Collaborative Visual Area Coverage
CCollaborative Visual Area Coverage (cid:73)
Sotiris Papatheodorou, Anthony Tzes, Yiannis Stergiopoulos Abstract
This article examines the problem of visual area coverage by a network of MobileAerial Agents (MAAs). Each MAA is assumed to be equipped with a downwards fac-ing camera with a conical field of view which covers all points within a circle on theground. The diameter of that circle is proportional to the altitude of the MAA, whereasthe quality of the covered area decreases with the altitude. A distributed control lawthat maximizes a joint coverage-quality criterion by adjusting the MAAs’ spatial co-ordinates is developed. The effectiveness of the proposed control scheme is evaluatedthrough simulation studies.
Keywords:
Cooperative Control, Autonomous Systems, Area Coverage, RoboticCamera Networks
1. Introduction
Area coverage over a planar region by ground agents has been studied extensivelywhen the sensing patterns of the agents are circular [1, 2]. Most of these techniquesare based on a Voronoi or similar partitioning [3, 4, 5] of the region of interest and usedistributed optimization, model predictive control [6, 7] or game theory [8] among other techniques. There is also significant work concerning arbitrary sensing pat-terns [9, 10, 11] avoiding the usage of Voronoi partitioning [12, 13]. Both convexand non-convex domains have been examined [14, 15]. (cid:73) This work has received funding from the European Union’s Horizon 2020 Research and InnovationProgramme under the Grant Agreement No.644128, AEROWORKS. A shorter version, without the stability–notion and subsequent proof has been submitted for possible inclusion at the ICRA 2017 proceedings. The authors are with the Electrical & Computer Engineering Department, University of Patras, Rio,Achaia 26500, Greece. Corresponding author’s email: [email protected]
Preprint submitted to Robotics and Autonomous Systems September 22, 2018 a r X i v : . [ c s . S Y ] D ec any algorithms have been developed for mapping by MAAs [16, 17, 18, 19] re-lying mostly in Voronoi-based tessellations or path–planning. Extensive work has also been done in area monitoring by MAAs equipped with cameras [20, 21]. In these pio-neering research efforts, there is no maximum allowable height that can be reached bythe MAAs and the case where there is overlapping of their covered areas is consideredan advantage as opposed to the same area viewed by a single camera. There are alsostudies on the connectivity and energy consumption of MAA networks [22, 23]. In this paper the persistent coverage problem of a convex planar region by a networkof MAAs is considered. The MAAs are assumed to have downwards facing visualsensors with a conical field of view, thus creating a circular sensing footprint. Thecovered area as well as the coverage quality of that area are dependent on the altitudeof each MAA. MAAs at higher altitudes cover more area but the coverage quality is smaller compared to MAAs at lower altitudes. A partitioning scheme of the sensedregion, similar to [12], is employed and a gradient based control law is developed.This control law leads the network to a locally optimal configuration with respect toa combined coverage-quality criterion, while also guaranteeing that the MAAs remainwithin a desired range of altitudes. The main contribution of this work is the guarantee it offers that all MAAs will remain within a predefined altitude range. In addition tothat, overlapping between the sensed regions of different MAAs is avoided if possible,in contrast to previous works which consider it an advantage.The problem statement and the joint coverage–quality criterion are presented inSection 2. The chosen quality function is defined in Section 3 and the resulting sensed space partitioning scheme in Section 4. The distributed control law is derived and itsmost notable properties explained in Section 5. The stability of the altitude control lawand its property to restrict the nodes’ altitude is examined in Section 6. Simulationstudies highlighting the efficiency of the proposed control law are provided in Section7 followed by concluding remarks. . Problem Statement Let Ω ⊂ R be a compact convex region under surveillance. We assume a swarmof n MAAs, each positioned at the spatial coordinates X i = [ x i y i z i ] T , i ∈ I n , where I n = { , . . . , n } . We also define the vector q i = [ x i y i ] T , q i ∈ Ω to note the projectionof the center of each MAA on the ground. The minimum and maximum altitudes each MAA can fly to are z min i and z max i respectively, thus z i ∈ [ z min i , z max i ] , i ∈ I n . It isalso assumed that z min i > , ∀ i ∈ I n , since setting the minimum altitude to zero couldpotentially cause some MAAs to crash.The simplified MAA’s kinodynamic model is˙ q i = u i , q , q i ∈ Ω , u i , q ∈ R , ˙ z i = u i , z , z i ∈ [ z min i , z max i ] , u i , z ∈ R . (1)where [ u i , q , u i , z ] is the corresponding ‘thrust’ control input for each MAA (node). The minimum altitude z min i is used to ensure the MAAs will fly above ground obstacles,whereas the maximum altitude z max i guarantees that they will not fly out of range oftheir base station. In the sequel, all MAAs are assumed to have common minimum z min and maximum z max altitudes.As far as the sensing performance of the MAAs (nodes) is concerned, all mem-bers are assumed to be equipped with identical downwards pointing sensors with conicsensing patterns. Thus the region of Ω sensed by each node is a disk defined as C si ( X i , a ) = { q ∈ Ω : (cid:107) q − q i (cid:107)≤ z i tan a } , i = , . . . , n , (2)where a is half the angle of the sensing cone. As shown in Figure 1, the higher the altitude of an MAA, the larger the area of Ω surveyed by its sensor.The coverage quality of each node is a function f ( z i ) : [ z min , z max ] → [ , ] whichis dependent on the node’s altitude constraints z min and z max . The coverage qualityof node i is assumed to be uniform throughout its sensed region C si . The higher thevalue of f ( z i ) , the better the coverage quality. It is assumed that as the altitude of a node increases, the visual quality of its sensed area decreases. The exact definition andproperties of f ( z i ) are presented in Section 3.3or each point q ∈ Ω , an importance weight is assigned via the space density func-tion φ : Ω → R + , encapsulating any a priori information regarding the region of inter-est. Thus the coverage-quality objective is H (cid:52) = (cid:90) Ω max i ∈ I n f ( z i ) φ ( q ) dq . (3)In the sequel, we assume φ ( q ) = , ∀ q ∈ Ω but the expressions can be easily altered totake into account any a priori weight function.Figure 1: MAA-visual area coverage concept
3. Coverage quality function A uniform coverage quality throughout the sensed region C si can be used to modeldownward facing cameras [24, 25] that provide uniform quality in the whole image.The uniform coverage quality function f ( z i ) : [ z min , z max ] → [ , ] was chosen to be f ( z i ) = (cid:16)(cid:0) z i − z min (cid:1) − (cid:0) z max − z min (cid:1) (cid:17) ( z max − z min ) , q ∈ C si , q / ∈ C si A plot of this function can be seen in Figure 2 [Left]. This function was chosenso that f ( z min ) = f ( z max ) =
0. In addition, f ( z i ) is first order differentiable withrespect to z i , or ∂ f ( z i ) ∂ z i exists within C si , which is a property that will be required whenderiving the control law in Section 5. 4 Figure 2: Uniform coverage quality function [Left] and its derivative [Right].The derivative ∂ f ( z i ) ∂ z i : [ z min , z max ] → [ f min d , ] is evaluated as f d ( z i ) (cid:52) = ∂ f ( z i ) ∂ z i = (cid:0) z i − z min (cid:1) (cid:104)(cid:0) z i − z min (cid:1) − (cid:0) z max − z min (cid:1) (cid:105) ( z max − z min ) , q ∈ C si , q / ∈ C si where f min d = f d (cid:16) z min + √ (cid:0) z max − z min (cid:1)(cid:17) = − √ ( z max − z min ) . A plot of this function can be seen in Figure 2 [Right]. f ( z i ) and f d ( z i ) are 4th and 3rd degree polynomials respectively and as a resultcontinuous functions of z i . It should be noted that any strictly decreasing and dif-ferentiable with a continuous derivative function f ( z i ) : [ z min , z max ] → [ , ] can bepotentially used.
4. Sensed space partitioning
The assignment of responsibility regions to the nodes is achieved in a manner sim-ilar to [12], where only the subset of Ω sensed by the nodes is partitioned. Each nodeis assigned a cell W i (cid:52) = (cid:8) q ∈ Ω : f ( z i ) ≥ f ( z j ) , j (cid:54) = i (cid:9) (4)with the equality holding true only at the boundary ∂ W i , so that the cells W i comprise acomplete tessellation of the sensed region.5ecause the coverage quality is uniform, ∂ W j ∩ ∂ W i is either an arc of ∂ C i if z i < z j or of ∂ C j if z i > z j . In the case where z i = z j , ∂ W j ∩ ∂ W i is chosen arbitrarily as the line segment defined by the two intersection points of ∂ C i and ∂ C j . Hence, the resultingcells consist of circular arcs and line segments.If the sensing disk of a node i is contained within the sensing disk of another node j , i.e. C si ∩ C sj = C si , then W i = C si and W j = C sj \ C si . An example partitioning withall of the aforementioned cases illustrated can be seen in Figure 3 [Left], where the boundaries of the sensing disks ∂ C si are in dashed and the boundaries of the cells ∂ W i in solid black. Nodes 1 and 2 are at the same altitude so the arbitrary partitioningscheme is used. The sensing disk of node 3 contains the sensing disk of node 4 andnodes 5 , H = ∑ i ∈ I n (cid:90) W i f ( z i ) φ ( q ) dq . (5) Definition 1.
