Nonlinear Oscillations: Exact Solutions and their Approximations

This work develops the geometry and dynamics of mechanical systems with nonholonomic constraints and symmetry from the perspective of Lagrangian mechanics and with a view to control theoretical applications. The basic methodology is that of geometric mechanics applied to the formulation of Lagrange d'Alembert, generalizing the use of connections and momentum maps associated with a given symmetry group to this case. We begin by formulating the mechanics of nonholonomic systems using an Ehresmann connection to model the constraints, and show how the curvature of this connection enters into Lagrange's equations. Unlike the situation with standard configuration space constraints, the presence of symmetries in the nonholonomic case may or may not lead to conservation laws. However, the momentum map determined by the symmetry group still satisfies a useful differential equation that decouples from the group variables. This momentum equation, which plays an important role in control problems, involves parallel transport operators and is computed explicitly in coordinates. An alternative description using a ``body reference frame'' relates part of the momentum equation to the components of the Euler-Poincar\'{e} equations along those symmetry directions consistent with the constraints. One of the purposes of this paper is to derive this evolution equation for the momentum and to distinguish geometrically and mechanically the cases where it is conserved and those where it is not. An example of the former is a ball or vertical disk rolling on a flat plane and an example of the latter is the snakeboard, a modified version of the skateboard which uses momentum coupling for locomotion generation. We construct a synthesis of the mechanical connection and the Ehresmann connection defining the constraints, obtaining an important new object we call the nonholonomic connection. When the nonholonomic connection is a principal connection for the given symmetry group, we show how to perform Lagrangian reduction in the presence of nonholonomic constraints, generalizing previous results which only held in special cases. Several detailed examples are given to illustrate the theory. September 1994 Revised, March 1995 Revised, June 1995

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Nonholonomic Mechanical Systems with Symmetry

The goal of this final chapter is to show how the boundary value problems of mathematical physics can be solved by the methods of the preceding chapters. This will be done by solving a variety of specific problems that illustrate the principal types of problems that were formulated in Chapter 7. Additional applications are developed in the Exercises. The primary solution method is Fourier’s method of separation of variables and the associated Sturm-Liouville theory of Chapter 8.

Boundary Value Problems of Mathematical Physics

This paper is concerned with the design of a robotic fish and its motion control algorithms. A radio-controlled, four-link biomimetic robotic fish is developed using a flexible posterior body and an oscillating foil as a propeller. The swimming speed of the robotic fish is adjusted by modulating joint's oscillating frequency, and its orientation is tuned by different joint's deflections. Since the motion control of a robotic fish involves both hydrodynamics of the fluid environment and dynamics of the robot, it is very difficult to establish a precise mathematical model employing purely analytical methods. Therefore, the fish's motion control task is decomposed into two control systems. The online speed control implements a hybrid control strategy and a proportional-integral-derivative (PID) control algorithm. The orientation control system is based on a fuzzy logic controller. In our experiments, a point-to-point (PTP) control algorithm is implemented and an overhead vision system is adopted to provide real-time visual feedback. The experimental results confirm the effectiveness of the proposed algorithms.

Development of a biomimetic robotic fish and its control algorithm

The bodies and brains of fish have evolved to achieve control objectives beyond the capabilities of current underwater vehicles. One route toward designing underwater vehicles with similar capabilities is to better understand fish physiological design and control strategies. This paper has two objectives: 1) to review clues to artificial swimmer design taken from fish physiology and 2) to formalize and review the control problems that must be solved by a robot fish. The goal is to exploit fish locomotion principles to address the truly difficult control challenges of station keeping under large perturbations, rapid maneuvering, power-efficient endurance swimming, and trajectory planning and tracking. The design and control of biomimetic swimming machines meeting these challenges will require state-of-the-art engineering and biology.

Mechanics and control of swimming: a review

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 robot's 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

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Geometric Phases and Robotic Locomotion

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.

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

Modelling efficient pisciform swimming for control

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.

Source Seeking for Two Nonholonomic Models of Fish Locomotion

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.

Scott David Kelly

Papers

Geometric Phases and Robotic Locomotion

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

Modelling efficient pisciform swimming for control

Source Seeking for Two Nonholonomic Models of Fish Locomotion

Inertial particle trapping in viscous streaming