Controlling Mobile Sensors for Monitoring Events with Coverage Constraints

Transcription

1 Controlling Mobile Sensors for Monitoring Events with Coverage Constraints Zack Butler and Daniela Rus Institute for Security Technology Studies & Dept. of Computer Science Dartmouth College Computer Science and Artificial Intelligence Laboratory MIT Abstract Sensor networks are systems of many small units that work together to monitor a given environment. Endowing such sensor units with mobility can allow them to reactively converge on more interesting portions of their environment. This enables the concentration of sensing resources where they are most useful and provides robustness by delivering redundancy at the point of interest. However, when converging, in general the sensors should not leave any portion of the environment unsensed. In this paper, we review distributed methods for controlling the sensors and describe a family of distributed methods for retaining coverage while allowing the convergence to proceed where possible. The coverage methods are based on the Voronoi diagram of the sensors positions, and can use different amounts of communication and computation to produce complete coverage of the environment. We also describe extensions that serve to make coverage more uniform or allow specific areas to be left uncovered. We present implementations of these algorithms in simulation and describe results and avenues of future work. I. INTRODUCTION Sensor networks are systems of many simple sensing elements endowed with computation and communication that can work together to provide information about an environment. By giving these sensors motion capability, we can reactively focus the sensing resources on particular areas of interest within the overall environment. For example, sensors can be deployed throughout a forest and converge near fires when they occur, or be deployed in a building and move toward any intruders to get as much data about the event as possible. Such reactive motion also gives the system robustness in case of sensor failure, in that other sensors will be readily available at the locations of interest. However, when the sensors mobilize to converge on an event, it is also generally important that they do not leave portions of the environment completely unsensed. One method to ensure complete coverage would be to use a secondary network of immobile sensors in addition to a smaller number of mobile units, but this may be cost-prohibitive for many sensing modalities. Instead, we propose that the mobile sensors move toward the areas of interest while also checking for sensor coverage of their local neighborhood. The sensor will then continue to move only when this does not leave any portion of the environment unsensed. In particular, we note that if the sensors are all moving according to a simple control algorithm, it can be easy to predict the motion of neighboring sensors. This information can then be used to inform a coverage algorithm. Using computation in this way alleviates the need for mutual sensing or communication to distribute position information. In this paper, we first review two simple algorithms for positioning mobile sensors in response to a series of events. We then develop a set of algorithms based on Voronoi diagrams of the sensor positions that enable the sensors to maintain coverage of their environment while also using either positioning algorithm to converge on events. We also evaluate the computation, communication and accuracy tradeoffs for the different coverage algorithms and discuss future extensions. II. RELATED WORK We are inspired by previous work in sensor networks and build on our previous work on routing in ad-hoc networks [1] and reactive sensor networks []. Important contributions by other groups on which we build include [3], [], [5]. Massively distributed sensor networks are becoming a reality, largely due to the availability of the Mote hardware []. In [7], Cerpa and Estrin propose an adaptive self-configuring sensor network topology in which sensors can choose to join the network or not based on the network condition, the loss rate, the connectivity, etc. The sensors do not move, but the overall structure of the network adapts to the situation by having the sensors activate and deactivate. Our work examines control of mobile sensors where we are interested in using redundancy to improve sensing rather than optimize power consumption. Recently, mobile sensor networks have come under more investigation. The addition of motion capability to the Mote sensors, creating Robomotes, was described in []. Algorithmic work has included even dispersal of sensors from a source point and redeployment for network rebuilding [9], []. The even dispersal methods use potential fields and give similar results to our coverage algorithms, however under our algorithms the sensors attempt to reactively achieve a non-uniform distribution in addition to coverage. Also unlike potential field algorithms, our technique switches explicitly between the event-driven control and the coverage task, simplifying prediction of sensor motion and allows the sensors to adhere exactly to the convergence algorithm when not required for coverage. Related recent work by Bullo et al [11] uses Voronoi methods to arrange mobile sensors in particular distributions, but in an analytic way that requires that the distributions be

