Geometry And Robotics

* Provides an elegant introduction to the geometric concepts that are important to applications in robotics * Includes significant state-of-the art material that reflects important advances, connecting robotics back to mathematical fundamentals in group theory and geometry * An invaluable reference that serves a wide audience of grad students and researchers in mechanical engineering, computer science, and applied mathematics

The role played by hormones in the development and treatment of malignant tumors has been controversial for nearly 50 years. The present volume concentrates on substantiated data obtained from the study of tumors developing from hormone-related or hormone-producing tissue, for example the thyroid, adrenal glands, prostate, and the female genital tract. Combining expertise from the fields of molecular biology, biochemistry, and histopathology, advances in the management of these tumors are elaborated. The book also provides information on the endonuclear diagnosis of adrenal tumors. Antihormones have proved to be important as they exhibit a destructive effect on prostate carcinomas and breast cancer. In addition, a special chapter discusses the diffuse endocrine cell system (DECS). Bridging the gap between molecular biology and endocrine therapy, the editors present innovative data on many aspects of hormone-related malignant tumors and offer both a survey of present knowledge and a basis for further research.

The main aim of this book is to introduce Lie groups and allied algebraic and geometric concepts to a robotics audience. These topics seem to be quite fashionable at the moment, but most of the robotics books that touch on these topics tend to treat Lie groups as little more than a fancy notation. I hope to show the power and elegance of these methods as they apply to problems in robotics. A subsidiary aim of the book is to reintroduce some old ideas by describing them in modem notation, particularly Study's Quadric-a description of the group of rigid motions in three dimensions as an algebraic variety (well, actually an open subset in an algebraic variety)-as well as some of the less well known aspects of Ball's theory of screws. In the first four chapters, a careful exposition of the theory of Lie groups and their Lie algebras is given. Except for the simplest examples, all examples used to illustrate these ideas are taken from robotics. So, unlike most standard texts on Lie groups, emphasis is placed on a group that is not semi-simple-the group of proper Euclidean motions in three dimensions. In particular, the continuous subgroups of this group are found, and the elements of its Lie algebra are identified with the surfaces of the lower Reuleaux pairs. These surfaces were first identified by Reuleaux in the latter half of the 19th century.

Robotics has come to attract the attention of mathematicians and theoretical computer scientists to a rapidly increasing degree. Initial investigations have shown that robotics is a rich source of deep theoretical problems, which range over computational geometry, control theory, and many aspects of physics, and whose solutions draw upon methods developed in subjects as diverse as automata theory, algebraic topology, and Fourier analysis.

The development of inexpensive and fast computers, coupled with the discovery of efficient algorithms for dealing with polynomial equations, has enabled exciting new applications of algebraic geometry and commutative algebra. Algebraic Geometry for Robotics and Control Theory shows how tools borrowed from these two fields can be efficiently employed to solve relevant problem arising in robotics and control theory.After a brief introduction to various algebraic objects and techniques, the book first covers a wide variety of topics concerning control theory, robotics, and their applications. Specifically this book shows how these computational and theoretical methods can be coupled with classical control techniques to: solve the inverse kinematics of robotic arms; design observers for nonlinear systems; solve systems of polynomial equalities and inequalities; plan the motion of mobile robots; analyze Boolean networks; solve (possibly, multi-objective) optimization problems; characterize the robustness of linear; time-invariant plants; and certify positivity of polynomials.

This book describes visual perception and control methods for robotic systems that need to interact with the environment. Multiple view geometry is utilized to extract low-dimensional geometric information from abundant and high-dimensional image information, making it convenient to develop general solutions for robot perception and control tasks. In this book, multiple view geometry is used for geometric modeling and scaled pose estimation. Then Lyapunov methods are applied to design stabilizing control laws in the presence of model uncertainties and multiple constraints.

This book highlights the mathematical depth and sophistication of techniques used in different areas of robotics. Each chapter is a peer-reviewed version of a paper presented during the 2021 IMA Conference on the Mathematics of Robotics, held online September 8–10, 2021. The conference gave a platform to researchers with fundamental contributions and for academic and to share new ideas. The book illustrates some of the current interest in advanced mathematics and robotics such as algebraic geometry, tropical geometry, monodromy and homotopy continuation methods applied to areas such as kinematics, path planning, swam robotics, dynamics and control. It is hoped that the conference and this publications will stimulate further related mathematical research in robotics.

First published in 1987, the seven chapters that comprise this book review contemporary work on the geometric side of robotics. The first chapter defines the fundamental goal of robotics in very broad terms and outlines a research agenda each of whose items constitutes a substantial area for further research. The second chapter presents recently developed techniques that have begun to address the geometric side of this research agenda and the third reviews several applied geometric ideas central to contemporary work on the problem of motion planning. The use of Voronoi diagrams, a theme opened in these chapters, is explored further later in the book. The fourth chapter develops a theme in computational geometry having obvious significance for the simplification of practical robotics problems — the approximation or decomposition of complex geometric objects into simple ones. The final chapters treat two examples of a class of geometric ‘reconstruction’ problem that have immediate application to computer-aided geometric design systems.