We define the neighbors N i of node i asN i (cid:52) = (cid:8) j (cid:54) = i : C sj ∩ C si (cid:54) = /0 (cid:9) . The neighbors of node i are those nodes that sense at least a part of the region that node i senses. It is clear that, due to the partitioning scheme used, only the nodes in N i need to be considered when creating W i . Remark 1.
The aforementioned partitioning is a complete tessellation of the sensedregion (cid:83) i ∈ I n C si . However it is not a complete tessellation of Ω . The neutral region notassigned by the partitioning scheme is denoted as O = Ω \ (cid:83) i ∈ I n W i . Remark 2.
The resulting cells W i are compact but they are not always convex. It isalso possible that a cell W i consists of multiple disjoint regions, such as the cell of node1 shown in red in Figure 3 [Right]. In addition it is possible that the cell of a node isempty, such as the cell of node 8 in Figure 3 [Right]. Its sensing circle ∂ C s is shown ina solid red line.
5. Spatially Distributed Coordination Algorithm
Based on the nodes kinodynamics (1), their sensing performance (2) and the cover-age criterion (5), a gradient based control law is designed. The control law utilizes thepartitioning (4) and result in monotonous increase of the covered area.
Theorem 1.
In an MAA visual network consisting of nodes with sensing performance as in (2), governed by the kinodynamics in (1) and the space partitioning described inSection 4, the control law u i , q = α i , q (cid:90) ∂ W i ∩ ∂ O n i f ( z i ) dq + ∑ j (cid:54) = i (cid:90) ∂ W i ∩ ∂ W j υ ii n i ( f ( z i ) − f ( z j )) dq (6) u i , z = α i , z (cid:90) ∂ W i ∩ ∂ O tan ( a ) f ( z i ) dq + f d ( z i ) (cid:90) W i dq + ∑ j (cid:54) = i (cid:90) ∂ W i ∩ ∂ W j ν ii · n i ( f ( z i ) − f ( z j )) dq (7) where α i , q , α i , z are positive constants, υ ii and ν ii are the Jacobian matrices of the pointsq ∈ ∂ W i with respect to q i and z i respectively and n i the outward pointing normalvector of W i , maximizes the performance criterion (5) monotonically along the nodes’ trajectories, leading in a locally optimal configuration. P ROOF . Initially we evaluate the time derivative of the optimization criterion H d H dt = ∑ i ∈ I n (cid:20) ∂ H ∂ q i ˙ q i + ∂ H ∂ z i ˙ z i (cid:21) = ∑ i ∈ I n (cid:20) ∂ H ∂ q i u i , q + ∂ H ∂ z i u i , z (cid:21) . . 7he usage of a gradient based control law in the form u i , q = α i , q ∂ H ∂ q i , u i , z = α i , z ∂ H ∂ z i will result in a monotonous increase of H .By using the Leibniz integral rule [26] we obtain ∂ H ∂ q i = ∑ i ∈ I n (cid:90) ∂ W i υ ii n i f ( z i ) dq + (cid:90) W i ∂ f ( z i ) ∂ q i dq = (cid:90) ∂ W i υ ii n i f ( z i ) dq + (cid:90) W i ∂ f ( z i ) ∂ q i dq + ∑ j (cid:54) = i (cid:90) ∂ W j υ ij n j f ( z j ) dq + (cid:90) W j ∂ f ( z j ) ∂ q i dq where υ ij stands for the Jacobian matrix with respect to q i of the points q ∈ ∂ W j , υ ij ( q ) (cid:52) = ∂ q ∂ q i , q ∈ ∂ W j , i , j ∈ I n . (8)Since ∂ f ( z i ) ∂ q i = ∂ f ( z j ) ∂ q i = ∂ H ∂ q i = (cid:90) ∂ W i υ ii n i f ( z i ) dq + ∑ j (cid:54) = i (cid:90) ∂ W j υ ij n j f ( z j ) dq whose two terms indicate how a movement of node i affects the boundary of its celland the boundaries of the cells of other nodes. It is clear that only the cells W j whichhave a common boundary with W i will be affected and only