research article articulated arm coordinate measuring machine … · 2019. 7. 31. · articulated...

Research ArticleArticulated Arm Coordinate Measuring Machine Calibration byLaser Tracker Multilateration

Jorge Santolaria, Ana C. Majarena, David Samper, Agustín Brau, and Jesús Velázquez

Departamento de Ingenieŕıa deDiseño y Fabricación, Edificio Torres Quevedo, EINA,Universidad de Zaragoza, 50018 Zaragoza, Spain

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

Received 27 August 2013; Accepted 3 November 2013; Published 29 January 2014

Academic Editors: G. Huang and D. Veeger

Copyright © 2014 Jorge Santolaria et al.This is an open access article distributed under the Creative CommonsAttribution License,which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

A new procedure for the calibration of an articulated arm coordinate measuring machine (AACMM) is presented in this paper.First, a self-calibration algorithm of four laser trackers (LTs) is developed. The spatial localization of a retroreflector target, placedin different positions within the workspace, is determined by means of a geometric multilateration system constructed from thefour LTs. Next, a nonlinear optimization algorithm for the identification procedure of the AACMM is explained. An objectivefunction based on Euclidean distances and standard deviations is developed. This function is obtained from the captured nominaldata (given by the LTs used as a gauge instrument) and the data obtained by the AACMM and compares the measured andcalculated coordinates of the target to obtain the identified model parameters that minimize this difference. Finally, results showthat the procedure presented, using the measurements of the LTs as a gauge instrument, is very effective by improving the AACMMprecision.

1. Introduction

In recent years, there has been an increasing interest inAACMM’s because of their advantages in terms of accuracy,portability and suitability for inspection and quality controltasks in machining tool processes and in the automotive andaerospace industry [1].

Nevertheless, few researches have focused on the cali-bration of these mechanisms. Moreover, there is an absenceof standards on verification and calibration procedures. Forthat reason, AACMMmanufacturers have developed its ownevaluation procedures. These evaluation methods are basedon the three main standards for performance evaluation incurrent CMM’s, UNE-EN ISO 10360, ASME B89.4.1, andVDI/VDE 2617 and are still carried out today to compare andevaluate the accuracy of an arm from the point of view ofthe CMM’s. In [1], the author presented a procedure to checkthe performance of coordinatemeasuring arms by calculatingthe distances between the centers of different spheres. Theresults obtained were compared with the application of theANSI/ASME B89 volumetric performance test showing goodagreement between the two approaches and a cost reduction.In [2], Shimojima et al. presented a new method to estimate

the uncertainty of a measuring arm using a tridimensionalgauge.This method consists of a fat plate with 9 spheres fixedat three different heights with respect to the metallic surfaceof the plate. Then 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 performance ofthe arm. Other works have been found in the literature whosemain goal is also to evaluate the performance of measuringarms [3–5].

However, the AACMM presents different characteristics,and different verification procedures are therefore required.A point clearly defines a position of the three machineaxes for a CMM. Nevertheless, the possible positions of theAACMM elements to achieve a fixed point defined in themeasurement volume are practically infinite. Moreover, forCMMs, evaluation tests can be performed to extract thepositioning errors, allowing correction models to be imple-mented [1].Thus, a high level of maintenance of the physical-mathematical relations between the error model parametersand the error physically committed by the machine can beachieved. However, the application of these models does notmake sense in AACMM’s, given the difficulty of directly

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

2 The Scientific World Journal

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

The calibration procedure consists of identifying the geo-metric parameters in order to improve the model accuracy.This procedure allows us to obtain correction models toestablish corrections in the measurements results and toquantify the effects of the influence variables in the finalmeasurement. To achieve this goal, the following five steps areusually carried out: determination of the kinematic model bymeans of nonlinear equations, data acquisition, optimizationor geometric parameter identification,model evaluation, and,finally, identification of the error sources and implementationof correction models.

The first step, determination of the kinematicmodel, con-sists of obtaining the non-linear equations that relate the jointvariables to the position and orientation of the end-effectorand the initial values of nominal geometric parameters. Oneof the most widely used geometric methods for modelling amechanism is the well-known Denavit-Hartenberg method[6], whichmodels the joints with four parameters. One of thelimitations of this method appears in those mechanisms thatpresent two consecutive parallel joint axes. In this case, aninfinite number of common normals of the same length exist,and the location of the axis coordinate systemmay be definedarbitrarily. Some studies [7–10] present methods to obtain acomplete, equivalent, and proportional model.

