Post on 08-Jul-2018
Research ArticleArticulated Arm Coordinate Measuring Machine Calibration byLaser Tracker Multilateration
Jorge Santolaria Ana C Majarena David Samper Agustiacuten Brau and Jesuacutes Velaacutezquez
Departamento de Ingenierıa deDiseno y Fabricacion Edificio Torres Quevedo EINAUniversidad de Zaragoza 50018 Zaragoza Spain
Correspondence should be addressed to Jorge Santolaria jsmazounizares
Received 27 August 2013 Accepted 3 November 2013 Published 29 January 2014
Academic Editors G Huang and D Veeger
Copyright copy 2014 Jorge Santolaria et alThis is an open access article distributed under the Creative CommonsAttribution Licensewhich 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 paperFirst 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 inAACMMrsquos because of their advantages in terms of accuracyportability 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 CMMrsquos UNE-EN ISO 10360 ASME B8941 andVDIVDE 2617 and are still carried out today to compare andevaluate the accuracy of an arm from the point of view ofthe CMMrsquos 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 theANSIASME B89 volumetric performance test showing goodagreement between the two approaches and a cost reductionIn [2] Shimojima et al presented a new method to estimate
the uncertainty of a measuring arm using a tridimensionalgaugeThis 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 orientationsand 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 [3ndash5]
However the AACMM presents different characteristicsand different verification procedures are therefore requiredA 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 AACMMrsquos given the difficulty of directly
Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 681853 11 pageshttpdxdoiorg1011552014681853
2 The Scientific World Journal
relating the error committed with the model parameterswhich are obtained by using optimization procedures
The calibration procedure consists of identifying the geo-metric parameters in order to improve the model accuracyThis 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 identificationmodel evaluation andfinally 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 existand the location of the axis coordinate systemmay be definedarbitrarily Some studies [7ndash10] 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 axesNongeometric 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 establishedproviding the global reference system can be measured bymeans of an external measurement instrument such as alaser tracker or a coordinate measuring machine In this waythis transformation obtains the coordinates of the measuredpoints in the global reference system of the mechanismThus 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 anothersince the errors committed by each arm will depend on theirconfiguration and assembly defectsThe performance of teststhat characterize the influence of each joint on the final errorto finally choose positions in accordance with this influencemust 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 performedmeasurementsThis 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 algorithmThis method usually presents lower computational costproviding 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-NewtonA multiobjective optimization scheme is developed in [17] tosolve the nonlinear optimization problem In [18] Santolariaet al presented a kinematic parameter estimation techniquewhich 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 002mm anda measurement uncertainty from 002 to 164 120583m in pointmeasurements to evaluate the measurement performance ofAACMMrsquos The AACMM performance test using the virtual
The Scientific World Journal 3
sphere plate resulted in a mean error of 0023mm and astandard deviation of 0039mm
A laser tracker is a large-scale measuring instrument withhigh accuracyThesemechanisms 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 trackingtesting calibration and maintenance These systems useinterferometry for measuring relative distances and opticalencoders in order to measure azimuth and elevation anglesof a beam-steering mirrorThe 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 retroreflectorthere is no parallel displacement between the emitted andthe reflected beam However when the retroreflector startsmoving there is a displacement of the reflected laser beamsince 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 ASMEB554 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 themechanismaccuracyThismethod 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 propagationobtaining an average standard deviation of length mea-surements around 12 120583m Hughes et al [26] presented alaser-interferometric measuring station and obtained thedisplacement measurement uncertainty which was used topredict a volumetric uncertainty for a multilateration systemconsisting 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 120583m
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
119910 = 119891 (120579119894 119902) (1)
with 119894 = 1 119899 for an arm with 119899 rotating joints 120579119894is given
by the vector of the joint variables and119891 represents themodeldefined which depends on the parameter vector 119902
The non-linear equation system to model the mechanismcan be developed by applying the Denavit-Hartenberg (D-H)method [6] to every chain in themechanismThismethodhasbeen widely used in mechanism modelling [28 29] and usesfour parameters (distances 119889
119894 119886119894 and angles 120579
119894 120572119894) to model
the coordinate transformation between successive referencesystems The homogenous transformation matrix betweenframe 119894 and 119894 minus 1 depends on these four parameters
119894minus1119860119894
= 119879119911119889
sdot 119877119911120579
sdot 119879119909119886
sdot 119877119909120572
=
[[[[
[
cos 120579119894minus cos120572
119894sdot sin 120579
119894sin120572119894sdot sin 120579
119894119886119894sdot cos 120579
119894
sin 120579119894
cos120572119894sdot cos 120579
119894minus sin120572
119894sdot cos 120579
119894119886119894sdot sin 120579
119894
0 sin120572119894
cos120572119894
119889119894
0 0 0 1
]]]]
]
(2)
The AACMMmodel used in this work is a Faro PlatinumArm having seven axes with a nominal value of 2120590 in thesingle point articulation and performance test of 0030mmaccording 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 oneas shown in the following
[119883 119884 119885 1]1015840
AACMM =01198796sdot [119883 119884 119885 1]
1015840
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 probeHowever the aim of this study is to obtain the sphere center
4 The Scientific World Journal
X0Y0
Z0
X4
Y4Z4
X5
Y5Z5
X6
Y6
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 Thusthe 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 measurements119898119894 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 trackerrespectively Consider
1198982
0= (119909 minus 119909
1)2+ (119910 minus 119910
1)2+ (119911 minus 119911
1)2 (4)
1198982
1= (119909 minus 119909
2)2+ (119910 minus 119910
2)2+ (119911 minus 119911
2)2 (5)
1198982
2= (119909 minus 119909
3)2+ (119910 minus 119910
3)2+ (119911 minus 119911
3)2 (6)
The knowledge of the locations of the reference systems ofthe laser trackers allows us to obtain the following equations
119909 =(1198982
0minus 1198982
1+ 1199092
2)
21199091
(7)
Reflector (XY Z)
m3
P3(X3 Y3 Z3)
m0
m2
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
119910 =(1198982
0minus 1198982
2+ 1199092
2+ 1199102
2minus 2119909119909
2)
21199102
(8)
119911 = plusmn(1198982
0minus 1199092minus 1199102)12
(9)
In thiswork for simplicity the unknown reference systemlocation of the three laser trackers has been defined as(0 0 0) (119909
1 0 0) and (119909
2 1199102 0) with respect to the multilat-
eration reference systemA fourth laser tracker can be used to avoid the sign
ambiguity in the 119885 coordinate obtained in (9) The referencesystem of this laser tracker is given by (119909
3 1199103 1199113) This point
should belong to a plane different from the plane119883119884 formedby the other three laser tracker reference systems as shown inFigure 2 In this figure 119875
119894represents LT
119894position system for
the 119894 laser trackers The origin of the multilateration globalreference system is given by 119875
0
The (119909 119910 119911) target coordinates are obtained in linearmatrix form by operating (7) (8) and (9) as expressed in thefollowing
[
[
119909
119910
119911
]
]
= minus 05 sdot
[[[[[[[[[[
[
1
1199091
0 0
minus1199092
11990911199102
1
1199102
0
minus(1199093
11990911199113
) + (11990921199103
119909111991021199113
) minus1199103
11991021199113
1
1199113
]]]]]]]]]]
]
sdot
[[[[
[
1198982
1minus 1198982
0minus 1199092
1
1198982
2minus 1198982
0minus 1199092
2minus 1199102
2
1198982
3minus 1198982
0minus 1199092
3minus 1199102
3minus 1199112
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
The Scientific World Journal 5
Table 1 Initial values for the AACMMD-H parameters
Joint 120579119894(∘) 120572
119894(∘) 119886
119894(mm) 119889
119894(mm)
1 minus90 minus90 minus40 3002 135 minus90 minus40 03 180 minus90 minus34 5904 minus90 90 34 05 minus90 minus90 minus34 5886 180 minus90 minus34 07 0 0 0 3182
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 119883 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
119894positions 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
120ordm
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 15inches 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 120579
119894 These
angles and the AACMM parameters are the input to thekinematic modelThe 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 119889
119894119895 AACMM where 119894 and 119895
represent two measured positionsAt the same time the four LTs simultaneously measured
the distance from the captured point to the local coordinatesystem
6 The Scientific World Journal
Xc Yc Zc =
Xi Yi Zi
Xi Yi Zi
Figure 5 Retroreflector center
Diagonal distances are obtained according to the follow-ing
119903 = 119898 + 119897 (11)
where 119898 is the incremental output of the displacementtransducer used and 119897 is the offset which should be calibratedbefore the multilateration implementation
The self-calibration and determination of the parameter119897 can be carried out by capturing some additional objectivemeasurements in some different positions By performingthe quadrilateration 119896 objective point times the number ofequations is given by 4119896 as shown in the following
(119903119896
119894)2
= (119898119896
119894+ 119897119894)2
= (119909119896minus 119909119894)2
+ (119910119896minus 119910119894)2
+ (119911119896minus 119911119894)2
(12)
for 119894 = 0 3 and 119896 = 0 119896 minus 1The system unknowns are the four offsets 119897
119894 and the
position of the coordinates 119909119896 119910119896 119911119896 (3119896 unknowns) Toidentify the four references 12 unknowns correspondingto 119909119894 119910119894 119911119894 must be added Thus the number of system
unknowns is given by 3119896 + 16The unknowns can be obtained by the least-square
method to minimize the errorThe multilateration technique must solve the objective
function defined by
120601 =
21
sum
119894=1
4
sum
119895=1
[(119909119894minus 119909LT119895)
2
+ (119910119894minus 119910LT119895)
2
+ (119911119894minus 119911LT119895)
2
minus (119898119894119895+ 119897LT119895)
2
]
(13)
where (119909119894 119910119894 119911119894) represents the measured point coordinates
in the multilateration reference system (119909LT119895 119910LT119895 119911LT119895) isthe origin of the LT
119895reference system 119898
119894119895are the measured
distances for every point in the LT119895reference system and 119897LT119895
represents the LT119895offset 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(119909119894 119910119894 119911119894)|SR Multi119897
(ii) the laser tracker offsets 119897LT119895
(iii) the origin of the reference systems for the four lasertrackers in the multilateration system (119909LT119895 119910LT119895
119911LT119895)|SR Multi119897
The calculated distances between each pair of retroreflec-tor positions can be obtained from the point coordinates inthe multilateration system obtaining 119889
119894119895 Multi119897 where 119894 and 119895
represent 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 procedureThe origin of the laser trackers wasmeasuredwith respect
to the reference system of LT1This reference system has been
considered themultilateration reference system for obtainingthe initial values of the LT
119895reference 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(119886119894 120572119894 119889119894 1205790119894 120579119894Enc) where 120579
119894Enc are the angles measured bythe encoder The model obtained is a non-linear equationsystem and it was solved by the L-M algorithmThe objectivefunction defined in this step considers both themeasured andcalculated distances as shown in
120601 =
119901
sum
119894=1119895=1
[(119889119894119895AACMM
minus 119889119894119895Multi119897
)2
] +
119899
sum
119894=1
[1205902
119909119894+ 1205902
119910119894+ 1205902
119911119894]
(14)
where 119901 represents the number of positions consideredin the parameter identification procedure and (120590
119883119894119895 120590119884119894119895
120590119885119894119895
) 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 procedurenull 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 locatedIn Figure 7 the coordinates of the captured points by
each LT and the coordinates of the multilaterated points
The Scientific World Journal 7
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 120579i(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 lt tolerance)
For i j = pos ini to pos final
120601 =21sum
i=1j=1[(dijAACMM minus dijmultil )
2 +nsumi=1
[1205902x119894 + 1205902y119894 + 1205902z119894 ]
Optimisation to minimize Ferror
Calculation of the geometric parameters Si
values
120601 =21sumi=1
4sumj=1
[(xi minus xLTj)2 + (yi minus yLTj)
2 + (zi minus zLTj)2 minus (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 Thuseach 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 120583m while the mean range for all set ofdistances was 43 120583m
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 MoreoverTable 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 119901 = 21 reflector points it isnecessary to consider the elimination of the termswhere 119894 = 119895
to avoid both the inclusion of null terms and duplicate of
8 The Scientific World Journal
Table 2 Multilaterated offsets and origin coordinates obtained in multilateration reference system 2798 iterations and objective functionvalue below 1 120583m
Offsets (mm) Origin coordinates (mm)119909LT 119910LT 119911LT
LT1 119897LT1 151695 0 0 0LT2 119897LT2 089275 226233138 0 0LT3 119897LT3 018749 171300335 209893521 0LT4 119897LT4 213351 191300558 minus34341959 301023578
Table 3 Quality indicators for the initial values of model parameters over 21 SMR locations (105 AACMM positions)
Distance error (mm) 2120590 by retroreflector point (mm)Maximum 3850723 Maximum 2200413Causing dist 4ndash10 Causing point 17Medium 1728859 Causing coord X
Medium 817533
Table 4 Identified values for the AACMMmodel parameters by L-M algorithm
Joint 120579119894(∘) 120572
119894(∘) 119886
119894(mm) 119889
119894(mm)
1 minus891269 minus899216 minus424313 3002 1347734 minus897178 minus417885 123373 1842704 minus901202 minus289084 59124234 minus954182 895364 294449 063005 minus880773 minus899482 minus285745 59152846 1810832 925750 275479 1221437 02117 0 04124 2540575
distance errors noting that 119889119894119895= 119889119895119894 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 positionsIn 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 reasonthe 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 120583m and a mean error of 48 120583m 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 arewhen 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 4927164mm The distanceobtained using the mean points expressed in the AACMMreference system of the 10 probed points of each point and theidentified parameters was 4927695mm obtaining an errorin this case with respect to the nominal of 53 120583m while themaximum standard deviation for the two probed points hasbeen calculated in the119883 coordinate of the first probed pointwith a value of 01426mm for all the 10 captured positionsIt is therefore possible to conclude that the obtained errorvalue for the identified parameter set can be generalizable to
The Scientific World Journal 9
Table 5 Quality indicators for the identified values of model parameters over 21 SMR locations (105 AACMM positions)
Distance error (mm) 2120590 by retroreflector point (mm)Maximum 011824 Maximum 020644Causing dist 2ndash12 Causing point 4Medium 004825 Causing coord Y
Medium 011545
500
1000
1500
30003500
minus2000
minus1800
minus1600
minus1400
minus1200
Identification points
Evaluation points
X(m
m)
Z(m
m)
Y (mm)
(a)
minus20000
20004000
2000
4000
minus1000
minus500
0
500
1000
X(mm)
Z(m
m)
LT1
LT2
LT3
LT4
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 (120583
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 LTrsquos
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 LTsAlthough 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 uncertaintytaking as nominal data only the measurements of the LTsBy combining the aforementioned measurements and after
10 The Scientific World Journal
minus10000
1000
minus10000
1000
minus1500
minus1000
minus500
0
500
1000
1500
X (mm)
Z(m
m)
Y (mm)
(a)
X (mm)
Z(m
m)
200400
600800
1000
minus2000200400600800
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 119885 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 systemThis 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
References
[1] A Piratelli-Filho and G R Lesnau ldquoVirtual spheres gauge forcoordinate measuring arms performance testrdquo Measurementvol 43 no 2 pp 236ndash244 2010
[2] K Shimojima R Furutani K Takamasu and K Araki ldquoTheestimation method of uncertainty of articulated coordinatemeasuring machinerdquo in Proceedings of the IEEE InternationalConference on Industrial Technology pp 411ndash415 DubrovnikCroatia 2002
[3] H Hamana M Tominaga M Ozaki and R FurutanildquoCalibration of articulated arm coordinate measuring machine
considering measuring posturerdquo International Journal ofAutomation Technology vol 5 no 2 pp 109ndash114 2011
[4] J Sładek K Ostrowska and A Gąska ldquoModeling and identifi-cation of errors of coordinate measuring arms with the use of ametrological modelrdquo Measurement vol 46 no 1 pp 667ndash6792013
[5] J Sładek K Ostrowska and K Gacek ldquoKinematic metrologicalmodel of the coordinate measuring arm (MCMA)rdquo in Pro-ceedings of the 19th IMEKO World Congress Fundamental andApplied Metrology pp 1988ndash1992 Lisbon Portugal 2009
[6] J Denavit and R S Hartenberg ldquoA kinematic notation forlower-pair mechanisms based on matricesrdquo Transactions ofASME-Journal of Applied Mechanics vol 22 no 1 pp 215ndash2211955
[7] M Miroir Y Nguyen J Szewczyk O Sterkers and A BozorgGrayeli ldquoDesign kinematic optimization and evaluation of ateleoperated system for middle ear microsurgeryrdquoThe ScientificWorld Journal vol 2012 Article ID 907372 19 pages 2012
[8] T W Hsu and L J Everett ldquoIdentification of the kinematicparameters of a robot manipulator for positional accuracyimprovementrdquo in Proceedings of the Computation in EngineeringConference pp 263ndash267 Boston Mass USA August 1985