at that common boundary.The boundary ∂ W i can be decomposed in disjoint sets as ∂ W i = { ∂ W i ∩ ∂ Ω } ∪ { ∂ W i ∩ ∂ O } ∪ { (cid:91) j (cid:54) = i ( ∂ W i ∩ ∂ W j ) } . (9)These sets represent the parts of ∂ W i that lie on the boundary of Ω , the boundary of thenode’s sensing region and the parts that are common between the boundary of the cell of node i and those of other nodes. This decomposition can be seen in Figure 4 with thesets ∂ W i ∩ ∂ Ω , ∂ W i ∩ ∂ O and ∂ W i ∩ (cid:83) j (cid:54) = i ∂ W j appearing in solid red, green and bluerespectively.At q ∈ ∂ Ω it holds that υ ii = × since we assume the region of interest is static.Additionally, since only the common boundary ∂ W j ∩ ∂ W i of node i with any other node j is affected by the movement of node i , ∂ H ∂ q i can be simplified as ∂ H ∂ q i = (cid:90) ∂ W i ∩ ∂ O υ ii n i f ( z i ) dq + ∑ j (cid:54) = i (cid:90) ∂ W i ∩ ∂ W j υ ii n i f ( z i ) dq + ∑ j (cid:54) = i (cid:90) ∂ W j ∩ ∂ W i υ ij n j f ( z j ) dq . ∂ W i -decomposition into disjoint setsThe evaluation of υ ii can be found in Appendix A. Because the boundary ∂ W i ∩ ∂ W j is common among nodes i and j , it holds true that υ ij = υ ii when evaluated over itand that n j = − n i . Finally, the sums and the integrals within them can be combined,producing the final form of the planar control law ∂ H ∂ q i = (cid:90) ∂ W i ∩ ∂ O n i f ( z i ) dq + ∑ j (cid:54) = i (cid:90) ∂ W j ∩ ∂ W i υ ii n i ( f ( z i ) − f ( z j )) dq . Similarly, by using the same ∂ W i decomposition and defining ν ij ( q ) (cid:52) = ∂ q ∂ z i , q ∈ ∂ W j , i , j ∈ I n , the altitude control law is ∂ H ∂ z i = (cid:90) ∂ W i ∩ ∂ O ν ii · n i f ( z i ) dq + (cid:90) W i ∂ f ( z i ) ∂ z i dq + ∑ j (cid:54) = i (cid:90) ∂ W j ∩ ∂ W i ν ii · n i (cid:0) f ( z i ) − f ( z j ) (cid:1) dq where the evaluation of ν ii ( q ) · n i on ∂ W i ∩ ∂ O and ∂ W j ∩ ∂ W i can also be found inAppendix A. Because ∂ f ( z i ) ∂ z i is constant over W i and using the expression for ν ii ( q ) · n i from Appendix A, the control law can be further simplified into ∂ H ∂ z i = (cid:90) ∂ W i ∩ ∂ O tan ( a ) f ( z i ) dq + f d ( z i ) (cid:90) W i dq + ∑ j (cid:54) = i (cid:90) ∂ W j ∩ ∂ W i ν ii · n i (cid:0) f ( z i ) − f ( z j ) (cid:1) dq . Remark 3.
The cell W i of node i is affected only by its neighbors N i thus resulting in adistributed control law. The discovery of the neighbors N i depends on their coordinates X j , j ∈ N i and does not correspond to the classical 2D-Delaunay neighbor search. Thecomputation of the N i set demands node i to be able to communicate with all nodeswithin a sphere centered around X i and radius r ci r ci = max (cid:26) z i tan a , (cid:16) z i + z min (cid:17) tan a + (cid:16) z i − z min (cid:17) , ( z i + z max ) tan a + ( z i − z max ) (cid:27) . z min , z and z max respectively.These are the worst case scenario neighbors of node 1 , the farthest of which dictatesthe communication range r c . Figure 5: N i neighbor set Remark 4.
When z i = z max , both the planar and altitude control laws are zero because f ( z i ) =
0. This results in the MAA being unable to move any further in the future and additionally its contribution to the coverage-quality objective being zero. However thisdegenerate case is of little concern, as shown in Sections 6.3 and 6.4.
Remark 5.
