Transverse Linearization for Controlled Mechanical Systems With Several Passive Degrees of Freedom
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.
Conference Proceeding: Stabilization of hybrid periodic orbits with application to bipedal walking[show abstract] [hide abstract]
ABSTRACT: This work describes a general computational framework for robust stabilization of periodic orbits for hybrid systems with impact effects. We demonstrate that for such systems dynamics can be decomposed into the transverse and tangential components if a proper orthogonalizing transform is applied before the decomposition. Subsequently, we show that the robust control synthesis problem can be cast as a semi-definite program which implies that computationally efficient linear matrix inequality (LMI) solvers can be used to find the controllers. The methodology is verified through the simulation on a five-link planar under-actuated biped robot, an example often used by other researchersAmerican Control Conference, 2006; 07/2006
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ABSTRACT: Several new controller design techniques for global stabilization of nonlinear systems have recently been reported. Typically, these methodologies provide the designer with various degrees of freedom; consequently, their application in specific examples leads to the definition of various control schemes. One question of interest is the relationship between these schemes, or whether one contains the other. Further, since these schemes will, in general, exhibit different transients and possess different robustness properties, another challenging research problem is to establish some common framework to compare their robustness and performance properties. In this paper we investigate these questions for three different controller design techniques as applied to the problem of global tracking of robots with flexible joints. The connection between the various controllers are investigated. Further, they are compared using the following performance indicators: continuity properties vis-à-vis the joint stiffness, availability of adaptive implementations when the robot parameters are unknown, and robustness to ‘energy-preserving’ (i.e. passive) unmodelled effects. Complete stability proofs of all the resulting controllers are given.Automatica. 01/1995;
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ABSTRACT: The paper studies the properties of a sinusoidally vibrating wedge billiard as a model for 2-D bounce juggling. It is shown that some periodic orbits that are unstable in the elastic fixed wedge become exponentially stable in the nonelastic vibrating wedge. These orbits are linked with certain classical juggling patterns, providing an interesting benchmark for the study of the frequency-locking properties in human rhythmic tasks. Experimental results on sensorless stabilization of juggling patterns are described.IEEE Transactions on Robotics 03/2006; · 2.57 Impact Factor