We consider large-scale, implicit-search-based solutions to the Shortest Path Problems on Graphs of Convex Sets (GCS). We propose GCS*, a forward heuristic search algorithm that generalizes A* search to the GCS setting, where a continuous-valued decision is made at each graph vertex, and constraints across graph edges couple these decisions, influencing costs and feasibility. Such mixed discrete-continuous planning is needed in many domains, including motion planning around obstacles and planning through contact. This setting provides a unique challenge for best-first search algorithms: the cost and feasibility of a path depend on continuous-valued points chosen along the entire path. We show that by pruning paths that are cost-dominated over their entire terminal vertex, GCS* can search efficiently while still guaranteeing cost optimality and completeness. To find satisficing solutions quickly, we also present a complete but suboptimal variation, pruning instead reachability-dominated paths. We implement these checks using polyhedral-containment or sampling-based methods. The sampling-based implementation is probabilistically complete and asymptotically cost optimal, and performs effectively even with minimal samples in practice. We demonstrate GCS* on planar pushing tasks where the combinatorial explosion of contact modes renders prior methods intractable and show it performs favorably compared to the state-of-the-art. Project website: https://shaoyuan.cc/research/gcs-star/

翻译：本文研究基于隐式搜索的大规模凸集图最短路径问题（GCS）求解方法。我们提出GCS*算法，这是一种将A*搜索推广至GCS场景的前向启发式搜索算法。在该场景中，每个图顶点需进行连续值决策，且跨越图边的约束会耦合这些决策，从而影响路径成本与可行性。此类离散-连续混合规划在众多领域具有需求，包括避障运动规划与接触式运动规划。该场景为最佳优先搜索算法带来了独特挑战：路径的成本与可行性取决于整条路径上选取的连续值点。我们证明，通过剪枝在终端顶点处全程受成本支配的路径，GCS*能够在保证成本最优性与完备性的同时实现高效搜索。为快速获取满意解，我们还提出一种完备但次优的变体算法，转而剪枝可达性支配路径。我们采用多面体包含检测或基于采样的方法实现这些检查机制。基于采样的实现具有概率完备性与渐近成本最优性，在实践中即使使用极少样本仍能有效运行。我们在平面推动任务中验证GCS*算法——该任务中接触模式的组合爆炸导致现有方法难以求解，实验表明GCS*相较最先进方法具有优越性能。项目网站：https://shaoyuan.cc/research/gcs-star/