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Workshop on Swarming in Natural and Engineered Systems Napa Valley, California

August 3-4, 2005 Abstracts: Designing a Distributed Intelligence: Are Fish Schools an Appropriate Model? ..............................3 Julia K. Parrish Analysis of Vicsek’s System with Integer-Valued Measurement Delays………………………………4 A. S. Morse Teamwork vs. congestion: on the role of scale in multi-agent mobile systems……………………….5 Emilio Frazzoli A tunable algorithm for collective decision-making by ant colonies…………………………………....6 Stephen Pratt, David J.T. Sumpter The Road to Catastrophe: Stability and Collapse in 2D Driven Particle Systems…………………....7 A. Bertozzi, M. D'Orsogna, Y. Chuang, L. Chayes Swarm Collaborative Intelligence: From Networked Control to Trust in MANET……………………..8 John S. Baras Programming Collaborative Behavior and Pattern Formation in Bacterial Communities…………....9 Ron Weiss Hierarchical abstractions for robotic swarms…………………………………………………………..…10 Calin Belta Controlling Sensor Network Deployment for Coverage and Connectivity…………………………….11 Gaurav S. Sukhatme Stable Agent Distributions……………………………………………………………………………….…12 Kevin M. Passino Distributed Receding Horizon Control of Spatially Invariant Systems ……………………………..…13 Nader Motee, Ali Jadbabaie Geodesic Control Laws for Distributed Velocity Alignment in Kinematic Agents………………….…14 Nima Moshtagh, Ali Jadbabaie Temporal Logic Motion Planning for Mobile Robots……………………………………………….……15 Georgios E. Fainekos, George Pappas Robust Distributed Localization and Covariance Estimation……………………………………….…..16 David Moore, D. Rus Adaptation and Learning at All Levels (AL^2) in Intelligent Robot Teams…………………………….17 Rafael Fierro, Dean Hougen, and Sesh Commuri Decentralized Estimation and Control of Swarm Formation Statistics………………………………...18 Peng Yang, Randy Freeman, Kevin Lynch

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Stability of discrete-time Flocks with Time Delays …………………………………………………….19 Herbert G. Tanner, Dimitrios Christodoulakis Coordinated Patterns of Moving Particles on Simple Orbits………………………………………….20 Fumin Zhang, Derek Paley, Naomi Leonard, Rodolphe Sepulchre Getting even: Regulation of ant worker allocation……………………………………………………..21 Frederick R. Adler The Statistical Dynamics of Programmed Robotic Self-Assembly……………………………………22 Eric Klavins Dynamic systems over random networks……………………………………………………………….23 Mehran Mesbahi Information and Control Architectures for Formation Maintenance…………………………………..24 Brian D. O. Anderson Resolving individual-level behaviors underlying distinct group search strategies in fish…………..25 Daniel Grunbaum Spatial Patterns in Cooperation and Conflict……………………………………………………………26 Eric Justh, P. S. Krishnaprasad What Can Higher Order Laplacians Do for Networked Control Systems? ....................................27 Abubakr Muhammad, Magnus Egerstedt Formation Control with Virtual Leaders and Reduced Communications…………………………….28 Eyad H. Abed A One-Dimensional Model for Fish Schooling………………………………………………………….29 Jeff Moehlis Group Behavior in Biology………………………………………………………………………………..30 David K. Skelly

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Designing a Distributed Intelligence: Are Fish Schools an Appropriate Model?

Julia K. Parrish

University of Washington

Animal aggregation is a pervasive phenomenon. There are a range of group types, from passive aggregations of organisms brought together by simultaneous attraction to a limiting source, to active congregation in which the attraction is social and the group has structure and function.

Within congregations, organisms can be related – the social insects are ultimate examples – or unrelated: e.g., schools of pelagic mass-spawning fish. Regardless of relatedness, individual members must make decisions about when and how to respond based on a complex combination of internal and external state.

There are two fundamental questions biologists ask about animal groups: Why do they do it, and How do they do it? The former necessitates an evolutionary framework in which the individual organism makes ‘selfish’ decisions to optimize long-term rewards (fitness, in evolutionary terms), potentially incurring short-term costs. The latter supposes a degree of cooperation and coordination among group members – an agreed upon rule set. Thus, there is a dynamic between the competitive nature of selfish individuals and the cooperative necessities of coordinated group activities.

