Signal Strength Coordination for Cooperative Mapping

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1 Signal Strength Coordination or Cooperative Mapping Bryan J. Thibodeau Andrew H. Fagg Brian N. Levine Department o Computer Science University o Massachusetts Amherst {thibodea,agg,brian}@cs.umass.edu Abstract Many mobile robot tasks can be perormed in parallel. Multi-robot teams have the potential to complete the task more quickly than a single robot. Communication and coordination can prevent robots rom duplicating the eort o other robots, allowing the team to address the task more eiciently. In non-trivial environments, maintaining communication can be diicult due to the unpredictable nature o wireless signal propagation. We propose a multi-robot coordination method based on perceived wireless signal strength between cooperating robots or exploration in maze-like environments. The robot ormation will be determined by and coordinated using the signal strength between pairs o robots. This new method is tested and compared to an existing method that relies on preserving a clear line-o-sight between robots to maintain communication. I. INTRODUCTION Teams o cooperating robots have the potential to perorm many useul tasks or urban search and rescue, military reconnaissance, and planetary exploration. An important component o cooperation is communication between team members. For tasks where dierent portions can be accomplished in parallel, such as reconnaissance or exploration, a team o robots can complete the task in a shorter amount o time than a single robot. I a team o robots cooperates and shares inormation among its members, then the task can be addressed even more eiciently since robots can avoid duplicating the eort o other robots. An example o this is sharing map inormation so multiple robots do not explore the same area o an environment [,, 7]. However, robots can only exchange inormation when they are in communication range o each other. Maintaining wireless communication among a team o robots moving through an unknown environment is a undamental problem in multi-agent mobile robotics. The unpredictable and time-varying nature o signal propagation can make it diicult to determine i two robots will be able to communicate in the near uture. Previous work oten relies on conservative coordination methods that successully keep robots in communication with each other, but which can overconstrain the relative movement o the robots [, ]. These methods rely on maintaining a clear line-o-sight between communicating robots, which means that a line drawn between the two robots cannot intersect any other object. However wireless communication does not oten require a line-o-sight. Line-o-sight methods also require that the distance between Now at University o Oklahoma School o Computer Science communicating robots not exceed some maximum distance. We assume this distance is equal to twice the sensing radius o a robot, as this is the maximum separation or which two communicating robots can veriy that there are no obstacles present along the line-o-sight between them. This can cause the robot team to be over-constrained since the range o wireless communication is oten much greater than the range o sensors such as cameras or range inders and, unlike the sensors, wireless communication can propagate through obstacles. In this paper, we develop a coordination method to maintain communication between robots using the perceived wireless signal strength among robots. Our method allows a robot to address task objectives as long as the wireless signal strength among the robots remains above some threshold. When the signal strength drops below the threshold, one or more robots cease to address task objectives while they take action to increase the signal strength. Compared to coordination methods that require a line-o-sight between robots, our method allows or greater lexibility since robots are not restricted to conigurations where there are no obstacles between them. This greater lexibility leads to a large increase in task perormance. Using signal strength to coordinate teams o robots presents a number o challenges. Not only does signal strength behave stochastically, it also doesn t directly correspond to the spatial relationship between robots. The lack o a deterministic mapping rom signal strength to spatial relationship complicates repairing the network o robots when the signal strength between robots becomes too low. In simulation, we compare a method that relies on maintaining a line-o-sight between robots to our method which relies on maintaining suitable signal strength between robots. In some situations the average time to explore an environment using signal strength coordination is less than one third o the average time achieved using the line-o-sight based method. Using our proposed coordination method, robots are more prone to temporary loss o communication with their teammates than i they used a coordination method that enorces a line-o-sight constraint. Depending on task requirements, these temporary losses o connectivity may be acceptable given the increase in task perormance. II. RELATED WORK The use o robot teams to explore an initially unknown environment has been the subject o much work [,,, 5].

