HOW TO REDISTRIBUTE ENERGY BETWEEN DIFFERENT LINKS OF THE PENDUBOT A.S. Shiriaev *a1 0. Kolesnichenko * L. Paramonov * * The Maersk Me-Kinney Moller Institute for Production Technology University of Southern Denmark Campusvej 55, DK-5230 Odense M, DENMARK {anton,kolesya,leonid}@mip.sdu.dk Abstract: This paper considers a new stabilization problem for the Pendubot. Namely how to construct and stabilize via feedback the following trajectory: the first (controlled) link of the Pendubot remains at rest having a given angle with horizontal, while the second (freely moving) link duplicates a given motion of the 1-d.0.f. plane pendulum. Such a stabilization could be also seen as re-distributing the energy of the Pendubot between actuated and non-actuated parts, or saving the energy in the motion of the non-actuated part. The main result of the paper is the description of a wide family of the state feedback controllers, which solve the problem. In addition, the delicate issue of the convergence’s rate of the closed loop system solutions to the desired trajectory is discussed in details. Keywords: passivity, feedback transformation, the Pendubot , constructing periodic motion via feedback 1. INTRODUCTION Controlling an underactuated nonlinear system is inherently difficult problem. There are a few analytical methods which are able to tackle such a problem, most of them are based on structural properties of the system. The reader, for exam- ple, can check the papers (Ortega et al., n.d.) and (Bloch et al., 2000), where conserved quan- tities and symmetries of the system play a major role in constructing a stabilizing controller for an equilibrium. Another example, where conserved quantities are important, is related to the problem of stabilization of some particular subset of the state space, possibly different from an equilibrium. The reader can check, for example, the results of (Fradkov, 1996; Astrom and Furuta, 2000; Shiri- aev et al., 200l), where a stabilization of homo- The work is supported by the Danish Technical Research Council, the grant 26-01-0164 clinic curves of the plane pendulum was made; and the results of (Ludvigsen et al., 1999; Al- bouy and Praly, 2000), where a stabilization of the stable manifold of the spherical pendulum (a 2-dimensional subset of the 4-dimensional state space), was done. This paper is concerned with an two-link under- actuated robot called the Pendubot. It has an ac- tuator at the shoulder (link 1) and no actuator at the elbow (link 2). One of standard control prob- lems related to the Pendubot is a stabilization of one of its equilibria. Among other papers the reader can check (Spong and Block, 1995; Fantoni et al., 2000; Kolesnichenko and Shiriaev, 2002), where some methods for stabilizing the upper equilibrium are suggested. Another interesting control problem related to the Pendubot is a construction and orbital sta- bilization of periodic motions via feedback. This paper is aimed at constructing and stabilizing Copyright © 2002 IFAC www.elsevier.com/locate/ifac Copyright © 2002 IFAC 15th Triennial World Congress, Barcelona, Spain 545