Control Barrier Functions via Minkowski Operations for Safe Navigation among Polytopic Sets
Published in IEEE CDC 2025, 2025
Recommended citation: Chen, Yi-Hsuan, et al. "Control Barrier Functions via Minkowski Operations for Safe Navigation among Polytopic Sets." arXiv preprint arXiv:2504.00364 (2025). https://arxiv.org/pdf/2504.00364
This work is inspired by the Gilbert-Johnson-Keerthi (GJK) algorithm; We formulate both (i) minimum distance and (ii) penetration depth between polytopic sets as convex optimization problems in the space of Minkowski difference operations (the $\mathcal{MD}$-space). By leveraging the geometric properties of the $\mathcal{MD}$-space, we show that the derivatives of the implicit Signed Distance Function (SDF) can be efficiently computed via differentiable optimization. We demonstrate the effectiveness of the proposed approach on two dynamical systems: (i) single-integrator and (ii) unicycle models, showcasing its capability to maintain safety for non-spherical robots in tight spaces.