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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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References

Mukherjee, R., and Chen, D., 1993, “Control of Free-Flying Underactuated Space Manipulator to Equilibrium Manifolds,” IEEE Trans. Rob. Autom., 9(5), pp. 561–570. [CrossRef]
Tortopidis, I., and Papadopoulos, E., 2007, “On Point-To-Point Motion Planning for Underactuated Space Manipulator Systems,” Rob. Auton. Syst., 55, pp. 122–131. [CrossRef]
Papadopoulos, E., and Dubowsky, S., 1991, “Failure Recovery Control For Space Robotic Systems,” Proceedings of the American Control Conference, 2, pp. 1485–1490. [CrossRef]
Roy, B., and Asada, H., 2009, “Nonlinear Feedback Control of a Gravity-Assisted Underactuated Manipulator With Application to Aircraft Assembly,” IEEE Trans. Rob., 25(5), pp. 1125–1133. [CrossRef]
Zhao, T. S., and Dai, J. S., 2003, “Dynamics and Coupling Actuation of Elastic Underactuated Manipulators,” J. Rob. Syst., 20(3), pp. 135–146. [CrossRef]
Arai, H., and Tachi, S., 1991, “Position Control of a Manipulators With Passive Joints Using Dynamic Coupling,” IEEE Trans. Rob. Autom., 7(4), pp. 528–534. [CrossRef]
Gilbert, J. M., 2007, “Gyrobot: Control of Multiple Degree of Freedom Underactuated Mechanisms Using a Gyrating Link and Cyclic Braking,” IEEE Trans. Rob., 4, pp. 822–827. [CrossRef]
Arai, H., Tanie, K., and Shiroma, N., 1998, “Nonholonomic Control of a Three-DOF Planar Underactuated Manipulator,” IEEE Trans. Rob. Autom., 14(2), pp. 681–695. [CrossRef]
Yu, K. H., Shito, Y., and Inooka, H., 1998, “Position Control of an Underactuated Manipulator Using Joint Friction,” Int. J. Nonlinear Mech., 33(4), pp. 607–614. [CrossRef]
Agrawal, S. K., and Sangwan, V., 2008, “Differentially Flat Designs of Underactuated Open-Chain Planar Robots,” IEEE Trans. Rob., 24(6), pp. 1445–1451. [CrossRef]
Mahindrakar, A. D., Banavar, R. N., and Reyhanoglu, M., 2005, “Controllability and Point-To-Point Control of 3-DOF Planar Horizontal Underactuated Manipulators,” Int. J. Control, 78(1), pp. 1–13. [CrossRef]
Xin, X., and Kaneda, M., 2007, “Swing-Up Control for a 3-DOF Gymnastic Robot With Passive First Joint: Design and Analysis,” IEEE Trans. Rob., 23(6), pp. 1277–1285. [CrossRef]
Hong, K. S., 2002, “An Open-Loop Control for Underactuated Manipulators Using Oscillatory Input: Steering Capability of an Unactuated Joint,” IEEE Trans. Control Syst. Technol., 10(3), pp. 469–480. [CrossRef]
Yabuno, H., Goto, K., and Aoshima, N., 2004, “Swing-Up and Stabilization of an Underactuated Manipulator Without State Feedback of Free Joint,” IEEE Trans. Rob. Autom., 20(2), pp. 359–365. [CrossRef]
Yabuno, H., and Hattori, M., 2008, “Reachable Area of an Underactuated Space Manipulator Subjected to Simple Spinning (Application of Bifurcation Control Under High-Frequency Excitation),” Nonlinear Dyn., 51, pp. 345–353. [CrossRef]
Kapitza, P. L., 1965, “Dynamical Stability of a Pendulum When its Point of Suspension Vibrates, and Pendulum With a Vibrating Suspension,” Collected Papers of P. L. Kapitza, D. T.Haar, ed., Pergamon Press, London, Vol. 2, pp. 714–737.
Stephenson, A., 1908, “On a New Type of Dynamical Stability,” Mem. Proc. Manch. Lit. Phil. Sci., 52, pp. 1–10.
Bellman, R. E., Bentsman, J., and Meerkov, S. M., 1986, “Vibration Control of Nonlinear Systems: Vibration Stabilizability,” IEEE Trans. Autom. Control, AC-31(8), pp. 710–716. [CrossRef]
Weibel, S. P., and Baillieul, J., 1998, “Open-Loop Oscillatory Stabilization of an n-Pendulum,” Int. J. Control, 16(11), pp. 931–957. [CrossRef]
Schmitt, J. M., and Bayly, P. V., 1998, “Bifurcations in the Mean Angle of a Horizontally Shaken Pendulum: Analysis and Experiment,” Nonlinear Dyn., 15, pp. 1–14. [CrossRef]
Yabuno, H., Miura, M., and Aoshima, N., 2004, “Bifurcation in an Inverted Pendulum With Tilted High Frequency Excitation: Analytical and Experimental Investigations on the Symmetry-Breaking of the Bifurcation,” J. Sound Vib., 273, pp. 493–513. [CrossRef]
Yabuno, H., Matsuda, T., and Aoshima, N., 2004, “Reachable and Stabilizable Area of Underactuated Manipulator Without State Feedback Control,” IEEE Trans. Rob. Autom., 10(4), pp. 397–403. [CrossRef]

Figures

Grahic Jump Location
Fig. 1

Analytical model of the three-link underactuated manipulator

Grahic Jump Location
Fig. 2

Root locus of the trivial equilibrium point

Grahic Jump Location
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.

Grahic Jump Location
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).

Grahic Jump Location
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)

Grahic Jump Location
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).

Grahic Jump Location
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. 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).

Grahic Jump Location
Fig. 5

Experimental apparatus

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