Higher order Voronoi based mobile coverage control
Abstract
Most current results on coverage control using mobile sensors require that one partitioned cell is the sole responsibility of one sensor. In this paper, we consider a class of generalized Voronoi coverage control problems by using higher order Voronoi partitions, motivated by applications that more than one senor is required to monitor and cover onecell. We introduce a framework depending on a coverage performance function incorporating higher order Voronoi cells and then design a gradientbased controller which allows the multisensor system to achieve a local equilibrium in a distributed manner. In addition, we provide a number of real world scenarios where our framework can be applied. Simulation results are also provided to show the controller performance.
I Introduction
In recent years, the performance of distributed multiagent systems (MASs) in various tasks involving cooperation has been studied with increasing intensity [1]. One fundamental problem is that optimal positioning of agents (where agents may refer to mobile sensors, vehicles, etc.) to cover an area in a way that some predefined coverage performance function can be optimized. This performance function can be related to the quality of service of a mobile sensing network, or the cumulative probability of certain events detected by sensors in the area of interest. In order to achieve the optimization goal, it is usually required that the control algorithms for different agents are carried out in a decentralized manner [2]. That is to say, agents are autonomous, capable of making decisions based on their own information, some of which could be obtained from neighbour agents. This type of mobile coverage control problem has been studied extensively in the literature since [3]. There are many papers on various extensions on this problem including obstacle avoidance [4], nonconvex area coverage control [5], coverage control with timevarying density functions [6], etc.
Typical current mobile coverage control problem is usually solved by using the geometry tool of Voronoi partitions [7]. When using this tool, each agent is responsible for monitoring a convex area determined in part by information from its neighbouring agents. Furthermore, each agent moves according to some gradientbased control law to achieve optimization of the coverage performance criterion.
In the framework of coverage control reported in most current literature, each cell in the Voronoi partition is only monitored by one agent. In this paper we consider a generalization of the classical coverage problem. The starting idea is to extend ‘one agent being responsible for one cell’ to ‘two or more agents being responsible for one single cell’. This generalization is motivated by many realworld applications where, in a coverage task, more than one agents are required to cooperatively monitor one cell. One example is bistatic radar. When deploying the bistatic radar, it is required that the transmitter and receiver are at different locations [8]. Though the transmitter is not a sensor, both the transmitter and the receiver need to be reasonably close to a potential target in order to have satisfactory detection; thus the transmitter from the coverage point of view is rather like a sensor. Another example is the geolocalization problem using TDOA sensors. In that example, at least three sensors at different noncollinear locations are required to localize an object. The current framework on coverage control is not suitable for the above applications. In this paper, we will use the concept of a higher order Voronoi partition as the main tool. Although this concept is not a recent idea, mobile coverage control using higherorder Voronoi partitions is novel to the best of our knowledge.
The rest of this paper is organized as follows. Section II reviews the problem settings and current results on the coverage control problem, and further presents some brief introduction to order Voronoi partition. Section III discusses mainly the order 2 coverage control problem, the controller design, and the stability analysis of the coverage sensor system. Simulation results are provided in Section IV to show the coverage properties of the controller. In Section V, we shows some potential applications of the high order coverage framework. Finally, Section VI concludes this paper.
Ii Background literature
Iia Order one Voronoi partition and coverage control
Suppose there is a 2D convex area to be covered by mobile sensors in this area. Note the convexity assumption we make in this paper is common in many research papers on coverage control, e.g. [3, 9]; there are also some papers which focus on nonconvex region coverage control (see e.g. [5]). A point in is denoted as and sensor ’s position is denoted by . A coverage performance function is defined as follows
(1) 
where is a distribution density function known to all sensors, denotes the Euclidean distance between and , and the function describes the measurement cost or measurement quality at a point by a sensor at . We also suppose that should be monotonically increasing and differentiable. The coverage control aims to minimize the above performance function and to find the corresponding optimal positions of mobile agents.
The minimum inside the integral of the performance function (1) induces a partition of into nonoverlapping cells. These cells are called the Voronoi partition of generated by the points defined as
(2) 
For a given set of , if and are adjacent (i.e. two cells which have a boundary either comprising an interval of nonzero length or which may just be a single point; Whichever is appropriate should be used), then agents and are defined as neighbours. The neighbor set of agent is denoted as . By using the Voronoi partition, the performance function can be transformed as
(3) 
The controller proposed in e.g. [3] is a gradientbased controller minimizing the performance function. An optimal coverage performance can be obtained by moving mobile sensor positions in accordance with the gradientbased law. The optimum may be local, not global.