The number of parameters is fixed when the kinematicmodel is determined, and this value will depend on theselected method. Moreover, there are a maximum number ofparameters that must be identified, and the model accuracydoes not have to improve by adding extra parameters [11].In this work, the author determined that four parametersmust be considered for each revolute joint, two of whichmustbe orientational, and for a prismatic joint two orientationalparameters are necessary, applied about the noncollinearaxes before and perpendicular to the translational joint axes.Nongeometric errors are usually compensated by addingparameters in the geometric model [12].

The second step is data acquisition. Any measurementerror of the external instrument is propagated to the resultsof the identified parameters. For that reason, an instrumentfor data acquisition should be, at least, one magnitude ordermore accurate than the mechanism whose parameters aregoing to be identified.

A direct geometric transformation can be established,providing the global reference system can be measured bymeans of an external measurement instrument such as alaser tracker or a coordinate measuring machine. In this way,this transformation obtains the coordinates of the measuredpoints in the global reference system of the mechanism.Thus, direct comparisons in the objective function, 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 of least-square methods. These methods allow usto obtain an approximation of this transformation, and thisapproximationwill depend on the configurations used in dataacquisition and on the mechanism error in the evaluatedpoints [13].

The home position is a position, within the AACMMworking range, where all joint angles have a predefinedvalue. The displacements of the probe are usually measuredwith respect to this defined position. In [14], Kovac andFrank developed a high precision gauge instrument for theparameter identification procedure and evaluation tests.

The determination of the number of required specificpositions is not generalizable from one AACMM to another,since the errors committed by each arm will depend on theirconfiguration and assembly defects.The performance of teststhat characterize the influence of each joint on the final error,to finally choose positions in accordance with this influence,must therefore be carried out before defining the capturepositions to identify the kinematic parameters. The positionsselected for the identification procedure should cover themaximum joint rotation range to cover the influences of allthe measuring arm elements in the workspace. In [15], Zhenget al. obtained the spatial error distribution model by usingsupport vector machine theory.

The third step is optimization or geometric parameteridentification, and the objective is to search for the optimumvalues of all parameters included in themodel thatminimizesthe error in the performedmeasurements.This step is usuallycarried out by means of approximation procedures basedon least-square fitting. This function can be defined as thequadratic difference of the error (obtained between the mea-sured value and the value computed by the kinematicmodel).The increment established for parametersmust be defined foreach iteration. In most of the cases, numerical optimizationtechniques are used to minimize the error. The Levenberg-Marquardt (L-M) [16] method is one of the most widely usedtechniques to solve the numerical optimization algorithm.This method usually presents lower computational cost,providing a solution closer to the optimum solution for theset of 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 [17] tosolve the nonlinear optimization problem. In [18], Santolariaet al. presented a kinematic parameter estimation technique,which allows us to improve the repeatability of the AACMMby more than 50%. This technique uses a ball bar gauge toperform the data acquisition procedure.

The fourth step, the model evaluation, consists of evalu-ating the mechanism behavior with the set of the identifiedparameters obtained in the geometric parameter identifica-tion procedure, in configurations different from those usedin the optimization process. In [19], Koseki et al. evaluatedthe accuracy of a mechanism by means of a laser trackingcoordinate measuring system.

Finally, an identification of the error sources and amodelling and implementation of the correction models canbe optionally performed.

In [20], Piratelli-Filho et al. developed a virtual sphereplate, having a standard deviation of around 0.02mm anda measurement uncertainty from 0.02 to 1.64 𝜇m in pointmeasurements, to evaluate the measurement performance ofAACMM’s. The AACMM performance test using the virtual

The Scientific World Journal 3

sphere plate resulted in a mean error of 0.023mm and astandard deviation of 0.039mm.

A laser tracker is a large-scale measuring instrument withhigh accuracy.Thesemechanisms are considered very reliable[21]. Consequently, laser trackers have been used recentlyinstead of other traditional methods such as theodolites orcollimators in multiple applications such as robot tracking,testing, calibration, and maintenance. These systems useinterferometry for measuring relative distances and opticalencoders in order to measure azimuth and elevation anglesof a beam-steering mirror.The interferometer measurementsare obtained relative to the starting point. Besides, this beammust track the positions of a retroreflector. A plane mirrormounted on a high precision universal joint deflects thebeam and hits the retroreflector. This element consists ofthree perpendicularly oriented planemirrors, and the beam isreflected parallel. Theoretically, the laser beam hits the centerpoint of the retroreflector. When there is no relative move-ment between the laser tracker head and the retroreflector,there is no parallel displacement between the emitted andthe reflected beam. However, when the retroreflector startsmoving, there is a displacement of the reflected laser beam,since, in this case, the laser beam does not hit the centerpoint of the retroreflector. In [22], Lin and Her used a lasertracker system tomeasure the volumetric errors of a precisionmachine. The technique presented is based on the ASMEB5.54 standards and offers a quick method to characterizethese errors. This measuring instrument can also be used inmultilateration for verification of machine tools, as presentedin [23], where the error order can also be in the error range ofa laser tracker. In [21], a laser tracker was used also to analysethe performance of an indoor GPS.

