Related papers: Motion planning in spaces with small fundamental g…
We investigate motion planning algorithms for the assembly of shapes in the \emph{tilt model} in which unit-square tiles move in a grid world under the influence of uniform external forces and self-assemble according to certain rules. We…
We develop the theory of locally small spaces in a new simple language and apply this simplification to re-build the theory of locally definable spaces over structures with topologies.
Wheeler emphasized the study of Superspace - the space of 3-geometries on a spatial manifold of fixed topology. This is a configuration space for GR; knowledge of configuration spaces is useful as regards dynamics and QM.In this Article I…
We derive the continuous-time limit of discrete quantum walks with topological phases. We show the existence of a continuous-time limit that preserves their topological phases. We consider both simple-step and split-step walks, and derive…
We prove that Hausdorff limit of topological minimal sets (with finitely generated coefficient group) are topologically minimal. The key idea is to reduce the homology group on the space to the homology group on the sphere, and reduce the…
We define a discrete closure operation for definably complete locally o-minimal structures $\mathcal M$. The pair of the underlying set of $\mathcal M$ and the discrete closure operation forms a pregeometry. We define the rank of a…
How hard is it to program $n$ robots to move about a long narrow aisle such that only $w$ of them can fit across the width of the aisle? In this paper, we answer that question by calculating the topological complexity of $\text{conf}(n,w)$,…
This paper presents a new multi-layered algorithm for motion planning under motion and sensing uncertainties for Linear Temporal Logic specifications. We propose a technique to guide a sampling-based search tree in the combined task and…
Navigating mobile robots through environments shared with humans is challenging. From the perspective of the robot, humans are dynamic obstacles that must be avoided. These obstacles make the collision-free space nonconvex, which leads to…
We compute the integral homology and cohomology groups of configuration spaces of two distinct points on a given real projective space. The explicit answer is related to the (known multiplicative structure in the) integral cohomology---with…
State-of-the-art generalist manipulation policies have enabled the deployment of robotic manipulators in unstructured human environments. However, these frameworks struggle in cluttered environments primarily because they utilize auxiliary…
In this paper, we introduce the notion of transversal topological complexity (TTC) for a smooth manifold $X$ with respect to a submanifold of codimension 1 together with basic results about this numerical invariant. In addition, we present…
Some boundedness properties of function spaces (considered as topological groups) are studied.
Sequential parametrized topological complexity is a numerical homotopy invariant of a fibration, which arose in the robot motion planning problem with external constraints. In this paper, we study sequential parametrized topological…
We present a method to solve planning problems involving sequential decision making in unpredictable environments while accomplishing a high level task specification expressed using the formalism of linear temporal logic. Our method…
This paper presents a minimum displacement motion planning problem wherein obstacles are displaced by a minimum amount to find a feasible path. We define a metric for robot-obstacle intersection that measures the extent of the intersection…
Given a compact, connected Lie group $K$, we use principal $K$-bundles to construct manifolds with prescribed finite-dimensional algebraic models. Conversely, let $M$ be a compact, connected, smooth manifold which supports an almost free…
Topological complexity $\TC{B}$ of a space $B$ is introduced by M. Farber to measure how much complex the space is, which is first considered on a configuration space of a motion planning of a robot arm. We also consider a stronger version…
In order to make the fundamental group, one of the most well known invariants in algebraic topology, more useful and powerful some researchers have introduced and studied various topologies on the fundamental group from the beginning of the…
The author proposes a method for investigating actions of finite groups on aspherical spaces. Complete homotopy classification of free actions of finite groups on aspherical spaces is obtained. Also there are some results about non-free…