articulated arm coordinate measuring machine calibration by laser tracker multilateration

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Research ArticleArticulated Arm Coordinate Measuring Machine Calibration byLaser Tracker Multilateration

Jorge Santolaria, Ana C. Majarena, David Samper, Agustn Brau, and Jess Velzquez

Departamento de IngenieradeDiseno y Fabricacion, Edicio orres Quevedo, EINA, Universidad de Zaragoza, Zaragoza, Spain

Correspondence should be addressed to Jorge Santolaria; [email protected]

Received August ; Accepted November ; Published January Academic Editors: G. Huang and D. Veeger

Copyright Jorge Santolaria et al. Tis is an open access article distributed under the Creative Commons Attribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A new procedure or the calibration o an articulated arm coordinate measuring machine (AACMM) is presented in this paper.First, a sel-calibration algorithm o our laser trackers (Ls) is developed. Te spatial localization o a retroreector target, placedin different positions within the workspace, is determined by means o a geometric multilateration system constructed rom theour Ls. Next, a nonlinear optimization algorithm or the identication procedure o the AACMM is explained. An objectiveunction based on Euclidean distances and standard deviations is developed. Tis unction is obtained rom the captured nominaldata (given by the Ls used as a gauge instrument) and the data obtained by the AACMM and compares the measured andcalculated coordinates o the target to obtain the identied model parameters that minimize this difference. Finally, results showthat the procedure presented, using the measurements o the Ls as a gauge instrument, is very effective by improving the AACMM

precision.

1. Introduction

In recent years, there has been an increasing interest inAACMMs because o their advantages in terms o accuracy,portability and suitability or inspection and quality controltasks in machining tool processes and in the automotive andaerospace industry [].

Nevertheless, ew researches have ocused on the cali-bration o these mechanisms. Moreover, there is an absenceo standards on verication and calibration procedures. For

that reason, AACMM manuacturers have developed its ownevaluation procedures. Tese evaluation methods are basedon the three main standards or perormance evaluation incurrent CMMs, UNE-EN ISO , ASME B.., andVDI/VDE and are still carried out today to compare andevaluate the accuracy o an arm rom the point o view othe CMMs. In [], the author presented a procedure to checkthe perormanceo coordinate measuring arms by calculatingthe distances between the centers o different spheres. Teresults obtained were compared with the application o theANSI/ASME B volumetric perormance test showing goodagreement between the two approaches and a cost reduction.In [], Shimojima et al. presented a new method to estimate

the uncertainty o a measuring arm using a tridimensionalgauge. Tis method consists o a at plate with spheres xedat three different heights with respect to the metallic suraceo the plate. Ten the spheres centers are measured withthe measuring arm at different locations and orientations,and distances between spheres centers are compared to thenominal distances to evaluate the measuring perormance othe arm. Other works have been ound in the literature whosemain goal is also to evaluate the perormance o measuringarms [].

However, the AACMM presents different characteristics,and different verication procedures are thereore required.A point clearly denes a position o the three machineaxes or a CMM. Nevertheless, the possible positions o theAACMM elements to achieve a xed point dened in themeasurement volume are practically innite. Moreover, orCMMs, evaluation tests can be perormed to extract thepositioning errors, allowing correction models to be imple-mented []. Tus, a high level o maintenance o the physical-mathematical relations between the error model parametersand the error physically committed by the machine can beachieved. However, the application o these models does notmake sense in AACMMs, given the difficulty o directly

Hindawi Publishing Corporatione Scientific World JournalVolume 2014, Article ID 681853, 11 pageshttp://dx.doi.org/10.1155/2014/681853
http://dx.doi.org/10.1155/2014/681853http://dx.doi.org/10.1155/2014/681853



Te Scientic World Journal

relating the error committed with the model parameters,which are obtained by using optimization procedures.

Te calibration procedure consists o identiying the geo-metric parameters in order to improve the model accuracy.Tis procedure allows us to obtain correction models toestablish corrections in the measurements results and to

quantiy the effects o the inuence variables in the nalmeasurement. o achieve thisgoal, the ollowing vestepsareusually carried out: determination o the kinematic model bymeans o nonlinear equations, data acquisition, optimizationor geometric parameteridentication,model evaluation, and,nally, identication o the error sources and implementationo correction models.

