Compressive and Collaborative Mobile Sensing forScalar Field Mapping in Robotic Networks

Minh T. Nguyen † , Hung M. La ‡ and Keith A. Teague †† Oklahoma State University, Stillwater, OK 74078

‡ University of Nevada, Reno, NV 89557Emails: {tuanminh.nguyen, keith.teague}@okstate.edu and hla@unr.edu

Abstract—In this paper, we propose a compressive andcollaborative sensing (CCS) algorithm for distributed roboticnetworks to build scalar field map. A collaborative controllaw is utilized to steer the robots to move on the field whileavoiding collision with each other and with obstacles. At eachtime instant, the robots collect, add measurements within theirsensing range and exchange data with their neighbors to formcompressive sensing (CS) measurements at each robot. After acertain times of moving and sampling, each robot can achievethat number of CS measurements to be able to reconstruct allsensory readings from the positions that the group of robotsvisited to build a scalar map. We further analyze and formulatethe total communication power consumption associated withthe number of robots, sensor communication range and providesuggestions for more energy saving.

Keywords—Robotic networks, compressive sensing, scalarfield mapping, collaborative control

I. INTRODUCTION

A. Motivation

Mobile sensor networks (MSNs) facilitate many existingapplication areas, such as monitoring temperature, humidity,acoustic, vibration or detecting target or special events(chemical leak, vehicle passing). The networks are combinedfrom sensors, control algorithms and other dynamic factorswhich depend on specific purposes or application scenar-ios [1], [2].

Scalar field mapping is one of the most common ap-plications for MSNs [3], [4]. In each map, a vector X =[x1 x2 . . . xN ]T represents all the unknown sensory readingsfrom N random positions in a sensing area needed to beobserved. They are temperature, humidity, etc. and are oftenhighly correlated. In general, the sensors are attached tomobile robots or vehicles and their movements follow acontrol algorithm [5]. All sensors need to move to visit theentire sensing area in order to build a full scalar map thatcost a huge power consumption for either movements andcommunications [4], [6]. There exist many research resultsshown to improve control algorithms, physical sensors, data

Minh T. Nguyen is a lecturer with Thai Nguyen University of Technology(TNUT), Thai Nguyen city, Vietnam. This work is supported by theVietnamese Ministry of Education and Training (MOET) and School ofElectrical and Computer Engineering (ECE), Oklahoma State University,University of Nevada, Reno, and NSF-NRI grant # 1426828.

processing methods and communication connections forsaving power for the networks [7], [8], [9], [10], [11].

Compressive sensing (CS) [12], [13], [14], [15] providesa novel framework to compress sparse or compressiblesignals that is perfectly applicable to the data collectedfrom sensor networks. The technique offers to reconstructall sensor readings based on a much smaller number ofCS measurements (Y = [y1 y2 . . . yM ]T ) compared to thenumber of positions (N ). The CS measurements are col-lected as Y = ΦX, where Φ is the measurement matrix,also called routing matrix in wireless networks that showshow a network collects data from a sensing area which ismentioned in this work.

In this paper, a team of robots are led by a collaborativecontrol with virtual leaders to sample sparsely a sensingarea. At instant time t, each robot adds all readings withinits sensing range and shares the accumulated scalar valueattached with the position indices to others. Each CS mea-surement is a sum or is collected from linear combinationsof the sensory data measured by the group of robots.After M periods of time, at different positions, each robotachieves M CS measurements and reconstructs N sensoryreadings from the area. Different from other methods suchas the consensus methods [16], [17] that usually require aconnected network, our robot networks maybe disconnectedsometimes due to the limited sensor communication rangeand obstacle avoidance on the field. This results in unequalCS measurements between each separated group of robotsthat does not affect the CS performance. Furthermore, thesensory readings transmitted between robots are added asa scalar value that saves a huge energy for communicationcompared to separated measurements transmitting in othermethods [16], [17].

B. Related Work

Collaborative sensing in MSNs has been studied in [3],[4], [8], [9] with many applications such as target track-ing, environmental mapping or monitoring. The distributedmobile robots are connected to share their information forspecific purposes.