[9] B W Mooring ldquoThe effect of joint axis misalignment onrobot positioning accuracyrdquo in Proceedings of the Computers inengineering International Computers in Engineering Conferenceand Exhibit pp 151ndash155 Chicago Ill USA August 1983
[10] H Stone A Sanderson and C Neuman ldquoArm signature identi-ficationrdquo in Proceedings of the IEEE International Conference onRobotics and Automation pp 41ndash48 San Francisco Calif USAApril 1986
[11] L J Everett and A H Suryohadiprojo ldquoA study of kinematicmodels for forward calibration of manipulatorsrdquo in Proceedingsof the IEEE International Conference on Robotics and Automa-tion pp 798ndash800 Philadelphia Pa USA April 1988
The Scientific World Journal 11
[12] J Chen and L M Chao ldquoPositioning error analysis for robotmanipulators with all rotary jointsrdquo IEEE Journal of Roboticsand Automation vol RA-3 no 6 pp 539ndash545 1987
[13] A C Majarena J Santolaria D Samper and J J AguilarldquoAn overview of kinematic and calibration models using inter-nalexternal sensors or constraints to improve the behavior ofspatial parallel mechanismsrdquo Sensors vol 10 no 11 pp 10256ndash10297 2010
[14] I Kovac and A Frank ldquoTesting and calibration of coordinatemeasuring armsrdquo Precision Engineering vol 25 no 2 pp 90ndash99 2001
[15] D Zheng CDu andYHu ldquoResearch on optimalmeasurementarea of flexible coordinate measuring machinesrdquoMeasurementvol 45 no 3 pp 250ndash254 2012
[16] K Levenberg ldquoA method for the solution of certain non-linearproblems in least squaresrdquo Quarterly of Applied Mathematicsvol 2 no 2 pp 164ndash168 1944
[17] Q Gao J Chen L Wang S Xu and Y Hou ldquoMultiobjectiveoptimization design of a fractional order PID controller fora gun control systemrdquo The Scientific World Journal vol 2013Article ID 907256 8 pages 2013
[18] J Santolaria J Aguilar J Yague and J Pastor ldquoKinematicparameter estimation technique for calibration and repeata-bility improvement of articulated arm coordinate measuringmachinesrdquo Precision Engineering vol 32 no 4 pp 251ndash2682008
[19] Y Koseki T Arai K Sugimoto T Takatuji and M GotoldquoDesign and accuracy evaluation of high-speed and highprecision parallel mechanismrdquo in Proceedings of the 1998 IEEEInternational Conference on Robotics and Automation Part 1 (of4) pp 1340ndash1345 May 1998
[20] A Piratelli-Filho F H T Fernandes and R V ArencibialdquoApplication of virtual spheres plate for AACMMs evaluationrdquoPrecision Engineering vol 36 no 2 pp 349ndash355 2012
[21] Z Wang L Mastrogiacomo F Franceschini and P Maropou-los ldquoExperimental comparison of dynamic tracking perfor-mance of iGPS and laser trackerrdquo International Journal ofAdvanced Manufacturing Technology vol 56 no 1ndash4 pp 205ndash213 2011
[22] C Lin and J Her ldquoCalibrating the volumetric errors of aprecision machine by a laser tracker systemrdquo InternationalJournal of Advanced Manufacturing Technology vol 26 no 11-12 pp 1255ndash1267 2005
[23] S Aguado D Samper J Santolaria and J J Aguilar ldquoIdentifi-cation strategy of error parameter in volumetric error compen-sation of machine tool based on laser tracker measurementsrdquoInternational Journal of Machine Tools andManufacture vol 53no 1 pp 160ndash169 2012
[24] T Takatsuji M Goto A Kirita T Kurosawa and Y TanimuraldquoThe relationship between the measurement error and thearrangement of laser trackers in laser trilaterationrdquo Measure-ment Science and Technology vol 11 no 5 pp 477ndash483 2000
[25] D F Zhang S Rolt and P G Maropoulos ldquoModelling andoptimization of novel laser multilateration schemes for high-precision applicationsrdquo Measurement Science and Technologyvol 16 no 12 pp 2541ndash2547 2005
[26] E B Hughes A Wilson and G N Peggs ldquoDesign of a high-accuracy CMM based on multi-lateration techniquesrdquo CIRPAnnals-Manufacturing Technology vol 49 no 1 pp 391ndash3942000
[27] S W Kim H G Rhee and J Y Chu ldquoVolumetric phase-measuring interferometer for three-dimensional coordinate
metrologyrdquo Precision Engineering vol 27 no 2 pp 205ndash2152003
[28] J Santolaria A Brau J Velzquez and J J Aguilar ldquoA self-centering active probing technique for kinematic parameteridentification and verification of articulated arm coordinatemeasuring machinesrdquo Measurement Science and Technologyvol 21 no 5 Article ID 055101 2010
[29] W K Kim Y K Byun and H S Cho ldquoClosed-form forward-position solution for a 6-DoF 3-PPSP parallel mechanism andits implementationrdquo International Journal of Robotics Researchvol 20 no 1 pp 85ndash99 2001
[30] S Aguado J Santolaria D Samper and J Aguilar ldquoInfluenceof measurement noise and laser arrangement on measurementuncertainty of laser tracker multilateration in machine toolvolumetric verificationrdquo Precision Engineering vol 37 pp 929ndash943 2013
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2 The Scientific World Journal
relating the error committed with the model parameterswhich are obtained by using optimization procedures
The calibration procedure consists of identifying the geo-metric parameters in order to improve the model accuracyThis 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 identificationmodel evaluation andfinally 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 existand the location of the axis coordinate systemmay be definedarbitrarily Some studies [7ndash10] 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 axesNongeometric 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 establishedproviding the global reference system can be measured bymeans of an external measurement instrument such as alaser tracker or a coordinate measuring machine In this waythis transformation obtains the coordinates of the measuredpoints in the global reference system of the mechanismThus 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 anothersince the errors committed by each arm will depend on theirconfiguration and assembly defectsThe performance of teststhat characterize the influence of each joint on the final errorto finally choose positions in accordance with this influencemust 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 performedmeasurementsThis 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 algorithmThis method usually presents lower computational costproviding 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-NewtonA multiobjective optimization scheme is developed in [17] tosolve the nonlinear optimization problem In [18] Santolariaet al presented a kinematic parameter estimation techniquewhich 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 002mm anda measurement uncertainty from 002 to 164 120583m in pointmeasurements to evaluate the measurement performance ofAACMMrsquos The AACMM performance test using the virtual
The Scientific World Journal 3
sphere plate resulted in a mean error of 0023mm and astandard deviation of 0039mm
A laser tracker is a large-scale measuring instrument withhigh accuracyThesemechanisms 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 trackingtesting calibration and maintenance These systems useinterferometry for measuring relative distances and opticalencoders in order to measure azimuth and elevation anglesof a beam-steering mirrorThe 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 retroreflectorthere is no parallel displacement between the emitted andthe reflected beam However when the retroreflector startsmoving there is a displacement of the reflected laser beamsince 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 ASMEB554 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 themechanismaccuracyThismethod 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 propagationobtaining an average standard deviation of length mea-surements around 12 120583m Hughes et al [26] presented alaser-interferometric measuring station and obtained thedisplacement measurement uncertainty which was used topredict a volumetric uncertainty for a multilateration systemconsisting 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 120583m
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
119910 = 119891 (120579119894 119902) (1)
with 119894 = 1 119899 for an arm with 119899 rotating joints 120579119894is given
by the vector of the joint variables and119891 represents themodeldefined which depends on the parameter vector 119902
The non-linear equation system to model the mechanismcan be developed by applying the Denavit-Hartenberg (D-H)method [6] to every chain in themechanismThismethodhasbeen widely used in mechanism modelling [28 29] and usesfour parameters (distances 119889
119894 119886119894 and angles 120579
119894 120572119894) to model
the coordinate transformation between successive referencesystems The homogenous transformation matrix betweenframe 119894 and 119894 minus 1 depends on these four parameters
119894minus1119860119894
= 119879119911119889
sdot 119877119911120579
sdot 119879119909119886
sdot 119877119909120572
=
[[[[
[
cos 120579119894minus cos120572
119894sdot sin 120579
119894sin120572119894sdot sin 120579
119894119886119894sdot cos 120579
119894
sin 120579119894
cos120572119894sdot cos 120579
119894minus sin120572
119894sdot cos 120579
119894119886119894sdot sin 120579
119894
0 sin120572119894
cos120572119894
119889119894
0 0 0 1
]]]]
]
(2)
The AACMMmodel used in this work is a Faro PlatinumArm having seven axes with a nominal value of 2120590 in thesingle point articulation and performance test of 0030mmaccording 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 oneas shown in the following
[119883 119884 119885 1]1015840
AACMM =01198796sdot [119883 119884 119885 1]
1015840
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 probeHowever the aim of this study is to obtain the sphere center
4 The Scientific World Journal
X0Y0
Z0
X4
Y4Z4
X5
Y5Z5
X6
Y6
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 Thusthe 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 measurements119898119894 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 trackerrespectively Consider
1198982
0= (119909 minus 119909
1)2+ (119910 minus 119910
1)2+ (119911 minus 119911
1)2 (4)
1198982
1= (119909 minus 119909
2)2+ (119910 minus 119910
2)2+ (119911 minus 119911
2)2 (5)
1198982
2= (119909 minus 119909
3)2+ (119910 minus 119910
3)2+ (119911 minus 119911
3)2 (6)
The knowledge of the locations of the reference systems ofthe laser trackers allows us to obtain the following equations
119909 =(1198982
0minus 1198982
1+ 1199092
2)
21199091
(7)
Reflector (XY Z)
m3
P3(X3 Y3 Z3)
m0
m2
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
119910 =(1198982
0minus 1198982
2+ 1199092
2+ 1199102
2minus 2119909119909
2)
21199102
(8)
119911 = plusmn(1198982
0minus 1199092minus 1199102)12
(9)
In thiswork for simplicity the unknown reference systemlocation of the three laser trackers has been defined as(0 0 0) (119909
1 0 0) and (119909
2 1199102 0) with respect to the multilat-
eration reference systemA fourth laser tracker can be used to avoid the sign
ambiguity in the 119885 coordinate obtained in (9) The referencesystem of this laser tracker is given by (119909
3 1199103 1199113) This point
should belong to a plane different from the plane119883119884 formedby the other three laser tracker reference systems as shown inFigure 2 In this figure 119875
119894represents LT
119894position system for
the 119894 laser trackers The origin of the multilateration globalreference system is given by 119875
0
The (119909 119910 119911) target coordinates are obtained in linearmatrix form by operating (7) (8) and (9) as expressed in thefollowing
[
[
119909
119910
119911
]
]
= minus 05 sdot
[[[[[[[[[[
[
1
1199091
0 0
minus1199092
11990911199102
1
1199102
0
minus(1199093
11990911199113
) + (11990921199103
119909111991021199113
) minus1199103
11991021199113
1
1199113
]]]]]]]]]]
]
sdot
[[[[
[
1198982
1minus 1198982
0minus 1199092
1
1198982
2minus 1198982
0minus 1199092
2minus 1199102
2
1198982
3minus 1198982
0minus 1199092
3minus 1199102
3minus 1199112
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
The Scientific World Journal 5
Table 1 Initial values for the AACMMD-H parameters
Joint 120579119894(∘) 120572
119894(∘) 119886
119894(mm) 119889
119894(mm)
1 minus90 minus90 minus40 3002 135 minus90 minus40 03 180 minus90 minus34 5904 minus90 90 34 05 minus90 minus90 minus34 5886 180 minus90 minus34 07 0 0 0 3182
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 119883 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
119894positions 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
120ordm
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 15inches 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 120579
119894 These
angles and the AACMM parameters are the input to thekinematic modelThe 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 119889
119894119895 AACMM where 119894 and 119895
represent two measured positionsAt the same time the four LTs simultaneously measured
the distance from the captured point to the local coordinatesystem
6 The Scientific World Journal
Xc Yc Zc =
Xi Yi Zi
Xi Yi Zi
Figure 5 Retroreflector center
Diagonal distances are obtained according to the follow-ing
119903 = 119898 + 119897 (11)
where 119898 is the incremental output of the displacementtransducer used and 119897 is the offset which should be calibratedbefore the multilateration implementation
The self-calibration and determination of the parameter119897 can be carried out by capturing some additional objectivemeasurements in some different positions By performingthe quadrilateration 119896 objective point times the number ofequations is given by 4119896 as shown in the following
(119903119896
119894)2
= (119898119896
119894+ 119897119894)2
= (119909119896minus 119909119894)2
+ (119910119896minus 119910119894)2
+ (119911119896minus 119911119894)2
(12)
for 119894 = 0 3 and 119896 = 0 119896 minus 1The system unknowns are the four offsets 119897
119894 and the
position of the coordinates 119909119896 119910119896 119911119896 (3119896 unknowns) Toidentify the four references 12 unknowns correspondingto 119909119894 119910119894 119911119894 must be added Thus the number of system
unknowns is given by 3119896 + 16The unknowns can be obtained by the least-square
method to minimize the errorThe multilateration technique must solve the objective
function defined by
120601 =
21
sum
119894=1
4
sum
119895=1
[(119909119894minus 119909LT119895)
2
+ (119910119894minus 119910LT119895)
2
+ (119911119894minus 119911LT119895)
2
minus (119898119894119895+ 119897LT119895)
2
]
(13)
where (119909119894 119910119894 119911119894) represents the measured point coordinates
in the multilateration reference system (119909LT119895 119910LT119895 119911LT119895) isthe origin of the LT
119895reference system 119898
119894119895are the measured
distances for every point in the LT119895reference system and 119897LT119895
represents the LT119895offset 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(119909119894 119910119894 119911119894)|SR Multi119897
(ii) the laser tracker offsets 119897LT119895
(iii) the origin of the reference systems for the four lasertrackers in the multilateration system (119909LT119895 119910LT119895
119911LT119895)|SR Multi119897
The calculated distances between each pair of retroreflec-tor positions can be obtained from the point coordinates inthe multilateration system obtaining 119889
119894119895 Multi119897 where 119894 and 119895
represent 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 procedureThe origin of the laser trackers wasmeasuredwith respect
to the reference system of LT1This reference system has been
considered themultilateration reference system for obtainingthe initial values of the LT
119895reference 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(119886119894 120572119894 119889119894 1205790119894 120579119894Enc) where 120579
119894Enc are the angles measured bythe encoder The model obtained is a non-linear equationsystem and it was solved by the L-M algorithmThe objectivefunction defined in this step considers both themeasured andcalculated distances as shown in
120601 =
119901
sum
119894=1119895=1
[(119889119894119895AACMM
minus 119889119894119895Multi119897
)2
] +
119899
sum
119894=1
[1205902
119909119894+ 1205902
119910119894+ 1205902
119911119894]
(14)
where 119901 represents the number of positions consideredin the parameter identification procedure and (120590
119883119894119895 120590119884119894119895
120590119885119894119895
) 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 procedurenull 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 locatedIn Figure 7 the coordinates of the captured points by
each LT and the coordinates of the multilaterated points
The Scientific World Journal 7
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 120579i(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 lt tolerance)
For i j = pos ini to pos final
120601 =21sum
i=1j=1[(dijAACMM minus dijmultil )
2 +nsumi=1
[1205902x119894 + 1205902y119894 + 1205902z119894 ]
Optimisation to minimize Ferror
Calculation of the geometric parameters Si
values
120601 =21sumi=1
4sumj=1
[(xi minus xLTj)2 + (yi minus yLTj)
2 + (zi minus zLTj)2 minus (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 Thuseach 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 120583m while the mean range for all set ofdistances was 43 120583m
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 MoreoverTable 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 119901 = 21 reflector points it isnecessary to consider the elimination of the termswhere 119894 = 119895
to avoid both the inclusion of null terms and duplicate of
8 The Scientific World Journal
Table 2 Multilaterated offsets and origin coordinates obtained in multilateration reference system 2798 iterations and objective functionvalue below 1 120583m
Offsets (mm) Origin coordinates (mm)119909LT 119910LT 119911LT
LT1 119897LT1 151695 0 0 0LT2 119897LT2 089275 226233138 0 0LT3 119897LT3 018749 171300335 209893521 0LT4 119897LT4 213351 191300558 minus34341959 301023578
Table 3 Quality indicators for the initial values of model parameters over 21 SMR locations (105 AACMM positions)
Distance error (mm) 2120590 by retroreflector point (mm)Maximum 3850723 Maximum 2200413Causing dist 4ndash10 Causing point 17Medium 1728859 Causing coord X
Medium 817533
Table 4 Identified values for the AACMMmodel parameters by L-M algorithm
Joint 120579119894(∘) 120572
119894(∘) 119886
119894(mm) 119889
119894(mm)
1 minus891269 minus899216 minus424313 3002 1347734 minus897178 minus417885 123373 1842704 minus901202 minus289084 59124234 minus954182 895364 294449 063005 minus880773 minus899482 minus285745 59152846 1810832 925750 275479 1221437 02117 0 04124 2540575
distance errors noting that 119889119894119895= 119889119895119894 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 positionsIn 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 reasonthe 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 120583m and a mean error of 48 120583m 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 arewhen 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 4927164mm The distanceobtained using the mean points expressed in the AACMMreference system of the 10 probed points of each point and theidentified parameters was 4927695mm obtaining an errorin this case with respect to the nominal of 53 120583m while themaximum standard deviation for the two probed points hasbeen calculated in the119883 coordinate of the first probed pointwith a value of 01426mm for all the 10 captured positionsIt is therefore possible to conclude that the obtained errorvalue for the identified parameter set can be generalizable to
