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.
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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.IEEE Transactions on Automatic Control 01/2013; 58(4):1001-1008. · 2.72 Impact Factor
Conference Paper: Control lyapunov functions and hybrid zero dynamics[Show abstract] [Hide abstract]
ABSTRACT: Hybrid zero dynamics extends the Byrnes-Isidori notion of zero dynamics to a class of hybrid models called systems with impulse effects. Specifically, given a smooth submanifold that is contained in the zero set of an output function and is invariant under both the continuous flow of the system with impulse effects as well as its reset map, the restriction dynamics is called the hybrid zero dynamics. Prior results on the stabilization of periodic orbits of the hybrid zero dynamics have relied on input-output linearization of the transverse variables. The principal result of this paper shows how control Lyapunov functions can be used to exponentially stabilize periodic orbits of the hybrid zero dynamics, thereby significantly extending the class of stabilizing controllers. An illustration of this result on a model of a bipedal walking robot is provided.Decision and Control (CDC), 2012 IEEE 51st Annual Conference on; 01/2012
Conference Paper: Synchronizing N cart-pendulums using virtual holonomic constraints[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; 06/2012