ASME B89.4.19 Performance Evaluation Tests and Geometric Misalignments in Laser Trackers Volume

1. Introduction

A spherical coordinate measurement instrument suchas a laser tracker is a mechanical assembly of severalcomponents that may contain misalignments within itsconstruction. These geometric offsets, tilts and eccen-tricities introduce systematic errors in the measuredspherical coordinates (angles and range readings) andpossibly in the calculated lengths of reference artifacts.It is general practice to compensate for these errorsthrough software error models [2-4] in a manner some-what similar to that performed for Cartesian coordinatemeasuring machines (CMMs).

Early designs for laser trackers used a mirror mount-ed on a two-axis gimbal mechanism to steer the beamto the target. Several sources of geometric misalign-ments for this configuration were identified by Loser

and Kyle [5], and an error model was developed andpublished. However, the presence of a geometric errormodel within the system does not necessarily imply theabsence of systematic errors. Incorrect compensation ormisalignments after compensation are possible andhave to be identified during performance evaluation.

The recently introduced ASME B89.4.19 Standardproposes a common set of ranging tests, length meas-urement system tests and two-face system tests, whichcan be performed by manufacturers and users to assessthe performance of their instrument. It is desirable thatthe tests described in the B89.4.19 Standard be sensi-tive to the different potential error sources, includingerrors resulting from geometric misalignments withinthe tracker. This paper addresses the issue of the sensi-tivity of the tests described in the B89.4.19 Standard togeometric misalignments within a tracker.

Volume 114, Number 1, January-February 2009Journal of Research of the National Institute of Standards and Technology

21

[J. Res. Natl. Inst. Stand. Technol. 114, 21-35 (2009)]

ASME B89.4.19 Performance Evaluation Testsand Geometric Misalignments

in Laser Trackers

Volume 114 Number 1 January-February 2009

B. Muralikrishnan, D. Sawyer,C. Blackburn, S. Phillips,B. Borchardt, and W. T. Estler

Precision Engineering Division,National Institute of Standardsand Technology,Gaithersburg, MD 20899

[email protected]@[email protected]@[email protected]@nist.gov

Small and unintended offsets, tilts, andeccentricity of the mechanical and opticalcomponents in laser trackers introducesystematic errors in the measured sphericalcoordinates (angles and range readings)and possibly in the calculated lengths ofreference artifacts. It is desirable that thetests described in the ASME B89.4.19Standard [1] be sensitive to thesegeometric misalignments so that anyresulting systematic errors are identifiedduring performance evaluation. In thispaper, we present some analysis, usingerror models and numerical simulation, ofthe sensitivity of the length measurementsystem tests and two-face system testsin the B89.4.19 Standard to misalignmentsin laser trackers. We highlight key attributes of the testing strategy adoptedin the Standard and propose new length

measurement system tests that demonstrateimproved sensitivity to some misalign-ments. Experimental results with a trackerthat is not properly error corrected for theeffects of the misalignments validateclaims regarding the proposed new lengthtests.

Key words: ASME B89.4.19; geometricerrors; laser tracker; misalignments;performance evaluation; sensitivityanalysis.

Accepted: November 3, 2008

Available online: http://www.nist.gov/jres

2. Approach

The corrected (or true) range (Rc) and angles (Hc,Vc) of any coordinate in space are functions of severalmisalignment parameters within the construction of thetracker and also of the measured coordinate values atthat location (Rm, Hm, Vm). The corrections ΔRm,ΔHm and ΔVm in Rm, Hm and Vm respectively may beexpressed as

where xi (i = 1 to n) are n misalignment parameters. Theabove constitutes an error model for a laser tracker.Loser and Kyle’s model containing 15 parameters isone such error model applicable to laser trackers with abeam steering mirror. Since the design of such trackersin the 1980s, trackers with other mechanical configura-tions have emerged. These trackers have different errormodels, but manufacturers have been reluctant topublish them. We have extended Loser and Kyle’smodel to two other common configurations also.

The development of an error model for a lasertracker immediately suggests a numerical approach tosensitivity analysis, where misalignment parametersare perturbed while determining the impact on each ofthe performance evaluation tests, which are alsonumerically simulated. We have performed such analy-sis for three different mechanical constructions oftrackers: a) a tracker with a beam steering mirror forwhich the Loser and Kyle model is applicable, b) atracker with the laser source in the rotating head andc) a scanner with source mounted on the transit axiswith a rotating prism mirror that steers the beam to thetarget.

In this paper, we describe our analysis for only thetracker with the laser source in the rotating head of theinstrument (or other constructions such as those with afiber coupled laser that may also be modeled in thesame manner). Such a tracker can be imagined to havea theodolite-type construction where the telescopein the rotating head is replaced by a laser source. Thesimple geometric design enhances clarity in our presen-tation; the method is applicable to the other designs aswell.

The organization of the remainder of the paper is asfollows. We briefly describe the ASME B89.4.19 testsin Sec. 3. We describe the coordinate system adopted in

Sec. 4. In Sec. 5, we detail geometric misalignmentparameters relevant to the chosen configuration of thetracker, and the different two-face and length measure-ment system tests that are sensitive to those terms. Wepresent an error model for the tracker in Sec. 6. Thesensitivity of any geometric misalignment parameter toeach of the tests described in the Standard is compiledin the form of a matrix and presented in Sec. 7. Asummary of our observations from the analysis inSec. 5 and the sensitivity matrix in Sec. 7 is presentedin Sec. 8. We propose a new set of length measurementsystem tests that demonstrate improved sensitivity inSec. 9. In Sec. 10, we discuss experimental results thatvalidate some of our proposed new length tests. Ourconclusions are summarized in Sec. 11.

3. The ASME B89.4.19 Standard

The B89.4.19 Standard describes three kinds of teststo be performed on laser trackers—ranging tests, lengthmeasurement system tests and two-face system tests.The ranging tests assess the distance measuring capa-bility of the tracker along a purely radial direction. Wedo not consider ranging tests in this paper as these arenot diagnostic of geometric misalignments in a tracker.

Length measurement system tests are performed toassess the tracker’s ability to measure different lengthswithin the work volume. Because these tests exercisethe kinematic links in the tracker, they are sensitive tomost of the tracker’s geometric misalignments. TheStandard requires length measurement system tests tobe performed in 33 predetermined and two user-definedpositions.

There are a number of geometric misalignments thatproduce angle errors that reverse in sign between afront face and back face measurement of the tracker.Two-face tests are therefore excellent diagnostics ofthese geometric misalignments. The Standard requiresthat two-face errors be measured at 36 predeterminedpositions.

