CONCEPT OF MEASURING AND CALCULATION SYSTEM, BASED ON ELECTRONIC CLINOMETERS TO DETERMINE THE GEOMETRY OF RAILWAY TRACK IN THE VERTICAL PLANE

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1 Archves of Photogrammetry, Cartography and Remote Sensng, Vol. 22, 2011, pp ISSN CONCEPT OF MEASURING AND CALCULATION SYSTEM, BASED ON ELECTRONIC CLINOMETERS TO DETERMINE THE GEOMETRY OF RAILWAY TRACK IN THE VERTICAL PLANE Mchał Strach 1, Grzegorz Lenda 2 1, 2 Department of Engneerng Surveyng and Cvl Engneerng, Faculty of Mnng Surveyng and Envronmental Engneerng, AGH Unversty of Scence and Technology, Kraków, al. Mckewcza 30 strach@agh.edu.pl, lenda@agh.edu.pl KEY WORDS: Surveyng, Applcaton, Automaton, Measurement, Algorthms, Infrastructure, Systems ABSTRACT: Geodetc nvestments support n the scope of modernzaton the exstng sectons of ralways and buldng new ones s assocated wth the work of hgh accuracy. For ths reason, surveyng teams use the latest technologcal solutons, both n terms of hardware and software. Among the ral surveyng systems dedcated for ralway areas, there are trolleys equpped wth GNSS satellte recevers. Unfortunately, the GNSS system s not suffcently accurate n determnng the heght of the ralway track. Therefore, the surveyng work s often broken down nto two stages. In the stuatonal surveys total statons or GNSS recevers are used, and n the surveys of heght, geometrc levellng s appled. The artcle descrbes the possblty of usng an electronc level n order to elmnate cumbersome levellng surveys. A clnometer used n the surveys frstly had been subjected to author's laboratory tests verfyng ts accuracy and allowng ts calbraton. Durng the feld tests, the selected secton of the track was measured usng a level, and addtonal standard surveys were carred out usng geometrc precson levellng. A computatonal method was also suggested, allowng for the determnaton of the heghts of ponts, based on the measured slopes. The method s based on ntegratng a contnuous model of slopes, nterpolated by a set of polygonal chans. The study conducted accuracy tests both of the computatonal method tself as well as combned, survey and computatonal method for determnng the heght. 1. INTRODUCTION Management of the state ralway network s the responsblty of PKP Polsh Ralway Lnes S.A. Wthn the scope of ral nfrastructure mantenance works, the company s requred to conduct an ongong revew of ts condton. The company carres out nspectons and surveys, and records any faults or defects, as well as t conducts mantenance works, ongong and man repars. The purpose of ths actvty s to ensure safety on the ralway lnes and to mantan techncal and operatng parameters of a lne at an adequate level. Among numerous works related to the mantenance of the lnes, there s an nventory survey of the track geometry and locaton. Qualfed dagnostc and surveyng professonals perform works wthn the scope specfed n the Instructon Id-14 (Instructon on carryng out surveys, testng and evaluaton of tracks condton, 2005). The Instructon ponts to the necessty of carryng out surveys of the parameters related to the geometry of the track 389

