We consider large-scale, implicit-search-based solutions to 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 former implementation is complete and cost-optimal, while the latter 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：本文研究凸集图最短路径问题的大规模隐式搜索解决方案。我们提出GCS*算法，这是一种将A*搜索推广至凸集图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，而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-连续混合规划在诸多领域具有需求，包括障碍物避让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沿整条路径选取的连续值点。我们证明，通过剪枝在终端顶点上完全成本占优的路径，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，通过剪枝可达性占优路径实现加速。我们采用多面体包含检测或基于采样的方法实现上述检查。前者具有完备性与成本最优性，后者具有概率完备性与渐近成本最优性，且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上验证GCS*算法，其中接触模式的组合爆炸使得现有方法难以处理，实验表明其性能优于当前最优方法。项目网站：https://shaoyuan.cc/research/gcs-star/