4. Coordinate System Definition

In Fig. 1, the X, Y, and Z axes form a mutuallyorthogonal fixed Cartesian system with its origin at O.We define two axes—the transit axis OT and its normalin the XY plane ON—attached to the instrument headthat can rotate about the standing axis OZ. The target islocated at P. In addition, we also define an axis OMattached to the instrument head. OM is orthogonal to


22

1 2( , , , , ,..., )r nRc Rm Rm f Rm Hm Vm x x x− = Δ =

1 2( , , , , ,..., )h nHc Hm Hm f Rm Hm Vm x x x− = Δ =

1 2( , , , , ,..., )v nVc Vm Vm f Rm Hm Vm x x x− = Δ =

the transit OT and the beam path OP. Horizontal anglesare measured from the Y axis to ON. Vertical angles aremeasured from the Z axis to OP. It should be noted thatin an ideal case, transit axis OT intersects, and is per-pendicular to the standing OZ. The beam emerges fromO and its path to the target OP is perpendicular to thetransit axis OT. In reality, any or all of these conditionsmay be violated, resulting in systematic errors in themeasured range and angles.

5. Geometric Misalignment Parametersand Sensitivity to B89.4.19 Tests

In this section, we discuss the effect of several geo-metric misalignment parameters in a theodolite-typetracker on the measured range and angles. We alsodiscuss two-face and length measurement systemtests that are sensitive to each of the misalignmentparameters.

5.1 Beam Offset (x1)

Description: The beam originating from the source(at O) may be displaced from its ideal position by aconstant offset (OA in Fig. 2) to emerge from A, a mis-alignment parameter referred to as beam offset. Theoffset can be resolved into components along M and Taxes (x1m and x1t) because the beam originates in therotating head traveling parallel to OP and perpendicu-lar to the MOT plane in the MTP coordinate system.The offset component along the transit (x1t) produces an

error in the measured horizontal angle. The correctionfor the beam offset is given by

(positive x1t is as shown in Fig. 2; therefore the meas-ured horizontal angle is smaller than the true angle,which produces a positive correction). The componentalong its normal (x1m) produces an error in themeasured vertical angle, and its correction is given by

Two-face system tests: The corrections for themeasured horizontal and vertical angles of a targetplaced distance Rm away from the front face of thetracker are given in the preceding sub-section. Thesecorrections reverse in sign in the back face of thetracker. The apparent distance E in a two-face systemtest is therefore given by E = 2ΔHmRmsin(Vm) = 2x1t

for an offset along the transit axis and E =2ΔVmRm = 2x1m for an offset along OM. Every two-face system test described in the Standard is thereforesensitive to both beam offset parameters by the samesensitivity factor of 2.

Length measurement system tests: Systematicerrors in measured range and angles lead to an error inthe determination of the coordinates of each end of thereference length. This however does not necessarilyimply an error in the calculated length between the twoends, because the error vectors at the two ends may


23

Fig. 1. Coordinate system definition.

1

sin( )tx

HmRm Vm

Δ =

1 .mxVm

Rm−

Δ =

Fig. 2. Beam offset.

simply result in translation and/or rotation of the lengthwithout a change in its magnitude (the error vectoris the vector between true coordinate and measuredcoordinate). Sensitivity to length measurement test isachieved primarily if the error vectors at the two endsproduce components along the length with a non-zerosum. Components perpendicular to the length generallydo not produce a significant change in length for thetest to be sensitive to the geometric misalignment underconsideration.

Any symmetrically placed reference length (such asthe horizontal, vertical or diagonal length tests in theStandard) is not sensitive to beam offset, because theyonly serve to translate and rotate the length. The defaultposition for the first user-defined test (asymmetricalvertical length test) is sensitive to beam offset along theM axis because the asymmetrical positioning of thereference length creates unequal error components atthe two ends of the reference length, which do notcompletely cancel each other. We discuss such asym-metrical positioning in Sec. 9.

5.2 Transit Offset (x2)

Description: The transit axis may be offset from itsideal location and therefore not intersect the standingaxis. This offset is referred to as the transit offset(OA in Fig. 3). The transit offset vector of consequenceis along the N axis. The offset along Z is simply a trans-lation of the coordinate system. Transit offset producesan error in the measured range (OB). The correction inthe range is given by ΔRm = x2 sin(Vm). Additionally,the transit offset produces an error in the measuredvertical angle (vertical angle error = AB/AP). The cor-

rection in vertical angle is given by

For a positive offset as shown in Fig. 3, the measuredrange and vertical angle are smaller than the truevalues, hence a positive correction.

Two-face system tests: The transit offset is sensi-tive to two-face measurements, because the error inthe measured vertical angle changes in sign betweenthe two faces. The two-face error E is given byE = 2RmΔVm = 2 x2 cos(Vm). The test is not sensitivein the horizontal plane (Vm = 90°, cos(Vm) = 0).Sensitivity is higher near the pole and also near thefloor but decreases farther away from the tracker(because the target continues to remain at the sameheight and therefore Vm increases). Note that theoreti-cally, range measurements are smaller in one face andlarger in the other and therefore might seem to alsocontribute to a two-face error. In practice, range

measurements are not recorded in the back face (forinterferometric systems, a break in the beam precludessuch measurements) and therefore range related effectsare not diagnosed in these tests.

Length measurement system tests: A ranging testwill not detect transit offset because point-to-pointdistances are unaffected along the radial direction. Ahorizontal length test, however, captures some of thisterm. Higher sensitivity is obtained when the referencelength is placed as close as possible to the tracker. Infact, if the tracker is placed between the targets and inline with them, we achieve a maximum sensitivity of 2.More generally, the sensitivity to horizontal length testsas described in the B89.4.19 Standard can be given by

where L is the value of the reference length

(2.3 m) and D is the distance between the referencelength and the tracker.

5.3 Vertical Index Offset (x3 )

Description: A shift in the zero of the verticalangle encoder from the pole results in a constant errorin the measured vertical angle, a misalignment para-meter referred to as vertical index offset.

Two-face system tests: The vertical angle errorchanges in sign between front face and back facemeasurements. The apparent distance between front


24

2 cos( ) .x VmVmRm

Δ =

Fig. 3. Transit offset.

22

,

4

L

L D+

face and back face measurements of the target scaleswith distance and is given by E = 2Rmx3 . There isincreased sensitivity farther away from the tracker.

Length measurement system tests: Vertical indexoffset results in a constant vertical angle error in everymeasured point. None of the length measurementsystem tests described in the Standard will capture thiserror because it manifests primarily as a coordinaterotation for the positions and orientations of the lengthsdescribed in the Standard. In Sec. 9, we describe alength test that is sensitive to this parameter.

