Connectivity aware Coordination of Robotic Networks for Area Coverage Optimization

Transcription

1 Connectivity aware Coordination of Robotic Networks for Area Coverage Optimization Yiannis Stergiopoulos Yiannis Kantaros Anthony Tzes Abstract The scope of this paper is the development of a distributed control scheme for mobile sensor networks in order to optimize coverage performance over a region of interest, while simultaneously guarantee information flow among neighbouring nodes via valid communication links. The nodes evolve in time in order to increase network coverage via distributed information acquired from their neighbours. Unlike previous works, the communication radii of the nodes are assumed fixed and less than twice the sensing ones, imposing an extra constraint concerning connectivity preservation in the overall problem formulation. Efficiency of the proposed scheme is further confirmed via simulation studies. Index Terms Mobile sensor networks, distributed optimization, Voronoi partitioning, network connectivity I. INTRODUCTION Continuous surveillance of certain regions of interest has been an issue of major importance in the last years, due to its application in security, data collection and/or military scenarios. Assuming robotic applications, it is apparent that a lot of tasks are impossible to be carried out by a single robot, while a large group of them can provide robustness to failures of single agents. The main objective in swarm coordination is the development of control laws that base their action on distributed information, so that they are directly applicable, while at the same time lead the network in a state where an aggregate objective function is optimized [ 3]. Thus, it is essential that network connectivity is ensured throughout the deployment stage, so that information is transferred from any node to another. However it is evident that demand for connectivity ensurance and area coverage optimality cannot be achieved simultaneously, and thus there is trade off to be balanced [, 5]. Distributed coordination of mobile networks via intuitive nearest neighbour rules has been proposed by the authors in [6]. Connectivity control of networks has been examined in [7, 8], while more generalized coordinate free theoretical approaches have been developed in [9, 0]. In the majority of works presented in the existing literature, the communication range of the nodes antennas is assumed either variable but unbounded(in terms of no upper limit) [, ],orboundedbutgreaterthantwicethesensingrange[,3, ]. This dependence of the radio range on the sensing one, although not met in practice, remarkably lets us surpass any network connectivity issues, and concentrate on optimization of the covered area. In this article, though, the nodes radio range is assumed fixed and can be less that the aforementioned bound (i.e. twice the sensing range). This restriction imposition, although met most often in practical scenarios (where the sensors and antennas ranges are uncorrelated), leads in inability to apply already presented area coverage oriented coordination schemes [, 6,, 5]. Despite the aforementioned radio range constraint, the algorithm proposed in this article guarantees network connectivity, while leading the nodes in an optimal state, considering coverage terms. The rest of the article is organized as follows. In section II the area coverage problem in mobile sensor networks is introduced, along with the main preliminaries on Voronoi partitioning. In section III the background concerning network connectivity is presented, while network s RF connectivity is analysed from a graph perspective. The proposed coordination scheme that takes into account both the network s coverage performance, along with communication constraints imposed due to radio restrictions is presented in section IV. Simulation results in section V further confirm efficiency of the proposed scheme, while concluding remarks are provided in the last section. II. AREA COVERAGE PROBLEM Consider n in number mobile robotic agents responsible for the sensing coverage of an area of interest D, defined as a convex compact set in R. LetI n ={,,...n} be the set of unique identifiers of the nodes, while their positions on the Euclidean plane are denoted asx i, i I n. The robots are considered to evolve in the interior ofdin discrete time via the control inputsu i, as x k+ i =x k i +uk i, u i R, x i D, i I n, () where the superscript k denotes the current time step, k = 0,,.... In this article, at each time step it is assumed that onlyonenodecanmove;thus,atthe firststep,k=,nodei= will move, while afterwards the node to move is determined in a random manner //$ IEEE 36 ICIT 0

