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A task that robotic manipulators most frequently perform is motion between specified points in the working space. It is therefore important that these motions are efficient. The presence of the obstacles and other requirements of the task often require that the path is specified in advance. Robot actuators cannot generate unlimited forces/torques so it is reasonable to ask how to traverse the prespecified path in minimum time so that the limits on the actuator torques are not violated.
It can be shown that the motion which requires least time to traverse a path requires at least one actuator to operate on the boundary (maximum or minimum). Furthermore, if the path is parameterized, the equations describing the robot dynamics can be rewritten as functions of the path parameter and its first and second derivatives. In general, the actuator bounds will be transformed into the bounds on the acceleration along the path. These bounds will be functions of the velocity and position. It is possible to demonstrate that the optimal motion will be almost always bang-bang in acceleration. The task of finding the optimal torques thus reduces to finding the instants at which the acceleration will switch between the boundaries.
An algorithm for finding the time optimal motion along prespecified paths that explores this idea will be presented. It will be shown that so called singular arcs exist on which the algorithm will fail. Modification of the algorithm for such situations will be presented. Also, some properties of the solutions of the more general problem when the path is not known will be discussed. Lie-algebraic techniques will be shown to be a convenient tool for the study of such problems.
Date Posted: 19 September 2006