Safe Cooperative Robotic Patterns via Dynamics on Graphs

This paper explores the possibility of using vector fields to design and implement reactive schedules for safe cooperative robot patterns on graphs. The word safe means that obstacles - designated illegal portions of the configuration space - are avoided. The word cooperative connotes situations wherein physically distributed agents are collectively responsible for executing the schedule. The word pattern refers to tasks that cannot be encoded simply in terms of a point goal in the configuration space. The word reactive will be interpreted as requiring that the desired pattern be asymptotically stable: conditions close but slightly removed from those desired remain close and converge toward the desired pattern. We consider Automated Guided Vehicles (AGV\u27s) operating upon a predefined network of pathways, contrasting the simple cases of locally Euclidean configuration spaces with the more topologically intricate non-manifold cases. The focus of the present inquiry is the achievement of safe cooperative patterns by means of a succession of edge point fields combined with a circulating field to regularize collisions at non-manifold vertices. For more information: Kod*La

Toward the Regulation and Composition of Cyclic Behaviors

Many tasks in robotics and automation require a cyclic exchange of energy between a machine and its environment. Since most environments are under actuated —that is, there are more objects to be manipulated than actuated degrees of freedom with which to manipulate them—the exchange must be punctuated by intermittent repeated contacts. In this paper, we develop the appropriate theoretical setting for framing these problems and propose a general method for regulating coupled cyclic systems. We prove for the first time the local stability of a (slight variant on a) phase regulation strategy that we have been using with empirical success in the lab for more than a decade. We apply these methods to three examples: juggling two balls, two legged synchronized hopping and two legged running—considering for the first time the analogies between juggling and running formally

Knots and Links in Three-Dimensional Flows

The closed orbits of three-dimensional flows form knots and links. This book develops the tools - template theory and symbolic dynamics - needed for studying knotted orbits. This theory is applied to the problems of understanding local and global bifurcations, as well as the embedding data of orbits in Morse-smale, Smale, and integrable Hamiltonian flows. The necesssary background theory is sketched; however, some familiarity with low-dimensional topology and differential equations is assumed

Rips Complexes of Planar Point Sets

Fix a finite set of points in Euclidean $n$-space \euc^n, thought of as a point-cloud sampling of a certain domain D\subset\euc^n. The Rips complex is a combinatorial simplicial complex based on proximity of neighbors that serves as an easily-computed but high-dimensional approximation to the homotopy type of $D$. There is a natural ``shadow'' projection map from the Rips complex to \euc^n that has as its image a more accurate $n$-dimensional approximation to the homotopy type of $D$. We demonstrate that this projection map is 1-connected for the planar case $n=2$. That is, for planar domains, the Rips complex accurately captures connectivity and fundamental group data. This implies that the fundamental group of a Rips complex for a planar point set is a free group. We show that, in contrast, introducing even a small amount of uncertainty in proximity detection leads to `quasi'-Rips complexes with nearly arbitrary fundamental groups. This topological noise can be mitigated by examining a pair of quasi-Rips complexes and using ideas from persistent topology. Finally, we show that the projection map does not preserve higher-order topological data for planar sets, nor does it preserve fundamental group data for point sets in dimension larger than three.Comment: 16 pages, 8 figure

Euler-Bessel and Euler-Fourier Transforms

We consider a topological integral transform of Bessel (concentric isospectral sets) type and Fourier (hyperplane isospectral sets) type, using the Euler characteristic as a measure. These transforms convert constructible \zed-valued functions to continuous $\real$-valued functions over a vector space. Core contributions include: the definition of the topological Bessel transform; a relationship in terms of the logarithmic blowup of the topological Fourier transform; and a novel Morse index formula for the transforms. We then apply the theory to problems of target reconstruction from enumerative sensor data, including localization and shape discrimination. This last application utilizes an extension of spatially variant apodization (SVA) to mitigate sidelobe phenomena

Branched two-manifolds supporting all links

We resolve several conjectures of J. Birman and R. F. Williams concerning the knotting and linking of closed orbits of ows on 3-manifolds. Our methods center on the symbolic dynamics of semi ows on branched 2-manifolds, or templates. By proving the existence of \universal templates, " or embedded branched 2-manifolds supporting all nite links, we conclude that the set of closed orbits of any ow transverse to the bration of the gure-eight knot complement inS 3 contains representatives of every (tame) knot and link isotopy class. In these notes, we will answer some questions raised by Birman and Williams in their original examination of the link of closed orbits in the ow onS 3 induced by the bration of the complement of a knot or link [5] (see x4 for de nitions). In this work, they proposed the following conjecture: Conjecture 1 (Birman and Williams, 1983) The gure-eight knot does not appear as a closed orbit of the ow induced by the bration of the complement of the gure-eight knot in S 3

"Chaotic" knots and "wild" dynamics

The delicate interplay between knot theory and dynamical systems is surveyed. Numerous bridges between these fields allow us to apply dynamical perspectives (entropy, "chaotic") to knot theory, as well as knot-theoretic perspectives (cablings, "wild") to dynamical phenomena and bifurcations thereof. This intricate relationship has opened new doors in the study of ODE models of physical systems, while conversely yielding interesting topological objects from dynamical flows. 1 The topology and dynamics of flows The concept of flow is central to several fields of inquiry: fluid dynamicists consider the motion of liquids and gasses, plasma physicists work with trajectories of particles in magnetic fields, chemical engineers are concerned with the mixing of media through stirred agents. Mathematicians, pure and applied, are also deeply involved in examining flows which arise as solutions to differential equations. Apart from these physical applications, the topological and dynamical proper..