Abubakr Muhammad

Coverage and hole-detection in sensor networks via homology

We consider coverage problems in sensor networks of stationary nodes with minimal geometric data. In particular, there are no coordinates and no localization of nodes. We introduce a new technique for detecting holes in coverage by means of homology, an algebraic topological invariant. The impetus for these techniques is a completion of network communication graphs to two types of simplicial complexes: the nerve complex and the Rips complex. The former gives information about coverage intersection of individual sensor nodes, and is very difficult to compute. The latter captures connectivity in terms of inter-node communication: it is easy to compute but does not in itself yield coverage data. We obtain coverage data by using persistence of homology classes for Rips complexes. These homological invariants are computable: we provide simulation results.

Blind Swarms for Coverage in 2-D

We consider coverage problems in robot sensor networks with minimal sensing capabilities. In particular, we demonstrate that a “blind” swarm of robots with no localization and only a weak form of distance estimation can rigorously determine coverage in a bounded planar domain of unknown size and shape. The methods we introduce come from algebraic topology. I. COVERAGE PROBLEMS Many of the potential applications of robot swarms require information about coverage in a given domain. For example, using a swarm of robot sensors for surveillance and security applications carries with it the charge to maximize, or, preferably, guarantee coverage. Such applications include networks of security cameras, mine field sweeping via networked robots [18], and oceanographic sampling [4]. In these contexts, each robot has some coverage domain, and one wishes to know about the union of these coverage domains. Such problems are also crucial in applications not involving robots directly, e.g., communication networks. As a preliminary analysis, we consider the static “field” coverage problem, in which robots are assumed stationary and the goal is to verify blanket coverage of a given domain. There is a large literature on this subject; see, e.g., [7], [1], [16]. In addition, there are variants on these problems involving “barrier” coverage to separate regions. Dynamic or “sweeping” coverage [3] is a common and challenging task with applications ranging from security to vacuuming. Although a sensor network composed of robots will have dynamic capabilities, we restrict attention in this brief paper to the static case in order to lay the groundwork for future inquiry. There are two primary approaches to static coverage problems in the literature. The first uses computational geometry tools applied to exact node coordinates. This typically involves ‘ruler-and-compass’ style geometry [10] or Delaunay triangulations of the domain [16], [14], [20]. Such approaches are very rigid with regards to inputs: one must know exact node coordinates and one must know the geometry of the domain precisely to determine the Delaunay complex. To alleviate the former requirement, many authors have turned to probabilistic tools. For example, in [13], the author assumes a randomly and uniformly distributed collection of nodes in a domain with a fixed geometry and proves expected area coverage. Other approaches [15], [19] give percolationtype results about coverage and network integrity for randomly distributed nodes. The drawback of these methods is the need for strong assumptions about the exact shape of the domain, as well as the need for a uniform distribution of nodes. In the sensor networks community, there is a compelling interest (and corresponding burgeoning literature) in determining properties of a network in which the nodes do not possess coordinate data. One example of a coordinate-free approach is in [17], which gives a heuristic method for geographic routing without coordinate data: among the large literature arising from this paper, we note in particular the mathematical analysis of this approach in [11]. To our knowledge, noone has treated the coverage problem in a coordinate-free setting. In this note, we introduce a new set of tools for answering coverage problems in robotics and sensor networks with minimal assumptions about domain geometry and node localization. We provide a sufficiency criterion for coverage. We do not answer the problem of how the nodes should be placed in order to maximize coverage, nor the minimum number of such nodes necessary; neither do we address how to reallocate nodes to fill coverage holes.

Decentralized Computation of Homology Groups in Networks by Gossip

In this paper, we present an approach towards the computation of certain topological invariants in real sensor networks. As shown by many researchers, these invariants are relevant for modeling certain properties of the network such as coverage and routing. What has been lacking so far is a concrete decentralized method to compute these invariants for proper implementation. In this paper, we give an approach towards such an implementation. The main tools being used here are the the so-called higher order Laplacian operators and distributed methods for their spectral analysis that resemble gossip algorithms.

Connectivity graphs as models of local interactions

In this paper, we study graphs that arise from certain sensory and communication limitations on the local interactions in multi-agent systems. In particular, we show that the set of graphs that can represent formations corresponds to a proper subset of all graphs and we denote such graphs as connectivity graphs. Such graphs have a special structure that allows them to be composed from a small number of atomic crossing generators using a certain kind of graph amalgamation. This structure allows us to give connectivity graphs a useful topological characterization in terms of their simplicial complexes.

