scholarly journals The combinatorics of CAT(0) cubical complexes

2013 ◽  
Vol DMTCS Proceedings vol. AS,... (Proceedings) ◽  
Author(s):  
Federico Ardila ◽  
Tia Baker ◽  
Rika Yatchak
Keyword(s):  
Remote Control ◽  
Shortest Path ◽  
Robotic Arm ◽  
General Strategy ◽  
Cubical Complex ◽  
Cubical Complexes ◽  
Square Grid ◽  
Reconfigurable System ◽  
International Audience ◽  
Simultaneous Moves

International audience Given a reconfigurable system $X$, such as a robot moving on a grid or a set of particles traversing a graph without colliding, the possible positions of $X$ naturally form a cubical complex $\mathcal{S}(X)$. When $\mathcal{S}(X)$ is a CAT(0) space, we can explicitly construct the shortest path between any two points, for any of the four most natural metrics: distance, time, number of moves, and number of steps of simultaneous moves. CAT(0) cubical complexes are in correspondence with posets with inconsistent pairs (PIPs), so we can prove that a state complex $\mathcal{S}(X)$ is CAT(0) by identifying the corresponding PIP. We illustrate this very general strategy with one known and one new example: Abrams and Ghrist's ``positive robotic arm" on a square grid, and the robotic arm in a strip. We then use the PIP as a combinatorial ``remote control" to move these robots efficiently from one position to another.

scholarly journals The configuration space of a robotic arm in a tunnel of width 2

2020 ◽  
Vol DMTCS Proceedings, 28th... ◽  
Author(s):  
Federico Ardila ◽  
Hanner Bastidas ◽  
Cesar Ceballos ◽  
John Guo
Keyword(s):  
Configuration Space ◽  
Robotic Arm ◽  
Configuration Spaces ◽  
Cubical Complex ◽  
International Audience ◽  
Rectangular Tunnel

International audience We study the motion of a robotic arm inside a rectangular tunnel of width 2. We prove that the configuration space S of all possible positions of the robot is a CAT(0) cubical complex. Before this work, very few families of robots were known to have CAT(0) configuration spaces. This property allows us to move the arm optimally from one position to another.

scholarly journals Topology of random -dimensional cubical complexes

2021 ◽  
Vol 9 ◽  
Author(s):  
Matthew Kahle ◽  
Elliot Paquette ◽  
Érika Roldán
Keyword(s):  
Fundamental Group ◽  
Random Graphs ◽  
High Probability ◽  
Cubical Complex ◽  
Sharp Threshold ◽  
Cubical Complexes ◽  
Simple Connectivity ◽  
Dimensional Cube ◽  
Dimensional Face ◽  
Connectivity Threshold

Abstract We study a natural model of a random $2$ -dimensional cubical complex which is a subcomplex of an n-dimensional cube, and where every possible square $2$ -face is included independently with probability p. Our main result exhibits a sharp threshold $p=1/2$ for homology vanishing as $n \to \infty $ . This is a $2$ -dimensional analogue of the Burtin and Erdoős–Spencer theorems characterising the connectivity threshold for random graphs on the $1$ -skeleton of the n-dimensional cube. Our main result can also be seen as a cubical counterpart to the Linial–Meshulam theorem for random $2$ -dimensional simplicial complexes. However, the models exhibit strikingly different behaviours. We show that if $p> 1 - \sqrt {1/2} \approx 0.2929$ , then with high probability the fundamental group is a free group with one generator for every maximal $1$ -dimensional face. As a corollary, homology vanishing and simple connectivity have the same threshold, even in the strong ‘hitting time’ sense. This is in contrast with the simplicial case, where the thresholds are far apart. The proof depends on an iterative algorithm for contracting cycles – we show that with high probability, the algorithm rapidly and dramatically simplifies the fundamental group, converging after only a few steps.

scholarly journals Random assignment and shortest path problems

2006 ◽  
Vol DMTCS Proceedings vol. AG,... (Proceedings) ◽  
Author(s):  
Johan Wästlund
Keyword(s):  
Limit Theorem ◽  
Shortest Path ◽  
Complete Graph ◽  
Assignment Problem ◽  
Shortest Path Problem ◽  
Random Assignment ◽  
Shortest Path Problems ◽  
Parisi Formula ◽  
International Audience ◽  
Random Assignment Problem

International audience We explore a similarity between the $n$ by $n$ random assignment problem and the random shortest path problem on the complete graph on $n+1$ vertices. This similarity is a consequence of the proof of the Parisi formula for the assignment problem given by C. Nair, B. Prabhakar and M. Sharma in 2003. We give direct proofs of the analogs for the shortest path problem of some results established by D. Aldous in connection with his $\zeta (2)$ limit theorem for the assignment problem.

