Mobility Increases the Surface Coverage of Distributed Sensor Networks

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Mobility Increases the Surface Coverage of Distributed Sensor Networks

Xiao-Yang Liuâ^∗, Kai-Liang Wu^b, Yanmin Zhuâ, Linghe Kongâ, Min-You Wuâ

aDepartment of Computer Science and Engineering, Shanghai Jiao Tong University, Shanghai, China

bSchool of Mathematical Science, Peking University, Beijing, China

Abstract

Coverage is a fundamental problem in sensor networks, which usually dictates the overall network performance. Previous studies on coverage issues mainly focused on sensor networks deployed on a 2D plane or in 3D space. However, in many real world applications, the target fields can be complex 3D surfaces where the existing coverage analysis methodology cannot be applied. This paper investigates the coverage of mobile sensor networks deployed over convex 3D surfaces. This setting is highly challenging because this dynamic type of coverage depends on not only sensors’ movement but also the characteristics of the target field. Specifically, we have made three major contributions. First, we generalize the previous analysis of coverage in the 2D plane case. Second, we derive the coverage characterization for the sphere case. Finally, we next consider the general convex 3D surface case and derive the coverage ratio as a function of sensor mobility, sensor density and surface features. Our work timely fills the blank of coverage characterization for sensor networks and provides insights into the essence of the coverage hole problem.

Numerical simulation and real-world evaluation verify our theoretical results. The results can serve as basic guidelines for mobile sensor network deployment in applications concerning complex sensing fields.

Keywords: Mobility, Surface Coverage, Distributed Sensor Networks

1. Introduction

Sensor networks are widely deployed in many commercial and military scenarios because of their unique advantages, such as low cost, ease of deployment and unattended operation. Typ- ical applications include tracking wild animals [1][2], forest fire detection [3], forest carbon monitoring [4], volcano surveillance [5] and environmental data reconstruction [6][7].

Ensuring coverage is a fundamental problem in sensor networks and one of the main considerations for system designers [8]. The coverage of a sensor network answers important ques- tions, e.g., what are the traces of the moving objects? What are the chances that an abnormal event like an intrusion will be detected during its lifetime? How well can the sensor network monitor a target field? How accurate is it if the sampled data are used to virtually reconstruct the environmental conditions of the field? Furthermore, the coverage property closely relates to the surveillance quality of a sensor network, the monitoring ability of an intrusion detection system and the connectivity of a k-hop clustered mobile wireless network [9]. Thus, it is importan- t to understand the relationship between coverage and system parameters including the sensor density, sensors’ mobility and

∗Corresponding author

Email addresses: yanglet@sjtu.edu.cn (Xiao-Yang Liuâ^∗), wukl@pku.edu.cn(Kai-Liang Wu^b), yzhu@sjtu.edu.cn (Yanmin Zhuâ), linghe.kong@sjtu.edu.cn (Linghe Kongâ),

mwu@sjtu.edu.cn(Min-You Wu^a)

field’s properties. This will help designers better deploy sensor networks for various practical applications concerning complex sensing fields.

Recent years have witnessed the increasing adoption of mobile sensor networks. Sensors can be mounted on autonomous robots, such as the Pioneer 3DX [10] and the Starburg [11], or be mounted on wild animals [1][2]. System designers em- brace mobile sensors since mobility enables self-deployment and adaptability. For example, in a hostile environment where sensors cannot be manually deployed, mobile sensors can move to the desired positions during the redeployment phase [12][13][14]; in ocean environments where sensors move with the surrounding ocean currents [11], mobile sensors can adap- t to the floating water. Moreover, mobility can be exploited to compensate for the insufficient number of sensors, to improve the area coverage of a randomly deployed sensor network [18][19] and to optimize the data collection operation [6]. Re- cent studies have already shown that mobility can increase communication capacity [20], network connectivity [9] and security [21] in ad hoc networks.

1.1. Motivation

For the coverage characterization, most existing works assume that the target field is a 2D plane or 3D space. However, in many real world applications, the fields of interest (FoI s) are complex 3D surfaces (Fig 1(a)). Such examples appear in the ZebraNet project [1], the GreenOrbs project [4], and the Tun- gurahua volcano monitoring project [5]. In a 2D plane or 3D

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(a)

Coverage Hole

Surface

2D Plane

(b)

Figure 1: a). An example of a mobile sensor network deployed over a 3D surface; b). A cross section view of the coverage hole problem: model the surface as a 2D plane, the deployment strategy that achieves full coverage on the 2D plane, however, will leave holes for the surface because sensors can only be positioned on the exposed area of the surface.

space, sensors can move freely within the whole FoI, but for 3D surfaces, sensors are restricted to move only on the surface.

This implies that existing results derived under the 2D plane or 3D space model may be inappropriate when being applied to the 3D surface case.

Surface coverage is first introduced by M. Zhao et al. [22]

and extended in [23]. They point out that coverage strate- gies derived on 2D planes do not work for 3D surfaces since they will encounter the coverage hole problem (Fig 1(b)). In [22][23], however, only surface coverage of static sensor networks is studied. The coverage ratio is determined by the initial network configuration and remains unchanged over time.

B. Liu et al. [18] have analyzed the coverage of mobile sensor networks, but they concentrate on the 2D plane case and their results cannot be directly applied to the 3D surface case since their analysis fails to take into account the properties of the FoI.

As a result, it is still not clear how mobility affects the coverage of mobile sensor networks on 3D surfaces.

