Collaborative Visual Area Coverage using Unmanned Aerial Vehicles
CCollaborative Visual Area Coverage using Unmanned Aerial Vehicles
Sotiris Papatheodorou, Anthony Tzes and Yiannis Stergiopoulos
Abstract — This article addresses the visual area coverageproblem using a team of Unmanned Aerial Vehicles (UAVs).The UAVs are assumed to be equipped with a downward facingcamera covering all points of interest within a circle on theground. The diameter of this circular conic-section increasesas the UAV flies at a larger height, yet the quality of theobserved area is inverse proportional to the UAV’s height. Theobjective is to provide a distributed control algorithm thatmaximizes a combined coverage-quality criterion by adjustingthe UAV’s altitude. Simulation studies are offered to highlightthe effectiveness of the suggested scheme.
Index Terms — Cooperative Control, Autonomous Systems,Unmanned Aerial Vehicles
I. I
NTRODUCTION
Area coverage over a planar region by ground agents hasbeen studied extensively when the sensing patterns of therobots are circular [1, 2]. Most of these techniques are basedon a Voronoi or similar partitioning [3, 4] of the regionof interest. However there have been studies concerningarbitrary sensing patterns [5, 6] where the partitioning is notVoronoi based [7]. Both convex and non-convex domainshave been examined [8].Many algorithms have been developed for the mapping byUAVs [9–11] relying mostly in Voronoi-based tessellations.Extensive work has also been done in area monitoring byUAVs equipped with cameras [12, 13]. In these pioneeringresearch efforts, there is no maximum allowable height thatcan be reached by the UAVs and for the case where thereis overlapping of the covered areas, this is considered anadvantage compared to the same area viewed by a singlecamera.In this paper the persistent coverage problem of a planarregion by a network of UAVs is considered. The UAVs areassumed to have downwards facing visual sensors with acircular sensing footprint, while no assumptions are madeabout the type of the sensor. The covered area as well asthe coverage quality of that area depend on the altitudeof each UAV. A partitioning scheme of the sensed region,similar to [7], is employed and a gradient based controllaw is developed. This control law leads the network to alocally optimal configuration with respect to a combinedcoverage-quality criterion, while also guaranteeing that theUAVs remain within a desired range of altitudes. This article,compared to [13] assumes the flight of the UAVs within a
This work has received funding from the European Union’s Horizon2020 Research and Innovation Programme under the Grant AgreementNo.644128, AEROWORKS.The authors are with the Electrical & Computer Engineering Department,University of Patras, Rio, Achaia 26500, Greece. Corresponding author’semail: [email protected] given regime and attempts to provide visual information ofan area using a single UAV-camera.The problem statement, along with the definition of thecoverage-quality criterion and the sensed space partitioningscheme are presented in Section II. In Section III, thedistributed control law is derived and its main properties areexplained. In Section IV, the partitioning scheme and thecontrol law are examined for two particular coverage qualityfunctions. Finally, in Section V, simulations are provided toshow the efficiency of the proposed technique.II. P
ROBLEM S TATEMENT
A. Main assumptions - Preliminaries