Upgrading from data collection methods using CS inwireless sensor networks [18], [19], [20], [21], recently,there have been some research studies that exploit the com-bination between the mobility of sensors and CS. Wang [22]

978-1-5090-1824-6/15/$31.00 ©2015 IEEE 873

Fifty-third Annual Allerton ConferenceAllerton House, UIUC, Illinois, USASeptember 29 - October 2, 2015

monitors vehicle networks based on CS to reduce the com-munication cost. Mostofi builds maps of obstacles/objectsusing robot networks [23] while the robots are deployedoutside the areas. In [24], mobile sensors are deployedrandomly in a sensing area to form CS measurements andto share the measurements with each other for CS recoveryprocess at each sensor.

To our best knowledge, the existing work has not focusedon exploring the collaboration and the mobility of roboticsensors to sample sparsely sensing areas and using CS toreconstruct all data at each distributed mobile robot. Insummary, the contributions of our paper are

1) A new distributed compressive and collaborativesensing algorithm for MSNs to build scalar fieldmaps at each mobile robot is proposed.

2) The total communication power consumptions forthe network are analyzed and formulated.

3) Some important factors such as the number ofmobile robots, the convergence time and the robotcommunication range are analyzed and simulatedto minimize the network power consumption.

Unlike the existing consensus filter algorithms, our pro-posed framework does not require the sensor network tomaintain connectivity all the time during its movement. Thisfeature could enable real world applications of the complexfield mapping and exploration where obstacles may existand interrupt the network connectivity. Moreover, instead offull field coverage, the sensor network just needs to moveand sample certain target locations on the field, but canefficiently reconstruct the map of the scalar field due to thehigh correlation of the sensory data.

The remainder of this paper is organized as follows. Thebackground of CS and the collaborative control algorithmand the Problem Formulation are addressed in Section II.The distributed compressive and collaborative sensing (CCS)algorithm and some discussions are mentioned Section III.The communication power consumptions is analyzed andformulated heuristically in Section IV. Simulation results areshown in Section V and finally, conclusion and suggestionsfor the future work are presented in Section VI.

II. BACKGROUND AND PROBLEM FORMULATION

A. Network Model

We assume a sensing area has N random positionscorresponding to N unknown sensor readings for mapping.There are L distributed mobile robots working as a teamin the area. The robots can connect to each other basedon the communication range, denoted as Rc. The sensingrange of the robots, denoted as Rs, can be selected smallerthan Rc. In this paper we choose Rs ≤ 1

2Rc. Due to thelimited power, we choose both Rs and Rc of the robots aremuch smaller than the sensing area. In order to lead therobots to measure data and avoid obstacles, a collaborativecontrol algorithm [5], [25] is chosen with predefined virtualleaders to to sample the entire scalar field. The obstacles andthe limitation of Rc cause the network not being alwaysconnected as shown in Figure 1 that may not affect the

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(b)

(a)

Fig. 1. Distributed mobile robots leading by the collaborative controlalgorithm in a sensing area can be (a) separated as three groups and (b)fully connected.

data collection method which will be discussed in the nextsections.

B. Distributed Collaborative Control Algorithm

As mentioned in our network model, we have a dynamictopology of an L-robot team based on an information graphG(V ,E). The vertex set V = {1, 2, ..., L} present all theactive mobile robots. The edge set E = {(i, j) : i, j ∈V , i = j} is defined by the communication links betweenrobots based on a robot communication range Rc.

Let pi,vi ∈ ℜ2 denote the position and velocity vectorsof the ith robot. For simplicity, the kinematic motion of eachrobot is modeled as a particle:{

pi = vi

vi = ui, i = 1, 2, ..., L.(1)

here ui is the control input vector of robot ith. Equation(1) can be used to model mobile robots that have omni-directional motion capability such as the Rovio robots [26].