Themultilateration technique is a procedurewhich can beused to improve themechanismaccuracy.Thismethod allowsus to reduce the measurement uncertainty by eliminating theangular noise. To achieve this, the multilateration obtainsthe half point position starting from data of several lasertrackers, located in different positions [24]. In [25], Zhanget al. presented specific recommendations for optimizationof multilateration set-ups and measurement plans and forminimizing measurement uncertainty. Besides, the authorsevaluated the volumetric measurement error propagation,obtaining an average standard deviation of length mea-surements around 1.2 𝜇m. Hughes et al. [26] presented alaser-interferometric measuring station and obtained thedisplacement measurement uncertainty, which was used topredict a volumetric uncertainty for a multilateration system,consisting of eight measuring stations and four targets. Thecombined uncertainty of the measuring station displacementmeasurements and the CMM repeatability obtained wasaround 200 nm. In [27], Kim et al. developed a volumetricinterferometer system and minimized least-square errorsby fitting the measured values to a geometric model ofmultilateration, obtaining a volumetric uncertainty of lessthan 1 𝜇m.

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

accuracy, the multilateration techniques allow us to capturepoints with an uncertainty much smaller than the oneobtained with an AACMM. The points captured from themultilateration are therefore considered as nominal data inthe calibration procedure of the AACMM.

2. AACMM Kinematic Model

The AACMM kinematics relates the joint variables and theprobe position for any arm posture.

The direct kinematic model is used to calculate thepositioning and orientation of the AACMM probe on thebasis of certain values of the joint variables, according to thefollowing:

𝑦 = 𝑓 (𝜃𝑖, 𝑞) , (1)

with 𝑖 = 1, . . . , 𝑛 for an arm with 𝑛 rotating joints. 𝜃𝑖is given

by the vector of the joint variables and𝑓 represents themodeldefined, which depends on the parameter vector 𝑞.

The non-linear equation system to model the mechanismcan be developed by applying the Denavit-Hartenberg (D-H)method [6] to every chain in themechanism.Thismethodhasbeen widely used in mechanism modelling [28, 29] and usesfour parameters (distances 𝑑

𝑖, 𝑎𝑖, and angles 𝜃

𝑖, 𝛼𝑖) to model

the coordinate transformation between successive referencesystems. The homogenous transformation matrix betweenframe 𝑖 and 𝑖 − 1 depends on these four parameters:

𝑖−1𝐴𝑖

= 𝑇𝑧,𝑑

⋅ 𝑅𝑧,𝜃

⋅ 𝑇𝑥,𝑎

⋅ 𝑅𝑥,𝛼

=

[[[[

[

cos 𝜃𝑖− cos𝛼

𝑖⋅ sin 𝜃

𝑖sin𝛼𝑖⋅ sin 𝜃

𝑖𝑎𝑖⋅ cos 𝜃

𝑖

sin 𝜃𝑖

cos𝛼𝑖⋅ cos 𝜃

𝑖− sin𝛼

𝑖⋅ cos 𝜃

𝑖𝑎𝑖⋅ sin 𝜃

𝑖

0 sin𝛼𝑖

cos𝛼𝑖

𝑑𝑖

0 0 0 1

]]]]

]

.

(2)

The AACMMmodel used in this work is a Faro PlatinumArm having seven axes, with a nominal value of 2𝜎 in thesingle point articulation and performance test of 0.030mmaccording to the specifications of the manufacturer.

Figure 1 shows the reference systems used in theAACMMmodel.

The armglobal transformationmatrix allows us to expressthe probe sphere center coordinates with respect to the baseof the AACMM. This matrix can be obtained by calculatingsuccessive coordinate transformations by premultiplying thetransformationmatrix between a frame and the previous one,as shown in the following:

[𝑋, 𝑌, 𝑍, 1]

AACMM =0𝑇6⋅ [𝑋, 𝑌, 𝑍, 1]

Probe. (3)

In this equation, 0 is the global reference system of thebase and 6 corresponds to the reference frame that moveswith the rotation of the last joint.

A reference system is usually defined in the probe.However, the aim of 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

Figure 1: AACMM reference systems.

coordinates of the static probe, so this reference system isnot necessary. Seven reference systems are used to model theAACMM. The last reference system is located in the centerof the reflector and is oriented as reference system six. Thus,the number of parameters for the 6 degrees of freedom (Dof)AACMM is 28. The kinematic model is fully described in[18]. Table 1 shows the initial values for the AACMM D-Hparameters.