2 Mchał Strach, Grzegorz Lenda (.e.: gauge, cant, unevenness of rals n horzontal and vertcal planes). It also mposes an oblgaton to dentfy the locaton of the track n the horzontal and vertcal planes n relaton to the track axs reference ponts. These varables can be measured drectly usng a portable surveyng equpment, as well as wth ndrect methods wth the use of heavy surveyng vehcles. The process of upgradng of the exstng sectons of ralway tracks and buldng new ones s supported by geodetc surveyng teams at many stages. They are supposed to conduct both the nventory surveys, as well as geodetc support for the realzaton works. The most common surveyng works performed n ralway areas nclude: drawng maps for desgn purposes, nventory surveys to dentfy track geometry as well as realzaton surveys, ncludng tampng machnes surveyng servces. An essental element assocated wth the works whch are amed at determnng the geometry of the track s executng a longtudnal profle or ts updatng. The data from these surveys s often used to make adjustments of track axes (Instructon D-19, 2000). The surveyng works can use a varety of technologcal solutons. The most precse and effcent ones, allowng smultaneous surveys of multple geometrcal parameters of the track, may nclude track measurement trolleys. These are mult-sensor devces provdng nformaton on ground clearance, cant and track statonng. Addtonally equpped wth geodetc equpment they guarantee the possblty of determnng the locaton of the track n the adopted spatal coordnate system. Recently, desgners of the measurng trolleys equp them wth GNSS precson satellte recevers. Also at AGH Unversty of Scence and Technology n Kraków such a trolley was bult (Strach, 2003). The conducted measurement tests ndcate that the technque RTK GNSS allows to specfy a ralroad track geometry wth suffcent accuracy only n the horzontal plane. GNSS measurements wth the antenna placed on the trolley are not able to provde suffcently precse heghts of the observed ponts. Ordnates of the tracks under nventory should be determned by means of geometrc levellng. Geodetc network ponts of heght along the track are used to engage n surveys. Great help n carryng out the surveys are ponts stablzed on tracton current pylons (Fg. 1), actng as both the reference ponts and the ralway geodetc control network. Breakdown of the works nto two ndependent technologcal stages (stuatonal and heght) prolongs the tme of carryng out surveys and ncreases the costs of ther mplementaton. The devce, whch can replace the tedous levellng method s a precse electronc level (clnometer) mounted on the measurement trolley. The clnometer should be placed on the constructon beam of the trolley parallel to the axs of the track. Ths postonng wll allow the measurement of longtudnal ral slopes. Before decdng to expand the track measurement trolley wth the aforementoned devce, approprate tests and analyss of the results were carred out. The tests of the constructed electronc clnometer (Fg. 2) were conducted n the laboratory and n the feld. Precse model surveys were conducted allowng the analyss of accuracy of the obtaned results from the surveys wth the clnometer. Also a mathematcal algorthm was proposed allowng to determne the heght of the ponts of track grade lne, based on the observed slopes. 390

Concept of measurng and calculaton system, based on electronc clnometers Fg. 1.

tracton current pylon Fg. 2. Electronc level durng the feld tests 2.

OBSERVED SLOPES The result of the carred out surveys are sets of two values: the

3 Concept of measurng and calculaton system, based on electronc clnometers Fg. 1. Track axs reference pont and at the same tme ralway geodetc network pont fxed to a tracton current pylon Fg. 2. Electronc level durng the feld tests 2. MATHEMATICAL PRINCIPLES OF DETERMINING THE HEIGHT OF GRADE LINE POINTS BASED ON THE OBSERVED SLOPES The result of the carred out surveys are sets of two values: the slopes of the track and the mutual dstances of ponts at whch the records of the slopes were made. Addtonally, opposte the selected tracton pylons, by the use of track reference ponts stablzed on them, the heghts of the tracks shall be specfed. Based on these values, the heghts of the 391

4 Mchał Strach, Grzegorz Lenda track should be determned at the slope observaton ponts. Track reference ponts on the pylons are arranged along the track every meters on straght sectons, whle on curved sectons the densty of the pylons s hgher. Observatons carred out wth the level are performed whle the trolley s not movng, whch takes place every few meters. As a result, based on the heght of the track transferred from the pylon, the heghts of over a dozen ponts on average should be determned, where the slope was observed. The process s repeated perodcally, for sectons between successve tracton current pylons, whch at all tmes ensure the ntalzaton of the determnaton of the track heght. To solve the task, the method of ntegraton of functons descrbng the slopes may be appled, whch was prevously used n (Gocał, 1981; Gocał, Lenda, 2003). The dea of the method goes down to the followng observatons. The observed slope at a gven pont P s a boundary poston of the secants of the track grade lne, connectng successve pars of ponts,.e. P-1 - P and P - P +1, thereby consttutng the slope of the tangent to the grade lne at ths pont. The gradent of the tangent to the curve k at a gven pont s determned by settng the value of ts dervatve at ths pont. Determnaton of the value of the functon k tself, that s the heght of the grade lne ponts, wll be subject to a reverse process,.e. ntegratng of the functons representng a set of slopes of the tangents. For ths purpose, an approxmatng functon should be ftted nto a set of slopes observed wth the level, whose prmary functon, assumng the approprate ntal condtons, shall appont the searched heghts. From the standpont of the accuracy of the results obtaned n ths manner, the type of functon descrbng the slope s essental. In the above-mentoned studes approxmatng polynomals were used for ths purpose. In general, the polynomals can be represented n the nterpolatng or approxmatng form. In the case of a greater number of ponts, nterpolaton requres the use of polynomals of hgher degrees, whose determnng process, wth an ncreasng degree, s fast becomng an ll-condtoned numercal task (Fortuna, Macukow, Wąsowsk, 2001; Cheney, Kncad, 2006), generatng sgnfcant dstortons of the resultng functon. Approxmaton n turn, made wth lowdegree polynomals, does not allow a good ft of a functon to the set of slopes, whch also has an adverse effect on the accuracy of the desgnated heghts of the grade lne. However, due to the fact that the track grade lne does not demonstrate large curvatures, good accuracy can be acheved even wth the replacement of pont model of the slope wth the smplest contnuous model n the form of a polygonal chan. Ths paper presents expermental results acheved usng an algorthm based on such a method. Each secton connectng consecutve ponts of the observed slopes s descrbed wth the equaton: ' F ( x) = a1 + a 2 x (1) Determnaton of the functon descrbng the track grade lne goes down to ntegratng all of such equatons: ' 1 2 F ( x) = F ( x) = a1 x + a2 x + c (2) 2 392

