65 research outputs found
Stabilization of relative equilibria
This paper discusses the problem of obtaining feedback laws to asymptotically stabilize relative equilibria of mechanical systems with symmetry. We show how to stabilize an internally unstable relative equilibrium using internal actuators. The methodology is that of potential shaping, but the system is allowed to be underactuated, i.e., have fewer actuators than the dimension of the shape space. The theory is illustrated with the problem of stabilization of the cowboy relative equilibrium of the double spherical pendulum
Stabilization of Relative Equilibria II
In this paper, we obtain feedback laws to asymptotically stabilize relative equilibria of mechanical systems with symmetry. We use a notion of stability ‘modulo the group action’ developed by Patrick [1992]. We deal with both
internal instability and with instability of the rigid motion. The methodology is that of potential shaping, but the system is allowed to be internally underactuated,
i.e., have fewer internal actuators than the dimension of the shape space
Asymptotic Stability, Instability and Stabilization of Relative Equilibria
In this paper we analyze asymptotic stability, instability and stabilization for the relative equilibria, i.e. equilibria modulo a group action, of natural mechanical systems. The practical applications of these results are to rotating mechanical systems where the group is the rotation group. We use a modification of the Energy-Casimir and Energy-Momentum methods for Hamiltonian systems to analyze systems with dissipation. Our work couples the modern theory of block diagonalization to the classical work of Chetaev
Stability and Stabilization of Relative Equilibria of Dumbbell Bodies in Central Gravity
Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/77368/1/AIAA-10546-792.pd
Controlled Lagrangians and Stabilization of the Discrete Cart-Pendulum System
Matching techniques are developed for discrete
mechanical systems with symmetry. We describe new phenomena
that arise in the controlled Lagrangian approach for mechanical
systems in the discrete context. In particular, one needs
to either make an appropriate selection of momentum levels or
introduce a new parameter into the controlled Lagrangian to
complete the matching procedure. We also discuss digital and
model predictive control
Flat Nonholonomic Matching
In this paper we extend the matching technique to a
class of nonholonomic systems with symmetries. Assuming
that the momentum equation defines an integrable
distribution, we introduce a family of reduced
systems. The method of controlled Lagrangians is then
applied to these systems resulting in a smooth stabilizing
controller
Controlled Lagrangians and Potential Shaping for Stabilization of Discrete Mechanical Systems
The method of controlled Lagrangians for discrete mechanical systems is
extended to include potential shaping in order to achieve complete state-space
asymptotic stabilization. New terms in the controlled shape equation that are
necessary for matching in the discrete context are introduced. The theory is
illustrated with the problem of stabilization of the cart-pendulum system on an
incline. We also discuss digital and model predictive control.Comment: IEEE Conference on Decision and Control, 2006 6 pages, 4 figure
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
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