Transverse Linearization for Controlled Mechanical Systems With Several Passive Degrees of Freedom

Dept. of Appl. Phys. & Electron., Umea Univ., Umea, Sweden
IEEE Transactions on Automatic Control (Impact Factor: 2.72). 05/2010; DOI: 10.1109/TAC.2010.2042000
Source: IEEE Xplore

ABSTRACT This study examines the mechanical systems with an arbitrary number of passive (non-actuated) degrees of freedom and proposes an analytical method for computing coefficients of a linear controlled system, solutions of which approximate dynamics transverse to a feasible motion. This constructive procedure is based on a particular choice of coordinates and allows explicit introduction of a moving Poincare?? section associated with a nontrivial finite-time or periodic motion. In these coordinates, transverse dynamics admits analytical linearization before any control design. If the forced motion of an underactuated mechanical system is periodic, then this linearization is an indispensable and constructive tool for stabilizing the cycle and for analyzing its orbital (in)stability. The technique is illustrated with two challenging examples. The first one is stabilization of a circular motions of a spherical pendulum on a puck around its upright equilibrium. The other one is creating stable synchronous oscillations of an arbitrary number of planar pendula on carts around their unstable equilibria.

1 Bookmark
  • [Show abstract] [Hide abstract]
    ABSTRACT: A solution is presented to the problem of synchronizing a chain of N cart-pendulums using virtual holonomic constraints. The approach is based on a master-slave configuration whereby the first cart-pendulum is controlled so as to stabilize a desired oscillation around its unstable equilibrium. Then, each remaining cart-pendulum is controlled so as to fully synchronize it to the previous pendulum.
    American Control Conference (ACC), 2012; 01/2012
  • [Show abstract] [Hide abstract]
    ABSTRACT: The paper aims at solving the walking control problem of a compass-like biped robot with underactuated ankle on the level ground or even uphill environment. The compass-like biped robot is equipped with a constraint mechanism to lock the hip angle when the swing leg retracts to a desired angle. For this system, an angular momentum based control is presented in order to make the biped robot walk on little downhill slope or even uphill. Existence conditions of limit cycle under angular momentum based control are presented. They can be used to determine whether there exists a gait on the slope or not. Furthermore, we apply the method of Poincare return map to analyze the stability property of the gait with angular momentum based control. Finally, an event-based control is adopted to make the walking gait of compass-like biped robot asymptotically stable.
    Proceedings of the 10th World Congress on Intelligent Control and Automation; 01/2012
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: A quasi-continuous high-order sliding mode (QC-HOSM) control is developed to solve the tracking control problem for an inertia wheel pendulum. A first step towards the solution of the tracking control problem in underac-tuated systems is to find the set of reference trajectories. A reference model based on the two relay controller idea is then developed for generating a set of desired periodic trajectories for the pendulum centered at its upright posi-tion. The two relay controller produces oscillations at the scalar output of the reference underactuated system where the desired amplitude and frequency are reached by choosing its gains. The HOSM will be capable of making the pendulum move, tracking the prescribed reference signals determined by the trajectory generator. Performance issues of the controller constructed are illustrated in an experimental study.
    Asian Journal of Control 04/2010; 14(9). · 1.41 Impact Factor