The application of command shaping to the tracking problem

This paper shows how the framework of a time-optimal point-to-point motion control problem can be extended to a single-axis trajectory tracking problem. Thus, the gap between these very different problems will be bridged in a way that has never been shown. The benchmark floating oscillator (as a rotational system) is considered and equations are derived that enforce tracking, acceleration and actuator rate limitations. By forming the discrete model, the time-optimal tracking problem can be solved with a least squares programming scheme. Tolerances on tracking, acceleration amplitude throughout the maneuver and actuator rate limits are then expressed explicitly into the optimization problem. Unlike most techniques that rely on weighting matrices that have little physical meaning, this paper develops tracking, vibration and actuator rate tolerances that come directly from the described tolerances. Results serve to demonstrate each of the constraints.