2 One o the challenges aced in this task is to maintain communication between members o a team. Exploration tasks can be completed more eiciently when robots share inormation with each other []. Communicating robots are also better able to address tasks requiring explicit coordination such as cooperative transport [6, ], where two or more robots must cooperate to move an object that cannot be moved by a single robot. The problem o maintaining wireless communication among a team o robots has been addressed by constraining robots to be within line-o-sight o each other [,, 7, ]. This method o coordination is very successul at maintaining communication between robots, but does not allow the robots to take advantage o the act that wireless signals (such as those used by consumer wireless networking products) typically do not require a direct line-o-sight between the transmitter and receiver. Thus, the robotic team may be overlyconstrained and not able to address its task as eiciently as possible. Clearly it is possible or all robots to share inormation even i every robot cannot communicate directly (i.e., in one hop) with every other robot in the team. To coordinate a team in this manner, some method is required to determine which pairs o robots must maintain direct communication with each other. Leader-ollower relationships between robots [, 5, 5,, ] are commonly used to coordinate robot teams. In exploration and ormation keeping tasks, when two robots are in a leaderollower relationship, typically the leader is ree to make progress towards task objectives, such as exploration, while the ollower is restricted to move within some area relative to the leader (e.g., the area in which it can communicate with the leader). Previous work on maintaining communication in a team o mobile robots exploits leader-ollower relationships to determine which robots need to maintain a clear line-o-sight to one another so they can communicate []. In this paper, we use leader-ollower relationships to determine which pairs o robots must maintain wireless links to each other. The leader-ollower style o coordination has traditionally assumed a ixed topology o robots. This assumption can be overly-restrictive: the breaking o a leader-ollower relationship does not necessarily mean that the network has become partitioned. Rather than trying to repair a lost leader-ollower relationship under conditions in which the network is still connected, we propose allowing a new leader-ollower relationship to orm. Such a luid topology approach enables the robots to expend less eort maintaining network connectivity. Sheng et al. () show that perormance in a multi-robot exploration task can be improved by increasing connectivity among members o a robot team []. In their work, robots share inormation whenever they are within a certain radius o each other, but no explicit attempt is made to maintain connectivity between members o the team. Sheng et al. () compare the case when the robots are biased toward exploring areas near other robots (in general improving connectivity) to the case where no such bias exists. They ound that the improved connectivity resulting rom such a bias decreases the amount o time required to explore an environment. Wagner and Arkin () propose an approach combining planning and reactive behavior to maintain communication in a team o robots perorming a reconnaissance task []. In this approach, they use plans designed by hand to help maintain communication in the team. Contingency plans are designed that can be used in the event that wireless communication is close to ailing or ails due to insuicient signal strength. Results are provided or teams o up to our robots utilizing various conigurations and control schemes. Hand designed plans allow or sophisticated strategies, but require a priori map knowledge, and thus are not suitable or exploration tasks in unknown environments, Powers and Balch () describe a method called Value- Based Communication Preservation or moving a team o robots to a goal location while maintaining communication among the team members [7]. Control decisions are based upon the perceived and predicted signal strength o communication with neighboring robots. It is assumed or the purpose o control that wireless communication is line-o-sight only. The results presented by Powers and Balch () demonstrate that it is easible to use the signal strength o wireless signals to maintain communication in a team o robots, but the potential beneits o non-line-o-sight communications is not considered. A. Task III. TASK AND ENVIRONMENT MODEL In this work, we address the task o cooperative mapping. The objective o the cooperative mapping task is to explore an initially unknown environment and to map all o the reachable obstacles and ree space in that environment. The map is represented in simulation by a discrete grid in which each square is marked reespace, obstacle, or unexplored. A grid square is marked as obstacle i there is an obstacle in any portion o the world represented by that grid square. Each grid square is m m. Even though the map is discretized, robots move continuously though the world. The task is complete when the environment has been explored; i.e., when all grid squares in the map that correspond to regions o the environment accessible rom the robots starting location are marked reespace or obstacle. In this work, we compare the perormance o signal strength coordination to line-o-sight coordination in the context o the cooperative mapping task. We observe both the task perormance and network connectivity o robot teams o varying sizes addressing the cooperative mapping task in a variety o environments. Since all o the experiments are carried out in simulation, we need to deine appropriate models or the environment, robots, and communication between the robots. B. Environments We perorm all experiments in a simulated m-by-m square environment. We use sparse and dense environments or experiments. The sparse environment has line-segment obstacles placed uniormly at random throughout the environment. The obstacles have a randomly chosen length uniormly distributed between m and m. With equal probability, obstacles are oriented parallel to one o the axes in the environment. The dense environment is similar except it contains