3. Multilateration System

The multilateration technique allows us to reduce the mea-surement uncertainty by eliminating the angular noise. Toachieve this, the multilateration obtains the weighted-pointposition by means of several laser tracker data, located indifferent positions. A minimum of three measurements ofeach point is necessary. Each laser tracker measures thedistance from the laser tracker to the target points. Thesemeasurements present a noise having a radial component andtwo angular components. The aim of the multilateration is todecrease themeasurement uncertainty, so this technique onlyuses the radial component of the laser tracker measurements,𝑚𝑖, thus decreasing the measurement noise influence, which

allows us to decrease the global uncertainty. These compo-nents define a sphere. The intersection of the three spheresobtained by measuring the same point by three laser trackersprovides two points. Equations from (4) to (6) provide thespheres obtained by the measurements of each laser tracker,respectively. Consider

𝑚2

0= (𝑥 − 𝑥

1)2+ (𝑦 − 𝑦

1)2+ (𝑧 − 𝑧

1)2, (4)

𝑚2

1= (𝑥 − 𝑥

2)2+ (𝑦 − 𝑦

2)2+ (𝑧 − 𝑧

2)2, (5)

𝑚2

2= (𝑥 − 𝑥

3)2+ (𝑦 − 𝑦

3)2+ (𝑧 − 𝑧

3)2. (6)

The knowledge of the locations of the reference systems ofthe laser trackers allows us to obtain the following equations:

𝑥 =(𝑚2

0− 𝑚2

1+ 𝑥2

2)

2𝑥1

, (7)

Reflector (X,Y, Z)

m3

P3(X3, Y3, Z3)

m0m2

m1

P2(X2, Y2, 0)

P0(0, 0, 0)

P1(X1, 0, 0)

Figure 2: Coordinates of the four laser tracker reference systemsexpressed in the multilateration system.

𝑦 =(𝑚2

0− 𝑚2

2+ 𝑥2

2+ 𝑦2

2− 2𝑥𝑥

2)

2𝑦2

, (8)

𝑧 = ±(𝑚2

0− 𝑥2− 𝑦2)1/2

. (9)

In thiswork, for simplicity, the unknown reference systemlocation of the three laser trackers has been defined as(0, 0, 0), (𝑥

1, 0, 0), and (𝑥

2, 𝑦2, 0) with respect to the multilat-

eration reference system.A fourth laser tracker can be used to avoid the sign

ambiguity in the 𝑍 coordinate obtained in (9). The referencesystem of this laser tracker is given by (𝑥

3, 𝑦3, 𝑧3). This point

should belong to a plane different from the plane𝑋𝑌 formedby the other three laser tracker reference systems as shown inFigure 2. In this figure, 𝑃

𝑖represents LT

𝑖position system for

the 𝑖 laser trackers. The origin of the multilateration globalreference system is given by 𝑃

0.

The (𝑥, 𝑦, 𝑧) target coordinates are obtained in linearmatrix form by operating (7), (8), and (9), as expressed in thefollowing:

[

[

𝑥

𝑦

𝑧

]

]

= − 0,5 ⋅

[[[[[[[[[[

[

1

𝑥1

0 0

−𝑥2

𝑥1𝑦2

1

𝑦2

0

−(𝑥3

𝑥1𝑧3

) + (𝑥2𝑦3

𝑥1𝑦2𝑧3

) −𝑦3

𝑦2𝑧3

1

𝑧3

]]]]]]]]]]

]

⋅

[[[[

[

𝑚2

1− 𝑚2

0− 𝑥2

1

𝑚2

2− 𝑚2

0− 𝑥2

2− 𝑦2

2

𝑚2

3− 𝑚2

0− 𝑥2

3− 𝑦2

3− 𝑧2

3

]]]]

]

.

(10)

Themultilateralized coordinates, obtained in (10), will beconsidered as the nominal coordinates in the identificationparameter procedure.

4. Data Acquisition

The data acquisition step consists of capturing the nominalcoordinates in the workspace of the AACMM. The suitablenumber of positions is not generalizable from one measuring


Table 1: Initial values for the AACMMD-H parameters.

Joint 𝜃𝑖(∘) 𝛼

𝑖(∘) 𝑎

𝑖(mm) 𝑑

𝑖(mm)

1 −90 −90 −40 3002 135 −90 −40 03 180 −90 −34 5904 −90 90 34 05 −90 −90 −34 5886 180 −90 −34 07 0 0 0 318.2

LT4: FARO ION

LT2: API T3

LT1: FARO X

LT3: LEICA LT-600

Retroreflector

AACMM

Figure 3: LTs and AACMM distribution used in tests.

arm to another, since each measuring arm error will dependon their configuration and assembly defects. The identifica-tion procedure should cover the maximum range of jointrotation to consider all the influences of the measuring armelements.

