Robotics team's exact SDF via Minkowski ops beats conservative CBF navigation

A team led by Yi-Hsuan Chen (with Shuo Liu, Wei Xiao, Calin Belta, and Michael Otte) published arXiv:2608.02886, presenting a new control barrier function (CBF) method for safe robot navigation among polytopic obstacles. Existing CBF approaches often approximate obstacles and robots as spheres or ellipsoids to obtain differentiable distance functions, leading to conservative behavior. The authors instead compute the exact signed distance function (SDF) for polytope-shaped bodies using Minkowski operations, solved via companion convex programs for both collision-free (positive SDF) and in-collision (negative SDF) cases. This preserves the true geometry, allowing the robot to exploit narrow corridors and tight clearances that smooth approximations would reject.

The key theoretical contribution is a unified analytical gradient of the exact SDF derived through sensitivity analysis of the companion programs, enabling efficient nonsmooth CBF-based control. Notably, the exact rotational gradient exposes a previously hidden class of local minima caused by the coupling between polytope geometry and nonholonomic kinematics. The framework is validated in 16 pages with 13 figures, including pure-translation scenarios and three unicycle-model tests: recovery from an unsafe initialization, single-obstacle, and multiple-obstacle avoidance. Compared to baseline spherical/ellipsoidal CBF methods, the proposed approach produces non-conservative maneuvers and safely recovers from states that would cause baseline controllers to fail. An expanded version of an IEEE CDC 2025 paper, this work provides practical tools for real-time navigation in cluttered, geometrically exact environments.