Showing papers by "Brian D. O. Anderson published in 2011"

Control of Minimally Persistent Leader-Remote-Follower and Coleader Formations in the Plane

Abstract: This paper solves an n -agent formation shape control problem in the plane. The objective is to design decentralized control laws so that the agents cooperatively restore a prescribed formation shape in the presence of small perturbations from the prescribed shape. We consider two classes of directed, cyclic information architectures associated with so-called minimally persistent formations: leader-remote-follower and coleader. In our framework the formation shape is maintained by controlling certain interagent distances. Only one agent is responsible for maintaining each distance. We propose a decentralized control law where each agent executes its control using only the relative position measurements of agents to which it must maintain its distance. The resulting nonlinear closed-loop system has a manifold of equilibria, which implies that the linearized system is nonhyperbolic. We apply center manifold theory to show local exponential stability of the desired formation shape. The result circumvents the non-compactness of the equilibrium manifold. Choosing stabilizing gains is possible if a certain submatrix of the rigidity matrix has all leading principal minors nonzero, and we show that this condition holds for all minimally persistent leader-remote-follower and coleader formations with generic agent positions. Simulations are provided.

Maintaining a Directed, Triangular Formation of Mobile Autonomous Agents

Abstract: This paper analyzes a class of distributed control laws which encompasses and generalizes three previously considered types of control laws for maintaining a triangular formation in the plane consisting of three point-modeled, mobile autonomous agents. It is shown that the control laws considered can cause any initially non-collinear, positively-oriented {resp. negatively-oriented} three agent formation to converge exponentially fast to a desired positively-oriented {resp. negatively-oriented} triangular formation. These findings extend earlier results and provide an alternative perspective.

Deterministic Gossiping

Abstract: For the purposes of this paper, “gossiping” is a distributed process whose purpose is to enable the members of a group of autonomous agents to asymptotically determine, in a decentralized manner, the average of the initial values of their scalar gossip variables. This paper discusses several different deterministic protocols for gossiping which avoid deadlocks and achieve consensus under different assumptions. First considered is T-periodic gossiping which is a gossiping protocol which stipulates that each agent must gossip with the same neighbor exactly once every T time units. Among the results discussed is the fact that if the underlying graph characterizing neighbor relations is a tree, convergence is exponential at a worst case rate which is the same for all possible T -periodic gossip sequences associated with the graph. Many gossiping protocols are request based which means simply that a gossip between two agents will occur whenever one of the two agents accepts a request to gossip placed by the other. Three deterministic request-based protocols are discussed. Each is guaranteed to not deadlock and to always generate sequences of gossip vectors which converge exponentially fast. It is shown that worst case convergence rates can be characterized in terms of the second largest singular values of suitably defined doubly stochastic matrices.

Stabilization of rigid formations with direction-only constraints

Abstract: Direction-based formation shape control for a collection of autonomous agents involves the design of distributed control laws that ensure the formation moves so that certain relative bearing constraints achieve, and maintain, some desired value. This paper looks at the design of a distributed control scheme to solve the direction-based formation shape control problem. A gradient control law is proposed based on the notion of bearing-only constrained graph rigidity and parallel drawings. This work provides an interesting and novel contrast to much of the existing work in formation control where distance-only constraints are typically maintained. A stability analysis is sketched and a number of illustrative examples are also given.

On the Information Propagation Process in Mobile Vehicular Ad Hoc Networks

Abstract: In this paper, we study the information propagation process in a 1-D mobile ad hoc network formed by vehicles Poissonly distributed on a highway and traveling in the same direction at randomly distributed speeds that are independent between vehicles. Considering a model in which time is divided into time slots of equal length and each vehicle changes its speed at the beginning of each time slot, independent of its speed in other time slots, we derive analytical formulas for the fundamental properties of the information propagation process and the information propagation speed (IPS). Using the formulas, one can straightforwardly study the impact on the IPS of various parameters such as radio range, vehicular traffic density, and time variation of vehicle speed. The accuracy of the results is validated using simulations. The research provides useful guidelines on the design of vehicular ad hoc networks (VANETs).

