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Mobility-based topology control of robot networks

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Mobility-based topology control of robot networks

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Content MOBILITY-BASED TOPOLOGY CONTROL OF ROBOT NETWORKS by Sameera Poduri A Dissertation Presented to the FACULTY OF THE GRADUATE SCHOOL UNIVERSITY OF SOUTHERN CALIFORNIA In Partial Fulfillment of the Requirements for the Degree DOCTOR OF PHILOSOPHY (COMPUTER SCIENCE) August 2008 Copyright 2008 Sameera Poduri Dedication This dissertation is dedicated to Amma, Nanna and Sundeep. ii Table of Contents Dedication ii List Of Tables v List Of Figures vi Abstract ix Chapter 1: Introduction 1 Chapter 2: Related Work 8 2.1 Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.2 Coverage Optimization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.3 Power Control for Network Connectivity . . . . . . . . . . . . . . . . . . . . . . 16 2.4 Network Configuration in three dimensions . . . . . . . . . . . . . . . . . . . . 18 2.5 Rendezvous . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 Chapter 3: Local Geometric Conditions for Global Connectivity 21 3.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24 3.2 Neighbor-Every-Theta (NET) Graphs . . . . . . . . . . . . . . . . . . . . . . . 25 3.2.1 Connectivity Analysis of NET Graphs . . . . . . . . . . . . . . . . . . . 27 3.2.2 Proximity Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 3.2.3 Coverage Analysis of NET Graphs . . . . . . . . . . . . . . . . . . . . . 40 3.3 NET graphs in three dimensions . . . . . . . . . . . . . . . . . . . . . . . . . . 43 3.4 Power Control with NET Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . 46 3.5 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50 Chapter 4: Distributed Control Algorithms 52 4.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53 4.2 Design of the Control Law . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.3 Experiments and Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59 4.3.1 k-degree Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60 4.3.2 NET constraint . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61 4.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66 iii Chapter 5: Connectivity Maintenance and Coalescence 72 5.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.1.1 Communication Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 75 5.1.2 Mobility Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.1.3 Domain Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76 5.2 Coalescence Time Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.2.1 Hitting Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 5.2.2 Meeting Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 5.2.3 Coalescence Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 5.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86 5.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 Chapter 6: Coverage Optimization in Complex Environments 88 6.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92 6.1.1 Partial derivative of the overall utility function . . . . . . . . . . . . . . 93 6.2 Distributed coverage control . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 6.2.1 Control Law for each sensor . . . . . . . . . . . . . . . . . . . . . . . . 94 6.2.2 Distributed Coverage Algorithm . . . . . . . . . . . . . . . . . . . . . . 95 6.2.3 Convergence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.3 Deployment of Cyclops camera networks . . . . . . . . . . . . . . . . . . . . . 96 6.3.1 Sensing Performance Function . . . . . . . . . . . . . . . . . . . . . . . 96 6.3.2 Coverage Algorithm for 1D loop . . . . . . . . . . . . . . . . . . . . . . 100 6.3.3 Simulation Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103 6.3.4 Cyclops Experiments and Results . . . . . . . . . . . . . . . . . . . . . 109 6.4 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 Chapter 7: Conclusions and Future Work 116 Bibliography 120 iv List Of Tables 2.1 Classification of related work . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20 4.1 Parameter values used in our implementation of the distributed deployment algo- rithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64 v List Of Figures 1.1 A network of 10 Create Robots deployed at USC . . . . . . . . . . . . . . . . . 2 1.2 Illustration of global connectivity of a robot network . . . . . . . . . . . . . . . 3 1.3 Illustration of global sensing coverage of a robot network . . . . . . . . . . . . . 4 2.1 Hexagonal arrangement of nodes for optimal coverage with (a) Overlaping disks and (b) Non-overlaping disks . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14 3.1 Illustration of the NET condition . . . . . . . . . . . . . . . . . . . . . . . . . . 26 3.2 Proof of Theorem 3.2.6 forkV 1 k= 1 andkV 1 k= 2 . . . . . . . . . . . . . . . 29 3.3 Proof of Theorem 3.2.6 forkV 1 k= 3 . . . . . . . . . . . . . . . . . . . . . . . 30 3.4 A finite graph where all non-boundary nodes satisfy the NET condition . . . . . . 31 3.5 Degenerate cases of cuts through NET graphs that result in poor connectivity . . 33 3.6 RNG sector condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 3.7 = 2 3 bound for RNG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 3.8 Sector condition for GG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 3.9 Sector condition for DelG . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39 3.10 Coverage withk nodes placed on the communication perimeter of nodex . . . . 42 3.11 NET3D graph: each cone of angle must have at least one neighbor. . . . . . . . 44 3.12 RNG cone angle condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45 vi 3.13 Comparison of NET graphs and CBTC() for a uniform random network of 500 nodes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49 3.14 Simulation results of power control based on NET graphs for a uniform random network of 500 nodes averaged over 100 runs . . . . . . . . . . . . . . . . . . . 50 4.1 Typical Network Configurations for a 64 robot deployment withk = 2 . . . . . . 67 4.2 Coverage and average degree forN = 49 andR s = 4 for different values ofR c . 68 4.3 Variation of coverage with network size . . . . . . . . . . . . . . . . . . . . . . 69 4.4 Sample deployments with 100 robots andR c =R s = 8:0. . . . . . . . . . . . . 69 4.5 Sample deployments in the presence of obstacles with 100 robots andR c =R s = 8:0. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.6 Performance of the Distributed deployment Algorithms for a deployment of 100 robots in terms of (a) Coverage, (b) Edge Connectivity (c) Average Degree and (d) NET condition satisfaction . . . . . . . . . . . . . . . . . . . . . . . . . . . 70 4.7 For 2 3 , the communication graph is a supergraph of RNG . . . . . . . . . . 71 5.1 Illustration of Coalescence forN = 5 . . . . . . . . . . . . . . . . . . . . . . . 74 5.2 Illustration of random direction mobility model . . . . . . . . . . . . . . . . . . 75 5.3 Verification of analytical model fork th meeting time . . . . . . . . . . . . . . . 79 5.4 Experimental validation of analytical bounds on Coalescence Time. The graph shows the coalescence time as a function of network size on a linear scale . . . . 82 5.5 Experimental validation of analytical bounds on Coalescence Time. The graph shows the coalescence time as a function of network size on a log scale . . . . . 83 6.1 (a) An aerial view of the garden (b) the pathway to be covered . . . . . . . . . . 89 6.2 Cyclops camera with an attached Mica2 Mote . . . . . . . . . . . . . . . . . . . 89 6.3 Node locations for (a) uniform deployment and (b) optimized locations and their detection results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90 6.4 Perspective projection model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 vii 6.5 Illustration of pixels on target not detected . . . . . . . . . . . . . . . . . . . . . 98 6.6 Validation of the proposed model for pixels detected vs. distance for three differ- ent environments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101 6.7 Performance of the distributed control law for three types of parameter functions on a 1D loop which is a 250 feet long smooth curve . . . . . . . . . . . . . . . . 104 6.8 Variation in coverage utility and convergence time for different network sizes averaged over 50 iterations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.9 Performance of the distributed control law in the presence of occlusions - (1) obstacles and (2) sharp corners. . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 6.10 Performance of the distributed control law with partial knowledge ofk 1 (x) and k 2 (x). . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 6.11 Cyclops setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110 6.12 Performance of the distributed control law in an indoor rectangular corridor . . . 112 6.13 Performance of the distributed control law in an outdoor environment . . . . . . 113 viii Abstract A fundamental problem that arises in realizing large-scale wireless networks of mobile robots is that of controlling the communication topology. This thesis makes three contributions to the area of mobility- based topology control in robot networks. First, this work proposes local, geometric conditions on robot positions that guarantee global network connectivity. When combined with distributed controllers for mobile robots, these conditions maximize sensing coverage while maintaining connectivity. The key idea is the introduction of a new construct - a Neighbor-Every-Theta (NET) graph - in which each node has at least one neighbor in every angular sector of size. We prove that for < , NET graphs are guaranteed to have an edge-connectivity of at leastb 2 c, even with an irregular radio communication range. The NET conditions are integrated into an artificial potential field-based controller for distributed deployment. Second, for robots communicating over unreliable wireless links, a coalescence behavior is intro- duced, in which disconnected robots recover connectivity by searching for others. Our contribution is an asymptotic analysis of coalescence time which quantifies the trade-off between the cost of maintaining connectivity and that of repairing connectivity - the first such guideline for designing controllers for robot networks. Finally, having shown how to construct and maintain robot networks with maximal coverage, we address a problem motivated by the performance of cheap, low-resolution cameras in complex outdoor environments. This variation of the sensor coverage problem is the first to address the case where the sensing performance of each robot carrying a camera depends on the local environmental conditions. We ix present a distributed reconfiguration algorithm for this case, and validate it using a network of real Cyclops cameras placed in environments with varying illumination and backgrounds. x Chapter 1 Introduction Networked Robotics is emerging as a new frontier of Robotics. Groups of robots that are in- dividually limited can use local communication to collectively achieve complex tasks involving sensing and sharing of information, collaborative reasoning and inference, and manipulation. Compared to traditional sensing technologies, the large-scale nature of these networks will allow high-fidelity sensing over vast terrains and robustness to failures in the sensors. Further, such networks can leverage mobility to adaptively configure in response to temporal changes in the sensing field, application requirements, network properties, etc. Currently, several robots have been designed, fabricated, and deployed as part of networked experimental testbeds. Examples include the Robomote from the University of Southern California [Dantu et al., 2005], Swarm- bot and Create (figure 1.1) from iRobot, Scout from the University of Minnesota [Hougen et al., 2000], Alice and e-puck from EPFL [Caprari and Siegwart, 2005], and Scarab from the University of Pennsylvania [Michael et al., 2008]. These deployments consist of 10 100 robots operat- ing over short periods of time. Going by the rate of advancement in electronic fabrication and communication abilities, we expect that in the near future swarms of several hundreds of robots 1 Figure 1.1: A network of 10 Create Robots deployed at USC. Each robot is equipped with an infrared bump sensor, 802:11 wireless radio and an embedded computer. will become available. Such systems will significantly change the way we interact with our sur- roundings. Consider for example, a robot network being deployed in the aftermath of disasters like hurricane Katrina to discover victims and provide communication infrastructure to rescue teams, or aquatic networks in oceans to detect and mitigate harmful algal blooms or oil spills, or a swarm of aerial robots that can track soldiers on a battlefield and form a routing network that dynamically reconfigures in response to the communication requirements of the soldiers. A fundamental challenge in realizing robot networks is developing algorithms that enable intelligent behavior. Since the networks will be very large, the algorithms must necessarily be de-centralized i.e. they must guarantee global behavior using only local information and local control. Each robot can typically sense information (such as the relative positions of other robots and obstacles) within a fixed range to itself and can communicate with robots that are close to it. Using this limited information, it must decide its motion strategy such that one or more global properties of the network (such as sensing coverage) are optimized and one or more global network constraints (such as connectivity) are satisfied. This is an instance of the classic local-to- global problem that arises in several complex systems. Coverage and connectivity are complex 2 (a) (b) Figure 1.2: Network connectivity is a global property of robot positions. The filled circles are robots and the black lines indicate communication links between pairs of robots. In (a) the net- work is connected. When the highlighted robot moves (b), it improves its local communication links (dotted edges) but the global connectivity is broken. functions of robot locations and controlling them through local interactions is not trivial. Con- sider the example in figure 1.2, where the network is initially connected. The highlighted robot moves and improves its local connectivity but in the process breaks one edge that results in the network being disconnected. Similarly, even if each robot optimizes its local sensing coverage, the network as a whole can have poor coverage (figure 1.3). The objective of this thesis is to develop distributed algorithms for large-scale robot networks to maximize sensing coverage while maintaining network connectivity. The problem of maintaining the topological properties of networks is called topology con- trol and has been studied extensively in the context of static and mobile ad-hoc networks such as packet radio networks, vehicular networks, cell phone networks, and, more recently, sensor networks [Santi, 2005]. In these networks, topology control can be achieved by varying the transmission power of the radios (power control) and selectively turning nodes on or off (sleep scheduling). The traditional approach is to assume that the network is connected when the radios operate at full power (or all nodes are turned on) and find the minimum transmission power at each node (or maximum number of nodes that can be turned off) while preserving the network 3 Figure 1.3: Sensing coverage is a global property of robot positions. The disk around each robot is its sensing range. Each robot choses a locally optimal location but the net coverage is far from optimal. connectivity. Robot networks present a very different and interesting scenario for topology con- trol. Unlike traditional static and mobile ad-hoc networks, robot networks allow control over positions. Since connectivity properties directly depend on the positions of nodes, position con- trol can be leveraged for effective topology control. 4 Our thesis is that “Mobility can be effectively used for topology control of robot networks” To establish the thesis, we present solutions to the following three problems: 1. Local conditions to guarantee global connectivity We introduce Neighbor-Every-Theta (NET) graphs defined as graphs in which each node (except those at the boundary) has at least one neighbor in every angular sector around it. Our key theo- retical result is that for<, a NET graph has an edge-connectivity of at leastb 2 c irrespective of the edge-lengths. Based on these conditions we design a distributed navigation controller and power controller. We implement a typical power control protocol using satisfaction of NET con- dition at each non-boundary node as the termination criterion and show that a power-efficient network with edge connectivityb 2 c can be achieved even with realistic, irregular links. We also present extensions of our results to 3D using a cone in place of the 2D sector. Topol- ogy control in three dimensions (3D) is much more complex compared to 2D. As a result, there is very little work on topology control algorithms in 3D. We propose an efficient algorithm for identifying the largest empty cone of a node’s communication range. This algorithm can be used in conjunction with several configurations mechanisms for implementing their 3D extensions. 2. Controllers for distributed deployment We present a distributed algorithm for self-deployment of robot networks based on artificial po- tential fields. The pair-wise interactions between the robots are govrened by two kinds of forces - one causes the robots to repel each other to improve their coverage and the other is an attractive 5 force that restricts the inter-robot distances and thereby imposes local edge constraints. By using a combination of these forces, every robot tries to maximize its coverage while maintaining local constraints. Two types of constraints are considered - (1) minimum number of neighbors at each robot and (2) NET condition. Simulations in Player/Stage simulation platform [Gerkey et al., 2001] show that the controller achieves good sensing coverage and successfully maintains local constraints even in the presence of obstacles. The above potential fields based controller assumes that the network is initially connected. To address the case where robots start in a disconnected state or accidentally lose connectivity due to failures, we propose a ‘coalescing’ behavior where robots search for their peers and form a connected network. We study the time taken for coalescence for a worst-case scenario where the robots are memoryless (do not build a map) and do not have any knowledge of the environment or positions of the other robots. Each robot performs an independent random search and when two robots meet, they coalesce into a cluster and move together. Using the random direction mobility model, we show that coalescence time has an exponential distribution which is a function of the number of robots, speed, communication range, and size of the domain. Further, as the number of robots (N) increases, coalescence time decreases asO( 1 p N ) and ( log(N) N ). Simulations validate our analysis and also suggest that the lower bound is tight. 3. Coverage optimization in complex environments We address a variation of the coverage problem for autonomous mobile sensors where the sens- ing performance of each sensor depends on the local environmental conditions. We propose a 6 distributed control algorithm by which mobile sensors reconfigure their locations to provide op- timal coverage. As an application we consider the problem of covering loopy pathways in indoor and outdoor environments using cyclops camera. Local lighting conditions in these environments significantly affect the sensing performance of each camera. Based on several experiments we propose a location dependent sensing model for the camera. We present simulation results for various scenarios and demonstrate convergence of our algorithm. We also present simulation results on data collected in the indoor and outdoor settings. Outline of Dissertation The dissertation is organized as follows. The next chapter surveys related literature in geometric optimization, sensor network topology control, and the Rendezvous problem in Robotics. Chapter 3 presents NET graphs and analysis of their connectivity properties in 2D and extensions to 3D. Self-deployment algorithms for mobile nodes that maintain a local constraint are presented in Chapter 4 followed by an analysis of Coalescence time in Chapter 5. In chapter 6, we describe deployment in the presence of non-idealities in the sensor models. Finally, chapter 7 summarizes the contributions and presents directions for future research. 7 Chapter 2 Related Work The sensor coverage problem is related to classical geometric optimization problems such as disk- covering, filling, voronoi partitions, set-cover [Hall, 1988], and the art-gallery problem [O’Rourke, 1987] and also the problem of multi-robot coverage [Gage, 1992]. The key factors that limit the utility of the solutions of these problems to robot networks are 1) they do not provide for net- work connectivity constraints and/or 2) they are centralized and assume global knowledge of the domain. Not surprisingly, the sensor coverage problem has been an active research problem in robotics. In this section, we review results from classical computational geometry as well as those from recent research for location optimization in robot networks. The optimal locations can be realized by using robots or humans to deploy networks [Batalin and Sukhatme, 2007; Corke et al., 2004] or with networks of mobile robots [Howard et al., 2002; Cortes et al., 2004; Batalin and Sukhatme, 2002]. In addition to controlling the locations of sensors, sensor coverage is also studied in the context of sleep scheduling. Sleep scheduling techniques assume an initial high- density deployment - the problem being to find the smallest subset of nodes that is sufficient to provide the required coverage and connectivity. This is a discrete version of the sensor placement 8 problem - in the extreme scenario, if the sensor density is infinite, sleep scheduling is equivalent to finding optimal sensor locations. In the next section, we present an overview of the various models used for sensing, coverage and communication. Later, we will discuss algorithms based on the models used (Table 2.1). 2.1 Models Sensor Model Letp =fp 1 ;p 2 ;:::;p N g;p i 2 Q be locations ofn sensors. The sensing performance of each sensori at pointq is given byf(i;q). Depending on the nature off(i;q), we can classify sensors as follows: Isotropic Binary Disk Model: f(i;q) = 1 ifjjp i qjjR s (i), and 0 otherwise. R s (i) is called the sensing range of sensori. e.g. ranging sensors, temperature, acoustic, radio, omnidirectional cameras, etc. Isotropic Continuous Model: f(i;q) = g(jjp i qjj) where g is a monotonically de- creasing, continuous function. Visibility Model: f(i;q) = 1 if the line segmentpq Q, and 0 otherwise. Also called line-of-sight sensors. Limited range, visibility Model: Iff 1 ; 2 ;::: n g are the orientations of the sensors, then f(i;q) = 1 if (1)jjp i qjj R s (i), (2) angle(i;q) i , and (3) pq Q, and 0 otherwise. i is the field-of-view of sensori. e.g. ranging sensors, cameras, etc. 9 A point in the domain, q is set to be covered if it is within the sensing region of at least one node, i.e., if f(i;q) for at least one i, where is an application-dependent coverage threshold. Similarly, a pointq in the domain iskcovered if it is simultaneously covered byk nodes. For sensors (like temperature sensors) that measure at a point, the sensing range can be interpreted as a measure of sampling granularity. Coverage Definition Depending on the application, the desired sensing coverage can be characterized in one of 3 ways (extension of the taxonomy proposed in [Gage, 1992]): 1. Blanket Coverage: (a) Max type: If (q);q2 Q, is the event density function, find an arrangement of sensors such that the number of event detections per unit time is maximized, i.e. C = i=n X i=1 Z Q (q):f(i;q)dq is maximized. If is constant on Q, this is equivalent to maximizing the net area covered givenn sensors. (b) Complete type: Find the minimum number of sensors such that each point in the domain is covered, i.e. , fq2Qjf(i;q); 1ingQ 10 2. Barrier Coverage: Find an arrangement of a barrier of static nodes that minimizes the probability of a moving target penetrating an area undetected. 