Coordinate uncertainty analyses of coupled multiple measurement systems

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IOP PUBLISHING MEASUREMENT SCIENCE AND TECHNOLOGY

Meas. Sci. Technol. 21 (2010) 065103 (6pp) doi:10.1088/0957-0233/21/6/065103

Coordinate uncertainty analyses ofcoupled multiple measurement systemsWanli Liu1,3 and Zhankui Wang2

1 School of Mechanical and Electrical Engineering, China University of Mining and Technology,Xuzhou 221116, People’s Republic of China2 Henan Institute of Science and Technology, Xinxiang, Henan 453003, People’s Republic of China

E-mail: liuwanli218@126.com

Received 23 February 2010, in final form 8 March 2010Published 28 April 2010Online at stacks.iop.org/MST/21/065103

AbstractPrecision three-dimensional measurement of large-scale workpiece frequently involves thecombination of several different types of measurement systems; they may be laser tracker, totalstation, laser scanner and portable coordinate measuring machines, etc. In order to provideoptimization measuring results, a new method, called the isolated variable sub-system (IVSS),has been developed to deal with variable coupling that fully takes into account the uncertaintyof each measurement individual system. The IVSS method is a combination of pattern searchand singular value decomposition. It can effectively determine the optimized location andorientation of each measurement system and minimizes the coordinate combined uncertaintyby multivariate statistics to the measured data. Intensive experimental studies have been madeto check the validity of the proposed method; the results show that using this technology themeasuring accuracy of coupled multiple measurement systems can be improved by about 49%and can accommodate missing data points from some of the measurement systems.

Keywords: multiple measurement system, uncertainty, isolated variable sub-system (IVSS),analysis

(Some figures in this article are in colour only in the electronic version)

1. Introduction

Nowadays, large-scale manufacturing operations areincreasing their reliance on portable coordinate measurementdevices. These devices include laser tracking systems, laserscanning systems, portable coordinate measuring machines,total station systems, video photogrammetry systems andelectronic theodolite networks [1–3]. This increase in portablemetrology is driven by cost and efficiency. Commercialairframe construction is a good example of this trend. Insteadof relying on elaborate holding fixtures and precise tooling,manufacturers are using the component parts as the tooling,and verifying design conformance with portable coordinatemeasurement devices. This design paradigm shift eliminatesthe need to manufacture and maintain complex, costly check-fixtures. It does, however, introduce portable metrologyinto the production process and therefore requires a rigorous

3 Author to whom any correspondence should be addressed.

accounting of the uncertainty in the design conformancemeasurements.

Much work has been done to understand andquantify the performance of various measurement systems.The manufacturers of the measurement devices publishperformance specifications. These laboratory specificationsare not, however, representative of the actual performanceof the instrument under typical in situ conditions. Manysignificant effects are ignored including operator contributionsto the uncertainty and the variability of real-world shop floormeasurement environments. In addition, the manufactureruncertainty statements often do not address the geometricnature of the coordinate uncertainty but instead provide avolumetric statement based on a spherical uncertainty at eachpoint [4–6].

Many large-scale measurement processes require morethan a single measurement instrument. Examplesinclude commercial giant antenna, airplane productionand shipbuilding. These applications necessitate either a

0957-0233/10/065103+06$30.00 1 © 2010 IOP Publishing Ltd Printed in the UK & the USA

Meas. Sci. Technol. 21 (2010) 065103 W Liu and Z Wang

combination of various measurement devices or the relocationof a single instrument throughout the measurement volumein order to acquire the necessary data. This combination ofmultiple measurement systems is the focus of this research.Users often combine measurement systems by tying individualmeasurement systems together based on common referencepoints, and then assume that they are still working within theinstrument’s published uncertainty. Alternatively, many usersapply heuristics to determine the uncertainty as they progressalong a chain of measurements [7–10]. These methods providevery poor approximations of uncertainty in all but the mostsimplistic cases. Even in cases where only a single placementof an instrument is used, its measurements are typically tiedinto a reference coordinate system. The uncertainty from thistie-in process is often ignored.