Te rst step, determination o the kinematic model, con-sists o obtaining the non-linear equationsthat relate the joint

variables to the position and orientation o the end-effectorand the initial values o nominal geometric parameters. Oneo the most widely used geometric methods or modelling amechanism is the well-known Denavit-Hartenberg method

[], which models the joints with our parameters. One o thelimitations o this method appears in those mechanisms thatpresent two consecutive parallel joint axes. In this case, aninnite number o common normals o the same length exist,and the location o the axis coordinate system may be denedarbitrarily. Some studies [] present methods to obtain acomplete, equivalent, and proportional model.

Te number o parameters is xed when the kinematicmodel is determined, and this value will depend on theselected method. Moreover, there are a maximum number oparameters that must be identied, and the model accuracydoes not have to improve by adding extra parameters [ ].In this work, the author determined that our parametersmust be considered or each revolute joint, two o which mustbe orientational, and or a prismatic joint two orientationalparameters are necessary, applied about the noncollinearaxes beore and perpendicular to the translational joint axes.Nongeometric errors are usually compensated by addingparameters in the geometric model [].

Te second step is data acquisition. Any measurementerror o the external instrument is propagated to the resultso the identied parameters. For that reason, an instrumentor data acquisition should be, at least, one magnitude ordermore accurate than the mechanism whose parameters aregoing to be identied.

A direct geometric transormation can be established,providing the global reerence system can be measured bymeans o an external measurement instrument such as alaser tracker or a coordinate measuring machine. In this way,this transormation obtains the coordinates o the measuredpoints in the global reerence system o the mechanism.Tus, direct comparisons in the objective unction, betweennominal and measured data, can be made.

However, this relation is not usually easy to obtainthrough a direct measure, and the problem is usually solvedby means o least-square methods. Tese methods allow usto obtain an approximation o this transormation, and thisapproximationwill dependon the congurations used in dataacquisition and on the mechanism error in the evaluatedpoints [].

Te home position is a position, within the AACMMworking range, where all joint angles have a predened

value. Te displacements o the probe are usually measuredwith respect to this dened position. In [], Kovac andFrank developed a high precision gauge instrument or theparameter identication procedure and evaluation tests.

Te determination o the number o required specicpositions is not generalizable rom one AACMM to another,since the errors committed by each arm will depend on theirconguration and assembly deects. Te perormance o teststhat characterize the inuence o each joint on the nal error,to nally choose positions in accordance with this inuence,must thereore be carried out beore dening the capturepositions to identiy the kinematic parameters. Te positionsselected or the identication procedure should cover themaximum joint rotation range to cover the inuences o allthe measuring arm elements in the workspace. In [], Zhenget al. obtained the spatial error distribution model by usingsupport vector machine theory.

Te third step is optimization or geometric parameteridentication, and the objective is to search or the optimumvalues o all parameters included in the model that minimizesthe error in the perormed measurements. Tis step is usuallycarried out by means o approximation procedures basedon least-square tting. Tis unction can be dened as thequadratic difference o the error (obtained between the mea-sured value andthe value computed by the kinematic model).Te increment established or parameters must be dened oreach iteration. In most o the cases, numerical optimizationtechniques are used to minimize the error. Te Levenberg-Marquardt (L-M) [] method is one o the most widely usedtechniques to solve the numerical optimization algorithm.Tis method usually presents lower computational cost,providing a solution closer to the optimum solution or theset o parameters considered. Moreover, the L-M algorithmsolves numerical problems that appear in other numericaloptimization techniques such as those based on the gradientor on the least-square methods such as the Gauss-Newton.A multiobjective optimization scheme is developed in [] tosolve the nonlinear optimization problem. In [], Santolariaet al. presented a kinematic parameter estimation technique,which allows us to improve the repeatability o the AACMMby more than %. Tis technique uses a ball bar gauge toperorm the data acquisition procedure.

