Summary. This paper is concerned with modeling the dynamics of N articulated solid bodies submerged in an ideal fluid. The model is used to analyze the locomotion of aquatic animals due to the coupling between their shape changes and the fluid dynamics in their environment. The equations of motion are obtained by making use of a two-stage reduction process which leads to significant mathematical and computational simplifications. The first reduction exploits particle relabeling symmetry: that is, the symmetry associated with the conservation of circulation for ideal, incompressible fluids. As a result, the equations of motion for the submerged solid bodies can be formulated without explicitly incorporating the fluid variables. This reduction by the fluid variables is a key difference with earlier methods, and it is appropriate since one is mainly interested in the location of the bodies, not the fluid particles. The second reduction is associated with the invariance of the dynamics under superimposed rigid motions. This invariance corresponds to the conservation of total momentum of the solid-fluid system. Due to this symmetry, the net locomotion of the solid system

We address the problem of motion planning for nonholonomic cooperating mobile robots manipulating and transporting objects while holding them in a stable grasp. We present a general approach for solving optimal control problems based on the calculus of variations. We specialize this approach to solving the motion planning problem and obtaining trajectories and actuator forces/torques for any maneuver in the presence of obstacles. The approach allows geometric constraints such as joint limits, kinematic constraints such as nonholonomic velocity constraints, and dynamic constraints such as frictional constraints and contact force constraints to be incorporated into the planning scheme. The application of this method is illustrated by computing motion plans for several examples and these motions plans are implemented on an experimental testbed.

We address the problem of coupling cyclic robotic tasks to produce a specified coordinated behavior. Such coordination tasks are common in robotics, appearing in applications like walking, hopping, running, juggling and factory automation. In this paper we introduce a general methodology for designing controllers for such settings. We introduce a class of dynamical systems defined over n-dimensional tori (the cross product of n oscillator phases) that serve as reference fields for the specified task. These dynamical systems represent the ideal flow and phase couplings of the various cyclic tasks to be coordinated. In particular, given a specification of the desired connections between oscillating subsystems, we synthesize an appropriate reference field and show how to determine whether the specification is realized by the field. In the simplest case that the oscillating components admit a continuous control authority, they are made to track the phases of the corresponding components of the reference field. We further demonstrate that reference fields can be applied to the control of intermittent contact systems, specifically to the task of juggling balls with a paddle and to the task of synchronizing hopping robots.

This paper surveys selected recent progress in geometric mechanics, focussing on Lagrangian reduction and gives some new applications to nonholonomic systems, that is, mechanical systems with constraints typified by rolling without slipping. Reduction theory for mechanical systems with symmetry has its roots in the classical works in mechanics of Euler, Jacobi, Lagrange, Hamilton, Routh, Poincaré and others. The modern vision of mechanics includes, besides the traditional mechanics of particles and rigid bodies, field theories such as electromagnetism, fluid mechanics, plasma physics, solid mechanics as well as quantum mechanics, and relativistic theories, including gravity. Symmetries in mechanics ranges from obvious translational and rotational symmetries to less obvious particle relabeling symmetries in fluids and plasmas, to subtle symmetries underlying integrable systems. Reduction theory concerns the removal of symmetries and utilizing their associated conservation laws. Reduction theory has been extremely useful in a wide variety of areas, from a deeper understanding of many

Some interesting aspects of motion and control for systems such as those found in biological and robotic locomotion, attitude control of spacecraft and underwater vehicles, and steering of cars and trailers, involve geometric concepts. When an animal or a robot moves its joints in a periodic fashion, it can move forward or rotate in place. When the amplitude of the motion increases, the resulting net displacements normally increase as well. These observations lead to the general idea that when certain variables in a system move 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. Geometric tools that have been useful to describe this phenomenon are \connections", mathematical objects that are extensively used in general relativity and other parts of theoretical physics. The theory of connections, which isnow part of the general subject of geometric mechanics, has also been helpful in the study of the stability or instability ofa system and in its bifurcations under parameter variations. This approach, currently in a period of rapid evolution, has been used, for example, to design stabilizing feedback control systems in the attitude dynamics of spacecraft and

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 st...

Abstract — In this paper we present a novel gait analysis technique which can directly be used to synthesize gaits for a broad class of mechanical systems. We build upon prior work in locomotion mechanics, however we take a different approach to generate gaits that yield absolute motion of the mechanical system. We present a systematic analysis to control all parameters of a proposed type of gait which eliminates the need for intuition and guesswork as was required in the prior work. The main contribution of the paper is relating position change or motion in the ber space to a volume integral bounded by closed curves on a two dimensional manifold embedded in the base space or shape space of the robot. Not only does our method remove the restriction of using sinusoidal gaits as was the case in the prior work but it also allows for generating optimal gaits by solving a variational problem rather than solving a dynamic programming problem as was the case in the prior work. I.

Abstract — In this paper, we build off results from geometric mechanics to gain fresh insight into the locomotion of underactuated systems. More specifically, we use the connection, which relates body velocity to internal shape changes, to create a set of vector fields on the shape space. Each of these fields corresponds to one component of the body velocity (forward, lateral, rotational), and together they highlight how a given shape change will move the system through ambient space. To demonstrate this approach, we use it to analyze the motion of several model systems, both simulated and physically instantiated. I.

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 generation. 1 Symplectic and Poisson Geometry of Nonholonomic Systems Bloch, Krishnaprasad, Marsden and Murray [1996], hereafter denoted [BKMM], applied methods of geometric mechanics to the Lagrange-d'Alembert formulation and extended the use of connections and momentum maps a...

I would like to express my gratitude to the people who made an important difference and played a vital role in the successful completion of my Doctoral studies at the University of Pennsylvania. I am grateful to my advisor Professor Vijay Kumar without whose competence, dedication, generosity, and vision this dissertation would not have been possible. Professor Kumar has been my mentor and an inspiring role model. His energy and relentless support encouraged me towards the completion of my Masters in Mathematics. Professor James Ostrowski, my co-advisor since 1996, provided valuable advice and insight towards my work. I am very thankful for his opinion and guidance during the course of my dissertation. I am very thankful to Professor Joel Burdick (Caltech), Chairman of the committee, Professor G. K. Ananthasuresh, Professor Ruzena Bajcsy, Professor Vijay Kumar and Professor James Ostrowski for agreeing to be the members of my dissertation committee and for taking the time to read my thesis and provide valuable suggestions. This dissertation has allowed me to work with people who are both competent and compassionate. I have been fortunate to receive constant guidance and support from Professor