Research Papers

Asymptotic Stability and Chaotic Motions in Trajectory Following Feedback Controlled Robots

[+] Author and Article Information
B. Sandeep Reddy

Department of Mechanical Engineering,
Indian Institute of Science,
Bangalore 560012, India
e-mail: bsandeep@mecheng.iisc.ernet.in

Ashitava Ghosal

Department of Mechanical Engineering,
Indian Institute of Science,
Bangalore 560012, India
e-mail: asitava@mecheng.iisc.ernet.in

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received October 4, 2015; final manuscript received December 22, 2015; published online February 3, 2016. Assoc. Editor: Haiyan Hu.

J. Comput. Nonlinear Dynam 11(5), 051012 (Feb 03, 2016) (11 pages) Paper No: CND-15-1325; doi: 10.1115/1.4032389 History: Received October 04, 2015; Revised December 22, 2015

A feedback controlled robot manipulator with positive controller gains is known to be asymptotically stable at a set point and for trajectory following in the sense of Lyapunov. However, when the end-effector of a robot or its joints are made to follow a time-dependent trajectory, the nonlinear dynamical equations modeling the feedback controlled robot can also exhibit chaotic motions and as a result cannot follow a desired trajectory. In this paper, using the example of a simple two-degree-of-freedom robot with two rotary (R) joints, we take a relook at the asymptotic stability of a 2R robot following a desired time-dependent trajectory under a proportional plus derivative (PD) and a model-based computed torque control. We demonstrate that the condition of positive controller gains is not enough and the gains must be large for chaos not to occur and for the robot to asymptotically follow a desired trajectory. We apply the method of multiple scales (MMS) to the two nonlinear second-order ordinary differential equations (ODEs), which describes the dynamics of the feedback controlled 2R robot, and derive a set of four first-order slow flow equations. At a fixed point, the Routh–Hurwitz criterion is used to obtain values of proportional and derivative gains at which the controller is asymptotically stable or indeterminate. For the model-based control, a parameter representing model mismatch is used and the controller gains for a chosen mismatch parameter value are obtained. From numerical simulations with controller gain values in the indeterminate region, it is shown that for some values, the nonlinear dynamical equations are chaotic, and hence, the 2R robot cannot follow the desired trajectory and be asymptotically stable.

