Analytical Tunnel-boring Machine Pose Precision and Sensitivity Evaluation for Underground Tunnelling

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1 Aalytical Tuel-borig Machie Pose Precisio ad Sesitivity Evaluatio for Udergroud Tuellig D. Li a, S. Mao a, M. Lu a ad X. She b a Departmet of Civil & Evirometal Egieerig, Uiversity of Alberta, Caada b School of Civil ad Evirometal Egieerig, Uiversity of New South Wales, Australia duashu@ualberta.ca, mao1@ualberta.ca, mlu6@ualberta.ca, x.she@usw.edu.au ABSTRACT Practical Tuel-borig Machie (TBM) guidace systems usually use sophisticated istrumets to obtai the orietatio of the TBM. These systems require frequet calibratio ad work shutdow, makig the procedure complex ad time cosumig. Besides, the majority of curret TBM guidace solutios caot provide the positio of the ivisible cutter head aalytically icludig commoly used laser poit trackig system. This makes it impossible to check the accuracy of the TBM positioig for esurig the quality of the tuel beig istalled i the groud. To address the problem, this research proposes a aalytical approach to quatify the accuracy of the TBM positio ad orietatio estimatio based o a previously proposed survey-based TBM guidace system. The coordiates for the cutter head ceter are derived through error propagatio based o defiig geometric costraits ad applyig computig algorithms, such as the layout cofiguratio of the prisms mouted o the TBM ad the relative positio of the cutter head ceter. The paper also provides a thorough aalysis of the sesitivity of the solutio with regards to certai cofiguratios based o uderlyig formulas. To verify the results, a tuellig experimet was also coducted based o a 1:20 scaled TBM model with its cutter head visible. The proposed method provides a valuable approach to evaluate the accuracy of the cutter head positio estimatio ad potetially eable the operators to cotrol ad guide the tuellig process i the field. Keywords - TBM; Navigatio; Aalytical; Error Propagatio, Accuracy; Sesitivity 1 Itroductio Costructio is the implemetatio process i the field, trasformig the egieerig desig ito the fial product i the real world. I tuel costructio, the desig makes rules about start/ed locatios, depth, tuel diameter, lier material, ad grades; cotractor must devise effective methods for both implemetig desig ad checkig desig i the field i order to satisfy error toleraces i techical specificatios. I tuel costructio, the geospatial-related desig factors (start/ed locatios, depth, diameter, ad grades) are represeted by a ivisible as-desiged tuel aligmet, ad the TBM positio is preseted by vertical/horizotal offsets (grade/lie deviatios) from this aligmet. TBM guidace systems are resposible to iterpret the ivisible aligmet ito visible idicators, ad guide the TBM operator steerig o the right track. A widely accepted visible idicator is a visible laser beam, which is parallel to the as-desiged aligmet. The laser projects at a receivig target mouted o the TBM, therefore the ivisible tuel aligmet is marked as a visible laser dot o the target board [1]. The surveyor team of the tuel costructio has two critical tasks: traslatig the desig specificatio ito as-desiged aligmet for the guidace systems; ad checkig coformity betwee the aligmet ad the idicator give by the guidace system. Due to the complex o-site eviromet of a tuel uder costructio, the guidace system caot reliably maitai the coformity with the as-desiged aligmet for a log time period [1]. Both the guidace istructios ad steerig operatios must be wellmaaged to eable the tuel s as-built aligmet to be fully fuctioal, for either large-diameter traffic tuels or small-diameter draiage tuels. For example, whe a seriously misaliged TBM breaks through to the exit shaft of a draiage tuel, the as-built tuel may ot be able coect to existig draiage pipelies. Or a

