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International Journal of Machine Tools & Manufacture 40 (2000) 427443

On the selection of flatness measurement points incoordinate measuring machine inspection

Weon-Seok Kim, Shivakumar Raman *

School of Industrial Engineering, University of Oklahoma, Norman, OK 73019, USA

Received 15 October 1998; accepted 23 May 1999

Abstract

Inspection of form tolerances using the coordinate measuring machine (CMM) presents two distinctproblems: data collection and data fitting. The former problem deals with the selection of the sample sizeand the sample point location while the latter involves the determination of the tolerance zone envelopingthese points.

Four types of strategies and five different sample sizes were studied in this work to address the formerproblem. The accuracy of flatness measurement was investigated using realtime experiments with respect

to the above two factors and their respective levels. In addition, the length of the probe path was studiedwith respect to the two factors using a simulation study. A priority coefficient was developed to combinethe influence of accuracy and path while selecting sampling strategies and sample size. Preliminary obser-vations made suggest that any one sampling method may not be the best solution in all cases, whileconsidering the accuracy of flatness and the shortest CMM probe path. 1999 Elsevier Science Ltd. Allrights reserved.

Keywords: Coordinate flatness measurement; Sample size; Sampling method; Sampling strategy; Travelling salesmanproblem

1. Introduction

Sampling strategies in measurement consider the sampling method and the sample size thatcan obtain the maximum representative information from a population, for a given specification,in terms of time and cost. Several types of sampling strategies are employed such as simplerandom sampling, stratified random sampling, and systematic sampling. The appropriate sampling

* Corresponding author.

0890-6955/00/$ - see front matter 1999 Elsevier Science Ltd. All rights reserved.PII: S 0 8 9 0 - 6 9 5 5 ( 9 9 ) 0 0 0 5 9 - 0


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428 W.-S. Kim, S. Raman / International Journal of Machine Tools & Manufacture 40 (2000) 427443

method is selected according to the accuracy requirement, the geometry features of the workpiece,and the condition of the machine producing the workpiece.

A few research efforts are evident in the development of formal procedures for efficient sam-pling in coordinate measuring machine (CMM) measurement. Sample size (the number of pointsmeasured) is typically proportional to time and cost and for a given sampling strategy; savingsin time may be achieved through a reduction of the sample size. Moreover, it maybe worthwhileto attempt a minimization of the length of CMM probe paths in addition to reducing the samplesize for achieving further time reductions. This work sought to examine alternative samplingstrategies in the context of accuracy, number of points inspected and the length of the CMM probetool path. The combined consideration of CMM path minimisation and accuracy enhancementin the light of alternative sampling strategies and sample sizes has not been addressed in theCMM literature.

Four kinds of sampling sequences (the Hammersley sequence sampling [1], the HaltonZar-

emba sequence sampling, the aligned systematic sampling, and the systematic random sampling)were investigated for each of five sample sizes (4, 8, 16, 32 and 64). The sample sizes for testingwere arbitrarily selected to represent typical small sizes used in such testing and also to providesome consistent basis for comparing alternate sampling sequences. Four kinds of tour constructionalgorithms and two kinds of tour improvement algorithms were applied to find the shortest CMMprobe path for each combination of sampling method and sample size. The employed models fortour construction were the nearest neighbor insertion, the random tour, the cheapest insertion, andthe arbitrary insertion. The models used for the tour improvement were the LinKernighan [2]method and the two-opt heuristic.

Thirty square plates (3.03.00.5 in) fabricated using cold rolling followed by milling wereemployed in the flatness measurement experiments (replicates). An analysis of data obtained wasused to make some preliminary conclusions regarding the CMM probe path and accuracy.

It was found that the systematic random sampling method possessed the highest accuracy witha discrepancy rate of 23.9 at a sample size of 32 while measuring the accuracy of flatness. Thealigned systematic sampling method had the shortest length rate at the sample sizes of 4 and 32.Considering the total sampling efficiency through trade-off between the accuracy of flatness andthe shortest CMM probe path, the HaltonZaremba sequence and the systematic random samplingmethod exhibited the best efficiency at higher ranges of priority coefficient (accuracy highlightedpriority). On the other hand, at lower ranges of priority coefficient (path highlighted priority), thealigned systematic and the systematic random sampling methods performed the best. These resultsstrengthen the need for consideration of multiple factors in CMM sampling.

2. Literature review

The sampling method is a procedure providing how to choose units from the population scien-tifically and objectively and provide a sample that can estimate the population totals and averages[3]. In order to make sampling more efficient, it is important to develop the sampling methodsthat provide true and accurate estimates at the minimum cost.

