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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;
sandeepreddy@iisc.ac.in

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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References

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Figures

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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