In fish, over half the 20,000 known species school at some point in their life history. Like any organism, fish engage in a broad range of behaviors which allow them to negotiate: (1) moment-by-moment change in their immediate environment, and (2) ultimate change in their fitness. Behavioral flexibility - the ability to alter latency to, or even type of, response - allows individuals to innovate behavioral pathways in the face of new situations. Innovation is also a function of the degree to which an organism can decipher useful information in a novel situation (that is, perception versus reality), and the degree to which the organism retains that information for use the next time (that is, learning). In schools, individuality and behavioral flexibility may be constrained by the actions of neighbors. Physical constraints are imposed by the group architecture - an individual simply may not be able to move into an occupied location. Social constraints are imposed by the collective memory of the group - a function of group size and turnover, information transfer, and individual learning. Thus, behavioral choices at the individual level - constrained by the actions and architecture of the group - collectively become the group’s response. Group behavior, in turn, sums in some non-linear fashion to population-level response.

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Analysis of Vicsek’s System with Integer-Valued Measurement Delays1

A. S. Morse

Yale University

Current interest in cooperative control of groups of mobile autonomous agents has led to the rapid increase in the application of graph theoretic ideas together with more familiar dynamical systems concepts to problems of analyzing and synthesizing a variety of desired group behaviors such as maintaining a formation, swarming, rendezvousing, or reaching a consensus. While this in-depth assault on group coordination using a combination of graph theory and system theory is in its early stages, it is likely to significantly expand in the years to come. One line of research which “graphically” illustrates the combined use of these concepts, is the recent theoretical work by a number of individuals which successfully explains the heading synchronization phenomenon observed in simulation by Vicsek, Reynolds and others more than a decade ago. Vicsek and coauthors consider a simple discrete-time model consisting of n autonomous agents or particles all moving in the plane with the same speed but with different headings. Each agent’s heading is updated using a local rule based on the average of its own heading plus the current headings of its “neighbors.” Agent i’s neighbors at time t, are those agents which are either in or on a circle of pre-specified radius ri centered at agent i’s current position. In their paper, Vicsek et al. provide a variety of interesting simulation results which demonstrate that the nearest neighbor rule they are studying can cause all agents to eventually move in the same direction despite the absence of centralized coordination and despite the fact that each agent’s set of nearest neighbors can change with time. A theoretical explanation for this observed behavior has recently been given. The explanation exploits ideas from graph theory and from the theory of non-homogeneous Markov chains. With the benefit of hind-sight it is now reasonably clear that it is more the graph theory than the Markov chains which will prove key as this line of research advances. The aim of this talk is to outline a problem which illustrates this point.

The problem is to analyze a modified version of the Vicsek system in which integer-valued delays occur in sensing the values of headings which are available to agents. More precisely we suppose that at each time t∈{0,1,2,...}, the value of neighboring agent j’s headings which agent i senses is θj (t - dij(t)) where dij (t) is a delay whose value at t is some integer between 0 and mi - 1; here mi is a pre-specified positive integer. While well established principles of feedback control suggest would that delays should be dealt with using dynamic compensation, in this talk we will consider the situation in which the delayed value of agent j’s heading sensed by agent i at time t is the value which will be used in the heading update law for agent i. Analysis of this system is carried out in much the same way as in the delay-free case. For example, just as in the delay-free case, graphs play a central role in the modeling process. While in the delay-free case graphs represent existing spacial relationships, in the version of the problem under consideration here, both temporal and spacial relationships are captured by the graphs. This leads to the in study of several specially structured families of directed graphs appropriate to the problem at hand, including the notion of a “hierarchical graph” which is of interest in its own right. 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 explain how convergence to a common heading is achieved.

1This work was done in collaboration with M. Cao {Yale University} and B. D. O. Anderson

{National Australian University}.

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Teamwork vs. congestion: on the role of scale in multi-agent mobile systems

Emilio Frazzoli

University of California, Los Angeles

A very active research area today addresses coordination of several robots: robotic teams and large-scale swarms are being considered for a broad class of applications, ranging from environmental monitoring to national security. While these concepts are undoubtedly fascinating, it is not clear what the benefits of scale are in such man-made systems. In this talk, recent work will be presented on two related classes of problems involving the coordination of possibly large numbers of robots. The focus of this work is to investigate the relation between the number of robots and the collective performance; as an additional outcome, we design coordination algorithms with provable performance guarantees.

The first part of the talk will address the following problem. Consider a team of robots free to move within a convex and bounded region in the plane. Assume that "service requests" are generated within the region by a Poisson point process; a service request is fulfilled when one of the robots visits the corresponding point. We present an algorithm that provably minimizes the expected waiting time between the issuance and fulfillment of a service request, in the asymptotic cases of light and heavy load (i.e., when the service requests are issued very rarely or very often). The proposed algorithm is spatially decentralized, and combines tools and results from geometric optimization, combinatorial optimization, and nonlinear control theory. In particular, we will show that the performance of the team increases at least quadratically with the number of robots in the heavy-load case. This provides a strong motivation for large-scale robotic networks.

