用ADMM求解非凸环境中的连续轨迹，效率更高且易收敛。
ADMM-based Continuous Trajectory Optimization in Graphs of Convex Sets
- 将轨迹参数化为多项式，闭式求解控制能耗最小问题。
- 构建时空分配图，将松弛变量更新转化为最短路径搜索。
- 可从简单初始值稳定收敛，适合复杂非凸场景的轨迹规划。
本文提出一种用于计算非凸环境中连续轨迹的数值求解器。该方法基于定制化的交替方向乘子法（ADMM），核心包含两点：首先，将轨迹参数化为多项式，使原始变量更新可通过闭式解求解，等价于最小控制努力问题；其次，引入基于混合整数规划的时空分配图，并将松弛变量更新建模为最短路径搜索。上述设计使求解器在多个方面优于现有方法：通过联合优化离散空间与连续时间域，探索更大搜索空间，发现更优轨迹；同时，其结构鲁棒性确保了从简单初始值即可可靠收敛，避免了非凸环境下复杂热启动的瓶颈。
原文摘要 · Abstract (English)
This paper presents a numerical solver for computing continuous trajectories in non-convex environments. Our approach relies on a customized implementation of the Alternating Direction Method of Multipliers (ADMM) built upon two key components: First, we parameterize trajectories as polynomials, allowing the primal update to be computed in closed form as a minimum-control-effort problem. Second, we introduce the concept of a spatio-temporal allocation graph based on a mixed-integer formulation and pose the slack update as a shortest-path search. The combination of these ingredients results in a solver with several distinct advantages over the state of the art. By jointly optimizing over both discrete spatial and continuous temporal domains, our method accesses a larger search space than existing decoupled approaches, enabling the discovery of superior trajectories. Additionally, the solver's structural robustness ensures reliable convergence from naive initializations, removing the bottleneck of complex warm starting in non-convex environments.
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