A Least Squares approach for learning gasdistribution maps from a set of integral gasconcentration measurements obtained with a

TDLAS sensorMarco Trincavelli

AASS Research CenterSchool of Science and Technology

Orebro UniversityEmail: marco.trincavelli@ieee.org

Victor Hernandez BennettsAASS Research Center

School of Science and TechnologyOrebro University

Email: victor.hernandez@oru.se

Achim J. LilienthalAASS Research Center

School of Science and TechnologyOrebro University

Email: achim.lilienthal@oru.se

Abstract—Applications related to industrial plant surveillanceand environmental monitoring often require the creation of gasdistribution maps (GDM). In this paper an approach for creatinga gas distribution map using a Tunable Diode Laser AbsorptionSpectroscopy (TDLAS) sensor and a laser range scanner mountedon a pan tilt unit is presented. The TDLAS sensor can remotelysense the target gas, in this case methane, requiring novel GDMalgorithms compared to the ones developed for traditional in-situ chemical sensors. The presented setup makes it possible tocreate a 3D model of the environment and to calculate the pathtravelled by the TDLAS beam. The knowledge of the beam path isof crucial importance since a TDLAS sensor provides an integralmeasurement of the gas concentration over that path. An efficientGDM algorithm based on a quadratic programming formulationis proposed. The approach is tested in an indoor scenario wheretransparent bottles filled with methane are successfully localized.

I. INTRODUCTION

Gas distribution mapping and source localization are tasksthat arise in various applications ranging from inspection ofindustrial plants or warehouses [1], to monitoring of landfillsor agricultural sites [2] and search for explosives or poisonousgases at harbours or airports [3]. The knowledge of the locationof the chemicals eventually present in the environment and, ifpossible, of their source is of great importance to carry outeventual recovery operations.

Gas distribution mapping and gas source localization arevery challenging for various reasons related to gas dispersaland to chemical sensing mechanisms. Indeed, gas dispersal innatural environments characterized by high Reynolds numberis turbulent [4]. This causes the gas distribution to assumea shape with an unpredictable fine structure without smoothgradients that can be used to navigate to the source or toassume a tractable functional form for the gas distribution.Moreover traditional chemical sensors are based on chemicalreactions that take place between the sensing surface and thegas to be analyzed. Therefore traditional sensors are in situsensors, which means that they provide information aboutthe chemical composition of a small area of few centimeters

around the gas sensor itself. Algorithms that can build a mapof the gas gas concentration using in situ gas sensors likemetal oxide gas sensors and photo ionization detectors havebeen already described in literature [5]. In the 1990s TDLASsensors started emerging as remote sensing gas analyzers thatcan be used to detect trace concentrations of gases in a widerange of applications. Depending on the setup, TDLAS sensorcan measure gases over distances up to 30m or even more than100 m if reflective mirrors are deployed. Moreover TDLASsensors, through accurate tuning of the emitted laser beam, canbe made perfectly selective to gases like HF , H2S, NH3,H2O, CH4, HCl [6]. However, TDLAS measurements arenot true gas concentration measurements but concentrationmeasurements integrated over the path of the laser beam.In order to infer local concentrations it is therefore neededto make assumptions on the gas distribution or to collectmultiple measurements and try to reconstruct the underlyinggas distribution from them. In particular, in this work wepropose a GDM algorithm that can infer a gas concentrationgrid map from a set of TDLAS readings and the respectivepaths of the beams. The algorithm is based on a non-negativeleast squares formulation, which is computationally efficient.

Previous works that aim at detecting increased methaneconcentrations through the use of a TDLAS sensors are [6],[7]. In [6] the TDLAS sensor is mounted on a car and on anairplane and it is demonstrated to be capable of detecting areasof increased methane concentration. However, no algorithm formeasurement aggregation is presented. In [7], an open pathTDLAS unit mounted on a pan tilt unit is used for estimatingthe emissions of CH4 from agricultural sites. The focus ison the quantification of the total amount of emitted methane,while in this work we focus on the accurate spatial localizationof high methane concentrations.

The remainder of the paper is organized as follows: Sec-tion II presents the experimental setup and Section III de-scribes the GDM algorithm, Section IV presents the results,and Section V concludes the paper with a discussion and

Fig. 1. Experimental setup. The TDLAS sensor and a laser range scannerare mounted on top of a pan tilt unit. In the background the two bottles filledwith methane are visible.

suggestions for future works.

