Given a convex potential in a space with convex obstacles, an artificial potential is used to navigate to the minimum of the natural potential while avoiding collisions. The artificial potential combines the natural potential with potentials that repel the agent from the border of the obstacles. This is a popular approach to navigation problems because it can be implemented with spatially local information that is acquired during operation time. Artificial potentials can, however, have local minima that prevent navigation to the minimum of the natural potential. This paper derives conditions that guarantee artificial potentials to have a single minimum that is arbitrarily close to the minimum of the natural potential. The qualitative implication is that artificial potentials succeed when either the condition number—the ratio of the maximum and the minimum eigenvalue—of the Hessian of the natural potential is not large and the obstacles are not too flat, or when the destination is not close to the border of an obstacle. Numerical analyses explore the practical value of these theoretical conclusions.

中文翻译：

---

具有凸障碍物的空间中凸势的导航函数


给定具有凸障碍物的空间中的凸势，可以使用人工势来导航到自然势的最小值，同时避免碰撞。人工势将自然势与将智能体排斥出障碍物边界的势相结合。这是解决导航问题的一种流行方法，因为它可以利用在操作期间获取的空间局部信息来实现。然而，人工势可能具有局部最小值，从而阻止导航至自然势的最小值。本文推导了保证人工势具有任意接近自然势最小值的单个最小值的条件。定性的含义是，当自然势的 Hessian 矩阵的条件数（最大特征值与最小特征值之比）不大并且障碍物不是太平坦时，或者当目的地不接近时，人工势就会成功。障碍物的边界。数值分析探讨了这些理论结论的实用价值。