Noname manuscript No. (will be inserted by the editor) Neural network interpolation of the magnetic field for the LISA Pathfinder Diagnostics Subsystem Marc Diaz-Aguilo · Alberto Lobo · Enrique Garc´ıa–Berro Received:January21,2011/Accepted:January21,2011 1 1 Abstract LISAPathfinderisascienceandtechnologydemonstratoroftheEuropean 0 Space Agency within the framework of its LISA mission, which aims to be the first 2 space-bornegravitationalwaveobservatory.ThepayloadofLISAPathfinderistheso- n called LISA Technology Package, which is designed to measure relative accelerations a betweentwotestmassesinnominalfreefall.Itsdisturbancesaremonitoredanddealt J by the diagnostics subsystem. This subsystem consists of several modules, and one of 0 theseisthemagneticdiagnosticssystem,whichincludesasetoffourtri-axialfluxgate 2 magnetometers, intended to measure with high precision the magnetic field at the positionsofthetestmasses.However,sincethemagnetometersarelocatedfarfromthe ] c positionsofthetestmasses,themagneticfieldattheirpositionsmustbeinterpolated.It q hasbeenrecentlyshownthatbecausetherearenotenoughmagneticchannels,classical - interpolation methods fail to derive reliable measurements at the positions of the test r g masses, while neural network interpolation can provide the required measurements [ at the desired accuracy. In this paper we expand these studies and we assess the reliability and robustness of the neural network interpolation scheme for variations 1 v of the locations and possible offsets of the magnetometers, as well as for changes in 5 environmental conditions. We find that neural networks are robust enough to derive 5 accuratemeasurementsofthemagneticfieldatthepositionsofthetestmassesinmost 9 circumstances. 3 . 1 MarcDiaz–AguiloandEnriqueGarc´ıa–Berro 0 DepartamentdeF´ısicaAplicada, 1 UniversitatPolit`ecnicadeCatalunya, 1 c/EsteveTerrades5,08860Castelldefels,Spain : Institutd’EstudisEspacialsdeCatalunya, v c/GranCapit`a2–4,Edif.Nexus104,08034Barcelona,Spain i E-mail:[email protected],[email protected] X AlbertoLobo r a InstitutdeCi`enciesdel’Espai,CSIC, CampusUAB,FacultatdeCi`encies, TorreC-5,08193Bellaterra,Spain Institutd’EstudisEspacialsdeCatalunya, c/GranCapit`a2-4,Edif.Nexus104,08034Barcelona,Spain E-mail:[email protected] 2 Keywords LISAPathfinder·magnetometers·on-boardinstrumentation·spaceborne and space-research instruments · neural networks 1 Introduction LISA(LaserInterferometerSpaceAntenna)isajointESA/NASAspacemissionaimed atdetectinglowfrequencygravitationalwaves,intherangebetween10−4Hzand1Hz. LISA will be a constellation of three spacecraft which occupy the vertexes of an equi- lateral triangle, with sides of 5 million kilometers. The barycenter of the constellation will revolve around the Sun in a quasi circular orbit, inclined 1◦ with respect to the ecliptic, and trailing the Earth by some 20◦, about 45 million kilometres. Each space- craftharborstwoproofmasses,carefullyprotectedagainstexternaldisturbancessuch as solar radiation pressure and charged particles, which ensures they are in nominal free-fall in the interplanetary gravitational field. Gravitational waves show up as dif- ferential accelerations between pairs of proof masses in distant spacecrafts, and the working principle of LISA is to measure such accelerations using picometer precision laserinterferometry.Theinterestedreaderisreferredtoreferences[1]and[2]formore extensiveinformation,aswellastotheLISAInternationalScienceTeam(LIST)web- page [3]. The technologies required for the LISA mission are many and very challenging. This, coupled with the fact that some flight hardware cannot be tested on ground since free fall conditions cannot be maintained during periods of hours, led to set up a technology demonstrator to test critical LISA technologies in a flight environment. Thesetechnologieswillbetestedinaprecursormission,whichiscalledLISAPathfinder (LPF).ThismissionisframedwithintheScientificProgramofESA,anditisexpected tobelaunchedtowardsearly2012.TheideaofLPFistosqueezeoneLISAarmfromfive millionkilometersto35centimeters,thendeterminethenoiseofthemeasurementsin afrequencyrangewhichisslightlylessdemandingthanthatofLISA.Morespecifically, the requirement is formulated in terms of spectral density of acceleration noise as (cid:34) (cid:18) (cid:19)2(cid:35) S1/2 ≤3×10−14 1+ ω/2π ms−2Hz−1/2 (1) δa 3mHz for 1mHz≤ω/2π≤30mHz, where ω/2π is the frequency in Hz. The payload on board LPF is called the LISA Technology Package (LTP) [4]. Its