2 misaliged railway tuel may cause steeply curved tracks, ad trais eed to reduce speed to safely pass by [2]. Moreover, if the guidace istructios or the steerig operatios are ot reliable ad cosistet, the as-built draiage tuel ca be bumpy, which will cause trouble i chaellig storm water flow [3]. Therefore, the error at the breakthrough poit should geerally be cotrolled with 10mm to 20mm per 1km legth for traffic tuel, ad draiage tuel should ot be misaliged more tha 150 mm horizotally or 89 mm vertically at ay sectio of the tuel beig built [4,5]. Surveyors utilize traverse ad levellig methods to fix the curret horizotal ad vertical positio of the TBM [2,5,6]. Due to the geometry of tuellig, both traverse ad levellig deals with ope-ed loop, which meas errors i survey caot be easily detected ad compesated [5]. Chrzaowski ad Stiros thoroughly studied the errors i traverse ad levellig survey o tuel breakthrough accuracy [2,5]. As surveyors iterpret tuel aligmet for guidace systems usig the traverse ad levellig results, the errors accumulated i survey procedures will itroduce errors betwee asdesiged ad as-surveyed aligmets. Moder laser-based TBM guidace systems [7] utilize motorized laser to calibrate the parallelism of the laser beam ad computerized the processig of receivig targets for attitudes calculatio [8 10]. These methods simplify the calibratio of laser beam, however, they rely o parallelism betwee laser ad aligmet to guide TBM, ad they are all affected by the survey errors i the as-surveyed aligmet. A improved method, the Virtual Laser Target Board (VLTB) [7,11], attempts to reduce the complexity i avigatig the costructio of 2.4~3 m diameter draiage tuels. The VLTB system utilizes three-poit positioig algorithms [12] to locate the cutter head from three visible prisms situated withi a visible widow at the rear ed of TBM, istead of parallelism of laser beam. VLTB still caot dodge the errors from survey data, ad this paper studies the error model based o threepoit positioig algorithm, ad provides a valuable approach to evaluate the accuracy of the cutter head positio estimatio. 2 Theory of Errors 2.1 True Values, Samples, ad Errors A surveyig device is supposed to retrieve true measuremet values; however, due to all ucotrollable maufacturig ad techological limits, the true value caot be foud. Therefore, to calibrate a surveyig device, maufacturers should establish a bechmark, utilizig some fier (more accurate) survey equipmet. There are two mai idicators of the so-called accuracy of survey device: accuracy ad precisio. As show i Figure 1: Figure 1 Shootig at the ceter of the cross: (from left to right) precise ot accurate, accurate ot precise, accurate ad precise, ad ot accurate ot precise The accuracy suggests the deviatio betwee the average of all measuremets ad the true value, ad precisio meas the cosistecy of all measuremets. Mathematically, for a set of measuremets X 1 X o a true value X, the accuracy ca be quatified by X 1++X X, ad precisio ca be quatified as stadard deviatio (X 1 X 1++X ) 2 ++(X X 1++X ) 2 1 [13]. A well-calibrated survey device should produce ubiased measuremets ( lim X 1++X X = 0), ad the precisio should be quatified, for istace expressed as stadard deviatio σ. Later i usage, the device should be able to measure ay target with a error, which ca be expressed as a ormal distributed variable with 0 as mea ad σ as stadard deviatio. I the followig sectios, all true values of the targets are ot kow ad errors, either from measuremet or error propagatio calculatio, are expressed as ormal distributed variables with 0 as mea ad a give stadard deviatio. 2.2 Error Propagatio I real world survey or other experimets, it is ot always practical to directly observe or measure the target, ad the uobservable poit will be deduced from a set of observable idicators with fuctio models ad costraits. For example, the bucket of the excavator is ot always observable durig costructio, ad by usig kiematics, the bucket positio ca be deduced from the kiematic chai model plus certai legth measuremets of hydraulic rods. However, legth measuremets cotai radom errors, which i tur will propagate the errors to the kiematic result, through a chai of liear ad o-liear computatios. Suppose a set of observatios X = [x 1 x ] T with a couterpart set of true values as X true = [x 1 true x true ] T. The true values caot be directly measured due to radom oise, ad the

