Improving Accuracy of Inertial Measurement Units using Support Vector Regression

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1 Improving Accuracy of Inerial Measuremen Unis using Suppor Vecor Regression Saran Ahuja, Wisi Jiraigalachoe, and Ar Tosborvorn Absrac Inerial measuremen uni (IMU) is a sensor ha measures acceleraion and angular velociy rae. I has become increasingly popular due o is small size and low cos comparing o ypical marker-based moion capure sysem. Noneheless, IMUs face considerable challenges, in paricular noiceable inaccuracy from accumulaed inegraion errors. In his projec, we aemped o improve accuracy of IMUs in measuring clinical knee angles using supervised machine learning, specifically suppor vecor regression. By employing highaccuracy daa from marker-based moion capure sysem as raining samples, we were able o see improvemen in he performance of he IMUs esimaion of clinical knee angles in walking moion. This resul is promising and encourages furher works ha migh exend he model o include a more general se of movemens. I. INTRODUCTION Since is inroducion in 199, moion capure is used no only in he enerainmen business such as films and games, bu i is also used exensively in spors, medicine, and miliary. Tradiionally, moion capure is done using images capured from cameras o riangulae posiions of each reflecive marker in he 3-dimensional space. While his marker-based sysem performs exremely well in erms of posiional accuracy and coninue o hold is saus as he indusrial sandard, here is a number of undesirable feaures, including high cos, seup complexiy, and space limiaion. Recenly, an alernaive mehod has gained populariy. Inerial moion capure uses sensors inerial measuremen unis, IMUs aached o differen pars of he body, much like reflecive markers used in he opical sysem, o measure acceleraion and angular velociy. This raw daa may hen be used o calculae parameers of ineres, such as join angles and oher posiional daa. Due o is small form facor, minimal seup requiremens, and low cos, inerial moion capure has come o be an appealing alernaive o he marker-based sysem. In addiion, since inerial moion capure does no require camera seup, i is a viable soluion for capuring oudoor moions like skiing or mounain biking ha oherwise may never be achieved hrough radiional marker-based moion capure. As good as i may sound, inerial moion capure has one big disadvanage. Unlike he marker-based sysem, inerial moion capure lacks absolue posiional daa and relies heavily on inegraing acceleraion as well as angular velociy vecors in order o compue is absolue posiion. Taking ino accoun sensor errors and moion approximaion, he errors from his inegraion procedure quickly add up, and he resul becomes more inaccurae as ime passes. In his projec, we aemped o improve he accuracy of inerial moion capure using supervised machine learning. Using boh sysems o simulaneously capure a subjec s moion, we rained he inerial moion capure model o more accuraely esimae he valuable clinical knee angles of he subjec during a walking moion. II. SYSTEM & DATA PROCESSING A. Inerial Measuremen Uni (IMU) In his projec, we used a sysem of low cos wireless inerial measuremen unis. The sysem is shown in Fig. 1. The sysem consised of wo IMUs wih he capabiliy of expanding up o 18 unis. Each IMU has a 3- axis acceleromeer wih ±8g range, a 3-axis gyroscope wih 16 deg/s range, and a 3-axis magneomeer wih ±4 Gauss range, all in 16-bi resoluion. The daa was sampled a 1 Hz and ransferred o mobile phone for processing and daa sorage via Blueooh connecion. B. Marker-based Moion Capure Sysem The marker-based moion capure sysem is widely acceped as he indusry s gold sandard. In our experimen, he reference sysem was he moion capure sysem wih 8 cameras seup, running a 6 frames per second wih sub-mm accuracy. C. Experimenal Proocol We colleced wo sandard saic rials in order o esimae he roaion marices from each IMU s body

