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Research Papers

Optimal FIR Input Shaper Designs for Motion Control With Zero Residual Vibration

[+] Author and Article Information
Matthew O. T. Cole

Department of Mechanical Engineering, Faculty of Engineering, Chiang Mai University, Chiang Mai 50200, Thailandmotcole@chiangmai.ac.th

Theeraphong Wongratanaphisan

Department of Mechanical Engineering, Faculty of Engineering, Chiang Mai University, Chiang Mai 50200, Thailandwong@dome.eng.cmu.ac.th

J. Dyn. Sys., Meas., Control 133(2), 021008 (Mar 09, 2011) (9 pages) doi:10.1115/1.4003097 History: Received December 16, 2009; Revised August 05, 2010; Published March 09, 2011; Online March 09, 2011

This paper considers the design of input shaping filters used in motion control of vibratory systems. The filters preshape a command or actuation signal in order to negate the effect of vibratory modes. A class of finite impulse response filter satisfying a set of orthogonality conditions that ensure zero residual vibration is introduced. Filter solutions having minimum quadratic gain, both with and without the inclusion of non-negativity (peak gain) constraints, are presented. Unlike impulse-based shapers, the filters have impulse responses with no singularities and therefore automatically remove discontinuities from an input signal. Minimum duration impulse response solutions are also presented. These contain singularities but may also have smooth components. Discrete-time designs can be obtained numerically from system modal parameters, accounting for all modes simultaneously so that convolving single-mode solutions, which leads to suboptimality of the final design, is not required. Selected designs are demonstrated experimentally on a flexible link planar manipulator.

Copyright © 2011 by American Society of Mechanical Engineers
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Figures

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Figure 1

Input shaping of a command signal to drive a vibratory system

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Figure 2

Characteristics of MQG input shapers of different durations

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Figure 3

Comparison of ZV impulse shaper (—) and MQG shaper (- - - - -) of same duration (0.5 cycles)

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Figure 4

Variation in residual vibration with natural frequency of mode normalized by the design frequency: comparison of ZV impulse shaper and MQG shaper of the same duration (0.5 cycles)

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Figure 5

Characteristics of robust quadratic input shapers (J=∫h2dt, ∫hdt=1) of different durations. Orthogonality conditions include the first derivatives with respect to changes in natural frequency.

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Figure 6

Residual vibration due to changes in system natural frequency from nominal value: comparison of robust (—-) and nonrobust (- - - - -) quadratic input shapers of same duration (0.5 cycles)

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Figure 7

Minimum quadratic gain input shapers satisfying a non-negativity constraint (∫hdt=∫|h|dt=1). The solution with minimum duration of 0.5 cycles is two impulses. For durations greater than 0.68 cycles, the non-negativity constraint has no influence on the solution.

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Figure 8

Robust minimum quadratic gain shapers satisfying a non-negativity constraint (∫hdt=∫|h|dt=1). The solution with minimum duration of 1.0 cycles is three impulses. For durations greater than 1.18 cycles, the non-negativity constraint has no influence on the solution.

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Figure 9

Input shapers for a two-mode system: (—) h formed by convolving separate MQG designs for each mode; (- - - - -) h synthesized from combined modal data

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Figure 10

Non-negative shapers for a two-mode system: (—) h formed by convolving separate minimum duration designs for each mode; (- - - - -) true minimum duration solution synthesized from combined modal data

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Figure 11

Variation in duration of time-optimal non-negative shaper with ratio of natural frequencies for a two-mode system

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Figure 12

Two-link manipulator with lightweight flexible end link

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Figure 13

Behavior of the end link when a ramped reference command drives the joint angle control loop

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Figure 14

Behavior of the end link when an input shaper is used to filter a ramped reference command driving the joint angle control loop: (a) ZV impulse shaper and (b) MQG shaper of the same duration

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Figure 15

Behavior of end link when a two-mode MQG shaper is used to filter a ramped reference command driving the joint angle control loop

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Figure 16

Behavior of the end link when a short duration (0.3 cycles) two-mode MQG shaper is used to filter a ramped reference command driving the joint angle control loop

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