In this study, a cloud of points located within thearm workspace was measured simultaneously with both theAACMM (measured values) and four LTs which conformto the multilateralized system (nominal values). The lasertracker FARO 𝑋 model was used as LT

1, API T3 as LT

2,

LEICA LT-600 as LT3, and FARO ION as LT

4. The positions

were distributed throughout the workspace of the arm andreached different arm angle values. The LTs were distributedforming the multilateration global coordinate system, as itwas detailed in Section 3. LT

𝑖positions have been chosen in

function of the reflector visibility, thus forming a spatial angleas near as possible to 90∘ between them. The measurementuncertainty is lower in this position, as demonstrated in aprevious work [30]. The AACMM has been arranged in aposition that maximizes the visibility of the LTs, as shown inFigure 3.

The AACMM data acquisition technique is usually per-formed bymeans of discrete contact probing of surface pointsof the gauge in order to obtain the center of the spheresfrom several surface measurements. The time required forthe capture of positions is high, and therefore, identification

120º

Xcenter , Ycenter , Zcenter

LProb

e 45∘

120∘

Figure 4: Probe used in the data acquisition procedure.

is generally carried out with a relatively low number of armpositions. In this work, a probe presented in [18], capableof directly probing the center of the spheres of the gaugewithout having to probe surface points, has been used. Thisprobe consists of three tungsten carbide spheres of 6mm indiameter, laid out at 120∘ on the probe, as can be seen inFigure 4.

Theprobe used allows us to define a probewith zero probesphere radius and with a distance from the position of thehousing to the center of the probed reflector sphere of 1.5inches, allowing direct probing of the sphere center when thethree spheres of the probe and the sphere are in contact.

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

23 positions of the retroreflector were measured, fromwhich 21 positions were considered in the parameter iden-tification process (identification positions) and the other 2positions were kept for the parameter evaluation procedure(test positions). A software developed captured the AACMMmeasurements, saving the AACMM joint angles, 𝜃

𝑖. These

angles and the AACMM parameters are the input to thekinematic model.The solution of the non-linear equations bythe L-M algorithm obtains themeasured point coordinates inthe AACMM reference system.

The data acquisition procedure was performed trying tocapture data in symmetrical trajectories in the retroreflectorto minimize the effect of probing force on the gauge.

Although the measuring of the retroreflector center withthe kinematic mount probe from different arm orientationsshould result in the same point measured, the unsuitablevalue of the nominal kinematic parameters of the model willbe shown by way of a probing error, resulting in differentcoordinates for the same measured point in different armorientations. For that reason, five measurements were takenfor each point and themean point of the set of points capturedwas considered as the center of the retroreflectormeasured inthe AACMM reference system, as shown in Figure 5.

The distance between the different retroreflector mea-sured positions can be obtained from the points capturedby means of the Euclidean distance between each pair ofretroreflector positions, obtaining 𝑑

𝑖𝑗 AACMM, where 𝑖 and 𝑗represent two measured positions.

At the same time, the four LTs simultaneously measuredthe distance from the captured point to the local coordinatesystem.


Xc, Yc, Zc =

Xi, Yi, Zi

Xi, Yi, Zi

Figure 5: Retroreflector center.

Diagonal distances are obtained according to the follow-ing:

𝑟 = 𝑚 + 𝑙, (11)

where 𝑚 is the incremental output of the displacementtransducer used and 𝑙 is the offset, which should be calibratedbefore the multilateration implementation.

The self-calibration and determination of the parameter𝑙 can be carried out by capturing some additional objectivemeasurements in some different positions. By performingthe quadrilateration 𝑘 objective point times, the number ofequations is given by 4𝑘 as shown in the following:

(𝑟𝑘

𝑖)2

= (𝑚𝑘

𝑖+ 𝑙𝑖)2

= (𝑥𝑘− 𝑥𝑖)2

+ (𝑦𝑘− 𝑦𝑖)2

+ (𝑧𝑘− 𝑧𝑖)2

,

(12)

for 𝑖 = 0, . . . , 3 and 𝑘 = 0, . . . , 𝑘 − 1.The system unknowns are the four offsets, 𝑙

𝑖, and the

position of the coordinates, 𝑥𝑘, 𝑦𝑘, 𝑧𝑘 (3𝑘 unknowns). Toidentify the four references, 12 unknowns, correspondingto 𝑥𝑖, 𝑦𝑖, 𝑧𝑖, must be added. Thus, the number of system

unknowns is given by 3𝑘 + 16.The unknowns can be obtained by the least-square

method to minimize the error.The multilateration technique must solve the objective

function defined by

𝜙 =

21

∑

𝑖=1

4

∑

𝑗=1

[(𝑥𝑖− 𝑥LT𝑗)

