Optimized Kalman Filter versus Rigorous Method in Deformation Analysis

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1 Optmzed Kalma Flter versus Rgorous Method Deformato Aalss N. Aharzad a, H. Seta b, * a Departmet of Geomatc Egeerg, Facult of Geoformato ad Real Estate, Uverst ekolog Malasa, 8131 Skuda, Johor Bahru, Malasa aezhla@lve.utm.m b Departmet of Geomatc Egeerg, Facult of Geoformato ad Real Estate, Uverst ekolog Malasa, 8131 Skuda, Johor Bahru, Malasa - KEY WORDS: Kalma flter, o-lear Kalma flter, Optmzed Kalma flter, Deformato Aalss, Deformato Motorg ABSRAC: Kalma flterg s a multple-put, multple-output flter that ca optmall estmate the states of a sstem, so t ca be cosdered a sutable meas for deformato aalss. he states are all the varables eeded to completel descrbe the sstem behavor of the deformato process as a fucto of tme (such as posto, veloct etc.). he stadard Kalma flter estmates the state vector where the measurg process s descrbed b a lear sstem. Whle, order to process a o-lear sstem a optmzed aspect of Kalma flter s approprate. he ma purpose of ths research s to evaluate the Optmzed Kalma flter as a o-robust method versus the IWS (Iteratve Weghted Smlart rasformato) as a rgorous (also called robust) method. o satsf ths objectve, frst a detaled descrpto o eecutg the Eteded Kalma flter usg the observato of agles ad dstaces drectl s provded. Later o, fve sets of -D otal Stato data clude dstaces ad agles are used to demostrate the Optmzed Kalma Flter. For detectg the deformato, sgle pot test for ever pot s appled compoet b compoet as a local test. Later o, the fdgs from Optmzed Kalma Flter are compared ad evaluated agast the results from IWS testg. I geeral, the outcome of Kalma flter algorthm s close to the prelmar results from IWS testg. he mamum ad mmum dffereces computed dsplacemets are equal to. ad. meters respectvel. Fall, Kalma flter approaches, havg some propertes, are recogzed as sutable techques for deformato aalss. 1.1 Itroducto 1. INRODUCION I cosequece of some factors ether atural (e.g. water pressure ad earthquake) or artfcal (e.g. the weght of the structure tself), the deformable object s subjected to var from ormal posto ad dmeso. Deformato motorg refers to a regular observato of the alteratos of a deformable object, whereas deformato aalss s the task of processg the avalable data from motorg phase ad detectg the magtude ad locato of the dsplacemet (Chrzaowsk et al., 7; USACE, ). here are several recogzed data processg techques whch are categorzed to two ma classes,.e. Robust methods, such as IWS (Iteratve Weghted Smlart rasformato) ad No-robust methods, such as Kalma Flterg ad etc. (asc, 1). he ordar Kalma Flter combes all the avalable formato to optmze the estmated state vector mmzg estmated error covarace where the measurg process s descrbed b a lear sstem. I ths stud, amog the varous detfed data processg techques the sutablt of the Optmzed Kalma Flter (OKF) for deformato aalss whe the observatos are appled drectl ad the stochastc dfferece equato ad/or the measuremet equato are olear s evaluated. I order to llustrate the OKF fve sets of - D otal Stato data whch cludes dstaces ad agles are used. For detectg the deformato, the sgle pot test s appled as a local test. Beg dfferet from the ordar sgle pot test (Md Som et al., 4; Lm et al., 1) of the sgle pot test whch has bee used Ice ad Sah (), the sgle pot test used ths stud eame ever compoets of the posto of each pots. For ever compoet f (computed) F 1, t, f (from table), the relevat pot s cosder as ustable pot (Ice ad Sah, )..1 Ordar Kalma flter. KALMAN FILER I 196, R.E. Kalma troduced hs ew approach whch s a powerful optmal recursve process to flterg, predcto ad smoothg the parameters whch var wth tme (Welch ad Bshop, 6; Cross, 1983). hs approach was frst used electrcal cotrol sstems (Kalma, 196). It has also bee used as oe of the methods used deformato aalss. he ordar Kalma flter also called Classcal Kalma flter or Dscrete Kalma flter s a sutable tool for deformato aalss where the measurg process s able to be govered b a lear sstem. he ma equatos of ordar Kalma flter are detaled Aharzad ad Seta (11). he Optmzed Kalma flter s recommeded where the dstace ad agle measuremets are drectl processed, the process to be estmated ad (or) the measuremet relatoshp to the process s o-lear (Welch ad Bshop, 6).. Optmzed Kalma flter Assume the o-lear fucto f relates the state at the earler tme step 1 to the state at the preset tme step (Welch ad Bshop, 1): * Correspodg author. hs s useful to kow for commucato wth the approprate perso cases wth more tha oe author.