3 Fig.. Example o a sparse environment team members they can currently communicate. Competition or access to the network s physical medium is not modeled. When reerring to the strength o a signal, we do so in terms o the path loss that occurs between the transmitter and receiver. Path loss is the amount o power that a signal loses between the transmitter and the receiver measured in decibels (db). The path loss between a pair o robots depends upon the distance between the robots, the number o obstacles between them, and the properties (such as material or density) o the obstacles between them. We assume a link exists between two robots when the path loss between the robots is less than some parameter R. To simulate path loss in an environment with obstacles, we use the ollowing model rom Rappaport () [9]: ( ) d PL(d) = PL(d ) + log + αd + PAF i ; () d i Fig.. Example o a dense environment line-segment obstacles. Examples o a sparse and a dense environment are shown in Figures and. All robots are located in the lower let-hand corner o the environment at the beginning o an experiment. C. Inter-robot communication For our experiments conducted in simulation to have signiicance or real robots communicating wirelessly, we need to take into account the wireless signal propagation characteristics o real signals. It is very diicult to predict how a signal will propagate in an environment, especially i there is no line-o-sight path between the transmitter and the receiver. Thus, we must rely on extremely simpliied models o signal propagation. We make the ollowing assumptions about the wireless channel between robots. I the signal between two robots is suiciently strong, then a link exists. I a link exists, then the robots have enough bandwidth to exchange inormation regarding their position, and any new map inormation every simulation time step (once per second). We also assume that the robots can accurately measure the signal strength o any signal they receive and can determine with which o their where: d is the distance between the transmitter and the receiver; PL(d ) is the path loss in db at a small distance d rom the transmitter; ( ) d log d + αd is the path loss due to the distance the signal must travel in ree space to reach the receiver; α is a constant that depends on the type o environment in which the signal is traveling (i.e., oice building, warehouse, or outdoors); PAF i is the partition attenuation actor or the i th obstacle between the transmitter and receiver. The partition attenuation actor o an obstacle is the amount o power a signal loses (db) by passing through that obstacle and depends on the material and density o the obstacle. We state the parameters used in our simulation below. In the model o path loss given in Eq., the path loss smoothly decreases as the receiver moves away rom the transmitter, except at the boundaries o shadows cast by obstacles. Within these shadows the predicted path loss changes smoothly. In this respect, this model does not match the reality o signal propagation. In practice, when there is no line-osight between a transmitter and a receiver, path loss can change radically even or very small physical displacements that do not introduce or remove occlusions between the transmitter and receiver. Also, i no line-o-sight exists between the transmitter and the receiver, the path loss may vary signiicantly even or ixed positions o the transmitter and receiver i the environment is not completely static. These behaviors are due to the eects o multi-path propagation, where multiple copies o the transmitted signal reach the receiver at slightly dierent times and rom dierent directions [9]. Multi-path propagation occurs because objects in the environment relect and scatter the transmitted signal in ways that can be diicult to predict. When a line-o-sight exists between the transmitter and the receiver, the signal propagating along the line-o-sight tends to dominate any multi-path eects, and signal strength is much easier to predict. We modeled multi-path eects in our simulation by adding noise to the model in Eq. when the transmitter and receiver are not within line-o-sight o each other. Since

4 Fig.. The path loss (in db) or a transmitter located at the center o the environment as determined by q.. The environment shown is 88m 96m. multi-path eects are oten dominated by propagation along the line-o-sight, a very small amount o noise is added to Eq. when the transmitter and receiver are within line-osight o each other. This model will not necessarily predict signal strength luctuations accurately, rather it is intended to complicate the coordination o a communicating robot team in the same manner that actual multi-path eects would. D. Communication Parameters In order to simulate wireless communication, we use the ollowing parameters: d = m, PL(d ) = db, and α =.5. We chose these parameters by comparing the path loss predicted by the model at various distances to the path loss predicted at the same distances in the speciication or a common 8.b card (the parameters chosen were not empirically veriied) [8]. The model generated by these parameters does not match the speciication exactly or any environment type, rather it yields path losses that are between those given or semi-open and closed environments. In most experiments we assume that a signal passing through an obstacle loses 5dB o signal strength. This could be expected rom obstacles such as cardboard boxes, storage racks, or other similar objects [9]. Figure shows the path loss determined by the above model or a transmitter at the center o a sample environment. Lighter shades represent lower path loss and black grid squares contain obstacles. No noise was added in this case. Figure shows the result o adding noise equal to N, where N is a Gaussian random variable with µ = db and σ = db, to the value given by Eq. or grid squares not within line-o-sight o the center o the environment. Noise is added to the squares that are within line o sight o the center o the environment in a similar manner, except the Gaussian distribution has σ =.5dB. Two robots can communicate when the path loss between them is less than R = 8.5dB. Even though actual wireless communication hardware (such as 8.b equipment) can maintain a link when path loss exceeds 8.5dB, we chose this value as a upper limit on communication range to make communication maintenance suiciently diicult given the environments that can be reasonably simulated. I there are no obstacles obstructing the signal, a path loss o 8.5dB corresponds to a distance o about 5m. Fig.. The path loss (in db) or a transmitter located at the center o the environment. Here noise is added to the path loss determined by Eq.. The environment shown is 88m 96m. E. Ad