The Scientific World Journal 9
Table 5 Quality indicators for the identified values of model parameters over 21 SMR locations (105 AACMM positions)
Distance error (mm) 2120590 by retroreflector point (mm)Maximum 011824 Maximum 020644Causing dist 2ndash12 Causing point 4Medium 004825 Causing coord Y
Medium 011545
500
1000
1500
30003500
minus2000
minus1800
minus1600
minus1400
minus1200
Identification points
Evaluation points
X(m
m)
Z(m
m)
Y (mm)
(a)
minus20000
20004000
2000
4000
minus1000
minus500
0
500
1000
X(mm)
Z(m
m)
LT1
LT2
LT3
LT4
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 (120583
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 LTrsquos
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 LTsAlthough 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 uncertaintytaking as nominal data only the measurements of the LTsBy combining the aforementioned measurements and after
10 The Scientific World Journal
minus10000
1000
minus10000
1000
minus1500
minus1000
minus500
0
500
1000
1500
X (mm)
Z(m
m)
Y (mm)
(a)
X (mm)
Z(m
m)
200400
600800
1000
minus2000200400600800
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 119885 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 systemThis 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
References
[1] A Piratelli-Filho and G R Lesnau ldquoVirtual spheres gauge forcoordinate measuring arms performance testrdquo Measurementvol 43 no 2 pp 236ndash244 2010
[2] K Shimojima R Furutani K Takamasu and K Araki ldquoTheestimation method of uncertainty of articulated coordinatemeasuring machinerdquo in Proceedings of the IEEE InternationalConference on Industrial Technology pp 411ndash415 DubrovnikCroatia 2002
[3] H Hamana M Tominaga M Ozaki and R FurutanildquoCalibration of articulated arm coordinate measuring machine
considering measuring posturerdquo International Journal ofAutomation Technology vol 5 no 2 pp 109ndash114 2011
[4] J Sładek K Ostrowska and A Gąska ldquoModeling and identifi-cation of errors of coordinate measuring arms with the use of ametrological modelrdquo Measurement vol 46 no 1 pp 667ndash6792013
[5] J Sładek K Ostrowska and K Gacek ldquoKinematic metrologicalmodel of the coordinate measuring arm (MCMA)rdquo in Pro-ceedings of the 19th IMEKO World Congress Fundamental andApplied Metrology pp 1988ndash1992 Lisbon Portugal 2009
[6] J Denavit and R S Hartenberg ldquoA kinematic notation forlower-pair mechanisms based on matricesrdquo Transactions ofASME-Journal of Applied Mechanics vol 22 no 1 pp 215ndash2211955
[7] M Miroir Y Nguyen J Szewczyk O Sterkers and A BozorgGrayeli ldquoDesign kinematic optimization and evaluation of ateleoperated system for middle ear microsurgeryrdquoThe ScientificWorld Journal vol 2012 Article ID 907372 19 pages 2012
[8] T W Hsu and L J Everett ldquoIdentification of the kinematicparameters of a robot manipulator for positional accuracyimprovementrdquo in Proceedings of the Computation in EngineeringConference pp 263ndash267 Boston Mass USA August 1985
[9] B W Mooring ldquoThe effect of joint axis misalignment onrobot positioning accuracyrdquo in Proceedings of the Computers inengineering International Computers in Engineering Conferenceand Exhibit pp 151ndash155 Chicago Ill USA August 1983
[10] H Stone A Sanderson and C Neuman ldquoArm signature identi-ficationrdquo in Proceedings of the IEEE International Conference onRobotics and Automation pp 41ndash48 San Francisco Calif USAApril 1986
[11] L J Everett and A H Suryohadiprojo ldquoA study of kinematicmodels for forward calibration of manipulatorsrdquo in Proceedingsof the IEEE International Conference on Robotics and Automa-tion pp 798ndash800 Philadelphia Pa USA April 1988
The Scientific World Journal 11
[12] J Chen and L M Chao ldquoPositioning error analysis for robotmanipulators with all rotary jointsrdquo IEEE Journal of Roboticsand Automation vol RA-3 no 6 pp 539ndash545 1987
[13] A C Majarena J Santolaria D Samper and J J AguilarldquoAn overview of kinematic and calibration models using inter-nalexternal sensors or constraints to improve the behavior ofspatial parallel mechanismsrdquo Sensors vol 10 no 11 pp 10256ndash10297 2010
[14] I Kovac and A Frank ldquoTesting and calibration of coordinatemeasuring armsrdquo Precision Engineering vol 25 no 2 pp 90ndash99 2001
[15] D Zheng CDu andYHu ldquoResearch on optimalmeasurementarea of flexible coordinate measuring machinesrdquoMeasurementvol 45 no 3 pp 250ndash254 2012
[16] K Levenberg ldquoA method for the solution of certain non-linearproblems in least squaresrdquo Quarterly of Applied Mathematicsvol 2 no 2 pp 164ndash168 1944
[17] Q Gao J Chen L Wang S Xu and Y Hou ldquoMultiobjectiveoptimization design of a fractional order PID controller fora gun control systemrdquo The Scientific World Journal vol 2013Article ID 907256 8 pages 2013
[18] J Santolaria J Aguilar J Yague and J Pastor ldquoKinematicparameter estimation technique for calibration and repeata-bility improvement of articulated arm coordinate measuringmachinesrdquo Precision Engineering vol 32 no 4 pp 251ndash2682008
[19] Y Koseki T Arai K Sugimoto T Takatuji and M GotoldquoDesign and accuracy evaluation of high-speed and highprecision parallel mechanismrdquo in Proceedings of the 1998 IEEEInternational Conference on Robotics and Automation Part 1 (of4) pp 1340ndash1345 May 1998
[20] A Piratelli-Filho F H T Fernandes and R V ArencibialdquoApplication of virtual spheres plate for AACMMs evaluationrdquoPrecision Engineering vol 36 no 2 pp 349ndash355 2012
[21] Z Wang L Mastrogiacomo F Franceschini and P Maropou-los ldquoExperimental comparison of dynamic tracking perfor-mance of iGPS and laser trackerrdquo International Journal ofAdvanced Manufacturing Technology vol 56 no 1ndash4 pp 205ndash213 2011
[22] C Lin and J Her ldquoCalibrating the volumetric errors of aprecision machine by a laser tracker systemrdquo InternationalJournal of Advanced Manufacturing Technology vol 26 no 11-12 pp 1255ndash1267 2005
[23] S Aguado D Samper J Santolaria and J J Aguilar ldquoIdentifi-cation strategy of error parameter in volumetric error compen-sation of machine tool based on laser tracker measurementsrdquoInternational Journal of Machine Tools andManufacture vol 53no 1 pp 160ndash169 2012
[24] T Takatsuji M Goto A Kirita T Kurosawa and Y TanimuraldquoThe relationship between the measurement error and thearrangement of laser trackers in laser trilaterationrdquo Measure-ment Science and Technology vol 11 no 5 pp 477ndash483 2000
[25] D F Zhang S Rolt and P G Maropoulos ldquoModelling andoptimization of novel laser multilateration schemes for high-precision applicationsrdquo Measurement Science and Technologyvol 16 no 12 pp 2541ndash2547 2005
[26] E B Hughes A Wilson and G N Peggs ldquoDesign of a high-accuracy CMM based on multi-lateration techniquesrdquo CIRPAnnals-Manufacturing Technology vol 49 no 1 pp 391ndash3942000
[27] S W Kim H G Rhee and J Y Chu ldquoVolumetric phase-measuring interferometer for three-dimensional coordinate
metrologyrdquo Precision Engineering vol 27 no 2 pp 205ndash2152003
[28] J Santolaria A Brau J Velzquez and J J Aguilar ldquoA self-centering active probing technique for kinematic parameteridentification and verification of articulated arm coordinatemeasuring machinesrdquo Measurement Science and Technologyvol 21 no 5 Article ID 055101 2010
[29] W K Kim Y K Byun and H S Cho ldquoClosed-form forward-position solution for a 6-DoF 3-PPSP parallel mechanism andits implementationrdquo International Journal of Robotics Researchvol 20 no 1 pp 85ndash99 2001
[30] S Aguado J Santolaria D Samper and J Aguilar ldquoInfluenceof measurement noise and laser arrangement on measurementuncertainty of laser tracker multilateration in machine toolvolumetric verificationrdquo Precision Engineering vol 37 pp 929ndash943 2013
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The Scientific World Journal 3
sphere plate resulted in a mean error of 0023mm and astandard deviation of 0039mm
A laser tracker is a large-scale measuring instrument withhigh accuracyThesemechanisms 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 trackingtesting calibration and maintenance These systems useinterferometry for measuring relative distances and opticalencoders in order to measure azimuth and elevation anglesof a beam-steering mirrorThe 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 retroreflectorthere is no parallel displacement between the emitted andthe reflected beam However when the retroreflector startsmoving there is a displacement of the reflected laser beamsince 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 ASMEB554 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 themechanismaccuracyThismethod 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 propagationobtaining an average standard deviation of length mea-surements around 12 120583m Hughes et al [26] presented alaser-interferometric measuring station and obtained thedisplacement measurement uncertainty which was used topredict a volumetric uncertainty for a multilateration systemconsisting 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 120583m
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
119910 = 119891 (120579119894 119902) (1)
with 119894 = 1 119899 for an arm with 119899 rotating joints 120579119894is given
by the vector of the joint variables and119891 represents themodeldefined which depends on the parameter vector 119902
The non-linear equation system to model the mechanismcan be developed by applying the Denavit-Hartenberg (D-H)method [6] to every chain in themechanismThismethodhasbeen widely used in mechanism modelling [28 29] and usesfour parameters (distances 119889
119894 119886119894 and angles 120579
119894 120572119894) to model
the coordinate transformation between successive referencesystems The homogenous transformation matrix betweenframe 119894 and 119894 minus 1 depends on these four parameters
119894minus1119860119894
= 119879119911119889
sdot 119877119911120579
sdot 119879119909119886
sdot 119877119909120572
=
[[[[
[
cos 120579119894minus cos120572
119894sdot sin 120579
119894sin120572119894sdot sin 120579
119894119886119894sdot cos 120579
119894
sin 120579119894
cos120572119894sdot cos 120579
119894minus sin120572
119894sdot cos 120579
119894119886119894sdot sin 120579
119894
0 sin120572119894
cos120572119894
119889119894
0 0 0 1
]]]]
]
(2)
The AACMMmodel used in this work is a Faro PlatinumArm having seven axes with a nominal value of 2120590 in thesingle point articulation and performance test of 0030mmaccording 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 oneas shown in the following
[119883 119884 119885 1]1015840
AACMM =01198796sdot [119883 119884 119885 1]
1015840
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 probeHowever the aim of this study is to obtain the sphere center
4 The Scientific World Journal
X0Y0
Z0
X4
Y4Z4
X5
Y5Z5
X6
Y6
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 Thusthe 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 measurements119898119894 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 trackerrespectively Consider
1198982
0= (119909 minus 119909
1)2+ (119910 minus 119910
1)2+ (119911 minus 119911
1)2 (4)
1198982
1= (119909 minus 119909
2)2+ (119910 minus 119910
2)2+ (119911 minus 119911
2)2 (5)
1198982
2= (119909 minus 119909
3)2+ (119910 minus 119910
3)2+ (119911 minus 119911
3)2 (6)
The knowledge of the locations of the reference systems ofthe laser trackers allows us to obtain the following equations
119909 =(1198982
0minus 1198982
1+ 1199092
2)
21199091
(7)
Reflector (XY Z)
m3
P3(X3 Y3 Z3)
m0
m2
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
119910 =(1198982
0minus 1198982
2+ 1199092
2+ 1199102
2minus 2119909119909
2)
21199102
(8)
119911 = plusmn(1198982
0minus 1199092minus 1199102)12
(9)
In thiswork for simplicity the unknown reference systemlocation of the three laser trackers has been defined as(0 0 0) (119909
1 0 0) and (119909
2 1199102 0) with respect to the multilat-
eration reference systemA fourth laser tracker can be used to avoid the sign
ambiguity in the 119885 coordinate obtained in (9) The referencesystem of this laser tracker is given by (119909
3 1199103 1199113) This point
should belong to a plane different from the plane119883119884 formedby the other three laser tracker reference systems as shown inFigure 2 In this figure 119875
119894represents LT
119894position system for
the 119894 laser trackers The origin of the multilateration globalreference system is given by 119875
0
The (119909 119910 119911) target coordinates are obtained in linearmatrix form by operating (7) (8) and (9) as expressed in thefollowing
[
[
119909
119910
119911
]
]
= minus 05 sdot
[[[[[[[[[[
[
1
1199091
0 0
minus1199092
11990911199102
1
1199102
0
minus(1199093
11990911199113
) + (11990921199103
119909111991021199113
) minus1199103
11991021199113
1
1199113
]]]]]]]]]]
]
sdot
[[[[
[
1198982
1minus 1198982
0minus 1199092
1
1198982
2minus 1198982
0minus 1199092
2minus 1199102
2
1198982
3minus 1198982
0minus 1199092
3minus 1199102
3minus 1199112
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
The Scientific World Journal 5
Table 1 Initial values for the AACMMD-H parameters
Joint 120579119894(∘) 120572
119894(∘) 119886
119894(mm) 119889
119894(mm)
1 minus90 minus90 minus40 3002 135 minus90 minus40 03 180 minus90 minus34 5904 minus90 90 34 05 minus90 minus90 minus34 5886 180 minus90 minus34 07 0 0 0 3182
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 119883 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
119894positions 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
120ordm
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 15inches 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 120579
119894 These
angles and the AACMM parameters are the input to thekinematic modelThe 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 119889
119894119895 AACMM where 119894 and 119895
represent two measured positionsAt the same time the four LTs simultaneously measured
the distance from the captured point to the local coordinatesystem
6 The Scientific World Journal
Xc Yc Zc =
Xi Yi Zi
Xi Yi Zi
Figure 5 Retroreflector center
Diagonal distances are obtained according to the follow-ing
119903 = 119898 + 119897 (11)
where 119898 is the incremental output of the displacementtransducer used and 119897 is the offset which should be calibratedbefore the multilateration implementation
The self-calibration and determination of the parameter119897 can be carried out by capturing some additional objectivemeasurements in some different positions By performingthe quadrilateration 119896 objective point times the number ofequations is given by 4119896 as shown in the following
(119903119896
119894)2
= (119898119896
119894+ 119897119894)2
= (119909119896minus 119909119894)2
+ (119910119896minus 119910119894)2
+ (119911119896minus 119911119894)2
(12)
for 119894 = 0 3 and 119896 = 0 119896 minus 1The system unknowns are the four offsets 119897
119894 and the
position of the coordinates 119909119896 119910119896 119911119896 (3119896 unknowns) Toidentify the four references 12 unknowns correspondingto 119909119894 119910119894 119911119894 must be added Thus the number of system
unknowns is given by 3119896 + 16The unknowns can be obtained by the least-square
method to minimize the errorThe multilateration technique must solve the objective
function defined by
120601 =
21
sum
119894=1
4
sum
119895=1
[(119909119894minus 119909LT119895)
2
+ (119910119894minus 119910LT119895)
2
+ (119911119894minus 119911LT119895)
2
minus (119898119894119895+ 119897LT119895)
2
]
(13)
where (119909119894 119910119894 119911119894) represents the measured point coordinates
in the multilateration reference system (119909LT119895 119910LT119895 119911LT119895) isthe origin of the LT
119895reference system 119898
119894119895are the measured
distances for every point in the LT119895reference system and 119897LT119895
represents the LT119895offset 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(119909119894 119910119894 119911119894)|SR Multi119897
(ii) the laser tracker offsets 119897LT119895
(iii) the origin of the reference systems for the four lasertrackers in the multilateration system (119909LT119895 119910LT119895
119911LT119895)|SR Multi119897
The calculated distances between each pair of retroreflec-tor positions can be obtained from the point coordinates inthe multilateration system obtaining 119889
119894119895 Multi119897 where 119894 and 119895
represent 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 procedureThe origin of the laser trackers wasmeasuredwith respect
to the reference system of LT1This reference system has been
considered themultilateration reference system for obtainingthe initial values of the LT
119895reference 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(119886119894 120572119894 119889119894 1205790119894 120579119894Enc) where 120579
119894Enc are the angles measured bythe encoder The model obtained is a non-linear equationsystem and it was solved by the L-M algorithmThe objectivefunction defined in this step considers both themeasured andcalculated distances as shown in
120601 =
119901
sum
119894=1119895=1
[(119889119894119895AACMM
minus 119889119894119895Multi119897
)2
] +
119899
sum
119894=1
[1205902
119909119894+ 1205902
119910119894+ 1205902
119911119894]
(14)
where 119901 represents the number of positions consideredin the parameter identification procedure and (120590
119883119894119895 120590119884119894119895
120590119885119894119895
) 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 procedurenull 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 locatedIn Figure 7 the coordinates of the captured points by
each LT and the coordinates of the multilaterated points
The Scientific World Journal 7
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 120579i(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 lt tolerance)
For i j = pos ini to pos final
120601 =21sum
i=1j=1[(dijAACMM minus dijmultil )
2 +nsumi=1
[1205902x119894 + 1205902y119894 + 1205902z119894 ]
Optimisation to minimize Ferror
Calculation of the geometric parameters Si
values
120601 =21sumi=1
4sumj=1
[(xi minus xLTj)2 + (yi minus yLTj)
2 + (zi minus zLTj)2 minus (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 Thuseach 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 120583m while the mean range for all set ofdistances was 43 120583m
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 MoreoverTable 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 119901 = 21 reflector points it isnecessary to consider the elimination of the termswhere 119894 = 119895
to avoid both the inclusion of null terms and duplicate of
8 The Scientific World Journal
Table 2 Multilaterated offsets and origin coordinates obtained in multilateration reference system 2798 iterations and objective functionvalue below 1 120583m
Offsets (mm) Origin coordinates (mm)119909LT 119910LT 119911LT
LT1 119897LT1 151695 0 0 0LT2 119897LT2 089275 226233138 0 0LT3 119897LT3 018749 171300335 209893521 0LT4 119897LT4 213351 191300558 minus34341959 301023578
Table 3 Quality indicators for the initial values of model parameters over 21 SMR locations (105 AACMM positions)