2 defined a priori. Our work focuses on distributed reactive algorithms for convergence to unknown distributions. Related research also includes mobile robotics work focused on distributed formation control. Control algorithms have shown how to generate specific configurations of a team of robots and keep that configuration fixed while the team moves [], [13]. However, these algorithms rely on specifically identified robots such as leaders and followers. Other work has looked into larger homogeneous groups, but has tended to produce simple shapes such as through the use of potential fields [] or local connection schemes [15], rather than changing the configuration of the group over time. Voronoi diagrams have been used in a variety of mobile robot research, but almost always it is the Voronoi diagram of the environment that is considered, whereas we consider the Voronoi diagram of robot positions as a simple way of measuring neighbor relationships. III. CONTROL OVERVIEW Our primary goal in this work is to control the sensors so that they converge on events occurring in their environment. Here we summarize two event-based control methods for a group of mobile sensors. In both methods, we assume that the sensors begin in a uniform distribution (either regular or random) over the environment, and that the events can be represented discretely in space and time. These assumptions allow the sensors to converge toward the distribution of a series of events without explicit consideration of the positions of their neighbors. The intuition behind this is that a sensor s role in the system is determined by its initial position within the environment and that the system as a whole will move from the uniform distribution to the distribution of events without having to consider the exact location of each sensor. The first method is purely reactive, in that it uses only the current position of the sensor and the position of an event to determine the motion of the sensor. At a high level, each sensor moves toward each event a small distance. The amount of motion after each event is the critical factor in this strategy. Moving a small constant or proportional distance leads to clustering at the mean of all events. Instead, we define a function f (d) of the distance d between a sensor and a event that supports proper clustering of sensors. This function takes the form f (d) = αd β e γd, and a sensor will move toward each event by the amount given by f. f is constructed so that f ( ) =, allowing the sensors to separate if the events are themselves in multiple clusters, and so that d f (d) is monotonic, meaning that sensors will not have to pass each other (wasting energy) when reacting to events. The reactive method is very simple to calculate, but is heuristic in nature. We therefore have developed a second algorithm that uses a small amount of history and more carefully analyzes the distribution of events. This algorithm does not keep the position of every event, but rather uses a coarse histogram of the events as a basis for determining its correct position. The history-based method works for (a) (c) (b) Fig. 1. Performance of reactive and history-based control algorithms. (a) Initial sensor positions (b) Event positions (c) Final positions after using the reactive algorithm (d) Final positions after using the history-based algorithm. one-dimensional and two-dimensional systems. In the onedimensional case, each sensor maintains an approximation of the distribution of the event positions. We then note that for each interval of the environment, the number of events sensed should be proportional to the number of sensors present. By scaling the event distribution appropriately, we can calculate the proportion of sensors that should be present in each interval at the present time. Because we assume an initially uniform distribution, each sensor can then determine its role purely based on its initial position, without needing to communicate with the other sensors. Each sensor computes the cumulative distribution (CDF) of the scaled event distribution, which describes what fraction of the sensors should be to the left of each point. The sensor then finds the point in the scaled CDF corresponding to its initial position, and moves to that location. This is extended to the two-dimensional case by keeping two sets of CDFs, one for each dimension. Each sensor then solves for a set of valid positions along each dimension, and chooses a location where