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Nonlinear regular dynamics of a single-degree robot model

Published online by Cambridge University Press:  09 March 2009

V. Paar ,
N. Pavin ,
N. Paar  and
B. Novaković
V. Paar
Affiliation:
Department of Physics, Faculty of Science, University of Zagreb, Zagreb (Croatia).
N. Pavin
Affiliation:
Department of Physics, Faculty of Science, University of Zagreb, Zagreb (Croatia).
N. Paar
Affiliation:
Department of Physics, Faculty of Science, University of Zagreb, Zagreb (Croatia).
B. Novaković
Affiliation:
† Faculty of Mechanical Engineering and Naval Architecture, University of Zagreb, Zagreb (Croatia).
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Summary

This paper presents a mathematical model of a robot with one degree of freedom and numerical investigation of its dynamics in a particular parameter scan which is close to the upper boundary of the estimates for the parameters of rigidity and friction, while the length parameter L is treated as a free control parameter. In this L-scan the quasiperiodic and frequency locked solutions, their pattern and order of appearance are studied in the interval from the parameter range of immediate engineering significance to the point of appearance of transient chaos. In particular, a fractaltype multiple splitting of Arnold tongues is found in the parameter region bordering the range of engineering significance.

Keywords

Robot modelsDampingNonlinear dynamicsQuasiperiodicityFrequency locking.
Type
Article
Information
Robotica , Volume 14 , Issue 4 , July 1996 , pp. 423 - 431
Copyright
Copyright © Cambridge University Press 1996