2 Assuming surveillance purposes, sensors are embedded on the robotic platforms that sense the area in range ofr around the nodes, denoted asb i, i.e. B i = { x R : x x i r }, i I n. () In an area coverage application, the aggregate objective function under optimization can be expressed as the area of the union of the nodes sensing regions over theddomain, i.e. H = ds, D i In B i where ds is the elementary surface for integration purposes. Aquitecommonmethodtodealwithsuchkindofproblems in swarm robotics is to tessellate the space into subsets via a distance based metric and assign them among the nodes. Voronoi diagram [6], V ={V i, i I n }, is the most common partitioning amongndistinct pointsx i, i I n, defined as V i = { x D: x x i x xj, j In }, i In. (3) In other words,v i is the set of the points ofdthat are closer tox i than any other points in { x j, j I n }. We refer tovi as the Voronoi cell of nodei. Let G D be the Delaunay graph associated with the corresponding Voronoi partitioning. We assume that the reader is familiar with the main preliminaries on graph theory[7]. Two nodes that share an edge of their Voronoi cells are considered as neighbours in G D. The Delaunay neighbours N i of an arbitrary nodeiare then defined as N i = { j I n :V i V j /0, j i }, i I n. () Apparently, if j N i, theni N j. Utilizing the setsv i and considering H, one can define in an equivalent manner the r limited Voronoi cell of an arbitrary nodei,vi r, as the parts of the corresponding Voronoi cell that are simultaneously sensed by that node, i.e. V r i =V i B i, i I n. (5) It is easily proventhat via this definition, the total area sensed by the network, H, can be expressedas the summationof the r limited Voronoi cells of the nodes, i.e. H = i I n Vi r ds. (6) Equivalently to the Delaunay graph G D, one can define the r limited Delaunay one, denoted as GD r, where the neighbours of a node i in this graph are the nodes whose r limited Voronoi cells share an edge withvi r, N r i = { j I n :V r i V r j /0, j i }, i I n. (7) Apparently, G r D consists a subgraph of G D. Figure shows graphically the neighbouring relationships among the nodes in the Delaunay, G D, and r limited Delaunay graphs, G r D. It should be noted that if two nodes are neighbours in G r D graph, then their distance is less than r; however, the reverse does not always hold, as seen by the numbered bottom right nodes of Fig.. Although the circles of these nodes intersect (dashed line), this does not hold for their r limited Voronoi cells. Fig.. Graphical representation of the Delaunay [upper] and r limited Delaunay [lower] neighbours in a sensor network. III. RADIO CONNECTIVITY ISSUES An issue of major importance in coordination of mobile sensor networks is the distributed nature of the designed control schemes. In other words, the nodes should organize their action without global knowledge of the network s state, but via local information from neighbouring nodes, instead. Each node is assumed to be equipped with radio transceivers in order to be able to exchange spatial information with other neighbouring nodes in range. The antennas are assumed to transmit omnidirectionally aroundx i up to a radiusr. Unlike the majority of previous works in the existing literature, where the antennas radii are considered variable, in this article the latter is assumed fixed, same for all nodes, and not demanded tobeatleasttwicethesensingrange.infact,thecaser ris examined in detail in the literature, since network connectivity is trivially guaranteed that way, in the network s area optimal state. Apparently, a bidirectional communication link exists among any two nodes i,j iff xi x j R, for spatial information exchange purposes. Graphically, the neighbouring relationships among the nodes from a communication aspect (i.e. ignoring the nodes sensing abilities) can be represented in the communication graph of the network, denoted as G c, where an edge exists among two nodes iff one is in radio range of the other, and vice versa. It should be noted that, although the Delaunay and r limited Delaunay graphs are somehow correlated, the communication graph is totally independent, since it does not rely on the nodes sensory domains, B i, but is determined only by the nodes positions and the common communication radius R. As far as the neighbouring relationships in the graph G c are considered, let us state the following definition. 37

