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 letter, we propose a framework that uses this scaling factor, with an offset, to systematically define a CBF for obstacle avoidance tasks. We provide theoretical analyses of the continuity and continuous differentiability of the proposed CBF. We empirically evaluate the proposed CBF's behavior and show that 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（CBFs）已广泛应用于安全关键的机器人应用场景。然而，针对机器人系统构建控制障碍函数仍然是一项具有挑战性的任务。近期，利用可微优化进行碰撞检测的方法提供了一种途径：既能计算使两个凸形产生交集的最小均匀缩放因子，又能计算该缩放因子的雅可比矩阵。本文提出一种框架，通过引入偏移量的缩放因子，系统性地定义用于避障任务的CBF。我们从理论上分析了所提CBF的连续性与连续可微性，并通过实验评估其行为特性，表明由此产生的最优控制问题具有计算高效性，可适用于实时机器人控制。我们首先通过二维移动机器人实例验证该方法，随后在Franka-Emika Research 3（FR3）机器人机械臂上开展仿真与实验验证。