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

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

Enumeration and Motion Planning for Modular Mobile Robots

This report focuses on two different aspects of modular robots, the enumeration of distinct configurations of a modular robot and the generation of gaits for hybrid robots with wheels and legs. Given a particular set of modules from which the robot can be formed, a modular robot made up of these modules can attain a number of different configurations based on the relative attachment of the modules. The distinct configurations possible are enumerated for a locomotion system consisting of a base with multiple ports where wheel or leg modules can be attached. Given a particular configuration of the modular robot, we would like to generate a set of inputs that would drive the robot from an initial position to a desired position. The method used for this must be applicable to different kinds of modules that may be used for locomotion. The method presented here involves generating a set of constant inputs that will drive a drift-free system from an initial to a final desired position. Simulation results are generated for translation and rotation of the robot and motion along a Lie Bracket direction (sideways motion) for the hybrid mobile robot

Motion Planning in Humans and Robots

We present a general framework for generating trajectories and actuator forces that will take a robot system from an initial configuration to a goal configuration in the presence of obstacles observed with noisy sensors. The central idea is to find the motion plan that optimizes a performance criterion dictated by specific task requirements. The approach is motivated by studies of human voluntary manipulation tasks that suggest that human motions can be described as solutions of certain optimization problems