Browsing by Author "Elkin, Yury"

In this thesis we extend topological model of planar robotic hands emerging in the field of topological robotics. This research elaborates further recent works of Robert Ghrist and others. The main purpose of this thesis is to classify configuration spaces in terms of topological and algebraic invariants, which among others provides complexity estimates for potential optimization algorithms. The thesis is split into two parts. In the first part we investigate a robotic system consisting of a single hand which can occupy any position as long as it doesn't self-intersect. Using a new innovative representation of positions we are able to treat two basic movements of the robotic arm: the 'claw' and the 'swap' movements separately. The main appliance of this part is the nerve theorem, which helps to establish that under some restrictions the configuration space of such robotic hand has the homotopy type of S^1. In the second part we investigate systems consisting of multiple hands. This time we are dealing with hands limited to length one whose positions satisfy the two conditions: each pairwise hand trace intersection is contractible and the hand intersection graph is a forest. As the local main result we prove that the fundamental group of such robotic system is isomorphic to the Artin right-angeled group, where the set of generators is in bijection with the set of all hands and relations are determined by the intersection graph. The main tool exploited in this chapter is the Seifert-van Kampen theorem. Although the results are proven only for some special cases, the thesis introduces methodology that can drive their generalization further. In the final chapter we give a few sophisticated research directions.