5.4 Beam Tilt (x4 )

Description: The beam emerging from the sourcemay be tilted (not normal to the transit axis) from itsideal path. This is a misalignment referred to as beamtilt. Although shown in Fig. 4 as an offset, we define x4

as the offset per unit length (along the beam path totarget), and it therefore may be expressed in units ofangle (small angle approximations will be valid). Beamtilt may be resolved into components in a similar man-ner to beam offset (x4 m and x4 t). However, the com-ponent along M is indistinguishable from the verticalindex offset and is therefore not considered. The com-ponent along T is equivalent to collimation error intheodolites. The correction in the measured horizontal

angle is given by

as shown in Fig. 4, the measured horizontal angle issmaller, hence a positive correction.

Two-face system tests: The apparent distanceE in a two-face system test is given by E =2Rm sin (Vm) ΔHm = 2Rmx4t , and scales with distanceas expected. There is increased sensitivity farther awayfrom the tracker.

Length measurement system tests: None of thelength measurement system tests described in theStandard is sensitive to this parameter. A referencelength placed such that both ends are at the same Zheight (horizontal length test) will have identical errorsat the two ends and therefore be insensitive to thisparameter. An asymmetric vertical length test is alsoinsensitive because the horizontal angle errors,although different at the two ends, are directed normalto the length; errors directed axially (along the length)will be most sensitive. A diagonal test where the twoends are at different Z heights is desirable. However,symmetrically placed diagonals such as in the B89.4.19Standard are not sensitive, because the errors areidentical at the two ends (sin(Vm) = sin(π–Vm)).Asymmetric diagonals that are sensitive to this para-meter are described in Sec. 9.

5.5 Transit Tilt (x5 )

Description: The transit axis, although intersectingthe standing axis, may be tilted (not at a right angle)relative to the standing axis. This lack of squarenessbetween the axes is referred to as transit tilt and is alsoa common theodolite error source. Again, as in beamtilt, the parameter is a ratio of offset per unit length. Itis expressed in radians using the small angle approxi-mation. Lack of squareness between the two axes pro-duces an error in the measured horizontal angle. The

correction is given by

Two-face system tests: The transit tilt produces anerror in the horizontal angle that is sensitive to two-face measurements. The apparent distance in a two-face measurement is given by E = 2 Rm sin (Vm)ΔHm =

system tests described in the Standard, there is no sensi-tivity when a test is performed at the tracker height(Vm = 90°, cos (Vm) = 0). For all other positions asdescribed in the Standard, the sensitivity is a constant,because Rm cos (Vm), which is the projection of thebeam vector along the Z direction, is a constant.

Length measurement system tests: The transit tiltis analogous to squareness error in an XY stage. Thediagonal length tests described in the Standard are sen-sitive to this parameter. The left and right diagonal tests


25

4 [6]. For positive tiltsin( )

txHm

VmΔ =

Fig. 4. Beam tilt.

5 [6].tan( )

xHm

VmΔ =

552 sin( ) 2 cos( ). For the two-face

tan( )x

Rm Vm Rmx VmVm

=

provide errors that are equal in magnitude but oppositein sign as a consequence of transit tilt. The sensitivitydoes not change with distance from the tracker becausealthough the range to the targets increases, the verticalangles also change (the reference length is same).

5.6 Encoder Eccentricity (x6 and x7)Description: The horizontal and vertical angle

encoders may be eccentrically mounted, parameters werefer to as horizontal and vertical angle encoder eccen-tricity. We describe the terms as a ratio of the actualeccentricity to the radius of the encoder so that x6 andx7 are dimensionless quantities. The horizontal angleencoder is fixed to the base and its eccentricity (OA)can therefore be resolved into X and Y components(x6 x and x6 y). The vertical angle encoder is fixed to themoving transit and, consequently, the physical encodereccentricity can be resolved into N and Z components(x7 n and x7 z ). For the coordinate system in Fig. 5, it canbe shown that the correction in the measured horizontalangle due to the horizontal angle encoder eccentricity isΔHm = x6 x cos (Hm) – x6 y sin (Hm), and the correctionin the measured vertical angle due to the vertical angleencoder eccentricity is ΔVm = x7 n cos (Vm) – x7 z sin (Vm).

Two-face system tests: The apparent distancemeasured between the two faces of the tracker in thecase of horizontal angle encoder eccentricity is givenby 2(Rm sin (Vm)) x6 x cos (Hm) for eccentricity along Xand 2(Rm sin (Vm)) x6 y sin (Hm) for eccentricity along Y.

The error increases farther away from the tracker andmay drop to zero depending on the azimuthal angle.The apparent distance in the case of vertical angleencoder eccentricity along N is also given by a similarexpression, 2Rmx7 n cos (Vm). Note, however, that verti-cal angle encoder eccentricity along Z is not sensitive totwo-face measurements. It is of interest to note thattwo-face system tests, if applicable, are sensitive toboth the azimuth and the distance of the target from thetracker.

Length measurement system tests: A horizontallength test as described in the B89.4.19 Standard posi-tioned parallel to the X axis is sensitive to the horizon-tal angle encoder eccentricity along the Y axis (x6 y )because the horizontal angle subtended by the length atthe origin is different from the true angle. Similarly, ahorizontal length positioned parallel to the Y axis issensitive to the horizontal angle encoder eccentricityalong the X axis. A vertical length test as described inthe Standard is sensitive to vertical angle encodereccentricity along N for the above reason. None of thepre-determined positions for length measurement sys-tem tests in the Standard is sensitive to vertical angleencoder eccentricity along Z, because the symmetricalpositioning produces error vectors that onlytranslate/rotate the length. The asymmetrical position-ing of the vertical length, such as in the user-defineddefault position 1, is somewhat sensitive. We discussthis in Sec. 9.


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Fig. 5. Horizontal and vertical angle encoder eccentricity.

5.7 Bird Bath Error (x8)

Description: The bird bath error is a constant errorin the range caused by incorrect calibration of thedistance to the fixed reference point (bird bath) withinthe tracker.

Two-face system tests: This term is not sensitive totwo-face system tests.

Length measurement system tests: Because rangemeasurements are either smaller or larger than the truevalue, all length measurement system tests producelength results that are either smaller or larger than thetrue value. The sensitivity is higher when the referencelength is closer to the tracker because a larger compo-nent of the error vector is along the length. The highestsensitivity (twice the bird bath error) is obtained whenthe tracker is placed in line between the targets, as inthe case of transit offset.

