A convergence proof of a fusion algorithm of global and local path-planning for multiple mobile robots

First of all, a global path-planning algorithm synchronously selects the shortest paths of multiple mobile robots. Therefore, it is time consuming because of the hard combination search. On the other hand, a local path-planning algorithm independently selects a shorter path of each mobile robot. Therefore, it is not time consuming because of the non-combination search. However, we must consider the following two problems. (1) Several types of collisions of two or more robots should be solved. (2) Convergence of all robots to their destinations should be theoretically ensured. To solve these problems, we design a fusion algorithm by mixing global and local path-planning algorithms, and then we give a proof such that all robots arrive at their destinations