Steering for a Class of Dynamic Nonholonomic Systems

In this paper we derive control algorithms for a class of dynamic nonholonomic steering problems, characterized as mechanical systems with nonholonomic constraints and symmetries. Recent research in geometric mechanics has led to a single, simplified framework that describes this class of systems, which includes examples such as wheeled mobile robots; undulatory robotic and biological locomotion systems, such as paramecia, inchworms, and snakes; and the reorientation of satellites and underwater vehicles. This geometric framework has also been applied to more unusual examples, such as the snakeboard robot, bicycles, the wobblestone, and the reorientation of a falling cat. We use this geometric framework as a basis for developing two types of control algorithms for such systems. The first is geared towards local controllability, using a perturbation approach to establish results similar to steering using sinusoids. The second method utilizes these results in applying more traditional steering algorithms for mobile robots, and is directed towards generating more non-local control methods of steering for this class of systems

The Mechanics and Control of Undulatory Robotic Locomotion

In this dissertation, we examine a formulation of problems of undulatory robotic locomotion within the context of mechanical systems with nonholonomic constraints and symmetries. Using tools from geometric mechanics, we study the underlying structure found in general problems of locomotion. In doing so, we decompose locomotion into two basic components: internal shape changes and net changes in position and orientation. This decomposition has a natural mathematical interpretation in which the relationship between shape changes and locomotion can be described using a connection on a trivial principal fiber bundle. We begin by reviewing the processes of Lagrangian reduction and reconstruction for unconstrained mechanical systems with Lie group symmetries, and present new formulations of this process which are easily adapted to accommodate external constraints. Additionally, important physical quantities such as the mechanical connection and reduced mass-inertia matrix can be trivially determined using this formulation. The presence of symmetries then allows us to reduce the necessary calculations to simple matrix manipulations. The addition of constraints significantly complicates the reduction process; however, we show that for invariant constraints, a meaningful connection can be synthesized by defining a generalized momentum representing the momentum of the system in directions allowed by the constraints. We then prove that the generalized momentum and its governing equation possess certain invariances which allows for a reduction process similar to that found in the unconstrained case. The form of the reduced equations highlights the synthesized connection and the matrix quantities used to calculate these equations. The use of connections naturally leads to methods for testing controllability and aids in developing intuition regarding the generation of various locomotive gaits. We present accessibility and controllability tests based on taking derivatives of the connection, and relate these tests to taking Lie brackets of the input vector fields. The theory is illustrated using several examples, in particular the examples of the snakeboard and Hirose snake robot. We interpret each of these examples in light of the theory developed in this thesis, and examine the generation of locomotive gaits using sinusoidal inputs and their relationship to the controllability tests based on Lie brackets

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

Symmetries in Motion: Geometric Foundations of Motion Control

Some interesting aspects of motion and control, such as those found in biological and robotic locomotion and attitude control of spacecraft, involve geometric concepts. When an animal or a robot moves its joints in a periodic fashion, it can rotate or move forward. This observation leads to the general idea that when one variable in a system moves in a periodic fashion, motion of the Whole object can result. This property can be used for control purposes; the position and attitude Of a satellite, for example, are often controlled by periodic motions of parts of the satellite, such as spinning rotors. One of the geometric tools that has been used to describe this phenomenon is that of connections, a notion that is used extensively in general relativity and other parts of theoretical physics. This tool, part of the general subject Of geometric mechanics, has been helpful in the study of both the stability and instability of a system and system bifurcations, that is, changes in the nature of the system dynamics, as some parameter changes. Geometric mechanics, currently in a period of rapid evolution, has been used, for example, to design stabilizing feedback control systems in attitude dynamics. Theory is also being developed for systems with rolling constraints such as those found in a simple rolling wheel. This paper explains how some of these tools of geometric mechanics are used in the study of motion control and locomotion generation

Optimal Control for Holonomic and Nonholonomic Mechanical Systems with Symmetry and Lagrangian Reduction

In this paper we establish necessary conditions for optimal control using the ideas of Lagrangian reduction in the sense of reduction under a symmetry group. The techniques developed here are designed for Lagrangian mechanical control systems with symmetry. The benefit of such an approach is that it makes use of the special structure of the system, especially its symmetry structure and thus it leads rather directly to the desired conclusions for such systems. Lagrangian reduction can do in one step what one can alternatively do by applying the Pontryagin Maximum Principle followed by an application of Poisson reduction. The idea of using Lagrangian reduction in the sense of symmetry reduction was also obtained by Bloch and Crouch [1995a,b] in a somewhat different context and the general idea is closely related to those in Montgomery [1990] and Vershik and Gershkovich [1994]. Here we develop this idea further and apply it to some known examples, such as optimal control on Lie groups and principal bundles (such as the ball and plate problem) and reorientation examples with zero angular momentum (such as the satellite with moveable masses). However, one of our main goals is to extend the method to the case of nonholonomic systems with a nontrivial momentum equation in the context of the work of Bloch, Krishnaprasad, Marsden and Murray [1995]. The snakeboard is used to illustrate the method

The geometric structure of nonholonomic mechanics

Many important problems in multibody dynamics, the dynamics of wheeled vehicles and motion generation, involve nonholonomic mechanics. Many of these systems have symmetry, such as the group of Euclidean motions in the plane or in space and this symmetry plays an important role in the theory. Despite considerable advances on both Hamiltonian and Lagrangian sides of the theory, there remains much to do. We report on progress on two of these fronts. The first is a Poisson description of the equations that is equivalent to those given by Lagrangian reduction, and second, a deeper understanding of holonomy for such systems. These results promise to lead to further progress on the stability issues and on locomotion generatio