5.8 Scale Errors in the Encoder (x 9 and x 10)Description: If scale errors are present, lower order

harmonics may be the largest contributors to errors inmeasured angles. The correction in the measured anglesdue to the presence of mth order harmonic errors in thescales may be expressed as ΔHm = x9 a sin(m.Hm) +x9 b cos(m.Hm) for the horizontal angle encoder andΔVm = x10 a sin(m.Vm) + x10 b cos(m.Vm) for the verticalangle encoder. The first order (m = 1) is indistinguish-able from encoder eccentricity. We only considersecond order harmonics in the discussion below and inthe error model in the next section.

Two-face system tests: Two-face system tests aresensitive to odd order harmonic errors while they arenot sensitive to even order harmonic errors. Therefore,length measurement system tests are important toidentify second and higher order even harmonics.

Length measurement system tests: Horizontallength tests as described in the Standard are sensitiveto one component of the second order harmonic error,x9 a sin(2Hm). Horizontal length tests performed 45°away in azimuth will be sensitive to the other compo-nent, x9 b cos(2Hm). The Standard however does notspecify length measurement system tests at anyazimuthal angle other than 0°, 90°, 180° and 270°. Asan alternative, an asymmetrical length test as describedin Sec. 9 may be performed as a diagnostic for thisparameter. Vertical length tests described in theStandard are sensitive to one component of the secondorder harmonic error, x10 a sin(2Vm). The user defineddefault position 1, which is the asymmetrical verticallength test described in Sec. 9, is sensitive to the othercomponent, x10 b cos(2Vm).

6. Error Model

The above described terms can be combined into thefollowing error model, which is an adaptation of theLoser and Kyle model [5] for trackers with a beamsteering mirror (model applicable for front face meas-urements only):

7. Sensitivity Matrices

The preceding analysis and error model suggest anumerical approach to sensitivity analysis where eachmisalignment parameter is individually perturbed toassess its impact on all performance evaluation tests,which are also numerically simulated. We performedthis analysis, and the results are provided in Tables 1and 2. The information is presented in matrix form, sothe relationship between the sensitivity for each mis-alignment parameter and the performance tests is easi-ly obtained.

Care must be taken in interpreting the sensitivityvalues. For geometric offsets, sensitivity representserror in micrometers for 1 μm of offset. For tilt terms,sensitivity represents error in micrometers for 1 μrad oftilt. For eccentricity terms, sensitivity represents errorin micrometers for one non-dimensional unit of eccen-tricity (eccentricity itself is a ratio of the offset inmicrometers to radius of the encoder, also in micro-meters).

Columns in the table correspond to the misalignmentparameters while rows correspond to tests described inthe Standard. Thus, the entry in row 3 (corresponding toTest # 3), column 3 (corresponding to x 2) implies asensitivity of – 0.7 for the horizontal length test (3 m,90° azimuth) to transit offset parameter (see note belowtables to interpret rows).


27

2 8sin( )Rc Rm x Vm x= + +

1 4 5

6 6 9

9

.sin( ) sin( ) tan( )cos( ) sin( ) sin(2 )

cos(2 )

t t

x y a

b

x x xHc Hm

Rm Vm Vm Vmx Hm x Hm x Hm

x Hm

= + + +

+ − +

+

1 23 7

7 10 10

cos( )cos( )

sin( ) sin(2 ) cos(2 )

mn

z a b

x x VmVc Vm x x Vm

Rm Rm x Vm x Vm x Vm

= − + + +

− + +


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Table 1. Sensitivity matrix for length measurement system tests (shaded boxes show non-zero sensitivity)a

X1t X1m X2 X3 X4t X5 X6x X6y X7n X7z X8 X9a X9b X10a X10b

HOR 1 m 0 1 0.0 0.0 –1.5 0.0 0.0 0.0 0.0 1.5 0.0 0.0 –1.5 –2.0 0.0 0.0 0.03 m 0 2 0.0 0.0 –0.7 0.0 0.0 0.0 0.0 2.1 0.0 0.0 –0.7 –4.0 0.0 0.0 0.0

90 3 0.0 0.0 –0.7 0.0 0.0 0.0 2.1 0.0 0.0 0.0 –0.7 4.0 0.0 0.0 0.0180 4 0.0 0.0 –0.7 0.0 0.0 0.0 0.0 –2.1 0.0 0.0 –0.7 –4.0 0.0 0.0 0.0270 5 0.0 0.0 –0.7 0.0 0.0 0.0 –2.1 0.0 0.0 0.0 –0.7 4.0 0.0 0.0 0.0

6 m 0 6 0.0 0.0 –0.4 0.0 0.0 0.0 0.0 2.3 0.0 0.0 –0.4 –4.4 0.0 0.0 0.090 7 0.0 0.0 –0.4 0.0 0.0 0.0 2.3 0.0 0.0 0.0 –0.4 4.4 0.0 0.0 0.0

180 8 0.0 0.0 –0.4 0.0 0.0 0.0 0.0 –2.3 0.0 0.0 –0.4 –4.4 0.0 0.0 0.0270 9 0.0 0.0 –0.4 0.0 0.0 0.0 –2.3 0.0 0.0 0.0 –0.4 4.4 0.0 0.0 0.0

VER 3 m 0 10 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.1 0.0 –0.7 0.0 0.0 4.0 0.090 11 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.1 0.0 –0.7 0.0 0.0 4.0 0.0

180 12 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.1 0.0 –0.7 0.0 0.0 4.0 0.0270 13 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.1 0.0 –0.7 0.0 0.0 4.0 0.0

6 m 0 14 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.3 0.0 –0.4 0.0 0.0 4.4 0.090 15 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.3 0.0 –0.4 0.0 0.0 4.4 0.0

180 16 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.3 0.0 –0.4 0.0 0.0 4.4 0.0270 17 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 2.3 0.0 –0.4 0.0 0.0 4.4 0.0

RD 3 m 0 18 0.0 0.0 –0.4 0.0 0.0 –1.1 0.0 1.1 1.0 0.0 –0.7 –2.1 0.0 2.0 0.090 19 0.0 0.0 –0.4 0.0 0.0 –1.1 1.1 0.0 1.0 0.0 –0.7 2.1 0.0 2.0 0.0

180 20 0.0 0.0 –0.4 0.0 0.0 –1.1 0.0 –1.1 1.0 0.0 –0.7 –2.1 0.0 2.0 0.0270 21 0.0 0.0 –0.4 0.0 0.0 –1.1 –1.1 0.0 1.0 0.0 –0.7 2.1 0.0 2.0 0.0

