Navigating Polytopes with Safety: A Control Barrier Function Approach
Tamas G. Molnar
TL;DR
This work addresses safe navigation for polytopes by deriving a closed-form control barrier function (CBF) candidate that guarantees collision-free motion for polytope-shaped agents in polytope environments. The method leverages smooth max/min approximations (e.g., log-sum-exp) to compose barriers for unions and intersections of half-spaces, enabling explicit, real-time-safe controllers for point and polytope agents in 2D and 3D, including time-varying obstacles. Key contributions include a unified, closed-form CBF formulation for half-spaces, convex polytopes, and general polytopes, extended to rigid-body agents via vertex representations, with provable safety under a conservative under-approximation and a tunable smoothing parameter ${\kappa}$ and safety buffer ${b}$. The approach emphasizes simplicity and real-time applicability over optimization-based planning, showing successful simulations in various obstacle configurations, including dynamic environments and non-convex geometries. This enables safer, reactive navigation using polytope approximations in practical robotic settings where geometry dominates collision risks.
Abstract
Collision-free motion is a fundamental requirement for many autonomous systems. This paper develops a safety-critical control approach for the collision-free navigation of polytope-shaped agents in polytope-shaped environments. A systematic method is proposed to generate control barrier function candidates in closed form that lead to controllers with formal safety guarantees. The proposed approach is demonstrated through simulation, with obstacle avoidance examples in 2D and 3D, including dynamically changing environments.