Frederick R. Cohen

Sensor Beams, Obstacles, and Possible Paths

This paper introduces a problem in which an agent (robot, human, or animal) travels among obstacles and binary detection beams. The task is to determine the possible agent path based only on the binary sensor data. This is a basic filtering problem encountered in many settings, which may arise from physical sensor beams or virtual beams that are derived from other sensing modalities. Methods are given for three alternative representations: 1) the possible sequences of regions visited, 2) path descriptions up to homotopy class, and 3) numbers of times winding around obstacles. The solutions are adapted to the minimal sensing setting; therefore, precise estimation, distances, and coordinates are replaced by topological expressions. Applications include sensor-based forensics, assisted living, security, and environmental monitoring.

Commuting elements, simplicial spaces and filtrations of classifying spaces

Using spaces of homomorphisms and the descending central series of the free groups, simplicial spaces are constructed for each integer q>1 and every topological group G, with realizations B(q,G) that filter the classifying space BG. In particular for q=2 this yields a single space B(2,G) assembled from all the n-tuples of commuting elements in G. Homotopy properties of the B(q,G) are considered for finite groups. Cohomology calculations are provided for compact Lie groups. The spaces B(2,G) are described in detail for transitively commutative groups.

On configuration spaces, their homology, and Lie algebras

Abstract Homological and geometric properties of configuration spaces of points in Euclidean space are described. Connections to non-abelian analogues of exterior algebras and mapping class groups are also given.

Gait Transitions for Quasi-Static Hexapedal Locomotion on Level Ground

As robot bodies become more capable, the motivation grows to better coordinate them–whether multiple limbs attached to a body or multiple bodies assigned to a task. This paper introduces a new formalism for coordination of periodic tasks, with specific application to gait transitions for legged platforms. Specifically, we make modest use of classical group theory to replace combinatorial search and optimization with a computationally simpler and conceptually more straightforward appeal to elementary algebra.

Stable splittings, spaces of representations and almost commuting elements in Lie groups

In this paper the space of almost commuting elements in a Lie group is studied through a homotopical point of view. In particular a stable splitting after one suspension is derived for these spaces and their quotients under conjugation. A complete description for the stable factors appearing in this splitting is provided for compact connected Lie groups of rank one.By using symmetric products, the colimits

Cup-products for the polyhedral product functor

\Rep(\Z^n, SU)

Commuting elements in central products of special unitary groups

,

On braid groups and homotopy groups

\Rep(\Z^n,U)

On spaces of commuting elements in Lie groups

and

Combinatorial Filters: Sensor Beams, Obstacles, and Possible Paths

\Rep(\Z^n, Sp)