Let Ω ⊂ R be a compact convex region under surveil-lance. Assume a swarm of n UAVs, each positioned atthe 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 tonote the projection of the center of each UAV on the ground.The minimum and maximum altitudes each UAV can fly toare z min i and z max i respectively, thus z i ∈ [ z min i , z max i ] , i ∈ I n .The simplified UAV’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 inputfor each robot (node). The minimum altitude z min i is used toensure the UAVs will fly above ground obstacles, whereasthe maximum altitude z max i guarantees that they will not flyout of range of their base station. In the sequel, all UAVsare assumed to have common minimum z min and maximum z max altitudes.As far as the sensing performance of the UAVs (nodes)is concerned, all members are assumed to be equipped withidentical downwards pointing sensors with conic sensingpatterns. Thus the region of Ω sensed by each node is adisk 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 inFigure 1, the higher the altitude of a UAV, the larger the areaof Ω surveyed by its sensor.The coverage quality of each node is in the general case afunction f i ( q ) : R → [ , ] which is dependent on the node’sposition X i as well as its altitude constraints z min , z max andthe angle a of its sensor. The higher the value of f i ( q ) , thebetter the coverage quality of point q ∈ R , with f i = z i = z min and f i = z i = z max . It is assumed that as thealtitude of a node increases, the visual quality of its sensed a r X i v : . [ c s . S Y ] D ec rea decreases. The properties of f i are examined in SectionIV along with some example functions.For each point q ∈ Ω , an importance weight is assigned viathe space density function φ : Ω → R + , encapsulating any apriori information regarding the region of interest. Thus thecoverage-quality objective is H (cid:52) = (cid:90) Ω max i ∈ I n f i ( q ) φ ( q ) dq . (3)In the sequel, we assume φ ( q ) = , ∀ q ∈ Ω but the expres-sions can be easily altered to take into account any a prioriweight function.Fig. 1: UAV-visual area coverage concept B. Sensed space partitioning
The assignment of responsibility regions to the nodes isdone in a manner similar to [7]. Only the subset of Ω sensedby the nodes is partitioned. Each node is assigned a cell W i (cid:52) = (cid:8) q ∈ Ω : f i ( q ) ≥ f j ( q ) , 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 a complete tessellation of thesensed region. Definition 1:
We define the neighbors N i of node i as N 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 leasta part of the region that node i senses. It is clear that onlythe nodes in N i need to be considered when creating W i . Remark 1:
The aforementioned partitioning is a completetessellation of the sensed region (cid:83) i ∈ I n C si , although it is not acomplete tessellation of Ω . Let us denote the neutral regionnot assigned by the partitioning scheme as O = Ω \ (cid:83) i ∈ I n W i . Remark 2:
The resulting cells W i are compact but notnecessarily convex. It is also possible that a cell W i consistsof multiple disjoint regions, such as the cell of node 1 shownin yellow in Figure 2. It is also possible that the cell of anode is empty, such as node 8 in Figure 2. Its sensing circle ∂ C s is shown in a dashed black line.By utilizing this partitioning scheme, the network’s cov-erage performance can be written as H = ∑ i ∈ I n (cid:90) W i f i ( q ) φ ( q ) dq . (5) Fig. 2: Examples of an empty cell (node 8) and one consist-ing of two disjoint regions (node 1).III. S PATIALLY D ISTRIBUTED C OORDINATION A LGORITHM
Based on the nodes kinodynamics (1), their sensing per-formance (2) and the coverage criterion (5), a gradientbased control law is designed. The control law utilizes thepartitioning (4) and result in monotonous increase of thecovered area.
Theorem 1:
In a UAV visual network consisting of nodeswith sensing performance as in (2), governed by the kinody-namics in (1) and the space partitioning described in SectionII-B, the control law u i , q = α i , q (cid:90) ∂ W i ∩ ∂ O n i f i ( q ) dq + (cid:90) W i ∂ f i ( q ) ∂ q i dq + ∑ j (cid:54) = i (cid:90) ∂ W i ∩ ∂ W j υ ii n i ( f i ( q ) − f j ( q )) dq (6) u i , z = α i , z (cid:90) ∂ W i ∩ ∂ O tan ( a ) f i ( q ) dq + (cid:90) W i ∂ f i ( q ) ∂ z i dq + ∑ j (cid:54) = i (cid:90) ∂ W i ∩ ∂ W j ν ii · n i ( f i ( q ) − f j ( q )) dq (7)where α i , q , α i , z are positive constants and n i the outwardpointing normal vector of W i , maximizes the performancecriterion (5) monotonically along the nodes’ trajectories,leading in a locally optimal configuration. Proof:
Initially we evaluate the time derivative of theoptimization criterion H d H dt = ∑ i ∈ I n (cid:20) ∂ H ∂ q i ˙ q i + ∂ H ∂ z i ˙ z i (cid:21) . . The 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 [14] we obtain H ∂ q i = ∑ i ∈ I n (cid:90) ∂ W i υ ii n i f i ( q ) dq + (cid:90) W i ∂ f i ( q ) ∂ q i dq = (cid:90) ∂ W i υ ii n i f i ( q ) dq + (cid:90) W i ∂ f i ( q ) ∂ q i dq + ∑ j (cid:54) = i (cid:90) ∂ W j υ ij n j f j ( q ) dq + (cid:90) W j ∂ f j ( q ) ∂ 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 j ( q ) ∂ q i = ∂ H ∂ q i = (cid:90) ∂ W i υ ii n i f i ( q ) dq + (cid:90) W i ∂ f i ( q ) ∂ q i dq + ∑ j (cid:54) = i (cid:90) ∂ W j υ ij n j f j ( q ) dq whose three terms indicate how a movement of node i affectsthe boundary of its cell, its whole cell and the boundariesof the cells of other nodes. It is clear that only the cells W j which have a common boundary with W i will be affectedand 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 boundaryof Ω , the boundary of the node’s sensing region and the partsthat are common between the boundary of the cell of node i and those of other nodes. This decomposition can be seenin Figure 3 with the sets ∂ W i ∩ ∂ Ω , ∂ W i ∩ ∂ O and ∂ W i ∩ (cid:83) j (cid:54) = i ∂ W j appearing in solid red, green and blue respectively.Fig. 3: ∂ W i -decomposition into disjoint setsAt q ∈ ∂ Ω it holds that υ ii = × since we assumethe 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 simplifiedas ∂ H ∂ q i = (cid:90) ∂ W i ∩ ∂ O υ ii n i f i ( q ) dq + (cid:90) W i ∂ f i ( q ) ∂ q i dq + ∑ j (cid:54) = i (cid:90) ∂ W i ∩ ∂ W j υ ii n i f i ( q ) dq + ∑ j (cid:54) = i (cid:90) ∂ W j ∩ ∂ W i υ ij n j f j ( q ) dq . The evaluation of υ ii on ∂ W i ∩ ∂ O can be found in theAppendix, whereas its evaluation on ∂ W j ∩ ∂ W i depends onthe choice of f i ( q ) and is not examined in this section.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 withinthen can be combined, producing the final form of the planarcontrol law ∂ H ∂ q i = (cid:90) ∂ W i ∩ ∂ O n i f i ( q ) dq + (cid:90) W i ∂ f i ( q ) ∂ q i dq + ∑ j (cid:54) = i (cid:90) ∂ W j ∩ ∂ W i υ ii n i ( f i ( q ) − f j ( q )) dq . Similarly, by using the same ∂ W i decomposition anddefining ν ij ( q ) (cid:52) = ∂ q ∂ z i , q ∈ ∂ W j , i , j ∈ I n , the altitude controllaw is ∂ H ∂ z i = (cid:90) ∂ W i ∩ ∂ O tan ( a ) f i ( q ) dq + (cid:90) W i ∂ f i ( q ) ∂ z i dq + ∑ j (cid:54) = i (cid:90) ∂ W j ∩ ∂ W i ν ii · n i ( f i ( q ) − f j ( q )) dq where the evaluation of ν ii ( q ) · n i on ∂ W i ∩ ∂ O and ∂ W j ∩ ∂ W i can be found in the Appendix. Remark 3:
The cell W i of node i is affected only by itsneighbors N i thus resulting in a distributed control law. Thefinding of the neighbors N i depends on their coordinates X j , j ∈ N i and does not correspond to the classical 2D-Delaunay neighbors. The computation of the N i set demandsnode i to be able to communicate with all nodes within asphere 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 − z min (cid:17) , ( z i + z max ) tan a + ( z − z max ) (cid:111) . Figure 4 highlights the case where nodes 2, 3 and 4 areat z min , z and z min respectively. These are the worst casescenario neighbors of node 1 , the farthest of which dictatesthe communication range r c . Remark 4:
When z i = z max , both the planar and altitudecontrol laws are zero because f i ( q i ) =
0. This results in theUAV being unable to move any further in the future andadditionally its contribution to the coverage-quality objectiveig. 4: N i neighbor setbeing zero. However this degenerate case is of little concern,as shown in Sections III-A and III-C. A. Stable altitude