IiB Order Voronoi partition
Most literature on coverage control, such as the works reviewed in Section II.A, assumes that each sensor is responsible for sensing or monitoring its own region. We term this an order one coverage control problem. Now we are going to generalize the problem to a higher order coverage problem such that each cell is defined by two or more sensors. The general literature of order Voronoi partitions includes [10, 11]. Note there are mature algorithms to compute the order Voronoi partition of a given area; see the survey paper [10] or [12].
The definition of an order Voronoi partition of a convex area is given below following [12]. Let be the set of sensors’ positions in . Suppose further that is a subset of and there are elements in . The generalized Voronoi partition is defined as
(4) 
where denotes the relative complement of with respect to . For each point in , is not further to any sensor in than to any sensor not in .
In this paper we largely focus on the order 2 coverage problem. As an example, Figure 1 shows an order 1 Voronoi partition and Figure 2 shows a corresponding order 2 Voronoi partition with the same sensor positions as Figure 1. There are some similar properties in higherorder partitions in comparison to order 1 Voronoi partitions, and we list some: (i) no two cells of overlap; (ii) the union of the cells, which are all closed, covers the convex region , and (iii) not every combination of sensors necessarily defines a cell in the partition.
For all , there are some that contain . We put all these into a set so that . Further suppose that . It is noticeable that when we put these together, we will not obtain the same cell containing in the order 1 partition. In fact, there holds . In addition, according to the definition of higher order Voronoi partition, and are both always convex but may not be convex. Other properties of higher order Voronoi partitions can be found in [13] and [14].
Iii Order two coverage control
Iiia Performance function and its relationship with generalized Voronoi partition
In this section, we are going to discuss order two coverage control so throughout this section we assume that . Define a set . We also note that the set designation of a particular as means points and . The generalized sensing performance function is constructed as follows
(5) 
We are trying to find optimal positions of the sensors that can minimize the above performance function. Similarly to an order 1 Voronoi coverage problem, the function indicates the measurement quality of a point but now by a pair of agents. Therefore, for a set of fixed sensor positions, we might measure the quality of sensing associated with those positions by . In fact, there is a broad set of for which it makes sense to formulate such a measure. We shall in fact impose certain properties on analogous to the order 1 mobile coverage control case, where being monotonically increasing and differentiable is a basic requirement. In the order 2 case, besides requiring that the function should be differentiable, it should also have the following properties


, and

The first two properties correspond to the monotonically increasing property in the order 1 case. We note that the higher order Voronoi partition is defined as (4), where the order of the elements in should not affect the actual partition. As a result, the order of independent variables should not affect the value of , either. The third requirement in the above condition ensures this property.
Based on the above performance function (5) and the distance function, we can obtain the following two lemmas.
Lemma 1
With the definitions of , , , and showed above, for all in the set and , there holds
(6) 
According to the definition of , we know each and is less than or equal to both and . Because , there holds .
Further because , there also holds . Thus the lemma is proved.
Lemma 2
By using the higher order Voronoi partition and the distance function defined above, the performance function can be further transformed as
(7) 
This lemma is a straightforward consequence of Lemma 1.
According to Lemma 1, as long as the three properties of hold, one can transform the original performance function (5) into (7) by using the order2 Voronoi concept. Different in the performance function will affect directly the optimization of the performance function, and thus the final configuration of sensors’ positions. The choice of depends on specific requirements for different applications. Here are some typical , expressed in terms of norm.

When , the sensing performance for each is related to the distance sum from to two sites and . The detailed explanation of these applications will be discussed later in Section V.

When , the sensing performance is expressed by the squared distance sum to two sites and . In this case, as we are going to show in the next section, the controller expression has a strong relationship with the centroid of each cell.

Suppose . As , there holds .
The three examples just listed correspond to the order 2 case but the idea can be generalized to higher order cases. For the generalized performance function (5), the optimal positions of sensors are dependent. In Section V, we will provide more discussion on specific applications by using different .
IiiB Controller design
IiiB1 Gradientbased controller
In order to minimize the performance function (7), we can design a controller for each sensor with position as
The mobile sensor system with the above controller defines a gradient flow of the performance function (5). According to the property of gradient systems [15], the above gradient controller provides a natural choice to optimize the performance function. Furthermore, in order to implement the above controller, each agent needs to know the position of its neighbouring agents. The neighbouring agents of are , which are agents that monitor the cells and the cells with common boundaries with these .