6 m 0 22 0.0 0.0 –0.2 0.0 0.0 –1.1 0.0 1.1 1.1 0.0 –0.4 –2.3 0.0 2.2 0.090 23 0.0 0.0 –0.2 0.0 0.0 –1.1 1.1 0.0 1.1 0.0 –0.4 2.3 0.0 2.2 0.0

180 24 0.0 0.0 –0.2 0.0 0.0 –1.1 0.0 –1.1 1.1 0.0 –0.4 –2.3 0.0 2.2 0.0270 25 0.0 0.0 –0.2 0.0 0.0 –1.1 –1.1 0.0 1.1 0.0 –0.4 2.3 0.0 2.2 0.0

LD 3 m 0 26 0.0 0.0 –0.4 0.0 0.0 1.1 0.0 1.1 1.0 0.0 –0.7 –2.1 0.0 2.0 0.090 27 0.0 0.0 –0.4 0.0 0.0 1.1 1.1 0.0 1.0 0.0 –0.7 2.1 0.0 2.0 0.0

180 28 0.0 0.0 –0.4 0.0 0.0 1.1 0.0 –1.1 1.0 0.0 –0.7 –2.1 0.0 2.0 0.0270 29 0.0 0.0 –0.4 0.0 0.0 1.1 –1.1 0.0 1.0 0.0 –0.7 2.1 0.0 2.0 0.0

6 m 0 30 0.0 0.0 –0.2 0.0 0.0 1.1 0.0 1.1 1.1 0.0 –0.4 –2.3 0.0 2.2 0.090 31 0.0 0.0 –0.2 0.0 0.0 1.1 1.1 0.0 1.1 0.0 –0.4 2.3 0.0 2.2 0.0

180 32 0.0 0.0 –0.2 0.0 0.0 1.1 0.0 –1.1 1.1 0.0 –0.4 –2.3 0.0 2.2 0.0270 33 0.0 0.0 –0.2 0.0 0.0 1.1 –1.1 0.0 1.1 0.0 –0.4 2.3 0.0 2.2 0.0

UD1 34 0.0 0.6 0.0 0.0 0.0 0.0 0.0 0.0 0.9 0.6 –0.9 0.0 0.0 0.7 1.7UD2 35 –0.1 –0.2 –0.1 0.0 0.0 –0.8 0.0 0.6 1.1 0.0 –0.3 –1.1 0.0 2.2 0.1

A1 0.8 0.0 –1.0 0.0 0.0 0.0 0.4 0.5 0.0 0.0 –1.0 –0.2 1.0 0.0 0.0A2 –0.8 0.0 –1.0 0.0 0.0 0.0 –0.4 0.5 0.0 0.0 –1.0 –0.2 –1.0 0.0 0.0B1 0.0 0.6 0.0 0.0 0.0 0.0 0.0 0.0 0.9 0.6 –0.9 0.0 0.0 0.7 1.7B2 0.0 –0.6 0.0 0.0 0.0 0.0 0.0 0.0 0.9 –0.6 –0.9 0.0 0.0 0.7 –1.7C1 0.5 0.5 –0.7 –0.8 0.8 1.2 0.2 0.3 0.3 1.1 –1.4 –0.2 0.6 0.2 1.5C2 –0.5 –0.5 –0.7 0.8 –0.8 1.2 –0.2 0.3 0.3 –1.1 –1.4 –0.2 –0.6 0.2 –1.5D1 0.0 1.3 –2.0 –2.0 0.0 0.0 0.0 0.0 –1.3 1.5 –1.5 0.0 0.0 –2.0 0.3D2 0.0 1.1 –1.8 –1.8 0.0 0.0 0.0 0.9 –1.1 1.4 –1.4 –0.7 0.0 –1.8 0.4E 0.0 0.0 –2.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 –2.0 0.0 0.0 0.0 0.0

aThe 35 length measurement system tests in Table 1 are in the order in which they appear in the Standard. Test 1 is the horizontal length test at

the near position (1 m away, azimuthal angle of 0°). Tests 2 through 5 are the horizontal lengths at four orientations of the tracker (0°, 90°, 180°and 270°) at the 3 m distance. Tests 6 through 9 are the horizontal lengths at four orientations of the tracker at the 6 m distance. Tests 10 through17 are the vertical length tests. Tests 18 through 25 are the right diagonal lengths and tests 26 through 33 are the left diagonal lengths. Tests 34and 35 are the user-defined positions. The subsequent 9 rows marked as tests A1, A2, B1, B2, C1, C2, D1, D2 and E are not described in theStandard; these are tests we propose in Sec. 9.

In the next section, we make several observationsbased on the preceding analysis and the sensitivitymatrices shown. It should be noted that the sensitivitymatrices are different for different tracker constructions(different error models) and we therefore cautionagainst extensive generalization from the one reportedinstance here. The observations in the next sectioncome from an analysis of the three commonly foundtracker configurations, which we have modeled.

8. Discussion

Two-face system tests

A large number of geometric misalignment para-meters are sensitive to two-face measurements. Thesetests are easy to perform and require no calibratedreference artifacts. Two-face system tests are thereforea critical component in a performance evaluation oflaser trackers.


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Table 2. Sensitivity matrix for two-face system tests (shaded boxes show non-zero sensitivity)a

X1t X1m X2 X3 X4t X5 X6x X6y X7n X7z X8 X9a X9b X10a X10b

1 m LOW 0 1 2.0 2.0 1.7 3.6 3.6 3.0 2.0 0.0 3.0 0.0 0.0 0.0 0.0 0.0 0.090 2 2.0 2.0 1.7 3.6 3.6 3.0 0.0 2.0 3.0 0.0 0.0 0.0 0.0 0.0 0.0

180 3 2.0 2.0 1.7 3.6 3.6 3.0 2.0 0.0 3.0 0.0 0.0 0.0 0.0 0.0 0.0270 4 2.0 2.0 1.7 3.6 3.6 3.0 0.0 2.0 3.0 0.0 0.0 0.0 0.0 0.0 0.0

MED 0 5 2.0 2.0 0.0 2.0 2.0 0.0 2.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.090 6 2.0 2.0 0.0 2.0 2.0 0.0 0.0 2.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

180 7 2.0 2.0 0.0 2.0 2.0 0.0 2.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0270 8 2.0 2.0 0.0 2.0 2.0 0.0 0.0 2.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

