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: 2.78). 05/2010; 55(4):893 - 906. DOI: 10.1109/TAC.2010.2042000
Source: IEEE Xplore


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.

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    • "[13], a detailed investigation of the integrals of motion is presented. In [15], the transverse linearization technique is generalized to the case when the degree of underactuation is greater than one. In [14], [17], VHC's are used to select and stabilize desired oscillations of the Furuta pendulum and the pendubot. "
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    ABSTRACT: This technical brief investigates virtual holonomic constraints for Euler-Lagrange systems with n degrees-of-freedom and n-1 controls. In our framework, a virtual holonomic constraint is a relation specifying n-1 configuration variables in terms of a single angular configuration variable. The enforcement by feedback of such a constraint induces a desired repetitive behavior in the system. We give conditions under which a virtual holonomic constraint is feasible, i.e, it can be made invariant by feedback, and it is stabilizable. We provide sufficient conditions under which the dynamics on the constraint manifold correspond to an Euler- Lagrange system. These ideas are applied to the problem of swinging up an underactuated pendulum while guaranteeing that the second link does not fall over.
    Full-text · Article · Apr 2013 · IEEE Transactions on Automatic Control
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    • "dynamic, with 2N states, and it stabilizes the manifold ¯ Γ in (29), thus enforcing the dynamic VHC (28). While in [1] one needs to stabilize an LTV system of order 2N − 1, our control design relies on the stabilization of a fourth-order LTI system, no matter how large N is. Thus, our control design is much simpler. "
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    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.
    Full-text · Conference Paper · Jun 2012
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    • "Spring-Manipulator system , which makes motion planning and control a challenging task. The virtual holonomic constraints approach [8], [9] is used as generic tool for shaping a desired ball-dribbling orbit (Section III) and its feedback stabilization (Section IV). Numerical simulations in Section V demonstrate that the structure and performance of the control system are suited for experimental tests. "
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    ABSTRACT: Ball dribbling is a central element of basketball. One main challenge for realizing basketball robots is to stabilize periodic motion of the ball. The task is nontrivial due to the discrete-continuous nature of the corresponding dynamics. This paper proposes to add an elastic element to the manipulator so the ball can be controlled in a continuous-time phase instead of an intermittent contact. Optimal catching and pushing trajectories are planned for the underactuated system based on the virtual holonomic constraints approach. First experimental studies are presented to evaluate the approach.
    Full-text · Conference Paper · Nov 2010
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