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Interaction and topology in distributed multi-agent coordination
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Interaction and topology in distributed multi-agent coordination
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INTERACTION AND TOPOLOGY IN DISTRIBUTED MULTI-AGENT COORDINATION by Ryan K. Williams A Dissertation Presented to the FACULTY OF THE USC GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (ELECTRICAL ENGINEERING) August 2014 Copyright © 2014 Ryan K. Williams Dedication For Mom. Your Courage, Optimism, and Brilliance will Not be Forgotten. ii Acknowledgments This journey, like most, has been long, treacherous, and at times, tragic. Survival has been the work of many, to whom I owe a great debt. To my parents, Clay and Diane, for unwavering support, and for commitment far beyond the norm. To my brothers, Cole and Keith, for making my life fun with an insatiable desire to explore, in thought and in action, and for an unfailing will to help in times of need. To my grandmother, Margaret, for continued aid, and to my extended family for ongoing moral support. To my advisor, Gaurav Sukhatme, and the University of Southern California for encouraging and supporting a Ph.D. under circumstances that were far outside the lines. Finally to my friends, in particular Geordan, and for any unnamed donators or volunteers, I sincerely appreciate your support. iii Contents Dedication ii Acknowledgments iii List of Figures vii Abstract x 1 Introduction 1 1.1 Interaction and Topology . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Thesis Outline and Contributions . . . . . . . . . . . . . . . . . . . 4 1.3 Notational Conventions . . . . . . . . . . . . . . . . . . . . . . . . . 19 2 Elements from the Theories of Graphs and Systems 21 2.1 Graph Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.1 Basic Definitions . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.2 Algebraic Graph Theory . . . . . . . . . . . . . . . . . . . . 23 2.2 The Agreement Problem . . . . . . . . . . . . . . . . . . . . . . . . 26 3 Probabilistic Mapping and Tracking: A Motivating Example 31 3.1 Decentralized Probabilistic Mapping . . . . . . . . . . . . . . . . . 33 3.1.1 The Multi-Agent Markov Random Field . . . . . . . . . . . 33 3.1.2 Distributed Loopy Belief Propagation . . . . . . . . . . . . . 36 3.1.3 Simulation Results in an Environmental Dataset . . . . . . . 40 3.2 Tracking Probabilistic Level Curves . . . . . . . . . . . . . . . . . . 46 3.2.1 Stable Curve Tracking in the Plane . . . . . . . . . . . . . . 46 3.2.2 Localized Gradient and Hessian Estimation . . . . . . . . . 50 3.2.3 Mapping and Tracking Simulation Results . . . . . . . . . . 53 3.3 Implications of Interaction and Topology . . . . . . . . . . . . . . . 56 4 Spatial Interaction and Mobility-based Topology Control 58 4.1 Models of Spatial Interaction and Dynamics . . . . . . . . . . . . . 59 4.2 Spatial Interaction Under Topological Constraints . . . . . . . . . . 65 iv 4.2.1 Switching Model for Discriminative Interaction . . . . . . . 66 4.2.2 Constraint-Aware Mobility Control . . . . . . . . . . . . . . 74 4.2.3 Aggregation Under Max Degree Constraints . . . . . . . . . 82 4.2.4 Dispersion Under Min Degree Constraints . . . . . . . . . . 90 4.2.5 Topological Control Simulations . . . . . . . . . . . . . . . . 95 4.3 Non-Local Constraints and an Illustration through Flocking . . . . 98 4.3.1 Mobility for Constrained Flocking . . . . . . . . . . . . . . . 99 4.3.2 Coordinating Non-Local Constraints . . . . . . . . . . . . . 101 4.3.3 Constrained Flocking in Simulation . . . . . . . . . . . . . . 104 4.4 Estimating and Controlling Algebraic Connectivity . . . . . . . . . 105 4.4.1 Distributed Connectivity Estimation . . . . . . . . . . . . . 106 4.4.2 Connectivity Maximizing, Constraint-Aware Mobility . . . . 109 4.4.3 Simulated Connectivity Estimation and Control . . . . . . . 112 4.5 Route Swarm: A Case Study of Heterogenous Mobility . . . . . . . 114 4.5.1 Information Control Plane (ICP) . . . . . . . . . . . . . . . 117 4.5.2 The Route Swarm Heuristic . . . . . . . . . . . . . . . . . . 122 4.5.3 Physical Control Plane (PCP) . . . . . . . . . . . . . . . . . 125 4.5.4 Maintaining Flow Connectivity . . . . . . . . . . . . . . . . 127 4.5.5 Route Swarm Simulations . . . . . . . . . . . . . . . . . . . 135 5 Rigidity Theory: A Primer 138 5.1 Characterizing the Rigidity Property . . . . . . . . . . . . . . . . . 138 5.2 A Pebble Game for Evaluating Rigidity . . . . . . . . . . . . . . . . 146 6 Evaluating, Constructing, and Controlling Rigid Networks 150 6.1 An Asynchronous Decentralized Pebble Game . . . . . . . . . . . . 151 6.1.1 Leader Election for Decentralization . . . . . . . . . . . . . . 155 6.1.2 Inter-Agent Messaging . . . . . . . . . . . . . . . . . . . . . 159 6.1.3 Exploiting Structure Towards Parallelization . . . . . . . . . 162 6.1.4 Gossip-like Messaging for Parallelization . . . . . . . . . . . 165 6.1.5 A Rigidity Control Scenario . . . . . . . . . . . . . . . . . . 171 6.1.6 Contiki Implementation: Real-World Feasibility Results . . . 174 6.1.7 The Serialized Algorithm . . . . . . . . . . . . . . . . . . . . 175 6.1.8 The Parallelized Algorithm . . . . . . . . . . . . . . . . . . . 178 6.2 Constructing Optimally Rigid Networks . . . . . . . . . . . . . . . 185 6.2.1 Optimal Greedy Algorithm . . . . . . . . . . . . . . . . . . . 188 6.2.2 Sub-Optimal Algorithm . . . . . . . . . . . . . . . . . . . . 195 6.2.3 Relative Sensing: A Case Study . . . . . . . . . . . . . . . . 198 6.3 Distributed and Efficient Rigidity Control . . . . . . . . . . . . . . 204 6.3.1 Predicates for Rigidity Preservation . . . . . . . . . . . . . . 205 6.3.2 Local Conservatism and Extensions . . . . . . . . . . . . . . 211 6.3.3 Simulated Rigidity Control . . . . . . . . . . . . . . . . . . . 214 v 7 Conclusions and Future Work 217 7.1 Thesis Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217 7.2 Directions for Future Work . . . . . . . . . . . . . . . . . . . . . . . 221 7.3 Closing Thoughts . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230 Reference List 231 vi List of Figures 2.1 Example of a (connected) graph topology. . . . . . . . . . . . . . . 23 3.1 3× 3 grid-structured pairwise multi-agent Markov random field. . . 36 3.2 Generalized cluster graph for approximate inference overH 1 . . . . . 39 3.3 MODIS data used for measurement sampling in mapping simulations. 42 3.4 Three agent system performing exact and approximate inference. . . 43 3.5 Average cell-wise KL-Div and plume membership error rate. . . . . 44 3.6 Level curve model for tracking control design. . . . . . . . . . . . . 49 3.7 Model for gradient and Hessian estimation over probabilistic grids. . 52 3.8 Spatial behavior of a 30 agent system mapping and tracking proba- bilistic level curves. . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 3.9 Tracking behavior of a 30 agent system mapping and tracking prob- abilistic level curves. . . . . . . . . . . . . . . . . . . . . . . . . . . 55 3.10 Mapping performance of a 30 agent system mapping and tracking probabilistic level curves. . . . . . . . . . . . . . . . . . . . . . . . . 56 4.1 Agent communication, sensing, and spatial interaction model. . . . 63 4.2 Switching model for discriminative spatial interaction. . . . . . . . . 71 4.3 Inter-agent potentials for aggregation, collision avoidance, and link retention. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85 vii 4.4 Inter-agent potentials for dispersion, link retention, and link denial. 92 4.5 Degree-constrained aggregation simulation of an n = 10 agent system. 96 4.6 Degree-constrained dispersion simulations of an n = 10 agent system. 97 4.7 Node degree bounds and swarm size for aggregation and dispersion simulations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 4.8 Flocking simulation composed of stationary hubs and mobile agents. 104 4.9 Distributed inverse iteration versus power iteration. . . . . . . . . . 113 4.10 Aggregation simulation of a 15-agent system with connectivity max- imization under degree constraint. . . . . . . . . . . . . . . . . . . . 114 4.11 Leader-following simulation, withn = 10 and connectivity maximiza- tion under|N i |≤ 7. . . . . . . . . . . . . . . . . . . . . . . . . . . . 115 4.12 Example supergraph construction for a network with f = 2 flows, s = 4 static nodes, and m = 6 mobile robots. . . . . . . . . . . . . . 123 4.13 The flexible detachment maneuver for shifting flow redundancy in response to flow detachment commands. . . . . . . . . . . . . . . . 131 4.14 Network progression for the simulated route swarming. . . . . . . . 136 4.15 Flow utilities for the simulated route swarming. . . . . . . . . . . . 137 5.1 Example graphs demonstrating several embodiments of rigidity. . . 141 5.2 The Henneberg operations. . . . . . . . . . . . . . . . . . . . . . . . 144 5.3 A Henneberg sequence. . . . . . . . . . . . . . . . . . . . . . . . . . 145 5.4 An example of the pebble game for a rigid graph with n = 3. . . . . 147 6.1 Illustration of leader-based decentralization for n = 4 agents. . . . . 157 6.2 Illustration of pebble covering attempts for a quadrupling an edge. . 159 6.3 Illustration of a sequence of the endpoint expansion rule and the two incident edge rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . 164 viii 6.4 Block diagram of the parallelized pebble game algorithm. . . . . . . 165 6.5 Parallel messaging for a minimally rigid graph with n = 4. . . . . . 171 6.6 Rigidity control simulation for n = 9 mobile agents applying a dispersive objective with a rigidity maintenance constraint. . . . . . 172 6.7 Swarm size and edge set cardinality for the rigidity control simulation.173 6.8 Monte Carlo results from a Contiki networking environment demon- strating execution time and message complexity. . . . . . . . . . . . 175 6.9 A weighted graph and its optimal Laman subgraph. . . . . . . . . . 186 6.10 An illustration of the relationship between edge consideration, redun- dancy, and optimality. . . . . . . . . . . . . . . . . . . . . . . . . . 194 6.11 Rigid network composed of 7 vertexes running optimal rigidity algo- rithms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200 6.12 Monte Carlo analysis comparing optimal rigidity algorithms. . . . . 202 6.13 Mobile RSN preserving rigidity and minimizing theH 2 norm. . . . 203 6.14 An illustration of conservatism in the local redundancy rule. . . . . 213 6.15 Monte Carlo simulations of the local rigidity control rule and a consensus-based extension. . . . . . . . . . . . . . . . . . . . . . . . 214 6.16 Monte Carlo simulations demonstrating conservative behavior of the local rigidity rule. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216 ix Abstract Interconnected systems have become the recent focus of intense investigation, particularly in the context of autonomous coordination, yielding fundamental advantages in adaptability, scalability, and efficiency compared to single-agent solutions. In this thesis, we investigate the topological assumptions that underly distributed multi-agent coordination, i.e., those properties defining interaction between agents in a network. We focus specifically on the properties of network connectivity and graph rigidity, which exhibit strong influence on fundamental multi-agent behaviors, e.g., joint decision-making, cooperative estimation, formation control, and relative localization. These bases of coordination contribute to the construction of increasingly complex multi-agent systems, and in harnessing the underlying threads of topology, there is hope in solving the future challenges of coordination in a world of robotic ubiquity. Thus, this thesis aims to strike distributed autonomy at its core, by treating assumptions which render theoretical treatments feasible, but which leave implementation relegated to the laboratory. Motivated by a case study that illustrates the topological assumptions necessary to solve the probabilistic mapping and tracking problem, we extend the state of the art in mobility control by regulating topology through preemptive mobility, discriminating link addition and deletion to shape spatial interaction under topo- logical constraints. Adopting realistic models of proximity-limited coordination, x local controllers are constructed with discrete switching for link discrimination, and attract-repel potential fields which yield constraint satisfying motion. Our mobility scheme acts as a full generalization of classical swarm-like controllers, yielding decision-based, topology-driven coordination. When topological constraints are non-local, as is the case for both connectivity and rigidity, we illustrate how consensus-based decision-making can preserve preemption while maintaining the feasibility of topological constraints. To evaluate the specific constraint of net- work connectedness, we propose an inverse iteration algorithm that estimates the eigenpair associated with algebraic connectivity. Our solution is fully distributed, scalable, and it improves on the convergence rate issues of the state of the art. Finally, as a case study of heterogeneity, we introduce a hybrid architecture in which a robotic network is dynamically reconfigured to ensure high quality information flow between static nodes while preserving connectivity. In solving this problem, we propose components that couple connectivity-preserving robot-to-flow allocations, with distributed communication optimizing mobility; a heuristic we call Route Swarm. The second half of the thesis focuses on the rigidity property of a multi-agent network. We provide a decentralized algorithm that determines a spanning edge set defining the minimally rigid subcomponent of a graph. Leader election manages sequential execution, and an asynchronous messaging scheme preserves local agent interaction, as well as robustness to delays, failures, etc. We extend our decentral- ization by parallelizing rigidity evaluation, taking inspiration from gossip messaging, yielding significant improvements in execution time and communication. Exploiting leader election and edge cost metrics, we further demonstrate the discovery of the optimal rigid subgraph embedded in a graph, with a tuning parameter for control- ling the complexity-optimality tradeoff. To close the theoretical contributions, we xi demonstrate the integration of our spatial topology control, with our methods for evaluating network rigidity, to achieve rigidity control in a multi-agent team. Rules for link addition and deletion are proposed that preserve combinatorial rigidity by exploiting neighbor-wise communication and local examination of edge redundancy. All thesis propositions are validated in extensive simulations and Monte Carlo studies, across various relevant simulation platforms. Finally, the thesis concludes with a discussion of future concerns for topology in distributed robotics, including heterogeneity, asymmetry in interaction, integration with classical notions of robotic intelligence, and the topological underpinnings of intersecting objectives. xii Chapter 1 Introduction 1.1 Interaction and Topology There has been significant interest recently in the study of distributed multi-agent systems, coordinating to achieve a common objective. The ubiquity of advanced communication technologies coupled with the continued scaling of processing capa- bilities has generated intense interest in the analysis and control of networked systems, and particularly in systems of mobile agents. Specifically, networks of cooperating intelligent agents have become a natural evolution of centralization, yielding gains in efficiency, scalability, and robustness when compared to classical solutions [1]. Further, there exists significant biological, ecological, and social evidence indicating the inherent strength of systems composed of locally interacting agents, and given recent advancements, the reality of mimicking such systems has motivated extensive studies of local interactions that can give rise to intelligent coordination, e.g., emergent phenomena such as aggregation, cohesion, and school- ing [2–4]. Research interests in the context of multi-agent systems range from mutual localization [5,6] and map-building [7], to the development of decentral- ized coordination algorithms [8], the design of decentralized interaction control frameworks [9,10], and finally to issues such as provably fast data collection for sensor networks [11] or joint scheduling and power control in wireless networks [12]. Remarkable capabilities have been demonstrated in various application contexts, including environmental exploration [13–16], search and rescue operations [17], 1 sampling, tracking, and coverage [18–20], biologically-inspired [21] or cooperative systems [22], target tracking [23–25], agent agreement problems [26–28], distributed monitoring [29], dynamic network optimization [30], and surveillance [31,32]. Although definitions will very based on context, we consider the following broad notion of an agent: Definition 1.1.1 (Agent). An agent is an entity that senses or acts in its envi- ronment, consumes or conveys information, or interacts with other agents. Our typical agent of choice will be robotic and mobile in nature (e.g., quadrotors, underwater vehicles, etc.), however many developments will be applicable beyond robotic systems, such as in fixed sensor networks. Of course robotic systems offer a natural context for studying collaborative autonomy, particularly when the focus lies on mobility as in this work. In studying teams of agents, and particularly coordination, we will often refer to the multi-agent network, an interconnection of agents over some medium of collaboration, and the network topology, which describes the agent-to-agent interactions that are occurring at a given instant. Inter- action manifests in many ways for multi-agent systems; communication, sensing, indirectly through motion or environmental percepts, physical interaction (end manipulators), higher level notions such as packet routing, and through vehicle dynamics in physically coupled systems (distributed flight arrays, collective trans- port). Central to the success of distributed systems is the modeling of coordinated objectives and the derivation of control or decision schemes that rely only on limited local information and/or interaction. While the range of research challenges in characterizing collaboration is vast, an underlying thread is the reliance on topol- ogy, i.e., a necessity for interaction and information exchange to achieve cohesion. In studying the topological underpinnings of collaboration, we derive motivation taxonomically by characterizing the need for topology into two arenas: spatial and 2 information-driven collaborations. In spatial collaboration, we can cite aggregative or swarming behaviors, area coverage, flocking, leader-following, formation and shape control, and rendezvous as important objectives where cohesion is achieved in team motion. Informational collaboration which acts on sensed or communicated system state, includes as examples consensus and agreement, joint estimation and filtering, task assignment, localization, and cooperative inference; objectives which form the foundation of intelligence in collaboration. The above behaviors we believe act as building blocks which may form the basis of increasingly complex multi-agent systems and joint objectives. In study- ing these building blocks, the community strives to identify the theoretical and application-driven novelties for each, and specifically the topological shortcomings or assumptions, which join theory and application. This thesis lies exactly in the gap between theory and practice, in providing methodologies for evaluating and controlling topology in distributed multi-agent systems. To be clear, the topological issues for collaboration are exacerbated in a distributed context; very generally, dif- ficulties arise in global state recreation, which taxes the extents of local interaction and communication. At the current scales of real-world multi-agent systems, notions of fully autonomous distributed systems may seem insignificant or unnecessary; why complicate or render infeasible problems which can be solved in a centralized manner? However, we argue that an ignorance to the deep scalability issues inherent in centralized systems is a mistake we as a community can not make. Particularly given the inevitable future of robotic ubiquity, the likelihood that the future is a centralizable vision is certainly low. Investigating and evolving the theories of centralized and decentralized autonomy in parallel is the only reasonable strategy; it is apparent that a symbiosis of centralized control and decentralized autonomy will be necessary to solve the future problems in multi-agent/robot systems. This 3 thesis thus aims to strike many of the issues of distributed autonomy at its core; to treat the common assumptions of interaction and topology which render theoretical treatments feasible, but which leave truly distributed implementations relegated to the laboratory. 1.2 Thesis Outline and Contributions In the following sections, we motivate the topics of this thesis, discuss our research methodology, and review the relevant literature. Probabilistic Mapping and Tracking: A Motivating Example The first contributions in the thesis are found in Chapter 3, where we perform an in-depth case study of probabilistic mapping and tracking of spatial processes. The purpose of the case study is to both contribute novelty to the literature, while motivating through illustration the shortcomings of topological assumptions in multi-robot autonomy. We will demonstrate that deep and meaningful topolog- ical assumptions are necessary to solve the mapping and tracking problem in a distributed way. The connectedness of the multi-agent network, the structure of the inference graph, and the cohesion of agent motion all rest on properties of the dynamic interactions of the team. It is commonplace for such properties to be assumed quantities, a fact that guides the developments on agent interaction and topological control this thesis presents. Probabilistic networks are an elegant framework for modeling and managing uncertainty in a single-agent setting. Well-studied algorithms exist for inference and learning in probabilistic networks and many real-world implementations have been proven highly effective [33–35]. Extending probabilistic networks to multi-agent 4 systems (MASs) is particularly attractive as such systems can afford significant gains in computational efficiency through concurrency, enhanced scale by spatiotemporal distribution, and robustness to failure via redundancy. The extension of the directed Bayesian network (BN) to MASs is most common in the literature, with applications ranging from coordinated tracking and surveillance [36] to distributed fault detection [37]. While BNs share some commonality with undirected Markov random fields (MRFs) in the probability distributions that they can represent, there exist certain classes of distributions that cannot be modeled by a BN structure (e.g. distributions that are characterized by cyclical dependencies). For this reason, we study the representation of probabilistic uncertainty and the inference process in MRFs over MASs, which we term multi-agent Markov random fields (MAMRFs). In the MAMRF framework, each agent maintains and exchanges its beliefs over a subset of domain variables with neighboring agents, where the agent communication topology is defined by a hypertree graph structure over the system. Local and global evidence obtained via inter-agent communication is used to perform intra-agent inference in order to answer queries and perform actions (see [38] for closely related work). The objective for the system is to cooperatively and efficiently map a spatial process over a grid workspace for the purposes of sampling. In the literature common methods for achieving such goals include occupancy grids and level curve tracking algorithms (e.g. [39] and [40]); however, these approaches exhibit several shortcomings. Such methods are generally inefficient as the spatial structure of the process is ignored [41] and a reliance on directly sampled gradient and Hessian information (e.g. in [42–45]) can present difficulties in practice. Instead, we propose a method that assumes a probabilistic model of spatial dependence in the process to achieve efficient mapping, and a Lyapunov stable level 5 curve tracking control that operates not through direct process observation, but by estimating gradient and Hessian information over the resultant process map. The central example will be the widely relevant pairwise MRF with grid structure. One of the primary strengths of this formulation of MRF is its simplicity in modeling localized spatial interactions amongst variables in complex domains. In practice, such interactions are prevalent, with applications including image segmentation [46], spatial mapping [41], and object classification [47]. A partitioning of the grid structured MRF over multiple agents will be defined in the context of the MAMRF framework and a presentation of inference therein will be given. The intractability of exact inference over the grid structured MRF motivates an approximate approach based on loopy belief propagation (LBP) in generalized cluster graphs. To encourage grid scale-invariance, an intelligent message passing scheme based on regions of influence (ROIs) in a cluster graph will be contrasted to a synchronous approach in the application of loopy belief propagation. Simulations of autonomous surface vehicles (ASVs) over a real-world dataset demonstrate the efficacy of the proposed methods in mapping and sampling a plume-like oceanographic process (data from NASA’s Moderate Resolution Imaging Spectroradiometer (MODIS) [48]). It is shown that the agents are able to map the plume-like process over a distributed set of heterogeneous observations in an efficient, accurate, and convergent manner. The feasibility of the approach in realistic multi- agent deployments is suggested by acceptable scaling and computational complexity over reasonably sized agent networks and workspaces. Spatial Interaction under Topological Constraints The gap exposed by our motivating example in topological evaluation and control underlies the material of Chapter 4, where we begin by studying the relationship 6 between mobility, spatial interaction, and topology. In particular, we consider a realistic model of interaction consisting of agents with proximity-limited communica- tion and local sensing capabilities (Section 4.1). When coupled with agent mobility, such interactions induce topological variations that can be modeled by the dynamic network (or dynamic graph) framework (Chapter 2). We are concerned with con- trolling the topology of such a dynamic network of agents, while respecting a set of constraints relevant to system objectives or some performance metric. The topology control problem has been addressed primarily in two manners, by maintaining the connectivity of initially connected networks, or through the regulation of global and local connectivity measures or discrete topological properties. In [49] a robust local measure of connectivity is proposed along with connectivity-preserving controls for solving broadcast optimization and sensor coverage problems. Alternately, [50] takes a hybrid automata approach, incorporating market-based auctions and a local connectivity measure to define distributed controls for connectivity preserving link deletion. Lying beyond simple connectedness are the implications of topology on the performance and robustness of networked processes (e.g. consensus). As demonstrated in [26,51–53], with biological insight [4], distinct tradeoffs exist in network composition, motivating the control of agent interaction beyond what guarantees of connectedness alone can provide. This viewpoint contrasts with works seeking direct enhancements to networked algorithms [54]; in such cases, performance remains at the mercy of network topology. As opposed to previous work that is centralized in nature [55,56], solutions that limit connectivity control to link retention [57–59] or unconstrained maximization [60,61], and approaches that regulate network links and not agent configuration [50], we propose a distributed 1 control framework that modulates topology through agent 1 Allowing for efficient scaling in network size and communication cost. 7 mobility, and manages discriminatively the addition and deletion of network links in order to shape agent interaction under constraints 2 . Works that treat link addition [50,64], act by switching links directly in the space of discrete graphs, yielding a divide in communication and agent configuration, and introducing the potential for issues such as interference in aggregating behaviors. Assuming proximity-limited communication and sensing, the treatment of link addition, i.e., interaction with previouslyundetectedagents, introducesdifficultiesinenforcingspatialdisplacement between constraint violating agents. Thus, to extend connectivity preserving formulations such as [50,56,59,60,64,65], we couple a hysteresis model of spatial interaction [59], with a switching framework that preemptively selects candidates for link deletion and addition, regulating network links through mobility and dictating constraint satisfying spatial configurations. Specifically, we construct local controllers with discrete switching for link discrimination, whereby adjacent agents are classified as either candidates for constraint-aware link removal or addition, or as constraint violators requiring either the retention or denial of communication links. Attractiveandrepulsivepotentialsfieldsestablishthelocalcontrolsnecessaryforlink discrimination and constraint violation predicates form the basis for discernment. We examine constrained interaction by considering two common coordination objectives, aggregation and dispersion, under two interaction constraints: maximum and minimum node degree. For each task, we propose predicates and continuity- preserving potential fields, we analyze the dynamical properties of the resulting hybrid system, and we present simulation results demonstrating the correctness of our control formulation. The primary motivations for our fully mobility-based formulation are the impli- cations of proximity-limited systems, particularly with regards to a recent surge 2 Previous work on constrained connectivity is relatively sparse, examples include [62,63]. 8 in swarm robotics (e.g. [66]), in terms of modes of interaction. Such realistic systems rely intimately on spatial interaction and communication that is effectively broadcast-based, rendering the direct switching of links between adjacent agents difficult (the approach taken by [50]). Further motivation can be derived from the implications of spatial control over constraints on the swarm behaviors of the collective. As the constraints are rendered spatially for both link addition and deletion, there exists a direct correspondence between discrete topological constraints and the agent configuration in space. Thus, it appears promising the possible applications of our spatial controls on the behavior of the collective beyond the analyzed aggregation and dispersion behaviors. Furthering our spatial topology control in Section 4.3, we consider the case where our desired topological constraint is non-local in nature. Many important topological properties exhibit non-locality, such as algebraic connectivity, rigidity, multi-hop adjacency, or graph loopiness. Choosing the collaborative primitive of flocking as a vehicle for analysis, we propose consensus-based coordination over topology changes in order to remain both preemptive and feasible with respect to a non-local constraint composition. Such a construction allows the agents to agree on link additions and deletions, when such changes may not be communicated locally. An analysis of the topology consensus and a Lyapunov-like convergence argument guarantee the flocking, collision avoidance, and constraint satisfaction properties of the system, while simulation results in a novel coordination scenario reinforce our claims and demonstrate the applicability of our methods in a realistic setting. Estimating and Controlling Algebraic Connectivity While spatial topology control yields a framework for sound decision-making and mobility for discrete constraints, we are further interested in evaluating continuous 9 topological constraints. In addition, there may be cases where maintaining a constraint is not sufficient, as one may wish to improve some topological metric. A constraint that is ubiquitous in collaboration is the connectivity property of a proximity-limited network (Section 4.4). In particular, the feasibility of col- laborative processes, such as consensus or flocking, rests on connectedness, while the connectivity metric when improved can yield gains in collaborative efficiency. Consensus algorithms are particularly interesting as they underly many distributed processes including filtering [67–69], optimization [70–72], and estimation [73–75]. Olfati-Saber showed in [22,26,76] that network connectivity, expressed as the spectrum of the graph Laplacian matrix, fundamentally impacts the convergence rate, time-delay stability, and robustness of consensus, yielding a distinct tradeoff in network topology design. It is then natural to consider controlling connectivity in order to yield performance gains in networked processes like consensus, or to otherwise shape network information flow [77]. Connectivity control has been addressed in recent works by considering vari- ous connectivity measures together with both distributed and centralized control schemes. Admissible control sets for double integrator agents to maintain disk graph connectivity are derived in [78], along with a distributed algorithm for com- puting controls nearest a desired polytope. In [55] centralized potential fields over the determinant of a reduced Laplacian are employed to guarantee connectivity. Distributed controls and algebraic connectivity estimation are considered in [60] and [79], while in [61] a distributed power iteration is formulated together with a distributed gradient controller for connectivity maximization. Works considering inter-agent constraints in the topology control problem include [62,65]. Finally, the impact of topology on consensus-like algorithms is examined in works such as [53], however not from the perspective of mobility control. 10 As opposed to previous work that is centralized [65,80], solutions that guarantee connectedness alone [59], or approaches employing unconstrained connectivity maximization [61] that yields fully connected topologies (nullifying a distributed design), we propose a distributed formulation that maximizes connectivity under local constraints, taking a step towards agent configurations that respect known tradeoffs in network topology. First, we consider an inverse iteration algorithm that enables each agent to estimate a component of the eigenvector associated with network algebraic connectivity. Our solution is fully distributed, scalable, and it improves on the convergence rate issues of [61]. Second, we construct distributed potential-based controls for agent mobility that constrain locally the connectedness of each agent by enforcing a maximal neighborhood size, while simultaneously driving the system to maximize connectivity, maintain links (and thus connectivity), and avoid collisions. Extensive simulations verify the correctness of connectivity control, while demonstrating performance beyond the state of the art in distributed connection estimation. Route Swarm: A Case Study of Heterogeneous Mobility The development of our spatial topology control rests on the underlying condition that the network is homogeneous, i.e., all agents share identical modes of com- munication, sensing, and mobility. However, as multi-robot systems increase in complexity and the scope of problems to address expands, heterogeneity will become a necessity in collaborative systems. As a case study of heterogeneity, particularly the integration of low-level spatial interaction and high-level network objectives, we introduce at a high level a novel hybrid architecture for command, control, and coordination of networked robots for sensing and information routing applications, called INSPIRE (for INformation and Sensing driven PhysIcally REconfigurable 11 robotic network). In the INSPIRE architecture, we propose two levels of control. At the low level there is a Physical Control Plane (PCP), and at the higher level is an Information Control Plane (ICP). At the PCP, iterative local communications between neighboring robots is used to shape the physical network topology by manipulating agent position through motion, using our developments on constrained interaction. At the ICP, more sophisticated multi-hop network algorithms enable efficient sensing and information routing (e.g., shortest cost routing computation, time slot allocation for sensor data collection, task allocation, clock synchronization, network localization, etc.). Unlike traditional approaches to distributed robotics, the introduction of the ICP provides the benefit of being able to scalably configure the sensing tasks and information flows in the network in a globally coherent manner even in a highly dynamical context by using multi-hop communications. This architecture is an early illustration of the centralized-decentralized coupling we see as the future of multi-agent/robot autonomy. As a proof of concept of the INSPIRE architecture, we detail a simple instantia- tion, in which a robotic network is dynamically reconfigured in order to ensure high quality routes between a set of static wireless nodes (i.e., a flow) while preserving connectivity. To complicate matters, we allow the number and composition of information flows in the network to change over time, demanding dynamic reconfig- urability. In solving this problem, we propose ICP and PCP components that couple connectivity-preserving robot-to-flow allocations, with communication optimizing positioning through distributed mobility control; a heuristic we call Route Swarm. We demonstrate our propositions through simulation, illustrating the INSPIRE architecture and the Route Swarm heuristic in a realistic wireless network regime. The integration of mobile robotics and wireless networking is an emerging domain. Researchers have previously investigated deploying mobile nodes to 12 provide sensor coverage in wireless sensor networks [81,82]. In [83], the authors present a work to ensure connectivity of a wireless network of mobile robots while reconfiguring it towards generic secondary objectives. Going beyond connectivity, recently, research has also addressed how to control a team of robots to maintain certain desired end-to-end rates while moving robots to do other tasks, referred to as the problem of maintaining network integrity [84]. This is done by interleaving potential-field based motion control and at the higher level an iterative primal-dual algorithm for rate optimization. All of these works point to the need for a hybrid control framework where low-level motion control can be integrated with a higher- level network control plane such as the INSPIRE architecture illustrated in this work. Closely related to our work is an early paper that advocated motion control as a network primitive in optimizing network information flows [85]. Although related in spirit, we provide in this work fundamental advances in flow-to-flow reallocations, dynamic and flexible connectivity maintenance allowing network reconfigurabil- ity, and refined potential-based control that requires only inter-agent distance in optimizing intra-flow positioning. Another, more recent work [86], focuses on a single-flow setting, but considers a more detailed fading model communication environment, and a slightly different path metric. In contrast to [86], we make novel contributions in multi-flow optimization which we have shown requires a more sophisticated network-layer information control plane. Moreover, the motion control presented in [86] can also be integrated with and adopted as a component of the PCP in the INSPIRE architecture presented here. In particular, our focus is on network flexibility while seeking optimality, allowing adaptation to changing environmental parameters, shifts in network objectives, and robustness to failure. 13 An Asynchronous Decentralized Pebble Game The second half of the thesis focuses on the rigidity property of a multi-agent network, with primary contributions in Chapter 6. While rigidity shares some relationshipwithconnectivity, e.g., allrigidnetworksareconnected, itissignificantly stricter in constraining a network and provides deeper implications for collaboration. Inmodelingandanalyzingmulti-agentnetworks, researchbalancesbetweenaccuracy in approximating realistic systems, and ease of technical analysis, most typically in understanding mobility, communication, and sensing. Here we take the former approach, considering a problem that underlies fundamental objectives in multi- robot research, while operating under the commonly desired parameters of real- world systems: decentralized implementation where information is exchanged only in local neighborhoods, asynchronicity in communication, and parallelization of agent actions to maximize efficiency. Our problem of interest in Section 6.1 is the evaluation of the rigidity property of an interconnected system of intelligent agents, e.g., robots, sensors, etc. A relatively under-explored topic in the area of multi-agent systems, rigidity has important implications particularly for mission objectives requiring collaboration. For example, its relevance is clear in the context of controlling formations of mobile nodes when only relative sensing information is available [87–92]. Specifically, the asymptotic stability of a formation is guaranteed when the graph that defines the formation is rigid by construction. Thus, in achieving or maintaining a network’s rigidity, it becomes possible to extend the traditional pre-defined formation methodology, to that of dynamic formations, precisely as rigidity guarantees correctness over time. The idea of formation persistence, i.e., the ability of a formation to remain stable in the face of external perturbation, is also supported by rigidity. As demonstrated for example in [93], minimally persistent co-leader formations are achieved if certain properties on the 14 minors of the rigidity matrix are maintained. Rigidity becomes a necessary (and in certain settings sufficient) condition for localization tasks with distance or bearing- only measurements [94–97]. The ability of a network to self-localize is of clear importance across various application contexts, and for example in [96] it is shown that if the rigidity conditions for localizability for traditional noiseless systems are satisfied, and measurement errors are small enough, then the network will be approximately localizable, providing a connection between robustness and rigidity. Finally, the flavor of rigidity studied here is also a necessary component of global rigidity [98–100], which can further strengthen the guarantees of formation stability and localizability, as the uniqueness of a given topological embedding is more easily characterized. It is clear then that network rigidity acts as a fundamental precursor to both important spatial behaviors and information-driven objectives, making it a strong motivation of this work. We point out that it is typical in the literature to assume rigidity properties of the network in order to achieve multi-agent behaviors, however few works provide means of evaluating or achieving network rigidity in a dynamic manner, or under the network conditions considered here. The general study of rigidity has a rich history in various contexts of science, mathematics, and engineering [99–106]. In [105], combinatorial operations are defined which preserve rigidity, with works such as [87,89] extending the ideas to multi-robot formations. In [107] an algorithm is proposed for generating rigid graphs in the plane based on the Henneberg construction [105], however from a centralized perspective. Similarly, [108] defines decentralized rigid constructions that are edge length optimal, however provide no means of determining an unknown graph’s rigidity properties. The work [109] defines a rigidity eigenvalue for infinitesimal rigidity evaluation and control, however such efforts remain centralized and require continuous communication and computational resources. 15 As opposed to previous work, we propose a decentralized method of evaluating generic graph rigidity in the plane, without a priori topological information, to our knowledge the first such effort, particularly in a multi-agent context. To this end, we decentralize in an asynchronous manner the pebble game proposed by Jacobs and Hendrickson in [106], an algorithm that determines in O(n 2 ) time the combinatorial rigidity of a network, and a spanning edge set defining the minimally rigidsubcomponentofthegraph. Specifically, weproposealeaderelectionprocedure based on distributed auctions that manages the sequential nature of the pebble game in a decentralized setting, together with a distributed memory architecture. Further, an asynchronous messaging scheme preserves local-only agent interaction, as well as robustness to delays, failures, etc. Towards network efficiency, we extend our decentralization by parallelizing a portion of the rigidity evaluation, taking inspiration from gossip messaging (Chapter 2), yielding significant improvements in execution time and communication. To illustrate our contributions, we provide a thorough analysis of the correctness, finite termination, and complexity of our propositions, along with an illustration of decentralized rigidity control. Finally, we provide Monte Carlo analysis of our algorithms in a Contiki networking environment, illustrating the real-world applicability of our methods. Although a few recent works have begun to investigate rigidity evaluation or control [107–110], they provide graph constructions or centralized control relying on expensive estimation techniques. We seek decentralization specifically to enhance scalability and robustness as network size increases, and to serve systems where centralized operation may be difficult or impossible. Our contributions therefore aim to bridge the gap between fundamentally important multi-agent behaviors and realistic rigidity evaluation in networked systems, ultimately moving towards robotic/sensor systems with achievable rigidity-based behaviors. 