In order to provide optimization measuring results usingcoupled multiple measurement systems, this paper proposesa new method of isolated variable sub-system (IVSS) todeal with variable coupling and poor initial guesses in arobust manner, and fully takes into account the uncertaintyof each measurement individual system. The IVSS methodcombines the pattern search method and a nonlinear least-squares optimization of a subset of the variables usingsingular value decomposition. It can effectively determinethe optimized location and orientation of each measurementsystem, and minimizes the coordinate combined uncertaintyby multivariate statistics to the measured data. Thus, it isfeasible and beneficial to use the proposed technology todesign the optimized measurement process and minimize thefinal coordinate uncertainty. This research will be widely usedin large-scale manufacturing measurement in the future.

2. Coordinate measurement uncertaintycharacterization

A coupled spatial measurement system is defined as ametrology system in which one or more metrology instrumentsare combined in an interdependent fashion, often withcorresponding nominal or CAD design data. In other words,the observations of one instrument are used to locate, eitherdirectly or indirectly, itself, other instruments, or some formof nominal data [11–13]. The uncertainty in coupled spatialmeasurement systems is determined through the combinationof the uncertainties in the various coordinate acquisitionsystems that comprise the measurement network. In thisnetwork, if all coordinate measurement devices were capableof producing measurements with uncertainties conformingnicely to X-, Y-, and Z-axis representations, this processwould have been much simpler. Given the wide array ofmeasurement devices in use today, however, this is not thecase. Several different types of sensors are commonly usedin measurement devices, and each type has its own errorcharacteristics. Polar devices such as theodolites and lasertrackers provide two angular values and a distance value.Kinematic devices such as portable and fixed CMM armsprovide a series of joint values. Laser scanning devicesprovide a raster grid with range values determined by an

Fixed target

Position 4

Position 2 Position 3

Position 1

Laser tracker

Figure 1. The evaluation method of the individual measurementsystem.

offset laser and camera arrangement. There are many others.Each system then provides coordinates through a computationwhich converts the native device values into XYZ coordinates.The resulting coordinates have uncertainties resulting from theuncertainties in the native device values and the model appliedin the conversion. Therefore, before dealing with the entirenetwork, we must first address the individual measurementsystem uncertainty. The individual uncertainties may then becombined with those of other systems.

Currently the individual measurement system uncertaintyevaluation methods typically make use of known artifactssuch as scale bars, tetrahedra and linear interferometer scales.These artifacts are expensive and difficult to maintain andtransport. In this paper, we propose a method to use a series offixed, but unknown targets. The performance of the instrumentis evaluated based on the fact that the points remain fixedthroughout the measurement process. The degree to which theinstrument shows the points to move when they are measuredfrom different instrument locations is directly related to theuncertainty of the system. If the system had zero uncertainty,the relative measurements of the points would be identical forall instrument locations.

The proposed approach is summarized as follows:(1) establish a field of unknown yet fixed points; (2) measurethese points from several locations of the instrument;(3) perform an optimization on all the measurementsto determine the transformations of the instruments thatminimize the residual discrepancies in the redundant pointmeasurements; and (4) statistically process the residual errorsto determine uncertainty values for the individual instrumentcomponents. In order to explain the proposed method, anexample using the laser tracker layout is shown in figure 1where four instrument positions are distributed around a fieldof ten fixed targets.

3. Isolated variable sub-system method

Given the measurements of each point in the target fieldfrom multiple instrument locations, the next task is todetermine the transformations of the instruments that minimize

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Meas. Sci. Technol. 21 (2010) 065103 W Liu and Z Wang

the discrepancy between the observations. Once this isaccomplished, the remaining residual errors will be analyzed todetermine the uncertainty of the instrument. Before describingthe instrument transformation optimization problem, we mustfirst address the point computation optimization since it isnested within the total problem.

3.1. Data analysis and optimization

Each one of the measurements is represented in terms of theinternal measurement device values. For a laser tracker, thesevalues are the horizontal angle, vertical angle and distance. Fora portable CMM, the joint values are used. For a conventionaltheodolite, there are only horizontal and vertical angles withno distance information. In any case, there are a series ofmeasurement values for each individual observation such that[14–16]

Pki = f (Mki , μk) (1)

where Pki is the vector location of point i based on instrumentk usually expressed in XYZ coordinates and Mki is themeasurement from instrument k to point i. This vector isof dimension τ , the number of output values the measurementdevice produces for each observation. μk is a set of calibrationand kinematic parameters for instrument k; f (·) is a functionconverting the measurement into the observed coordinaterelative to the instrument’s reference frame.