Close target reconnaissance with guaranteed collision avoidance

Abstract: This manuscript considers the problem of close target reconnaissance by a group of autonomous agents. The overall close target reconnaissance (CTR) involves subtasks of avoiding inter-agent collisions, reaching a close vicinity of a specific target position, and forming an equilateral polygon formation around the target. The agents performing the task fly at a constant speed to mimic the velocity behavior of small fixed-wing unmanned aerial vehicles (UAV). A decentralized control scheme is developed for this overall task and the finite-time convergence of the system under the proposed control law is established. Furthermore, it is guaranteed that no collision occurs among the agent. The relevant analysis and simulation test results are provided. Copyright © 2010 John Wiley & Sons, Ltd.

Cooperative Self-Localization of Mobile Agents

Abstract: This paper considers the problem of localizing multiple agents, e.g. unmanned aerial vehicles (UAVs), robots, etc., moving in two-dimensional space when the known data comprise 1) the inter-agent distances, and 2) the angle subtended at each agent by lines drawn from two landmarks at known positions. Later it is shown that this result has direct application in a different general robotic problem, viz. robot-to-robot relative pose determination (relative reference frame determination), using measured distances. The methods proposed are validated through simulations and experiments.

On the asymptotic connectivity of random networks under the random connection model

Abstract: Consider a network where all nodes are distributed on a unit square following a Poisson distribution with known density ρ and a pair of nodes separated by an Euclidean distance x are directly connected with probability g(x over r ρ ), where g : [0,∞) → [0,1] satisfies three conditions: rotational invariance, qnon-increasing monotonicity and integral boundedness, equation and b is a constant, independent of the event that another pair of nodes are directly connected. In this paper, we analyze the asymptotic distribution of the number of isolated nodes in the above network using the Chen-Stein technique and the impact of the boundary effect on the number of isolated nodes as ρ → ∞. On that basis we derive a necessary condition for the above network to be asymptotically almost surely connected. These results form an important link in expanding recent results on the connectivity of the random geometric graphs from the commonly used unit disk model to the more generic and more practical random connection model.

Morse theory and formation control

Abstract: Formation shape control for a collection of point agents is concerned with devising decentralized control laws which will ensure that the formation will move so that certain inter-agent distances assume prescribed values. A number of algorithms based on steepest descent of an error function have been suggested for various problems, and all display the existence of incorrect equilibria, though often the equilibria are saddle points or unstable. This paper introduces Morse theory as a tool for analyzing the number of such equilibria. A key conclusion is that for two-dimensional rigid formations of point agents, there will always be incorrect equilibria associated with any steepest descent law.

Solutions of Yule-Walker equations for singular AR processes

Abstract: A study is presented on solutions of the Yule-Walker equations for singular AR processes that are stationary outputs of a given AR system. If the Yule-Walker equations admit more than one solution and the order of the AR system is no less than two, the solution set includes solutions which define unstable AR systems. The solution set also includes one solution, the minimal norm solution, which defines an AR system whose characteristic polynomial has either only stable zeros (implying that only one stationary output exists for this system and it is linearly regular) or has stable zeros as well as zeros of unit modulus, (implying that stationary solutions of this system are a sum of a linearly regular process and a linearly singular process). The numbers of stable and unit circle zeros of the characteristic polynomial of the defined AR system can be characterized in terms of the ranks of certain matrices, and the characteristic polynomial of the AR system defined by the minimal norm solution has the least number of unit circle zeros and the most number of stable zeros over all possible solutions.

Controlling rectangular formations

Abstract: We consider formation shape control of four point agents in the plane. Control laws based on specified interagent distances are used. For a complete graph, specification of all interagent distances determines the formation shape uniquely. Krick, Broucke and Francis showed that for a standard control law, there may exist equilibrium formations with incorrect interagent distances. This paper studies such equilibria and derives two main results. That each such incorrect equilibrium is attached to at most one desired formation. That all such equilibria are unstable if the desired formation is a rectangle.

