Controlled Lagrangians and the stabilization of mechanical systems. II. Potential shaping

For pt.I, see ibid., vol.45, p.2253-70 (2000). We extend the method of controlled Lagrangians (CL) to include potential shaping, which achieves complete state-space asymptotic stabilization of mechanical systems. The CL method deals with mechanical systems with symmetry and provides symmetry-preserving kinetic shaping and feedback-controlled dissipation for state-space stabilization in all but the symmetry variables. Potential shaping complements the kinetic shaping by breaking symmetry and stabilizing the remaining state variables. The approach also extends the method of controlled Lagrangians to include a class of mechanical systems without symmetry such as the inverted pendulum on a cart that travels along an incline

Controlled Lagrangians and the stabilization of mechanical systems. I. The first matching theorem

We develop a method for the stabilization of mechanical systems with symmetry based on the technique of controlled Lagrangians. The procedure involves making structured modifications to the Lagrangian for the uncontrolled system, thereby constructing the controlled Lagrangian. The Euler-Lagrange equations derived from the controlled Lagrangian describe the closed-loop system, where new terms in these equations are identified with control forces. Since the controlled system is Lagrangian by construction, energy methods can be used to find control gains that yield closed-loop stability. We use kinetic shaping to preserve symmetry and only stabilize systems module the symmetry group. The procedure is demonstrated for several underactuated balance problems, including the stabilization of an inverted planar pendulum on a cart moving on a line and an inverted spherical pendulum on a cart moving in the plane

Asymptotic stabilization of the hanging equilibrium manifold of the 3D pendulum

The 3D pendulum consists of a rigid body, supported at a fixed pivot, with three rotational degrees of freedom; it is acted on by gravity and it is fully actuated by control forces. The 3D pendulum has two disjoint equilibrium manifolds, namely a hanging equilibrium manifold and an inverted equilibrium manifold. This paper shows that a controller based on angular velocity feedback can be used to asymptotically stabilize the hanging equilibrium manifold of the 3D pendulum. Lyapunov analysis and nonlinear geometric methods are used to assess the global closed-loop properties. We explicitly construct compact sets that lie in the domain of attraction of the hanging equilibrium of the closed-loop. Finally, this controller is shown to achieve almost global asymptotic stability of the hanging equilibrium manifold. An invariant manifold of the closed-loop that converges to the inverted equilibrium manifold is identified. Copyright © 2007 John Wiley & Sons, Ltd.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/56146/1/1178_ftp.pd

On the integration of singularity-free representations of SO(3) for direct optimal control

In this paper we analyze the performance of different combinations of: (1) parameterization of the rotational degrees of freedom (DOF) of multibody systems, and (2) choice of the integration scheme, in the context of direct optimal control discretized according to the direct multiple-shooting method. The considered representations include quaternions and Direction Cosine Matrices, both having the peculiarity of being non-singular and requiring more than three parameters to describe an element of the Special Orthogonal group SO(3). These representations yield invariants in the dynamics of the system, i.e., algebraic conditions which have to be satisfied in order for the model to be representative of physical reality. The investigated integration schemes include the classical explicit Rungeâ\u80\u93Kutta method, its stabilized version based on Baumgarteâ\u80\u99s technique, which tends to reduce the drift from the underlying manifold, and a structure-preserving alternative, namely the Rungeâ\u80\u93Kutta Munthe-Kaas method, which preserves the invariants by construction. The performances of the combined choice of representation and integrator are assessed by solving thousands of planning tasks for a nonholonomic, underactuated cart-pendulum system, where the pendulum can experience arbitrarily large 3D rotations. The aspects analyzed include success rate, average number of iterations and CPU time to convergence, and quality of the solution. The results reveal how structure-preserving integrators are the only choice for lower accuracies, whereas higher-order, non-stabilized standard integrators seem to be the computationally most competitive solution when higher levels of accuracy are pursued. Overall, the quaternion-based representation is the most efficient in terms of both iterations and CPU time to convergence, albeit at the cost of lower success rates and increased probability of being trapped by higher local minima

Remarks on stability of inverted pendula

Using linearization principle and tools from chronological calculus one establishes a criteria for stabilization of, usually unstable, equilibrium position of Double Inverted Pendula when subject an arbitrary fast oscillation. Both, planar and spherical cases are considered.Fundação para a Ciência e a Tecnologia (FCT)Dipartimento di Matematica per le Decisione, Università di Firenze, Itali