Control barrier functions (CBFs) have been widely applied to safety-critical robotic applications. However, the construction of control barrier functions for robotic systems remains a challenging task. Recently, collision detection using differentiable optimization has provided a way to compute the minimum uniform scaling factor that results in an intersection between two convex shapes and to also compute the Jacobian of the scaling factor. In this paper, we propose a framework that uses this scaling factor, with an offset, to systematically define a CBF for obstacle avoidance tasks. We provide a theoretical analysis that proves the continuity of the proposed CBF. Empirically, we show that the proposed CBF is continuously differentiable, and the resulting optimal control problem is computationally efficient, which makes it applicable for real-time robotic control. We validate our approach, first using a 2D mobile robot example, then on the Franka-Emika Research~3 (FR3) robot manipulator both in simulation and experiment.

翻译：控制障碍函数已广泛应用于安全关键的机器人应用。然而，为机器人系统构建控制障碍函数仍是一项具有挑战性的任务。近年来，利用可微优化的碰撞检测提供了一种计算两个凸形状间相交所需的最小均匀缩放因子的方法，并可同时计算该缩放因子的雅可比矩阵。本文提出了一种框架，利用带偏移量的缩放因子系统地定义用于避障任务的控制障碍函数。我们提供了理论分析，证明了所提控制障碍函数的连续性。实验表明，该控制障碍函数是连续可微的，且由此产生的最优控制问题计算效率高，适用于实时机器人控制。我们首先通过二维移动机器人示例验证了该方法，随后在Franka-Emika Research~3 (FR3)机器人机械臂上进行了仿真与实验验证。