移动性提升分布式传感器网络的表面覆盖

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"本文研究了在复杂3D曲面上移动传感器网络的覆盖问题，强调了在现实世界应用中，现有2D或3D空间覆盖分析方法的局限性，并提出移动性可以提高分布式传感器网络的表面覆盖能力。" 文章标题与描述所涉及的主要知识点包括： 1. **传感器网络覆盖**：在传感器网络中，覆盖是指传感器节点能够感知其周围环境的能力，是评估网络性能的基础。当传感器节点分布在一个区域内时，确保整个目标区域的有效覆盖至关重要，因为它直接影响到数据收集的完整性和网络的效率。 2. **2D与3D部署的差异**：传统研究通常假设传感器网络部署在二维平面上或三维空间中，但在实际应用中，如地形监测、建筑物内部监控等，目标区域可能是复杂的3D表面。这给传统的覆盖分析带来了挑战，因为这些表面的几何特性使得简单的平面或空间模型不再适用。 3. **移动性的影响**：本文指出，传感器节点的移动性可以增加网络的表面覆盖。移动传感器可以更灵活地调整位置，适应不断变化的环境需求，从而提高覆盖质量。这种动态覆盖依赖于传感器的运动模式以及环境特征。 4. **3D曲面覆盖**：在凸形3D表面上部署的移动传感器网络的研究是一个新的挑战。在这种环境下，必须考虑传感器的移动路径、曲面的几何形状以及它们相互作用的方式，以优化覆盖效果。 5. **分布式传感器网络**：传感器网络通常由大量独立工作的节点组成，每个节点都能感知和处理数据。分布式部署允许网络具有自我组织和自适应性，能更好地应对环境变化和故障。 6. **关键技术与方法**：解决3D曲面上的移动传感器网络覆盖问题可能需要新的算法和模型，包括运动规划、覆盖策略优化、传感器协作机制等。这些技术需要考虑到曲面的物理特性，如高度变化、障碍物分布等，以及传感器的通信和能量限制。这篇论文探讨了在复杂3D环境中移动传感器网络的覆盖问题，揭示了移动性在提高覆盖范围中的潜在优势，这对于设计和优化实际应用中的传感器网络具有重要的理论和实践价值。

deﬁne a function G

ðk

; r; sÞ to characterize the properties of

the FoI and the mobility of a sensor, with the following

form:

ðk

; r; sÞ¼

ðsÞ



ðsÞ

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

4 ð



ðsÞrÞ

þ 4



ðsÞk

ðsÞ



ðsÞk

ðsÞ

arcsin



ðsÞr

: ð1Þ

Intuitively, G

ðk

; r; sÞ indicates the local smoothness of the

surface 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. G

ðk

; r; sÞ has small val-

ues in sharp regions, and big values in smooth regions.

Our main theoretical results are:

 We generalize the results in the 2D plane case [18] by

considering sensors moving along general curves. The

expected area of the region covered by a mobile sensor

over the time period [0,

)iskG

c;s

k¼

þ 2r

long as r 6 min

s2[0, v

)

1/k(s) when sensors move along

straight lines (the SL Walk), circular arcs (the CA Walk)

or general curves (the GC Walk). The coverage ratio,

), is 1  e

kð

þ2r

 The coverage ratio of mobile sensor networks on a

sphere is studied as a special case, i.e., a sphere is a con-

vex 3D surface with constant Gaussian curvature K. The

expected area of the region covered by a sensor s

over

the time period [0,v

)iskG

c;s

k¼

þ r

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

4  Kr

long as r 6

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2ðk  k

Þ=ðKkÞ

under the CA Walk and

the GC Walk, and Fð

Þ¼1  e

k½

þr

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

4Kr



, 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

c;s

k¼

Gðk

; r; sÞds þ cðrÞ as long as

r 6 min

s2½0;

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

2ðkðsÞk

ðsÞÞ=ð



ðsÞkðsÞÞ

under the GC

Walk; we then obtain a formula on the transformation

from the area measure of G

c;s

to the coverage ratio;

ﬁnally, let h

Þ¼lim

n!1

i¼1

ðk

; r; sÞds,ifit

exists, and then Fð

Þ¼1  e

k½

þh

ÞþcðrÞ

. G

ðk

; r; sÞ,

deﬁned in Eq. (1), characterizes the properties of the

FoI and the mobility of sensor, and the function c(r) sat-

isﬁes lim

r!0

cðrÞ

¼ c; ðjcj < 1Þ.

These results are consistent with previous results on the

stationary 2D plane, mobile 2D plane and stationary sur-

face scenarios [24,18,22]. We present the main implica-

tions 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

ðsÞ¼



ðsÞ¼

,so

we have

ðk

; r; sÞ¼r

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

4 



¼ r

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

4  Kr

;

Þ¼r

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

4  Kr

;

and c(r) = 0 for the sphere case. Therefore, the coverage ra-

tio of a sphere can be reduced from that in the general sur-

face case:

Fð

Þ¼1  e

k½

þh

ÞþcðrÞ

¼ 1  e

k½

þr

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

4Kr



Remark 3.2. When the sphere expands to a 2D plane, i.e.

R ? 1 (K ? 0), then h

+2rv

. Therefore, the cov-

erage 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

Fð

Þ¼1  e

k½

þh

ÞþcðrÞ

for the mobility case, and

Fð0Þ¼1  e

k

þcðrÞ

for the stationary case. Since h

)is

always positive, and —c(r)— is smaller than h

), so

Fð

Þ¼1  e

k½

þh

ÞþcðrÞ

P 1  e

k

¼ Fð0Þ:

Then we always have that mobility increases the surface

coverage of sensor networks. Furthermore, since h

) in-

creases with v

, there are two ways for increasing F(

increasing the sensors’ moving speed or prolonging the

time interval.

Remark 3.4. From function h

) and Gðk

; r; sÞ, we know

that sensors moving at positions with bigger Gaussian cur-

vature will cover a region with less area. It is not difﬁcult to

check that the inequality Gðk

; r; sÞ 6 2r always holds, with

equality holding 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 gen-

eral 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, deploy-

ment and mobility pattern, respectively, and presents sev-

eral 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 geome-

try 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 C

in 3D space.

S can be expressed as a single valued

function z = h(x,y) in a Cartesian coordinate system. In par-

ticular, 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 s

is said to be placed on S if the coordinates

of s

satisfy the equation of S, which is denoted as s

2 S.

Convex surface, Gaussian curvature, surface of class C

are considered

as a prior knowledge. Refer to Appendix A or [31] for detailed deﬁnitions.

X.-Y. Liu et al. / Computer Networks 57 (2013) 2348–2363

2351

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