The control law essentially maximizes the volume contained by the unionof all the cylinders defined by f ( z i ) , i ∈ I n , under the constraints imposed by the net-work and area of interest.
6. MAA Altitude Stability
In this section we examine the stability of the nodes’ altitude z i and show that italways remains in the interval [ z min , z max ] . The system under examination is˙ z i = u i , z , u i , z ∈ R . We will first find and characterize its equilibrium points for the case of a single nodeand then generalize to the case of multiple nodes.10 .1. Optimal altitude for a single MAA
It is useful to define an optimal altitude z opt as the altitude a node would reachif: 1) it had no neighbors ( N i = ∅ ) , and 2) its whole cell was inside the region ofinterest ( Ω ∩ W i = W i ) . When the aforementioned requirements are met it holds truethat W i = C si . This optimal altitude is the stable equilibrium point of the system˙ z i = u opti , z , u opti , z ∈ R where u opti , z = (cid:90) ∂ C si tan ( a ) f ( z i ) dq + f d ( z i ) (cid:90) C si dq = π tan ( a ) z i f ( z i ) + π tan ( a ) z i f d ( z i ) Its value and stability are examined in the following section. This altitude is con-stant and depends solely on the network’s parameters z min and z max . Had we allowedthe nodes to have different minimum and maximum altitudes, each node would have adifferent constant optimal altitude z opti .Additionally, let us denote the sensing region of a node i at z opt as C si , opt (cid:0) [ x i y i z opti ] T , a (cid:1) and H opt the value of the criterion when all nodes are located at z opt .If Ω = R and because the planar control law u i , q results in the repulsion of thenodes, the network will reach a state in which no node will have neighbors and all nodeswill be at z opt . In that state, the coverage-quality criterion (5) will have attained itsmaximum possible value H opt for that particular network configuration and coverage quality function f . This network configuration will be globally optimal.When Ω is a convex compact subset of R , it is possible for the network to reach astate where all the nodes are at z opt only if n C si , opt disks can be packed inside Ω . Thisstate will be globally optimal. If that is not the case, the nodes will converge at somealtitude other than z opt and in general different among nodes. It should be noted that although the nodes do not reach z opt , the network configuration is locally optimal. We will now evaluate z opt and its stability properties. The system under examina-tion is ˙ z i = u opti , z .
11n Appendix B it is shown that out of the five equilibrium points of this system, only two reside in the interval [ z min , z max ] . Those are z eq = z max z eq = z min + (cid:112) Q where Q = z max2 − z max z min + z min2 = (cid:16) z max − z min (cid:17) + z min2 = P + z min2 > . (10)Because the system is scalar, in order to evaluate the stability of those two equilib-rium points, it is sufficient to consider the sign of u opti , z in the interval [ z min , z max ] . Since u opti , z is continuous in [ z min , z max ] , its sign will be constant between consecutive roots of u opti , z =
0. It is shown in Appendix C that u opti , z > , ∀ z i ∈ (cid:104) z min , z eq (cid:17) u opti , z < , ∀ z i ∈ (cid:0) z eq , z max (cid:1) . This can also be seen in Figure 6 where u opti , z ( z i ) is shown in blue, the integral over W i in green and the integral over ∂ W i in red.Figure 6: Plot of u opti , z and its terms over W i and ∂ W i with respect to z i .It can now be shown that the equilibrium point z max is unstable because a smallnegative disturbance dz will result in u opti , z <
0, thus leading the node to a lower altitudeand away from z max . z eq is asymptotically stable. This is because a smallnegative disturbance dz will result in u opti , z >
0, thus leading the node to a higher altitudeand closer to z eq . Conversely, a small positive disturbance dz will result in u opti , z < z eq .To conclude, when a node has no neighbors and its whole cell is inside Ω , the only stable equilibrium point is z opt = z eq which has a domain of attraction [ z min , z max ) . In the general case, each node will move towards an altitude which is an equilibriumpoint of the system ˙ z i = u i , z , u i , z ∈ R (11)where u i , z = (cid:90) ∂ W i ∩ ∂ O tan ( a ) f ( z i ) dq + f d ( z i ) (cid:90) W i dq + ∑ j (cid:54) = i (cid:90) ∂ W i ∩ ∂ W j tan ( a ) ( f ( z i ) − f ( z j )) dq (12) We call this the stable altitude z stbi . The stable altitude is not common among nodesas it depends on one’s neighbors N i and is not constant over time since the neighborschange over time. We will attempt to generalize the proof of Section 6.2 in the case of a node