Validation of a fast and accurate magnetic tracker operating in the environmental field
Valerio Biancalana, Roberto Cecchi, Piero Chessa, Marco Mandal?, Giuseppe Bevilacqua, Yordanka Dancheva, Antonio Vigilante
VValidation of a fast and accurate magnetic tracker operating in the environmental field
Valerio Biancalana, Roberto Cecchi, Piero Chessa, Marco Mandalà, Giuseppe Bevilacqua, Yordanka Dancheva, and Antonio Vigilante DIISM, University of Siena – Via Roma 56, 53100 Siena, Italy ∗ DSFTA, University of Siena – Via Roma 56, 53100 Siena, Italy Department of Physics, Pisa University, Largo Pontecorvo, 3, 56127 Pisa, Italy DSMCN, Siena University, Viale Bracci 16 53100 Siena, Italy Aerospazio Tecnologie srl, Strada di Ficaiole, 53040 Rapolano Terme (SI), Italy Department of Physics and Astronomy, University College London,Gower Street, London WC1E 6BT, United Kingdom
We characterize the performance of a system based on a magnetoresistor array. This instrumentis developed to map the magnetic field, and to track a dipolar magnetic source in the presence of astatic homogeneous field. The position and orientation of the magnetic source with respect to thesensor frame is retrieved together with the orientation of the frame with respect to the environmentalfield. A nonlinear best-fit procedure is used, and its precision, time performance, and reliability areanalyzed. This analysis is performed in view of the practical application for which the system isdesigned that is an eye-tracking diagnostics and rehabilitative tool for medical purposes, whichrequire high speed ( ≥
100 Sa/s) and sub-millimetric spatial resolution. A throughout investigationon the results makes it possible to list several observations, suggestions, and hints, which will beuseful in the design of similar setups.
INTRODUCTION
Tracking methodologies based on magnetic field mea-surements have been widely studied in the past decadesand find application in several areas, including low-invasivity medical diagnostics [1, 2]. These methodolo-gies are based on the accurate mapping of the mag-netic field generated by one or more magnetic sources,and on mathematical analysis aimed to reconstruct ge-ometrical features (position and orientation) of thosesources. Tracking apparatuses based on permanent-magnet sources constitute an excellent tool for wireless(and thus minimally invasive) medical applications. Theavailability of strongly magnetizable materials, such asNeodymium-Iron-Boron alloys, permits the generation ofwell detectable magnetic signals by very small size de-vices, with further reduction of the invasivity level.Compared to other state-of-the-art tracking and/orimaging techniques based on optical detection or ul-trasonic measurements, magnetic tracking provides anocclusion-free scheme to estimate the target position andis intrinsically responsive to its orientation. This en-ables a reliable localization, and accurate trajectory re-construction, in more than three dimensions. In the sim-plest case of a dipolar target, five-dimensional informa-tion is normally retrieved, consisting in (3D) position and(2D) angular orientation: the system has a blind angularcomponent, which corresponds to rotations around thedipole direction.The problem of retrieving position and orientation of amagnetic dipole has been successfully faced both on thebasis of linear, deterministic methods [3] and non-linear ∗ [email protected] best-fit approaches [4, 5], as well as with a combinationof the two, where the linear solution is used as an input-guess for the non-linear calculation [6].At the expense of a heavier computation, the non-linear best fit approach permits more accurate evalua-tion, also thanks to the possibility of using larger datasets. Its main drawback is constituted by the need ofproviding an appropriate initial guess to the numericalalgorithm, i.e. on a limited reliability caused by the non-convexity of the strongly non-linear function to be min-imized. The typical working principle of best-fit proce-dures is based on finding a path in the parameter space,which efficiently brings from an initial, arbitrarily as-signed guess to the best fitting parameter set. The lat-ter is identified as the one that minimizes the differencebetween measured quantities and their theoretical val-ues. The assignment of an appropriate initial guess has atwofold relevance. It helps accelerating the convergenceof the algorithm, and –more important– helps prevent-ing that the algorithms converge to local minima, i.e. towrong solutions. Unfortunately, the mathematical com-plexity which renders these numerical approaches a fa-vorite choice, makes also difficult to elaborate reliablepredictions of the guess appropriateness. Empiric testsare necessary to quantify the robustness of the best-fitprocedure with respect to the