Article

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
 · 
64 Views
  • [Show abstract] [Hide abstract]
    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
  • Source
    [Show abstract] [Hide abstract]
    ABSTRACT: A new approach to trajectory planning for underactuated mechanical systems is presented and discussed based on analysis of feasible behaviors of a standard 2-DOF benchmark example — the cart–pendulum system. Following the Controlled Lagrangians approach of Bloch et al. (2000) [7], we present and re-establish known conditions and forms of feedback control laws for this example, which are leading to an equivalent completely integrable closed-loop Euler–Lagrange system; and then extend them. As shown, full integrability and, in particular, the presence of a linear in velocities first integral of dynamics plays the key role in an elegant new procedure for trajectory planning.
    European Journal of Control 01/2013; 19(6):438–444. · 1.25 Impact Factor
  • [Show abstract] [Hide abstract]
    ABSTRACT: We consider a benchmark example of a three-link planar biped walker with torso, which is actuated in between the legs. The torso is thought to be kept upright by two identical torsional springs. The mathematical model reflects a three-degree-of-freedom mechanical system with impulse effects, which describe the impacts of the swing leg with the ground, and the aim is to induce stable limit-cycle walking on level ground. The main contribution is a novel systematic trajectory planning procedure for solving the problem of gait synthesis. The key idea is to find a system of ordinary differential equations for the functions describing a synchronization pattern for the time evolution of the generalized coordinates along a periodic motion. These functions, which are known as virtual holonomic constraints, are also used to compute an impulsive linear system that approximates the time evolution of the subset of coordinates that are transverse to the orbit of the continuous part of the periodic solution. This auxiliary system, which is known as transverse linearization, is used to design a nonlinear exponentially orbitally stabilizing feedback controller. The performance of the closed-loop system and its robustness with respect to various perturbations and uncertainties are illustrated via numerical simulations.
    IEEE Transactions on Robotics 01/2013; 29(3):589-601. · 2.57 Impact Factor