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This paper introduces an exact Signed Distance Function (SDF) formulation for polytopic robots navigating polytopic obstacles, overcoming limitations of existing control barrier function (CBF) methods that rely on conservative approximations. By utilizing Minkowski operations, the authors compute the SDF through companion convex programs, achieving precise control policies that maintain safety while allowing for non-conservative maneuvers. The results demonstrate significant improvements in safety recovery and obstacle avoidance compared to baseline methods, particularly in scenarios involving nonholonomic kinematics.
Exact Signed Distance Functions enable non-conservative navigation strategies that enhance safety recovery in complex polytopic environments.
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.