withneighbors, which is the general case. The system under examination is derived fromequations (11) and (12). The integrals over ∂ W i are non–negative whereas the integralover W i is non–positive. The integrals over ∂ W i of a node with neighbors will alwaysbe smaller than the same integral of a node without neighbors. This is because the neighbors will either remove some arcs of W i from the integral or reduce their influencedue to the term f ( z i ) − f ( z j ) . Similarly, the absolute value of the integral over W i ofa node with neighbors will not be greater than the same integral of a node withoutneighbors. This is due to the area of W i possibly being reduced because part of C si hasbeen assigned to neighbors with higher coverage quality. Thus we conclude that z stbi will attain its minimum value when the integrals over ∂ W i are zero and its maximumvalue when the integral over W i is zero.When the integrals over ∂ W i are both zero, the control law u i , z has a negative value.This will lead to a reduction of the node’s altitude and in time the node will reach13 stbi = z min , provided the integrals over ∂ W i remain zero. Once the node reaches z min its altitude control law will be 0 until the integral over ∂ W i stops being zero. The planarcontrol law u i , q however is unaffected, so the node’s performance in the future is notaffected. This situation may arise in a node with several neighboring nodes at loweraltitude that result in ∂ C si ∩ ∂ W i = /0.When the integral over W i is zero, the control law u i , z has a positive value. This will lead to an increase of the node’s altitude and in time the node will reach z stbi = z max andas shown in Remark 4 the node will be immobilized from this time onwards. Howeverthis situation will not arise in practice as explained in Section 6.4.When the integral over W i and at least one of the integrals over ∂ W i are non-zero,then z stbi ∈ (cid:0) z min , z max (cid:1) . The stability of z stbi is shown similarly to the stability of z opt , by using the sign of u i , z . It is possible due to the nodes’ initial positions that the sensing disk of some node i is completely contained within the sensing disk of another node j , i.e. C si ∩ C sj = C si . In such a case, it is not guaranteed that the control law will result in separation of thenodes’ sensing regions and thus it is possible that the nodes do not reach z opt . Instead,node j may converge to a higher altitude and node i to a lower altitude than z opt , whiletheir projections on the ground q i and q j remain stationary. Because the region coveredby node i is also covered by node j , the network’s performance is impacted negatively. Since this degenerate case may only arise at the network’s initial state, care must betaken to avoid it during the agents’ deployment. Such a degenerate case is shown inFigure 3 [Left] where the sensing disk of node 4 is completely contained within that ofnode 3.Another case of interest is when some node i is not assigned a region of respon- sibility, i.e. W i = /0. This is due to the existence of other nodes at lower altitude thatcover all of C si with better quality than node i . This is the case with node 8 in Figure3 [Right]. This situation is resolved since the nodes at lower altitude will move awayfrom node i and once node i has been assigned a region of responsibility it will also14ove. It should be noted that the coverage objective H remains continuous even when node i changes from being assigned no region of responsibility to being assigned someregion of responsibility.In order for a node to reach z max , as explained in Section 6.3, the integral over W i of its altitude control law u i , z must be zero, that is its cell must consist of just a closedcurve without its interior. In order to have W i = ∂ W i , a second node j must be directly below node i at an infinitesimal distance. However just as node i starts moving upwardsthe integral over W i will stop being zero thus changing the stable altitude to some value z stbi < z max . In other words, in order for a node to reach z max , the configuration describedmust happen at an altitude infinitesimally smaller than z max . So in practice, if all nodesare deployed initially at an altitude smaller than z max , no node will reach z max in the future.
7. Simulation Studies
Simulation results of the proposed control law using the uniform coverage qualityfunction f are presented in this section. The region of interest Ω is the same as theone used in [3] for consistency. All nodes are identical with a half sensing cone angle a = ◦ and z i ∈ [ . , . ] , ∀ i ∈ I n . The boundaries of the nodes’ cells are shown insolid black and the boundaries of their sensing disks in dashed red lines. Remark 6.