Te ourth step, the model evaluation, consists o evalu-ating the mechanism behavior with the set o the identiedparameters obtained in the geometric parameter identica-tion procedure, in congurations different rom those usedin the optimization process. In [], Koseki et al. evaluatedthe accuracy o a mechanism by means o a laser trackingcoordinate measuring system.

Finally, an identication o the error sources and amodelling and implementation o the correction models canbe optionally perormed.

In [], Piratelli-Filho et al. developed a virtual sphereplate, having a standard deviation o around . mm anda measurement uncertainty rom . to . m in pointmeasurements, to evaluate the measurement perormance oAACMMs. Te AACMM perormance test using the virtual




sphere plate resulted in a mean error o . mm and astandard deviation o . mm.

A laser tracker is a large-scale measuring instrument withhigh accuracy. Tese mechanismsare considered very reliable[]. Consequently, laser trackers have been used recentlyinstead o other traditional methods such as theodolites or

collimators in multiple applications such as robot tracking,testing, calibration, and maintenance. Tese systems useintererometry or measuring relative distances and opticalencoders in order to measure azimuth and elevation angleso a beam-steering mirror. Te intererometer measurementsare obtained relative to the starting point. Besides, this beammust track the positions o a retroreector. A plane mirrormounted on a high precision universal joint deects thebeam and hits the retroreector. Tis element consists othree perpendicularly orientedplane mirrors, andthe beam isreected parallel. Teoretically, the laser beam hits the centerpoint o the retroreector. When there is no relative move-ment between the laser tracker head and the retroreector,

there is no parallel displacement between the emitted andthe reected beam. However, when the retroreector startsmoving, there is a displacement o the reected laser beam,since, in this case, the laser beam does not hit the centerpoint o the retroreector. In [], Lin and Her used a lasertracker system to measure the volumetricerrors o a precisionmachine. Te technique presented is based on the ASMEB. standards and offers a quick method to characterizethese errors. Tis measuring instrument can also be used inmultilateration or verication o machine tools, as presentedin [], where the error order can also be in the error range oa laser tracker. In [], a laser tracker was used also to analysethe perormance o an indoor GPS.

Te multilateration techniqueis a procedure which canbeused to improve themechanism accuracy. Tismethodallowsus to reduce the measurement uncertainty by eliminating theangular noise. o achieve this, the multilateration obtainsthe hal point position starting rom data o several lasertrackers, located in different positions []. In [], Zhanget al. presented specic recommendations or optimizationo multilateration set-ups and measurement plans and orminimizing measurement uncertainty. Besides, the authorsevaluated the volumetric measurement error propagation,obtaining an average standard deviation o length mea-surements around .m. Hughes et al. [] presented alaser-intererometric measuring station and obtained thedisplacement measurement uncertainty, which was used topredict a volumetric uncertainty or a multilateration system,consisting o eight measuring stations and our targets. Tecombined uncertainty o the measuring station displacementmeasurements and the CMM repeatability obtained wasaround nm. In [], Kim et al. developed a volumetricintererometer system and minimized least-square errorsby tting the measured values to a geometric model omultilateration, obtaining a volumetric uncertainty o lessthan m.

Te aim o this work is to improve the accuracy o anAACMM by means o a new calibration procedure basedon laser tracker multilateration. Although both instrumentspresent the same order o magnitude with respect to the

accuracy, the multilateration techniques allow us to capturepoints with an uncertainty much smaller than the oneobtained with an AACMM. Te points captured rom themultilateration are thereore considered as nominal data inthe calibration procedure o the AACMM.

2. AACMM Kinematic Model

Te AACMM kinematics relates the joint variables and theprobe position or any arm posture.

Te direct kinematic model is used to calculate thepositioning and orientation o the AACMM probe on thebasis o certain values o the joint variables, according to theollowing:

= , , ()with = 1, . . . , or an arm withrotating joints.is givenby the vector o the joint variables and represents the modeldened, which depends on the parameter vector.Te non-linear equation system to model the mechanismcan be developed by applying the Denavit-Hartenberg (D-H)method [] toevery chain in the mechanism.Tis method hasbeen widely used in mechanism modelling [,] and usesour parameters (distances,, and angles,) to modelthe coordinate transormation between successive reerencesystems. Te homogenous transormation matrix betweenrameand 1depends on these our parameters:

1 = , , , ,

=

cos cossin sinsin cos sin coscos sincos sin 0 sin cos 0 0 0 1.