Distance error (mm) 2120590 by retroreflector point (mm)Maximum 3850723 Maximum 2200413Causing dist 4ndash10 Causing point 17Medium 1728859 Causing coord X
Medium 817533
Table 4 Identified values for the AACMMmodel parameters by L-M algorithm
Joint 120579119894(∘) 120572
119894(∘) 119886
119894(mm) 119889
119894(mm)
1 minus891269 minus899216 minus424313 3002 1347734 minus897178 minus417885 123373 1842704 minus901202 minus289084 59124234 minus954182 895364 294449 063005 minus880773 minus899482 minus285745 59152846 1810832 925750 275479 1221437 02117 0 04124 2540575
distance errors noting that 119889119894119895= 119889119895119894 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 positionsIn 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 reasonthe 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 120583m and a mean error of 48 120583m 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 arewhen 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 4927164mm The distanceobtained using the mean points expressed in the AACMMreference system of the 10 probed points of each point and theidentified parameters was 4927695mm obtaining an errorin this case with respect to the nominal of 53 120583m while themaximum standard deviation for the two probed points hasbeen calculated in the119883 coordinate of the first probed pointwith a value of 01426mm for all the 10 captured positionsIt is therefore possible to conclude that the obtained errorvalue for the identified parameter set can be generalizable to
The Scientific World Journal 9
Table 5 Quality indicators for the identified values of model parameters over 21 SMR locations (105 AACMM positions)
Distance error (mm) 2120590 by retroreflector point (mm)Maximum 011824 Maximum 020644Causing dist 2ndash12 Causing point 4Medium 004825 Causing coord Y
Medium 011545
500
1000
1500
30003500
minus2000
minus1800
minus1600
minus1400
minus1200
Identification points
Evaluation points
X(m
m)
Z(m
m)
Y (mm)
(a)
minus20000
20004000
2000
4000
minus1000
minus500
0
500
1000
X(mm)
Z(m
m)
LT1
LT2
LT3
LT4
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 (120583
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 LTrsquos
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 LTsAlthough 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 uncertaintytaking as nominal data only the measurements of the LTsBy combining the aforementioned measurements and after
10 The Scientific World Journal
minus10000
1000
minus10000
1000
minus1500
minus1000
minus500
0
500
1000
1500
X (mm)
Z(m
m)
Y (mm)
(a)
X (mm)
Z(m
m)
200400
600800
1000
minus2000200400600800
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 119885 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 systemThis 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
References
[1] A Piratelli-Filho and G R Lesnau ldquoVirtual spheres gauge forcoordinate measuring arms performance testrdquo Measurementvol 43 no 2 pp 236ndash244 2010
[2] K Shimojima R Furutani K Takamasu and K Araki ldquoTheestimation method of uncertainty of articulated coordinatemeasuring machinerdquo in Proceedings of the IEEE InternationalConference on Industrial Technology pp 411ndash415 DubrovnikCroatia 2002
[3] H Hamana M Tominaga M Ozaki and R FurutanildquoCalibration of articulated arm coordinate measuring machine
considering measuring posturerdquo International Journal ofAutomation Technology vol 5 no 2 pp 109ndash114 2011
[4] J Sładek K Ostrowska and A Gąska ldquoModeling and identifi-cation of errors of coordinate measuring arms with the use of ametrological modelrdquo Measurement vol 46 no 1 pp 667ndash6792013
[5] J Sładek K Ostrowska and K Gacek ldquoKinematic metrologicalmodel of the coordinate measuring arm (MCMA)rdquo in Pro-ceedings of the 19th IMEKO World Congress Fundamental andApplied Metrology pp 1988ndash1992 Lisbon Portugal 2009
[6] J Denavit and R S Hartenberg ldquoA kinematic notation forlower-pair mechanisms based on matricesrdquo Transactions ofASME-Journal of Applied Mechanics vol 22 no 1 pp 215ndash2211955
[7] M Miroir Y Nguyen J Szewczyk O Sterkers and A BozorgGrayeli ldquoDesign kinematic optimization and evaluation of ateleoperated system for middle ear microsurgeryrdquoThe ScientificWorld Journal vol 2012 Article ID 907372 19 pages 2012
[8] T W Hsu and L J Everett ldquoIdentification of the kinematicparameters of a robot manipulator for positional accuracyimprovementrdquo in Proceedings of the Computation in EngineeringConference pp 263ndash267 Boston Mass USA August 1985
[9] B W Mooring ldquoThe effect of joint axis misalignment onrobot positioning accuracyrdquo in Proceedings of the Computers inengineering International Computers in Engineering Conferenceand Exhibit pp 151ndash155 Chicago Ill USA August 1983
[10] H Stone A Sanderson and C Neuman ldquoArm signature identi-ficationrdquo in Proceedings of the IEEE International Conference onRobotics and Automation pp 41ndash48 San Francisco Calif USAApril 1986
[11] L J Everett and A H Suryohadiprojo ldquoA study of kinematicmodels for forward calibration of manipulatorsrdquo in Proceedingsof the IEEE International Conference on Robotics and Automa-tion pp 798ndash800 Philadelphia Pa USA April 1988
The Scientific World Journal 11
[12] J Chen and L M Chao ldquoPositioning error analysis for robotmanipulators with all rotary jointsrdquo IEEE Journal of Roboticsand Automation vol RA-3 no 6 pp 539ndash545 1987
[13] A C Majarena J Santolaria D Samper and J J AguilarldquoAn overview of kinematic and calibration models using inter-nalexternal sensors or constraints to improve the behavior ofspatial parallel mechanismsrdquo Sensors vol 10 no 11 pp 10256ndash10297 2010
[14] I Kovac and A Frank ldquoTesting and calibration of coordinatemeasuring armsrdquo Precision Engineering vol 25 no 2 pp 90ndash99 2001
[15] D Zheng CDu andYHu ldquoResearch on optimalmeasurementarea of flexible coordinate measuring machinesrdquoMeasurementvol 45 no 3 pp 250ndash254 2012
[16] K Levenberg ldquoA method for the solution of certain non-linearproblems in least squaresrdquo Quarterly of Applied Mathematicsvol 2 no 2 pp 164ndash168 1944
[17] Q Gao J Chen L Wang S Xu and Y Hou ldquoMultiobjectiveoptimization design of a fractional order PID controller fora gun control systemrdquo The Scientific World Journal vol 2013Article ID 907256 8 pages 2013
[18] J Santolaria J Aguilar J Yague and J Pastor ldquoKinematicparameter estimation technique for calibration and repeata-bility improvement of articulated arm coordinate measuringmachinesrdquo Precision Engineering vol 32 no 4 pp 251ndash2682008
[19] Y Koseki T Arai K Sugimoto T Takatuji and M GotoldquoDesign and accuracy evaluation of high-speed and highprecision parallel mechanismrdquo in Proceedings of the 1998 IEEEInternational Conference on Robotics and Automation Part 1 (of4) pp 1340ndash1345 May 1998
[20] A Piratelli-Filho F H T Fernandes and R V ArencibialdquoApplication of virtual spheres plate for AACMMs evaluationrdquoPrecision Engineering vol 36 no 2 pp 349ndash355 2012
[21] Z Wang L Mastrogiacomo F Franceschini and P Maropou-los ldquoExperimental comparison of dynamic tracking perfor-mance of iGPS and laser trackerrdquo International Journal ofAdvanced Manufacturing Technology vol 56 no 1ndash4 pp 205ndash213 2011
[22] C Lin and J Her ldquoCalibrating the volumetric errors of aprecision machine by a laser tracker systemrdquo InternationalJournal of Advanced Manufacturing Technology vol 26 no 11-12 pp 1255ndash1267 2005
[23] S Aguado D Samper J Santolaria and J J Aguilar ldquoIdentifi-cation strategy of error parameter in volumetric error compen-sation of machine tool based on laser tracker measurementsrdquoInternational Journal of Machine Tools andManufacture vol 53no 1 pp 160ndash169 2012
[24] T Takatsuji M Goto A Kirita T Kurosawa and Y TanimuraldquoThe relationship between the measurement error and thearrangement of laser trackers in laser trilaterationrdquo Measure-ment Science and Technology vol 11 no 5 pp 477ndash483 2000
[25] D F Zhang S Rolt and P G Maropoulos ldquoModelling andoptimization of novel laser multilateration schemes for high-precision applicationsrdquo Measurement Science and Technologyvol 16 no 12 pp 2541ndash2547 2005
[26] E B Hughes A Wilson and G N Peggs ldquoDesign of a high-accuracy CMM based on multi-lateration techniquesrdquo CIRPAnnals-Manufacturing Technology vol 49 no 1 pp 391ndash3942000
[27] S W Kim H G Rhee and J Y Chu ldquoVolumetric phase-measuring interferometer for three-dimensional coordinate
metrologyrdquo Precision Engineering vol 27 no 2 pp 205ndash2152003
[28] J Santolaria A Brau J Velzquez and J J Aguilar ldquoA self-centering active probing technique for kinematic parameteridentification and verification of articulated arm coordinatemeasuring machinesrdquo Measurement Science and Technologyvol 21 no 5 Article ID 055101 2010
[29] W K Kim Y K Byun and H S Cho ldquoClosed-form forward-position solution for a 6-DoF 3-PPSP parallel mechanism andits implementationrdquo International Journal of Robotics Researchvol 20 no 1 pp 85ndash99 2001
[30] S Aguado J Santolaria D Samper and J Aguilar ldquoInfluenceof measurement noise and laser arrangement on measurementuncertainty of laser tracker multilateration in machine toolvolumetric verificationrdquo Precision Engineering vol 37 pp 929ndash943 2013
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DistributedSensor Networks
International Journal of
4 The Scientific World Journal
X0Y0
Z0
X4
Y4Z4
X5
Y5Z5
X6
Y6
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 Thusthe 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 measurements119898119894 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 trackerrespectively Consider
1198982
0= (119909 minus 119909
1)2+ (119910 minus 119910
1)2+ (119911 minus 119911
1)2 (4)
1198982
1= (119909 minus 119909
2)2+ (119910 minus 119910
2)2+ (119911 minus 119911
2)2 (5)
1198982
2= (119909 minus 119909
3)2+ (119910 minus 119910
3)2+ (119911 minus 119911
3)2 (6)
The knowledge of the locations of the reference systems ofthe laser trackers allows us to obtain the following equations
119909 =(1198982
0minus 1198982
1+ 1199092
2)
21199091
(7)
Reflector (XY Z)
m3
P3(X3 Y3 Z3)
m0
m2
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
119910 =(1198982
0minus 1198982
2+ 1199092
2+ 1199102
2minus 2119909119909
2)
21199102
(8)
119911 = plusmn(1198982
0minus 1199092minus 1199102)12
(9)
In thiswork for simplicity the unknown reference systemlocation of the three laser trackers has been defined as(0 0 0) (119909
1 0 0) and (119909
2 1199102 0) with respect to the multilat-
eration reference systemA fourth laser tracker can be used to avoid the sign
ambiguity in the 119885 coordinate obtained in (9) The referencesystem of this laser tracker is given by (119909
3 1199103 1199113) This point
should belong to a plane different from the plane119883119884 formedby the other three laser tracker reference systems as shown inFigure 2 In this figure 119875
119894represents LT
119894position system for
the 119894 laser trackers The origin of the multilateration globalreference system is given by 119875
0
The (119909 119910 119911) target coordinates are obtained in linearmatrix form by operating (7) (8) and (9) as expressed in thefollowing
[
[
119909
119910
119911
]
]
= minus 05 sdot
[[[[[[[[[[
[
1
1199091
0 0
minus1199092
11990911199102
1
1199102
0
minus(1199093
11990911199113
) + (11990921199103
119909111991021199113
) minus1199103
11991021199113
1
1199113
]]]]]]]]]]
]
sdot
[[[[
[
1198982
1minus 1198982
0minus 1199092
1
1198982
2minus 1198982
0minus 1199092
2minus 1199102
2
1198982
3minus 1198982
0minus 1199092
3minus 1199102
3minus 1199112
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
The Scientific World Journal 5
Table 1 Initial values for the AACMMD-H parameters
Joint 120579119894(∘) 120572
119894(∘) 119886
119894(mm) 119889
119894(mm)
1 minus90 minus90 minus40 3002 135 minus90 minus40 03 180 minus90 minus34 5904 minus90 90 34 05 minus90 minus90 minus34 5886 180 minus90 minus34 07 0 0 0 3182
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 119883 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
119894positions 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
120ordm
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 15inches 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 120579
119894 These
angles and the AACMM parameters are the input to thekinematic modelThe 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 119889
119894119895 AACMM where 119894 and 119895
represent two measured positionsAt the same time the four LTs simultaneously measured
the distance from the captured point to the local coordinatesystem
6 The Scientific World Journal
Xc Yc Zc =
Xi Yi Zi
Xi Yi Zi
Figure 5 Retroreflector center
Diagonal distances are obtained according to the follow-ing
119903 = 119898 + 119897 (11)
where 119898 is the incremental output of the displacementtransducer used and 119897 is the offset which should be calibratedbefore the multilateration implementation
The self-calibration and determination of the parameter119897 can be carried out by capturing some additional objectivemeasurements in some different positions By performingthe quadrilateration 119896 objective point times the number ofequations is given by 4119896 as shown in the following
(119903119896
119894)2
= (119898119896
119894+ 119897119894)2
= (119909119896minus 119909119894)2
+ (119910119896minus 119910119894)2
+ (119911119896minus 119911119894)2
(12)
for 119894 = 0 3 and 119896 = 0 119896 minus 1The system unknowns are the four offsets 119897
119894 and the
position of the coordinates 119909119896 119910119896 119911119896 (3119896 unknowns) Toidentify the four references 12 unknowns correspondingto 119909119894 119910119894 119911119894 must be added Thus the number of system
unknowns is given by 3119896 + 16The unknowns can be obtained by the least-square
method to minimize the errorThe multilateration technique must solve the objective
function defined by
120601 =
21
sum
119894=1
4
sum
119895=1
[(119909119894minus 119909LT119895)
2
+ (119910119894minus 119910LT119895)
2
+ (119911119894minus 119911LT119895)
2
minus (119898119894119895+ 119897LT119895)
2
]
(13)
where (119909119894 119910119894 119911119894) represents the measured point coordinates
in the multilateration reference system (119909LT119895 119910LT119895 119911LT119895) isthe origin of the LT
119895reference system 119898
119894119895are the measured
distances for every point in the LT119895reference system and 119897LT119895
represents the LT119895offset 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(119909119894 119910119894 119911119894)|SR Multi119897
(ii) the laser tracker offsets 119897LT119895
(iii) the origin of the reference systems for the four lasertrackers in the multilateration system (119909LT119895 119910LT119895
119911LT119895)|SR Multi119897
The calculated distances between each pair of retroreflec-tor positions can be obtained from the point coordinates inthe multilateration system obtaining 119889
119894119895 Multi119897 where 119894 and 119895
represent 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 procedureThe origin of the laser trackers wasmeasuredwith respect
to the reference system of LT1This reference system has been
considered themultilateration reference system for obtainingthe initial values of the LT
119895reference 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(119886119894 120572119894 119889119894 1205790119894 120579119894Enc) where 120579
119894Enc are the angles measured bythe encoder The model obtained is a non-linear equationsystem and it was solved by the L-M algorithmThe objectivefunction defined in this step considers both themeasured andcalculated distances as shown in
120601 =
119901
sum
119894=1119895=1
[(119889119894119895AACMM
minus 119889119894119895Multi119897
)2
] +
119899
sum
119894=1
[1205902
119909119894+ 1205902
119910119894+ 1205902
119911119894]
(14)
where 119901 represents the number of positions consideredin the parameter identification procedure and (120590
119883119894119895 120590119884119894119895
120590119885119894119895
) 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 procedurenull 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 locatedIn Figure 7 the coordinates of the captured points by
each LT and the coordinates of the multilaterated points
The Scientific World Journal 7
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 120579i(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 lt tolerance)
For i j = pos ini to pos final
120601 =21sum
i=1j=1[(dijAACMM minus dijmultil )
2 +nsumi=1
[1205902x119894 + 1205902y119894 + 1205902z119894 ]
Optimisation to minimize Ferror
Calculation of the geometric parameters Si
values
120601 =21sumi=1
4sumj=1
[(xi minus xLTj)2 + (yi minus yLTj)
2 + (zi minus zLTj)2 minus (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 Thuseach 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 120583m while the mean range for all set ofdistances was 43 120583m
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 MoreoverTable 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 119901 = 21 reflector points it isnecessary to consider the elimination of the termswhere 119894 = 119895
to avoid both the inclusion of null terms and duplicate of
8 The Scientific World Journal
Table 2 Multilaterated offsets and origin coordinates obtained in multilateration reference system 2798 iterations and objective functionvalue below 1 120583m
Offsets (mm) Origin coordinates (mm)119909LT 119910LT 119911LT
LT1 119897LT1 151695 0 0 0LT2 119897LT2 089275 226233138 0 0LT3 119897LT3 018749 171300335 209893521 0LT4 119897LT4 213351 191300558 minus34341959 301023578
Table 3 Quality indicators for the initial values of model parameters over 21 SMR locations (105 AACMM positions)
Distance error (mm) 2120590 by retroreflector point (mm)Maximum 3850723 Maximum 2200413Causing dist 4ndash10 Causing point 17Medium 1728859 Causing coord X
Medium 817533
Table 4 Identified values for the AACMMmodel parameters by L-M algorithm
Joint 120579119894(∘) 120572
119894(∘) 119886
119894(mm) 119889
119894(mm)
1 minus891269 minus899216 minus424313 3002 1347734 minus897178 minus417885 123373 1842704 minus901202 minus289084 59124234 minus954182 895364 294449 063005 minus880773 minus899482 minus285745 59152846 1810832 925750 275479 1221437 02117 0 04124 2540575
distance errors noting that 119889119894119895= 119889119895119894 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 positionsIn 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 reasonthe 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 120583m and a mean error of 48 120583m 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 arewhen 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 4927164mm The distanceobtained using the mean points expressed in the AACMMreference system of the 10 probed points of each point and theidentified parameters was 4927695mm obtaining an errorin this case with respect to the nominal of 53 120583m while themaximum standard deviation for the two probed points hasbeen calculated in the119883 coordinate of the first probed pointwith a value of 01426mm for all the 10 captured positionsIt is therefore possible to conclude that the obtained errorvalue for the identified parameter set can be generalizable to
The Scientific World Journal 9
Table 5 Quality indicators for the identified values of model parameters over 21 SMR locations (105 AACMM positions)
