Sampling-based motion planning has emerged as a powerful approach for robotics, enabling exploration of complex, high-dimensional configuration spaces. When combined with Signal Temporal Logic (STL), a temporal logic widely used for formalizing interpretable robotic tasks, these methods can address complex spatiotemporal constraints. However, traditional approaches rely on min-max robustness measures that focus only on critical time points and subformulae, creating non-smooth optimization landscapes with sharp decision boundaries that hinder efficient tree exploration. We propose RRT$^η$, a sampling-based planning framework that integrates the Arithmetic-Geometric Mean (AGM) robustness measure to evaluate satisfaction across all time points and subformulae. Our key contributions include: (1) AGM robustness interval semantics for reasoning about partial trajectories during tree construction, (2) an efficient incremental monitoring algorithm computing these intervals, and (3) enhanced Direction of Increasing Satisfaction vectors leveraging Fulfillment Priority Logic (FPL) for principled objective composition. Our framework synthesizes dynamically feasible control sequences satisfying STL specifications with high robustness while maintaining the probabilistic completeness and asymptotic optimality of RRT$^\ast$. We validate our approach on three robotic systems. A double integrator point robot, a unicycle mobile robot, and a 7-DOF robot arm, demonstrating superior performance over traditional STL robustness-based planners in multi-constraint scenarios with limited guidance signals.

翻译：基于采样的运动规划已成为机器人学中的一种强大方法，能够探索复杂的高维构型空间。当与信号时序逻辑（STL）——一种广泛用于形式化可解释机器人任务的时序逻辑——相结合时，这些方法能够处理复杂的时空约束。然而，传统方法依赖于仅关注关键时间点和子公式的最小-最大鲁棒性度量，从而产生具有尖锐决策边界的非光滑优化空间，阻碍了树结构的高效探索。我们提出了RRT$^η$，一种基于采样的规划框架，它集成了算术-几何平均（AGM）鲁棒性度量，以评估所有时间点和子公式的满足程度。我们的主要贡献包括：（1）用于在树构建过程中对部分轨迹进行推理的AGM鲁棒性区间语义；（2）计算这些区间的高效增量监控算法；（3）利用满足优先级逻辑（FPL）增强的“满意度提升方向”向量，以实现有原则的目标组合。我们的框架能够合成满足STL规约且具有高鲁棒性的动态可行控制序列，同时保持RRT$^\ast$的概率完备性和渐近最优性。我们在三个机器人系统上验证了我们的方法：一个双积分点机器人、一个独轮车移动机器人和一个7自由度机械臂，结果表明在具有有限引导信号的多约束场景中，其性能优于传统的基于STL鲁棒性的规划器。