Safe Robotics Control with Directional Projection Control Barrier Functions Via Differentiable Optimization

Abstract

Collision avoidance is essential for robotic systems. This paper presents a method for designing directional projection control barrier functions (CBFs) based on differentiable optimization for second-order robotic systems. The approach reduces high-order CBFs to first-order ones and estimates collision risk by examining the intersection of projections along the relative velocity direction. Under the assumption that both the target and obstacles are convex polyhedra whose projections yield convex polygons, a tunable uniform scaling function, centered at the centroid, is introduced to pad the convex polygon. The strict convexity of this padded region is rigorously proven. Using the minimum scaling factor that leads to intersection between two projected convex polygons, a CBF is constructed and incorporated into a tracking controller to ensure collision avoidance. The effectiveness of the proposed method is validated through simulations with a 2D mobile robot and a 7-DOF Franka manipulator.