Motion planning in spaces with small fundamental group
Armindo Costa, Michael Farber
在具有小基本群的空间中，即当空间是小阶有限的或具有小的上同调维数时，我们建立了运动规划问题拓扑复杂性的精确上界。
We establish sharp upper bounds for the topological complexity of motion planning problem in spaces with small fundamental group, i.e. when it is finite of small order or has small cohomological dimension.
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Collision-free motion planning on manifolds with boundary
本文研究了带边界流形的序位形空间的同伦型，作为应用，我们将研究带边界流形上的无碰撞运动规划问题。
Topological complexity of collision-free multi-tasking motion planning on orientable surfaces
本文计算了可定向曲面的序位形空间的高拓扑复杂性，从而推广了Cohen-Farber对这些空间的一般拓扑复杂性的描述.
Motion planning algorithms for Configuration Spaces
我们提供了明确的运动规划Euiclidean配置空间。这使我们能够恢复这些空间的拓扑复杂性和Lusternik-Schinirelman范畴的一些已知值。
Sequential motion planning of non-colliding particles in Euclidean spaces
在Rudyak的概括法伯的拓扑复杂性的路径运动规划问题的机器人技术，我们给出了一个完整的描述的拓扑不稳定性在任何顺序的运动规划算法的系统组成的非碰撞的自主实体执行任务的空间，同时避免碰撞与几个移动的障碍物。从流形拓扑的同位素延伸定理意味着，有点令人惊讶的是，这个问题的复杂性与相应的问题的复杂性相吻合，其中障碍物是固定的。
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…
A Discrete Topological Complexity of Discrete Motion Planning
本文通过推广离散同伦概念，为离散移动机器人引入离散运动规划，并证明其拓扑复杂度降低且仅依赖于空间的严格离散同伦型。
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.
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.
Configuration Spaces and Robot Motion Planning Algorithms
本文综述了与机器人运动规划问题相关的拓扑问题，并包括一些新的结果和结构。首先，我们分析的概念，拓扑复杂性的配置空间，负责机器人导航算法的不连续性。然后，我们提出了明确的运动规划算法协调碰撞自由控制的多个粒子在欧几里得空间或图形上移动。这些算法是最佳的意义上说，他们有最小数量的区域的连续性。此外，我们详细描述了树上两个粒子的位形空间的拓扑结构，并利用它构造了树上n个粒子的位形空间中的一些顶维上同调类。
In this paper we develop theory of sequential parametrized motion planning which generalises the approach of parametrized motion planning, which was introduced recently in [3]. A sequential parametrized motion planning algorithm produced a motion of the system which is required to visit a prescribed sequence of states, in certain order, at specified moments of time. The sequential parametrized algorithms are universal as the external conditions are not fixed in advance but rather constitute part of the input of the algorithm. The second part of this article consists of a detailed analysis of t…
Lusternik-Schnirelmann category and based topological complexities of motion planning
Farber和Rudyak引入了运动规划的拓扑复杂度$\mathbf{TC}(X)$及其高级类似物$\mathbf{TC}_n(X)$来度量将路径分配给点元组的复杂度。出于运动规划的机器人系统开始在家庭配置，并可能回来后，通过一系列的位置，我们定义了其他三个类的拓扑复杂性$\mathbf{LTC}_n(X)$，$\mathbf{ltc}_n(X)$和$\mathbf{tc}_n(X)$。我们将比较这些概念，并计算一些熟悉的空间类的后者。
Simplicial Complexity: piecewise linear motion planning in robotics
利用单纯映射的邻接性概念，我们将Farber的拓扑复杂性引入到单纯复形的领域。我们表明，对于一个有限的单纯复形$K$，我们的离散概念恢复的拓扑复杂性的实现$\|K\|$。我们的方法非常适合设计和实现算法，搜索最佳的运动规划器的自主系统在现实生活中的应用。
Symmetric Motion Planning
在本文中，我们研究对称运动规划算法，即，使得从一个状态A到另一个B的运动，由算法规定，是从B到A的运动的时间反转。我们实验了几种不同的概念，拓扑复杂性的算法，并将它们相互比较，并与通常的（非对称）概念的拓扑复杂性。利用等变上同调和Schwarz亏格理论，得到了对称拓扑复杂度的上同调下界.我们的一个主要结果指出，在非球面流形的情况下，对称运动规划算法的复杂性与固定的中点地图超过两倍的杯长。本文引入了一个新的概念--上同调类的区间范畴权，它推广了E. Fadell和S.侯赛尼我们应用这个概念来研究非球面流形的对称拓扑复杂性。
Collision Free Motion Planning on Graphs
作者最近提出的一种拓扑理论，用代数拓扑的方法对运动规划算法中出现的不稳定性进行了数值估计。本文研究了随机运动规划算法，揭示了机器人位形空间的拓扑结构如何影响其结构。我们证明了运动规划的拓扑复杂度TC（X）与最小n一致，使得系统存在n值随机运动规划算法，这里$X$表示配置空间.本文详细研究了图G上多个物体的无碰撞运动问题。我们描述了一个明确的运动规划算法，这个问题。我们证明了如果G是一棵树，且目标数目足够大，则该运动规划问题的拓扑复杂度等于2m（G）+1，其中m（G）是G的本质顶点数.结果表明（与早先与S. Yuzvinsky）的拓扑复杂性是独立的粒子的数量。
Topological complexity of collision free motion planning algorithms in the presence of multiple moving obstacles
我们研究的运动规划算法的无碰撞控制的多个对象中存在的移动障碍物。我们计算的拓扑复杂性的算法解决这个问题。我们应用拓扑工具，并使用配置空间的上同调代数的信息。本文的结果可能会被用于自动交通控制系统。
Computing braid groups of graphs with applications to robot motion planning
我们设计了一个算法，写下图辫群的表示。发电机表示的实际运动的机器人移动没有碰撞在一个给定的图形。一个关键因素是一个新的运动规划算法，其复杂性是线性的边缘和机器人的数量平方。该算法表明所有轻可平面图的2-点辫群都有关系子是交换子的表示。
Motion planning in tori
设X是n维环面的标准CW-分解的子复形。我们展示了一个显式的最优运动规划算法的X。这种构造用于计算一般位置排列的补空间和与直角Artin群相关的Eilenberg-Mac Lane空间的拓扑复杂性。
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.
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…
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…
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…
Topological complexity and efficiency of motion planning algorithms
我们介绍了一个变种的法伯的拓扑复杂性，定义为光滑紧凑定向黎曼流形，它只考虑到运动规划与最低可能的“平均长度”的输出路径。我们证明，它从来没有不同的拓扑复杂性超过$1$，从而表明，后者不变量解决的问题，存在的运动规划是“有效的”。
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