these positions intersect as its new position. Details of this algorithm can be found in []. IV. MAINTAINING COVERAGE Under the algorithms presented in Sec. III, sensor networks may lose network connectivity or sensor coverage of their environment. The ability to maintain this type of coverage while still reacting to events is an important practical constraint because it can guarantee that the network remains connected and provides monitoring of the entire space. This way, new events that appear in currently quiet areas will still be detected and responded to. For example, consider the case of two clusters of events such as the ones shown in Fig. 1b. If these clusters are sufficiently far away, the sensors in the middle will start to pull apart from each other, and could partition the network. It is also possible to leave an area in the middle (d)

3 Fig.. (a) (b) (c) Representative result of predictive coverage maintenance. (a) Event positions (b) Final sensor positions (c) Voronoi diagram of sensor positions. uncovered while maintaining network connectivity via other sensors around the boundary of the space. We assume that each sensor has a limited sensing range R s, and every point in the environment should be sensed by at least one sensor. Every sensor will halt to maintain coverage or will follow the event distribution exactly if not required for coverage purposes. The coverage motions are similar to spacefilling coverage methods, such as those that use potential fields [], in which each robot moves away from its colleagues to produce a regular pattern in the space and thereby complete coverage. These methods could be extended to the variable distribution case by changing the potential field strengths based on the event distribution. In our work, we instead have the sensors follow the event distribution exactly until required for coverage, so that they can achieve a good approximation to the event distribution in high density areas and good coverage in low density areas. This switching technique also simplifies prediction of others motions. In both methods presented in Sec. III, each sensor moves according to a simple known control function. Each sensor can therefore predict the motion of other sensors, and use this information to ensure that coverage is maintained. Prediction of other sensor positions requires additional computation, which can be significant if the update algorithm is complex or the number of sensors to be tracked is large. This computation can be avoided by using communication, such that each sensor broadcasts its position to nearby sensors. However, more communication also has potential drawbacks in terms of power usage. In this section we present three related methods for maintaining coverage that use different amounts of communication and computation, and present comparisons of their performance. A. Complete Voronoi algorithm In this section we describe a simple version of the coverage algorithm which also forms the basis for the subsequent variations. All versions of the algorithm involve constructing Voronoi diagrams of the sensor positions to determine if coverage exists. Each sensor uses the geometry of its own Voronoi region to decide whether it is required for coverage. Later variations approximate the sensor positions or Voronoi diagram in different ways, allowing a tradeoff to be made between computation, communication, and accuracy of coverage. The simplest version of the algorithm, referred to as the Complete Voronoi protocol, is also the most computationally intensive. In this protocol, each sensor calculates the motion of every other sensor and uses this information to compute its Voronoi region after each event. This ensures the best performance, in that each sensor will know exactly what area it should consider for coverage purposes. If any part of the sensor s Voronoi region is farther away than R s (note that only vertices of the region need to be checked, since it is always polygonal), it knows that no other sensor is closer to this point, and it should not move away from that point. As long as the sensor maintains its Voronoi region in this way, coverage is assured. Figure shows a typical result of this technique. The Voronoi diagram of the sensors shows no region that is larger than R s = 3 units from the sensor at its center. To make this prediction correctly, we note that once a sensor has stopped, it is no longer following the control algorithm used to predict its position. Therefore, in order for a sensor to accurately predict the state of the network, it must also know which sensors have stopped. This can be done in two ways, presented side by side as Algorithm 1. If we wish to have no additional communication, then each sensor predicts whether other sensors will stop, based on the same Voronoi region calculation. This is presented as Algorithm 1 (left). However, this is a large amount of computation, and can be avoided with a small