We define an actual neighborhood set of robot i at timet as follows:

Nai (t) =

{j ∈ V : ∥pj − pi∥ ≤ Rc, j = i

}, (2)

where ∥.∥ is the Euclidean distance. The superscript aindicates the actual neighbors of robot i that is used todistinguish with virtual neighbors in the case of obstacleavoidance. The set of virtual neighbors of robot i at time twith K obstacles is defined as

Nvi (t) = {k ∈ V o : ∥pik − pi∥ ≤ R0,V o = {1, 2, ...,K}}

(3)where R0 is the obstacle detection range. V o is a set ofobstacles. pik is the position of robot i projecting on thekth obstacle. For convenience from now on we just use Na

iand Nv

i by dropping t. The virtual neighbors are used togenerate the repulsive force to push the robots away fromthe obstacles.

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Mobile robots in the team will work together withneighbors to form a certain geometry formation (e.g quasilattice formation) while avoiding collision with each otherand static/moving obstacles. The distributed collaborativecontrol law ui for each robot is introduced [25], [27], [28]consisting of three terms: formation control ff

i , obstacleavoidance fo

i , and navigation fni , namely,

ui = ffi + fo

i + fni . (4)

The term ffi is the pair-wise attractive/repulsive action

function to allow robots to maintain a certain distance withits neighbors as well as avoid collision among them. Thisfunction is designed as:

ffi = ca1

∑j∈Na

i

ϕa(∥pj−pi∥σ)nij+ca2∑j∈Na

i

aij(p)(vj −vi),

where ϕa(.) is the action function defined in [25]. The σ- norm, ∥.∥σ , defined as ∥x∥σ = 1

ϵ [√1 + ϵ∥x∥2 − 1] with

ϵ > 0, unlike the Euclidean norm ∥z∥, is differentiable everywhere.

During the scalar field exploration, one may have obsta-cles distributed along the field, and the obstacle avoidancefunction needs to be designed to enable obstacle avoidancecapability of mobile robots. With this purpose the term fo

i inEquation (4) is the repulsive action function to allow robotsto avoid obstacles, and it is designed as:

foi = co1

∑j∈Nv

i

ϕo(∥pik−pj∥σ)nik+co2∑j∈Nv

i

bik(p)(vik−vj),

where ϕo(.) is a repulsive action function to allow robots toavoid obstacles defined in [25]. nij and nik are vectors alongthe line connecting pj to pj and pik to pi, respectively.Matrices A = [aij ] and B = [bik] are the adjacency matricesdefined by graph G [29] .

Since the scalar field is very large comparing to thelimited sensing of each robot, all mobile robots need tomove and collaborate with each other to sample certainlocations and reconstruct the map of the field. To allow thesensor/robot network to travel to these certain locations, thenavigation function is designed as

fni = −cn1 (pi − pt)− cn2 (p(Na

i ∪{i}) − pt), (5)

where cn1 and cn2 are positive constants, and (p(Nai ∪{i}) is

defined as p(Nai ∪{i}) =

1|Na

i ∪{i}|∑|Na

i ∪{i}|i=1 pi, where |Na

i ∪{i}| is the number of robots in robot i’s local neighborhoodincluding robot i itself. pt is a target location that the robotnetwork wants to move there to collect data. Certainly, thelarger scalar field the more target locations are needed toallow the robot network to cover the field.

C. Compressive Sensing (CS)

1) Sparse signal presentation: Compressed sensing tech-niques provide us a sound approach to recover a sparse orcompressible signal from undersampled random projections,also refer to as measurements [12], [13]. In order to applyCS, signals are supposed to be sparse or compressible. A

vector signal X ∈ RN (X = [x1 x2 . . . xN ]T ) is k-sparse ifit has k non-zero elements. In another case, X may be dense(most elements of X are non-zeros) but can be considered assparse in Ψ domain if X = ΨΘ and Θ is a k-sparse vector.

When employing CS to a sensor/robot network formonitoring purposes, vector X represents all N unknownreadings from the sensing area. The gain obtained from usingCS is that the number of CS measurements required (M ) aremuch less than the number of the sensory readings (N ).