main components are the two gravitational reference sensors, which are the two large vertical cylinders in figure 1. In this figure it can also be seen the optical metrology subsystem, which provides picometer precision measurements of the relative position and acceleration of the two test masses, using precise interferometry. This system is located between the two gravitational reference sensors. Also visible, and specially relevant here, are the four tri-axial magnetometers, which are represented as floating boxes.Theiractualphysicalsupportisthelateralwallofa(notdrawn)largercylinder which encloses the entire LTP. MagneticnoiseintheLTPisrequiredtobenotmorethan40%ofthetotalreadout noise, i.e., 1.2×10−14ms−2Hz−1/2 out of 3.0×10−14ms−2Hz−1/2 in the measure- ment bandwidth, see Eq. (1). This noise appears because the residual magnetization andsusceptibilityofthetestmassescouplewiththesurroundingmagneticfield,giving 3 Fig. 1 AschematicviewofthepayloadofLISAPathfinder,theLTP. rise to a force in each of them (cid:28)(cid:20)(cid:18) (cid:19) (cid:21) (cid:29) χ F= M+ B ·∇ B V. (2) µ 0 In this expression B is the magnetic field in the test mass, M is its density of mag- netic moment (magnetization), V is the volume of the test mass, χ is its magnetic susceptibility, µ is the vacuum magnetic constant (4π×10−7m kg s−2 A−2), and 0 (cid:104)···(cid:105) indicates the volume average of the enclosed quantity. As clearly seen from Eq. (2),therearetwosourcesofmagneticnoise.Thefirstoneisduetothefluctuationsof the magnetic field and its gradient in the regions occupied by the test masses [5]. The secondonecomesfromthesusceptibilityofthetestmassandthemagneticremanence fluctuations [6]. This additional noise is expected to be much less important, and it is usually disregarded. Clearly, a quantitative assessment of magnetic noise in the LTP requires real-time monitoring of the magnetic field and its gradient [7]. This is the ultimate goal of the tri-axialfluxgatemagnetometers[8].Thesedeviceshaveahighpermeabilitymagnetic core,whichdrivesadesignconstrainttokeepthemsomewhatfarfromthetestmasses. Thus, their readouts do not provide a direct measurement of the magnetic field at the position of the test masses, complicating the task of inferring the field at their position, and forcing the implementation of an interpolation method to overcome this shortcoming.However,suchinterpolationprocessfacesseriousdifficulties.Indeed,the sizeoftheinterpolationregion,thatis,theinterioroftheLTPCoreAssembly(LCA), is too large for a linear interpolation scheme to be reliable. Additionally, the number of magnetometer channels does not provide sufficient data to go beyond a poor linear approximation[9].However,thestructureofthemagneticfieldisrathercomplex,asthe sourcesofmagneticfieldareessentiallytheelectroniccomponentsinsidethespacecraft. 4 Thenumberofidentifiedsourcesisabout50,andtheybehaveasmagneticdipoles,the onlyexceptionbeingthesolarpanel,whichisbestapproximatedbyaquadrupole.The positions of the sources are dictated by the architecture of the satellite, which defines the exact position of each electronic subsystem. Fortunately, there are no sources of magnetic field inside the LCA, all being placed within the spacecraft, but outside the LCA walls. Adequate processing of all the available information shows that the magnetic field is smaller towards the center of the spacecraft (where the test masses arelocated)thanitisinitsperiphery(wherethemagnetometerstakemeasurements). Ithasbeenrecentlyshown[9]thatsincethestandardinterpolationscheme,which isbasedinmultipoleexpansionofthemagneticfieldinsidetheLCAvolume,doesnot go beyond quadrupole order, its performance in estimating the magnetic field and its gradientsisverypoor.Onthecontrary,artificialneuralnetworkshavebeenshowntobe areliablealternativetoestimatetherequiredfieldandgradientvaluesatthepositions of the test masses. The reasons for this are multiple. Firstly, the multipole expansion onlytakesintoaccountthereadingsofthemagnetometers,whereastheartificialneural network also uses the actual value of the magnetic field at the position of the test masses to train the network. This is a crucial issue since the interpolation algorithm is fed with additional information. Secondly, the classical interpolation method seeks for a global solution of the magnetic field. That is, the multipole expansion models the magnetic field inside the entire volume of the LCA. Clearly, since the available information for the