Europan Scintific Journal Sptmbr 24 dition vol., No.27 ISSN: 87 788 (rint) - ISSN 87-743 DETERMINATION OF ELECTRONIC DISTANCE MEASUREMENT ZERO ERROR USING KALMAN FILTER Onuwa Owuashi, hd Dpartmnt of Goinformatics & Survying, Univrsity of Uyo, Nigria Dup Nihinlola Olayina, hd Dpartmnt of Survying & Goinformatics,Univrsity of Lagos, Nigria Abstract Kalman filtr has gaind popularity in th adjustmnt of fild masurmnts sinc it was invntd in 96. Unli th last squars tchniqu it is rlativly not rnownd in th adjustmnt of survying masurmnts. This wor prsnts how it can b usd in th dtrmination of th lctronic distanc masurmnt zro rror. First th valus of th unnown paramtrs and thir corrsponding covariancs ar prdictd. Subsquntly, updats on th prdictd paramtrs and covariancs ar stimatd in svral itrations. Th rsiduals ar stimatd to dtrmin th accuracy of th xprimnt. Th stimatd lctronic distanc masurmnt zro rrors for thr pochs ar.3322833m,.34248732m and.3298963m. Kywords: Elctronic distanc masurmnt, Kalman filtr, Zro rror Introduction Survying masurmnts ar usually compromisd by rrors in fild obsrvations and thrfor rquir mathmatical adjustmnt. In th first half of th 9th cntury th Last Squars (LS) (Gauss, 823) adjustmnt tchniqu was dvlopd. LS is th convntional tchniqu for adjusting survying masurmnts. Th LS tchniqu minimiss th sum of th squars of diffrncs btwn th obsrvation and stimat. LS has a disadvantag of rquiring matrix invrsions that tnd to slow down th procss of adjusting masurmnts. In this rsarch th us of th Kalman Filtr (KF) (Kalman, 96) tchniqu for dtrmining th zro rror of th Elctronic Distanc Masurmnt (EDM) will b prsntd. KF asily rsolvs th adjustmnt of fild masurmnts by using a rcursiv algorithm utilising part of its output as an input for th nxt itration (Bzruca, 2). 8
Europan Scintific Journal Sptmbr 24 dition vol., No.27 ISSN: 87 788 (rint) - ISSN 87-743 Th principls of Kalman filtr Th KF is a rcursiv algorithm (Maybc, 979) that stimats th currnt stat of th dynamic systm out of incomplt noisy indirct masurmnts. KF is suitabl for both linar and nonlinar procsss. Th principl of KF is basd on two basic phass of th procss: prdiction and updat (Bzruca, 2). KF prdicts or stimats th stat of a dynamic systm from a sris of incomplt and /or noisy masurmnts. Suppos w hav a noisy linar systm that is dfind by th following quations: ˆ A w () Z H v (2) Whr is stimatd stat at tim, A is th stat transition matrix, is stimatd stat for prcding tim, w is procss nois at tim, Z is th masurmnt, H is th masurmnt dsign matrix and v is th masurmnt nois. rdiction Th prviously stimatd stat ˆ can b usd to prdict th currnt stat at tim, as shown by th following quation: ˆ A (3) T Aˆ A Q (4) Whr and ˆ ar stimatd rror covarianc matrics at tims and rspctivly; whil Q is th procss nois covarianc matrix. Updat Th Kalman gain K H ( H H R) K is stimatd as, T T () R is th masurmnt nois covarianc matrix. Equation shows that a mor prcis masurmnt (i.. th lowr covarianc matrix lmnts) raiss its wight (Wlch and Bishop, 2). K H (6) lim R H is gnrally a nonsquar matrix, and thus cannot b invrtd. Equation 6 should b statd in th following form, 9
Europan Scintific Journal Sptmbr 24 dition vol., No.27 ISSN: 87 788 (rint) - ISSN 87-743 R K H I lim (7) Th a priori covarianc matrix approaching zro valus mans th low wight of th obsrvation and a priori rsidual (Wlch and Bishop, 2), limk (8) Assuming th apriori rsidual as th diffrnc btwn th currnt obsrvation and th xpctd obsrvation dtrmind in th last paramtr stimat is (Bzruca, 2), Z Hˆ (9) Thrfor updatd stimat of stat is dtrmind as, ˆ ˆ K () and its updatd covarianc matrix is dtrmind as, ( I K H ) () Application In ordr to dtrmin th instrumnt constant or zro rror ( ) of a short-rang EDM a straight baslin dsign mthod was implmntd (Ayni, 2). Twlv masurmnts wr mad using points A, B, C and D (Figur and Tabl ). Figur. Baslin masurmnts (Ayni, 2) Tabl. Obsrvations (Ayni, 2)
