We consider geometric collision-detection problems for modular reconfigurable robots. Assuming the nodes (modules) are connected squares on a grid, we investigate the complexity of deciding whether collisions may occur, or can be avoided, if a set of expansion and contraction operations is executed. We study both discrete- and continuous-time models, and allow operations to be coupled into a single parallel group. Our algorithms to decide if a collision may occur run in $O(n^2\log^2 n)$ time, $O(n^2)$ time, or $O(n\log^2 n)$ time, depending on the presence and type of coupled operations, in a continuous-time model for a modular robot with $n$ nodes. To decide if collisions can be avoided, we show that a very restricted version is already NP-complete in the discrete-time model, while the same problem is polynomial in the continuous-time model. A less restricted version is NP-hard in the continuous-time model.

翻译：我们研究了模块化可重构机器人的几何碰撞检测问题。假设节点（模块）是网格上相连的正方形，我们探讨在执行一组扩张和收缩操作时，判断是否可能发生碰撞或能否避免碰撞的计算复杂度。我们研究了离散时间模型和连续时间模型，并允许将操作耦合为单个并行组。在连续时间模型中，对于具有n个节点的模块化机器人，根据是否存在耦合操作及其类型，判断是否可能发生碰撞的算法时间复杂度分别为$O(n^2\log^2 n)$、$O(n^2)$或$O(n\log^2 n)$。对于判断能否避免碰撞的问题，我们证明在离散时间模型中，一个非常受限的版本已是NP完全的，而相同问题在连续时间模型中为多项式时间可解。在连续时间模型中，一个限制较少的版本是NP难的。