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Slotine, J. J. E. , and Li, W. , 1991, Applied Nonlinear Control, Prentice Hall, Englewood Cliffs, NJ, pp. 40–94.
Antonio, L. , Erjen, L. , and Henk, N. , 2000, “ Global Asymptotic Stability of Robot Manipulators With Linear PID and PI2D Control,” SACTA, 3(2), pp. 138–149.
Rafael, K ., 1995, “ A Tuning Procedure for Stable PID Control of Robot Manipulators,” Robotica, 13(2), pp. 141–148. [CrossRef]
Jose, A. H. , and Wen, Y. , 2000, “ A High-Gain Observer-Based PD Control for Robot Manipulator,” American Control Conference, Chicago, IL, June 28–30, pp. 2518–2522.
Ge, S. S. , Lee, T. H. , and Zu, G. , 1997, “ Non-Model-Based Position Control of a Planar Multi-Link Flexible Robot,” Mech. Syst. Signal Process., 11(5), pp. 707–724. [CrossRef]
Amol, A. K. , Gopinathan, L. , and Goshaidas, R. , 2011, “ An Adaptive Fuzzy Controller for Trajectory Tracking of a Robot Manipulator,” Intell. Control Autom., 2, pp. 364–370. [CrossRef]
Antonio, Y. , Victor, S. , and Javier, M. V. , 2011, “ Global Asymptotic Stability of the Classical PID Controller by Considering Saturation Effects in Industrial Robots,” Int. J. Adv. Rob. Syst., 8(4), pp. 34–42.
Vincente, P. G. , Suguru, A. , Yun, H. L. , Gerhard, H. , and Prasad, A. , 2003, “ Dynamic Sliding PID Control for Tracking of Robot Manipulators: Theory and Experiments,” IEEE Trans. Rob. Autom., 19(6), pp. 967–976. [CrossRef]
Ruvinda, G. , and Fathi, G. , 1997, “ PD Control of Closed-Chain Mechanical Systems: An Experimental Study,” 5th IFAC Symposium of Robot Control, Nantes, France, Sept. 3–5, pp. 79–84.
Liang, C.-H. , 2007, “ Lyapunov Based Control of a Robot and Mass Spring System Undergoing an Impact Collision,” M.S. thesis, Department of Mechanical and Aerospace Engineering, University of Florida, Gainesville, FL.
Asada, H. , and Slotine, J. J. E. , 1986, Robot Analysis and Control, Wiley, New York, pp. 133–157.
Khalil, K. H. , 1996, Nonlinear Systems, Prentice Hall, Upper Saddle River, NJ, pp. 191–196.
Burov, A. A. , 1986, “ On the Non-Existence of a Supplementary Integral in the Problem of a Heavy Two-Link Plane Pendulum,” Prikl. Matem. Mekhan. USSR, 50(1), pp. 123–125.
Yu, P. , and Bi, Q. , 1988, “ Analysis of Nonlinear Dynamics and Bifurcations of a Double Pendulum,” J. Sound Vib., 217(4), pp. 691–736. [CrossRef]
Lankalapalli, S. , and Ghosal, A. , 1996, “ Possible Chaotic Motions in a Feedback Controlled 2R Robot,” IEEE International Conference on Robotics and Automation, N. Caplan , and T. J. Tarn , eds., Minneapolis, MN, Apr. 22–28, IEEE Press, New York, pp. 1241–1246.
Lankalapalli, S. , and Ghosal, A. , 1997, “ Chaotic Motion of a Planar 2-DOF Robot,” Int. J. Bifurcation Chaos, 7(3), pp. 707–729. [CrossRef]
Li, K. F. , Li, L. , and Chen, Y. , 2002, “ Chaotic Motion of a Planar 2-DOF Robot,” Mech. Sci. Technol., 29(1), pp. 6–8.
Hilborn, R. C. , 2000, Chaos and Nonlinear Dynamics: An Introduction to Scientists and Engineers, Oxford University Press, New York, pp. 3–61.
Nayfeh, A. H. , 1993, Introduction to Perturbation Techniques, Wiley, New York, pp. 388–401.
Arthur, M. G. O. , 1999, Design and Analysis of Control Systems, CRC Press, Boca Raton, FL, pp. 323–332.
Parker, T. S. , and Chua, O. S. , 1989, Practical Numerical Algorithms for Chaotic Systems, Springer-Verlag, New York, pp. 57–81.
Ghosal, A. , 2006, Robotics: Fundamental Concepts and Analysis, Oxford University Press, New Delhi, India, pp. 183–196.
Ravishankar, A. S. , and Ghosal, A. , 1999, “ Nonlinear Dynamics and Chaotic Motions in Feedback Controlled Two and Three-Degree-of-Freedom Robots,” Int. J. Rob. Res., 18(1), pp. 93–108.
Nayfeh, A. H. , and Balachandran, B. , 2004, Applied Nonlinear Dynamics: Analytical, Computational and Experimental Methods, Wiley-VCH Verlag, Weinheim, Germany, pp. 108–121.
El-Bassiouny, A. F. , 1999, “ Response of a Three-Degree-of-Freedom System With Cubic Nonlinearities to Harmonic Excitation,” Appl. Math. Comput., 104(1), pp. 65–84. [CrossRef]
Jinchen, J. , and Yushu, C. , 1999, “ Bifurcation in a Parametrically Excited Two-Degree-of-Freedom Nonlinear Oscillating System With 1:2 Internal Resonance,” Appl. Math. Mech., 20(4), pp. 350–359. [CrossRef]
Reddy, B. S. , and Ghosal, A. , 2015, “ Nonlinear Dynamics of a Rotating Flexible Link,” ASME J. Comput. Nonlinear Dyn., 10(6), p. 061014. [CrossRef]
Cartmell, M. P. , Ziegler, S. W. , Khanin, R. , and Forehand, D. I. M. , 2003, “ Multiple Scales Analyses of the Dynamics of Weakly Nonlinear Mechanical Systems,” ASME Appl. Mech. Rev., 56(5), pp. 455–492. [CrossRef]
Sartorelli, J. C. , and Lacarbonara, W. , 2012, “ Parametric Resonances in a Base-Excited Double Pendulum,” Nonlinear Dyn., 69(4), pp. 1679–1692. [CrossRef]
Wall, H. S. , 1945, “ Polynomials Whose Zeros Have Negative Real Parts,” Am. Math. Mon., 52(6), pp. 308–322. [CrossRef]
Gupta, P. D. , Majee, N. C. , and Roy, A. B. , 2008, “ Asymptotic Stability, Orbital Stability of Hopf-Bifurcating Periodic Solution of a Simple Three-Neuron Artificial Neural Network With Distributed Delay,” Nonlinear Anal., 13(1), pp. 9–30.
Mathworks, 2012, Matlab Version 8.0 (R2012Rb), Mathworks, Inc., Natick, MA.


Grahic Jump Location
Fig. 3

Spectra of Lyapunov exponents of the 2R robot equations for PD control: (a) (Kp, Kv) = (54, 1)—chaotic and (b) (Kp, Kv) = (54, 4)—asymptotically stable

Grahic Jump Location
Fig. 2

Chaos maps in (Kp, Kv) space for PD control for various values of forcing frequency Ω

Grahic Jump Location
Fig. 4

Chaos maps in (Kp, Kv) space for model-based control for various values of mismatch parameter e

Grahic Jump Location
Fig. 5

Spectra of Lyapunov exponents of the 2R robot equations (11) for model-based control: (a) (Kp, Kv) = (46, 1)—chaotic and (b) (Kp, Kv) = (46, 8)—asymptotically stable



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