this paper is as follows: in Section 2, we collect some mathematical preliminaries from the literature on controllability of nonlinear systems and on classification of free Lie algebras. These are drawn from classical references in control theory [4, 17, 18, 36, 40] and Lie algebras [15, 43]. In Section 3, using some outstanding results of Brockett on optimal steering of certain classes of systems as motivation [5], we discuss the use of sinusoidal inputs for steering systems of first order, i.e., systems where controllability is achieved after just one level of Lie brackets of the input vector fields. Section 4 attempts to expand the domain of applicability of these results to more complex systems, where several orders of Lie brackets are needed to obtain the full Lie algebra associated with the input distribution. The 4 MURRAY AND SASTRY

In this paper we address the constructive controllability problem for drift-free, leftinvariant systems on finite-dimensional Lie groups with fewer controls than state dimension. We consider small (ffl) amplitude, low-frequency, periodically time-varying controls and derive average solutions for system behavior. We show how the pth-order average formula can be used to construct open-loop controls for point-to-point maneuvering of systems that require up to (p \Gamma 1) iterations of Lie brackets to satisfy the Lie algebra controllability rank condition. In the cases p = 2; 3, we give algorithms for constructing these controls as a function of structure constants that define the control authority, i.e., the actuator capability, of the system. The algorithms are based on a geometric interpretation of the average formulas and produce sinusoidal controls that solve the constructive controllability problem with O(ffl ) accuracy in general (exactly if the Lie algebra is nilpotent). The methodology is applicable to a variety of control problems and is illustrated for the motion control problem of an autonomous underwater vehicle with as few as three control inputs.

This dissertation lays the foundation for practical exponential stabilization of driftless control systems. Driftless systems have the form, x = X 1 (x)u 1 + \Delta \Delta \Delta + Xm (x)um ; x 2 R n : Such systems arise when modeling mechanical systems with nonholonomic constraints. In engineering applications it is often required to maintain the mechanical system around a desired configuration. This task is treated as a stabilization problem where the desired configuration is made an asymptotically stable equilibrium point. The control design is carried out on an approximate system. The approximation process yields a nilpotent set of input vector fields which, in a special coordinate system, are homogeneous with respect to a non-standard dilation. Even though the approximation can be given a coordinate-free interpretation, the homogeneous structure is useful to exploit: the feedbacks are required to be homogeneous functions and thus preserve the homogeneous structure in the close...

This report is concerned with the problem of motion generation via cyclic variations in selected degrees of freedom (usually referred to as shape variables) in mechanical systems subject to nonholonomic constraints (here the classical one of a disk rolling without sliding on a at surface). In earlier work, we identi ed an interesting class of such problems arising in the setting of Lie groups, and investigated these under a hypothesis on constraints, that naturally led to a purely kinematic approach. In the present work, the hypothesis on constraints does not hold, and as a consequence, it is necessary to take into account certain dynamical phenomena. Speci cally we concern ourselves with the group SE(2) of rigid motions in the plane and a concrete mechanical realization dubbed the 2{node, 1{module SE(2){snake. In a restricted version, it is also known as the Roller Racer (a patented ride/toy). Based on the work of Bloch, Krishnaprasad, Marsden and Murray, one recognizes in the example of this report a balance law called the momentum equation, which is a direct consequence of the interaction of the SE(2){symmetry of the problem with the

Pairs of bodies with regular rigid surfaces rolling onto each other in space form a nonholonomic system of a rather general type, posing several interesting control problems of which not much is known. The nonholonomy of such systems can be exploited in practical devices, which is very useful in robotic applications. In order to achieve all potential benefits, a deeper understanding of these types of systems and more practical algorithms for planning and controlling their motions are necessary. In this paper, we study the controllability aspect of this problem, giving a complete description of the reachable manifold for general pairs of bodies, and a constructive controllability algorithm for planning rolling motions for dexterous robot hands. Index Terms---Nonholonomic systems, nonlinear controllability theory, robotic manipulation. I. INTRODUCTION N ON-HOLONOMIC systems have been attracting much attention in control literature recently, due to both their relevance to practical ap...

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

A paradigm for isothermal, mechanical rectification of stochastic fluctuations is introduced in this paper. The central idea is to transform energy injected by random perturbations into rigid-body rotational kinetic energy. The prototype considered in this paper is a mechanical system consisting of a set of rigid bodies in interaction through magnetic fields. The system is stochastically forced by white noise and dissipative through mechanical friction. The Gibbs-Boltzmann distribution at a specific temperature defines the unique invariant measure under the flow of this stochastic process and allows us to define “the temperature ” of the system. This measure is also ergodic and weakly mixing. Although the system does not exhibit global directed motion, it is shown that global ballistic motion is possible (the mean-squared displacement grows like t 2). More precisely, although work cannot be extracted from thermal energy by the second law of thermodynamics, it is shown that ballistic transport from thermal energy is possible. In particular, the dynamics is characterized by a meta-stable state in which the system exhibits directed motion over random time scales. This phenomenon is caused by interaction of three attributes of the system: a non flat (yet bounded) potential energy landscape, a rigid body effect (coupling translational momentum and angular momentum through friction) and the degeneracy of the noise/friction tensor on the momentums (the fact that noise is not applied to all degrees of freedom). 1

Abstract—Rolling a ball on a plane is a standard example of nonholonomy reported in many textbooks, and the problem is also well understood for any smooth deformation of the surfaces. For nonsmoothly deformed surfaces, however, much less is known. Although it may seem intuitive that nonholonomy is conserved (think e.g. to polyhedral approximations of smooth surfaces), current definitions of “nonholonomy ” are inherently referred to systems described by ordinary differential equations, and are thus inapplicable to such systems. In this paper, we study the set of positions and orientations that a polyhedral part can reach by rolling on a plane through sequences of adjacent faces. We provide a description of such reachable set, discuss conditions under which the set is dense, or discrete, or has a compound structure, and provide a method for steering the system to a desired reachable configuration, robustly with respect to model uncertainties. Based on ideas and concepts encountered in this case study, and in some other examples we provide, we turn back to the most general aspects of the problem and investigate the possible generalization of the notion of (kinematic) nonholonomy to nonsmooth, discrete, and hybrid dynamical systems. To capture the essence of phenomena commonly regarded as “nonholonomic, ” at least two irreducible concepts are to be defined, of “internal ” and “external ” nonholonomy, which may coexist in the same system. These definitions are instantiated by examples. Index Terms—Hybrid systems, motion planning, nonholonomic systems, quantized control systems, reachability analysis. I.

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 extensively used 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 the stability or instability of a system and in its bifurcations, that is, changes in the nature of the systems 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. The theory is also being developed for systems with rolling constraints such as those found in a simple rolling wheel. This article explains how some of these tools of geometric mechanics are used in the study of motion control and locomotion generation. 1

In this paper we consider some new avenues that the design and control of versatile robotic end-effectors, or "hands", are taking to tackle the stringent requirements of both industrial and servicing applications. A point is made in favour of the so--called minimalist approach to design, consisting in the reduction of the hardware complexity to the bare minimum necessary to fulfill the specifications. It will be shown that to serve this purpose best, more advanced understanding of the mechanics and control of the hand--object system is necessary. Some advancements in this direction are reported, while few of the many problems still open are pointed out.