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Research Papers

Swing-Up Control of a Three-Link Underactuated Manipulator by High-Frequency Horizontal Excitation

[+] Author and Article Information
Kazuya Endo

e-mail: rfschwarexet@a3.keio.jp

Hiroshi Yabuno

Professor
e-mail: yabuno@mech.keio.ac.jp
Department of Mechanical Engineering,
Keio University,
3-14-1, Hiyoshi,
Kouhoku-ku, Yokohama,
Kanagawa 223-8522 Japan

Contributed by the Design Engineering Division of ASME for publication in the JOURNAL OF COMPUTATIONAL AND NONLINEAR DYNAMICS. Manuscript received May 20, 2011; final manuscript received February 28, 2012 published online June 14, 2012. Assoc. Editor: Yoshiaki Terumichi.

J. Comput. Nonlinear Dynam 8(1), 011002 (Jun 14, 2012) (8 pages) Paper No: CND-11-1070; doi: 10.1115/1.4006251 History: Received May 20, 2011; Revised February 28, 2012

In the present paper, we consider a three-link underactuated manipulator, the first joint of which is active and the second and third joints of which exhibit passive motion, on a plane inclined at slight angle from horizontal the plane. We analytically investigate changes in the stability of equilibrium points of the free links connected to the passive joints using high-frequency horizontal excitation of the first link. We derive autonomous averaged equations from the dimensionless equations of motion using the method of multiple scales. We clarify that the two free links can be swung up through pitchfork bifurcations and stabilized at some configurations by producing nontrivial and stable equilibrium points due to the high-frequency excitation. Furthermore, it is experimentally verified that increasing the excitation frequency multiplies stable and nontrivial equilibrium points.

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Figures

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Fig. 1

Analytical model of the three-link underactuated manipulator

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Fig. 2

Root locus of the trivial equilibrium point

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Fig. 3

Bifurcation diagram (branches related to nontrivial and stable equilibrium points bifurcate continuously at the supercritical pitchfork bifurcation points σ1). The solid and dashed lines denote stable and unstable equilibrium points, respectively. The black line (trivial line in the case of σ>σ1) denotes the trivial equilibrium point (initial rest state). When the relative angles θ2 of the second link can be swung up and stabilized (as indicated by the blue and red solid lines: nontrivial lines in the case of σ>σ1), the relative angle θ3 of the third link can also be swung up and stabilized (as indicated by the blue and red solid lines: nontrivial lines in the case of σ<σ1).

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Fig. 4

Bifurcation diagram (branches are related to nontrivial and unstable equilibrium points, which bifurcate at the bifurcation point σ2, other than those considered in Fig. 3). The relative angles θ2 and θ3 of the second and third links, respectively, are unstable in the region in which σ < σ2. The configurations indicated by the black and green dashed lines cannot be realized.

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Fig. 5

Experimental apparatus

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Fig. 7

Swing-up and stabilization angles of θ2 and θ3 by high-frequency horizontal excitation from ν/2π = 3 Hz to ν/2π = 42 Hz. The black circles indicate rest positions. The blue and red squares indicate the relative angles of θ2 and θ3 corresponding to the case in which σ < σ1, as shown in the combination of the surpercritical pitchfork bifurcation in Figs. 3(a) and 3(c). The blue and red closed triangles indicate the relative angles of θ2 and θ3 corresponding to the case in which σ < σ1 in the combination of Figs. 3(a) and 3(c). The blue and red opened triangles indicate the relative angles of θ2 and θ3 corresponding to the case in σ < σ1 of the combination of Figs. 3(a) and 3(b).

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Fig. 8

Stable configuration for the case in which ν/2π = 8 (Hz). This situation corresponds to σ < σ1 in the combination of Figs. 3(a) and 3(c).

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Fig. 9

Stable configuration in the case of in which ν/2π = 15 (Hz). This situation corresponds to σ < σ1 in the combination of Figs. 3(a) and 3(b).

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Fig. 10

Stable configuration in the case of in which ν/2π = 15 (Hz). This situation corresponds to σ < σ1 in the combination of Figs. 3(a) and 3(b)

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