multipole expansion is rather limited, the quality of the global solutionisverypoor.Insharpcontrast,theartificialneuralnetworkfirstfindsandthen uses the correlation between the magnetic field at the positions of the magnetometers and the test masses to obtain reliable values of the magnetic field for any magnetic configuration. As a matter of fact, the artificial neural network performs a point-to- point interpolation and it is not aimed at reproducing the highly non-linear magnetic field well at any arbitrary position within the volume of the LCA. Finally, artificial neural networks are trained using a large number of magnetic field realizations, thus the interpolating algorithm uses a statiscally elaborated information. In this sense, it is important to realize that artificial neural networks have been shown to be a robust and easily implementable technique among numerous statistical modeling tools [10]. On the contrary, the multipole expansion does not use statistical information. Once thereadingsofthemagnetometersareknown,thetheoreticalsolutionforthemagnetic field within the entire volume of the LCA is determined in a straightforward way. Nevertheless, an in depth study of how the results of the interpolation procedure dependonthespecificcharacteristicsoftheneuralnetworkremainstobedone.Italso remains to further investigate why the neural network — which uses lineal transfer functions — obtains such good results interpolating the value of the magnetic field at the positions of the test masses, which are well inside a deep well of magnetic field. Finally,anassessmentoftherobustnessoftheneuralnetworkinterpolatingschemein front of the unavoidable errors in the positions of the magnetometers, or in front of low-frequency variations of the magnetic environment and, more importantly, in front of offsets in the readings of the magnetometers still is needed. These are precisely the goals of this paper. The paper is organized as follows. In Sect. 2 we discuss the appropriateness of our neural network approach to measure the magnetic field and its gradients at the positions of the test masses, and we discuss which are the accuracies obtained when different architectures of the neural network are adopted. It follows Sect. 3, where we discuss how the unavoidable errors in the onground measurements of the magnetic 5 dipoles of each electronic box affect the performance of the adopted neural network. InSect.4weevaluatetheexpectederrorsintheestimateofthemagneticfieldandits gradients due to a possible offset in the readings of the magnetometers due to launch stresses, whereas in Sect. 5 we study how the mechanical precision of the positions of the tri-axial magnetometers and their spatial resolution affect the determination of the magnetic field and its gradients. Sect. 6 is devoted to assess the reliability of our neural network approach in front of a slowly varying magnetic environment. Finally, in Sect. 7 we summarize our main findings, we discuss the significance of our results and we draw our conclusions. 2 The neural network architecture Although neural networks have been used in different space applications [11,12], to the best of our knowledge this is the first application of neural networks to analyze inflightoutputsinspacemissions.Hence,studyingtherobustnessoftheneuralnetwork architecture proposed to estimate the magnetic field inside the LCA is a mandatory task. 2.1 The fiducial neural network architecture Fig. 2 shows a simplified version of the fiducial architecture of our neural network. As can be seen, the number of inputs is twelve — one for each magnetometer readout — correspondingtothefourtri-axialmagnetometersplacedinthespacecraft.Theseread- ingsaretheonlyvaluableinformationwhichcanbeusedtoestimatethemagneticfield atthepositionsofthetestmasses,andconstitutetheinputlayeroftheneuralnetwork. On the other hand, to estimate the magnetic field three outputs will be required — corresponding to the three field components per test mass — whereas to estimate the gradientonlyfiveadditionaloutputsareneeded.Thisisbecausethemagneticfieldhas zero divergence and zero rotational. Thus, the gradient matrix ∂B /∂x is a traceless i j symmetricmatrix,andthereforeonly5outofits9componentsareindependent.These outputs are the output layer of the neural network. In addition to the two previously described layers, there is only one intermediate layer, which constitutes the hidden layer.Thislayerismadeof15neurons.Usingthisarchitecturefortheneuralnetwork the magnetic field estimates typically have standard deviations on the order of ∼2% [9], a value to which we compare the results of our analysis. 