hoc network We assume that the robots maintain an ad hoc network among themselves to the extent that the path loss between them permits. The simulation does not consider the details o such a network, but only determines which subsets o the robot team can currently communicate. The robot team is represented as a graph in which the edges represent path loss. Prim s algorithm [] is used to construct a minimum spanning tree o the set o robots. The number o partitions in the network can be determined by counting the number o links in the minimum spanning tree tree that have a path loss greater than R = 8.5dB. It is important to note that the minimum spanning tree does not necessarily represent the complete routing topology o the ad hoc network, rather it is used to determine the network connectivity. The robots also use the minimum spanning tree to determine the leaderollower relationships between robots. We provide the details o the minimum spanning tree construction in Section IV. F. Robot Model The robots are assumed to be holonomic point robots. We assume a vision or range inder sensor that can see S = 8 meters. I any part o a map grid square is observed, it is assumed that the entire contents o the grid square are observed. Thereore, each robot can detect obstacles and other robots within a range o 8m. This means that or two robots to be in line-o-sight they must be no more than 6m apart. The robots move at a constant speed o.5m per time step. A simulation time step is equal to second. The robots are localized and always know their current position in the world. IV. ALGORITHM In this section, we propose a simple coordination method to preserve communication in a team o robots addressing the cooperative mapping task. This coordination method utilizes leader-ollower relationships between robots. The leaderollower relationships are determined by a team topology that adapts based upon the path loss between team members. A robot s active controller is determined by the perceived signal strength between that robot and its leader. The purpose o this coordination method is to maintain communication in

5 a team o cooperating robots, while, relative to line-o-sight coordination, allowing the team more reedom to address task objectives such as mapping the environment. The team uses the ability to communicate with each other to complete the mapping task more eiciently. When robots can communicate with the team leader, they share map data with the team leader. This allows all team members able to communicate with the team leader to use the same map. When map data is shared, a robot will not unnecessarily explore a region that a teammate has already explored. A. Team Topology The task o coordinating the robotic team can be simpliied by using a team topology with the the ollowing properties. It should be the case that i every ollower is in direct communication with its leader, then the network o robots is connected. This simpliies the problem o global connectivity by reducing it to one o maintaining pairwise relationships between robots. It is also important that every robot have only one leader. I a robot has more than one leader, unless those leaders explicitly coordinate their actions, it may not be possible or the ollower to maintain communication with all o its leaders, which could cause the network to become partitioned. Another desired property o the team s topology is that there be a robot that is the team leader. The team leader has no leader (thus it is ree to address task objectives) and every other robot should ultimately be a ollower o the team leader (i.e., every robot is a ollower o the leader, or a ollower o a ollower o the team leader, and so on). The presence o a team leader helps ensure that the team will always be making some amount o progress toward task objectives. To meet these criteria, the robot team uses an adaptive topology based upon a minimum spanning tree that is updated every k simulation time steps. At the beginning o the task, one member o the team, robot r, is chosen as the team leader. r will be the team leader or the remainder o the task. To orm the topology or the team, a minimum spanning tree is built where each robot is a node, r is the root node, and the link costs between robots is the path loss o a signal between them. Recall that path loss is the amount o power a signal loses between the transmitter and receiver, and is determined in our simulation environment by the methods described in Section III-C. We use Prim s algorithm to build the minimum spanning tree []. The tree starts as just the team leader, r. At each step o the algorithm, we ind the robot, i, not in the tree that has the smallest minimum cost link to any robot already in the tree. Let the robot already in the tree with the minimum cost link to robot i be robot j. Robot i will be added to the tree by adding the link between robot i and robot j to the tree. Robot j will be robot i s leader. The above algorithm guarantees that every robot, except the team leader r, has exactly one leader, and that the leader relations propagate such that every robot in the team is ultimately ollowing the team leader. As discussed above, i all o the links in the minimum spanning tree have a cost less than R = 8.5dB, the network o robots will be connected. Thereore, i every ollower can communicate directly with its leader, then the minimum spanning tree is intact and the ad hoc network is connected. We have ound empirically that updating the topology every k = 5 simulation time steps works well since it prevents thrashing when the path loss between various pairs o robots is similar. B. Harmonic Path Planners We use a set o controllers based on harmonic path planners to implement the above coordination method. Harmonic path planners generate trajectories using a harmonic unction, which is a solution to Laplace s equation. Harmonic unctions generate an artiicial potential in the robot s coniguration space and have a number o properties that make them desirable as path planners. Steepest gradient descent o the artiicial potential generated by a harmonic unction results in the minimum hitting probability path to a goal location. Harmonic unctions are resolution complete and ree o local minima [9, 8]. In this work, a robot s coniguration consists o its coordinates in a planar world. For the purposes o computing harmonic unctions, we represent coniguration space as a discrete grid where every grid square is designated as reespace, goal, or obstacle. Steepest descent o the