Invariants for homology classes with application to optimal search and planning problem in robotics

Bhattacharya, Subhrajit; Lipsky, David B.; Kumar, Vijay

doi:10.1007/s10472-013-9357-7

Cited by 31 publications

(30 citation statements)

References 22 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Gathering algorithms for the case where at most one robot may crash, or behave in a Byzantine way, was proposed in [1], and for multiple crash failures in [10]. However, we are not aware of the use of algebraic topology techniques in the style of [21] (work about computing topological properties of a space is of a different nature, e.g., [6]). In our setting, gathering is impossible ("Impossibility results" section) because, in contrast to other settings, robots cannot observe directly the positions of other robots, they need to communicate with each other to find them.…”

Section: Related Workmentioning

confidence: 99%

Convergence and covering on graphs for wait-free robots

2018

View full text Add to dashboard Cite

The class of robot convergence tasks has been shown to capture fundamental aspects of fault-tolerant computability. A set of asynchronous robots that may fail by crashing, start from unknown places in some given space, and have to move towards positions close to each other. In this article, we study the case where the space is uni-dimensional, modeled as a graph G. In graph convergence, robots have to end up on one or two vertices of the same edge. We consider also a variant of robot convergence on graphs, edge covering, where additionally, it is required that not all robots end up on the same vertex. Remarkably, these two similar problems have very different computability properties, related to orthogonal fundamental issues of distributed computations: agreement and symmetry breaking. We characterize the graphs on which each of these problems is solvable, and give optimal time algorithms for the solvable cases. Although the results can be derived from known general topology theorems, the presentation serves as a self-contained introduction to the algebraic topology approach to distributed computing, and yields concrete algorithms and impossibility results.

show abstract

Section: Related Workmentioning

confidence: 99%

Convergence and covering on graphs for wait-free robots

2018

View full text Add to dashboard Cite

show abstract

“…Similarly to our work, the authors argue that homological information is useful and computationally favorable to more general homotopy invariants in robotics. In [7], a generalization to arbitrary dimension is proposed and an integration of differential 1-forms over cycles is shown to be sufficient to determine topological classes using the classical language of de Rham cohomology theory. In [21], motion planning in 2D with homology constraints is formulated as a mixed-integer quadratic program by endowing path segments with binary labels that identify their relation to the domain obstacles.…”

Section: Motivation and Related Workmentioning

confidence: 99%

Multiscale Topological Trajectory Classification with Persistent Homology

Pokorny

Hawasly

Ramamoorthy

2014

Robotics: Science and Systems X

View full text Add to dashboard Cite

Abstract-Topological approaches to studying equivalence classes of trajectories in a configuration space have recently received attention in robotics since they allow a robot to reason about trajectories at a high level of abstraction. While recent work has approached the problem of topological motion planning under the assumption that the configuration space and obstacles within it are explicitly described in a noise-free manner, we focus on trajectory classification and present a samplingbased approach which can handle noise, which is applicable to general configuration spaces and which relies only on the availability of collision free samples. Unlike previous samplingbased approaches in robotics which use graphs to capture information about the path-connectedness of a configuration space, we construct a multiscale approximation of neighborhoods of the collision free configurations based on filtrations of simplicial complexes. Our approach thereby extracts additional homological information which is essential for a topological trajectory classification. By computing a basis for the first persistent homology groups, we obtain a multiscale classification algorithm for trajectories in configuration spaces of arbitrary dimension. We furthermore show how an augmented filtration of simplicial complexes based on a cost function can be defined to incorporate additional constraints. We present an evaluation of our approach in 2, 3, 4 and 6 dimensional configuration spaces in simulation and using a Baxter robot.

show abstract

“…However, the possible choice of such invariants has been broadened in [6], where the choice of the vector of differential 1-forms, which needs to be integrated over γ to obtain the invariant, has been proven to be any complete set of generators of the de Rham cohomology group,…”

Section: B Homology and Homotopy Invariantsmentioning

confidence: 99%

A Topological Approach to Using Cables to Separate and Manipulate Sets of Objects

Kim

Bhattacharya

Heidarsson

et al. 2013

Robotics: Science and Systems IX

Self Cite

View full text Add to dashboard Cite

Abstract-In this paper we study the problem of manipulating and transporting multiple objects on the plane using a cable attached at each end to a mobile robot. This problem is motivated by the use of boats with booms in skimming operations for cleaning oil spills or removing debris on the surface of the water. The goal in this paper is to automate the task of separating the objects of interest from a collection of objects by manipulating them with cables that are actuated only at the ends, and then transporting them to specified destinations. Because the cable is flexible, the shape of the cable must be explicitly modeled in the problem. Further, the robots must cooperatively plan motions to achieve the required cable shape and gross position/orientation to separate the objects of interest and then transport them as specified. The theoretical foundation for the problem is derived from topological invariants, homology and homotopy. We first derive the necessary topological conditions for achieving the desired separation of objects. We then propose a distributed search-based planning technique for finding optimal robot trajectories for separation and transportation. We demonstrate the applicability of this method using a dynamic simulation platform with explicit models of the cable dynamics, the contact between the cable and one or more objects, and the surface drag on the cable and on the objects. We also describe our preliminary efforts to develop an experimental platform consisting of a system of two cooperating autonomous boats.

show abstract

Invariants for homology classes with application to optimal search and planning problem in robotics

Cited by 31 publications

References 22 publications

Convergence and covering on graphs for wait-free robots

Convergence and covering on graphs for wait-free robots

Multiscale Topological Trajectory Classification with Persistent Homology

A Topological Approach to Using Cables to Separate and Manipulate Sets of Objects

Contact Info

Product

Resources

About