2 he roaion marix can be compued as follow: T λ)i + λ λ T q [λ ], R( q) = (q λ (1) where I is he ideniy marix and [λ ] is sandard vecor cross produc. Then he following differenial moion describes he dynamic of body angular moion: d q() = Aω () q, d () where ωz ωy ωx 1 ωz ωx ωy Aω () = ωz ωy ωx ωx ωy ωz Fig. 1. Inerial measuremen uni (IMU) sysem wih Blueooh capabiliy. frame o he reference frame. Afer he iniial seup, we colleced daa of 4 normal walking rials on a readmill a 1. m/s. For each rial, we colleced he daa using boh IMUs and sysems simulaneously. A he beginning of each rial, he subjec jumped and sood sill for a few seconds in order for us o be able o synchronize he iming of boh sysems. Of he 4 rials, hree rials were dropped due o he corruped daa from he IMUs, leaving us wih 39 rials. D. Clinical Knee Angle Esimaion Some of he key of parameers of a walking moion are he clinical knee angles. These parameers are very helpful for a physician o analyze knee-relaed injuries in paiens. The clinical knee angle measuremens consis of hree parameers: flexion/exension, adducion/abducion, and inernal/exernal roaion angles. Before processing any daa, we synchronized he daa from boh IMU and ogeher using he jumping even a he beginning of he rial. We hen only used he daa afer jumping for raining and reporing resuls. From he IMUs daa, we performed a quaernionbased srap-down inegraion mehod [1], [], [3] in order o esimae he roaion marices. Our raw daa consised of gyroscopic raes, denoed by ω(), and acceleraion wih respec o he body frame B. We need o find a roaional marix ha ransformed an IMU body frame ino he world reference frame W. We will denoe his roaion marix by R(). A convenien way o keep rack of R() is o use he quaernion λ T ], where q is a scalar and represenaion q = [q, λ is a 3 1 vecor. Given he quaernion represenaion, λ and ω() = [ωx, ωy, ωz ], he angular velociy of B. Le denoe he sysem s sampling inerval and assume ha q () is consan for [k, (k + 1) ]. Then we can solve equaion () explicily and ge a discree-ime quaernion q k+1 = eak q k, where q k is he quaernion a ime k and Ak = Aω (k ). q is compued so ha he roaion marix is consisen wih our iniial saic posiion. Once he discree sysem above is solved, he roaion marix is updaed using equaion (1). Afer we found he roaion marix represening he roaion from IMUs body frame o he world frame, we used a join coordinae sysem (JCS) as recommended by ISB and defined by Grood and Sunay [4], [5] o calculae he clinical knee angles. Fig.. Experimenal Seup: A subjec is wearing boh IMUs and reflecive markers.

3 III. METHODOLOGY As menioned earlier, inerial moion capure suffers from buildup of errors. In his secion, we describe he use of supervised machine learning procedure, in paricular suppor vecor regression, o improve he esimaion from measuremen of IMUs. Afer processing he daa as described in he previous secion, he clinical knee angles esimaion θ IMU () are compued. Using relaively accurae posiion daa from marker-based sysem, we were able o compue hese angles wih grea accuracy. This se of daa served as he arge value. We hen rain he model using ν-svr wih he pairs (θ IMU (), θ Marker ()). A. Model Acceped as he gold sandard for posiion capuring, θ Marker () is reaed as acual knee join angle and we have he following model: θ IMU = θ Marker + ε, where ε is he error from he measuremen (sensor noise) and inegraion. Since our main applicaion is o measure knee join angle relaed o a specific body movemen (walking, running, ec.), one migh expec he error o be correlaed wih he angle θ Marker (). Le so ha θ IMU = θ Marker ε = φ(θ Marker ) + ε, + φ(θ Marker Marker ) + ε = φ(θ ) + ε. Invering his relaionship, we have θ Marker = ψ(θ IMU ) + ε, which