16 Constructing Optimally Rigid Networks The decentralized rigidity evaluation result yields an answer to whether a graph is rigid, and as a statement of proof, some rigid spanning subcomponent of the graph. However, in general a non-minimally rigid graph may possess a set of rigid subcomponents, from which our evaluation chooses one based on the order of leader election. As we control leader election, one would expect that if we were concerned with exactly what edges constituted the resultant rigid subcompo- nent, we could harness leader election to find it. Towards this goal of identifying a particular rigid subgraph, in Section 6.2 we assume a generalized node/edge metric measuring network performance is associated with inter-node connections, allowing us to formulate an optimization problem within the rigidity framework. In particular, this work aims at finding in a decentralized way the optimal rigid subgraph embedded in the graph encoding node interactions and link metrics. Such an optimization is attractive as the resulting graph not only holds the guarantees associated with rigidity, but also considers network cost, e.g. for localizable and edge optimal sensor embeddings or mobile networks. Despite its great theoretical and application oriented appeal, this problem has been largely overlooked in the recent literature. Again, in [107] an algorithm is proposed for generating optimally rigid graphs based on the Henneberg construction [105], however the proposed procedure is centralized, generates only locally optimal solutions, and requires certain neighborhood assumptions to ensure convergence. The major novelty of this work is therefore represented by two decentralized auction-based algorithms to find the optimal rigid subgraph embedded in a rigid graph. Resting on the combinatorial Laman conditions which characterize the contribution of each network edge to rigidity, and a leader election procedure to control rigidity evaluation and decentralization, the first approach finds an 17 optimal solution at the cost of higher communication complexity. In considering the combinatorial and constrained optimization necessary to discover an optimally rigid network subcomponent, it appears as if one approaches a problem that will be solvable only approximately or locally. However, as we will demonstrate, this intuition is actually misleading, as the properties of the Laman conditions for network rigidity and a greedy edge consideration ultimately yields a provably optimal solution. Such a solution is then quite convenient for decentralization, as a simple and locally computable edge ordering dictates algorithm execution. The second approach then provides a sub-optimal solution while reducing the computational burden according to a sliding parameter ξ, controlling the balance of optimality and complexity. A theoretical characterization of the optimality of the first algorithm is provided. Furthermore, a closed form of the maximum gap between the optimal solution and the sub-optimal solution provided by the second algorithm expressed in terms of the tuning parameter ξ is also provided. We then close the work by providing a case study of our approach in the context of an H 2 metric defining the performance of RSN proposed in [111]. It is shown that our decentralized algorithms outperform the state of the art, which remains both centralized and locally optimal in nature. Distributed and Efficient Rigidity Control To close the thesis, we demonstrate the integration of our spatial topology control, with our methods for evaluating network rigidity, to achieve rigidity control in a multi-agent team. In contrast to previous work, we propose a distributed rigidity controller that preserves the combinatorial rigidity of a dynamic network topology in the plane, through mobility control. Our motivation rests on the notion that rigidity is a generic property of a network topology [112], eliminating the need 18 to examine all possible realizations. Thus we require rigidity control only during transitions innetworktopology, resultinginasolutionthatyieldsguaranteedgeneric rigidity (with direct applicability, as in [96]), infinitesimal rigidity in almost all realizations [112], and by-construction complexity and robustness advantages due to non-continuous operation. Local link addition and deletion rules are proposed that preserve combinatorial rigidity by exploiting neighbor-wise communication. The decentralized pebble game algorithm is applied to local subgraphs in the region of link propositions to determine changes in topology that preserve rigidity. It is shown that local redundancy of a link in some subgraph is sufficient for redundancy in the entire network, and thus applying minimal communication, and computation that scales likeO(n 2 ), the generic topological rigidity of a network can be preserved. An analysis of the properties of the local rigidity rule, rigidity maintenance guarantees, and a simple consensus-based extension are explored, while Monte Carlo simulations illustrate complexity results. Finally a dynamic agent simulation demonstrates the distributed rigidity maintenance controller in a realistic coordination scenario. 1.3 Notational Conventions We now introduce some notation which we will use in the remainder of the thesis. First, variables in lower case will generally refer to scalars, vectors, or elements of sets. Matrices will be denoted by variables in uppercase, while calligraphed letters refer to sets or graphs. When enumerating or referencing the elements of a vector v, v i refers to the ith element of that vector, and likewise for set elements. When vector v represents the state of a multi-agent system, we will make this distinction with boldface v.|S| denotes the cardinality of the set S. Further, 1 is the vector 19 of all ones and 0 is the vector of all zeros, respectively, with vector size indicated by context. In iterative algorithms or switched systems, on occasion we will use notation t − and t + to refer to the transitions of a state or a switching signal. In representing decision-making processes, we denote by∧,∨,¬,⊕, the boolean AND, OR, NOT, XOR operations, respectively. Further, when the operands are set-valued, we refer to the boolean operations over sets defined in the standard manner. Finally, for compactness and clarity we will drop notation for time dependence after the introduction of a topic, for example in agent state or graph topologies. It will be clear from context the time-varying nature of all quantities. The same rule will apply for state dependence where the relationship is apparent. 20 Chapter 2 Elements from the Theories of Graphs and Systems In this chapter, we introduce select concepts from both graph theory and systems theory that will prove useful in our developments on interaction and topology in multi-agent systems. In Section 2.1, we introduce basic definitions related to the graph object, summarize pertinent ideas from algebraic graph theory, and highlight discrete topological representations which will be relevant. Of particular note will be the Laplacian matrix, which underlies significant recent developments in multi- agent theory, and also drives many of the concepts of the networked agreement problem which we later identify in Section 2.2. The literature on graph theory is quite deep and there are various texts to which we can direct the reader; examples include [113,114], with results focused on algebraic graph theory found in [115,116]. Our discussion of systems theory is limited to the intersection it finds with graph theory in the agreement problem; important works in this context include [22,117]. 2.1 Graph Theory 2.1.1 Basic Definitions A dynamic directed graph is defined as the pairG(t) = (V,E(t)), with vertices or nodesV ={1,...,n} (with graph size|G| = n), and edgesE(t)⊆V×V, where 21 e = (i,j)∈E(t) and i,j ∈V. The time-varying nature ofG, i.e., edges may be gained or lost over time, can be due to a number of systematic influences, depending on the application context. Communication links or sensors may fail, routing or energy control may disable or enable edges on demand, etc. Of particular interest for us is the relationship between mobile agents and proximity-limited communication and sensing (Section 4.1), which compels us to make the dynamic graph assumption. The first element of an edge e is denoted by head(e), while the second element is referred to by tail(e). It is assumed (i,i) / ∈E,∀i∈V, i.e., head(e)6= tail(e) meaning there are no self loops, and further that eache∈E is unique. Of particular interest in this thesis will be the undirected dynamic graph, defined as a graph with the property that for any (i,j)∈E, it must hold also that (j,i)∈E. The in(out)-degree of a vertex v is the cardinality of edges with v as the head (tail), where for undirected graphs the in-degree and out-degree are equal and is simply called degree. Degree is alternatively captured by the ideas of neighbors of a vertex and neighborhoods of a graph. Nodes that are connected by an edge are called neighbors and the set of neighbors of node i (a neighborhood) is given by N i ={j∈V| (i,j)∈E}, which for an undirected graph induces the symmetry property j∈N i ⇔ i∈N j . If we have subsets ¯ V ⊂V and ¯ E ⊂E, the graph ¯ G = ( ¯ V, ¯ E) is called a subgraph ofG. A path onG is an ordered set of distinct vertices{v 0 ,v i ,...,v N } such that (v i−1 ,v i )∈E for all i∈ [1,N]. One of the motivating ideas of this thesis is that agents can in some manner influence the state, actions, beliefs, etc., of all other collaborators in a network or team (i.e., over paths). One certainly expects that cohesion or collaboration is unlikely to occur in networks without this connectedness property, a definition for which is given below. 22 (1,2) (1,4) (2,3) (3,4) (2,4) 1 2 3 4 Figure 2.1: Example of a (connected) graph topology. Definition 2.1.1 (Graph Connectivity). The (directed) undirected graphG(t) is (strongly) connected at time t if for every pair of vertices i,j∈V there exists a path from i to j. We denote byC n the set of (strongly) connected graphs of n nodes. A graph for which this property does not hold is called disconnected. We refer the reader to Figure 2.1 for a basic illustration of the concepts of this section. In the depicted undirected graph we have n = 4, with edges indicated by (i,j). Nodes 1 and 3 have degree 2, while nodes 2 and 4 have degree 3. Finally, a path exists between all vertices and thus the graph is connected. 2.1.2 Algebraic Graph Theory Ultimately we will be concerned with evaluating and controlling the topological properties of a network of agents described by an underlying dynamic graphG. 23 Of particular significance in studying the relationship between graph structure and matrix representations is the area of algebraic graph theory. In defining a graph’s structure, we employ an algebraic representation ofG by first considering the adjacency matrix A(t)∈{0, 1} n×n having symmetric elements A ij (t) = 1 if (i,j)∈E(t) and A ij (t) = 0 otherwise, where we denote by A i (t) the ith row of A(t). The adjacency matrix uniquely describes a graph and as such the eigenvalue spectrum of A are uniquely specified by a graph, which yields deep insights into the graph-theoretic propertiesG. While graph-theoretic investigations have previously focused on the adjacency matrix and its spectrum, recent results have focused on an alternative representation ofG, the Laplacian matrix. LetD(t) = diag i ( P j A ij (t)) be the degree matrix having on its diagonal the node degrees of each agent. We can define the Laplacian matrix (or graph Laplacian) L(t) =D(t)−A(t) (2.1) having ordered eigenvalues λ 1 = 0≤λ 2 ≤···≤ λ n , and associated eigenvectors {v 1 ,v 2 ,...,v n }. To illustrate, consider again the graph in Figure 2.1. For this simple graph, the degree, adjacency, and Laplacian matrices are given as follows: D = 2 0 0 0 0 3 0 0 0 0 2 0 0 0 0 3 A = 0 1 0 1 1 0 1 1 0 1 0 1 1 1 1 0 L = 2 −1 0 −1 −1 3 −1 −1 0 −1 2 −1 −1 −1 −1 3 (2.2) where the Laplacian L possesses eigenvalue spectrum [λ 1 = 0, 2, 4, 4]. The properties of the Laplacian matrix and its eigenvalues with respect to graph G are well-studied; see surveys including [118,119]. A deep accounting of these 24 properties is out of scope in this thesis; we instead provide a broad overview of those properties which will best serve our analyses (in particular Section 4.4). First, notice that by construction the rows ofL sum to zero and thus we can conclude that λ 1 = 0. Furthermore, it must also hold that v 1 = 1 T is the eigenvector associated with the zero eigenvalue. While the spectrum{λ 2 ,...,λ n } is generally useful in characterizing topological properties ofG, we focus onλ 2 , the algebraic connectivity, specifically as it has been shown that λ 2 > 0 ensures graph connectedness [115], while λ 2 = 0 indicates disconnectedness. An alternative definition of algebraic connectivity is as a solution to a minimization problem: λ 2 = inf y∈1 ⊥ y T Ly = inf y∈1 ⊥ n X i=1 X j∈N i A ij (y i −y j ) 2 (2.3) where y6= 0, and we assumekyk = 1. From (2.3) it is clear that if, for example, A ij were a monotonic function of inter-agent distance, λ 2 represents the degree to which a graph is connected, a property we will exploit in designing connectivity controls in Section 4.4. Discrete Properties The continuous metric of graph connectivity given by λ 2 is useful but it generally provides a globally coupled picture of the graph topology, that is, we are provided with no information related to discrete agent interactions or paths in the network. With this idea in mind, we can then construct the matrix C p k (t) =A(t) +A 2 (t) +··· +A k (t) (2.4) withelements{c (k,p) ij }representingthenumberofpathsk-hopsorlessfromitoj, and matrix C k (t) with elements{c (k) ij =c (k,p) ij > 0} indicating the existence of a path of 25 k-hops or less betweeni andj [56,115]. Notice that we then have|N (k,p) i | = P j c (k,p) ij and|N (k) i | = P j c (k) ij . The relationships between discrete topological properties and multi-agent collaboration is still far from well understood. However, very recent studies, for example the consensus problem of the following section, indicate that discrete properties such as node degree, multi-hop adjacency, and cycles are relevant in collaboration, primarily as it relates to robustness is such processes. 2.2 The Agreement Problem As we move towards methods to control the topology of a mobile system (Chapter 4), it will often be useful for the agents to agree on a quantity of interest. In making joint decisions or, more generally, in computing in a distributed way, networked agreement provides a theoretical vehicle for collaboration, with applications ranging from cohesive collective motion [120–122], to cooperative estimation, filtering, and optimization [27,61,70]. We provide here a brief overview of important results and algorithms addressing the agreement problem that will be employed in this thesis; we direct the reader to [22,123–126] for a more comprehensive overview of the subject. Consensus Dynamics Consider the case in which n decision-making agents possess a state x that evolves according to the single integrator model, ˙ x i =u i . A simple consensus dynamics to reach an agreement on the state of the integrator agents is then given by ˙ x i (t) = X j∈N i (x j (t)−x i (t)) +b i (t), x i (0) =z i ∈R, b i (t) = 0 (2.5) 26 The so-called collective dynamics of the protocol (2.5) can be written as ˙ x =−Lx (2.6) with respect to the stacked state vector x and the graph Laplacian L. As L has rows that sum to zero, the eigenvector v 1 = 1 T belongs to the null-space of (2.6), L1 = 0. Thus, an equilibrium of dynamics (2.6) takes the form x ∗ = (α,...,α) T . It can be shown that for the case of (2.6) the consensus value is α = 1 n n X i=1 z i (2.7) the average of the initial states z i . Moreover, this average consensus is globally exponentially stable. AssumingG is connected for all time, the convergence rate of (2.5) is proportional to λ 2 , while the delay τ that preserves stability is bounded by 0≤τ <π/2λ n [22]. Thus, there exists a distinct tradeoff between robustness and convergence as networks become highly connected and node degree increases. Such themes are further illustrated in [4,52] where the virtues of topology control and specifically limiting node connectedness are argued. Such topology-collaboration relationships in many ways have motivated this thesis. It is clear that collaborative processes such as agreement have performance and feasibility which are tightly coupledwiththeunderlyingtopology. Inwaysways, thisisanexpectedandintuitive conclusion. We certainly expect that processes which rely upon agent collaboration would be profoundly influenced by how collaboration, e.g., communication or spatial motion, proceeds; precisely the role of network topology. Understanding, evaluating, and controlling topology thus provides a means of regulating contexts in which collaboration may occur, and the efficacy of such collective processes. 27 Distributed Auctions A class of agreement problems that are closely related to consensus averaging are auction algorithms. In the auction problem, instead of desiring an agreement on some (often continuous) state variable, we desire our agents to vote or bid on a proposition in the network. Auctions are also an intuitive means of solving the classical assignment problem. For example, in our decentralized rigidity evaluation algorithm (Section 6.1), auctions allow the agents to elect leaders in the network to control evaluation, where the agreed upon proposition is the new lead agent. Thus, auctions will serve as an agreement tool that can act to arbitrate discrete decisions in a network, which often couples well with continuous actions such as mobility (as in Section 4.2). Our brief overview of distributed auctions will now follow the seminal work of [127] and more recent work [128]. We specialize the market-based auctions of [127] to the case where there exists a single proposition that we wish the network to execute or decide on, and for which each agent can submit a bid, b i > 0. Letting now x i be the auction state for the ith agent with x i (0) =b i , and assuming that we consider larger bids as better, the discrete-time auction dynamics are given by x i (k + 1) = argmax j∈N i (x j (k)) (2.8) where at each time step an agent updates its current estimate of the largest network bid for the desired proposition 1 . IfG is connected, iterations (2.8) converge in at most n− 1 steps to x ∗ i = argmax i=1,...,n (b i ) for each agent i, that is uniformly to the largest bid submitted to the auction. The auction winner is then easily agreed 1 Note that the auction (2.8) holds equivalently for cost-centric auctions, where we may need to choose the smallest bid (using argmin). 28 upon as each agent is aware of both the largest (or smallest) bid and their own submission to the auction. Gossip Algorithms Gossip-based algorithms are important asynchronous alternatives to the Laplacian- based consensus algorithm described above. Markov processes establish an interest- ing connection between the information propagation speed in these two categories of algorithms proposed by computer scientists and control theorists [129]. For our purposes, in deriving a decentralized algorithm for rigidity evaluation (Section 6.1), we will take inspiration from the messaging style of gossip, specifically for parallelizing agent actions. It is thus relevant to provide an overview of pertinent gossip formulations, following the exposition of [129]. Like the average consensus algorithm, the goal for the presented gossip algorithm will be to determine, in a collaborative way, the average of the initial agents states x i . Each agent has a clock that ticks at times of a rate 1 Poisson process, or equivalently, we define a global discretization of time into slots between local clock ticks, as these times represent the only instances when a state x i may change. The gossip averaging algorithm can then be described as follows. Assume we have a stochastic matrix P with non-negative entries with the condition that P ij > 0 only if (i,j)∈E. Now, in the kth time slot, let agent i’s clock tick and let it contact some neighboring node j∈N i with probability P ij . When this occurs both agents set their states equal to the average of their current states. More formally, x i (k) =W (k)x(k− 1) (2.9) 29 where with probability 1 n P ij the random matrix W (k) is W ij =I− (e i −e j )(e i −e j ) T 2 (2.10) where e i = [0··· 0 1 0··· 0] T is an n× 1 unit vector with the ith component equal to 1. Again, we can cite strong topological implications for the convergence of this algorithm. In particular, the averaging time T ave of the described gossip averaging algorithm, with some convergence threshold , is bounded by 0.5 log −1 logλ 2 (W ) −1 ≤T ave ≤ 3 log −1 logλ 2 (W ) −1 (2.11) relating the algebraic connectivity of the gossip matrix W, and thus the network topology, to the speed of averaging. 30 Chapter 3 Probabilistic Mapping and Tracking: A Motivating Example In this chapter, we begin our investigation into the relationship between topology and collaboration by detailing a motivating application: probabilistically mapping and tracking a spatially varying process. The case study will require that a team of mobile agents, each of which can measure the environment and intercommunicate, produce a grid-structured map of a process given a distributed set of heterogeneous measurements. The inferred map will then be used to inform agent motion and thus future measurements; in our case we will propose tracking probabilistic level curves to identify boundaries of saliency in the process. As we will demonstrate, our methods produce novel and effective collaborative mapping and tracking behaviors, and as the literature reflects, the probabilistic mapping problem has significant impact in autonomy, e.g., for environmental monitoring. However, our primary goal in this chapter is to provide a vehicle for illustrating the assumptions of topology and agent interaction that are made ubiquitously in collaborative systems. For example, as we develop our mapping and tracking results, we will leverage assumptions on connectedness to perform inference and decision-making, tree structured belief graphs for probabilistic feasibility, etc. These necessities of theory will motivate the primary contributions of this thesis, particularly in evaluating, constructing, and controlling topologically constrained networks, with the ultimate goal of moving multi-agent collaboration from the laboratory to reality. 31 The mathematical tools necessary for the results in this chapter are rooted in the area of probabilistic graphical models, a vibrant and impactful field which has driven many recent advances in machine learning. Although there is novelty to be found in this chapter, we use the developments primarily to motivate our investigation into the deeper issues of multi-agent interaction and topology. As such, we find a thorough treatment of probabilistic graphical models to be beyond the scope of this thesis; we rely on Section 3.3 to elucidate the motivations we derive from the mapping problem, and we refer the reader to the excellent text by Koller and Friedman [130] for a deep investigation of probabilistic graphical models. Chapter Outline The outline of this chapter is as follows. The proposed multi-agent Markov random field (MAMRF) framework is presented in Section 3.1.1, where we analyze the grid structured MAMRF and the necessary network representation for distributed implementation. In Section 3.1.2, a distributed algorithm for approximate inference is presented, along with a discussion of loopy belief propagation message passing schemes. Simulation results are then provided in Section 3.1.3, which demonstrate our techniques over an environmental dataset. Process tracking is then considered, beginning in Section 3.2.1 where we provide a curve model and a Lyapunov stable control law for tracking levels curves in the plane. Section 3.2.2 then discusses gradient and Hessian estimation for tracking the level curves of a process over a discrete grid. Simulation results are provided in Section 3.2.3, which demonstrates our mapping and tracking results, again over an environmental dataset. To close the chapter, we provide in Section 3.3 a summary of the underlying topological 32 issues that our mapping example uncovers, which acts to drive the remainder of the thesis. 3.1 Decentralized Probabilistic Mapping In this section, we detail the probabilistic tools and our multi-agent formulation for solving the cooperative mapping problem. Fundamental to our developments will be a grid-based probabilistic network, together with distributed inference methodologies for mapping with a multi-agent team. 3.1.1 The Multi-Agent Markov Random Field A Markov random field is a tripletM = (X,H,P), whereX ={X 1 ,...,X N } is a set of N domain variables,H = (X,E,φ) is an undirected graph with nodes labeled byX, edgesE, and potentials φ, and P is a probability distribution overX. The graphH encodes a set of independencies present in P overX. In an MAMRF, a set of K agentsA ={A 1 ,...,A K } each maintains belief over a subset of domain variablesX i ⊂X represented locally as a Markov subnet M i = (X i ,H i ,P i ). The set of independencies implied byH are then present as a partitioningH ={H 1 ∪H 2 ...∪H K } across the local graph structures. This partitioning allows each agent to reason over their subset of domain variables and exchange local belief over only a shared set of domain variables with neighboring agents. In this way, agents must cooperate via structured communication to reason over the global distribution P. InordertoensurevalidprobabilisticinferenceoveranMAMRF,thelocalsubnets along with the subnet topology must obey certain conditions. Specifically, letH be partitioned into subgraphsH i = (X i ,E i ,φ i ). Let the subgraphs be organized into a 33 tree Ψ = (H,L) where each node in Ψ, called a hypernode, is labeled byH i and each edge inL, called a hyperlink, is labeled with the interfaceX i ∩X j betweenH i andH j . Agent A j is considered a neighbor of agent A i if l ij ∈L, and the set of neighbors of agent A i is denoted by Nb i . Assume also that Ψ satisfies the running intersection property, that is for each pair of hypernodesH i andH j ,X i ∩X j is contained in each hypernode on the path fromH i toH j . The tree Ψ is called a hypertree overH [131]. The joint probability distribution P (X ) = 1 Z Y i φ i (X i ) (3.1) is then the product of the potentials associated with eachH i (with duplication ignored), where Z is the standard normalizing function. The hyperlinks serve as the communication channels between adjacent agents, while the hypertree structure defines the communication topology of the multi-agent system. The running intersection property enforces d-separation of hypernodes by the hyperlinks and thus ensures that message-passing operations over the hypertree are probabilistically sound. Grid Structured Multi-Agent Markov Random Fields Given the generalization above for describing a probabilistic network over a multi- agent team, we propose now a grid-based structure that is appropriate for mapping with distributed measurements. Specifically, consider a system ofK agents indexed by{1,...,K} operating over a fixed spatial grid of N×N cells. To each cell we assign a random variable X rs , termed a process variable, with row and column indices r,s = 1,...,N. We associate with each process variable a set of random variables Y k rs , termed model variables, with k = 1,...,M. The model variables 34 represent a measurable realization of an unobservable (or hidden) process given by the X rs ’s, with M observables per grid cell. The joint probability distribution over the domainX ={X rs }∪{Y k rs } is represented by the Markov random field M = (X,H,P), whereH = (X,E, Φ) is an undirected graph with nodes labeled byX, edgesE, and potential set Φ, and P is the probability distribution over X. We assume a pairwise dependence model over the process variables defined by node potentials φ(X rs ) and edge potentials φ(X rs ,X lm ). The system observation models are heterogeneous in nature and are defined by edge potentials φ(X rs ,Y k rs ). Our choice of spatial model is motivated by the generality of the Markov random field in representing arbitrary process and model distributions, enabling both agent heterogeneity and application specificity. Additionally, the computational properties of inference in such networks make them well suited for distributed implementation and efficient parallel processing. The local domainsX i of the MAMRF are then defined as follows. To all agents we assign the set of process variables, requiring that the probabilistic interfaces X i ∩X j are uniformly{X rs }. The model variables are then partitioned in an application-specific manner to meet system goals or constraints (e.g. assigning each agent a unique sensor and associated model). Such a configuration allows the agents to retain a homogeneous view of the workspace (via the process variables), while simultaneously enabling varied observation models, and ultimately fusion. Given a distributed set of heterogeneous observations, the agents must cooperate via local communication to reason over the global distribution P. In particular the joint probability distribution P (X ) = 1 Z K Y i=1 Y φ j ∈Φ i φ j (3.2) is the product of the potentials associated with each H i = (X i ,E i , Φ i ) (with duplication ignored), where Z is a normalizing constant. Figure 3.1 illustrates 35 X 11 X 12 X 21 X 22 X 13 X 23 X 31 X 32 X 33 X 11 X 12 X 21 X 22 X 13 X 23 X 31 X 32 X 33 Y 1 11 Y 2 21 Y 2 32 H 1 H 2 Y 1 21 X rs : Process variable Y k rs : Model variable (observations) { X rs } Figure 3.1: 3× 3 grid-structured pairwise multi-agent Markov random field. a novel 3× 3 grid structured multi-agent Markov random field with two agents and one set of observables per agent (not all shown for clarity). 3.1.2 Distributed Loopy Belief Propagation Our goal is to infer a set of marginal distributions{P(X ij |Z), i,j = 1,...,N p } over the process variables that represents the collective conditional beliefs of the multi-agent system given a distributed set of observed model variablesZ = {Z 1 ∪Z 2 ∪...∪Z K }. To that end, we decompose the global inference problem into a set of inter-agent belief exchanges over the hypertree and a set of local inference problems performed per-agent over eachH i . Exact inference over the local grid structured MRFs using a clique tree message passing algorithm has a complexity that grows exponentially inN p [130]. To alleviate this problem for realistic domains we propose an approximate inference algorithm based on the extension of loopy belief propagation (LBP) over generalized cluster graphs to MASs. LBP algorithms have been shown to be an invaluable resource in practice, particularly in the coding theory community (LDPC codes, turbocodes, etc.). For our purposes we 36 choose the LBP framework as it allows for significant computational savings in the inference process over the grid structured MRF; a structure that has powerful representational qualities but presents difficulties for exact inference due to the tight variable coupling found in the network. To facilitate the extraction of process marginals, we replace the local graphsH i with generalized cluster graphs (denotedC i ). Generalized cluster graphs are similar to clique trees in that they still must satisfy the running intersection property, but they relax the tree structure requirement, and can be used directly for inference without further preprocessing (e.g., chordalization). For each potential in the original networkH i we introduce a corresponding cluster and connect clusters with overlapping scope [130]. For a simple 2× 2 MAMRF with a single observable per cell, the described method generates the cluster graph shown in Fig. 3.2 for local graphH 1 . Notice that there now exists univariate clusters from which to extract the set of process marginals without the need for marginalization. Inference begins with each agent initializing their local cluster graphs by com- puting initial cluster potentials, denoted ψ, by reducing assigned potentials by the set of local evidence Z i . The agents then perform message passing over the hypertree to exchange beliefs over the shared set of process variables, induced by local observations on the hidden model variables. We apply a structured col- lect/distribute message passing scheme for the purposes of clarity (an asynchronous scheme is equally appropriate [130]). During message collection, message sets are passed between neighbors upward through the hypertree towards an arbitrarily chosen root node. Each message set, defined by {δ k i→j } =ψ k Y m∈Nb i −{j} δ k m→i , ∀k∈Z i (3.3) 37 consists of messages generated by observed model clusters (the leaves inC i ), sent from agent A i to agent A j . The domain of each message is the process variable associated with kth observed cluster. The message distribution step is analogous to the collection phase, with message sets flowing downward from the root. Attheconclusionofhypertreemessagepassing, eachagenthasreached consensus with respect to the set of model-induced beliefs over the shared process variables. The final phase in the hypertree-based inference algorithm is the injection step, where each agent injects the messages received from neighboring agents over the hypertree into a local LBP inference process. Each agent A i generates an injection message given by η j i = Y k∈Nb i δ j k→i (3.4) for each observed model variable (indexed by j) by multiplying the messages received over the hypertree that agree on message scope. The resulting injection messages, whose domains are each in the set of process variables, are assigned to the lightest cluster in the agent’s local cluster graph containing η’s domain (i.e. the cluster with the fewest assigned potentials). After injection, each agent then runs an LBP inference process on their injected local cluster graph and extracts a set of marginal distributions over the process variables. Message Passing in Loopy Belief Propagation The primary drawbacks of the LBP algorithm are non-convergence and message passing complexity. There are various heuristic methods that can be applied to alleviate the issues, several of which are related to intelligent message passing approaches [130]. One such scheme is residual belief propagation (RBP) wherein messages are propagated between clusters that disagree most over inter-cluster belief. For a general network RBP must compute inter-cluster belief for all clusters to find 38 X 11 X 12 X 21 X 22 X 11 ,X 12 X 11 ,X 21 X 12 ,X 22 X 21 ,X 22 X 11 ,Y 1 11 X 12 ,Y 1 12 X 21 ,Y 1 21 X 22 ,Y 1 22 Figure 3.2: Generalized cluster graph for approximate inference overH 1 . the maximal disagreement. As the grid size increases, this search becomes infeasible, especially when convergence is slow. To address computational complexity, we propose an intelligent message passing scheme that exploits the underlying structure of the proposed grid structured MRF to determine regions in the graph where messages would the most useful, as opposed to the synchronous approach. Noting that messages generated by non-observed model clusters are unitary and thus ineffectual in the inference process, we make the fundamental assumption that messages would be most useful in regions surrounding grid clusters with observed model cluster neighbors. We call these areas of the graph regions of influence (ROIs). The ROIs are characterized by a fixed radius R over which standard sum- product messages are recursively passed to neighboring clusters. Message passing over the ROIs eliminates the need to search for regions of potential influence and also reduces significantly the total number of messages passed during an iteration of LBP when the ROIs do not fully overlap. As an example consider a large spatial grid 1000×1000 in size, over which two teams of agents are to operate. Let us 39 assume that the ROI radii are chosen and the teams are arranged such that there exist two effective ROIs of radius 50 over which LBP message passing will occur. In such a scenario, each local cluster graph will contain 3N 2 p − 2N p = 2.998× 10 6 non-leaf clusters over which messages are passed in each iteration of synchronous LBP. In comparison the ROI assumption reduces the effective cluster count to 5.96× 10 4 , a reduction of about 50-to-1 per iteration. In particular, intelligent ROI message passing is independent of grid size and scales only according to the size of the evidence set (which is proportional to the number of agents) and the radius of the ROIs, enabling mission dependent tuning for achieving feasibility in otherwise infeasible workspaces. The primary sacrifice that is made by the ROI assumption is that of accuracy. Since the ROIs are of fixed width, regions that do not overlap are rendered inde- pendent since messages will not propagate from one region to the other. While this property would certainly be inappropriate in some problem domains, there are others where it could be a benefit, for example in large spatial grids where the independence of distant features is quite a natural assumption. 3.1.3 Simulation Results in an Environmental Dataset Wepresentthesimulationresultsfortheproposedapproximatemulti-agentinference algorithm for a dynamic three agent system (K = 3) over a 10× 10 spatial grid (N p = 10). Agent dynamics are introduced by applying simple waypoint-based locomotion over the grid cells. At a predefined interval, the agents capture local observations and perform cooperative inference to iteratively compute a set of process marginals. The overriding goal of the system is to probabilistically map a process of interest over the grid. For the purposes of our simulations we chose to consider the problem of mapping oceanographic phenomenon that are typified by 40 plume-like spatial behavior (e.g. harmful algal blooms (HABs) [132]; more generally [133]). Towards this goal, we simulate observations over the workspace by sampling from sea surface temperature (SST, in ◦ C) and chlorophyll concentration (CC, in mg/m 3 ) data taken from the Moderate Resolution Imaging Spectroradiometer (MODIS) instrument aboard NASA satellites Terra and Aqua [48]. In particular we selected data captured from a MODIS sampling taken in 2007 over the coast of California near Santa Cruz. Fig. 3.3 shows images of SST and CC data for a plume-like region which we select for sampling and simulation. Each agent notionally represents an ASV on the ocean surface. A single model variable is assigned to each agent per grid cell, where agents A 1 and A 3 are assigned SST and agent A 2 is assigned CC. As we are interested in plume detection, we choose binary process variables with potentials φ(X ij ) = 0.5 0.5 , φ(X ij ,X lm ) = 0.7 0.3 0.3 0.7 (3.5) The model variables are assumed to be Gaussian distributed with P (Y k ij |x)∼N (μ x ;σ 2 x ) (3.6) where for every x∈ Val(X ij ) we have an associated mean and variance, μ x and σ 2 x . We determine the appropriate conditional Gaussian potentials empirically by selecting a portion of the workspace as being in-plume a priori and calculating mean and variance vectors μ sst = (14.5 13.75), σ 2 sst = (0.5 0.25) μ cc = (1.2 1.9), σ 2 cc = (0.3 0.2) (3.7) 41 12 12.5 13 13.5 14 14.5 15 Sea Surface Temperature (ºC) (a) 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Chlorophyll Concentration (mg/m 3 ) (b) Figure 3.3: MODIS data used for measurement sampling in mapping simulations. The model parameters define the expected in plume and out of plume distribution of SST and CC measurements, while the process potentials reflect both an unbiased predisposition towards cell-wise plume membership and a desire for smoothness of the generated process probabilities. Although we choose a uniform and symmetric set of process potentials for simplicity, varied potentials afford flexibility in modeling process features that are significantly more complex, including spatial biasing and localized asymmetries. Fig. 3.4 depicts an example of a 100 time unit simulation of the 3-agent system dynamically performing exact and approximate inference over the grid. The agents were assigned initial positions x 1 = (0.5, 0.5),x 2 = (9.5, 9.5),x 3 = (0.5, 9.5), and a list of waypoints that define lawnmower patterned cell traversals (a standard pattern in ocean sampling). Agent measurements were sampled from the MODIS dataset with additive Gaussian noise defined byN (0; 0.1), to simulate the effects of imperfect sensors. To assess the relative performance of the proposed inference algorithms, we generated 100 simulation runs with randomized initial positions and sampling 42 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Exact −− Time = 40.0 / 100.0 (a) 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Sync −− Time = 40.0 / 100.0 (b) 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Intel −− Time = 40.0 / 100.0 (c) 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Exact −− Time = 100.0 / 100.0 (d) 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Sync −− Time = 100.0 / 100.0 (e) 1 2 3 4 5 6 7 8 9 10 1 2 3 4 5 6 7 8 9 10 Intel −− Time = 100.0 / 100.0 (f) Figure 3.4: Three agent system performing exact and approximate inference. patterns over the grid. For each simulation we then computed the Kullback– Leibler divergence (KL-Div) for each grid cell marginal distribution versus a set of baseline distributions generated by performing exact inference over the grid with full evidence (i.e. all model variables are observed). The error of a grid of inferred process marginals is then given by = Np X i,j X x P ij (X ij =x) ln ( P ij (X ij =x) Q ij (X ij =x) ) (3.8) where P ij (·) is the baseline distribution and Q ij (·) is the estimated distribution for a cell in row i and column j of the grid. We refer to this error as the cell-wise KL-Div. 43 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 20 40 60 80 100 % Samples Taken Cell−wise KL−Divergence vs. Samples Taken Cell−wise KL−Divergence Exact Sync Intel−4 Intel−8 Intel−12 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 0 10 20 30 40 50 60 Classification Error Rate vs. Samples Taken % Samples Taken Classification Error Rate Exact Sync Intel−4 Intel−8 Intel−12 Figure 3.5: Average cell-wise KL-Div and plume membership error rate. The impact of cell-wise error on agent decision making is assessed by calculating the cell-wise classification error by applying a threshold decision rule to each estimated cell and comparing the result to the baseline classification. Fig. 3.5 shows a comparison averaged over the 100 simulation runs of inference error and classification error (with a threshold of 0.7) as a function of the percentage of samples taken for the exact method, the approximate method with synchronous message passing, and the approximate method with intelligent message passing with ROI radii of 4, 8, and 12. 44 Discussion We now remark on the efficacy of the inference algorithms based on our simulation results. With only a small number of model variable observations over the grid (20-30%), the agents are able to infer a reasonably accurate map of the underlying process in a cooperative and distributed way. It is clear that the exact approach is superior in terms of accuracy and smoothness, an unsurprising result. The approximation methods require more samples to generate accurate results and the outcome exhibits less smoothness, especially in the case of the localized ROI message passing scheme. Also note that synchronous message passing generates marginally better solutions than the ROI scheme, where the ROI radius acts to modulate the estimate accuracy. Given a threshold classification rule, the exact and approximate (synchronous and intelligent with ROI radius of 12) inference methods have 10-15% error rates after the first 25-35% of samples, with the exact method outperforming the approxi- mate methods on average by only 5%. The strength of the approximation methods, especially ROI message passing, lies in computational complexity. In our sim- ulations the computational burden of exact inference became obvious as grids with N p >> 10 were rendered infeasible by the exponentially rising cost of graph triangulation, cluster size, and message generation. The approximate methods were simulated on grids orders of magnitude greater in size with computation times approaching those required for real-time feasibility. For example, we have achieved LBP computation rates of 5-10 Hz on 100× 100 grids in simulation with minimal optimization. We believe that with the recent explosion in computational power and parallelism our approximate algorithms represent a viable option for efficient spatial mapping in realistic multi-agent deployments. 45 As inter-agent communication is neighbor-wise and not all-to-all or centralized, we expect the proposed algorithms to scale well as the number of agents increases. The number of inter-agent messages is approximately linear in agent count and the computational burden of injecting increased global beliefs by each agent is negligible in comparison to the local LBP process, which scales only according to grid size. The advantages of agent scaling are clear: large workspaces become manageable due to enhanced spatial distribution, measurement capabilities and thus process modeling are heightened through flexibility in agent composition, and the impact of agent failures is mitigated by redundancy. Finally, the proposed MAMRF formulation also has several qualitative benefits compared to other plume detection/tracking implementations (e.g. [42–44]). As opposed to gradient based approaches that require local environmental gradients that are difficult to obtain in practice, our method exploits spatial dependence to achieve results based on scalar observations. In addition, our approach leverages distributed global observations in contrast to methods that only incorporate local information to drive control goals. Finally, our algorithm benefits from the flexibility and robustness of probabilistic outputs versus hard decisions or controls. 