amount of communication. When one sensor stops to avoid loss of coverage, it sends a broadcast message with the position at which it stopped. Other sensors can then assume adherence to the motion algorithm unless such a message is received. This is detailed as Algorithm 1 (right). Using this communication motivates a choice in the algorithm that each sensor will stop only once. This limits the accuracy of coverage in that sensors may no longer be needed in their current position for coverage after some time has passed, and they are not allowed to move. However, if the event distribution remains constant, this is not likely to be a major concern, and having each sensor stop only once limits communication under this algorithm to O(s) broadcasts over the length of the task, rather than s per event. Properties of the Complete Voronoi Algorithm without the use of communication (C c is the amount of computation required by the control algorithm): Communication: Zero

4 Algorithm 1 Complete Voronoi algorithm for ensuring coverage left, an algorithm that uses no communication; right, an algorithm that uses some communication. 1: for each other sensor s i do : if Fixed(i) = then 3: Compute new position x i : else 5: Use sensor s stored position : Compute complete Voronoi diagram V of sensor positions X 7: for each region v i of V do : if Fixed(i) = then 9: for each vertex v ik of v i do : if v ik x i > R s then 11: Fixed(i) = 1 : Store position x i for sensor i 1: for each other sensor s i do : if Fixed(i) = then 3: Compute new position x i : else 5: Use sensor s fixed position : Compute Voronoi region v containing my position x 7: for each vertex v k of v do : if v k x > R s then 9: Fixed = 1 : Broadcast fixed message (a) Fig. 3. Comparison of using additional sensor motion toward uncovered area. (a) Results from Complete Voronoi algorithm (b) Results when sensors move toward the farthest vertex in their Voronoi region. Computation: O(s log s) (coverage) + O(sn) (stopping prediction) + sc c (position prediction) Properties of Complete Voronoi with communication: Communication: O(s ) for all time Computation: O(slogs) + sc c 1) Relaxation: One extension we have explored involves the behavior of a sensor that has discovered danger in leaving an area uncovered. Rather than simply stopping, the sensor can move a small distance toward the most distant vertex of its Voronoi region. This can lead to more uniform coverage in sparse areas, and in many cases reduces the number of sensors required to stop to assist coverage, but does have some drawbacks. If communication is not used, calculating this extra step for each other sensor during the predictive calculations is an additional burden. Also, sensors that are at the edge of a dense portion of the environment may be hurting the event distribution approximation by moving away from the dense area when not strictly necessary. We show the results of using this small step in Fig. 3. B. Local Voronoi algorithm Using the complete Voronoi diagrams requires a large amount of computation, both to track all the sensors in the network and to compute the diagram itself. To develop a more scalable protocol, we trade off a small amount of coverage accuracy for a large reduction in computation. After an initialization phase in which all sensors discover the location of (b) all other sensors, each sensor computes its Voronoi region. As the task progresses, each sensor tracks only those others which were its neighbors in the original configuration. It then calculates its Voronoi region after each event based only on this subset of the network. It examines its Voronoi region in the same way as in the Complete Voronoi protocol to determine whether or not to stop to maintain coverage. This algorithm is listed as Algorithm. Algorithm Local Voronoi algorithm for ensuring coverage 1: for each neighboring sensor s i s nbrs do : Compute new position x i 3: Compute Voronoi region v among x nbrs containing my position x : if v is semi-infinite then 5: Fixed = 1 : Broadcast fixed message 7: else : for each vertex v k of v do 9: if v k x > R s then : Fixed = 1 11: Broadcast fixed message Properties of the Local Voronoi Algorithm: Communication: O(s ) for all time Computation: O(nlogn) + nc c As long as the neighbor relationships remain fairly constant, this algorithm will produce similar results to the complete Voronoi algorithm. In addition, it will always make the sensors more conservative about coverage than the complete algorithm. This can be seen by comparing a sensor s true Voronoi region to the one it calculates. If we start with the true region and add sensors to the calculation, by definition the region will not change. Deleting sensors (representing sensors that were not neighbors in the original configuration) will only increase the size of the calculated region. In cases where movement is small or generally in a single direction, the neighbor relationships will remain fairly constant. If the motion is large, or if nearby sensors are moving in different directions, the neighbor relationships may