The second part of the talk will address the other face of the coin, showing how traffic congestion imposes severe limitations on the performance of large multi-robot systems. More specifically, we will address the following basic problem: Given n robots and n origin-destination pairs in the plane, what is the minimum time needed to transfer each robot from its origin to its destination, avoiding conflicts with other robots? The environment is free of obstacles and a conflict occurs when the distance between any two robots is smaller than a velocity-dependent safety distance. We will show that, even in the case in which the size of the robots vanishes as n increases, the transfer requires Theta(n^(1/2)) time in the best possible case (i.e., the average speed of the robots decreases at least as 1/n^(1/2)). Moreover, we will present an algorithm that solves the problem for a random choice of origin/destination points in O(n^(1/2)) time, with high probability.

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A tunable algorithm for collective decision-making by ant colonies

Stephen Pratt, David J.T. Sumpter

Princeton University

The decision-making mechanisms of social insects must not only integrate colony labor without central control, but also adaptively tune colony behavior to a variable environment. We have found that house-hunting colonies of the ant Temnothorax curvispinosus can adjust the parameters of a decision algorithm to emphasize either speed or accuracy, depending on the urgency with which they need to find a new home. The colony’s challenge is to choose the best among several potential new homes, even when few of the scouts who organize the move assess more than one of the options. The ants’ solution is a strategy of graded commitment to each potential home. On finding a site, a scout proceeds from independent assessment, to slow recruitment of fellow scouts, to rapid transport of the passive bulk of the colony. Assessment duration varies inversely with site quality, and the switch from slow to fast recruitment requires that a quorum of ants first be summoned to the site. These rules generate a collective decision, by creating and amplifying differential population growth rates among sites. They also allow adaptive tuning of the algorithm according to emigration urgency. By lengthening assessment duration and raising their quorum size, the ants cut emigration time significantly, but at the cost of increasing the likelihood that the colony will split between alternative new sites of different quality, rather than moving unanimously into the best one. These findings show how a single collective algorithm, through adaptive tuning of a few parameters, can be used to solve distinct problems.

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The Road to Catastrophe: Stability and Collapse in 2D Driven Particle Systems

Andrea Bertozzi, Maria D'Orsogna, Yao-Li Chuang, Lincoln Chayes University of California, Los Angeles

Self-organization of driven particle systems is an emerging research area that spans many fields, from complex biological systems to cooperative control of multiple unmanned vehicles. There is a growing interest in developing a quantitative theoretical understanding of such systems, first to understand ones that arise in nature and then to use this knowledge as a building block for the design of artificial systems. In this paper we focus on a class of models in which individuals have inherent self-propulsion (be they mechanical or biological motors) and pairwise interaction between members of the group with simple attractive and repulsive components. We classify different morphologies of organization based on parameter regimes of the model. Although many specific cases have been studied in the recent literature, and in some studies specific phase transitions have been observed, a complete picture with predictive design capabilities has been lacking. Here, we present a coherent theory, based on fundamental statistical mechanics, for all possible phases of motion.

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Swarm Collaborative Intelligence: From Networked Control to Trust in MANET

John S. Baras

Institute for Systems Research University of Maryland College Park

We consider two seemingly unrelated problems: collaborative vehicle control and collaborative behavior in infrastructureless networks (mobile ad hoc networks). The first problem addresses the development and analysis of distributed autonomous coordination of vehicles that are trying to accomplish a specific goal (objective), while at the same time avoiding collisions with each other, avoiding obstacles and avoiding moving opponents that can destroy them. We develop a formulation as a Markov Random Field control problem and develop several algorithms for achieving coordination based on local Interactions alone. Distributed self-organization of autonomous vehicles is achieved through a parallel implementation of Gibbs samplers-based simulated annealing. The impact of the Gibbs potential functions on the convergence speed is also investigated, which provides insight into the design of these functions. We demonstrate convergence of the algorithms in specific instances and demonstrate avoidance of local optima. We discuss the challenges posed by the dynamic nature of the underlying graph in the convergence analysis of the full parallel algorithm. The second problem addresses how collaboration and self-organization can be induced, analyzed and evaluated in MANET, and more general autonomic networks. We use distributed trust establishment and maintenance in MANET as a specific emergent property. We describe a new framework within which this problem can be formulated and analyzed. We develop and analyze distributed methods that use local interactions. We also develop and analyze a cooperative game framework and demonstrate how collaboration can be induced. We show that negotiation between the mobile agents is an important component for achieving collaboration within this framework. We next develop a model for establishing, propagating and managing trust within a MANET. We show that MRF methods lead to a statistical trust evaluation rule, prove its convergence and investigate its characteristics when the system is at the steady state. Our investigation gives several conclusions for the design of trust evaluation rules, some of which are quite unexpected. We show that such trust mechanisms can also establish collaboration, even without negotiations between the mobile agents. We investigate both the dynamics of games as well as of trust propagation as a means for quantifying the degree of collaboration achieved among the agents and of the speed by which this collaboration spreads in a large part of the network agents. In the context of our research reported here, for both problems, we have drawn inspiration from analytical methods used in statistical mechanics investigations of the Ising model and spin glasses. These include the existence and investigation of phenomena analogous to phase transitions.