The altitude control law u i , z moves each UAV towardsan altitude in which u i , z = z stbi and it is the solution with respect to z i ofthe equation: u i , z = ⇒ (cid:82) ∂ W i ∩ ∂ O tan ( a ) f i ( q ) dq + (cid:82) W i ∂ f i ( q ) ∂ z i dq + ∑ j (cid:54) = i (cid:82) ∂ W i ∩ ∂ W j ν ii · n i ( f i ( q ) − f j ( q )) dq = . Both integrals over ∂ W i are non-negative since their inte-grands are non-negative, whereas the integral over W i is neg-ative since coverage quality decreases as altitude increases.The stable altitude is not common among nodes as itdepends on one’s neighbors N i and is not constant over timesince the neighbors change over time.When the integrals over ∂ W i are both zero, the resultingcontrol law has a negative value. This will lead to a reductionof the node’s altitude and in time the node will reach z stb = z min . Once the node reaches z min its altitude control law willbe 0 until the integral over ∂ W i stops being zero. The planarcontrol law however is unaffected, so the node’s performancein the future is not affected. This situation may arise in a nodewith several neighboring nodes at lower altitude that resultin ∂ C si ∩ ∂ W i = /0.When the integral over W i is zero, the resulting controllaw has a positive value. This will lead to an increase of thenode’s altitude and in time the node will reach z stb = z max andas shown in Remark 4 the node will be immobilized fromthis time onwards. However this situation will not arise inpractice as explained in Section III-C.When the integral over W i and at least one of the integralsover ∂ W i are non-zero, then z stb ∈ (cid:0) z min , z max (cid:1) . B. Optimal altitude
It is useful to define an optimal altitude z opt as the altitudea node would reach if: 1) it had no neighbors ( N i = ∅ ) , and 2)no part of W i on ∂ Ω , ( ∂ Ω ∩ ∂ C si = ∅ ) . This optimal altitudeis the solution with respect to z i of the equation (cid:90) ∂ C si tan ( a ) f i ( q ) dq + (cid:90) W i ∂ f i ( q ) ∂ z i dq = . Additionally, let us denote the sensing region of a node i at z opt as C si , opt and H opt the value of the criterion when allnodes are located at z opt .If Ω = R and because the planar control law u i , q results inthe repulsion of the nodes, the network will reach a state inwhich no node will have neighbors and all nodes will be at z opt . In that state, the coverage-quality criterion (5) will haveattained it’s maximum possible value H opt for that particularnetwork configuration and coverage quality function f i . Thisnetwork configuration will be globally optimal.When Ω is a convex compact subset of R , it is possiblefor the network to reach a state where all the nodes are at z opt only if n C si , opt disks can be packed inside Ω . This statewill be globally optimal. If that is not the case, the nodeswill converge at some altitude other than z opt and in generaldifferent among nodes. It should be noted that although thenodes do not reach z opt , the network configuration is locallyoptimal. C. Degenerate cases
It is possible due to the nodes’ initial positions that thesensing disk of some node i is completely contained withinthe sensing disk of another node j , i.e. C si ∩ C sj = C si . In sucha case, it is not guaranteed that the control law will resultin separation of the nodes’ sensing regions and thus it ispossible that the nodes do not reach z opt . Instead, node j may converge to a higher altitude and node i to a loweraltitude than z opt , while their projections on the ground q i and q j remain stationary. Because the region covered by node i is also covered by node j , the network’s performance isimpacted negatively. Since this degenerate case may onlyarise at the network’s initial state, care must be taken toavoid it during the robots’ deployment. Such a degeneratecase is shown in Figure 5 where