accuracy level of the ini-tial guess.The non-linear approach can be easily extended tomore complex problems, as in the cases of multi-targettracking [7] or when operating in the presence of a back-ground field, which is the case considered here. The latterfeature characterizes the device described in this work,because of two main reasons: firstly it makes possibleto operate with small targets in the presence of Earthfield without suffering of tracking distortions induced bythe latter, secondly it allows the simultaneous tracking a r X i v : . [ phy s i c s . a pp - ph ] J a n of the magnetic target with respect to the sensors andof the sensors with respect to the ambient field. Thisis of interest in diagnostic measurements focused on thecorrelation of head and eye motion, which, with instru-mentation based on alternative kind of measurements,requires the fusion of data recorded by detectors of dif-ferent nature [8, 9].Up to date the gold standard for eye-tracking system inmedical field is represented by magnetic search coil sys-tems for research purposes and infrared head mountedcameras for clinical use. Typical angular and temporalresolution of the two different techniques are respectively < . ◦ and > . ◦ and 100 −
200 Hz (high-speed, head mounted infraredcameras) [10]. The last option is the most widely used inthe medical field since it is less expensive, non-invasiveand portable. Despite of that, infrared cameras haveseveral drawbacks in terms of eye movement recording:lower spatiotemporal resolution, difficulties in recordingvertical and torsional eye movements, difficulties withlight eyes, artifacts due to blinks, need the camera tobe strongly tightened to the head, need of data fusion torecord also head movements, heavy computational bur-den for image analysis. An overview about speed andangular resolution of low-cost eye trackers developed inthe last decade can be found in ref.[11].The paper is organized as follows: Sec.I provides abrief description of the hardware setup and of its spec-ifications; Sec.II describes the methodology applied toextract tracking information from magnetometric data;Sec.III reports a set of tracking outputs and estimates oftracking accuracy; Sec.IV describes an analysis devotedto quantify the achieved speed under different operationconditions of the numerical algorithm application; Sec.Vdeepens the analysis of the time performance and stud-ies the relevance of providing an appropriate initial guessto the numerical algorithm. Finally, a synthesis of theachievements and potentialities is provided in the follow-ing two sections.
I. SETUP
The hardware of the setup is extensively described inthe Ref.[12]. Briefly, eight triaxial magnetoresistive sen-sor chips (Isentek IST 8308) are mounted on two par-allel, rigidly assembled printed circuit boards (PCBs).The chips are driven by a microcontroller, which syn-chronizes their acquisition and makes possible to memo-rize their readings (for calibration purpose) or to trans-mit them in real time to a computer, via USB interface(for data storage and/or online tracking). Beside thethree sensors, each chip contains a triple 14 bit, 200 Sa/sanalog-to-digital converter and numerical filters for noisereduction. Prior to normal (tracking) operation, calibra-tion data are initially collected in a uniform field andanalyzed to define a set of conversion coefficients savedinto the computer. These conversion coefficients are used
Figure 1. The sensor array contains 8 sensors distributed ontwo parallel PCBs (three of them, highlighted with blue circlesare a z = 0 and other five (orange circles) at z = 16 . during the normal operation to:• Subtract the individual sensor offsets;• Make the responses isotropic;• Refer to a unique co-ordinate system, oriented inaccordance with that of a sensor selected as thereference one;that is, in synthesis, to convert the raw readings in accu-rate and consistent magnetometric data.During the tracking measurements, the resulting ho-mogeneous, calibrated and equalized magnetometric dataare numerically analyzed by means of a Levemberg-Marquardt algorithm [13, 14] to localize the magneticsource and to determine vectorially the environmentalfield and the magnetic dipole of the target.The permanent magnets used in this experiment aredisks 2 mm in diameter and 0.5 mm in thickness.Their volume –considering the typical magnetization ofNeodymium-Iron-Boron materials ( M ≈ µ )– givesa magnetic dipole m ≈ − Am . Such a dipole pro-duces a magnetic field comparable with the Earth one( ≈ µ T) at centimetric distances. The tests presentedin this work are performed with the magnet attached onthe surface of the white cylinder visible in Figure 1. Thecylinder rotates around its axis which can be alterna-tively plain or threaded, as to produce circular or helixtrajectories.