For example, suppose we have a single laser trackerproviding compensated measurements of horizontal angle,vertical angle and distance for a single point. The actualconversion process from instrument encoder values to thesecompensated measurement values is a complex process usuallyhandled by the instrument manufacturer. This complex set ofcompensation parameters is represented by μk in equation (1).In this case, however, we deal with data that are alreadycompensated. The compensated measurement output valuesfrom the laser tracker are

M = [l θ ϕ 1] (2)

where l is the distance measurement; θ is the horizontal anglemeasurement and ϕ is the vertical angle measurement.

In this case, the dimension of M is τ = 3. We use thefollowing functions to compute the XYZ location of the pointgiven only the measurements from a single instrument:⎧⎨

⎩Pxik

= l sin ϕ cos θ

Pyik= l sin ϕ sin θ

Pyik= l cos ϕ

(3)

The point Pik represents the measurement from a singleinstrument. If, however, multiple instruments measure acommon point, there is an optimization process that isperformed to determine the point location given the redundantdata. This process yields a point Pi that is based onall the measurements in the network for this point. Inthis optimization process, it is assumed that the instrumenttransformations are already known. The optimization beginswith an assumed point location. This can be a simple averageof the point values for each instrument’s measurement ofthe point or a more elaborate function. Starting with the

*

θε

lε

1

2

3

Figure 2. Point computation residual errors.

guessed point value P∗i , the objective function is evaluated.

This evaluation is performed in terms of the measurementvector Mki . This is done by converting the current point P∗

i

to a nominal measurement vector M∗i . For the case of a polar

system, this conversion is the inverse of equation (3):⎧⎪⎨⎪⎩

l =√

P 2x + P 2

y + P 2z

θ = arctan(Py/Px)

ϕ = arccos(Pz/Px)

(4)

At this point, we have M∗i representing the current guess

at the optimum point and a measurement, Mi , for eachkth observation. The objective function values for eachinstrument’s measurement are

εik = M∗i − Mik (5)

The complete objective function vector εi is formed byconcatenating each εik sequentially. The result is a vectorof dimension N = 3k, containing all the residual errors forpoint i:

εi = [εli1 εθi1 εϕi1 · · · εlik εθik εϕik

](6)

where εlik is the error in the length component for point i,measured by instrument k.

A two-dimensional representation of the residual errors isshown in figure 2.

The point computation optimization for point i is statedas find P∗

i and minimize∑

n ε2in

.

3.2. Coupled multiple measurement system transformationoptimization

For measuring data taken in the j th, j ∈ {1, 2, . . . , k},according to equation (2), the homogeneous coordinate vectorand its components are written with a pre-superscript asj M = [j l j θ jϕ 1]T .

The homogeneous transformation j2j1T from the coordinate

system j1 to the coordinate system j2 is defined as follows:

j2 M = j2j1T · j1 M (7)

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Meas. Sci. Technol. 21 (2010) 065103 W Liu and Z Wang

where

j2j1T =

[j2j1R

j2j1

P0 0 0 1

],

and the pure rotation matrix j2j1R is a function of the three fixed

angles α, β and γ , which correspond to roll, pitch and yawabout the fixed x, y and z axes respectively:

j2j1

R =⎡⎣cos β cos γ sin α sin β cos γ − cos α sin γ cos α sin β cos γ + sin α sin γ

sin β sin γ sin α sin β sin γ + cos α cos γ cos α sin β sin γ − sin α cos γ

−sin β sin α cos β cos α cos β

⎤⎦

(8)

The translation vectors are given by

j2j1

P = [j2j1x

j2j1y

j2j1z 1]T (9)

The k measurement systems have their own local coordinatesystem j ∈ {1, 2, . . . , k}. Each of the k measurementsystems takes data at n points in three dimensions. Forthree-dimensional data at the ith point i ∈ {1, 2, . . . , n}taken by the j th measurement system, the following notationis used for the data in homogeneous coordinates: j Mj,i =[j l1,j,i

j θ1,j,ij ϕ1,j,i 1]T

Each set of n data points taken by the j th measurementsystem are grouped into a 4 × n data matrix j Dj by groupingthe column vectors, as shown in

j Dj =

⎡⎢⎢⎣

j l1,j,1j l1,j,2 · · · j l1,j,n

j θ1,j,1j θ1,j,2 · · · j θ1,j,n

jϕ1,j,1jϕ1,j,2 · · · jϕ1,j,n

1 1 · · · 1

⎤⎥⎥⎦ (10)

One of the measurement instruments is selected as thereference for all the data and is set to system number1. Typically, this would be the system with the lowestuncertainties, the most recently calibrated, or the mostcentrally located system with respect to the object beingmeasured.

