M. Cao

Reaching a Consensus in a Dynamically Changing Environment: A Graphical Approach

This paper presents new graph-theoretic results appropriate for the analysis of a variety of consensus problems cast in dynamically changing environments. The concepts of rooted, strongly rooted, and neighbor-shared are defined, and conditions are derived for compositions of sequences of directed graphs to be of these types. The graph of a stochastic matrix is defined, and it is shown that under certain conditions the graph of a Sarymsakov matrix and a rooted graph are one and the same. As an illustration of the use of the concepts developed in this paper, graph-theoretic conditions are obtained which address the convergence question for the leaderless version of the widely studied Vicsek consensus problem.

Reaching a Consensus in a Dynamically Changing Environment: Convergence Rates, Measurement Delays, and Asynchronous Events

This paper uses recently established properties of compositions of directed graphs together with results from the theory of nonhomogeneous Markov chains to derive worst case convergence rates for the headings of a group of mobile autonomous agents which arise in connection with the widely studied Vicsek consensus problem. The paper also uses graph-theoretic constructions to solve modified versions of the Vicsek problem in which there are measurement delays, asynchronous events, or a group leader. In all three cases the conditions under which consensus is achieved prove to be almost the same as the conditions under which consensus is achieved in the synchronous, delay-free, leaderless case.

Localization in sparse networks using sweeps

Determining node positions is essential for many next-generation network functionalities. Previous localization algorithms lack correctness guarantees or require network density higher than required for unique localizability. In this paper, we describe a class of algorithms for fine-grained localization called Sweeps. Sweeps correctly finitely localizes all nodes in bilateration networks. Sweeps also handles angle measurements and noisy measurements. We demonstrate the practicality of our algorithm through extensive simulations on a large number of networks, upon which it consistently localizes one-thousand-node networks of average degree less than five in less than two minutes on a consumer PC.

A Lower Bound on Convergence of a Distributed Network Consensus Algorithm

This paper gives a lower bound on the convergence rate of a class of network consensus algorithms. Two different approaches using directed graphs as a main tool are introduced: one is to compute the scrambling constants of stochastic matrices associated with neighbor shared graphs and the other is to analyze random walks on a sequence of graphs. Both approaches prove that the time to reach consensus within a dynamic network is logarithmic in the relative error and is in worst case exponential in the size of the network.

Sensor Network Localization with Imprecise Distances

An approach to formulate geometric relations among distances between nodes as equality constraints is introduced in this paper to study the localization problem with imprecise distance information in sensor networks. These constraints can be further used to formulate optimization problems for distance estimation. The optimization solutions correspond to a set of distances that are consistent with the fact that sensor nodes live in the same plane or 3D space as the anchor nodes. These techniques serve as the foundation for most of the existing localization algorithms that depend on the sensors’ distances to anchors to compute each sensor’s location.

Controlling a triangular formation of mobile autonomous agents

This paper proposes a distributed control law for maintaining a triangular formation in the plane consisting of three mobile autonomous agents. It is shown that the control law can cause any initially non-collinear, positively-oriented {resp. negatively-oriented} triangular formation to converge exponentially fast to a desired positively-oriented {resp. negatively- oriented} triangular formation. It is also shown that there is a thin set of initially collinear formations which remain collinear and may drift off to infinity as t rarr infin. These findings complement and extend earlier findings cited below.

Reaching an Agreement Using Delayed Information

This paper studies a modified version of the Vicseks problem, also known as the consensus problem. Vicsek et al. (1995) consider a discrete-time model consisting of n autonomous agents all moving in the plane with the same speed but with different headings. Each agents heading is updated using a local rule based on the average of the headings of its neighbors. We consider a modified version of the Vicseks problem in which integer valued delays occur in sensing the values of headings which are available to agents. By appealing to the concept of graph composition, we side-step most issues involving products of stochastic matrices and present a variety of graph theoretic results which explains how convergence to a common heading is achieved

Localization with Imprecise Distance Information in Sensor Networks

An approach to formulate geometric relations among distances between nodes as equality constraints is introduced in this paper to study the localization problem with imprecise distance information in sensor networks. These constraints can be further used to formulate optimization problems for distance estimation. The optimization solutions correspond to a set of distances that are consistent with the fact that sensor nodes live in the same plane or 3D space as the anchor nodes. These techniques serve as the foundation for most of the existing localization algorithms that depend on the sensors distances to anchors to compute each sensor’s location.

Station Keeping in the Plane with Range-Only Measurements

Using concepts from switched adaptive control theory, provably correct solution is given to the three landmark station keeping problem in the plane in which range measurements are the only sensed signals upon which station keeping is to be based. The performance of the resulting system degrades gracefully in the face of measurement and miss- alignment errors, provided the measurement errors are not too large.

Agreeing Asynchronously: Announcement of Results

This paper formulates and solves a continuous-time version of the widely studied Vicsek consensus problem in which each agent independently updates its heading at times determined by its own clock. It is not assumed that the agents clocks are synchronized or that the event times between which any one agent updates its heading are evenly spaced. Heading updates need not occur instantaneously. Using the concept of analytic synchronization together with several key results concerned with properties of compositions of directed graphs, it is shown that the conditions under which a consensus is achieved are essentially the same as those applicable in the synchronous discrete-time case provided the notion of an agents neighbor between its event times is appropriately defined