Distance error (mm) 2120590 by retroreflector point (mm)Maximum 011824 Maximum 020644Causing dist 2ndash12 Causing point 4Medium 004825 Causing coord Y
Medium 011545
500
1000
1500
30003500
minus2000
minus1800
minus1600
minus1400
minus1200
Identification points
Evaluation points
X(m
m)
Z(m
m)
Y (mm)
(a)
minus20000
20004000
2000
4000
minus1000
minus500
0
500
1000
X(mm)
Z(m
m)
LT1
LT2
LT3
LT4
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 (120583
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 LTrsquos
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 LTsAlthough 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 uncertaintytaking as nominal data only the measurements of the LTsBy combining the aforementioned measurements and after
10 The Scientific World Journal
minus10000
1000
minus10000
1000
minus1500
minus1000
minus500
0
500
1000
1500
X (mm)
Z(m
m)
Y (mm)
(a)
X (mm)
Z(m
m)
200400
600800
1000
minus2000200400600800
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 119885 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 systemThis 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
References
[1] A Piratelli-Filho and G R Lesnau ldquoVirtual spheres gauge forcoordinate measuring arms performance testrdquo Measurementvol 43 no 2 pp 236ndash244 2010
[2] K Shimojima R Furutani K Takamasu and K Araki ldquoTheestimation method of uncertainty of articulated coordinatemeasuring machinerdquo in Proceedings of the IEEE InternationalConference on Industrial Technology pp 411ndash415 DubrovnikCroatia 2002
[3] H Hamana M Tominaga M Ozaki and R FurutanildquoCalibration of articulated arm coordinate measuring machine
considering measuring posturerdquo International Journal ofAutomation Technology vol 5 no 2 pp 109ndash114 2011
[4] J Sładek K Ostrowska and A Gąska ldquoModeling and identifi-cation of errors of coordinate measuring arms with the use of ametrological modelrdquo Measurement vol 46 no 1 pp 667ndash6792013
[5] J Sładek K Ostrowska and K Gacek ldquoKinematic metrologicalmodel of the coordinate measuring arm (MCMA)rdquo in Pro-ceedings of the 19th IMEKO World Congress Fundamental andApplied Metrology pp 1988ndash1992 Lisbon Portugal 2009
[6] J Denavit and R S Hartenberg ldquoA kinematic notation forlower-pair mechanisms based on matricesrdquo Transactions ofASME-Journal of Applied Mechanics vol 22 no 1 pp 215ndash2211955
[7] M Miroir Y Nguyen J Szewczyk O Sterkers and A BozorgGrayeli ldquoDesign kinematic optimization and evaluation of ateleoperated system for middle ear microsurgeryrdquoThe ScientificWorld Journal vol 2012 Article ID 907372 19 pages 2012
[8] T W Hsu and L J Everett ldquoIdentification of the kinematicparameters of a robot manipulator for positional accuracyimprovementrdquo in Proceedings of the Computation in EngineeringConference pp 263ndash267 Boston Mass USA August 1985
[9] B W Mooring ldquoThe effect of joint axis misalignment onrobot positioning accuracyrdquo in Proceedings of the Computers inengineering International Computers in Engineering Conferenceand Exhibit pp 151ndash155 Chicago Ill USA August 1983
[10] H Stone A Sanderson and C Neuman ldquoArm signature identi-ficationrdquo in Proceedings of the IEEE International Conference onRobotics and Automation pp 41ndash48 San Francisco Calif USAApril 1986
[11] L J Everett and A H Suryohadiprojo ldquoA study of kinematicmodels for forward calibration of manipulatorsrdquo in Proceedingsof the IEEE International Conference on Robotics and Automa-tion pp 798ndash800 Philadelphia Pa USA April 1988
The Scientific World Journal 11
[12] J Chen and L M Chao ldquoPositioning error analysis for robotmanipulators with all rotary jointsrdquo IEEE Journal of Roboticsand Automation vol RA-3 no 6 pp 539ndash545 1987
[13] A C Majarena J Santolaria D Samper and J J AguilarldquoAn overview of kinematic and calibration models using inter-nalexternal sensors or constraints to improve the behavior ofspatial parallel mechanismsrdquo Sensors vol 10 no 11 pp 10256ndash10297 2010
[14] I Kovac and A Frank ldquoTesting and calibration of coordinatemeasuring armsrdquo Precision Engineering vol 25 no 2 pp 90ndash99 2001
[15] D Zheng CDu andYHu ldquoResearch on optimalmeasurementarea of flexible coordinate measuring machinesrdquoMeasurementvol 45 no 3 pp 250ndash254 2012
[16] K Levenberg ldquoA method for the solution of certain non-linearproblems in least squaresrdquo Quarterly of Applied Mathematicsvol 2 no 2 pp 164ndash168 1944
[17] Q Gao J Chen L Wang S Xu and Y Hou ldquoMultiobjectiveoptimization design of a fractional order PID controller fora gun control systemrdquo The Scientific World Journal vol 2013Article ID 907256 8 pages 2013
[18] J Santolaria J Aguilar J Yague and J Pastor ldquoKinematicparameter estimation technique for calibration and repeata-bility improvement of articulated arm coordinate measuringmachinesrdquo Precision Engineering vol 32 no 4 pp 251ndash2682008
[19] Y Koseki T Arai K Sugimoto T Takatuji and M GotoldquoDesign and accuracy evaluation of high-speed and highprecision parallel mechanismrdquo in Proceedings of the 1998 IEEEInternational Conference on Robotics and Automation Part 1 (of4) pp 1340ndash1345 May 1998
[20] A Piratelli-Filho F H T Fernandes and R V ArencibialdquoApplication of virtual spheres plate for AACMMs evaluationrdquoPrecision Engineering vol 36 no 2 pp 349ndash355 2012
[21] Z Wang L Mastrogiacomo F Franceschini and P Maropou-los ldquoExperimental comparison of dynamic tracking perfor-mance of iGPS and laser trackerrdquo International Journal ofAdvanced Manufacturing Technology vol 56 no 1ndash4 pp 205ndash213 2011
[22] C Lin and J Her ldquoCalibrating the volumetric errors of aprecision machine by a laser tracker systemrdquo InternationalJournal of Advanced Manufacturing Technology vol 26 no 11-12 pp 1255ndash1267 2005
[23] S Aguado D Samper J Santolaria and J J Aguilar ldquoIdentifi-cation strategy of error parameter in volumetric error compen-sation of machine tool based on laser tracker measurementsrdquoInternational Journal of Machine Tools andManufacture vol 53no 1 pp 160ndash169 2012
[24] T Takatsuji M Goto A Kirita T Kurosawa and Y TanimuraldquoThe relationship between the measurement error and thearrangement of laser trackers in laser trilaterationrdquo Measure-ment Science and Technology vol 11 no 5 pp 477ndash483 2000
[25] D F Zhang S Rolt and P G Maropoulos ldquoModelling andoptimization of novel laser multilateration schemes for high-precision applicationsrdquo Measurement Science and Technologyvol 16 no 12 pp 2541ndash2547 2005
[26] E B Hughes A Wilson and G N Peggs ldquoDesign of a high-accuracy CMM based on multi-lateration techniquesrdquo CIRPAnnals-Manufacturing Technology vol 49 no 1 pp 391ndash3942000
[27] S W Kim H G Rhee and J Y Chu ldquoVolumetric phase-measuring interferometer for three-dimensional coordinate
metrologyrdquo Precision Engineering vol 27 no 2 pp 205ndash2152003
[28] J Santolaria A Brau J Velzquez and J J Aguilar ldquoA self-centering active probing technique for kinematic parameteridentification and verification of articulated arm coordinatemeasuring machinesrdquo Measurement Science and Technologyvol 21 no 5 Article ID 055101 2010
[29] W K Kim Y K Byun and H S Cho ldquoClosed-form forward-position solution for a 6-DoF 3-PPSP parallel mechanism andits implementationrdquo International Journal of Robotics Researchvol 20 no 1 pp 85ndash99 2001
[30] S Aguado J Santolaria D Samper and J Aguilar ldquoInfluenceof measurement noise and laser arrangement on measurementuncertainty of laser tracker multilateration in machine toolvolumetric verificationrdquo Precision Engineering vol 37 pp 929ndash943 2013
International Journal of
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The Scientific World Journal 5
Table 1 Initial values for the AACMMD-H parameters
Joint 120579119894(∘) 120572
119894(∘) 119886
119894(mm) 119889
119894(mm)
1 minus90 minus90 minus40 3002 135 minus90 minus40 03 180 minus90 minus34 5904 minus90 90 34 05 minus90 minus90 minus34 5886 180 minus90 minus34 07 0 0 0 3182
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 119883 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
119894positions 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
120ordm
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 15inches 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 120579
119894 These
angles and the AACMM parameters are the input to thekinematic modelThe 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 119889
119894119895 AACMM where 119894 and 119895
represent two measured positionsAt the same time the four LTs simultaneously measured
the distance from the captured point to the local coordinatesystem
6 The Scientific World Journal
Xc Yc Zc =
Xi Yi Zi
Xi Yi Zi
Figure 5 Retroreflector center
Diagonal distances are obtained according to the follow-ing
119903 = 119898 + 119897 (11)
where 119898 is the incremental output of the displacementtransducer used and 119897 is the offset which should be calibratedbefore the multilateration implementation
The self-calibration and determination of the parameter119897 can be carried out by capturing some additional objectivemeasurements in some different positions By performingthe quadrilateration 119896 objective point times the number ofequations is given by 4119896 as shown in the following
(119903119896
119894)2
= (119898119896
119894+ 119897119894)2
= (119909119896minus 119909119894)2
+ (119910119896minus 119910119894)2
+ (119911119896minus 119911119894)2
(12)
for 119894 = 0 3 and 119896 = 0 119896 minus 1The system unknowns are the four offsets 119897
119894 and the
position of the coordinates 119909119896 119910119896 119911119896 (3119896 unknowns) Toidentify the four references 12 unknowns correspondingto 119909119894 119910119894 119911119894 must be added Thus the number of system
unknowns is given by 3119896 + 16The unknowns can be obtained by the least-square
method to minimize the errorThe multilateration technique must solve the objective
function defined by
120601 =
21
sum
119894=1
4
sum
119895=1
[(119909119894minus 119909LT119895)
2
+ (119910119894minus 119910LT119895)
2
+ (119911119894minus 119911LT119895)
2
minus (119898119894119895+ 119897LT119895)
2
]
(13)
where (119909119894 119910119894 119911119894) represents the measured point coordinates
in the multilateration reference system (119909LT119895 119910LT119895 119911LT119895) isthe origin of the LT
119895reference system 119898
119894119895are the measured
distances for every point in the LT119895reference system and 119897LT119895
represents the LT119895offset 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(119909119894 119910119894 119911119894)|SR Multi119897
(ii) the laser tracker offsets 119897LT119895
(iii) the origin of the reference systems for the four lasertrackers in the multilateration system (119909LT119895 119910LT119895
119911LT119895)|SR Multi119897
The calculated distances between each pair of retroreflec-tor positions can be obtained from the point coordinates inthe multilateration system obtaining 119889
119894119895 Multi119897 where 119894 and 119895
represent 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 procedureThe origin of the laser trackers wasmeasuredwith respect
to the reference system of LT1This reference system has been
considered themultilateration reference system for obtainingthe initial values of the LT
119895reference 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(119886119894 120572119894 119889119894 1205790119894 120579119894Enc) where 120579
119894Enc are the angles measured bythe encoder The model obtained is a non-linear equationsystem and it was solved by the L-M algorithmThe objectivefunction defined in this step considers both themeasured andcalculated distances as shown in
120601 =
119901
sum
119894=1119895=1
[(119889119894119895AACMM
minus 119889119894119895Multi119897
)2
] +
119899
sum
119894=1
[1205902
119909119894+ 1205902
119910119894+ 1205902
119911119894]
(14)
where 119901 represents the number of positions consideredin the parameter identification procedure and (120590
119883119894119895 120590119884119894119895
120590119885119894119895
) 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 procedurenull 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 locatedIn Figure 7 the coordinates of the captured points by
each LT and the coordinates of the multilaterated points
The Scientific World Journal 7
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 120579i(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 lt tolerance)
For i j = pos ini to pos final
120601 =21sum
i=1j=1[(dijAACMM minus dijmultil )
2 +nsumi=1
[1205902x119894 + 1205902y119894 + 1205902z119894 ]
Optimisation to minimize Ferror
Calculation of the geometric parameters Si
values
120601 =21sumi=1
4sumj=1
[(xi minus xLTj)2 + (yi minus yLTj)
2 + (zi minus zLTj)2 minus (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 Thuseach 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 120583m while the mean range for all set ofdistances was 43 120583m
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 MoreoverTable 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 119901 = 21 reflector points it isnecessary to consider the elimination of the termswhere 119894 = 119895
to avoid both the inclusion of null terms and duplicate of
8 The Scientific World Journal
Table 2 Multilaterated offsets and origin coordinates obtained in multilateration reference system 2798 iterations and objective functionvalue below 1 120583m
Offsets (mm) Origin coordinates (mm)119909LT 119910LT 119911LT
LT1 119897LT1 151695 0 0 0LT2 119897LT2 089275 226233138 0 0LT3 119897LT3 018749 171300335 209893521 0LT4 119897LT4 213351 191300558 minus34341959 301023578
Table 3 Quality indicators for the initial values of model parameters over 21 SMR locations (105 AACMM positions)
Distance error (mm) 2120590 by retroreflector point (mm)Maximum 3850723 Maximum 2200413Causing dist 4ndash10 Causing point 17Medium 1728859 Causing coord X
Medium 817533
Table 4 Identified values for the AACMMmodel parameters by L-M algorithm
Joint 120579119894(∘) 120572
119894(∘) 119886
119894(mm) 119889
119894(mm)
1 minus891269 minus899216 minus424313 3002 1347734 minus897178 minus417885 123373 1842704 minus901202 minus289084 59124234 minus954182 895364 294449 063005 minus880773 minus899482 minus285745 59152846 1810832 925750 275479 1221437 02117 0 04124 2540575
distance errors noting that 119889119894119895= 119889119895119894 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 positionsIn 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 reasonthe 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 120583m and a mean error of 48 120583m 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 arewhen 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 4927164mm The distanceobtained using the mean points expressed in the AACMMreference system of the 10 probed points of each point and theidentified parameters was 4927695mm obtaining an errorin this case with respect to the nominal of 53 120583m while themaximum standard deviation for the two probed points hasbeen calculated in the119883 coordinate of the first probed pointwith a value of 01426mm for all the 10 captured positionsIt is therefore possible to conclude that the obtained errorvalue for the identified parameter set can be generalizable to
The Scientific World Journal 9
Table 5 Quality indicators for the identified values of model parameters over 21 SMR locations (105 AACMM positions)
Distance error (mm) 2120590 by retroreflector point (mm)Maximum 011824 Maximum 020644Causing dist 2ndash12 Causing point 4Medium 004825 Causing coord Y
Medium 011545
500
1000
1500
30003500
minus2000
minus1800
minus1600
minus1400
minus1200
Identification points
Evaluation points
X(m
m)
Z(m
m)
Y (mm)
(a)
minus20000
20004000
2000
4000
minus1000
minus500
0
500
1000
X(mm)
Z(m
m)
LT1
LT2
LT3
LT4
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 (120583
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 LTrsquos
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 LTsAlthough 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 uncertaintytaking as nominal data only the measurements of the LTsBy combining the aforementioned measurements and after
10 The Scientific World Journal
minus10000
1000
minus10000
1000
minus1500
minus1000
minus500
0
500
1000
1500
X (mm)
Z(m
m)
Y (mm)
(a)
X (mm)
Z(m
m)
200400
600800
1000
minus2000200400600800
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 119885 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 systemThis 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
References
[1] A Piratelli-Filho and G R Lesnau ldquoVirtual spheres gauge forcoordinate measuring arms performance testrdquo Measurementvol 43 no 2 pp 236ndash244 2010
[2] K Shimojima R Furutani K Takamasu and K Araki ldquoTheestimation method of uncertainty of articulated coordinatemeasuring machinerdquo in Proceedings of the IEEE InternationalConference on Industrial Technology pp 411ndash415 DubrovnikCroatia 2002
[3] H Hamana M Tominaga M Ozaki and R FurutanildquoCalibration of articulated arm coordinate measuring machine
considering measuring posturerdquo International Journal ofAutomation Technology vol 5 no 2 pp 109ndash114 2011
[4] J Sładek K Ostrowska and A Gąska ldquoModeling and identifi-cation of errors of coordinate measuring arms with the use of ametrological modelrdquo Measurement vol 46 no 1 pp 667ndash6792013
[5] J Sładek K Ostrowska and K Gacek ldquoKinematic metrologicalmodel of the coordinate measuring arm (MCMA)rdquo in Pro-ceedings of the 19th IMEKO World Congress Fundamental andApplied Metrology pp 1988ndash1992 Lisbon Portugal 2009
[6] J Denavit and R S Hartenberg ldquoA kinematic notation forlower-pair mechanisms based on matricesrdquo Transactions ofASME-Journal of Applied Mechanics vol 22 no 1 pp 215ndash2211955
[7] M Miroir Y Nguyen J Szewczyk O Sterkers and A BozorgGrayeli ldquoDesign kinematic optimization and evaluation of ateleoperated system for middle ear microsurgeryrdquoThe ScientificWorld Journal vol 2012 Article ID 907372 19 pages 2012
[8] T W Hsu and L J Everett ldquoIdentification of the kinematicparameters of a robot manipulator for positional accuracyimprovementrdquo in Proceedings of the Computation in EngineeringConference pp 263ndash267 Boston Mass USA August 1985
[9] B W Mooring ldquoThe effect of joint axis misalignment onrobot positioning accuracyrdquo in Proceedings of the Computers inengineering International Computers in Engineering Conferenceand Exhibit pp 151ndash155 Chicago Ill USA August 1983
[10] H Stone A Sanderson and C Neuman ldquoArm signature identi-ficationrdquo in Proceedings of the IEEE International Conference onRobotics and Automation pp 41ndash48 San Francisco Calif USAApril 1986
[11] L J Everett and A H Suryohadiprojo ldquoA study of kinematicmodels for forward calibration of manipulatorsrdquo in Proceedingsof the IEEE International Conference on Robotics and Automa-tion pp 798ndash800 Philadelphia Pa USA April 1988
The Scientific World Journal 11
[12] J Chen and L M Chao ldquoPositioning error analysis for robotmanipulators with all rotary jointsrdquo IEEE Journal of Roboticsand Automation vol RA-3 no 6 pp 539ndash545 1987
[13] A C Majarena J Santolaria D Samper and J J AguilarldquoAn overview of kinematic and calibration models using inter-nalexternal sensors or constraints to improve the behavior ofspatial parallel mechanismsrdquo Sensors vol 10 no 11 pp 10256ndash10297 2010
