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

International audienceWe 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

The combinatorics of CAT(0) cubical complexes

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

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

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

Why CAT(0) cube complexes should be replaced with median graphs

In this note, we discuss and motivate the use of the terminology ``median graphs'' in place of ``CAT(0) cube complexes'' in geometric group theory.Comment: 9 pages. Comments are welcome

Los complejos cúbicos CAT(0) en la robótica

Los "complejos cúbicos CAT(0)" son espacios de curvatura no positiva construidos a partir de cubos. Estos espacios juegan un papel importante en la matemática pura (teóricaa geométrica de grupos) y aplicada (filogenética, robótica). En particular, el "mapa" de posibles posiciones de un robot discreto muchas veces es un complejo cúbico CAT(0). En esta charla mostraremos cómo los complejos cúbicos CAT(0), que en principio son objetos geométricos, pueden ser descritos de una manera completamente combinatoria. Para muchos robots, esto nos permite construir un "control remoto" que usamos para encontrar la manera más rápida de llegar de una posición a otra. Este es un trabajo conjunto con Tia Baker (SFSU), Megan Owen (Waterloo), Seth Sullivant (NCSU) y Rika Yatchak (Linz). La charla no asumira ningun conocimiento previo de estos tema

Geometry and Topology in Memory and Navigation

Okinawa Institute of Science and Technology Graduate UniversityDoctor of PhilosophyGeometry and topology offer rich mathematical worlds and perspectives with which to study and improve our understanding of cognitive function. Here I present the following examples: (1) a functional role for inhibitory diversity in associative memories with graph- ical relationships; (2) improved memory capacity in an associative memory model with setwise connectivity, with implications for glial and dendritic function; (3) safe and effi- cient group navigation among conspecifics using purely local geometric information; and　(4) enhancing geometric and topological methods to probe the relations between neural activity and behaviour. In each work, tools and insights from geometry and topology are used in essential ways to gain improved insights or performance. This thesis contributes to our knowledge of the potential computational affordances of biological mechanisms (such as inhibition and setwise connectivity), while also demonstrating new geometric and topological methods and perspectives with which to deepen our understanding of cognitive tasks and their neural representations.doctoral thesi