3.2 Tracking Probabilistic Level Curves 3.2.1 Stable Curve Tracking in the Plane In addition to mapping the process of interest, we also desire our agents to actively sample the process in a coordinated manner. Towards this goal we consider tracking the level curves of a planar process p, which will ultimately allow us to track probabilistic level curves for the purpose of tracking. Assume for now that p satisfies the following assumption: 46 Assumption 1. The process function p is a C 2 smooth function on some bounded open set B⊂R 2 . There exists a set of closed curvesC(p); the level curves of p. On the set B we have||∇p||6= 0 [40]. We construct a planar curve model by considering the time derivative ofp along the trajectory of a moving agent. Applying the directional derivative and assuming the common double integrator agent model, our multi-agent dynamical system is given by ˙ r i = v i , ˙ v i = u i , ˙ p i =∇p i · v i (3.9) with agent position and velocity r i , v i ∈R 2 , and process value at the position of the ith agent, p(r i )∈R, denoted p i . We require inputs u i that will drive the system (3.9) to a desired configuration asymptotically in time, i.e. one in which the agents stably track a specified set of level curves. Towards that goal, consider the following desired system configuration: p i =C i , v i =μ i x p,i (3.10) where for each agenti,C i ∈B is the desired level curve value, μ i > 0 is the desired tracking speed, and x p,i is the curve tangent vector with y p,i = ∇p i ||∇p i || x p,i = Ry p,i (3.11) where rotation matrix R is chosen such that x p,i and y p,i form a right-handed frame. Figure 3.6 depicts our assumed level curve tracking model [40]. 47 To design a control law to satisfy configuration (3.10) we consider the following Lyapunov candidate function V = 1 2 K X i=1 η i (C i −p i ) 2 + 1 2 K X i=1 ||v i −μ i x p,i || 2 (3.12) where η i > 0. This function has been constructed such that the desired system configuration is the unique critical point. The time derivative of the Lyapunov function along the trajectories of system (3.9) is given by ˙ V =− K X i=1 η i (C i −p i )∇p i · v i + K X i=1 (v i −μ i x p,i )· u i − μ i ||∇p i || R∇ 2 p i v i − y p,i ·∇ 2 p i v i x p,i ! (3.13) where∇ 2 p i is the Hessian matrix of the process function p(r i ). The agent controls u i are chosen to ensure the negative semi-definiteness of (3.13) as required by standard Lyapunov control design. Considering control laws for i = 1,...,K u i = μ i ||∇p i || R∇ 2 p i v i − y p,i ·∇ 2 p i v i x p,i +η i (C i −p i )∇p i −α i (v i −μ i x p,i ) (3.14) with α i > 0 gives ˙ V =− K X i=1 α i ||v i −μ i x p,i || 2 ≤ 0 (3.15) our desired result. Scalarsη i andα i are tunable control parameters for descent rate and velocity alignment, respectively, while the choice of R controls the direction of curve traversal (i.e. clockwise or counter-clockwise). Assuming finite initial conditions and that Assumption 1 is satisfied, the desired system configuration 48 r i ∇p i x p,i y p,i v i p Figure 3.6: Level curve model for tracking control design. (3.10) is achieved asymptotically under controls (3.14). This fact follows directly from (3.15), LaSalle’s Invariance Principle, and Barbalat’s lemma [134]. We also mention briefly collision avoidance between agents in the system. In realistic environments and particularly in systems with many agents, avoiding collisions is critical. Assume that each agent has a circular collision region of radius d i within which other agents are detected and avoided. We consider the following collision control based on a logarithmic potential between agents [135]: u i,col =−β i R X j∈Ω i 1 ||r ij || − d i ||r ij || 2 ! r ij (3.16) where r ij = r i − r j , β i > 0, and Ω i ={j6= i :||r ij ||≤ d i } is the set of detected collision threats. The total control input for the ith agent is then the sum of the tracking control (3.14) and the collision avoidance control (3.16). 49 3.2.2 Localized Gradient and Hessian Estimation The controls derived above require both the gradient and the Hessian of the process of interest in order to perform level curve tracking. Acquiring both quantities in a directly sampled environment can be difficult and costly, or even impossible. Alternatives to direct sampling have been proposed in [44,45] wherein localized groups of agents are used to generate gradient and Hessian estimates in order to track level curves. However, such a technique encumbers the agents spatially and mitigates globally distributed sampling and exploration. We instead consider track- ing probabilistic level curves over the spatial map generated using our distributed MAMRF inference approach. Not only is this method more robust due to its probabilistic underpinnings (i.e. an informed process/observation model forms its foundation), each agent can also generate gradient and Hessian estimates directly from the map without the need for coordinating agent localization. The process functionp is now assumed to be the underlying probability function for which our estimated probability grid is a discrete realization. Considering only a single agent and dropping subscripts for clarity, we examine the second order Taylor expansion of p at a point in the neighborhood of r given by p(r + ¯ r) =p(r) + ¯ r T ∇p(r) + 1 2 ¯ r T ∇ 2 p(r)¯ r (3.17) where we assume r + ¯ r is sufficiently close to r such that (3.17) is a reasonable local approximation. Given a set of points in the neighborhood of r we can solve a system of equations directly for local approximations of∇p(r) and∇ 2 p(r). Considering a grid of points around r with a scale h as depicted by Figure 3.7, we form the linear system Ax = b, where the values of p(r) and the neighboring points are 50 interpolated from the probability grid using standard techniques (e.g. bilinear interpolation). Specifically we have A = 0 −h 0 0 0 1 2 h 2 0 h 0 0 0 1 2 h 2 −h 0 1 2 h 2 0 0 0 h 0 1 2 h 2 0 0 0 −h h 1 2 h 2 − 1 2 h 2 − 1 2 h 2 1 2 h 2 h h 1 2 h 2 1 2 h 2 1 2 h 2 1 2 h 2 h −h 1 2 h 2 − 1 2 h 2 − 1 2 h 2 1 2 h 2 −h −h 1 2 h 2 1 2 h 2 1 2 h 2 1 2 h 2 x = ∇ x p(r) ∇ y p(r) ∇ 2 xx p(r) ∇ 2 xy p(r) ∇ 2 yx p(r) ∇ 2 yy p(r) b = p(r−hˆ e y ) p(r +hˆ e y ) p(r−hˆ e x ) p(r +hˆ e x ) p(r−hˆ e x−y ) p(r +h1) p(r +hˆ e x−y ) p(r−h1) −p(r)1 (3.18) where 1 = [1, 1,..., 1] T and ˆ e x and ˆ e y are the standard Cartesian basis vectors with ˆ e x−y = ˆ e x − ˆ e y . Solving the overdetermined, rank deficient system (3.18) can be accomplished by orthogonal decomposition of the coefficient matrix A [136]. Considering the singular value decomposition where A = UΣV the gradient and 51 ∇p h p( r ) { Figure 3.7: Model for gradient and Hessian estimation over probabilistic grids. Hessian are found by computing x = V T Σ + U T b where Σ + is the pseudo-inverse of Σ. Precomputation of V T Σ + U T ∈R 6×8 allows the agents to efficiently estimate gradient and Hessian information of the spatial grid and track probabilistic level curves using controls (3.14). Mapping and Tracking Algorithm The integration of the mapping process, probabilistic curve tracking, and the above gradient and Hessian estimation is performed as follows. The process is iterative in nature; at arbitrary time instances each agent samples the workspace, evaluates their local observation model, exchanges beliefs, and runs their local inference process to generate a process map. From the map, gradient and Hessian information is extracted and a time step of the tracking and collision avoidance controls is executed. This process is then repeated in order to map the workspace and track probabilistic level curves of the process. The choice of time steps, the 52 agent sampling rates, and the interval at which information is exchanged can be chosen to suit mission parameters or satisfy computational and communication constraints. 3.2.3 Mapping and Tracking Simulation Results To evaluate the proposed spatial mapping and curve tracking algorithms we recon- sider the problem of mapping oceanographic processes that are plume-like in nature. Figures 3.8-3.9 depicts an example of a 100 time unit simulation of a 30 agent system performing simultaneous probabilistic mapping and level curve tracking of the process shown in Figure 3.3. The agents are divided into two teams; one is assigned sea surface temperature, C i = 0.95, and μ i = 1.0, while the other is assigned chlorophyll concentration, C i = 0.87, and μ i = 1.5. The initial positions of the agents are random and the control parameters are given by η i = 10.0 and α i = 1.0, with collision avoidance parameters d i = 2 and β i = 2.0. Over the simulation approximately 30% of the workspace is measured. To assess the performance of our proposed algorithm we consider three metrics: the percentage of agent samples that are in-process (detection performance), the threshold classification error over sampled cells (decisioning performance), and the percentage of in-process cells sampled (mapping performance). Figure 3.10a depicts detection (top) and decisioning (bottom) performance over time for varying level curve targets. For each target we generated 20 randomized simulation runs of 100 time units, for a 30 agent system with C i ∈{0.95, 0.90, 0.85, 0.80} and μ i = 1.0. Figure 3.10b shows mapping (top) and detection (bottom) performance over time for a system with a varying number of agents. Again we generated 20 random simulations for each K∈{50, 40, 30, 20, 10}, with C i = 0.9 and μ i = 1.0. 53 Dynamic Tracking -- Time = 10.0 / 100.0 (a) Dynamic Tracking -- Time = 100.0 / 100.0 (b) Figure 3.8: Spatialbehaviorofa30agentsystemmappingandtrackingprobabilistic level curves. Discussion The performance of the proposed algorithm is apparent from our simulations. The agents map the process over a set of distributed observations in an efficient, accurate, and convergent manner (c.f. Figure 3.8 and Figure 3.9). The probabilistic level curves of the process are quickly detected and stably tracked, generating a coordinated, radial sampling pattern (c.f. Figure 3.9b). In comparison to methods that track the level curves of a scalar process directly using gradient controls (e.g. [42–45]), our probabilistic approach offers several benefits. It does not require environmental gradients and it operates according to global information through an informed model as opposed to only local process information. Furthermore, our approach seamlessly fuses observations from heterogeneous agents (sensors) and offers the extensibility for arbitrarily complex process and observation distributions. We see from Figure 3.10b that mapping performance increases according to the size of the multi-agent system, an expected result. Increased network size leads to 54 0 20 40 60 80 100 0 0.5 1 Simulation Time Belief Value Belief Tracking vs. Time 0 20 40 60 80 100 0 2 4 6 8 Simulation Time Speed Tracking Speed vs. Time C = 0.87 C = 0.95 µ = 1.5 µ = 1.0 (a) Agent Paths over Time (b) Figure 3.9: Tracking behavior of a 30 agent system mapping and tracking proba- bilistic level curves. better spatial distribution and more observations, leading to increasingly informed process beliefs. Figure 3.10b also shows that detection (bottom) is marginally impacted in larger systems as there is greater opportunity for observations that are misleading (e.g. agents that are localized in areas that are out of process), a natural tradeoff. Communication cost should scale well as network size increases given the neighbor-wise agent topology (as opposed to an all-to-all or centralized scheme). In terms of complexity, we have achieved computation rates of 5-10 Hz on 100× 100 grids with minimal optimization (similar to [41], where spatial dependence is not considered). 55 0 20 40 60 80 100 40 50 60 70 80 Simulation Time % Process Samples In Process Samples vs. Time 0 20 40 60 80 100 10 20 30 40 50 60 Simulation Time Classification Error Sampled Classification Error vs. Time C = 0.95 C = 0.90 C = 0.85 C = 0.80 (a) 0 20 40 60 80 100 0 20 40 60 Simulation Time % Process Sampled Process Sampling vs. Time 0 20 40 60 80 100 40 50 60 70 80 Simulation Time % Process Samples In Process Samples vs. Time K = 10 K = 20 K = 30 K = 40 K = 50 (b) Figure 3.10: Mapping performance of a 30 agent system mapping and tracking probabilistic level curves. 3.3 Implications of Interaction and Topology Thepropositionsof theprecedingsections, while illustrating anovelmethodologyfor probabilistic mapping and tracking, yield much deeper insights into the fundamental necessity of interaction and topology in multi-agent collaboration. In a centralized implementation of our techniques, concern for agent-to-agent interaction is absent; measurements could be communicated to a single control point, where a map would be generated, yielding new commands for agent mobility or action. However, for reasons of scalability and robustness, we seek autonomy that acts in distributed ways; the agents map, track, and act without external influence, requiring cooperative control, communication, sensing, and ultimately, decision-making. It is here that the fundamental assumptions of this chapter are clearly seen. Consider first the inter-agent communication required for inferring the probabilistic map. Clearly it must be the case that for sound inference, the topology over which belief is communicated must be connected. This connectedness requirement if of course 56 placed in jeopardy by agent mobility, and specifically by the level curve tracking behavior. However, up to this point we have simply assumed that a connected network is maintained by the mobile system; in reality this is far from the case. In fact, the possibility of stably tracking level curves may be questionable when connectivity is properly treated through mobility. Similar conclusions can be made when we consider the necessity of a tree-structured inference topology, the inherent stability of the curve tracking motion, or the decision-making necessary for adaptation as the map is built. No care is taken for spatial interaction, finite ranged sensing or communication, decisions and agreement, or connected topologies. In essence, the assumptions on interaction and topology render a useful piece of distributed theory feasible, but yields in reality an implementation that would be necessarily centralized or all-to-all in terms of communication. This relationship between interaction and topological assumptions, and the feasibility of distributed collaboration is therefore the strongest motivator of this work. Too often, profound assumptions are made to serve theory, yet in practice regressions must be made to fill the gaps left in guaranteeing topological properties. The remainder of this thesis focuses precisely on this gap, by illustrating methods for evaluating, constructing, and controlling multi-agent networks under the most ubiquitous topological constraints:, network connectivity and graph rigidity, all while faithfully treating realistic issues of agent interaction. 57 Chapter 4 Spatial Interaction and Mobility-based Topology Control In this chapter, with the mapping and tracking case study as motivation, we derive interaction models, methods of topological evaluation, and agent motion for solving various facets of the distributed topology control problem in mobile multi-agent systems. Following the theme of realistically interacting systems, we assume the agents are proximity-limited in sensing and communication, and further, that topology manifests through agent configuration in space. For systems that sense and communicate over finite areas, it is intuitive that inter-agent distance, and thus team configuration, plays a key role in the efficacy and robustness of collaboration. A configuration-centric formulation thus allows us to approach topology control completely through agent mobility, without routing or link switching. In addition, viewing topology control from the context of mobility allows us to accommodate systems that lack highly structured communication, e.g., in swarm style robotic teams or underwater platoons, where communication is highly limited, broadcast- based, or costly. Our developments will focus on several embodiments of the network connectivity property, with motivations rooted in the relationship between multi-agent collaboration and connectivity constraints. We will investigate mobility for controlling discrete link addition and deletion, decision-making under non-local constraints, estimation and control schemes for algebraic connectivity, and finally, heterogeneous mobility with the integration of higher-level agent intelligence. 58 Chapter Outline The outline of this chapter is as follows. In Section 4.1, we provide preliminary materials including agent and network models, a spatial model of interaction, and problem statements. A general framework for constrained agent interaction, includ- ing a switching model for discriminative connectivity and constraint-aware mobility control, is presented in Section 4.2. Further, for illustration purposes, Sections 4.2.3 and 4.2.4 present an analysis of constrained aggregation and dispersion behaviors, respectively. Simulation results are provided in Section 4.2.5, demonstrating the constrained interaction framework in several topology control scenarios. In the context of flocking, Section 4.3 discusses a consensus approach to coordinating topology change relative to non-local constraints and its relationship to the mobility control, including a systematic proof of constraint satisfaction. Simulation results in Section 4.3.3 then verify constraint satisfaction and illustrate topological flocking in a novel coordination scenario. Next, distributed algebraic connectivity estima- tion is presented in Section 4.4.1, while Section 4.4.2 discusses potential fields for constrained connectivity control. Section 4.4.3 then depicts simulation results that corroborate performance beyond the state of the art. To close the chapter, Section 4.5 provides a full case study of how topology control can integrate with high level communication objectives in a network with heterogeneous mobility. 4.1 Models of Spatial Interaction and Dynamics Before detailing methods for topology control in mobile systems, we first outline the models for agent dynamics and spatial interaction that we will assume in our various developments. First, to ease technical analysis we will assume that our 59 team of n agents possess mobility that follows single integrator dynamices (except where specified otherwise), i.e., ˙ x i (t) =u i (t) (4.1) where x i (t)∈R m is the position of the ith agent at time t∈R + and u i (t)∈R m is the control input for the ith agent. We can alternatively view the system in terms of the composite network dynamics given by ˙ x(t) = u(t) (4.2) where ˙ x(t) = [ ˙ x 1 (t),..., ˙ x n (t)] T ∈R mn and u(t) = [u 1 (t),...,u n (t)] T ∈R mn are formed by stacking vector components. It is assumed that the agents have some primary coordination objective (such as aggregation, dispersion, flocking, coverage, etc.) together with interaction constraints (such as a topological constraint) that form the basis of the local control inputs. While dynamics (4.1) generally represent a vastly simplified view of agent dynamics, our primary concern is to investigate interaction and topology in collaboration, and such an assumption allows us to focus technical machinery on pertinent issues. There are, of course, cases where single integrators are reasonably faithful as a representation, as in linearized unicycle models, and it is not beyond reason to extend this work to more complex vehicle dynamics. The coordination of system (4.2) lies in agent interaction and in the exchange of information, either by means of passive sensing or explicit communication. The literature examines a broad range of dynamically interacting systems, from central- ized scenarios to all-to-all communication assumptions. Here we adopt models of 60 interaction that assume the agents operate without contact to a centralized con- troller and that sensing and communication occurs only within limited proximities. To begin, we outline our assumed model of discrete interaction in proximity-limited multi-agent systems. Discrete Interaction Models For the purposes of integration into a motion control architecture it is assumed that each agent can sense other nearby agents or obstacles, yielding the displacement d ij ∈ R,kx ij k,kx i −x j k. The spatial neighborhood of each agent is then partitioned as follows (Figure 4.1): •kx ij k∈ [0,ρ 2 ]: communication over established link (i,j) and sensing of displacementkx ij k can occur. •kx ij k ∈ (ρ 1 ,ρ 2 ]: spatial discernment of adjacent agent j for topological constraint satisfaction. •kx ij k∈ [0,ρ 1 ]: communication link (i,j) established 1 . •kx ij k∈ [0,ρ 0 ]: collision avoidance initiated. where we refer to ρ 2 and ρ 1 as the interaction and connection (or collaboration) radii, respectively. We assume that edges (i,j) represent either communication or sensing between agents in a network and will refer to nodesi andj with (i,j)∈E as interacting. Such a model captures a relevant theoretical view of interacting systems: finite-ranged communication and sensing, and spatially discerned interaction. Agent mobility coupled with the described model of interaction induces a time- varying (or switching) element to the network topology, i.e, as the system evolves 1 This area can also be seen as a region of optimal collaboration, which must be kept spatially distinct relative to topological constraints. 61 agents may dynamically enter or exit the interaction radii of others and establish communication (or interact optimally). As is common, we associate a dynamic graph object with the underlying topology of the network 2 : G(t) = (V,E(t)) (4.3) with vertices (nodes)V ={1,...,n} indexed by the set of agents and time-varying edge setE ={(i,j)|i,j∈V} defined by (i,j)∈E⇔ (kx ij k≤ρ 2 ) ∧ σ ij (t) (4.4) with switching signals [50,59,138] σ ij (t + ) = 0, (i,j) / ∈E ∧kx ij k>ρ 1 1, otherwise (4.5) Switching signals (4.5) define our assumed mechanism of delayed link addition at the connection boundary and induce a lag in topology changes. The further implication is that agents detect potential collaborators within some region of sufficient communication, yet remain in a disconnected state. As discussed in [59], such a construction generates a hysteresis effect in link addition and loss, with a switching threshold 3 modulated by ρ 2 −ρ 1 . The hysteresis in topology switching is vital to properly formalizing controls for link discrimination. Specifically, our eventual application of infinite potential fields in controlling mobility will require a lag in topology change to guarantee that the controls attain infinity only in 2 Also referred to as a proximity graph or disk graph [10,137]. 3 Specifically, a dwell time between topology switches is introduced which is fundamental to certain systematic stability proofs [50]. 62 i j k l Rep el Con n ect i on Av oi d ρ 2 ρ 1 ρ 0 A ttract D i scer n men t Figure 4.1: Agent communication, sensing, and spatial interaction model. undesired states (link loss or link addition, relative to constraints). Further, such a dwell time allows a proximity-limited agent to engage in detect-react behavior, a foundation of the proposed constraint-aware interaction framework. Weighted Interaction Models In the above model, we provide an abstract formulation of interaction that defines, in a discrete way, when agents are either communicating (collaboration) or sensing. While such a discrete model will induce a correspondence between communication or sensing, and inter-agent distance, we can make such a relationship explicit by assuming weighted models of interaction. Weighted models assign a continuous metric that relates inter-agent distance to the strength or quality of an interaction, which can facilitate the control of interaction towards an optimum, such as in connectivity control (Section 4.4) or wireless network optimization (Section 4.5). 63 Now, we consider proximity-limited communication by assigning to each link (i,j) between agents i and j a weight w ij = e −kx ij k 2 /2α 2 , kx ij k≤ρ 2 0, otherwise (4.6) where we assume w ii = 0, with α> 0. Define switching signals σ ij =σ ji ∈{0, 1} having elements that determine the inclusion of link (i,j) through the switched weight w σ ij = w ij σ ij [50,59]. The state dependent, switched system induces the weighted dynamic graphG = (V,W,E), with verticesV ={1,...,n} indexed by the set of agents, weightsW ={w σ ij |i,j∈V}, and proximity dependent, switched edgesE ={(i,j)|w σ ij > 0}. Given the symmetry of (4.6) and σ ij ,E will contain only undirected edges andG will itself be undirected. The neighbor set of an agent i is given byN i ={j∈V| (i,j)∈E}, with symmetry property j∈N i ⇔i∈N j . Define the adjacency matrix A∈R n×n of the graphG having symmetric elements A ij =w σ ij . Further consider the weighted node degreesd i = P j w σ ij organized as the diagonalelementsofthedegreematrixD = diag(d i ). TheweightedLaplacian matrix ofG is then given byL =D−A with eigenvalues 0≤λ 1 ≤λ 2 ≤...≤λ n ≤n, and corresponding unit eigenvectors{v 1 ,v 2 ,...,v n }, where by constructionv 1 = 1∈R n , the vector of all ones. Finally, similar to the above weighted model, we can assume an alternative model that has deeper roots in wireless communication. To measure link quality towards optimizing wireless information flow, we can assume that each (i,j)∈E has a weight parameter w ij that describes their cost with respect to transmitting information. A commonly used metric for link quality is ETX, i.e. the expected number of transmissions per successfully delivered packet. This can be modeled as the inverse of the successful packet reception rateλ ij over the link. As the expected 64 packet reception rate has been empirically observed and analytically shown to be a sigmoidal function of distance decaying from 1 to 0 as distanced ij is increased [139], it can be modeled as follows: λ ij ≈ 1− 1 (1 +e −a(d ij −b) ) (4.7) where a,b∈R + are shape and center parameters depending on the communication range and the variance of environmental fading. Accordingly, the link weights w ij , if chosen to represent ETX, can be modeled as a convex function of the inter-node distance d ij : w ij = 1 λ ij = 1 1− 1 1+e −a(d ij −b) = 1 +e a(d ij −b) (4.8) 4.2 Spatial Interaction Under Topological Con- straints Though intuitive, it has been made clear in the literature that topology profoundly impacts the diffusion of information in networks, the stability of coordinated control processes, etc. [22,77,140]. Quite simply, the issue of interaction lies at the core of most dynamic network processes. In this section, we examine constrained interaction by considering two canonical coordination objectives, aggregation and dispersion, under two interaction constraints: maximum and minimum node degree, respectively. The specific problems considered are summarized as follows: Problem 1 (Max degree aggregation). Assuming an initial topologyG(0)∈C n , design agent controls u i such that inter-agent displacements are decreasing, i.e. aggregation occurs. Further, we requireG(t)∈C n for all t≥ 0, while satisfying constraints|N i |≤β i ∀i∈V. 65 Problem2 (Mindegreedispersion). Assuming an initial topologyG(0)∈C n , design agent controls u i such that inter-agent displacements are increasing, i.e. dispersion occurs. Further,G(t) for all t≥ 0 must satisfy constraints|N i |≥ω i ∀i∈V while establishing no new communication links. In the sequel we will propose a general formulation for constraining the inter- actions of proximity-limited agents and the examination of methods and related proofs for solving the proposed constrained coordination problems. 4.2.1 Switching Model for Discriminative Interaction In a system comprising information sharing agents, interaction is fundamentally a matter of communication. The implication is then in order to constrain interaction one must control the underlying communication topology, that is discriminately create and delete links, and act to deny or retain new or established links, respec- tively. Our proposition is the notion of discriminative connectivity among agents in the system, leveraging a spatial model of interaction and agent mobility to generate an environment in which agents actively regulate their neighbors based on inter-agent constraints. Generally, the constraints may not be explicitly spatial in nature (e.g. a constraint on inter-agent mutual information), however we consider the satisfaction of constraints to be a spatial task; that is, unlike works that simply activate or deactivate links (e.g. [50]), we force link discrimination through spatial organization. In many ways, this formulation can further be seen as a generalization of classical swarming controllers, e.g., as explored in [141], enabling an intelligent, decision-making swarm. To achieve discriminative connectivity, we begin by describing further the previ- ously introduced spatial model of interaction. Specifically, consider the region of 66 widthρ 2 −ρ 1 between the interaction and connection radii, which we deem the dis- cernment region. Within this zone each agent discerns, relative to local constraints, between candidates for link addition (non-neighbors) or deletion (neighbors), and agents which must be either attracted (link retention) or repelled (link denial). The discernment region is an intuitive construction, as in a realistic interacting system communication does not occur at the instant of discovery, it is typical to determine or negotiate a connection. Our model is simply an analog of this notion applied to a proximity-limited mobile system, that is interactions should always be tempered. To reiterate, Figure 4.1 illustrates our proposed model and the varying scenarios that may occur during discernment 4 . We are now prepared to present a formal construction for the satisfaction of interaction constraints through agent mobility. First consider the following notion of a predicate [50]: Definition 4.2.1 (Predicate). LetX be a finite set of variables. A predicate over X, denoted P(X ):X→{0, 1}, is a boolean-valued function defined as the finite conjunction of strict or non-strict inequalities overX. That is, P(X ) is a logical statement over the elements ofX. For example, a relevant predicate for this work is P(X ),|N i |≤β i over the setX∈Z + , which returns 1 if agent i has a degree bounded above by β i and 0 otherwise. The predicate forms the basis of discernment for each agent, where for brevity we omit the predicate domain notation in subsequent usage 5 . 4 The scenarios depicted by Figure 4.1 would occur for example if the agents were under maximum and minimum degree constraints (as will be examined in the sequel). 5 Further, we use varying typefaces and subscripts/superscripts to distinguish global and local predicates and predicate purpose. We also use the notation P(t) to indicate a predicate evaluated at some time t. 67 Assume there exists some global constraint on agent interaction that is defined as the finite conjunction of local constraint predicates assigned to each agent. Specifically we have a global constraint predicate P, n ^ i=1 p i ^ k=1 P k i (4.9) that indicates the constraint satisfaction status of the system, where each agent possesses p i local constraint predicates, denoted byP k i and indexed by k. Thus, the global constraint is assumed to have a separability quality across the system or there exists an agreement mechanism in the local constraint formulation, e.g. consensus or an auction. The local predicatesP k i indicate the satisfaction of thekth local constraint and the conjunction V p i k=1 P k i represents the constraint satisfaction status of the ith agent. It is assumed that the predicates are well-posed and that they are topological in nature, i.e. they can be manipulated directly or indirectly through network link addition or removal. Controlling agent interaction under topological constraints requires action that regulates network transitions, by discriminately allowing or disallowing link creation or deletion. As opposed to switching links directly (as in [50,64]), we propose a spatial construction that is preemptive and discerning; viewingkx ij k≤ ρ 1 as a region of optimal communication, we exert spatial control over agents that constitute a constraint violating interaction. Specifically, within the discernment regionkx ij k∈ (ρ 1 ,ρ 2 ] an agent i determines relative to system constraints whether agent j is a candidate for link addition (j / ∈N i ) or deletion (j∈N i ), or if agent j should be attracted (retain (i,j)∈E(t)) or repelled (deny (i,j) / ∈E(t)). Figure 4.1 depicts the scenarios of our model of agent interaction and link discrimination. Additionally, we make the following assumption concerning all agent interaction: 68 Assumption 2 (Synchronization). For clarity we assume agent synchronization in the discernment of network links, and in the application of consensus required for decentralization. We direct the reader to work such as [142] that considers, in a similar setting, certain issues related to asynchronicity. In order to appropriately regulate topology, each agent is assigned predicates for link addition and deletion, P a ij and P d ij , that indicate possible constraint violations by the addition or deletion of link (i,j). Necessitated by the hysteresis in topology switching, these discernment predicates enable the agents to properly determine constraint-aware topology changes and act through mobility. We then associate switching signals{ϕ i } n i=1 ,{γ i } n i=1 ,{ξ i } n i=1 , and{ζ i } n i=1 with addition candidacy, deletion candidacy, attraction, and repulsion, where ϕ i ∈{0, 1} n , γ i ∈{0, 1} n , ξ i ∈{0, 1} n , and ζ i ∈{0, 1} n are associated with agent i. We denote by ϕ j i , γ j i , ξ j i , and ζ j i the jth element of the respective signals, where we assume ϕ i i = 0, γ i i = 0, ξ i i = 0, andζ i i = 0. The signals indicate for theith agent the membership of nearby agents j in link addition and deletion candidate sets C a i ={j∈V|ϕ j i = 1} C d i ={j∈V|γ j i = 1} (4.10) and attraction and repulsion sets D a i ={j∈V|ξ j i = 1} D r i ={j∈V|ζ j i = 1} (4.11) dependent on constraint violation. Discriminative connectivity is then achieved by defining constraint dependent dynamics for the sets (4.10), (4.11) through 69 their respective signals in order to regulate the system topology by attraction and repulsion. The proposed candidate signal updates (per element) are then ϕ j i (t + ) = I z }| { ((kx ij k≤ρ 2 )∧ (j / ∈N i ))∧ ϕ j i (t − )∨ ¬ζ j i (t − )∧¬P a ij ∧¬P a ji | {z } II (4.12) for link addition, and γ j i (t + ) = III z }| { ((kx ij k>ρ 1 )∧ (j∈N i ))∧ γ j i (t − )∨ ¬ξ j i (t − )∧¬P d ij ∧¬P d ji | {z } IV (4.13) for link deletion. The attraction and repulsion signal updates take forms compli- mentary to (4.12), (4.13) given by ξ j i (t + ) = ((kx ij k>ρ 1 )∧ (j∈N i ))∧ ξ j i (t − )∨ ¬γ j i (t − )∧ P d ij ∨P d ji (4.14) and ζ j i (t + ) = ((kx ij k≤ρ 2 )∧ (j / ∈N i ))∧ ζ j i (t − )∨ ¬ϕ j i (t − )∧ P a ij ∨P a ji (4.15) respectively. Notice that the updates (4.12)-(4.15) take a straightforward form. Considering for illustration the candidate signals (4.12), (4.13), Terms I and III represent discernment management, that is, nearby agents are determined for candidacy only if they lie within the discernment region and have the appropriate 70 j ∈N i A ttract Candidacy Rep el { Connected Disconnected j/ ∈N i d ij ≤ ρ 1 d ij > ρ 2 ¬P d ij d ij > ρ 1 P d ij d ij > ρ 1 j ∈D a i j ∈ C d i j ∈ C a i j ∈D r i d ij ≤ ρ 2 d ij ≤ ρ 2 P a ij ¬P a ij Figure 4.2: Switching model for discriminative spatial interaction. neighbor status. Terms II and IV then operate to assign agents within the discern- ment region to the appropriate candidate set on the basis of constraint violation. The dynamics of (4.12)-(4.15) are illustrated in Figure 4.2. The salient properties of the candidate, attraction, and repulsion sets are summarized in the following. The Lyapunov arguments that underlie Theorem 4.2.2 leverage a fundamental assumption on the construction of P a ij and P d ij , which requires first the notion of a worst-case graph, a requirement due to the hysteresis in agent interaction and decisioning: Definition 4.2.2 (Worst-case graph). Assume we have a topological constraintP, and a known space of reachable topologiesG C (t) in which the network graph must lie,G(t)∈G C (t)⊂G n , for all t≥ 0. A worst-case graph relative to constraintP is a graphG w ∈G C such thatP(G w ) = 1⇒P(G) = 1,∀G∈G C . A simple example of worst-case topologies can be given for node degree con- straints. Suppose we have a graphG and we know that for all time only link (i,j) / ∈E may be added and only link (k,l)∈E may be removed. Then, for a 71 maximum degree constraint the graph with (i,j), (k,l)∈E is a worst-case, while for a minimum degree constraint the graph with (i,j), (k,l) / ∈E is a worst-case. Finally, the predicates must be constructed along the following restrictions: Assumption 3 (Preemption, [9,143]). By construction, transitions inG(t) due to link addition or deletion occur over only candidate links, i.e. new communication links to be added toN i must first be members ofC a i and deleted links, that is those removed fromN i , must first be members ofC d i as in Figure 4.2. Thus, at time t,C a i (t),C d i (t), andG(t) define the space of reachable network topologies, G C (t)⊂G n . If inG C we can identify a worst-case graph, we can choose P a ij and P d ij (and by extension, the candidates) such that P(G w ) = 1, which implies that P(G) = 1,∀G∈G C , and the network constraints are satisfied. The above assumption dictates that predicates P a ij and P d ij preemptively deter- mine the worst-case topology with respect to network constraints and proposed link (i,j), only allowing transitions inG w that are constraint satisfying.We now possess the tools necessary to solve Problem 6, specifically we must equip each agent with predicates P a ij and P d ij that satisfy Assumption 3 in terms of a graph rigidity constraint, or equivalently, disallow the removal of independent edges in the graph. Proposition 4.2.1 (Discernment set properties). The sets (4.10), (4.11) under dynamics (4.12)-(4.15) possess the following properties: 1. Symmetry: The inclusion of P a ji and P d ji in the set dynamics of discerning agent i induces a symmetry in the discernment sets of the interacting agents: j∈C a i ⇔i∈C a j j∈C d i ⇔i∈C d j j∈D a i ⇔i∈D a j j∈D r i ⇔i∈D r j (4.16) 72 That is, there exists a notion of agreement or cooperation in interactions. 2. Exclusivity: For a discerning agent i, a nearby agent j with ρ 1 <kx ij k≤ρ 2 belongs to one and only one discernment set of i: j∈C a i ⇒j / ∈ C d i ∪D a i ∪D r i j∈C d i ⇒j / ∈ (C a i ∪D a i ∪D r i ) j∈D a i ⇒j / ∈ C a i ∪C d i ∪D r i j∈D r i ⇒j / ∈ C a i ∪C d i ∪D a i (4.17) 3. Persistence: For a discerning agent i, a nearby agent j with ρ 1 <kx ij k≤ρ 2 that belongs to a discernment set ofi remains a member of the set for all time until eitherkx ij k≤ ρ 1 orkx ij k > ρ 2 , i.e. j leaves the discernment region. Together with the exclusivity property, it is implied that during this period of persistence agent j does not become a member of any other discernment set of i, i.e. switching between discernment sets does not occur. We point out several important characteristics of the proposed constraint satisfaction formulation. In the construction of discernment set dynamics (4.12)- (4.15) we have forced the discrimination of links to occur on the boundaries of the discernment region, before the actual addition or deletion of a link. Specifically, switching in the elements of sets (4.10), (4.11) occurs only when displacement kx ij k transits the ρ 1 or ρ 2 radii. As will be shown in Section 4.2.2, the constraint violation controls approach infinity at the boundaries of the discernment region and thus our preemptive policy ensures a hysteresis effect similar to the switching signal (4.5) for link addition. Further, without preemptively determining violators and applying appropriate mobility controls, a situation could arise where multiple agents reach the connection or interaction boundary simultaneously, generate a 73 constraint violation, and experience infinite control force. Finally, in the vein of preemption we require the following general assumption on the construction of discernment predicates. Assumption 4 (Preemption). As the invocation of P a ij and P d ij occurs on the boundaries of the discernment region, while possible link addition occurs at some future time relative to ρ 2 −ρ 1 , we require the following assumption on predicate formulation to hold for all agents i: n (∃t 2 |j∈D r i (t 2 ))⇔ ∃t 1 |P a ij (t 1 )∨P a ji (t 1 ) o ∧ {(j∈D r i (t))⇒∃ r > 0|kx ij (t)k≥ρ 1 + r }∧ n (∃t 2 |j∈D a i (t 2 ))⇔ ∃t 1 |P d ij (t 1 )∨P d ji (t 1 ) o ∧ {(j∈D a i (t))⇒∃ a > 0|kx ij (t)k≤ρ 2 − a } ⇒ p i ^ k=1 P k i (4.18) where t 1 ≤t 2 , and separate instances of t 1 ,t 2 are treated independently. In words, the discernment predicates must be constructed to be preemptive in nature, i.e. in tandem with appropriate link discrimination, they predict the potential for con- straint violation upon link addition or deletion before it occurs, ensuring topological constraint satisfaction 6 . 4.2.2 Constraint-Aware Mobility Control The discernment of interacting agents into either candidates or constraint violators allows us now to define attractive and repulsive controls between agents comprising 6 An implication of the predicate formulation is that it will require the inclusion of candidate set state, as will be shown. We make no claims concerning a systematic way of constructing predicates, this is reserved for future work. 74 a violating interaction (one or both agents violate a predicate). First, consider an attractive potential field ψ a ij :R + →R + for the purpose of retaining established neighbors that present a constraint violation risk if lost. We consider here the case of an unbounded potential and require ψ a ij to be constructed with the following properties [58]: 1. ψ a ij is a function of the distance between adjacent agents i an j, i.e. ψ a ij , ψ a ij (kx ij k), ensuring that interactions require the knowledge of only local information. 2. ψ a ij →∞ askx ij k→ ρ 2 , guaranteeing neighboring agents that present a constraint violation risk are retained. 3. The potential is continuously differentiable overkx ij k∈ (ρ 1 ,ρ 2 ]. 4. Forkx ij k≤ρ 1 , we have ψ a ij = 0 and ∂ψ a ij /∂x i = 0. 5. Forkx ij k > ρ 1 we have ∂ψ a ij /∂kx ij k 2 > 0, and forkx ij k≤ ρ 1 , we have ∂ψ a ij /∂kx ij k 2 = 0. That is, the potential is attractive within the discernment region and ineffectual otherwise. The dependence of ψ a ij onkx ij k induces the symmetry properties ψ a ij = ψ a ji and ∂ψ a ij /∂kx ij k 2 =∂ψ a ji /∂kx ji k 2 . An appropriate attractive potential which we adopt for this work takes the following form: ψ a ij = 0, kx ij k∈ (0,ρ 1 ) 1 ρ 2 2 −kx ij k 2 + Ψ a , kx ij k∈ [ρ 1 ,ρ 2 ) (4.19) where Ψ a (kx ij k) =akx ij k 2 +bkx ij k +c is chosen such that (4.19) is smooth over the ρ 1 transition, i.e. Ψ a (ρ 1 ) =∂Ψ a /∂kx ij k(ρ 1 ) = 0. 75 Now, consider a repulsive potential field ψ r ij :R + → R + for the purpose of repelling non-neighbors that present a constraint violation risk upon link creation. We consider again the case of an unbounded potential and require ψ r ij possess the following properties: 1. ψ r ij is a function of the distance between adjacent agents i an j, i.e. ψ r ij , ψ r ij (kx ij k). 2. ψ r ij →∞ askx ij k→ρ 1 , guaranteeing that communication is not established with non-neighboring agents that present a constraint violation risk. 3. The potential is continuously differentiable overkx ij k∈ [ρ 1 ,ρ 2 ). 4. Forkx ij k>ρ 2 , we have ψ r ij = 0 and ∂ψ r ij /∂x i = 0. 5. Forkx ij k≤ ρ 2 we have ∂ψ r ij /∂kx ij k 2 < 0, and forkx ij k > ρ 2 , we have ∂ψ r ij /∂kx ij k 2 = 0. That is, the potential is repulsive within the discernment region and ineffectual otherwise. Again, the dependence of ψ r ij onkx ij k induces the symmetry properties ψ r ij =ψ r ji and ∂ψ r ij /∂kx ij k 2 = ∂ψ r ji /∂kx ji k 2 . Similar to the attractive potential (4.19), the repulsive potential takes the form ψ r ij = 1 kx ij k 2 −ρ 2 1 + Ψ r , kx ij k∈ (ρ 1 ,ρ 2 ) 0, kx ij k∈ [ρ 2 ,∞) (4.20) where Ψ r is again a second order polynomial inkx ij k chosen to guarantee ψ r ij is smooth over the ρ 2 transition. 76 Finally, our proposed constraint-aware mobility controls are given by u i =−∇ x i X j∈I i ψ o ij + X j∈D a i ψ a ij + X j∈D r i ψ r ij =−∇ x i ψ(kx ij k) (4.21) where we consider for now a generalized coordination objective driven by potential field ψ o ij :R + →R + over agents in the interaction set. We assume the coordination potential has the following properties: 1. ψ o ij is a function of the distance between adjacent agents i an j, ψ o ij , ψ o ij (kx ij k), i.e. it constitutes a distributed coordination protocol. This also guarantees the symmetry properties ψ o ij = ψ o ji and ∂ψ o ij /∂kx ij k 2 = ∂ψ o ji /∂kx ji k 2 . 2. The potential is continuously differentiable overkx ij k∈ [0,ρ 2 ), specifically with respect to switches over the interaction set (or any further partitioning of the interaction set, as we will see). 