Woo and Liang [4] investigated the number and location of the discrete samples for the dimen-sional measurement of machined surfaces. Accuracy and time were considered as the criteria for


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429W.-S. Kim, S. Raman / International Journal of Machine Tools & Manufacture 40 (2000) 427443

assessing sampling error. It was hence proposed that the accuracy be expressed by the discrepancyof a finite set ofNpoints for which a lower bound exists and the time be quantified in terms of

N. For the sampling strategy, the Hammersley sequence was compared against the uniform sam-pling. The results showed a remarkable improvement in reducing the number of samples and units

of time, while maintaining the same level of accuracy. The Hammersley points were of the sameorder of accuracy as the uniform points, despite significant reduction in number.

Woo et al. [5] investigated two kinds of basic questions regarding the relationship between thesample size and the error in measurement. The first question dealt with raising the accuracy ofsampling for a given sample size. The second question dealt with the reduction of the size of thesample for a given accuracy. They suggested that the key to both questions were directly relatedto the sample point distribution. To experiment this, the used sequences were the Hammersleysequence sampling, the HaltonZaremba sequence sampling, and the uniform sampling. Throughexperiments, it was proved that there was no discernable difference in the performance between

the Hammersley and the HaltonZaremba in 2-D space.Lee et al. [6] created a feature-based methodology, which coordinates the Hammersley sequence

and the stratified sampling method. They tried to compare the effectiveness of the Hammersleysequence sampling, the uniform sampling and the random sampling during the dimensionalmeasurement of the part.

Most of the studies in this area have relied on simulation of pseudo-random surfaces and donot verify their results with respect to measurement of actual surfaces.

The CMM probe path planning allows the determination of the inspection path joining theCMM measurement points based on the geometry of an existing part model and the specificationfor inspection. Few works have been done in the development of CMM probe path planning.Moreover, majority of the CMM probe path studies has concentrated on generating the path forcollision-free inspection of parts having multiple surfaces.

Lim and Menq [7] introduced the notion of path generation in CMM dimensional inspection.Feasible probe orientations were determined through which collision was avoided between theworkpiece and the touch probe and probe stylus. Lu et al. [8] developed an algorithm for generat-ing an optimum CMM inspection path to improve the throughput of CMMs. Yau and Menq [9]presented a hierarchical planning system for path planning in dimensional inspection using CMM.The proposed system was designed to generate inspection paths efficiently for geometrically com-plex parts having multiple surfaces.

This work has studied the issues of accuracy of flatness measurement, as determined by theCMM, and the length of tool path with reference to the sampling strategy and sample size. Thus,

two measures of sampling time reduction were investigated. Furthermore, actual manufacturedparts were used in experimental validation. The CMM probe path problem was formulated as atravelling salesman problem in this work.

The travelling salesman problem (TSP) is a classical problem to find a path that minimises thetotal distance while visiting N cities and returning to the starting city. The assignment problemis to ensure that the salesman visits all the cities only once and finishes at the same city wherehe began. Many heuristics have been developed to find appropriate solutions to this problem, bymany researchers, working on a diverse set of applications. However, it has been difficult toguarantee an optimal solution in a polynomial time and the TSP has hence been considered a NPcomplete problem.


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For TSP, Golden et al. [10], Bozer et al. [11], and Bentley [12] have mentioned two broadcategories of algorithms; tour construction and tour improvement. Tour construction algorithms

construct a route of all the tour points, while leaving each point exactly once and arriving at eachpoint exactly once. And then, tour improvement algorithms provide the improved tour by switch-ing the position of points in the tour.

3. Experimental design

To experiment the efficiency of sampling strategy relevant to the CMM probe path, two experi-mental objectives were considered. The first objective sought to evaluate the model of the sam-pling strategy for minimising the sample size. The second objective was to investigate alternative

optimisation models for minimizing the CMM probe path.The selected models of sampling methods chosen were those commonly employed in CMM

inspection literature. Four kinds of sampling methods were investigated to compare the effective-ness of the sampling method in CMM measurement: Hammersley sequence sampling, HaltonZaremba sequence sampling, aligned systematic sampling, and systematic random sampling.

3.1. Hammersley sequence sampling

Lee et al. [6], Woo et al. [5], and Woo and Liang [4] used a sampling methodology thatintegrated the Hammersley sequence and a stratified sampling method. This sampling was derivedin 2-D, so the coordinates of Hammersley point were made in accordance with the following:

Pii/N (1)

Qik1

j0

bij2j1 (2)

where N is the total number of sample points; i[0,N1]; bi is the binary representation of theindex i; bij is the jth bit in bi; k is log2N.

