Invitation to Topological Robotics
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.
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admissible configuration affine function An~l assume bars cell coefficients cohomology class collinear collinear configurations cone configuration space Consider continuous section Corollary 4.40 critical points defined denote diffeomorphic disjoint edges equals equation equilibrium stress Euler characteristic exists expansive motion fibration finite G R2 G Xe given graph Hence homeomorphic homology homotopy inequality integer intersection involution knot theory Lemma length vector lens spaces linkages long subset lying manifold median subset moduli space Morse index motion planning algorithms navigation function neighborhood nonzero obtain open cover open subset pair path pi+i polyhedron problem proof of Theorem result satisfying Schwarz genus sequence short subsets shown on Figure simplex statement strut submanifold subset J C tangent vector TC(X TC(Y Theorem Theorem 2.3 topological complexity topological space torus total Betti number unstable manifolds vector field vertex vertices zero-divisors