3. The potential is collision avoiding, i.e. ψ o ij →∞ askx ij k→ 0. The overall system consisting of proximity-limited agent dynamics (4.1), along with interaction constraints and objectives driven by potential field controls (4.21), can be treated as a hybrid system 7 in which discrete transitions occur at times when new network edges are added or removed, or when switches in the discernment sets occur. The analysis of such a system can be treated using the common Lyapunov function and an extension of LaSalle’s invariance principle from hybrid system theory [144–147]. First, the common Lyapunov function can be defined as follows: 7 In this work we have avoided the specific nomenclature and representations of formal hybrid automata. We reserve such a treatment for future work. 77 Definition 4.2.3 (Common Lyapunov function). If we denote the switching sys- tem generally by the differential equation ˙ x =f s (x), where s indexes the system transitions, a common Lyapunov function is a positive definite, smooth function V such that∇V (x)f s (x)< 0 for all s [147]. Additionally, we make the following assumption concerning the switching behav- ior of the system, as induced by the constraint and predicate construction, and agent dynamics: Assumption 5 (Zeno behavior). An execution of a hybrid system is called finite if there exists a finite sequence of switching times ending with a compact interval, and infinite if the sequence of switches is either an infinite sequence, or if the time between switches is infinite. An execution is called Zeno if it takes an infinite number of discrete transitions in a finite amount of time [144]. In this work, we assume that the composition of agent dynamics with the system constraints and predicates generates switching without Zeno behavior. We begin our analysis by defining the set of feasible initial conditions F,{x∈R mn |kx ij k∈ [0,ρ 1 ),∀ (i,j)∈E}∩ {x∈R mn |kx ij k∈ [ρ 2 ,∞],∀ (i,j) / ∈E}∩ {x∈R mn |P = 1} (4.22) where we specifically denote the constraint satisfaction set byF p ,{x∈R mn |P = 1}6=∅. Note that it is implied by (4.22) and dynamics (4.12)-(4.15) that inF we haveC a i =C d i =D a i =D r i =∅,∀i∈V. The first result of this paper now follows. Theorem 4.2.1. Consider the system (4.1) under the previously described inter- action model, driven by local controls (4.21), and starting from feasible initial 78 conditions (4.22). Then, the system converges to an equilibrium configuration where u i = 0,∀i∈V. Proof. First, consider the Lyapunov function V :R mn →R + given by V = 1 2 n X i=1 X j∈I i ψ o ij + X j∈D a i ψ a ij + X j∈D r i ψ r ij (4.23) and observe that at times where the system switches, V and controls u are continu- ously differentiable by construction, with values that remain constant 8 . Specifically, as ψ o ij is smooth over switches inI i , ψ a ij is smooth over switches inD a i , and ψ r ij is smooth over switches inD r i , V is smooth in x over all system transitions, satisfy- ing the requirements of Definition 4.2.3. Thus, V serves as a common Lyapunov function for the hybrid system and analysis can proceed in the standard manner. Now, for any c > 0, define the set Ω V ={x∈R mn |V ≤ c}, noticing that Ω V is compact due to the continuity of V and as a consequence of potential construction. The time derivative of V in Ω V is ˙ V = 1 2 n X i=1 X j∈I i ˙ ψ o ij + X j∈D a i ˙ ψ a ij + X j∈D r i ˙ ψ r ij (4.24) where n X i=1 X j∈I i ˙ ψ o ij = 2 n X i=1 X j∈I i ˙ x T i ∇ x i ψ o ij n X i=1 X j∈D a i ˙ ψ a ij = 2 n X i=1 X j∈D a i ˙ x T i ∇ x i ψ a ij n X i=1 X j∈D r i ˙ ψ r ij = 2 n X i=1 X j∈D r i ˙ x T i ∇ x i ψ r ij (4.25) 8 This analysis is inspired by the application of a smooth, switching Lyapunov function to a hybrid system in a similar context by [58]. 79 due to symmetry of potentials ψ o ij , ψ a ij , and ψ r ij , and setsI i ,D a i andD r i . Thus ˙ V = n X i=1 ˙ x T i ∇ x i X j∈I i ψ o ij + X j∈D a i ψ a ij + X j∈D r i ψ r ij =− n X i=1 k∇ x i ψ(kx ij k)k 2 ≤ 0 (4.26) which implies the positive invariance of the level sets Ω V of V. The extension of LaSalle’s invariance principle for hybrid systems 9 together with (4.26) and initial conditions (4.22) guarantees that the system converges to the largest invariant subset of{x∈R mn |∇ x i ψ(kx ij k) = 0}. As u i =−∇ x i ψ(kx ij k) we must have at equilibrium u i = 0,∀i∈V. The collision avoidance and discernment set reachability properties are estab- lished in the following lemma. Lemma 4.2.1. Assume the system (4.1) evolves according to control laws (4.21) and starts from initial conditions (4.22). Then, the setsF c ,{x∈R mn |kx ij k> 0,∀i∈V,j ∈I i },F a ,{x ∈ R mn |kx ij k≤ ρ 2 − a ,∀i∈V,j ∈D a i }, and F r ,{x∈R mn |kx ij k≥ρ 1 + r ,∀i∈V,j∈D r i } are invariant for the trajectories of the closed-loop hybrid system between instances of switching transitions. Further, invariance is preserved at the instance of a system transition, i.e. the switch of a discernment set or neighbor set. Proof. It follows from (4.26) that the time derivative of V remains non-positive for every x(0)∈F and for all t≥ 0. Since V (x(0)) <∞, we must have V (x(t)) < ∞,∀t≥ 0. Askx ij k→ 0 + for j∈I i ,kx ij k→ ρ − 2 for j∈D a i , andkx ij k→ ρ + 1 for j∈D r i , we have V→∞. Thus, we can conclude that there exists a > 0 and r > 0 such that x(t)∈F c ∩F a ∩F r for all t with 0<t 1 <t<t 2 , where t 1 ,t 2 are 9 We can invoke this extension as the system exhibits no Zeno behavior [144]. 80 switching instances. Now, upon a switch inN i ,D a i , orD r i , we have by construction of the interaction regionkx ij k≥ρ 1 ,kx ij k =ρ 1 , andkx ij k =ρ 2 , respectively. Thus, by inspection the invariance ofF c ,F a , andF r holds upon transitions inN i ,D a i , andD r i . We now come to our primary result concerning the constraint satisfaction properties of the system. Theorem 4.2.2. Consider the multi-agent system (4.1) controlled by (4.21) and starting from initial conditions (4.22). Further, assume predicates P a ij and P d ij are constructed according to Assumption 4. Then the constraint satisfaction setF p is invariant for the trajectories of the closed-loop hybrid system. Proof. By dynamics (4.14), initial conditionD a i (0) =∅,∀i∈V, and the exclusivity and persistence properties ofD a i , we conclude that j∈D a i (t 2 ) at time t 2 for some agent i if and only if there exists a time t 1 ≤t 2 such that P d ij (t 1 )∨P d ji (t 1 ) = 1 with kx ij (t 1 )k =ρ 1 andj∈N i (t 1 ), i.e. a link deletion predicate for established link (i,j) was violated on the transit of a neighboring agent j over the connection boundary ρ 1 . Similarly, by dynamics (4.15), initial conditionD r i (0) =∅,∀i∈V, and the exclusivity and persistence properties ofD r i , we conclude thatj∈D r i (t 2 ) at timet 2 forsomeagentiifandonlyifthereexistsatimet 1 ≤t 2 suchthatP a ij (t 1 )∨P a ji (t 1 ) = 1 withkx ij (t 1 )k =ρ 2 and j / ∈N i (t 1 ), i.e. a link addition predicate for potential link (i,j) was violated on the transit of a non-neighboring agent j over the interaction boundary ρ 2 . Now, invoking Lemma 4.2.1, we conclude proper link discrimination occurs, i.e. there exists no member j∈D a i that becomes disconnected from an agent i, and likewise there exists no member j∈D r i that becomes a neighbor of agent i. Finally, having satisfied the requirements of Assumption 4 concerning predicate formulation, we can conclude V p i k=1 P k i (t) = 1,∀i∈V,t≥ 0, and the result follows. 81 It is clear that the proper execution of the system rests primarily on the con- struction of the discernment predicates, specifically with respect to the requirements imposed by Assumption 4. We dedicate the remainder of this paper to the investi- gation of predicates for realistic multi-agent problems, related system analysis, and simulation. 4.2.3 Aggregation Under Max Degree Constraints Our first coordination objective calls for the distributed system of agents to aggre- gate 10 , that is, from some initial position the group of agents must enter asymp- totically into a bounded region of R m . Further, in the spirit of this work, we require that the agents respect upper bounds on the degree of local connectivity, i.e.|N i |≤ β i ,∀i∈V, while maintaining overall network connectedness 11 . We suggest that by constraining dynamically the connectivity of each agent, we yield realistic aggregation behaviors in terms of limiting local communication burden and bolstering the performance of information diffusion in the network 12 . Consider first a potential field ψ agg ij :R + →R + for achieving the aggregation objective: ψ agg ij =K 1 kx ij k 2 (4.27) where K 1 > 0 is a gain parameter. Our desire is to apply ψ agg ij over only j∈N i , however note that as ψ agg ij (ρ 1 )6= 0 a discontinuity occurs upon the network switch 10 This behavior is also referred to as swarming and the agent collective as a swarm. Without some collision avoidance requirement, aggregation is typically treated as the common rendezvous problem. 11 Our contributions here go beyond the investigation in [58], which considers unconstrained aggregating behaviors. 12 In [26] it is shown that large node degrees induce time-delay sensitivity in networked consensus processes. 82 on link addition atρ 1 . We thus require an additional potential fieldψ can ij :R + →R + that stitches together the aggregation potential for link addition candidates in order to preserve continuity over the topology switch at ρ 1 . That is, we define ψ can ij =akx ij k 2 +bkx ij k +c (4.28) with constants a,b,c chosen such that the following properties apply: 1. To ensure that the transition over ρ 1 is sufficiently smooth we require ψ can ij (ρ 1 ) =ψ agg ij (ρ 1 ) and ∂ψ can ij /∂kx ij k(ρ 1 ) =∂ψ agg ij /∂kx ij k(ρ 1 ). 2. ∂ψ can ij /∂kx ij k(ρ 2 ) = 0, ensuring continuity over switches in the ρ 2 transition. For the purposes of collision avoidance, consider the potential field ψ col ij :R + → R + defined as [58]: ψ col ij = K 2 log 1 kx ij k 2 ! , kx ij k∈ (0,z) Ψ col (kx ij k), kx ij k∈ [z,ρ 0 ) (4.29) where 0<z <ρ 0 and Ψ col =akx ij k 3 +bkx ij k 2 +ckx ij k +d, with constantsa,b,c,d chosen such that: 1. To ensure continuity over the potential transition at z we require Ψ col (z) = K 2 log (1/z 2 ) and ∂Ψ col /∂kx ij k(z) =−2K 2 /z. 2. Ψ col (ρ 0 ) = 0 and ∂Ψ col /∂kx ij k(ρ 0 ) = 0, ensuring continuity of the potential over switches in the ρ 0 transition. As we will see in the sequel, this seemingly peculiar definition of ψ col ij allows us to bound the potential derivative and derive bounds on the swarm size of the system. 83 The final control laws are now given by u i =−∇ x i X j∈N i ψ agg ij + X j∈C a i ψ can ij + X j∈Π i ψ col ij + X j∈D a i ψ a ij + X j∈D r i ψ r ij (4.30) where Π i ={j ∈V|kx ij k≤ ρ 0 } is the collision avoidance set for agent i. A composite of the potentials comprising control laws (4.30) is depicted in Figure 4.3. It is important to note that the controls (4.30) are continuously differentiable over switches in the network topology and transitions in the discernment sets as required for analysis with a common Lyapunov function. Predicate Formulation for Max Degree Constraints Together with controls (4.30) we require predicates P a ij and P d ij to dictate the switching behavior of discernment setsD a i andD r i in order to satisfy the maximum degree constraints|N i |≤β i of Problem 1. In particular we must choose predicates that meet the specifications of Assumption 4 to guarantee constraint satisfaction under the dynamic network topology. It is implied by the predicate formulation requirements that the agents possess the capability to either evaluate or predict the impact of a link addition or deletion before it occurs and be able to appropriately temper that knowledge against local constraints. In the case of bounding node degree, the interaction constraint is explicit to the existence of the link itself, and thus we can formulate in a straightforward manner constraints that are compliant with Assumption 4. Specifically, we consider a worst-case network topology as it relates to the max degree constraints, yielding link addition predicate P a ij ,|N i | +|C a i |≥β i (4.31) 84 ρ 0 ρ 2 ρ 1 ψ(||x ij ||) ||x ij || 0 0.5 1 1.5 2 0 2 4 6 8 10 Figure 4.3: Inter-agent potentials for aggregation, collision avoidance, and link retention. In terms of a maximal degree bound, predicate (4.31) evaluates a worst-case network topology by considering all candidates for connection, i.e. j ∈C a i , as virtual neighbors of discerning agent i. In this way, the preemptive property is guaranteed as discernments always evaluate over a worst-case degree bounded network topology. Problem 1 also dictates that connectivity be preserved over the trajectories of the system. For this purpose, we define the link deletion predicate P d ij = 1 (4.32) such that all links in the network are retained and initial connectivity implies dynamic connectedness. Although the maintenance of all established links for the purposes of guaranteeing connectivity is by nature conservative, we point out that in an aggregation objective the addition of links is of primary concern. 85 Stability Analysis of Constrained Aggregation We are now prepared to analyze the behavior of dynamics (4.1) under controls (4.30) and predicates (4.31), (4.32), with initial conditions inF∩F C , where F C ,{x∈R mn |G∈C n } is the set of agent configurations that are connected. The equilibrium, collision avoidance, and discernment set reachability properties of the system can be determined as follows. Theorem 4.2.3. Assume that the agents (4.1) are subject to controls (4.30) starting from initial conditions x(0)∈F∩F C . Then, the system converges to an equilibrium configuration where u i = 0,∀i∈V, and the setsF c andF d are invariant for the trajectories of the closed-loop hybrid system. Proof. By construction the objective potential field X j∈I i ψ o ij , X j∈N i ψ agg ij + X j∈C a i ψ can ij + X j∈Π i ψ col ij (4.33) meets the standards of a generalized coordination objective, i.e. (4.33) is contin- uously differentiable overkx ij k∈ [0,ρ 2 ) with respect to switching inN i ,C a i , and Π i . It is clear then that aggregation controls (4.30) are equivalent in nature to (4.21), and we can invoke Theorem 4.2.1 and Lemma 4.2.1, yielding our desired result 13 . Now, we establish the constraint satisfaction properties of the system in the following. Theorem 4.2.4. Consider the system (4.1) under control laws (4.30) and discern- ment predicates (4.31), (4.32), with x(0)∈F∩F C . Further, assume the local 13 Notice that the aggregating application satisfies Assumption 5 as all links are retained, i.e. the switching is finite. 86 constraint predicates areP 1 i ,|N i |≤β i ,∀i∈V. Then the constraint satisfaction setF p is invariant for the trajectories of the closed-loop hybrid system and the network remains connected for all t≥ 0. Proof. Following Theorem 4.2.2 with link addition predicate (4.31) we can conclude that non-neighboring agents i and j become mutual candidates for link addition only when|N i | +|C a i |<β i and|N j | +|C a j |<β j hold. In conjunction with proper discrimination of link additions (guaranteed by Theorems 4.2.3 and 4.2.2) and the fact that|N i (0)| +|C a i (0)|≤β i , it follows that|N i (t)|≤β i −|C a i (t)|,∀i∈V,t≥ 0. Further, considering link deletion predicate (4.32) it is clear that noj∈N i becomes a candidate for link deletion, and all established links are maintained. Noting that link retention has no effect onP 1 i , we can finally conclude that V n i=1 P 1 i (t) = 1 and x(t)∈F C (due to link maintenance over an initially connected network) for all t≥ 0, our desired result. From the preceding analysis we can conclude that the system is convergent to a static state and that the desired collision avoidance and constraint satisfaction requirements are abided. However, it remains to determine whether the system behavior constitutes aggregation, that is, at equilibrium have the agents entered into a bounded region of the workspace. We begin our analysis of the coordination behavior by defining the center of the agent collective, known as the swarm center, given by ¯ x, 1 n P n i=1 x i . Notice that as u i = P j f(x i −x j ), and by construction anti-symmetry f(x i −x j ) =−f(x j −x i ) holds, it follows that ˙ ¯ x = 0, that is the swarm center is invariant over the trajectories of the system [58,59]. This property holds due to the symmetry of all sets that undergo switching (neighbor and discernment sets), guaranteeing that despite switching, the contribution of agent 87 controls is balanced and the swarm center remains fixed 14 . Therefore, without loss of generality, for the following analysis we choose ¯ x as the origin of our coordinate system and define the swarm energy function V s = 1 2 n X i=1 kx i k 2 (4.34) Note that V s is positive semi-definite and V s → 0 as x i → ¯ x,∀i∈V, i.e. upon agent rendezvous. The time derivative of V s is then ˙ V s = 1 2 n X i=1 ∂kx i k 2 ∂kx i k ∂kx i k ∂x i ! T ˙ x i = n X i=1 x T i u i (4.35) Applying the chain rule to controls (4.30), plugging into (4.35), and exploiting the potential and discernment set symmetry then yields ˙ V s =− n X i=1 X j∈N i δ agg kx ij k 2 + X j∈C a i δ can kx ij k 2 − X j∈Π i |δ col |kx ij k 2 + X j∈D a i δ a kx ij k 2 − X j∈D r i |δ r |kx ij k 2 (4.36) whereforclarityweletδ ,∂ψ ij /∂kx ij k 2 , andweusethefactthatδ col ≤ 0,∀j∈ Π i and δ r ≤ 0,∀j∈D r i . Now consider the derivative bounds, δ agg = K 1 , δ can ≥ 0, δ a ≥ 0,|δ col |≤ K 2 /kx ij k 2 and|δ r |≤ κ, and agent displacement boundkx ij k≤ 14 Compare for example to the convergence of average consensus in a dynamic graph as is reviewed in [59]. 88 ρ 2 ,∀j∈D r i , which when applied to (4.36) gives the following bound on the time derivative of the swarm energy: ˙ V s ≤−K 1 n X i=1 X j∈N i kx ij k 2 + n X i=1 K 2 |Π i | +κρ 2 2 |D r i | (4.37) Now, we can derive an upper bound on the swarm size by determining the conditions under which the swarm energy is decreasing, ˙ V s < 0. That is, n X i=1 X j∈N i kx ij k 2 > 1 K 1 n X i=1 K 2 |Π i | +κρ 2 2 |D r i | (4.38) By Theorem 4.2.4 we know that for all timeG is connected and thus we can assume ˆ x, max i,j∈V kx ij k 2 ≤ n X i=1 X j∈N i kx ij k 2 (4.39) where ˆ x is the squared swarm size. We then have ˆ x> 1 K 1 n X i=1 K 2 |Π i | +κρ 2 2 |D r i | ←→ ˙ V s < 0 (4.40) It is important to notice that the left-hand side of (4.40) switches with the dynamics of the network topology and discernment sets. To obtain a bound over all possible network and discernment topologies, we consider the worst case of (4.40) yielding the final swarm bound ˆ x≤ 1 K 1 n X i=1 max G K 2 |Π i | +κρ 2 2 |D r i | (4.41) As the terms of (4.41) are finite, we can conclude that the agents enter into a bounded region of the workspace, that is the desired aggregation behavior is achieved. 89 Remark 4.2.1 (Bounded repulsion). The bound κ on δ r can be derived using the method proposed by Ji and Egerstedt in [59]. Briefly, we consider the Lyapunov function as in (4.23) and determine the maximum energy that can be generated over the possible network and discernment topologies. We then allocate this maximal energy toψ r ij (d min ), whered min is the minimum distance fromρ 1 that can be achieved. That is, we solve max G (V ) =ψ r ij (d min ) (4.42) for d min , yielding bound κ =δ r (d min ). Of course, one could also apply a bounded potential field for repulsion, however constraint satisfaction would no longer be guaranteed. Remark 4.2.2 (Explicit bound). To obtain an explicit form of bound (4.41), the problem max G (K 2 |Π i | +κρ 2 2 |D r i |) can be solved as follows. Notice that by Theorem 4.2.4 we have|N i (t)| ≤ β i ,∀i ∈ V,t > 0, requiring|Π i | ≤ β i and |D r i (t)|≤n− 1−β i ,∀i∈V,t> 0. We then have directly the solution max G K 2 |Π i | +κρ 2 2 |D r i | =K 2 β i +κρ 2 2 (n− 1−β i ) (4.43) which yields an explicit bound when applied to (4.41). 4.2.4 Dispersion Under Min Degree Constraints We now consider solving Problem 2 by defining local controls that ensure the agents disperse, while respecting lower bounds on the degree of local connectedness,|N i |≥ ω i ,∀i∈V. Analogous to the aggregation behavior, realistic dispersive dynamics are generated by forcing a minimal level of agent connectivity, as information exchange 90 processes are guaranteed a minimum level of performance 15 . More generally, we can also envision a system in which the agents act to maximize coverage area through dispersion (e.g. threat monitoring, environmental sampling, etc.), while maintaining certain critical links in the network as defined by some mission objective or metric. The dispersion objective is achieved by defining potential field ψ dis ij :R + →R + given by ψ dis ij = K 3 kx ij k 2 +akx ij k 2 +bkx ij k +c (4.44) where K 3 > 0 is a gain parameter and we choose constants a,b,c such that ψ dis ij (ρ 2 ) = 0 and ∂ψ dis ij /∂kx ij k(ρ 2 ) = 0, ensuring sufficient smoothness of the potential on link loss at ρ 2 . Notice that in this case, collision avoidance is implicit to the potential (4.44). Our control laws for constrained dispersion are then u i =−∇ x i X j∈N i ψ dis ij + X j∈D a i ψ a ij + X j∈D r i ψ r ij (4.45) Figure 4.4 depicts a composite of the potentials composing control laws (4.45). Predicates for constraint satisfaction are derived in a manner similar to aggrega- tion, by considering worst-case network topologies relative to the minimum degree constraint. Specifically, the link deletion predicate takes the form P d ij ,|N i |−|C d i |≤ω i (4.46) By considering all candidates for deletion, i.e. j∈C d i , as virtual non-neighbors, predicate (4.46) discerns link loss over a worst-case network topology with respect 15 In [22] it is shown that increased connectedness is correlated with improved consensus in networked systems. 91 ρ 0 ρ 2 ρ 1 ψ(||x ij ||) ||x ij || 0 0.5 1 1.5 2 0 2 4 6 8 10 Figure 4.4: Inter-agent potentials for dispersion, link retention, and link denial. to the minimum degree constraint. Problem 2 also requires that no links are added during the system execution. For that purpose we define link addition predicate P a ij = 1 (4.47) guaranteeing that all non-neighbor agents that enter the discernment region are repelled. Stability Analysis of Constrained Dispersion We now determine the behavior of dynamics (4.1) under controls (4.45) and pred- icates (4.46), (4.47). The equilibrium, collision avoidance, and discernment set reachability properties of the system can be determined as follows. Theorem 4.2.5. Assume that the agents (4.1) are subject to controls (4.45) starting from initial conditions x(0)∈F∩F C . Then, the system converges to an equilibrium 92 configuration where u i = 0,∀i∈V, and the setsF c andF d are invariant for the trajectories of the closed-loop hybrid system. Proof. By construction the objective potential field P j∈I i ψ o ij , P j∈N i ψ dis ij meets the standards of a generalized coordination objective, i.e. ψ dis ij is continuously differentiable overkx ij k∈ [0,ρ 2 ) with respect to switching inN i . It is clear then that dispersion controls (4.45) are equivalent in nature to (4.21), and invoking Theorem 4.2.1 and Lemma 4.2.1, the result follows 16 . Now, we establish the constraint satisfaction properties of the system in the following. Theorem 4.2.6. Consider the system (4.1) under control laws (4.45) and dis- cernment predicates (4.46), (4.47), with x(0)∈F∩F C . Further, assume the goal constraint predicates areP 1 i ,|N i |≥ω i ,∀i∈V. Then the constraint satisfaction setF p is invariant for the trajectories of the closed-loop hybrid system 17 . Proof. Following Theorem 4.2.2 with link deletion predicate (4.46) we can conclude that neighboring agents i and j become mutual candidates for link deletion only when|N i |−|C d i | > ω i and|N j |−|C d j | > ω j hold. In conjunction with proper discrimination of link deletions (guaranteed by Theorems 4.2.5 and 4.2.2) and the fact that|N i (0)|−|C d i (0)|≥ω i , it follows that|N i (t)|≥ω i +|C d i (t)|,∀i∈V,t≥ 0. Further, considering link addition predicate (4.47) it is clear that noj / ∈N i becomes a candidate for link addition, and thus no new links are established. Noting that link denial has no effect onP 1 i , we can finally conclude that V n i=1 P 1 i (t) = 1 for all t≥ 0, our desired result. 16 Notice that the dispersion application satisfies Assumption 5 as no links are added and thus switching must be finite. 17 Notice that unlike Theorem 4.2.4, there is no guarantee of network connectedness. Inves- tigations into initial conditions and interaction constraints that guarantee connectedness in a dispersive objective is a line of our future research. 93 With knowledge that the system is convergent and respects our desired collision avoidance and constraint satisfaction requirements, we now determine whether the induced coordination behavior is indeed dispersive. Consider once again the swarm energy V s , in this context having time derivative ˙ V s = n X i=1 X j∈D a i (|δ dis |−δ a )kx ij k 2 + X j∈Δ i |δ dis |kx ij k 2 + X j∈Θ i |δ dis |kx ij k 2 + X j∈D r i |δ r |kx ij k 2 (4.48) where we further partition the interaction region by defining sets Δ i ={j∈V|j∈N i ,kx ij k≤ρ 1 } Θ i ={j∈V|j∈N i ∩D a i ,kx ij k>ρ 1 } (4.49) withD a i denoting the complement ofD a i . Now consider derivative bounds,|δ dis |≥ 0,δ a ≤ τ,∀j∈D a i ,|δ dis |≥|δ dis (ρ 1 )|,∀j∈ Δ i ,|δ dis |≥ 0,∀j∈ Θ i , and|δ r |≥ 0,∀j∈D r i , together with agent displacement boundkx ij k≤ρ 2 ,∀j∈D a i , yielding ˙ V s ≥−τρ 2 2 n X i=1 |D a i | +|δ dis (ρ 1 )| n X i=1 X j∈Δ i kx ij k 2 (4.50) Thus, for dispersive action, i.e. ˙ V s > 0, we require n X i=1 X j∈Δ i kx ij k 2 > τρ 2 2 |δ dis (ρ 1 )| n X i=1 |D a i | (4.51) 94 As the network is assumed to be initially connected with|D a i |(0) = 0,∀i∈V and 0<kx ij (0)k≤ρ 1 ,∀i,j∈E, we have ˙ V s > 0 if n X i=1 X j∈Δ i kx ij k 2 > 0 (4.52) which always holds. We therefore have the following intuitive result. For any valid initial configuration x∈F∩F C , the agents are guaranteed to disperse until a point when some agent j becomes a member of a setD a i , an occurrence that results from j being a minimum degree violation risk. Remark 4.2.3 (Bounded attraction). The bound τ on δ a can be derived using the steps described in Remark 4.2.1, that is by applying the method proposed by Ji and Egerstedt in [59]. In this case however, we would determine a maximal distance d max achievable between an agent i and a neighbor j under attractive potential ψ a ij . 4.2.5 Topological Control Simulations In this section, we present simulation results of our proposed control scheme for the constrained interaction of proximity-limited mobile agents, and show that the desired interactivity constraints, collision avoidance and connectivity properties hold. In particular, we consider the coordination scenarios of aggregation and dispersion analyzed in Sections 4.2.3 and 4.2.4, subject to maximum and minimum degree constraints, respectively. For all simulations we consider n = 10 agents operating in a bounded workspace over R 2 , with interaction model parameters ρ 2 = 10, ρ 1 = 7, and ρ 0 = 2. First, we consider the agents driven by constrained aggregation controls (4.30) and subject to discernment predicates (4.31), (4.32). Two initial agent configurations are considered, a random placement of agents in the workspace (Figure 4.5a), and a ring network of radius 10 (Figure 4.5d), 95 15 20 25 30 24 26 28 30 32 34 36 38 40 42 (a) 15 20 25 30 24 26 28 30 32 34 36 38 40 42 (b) 15 20 25 30 24 26 28 30 32 34 36 38 40 42 (c) 10 20 30 40 10 15 20 25 30 35 40 (d) 10 20 30 40 10 15 20 25 30 35 40 (e) 10 20 30 40 10 15 20 25 30 35 40 (f) Figure 4.5: Degree-constrained aggregation simulation of an n = 10 agent system. each satisfying initial condition requirement x∈F∩F C . To the randomly placed network we apply maximal degree bound β i = 6,∀i∈V, yielding intermediate configuration (Figure 4.5b) and final configuration (Figure 4.5c). For the ring network, we assume the cases of β i = 4,∀i∈V and β i = 5,∀i∈V, yielding final configurations (Figure 4.5e) and (Figure 4.5f), respectively. As illustrated by the maximum node degree and swarm size over the simulations (Figure 4.7a), the switching discernment controls satisfy the desired interaction constraints, avoid collisions, and maintain the connectedness of the resulting network, while driving the agents to perform a coordinated aggregation objective. Now, we consider the agents driven by constrained dispersion controls (4.45) and subject to discernment predicates (4.46), (4.47). In this case, a fully connected ring network of radius 2.5 is considered for the initial agent configuration (Figure 4.6a), 96 15 20 25 30 35 15 20 25 30 35 (a) 15 20 25 30 35 15 20 25 30 35 (b) 15 20 25 30 35 15 20 25 30 35 (c) Figure 4.6: Degree-constrained dispersion simulations of an n = 10 agent system. 5 Max 4 Max 6 Max Swarm Size Time Max Degree 0 10 20 30 40 50 0 10 20 30 40 50 5 10 15 20 25 2 4 6 (a) 5 Min 3 Min Swarm Size Time Min Degree 0 10 20 30 40 50 0 10 20 30 40 50 0 5 10 15 20 2 4 6 8 10 (b) Figure 4.7: Node degree bounds and swarm size for aggregation and dispersion simulations. again satisfying initial condition x∈F∩F C . We assume the cases of minimal degree bounds ω i = 5,∀i∈V and ω i = 3,∀i∈V, yielding final configurations (Figure 4.6b) and (Figure 4.6c), respectively. From the minimum node degree and swarm size over the simulations (Figure 4.7b), we see that the controls satisfy our desired interaction constraints while avoiding collisions, resulting in a topology-aware, coordinated dispersion objective. 97 4.3 Non-Local Constraints and an Illustration through Flocking In the developments of the previous section, it is implicitly assumed that our desired constraints can be evaluated locally, as in node degree. However, many important topological properties exhibit non-locality, such as algebraic connectivity, multi-hop adjacency, or graph loopiness. In studying non-local constraints, we will consider the objective of flocking, necessitating a switch to double integrator dynamics given by ˙ x i (t) =v i (t) ˙ v i (t) =u i (t) (4.53) where x i (t),v i (t)∈ R m are the position and velocity of the ith agent at time t∈R + , respectively, and u i (t)∈R m is an acceleration control input. We assume that each agent possesses a set of non-local topological constraints, represented by well-posed logical predicates taking the generic formsP u (A(t)),f u (A(t))≤β orP l (A(t)), f l (A(t))≥ ω, where each f is some non-decreasing function of A(t). Specifically, the constraint satisfaction of each agent is given byP i (t), (∧ k P k i,u (A(t)))∧ (∧ k P k i,l (A(t))), whereP k i,u andP k i,l are the kth constraints having formP u andP l , respectively. By considering constraints over C p k (t),C k (t), and λ 2 , we can regulate both general network connectivity and the individual interactions of agents in a non-local manner 18 , motivating the consideration of the following problem: 18 Note that C p k (t),C k (t), and λ 2 are non-decreasing in A(t). In this context, non-decreasing relates link addition and removal to increases or decreases in the associated metric, e.g. algebraic connectivity λ 2 is non-decreasing on link addition. 98 Problem 3 (Topology-constrained flocking). Given an initial networkG(0) such that P(0),∧ i P i (0) = 1, design controls u i (t) such that P(t) = 1 for all t > 0, agent velocities are asymptotically aligned, and collisions are avoided. 4.3.1 Mobility for Constrained Flocking Exploiting our spatial topology control, by adding a flocking term and a generalized secondary objective, our proposed constraint-aware control laws are given by u i =− X j∈N i (v i −v j )− X j∈N i ∇ x i ψ o ij − X j∈D a i ∇ x i ψ a ij − X j∈D r i ∇ x i ψ r ij − X j∈Π i ∇ x i ψ c ij =− X j∈N i (v i −v j )−∇ x i ψ i (kx ij k) (4.54) where the objective potential ψ o ij is a C 2 smooth function ofkx ij k over [0,ρ 2 ], and Π i ={j∈V|kx ij k≤ρ 0 } is the collision avoidance set for agent i. The overall system consisting of dynamics (4.53) driven by controls (4.54) can be viewed generally as a switched system ˙ z =g α (z),α(t) : [0,∞)→S, whereS is a finite set indexing the switched dynamics, and we let t p , for p = 1, 2,... denote the times when the switching signal α(t) changes [64,148]. It is important to note that discontinuities in (4.54) are due only to the flocking term, arising from switches inG(t), as a consequence of potential smoothness with respect to discernment set switching. Thus, the link dynamics induce a dwell time τ > 0 between transitions in α(t). The first result concerning topology-constrained flocking now follows. Theorem 4.3.1 (Topology-Constrained Flocking). Consider dynamics (4.53) subject to local controls (4.54) and assume that t p − t p−1 > τ for all switch- ing times t p . Then, kx ij k > 0,∀i,j ∈ V, kx ij k < ρ 2 ,∀i ∈ V,j ∈ D a i , and 99 kx ij k > ρ 1 ,∀i∈V,j∈D r i , for all t > 0. Further, v i → v j as t→∞,∀i,j∈V. That is, constraint violations are avoided as repelled agents remain outside of the connection boundary, and attracted agents remain within the disconnection boundary, while collision avoidance and flocking are achieved. Proof. Our analysis is generally along the lines of [149]. Denote byQ s the union of the time intervals when α(t) = s, for s∈S, and let ˆ x∈R mn(n−1)/2 , v∈R mn , be the vectors formed by stacking x ij and v i , respectively [148]. Consider the Lyapunov-like function V α :R mn(n−1)/2 ×R mn →R + defined as V α = 1 2 kvk 2 + n X i=1 ψ i ! (4.55) and the associated level sets Ω α = {(ˆ x, v) ∈ R mn(n−1)/2 ×R mn |V α ≤ c}, for any c > 0. Now, noting that 1/2 P n i=1 ˙ ψ i = P n i=1 v T i ∇ x i ψ i due to the symmetry of potentials ψ o ij ,ψ a ij ,ψ r ij ,ψ c ij and setsN i ,D a i ,D r i , Π i , the time derivative of V α is then [148] ˙ V α =− n X i=1 v T i X j∈N i (v i −v j ) +∇ x i ψ i + n X i=1 v T i ∇ x i ψ i =− n X i=1 v T i X j∈N i (v i −v j ) =−v T (L α ⊗I)v≤ 0 (4.56) as the Laplacian matrixL α is always positive semi-definite assuming connectedness (⊗ denotes the Kronecker product). Thus, for any dynamics indexed by α(t), the level sets Ω α are positively invariant, which implies that the potentials ψ i (kx ij k) remain bounded. We can therefore conclude that since ψ i (kx ij k)→∞, askx ij k→ 0 + for j∈ Π i ,kx ij k→ρ − 2 for j∈D a i , andkx ij k→ρ + 1 for j∈D r i , it follows from the continuity of V α thatkx ij k > 0,∀i,j∈V,kx ij k < ρ 2 ,∀i∈V,j∈D a i , and kx ij k>ρ 1 ,∀i∈V,j∈D r i , for t in an interval ofQ s , i.e. between switching times. 100 By the switching conditions of Figure 4.2, it follows that at the transition to any switched state α(t)∈S the previous relationships continue to hold, and thus they hold in the uniont∈Q s for anys∈S, and thus over allt> 0. Now, as the level sets Ω α are positively invariant and compact (due to (4.56) and continuity ofV α ) and the stacked dynamics are locally Lipschitz (a consequence ofV α ∈C 2 ), it is implied that v T (L α ⊗I)v is uniformly continuous inQ s [64]. These conditions together with the dwell time assumption are then sufficient to ensure that v T (L α ⊗I)v→ 0 ast→∞, which for a connected L α implies v∈ span(1) and flocking is achieved [149]. 4.3.2 Coordinating Non-Local Constraints Having guaranteed the desired dynamical properties of the system under the switching formulation of Figure 4.2, we now require local predicates P a ij and P d ij that properly select candidates j∈C a i ,C d i that are feasible with respect to possibly non-local network constraints. Define by A i (t),W i (t),Y i (t)∈{0, 1} n×n , the local adjacency, addition candidacy, and deletion candidacy information known by agent i, respectively. A proposed topology change is denoted by r, [(i,j),ς,b] T ∈R 4 , where (i,j)∈V×V is the proposed link, ς∈{0, 1} indicates link addition or deletion, and b∈R + is a bid associated with the proposition. The set of topology propositions known toi isR i (t), while the set of propositions available through local communication isR N i ,∪ l∈{N i (t),i} R l (t). The transition of somej / ∈N i ,kx ij k>ρ 2 across ρ 2 (proposed addition), or some j∈N i ,kx ij k≤ ρ 1 across ρ 1 (proposed deletion) for an agent i then initiates the following consensus processes [50]: A i (t + 1) =∨ j∈N i (A i (t)∨A j (t)) (4.57a) W i (t + 1) =∨ j∈N i (W i (t)∨W j (t)) (4.57b) Y i (t + 1) =∨ j∈N i (Y i (t)∨Y j (t)) (4.57c) 101 R i (t + 1) = argmax r k ∈R N i (r k,4 ) (4.57d) where each agenti initializesA i (0),W i (0),Y i (0) witha i (t),ϕ i (t),γ i (t) for rowi and remaining rows to 0 T . The propositionsR i (0) are initialized with local links induced by the transition into the discernment region, propositions that were unfulfilled in previous discernments, or current members ofD a i ,D r i for reevaluation, with bid b> 0 indicating some valuation of the link. AssumingG(t) is connected and does not switch during execution 19 , iterations (6.27) converge in at most n− 1 steps to A i =A(t),W i =W(t),Y i =Y (t), and r∈R i (t) =r max = argmax r k ∈∪ l∈V R l (t) (r k,4 ) for each agent i, that is uniformly to the global topology and candidacy state, and a winning proposition upon which constraint evaluation will be based [50]. Considering the discernment switching of Figure 4.2 and the guarantees of Theorem 4.3.1 concerning the dynamics of members j∈D a i ,D r i , it follows that A(t),W(t) and Y (t) capture fully the space of possible network topologies that could occur between discernments. Thus, each agent can evaluate their local constraints over worst-case topologies and vote on proposition r max . In particular, as f(A(t)) is non-decreasing in A(t), the worst-case topologies with respect to predicate formulationsP u andP l are the maximally and minimally connected networks described by A max =A i ∨W a i and A min =A i ⊕Y d i , respectively, where W a i isW i augmented with linkr max ifς = 0 (link addition), andY d i isY i augmented with link r max if ς = 1 (link deletion). The voting process then proceeds through the following consensus q i (t + 1) =∧ j∈N i (q i (t)∧q j (t)) (4.58) 19 We can guarantee this condition by assuming all undecided propositions are virtual constraint violations, thereby disallowing topology transitions. 102 where q i (t) is the vote on proposition r max for the ith agent, with initialization q i (0) = (∧ k P k i,u (A max ))∧ (∧ k P k i,l (A min )). Like (6.27), voting process (4.58) is con- vergent withinn− 1 steps toq i =∧ j∈V q j (0). Finally, for proposed link (i,j)∈r max , the associated predicates are given by¬P a ij =q i for ς = 0 or¬P d ij =v i for ς = 1, while non-winning propositions are reentered into future rounds of discernment. This now brings us to our main result concerning constraint satisfaction. Theorem 4.3.2 (Constraint satisfaction). Assume system (4.53) evolves according to controls (4.54), with switching illustrated by Figure 4.2, and link discernment predicates P a ij ,P d ij constructed by the consensus processes (6.27), (4.58). Further, assume we have initiallyP(0) = 1 andC a i (0) =C d i (0) =D a i (0) =D r i (0) =∅,∀i∈V. Then P(t) = 1 for all t > 0. That is, for the extent of execution, the network topology of the system (i.e. agent interaction) satisfies the composition of constraints P(t) having a potentially non-local nature. Proof. ByTheorem4.3.1, theexclusivityandpersistencepropertiesofC a i ,C d i ,D a i ,D r i (c.f. Figure 4.2), and the assumed initial condition, it follows that transitions in G(t), due to link addition or deletion, occur over only candidate links, i.e. (i,j)∈ E(t + )∧(i,j) / ∈E(t − )⇒j∈C a i (t − )∧i∈C a j (t − ) and (i,j) / ∈E(t + )∧(i,j)∈E(t − )⇒ j∈C d i (t − )∧i∈C d j (t − ) hold. Now, fromP a ij ,P d ij based on voting (4.58) we conclude thatj∈C a i orj∈C d i if and only if∧ n w=1 ((∧ k P k w,u (A max ))∧(∧ k P k w,l (A min ))) = 1, and as local votesq i agree, we are guaranteed the symmetry ofC a i ,C d i . Finally, since the constraint functionsf(A(t)) are non-decreasing inA(t), we havef(A(t))≤f(A max ) and f(A(t))≥ f(A min ), from which we concludeP k w,u (A max ))⇒P k w,u (A(t)) and P k w,l (A min ))⇒P k w,l (A(t)) for all w∈V and t> 0, and the result follows. 103 4 3 2 1 0 20 40 60 −20 −10 0 10 20 30 40 50 60 (a) 4 3 2 1 0 20 40 60 −20 −10 0 10 20 30 40 50 60 (b) 4 3 2 1 0 20 40 60 −20 −10 0 10 20 30 40 50 60 (c) Swarm Size Time ||¯ v− v i || 0 10 20 30 40 0 10 20 30 40 30 40 50 60 0 0.5 1 1.5 (d) |N i | : Agent Time |N i | : Hub 0 10 20 30 40 0 10 20 30 40 2 3 4 5 0 1 2 3 (e) |H (2,p) i | Time |H (2) i | 0 10 20 30 40 0 10 20 30 40 2 4 6 1 2 3 4 (f) Figure 4.8: Flocking simulation composed of stationary hubs and mobile agents. 4.3.3 Constrained Flocking in Simulation In this section, we present simulation results of our proposed control formulation, and show that the desired constraint, flocking, and collision avoidance properties holdunderanovelcoordinationscenario. Supposewehaveasystemofn = 10agents operating in a bounded workspace over R 2 , with interaction model parameters (ρ 2 ,ρ 1 ,ρ 0 ) = (20, 15, 5). We assign 6 agents the flocking controller (4.54) together with an additional dispersive control u dis =− P j∈N i ∇ x i (1/kx ij k 2 ) for the purposes of forcing link deletions in the network 20 . The remaining 4 agents we designate as stationaryhubs withnoassignedmobilitycontrol. Thehubscanbeviewedinvarious contexts, e.g. as data repositories with which the mobile agents require connection, relay switches to external networks, or as markers partitioning subregions of the 20 See our previous work [9] for analysis of dispersion under constraints. 104 workspace. We assign to the hub agents max degree constraints|N 1 |,|N 2 |≤ 2 and |N 3 |,|N 4 |≤ 3, while the mobile agents are assigned connectivity constraintsλ 2 > 0, |N i |≥ 2, and min hub constraints|H (2) i |≥ 1,|H (2,p) i |≥ 2, where we denote byH (k) i andH (k,p) i the k-hop neighbors and paths that correspond to hubs, respectively. Constraints of this nature may reflect spatial limitations on hub availability, and mobile clients that require a minimum level of interconnectivity and connection to a hub with path redundancy. Figure 4.8 depicts a simulation of the coordination scenario with an initial configuration satisfyingP(0) = 1 as shown in Figure 4.8a, where the mobile agent velocities are random (black arrows) and connectivity is represented by black links. Hubs are depicted in green (with radii ρ 2 ,ρ 1 shown) and mobile agents in red, where constrained interaction is depicted by red solid links (attraction) and blue dashed links (repulsion). The spatial progression of the system is given by the intermediate and final configurations Figs. 4.8b and 4.8c, with velocity alignment clearly achieved. Figure 4.8d reinforces flocking behavior with converged velocity deviationk¯ v−v i k (top), where constraint action is evident through fluctuations in velocity and swarm size (bottom). Finally, from Figs. 4.8e and 4.8f we can conclude constraint satisfaction is achieved: hub max degree (4.8e, top), agent min degree (4.8e, bottom), agent-hub connectivity (4.8f, top), and agent-hub paths (4.8f, bottom). 4.4 Estimating and Controlling Algebraic Con- nectivity While spatial topology control yields a framework for sound decision-making and mobility, we still require methods to evaluate important topological constraints. In 105 addition, there may be cases where maintaining a constraint is not sufficient, as one may wish to improve some topological metric. A constraint that is ubiquitous in collaboration is the connectivity property of a proximity-limited network. In particular, the feasibility of collaborative processes, such as consensus or flocking, rests on connectedness, while the connectivity metric when improved can yield gains in collaborative efficiency. We are therefore motivated to consider the following problem: Problem 4. Design mobility controls u i such that ifG is initially connected it remains connected for all time, and λ 2 is maximized while respecting constraints on agent node degree,|N i |≤β i , with β i ≥ 1. Notice that in the above problem we retain a discrete topological constraint. We do so to illustrate the inclusion of continuous and discrete constraints in controlling a collaborative system. 4.4.1 Distributed Connectivity Estimation To effectively control the connectivity of our system, we establish a feedback loop by generating, in a distributed manner, an estimate of the algebraic connectivity λ 2 of the network. Yang et. al. in [61] proposed a distributed eigenvalue/vector (eigenpair) estimation scheme based on the power iteration algorithm. However, power iteration converges with a rate that is linear in|λ n−1 /λ n |, the ratio of the dominant eigenvalues of the target matrix [150]. Therefore, when the dominant eigenvalues are close in magnitude, the power iteration algorithm converges very slowly. We alleviate the issue of slow convergence by considering the closely related inverse iteration algorithm (also called the inverse power iteration) [150]. First, 106 considerthefollowingdeflationofL, aimedatremovingtheinfluenceoftheeigenpair associated with λ 1 : ˆ L =L + n +δ n 11 T (4.59) where δ > 0. The deflated matrix ˆ L has eigenvalues{λ 2 ,...,λ n ,n +δ}, with associated eigenvectors{v 2 ,...,v n , 1/ √ n}. Defining the matrix ( ˆ L−μI) −1 , with μ∈R, it follows that the eigenvectors of ( ˆ L−μI) −1 are the same as those of ˆ L, while the eigenvalues are precisely {(λ 2 −μ) −1 ,..., (λ n −μ) −1 , (n +δ−μ) −1 } (4.60) Thus if we choose μ close to λ 2 (i.e. μ is an initial estimate of our target λ 2 ), the dominant eigenvalue of ( ˆ L−μI) −1 will be quite large, meaning the power iteration applied to ( ˆ L−μI) −1 will converge quickly to the network algebraic connectivity. This idea underlies the inverse iteration algorithm as summarized in the following. Proceeding iteratively with index k, we first solve the linear system ( ˆ L−μI)y = ˜ v (k) 2 (4.61) for the vector y∈R n , where ˜ v (k) 2 is the eigenvector estimate at step k. To do so we consider the overrelaxed Jacobi algorithm as it is well suited for distributed implementation [126]. Defining the relaxation factor 0<γ < 1 and iterate index p, the over-relaxed Jacobi iteration applied to (4.61) in distributed form is given by: y (p+1) i = (1−γ)y (p) i − γ d i +δ/n + 1−μ − X j∈N i A ij y (p) j + δ n + 1 ! X j6=i y (p) j − ˜ v (k) 2,i (4.62) 107 where y i and ˜ v 2,i are per-agent components of y and ˜ v 2 . The iterates (4.62) incorporate mostly local information, requiring global information only to compute the second summation. For a distributed implementation of (4.62) we consider the consensus algorithm (2.5) with z a i (0) =y (p) i , giving the second summation in (4.62) as (δ/n + 1)(n¯ z a −y (p) i ). The iteration (4.62) converges to a solution y of (4.61) if ( ˆ L−μI) is positive definite and symmetric, and γ is chosen sufficiently small. By construction ( ˆ L−μI) is symmetric and positive definiteness is ensured by choosing 0≤μ<λ 2 . We note that iteration (4.62) generally converges quickly in practice [126], a convenient property for our purposes 21 . Thenormalization ˜ v (k+1) 2 =y/kykthenyieldsaneigenvectorestimate. Assuming convergence of iteration (4.62) after ¯ p iterations, the normalization is given in distributed form by ˜ v (k+1) 2,i =y (¯ p) i / √ n¯ z b , where consensus is applied with z b i (0) = (y (¯ p) i ) 2 . Finally, an estimate of the eigenvalue associated with ˜ v (k+1) 2 is given by the Rayleigh quotient ˜ λ (k+1) 2 = (˜ v (k+1) 2 ) T L˜ v (k+1) 2 [150]. The local eigenvalue estimates are computed once more using consensus with z c i (0) =−˜ v (k+1) 2,i X j∈N i A ij ˜ v (k+1) 2,j (4.63) yielding ˜ λ (k+1) 2,i =n¯ z c . The distributed inverse iteration as constructed converges to the dominant eigenpair of ( ˆ L−μI) −1 , that is{λ 2 ,v 2 } of L, our desired result. Convergence is linear in the ratio|(μ−λ 2 )/(μ−λ 3 )|, thus, with μ chosen close to λ 2 , it is far superior to power iteration [150]. In a realistic application inverse iteration would run in conjunction with motion control, and after a seeding period where we assume 21 This fact is reflected in our extensive simulation studies of applying the over-relaxed Jacobi iteration to Laplacian matrices. 