3 Definition. Two nodesi,j in the communication graph G c associated with a wireless sensor network are called (directly) connected, while denoted asi j, iff xi x j R, wherer is the common radio range. Given a sensor network and the communication graph G c, one can conclude (global) connectivity of the latter if there existsaroutingpathfromanynodeofthenetworktoanyother, a.k.a. end to end connectivity. This can also be expressed in algebraic terms via examining positiveness of the second smallest eigenvalue of the Laplacian matrix that corresponds to the aforementioned graph. For more information on algebraic graph theory the reader is encouraged to refer to [7]. The main issue, however, in controlling network connectivity via the graph s Laplacian is the fact that it is a centralized approach, and thus inapplicable in cases of communication range constraints imposed by the nodes antennas physical characteristics. Assumption. Initially, each node is connected with all its r limited Delaunay neighbours, i.e.i j, j N r i, i I n. The reason for insisting in RF connectivity among the r limited Delaunay neighbours in the sensor network is the fact that these are the subset of nodes that can possibly affect network s coverage performance via their r limited Voronoi cells alteration, as depicted graphically in Fig.. Consequently, distributed approaches can be developed that 3 5 Fig.. Graphical depiction of RF connectivity demand over the GD r graph for coverage applications. Infinitesimal maneuvres of node 5 may alter coverage contribution of nodes 3, and 6. are based on the aforementioned set for the node to move at each step, without requiring global knowledge of the state of the network, while contributing in network coverage increase. In fact, it is apparent that, if R r then each node is directly connected (from a communication aspect) with all of its r limited Delaunay neighbours, i.e. j Ni r j i. The scope of the following section is the development of a distributed control law, such that the network is leaded in an area optimal configuration, given communication range constraint R, and RF connectivity demand over the r limited Delaunay sets. IV. CONNECTIVITY AWARE COORDINATION SCHEME Recalling H as defined in (6), the primary objective of the nodes in the network should be to self deploy themselves 6 7 in a way that the area of the sensed part of D (expressed as the summation of the areas of the corresponding r limited Voronoi cells) is as high as possible, assuming bounded sensing domains. Furthermore, coordination at this stage should be performed in a distributed manner via local information, as exchanged among the nodes considering their communicational capabilities. As far as the coverage part of the objective is concerned, let us first state the following lemma which will form the basis of the coordination stage. Lemma. [5] The area covered by a set of nodes, where each lies at the centroid of its ownr limited Voronoi cell, is locally maximum. In this article, since coordination of the network is assumed to be performed in discrete time steps, the nodes are selected to move towards the centroid of their corresponding r limited Voronoi cells via fixed step sizes, until they reach optimum configuration where the total sensed area is maximum. More specifically, considering (), the control lawu i for the corresponding node to move at each step can be selected as u i = σ centr(vr i ) x i centr(v r i ) x i, (8) where the sample instance superscripts have been omitted to avoid notation complexity. In the above expression, centr( ) stands for the centroid of the compact set argument, while σ can be selected as σ = min(ε, centr(v r i ) x i ), for any arbitrarily small ε >0. Itshouldbenotedthatmonotonicityof H isnotguaranteed during the transition of the network towards the r limited centroidal Voronoi configuration; however, since one node moves at each time step (as stated in section II), the network will reach asymptotically area optimal configuration. However, in order to characterize the control scheme (8) as decentralized, in the sense that the node to move does not require global network knowledge to apply it, the aforementioned node should be able to evaluate its r limited Voronoi cell via information from the nodes in range. In most of previous works in the literature, this communication issue is surpassed by allowing the nodes communication range to be at least twice the sensing one, since only the nodes in range of r are needed for distributed evaluation ofv r i []. In this article, however, an extra constraint is considered via limiting the communication range R to any arbitrary value, which in practice is imposed by the antennas radio characteristics. The proposed control action is based mainly on preserving radio connectivity among the r limited Delaunay neighbours, so that (8) is distributively evaluated. Letusdenoteasithenode to moveatanarbitrarytimestep k. The latter decides on making a move towards the centroid of its r limited Voronoi cell, in order to increase network s coverage performance, via (8). However, node i is assumed to have spatial information acquired from the nodes that it is connected with. At this stage, nodeichecks how network connectivity will be affected by its motion, before making the motion (i.e. via numerical evaluations, as if it was at that 38

