Showing papers by "N.H. McClamroch published in 2005"

Stabilization of a 3D rigid pendulum

Abstract: We introduced models for a 3D pendulum, consisting of a rigid body that is supported at a frictionless pivot, in a 2004 CDC paper [Shen, J, Dec. 2004]. In that paper, several different classifications were given and models were developed for each classification. Control problems were posed based on these various models. This paper continues that line of research by studying stabilization problems for a reduced model of the 3D pendulum. Two different stabilization strategies are proposed. The first controller, based on angular velocity feedback only, asymptotically stabilizes the hanging equilibrium. The domain of attraction is shown to be almost global. The second controller, based on angular velocity and reduced attitude feedback, asymptotically stabilizes the inverted equilibrium, providing an almost global domain of attraction. Simulation results are provided to illustrate closed loop properties.

Stabilization of a 3D Axially Symmetric Rigid Pendulum

Abstract: Models for a 3D pendulum, consisting of a rigid body that is supported at a frictionless pivot, were introduced in a recent 2004 CDC paper [1]. Control problems were posed based on these models. A subsequent paper, in the 2005 ACC [2], developed stabilizing controllers for a 3D rigid pendulum assuming three independent control inputs. In the present paper, stabilizing controllers are developed for a 3D rigid pendulum assuming that the pendulum has a single axis of symmetry that is uncontrollable. This assumption allows development of a reduced model that forms the basis for controller design and closed loop analysis; this reduced model is parameterized by the constant angular velocity component of the 3D pendulum about its axis of symmetry. Several different controllers are proposed. The first controller, based on angular velocity feedback only, asymptotically stabilizes the hanging equilibrium. Then controllers are introduced, based on angular velocity and reduced attitude feedback, that asymptotically stabilize either the hanging equilibrium or the inverted equilibrium. These problems can be viewed as stabilization of a Lagrange top. Finally, if the angular velocity about the axis of symmetry is assumed to be zero, controllers are introduced, based on angular velocity and reduced attitude feedback, that asymptotically stabilize either the hanging equilibrium or the inverted equilibrium. This problem can be viewed as stabilization of a spherical pendulum.

Optimal fuel-image motion planning for a class of dual spacecraft formations

Abstract: In this paper we study a class of dual spacecraft formations for imaging applications. After motivating the problem, we discuss the general goals of an imaging formation. We then specialize the discussion to a class of dual spacecraft formations and introduce the geometric constraints imposed on the formation. The first main contribution of this paper is that we combine two ideas introduced separately in the literature and propose a maneuver that offers improved imaging performance. We then formulate an optimal control problem to minimize fuel consumption and further maximize image quality by minimizing the relative speed, which is proportional to the signal-to-noise ratio of the reconstructed image. We use the maximum principle to derive the necessary optimality conditions und show that they are also sufficient and that the resulting control law is unique. Finally, we apply a continuation method to solve for the unique optimal trajectory.

Control of Mechanical Systems with Cyclic Coordinates using Higher Order Averaging

Abstract: The control and dynamics of complex mechanical systems with unactuated cyclic coordinates, using only internal controls, is treated here. The goal is to achieve full control-liability of the reduced dynamics obtained by eliminating the cyclic coordinates using standard Routh reduction. The reduced system is also underactuated. We use high frequency, high amplitude periodic inputs and the framework of chronological calculus and averaging theory, for this purpose. A feedback scheme based on this approach is applied to the example of a dumbbell body in planar motion with an attitude control input in a central gravitational field. From our earlier work on this model, based on linearization, we know that the system is controllable at its relative equilibria. This work supplements earlier research on the possible use of internal controls for orbital maneuvers of underactuated spacecraft.

Experiments on stabilization of the hanging equilibrium of a 3D asymmetric rigid pendulum

Abstract: We introduced a theory for stabilization of a 3D pendulum, consisting of a rigid body that is supported at a frictionless pivot, in a 2005 ACC paper. One of the controllers proposed in that paper, based on angular velocity feedback only, asymptotically stabilizes the hanging equilibrium of the pendulum. This paper continues this line of research, providing a description of an experimental setup and a sample of experimental results illustrating the closed loop properties of the 3D pendulum. Experimental results are discussed and compared with the theory presented in the previous paper