108 μ = 0, a reasonable estimate of λ 2 would be available to initialize μ for future estimation. Remark 4.4.1. Although the proposed inverse iteration requires consensus for distribution, the power iteration of [61] exploits consensus to similar effect. Thus, the Jacobi iterations (4.62) do not negatively impact performance when compared to the power iteration. In fact, as will be demonstrated in simulation, the deflation employed by [61],I−αL for sufficiently smallα, generates poor convergence scaling in network size (as α must scale like 1/n), an effect that our proposed formulation does not exhibit. 4.4.2 Connectivity Maximizing, Constraint-Aware Mobil- ity As constraint satisfaction, link maintenance, and collision avoidance are passive objectives, we introduce a final potential with the objective of connectivity max- imization. Connectivity impacts directly the rate at which a network diffuses information among agents (e.g. in a consensus process), and thus applications requiring the observation and exchange of significant data in a network (e.g. sen- sor networks) could benefit greatly from aggregative, connectivity maximizing maneuvers. To begin, recall the alternative definition of algebraic connectivity: λ 2 = inf y∈1 ⊥ y T Ly = inf y∈1 ⊥ n X i=1 X j∈N i A ij (y i −y j ) 2 (4.64) where y6= 0, and we assumekyk = 1. It is clear from (4.64) that decreasing inter-agent spacing (increasing A ij ) and adding new links, A ij > 0, acts to increase network connectivity, an intuitive result. Given the dependence of A ij on agent 109 displacement, we derive mobility controls that drive connectivity maximization by considering the composite potential φ = 1 (λ 2 ) η = 1 (v T 2 Lv 2 ) η (4.65) with η > 0. The final agent controls are then derived by taking the negative gradient over the control potentials as follows: u i =−∇ x i (ψ i +φ) (4.66) where ψ i = X j∈R i ψ v ij + X j∈N i ψ l ij + X j∈Ω i ψ c ij (4.67) with collision avoidance set Ω i ={j∈V|kx ij k≤ ρ 0 }. In particular, applying (4.64) the connectivity maximizing term is given by ∇ x i φ =∇ λ 2 φ X j∈N i ∇ x i A ij (v 2,i −v 2,j ) 2 (4.68) where we have used the fact that∇ x i λ 2 =∇ x i v T 2 Lv 2 = v T 2 (∇ x i L)v 2 due to L T =L andkv 2 k = 1. We can then substitute the connectivity estimates ˜ v 2,i and ˜ λ 2,i from Section 4.4.1 into (4.68) to obtain the final connectivity maximization controls 22 . Notice that (4.68) takes on the desired distributed form, requiring only the local exchange of agent state and connectivity estimates to implement. Theorem 4.4.1. Assume the closed loop system (4.1), (4.66) is such that initially G is connected and|N i |≤ β i . Then for all future timeG remains connected, |N i |≤β i is satisfied, and collisions between agents are avoided. 22 Note that in following−∇ xi φ the system may only reach local maxima in connectivity, a consequence of the non-concavity of λ 2 in x i . 110 Proof. Consider the total system energy V = 1 2 n X i=1 ψ i +φ ! (4.69) and first observe that n X i=1 X j∈R i ˙ ψ v ij = 2 n X i=1 X j∈R i ˙ x T i ∇ x i ψ v ij (4.70) due to symmetry ofR i , with analogous relationships holding for ˙ ψ l ij and ˙ ψ c ij . Further, we have ˙ φ =∇ λ 2 φ n X i=1 X j∈N i ˙ x T ij ∇ x ij A ij (v 2,i −v 2,j ) 2 = 2 n X i=1 ˙ x T i ∇ x i φ (4.71) by (4.68) and symmetry of elements A ij . Therefore ˙ V =− n X i=1 k∇ x i (ψ i +φ)k 2 ≤ 0 (4.72) which implies the positive invariance of the level sets V −1 ([0,c]). Askx ij k→ρ + 1 for j∈R i ,kx ij k→ρ − 2 for j∈N i , andkx ij k→ρ + 0 for j∈ Ω i , we have ψ v ij →∞, ψ l ij →∞, and ψ c ij →∞. Thus we conclude that all links are preserved and connectivity is guaranteed 23 , there exists no j∈R i that becomes a neighbor of an agent i (link discrimination), and collisions are avoided. Finally, as agents i and j become mutual candidates for connection only when|N i | +|C i |<β i , we conclude that|C i |≤ β i −|N i | for all time. Since only j∈C i can become neighbors due 23 We choose here to employ a link retention policy as in an aggregative objective, link deletion is of lesser concern. 111 to proper link discrimination, we have max G (|N i |) =|N i | +|C i |≤β i , our desired result. 4.4.3 Simulated Connectivity Estimation and Control In this section, we present simulation results of our proposed connectivity estima- tion method and constrained connectivity control. First, to evaluate connectivity estimation we compare our proposed inverse iteration against a discrete implemen- tation of the power iteration proposed in [61]. Specifically, a set of 20000 Monte Carlo scenarios is simulated over randomly generated connected networks (λ 2 > 0) having size n∈ [5, 50] uniformly. For each test graph, the power iteration and inverse iteration are applied with random initial conditions, where we consider μ∈ (0, 0.4λ 2 , 0.8λ 2 ) to demonstrate the impact of varying the quality of initial connectivity estimates. To capture complexity, we track for each test the average inter-agent messages in the network during execution, per agent; a relevant metric that reflects the effect of convergence rate and its coupling with the embedded consensus processes required for decentralization. The results of our testing are depicted in Figure 4.9, from which we can conclude the following. First, notice that in small networks where initial connectivity estimates are poor, the algorithms perform equivalently in terms of complexity. Such a result can be attributed to the effects of the power iteration deflation, which degrades convergence sufficiently to account for the additional complexity of the Jacobi algorithm required for the inverse iteration. However, when the quality of μ increases we see an expected advantage for the inverse iteration, as the convergence ratio|(μ−λ 2 )/(μ−λ 3 )| drives performance. As network size increases, similar behavior is observed, however power iteration fairs worse due to poor scaling in the Laplacian deflation and thus the convergence ratio of power iteration. A reasonable result also lies in increasing 112 Inverse (μ = 0.8λ 2 ) Inverse (μ = 0.4λ 2 ) Inverse (μ = 0) Power Network Size (n) Communication Load 5 10 15 20 25 30 35 40 45 50 ×10 5 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 2 Figure 4.9: Distributed inverse iteration versus power iteration. the quality of μ, that is the effects of increased network size are dampened due to an improved convergence ratio. To demonstrate the aggregation properties of our control laws we consider a 15-agent system with random and connected initial positions, as depicted by Figure 4.10a. The interaction model is defined by radii (20, 18, 5), where we assume μ = 0 and γ = 0.2. Figure 4.10b depicts the final converged configuration of the system under connectivity maximization with constraints|N i |≤ 9, where agent paths are depicted by dashed lines. As desired, connectivity is maintained and increased, while the local constraints remain satisfied and collisions are avoided. Figure 4.10c demonstrates the regulation of network connectivity and node degree under varying maximal node constraints, with our illustrated test case shown in black. 113 0 10 20 30 40 50 0 10 20 30 40 50 (a) 0 10 20 30 40 50 0 10 20 30 40 50 (b) 9 Max 8 Max 7 Max 6 Max 5 Max Max Degree Time Connectivity 0 50 100 150 200 0 50 100 150 200 4 6 8 10 0 0.1 0.2 0.3 0.4 (c) Figure 4.10: Aggregation simulation of a 15-agent system with connectivity maxi- mization under degree constraint. Our final simulation consists of adding a secondary objective to the constrained connectivity objectives. Consider the case of a 10-agent system under local con- straint|N i |≤ 7, where one of the agents is a leader that tracks a preplanned trajectory through the workspace, while the remaining agents apply controls (4.66). Figure 4.11 depicts a composite of snapshots of the agent configurations throughout the simulation, with the leader and its trajectory shown in green and a detailed zoom showing the final configuration with node degree highlighted by agent color. Observe that like the aggregation behavior, the desired system specifications are met for the extent of the simulation, and the agents successfully follow the lead agent. 4.5 Route Swarm: A Case Study of Heteroge- nous Mobility The development of our spatial topology control rests on the underlying condi- tion that the network is homogeneous, i.e., all agents share identical modes of communication, sensing, and mobility. However, as multi-robot systems increase 114 Zoom End Start −10 0 10 20 30 40 50 60 70 80 −100 −80 −60 −40 −20 0 20 40 Figure 4.11: Leader-following simulation, with n = 10 and connectivity maximiza- tion under|N i |≤ 7. in complexity and the scope of problems to address expands, heterogeneity will become a necessity in collaborative systems. As a case study of heterogeneity, particularly the integration of low-level spatial interaction and high-level network objectives, we introduce at a high level a novel hybrid architecture for command, control, and coordination of networked robots for sensing and information routing applications, called INSPIRE (for INformation and Sensing driven PhysIcally REconfigurable robotic network). In the INSPIRE architecture, we propose two levels of control. At the low level there is a Physical Control Plane (PCP), and at the higher level is an Information Control Plane (ICP). At the PCP, iterative local communications between neighboring robots is used to shape the physical 115 network topology by manipulating agent position through motion. At the ICP, more sophisticated multi-hop network algorithms enable efficient sensing and information routing. Consider a system of n =m +s agents consisting of m mobile robots indexed byI M ,{1,...,m}, and s static sensors indexed byI S ,{m + 1,...,m +s}. From the set of static sensorsI S we construct f information flows indexed by I F ,{n + 1,...,n +f}, each consisting of source-destination pairs defining a desired flow of network information. For a given flow i∈I F , we use the following notation:F i ∈I S ×I S represents the source and destination nodes for flow i, with F s i ∈I S andF d i ∈I S representing source and destination indices, respectively. Further, for convenience we use notation x s i ,x d i ∈R 2 to represent the position of the source and destination for the flow i∈I F . The set of flow pairs is denoted F,{F 1 ,...,F f }. The cost for flowk∈I F is then taken to be the sum of ETX values on the path of the flow, i.e.: W k = X (i,j)∈E k F w ij (4.73) with w ij given by (4.8), where we apply notation G k F = (V k F ,E k F ) as the graph defining the interconnection over flow k∈I F (we give a concrete definition of flow membership in the sequel). At any time, a subset of these static pairs is active, forming the set of active flowsI F , calling for dynamic configurability of the hybrid network, our contribution in this work. Thus, at a high level our system objective is to facilitate information flow for each source/destination pair by configuring the mobile robots such that each flow is connected and is at least approximately optimal in terms of data transmission, and that the entire network itselfG is connected 116 to guarantee complete network collaboration. Our problem in this work is then formalized as follows: Problem 5 (Multi-flow optimization). The network-wide goal then is to find an allocation and configuration of mobile agents so as to minimize the total cost function 24 P f k=1 W k while maintaining both intra-flow connectivity (to guarantee information delivery) and inter-flow connectivity (to guarantee flow-to-flow infor- mation passage/collaboration). To guarantee the latter, note that some of the nodes will need to be positioned to act as bridges to preserve global connectivity and may not necessarily participate in active flows. Such a connectivity constrained global optimization is difficult to solve exactly; thus we propose the Route Swarm heuristic to generate our desired dynamic flow optimizing configurations; the heuristic tries to approximate the solution to the unconstrained problem, which we show in the following can be formulated in terms of convex optimization, but has additional mechanisms to preserve connectivity. 4.5.1 Information Control Plane (ICP) There are two key elements in solving Problem 5: on the one hand, within a given flow k∈I F , for a given allocation of a certain number of mobile nodes to that flow, node configuration should minimize the flow cost W k . On the other hand, the number of mobile nodes allocated to each flow should minimize the overall cost P k W k . We first consider these optimizations ideally, in the absence of connectivity constraints. The first, per-flow element of the network optimization dictates the desired spatial configuration of allocated mobile nodes within a flow. 24 The negative of the cost could be treated as a utility function. We therefore equivalently talk about cost minimization or utility maximization. 117 Theorem 4.5.1 (Equidistant optima). For a fixed number of mobile nodes m k , |V k F | allocated to a given flow and arranged on the line between the source and destination of that flow, the arrangement which minimizes (4.73) is one where the nodes are equally spaced. Proof. This follows from the following general result: to minimize a convexy( − → z ) s.t. P z i =c, the first order condition (setting the partial derivative of the Lagrangian with respect to each element z i to 0) yields that ∂y z i =μ where μ is the Lagrange multiplier. Now if ∂y ∂z i is the same for all z i , then the solution to this optimization is to set z i = c |z| , i.e. all variables are made to be the same. The z i in the above correspond to the inter-node distances d ij , and the y correspondsto(4.73). Sincew ij isaconvexfunctionofdistancebetweenneighboring nodes, the path metricW k is a convex function of the vector of inter-node distances. Further, since the sum of all inter-node distances is the total distance between the source and destination of the flow, which are static, it is constrained to be a constant d k . Since the weight of each link is the same function of the inter-node distance, ∂W k d ij is the same for all pairs of neighboring nodes. Therefore the intra-flow optimization (i.e., choosing node positions to minimize W k ) is achieved by a equal spacing of the nodes. The second, global cross-flow element of the network optimization dictates the number of mobile nodes allocated to each flow. The goal is to minimize the total network cost P k W k . Our approach to solving this optimization is motivated by the following observation: When the intra-flow locations of the robots are optimized to be equally spaced, W k is a function of the number of nodes m k allocated to flow k, and the total number of nodes allocated to all flows is constrained by the total number of mobile nodes. If we could show that W k is a convex function, then to minimize the total network cost we need to identify the allocation at which the 118 marginal costs for all flows are as close to equal as possible (ideally, if m k was a continuous quantity, they would all be equal at the optimum point, but due to the discrete nature of m k this is generally not possibly). In the following, for ease of analysis, we consider the continuous relaxation of the problem, allowing m k to be a real number, and hence W k to be a continuous function. Theorem 4.5.2 (Convexity). For any flow k that has been optimized to have the lowest possible costW k (i.e. allm k mobile nodes are equally spaced),W k is a convex function of m k . Proof. First note that κ k =h k (W k ) =h k ( P ij w ij ). For equally spaced nodes, then κ k = h k (m k w( d k m k +1 )). Since h k is a monotone convex function, then κ k will be convex in m k so long as its argument W k is convex in m k . This can be shown as follows: W k = m k w d k m k + 1 ! (4.74) ⇒W 0 k = w d k m k + 1 ! − 1 (m k + 1) 2 w 0 d k m k + 1 ! (4.75) ⇒W 00 k = w 00 d k m k + 1 ! 1 (m k + 1) 3 (4.76) Since the link weight function w(·) is a convex function of the internode distance, we have thatw 00 ( d k m k +1 ) is positive, and therefore we have thatW 00 k is positive, hence W k is convex in m k . Note that in realitym k is a discrete quantity, but the above argument suffices to show thatW k is a discrete convex function of m k (since convexity over a single real variable implies discrete convexity over the integer discretization of the variable). 119 Theorem 4.5.3 (Optimality). The problem of minimizing P k W k (m k ) subject to a constraint on P k m k can be solved optimally in an iterative fashion by the following greedy algorithm: at each iteration, move one node from the flow where the removal induces the lowest increase in cost to the flow where its addition would yield the highest decrease in cost, so long as the latter’s decrease in cost is strictly higher in absolute value than the former’s increase in cost (i.e. so long as the move serves to reduce the overall cost). Intuitively, the above algorithm works by moving the system iteratively towards the optimum by following the steepest gradient in terms of cost reduction for the movement of each node. Since the overall optimization problem is convex, there is only a single optimum, to which this algorithm will converge. Moreover, since there is a strict improvement in each step and there are a finite number of nodes, the algorithm reaches the optimal arrangement in a finite number of steps. Thus far, we have described both the intra-flow and inter-flow optimization problems in an ideal setting where both problems are convex optimization problems and as such can be solved exactly. However, in the robotic system we are considering there is one significant source of non-ideality/non-convexity, which is that the network must be maintained at all times in a connected configuration. This has two consequences. First, some of the mobile nodes may be needed as bridge nodes that do not participate in any flow and are instead used to maintain connectivity across flows. Second, the locations of some of the nodes even within each flow may be constrained in order to maintain the connectivity requirement. The solution for the constrained problem therefore may not correspond exactly to the solutions of the ideal optimization problems described above. We therefore develop a heuristic solution that we refer to as Route Swarm, which is inspired by and approximates the ideal optimizations above but is adapted 120 Algorithm 1 ICP optimization algorithm. 1: procedureInformationControlPlane 2: . Detect initial flow members based on shortest paths: 3: for i∈I F do 4: M←ShortestPath(G,F i ) 5: end for 6: . Detect flow-to-flow bridges for connectivity: 7: b←DetectBridges(G,F,M) 8: . Detect best connectivity-preserving flow detachment: 9: d←BestDetachment(G,F,M,b) 10: . Compute flow attachment with most utility: 11: a←BestAttachment(F,M) 12: . Ensure optimizing reconfiguration exists (weighted by β∈R + ): 13: if a>βd then 14: . Optimal command is best detach/attach pair: 15: returnC d ←a 16: end if 17: end procedure to maintain connectivity. Route Swarm has both an intra-flow and inter-flow component. As per the INSPIRE architecture, the intra-flow function is performed by the PCP, while the inter-flow function is performed by ICP. The ICP algorithm, shown as pseudocode as Algorithm 1, approximates the ideal iterative optimization described above by allocating mobile robots between flows greedily on the basis of greatest cost-reduction; to handle the inter-flow connectivity constraint, it incorporates a subroutine (Algorithm 2) to detect which nodes are not mobile because they must act as bridge nodes. Moreover, it also allocates any nodes that are no longer required to support an inactive flow to join active flows. And within each flow, the PCP algorithm (Algorithm 7) attempts to keep the robots as close to evenly spaced as possible while taking into account the inflexibility of the bridge nodes. The details of the route swarm algorithm are given below. 121 Algorithm 2 ICP bridge detection. 1: procedureDetectBridges(G,F,M) 2: . Initialize supergraph with nodes for each flow: 3: S← ({i∈I F },∅) 4: . Append nodes/edges for non-flow members: 5: S←AddNonFlowMembers(S,F,M) 6: . Append edges for flow members: 7: S←AddFlowMembers(S,F,M) 8: . Bridges lie on shortest path between all flow pairs: 9: return b←ShortestPaths(S,F) 10: end procedure 4.5.2 The Route Swarm Heuristic In solving the connectivity constrained version of the flow optimization of Theorems 4.5.1, 4.5.2, and 4.5.3, due to problem complexity we provide a heuristic algorithm to solve the inter-flow allocation problem, leaving the intra-flow optimization to the PCP (Section 4.5.3). Algorithm 1 depicts the high-level components of the proposed heuristic, each of which is detailed in the sequel. To begin, we define the flow membership for an agent as i∈I M asM i ∈I F , where the set of memberships is denoted asM,{M 1 ,...,M n }. We denote with G i F = (V i F ,E i F ) the graph defining the interconnection over flow i∈I F of agents j∈I S ∪I M with i∈M j . Thus we haveV i F ={j∈I S ∪I M |i∈M j } and (j,k)∈E i F ⇔ (i∈M j ∩M k )∧ (j∈N k ), with k∈I S ∪I M . Notice that by definitionG i F ⊆G. We will also refer to the collection of flow graphs simply byG F . We also have flow neighbors defined asN F i ={j∈N i |M j ∩M i 6=∅}. Furthermore, we denote withD i ∈I M the detachable agentsj∈I M for flowi∈I F , i.e. those agents for which reconfiguration does not impact network connectedness. The set of detachments is denoted byD,{D 1 ,...,D f }. Due to the system connectivity constraint, the ICP is not free to select any mobile agent for flow reallocation, specifically as intra-flow connectivity or overall 122 } b i =1 ⇒ G S Figure 4.12: Example supergraph construction for a network with f = 2 flows, s = 4 static nodes, and m = 6 mobile robots. flow-to-flow connectivity may be lost. Thus, the ICP must detect bridges, i.e., mobile nodes whose reconfiguration might break flow-to-flow connectivity over the network, and also consider safe detachments (flow-to-flow motion), i.e., a node whose reconfiguration in the workspace does not impact the connectivity of the source flow. Respecting our connectivity constraint, we first detect initial flow membership by computing the shortest path (in terms of link cost) for each source/destination pair per flow by using for example Dijkstra’s algorithm (lines 3-5 of Algorithm 1). This defines the connected backbone for each flow implicitly identifying the mobile nodes required to maintain flow-connectivity. Additionally, we have optimality of the connected flow backbones as we maximize the link utility (or minimize path costs) between source and destination nodes. In detecting bridge agents we follow the process outlined by Algorithms 2, 3, and 4, where we denote by b i ∈{0, 1} the status of agent i∈I S ∪I M as a connectivity preserving bridge, with b ={b 1 ,...,b n }. Briefly speaking, the primary problem of this process is the construction of a supergraph, denotedS = (V S ,E S ), defining the interconnection of the flows. In this way, we can identify nodes which are critical in defining the flow-to-flow connectivity over the system. Figure 4.12 123 depicts an example of supergraph construction. We first add one node for each flow in the system, and then we add a node for each non-flow member in the system that represents potential bridge candidates. Edges between non-flow members are preserved, while there is only one edge between any given non-flow member and the node which represents a flow in the supergraph. Bridges can then be simply detected as the members of the shortest path between any pair of nodes representing flows in the supergraph (e.g. Figure 4.12), where connectivity is guaranteed by construction: Proposition 4.5.1 (Bridge detection). Consider the graph G c ⊆G obtained by including all the flow-members for any flow and the non-flow members belonging to any shortest path of the supergraphS. ThenG c is a spanning-graph representing a connected component of the graphG. Proof. In order to prove this result we must show that both intra-flow and inter-flow connectedness is ensured. The former follows directly from the flow-membership definition while the latter follows from the connectedness between any pair of flows. After identifying the bridge agents that maintain flow-to-flow connectivity and the safely detachable agents per-flow (those which are not on the backbone), the ICP issues reconfiguration commandsC i ∈I F to mobile agents i∈I M indicating a desired flow membership towards optimizing inter-flow agent allocation. Specifically, as detailed in Algorithms 5 and 6, and as motivated by Theorem 4.5.3, we compute the safe detachment having the least in-flow utility, and couple it with the flow attachment (i.e. the addition of a contributing flow member) which improves most in terms of link utility and flow alignment (the primary contributions a mobile agent can have in information flow). This decision, if feasible (utility of attachment 124 Algorithm 3 ICP supergraph non-flow member nodes/edges. 1: procedureAddNonFlowMembers(S,F,M) 2: . Add nodes for each non-leaf, non-flow member: 3: V + ←{i∈I M | (M i =∅)∧ (N i ≥ 2)} 4: V S ←V S ∪V + 5: . Add edges between non-flow members: 6: E S ←E S ∪{(i,j)| (i,j∈V + )∧ (j∈N i )} 7: . Add non-flow member to flow member edges: 8: E S ←E S ∪{(i,j)| (i∈V + )∧ (j∈N i )∧ (M j 6=∅)} 9: returnS 10: end procedure Algorithm 4 ICP supergraph flow member edges. 1: procedureAddFlowMembers(S,F,M) 2: . Add edges due to multiple flow memberships: 3: for i∈I M ||M i |≥ 2 do 4: . Add an edge for every membership pair: 5: E S ←E S ∪{(i,j)|j∈I F ∩M i } 6: end for 7: returnS 8: end procedure outweighs cost of detachment), is then passed to the PCP to execute the mobility necessary for reconfiguration, achieving our goal of dynamic utility improving and connectivity preserving network configurations. 4.5.3 Physical Control Plane (PCP) The complementary component to the ICP in the INSPIRE architecture is the PCP which coordinates via state feedback, i.e.{x,G,G F }, to generate swarming behaviors that optimize the network dynamically in response to ICP commands. Our desire for generality in coordinating behaviors dictates that the PCP takes on 125 Algorithm 5 ICP best flow detachment. 1: procedureBestDetachment(G,F,M,b) 2: . Non-member, non-bridges are detachable: 3: for (i∈I M )∧ (M i =∅)∧ (¬b i ) do 4: j← Flow most contributed to by agenti (utility) 5: M i ←j 6: D j ←D j ∪i 7: end for 8: . Best detachment has least in-flow utility: 9: return argmin i∈D ( P j∈N F i w ij ) 10: end procedure Algorithm 6 ICP best flow attachment. 1: procedureBestAttachment(F,M) 2: . Determine utility of attachment per flow: 3: for i∈I F do 4: . Weigh added node utility against flow path alignment: 5: a i ← (|V i F | + 1) P j∈V i F 1/kx j − 1/2(x s i +x d i )k 6: end for 7: return argmax i∈I F (a i ) 8: end procedure a switching nature, associating a distinct behavior controller with each of a finite set of discrete agent states. Specifically, define S i ∈B,{SWARMING,RECONFIGURE} (4.77) as the behavior state of a mobile robot i∈I M , whereB is the space of discrete agent behaviors. Here, whenS i = SWARMING, agent i acts to optimize its assigned flowM i . Otherwise, whenS i = RECONFIGURE, agent i traverses the workspace fulfilling global allocation commandsC i from the ICP. In this work, the state machine that drives the switching of the behavior controllers is depicted in Algorithm 7. Each component comprising the PCP switching is detailed in the sequel. 126 Algorithm 7 PCP switching logic. 1: procedurePhysicalControlPlane 2: for i∈I M do . Control mobile agents 3: . Reconfiguration Commanded: 4: ifC i 6=∅ then 5: Set waypoint to target flowC i ∈I F 6: S i ← RECONFIGURE 7: end if 8: . Flow members, to-flow maneuver: 9: ifM i 6=∅∧S i 6= RECONFIGURE then 10: if¬(On Path Connecting FlowM i ) then 11: Set waypoint to flowM i ∈I F 12: S i ← RECONFIGURE 13: else 14: S i ← SWARMING 15: end if 16: end if 17: . Agent Behaviors: 18: ifS i = SWARMING then 19: Run dispersion controller optimizing flowM i 20: end if 21: ifS i = RECONFIGURE then 22: Run waypoint controller for reconfiguration 23: if At Waypoint then 24: S i ← SWARMING 25: end if 26: end if 27: end for 28: end procedure 4.5.4 Maintaining Flow Connectivity Notice that in maintaining network connectivity, we require only link retention action, allowing us to immediately choose P a ij , 0, ∀i∈I M ,j∈I M ∪I S , (S i ,S j )∈B×B (4.78) 127 for the link addition predicates, effectively allowing link additions to occur between all interacting agents across all network states 25 . Now, in accordance with Algorithm 7, the link deletion predicates are given by: P d ij , 1, (S i ∨S j = SWARMING)∧ ((M i ∩M j 6=∅)∨ (b i ∨b j = 1)) 0, otherwise (4.79) where by assumptioni,j∈I M , i.e. only mobile agents apply controllers. We choose link retention (6.25) to guarantee that connectivity is maintained both within flows acrossG F , and from flow to flow across bridge agents over the supergraphS, noting that idle agents withM i =∅ and reconfiguring agents withS i = RECONFIGURE are free to lose links as they have been deemed redundant by the ICP with respect to network connectivity. Flow Reconfiguration Maneuvers While maintaining connectivity as above, each agent further acts according to ICP reconfiguration commands towards optimizing inter-flow allocations. Specif- ically, in response to command C i , agent i enters the reconfiguration state S i ← RECONFIGURE, and begins to apply a waypoint controller as follows 25 Although in this work we allow all link additions, link addition control could be useful for example in regulating neighborhood sizes to mitigate spatial interference, or to disallow interaction between certain agents. 128 (c.f. lines 4-7 of Algorithm 7). WhenS i = RECONFIGURE, agent i applies exogenous objective controller u e i , x w −x i kx w −x i k − ˙ x i (4.80) where x w ∈R 2 is the target waypoint calculated as the midpoint of target flow C i . The input (4.80) is a velocity damped waypoint seeking controller, having unique critical point x i →x w (i.e. a point at which u e i = 0), guaranteeing that the target intra-flow positioning (and thus membership) for agent i is achieved. As the convergence of x i →x w is asymptotic in nature, to guarantee finite convergence and state switching, we apply a saturationkx w −x i k≤ w with 0< w << 1 to detect waypoint convergence, initiating a switch toS i ← SWARMING as in lines 23-25 of Algorithm 7. Although by design the ICP commandsC i select agents i∈I M such that both the per-flow graphsG F and the system-wide supergraphS remain connected, we can exploit the ICP↔ PCP feedback loop to alternatively provide to the ICP the flexibility to choose any agent for safe detachment from a flow, given such a detachment exists. Towards this goal, we first have simple geometric conditions that guarantee a given flow admits a safe detachment: Proposition 4.5.2 (Detachment existence). Consider first a flow i∈I F with associated flow graphG i F , and assume that we have a static configuration of mobile agents (i.e. eliminating hysteresis in interaction). Further assuming that for all flow members j∈I M |i∈M j , we have x j = x s i +τx d i with τ∈ [0, 1], it follows that if |V i F |≥ & kx s i −x d i k 2ρ 1 ' + 1 (4.81) 129 whered·e is the standard ceiling operator, thenG i F is connected and there exists some j∈V i F that can be safely detached from flow i. Alternatively, if we have the communication radii such that ρ 1 ≥ kx s i −x d i k 2(|V i F |− 1) (4.82) for a fixed flow cardinality|V i F |≥ 2, then an equivalent result follows 26 . Proof. As the flow members are assumed to lie on the line betweenF s i andF d i , we simply derive a flow cardinality under which a connected backbone emerges, and add a single flow member to this cardinality representing the guaranteed detachable agent. Specifically, to span the line x s i +τx d i with radii ρ 1 we requirekx s i −x d i k/ρ 1 elements, which corresponds to at leastd1/2(kx s i −x d i k/ρ 1 )e agents, and the result follows. The condition (4.82) is then a simple consequence of (4.81). Assuming the conditions of Proposition 4.5.2 hold, we propose the novel intra- flow maneuver depicted in Figure 4.13, to allow an ICP commandC i to select any i∈I M , by reconfiguring mobile flow members such that the target detachment of agent i is rendered safe. Denote by j∈I M the agent that can be safely detached from flowC i , and assume that agenti has a positionx i such that it cannot be safely detached 27 . In such a scenario a commandC i then initiates a state change for the current detachable agentS j ← RECONFIGURE, where the waypoint controller (4.80) is applied with x w ,x i . Agent j thus reconfigures to the position of desired detachment i, taking on the neighbor setN j ←N i , effectively transferring the 26 Extensions of this result to flows possessing multiple source/destination pairs or a planar construction are straightforward considerations of the convex hull of the static members of a flow. Such an analysis is beyond the scope of the assumptions of this work and is left for future investigation. 27 This position dependence is a direct consequence of the assumed proximity limitations of both the static sensors and the mobile robots. 130 Figure 4.13: The flexible detachment maneuver for shifting flow redundancy in response to flow detachment commands. detachability ofj to the targeti. Such a maneuver can be seen as reconfiguring the spatial slack in a flow to enable targeted detachments, yielding maximal flexibility in the feedback loop with the ICP. Remark4.5.1 (Flexibledetachability). Providing the ICP with flexibility in detach- ment commands, while more costly due to the additional intra-flow reconfiguration, promotes heterogeneity in agent capability. Although the ICP could integrate agent heterogeneity into its decisioning processes (e.g. choosing a safe detachment with minimal cost capable of providing some minimal flow bandwidth), eliminating the need to consider such multi-objective optimizations through simple mobility, appears as a reasonable heuristic to lessen computational requirements of the ICP. Remark 4.5.2 (Detecting detachability). A critical element of the above maneuver is the capability of flow members i∈I M to detect their own detachability from a flow in f∈M i . We point out that given the assumption of flow paths lying inR 131 and homogeneous communication radii, detachability can be detected locally by an agent i by observing the condition (j∈N k )∧ (k∈N j ), for neighbors j,k∈N i , such that j is the nearest neighbor on the line segment x s f +τx i and k is the nearest neighbor on the line segment x d f +τx i . Intra-Flow Controllers Once the ICP has assigned flow membershipsM i ∀i∈I M and all commanded reconfigurationsC i have been completed, the mobile agents begin to seek to optimize the flow to which they are a member. First, we assume that flow members must configure along the line segment connecting flow source/destination pairs, yielding in the case of proximity-limited communication, a line-of-sight or beamforming style heuristic. The membership of an agent i∈I M to a flow j∈M i thus initiates a check to determine ifx i lies on the flow pathx s j +τx d j , within a margin 0< F << 1 (c.f. lines 9-16, Algorithm 7). To do so, the projection of x i onto x s j +τx d j is determined first by computing τ, (x i −x s j )· (x d j −x s j ) kx d j −x s j k 2 (4.83) defining whether the projection will lie within or outside of the flow path. Then we have the saturated projection x i→F j = x s j −ατ(x d j −x s j ), τ < 0 x d j −ατ(x d j −x s j ), τ > 1 x s j +τ(x d j −x s j ), τ∈ (0, 1) (4.84) 132 where α> 0 is a biasing term such that the projection does not intersect x s j or x d j . We then have the state transition condition kx i→F j −x i k≤ F (4.85) which when satisfied gives S i ← SWARMING (line 14, Algorithm 7, and described below). If condition (4.85) is not satisfied, agent i transitions to state S i ← RECONFIGURE, applying waypoint controller (4.80) with x w , x i→F j , guaranteeing a reconfiguration, in a shortest path manner, to a point on the line segment defining its assigned flowM i . Finally, when an agenti is in the swarming stateS i = SWARMING (lines 18-20, Algorithm 7), after all necessary reconfigurations have been made (either by the ICP viaC i or internally by flow alignment), a dispersive inter-neighbor controller is applied in order to optimize the assigned flow. Specifically, each swarming agent i∈I M applies a coordination controller (regardless of bridge status b i ): u o i ,−∇ x i X j∈N A i 1 kx ij k 2 − X j∈N S i ∇ x j 1 kx ji k 2 (4.86) where N A i ,{j∈N i | (M j ∩M i 6=∅)∧ [(S j = SWARMING)∨ (j∈I S )]} (4.87) is the set of neighbors that share membership in flowM i , and who are either in flow and actively swarming (i.e. by condition (4.85)), or are a static source/destination node. Further, we define N S i ,{j∈N A i |j∈I S } (4.88) 133 as the set of static in flow neighbors for which compensation (Remark 4.5.3, below) must be applied. Controller (4.86) dictates that mobile flow members disperse equally only with fellow flow members and also with the source/destination nodes of their assigned flowM i . Remark 4.5.3 (Energy compensation). The inclusion of supplementary control terms for interactions with static neighbors j∈N i ∩I S in (4.86) acts to retain the inter-agent symmetry required for the application of constrained interaction [9], specifically as static agents do not contribute to the system energy. We refer to this control action as energy compensation, an idea that will evolve in future work to treat systems with asymmetry in sensing, communication, or mobility. While dispersive controllers generally yield equilibria in which inter-agent distant ismaximized(uptoρ 2 )[151], aseachflowisconstrainedbystaticsource/destination nodes, the dispersion (4.86) generates our desired equidistant intra-flow configura- tion as formalized below: Proposition 4.5.3 (Equidistant dispersion). Consider the application of coordina- tion objective (4.86) to a set of mobile agentsi∈I M within the context of interaction controller (4.21), each sharing membership to a flow k∈I F , i.e.M i = k,∀i. It follows that at equilibrium the agents are configured such that the equidistant spacing condition kx ij k→ kx d k −x s k k |V k F |− 2 , ∀i|j∈N F i (4.89) holds asymptotically over flow k. The above result is a simple consequence of an energy balancing argument and a standard Lyapunov analysis. Such a control formulation results in swarming behavior that optimizes network information flows with a line-of-sight style heuristic, 134 aligning flow members with equal spatial distribution along the path connecting each flow’s static endpoints. 4.5.5 Route Swarm Simulations In this section, we present a simulated execution of our described INSPIRE proof- of-concept, Route Swarm. Consider a system operating over a workspace inR 2 , having n = 15 total agents, m = 9 of which are mobile and s = 6 of which are static information source/destinations. Assume we have f = 3 flows (green, red, and blue indicate flow membership), with the initial system configuration depicted as in Figure 4.14a (notice thatG is initially connected), with the system dynamics shown in Figure 4.14b through 4.14f. We simulate a scenario in whichF 3 is initially inactive (gray), allowing the ICP to optimize agent allocation over onlyf = 2 flows, as in Figure 4.14b to 4.14c. By Figure 4.14c, flowsF 1 andF 2 have been assigned an evenly distributed allocation of mobile agents, where the PCP has provided equidistant agent spacing for each flow. At this same time (650 time steps), the flowF 3 activates, initiating a reconfiguration by the ICP to optimize the newly added flow, as in Figure 4.14d, noting that initially in Figure 4.14c,F 3 is poorly served by the network configuration. Finally, in Figure 4.14e, flowF 2 is deactivated, forcing another reconfiguration yielding the equilibrium shown in Figure 4.14f. The per-flow utility over the simulation, given for a flow i∈I F by P (j,k)∈E|i∈M j ∩M k w ij (the sum of the link utilities associated with each flow), is depicted in Figure 4.15. Finally, to better illustrate the dynamics of our proposed algorithms, we direct the reader to http://anrg.usc.edu/www/Downloads for the associated simulation video. Remark 4.5.4 (Dynamic vs. static). The optimizations proposed in this work are advantageous in terms of dynamic information flow needs and changing system 135 Time = 0.00 / 1500.00 30 40 50 60 50 55 60 65 70 75 80 (a) Time = 100.00 / 1500.00 30 40 50 60 50 55 60 65 70 75 80 (b) Time = 450.00 / 1500.00 30 40 50 60 50 55 60 65 70 75 80 (c) Time = 650.00 / 1500.00 30 40 50 60 50 55 60 65 70 75 80 (d) Time = 850.00 / 1500.00 30 40 50 60 50 55 60 65 70 75 80 (e) Time = 1500.00 / 1500.00 30 40 50 60 50 55 60 65 70 75 80 (f) Figure 4.14: Network progression for the simulated route swarming. objectives, when compared to static solutions. On flow switches, static placements fail to fulfill the information flow needs of the altered system configuration. Addi- tionally, our methods allow for dynamics inI M itself, as the ICP can adaptively reconfigure the system to utilize the available agents across the network flows. 136 F 3 F 2 F 1 Flow Utility Simulation Time 0 500 1000 1500 0 2 4 6 8 10 12 Figure 4.15: Flow utilities for the simulated route swarming. 137 Chapter 5 Rigidity Theory: A Primer In this chapter, we introduce pertinent concepts from rigidity theory that will allow us in the remainder of this thesis to develop distributed methods to evaluate, construct, and control rigid networks of collaborating agents. In Section 5.1, we introduce basic characterizations of the rigidity property, both combinatorial and algebraic in nature, and we describe a simple method for constructing rigid graphs in the plane. Of particular note will be the combinatorial Laman conditions, which underly many of our developments in rigidity evaluation and control (Chaper 6), and upon which a centralized algorithm for evaluating planar rigidity, the pebble game (Section 5.2), is based. Similar to the theories of graphs and systems, the literature on rigidity theory is varied and deep; we direct the reader to [101–105] for an encompassing review of rigidity theory. 5.1 Characterizing the Rigidity Property The primary concern in the reminder of this thesis is the rigidity property of the underlying graphG describing the network topology, specifically as rigid graphs imply guarantees for example in both localizability and formation stability of multi- robot systems [89]. To begin, we recall the intuition of how rigidity is recognized in a planar graph, following the exposition of [106]. Clearly, graphs with many edges are more likely to be rigid than those with only a few, specifically as each edge acts 138 to constrain the degrees of freedom of motion of the agents in the graph. InR 2 , there are 2n degrees of freedom in a network of n agents, and when we remove the three degrees associated with rigid translation and rotation, we arrive at 2n− 3 degrees of freedom we must constrain to achieve rigidity. Each edge in the graph can be seen as constraining these degrees of freedom, and thus we expect 2n− 3 edges will be required to guarantee a rigid graph. In particular, if a subgraph containing k vertices happens to contain more than 2k− 3 edges, then these edges cannot all be required for constraining the degrees of motion, i.e., they cannot be all independent. Our goal in evaluating rigidity is thus to identify the 2n− 3 edges that independently constrain the motion of our agents, precisely describing a network’s underlying rigid component. More formally, we first require the notion of a graph embedding in the plane, captured by the frameworkF p , (G,p) comprising graphG together with a mapping p:V→R 2 , assigning to each node inG, a location inR 2 . The natural embedding for us is to assign each node the position x i associated with each agent, defined by the mapping p(i) =x i , otherwise known as a realization ofG inR m . Therefore, a framework describes both the communication topology of a multi-agent system, and the spatial configuration of each agent in the plane. The infinitesimal motion ofF p can be described by assigning to the vertices of G, a velocity ˙ p i , ˙ x i ∈R 2 such that ( ˙ x i − ˙ x j )· (x i −x j ) = 0, ∀ (i,j)∈E (5.1) where· is the standard dot product overR m . That is, edge lengths are preserved, implying that no edge is compressed or stretched over time. The framework is said to undergo a finite flexing if p i is differentiable and edge lengths are preserved, 139 with trivial flexings defined as translations and rotations of R 2 itself. If for F p all infinitesimal motions are trivial flexings, thenF p is said to be infinitesimally rigid. Otherwise, the framework is called infinitesimally flexible, as in Figure 5.1a, where v 1 and v 3 can move inward with v 2 and v 4 moving outward, while preserving edge lengths [105]. In the context of a robotic network, rigid infinitesimal motion corresponds to movement of the ensemble in which the distances between robots remain fixed over time. Additionally, infinitesimal rigidity can be evaluated analyt- ically by rewriting the equations (5.1) in matrix form yielding R(x) ˙ x = 0. where R∈R |E|×mn is called the rigidity matrix. It can then be shown that the framework F x is infinitesimally rigid for m = 2 if and only if rank[R(x)] = 2|V|− 3, [105]. In determining and controlling the rigidity of the frameworkF x over time, one could continuously check infinitesimal rigidity through the rigidity matrix, and take suitable action towards preservation, as is explored in [109]. The infinitesimal rigidity of F p is tied to the specific embedding ofG in R 2 , however it has been shown that the notion of rigidity is a generic property ofG, specifically as almost all realizations of a graph are either infinitesimally rigid or flexible, i.e., they form a dense open set inR 2 [112]. Thus, we can treat rigidity from the perspective ofG, abstracting away the necessity to check every possible realization. The first such combinatorial characterization of graph rigidity was described by Laman in [101], and is summarized as follows 1 : Theorem 5.1.1 (Graph rigidity, [101]). A graphG = (V,E) with realizations in R 2 having n≥ 2 nodes is rigid if and only if there exists a subset ¯ E⊆E consisting of| ¯ E| = 2n− 3 edges satisfying the property that for any non-empty subset ˆ E⊆ ¯ E, 1 The extension of Laman’s conditions to higher dimensions is at present an unresolved problem in rigidity theory. 