To fill this gap, we study the surface coverage of a mobile sensor network deployed over a convex 3D surface. Specifical- ly, we are interested in the surface coverage ratio over a time period, which is achieved by the continuous movement of sensors.

Unlike traditional approaches that aim to provide simultaneous coverage of all locations at each time instant, or exploit mobility to obtain a new stationary configuration that improves the coverage ratio after the sensors move to the desired positions, the surface coverage in mobile scenarios aims at covering the locations once during an event’s lifetime. This kind of surface coverage can be reduced to the scenario in [22][23] by making the sensors move at zero speed. Characterizing the surface coverage ratio of a mobile sensor network requires comprehensive consideration of the initial network configuration, features of the convex 3D surface and dynamic aspects of sensors’ movement. In contrast, the stationary 2D plane coverage, mobile 2D plane coverage and stationary surface coverage consider only one or two of those three aspects.

1.2. Our Contributions

The main contributions in this paper are summarized as following:

• To the best of our knowledge, this is the first attempt at characterizing the surface coverage of distributed mobile

sensor networks. We propose a theoretical analysis framework for coverage studies on general convex 3D surfaces.

• We derive theoretical results for 2D planes, spheres and general convex 3D surfaces under three mobility models.

Our results show that mobility increases the surface coverage.

• Numerical simulation and real-world evaluation testify the accuracy of our theoretical results. Results using a 2D plane model perform poorly for 3D surfaces due to the coverage hole problem, which verifies our motivation.

• Our theoretical results provide insights into the essence of the coverage hole problem: the nonzero Gaussian curvature is the root cause for the invalidity of the 2D plane model for the surface coverage case.

The paper is organized as follows. Section 2 gives a brief review of related works, then in Section 3 we summarize our main results and give corresponding interpretations. The network model and coverage metrics are introduced in Section 4.

Section 5 is devoted to consider the 2D plane case, while the sphere case is presented in Section 6. Section 7 shows our analysis framework for general surfaces, followed by our simulation and evaluation results in Section 8. In Section 9, we give a brief discussion, then conclude our work and point out possible directions for future work.

2. Related Works

Coverage of sensor networks has been extensively studied.

Existing works on coverage can be divided into two categories:

those focusing on stationary sensor networks and those focusing on mobile sensor networks. For the first class, various type- s of coverage have been investigated, such as area coverage [24][8][25][26], barrier coverage [27] and path coverage [28].

For the second class, two mobility models have been investigated: limited mobility [10][12][19][13][14], which assumes that the sensors can move only once over a short distance, and continuous mobility [11][18][29]. More thorough surveys on the coverage problem are provided by [28][30].

For stationary sensor networks, there are mainly four kinds of FoI models used: strip-shaped barrier, 3D space, 2D plane and 3D surface. Barrier coverage seeks to minimize the probability of undetected network penetrations [27]. 3D full space coverage [34][26] differs fundamentally from 3D complex surface coverage, because in the latter case sensors can only be deployed on the exposed surface area, not freely within the w- hole target FoI. For 2D plane coverage, Meguerdichian et al. [8] consider the coverage as a measure of the quality of service (QoS) of the sensor networks and design a robust, efficient, and scalable polynomial-time algorithm for connectivity and coverage based on Voronoi diagrams and Delaunay triangulations;

P.-J. Wan and C.-W. Yi [24] address the asymptotic k-coverage of a randomly deployed sensor network. Still, proposed solu- tions under the 2D plane model have found a wide range of applications and some of them can be easily adapted to the 3D

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full space case. However, all of these results derived from the 2D plane and then applied to 3D complex surface, suffer from the coverage hole problem [22][23]. Thus, M.-C. Zhao et al. [22] propose using the concept of surface coverage and provide analytical results for the coverage ratio.

For mobile sensor networks, some work focuses on redeploy- ing sensors to achieve a static configuration with better cov- erage. Based on potential fields, A. Howard et al. [12] model sensor nodes as virtual particles under the control of virtual forces. The virtual forces repel sensors away from each other and make sensors spread out, leading to a maximized cover- age area. Y. Zou et al. [13] propose another virtual-force-based algorithm to improve the coverage of an initial randomly deployed sensor network and this virtual force is defined according to the distance between the sensor and the other sensors or obstacles. G. Wang et al. [14] propose several algorithms to identify coverage holes and compute the desired locations that can increase the coverage ratio the most. Those proposed algorithms strive to maximize the covered area in a redeployment phase which ends with sensors moving to form a new static configuration. The main difference is how exactly this new configuration is computed in an efficient manner.

Besides the work in [22][23], several recent papers also study the surface coverage problem. A distributed algorithm [15] is proposed to produce triangulations for arbitrary 3D surfaces.

Further, [16] studies the optimal solution for 3D surface sensor deployment with minimized overall unreliability. It also design- s a series of excellent algorithms for practical implementation.

This study focused on sensor networks under the deterministic deployment. Liu et al. [17] derive the expected coverage ratio for irregular terrains using the cone/cos model and for irregular terrains using the digital elevation model. It focuses on sensor networks under the stochastic deployment. All of them target at stationary sensor networks.

B. Liu et al. [18] consider the coverage over an time interval resulting from the continuous movements of sensors, only on 2D planes. For 3D surfaces, the dynamic characteristics of the surface coverage of mobile sensor networks are left untouched.