II. MODEL AND ALGORITHMS
Each measurement produces 3 K ( K = 8) magneto-metric data corresponding to the three components ofthe magnetic field detected in the (nominally known) po-sitions ~r k of the K sensors. A. Field model
The field measured by the k th sensor when the dipole ~m is localized in ~r is modeled as: ~B k ( ~r, ~m ) = µ π (cid:18) ~m · ( ~r k − ~r )]( ~r k − ~r ) | ~r k − ~r | − ~m | ~r k − ~r | (cid:19) + ~B g , (1)where ~r k ( k = 0 , . . . , K −
1) is the position of the k th sen-sor and ~B g is the environmental field, which is assumedto be homogeneous over the sensor-array volume.Despite the highly non linear ~B k ( ~r, ~m ) dependence, theinverse problem of determining ~r and ~m from a minimalset of ~B k ( ~r ) measurements performed in known ~r k posi-tions has been successfully approached, under conditionsof negligible ~B g [3].However, also in those conditions, a non-linear bestfit approach, making use of larger data sets from manysensors, helps reduce noise and imperfection effects, andproduce better tracking results at the expense of a heav-ier computation burden. The latter problem is efficientlycircumvented by the currently available computers, andits main drawback actually consists in the risk that best-fit convergence conditions are not fulfilled. B. Limits
Figure 2. The field components are referred to co-ordinatesdefined within the reference sensor chip S ref (in black), whichmight be slightly misaligned with respect to the PCB frame(in red). This can lead to inaccuracy in the position of theother chips ( S k ). In addition, each chip contains three sensors(represented by colored dots), which are (submillimetrically)displaced with respect to each other. One of the possible imperfections is related to the lim-ited precision with which the sensor positions are known.Firstly, each chip contains three separate magnetoresis-tive sensors (represented with dots in the Figure 2) for the measurement of the three field components, and theyare not located in one point within the chip. This limitcould be (at least partially) mitigated assigning the pre-cise locations; however the relative displacements aresub-millimetric and do not play a substantial role, unlessthe system to be built is scaled down to sub-centimetricsize. Secondly, there could be a misalignment betweenthe co-ordinate axes within the reference sensor (thatused to define the field components) and the PCB refer-ence frame (that used to define the positions of the othersensors with respect to the reference one). As schemat-ically shown in Figure 2, an angular misalignment δθ would produce systematic errors L k δθ in the determina-tion of the k th sensor position, L k = | ~r k − ~r | being itsdistance from the reference one. The problem is 3-D andthree angular uncertainties concur to determine the sen-sor position errors. In our implementation the L k are inthe cm scale, and δθ might be of the order of one-degree,which leads –again– to submillimetric errors. Additional(but even smaller) uncertainties are related to minor chiptranslations with respect to their ideal positions on thePCB.Best-fit procedures (based on data consistency) mayhelp determine the sensor displacements with respect totheir nominal positions. We have successfully attemptedto face the problem with this approach, obtaining onlysubmillimetric corrections and barely appreciable im-provements in terms of tracking accuracy: this activitywill not be discussed further in this paper.Another assumption made to derive eq.1 is the punc-tual nature of the source, which is indeed modeled as apure dipole. This assumption is very well justified bythe small size of the magnetic source ( ≤ > B g variations, which demonstrates the feasibility and relia-bility of this approach. C. Best-fit degrees-of-freedom