gives a seing where he suppor vecor regression can be applied. This derivaion is no mean o be rigorous, bu raher o moivae our use of suppor vecor regression. B. Learning Mehod From each rials, we colleced and processed our daa o ge a sample pah of angles measuremen from boh IMUs and marker-based moion capure as a funcion of ime. The processed daa consised of riples (, θ,i IMU, θ,i Marker ), where = 1,,..., T, and i = 1,,..., N. T is he number of ime seps and N is he number of rials (39 in our experimen). Our raining size is N T where (, θ,i IMU ) gives us inpu samples o be rained wih arge θ,i Marker. We applied ν-svr wih various kernels o he raining se and compared he performance among differen TABLE I ROOT MEAN SQUARE ERROR Clinical Knee Angle Linear Polynomial RBF Flexion/Exension Abducion/Adducion Inernal/Exernal Roaion kernels. To evaluae he generalizaion capabiliy of he model, we used leave-five-ou cross validaion mehod wih roo mean square error as a crieria. More concreely, we performed 8 rainings and rained he model using 34 walks and cross validaed wih 5 walks (excep for he las raining in which we used 35 walks o rain and 4 walks o cross validae). The learning procedure can be summarized as follows: Normalize he raining sample o have range [,1]. Le θ m, θ M be he minimum and maximum of our sample angles, hen ( ) x,i = T, θimu,i θ m θ M θ m, y,i = θmarker,i θ m. θ M θ m Apply ν SVR wih regularizaion parameer C = 1 o he normalized raining se {(x,i, y,i )}, i = 1,,..., N, = 1,,..., T }. We use libsvm oolbox [6] which provides all necessary Malab funcions. Compue leave-five-ou cross validaion error using roo mean square error as a meric. IV. RESULTS For kernel selecion in our suppor vecor regression mehod, we ried linear, polynomial, and radio-basis funcion (RBF) kernels. The resuls showed no significan advanage of one kernel over ohers across all hree angle measuremens, as can be seen in Table I. For knee flexion/exension angle, we found he RMS error o be approximaely 6.1 degrees. Knee adducion/abducion angle had an RMS error around.63 degrees, and for he knee inernal/exernal roaion angle, he RMS error was.35 degrees. By using our model from SVR, we found a significan improvemen in all hree knee angles esimaion from IMUs. As seen in a sample walking rial in Fig. 3, he (i.e. IMU daa processed wih SVR) was closer o he reference sysem, on average of more han 5%. In addiion, our esimaed model reduced he RMS errors over ime across all rials a leas in half, as seen in Fig. 4.

4 8 6 4 flexion/exension adducion/abducion inernal/exernal roaion flexion/exension adducion/abducion inernal/exernal roaion Fig. 3. Esimaions of hree clinical knee angles using daa from IMUs improve by more han 5% using SVR, as we can see is close o he reference sysem (). V. DISCUSSION Each kernel (linear, polynomial, and RBF) showed comparable performance across all hree ypes of knee angles, as seen in Table I. This is possibly due o he srucure of he error ha was no specific o any kind of kernel. These relaively low cross validaion errors allowed he model o fi he acual angle nicely wihou over fiing he daa. Also noe ha he magniudes of hese errors are relaively good comparing o wha have been found in a few oher similar sudies [7], [8], [9], [1]. Our model esimaion has an advanage of very lile compuaion ime requiremen (around 15 seconds for a sample of 34 rials, polynomial kernel). However, our model only considered a specific normal walking condiion on a readmill. Thus furher experimens involving various moions and speed will be required o improve he generalizaion of he model. Suppor vecor regression was able o reduce he error in he knee angle esimaion significanly, as seen in Fig. 3. One observaion from he daa is ha he error increased in ime due o accumulaion of error from each ime sep discreizaion. By