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Research Papers

Stability of Controlled Mechanical Systems With Ideal Coulomb Friction

[+] Author and Article Information
Soo Jeon

Department of Mechanical Engineering, University of California at Berkeley, Berkeley, CA 94720soojeon@newton.berkeley.edu

Masayoshi Tomizuka

Department of Mechanical Engineering, University of California at Berkeley, Berkeley, CA 94720tomizuka@me.berkeley.edu

J. Dyn. Sys., Meas., Control 130(1), 011013 (Jan 14, 2008) (9 pages) doi:10.1115/1.2807069 History: Received May 12, 2006; Revised April 25, 2007; Published January 14, 2008

When a mechanical system with Coulomb friction is under feedback control, the closed-loop system may asymptotically converge to a point in the equilibrium set or generate nonlinear oscillations such as limit cycles depending on the control algorithm. Thus, it is important to know how to guarantee the stability in the presence of Coulomb friction. This paper presents the stability analysis of controlled mechanical systems with multiple ideal Coulomb friction sources. Common properties of controlled mechanical systems with multiple ideal Coulomb friction sources have been explored and generalized into the state space formulation leading to a class of ideal relay feedback systems. Various stability criteria are considered and a new sufficient condition for the pointwise global stability is suggested. Simulation results for a single mass system and experimental results for a single link flexible joint mechanism are presented to confirm the analysis and to illustrate various aspects of stability conditions for controlled mechanical systems with ideal Coulomb friction. The results given in this paper can be useful for the design of mechanical systems free from the limit cycle.

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Copyright © 2008 by American Society of Mechanical Engineers
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Figures

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Figure 1

Schematic diagram of a controlled linear system with multiple Coulomb friction forces

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Figure 2

Diagram for typical relations between index sets

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Figure 3

The existence of attractive regular sliding mode: (a) cibi>0 and (b) cibi<0

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Figure 4

Equilibrium set for two Coulomb friction sources

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Figure 5

State trajectory of the single mass system with position loop PID control

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Figure 6

State trajectory of the single mass system with the controller in Eq. 61

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Figure 7

Schematic of a single link flexible joint mechanism

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Figure 8

Block diagram for simulation and experiment

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Figure 9

Experimental result of control input u(t) for all the desired frequency pairs

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Figure 10

Comparison of simulation and experimental result for frequency pair (f1,f2)a: (a) control input u(t) and (b) motor position θm(t)

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Figure 11

Comparison of simulation and experimental result for frequency pair (f1,f2)b: (a) control input u(t) and (b) motor position θm(t)

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