2) Signal sampling: Vector Y, called the measurementvector, contains data sampled from X with length N ; Y ∈RM (Y = [y1 y2 . . . yM ]T ), and M ≪ N . The randommeasurements are generated by Y = ΦX where Φ ∈ RM×N

is often a full-Gaussian matrix or binary matrix [30], calledthe measurement matrix. Y ∈ RM is the measurement vectorwith yi =

∑nj=1 φi,jxj , where φi,j are all entries on the ith

row of the projection matrix Φ.

3) Signal recovery: The number of measurements re-quired to reconstruct the original signals perfectly withhigh probability is M = O (k log N/k) following the l1optimization problem given by [14]. The unknown signalsare reconstructed based on the CS measurements as follows

Θ = argmin ∥ Θ ∥1, subject to Y = ΦΨΘ, (6)

where ||Θ ||1 =∑N

i=1 |θi| and X = ΨΘ. The l1 opti-mization problem can be solved with linear programmingtechniques such as Basis Pursuit (BP) [12]. In reality, noisyCS measurements should be considered while collecting asY = ΦX + e , with ||e||2 < ϵ. The recovery algorithm isaddressed as follows

Θ = argmin ∥ Θ ∥1, subject to ∥ Y−ΦΨΘ ∥2 < ϵ. (7)

D. Problem Formulation

As mentioned in the network model, given the unknownscalar field with N values needed to be observed andrepresented as X ∈ RN (X = [x1 x2 . . . xN ]T ). Aftera period of time T , the collaborative control leads the Lconnected mobile robots to sample almost the sensing area.We divide the period T into M time slots as t = T/M . Thedata collection processes are steps as follow

1) Collecting data: At time instant t, each mobile robotcan measure several positions covered by its sensing rangeRs. Due to the random distribution, on average, the numberof positions sensed by a robot are

γ =N

SπR2

s, (8)

where S is the area of the sensing area. If there is nooverlapped sensing area between L robots, at a time instantt, there are about γL positions sampled. Each mobile robotsimply sums all sensory data as

∑γi=1 xi to be exchanged

between connected robots. As shown in Figure 2, there arenon-overlapped circular regions sensed by L mobile robots.There is no region sampled twice at the same time.

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Sensing range Rs = 5; Communication range R

c = 11

Fig. 2. An illustration of communication links (blue lines) and sensingrange (within red circles) within a group of robots sampling in a smallregion of the sensing area

2) Exchanging the accumulated sensing information:Connected mobile robots exchange their accumulated dataattached with the sensing position indices sampled by eachrobot through their neighbors based on the robot commu-nication range Rc. After a certain numbers of sharing data,called convergence time, each mobile robot achieves one CSmeasurement which is a sum of all readings sampled by Lmobile robots as

yt =L∑

j=1

γ∑i=1

xi. (9)

We have two definitions at this point:Definition 1: Two robots are considered neighbors if theEuclidean distance between them is less than or equal to thecommunication range Rc.Definition 2: Convergence time, denoted as I , is the averagenumber of times each robot updates data from the neighborsuntil it achieves a CS measurement collected from all robotsin the team-work. It measures of how fast a group of robotscan create one CS measurement.

3) Moving for more CS measurements: Being led by thecollaborative control, the group of L mobile robots moveto different regions for sensing. At another time (t + 1),one more CS measurement is created following step 1 and2. After T or M periods of time (T = Mt), each robotachieves M CS measurements for the CS recovery processas Y = [y1 y2 . . . yM ].

4) Data reconstruction at each distributed robot: Eachrobot implements the CS recovery algorithm to reconstructall N readings to build a scalar map itself following Equa-tion (6) or (7) with noiseless or noisy CS measurements,respectively.

E. Analysis of Measurement Matrix Φ

By adding the readings together, the measurement matrixcreated by collecting M CS measurements (Y = ΦX) is asparse binary matrix as shown in Equation (10). We need toensure that this type of matrix can work as well as Gaussian

matrices in which all entries are i.i.d. zero-mean randomvariables with variance 1

M .