Our work fills the gap. We share a similar proof technique with [18], but the 3D surfaces are mathematically much more different and difficult compared with the 2D plane. Fortunately, the results in this paper are consistent with those on the static 2D plane case [24], the mobile 2D plane case [18] and the stationary 3D surface case [22]. This paper is the first to verify that the intuition introduced by [18] on the 2D plane also holds on the convex 3D surface: mobility increases the coverage of sensor networks.

3. Main Results

This section presents the main results and the corresponding implications. The notations are listed in TABLE 1, and read- ers can refer to it easily. Throughout the paper,∥·∥ denotes the area of a region, or the arc length of a curve; d(x1, x2) denotes the Euclidean distance between points x1and x2; A^′denotes the complement of set A. We define a functionGi(kn, r, s) to char- acterize the properties of the FoI and the mobility of a sensor,

Table 1: Notations Symbol Definition

S The FoI, being a convex 3D surface or 2D plane.

C The whole network or the locations of sensors.

si The i^thsensor.

r Sensing range.

λ Network density.

v Sensors’ speed, being a constant.

K Gaussian curvature of a surface.

t A time instant in [0, τ).

[0, τ) The time interval of interest.

[0, vτ) The trace of a sensor over [0, τ).

k(s) Curvature of a curve.

kg(s) Geodesic curvature at s.

kn(s) Normal curvature at s.

kn(s) Conjugated normal curvature at s.

Gc,si, G^t_c,s_i The region covered by siat t.

Gc, G^tc The covered region and uncovered region at t.

G^τ_c_,s_i The region covered by siover [0, τ).

G^τ_c The covered region over [0, τ).

f (t), F(τ) Coverage ratio at t, over [0, τ).

K, U Surface convex sets.

with the following form:

Gi(kn, r, s)

=kn(s) kn(s)

r

È

4− (kn(s)r)²+ 4 kn(s)− kn(s) kn(s)kn(s) arcsin

_k

n(s)r 2

. (1)

Intuitively,Gi(k_n, r, s) indicates the local smoothness of the sur- face at the nearby region of point s. It quantitatively measures the bending degree of the surface area within the sensing range of a sensor node. Gi(kn, r, s) has small value in sharp regions, and big value in smooth regions.

Our main theoretical results are:

• We generalize the results in the 2D plane case [18] by con- sidering sensors moving along general curves. The ex- pected area of the region covered by a mobile sensor si

over the time period [0, τ) is ∥G^τc,si∥ = πr²+ 2rvτ as long as r ≤ min

s∈[0,vτ)1/k(s) when sensors move along straight lines (the SL Walk), circular arcs (the CA Walk) or gen- eral curves (the GC Walk). The coverage ratio, F(τ), is 1− e^−λ(πr²^+2rvτ).

• The coverage ratio of mobile sensor networks on a sphere is studied as a special case, i.e., a sphere is a convex 3D surface with constant Gaussian curvature K. The expected area of the region covered by a sensor si over the time period [0, vτ) is ∥G^τc,si∥ = πr²+ rvτ√

4− Kr² as long as r≤ p

2(k− kg)/(Kk)under the CA Walk and the GC Walk, and F(τ) = 1 − e^−λ[πr²^+rvτ^√^4−Kr²^], which gives us intuitions for the general surface case.

• We derive, in closed form, the coverage ratio on general convex 3D surfaces. We first identify that

∥G^τc,si∥ = πr² +Rvτ

0 G(kn, r, s)ds + c(r) as long as r ≤

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s∈[0,vτ)min È

2(k(s)− kg(s))/(k²n(s)k(s)) under the GC Walk; we then obtain a formula on the transformation from the area measure of G^τ_c_,s

i to the coverage ratio; finally, let hr(vτ) =

nlim→∞

1 n

Pn i=1

Rvτ

0 Gi(kn, r, s)ds, if it exists, and then F(τ) = 1−e^−λ[πr²^+h^r^(v^τ)+c(r)].Gi(k_n, r, s), defined in Equ(1), charac- terizes the properties of the FoI and the mobility of sensor, and the function c(r) satisfies lim

r→0 c(r)

r³ = c, (|c| < ∞).

These results are consistent with previous results on the stationary 2D plane, mobile 2D plane and stationary surface scenarios [24][18][22]. We present the main implications of the above results below. From an abstract point of view, the first two reflect the inner consistency of our analysis methodology and thus verify the correctness of our results.

Remark 3.1. For a sphere of radius R, k_n(s)= kn(s)= _R¹, so we have

Gi(kn, r, s) = r Ê

4−

r R

2

= r√ 4− Kr², h_r(vτ) = rvτ√

4− Kr²,

and c(r)= 0 for the sphere case. Therefore, the coverage ratio of a sphere can be reduced from that in the general surface case:

F(τ) = 1 − e^−λ[πr²^+h^r^(v^τ)+c(r)]= 1 − e^−λ[πr²^+rvτ^√⁴^−Kr²^]. Remark 3.2. When the sphere expands to a 2D plane, i.e. R→

∞ (K → 0), then hr(vτ) = πr²+ 2rvπ. Therefore, the coverage ratio of the 2D plane can be reduced from these in both the sphere case and the general surface case.

Remark 3.3. The coverage ratio has the general form F(τ) = 1 − e^−λ[πr²^+h^r^(v^τ)+c(r)] for the mobility case, and F(0) = 1 − e^−λπr²^+c(r)for the stationary case. Since h_r(vτ) is always pos- itive, and|c(r)| is smaller than hr(vτ), so

F(τ) = 1 − e^−λ[πr²^+h^r^(v^τ)+c(r)]≥ 1 − e^−λπr²= F(0).