the sensing disk of node 4is completely contained within that of node 3.Another case of interest is when some node i is notassigned a region of responsibility, i.e. W i = /0. This is due tothe existence of other nodes at lower altitude that cover all of C si with better quality than node i . This is the case with node8 in Figure 2. This situation is resolved since the nodes atlower altitude will move away from node i and once node i has been assigned a region of responsibility it will also move.It should be noted that the coverage objective H remainscontinuous even when node i changes from being assignedno region of responsibility to being assigned some region ofresponsibility.In order for a node to reach z max , as explained in SectionIII-A, the integral over W i of its altitude control law mustbe zero, that is its cell must consist of just a closed curvewithout 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 upwards the integralover W i will stop being zero thus changing the stable altitudeto some value z stb < z max . In other words, in order for a nodeto reach z max , the configuration described must happen at analtitude infinitesimally smaller than z max . So in practice, ifll nodes are deployed initially at an altitude smaller than z max , no node will reach z max in the future.Fig. 5: Sensed space partitioning using the coverage qualityfunctions f ui [Left] and f pi [Right].IV. C OVERAGE QUALITY FUNCTIONS
The choice of coverage quality function affects both thesensed space partitioning (4) and the control law (7). Thefunction f i , i ∈ I n is required to have the following properties1) f i ( q ) = , ∀ q / ∈ C si , f i ( q ) ≥ , ∀ q ∈ ∂ C si , f i ( q ) is first order differentiable with respect to q i and z i , or ∂ f i ( q ) ∂ q i and ∂ f i ( q ) ∂ z i exist within C si ,4) f i ( q ) is symmetric around the z -axis,5) f i ( q ) is a decreasing function of z i ,6) f i ( q ) is a non-increasing function of (cid:107) q − q i (cid:107) ,7) f i ( q i ) = z i = z min and f i ( q i ) = z i = z max .The need for the first property is clear since the coveragequality with respect to node i should be zero outside C si .The second guarantees that the integrals of the control lawover some part of ∂ C si are not zero and the third is requiredso that the integrals over W i can be defined. The fourthproperty is due to the fact that the sensing patterns arerotation invariant. Two indicative coverage quality functionsare examined in the sequel along with the properties of theresulting partitioning schemes and control laws. A. Uniform coverage quality
The simplest coverage quality function that can be chosenis the uniform one, i.e. the coverage quality is the same for allpoints in C si . This function can effectively model downwardfacing cameras [15, 16] that provide uniform quality in thewhole image. This function will be referred to as f ui fromnow on and is defined as f ui ( q ) = (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 It should be noted that the above definition of a uniformcoverage quality function is not unique, just the simplest ofthe possible ones.A projection of this function on the y − z plane can be seenin the left portion of Figure 6 for a node at X i = (cid:2) , , z min (cid:3) T .It can be shown that f ui ( q ) = (cid:107) q − q i (cid:107) > z i tan a , i.e.when q is outside the sensing disk. −0.15 −0.1 −0.05 0 0.05 0.1 0.1500.20.40.60.81 x f u i −0.15 −0.1 −0.05 0 0.05 0.1 0.1500.20.40.60.81 x f p i Fig. 6: Coverage quality functions: f ui ( q ) [Left], f pi ( q ) [Right] for different values of b .This choice of quality function simplifies the space parti-tioning since ∂ 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 chosenarbitrarily as the line segment defined by the two intersectionpoints of ∂ C i and ∂ C j . The resulting cells consist of circulararcs and line segments.If the sensing disk of a node i is contained within thesensing 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 with all of theabove cases shown can be seen in Figure 5 [Left], wherethe boundaries of the sensing disks ∂ C si are in red and theboundaries of the cells ∂ W i in black. Nodes 1 and 2 are atthe same altitude so the arbitrary partitioning scheme is used.The sensing disk of node 3 contains the sensing disk of node4 and nodes 5 , ∂ f i ( q ) ∂ q i =