The uncertainty estimate of the transformation from eachof the other coordinate systems to coordinate system 1 is found.Extending equation (10) for the homogeneous transformationto the entire set of data j Dj the transformation 1

j T from thecoordinate system j to coordinate system 1 is defined as

1Dj = 1j T · j Dj (11)

where the 1Dj and j Dj data matrices are for the first data setviewed in the j th coordinate system and for the j th data setviewed in the j th coordinate system respectively.

The initial estimate 1j Test of the transformation 1

j T is foundby assuming equal uncertainties in each dimension for all themeasurements. Then, according to equation (7), the estimatedtransformation is calculated as

1j Test = 1Dj · (

j Dj)−1

(12)

Estimates for the displacements between coordinate system j

and coordinate system 1 are found from the fourth column of

the transformation matrix 1j Test as was defined in equation (7):⎧⎪⎨

⎪⎩1jxest,j = 1

j Test,x,j

1jyest,j = 1

j Test,y,j

1jzest,j = 1

j Test,z,j

(13)

Using equation (8) for the definition of the rotation matrix, 1jR

initial estimates for the angles 1jαest , 1

jβest and 1j γest are derived

from⎧⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎩

1j αest1 = atan 2

[1j Test,y,j ,

1j Test,x,j

]

1j βest1 = atan 2

[−

(1j Test,z,j

),

√(1j Test,x,j

)2+

(1j Test,y,j

)2]

1j γest1 = atan 2

[1j Test,z,j ,

√(1j Test,x,j

)2+

(1j Test,y,j

)2+

(1j Test,z,j

)2](14)

where the atan2(y, x) function returns an angle in the range−π to π .

Each measurement has an associated uncertainty scopein the (l, ϕ, θ) principal axis coordinate system. For ameasurement by the j th measurement system on the ithmeasurement point, the uncertainties in l, ϕ and θ are⎧⎪⎨

⎪⎩ξl = M̂l(ulmin,j + li,j · ul,j )

ξϕ = M̂ϕ · (li,j · uϕ,j )

ξθ = M̂θ · (li,j · uθ,j )

(15)

where the uncertainty factors ulmin,j , ul,j , uϕ,j and uθ,j are onlydependent on the measurement system j , not on the particularpoint that is being measured. Also the uncertainties inequation (15) are only a function of the measurement pointi through li,j , which is the range between the measurementstation and target.

After these variables l, ϕ, θ , α, β and γ arecalculated, the homogeneous spatial transformation matrix j2

j1T

(equation (7)) may be specified using a vector containing sixvariables as follows:

j2j1

T =[

j2j1

Pxj2j1

Pyj2j1

Pzj2j1

Rαj2j1

Rβj2j1

Rγ

]T

(16)

The equations for converting these six variables into thetransformation matrix components and reversing that processto extract the variables is well understood.

In total, we have variables for the entire network. Oneof the instruments (usually the first location, j = 1) shouldremain fixed. If all the instruments are allowed to move,the entire system will tend to float in the world coordinatesystem. This introduces an unnecessary infinity of solutions.By leaving one instrument fixed, however, this is avoided.This brings the actual total number of unknown variables to6 × (j −1). The unknown variable vector for the optimizationis the combination of equation (16) for all instruments,skipping the first, which is the reference:

T = 21T · 3

2T · · · · · j

j−1T (17)

where T is the unknown variable vector for the optimization,dimension 6 × (j − 1).

As with all optimization processes, it is important to makea reasonable initial guess, this shortens the solution time andhelps to avoid convergence to the wrong solution. In this case,making a good initial guess is straightforward.

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Meas. Sci. Technol. 21 (2010) 065103 W Liu and Z Wang

The equations governing the optimization process are theresidual errors in the redundant measurements of each point.These residuals were presented in equation (6) as εi . Theyresult from the point computation optimization process. It isimportant that these point computation residuals are extractedafter a full point optimization for each evaluation of theinstrument transformation optimization. They are only validif they are representative of the current instrument transformcondition for the entire system.

