Nonlinear Dynamics and Chaotic Motions in Feedback-Controlled Two-and Three-Degree-of-Freedom Robots

Abstract

The dynamics of a feedback-controlled rigid robot is most commonly described by a set of nonlinear ordinary differential equations. In this paper we analyze these equations, representing the feedback-controlled motion of twoand three-degrees-of-freedom rigid robots with rnvolute (R) and prismatic (P) joints in the absence of compliance, friction, and potential energy for the possibility of chaotic motions.

We first study the unforced or inertial motions of the robots, and show that when the Gaussian or Riemannian curvature of the configuration space of a robot is negative, the robot equations can exhibit chaos. If the curvature is zero or positive, then the robot equations cannot exhibit chaos. We show that among the two-degrees-of-freedom robots, the PP and the PR robot have zer Gaussian curvature while the RP and RR rohots have negative Gaussian curvatures. For the three-degrees-offireedom robots, we analyze the two well-known RRP and RRR configurations of the Stanford arm and the PUMA manipulator, respectively, and derive the conditions for negative curvature and possible chaotic motions. The criteria of negative curvature cannot be usedfor the forced or feedback-controlled motions. For the foired motion, we resort to the well-known numerical techniques and compute chaos maps, Poincard maps, and bifurcation diagrams. Numerical results are presentedfor the two-degreesf-ffreedom RP and RR robots, and we show that these robot equations can exhibit chaos for low controller gains and for large underestimated models. From the bifurcation diagrams, the mute to chaos appears to he through period doubling.

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References

Armstrong, B., Khatib, O., and Burdick, J. W. 1986. The explicit dynamic model and inertial parameters of the PUMA 560 arm. Proc. of the Int. Conf. on Robot. and Automat. Los Alamitos, CA: IEEE , pp. 510-517.

Arnold, V. I.. 1989. Mathematical Methods in Classical Mechanics, 2nd ed. New York: Springer-Xerlag .

Asada, H., and Slotline, J.-J.E. 1986. Robot Analysis and Control. New York: Wiley .

Biihler, M., and Koditschek, D. E. 1991. Analysis of a simplified hopping robot . Int. J. Robot. Res. 10(6): 587-605 .

Craig, J. J. 1988. Adaptive Control of Mechanical Manipulators. Reading, MA: Addison-Wesley .

Craig, J. J. 1989. Introduction to Robotics: Mechanics and Control, 2nd ed. Reading, MA: Addison-Wesley .

D'Humieres, D., Beasely, M. R., Huberman, B. A., and Lichaber, A. 1982. Chaotic states and routes to chaos in a forced pendulum . Physics Rev. Lett. 65(4): 3211-3214 .

Dowell, E. H., and Pezeshki, C. 1986. On the understanding of chaos in Duffing's equation including a comparison with experiment . ASAE J. Appl. Mech. 53(3): 5-9 .

Gordon, M. K., and Shampine, L. F. 1975. Computer Solutions of Ordinary Diferential Equations: The Initial Jfilue Problem. San Francisco: W H. Freeman .

10.

Guckenheimer, J., and Holmes, P. 1983. Nonlinear Oscillations, Dynamical Systems, and Bifiurcations of Vector Fields. New York: Springer-Nrlag .

11.

Hubinger, B., Doerner, R., Heng, H., and Martienssen, W. 1994a. Approaching nonlinear dynamics by studying the motion of a pendulum: I. Observing trajectories in phase space . Int. J. Bifurcation Chaos 4(4): 751-760 .

12.

Hubinger, B., Doemner, R., Heng, H., and Martienssen, W. 1994b. Approaching nonlinear dynamics by studying the motion of a pendulum. II. Analyzing chaotic motions . Int. J. Bifurcation Chaos 4(4): 761-771 .

13.

Hibinger, B., Doemer, R.Heng, H., and Martienssen, W. 1994c. Approaching nonlinear dynamics by studying the motion of a pendulum. Ill. Predictability and control of chaotic motion . Int. J. Bifurcation Chaos 4(4): 773-784 .