0, the corresponding integral of the control law (7)is 0. Additionally, υ ii and ν ii n i are evaluated as υ ii = (cid:40) I , z i ≤ z j , z i > z j (10) ν ii · n i = (cid:40) tan ( a ) , z i ≤ z j , z i > z j (11)Because both υ ii and ν ii · n i are zero when z i > z j ⇒ f ui < f uj ,the integrals over ∂ W i ∩ ∂ W j are zero for both the planar andaltitude control laws when f ui < f uj and as a result need onlybe evaluated when f ui > f uj .The control law essentially maximizes the volume con-tained by all the cylinders defined by f ui , i ∈ I n , under theconstraints imposed by the network and area of interest.While the control law in the general case guarantees that z i < z max , ∀ i ∈ I n , this particular one also guarantees that z i > z min , ∀ i ∈ I n . If z i = z min , then it follows that ∂ f i ( q ) ∂ z i = W i being zero. Since f ui ispositive and only arcs of ∂ W i where f ui > f uj are integratedover, the remaining part of the control law is positive,resulting in the node moving to a higher altitude. B. Decreasing coverage quality
A more realistic assumption for some types of visualsensors would be for the coverage quality to decrease asdistance from the node increases, to take into account lensdistortion for example. The coverage quality of a point q ∈ C si is maximum directly below the node and decreasesas (cid:107) q − q i (cid:107) increases. One such function is an invertedaraboloid whose maximum value depends on z i and couldbe defined as f pi ( q ) = (cid:34) − − b [ z i tan ( a )] (cid:104) ( x − x i ) + ( y − y i ) (cid:105)(cid:35) f ui , q ∈ C si , q / ∈ C si where b ∈ ( , ) is the coverage quality on ∂ C i as a per-centage of the coverage quality on q i and f ui is the uniformcoverage quality function defined previously and is used toset the maximum value of the paraboloid. A projection of thisfunction on the y − z plane can be seen in the right portionof Figure 6 for a node at X i = [ , , z max ] T and various b -values; b ∈ { , . , . , } corresponds to blue, red, greenand black colored curves, respectively. It can be shown that f pi ( q ) = (cid:107) q − q i (cid:107) > z i tan a , i.e. when q is outsidethe sensing disk. Additionally, when (cid:107) q − q i (cid:107) = z i tan a , f pi = b f ui .For the space partitioning it is necessary to solve f pi ( q ) ≥ f pj ( q ) in order to find part of ∂ W j ∩ W i . The resulting solutionon the x − y plane is a disk C ii , j when z i (cid:54) = z j and the halfplanedefined by the perpendicular bisector of q i and q j when z i = z j . Finally, the cell of node i with respect to node j is W i = C si ∩ C ii , j if z i < z j and W i = C si \ (cid:16) C sj ∩ C ii , j (cid:17) if z i > z j . Figure7 shows the intersection of two paraboloids f pi and f pj with z i < z j and the boundaries of the resulting cells above them.The red line shows ∂ C ii , j ∩ (cid:16) C si ∩ C sj (cid:17) , i.e. the boundary ofthe intersection disk constrained within the sensing disks ofthe two nodes, since only for q ∈ C si ∩ C sj can f pi ( q ) = f pj ( q ) .The green arcs represent ∂ W i ∩ ∂ C si ∩ W j while the blue onesrepresent ∂ W i ∩ ∂ O . Thus the boundary of the cell of anynode i can be decomposed into disjoint sets as ∂ W i = (cid:40) ( ∂ W i ∩ ∂ O ) (cid:91) j ∈ N i (cid:16) ( ∂ W i ∩ ∂ C ii , j ) ∩ ( ∂ W i \ ∂ C ii , j ∩ W j ) (cid:17)(cid:41) . Fig. 7: ∂ W i -decomposition when f pi is used.If the sensing disk of a node i is contained within thesensing