ForN=8, thePicould be denoted by 0/8, 1/8, 2/8, 7/8. And Qicould be obtained by multiply-

ing the term bij to 2

j1

and summed from j=

0 to j=

k

1, whereklog2N

is the smallest integer greater than or equal to (log2N). bi denotes the binary notification in termsof the index i. Computed Qi, the coordinates for N=8 could be expressed as 0/8, 4/8, 2/8, 6/8,1/8, 5/8, 3/8, and 7/8 by taking the mirror image of the binary representation for i about thedecimal point. For example, if i is equal to 1 or Pi is equal to 1/8 by Eq. (1), bi would be (0 0 1)and the mirror image would be expressed by bij=(1 0 0). Hence, 2

j1 could be calculated by(212223). So, Qi could be obtained by Eq. (2). That is, Qi would become121+022+023=1/2. The coordinates for N=8 is shown in Fig. 1(a).


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Fig. 1. Coordinates of each sampling method: (a) Hammersley sequence sampling; (b) HaltonZaremba sequencesampling; (c) aligned systematic sampling; (d) systematic random sampling.

3.2. HaltonZaremba sequence sampling

Woo et al. [5] used a sampling methodology which integrated the HaltonZaremba sequenceand a stratified sampling method. This sampling was derived in 2-D and the coordinates of theHaltonZaremba point were found in accordance with the following procedures. However, to usethis method, there is a restriction on the sample points that the number be a power of 2. Forexample, Ncould be 2k=2, 4, for k1.

Pii/Nk1

j0

bij2(kj) (3)

Qik1

j0

bij2j1 (4)

where N is the total number of sample points; i[0,N1]; bi is the binary representation of theindex i; bij is the jth bit in bi; bij is 1bij for j odd, and is bij otherwise; k is [log2N].

ForN=8, thePicould be denoted by 0, 1/8, 2/8, 7/8, and Qicould be obtained by multiplyingthe term bij to 2

j1 and summed from j=0 to j=k1, where k=log2N. And bi denotes the binary


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notification in terms of the index i. Computed Qi, the coordinates for N=8 could be expressed as1/4, 3/4, 0, 1/2, 3/8, 7/8, 1/8, 5/8. For example, if i=1 and bij=(0 0 1), j would be from 0 tok

1=

2. And then, b

ij

would be determined by the value of j

. If j=

0, b

ij=

1. However, b

ij=

1becausej is even. Ifj=1, bij=0. In the same way, bij=1 becausej is odd. Ifj=2, bij=bij=0 becausej is even. Therefore bij=(1 1 0). And Pi is equal to 1/8 by Eq. (3). By the way, 2

j1 couldbe calculated as (212223). So, Qi could be obtained by Eq. (4). That is, Qi would become121+122+023=3/4. The coordinates for N=8 are shown in Fig. 1(b).

3.3. Aligned systematic sampling

The aligned systematic sampling is a uniform sampling method. For the systematic samplingin two dimensions, the square grid pattern is a representative type of aligned sample as shownin Fig. 1(c). The sample was first determined by the choice of a pair of random numbers to makea decision for the coordinates of the upper left unit. The same interval and same location determ-ined the remaining column of strata and the row of strata. Suppose that a population is arrangedin the form ofzrrows and each row consists ofxy units. When a systematic sample ofxz unitsis selected, the basic procedure is as follows. A pair of random numbers (p,q) needs to be obtainedfirst so that prand qy. These numbers would decide the coordinates of the upper left unit bytheqth unit in the pth row. Then the rows consist ofp, p+r,p+2r,,p+(z1)r, while the columnsconsist of q, q+y, q+2y,,q+(x1)y. The position of the xz selected units were determined bythe point at which the x selected rows and z selected columns intersect [13]. Fig. 1(c) shows anexample of aligned systematic sampling sequence for N=8 in the case that x=2,y=4,r=0.5,z=0.25,

p=0.1 and q=0.1.

3.4. Systematic random sampling

This is a newly developed sampling method for experimentation in this study. The basic prin-ciple is to mix systematic sampling with random sampling. To fix the coordinate of the upperleft unit in strata, a pair of random numbers are first selected like the aligned systematic sampling.The selection of two random numbers is to determine the horizontal and the vertical location ofthat coordinate. Then the locations of the horizontal coordinates of the remaining units in allcolumns of strata are fixed by the selection of additional random numbers. So are the locationsof the vertical coordinates of the remaining units in all rows of strata. However, the interval of

each column and each row is determined like the aligned systematic sampling in fixing thelocations of all the points.

