Dong Eui Chang

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.

Collision avoidance for multiple agent systems

Techniques using gyroscopic forces and scalar potentials are used to create swarming behaviors for multiple agent systems. The methods result in collision avoidance between the agents as well as with obstacles.

Normal forms and bifurcations of control systems

We present the quadratic and cubic normal forms of a nonlinear control system around an equilibrium point. These are the normal forms under change of state coordinates and invertible state feedback. The system need not be linearly controllable. A control bifurcation of a nonlinear system occurs when its linear approximation loses stabilizability. We study some important control bifurcations, the analogues of the classical fold, transcritical and Hopf bifurcations.

GEOMETRIC DERIVATION OF THE DELAUNAY VARIABLES AND GEOMETRIC PHASES

We derive the classical Delaunay variables by finding a suitable symmetry action of the three torus T3 on the phase space of the Kepler problem, computing its associated momentum map and using the geometry associated with this structure. A central feature in this derivation is the identification of the mean anomaly as the angle variable for a symplectic S1 action on the union of the non-degenerate elliptic Kepler orbits. This approach is geometrically more natural than traditional ones such as directly solving Hamilton–Jacobi equations, or employing the Lagrange bracket. As an application of the new derivation, we give a singularity free treatment of the averaged J2-dynamics (the effect of the bulge of the Earth) in the Cartesian coordinates by making use of the fact that the averaged J2-Hamiltonian is a collective Hamiltonian of the T3 momentum map. We also use this geometric structure to identify the drifts in satellite orbits due to the J2 effect as geometric phases.

Asymptotic Stabilization of Euler-Poincaré Mechanical Systems

Stabilization of mechanical control systems by the method of controlled Lagrangians nand matching is used to analyze asymptotic stabilization of systems whose nunderlying dynamics are governed by the Euler-Poincar´e equations. In particular, we nanalyze asymptotic stabilization of a satellite.

Asymptotic stabilization of the heavy top using controlled Lagrangians

We extend previous work on the asymptotic stabilization of pure Euler-Poincare mechanical systems using controlled Lagrangians to the study of asymptotic stabilization of Euler-Poincare mechanical systems such as the heavy top.

Normal Forms of Linearly Uncontrollable Nonlinear Control Systems with a Single Input

Abstract We derive the controller normal form and the dual normal form for linearly uncontrollable nonlinear control systems with a single input. The invariants under the state and feedback transformation of degree d are found for d ≥ 2.

Potential and kinetic shaping for control of underactuated mechanical systems

This paper combines techniques of potential shaping with those of kinetic shaping to produce some new methods for stabilization of mechanical control systems. As with each of the techniques themselves, our method employs the energy methods and LaSalles invariance principle. We give explicit criteria for asymptotic stabilization of equilibria of mechanical systems which, in the absence of controls, have a kinetic energy function that is invariant under an Abelian group.