A topological theory initiated recently by the author uses methods of algebraic topology to estimate numerically the character of instabilities arising in motion planning algorithms. The present paper studies random motion planning algorithms and reveals how the topology of the robot's configuration space influences their structure. We prove that the topological complexity of motion planning TC(X) coincides with the minimal n such that there exists an n-valued random motion planning algorithm for the system; here $X$ denotes the configuration space. We study in detail the problem of collision free motion of several objects on a graph G. We describe an explicit motion planning algorithm for this problem. We prove that if G is a tree and if the number of objects is large enough, then the topological complexity of this motion planning problem equals 2m(G)+1 where m(G) is the number of the essential vertices of G. It turns out (in contrast with the results on the collision free control of many objects in space obtained earlier jointly with S. Yuzvinsky) that the topological complexity is independent of the number of particles.
This paper concerns the study of the homotopy type of the ordered configuration space for manifolds with boundary and as an application we will study the collision free motion planning problem on manifolds with boundary.
We compute the higher topological complexity of ordered configuration spaces of orientable surfaces, thus extending Cohen-Farber's description of the ordinary topological complexity of those spaces.
In robotics, a topological theory of motion planning was initiated by M. Farber. The multitasking motion planning problem is new and its theoretical part via topological complexity has hardly been developed, but the concrete implementations are still non-existent, and in fact this work takes the first step in this last direction (producing explicit algorithms.) We present optimal motion planning algorithms which can be used in designing practical systems controlling objects moving in Euclidean space without collisions between them and avoiding obstacles. Furthermore, we present the multitaskin…
Louisiana State University（路易斯安那州立大学）；Queen Mary University of London（伦敦大学玛丽皇后学院）；The University of Chicago（芝加哥大学）
Parametrized motion planning algorithms have high degrees of universality and flexibility, as they are designed to work under a variety of external conditions, which are viewed as parameters and form part of the input of the underlying motion planning problem. In this paper, we analyze the parameterized motion planning problem for the motion of many distinct points in the plane, moving without collision and avoiding multiple distinct obstacles with a priori unknown positions. This complements our prior work [arXiv:2009.06023], where parameterized motion planning algorithms were introduced, and…
We exhibit an algorithm with continuous instructions for two robots moving without collisions on a track shaped as a wedge of three circles. We show that the topological complexity of the configuration space associated with this problem is 3. The topological complexity is a homotopy invariant that can be thought of as the minimum number of continuous instructions required to describe the movement of the robots between any initial configuration to any final one without collisions. The algorithm presented is optimal in the sense that it requires exactly 3 continuous instructions.
We provide explicit motion planners for Euiclidean configuration spaces. This allows us to recover some known values of the topological complexity and the Lusternik-Schinirelman category of these spaces.
In terms of Rudyak's generalization of Farber's topological complexity of the path motion planning problem in robotics, we give a complete description of the topological instabilities in any sequential motion planning algorithm for a system consisting of non-colliding autonomous entities performing tasks in space whilst avoiding collisions with several moving obstacles. The Isotopy Extension Theorem from manifold topology implies, somewhat surprisingly, that the complexity of this problem coincides with the complexity of the corresponding problem in which the obstacles are stationary.
In this paper we introduce and study a new concept of parametrised topological complexity, a topological invariant motivated by the motion planning problem of robotics. In the parametrised setting, a motion planning algorithm has high degree of universality and flexibility, it can function under a variety of external conditions (such as positions of the obstacles etc). We explicitly compute the parameterised topological complexity of obstacle-avoiding collision-free motion of many particles (robots) in 3-dimensional space. Our results show that the parameterised topological complexity can be s…
We study higher analogues of effective and effectual topological complexity of spaces equipped with a group action. These are $G$-homotopy invariant and are motivated by the (higher) motion planning problem of $G$-spaces for which their group action is thought of as an external system assisting the motion planning. Related to this interpretation we define what we call orbital topological complexity, which is also a $G$-homotopy invariant that provides an upper bound for the topological complexity of the quotient space by the group action. We apply these concepts to actions of the group of orde…
We define a simpler notion of symmetric topological complexity more ad hoc to the motion planning problem which was the original motivation for the definition of topological complexity. This is a homotopy invariant that we call bidirectional topological complexity. We prove properties of this invariant and show specific instances for which the symmetrized topological complexity can be relaxed to the bidirectional setting. This approach allows us to estimate higher values of symmetrized topological complexities.
Universidade de São Paulo（圣保罗大学）；Instituto de Ciências Matemáticas e de Computação - USP（USP数学与计算机科学研究所）；Centro de Investigación y de Estudios Avanzados del I. P. N.（墨西哥国立理工学院高级研究中心）
We present optimal motion planning algorithms which can be used in designing practical systems controlling objects moving in Euclidean space without collisions. Our algorithms are optimal in a very concrete sense, namely, they have the minimal possible number of local planners. Our algorithms are motivated by those presented by Mas-Ku and Torres-Giese (as streamlined by Farber), and are developed within the more general context of the multitasking (a.k.a.~higher) motion planning problem. In addition, an eventual implementation of our algorithms is expected to work more efficiently than previou…
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 several examples of explicit transversal algorithms.
In this short note we observe that the higher topological complexity of an iterated connected sum of real projective spaces is maximal possible. Unlike the case of regular TC, the result is accessible through easy mod 2 zero-divisor cup-length considerations.
The paper surveys topological problems relevant to the motion planning problem of robotics and includes some new results and constructions. First we analyse the notion of topological complexity of configuration spaces which is responsible for discontinuities in algorithms for robot navigation. Then we present explicit motion planning algorithms for coordinated collision free control of many particles moving in Euclidean spaces or on graphs. These algorithms are optimal in the sense that they have minimal number of regions of continuity. Moreover, we describe in full detail the topology of conf…
We design an algorithm writing down presentations of graph braid groups. Generators are represented in terms of actual motions of robots moving without collisions on a given graph. A key ingredient is a new motion planning algorithm whose complexity is linear in the number of edges and quadratic in the number of robots. The computing algorithm implies that 2-point braid groups of all light planar graphs have presentations where all relators are commutators.
We consider the problem of robot motion planning in an oriented Riemannian manifold as a topological motion planning problem in its oriented frame bundle. For this purpose, we study the topological complexity of oriented frame bundles, derive an upper bound for this invariant and certain lower bounds from cup length computations. In particular, we show that for large classes of oriented manifolds, e.g. for spin manifolds, the topological complexity of the oriented frame bundle is bounded from below by the dimension of the base manifold.
Queen Mary University of London（伦敦玛丽女王大学）；The University of Chicago（芝加哥大学）
In this paper we study paramertized motion planning algorithms which provide universal and flexible solutions to diverse motion planning problems. Such algorithms are intended to function under a variety of external conditions which are viewed as parameters and serve as part of the input of the algorithm. Continuing a recent paper, we study further the concept of parametrized topological complexity. We analyse in full detail the problem of controlling a swarm of robots in the presence of multiple obstacles in Euclidean space which served for us a natural motivating example. We present an expli…
In this paper we study a notion of topological complexity for the motion planning problem. The topological complexity is a number which measures discontinuity of the process of motion planning in the configuration space X. More precisely, it is the minimal number k such that there are k different motion planning rules, each defined on an open subset of XxX, so that each rule is continuous in the source and target configurations. We use methods of algebraic topology (the Lusternik - Schnirelman theory) to study the topological complexity . We give an upper bound (in terms of the dimension of th…
