Abstract not found

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

Nonholonomic constraints in robotic systems are the source of some di#culties in planning and control; however, they also introduce interesting properties that can be practically exploited. In this paper we consider the design of a robot hand that achieves dexterity #i.e., the ability to arbitrarily locate and reorient manipulated objects# through rolling. Some interesting issues arising in planning and controlling motions of such device are considered, including exact planning for a spherical object and approximate planning for general objects. An experimental prototype of a three#plus#one d.o.f. hand achieving dexterous manipulation capabilities is described along with experimental results from manipulation. 1 Introduction Dexterous hands, i.e. cooperating multilimb robots with the capability of manipulating an object so as to arbitrarily steer its con#guration in space, have attracted muchinterest in the robotics literature. However, the high degree of sophistication in their mech...

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...

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

In this paper we describe an experimental apparatus developed in our laboratory for research and advanced teaching purposes. The device consists in an untethered spherical vehicle that autonomously rolls on the laboratory floor, and can reach arbitrary positions and orientations in the environment. The kinematics of the vehicle are nonholonomic, and result from the combination of the kinematics of two classical nonholonomic systems, namely, a unicycle and a plate--ball system. The "sphericle" however introduces features that are new with respect to both, which fact renders its study particularly interesting. 1 Introduction Nonholonomy in the kinematics and dynamics of mechanical systems has been attracting much attention in the robotics and control communities in recent years. Nonholonomic systems arise very often in practice, as for instance in car--like vehicles, tractor--trailer systems, airborne and underwater vehicles. Nonholonomic behaviours are sometimes introduced on purpose ...

Rolling between rigid surfaces in space is a well-- known nonholonomic system, whose mathematical model has some interesting features that make it a paradigm for the study of some very general systems. It also turns out that the nonholonomic features of this system can be exploited in practical devices with some appeal for engineers. However, in order to achieve all potential benefits, a greater understanding of these rather complex systems and more practical algorithms for planning and controlling their motions are necessary. In this paper, we will consider some geometric and control aspects of the problem of arbitrarily displacing and reorienting a body which rolls without slipping among other bodies. 1 Introduction Nonholonomic systems have been attracting much attention in the control literature recently, due to both their relevance to practical applications (in particular, to Robotics) and to the challenges that arise in planning and controlling them. Nonholonomic systems commonl...

A solution of the motion planning without obstacles for the standard n-trailer system is proposed. This solution relies basically on the fact that the system is flat with the Cartesian coordinates of the last trailer as a linearizing output. The Frénet formulae are used to simplify the calculations and permit to deal with angle constraints. The general 1-trailer system, where the trailer is not directly hitched to the car at the center of the rear axle, is also flat. The geometric construction used for the standard 1-trailer system can be extended to this more realistic system. MATLAB simulations illustrate this method. 1

In this paper we consider a complete dynamic model for the #Sphericle", a spherical vehicle that has been designed and realized in our laboratory. The sphericle is able to roll on the #oor of the laboratory and reach arbitrary positions and orientations, through the use of only two motors placed within the rolling sphere. In this paper, we report on the derivation of the kinematic model of the Sphericle, which incorporates two types of nonholonomic constraints, and its dynamic model. 1 Introduction In recentyears, the study of systems with nonholonomic constraints has attracted a lot of attention for several reasons #see e.g. #1##. Such constraints arise naturally in many mechanical devices: typical cases are car#like vehicles #2,3#,underwater vehicles,underactuated satellites,or dexterous robotic hands #4, 5, 6,7,8,9,10,11,12#. Sometimes nonholonomic constraints are introduced on purpose to obtain a better behaviour of the system. An interesting aspect of such systems is due to the fa...

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.