Φ =

1 0 0 1 0 1 ... 00 1 1 0 0 0 ... 10 0 0 1 1 0 ... 1... ... ... ... ... ... ... ...1 0 1 0 0 1 ... 0

M×N

(10)

If L mobile robots are connected all the time of sampling,the matrix has the average row weight of γL. It means thatthe matrix has γL non-zeros elements (’1’) on average ineach row. The remainders of (N−γL) in each row are zeros.

Based on the way positions are indexed and the waymobile robots sampling data, the measurement matrix isasymptotic to random sparse binary one which is wellstudied in [30]. It has been shown that this type of ma-trix can satisfy the restricted isometry property (RIP) andtherefore can be used as an energy-efficient measurementmatrix. In the simulation section, we will be showing thatour measurement matrix (ΦM×N ) can work as well as aGaussian matrix for the CS recovery process. It means thatwe only need to sample the sensing area with a few robotsat a time to form 1 CS measurement that does not affect CSperformance.

III. THE COMPRESSIVE AND COLLABORATIVE MOBILESENSING ALGORITHM

The proposed compressive and collaborative mobilesensing algorithm is summarized in Algorithm 1. At theInitialization phase, each robot stores its own data collectedwithin its sensing range Rs which is added as only onescalar value. The Exchange data phase shows the collabo-ration between connected robots to build a CS measurementbetween them by exchanging data at a time instant t. Thedistributed robots only communicate to their neighbors bythe time of this phase. Note that each robot only adds newdata that it has not stored from robots identified by theirindices. Based on these indices, a sparse binary measurementmatrix is created at each robot for the CS recovery process.

In the Moving phase, the group of robots move followingthe algorithm as mention in Section II-B. Depending on theaccuracy of the reconstructed map we are expecting, therobots move a certain number of times for those numberof CS measurements. The number of measurements couldbe pre-defined for each robot. The Data recovery phasecompletes the task of building a map at each robot.

In some cases, mobile robots maybe disconnected fromthe others to avoid obstacles. It means that the CS mea-surements collected at that time are not contributed by allL robots and are different. This results in the measurementmatrix that the corresponding rows may have different rowweights. As analyzed in [30] this does not affect the CSrecovery algorithm. By applying CS, each mobile robot onlyhas to collect M measurements to build a scalar field map(M ≪ N ). This reduces significantly consumed power notonly for communications but also for robot movements.

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Algorithm 1: Distributed Compressive and Collabora-tive Mobile Sensing Algorithm

1. Initialization phase:- X = [x1 x2 . . . xN ]T represents N unknown values.- Y = [y1 y2 . . . xM ]T represents CS measurementscreated at each robot for the CS recovery process.- S = [s1 s2 . . . sL] represents data stored at L robots.In this phase, each robot stores its own readings assj =

∑γi=1 xi.

- Convergence time at each robotIS = [IS1 IS2 . . . ISL] = zeros(1, L);

- A communication range Rc is chosen for all Lrobots.- A neighborhood matrix Q is created based onconnections among L robots as

Q[i, j] =

{1, if r(i, j) ≤ Rc

0, others(11)

2. Exchange data phase between L robots to form 1CS measurement at each robot:for k = 1 to L do

- Finding neighbors of each robotnei = find (Q(k, :) == 1);

- A robot adds all received data togetherincluding its own readingss(k) = s(k) + s(nei);

- The convergence time at each robot is countedasIS(k) = 1;

end- The ”new data” delivered to robots is recognized bythe attached sensing positions.while No new data received do

for i = 1 to L doif New data received from any neighbor (j)then

- Add new data to the previous storeddata ass(i) = s(i) + s(j);- Forward new s(j) to its nei;- Convergence time at each robotincreases asIS(i) = IS(i) + 1;

endend- Repeat updating new data at each robot

end- The first CS measurement is created at robot ith asy1 = si;

- The convergence time is calculated asI = mean(IS);

3. Moving phase- The mobile robots move following the collaborativealgorithm in Equation (4).- After M periods of time, each robot collects andstores M measurements required.4. Data recovery phaseAll unknown values (X) are reconstructed based onthe CS measurements (Y) stored at each robotfollowing Equation (6) or (7).

