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Research Papers

# Nonlinear Bifurcation Analysis of a Single-DoF Model of a Robotic Arm Subject to Digital Position Control

[+] Author and Article Information
Giuseppe Habib

e-mail: giuseppe.habib@uniroma1.it

Giuseppe Rega

e-mail: giuseppe.rega@uniroma1.it
Dipartimento di Ingegneria,
Strutturale e Geotecnica,
Sapienza University of Rome,
Rome, Italy

Gabor Stepan

Department or Applied Mechanics,
Budapest University of Technology and Economics,
H-1521, Budapest, Hungary
e-mail: stepan@mm.bme.hu

1Corresponding author.

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received November 7, 2011; final manuscript received March 6, 2012; published online June 14, 2012. Assoc. Editor: Claude-Henri Lamarque.

J. Comput. Nonlinear Dynam 8(1), 011009 (Jun 14, 2012) (9 pages) Paper No: CND-11-1209; doi: 10.1115/1.4006430 History: Received November 07, 2011; Revised March 06, 2012

## Abstract

Precision and stability in position control of robots are critical parameters in many industrial applications where high accuracy is needed. It is well known that digital effect is destabilizing and can cause instabilities. In this paper, we analyze a single DoF model of a robotic arm and we present the stability limits in the parameter space of the control gains. Furthermore we introduce a nonlinearity relative to the saturation of the control force in the model, reduce the dynamics of the nonlinear map to its local center manifold, study the bifurcation along the stability border and identify conditions under which a supercritical or subcritical bifurcation occurs. The obtained results explain some of the typical instabilities occurring in industrial applications. We verify the obtained results through numerical simulations.

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## References

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## Figures

Fig. 1

Mechanical model

Fig. 2

Transformation for stability analysis

Fig. 3

Stability chart (a) and vibration frequencies on the stability limit (b)

Fig. 12

Zone with phase locking in the q0 and p parameter space. The shaded area indicates the existence of an attractor with phase locking. The numbers in the circles indicate the ratio between the period of motion and τ. In the figure τ = 0.01s and Q0 = 10 m/s2.

Fig. 11

Attractors of the map (Eqs. (21), (26) and (27)) for increasing asymmetry of the control force. The diagram has been obtained fixing p = 0.2501, τ = 0.01s and Q0 = 10 m/s2. All figures have the displacement [m] in the x-axis and the velocity [m/s] in the y-axis.

Fig. 10

Diagram of the control force for different values of q0 with increasing asymmetry

Fig. 9

Bifurcation diagram for p = 0.2492, Q0 = 10 m/s2 and τ = 0.01s. Solid line: stable solutions, dashed line: unstable solutions, thin dashed-dotted line: analytical results.

Fig. 7

Bifurcation diagram for different values of q0. Q0 = 10 m/s2 and τ = 0.01s. Solid lines: stable solutions, dashed lines: unstable solutions.

Fig. 6

Existing attractors in the q0 and p parameter space. F: stable focus, P: stable periodic motion, C: chaotic motion. In brackets the unstable solutions. Q0 = 10 m/s2 and τ = 0.01s.

Fig. 5

Bifurcation diagram. Solid lines: analytical results; dashed lines: numerical results. The diagrams have been obtained fixing τ = 0.01s and Q0 = 10 m/s2 and using Eq. (52) before approximation.

Fig. 4

Asymmetric control force

Fig. 13

(a): chaotic attractor of the stroboscopic map (Eqs. (21), (26) and (27)), (b): detail of the attractor to show its fractal nature, (c): Lyapunov exponent and (d): FFT of the response. q0 = 0.97, p = 0.2501Q0 = 10 m/s2, τ = 0.01s.

Fig. 8

Bifurcation diagram for q0 = 0.67, Q0 = 10 m/s2 and τ = 0.01s. Solid line: stable solutions, dashed line: unstable solutions, dashed-dotted line; analytical results.

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