3. Sweep Coverage: Find a motion strategy for a group of mobile sensors such that the num- ber of event detection per time is maximized or the number of event misses per area is minimized. This can be treated as a moving barrier where the targets are static. Robot network coverage is generally of the blanket type because most of the applications involve gathering spatial data from the environment. In this work, we consider blanket coverage optimization. Communication Model Wireless communication depends on relative positions of nodes, radio characteristics and envi- ronmental conditions. The probability that a message from nodei reaches nodej can be repre- sented asPRR(i;j). In general,PRR(i;j)6=PRR(j;i). Binary Disk Model: PRR(i;j) = 1 ifjjp i p j jj R c (i), 0 otherwise. R c (i) is called the Communication range of sensori. Probabilistic Model: In real scenarios,PRR(i;j) is not binary due to non-idealities in- troduced by hardware and wireless propogation channel. It can be estimated empirically as a probability distribution. [Zhao and Govindan, 2003; Zuniga and Krishnamachari, 2004; Cerpa et al., 2005; Lymberopoulos et al., 2006]. 11 Connectivity Constraints A network is said to be connected if there exists at least one path between any two nodes. If there exist at least k paths between any two nodes, then the network is kconnected. Since robot networks must mostly operate unattended, connectivity is a necessary requirement to ac- cess information from the network. Sometimes kconnectivity is desirable for increased fault tolerance. Since our work deals with blanket coverage, we first present an extensive survey of blanket coverage optimization algorithms categorized by the sensor models they use. Several of these techniques do not address connectivity. In section 2.3 we look at topology control techniques for a sensor network whose main focus is on maintaining connectivity. Lastly, we review rendezvous algorithms for groups of mobile robots. 2.2 Coverage Optimization Binary Disk Sensor Model Given a domain, D, and disks of fixed radius, R, disk-packing is the problem of finding the densest arrangement of non-overlapping disks. For simple domains without holes/obstacles, the hexagonal arrangement (fig. 2.1(a)) is optimal [Williams, 1979]. This is the densest arrangement of binary disk sensors such that the area covered per sensor is maximum. In the Disk Covering problem, the disks can overlap and the problem is to find the minimum number of disks that cover every point inD. Again, a hexagonal arrangement (fig. 2.1(b)) is optimal. If the communication 12 model is a disk with radius at least twice the sensing range, then disk covering implies connectiv- ity [Xing et al., 2005; Zhang and Hou, 2005]. ForR c 2R s , sleep scheduling protocols can be based only on coverage and a node can be turned off if it does not cover any point that is already covered by other sensors. ForR c < 2R s , CCP [Xing et al., 2005] and OGDC [Zhang and Hou, 2005] present heuristics to decide if a node can be turned off. They improve the efficiency of the network by decreasing the number of active nodes but it is not necessarily the optimal, i.e. the configuration with the minimum number of active nodes. Several algorithms have been proposed for dispersion of mobile sensors in unknown environ- ments [Hsiang et al., 2002; Howard et al., 2002; Batalin and Sukhatme, 2002]. These algorithms do not consider the connectivity constraint. Potential Field techniques for robot applications were first introduced by Khatib [Khatib, 1986] and ever since have been used extensively to develop elegant solutions for path planning. Social potential fields have been introduced for robot for- mations [Reif and Wang, 1999]. The deployment algorithm we propose is an extension of the potential field-based deployment algorithm proposed in [Howard et al., 2002] where coverage is achieved as an emergent property of the system. Continuous Disk Model There is a rich body of literature on optimization of facility locations such as locations of mail- boxes in a city. The problem is to find locations for a limited number of facilities, that provide a service demanded by users, such that the average (or maximum) distance from the demand points to the facilities is minimized. The facilities can be modeled as points, lines, circles, or polygons and the distance metrics can be euclidean, geodesic, minkowski, etc. For the sensor coverage 13 (a) (b) Figure 2.1: Hexagonal arrangement of nodes for optimal coverage with (a) Overlaping disks and (b) Non-overlaping disks problem, we are interested in the setting with point-like facilities and euclidean distance metric. The distance to the facility (or a function of it) gives us the sensor’s performance function, f, which is isotropic and monotonically decreasing. If the demand points are uniformly distributed in the domain, the voronoi diagram [Okabe et al., 1992], of a given arrangement of facilities, par- titions the domain by allocating each domain-point to its nearest facility. The voronoi partition of an arrangement of sensors can be used to compute the maximal breach path [Megerian et al., 2005] which is the path between two points that maximizes the minimum distance to any sensor. For point-like facilities, the Centroidal V oronoi Partition minimizes the average distance to the facilities [Du et al., 1999]. Llyod’s Method [Llyod, 1982] is an iterative technique to construct Centroidal V oronoi Partitions. Given an initial arrangement ofk points (or facilities), the V oronoi cells are constructed and their centroids are used as the new locations. This process is repeated until the locations coincide with the centroids. Llyods method has been implemented as a dis- tributed algorithm for a group of mobile facilities where each facility computes its local voronoi cell and moves towards the centroid [Drezner, 1995; Cortes et al., 2004, 2005]. A non-uniform 14 demand density function can also be incorporated into this framework so that the distribution of sensors follows the density function. It has been shown that a local estimate of the density func- tion suffices for this algorithm [Schwager et al., 2006]. A key drawback of these methods is that they do not consider obstacles in the domain. The optimality and convergence is proved only for cases without obstacles. Visibility Model With the recent development of inexpensive image sensors [Rahimi et al., 2005], large camera networks will soon be possible. Optimal positioning of visibility sensors is a different instance of the locational optimization problem because they require line-of-sight. The art-gallery prob- lem [O’Rourke, 1987] is to find an arrangement of cameras (that have infinite range and 360 o field of view) that can (blanket) cover a given polygon. In 2D, this problem can be solved in polynomial time even when the polygons have holes. However, finding the arrangement with a minimum number of cameras is NP-hard [Aggarwal, 1984]. A similar problem is also addressed in motion-planning for mobile robots, to find the shortest trajectory or large workspace such that the robot has an unobstructed view of the target [Briggs and Donald, 2000]. The above algorithms are centralized and do not consider network connectivity constraints. Finite range, visibility Model Many of the real world cameras and ranging sensors have both limited field-of-view and limited range. Finding the smallest number of such sensors to cover an arbitrarily complex domain is a very hard problem. Recently, a number of approximation algorithms have been proposed. The 15 solutions are either based on randomized approaches [Danner and Kavraki, 2000; Banos and Latombe, 1998] or a decomposition of the domain into smaller convex polygons [Kazazakis and Argyros, 2002]. All of these algorithms are centralized and to the best of our knowledge, a distributed algorithm has not been proposed yet. Instead of repositioning cameras, configuring the settings such as pan, tilt, and zoom of a static arrangement of cameras can lead significant improvement in sensing coverage and can be achieved in a distributed manner [Kansal et al., 2007]. 2.3 Power Control for Network Connectivity Topology Control is considered a canonical problem in static and mobile ad-hoc networks (see [Santi, 2005] for a detailed survey). Power control algorithms deal with maintaining connectivity and do not address the problem of sensing coverage. In Cone Based Topology Control (CBTC) [Wattenhofer et al., 2005], every node divides the space around it into sectors of angle and increases its transmission power till in each sector it can reach a neighbor or till it reaches its maximum transmission power. There are two key differences between CBTC and the NET based structured deployment: (1) NET condition does not assume the existence of a connected graph when nodes operate at full power but provide ways to construct such a graph and (2) In CBTC, it is possible that a non-boundary node operating at full power does not have a neighbor in an sector while in a NET graph every non-boundary node must have at least one neighbor in each sector. The largest sector angle that guarantees connectivity in case of the NET graph is [D’Souza et al., 2006] as against 5 6 in CBTC. Several authors have proposed pruning the communication graph to proximity graphs such as the RNG, GG, and Yao Graphs as a way to 16 achieve desirable connectivity and sparseness properties. Jennings, et. al., present a distributed algorithm for constructing an RNG [Jennings and Okino, 2002] where each node increases it’s transmit power till its neighbors cover the 2 region around it. We show that to obtain an RNG it is sufficient to have one neighbor in each 2 3 sector and present sector based conditions for other proximity graphs. In a similar spirit, D’Souza, et al.,[D’Souza et al., 2006] have shown that if each non-boundary node in a network has at least one neighbor in every sector around it, the graph is guaranteed to be connected and is also sparse. Bredin, et. al., [Brendin et al., 2005] have recently proposed a network repair strategy that finds additional node locations required to ensure k-connectivity in any given network. The algorithm is proven to be within a constant factor of the absolute minimum number of additional nodes. Since sensing coverage and network connectivity are both related to the spatial configura- tion of nodes, the relation between the two has generated a lot of interest. It has been shown that if the communication model is a disk with radius at least twice the sensing range, then disk covering implies connectivity [Xing et al., 2005; Zhang and Hou, 2005; Tian and Georganas, 2005]. Miorandi and Altman [Miorandi and Altman, 2005] performed a probabilistic analysis of the relationship between radio connectivity and sensing coverage assuming a random uniform distribution of the nodes. Recently, sensor placement problem has been studied for non-ideal sen- sor and communication models [Krause et al., 2006; Zhang and Sukhatme, 2007]. The proposed algorithm uses a pilot deployment to gather data and build non-parametric probabilistic models for sensing and communication which are then used to find optimal sensor locations. 17 2.4 Network Configuration in three dimensions Coverage optimization in 3D is significantly more challenging that in 2D. In fact, most of the related problems like sphere-packing, art-gallery, and voronoi tetrahedralization are extremely complex in 3D. Not surprisingly, there have been few efforts to address 3D sensor network prob- lems like geographic routing [Kao et al., 2005] and localization [Biswas et al., 2006]. In [Bahram- giri et al., 2006], an algorithm has been presented to extend CBTC [Wattenhofer et al., 2005]. The computational complexity at each node isO(d 3 log(d)) as againstO(dlog(d)) in 2D,d being the average node degree. XTC [Wattenhofer and Zollinger, 2004] does not use the disc assumption or angular information; given an initially connected network in 3D, it can retain connectivity. 2.5 Rendezvous Rendezvous is the problem of multiple mobile robots meeting at a point in an unknown environ- ment. In Coalescence, the robots do not stay collocated but try to spread out. Several distributed Rendezvous algorithms have been proposed. Some of these assume connectivity [Ando et al., 1999; Lin et al., 2003]. If the communication graph is an RNG then rendezvous algorithms have been proposed that can deal with partial communication loss over some of the links [Cortes et al., 2006]. In the absence of connectivity, the robots can explore the environment building a map of the landmarks [Roy and Dudek, 2001] where other robots are likely to visit. This is similar in spirit to the Coalescence problem where the base station is a landmark whose location is un- known. However, our focus is on forming a connected component while allowing the robots to remain “spread out” so that the time for coalescence is reduced. 18 Several algorithms have been proposed to maintain connectivity of a wireless network [Za- vlanos and Pappas, 2005; Spanos and Murray, 2004] given that the initial network is connected. Coalescence algorithms complement these by providing a way to regain connectivity in the event of failures. In [Porfiri et al., 2007], the variation in connectivity of a network of robots performing random walks on a lattice has been analyzed. Recently, the role of mobility in increasing sensing coverage of a network has been analyzed [Liu et al., 2005]. In particular, it has been shown that the time taken to detect an intruder decreases in comparison to a static network. In delay tolerant networks, mobile nodes are disconnected most of the time and communicate when they are within communication range of each other. To study the routing delay, the motion of the mobile nodes is modeled as a random mobility model. In both the above cases, achieving connectivity is not the goal and the nodes continue moving after an intruder is detected or message is exchanged. In random walks literature, a coalescing random walk refers to a system of particles that coalesce when they hit other particles while performing a random walk [Dykiel, 2005]. Several asymptotic properties such as the time for convergence have been analyzed. Here, a group of coalesced particles is equivalent to one particle and this analysis does not capture the increase in communication spread. 19 Reference Mechanism Coverage Model Comm Model & Connectivity Distributed Comment Isotropic Binary Model Williams [1979] placement Blanket, Max. type None No Optimal Williams [1979] placement Blanket, Com- plete None No Optimal Howard et al. [2002]; Batalin and Sukhatme [2002] Mobile Blanket, Max. type None Yes Heuristic Zhang and Hou [2005]; Xing et al. [2005] Sleep schedule Blanket, Com- plete Binary disk & 1- Conn Yes Heuristic NET graphs placement Blanket, Max. type Binary Disk & tunable k-Conn Yes Heuristic Isotropic Continuous Model Okabe and Suzuki [1997] placement Blanket; Max None No Sufficient Du et al. [1999] placement Blanket; Max None Yes Optimal Cortes et al. [2004]; Schwager et al. [2006] mobile Blanket; Max None Yes Optimal Infinite Range Visibility Chvatal [1975]; Aggarwal [1984] placement Blanket; Com- plete None No Heuristic Finite Range Visibility Kazazakis and Ar- gyros [2002] placement Blanket, Com- plete None No Approximate Gerkey et al. [2006] mobile Sweep None No Heuristic Kansal et al. [2007] “motility” Blanket, Max type None Yes Optimal Probabilistic Model Krause et al. [2006] placement random random No Optimal Data-centric deployment placement random no Yes Locally Opti- mal Table 2.1: Classification of related work. The entries in bold refer to our work. 20 Chapter 3 Local Geometric Conditions for Global Connectivity Topology control for ad-hoc wireless and sensor networks has traditionally only dealt with uncon- trolled deployments, where there is no explicit control on positions of nodes [Santi, 2005]. The primary mechanisms proposed are power control and sleep scheduling. These methods involve removing edges from an existing, well connected communication graph in order to save power while ensuring that the resultant sub-graph preserves connectivity. Robot networks present a dif- ferent and interesting scenario for topology control. Since connectivity properties directly depend on the positions of nodes, position control can be leveraged for effective topology control. Tradi- tional approaches are ill-suited in such scenarios since they are not designed to exploit control of the motion and placement of the nodes. We develop a general approach that supports traditional methods like power control and also allows new designs for controlled deployments. Our main contributions are: ak-connectivity result, based on local conditions and independent of communication model topology control algorithms based on thek-connectivity result for a) power control with static nodes, and b) distributed deployment of mobile nodes 21 3D extensions of 2D results and an efficient algorithm for topology control in 3D. We define Neighbor-Every-Theta (NET) graphs in which each node (except those at the boundary) have at least one neighbor in every sector of its communication range. Our key theoretical result is that for < , a NET graph has an edge-connectivity of at leastb 2 c irre- spective of the communication model. This implies that the condition only depends on the angles between the neighbors of each node and holds for arbitrary edge-lengths. This feature is par- ticularly relevant for sensor networks using low-power radios that have irregular communication range. For the special case with an idealized disk communication model, we derive conditions for maximizing the sensing coverage area (defined as the total area sensed by at least one node) while satisfying thek-connectivity constraint, and for obtaining proximity graphs such as the Relative Neighborhood Graph. It should be emphasized that the NET condition is local since each node only requires relative position information of its (communicating) neighbors. NET graphs are naturally suited for distributed power control. We implement a typical power control protocol using satisfaction of NET condition at each non-boundary node as the termination criterion and show that a power-efficient network with edge connectivityb 2 c can be achieved even with realistic, irregular links. This is in contrast to the sector based topology control algo- rithms in literature including CBTC [Wattenhofer et al., 2005; Li et al., 2001], that rely on an idealized disk communication model to guarantee connectivity properties. In CBTC based power control [Bahramgiri et al., 2006], for a network to bek-connected, each node must either have a neighbor in every = 2 3k sector or operate at full power. Our results imply that a sector angle of =b 2 k c is sufficient fork-connectivity. A three times larger value of sector angle implies a much lower power requirement. We present simulation results to establish this in Section VI. 22 Even though we do not consider sleep-scheduling mechanisms in depth, it is possible to think of designs where densely deployed nodes make sleep/wake decisions in a distributed manner by using local, pair-wise negotiations based on combinations of NET condition satisfaction and other criteria. Position control is ideal for obtaining topologies that are NET graphs. Given the properties guaranteed by the NET condition, an external agent deploying a static sensor network can make decisions on the best positions for deployment of new nodes based on the geometry of the existing network. Self-deploying mobile nodes can use this condition to decide their motion strategy. We present a distributed algorithm for self-deployment of mobile nodes to concretely demonstrate the use of the local and geometric conditions to implement coverage and connectivity tradeoffs. The algorithm involves achieving NET condition satisfaction through purely pair-wise negotiations between neighbors. The performance of this algorithm is studied through an implementation on the Player/Stage simulation platform [Gerkey et al., 2001]. Results illustrate the tunable tradeoff between connectivity and coverage and show that the distributed algorithm can achieve approxi- mations of some coverage-maximizing tiling structures that also satisfy NET conditions. Topology control in three dimensions (3D) is much more complex. There are several oper- ations that are common to a wide range of topology control algorithms that have increased and sometimes prohibitively high complexity in 3D. Examples include obtaining angular information of and ordering neighbors and checking for intersections of overlapping regions [Santi, 2005; Zhang and Hou, 2005; Xing et al., 2005; Cortes et al., 2005]. As a result, there is very little work on topology control algorithms in 3D. We argue that controlled deployment and NET graphs are well suited for sensor networks in 3D and present extensions to our 2D results. We propose an 23 efficient algorithm for identifying the largest empty cone of a node’s communication range. This algorithm can be used in conjunction with several deployment and topology control mechanisms for implementing their 3D extensions. This chapter is organized as follows. The next section defines the problem and section 3.2 presents NET graphs and analysis of their connectivity properties which are extended to 3D in section 3.3. Sections3.4 presents application of NET graphs for distributed power control respec- tively and conclude in section 3.5. 3.1 Problem Formulation Local conditions that guarantee global network properties are the key to designing distributed topology control algorithms. In particular, we study local conditions that guarantee global k- edge-connectivity of the network. Once such conditions are found, they can be integrated with controls available in order to design topology control algorithms. Further, it is important to understand if the results can be extended to 3D networks in an efficient manner. A detailed description of these three problems follows. Problem 1: k-connectivity certificates Given a network, find local geometric conditions between node positions that can guarantee global k-edge-connectivity. A graph is said to be k-edge-connected if at least k edges must be removed to disconnect it Diestel [2000]. By Menger’s theorem, this is equivalent to saying that there exist at least k edge-disjoint paths between any two nodes in the graph. The graph that we consider is the 24 communication network where two nodes are said to have an edge between them if they can communicate. In this case, high edge-connectivity is desirable because it implies high fault- tolerance and path diversity. By ‘local’ we mean that each node only has information about positions of its communicating neighbors relative to its own position. The global coordinates of nodes are not available. The con- ditions we seek must be based only on this local information so that each node can independently decide whether it satisfies the condition. Problem 2: Topology control How can the local geometric conditions fork-connectivity be integrated power control? Given a static network ofN nodes, how should they choose their communication power such that the network is k-connected and the energy is minimized? Lastly, we study the extension to networks in three dimensions. Problem 3: 3D networks Do the local geometric conditions for 2D networks extend to 3D networks? 3.2 Neighbor-Every-Theta (NET) Graphs In this section, we will define Neighbor-Every-Theta graphs and show that they give us k- connectivity certificates. 25 (a) (b) Figure 3.1: Illustration of the NET condition. Filled circles are symmetric neighbors. In a) node does not satisfy NET condition. It has one sector greater than with no neighbor. In b) node satisfies NET condition, the largest sector with no neighbors is smaller than. Definition 3.2.1 The Neighbor-Every-Theta (NET) condition (Fig. 3.1) for a node embedded in the 2D plane is defined as requiring at least one symmetric neighbor in every sector of its communication range. NodesA andB are symmetric neighbors ifA can communicate toB andB can communicate to A. For finite networks, nodes on the network boundary cannot satisfy such a condition. The boundary of a network can be defined as a cycle of nodes such that every other node lies inside the cycle [Kroller et al., 2006]. A node that does not belong to the boundary is called an interior node. Definition 3.2.2 A NET graph is one in which every interior node satisfies the NET condition for a given. 26 We now analyze the connectivity and coverage properties of NET graphs. For large networks where the number of boundary nodes is small compared to the network size, we show that NET graphs have an edge connectivity of at leastb 2 c, independent of the communication model. With the stronger assumption of a idealized disk communication model, NET graphs, for specific values of, contain proximity graphs such as the Relative Neighborhood Graph (RNG). An upper bound on the sensing coverage is shown to be obtained from a symmetric arrangement of nodes and can be computed as a function of . NET graphs form a family of graphs based on the single parameter - as becomes smaller, the graphs become denser with an increasing level of connectivity. 