()

Te AACMM model used in this work is a Faro PlatinumArm having seven axes, with a nominal value o2 in thesingle point articulation and perormance test o . mmaccording to the specications o the manuacturer.

Figure shows the reerence systems used in theAACMMmodel.

Te armglobaltransormationmatrix allowsus to expressthe probe sphere center coordinates with respect to the baseo the AACMM. Tis matrix can be obtained by calculatingsuccessive coordinate transormations by premultiplying thetransormation matrix between a rame andthe previous one,as shown in the ollowing:

,,,1AACMM= 06 ,,,1Probe. ()In this equation,0is the global reerence system o the

base and corresponds to the reerence rame that moveswith the rotation o the last joint.

A reerence system is usually dened in the probe.However, the aim o this study is to obtain the sphere center




X0Y0

Z0

X4

Y4Z4

X5

Y5Z5

X6Y6

Z6

X7

Y7

Z7

X3

Y3Z3

X2

Y2

Z2

X1Y1

Z1

F : AACMM reerence systems.

coordinates o the static probe, so this reerence system is

not necessary. Seven reerence systems are used to model theAACMM. Te last reerence system is located in the centero the reector and is oriented as reerence system six. Tus,the number o parameters or the degrees o reedom (Do)AACMM is . Te kinematic model is ully described in[].able shows the initial values or the AACMM D-Hparameters.

3. Multilateration System

Te multilateration technique allows us to reduce the mea-surement uncertainty by eliminating the angular noise. oachieve this, the multilateration obtains the weighted-point

position by means o several laser tracker data, located indifferent positions. A minimum o three measurements oeach point is necessary. Each laser tracker measures thedistance rom the laser tracker to the target points. Tesemeasurementspresent a noise having a radial component andtwo angular components. Te aim o the multilateration is todecrease the measurement uncertainty, so this technique onlyuses the radial component o the laser tracker measurements,, thus decreasing the measurement noise inuence, whichallows us to decrease the global uncertainty. Tese compo-nents dene a sphere. Te intersection o the three spheresobtained by measuring the same point by three laser trackersprovides two points. Equations rom () to () provide the

spheres obtained by the measurements o each laser tracker,respectively. Consider

20= 12 + 12 + 12, ()21= 22 + 22 + 22, ()22= 32 + 32 + 32. ()

Te knowledge o the locations o the reerence systems othe laser trackers allows us to obtain the ollowing equations:

=20 21+ 22

21

, ()

Reflector (X,Y, Z)

m3

P3(X3, Y3, Z3)

m0m2

m1

P2(X2, Y2, 0)

P0(0, 0, 0)

P1(X1,0,0)

F : Coordinates o the our laser tracker reerence systemsexpressed in the multilateration system.

=20 22+ 22+ 22 22

22 , () = 20 2 21/2. ()

In thiswork, orsimplicity, the unknown reerencesystemlocation o the three laser trackers has been dened as(0,0,0),(1,0,0), and(2, 2, 0)with respect to the multilat-eration reerence system.

A ourth laser tracker can be used to avoid the signambiguity in thecoordinate obtained in (). Te reerencesystem o this laser tracker is given by(3, 3, 3). Tis pointshould belong to a plane different rom the planeormedby the other three laser tracker reerence systems as shown inFigure . In this gure,represents Lposition system orthe laser trackers. Te origin o the multilateration globalreerence system is given by0.

Te(,, ) target coordinates are obtained in linearmatrix orm by operating (), (), and (), as expressed in theollowing:

= 0,5

11 0 0

21212 0

313 + 23123

32313

21 20 21

22

20

22

22

23 20 23 23 23 .

()

Te multilateralized coordinates, obtained in (), will beconsidered as the nominal coordinates in the identicationparameter procedure.