Then we always have that mobility increases the surface cov- erage of sensor networks. Furthermore, since hr(vτ) increases with vτ, there are two ways for increasing F(τ): increasing the sensors’ moving speed or prolonging the time interval.

Remark 3.4. From function h_r(vτ) and G(kn, r, s), we know that sensors moving at positions with bigger Gaussian curvature will cover a region with less area. It is not difficult to check that the inequalityG(kn, r, s) ≤ 2r always holds, with equality hold- ing for the 2D plane case. Therefore, given the speed, the area covered by sensors moving on a 2D plane over an equivalent time interval is larger than that on general 3D surfaces. This lead us to the conclusion that the nonzero Gaussian curvature leads to the invalidity of the 2D plane model for 3D surfaces, i.e., the coverage hole problem.

4. Network Models and Metrics

This section describes models for FoI, sensing, deploymen- t and mobility pattern, respectively, and presents several measures to assess the surface coverage performance of mobile sensor networks.

To understand our work, the reader must be familiar with preliminaries of the integral and differential geometry theories.

For convenience, Appendix A lists the related definitions and theorems.

4.1. The Unit Ball Sensing Model

We assume that the target FoI is a convex surface S of class C²in 3D space¹. S can be expressed as a single valued function z= h(x, y) in a Cartesian coordinate system. In particular, S is a plane if and only if the function is z= c where c is a constant, for an appropriate selection of the coordinate system. A sensor si is said to be placed on S if the coordinates of sisatisfy the equation of S , which is denoted as si∈ S .

We use a unit ball sensing model, i.e., assume that each sen- sor has the same sensing radius r in 3D Euclidean space and that a sensor can sense and detect events within its sensing range², thus the sensing region forms a ball of radius r centered at siin 3D space (or a disk on a 2D plane).

Let Gc,sidenote the region covered by sensor sion S , we have Gc,si⊆ S with

G_c_,s_i= {x | d(si, x) ≤ r, x ∈ S } . (2) A point p ∈ S is said to be covered by sensor si if p∈ Gc,si. After n sensors are deployed, the FoI is thus partitioned into two kinds of regions: the covered region Gcand the uncovered region G^′_c:

Gc= [n i=1

Gc,si. (3)

Every point in Gc is covered by at least one sensor; G^′_cis the complement of Gc. An event happening in Gc or an intruder appearing in this region will be detected immediately.

4.2. Sensor Deployment

Definition 4.1. Surface Poisson Point Process (SP3). Assume that sensors are distributed uniformly, both n, ∥S ∥ → ∞ in such a way that_{∥S ∥}ⁿ → λ (which is a positive constant), the probabil- ity that there are m sensors lie in a set G is

n→∞lim

∥S ∥→∞

(λ∥G∥)^m

m! e^−λ∥G∥. (4)

The right-hand side of Equ.(4) is the probability function of the Poisson distribution; it depends only on the productλ∥G∥,

1Convex surface, Gaussian curvature, surface of classC^kare considered as a prior knowledge. Refer to Appendix A or [31] for detailed definitions.

2This assumption is a bit strong but quite satisfying as S is quite large and r is small in real world scenarios; it simplifies the analysis a lot. Refer to Section 9 for further discussions.

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which is called the parameter of the distribution. This proba- bility model for points on the surface is said to be a Surface Poisson Point Process (SP3) of intensityλ. Note that λ is the network density.

We concentrate on large-scale mobile sensor networks deployed over vast 3D surfaces³. For the initial configuration, we assume that the locations of these sensors are uniformly and independently distributed according to the S P3 model at time t= 0, as in [22][23].

We adopt the Surface Poisson Point Process (S P3) due to the following two reasons: (1) S P3 provides a uniform distribution for the initial locations of sensors; (2) If sensors move by choos- ing its forwarding directions independently and randomly, S P3 holds in the whole process. This uniform property simplifies our analysis greatly and we believe that it might serve as a basis for analyzing other kinds of distribution patterns, since many more complicated distributions can be reduced to uniform distributions as Gaussian distribution and exponential distribution giving proper parameters.

4.3. Mobility Model

In this work, we consider the following three simple sensor mobility models: SL Walk, CA Walk and GC Walk. The movement of a sensor is characterized by its speed and direction. We assume that sensors move independently of each other, as in [18]. All sensors move with constant velocity v. Ignoring the edge effect as S is vast or infinite, then the S P3 distribution always holds during time interval [0, τ).

• Straight line walk (SL Walk): On 2D planes, each sensor randomly chooses a direction from [0, 2π) uniformly at t = 0 and never change over [0, τ). It is quite similar to the Random Waypoint Model.

• Circular arc walk (CA Walk): On 2D planes/spherical surfaces, each sensor randomly chooses a direction α ∈ [0, 2π) uniformly at t = 0 and then performs uniform circular motionwith radius of 1/k.

• General curve walk (GC Walk): Divide time into appro- priate slots. On 2D planes/sphere/general surfaces, each sensor randomly chooses a direction form all possible directions uniformly at the beginning of each slot and then move forward with constant velocity v.

The SL and CA Walk are special cases of the GC Walk, and the results of the SL and CA Walk are intermediate results for the proof of the GC Walk case. More specifically, in the 2D plane case, sensors can perform straight line walk, circular arc walk and general curve walk; in the sphere case, sensors can perform circular arc walk and general curve walk; while in the

3We concentrate on large-scale homogeneous mobile sensor networks:

Large-scale is reflected in both the number of sensors and the area of the FoI are infinite (as n, ∥S ∥ → ∞); homogeneous means that the capability of the sensors are the same, and that the sensors are distributed uniformly, e.g. de- ploying sensors by vehicles running on the surface or even by aircraft, humans and robots.

general convex 3D surface case, sensors can perform only general curve walk. Furthermore, the results obtained on 2D planes and 3D spheres are then extended to the general convex 3D surfaces; the results of straight line walk are extended to derive results for the circular arc walk, then for the general curve walk.