[14] I Kovac and A Frank ldquoTesting and calibration of coordinatemeasuring armsrdquo Precision Engineering vol 25 no 2 pp 90ndash99 2001
[15] D Zheng CDu andYHu ldquoResearch on optimalmeasurementarea of flexible coordinate measuring machinesrdquoMeasurementvol 45 no 3 pp 250ndash254 2012
[16] K Levenberg ldquoA method for the solution of certain non-linearproblems in least squaresrdquo Quarterly of Applied Mathematicsvol 2 no 2 pp 164ndash168 1944
[17] Q Gao J Chen L Wang S Xu and Y Hou ldquoMultiobjectiveoptimization design of a fractional order PID controller fora gun control systemrdquo The Scientific World Journal vol 2013Article ID 907256 8 pages 2013
[18] J Santolaria J Aguilar J Yague and J Pastor ldquoKinematicparameter estimation technique for calibration and repeata-bility improvement of articulated arm coordinate measuringmachinesrdquo Precision Engineering vol 32 no 4 pp 251ndash2682008
[19] Y Koseki T Arai K Sugimoto T Takatuji and M GotoldquoDesign and accuracy evaluation of high-speed and highprecision parallel mechanismrdquo in Proceedings of the 1998 IEEEInternational Conference on Robotics and Automation Part 1 (of4) pp 1340ndash1345 May 1998
[20] A Piratelli-Filho F H T Fernandes and R V ArencibialdquoApplication of virtual spheres plate for AACMMs evaluationrdquoPrecision Engineering vol 36 no 2 pp 349ndash355 2012
[21] Z Wang L Mastrogiacomo F Franceschini and P Maropou-los ldquoExperimental comparison of dynamic tracking perfor-mance of iGPS and laser trackerrdquo International Journal ofAdvanced Manufacturing Technology vol 56 no 1ndash4 pp 205ndash213 2011
[22] C Lin and J Her ldquoCalibrating the volumetric errors of aprecision machine by a laser tracker systemrdquo InternationalJournal of Advanced Manufacturing Technology vol 26 no 11-12 pp 1255ndash1267 2005
[23] S Aguado D Samper J Santolaria and J J Aguilar ldquoIdentifi-cation strategy of error parameter in volumetric error compen-sation of machine tool based on laser tracker measurementsrdquoInternational Journal of Machine Tools andManufacture vol 53no 1 pp 160ndash169 2012
[24] T Takatsuji M Goto A Kirita T Kurosawa and Y TanimuraldquoThe relationship between the measurement error and thearrangement of laser trackers in laser trilaterationrdquo Measure-ment Science and Technology vol 11 no 5 pp 477ndash483 2000
[25] D F Zhang S Rolt and P G Maropoulos ldquoModelling andoptimization of novel laser multilateration schemes for high-precision applicationsrdquo Measurement Science and Technologyvol 16 no 12 pp 2541ndash2547 2005
[26] E B Hughes A Wilson and G N Peggs ldquoDesign of a high-accuracy CMM based on multi-lateration techniquesrdquo CIRPAnnals-Manufacturing Technology vol 49 no 1 pp 391ndash3942000
[27] S W Kim H G Rhee and J Y Chu ldquoVolumetric phase-measuring interferometer for three-dimensional coordinate
metrologyrdquo Precision Engineering vol 27 no 2 pp 205ndash2152003
[28] J Santolaria A Brau J Velzquez and J J Aguilar ldquoA self-centering active probing technique for kinematic parameteridentification and verification of articulated arm coordinatemeasuring machinesrdquo Measurement Science and Technologyvol 21 no 5 Article ID 055101 2010
[29] W K Kim Y K Byun and H S Cho ldquoClosed-form forward-position solution for a 6-DoF 3-PPSP parallel mechanism andits implementationrdquo International Journal of Robotics Researchvol 20 no 1 pp 85ndash99 2001
[30] S Aguado J Santolaria D Samper and J Aguilar ldquoInfluenceof measurement noise and laser arrangement on measurementuncertainty of laser tracker multilateration in machine toolvolumetric verificationrdquo Precision Engineering vol 37 pp 929ndash943 2013
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
6 The Scientific World Journal
Xc Yc Zc =
Xi Yi Zi
Xi Yi Zi
Figure 5 Retroreflector center
Diagonal distances are obtained according to the follow-ing
119903 = 119898 + 119897 (11)
where 119898 is the incremental output of the displacementtransducer used and 119897 is the offset which should be calibratedbefore the multilateration implementation
The self-calibration and determination of the parameter119897 can be carried out by capturing some additional objectivemeasurements in some different positions By performingthe quadrilateration 119896 objective point times the number ofequations is given by 4119896 as shown in the following
(119903119896
119894)2
= (119898119896
119894+ 119897119894)2
= (119909119896minus 119909119894)2
+ (119910119896minus 119910119894)2
+ (119911119896minus 119911119894)2
(12)
for 119894 = 0 3 and 119896 = 0 119896 minus 1The system unknowns are the four offsets 119897
119894 and the
position of the coordinates 119909119896 119910119896 119911119896 (3119896 unknowns) Toidentify the four references 12 unknowns correspondingto 119909119894 119910119894 119911119894 must be added Thus the number of system
unknowns is given by 3119896 + 16The unknowns can be obtained by the least-square
method to minimize the errorThe multilateration technique must solve the objective
function defined by
120601 =
21
sum
119894=1
4
sum
119895=1
[(119909119894minus 119909LT119895)
2
+ (119910119894minus 119910LT119895)
2
+ (119911119894minus 119911LT119895)
2
minus (119898119894119895+ 119897LT119895)
2
]
(13)
where (119909119894 119910119894 119911119894) represents the measured point coordinates
in the multilateration reference system (119909LT119895 119910LT119895 119911LT119895) isthe origin of the LT
119895reference system 119898
119894119895are the measured
distances for every point in the LT119895reference system and 119897LT119895
represents the LT119895offset 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(119909119894 119910119894 119911119894)|SR Multi119897
(ii) the laser tracker offsets 119897LT119895
(iii) the origin of the reference systems for the four lasertrackers in the multilateration system (119909LT119895 119910LT119895
119911LT119895)|SR Multi119897
The calculated distances between each pair of retroreflec-tor positions can be obtained from the point coordinates inthe multilateration system obtaining 119889
119894119895 Multi119897 where 119894 and 119895
represent 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 procedureThe origin of the laser trackers wasmeasuredwith respect
to the reference system of LT1This reference system has been
considered themultilateration reference system for obtainingthe initial values of the LT
119895reference 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(119886119894 120572119894 119889119894 1205790119894 120579119894Enc) where 120579
119894Enc are the angles measured bythe encoder The model obtained is a non-linear equationsystem and it was solved by the L-M algorithmThe objectivefunction defined in this step considers both themeasured andcalculated distances as shown in
120601 =
119901
sum
119894=1119895=1
[(119889119894119895AACMM
minus 119889119894119895Multi119897
)2
] +
119899
sum
119894=1
[1205902
119909119894+ 1205902
119910119894+ 1205902
119911119894]
(14)
where 119901 represents the number of positions consideredin the parameter identification procedure and (120590
119883119894119895 120590119884119894119895
120590119885119894119895
) 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 procedurenull 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 locatedIn Figure 7 the coordinates of the captured points by
each LT and the coordinates of the multilaterated points
The Scientific World Journal 7
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 120579i(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 lt tolerance)
For i j = pos ini to pos final
120601 =21sum
i=1j=1[(dijAACMM minus dijmultil )
2 +nsumi=1
[1205902x119894 + 1205902y119894 + 1205902z119894 ]
Optimisation to minimize Ferror
Calculation of the geometric parameters Si
values
120601 =21sumi=1
4sumj=1
[(xi minus xLTj)2 + (yi minus yLTj)
2 + (zi minus zLTj)2 minus (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 Thuseach 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 120583m while the mean range for all set ofdistances was 43 120583m
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 MoreoverTable 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 119901 = 21 reflector points it isnecessary to consider the elimination of the termswhere 119894 = 119895
to avoid both the inclusion of null terms and duplicate of
8 The Scientific World Journal
Table 2 Multilaterated offsets and origin coordinates obtained in multilateration reference system 2798 iterations and objective functionvalue below 1 120583m
Offsets (mm) Origin coordinates (mm)119909LT 119910LT 119911LT
LT1 119897LT1 151695 0 0 0LT2 119897LT2 089275 226233138 0 0LT3 119897LT3 018749 171300335 209893521 0LT4 119897LT4 213351 191300558 minus34341959 301023578
Table 3 Quality indicators for the initial values of model parameters over 21 SMR locations (105 AACMM positions)
Distance error (mm) 2120590 by retroreflector point (mm)Maximum 3850723 Maximum 2200413Causing dist 4ndash10 Causing point 17Medium 1728859 Causing coord X
Medium 817533
Table 4 Identified values for the AACMMmodel parameters by L-M algorithm
Joint 120579119894(∘) 120572
119894(∘) 119886
119894(mm) 119889
119894(mm)
1 minus891269 minus899216 minus424313 3002 1347734 minus897178 minus417885 123373 1842704 minus901202 minus289084 59124234 minus954182 895364 294449 063005 minus880773 minus899482 minus285745 59152846 1810832 925750 275479 1221437 02117 0 04124 2540575
distance errors noting that 119889119894119895= 119889119895119894 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 positionsIn 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 reasonthe 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 120583m and a mean error of 48 120583m 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 arewhen 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 4927164mm The distanceobtained using the mean points expressed in the AACMMreference system of the 10 probed points of each point and theidentified parameters was 4927695mm obtaining an errorin this case with respect to the nominal of 53 120583m while themaximum standard deviation for the two probed points hasbeen calculated in the119883 coordinate of the first probed pointwith a value of 01426mm for all the 10 captured positionsIt is therefore possible to conclude that the obtained errorvalue for the identified parameter set can be generalizable to
The Scientific World Journal 9
Table 5 Quality indicators for the identified values of model parameters over 21 SMR locations (105 AACMM positions)
Distance error (mm) 2120590 by retroreflector point (mm)Maximum 011824 Maximum 020644Causing dist 2ndash12 Causing point 4Medium 004825 Causing coord Y
Medium 011545
500
1000
1500
30003500
minus2000
minus1800
minus1600
minus1400
minus1200
Identification points
Evaluation points
X(m
m)
Z(m
m)
Y (mm)
(a)
minus20000
20004000
2000
4000
minus1000
minus500
0
500
1000
X(mm)
Z(m
m)
LT1
LT2
LT3
LT4
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 (120583
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 LTrsquos
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 LTsAlthough 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 uncertaintytaking as nominal data only the measurements of the LTsBy combining the aforementioned measurements and after
10 The Scientific World Journal
minus10000
1000
minus10000
1000
minus1500
minus1000
minus500
0
500
1000
1500
X (mm)
Z(m
m)
Y (mm)
(a)
X (mm)
Z(m
m)
200400
600800
1000
minus2000200400600800
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 119885 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 systemThis 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
References
[1] A Piratelli-Filho and G R Lesnau ldquoVirtual spheres gauge forcoordinate measuring arms performance testrdquo Measurementvol 43 no 2 pp 236ndash244 2010
[2] K Shimojima R Furutani K Takamasu and K Araki ldquoTheestimation method of uncertainty of articulated coordinatemeasuring machinerdquo in Proceedings of the IEEE InternationalConference on Industrial Technology pp 411ndash415 DubrovnikCroatia 2002
[3] H Hamana M Tominaga M Ozaki and R FurutanildquoCalibration of articulated arm coordinate measuring machine
considering measuring posturerdquo International Journal ofAutomation Technology vol 5 no 2 pp 109ndash114 2011
[4] J Sładek K Ostrowska and A Gąska ldquoModeling and identifi-cation of errors of coordinate measuring arms with the use of ametrological modelrdquo Measurement vol 46 no 1 pp 667ndash6792013
[5] J Sładek K Ostrowska and K Gacek ldquoKinematic metrologicalmodel of the coordinate measuring arm (MCMA)rdquo in Pro-ceedings of the 19th IMEKO World Congress Fundamental andApplied Metrology pp 1988ndash1992 Lisbon Portugal 2009
[6] J Denavit and R S Hartenberg ldquoA kinematic notation forlower-pair mechanisms based on matricesrdquo Transactions ofASME-Journal of Applied Mechanics vol 22 no 1 pp 215ndash2211955
[7] M Miroir Y Nguyen J Szewczyk O Sterkers and A BozorgGrayeli ldquoDesign kinematic optimization and evaluation of ateleoperated system for middle ear microsurgeryrdquoThe ScientificWorld Journal vol 2012 Article ID 907372 19 pages 2012
[8] T W Hsu and L J Everett ldquoIdentification of the kinematicparameters of a robot manipulator for positional accuracyimprovementrdquo in Proceedings of the Computation in EngineeringConference pp 263ndash267 Boston Mass USA August 1985
[9] B W Mooring ldquoThe effect of joint axis misalignment onrobot positioning accuracyrdquo in Proceedings of the Computers inengineering International Computers in Engineering Conferenceand Exhibit pp 151ndash155 Chicago Ill USA August 1983
[10] H Stone A Sanderson and C Neuman ldquoArm signature identi-ficationrdquo in Proceedings of the IEEE International Conference onRobotics and Automation pp 41ndash48 San Francisco Calif USAApril 1986
[11] L J Everett and A H Suryohadiprojo ldquoA study of kinematicmodels for forward calibration of manipulatorsrdquo in Proceedingsof the IEEE International Conference on Robotics and Automa-tion pp 798ndash800 Philadelphia Pa USA April 1988
The Scientific World Journal 11
[12] J Chen and L M Chao ldquoPositioning error analysis for robotmanipulators with all rotary jointsrdquo IEEE Journal of Roboticsand Automation vol RA-3 no 6 pp 539ndash545 1987
[13] A C Majarena J Santolaria D Samper and J J AguilarldquoAn overview of kinematic and calibration models using inter-nalexternal sensors or constraints to improve the behavior ofspatial parallel mechanismsrdquo Sensors vol 10 no 11 pp 10256ndash10297 2010
[14] I Kovac and A Frank ldquoTesting and calibration of coordinatemeasuring armsrdquo Precision Engineering vol 25 no 2 pp 90ndash99 2001
[15] D Zheng CDu andYHu ldquoResearch on optimalmeasurementarea of flexible coordinate measuring machinesrdquoMeasurementvol 45 no 3 pp 250ndash254 2012
[16] K Levenberg ldquoA method for the solution of certain non-linearproblems in least squaresrdquo Quarterly of Applied Mathematicsvol 2 no 2 pp 164ndash168 1944
[17] Q Gao J Chen L Wang S Xu and Y Hou ldquoMultiobjectiveoptimization design of a fractional order PID controller fora gun control systemrdquo The Scientific World Journal vol 2013Article ID 907256 8 pages 2013
[18] J Santolaria J Aguilar J Yague and J Pastor ldquoKinematicparameter estimation technique for calibration and repeata-bility improvement of articulated arm coordinate measuringmachinesrdquo Precision Engineering vol 32 no 4 pp 251ndash2682008
[19] Y Koseki T Arai K Sugimoto T Takatuji and M GotoldquoDesign and accuracy evaluation of high-speed and highprecision parallel mechanismrdquo in Proceedings of the 1998 IEEEInternational Conference on Robotics and Automation Part 1 (of4) pp 1340ndash1345 May 1998
[20] A Piratelli-Filho F H T Fernandes and R V ArencibialdquoApplication of virtual spheres plate for AACMMs evaluationrdquoPrecision Engineering vol 36 no 2 pp 349ndash355 2012
[21] Z Wang L Mastrogiacomo F Franceschini and P Maropou-los ldquoExperimental comparison of dynamic tracking perfor-mance of iGPS and laser trackerrdquo International Journal ofAdvanced Manufacturing Technology vol 56 no 1ndash4 pp 205ndash213 2011
[22] C Lin and J Her ldquoCalibrating the volumetric errors of aprecision machine by a laser tracker systemrdquo InternationalJournal of Advanced Manufacturing Technology vol 26 no 11-12 pp 1255ndash1267 2005
[23] S Aguado D Samper J Santolaria and J J Aguilar ldquoIdentifi-cation strategy of error parameter in volumetric error compen-sation of machine tool based on laser tracker measurementsrdquoInternational Journal of Machine Tools andManufacture vol 53no 1 pp 160ndash169 2012
[24] T Takatsuji M Goto A Kirita T Kurosawa and Y TanimuraldquoThe relationship between the measurement error and thearrangement of laser trackers in laser trilaterationrdquo Measure-ment Science and Technology vol 11 no 5 pp 477ndash483 2000
[25] D F Zhang S Rolt and P G Maropoulos ldquoModelling andoptimization of novel laser multilateration schemes for high-precision applicationsrdquo Measurement Science and Technologyvol 16 no 12 pp 2541ndash2547 2005
[26] E B Hughes A Wilson and G N Peggs ldquoDesign of a high-accuracy CMM based on multi-lateration techniquesrdquo CIRPAnnals-Manufacturing Technology vol 49 no 1 pp 391ndash3942000
[27] S W Kim H G Rhee and J Y Chu ldquoVolumetric phase-measuring interferometer for three-dimensional coordinate
metrologyrdquo Precision Engineering vol 27 no 2 pp 205ndash2152003
[28] J Santolaria A Brau J Velzquez and J J Aguilar ldquoA self-centering active probing technique for kinematic parameteridentification and verification of articulated arm coordinatemeasuring machinesrdquo Measurement Science and Technologyvol 21 no 5 Article ID 055101 2010
[29] W K Kim Y K Byun and H S Cho ldquoClosed-form forward-position solution for a 6-DoF 3-PPSP parallel mechanism andits implementationrdquo International Journal of Robotics Researchvol 20 no 1 pp 85ndash99 2001
[30] S Aguado J Santolaria D Samper and J Aguilar ldquoInfluenceof measurement noise and laser arrangement on measurementuncertainty of laser tracker multilateration in machine toolvolumetric verificationrdquo Precision Engineering vol 37 pp 929ndash943 2013
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
The Scientific World Journal 7
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 120579i(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 lt tolerance)
For i j = pos ini to pos final
120601 =21sum
i=1j=1[(dijAACMM minus dijmultil )
2 +nsumi=1
[1205902x119894 + 1205902y119894 + 1205902z119894 ]
Optimisation to minimize Ferror
Calculation of the geometric parameters Si
values
120601 =21sumi=1
4sumj=1
[(xi minus xLTj)2 + (yi minus yLTj)
2 + (zi minus zLTj)2 minus (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 Thuseach 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 120583m while the mean range for all set ofdistances was 43 120583m
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 MoreoverTable 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 119901 = 21 reflector points it isnecessary to consider the elimination of the termswhere 119894 = 119895
to avoid both the inclusion of null terms and duplicate of
8 The Scientific World Journal
Table 2 Multilaterated offsets and origin coordinates obtained in multilateration reference system 2798 iterations and objective functionvalue below 1 120583m
Offsets (mm) Origin coordinates (mm)119909LT 119910LT 119911LT
LT1 119897LT1 151695 0 0 0LT2 119897LT2 089275 226233138 0 0LT3 119897LT3 018749 171300335 209893521 0LT4 119897LT4 213351 191300558 minus34341959 301023578
