In this paper, we discuss an efficient algorithm for computing the growth distance between two compact convex sets with representable support functions. The growth distance between two sets is the minimum scaling factor such that the sets intersect when scaled about some center points. Unlike the minimum distance between sets, the growth distance provides a unified measure for set intersection and separation. We first reduce the growth distance problem to an equivalent ray intersection problem on the Minkowski difference set. Then, we propose an algorithm to solve the ray intersection problem by iteratively constructing inner and outer polyhedral approximations of the Minkowski difference set. We show that our algorithm satisfies several key properties, such as primal and dual feasibility and monotone convergence. We provide extensive benchmark results for our algorithm and show that our open-source implementation achieves state-of-the-art performance across a wide variety of convex sets. Finally, we demonstrate robotics applications of our algorithm in motion planning and rigid-body simulation.

翻译：本文讨论了一种高效算法，用于计算具有可表示支撑函数的两个紧凸集之间的生长距离。两个集合之间的生长距离是使集合在绕某中心点缩放后相交的最小缩放因子。与集合间最小距离不同，生长距离为集合的相交与分离提供了统一度量。我们首先将生长距离问题简化为闵可夫斯基差集上的等价射线相交问题。随后，提出了一种通过迭代构建闵可夫斯基差集的内外多面体近似来求解射线相交问题的算法。我们证明了该算法具有原始可行性与对偶可行性及单调收敛等关键性质。通过广泛的基准测试结果，我们展示了该开源实现能在多种凸集上达到最先进的性能。最后，我们演示了该算法在运动规划与刚体仿真中的机器人应用。