scholarly journals Combinatorial aspects of Escher tilings

2010 ◽  
Vol DMTCS Proceedings vol. AN,... (Proceedings) ◽  
Author(s):  
Alexandre Blondin Massé ◽  
Srecko Brlek ◽  
Sébastien Labbé
Keyword(s):  
Mathematical Community ◽  
Combinatorics On Words ◽  
Combinatorial Object ◽  
Square Grid ◽  
Letter Alphabet ◽  
International Audience ◽  
Escher Tilings

International audience In the late 30's, Maurits Cornelis Escher astonished the artistic world by producing some puzzling drawings. In particular, the tesselations of the plane obtained by using a single tile appear to be a major concern in his work, drawing attention from the mathematical community. Since a tile in the continuous world can be approximated by a path on a sufficiently small square grid - a widely used method in applications using computer displays - the natural combinatorial object that models the tiles is the polyomino. As polyominoes are encoded by paths on a four letter alphabet coding their contours, the use of combinatorics on words for the study of tiling properties becomes relevant. In this paper we present several results, ranging from recognition of these tiles to their generation, leading also to some surprising links with the well-known sequences of Fibonacci and Pell. Lorsque Maurits Cornelis Escher commença à la fin des années 30 à produire des pavages du plan avec des tuiles, il étonna le monde artistique par la singularité de ses dessins. En particulier, les pavages du plan obtenus avec des copies d'une seule tuile apparaissent souvent dans son œuvre et ont attiré peu à peu l'attention de la communauté mathématique. Puisqu'une tuile dans le monde continu peut être approximée par un chemin sur un réseau carré suffisamment fin - une méthode universellement utilisée dans les applications utilisant des écrans graphiques - l'objet combinatoire qui modèle adéquatement la tuile est le polyomino. Comme ceux-ci sont naturellement codés par des chemins sur un alphabet de quatre lettres, l'utilisation de la combinatoire des mots devient pertinente pour l'étude des propriétés des tuiles pavantes. Nous présentons dans ce papier plusieurs résultats, allant de la reconnaissance de ces tuiles à leur génération, conduisant à des liens surprenants avec les célèbres suites de Fibonacci et de Pell.

scholarly journals Internet Remote Control Interface for a Multipurpose Robotic Arm

10.5772/5741 ◽  
2006 ◽  
Vol 3 (2) ◽  
pp. 27
Author(s):  
Cyprian M. Wronka ◽  
Matthew W. Dunnigan
Keyword(s):  
Remote Control ◽  
Robotic Arm ◽  
Control Interface

scholarly journals Shortest path poset of finite Coxeter groups

2009 ◽  
Vol DMTCS Proceedings vol. AK,... (Proceedings) ◽  
Author(s):  
Saúl A. Blanco
Keyword(s):  
Shortest Path ◽  
Coxeter Group ◽  
Coxeter Groups ◽  
Shortest Paths ◽  
Combinatorial Interpretation ◽  
Nous Permet ◽  
Finite Coxeter Group ◽  
International Audience ◽  
Absolute Length ◽  
Bruhat Graph

International audience We define a poset using the shortest paths in the Bruhat graph of a finite Coxeter group $W$ from the identity to the longest word in $W, w_0$. We show that this poset is the union of Boolean posets of rank absolute length of $w_0$; that is, any shortest path labeled by reflections $t_1,\ldots,t_m$ is fully commutative. This allows us to give a combinatorial interpretation to the lowest-degree terms in the complete $\textbf{cd}$-index of $W$. Nous définissons un poset en utilisant le plus court chemin entre l'identité et le plus long mot de $W, w_0$, dans le graph de Bruhat du groupe finie Coxeter, $W$. Nous prouvons que ce poset est l'union de posets Boolean du même rang que la longueur absolute de $w_0$; ça signifie que tous les plus courts chemins, étiquetés par réflexions $t_1,\ldots, t_m$ sont totalement commutatives. Ça nous permet de donner une interprétation combinatoire aux termes avec le moindre grade dans le $\textbf{cd}$-index complet de $W$.

scholarly journals A Matlab-based framework for the remote control of a 6-DOF robotic arm for education and research in control theory

2012 ◽  
Vol 45 (11) ◽  
pp. 366-371 ◽  
Author(s):  
Michael Ruderman ◽  
Frank Hoffmann ◽  
Torsten Bertram
Keyword(s):  
Remote Control ◽  
Control Theory ◽  
Robotic Arm
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