Scott David Kelly

Geometric Phases and Robotic Locomotion

Robotic locomotion is based in a variety of instances upon cyclic changes in the shape of a robot mechanism. Certain variations in shape exploit the constrained nature of a robots interaction with its environment to generate net motion. This is true for legged robots, snakelike robots, and wheeled mobile robots undertaking maneuvers such as parallel parking. In this paper we explore the use of tools from differential geometry to model and analyze this class of locomotion mechanisms in a unified way. In particular, we describe locomotion in terms of the geometric phase associated with a connection on a principal bundle, and address issues such as controllability and choice of gait. We also provide an introduction to the basic mathematical concepts which we require and apply the theory to numerous example systems.

The Hamiltonian structure of a two-dimensional rigid circular cylinder interacting dynamically with N point vortices

This paper studies the dynamical fluid plus rigid-body system consisting of a two-dimensional rigid cylinder of general cross-sectional shape interacting with N point vortices. We derive the equations of motion for this system and show that, in particular, if the vortex strengths sum to zero and the rigid-body has a circular shape, the equations are Hamiltonian with respect to a Poisson bracket structure that is the sum of the rigid body Lie–Poisson bracket on Se(2)*, the dual of the Lie algebra of the Euclidean group on the plane, and the canonical Poisson bracket for the dynamics of N point vortices in an unbounded plane. We then use this Hamiltonian structure to study the linear and nonlinear stability of the moving Foppl equilibrium solutions using the energy-Casimir method.

Modelling and experimental investigation of carangiform locomotion for control

We propose a model for planar carangiform swimming based on conservative equations for the interaction of a rigid body and an incompressible fluid. We account for the generation of thrust due to vortex shedding through controlled coupling terms. We investigate the correct form of this coupling experimentally with a robotic propulsor, comparing its observed behavior to that predicted by unsteady hydrodynamics. Our analysis of thrust generation by an oscillating hydrofoil allows us to characterize and evaluate certain families of gaits. Our final swimming model takes the form of a control-affine nonlinear system.

Source Seeking for Two Nonholonomic Models of Fish Locomotion

In this paper, we present a method of locomotion control for underwater vehicles that are propelled by a periodic deformation of the vehicle body, which is similar to the way a fish moves. We have developed control laws employing ldquoextremum seekingrdquo for two different ldquofishrdquo models. The first model consists of three rigid body links and relies on a 2-degree-of-freedom (DOF) movement that propels the fish without relying on vortices. The second fish model uses a Joukowski airfoil that has only 1 DOF in its movement and, thus, relies on vortex shedding for propulsion. We achieve model-free and position-free ldquosource seeking,rdquo and, if position is available, navigation along a predetermined path.

The geometry and control of dissipative systems

We regard the internal configuration of a deformable body, together with its position and orientation in ambient space, as a point in a trivial principal fiber bundle over the manifold of body deformations. In the presence of a symmetry which leads to a conservation law the self-propulsion of such a body due to cyclic changes in shape is described by the corresponding mechanical connection on the configuration bundle. In the presence of viscous drag sufficient to negate inertial effects, the Stokes connection takes the place of the mechanical connection. Both connections may be represented locally in terms of the variables describing the bodys shape. In the presence of both inertial and viscous effects, the equations of motion may be written in terms of the two local connection forms as an affine control system with drift on the manifold of configurations and body momenta. We apply techniques from nonlinear control theory to the equations in this form to obtain criteria for a particular form of accessibility.

Inertial particle trapping in viscous streaming

The motion of an inertial particle in a viscous streaming flow of Reynolds number order 10 is investigated theoretically and numerically. The streaming flow created by a circular cylinder undergoing rectilinear oscillation with small amplitude is obtained by asymptotic expansion from previous work, and the resulting velocity field is used to integrate the Maxey–Riley equation with the Saffman lift for the motion of an inertial spherical particle immersed in this flow. It is found that inertial particles spiral inward and become trapped inside one of the four streaming cells established by the cylinder oscillation, regardless of the particle size, density and flow Reynolds number. It is shown that the Faxen correction terms divert the particles from the fluid particle trajectories, and once diverted, the Saffman lift force is most responsible for effecting the inward motion and trapping. The speed of this trapping increases with increasing particle size, decreasing particle density, and increasing oscillation Reynolds number. The effects of Reynolds number on the streaming cell topology and the boundaries of particle attraction are also explored. It is found that particles initially outside the streaming cell are repelled by the flow rather than trapped.

Proportional heading control for planar navigation: The Chaplygin beanie and fishlike robotic swimming

We present a simple method for the coupled propulsion and steering of certain single-input planar vehicles. We demonstrate the applicability of this method to two distinct systems, one a hybridization of canonical systems from the mechanics literature and the other a fishlike robot. The first system is examined analytically, the second experimentally.

Mechanics, Dynamics, and Control of a Single-Input Aquatic Vehicle With Variable Coefficient of Lift

We describe basic considerations in the Lagrangian modeling of aquatic vehicles developing liftlike forces in a controlled way. We introduce the aquatic Flettner rotor as prototypical of this class of vehicles, and demonstrate the compatibility of Lagrangian formalism with experimental data describing a laboratory rotor. We analyze the controllability of a model for the rotor, connect its structure to that of models for Lagrangian systems subject to nonholonomic constraints, and present numerical evidence that our model can behave chaotically given physically motivated inputs and disturbances. We conclude with a description of a fishlike robotic vehicle employing a rotor in place of a caudal fin

Controlled Hydrodynamic Interactions in Schooling Aquatic Locomotion

We present experimental data elucidating the effects of hydrodynamic coupling on the propulsive efficiency of an array of three oscillating hydrofoils. We simulate this system using an inviscid flow model; this model duplicates certain key features of our experimental data but fails to consider the effects of wake vortex generation and interaction. We present a qualitative model for the role played by wake vortex dynamics in the cooperative locomotion of fish schools, and derive a mathematical model in the form of a nonlinear control system describing the interaction of a single deformable body with a single nearby vortex. We present simulations based on the latter to illustrate the capture of vortices shed from one fish in a school by a second, trailing fish; vortex capture in this sense is the control problem central to cooperative swimming.

Dynamics and self-propulsion of a spherical body shedding coaxial vortex rings in an ideal fluid

We describe a model for the dynamic interaction of a sphere with uniform density and a system of coaxial circular vortex rings in an ideal fluid of equal density. At regular intervals in time, a constraint is imposed that requires the velocity of the fluid relative to the sphere to have no component transverse to a particular circular contour on the sphere. In order to enforce this constraint, new vortex rings are introduced in a manner that conserves the total momentum in the system. This models the shedding of rings from a sharp physical ridge on the sphere coincident with the circular contour. If the position of the contour is fixed on the sphere, vortex shedding is a source of drag. If the position of the contour varies periodically, propulsive rings may be shed in a manner that mimics the locomotion of certain jellyfish. We present simulations representing both cases.