Technical Brief

Robustness Analysis of a Simple and Augmented Proportional Plus Derivative Controller in Trajectory Following Robots Using the Floquet Theory

[+] Author and Article Information
B. Sandeep Reddy

Center for Nano Science and Engineering,
Indian Institute of Science,
Bangalore 560012, India
e-mails: bsandeepr07@gmail.com;

Ashitava Ghosal

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

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received November 27, 2017; final manuscript received April 9, 2018; published online May 17, 2018. Assoc. Editor: Firdaus Udwadia.

J. Comput. Nonlinear Dynam 13(7), 074501 (May 17, 2018) (6 pages) Paper No: CND-17-1524; doi: 10.1115/1.4040022 History: Received November 27, 2017; Revised April 09, 2018

This paper deals with the issue of robustness in control of robots using the proportional plus derivative (PD) controller and the augmented PD controller. In the literature, a variety of PD and model-based controllers for multilink serial manipulator have been claimed to be asymptotically stable for trajectory tracking, in the sense of Lyapunov, as long as the controller gains are positive. In this paper, we first establish that for simple PD controllers, the criteria of positive controller gains are insufficient to establish asymptotic stability, and second that for the augmented PD controller the criteria of positive controller gains are valid only when there is no uncertainty in the model parameters. We show both these results for a simple planar two-degrees-of-freedom (2DOFs) robot with two rotary (R) joints, following a desired periodic trajectory, using the Floquet theory. We provide numerical simulation results which conclusively demonstrate the same.

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Slotine, J. J. E. , and Li, W. , 1991, Applied Nonlinear Control, Prentice Hall, Englewood Cliffs, NJ.
Udwadia, F. E. , and Koganti, P. B. , 2015, “ Dynamics and Control of a Multi-Body Planar Pendulum,” Nonlinear Dyn., 81(1–2), pp. 845–866. [CrossRef]
Udwadia, F. E. , and Koganti, P. B. , 2015, “ Optimal Stable Control for Nonlinear Dynamical Systems: An Analytical Dynamics Based Approach,” Nonlinear Dyn., 82(1–2), pp. 547–562. [CrossRef]
Udwadia, F. E. , and Wanichanon, T. , 2014, “ Control of Uncertain Nonlinear Multibody Mechanical Systems,” ASME J. Appl. Mech., 81(4), p. 041020. [CrossRef]
Udwadia, F. E. , and Wanichanon, T. , 2014, “ A New Approach to the Tracking Control of Uncertain Nonlinear Multi-Body Mechanical Systems,” Nonlinear Approaches in Engineering Applications, Vol. 2, Springer, New York, pp. 101–136. [CrossRef]
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. http://mate.tue.nl/mate/pdfs/4015.pdf
Rafael, K. A. , 1995, “ 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 (ACC), 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. [CrossRef]
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,” Fifth IFAC Symposium of Robot Control, Nantes, France, Sept. 3–5, pp. 79–84.
Chien, H. L. , 2007, “ Lyapunov-Based Control of a Robot and Mass-Spring System Undergoing an Impact Collision,” MS thesis, University of Florida, Gainesville, FL. http://ncr.mae.ufl.edu/thesis/chienhao.pdf
Murray, R. M. , Li, Z. , and Sastry, S. S. , 1994, A Mathematical Introduction to Robotic Manipulation, CRC Press, Boca Raton, FL.
Chen, Q. , Chen, H. , Wang, Y. , and Woo, P. , 2000, “ Global Stability Analysis for Some Trajectory-Tracking Control Schemes of Robotic Manipulators,” American Control Conference (ACC), Chicago, IL, June 28–30, pp. 3343–3347.
Lankalapalli, S. , and Ghosal, A. , 1997, “ Chaos in Robot Control Equations,” Int. J. Bifurcation Chaos, 7(3), pp. 707–720. [CrossRef]
Sandeep, R. B. , and Ghosal, A. , 2016, “ Asymptotic Stability and Chaotic Motions in Trajectory Following Feedback Controlled Robots,” ASME J. Comput. Nonlinear Dyn., 11(5), p. 051012. [CrossRef]
Ghosal, A. , 2006, Robotics: Fundamental Concepts and Analysis, Oxford University Press, Oxford, UK.
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.
Coddington, E. A. , and Carlson, R. , 1997, Linear Ordinary Differential Equations, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA. [CrossRef]
Bittanti, S. , and Colaneri, P. , 2009, Periodic Systems Filtering and Control, Communications and Control Engineering, Springer-Verlag, London.
Thomas, O. , Lazarus, A. , and Touze, C. , 2010, “ A Harmonic-Based Method for Computing the Stability of Periodic Oscillations of Non-Linear Structural Systems,” ASME Paper No. DETC2010-28407.
Kurt, L. , 2001, “ Improved Numerical Floquet Multipliers,” Int. J. Bifurcation Chaos, 11(9), pp. 2389–2410. [CrossRef]
Slane, J. , and Tragesser, S. , 2011, “ Analysis of Periodic Nonautonomous Inhomogeneous Systems,” Nonlinear Dyn. Syst. Theory, 11(2), pp. 183–198. http://sunrise-0014438.e-ndst.kiev.ua/v11n2/8(35).pdf
Thomsen, J. J. , 2003, Vibrations and Stability: Advanced Theory, Analysis and Tools, Springer Verlag, Berlin.
Parker, T. S. , and Chua, L. O. , 1989, Practical Numerical Algorithms for Chaotic Systems, Springer Verlag, New York. [CrossRef]


Grahic Jump Location
Fig. 2

Maps showing instability in (Kp, Kv) space for the simple PD controller for different Ω for Af=πrad: (a) Ω = 2 rad/s and (b) Ω = 5 rad/s

Grahic Jump Location
Fig. 3

Maps showing instability in (Kp, Kv) space for the augmented PD controller for ε=−0.3 at varying values of Ω and Af: (a) Ω = 5 rad/s and Af = π rad, (b) Ω = 5 rad/s and Af = 2π rad, and (c) Ω = 2 rad/s and Af = 2π rad

Grahic Jump Location
Fig. 4

Poincaré map showing chaos for (Kp, Kv) = (50,1) for the augmented PD controller at Ω = 5 rad/s and Af=2πrad



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