2

+ (𝑦𝑖− 𝑦LT𝑗)

2

+ (𝑧𝑖− 𝑧LT𝑗)

2

− (𝑚𝑖𝑗+ 𝑙LT𝑗)

2

] ,

(13)

where (𝑥𝑖, 𝑦𝑖, 𝑧𝑖) represents the measured point coordinates

in the multilateration reference system, (𝑥LT𝑗 , 𝑦LT𝑗 , 𝑧LT𝑗) isthe origin of the LT

𝑗reference system, 𝑚

𝑖𝑗are the measured

distances for every point in the LT𝑗reference system, and 𝑙LT𝑗

represents the LT𝑗offset. The non-linear system obtained by

the multilateration technique was solved by means of the L-M algorithm, and the solution obtained gives the followinginformation:

(i) the point coordinates in the multilateration system:(𝑥𝑖, 𝑦𝑖, 𝑧𝑖)|SR Multi𝑙,

(ii) the laser tracker offsets: 𝑙LT𝑗 ,

(iii) the origin of the reference systems for the four lasertrackers in the multilateration system: (𝑥LT𝑗 , 𝑦LT𝑗 ,𝑧LT𝑗)|SR Multi𝑙.

The calculated distances between each pair of retroreflec-tor positions can be obtained from the point coordinates inthe multilateration system, obtaining 𝑑

𝑖𝑗 Multi𝑙, where 𝑖 and 𝑗represent the two positions.

5. Parameter Identification Procedure

Once the data acquisition technique has been carried out, theparameter identification procedure can be performed.

Figure 6 shows a scheme of the calibration procedure.The origin of the laser trackers wasmeasuredwith respect

to the reference system of LT1.This reference system has been

considered themultilateration reference system for obtainingthe initial values of the LT

𝑗reference systems.

The kinematic parameter identification procedure can beperformed starting from the measured and calculated dis-tances (obtained as explained in Section 4). The parametersconsidered in the AACMM kinematic model are given by(𝑎𝑖, 𝛼𝑖, 𝑑𝑖, 𝜃0𝑖, 𝜃𝑖Enc), where 𝜃𝑖Enc are the angles measured by

the encoder. The model obtained is a non-linear equationsystem, and it was solved by the L-M algorithm.The objectivefunction defined in this step considers both themeasured andcalculated distances, as shown in

𝜙 =

𝑝

∑

𝑖=1,𝑗=1

[(𝑑𝑖𝑗AACMM

− 𝑑𝑖𝑗Multi𝑙

)2

] +

𝑛

∑

𝑖=1

[𝜎2

𝑥𝑖+ 𝜎2

𝑦𝑖+ 𝜎2

𝑧𝑖] ,

(14)

where 𝑝 represents the number of positions consideredin the parameter identification procedure and (𝜎

𝑋𝑖𝑗, 𝜎𝑌𝑖𝑗,

𝜎𝑍𝑖𝑗

) represents the standard deviation of the points mea-sured in each position and each coordinate, showing theinfluence of the volumetric accuracy and point repeatability.

6. Calibration Results

In the optimization process, the distances from each point toevery point are taken into account, obtaining 253 distancesbetween the 23 points.

The multilateration procedure described above was car-ried out based on the 23 captured points for each LTcorresponding to the reflector positions used. Table 2 showsthe results obtained. As the initial values for this procedure,null values were assigned for the offsets of all LTs, and for theorigin points the corresponding coordinates were those thatbetter fit the distribution observed taken from the measuredhome points of LT

2, LT3, and LT

4fromLT

1, where the origin

of the multilaterized reference system is located.In Figure 7, the coordinates of the captured points by

each LT and the coordinates of 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 identification

mij

(xLTj, yLTj, zLTj)SR LT1

(xLTj, yLTj, zLTj)SR multil.

ILTj

ILTj

initial

AACMM: 𝜃i(21 positions, 5 measurements

for each position)

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

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

For i, j = pos ini to pos final

𝜙 =21∑

i=1,j=1[(dijAACMM − dijmultil )

2 +n∑i=1

[𝜎2x𝑖 + 𝜎2y𝑖+ 𝜎2z𝑖 ]

Optimisation: to minimize Ferror

Calculation of the geometric parameters: Si

values

𝜙 =21∑i=1

4∑j=1

[(xi − xLTj)2 + (yi − yLTj)

2 + (zi − zLTj)2 − (m +ij lLTj)

2]

Ferror = [E1 . . . Epos final]

Figure 6: Scheme of the calibration procedure.

used as nominal coordinates in the parameter identificationprocedure of the AACMM, are shown graphically.