2 f, u, w 1 1 (1) f A, ˆ, u, j 1 (1) j ad the o-lear fucto h relates the state measuremet z : to the f W, ˆ, u, j 1 (11) w j z h, v () I practce the eact values of the ose w ad v at each tme step ma ot be kow. So, the appromato of the state ad measuremet vector wll be used wthout them as:,, f 1 u (3) ad the o-lear fucto h relates the state measuremet z : to the z h, (4) he the EKF Predcto equatos or the tme update equatos, ad the flterg equatos, also called the measuremet update equatos are gve as Eq (5) to Eq (9) that order (Welch ad Bshop, 1): ˆ f ˆ, u, (5) 1 C M C M WQ W ˆ ˆ 1, 1, 1 1 G C A AC A V RV (7) ˆ ˆ ˆ ˆ G z h ˆ, (8) C I G A C (9) ˆ ˆ 1 (6) h M, ˆ, j (1) j h V, ˆ, j (13) v j Note that for smplct the otato we do ot use the tme step subscrpt wth the Jacobas A, W, M ad V, eve though the are fact dfferet at each tme step. Where: ad z = he state ad measuremet vectors at epoch respectvel; u = he optoal cotrol put; w = he process ose; v = he measuremet ose; = he predcted state vector also called the a pror state ˆ estmate at epoch ; = he fltered state vector also called the a posteror ˆ state estmate at epoch ; C = he a pror estmate error covarace at epoch ; ˆ C = he a posteror estmate error covarace at epoch ; ˆ Q = he process ose covarace at epoch ; G = he Kalma ga matr at epoch ; R = he measuremet ose covarace at epoch. 3. SAISICAL ES AND DEFORMAION DEECION he sgle pot test as the local statstcal test s carred out for each pot compoet b compoet at the sgfcace level.5. he crtcal value s computed va Eq. (14) to Eq. (18) (Ice ad Sah, ; Aharzad ad Seta, 11). he the A, W, M ad V matrces ca be derved usg partal dervatves as: d d 1, 1 1, 1 (14)

3 dv d d (15) 1, 1, 1, he zero ad alteratve hpotheses are: trace techque s performed (Seta, 1995). So, the relevat compoets of matr W (Eq (19)) to the fve metoed cotrol statos are all oe. 1 S I G G W G G W (19) H H : E d H : E d, : E : E 1, a 1, d H 1, a 1, d (16) Q j Geerall, j S S Q S G Eq (19) for -D etwork s defed as: () 1 d 1, 1 d 1, d 1, 1, d 1, 1, d 1, d 1, (17) (18) G (1) Where frst two rows of G defe the traslato alog ad aes respectvel. he thrd ad fourth rows are related to the rotatos aroud the z as ad scale of the etwork respectvel. Sce, ths stud the observed data cludes dstace ad bearg the thrd ad fourth rows are elmated (Seta, 1995). I Eq (3.33), represets umber of statos ad: Where d ad d 1, = the (East) ad (North) 1, compoets of dsplacemet vector at epoch respectvel. d 1, ad d 1, = he varace of the dfferece vector for ad compoets respectvel; 1 ad = he varace of the estmato at 1 ad epochs respectvel; 1 ad = he varace of the estmato at 1 ad epochs respectvel; he computed 1, ad 1, values are compared wth crtcal value derved from t-test table based o the ad degrees of freedom f. For ever pot f ether 1,, or F 1, t, f F t f 1,, t s 1, dv cosdered that the dfferece vector, 1,, s sgfcat ad the relevat pot s segregated as ustable pot. 4. S-RANSFORMAION he method of S-trasformato s appled to avod the problem of datum depedet dsplacemet ad to make the results comparable wth the prelmar results from IWS testg. For ths purpose the fve cotrol statos (pot 1 to pot5) are take to accout as the datum pots ad partal mmum 1 1 & 5. DAA ANALYSIS AND RESULS 5.1 otal Stato Data () I ths stud, fve sets of otal Stato data whch were acqured from motorg of a cocrete block subjected to load s used the deformato process drectl. A mcro tragulato etwork cossts of 1 statos, 5 referece pots ad 7 object pots, was establshed sde a laborator. Pots 1 to 5 were located wth a area of about 64 m as the referece pots (Fgure 1). he object pots, pots 6 to 1, were sted o oe facade of the block (Fgure ). A -D tragulato measuremet of dstace ad bearg were doe usg Sokka Set3 otal Stato (Md Som et al., 4). As data cludes dstaces ad azmuths the sstem models used as prmar ad secodar models wll be o-lear. So, Optmzed Kalma flter approach has to be used to aalze the data ad compute the deformato vector.