It is possible to observe jittering on the cells of some nodes i and j . Thiscan happen when z i = z j and the arbitrary boundary ∂ W i ∩ ∂ W j is used. Once the alti-tude of one of the nodes changes slightly, the boundary between the cells will change instantaneously from a line segment to a circular arc. The coverage-quality objective H however will present no discontinuity when this happens. In this simulation three nodes start grouped as seen in Figure (7) [Left]. Since theregion of interest is large enough for three optimal disks C si , opt to fit inside, all the nodes converge at the optimal altitude z opt . As it can be seen in Figure 10, the area covered bythe network is equal to A (cid:16) (cid:83) i ∈ I n C si , opt (cid:17) and the coverage-quality criterion has reached15 opt = A (cid:16) (cid:83) i ∈ I n C si , opt (cid:17) . However since all nodes converged at z opt , the addition ofmore nodes will result in significantly better performance coverage and quality wise,as is clear from Figure 7 [Right] and Figure 10 [Left]. Figure 8 shows a graphical representation of the coverage quality at the initial and final stages of the simulation. Itis essentially a plot of all f ( z i ) inside the region of interest. The volume of the cylindersin Figure 8 [Right] is the maximum possible. The trajectories of the MAAs in R canbe seen in Figure 9 in red and their projections on the region of interest in black. Theinitial positions of the MAAs are marked by squares and their final positions by circles. Figure 7: Initial [Left] and final [Right] network configuration and space partitioning. x y f x y f Figure 8: Initial [Left] and final [Right] coverage quality.16 .521.5 x y z Figure 9: Node trajectories (blue) and their projections on the sensed region (black).
Time ( s ) A c ov ( % ) Time ( s ) HH o p t ( % ) Figure 10: A ( (cid:83) i ∈ In C si ) A ( Ω ) [Left] and HH opt [Right].17 .2. Case Study II A network of nine nodes, identical to those in Case Study I, is examined in thissimulation with an initial configuration as seen in Figure 11 [Left]. The region Ω is notlarge enough to contain these nine C si , opt disks and so the nodes converge at differentaltitudes below z opt . This is why the covered area never reaches A (cid:16) (cid:83) i ∈ I n C si , opt (cid:17) , which is larger than A ( Ω ) and why H never reaches H opt , as seen in Figure 14. Itcan be clearly seen though from Figure 11 [Right] and Figure 14 [Left] that the networkcovers a significant portion of Ω with better quality than Case Study I. The volume ofthe cylinders in Figure 12 [Right] has reached a local optimum. The trajectories ofthe MAAs in R can be seen in Figure 13 in red and their projections on the region of interest in black. The initial positions of the MAAs are marked by squares and theirfinal positions by circles. It can be seen from the trajectories that the altitude of somenodes was not constantly increasing. This is expected behavior since nodes at loweraltitude will increase the stable altitude of nodes at higher altitude they share sensedregions with. Once they no longer share sensed regions, or share a smaller portion, the stable altitude of the upper node will decrease, leading to a decrease in their altitude.Figure 11: Initial [Left] and final [Right] network configuration and space partitioning.
8. Conclusions
Area coverage by a network of MAAs has been studied in this article by use of acombined coverage-quality metric. A partitioning scheme based on coverage quality18 x y f x y f Figure 12: Initial [Left] and final [Right] coverage quality. x y z Figure 13: Node trajectories (blue) and their projections on the sensed region (black).19 ime ( s ) A c ov ( % ) Time ( s ) HH o p t ( % ) Figure 14: A ( (cid:83) i ∈ In C si ) A ( Ω ) [Left] and HH opt [Right].is employed to assign each MAA an area of responsibility. The proposed control law leads the network to a locally optimal configuration which provides a compromisebetween covered area and coverage quality. It also guarantees that the altitude of allMAAs will remain within a predefined range, thus avoiding potential obstacles whilealso keeping the MAAs below their maximum operational altitude and in range of theirbase station. Simulation studies are presented to indicate the efficiency of the proposed control algorithm. APPENDIX A - Evaluation of Jacobian matrices
