretical illustrations of its performance in specific scenarios.