These above random mobility models enable us to prove the final argument that “Mobility increased the surface coverage of distributed sensor networks”, because even this simply and naive motion ability increases the network performance greatly, then elaborate, intelligent and collective mobility is bound to promote network performance more. This methodology is also adopted by [20][9][21][18], which assume random mobility model to prove that mobility increases communication capacity, network connectivity, security and 2D coverage, respectively.

4.4. Coverage Metrics

In [22][23], the authors consider the full surface coverage in stationary scenario, which can be formally defined as: find a deployment strategy using the minimum number of sensors while providing full coverage of the FoI. Here, we take a dif- ferent approach and establish a theoretical analysis framework for partial surface coverage by characterizing the coverage ratio, i.e., studying how initial network configuration, properties of the surface and mobility pattern together affect the coverage ratio.

To study the surface coverage, we use the following three coverage measures: area coverage ratio at time instant t, f (t), the area covered by one sensor s_i over a time interval [0, τ),

∥G^τc,si∥, and area coverage ratio over [0, τ), F(τ). Let G^tcdenote the covered region of the FoI at t and G^τ_c denote the region covered over [0, τ), which is defined by

G^τ_c,s_i ₌ [

t∈[0,τ)

G^t_c,s_i _, G^τ_c ₌ [ⁿ

i=1

G^τ_c,s_i _. ₍₅₎ Area coverage ratio at t is defined by the probabili- ty that a randomly selected point from S lies in G^t_c, i.e.

P p ∈ G^tc| p ∈ S

, as both n, ∥S ∥ → ∞ in such a way that

∥S ∥n → λ, if the following limit f (t)= lim_n_→∞

∥S ∥→∞

P p ∈ G^tc| p ∈ S

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exists, then f (t) is the corresponding coverage ratio. Similarly, we define F(τ) as

F(τ) = lim_n_→∞

∥S ∥→∞

P p ∈ G^τc| p ∈ S

. (7)

All three coverage measures depend not only on the network configuration, but also on the sensor mobility pattern. f (t) is the fraction of the geographical area covered by at least one sensor at time instant t; it measures the coverage ratio achieved by a sensor network at a snapshot view. Specifically, for a stationary scenario, f (t) remains unchanged, i.e., f (t)= f (0), ∀t ∈ [0, τ), and mainly depends on the network’s initial configuration (e.g., the sensor distribution, network density and sensing range) and the properties of the FoI. Since our networks are homogeneous

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and sensors move under an identical random and independent way on surface, the expectation of the area covered during any given time period by each sensor is the same. Thus without loss of generality, we denote it by a common symbol, as G^t_c_,s

1

for G^t_c_,s

i, G^τ_c_,s

1for G^τ_c_,s

i,∀i ∈ [2, n]. F(τ) measures the area cov- erage ratio of a mobile sensor network during the time interval [0, τ), which is the fraction of the geographical area covered by at least one sensor at least once at some time instant t. As pointed out by B. Liu et al [18], the characterization of area coverage ratio at specific time instant is useful for application- s that require simultaneous coverage, while the area coverage ratio over a time interval is appropriate for applications that do not require simultaneous coverage of all positions at specific time instant, but rather prefer partial-time coverage.

5. Mobility on a 2D Plane

The 2D plane is a special case of complex surface with Gaus- sian curvature of zero. The results seem to be straightforward, but can provide a brief overview of the proving process in the following sections. B. Liu et al. [18] discussed the situation where sensors move under the SL Walk. In this section, we further study the area coverage achieved under the CA and GC Walk. As the SL and CA Walk are special cases of the GC Walk, we thus generalize the results of [18].

Lemma 5.1. The SP3 distribution model can be reduced to the Poisson Point Process (PP3) on a 2D plane.

Proof. For a 2D plane, the Gaussian curvature of S is zero.

Combining Definition 4.1 and Lemma Appendix A.4, it can be

immediately obtained.

Theorem 5.2. Consider the mobile sensor network, C, de- ployed under the SP3 model on plane S at time instant t0 = 0.

Sensors move under the SL, SA and GC Walk over [0, τ). If r≤ min

s∈[o,vτ)1/k(s), then we have:

f (t)= 1 − e^−λπr², ∀t ∈ [0, τ),

∥G^τc∥ = πr²+ 2rvτ, F(τ) = 1 − e^−λ(πr²^+2rvτ).

Proof. For the SL walk, shown in Fig 2(a), the results were presented in [18]. They hold because under the SL Walk, at each time instant t ∈ [0, τ), the locations of the sensors still follow the Poisson Point Process (PP3) of the same density [32].

Next we study the CA and GC Walk during [0, τ).

1) The CA Walk is shown in Fig 2(b). The covered region forms a circular race track with radius R. G^τ_c_,s

i, which is the initial covered region plus the region covered by the diameter of the sensor, has the area:

G^τ_c_,s_i = πr²+ Rθ

2πR· π[(R + r)²− (R − r)²]

= πr²+ vτ

2πR· 4πRr = πr²+ 2rvτ.

