Causal end-effector inversion of a flexible link manipulator

A new method to obtain the required base torque for the rest-to-rest manoeuvre of a single flexible link manipulator as its end-effector moves along a desired trajectory is introduced. Contrary to available causal and non-causal end-effector inversion techniques, this new approach does not require pre- and/or post-actuation and works even in the presence of the purely imaginary zeros for the transfer function. In this approach, the desired end-effector trajectory is divided into a finite number of segments. In each segment, but the last one, the desired trajectory is redefined so that a bounded continuous torque can be obtained using causal dynamic inversion. For the last segment the desired trajectory is redefined so that not only the causal inversion is achieved but also the final conditions are satisfied; that is, the end-effector reaches its desired position and the manipulator comes to rest at a given time. The redefinition of the desired trajectory at each segment employs summation of the stable exponential functions, which leads to a family of possible answers for the redefined trajectory. To obtain the best member of this family which minimizes the difference between the desired trajectory and redefined trajectory, an efficient search algorithm is proposed and used. Simulation results show the effectiveness of this new method.