3.2.1 Connectivity Analysis of NET Graphs The edge-connectivity of any graph is at most its minimum node degree and can in general be ar- bitrarily low irrespective of the node degree [Diestel, 2000]. For some special graphs, higher node degree implies higher edge-connectivity. One example of such a graph is the random geometric graph, where an average node degree ofO(logN) (whereN is the network size) guarantees an edge-connectivity of 1 with high probability [Xue and Kumar, 2004]. It turns out that for em NET graphs also there is relation between minimum node degree and edge-connectivity. In NET graphs every interior node has a degree of at leastb 2 c. We will show that the edge-connectivity is also at leastb 2 c (except in pathological cases where boundary nodes can be disconnected by removing fewer edges). This means that the easiest way to disconnect a NET graph is to discon- nect a single interior node by removingb 2 c edges. We will now formally establish this property 27 by first considering the simple case of graphs that are very large so that edges close to the bound- ary are not significant. An extreme case of such graphs are the “boundary-less” graphs defined below. Later, we will analyze the impact of boundary nodes on connectivity. Definition 3.2.3 A NET graph is boundary-less if every node satisfies the NET condition. For <, such a graph must span the entire 2D plane. Definition 3.2.4 Given a graph G = fV;Eg, a cut set is a set of edges E 0 E such that G 0 =fV;EE 0 g has more than one component. The edge-connectivity a graph,, is the size of its smallest cut set,C. Note that ifG is connected,C must divide it into exactly two components. Definition 3.2.5 For a graph embedded in the Euclidean plane a cut is a curve that partitions the graph into two or more components. Theorem 3.2.6 For<, a boundary-less NET graph has an edge-connectivityb 2 c Proof: Consider a NET graph G =fV;Eg. Let C E be the smallest cut set of G so that =kCk. Let the two components ofG 0 =fV;EE 0 g beG 1 =fV 1 ;E 1 g andG 2 =fV 2 ;E 2 g. WLOG,kV 1 kkV 2 k. Ifk V 1 k= 1 thenk Ck MinDegreeb 2 c (Fig 3.2(a)). Ifk V 1 k= 2 thenk Ck 2 b 2 c 1 b 2 c since< (Fig. 3.2(b)). Supposek V 1 k> 2. Construct the convex hull,H 1 ofV 1 . ThenH 1 V 1 . Define angle i ath i 2 H 1 as shown in Fig 3.3. There will exist at least 3 vertices 1 h i such that i > i.e. , \h i < . WLOG, assume that 1 ; 2 and 3 > . Since G =fV;Eg is a NET graph, for i = 1; 2; 3, i must containb c edges2C. Therefore,kCk 3b cb 2 c.2 1 SinceH is a convex polygon,\hi;8i. Suppose all except 2 internal angles are=, say0<\h1;\h2 < and\hi = fori = 3;4;:::k, wherek is number of vertices ofH, then P k 1 \hi = ((k 2) +\h1 +\h2) > (k2). Contradiction since P k 1 \hi = (k2). 28 (a)kV 1 k= 1 (b)kV 1 k= 2 Figure 3.2: The filled circles and edges representG 1 , dotted edges form the cut setC. The above result holds for the general case where the graph has boundary nodes provided the cut is completely in the interior of the network, i.e. if all the nodes in V 1 satisfy the NET condition. If on the other hand, the cut mostly consists of boundary edges then it is not inter- esting because it is unlikely to impact network performance. Now consider cuts that intersect the network boundary twice - when entering and leaving the graph (fig. 3.4(a)). WLOG, assume that the cut is minimal in length i.e. it is the shortest curve corresponding to its cut set. The case when the cut intersects itself is covered by the above theorem - the number of edges cut inside the loop alone must be at leastb 2 c. Assume that the cut does not intersect itself and by way of contradiction, that it disconnects the network by cutting less thanb 2 c edges. We will now analyze the nature of such a cut and prove that away from the network boundary, the distance along the cut between two consecutive edges must be less than R c :sec(=2), where R c is an upper bound on the edge-length. This implies that away from the boundary the cut cannot be longer thanb 2 cR c :sec(=2). If is not close to this expression is bounded. For example, for = 0:9 it is 13R c and for = 2 3 it is 6R c . For large networks, a “short” cut like this 29 Figure 3.3:k V 1 k 3 The red nodes and edges representG 1 , the dotted edges form the cut set C. The corresponding convex hullH is shown with solid black lines. can only exist close to the boundary (fig. 3.5(a)) or if the boundary is “pinched” [de Silva et al., 2005](fig. 3.5(b)). Lemma 3.2.7 For < , the length of a minimal cut between two consecutive edges in a boundary-less NET graph is less thanR c sec(=2). Proof: LetL be the cut andl its length between two consecutive edges it cuts, say (u 1 ;v 1 ) and (u 2 ;v 2 ) (figure 3.4(b)). Thenl maxfd(u 1 ;u 2 );d(u 1 ;v 2 );d(v 1 ;u 2 );d(v 1 ;v 2 )g, whered() is the euclidean distance function. Let the node pair that leads to maximum distance be P 1 ;P 2 . NowP 1 must have at least one neighbor in each of the two sectors adjoining ! P 1 P 2 . Let these be Q 1 at an angle 0 1 andQ 0 1 at an angle 0 0 1 with ! P 1 P 2 . 0 1 + 0 1 <. WLOG, 1 < =2. SimilarlyQ 1 must have a neighborQ 2 at an angle 2 , and so on till some Q m1 Q m intersects ! P 1 P 2 . Note that the point of intersection cannot be in between P 1 30 (a) (b) Figure 3.4: (a) A finite graph where all non-boundary nodes satisfy the NET condition. The thick curved lineL is a cut through the graph (b)L cuts polygons andP 2 because otherwiseL would cutQ m1 Q m contradicting the assumption that (u 1 ;v 1 ) and (u 2 ;v 2 ) are consecutive edges. We have, 31 l = kP 1 P 2 k r 1 cos( 1 ) +r 2 cos( 1 + 2 ) + +r m cos( 1 + 2 + + m (m 1)) (where=2> 1 > 1 + 2 > > 1 + 2 + + m (m 1))>=2) R c (cos( 1 ) +cos( 1 + 2 ) + +cos( 1 + 2 + + m (m 1))) (whereR c is an upper bound on edge length) R c ((cos( 1 ) +cos( 1 +) + +cos( 1 (m 1)())) (sincecos is a concave function in [=2;=2]) = R c sin m() 2 cos 1 + (m1)() 2 sin 2 R c 1 cos(=2) 2 It must be emphasized that the connectivity result only needs the largest edge in the network to be bounded and holds even for a non-ideal and irregular communication models. In the following subsections, we present further interesting properties of NET graphs with a stronger assumption of an idealized disk communication model. 32 (a) (b) Figure 3.5: Degenerate cases of cuts through NET graphs that result in poor connectivity. (a) A cut close to the boundary (b) pinched boundary 3.2.2 Proximity Graphs Proximity graphs such as the Relative Neighborhood Graph (RNG), Gabriel Graph (GG) and the Delaunay Graph (DelG) have several properties such as connectivity, sparseness, efficient network routes, etc. that make them desirable network topologies [Li, 2003; Jennings and Okino, 2002; Rajaraman, 2002; Santi, 2005]. We will now analyze conditions under which NET graphs will contain these proximity graphs. Here we assume an idealized disk communication model but allow each node to have a different communication range. Further, the number of boundary nodes is assumed to be small compared to the network size. The analysis presented is non-trivial since the edge-lengths in NET graphs are restricted by the communication range of nodes, while in proximity graphs they depend only on the relative positions of nodes and very long edges are possible. The set of edgesE for various proximity graphs is defined as follows [Diestel, 2000]. In what follows, we use the name and location of a node interchangeably. 33 Figure 3.6: RNG sector condition. a) The circle of radiusd(x;y)>R c subtends an angle of 2 3 at x. The lune contains an area larger than a 2 3 sector. b) The limiting case, whend(x;y) =R(x). Disk graph (DG(V;E;R)): The directed graph containing all outgoing edges of a node, u2V , not longer thanR(u). E =f(u;v)ju;v2V andd(u;v)R(u)g Given positiver2R, letC(p;r) be the circle consisting of points whose distance from point p is strictly less thanr. Define the lune, denotedL(p;q), to be the intersection of two circles, both of radiusd(p;q), centered at these points, that is,L(p;q) =C(p;d(p;q))\C(q;d(p;q)). 34 Relative Neighborhood Graph (RNG(V)): The undirected graph containing an edge (u;v) if there is no pointw2V that is simultaneously closer to bothu andv. Equivalently, (p;q) is an edge ifL(p;q)\V =;. E =f(u;v)ju;v2V and9 now2V 3 d(u;w)<d(u;v) andd(v;w)<d(u;v)g Gabriel Graph (GG): The undirected graph containing an edge (u;v) if the disk whose diameter is edge (u;v) does not contain any other points of V , that is, ifC( u+v 2 ; d(u;v) 2 )\V =;: E =f(u;v)ju;v2V andC( u +v 2 ; d(u;v) 2 )\V =;g Delaunay Graph (DelG): The undirected graph containing an edge (u;v) if the V oronoi regions of u and v have non-empty intersection. From the properties of V oronoi diagrams it follows the edges of triangle (u;v;w) are inDelG(V ) if their circumcircle does not contain any other points ofV . These graphs are hierarchically related: RNG(V )GG(V )DelG(V ). In the proofs that follow, we assume that each node,u, in the network has a communication rangeR(u) so that the communication graph of the network is a disk graph. Our next step is to find conditions under which the RNG of the network is contained in the communication graph. 35 This is equivalent to finding conditions under which no edge (u;v) of an RNG is longer than either R(u) or R(v). The following theorem presents this condition. Later, we prove similar results for the GG and DelG. Theorem 3.2.8 If each nodex2V has at least one neighbor in every 2 3 sector ofC(x;R(x)), the communication graph is a supergraph ofRNG(V ). Moreover, 2 3 is the largest angle that satisfies this property. Proof: Consider any node x2 V . Suppose x has at least one neighbor in every 2 3 sector of C(x;R(x)). We first show that for any nodey outsideC(x;R(x)), the edge (x;y) = 2RNG(V ). The luneL(x;y) will contain a sector of at least 2 3 (Fig.3.6). By premise,9 a node inL(x;y). This implies thatRNG(V ) does not have any edges incident onx that are longer thanR(x). Next we show that for any nodez2V insideC(x;R(x)) such thatd(x;z)>R(z) i.e. (x;z)2 DG(V;E;R) but (z;x) = 2 DG(V;E;R), the edge (x;z) = 2 RNG(V ). Since z also has a neighbor in every 2 3 sector of C(z;R(z)) and d(x;z) > R(z), from the above argument (x;z) = 2RNG(V ). ThereforeRNG(V )DG(V;E;R(x)). Now suppose a nodex 0 2V 0 has two neighbors with a sector angle of 2 3 + (> 0) between them (Fig.3.7). We can place a nodey 0 outsideC(x 0 ;R c ) such that the edge (x 0 ;y 0 )2RNG(V 0 ). Therefore, 2 3 is the largest angle for which this condition holds.2 We have shown that for 2 3 , NET graph contains the RNG assuming an idealized disk communication model. We can prove a similar property for GG and DelG, except that in this case the value of is a function of the distance to neighbors. 36 Figure 3.7: = 2 3 bound for RNG -L(x;y) is empty Lemma 3.2.9 If each nodex2V has at least one neighbor in every = 2 arccos( r R(x) ) sector ofC(x;r) (rR(x)), the communication graph is a supergraph ofGG(V ). Proof:Consider a nodeX and a nodeY outsideC(X;R c ). For anyr R c , let be the angle subtended by the intersection areaC(X;r)\C( X+Y 2 ; d(X;Y ) 2 ) atX. From Fig.3.8 we have, r 2 = d(X;Y ) 2 cos( 2 )) = 2 arccos( r d(X;Y ) ): Since arccos is strictly decreasing in [0; 1] andd(X;Y )2 (R c ;1), the smallest angle subtended is = inf = 2 arccos( r R c ): (3.1) Supposex has at least one neighbor in every = 2 arccos( r R c ) 37 Figure 3.8: Sector condition for GG sector ofC(X;r). By definition, the edge (X;Y ) = 2GG(V ) if C( d(X;Y ) 2 ; d(X;Y ) 2 ; )\V 6=;: (3.2) (3.2) is satisfied for any choice of Y if there is a node Z2 V in the sector of C(X;r). By premise such aZ exists and hence the GG will not have any edge lengths greater thanR c . This implies thatGG(V )DG(V;E;R c ):2 Corollary 3.2.10 If a node x has at least one neighbor in every = 2 arccos( r R(x) ) sector of C(x;r) (r R(x)), the communication graph is a supergraph ofDelG(V ). Proof: Consider all circles passing throughX andY . The smallest isC( X+Y 2 ; d(X;Y ) 2 ) and the largest are two circles of infinite radius of which the straight line throughX andY is an arc. From Fig.3.9, it 38 Figure 3.9: Sector condition for DelG can be seen that the intersections of these circles withC(X;r) subtend an angle which is at least atX, where = inf = 2 arccos( r R c ): By premise, there is at least one node in every sector ofC(X;r). Therefore, every circle passing throughX andY will contain another node. So, 1. there is noZ2V such that the4XYZ2DelG(V ) 2. triviallyC( X+Y 2 ; d(X;Y ) 2 ) also contains another node. By 1,2 and definition of DelG, the edge (X;Y ) = 2DelG(V ). Since the choice ofY is arbitrary, there are no edge lengths greater thanR c in the DelG. HenceDelGDG(V;E;R c ). In real networks, it is not possible for the boundary nodes to satisfy the conditions required by the theorems. We show using simulated deployments (in section 4.3.2) that the assertions can be validated in spite of these exceptions. 39 3.2.3 Coverage Analysis ofNET Graphs Having listed the conditions that guarantee global connectivity properties, we now turn to the problem of maximizing coverage. We assume that all nodes have a idealized disk sensing model with a sensing radius ofR s . Maximizing sensing coverage is equivalent to packing problem with a constraint placed on the communication neighbors. This is a very hard problem given an irreg- ular communication model and even harder to solve locally. Therefore, we make a simplifying assumption that the communication model is also an idealized disk with a radiusR c for all nodes. Suppose that in order to satisfy the NET sector conditions, a node must havek =b 2 c neighbors. From the node’s local perspective, all neighbors must be located on the perimeter of the commu- nication range to maximize coverage. Intuitively, the nodes must also be placed symmetrically on the perimeter. We prove this result for the special case whenR c =R s . Lemma 3.2.11 ForR s =R c , the area coverage is maximized when thek 3 nodes are placed at the edges ofk disjoint 2 k sectors ofC(x;R c ). Proof: Consider a nodex andk nodesy 1 ;y 2 ;::::;y k placed on the perimeter ofC(x;R c ). Let these nodes be placed in anticlockwise order at angles 1 = 0; 2 ;::::; k respectively. Define i = i+1 i ; 1ik 1 k = 2 k (3.3) We need to find i such that the total area covered by thesek + 1 nodes is maximized. The open disks of all nodes lie withinC(x; 2R c ) and are tangent to this disk at exactly one point (T i ) each. 40 The disks of adjacent nodesi andi + 1 intersect at pointI i (and atX). The total coverage lies betweenR 2 c and 4R 2 c and is maximized when the area k P i=1 T i I i T i+1 is minimum (Fig.3.10). T i I i T i+1 =T i XT i+1 I i Y i T i I i Y i+1 T i+1 I i Y i XY i+1 = (2R c ) 2 i 2 2R 2 c i 2 1 2 (2R c sin i 2 )(2R c cos i 2 ) =R 2 c ( i sin i ) k X i=1 T i I i T i+1 = R 2 c ( k X i=1 k X i=1 sin i ) = R 2 c (2 k X i=1 sin i ) (3.4) The problem now reduces to finding max k P i=1 sin i subject to k P i=1 i = 2 and 0 i 2. Since sin is non-negative and concave in [0;] and non-positive in (; 2], it follows that the solution is i = 2 k ; 1ik.2 For k = 3; 4; 6, it is possible to place nodes such that each node has its neighbors placed symmetrically on its communication range. The resulting communication graph will be a tiling of the space: hexagonal, square and triangle for k = 3; 4 and 6 respectively. In these cases, since each node maximizes coverage locally, the total coverage of the network will also be the maximized. For values of k other than 3; 4 and 6 such an arrangement is not possible since the corresponding tiling structures do not exist. Therefore in these cases, the local condition for optimizing coverage will not necessarily optimize global coverage. Due to their symmetry, 41 Figure 3.10: Coverage with k nodes placed on the communication perimeter of node x. The shaded area is the total coverage. tiling graphs possess some other desirable properties. It can be verified that the hexagonal tiling ( = 2 3 ) is an RNG and in this case, GG RNG. This holds for the square tiling ( = 2 ) as well. The triangle tiling ( = 3 ) is a DelG. Based on the above theorem, we can compute an upper bound on the maximum achievable coverage of a NET graph as a function of. Consider the arrangement in Fig. 3.10 with all the neighbors placed symmetrically aroundx. If each overlap area within the sensing range ofx is divided by the number of nodes that cover it, then the sum of these weighted areas will give the maximum possible per-node coverage. In this computation we ignore the boundary nodes which cannot have a symmetric placement of neighbors. Therefore, this is an asymptotic bound. This upper bound on coverage is plotted in Fig. 4.6 along with the lower bound for edge-connectivity derived in theorem 3.2.11. 42 In the next section we extend NET graphs to 3D and study their connectivity and coverage properties. 3.3 NET graphs in three dimensions Network configuration in 3D is significantly more complex than in 2D. There are several oper- ations that are common to a wide range of sensor network configuration algorithms in 2D that have increased and sometimes prohibitively high complexity in 3D. We argue that controlled deployment and NET graphs are well suited for sensor networks in 3D and present extensions to our 2D results. We propose an efficient algorithm for identifying the largest empty cone of a node’s communication range. This algorithm can be used in conjunction with a number of generic deployment and topology construction mechanisms for implementing their 3D extensions. A node satisfies the NET3D condition if it has at least one symmetric neighbor in every cone of solid angle. A NET3D graph is one in which every node except those on the boundary satisfy the NET3D condition. If the network is very large compared to the size of its boundary, we make the following conjecture for the edge-connectivity. Conjecture 3.3.1 For < 2, a boundary-less NET3D graph has an edge-connectivity b 2 c2 Note that, like in 2D, the connectivity result is independent of the communication model. For specific values of, NET3D graphs will contain proximity graphs such as RNG, GG, and DelG. Lemma 3.3.2 If each nodex2V has at least one neighbor in every = cone ofS(X;R(x)), the communication graph is a supergraph ofRNG(V ). 43 Figure 3.11: NET3D graph: each cone of angle must have at least one neighbor. Lemma 3.3.3 If each node x2 V has at least one neighbor in every = 2(1 r Rc ) cone ofS(X;R(x)), the communication graph is a supergraph ofGG(V ) andDelG(V ). Moreover, 2(1 r R(x) ) is the largest angle that satisfies this property. The proofs follow from the corresponding proofs in 2D and the fact that a cone with apex angle will contain a solid angle of = 2(1cos()). For example in the proof for RNG, the 3D lune will contain a cone with apex angle = 2 3 and solid angle of =. Algorithm to check NET condition The local conditions described above can be integrated with node placement to construct efficient topologies. A key requirement is an algorithm to check for empty cones larger than a given. 44 Figure 3.12: RNG cone angle condition For instance, to check for formation of RNG, =. This step is non-trivial in 3D because there exists no natural “order” of neighbors. We propose the following algorithm for finding the largest empty cone around a given node. Algorithm 1: largestCone(G = (V;E),v2V ) letS be the unit sphere centered atv for eachu2Neighbor(v), let ! vu be the direction vector fromv tou letc u be the intersection of ! vu withS letDT be spherical delaunay triangulationc u 8u finda i;j;k = area of circumcircle of triangle (u i ;u j ;u k )2DT returnmax(a i;j;k ) Algorithm 1: Find largest empty cone around a node 45 Theorem 3.3.4 For a given graphG = (V;E) and nodev,largestCone returns the largest empty cone aroundv. Proof: The circumcircle of every (spherical) triangle in the spherical delaunay triangulation is empty [Na et al., 2002]. Therefore the cone returned bylargestCone is certainly empty. Suppose there exists an empty cone (whose image on the unit sphere is the circle c) that is larger than the one returned by largestCone. Then the center of c lies in some triangle t of the delaunay triangulation. Sincec is empty, none of the vertices oft must lie insidec. This implies that the circumcircle oft will be larger thanc which is a contradiction. Therefore largestCone correctly returns the largest empty cone.2. The computational complexity of largestCone isO(dlog(d)) whered is the number of neigh- bors of a node. This is because, the complexity of spherical triangulation is O(dlog(d)), the number of triangles generated isO(d), and sorting them will also takeO(dlog(d)) time. Several topology control algorithms [Santi, 2005; Wattenhofer et al., 2005; Jennings and Okino, 2002; Xing et al., 2005; Zhang and Hou, 2005] in 2D rely on directional information and in particular use the angle between adjacent neighbors. This algorithm can be used as a primitive for extending such algorithms to 3D. 3.4 Power Control withNET Graphs In this section we illustrate how NET graphs can be used for topology control of a static network by varying the communication power. The result is a power control mechanism that can guaran- teek-edge-connectivity without any restriction on the communication model. In contrast, most 46 existing power control algorithms, including the well-known CBTC [Wattenhofer et al., 2005; Bahramgiri et al., 2006] algorithm, require an idealized binary disk communication model. In the power control problem, we are given a network deployed with a sufficiently large density so that connectivity (k-connectivity) is guaranteed when nodes operate at full power. The objective is to find an assignment of transmission powers to nodes so that the reduced graph retains connectivity (k-connectivity) while expending minimum energy. In some applications it is also desirable to maintain other properties such as planarity, sparseness, spanner, etc. NET graphs are naturally suited for distributed power control because they are based on local geometric conditions that guarantee globalk-connectivity, communication range for the low power radios used in sensor network applications is highly irregular [Zhao and Govindan, 2003; Zuniga and Krishnamachari, 2004]. The con- nectivity properties of NET graphs are independent of the communication model. Consider a sensor network such that when nodes operate at full power, they satisfy the NET condition for a given. Using a typical power control protocol and satisfaction of NET condition at each internal node as the termination criterion, a power-efficientk-connected network can be achieved. We have implemented a completely distributed power control algorithm as follows. Starting with a minimum value, each node incrementally increases its transmission power till the NET condition is satisfied. We assume that the orientation of neighbors can be computed either from angle-of-arrival of messages or localization information. Each node computes its empty sectors and increments power if 1) NET condition is not satisfied or 2) if it has an asymmetric incoming link from a node that does not satisfy NET condition. Since boundary nodes (detected using 47 an algorithm such as [Kroller et al., 2006]) are not required to satisfy NET condition, they only increase power in response to incoming asymmetric links. This is a simple implementation with scope for further power optimization which we plan to pursue in future. Our objective here is to demonstrate that for a given the resulting topologies preserveb 2 c edge-connectivity. In simulations, we used realistic statistical models for wireless links developed in [Zuniga and Krishnamachari, 2004] for which the authors have made code available online [Zuniga et al., 2005]. Fig. 3.4 shows typical topologies resulting with 500 nodes distributed uniformly at random on a 80m x 80m square. Nodes that cannot satisfy the NET condition at full power are identified as boundary nodes (shown as red squares). The initial transmission power was set to10dB and incremented in steps of 1dB. Links that had a packet reception rate of at least 90% were considered active. Note that the figures only show bidirectional links. The simulations were terminated when all non-boundary nodes satisfied NET condition. Fig. 3.4 (a) and (b) show the comparison the topologies resulting from NET based power control and CBTC() [Bahramgiri et al., 2006] for 2-edge-connectivity. According to CBTC() a node must either have a neighbor in every = 2 3k or operate at full power to guarantee k- edge-connectivity. We show that having a neighbor in every = 2 k is sufficient to guarantee k-edge-connectivity. As a result, NET condition results in a significantly sparser, and hence efficient, topologies compared to CBTC(). Fig. 3.14 shows the edge-connectivity and average power over 100 iterations. Edge-connectivity was computed as the minimum number of paths consisting of symmetric links between any two internal nodes. In all experiments, the edge-connectivity was always greater thanb 2 c. This 48 validates our key theoretical result, Theorem 3:4. The average power used increases quickly as decreases (fig. 3.14(b)). This implies that for efficient topology control it is important to choose the largest value of possible depending on the application requirements. Because NET graph requires an angle ( = 2=k) that is three times bigger than the angle required by CBTC() ( = 2=3k), the power saved will very large. For example, for 2-edge-connectivity, CBTC() requires an average power of 4dB while NET graph requires6dB. These initial results are promising because we are able to achieve power-efficient topologies that guaranteek-connectivity through distributed power control even with realistic, irregular link models. (a) = (b) = 2 6 Figure 3.13: Comparison of NET graphs and CBTC() for a uniform random network of 500 nodes. For 2-edge-connectivity, (a) NET graph requires sector angle = and average power 5:91dB while (b) CBTC requires = 2 6 and average power of 4:10dB. 49 (a) edge-connectivity (b) average power Figure 3.14: Simulation results of power control based on NET graphs for a uniform random network of 500 nodes averaged over 100 runs. The edge-connectivity is greater than the 2 k lower bound derived in Theorem 3:4. The average communication power for achieving 2-connectivity using CBTC() is 4dB (red square) compared to6dB using NET graphs. 