2.2 Training and testing Trainingandtestingdatasetsweresimulatedusingthemostcompleteandup-to-date information about the magnetic configuration within the spacecraft. The complete magneticconfigurationofthesatellitehasnotbeenmeasuredyet,becausesomeunits have not been delivered yet to the prime contractor. Nevertheless, the exact position of each unit in the spacecraft reference frame is known. On the other hand, the mag- netic moments used in our simulations are those reported by the constructors of each subsystem. Unfortunately, this data is not available yet for all units, and moreover although the moduli of the dipoles are known for all the subsystems their directions 6 i1 i2 i3 i4 MAG1 MAG2 MAG3 MAG4 H1 H2 H3 H4 ... H15 TM1_B TM1_G TM2_B TM2_G o1 o2 o3 o4 Fig.2 Thefiducialfeed-forwardneuralnetworkarchitecture.Thereadingsofthemagnetome- tersarethesysteminputs(4magnetometers,eachonewith3datachannels).Theoutputsof thesystemarethemagneticfieldandgradientcomponentsatthepositionsofthetestmasses (3 field components and 5 gradient estimates for each test mass). For the sake of simplicity, all the field and gradient channels have been grouped into a single neuron. Moreover, not all theneuronsinthehiddenlayerareshowninthisfigure. are not known yet for most of the units. The three-dimensional values of the mag- netic dipoles of each unit will be accurately measured in the final testing campaign to be performed on each subsystem before assembling. This campaign is expected to be performed on the assembled spacecraft during 2011. The training and validation of the neural network using the measured values of the magnetic dipoles will be done afterthecampaignbutthespecificdetailsoftheprocessingalgorithmareexpectedto remain unchanged. Moreover, the magnetic field inside the LCA is expected to vary substantiallybetweenthedifferentoperationalmodes.Accordingly,sincethemagnetic configuration of the spacecraft may have different characteristics for different opera- tional modes, it is foreseen that a different neural network will be trained for each of these configurations. Given the unknown orientations of the magnetic dipoles we generate several mag- neticconfigurationsassigningrandomlytheorientationsofthe46dipoles.Anexample scenarioisthuscharacterizedbyaselectionofthe46dipoleswithrandomorientations. 7 Table 1 Positionsofthetestmassesandpositionsofthemagnetometersreferredtoacoor- dinatesystemfixedtothespacecraft. Testmasses x[m] y [m] z [m] 1 −0.1880 0 0.4784 2 0.1880 0 0.4784 Magnetometers x[m] y [m] z [m] 1 −0.0758 0.3694 0.4784 2 −0.3765 0 0.4784 3 0.0758 −0.3694 0.4784 4 0.3765 0 0.4784 Therandomcharacteroftheproceduremayseemunrealistic,sincetheactualsatellite configuration is not random. In this context, however, randomness is an efficient way of mimicking our lack of knowledge of all the directions of the sources of magnetic field. Nevertheless, it is worth mentioning that the produced sets are consistent with our expectations and the mission requirements [5,8] since at the positions of the test masses we obtain magnetic fields ∼300 nT, while the readings at the magnetometers areoftheorderof4to10µT.Withthisapproachthemagneticfieldgeneratedbythe dipole distribution at a generic point x and time t is therefore given by n B(x,t)= µ0 (cid:88) 3[ma(t)·na] na−ma(t) (3) 4π a=1 |x−xa|3 wherena= (x−xa)/|x−xa|areunitvectorsconnectingthethea-thdipolema with thefieldpointx,andnisthenumberofdipoles.Inordertosimulaterealisticmagnetic environments, we compute the magnetic field at the positions of the magnetometers and at the positions of the test masses using Eq. (3). The positions of the test masses and of the magnetometers are shown in table 1. Two different batches of 103 samples aregenerated.Thefirstbatchwasusedasthetrainingsetforaneuralnetworkwiththe architectureofFig.2.Thisbatchconsistsin12inputs(3inputsforeachofthe4vector magnetometers) and 16 outputs representing the field information at the position of the two test masses (3 field plus 5 gradient components per test mass). The second batch has been used for validation to assess the performance of the neural network. 