II. EXPERIMENTAL SETUP

The sensor unit used in this work consists of a TDLASsensor (RMLD, Sewerin), a laser range scanner (LMS 200,SICK AG) and a pan-tilt unit (PTU-D46-70, Flir). The TDLASsensor and a laser range scanner are mounted on the top ofa pan tilt unit as shown in Figure 1. The PTU-D46-70 is acomputer-controlled pan-tilt unit that measures 0.0762 m ×0.13 m × 0.1079 m and weights 1.36 kg. According to themanufacturer’s specifications, the PTU-D46-70 has pan andtilt ranges of (78◦, 180◦) at a 0.003◦ resolution. The LMS200is a non contact range measurement system built for industrialapplications. By measuring the time of flight of a set ofinfra-red laser beams, the LMS200 is able to produce twodimensional scans with a field of view of 180◦ at a resolutionof 1◦. According to the manufacturer’s specifications, theLMS200 has a maximum operating range of 80 m. TheTDLAS is a spectroscopy based gas sensor. It is based on theprinciple that gas molecules absorb energy in narrow bandsof the electromagnetic spectrum. Outside these bands thereis no absorption. Devices based on Wavelength ModulationSpectroscopy (WMS), like the one used in this work, emit alaser beam which is tuned on the absorption band of the gasof interest. The wavelength is rapidly increased and decreasedso that the beam is in the absorption line for some time andthen off. While this scanning procedure is repeated, the powerof the transmitted laser beam is measured continuously. Themeasurement of the difference of the transmitted power whenthe laser is on or off the absorption line provides a precisehighly selective and highly sensitive measure of the target gas.From the user perspective, this sensing principle entails twomajor differences, with respect to traditional chemical sensors,which are based on a sensing surface that reacts with the

Fig. 2. Example measurement with a TDLAS sensor. In this case the beamtravels for 10m, of which one meter has a methane concentration of 500ppmand the other 9 meters of 10ppm. The reading of the TDLAS sensor in thisexample is (500ppm× 1m+ 10ppm× 9m) = 590ppm×m.

target gas and one or more electrical properties of the surfaceare measured (e.g. conductance for metal oxide gas sensors).The first difference is that, in case of a TDLAS sensor, thesensor itself does not need to enter in contact with the gas butremote detection is performed. The second difference is thatinstead of an absolute gas concentration or a measure that canbe directly related to absolute gas concentration, the TDLASsensor provides an integral measurement of concentration.Each measurement of a Tunable Diode Laser AbsorptionSpectroscopy (TDLAS) sensor is a line integral of the gasconcentration over the distance travelled by the beam, seeFigure 2.

III. ALGORITHM DESCRIPTION

The task of a GDM algorithm, which takes as input TDLASreadings, is to reconstruct the underlying gas distribution froma set of line integral measurements. This scenario is similarto the computer assisted tomography scenario in medicalinstrumentation, where the image of a body of interested isreconstructed from a set of attenuation measurements of e.g.X-rays beams [8].

As a first step of the algorithm, the target region is dividedinto a uniform grid of cubic cells. It is assumed that thegas concentration in each cell is uniform and constant. Theproposed algorithm estimates the concentration of each cell inthe lattice. Since each TDLAS measurement is a line integralof the gas concentration over the path travelled by the beam,the measurement can be decomposed in the following way:

y =

M∑i=1

lixi + ε (1)

where y is the TDLAS reading, M is the number of cells,li is the distance travelled by the beam in cell i, xi is the gasconcentration in cell i, and ε is a measurement noise term. Atwo dimensional example is displayed in Figure 3.

The problem of gas distribution mapping can now be formu-lated as the problem to estimate the vector of concentrationsx which best explains the measurements, or in other words,that maximizes the likelihood of the measurements. Given a

Fig. 3. Example scenario in which a measurement has been taken in an areadescribed by a 4×6 lattice. The dashed line is the path travelled by the beamin the measurement at hand. In this case the measure obtained by the TDLASsensor isy = l6x6+ l10x10+ l11x11+ l15x15+ l16x16+ l20x20+ ε whereε is the measurement noise.

set of N measurements, we can define a column vector y oflength N that contains all the measurement values, a columnvector x of length M that contains the concentration values ofthe cells, and a matrix L of dimensions N ×M that containsthe lines that each measurement travelled. Therefore the set ofmeasurement has to fulfill the following matrix equation:

y = Lx+ ε1 (2)

where 1 is a column vector of ones of length N . In orderto calculate the likelihood of the measurements we need toassume an appropriate distribution for the noise ε. If no sensormeasurement model is available, the most natural choice forthe noise distribution is the Normal distribution. If the noisedistribution is assumed to have zero mean (that means thatthe noise in the TDLAS sensor is unbiased) the maximumlikelihood solution for x is the least squares solution. Sincenegative gas concentrations do not make sense, the leastsquares problem has to be solved under the constraint thatx � 0. Moreover, since the measurements often do not coverentirely the map there will be many cells which are neverobserved. Also, it will make the the trajectories nearly linearlydependent, if the rays originates from very few observationpoints. This causes the least squares problem to have an infinitenumber of solutions, which is undesirable. It is therefore usefulto define a prior distribution for x. As in the case of themeasurement noise, the most natural choice for the prior of xis the Normal distribution, which turns into adding a quadraticregularization term to the objective function. Therefore, thegeneral problem of gas distribution mapping from a set ofTDLAS measurements can be formalized as the followingoptimization problem:

minimizex

‖Lx− y‖2 + λ‖x‖2 (3)

subject to x � 0

Fig. 4. Close up on the experimental area that shows the position of the twobottles that contain methane.

where λ is a parameter that determines the strength of theprior. Exploiting the fact that the square root is a monothonicfunction the optimization problem can be rewritten in thefollowing equivalent form:

minimizex

x′(L′L+ λI)x− y′Lx+ y′y (4)

subject to x � 0

where I is the identity matrix of dimension M . It isimportant to notice that, with this formulation of the problem,neither the number of variables nor the number of constraintsdepend on the number of measurements. As a result, thecomplexity of the problem to be solved depends only on thenumber of cells in the map and not on the number of collectedmeasurements. Therefore the cell size can be chosen as aresult of a trade-off between the resolution of the map andthe computational complexity of the optimization algorithm.

IV. RESULTS

The sensor unit displayed in Figure 1 was placed in fourdifferent measurement positions to allow the sensors to acquiredata about the gas distribution from different points of view.For each measurement position the pan tilt unit performs firsta tilt sweep in order to collect a set of laser range scannerreadings that are used to build a 3D model of the environmentusing the Octomap library [9]. The 3D occupancy model isthen used to calculate the path of the TDLAS beams for eachmeasurement. Then, the pan tilt unit performs a sweep in apredefined range of pan and tilt that covers the whole areaof inspection in order to collect TDLAS measurements. Oncethe movement of the pan tilt unit is completed, the sensorunit is moved to another measurement point. The total numberof TDLAS measurements collected in the four measurementpositions was 18164. The area to be mapped, whose dimension

(a) 3D view (b) Top view (c) Front view

Fig. 5. GDM obtained for the considered experimental setup, when λ = 0.1. The grey background represents the 3D occupancy model. Estimated methaneconcentration is denoted by dots. Small yellow dots represent low concentrations, while large red dots represent high concentration.

are 2.25 m × 1.63 m × 7.95 m is divided into a uniformgrid with cells of 0.2m3, giving a total of 4320 cells. In theexperiments, the solution did not show large sensitivity to thevalue of the regularization constant λ, therefore its value is setheuristically to 0.1. The quadratic program has been solvedusing the commercial package Gurobi [10]. The solution timewas roughly 4 s. Figure 4 shows a close up on the bottlescontaining methane and Figure 5 shows the gas distributionmap obtained in the test scenario. We can observe that theposition of the maxima of gas concentration in the map isvery close to the position of the bottles containing methane.

V. CONCLUSIONS AND FUTURE WORKS

This paper presented a sensor setup and an algorithm forcreating a gas distribution map from integral gas concentrationmeasurements. The measurement unit consist of a TDLASsensor and a laser range scanner mounted on top of a pantilt unit, so that they can scan a whole area. The algorithmreconstructs the gas distribution that best explains the collectedmeasurements. Gas distribution mapping is formulated as anoptimization problem and solved efficiently. The proposedapproach has been tested in a simplified setup where twobottles filled with methane were correctly localized.

Further analysis are not present at the moment, and manypossibilities of future work are open. The most evident lim-itation is that, given the fact that the exact concentrationof methane in the bottles was unknown, it was possible toevaluate the GDM only with respect to the spatial displacementof the increased methane concentration, but the accuracy of theconcentration estimation could not be established. Moreover,in the presented setup, given the fact that the methane gas wascontained into bottles, the gas distribution was static for thewhole duration of the experiment. In a more realistic scenariothe methane would be leaked in the environment and for aturbulent plume.

Another aspect to be improved in future works is the factthat the sensor unit could be placed on a mobile robot in orderto address scenarios like indoor inspection of corridors withpipes or outdoor landfill or agricultural site monitoring. Thissetup would introduce additional problems like uncertainty inthe position of the robot, which would turn into uncertaintyin the path of the beam. Finally, an interesting aspect to

investigate is the sensor planning problem, which is, whereshould the robot go to measure next in order to acquiremeasurement which are most useful for improving the GDMmodel.

ACKNOWLEDGMENT

The authors would like to thank Robotdalen and KKS forproviding the financial resources for the development of theGasbot project.

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