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This paper addresses the challenging problem of finding collision-free trajectories for many robots moving toward individual goals within a common environment. Most popular algorithms for multirobot planning manage the complexity of the problem by planning trajectories for robots individually; such decoupled methods are not guaranteed to find a solution if one exists. In contrast, this paper describes a multiphase approach to the planning problem that uses a graph and spanning tree representation to create and maintain obstacle-free paths through the environment for each robot to reach its goal. The resulting algorithm guarantees a solution for a well-defined number of robots in a common environment. The computational cost is shown to be scalable with complexity linear in the number of the robots, and demonstrated by solving the planning problem for 100 robots, simulated in an underground mine environment, in less than 1.5 s with a 1.5 GHz processor. The practicality of the algorithm is demonstrated in a real-world application requiring coordinated motion planning of multiple physical robots.