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Research Papers

# Damping-Induced Self Recovery Phenomenon in Mechanical Systems With an Unactuated Cyclic Variable

[+] Author and Article Information
Dong Eui Chang

Department of Applied Mathematics,
University of Waterloo,
e-mail: dechang@uwaterloo.ca

Soo Jeon

Department of Mechanical and Mechatronics Engineering,
University of Waterloo,
e-mail: soojeon@uwaterloo.ca

Some books have a different definition of a cyclic variable. For example, in Ref. [12] a variable is defined to be cyclic if the kinetic energy is independent of the variable, not requiring that the potential energy be also independent of the variable. In this paper, we follow the definition in Ref. [4] since it is the one more widely accepted. Cyclic variables are sometimes called ignorable coordinates [6].

A different configuration space might be considered such as $=Tk×Rn-k$ where $Tk$ is the k– torus, i.e., the k-fold Cartesian product of the unit circle $S1$, but for the sake of simplicity we have chosen $Rn$ as our configuration space.

If an index is repeated, a summation is made over the index.

A function f is called the piecewise continuous if it is continuous at all but a finite number of points and, at each point of discontinuity, say $x0$, both $limx→x0-f(x)$ and $limx→x0+f(x)$ exist.

Contributed by the Dynamic Systems Division of ASME for publication in the Journal of Dynamic Systems, Measurement, and Control. Manuscript received December 2, 2011; final manuscript received June 26, 2012; published online November 7, 2012. Assoc. Editor: Won-jong Kim.

J. Dyn. Sys., Meas., Control 135(2), 021011 (Nov 07, 2012) (6 pages) Paper No: DS-11-1369; doi: 10.1115/1.4007556 History: Received December 02, 2011; Revised June 26, 2012

## Abstract

Conservation of momentum is often used in controlling underactuated mechanical systems with symmetry. If a symmetry-breaking force is applied to the system, then the momentum is not conserved any longer in general. However, there exist forces linear in velocity such as the damping force that break the symmetry but induce a new conserved quantity in place of the original momentum map. This paper formalizes a new conserved quantity which can be constructed by combining the time integral of a general damping force and the original momentum map associated with the symmetry. From the perspective of stability theories, the new conserved quantity implies the corresponding variable possesses the self recovery phenomenon, i.e., it will be globally attractive to the initial condition of the variable. We discover that what is fundamental in the damping-induced self recovery is not the positivity of the damping coefficient but certain properties of the time integral of the damping force. The self recovery effect and theoretical findings are demonstrated by simulation results using the two-link planar manipulator and the torque-controlled inverted pendulum on a passive cart. The results in this paper will be useful in designing and controlling mechanical systems with underactuation.

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## Figures

Fig. 1

Rotating stool and a bicycle wheel

Fig. 2

Schematic of a two link planar manipulator

Fig. 3

Joint trajectories with a controlled step input for q2

Fig. 4

Joint trajectories of the two link arm with the large displacement of q2

Fig. 5

Torque-controlled inverted pendulum on a passive cart

Fig. 6

Trajectories of torque-controlled pendulum on a passive cart

Fig. 7

Trajectories of torque-controlled pendulum on a passive cart with kv=0.5(1+cos(q1/rω))N·s/m and Coulomb friction force

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