The detailed procedure is as follows: independently x random integers p11,p12,,pxy, are selec-ted and each of them is less than or equal to r. And then y random integers p11,p12,,pxy, areselected in the same manner such that each of them is less than or equal to z. Then the sampleunits locate the coordinates: (p11+wr,q11+kz), (p12+wr,q12+kz), (p13+wr,q13+kz),,(p(w+1)(k+1) +wr,q(w+1)(k+1)+kz); w=0,1,2,,(x1) and k=0,1,2,,(y1). One may note that this is a modifiedunaligned systematic sampling.

Fig. 1(d) shows this systematic random sampling for N=9 when x=2, y=4, r=0.5, and z=0.25.It is an example of coordinates of systematic random sampling in the case that the selected random


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numbers are p11=0.1, p12=0.3, p13=0.2, p14=0.1, p21=0, p22=0.2, p23=0.1, p24=0, q11=0, q12=0.1,q13=0.1, q14=0.05, q21=0, q22=0.1, q23=0.2 and q24=0.1 from the random table.

3.5. Preparation of software

Three computer programs were used to conduct the experimentation. The first software wasprepared to designate the location of sampling points on the workpiece according to each samplingmethod using Matlab. A computer numerical control (CNC) program was then prepared tooperate the CMM according to the sampling strategy. The third set of software used to measurethe distance between each sample point for finding the shortest CMM probe path were the onesdeveloped by Soh [14].

4. Experimental procedure

Thirty sample plates were employed in experimentation of the efficient sampling strategieswhile using the CMM for determination of accuracy of flatness. The selection of the number ofplates was made in accordance with the minimum requirement for the assumption of normaldistribution. Four kinds of sampling methods (the Hammersley sequence sampling, the HaltonZaremba sequence sampling, the aligned systematic sampling, and the systematic randomsampling) and five kinds of sample sizes (4, 8, 16, 32 and 64) were used in this experiment. Theresponse variables evaluated were the accuracy of flatness and the shortest length of CMM probepath. A two-factor factorial experiment was introduced. A different experiment was conducted

for each of the two response variables: one experimental and the other analytical. To developestimators for the parameters in the two-factor model (for accuracy), let yijk be the observedresponse when factor A (sampling method) is at the ith level (i=1,2,3,4), factor B (sample size)is at thejth level (j=1,2,,5) for thekth replicate (k=1,2,,30). The observations can be describedby the linear statistical model as:

yijkmtibj(tb)ijeijk

wherem is the overall mean effect, tiis the effect of the ith level of the column factor A (samplingmethod), bj is the effect of the jth level of the row factor B (sample size), (tb)ij is the effect ofthe interaction between factors A and B and eijk is the random error component. k representsreplicates and there are a total of (4)(5)(30)=600 observations.

The accuracy of flatness was measured using a commercial CMM (Brown and Sharpe.PFx454). The shortest length of the CMM probe path was computed using a program developedby Soh [14]. While determining the shortest length of CMM probe path, only the XYplane wasconsidered. The detailed measurement procedure used is listed as follows:

1. The first step was to select randomly the replicates; the plate samples were numbered seriallyfrom 1 to 30.

2. The coordinates of sample points on the plates were then generated. This step was to designatethe position of the sample points for measurement of flatness on the plate according to thesampling strategy.


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3. The sample plate was installed on the CMM worktable by fixing it with the clamping tool suchthat the plate could not be moved. This was done to minimize the measurement error included

during measurement. Thus the measurement was initialised.4. Actual measurement started with calibration, to teach the probe data to the CMM. After cali-bration of the CMM probe, the reference system was set up to teach the standard 3-dimensionaldata to the CMM. Two data files were used in the CNC part program to measure the flatnessof the surface of a sample plate.

5. The CNC part program was written and run in the Matlabsoftware to create the source filesadaptable for the program compiler of the CMM manufacturer. For covering the general samplesizes and sampling methods, this procedure was repeated.

6. The accuracy data was collected from the CMM during measurement of flatness on each sam-ple plate.

The CNC part programs were tailored for the appropriate combination of sample sizes andsampling methods. A new sample plate was loaded on the worktable of the CMM after allmeasurements on the previous plate were finished.