3 radom oise fits to a ormal distributio(μ, σ), with μ as mea ad σ as stadard deviatio. I order to elimiate radom oise, repeat measuremets o variables for m times, ad the results are marked as X 1 X m. The expectatio of these variables are E(X) ad if m, the μ X = E(X) X^true. I additio, the covariace of all variables is: σ XX = C XX = σ x1 x 1 σ x1 x [ ]. σ x x 1 σ x x For a set of deduced variables Z = [z 1 z m] T, K 11 K 1 ad Z = KX + K 0 with K = [ ] ad K m1 K m K 0 0 = [k 1 k 0 ] T beig a liear expressio of coefficiet K ad observable variables X, ad K 0 are costats. The expectatio, variace-covariace matrix of Z after propagatio ca be expressed as [13]: E(Z) = E(KX + K 0 ) = KE(X) + K 0 = Kμ X + K 0 C ZZ = KC XX K T = Kσ XX K T As for oliear expressios, Z = [ f 1 (X 1,, X ) ], f m (X 1,, X ) with each f i (X 1,, X ) as a o-liear fuctio. I this case, the error aalysis is performed o local derivable segmets, ad all fuctios f i (X 1,, X ) eed to be 0 liearized at some give poit [x 1 x 0 ] T. Apply total derivative o each fuctio f i (X 1,, X ), the dz i = ( df i dx dx ( 1 df i dx dx, therefore the mea ad covariace of oliear fuctios at poit 0 [x 1 x 0 ] T ca be expressed as [13]: dz = K dx dz 1 dx 1 dz = [ ], dx = [ ] ad K dz m dx m = ( df 1 ( df 1 dx 1 dx ( df m [ dx 1 ( df m dx ] μ Z = E(Z) = [ f 1 (E(X 1 ),, E(X )) ] ad σ ZZ f m (E(X 1 ),, E(X )) = KC XX K T = Kσ XX K T 3 Aalytical Accuracy of TBM Navigatio 3.1 Tuel-borig Machie Three-poit Positioig Accuracy The three-poit positioig algorithm for TBM [12] is based o the similar idea as kiematics models, which calculate the coordiates of the target (e.g. the cutter head ceter) through limited measurable parameters. I a typical three-poit positioig problem, the affectig factors iclude: Prior covariace of referece poits (istalled o the completed tuel as tuel survey cotrol poits) Total statio o Stadard deviatio of agle measuremet error o Stadard deviatio of distace measuremet error o Stadard deviatio of prism-ceter locatig error o Parts per millio error i distace measuremet Three-poit rotatio ad trasformatio o Error due to rotatio/trasformatio strategy Figure 2 Three measurable poits ad cutter head Despite the multiple factors i the above list, the error aalysis ca be modelled through two steps: 1) determie the covariace matrix for poit survey; 2) determie the rotatio error due to poit survey error. The total statio surveys a target by electroic distace measuremet ad horizotal/vertical agle measuremet. As show i Figure 3, the slope distace betwee total statio ad target is d, ad vertical/horizotal agles are θ ad α separately. The X Q Y Q coordiate of the total statio is Q = [ ], the the Z Q coordiate of the target is P =

4 X Q + cos(θ) d cos(α) [ Y Q + cos(θ) d si(α) ]. All six parameters Z Q + si(θ) d X Q, Y Q, Z Q, d, θ, α cotai errors, the applyig total X Q X Q dp X dp X Y dx Q Y Q dx Q dα derivative to [ dy] = Z Q D Z Q. dp dz Z dp Z d d [ dx Q dα ] θ θ [ α ] [ α ] Assume the covariace of all six parameters X Q, Y Q, Z Q, d, θ, α is C XQ,Y Q,Z Q,d,θ,α, the the C XYZ = D C XQ,Y Q,Z Q,d,θ,α D T. Figure 3 Survey model of total statio I the secod step, the total statio measures the three visible poits P 0, P 1, P 2 with the covariace give above. Before the TBM is take udergroud, the relative positio betwee the cutter head ad the three visible poits is registered i a local coordiate system P 0 XYZ. The coordiate system is established by the followig steps (as show i Figure 2): Temporary axis: Z orm(p 1 P 0 ) Axis: X orm(p 2 P 0 ) Axis: Y Z X Axis: Z = X Y Traslatio: T = P 0 X T Rotatio matrix: R = [ Y T ] Z T The there should be P h = R(P h P 0 ) + T Whe TBM is take udergroud, the robotic total statio keeps trackig the three poits, ad surveys their curret coordiates P 0, P 1, P 2. By applyig the same coordiate-fixig strategy, the ew traslatio T ad rotatio R ca be calculated, ad also the curret ivisible cutter head coordiate is P h = R (P h P 0 ) + T. The coordiate of the cutter head is affected by ie parameters