The parametric equation of the boundary of the sensing disk C si ( X i , a ) defined in(2) is γ i ( k ) : xy = x i + z i tan ( a ) cos ( k ) y i + z i tan ( a ) sin ( k ) , k ∈ [ , π ) We will first evaluate n i , υ ii ( q ) and ν ii ( q ) on ∂ W i ∩ ∂ O which is always an arc ofthe circle γ i ( k ) because of the partitioning scheme (4). The normal vector n i is givenby n i = cos ( k ) sin ( k ) , k ∈ [ , π ) . It can be shown that υ ii ( q ) = ∂ x ∂ x i ∂ x ∂ y i ∂ y ∂ x i ∂ y ∂ y i = = I ν ii ( q ) = ∂ x ∂ z i ∂ y ∂ z i = tan ( a ) cos ( k ) tan ( a ) sin ( k ) , k ∈ [ , π ) resulting in ν ii ( q ) · n i = tan ( a ) . We will now evaluate n i , υ ii ( q ) and ν ii ( q ) on ∂ W j ∩ ∂ W i .If f ( z i ) = f ( z j ) , the evaluation of n i , υ ii ( q ) and ν ii ( q ) is irrelevant since the corre-sponding integral will be 0 due to the f ( z i ) − f ( z j ) term. If f ( z i ) > f ( z j ) , then according to the partitioning scheme (4), ∂ W j ∩ ∂ W i will bean arc of γ i ( k ) . Thus the evaluation of n i , υ ii ( q ) and ν ii ( q ) is the same as it was over ∂ W i ∩ ∂ O .If f ( z i ) < f ( z j ) , then according to the partitioning scheme (4), ∂ W j ∩ ∂ W i will bean arc of γ j ( k ) . Thus both υ ii ( q ) and ν ii ( q ) will be 0, since C sj ( X j , a ) is not dependent on X i .To sum up, the evaluation of υ ii ( q ) and ν ii ( q ) over ∂ W j ∩ ∂ W i are the following υ ii = I , z i < z j , z i ≥ z j ν ii · n i = tan ( a ) , z i < z j , z i ≥ z j where = . It is thus concluded that for the integrals over ∂ W j ∩ ∂ W i for the control law of node i ,only arcs where f ( z i ) > f ( z j ) need to be considered. APPENDIX B - Equilibrium points
The dynamical system can be written as˙ z i = u opti , z = π tan ( a ) z i [ f ( z i ) + z i f d ( z i )] . f ( z i ) and f d ( z i ) are 4th and 3rd degree polynomials respectively, the system hasfive equilibrium points, one of them being z eq = . (13)The other four are the solutions of the 4th degree polynomial 2 f ( z i ) + z i f d ( z i ) = z eq = z max z eq = z min − z max z eq = z min − (cid:112) Qz eq = z min + (cid:112) Q where Q is defined in (10), thus all equilibrium points are real.We will examine which of these equilibrium points reside in the interval D = [ z min , z max ] .Equilibrium point z eq = / ∈ D since z min > z eq = z max ∈ D .Equilibrium point z eq = z min − z max < z min thus z eq / ∈ D .Equilibrium point z eq = z min − √ Q < z min thus z eq / ∈ D . Equilibrium point z eq = z min + √ Q ∈ D since z eq > z min and z eq < z max .Thus the only equilibrium points in the interval [ z min , z max ] are z eq = z max z eq = z min + (cid:112) Q . APPENDIX C - Sign of u opti , z Since u opti , z ( z i ) is a fifth degree polynomial function, thus both u opti , z and its derivative ∂ u opti , z ∂ z i are continuous functions. As a result the sign of u opti , z will be constant between consecutive roots of u opti , z =
0. Since we are interested in the sign of u opti , z in the interval [ z min , z max ] and the only roots in that interval are z max and z eq ∈ ( z min , z max ) , as shown22n Appendix B, we just need to evaluate the sign of u opti , z in the intervals (cid:2) z min , z eq (cid:1) and (cid:0) z eq , z max (cid:1) .We will show that u opti , z > , ∀ z i ∈ (cid:2) z min , z eq (cid:1) by substituting z pi = z min + z eq into u opti , z . After tedious algebraic manipulations it can be shown that the inequality u opti , z ( z pi ) > (cid:16) z max2 − z max z min + z min2 − z min (cid:112) Q (cid:17)(cid:16) z max2 − z max z min + z min2 + z min (cid:112) Q (cid:17) > ⇒ ( P − R ) · ( P + R ) > where R (cid:52) = z min √ Q − z min2 . Since R > P > P + R > P − R > P − R = z max2 − z max z min + z min2 − z min (cid:112) Q + z min2 > ⇒ (cid:16) z max2 − z max z min + z min2 (cid:17) > z min2 Q . Substitution of Q from (10) yields P + z min2 > u opti , z ( z pi ) > u opti , z > , ∀ z i ∈ [ z min , z eq ] .We will show that u opti , z < , ∀ z i ∈ (cid:0) z eq , z max (cid:1) by evaluating the derivative of u opti , z at z max ∂ u opti , z ∂ z i ( z max ) = π ( tan a ) z max2 ( z max − z min ) > . Hence u opti , z ( z max ) = ∂ u opti , z ∂ z i ( z max ) > ∂ u opti , z ∂ z i is a continuous function and ∂ u opti , z ∂ z i ( z max ) >
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