GJK-CBF: Control Barrier Functions for Convex Rigid Body Collision Avoidance on SE(3)

2026-10-04

AI Summary

This study addresses the high computational overhead and excessive conservatism of existing control barrier function (CBF) methods for convex obstacle avoidance, which typically rely on differentiable optimization. To overcome these limitations, this work proposes the GJK-CBF framework. By leveraging the Gilbert–Johnson–Keerthi (GJK) algorithm to extract witness point pairs, the method directly constructs CBF gradients through relative kinematics. This formulation eliminates the need for differentiable optimization, enabling efficient, real-time collision avoidance for convex rigid bodies on the SE(3) manifold. Experimental evaluations across 2D and 3D multi-robot position swapping, narrow-gap navigation, and robotic manipulator scenarios demonstrate that the proposed framework guarantees collision-free motion while significantly reducing conservatism and improving computational efficiency compared to conventional approaches.

Abstract

Collision avoidance among convex bodies is a fundamental problem in robotics. Control Barrier Functions (CBFs) provide a practical framework for real-time safety filtering due to their computational efficiency. For general convex bodies, exact separation measures, such as distance or scaling factor, are typically computed through optimization. Existing CBF formulations often obtain the required gradient via differentiable optimization (diffOpt), adding computational overhead. In contrast, we leverage the Gilbert-Johnson-Keerthi (GJK) algorithm to obtain the current witness pair---the pair of points realizing the minimum distance or penetration depth---and formulate a CBF, termed GJK-CBF, whose gradient is constructed directly from the relative rigid-body motion of the witness pair, without resorting to diffOpt. This formulation applies to both 2D and 3D environments across a broad class of convex body pairs, provided that at least one in each one-to-one interaction is strictly convex. The proposed GJK-CBF is validated in various scenarios, including multi-robot position swapping in both 2D and 3D, navigation through a vertical slit in 3D, and its applicability to manipulators. The results demonstrate collision-free motion across all scenarios while reducing the conservativeness introduced by geometric approximations, particularly in narrow environments.

Problem

• Collision Avoidance
• Control Barrier Functions
• Convex Rigid Bodies
• GJK Algorithm
• SE(3)

Innovation

• Control Barrier Functions
• GJK algorithm
• Collision Avoidance
• SE(3)
• Convex Rigid Body