140 v 1 v 2 v 3 v 4 (a) Non-rigid. v 1 v 2 v 3 v 4 (b) Minimally rigid. v 1 v 2 v 3 v 4 (c) Non-minimally rigid. v 1 v 2 v 3 v 4 v 5 (d) Non-infinitesimally rigid. Figure 5.1: Example graphs demonstrating several embodiments of rigidity. we have| ˆ E|≤ 2k− 3, where k is the number of nodes inV that are endpoints of (i,j)∈ ˆ E. Laman’s notion of graph rigidity is also referred to as generic rigidity, and is characterized by the Laman conditions on the network’s subgraphs. Intuitively, the concept of rigidity can be thought of in a physical way, that is if the graph were a bar and joint framework, it would be mechanically rigid against external and internal forces. However, we point out that the rigidity of an underlying graph is 141 purely a topological property. A network of agents that is described by a rigid graph is not necessarily mechanically rigid. Instead, the structure of its interconnections, in our case robot-to-robot communication, possesses the combinatorial properties of the above Laman conditions. We point out further that Laman’s theorem places restrictions on the network edges, and thus we must only evaluate the Laman conditions during transitions inG. When compared to continuum methods such as [109], exploiting Theorem 5.1.1 in rigidity control yields a fundamentally more efficient and robust solution, as continuous computation and communication resources are not required for implementation. Denote byG R the set of all rigid graphs in R 2 , and the graphS = (V, ¯ E) satisfying Theorem 5.1.1 a Laman subgraph ofG. It follows from Theorem 5.1.1 that any rigid graph in the plane must then have|E|≥ 2n− 3 edges, with equality for minimally rigid graphs. The impact of each edge on the rigidity ofG is captured in the notion of edge independence, a direct consequence of Theorem 5.1.1: Definition5.1.1 (Edgeindependence,[106]). Edges (i,j)∈E of a graphG = (V,E) are independent inR 2 if and only if no subgraph ¯ G = ( ¯ V, ¯ E) has| ¯ E|> 2| ¯ V|− 3. A set of independent edges will be denoted byE ∗ , while the graph overE ∗ is denoted byG ∗ . The above conditions imply that a graph is rigid inR 2 if and only if it possesses |E ∗ | = 2n− 3 independent edges, where edges that do not meet the conditions of Definition 5.1.1 are called redundant. Thus, in determining the rigidity ofG, we must verify the Laman conditions to discover a suitable set of independent edgesE ∗ . We refer the reader to Figure 5.1 for a depiction of graph rigidity. Notice that the graph in Figure 5.1a is non-rigid as it does not fulfill the basic 2n− 3 edge condition of Laman. In adding edge (v 2 ,v 4 ) we then generate the minimally rigid graph of Figure 5.1b as every subgraph of k vertices has at most 2k− 3 edges. Further 142 addition of (v 1 ,v 3 ) yields the non-minimally rigid graph in Figure 5.1b, precisely as the graph possesses greater than 2n− 3 edges. Finally, Figure 5.1d represents a non-infinitesimally rigid graph as there exist non-trivial motions that preserve edge lengths, i.e.,v 5 moves independently while the remaining nodes rotate together, but it also is a generically rigid graph as the underlying Laman conditions are satisfied. The Henneberg Construction The above characterizations of rigidity tell us ways in which we can understand the rigidity properties of some known graph. If instead we wish to construct a rigid graph, we can look towards the Henneberg construction, detailed in [105], for building minimally rigid graphs in the plane. First, consider the 2-valent operation: Proposition 5.1.1 (2-valent addition, Prop 3.1, [105]). LetG be a planar graph with two distinct vertices a and b andG ∗ be the planar graph obtained by attaching a new vertex q with edges (q,a) and (q,b) toG. ThenG is generically rigid if and only ifG ∗ is generically rigid. This operation appends to a minimally rigid graph a new node with two new edges, as depicted in Figure 5.2a. We also require a 3-valent operation described as follows. Proposition 5.1.2 (3-valent addition, Prop 3.2, [105]). LetG be a generically rigid planar graph with three distinct vertices a, b, and c andG ∗ be the planar graph obtained by removing the edge (a,b) fromG and attaching a new vertex q with edges (q,a), (q,b), and (q,c) toG. ThenG ∗ is generically rigid. This operation appends to a minimally rigid graph a new node with three new edges by removing a single edge from the original graph, as depicted in Figure 5.2b. 143 v 1 v 2 v 3 v 1 v 2 v 3 v 4 (a) 2-valent operation. v 1 v 2 v 3 v 1 v 2 v 3 v 4 (b) 3-valent operation. Figure 5.2: The Henneberg operations. Now, we can define a sequence of node addition operations that constructs a planar graph: Definition 5.1.2 (Henneberg sequence, [105]). A Henneberg sequence is a sequence of graphsG 2 ,...,G k , whereG 2 is a single bar graph, andG i+1 comes fromG i through either a 2-valent or 3-valent Henneberg operation. A Henneberg sequence, which constructs graphs in a manner depicted by Figure 5.3, is then useful in generating members of the space of minimally rigid planar graphs: 144 Figure 5.3: A Henneberg sequence. Proposition 5.1.3 (Henneberg construction, Prop 3.4, [105]). A planar graph G is minimally generally rigid if and only if there is a Henneberg sequence that constructsG. We will eventually take inspiration from the Henneberg construction to derive methodologies for building optimally rigid graphs, as detailed in Section 6.2. 145 5.2 A Pebble Game for Evaluating Rigidity To lessen the exponential complexity of the Laman conditions one can consider the pebble game proposed by Jacobs and Hendrickson in [106]. A brief overview of the centralized pebble game will be given here, beginning with a useful characterization of the Laman conditions and edge independence: Theorem 5.2.1 (Laman restated, [106]). For graphG = (V,E), the following statements are equivalent: • All (i,j)∈E are independent inR 2 . • For each (i,j)∈E, the graph formed by quadrupling (i,j), i.e., adding 4 virtual copies of (i,j) toE, has no subgraph ¯ G = ( ¯ V, ¯ E) in which| ¯ E|> 2| ¯ V|. Theorem 5.2.1 represents the Laman condition, i.e., a subgraph of k vertices can possess at most 2k− 3 edges, through the simple quadrupling operation. In other words, if we add 3 virtual copies of an edge to some subgraph, and the condition| ¯ E|≤ 2k is met in this subgraph, it must be the case that the property | ¯ E|≤ 2k− 3 holds in the original subgraph. This intuition can be further extended to incrementally evaluate edge independence: Lemma 5.2.1 (Edge quadrupling, [106]). Given an independent edge setE ∗ , an edge (i,j) / ∈E ∗ is independent ofE ∗ if and only if the graph formed by the union of E ∗ and quadrupled edge (i,j) has no subgraph ¯ G = ( ¯ V, ¯ E) in which| ¯ E|> 2| ¯ V|. The above Lemma provides us with a simple process for testing rigidity: we incrementally quadruple edges in the graph, check the induced subgraph property, and continue until we have either discovered 2n− 3 independent edges or we have exhaustedE. However, this process alone does not save us from the exponential 146 Figure 5.4: An example of the pebble game for a rigid graph with n = 3. complexity of verifying the subgraph property. To this end, [106] provides a natural simplification in the following pebble game: Definition 5.2.1 (The pebble game, [106]). Considering a graphG = (V,E) where we associate an agent with each v∈V, give to each agent two pebbles which can be assigned to an edge inE. Our goal in the pebble game is to assign the pebbles inG such that all edges are covered, i.e., a pebble covering. In finding a pebble covering, we allow the assignment of pebbles by agent i only to edges incident to v i inG. Further we allow pebbles to be rearranged only by removing pebbles from edges which have an adjacent vertex with a free pebble, such that the free pebble is shifted to the assigned pebble, freeing the assigned pebble for assignment elsewhere. Thus, if we consider pebble assignments as directed edges exiting from an assigning agent i, when a pebble is needed in the network to cover an edge (i,j), a pebble search over a directed network occurs. If a free pebble is found, the rules for local assignment and rearranging then dictate the pebble’s return and assignment to (i,j). 147 Lemma 5.2.2 (Pebble covering, [106]). In the context of the pebble game of Definition 5.2.1, if there exists a pebble covering for an independent edge setE ∗ with a quadrupled edge (i,j) / ∈E ∗ , there is no subgraph violating the conditions of Lemma 5.2.1, and the setE ∗ ∪ (i,j) is independent. Rigidity evaluation now operates as follows: every edgee∈E is quadrupled, and an attempt to expand the current pebble covering forE ∗ to each copy of e is made, with success resulting inE ∗ ←E ∗ ∪e and termination coming when|E ∗ | = 2n− 3. Intuitively, an agent’s pebbles represent its possible commitments to the network’s subgraphs, while maintaining the subgraph conditions of Lemma 5.2.1, or in a physical way the degrees of freedom of motion inR 2 . Further, the edge quadrupling operation and the pebble game effectively cast the Laman conditions on subgraphs in terms of a matching problem. That is, as each agent is given 2 pebbles, and each of 4 instances of a considered edge must be assigned a pebble, we implicitly verify the 2k− 3 edge condition of Laman when these 4 pebbles are found in a subgraph containing k vertices. This is the case precisely because each previously considered edge is assigned a single pebble. The centralized pebble game of Jacobs is depicted in Algorithm 8, with an illustration of the quadrupling and pebble search procedure depicted in Figure 5.4. In the simple three node graph shown, there are six available pebbles that can only be assigned locally. Thus, in quadrupling each edge and finding four pebbles, the subgraph conditions of Laman are incrementally verified. The progression is given from left to right, with pebbles given by black dots, quadrupled edges by thick links (blue), pebble shifts by dashed arrows, and the local assignment of pebbles by black arrows. Graph edges that remain to be quadrupled are dashed. Notice that in discovering the pebble to cover the final copy of the last quadrupled edge 148 Algorithm 8 The centralized pebble game [106]. 1: procedurePebbleGame(G = (V,E)) 2: Assign each v i two pebbles,∀i∈I 3: E ∗ ←∅ 4: for all (i,j)∈E do 5: Quadruple (i,j) overG 6: Search for 4 pebbles, originating from v i and v j 7: if found then 8: Rearrange pebbles to cover quadrupled (i,j) 9: . Expand independent set, check rigidity: 10: E ∗ ←E ∗ ∪ (i,j) 11: if|E ∗ | = 2|V|− 3 then 12: returnE ∗ 13: end if 14: end if 15: end for 16: end procedure (bottom middle pane), a search occurs over the directed graph formed by previous pebble assignments. 149 Chapter 6 Evaluating, Constructing, and Controlling Rigid Networks In this chapter, we close our contributions with a thorough treatment of the rigidity property of a networked multi-agent system. In comparison to the connectivity prop- erty, rigidity has only very recently become a focus of efforts in graph construction or control. While previous work illustrated the necessity for the rigidity assumption, e.g., in highly impactful areas such as formation control and relative localization, dynamically rigid or rigid-by-construction networks were ubiquitously assumed. Thus, in our continued theme of eliminating the assumptions of collaboration, we begin the chapter with the problem of evaluating the rigidness of an unknown network in the plane. In particular, we provide a full decentralization and partial parallelization of the pebble game, an efficient centralized algorithm for rigidity evaluation. Next, we demonstrate how our specific method of decentralization, i.e., leader election, can be exploited to construct rigid networks in an optimal way, given edge and node costs. Finally, resting on the spatial topology control derived in Chapter 4, and applying our methods for decentralized rigidity evaluation, we propose agent predicates that yield efficient and localized rigidity control. The materials of this chapter thus represent a full spectrum solution to guaranteeing the rigidity property in planar multi-agent teams, yielding the vast benefits of collaboration that rigidity implies. 150 Chapter Outline The outline of this chapter is as follows. In Section 6.1, we begin with preliminary materials including models of asynchronous time, algorithm execution, and inter- agent messaging. A decentralization of the pebble game is presented in Sections 6.1.1 and 6.1.2, with a parallelization given in Sections 6.1.3 and 6.1.4. Simulation results are provided in Sections 6.1.5 and 6.1.6, where we demonstrate a dynamic rigidity control scenario, along with Monte Carlo results in the Contiki environment, a real-world networking simulator. Finally, technical algorithm details and related proofs are given in Sections 6.1.7 and 6.1.8. Our rigidity evaluation results are then extended in Section 6.2, where two approaches to find an optimal and sub- optimal rigid subgraphs are presented and theoretically characterized. Simulations to corroborate the theoretical findings are reported in Section 6.2.3, in the context of a relative sensing application. Finally, a local rule for dynamically maintaining graph rigidity is studied in Section 6.3.1, together with a consensus-based extension and an analysis of conservative behaviors in Section 6.3.2. Simulation results are provided in Section 6.3.3, which characterize the rule behavior in Monte Carlo and depict the controller in a rigidity control application. 6.1 An Asynchronous Decentralized Pebble Game Our problem of interest in this section is the evaluation of the rigidity property of an interconnected system of intelligent agents, e.g., robots, sensors, etc. A relatively under-explored topic in the area of multi-agent systems, rigidity has important implications particularly for mission objectives requiring collaboration. To begin, we require a basic assumption on the multi-agent network that evaluates rigidity: 151 Assumption 6 (Connectedness). We assume the network topologyG is connected for all time to guarantee all agents can participate in rigidity evaluation. Notice that this assumption is trivially satisfied in rigid networks. As our concern in this work is to operate under the typical parameters of realistic interacting systems, we assume an asynchronous model of time, where each agent i∈I has a clock which ticks according to some discrete distribution with finite support 1 , independently of the clocks of the other agents [152], allowing also for the possibility of delayed communication over links (i,j)∈E. Equivalently, this corresponds to a global clock having time-slots [t k ,t k+1 ) which discretizes system time according to clock ticks, where for convenience we will use simply t to denote time [129]. Such assumptions induce asynchronicity in both agent computation and the broadcast and reception of inter-agent messages. First, we make the following assumptions concerning agent execution: Assumption 7 (Agent execution). Each agent i∈I executes according to an algorithm on ticks of their clock, handling messages from neighbors j∈N i and sending messages if dictated by the execution. Local execution is assumed to consist of atomic logic and message handling, that is all local algorithmic state, denoted X i , is assumed to be without race conditions due to asynchronicity. A coordinated algorithm execution with associated stopping condition can then be defined as follows: 1 Notice that such an assumption allows us to appropriately characterize finite algorithm termination. 152 Definition 6.1.1 (Coordinated execution). A coordinated algorithm execution is given as a sequence of ticks t k and therefore local execution and asynchronous messaging, yielding a terminal state upon some discrete network stopping condition f stop ∈{0, 1},f({X 1 ,..., X n }) (6.1) dependent on the execution states of the network agents. It is assumed that (6.1) can be computed using distributed techniques, e.g., consensus [22], as will be demon- strated in our proposed algorithms. After the stopping condition is observed the agents enter into an idle state where no execution occurs. Finally, to guarantee soundness with respect to network communication and asynchronicity, we make the following assumptions: Assumption 8 (Asynchronous messaging). We assume each agent i∈I treats messages received from neighbors j∈N i in a first-in-first-out (FIFO) manner, guaranteeing soundness with respect to our proposed algorithm executions. Further, the possibility of communication failure is handled with best-effort messaging, i.e., there exists an underlying communication control layer where a best effort is made to deliver packets in the network. Specifically, we assume that the best effort guarantees message reception in finite time, or equivalently a message failure can be handled appropriately with respect to the algorithms that will be discussed. The primary considerations in decentralizing the pebble game of [106] lie in the sequential building of the independent edge setE ∗ , the storage ofE ∗ and associated pebble assignments over a distributed network, and the search and rearranging of pebbles throughout the network. To deal with these issues, we summarize the high level components of our decentralization: 153 • Leader election: to control the sequential building ofE ∗ , lead agents are elected through auctions to examine their incident edges for independence. In determining edge independence, pebbles are queried from the network through inter-agent messaging in order to cover each copy of a quadrupled incident edge. Leadership then transfers to the next auction winner when the current leader’s neighborhood has been exhausted. • Distributed storage: independent edges and pebble assignments are local- ized to each agent, effectively distributing network storage. We then rely on messaging and proper agent logic to support pebble searches and shifts. • Local messaging: as opposed to searching a centralized graph object for pebbles to establish edge independence, we endow the network with a pebble request/response messaging protocol to facilitate pebble searches. Intuitively, our leader-based decentralization is an incremental rooting of pebble searches at appropriately elected network leaders, effectively partitioning rigidity evaluation as in Figure 6.1. For convenience we will denote byS our decentralization of the serial pebble game of Algorithm 8. In describing our algorithm we associate with each agent i∈I the following variables, with initialization indicated by←: •P i ←∅: Pebble assignment set containing at most two edges{(i,j)∈ E|j∈N i }, that is incident edges (i,j) to which a pebble is associated. For convenience, we letp i = 2−|P i |∈{0, 1, 2} denote agenti’s free pebble count. •E ∗ i ←∅: Local independent edge set, containing edges{(i,j)∈E|j∈N i } for which quadrupling and pebble covering succeeds. By constructionE ∗ = S i E ∗ i . 154 6.1.1 Leader Election for Decentralization An execution of theS algorithm begins when an agent detects network conditions that require rigidity evaluation, e.g., verifying link deletion to preserve rigidity. The initiating agent begins by triggering an auction for electing an agent in the network to become the leader. Specifically, to each agent i∈I we associate a bid for leadership r i = [i,b i ] with b i ∈R ≥0 indicating the agent’s fitness in becoming the new leader, with b i = 0 if agent i has previously been a leader, and b i ∈R + otherwise. Denoting the local bid set byR i ={r j |j∈N i ∪{i}}, the auction then operates according the following agreement process: r i (t + ) = argmax r j ∈R i (b j ) (6.2) where the notation t + indicates a transition in r i after all neighboring bids have been collected through messaging. AsG is assumed connected for all time, (6.2) converges uniformly to the largest leadership bid r i = argmax r j (0) (b j (0)), ∀i,j∈I (6.3) after some finite time [128,153]. After convergence of (6.2) the winning agent then takes on the leadership role, with the previous leader relinquishing its status. The proposed auction mechanism allows us to decentralize the pebble game by assigning to each leader the responsibility of expandingE ∗ i by evaluating only their incident edges for independence. Also, notice that previous leaders are never reelected due to b i = 0 for such agents, and that the condition b i = 0,∀i∈I allows termination of the algorithm. 155 One would expect that as each leader expands the independent set sequentially that the order of election is meaningful. We characterize that relationship in the following: Proposition 6.1.1 (Initial leader edges). All incident edges{(i,j)∈E|j∈N i } belonging to an initial leader i are members of the independent set (i,j)∈E ∗ . Proof. For each edge, a new node j6= i must be considered as no two edges of i can have the same endpoint andE ∗ is empty due to i being the initial leader. Therefore, every subgraph containing the edges and nodes incident to i must have |E s |≤ 2|V s |− 3 edges, whereV s are the nodes of the considered subgraph and |E s | =|V s |− 1 due to the subgraph’s implicit tree structure. Thus, as there exists no subgraph violating Definition 5.1.1, the result follows. As the agent with the largest bid is elected, the bids dictate the order of elected leaders and thus the edges that constitute the identified rigid subgraph. In other words, the bids can be applied based on the application. For example, if we assume each edge is assigned a weight which indicates its value in sensing or information, we could choose leader bids that are the sum of incident edge value. Then the resulting rigid subgraph would possess those edges that both establish rigidity and are the most valuable in the given application. Bids could also be chosen to reflect agent availability, processing capability, or the cardinality of incident edges, or they can be leveraged in terms of metrics related to mission objectives. The proposed auction technique therefore affords us control overE ∗ that goes beyond simply discovering the network’s rigidity property. 156 v 1 v 2 v 3 v 4 v 1 v 2 v 3 Figure 6.1: Illustration of leader-based decentralization for n = 4 agents. Leader Tasks After election, the primary task of the leader i is to continue the expansion ofE ∗ by evaluating the independence of each edge (i,j)∈E i ,{N i |¬beenLeader(j)}, i.e., the set of unevaluated incident edges. In initializingE i in such a way, incident edges (i,j) are considered only when the neighborj∈N i has not yet been a leader, as edges incident to a previous leaderj have already been checked. This guarantees that network edges are considered only once for quadrupling and pebble covering. Also, note that each leader receives the current size of the independent edge set |E ∗ (t)| in initialization, by embedding|E ∗ (t)| in the leadership auction. This allows a leader to terminate the algorithm when 2n− 3 independent edges have been identified. The leader executes the procedureLeaderRun depicted in Algorithm 9, given in the Appendix, to accomplish the task of evaluating its incident edges. First, recall that in checking independence a pebble covering for each quadrupled edge e i ∈E i must be determined. As the pebble information is distributed across the network, the lead agent must therefore request pebbles through messaging in an attempt to assign pebbles toe i . After making a pebble request, the lead agent then 157 pauses execution and waits for pebble responses before continuing; a method often referred to as blocking. When there exists no unfulfilled pebble requests, the lead agent starts or resumes the quadrupling procedure on the current incident edgee i ∈E i , lines 3–11. For each copy of e i , the leader searches for a pebble to cover e i , first by looking locally for free pebbles, assigning e i toP i if found. If no local pebbles are available, the agent then sends a PebbleRequestMsg to the endpoint of the first edge to which a pebble is assigned, requesting a free pebble. If a pebble is received from this request, the quadrupling process continues, otherwise another request is sent to the endpoint of the second edge to which a pebble is assigned. In sending requests only along (i,j)∈P i , we properly evaluate independence with respect toE ∗ , as each (i,j)∈E ∗ must have an assigned pebble from previous evaluations of independence. As established by Lemma 5.2.2, the outcome of the quadrupling process, i.e., the existence of 4 free pebbles in the network, dictates the independence of edge e i . If the leader fails to receive 4 pebbles to covere i , the edge is deemed redundant and evaluation moves to the next member ofE i . On the other hand, for any edgee i with a pebble covering, obtained through a combination of local assignment and pebble responses, the following actions are taken, lines 13–24. First, we return 3 pebbles to the endpoints ofe i leaving a single pebble one i to establish independence, and then add e i toE ∗ i . If in adding e i , 2n− 3 independent edges have been identified, the leader sends a simple message to the network indicating that the graph is rigid, and the algorithm terminates. Otherwise, the leader moves to the next member ofE i and begins a new quadrupling process. When all members ofE i have been evaluated, the leader initiates the auction (6.2) to elect the next leader. The process of edge quadrupling, pebble requests, edge covering, and expansion of the independent set 158 v 1 v 2 v 3 v 4 (a) First edge of quadrupling. v 2 v 1 v 4 v 3 (b) Second edge of quadrupling. Figure 6.2: Illustration of pebble covering attempts for a quadrupling an edge. then continues from leader to leader until either the network is found to be rigid, or every agent has been a leader, indicating non-rigidity. 6.1.2 Inter-Agent Messaging As each leader attempts to expandE ∗ i through quadrupling each of its members, free pebbles are needed to establish a pebble covering. We facilitate the pebble search by defining asynchronous message PebbleRequestMsg, accompanied by response messagesPebbleFoundMsg andPebbleNotFoundMsg, indicating the existence of free pebbles. The arrival of these messages then triggers message handlers that form the foundation of the pebble search mechanism. For technical details of the protocol, see the pseudocode given in the Appendix. The reception of a PebbleRequestMsg initiates the handler HandlePebbleRequest depicted in Algorithm 10. Each pebble request is marked with a unique identifier, originating from the lead agent, defining the 159 pebble search to which the request is a member and ensuring proper message flow in the network, lines 2–5. For unique requests, the receiving agent first attempts to assign local pebbles to the edge connecting the pebble requester, i.e., a pebble shift operation, lines 7–9. If a free pebble is available for the shift, a PebbleFoundMsg is sent in response, allowing the requester to free an assigned pebble for either local assignment or to itself respond to a pebble request. If instead the request recipient has no free pebbles, the agent forwards the request to the endpoints of its assigned pebbles, recording the original pebble requester such that responses can be properly returned, lines 11–12. Notice that this messaging logic not only facilitates the pebble shift and assignment rules of the original pebble game, but also eliminates the need for explicit message routing. Instead, it is previous pebble assignments that dictate message routing. When the PebbleFoundMsg response to a pebble request is received it triggers the handler HandlePebbleFound depicted in Algorithm 11. Similar to the shifting action of HandlePebbleRequest, the agent first frees the local pebble assigned to the edge connecting the responder, line 2, and then uses the newly freed pebble depending on leader status. If the agent is currently the leader, line 4, the freed pebble is assigned locally to e i , continuing the edge quadrupling process and relieving the request blocking condition. For non-lead agents, a pebble shift is performed to again free a pebble for a requesting agent, indicating the shift by returning aPebbleFoundMsg to the requester, lines 6–7. Finally, thePebbleNotFoundMsg response to a pebble request initiates the handlerHandlePebbleNotFound depicted in Algorithm 12. For both leaders and non-leaders, the lack of a free pebble initiates a further search in the network, along untraversed incident edges to which a pebble is assigned, line 3. However, if both available search paths have been exhausted, the leadership status of the 160 receiver dictates the action taken. In the case of a non-leader, line 11, the response is simply returned to the original requester in order to initiate further search rooted from the requester. For a leader, lines 6–9, the lack of free pebbles in the network indicates precisely that the conditions of Lemma 5.2.1 do not hold, implying the currently considered edgee i is redundant. The edgee i is removed from consideration by returning all pebbles assigned during the covering attempt to the endpoints of e i , and the process is moved to the next incident edge. A basic illustration of a snapshot of theS algorithm is given in Figure 6.2. Complexity Analysis The complexity ofS is promising for realistic decentralized operation: Proposition 6.1.2 (S complexity). By construction, executions of theS algorithm have worst-case O(n 2 ) messaging complexity and O(n) storage scaling. Proof. As the pebble game exhibits O(n 2 ) complexity [106], our pebble messaging scales like O(n 2 ). In applying leader auction (6.2) we incur O(n 2 ) as we expend O(n) auction messaging for O(n) leaders. Equivalently, the centralized execution takesO(n 2 ) and we simply apply anO(n 2 ) decentralization to provide the algorithm with the appropriate runtime information. Thus, our overall algorithm will run with O(n 2 ) complexity. Finally, the per-agent storage complexity scales like O(n), the maximal cardinality ofN i , as assignments toE i occur over only edges incident to i, line 23 Algorithm 9. The above result demonstrates that our proposedS algorithm represents a fully decentralized and efficient solution to the planar generic rigidity evaluation problem, providing the opportunity to exploit the vast advantages of network rigidity in realistic robotic networks. 161 6.1.3 Exploiting Structure Towards Parallelization To fully exploit a distributed multi-agent system, we seek a parallelization of the serialized algorithm previously proposed, with the goal of reducing the overall execution time of rigidity evaluation, and rendering real-world application feasible. It turns out that evaluating network rigidity is intrinsically serial and centralized in nature, making it difficult to asymptotically reduce the computational complexity through parallelization. Instead, we aim to provide a parallelization that is advanta- geous under realistic circumstances, yielding both non-trivial runtime improvements and uses for building rigid networks, with no additional hardware or communication requirements. At a high level, our scheme consists of identifying local edge addition operations that preserve independence, and allowing the agents to apply these rules simultaneously to build a set of independent edges. We will develop these ideas in the sequel and direct the reader to Remark 6.1.2 for a complete summary of the advantages of parallelization. Independence Preserving Operations Let us begin by formally defining addition and subtraction operations for graph edges as follows. Definition 6.1.2 (Edge addition/subtraction [87]). Consider a graphG = (V,E) and let the graph augmented with an edgee be denotedG + = (V,E∪{e}). Similarly, the graphG with e removed is denoted byG − = (V,E\{e}). We refer to the operation [·] + e such thatG + = [G] + e as edge addition. Likewise, the operation [·] − e such thatG − = [G] − e as edge subtraction (or deletion). Now we are prepared to consider independence preserving graph operations. Specifically: 162 Definition6.1.3 (Independencepreservingoperations). We call the edge operations [·] + e and [·] − e over graphG = (V,E) having independent edgesE satisfying Definition 5.1.1 independence preserving (IP) ifE∪{e} andE\{e} are themselves independent, respectively. Clearly, all operations [·] − e over independent edgesE are independence preserving. Also, we have that addition operations [·] + e that are not independence preserving imply e is redundant with respect toE. Thus, we seek IP edge addition operations that enable the construction ofE ∗ in a parallel fashion. Then, given an initial independent setE ∗ (0) =∅ with associated graphG ∗ = (V,E ∗ ), we can generate the sequence G ∗ (0) = (V,∅) G(k) = [G(k− 1)] + e , k = 1,...,m (6.4) where if each [·] + e is independence preserving, the resulting graphG ∗ (m) possesses independent edges. Then, if each operation [·] + e is local to the endpoints of edge e, sequence (6.4) can be achieved in parallel. To identify such operations, we take inspiration from the Henneberg construction, a sequence of node and edge additions that iteratively builds a minimally rigid graph [105]. First, consider a simple rule based on the membership of v i or v j as endpoints inG ∗ (k): Definition 6.1.4 (Endpoint expansion rule). Consider the graphG = (V,E) and the associated node set b V ={v i ∈V|∃j∈I, (i,j)∈E}, containing the nodes in V which are endpoints of edges inE. Define the endpoint expansion rule (EER) as the edge addition operation [G] + e possessing the property that|[ b V] + e |>| b V|, where [ b V] + e are the endpoints ofG + . Notice that for an EER operation it trivially holds that 2≥|[ b V] + e |−| b V|≥ 1. 163 (a) The endpoint expansion rule (Definition 6.1.4). (b) The two incident edge rule (Definition 6.1.5). Figure 6.3: Illustration of a sequence of the endpoint expansion rule and the two incident edge rule. Clearly the endpoint expansion rule 6.1.4 is limiting in terms of the identified set E ∗ , specifically as the identified set can be described as the union of spanning trees overG, a direct consequence of expanding endpoints in a graph, as in Figure 6.3a. Thus, we can further consider a two edge rule that is also independence preserving: Definition 6.1.5 (Two incident edge rule). Consider the graphG = (V,E) and the augmented graphG + through addition of edge e, (i,j). The two incident edge rule (TIER) is an edge addition operation [G] + e where it holds that (N + i ≤ 2)∨ (N + j ≤ 2) overG + , with (N i ≥ 1)∧ (N j ≥ 1) overG, otherwise the edge operation would constitute an endpoint expansion. We illustrate the rules of Definitions 6.1.4 and 6.1.5 in Figs. 6.3a and 6.3b, respectively, with the independence preservation of the proposed rules given by 164 P G S E ∗ 1 E ∗ n . . . E ∗ PebbleAssign |E ∗ P | {P i ,p i } GossipAverage Figure 6.4: Block diagram of the parallelized pebble game algorithm. Proposition6.1.8intheAppendix. TheEERandTIERoperationsindicatefirstthat an independent set could be built incrementally as in (6.4), much like the original pebble game. However, instead of requiring inherently global pebble searches, the EER and TIER operations are distinctly local in nature, making them amenable to parallel implementation. 6.1.4 Gossip-like Messaging for Parallelization We now take inspiration from the randomized communication scheme typical of gossip algorithms, e.g., [154,155], to define the execution and messaging structure for partial parallelization of rigidity evaluation. Each agent i exchanges inter-neighbor messages in an attempt to assign incident edges (i,j),∀j∈N i toE ∗ i according to the EER and TIER rules. Such a construction, denoted as algorithmP, allows the network to determine a subset of independent edgesE ∗ P ⊆E ∗ with significantly reduced execution time and messaging, as will be verified in Section 6.1.6. TocharacterizetheEERandTIERoperationsontheedgesassignedtoanagent’s independent set, we associate with each agent i the variable committed(i)∈Z ≥0 . This commitment variable stores the cardinality of i as an endpoint of edges in the distributed independent edge set, or more formally, the node degree of v i in the graphG ∗ = (V,∪ i E ∗ i ). As the EER and TIER rules are effectively conditions on node degrees, as one would expect given the nature of the Laman conditions, the 165 commitment variables guarantee that all edges added to the independent set are in fact independent. As opposed to the leader-based execution of algorithm S, here every agent executes concurrently according to Algorithm 13, with initialization E i (0)←N i , e i 8 (i,j)∈E i (0) ∀i∈I (6.5) where we use notation8 to represent a random assignment of a link (i,j)∈E i . EachagentattemptstoexpanditslocalindependencesetE ∗ i byexploitingmessaging to determine a neighbor j’s feasibility as an endpoint inG ∗ (k), ensuring that the EER and TIER conditions are fulfilled. Thus, for each edge e i , (i,j) chosen randomly fromE i , agenti requests neighborj’s commitment toG ∗ through message EdgeRequestMsg(i,j), showninAlgorithm13, lines3–4, andblocksfurtheredge consideration, similartotheleaderlogicdescribedinSection6.1.1. Whenalledgesin E i have been considered, line 8, the agent enters into the idle state until the network stopping condition is met. Algorithm termination then occurs when all agents have entered into the idle state. The remaining logic for independent edge selection and rule checking resides in the message handlers related to EdgeRequestMsg(i,j) of Algorithm 14, as will be discussed in the following. Inter-Agent Messaging The reception of an edge request message by agent i from neighbor j∈N i triggers the message handling logicHandleEdgeRequest(i,j) depicted in Algorithm 14. Recall that when an agent i receives such a request from j it indicates an attempt by j to make the assignment of edge (j,i) to its independent setE ∗ j . Requests for an edge assignment must therefore be tested for fulfillment of the EER and TIER 166 rules. However, as all agents are trying to add incident edges to the independent edge set simultaneously, it may be the case that two agents sharing an edge conflict on which agent takes the edge, specifically as both cannot. Thus, receiving agent i first determines whether a request represents a conflict over edge (i,j). If there is a conflict, agents i and j resolve the conflict as follows: Remark 6.1.1 (Conflict resolution). We define an edge conflict in the exe- cution of algorithm P as the state in which an agent i ∈ I has received an EdgeRequesMsg(i,j) fromj∈N i , where fori it holds that requestedFrom(i) =j. Intuitively, this state indicates that both i and j are attempting to add edge (i,j) toE ∗ , a condition that would introduce inconsistencies in|E ∗ |. In such scenarios we assume there exists a function ResolveConflict(i,j) :E→I indicating the conflict winner, such that ResolveConflict(i,j) =ResolveConflict(j,i) (6.6) for alli6=j∈I. Now, resolving a conflict is simply a matter of first deciding which agent wins and receives the edge, and then ensuring that the losing agent does not take the edge. The winner of a conflict is decided according to the predetermined policy, ResolveConflict, which is chosen as a design parameter, and the loser is denied the edge by rejecting its edge request, lines 2–8. This is accomplished by saturating the commitments of the winning agent, such that the edge request of the losing agent cannot be fulfilled due to violation of the EER and TIER rules. In achieving condition (6.6), we could consider various options, e.g., balancing |E ∗ i | and|E ∗ j | in choosing the winner, applying a simple predetermined condition such as agent label, or perhaps a more complex method such as considering optimal assignments based on some cost or utility function over (i,j). 167 After properly handling potential conflicts over (i,j), receiver i then simply responds to j via EdgeResponseMsg(i,j,response) indicating its current com- mitment toE ∗ , increases its own commitment count to guarantee independence preservation, and removes (i,j) fromE i to avoid double checking, lines 8–13. In coherence with the edge request logic, in receiving an EdgeResponseMsg(i,j,response), an agent i acts according to Algorithm 15. Again, as an edge response conveys agent j’s commitment toE ∗ , it is validated against the EER and TIER rules, with success resulting in the assignment of (i,j) toE i and an incrementing of committed(i) as i is now an endpoint ofG ∗ , lines 3–4. After independence preserving assignment, agent i simply moves to consider its next incident edge by choosing a random (i,j)∈E i , lines 7–8. At the conclusion of algorithmP, a distributed independent edge setE ∗ P has been computed across eachE ∗ i . However, by the EER and TIER definitions, it must hold generally thatE ∗ P ⊆E ∗ . Thus, to fully determineE ∗ we generate a composite algorithm by passing the terminal state ofP to the serialized algorithmS in order to apply global network information in completingE ∗ . The initial conditions for the serialized algorithm are generated from the output of the parallel algorithm in a way that it is feasible that the serial algorithm itself had run up to that point. Thus, the output of the parallel algorithm must be shaped to mimic the conditions of the serialized algorithm. To do so we simply apply a network summing algorithm to determine the overall size of the independent edge set (i.e., the sum of the local set sizes), and then assign pebbles to cover the independent edges as required for the pebble game. Specifically, we first apply gossip averaging [154], to yield|E ∗ P |. Then, local pebble assignments can easily be determined locally, as verified by Proposition 6.1.12. This composite construction, which we will denoteP +S, is 168 illustrated in Figure 6.4. An example execution of the parallel algorithm is depicted in Figure 6.5. Complexity Analysis Towards the real-world applicability of our propositions, we have complexity that scales well with network size: Proposition 6.1.3 (P +S complexity). By construction, executions of theP +S algorithm have worst-case O(n 2 ) messaging complexity, and O(n) storage scaling. Proof. For theP portion of theP +S algorithm, we have that each of n agents communicates with at most n− 1 neighbors, yielding O(n 2 ) messaging. The gossip averaging for determining|E ∗ P | exhibits O(n logn) messaging [154], while a local pebble assignment trivially requires O(n 2 ) operations. Thus we have an overall worst-case message complexity of O(n 2 ) for aP +S asS also has O(n 2 ) messaging by Proposition 6.1.2. The overall storage complexity follows directly from the fact that assignments toE ∗ can be made only locally by each agent i∈I, and thus scales like O(n). Finally, to close we can also roughly evaluate the expected improvement provided by the partial parallelization due toP: Proposition 6.1.4 (Parallel identification). An execution ofP, applied to a graph G = (V,E) (where we assume every v i ∈V is an endpoint inE), with n≥ 3 must result in the terminal condition |E ∗ P ( ¯ t P )|, [ i E ∗ P,i ( ¯ t P ) ≥ n− 1 2 + 1 (6.7) whered·e is the standard ceiling operator, yielding a lower bound on the independent edges identified by theP algorithm. 169 Proof. First, notice that we disregard the case of n = 1 (as|E| = 0), and for n = 2 we can always identify the single member ofE ∗ due to symmetric conflict resolution. Now, observing that the single n = 2 graph is a worst-case in terms of detectable independent edges , with|E ∗ | =dn/2e, for n≥ 3 we can construct a similar worst-case graph by appending a single node and edge to the n− 1 worst-case, as the appended edge will always be detectable by an EER operation. As we add a single detectable edge to the previous worst-case, a simple inductive argument yields the result. The above result states directly that we can always detect a spanning tree over G using our proposed parallel and distributed interactions. Further, the number of edges that the parallelization identifies directly affects the number of edges left to be found, and thus the complexity of the serial execution. A summary of the advantages of our parallelization are given in the following remark, while technical analysis and detailed pseudocode can be found in the Appendix. Remark 6.1.2 (Parallelization Benefits). First, as the parallel algorithm identifies at least a spanning tree in its execution by Proposition 6.1.4, it can quickly help to determine when the network is non-rigid without having to run the serialized step, yielding significant speed advantages in those scenarios. Additionally, as the parallel algorithm takes advantage of independence preserving operations, one could leverage it to build a rigid network autonomously and efficiently, e.g., in constructing a localizable sensor network topology. From a practical perspective, the parallelization is simply a more intelligent use of available network resources, i.e., it can be implemented without any additional communication or hardware requirements, so even a factor of two speedup may be convenient in practice. Also, such speedups relate well to realistic scenarios such as the time scales of external network influences and the speed of rigidity evaluation. Even constant factor speedups can expand the 170 v 2 v 1 v 4 v 3 (a) Initial requests. v 2 v 1 v 4 v 3 (b) Initial responses. v 2 v 1 v 4 v 3 (c) Final requests. v 2 v 1 v 4 v 3 (d) Final responses. Figure 6.5: Parallel messaging for a minimally rigid graph with n = 4. applicability of our algorithms under faster switching topologies or environmental conditions. 6.1.5 A Rigidity Control Scenario We wish to demonstrate here a scenario where network rigidity can be controlled in a dynamic multi-robot system. To begin, consider that in controlling generic rigidity we need only to disallow the loss of independent edges (i,j)∈E ∗ . For this purpose, we can employ the constrained interaction framework previously detailed 171 −20 −10 0 10 20 30 −20 −10 0 10 20 30 (a) −20 −10 0 10 20 30 −20 −10 0 10 20 30 (b) −20 −10 0 10 20 30 −20 −10 0 10 20 30 (c) −20 −10 0 10 20 30 −20 −10 0 10 20 30 (d) Figure 6.6: Rigiditycontrolsimulationforn = 9mobileagentsapplyingadispersive objective with a rigidity maintenance constraint. in this thesis. Thus, in applying our proposed decentralized pebble game to identify local setsE ∗ i ∀i∈I, we arrive directly at rigidity preserving predicates. That is, P a ij , 0, P d ij , (i,j)∈ (E ∗ i ∪E ∗ j ) (6.8) Now, we are prepared to present our rigidity control simulation results. We assumeasystemofn = 9mobileagents, eachwithproximity-limitedcommunication 172 Time |E| Swarm Size 0 100 200 300 400 500 0 100 200 300 400 500 15 20 25 30 35 40 0 10 20 30 40 50 Figure 6.7: Swarm size and edge set cardinality for the rigidity control simulation. and sensing, applying for the sake of link deletion, a dispersive objective controller, yielding agent controllers with generic form: u i =u CI −∇ x i X i∈N i 1 kx i −x j k 2 ∀i∈I (6.9) where u CI is the control contribution due to the constrained interaction framework and predicates (6.8). The agents begin in the fully connected initial configuration given by the ring network depicted in Fig. 6.6a, satisfying the initial condition G(0)∈G R . Through controllers (6.9) the agents reach intermediate configurations given by Figs. 6.6b and 6.6c, ultimately terminating in the final configuration in Fig. 6.6d. Fig. 6.7 depicts the spatial size of the swarm, i.e., the largest distance between any two agents, and the size of the independent edge set, which dictates network rigidity. Thus Fig. 6.7 demonstrates that the dispersive objective is achieved 173 through increasing swarm size, and that the network remains rigid as the size of the independent set is bounded below by 2n− 3. 