3.5 Conclusions This chapter addresses distributed topology control using a construct called Neighbor-Every- Theta (NET) graphs. These graphs have the following two properties that allow for simple and practical algorithm design: tunable connectivity based on a single parameter, the sector angle connectivity guarantees do not depend on communication model NET graphs are such that each node has at least one neighbor in every sector around it. We prove that for a given <, NET graphs have an edge connectivity of at leastb 2 c, except in pathological cases where the network can be partitioned close to its boundary. This property holds even in cases when the communication model is irregular. For the special case of an ide- alized disk communication model, it is shown that for specific values of, NET graphs contain 50 proximity graphs such as the relative neighborhood graphs that are well known to be desirable communication topologies. We also prove the symmetric neighbor placement condition for max- imizing sensing coverage under the NET condition. To concretely demonstrate the use of NET graphs for tunable power control, we implement a typical protocol using satisfaction of NET condition at each internal node as the termination criterion and show that a power-efficient network with edge connectivity 2 can be achieved even with realistic, irregular links. Lastly, we consider NET graphs in three dimensions and study their connectivity properties. It is our conjecture that for a given solid angle< 2, NET3D graphs have an edge connectivity of at leastb 4 c when partitions close to the boundary are ignored. There is very little earlier work on topology control in 3D. Several geometric computations become complex and even intractable when extended from 2D to 3D. We have developed an efficient algorithm for deter- mining the largest empty cone around a node in 3D based on spherical Delaunay triangulations. The running time is O(dlogd) for average node density d, which is a significant improvement over the earlierO(d 3 logd) algorithm proposed in the context of CBTC [Bahramgiri et al., 2006]. The new algorithm can be used to extend several sector based topology control algorithms to 3D. Topology control in 3D is an important area for further research. In the next chapter, we develop distributed deployment algorithms for robot networks to achieve NET conditions. 51 Chapter 4 Distributed Control Algorithms In this chapter we propose deployment algorithms for robot networks that maximize sensing coverage while maintaining a local constraint. We consider two constraints 1. kdegree: each robot has degree at leastk wherek is a parameter to the algorithm 2. NET condition: each robot satisfies the NET constraint, i.e. has at least one neighbor in every sector of its communication range, where is a parameter to the algorithm. We assume that the robots do not have global localization or a global map of the environment but can sense the distance and orientation of their neighboring robots. Our approach is based on virtual potential fields. We treat each robot in the network as a vir- tual charged particle and define simple force laws to govern the interaction between neighboring robots. These laws incorporate two different kinds of forces. One is a repulsive force that tries to maximize coverage while the other is an attractive force that imposes the local constraint. As a result of these forces, a group of robots placed close together spreads out into a network to maxi- mize the coverage while satisfying the local constraint. The deployment proceeds by starting in a state where all the constraints are satisfied and at each step robots move such that no constraints 52 are broken. In the next chapter, we consider the case where the robots are disconnected at the start of deployment. 4.1 Problem Formulation Problem: GivenN mobile robots with isotropic radial sensors of rangeR s and isotropic radio communication of rangeR c , how should they deploy themselves so that the resulting configura- tion maximizes the total sensor coverage of the network while maintaining a local constraint at each robot? We make the following assumptions: 1. The robots have isotropic binary disk sensing and communication, 2. The robots are capable of omni-directional motion, and 3. Each robot can sense the exact relative range and bearing of robots within its sensing range We impose certain desiderata on the problem solution. The deployment algorithm should: be distributed and scalable not require a prior map or localization of robots adapt to changes in the environment and the network itself We use the following three metrics to evaluate the performance of the deployment algorithm. 1. the normalized per-robot coverage. This is defined as: Coverage = (Net Area Covered by the Network) NR 2 s 53 In the remainder of the chapter, we use the term ‘coverage’ to mean the normalized per- robot coverage as defined above. 2. the percentage of robots in the network that satisfy the local constraint, and 3. the average degree of the network. 4.2 Design of the Control Law Potential field-based techniques have been used extensively to solve navigation problems in mo- bile robotics. In these methods, virtual potential fields are used to represent goals and constraints and the control law for the robot’s motion is formulated such that it moves from a high potential state to a low potential state similar to the way in which a charged particle would move in an electrostatic field. In our deployment algorithm, we construct virtual forces between robots so that each robot can attract or repel its neighbors. The forces are of two kinds. The first,F cover , causes a pair of robots to repel each other to increase their sensing coverage. The second,F constraint constrains a pair robots by making them attract each other when they are on the verge of being disconnected. By using a combination of these mutually opposing forces, each robot maximizes its coverage while maintaining the local constraint. The deployment proceeds by starting in a state where all the constraints are satisfied and at each step robots move such that no constraints are broken. A robot whose constraints are fully satisfied, repels all of its neighbors usingF cover till its local constraint is about to be broken. The resulting neighbors are called the robot’s critical neighbors and the connections between them and the robot are called critical connections. The 54 robot now communicates to all its critical neighbors that its connection with them is critical and therefore should not be broken. It then continues to repel all its neighbors using F cover but as the distance between the robot and its critical neighbor increases,kF cover k decreases and kF constraint k increases. As a result, at some distance R c , where 0 < < 1, the net force kF cover +F constraint k between the robot and its neighbor is zero. At this distance, the robot and its neighbor are in equilibrium with respect to each other. We call 1 the safety factor because the larger its value, the smaller the probability of critical neighbors losing connectivity. The forces are constructed as inverse square law profiles -kF cover k tends to infinity when the distance between robots is zero so that collisions are avoided. Similarly,kF constraint k tends to infinity when the distance between the critical neighbors is R c so that loss of connectivity between them is prevented. Mathematically, the forces can be expressed as follows. Consider a network of n robots 1; 2; 3;:::;n at positionsx 1 ;x 2 ;:::;x n respectively. Let x ij represent the Euclidean distance between robotsi andj, i.e. x ij =kx i x j k F cover andF constraint are defined as follows. F cover (i;j) = K cover x ij 2 x i x j x ij F constraint (i;j) = 8 > > > > < > > > > : K constraint (x ij Rc) 2 x i x j x ij if critical connection; 0 otherwise. whereK cover andK constraint are the force constants. 55 The resultant force between the robotsi andj is F(i;j) =F cover (i;j) +F constraint (i;j) and roboti will experience a net force of F i = X all neighbors j F(i;j) The equation of motion for roboti is formulated as: x i (t) = F i _ x i m where is a damping factor andm is the virtual mass of the robot which is assumed to be 1. Computational Details Having described the equation of motion for the robot, we discuss our choices of the four param- etersK cover ,K constraint , and. K cover : Consider two robots that are repelling each other. As the distanced between them increases, the combined coverage of the robots increases, reaches a maximum of 2R 2 s at d = 2R s and remains constant after that. This implies that ford> 2R s repelling does not improve coverage. We therefore pick a value forK cover such that at d = 2R s ;kF cover k 0 56 K constraint : Atd =R c we wantkF cover +F constraint k = 0, i.e. K cover 2 R c 2 + K constraint ( 1) 2 R c 2 = 0 K constraint = (1) 2 2 K cover : A large increases the probability of critical neighbors getting disconnected while a small results in lesser coverage. We used a value of 0.8 for. This is a heuristic choice based on experimental experience. : We conducted a simple ‘two-body’ variant of our scenario by implementing our algo- rithm on two robots to study the variation in their interaction for different values of. For these experiments we fixed the values ofK cover ,K constraint and as explained above. We found that for small values of the system oscillates. We picked the smallest value of that does not lead to oscillations. This value corresponds roughly to the critically dampedcase for our system. In our experiments, we used = 0:25. In the following, we analyze the coverage achievable by uniform random and symmetrically tiled network configurations that satisfy the constraint ofk-degree. These will serve as reference points to evaluate the performance of our algorithm. Coverage of Uniform Random Networks A uniform random network is one in which the robots are distributed randomly and with a uniform density. For such networks, the probability of findingi robots in a specified domain depends only 57 on the area of the domain and not on its shape or location. Given any areaS, the probability that it will contain exactlyi robots is given by P (i) = (S) i i! e S where is the density of robot deployment [Hall, 1988]. The probability that a randomly chosen robot will have a degree of at leastk is the probability that there will be at leastk robots in the areaR 2 c around it. P (iK) = 1 i=(K1) X i=0 (R 2 c ) i i! e R 2 c As the density of deployment increases, the probability that each robot will have a degree of at leastk, increases but the per-robot coverage of the network decreases. Therefore the best coverage achieved by a random network that satisfies the constraint of k-degree with a high probability, say 0.95, will correspond to the smallest density for whichP (i K) 0.95. Let this density be 0 . The coverage of a uniform random network is a function of its density andR s . For a network with density 0 the normalized per robot coverage is: Coverage = 1e 0 R 2 s 0 R 2 s Note that this expression is independent of the number of robots in the network. 58 Coverage of Symmetrically Tiled Networks In chapter 3, we proved that for R c = R s , symmetrically tiled networks give the maximum coverage achievable by networks satisfying thek-degree constraint. Given a symmetrically tiled configuration, we can compute the per robot coverage as a function ofR s andR c . IfR c 2R s , Coverage = 1 because there will be no overlap between the robots. For the case whenR c < 2R s and K = 1, Coverage = =2+sin(=2) where = cos 1 ( Rr 2Rs ). We can derive similar expressions for the other values ofk. 4.3 Experiments and Results This section presents a set of experiments designed to study the performance of the proposed system for different values of the input parameters. The simulations were conducted using the Player/Stage 1 software platform which simulates the behavior of real sensors and actuators with a high degree of fidelity [Gerkey et al., 2001]. Each of the robots in our simulations is capable of omni-directional motion and sensing (using a laser range finder). Further, each robot has a retro-reflective beacon so that it can be distinguished from the obstacles in the environment. In our simulations we used a 2D obstacle-less environment. Fig. 4.1 shows the initial and final network configuration for a typical deployment. The circles in the figure represent the coverage areas of individual robots. The robots start in a compact grid like configuration at the center of the environment and spread out to cover a large portion of the 1 Player/Stage was initially developed jointly at the USC Robotics Research Labs and is freely available under the GNU General Public License from http://playerstage.sourceforge.net 59 environment. In the particular instance shown, the sensing range of the robots is equal to their communication range. 4.3.1 k-degree Constraint Fig. 4.1 shows the variation of the coverage and average degree of the network with time for different values ofk. The coverage (average degree) increases (decreases) rapidly in the first 1-2 minutes and then saturates to a stable value within 4-5 minutes. This is because, initially all the robots have more thank neighbors and so they spread out uninhibitedly to improve the coverage until the degree constraints activate and restrict their motion. Further, since these constraints activate at different stages for different values ofk, the coverage (average degree) graphs for the different values ofk, start off identically but branch off at different points and settle at different values of final coverage (degree). Fig. 4.2 compares the performance of our algorithm in terms of the coverage and average robot degree with the uniform random and symmetrically tiled network configurations for three different regimes -R c > 2R s ,R c = 2R s andR c < 2R s . Clearly, the configurations we obtain outperform the random network. Note that while computing the coverage of the symmetrically tiled configurations we assume that the size of the network is infinite. The values thus obtained are in reality an upper bound on the coverage that can be achieved with finite networks because in the latter case, we will have to take into account the edge effects. Our third performance metric is the percentage of the robots in the network that have a min- imum degree ofk. This we found was at least 95% in all the network configurations resulting form our deployment algorithm. This is also the case with the random networks. Recall that 60 while finding the density of deployment for the random network we explicitly imposed the con- straint that each robot should have at leastk neighbors with a probability of at least 0:95. In the symmetrically tiled configurations however this probability is 1 since all the robots have exactly k neighbors. An appealing and unexpected result is that there is no significant change in the per-robot coverage obtained when the size of the network N was varied from 49 to 81 (fig. 4.3). One would expect that for smaller values ofN the edge effects will be more significant and therefore asN increases the per-robot coverage will increase. We speculate that either the edge effects do not vary significantly with the network size or a 49 robot network is large enough to make edge effects negligible. 4.3.2 NET constraint In a typical mobile deployment scenario, all robots start in positions close to each other so that the initial network is highly well connected and the NET condition is trivially satisfied. Each robot begins by repelling all neighbors to increase its sensing coverage. In the process, it loses communication with some neighbors that move farther thanR c . When the number of neighbors is close to the number required to satisfy the NET condition, the robot assigns priority values to each of its neighbors based on their contribution towards satisfying its NET condition. This is done by computing sector angles between adjacent neighbors. The neighbors contributing to larger sector angles have a higher priority and the robot applies the attractive force to hold them within its communication range. Nodes on the boundary designate all neighbors as low priority and hence allow the decision to be made based on the requirement of their neighbors. The pseudocode for 61 this algorithm is shown below. c 1 ;c 2 are parameters that can be tuned for how strictly the NET condition is required to be satisfied. Algorithm 2: assignPriority(,robot,neighborList) ifrobot is on boundary forq2neighborList priority[robot][q] = 1 else forq2neighborList sectorVoid[q] =angle(q prev ;q) +angle(q;q next ) ifsectorVoid[q]<c 1 priority[robot][q] = 0 (redundant) else ifc 1 sectorVoid[q]<c 2 priority[robot][q] = 1 (low priority) elsepriority[robot][q] = 2 (high priority) The condition used for classifying boundary robots plays an important role in determining the final network structure. During deployment, the boundary of the network grows and robots that were previously interior robots become boundary robots. Once a robot becomes a boundary robot, it is not required to satisfy the NET condition. If interior robots are allowed to easily switch to becoming boundary robots, the coverage will increase because robots will spread out but the connectivity properties will suffer. On the other hand, if interior robots are constrained to always satisfy NET condition and are not able to switch to the boundary then the network cannot spread out and coverage will be poor. We use a heuristic based on the observation that boundary robots 62 have large empty sectors. Initially all robots are designated as non-boundary. If a robot has an empty sector greater than b for time, then it declares itself as a boundary robot. Similarly, if a boundary robot has no empty sector greater than b for time, then it becomes an interior robot. Algorithm 3: distributedDeployment() findneighborList degree =sum(neighborList) if (degree> ( 2 + 1)) repel all neighbors else if (degree> 2 orboundary) priority = assignPriority(,self,neighborList) forq2neighborList, if (priority[self][q] +priority[q][self]) 1 repel + attractq else repelq else repel + attract all neighbors The parametersc 1 andc 2 in assignPriority can be tuned depending on how strictly NET sat- isfaction is desired. For the results presented in this section, a conservative choice was sought and the empirically determined values are shown in Table. 4.1. The sector parameter b is used by individual robots for boundary detection. Choosing a large value for b will result in many boundary robots considering themselves interior robots trying to ensure NET satisfaction and 63 2 3 2 2 5 3 2 7 4 c 1 1.15 1.15 1.25 1.0 1.1 1.1 c 2 1.45 1.5 1.6 1.3 1.3 1.3 b 21 20 11 12 5 6 5 6 5 6 5 6 Table 4.1: Parameter values used in our implementation of the distributed deployment algorithm impedes the spreading. On the other hand, choosing a small value leads to interior robots switch- ing to boundary robots and decreases network connectivity. The appropriate setting of boundary sector b depends on the value and the empirically determined settings are shown in Table. 4.1. Results are shown for deployments of 100 robots with each experiment repeated 10 times. Fig.4.4 show the final network configuration from sample runs of the distributed algorithm for varying values. For = 2 3 , hexagonal tiling is difficult to achieve in a distributed manner and combined with the conservative choice ofc 1 andc 2 , the algorithm pushes the deployment to a square tiling (Fig.4.4a). Large blocks of square tiling are achieved for = 2 (Fig. 4.4b). For the non-tiling angle = 2 5 (at least 5 neighbors required), in most cases it is actually beneficial to settle for 6 neighbors and tile triangularly. Fig. 4.4c shows that the algorithm correctly allows for the highly stable tiling in large portions of the final configuration. This is also obtained for the tiling angle = 3 , as shown in figure Fig. 4.4d. Samples for = 2 7 and = 4 are also shown. No tiling is possible for these angles. Fig.4.5 shows that the deployment algorithm adapts well to obstacles. On encountering an obstacle, the robots continue to spread while avoiding it and surround it with little impact on the NET graph. Obstacles have the effect of increasing the number of boundary robots in the network and as a result, the network structure is not as uniform as before. Fig. 4.6a compares the coverage obtained from the deployment algorithms to an asymptotic upper bound on coverage obtained from Lemma 3.2.11 by considering robot overlaps as described 64 in Section.4.3.2. The algorithm’s coverage is close to the bound for smaller values of and the difference grows with. The difference can be attributed to a conservative choice of parameters c 1 andc 2 . This is reflected in the fact that the average robot degree resulting from the deployment algorithm is always greater thanb 2 c (Fig. 4.6c) even though the boundary robots have smaller robot degrees. Also note that the coverage upper bound is tight only in case of tiling angles. For non-tiling angles, it is not clear what the optimal deployment is. Fig. 4.6b shows the edge connectivity values. For the k-connectivity calculation, boundary robots and one hop neighbors of boundary robots are not considered. For the chosenc 1 andc 2 values a connectivity ofk =b 2 c is not guaranteed, but it is very often achieved and connectivity b 2 c 1 is almost always assured. Using lower values forc 1 andc 2 can provide a guaranteed level of connectivity at the cost of some coverage. Fig. 4.6d shows the number of sectors of interior robots that violate the NET condition. A 10 % leeway was allowed for deviation from, i.e., a violation occurs if sector angle> 1:1. There is a drop in violations for = 2 5 since it defaults to the 6-neighbor triangular tiling while there is a sudden increase for = 2 7 and = 4 since there are no tiling angles in sight making it difficult for the pair-wise negotiations to attain symmetrical placement of neighbors. The asymmetry and small value of result in increased violations. Note that this does not adversely affect thek-connectivity. Fig. 4.7 shows a comparison of the communication graph obtained from deployment with = 3 with its corresponding RNG. In Fig. 4.7c, the dark lines represent the edges in the RNG that are absent in the communication graph. The communication graph differs from the RNG in exactly 3 edges at the boundary of the network. Even in the presence of boundary robots in real networks, these results validate the assertion in Theorem 3.2.8 that the communication graph will 65 contain the RNG for 2 3 . Also, the communication graph contains only a few non-boundary edges that are not in the RNG and will therefore inherit the sparseness properties of the RNG. 4.4 Conclusions In this chapter we have presented potential field based self-deployment algorithms for groups of mobile robots. The aim of the algorithm to to maximize the total sensing coverage while maintaining a local constraint at each robot. We demonstrated the algorithm with two types of constraints - 1) number of neighbors of each robot, and 2) NET conditions. The algorithms pre- sented prove that local constraints can be achieved using potential field algorithms. They do not optimize the coverage due to over-deployment. However, the insights gained can be used to de- sign more sophisticated algorithms. The deployment proceeds by starting in a state where all the constraints are satisfied and at each step robots move such that no constraints are broken. In the next chapter, we consider the case where the robots are disconnected at the start of deployment. 