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vt

(a)

R

(b) (c)

Figure 2: Sensor moves on a 2D plane. (a). The SL Walk case; (b). The CA Walk case; (c). The GC Walk case.

ds

˄ ˅S

Figure 3: The infinitesimal dividing method in the GC Walk case.

2) The GC Walk is shown in Fig 2(c). The covered region has a curly ring shape. Denote the curve asL(s), (0 ≤ s < vτ) and the radius at s asρ(s) = 1/k(s). Let r ≤ min

s∈[0,vτ)1/k(s). The infinitesimal arc element [s, s + ds] (see Fig 3) can be approx- imated by an elementary circular arc with radius ofρ(s). The expected area covered by the diameter of a sensor moving along [s, s + ds] is dS = 2rds. By integration, we get:

G^τ_c_,s

i = πr²+Z

L(s)dS = πr²+Z vτ 0

2rds

= πr²+ 2rvτ.

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3) It was pointed out in [33] that area coverage ratio depends on the distribution of the random shapes only through its expected area measure, thus we have:

F(τ) = 1 − e^−λ∥G^τ^c^,si^∥= 1 − e^−λ(πr²^+2rvτ). (10)

6. Mobility on a Sphere

A sphere is another special case of 3D surfaces. A sphere with radius R has constant Gaussian curvature K = 1/R². In this section, we study the scenario when sensors move under the CA and GC Walk.

Lemma 6.1. On a sphere of radius R, the geodesic curvature k_gof a circular arc curve with radiusρ satisfies:

kg= ±

pR²− ρ²

Rρ . (11)

Proof. From Lemma Appendix A.1, and kn= 1/R, we have:

kg = ±È

k²− k²n= ± Ê1

ρ² − 1 R² = ±

pR²− ρ² Rρ .

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The sign is determined by the projection direction of the curve on it tangent plane: if the projection directs to the inner of the region, it is positive; otherwise, it is negative. Theorem 6.2. Consider the mobile sensor network, C, de- ployed under the S P3 model on sphere S at time instant t0= 0.

Sensors move under the CA Walk over [0, τ); simoves along a circular arc of curvature k. We have:

G^τ_c_,s_i = πr²+ rvτ√ 4− Kr², if r≤ È

2(k−kg) Kk .

Refer to the appendix for the corresponding proof.

Remark 6.1. For the CA Walk, the proving process assumes that the sensing range is relatively small, such that

r≤ ∥DC∥ = Ê

ρ²+

R−È

R²− ρ²

2

=

r2(k− kg) Kk . Remark 6.2. For the CA Walk, lettingτ → 0, we get the result for stationary network scenarios:

G⁰_c_,s

i =lim

τ→0G^τ_c_,s

i = lim

τ→0

πr²+ rvτ√

4− Kr²

= πr².

Remark 6.3. For the CA Walk, over[0, τ), let Ωc,si denote the additional region covered by the diameter of sensor sidue to its movement, then

Ωc,si = G^τ_c_,s

i − G⁰_c_,s

i =rvτ√

4− Kr².

This indicates that on spherical surface, at each snapshot, the area covered by a sensor is constant as long as r ≤ p2(k− kg)/(Kk). Furthermore, as long as this condition holds along the moving trace, the expected area covered by the mobile sensor over [0, τ) is independent of the trajectory.

Theorem 6.3. Consider the mobile sensor network C deployed under the S P3 model on sphere S at time instant t0 = 0. Sen- sors move under GC Walk; s_imoves along a general curve of curvature k(s), 0≤ s < vτ. We have

G^τ_c_,s_i = πr²+ rvτ√ 4− Kr², if r≤ min

0≤s<vτ

q2(k(s)−kg(s)) Kk(s)

.

Proof. By utilizing a similar methodology as in the 2D plane case, for the GC Walk, we approximate the infinitesimal arc element dS as an elementary circular arc; by integration we obtain the final result.

Denote the curve as L(s), 0 ≤ s < vτ, and the radius at s is ρ(s) = 1/k(s). For a infinitesimal element [s, s + ds], which can be approximated as an elementary circular arc with radius of ρ(s), from Remark 6.3, we know that the area cov- ered by the diameter of a sensor moving along [s, s + ds] is dS = r√

4− Kr²ds. By integration, we get _Ω^τ_c_,s

i ₌Z

L(s)

dS = Z _vτ

0

r√

4− Kr²ds= rvτ√

4− Kr², (12)

G^τ_c_,s_i = Ω^τc,si + G⁰_c_,s_i = πr²+ rvτ√

4− Kr². (13)

Theorem 6.4. Consider the mobile sensor network, C, de- ployed under the SP3 model on S at time instant t₀ = 0. Sen- sors move under the GC Walk and along general curvesL(s), 0≤ s < vτ over [0, τ). We have

P p∈

[n i=1

G^τ_c_,s

i | p ∈ S

= 1 −

1−πr²+ rvτ√ 4− Kr² 4π/K

n

,

F(τ) = 1 − e^−λ(πr²^+rvτ^√⁴^−Kr²⁾, if r≤ min

0≤s<vτ

q2(k(s)−kg(s)) Kk(s)

.