Table 3 Quality indicators for the initial values of model parameters over 21 SMR locations (105 AACMM positions)
Distance error (mm) 2120590 by retroreflector point (mm)Maximum 3850723 Maximum 2200413Causing dist 4ndash10 Causing point 17Medium 1728859 Causing coord X
Medium 817533
Table 4 Identified values for the AACMMmodel parameters by L-M algorithm
Joint 120579119894(∘) 120572
119894(∘) 119886
119894(mm) 119889
119894(mm)
1 minus891269 minus899216 minus424313 3002 1347734 minus897178 minus417885 123373 1842704 minus901202 minus289084 59124234 minus954182 895364 294449 063005 minus880773 minus899482 minus285745 59152846 1810832 925750 275479 1221437 02117 0 04124 2540575
distance errors noting that 119889119894119895= 119889119895119894 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 positionsIn 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 reasonthe 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 120583m and a mean error of 48 120583m 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 arewhen 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 4927164mm The distanceobtained using the mean points expressed in the AACMMreference system of the 10 probed points of each point and theidentified parameters was 4927695mm obtaining an errorin this case with respect to the nominal of 53 120583m while themaximum standard deviation for the two probed points hasbeen calculated in the119883 coordinate of the first probed pointwith a value of 01426mm for all the 10 captured positionsIt is therefore possible to conclude that the obtained errorvalue for the identified parameter set can be generalizable to
The Scientific World Journal 9
Table 5 Quality indicators for the identified values of model parameters over 21 SMR locations (105 AACMM positions)
Distance error (mm) 2120590 by retroreflector point (mm)Maximum 011824 Maximum 020644Causing dist 2ndash12 Causing point 4Medium 004825 Causing coord Y
Medium 011545
500
1000
1500
30003500
minus2000
minus1800
minus1600
minus1400
minus1200
Identification points
Evaluation points
X(m
m)
Z(m
m)
Y (mm)
(a)
minus20000
20004000
2000
4000
minus1000
minus500
0
500
1000
X(mm)
Z(m
m)
LT1
LT2
LT3
LT4
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 (120583
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 LTrsquos
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 LTsAlthough 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 uncertaintytaking as nominal data only the measurements of the LTsBy combining the aforementioned measurements and after
10 The Scientific World Journal
minus10000
1000
minus10000
1000
minus1500
minus1000
minus500
0
500
1000
1500
X (mm)
Z(m
m)
Y (mm)
(a)
X (mm)
Z(m
m)
200400
600800
1000
minus2000200400600800
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 119885 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 systemThis 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
References
[1] A Piratelli-Filho and G R Lesnau ldquoVirtual spheres gauge forcoordinate measuring arms performance testrdquo Measurementvol 43 no 2 pp 236ndash244 2010
[2] K Shimojima R Furutani K Takamasu and K Araki ldquoTheestimation method of uncertainty of articulated coordinatemeasuring machinerdquo in Proceedings of the IEEE InternationalConference on Industrial Technology pp 411ndash415 DubrovnikCroatia 2002
[3] H Hamana M Tominaga M Ozaki and R FurutanildquoCalibration of articulated arm coordinate measuring machine
considering measuring posturerdquo International Journal ofAutomation Technology vol 5 no 2 pp 109ndash114 2011
[4] J Sładek K Ostrowska and A Gąska ldquoModeling and identifi-cation of errors of coordinate measuring arms with the use of ametrological modelrdquo Measurement vol 46 no 1 pp 667ndash6792013
[5] J Sładek K Ostrowska and K Gacek ldquoKinematic metrologicalmodel of the coordinate measuring arm (MCMA)rdquo in Pro-ceedings of the 19th IMEKO World Congress Fundamental andApplied Metrology pp 1988ndash1992 Lisbon Portugal 2009
[6] J Denavit and R S Hartenberg ldquoA kinematic notation forlower-pair mechanisms based on matricesrdquo Transactions ofASME-Journal of Applied Mechanics vol 22 no 1 pp 215ndash2211955
[7] M Miroir Y Nguyen J Szewczyk O Sterkers and A BozorgGrayeli ldquoDesign kinematic optimization and evaluation of ateleoperated system for middle ear microsurgeryrdquoThe ScientificWorld Journal vol 2012 Article ID 907372 19 pages 2012
[8] T W Hsu and L J Everett ldquoIdentification of the kinematicparameters of a robot manipulator for positional accuracyimprovementrdquo in Proceedings of the Computation in EngineeringConference pp 263ndash267 Boston Mass USA August 1985
[9] B W Mooring ldquoThe effect of joint axis misalignment onrobot positioning accuracyrdquo in Proceedings of the Computers inengineering International Computers in Engineering Conferenceand Exhibit pp 151ndash155 Chicago Ill USA August 1983
[10] H Stone A Sanderson and C Neuman ldquoArm signature identi-ficationrdquo in Proceedings of the IEEE International Conference onRobotics and Automation pp 41ndash48 San Francisco Calif USAApril 1986
[11] L J Everett and A H Suryohadiprojo ldquoA study of kinematicmodels for forward calibration of manipulatorsrdquo in Proceedingsof the IEEE International Conference on Robotics and Automa-tion pp 798ndash800 Philadelphia Pa USA April 1988
The Scientific World Journal 11
[12] J Chen and L M Chao ldquoPositioning error analysis for robotmanipulators with all rotary jointsrdquo IEEE Journal of Roboticsand Automation vol RA-3 no 6 pp 539ndash545 1987
[13] A C Majarena J Santolaria D Samper and J J AguilarldquoAn overview of kinematic and calibration models using inter-nalexternal sensors or constraints to improve the behavior ofspatial parallel mechanismsrdquo Sensors vol 10 no 11 pp 10256ndash10297 2010
[14] I Kovac and A Frank ldquoTesting and calibration of coordinatemeasuring armsrdquo Precision Engineering vol 25 no 2 pp 90ndash99 2001
[15] D Zheng CDu andYHu ldquoResearch on optimalmeasurementarea of flexible coordinate measuring machinesrdquoMeasurementvol 45 no 3 pp 250ndash254 2012
[16] K Levenberg ldquoA method for the solution of certain non-linearproblems in least squaresrdquo Quarterly of Applied Mathematicsvol 2 no 2 pp 164ndash168 1944
[17] Q Gao J Chen L Wang S Xu and Y Hou ldquoMultiobjectiveoptimization design of a fractional order PID controller fora gun control systemrdquo The Scientific World Journal vol 2013Article ID 907256 8 pages 2013
[18] J Santolaria J Aguilar J Yague and J Pastor ldquoKinematicparameter estimation technique for calibration and repeata-bility improvement of articulated arm coordinate measuringmachinesrdquo Precision Engineering vol 32 no 4 pp 251ndash2682008
[19] Y Koseki T Arai K Sugimoto T Takatuji and M GotoldquoDesign and accuracy evaluation of high-speed and highprecision parallel mechanismrdquo in Proceedings of the 1998 IEEEInternational Conference on Robotics and Automation Part 1 (of4) pp 1340ndash1345 May 1998
[20] A Piratelli-Filho F H T Fernandes and R V ArencibialdquoApplication of virtual spheres plate for AACMMs evaluationrdquoPrecision Engineering vol 36 no 2 pp 349ndash355 2012
[21] Z Wang L Mastrogiacomo F Franceschini and P Maropou-los ldquoExperimental comparison of dynamic tracking perfor-mance of iGPS and laser trackerrdquo International Journal ofAdvanced Manufacturing Technology vol 56 no 1ndash4 pp 205ndash213 2011
[22] C Lin and J Her ldquoCalibrating the volumetric errors of aprecision machine by a laser tracker systemrdquo InternationalJournal of Advanced Manufacturing Technology vol 26 no 11-12 pp 1255ndash1267 2005
[23] S Aguado D Samper J Santolaria and J J Aguilar ldquoIdentifi-cation strategy of error parameter in volumetric error compen-sation of machine tool based on laser tracker measurementsrdquoInternational Journal of Machine Tools andManufacture vol 53no 1 pp 160ndash169 2012
[24] T Takatsuji M Goto A Kirita T Kurosawa and Y TanimuraldquoThe relationship between the measurement error and thearrangement of laser trackers in laser trilaterationrdquo Measure-ment Science and Technology vol 11 no 5 pp 477ndash483 2000
[25] D F Zhang S Rolt and P G Maropoulos ldquoModelling andoptimization of novel laser multilateration schemes for high-precision applicationsrdquo Measurement Science and Technologyvol 16 no 12 pp 2541ndash2547 2005
[26] E B Hughes A Wilson and G N Peggs ldquoDesign of a high-accuracy CMM based on multi-lateration techniquesrdquo CIRPAnnals-Manufacturing Technology vol 49 no 1 pp 391ndash3942000
[27] S W Kim H G Rhee and J Y Chu ldquoVolumetric phase-measuring interferometer for three-dimensional coordinate
metrologyrdquo Precision Engineering vol 27 no 2 pp 205ndash2152003
[28] J Santolaria A Brau J Velzquez and J J Aguilar ldquoA self-centering active probing technique for kinematic parameteridentification and verification of articulated arm coordinatemeasuring machinesrdquo Measurement Science and Technologyvol 21 no 5 Article ID 055101 2010
[29] W K Kim Y K Byun and H S Cho ldquoClosed-form forward-position solution for a 6-DoF 3-PPSP parallel mechanism andits implementationrdquo International Journal of Robotics Researchvol 20 no 1 pp 85ndash99 2001
[30] S Aguado J Santolaria D Samper and J Aguilar ldquoInfluenceof measurement noise and laser arrangement on measurementuncertainty of laser tracker multilateration in machine toolvolumetric verificationrdquo Precision Engineering vol 37 pp 929ndash943 2013
International Journal of
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RoboticsJournal of
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Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
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Shock and Vibration
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Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
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Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
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DistributedSensor Networks
International Journal of
8 The Scientific World Journal
Table 2 Multilaterated offsets and origin coordinates obtained in multilateration reference system 2798 iterations and objective functionvalue below 1 120583m
Offsets (mm) Origin coordinates (mm)119909LT 119910LT 119911LT
LT1 119897LT1 151695 0 0 0LT2 119897LT2 089275 226233138 0 0LT3 119897LT3 018749 171300335 209893521 0LT4 119897LT4 213351 191300558 minus34341959 301023578
Table 3 Quality indicators for the initial values of model parameters over 21 SMR locations (105 AACMM positions)
Distance error (mm) 2120590 by retroreflector point (mm)Maximum 3850723 Maximum 2200413Causing dist 4ndash10 Causing point 17Medium 1728859 Causing coord X
Medium 817533
Table 4 Identified values for the AACMMmodel parameters by L-M algorithm
Joint 120579119894(∘) 120572
119894(∘) 119886
119894(mm) 119889
119894(mm)
1 minus891269 minus899216 minus424313 3002 1347734 minus897178 minus417885 123373 1842704 minus901202 minus289084 59124234 minus954182 895364 294449 063005 minus880773 minus899482 minus285745 59152846 1810832 925750 275479 1221437 02117 0 04124 2540575
distance errors noting that 119889119894119895= 119889119895119894 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 positionsIn 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 reasonthe 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 120583m and a mean error of 48 120583m 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 arewhen 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 4927164mm The distanceobtained using the mean points expressed in the AACMMreference system of the 10 probed points of each point and theidentified parameters was 4927695mm obtaining an errorin this case with respect to the nominal of 53 120583m while themaximum standard deviation for the two probed points hasbeen calculated in the119883 coordinate of the first probed pointwith a value of 01426mm for all the 10 captured positionsIt is therefore possible to conclude that the obtained errorvalue for the identified parameter set can be generalizable to
The Scientific World Journal 9
Table 5 Quality indicators for the identified values of model parameters over 21 SMR locations (105 AACMM positions)
Distance error (mm) 2120590 by retroreflector point (mm)Maximum 011824 Maximum 020644Causing dist 2ndash12 Causing point 4Medium 004825 Causing coord Y
Medium 011545
500
1000
1500
30003500
minus2000
minus1800
minus1600
minus1400
minus1200
Identification points
Evaluation points
X(m
m)
Z(m
m)
Y (mm)
(a)
minus20000
20004000
2000
4000
minus1000
minus500
0
500
1000
X(mm)
Z(m
m)
LT1
LT2
LT3
LT4
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 (120583
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 LTrsquos
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 LTsAlthough 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 uncertaintytaking as nominal data only the measurements of the LTsBy combining the aforementioned measurements and after
10 The Scientific World Journal
minus10000
1000
minus10000
1000
minus1500
minus1000
minus500
0
500
1000
1500
X (mm)
Z(m
m)
Y (mm)
(a)
X (mm)
Z(m
m)
200400
600800
1000
minus2000200400600800
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 119885 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 systemThis 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
References
[1] A Piratelli-Filho and G R Lesnau ldquoVirtual spheres gauge forcoordinate measuring arms performance testrdquo Measurementvol 43 no 2 pp 236ndash244 2010
[2] K Shimojima R Furutani K Takamasu and K Araki ldquoTheestimation method of uncertainty of articulated coordinatemeasuring machinerdquo in Proceedings of the IEEE InternationalConference on Industrial Technology pp 411ndash415 DubrovnikCroatia 2002
[3] H Hamana M Tominaga M Ozaki and R FurutanildquoCalibration of articulated arm coordinate measuring machine
considering measuring posturerdquo International Journal ofAutomation Technology vol 5 no 2 pp 109ndash114 2011
[4] J Sładek K Ostrowska and A Gąska ldquoModeling and identifi-cation of errors of coordinate measuring arms with the use of ametrological modelrdquo Measurement vol 46 no 1 pp 667ndash6792013
[5] J Sładek K Ostrowska and K Gacek ldquoKinematic metrologicalmodel of the coordinate measuring arm (MCMA)rdquo in Pro-ceedings of the 19th IMEKO World Congress Fundamental andApplied Metrology pp 1988ndash1992 Lisbon Portugal 2009
[6] J Denavit and R S Hartenberg ldquoA kinematic notation forlower-pair mechanisms based on matricesrdquo Transactions ofASME-Journal of Applied Mechanics vol 22 no 1 pp 215ndash2211955
[7] M Miroir Y Nguyen J Szewczyk O Sterkers and A BozorgGrayeli ldquoDesign kinematic optimization and evaluation of ateleoperated system for middle ear microsurgeryrdquoThe ScientificWorld Journal vol 2012 Article ID 907372 19 pages 2012
[8] T W Hsu and L J Everett ldquoIdentification of the kinematicparameters of a robot manipulator for positional accuracyimprovementrdquo in Proceedings of the Computation in EngineeringConference pp 263ndash267 Boston Mass USA August 1985
[9] B W Mooring ldquoThe effect of joint axis misalignment onrobot positioning accuracyrdquo in Proceedings of the Computers inengineering International Computers in Engineering Conferenceand Exhibit pp 151ndash155 Chicago Ill USA August 1983
[10] H Stone A Sanderson and C Neuman ldquoArm signature identi-ficationrdquo in Proceedings of the IEEE International Conference onRobotics and Automation pp 41ndash48 San Francisco Calif USAApril 1986
[11] L J Everett and A H Suryohadiprojo ldquoA study of kinematicmodels for forward calibration of manipulatorsrdquo in Proceedingsof the IEEE International Conference on Robotics and Automa-tion pp 798ndash800 Philadelphia Pa USA April 1988
The Scientific World Journal 11
[12] J Chen and L M Chao ldquoPositioning error analysis for robotmanipulators with all rotary jointsrdquo IEEE Journal of Roboticsand Automation vol RA-3 no 6 pp 539ndash545 1987
[13] A C Majarena J Santolaria D Samper and J J AguilarldquoAn overview of kinematic and calibration models using inter-nalexternal sensors or constraints to improve the behavior ofspatial parallel mechanismsrdquo Sensors vol 10 no 11 pp 10256ndash10297 2010
[14] I Kovac and A Frank ldquoTesting and calibration of coordinatemeasuring armsrdquo Precision Engineering vol 25 no 2 pp 90ndash99 2001
[15] D Zheng CDu andYHu ldquoResearch on optimalmeasurementarea of flexible coordinate measuring machinesrdquoMeasurementvol 45 no 3 pp 250ndash254 2012
[16] K Levenberg ldquoA method for the solution of certain non-linearproblems in least squaresrdquo Quarterly of Applied Mathematicsvol 2 no 2 pp 164ndash168 1944
[17] Q Gao J Chen L Wang S Xu and Y Hou ldquoMultiobjectiveoptimization design of a fractional order PID controller fora gun control systemrdquo The Scientific World Journal vol 2013Article ID 907256 8 pages 2013
[18] J Santolaria J Aguilar J Yague and J Pastor ldquoKinematicparameter estimation technique for calibration and repeata-bility improvement of articulated arm coordinate measuringmachinesrdquo Precision Engineering vol 32 no 4 pp 251ndash2682008
[19] Y Koseki T Arai K Sugimoto T Takatuji and M GotoldquoDesign and accuracy evaluation of high-speed and highprecision parallel mechanismrdquo in Proceedings of the 1998 IEEEInternational Conference on Robotics and Automation Part 1 (of4) pp 1340ndash1345 May 1998
[20] A Piratelli-Filho F H T Fernandes and R V ArencibialdquoApplication of virtual spheres plate for AACMMs evaluationrdquoPrecision Engineering vol 36 no 2 pp 349ndash355 2012
[21] Z Wang L Mastrogiacomo F Franceschini and P Maropou-los ldquoExperimental comparison of dynamic tracking perfor-mance of iGPS and laser trackerrdquo International Journal ofAdvanced Manufacturing Technology vol 56 no 1ndash4 pp 205ndash213 2011
[22] C Lin and J Her ldquoCalibrating the volumetric errors of aprecision machine by a laser tracker systemrdquo InternationalJournal of Advanced Manufacturing Technology vol 26 no 11-12 pp 1255ndash1267 2005
[23] S Aguado D Samper J Santolaria and J J Aguilar ldquoIdentifi-cation strategy of error parameter in volumetric error compen-sation of machine tool based on laser tracker measurementsrdquoInternational Journal of Machine Tools andManufacture vol 53no 1 pp 160ndash169 2012
[24] T Takatsuji M Goto A Kirita T Kurosawa and Y TanimuraldquoThe relationship between the measurement error and thearrangement of laser trackers in laser trilaterationrdquo Measure-ment Science and Technology vol 11 no 5 pp 477ndash483 2000
[25] D F Zhang S Rolt and P G Maropoulos ldquoModelling andoptimization of novel laser multilateration schemes for high-precision applicationsrdquo Measurement Science and Technologyvol 16 no 12 pp 2541ndash2547 2005
[26] E B Hughes A Wilson and G N Peggs ldquoDesign of a high-accuracy CMM based on multi-lateration techniquesrdquo CIRPAnnals-Manufacturing Technology vol 49 no 1 pp 391ndash3942000
[27] S W Kim H G Rhee and J Y Chu ldquoVolumetric phase-measuring interferometer for three-dimensional coordinate
metrologyrdquo Precision Engineering vol 27 no 2 pp 205ndash2152003
[28] J Santolaria A Brau J Velzquez and J J Aguilar ldquoA self-centering active probing technique for kinematic parameteridentification and verification of articulated arm coordinatemeasuring machinesrdquo Measurement Science and Technologyvol 21 no 5 Article ID 055101 2010
[29] W K Kim Y K Byun and H S Cho ldquoClosed-form forward-position solution for a 6-DoF 3-PPSP parallel mechanism andits implementationrdquo International Journal of Robotics Researchvol 20 no 1 pp 85ndash99 2001