IV. COMMUNICATION POWER CONSUMPTION ANALYSIS

The total consumed power for robot communicationscontains main elements: the consumed power for commu-nications within all robot neighborhoods denoted as Pnei,the convergence time, denoted as I , and the number ofmeasurements required denoted as M . In order to obtain thenumber of CS measurements, each mobile robot has to gothrough M time slots to collect data. The more the numberof measurements, the more accurate the reconstructed data.So, the total power consumption can be calculated as

Ptotal = Pnei × I ×M. (12)

Pnei represents the total consumed power to transmit databetween L neighborhoods that can be calculated as

Pnei = ω ×Rαc × L, (13)

where ω represents the number of neighbors of each robotthat forms the number of communications in each neighbor-hood. Rc is the robot communication range. The path lossexponent could be between 2 ≤ α ≤ 4 in free space andmulti-path fading channels, respectively [31]. For simplicity,we choose α = 2 throughout this paper. Hence

Ptotal = ωR2cLIM. (14)

The convergence time I generally depends on both therobot density and the communication connections. We preferto have small number of robots deployed in the area thatcan provide the small value of I . Since the robots workingunder the collaborative control algorithm are organized asa lattice network, the number of connections between themare limited. This does not change I much as we changecommunication range Rc. The convergence time can beconsidered as a constant with a fixed number of robots. Thisis shown in Figure 5 in the simulation section.

V. SIMULATION RESULTS

In this section, we chose a fixed dimension square sens-ing area 300× 300 square units. We also assume that thereare about 5000 positions need to be observed in the area.These positions are generated randomly and uniformly in thearea. We deploy about 50 or 100 mobile robots led by thecollaborative control algorithm. The real sensor readings tobuild the scalar map are collected from Sensorscope: SensorNetworks for Environmental Monitoring [32]. Without lossof generality, it is assumed that the power for transmitting1 unit of data is 1 unit of power.

The result of the 50 robots flocking/moving togetherbased on the control law (4) is presented in Figure 3.Parameters for the control law (4) are: ca1 = 30, ca2 = 2

√ca1 ,

co1 = 1500, c02 = 2√co1, cn1 = 1.1 and cn2 = 2

√cn1 .

Three virtual target locations (see Equ. 5) selected for ptare: [70 50], [250 70] and [250 250] to navigate the robotsto visit entire scalar field. As can be seen 50 robots canmove together to cover the scalar field while collecting CSmeasurements. The robots can form a quasi-lattice networkformation and avoid collision with each other, and they mayget disconnected sometimes when avoiding obstacles.

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Fig. 3. The large scalar field has its size of 300 × 300 with 5000 random sample locations plotted in magenta color. Since the field is much largerthan the sensing range of each robot/mobile sensor (Rs = 5), the group of 50 robots (black triangles) with sensing range that covers circular regions inthe red circles have to move together along the field and take measurements. Three target locations for the navigation function defined in Equation (5) asp1 = [70 50]T , p2 = [250 70]T and p3 = [250 250]T are pre-determined to allow the mobile robot network to cover the field.

0 2 4 6 8 10 12 140

500

1000

1500

2000

2500

Number of positions sampled by each sensor

Sensing range Rs = 5

Fig. 4. Histogram of number of positions sampled by each robot.

As we chose the sensing range Rs = 5, Figure 4 showsthe histogram of the number of positions sampled by eachsensor in the area. These numbers will provide the numberof positions sampled by L sensors at each sampling time. Inthe figure, the average value equals the one calculated fromEquation 8 as γ = 5000

300×3003.14× 52 = 4.36.

