The distance between sets is a long-standing computational geometry problem. In robotics, the distance between convex sets with Minkowski sum structure plays a fundamental role in collision detection. However, it is typically nontrivial to be computed, even if the projection onto each component set admits explicit formula. In this paper, we explore the problem of calculating the distance between convex sets arising from robotics. Upon the recent progress in convex optimization community, the proposed model can be efficiently solved by the recent hot-investigated first-order methods, e.g., alternating direction method of multipliers or primal-dual hybrid gradient method. Preliminary numerical results demonstrate that those first-order methods are fairly efficient in solving distance problems in robotics.

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Minkowski和结构的凸集之间的距离：在碰撞检测中的应用

组之间的距离是一个长期存在的计算几何问题。在机器人技术中，具有Minkowski和结构的凸集之间的距离在碰撞检测中起着基本作用。但是，即使投影到每个组件集上都包含显式公式，通常也很容易计算。在本文中，我们探讨了计算机器人引起的凸集之间距离的问题。随着凸优化社区的最新发展，所提出的模型可以通过最近热研究的一阶方法（例如乘数的交替方向方法或原始对偶混合梯度法）有效地求解。初步的数值结果表明，这些一阶方法在解决机器人技术中的距离问题方面相当有效。