Andrew D. Lewis

Controllability and motion algorithms for underactuated Lagrangian systems on Lie groups

We provide controllability tests and motion control algorithms for underactuated mechanical control systems on Lie groups with Lagrangian equal to kinetic energy. Examples include satellite and underwater vehicle control systems with the number of control inputs less than the dimension of the configuration space. Local controllability properties of these systems are characterized, and two algebraic tests are derived in terms of the symmetric product and the Lie bracket of the input vector fields. Perturbation theory is applied to compute approximate solutions for the system under small-amplitude forcing; in-phase signals play a crucial role in achieving motion along symmetric product directions. Motion control algorithms are then designed to solve problems of point-to-point reconfiguration, static interpolation and exponential stabilization. We illustrate the theoretical results and the algorithms with applications to models of planar rigid bodies, satellites and underwater vehicles.

Nonholonomic mechanics and locomotion: the snakeboard example

Analysis and simulations are performed for a simplified model of a commercially available variant of the skateboard, known as the Snakeboard. Although the model exhibits basic gait patterns seen in a large number of locomotion problems, the analysis tools currently available do not apply to this problem. The difficulty lies primarily in the way in which the nonholonomic constraints enter into the system. As a first step towards understanding systems represented by their model the authors present the equations of motion and perform some controllability analysis for the snakeboard. The authors also perform numerical simulations of possible gait patterns which are characteristic of snakeboard locomotion.<<ETX>>

Variational principles for constrained systems: Theory and experiment

In this paper we present two methods, the nonholonomic method and the vakonomic method, for deriving equations of motion for a mechanical system with constraints. The resulting equations are compared. Results are also presented from an experiment for a model system: a ball rolling without sliding on a rotating table. Both sets of equations of motion for the model system are compared with the experimental results. The effects of various forms of friction are considered in the nonholonomic equations. With appropriate friction terms, the nonholonomic equations of motion for the model system give reasonable agreement with the experimental observations.

Simple mechanical control systems with constraints

We apply some recently developed control theoretic techniques to the analysis of a class of mechanical systems with constraints. Certain simple aspects of the theory of affine connections play an important part in our presentation. The necessary background is presented in order to illustrate how the methods may be applied. The bulk of this paper is devoted to a detailed analysis of some examples of nonholonomic mechanical control systems. We look at the Heisenberg system, the upright rolling disk, the roller racer, and the snakeboard.

Affine connections and distributions with applications to nonholonomic mechanics

Abstract Motivated by nonholonomic mechanics, we investigate various aspects of the interplay of an affine connection with a distribution. When the affine connection restricts to the distribution, we discuss torsion, curvature, and holonomy of the affine connection. We also investigate transformations which respect both the affine connection and the distribution. A stronger notion than that of restricting to a distribution is that of geodesic invariance. This is a natural generalisation to a distribution of the idea of a totally geodesic submanifold. We provide a product for vector fields which allows one to test for geodesic invariance in the same way one uses the Lie bracket to test for integrability. If the affine connection does not restrict to the distribution, we are able to define its restriction and in the process generalise the notion of the second fundamental form for submanifolds. We characterise some transformations of these restricted connections and derive conservation laws in the case when the original connection is the Levi-Civita connection associated with a Riemannian metric.

Kinematic controllability and motion planning for the snakeboard

The snakeboard is shown to possess two decoupling vector fields, and to be kinematically controllable. Accordingly, the problem of steering the snakeboard from a given configuration at rest to a desired configuration at rest is posed as a constrained static nonlinear inversion problem. An explicit algorithmic solution to the problem is provided, and its limitations are discussed. An ad hoc solution to the nonlinear inversion problem is also exhibited.

The mechanics of undulatory locomotion: the mixed kinematic and dynamic case

This paper studies the mechanics of undulatory locomotion. This type of locomotion is generated by a coupling of internal shape changes to external non-holonomic constraints. Employing methods from geometric mechanics, the authors use the dynamic symmetries and kinematic constraints to develop a specialized form of the dynamic equations which govern undulatory systems. These equations are written in terms of physically meaningful and intuitively appealing variables that show the role of internal shape changes in driving locomotion.

Low-Order Controllability and Kinematic Reductions for Affine Connection Control Systems

Controllability and kinematic modeling notions are investigated for a class of mechanical control systems. First, low-order controllability results are given for the class of mechanical control systems. Second, a precise connection is made between those mechanical systems which are dynamic (i.e., have forces as inputs) and those which are kinematic (i.e., have velocities as inputs). Interestingly and surprisingly, these two subjects are characterized and linked by a certain intrinsic vector-valued quadratic form that can be associated to an affine connection control system.

Approximating a-cuts with the vertex method

If f:Rn → R is continuous and monotonic in eachvariable, and if μi is a fuzzy number on the ith coordinate, then the membership on R induced by ƒ and by the membership onfRn given by μ(x) = min(μ1(x1), …, μn(xn)) can be evaluated by determining the membership at the endpoints of the level cuts of each μi. Here more general conditions are given for both the function ƒ and the manner in which the fuzzy numbers {μi} are combined so that this simple method for computing induced membership may be used. In particular, a geometric condition is given so that the α-cuts computed when the fuzzy numbers are combined using min is an upper bound for the actual induced membership.

Notes on energy shaping

The problem of shaping the kinetic and potential energy of a mechanical system by feedback is cast in a differential geometric framework. The nature of the set of solutions to the potential energy shaping problem is described. The kinetic energy shaping problem is posed in (1) an affine differential geometric framework and (2) a manner where the geometric integrability theory for partial differential equations can be applied.