66 (a) (b) (c) (d) Figure 4.1: Typical Network Configurations for a 64 robot deployment with k = 2 (a) Initial Configuration, (b) Final Configuration, The temporal performance for N = 64, R s = 4 and R c = 8 (averaged over 10 trials) (c) Coverage vs. Time and (d)Average Node Degree vs. Time 67 (a) (b) (c) (d) (e) (f) Figure 4.2: Coverage and average degree for N = 49 and R s = 4 for different values of R c (averaged over 10 trials) (a), (b) R c = 10 (R c > 2R s ) (c), (d) R c = 8 (R c = 2R s ), (e), (f) R c = 6 68 Figure 4.3: Variation of coverage with network size (N) forR s = 4 andR c = 8 (averaged over 10 trials) (a) = 2 3 (b) = 2 (c) = 2 5 (d) = 3 (e) = 2 7 (f) = 4 Figure 4.4: Sample deployments with 100 robots andR c =R s = 8:0. 69 (a) = 2 3 (b) = 2 5 (c) = 3 Figure 4.5: Sample deployments in the presence of obstacles with 100 robots andR c =R s = 8:0. (a) (b) (c) (d) Figure 4.6: Performance of the Distributed deployment Algorithms for a deployment of 100 robots in terms of (a) Coverage, (b) Edge Connectivity (c) Average Degree and (d) NET condition satisfaction 70 (a) (b) (c) (d) (e) (f) Figure 4.7: For 2 3 , the communication graph is a supergraph of RNG. (a) Communication Graph (b) RNG (c) Difference for = 2 3 . (d) Communication Graph (e) RNG (f) Difference for = 2 4 71 Chapter 5 Connectivity Maintenance and Coalescence Communication connectivity is both essential and challenging for robot networks. While on one hand robot groups rely on network-wide connectivity for coordination, on the other hand maintaining connectivity in a large group of moving robots is hard because it imposes constraints on robot motion. Another difficulty is that the communication range for most low power radios is highly irregular [Zhao and Govindan, 2003] which adds to the unpredictability of connectivity. Given the high cost of maintaining connectivity, we are interested in the question - what happens if connectivity is lost? This situation could arise due to several reasons - robots could start in a disconnected state, or accidentally lose connectivity due to failures, or intentionally disconnect for efficient task completion. The cost of regaining connectivity, in terms of time and energy spent, should be taken into account while designing the motion coordination algorithm. If the cost is high the coordination algorithm should minimize the chances of disconnections and if the cost is low it should allow the robots to occasionally disconnect and rejoin the network after task completion. 72 Coalescence is the problem of isolated mobile robots independently searching for peers with the goal of forming a single connected network. In this chapter, we study the time taken for coalescence. We consider the scenario where the robots are memoryless (do not build a map) and do not have any knowledge of the environment or positions of the other robots. Each robot performs an independent random search and when two robots meet, they coalesce into a cluster and the cluster thereafter sticks together and moves as a single random walk. This is illustrated in figure 5.1. While the memoryless model is realistic for large swarms of simple robots, in several applications robots will be capable of performing systematic exploration and mapping to speed up coalescence. The coalescence time in this case is tied to the exploration algorithm and structure of the domain and the memoryless random walk provides a simplified worst case study. While coalescence can be considered a variation of the rendezvous problem, we use the term coalescence to emphasize that the communication spread of the clusters increases as more robots join. As we will show in section 5.2.3, the communication spread plays an important role in determining latency. The basic idea is that robots need not be collocated to remain connected. If they remain spread out, the disconnected robots will have a better chance of discovering them. The coalescence time can be further improved if the clusters form an optimal shape such as a line and move in a direction that maximizes the sweep area. This will involve a significant coordination overhead because maintaining a formation while moving through an environment with obstacles is challenging [Muhammad and Egerstedt, 2004]. Therefore we do not make any assumption on the shape of the cluster. The outline of the chapter is as follows. Section 5.1 presents the problem formulation and mobility model. In section 5.2 we derive the coalescence time based on the meeting time of the 73 (a)t = 0 (b)t =MT 5 1 (c)MT 5 2 (d)t =MT 5 2 <t<MT 5 3 (e)t =MT 5 3 (f)t =MT 5 4 =CT (N = 5) Figure 5.1: Illustration of Coalescence for N = 5. The dark red circles represent robots and the yellow discs around them are their communication regions. The arrows show the random walk steps with the faded discs being the robot positions at the previous time instant. As robots coalesce, the communication spread of clusters grows and the robots that are still searching have a better chance of hitting it. mobility model. The coalescence time has an exponential distribution which is a function of the number of robots, speed, communication range, and size of the domain. The main challenge in the analysis is that the time between two clusters hitting each other depends on the size and shape of the clusters which are both random and difficult to model. We make several simplifying assumptions to make the analysis tractable which are validated using simulations in section 5.3. We are particularly interested in the asymptotic behavior of the coalescence time with respect to the size of the network,N. In section 5.2.3 we consider three extreme scenarios and show that coalescence time isO( 1 p N ) and ( log(N) N ). Simulations in MATLAB (section 5.3) verify these bounds and also suggest that the lower bound is tight i.e. the coalescence time is (log(N)=N). 74 (a)t = 0 (b)t = 1 (c)t = 7 (d)HT =t = 11 Figure 5.2: Illustration of random direction mobility model, RM. The blue cone and dark red circle are a stationary node and the isolated robot respectively along with their communication discs. The light yellow region is the area that the robot has “covered” in its search for the station- ary node. 5.1 Problem Formulation We are interested in the following problem. Given N isolated mobile robots placed at unknown locations within a bounded domain D< 2 , how long will it take for them to coalesce into a single connected component? The goal of coalescence is to reach a state where all robots can communicate with each other either directly or via other robots. Clearly, coalescence time will be depend on the communication and mobility models of the robots. We use the following models. 5.1.1 Communication Model All robots have a uniform binary disk communication with rangeR i.e. two robots can commu- nicate with each other if and only if the distance between them is less thanR. We say that two robots “encounter” or “hit” each other when they are within communication rangeR and at some 75 timet 0 the area that a robot has “covered” is the union of the area that has been within its communication range at 0t 0 t (figure 5.2). 5.1.2 Mobility Model Each robot performs an independent random search for other robots. The motion strategy is a simplified version of the Random Direction model [Aldous and Fill]. It is defined as follows (figure 5.2). At each time step the robot 1. chooses a direction uniformly in [0; 2) 2. moves in direction with a constant speedv for a constant distancel. The random direction model results in a uniform stationary distribution. At any point of time if two robots meet, they form a single cluster i.e. stay connected to each other and move together following the same path. Similarly when two clusters meet they coalesce to form a single cluster. Note that this interaction model does not affect the uniform stationary distribution of the clusters. 5.1.3 Domain Model For ease of analysis we assume that the domain is a torus. A more realistic boundary condition is the reflecting model where the robots bounce back on hitting a boundary. However, it is known that both the torus boundary condition and reflecting boundary condition retain the stationary distribution of robots [Bettstetter, 2001]. Since stationary distribution is the only condition we require of the torus boundary condition our analysis is valid for the reflecting boundary condition. 76 5.2 Coalescence Time Analysis Coalescence time (CT ) is related to the time elapsed between robots meeting each other. There- fore to model coalescence time, we must first understand the statistics of encounter times between robots, called hitting time (HT ) and meeting time (MT ). Definition 5.2.1 The hitting time of a mobility model is the time until a robot following the mobility model encounters a static robot. If roboti moves according to the mobility model and robotj is static,HT = min t ft :kX i (t)X j kRg. Definition 5.2.2 The meeting time is the time until two robots following the same mobility model meet. If j also follows the mobility model described above and starts from its station- ary distribution, the meeting time between i and j is the time before their first encounter. MT = min t ft :kX i (t)X j (t)kRg We will now derive the hitting time and meeting time for our mobility model. 5.2.1 Hitting Time Let roboti move according to the random mobility model and let(t) represent the fraction of domain area covered by the communication disk ofi from time 0 tot. We can assume that this area is uniformly distributed in the domain. Then the new area covered in the (t + 1) th time step is proportional to the fraction of uncovered area inD and is given by (t + 1)(t) (1(t)) 2R` A(D) (5.1) 77 whereA(D) is the area of the domain. Solving the recursion, we get (t) 1 1 2R` A(D) t (5.2) If j is a stationary robot placed at a random position chosen uniformly in the domain, the probability of hittingj at timet is equal to the fraction of the domain area covered byi, (t). Therefore the hitting time has the same exponential distribution as(t) with mean E[HT ] = A(D) 2R` (5.3) Note that as expected, the hitting time is a function of the area of the domain, communication range, and speed (step length). 5.2.2 Meeting Time If bothi andj start at a random positions chosen uniformly in the domain and follow the random mobility model, then the meeting time will follow the same exponential distribution as the hit- ting time with meanE[MT ] = E[HT ]=^ v [Spyropoulos et al., 2006] where ^ v is the normalized relative speed given by ^ v = k~ v i ~ v j k v (5.4) If is the angle between the velocity vectors ofi andj, then ^ v can be computed as 1 2 Z 2 0 p (1 +cos) 2 +sin 2 d = 1:27 (5.5) 78 Figure 5.3: Verification of analytical model fork th meeting time,MT N k derived in equation (9) forN = 100,l = 100, andR = 20 averaged over 100 runs For N = 2, the coalescence time CT is the meeting time. E[CT (n = 2)] = E[MT ] = E[HT ]=1:27. 5.2.3 Coalescence Time LetMT N 1 MT N 2 MT N k MT N N1 be the 1 st ; 2 nd ; ;k th ; (N 1) th meeting times in a network of sizeN. The coalescence timeCT (n =N) =MT N N1 . We will now find an expression for the k th meeting time MT N k . The first hitting time is the minimum of N 2 meeting times between the different pairs of robots. Since each of these is independent and has an exponential distribution, 79 E[MT N 1 ] = 1 N 2 E[MT ] = 2 N(N 1) 1 1:27 A(D) 2R` (5.6) t =MT N 1 , there areN1 clusters with 1 cluster consisting of two robots andN2 clusters consisting of 1 robot each. Att = MT N k , there areNk clusters in the domain. The number of robots in each cluster is a random variable. At t = MT N k+1 two of these Nk clusters meet and coalesce to form a bigger cluster. We expect that the time elapsed between clusters meeting will also have an exponential distribution similar to the first meeting timeMT N 1 . The challenge is that this meeting time will depend on the size and shape of the clusters which is difficult to estimate. Even if we know the number of robots in each cluster, the shape will still be random. We will make a simplifying assumption that all clusters have the same size. Further, we assume that the expected shape of the cluster is a disc since clusters coalesce in all directions with equal probability. We validate these assumptions using simulations in MATLAB described in section 5.3. The meeting time derived based on these assumptions is surprising close to the simulation results (figure 5.3). Based on equation (6) we write the following MT N k+1 =MT N k+1 MT N k E[MT N k+1 ] = 2 (Nk)(Nk 1) 1 1:27 A(D) 2R eff ` (5.7) whereR eff is the effective radius of the clusters. R eff depends not only on the size of the clusters but also the shapes. Att =MT N k , there areNk clusters. If we assume that the area of a cluster grows linearly with the number of robots the the total area of all clusters att = MT N k 80 isNR 2 kR 2 . Therefore the area of each clusterC att = MT N k isA[C k ] = R 2 ( Nk Nk ) where 0 < 1 is a constant. For the optimal case when robots are at a distance fromR of each other, 0:2. Approximating the shape of each cluster as a disc, we getR eff = q Nk Nk . Substituting in the above equations we have E[MT N k+1 ] 2 (Nk)(Nk 1) E[MT ] q ( Nk Nk ) (5.8) E[MT N k ] = k1 X 0 E[MT N k+1 ] = k1 X 0 2 (Nk 1) (Nk) E[MT ] q ( Nk Nk ) (5.9) We now have an expression for coalescence time in terms of the number of robots,N and the expected meeting timeE[MT ] which in turn is a function of size of the domainD, communica- tion range,R, and step length or speed`. E[CT (n =N)] =E[MT N N1 ] = N2 X 0 E[MT N k+1 ] = N2 X 0 2 (Nk 1) (Nk) E[MT ] q ( Nk Nk ) (5.10) The asymptotic behavior of the coalescence time as the number of robots N increases is not clear from the above expression. Therefore we will consider three cases where the analysis is tractable. In the first case, we will assume that the communication area of clusters does not 81 Figure 5.4: The mean Coalescence Time is bounded byO( 1 p N ) and ( log(N) N ). The graph shows the coalescence time(CT ) as a function of network size(N) averaged over 200 runs withl = 100 andR = 20 on a linear scale. change when robots join i.e. every cluster is a disk ofR 2 irrespective of the number robots it contains. As expected this case gives a very loose upper bound on the coalescence time. Second, we will assume that there is only one cluster that robots coalesce to upon hitting and ignore interactions between all other clusters. This case is equivalent to the stationary base station case and gives us aO( 1 p N ) upper bound. Lastly, we will derive a lower bound using an idealized case where the number of clusters decreases exponentially with time i.e. at each meeting time, every cluster coalesces with exactly one other cluster. 82 Figure 5.5: The mean Coalescence Time is bounded byO( 1 p N ) and ( log(N) N ). The graph shows the coalescence time(CT ) as a function of network size(N) averaged over 200 runs withl = 100 andR = 20 on a log scale. Case 1: Ignoring area growth Att =MT N k there areNk clusters each of which has a disc shape of radiusR. Substituting R eff R in equation (8) we get 83 E[MT N k+1 ] 2 (Nk)(Nk 1) E[MT ] E[MT N N1 ] = N2 X 0 E[MT N k+1 ] N2 X 0 2 (Nk 1) (Nk) E[MT ] = N2 X 0 2( 1 Nk 1 1 Nk )E[MT ] = 2(log(2) log(N) + log(Nk + 1))E[MT ] (5.11) We get,E[CT (n = N)] = E[MT N N1 ] log(2)MT i.e. the coalescence time is at most a constant times the meeting time irrespective of the size of the network. As we will see in the simulation results, this is a very loose bound for the coalescence time which means that the communication area growth plays an important role in decreasing the coalescence time. Case 2: Single cluster Suppose there is one special robot in the network that robots coalesce with upon meeting. En- counters between all other clusters do not affect the random motion of robots and in particular, do not result in robots coalescing. This simplifies analysis because the number of robots in different clusters is no longer random. At timet =MT N k , there is 1 cluster consisting ofk + 1 robots and Nk 1 single-robot clusters. Again, we assume that the area of a cluster grows linearly with the number of robots. Att =MT N k , the clusterC k containsk robots andA[C k ] =R 2 (1 +k) 84 where 0< 1 is a constant. Since robots can join the cluster in all directions with equal prob- ability, we approximate the shape of the cluster as a disk. This give usR eff = p 1 +kR. At t = MT N k , each of the Nk 1 isolated robots are uniformly distributed and have an equal chance of meeting the cluster. Therefore the time before theMT N k+1 is the minimum of the Nk 1 meeting times. E[MT N k+1 ] 1 (Nk 1) E[MT ] p 1 +k E[MT N N1 ] = N2 X 0 E[MT N k+1 ] N2 X 0 1 (Nk 1) 1 p 1 +k E[MT ] Z N1 0 1 (Nx 1) 1 p 1 +x E[MT ] dx = 0 @ 2tanh 1 ( p x+1 p N+1 ) p N + 1 1 A N1 0 = 1 p N (5.12) 85 Case 3: Simultaneous meeting We will now derive a lower bound on coalescence time by considering an extreme scenario where at each meeting time, every cluster simultaneously hits one other cluster. LetN = 2 p . E[MT N k+1 ] = 1 2 pk 2 1 2 k E[MT ] = 1 2 pk (2 pk 1) 1 2 k1 E[MT ] E[MT N N1 ] = p1 X 0 1 2 pk (2 pk 1) 1 2 k1 E[MT ] = 2 p p (log(2) 1)E[MT ] = log(N) N (5.13) In the next section we will verify these bounds through simulations. 5.3 Simulation Results Simulations were conducted in MATLAB to verify the analysis. The domain used is a 2D torus. The parameters to the program are number of robotsN, step size`, communication rageR and torus side T . The robots start at random positions uniformly distributed over the torus. Every robot picks a random direction and moves with a uniform speed of 1 unit per time step. At each time step, the robot checks if it is within a distanceR of another robot, in which case, the robots coalesce and follow the motion of the robot with the smallest ID. After` time steps, each cluster picks a new direction. The simulation stops when there is only one cluster. 86 We first verify the model for meeting time (equation (9)). This expression was was based on three simplifying assumptions 1) the clusters sizes are close to uniform, 2) the shape of the clusters is close to circular and 3) the area of a cluster grows linearly with the number of nodes. The results from simulation plotted in figure 5.3 are very close to the analytical expression and validate our model. Our next set of experiments simulated Coalescence and measured Coalescence Time as a function of N. Figure 5.4 compares the mean Coalescence Time to the analytical expressions derived above. Our objective is to compare the shapes of the curves and the constants in the expression were found empirically. We see that the coalescence time has bounds ofO( 1 p N ) and ( log(N) N ). 5.4 Conclusions We consider the problem of coalescence, i.e. isolated robots independently searching for peers and forming a single connected component. Coalescence strategies can complement algorithms for robot collaboration that require a connected network. In the absence of localization or any information about the environment, the robots can perform a simple random walk search till they are within the communication range of other robots. We show using probabilistic analysis that for such a strategy, Coalescence Time has an exponential distribution which is a function of the number of robots, speed, communication range, and size of the domain. Further, it decreases as O( 1 p N ) and ( log(N) N ) with the number of robots,N. Even though the analysis is approximate, there is a surprisingly good match with the simulation results. 87 Chapter 6 Coverage Optimization in Complex Environments In this chapter, we address the coverage problem for mobile sensors with location dependent sensing models. In reality, the performance of each sensor is impacted by the environmental conditions in which the sensor is located. We observed this while exploring the possibility of deploying a camera sensor network involving cyclops image sensors [Rahimi et al., 2005] to monitor indoor and outdoor pathways. We conducted the following simple experiment. Figure 6.2(a) shows an aerial view of a botanical garden. We placed 11 cyclops image sensors(Figure 6.2(b)) uniformly along a path in the garden as shown in Figure 6.3(a). We used the standard background-subtraction-based detection algorithm in [Horprasert et al., 2000] to test the number of times a person walking along the path was detected by this network. The results are shown in Figure 6.3(c). We repeated the same experiment for an alternate sensor arrangement as shown in Figure 6.3(b) – this time used sensing performance data from various locations along the path. The detection results in this case were far superior (Figure 6.3(d)).This is due to the fact that background subtraction is sensitive to lighting conditions that vary from location to location in the deployment domain. As a result the sensing performance of the image sensors varies from 88 (a) (b) Figure 6.1: (a) An aerial view of the garden (b) the pathway to be covered Figure 6.2: Cyclops camera with an attached Mica2 Mote location to location. In a recent work in [Shi et al., 2007], the authors report that underwater sound propagation is susceptible to changes in water salinity and temperature and therefore the listening range, or coverage, of a sound detector is largely influenced by its location underwater. Currently, coverage algorithms do not address such location dependence. This leads to inefficient usage of resources. 89 (a) (b) (c) (d) Figure 6.3: Node locations for (a) uniform deployment and (b) optimized locations and their detection results. The graphs illustrate detection events by node numbers (y-axis) at a given time step (x-axis). The red squares indicate false-positives.The optimized placements use 9 nodes while uniform placements use 11 nodes For this variation of the coverage problem, we develop a distributed control algorithm based on the generalized V oronoi partitioning to obtain optimal sensor arrangements with guaranteed convergence. Cyclops camera sensors provide a wonderful playground to apply our algorithms. We consider the problem of covering indoor and outdoor pathways with Cyclops camera sen- sors. Cheap and low-power imaging sensors such as Cyclops and CMUCam [Rowe et al., 2002] along with Motes have been demonstrated to have many potential applications in monitoring and surveillance. Because of the constraints on memory, computation and energy resources available to the imagers, they will most likely use simple image processing such as background subtraction for object detection. It is is well known that background subtraction is sensitive to changes in the background and lighting conditions making the sensing performance location-dependent. 90 Our contributions We formulate the coverage problem as the locational optimization problem with the sensing per- formance function that is location dependent, i.e., it is a function of the distance between the sensor and the sampling location as well as the location of the sensor. In the same spirit of the coverage algorithms in [Cortes et al., 2004], we propose a distributed control and coordination algorithm for the new formulation. The convergence of the algorithm is proved. We apply these ideas to the problem of covering indoor and outdoor pathways using Cyclops cameras. Based on several indoor and outdoor measurements we propose a 1D empirical function for the sensing performance for the Cyclops sensor. The sensing performance function corresponds to the num- ber of pixels detected of the test object at different locations based on the standard background- subtraction-based algorithm [Horprasert et al., 2000]. The function is characterized by two pa- rameters that vary with the location of the camera sensor thus capturing the location dependence of the sensing performance. Based on this, we formulate the problem of covering pathways as a 1D coverage algorithm. We present simulation results for various forms of the functions for the parameters and demonstrate convergence of our algorithm. In reality, these functions are not known a priori. Using the history of measurements as camera locations evolve, we estimate the functions using a piecewise linear form. We present simulation results using this approach. We conducted experiments in indoor setting under controlled lighting as well as in outdoor settings at fine scales to find values of the parameters in the performance functions. We present simulation results on this physical data and demonstrate the performance of our algorithm. The rest of the chapter is organized as follows. In Section 6.1 we formulate the coverage with problem with the location dependent sensing performance function. A distributed control 91 algorithm is proposed in Section 6.2 and its convergence is proved. In Section 6.3, we study ap- plication of the control algorithm to the problem of covering indoor and outdoor pathways using cyclops cameras. We propose a sensing performance function of the cyclops camera capturing the location dependence and present simulation and experimental results. 