This book discusses several selected topics of a new emerging area of research on the interface between topology and engineering. The first main topic is topology of configuration spaces of mechanical linkages. These manifolds arise in various fields of mathematics and in other sciences, e.g., engineering, statistics, molecular biology. To compute Betti numbers of these configuration spaces the author applies a new technique of Morse theory in the presence of an involution. A significant result of topology of linkages presented in this book is a solution of a conjecture of Kevin Walker which states that the relative sizes of bars of a linkage are determined, up to certain equivalence, by the cohomology algebra of the linkage configuration space. This book also describes a new probabilistic approach to topology of linkages which treats the bar lengths as random variables and studies mathematical expectations of Betti numbers. The second main topic is topology of configuration spaces associated to polyhedra. The author gives an account of a beautiful work of S. R. Gal, suggesting an explicit formula for the generating function encoding Euler characteristics of these spaces. Next the author studies the knot theory of a robot arm, focusing on a recent important result of R. Connelly, E. Demain, and G. Rote. Finally, he investigates topological problems arising in the theory of robot motion planning algorithms and studies the homotopy invariant TC(X) measuring navigational complexity of configuration spaces. This book is intended as an appetizer and will introduce the reader to many fascinating topological problems motivated by engineering.

Ever since the literary works of Capek and Asimov, mankind has been fascinated by the idea of robots. Modern research in robotics reveals that along with many other branches of mathematics, topology has a fundamental role to play in making these grand ideas a reality. This volume summarizes recent progress in the field of topological robotics--a new discipline at the crossroads of topology, engineering and computer science. Currently, topological robotics is developing in two main directions. On one hand, it studies pure topological problems inspired by robotics and engineering. On the other hand, it uses topological ideas, topological language, topological philosophy, and specially developed tools of algebraic topology to solve problems of engineering and computer science. Examples of research in both these directions are given by articles in this volume, which is designed to be a mixture of various interesting topics of pure mathematics and practical engineering.

A modern and unified treatment of the mechanics, planning, and control of robots, suitable for a first course in robotics.

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.

This book focuses on the topology theory of mechanisms developed by the authors and provides a systematic method for the topology design of robot mechanisms. The main original theoretical contributions of this book include: A. Three basic concepts · The “geometrical constraint type of axes” is introduced as the third element of the topological structure of a mechanism. When it is combined with the other two elements, the kinematic pair and the connection of links, the symbolic expression of the topological structure is independent of the motion positions (except for the singularity positions) and the fixed coordinate system (Chapter 2). · The position and orientation characteristic (POC) set is used to describe the POC of the relative motion between any two links. The POC set, derived from the unit vector set of the velocity of a link, is only depend on the topological structure of a mechanism. Therefore, it is also independent of the motion positions and the fixed coordinate system (Chapter 3). · The single open chain (SOC) unit is the base unit of the topological structure used to develop the four basic equations of the mechanism topology (Chapters 2, 4–6). B. The mechanism composition principle based on the SOC units This book proposes a mechanism composition principle, based on the SOC units, to establish a systematic theory for the unified modeling of the topology, kinematics, and dynamics of mechanisms based on the SOC units (Chapter 7). C. Four basic equations • The POC equation of serial mechanisms with 10 symbolic operation rules (Chapter 4). • The POC equation of parallel mechanisms with 14 symbolic operation rules (Chapter 5). • The general DOF formula for spatial mechanisms (Chapter 6). • The coupling degree formula for the Assur kinematic chain (Chapter 7). D. One systematic method for the topology design of robot mechanisms (Chapters 8–10) Based on the three basic concepts and the four basic equations addressed above, this book puts forward a systematic method for the topology design of parallel mechanisms, which is fundamentally different from all existing methods. Its main characteristics are as follows: • The design process includes two stages: the first is structure synthesis, which derives many structure types; the second involves the performance analysis, classification and optimization of structure types derived from the first stage. • The design operation is independent of the motion positions and the fixed coordinate system. Therefore, the proposed method is essentially a geometrical method, which ensures the full-cycle DOF and the generality of geometric conditions of mechanism existence. • Each individual design step follows an explicit formula or the guidelines for design criteria, making the operation simple, feasible and reproducible. In addition, the topology design of the SCARA PMs is studied in detail to demonstrate the proposed method (Chapter 10).

With applications in mind, this self-contained monograph provides a coherent and thorough treatment of the configuration spaces of Euclidean spaces and spheres, making the subject accessible to researchers and graduates with a minimal background in classical homotopy theory and algebraic topology.

Discover a unique and modern treatment of topology employing a cross-disciplinary approach Implemented recently to understand diverse topics, such as cell biology, superconductors, and robot motion, topology has been transformed from a theoretical field that highlights mathematical theory to a subject that plays a growing role in nearly all fields of scientific investigation. Moving from the concrete to the abstract, Topology and Its Applications displays both the beauty and utility of topology, first presenting the essentials of topology followed by its emerging role within the new frontiers in research. Filling a gap between the teaching of topology and its modern uses in real-world phenomena, Topology and Its Applications is organized around the mathematical theory of topology, a framework of rigorous theorems, and clear, elegant proofs. This book is the first of its kind to present applications in computer graphics, economics, dynamical systems, condensed matter physics, biology, robotics, chemistry, cosmology, material science, computational topology, and population modeling, as well as other areas of science and engineering. Many of these applications are presented in optional sections, allowing an instructor to customize the presentation. The author presents a diversity of topological areas, including point-set topology, geometric topology, differential topology, and algebraic/combinatorial topology. Topics within these areas include: Open sets Compactness Homotopy Surface classification Index theory on surfaces Manifolds and complexes Topological groups The fundamental group and homology Special "core intuition" segments throughout the book briefly explain the basic intuition essential to understanding several topics. A generous number of figures and examples, many of which come from applications such as liquid crystals, space probe data, and computer graphics, are all available from the publisher's Web site.