5 Concept of measurng and calculaton system, based on electronc clnometers and to defnng the approprate constants of ntegraton c. The value of the frst of these can be determned usng the known heght of the track H 0, referred to the frst pont located opposte the reference pont on the tracton current pylon. 1 2 c 0 = H 0 a10 x0 + a 20 x0 (3) 2 The frst secton of the functon F 0 (x) connects the two ntal grade lne ponts, and therefore t enables to calculate the heght of the next pont at whch the slope was observed. Ths heght wll be the bass to desgnate the constant of ntegraton of the next secton of the functon F (x) n the same way as for the formula (3). The whole algorthm s repeated untl reachng the ordnate of the next tracton current pylon equpped wth track reference pont, from whch the exact heght wll be transferred onto the track. From ths pont the algorthm s nvoked agan, settng the grade lne heght for the next secton located between the track reference ponts. 3. METHODOLOGY OF THE RESEARCH AND RESULTS OF THE PRECISION ACCURACY TESTS In order to verfy the accuracy acheved by the proposed method, tests were performed n three categores: the accuracy of the surveyng method (electronc level), the accuracy of the computatonal method and the total accuracy of the system. 3.1 Accuracy of the electronc level The bass for the realzaton of expermental studes on the use of clnometers n determnng the heght of the track was to fnd a suffcently precse electronc level. The autors used an electronc clnometer (Fg. 1), bult for the purpose of levellng of machnery and technologcal equpment. It s a sngle-axs analogue nclnometer made by the company VTI Technologes Oy, mounted on the system SCA103T. In accordance wth the techncal specfcaton, the devce operates wth a resoluton of 0.001, and the sensor measurng range falls wthn ± 15 ( 2011). Each nstrument s characterzed by ts own dstrbuton of errors, knowledge of whch becomes necessary n the case of hgh-precson measurements. The verfcaton of accuracy and calbraton of the electronc level were carred out n geodetc metrology laboratory of AGH n Kraków. In laboratory tests, a precse electronc total staton Leca TC2002 was used, wth the accuracy of measurng vertcal angles of 0.5". The level was attached to the total staton telescope set horzontally (Fg. 3). The dea of the test conssted of recordng the consecutve pars of readngs from the slopng of the level and the total staton telescope. After each survey, the telescope was rotated wth an nterval equal to approxmately 0.5 mm / m, wchch was read from the level. The values recorded from both of the devces were normalzed to be able to compare and evaluate the accuracy of the clnometer. The graph llustrated n Fgure 4 presents the characterstcs of the clnometer operaton compared wth the aforementoned total staton. 393

Mchał Strach, Grzegorz Lenda Wth the full range of work of the level (-10 +10 mm/m) the dfferences between the ndcatons of these two devces reach 0.31 mm/m.

Once they are ncluded n the results, we can conclude that the accuracy of the clnometer s no worse than 0.1 mm/m. Fg. 3.

6 Mchał Strach, Grzegorz Lenda Wth the full range of work of the level ( mm/m) the dfferences between the ndcatons of these two devces reach 0.31 mm/m. The values of the dfferences n readngs from both devces enabled the calculaton of the calbraton correctons for the controlled level. Once they are ncluded n the results, we can conclude that the accuracy of the clnometer s no worse than 0.1 mm/m. Fg. 3. Electronc level wth electronc total staton Leca TC2002 durng testng n geodetc metrology laboratory of AGH Unversty of Scence and Technology 394 Fg. 4. Clnometer work characterstcs compared wth total staton Leca TC2002