4 position). It is clear that, if at the new/candidate position a new node joins the set of the nodes thatiis connected with, then no issue arises, since this just provides an extra link in the G c graph, as long as H is increased at the new position. In the case, however, where a link is about to break if motion is performed, then the node to move checks if that link concerns one of his r limited Delaunay neighbours. If so,thenmotionistriedtobeperformedwithhalfthestep size σ, and so on, till a predefined search depth. In the case where neither of the examined cases is able to preserve connectivity among the aforementioned set of nodes and simultaneously increase network s coverage, then the node stays idle and does not perform any action. The procedure described above is depicted graphically in Fig. 3, presenting a fully connected network of four nodes. Let it be node 3 to move at the specified time step. The latter Fig. 3. Graphical representation of the proposed coordination scheme for a small sized network. can evaluate its r limited Voronoi cell since it has adequate information of its neighbouring nodes. For a specified σ, the node is about to examine the candidate position denoted by grey color and enumeratedas 3, where the sign corresponds to a candidate position for the corresponding node. At this stage, the node performs evaluations on its new position (while still remaining at 3). The corresponding sensed space partitioning, shown in the right part of the Figure, indicates that (i) the sensing coverage performance of the set is increased compared to the initial state (from a numerical aspect), while (ii) RF connectivity links among the nodes will be perturbed (if motion is performed). The link that is about to break corresponds to that with node ; however, as seen from the evaluated topology [right part], node does not belong to the r limited Delaunay neigbours set of node 3 (at 3 coordinates), and hence the link can be abandoned/dropped without affecting connectivity of the network. Although the algorithm described above incorporates some kind of conservatism, it is still able to (i) increase network s coverage performance from one step to another monotonically, while (ii) preserving radio connectivity among the r limited Delaunay neighbours, as required for distributed evaluation of network s coverage increase. The fact that no motion is performed if a link that is about to break determines connectivity of the communication graph, is the reason for the existence of overlapping among the nodes sensory domains in the optimal state. V. SIMULATION RESULTS Simulations were conducted in order to further verify efficiency of the proposed coordination scheme. The region D to be surveyed is considered as a convex compact planar set of total area D ds=6.05 units. The latter is identical to the one used in [8]. The network consists ofn=0 nodes with r=0.5 units, while the communication radiusrwas selected equaltor=.5r=0.75 units, so thatrf connectivityin GD r is not trivially guaranteed. The nodes are initially deployed randomly in D, so that Assumption holds. Given the set D, the maximum possible coverage ratio achieved, evaluated as the summation of n circles (best case scenario) is equal to 00% of D. The network s initial configuration, evolution through time, along with the final network s state, are shown in Fig., in this order. It is apparent that the nodes tend to spread in the interior of the domain of interest in order to increase their coverage performance the network achieves optimum coverage, while radio connectivity among the r limited Delaunay neighbours is attained throughout the whole evolution stage. Figure 5 depicts the evolution of the normalized network s coverage, i.e. H as a ratio of the area ofdwhen coordination scheme ofsection IVis applied[blue line].the redline representsthe maximum possible coverage ratio, i.e. 00% in that case. The sensed