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References

1.Thompson, J.M.T. and Stewart, H.B., Nonlinear Dynamics and Chaos (Wiley, Chichester, 1988).Google Scholar
2.Hayashi, C., Nonlinear Oscillations in Physical Systems (Princeton University Press, Princeton, NJ., 1985).Google Scholar
3.Moon, F.C., Chaotic Vibrations (John Wiley. New York, 1987).Google Scholar
4.Szemplinska-Stupnicka, W., “Chaotic and regular motion in nonlinear vibrating systems” In: (Szemplinska-Stupnicka, W., Iooss, G. and Moon, F.C.: Eds.) Chaotic Motions in Nonlinear Dynamical Systems (Springer, Wien, 1988) pp. 51–136.CrossRefGoogle Scholar
5.Guckenheimer, J. and Holmes, P., Nonlinear Oscillations, Dynamical Systems, and Bifurcation of Vector Fields (Springer, New York, 1990).Google Scholar
6.Nayfeh, A.H. and Mook, D.T., Nonlinear Oscillations (Wiley. New York. 1979).Google Scholar
7.Holmes, P.J. and Moon, F.C., “Strange attractors and chaos in nonlinear mechanics” J. Appl. Mech. 50, 1021–1032 (1983).CrossRefGoogle Scholar
8.Parlitz, U. and Lauterborn, W., “Superstructure in the bifurcation set of the Duffing equation X + dx+.x+.xx = fcoswt” Phys. Lett. A107, 351–355 (1985).CrossRefGoogle Scholar
9.Pezeshki, C. and Dowell, E.H., “On chaos and fractal behavior in a generalized Duffing's system” Phvsica D32, 194–209 (1988).Google Scholar
10.Vance, W. and Ross, J., “A detailed study of forced chemical oscillator: Arnold tongues and bifurcation sets” J. Chem. Phys. 91, 7654–7670 (1989).CrossRefGoogle Scholar
11.Glass, L. and Sun, J., “Periodic forcing of a limit-cycle oscillator: Fixed points, Arnold tongues, and the global organization of bifurcations” Phys. Rev. E50, 5077–5084 (1994).Google Scholar
12.Heimann, B., Loose, H. and Schuster, G., “Contribution to optimal control of an industrial robot” Proc. 4th CISM-IFTOMM Symp. on Theory and Pract. of Robots and manipulators, Warsaw (1981) pp. 211–219.Google Scholar
13.Paul, R., Robot Manipulators: Mathematics, Programming, and Control (MIT Press, Cambridge, Mass., 1981).Google Scholar
14.Novakovic, B., “A time and energy optimal control of industrial robots” In: (Kopacek, P., Troch, I. and Desoyer, K.; Eds.) Theory of Robots, IF AC Proc. Ser. 205–210 (Pergamon Press, New York, 1988).Google Scholar
15.Novaković, B., “An algorithm for the nonlinear robot control synthesis” Proc. Int. AMSE Conf: Signals and Systems Vol. 2. Brighton (1989) pp. 12–14.Google Scholar
16.Novaković, B.. Control Methods of Technical Svstems (Školska Knjiga. Zagreb, 1990).Google Scholar
17.Isidori, A. and Ruberti, A., “On the synthesis of linear input-output responses for nonlinear systems” Syst. Contr. Lett. 4, 17–22 (1984).CrossRefGoogle Scholar
18.Hemp, G. W., “Dynamic analysis of the runaway escapement mechanism” Shock. Vib. Bull. 42, 125–133 (1972).Google Scholar
19.Csaki, F., State Space Methods for Control Systems (Akademiai Kiado, Budapest, 1977).Google Scholar
20.Ueda, Y., “Random phenomena resulting from nonlinearity in the system described by Duffing's equation” Int. J. Non-Linear Mechanics 20, 481–491 (1985).CrossRefGoogle Scholar
21.Parlitz, U. and Lauterborn, W., “Resonance and torsion numbers of driven dissipative nonlinear oscillators” Z. Naturforsch. 41a, 605–614 (1985).Google Scholar
22.Leven, R.W., Pompe, B., Wilke, C. and Koch, B.P., “Experiments on periodic and chaotic motions of a parametrically forced pendulum” Phvsica 16D, 371–384 (1985).Google Scholar
23.Narayanan, S. and Jayaraman, K., “Chaotic motion in nonlinear system with Coulomb damping” In: (Schielen, W., Ed.) Nonlinear Dynamics in Engineering Svstems (Springer, Berlin, 1990) pp. 217–224.CrossRefGoogle Scholar
24.Popp, K. and Stelter, P., “Nonlinear oscillations of structures induced by dry friction” In: (Schielen, W. Ed.) Nonlinear Dynamics in Engineering Systems (Springer, Berlin, 1990, pp. 233–240); K. Popp, “Chaotische Bewegungen beim Reibschwinger mit simultaner selbst— und Fremderregung” ZAMM-Z. angew. Math. Mech. 71, T71–73 (1991).CrossRefGoogle Scholar
25.Stelter, P. and Sextro, W., “Bifurcations in Dynamic Systems with Dry Friction” Int. Ser. Numerical Math. 47, 343–347 (1991).Google Scholar
26.Haucke, H., and Ecke, R., “Mode-locking and chaos in Rayleigh-Benard convection” Physica 25D, 307–329 (1987).Google Scholar
27.Jensen, M.H., Bak, P. and Bohr, T., “Transition to chaos by interactions of resonances in dissipative systems. I. Circle maps” Phys. Rev. A30, 1960–1969 (1984).CrossRefGoogle Scholar
28.He, D.R., Yeh, W.J. and Kao, Y.H., “Studies of return maps, chaos, and phase-locked states in a current-driven Josephson-junction simulator” Phys. Rev. B31, 1359–1373 (1985).CrossRefGoogle Scholar
29.Ding, E.J., “Analytic treatment of a driven oscillator with a limit cycle” Phys. Rev. A3S, 2669–2683 (1987).CrossRefGoogle Scholar
30.D'Humieres, D., Beasly, M.R., Huberman, B.A. and Libchaber, A., “Chaotic states and routes to chaos in the forced pendulum” Phys. Rev. A26, 3483–3496 (1982).CrossRefGoogle Scholar
31.Swift, J.W. and Wiesenfeld, K., “Suppression of period doubling in symmetric systems” Phys. Rev. Lett. 52, 705–708 (1984).CrossRefGoogle Scholar