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Programming Collaborative Behavior and Pattern Formation in Bacterial Communities

Ron Weiss

Princeton University

Cell-cell communication is a pervasive activity common to both single cell and multicellular organisms, and is used in coordinating cell behavior for a variety of tasks ranging from quorum sensing in bacteria to embryogenesis in mammalian cells. Engineering synthetic multicellular systems to exhibit desired functions will improve our quantitative understanding of naturally occurring cell-cell communication, and will also have biotechnology applications in areas such as biosensing, biomaterial fabrication, and tissue engineering. Here we will present theoretical and experimental results from synthetic systems implemented in bacteria. We will begin by describing how information flows through synthetic transcriptional cascades in single cells by examining noise propagation, ultrasensitivity, and impedance matching. Understanding these issues is critical for the analysis and de novo engineering of complex gene networks We will then discuss several synthetic multicellular systems that have been programmed to exhibit unique coordinated cell behavior. The first system is the pulse generator where sender cells communicate to nearby receiver cells, which then respond with a transient burst of gene expression whose amplitude and duration depends on the distance from the senders. In the second system, receiver cells have been engineered to respond to cell-cell communication signals only within prespecified ranges. We will demonstrate how this system can be used to generate a variety of interesting spatial patterns. In the third system, cells have been engineered to "play" Conway's Game of Life, where cells live or die based on the density of their neighbors. This system exhibits complex global emergent behavior that arises from the interaction of cells based on simple local rules. In this talk, we will correlate experimental results from observing the behavior of these systems with our quantitative spatiotemporal models.

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Hierarchical abstractions for robotic swarms Calin Belta

Boston University We develop a framework for planning and control of arbitrarily large groups of

fully actuated robots with polyhedral velocity bounds moving in polygonal environments with polygonal obstacles. Central to our approach is the notation of abstraction, which is based on a special type of equivalence relation, called bisimulation. At the first level of abstraction, this is used to correctly aggregate the high dimensional control system of the swarm into a small dimensional control system capturing its essential features. At the second level, bisimulation relations are used to reduce the problem of controlling the set of essential features to a model checking problem. In the obtained hierarchical framework, high level specifications given in natural language such as linear temporal logic formulas over linear predicates in the essential features are automatically mapped to provably correct robot control laws. Example of such high level specifications include "always avoid obstacles", "always stay inside the environment", "always maintain a pre-defined minimum pairwise distance between the robots", "eventually reach a given position and shape", "maintain given position and shape before reaching desired position and shape", etc.

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Controlling Sensor Network Deployment for Coverage and Connectivity Gaurav S. Sukhatme

University of Southern California

Topology control, primarily concerned with ensuring desired levels of coverage and connectivity, is a vital self-configuration operation in unattended sensor networks. We present a classification of sensor network topologies and discuss their implications for topology control. Our main contribution is a unifying framework that forms a basis for tunable topology control in all classes of topologies. It is based on a simple, local condition of ensuring a neighbor in every sector (of given size theta) of each node's communication range. We present analysis to establish that varying the single parameter theta can indeed provide a wide range of coverage and connectivity tradeoffs. For specific values of theta, we show that the Neighbor-Every-Theta (NET) condition guarantees various proximity graphs such as the relative neighborhood graph. The problem of maximizing coverage given such a condition is also addressed. Algorithms for controlled deployment are presented to demonstrate how the NET condition can be integrated with positioning of nodes for tunable topology control in swarms of robots.

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Stable Agent Distributions

Kevin M. Passino Ohio State University

Local agent sensing, coordination, and motion requirements for stable spatial

agent distributions are derived. Applications to cooperative control of autonomous air vehicles and analysis of honey bee social foraging are discussed. Our initiation of an effort to develop an experimentally validated model of honey bee swarm flight is briefly overviewed.