Referring to eq.1, the best fit procedure is in chargeof retrieving n p = 9 parameters, and namely the compo-nents of the 3D vectors ~r, ~B g , ~m . In principle, being theintensity of ~B g and ~m assigned, a reduced fitting param-eter set could be used. In our experiments we had theevidence that setting a fixed B g modulus severely wors-ens the performance (accuracy and reliability), as soon asits real value changes because of sensors displacements.The reason for that is the typical presence of smoothbut non-negligible ambient field inhomogeneities caused,e.g. by ferromagnetic materials of buildings and furni-ture. In contrast, it can be definitely advantageous usinga fixed | ~m | , as to reduce the best fit to determine only thedipole orientation. Provided that m is accurately eval-uated, this reduction of n p from 9 down to 8 improvesthe tracking performance, as discussed below in Sec.III B.Differing from what one could expect, passing from a 9-parameter procedure (9P) to an 8-parameter one (8P),does not help reduce the number of iterations or shortenthe computation time, as discussed in Sec.IV.The Levemberg-Marquardt algorithm used for theanalyses here presented has a termination conditionbased on the comparison of a tolerance parameter ( T )with the variation of a mean residual R , which is definedas the root of the mean squared deviation between theestimated and the measured values of the fields. Thefitting procedure is terminated and a parameter estima-tion is output as soon as an iteration produces a rela-tive variation ∆ R/R < T . All the results reported inthis paper are obtained with T set at 10 − . However,we verified that selecting a 100 times tighter condition( T = 10 − ) increases the number of necessary iterationsby a few units (typically less than 10), both in the 8Pand in the 9P case. D. Scalability
The equation 1 inherently expresses an importantscale-law. For a given material remanence, the dipoleintensity m scales with the volume of the magnet, i.e.with the third power of the magnet size. For a givenspatial positioning (location and orientation) the dipolefield scales with a 1 /r law, while B g is obviously notdependent on the system size. In conclusion, the samemeasurement results will be obtained if the whole systemis stretched or shrunk by a given scale factor. Of coursethis scale-law would fail when the shrinking factor wouldbe so large to make the punctual approximation of thesensor size inappropriate, while stretching can result ex-cessive whenever the increase of the array size causes B g quantity x y z ~r
20 mm −
20 mm 40 mm ~B g µ T 20 µ T 20 µ T ~m µ Am µ Am µ Am Table I. Default values used as starting guesses (unless differ-ently specified) for the target position ~r , the environmentalfield ~B g , and the magnetic dipole ~m . inhomogeneities play a relevant role (in other terms, thementioned scale-law would persist only if the distance offerromagnetic disturbers was stretched as well). III. TRAJECTORY PERFORMANCE
The precision with which position and trajectoriescan be retrieved depends on diverse parameters, amongwhich:• accuracy of the calibration and subsequent consis-tency of the magnetometric data• accuracy of knowledge of the sensor positions• presence of time-dependent noise or other distur-bances preventing accurate measurements• presence of relevant inhomogeneities of the ambientfield.In addition, the condition of single measurement maygreatly affect the tracking accuracy, in particular, a cru-cial role is played by:• intensity of the target dipole | ~m | ;• distance of the target from the sensor array• position and orientation of the target with respectthe sensor array (the performance is not isotropicaround the array center, and the anisotropy de-pends also on the dipole orientation)In contrast, the initial guess has no effects on the trackingprecision, provided that it is suitably assigned to guar-antee correct convergence, that is to let the best-fit pro-cedure get the global minimimum. A target trajectoryis reconstructed and sampled as a sequence of trackedpoints. The trajectories reported in this section are ob-tained using a default guess (see Tab.I) to determine thefirst point, and then assigning each tracked point as aninitial guess for the subsequent one. The latter detail hasrelevance for the computation time (see Secs.IV and V),but not for the tracking accuracy and precision.A general and throughout study of the possible scenar-ios is not feasible, while having in mind a specific use letsreduce the number of possible configurations and makesa focused characterization possible.Our system is designed for eye-tracking and, consider-ing the size of the "lenses like" frame shown in Figure 1,we are interested in cases where the target is located ona given side of the array (in particular z > z , having set z = 0 for the three-sensor PCM and z = 16 . x, y ) will not go more thansome tens of mm away from the sensor axis. The whitecylinder in Figure 1 constitutes a reasonable representa-tion of that region of interest (RoI).Any quantification of the system performance, will de-pend on the specific positions/trajectories selected. Onthe other hand, any estimate performed with the targetin the mentioned RoI will be a significant example, suchto provide a performance evaluation. In this section, weare reporting a set of such significant examples, providingquantitative or qualitative (graphical) information aboutthe precision achieved. A. Circle