7 Concept of measurng and calculaton system, based on electronc clnometers 3.2 Accuracy of the computatonal method The accuracy of the computatonal method was tested based on the model data generated on the base of levellng surveys of the track grade lne. Precse levellng provded a set of grade lne heghts of the ponts wth known mutual dstances along the track. Splne functon was ftted nto ths set, allowng the replacement of a pont model wth a smooth model of a contnuous, and also a mnmum total curvature. For the analyzed case, the so called natural splne functon wll have the desrable propertes, whch accordng to the Holladay's theorem (Ahlberg, Nlson, Walsh; 1967) offers the best smoothness. To create a contnuous model, a functon n the form of B-splne (Kcak, 2000) was used: n 4 3 S ( x) = d N ( x) x = { x 0,.., x n } (4) = 0 where: d = coeffcents of lnear combnaton of bass functons N, defned by the formula Mansfeld-De Boor-Cox (Kcak, 2000) Then, at the heght measurement ponts (nterpolaton), tangents to ths functon were determned, specfyng ther slope, whch are a model value of the measurements whch should be made n practce usng the level. The slopes were read from the model wth an accuracy approprate for the descrbed level,.e. at the level of 0.1 mm/m. The slopes obtaned n ths way were then used to determne the heght of the track grade lne ponts accordng to the proposed method. The results were compared wth the heghts of the grade lne selected to create a model (from levellng). In ths way, the qualty of the mathematcal method operatng on the measurement data wth a gven accuracy was defned. The tests were conducted for three selected sectons measured by precse levellng of the track, each consstng of twenty ponts. The sectons (of ten meter length), whch were selected for the test, wth the vertcal scale ncreased respectvely (100-fold), have been shown n Fgure 5. The frst of them (m1) has a relatvely constant slope, the second one (m2) has a varable slope and small changes n curvature, and the thrd one (m3) - has a varable slope and sgnfcant changes n curvature. m1 m2 m3 Fg. 5. Sectons selected for testng for the study of the accuracy of the computatonal method. The vertcal scale n relaton to the horzontal scale: 100: 1. Horzontal dstance between ponts equals 10 [m] 395

8 Mchał Strach, Grzegorz Lenda The results of the carred out tests have been compled n Table 1. Tab. 1. Determnaton of the accuracy of the computatonal method for the model data. The dfferences n heghts dh from levellng, and determned by ntegratng the functon descrbng the slope Accuracy of determnng heght dh [mm] m1 m2 m dh max dh mean The total accuracy of the method In order to obtan a total measurement and computatonal estmate of the accuracy of the proposed system, surveys were conducted on a secton of the track lne Warsaw Kraków, of the route Słomnk - Nedźwedź at the statonng of The observatons were carred out by two ndependent methods: usng a precse geometrc levellng, whch s the reference method, and usng the electronc level. The level was appled twce, ndependently, on the surface of the ral head, parallel to ts axs. The places of the level applcaton were the prevously determned and surveyed (wth the use of geometrc levellng) cross-sectons on the track. Geometrc levellng was carred out wth the use of precse leveller DNA03 wth a precse dgtal levellng rod. Ths set has an accuracy of 0.3 mm/km. In the reported surveys closed levellng traverse was appled. One of the reference ponts was used to determne the heght. The surveyed levellng traverse s characterzed by an error of 0.8 mm for the length of 800 m. As a result of the surveys, the heghts of all of the track reference ponts were determned. Also the heghts of the track opposte the reference ponts and n cross-sectons every 2 meters were determned. The tests were conducted n an analogous manner as descrbed above, on the same track sectons (Fg. 5). For each of them the heghts of the grade lne ponts were calculated based on the ntegraton of the functons ftted nto the surveyed slopes, and afterwards they 396

9 Concept of measurng and calculaton system, based on electronc clnometers were compared wth the heghts obtaned by precse levellng. The dfferences between the two methods have been presented n Table 2. Tab. 2. Determnaton of total, measurement and computatonal accuracy of the method. The dfferences n heghts dh from levellng, and determned by ntegratng the functon descrbng the slope Accuracy of determnng heght dh [mm] m1 m2 m dh max dh mean Accuracy of the level The obtaned test results allow the concluson that the analyzed level enables surveyng wth an accuracy of 0.1mm/m. In the geodetc metrology laboratory of AGH n Kraków, a laser nterferometer HP 5529 s also avalable. The accuracy of the surveys performed wth ths devce reaches the µm/m range. In the future t s planned to calbrate the level usng the aforementoned nterferometer The accuracy of the computatonal method The proposed computatonal method, assumng a slope measurement accuracy of 0.1 mm/m, acheved satsfactory results (Table 1). Maxmum devatons for the two tested sectons were at the level of 3 mm, for the thrd one - 5 mm. The obtaned results are wthn the standards of measurement accuracy, whch are specfed n Instructon D-19, 6 p.96). It ncludes a clause that the heghts of the ponts of the track system and terrestral fttngs of underground utltes should be determned by total staton measurements, and addtonally wth geometrc levellng method. Devaton between the ordnates of the two methods of surveyng cannot exceed ± 1 cm. Devatons greater than ths value make these surveys to be repeated. For the analyzed set of observatons the average devatons n no 397

