On the selection of flatness measurement points in coordinate measuring machine inspection

International Journal of Machine Tools & Manufacture 40 (2000) 427–443

On the selection of flatness measurement points in coordinate 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 distinct problems: data collection and data fitting. The former problem deals with the selection of the sample size and the sample point location while the latter involves the determination of the tolerance zone enveloping these points. Four types of strategies and five different sample sizes were studied in this work to address the former problem. 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 studied with respect to the two factors using a simulation study. A priority coefficient was developed to combine the influence of accuracy and path while selecting sampling strategies and sample size. Preliminary observations made suggest that any one sampling method may not be the best solution in all cases, while considering the accuracy of flatness and the shortest CMM probe path.  1999 Elsevier Science Ltd. All rights reserved. Keywords: Coordinate flatness measurement; Sample size; Sampling method; Sampling strategy; Travelling salesman problem

1. Introduction Sampling strategies in measurement consider the sampling method and the sample size that can 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 simple random 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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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 sampling in coordinate measuring machine (CMM) measurement. Sample size (the number of points measured) is typically proportional to time and cost and for a given sampling strategy; savings in time may be achieved through a reduction of the sample size. Moreover, it maybe worthwhile to attempt a minimization of the length of CMM probe paths in addition to reducing the sample size for achieving further time reductions. This work sought to examine alternative sampling strategies in the context of accuracy, number of points inspected and the length of the CMM probe tool path. The combined consideration of CMM path minimisation and accuracy enhancement in the light of alternative sampling strategies and sample sizes has not been addressed in the CMM literature. Four kinds of sampling sequences (the Hammersley sequence sampling [1], the Halton–Zaremba 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 testing were arbitrarily selected to represent typical small sizes used in such testing and also to provide some consistent basis for comparing alternate sampling sequences. Four kinds of tour construction algorithms and two kinds of tour improvement algorithms were applied to find the shortest CMM probe path for each combination of sampling method and sample size. The employed models for tour construction were the nearest neighbor insertion, the random tour, the cheapest insertion, and the arbitrary insertion. The models used for the tour improvement were the Lin–Kernighan [2] method and the two-opt heuristic. Thirty square plates (3.0×3.0×0.5 in) fabricated using cold rolling followed by milling were employed in the flatness measurement experiments (replicates). An analysis of data obtained was used 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 with a discrepancy rate of 23.9 at a sample size of 32 while measuring the accuracy of flatness. The aligned 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 and the shortest CMM probe path, the Halton–Zaremba sequence and the systematic random sampling method exhibited the best efficiency at higher ranges of priority coefficient (accuracy highlighted priority). On the other hand, at lower ranges of priority coefficient (path highlighted priority), the aligned systematic and the systematic random sampling methods performed the best. These results strengthen 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 scientifically 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 methods that provide true and accurate estimates at the minimum cost. Woo and Liang [4] investigated the number and location of the discrete samples for the dimensional measurement of machined surfaces. Accuracy and time were considered as the criteria for

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assessing sampling error. It was hence proposed that the accuracy be expressed by the discrepancy of a finite set of N points 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 sampling. 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 same order 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 the sample size and the error in measurement. The first question dealt with raising the accuracy of sampling for a given sample size. The second question dealt with the reduction of the size of the sample for a given accuracy. They suggested that the key to both questions were directly related to the sample point distribution. To experiment this, the used sequences were the Hammersley sequence sampling, the Halton–Zaremba sequence sampling, and the uniform sampling. Through experiments, it was proved that there was no discernable difference in the performance between the Hammersley and the Halton–Zaremba 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 Hammersley sequence sampling, the uniform sampling and the random sampling during the dimensional measurement of the part. Most of the studies in this area have relied on simulation of pseudo-random surfaces and do not verify their results with respect to measurement of actual surfaces. The CMM probe path planning allows the determination of the inspection path joining the CMM measurement points based on the geometry of an existing part model and the specification for 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 for collision-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 the workpiece and the touch probe and probe stylus. Lu et al. [8] developed an algorithm for generating 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 complex parts having multiple surfaces. This work has studied the issues of accuracy of flatness measurement, as determined by the CMM, 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 manufactured parts were used in experimental validation. The CMM probe path problem was formulated as a travelling salesman problem in this work. The travelling salesman problem (TSP) is a classical problem to find a path that minimises the total distance while visiting N cities and returning to the starting city. The assignment problem is to ensure that the salesman visits all the cities only once and finishes at the same city where he began. Many heuristics have been developed to find appropriate solutions to this problem, by many researchers, working on a diverse set of applications. However, it has been difficult to guarantee an optimal solution in a polynomial time and the TSP has hence been considered a NP complete problem.