X P0, Y P0, Z P0, X P1, Y P1, Z P1, X P2, Y 2, Z P2, the the total derivative is D = dx P h dz P 2. The [ dz P 2 ] covariace of coordiate of the cutter head ca be preseted as C = [ dz P 2 dz P 2] [C XYZ C XYZ 4 Sesitivity Aalysis C XYZ ] [ dz P 2 dz P 2] I the real VLTB avigatio practice for tuel costructio, two kids of survey should be doe to make it complete. The shop survey aims to fid the layout of the three prisms fixed at the ed of the TBM ad the cutter head. Thus aother prism is mouted at the cutter head of the TBM. As the cutter head will ot be visible i the tuel durig tuel costructio, the shop survey has to be doe before the istallatio of the TBM. All of the positios of four prisms have to be measured by the total statio i this stage. The field survey is used to fid the directio of the TBM, or the positio of the cutter head i the field coordiate system. Differet from shop survey, i the field survey, the cutter head is ot measured directly by the total statio but through the trasformatio estimated from the three poits at the ed of the TBM. I real tuel applicatios, it is ot possible to measure the cutter head of the TBM, thus the validatio is doe i lab with the scaled TBM model with its cutter head ceter poit beig visible. The procedure used to evaluate the accuracy is made up of two stages i order to imitate the shop survey ad the field survey. I practice, compared to the shop survey, the eviromet of field survey ad the positio of the total statio i relatio to the TBM i the tuel chages sigificatly. To make the imitatio much more realistic, the positio of the total statio is moved to aother place after the first stage shop survey i lab. After the total statio is chaged to aother locatio, the total statio has to do the resectio to set up its ow locatio, just like the case of field survey. The chage of locatio of the total statio makes the validatio much more realistic, while also itroducig additioal error caused by the chage of eviromet ad the chage of the relative positio cofiguratio of the total statio ad the prisms. These chages will affect the accuracy of prism detectio ad targetig ad the accuracy of the total statio itself ad the measuremets. The validatio procedure is much differet from the case whe the same coordiate system is used without movig the total statio. T

5 Therefore the system is expected to result i lower accuracy. However the validatio is much more reliable ad realistic. I the experimet, two cycles are coducted i each stage ad all of the 4 prisms are measured i each cycle; the coordiates of the three prisms at the back ed are used for computig the cutter head ceter while the surveyed coordiate of the cutter head ceter is used for cross-checkig purpose. First, a survey similar to the shop survey for registratio of critical poits o TBM is doe to determie the layout of the prisms. For coveiece, it will be referred as shop survey. I the lab setup, the local field coordiate system is defied with a sigle large prism. The approach is differet from the practical case where resectio with two kow cotrol poits is required to determie the locatio ad orietatio of the total statio i the local field coordiate system, however the differece will be oly reflected as a mior traslatio which does ot affect the applicatio. The geometry layout of the prisms is give i Figure 4. The large prism is used to set the orietatio (Northig or Y i the local coordiate system) which ca be replaced with ay other target. P1, P2, ad P3 are three prisms at the ed of the TBM (model) ad P4 is the prism at the cutter head of the TBM model which is oly visible at the mechaical shop for a real TBM. The statio poit is fixed at coordiates (0.0456, , ) i (East, Northig, Elevatio) system for all of the cycles of survey. Figure 4 Geometry layout of the prisms To evaluate the accuracy of the algorithm, measuremets of the 4 prisms at 5 differet statios are used to simulate the shop survey ad field survey alteratively. The data is preseted i Table 1. The data collected from differet statios were used as shop survey ad field survey alteratively to mimic the real