INPUT SHAPING FOR VIBRATION REDUCTION WITH SPECIFIED INSENSITIVITY TO MODELING ERRORS

William E. Singhose andWarren P. SeeringDepartment of Mechanical EngineeringMassachusetts Institute of TechnologyCambridge, MANeil C. SingerConvolve, Inc.New York, NYABSTRACTInput shaping reduces residual vibration by generating acommand signal that is self canceling. That is, any vibrationcaused by a portion of the command is canceled by a laterportion of the command. A command with this special propertyis formed by convolving a sequence of impulses with theoriginal, unmodified system command. Due to modeling errorsand disturbances, the residual vibration will never be exactlyzero on real systems. Robustness constraints must be enforcedto ensure effective input shaping when the system frequenciesare not known precisely or are expected to vary with time.Traditionally, the degree of robustness has not been specified,rather robustness was obtained by enforcing a reasonablerobustness constraint. Procedures for precisely specifying thedegree of robustness are presented. Characteristics of theimpulse sequence as a function of robustness and systemdamping are discussed.INTRODUCTIONInput shaping improves throughput and positioning accuracyby reducing residual vibration in computer controlled machines.Input shaping is implemented by convolving a sequence ofimpulses, an input shaper, with a desired system command. Theconvolved signal is then used to drive the system. Theamplitudes and time locations of the impulses are calculatedusing a simple system model consisting of estimates of thenatural frequencies and damping ratios. The input shapingprocess is demonstrated in Figure 1 with a two-impulse shaper.The duration of the shaped input is the duration of the unshapedinput plus the duration of the shaper. The shaping processincreases the rise-time by the duration of the input shaper.In general, a shaper can contain any number of impulses andhave any duration. The challenge of input shaping is to designan input shaper so that a desired level of performance isachieved. A shaper is designed by generating a set of dynamicalconstraint equations which limit the residual vibration andensure some level of robustness to modeling errors. By solvingthe set of constraints, the amplitudes and time locations of theimpulses are determined.If the dynamical constraints only require zero residualvibration, then the resulting shaper is called a zero vibration(ZV) shaper. A ZV shaper will not work well on most systemsbecause it will be sensitive to modeling errors. The earliestincarnation of ZV shaping was the technique of posicast controldeveloped in the 1950’s (Smith, 1957; Smith, 1958; Tallman andSmith, 1958; Cook, 1966; Mee, 1974).To design a shaper that is likely to work well on real systems,the dynamical constraints must ensure robustness to modelingerrors. The earliest form of robust input shaping was achievedby setting the derivative with respect to the frequency of theresidual vibration equal to zero (Singer, 1989; Singer andSeering, 1990). The resulting shaper is called the zero vibrationand derivative (ZVD) shaper.To compare the performance of the robust and non-robustapproaches, we can plot the amplitude of residual vibration as a

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