We study the problem of determining coordinated motions, of minimum total length, for two arbitrary convex centrally-symmetric (CCS) robots in an otherwise obstacle-free plane. Using the total path length traced by the two robot centres as a measure of distance, we give an exact characterization of a (not necessarily unique) shortest collision-avoiding motion for all initial and goal configurations of the robots. The individual paths are composed of at most six convex pieces, and their total length can be expressed as a simple integral with a closed form solution depending only on the initial and goal configuration of the robots. The path pieces are either straight segments or segments of the boundary of the Minkowski sum of the two robots (circular arcs, in the special case of disc robots). Furthermore, the paths can be parameterized in such a way that (i) only one robot is moving at any given time (decoupled motion), or (ii) the orientation of the robot configuration changes monotonically.

翻译：本文研究在无障碍平面上，为两个任意凸中心对称机器人确定总长度最小的协同运动问题。以两个机器人中心点轨迹的总路径长度作为距离度量，我们精确刻画了机器人所有初始与目标配置下的一种（不一定唯一）最短无碰撞运动。各机器人路径最多由六段凸曲线段组成，其总长度可表示为仅依赖于机器人初始与目标配置的闭式解简单积分。这些路径段或是直线段，或是两个机器人闵可夫斯基和边界的曲线段（在圆盘形机器人的特殊情况下为圆弧段）。此外，这些路径可通过参数化实现：（i）任意时刻仅有一个机器人运动（解耦运动），或（ii）机器人配置的朝向单调变化。