disk of another node j , i.e. C si ∩ C sj = C si , then W i ⊂ C si and W j = C sj \ W i . It is important to note that in the generalcase, ∂ W j ∩ ∂ W i consists of circular arcs from both ∂ C ii , j and ∂ C si or ∂ C sj (depending on the relationship of z i and z j ).An example partitioning with all of the above cases showncan be seen in Figure 5 [Right], where the boundaries of the sensing disks ∂ C si are in red and the cells W i in black. Thesensing disk of node 3 contains the sensing disk of node 4and nodes 5 , υ ii and ν ii · n i the partitioning of ∂ W i explained previously will be used. Since ∀ q ∈ ∂ C ii , j ⇒ f pi ( q ) = f pj ( q ) , the integral over ∂ C ii , j is zero. For the remain-ing part of ∂ W i ∩ W j which is ∂ W i \ ∂ C ii , j ∩ W j , the evaluationof υ ii and ν ii · n i is the same as in the case of the uniformcoverage function f ui . For the same reasons as in the uniformcoverage quality, the integrals over ∂ W i ∩ ∂ W j are zero forboth the planar and altitude control laws when f pi < f pj andas a result need only be evaluated when f pi > f pj .The control law essentially leads to the maximization ofthe volume contained by all the paraboloid-like functions f pi , i ∈ I n , under the constraints imposed by the network andarea of interest.This control law also guarantees that that z i > z min , ∀ i ∈ I n in addition to z i < z max , ∀ i ∈ I n . Because f pi is positive andonly arcs of ∂ W i where f pi > f pj are integrated over, theintegral over ∂ W i is always positive when z i = z min . It canalso be shown that ∂ f p ( q ) ∂ z i > , ∀ q ∈ W i when z i = z min and thusthe integral over W i of u i , z will also be positive. As a resultthe altitude control law is always positive when z i = z min resulting in the altitude increase of node i .V. S IMULATION S TUDIES
Simulation results of the proposed control law using theuniform coverage quality function f ui are presented in thissection. The region of interest Ω is the same as the one usedin [3] for consistency. All nodes are identical with a halfsensing cone angle a = ◦ and z i ∈ [ . , . ] , ∀ i ∈ I n . Theboundaries of the nodes’ cells are shown in black and theboundaries of their sensing disks in red. Remark 5:
When using f ui ( q ) it is possible to observejittering on the cells of some nodes i and j . This can happenwhen z i = z j and the arbitrary boundary ∂ W i ∩ ∂ W j is used.Once the altitude of one of the nodes changes slightly,the boundary between the cells will change instantaneouslyfrom a line segment to a circular arc. The coverage-qualityobjective H however will present no discontinuity when thishappens. A. Case Study I
In this simulation three nodes start grouped as seen inFigure (8) [Left]. Since the region of interest is large enoughfor three optimal disks C si , opt to fit inside, all the nodesconverge at the optimal altitude z opt . As it can be seenin Figure 11, the area covered by the network is equal to A (cid:0) (cid:83) i ∈ I n C si (cid:1) and the coverage-quality criterion has reached H opt = A (cid:16) (cid:83) i ∈ I n C si , opt (cid:17) . However since all nodes convergedat z opt , the addition of more nodes will result in significantlybetter performance coverage and quality wise, as is clearfrom Figure 8 [Right] and Figure 11 [Left]. Figure 9 shows agraphical representation of the coverage quality at the initialand final stages of the simulation. It is essentially a plot ofall f ui ( q ) inside the region of interest. The volume of theig. 8: Initial [Left] and final [Right] network configurationand space partitioning.cylinders in Figure 9 [Right] is the maximum possible. Thetrajectories of the UAVs in R can be seen in Figure 10 inblue and their projections on the region of interest in black.Fig. 9: Initial [Left] and final [Right] coverage quality. z Fig. 10: Node trajectories (blue) and their projections on thesensed region (black).
