Recently, many reactive trajectory planning approaches were suggested in the literature because of their inherent immediate adaption in the ever more demanding cluttered and unpredictable environments of robotic systems. However, typically those approaches are only locally reactive without considering global path planning and no guarantees for simultaneous collision avoidance and goal convergence can be given. In this paper, we study a recently developed circular field (CF)-based motion planner that combines local reactive control with global trajectory generation by adapting an artificial magnetic field such that multiple trajectories around obstacles can be evaluated. In particular, we provide a mathematically rigorous analysis of this planner in a planar environment to ensure safe motion of the controlled robot. Contrary to existing results, the derived collision avoidance analysis covers the entire CF motion planning algorithm including attractive forces for goal convergence and is not limited to a specific choice of the rotation field, i.e., our guarantees are not limited to a specific potentially suboptimal trajectory. Our Lyapunov-type collision avoidance analysis is based on the definition of an (equivalent) two-dimensional auxiliary system, which enables us to provide tight, if and only if conditions for the case of a collision with point obstacles. Furthermore, we show how this analysis naturally extends to multiple obstacles and we specify sufficient conditions for goal convergence. Finally, we provide a challenging simulation scenario with multiple non-convex point cloud obstacles and demonstrate collision avoidance and goal convergence.

翻译：近期，文献中提出了多种反应性轨迹规划方法，因其能在机器人系统日益复杂、不可预测的杂乱环境中实现即时自适应调整。然而，这类方法通常仅具有局部反应性，未考虑全局路径规划，无法同时保障避障与目标收敛。本文研究了一种基于圆形场（CF）的运动规划器，该方法通过调整人工磁场以评估障碍物周围的多种轨迹，将局部反应控制与全局轨迹生成相结合。具体而言，我们在平面环境中对该规划器进行了严谨的数学分析，以确保受控机器人的安全运动。与现有结果不同，本文推导的避障分析覆盖了整个CF运动规划算法（包括用于目标收敛的吸引力），且不局限于特定旋转场的选择（即我们的保障不局限于某条潜在次优轨迹）。我们的李雅普诺夫型避障分析基于（等效）二维辅助系统的定义，从而能够为点障碍物碰撞情形提供紧致的充要条件。此外，我们展示了该分析如何自然扩展至多障碍物场景，并给出了目标收敛的充分条件。最后，我们通过包含多个非凸点云障碍物的挑战性仿真场景，验证了避障与目标收敛性能。