applicatio. After the trasformatio is determied, the coordiates of P4 at the shop survey statio is trasformed to the field coordiate system. The deviatio of the coordiates betwee the trasformed P4 from the shop survey statio ad the measured P4 at the field survey statio were used to evaluate the accuracy of the trasform algorithm. The experimet ca be divided ito two categories. Oe category uses sigle statio data to simulate the process, i.e. the data from oe statio but differet cycles will be used to simulate the shop survey ad the field survey. I this case, cycle 1 of each statio was used as shop survey ad cycle 2 was treated as field survey. The other category which is the focus of this study follows the procedure stated i previous sectios. This category uses data i differet statios to simulate the shop survey ad the field survey ad thus is much more realistic. I our experimet, the cycle 1 of each statio was used as shop survey ad field survey alteratively to geerate multiple cases. Cosiderig the procedure of calculatio has bee give i previous Table 1 Coordiates of P1~P4 i Five Differet Total Statio Setup Uit Statio 1 Statio 2 Statio 3 Statio 4 Statio 5 (meter) X Y Z X Y Z X Y Z X Y Z X Y Z P Cycle 1 Cycle 2 P P P P P P P

6 Sigle Statio Vary Statios sectios, oly the result will be show i this sectio. Statio Type Table 2 Error Evaluatio Triad Error (mm) Quaterio Error (mm) Shop Field X Y Z X Y Z The result shows that the quaterio method does ot improve the result very much because o costraits are available to restrict the solutio to the real solutio if oly three prism are used to estimate the trasformatio. Aother fidig is that the error ca be sigificat eve the total statio is ot moved accordig to the result whe cycle 1 of statio 1 was take as shop survey ad cycle 2 of statio 1 as field survey. I this case, the mere differece is the mior differece ( 1 mm ) of P2, however, the result produces quite a large error. To verify this problem, the sesitivity of the algorithm due to the measuremet error of each coordiate of the poits is evaluated through a aalytical approach which calculates the distace betwee the estimated cutter head ad the measured oe with varyig error o oe coordiate of P2. I the experimet, the data collected i cycle 1 at statio 1 is used as the shop survey. The field survey data is the cycle 1 data collected at statio 1 as well but with a varyig mior error added o to the X, Y ad Z coordiates separately. The deviatio of the cutter head vs the error i the coordiates of the surveyed poits is give i Figure 5 to Figure 13. The gradiet of the curve at 0 will be the partial derivative of the Y value correspodig to the X value which ca be foud i the Jacobia matrix derived i previous sectios. Figure 5 Deviatio vs error of X coordiate of P1 Figure 6 Deviatio vs error of Y coordiate of P1 Figure 7 Deviatio vs error of Z coordiate of P1

7 Figure 8 Deviatio vs error of X coordiate of P2 Figure 11 Deviatio vs error of X coordiate of P3 Figure 9 Deviatio vs error of Y coordiate of P2 Figure 12 Deviatio vs error of Y coordiate of P3 Figure 10 Deviatio vs error of Z coordiate of P2 Figure 13 Deviatio vs error of Z coordiate of P3 Based o results, it is observed that the measuremet error of the Y coordiate of P2 has the largest ifluece o the result uder the geometry layout of the TBM model. A 1 mm error o the Y coordiate of P2 ca result i as large as 18 mm error i the positio error of the estimated cutter head. The error i Y directio of P1