For each of the n points in the target field, there is ameasurement from each of the k instruments. If the dataset is incomplete due to occlusions in the target field orother considerations, there are fewer data for the uncertaintycomputation. For this discussion, a full measurement data setis assumed. This means the number of equations contributingto the residual error N is given by

N = τnk (18)

where N is the number of elements in the residual error vector;τ is the number of components in each measurement (2 fora theodolite, 3 for a laser tracker, the number of joints for aCMM, etc); n is the number of points in the field and k is thenumber of instrument locations.

For this optimization, the residual error vector is ofdimension N and resembles

E = [ε1 ε2 · · · εi · · · εk

](19)

where E is the residual error vector (dimension M) for theoptimization; εi is the point computation residual error forpoint i as stated in equation (6).

The residual error for the instrument transformationoptimization E is a combination of all the individual pointcomputation residuals. As stated previously, the pointcomputation optimization must run to completion beforeequation (19) is formed. This results in a nested optimizationcomputation. For each exploratory move in the instrumenttransformation space, Tr, the point computation optimizationis used to compute the optimal P ∗

i for all i.The instrument transform optimization problem is stated

as

find T and minimize∑

n

E2n.

Once the optimization is complete T represents thetransformations of the instruments that minimize the residualerrors in the point computation optimizations for each pointP ∗

i in the target field.

4. Experiments

In order to check the validity of our proposed method, thegiant antenna with the diameter 50 m is used to verifythe conformance of coupled multiple measurement systems.During the various fabrication stages of a giant antenna, one ofthe metrology tasks is the alignment of the surface reflectionsegments to receive signal. In the construction process, itis necessary to quantify the alignment of the key supportpoints along the giant antenna for the purposes of determiningthe reflector thickness at various locations needed to provide

Figure 3. The measurement layout of the giant antenna.

optimal alignment. This alignment is critical since it reducesthe frequency of giant antenna component maintenance.

Given the geometry of the giant antenna, the number ofpoints that must be measured, and the desired accuracy, thelaser tracker and total station theodolites were chosen as thebest type of instrument for the application. Due to the largequantity of measurements required and the physical constraintsof the measurement environment, it was necessary to use twolaser trackers and two total station theodolites to measure thefinal assembly and verify conformance. Figure 3 shows themeasured sections along with the locations of the instrumentstations.

The traditional analysis method used for this measurementjob was to combine all of the stations into a commonreference frame using sequential best-fit transformations.Each measurement device must be coupled to the adjacentdevice by a set of common points that both systems canmeasure. No points are shared between non-adjacent systems,and the choice of reference coordinate system is arbitrary.The measuring results must be exported in some manner fromeach measurement system and then imported into a commonanalysis platform for best-fit operations. The process must berepeated with the next measurement system in the chain untilall systems have been transformed. Therefore, the measuringaccuracy and alignment of the traditional measurement methodare not high; also it cannot handle loops in the measurementchain.

The measuring results are as shown in figure 4. Therewere a total of 100 measured times of giant antenna diameterin this network. The horizontal axis represents the numberof measurements and the vertical axis denotes the measuringerror. The red mark line denotes the results achieved usingthe traditional method as mentioned above. The blue linedenotes the results achieved using our proposed technique.The measuring average error of using our proposed techniqueis 0.0389 mm and the traditional method is 0.0771 mm.We obtain about 49% improvement for the measurementaccuracy.

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Meas. Sci. Technol. 21 (2010) 065103 W Liu and Z Wang

Figure 4. Measured accuracy using the traditional and proposedtechnique.

5. Conclusions

We develop a new method of the isolated variable sub-system to evaluate the uncertainty fields in coupled multiplemetrology systems. The proposed method can effectivelydetermine the optimized location and orientation of eachmeasurement system, and minimizes the coordinate combineduncertainty by multivariate statistics to the measured data.Not only does this provide a more realistic measurementresult, but it also provides a better fit between data sets fromdifferent instruments by taking full advantage of the instrumentcharacteristics as opposed to the current practice of matchingup XYZ values. The experimental results have demonstratedthat using this technique to evaluate the uncertainty fields incoupled multiple metrology systems, the measuring accuracycan be improved by 49%.

The future applications of this research could include anadditional layer of optimization around the entire process. Thiscould include optimization of instrument type and placement inorder to minimize the uncertainty in the measurement results.Another future application of this research is an expandedmodel for internal instrument parameters.

Acknowledgments

The China Postdoctoral Science Foundation (20090461155)and the Youth Foundation of China University of Mining andTechnology (2009A011) sponsored this project. The authorswould like to express their sincere thanks to them.

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