2.3 Varying the number of neurons Assesingthecorrectchoiceofthenumberofneuronsofaneuralnetworkisnotasimple task. When the neural network is composed by only one hidden layer, the input layer containsasmanyinputs-neuronsastheinformationweprovidetothenetworkandas many output-neurons as the target information we want to reconstruct. Nevertheless, as far as the number of neurons of the hidden layer is concerned, it is not guaranteed thatthearchitectureoftheselectedneuralnetworkisoptimalnorthereisanalgorithm inthecurrentliteraturetodeterminetheoptimalnumberofneurons[13,14].Normally, to obtain good results, the smallest system obtained after prunning that is capable to fitthedatashouldbeused.Unfortunately,itisnotobviouswhatsizeisbest.Asystem 8 10 15 σ 0 2 4 6 8 10 12 14 16 N 10 25 σ 0 2 4 6 8 10 12 14 16 N Fig. 3 Quality of the estimate of the magnetic field as a function of the number of neurons in the hidden layer. The maximum interpolation error remains almost constant for neural networkslargerthan∼15neuronsinthehiddenlayer.ThesolidlinecorrespondstotheBx- component,thedottedlinetotheBy-componentand,thedashedlinetotheBz-component. with a small number of neurons will not be able to learn from the data, while one with a large number of neurons may learn very slowly and, moreover, it will be very sensitive to the initial conditions and learning paramaters. Additionally, it should be takenintoaccountthatoneofthebiggestproblemsoflargenetworksforsomespecific problemsisthefactthatintheearlystagesoftraining,theerroronboththetraining and tests tends to decrease with time as the network generalizes for the examples to theunderlyingfunction.However,atsomepoint,theerroronthetestingsetreachesa minimum and begins to increase again as the network starts to adapt to artifacts and specific details in the training data, while the training error asymptotically decreases. Thisproblem,knownasoverfitting,occursmorefrequentlyinlargenetworksduetothe excessivenumberofdegreesoffreedomincomparisontothetrainingset[15].Toavoid this, we have used the early stopping technique, which overcomes this shortcoming. In the early stopping technique the available data is divided into three subsets [10]. The first subset is the training set, which is used for computing the gradient and updating the network weights and biases. The second subset is the validation set. The error on the validation set is monitored during the training process. The validation error normally decreases during the initial phase of training, as does the training set error. However, when the network begins to overfit the data, the error on the validation set typically begins to rise. When the validation error increases for a specified number of iterations, the training is stopped, and the weights and biases at the minimum of the validation error are returned to the values obtained at the minimum. All these precautionary measures avoid overfitting. Therefore, the analysis 9 Table 2 Qualityoftheestimateforthemostcommonneuronactivationfunctions. TM1 TM2 Function σx σy σz σx σy σz Tangentsigmoid 4.1 3.8 2.5 5.9 5.2 4.5 Linear 3.8 3.5 2.3 5.7 5.4 4.2 Logarithmicsigmoid 4.2 3.8 2.5 6.2 5.1 4.5 Radialbase 4.2 4.3 3.9 6.3 6.0 4.8 Step 7.5 7.6 4.9 12.3 8.2 7.9 of the number of neurons needed for the hidden layer can be made analyzing the evolutionoftheestimationerroronthetestingsetasthenumberofneuronsincreases. The results of such an analysis are depicted in figure 3, which shows the standard deviation of the estimate for both test mass 1, σ , and test masss 2, σ as a function 1 2 of the number of neurons in the hidden layer, N. Ascanbeseeninthisfigure,whenareducednumberofneuronsisusedthemodel cannotaccuratelyestimatetheunderlyingfunctionduetothelackoftunnableparam- eters. As the number of neurons in the hidden layer is increased, the neural network performs better and for a number of neurons larger than 15 the error is not further reduced. Consequently, we conclude that for this specific application the adequate number of neurons for the hidden layer lies between 10 and 15. This choice ensures a network large enough to be capable of estimating the underlying relationship and not excessively large to consume excessive training time, learn slowly and be dependent onthelearningalgorithm.Wehavealsocheckedthatincreasingthenumberofhidden layersoftheneuralnetworkdoesnotresultinabetterperformanceoftheinterpolating algorithm, but for the sake of conciseness we do not show the results here. All in all, it seems that our fiducial architecture seems to work best. 2.4 Changing the type of neuron Mostofthefeed-forwardnetworksaretrainedwiththeback-propagationalgorithmand gradient descent techniques are used to