14.

Kawakami, H. 1984. Bifurcations of periodic responses in forced dynamic nonlinear circuits: Computation of bifurcation values of the system parameters . IEEE Trans. Circuits Systems 31(4): 248-260 .

15.

Khosla, P K. 1986. Real-time control and identification of direct-drive manipulators. PhD thesis, Department of Electrical and Computer Engineering, Carnegie-Mellon University, USA.

16.

Mahout, V 1994. Recurrences and complex behaviours within robotics in the context of repetitive tasks-application to a SCARA manipulator. PhD thesis, INSA de Toulouse. (Available as Technical Report 267.)

17.

Mahout, V., Lopez, P., Carcasses, J. P., and Mira, C. 1993. Complex behaviours of a two-revolute-joint robot: Harmonic, subharmonic, higher harmonic, fractional harmonic, chaotic responses. Proc. of the Int. Conf on Sys., Man and Cyhernet., vol. 5. Los Alamitos, CA: IEEE , pp. 201-205.

18.

M'Closkey, R. T., and Burdick, J. W. 1993. Periodic motions of a hopping robot with vertical and forward motion . Int. J. Robot. Res. 12(6): 197-218 .

19.

Moon, F. C. 1987. Chaotic Ubrations: An Introduction for Applied Scientists and Engineers. New York: Wiley .

20.

Parker, T. S., and Chua, L. O. 1989. Practical Numerical Algorithms for Chaotic Systems. New York: Springer-Verlag .

21.

Paul, R. P. 1981. Robot Manipualtors: Mathematics, Programming and Control. Cambridge, MA: MIT Press .

22.

Sekar, P., and Narayanan, S. 1995. Chaos in mechanical systems: A review . Sadhana 20(2-4): 529-582 .

23.

Shrinivas, L., and Ghosal, A. 1996. Possible chaotic motions in a feedback controlled 2-R robot. Proc. of the IEEE Conf on Robot. and Automat. Washington, DC: IEEE , pp. 1241-1246.

24.

Spong, M. W. 1992. Remark on robot dynamics: Canonical transformations and Riemannian geometry. Proc. ofthe Int. Conf. on Robot. and Automat. Los Alamitos, CA: IEEE , pp. 554-559.

25.

Stoker, J. J. 1969. Differential Geometry. New York: Wiley .

26.

Streit, D. A., Krousgrill, C. M., and Bajaj, A. K. 1986. A preliminary investigation of the dynamic stability of flexible manipulators performing repetitive tasks . Trans. ASME J. Dyn. Sys. Meas. Control 108(6): 206-214 .

27.

Tongue, B. H. 1986. On necessary and sufficient conditions for chaos in Duffing's equation: A heuristic approach . J. Sound Jibration 110(l): 59-78 .

28.

Vakakis, A. E., and Burdick, J. W. 1990. Chaotic motions in the dynamics of a hopping robot. Proc. of the IEEE Int. Conf. on Robot. and Automat. Los Alamitos, CA: IEEE , pp. 1464-1469.

29.

Vakakis, A. F., and Burdick, J. W. 1991. An "interesting" strange attractor in the dynamics of a hopping robot . Int. J. Robot. Res. 10(6): 606-618 .

30.

Xarghese, M., Fuchs, A., and Mukundan, R. 1991. Chaotic zero dynamics in kinematically redundant robots . IEEE Trans. Aerospace Electronic Sys. 27(5): 784-796 .

31.

Wblf, A., Swift, J. B., Swinney, H. A., and Mistano, J. A. 1985. Determining Lyapunov exponents from a time series . Physica 16D: 285-317 .

32.

Zak, M. 1985a. Criteria for chaos in nonlinear mechanics . Int. J. Nonlinear Mech. 21(3): 175-182 .

33.

Zak, M. 1985b. Two types of chaos in nonlinear mechanics . Int. J. Nonlinear Mech. 20(4): 297-308 .