area percentage, starting from an initial value of 3% (dependent on the initial network configuration), increases as network evolves, until it converges to 97%, which is more than satisfactory considering non trivial communication constraints imposed. Important is the fact, that network s coverage performance is leaded to the optimum in a monotonic manner, since the corresponding node to move at each step performs the motion iff it will contribute to coverage increase, evaluated by itself in the decision making stage. Fig. 5. Percentage of sensed area during network evolution. The red upper limit represents the maximum possible coverage ratio. VI. CONCLUSIONS In this paper, a spatially distributed approach was proposed for leading the mobile nodes of a sensor network towards an 39

5 Fig.. Coordination results derived via the proposed control scheme. [Left] Initial network configuration. [Middle] Network evolution. The green (red) circles represent the nodes final (initial) positions. [Right] Final network state. Communication graph indicates RF connectivity among the r limited Delaunay neighbours. area optimal configuration, while simultaneously preserving radio connectivity among the nodes. The scheme was developed for the general case, where the radio range of the nodes antennas can be less that twice the sensing range. Each node that moves is ensured to attain RF connectivity with the subset of nodes needed for decision making about the spot to move, so that the area coverage of the network is increased. Simulation studies were conducted verifying the efficiency of the proposed scheme. REFERENCES [] J. Cortés, S. Martinez, T. Karatas, and F. Bullo, Coverage control for mobile sensing networks, IEEE Transactions on Robotics and Automation, vol. 0, no., pp. 3 55, 00. [] J. Le Ny and G. Pappas, Sensor based robot deployment algorithms, in Proc. IEEE Conference on Decision and Control, Atlanta, GA, 00. [3] A. Panousopoulou and A. Tzes, On mobile agent positioning for wireless network reconfiguration, in Proc. 6th International Symposium on Modeling and Optimization in Mobile, Ad Hoc, and Wireless Networks and Workshops WiOPT 008, Berlin, Germany, 008, pp [] A. Panousopoulou and A. Tzes, RF-power overlapping control for connectivity awareness in wireless ad-hoc and sensor networks, in Proc. nd IFAC Workshop on Distributed Estimation and Control in Networked Systems, Annecy, France, 00, pp [5] M. Zavlanos, H. Tanner, A. Jadbabaie, and G. Pappas, Hybrid control for connectivity preserving flocking, IEEE Transactions on Automatic Control, vol. 5, no., pp , 009. [6] A. Jadbabaie, J. Lin, and A. Morse, Coordination of groups of mobile autonomous agents using nearest neighbor rules, IEEE Transactions on Automatic Control, vol. 8, no. 6, pp , 003. [7] M. Zavlanos and G. Pappas, Distributed connectivity control of mobile networks, IEEE Transactions on Robotics, vol., no. 6, pp. 6 8, 008. [8] M. Zavlanos, V. Preciado, A. Jadbabaie, and G. Pappas, Spectral control of mobile robot networks, in Proc. 0 American Control Conference, San Francisco, CA, 0. [9] M. Zavlanos, A. Tahbaz-Salehi, A. Jadbabaie, and G. Pappas, Distributed topology control of dynamic networks, in Proc. 008 American Control Conference, Seattle, WA, 008. [0] M. Zavlanos, M. Egerstedt, and G. Pappas, Graph theoretic connectivity control of mobile robot networks, Proc. of the IEEE, vol. 99, no. 9, pp , 0. [] Y. Stergiopoulos and A. Tzes, Decentralized swarm coordination: A combined coverage/connectivity approach, Journal of Intelligent and Robotic Systems, vol. 6, no. 3-, pp , 0. [] J. Stergiopoulos and A. Tzes, Decentralized communication range adjustment issues in multiagent mobile networks, in Proc. 00 American Control Conference, Baltimore, USA, 00, pp [3] J. Cortés, S. Martinez, and F. Bullo, Spatially-distributed coverage optimization and control with limited-range interactions, ESAIM: Control, Optimisation and Calculus of Variations, vol., no., pp , 005. [] S. Martinez, J. Cortés, and F. Bullo, Motion coordination with distributed information, IEEE Control Systems Magazine, vol. 7, no., pp , 007. [5] F. Bullo, J. Cortés, and S. Martinez, Distributed Control of Robotic Networks. Princeton University Press, 009. [6] F. Aurenhammer and R. Klein, Handbook of Computational Geometry. Elsevier Publishing House, 999, ch. 5: Voronoi Diagrams, pp [7] C. Godsil and G. Royle, Algebraic Graph Theory. New York: Springer- Verlag, 00. [8] J. Cortés and F. Bullo, Coordination and geometric optimization via distributed dynamical systems, SIAM Journal on Control and Optimization, vol., no. 5, pp ,