4. Data Acquisition

Te data acquisition step consists o capturing the nominalcoordinates in the workspace o the AACMM. Te suitablenumber o positions is not generalizable rom one measuring




: Initial values or the AACMM D-H parameters.

Joint () () (mm) (mm) .

LT4: FARO ION

LT2: API T3

LT1: FAROX

LT3: LEICA LT-600

Retroreflector

AACMM

F : Ls and AACMM distribution used in tests.

arm to another, since each measuring arm error will dependon their conguration and assembly deects. Te identica-tion procedure should cover the maximum range o jointrotation to consider all the inuences o the measuring armelements.

In this study, a cloud o points located within thearm workspace was measured simultaneously with both theAACMM (measured values) and our Ls which conormto the multilateralized system (nominal values). Te lasertracker FARO model was used as L1, API as L2,LEICA L- as L3, and FARO ION as L4. Te positionswere distributed throughout the workspace o the arm andreached different arm angle values. Te Ls were distributedorming the multilateration global coordinate system, as itwas detailed inSection . L positions have been chosen inunction o the reector visibility, thus orming a spatial angleas near as possible to between them. Te measurementuncertainty is lower in this position, as demonstrated in aprevious work []. Te AACMM has been arranged in aposition that maximizes the visibility o the Ls, as shown inFigure .

Te AACMM data acquisition technique is usually per-ormed by means o discrete contact probing o surace pointso the gauge in order to obtain the center o the spheresrom several surace measurements. Te time required orthe capture o positions is high, and thereore, identication

120

Xcenter ,Ycenter, Zcenter

LProbe

45

120

F : Probe used in the data acquisition procedure.

is generally carried out with a relatively low number o armpositions. In this work, a probe presented in [], capableo directly probing the center o the spheres o the gaugewithout having to probe surace points, has been used. Tisprobe consists o three tungsten carbide spheres o mm indiameter, laid out at on the probe, as can be seen inFigure .

Te probe used allowsus todene a probe with zeroprobesphere radius and with a distance rom the position o thehousing to the center o the probed reector sphere o .inches, allowing direct probing o the sphere center when thethree spheres o the probe and the sphere are in contact.

One o the advantages o this type o probe is thatthe massive capture o arm positions can be perormedcorresponding to several points o theworkspace, which leadsto save a considerably amount o time.

positions o the retroreector were measured, romwhich positions were considered in the parameter iden-tication process (identication positions) and the other positions were kept or the parameter evaluation procedure(test positions). A sofware developed captured the AACMMmeasurements, saving the AACMM joint angles,. Teseangles and the AACMM parameters are the input to thekinematic model. Te solution o the non-linear equations bythe L-M algorithm obtains the measured point coordinates inthe AACMM reerence system.

Te data acquisition procedure was perormed trying tocapture data in symmetrical trajectories in the retroreectorto minimize the effect o probing orce on the gauge.

Although the measuring o the retroreector center withthe kinematic mount probe rom different arm orientationsshould result in the same point measured, the unsuitable

value o the nominal kinematic parameters o the model willbe shown by way o a probing error, resulting in differentcoordinates or the same measured point in different armorientations. For that reason, ve measurements were takenor eachpoint and the meanpoint othe set o pointscapturedwas considered as the center o the retroreector measured inthe AACMM reerence system, as shown inFigure .

Te distance between the different retroreector mea-sured positions can be obtained rom the points capturedby means o the Euclidean distance between each pair oretroreector positions, obtainingAACMM, where andrepresent two measured positions.

At the same time, the our Ls simultaneously measuredthe distance rom the captured point to the local coordinatesystem.




Xc,Yc,Zc =

Xi , Yi, Zi

Xi , Yi, Zi

F : Retroreector center.

Diagonal distances are obtained according to the ollow-ing:

= + , ()

where is the incremental output o the displacementtransducer used and is the offset, which should be calibratedbeore the multilateration implementation.