IiiB2 Cancellation of boundary terms
In the following, we will present an explicit formula for the controller. In our order two problem, for each , we are going to show
(8) 
In the expression of as shown in (7), the domain of integration is a function of . As a result, when one calculates the partial derivative of with respect to , one needs to deal with the problem of differentiation under the integral sign. Some basic facts about the problem of differentiating under the integral sign are given in the Appendix.
Now we are going to define some quantities in relation to order 2 Voronoi cells. Figure 3 shows illustrative representations of the notations used in the derivation below.
Let and be the boundary of and respectively. Further suppose that is the common boundary of and . Let be a point on the boundary of , and be a point on the common boundary of and . We also define as the outward facing unit normal vector of , and as the unit normal vector of from cell to .
In our problem, if we calculate the partial derivatives of with respect to each sensor position under the integral sign, we have
(9) 
Note the second line in (9) is always zero because is always stationary and its partial derivatives with respect to the entries of are always zero. By the definition of higher order Voronoi cell [16], we know and . As a result, the third and fourth lines in (9) cancel out with each other. Therefore (8) holds.
IiiB3 Relationship with cell centroids
For any given , it is not easy to find the general relationship of (8) with cell centroids of , so we first consider the case when . In this case, the performance function becomes
Note the centroid and mass of a Voronoi cell are
and
respectively. In this case,
(10) 
Suppose further that the centroid and mass of are
and
respectively. Now we have
(11) 
The above result indicates that is moving towards the centroid of . Note that this centroid in general moves when the sensors move, since the Voronoi diagram will change with moving sensors. This result is very similar to the order 1 centroid coverage control case when is defined as . Now we summarize the convergence results concerning the above gradient system as follows.
Lemma 3
For a group of mobile agents with the closedloop system induced by (8), all the agents’ positions will converge to the set of critical points of . Furthermore, by taking and designing the controller as where is a positive gain, agents’ locations will converge to the cell centroids which result in a higher order centroidal Voronoi configuration.
The proof follows directly from the properties of gradient systems (see e.g. [15]) and is omitted here.
Remark 1
In this section we mainly focus on generalized coverage control using an order 2 Voronoi partition. We mention that the coverage scheme can be extended to higher order Voronoi partitions by modifying the performance function stated in (5) with reasonable distance functions. Also, the main analysis for the 2D case can be extended straightforwardly to 3D space coverage. The controller for the 3D case takes the same form as in (8) but the derivation requires a more sophisticated analysis of boundary issues.
Remark 2
The discretized version of the original order 1 coverage control problem in the field of data analysis and image processing is called kmean clustering [17]. To the best of our knowledge, there is no previous literature on the kmean clustering problem corresponding to the ‘order 2’ coverage problem in the continuous case. In the future, we intend to use the idea of higher order Voronoi partition in the research of kmean clustering methods.
Iv Simulation results and discussion
A simulation with 50 agents is shown in Figure 4. Figure 4(a) shows the initial positions of a group of agents. The initial positions of these agents are generated randomly with a uniform distribution in the green square area. This figure also shows the order 2 Voronoi partition with these initial positions. Apart from that, Figure 4(b) shows the moving trajectory of these agents using the controller designed in Section IIIB1. Note that and . Each blue curve represents the trajectory of one agent and each red circle represents the final position of the agent. In addition, Figure 4(c) shows the final positions of the group of agents and also the final order 2 Voronoi partitions. Furthermore, Figure 4(d) shows the evolution of the value of the performance function from time to time . As shown in this figure, the agent positions reach an equilibrium in the end.
Due to the gradientbased nature of our controller, sometimes the obtained solution may be an undesired local optimum where some pair of agents are collocated. In order to avoid these undesired local optima, one can use modified controller functions to prevent collocation. One possible way is to modify the controller as , where is a control term to prevent the collocation (see e.g. [18]). Such modifications also have much practical significance. For example, the performance of a bistatic radar is very poor when the transmitter is collocated with the receiver. Furthermore, for mobile agents, it is always desirable to keep a safe distance between each pair of agents so as to avoid collision.
V Real world applications
Va Supermarket problem
As an extension of the wellknown postoffice problem [19], the supermarket problem brings in the idea of more than one supermarket serving one cell which enables competition. When selecting sites for supermarkets, it has to be considered important to ensure enough competition among supermarkets [20]. In fact, in the Australian Capital Territory, the government issued a policy saying that supermarkets should be located so that everybody is as near as possible to at least two supermarkets. Although supermarkets are not mobile, the optimization technique is still applicable. Denote the set of positions of supermarkets as in a region . For each person at position , the function is
By interpreting as the population density at each position , one can formulate the following coverage performance function
where the definition of is the same with Section III. To optimize the supermarkets locations, one should minimize the above performance function.