Contractions for consensus processes

Abstract: Many distributed control algorithms of current interest can be modeled by linear recursion equations of the form x(t + 1) = M(t)x(t), t ≥ 1 where each M(t) is a real-valued “stochastic” or “doubly stochastic” matrix. Convergence of such recursions often reduces to deciding when the sequence of matrix productsM(1), M(2)M(1), M(3)M(2)M(1), … converges. Certain types of stochastic and doubly stochastic matrices have the property that any sequence of products of such matrices of the form S 1 , S 2 S 1 , S 3 S 2 S 1 , … converges exponentially fast. We explicitly characterize the largest classes of stochastic and doubly stochastic matrices with positive diagonal entries which have these properties. The main goal of this paper is to find a “semi-norm” with respect to which matrices from these “convergability classes” are contractions. For any doubly stochastic matrix S such a semi-norm is identified and is shown to coincide with the second largest singular value of S.

On the zeros of blocked linear systems with single and mixed frequency data

Abstract: This paper studies properties of blocked systems resulting from blocking discrete linear systems with mixed frequency data The focus is on the zeros of the blocked systems We first establish results on the simpler single frequency case, where the unblocked linear systems have all data at the same frequency In particular, an explicit relation between the system matrix of the unblocked linear systems and that of their corresponding blocked systems is derived Based on this relation, it is shown that the blocked systems are zero free if and only if the related unblocked systems are zero free Furthermore, it is illustrated that square systems have zeros generically, ie for generic parameter matrices, and the corresponding kernel is of dimension one With the help of the results obtained for the single frequency case, we then identify a situation in which the blocked systems can be zero free

Analysis of target velocity and position estimation via doppler-shift measurements

Abstract: This paper outlines the problem of doppler-based target position and velocity estimation using a sensor network. The minimum number of doppler shift measurements at distinct generic sensor positions to have a finite number of solutions, and later, a unique solution for the unknown target position and velocity is stated analytically, for the case when no measurement noise is present. Furthermore, we study the same problem where not only doppler shift measurements are collected, but also other types of measurements are available, e.g. bearing or distance to the target from each of the sensors. Subsequently, allowing nonzero measurement noise, we present an optimization method to estimate the position and the velocity of the target. An illustrative example is presented to show the validity of the analysis and the performance of the estimation method proposed. Some concluding remarks and future work directions are presented in the end.

Range-only sensing for formation shape control and easy sensor network localization

Abstract: This paper is concerned with two related problems using graph theoretic methods for their solution. The central contribution to formation control is to demonstrate that if each agent senses a modest number of additional distances beyond those which are actively being controlled, and with limited message passing between neighboring agents, each agent can infer the relative positions of its neighbors in its own coordinate basis. Additional effort to related this to the basis used for viewing its controls may be required. Each agent can thus apply the known algorithms for formation shape control based on distance preservation, without needing to actually sense bearings. The contribution to sensor network localization is to identify circumstances in which the complexity is effectively linear in the number of nodes.

A Statistically Modelling Method for Performance Limits in Sensor Localization

Abstract: In this paper, we study performance limits of sensor localization from a novel perspective. Specifically, we consider the Cramer-Rao Lower Bound (CRLB) in single-hop sensor localization using measurements from received signal strength (RSS), time of arrival (TOA) and bearing, respectively, but differently from the existing work, we statistically analyze the trace of the associated CRLB matrix (i.e. as a scalar metric for performance limits of sensor localization) by assuming anchor locations are random. By the Central Limit Theorems for U -statistics, we show that as the number of the anchors increases, this scalar metric is asymptotically normal in the RSS/bearing case, and converges to a random variable which is an affine transformation of a chi-square random variable of d egree 2 in the TOA case. Moreover, we provide formulas quantitatively describing the relations hip among the mean and standard deviation of the scalar metric, the number of the anchors, the parameters of communication channels, the noise statistics in measurements and the spatial distribution of the anchors. These formulas, though asymptotic in the number of the anchors, in many cases turn out to be remarkably accurate in predicting performance limits, even if the number is small. Simulations are carried out to confirm our results.

