Applications of Algebraic Topology

代数拓扑的应用

• 批准号: EP/D035759/1
• 负责人: Michael Farber
• 金额: $ 18.74万
• 依托单位: Durham University
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2006
• 资助国家: 英国
• 起止时间: 2006 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FD035759%2F1
• 关键词: Applications; Algebraic; Topology

In this research we will study problems of algebraic topology which are inspired by applications. Consider a mechanical system which functions in the following regime: an operator introduces the current and the desired state of the system and the computer of the system (using an apriori designed motion planning algorithm) determines a continuous motion of the system from its current state to the desired state. In this research we study topological properties of such motion planning algorithms. In most cases such motion planning algorithms are discontinuous, i.e. small perturbations of the input data may lead to significant changes in the motion curve. The theory we plan to develop further in this reserach allows to use methods of algebraic topology to describe instabilities in the motion planning algorithms and to minimize them. We will also tackle several other interesting topological problems inspired by robotics applications.

在这项研究中，我们将研究受应用启发的代数拓扑问题。考虑一个按以下方式运行的机械系统：操作员引入系统的当前状态和期望状态，系统的计算机（使用先验设计的运动规划算法）确定系统从当前状态到期望状态的连续运动。在这项研究中，我们研究了此类运动规划算法的拓扑特性。在大多数情况下，这种运动规划算法是不连续的，即输入数据的小扰动可能会导致运动曲线的显着变化。我们计划在本研究中进一步发展的理论允许使用代数拓扑方法来描述运动规划算法中的不稳定性并将其最小化。我们还将解决受机器人应用启发的其他几个有趣的拓扑问题。

项目成果

Topological complexity of collision-free motion planning on surfaces

曲面上无碰撞运动规划的拓扑复杂性

• DOI: 10.1112/s0010437x10005038
• 发表时间: 2010
• 期刊: Compositio Mathematica
• 影响因子: 1.8
• 作者: Cohen D

MOTION PLANNING IN SPACES WITH SMALL FUNDAMENTAL GROUPS

具有小型基本组的空间中的运动规划

• DOI: 10.1142/s0219199710003750
• 发表时间: 2011
• 期刊: Communications in Contemporary Mathematics
• 影响因子: 1.6
• 作者: COSTA A

Topology of configuration space of two particles on a graph, I

图上两个粒子的配置空间的拓扑，I

• DOI: 10.2140/agt.2009.9.593
• 发表时间: 2009
• 期刊: Algebraic & Geometric Topology
• 影响因子: 0.7
• 作者: Barnett K