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Distributed Receding Horizon Control of Spatially Invariant Systems

Nader Motee and Ali Jadbabaie University of Pennsylvania

The class of spatially invariant systems covers all systems governed by PDEs with constant coefficients, lattices of identical vehicles (e.g. UAVs and UGVs), flow control problems, microelectromechanical systems (MEMS), and any other distributed dynamical system in which the spatial variable is indexed over a locally compact group G (e.g. G ∈ R (Real numbers), Z (integer numbers), Zp (integers module p), ∂D (unit disk), G is called the spatial domain). Recently Bamieh et al. have shown that optimal (LQR, H2, H∞) controllers of the unconstraint spatially invariant systems to some extent inherit the structure of the system itself. Specifically, they show that the kernel of the optimal control decays exponentially on spatial domain, making the control distributed. This offers a natural way to reduce the complexity of the design from an infinite--dimensional optimal control problem to finitely many, tractable finite-dimensional sub-problems by decomposition on the spatial domain. In this poster, it is shown that a similar result holds for Receding Horizon controllers. It is shown that the kernel of the optimal solution as well as the corresponding Lagrange multipliers decay exponentially in the spatial domain, implying that the receding horizon controller has an inherent degree of decentralization, similar to infinite horizon problems. Some preliminary results for the constraint Receding Horizon control of Spatially Invariant Systems is also presented.

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Geodesic Control Laws for Distributed Velocity Alignment in Kinematic Agents

Nima Moshtagh, Ali Jadbabaie University of Pennsylvania

We study the problem of flocking and coordination of a group of kinematic nonholonomic agents in 2 and 3 dimensions. By analyzing the velocity vectors of agents on a circle (for planar motion) or sphere (for 3D motion), we develop a series of geodesic control laws that minimize a certain misalignment potential and result in velocity alignment. The proposed control laws are distributed and will provably result in flocking when the underlying proximity graph which represents the neighborhood relation among agents is connected. We further show that flocking is possible even when the topology of the proximity changes over time, so long as a weak notion of connectivity over time is maintained. Furthermore, we develop a vision based control law that does not rely on heading measurements, but only requires measurement of bearing, optical flow and time to collision, all of which can be efficiently measured. Finally, some interesting connections between the proposed control law and the Kuramoto model of coupled nonlinear oscillators will also be presented.

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Temporal Logic Motion Planning for Mobile Robots

Georgios E. Fainekos, George Pappas

University of Pennsylvania

In our poster, we will present the problem of robot motion planning for temporally extended goals expressible in temporal logics. Temporal logics naturally express traditional robot specifications such as reaching a goal or avoiding an obstacle, but also more sophisticated specifications such as sequencing, coverage, or temporal ordering of different tasks. In order to provide computational solutions to this problem, we first construct discrete abstractions of robot motion based on some environmental decomposition. We then generate discrete plans satisfying the temporal logic formula using powerful model checking tools, and finally translate the discrete plans to continuous trajectories using hybrid control. Critical to our approach is providing formal guarantees ensuring that if the discrete plan satisfies the temporal logic formula, then the continuous motion also satisfies the exact same formula.

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Robust Distributed Localization and Covariance Estimation

David Moore

MIT

We describe a general localization algorithm for a network of mobile sensors with an emphasis on the case of range-only localization with noisy distance measurements. The primary characteristic of the approach is robustness against errors such as those caused by flip ambiguities, which can often corrupt localization results. This robustness is achieved by constructing local subgraphs using the novel principle of robust quadrilaterals. In addition, the algorithm estimates the covariance of the localized positions on-line, and operates in a fully distributed and scalable fashion that does not require beacons attached to a global coordinate system.

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Adaptation and Learning at All Levels (AL^2) in Intelligent Robot Teams

Rafael Fierro, Dean Hougen*, and Sesh Commuri*

Oklahoma State University, *University of Oklahoma

Swarms of robots are crucial to future information-gathering missions such as reconnaissance, surveillance, and battlefield assessment. In order to complete these missions successfully, these robots will need to:

• operate in hostile, unstructured, and unknown environments; • integrate information from disparate sensors in real time; • maintain high levels of reliability at all times; • function with variable autonomy; • perform intelligently alone, alongside humans, and in networks of

multiple teams; • coordinate, cooperate, and communicate with teammates; • be dynamically reconfigurable; • make efficient use of their resources; and • be acquired and maintained cost-effectively.