adding ime since las saic posiion parameer ino our model, we were able o reduce accumulaing errors significanly. Neverheless, SVR is sill unable o compleely remove Fig. 4. Roo-mean-squared errors due o ime effec from IMUs are reduced by more han 4% on average. non-sysemaic errors such as hardware sensor noises. Wih regards o he naure of he subjec s movemen, we noice ha when he subjec s foo hi he ground, he IMU regisered significan noise in acceleraion. This would sugges ha we migh observe periodic spikes in errors from he IMU. However, Fig. 4 showed ha he roo-mean-squared error from he IMUs does no appear o be periodic as expeced. This was due in par by he no compleely synchronized sep cycles from our pre-processed daa. Therefore hese spikes were averaged ou over 39 rials. The residuals, on he oher hand, exhibis higher degree of periodiciy. This migh be due o SVR, which akes ino accoun of he ime parameer, was able o noice he cyclical rend. VI. CONCLUSION In his paper, we explored suppor vecor regression mehod o help improve knee angle esimaion from muliple inerial measuremen unis. We used a gold sandard marker-based moion capure sysem as our learning examples. We hen uilized cross validaion mehod o find an appropriae kernel o rain he model. The resuls demonsraed a vas improvemen of clinical knee angles esimaion over he ypical IMUs measuremen. This would allow us o capure he essence of knee moion in a near-real-ime calculaion, suiable for aciviies ha require oudoor seing and

5 insan feedback. Fuure work would require esing and learning of muliple moions and join angles in order o improve he generalizaion of he model. REFERENCES [1] A. M. Sabaini, Quaernion-based srap-down inegraion mehod for applicaions of inerial sensing o gai analysis. Medical and Biological Engineering and Compuing, vol. 43, no. 1, pp , 5. [] J. Favre, B. M. Jolles, O. Siegris, and K. Aminian, Quaernion-based fusion of gyroscopes and acceleromeers o improve 3D angle measuremen, Elecronics Leers, vol. 4, no. 11, pp , 6. [3] J. C. K. Chou, Quaernion kinemaic and dynamic differenial equaions, Roboics and Auomaion, IEEE Transacions on, vol. 8, no. 1, pp , Feb [4] G. Wu, S. Siegler, P. Allard, C. Kirley, A. Leardini, D. Rosenbaum, M. While, D. D. D?Lima, L. Crisofolini, H. Wie, and e al., Isb recommendaion on definiions of join coordinae sysem of various joins for he reporing of human join moion par i: ankle, hip, and spine. inernaional sociey of biomechanics. Journal of Biomechanics, vol. 35, no. 4, pp ,. [5] E. S. Grood and W. J. Sunay, A join coordinae sysem for he clinical descripion of hree-dimensional moions: applicaion o he knee. Journal of Biomechanical Engineering, vol. 15, no., pp , [6] C.-C. Chang and C.-J. Lin, LIBSVM: A library for suppor vecor machines, ACM Transacions on Inelligen Sysems and Technology, vol., pp. 7:1 7:7, 11. [7] J. Favre, R. Aissaoui, B. M. Jolles, O. Siegris, J. A. de Guise, and K. Aminian, 3D join roaion measuremen using MEMs inerial sensors: Applicaion o he knee join, in Ninh Inernaional Symposium on he 3D Analysis of Human Movemen. Inernaional Sociey of Biomechanics (ISB) Technical Group on he 3-D Analysis of Human Movemen, June 6. [8] R. Takeda, S. Tadano, M. Todoh, M. Morikawa, M. Nakayasu, and S. Yoshinari, Gai analysis using graviaional acceleraion measured by wearable sensors. Journal of Biomechanics, vol. 4, no. 3, pp. 3 33, 9. [9] G. Cooper, I. Shere, L. McMillian, K. Siliverdis, N. Sha, D. Hodgins, L. Kenney, and D. Howard, Inerial sensorbased knee flexion/exension angle esimaion, Journal of Biomechanics, Sep. 9. [1] K. J. ODonovan, R. Kamnik, D. T. OKeeffe, and G. M. Lyons, An inerial and magneic sensor based echnique for join angle measuremen, Journal of Biomechanics, vol. 4, no. 1, pp , 7. [Online]. Available: hp://www. sciencedirec.com/science/aricle/pii/s199713