We present a test performed by imposing a circulartrajectory, on a plane nearly parallel to the PCBs.The Figure 3 shows its xy and zy projections, togetherwith best-fitting curves based on an elliptical model. A8P tracking is applied, where m = | ~m | is previously esti-mated as the mean value on the whole trajectory recon-structed by a 9P one ( h m i = (995 ± µ Am ). The 9Presult (not reported) is indistinguishable in the xy pro-jection and worse in the zy one. The latter is a typicalbehavior, and there is evidence that a less accurate track-ing is obtained –particularly with 9P– along the directionperpendicular to the dipole ( z in the present case, thedipole being radially oriented). The elliptic fit enablesan estimate of the major axis (19.9mm), which matchesthe cylinder diameter within a 0.1 mm accuracy.The data analysis that produces these spatial track-ings let also determine the dipole components and quan-tify the uncertainty δθ affecting the angular estimateof the dipole orientation. The measurement and dataelaboration performed to this end are as follows. Theangle describing the dipole orientation around the rota-tion axis (which is nearly parallel to z ) is estimated as θ = arctan( m y /m x ). After a phase unwrapping proce-dure, the angles θ n = θ ( t n ) are modeled and fitted with alinear function, which correspond to consider a constantangular velocity ω of the magnet driver. The residual ofthis linear regression provides the mean squared devia-tion between the estimated ( ωt n + θ ) and the measured θ n values of the dipole orientation. This evaluation gives δθ P = 0 . ◦ and δθ P = 0 . ◦ , with the 8P and 9Palgorithms, respectively. It is worth mentioning that inthe eye tracking application the eye orientation can beretrieved both directly from the orientation of ~m and in-directly from ~r , assuming that the magnet is constrainedto move on a spherical surface. This redundancy couldbe used to improve the angular resolution. B. Helix
A more complex trajectory with 3D reconstruction per-mits performance evaluations that are qualitative, but–nevertheless– significant. To this aim, we apply 3D tra-jectories to the target magnet. In this case, the magnetholder is mounted on a screw-axis, with a 1 mm pitchthread. The trajectory has consequently an helix shape,20 mm in diameter and 1mm in pitch. The sample3D trackings shown in the Figure 4 represent the recon-structed trajectories obtained on the basis of a 9P and 8P,respectively. Also in this case, the dipole is radially ori-ented. The 9P produces evident distortions, particularlydue to undesired fluctuations of the z co-ordinate. Re-markably, these z distortions occur simultaneously withunexpected variations of the m estimates. This feature islikely due to the imperfections mentioned in the Sec.II B.The 8P greatly helps improve the reconstructed trajec-tory in this respect. C. Tuning the 8-parameter fit
The better performance of the 8P comes at the expenseof a computationally heavier model formula (it includesseveral trigonometric functions, which partially compen-sates the reduction of the parameter space dimensions)and the need of providing the procedure with an accurateestimate of the modulus | ~m | . To this end, it is possibleto start with a 9P evaluation of the mean value of m over a trajectory, and subsequently use that estimate of m for 8P tracking. The Figure 5 shows the effects of pro-viding 8P with wrong estimations of m . We report xy projections of five trackings obtained with correct (9Pestimated), overestimated and underestimated m values,respectively: evident trajectory distortions occur, whichgo rapidly in excess of 1 mm. IV. TIME PERFORMANCE
