The Shortest-Path Problem in Graph of Convex Sets (SPP in GCS) is a recently developed optimization framework that blends discrete and continuous decision making. Many relevant problems in robotics, such as collision-free motion planning, can be cast and solved as an SPP in GCS, yielding lower-cost solutions and faster runtimes than state-of-the-art algorithms. In this paper, we are motivated by motion planning of robot arms that must operate swiftly in static environments. We consider a multi-query extension of the SPP in GCS, where the goal is to efficiently precompute optimal paths between given sets of initial and target conditions. Our solution consists of two stages. Offline, we use semidefinite programming to compute a coarse lower bound on the problem's cost-to-go function. Then, online, this lower bound is used to incrementally generate feasible paths by solving short-horizon convex programs. For a robot arm with seven joints, our method designs higher quality trajectories up to two orders of magnitude faster than existing motion planners.

翻译：凸集图最短路径问题是近期发展的一种融合离散与连续决策的优化框架。机器人学中的许多相关问题，例如无碰撞运动规划，均可建模并求解为凸集图最短路径问题，相比现有先进算法能获得更低成本的解与更快的运行时间。本文受机器人机械臂在静态环境中需快速运行的运动规划问题启发，研究了凸集图最短路径问题的多查询扩展，其目标在于高效预计算给定初始条件集与目标条件集之间的最优路径。我们的解决方案包含两个阶段：离线阶段使用半定规划计算问题成本至目标函数的粗略下界；在线阶段则利用该下界，通过求解短时域凸规划逐步生成可行路径。对于七关节机器人机械臂，本方法设计的轨迹质量更高，且规划速度比现有运动规划器快达两个数量级。