scholarly journals Topological complexity of motion planning and Massey products

2009 ◽  
Author(s):  
Mark Grant
Keyword(s):  
Motion Planning ◽  
Topological Complexity ◽  
Massey Products

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.

scholarly journals Topological complexity of motion planning in projective product spaces

2013 ◽  
Vol 13 (2) ◽  
pp. 1027-1047 ◽  
Author(s):  
Jesús González ◽  
Mark Grant ◽  
Enrique Torres-Giese ◽  
Miguel Xicoténcatl
Keyword(s):  
Motion Planning ◽  
Topological Complexity ◽  
Product Spaces

Sequential motion planning algorithms in real projective spaces: An approach to their immersion dimension

2018 ◽  
Vol 30 (2) ◽  
pp. 397-417 ◽  
Author(s):  
Natalia Cadavid-Aguilar ◽  
Jesús González ◽  
Darwin Gutiérrez ◽  
Aldo Guzmán-Sáenz ◽  
Adriana Lara
Keyword(s):  
Positive Integer ◽  
Projective Space ◽  
Motion Planning ◽  
Product Space ◽  
Projective Spaces ◽  
Topological Complexity ◽  
Real Projective Space ◽  
Binary Expansion ◽  
The Real

AbstractThes-th higher topological complexity{\operatorname{TC}_{s}(X)}of a spaceXcan be estimated from above by homotopical methods, and from below by homological methods. We give a thorough analysis of the gap between such estimates when{X=\operatorname{\mathbb{R}P}^{m}}, the real projective space of dimensionm. In particular, we describe a number{r(m)}, which depends on the structure of zeros and ones in the binary expansion ofm, and with the property that{0\leq sm-\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})\leq\delta_{s}(% m)}for{s\geq r(m)}, where{\delta_{s}(m)=(0,1,0)}for{m\equiv(0,1,2)\bmod 4}. Such an estimation for{\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})}appears to be closely related to the determination of the Euclidean immersion dimension of{\operatorname{\mathbb{R}P}^{m}}. We illustrate the phenomenon in the case{m=3\cdot 2^{a}}. In addition, we show that, for large enoughsand evenm,{\operatorname{TC}_{s}(\operatorname{\mathbb{R}P}^{m})}is characterized as the smallest positive integer{t=t(m,s)}for which there is a suitable equivariant map from Davis’ projective product space{\mathrm{P}_{\mathbf{m}_{s}}}to the{(t+1)}-st join-power{((\mathbb{Z}_{2})^{s-1})^{\ast(t+1)}}. This is a (partial, but conjecturally complete) generalization of the work of Farber, Tabachnikov and Yuzvinsky relating{\operatorname{TC}_{2}}to the immersion dimension of real projective spaces.

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 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 Topological complexity of real Grassmannians

2021 ◽  
pp. 1-17
Author(s):  
Petar Pavešić
Keyword(s):  
Motion Planning ◽  
Zero Divisor ◽  
Cohomology Ring ◽  
Detailed Knowledge ◽  
Topological Complexity ◽  
Grassmann Manifolds ◽  
Linear Subspaces ◽  
Lusternik Schnirelmann

We use some detailed knowledge of the cohomology ring of real Grassmann manifolds G k (ℝ n ) to compute zero-divisor cup-length and estimate topological complexity of motion planning for k-linear subspaces in ℝ n . In addition, we obtain results about monotonicity of Lusternik–Schnirelmann category and topological complexity of G k (ℝ n ) as a function of n.

scholarly journals Topological Complexity of Motion Planning

2003 ◽  
Vol 29 (2) ◽  
pp. 211-221 ◽  
Author(s):  
Michael Farber
Keyword(s):  
Motion Planning ◽  
Topological Complexity

scholarly journals Symmetric Bi-Skew Maps and Symmetrized Motion Planning in Projective Spaces

2018 ◽  
Vol 61 (4) ◽  
pp. 1087-1100
Author(s):  
Jesús González
Keyword(s):  
Positive Integer ◽  
Projective Space ◽  
Motion Planning ◽  
Cohomology Ring ◽  
Projective Spaces ◽  
Closed Surface ◽  
Topological Complexity ◽  
Embedding Dimension ◽  
The Core

AbstractThis work is motivated by the question of whether there are spacesXfor which the Farber–Grant symmetric topological complexity TCS(X) differs from the Basabe–González–Rudyak–Tamaki symmetric topological complexity TCΣ(X). For a projective space${\open R}\hbox{P}^m$, it is known that$\hbox{TC}^S ({\open R}\hbox{P}^{m})$captures, with a few potential exceptional cases, the Euclidean embedding dimension of${\open R}\hbox{P}^{m}$. We now show that, for allm≥1,$\hbox{TC}^{\Sigma}({\open R}\hbox{P}^{m})$is characterized as the smallest positive integernfor which there is a symmetric${\open Z}_{2}$-biequivariant mapSm×Sm→Snwith a ‘monoidal’ behaviour on the diagonal. This result thus lies at the core of the efforts in the 1970s to characterize the embedding dimension of real projective spaces in terms of the existence of symmetric axial maps. Together with Nakaoka's description of the cohomology ring of symmetric squares, this allows us to compute both TC numbers in the case of${\open R}\hbox{P}^{2^{e}}$fore≥1. In particular, this leaves the torusS1×S1as the only closed surface whose symmetric (symmetrized) TCS(TCΣ) invariant is currently unknown.

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