Figure 5 depicts two convergence times (I) for both 50-

robot and 100-robot networks versus different communica-tion ranges. As mentioned, when we change the communi-cation range Rc, these convergence times can be consideredas constants. The smaller convergence time correspondingto the group of 50 robots should be chosen for energysaving. Figure 6 depicts the the total power consumptionfor all communications in the 50-robot network to collectand share M = 800 CS measurements versus differentcommunication range Rc. The total power consumption isalso calculated versus different number of measurementswith a fixed Rc = 10, as shown in Figure 7.

As mentioned in [30], a sparse binary matrix shouldwork as well as the full Gaussian measurement matrixwhich corresponds to the way of sampling all positions foreach CS measurement. The normalized reconstruction errors(∥X−X∥2

∥X∥2) are calculated for comparison when both types of

the matrices are used in the recovery process. In Figure 8we show two sparse binary matrices created by two teams ofrobots, L = 50 and L = 100, compared to a full Gaussianmatrix in a CS recovery process. The figure shows that either50 or 100 can work as well as the case M CS measurementsare collected from all positions (N = 5000) correspondingto the full Gaussian matrix.

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Scalar field mapping based on the original data

Y

Temperature

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Scalar field mapping based on the reconstructed data

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Fig. 9. Comparison of the true data and the reconstructed data and the reconstruction error when each robot stores M = 800 CS measurements.

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Average convergence time versus communication range Rc

Communication range Rc [unit]

Average convergence time [update]

L = 50

L = 100

Fig. 5. The average convergence time versus different communicationranges in two team networks, 50-robot and 100-robot.

10 10.5 11 11.5 12 12.5 13 13.5 14 14.5 150.6

0.8

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1.2

1.4

1.6

1.8x 10

9 Total power consumption vs Rc (L = 50; M = 800; I = 45)

Communication range Rc [unit]

Total communication power consumption [unit]

Fig. 6. Total power consumption for communication in the 50-robotnetwork with different communication range Rc.

Figure 9 depicts the true and the reconstructed scalarmaps in 3-D and the error map representing the differencebetween the true values and the reconstructed data as (X −X). The error map looks quite uniform all over the sensingarea.

* Discussion: By combining CS and the collaboration,the group of robots can save a huge energy for transmittingdata in the network as shown in Table I. Besides, the robotscan also save energy from their motion and be flexible inconnection between each other.

600 650 700 750 800 850 900 950 10005.5

6

6.5

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7.5

8

8.5

9

9.5

10x 10

8 Total power consumption vs M (L = 50; Rc = 10; I = 45)

Number of measurements

Total communication power consumption [unit]

Fig. 7. Total power consumption for communication in the 50-robotnetwork with different number of measurements M .

600 650 700 750 800 850 900 950 10000.08

0.09

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Compare different measurement matrices

Number of CS measurements

Norm

alized Reconstruction Error

Full Gaussian matrix

Sparse binary matrix (L = 50)

Sparse binary matrix (L = 100)

Fig. 8. Compare the reconstruction errors provided by three measurementmatrices: full dense Gaussian and two sparse binary matrices created by 50and 100 mobile robots, respectively.

TABLE I. COMPARISON BETWEEN USING CS AND NOT USING CSIN COMMUNICATION

Numberof robots

Number ofpositions tobe observed

Samples sentfrom eachrobot

Totalmeasurementssent

Apply CS 50 5000 150 × 800= 40, 000

Not apply CS 50 5000 4.36 50 × 5000= 250, 000

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VI. CONCLUSION AND FUTURE WORK

In this paper we proposed a compressive and collabora-tive sensing algorithm for distributed mobile robots collect-ing data in MSNs to build scalar maps themselves. The novelcombination is that the robots can save energy for both theirmotion and communication. Instead of sampling and sendinga hug data among robots in order to build a map at eachrobot, a certain number of CS measurement are created to beable to reconstruct data needed. The communication powerconsumption and the coverage are analyzed and formulatedheuristically. Simulation results are provided. We suggestthat the smallest communication range and the smallestnumber of robots can minimize the total power consumptionfor the networks.

In future work, we find the boundary for either thenumber of robots and the communication range. This isdirectly related to the boundary of the sparsity of themeasurement matrix that we can explore for further energysaving.

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