HIGH 0 9 2.0 2.0 1.7 3.6 3.6 3.0 2.0 0.0 3.0 0.0 0.0 0.0 0.0 0.0 0.090 10 2.0 2.0 1.7 3.6 3.6 3.0 0.0 2.0 3.0 0.0 0.0 0.0 0.0 0.0 0.0

180 11 2.0 2.0 1.7 3.6 3.6 3.0 2.0 0.0 3.0 0.0 0.0 0.0 0.0 0.0 0.0270 12 2.0 2.0 1.7 3.6 3.6 3.0 0.0 2.0 3.0 0.0 0.0 0.0 0.0 0.0 0.0

3 m LOW 0 13 2.0 2.0 0.9 6.7 6.7 3.0 6.0 0.0 3.0 0.0 0.0 0.0 0.0 0.0 0.090 14 2.0 2.0 0.9 6.7 6.7 3.0 0.0 6.0 3.0 0.0 0.0 0.0 0.0 0.0 0.0

180 15 2.0 2.0 0.9 6.7 6.7 3.0 6.0 0.0 3.0 0.0 0.0 0.0 0.0 0.0 0.0270 16 2.0 2.0 0.9 6.7 6.7 3.0 0.0 6.0 3.0 0.0 0.0 0.0 0.0 0.0 0.0

MED 0 17 2.0 2.0 0.0 6.0 6.0 0.0 6.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.090 18 2.0 2.0 0.0 6.0 6.0 0.0 0.0 6.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

180 19 2.0 2.0 0.0 6.0 6.0 0.0 6.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0270 20 2.0 2.0 0.0 6.0 6.0 0.0 0.0 6.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

HIGH 0 21 2.0 2.0 0.9 6.7 6.7 3.0 6.0 0.0 3.0 0.0 0.0 0.0 0.0 0.0 0.090 22 2.0 2.0 0.9 6.7 6.7 3.0 0.0 6.0 3.0 0.0 0.0 0.0 0.0 0.0 0.0

180 23 2.0 2.0 0.9 6.7 6.7 3.0 6.0 0.0 3.0 0.0 0.0 0.0 0.0 0.0 0.0270 24 2.0 2.0 0.9 6.7 6.7 3.0 0.0 6.0 3.0 0.0 0.0 0.0 0.0 0.0 0.0

6 m LOW 0 25 2.0 2.0 0.5 12.4 12.4 3.0 12.0 0.0 3.0 0.0 0.0 0.0 0.0 0.0 0.090 26 2.0 2.0 0.5 12.4 12.4 3.0 0.0 12.0 3.0 0.0 0.0 0.0 0.0 0.0 0.0

180 27 2.0 2.0 0.5 12.4 12.4 3.0 12.0 0.0 3.0 0.0 0.0 0.0 0.0 0.0 0.0270 28 2.0 2.0 0.5 12.4 12.4 3.0 0.0 12.0 3.0 0.0 0.0 0.0 0.0 0.0 0.0

MED 0 29 2.0 2.0 0.0 12.0 12.0 0.0 12.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.090 30 2.0 2.0 0.0 12.0 12.0 0.0 0.0 12.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

180 31 2.0 2.0 0.0 12.0 12.0 0.0 12.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0270 32 2.0 2.0 0.0 12.0 12.0 0.0 0.0 12.0 0.0 0.0 0.0 0.0 0.0 0.0 0.0

HIGH 0 33 2.0 2.0 0.5 12.4 12.4 3.0 12.0 0.0 3.0 0.0 0.0 0.0 0.0 0.0 0.090 34 2.0 2.0 0.5 12.4 12.4 3.0 0.0 12.0 3.0 0.0 0.0 0.0 0.0 0.0 0.0

180 35 2.0 2.0 0.5 12.4 12.4 3.0 12.0 0.0 3.0 0.0 0.0 0.0 0.0 0.0 0.0270 36 2.0 2.0 0.5 12.4 12.4 3.0 0.0 12.0 3.0 0.0 0.0 0.0 0.0 0.0 0.0

a The 36 two-face system tests in Table 2 are in the order in which they appear in the B89.4.19 Standard. Therefore, tests 1 through 4 are the two-face system tests at the near position (1 m) with the target on the floor for four orientations of the tracker (0°, 90°, 180° and 270°). Tests 5 through8 are the two-face system tests at the near position (1 m) with the target at tracker height for four orientations of the tracker. Tests 9 through 12are the two-face system tests at the near position (1 m) with the target at twice the tracker height for four orientations of the tracker (0°, 90°, 180°and 270°). Tests 13 through 24 are a repetition of tests 1 through 12 but with the tracker 3 m away from the target. Tests 25 through 36 are arepetition of tests 1 through 12 but with the tracker 6 m away from the target.

Some geometric misalignment parameters areequally sensitive to two-face measurements at alldistances from the tracker while others show increasingsensitivity farther away from the tracker. Some para-meters are insensitive to the azimuth or elevation, whileothers are sensitive. The tests described in the B89.4.19Standard capture these influence parameters effectivelyby requiring the target be positioned at different dis-tances from the tracker, and at different horizontal andvertical angles.

It should be pointed out that the sensitivity valueswere generated by considering the influence of onlyone misalignment parameter at a time, and each of unitvalue. The error in any two-face system test in thepresence of two or more misalignment parameters at atime (each of unit value) is not necessarily given by thesum of the individual sensitivities for two reasons.

First, the distance error is determined as the root sumsquare of the errors in two orthogonal directions; onecomponent is due to the horizontal angle error andanother due to vertical angle error. Therefore, summa-tion without regard to direction will produce erroneousresults. Second, the two-face error, which is defined asthe magnitude of the vector joining the coordinatedetermined from the front face measurement to thatdetermined from the back face measurement, disre-gards the sign of the vector. Therefore, two or moreparameters that produce non-zero error individuallymay in combination result in cancellation of the error.

The convolution of the angle errors into a distanceerror may result in unification of the reporting methodwith length measurement system tests within theStandard. But the above observations motivate us tosuggest that angle errors be recorded in addition todistance errors in two-face system tests as they havesignificant diagnostic value.

For the tracker mechanical configuration we dis-cussed in this paper, there are only a few misalignmentparameters that are insensitive to two-face system tests(vertical angle encoder eccentricity along Z, bird bath,and second order scale error). For other configurationssuch as those with rotating mirrors, there are manyterms that are not sensitive to two-face system tests (see[5]). It is therefore desirable to have a set of lengthmeasurement system tests that are also sensitive tomost or all of the geometric misalignment parametersin a tracker.

