We present an efficient method for certifying non-collision for piecewise-polynomial motion plans in algebraic reparametrizations of configuration space. Such motion plans include those generated by popular randomized methods including RRTs and PRMs, as well as those generated by many methods in trajectory optimization. Based on Sums-of-Squares optimization, our method provides exact, rigorous certificates of non-collision; it can never falsely claim that a motion plan containing collisions is collision-free. We demonstrate that our formulation is practical for real world deployment, certifying the safety of a twelve degree of freedom motion plan in just over a second. Moreover, the method is capable of discriminating the safety or lack thereof of two motion plans which differ by only millimeters.

翻译：我们提出了一种高效方法，用于在构型空间的代数重参数化中验证分段多项式运动规划的无碰撞性。此类运动规划包括由RRT和PRM等主流随机方法生成的规划，以及许多轨迹优化方法生成的规划。基于平方和优化技术，本方法能够提供精确且严格的无碰撞性证明，绝不会错误地将含有碰撞的运动规划判定为无碰撞。实验表明，该公式化方法具备实际部署价值，能在略超一秒的时间内验证十二自由度运动规划的安全性。此外，该方法还能区分毫米级差异的两个运动规划是否存在碰撞风险。