harmonic potential in this space is guaranteed to result in trajectories that avoid all points designated as obstacle and eventually reach one o the grid squares designated as goal. Successive over relaxation is used to compute the potentials at each grid square, and bilinear interpolation is then used to compute the gradient at the robot s location [9]. Due to issues with numerical precision, in rare cases the gradient o a harmonic unction cannot be determined in some portions o the coniguration space. This can occur or regions o space that are very ar rom goals. When a robot cannot determine the local gradient o the harmonic unction, it relies on the NF navigation unction [] to determine the direction o motion. The NF unction computes a gradient based on the Manhattan distance rom a grid square in coniguration space to the nearest goal in coniguration space. C. Controllers Controllers using harmonic path planners are used to generate the dierent robot behavior necessary or completing the cooperative mapping task and or maintaining communication. We use two controllers to generate motions or our robots: one that moves a robot into a region where it is in line-o-sight o another robot; and a second that causes a robot to move toward unexplored areas o the environment. Both o these controllers use a harmonic path planner as described above and dier only in how they deine goals in coniguration space. We describe controllers using the notation φ g i, where: φ is an artiicial potential; g is sensory inormation used to determine the shape o φ; i is a set o eectors used to descend φ. g may reer to sensory inormation at any level o abstraction. We use sensory abstractions at the level o coniguration space

6 maps or speciic objectives. Harmonic unctions are used to generate the artiicial potential φ, or in cases where the local gradient o the harmonic unction cannot be determined, φ is determined using the NF unction. The eectors always consist o single robots. The controllers are similar to those described by Sweeney, et al. (,) [, ]. Robot r uses the controller φ EXPr r or exploration. The sensory abstraction EXP r marks all unobserved grid squares as goal, all observed grid squares containing obstacles as obstacle, and all other observed squares as reespace. A grid square is considered observed i robot r directly senses the grid square itsel or was inormed about the contents o that grid square by another robot. The boundaries o the coniguration space are always designated as obstacle. φ EXPr r will generate trajectories that avoid obstacles and move the robot toward unobserved areas o the world. We use the controller, φ i j, to bring robot j to a location where it is in line-o-sight o robot i (i j). The sensory abstraction i marks all known obstacles (and the boundaries o the coniguration space) as obstacle. Grid squares that are within some distance S o robot i, do not contain an obstacle, and are within line-o-sight o robot i are marked as goal. Thus, φ i j moves robot j toward the region o space within line-o-sight o robot i; i robot i is stationary, robot j is guaranteed to reach this region o space. D. Coordination Methods To achieve the desired robot behavior using the above controllers, we combine multiple controllers using a technique inspired by null space control. In systems with excess degrees o reedom, subordinate tasks can be addressed in the null space o superior tasks. Thus, one can guarantee that subordinate tasks will not aect the perormance o superior tasks. In general, null space control allows multiple goals to be addressed concurrently. When tasks are deined using n-dimensional artiicial potentials, a unique (one-dimensional) gradient direction can be computed locally. The n orthogonal subset o the potential maniold describes the null space o the potential ield a space in which subordinate actions do not alter the potential underlying the superior controller. Using this notion o a null space we can create compositions o controllers where the actions o subordinate controllers do not eect the progress o superior controllers [, 6]. The subject-to operator [] is used to combine the actions o disparate controllers. For controllers φ α and φ β, φ β φ α (read φ β subject-to φ α ) means that the actions o φ β are projected onto the equipotential maniold o controller φ α s artiicial potential. Thus, the actions generated by φ β do not interact destructively with the progress o φ α toward its minimum. In this work, we compose controllers in a way that approximates null space control. In particular, we consider systems that must preserve the equilibrium status o primary controllers while addressing secondary gradients. We use φ β φ α to mean that when the system is in a goal state o φ α, φ β is used to generate motion commands. When the system is not in a goal state o φ α, then φ α is used to generate motion commands R Fig. 5. S is the distance (in meters) in which a robot can see obstacles or other robots; R is the range (in db path loss) o wireless communication between robots; τ (in db path loss) is the threshold at which a ollower robot activates φ l when using signal strength control. Note that path loss cannot be directly translated to meters without taking into account speciic environmental eatures, but τ is chosen such that in almost all situations τ will correspond to a urther distance in meters than S. exclusively. This method o control composition requires that superior goals have been met beore subordinate goals are addressed and allows the subordinate controller to disturb the superior controller within bounds. We use the parameter τ deined below to determine the bound on disturbances. Leader-Follower Relationship or Line o Sight Coordination: Under line-o-sight coordination, robot uses φ EXP φ l, where robot l is robot s leader. This means that robot will explore the environment as long as it is within line-o-sight o robot l. When robot is not within line-o-sight o robot l, robot uses φ l to move toward the region where it would be within line-o-sight o robot l. For the team leader, is undeined (because the team leader has no leader), φ l and φ EXP is always used or control. Leader-Follower Relationship or Signal Strength Coordination: For signal strength coordination, we will need to deine one more controller. Let φ SIG l be a controller that is the same as φ l, except the goal region generated by SIG l is all points in coniguration space where the path loss to robot l is less than some threshold τ. τ is always less than R, the maximum path loss at which communication is still possible. For signal strength coordination, robot uses φ EXP φ SIG l or control. This means that robot will explore the environment as long as the path loss to robot s leader, robot l, is less than τ. Otherwise, robot will move toward the region o space where the path loss to robot l is less than τ. As discussed above, it is very diicult to predict the path loss or arbitrary locations o a transmitter and receiver. Thereore, it is not easible to compute the conigurations o robot where the path loss rom robot l to robot is less than τ. However, it is still possible or robot to directly sense (by measuring the strength o the signal rom robot l) whether it is in a goal state o φ SIG l τ S. Because we cannot