Length measurement system tests

From the discussion in the previous section and thesensitivity matrix in Table 1, it is clear that certainmisalignment parameters are detected by some of thelength measurement system tests described in theStandard (three out of four encoder eccentricity para-meters, transit tilt, transit offset, and some second orderscale error components). There however appears to bereduced or no sensitivity to other geometric misalign-ment parameters (beam offsets, beam tilt, vertical indexoffset, vertical encoder eccentricity along Z). Smallchanges in the position and orientation of the referencelengths greatly improve sensitivity as shown in the nextsection.

Another observation may be made from Table 1.From a geometric error modeling perspective alone, itappears that there is no value in performing verticallength tests at different orientations of the tracker (0°,90°, 180° and 270°) because the vertical length tests arenever a function of the azimuth. Further, the value inperforming vertical length tests at different distancesfrom the tracker is also not apparent.

Finally, we note that unlike in the case of two-faceerrors, the error in any length measurement system testin the presence of multiple geometric misalignmentparameters (each of unit value) is indeed given by thesum of the individual sensitivities. This is true becausethe sensitivity to any given geometric misalignmentparameter is determined only from the component ofthe error vector along the length (perpendicular compo-nents are not sensitive), and these scalar componentscan be summed algebraically.

9. Proposed New Length MeasurementSystem Tests

We propose in this section some new length meas-urement system tests that demonstrate improved sensi-tivity to some of the geometric misalignment parame-ters previously undetected. It should be pointed out thatthe tests described here may be sensitive to multiplemisalignment parameters, but we highlight only one ortwo misalignment parameters for each test for purpos-es of illustration. Further, we re-emphasize that sensi-tivity is a function of the mechanical design employed,and the description below pertains only to the theodo-lite-type tracker we have considered in this paper.


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9.1 Asymmetrical Horizontal Length Test (RowsA1, A2 in Table 1)

We mentioned earlier that beam offset along thetransit axis is insensitive to symmetrically placed refer-ence lengths because the error vectors at the two endsonly serve to translate and rotate the length, not changeits magnitude. In fact, we noted that there is no lengthmeasurement system test in the B89.4.19 Standard thatis sensitive to this term.

A small modification in the positioning of the refer-ence length in the horizontal length test addresses thisissue. Instead of being placed at the center of the refer-ence length, the tracker is placed near one end and asclose as possible (0.5 m) to the length. Figure 6 showsa schematic of the setup with the tracker at A1 and thetwo ends of the reference length at a and b. Also shownin the figure is the offset beam pointing at a1 instead ofat a. In rotating the tracker so that the beam points to a,the target appears to be at a2 in the tracker coordinatesystem. A similar effect occurs at target b also. This testis sensitive to beam offset, because the projections ofthe errors along the reference length are much larger ata than at b.

In addition to the position A1, we also define a mirrorposition A2 where the tracker can be placed and thelength re-measured. The error at this location of thetracker is equal in magnitude but opposite in sign tothat measured with the tracker at A1. Two quick meas-urements can therefore provide substantial diagnosticinformation.

This test, as mentioned earlier, is sensitive to severalgeometric misalignment parameters. Of these, it isworth mentioning that this test captures a component ofthe second order scale error, x9b cos (2Hm), not detectedby other tests.

9.2 Asymmetrical Vertical Length Test (Rows B1,B2 in Table 1)

An argument similar to the above may be made forbeam offset along the M axis where an asymmetricalvertical length test is sensitive, while other tests are not.As mentioned earlier, such a test is described in theStandard as the user defined default position 1. We sug-gest the tracker be placed as close as possible to the2.3 m reference length and level with the lower target(Vm = 90°). Our experiments suggest a 1 m distance ispractical; hence the lower sensitivity in Table 1 in com-parison with the asymmetrical horizontal length test forwhich we use a 0.5 m distance. Again, there is a mirrorposition (where the tracker is placed at the high point sothat Vm = 90° for the target at the top) for this testwhere the error reverses in sign, although in practice itmay be more difficult to mount the tracker at thisheight.

9.3 Asymmetrical Diagonal Test (Rows C1, C2 inTable 1)

As mentioned earlier the beam tilt parameter (colli-mation error in theodolites) is not captured by any ofthe B89.4.19 length measurement system tests. We sug-gested that an asymmetrical diagonal test is somewhatsensitive to this term. Fig. 7 shows a schematic of sucha test. In Fig. 7, with the tracker at C1 and target at a, letthe beam tilt result in the beam pointing at a1. Theeffect of this error is that the target appears to be at a2

in the tracker coordinate system. There is a similareffect for the target at b also. However, the error vectorsa-a2 and b-b2 have different directions and magnitudes


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Fig. 6. Asymmetrical horizontal length test (top view shown).

(the horizontal angle errors are not the same at a and bbecause of the different vertical angles; recall that the

utes a different component along the length. The resi-dual provides the sensitivity. Again, there exists amirror position C2 where the error reverses in sign.

9.4 Horizontal Length Above the Tracker (RowsD1, D2 in Table 1)

The horizontal length test above the tracker (trackerat D1 in Fig. 8) is particularly sensitive to two misalign-ment parameters generally not sensitive to the lengthmeasurement system tests described in the Standard -vertical encoder eccentricity along Z and vertical indexoffset.

With the tracker at D1 and target at a, assume eitherof the two misalignment parameters mentioned abovecauses the point to appear at a1. Rotation about thestanding axis results in the same effect with target at bappearing to be at b1. In case of vertical index offset,there is either a dead space in the encoder near the poleor an overlap of angle resulting in a length error. Thecomponents of the error vectors a-a1 and b-b1 along thereference lengths sum to make this test sensitive tothese parameters. A practical realization for this test, ifthe above were not feasible, would be to place thetracker as close as possible to the horizontal referencelength (say at D2).

9.5 Tracker in Line and in Between Targets(Row E in Table 1)

The standard requires the measurement of a horizon-tal length with the tracker as close as possible to thereference length. This test is particularly sensitive totransit offset and the bird bath error if the tracker isplaced in between the target nests and in line with thetargets.

10. Experimental Validation

The B89.4.19 length measurement system testresults and two-face system test results for a tracker arepresented in Fig. 9 (a) and Fig. 9 (b). Notice that thetracker passed the length measurement system testportion of the B89.4.19 Standard but showed largeerrors in the two-face system tests. We extracted theraw horizontal and vertical angle data for the two-facesystem tests; they are shown in Fig. 9 (c). Note that theangular errors scale inversely with distance.