Proof. From Theorem 6.3, we know that G^τ_c_,s_i = πr² + rvτ√

4− Kr². Since the FoI is the entire spherical surface S with∥S ∥ = 4πR² = ⁴_K^π and G^τ_c_,s

i ⊆ S , the probability that a random chosen point p∈ S lies in G^τc,siis

P p∈ G^τc,si

= G^τ_c_,s_i

∥S ∥ = πr²+ rvτ√ 4− Kr²

4π/K . (14)

Because each sensor moves independently, thus we get P

p∈

[n i=1

G^τ_c_,s_i

= 1 − P

p∈

[ⁿ

i=1

G^τ_c_,s_i

_′

= 1 − P

p∈

\n i=1

G^τ_c_,s_i_′

= 1 − Yn

i=1

P

p∈

G^τ_c_,s_i_′

= 1 − Yn

i=1

1− P

p∈ G^τc,si

= 1 −

1−πr²+ rvτ√ 4− Kr² 4π/K

n

.

(15)

Note that

1−¹_xx

x→∞

−−−−→ e⁻¹,_{∥S ∥}ⁿ = λ, thus

F(τ) = lim_n_→∞

∥S ∥→∞

P

p∈

[n i=1

G^τ_c_,s_i| p ∈ S

= lim_n_→∞

∥S ∥→∞

1−

1− λ ·πr²+ rvτ√ 4− Kr² n

n

= 1 − e^−λ(πr²^+rvτ^√⁴^−Kr²⁾.

(16)

7. Mobility on a General Convex 3D Surface

This section is devoted to analyzing the surface coverage ratio on general surfaces when sensors move under the GC Walk.

We follow a similar methodology as in the 2D plane case and the sphere cases, but is much more complicated. First we study the diameter and area of the region covered by a moving sensor;

(8)

S

A C

B D ,i

Gc s

x y

Figure 4: Diameter of the region covered by a moving sensor si.

then characterize the probability of the following event: a random chosen point from a surface convex set lies in a subset of that set; finally we are able to obtain the formula of F(τ) with full consideration of the initial sensor distribution, properties of the target surface field and mobility pattern of sensors.

Lemma 7.1. On a general surface, let G^τ_c_,s_i be the region cov- ered by a moving sensor over [0, τ), and |G^τc,si|Dbe the diameter of G^τ_c_,s_i. We have

|G^τc,si|D≤ 2r + vτ.

Proof. Randomly pick two points x,y from G^τ_c_,s_i, as shown in Fig 4. Let öAB denote the trace of sensor si. According to the definition of G^τ_c_,s

i, we know that∃ points C, D ∈ öAB, s.t. x ∈ B(C, r)∩S, y ∈ B(D, r)∩S , where B(C, r) denote a ball with ra- dius r and center point C. Then the following inequality holds:

d(x, y) ≤ d(x, C) + d(C, D) + d(D, y)

≤ r + ∥öCD∥ + r ≤ 2r + ∥öAB∥ ≤ 2r + vτ. (17)

Therefore,G^τ_c_,s_i

D= sup

∀x,y∈G^τc,si

§ d(x, y)

ª

≤ 2r + vτ.

Theorem 7.2. Consider the mobile sensor network, C, de- ployed under S P3 model on S at time instant t0 = 0. If sensors move under the GC Walk; sensor i is assumed to move along general curve of curvature k(s), 0≤ s < vτ. We have

G^τ_c_,s

i = πr²+ Z _v_τ

0

Gi(k_n, r, s)ds + c(r)

if r≤ min

0≤s<vτ

q_2(k(s)_−k

g(s)) k²_n(s)k(s)

,

where the function c(r) satisfies lim

r→0 c(r)

r³ = c, (|c| < ∞), and Gi(kn, r, s) characterizes the properties of the surface and mo- bility pattern of sensor i, with the following form:

Gi(kn, r, s)

= kn(s) kn(s)

r

È

4− (kn(s)r)²+ 4kn(s)− kn(s) kn(s)kn(s) arcsin

_k

n(s)r 2

.

Given sensing range r and the target FoI, G(kn, r, s) depend- s only on the trace of each sensor. So we call G(kn, r, s) the mobility function, which characterizes sensor’s mobility.

Remark 7.1. There are two special cases we would like to point out. If kn(s)= 0, the mobility function can be obtained by taking the limit, i.e., Gi(k_n, r, s) = 2r; If kn(s) , 0 and kn(s) = 0, it can also be obtained by taking the limit, i.e., Gi(k_n, r, s) =

4 k_n(s)arcsin

kn(s)r 2

.

Next, we characterize the coverage ratio over a time interval [0, τ). First, we need establish the random process on general surface, which is mathematically hard due to the lack of existing theoretical results. We utilize the rules of clipping and ap- proaching, which can produce satisfactory results for our analysis.

Theorem 7.3. Consider a general convex surface S : z = h(x, y), which satisfies max

S ∥ ▽ h∥ < ∞, and a surface convex setU. Randomly pick a surface convex set K ⊆ S such that U ∩ K , ∅, the probability that a randomly selected point in U is also located in K is given by

P(p ∈ U ∩ K|U ∩ K , ∅) = K U + ξ(U, K) with the functionξ(U, K) satisfies

0≤ ξ(U, K) ≤ M(∥∂U^′∥δ + πδ²), where M= max

S

p1+ ∥ ▽ h∥², andδ = sup

K⊆S|K|D, Uzdenotes the z-projection ofU on the plane xOy.

