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    体彩江苏7位数规则:Stabilization of periodic orbits in a wedge billiard

    江苏七位体彩开奖结果 www.jwbw.net Stabilization of periodic orbits in a wedge billiard
    Manuel Gerard, Rodolphe Sepulchre Department of Electrical Engineering and Computer Science Universit? de Li` ge, Belgium e e [email protected], [email protected]

    1 The wedge billiard system The wedge billiard system is depicted on the Figure 1. A point mass (ball) moves in the plane under the action of a constant gravitational ?eld. The ball undergoes collisions with two intersecting edges, an idealization of the juggler’s two arms. In the absence of control, the two edges form a g ? J
    J
    J
    θ J
    J
    J
    J
    J
    Figure 1: The wedge billiard. ?xed angle θ with the direction of gravity. Rotational actuation of the edges around their ?xed intersection point is used to stabilize one particular orbit of the uncontrolled system. When uncontrolled, the wedge billiard is a rich dynamical model leading to stabilization problems of various complexity. It was realized in [2], [6] that the wedge billiard displays a variety of dynamical phenomena as a function of the angle θ. For θ < 45? , the phase space exhibits stable and chaotic behavior associated with periodic orbits of any period. For θ > 45? , the motion appears completely chaotic. The value θ = 45? is very special and leads to a completely integrable system with a two-parameter family of unstable periodic orbits. 2 Stabilization problem The stabilization problem under consideration is viewed as a benchmark for theoretical investigations of impact control problems encountered in legged robotics. The dif?culty when studying these mechanisms comes from the underactuated and intermittent nature of the control. Active stabilization of juggling machines has been addressed by [1], [3],[4], [7]. From a theoretical point of view, the stabilization of periodic orbits through impact control is rephrased as the ?xed point discrete-time stabilization of the Poincar map. Hence our

    problem de?nes as the stabilization of a three-dimensional discrete-time nonlinear system. In [5], the problem of stabilizing a period-two orbit (as depicted on the left part of Figure 2) is addressed in the case θ = 45? . The present paper will address the stabilization of a periodone orbit (see right part of Figure 2) for an arbitrary wedge angle 0? < θ < 90? .

    45° 45°

    Figure 2: Period-two orbit (left, θ = 45? ) and period-one orbit
    (right).

    References [1] M. Buehler, D.E. Koditschek, and P.J. Kindlmann, Planning and control of robotic juggling and catching tasks, International Journal of Robotics Research 13 (1994), no. 2, 101–118. [2] Lehtihet, H. E. Miller, and B. N., Numerical study of a billiard in a gravitational ?eld, Physica 21D (1986), 93– 104. [3] K.M. Lynch and C.K. Black, Control of underactuated manipulation by real-time nonlinear optimization, 9th Int. Symposium Robotics Research (Snowbird, UT), October 1999. [4] A. A. Rizzi and D. E. Koditschek, Progress in spatial robot juggling, IEEE International Conference on Robotics and Automation (Nice, France), 1992, pp. 775–780. [5] R. Sepulchre and M. Gerard, Stabilization of periodic orbits in a wedge billiard, IEEE 42nd Conf. on Decision and Control (Maui, Hawaii-USA), 2003, pp. 1568–1573. [6] T. Szeredi and D.A. Goodings, Classical and quantum chaos of the wedge billiard. i. classical mechanics, Physical Review E 48 (1993), no. 5, 3518–3528. [7] A. Zavala-Rio and B. Brogliato, On the control of a one degree-of-freedom juggling robot, Dynamics and Control 9 (1999), 67–90.


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