As stated before, in order to introduce redundancy inthe objective function and thereby restrict to the nominalpoints the final points obtainedwith the identified parametersof the measuring arm, all possible distances between themultilaterated points are calculated. This way, we can obtain253 distances, which will be used as nominal data in theobjective function (14) in the identification procedure. InFigure 8, the range of each of the calculated distances in themeasurements made by the four LTs can be observed. Thus,each data represents the difference between the maximumand minimum values of each considered distance calculatedfrom the set of the 4 LTs measured data. The distances arearranged starting from position 1 of the retroreflector. Themaximum range value obtained by calculating the distanceswith the 4 LTs was 105 𝜇m, while the mean range for all set ofdistances was 43 𝜇m.

Following the scheme presented in Figure 6, we cancalculate all possible distances between spheres as well asthe standard deviations measured with the AACMM fromthe saved angular data obtained during the data capture

process and the set of initial parameters of the AACMMmathematical model shown in Table 1. In accordance withthe optimization for the identification scheme presented inFigure 6, the quality indicators for the set of initial parametersof the AACMM model are shown in Table 3. Moreover,Table 3 shows in its first column the distancesmaximumerrorobtained for all the reflector points and the index of the pointsthat determine the distance in which the maximum error iscalculated.

Analogously, the mean distance errors for all the evalu-ated distances are also shown. With respect to the standarddeviation, Table 3 shows, besides the maximum and meanvalues, the index of the point and its corresponding coordi-nate where the greater value is obtained, since in this case theobjective function considers the standard deviation for eachcoordinate independently. As expected, the obtained valuesare high, considering the initial set of parameters defined forthe AACMMmathematical model.

In the objective function proposed in (14), the datacapture setup described for 𝑝 = 21 reflector points, it isnecessary to consider the elimination of the termswhere 𝑖 = 𝑗to avoid both the inclusion of null terms and duplicate of


Table 2: Multilaterated offsets and origin coordinates obtained in multilateration reference system. 2798 iterations and objective functionvalue below 1 𝜇m.

Offsets (mm) Origin coordinates (mm)𝑥LT 𝑦LT 𝑧LT

LT1 𝑙LT1 1.51695 0 0 0LT2 𝑙LT2 0.89275 2262.33138 0 0LT3 𝑙LT3 0.18749 1713.00335 2098.93521 0LT4 𝑙LT4 2.13351 1913.00558 −343.41959 3010.23578

Table 3: Quality indicators for the initial values of model parameters over 21 SMR locations (105 AACMM positions).

Distance error (mm) 2𝜎 by retroreflector point (mm)Maximum 38.50723 Maximum 22.00413Causing dist. 4–10 Causing point 17Medium 17.28859 Causing coord. X

Medium 8.17533

Table 4: Identified values for the AACMMmodel parameters by L-M algorithm.

Joint 𝜃𝑖(∘) 𝛼

𝑖(∘) 𝑎

𝑖(mm) 𝑑

𝑖(mm)

1 −89.1269 −89.9216 −42.4313 3002 134.7734 −89.7178 −41.7885 1.23373 184.2704 −90.1202 −28.9084 591.24234 −95.4182 89.5364 29.4449 0.63005 −88.0773 −89.9482 −28.5745 591.52846 181.0832 92.5750 27.5479 12.21437 0.2117 0 0.4124 254.0575

distance errors noting that 𝑑𝑖𝑗= 𝑑𝑗𝑖. In regard to the standard

deviation, each coordinate deviation for every one of thereflector positions is considered. To express mathematicallythe optimization problem it is required to consider the sumof the calculated quadratic errors. This way, through theobjective function in (14), we obtain 231 terms correspondingto the distance errors for the 21 reflector points, plus 63standard deviation terms of these points, obtaining a totalof 294 terms to determine the objective function value ineach iteration of the optimization algorithm. This value willcontain the influence of the kinematic parameters and thearticulation variables considering for this the terms relatedon the one hand to volumetric precision and on the other torepeatability, for the 105 measuring arm captured positions.In Figure 9, it can be observed graphically the distribution ofthe captured points in the AACMM reference system, beforeand after the optimization procedure, while Table 4 showsthe identified parameters starting from the initial values ofTable 1.

From the identified parameters (Table 4), Table 5 showsthe error characteristics results obtained for each of thecaptured points considering in this case these parameters.