Based on our models, the likely cause for this behav-ior is an offset in the beam as it emerges from the head(a constant offset implies reducing angular errors withincreasing distance). The beam is likely offset alongboth the transit axis OT and along its normal OM (asdefined in Fig. 1). From the measured angle errors andrange values, we compute the actual beam offsets with-in the instrument head along the two directions to be


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Fig. 7. Asymmetrical diagonal length test.

4correction ). Each error term contrib-sin( )

txHm

VmΔ =

Fig. 8. Horizontal length above tracker.

473 μm and 359 μm using a least-squares best fit. Theprecise offset values are not relevant; approximatevalues are useful in understanding the impact on lengthmeasurements made in the working volume of thetracker.

We performed the asymmetrical horizontal and theasymmetrical vertical length tests as described in

Sec. 9 because these tests, according to our analysis, arediagnostic of beam offset. Our simulations based on theabove offset values suggest a length error of about250 μm, when a 2 m length is measured with the trackerabout 0.5 m away from the reference length for both thehorizontal and vertical orientations. With the trackerat the mirror position, we predict an error of equal


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Fig. 9. (a) Length measurement system test results (b) Two-face system test results (c) Angle errors in two-face data.

magnitude but opposite sign. Our experimental valueswere 350 μm and –250 μm for the horizontal lengthtests, and 320 μm and –280 μm for the vertical lengthtests. These values are comparable to the simulationresults; the slight discrepancy stems from other sourcesof misalignments that may also contribute to thesetests. But overall, the errors are much larger than any ofthe errors shown in Fig. 9 (a) and confirm the value ofasymmetrical tests in tracker testing.

We note the following two points from Fig. 9. First,an offset in the beam along the OM axis should gener-ally be sensitive to the length measurement system testat the user-defined default position 1, if the length ispositioned as close as possible to the tracker. The meas-ured error (see point # 34 in Fig. 9 (a)) at this positionis approximately 50 μm and within the MPE becausethe user-defined position 1 was performed at a non-optimal position of about 3 m from the tracker. Thissuggests a need for emphasizing the placement positionof the reference length in default position 1 in theStandard.

Second, in Fig. 9 (c), we note that the vertical angleplot shows a saw-tooth pattern indicating dependenceon the azimuthal angle. This pattern, when scaled bythe range, does indeed map on to the saw-tooth patternin Fig. 9 (b). While such dependence of two-face erroron azimuth might be indicative of horizontal angleencoder eccentricity, the absence of a similar behaviorin the length test results, particularly the horizontallength tests, suggests some other error source not mod-eled by the geometric error model might be the reasonfor the observed systematic error (for instance, internalstressing and relaxation of components such as cablesmay conceivably cause such observed behavior).

11. Conclusions

We have, in this paper, explored the relationshipbetween geometric misalignments in the constructionof laser trackers and performance evaluation testsdescribed in the B89.4.19 Standard. It is desirable thattests described in any performance evaluation Standardbe sensitive to all sources of systematic error includinggeometric misalignments. Our approach to the evalua-tion of the B89.4.19 tests uses geometric error modelsand numerical simulation. This requires the develop-ment of different error models for different mechanicaldesigns, but that effort provides a systematic yetrelatively straightforward numerical approach to under-standing sensitivities.

Summarized here are some of our primary observations:

• Two-face system tests described in the Standardoffer quick and fairly extensive diagnostic capa-bility. There is considerable diagnostic value inrecording actual errors in the angles in two-facesystem tests instead of convolving these errorsinto a distance error.

• Length measurement system tests described inthe Standard are sensitive to numerous geo-metric misalignment parameters, but for certaintracker configurations, there may be room forimprovement in the placement of the referencelength artifact. We have proposed some newplacement scenarios that improve sensitivity forcertain tracker constructions and these tests maybe considered as additional or optional tests.

• Two of the tests we have proposed already existin some form in the Standard. These are the user-defined default position 1 and the horizontallength test in the near position. The verbiage inthe Standard allows for flexibility in the place-ment of the reference lengths; non-optimalplacement will reduce sensitivity to some geo-metric misalignment parameters as we demon-strated in Sec. 10.

• From a geometrical misalignment perspective,there are some redundant tests in the Standard,but we do realize that other systematic sourcesexist, and these tests may prove valuable.Modeling other systematic sources of error andadditional testing is required prior to any recom-mendations for removing redundant tests.

As a final note, we realize that a number of interest-ing possibilities emerge as a result of the analysismethods described in this paper. Suitable artifact loca-tions may be identified using sensitivity analysis for thedetermination of geometric misalignment parametersby best-fitting. Monte Carlo simulation in combinationwith error models may be used in the determination ofthe component of uncertainty in a length measurementfrom uncertainty in the error model parameters them-selves. New international Standards development activ-ity conceivably may benefit from such sensitivityanalysis in optimal placement of artifacts within thework volume.


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Acknowledgments

We are grateful to members of the ASME B89.4.19committee for their comments; in particular, we thankRobert Bridges, Craig Shakarji, John Palmateer andEdward Morse for reviewing the manuscript and fortheir valuable feedback.

12. References

[1] ASME B89.4.19-2006 Standard—Performance Evaluation ofLaser-Based Spherical Coordinate Measurement Systems,www.asme.org.

[2] H. Zhuang and Z. Roth, Modeling gimbal axis misalignmentsand mirror center offset in a single-beam laser trackingmeasurement system, The International Journal of RoboticsResearch. 14 (3), 211-224 (1995).

[3] H. Jiang, S. Osawa, T. Takatsuji, H. Noguchi, and T. Kurosawa,High-performance laser tracker using an articulating mirror forthe calibration of coordinate measuring machine, OpticalEngineering. 41 (3), 632-637 (2002).

[4] P. D. Lin and C-H. Lu, Modeling and sensitivity analysis oflaser tracking system by skew-ray tracing method, ASMEJournal of Manufacturing Science and Engineering. 127 (3),654-662 (2005).

[5] R. Loser and S. Kyle, Alignment and field check procedures forthe Leica Laser Tracker LTD 500, Boeing Large Scale OpticalMetrology Seminar (1999 ).

[6] Fritz Deumlich, Surveying Instruments, Walter de Gruyter,Berlin/New York (1982).

About the authors: Bala Muralikrishnan is a guestresearcher, Daniel Sawyer is a mechanical engineer,Chris Blackburn is a physical science technician, StevePhillips, Bruce Borchardt, and Tyler Estler are physi-cists; all are with the Precision Engineering Division ofthe NIST Manufacturing Engineering Laboratory. TheNational Institute of Standards and Technology is anagency of the U.S. Department of Commerce.


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ASME B89.4.19 Performance Evaluation Tests and Geometric Misalignments in Laser Trackers Volume

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