VB TDOA sensors and bearingonly sensors
The above supermarket problem is not strictly a mobile agents problem. Now we are going to look at some mobile sensing problems using the same distance function with the above. If one uses bearing only sensors to localize targets, at least two sensors are required to obtain a measurement. Furthermore, for complete coverage, one actually needs three noncollinear bearingonly sensors to avoid the case when the sensors and target are almost collinear. Similar to bearingonly sensors, TDOA (Time Difference of Arrival) sensors are passive sensors and at least two sensors are required to obtain a branch of a hyperbola on which a target lies. Therefore, to localize a target using TDOA sensors, three sensors at different positions are required. In this case, an order three Voronoi partition may be used to solve the problem. ^{1}^{1}1Similar to the bearingonly sensors case, for complete coverage, one actually needs four TDOA sensors to avoid undesired geometry, in which case the use of even higher order Voronoi partition is required.
In the above example, sensors all have limited sensing range and their performance will degrade as the distance from the target to the sensor increases. If one wants to use sensor arrays to cover a large area, an order 3 coverage control problem is an appropriate tool. When detecting a target, all sensors in a cell need to receive signals from the target reliably, with the performance of monitoring a cell depends on the sensor with furthest distance to the target. Thus the performance of a group of sensors monitoring a target at can be expressed as
Suppose there is a target in a twodimensional region . Suppose further that the probability of the target appearing at position is . Then the expected value of this localization performance measure is
where . Minimizing the above performance function will give the optimal positions of sensors.
VC Bistatic radar
One possible application of our order two coverage control strategy involving a general function is the coverage design using bistatic radars [21]. When deploying the a bistatic radar, it is required that the transmitter and receiver are at different locations. We suppose that each agent with position has both transmitter and receiver on board but it can only receive signals from another agent instead of itself. In this case, the probability of detection in one radar pulse is expressed by
(12) 
where the term being integrated is just the probability density of the received signal strength using a common radar receiver model; see [22]. In particular, is a chosen detection threshold, is the signal amplitude, is the noise amplitude and is the modified Bessel function of the first kind and zeroth order. The choice of depends on the acceptable probability of false alarm denoted by , i.e. the acceptable probability that the signal magnitude exceeds the threshold value when noise alone is present [22].
For a typical bistatic radar, is the signaltonoise ratio, which is proportional to where is the distance between the transmitter and the target, is the distance between the receiver and the target and depends on the radar power, antenna gain etc. Note and are equivalent to and in Section III. It can be shown that
Suppose there are a group of agents to be deployed in a convex area ; each agent with position has both transmitter and receiver on board but it can only receive signal from another agent instead of itself. There is a target in the area and the probability of the target appearing at position is . Note there holds . Now the probability of detection of the target at is and the expected value of this probability of detection is
Now let , then the performance function in Section III is stated as
The above model is only about the first detection of targets. As for the localization part of this problem, a pair of bistatic radars only gives an eclipse showing the possible positions of a target. If one needs to obtain an exact position of the target, even higher order coverage control strategy is required.
Vi Conclusions
In this paper, we considered a class of generalized Voronoi coverage control problems by introducing the higher order Voronoi partition concept in the coverage performance functions. This coverage problem is motivated by many real life applications which require more than one sensor to cooperate in monitoring one single cell. We focused on the order 2 Voronoi based coverage problem, and provided detailed analysis on the performance function, the controller design and controller performance, supported by several simulations. In addition, we provided a number of real world scenarios where our framework can be applied.
In the future, we plan to extend our results to even higher order Voronoibased coverage control with more rigorous analysis on the convergence rate. Furthermore, because simulations suggest that away from boundaries of the overall region, the cells are of some standard shape and the area is tiled uniformly, it may be possible to provide some theoretical analysis on the final shape of each cell. Apart from that, we would also like to use the idea of higher order Voronoi partitions in the research of kmean clustering, the discretized version of our problem in the field of data analysis.
Here we recall some basic facts in the problem of differentiation under the integral sign. Suppose , the integrand, is a function of both and ; suppose further , the domain of integration, is a function of . Now the derivative of the integral with respect to at the point is
(13) 
where the second term is the boundary variation. Note the independent variable could be a vector instead of a scalar in this case. According to [23], the boundary variation equals
where denotes the boundary of , denotes the direction of moving of a point on , and denotes the outward unit normal vector of the point on .
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