scholarly journals CATEGORY AND TOPOLOGICAL COMPLEXITY OF THE CONFIGURATION SPACE

2019 ◽  
Vol 100 (3) ◽  
pp. 507-517
Author(s):  
CESAR A. IPANAQUE ZAPATA
Keyword(s):  
Motion Planning ◽  
Configuration Space ◽  
Rigid Bodies ◽  
Robot Motion ◽  
Planning Problem ◽  
Topological Complexity ◽  
Robot Motion Planning ◽  
Spatial Motion ◽  
Lusternik Schnirelmann ◽  
Homotopy Invariants

The Lusternik–Schnirelmann category cat and topological complexity TC are related homotopy invariants. The topological complexity TC has applications to the robot motion planning problem. We calculate the Lusternik–Schnirelmann category and topological complexity of the ordered configuration space of two distinct points in the product $G\times \mathbb{R}^{n}$ and apply the results to the planar and spatial motion of two rigid bodies in $\mathbb{R}^{2}$ and $\mathbb{R}^{3}$ respectively.

scholarly journals A Randomized Greedy Algorithm for Piecewise Linear Motion Planning

Mathematics ◽  
2021 ◽  
Vol 9 (19) ◽  
pp. 2358
Author(s):  
Carlos Ortiz ◽  
Adriana Lara ◽  
Jesús González ◽  
Ayse Borat
Keyword(s):  
Motion Planning ◽  
Piecewise Linear ◽  
Randomized Algorithm ◽  
Robot Motion ◽  
Topological Complexity ◽  
Robot Motion Planning ◽  
Automated Guided Vehicle ◽  
Linear Motion ◽  
Lusternik Schnirelmann ◽  
Robust To Noise

We describe and implement a randomized algorithm that inputs a polyhedron, thought of as the space of states of some automated guided vehicle R, and outputs an explicit system of piecewise linear motion planners for R. The algorithm is designed in such a way that the cardinality of the output is probabilistically close (with parameters chosen by the user) to the minimal possible cardinality.This yields the first automated solution for robust-to-noise robot motion planning in terms of simplicial complexity (SC) techniques, a discretization of Farber’s topological complexity TC. Besides its relevance toward technological applications, our work reveals that, unlike other discrete approaches to TC, the SC model can recast Farber’s invariant without having to introduce costly subdivisions. We develop and implement our algorithm by actually discretizing Macías-Virgós and Mosquera-Lois’ notion of homotopic distance, thus encompassing computer estimations of other sectional category invariants as well, such as the Lusternik-Schnirelmann category of polyhedra.

scholarly journals Topological complexity of collision-free motion planning on surfaces

2010 ◽  
Vol 147 (2) ◽  
pp. 649-660 ◽  
Author(s):  
Daniel C. Cohen ◽  
Michael Farber
Keyword(s):  
Motion Planning ◽  
Configuration Space ◽  
Main Tool ◽  
Orientable Surface ◽  
Configuration Spaces ◽  
Topological Complexity ◽  
Algebraic Varieties ◽  
Hodge Structures ◽  
Lusternik Schnirelmann ◽  
Mixed Hodge Structures

AbstractThe topological complexity$\mathsf {TC}(X)$is a numerical homotopy invariant of a topological spaceXwhich is motivated by robotics and is similar in spirit to the classical Lusternik–Schnirelmann category ofX. Given a mechanical system with configuration spaceX, the invariant$\mathsf {TC}(X)$measures the complexity of motion planning algorithms which can be designed for the system. In this paper, we compute the topological complexity of the configuration space ofndistinct ordered points on an orientable surface, for both closed and punctured surfaces. Our main tool is a theorem of B. Totaro describing the cohomology of configuration spaces of algebraic varieties. For configuration spaces of punctured surfaces, this is used in conjunction with techniques from the theory of mixed Hodge structures.

The Role of Motion Planning in Robotics

2015 ◽  
Vol 811 ◽  
pp. 311-317
Author(s):  
Ellips Masehian
Keyword(s):  
Motion Planning ◽  
Computer Science ◽  
Robotic Systems ◽  
Robot Motion ◽  
Planning Problem ◽  
Robot Motion Planning ◽  
Wide Range ◽  
Vast Literature ◽  
High Level

As robotic systems evolve and get more sophisticated, expectations of them to accomplish high-level tasks increase gradually and their motion planning becomes more complex and difficult. The motion planning problem has been studied for more than four decades from different aspects such that presently has a vast literature. This paper investigates different components of the robot motion planning (RMP) problem and presents a new comprehensive taxonomy for a wide range of RMP problems. The taxonomy is based on a survey of the literature on RMP problems and applications in robotics and computer science.

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