[30] S Aguado J Santolaria D Samper and J Aguilar ldquoInfluenceof measurement noise and laser arrangement on measurementuncertainty of laser tracker multilateration in machine toolvolumetric verificationrdquo Precision Engineering vol 37 pp 929ndash943 2013
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
The Scientific World Journal 9
Table 5 Quality indicators for the identified values of model parameters over 21 SMR locations (105 AACMM positions)
Distance error (mm) 2120590 by retroreflector point (mm)Maximum 011824 Maximum 020644Causing dist 2ndash12 Causing point 4Medium 004825 Causing coord Y
Medium 011545
500
1000
1500
30003500
minus2000
minus1800
minus1600
minus1400
minus1200
Identification points
Evaluation points
X(m
m)
Z(m
m)
Y (mm)
(a)
minus20000
20004000
2000
4000
minus1000
minus500
0
500
1000
X(mm)
Z(m
m)
LT1
LT2
LT3
LT4
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 (120583
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 LTrsquos
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 LTsAlthough 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 uncertaintytaking as nominal data only the measurements of the LTsBy combining the aforementioned measurements and after
10 The Scientific World Journal
minus10000
1000
minus10000
1000
minus1500
minus1000
minus500
0
500
1000
1500
X (mm)
Z(m
m)
Y (mm)
(a)
X (mm)
Z(m
m)
200400
600800
1000
minus2000200400600800
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 119885 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 systemThis 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
References
[1] A Piratelli-Filho and G R Lesnau ldquoVirtual spheres gauge forcoordinate measuring arms performance testrdquo Measurementvol 43 no 2 pp 236ndash244 2010
[2] K Shimojima R Furutani K Takamasu and K Araki ldquoTheestimation method of uncertainty of articulated coordinatemeasuring machinerdquo in Proceedings of the IEEE InternationalConference on Industrial Technology pp 411ndash415 DubrovnikCroatia 2002
[3] H Hamana M Tominaga M Ozaki and R FurutanildquoCalibration of articulated arm coordinate measuring machine
considering measuring posturerdquo International Journal ofAutomation Technology vol 5 no 2 pp 109ndash114 2011
[4] J Sładek K Ostrowska and A Gąska ldquoModeling and identifi-cation of errors of coordinate measuring arms with the use of ametrological modelrdquo Measurement vol 46 no 1 pp 667ndash6792013
[5] J Sładek K Ostrowska and K Gacek ldquoKinematic metrologicalmodel of the coordinate measuring arm (MCMA)rdquo in Pro-ceedings of the 19th IMEKO World Congress Fundamental andApplied Metrology pp 1988ndash1992 Lisbon Portugal 2009
[6] J Denavit and R S Hartenberg ldquoA kinematic notation forlower-pair mechanisms based on matricesrdquo Transactions ofASME-Journal of Applied Mechanics vol 22 no 1 pp 215ndash2211955
[7] M Miroir Y Nguyen J Szewczyk O Sterkers and A BozorgGrayeli ldquoDesign kinematic optimization and evaluation of ateleoperated system for middle ear microsurgeryrdquoThe ScientificWorld Journal vol 2012 Article ID 907372 19 pages 2012
[8] T W Hsu and L J Everett ldquoIdentification of the kinematicparameters of a robot manipulator for positional accuracyimprovementrdquo in Proceedings of the Computation in EngineeringConference pp 263ndash267 Boston Mass USA August 1985
[9] B W Mooring ldquoThe effect of joint axis misalignment onrobot positioning accuracyrdquo in Proceedings of the Computers inengineering International Computers in Engineering Conferenceand Exhibit pp 151ndash155 Chicago Ill USA August 1983
[10] H Stone A Sanderson and C Neuman ldquoArm signature identi-ficationrdquo in Proceedings of the IEEE International Conference onRobotics and Automation pp 41ndash48 San Francisco Calif USAApril 1986
[11] L J Everett and A H Suryohadiprojo ldquoA study of kinematicmodels for forward calibration of manipulatorsrdquo in Proceedingsof the IEEE International Conference on Robotics and Automa-tion pp 798ndash800 Philadelphia Pa USA April 1988
The Scientific World Journal 11
[12] J Chen and L M Chao ldquoPositioning error analysis for robotmanipulators with all rotary jointsrdquo IEEE Journal of Roboticsand Automation vol RA-3 no 6 pp 539ndash545 1987
[13] A C Majarena J Santolaria D Samper and J J AguilarldquoAn overview of kinematic and calibration models using inter-nalexternal sensors or constraints to improve the behavior ofspatial parallel mechanismsrdquo Sensors vol 10 no 11 pp 10256ndash10297 2010
[14] I Kovac and A Frank ldquoTesting and calibration of coordinatemeasuring armsrdquo Precision Engineering vol 25 no 2 pp 90ndash99 2001
[15] D Zheng CDu andYHu ldquoResearch on optimalmeasurementarea of flexible coordinate measuring machinesrdquoMeasurementvol 45 no 3 pp 250ndash254 2012
[16] K Levenberg ldquoA method for the solution of certain non-linearproblems in least squaresrdquo Quarterly of Applied Mathematicsvol 2 no 2 pp 164ndash168 1944
[17] Q Gao J Chen L Wang S Xu and Y Hou ldquoMultiobjectiveoptimization design of a fractional order PID controller fora gun control systemrdquo The Scientific World Journal vol 2013Article ID 907256 8 pages 2013
[18] J Santolaria J Aguilar J Yague and J Pastor ldquoKinematicparameter estimation technique for calibration and repeata-bility improvement of articulated arm coordinate measuringmachinesrdquo Precision Engineering vol 32 no 4 pp 251ndash2682008
[19] Y Koseki T Arai K Sugimoto T Takatuji and M GotoldquoDesign and accuracy evaluation of high-speed and highprecision parallel mechanismrdquo in Proceedings of the 1998 IEEEInternational Conference on Robotics and Automation Part 1 (of4) pp 1340ndash1345 May 1998
[20] A Piratelli-Filho F H T Fernandes and R V ArencibialdquoApplication of virtual spheres plate for AACMMs evaluationrdquoPrecision Engineering vol 36 no 2 pp 349ndash355 2012
[21] Z Wang L Mastrogiacomo F Franceschini and P Maropou-los ldquoExperimental comparison of dynamic tracking perfor-mance of iGPS and laser trackerrdquo International Journal ofAdvanced Manufacturing Technology vol 56 no 1ndash4 pp 205ndash213 2011
[22] C Lin and J Her ldquoCalibrating the volumetric errors of aprecision machine by a laser tracker systemrdquo InternationalJournal of Advanced Manufacturing Technology vol 26 no 11-12 pp 1255ndash1267 2005
[23] S Aguado D Samper J Santolaria and J J Aguilar ldquoIdentifi-cation strategy of error parameter in volumetric error compen-sation of machine tool based on laser tracker measurementsrdquoInternational Journal of Machine Tools andManufacture vol 53no 1 pp 160ndash169 2012
[24] T Takatsuji M Goto A Kirita T Kurosawa and Y TanimuraldquoThe relationship between the measurement error and thearrangement of laser trackers in laser trilaterationrdquo Measure-ment Science and Technology vol 11 no 5 pp 477ndash483 2000
[25] D F Zhang S Rolt and P G Maropoulos ldquoModelling andoptimization of novel laser multilateration schemes for high-precision applicationsrdquo Measurement Science and Technologyvol 16 no 12 pp 2541ndash2547 2005
[26] E B Hughes A Wilson and G N Peggs ldquoDesign of a high-accuracy CMM based on multi-lateration techniquesrdquo CIRPAnnals-Manufacturing Technology vol 49 no 1 pp 391ndash3942000
[27] S W Kim H G Rhee and J Y Chu ldquoVolumetric phase-measuring interferometer for three-dimensional coordinate
metrologyrdquo Precision Engineering vol 27 no 2 pp 205ndash2152003
[28] J Santolaria A Brau J Velzquez and J J Aguilar ldquoA self-centering active probing technique for kinematic parameteridentification and verification of articulated arm coordinatemeasuring machinesrdquo Measurement Science and Technologyvol 21 no 5 Article ID 055101 2010
[29] W K Kim Y K Byun and H S Cho ldquoClosed-form forward-position solution for a 6-DoF 3-PPSP parallel mechanism andits implementationrdquo International Journal of Robotics Researchvol 20 no 1 pp 85ndash99 2001
[30] S Aguado J Santolaria D Samper and J Aguilar ldquoInfluenceof measurement noise and laser arrangement on measurementuncertainty of laser tracker multilateration in machine toolvolumetric verificationrdquo Precision Engineering vol 37 pp 929ndash943 2013
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
10 The Scientific World Journal
minus10000
1000
minus10000
1000
minus1500
minus1000
minus500
0
500
1000
1500
X (mm)
Z(m
m)
Y (mm)
(a)
X (mm)
Z(m
m)
200400
600800
1000
minus2000200400600800
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 119885 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 systemThis 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
References
[1] A Piratelli-Filho and G R Lesnau ldquoVirtual spheres gauge forcoordinate measuring arms performance testrdquo Measurementvol 43 no 2 pp 236ndash244 2010
[2] K Shimojima R Furutani K Takamasu and K Araki ldquoTheestimation method of uncertainty of articulated coordinatemeasuring machinerdquo in Proceedings of the IEEE InternationalConference on Industrial Technology pp 411ndash415 DubrovnikCroatia 2002
[3] H Hamana M Tominaga M Ozaki and R FurutanildquoCalibration of articulated arm coordinate measuring machine
considering measuring posturerdquo International Journal ofAutomation Technology vol 5 no 2 pp 109ndash114 2011
[4] J Sładek K Ostrowska and A Gąska ldquoModeling and identifi-cation of errors of coordinate measuring arms with the use of ametrological modelrdquo Measurement vol 46 no 1 pp 667ndash6792013
[5] J Sładek K Ostrowska and K Gacek ldquoKinematic metrologicalmodel of the coordinate measuring arm (MCMA)rdquo in Pro-ceedings of the 19th IMEKO World Congress Fundamental andApplied Metrology pp 1988ndash1992 Lisbon Portugal 2009
[6] J Denavit and R S Hartenberg ldquoA kinematic notation forlower-pair mechanisms based on matricesrdquo Transactions ofASME-Journal of Applied Mechanics vol 22 no 1 pp 215ndash2211955
[7] M Miroir Y Nguyen J Szewczyk O Sterkers and A BozorgGrayeli ldquoDesign kinematic optimization and evaluation of ateleoperated system for middle ear microsurgeryrdquoThe ScientificWorld Journal vol 2012 Article ID 907372 19 pages 2012
[8] T W Hsu and L J Everett ldquoIdentification of the kinematicparameters of a robot manipulator for positional accuracyimprovementrdquo in Proceedings of the Computation in EngineeringConference pp 263ndash267 Boston Mass USA August 1985
[9] B W Mooring ldquoThe effect of joint axis misalignment onrobot positioning accuracyrdquo in Proceedings of the Computers inengineering International Computers in Engineering Conferenceand Exhibit pp 151ndash155 Chicago Ill USA August 1983
[10] H Stone A Sanderson and C Neuman ldquoArm signature identi-ficationrdquo in Proceedings of the IEEE International Conference onRobotics and Automation pp 41ndash48 San Francisco Calif USAApril 1986
[11] L J Everett and A H Suryohadiprojo ldquoA study of kinematicmodels for forward calibration of manipulatorsrdquo in Proceedingsof the IEEE International Conference on Robotics and Automa-tion pp 798ndash800 Philadelphia Pa USA April 1988
The Scientific World Journal 11
[12] J Chen and L M Chao ldquoPositioning error analysis for robotmanipulators with all rotary jointsrdquo IEEE Journal of Roboticsand Automation vol RA-3 no 6 pp 539ndash545 1987
[13] A C Majarena J Santolaria D Samper and J J AguilarldquoAn overview of kinematic and calibration models using inter-nalexternal sensors or constraints to improve the behavior ofspatial parallel mechanismsrdquo Sensors vol 10 no 11 pp 10256ndash10297 2010
[14] I Kovac and A Frank ldquoTesting and calibration of coordinatemeasuring armsrdquo Precision Engineering vol 25 no 2 pp 90ndash99 2001
[15] D Zheng CDu andYHu ldquoResearch on optimalmeasurementarea of flexible coordinate measuring machinesrdquoMeasurementvol 45 no 3 pp 250ndash254 2012
[16] K Levenberg ldquoA method for the solution of certain non-linearproblems in least squaresrdquo Quarterly of Applied Mathematicsvol 2 no 2 pp 164ndash168 1944
[17] Q Gao J Chen L Wang S Xu and Y Hou ldquoMultiobjectiveoptimization design of a fractional order PID controller fora gun control systemrdquo The Scientific World Journal vol 2013Article ID 907256 8 pages 2013
[18] J Santolaria J Aguilar J Yague and J Pastor ldquoKinematicparameter estimation technique for calibration and repeata-bility improvement of articulated arm coordinate measuringmachinesrdquo Precision Engineering vol 32 no 4 pp 251ndash2682008
[19] Y Koseki T Arai K Sugimoto T Takatuji and M GotoldquoDesign and accuracy evaluation of high-speed and highprecision parallel mechanismrdquo in Proceedings of the 1998 IEEEInternational Conference on Robotics and Automation Part 1 (of4) pp 1340ndash1345 May 1998
[20] A Piratelli-Filho F H T Fernandes and R V ArencibialdquoApplication of virtual spheres plate for AACMMs evaluationrdquoPrecision Engineering vol 36 no 2 pp 349ndash355 2012
[21] Z Wang L Mastrogiacomo F Franceschini and P Maropou-los ldquoExperimental comparison of dynamic tracking perfor-mance of iGPS and laser trackerrdquo International Journal ofAdvanced Manufacturing Technology vol 56 no 1ndash4 pp 205ndash213 2011
[22] C Lin and J Her ldquoCalibrating the volumetric errors of aprecision machine by a laser tracker systemrdquo InternationalJournal of Advanced Manufacturing Technology vol 26 no 11-12 pp 1255ndash1267 2005
[23] S Aguado D Samper J Santolaria and J J Aguilar ldquoIdentifi-cation strategy of error parameter in volumetric error compen-sation of machine tool based on laser tracker measurementsrdquoInternational Journal of Machine Tools andManufacture vol 53no 1 pp 160ndash169 2012
[24] T Takatsuji M Goto A Kirita T Kurosawa and Y TanimuraldquoThe relationship between the measurement error and thearrangement of laser trackers in laser trilaterationrdquo Measure-ment Science and Technology vol 11 no 5 pp 477ndash483 2000
[25] D F Zhang S Rolt and P G Maropoulos ldquoModelling andoptimization of novel laser multilateration schemes for high-precision applicationsrdquo Measurement Science and Technologyvol 16 no 12 pp 2541ndash2547 2005
[26] E B Hughes A Wilson and G N Peggs ldquoDesign of a high-accuracy CMM based on multi-lateration techniquesrdquo CIRPAnnals-Manufacturing Technology vol 49 no 1 pp 391ndash3942000
[27] S W Kim H G Rhee and J Y Chu ldquoVolumetric phase-measuring interferometer for three-dimensional coordinate
metrologyrdquo Precision Engineering vol 27 no 2 pp 205ndash2152003
[28] J Santolaria A Brau J Velzquez and J J Aguilar ldquoA self-centering active probing technique for kinematic parameteridentification and verification of articulated arm coordinatemeasuring machinesrdquo Measurement Science and Technologyvol 21 no 5 Article ID 055101 2010
[29] W K Kim Y K Byun and H S Cho ldquoClosed-form forward-position solution for a 6-DoF 3-PPSP parallel mechanism andits implementationrdquo International Journal of Robotics Researchvol 20 no 1 pp 85ndash99 2001
[30] S Aguado J Santolaria D Samper and J Aguilar ldquoInfluenceof measurement noise and laser arrangement on measurementuncertainty of laser tracker multilateration in machine toolvolumetric verificationrdquo Precision Engineering vol 37 pp 929ndash943 2013
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
The Scientific World Journal 11
[12] J Chen and L M Chao ldquoPositioning error analysis for robotmanipulators with all rotary jointsrdquo IEEE Journal of Roboticsand Automation vol RA-3 no 6 pp 539ndash545 1987
[13] A C Majarena J Santolaria D Samper and J J AguilarldquoAn overview of kinematic and calibration models using inter-nalexternal sensors or constraints to improve the behavior ofspatial parallel mechanismsrdquo Sensors vol 10 no 11 pp 10256ndash10297 2010
[14] I Kovac and A Frank ldquoTesting and calibration of coordinatemeasuring armsrdquo Precision Engineering vol 25 no 2 pp 90ndash99 2001
[15] D Zheng CDu andYHu ldquoResearch on optimalmeasurementarea of flexible coordinate measuring machinesrdquoMeasurementvol 45 no 3 pp 250ndash254 2012
[16] K Levenberg ldquoA method for the solution of certain non-linearproblems in least squaresrdquo Quarterly of Applied Mathematicsvol 2 no 2 pp 164ndash168 1944
[17] Q Gao J Chen L Wang S Xu and Y Hou ldquoMultiobjectiveoptimization design of a fractional order PID controller fora gun control systemrdquo The Scientific World Journal vol 2013Article ID 907256 8 pages 2013
[18] J Santolaria J Aguilar J Yague and J Pastor ldquoKinematicparameter estimation technique for calibration and repeata-bility improvement of articulated arm coordinate measuringmachinesrdquo Precision Engineering vol 32 no 4 pp 251ndash2682008
[19] Y Koseki T Arai K Sugimoto T Takatuji and M GotoldquoDesign and accuracy evaluation of high-speed and highprecision parallel mechanismrdquo in Proceedings of the 1998 IEEEInternational Conference on Robotics and Automation Part 1 (of4) pp 1340ndash1345 May 1998
[20] A Piratelli-Filho F H T Fernandes and R V ArencibialdquoApplication of virtual spheres plate for AACMMs evaluationrdquoPrecision Engineering vol 36 no 2 pp 349ndash355 2012
[21] Z Wang L Mastrogiacomo F Franceschini and P Maropou-los ldquoExperimental comparison of dynamic tracking perfor-mance of iGPS and laser trackerrdquo International Journal ofAdvanced Manufacturing Technology vol 56 no 1ndash4 pp 205ndash213 2011
[22] C Lin and J Her ldquoCalibrating the volumetric errors of aprecision machine by a laser tracker systemrdquo InternationalJournal of Advanced Manufacturing Technology vol 26 no 11-12 pp 1255ndash1267 2005
[23] S Aguado D Samper J Santolaria and J J Aguilar ldquoIdentifi-cation strategy of error parameter in volumetric error compen-sation of machine tool based on laser tracker measurementsrdquoInternational Journal of Machine Tools andManufacture vol 53no 1 pp 160ndash169 2012
[24] T Takatsuji M Goto A Kirita T Kurosawa and Y TanimuraldquoThe relationship between the measurement error and thearrangement of laser trackers in laser trilaterationrdquo Measure-ment Science and Technology vol 11 no 5 pp 477ndash483 2000
[25] D F Zhang S Rolt and P G Maropoulos ldquoModelling andoptimization of novel laser multilateration schemes for high-precision applicationsrdquo Measurement Science and Technologyvol 16 no 12 pp 2541ndash2547 2005
[26] E B Hughes A Wilson and G N Peggs ldquoDesign of a high-accuracy CMM based on multi-lateration techniquesrdquo CIRPAnnals-Manufacturing Technology vol 49 no 1 pp 391ndash3942000
[27] S W Kim H G Rhee and J Y Chu ldquoVolumetric phase-measuring interferometer for three-dimensional coordinate
metrologyrdquo Precision Engineering vol 27 no 2 pp 205ndash2152003
[28] J Santolaria A Brau J Velzquez and J J Aguilar ldquoA self-centering active probing technique for kinematic parameteridentification and verification of articulated arm coordinatemeasuring machinesrdquo Measurement Science and Technologyvol 21 no 5 Article ID 055101 2010
[29] W K Kim Y K Byun and H S Cho ldquoClosed-form forward-position solution for a 6-DoF 3-PPSP parallel mechanism andits implementationrdquo International Journal of Robotics Researchvol 20 no 1 pp 85ndash99 2001
[30] S Aguado J Santolaria D Samper and J Aguilar ldquoInfluenceof measurement noise and laser arrangement on measurementuncertainty of laser tracker multilateration in machine toolvolumetric verificationrdquo Precision Engineering vol 37 pp 929ndash943 2013
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
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Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
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