7 compute the goal set o φ SIG l, when robot is not in a goal set o φ SIG l (i.e., when the path loss between robots and l is greater than τ), φ l will be used or control. Note that φ l is not guaranteed to monotonically decrease the path loss between robots l and. However, as long as τ is greater than the path loss or an unobstructed signal traveling distance S (the maximum separation o two robots that are within lineo-sight), the path loss between robots l and will be less than τ or all conigurations in the goal set o φ l is a conservative approximation o φ SIG l decrease the path loss between robots l and.. Thus, φ l, and will in general It is possible or either o the coordination methods to ail to keep every leader-ollower pair in contact. When a ollower loses contact with its leader, the ollower moves toward the position the leader was at when communication was last possible. The only other robot whose behavior changes is the team leader, which immediately stops whenever the network o robots becomes partitioned. By having the team leader remain in place, we can guarantee that communication with the team will eventually be restored. I the robot that lost communication with its leader reaches the last known position o its leader without reestablishing contact with any team member, it will then move to the position o the team leader, which hasn t moved since communication was lost. Since we assume the environment is static, that all o the robots are localized, and that movement is error ree, the disconnected robot will eventually establish communication with the team leader i it does not encounter any other member o the team irst. Even though we guarantee that network partitions will always be temporary, the progress o the search task can be adversely eected by partitions in the network. Since the group leader stops whenever there is a partition, the area o the environment that is reachable by the group is limited until all partitions have been repaired. Also, since map inormation is not shared across partitions, the robots that are not connected to the group leader s partition may not act eiciently since they lack the map knowledge that other robots have discovered while the partition exists. V. RESULTS In this section, we present results demonstrating the perormance o the signal strength coordination method in a variety o conditions and compare the perormance o the signal strength coordination method to the perormance o the lineo-sight coordination method. The experiments are designed to test the scalability o the coordination methods, the robustness o the coordination methods to various environmental actors, and the sensitivity o signal strength coordination to algorithmic parameters. The adaptive topology is also compared to a number o ixed topologies to empirically veriy its perormance. Experiments were perormed with teams o,, 8, 6, and sometimes robots in both the sparse and dense environments. We used the same 5 randomly generated instances o each environment or every experiment. The PAF o the obstacles in the environment are varied depending on Partition Partitions Partitions Partitions SIG R= SIG R= SIG R= SIG R= SIG R= Fig. 6. Mean number o time steps to map sparse environment versus team size. Each bar indicates the amount o time spent by the network in,,, or partitions. indicates the line-o-sight coordination method, SIG denotes the signal strength based coordination method, and R= denotes the case where communication range is assumed to be ininite and signal strength based coordination is used. The error bars represent one standard deviation, and the numbers to the upper right o each bar indicate the maximum number o network partitions present at one time in any o the 5 trials represented by the bar. the experiment. In the ollowing graphs, each bar represents the average o 5 trials, one in each instance o the appropriate environment type. We ound that in most environment, the perormance o the signal strength coordination method compared avorably with that o the line-o-sight coordination method. Furthermore, the signal strength coordination method does not appear to be extremely sensitive to algorithmic parameters. We also ound that a luid topology was beneicial to both the lineo-sight and signal strength coordination methods. A. Number o Robots We irst conducted experiments to determine how team size eects the perormance o teams using either the line-o-sight or signal strength coordination methods. This is a crucial metric or algorithms designed or robot teams, since a good coordination method or a team o robots should make eicient use o all members o the team. The average time to ully search the environment using both the line-o-sight and signal strength coordination methods is shown in Figures 6 and 7. For signal strength coordination we set τ = db. The bars labeled correspond to lineo-sight coordination, and the bars labeled SIG correspond to signal strength coordination. The bars labeled R= correspond to the case where signal strength coordination is used and communication range is unlimited (τ is also set to ininity in this case). This case is included to provide a lower bound on the search time. The shading o each bar represents the number o partitions in the network. For example, or sixteen robots using signal strength coordination in the dense environment, the network has one partition (i.e., it is ully connected) or about 77 time steps; or about 95 time steps there were two partitions;