Te sel-calibration and determination o the parametercan be carried out by capturing some additional objectivemeasurements in some different positions. By perormingthe quadrilaterationobjective point times, the number oequations is given by4as shown in the ollowing:

2 = + 2

=

2

+

2

+

2

,

()

or = 0, . . . ,3 and = 0, . . . , 1.Te system unknowns are the our offsets,, and the

position o the coordinates,,, (3 unknowns). oidentiy the our reerences, unknowns, correspondingto,,, must be added. Tus, the number o systemunknowns is given by3 + 16.

Te unknowns can be obtained by the least-squaremethod to minimize the error.

Te multilateration technique must solve the objectiveunction dened by

= 21

=14

=1 L

2

+ L

2

+ L

2

+ L2 ,()

where(, , ) represents the measured point coordinatesin the multilateration reerence system,(L , L , L) isthe origin o the L reerence system,are the measureddistances or every point in the L reerence system, and Lrepresents the L offset. Te non-linear system obtained bythe multilateration technique was solved by means o the L-M algorithm, and the solution obtained gives the ollowinginormation:

(i) the point coordinates in the multilateration system:(, , )|SR Multi,(ii) the laser tracker offsets:L ,

(iii) the origin o the reerence systems or the our lasertrackers in the multilateration system:(L , L ,L)|SR Multi.Te calculated distances between each pair o retroreec-

tor positions can be obtained rom the point coordinates inthe multilateration system, obtainingMulti, whereandrepresent the two positions.

5. Parameter Identification Procedure

Once the data acquisition technique has been carried out, theparameter identication procedure can be perormed.

Figure shows a scheme o the calibration procedure.Te origin o the laser trackers was measured with respect

to the reerence system o L1. Tis reerence system has beenconsidered the multilateration reerence system or obtainingthe initial values o the Lreerence systems.

Te kinematic parameter identication procedure can beperormed starting rom the measured and calculated dis-tances (obtained as explained inSection ). Te parametersconsidered in the AACMM kinematic model are given by(, , , 0, Enc), whereEnc are the angles measured bythe encoder. Te model obtained is a non-linear equationsystem, and it was solved by the L-M algorithm. Te objectiveunction denedin this step considersboth the measuredandcalculated distances, as shown in

=

=1,=1 [AACMM Multi2

] +

=1 2 + 2 + 2 ,()

where represents the number o positions consideredin the parameter identication procedure and( , ,) represents the standard deviation o the points mea-sured in each position and each coordinate, showing theinuence o the volumetric accuracy and point repeatability.

6. Calibration Results

In the optimization process, the distances rom each point toevery point are taken into account, obtaining distancesbetween the points.

Te multilateration procedure described above was car-ried out based on the captured points or each Lcorresponding to the reector positions used. able showsthe results obtained. As the initial values or this procedure,null values were assigned or the offsets o all Ls, and or theorigin points the corresponding coordinates were those thatbetter t the distribution observed taken rom the measuredhome points o L2, L3,andL4rom L1, where the origino the multilaterized reerence system is located.

InFigure , the coordinates o the captured points byeach L and the coordinates o the multilaterated points,




Measuring each point with the

AACMM coordinates

Direct model(D-H)

End

End

Home

Initial values

Multilateration

Measuring each point with all the LT

AACMM coordinates

Parameter identication

mij

(xLTj, yLTj, zLTj)SR LT1

(xLTj, yLTj, zLTj)SR multil.

ILTj

ILTj

initial

AACMM:i(21 positions, 5 measurements

for each position)

(xi , yi, zi)SR multil. (xi, yi , zi)SR AACMM

dij multil. dijAACMMFor iter = 1 to (Ferror < tolerance)

Fori, j = pos ini to pos nal

=21

i=1,j=1[(dijAACMM dijmultil)

2 +ni=1

[2x + 2y

+ 2z ]

Optimisation: to minimizeFerror

Calculation of the geometric parameters:Si

values

=21i=1

4

j=1[(xi xLTj)

2 + (yi yLTj)2 + (zi zLTj)

2 (m +ij lLTj)2]

Ferror = [E1. . . Epos final]

F : Scheme o the calibration procedure.

used as nominal coordinates in the parameter identicationprocedure o the AACMM, are shown graphically.