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For TSP, Golden et al. [10], Bozer et al. [11], and Bentley [12] have mentioned two broad categories 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 each point exactly once. And then, tour improvement algorithms provide the improved tour by switching the position of points in the tour.

3. Experimental design To experiment the efficiency of sampling strategy relevant to the CMM probe path, two experimental objectives were considered. The first objective sought to evaluate the model of the sampling 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 effectiveness of the sampling method in CMM measurement: Hammersley sequence sampling, Halton– Zaremba 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 that integrated the Hammersley sequence and a stratified sampling method. This sampling was derived in 2-D, so the coordinates of Hammersley point were made in accordance with the following: Pi⫽i/N

冘

(1)

k⫺1

Qi⫽

bij 2−j−1

(2)

j⫽0

where N is the total number of sample points; i⑀[0,N⫺1]; bi is the binary representation of the index i; bij is the jth bit in bi; k is log2N. For N=8, the Pi could be denoted by 0/8, 1/8, 2/8, …7/8. And Qi could be obtained by multiplying the term bij to 2⫺j ⫺1 and summed from j=0 to j=k⫺1, where k⫽log2N is the smallest integer greater than or equal to (log2N). bi denotes the binary notification in terms of 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 the decimal 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⫺j ⫺1 could be calculated by (2⫺12⫺22⫺3). So, Qi could be obtained by Eq. (2). That is, Qi would become 1×2⫺1+0×2⫺2+0×2⫺3=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) Halton–Zaremba sequence sampling; (c) aligned systematic sampling; (d) systematic random sampling.

3.2. Halton–Zaremba sequence sampling Woo et al. [5] used a sampling methodology which integrated the Halton–Zaremba sequence and a stratified sampling method. This sampling was derived in 2-D and the coordinates of the Halton–Zaremba point were found in accordance with the following procedures. However, to use this method, there is a restriction on the sample points that the number be a power of 2. For example, N could be 2k=2, 4, … for k⬎1.

冘

k⫺1

Pi⫽i/N⫽

bij 2−(k−j)

(3)

j⫽0

冘

k⫺1

Qi⫽

bij ⬘2−j−1

(4)

j⫽0

where N is the total number of sample points; i⑀[0,N⫺1]; bi is the binary representation of the index i; bij is the jth bit in bi; bij⬘ is 1⫺bij for j odd, and is bij otherwise; k is [log2N]. For N=8, the Pi could be denoted by 0, 1/8, 2/8, … 7/8, and Qi could be obtained by multiplying the term bij⬘ to 2⫺j ⫺1 and summed from j=0 to j=k⫺1, 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 as 1/4, 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 to k⫺1=2. And then, bij⬘ would be determined by the value of j. If j=0, bij=1. However, bij⬘=1 because j is even. If j=1, bij=0. In the same way, bij⬘=1 because j is odd. If j=2, bij⬘=bij⬘=0 because j is even. Therefore bij⬘=(1 1 0). And Pi is equal to 1/8 by Eq. (3). By the way, 2⫺j ⫺1 could be calculated as (2⫺12⫺22⫺3). So, Qi could be obtained by Eq. (4). That is, Qi would become 1×2⫺1+1×2⫺2+0×2⫺3=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 sampling in two dimensions, the “square grid” pattern is a representative type of aligned sample as shown in Fig. 1(c). The sample was first determined by the choice of a pair of random numbers to make a decision for the coordinates of the upper left unit. The same interval and same location determined the remaining column of strata and the row of strata. Suppose that a population is arranged in the form of zr rows and each row consists of xy units. When a systematic sample of xz units is selected, the basic procedure is as follows. A pair of random numbers (p,q) needs to be obtained first so that pⱕr and qⱕy. These numbers would decide the coordinates of the upper left unit by the qth unit in the pth row. Then the rows consist of p, p+r, p+2r,…,p+(z⫺1)r, while the columns consist of q, q+y, q+2y,…,q+(x⫺1)y. The position of the xz selected units were determined by the point at which the x selected rows and z selected columns intersect [13]. Fig. 1(c) shows an example 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 principle is to mix systematic sampling with random sampling. To fix the coordinate of the upper left 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 of that coordinate. Then the locations of the horizontal coordinates of the remaining units in all columns of strata are fixed by the selection of additional random numbers. So are the locations of 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 the locations of all the points. The detailed procedure is as follows: independently x random integers p11,p12,…,pxy, are selected and each of them is less than or equal to r. And then y random integers p11,p12,…,pxy, are selected in the same manner such that each of them is less than or equal to z. Then the sample units 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,…,(x⫺1) and k=0,1,2,…,(y⫺1). One may note that this is a modified unaligned 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 was prepared to designate the location of sampling points on the workpiece according to each sampling method using Matlab. A computer numerical control (CNC) program was then prepared to operate the CMM according to the sampling strategy. The third set of software used to measure the distance between each sample point for finding the shortest CMM probe path were the ones developed by Soh [14].

