Vibration Reduction Using Multi-Hump Extra-Insensitive Input Shapers

Input shaping is a method for reducing residual vibrations in computer controlled machines. Vibration is eliminated by convolving an input shaper, which is a sequence of impulses, with a desired system command to produce a shaped input. The shaped input then becomes the command to the system. Requiring the vibration reduction to be insensitive to modeling errors and system nonlinearities is critical to the success of the shaping process on any real system. Input shapers can be made very insensitive to parameter uncertainty; however, increasing insensitivity usually increases system delays. A design process is presented that generates input shapers with insensitivity-totime-delay ratios that are much larger than traditionally designed input shapers. The advantages of the new shapers are demonstrated with simulations of a simple linear system and simulations of the MACE experimental apparatus. Introduction Input shaping is a method of reducing residual vibrations in computer controlled machines. The method requires only a simple system model consisting of estimates of the natural frequencies and damping ratios. Input shaping is implemented by convolving an input shaper, which is made up of a sequence of impulses, with a desired system command to produce a shaped input that is then used to command the system. The convolution process lengthens the command signal by an amount equal to the time duration of the input shaper. Therefore, it is desirable to make the input shaper as short as possible, so that system delays are minimized. However, traditional design methods require that the input shaper be lengthened if additional insensitivity to modeling errors is required. This paper will present an algorithm for increasing insensitivity without increasing shaper length. During its original presentation [13, 14], input shaping was explained by a variety of methods, including time domain analysis, vector diagram representation, frequency domain analysis, phase plane description, and pole-zero cancellation in the s-plane. The vector diagram was used to improve insensitivity [18] and cancel multiple modes of vibration [15]. The frequency domain and pole-zero cancellation representations have been investigated in several papers [2, 8, 10, 12, 16, 21]. Input shaping was shown to reduce residual vibration and maximum deflections during the slewing of a large nonlinear space-based antenna [1] and long-reach manipulators [7, 10]. Two-mode input shapers were used to increase the throughput of a silicon wafer handling robot [11]. Input shaping has been extended to systems equipped only with constant-amplitude actuators [9, 16, 17, 23]. Trajectory-following applications have also been shown to benefit from input shaping [4, 19]. A brief review of input shaping will be given in the next section. The new input shapers are then designed and their vibration-reducing properties are explained in the s-plane. Computer simulations are then used to demonstrate the advantages of the new shapers. Conclusions will be presented in the final section. -0.5 0 0.5 1 0 1 2 3 4 Response to A1 Response to A2 Response to A1 & A2