Time ( s ) A c ov ( % ) Time ( s ) HH o p t ( % ) Fig. 11: A ( (cid:83) i ∈ In C si ) A ( Ω ) [Left] and HH opt [Right]. B. Case Study II
A network of nine nodes, identical to those in Case StudyI, is examined in this simulation with an initial configurationas seen in Figure 12 [Left]. The region Ω is not large enoughto contain these nine C si , opt disks and so the nodes convergeat different altitudes below z opt . This is why the covered areanever reaches A (cid:0) (cid:83) i ∈ I n C si (cid:1) , which is larger than A ( Ω ) and why H never reaches H opt , as seen in Figure 15. It canbe clearly seen though from Figure 12 [Right] and Figure15 [Left] that the network covers a significant portion of Ω with better quality than Case Study I. The volume of thecylinders in Figure 13 [Right] has reached a local optimum.The trajectories of the UAVs in R can be seen in Figure14 in blue and their projections on the region of interest inblack. It can be seen from the trajectories that the altitude ofsome nodes was not constantly increasing. This is expectedbehavior since nodes at lower altitude will increase the stablealtitude of nodes at higher altitude they share sensed regionswith. Once they no longer share sensed regions, or share asmaller portion, the stable altitude of the upper node willdecrease, leading to a decrease in their altitude.Fig. 12: Initial [Left] and final [Right] network configurationand space partitioning.Fig. 13: Initial [Left] and final [Right] coverage quality. z Fig. 14: Node trajectories (blue) and their projections on thesensed region (black).VI. C
ONCLUSIONS
Area coverage by a network of UAVs has been studied inthis article by use of a combined coverage-quality metric. Apartitioning scheme based on coverage quality is employedto assign each UAV an area of responsibility. The proposed
Time ( s ) A c ov ( % ) Time ( s ) HH o p t ( % ) Fig. 15: A ( (cid:83) i ∈ In C si ) A ( Ω ) [Left] and HH opt [Right].control law leads the network to a locally optimal configu-ration which provides a compromise between covered areaand coverage quality. It also guarantees that the altitude of allUAVs will remain within a predefined range, thus avoidingpotential obstacles while also keeping the UAVs belowtheir maximum operational altitude. Simulation studies arepresented to indicate the efficiency of the proposed controlalgorithm. APPENDIXThe parametric equation of the boundary of the sensingdisk C si ( X i , a ) defined in (2) is γ ( k ) : (cid:20) xy (cid:21) = (cid:20) x i + z i tan ( a ) cos ( k ) y i + z i tan ( a ) sin ( k ) (cid:21) , k ∈ [ , π ) We will now evaluate n i , υ ii ( q ) and ν ii ( q ) on ∂ W i ∩ ∂ O which is always an arc of the circle γ ( k ) .The normal vector n i is given by n i = (cid:20) cos ( k ) sin ( k ) (cid:21) , k ∈ [ , π ) It can be easily shown that υ ii ( q ) = (cid:34) ∂ x ∂ x i ∂ x ∂ y i ∂ y ∂ x i ∂ y ∂ y i (cid:35) = I and similarly that ν ii ( q ) = (cid:34) ∂ x ∂ z i ∂ y ∂ z i (cid:35) = (cid:20) tan ( a ) cos ( k ) tan ( a ) sin ( k ) (cid:21) . It is now clear that ν ii ( q ) · n i = tan ( a ) . The evaluation of n i , υ ii ( q ) and ν ii ( q ) on ∂ W j ∩ ∂ W i depends on the choice of f ( q ) and the partitioning schemedescribed in Section II-B.When f i ( q ) = f j ( q ) , the evaluation of n i , υ ii ( q ) and ν ii ( q ) is irrelevant since the corresponding integral will be 0 dueto the f i ( q ) − f j ( q ) term.When f i ( q ) > f j ( q ) , due to the partition scheme, thecommon sensed region between nodes i and j will beassigned to node i . As a result, ∂ W i ∩ ∂ W j will be an arc of ∂ C si and will change when node i moves. Thus the evaluationof n i , υ ii ( q ) and ν ii ( q ) will be the same as their evaluationover ∂ W i ∩ ∂ O described previously. When f i ( q ) < f j ( q ) , due to the partition scheme, thecommon sensed region between nodes i and j will beassigned to node j . As a result, ∂ W i ∩ ∂ W j will be an arcof ∂ C sj which will not change when node i moves. Thus n i , υ ii ( q ) and ν ii ( q ) are both zero in this case.It is thus concluded that for the integrals over ∂ W i ∩ ∂ W j ofnode i , only arcs where f i ( q ) > f j ( q ) need to be consideredwhen integrating. R EFERENCES[1] J. Cort´es, S. Martinez, and F. Bullo, “Spatially-distributed coverageoptimization and control with limited-range interactions,”
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