4. Experimental procedure Thirty sample plates were employed in experimentation of the efficient sampling strategies while using the CMM for determination of accuracy of flatness. The selection of the number of plates was made in accordance with the minimum requirement for the assumption of normal distribution. Four kinds of sampling methods (the Hammersley sequence sampling, the Halton– Zaremba sequence sampling, the aligned systematic sampling, and the systematic random sampling) and five kinds of sample sizes (4, 8, 16, 32 and 64) were used in this experiment. The response variables evaluated were the accuracy of flatness and the shortest length of CMM probe path. 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 develop estimators for the parameters in the two-factor model (for accuracy), let yijk be the observed response when factor A (sampling method) is at the ith level (i=1,2,3,4), factor B (sample size) is at the jth level (j=1,2,…,5) for the kth replicate (k=1,2,…,30). The observations can be described by the linear statistical model as: yijk⫽m⫹ti⫹bj ⫹(tb)ij ⫹eijk where m is the overall mean effect, ti is the effect of the ith level of the column factor A (sampling method), bj is the effect of the jth level of the row factor B (sample size), (tb)ij is the effect of the interaction between factors A and B and eijk is the random error component. k represents replicates 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 developed by Soh [14]. While determining the shortest length of CMM probe path, only the XY plane was considered. The detailed measurement procedure used is listed as follows: 1. The first step was to select randomly the replicates; the plate samples were numbered serially from 1 to 30. 2. The coordinates of sample points on the plates were then generated. This step was to designate the position of the sample points for measurement of flatness on the plate according to the sampling strategy.

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3. The sample plate was installed on the CMM worktable by fixing it with the clamping tool such that 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 calibration of the CMM probe, the reference system was set up to teach the standard 3-dimensional data to the CMM. Two data files were used in the CNC part program to measure the flatness of the surface of a sample plate. 5. The CNC part program was written and run in the Matlab software to create the source files adaptable for the program compiler of the CMM manufacturer. For covering the general sample sizes and sampling methods, this procedure was repeated. 6. The accuracy data was collected from the CMM during measurement of flatness on each sample plate. The CNC part programs were tailored for the appropriate combination of sample sizes and sampling methods. A new sample plate was loaded on the worktable of the CMM after all measurements on the previous plate were finished. For computing the length of the shortest CMM probe path, computer simulation was used in two dimensions (XY). The program code used was written by Soh [14]. The coordinates of measuring points on the plates were generated using the same procedure as described in the first experiment.

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 measurement for each case was recorded and a discrepancy rate calculated based on the maximum achievable accuracy of measurement. The average discrepancy rate (r) was computed as: r⫽

(a−b) ⫻100 a

(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 of a, regression analysis was employed. In this work, the most accurate value of the flatness measurement of sample plates was assumed to be obtainable at a sample size of 300. This is an arbitrary assumption, based on the supposition that a very large sample size results in the highest achievable accuracy. It was also assumed that beyond 300 the accuracy does not increase further. The accuracy at a sample size of 300 was determined using extrapolation through linear 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 rate shown 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 1 Average discrepancy rate for accuracy of flatness of 30 sample platesa Sampling method

Hammersley sequence sampling Halton–Zaremba sequence sampling Aligned systematic sampling Systematic random sampling Sub-total mean a

Sample size

Sub-total mean

4

8

16

32

64

77.7 62.9 95.8 58.7 73.8

57.7 52.4 73.4 69.6 63.3

43.9 37.6 58.2 45.4 46.3

36.2 34.1 46.9 23.9 35.3

29.2 28.4 39.0 26.0 30.7

48.9 43.1 62.7 44.7 49.9

Unit: %.

