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A theory for planning collision-free paths of a moving object among obstacles is described. Using the concepts of state space and rotation mapping, the relationship between the positions and the corresponding collision-free orientations of a moving object among obstacles is represented as some set of a state space. This set is called the rotation mapping graph (RMG) of that object. The problem of finding collision-free paths for an object translating and rotating among obstacles is thus transformed to that of considering the connectivity of the RMG. Since the connectivity of the graph can be solved by topological methods, the problem of planning collision-free paths is easily solved in theory. Using this theory, a topological method for planning collision-free paths of a rod-object translating and rotating among obstacles is presented. If a nonrigid robotic arm is viewed as a composite rod with some degrees of freedom, the planning of collision-free paths of a robotic arm can be solved in a similar way to a rod.