Conjugate degradability and the quantum capacity of cloning channels

A quantum channel is conjugate degradable if the channel’s environment can be simulated up to complex conjugation using the channel’s output. For all such channels, the quantum capacity can be evaluated using a single-letter formula. In this article we introduce conjugate degradability and establish a number of its basic properties. We then use it to calculate the quantum capacity of N to N+1 and 1 to M universal quantum cloning machines as well as the quantum capacity of a channel that arises naturally when data are being transmitted to an accelerating receiver. All the channels considered turn out to have strictly positive quantum capacity, meaning they could be used as part of a communication system to send quantum states reliably.

Autonomous Formation Switching for Multiple, Mobile Robots

Abstract In this paper we investigate the question concerning what multi-agent formations to use in a given situation. In particular, we show how it is possible to produce a control strategy for teams of mobile robots that switches between different formations as a reaction to environmental changes. The feasibility of the approach is verified in simulation, where a steepest descent algorithm is combined with standard reactive behaviors that ensure that the individual robots avoid neighboring obstacles and robots, while approaching a desired target location

Control of Very-Large Scale Irrigation Networks: A CPS Approach in a Developing-World Setting

Abstract Many developing countries face severe water shortages, a large magnitude of which can be attributed to the inefficient operation of their water networks. In this paper, we study this issue from the point of view of implementing a cyber physical systems (CPS) infrastructure, in which networked embedded controllers can be used for local control of a very large-scale canal network. We take inspiration from existing literature and report on the suitability (or otherwise) of existing methods in a third world setting. We describe partial differential equation (PDE) models for open channel flows, system identification techniques to simplify the model, local control system design and strategies to implement a decentralized control network. The theoretical study is supported by simulations on large networks.

Dynamic Coverage Verification in Mobile Sensor Networks Via Switched Higher Order Laplacians.

In this paper, we study the problem of verifying dynamic coverage in mobile sensor networks using certain switched linear systems. These switched systems describe the flow of discrete differential forms on time-evolving simplicial complexes. The simplicial complexes model the connectivity of agents in the network, and the homology groups of the simplicial complexes provides information about the coverage properties of the network. Our main result states that the asymptotic stability the switched linear system implies that every point of the domain covered by the mobile sensor nodes is visited infinitely often, hence verifying dynamic coverage. The enabling mathematical technique for this result is the theory of higher order Laplacian operators, which is a generalization of the graph Laplacian used in spectral graph theory and continuous-time consensus problems.

Development of a Low-Power Smart Water Meter for Discharges in Indus Basin Irrigation Networks

To improve the sampling frequency of water diversion to distributary canals and to improve equity of distribution and data handling we have developed a smart electronic water meter based on ultrasonic sensors and GPRS modem to frequently record and transmit the water diversion data to a centralized server. The server processes the data to extract useful information for example seasonal cumulative water deliveries and discharge time series. The Wireless Sensor Node (WSN) inspired design is extremely low-power, field deployable and scalable with respect to cost and numbers. This paper, reports the first steps towards practical realization of a smart water grid in the Indus river basin, conceptualized by the authors in previous theoretical studies.

Model-Driven Performance Analysis of Large Scale Irrigation Networks

Irrigation networks play a fundamental part in the agriculture system of various countries. In the wake of global environmental challenges and economic competition, efficient use of water resources has become extremely important. This can only be achieved by developing smarter control infrastructures for irrigation networks, via the incorporation of communication and computation technologies. Thus, future irrigation networks represent a prime example of cyber physical systems. Effective operation of these complex cyber physical systems is not possible with conventional methods and requires unprecedented levels of automation and decision-support tools. We argue that these cyber physical systems will require a complete model-driven toolset for effective operation. As a first step towards that tool flow, we have developed a model-driven simulation infrastructure for irrigation networks. In the future, we propose to complete the toolset by developing a model-driven configuration infrastructure. Our contributions in this paper include the development of a domain-specific modeling language (DSML) for irrigation networks, implementation of this DSML in Generic Modeling Environment (GME), and automatic simulator M-file generation capability from the DSML-based case diagram of an arbitrary irrigation network. Moreover, we present case studies of water distribution and flood management to show the utility as well as the effectiveness of our approach. We also present the performance of our toolset for the realistic scenario of irrigation networks in Pakistan.