8 ad P3 ca reach as high as 10 mm error as well. To further aalyse the error, it is split i X, Y ad Z directios separately, the result with measuremet error i Y coordiate of P2 is give i the followig figures. Figure 14 Deviatio of cutter head o X directio vs error of Z coordiate of P3 directio of the as-desig tuel aligmet. Usually a aligmet tolerace will be set for the tuel depedig o the applicatio. To esure the quality of the tuel costructio ad aalyse the sesitivity of the algorithm i term of aligmet error, the positioig error is projected o to the pla by assumig that the curret directio where the TBM is headig to is exactly the directio of the as-built desig. Based o the assumptio, the aligmet error will be: e (P 4 C) Where C is the rear ceter of the TBM which ca be derived from P1, P2, ad P3; e is the deviatio of the estimated cutter head compared to the measured cutter head. The aligmet error correspodig to errors o Y directio of P2 is give i Figure 17. From the figure it ca be cocluded that the error i Y coordiates of P2 would result i a large error of the cutter head ad most of the error will cotribute to the aligmet error which is critical to tuel costructio. Figure 15 Deviatio of cutter head o Y directio vs error of Z coordiate of P3 Figure 16 Deviatio of cutter head o Z directio vs error of Z coordiate of P3 However, the error alog the directio of the asdesig tuel should be igored because what is critical is the aligmet error o the plae perpedicular to the Figure 17 Aligmet error vs error of Y coordiate of P2 5 Coclusios The paper discusses the system-wide error sources ad propagatio model of three-poit positioig algorithm for TBM guidace. To better simulate the differet eviromets of shop registratio ad site survey i real world case, the validatio test relocates the total statio after iitial coordiate registratio, ad calculates the error from the ew locatio. The result shows that, give three kow prisms, the quaterio method will ot have better result tha the triad method, as there is o redudat survey to adjust errors. I the sesitivity aalysis, due to the short distaces betwee prisms, eve 1 mm error o Y axis would result i 18 mm error i cutter-head positio calculatio. Although this error is large ad reflects the difficulty to estimate the cutter-head from a small survey widow, the error alog the directio of the as desig tuel should be igored because what really matters is the aligmet error o the plae perpedicular to the

9 directio of the as desig tuel aligmet. Moreover, the error is idepedet betwee every two surveys, ad the error will ot accumulate. This meas the error i deviatio measuremet is well cotrolled ad the deviatio measuremet is reliable. I the future, the authors will optimize the geometry of the setup of the prisms, ad itroduce a fourth prism as a survey redudat to further improve the accuracy for cutter head estimatio. Refereces [1] Mao S ad She X ad Lu M ad Wu X. Realtime Tablet-based Virtual Reality Implemetatio to Facilitate Tuel Borig Machie Steerig Cotrol i Tuel Costructio. Proceedigs of 13th Iteratioal Coferece O Costructio Applicatios of Virtual Reality, Lodo, Uited Kigdom, [2] Stiros SC. Aligmet ad Breakthrough Errors i Tuelig. Tuellig ad Udergroud Space Techology. (24) , [3] Wu X ad Lu M ad Mao S ad She X. Realtime As-built Tuel Product Modelig ad Visualizatio by Trackig Tuel Borig Machies. The Proceedigs of 30th Iteratioal Symposium of Automatio ad Robotics i Costructio ad Miig (ISARC 2013) Proceedigs, Motreal, Caada, [4] The City of Edmoto. The City of Edmoto - Desig & Costructio Stadards - Volume 3: Draiage [5] Chrzaowski A. Optimizatio of The Breakthrough Accuracy i Tuelig Surveys. The Caadia Surveyor. (35). 5 16, [6] Kavaagh BF. Surveyig with Costructio Applicatios. Upper Saddle River, N.J., Pretice Hall, [7] She X ad Lu M. Developmet of Virtual Laser Target Board for Tuel Borig Machie Guidace Cotrol. Computig i Civil Egieerig (2012), America Society of Civil Egieers, 2012, pages [8] Jardó A ad Victores JG ad Martíez S ad Balaguer C. Experiece Acquisitio Simulator for Operatig Microtuelig Borig Machies. Automatio i Costructio. (23) , [9] tacs GmbH. Tacs Acs +++ Summary of The Acs Guidace System ew.htm (accessed May 2, 2013). [10] VMT GmbH. SLS-T APD Tuel Guidace System [11] Mao S ad She X ad Lu M. Virtual Laser Target Board for Aligmet Cotrol ad Machie Guidace i Tuel-Borig Operatios. Joural of Itelliget & Robotic Systems. 1 16, [12] She X ad Lu M ad Che W. Computig Three-Axis Orietatios of A Tuel-Borig Machie Through Surveyig Observatio Poits. Joural of Computig i Civil Egieerig. (25) , [13] Mikhail EM ad Ackerma FE. Observatios ad Least Squares, with Cotributios by F. Ackerma. New York, IEP, 1976.