6.1 Problem Formulation We build on the notation used in [Cortes et al., 2004] to define our coverage problem. Let Q denote the bounded convex domain inR. We seek to cover Q using n mobile sensors. Let :Q!R denote the scalar density function. It captures the importance of covering a particular location. Let p 1 ;p 2 ; ;p n denote the current locations of n sensors. We assume a location- dependent sensing model for each sensor, i.e., the sensing performance at point q for the i th sensor not only depend onjjq p i jj but also on the location, p i itself. Here,jjjj denotes the Euclidean norm. We denote the sensing performance function for sensor i at point q by f(jjqp i jj;p i ), wheref :R + Q!R. Note that we explicitly indicate the dependence of the performance function on the sensor location. The performance function naturally induces a generalized Voronoi partition ofQ. Each sensor has its own dominance region, i.e., its V oronoi cell where the sensing performance is better than that for any other sensor. More formally, ifW i denotes the dominance region for thei th sensor, then W i =fq2Qjf(jjqp i jj;p i ) f(jjqp j jj;p j );8j6=ig: (6.1) 92 Now we define our coverage metric or net utility function as follows: H(p 1 ;p 2 ; ;p n ) = n X i=1 Z W i (q)f(jjqp i jj;p i )dq (6.2) The utility function is the sum of the sensing performance function of each sensor over its dominance region including the weighing density function. We refer to R W i (q)f(jjqp i jj;p i )dq as the utility of thei th sensor. We formulate the coverage problem as finding the optimal set of locations of the sensors that maximizes the net utility. arg maxH(p 1 ;p 2 ; ;p n ) (6.3) Note that it is possible to formulate other metrics such as maximizing the minimum over the utility functions of all the sensors, as suggested in the literature, e.g., [Cortes et al., 2004]. We would like to note that in the common literature [Cortes et al., 2004] the coverage objective function is known as the locational optimization function and the coverage problem is formulated as the minimization problem. In the spirit of the application we consider in this chapter, we formulate the problem as the maximization problem. 6.1.1 Partial derivative of the overall utility function We note that the result about the partial derivative of the net utility function based on the diver- gence theorem [Cortes et al., 2004] generalizes easily to our case. 93 Lemma 6.1.1 The partial derivative of the net utility function with respect to the position of the i-th sensor is given by, @H @p i = Z W i (q) @f(jjqp i jj;p i ) @p i dq (6.4) We refer readers to [Cortes et al., 2004] for the proof of the lemma. 6.2 Distributed coverage control We propose a distributed coverage control algorithm that is based on iterative local sub-gradient method. 6.2.1 Control Law for each sensor Let p i (n) be the position of the i-th sensor after n updates. We state the control law for the position update of thei-th sensor top i (n+1). During the time thei-th sensor updates its position from, our coverage algorithm does not allow the simultaneous update at the V oronoi neighbors of thei-th sensor. The local control law involves searching for the local optimal solution. Given that the positions of the V oronoi neighbors are fixed, the net utilityH and the local gradient, @H @p i are just the functions ofp i . We state the control law as follows. p i (n + 1) =p i ,where,p i = arg max p i 2W i H( ;p i ; ) (6.5) Essentially, the control law involves local search over the V oronoi partitionW i for the optimal solutionp i by solving the constrained optimization problem. Ifp i does not lie on the boundary ofW i , then it is the root of equationg(p i ) = 0. Since the search involves finding the optimal 94 location just for one sensor, it is computationally tractable. We state the following lemma about the increase in the net utility after every pose update. Lemma 6.2.1 The control law for an individual sensor in Equation 6.5 guarantees that the net utility increases after the position update of the sensor. Implementation of the above control law requires the exact knowledge of the sensing perfor- mance functionf(jjqp i jj;p i ) for each sensori and the density function(q). Later we show in our case study of camera sensor network thatf(jjqp i jj;p i ) can be estimated on the fly using measurements. We also note that in [Schwager et al., 2007] it is shown that the global knowledge of(q) is not required. 6.2.2 Distributed Coverage Algorithm Below we state the coverage algorithm that runs at each node. We assume discrete time steps, 1, 2, . The algorithm involves the implementation of the local control law that we described in the previous section. A node updates its position only if its V oronoi neighbors are not updating their positions. t : UpdateV i t+1: Check if anyj2V i is updating its position t+2: if yes, wait for random duration of time and repeat the above steps t+2: if no, schedule position update at t+3 and broadcast overV i t+3: start moving to optimalp i as obtained using Equation 6.5 t+k: After the move, wait for random duration before the next update Algorithm 1: Coverage algorithm at the i-th sensor: Let p i denote its position. Let V i denote the set of its V oronoi neighbors. 95 6.2.3 Convergence Theorem 6.2.2 The distributed coverage algorithm at each sensor as described in Algorithm 1 converges, as the number of position updates tends to infinity. The proof follows from the fact that the net coverage utility is bounded and by lemma 6.2.1, it increases monotonically with each iteration of the algorithm. The algorithm yields a local optimal solution to the coverage problem. Comments about the optimal solution. 6.3 Deployment of Cyclops camera networks In this section, we discuss implementation issues and results of our algorithm for the coverage problem using Cyclops camera sensors. 1 The Cyclops camera sensor consists of a CMOS im- ager with CIF image capability and an internal image processor unit for image interpretation and analysis. It couples with a Mote and periodically captures images (figure 6.2). Our objective is to cover loopy environments such as indoor corridors or outdoor pathways such that an intruder ob- ject can be detected. We show that this problem can be mapped into a suitable coverage problem. We first propose a model for the sensing performance function of a camera. 6.3.1 Sensing Performance Function Our choice of the sensing performance function of a camera is related the performance of the object detection algorithm that is based on background subtraction. Foreground object detection using background subtraction has been used extensively in video surveillance applications due 1 We do not consider any network related issues here. 96 to ease of implementation and effectiveness [ref]. This method has low resource requirements in terms of memory and computation and is therefore well suited for low complexity camera sensor networks. According to this method, the cyclops camera periodically captures images and maintains a moving average background model of the environment. A foreground model is also constructed as the absolute difference of the instantaneous image and the constructed background. The resulting foreground images are then are passed through a luminance and size filter, which identifies clusters of pixels that have a considerable change in illumination. See [ref] for the details of the algorithm. Figure 6.5(a) shows a cyclops camera set up in a corridor in a building. Figure 6.5(b) and 6.5(c) show the background model and the instantaneous image obtained by placing a white board at some distance. The foreground images as the difference between the two images is shown in Figure 6.5(d). The detectability of the object depends on the number of pixels identified after passing the foreground model through the filter – larger the number of pixels, better the detection probability. We therefore use the number of pixels identified as the sensing performance function for each camera sensor. Note that the number of pixels detected is also a function of distance between the camera and the object. The further away the object is, the lower the number of pixels detected will be. Suppose a rectangular object of dimension hw placed at a distance d on the camera’s central axis exactly facing the camera. Consider an ideal scenario in which every point on the object differs significantly in its illumination characteristics from the background image. Then, 97 Figure 6.4: Perspective projection model (a) (b) (c) Figure 6.5: Illustration of pixels on target not detected. (a) background image (b) original image and (c) foreground image. In the foreground image, some of the pixels in the target board are black. These are the pixels that are similar to the background and are therefore not detected by the camera. 98 according to a perspective projection camera model, the number of pixels, m, detected by the camera is approximately given by: m' fhw d 2 resolution sensor area =k hw d 2 resolution wheref is the focal length andk is a constant. Figure 6.4 shows the perspective projection model. Thus, the number of pixels detected is inversely proportional to square of the distance between the camera and the object. We indeed observed this relationship in practice when we experimented with a white board of dimension 3ft 2ft placed at different distances in front of a cyclops camera that was placed in a well-lit corridor. Figure 6.6.1(a) shows the experimental set-up and Figure 6.6.2(a) shows the number of pixels identified using the background subtraction algorithm as a function of distance between the camera and the white board. We indeed observe the trend that the number of pixels are inversely proportional to the squared distance. However these situations are rare in the environments where the lighting conditions and the object properties are far from the ideal situation. In practice, the number of pixels detected by the camera deviates from this ideal model because some pixels on the object have the same texture as that of the background and go undetected. This is illustrated in fig 6.5 where the target is a white rectangular board. In the foreground image, the pixels in white are the ones that are detected. Some of the pixels on the board have intensities close to their corresponding pixels in the background. This indicates that the sensing performance function of camera depends not only on the dis- tance but also on the location of the camera itself. This very behavior motivates the definition 99 of the coverage problem we address. We restrict ourselves to coverage of narrow pathways re- ducing the problem to coverage of 1D loops. We conducted a series of experiments both indoors and outdoors to extract the sensing performance function of a camera. We placed a white/black rectangular board in front of a camera along its central axis at increments of 5 feet to obtain the number of pixels detected. Based on several data sets we observed that the following empirical model for the sensing performance function best fits the data. f(jjxqjj;x) = k 1 (x) k 2 (x) +jjxqjj 2 (6.6) Here x denotes the location of the camera and q denote the sensing performance, i.e., the number of pixels detected at locationq. Note the dependence on the location ofx via functions k 1 (x) and k 2 (x). Figure 6.6 shows the best fit and the corresponding values of k 1 and k 2 for indoor and an outdoor instances. Note that the sensing performance function is not exact; it corresponds to the best fit with the data. 6.3.2 Coverage Algorithm for 1D loop In this section, we discuss the coverage problem formulation and algorithm for covering a loopy pathway using the cyclops cameras. We assume that the pathway is narrow enough so that we formulate the problem as a 1D coverage problem. We assume that each sensor is a bi-directional camera obtained by combining two cyclops camera modules aligned along the same axis. Thus each sensor can see on its left and right directions on the loop. The sensing performance function along each direction takes the form of Equation 6.6 with possibly different parameter values. For now we assume that the pathway is a smooth curve that can be parametrized with a single 100 1(a) 1(b) 1(c) 2(a) 2(b) 2(c) Figure 6.6: Validation of the proposed model for pixels detected vs. distance for three different environments. 1(a) is a well-lit corridor and close to an ideal environment. 2(a) is an outdoor pathway with shadows of trees and buildings. 3(a) is inside a lab where the lighting conditions are not uniform. The solid blue line is the number of pixels detected from experiments and the red dashed line is the minimum least squares fit using the model in Equation 6.6 101 parameter and does not contain any sharp corners, i.e., at any location the curvature of the loop is very large compared with the local detection range of the camera assuming that the sensor is always aligned on the loop such that its central axis is always tangential to the loop. Letn be the number of cameras and letp 1 ;p 2 ; ;p n denote their locations on the loop in the order such that p i is betweenp i1 andp i+1 . Further, letp n+1 =p 1 . Letf l (jjp i qjj;p i ) andf r (jjp i qjj;p i ) be the left and the right sensing performance functions respectively of thei-th camera located atp i . We assume that bothf l andf r have the empirical form described in Equation 6.6: f l (jjp i qjj;p i ) = k 1l (p i ) k 2l (p i ) +jjp i qjj 2 (6.7) f r (jjp i qjj;p i ) = k 1r (p i ) k 2r (p i ) +jjp i qjj 2 (6.8) Because the problem is 1D, the region of dominance of any sensor corresponds to a segment on the loop. We further assume that the region of dominance of thei-sensor strictly belongs to the interval (p i1 ;p i+1 ). In other words, there are no abrupt changes in the sensing behavior over the domain such that each sensor contributes to the net utility function. Let [l i ;r i ] be the dominance range of thei-th sensor. Thus,l i 2 (p i1 ;p i ) andr i 2 (p i ;p i+1 ). According to the definition of the dominance region, f l (jjp i l i jj;p i ) =f r (jjp i1 l i jj;p i1 ) and, f r (jjp i r i jj;p i ) =f l (jjp i+1 r i jj;p i+1 ): 102 Using the above notation, the net utility function in 6.2 assuming uniform density function ((q) = 1) can be expressed in the following form for the case of 1D loop. H(p 1 ;p 2 ; ;p n ) = n X i=1 f Z p i l i f l (jjp i qjj;p i )dq + Z r i p i f r (jjp i qjj;p i )g (6.9) 6.3.3 Simulation Results Simulation Environment We ran simulations in MATLAB to test the convergence and resulting coverage of our distributed control law. The path is a 1D loop that can have sharp corners and obstacles that cause the sensing performance function to vary abruptly. The sensors start at randomly chosen initial positions and in each iteration of the algorithm, they update their position according to Equation 6.5 using a constrained optimization routine in MATLAB. We assume that sensors are localized and adjacent sensors on the path can communicate with each other. The simulation converges when the no sensor changes its position. We test for different models ofk 1 (x) andk 2 (x), different shapes of the path, and different number of sensors. Global knowledge of sensor performance function We first consider the ideal scenario where sensors have perfect knowledge of the sensing perfor- mance function. In practice, the sensors will estimate the performance function based on observed data the performance of the coverage algorithm will depend on the accuracy of this estimation. 103 1(a) 1(b) 1(c) 2(a) 2(b) 2(c) 3(a) 3(b) 3(c) Utility Figure 6.7: Performance of the distributed control law for three types of parameter functions on a 1D loop which is a 250 feet long smooth curve. 1(a), 2(a) and 3(a) show the parameter values constant, sine and step. 1(b), 2(b) and 3(b) show the corresponding final positions of sensors (red circles) superimposed on the k 1 function. The crosses show the dominance region boundaries between sensors. 1(c), 2(c) and 3(c) show the variation in the net coverage utility. The utility increases monotonically with each iteration. 104 (a) (b) Figure 6.8: Variation in coverage utility and convergence time for different network sizes aver- aged over 50 iterations. 1(a) 1(b) 1(c) 2(a) 2(b) 2(c) Figure 6.9: Performance of the distributed control law in the presence of occlusions - (1) obstacles and (2) sharp corners. 105 Figure 6.7 shows the performance of the algorithm for three different sensing performance func- tions. In each case, the sensors start at random initial locations. As expected, for constant pa- rameter values (Figure 6.7.1(a)), the sensors converge to a uniform distribution(Figure 6.7.1(b)). When the parameters have significant variations (Figure 6.7.2(a), 6.7.3(a)), the distribution if non-uniform (Figure 6.7.2(b), 6.7.3(b)). Sensors try to move to positions where the value ofk 1 is large andk 2 is small so that their local utility is maximized. Note that in each case the global cov- erage utility increases monotonically even though the position updates at each sensor are local. This is in agreement with Theorem 4:2. To study the impact of the network size on the algorithm performance, varied the network size from 5 to 100. For each network size, we repeated the sim- ulation 50 times starting with a different random initial position each time. The coverage utility increases exponentially and eventually saturates (Figure 6.8(a)) since as the number of sensors increases, the marginal utility of each additional sensor decreases and tends to zero. The varia- tion in utility is very small indicating that the initial locations do not impact the final coverage. The convergence time initially increases with the number of sensors and then stops increasing (Figure 6.8(b)) proving that the algorithm is scalable. The large deviations indicate that unlike coverage, the convergence time varies significantly with the initial sensor locations. 106 Occlusions Occlusions can occur due to obstacles in the path or sharp corners. Since the camera cannot sense beyond an occlusion, it cause an abrupt drop in the sensing performance function. Equations 6.7, 6.8 can be modified as follows to account for occlusions. f l (jjp i qjj;p i ) = 8 > > > > < > > > > : k 1l (p i ) k 2l (p i )+jjp i qjj 2 ifq is not occluded fromp i 0; otherwise (6.10) f r (jjp i qjj;p i ) = 8 > > > > < > > > > : k 1r (p i ) k 2r (p i )+jjp i qjj 2 ifq is not occluded fromp i 0; otherwise (6.11) Because the sensing function takes on a constant value for points beyond an occlusion, it is possible that the equationsf l (jjp i l i jj;p i ) = f r (jjp i1 l i jj;p i1 ) and, f r (jjp i r i jj;p i ) = f l (jjp i+1 r i jj;p i+1 ) that define the dominance region have many solutions. This will happen, for example, when adjacent sensors have two or more obstacles between them. For such cases, we define the dominance region boundary to be the position of its closest occlusion i.e., if the value of the sensing function of a camera is zero at a point q then q is considered not covered by the camera and will not belong to its dominance region. Notice that with this definition, the dominance regions may not form a partition of the path and there could be segments of the path that do not belong to any sensor’s dominance region. However, since these segments have zero coverage utility for all the sensors, and the rest of the path is partitioned according to the V oronoi 107 property, this definition of dominance regions is an optimal assignment of points on the path to sensors. With these new definitions for performance function and dominance regions, we can apply the control law in equation 6.5 and Theorem 4:2 guarantees convergence. However, with this control law, sensors adjacent to occlusions will not cross the occlusions because they only pick new positions within their dominance regions. To further improve the coverage utility, we intro- duce an additional switching step where sensors can cross obstacles if the local utility gained by doing so is more than the utility gain of the optimal location within its dominance region. While computing the switching utility, we account for change in the dominance regions and the utility of the neighboring sensors. Note that the local utility of every other sensor remains unchanged. Figure 6.9.1 shows the result of the algorithm on a loop of length 250 feet with 3 obstacles. The parameters vary as sine functions. Figure 6.9.1(b) shows the positions of the obstacles (black squares) and final positions of the sensors (red circles). Figure 6.9.2 shows an example on a rectangular path with 4 sharp corners that cause occlusions. In both cases, the utility increases monotonically with each iteration and the algorithm converges. As expected the coverage ob- tained here is less than Figure 6.7.2(a) where there are no occlusions. We can generalize this approach to deal with more complex pathways with multiple junctions so at a sensor adjacent to a junction reasons about the coverage gain in each of the branches that meet at the junction. Online estimation of sensor performance function In the coverage algorithm in the previous section we assumed that each sensor has global knowl- edge of the sensing performance functionf(jjp i qjj;p i ). This permits a sensor to take the best 108 (a) (b) (c) Figure 6.10: Performance of the distributed control law with partial knowledge of k 1 (x) and k 2 (x). step according to Equation 6.5. However in practice this is not possible. Specifically, in case of camera sensors, functionsk 1 (x) andk 2 (x) are not globally known. This necessitates modifica- tion of the current coverage algorithm. We assume that at every step of the algorithm, a camera learns its sensing performance function, i.e., the empirical values ofk 1 andk 2 . In the simulations shown in Figure 6.10, each sensor builds piecewise linear models of k 1 andk 2 based on data from its neighboring sensors and its current data. To improve the estimation in the first iteration, we assume that sensors share data over multiple hops. We observe that the utility sometimes decreases because of erroneous estimation (Figure 6.10(c)). But even with this very simple estimation, the coverage utility is close to the case when global knowledge was assumed. 6.3.4 Cyclops Experiments and Results We conducted a series of controlled experiments with cyclops cameras to gather several images in two different environmental settings – indoor and outdoor. We used interpolation techniques 109 (a) indoor (b) outdoor Figure 6.11: Cyclops setup on the gathered data to reconstruct the sensing performance functions in these settings and tested the performance of our coverage algorithm for these pseudo-real functions. 