6.1.6 Contiki Implementation: Real-World Feasibility Results To determine the performance of our algorithms under realistic networking condi- tions, we consider the Contiki operating system, together with the Cooja network simulator 2 . We implemented both the serial and parallel decentralized pebble game for Contiki and tested our codebase against a range of emulated hardware platforms and communication stacks for correctness. A Monte Carlo set was sim- ulated by generating rigid and non-rigid networks for n∈{5, 29}, yielding the results depicted in Fig. 6.8. Specifically, Fig. 6.8 (top) compares the execution time (seconds) for the serial and parallel algorithms, while Fig. 6.8 (bottom) shows the per-agent messaging burden. It is clear that both of our algorithms exhibit feasible and efficient performance, with actual scaling that is approximately O(n) in both execution and messaging. The parallel version however represents our goal of real-world capability by outperforming the serial version by a factor 2, as even in reasonably sized networks, execution times for evaluation are under 1 second. An initial version of the base code for our proposed algorithms has also been released for application in the robotics community 3 . 2 See http://www.contiki-os.org. 3 See http://github.com/Attilio-Priolo/Rigidity_Check_Contiki 174 Parallel Serial Parallel Serial Network Size (n) Messages Execution (sec) 5 10 15 20 25 5 10 15 20 25 0 10 20 30 40 50 60 0 2 4 6 8 10 Figure 6.8: Monte Carlo results from a Contiki networking environment demon- strating execution time and message complexity. 6.1.7 The Serialized Algorithm To complement our discussion of theS algorithm, we analyze the correctness, finite termination, and cost properties of our algorithms. First, we formally establish the stopping condition for theS algorithm: 175 Definition 6.1.6 (S stopping condition). As previously discussed, theS algorithm terminates upon satisfaction of the following condition: f S stop , ( n X i=1 b i = 0 ! ∨ n X i=1 |E ∗ i | = 2n− 3 !) (6.10) where the P i b i = 0 indicates that all agents have been a leader, and P i |E ∗ i | = 2n−3 is detected by the lead agent on line 24 of Algorithm 9. Next, we verify that our formulation guarantees the entire network is evaluated for rigidity, with no edge reconsideration, and further that mutual exclusion of the local independent sets holds: Proposition 6.1.5 (Edge consideration, mutual exclusion). Disregarding algorithm termination when|E ∗ | = 2n− 3, every (i,j)∈E is eligible to be considered for independence. Further,E ∗ i ∩E ∗ j =∅ holds for all i6=j∈I. Proof. These results are a simple consequence of the guaranteed convergence of auction (6.2), b i = 0 for all beenLeader(i) = 1 guaranteeing no reelection, and the initialization ofE i with edges not shared with previous leaders. To ensure timely results, we must also have finite termination ofS: Proposition 6.1.6 (S termination). Consider the execution of theS algorithm as described in Sections 6.1.1, 6.1.1, and 6.1.2. By construction, it follows that the stopping condition f S stop of (6.10) is satisfied after a finite number of clock ticks. Proof. We can guarantee no message-induced race conditions by Assumptions 7 and 8, and that there exist no algorithmic race conditions due to the internal blocking on line 2 of Algorithm 9. From the request checking mechanism (line 2 Algorithm 10) and the guaranteed delivery of inter-agent messages by the best-effort 176 Assumption 8, we have that all pebble request messaging rooted at agent i with isLeader(i) = 1, is finite, i.e., every pebble request receives a response. Now, the finiteness of execution is a direct consequence of the finiteness of eachE i ,∀i∈I and the finite convergence of auction (6.2) [128], as there exists no leader reelection by construction. Now we come to our primary result concerning the correctness of theS algorithm: Proposition 6.1.7 (S correctness). Consider an execution of S, applied to a graphG = (V,E). It follows that by construction we are guaranteed to identify |E ∗ | = 2n− 3 independent edges whenG∈G R , and|E ∗ |< 2n− 3 otherwise, i.e.,S properly identifies the generic rigidity ofG. Proof. First, notice that by Proposition 6.1.5, we can ensure that every (i,j)∈E is eligible for quadrupling and pebble covering as dictated by the original pebble game [106], and further thatE ∗ i ∩E ∗ j =∅ holds for alli6=j∈I ensures that|E ∗ | is properly tracked by our distributed storage. Thus, correctness is shown by arguing that our leadership and messaging formulation is faithful to the rules of the pebble game. This result follows by observing that pebble assignments and shift operations are only made locally (line 8 of Algorithm 9, line 7 of Algorithm 10, and lines 2, 4, and 8 of Algorithm 11), and that the pebble search mechanism respects the network’s distributed pebble assignmentsP i ∀i∈I (line 12 of Algorithm 9, line 11 of Algorithm 10, and line 3 of Algorithm 12). Finally, as there is only one leader active at any time, each quadrupling operation (lines 6-17, Algorithm 9) is sound with respect to the current setE ∗ , and the result follows. The above result demonstrates that S is sound in terms of planar rigidity evaluation. 177 Algorithm 9 Leader execution logic. 1: procedureLeaderRun(i) 2: while e i , (i,j) do . Continue pebble covering 3: while Quadrupled Copies≤ 4 do 4: if p i > 0 then . Assign local pebble 5: P i ←P i ∪e i 6: p i ←p i − 1 7: else . Request pebble along first edge 8: PebbleRequestMsg(i,P i (1, 2)) 9: return 10: end if 11: end while 12: . Quadrupling success, return 3 pebbles: 13: P i ←∅ 14: p i ← 2 15: Return 1 pebble to v j 16: . Add independent edge and check rigidity: 17: E ∗ i ←E ∗ i ∪e i 18: if|E ∗ | = 2n− 3 then 19: Send network rigidity notification 20: return 21: end if 22: . Go to next incident edge: 23: E i ←E i −e i 24: e i ← (i,j)∈E i 25: end while 26: . All local edges checked: 27: Initiate leadership transfer auction 28: end procedure 6.1.8 The Parallelized Algorithm We begin with a proof of the independence preservation of the EER and TIER graph operations: Proposition 6.1.8 (Independence preservation). Consider the graphG = (V,E) having edgesE forming an independent set according to Definition 5.1.1. The edge addition operations [·] + e overG abiding by the EER and TIER requirements of 178 Algorithm 10 Pebble request handler for agent i. 1: procedureHandlePebbleRequest(from, i) 2: if Request Not Unique then . Already requested 3: PebbleNotFoundMsg(i, from) 4: return 5: end if 6: if p i > 0 then . Local pebble available 7: P i ←P i ∪ (i,from) . Shift free pebble 8: p i ←p i − 1 9: PebbleFoundMsg(i, from) 10: else . Request along first assigned edge 11: PebbleRequestMsg(i,P i (1, 2)) 12: requester(i)← from 13: end if 14: end procedure Algorithm 11 Pebble found handler for agent i. 1: procedureHandlePebbleFound(from, i) 2: P i ←P i − (i,from) . Free local pebble 3: if isLeader(i) then 4: P i ←P i ∪e i . Expand covering 5: else . Give free pebble to requester 6: P i ←P i ∪ (i,requester(i)) 7: PebbleFoundMsg(i, requester(i)) 8: end if 9: end procedure Definitions 6.1.4 and 6.1.5 are independence preserving in the sense of Definition 6.1.3, respectively. Further, considering a sequence of graphs{G(0),...,G(m)} generated by G(0) =G G(k) = [G(k− 1)] + e , k = 1,...,m (6.11) over EER and TIER operations yields graphG m having independent edgesE m . Proof. First, consider the case of an EER operation overG. By the independence ofE, we have by Definition 5.1.1 that for every subgraph ¯ G = ( ¯ V, ¯ E),| ¯ E|≤ 2| ¯ V|− 3. 179 Algorithm 12 Pebble not found handler for agent i. 1: procedureHandlePebbleNotFound(from, i) 2: if Paths Searched < 2 then . Search other path 3: PebbleRequestMsg(i,P i (2, 2)) 4: else . Search failed, no free pebbles 5: if isLeader(i) then . e i is redundant 6: Return pebbles assigned to e i 7: . Go to next incident edge: 8: E i ←E i −e i 9: e i ← (i,j)∈E i 10: else 11: PebbleNotFoundMsg(i, requester(i)) 12: end if 13: end if 14: end procedure In the augmented graph edge e introduces expanded subgraphs containing e all having the property | ¯ E| + 1≤ 2| ¯ V|− 3 + 1≤ 2(| ¯ V| + 1)− 3 (6.12) due to the node expansion property of the EER, all of which therefore abide by the independence subgraph property. The remaining subgraphs ofG + are independent by assumption. Thus, we conclude that the EER operation according to Definition 6.1.4 is independence preserving. Now, consider the application of the TIER operation overG. By Definition 6.1.5 there must exist an endpoint of e, (i,j), indexed by i∈I, with exactly N i = 1 overG. Thus, we can view the edges (i,j) and (i,k) with k∈N i , and the nodei as members of a two edge Henneberg operation, as in Section 3 of [105], e.g., adding vertex v 4 with edges (4,2) and (4,1) in Figure 6.3b. As the edge subtraction operation [·] − e is independence preserving, the graphG + described by applying the previous two edge Henneberg operation to [G] − (i,k) has independent edges by 180 Algorithm 13 Parallel execution logic for agent i. 1: procedureParallelRun(i) 2: if e i , (i,j)6= 0 then . Next incident edge 3: EdgeRequestMsg(i, j) 4: requestedFrom(i)←j 5: return 6: end if 7: . All local edges checked: 8: idle(i)← Yes 9: end procedure Algorithm 14 Parallel edge request handler for agent i. 1: procedureHandleEdgeRequest(i, j) 2: response← committed(i) 3: if requestedFrom(i) =j then . Edge contention 4: if ResolveConflict(i, j) =i then 5: response← 2 . Ensure i wins edge 6: end if 7: end if 8: EdgeResponseMsg(i, j, response) 9: if response < 2 then . Max of 2 incident edges 10: committed(i)← committed(i) + 1 11: end if 12: . Do not double check (i,j): 13: E i ←E i − (i,j) 14: end procedure Proposition 3.1 of [105]. Briefly, this result follows from the independence ofE and the relationships,|E| =|E + |− 2 and|V| =|V + |− 1. Thus, the TIER operation is also independence preserving. Finally, the independence preservation of a sequence of EER and TIER opera- tions is a trivial consequence of the initial independence ofE and the IP properties of each edge augmentation. 181 Algorithm 15 Parallel edge response handler for agent i. 1: procedureHandleEdgeResponse(i, j, response) 2: if response < 2 then . Independence guaranteed 3: E ∗ i ←E ∗ i ∪ (i,j) 4: committed(i)← committed(i) + 1 5: end if 6: . Go to next incident edge: 7: E i ←E i − (i,j) 8: e i 8 (i,j)∈E i 9: end procedure To complement our discussion of theP+S algorithm, we analyze the correctness, finite termination, and cost properties of our algorithms. First, we verify that E ∗ P , S i E ∗ P,i has a valid distributed construction: Proposition 6.1.9 (Parallel mutual exclusion). Consider the application of the parallel P algorithm to a graphG = (V,E). It follows that upon termination we have mutual exclusionE ∗ i ∩E ∗ j =∅,∀i6=j∈I. Proof. Given the assumptions of asynchronicity in messaging and the FIFO queuing of received messages (Assumption 8) and execution devoid of race conditions (Assumption 7), the following scenarios must be considered, viewed from the instant when agent i handles an EdgeResponseMsg from j, implying that e i = (i,j) and (i,j) / ∈E ∗ i (line 2, Algorithm 15): • (i,j) / ∈E ∗ j : We must consider two cases here, either e j = (i,j) or e j 6= (i,j). First, in the trivial case of e j 6= (i,j), it follows from reception of an EdgeResponseMsg from j, the atomic nature of execution, and line 13 of Algorithm 14, that regardless of assignment toE ∗ i , (i,j) / ∈E ∗ j for all execution t > 0. When e j = (i,j), the conflict resolution of line 3-7 in Algorithm 14, together with line 2 of Algorithm 15 ensures that simultaneous 182 requests made over (i,j) agree on assignment, specifically as by assumption ResolveConflict(i,j) =ResolveConflict(j,i). • (i,j)∈E ∗ j : Here it is implied that at some previous time agent i received and responded to aEdgeRequestMsg from j. As agent i being in a state of response reception over (i,j) is contradictory given line 13 of Algorithm 14, it must be the case that requests over (i,j) have been made in concert. However, as previously stated, the conflict resolution ensuresE ∗ i ∩E ∗ j =∅ in such scenarios. Notice that due to the uniformity of execution and messaging logic across i∈I, the previous scenarios hold equivalently from the perspective of agent j, and thus for all pairs{i6=j| (i,j)∈E}, and the result follows. Of course,E ∗ P must also fulfill the independence requirements of Definition 5.1.1, as is shown below. Proposition 6.1.10 (Parallel Correctness). Consider the algorithmP applied to a graphG = (V,E). For all execution t> 0 it follows that edge addition operations, E ∗ i (t + ) =E ∗ i (t)∪ (i,j) (line 3 Algorithm 15), are independence preserving and S i E ∗ i is independent with| S i E ∗ i |≤ 2n− 3. Proof. Resting again on Assumptions 7 and 8, and the uniform conflict resolution of ResolveConflict(i,j), this result is a consequence of the commitment counting rules (lines 10 and 5 of Algorithms 14 and 15), and the condition on line 2 of Algorithm 15 that enforces the cardinality of endpoint j in S i E ∗ i (t + ). In particular, when j / ∈ S i E ∗ i (t) (committed(j) = 0),E ∗ i (t + ) =E ∗ i (t)∪ (i,j) constitutes an EER operation (c.f. Definition 6.1.4), otherwise when j∈ S i E ∗ i (t) (committed(j) = 1), E ∗ i (t + ) =E ∗ i (t)∪ (i,j) constitutes a TIER operation (c.f. Definition 6.1.5). As 183 sequences of EER and TIER operations preserve independence by Proposition 6.1.8, S i E ∗ i is independent, with| S i E ∗ i |≤ 2n− 3 following directly from the Laman conditions Theorem 5.1.1. To ensure timely results, we must also have finite termination ofP +S: Proposition 6.1.11 (P termination). Consider the execution of theP algorithm as described in Section 6.1.4. By construction, it follows that the stopping condition, i.e., all agents are idle, is satisfied after a finite number of clock ticks. Proof. We can again guarantee no message-induced race conditions by Assumptions 7 and 8. Thus, the finiteness of execution is a direct consequence of the finiteness of eachE i ,∀i∈I, the internal blocking on line 2 of Algorithm 13, the guaranteed delivery of inter-agent messages by the best-effort Assumption 8, and finally the symmetric conflict resolution of Remark 6.1.1, disallowing conflict based race conditions. To constitute a valid initial condition forS, the pebble assignments applied to the terminal stateE ∗ P ofP must be sound: Proposition 6.1.12 (Pebble Assignments). Consider an execution ofP, applied to a graphG = (V,E). There must exist a local pebble covering for every (i,j)∈ E ∗ i ∀i∈I, that is a local assignment of a pebble by either i or j to (i,j). Proof. In guaranteeing that such an assignment exists, we rely on the properties of the EER and TIER operations. As each operationE ∗ i (t + ) =E ∗ i (t)∪ (i,j) respects Proposition 6.1.8 by Proposition 6.1.10, we have that for any (i,j)∈ S i E ∗ i there must exist an endpoint i or j with at most two incident edges. From this endpoint we can thus always select a pebble to cover (i,j) as each agent i∈I is initially assigned two pebbles. 184 Now we come to our primary result concerning the correctness and finite termination of theP +S algorithm: Proposition 6.1.13 (P +S correctness and termination). Consider an execution ofP +S, applied to a graphG = (V,E). It follows that the terminal system state: P( ¯ t P ), ( [ i E ∗ i ( ¯ t P ),P i ,p i ) ∀i∈I (6.13) is a valid initial condition for theS algorithm, the execution ofP +S terminates after a finite number of clock ticks, and properly identifies the generic topology rigidity ofG. Proof. First, we have directly from Propositions 6.1.6 and 6.1.11, the known finite convergence of gossip averaging [154], and the trivial finiteness of a local pebble assignment process, that the composite execution ofP +S terminates in finite time. Now, from Propositions 6.1.9 and 6.1.10 it follows that the stateP( ¯ t P ) represents a properly distributed and independent edge set, and from Proposition 6.1.12 that there must exist pebble assignmentsP i that are local shift operations relative to Definition 5.2.1, and thus constitute a valid pebble covering. Thus the application ofS with inputP( ¯ t P ) is correct by Proposition 6.1.7, and the result follows. 6.2 Constructing Optimally Rigid Networks Assume now that a weight w(e ij ) =ψ(v i ) +ψ(v j ) +φ(v i ,v j ) is associated to each edgee ij , whereψ(v i ) :V→R + is a cost function associated to each endpoint of the edge andφ(v i ,v j ) :V×V→R + is a metric associated toe ij . We consider the weight distributions to have known upper and lower limits, such thatw min ≤w(e ij )≤w max holds for all realizations of the weighted graphG (or at least is known to occur with 185 v 1 v 2 v 3 v 4 v 6 v 5 (a) Full weighted graph. v 1 v 2 v 3 v 4 v 6 v 5 (b) Optimal Laman subgraph. Figure 6.9: A weighted graph and its optimal Laman subgraph. high probability). Note that due to the symmetry ofG(V,E), e ij ∈E⇒e ji ∈E, w(e ij ) =w(e ji ). As the node with the smallest bid is elected in the previously described rigidity evaluation, the bids dictate the order of elected leaders and thus the edges that constitute the identified rigid subgraph. In other words, the bids can be applied based on the application. For example, if we assume each edge is assigned a weight which indicates its cost in sensing or information, we could choose leader bids that are the sum of incident edge cost. Then the resulting rigid subgraph would possess those edges that are the least costly in the given application, while ensuring rigidity. Bids could also be chosen to reflect node availability, processing capability, or the cardinality of incident edges, or they can be leveraged in terms of metrics related to mission objectives. The proposed auction technique therefore affords us control overE ∗ that goes beyond simply discovering the network’s rigidity property. Thus, based on our results so far, we assume that the network can apply a decentralized (and asynchronous) pebble game to determine with O(n 2 ) complexity the independence of any edge e ij inG. We will then exploit here the leader-based 186 mechanism to control the order and cardinality of considered edges, and ultimately optimality vs. complexity. Under the assumption of weighted graph edges, we will therefore discover optimal Laman subgraphs to construct cost effective rigid networks in a fully decentralized way. Let us begin by providing some preliminary assumptions and definitions which are instrumental for the considered optimization problem. Consider a generic objective function ρ(G) :G→R + of the form: ρ(G) = X e ij ∈E w ij (6.14) to be defined over the graphG(V,E). We are now ready to introduce the definition of an optimal subgraph ¯ G: Definition 6.2.1. A subgraph ¯ G(V, ¯ E)⊆G(V,E) is optimal if: @ ˆ G(V, ˆ E)⊆G(V,E) :ρ( ˆ G)<ρ( ¯ G), with ¯ G, ˆ G rigid. See Figure 6.9a and Figure 6.9b for an illustration of an optimal Laman subgraph. The identification of an optimal subgraph ¯ G can be expressed in terms of a minimization problem as follows: ¯ G = arg min G 0 ⊆G ρ(G 0 ). (6.15) Assumption 9. The graphG includes at least 2n− 3 independent edges, i.e.,G is rigid. Note that, Assumption 9 is a necessary condition for the existence of a feasible solution. Nevertheless, it should be highlighted that useful information could still 187 be extracted from the execution of the proposed algorithms even if this assumption does not hold, e.g., to check the existence of rigid subcomponents. Furthermore, it should be noticed that this optimization problem may not have a unique solution as there could exist several rigid subgraphs which minimize the objective function. In this work, no distinction will be made among such solutions (which we assume to be equivalent), however minor extensions of this work could provide such discernment, e.g., by introducing additional node or edge costs to further delineate equivalent optimal subcomponents. According to Definition 6.2.1 a naive approach to find an optimal rigid subgraph would be to inspect all the possible subgraphs ofG , thus leading to an impractical computational complexity. However, it has been proven in [107] that the optimal solution to (6.15) is a Laman subgraph ¯ S(V, ¯ E). Therefore in this work we propose a decentralized approach to build an optimal Laman subgraph ¯ S(V, ¯ E)⊆G(V,E) based on pairwise collaboration among the nodes. More specifically, we propose two alternative algorithms: the first approach finds an optimal solution to (6.15) at the cost of higher communication complexity; the second approach provides a sub-optimal solution while reducing the computational burden according to a design parameter ξ. 6.2.1 Optimal Greedy Algorithm TheOptimalGreedyalgorithm(OG)proposedinthisworkextendsthedecentralized version of the pebble game algorithm. Briefly speaking, major modifications are made to the bidding process for leader election and the number of edges each leader can check for independence at each iteration. Moving forward, the decentralized pebble game algorithm by PebbleGame in our pseudocode implementation. 188 Algorithm 16 The Optimal Greedy Algorithm. 1: procedureOptGreedy(G = (V,E)) . Init the data structures 2: {E ∗ i (0),E i (0)}←InitStructs(i,G),∀i∈I 3: k← 1 4: repeat 5: . Compute the bids for each i: 6: {b i (k),e min ij (k)}←ComputeBid(i,E i (k− 1)) 7: . Auction to select the leader with minimum bid: 8: l← arg min i b i (k) 9: . Leader checks e min lj (k) independence: 10: E ∗ l (k)←LeaderRun(l,E ∗ l (k− 1),e min lj (k)) 11: . Remove already considered edges: 12: E l (k) =E l (k− 1)\e min lj (k) 13: E j (k) =E j (k− 1)\e min jl (k) 14: k←k + 1 15: until|E ∗ (k)| =|∪ i E ∗ i (k)| = 2n− 3 16: end procedure The pseudo-code of the proposed OG algorithm is depicted in Algorithm 16. Generally speaking, the algorithm works as follows. In step 2, each node initializes according to Procedure 17. Then, the algorithm runs until the cardinality of the independent set is|E ∗ (k)| = 2n− 3, that is an optimal Laman subgraph has been found at thek th iteration 4 . In particular, the following steps are executed. First, the bidding process for the leader election detailed in Procedure 18 is run (step 6). Then, the chosen leader selects (and removes) from the local set of candidates the edge with minimum weight and checks its independence by running the decentralized pebble game according to Procedure 19. This edge is then added to the local independent edge set if the check succeeds, or it is removed otherwise. Let us now further detail the three procedures mentioned above. 4 Note that to compute|E ∗ (k)|, after each leadership auction, the nodes update a global counter embedded in the auction packages, decentralizing the process. 189 Procedure 17 Data structures initialization. 1: procedureInitStructs Input:i,G Output:E ∗ i (0),E i (0) 2: E ∗ i (0) ={∅},∀i∈I . Init independent set portion 3: . Sort the incident edges set: 4: E i (0)←{e ij :v j ∈N i } 5: Sort (E i (0)),∀i∈I 6: end procedure Procedure 17 performs the initialization of the required data structure. In particular, in step 2 the portion of the independent setE ∗ i owned by the i th node is initialized along with the set of edgesE i originating/ending with the i th node and sorted according to the chosen metric. Procedure 18 computes the bids for each node before the leader election step. The value of the bid is determined according to the cardinality of the setE i (k), i.e., the set of candidate edges at step k. If this set is empty, i.e., all the incident edges have been checked, an arbitrary large value is assigned to the bid (step 6). As it will be explained in the following, a leader is elected by the algorithm if it submits the smallest bid. Therefore, a large value in the bid prevents node i from becoming a leader. If|E i (k)|6= 0, then the minimum weight among the candidates edges is computed and used as the bid (step 3). Using the smallest weight as bid guarantees that the solution is incrementally built considering the graph edges in an increasing order with respect to their weights. The edge corresponding to the minimum weight inE i (k) is stored by each node i in e min ij (k) because if i is elected as the next leader, then e min ij (k) is the edge whose independence has to be checked (step 4). Procedure 19 describes the leader execution. In step 2, the decentralized pebble game originally introduced by the authors in [156] is used for checking the 190 Procedure 18 Bid and minimum weight edge computation. 1: procedureComputeBid Input:i,E i (k− 1) Output:b i (k), e min ij (k) 2: ifE i (k− 1)6={∅} then 3: b i (k) = min e ij ∈E i (k−1) w ij 4: e min ij (k) = arg min e ij ∈E i (k−1) w ij 5: else 6: b i (k) =∞ 7: end if 8: end procedure Procedure 19 The leader execution. 1: procedureLeaderRun Input:l,E ∗ l (k− 1), e min lj (k) Output:E ∗ l (k) 2: res←PebbleGame(e min lj (k)) 3: if res is true then 4: E ∗ l (k) =E ∗ l (k− 1)∪e min lj (k) 5: end if 6: end procedure independence of e min lj (k). If the check succeeds, then the edge is added to the i th local set of independent edges in step 4. Remark 6.2.1. Note that, from an implementation standpoint the algorithm requires a unique identifier to be associated to each node, e.g., the address of the node’s communication device. Let us now prove that the solution computed by Algorithm 16 is indeed optimal, that is the algorithm builds a Laman subgraph ¯ S⊆G with minimum cost. The following propositions are instrumental to prove such an optimality result. Proposition 6.2.1. Consider a Laman subgraphS 0 (V 0 ,E 0 )⊆G(V,E). To preserve the Laman condition the removal of a subset of independent edges ˆ E⊆E 0 with 191 cardinality| ˆ E| forces the insertion of only| ˆ E| independent edges with endpoints in V 0 . Proof. The proof of Proposition 6.2.1 is a direct consequence of the Laman condition given in Theorem 5.1.1. In fact, by definition the Laman subgraphS 0 has 2|V 0 |− 3 independent edges. Therefore, in order to preserve the condition imposed by the theorem, up to| ˆ E| edges with endpoints inV 0 can be inserted into the independent set after removing ˆ E. Note that, this proposition does not provide any information about how these | ˆ E| edges should be chosen. Indeed, it simply provides a conservation principle, that is the number of edges to be added has to match the number of edges that have been removed in order to preserve the Laman property of the subgraph. Proposition 6.2.2. Denote withG ∗ k (V ∗ (k),E ∗ (k)) the subgraph of Definition 5.1.1 and incrementally built by k iterations of Algorithm 16. Then, substituting an edge e ij ∈E ∗ (k) with the edge e min lh (q)∈E : v l ,v h ∈V ∗ (k) chosen at step q>k, leads to: ρ( ˆ G)≥ρ(G ∗ k ), (6.16) with ˆ G(V ∗ (k), ˆ E), ˆ E ={E ∗ (k)\e ij }∪e min lh (q). Proof. At each step, the current leader selects among the local set of residual candidate edges the one with the minimum weight for independence checking. Therefore, w min lh ≥w ij ,∀e ij ∈E ∗ (k). The proof then follows from the definition of the objective function in (6.14). Notice that in a minimally rigid graph it trivially follows that our algorithm identifies the optimal Laman subgraph, as such a subgraph constitutes the graph itself. Thus we can conclude that when considering the order of edge evaluation 192 and its relationship to subgraph optimality, we must consider the impact of edge redundancy in the process. By Theorem 5.1.1 and Definition 5.1.1 we know that redundancy is a subgraph property and thus we must only focus our reasoning for a redundant edge e ij on the subgraph against which e ij is redundant. First, recall that redundancy implies that a degree of freedom in the subgraph has already been satisfied and thus it follows that a redundant edge can be substituted with only a single edge already in the subgraph and achieve equivalent constraints on the subgraph. We can therefore view the choice of an independent edge as disallowing future edges from addition to the independent edge set spanning the subgraph. In terms of optimality, we therefore must only ask whether those edges chosen as independent disallow or displace edges of higher value in given subgraph. In Figure 6.10 we have illustrated why a greedy approach to edge consideration is reasonable. In (a) we have the original non-minimally rigid graph and in (b) we have a greedy sequence of edge considerations, where solid edges have been added to the independent set, dashed edges are redundant, and the bold edge is the final independent edge that establishes the graph’s rigidity. Notice that edge (2, 5) renders both (2, 6) and (3, 5) redundant when it is chosen, due to the Laman subgraph property. However, it is straightforward to reason that swapping either (2, 6) or (3, 5) for (2, 5) can only be correct if they possessed a lower weight. But such a case is captured also by a greedy ordering, i.e., the choice of (2, 6) or (3, 5) would occur first then rendering (2, 5) redundant. Thus, if our order of edge consideration is greedy, we achieve optimality since all previous edges identified as independent have lesser weight and thus a single swap of any displaced edges in the subgraph always violates optimality. Following the above reasoning, combined with the edge substitution result of Proposition 6.2.2, leads us directly to the optimal solution by considering the graph 193 v 1 v 2 v 3 v 4 v 5 v 6 (a) Original non-minimally rigid graph. v 1 v 2 v 5 v 6 v 2 v 3 v 4 v 5 (b) Edge consideration sequence and sub- graphs. Figure 6.10: An illustration of the relationship between edge consideration, redun- dancy, and optimality. edges in a greedy ordering. This optimality, while potentially counterintuitive, is informally a consequence of the embedded global information that is verified in the independence check of each edge. We solidify this intuition in the following inductive proof that formalizes the optimality of our proposed approach. Theorem 6.2.1. The solution ¯ S(V, ¯ E)⊆G(V,E) built incrementally according to Algorithm 16 is optimal. Proof. Let us consider the first iteration of the algorithm, i.e,k = 1. The bid choice in Algorithm 16 guarantees that the edge e min lj (1) with minimum weight is selected and added to the independent setE ∗ l (1) with l the index of the current leader. Therefore the value of the objective function computed usingE ∗ (1) is minimum. Let us now consider the generic k th iteration, with k> 1. Two cases are possible: C1 e min lj (k) is independent, C2 e min lj (k) is redundant. Let us analyze C1. First of all, note that|E ∗ (k− 1)|< 2n− 3, otherwise the edge would be redundant. Then, the insertion of e min lj (k) guarantees that the objective function computed using the edges inE ∗ (k) =E ∗ (k− 1)∪e min lj (k) is minimum because of the e min lj (k) definition. 194 Consider now C2. According to Proposition 6.2.1, the redundancy of e min lj (k) and the fact the optimal solution must be a Laman subgraph, denoted in the following with ˆ S( ˆ V, ˆ E), implies that an edge e ij should be removed from ˆ S to add e min lj (k). However, the replacement of a generic edgee ij ∈ ˆ E would increase the value of the solution according to Proposition 6.2.2. Therefore, at step k the edgee min lj (k) should not be inserted intoE ∗ i (k). Iterating this reasoning until|E ∗ (k)| = 2n− 3, it turns out that the optimality of the solution is preserved at each step. Note also that the existence of a solution is guaranteed by Assumption 9. As previously stated, the optimality of the solution comes at a price. In the following proposition, the communication complexity of the algorithm, in terms of number of exchanged messages, is characterized. Proposition 6.2.3. Algorithm 16 exhibits a O(n 3 ) worst case communication complexity for the leader election process. Proof. In the leader election process, O(n) messages are expended for each auction. In the worst case, all the edges of the network are analyzed, incurring in O(n 2 ) executions. Therefore, the overall messaging complexity becomes O(n 3 ). While small networks may be undeterred by the above complexity, scalability is of the utmost concern in larger networks. Thus, we consider a modification of the greedy approach that aims to balance optimality and complexity. 6.2.2 Sub-Optimal Algorithm In this section we introduce a modified algorithm, denoted as SO algorithm, by which the computational complexity of the OG algorithm can be mitigated. To this end, we modify the OG algorithm by introducing a tuning parameter ξ which represents the number of edges to be checked for independence at each leader 195 election. It will be shown that in this way, the communication complexity can be reduced up to O(n 2 ). In addition, we provide a closed form of the maximum gap between the optimal solution and the sub-optimal solution provided by this algorithm expressed in terms of the tuning parameter ξ. Let us now detail the major differences concerning the modified SO algorithm and the OG algorithm given in Algorithm 16. As in the previous section, let us assume that each node i is able to sort its incident edgesE i according to the associated weights w ij . Let us denote with w q ij the q th weight among the weights associated to node i edges after the sorting, such that w 1 ij ≤ w q ij ≤ w |E i | ij , where q ∈ [1,...,|E i |] and with e q ij ∈ E i the edge corresponding to w q ij . Before the algorithm execution, a parameter ξ∈ [1,...,n− 1] is chosen. It represents the number of edges that an elected leader is allowed to check for independence. This parameter is used by the modified ComputeBid and LeaderRun procedures depicted in Procedures 20 and 21, respectively. In Procedure 20, the bid for the leader auction process is computed by each node summing up the first ξ edges weights and the edges themselves are stored in E ξ i (steps 3-4). This represents a conservative approach to minimize the increase of the solution at step k + 1, even though this might lead to a sub-optimal solution. In Procedure 21, the edges in the setE ξ l are checked for independence (step 3). Each edge is inserted into the the i th portion of the independent set after a successful independence check, it is removed otherwise. In allowing each elected leader to insertξ edges we may deviate from the greedy edge ordering that leads us to the optimal rigid subgraph. The parameter ξ yields what is effectively a queued auction controlling the evaluation of edge independence. We expect that the network size n, the queue size ξ, and the known bounds of the weight distribution will factor into the optimality gap experienced by the SO 196 algorithm. To characterize this relationship, let us first understand the potential gap in optimality that may occur when a leaderi considersξ edges for independence, as in the following: ξ X q=1 w q ij −U(q) ≤ξΔ w (6.17) whereU(q) is the q th smallest weight in the set of unconsidered weightsU, and Δ w =w max −w min . Notice that it may be the case that not all members ofE ξ i are independent with respect to the current set of independent edges. In such a case, e q ij ∈E ξ i would not be eligible for addition to the independent set, and if w q ij were not the minimum weight in the graph, an error in optimality caused by selectinge q ij would be avoided. That is, redundancy always reduces the local optimality gap, and thus the form of (6.17) acts as a worst case bound for any elected leader as all edges are viewed as independent. Note that due to the abstract and combinatorial nature of edge redundancy inG, it is unlikely to find a closed form of (6.17) that both accounts for redundancy, and conveys more information about suboptimality. In other words, regardless of redundancy, suboptimality is certainly always a function of the variability of the weight metric, as is conveyed in (6.17). We can further apply the idea that redundancy aids in bounding suboptimality to extend (6.17) to a full execution of the SO algorithm. In particular, consider the overall bound d2n−3/ξe X k=1 ξΔ w ≤ξO(n)Δ w (6.18) whered2n− 3/ξe is the number of elections required to consider 2n− 3 edges, ξ at a time. Again, as redundancy aids in suboptimality, viewing the suboptimality of the first 2n− 3 edges as in (6.18) acts as an upper bound on the gap that would be experienced in considering the first 2n− 3 independent edges, the algorithm stopping condition. Finally, in applying (6.18) for system design, we can assume 197 that one would choose as a parameter an acceptable gap in optimality, denoted as (n). Thus, (6.18) yields the relationship ξ∼ (n) O(n)Δ w (6.19) possessing exactly the intuition we would expect from our queued auction approach. Precisely, the average election gap (n)/O(n) trades off directly with our choice of queue size ξ and the known variation of our cost metric. To conclude, let us now analyze the messaging complexity exhibited by the algorithm: Proposition 6.2.4. The modified algorithm exhibits a O(n 3 /ξ) worst-case messag- ing complexity for the leader election process. Proof. To perform a worst case analysis, assume the graphG to be fully connected, i.e.,|N i | =n− 1,∀i∈I. Therefore, to check all the incident edges, in the worst case each node has to be elected (n− 1)/ξ times as leader. Iterating this reasoning for the n nodes and keeping in mind that n messages are sent during each leader election step, then the overall messaging complexity for the leadership auction process is O(n 3 /ξ). It is worthy to note that if ξ→ 1, the messaging complexity is O(n 3 ) as the sub-optimal algorithm collapses to the greedy version. Instead, if ξ→n− 1 the complexity becomes quadratic, i.e., O(n 2 ). Thus ξ behaves exactly like a sliding mode control, trading off complexity with optimality. 6.2.3 Relative Sensing: A Case Study In this section, we consider Relative Sensing Networks (RSNs) introduced in [111] as an application scenario to evaluate the effectiveness of the proposed decentralized 198 Procedure 20 Modified bid and minimum weight edge computation. 1: procedureModComputeBid Input:i,E i (k− 1),ξ Output:b i (k),E ξ i (k) 2: ifE i (k− 1)6={∅} then 3: b i (k) = ξ P q=1 w q ij 4: E ξ i (k) ={e q ij ∈E i (k) :q≤ξ} 5: else 6: b i (k) =∞ 7: end if 8: end procedure Procedure 21 The modified leader execution. 1: procedureModLeaderRun Input:l,E ∗ l (k− 1),E ξ l (k) Output:E ∗ l (k) 2: for all e ij ∈E ξ l do 3: res←PebbleGame(e ij ) 4: if res is true then 5: E ∗ l (k) =E ∗ l (k− 1)∪e ij 6: end if 7: end for 8: end procedure algorithms. Generally speaking, RSNs denote collections of autonomous nodes that use sensed relative state information to collaboratively achieve higher level objectives. In such systems, a sensing topology (or graph) is induced by the spatial orientation of the nodes and the capabilities of the relative sensors. In this way, the underlying sensing topology couples the nodes at their outputs. As the exchange of information is crucial in these systems to achieve almost any collaborative objective, a key research problem is to understand how the network topology affects the overall behavior of the system and study its resilience to possible external disturbances. By assuming nodes to have linear (possibly heterogeneous) dynamics, two inter- esting metrics are proposed in [111], namely theH 2 andH ∞ norms, to investigate 199 ρ(G) = 18924 Starting Graph G (a) Starting GraphG. ρ(G) = 10036 OG-Algorithm (b) OG Algorithm. ρ(G) = 10386 SO-Algorithm ξ = ⌈N/2⌉ (c) SO Algorithm. ρ(G) = 10698 Zelazo et al. Algorithm (d) Zelazo Algorithm. Figure 6.11: Rigid network composed of 7 vertexes running optimal rigidity algorithms. the role of the underlying connection topology on the system norms mapping the exogenous inputs to the relative sensed output. In this way a characterization of the robustness of the network (in terms of network topology) against external dis- turbances is derived. The reader is referred to [111] for a comprehensive description of the overall framework and more specifically of the nodes dynamics and metric definitions. 200 Notably, theH 2 normisexploitedin[107]asanoptimalitycriterionforthedesign of RSNs where the underlying sensing graph is rigid. As mentioned before, rigidity is an important requirement for collaborative systems with several implications in different contexts ranging from localization to formation control. In particular, based on the observation that an optimalH 2 RSN must be a minimally rigid graph, a centralized algorithm for generating locallyH 2 optimal rigid graphs is outlined in [107]. Briefly, this algorithm consists of a variation of the Henneberg construction for generating rigid graphs in the plane by adding performance requirements and sensing constraints, yielding theH 2 optimal vertex addition and edge splitting procedures. Nevertheless, apart from being centralized, several assumptions on the underlying structure of the sensing graphs, which are required for the correct termination of the algorithm, render its applicability in the context of decentralized systems remarkably challenging. On the contrary, we point out how this problem formulation of optimally growing minimally rigid graphs fits well the optimization problem addressed in this work. In particular, for a given RSN whose interaction is described by the weighted graphG = (V,E) consider the following optimization problem as in [107]: min G kΣ(G)k 2 +c|E| subject to G is rigid (6.20) wherekΣ(G)k 2 is theH 2 performance of the RSN. Furthermore, such anH 2 performance in the context of heterogeneous RSNs can be detailed as follows: kΣ(G)k 2 = n X i=1 d i kΣ i k 2 2 (6.21) where d i =N i is the neighborhood of node i andkΣ i k 2 is itsH 2 norm. Notably, this problem formulation can be restated in terms of the optimization problem 201 Zelazo et al. SO ξ = N −1 SO ξ = ⌈N/2⌉ GO ρ(G) |G| 6 8 10 12 14 16 18 20 22 24 26 28 30 10 3.1 10 3.3 10 3.5 10 3.7 (a) Objective Function Comparison. SO ξ =N −1 SO ξ = ⌈N/2⌉ GO # Packets |G| 6 8 10 12 14 16 18 20 22 24 26 28 30 10 2 (b) Communication Cost Comparison. Figure 6.12: Monte Carlo analysis comparing optimal rigidity algorithms. given in eq. (6.14) considered in this work. In particular, a weight w ij for a pair of nodes i and j such that e ij ∈E is computed as: w ij =ψ(v i ) +ψ(v j ) =kΣ i k 2 2 +kΣ j k 2 2 . (6.22) Figure 6.11 depicts a typical run of the algorithms for a rigid network composed of 7 vertexes where the thickness of the edge denotes its weight normalized with respect to the largest possible. It can be noticed that the OG algorithm provides the solution with the lowest value of the optimization function ρ(G) = 10036, while the centralized algorithm given in [107] provides (as expected) a locally optimal solution ρ(G) = 10698, and finally the SO-algorithm withξ =dN/2e which allows to reduce the communication cost also provides a sub-optimal solution ρ(G) = 10386. We now provide a Monte Carlo analysis demonstrating applicability under realistic conditions of the proposed algorithms. Furthermore, we compare the performance of proposed SO algorithm against the centralized locally optimal algorithm given in [107] for different values of the tunable parameterξ. In particular, Figure 6.12 depicts the results of Monte Carlo analysis for a network with a number 202 Initial Configuration (a) Initially topology. t = 1 10 20 30 40 50 60 0 10 20 30 40 50 60 (b) t = 1. t = 50 10 20 30 40 50 60 0 10 20 30 40 50 60 (c) t = 50. t = 100 10 20 30 40 50 60 0 10 20 30 40 50 60 (d) t = 100. t = 300 10 20 30 40 50 60 0 10 20 30 40 50 60 (e) t = 300. t = 751 10 20 30 40 50 60 0 10 20 30 40 50 60 (f) t = 751. Figure 6.13: Mobile RSN preserving rigidity and minimizing theH 2 norm. of nodes ranging from 6 to 30. For a fixed network size, we consider 50 different graph topology realizations with the related weights. Fig. 6.12a depicts the average of the objective function for the OG algorithm, the SO algorithm with ξ =dN/2e, the SO algorithm with ξ =N− 1, and the centralized algorithm described in [107], respectively. It can be noticed that the OG algorithm as expected provides always the best (optimal) solution. Notably, the SO algorithm with different ξ values, althoughsub-optimalstillprovidessatisfactoryresults, comparablewiththe(locally) optimal solution given by the centralized algorithm described in [107]. Fig. 6.12b shows a comparison of the average communication cost in term of number of packets exchanged for the OG algorithm, the SO algorithm with ξ =dN/2e, and the SO 203 algorithm with ξ = N− 1, respectively. It can be noticed that, while providing sub-optimal solutions, the SO-algorithm scales better in terms of the number of packets exchanged compared to the OG-algorithm. Finally, we consider as an illustrative example a spatially interacting mobile RSN performing a dispersive behavior while preserving the rigidness of the network topology and minimizing theH 2 norm. Interestingly, it should be noticed that for this particular scenario, due to the spatial nature of the interaction, the optimal solution can be achieved by assuming the nodes to start fully connected while running the proposed algorithms to solve the optimization problem and then start spreading while preserving the connectedness of the optimal subgraph. Fig. 6.13 provides a series of screenshots illustrating the result of the constrained dispersive behavior for a mobile RSN composed of 7 nodes. In particular, Fig. 6.13a depicts the initial configuration where the thickness of the edges represents their normalized weight obtained according to eq. (6.22). Furthermore, a solid line describes an edge belonging to the optimal minimally rigid graph to be preserved computed by the OG-algorithm, while a dashed line describes redundant edges. By looking at Figs 6.13b-6.13f it can be noticed how the network shapes itself in such a way to maximize the dispersion among the nodes while preserving the edges belonging to the optimal minimally rigid graph. 