8 Partition Partitions Partitions Partitions 5 Partitions 6 Partitions 7 Partitions SIG R= SIG R= SIG R= SIG R= SIG R= Fig. 7. Mean number o time steps to map dense environment versus team size. Each bar indicates the amount o time spent by the network in,,,, 5, or 6 partitions. indicates the line-o-sight coordination method, SIG denotes the signal strength based coordination method, and R= denotes the case where communication range is assumed to be ininite and signal strength based coordination is used. The error bars represent one standard deviation, and the numbers to the upper right o each bar indicate the maximum number o network partitions present at one time in any o the 5 trials represented by the bar. or roughly 8 time steps there were three partitions; and or less than time steps there were our, ive, six, or seven partitions in the network. The total number o time steps in which the group contained more than three partitions is so low that the corresponding regions in the graph are not visible. The maximum number o partitions that occurred during any o the 5 trials is indicated by the number at the upper right o the bar. The results show that the signal strength coordination method outperorms the line-o-sight coordination method in every case, but line-o-sight coordination is more successul at keeping the network connected. This is to be expected since the signal strength coordination method allows the robots to spread out more than the line-o-sight coordination method does (even though both methods are subject to the same networking constraints), increasing the amount o the environment that can be covered in parallel, but also increasing the chance that the network connecting the robots will become partitioned. The search times are lower in general or the dense environment since it has a smaller area to be searched than the sparse environment. Our results indicate that signal strength coordination beneits much more rom additional robots than line-o-sight coordination does. When line-o-sight coordination is used, teams o robots complete the search task approximately.-. times quicker than teams o two robots. When signal strength coordination is used, teams o robots complete the search task approximately times quicker than teams o two robots. The signal strength coordination method makes better use o additional robots since it allows robots to disperse urther, increasing the likelihood that a robot is observing a part o the environment that has not been observed by another 7 robot. Also, these results demonstrate that it is more diicult to maintain a connected network in the dense environment due to the additional obstacles. The additional obstacles will tend to increase the chance that a motion could cause a large change in path loss (by introducing one or more obstacles between the transmitter and receiver) which makes it more diicult or the network to remain connected. The cases where the communication range is assumed to be ininite (R= ) provide an upper bound on the perormance o any coordination method given the particular search strategy employed. Figures 6 and 7 show that as the number o robots increases (and particularly or the cases where n = 6 or ) the perormance o signal strength coordination approaches the perormance achieved when the communication range is assumed to be ininite. The dierence in perormance between signal strength coordination and the R= case is statistically insigniicant (p =.58 using a paired t-test) or robots in the sparse environment. For every other case, the dierence in perormance between signal strength coordination and the R= case is statistically signiicant. No method or maintaining communication (that doesn t involve changing the search strategy) should be able to improve over the case where the communication range is ininite. Another interesting eect that can be seen in Figures 6 and 7 is that the network connectivity seems to worsen as n increases or n 8, but or n 8, the network connectivity appears to improve. We attribute this eect to the teams o 6 and robots saturating the environment. As the environment becomes saturated with robots, we would expect less change in search times as more robots are added since there is not enough room or the robots to spread out and cover independent portions o the environment. But we would expect the connectivity to continue improve as robots are added since a higher spatial density o nodes makes a network partition less likely to occur. B. Signal strength threshold The signal strength coordination method requires choosing a value or the threshold τ that determines when robots will switch rom the φ EXPr r controller to the φ i j controller. In order to evaluate the signal strength coordination method we need to ind both the optimum value or τ and how sensitive task perormance and network connectivity are to dierent choices o τ. The optimum value or τ in a given situation demonstrates the potential o the signal strength coordination method. I the method is to be used in practice, we need to know how sensitive perormance is to variations in τ as environmental characteristics change. I the sensitivity is too high then it may be diicult or a system designer to pick an appropriate value or τ, especially i the characteristics o the environment are unknown, which would reduce the utility o the method. Experiments were perormed in both the sparse and dense environments, varying the value o τ. Since communication is broken when the path loss exceeds 8.5dB, we only consider values o τ less than or equal to 8.5dB. Figures 8 and 9 show the search times in the sparse and dense environments, respectively. Figure shows the search

9 5 5 n= n= n=8 n=6 5 5 n= n= n=8 n= τ (db) Fig. 8. Mean number o time steps to map sparse environment versus τ (PAF = 5dB). τ is the threshold that determines when the controller φ l is used in the signal strength control method. Each line shows the search time or a speciic number o robots (,, 8, or 6) as τ increases. The letmost point o each line shows the perormance o the line-o-sight method or the corresponding number o robots τ (db) Fig.. Mean number o time steps to map dense environment versus τ (PAF = 5dB). τ is the threshold that determines when the controller φ l is used in the signal strength control method. Each line shows the search time or a speciic number o robots (,, 8, or 6) as τ increases. The letmost point o each line shows the perormance o the line-o-sight method or the corresponding number o robots. 