On the Information Propagation in Mobile Ad-Hoc Networks Using Epidemic Routing

Abstract: In this paper, we study information propagation in a 2D mobile ad-hoc network, where mobile nodes are randomly and independently distributed on a torus following a homogeneous Poisson process with a given density. Nodes in the network move following a random direction mobility model. A piece of information is broadcast from a source node to all other nodes in the network, using a Susceptible-Infectious-Recovered (SIR) epidemic routing protocol. A distinguishing feature of the SIR algorithm, which leverages the mobility of mobile users, is that a relay node carries and forwards a piece of information for a specified amount of time. We first propose a metric fundamentally characterizing the information propagation in mobile ad-hoc networks. Then analytical results are derived for the probability that a non-zero fraction of nodes receive the information in the limit of large network size and for the expected fraction of nodes that receive the information. The analytical results are verified using simulations. The research provides useful insights on the design of mobile ad-hoc networks.

Request-based gossiping

Abstract: By the distributed averaging problem is meant the problem of computing the average value of a set of numbers possessed by the agents in a distributed network using only communication between neighboring agents. Gossiping is a well-known approach to the problem which seeks to iteratively arrive at a solution by allowing each agent to interchange information with at most one neighbor at each iterative step. Crafting a gossiping protocol which accomplishes this is challenging because gossiping is an inherently collaborative process which can lead to deadlock unless careful precautions are taken to ensure that it does not. In this paper we present three gossiping protocols. We show by example that the first can deadlock. While the second cannot, it requires a degree of network-wide coordination which may not be possible to secure in some applications. The third protocol uses only local information, is guaranteed to avoid deadlock, and requires fewer transmissions per iteration than standard broadcast-based distributed averaging protocols.

Multi-agent rigid formations: A study of robustness to the loss of multiple agents

Abstract: In this paper we study the robustness of information architectures to control a formation of autonomous agents. If agents are expected to work in hazardous environments like battle-fields, the formations are prone to multiple agent/link loss. Due to the higher severity of agent loss than link loss, the main contribution of this paper is to propose information architectures for shape-controlled multi-agent formations, which are robust against the loss of multiple agents. A formation is said to be rigid if by actively maintaining a designated set of inter-agent distances, the formation preserves its shape. We will use the rigidity theory to formalize the robust architecture problem. In particular we study the properties of formation graphs which remain rigid after the loss of any set of up to k−1 vertices. Such a graph is called k-vertex rigid. We provide a set of distinct necessary and sufficient conditions for these graphs. We then show that 3-vertex rigidity is the highest possible robustness one can achieve by just adding a small number of edges to a minimally rigid graph, i.e. retention of rigidity given the loss of 3 or more agents of a formation requires many more inter-agent distances to be specified than when maintaining rigidity with no, one or two agent losses. Based on this result, we further focus on 3-vertex rigid graphs and characterize a class of information architectures (with minimum number of control links) which are robust against the loss of up to two agents.

On the performance limit of sensor localization

Abstract: In this paper, we analyze the performance limit of sensor localization from a novel perspective. We consider distance-based single-hop sensor localization with noisy distance measurements by Received Signal Strength (RSS). Differently from the existing studies, the anchors are assumed to be randomly deployed, with the result that the trace of the associated Cramer-Rao Lower Bound (CRLB) matrix becomes a random variable. We adopt this random variable as a scalar metric for the performance limit and then focus on its statistical attributes. By the Central Limit Theorems for U-statistics, we show that as the number of anchors goes to infinity, this scalar metric is asymptotically normal. In addition, we provide the quantitative relationship among the mean, the standard deviation, the number of anchors, parameters of communication channels and the distribution of the anchors. Extensive simulations are carried out to confirm the theoretical results. On the one hand, our study reveals some fundamental features of sensor localization; on the other hand, the conclusions we draw can in turn guide us in the design of wireless sensor networks.