5 Event Update frequency (every N events) distribution 3 Single Gaussian Two Gaussians Diagonal line TABLE II NUMBER OF FIXED SENSORS OUT OF A GROUP OF FOR THE LOCAL VORONOI ALGORITHM WITH DIFFERENT POSITION UPDATE MEANS THAT A SINGLE UPDATE WAS SENT OUT AFTER 5 EVENTS. Fig.. Results of the Complete Voronoi coverage for a situation with events from two clusters and one corner of the environment (below and to the left of the dashed line) allowed to be left uncovered. Sensors denoted with a * have stopped moving to ensure coverage. Note that the sensors were initially uniformly distributed over the entire square region. change. In the latter case, we can modify the algorithm slightly by repeating the initialization step at regular intervals. This will allow the sensors to discover their new neighborhood and improve the accuracy of the algorithm while keeping communication limited. In the experiments below, we show the effects of varying the frequency of this neighbor update. C. Selective Coverage We also note that near the boundary of the environment, the sensors can intersect their Voronoi region with the boundaries of the environment. This will improve coverage by allowing these sensors to move away from the boundary up to their sensing radius. This is not limited to regularly shaped environments, although the intersection calculation may need to be conservatively approximated if the boundaries are complex. In the same way, portions of the environment can be denoted as acceptable to lose coverage of. In these cases, the designated area may start out covered as part of the initial distribution of sensors, but if the control algorithm sends them away, they will not fix their position. To allow selective coverage, when performing the Voronoi check, a sensor will first check its own position, and if outside the area required to be covered, can ignore the remainder of the computation. It then intersects its Voronoi region and stops only if a portion within the area to be covered is beyond its sensor range. Figure shows the results for a scenario where one corner of the initially occupied environment has been deemed unnecessary to cover. V. ALGORITHM COMPARISON To compare these different protocols, we performed analysis and empirical tests to determine the amount of communication and computation required for each algorithm under various circumstances. Since each protocol can be used with either the history-based or historyless update algorithms, we present only the communication and computation required for the coveragerelated portion. In the predictive algorithms, the amount of computation depends on the update rule used, and this is noted in the results. In Table I we present the actual amount of computation used in the Matlab simulations of each algorithm. The difference between the last two algorithms (namely the use of occasional global position updates) is only partially reflected in the amount of computation and communication. Clearly the periodic updates require additional communication, but the advantage to be gained is that the coverage detection is more accurate. This can be seen graphically in Fig. 5, which shows the final positions after events for the three different algorithms. We use the number of sensors that have stopped to assist in coverage as a metric for comparison between these algorithms. Since in all cases complete coverage is achieved, lower numbers are better. Numerical results for these algorithms are shown in the rightmost columns of Table I, which report the number of sensors out of the group of that are required for coverage. This clearly shows that the use of the original neighbors for all time is not very effective. The use of periodic updates of the neighborhood can give almost as accurate results as the complete algorithm, while using far less computation than the complete algorithm and less communication than the communication-based algorithm. To further explore the utility of occasional position updates for recalculating neighbor relationships, we varied the frequency of the update for the same sets of initial positions and event locations. In Table II, we present data from simulations of events in various distributions. Here we see that for these cases, the accuracy of coverage degrades gradually as the updates get less frequent, indicating a fairly smooth tradeoff between communication and coverage accuracy. VI. CONCLUSION We have presented distributed control algorithms with which mobile sensors can track and converge on a series of events while also ensuring complete sensor coverage of their environment. The basic control algorithms are simple and cause the sensors to converge on the events using minimal levels of communication and computation. The coverage algorithms require sensors to interact, either through communication or prediction of their neighbors positions. Voronoi diagrams of the sensor positions are used to measure the local structure of the system, and approximations to the true diagram are used in order to lessen the computational burden on the sensors. The different techniques presented allow a tradeoff to be made between communication, computation and accuracy as appropriate to the particular task. In the future, we plan to investigate other techniques for