5 5 n= n= n=8 n= τ (db) Fig. 9. Mean number o time steps to map dense environment versus τ (PAF = 5dB). τ is the threshold that determines when the controller φ l is used in the signal strength control method. Each line shows the search time or a speciic number o robots (,, 8, or 6) as τ increases. The letmost point o each line shows the perormance o the line-o-sight method or the corresponding number o robots. times in the dense environment with a PAF o 5dB. Each line in these graphs corresponds to a particular team size, and shows the change in search perormance as τ is increased. The letmost point o each line shows the perormance o the line-o-sight method. The graphs indicate that perormance is similar or a large range o possible values or τ. Values o τ that are too too low can adversely aect the task perormance. The results also suggest that values o τ that are too high can also adversely eect task perormance, especially i τ is equal to the path loss where a wireless link is broken. In the environments tested, τ can be set around 77dB with essentially no adverse eect on task perormance. This demonstrates that signal strength coordination does not require extensive parameter tuning in order to outperorm the line-osight method in the three environments tested. These experiments also demonstrate how the value o τ eects connectivity in the network o robots. Figures,, and display the time to map the environment or each team size and value o τ, broken down by the number o partitions in the network. We can see rom Figure 8 that τ has less o an eect on network connectivity in the sparse environment. The network is partitioned only slightly more oten as τ increases, which is expected since a higher value o τ allows the robots to disperse more. Figures and show that in the dense environment (with PAF = 5 or 5dB) the value o τ has a much greater eect on the network connectivity. This suggests that in denser environments, or environments where the obstacles have a higher PAF, network connectivity may be more sensitive to the value o τ than task perormance. Thus, in some cases, τ may need to be chosen more careully i continuous network connectivity is a high priority. C. Obstacle Composition The partition absorption actor, or PAF, o an obstacle determines the amount o path loss that occurs due to that obstacle being between the transmitter and receiver. So ar, we have mostly considered obstacles with a PAF o 5 or 5dB. In real environments, robot teams are likely to encounter obstacles with a large range o PAFs, since the PAF o an obstacle is dependent upon the obstacle s composition and thickness. For this set o experiments, the mapping task was perormed in the dense environment with a team o 8 robots with the PAF o every obstacle set to 5, 5, 5, 5, or db. This range o PAFs covers many dierent materials, including those that are opaque to wireless signals. As stated beore, a PAF o 5dB might be expected rom obstacles such as empty cardboard boxes or storage racks, whereas PAFs rom -5dB might be

10 Partition Partitions Partitions Partitions 5 Partitions 6 Partitions Fig.. Mean number o time steps to map sparse environment versus τ (PAF = 5)dB. τ is the threshold that determines when the controller φ l is used in the signal strength control method. The bars are grouped by number o robots, and each bar is labeled with its corresponding threshold value τ, or or line-o-sight control. The error bars represent one standard deviation, and the numbers to the upper right o each bar indicate the maximum number o network partitions present at one time in any o the 5 trials represented by the bar. The data used to produce this graph is the same as that used to produce Figure Partition Partitions Partitions Partitions 5 Partitions 6 Partitions Fig.. Mean number o time steps to map dense environment versus τ (PAF = 5)dB. τ is the threshold that determines when the controller φ l is used in the signal strength control method. The bars are grouped by number o robots, and each bar is labeled with its corresponding threshold value τ, or marked as or line-o-sight control. The error bars represent one standard deviation, and the numbers to the upper right o each bar indicate the maximum number o network partitions present at one time in any o the 5 trials represented by the bar. The data used to produce this graph is the same as that used to produce Figure Partition Partitions Partitions Partitions 5 Partitions 6 Partitions Fig.. Mean number o time steps to map dense environment versus τ (PAF = 5)dB. τ is the threshold that determines when the controller φ l is used in the signal strength control method. The bars are grouped by number o robots, and each bar is labeled with its corresponding threshold value τ, or marked as or line-o-sight control. The error bars represent one standard deviation, and the numbers to the upper right o each bar indicate the maximum number o network partitions present at one time in any o the 5 trials represented by the bar. The data used to produce this graph is the same as that used to produce Figure. expected rom concrete block walls or metal obstacles [9]. The results o these experiments are shown in Figure. The perormance o the line-o-sight coordination method is nearly constant or all o the values o the PAF. When the robot team uses line-o-sight coordination, there is almost never more than one obstacle between a transmitter and a receiver, and as soon as an obstacle is introduced between a transmitter and receiver (a leader-ollower pair), the ollower takes action to reestablish line-o-sight. Thereore, the PAF o the obstacles is not expected to have a large eect on the line-o-sight method. Surprisingly, the perormance o the signal strength based coordination method degrades very little as the PAF increases. The perormance appears to be essentially constant or PAFs o 5dB and above since there is no statistically signiicant dierence between any o cases where τ 5 (p.7). This means that even when obstacles are opaque, the signal strength coordination method outperorms the line-o-sight method. This is due to the larger distances between team members allowed by the signal strength method. It is important to note that these results were achieved by using the same switching threshold or the signal strength method, τ = db. This suggests that careul optimization o τ is not necessary or the signal strength method to outperorm the line-o-sight method or a wide range o obstacle types. The drawback o the signal strength method is that it allows or many more temporary partitions to occur in the network o robots. For the task considered here, these intermittent breaks do not signiicantly harm the task perormance (or at least the harm caused by the intermittent breaks is outweighed by the beneits o a bigger coverage area).

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