On the k-hop partial connectivity in finite wireless multi-hop networks

Abstract: In this paper, we consider wireless multi-hop networks with a finite number of (ordinary) nodes randomly deployed in a given 2D area, and a finite number of gateways (infrastructure nodes) deterministically placed in the same area. We study the connectivity between the ordinary nodes and the gateways. In real applications, it is often desirable to limit the maximum number of hops between the ordinary nodes and the gateways in order to provide reliable services. On the other hand, requiring every ordinary node to be connected to at least one gateway imposes strong requirement on transmission range/power or the number of gateways. Therefore it is beneficial to allow a small fraction of ordinary nodes to be disconnected from the gateways so that the network is only partially connected. Based on the above two considerations, we propose the concept of k-hop partial connectivity, which is the fraction of ordinary nodes that are connected to at least one gateway in at most k hops. Analytical results are provided characterizing the k-hop partial connectivity. The research provides useful guidelines on the design of wireless multi-hop networks.

Consensus over Random Graph Processes: Network Borel-Cantelli Lemmas for Almost Sure Convergence

Abstract: Distributed consensus computation over random graph processes is considered. The random graph process is defined as a sequence of random variables which take values from the set of all possible digraphs over the node set. At each time step, every node updates its state based on a Bernoulli trial, independent in time and among different nodes: either averaging among the neighbor set generated by the random graph, or sticking with its current state. Connectivity-independence and arc-independence are introduced to capture the fundamental influence of the random graphs on the consensus convergence. Necessary and/or sufficient conditions are presented on the success probabilities of the Bernoulli trials for the network to reach a global almost sure consensus, with some sharp threshold established revealing a consensus zero-one law. Convergence rates are established by lower and upper bounds of the $\epsilon$-computation time. We also generalize the concepts of connectivity/arc independence to their analogues from the $*$-mixing point of view, so that our results apply to a very wide class of graphical models, including the majority of random graph models in the literature, e.g., Erdős-Renyi, gossiping, and Markovian random graphs. We show that under $*$-mixing, our convergence analysis continues to hold and the corresponding almost sure consensus conditions are established. Finally, we further investigate almost sure finite-time convergence of random gossiping algorithms, and prove that the Bernoulli trials play a key role in ensuring finite-time convergence. These results add to the understanding of the interplay between random graphs, random computations, and convergence probability for distributed information processing.

A generic bias-correction method with application to scan-based localization

Abstract: In previous work a method was proposed to determine the bias in localization algorithms using range or bearing data. In this paper the method is extended to be more generic; in particular, different types of measurement data are permitted, and there may be more measurements than there are variables to estimate. The method combines the Taylor series and Jacobian matrices to determine the bias, and leads to an easily calculated analytical bias expression, despite the general unavailability of analytic expressions for the solution of most localization problems. The method is used to estimate the bias in scan-based localization. Monte Carlo simulation results verify the performance of the proposed method in this context.

Analytical Bounds on the Critical Density for Percolation in Wireless Multi-Hop Networks

Abstract: In this paper we develop analytical bounds on the critical density for percolation in wireless multi- hop networks, but in contrast to other studies, under a random connection model and with nodes Poissonly distributed in the plane $\mathbb{R}^2$. The establishment of a direct connection between any two nodes follows a random connection model satisfying some intuitively reasonable conditions, i.e. rotational and translational invariance, non- increasing monotonicity and integral boundedness. It is well known that under the above network model and connection model there exists a critical density below which almost surely a fixed but arbitrary node is connected (via single or multi-hop path) to finite number of other nodes only, and above which the node is connected to an infinite number of other nodes with a positive probability. In this paper we investigate the bounds on the critical density. The result is compared with the existing results under a specific connection model, i.e. the unit disk communication model, and it is shown that our method generates bounds close to the known ones. The result provides valuable insight into the design of large- scale wireless multi-hop networks.