Controlled pendulum on a movable base

Mechanics of Solids - Tập 48 - Trang 6-18 - 2013
Yu. G. Martynenko1, A. M. Formal’skii1
1Institute of Mechanics, Lomonosov Moscow State University, Moscow, Russia

Tóm tắt

A plane motion of a multilink pendulum hinged to a movable base (a wheel or a carriage) is considered. The control torque applied between the base and the first link of the pendulum is independent of the base position and velocity and is bounded in absolute value. The coordinate determining the base position is cyclic. The mathematical model of the system permits one to single out the equations describing the pendulum motion alone, which differ from the well-known equations of motion of a pendulum with a fixed suspension point both in the structure and in the parameters occurring in these equations. The phase portrait of motions of a control-free one-link pendulum suspended on a wheel or a carriage is obtained. A feedback control ensuring global stabilization of the unstable upper equilibrium of the pendulum is constructed. Time-optimal control synthesis is outlined.

Tài liệu tham khảo

Yu. G. Martynenko and A. M. Formal’skii, “The Theory of the Control of a Monocycle,” Prikl. Mat. Mekh. 69(4), 569–583 (2005) [J. Appl. Math. Mech. (Engl. Transl.) 69 (4), 516–528 (2005)].

Yu.G. Martynenko and A.M. Formal’skii, “Problems of Control ofUnstable Systems,” Uspekhi Mekh. 3(2), 71–135 (2005).

Yu. G. Martynenko and A. M. Formal’skii, “A Control of the Longitudinal Motion of a Single-Wheel Robot on an Uneven Surface,” Izv. Ross. Akad. Nauk. Teor. Sist. Upr., No. 4, 165–173 (2005) [J. Comp. Syst. Sci. Int. (Engl. Transl.) 44 (4), 662–670 (2005)].

F. L. Chernousko, L. D. Akulenko, and B. N. Sokolov, Control of Oscillations (Nauka, Moscow, 1980) [in Russian].

Y. Aoustin and A. M. Formal’sky, “Simple Anti-Swing Feedback Control for a Gantry Crane,” Robotica 21, 655–666 (2003).

N. N. Bolotnik, I. M. Zeidis, K. Zimmermann, and S. F. Yatsun, “Dynamics of Controlled Motion of Vibration-Driven Systems,” Izv. Ross. Akad. Nauk. Teor. Sist. Upr., No. 5, 157–167 (2006) [J. Comp. Syst. Sci. Int. (Engl. Transl.) 45 (5), 831–840 (2006)].

A. M. Formal’skii, Displacement of Anthropomorphic Mechanisms (Nauka, Moscow, 1982) [in Russian].

Khaled Gamal Eltohamy and Chen-Yuan Kuo, “Nonlinear Generalized Equations of Motion for Multi-Link Inverted Pendulum Systems,” Int. J. Syst. Sci. 30(5), 505–513 (1999).

S. Lam and E. J. Davison, “The Real Stabilizability Radius of the Multi-Link Inverted Pendulum,” in Proc. 2006 American Control Conf.Minneapolis, Minnesota, USA (2006), pp. 1814–1819.

T. G. Strizhak, Methods for Studying ‘Pendulum’-Type Dynamical Systems (Nauka, Alma-Ata, 1981) [in Russian].

A.M. Formal’skii, “Stabilization of an Inverted Pendulumwith a Fixed or Movable Suspension Point,” Dokl. Ross. Akad. Nauk 406(2), 175–179 (2006) [Dokl. Math. (Engl. Transl.) 73 (1), 152–156 (2006)].

A.M. Formal’skii, “An Inverted Pendulum on a Fixed and a Moving Base,” Prikl. Mat. Mekh. 70(1), 62–71 (2006) [J. Appl. Math. Mech. (Engl. Transl.) 70 (1), 56–64 (2006)].

A.M. Formal’skii, “Stabilization of Unstable Mechanical Systems,” J. Optimiz. Theory Appl. 144(2), 227–253 (2010).

N. G. Chetaev, Stability of Motion (Izdat. AN SSSR, Moscow, 1962) [in Russian].

B. A. Smol’nikov, Problems ofMechanics and Robots Optimization (Nauka, Moscow, 1991) [in Russian].

A.M. Formal’skii, Controllability and Stability of Systems with Restricted Resources (Nauka, Moscow, 1974) [in Russian].

E. B. Lee and L. Markus, Foundations of Optimal Control Theory (Wiley, New York, 1967; Nauka, Moscow, 1972).

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, and E. F. Mishchenko, TheMathematical Theory of Optimal Processes (Nauka, Moscow, 1983; Gordon & Breach Sci. Publ., New York, 1986).

S. A. Reshmin and F. L. Chernous’ko, “Time-Optimal Control of an Inverted Pendulum in the Feedback Form,” Izv. Ross. Akad. Nauk. Teor. Sist. Upr., No. 3, 51–62 (2006) [J. Comp. Syst. Sci. Int. (Engl. Transl.) 45 (3), 383–394 (2006)].

S. A. Reshmin and F. L. Chernous’ko, “A Time-Optimal Control Synthesis for a Nonlinear Pendulum,” Izv. Ross. Akad. Nauk. Teor. Sist. Upr., No. 1, 13–22 (2007) [J. Comp. Syst. Sci. Int. (Engl. Transl.) 46 (1), 9–18 (2007)].

S. A. Reshmin and F. L. Chernousko, “Optimal in the Speed of Response Synthesis of Control in Problems of Swaying and Damping of Nonlinear Pendulum Oscillations,” in Proc. 9th Chetaev Conf. “Analytical Mechanics, Stability, and Control of Motion”, Vol. 3 (Irkutsk, 2007), pp. 179–196 [in Russian].

R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 1 (Gostekhizdat, Moscow-Leningrad, 1951) [in Russian].

F. R. Gantmakher, Theory of Matrices (Nauka, Moscow, 1967) [in Russian].

Ya. G. Panovko and I. I. Gubanova, Stability and Oscillations of Elastic Systems (Consultant Bureau, New York 1965; Nauka,Moscow, 1987).

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Mechanics of Solids

Tập 48

6-18

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