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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

Professor
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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Figures

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. 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. 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

Grahic Jump Location
Fig. 2

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

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