While component technologies aimed at realizing these needs have matured

in the past few years, the design of these sophisticated autonomous teams is not a matter of simple integration of previously developed systems. We propose to merge the strengths of hybrid control theory, machine learning theory, and distributed sensing and communication in a unified framework. This framework is instantiated in a new architecture for robot swarms called "Adaptation and Learning at All Levels (AL^2)". Adaptation and learning have been shown to be an effective way to enable robots to carry out complex behaviors that are difficult to directly program. Adaptation and learning with multiple robots may be even more crucial but also more difficult. Because of the hierarchical nature of the AL^2 architecture, adaptation and learning at one level can benefit, rather than confuse, adaptation and learning other levels. At each level of the hierarchy, each activity unit understands and incorporates the adaptation and learning capabilities of the next lower level, both those directly found in individual sub-units and those resulting from the aggregation of units at recursively lower levels.

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Decentralized Estimation and Control of Swarm Formation Statistics Peng Yang, Randy Freeman, Kevin Lynch

Northwestern University

In a large robot formation, it may not be necessary to specify the position of

each robot. Instead, we can describe the position, orientation, and shape of a swarm by a set of summary statistics. These serve as an abstraction allowing high-level human control, and the statistics can be optimized for the task (e.g., surveillance and search and rescue). The problem: each robot must move based only on local information to achieve the desired global statistics.

Using information from nearby robots, each robot in a swarm implements a dynamic consensus estimator to estimate first- and second-order moment statistics describing the global formation. Under weak assumptions on the communication graph, we show that these estimates converge to the correct values. Each robot also implements a nonlinear control law that guarantees the global statistics are driven to their desired values (except from a set of measure zero) when the estimates are perfect. By small-gain arguments, the robots' coupled estimators and controllers lead to the desired global statistics.

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Stability of discrete-time Flocks with Time Delays

Herbert G. Tanner and Dimitrios Christodoulakis

University of New Mexico

We extend our recent work on discrete-time synchronization of multiple agents (communication related) time delays and include cohesion and separation forces in the discrete time model. Having previously shown that the stability of the velocity synchronization scheme is not affected by the time delays under mild assumptions on the communication protocol, we now introduce a special type of inter-agent potentials that are designed to preserve stability, irrespectively of the time delays. Due to the introduction of the new interaction terms, we perform the stability analysis in the Lyapunov framework, and we exploit our previous algebraic stability results. The combination yields a globally stable flocking model in discrete time that is robust to communication delays.

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Coordinated Patterns of Moving Particles on Simple Orbits

Fumin Zhang, Derek Paley, Naomi Leonard, Rodolphe Sepulchre

Princeton University

Control of the relative spacing between the nodes of a mobile sensor network is critical to maximize array gain. We show methods to establish coordinated patterns of identical particles moving in the plane at unit speed. Each particle converges to travel along a closed curve from its initial position within a compact set in the plane. If the closed curve is circular, all particles will have fixed relative phase. For a general simple closed curve, the curve length between each pair of the particles is fixed. The stabilizing feedbacks derive from Lyapunov functions that prove exponential stability and suggest almost global convergence properties. The development is strongly motivated by application to adaptive sampling in the ocean using autonomous underwater vehicles.

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Getting even: Regulation of ant worker allocation

Frederick R. Adler

University of Utah

Ants need to achieve a variety of tasks in an unpredictable environment, including collection or capture of food and deterring or fighting with competitors. Given their well-known lack of centralized control, we propose that simple algorithms to equalize worker numbers in different patches provides a robust way to achieve these tasks efficiently, although not quite optimally. In particular, we show how this approach may be used by seed harvester ants to harvest slowly-renewing seed resources, by army ants to capture fast-moving insect prey, and by one species in the genus Formica to protect aphid food sources by deterring and defeating a larger species in that same genus. Maintaining an even distribution of workers may provide one basis for the diversity and ecological importance of ants.

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The Statistical Dynamics of Programmed Robotic Self-Assembly

Eric Klavins

University of Washington

We have recently demonstrated the use of simple local rules to direct the self-

assembly and self-organization of simple robotic parts. The parts are not self-motive, but are instead mixed randomly to produce collisions between parts. Upon colliding, two parts use simple rules to decide whether to attach to each other or not based only on their internal states. Using a formal description of the simple rules based on graph re-write systems, we can analyze the dynamics of rule sets and we can synthesize rules sets that produce any desired final assembly. We have focused on the correctness of the approach using a non-deterministic model of the dynamics of our system. We now extend the approach to a stochastic setting. In particular, we model the assembly process in terms of statistical dynamics (the extension of statistical mechanics dealing with non-equilibrium systems) and arrive at a Markov Process that is a function of the rule set used to direct the self-assembly and the basic "reaction rates" between reachable components of the graph re-write system. This allows us to analyze the rate at which assembly occurs, to understand the "usefulness" of a given rule, and to formalize a design process for generating fast rule sets in this setting. We illustrate the approach with our programmable parts testbed using both simulation and experiment.

