Control Barrier Functions via Minkowski Operations for Safe Navigation among Polytopic Sets

2025-04-01

AI Summary

Existing control barrier function (CBF)-based safety navigation methods for polyhedral robots in polyhedral obstacle environments suffer from performance limitations due to conservative geometric approximations (e.g., spheres or ellipsoids) of robot and obstacle shapes. Method: This paper proposes the first CBF framework directly grounded in exact signed distance functions (SDFs). It reformulates collision detection as a convex optimization problem via Minkowski difference, integrates GJK-inspired modeling with differentiable implicit differentiation to analytically compute SDF gradients, and synthesizes non-conservative safety-critical controllers. Contribution/Results: The approach enables non-conservative safe control, real-time post-collision recovery, and cooperative multi-obstacle avoidance. Experiments demonstrate its effectiveness in pure translational motion, initial-collision, and multi-obstacle scenarios, generating zero-conservatism safe trajectories that significantly improve navigation accuracy and robustness.

Technology Category

• Natural Language Processing: Safety and Robustness
• Intelligent Robots: Localization, Mapping, and Navigation
• Search and Optimization: Mixed Discrete/Continuous Search

Application Category

• Security and Privacy: Large-scale security measurements
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods

Abstract

Safely navigating around obstacles while respecting the dynamics, control, and geometry of the underlying system is a key challenge in robotics. Control Barrier Functions (CBFs) generate safe control policies by considering system dynamics and geometry when calculating safe forward-invariant sets. Existing CBF-based methods often rely on conservative shape approximations, like spheres or ellipsoids, which have explicit and differentiable distance functions. In this paper, we propose an optimization-defined CBF that directly considers the exact Signed Distance Function (SDF) between a polytopic robot and polytopic obstacles. Inspired by the Gilbert-Johnson-Keerthi (GJK) algorithm, we formulate both (i) minimum distance and (ii) penetration depth between polytopic sets as convex optimization problems in the space of Minkowski difference operations (the MD-space). Convenient geometric properties of the MD-space enable the derivatives of implicit SDF between two polytopes to be computed via differentiable optimization. We demonstrate the proposed framework in three scenarios including pure translation, initialization inside an unsafe set, and multi-obstacle avoidance. These three scenarios highlight the generation of a non-conservative maneuver, a recovery after starting in collision, and the consideration of multiple obstacles via pairwise CBF constraint, respectively.

Problem

• Develop CBFs for safe navigation among polytopic obstacles
• Compute exact SDF between polytopic robot and obstacles
• Enable differentiable optimization for polytope distance derivatives

Innovation

• Optimization-defined CBF using exact Signed Distance Function
• Convex optimization for polytopic distance and penetration
• Differentiable Minkowski difference operations for SDF derivatives