发表机构
University of Maryland; Boston University; Nanyang Technological University（马里兰大学; 波士顿大学; 南洋理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对多面体环境的机器人安全导航问题，提出结合非光滑控制障碍函数与闵可夫斯基运算的精确符号距离函数方法，可实现非保守机动与安全恢复。
AI 中文摘要
在机器人学中，在尊重底层系统的动力学、控制及精确几何特性的前提下，于多面体环境中安全导航是一项挑战。控制障碍函数（CBF）通过使安全集前向不变来合成安全控制策略，但许多现有基于CBF的方法会用保守的光滑形状（如球体或椭球体）近似多面体，以获取显式可微的距离函数。本文针对多面体机器人与多面体障碍物，提出一种精确的符号距离函数（SDF）公式，并将其与非光滑CBF相结合。利用闵可夫斯基运算，所提方法在无碰撞（正号）与碰撞（负号）两种情形下，通过配套凸规划计算精确SDF。此外，通过利用二维闵可夫斯基运算的便利几何特性及两个配套凸规划的最优性条件，我们借助敏感性分析推导出精确SDF梯度的统一解析表达式。该精确旋转梯度进一步揭示了由几何与非完整运动学耦合所引发的、此前未被发现的一类局部极小值。我们通过纯平移案例及三个涉及独轮车模型的场景验证所提框架的有效性，这些场景包括从不安全初始化状态恢复、单障碍物避障及多障碍物避障。与基线方法的对比凸显了所提框架如何实现非保守机动与安全恢复。
英文摘要
Safely navigating polytopic environments while respecting the dynamics, control, and exact geometry of the underlying system is a challenge in robotics. Control barrier functions (CBFs) synthesize safe control policies by rendering the safe set forward invariant, but many existing CBF-based methods approximate polytopes using conservative smooth shapes, such as spheres or ellipsoids, to obtain explicit differentiable distance functions. In this article, we propose an exact Signed Distance Function (SDF) formulation for a {\it polytopic} robot and {\it polytopic} obstacles and integrate it with nonsmooth CBFs. Leveraging Minkowski operations, the proposed method computes the exact SDF via companion convex programs in both the collision-free (positive-sign) and in-collision (negative-sign) cases. Furthermore, by exploiting the convenient geometric properties of 2D Minkowski operations and the optimality conditions of the two companion convex programs, we derive a unified analytical expression for the gradient of the exact SDF via sensitivity analysis. The exact rotational gradient further reveals a previously masked class of local minima induced by the coupling between geometry and nonholonomic kinematics. We demonstrate the effectiveness of the proposed framework through a pure-translation case and three scenarios with unicycle models involving recovery from an unsafe initialization and single- and multiple-obstacle avoidance. Comparisons with baseline methods highlight how the proposed framework enables non-conservative maneuvers and safety recovery.
Comments16 pages, 13 figures. Expanded version of a paper published in IEEE CDC 2025. Demo video: https://youtu.be/D0zVswzyxaE