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Dynamic systems over random networks

Mehran Mesbahi

University of Washington

In this talk we first consider the agreement problem over Erdos/Renyi type

random networks. In these networks, the existence of an information channel between a pair of units at each time instance is probabilistic and independent of other channels. In this venue, we delineate on the probabilistic asymptotic agreement for the networked units via supermartingales and stochastic stability. The rate of convergence of random agreement protocols as it relates to the spectral distribution of adjacency and Laplacian matrices will also be discussed. We conclude the presentation with an outline of an analysis framework for networked dynamic systems that is based on random and quasi-random graphs and the probabilistic method.

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Information and Control Architectures for Formation Maintenance

Brian D. O. Anderson

Australian National University and National ICT Australia

A paradigm for maintaining the shape of a moving formation in two or three dimensions of a collection of agents, assumed initially to be point agents for convenience, is to postulate that certain agent pairs measure the distance between each other, and maintain that distance constant. When enough of the inter-agent distances are explicitly maintained, the remaining ones will be consequentially maintained.

When for a given agent pair, both agents are tasked with maintaining the distance, the formation is akin to a physical structure that for each agent pair with inter-agent distance to be explicitly maintained inserts a bar between those agents, equal in length to the distance required to be maintained between the agents. One can then ask when the resulting structure will be rigid, and rigidity of the structure will correspond to maintenance of the formation shape. This question can be answered in two or three dimensions using linear algebra; a certain matrix is required to have a certain rank. The entries of the matrix are obtained from the coordinates of the various agents for a particular position of the formation. (This is unfortunate, in that the matrix is not straightforwardly constructible simply from the set of specified lengths).

In the case of two dimensions, an entirely graphical criterion (in the form of a necessary and sufficient condition) is also available for checking rigidity, or at least generic rigidity, i.e. rigidity for almost all values of nominated lengths. The underlying graph, which is undirected, has vertices corresponding to the agents of the formation, and bars between those vertices corresponding to agent pairs between which the length is specifically maintained. No result necessary and sufficient condition like this is however available for three-dimensional formations/graphs.

An alternative control structure involving agent pairs has only one member of the pair given the task of maintaining the prescribed distance. In this case, the notion of a physical structure and the modeling using an undirected graph is inadequate. In fact, a directed graph, unsurprisingly, can provide a convenient model. More to the point, one can for two dimensions construct a test to check whether the information/control architecture will maintain the formation. There are two components to the test: rigidity as before (for the associated undirected graph), and a property called constraint consistency, which is a requirement that a particular agent not be given an inconsistent set of distance constraints to maintain. When both rigidity and constraint consistency are present, we say the formation is persistent. An attractive theoretical result is that a directed graph is persistent if and only if each graph of a set of subgraphs formed following a certain rule from the original graph, without regard for directions, is itself a rigid graph.

In three (and higher) dimensions, rigidity and constraint consistency, or persistency, is not quite enough. A further property, which we term structural persistency, is required. This has a very simple equivalence in three dimensions: no two vertices of the graph can simultaneously have three degrees of freedom, i.e. there cannot be two totally unconstrained leaders.

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Resolving individual-level behaviors underlying distinct group search strategies in fish.

Daniel Grunbaum

University of Washington

The difficulty of designing effective control algorithms for coordinated group activities by autonomous agents has motivated interest in possible analogies with herds, flocks, schools and other natural social groups. These are of interest because they appear to operate under analogous constraints on locomotion and information exchange but nonetheless perform certain functions well, such as coordinated movement, foraging, and predator avoidance. However, few tools are available to deduce underlying behavioral algorithms from observations of natural social groups. We have been investigating how emergent characteristics of fish schools vary in response to environmental conditions and physiological states of members. After introducing some biological perspectives on grouping, I will present evidence that, in at least some fish schools, individuals are capable of multiple modes of interacting with neighbors that result in distinct group characteristics. Furthermore, individuals within a group appear to have a consensus behavioral mode expressed at a given time, suggesting that direct or indirect cues of modes adopted by neighbors are intrinsic to schooling dynamics. Interactions with neighbors occur on at least two levels: collective choice of group characteristics, and movement responses to neighbors to implement that choice.