4 Fgure 1. Mcro tragulato motorg etwork set up the laborator (Md Som et al., 4) D1, D1, D1, D1,3 D1,3 D1,3 D1,4 D1,4 D1,4 Az1, Az1, A1,,,, Az1,3 Az1,3 A1,3 b1 b b a a a a a a Az1,4 Az1,4 Az1,4 1 (3) Fgure 1. Object pots locato o the facade of coceret block (Md Som et al., 4) 5. Data Processg Steps he overall of data processg s fulflled four ma steps: 1. Kalma flter eecutg;. Deformato detecto usg Kalma flter outcomes drectl; 3. S-trasformato; 4. Deformato detecto usg the outcomes of S- trasformato phase. he Kalma flter step b tself cotas s modules: 1. Observato vector ( b ) troducg (Eq. (3));. State vector ( X ) defto (Eq. (4)); 3. Observato (prmar) models defto (Eq. (5) to Eq. (8)); 4. Damc (secodar) models defto (Eq. (9) to Eq. (3); X v v v v Dstace relates to posto model: f D : Dj j j (5) f Az : Az ta j Bearg relates to posto model: 1 j j Accelerato relates to veloct models: (6) f v v : dt (7) f v v : dt (8) (3) Relates posto at epoch to epoch -1: h 1 : 1 v dt 1 dt (9) 1 h : 1v dt a 1 dt (3)

5 h h Relates veloct at epoch to epoch -1: : v v a : v v a dt (31) dt (3) v 1 v 1 5. Desg matr (A) ad state trasto matr (M) dervato or Prmar ad Secodar models learzato; Matr A s computed b frst order dfferetato of prmar model wth respect to elemets of state vector (Eq. (1)). Matr M s derved b frst order dfferetato of 1, secodar model wth respect to elemets of state vector (Eq. (1)) ad secodar model s chaged to matr form Kalma flter Stato (before S-trasformato) (A) Dsplacemet status 1 m stable.1 m stable 3.1 m stable 4.1 m stable 5.1 m stable 6.9 m stable 7.6 m moved 8.43 m moved 9.48 m moved 1.14 m moved 11.8 m stable 1.6 m moved a M v 1, v a v v v v 1 1 v v Kalma tal value computato; 7. Predcto step (Eq. (5) ad Eq. (6)); 8. Flterg step (Eq. (7) to Eq. (9)). (33) able 1. Dsplacemet ad pot status before applg the S- trasformato Kalma flter Stato (after S-trasformato) (B) Dsplacemet status 1.1 m stable.1 m stable 3.4 m stable 4.1 m stable 5.3 m stable 6.1 m moved 7.8 m moved 8.44 m moved 9.49 m moved 1.16 m moved 11.8 m moved 1.64 m moved able. Dsplacemet ad pot status after applg the S- trasformato 5.3 Results Due to the page lmtato ths artcle, ol the results of frst two epochs are show ths artcle. able 1 shows the total dsplacemet of each pot before applg the S- trasformato (case A) alog wth the pot status resultg from the statstcal test phase. Whereas, the outcome of test statstc ad deformato aalss after mplemetato of s- trasformato (case B) that detfes whether a pot remas stable or ot s gve able. he results show able are comparable wth the prelmar results of IWS testg whch are represeted able 3. able 4 ad able 4 stad for the detals of sgle pot test for both cases ( case A ad case B) respectvel. Stato IWS testg Dsplacemet status 1.14 m stable.8 m stable 3.8 m stable 4.7 m stable 5.6 m stable 6.1 m moved 7.5 m moved 8.41 m moved 9.46 m moved 1.1 m moved 11.1 m moved 1.44 m moved able 3. Prelmar results of IWS testg

