In this paper, an elastic-deformation estimator is proposed for real time end-point tracking control of a flexible two-link manipulator. Due to the noncolocated characteristics of the system, the inverse model (from end-point motion to control torques) is divided into two subsystems, namely, the stable subsystem and the unstable one, corresponding to the causal part and noncausal part of the system’s elastic motion, respectively. A digital filter is formulated to replace the unstable subsystem so as to estimate the noncausal part of the elastic motion associated with a specified end-point motion. For the design of the filter, the frequency response ratio between the filter and the unstable subsystem is used as the criterion, the objective of which is to have the frequency response ratio have zero phase shift as well as unity gain within a specified frequency range. It is shown that due to the noncausal characteristics of the unstable subsystem, preview information of the input trajectory is required for implementing the proposed filter, and the estimation accuracy increases as increasing the preview steps. Based on the stable subsystem and the proposed digital filter, a time-varying estimator is designed to estimate the elastic motion of the system when the end-point motion is specified. A command feedforward controller is then used to calculate the required control torques based on the estimated elastic deformation and the desired end-point motion. The computed torques along with a feedback controller then form a control scheme making the flexible two-link manipulator become capable of precision end-point tracking. Simulation results are presented to show the performance of the proposed end-point tracking scheme as well as the elastic-deformation estimator.

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