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Research Papers

The Application of Command Shaping to the Tracking Problem

[+] Author and Article Information
Michael C. Reynolds

Mechanical Engineering, University of Arkansas–Fort Smith, 5210 Grand Avenue, P.O. Box 3649, Fort Smith, AR 72913mreynold@uafortsmith.edu

Peter H. Meckl

Mechanical Engineering, Purdue University, West Lafayette, IN 47907-2088meckl@purdue.edu

J. Dyn. Sys., Meas., Control 130(3), 031007 (Apr 24, 2008) (12 pages) doi:10.1115/1.2907379 History: Received February 13, 2006; Revised December 22, 2007; Published April 24, 2008

This work presents a novel technique for the solution of an optimal input for trajectory tracking. Many researchers have documented the performance advantages of command shaping, which focuses on the design of an optimal input. Nearly all research in command shaping has been centered on the point-to-point motion control problem. However, tracking problems are also an important application of control theory. The proposed optimal tracking technique extends the point-to-point motion control problem to the solution of the tracking problem. Thus, two very different problems are brought into one solution scheme. The technique uses tolerances on trajectory following to meet constraints and minimize either maneuver time or input energy. A major advantage of the technique is that hard physical constraints such as acceleration or allowable tracking error can be directly constrained. Previous methods to perform such a task involved using various weightings that lack physical meaning. The optimal tracking technique allows for fast and efficient exploration of the solution space for motion control. A solution verification technique is presented and some examples are included to demonstrate the technique.

Copyright © 2008 by American Society of Mechanical Engineers
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References

Figures

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Figure 1

Two-mass rotational system and free body diagram

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Figure 2

Control structure for the implementation of the time-optimal tracking controller

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Figure 3

Input energy versus maneuver time for system of Eq. 4

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Figure 4

Acceleration amplitude versus time for time-optimal input and time-optimal acceleration amplitude bounded input

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Figure 5

Input amplitude versus time for time-optimal input and time-optimal acceleration amplitude bounded input

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Figure 6

Time-optimal response without tracking constraint and the desired reference

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Figure 7

Time-optimal response with tracking constraint (2deg) and the desired reference

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Figure 8

Time-optimal input for response of Fig. 6

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Figure 9

Time-optimal tracking-constrained minimum-energy input for response of Fig. 7

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Figure 10

Acceleration response to time-optimal tracking and acceleration (15deg∕s2) constrained input

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Figure 11

Time-optimal tracking and acceleration (15deg∕s2) constrained input

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Figure 12

System response to true inverse (reference) and minimum quadratic input

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Figure 13

Constrained minimum-energy input for system of Eq. 4

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Figure 14

System response to constrained minimum-energy input and reference

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Figure 15

Simplified schematic of Quanser gantry crane

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Figure 16

Experimental and theoretical responses to optimal input

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