6 Stato est statstc ( case A) compoet compoet Crtcal value Status stable stable stable stable stable moved moved moved moved moved moved moved Stato able 4. Detals of sgle pot test (case A) est statstc ( case A) compoet compoet Crtcal value Status stable stable stable stable stable stable moved moved moved moved moved moved moved able 5. Detals of sgle pot test (case B) Addtoall, the state vector cludes the veloct compoets. So the varato of dsplacemet s computed both ad drectos. able 6 gves the dea about posto varatos betwee frst ad secod epochs ad drectos (V ad V that order). stato V (m/s) V (m/s) 1 assumed as f assumed as f able 6. Rate of varato 6. CONCLUSIONS he ma objectve of ths stud was the evaluatg of Optmzed Kalma Flter mplemetato ad sutablt deformato aalss agast a rgorous method (IWS). o satsf ths objectve fve set of -D data was utlzed to eecute Kalma Flter kematc mode. he results of Kalma Flter approach are verfed wth the prelmar results of IWS. Beg dfferet from IWS method of Kalama Flterg, Kalma Flterg combes all the formato to estmate the desred parameters. A smple sgle pot test ca be appled compoet b compoet to detect both the stable ad ustable pots. he elemets of the state vector are posto ad the varato of the posto. Hece, Kalma Flterg s sutable for stud the behavor of the deformato for vestgato of catastrophes. REFERENCES Aharzad, N. ad Seta, H., 11. Multdmesoal deformato aalss wth kalma flter. 11th South East Asa Surve Cogress. Co-sposored b FIG. -4 Jue. Kuala Lumpur, Malasa. Chrzaowsk, A., Szostak-Chrzaowsk, A., Bod, J., ad Wlks, R., 7. Icreasg publc ad evrometal safet through tegrated motorg ad aalss of structural ad groud deformatos. Geomatcs Solutos for Dsaster Maagemet (eds: J. L, S. Zlataova, A. Fabbr), Sprger, pp Cross, P. A., Advaced least squares appled to posto fg. Workg Paper No. 6. Departmet of Lad Surveg, North East Lodo Poltechc. Ice, C. D., ad Sah, M.,. Real-tme deformato motorg wth GPS ad Kalma flterg. Earth Plaet Space, 5, pp asc, L., 1. Aalss of dam deformato measuremets wth the robust ad o-robust methods. Scetfc Research ad Essas, 5(14), pp

7 Kalma, R. E. (196). A ew approach to lear flterg ad predcto problems. Joural of Basc Egeerg, 8D, pp Lm, M. C., Seta, H., Othma, R. ad Omar K., 1. Cotuous deformato detecto ad vsualsato of ISKANDARet. Iteratoal Smposum o GPS/GNSS. October 6-8. ape, awa. Md Som, Z. A., Seta, H. ad M Idrs, K. N., 4. A geodetc deformato surve to motor the behavor of a cocrete slab durg ts aal compresso testg. 1 st FIG Iteratoal Smposum o Egeerg Surves for Costructo Works ad Structural Egeerg. 8 Jul. Nottgham, Uted Kgdom. USACE,. Structural deformato surveg. Departmet of the Arm, US Arm Corps of Egeers, EM , Washgto, D.C. Welch, G. ad Bshop, G., 1. A troducto to the kalma flter. Proceedg ACM SIGGRAPH 1, Course 8, August Los Ageles, CA. Welch, G. ad Bshop, G., 6. A troducto to the kalma flter. Departmet of computer scece, Uverst of North Carola at Chapel Hll, R 95-41, Jul 4, 6.

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