As stated beore, in order to introduce redundancy inthe objective unction and thereby restrict to the nominalpoints thenal pointsobtained with the identiedparameterso the measuring arm, all possible distances between themultilaterated points are calculated. Tis way, we can obtain distances, which will be used as nominal data in theobjective unction () in the identication procedure. In

Figure , the range o each o the calculated distances in themeasurements made by the our Ls can be observed. Tus,each data represents the difference between the maximumand minimum values o each considered distance calculatedrom the set o the Ls measured data. Te distances arearranged starting rom position o the retroreector. Temaximum range value obtained by calculating the distanceswith the Ls was m, while the mean range or all set odistances was m.

Following the scheme presented in Figure , we cancalculate all possible distances between spheres as well asthe standard deviations measured with the AACMM romthe saved angular data obtained during the data capture

process and the set o initial parameters o the AACMMmathematical model shown in able . In accordance withthe optimization or the identication scheme presented inFigure , the quality indicators orthe set o initialparameterso the AACMM model are shown in able . Moreover,able shows in itsrst column thedistances maximum errorobtained orall the reector points andthe index o the pointsthat determine the distance in which the maximum error iscalculated.

Analogously, the mean distance errors or all the evalu-ated distances are also shown. With respect to the standarddeviation,able shows, besides the maximum and mean

values, the index o the point and its corresponding coordi-nate where the greater value is obtained, since in this case theobjective unction considers the standard deviation or eachcoordinate independently. As expected, the obtained valuesare high, considering the initial set o parameters dened orthe AACMM mathematical model.

In the objective unction proposed in (), the datacapture setup described or = 21 reector points, it isnecessary to consider the elimination o the terms where = to avoid both the inclusion o null terms and duplicate o


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: Multilaterated offsets and origin coordinates obtained in multilateration reerence system. iterations and objective unctionvalue below m.

Offsets (mm) Origin coordinates (mm)

L L LL L1 . L

L . .

L L . . . L L . . . .

: Quality indicators or the initial values o model parameters over SMR locations ( AACMM positions).

Distance error (mm) by retroreector point (mm)Maximum . Maximum .

Causing dist. Causing point

Medium . Causing coord. X

Medium .

: Identied values or the AACMM model parameters by L-M algorithm.

Joint () () (mm) (mm) . . . . . . . . . . . . . . . . . . . . . . .

. . .

distance errors noting that= . In regard to the standarddeviation, each coordinate deviation or every one o thereector positions is considered. o express mathematicallythe optimization problem it is required to consider the sumo the calculated quadratic errors. Tis way, through theobjective unction in (), we obtain terms correspondingto the distance errors or the reector points, plus standard deviation terms o these points, obtaining a totalo terms to determine the objective unction value ineach iteration o the optimization algorithm. Tis value willcontain the inuence o the kinematic parameters and thearticulation variables considering or this the terms relatedon the one hand to volumetric precision and on the other torepeatability, or the measuring arm captured positions.InFigure , it can be observed graphically the distribution othe captured points in the AACMM reerence system, beoreand afer the optimization procedure, while able showsthe identied parameters starting rom the initial values oable .

From the identied parameters (able ),able showsthe error characteristics results obtained or each o thecaptured points considering in this case these parameters.

Te validation and generalization o the error results cal-culated or the set o identied parameters over the captureddata, to the rest o the AACMM work volume, imply in themore restrictive case obtaining error anddeviation values less

than the maximum values obtained in this case (able ) orany evaluated position o the measuring arm. For this reason,the assessment o the AACMM error in different positionsto the ones used in the identication procedure is highlyrecommended. As shown in able , a maximum error om and a mean error o m or the measuring volumehave been obtained, considering the nominal points usedin the identication. In normal operation o the measuringarm in this work volume, it is expected to get error valuesclose to the mean value and obtaining maximum error valuesonly in certain arm congurations.

