| We consider the problem of input shaping, a general feedforward approach for reducing residual vibration in flexible structures, while allowing rapid maneuvers to be achieved. Input shaping techniques have been successfully applied and continue to be investigated for applications in a variety of systems such as robotic manipulators, disk-drive heads, and pointing systems. Since previous input shaping methods do not consider parameter uncertainty in their derivations, we propose two shaping methods that account for parameter variations.;We also analyze fuel-efficient shapers in the context of optimal control theory. We provide a mathematical justification for the existence of the different control profiles that arise in this type of problem. We demonstrate some interesting symmetry properties of optimal commands for undamped flexible structures and also discuss the characteristics of robust optimal commands that are less sensitive to parameter variations.;Finally, we provide a proof of the equivalence between two seemingly different problems: the time-optimal control problem and the minimum time input shaping problem. It has been conjectured for some time that these two problems are equivalent, but a rigorous proof of the conjecture has remained elusive. We make use of the well known Karush-Kuhn-Tucker optimality conditions to establish the equivalence conjecture. |
