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（CBFs）已被广泛应用于安全关键的机器人应用领域。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。近期，利用可微优化的碰撞检测方法为计算两个凸形状间相交所需的最小统一缩放因子及其雅可比矩阵提供了可行方案。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面向避碰任务的CBF。我们提供理论分析证明所提CBF的连续性。实验结果表明，该CBF具有连续可微性，且由其构成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案例验证该方法，随后在Frank-Emika研究3型（FR3）机器人操作臂上完成仿真与实验验证。