Invitation to Topological RoboticsEuropean Mathematical Society, 2008 - 132 pages 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. |
Table des matières
16 | 1 |
Poincaré polynomials of planar polygon spaces | 16 |
8 | 24 |
9 | 30 |
Euler Characteristics of Configuration Spaces | 41 |
Cut and paste Grothendieck ring | 55 |
Knot Theory of the Robot | 61 |
Navigational Complexity of Configuration Spaces | 87 |
61 | 122 |
Recommendations for further reading | 125 |
64 | 130 |
Expressions et termes fréquents
algebra applied assume bars cell chapter circle Clearly closed cohomology class configuration space connected Consider constant construction contains continuous section Corollary cover critical points defined DEFINITION denote describe determines dimension disjoint edges ending equals equation equivalent example exists expansive fibration field Figure finite fixed function given gives graph Hence homotopy inequality initial integer Lemma length vector linkages lying manifold motion planning algorithms navigation Note obtain pair path planar polygon polyhedron problem projections PROOF proof of Theorem Proposition prove represents respect result robot arm satisfying sequence short smooth statement stress strut studied submanifold subsets Suppose tangent tangent vector TC(X Theorem topological complexity U₁ union values vertex vertices viewed X x X