1) a rectangular corridor in an building with controlled lighting (figure 6.11(a)) a rectangular pathway surrounded by trees with uncontrolled lighting (figure 6.11(b)). For each of these set- tings, we picked sample positions for the camera and for each location, we estimated the sensing performance function (k 1 andk 2 ) by placing a target at different distances from the camera. Indoor experiments Corridors of buildings provide an ideal playground for testing our coverage algorithms with cy- clops cameras. We performed experiments under controlled lighting conditions in a rectangular corridor of which the central axis rectangle is of size 90ft 30ft. The corridor is 5ft wide and Figure 6.11(a) shows a cyclops camera set up along the length of the corridor. At every 20ft along the lengthwise central axis of the corridor, we placed a cyclops camera and captured on 110 both sides the images of the corridor which serve as background models. We also captured im- ages of a white or black board of dimension 3ft 2ft placed along the central line kept at every 5ft from the camera up to 25ft. These images serve as foreground models. Since the width of the corridor is 30ft, along the widthwise central axis, we placed a camera at the middle point and captured images on both sides in the similar fashion. We used these images to fit the sensing performance function to obtain the values of k 1 and k 2 at various locations. Along the entire corridor, we obtained 22 pairs of values ofk 1 andk 2 . We interpolate spline functions through these values ofk 1 andk 2 to estimate the pseudo-real global sensing performance function. Fig- ure 6.12(a) showsk 1 (x) andk 2 (x) along the entire length of the corridor. We use this form to test our coverage algorithm. In our simulations, we placedk sensors initially uniformly randomly. simulations results - comment about number of sensors, utility, global knowledge / vs online estimation, convergence Outdoor experiments We conducted a similar experiment as in indoor case around an outdoor rectangular pathway surrounded by trees without any control over lighting conditions. The pathway was of sizeab and Figure 6.11(b) shows a picture of a cyclops camera set up along the pathway. We captured background images and foreground images atn locations around the pathway in both directions. We observed that the lighting conditions in the outdoor setting change rapidly and because of this effect we collected data at relatively a few spots. Based on these images, we estimated the values ofk 1 andk 2 at the sampling points. We interpolate spline functions through these values 111 (a) (b) (c) (d) Figure 6.12: Performance of the distributed control law in an indoor rectangular corridor. (a) Sensor model parameters for left facing camera, (b) Sensor model parameters for right facing camera, (c) final locations (d) utility 112 (a) (b) (c) (d) Figure 6.13: Performance of the distributed control law in an outdoor environment. (a) Sensor model parameters for left facing camera, (b) Sensor model parameters for right facing camera, (c) final locations (d) utility 113 to obtain the sensing performance function over the domain and Figure 6.13(a) shows this model. We use this model to test our coverage algorithm. Dynamic Environments All the experiments we conducted involved manual positioning of the cameras and the test boards and was very time consuming. Unlike the indoor environments, the lighting conditions in outdoor environments change rapidly relative the time scale of collecting all the measurements. However we believe that with some degree of automation, our coverage algorithm can easily implemented in dynamic environments that demand frequent reconfiguration of the sensor positions. In partic- ular, we imagine that the cyclops cameras and the testing board be mounted on mobile platforms that will have localization ability. The images can be processed on board to estimate the sensing performance function at the current location. By means of wireless communication the sensors can implement distributed coverage algorithm. We restricted our work to a simple pixel detection algorithm. It is possible to implement other kinds of algorithms to improve the local sensing performance functions. But our coverage algorithm still holds in this scenario. 6.4 Conclusions We addressed the coverage problem where the performance of each sensor depends on its loca- tion. We formulate the coverage problem as a locational optimization problem and propose a distributed algorithm with guaranteed convergence. We applied our algorithm to the problem of covering indoor and outdoor pathways using Cyclops camera networks. Based on several mea- surements we proposed an empirical function for the sensing performance of the Cyclops camera. 114 We presented simulation results for various scenarios and demonstrated that our algorithm con- verges. We also presented simulation results on the data collected at fine scales in indoor and outdoor settings. As a part of future work we planning to apply our algorithm in 2D situations. 115 Chapter 7 Conclusions and Future Work The central thesis of this work is that “Mobility can be effectively used for topology control of robot networks. In the following we summarize the key contributions and discuss directions for future work. 1. Network Geometry: Local conditions that affect global topological properties aid scala- bility and can drastically reduce the complexity of robot controllers. Recently, it has been shown that for a disk graph in a two-dimensional plane, if each node has one neighbor in every < sector then the network is guaranteed to be 1-connected [Xue and Kumar, 2006]. We extended this result to show that < 2 k < guaranteesk-edge-connectivity. Surprisingly, the condition fork-connectivity applies to any general (non-disk) graph in a two-dimensional plane provided the graph is 1-connected. This implies that the condition only depends on the angles between the neighbors of each node and holds for arbitrary edge-lengths. This feature is particularly relevant for wireless networks using low-power radios that have irregular communication range. In re- lated work not shown here, we have also showed that for disk graphs, a weaker local information, 116 cyclic ordering of neighbors, is sufficient to solve the problem of computing a loop of nodes that encloses a given node. The results described above are a significant advance since the properties they achieve were formerly widely assumed to require global information. For robot networks, this line of research holds promise of further expansion in the scale and complexity of achievable tasks. In future, one could investigate local conditions for a range of desirable global properties including vertex-connectivity, spanner properties, and coverage holes. It is of considerable theoretical and practical significance to characterize a class of network properties that can be achieved through local information and interactions. A related issue of interest is complexity, in terms of mini- mality: given a global property, what is the minimum local information (conditions) sufficient to compute (guarantee) it? Further, these questions need to be studied for general non-disk graphs to ensure the validity of results to real-world wireless networks. 2. 3D Networks: There is an increasing demand for deploying robot networks in three dimensions in urban structures, underwater and aerial settings. Network configuration in such deployments is challenging because geometry in three dimensions is inherently more complex. Some of the algorithms developed for two-dimensional networks will directly generalize to three dimensions, many will require additional computational complexity and a few will not generalize at all. It has been proposed that sector-based configuration algorithms can be extended to cone- based algorithms in three dimensions [Bahramgiri et al., 2006]. We developed a novel, efficient algorithm based on spherical Delaunay triangulation to find the largest empty cone around any node. This algorithm has a run time ofO(dlogd) at each node of degreed, compared to the earlier 117 best ofO(d 3 logd) [Bahramgiri et al., 2006] and crucially, serves as a building block to extend other sector-based algorithms. Three-dimensional wireless networks are just beginning to receive attention and consider- able research efforts are required to develop practical and efficient algorithms. Our simulation experiments indicate that the cone-based network configuration algorithms, while being correct, are inefficient and result in overly dense topologies. It is possible that a different structure, such as the spherical triangle, will lead to more natural and efficient extensions of the sector-based conditions. We can explore this problem using tools from Spherical and Projective Geometry. Currently, there is also a lack of statistical models for wireless links in three-dimensions. To address this need, we plan to gather link data for a large set of configurations of a pair of receiver- transmitter nodes in various environments. 3. Communication sensitive control: While it is a necessity for coordination, maintain- ing connectivity in a large group of moving robots is challenging due to irregular and unreliable wireless links. We designed a virtual potential fields based switched robot controller which uses a combination of repulsive and attractive forces to maximize sensing coverage while maintaining constraints on link lengths. This is an example of a commonly used conservative approach in which only constraint-preserving movements are allowed, sometimes at the cost of performance. In some applications, it might be more efficient to allow the robots to intermittently break away and regain connectivity as required. We studied the problem of communication recovery, defining a behavior called Coalescence, where robots search for each other and form a single connected component. By modeling the search strategy as a random walk and the communication spread 118 of the connected component as a random shape, we showed that the worst case expected Coales- cence Time decreases asO(log(N)=N) with the sizeN of the network. The two controllers described above lie at the extremes of the classic exploitation versus ex- ploration tradeoff. Efficient controllers will be those that can achieve a balance, depending on the task at hand and the relative cost of maintaining and regaining connectivity. Future work includes establishing design principles for a range of such controllers based on relaxations of connectivity constraints, such as the time-lapse allowed. 4. Data-driven sampling and coverage: In many applications, the sensing utility of a robot depends on (i) the distribution of phenomenon of interest, and (ii) physical properties of the environment that affect the sensor’s performance, and is in general be a function of the robot’s po- sition. In such cases, a priori computation of robot placement can lead to poor performance. We proposed a reconfiguration algorithm for coverage optimization of a robot network with location- dependent utility. The robots iteratively estimate the local utility function based on their current measurements, and perform a gradient descent based control. This simple, greedy controller is guaranteed to converge to a local optimum. We evaluated the performance of the controller on a network of low-resolution Cyclops cameras for intruder detection along a loop in a complex outdoor environment where they exhibit significant variation in sensing range because of lighting conditions. Given such location-dependent utility functions, the problem of optimizing coverage while maintaining global connectivity constraints has not been addressed. 119 Bibliography A. Aggarwal. The art gallery problem: Its variations, applications, and algorithmic aspects. PhD thesis, Johns Hopkins University, Baltimore, MD, 1984. D. Aldous and J. A. Fill. Reversible markov chains and random walks on graphs. preprint available at http://www.stat.berkeley.edu/users/aldous/RWG/book.html. H. Ando, T. Oasa, J. Suzuki, and M. Yamashita. Distributed memoryless point convergence algo- rithm for mobile robots with limited visibility. IEEE Transactions on Robotics and Automation, pages 818–828, October 1999. M. Bahramgiri, M. Hajiaghayi, and V . S. Mirrokni. Fault-tolerant and 3-dimensional distributed topology control algorithms in wireless multi-hop networks. ACM/Kluwer Wireless Networks, 12(2):179–188, 2006. H. G. Banos and J. C. Latombe. Planning robot motions for range-image acquisition and auto- matic 3d model construction. In AAAI Fall Symposium Series, Orlando, Florida, USA, October 1998. Maxim Batalin and Gaurav S. Sukhatme. Spreading out: A local approach to multi-robot cover- age. In International Symposium on Distributed Autonomous Robotic Systems, pages 373–382, Fukuoka, Japan, June 2002. Maxim Batalin and Gaurav S. Sukhatme. The design and analysis of an efficient local algorithm for coverage and exploration based on sensor network deployment. IEEE Transactions on Robotics, 23(4):661–675, August 2007. Christian Bettstetter. Mobility modeling in wireless networks: categorization, smooth movement, and border effects. ACM SIGMOBILE Mobile Computing and Communications Review, 5(3): 55–66, 2001. P. Biswas, T/ Lian, T. Wang, and Y . Ye. Semidefinite programming based algorithms for sensor network localization. ACM Transactions on Sensor Networks, 2(2):188–220, May 2006. J. Brendin, E. Demaine, M. T. Hajiaghayi, and D. Rus. Deploying sensor nets with guaranteed capacity and fault tolerance. In ACM International symposium on Mobile ad hoc networking and computing (MobiHoc), Urbana-Champaign, IL, May 2005. A. Briggs and B. Donald. Visibility-based planning of sensor control strategies. Algorithmica, 26(3/4):364–388, 2000. 120 G. Caprari and R. Siegwart. Mobile micro-robots ready to use: Alice. In IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), Edmonton, Alberta, Canada, Aug 2005. A. Cerpa, J. L. Wong, L. Kuang, M. Potkonjak, and D. Estrin. Statistical model of lossy links in wireless sensor networks. In IEEE/ACM International Conference on Information Processing in Sensor Networks (IPSN), Los Angeles, California, USA, April 2005. V . Chvatal. A combinatorial theorem in plane geometry. Journal of Combinatorial Theory, 18: 39–41, 1975. P. I. Corke, S. E. Hrabar, R. Peterson, D. Rus, S. Saripalli, and G. S. Sukhatme. Autonomous deployment and repair of a sensor network using an unmanned aerial vehicle. In IEEE Inter- national Conference on Robotics and Automation, pages 3602–3609, April 2004. J. Cortes, S. Martinez, T. Karatas, and F. Bullo. Coverage control for mobile sensing networks. IEEE Transactions on Robotics and Automation, 20(2):243–255, 2004. J. Cortes, S. Martinez, and F. Bullo. Spatially-distributed coverage optimization and control with limited-range interactions. ESAIM. Control, Optimisation & Calculus of Variations, 11: 691–719, 2005. J. Cortes, S. Martinez, and F. Bullo. Robust rendezvous for mobile autonomous agents via prox- imity graphs in arbitrary dimensions. IEEE Transactions on Automatic Control, 51(8):1289– 1298, 2006. T. Danner and L. E. Kavraki. Randomized planning for short inspection paths. In IEEE Interna- tional Conference on Robotics and Automation (ICRA), San Francisco, California, USA, April 2000. K. Dantu, M. H. Rahimi, H. Shah, S. Babel, A. Dhariwal, and G. S. Sukhatme. Robomote: Enabling mobility in sensor networks. In IEEE/ACM International Conference on Information Processing in Sensor Networks (IPSN-SPOTS), pages 404–409, Los Angeles, California, USA, April 2005. V . de Silva, R. Ghrist, and A. Muhammad. Blind swarms for coverage in 2-D. In Robotics: Science and Systems, Cambridge, Massachusetts, USA, June 2005. R. Diestel. Graph Theory, Graduate texts in Mathematics, second edition, volume 173. Springer Verlag, New York, NY , 2000. Z. Drezner. Facility Location: A survey of applications and methods. Springer-Verlag, NY , 1995. Raissa M. D’Souza, D. Galvin, C. Moore, and D. Randall. Global connectivity from local geo- metric constraints for sensor networks with various wireless footprints. In ACM/IEEE Inter- national Conference on Information Processing in Sensor Networks (IPSN), Nashville, Ten- nessee, USA, April 2006. Q. Du, V . Faber, and M.Gunzburger. Centroidal voronoi tessellations: applications and algo- rithms. SIAM Review, 41(4):637–676, 1999. 121 P. Dykiel. Asymptotic properties of coalescing random walks. Technical Report 2005:15, Up- pasala University, December 2005. D. W. Gage. Command control for many-robot systems. Unmanned Systems, 10(4):28–34, Fall 1992. B. P. Gerkey, S. Thrun, and G. Gordon. Visibility-based pursuit-evasion with limited field of view. International Journal of Robotics Research, 25(4):299–316, April 2006. Brian P. Gerkey, Richard T. Vaughan, Kasper Stoy, Andrew Howard, Gaurav S. Sukhatme, and Maja J. Matari´ c. Most valuable player: A robot device server for distributed control. In IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), pages 1226– 1231, Maui, Hawaii, USA, October 2001. P. Hall. Introduction to the theory of coverage processes. Wiley Series, 1988. T Horprasert, D Harwood, and LS Davis. A robust background subtraction and shadow detection. In Asian Conference on Computer Vision, Taipei, Taiwan, Feb-March 2000. D. F. Hougen, S. Benjaafar, J. C. Bonney, J. R. Budenske, M. Dvorak, M. Gini, H. French, D. G. Krantz, P. Y . Li, F. Malver, B. Nelson, N. Papanikolopoulos, P. E. Rybski, S. A. Stoeter, R. V oyles, and K. B. Yesin. A miniature robotic system for reconnaissance and surveillance. In IEEE International Conference on Robotics and Automation (ICRA), pages 501–507, San Francisco, CA, USA, April 2000. A. Howard, M. J. Matari´ c, and G. S. Sukhatme. Mobile sensor network deployment using po- tential fields: A distributed, scalable solution to the area coverage problem. In International Symposium on Distributed Autonomous Robotic Systems (DARS), pages 299–308, Fukuoka, Japan, June 2002. T. Hsiang, E. M. Arkin, M. A. Bender, S. P. Fekete, and J. S. B. Mitchell. Algorithms for rapidly dispersing robot swarms in unknown environments. In International Workshop on Algorithmic Foundations of Robotics (WAFR), Nice, France, December 2002. E. H. Jennings and C. M. Okino. Topology control for efficient information dissemination in ad-hoc networks. In International Symposium on Performance Evaluation of Computer and Telecommunication Systems, San Diego, California, July 2002. A. Kansal, W. Kaiser, G. Pottie, M. Srivastava, and G. S. Sukhatme. Reconfiguration methods for mobile sensor networks. ACM Transactions on Sensor Networks, 3(4):22–23, 2007. G. Kao, T. Fevens, and J. Opatrny. Position-based routing on 3-d geometric graphs in mobile ad hoc networks. In Canadian Conference on Computational Geometry, pages 88–91, Ontario, Canada, August 2005. G. D. Kazazakis and A. A. Argyros. Fast positioning of limited-visibility guards for the inspection of 2d workspaces. In IEEE International Conference on Intelligent Robots and Systems (IROS), EPFL, Switzerland, October 2002. 122 O. Khatib. Real-time obstacle avoidance for manipulators and mobile robots. The International Journal of Robotics Research, 5(1):90–98, 1986. A. Krause, C. Guestrin, A. Gupta, and J. Kleinberg. Near-optimal sensor placements: maximizing information while minimizing communication cost. In ACM/IEEE International Conference on Information Processing in Sensor Networks (IPSN), Nashville, Tennessee, April 2006. A. Kroller, S. P. Fekete, D. Pfisterer, and S. Fischer. Deterministic boundary recognition and topology extraction for large sensor networks. In ACM-SIAM Symposium on Discrete algo- rithm (SODA), pages 1000–1009, Miami, Florida,USA, 2006. X-Y . Li, P-J. Wan, and Y . Wang. Power efficient and sparse spanner for wireless ad hoc net- works. In IEEE International Conference on Computer Communications and Networks (IC- CCN), pages 564–567, Scottsdale, Arizona, USA, Oct 2001. Xiang-Yang Li. Algorithmic, geometric and graphs issues in wireless networks. Wireless Com- munications and Mobile Computing, 3(2):119–140, 2003. J. Lin, A. S. Morse, and B. D. O. Anderson. The multi-agent rendezvous problem. In IEEE International Conference on Decision and Control (CDC), pages 1508–1513, Maui, Hawaii, USA, December 2003. Benyuan Liu, Peter Brass, Olivier Dousse, Philippe Nain, and Don Towsley. Mobility improves coverage of sensor networks. In ACM international symposium on Mobile ad hoc networking and computing (MobiHoc), pages 300–308, Urbana-Champaign, IL, USA, 2005. S. P. Llyod. Least squares quantization in pcm. IEEE Transactions on Information Theory, 28: 129–137, 1982. D. Lymberopoulos, Q. Lindsey, and A. Savvides. An empirical analysis of radio signal strength variability in ieee 802.15.4 networks using monopole antennas. In European Workshop on Sensor Networks, 2006. S. Megerian, F. Koushanfar, M. Potkonjak, and M. B. Srivastava. Worst and best-case coverage in sensor networks. IEEE Transactions on Mobile Computing, 04(1):84–92, Jan/Feb 2005. ISSN 1536–1233. N. Michael, J. Fink, and V . Kumar. Experimental testbed for large multi-robot teams: Verification and validation. IEEE Robotics and Automation Magazine, 15(1):53–61, March 2008. D Miorandi and E Altman. Coverage and connectivity of ad hoc networks in presence of chan- nel randomness. In Joint Conference of the IEEE Computer and Communications Societies (INFOCOM), pages 491–502, Miami, Florida, USA, March 2005. A. Muhammad and M. Egerstedt. On the structural complexity of multi-agent robot formations. In American Control Conference, pages 4957–4962, Boston, Massachusetts, USA, June-July 2004. 123 H. Na, C. Lee, and O. Cheong. V oronoi diagrams on the sphere. Computational Geometry Theory Applications, 23(2):183–194, 2002. A. Okabe and A. Suzuki. Locational optimization problems solved through voronoi diagrams. European Journal of Operations Research, 98(3):445–456, 1997. A. Okabe, B. Boots, and K. Sugihara. Spatial tessellations: Concepts and applications of voronoi diagrams. John Wiley & Sons Ltd., 1992. J. O’Rourke. Art gallery theorems and algorithms. Oxford University Press, 1987. M. Porfiri, D. J. Stilwell, E. M. Bollt, and J. D. Skufca. Stochastic synchronization over a moving neighborhood network. In American Control Conference, New York City, USA, July 2007. Mohammad Rahimi, Rick Baer, Obimdinachi Iroezi, Juan Garcia, Jay Warrior, Deborah Estrin, and Mani Srivastava. Cyclops: In situ image sensing and interpretation in wireless sensor networks. In ACM Conference on Embedded Networked Sensor Systems (SenSys), San Diego, California, USA, November 2005. Rajmohan Rajaraman. Topology control and routing in ad hoc networks: a survey. SIGACT News, 33(2):60–73, 2002. J. H. Reif and H. Wang. Social potential fields: A distributed behavioral control for autonomous robots. Robotics and Autonomous Systems, 27(3):171–194, 1999. A. Rowe, C. Rosenberg, and I. Nourbakhsh. A low cost embedded color vision system. In IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), EPFL, Switzer- land, October 2002. N. Roy and G. Dudek. Collaborative exploration and rendezvous: Algorithms, performance bounds and observations. Autonomous Robots, 11(2):117–136, September 2001. P. Santi. Topology control in wireless ad hoc and sensor networks. Wiley, 2005. M. Schwager, J. McLurkin, and D. Rus. Distributed coverage control with sensory feedback for networked robots. In Robotics: Science and Systems, Philadelphia, Pennsylvania, August 2006. M. Schwager, J. Slotine, and D. Rus. Decentralized, adaptive control for coverage with networked robots. In International Conference on Robotics and Automation, Rome, Italy, April 2007. H. Shi, D.Kruger, and J. Nickerson. Incorporating environmental information into underwater acoustic sensor coverage estimation in estuaries. In Military Communications Conference, MILCOM, October 2007. D. Spanos and R. M. Murray. Robust connectivity of networked vehicles. In IEEE International Conference on Decision and Control (CDC), pages 2893–2898, Atlantis, Bahamas, December 2004. 124 T. Spyropoulos, K. Psounis, and C. S. Raghavendra. Performance analysis of mobility-assisted routing. In ACM International symposium on Mobile ad hoc networking and computing (Mo- biHoc), pages 49–60, Florence, Italy, 2006. D. Tian and N. D. Georganas. Connectivity maintenance and coverage preservation in wireless sensor networks. Ad Hoc Networks, 2005. R. Wattenhofer and A. Zollinger. Xtc: A practical topology control algorithm for ad-hoc net- works. In International Workshop on Algorithms for Wireless, Mobile, Ad Hoc and Sensor Networks, 2004. R. Wattenhofer, L. Li, P. Bahl, and Y . M. Wang. A cone-based distributed topology control algorithm for wireless multi-hop networks. IEEE/ACM Transactions on Networking, 13(1): 147–159, Feb 2005. R. Williams. The geometrical foundation of natural structure: A source book of design. Dover Publications, Inc. New York, 1979. G. Xing, X. Wang, Y . Zhang, C. Lu, R. Pless, and C. Gill. Integrated coverage and connectiv- ity configuration for energy conservation in sensor networks. ACM Transactions on Sensor Networks, 1(1):36–72, Aug 2005. F. Xue and P. R. Kumar. On the theta-coverage and connectivity of large random networks. IEEE Transactions on Information Theory, 52(6):2289–2299, 2006. F. Xue and P. R. Kumar. The number of neighbors needed for connectivity of wireless networks. Wireless Networks, 10(2):169–181, 2004. M. M. Zavlanos and G. J. Pappas. Controlling connectivity of dynamic graphs. In IEEE In- ternational Conference on Decision and Control, pages 6388–6393, Seville, Spain, December 2005. B. Zhang and G. S. Sukhatme. Adaptive sampling for estimating a scalar field using a robotic boat and a sensor network. In IEEE International Conference on Robotics and Automation, pages 3295–3300, Rome, Italy, April 2007. H. Zhang and J. C. Hou. Maintaining sensing coverage and connectivity in large sensor networks. Ad Hoc & Sensor Wireless Networks, 1(1-2):89–123, Jan 2005. J. Zhao and R. Govindan. Understanding packet delivery performance in dense wireless sensor networks. In ACM Conference on Embedded Networked Sensor Systems (Sensys), pages 1–13, Los Angeles, California, November 2003. M. Zuniga and B. Krishnamachari. Analyzing the transitional region in low power wireless links. In IEEE International Conference on Sensor and Ad hoc Communications and Networks (SECON), pages 517–526, Santa Clara, California, October 2004. M. Zuniga, B. Krishnamachari, and R. Urgaonkar. Realistic wireless link quality model and generator. available online, Dec 2005. 125

