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This research advances safe robot navigation by replacing conservative geometric approximations with exact Signed Distance Functions derived from Minkowski operations for polytopic environments. By integrating these precise formulations into nonsmooth Control Barrier Functions, the method enables non-conservative maneuvers while analytically revealing previously hidden local minima caused by geometry-kinematics coupling. The framework demonstrates significant improvements in safety recovery and maneuverability compared to baseline approaches that rely on smooth shape approximations like spheres or ellipsoids.
Control Barrier Functions via Minkowski Operations for Safe Navigation among Polytopes
arXiv:2608.02886v1 Announce Type: new Abstract: Safely navigating polytopic environments while respecting the dynamics, control, and exact geometry of the underlying system is a challenge in robotics. Control barrier functions (CBFs) synthesize safe control policies by rendering the safe set forward invariant, but many existing CBF-based methods approximate polytopes using conservative smooth shapes, such as spheres or ellipsoids, to obtain explicit differentiable distance functions. In this article, we propose an exact Signed Distance Function (SDF) formulation for a {\it polytopic} robot and {\it polytopic} obstacles and integrate it with nonsmooth CBFs. Leveraging Minkowski operations, the proposed method computes the exact SDF via companion convex programs in both the collision-free (positive-sign) and in-collision (negative-sign) cases. Furthermore, by exploiting the convenient geometric properties of 2D Minkowski operations and the optimality conditions of the two companion convex programs, we derive a unified analytical expression for the gradient of the exact SDF via sensitivity analysis. The exact rotational gradient further reveals a previously masked class of local minima induced by the coupling between geometry and nonholonomic kinematics. We demonstrate the effectiveness of the proposed framework through a pure-translation case and three scenarios with unicycle models involving recovery from an unsafe initialization and single- and multiple-obstacle avoidance. Comparisons with baseline methods highlight how the proposed framework enables non-conservative maneuvers and safety recovery.
Abstract:Safely navigating polytopic environments while respecting the dynamics, control, and exact geometry of the underlying system is a challenge in robotics. Control barrier functions (CBFs) synthesize safe control policies by rendering the safe set forward invariant, but many existing CBF-based methods approximate polytopes using conservative smooth shapes, such as spheres or ellipsoids, to obtain explicit differentiable distance functions. In this article, we propose an exact Signed Distance Function (SDF) formulation for a {\it polytopic} robot and {\it polytopic} obstacles and integrate it with nonsmooth CBFs. Leveraging Minkowski operations, the proposed method computes the exact SDF via companion convex programs in both the collision-free (positive-sign) and in-collision (negative-sign) cases. Furthermore, by exploiting the convenient geometric properties of 2D Minkowski operations and the optimality conditions of the two companion convex programs, we derive a unified analytical expression for the gradient of the exact SDF via sensitivity analysis. The exact rotational gradient further reveals a previously masked class of local minima induced by the coupling between geometry and nonholonomic kinematics. We demonstrate the effectiveness of the proposed framework through a pure-translation case and three scenarios with unicycle models involving recovery from an unsafe initialization and single- and multiple-obstacle avoidance. Comparisons with baseline methods highlight how the proposed framework enables non-conservative maneuvers and safety recovery.
16 pages, 13 figures. Expanded version of a paper published in IEEE CDC 2025. Demo video: this https URL
Robotics (cs.RO); Systems and Control (eess.SY)
arXiv:2608.02886 [cs.RO]
(or arXiv:2608.02886v1 [cs.RO] for this version)
https://doi.org/10.48550/arXiv.2608.02886 Focus to learn more arXiv-issued DOI via DataCite (pending registration)
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