In this paper, we study a trajectory tracking problem for a class of time-delayed robotic manipulator systems. The approach described in this paper can also be applied to some more general time-delayed nonlinear control systems. We demonstrate the basic ideas and techniques by working through a specific flexible-joint robot arm model. We first linearize the nonlinear robot arm model in the controller design for trajectory tracking, where a closed-form analytic solution of the tracking problem is derived under a minimum control-energy criterion. We then combine all the unmodelled and unknown effects on the original system, such as the unmodelled nonlinearities and flexibilities, linearization errors, and unknown parameters as uncertainties, to study the robust stability of the control system subject to these uncertainties. We obtain conditions for maximum allowable variations of system parameters, including time delays, for robust stability of the linearized model. We also discuss two different controllers, the standard proportional-derivative controller and the Smith predictor, under two different conditions on the maximum allowable time-delay constant, for the trajectory tracking performance and robust stability analyses.
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