Safe Navigation in Dynamic Environments using Density Functions

TL;DR

This paper introduces a density-based framework for safe navigation in dynamic environments, providing analytical guarantees of safety for systems like robots and multi-agent systems using time-varying density functions and feedback control.

Contribution

It presents the first rigorous convergence proof ensuring almost-everywhere safety in dynamic environments using density functions for single-integrator systems.

Findings

Proposes a novel density function construction for dynamic environments.

Provides a convergence proof for safe navigation.

Demonstrates applicability to complex robotic systems.

Abstract

This work presents a density-based framework for safe navigation in dynamic environments characterized by time-varying obstacle sets and time-varying target regions. We propose an analytical construction of time-varying density functions that enables the synthesis of a feedback controller defined as the positive gradient of the resulting density field. The primary contribution of this paper is a rigorous convergence proof demonstrating almost-everywhere safe navigation under the proposed framework, specifically for systems governed by single-integrator dynamics. To the best of our knowledge, these are the first analytical guarantees of their kind for navigation in dynamic environments using density functions. We illustrate the applicability of the framework to systems with more complex dynamics, including multi-agent systems and robotic manipulators, using standard control design techniques such as backstepping and inverse dynamics. These results provide a foundation for extending density-based navigation methods to a broad class of robotic systems operating in time-varying environments.