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Spatial Patterns in Cooperation and Conflict

Eric Justh and P. S. Krishnaprasad

University of Maryland

In prior work of the authors, feedback laws for the creation of spatial patterns of interacting particles were derived that suggest means to create swarms of unmanned aerial vehicles and other cooperating robotic systems. Understanding pairwise interactions using methods from geometric mechanics and symmetry principles was key to our approach. Additionally, inspiration from biological swarms was clearly discernible in such feedback laws. In this talk we investigate other spatial patterns that arise in nature in the setting of conflict and competition. We show that such patterns (in pairwise interactions) can also be understood using methods from geometric mechanics. Using a high-gain limit, we argue that a proposed feedback law is biologically plausible. We suggest some technological applications of these ideas to adversarial encounters.

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What Can Higher Order Laplacians Do for Networked Control Systems?

Abubakr Muhammad and Magnus Egerstedt

Georgia Institute of Technology

Higher order combinatorial Laplacians of simplicial complexes have a natural relationship to some basic structural properties of networked control systems. This relationship will be highlighted and it will be shown how the higher order Laplacians can be used both as modeling tools and as a basis for control (e.g. for coverage control in distributed sensor networks). This construction represents a generalization of the use of algebraic graph theory for addressing consensus and averaging problems, where the graph Laplacian corresponds to the zero order combinatorial Laplacian.

28

Formation Control with Virtual Leaders and Reduced Communications

Eyad H. Abed

University of Maryland

Two formation control designs are presented for flocking of a group of mobile autonomous agents. Both designs use virtual leader(s) and two different interactive forces. The absence of an attractive force between neighboring agents is a new feature of this approach, meant to help reduce sensing and energy requirements. The talk will also include a discussion of the effects of noise and parameter variations on the behavior of the formation.

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A One-Dimensional Model for Fish Schooling

Jeff Moehlis

University of California at Santa Barbara

We describe a one-dimensional analog of the three-dimensional individual-based model for fish schooling from Couzin et al. At each time step an individual fish decides which direction (left or right) to swim based on the positions and directions of the fish surrounding it. The model incorporates the biologically realistic behavioral rules that the highest priority of an individual is to maintain a minimum distance between themselves and others, but if this is not a concern, the individual is attracted to other individuals to avoid isolation and tries to align its orientation with neighbors. This is accomplished through the definition of three non-overlapping zones: a zone of repulsion, a zone of orientation, and a zone of attraction. If another fish is in the zone of repulsion, a repulsive force is turned on to avoid collisions. If there are no fish in the zone of repulsion, each individual fish feels alignment forces from fish in the zone of orientation, and attractive forces from fish in the zone of attraction. Sensory and movement error are modeled by a probability that each individual fish randomly switches directions at each time step.

Depending on parameters, the population for this model can form one or more groups which translate uniformly in one direction, or stationary groups in which each individual finds itself within the zone of repulsion of its neighbors and must therefore change its direction at each time step. These correspond to one-dimensional analogs of the parallel group and torus behaviors found for the original three-dimensional model of Couzin et al. We are currently developing equation-free analysis techniques for the one-dimensional model, including the development of an equation-free effective free energy description of the ``stick-slip'' dynamics which exist near the boundary in parameter space between uniform translation and stationary grouping. Here the stochastic component of the model can cause the population to switch between the two behaviors.

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Group Behavior in Biology

David K. Skelly

Yale University

Most living organisms are found within groups during some part of their life history and many live in groups for virtually their entire lives. Biologists considering group behavior have uncovered a variety of benefits including enhanced growth, development, survival and reproduction. Support for an evolutionary basis for group behavior is provided by evidence of its heritable basis. These studies also reveal variation in behavior within populations possibly reflecting variable benefits and costs of such behavior. Studies of the mechanistic underpinnings of groups suggest that the benefits of group behavior are often mediated through the sharing of information or by specialization among members such that group member performance exceeds that of a solitary individual. In many cases, this benefit is complemented by the physical advantages of being within a group. An individual in a selfish herd can better avoid a predator. Equally, groups can moderate the impact of harsh physical conditions. Studies of larval amphibians have provided important insights into the nature of group behavior, providing one of the first examples of kin recognition and suggesting that frequently observed aggregations may be comprised of kin groups. More recent work has uncovered evidence of both kin attraction and kin repulsion. As with many aspects of group behavior it appears that strong context dependence operates on the benefits and costs of being within a group. The pervasiveness of group behavior in living organisms in spite of context dependence implies well developed abilities to dynamically assess the environment and alter behavior accordingly.