Europan Scintific Journal Sptmbr 24 dition vol., No.27 ISSN: 87 788 (rint) - ISSN 87-743 Th unnown paramtrs to b dtrmind ar th adjustd distancs AB, BC, CD and zro rror. Th obsrvation quations ar (Ayni, 2), Whr D ia ar th adjustd distancs;,, 2 3 rprsnt adjustd distancs AB, BC and CD rspctivly; whil rprsnts th adjustd EDM instrumnt zro rror. Th solution to th statd problm was implmntd using th MATLAB programming softwar. 2 Th solution is, furnishd by quation. 3
Europan Scintific Journal Sptmbr 24 dition vol., No.27 ISSN: 87 788 (rint) - ISSN 87-743 2 Th valus of,, Q, Z, H, A and R ar givn as, 4.2 3. 3.7. 3.6 6 4.3 6 3.9 8. 7. 2. 7.2 7 8.4 R,,.... Q, 67.3 32.948 22.747 34.2297.6222 22.747.24 67.87 34.222.7 Z, H and
Europan Scintific Journal Sptmbr 24 dition vol., No.27 ISSN: 87 788 (rint) - ISSN 87-743 Z is th masurmnt shown in Tabl. H was drivd from th obsrvation quations, whil th statd valus for,, Q, A and R wr assumd a priori. Rsults Thr itrations wr run to dtrmin th valus of,, and for,..., n, whr n 3 (Tabl 2). A postriori rsiduals wr calculatd for, 2 and 3 (Figur 2). Th rsiduals showd th rrors in computing th adjustd valus of th unnown paramtrs. Figur 3 was calculatd by summing th absolut valus of th rsiduals for, 2 and 3. From Figur 3, 2 yildd th lowst rsidual valu and thrfor th most accurat whil yildd th highst rsidual valu and thrfor th last accurat. From Tabl 2,.3322833m,.34248732m and.3298963m wr th stimatd EDM zro rrors for thr pochs, 2 and 3 rspctivly. Tabl 2. Estimatd rsults for th unnown paramtrs and thir covariancs.498.498.498.498.4866639826.434-4.868-4.7234-4 -.99439-4 22.694763624.868-4.3777-4 -.996-4 -.682-4 32.8636728992.7234-4 -.996-4.3-4 -.7337-4.3322833 -.99439-4 -.682-4 -.7337-4.69-4 2 2.4866639826..867778.723268 -.9949 22.6947636237.867768.38 -.992 -.6784 32.8636728992.723268 -.992.2 -.7336.33228326 -.9949 -.6784 -.7336.6 2.499776823.386-4.867688-4.722998-4 -.99389 22.74996236.867688-4.37738-4 -.9942-4 -.678-4 32.893873697.722998-4 -.9942-4.489-4 -.733329-4.34248732 -.99389 -.678-4 -.733329-4.6468-4 3 2 3 3
Europan Scintific Journal Sptmbr 24 dition vol., No.27 ISSN: 87 788 (rint) - ISSN 87-743.489626842893..8674.72284 -.993696 22.69472473493.8674.38 -.992 -.67 32.86993466.72284 -.992. -.73383.33737 -.993696 -.67 -.73383.6 3.499778849447.386-4.867688-4.722998-4 -.99389-4 22.7499627943.867688-4.37738-4 -.9942-4 -.678-4 32.893824382.722998-4 -.9942-4.489-4 -.733329-4.3298963 -.99389-4 -.678-4 -.733329-4.6468-4 3 2 3 -.. -. -. Rsiduals (m) -. -.2 2 9 8 7 6 4 3 2 Obsrvations Figur 2. Rsiduals.7.6.8. -.4.3 2 Itrations 3.2 Sum of absolut valus of th rsiduals (m). Figur 3. Sum of absolut valus of th rsiduals Various valus of R wr usd to comput th adjustd valus of th actual masurmnts (Figur 4). From Figur 4, R ( ) (whos matrix 4
Europan Scintific Journal Sptmbr 24 dition vol., No.27 ISSN: 87 788 (rint) - ISSN 87-743 was givn in th prvious sction) yildd th bst fit of th actual masurmnts. Whil R * y(2) yildd th last fit of th actual masurmnts. Actual masurmnt Adjustd masurmnt with R() Adjustd masurmnt with R-* Adjustd masurmnt with R-* Adjustd masurmnt with R-2* 8-2 3 4 6 7 8 9 2 2 4 6 Distancs (m) Obsrvations Figur 4. Computd adjustd masurmnts using various R valus Conclusion Th KF has th bnfit of furnishing svral solutions in succssiv itrations as shown in this wor. Othr advantags of th KF includ th possibility of altring th valus of th procss nois covarianc matrix Q and masurmnt nois covarianc matrix R in ordr to improv th accuracy of th prdiction. Rfrncs: Ayni, O. O. Statistical adjustmnt and analysis of data. A Manual, in Dpartmnt of Survying and Goinformatics, Faculty of Enginring, Univrsity of Lagos, Nigria, 2. Bzruca, J. Th us of a Kalman filtr in godsy and navigation. Slova Journal of Civil Enginring, 2:8-, 2. Gauss, C. F. Thoria combinationis obsvationum rroribus minimis obnoxia. Wr, Vol. 4. Göttingn, Grmany, 823. Kalman, R. E. A Nw Approach to Linar Filtring and rdiction roblms. Transactions of th ASME-Journal of Basic Enginring, (Sris D), 82:3-4, 96. Maybc,. S. Stochastic modls, stimation and control. Volum I, Nw Yor, pp.-6, 979. Wlch, G. and Bishop, G. (2). An Introduction to th Kalman Filtr. UNC Chapl Hill, 2.