Abstract (if available)

Abstract A fundamental problem that arises in realizing large-scale wireless networks of mobile robots is that of controlling the communication topology. This thesis makes three contributions to the area of mobility-based topology control in robot networks.

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Asset Metadata

Creator Poduri, Sameera (author)

Core Title Mobility-based topology control of robot networks

School Viterbi School of Engineering

Degree Doctor of Philosophy

Degree Program Computer Science

Publication Date 08/05/2008

Defense Date 05/06/2008

Publisher University of Southern California (original), University of Southern California. Libraries (digital)

Tag coverage optimization,deployment,OAI-PMH Harvest,robot networks,topology control

Language English

Contributor Electronically uploaded by the author (provenance)

Advisor Sukhatme, Gaurav S. (committee chair), Golubchik, Leana (committee member), Krishnamachari, Bhaskar (committee member)

Creator Email sameera.poduri@gmail.com,sameera@usc.edu

Permanent Link (DOI) https://doi.org/10.25549/usctheses-m1535

Unique identifier UC156644

Identifier etd-Poduri-2184 (filename),usctheses-m40 (legacy collection record id),usctheses-c127-115883 (legacy record id),usctheses-m1535 (legacy record id)

Legacy Identifier etd-Poduri-2184-0.pdf

Dmrecord 115883

Document Type Dissertation

Rights Poduri, Sameera

Type texts

Source University of Southern California (contributing entity), University of Southern California Dissertations and Theses (collection)

Repository Name Libraries, University of Southern California

Repository Location Los Angeles, California

Repository Email cisadmin@lib.usc.edu