For computing the length of the shortest CMM probe path, computer simulation was used intwo dimensions (XY). The program code used was written by Soh [14]. The coordinates of measur-ing points on the plates were generated using the same procedure as described in the first experi-ment.

5. Results and discussion

5.1. Average discrepancy rate for accuracy of flatness

Flatness data was collected using the CMM on 30 sample plates. Accuracy of measurementfor each case was recorded and a discrepancy rate calculated based on the maximum achievableaccuracy of measurement. The average discrepancy rate (r) was computed as:

r(ab)

a 100 (5)

where a is a value regarded as the highest achievable accuracy for a sample plate (large sample

size) and b is the actual data value obtained through the experiment for a specific sample size.To obtain the value ofa, regression analysis was employed. In this work, the most accurate valueof the flatness measurement of sample plates was assumed to be obtainable at a sample size of300. This is an arbitrary assumption, based on the supposition that a very large sample size resultsin the highest achievable accuracy. It was also assumed that beyond 300 the accuracy does notincrease further. The accuracy at a sample size of 300 was determined using extrapolation throughlinear regression, based on data obtained from experimentation at the sample sizes specified above.The corresponding discrepancy rate data for the 30 sample plates were calculated by using Eq.(5). The average discrepancy rateshown in Table 1 has been averaged over the 30 sample plates.The unconverted accuracy data of flatness for the 30 plates is shown in the Appendix.


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Table 1Average discrepancy rate for accuracy of flatness of 30 sample platesa

Sampling method Sample size Sub-total mean

4 8 16 32 64

Hammersley sequence sampling 77.7 57.7 43.9 36.2 29.2 48.9HaltonZaremba sequence sampling 62.9 52.4 37.6 34.1 28.4 43.1Aligned systematic sampling 95.8 73.4 58.2 46.9 39.0 62.7Systematic random sampling 58.7 69.6 45.4 23.9 26.0 44.7Sub-total mean 73.8 63.3 46.3 35.3 30.7 49.9

a Unit: %.

5.2. Length data of CMM probe path

The length data of the CMM probe path was collected using computer simulation. After runningseveral algorithms, the shortest path among them was chosen for each combination of samplingmethod and sample size. Table 2 shows the relative length rate for the CMM tool path. Therelative length rate(l) is the percentage ratio of the actual path to the longest path for each samplesize as:

ln

m

100 (6)

where m is the longest path among the lengths of all the sample sizes and n is the length of theactual path obtained through the experiment for a specific sample size. For example, while com-puting the relative length rate for the HaltonZaremba sequence sampling at a sample size of 32,the length of the actual path for this was 16.99218 and the length of the longest path was 24.15501.Therefore, the relative length rate (l) was computed by Eq. (6) as 70.3.

Table 2Relative length rate for CMM tool path by sampling strategiesa

Sampling method Sample size Sub-total mean

4 8 16 32 64

Hammersley sequence sampling 27.8 37.3 52.2 71.9 100.0 57.8HaltonZaremba sequence sampling 27.8 41.0 48.6 70.3 96.2 56.8Aligned systematic sampling 24.8 31.0 49.7 59.0 99.4 52.8Systematic random sampling 31.5 30.7 44.1 67.2 92.1 53.1Sub-total mean 28.0 35.0 48.7 67.1 96.9 55.1

a Unit: %.


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Table 3ANOVA summary for all factors

Source of variation Sum of squares Degree of Mean square F PrFfreedom

Sampling method 35619.05 3 11873.02 177.54 0.001Sample size 161701.69 4 40425.42 604.49 0.001Interaction 16234.31 12 1352.86 20.23 0.001Error 38787.76 580 66.88Total 252342.81 599

5.3. Analysis of the accuracy of flatness

A two-factor factorial, fixed effects model was used to analyse the gathered accuracy of flatnessfor the sample plates. The two independent variables included in the model were sample size andsampling method. The response variable was the accuracy of flatness. The ANOVA is presentedin Table 3. The statistical analysis system (SAS) was used to analyse the collected data. Table3 shows that the main effects of accuracy of flatness were highly significant for all the inde-pendent variables.

Considering this table, one can conclude that there is a significant interaction between samplesize and sampling method because F0.05,12,350=1.75 from the table of the percentage points of theFdistribution. Moreover, the main effects of sample size and sampling method are also significantbecause F0.05,3,580=2.60 and F0.05,3,580=2.37, respectively. A pictorial graph (Fig. 2) of the average

responses at each treatment combination is helpful in getting a better understanding of the individ-ual effects of the model.

