Organizations: Department of Aerospace Engineering, University of Maryland, College Park, MD, USA · Department of Mechanical Engineering, Boston University, Boston, MA 02215, USA · School of Electrical and Electronic Engineering, Nanyang Technological University, and with M3S, SMART, Singapore · Department of Electrical and Computer Engineering and Department of Computer Science, University of Maryland, College Park, MD, USA
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.
Figures & tables
Figure 1: Side (left) and top (right) views of a polytopic robot (blue) traversing a vertical slit in 3D. (Top) Conservative enclosing-ellipsoid approximations render the direct passage through the slit infeasible, forcing the robot to slightly detour above the obstacles [ 7 ] . (Bottom) In contrast, the proposed GJK-CBF operates directly on the original convex geometries and successfully guides the robot through. Its construction and gradient computation require only the pair of closest points—witness pair—between two convex bodies without using differentiable optimization . Only the initial and goal poses are given, with no intermediate poses or waypoints.
Figure 2: Left: Comparison between (top) the supporting-hyperplane-CBF [ 7 ] and (bottom) the proposed GJK-CBF for six ellipsoidal robots swapping positions. Dotted outlines denote the desired poses, and solid curves show the robot trajectories. Dash-dotted lines in the top row represent the supporting hyperplanes. Right: (top) minimum pairwise CBF values and (bottom) control-loop computation-time distributions, which are measured for the serial implementation and include all pairwise CBF evaluations and local QP solves for the six robots.
Figure 3: Top: Snapshots of four heterogeneous superellipsoidal robots exchanging diagonally opposite poses using the decentralized GJK-CBF ( 18 ). Bottom: The corresponding pairwise CBF values.
Figure 4: Collision avoidance for a velocity-controlled UR5 manipulator. Top: Side (left) and top (right) views of the motion from the start (green) to the goal (magenta). The blue and cyan curves denote the reference and filtered end-effector trajectories, respectively. Bottom: CBF values for the four links relative to superellipsoidal obstacles O1 and O2 .
Safe navigation often relies on well-defined conditions based on the shape of robots and obstacles, and can be challenging when they have irregular geometries. While Control Barrier Functions (CBFs) offer an efficient mechanism to enforce safe set forward invariance, common shape surrogates (e.g., spheres or super-ellipsoids) either are overly conservative in unstructured scenes or require many local primitives, which inflates constraint counts and degrades real-time performance. In this paper, we introduce a novel geometry-aware Control Barrier Function (CBF) based on Bernstein-Polynomial Signed Distance Fields (BP-SDFs). It provides a unified way to represent the obstacles and robots, so as to represent the barrier function with a unified minimum distance. Benefiting from the differentiability of the Bernstein polynomials, one can easily enforce the control constraints in a closed loop. We validate the method's efficiency and performance to guarantee safety in single-robot navigation and heterogeneous multi-robot collision avoidance via simulations under different environments.
Siwon Jo, Yanze Zhang, Yupeng Yang +1
GRASP Laboratory, University of Pennsylvania, Philadelphia, PA, USA · Department of Computer Science, University of Illinois Chicago, Chicago, IL, USA · Department of Computer Science, University of North Carolina at Charlotte, Charlotte, NC, USA
The collision cone used by velocity-space control barrier functions is conservative: it rejects every relative velocity aimed into an obstacle, however slow. We propose the shifted collision-cone CBF (SC3BF), which adds a state-dependent allowance to the cone condition, so the robot may approach the obstacle at a rate that grows with distance and with its own speed. SC3BF is enforced by an ordinary quadratic program, and its safe set is forward invariant under bounded inputs without a minimum forward speed or a clearance margin. We prove that a nonzero allowance preserving safety always exists, and derive one in closed form. Against three velocity-space baselines on a kinematic bicycle among up to 100 moving obstacles, SC3BF reaches the goal more often and modifies the nominal input less than half as much.
Amin Kashiri, Yasin Yazıcıoğlu
Department of Electrical and Computer Engineering at Northeastern University, Boston, MA · Departments of Mechanical and Industrial Engineering and Electrical and Computer Engineering at Northeastern University, Boston, MA
Collision avoidance for robotic manipulators requires enforcing full-body safety constraints in high-dimensional configuration spaces. Control Barrier Function (CBF) based safety filters have proven effective in enabling safe behaviors, but enforcing the high number of constraints needed for safe manipulation leads to theoretic and computational challenges. This work presents a framework for full-body collision avoidance for manipulators in dynamic environments by leveraging 3D Poisson Safety Functions (PSFs). In particular, given environmental occupancy data, we sample the manipulator surface at a prescribed resolution and shrink free space via a Pontryagin difference according to this resolution. On this buffered domain, we synthesize a globally smooth CBF by solving Poisson's equation, yielding a single safety function for the entire environment. This safety function, evaluated at each sampled point, yields task-space CBF constraints enforced by a real-time safety filter via a multi-constraint quadratic program. We prove that keeping the sample points safe in the buffered region guarantees collision avoidance for the entire continuous robot surface. The framework is validated on a 7-degree-of-freedom manipulator in dynamic environments.
Meg Wilkinson, Gilbert Bahati, Ryan M. Bena +3
Computing and Mathematical Sciences, California Institute of Technology. · Mechanical and Civil Engineering, California Institute of Technology.