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An algorithm for determining the minimum time trajectory (path and velocity along that path) for a robot arm with actuator constraints is presented. The method combines the control theory and the exhaustive search approaches to yield an algorithm that is faster than either approach. The method consists of first parameterizing the path in the configuration space, then, given a path, using control theory to determine the minimum time trajectory subject to the actuator torque constraints, and finally searching among all possible paths to find the minimum time path. The minimum time path is assumed to be smooth and hence is parameterized by splines. In presence of obstacles in the work-space, the algorithm can start with a path from a path planner and determine the minimum time path close to the given path. The results for a two degree of freedom arm are presented.