10 Mchał Strach, Grzegorz Lenda case exceed the value of 1.5 mm. For all the models, n the areas of the occurrence of maxmum devatons, oscllatons of ther values were recorded, based on the frequent change of the reference pont The overall measurement and calculaton accuracy The overall measurement and computatonal accuracy of the test results s often at an unsatsfactory level. Permssble devaton lmts at some ponts are exceeded by 1.5 tmes. Devatons are dstrbuted randomly, and ther occurrence s assocated wth nsuffcently accurate way of applyng the level to the ral. The devce has the accuracy of 0.1 mm / m. However, ts unequal applyng to the upper surface of the ralhead resulted n naccuraces, already apparent durng the survey. Sgnfcant dfferences n the slopes from double readngs of the level at each pont were observed. Placng the level on a measurng trolley whch ensures repeatable applcaton of the level to the track, elmnates the problem. 4. CONCLUSIONS AND FUTURE RESEARCH PLANS The used level meets accuracy requrements necessary for the mplementaton of the works on the ralroad tracks. Unfortunately, the way t s appled to the ral dd not assure obtanng the requred accuracy. Also, for practcal reasons and due to the nusance of ts applcaton, modfcaton should be ntroduced n ts constructon and n the manner of conductng surveys. Currently, ntensve works on mprovng both the equpment and the measurement technology are beng carred out. In the scope of the proposed computatonal method, t should be noted that for the total tests, there were no places of sgns oscllatons of devatons wth the largest values, occurrng for the tests of the computatonal method. It can therefore be estmated that these oscllatons are not necessarly related to the operaton of the proposed computatonal method. They may be partally due to the naccuracy of the model, created for the earler tests usng splne functons. The results from Table 1 evaluatng the accuracy of the computatonal method tself, as dsturbed by an addtonal modellng error, should therefore be evaluated as more optmstc. They can be treated as top-down estmate of the accuracy offered by the proposed computatonal method. The use of a polygonal chan allows for obtanng good accuracy of the determned heghts at track reference ponts of the slopes. In order to determne the heght at random ponts of the track grade lne, non-lnear functons should be used. For ths purpose, the authors provde for the use of splne functons for whch they are currently carryng out accuracy tests. Ths method s also lkely to allow an overall ncrease n computatonal accuracy. ACKNOWLEDGMENT The paper was prepared wthn the scope of the AGH Unversty of Scence and Technology statutory research no n

11 Concept of measurng and calculaton system, based on electronc clnometers BIBLIOGRAPHY References from Journals: Gocał, J., Określane kształtu elementu konstrukcyjnego maszyny sposobem aproksymacj odchyleń stycznych, Zeszyty Naukowe AGH, Geodezja, Vol. 62, Kraków. Gocał, J., Lenda, G. (2003). Determnaton of the shape of a ral course on the bass of changes of longtudnal nclnaton observed contnuously. Polsh Academy of Scences, Photogrammetry and Montorng of Envronment, Vol. 40, pp. 5 13, Kraków. References from Books: De Boor, C., A practcal gude to splnes. Sprnger-Verlag, New York Fortuna, Z., Macukow, B., Wąsowsk, J., Metody numeryczne. WNT, Warszawa. Kcak, P., Podstawy modelowana krzywych powerzchn. WNT, Warszawa. Kncad, D., Cheney, W., Analza numeryczna, WNT, Warszawa. References from Other Lterature: D-19, Instrukcja o organzacj wykonywanu pomarów w geodezj kolejowej. PKP S. A., Warszawa. Id-14 (D-75) (2005). Instrukcja o dokonywanu pomarów, badań oceny stanu torów. PKP PLK S. A., Warszawa. Strach, M., Ocena możlwośc wykorzystana technk sateltarnej RTK GPS do regulacj os torów kolejowych. PhD thess, AGH Kraków. References from webstes: (techncal data, accessed 01 Aprl 2011) 399

y and the total sum of

y and the total sum of Lnear regresson Testng for non-lnearty In analytcal chemstry, lnear regresson s commonly used n the constructon of calbraton functons requred for analytcal technques such as gas chromatography, atomc absorpton