6.3 Distributed and Efficient Rigidity Control To close the thesis, we demonstrate the integration of our spatial topology control, with our methods for evaluating network rigidity, to achieve rigidity control in a multi-agent team. Specifically, we consider the following problem: 204 Problem 6 (Graph rigidity control). Given an initial rigid 5 graphG(0) associated with agents having dynamics (4.1), design velocity controls u i (t) such thatG(t) is rigid for all t> 0. In solving Problem 6 we will therefore require knowledge of the impact of each edge on the rigidity of graphG, which is captured in the notion of edge independence, a direct consequence of Theorem 5.1.1, where we note that from Theorem 5.1.1 it follows that edges that are redundant with respect to an essential subgraph preserve rigidity upon removal (in Section 6.3.1 we will show that such reasoning extends to locally redundant edges as well). The above conditions imply that all rigid graphs have 2n− 3 independent edges, precisely members of a Laman subgraph. The solution to Problem 6 is now apparent: we simply disallow transitions (switches) inG that correspond to the removal of independent links, implying the preservation of graph rigidity. 6.3.1 Predicates for Rigidity Preservation Determining an independent edge set overG is clearly a global operation, yet we aim to define a distributed method with minimal communication load. Thus, we propose here a local method for discerning links in the graph by applying redundancy in subgraphs containing discerned links (i,j). Notice first that the Laman conditions, while elegant, exhibit exponential complexity when applied directly. Conveniently, many works have discovered polynomial time algorithms for combinatorial rigidity; for its simplistic appeal, we again cite here the pebble game proposed by Jacobs and Hendrickson in [106], which can determine graph 5 Such a condition can be relaxed to a connectedness requirement if we assume the agents perform a rendezvous maneuver at system initialization, specifically as the fully connected network is rigid. 205 rigidity with worst case complexity O(n 2 ), generating in the case of a rigid graph, a set of 2n− 3 independent edges through incremental application of Definition 5.1.1. In determining edge independence, we will therefore assume that each agent can apply efficiently the pebble game algorithm. Now, we need to understand how local knowledge of edge redundancy influences the rigidity of the entire network graph. The simple case of a minimally rigid graph is a direct consequence of the Laman conditions: inG there exists a single Laman subgraph,G itself, and thus in every subgraph ¯ G we must have| ¯ E|≤ 2| ¯ V|− 3, requiring that every edge (i,j)∈ ¯ E is independent in ¯ G. Thus, in the context of a minimally rigid graph, a local rule exploiting edge redundancy will properly preserve rigidity. We extend such reasoning for the case of non-minimally rigid graphs where there must exist redundant edges. Specifically, we are interested in the global implications of redundant edges identified in local subgraphs: Lemma 6.3.1 (Local redundancy). Assume graphG = (V,E) is non-minimally rigid in R 2 and consider a subgraph ¯ G = ( ¯ V, ¯ E)⊆G having an edge (i,j)∈ ¯ E that is redundant with respect to ¯ G. Then, there must exist a Laman subgraph S = (V,E s )⊂G such that (i,j) / ∈E s . Proof. This is again a consequence of Theorem 5.1.1, and we can provide a reasoning that relates well to our application of the pebble game in revealing independent edges. Clearly we have that the order of edge consideration for independence (i.e. applications of Definition 5.1.1) is irrelevant under the assumption that all edges are considered (specifically as edges in the independent edge set are mutually independent by Definition 5.1.1). AsG is rigid we know for any such ordering we must find 2n−3 independent edges, members of a Laman subgraphS = (V,E s )⊂G. Denote by ¯ S = ( ¯ V, ¯ E s ) a Laman subgraph of ¯ G, with (i,j) / ∈ ¯ E s by definition. Now consider a specific independence check ordering, where we begin by considering first 206 every (k,l)∈ ¯ E s , then every (k,l)∈E| (k,l) / ∈ ¯ E s , (k,l)6= (i,j), leaving then (i,j) for the final independence consideration. As we know all (k,l)∈ ¯ E s are independent by definition, we have ¯ E s ⊂E s by the consideration ordering. As (i,j) is redundant against ¯ E s it follows that (i,j) / ∈E s as then there would exist a subgraph inS that violates Definition 5.1.1. Thus, we have identified a Laman subgraph ofG of which (i,j) is not a member, and the result follows. 6 . Equivalently Lemma 6.3.1 implies that edge (i,j) is redundant with respect to G and thus the graphG −e = (V,E− (i,j)) is rigid. Such a conclusion allows us then to gain insight into the impact of arbitrary links on the rigidity of the graph by inspecting only local subgraphs of our choosing. This is powerful in the context of the constrained interaction framework, as we now can define predicates that preserve independent links in the network (or equivalently, allow the removal of redundant links), and guarantee dynamic rigidity maintenance. First, in order to comply with the predicate construction requirements of Assumption 3, we require a worst-case graphG w , with respect toC a i ,C d i . Consider the following graph G w = (V,E w ),E w ,{(i,j)∈E|∀v i ∈V,v j / ∈C d i } (6.23) corresponding to the graphG with all candidates for deletion removed (as addition candidates are not yet links, such links are also not present inG w ). Defining function f r (G):G n →{0, 1} representing the rigidity of a graphG, we then have the following result: Lemma 6.3.2 (Worst-case rigid graph). Consider a graphG = (V,E) with an associated worst-case graphG w = (V,E w ) defined as in (6.23). It follows that if 6 Notice that the key aspect in this proof is the discovery of a Laman subgraph not containing (i,j), andthusthepurposeofourspecificchoiceofordering; alternativeorderingsdonotnecessarily reveal such information. 207 f r (G w ) = 1, then f r (G) = 1 for allG∈G C , the space of reachable topologies over C a i ,C d i ,∀i∈I. Proof. Notice that as all deletion candidate links are removed fromG w , there exist no further edges that can be removed fromG as topological transitions can only occur over candidate links (c.f. Figure 4.2). Thus, fromG w we can construct anyG ∈G C through link addition, that is we simply add deletion candidates or addition candidates toG w to form all possible combinations. Thus, assuming f r (G w ) = 1 and noting that link addition preserves graph rigidity, it follows that f r (G) = 1,∀G∈G C , our desired result. We can now leverage Lemma 6.3.2 in conjunction with Lemma 6.3.1 to generate distributed predicates for rigidity preservation. Considering the proposition of the addition or deletion of link (i,j)∈E, we construct a local worst-case subgraph G w ij = (V w ij ,E w ij )⊆G w ⊆G as follows: V w ij ,{v i ,v j }∪N w i ∪{N w k |v k ∈N w i } E w ij ,{(k,l)∈E|v k ,v l ∈V w ij } (6.24) withN w k ,{v l ∈N k | (k,l)∈E w },∀k∈I, where we refer to agent i as the discerning agent, that is the agent to whom the responsibility has been assigned to determine the independence of link (i,j). Notice thatG w ij constitutes 2-hop information from the perspective of agent i, however under the assumption that all agents maintain an updated neighbor set, the construction ofG w ij requires only local communication with v k ∈N w i . It turns out that such a local graph construction is necessary to extract information that is meaningful in the global graph scope: Lemma6.3.3 (Localgraphconstruction). For an agenti, the local graph containing neighboring nodes and incident edges possesses only independent edges. 208 Proof. If we consider onlyN i , we generate discernment graphs with 1 +|N i | nodes and|N i | edges, and as rigid graphs must satisfy|N i |> 1,∀i∈I, we have |N i |< 2(|N i | + 1)− 3 and by Definition 5.1.1 all edges are independent. Despite the existence of redundant edges in a network, Lemma 6.3.3 indicates that a purely local inspection yields no insight into edge redundancy. We are finally prepared to define our link discernment predicates for dynamic rigidity preservation in the constrained interaction framework. Denoting f red ((i,j),G):V×V→{0, 1} as the function indicating the redundancy of a link (i,j) with respect to graphG, we have link deletion predicate P d ij ,¬f red ((i,j),G w ij ) (6.25) discerning links relative to the local worst-case graphs, and link addition predicate P a ij , 0 (6.26) allowing all link additions as they preserve graph rigidity. Our primary conclusion concerning dynamic graph rigidity preservation can now be stated: Theorem 6.3.1 (Rigidity preservation). Consider the multi-agent system defined by dynamics (4.1), with control inputs driven by the switching behavior depicted in Figure 4.2, and starting from a feasible initial condition, where our assumed constraint isP(G),f r (G), implying that the initial graphG(0) is rigid. Further, assume that P a ij and P d ij are defined as in (6.25), (6.26). Then for all t > 0, we haveP(G(t)) = 1 and graph rigidity is maintained. Proof. Assuming trivially that we haveC a i (0) =C d i (0) =∅, it follows thatG w (0) = G(0) and thus f r (G w (0)) = 1. Now, consider the possible transitions inG w that 209 can occur due to switching as in Figure 4.2. We have two cases to consider, either current members of j∈C a i ,C d i are lost due to d ij >ρ 2 or d ij ≤ρ 1 , respectively, or new members j∈C a i ,C d i are gained due to d ij ≤ ρ 2 ,P a ij = 0 or d ij > ρ 1 ,P d ij = 0, respectively. The first case is trivial as lost members j∈C a i were not members of G w and thus no transition occurs, while lost members j∈C d i represent the link (i,j) being added back toG w , preserving rigidity. Now, in the second case for new members j∈C a i the link (i,j) is not added toG w by definition, and again no transition occurs. Finally, for new members j∈C d i , we have f red ((i,j),G w ij ) = 1, that is the link (i,j) is redundant with respect toG w ij . AsG w ij ⊆G w it follows from Lemma 6.3.1 that (i,j) is also redundant with respect toG w , implying that the removal of (i,j) fromG w due to new membershipj∈C d i preserves the rigidity ofG w . Thus, for all possible transitions inG w rigidity is preserved andP a ij ,P d ij abide by the preemptive construction required by Assumption 3. As all conditions of Theorem 4.2.2 have been satisfied, we can then conclude thatP(G(t)),f r (G(t)) = 1 for all t> 0, our desired result. From the above result we thus arrive at a distributed controller for combinatorial rigidity preservation in mobile networks, when interaction is proximity-limited. Remark 6.3.1 (Local rule complexity). The discernment of a link (i,j) requires a single communication message of size O(n), between O(n) neighbors for the discerning agent i. The application of the pebble game by agent i requires O(n 2 ) operations in the worst-case, with average performance on the order ofO(n 1.15 ) [106]. Most importantly, our controller requires communication and computation only during proposed transitions in the network topology, not continuous operation as in [109]. Remark 6.3.2 (Local rigidity matrix). In determining link redundancy, a rigidity matrix [105] could also be applied in the context of the constrained interaction 210 framework to control graph rigidity. However, we choose the pebble game [106] for the following advantages: improved computational complexity (O(n 2 ) vs. O(n 3 )), implications in reusing pebble assignments for future discernments to improve dynamic complexity, and amenability to distributed implementation (as already demonstrated). It should be noted that applying the rank condition, while increasing the computational cost, could allow for operation in dimension m > 2, i.e. the arguments made for local redundancy checks hold perfectly valid in using the rigidity matrix. 6.3.2 Local Conservatism and Extensions To conclude, we provide a brief discussion of the tradeoffs involved in applying a local rigidity control rule, and a consensus-based extension to allow the discernment of links relative to the entire network. In applying deletion predicate (6.25) over local subgraphG w ij , notice that f red ((i,j),G w ij ) is affirmative only when, according to Definition 5.1.1, there exists some subgraph ¯ G w ij = ( ¯ V w ij , ¯ E w ij ) ⊆ G w ij having | ¯ E w ij |> 2| ¯ V w ij |− 3 edges. However, forG w rigid, such a condition need not hold in everyG w ij , leading to scenarios such as that depicted in Figure 6.14, in which the local rule can behave conservatively in link deletions 7 . In Figure 6.14a we have a non-minimally rigid graph in which edge (v 4 ,v 5 ) is redundant by construction. If we then assume that agents v 4 and v 5 move apart, triggering the discernment of (v 4 ,v 5 ) for deletion, we obtain the subgraph Figure 6.14b against which (v 4 ,v 5 ) is checked for redundancy. Clearly in this graph (v 4 ,v 5 ) is independent by Definition 5.1.1 and as we would then have predicate P d 45 = 1, edge (v 4 ,v 5 ) would become a member ofD a 4 ,D a 5 , denying deletion, despite the redundancy of (v 4 ,v 5 ) overG w . Such conservatism clearly diminishes with 7 Even in light of such behavior, network rigidity is still maintained. 211 increasing connectivity, due to shrinking network diameter (making a local check effectively global) and the increase in network links and thus possibilities for subgraphs with redundancy. As discussed by Remark 6.3.1 such conservatism trades off with minimal local communication and computation that scales well in n. Thus in rigid networks with localized redundancy, our proposed rule will uncover edge redundancy without a full network inquiry, becoming conservative only near the boundary of minimal rigidity. However, withoutmodifyingthelocalcommunicationconstraint, oneobservation is immediately useful in applying our local rule. If followed naively the local rule will consider redundancy against the entire subgraphG w ij , while in fact by Definition 5.1.1 we only need one subgraph against which the discerned edge is redundant. Thus, in applying the pebble game overG w ij (i.e. building an independent edge set), we simply compare (i,j) against the current independent set after each expansion, and check the edge condition of Definition 5.1.1. Such a test is O(1) for each expansion of the independent set, allows early stopping of the check for redundancy of (i,j) inG w ij , and can alleviate potential conservatism in link deletions. For example, If in Figure 6.14a we had an edge (v 4 ,v 6 ), then in Figure 6.14b, the subgraph over {v 4 ,v 5 ,v 6 ,v 8 } clearly reveals the redundancy of (v 4 ,v 5 ) despite the non-rigidity of the entire graph. If there are cases where the conservatism of a local check is undesirable, we can simply expand the checked subgraphs by applying consensus to generateG w instead ofG w ij . We can collect the candidacy information j∈C a i ,C d i ,∀i,j∈I into matrices W(t),Y (t)∈{0, 1} n×n , having elements w ij = 1 if j∈C a i and y ij = 1 if 212 v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 (a) Original rigid graph. v 1 v 2 v 3 v 4 v 5 v 6 v 7 v 8 (b) Non-rigid local subgraph. Figure 6.14: An illustration of conservatism in the local redundancy rule. j∈C d i , respectively. Now, inspired by previous work [50,143], the network can perform the following consensus iteration (per agent i): A i (t + 1) =∨ j∈N i (A i (t)∨A j (t)) (6.27a) W i (t + 1) =∨ j∈N i (W i (t)∨W j (t)) (6.27b) Y i (t + 1) =∨ j∈N i (Y i (t)∨Y j (t)) (6.27c) where each agent i initializes A i (0),W i (0),Y i (0) with their local knowledge only. AssumingG(t) is connected, iterations (6.27) converge in at most n− 1 steps to A i = A(t),W i = W(t), and Y i = Y (t), that is uniformly to the global topology and candidacy state [50]. Now we can constructG w locally for discerning agent i from the matrix A w =A i ⊕Y i , where⊕ is the matrix-wise XOR operation, noting that adjacency A w precisely describes the construction (6.23). Discernment of the redundancy of link (i,j) can then occur over the entire graphG w , eliminating the 213 Consensus Fit Consensus Local Fit Local Consensus Fit Consensus Local Fit Local Messages Network Size (n) Computation (ms) 4 6 8 10 12 14 16 18 20 4 6 8 10 12 14 16 18 20 0 200 400 600 800 0 20 40 60 Figure 6.15: Monte Carlo simulations of the local rigidity control rule and a consensus-based extension. previously discussed conservative behavior. Of course instead of a single communi- cation with O(n) neighbors, iterations (6.27) require O(n 2 ) communications in the worst case. 6.3.3 Simulated Rigidity Control In this section, we present simulation results of our proposed graph rigidity con- troller. First, to verify the technical contributions of Section 6.3.1, we randomly generated 1000 rigid graphs with n∈ [4, 20] uniformly. For each graph, we then applied our local redundancy rule and the consensus-based extension to every edge in the graph, determining if an edge could be removed and preserve rigidity, comparing the decision to the known redundancy of each edge. To demonstrate 214 complexity, we compared the average computation time (in milliseconds) for the local rule (6.25) and the consensus-based extension (6.27) versus network size n, as depicted in Figure 6.15. We see that the local rule fairs better in computation (top) as the network size increases, as local checks become less likely to be global (due to increasing network diameter). Further, the local rule vastly outperforms the consensus-based rule in communication as is clear by Figure 6.15 (bottom, measured in total inter-agent messages). Overall, in our simulations the local rule and consensus extension exhibited O(n 1.007 ) vs. O(n 1.065 ) computation scaling, respectively, and O(n 0.7212 ) vs. O(n 2.022 ) communication scaling, respectively. To demonstrate how the complexity advantages of the local rule tradeoff with conservative link deletions, we provide the results depicted in Figure 6.16. Specif- ically, we compared the ratio of known redundant edges to those identified by our local rule (i.e. 1 implies least conservative, 0 implies most conservative), to the network redundancy ratio, that is (|E|− 2n + 3)/(n(n− 1)/2− 2n + 3), the ratio of redundant edges to total possible redundant edges (i.e. 1 implies most redundant, 0 implies minimally rigid). We see immediately that near the region of minimally rigid graphs, the local rule becomes conservative in link deletions, as locally redundant subgraphs become less likely. However, such conservatism recovers quickly as network redundancy increases, with complete ignorance to redundant links only occurring very near minimal rigidity. Thus, the local rigidity controller allows topologies with near-optimal redundancy, with complexity that has demonstrated near-linear scaling. Finally, we refer the reader to link for a video of our methods in the Player/Stage environment. 215 Redundancy Ratio Deleted Edges Ratio 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1 Figure 6.16: Monte Carlo simulations demonstrating conservative behavior of the local rigidity rule. 216 Chapter 7 Conclusions and Future Work 7.1 Thesis Summary In the following, we summarize the primary contributions and methodologies we have presented in this thesis. Probabilistic Mapping and Tracking In order to motivate the study of interaction and topology in multi-agent coordina- tion, we detailed a novel method for cooperative inference in multi-agent systems where uncertainty was modeled by the grid structured pairwise Markov random field. The multi-agent Markov random field framework was proposed, wherein the global inference problem is decomposed into inter-agent belief exchanges and local intra-agent inference problems. An approximate inference algorithm inspired by loopy belief propagation was chosen due to the exponential complexity of exact inference. An intelligent message passing scheme based on the idea of regions of influence in a cluster graph in LBP was discussed. Finally, we presented simulation results for the inference methods over a spatial grid with the goal of identifying plume-like features in oceanographic data using multiple virtual robots. The results demonstrated the accuracy and smoothness of exact inference but also showed its intractability over large grids. The approximate inference methods were shown to be a reasonable alternative both in terms of computational efficiency and accuracy. 217 In particular, the intelligent ROI message passing scheme is attractive as it exhibits a lack of dependence on grid size. The probabilistic map generated by cooperative inference was then used as a base for tracking the probabilistic level curves of a spatial process. A Lyapunov stable curve tracking control was derived and a method for gradient and Hessian estimation was presented for applying the control in a discrete map of the process. Simulation results showed promising performance and feasibility of the algorithms in realistic multi-agent deployments. Most importantly however, probabilistic mapping and tracking illustrated the deep assumptions of interaction and topology that were necessary for collaboration. In the case study, we assumed properties such as connectedness for collective decision-making, map building, and cohesive motion, discrete network properties for feasible inference, and idealized communication and sensing for each agent. All too common in theoretical studies of collaboration, topological assumptions and agent interaction were identified as a weakness in the study of coordination, motivating the true contributions of this thesis. Spatial Interaction and Mobility-based Topology Control With the goal of generalizing topological control in multi-agent systems, we consid- ered the problem of controlling the interactions of a group of mobile agents, such that a set of topological constraints were satisfied. Assuming proximity-limited communication between agents, we leveraged mobility to enable adjacent agents to achieve discriminative connectivity, that is, to actively retain established links or reject link creation on the basis of constraint satisfaction. Specifically, we pro- posed a distributed scheme consisting of controllers with discrete switching for link discrimination, whereby adjacent agents were classified as either candidates for 218 constraint-aware link removal or addition, or as constraint violators requiring either link retention through attraction or link denial through repulsion. Attractive and repulsive potentials fields established the local controls necessary for link discrimi- nation and constraint violation predicates formed the basis for discernment. We analyzed the application of constrained interaction to two canonical coordination objectives in distributed multi-agent systems: aggregation and dispersion behaviors, with maximum and minimum node degree constraints, respectively. For each task, we proposed predicates and continuity-preserving potential fields and proved the dynamical and constraint satisfaction properties of the resulting hybrid system. Simulation results demonstrated the correctness of our proposed methods and the ability of our framework to generate topology-aware self organization. Our spatial topology results were then cast in the context of flocking, in order to study the system under topology constraints having a non-local composition. In tandem with consensus-based coordination over propositions in topology switches, it was shown that the satisfaction of constraints is guaranteed over the system trajectories, with the desired velocity alignment and collision avoidance achieved. Simulations of a novel constrained coordination scenario demonstrated the correct- ness and applicability of our proposed methods. To expand the scope of our topological tools, we moved towards continuous topology control by considering distributed algorithms for estimating the algebraic connectivity of a network of agents, as well as control laws for connectivity maxi- mization under local topology constraints. To estimate algebraic connectivity, the inverse iteration algorithm was extended to a distributed domain, along with a Jacobi iteration method for solving linear systems. A switching interaction model and potential field controls for connectivity maximization, constrained agent degree, link maintenance, and collision avoidance were presented and convergence was 219 proved. Simulation results demonstrated the correctness of our inverse iteration formulation, while agent aggregation and leader-following simulations showed the efficacy of our proposed potential fields for connectivity control under topological constraints. Completing our investigation of mobility-based topology control, we illustrated a novel hybrid architecture for command, control, and coordination of networked robots for sensing and information routing applications, called INSPIRE (for INfor- mation and Sensing driven PhysIcally REconfigurable robotic network). INSPIRE provided two control levels, namely the Information Control Plane and Physical Control Plane, so that a feedback between information and sensing needs and robotic configuration is established. An instantiation was provided as a proof of concept where a mobile robotic network is dynamically reconfigured to ensure high quality routes between static wireless nodes, which act as source/destination pairs for information flow. The case study illustrated how localized and distributed topology control can integrate with high-level decision-making, as well as beginning to cope with heterogeneity in agent mobility. Evaluating, Constructing, and Controlling Rigid Networks Along with network connectivity, graph rigidity stands as one of the most ubiquitous topological assumptions in multi-agent coordination. To close the thesis, we therefore considered the problem of evaluating the rigidity of a planar network under the requirements of decentralization, asynchronicity, and parallelization. We proposed the decentralization of the pebble game algorithm of Jacobs et. al., based on asynchronous inter-agent message-passing and distributed auctions for electing network leaders. Further, we provided a parallelization of our methods that yielded significantly reduced execution time and messaging burden. Finally, we provided a 220 simulated application in decentralized rigidity control, and Monte Carlo analysis of our algorithms in a Contiki networking environment, illustrating the real-world applicability of our methods. Expanding upon our rigidity evaluation technique, the problem of building an optimal rigid subgraph was considered. In particular, two decentralized approaches were presented. The first approach iteratively builds the optimal minimally rigid graph by electing leaders to check the independence of at most a single incident edge. To mitigate the messaging complexity required by this algorithm, a modified version was introduced. In particular, by choosing a parameter ξ governing the number of weights incorporated in the bids and the number of edges considered for the independence check, the messaging complexity has been scaled by a factor 1/ξ. This comes at a price, i.e. the modified approach generates a sub-optimal minimally rigid graph. However, a closed form to upper-bound the difference between the two algorithm solutions was derived. Finally, to solve the dynamic rigidity control problem, we proposed local rules for link addition and deletion that guaranteed the preservation of rigidity of a network graph through agent mobility. It was proven that local link redundancy is sufficient for global redundancy in preserving graph rigidity, and the tradeoffs of applying a localized rule were explored, together with a simple consensus-based extension. Finally, Monte Carlo analysis and a simulated agent coordination scenario exploiting generic rigidity demonstrated and confirmed our results. 7.2 Directions for Future Work The developments of this thesis, while approaching many of the open issues of multi-agent topology control, have also uncovered various directions for future 221 investigation. Again, in the vein of moving collaborative theory to reality, we will focus on identifying the outstanding assumptions that remain in the methods outlined in this work. Probabilistic Mapping and Tracking Although we used the mapping and tracking case study as a motivation, it is still worthy to briefly discuss extensions and improvements of the proposed framework. First, a significant advancement would lie in identifying less stringent requirements on agent communication, as maintaining a tree structured communication graph may place significant overhead on the system under realistic conditions. A superior system would allow arbitrary, possibly time-varying hyper-structures over which cooperative inference tasks would take place. There exist so-called belief consensus algorithms that may hold promise in this direction. We also aim to apply approx- imation methods to the multi-agent inference problem that do not exhibit the convergence issues of the LBP algorithm. Possible alternatives include structural approximations, cooperative sampling approaches, and approximate inter-agent and intra-agent message passing. Investigating varying process and model potentials may prove useful in mapping more complex spatial phenomenon. Finally, the integration of curve tracking with topological constraints could act to eliminate the network assumptions required for cohesive and stable agent motion. We also point out that this case study provides motivation to study the inter- section of objectives, e.g., how is mapping or tracking impacted by topological constraints, and can we quantify this relationship to aid in system design and deployment? Also, what style of heuristics can we adopt to treat intersecting objec- tives that have little hope of admitting optimality? In most non-trivial scenarios the inclusion of topological constraints yields optimization problems which are 222 difficult to solve optimally, especially in a distributed context. In fact, high-level objectives such as informative path planning can not be approximated efficiently when topological constraints are considered. Thus, it may be fruitful for example to pursue periodically breaking topological constraints relative to multi-objective pri- oritization, at least as a base strategy for building heuristic solutions to constrained deployment. Of course, constraints like connectivity may render collaboration infeasible if broken; issues of intersecting high-level objectives with topological constraints must be pursued by the multi-agent/robot community to characterize such contradictions. Spatial Interaction and Mobility-based Topology Control In guaranteeing topological constraints through mobility, we again question the relationship between topology and objectives of collaboration. Initial motivation for topology control is derived from feasibility and efficacy of singular objectives. If we desire cohesion of motion, as in rendezvous, or collective agreement, as in consensus, we desire connectedness. If we desire improved convergence of agreement, or associated processes such as algebraic connectivity estimation, we simply increase network connectivity. For formation stability or feasible localization, we ask for rigidity. These one-to-one relationships are reasonable motivators, but one would certainly predict that the complex autonomy problems of the coming decades, e.g., in multi-robot systems, will be solved by systems not composed of homogeneously capable agents, but through specialization and cooperation, and thus heterogeneity. Further, realistic robotic systems will operate over a hierarchy of algorithms and controls, dictating communication, routing, mobility, decision-making, sensing, etc. Thus, investigating the intersection of layers of agent intelligence and network 223 topology is a frontier which will be vitally important in realizing distributed multi- agent systems. Very little investigation has been directed towards understanding intersecting topological constraints with other team objectives such as sensing, communication, spatial objectives, etc. These multi-objective characterizations need the community’s attention and will likely be approachable for example by examining recentresultsinnon-convexoptimization, orpotentiallygametheoreticformulations. As we will discuss in our closing thoughts (Section 7.3), identifying the underlying relationships between commonly sought multi-agent objectives may lead us towards a taxonomy of interconnectedness, providing researchers with the building blocks to better understand both how to construct complex and topologically constrained multi-objective systems, but also to characterize for example how heterogeneous systems may behave in practice. An accessible starting point may be integrating topology control into classical robotic intelligence, such as path planning or SLAM, as there already exists deep community wisdom in such objectives, and a clear need for feasible multi-robot collaboration in such systems. We have made initial strides with our Route Swarm heuristics, where an extension to more complex routing requirements may continue to inform of the connections between topology and layered autonomy. While we have illustrated topology control through connectivity and rigidity, arguably the most important network properties for collaboration, it is likely that novelty will be found in further investigations of topological constraints. For exam- ple, determining existential guarantees for predicate formulations that solve certain classes of constraint problems, and exploring constraints that are not explicitly spa- tial in nature, e.g., mutual information, agent similarity or compatibility measures, etc. Further, it is conjectured that by examining compositions of local and non-local interaction constraints in our mobility framework, there exists a foundation for 224 rich coordinated behaviors that extend dynamical complexity beyond aggregation and dispersion. Just as the Laman conditions characterize the rigidity property through discrete constraints, it may be possible to characterize further topological constraints that achieve useful spatial configurations. Such a perspective could be seen as generalizing formation or shape control, choosing configurations from a class of desirable topologies. An example of such a characterization is found [157], where the authors identify topological conditions that allow for spatial motion that guarantees dynamic connectivity preservation, without explicit control. This fusing of agent dynamics, topological constraints, and a desired topological property, is promising in terms of our suggested topological characterizations. It may also prove useful to characterize the relationship between discrete topo- logical constraints, and realistic distance-based communication models. Recent work in [158] has investigated approximately optimizing communication models through mobility and agent configuration. This work compels us to ask: how well can continuum communication models approximate discrete constraints, and conversely how well can discrete constraints, such as node degree or multi-hop prop- erties, approximate continuous models of communication, routing, or interference? Understanding such relationships would allow us to choose appropriate frameworks for implementation, yielding a powerful collaborative tool for multi-agent systems. For example, the discrete control proposed in this work can be achieved with signif- icantly less overhead than continuous communication optimization. More directly, how poor can communication become before we are willing to expend additional energy, computation, communication, or mobility to improve the situation? Can discrete topology control not be seen as a piecewise approximation to expensive, continuous-time optimizations? Certainly there are cases where discrete constraints 225 will either suffice or will be necessary (e.g., in broadcast-based communication), and it is only through experimentation that these relationships will be truly revealed. There is also an exceptional amount of effort that must be dedicated towards coping with the conditions of realistic multi-agent systems that have not been addressed in this thesis. First, is extending our efforts of constraint maintenance to deployments with initial conditions that are topologically infeasible. Clearly, the deployment problem is significantly more complex than bounding a system away from constraint violation. Yet, in realistic systems, guaranteeing initial feasibility is counter-productive to autonomy, as human operators must account for this property when deploying a system. Thus, studying the constrained deployment problem is a vital step towards improved autonomy. Inspiration for solving this problem may lie in infeasibility methods for non-convex optimization, but there also may be hope in exploiting algorithmic consequences, for example in rigidity. When the pebble game indicates a non-rigid network, information concerning where rigidity fails is also conveyed through pebble assignment and the search mechanism. Exploiting this algorithmic output may identify local agent motion that projects the network towards rigidity. Further realistic considerations are centered around the notion of asymmetry in agent interaction. We have focused in this thesis on undirected sensing and communication, as such assumptions are reasonable in many homogeneous scenarios and ease technical analysis. However, even systems that are designed around undirected interaction will in practice exhibit asymmetry, typically due to failure, noise, time delays, etc. It is also reasonable for systems to have purposefully designed asymmetries, e.g., in teams that communicate or sense over different scales, or have varying locomotion. Asymmetry has been and continues to be addressed in collaboration, for example, the convergence properties of consensus 226 processes are fairly well understood in asymmetric contexts. However, much like the motivators of this thesis, the topological requirements of asymmetric collaboration remain assumptions. It is unclear how topology control translates to account for asymmetry, particularly in terms of mobility control, where sensing and communication are vital to maintaining feasibility. We can note however, that the preemptive formulation of the discernment predicates in constrained interaction appear promising in approaching directed interaction between agents, due to sensor errors, communication failure, etc. As the preemptive characteristic creates a spatial buffer in interaction, i.e., a dwell between decisioning and topology change, there may exist error/failure bounds from which constraint satisfaction can be recovered. Our results have also assumed agent dynamics of the most basic form. While linearization methods or trajectory tracking can in some cases accommodate more complex vehicle dynamics, the relationship between model faithful control techniques and our constraint satisfaction methodologies is indeed interesting. Although we argue that relying completely on continuous estimation or opti- mization is likely to threaten the scalability of distributed implementations, it still remainsrelevanttodiscussextensionstoouralgebraicestimationscheme. Asinverse iteration is a general method for estimating an eigenpair, it would be reasonable to identify further domains requiring spectral analysis, and extend our methods. An example would lie in estimating the so-called rigidity eigenvalue as in [109], where inverse iteration would yield significant advantages in evaluation time. Further, as the inverse iteration performs well with previous estimates, application in environ- ments where spectral properties are analyzed intermittently may prove useful. In these contexts, characterizing the expected speedup of inverse iteration versus the state of the art would inform design decisions, much like our previous suggestions. Also, one would expect that intermittence in connectivity control would be suitable 227 for communication limited systems, much like our conjectures of discrete versus continuous communication control. In this case, it would be compelling to identify bounds on intermittence which could approximate continuous connectivity control with a desired error. Not only would intermittence serve resource constrained systems, but it also accommodates failure and unforeseen conditions of real-world implementation which degrade the ability of a system to act in a continuous or synchronous way. Evaluating, Constructing, and Controlling Rigid Networks As our results in rigidity are based on spatial topology control, most of the future work previously suggested will also apply in the context of rigidity control. However, rigidity is reasonably strict in constraining a network, and thus we expect that extensions of rigidity control to realistic domains will require significant attention to robustness. Realistic applications present difficult and unpredictable influences, e.g., wind gusts or variability in ocean currents in autonomous surface vehicles (ASVs). These environmental variables and their timescales will directly impact the capability of the network to determine network rigidity in a timely fashion, as the network topology may change too often due to uncontrollable influences. Thus, the relationship between the parameters of an application, the properties of the employed communication network, and the numerical bounds on rigidity evaluation run-time must all be well understood by an implementer. Informally, the switching time of the topology, which is dictated by communication hardware and application-specific factors, and the network size become crucial design variables as they determine the feasibility of computing network rigidity in time to steer the network away from non-rigidity. In general, networks which exhibit short switching 228 times will require smaller network size or faster communication to combat external influences for feasible operation. There is also room to extend our algorithm for evaluating generic rigidity to stronger notions of rigidity, to three dimensional workspaces, and to more demanding networking conditions. The notion of global rigidity, which combines 3- connectednesswithredundantrigidity, isnecessary insolvingthegenerallocalization problem. In achieving global rigidity, our methods could be extended to identify redundant rigidity, by examining the implications of pebble game executions. In particular, redundancy should be indicated by the inability of a subgraph to yield four pebbles in a pebble search. By tracking and verifying that each searched subgraph has this property, redundant rigidity should emerge. Combined with methods from the literature for evaluating 3-connectedness, a controller for maintaining global rigidity of a networked team should be viable. Considering extensions to three dimensional operation will also be necessary for many of the emerging robotic platforms, such as the quadrotor. By fusing generic rigidity with continuous control of algebraic independence to enhance the current state of infinitesimal rigidity control, there may be hope in such workspaces. It is also interesting to consider the application of the pebble game to networks with anonymous agents. For the current generation of multi-robot implementations it may seem trivial to identify agents, however there are emerging control schemes and issues of scalability and dynamic team membership which motivate anonymity (privacy can also be seen as a motivation in longer term visions of robotics). Evaluating rigidity in such a context is certainly challenging and is quite compelling in terms of serving the novelties of future collaborative robotic systems. Finally, real-world experimentation for rigidity control is vital moving forward, particularly given the issues of implementation cited above. Rigidity appears to be 229 an excellent testbed for evaluating the realistic shortcomings of distributed topology control in robotic systems. Not only would rigidity control yield guarantees in formation stability and self-localization, vital aspects of multi-robot implementa- tions, the lessons learned from experimentation would be immediately applicable to connectivity control, a ubiquitous requirement for networked autonomy. Direct applications of rigidity also show promise. For example, a network that is capable of evaluating rigidity would surely be vital in self-assembly or cooperative construction where autonomous verification of load bearing is a necessity. 7.3 Closing Thoughts There is a strong suggestion in this work, that topology grips tightly the under- pinnings of collaboration. Of course, this is not necessarily a new notion, nor is the idea of topology control in robotic systems. 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Williams, Ryan K.
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Core Title
Interaction and topology in distributed multi-agent coordination
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Viterbi School of Engineering
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Doctor of Philosophy
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Electrical Engineering
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07/07/2014
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distributed robot systems,dynamic networks,graph connectivity,graph rigidity,multi‐robot coordination,OAI-PMH Harvest,sensor networks
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rkwillia@usc.edu,ryan.k.willia@gmail.com
Permanent Link (DOI)
https://doi.org/10.25549/usctheses-c3-432814
Unique identifier
UC11287603
Identifier
etd-WilliamsRy-2630.pdf (filename),usctheses-c3-432814 (legacy record id)
Legacy Identifier
etd-WilliamsRy-2630.pdf
Dmrecord
432814
Document Type
Dissertation
Format
application/pdf (imt)
Rights
Williams, Ryan K.
Type
texts
Source
University of Southern California
(contributing entity),
University of Southern California Dissertations and Theses
(collection)
Access Conditions
The author retains rights to his/her dissertation, thesis or other graduate work according to U.S. copyright law. Electronic access is being provided by the USC Libraries in agreement with the a...
Repository Name
University of Southern California Digital Library
Repository Location
USC Digital Library, University of Southern California, University Park Campus MC 2810, 3434 South Grand Avenue, 2nd Floor, Los Angeles, California 90089-2810, USA
Tags
distributed robot systems
dynamic networks
graph connectivity
graph rigidity
multi‐robot coordination
sensor networks