Fig. 2 shows a comparison of the average discrepancy rate by alternative sampling strategies.The lack of parallelism of the lines indicates significant interaction. Generally, increased samplesize shows low discrepancy through all the sampling methods. From a sample size of 4 to a

Fig. 2. Comparison of average discrepancy rate by sampling strategies.


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sample size of 8 and from a sample size of 32 to a sample size of 64, the discrepancy rate withthe systematic random sampling method increases, whereas that of the other sampling methods

decreases. The discrepancy rate

of the HaltonZaremba sequence sampling is a little less thanthat of the Hammersley sequence sampling. The HaltonZaremba sequence sampling gives betterresults than the Hammersley sequence sampling for all sample sizes. But the systematic randomsampling gives the lowest discrepancy rate in the range of the sample sizes of 32 and 64. Onthe other hand, the aligned systematic sampling shows the largest discrepancy rate through allthe sample sizes.

5.4. Analysis of the shortest length of CMM probe path

To help explain the results of the length of CMM probe path obtained through different sam-pling strategies, a graph (Fig. 3) of the average response at each treatment combination is useful.

Therelative length rate was obtained by taking the percentage of the actual path to the longestpath and setting the longest path to 100. From the above graph, the aligned systematic samplingmethod showed the shortest length at the sample sizes of 4 and 32. The Hammersley sequencesampling showed the longest length at the sample sizes of 16, 32, and 64. Generally, the alignedsystematic sampling and the systematic random sampling method showed a shorter length thanthe Hammersley or the HaltonZaremba sequence sampling.

5.5. Efficient sampling strategies

Accuracy is important during measurement to maintain a high level of quality. This accuracycould usually be improved by increasing the sample size. But, increasing the number of sampledpoints is time consuming, impacting the economy. Therefore, a trade-off must be establishedbetween the objectives.

In recent literature, it has been shown that the accuracy could be increased by using moreefficient sampling sequences, at a given sample size. Thus, when the same sample size is used

Fig. 3. Relative length rate of CMM probe path by sampling strategies.


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for general measurement, it looks as though an efficient sampling method may be selected thatprovides the highest accuracy. On the other hand, a manufactured product may not need a high

level of quality in all cases. In such cases, the most efficient sampling method would be one thatminimises the measuring time. Different sampling methods result in different lengths of CMMprobe path, even for a given sample size.

In this paper, a quantification of the trade-off was attempted for alternative path selectionthrough different sampling methods. A concept of a priority coefficient was developed, to deter-mine the most efficient sampling method in the light of the two objectives. Hence, the samplingefficiency (Et) was arbitrarily constructed as:

Et(100Ra)b(100Rp)(1b) (7)

where Ra is the discrepancy rate of accuracy of flatness and Rp is the relative length rate of thelength of CMM probe path. b is the priority coefficient which indicates the priority of accuracy

versus length for path selection. This priority coefficient could be decided based on the qualityphilosophy of the manufacturer. The sampling efficiency equation is arbitrarily selected to demon-strate the rationale for combining multiple factors in path planning. Hence, the simplest possiblecombining equation that is intuitive is chosen (remember the weights assigned to each factor musttotal to 1). Future research can concentrate on applying sturdier procedures such as the analytichierarchy process for determining (priority coefficients) weights for the equation. For example,given a discrepancy rate of flatness of 77.7, the relative length rate of CMM probe path of 27.8,and the priority coefficient of 0.8, the sampling efficiency could be obtained from Eq. (7) as

Et(10077.7)0.8(10027.8)(10.8)32.3

Table 4 shows the sampling efficiency computed as the result of the trade-off between theaccuracy of flatness and the length of CMM probe path according to a priority coefficient. Evaluat-ing the sampling efficiency in terms of the sample size and the priority coefficient, it was observedthat a smaller sample size and a lower priority coefficient exhibited a higher sampling efficiency.Also a larger sample size and a higher priority coefficient exhibited a higher sampling efficiency.The total mean illustrated the total sampling efficiency by the trade-off between the averageaccuracy of flatness and the average length of CMM probe path for all sample sizes.

Fig. 4(af) plots the sampling efficiency as a function of the trade-off between accuracy offlatness and CMM probe path length, against both the priority coefficient and sample size. Fig.4(a) shows that the efficiency of the systematic random sampling method at a sample size of 4

was highest at higher values of the priority coefficient, but had a lower value at the low rangesof the priority coefficient. On the other hand, the efficiency of the aligned systematic samplingmethod at a sample size of 4 was lowest at a higher range of the priority coefficient and highestat lower ranges of the priority coefficient. Considering the mid-range of the priority coefficientfrom 0.4 to 0.6, the HaltonZaremba sequence and the systematic random sampling methods ata sample size of 4 exhibit the highest efficiency. Similar inferences may be made at other samplesizes as shown in Table 4 and Fig. 4(be).

