On the feedforward torques and reference trajectory for flexible two-link arm

A method for synthesizing the feedforward torques and reference trajectory for a flexible two-link planar manipulator is proposed. These torques and the trajectory are not far from being time-optimal. The control law for each drive torque consists of two parts: a commanded feedforward torque and a linear angular position and velocity feedback. Initially, we theoretically prove that so-called 'fluent' control enables an individual link to reach the desired hub acceleration with little error and avoid unwanted large flexible vibrations. With this control, the dynamics of a flexible link is close to the dynamics of the associated rigid link. We begin the design of the feedforward torques and reference trajectory by computing the control torques for an arm with two rigid links which are near time-optimal control functions. These torques are discontinuous functions of time. The jumps in the control torques are not acceptable for an arm with flexible links because large elastic vibrations appear in the links. This conclusion follows from a theoretical study of individual links and also from experimentation with one-link and two-link flexible arms. Therefore, fluent control is considered here. Designing this fluent control, we transform the obtained discontinuous control into continuous functions of time, taking into account the elasticity of flexible links. During the experiments with flexible arms, these continuous functions are used as commanded feedforward torques. We also feed on-line these control torques to a mathematical model of the arm with two rigid links and compute the corresponding desired angular velocities and positions of the links as functions of time. Then we feed this reference trajectory to the linear feedback system (usual PD-controller) of the flexible arm. Designed control does not excite large elastic vibrations although, simultaneously, it is not far from being time-optimal. The designed control algorithm has been successfully implemented in the experiments.