Understanding the geometry of collision-free configuration space (C-free) in the presence of task-space obstacles is an essential ingredient for collision-free motion planning. While it is possible to check for collisions at a point using standard algorithms, to date no practical method exists for computing C-free regions with rigorous certificates due to the complexity of mapping task-space obstacles through the kinematics. In this work, we present the first to our knowledge rigorous method for approximately decomposing a rational parametrization of C-free into certified polyhedral regions. Our method, called C-IRIS (C-space Iterative Regional Inflation by Semidefinite programming), generates large, convex polytopes in a rational parameterization of the configuration space which are rigorously certified to be collision-free. Such regions have been shown to be useful for both optimization-based and randomized motion planning. Based on convex optimization, our method works in arbitrary dimensions, only makes assumptions about the convexity of the obstacles in the task space, and is fast enough to scale to realistic problems in manipulation. We demonstrate our algorithm's ability to fill a non-trivial amount of collision-free C-space in several 2-DOF examples where the C-space can be visualized, as well as the scalability of our algorithm on a 7-DOF KUKA iiwa, a 6-DOF UR3e and 12-DOF bimanual manipulators. An implementation of our algorithm is open-sourced in Drake. We furthermore provide examples of our algorithm in interactive Python notebooks.

翻译：理解含任务空间障碍的碰撞自由构型空间（C-free）的几何结构是实现无碰撞运动规划的核心要素。尽管可通过标准算法检测单点碰撞，但由于需通过运动学映射任务空间障碍的复杂性，目前尚无实用方法能够严格证明C-free区域的存在性。本研究首次提出一种基于严格证明的方法，将C-free的有理参数化近似分解为可认证的多面体区域。所提方法C-IRIS（基于半定规划的构型空间迭代区域膨胀法）可在构型空间的有理参数化中生成严格认证无碰撞的大型凸多面体。此类区域已被证明对基于优化和随机采样的运动规划均具有实用价值。该方法基于凸优化，适用于任意维度，仅需假设任务空间障碍的凸性，且运算速度足以应对实际操作问题中的复杂场景。我们通过多个二维自由度的可视化构型空间案例验证了算法填充非平凡无碰撞C-free空间的能力，并在7自由度KUKA iiwa、6自由度UR3e及12自由度双臂操作器上验证了算法的可扩展性。该算法已在Drake平台开源，并提供交互式Python笔记本示例。