Nonholonomic motion planning: steering using sinusoids

Methods for steering systems with nonholonomic constraints between arbitrary configurations are investigated. Suboptimal trajectories are derived for systems that are not in canonical form. Systems in which it takes more than one level of bracketing to achieve controllability are considered. The trajectories use sinusoids at integrally related frequencies to achieve motion at a given bracketing level. A class of systems that can be steered using sinusoids (claimed systems) is defined. Conditions under which a class of two-input systems can be converted into this form are given

Trajectory generation for the N-trailer problem using Goursat normal form

Develops the machinery of exterior differential forms, more particularly the Goursat normal form for a Pfaffian system, for solving nonholonomic motion planning problems, i.e., motion planning for systems with nonintegrable velocity constraints. The authors use this technique to solve the problem of steering a mobile robot with n trailers. The authors present an algorithm for finding a family of transformations which will convert the system of rolling constraints on the wheels of the robot with n trailers into the Goursat canonical form. Two of these transformations are studied in detail. The Goursat normal form for exterior differential systems is dual to the so-called chained-form for vector fields that has been studied previously. Consequently, the authors are able to give the state feedback law and change of coordinates to convert the N-trailer system into chained-form. Three methods for planning trajectories for chained-form systems using sinusoids, piecewise constants, and polynomials as inputs are presented. The motion planning strategy is therefore to first convert the N-trailer system into Goursat form, use this to find the chained-form coordinates, plan a path for the corresponding chained-form system, and then transform the resulting trajectory back into the original coordinates. Simulations and frames of movie animations of the N-trailer system for parallel parking and backing into a loading dock using this strategy are included

The relationship between fragility, configurational entropy and the potential energy landscape of glass forming liquids

Glass is a microscopically disordered, solid form of matter that results when a fluid is cooled or compressed in such a fashion that it does not crystallise. Almost all types of materials are capable of glass formation -- polymers, metal alloys, and molten salts, to name a few. Given such diversity, organising principles which systematise data concerning glass formation are invaluable. One such principle is the classification of glass formers according to their fragility\cite{fragility}. Fragility measures the rapidity with which a liquid's properties such as viscosity change as the glassy state is approached. Although the relationship between features of the energy landscape of a glass former, its configurational entropy and fragility have been analysed previously (e. g.,\cite{speedyfr}), an understanding of the origins of fragility in these features is far from being well established. Results for a model liquid, whose fragility depends on its bulk density, are presented in this letter. Analysis of the relationship between fragility and quantitative measures of the energy landscape (the complicated dependence of energy on configuration) reveal that the fragility depends on changes in the vibrational properties of individual energy basins, in addition to the total number of such basins present, and their spread in energy. A thermodynamic expression for fragility is derived, which is in quantitative agreement with {\it kinetic} fragilities obtained from the liquid's diffusivity.Comment: 8 pages, 3 figure

Steering nonholonomic systems in chained form

The authors introduce a nilpotent form, called a chained form, for nonholonomic control systems. For the case of a nonholonomic system with two inputs, they give constructive conditions for the existence of a feedback transformation which puts the system into chained form, and show how to steer the system between arbitrary states. Examples are presented for steering a car and a car with a trailer attached: other examples can be found in the areas of space robotics and multifingered robot hands. The present results also have applications in the area of nilpotentization of distributions of vector fields on R^n