# Department of Mathematics and Statistics: Dr. Andrew D. Lewis

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These are the preprint versions of the papers, and will differ somewhat from what you will see published. I do not accept responsibility for those items which have not been reviewed.

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Item An example with interesting controllability and stabilisation properties(2005) Hirschorn, Ron M.; Lewis, Andrew D.A simple three-state system with two inputs is considered. The system's controllability is determined using properties of vector-valued quadratic forms. The quadratic structure is then used as the basis for the design of a homogeneous, discontinuous, stabilising feedback controller. The paper should be seen as an attempt to relate controllability of a system from a point to stabilisability of the system to the same point.Item Reduction, linearization, and stability of relative equilibria for mechanical systems on Riemannian manifolds(2004) Bullo, Francesco; Lewis, Andrew D.Consider a Riemannian manifold equipped with an infinitesimal isometry. For this setup, a unified treatment is provided, solely in the language of Riemannian geometry, of techniques in reduction, linearization, and stability of relative equilibria. In particular, for mechanical control systems, an explicit characterization is given for the manner in which reduction by an infinitesimal isometry, and linearization along a controlled trajectory ``commute.'' As part of the development, relationships are derived between the Jacobi equation of geodesic variation and concepts from reduction theory, such as the curvature of the mechanical connection and the effective potential. As an application of our techniques, fiber and base stability of relative equilibria are studied. The paper also serves as a tutorial of Riemannian geometric methods applicable in the intersection of mechanics and control theory.Item Discussion on: ``Dynamic Sliding Mode Control for a Class of Systems with Mismatched Uncertainty''(Lavoisier, 2005) Lewis, Andrew D.This is an invited discussion paper on the paper ``Dynamic Sliding Mode Control for a Class of Systems with Mismatched Uncertainty'' by Xing-Gang Yan, Sarah K. Spurgeon, and Christopher Edwards, that will appear in the European Journal of Control.Item Rigid body mechanics in Galilean spacetimes(2004) bhand, Ajit; Lewis, Andrew D.An observer-independent formulation of rigid body dynamics is provided in the general setting of a Galilean spacetime. The equations governing the motion of a rigid body undergoing a rigid motion in a Galilean spacetime are derived on the basis of the principle of conservation of spatial momentum. The formulation of rigid body dynamics is then studied in the presence of an observer. It is seen that an observer defines a connection such that there exist rigid motions that are horizontal with respect to this connection that give the same physical motion of the rigid body, and for which the general equations of motion are exactly the usual Euler equations for a rigid body undergoing rigid motion.Item Kinematic controllability and motion planning for the snakeboard(IEEE, 2003) Bullo, Francesco; Lewis, Andrew D.The snakeboard is shown to be kinematically controllable. Associated with the two decoupling vector fields for the problem, a constrained static nonlinear programming problem is posed whose solutions provide a solution to the problem of steering the snakeboard from a given configuration at rest to a desired configuration at rest. An explicit solution to the problem is provided, and the limitations of this explicit solution are discussed. A few ad hoc solutions to the nonlinear programming problem are also exhibited.