minimize somespecific cost function, and this has been the case for the training algorithm used here. This means that all activation functionswithinthenetworkmustbedifferentiabletobeabletocomputethenetwork gradient for each learning step. Normally, the most commonly used type of functions are the tangent sigmoid or the logarithmic sigmoid [10], which can model any non- linear function if properly trained [16], whereas linear functions are usually employed for linear models with high dimensionality. We have studied several possibilities and the results are listed in table 2, where we show for the different types of neurons the standard deviations of the probability densityfunctionsoftheestimatesofthemagneticfieldforbothtestmass1and2(TM 1 andTM ,respectively).Inourcase,andasborneoutfromtable2,thelinearfunction 2 together with the tangent sigmoid and the logarithmic sigmoid are the most efficient choices,whiletheperformanceoftheradialbasefunctionisslightlyworse.Finally,the stepfunction(thepopularperceptron)doesnotyieldgoodresultsbecauseitisspecif- ically designed to be used for classification problems. Specifically, the linear neuron is 10 the one for which we obtain the best results. This could be surprising given that our problemishighlynon-linear.Thereasonisthatforeverymagneticconfigurationthere existsalargeandfairlystabledifferencebetweenthevalueofthemagneticfieldatthe location of the magnetometers (all of the components of the magnetic field are ∼ 10 µT)andthefieldatthepositionofthetestmasses(allthecomponentsareonthe100 nT level). For this reason, the weigths of the network happen to be the most relevant modeling factors. That is, the point-to-point interpolation can be understood in the linearcaseasasimpleweightedsumofthemagnetometersmeasurements.Accordingly, because of its simplicity and good results, we use the linear function as the basic unit in our regression study. It is worth noting at this point that similar results could be obtainedusingahigh-dimensionalityleastsquaresanalysis,butinourspecificcasewe have found matrix inversion problems because some magnetometer channels present highly correlated signals. 2.5 Underlying structures We have already shown that our neural network is highly reliable. Thus, it is normal to ask ourselves which is the ultimate reason of this behavior. The answer to this questionisthatduringthetrainingprocess,theneuralnetworkeventuallylearnsthat the magnetic field at the positions of the test masses is generally smaller than the magnetometers readouts — with occasional exceptions due to the rich and complex profile structure of the field inside the LCA. Moreover, the neural network is able to learn an inference procedure which is actually quite efficient. To better understand this, we found instructive to look into relationships between the data read by the magnetometersandtheestimatesofthemagneticfieldgeneratedbytheneuralnetwork. We chose to calculate correlation coefficients between input and output data, and the results are displayed in table 3. The test masses are labeled as TM and TM , 1 2 respectively, whilst the four magnetometers are listed as M , i=1,...,4. i Thefollowingmajorfeaturescanbeeasilyidentified.Firstly,eachcomponentofthe field is basically estimated from the magnetometers reading of the same component. Forexample,theinterpolationoftheBx-componentintestmass1ismostlydependent on the Bx-readings of the magnetometers. Secondly, the measurements of the magne- tometersclosertotheinterpolationpointshavelargerweights.Forinstance,whenthe field is estimated at the position of TM , to which the magnetometer M is the clos- 1 4 est magnetometer, the value it measures is the largest contributor to the interpolated field in TM . At the same time, magnetometers M and M being nearly equidistant 1 1 3 frombothtestmasses,theirweightsarealmostidentical(seetable1formoredetails). Finally, no apparent or easily deductible physical relationship is found between the estimated gradient at the positions of the test masses and the magnetometer inputs. 3 Variations of the magnetic dipoles The numerical experiments done so far indicate that the neural network interpolating scheme offers good performances when properly trained, irrespective of its specific architecture. However, we emphasize that the neural network has been trained using simulated data, while for the real spacecraft the neural network will be trained using ongroundmeasureddata.Thisdata,asalreadymentionedinsection2.2,isplannedto