From quadrotors delivering packages in urban areas to robot arms moving in confined warehouses, motion planning around obstacles is a core challenge in modern robotics. Planners based on optimization can design trajectories in high-dimensional spaces while satisfying the robot dynamics. However, in the presence of obstacles, these optimization problems become nonconvex and very hard to solve, even just locally. Thus, when facing cluttered environments, roboticists typically fall back to sampling-based planners that do not scale equally well to high dimensions and struggle with continuous differential constraints. Here, we present a framework that enables convex optimization to efficiently and reliably plan trajectories around obstacles. Specifically, we focus on collision-free motion planning with costs and constraints on the shape, the duration, and the velocity of the trajectory. Using recent techniques for finding shortest paths in Graphs of Convex Sets (GCS), we design a practical convex relaxation of the planning problem. We show that this relaxation is typically very tight, to the point that a cheap postprocessing of its solution is almost always sufficient to identify a collision-free trajectory that is globally optimal (within the parameterized class of curves). Through numerical and hardware experiments, we demonstrate that our planner, which we name GCS, can find better trajectories in less time than widely used sampling-based algorithms and can reliably design trajectories in high-dimensional complex environments.

中文翻译：

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使用凸优化围绕障碍物进行运动规划

从在城市地区运送包裹的四旋翼飞行器到在密闭仓库中移动的机器人手臂，围绕障碍物的运动规划是现代机器人技术的核心挑战。基于优化的规划器可以在满足机器人动力学的同时设计高维空间中的轨迹。然而，在存在障碍的情况下，这些优化问题变得非凸并且很难解决，即使只是局部解决。因此，当面对杂乱的环境时，机器人专家通常会求助于基于采样的规划器，这些规划器不能很好地扩展到高维度，并且难以应对连续的差分约束。在这里，我们提出了一个框架，使凸优化能够有效、可靠地规划围绕障碍物的轨迹。具体来说，我们专注于无碰撞运动规划，其成本和轨迹的形状、持续时间和速度受到限制。使用在凸集图（GCS）中寻找最短路径的最新技术，我们设计了规划问题的实用凸松弛。我们表明，这种松弛通常非常严格，以至于其解决方案的廉价后处理几乎总是足以识别全局最优的无碰撞轨迹（在参数化曲线类内）。通过数值和硬件实验，我们证明了我们的规划器（我们称之为 GCS）可以在比广泛使用的基于采样的算法更短的时间内找到更好的轨迹，并且可以在高维复杂环境中可靠地设计轨迹。