Fig. 4(f) shows the combined data for all sample sizes. For higher ranges of the priority coef-ficient, the HaltonZaremba sequence sampling and the systematic random sampling method pro-vided the highest efficiency, and the aligned systematic sampling method had the lowest


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Table 4Sampling efficiency by the trade-off between the accuracy of flatness and the length of CMM probe path, accordingto the priority coefficienta

Sample size Sampling method Priority coefficient

0 0.2 0.4 0.6 0.8 1

4 HM 72.2 62.2 52.2 42.3 32.3 22.3HZ 72.2 65.2 58.2 51.1 44.1 37.1AS 75.2 61.0 46.8 32.6 18.4 4.2SR 68.5 63.1 57.6 52.2 46.7 41.3

8 HM 62.7 58.6 54.5 50.5 46.4 42.3HZ 59.0 56.7 54.4 52.2 49.9 47.6AS 69.0 60.5 52.0 43.6 35.1 26.6SR 69.3 61.5 53.7 46.0 38.2 30.4

16 HM 47.8 49.5 51.1 52.8 54.4 56.1HZ 51.4 53.6 55.8 58.0 60.2 62.4AS 50.3 48.6 46.9 45.2 43.5 41.8SR 55.9 55.6 55.4 55.1 54.9 54.6

32 HM 28.1 35.2 42.4 49.5 56.7 63.8HZ 29.7 36.9 44.2 51.4 58.7 65.9AS 41.0 43.4 45.8 48.3 50.7 53.1SR 32.8 41.5 50.1 58.8 67.4 76.1

64 HM 0 14.2 28.3 42.5 56.7 70.8HZ 3.8 17.4 30.9 44.5 58.0 71.6AS 0.6 12.7 24.8 36.8 48.9 61.0SR 7.9 21.1 34.3 47.6 60.8 74.0

Total mean HM 42.2 44.0 45.8 47.5 49.3 51.1HZ 43.2 45.9 48.7 51.4 54.2 56.9AS 47.2 45.2 43.3 41.3 39.3 37.3SR 46.9 48.6 50.3 51.9 53.6 55.3

a HM, Hammersley sequence sampling; HZ, HaltonZaremba sequence sampling; AS, aligned systematic sampling;SR, systematic random sampling.

efficiency. At lower ranges of the priority coefficient, the sampling method that had the highestefficiency was the aligned systematic and systematic random sampling method, while the Ham-mersley and the HaltonZaremba sequence sampling method had the lowest efficiency.

Considering that the mid-range of the priority coefficient was from 0.4 to 0.6, the systematicrandom and the HaltonZaremba sequence sampling method resulted in the highest efficiency.

The selection of a path in CMM inspection must consider both accuracy and time. The samplingpath changes based on the method used and the sample size. This paper has thus shown thatsignificant interaction exists between each of the methods and sample sizes. Combining the twoobjectives and setting a priority for each indicates that different methods are efficient at differentsizes and different priority coefficients. This paper represents the first attempt in the literature tocombine the interaction between the two factors. It is hence the recommendation to measurementpersonnel to explore all methods, sample sizes, and their priorities while selecting a path duringinspection of tolerances using CMMs.


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Fig. 4. Plot of sampling efficiency (trade-off between accuracy of flatness and CMM probe path length): (a) at thesample size of 4; (b) at the sample size of 8; (c) at the sample size of 16; (d) at the sample size of 32; (e) at thesample size of 64; and (f) at the sample size of total mean.

6. Discussion, conclusions and recommendations

Statistically analysing the data obtained through experimentation, the sampling method and thesample size had a significant effect (a=0.05) on the accuracy of flatness of plates. Stated otherwise,the accuracy of flatness is directly affected by the sampling method and sample size.

It was observed that, as the sample size increased, the discrepancy rate of flatness decreasedlike a unimodal function through three kinds of sampling methods: the Hammersley sequence

sampling, the HaltonZaremba sequence sampling, and the aligned systematic sampling. The sys-tematic random sampling method behaved in a somewhat irregular function. The irregularity inthe systematic random sampling method may have been caused by the random number generation(no regular law) process.