The algorithm speed becomes a critical feature when-ever a real-time output or large data-sets analysis arerequired. Assigning an appropriate guess not only playsa role in guaranteeing convergence to the right solution,but helps reduce the convergence time. The Figure 6provides an example of time-performance estimation asa function of the guess accuracy. The plots represent thealgorithm iteration number necessary to track points ofa circular trajectory (same data as for Figure 3) sampledin N = 670 points, as a function of the guess accuracy.To this end, for the j th tracking point, we used as start-ing guess the tracking output j − q (or N + j − q , when q > j ), with q ranging from 1 (guess coming from theprevious point in the tracking list) to N − q = N/ Figure 3. Projections on the xy e zy planes of a 8P reconstructed circular trajectory. The red curves are elliptical best-fits.Residuals as small al 6.1 µ m and 92 µ m are obtained, respectively. The corresponding 9P analysis lead to results that aresimilar (5.6 µ m) in the xy projection and worse (95 µ m) in the xy projection. dipole orientation. Using both 8P and 9P, a correct con-vergence occurs in all the cases, and for both 8P and 9Pthe average iteration number increases by about 50% inthe worst case ( q = N/ n i isapproximately proportional to the computation burden.The number of function calls n c depends on n i accordingto n c = ( n i + 2)(2 n p + 1). The number n c is the quantitythat, together with the complexity of the model formula,mainly determines the best-fit computing time. The con-version factor from calls to time is machine-dependent. Inour case, with an Intel-I7 2.4 GHz CPU we obtain about14 and 13 µ s/call for the 8P and 9P respectively. Theconsequent time performance is summarized in Tab.II,which (with the exception of the 8P worst cases) showsthat both 8P and 9P require less than 10ms/tracking, asto enable a real-time 100 Sa/s acquisition. Further anal-yses of the convergence speed are provided at the end ofnext section, after some considerations about the condi-tions under which a correct convergence is obtained. quantity q=1 q=N/2 default
8P (iter. number) 30 55 489P (iter. number) 20 35 258P (ms/tracking) 7.5 13.8 129P (ms/tracking) 5.4 9.5 6.8Table II. Time performances of 8P and 9P, in the case of next-neighbor guess, in case of diametrically opposite guess and incase of default values assigned programmatically (see Tab-I)
V. GUESS CRITICALITY
In this section we study the criticality of providing anappropriate initial guess to make the algorithm convergeto the correct solution and we give some additional in-formation about the convergence speed.For each given configuration ( ~r, ~B g , ~m ) of the target,there exists a volume surrounding that configuration,which corresponds to guesses that will guarantee a cor-rect convergence. Guesses chosen out of that volume willproduce wrong solutions. We have verified that the con-vergence condition is weakly affected by the initial val-ues assigned to ~B g and ~m , while the good-convergencevolume V G in the subspace of the geometrical x, y, z co-ordinates presents a large variety of shapes.Intersecting of the guess volumes with planes parallel Figure 4. 3D views of reconstructed helix trajectory, the axes units are expressed in mm. The magnet is radially orientedand follows a helix trajectory 10 mm in radius and 1 mm in pitch, around the z axis. Beside the spatial position the trackerretrieves the dipole vector: red-blue dots are used to represent this data. The left tracking is obtained by 9P and suffers ofevident (nevertheless submillimetric) distortions of the z co-ordinate. The right tracking uses the same magnetometric data,but a 8P analysis.Figure 5. Tracking of xy projections as obtained with 8P,using for | ~m | its value resulting from 9P (black) and valuesincreased (blue, green) or decreased (red, magenta) by 20%and 40%, respectively. to the co-ordinate planes xy , xz or yz provides a 2Dvisualization of sections of the convergence volume V G and of its complement V R , corresponding to the spatialguesses that lead to wrong convergence.The figures here presented are 2D maps obtained using9P to identify the mentioned sections: two spatial param-eters of the initial guess are varied while the remaining 7are kept at fixed values. The good-guess regions V G arerepresented in green and the bad-guess regions V R arerepresented in red. A color scale is used to represent theiteration numbers: bright colors denote fast convergenceand dark colors denote slow convergence (the iteration Figure 6. Average iteration number as a function of the guessaccuracy for the 8P (blue) and 9P (red) cases. The cases withq=1 and q=670 correspond to using one of the nearest neigh-bors as an initial guess, while q=335 corresponds to usingdiametrically opposite points. numbers and the corresponding color scales are reportedin the bottom rules of each map). The target positionis represented by a white dot, and the sensors are repre-sented in colors: blue dots for the three on the z = 0 PCBand orange dots for the five ones on the z = 16 . ×