6 (a) (b) (c) Fig. 5. Typical results of coverage maintenance algorithms. Final sensor positions after a series of events along a diagonal, with a sensor radius of (in a environment). (a) Complete Voronoi (b) Local Voronoi with no neighbor update (c) Local Voronoi with neighbor update every events. Algorithm Communication Computation Number of fixed sensors (total) (per event) Gaussian Diagonal Two lines Complete Voronoi, no comm s initial k + sc c Complete Voronoi s initial, k + sc c Same as above with comm < s additional Local Voronoi s initial, + nc c no neighbor update < s additional Local Voronoi s per update + nc c 7 5 update every events TABLE I COMPARISON BETWEEN DIFFERENT COVERAGE PROTOCOLS, WHERE s IS THE TOTAL NUMBER OF SENSORS IN THE NETWORK, n THE NUMBER OF NEIGHBORS OF A GIVEN SENSOR AND C c THE COMPUTATION REQUIRED BY THE CONTROL LAW USED. NUMERICAL COMPUTATION RESULTS BASED ON MATLAB IMPLEMENTATIONS FOR COMMON SETS OF EVENTS OF DIFFERENT DISTRIBUTIONS IN A NETWORK OF SENSORS. THE NUMBER OF FIXED SENSORS FOR EACH ALGORITHM IS GIVEN FOR THREE DIFFERENT REPRESENTATIVE EVENT DISTRIBUTIONS. sensor positioning and extend these techniques to more complex tasks such as constrained sensor motion and non-circular regions of sensor or communication range. For the former, if the sensors have inherent motion constraints or obstacles are present, the sensors may be able to plan paths to achieve their correct position, or sensors could be allowed to switch roles (potentially just by switching their recorded initial positions) if this enables more efficient behavior. If the sensing elements are not isotropic, but their sensing range can be easily represented, this should be able to be handled by computing the Voronoi diagram in a transformed space that takes the anisotropy into account. These extensions will allow these algorithms to be applied to a wider variety of sensor systems. Acknowledgments Support for this work was provided through the Institute for Security Technology Studies, NSF awards EIA-99159, IIS-999, IIS-99193, EIA-79 and 5, ONR award N and DARPA Task Grant F REFERENCES [1] Qun Li and Daniela Rus, Sending messages to mobile users in disconnected ad-hoc wireless networks, in MOBICOM, Boston, August, pp. 55. [] Qun Li, Michael DeRosa, and Daniela Rus, Distributed algorithms for guiding navigation across sensor networks, in MOBICOM, 3. [3] Gregory J. Pottie, Wireless sensor networks, in IEEE Information Theory Workshop, 199, pp [] Jon Agre and Loren Clare, An integrated architeture for cooperative sensing networks, Computer, pp., May. [5] Deborah Estrin, Ramesh Govindan, John Heidemann, and Satish Kumar, Next century challenges: Scalable coordination in sensor networks, in ACM MobiCom 99, Seattle, USA, August [] Jason Hill, Robert Szewczyk, Alec Woo, Seth Hollar, David Culler, and Kristofer Pister, System architecture directions for network sensors, in ASPLOS,. [7] Alberto Cerpa and Deborah Estrin, Ascent: Adaptive self-configuring sensor networks topologies, in INFOCOM, New York, NY, June. [] G.T. Sibley, M.H. Rahimi, and G.S. Sukhatme, Robomote: A tiny mobile robot platform for large-scale sensor networks, in Proc. of IEEE ICRA,, pp. 13. [9] M.A. Batalin and G.S. Sukhatme, Spreading out: A local approach to multi-robot coverage, in Distributed Autonomous Robotic Systems 5,, pp [] A. Howard, M.J. Mataric, and G.S. Sukhatme, Mobile sensor network deployment using potential fields: A distributed, scalable solution to the area coverage problem, in Distributed Autonomous Robotic Systems 5,, pp [11] J. Cortes, S. Martinez, T. Karatas, and F. Bullo, Coverage control for mobile sensing networks: Variations on a theme, in Proc. of IEEE ICRA, 3, To appear. [] Tucker Balch and Ronald C. Arkin, Behavior-based formation control for multiple mobile robots, IEEE Transactions on Robotics and Automation, vol., no., pp , December 199. [13] Stefano Carpin and Lynne Parker, Cooperative motion coordination amidst dynamic obstacles, in Distributed Autonomous Robotic Systems 5,, pp [] John H. Reif and Hongyan Wang, Social potential fields: A distributed behavioral control for autonomous robots, Robotics and Autonomous Systems, vol. 7, no. 3, pp , May [15] Tucker Balch and Maria Hybinette, Social potentials for scalable multirobot formations, in Proc. of Int l Conf. on Robotics and Automation, San Francisco, April, pp. 73. [] Zack Butler and Daniela Rus, Event-based control for mobile sensor networks, IEEE Pervasive Computing, Nov. 3.