As a last step o the identication procedure, it isnecessary to evaluate the set o parameters obtained indifferent arm positions rom those considered in its ownidentication procedure, such that it is possible to concludethat the error results can be considered reliably within themeasuring arm work volume. Furthermore, it is expectedthat the more similar the measuring arm positions are,when probing the reector points, to the ones used in theidentication, the closer the error results should be to theones shown inable . Tereore, the points and positionsor evaluation must be different rom the ones used in theidentication procedure. o illustrate this characteristic, inthis case two extra positions o the reector have beenconsidered as test points (Figure ). For each one o thesenew reector positions, angle combinations have beencaptured corresponding to the center positions o each oneo them, captured in the same captured conditions compareto the rest o the points. Te nominal distance betweenthese two points, calculated as the Euclidean distance o themultilaterated coordinates, was . mm. Te distanceobtained, using the mean points expressed in the AACMMreerence system o the probed points o each point and theidentied parameters, was . mm, obtaining an errorin this case with respect to the nominal o m, while themaximum standard deviation or the two probed points hasbeen calculated in thecoordinate o the rst probed point,with a value o .mm or all the captured positions.It is thereore possible to conclude that the obtained error

value or the identied parameter set can be generalizable to


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: Quality indicators or the identied values o model parameters over SMR locations ( AACMM positions).

Distance error (mm) by retroreector point (mm)Maximum . Maximum .

Causing dist. Causing point

Medium . Causing coord. Y

Medium .

500

1000

1500

3000

3500

2000

1800

1600

1400

1200

Identication points

Evaluation points

X(m

m)

Z

(mm)

Y(mm)

(a)

20000

20004000

2000

4000

1000

500

0

500

1000

X(m

m)

Z

(mm)

LT1LT2

LT3LT4

Y(mm)

(b)

F : Points captured by Ls. (a) Multilaterated points used as nominal data or the parameter identication procedure; (b) pointscaptured by each L.

0 50 100 150 200 250

0

20

40

60

80

100

120

Distance number

Range(m)

F : Range o each distance considered in the identicationprocess. Each data represents the difference between the maximumand minimum values o each distance calculated with the measure-ments made by the our Ls.

the evaluated work volume. For this reason, the assessmento more evaluation positions different rom the one usedas an example is recommended when increasing the work

volume to be identied. Te ideal is to obtain error valuesalways below the identication maximum error, although

it is possible to set an acceptable error percentage abovethe obtained maximum error in order to set a characteristic

value o the measuring arm global error according to a lessrestrictive criterion.

7. Conclusions

In this work, a novel calibration technique or parameterkinematic identication o an AACMM is presented. Tiscalibration technique is based on an objective unction thatconsiders the volumetric error and repeatability by meanso the distance errors and the standard deviation o physicalprobed points, respectively. Moreover, a new procedure toobtain nominal gauge values or this calibration techniqueis carried out. Tis new procedure is based on the mea-surements o a calibrated spherical retroreector with Ls.Although the error range o this type o measuring instru-ments has an order o magnitude similar to the AACMMor this work distances, the use o multilateration techniquescan be o great help to reduce the measurement uncertainty,taking as nominal data only the measurements o the Ls.By combining the aorementioned measurements and afer


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1000

0

1000

1000

0

1000

1500

1000

500

0

500

1000

1500

X(mm)

Z

(mm)

Y(mm)

(a)

X(mm

)

Z

(mm)

200

400

600

800

1000

2000

200400

600800

150

200

250

300

350

400

450

Y(mm)

(b)

F : Points captured by the AACMM. (a) Nominal kinematic parameters; (b) identied kinematic parameters.

the described optimization procedure, is possible to obtainpoints that can be used as nominal points, in this casematerializing distances between them or their use in theAACMM parameteridentication procedure. Even thoughinthis work Ls have been used to eliminate thecoordinatesign ambiguity o the multilaterated points, it is possibleto realize this procedure using only Ls by making surethat all o the points have the same sign with respect to the

multilateration reerence system. Tis way, in the cases whenaccess to this type o measuring instruments is available, itis possible to carry out an AACMM identication procedurewithout the use o common physical gauges used in this typeo procedures.

Finally, the simplication o the calibration procedurepresented in this work can be achieved by applying sequentialmultilarization; thus the use o only one L is needed to carryout the adapted procedure, with the aim o reducing costs.

Conflict of Interests

Te authors declare that there is no conict o interests

regarding the publication o this paper.

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