300 mm areason the intersecting planes, such to be significantly largerthan the RoI for the eye-tracking application. The tableIII reports the co-ordinates of the sensors and of two an-alyzed positions of the target, both selected in our RoI,the first one in the vicinity of the sensor array axis andthe second one in a more peripheral location.The Figure 7 represents convergence maps obtained (a) (b)(c) (d)Figure 7. 2D convergence maps of xy (a,b) and xz (c,d) sections. The remaining 7 parameters are set to their exact values(a,c) or to their default values (b,d). The target (white dot) is in a nearly axial position, at z = 31 mm. Unexpectedly, usingwrong guesses for ~B g and ~m and wrong (larger) z makes the ( x, y ) convergence area larger than using exact values. All thesemaps describe a 300 mm ×
300 mm area, the yellow circle in (c) corresponds to a sphere (26 mm in diameter) that is the typicalsize of human eye bulb. object x (mm) y (mm) z (mm) sensor 0 0 0 0sensor 1 0 -27.05 0sensor 2 40.64 -19.28 0sensor 3 0 -19.28 16.6sensor 4 20.26 -45.74 16.6sensor 5 40.64 -32 16.6sensor 6 40.64 -6.6 16.6sensor 7 20.33 7.25 16.6 ~r c
13 -20 30.5 ~r p
33 -19.8 29.7Table III. Co-ordinates of the sensors. The last two linesreport the central and peripheral target positions (determinedby the tracker) considered in the analyses of this Section. when the dipole is located in ~r c , i.e. proximally to thesensor axis. The four maps correspond to xy and xz sections, obtained when assigning the fixed 7 parameterswith their correct values or with default values (detailsprovided in the caption). The sections of V G are quiteregular and symmetric. There is a convex volume, outof which the convergence is wrong and fast, and insideof which the V G part leads to homogeneously fast con-vergence, while the V R contains guesses that lead to slowconvergence (to a wrong solution).The Figure 8 represents convergence maps obtainedwhen the dipole is located more peripherally, in ~r p . Again xy and xz sections are shown. They are obtained by as-signing the fixed 7 parameters either their correct valuesor default guesses (details provided in the caption). Now,the shapes have more variegated aspects; however, alsoin this case, guesses with nearly axial ( x, y ) coordinateslead systematically to good convergence, provided that z is assigned in the correct interval, above z and not toofar from the sensors.A hint for an appropriate selection of z , is suggestedby the maps represented in Figure 9. This figure repre-sents a xy convergence maps obtained when the dipole islocated peripherally (the same as for Figure 8), the ~B g and ~m guesses are assigned their default values, while z co-ordinate is varied (details provided in the caption). AsFigs.7 c-d and 8 c-d suggest, excessive z values may leadto bad convergence, however intermediate values (e.g.z=60 mm to 80 mm) produce –with a larger number ofiterations– a wider convergence area in the xy section. Incontrast, wrong z (Figure 9-a) and small z (Figure 9-b)may reduce dramatically the convergence region.Despite the complexity of the possible scenarios (inten-sity and orientation of the dipole and of the environmen-tal field, accessible dipole positions with respect to thesensors, etc.) some general hints can be derived from theanalysis of results shown in this section and in Sec.III,IV, which can be summarized as follows:• the initial guess of ~m is not critical (just assignreasonable values);• the initial guess of ~B g is not critical (just assign reasonable values);• the initial guess of ~r is critical, but whenever it isknown that the target is on one side of the array (agiven sign of z ), and is not too displaced from thearray axis, an axial guess for ( x, y ) together