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DTSTART:20071026T210000Z
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UID:74598@illinois.edu
CREATED:20071012T000000Z
DESCRIPTION:ABSTRACT:Most people will agree that performing a pirouette is intrinsically challenging: for humans it takes both natural talent and years of training. Looking at it from the perspective of a control systems scientist does not necessarily make it any easier\, but does allow us to be more specific about what the problem is. For one\, the motion is periodic\, and it is well-known that stabilization of periodic motions provides many challenges over and above those found when stabilizing an equilibrium point. A second difficulty is that\, standing on tiptoes\, the dancer cannot directly maintain their upright position; that is\, the system is underactuated: there are less independent control inputs than dynamical degrees of freedom.This talk is about orbital stabilization of periodic motions of underactuated mechanical systems. In particular\, we will discuss a feedback control design based on construction of a "transverse linearization." Roughly speaking\, the transverse linearization is a periodic linear system of dimension one less than the nonlinear system such that stabilization of this system is in a certain sense equivalent to exponential orbital stabilization of a desired periodic motion of the original nonlinear system. We consider a large class of mechanical systems that includes many popular research benchmark set-ups (the Furuta pendulum\, the Acrobot\, a pendulum on a cart\, a spherical pendulum) and applications (bipeds\, ocean-going vessels). Remarkably\, for this class of nonlinear controlled systems\, the transverse linearization of any feasible orbit\, which in general is related to defining moving Poincar???????? sections\, can be introduced analytically. This fact opens a broad range of opportunities.The approach can be considered as an alternative to the standard Poincar?? first-return map\, the most frequently used tool for analysis of existence and stability of periodic trajectories. Calculation of the Poincar?? map of a nonlinear system typically requires numerical solution of the system dynamics for a large number of initial conditions\, which is computationally expensive and motivates investigation of alternative strategies.The most prominent remaining challenges that restrict wider application of this technique are numerical. In the end\, one must find a stabilizing controller for a periodic linear system. For example\, the theory behind the LQR approach for periodic systems is well-established\, but requires finding the stabilizing solution of a matrix Riccati differential equation with periodic coefficients. In certain cases this is achievable (we will show experimental results on several set-ups including the Furuta pendulum); however\, there is a strong need for more reliable numerical methods.
LAST-MODIFIED:20071012T000000Z
LOCATION:141 Coordinated Science Laboratory
SUMMARY:CAESAR Seminar and Decision\, Control\, and Optimization Seminar: A. Shiriaev: "Can We Make a Robot Ballerina Perform a Pirouette?: Orbital Stabilization of Periodic Motions of Underactuated Mechanical Systems"
ORGANIZER:Information Trust Institute and the Decision & Control group in the Coordinated Science Laboratory
CATEGORIES:Seminar
URL;VALUE=URI:http://illinois.edu/calendar/Calendar?calId=442&ACTION=VIEW_EVENT&eventId=74598
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