For the total mean accuracy of flatness, it was seen that the HaltonZaremba sequence samplingmethod was the most accurate at a mean discrepancy rate of 43.1. The systematic random samplingmethod, with a mean discrepancy rate of 44.7, was also very accurate. On the other hand, thealigned systematic sampling method was the worst in terms of accuracy with a mean discrepancyrate of 62.7.

Evaluating the accuracy of flatness through the total sample size, the systematic random sam-


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pling method showed the highest accuracy at a discrepancy rate of 23.9 at the sample size of 32.The next most accurate method was also the systematic random sampling method with a discrep-

ancy rate of 26.0 at the sample size of 64. The worst accuracy was seen with the aligned systematicsampling method, with a discrepancy rate of 95.8 at a sample size of 4.The aligned systematic sampling method showed the shortest length rate at the sample sizes

of 4 and 32. But the Hammersley sequence sampling method showed the longest length rate atthe sample sizes of 16, 32 and 64. In general, the aligned systematic and the systematic randomsampling method yielded paths shorter than the Hammersley and the HaltonZaremba sequencesampling methods.

The most accurate sampling method may not always be the most efficient sampling methodduring inspection of products. For example, in the case of mass-produced products that requireonly a reasonable accuracy, it will be important to reduce time. Therefore, in this work a newapproach is suggested to find the efficient sampling method through a trade-off between the accu-

racy and the shortest CMM probe path in measurement.The trade-off between the accuracy of flatness and the shortest CMM probe path was modelled

using a priority coefficient. The efficiency of a path can thus be evaluated in a proposed way thatintegrates the accuracy and path length considerations.

The most efficient sampling method was varied according to the priority coefficient and thesample size. The details are summarised in Table 5. Note that this table is derived purely basedon the observations during analysis.

Considering the total sampling efficiency, the HaltonZaremba sequence and the systematicrandom method possessed the highest efficiency at the high range of the priority coefficient.However, the aligned systematic and the systematic random sampling method had the highestefficiency at the low range of the priority coefficient. At the mid-range of the priority coefficient,the HaltonZaremba sequence and the systematic random sampling method possessed the high-est efficiency.

This experiment was conducted using rectangular plates. So, the response variable was selectedas the accuracy of flatness measurement of the plate surface. In future studies, it is recommendedto use other shapes of samples such as spheres, cylinders, and cones and determine relevant

Table 5Efficient sampling methodsa

Sample size Priority coefficient

Low accuracy priority Mid accuracy priority High accuracy priority

4 AS HZ or SR SR8 AS or SR HZ or HM HZ16 SR HZ or SR HZ32 AS SR SR64 SR SR SRTotal AS or SR HZ or SR HZ or SR

a HM, Hammersley sequence sampling; HZ, HaltonZaremba sequence sampling; AS, aligned systematic sampling;SR, systematic random sampling.


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accuracies of sphericity, cylindricity and conicity, respectively. Also, this study was focused onthe smaller ranges of sample size to give a reference for practical measurement of manufactured

products in industry. So, it would be advisable for future researchers to increase the sample sizebeyond 64 to investigate the behavior-relevant factors.The development of the priority coefficient using sturdy decision analysis methods to integrate

accuracy with time (length) is another challenging work that must be researched in the future. Amore challenging task would be to design an integrated program which can detect the efficientmethod and sampling size automatically and drive the CMM according to the requirement of theaccuracy specified and the time constraints of measurement.

Appendix A

Table A1

Table A1Accuracy of flatness for thirty sample plates

Sampling method Sample sizea

4 8 16 32 64

Hammersley sequence sampling Mean 0.000382 0.000722 0.000970 0.001098 0.001215

Std. dev 0.000162 0.000184 0.000228 0.000206 0.000215Halton-Zaremba sequence sampling Mean 0.000646 0.000820 0.001078 0.001134 0.001230

Std. dev 0.000167 0.000175 0.000227 0.000212 0.000222Aligned systematic sampling Mean 0.000074 0.000455 0.000719 0.000911 0.001048

Std. dev 0.000054 0.000200 0.000156 0.000182 0.000201Unaligned systematic sampling Mean 0.000711 0.000517 0.000934 0.001309 0.001270

Std. dev 0.000246 0.000116 0.000201 0.000256 0.000222

a Unit: inch.

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[6] G. Lee, J. Mou, Y. Shen, Sampling strategy design for dimensional measurement of geometric features usingcoordinate measuring machine, International Journal of Machine Tools and Manufacture 37 (7) (1997) 917934.

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Documents

Hammersley Sequence