with areasonable guess for z will work;• the larger is the z guess, the less critical is the( x, y ) choice, but large values make the convergenceslower, and too large values will prevent conver-gence;• in case of correct guess, the iteration number de-pends weakly of the termination conditions and onthe selected guess;• good guesses bring to convergence within few tensof iterations;• bad guesses with too large | r | bring to (wrong) con-vergence within a few iteration steps;• some wrong guesses with more reasonable | r | bringto (wrong) convergence quite slowly, with many it-eration steps (setting a limited number of iterationsteps will help avoid wasting time with useless cal-culations);• wrong convergence is easily detected, because the(local) minimum found is orders of magnitudelarger than the absolute one;• in case of a wrong convergence detected, tryingother initial guesses having different ~r will help;• in the considered application of eye-tracking, thelimited RoI size makes a certainly-good guess pos-sible;• the accuracy in reconstructing trajectories is gen-erally good, but the best performance is obtainedwhen the target moves on a surface nearly parallelto ~m : in eye-tracking application, a radial orienta-tion of ~m will be a favorite choice. VI. CONCLUSION
We have tested a new concept of an eye-motion trackerbased upon a 8-sensor magnetic tracker and a small mag-net. The performance in terms of accuracy, precisionand speed has been analyzed, under different operatingconditions. In particular, we have empirically studiedthe criticality (in terms of correctness and speed of theconvergence) of assigning an appropriate starting guessto the numerical algorithm that infers the tracking pa-rameters from the magnetometric measurements. Thefeasibility of a reliable sub-millimetric, 100Sa/s real-timetracking within a volume large enough to contain the RoIof eye-tracking experiments has been demonstrated.0 (a) (b)(c) (d)Figure 8. 2D convergence maps of xy (a,b) and xz (c,d) sections. The remaining 7 parameters are set to their exact values(a,c) or to their default values (b,d). The target (white dot) is in a more peripheral position (larger distance from the arrayaxis), with respect to the case of Figure 7, at z = 31mm. It is confirmed that using wrong guesses for ~B g and ~m and wrong(larger) z makes the ( x, y ) convergence area larger than using the exact values. PATENTS
The accuracy and speed analyses at the focus of thiswork constitute a performance assessment of the hard-ware described in Ref.[12]. A patent [15] is pending aboutinventions related to this research. [1] T. D. Than, G. Alici, H. Zhou, and W. Li, IEEE Trans-actions on Biomedical Engineering , 2387 (2012).[2] C. Di Natali, M. Beccani, and P. Valdastri, IEEE Trans-actions on Magnetics , 3524 (2013).[3] C. Hu, M. Q. . Meng, and M. Mandal, IEEE Transac-tions on Magnetics , 4096 (2007).[4] W. Weitschies, J. Wedemeyer, R. Stehr, and L. Trahms,IEEE Transactions on Biomedical Engineering , 192(1994).[5] V. Schlageter, P.-A. Besse, R. Popovic, and P. Kucera,Sensors and Actuators A: Physical , 37 (2001), se-lected Papers for Eurosensors XIV.[6] Chao Hu, Wanan Yang, Dongmei Chen, M. Q. . Meng,and Houde Dai, in (2008) pp. 2055–2058.[7] S. Song, C. Hu, and M. Q. . Meng, IEEE Transactionson Magnetics , 1 (2016).[8] H. Dai, C. Hu, S. Su, M. Lin, and S. Song, IEEE Trans-actions on Instrumentation and Measurement , 3379 (2019).[9] L. Wöhle and M. Gebhard, Sensors , 2759 (2020).[10] Y. Agrawal, M. C. Schubert, D. S. Migliaccio, AmericoA. ans Zee, E. Schneider, N. Lehnen, and J. P. Carey,Otology & Neurotology , 283 (2014).[11] I. Rakhmatulin, “A review of the low-cost eye-trackingsystems for 2010-2020,” (2020), arXiv:2010.05480[cs.AI].[12] V. Biancalana, R. Cecchi, P. Chessa, G. Bevilacqua,Y. Dancheva, and A. Vigilante, Instruments , 3 (2021).[13] K. Levenberg, Quarterly of Applied Mathematics , 164(1944).[14] D. W. Marquardt, Journal of the Society for Industrialand Applied Mathematics , 431 (1963).[15] V. Biancalana, R. Cecchi, P. Chessa, M. Mandalà, andD. Prattichizzo, “Patent pending: System for tracking anobject,” (2020), 102020000017776. (a) (b)(c) (d)Figure 9. 2D convergence maps of xy sections. The ~B g and ~m guess is set at the default, and the z guess is variously assigned:from 8 mm (a), to 20 mm (b), 60 mm (c) and 80 mm (d). The zz