Input Shaping With Negative Sequences for Reducing Vibrations in Flexible Structures

Input shaping by convolving system commands with impulse sequences has been shown to be an effective method of reducing residual vibrations in flexible systems [1, 3, 5, 6, 7]. The three-impulse sequence developed by Singer & Seering [6] extends the move duration by the period of the vibrational frequency while eliminating residual vibrations at that frequency. In Singer & Seering [6], the input shaping sequence is constrained so that frequencies other than the one being shaped are not excited and actuator limits are not exceeded. By carefully relaxing these constraints, we can generate input shaping sequences with move times shorter than those of the three-impulse sequence while providing comparable vibration reduction and insensitivity to modeling errors. This paper presents five different methods for calculating sequences with relaxed constraints. The first two methods permit the excitation of certain frequencies, but maintain the constraint on actuator limits. Both an analytic method and a method of optimizing using non-linear programming are presented. Third, a method is presented which uses the optimization routines to permit exceeding the steady-state actuator limits while staying within peak actuator limits. Fourth, an alternate constraint which enables greater insensitivity to system frequency variations is discussed. Finally, a method for constraining the excitation of specific higher frequencies is presented. The various techniques are demonstrated on a one-link flexible beam.