Theorem 7.4. Consider a general infinite convex surface S : z = h(x, y) satisfying⁴ max

S ∥ ▽ h∥ < ∞, and assume that sensors move under the GC Walk over [0, τ). Let hr(vτ) =

nlim→∞

1 n

Pn i=1

Rvτ

0 Gi(kn, r, s)ds, we have

nlim→∞P

p∈ [

1≤i≤n

G^τ_c_,s_i

→ 1 − e^−λ[πr²^+h^r^(v^τ)+c(r)]

if r≤ min

s∈[0,vτ)

É2(k(s)−kg(s))

k²_n(s)k(s) , or equivalently, F(τ) = 1 − e^−λ[πr²^+h^r^(v^τ)+c(r)],

whereGi(kn, r, s) characterizes the properties of the FoI and the mobility of sensor i, and the function c(r) satisfies lim

r→0 c(r)

r³ = c, (|c| < ∞).

Proof. As pointed out in Theorem 7.2 and Theorem 7.3, the area measure of the region covered by sensor siover [0, τ) is G^τ_c_,s_i and the probability that a random chosen point on S lies in this region isP(p ∈ G^τc,si), as:

G^τ_c_,s_i = πr²+ Z vτ

0 Gi(kn, r, s)ds + c(r), (18)

4Intuitively, this condition assures that the surface S has no sudden change.

(9)

P(p ∈ G^τc,si)= G^τ_c_,s

i

∥S ∥ + ξ(S, G^τc,si), (19) withξ(S, G^τc,si) satisfies

0≤ ξ(S, G^τc,si)≤ M(∥∂Sz∥δs_i+ πδ²s_i), (20)

0≤ξ(S, G^τc,si)

∥S ∥ ≤ M(∥∂Sz∥d + πd²)

∥S ∥

∥Sz∥→∞

−−−−−−→ 0. (21)

Then, we have P(p ∈ [

1≤i≤n

G^τ_c_,s

i)= 1 − Y

1≤i≤n

1− P(p ∈ G^τc,si)

= 1 − Y

1≤i≤n

1− G^τ_c_,s_i

∥S ∥ + ξ(S, G^τc,si)

= 1 − Y

1≤i≤n

1−βi

n

, (22) where

βi= n G^τ_c_,s

i

∥S ∥ + ξ(S, G^τc,si)

n→∞

−−−−→ λ G^τ_c_,s

i . (23)

Note that the following relations hold:

n→∞,∥S ∥→∞lim n

∥S ∥ = λ, lim

n→∞,∥S ∥→∞

ξ(S, G^τc,si)

∥S ∥ = 0. (24)

Since hr(vτ) = lim

n→∞

1 n

Pn i=1

Rvτ

0 G(kn, r, s)ds, we get

nlim→∞

1 n

Xn i=1

βi= λ lim

n→∞

1 n

Xn i=1

∥G^τc,si∥ = λ

πr²+ hr(vτ) + c(r) , (25)

n→∞lim 1

∥S ∥ − ∥G^τc,si∥ Xn

i=1

∥G^τc,si∥ = lim

n→∞

∥S ∥

∥S ∥ − ∥G^τc,si∥ n

∥S ∥ 1 n

Xn i=1

∥G^τc,si∥

= λ

πr²+ hr(vτ) + c(r) .

(26)

Therefore, for ∀ϵ > 0, we obtain the following relations as n→ ∞:

Y

1≤i≤n

1−βi

n

≤

"

1−1 n

1 n

Xn i=1

βi

#ⁿ

≤

1−1

n

λ

πr²+ hr(vτ) + c(r)

− ϵ

n

n→∞

−−−−→ e^−λ[πr²+hr(vτ)+c(r)]+ϵ,

(27)

Y

1≤i≤n

1− _G^τ_c_,s

i

∥S ∥ + ξ(S, G^τc,si)

≥Y

1≤i≤n

1− _G^τ_c_,s

i

∥S ∥

=

"

Y

1≤i≤n

1+ G^τ_c,s

i

∥S ∥ − _G^τ_c_,s

i

#−1

≥

1 n

Xn i=1

1+ _G^τ_c_,s

i

∥S ∥ − _G^τ_c_,s

i

−n

=

1+1

n

∥S ∥

∥S ∥ − ∥G^Tc,si∥ 1

∥S ∥ Xn

i=1

∥G^τ_c,s_i∥

_−n

≥

1+λ

πr²+ hr(vτ) + c(r) + ϵ n

−n

n→∞

−−−→ e^−λ[πr²+hr(vτ)+c(r)]−ϵ.

(28)

Hence, we get F(τ) = lim

n→∞P p∈ [

1≤i≤n

G^τ_c_,s_i

= 1 − e^−λ[πr²^+h^r^(v^τ)+c(r)], (29)

where c(r) satisfies lim

r→0 c(r)

r³ = c, (|c| < ∞).

8. Simulation and Evaluation 8.1. Surface Generation

In order to compare the coverage property of surfaces with different curvatures, we consider the following surfaces which can be expressed as a single valued function z= h(x, y):

z= 100 + 50 sinCπx 1000

sin

Cπy 1000

, (30)

where x, y ∈ [0, 3000] m, and the parameter C is taken as C = 1, 3, 9 to generate three surfaces with 9, 81 and 729 peaks and valleys in the region [0, 3000] m×[0, 3000] m, respectively. Fig 5 gives the contours of these surfaces. Here, the unit of length is the meter.

8.2. Numerical Results

The first simulation is implemented in the FoI with three d- ifferent surfaces S given by Equ.(30). n = 1000 sensors with the same sensing range r= 20m are randomly deployed on the surface according to S P3 distribution, and they could indepen- dently and randomly move at speed of v= 1m/s on the surface.

Figure 5: The contours of the surfaces with 9, 81, and 729 peaks and valleys in a region spanning over a [0, 3000] m × [0, 3000] m square, respectively.