The validation and generalization of the error results cal-culated for the set of identified parameters over the captureddata, to the rest of the AACMM work volume, imply in themore restrictive case obtaining error and deviation values less

than the maximum values obtained in this case (Table 5) forany evaluated position of the measuring arm. For this reason,the assessment of the AACMM error in different positionsto the ones used in the identification procedure is highlyrecommended. As shown in Table 5, a maximum error of118 𝜇m and a mean error of 48 𝜇m for the measuring volumehave been obtained, considering the 21 nominal points usedin the identification. In normal operation of the measuringarm in this work volume, it is expected to get error valuesclose to the mean value and obtainingmaximum error valuesonly in certain arm configurations.

As a last step of the identification procedure, it isnecessary to evaluate the set of parameters obtained indifferent arm positions from those considered in its ownidentification 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 reflector points, to the ones used in theidentification, the closer the error results should be to theones shown in Table 5. Therefore, the points and positionsfor evaluation must be different from the ones used in theidentification procedure. To illustrate this characteristic, inthis case two extra positions of the reflector have beenconsidered as test points (Figure 7). For each one of thesenew reflector positions, 10 angle combinations have beencaptured corresponding to the center positions of each oneof them, captured in the same captured conditions compareto the rest of the points. The nominal distance betweenthese two points, calculated as the Euclidean distance of themultilaterated coordinates, was 492.7164mm. The distanceobtained, using the mean points expressed in the AACMMreference system of the 10 probed points of each point and theidentified parameters, was 492.7695mm, obtaining an errorin this case with respect to the nominal of 53 𝜇m, while themaximum standard deviation for the two probed points hasbeen calculated in the𝑋 coordinate of the first probed point,with a value of 0.1426mm for all the 10 captured positions.It is therefore possible to conclude that the obtained errorvalue for the identified parameter set can be generalizable to


Table 5: Quality indicators for the identified values of model parameters over 21 SMR locations (105 AACMM positions).

Distance error (mm) 2𝜎 by retroreflector point (mm)Maximum 0.11824 Maximum 0.20644Causing dist. 2–12 Causing point 4Medium 0.04825 Causing coord. Y

Medium 0.11545

500

1000

1500

30003500

−2000

−1800

−1600

−1400

−1200

Identification points

Evaluation points

X(m

m)

Z(m

m)

Y (mm)

(a)

−20000

20004000

2000

4000

−1000

−500

0

500

1000

X(mm

)

Z(m

m)

LT1LT2

LT3LT4

Y (mm)

(b)

Figure 7: Points captured by LTs. (a) Multilaterated points used as nominal data for the parameter identification procedure; (b) pointscaptured by each LT.

0 50 100 150 200 2500

20

40

60

80

100

120

Distance number

Rang

e (𝜇

m)

Figure 8: Range of each distance considered in the identificationprocess. Each data represents the difference between the maximumand minimum values of each distance calculated with the measure-ments made by the four LT’s.

the evaluated work volume. For this reason, the assessmentof more evaluation positions different from the one usedas an example is recommended when increasing the workvolume to be identified. The ideal is to obtain error valuesalways below the identification maximum error, although

it is possible to set an acceptable error percentage abovethe obtained maximum error in order to set a characteristicvalue of the measuring arm global error according to a lessrestrictive criterion.

7. Conclusions

In this work, a novel calibration technique for parameterkinematic identification of an AACMM is presented. Thiscalibration technique is based on an objective function thatconsiders the volumetric error and repeatability by meansof the distance errors and the standard deviation of physicalprobed points, respectively. Moreover, a new procedure toobtain nominal gauge values for this calibration techniqueis carried out. This new procedure is based on the mea-surements of a calibrated spherical retroreflector with 4 LTs.Although the error range of this type of measuring instru-ments has an order of magnitude similar to the AACMMfor this work distances, the use of multilateration techniquescan be of great help to reduce the measurement uncertainty,taking as nominal data only the measurements of the LTs.By combining the aforementioned measurements and after


−10000

1000

−10000

1000

−1500

−1000

−500

0

500

1000

1500

X (mm)

Z(m

m)

Y (mm)

(a)

X (mm)

Z(m

m)

200400

600800

1000

−2000200

400600800

150

200

250

300

350

400

450

Y (mm)

(b)

Figure 9: Points captured by the AACMM. (a) Nominal kinematic parameters; (b) identified kinematic parameters.

the described optimization procedure, is possible to obtainpoints that can be used as nominal points, in this casematerializing distances between them for their use in theAACMMparameter identification procedure. Even though inthis work 4 LTs have been used to eliminate the 𝑍 coordinatesign ambiguity of the multilaterated points, it is possibleto realize this procedure using only 3 LTs by making surethat all of the points have the same sign with respect to themultilateration reference system.This way, in the cases whenaccess to this type of measuring instruments is available, itis possible to carry out an AACMM identification procedurewithout the use of common physical gauges used in this typeof procedures.

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

Conflict of Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper.

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