5.2. Length data of CMM probe path The length data of the CMM probe path was collected using computer simulation. After running several algorithms, the shortest path among them was chosen for each combination of sampling method and sample size. Table 2 shows the relative length rate for the CMM tool path. The relative length rate (l) is the percentage ratio of the actual path to the longest path for each sample size as: n l⫽ ⫻100 m

(6)

where m is the longest path among the lengths of all the sample sizes and n is the length of the actual path obtained through the experiment for a specific sample size. For example, while computing the relative length rate for the Halton–Zaremba 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 2 Relative length rate for CMM tool path by sampling strategiesa Sampling method

Hammersley sequence sampling Halton–Zaremba sequence sampling Aligned systematic sampling Systematic random sampling Sub-total mean a

Unit: %.

Sample size

Sub-total mean

4

8

16

32

64

27.8 27.8 24.8 31.5 28.0

37.3 41.0 31.0 30.7 35.0

52.2 48.6 49.7 44.1 48.7

71.9 70.3 59.0 67.2 67.1

100.0 96.2 99.4 92.1 96.9

57.8 56.8 52.8 53.1 55.1

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Table 3 ANOVA summary for all factors Source of variation

Sum of squares

Degree of freedom

Mean square

F

Pr⬎F

Sampling method Sample size Interaction Error Total

35619.05 161701.69 16234.31 38787.76 252342.81

3 4 12 580 599

11873.02 40425.42 1352.86 66.88

177.54 604.49 20.23

0.001 0.001 0.001

5.3. Analysis of the accuracy of flatness A two-factor factorial, fixed effects model was used to analyse the gathered accuracy of flatness for the sample plates. The two independent variables included in the model were sample size and sampling method. The response variable was the accuracy of flatness. The ANOVA is presented in Table 3. The statistical analysis system (SAS) was used to analyse the collected data. Table 3 shows that the main effects of accuracy of flatness were highly significant for all the independent variables. Considering this table, one can conclude that there is a significant interaction between sample size and sampling method because F0.05,12,350=1.75 from the table of the percentage points of the F distribution. Moreover, the main effects of sample size and sampling method are also significant because 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 individual 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 sample size 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 with the systematic random sampling method increases, whereas that of the other sampling methods decreases. The discrepancy rate of the Halton–Zaremba sequence sampling is a little less than that of the Hammersley sequence sampling. The Halton–Zaremba sequence sampling gives better results than the Hammersley sequence sampling for all sample sizes. But the systematic random sampling gives the lowest discrepancy rate in the range of the sample sizes of 32 and 64. On the other hand, the aligned systematic sampling shows the largest discrepancy rate through all the 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 sampling strategies, a graph (Fig. 3) of the average response at each treatment combination is useful. The relative length rate was obtained by taking the percentage of the actual path to the longest path and setting the longest path to 100. From the above graph, the aligned systematic sampling method showed the shortest length at the sample sizes of 4 and 32. The Hammersley sequence sampling showed the longest length at the sample sizes of 16, 32, and 64. Generally, the aligned systematic sampling and the systematic random sampling method showed a shorter length than the Hammersley or the Halton–Zaremba sequence sampling. 5.5. Efficient sampling strategies Accuracy is important during measurement to maintain a high level of quality. This accuracy could usually be improved by increasing the sample size. But, increasing the number of sampled points is time consuming, impacting the economy. Therefore, a trade-off must be established between the objectives. In recent literature, it has been shown that the accuracy could be increased by using more efficient 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 that provides 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 that minimises the measuring time. Different sampling methods result in different lengths of CMM probe path, even for a given sample size. In this paper, a quantification of the trade-off was attempted for alternative path selection through different sampling methods. A concept of a priority coefficient was developed, to determine the most efficient sampling method in the light of the two objectives. Hence, the sampling efficiency (Et) was arbitrarily constructed as: Et⫽(100⫺Ra)∗b⫹(100⫺Rp)∗(1⫺b)

(7)

where Ra is the discrepancy rate of accuracy of flatness and Rp is the relative length rate of the length 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 quality philosophy of the manufacturer. The sampling efficiency equation is arbitrarily selected to demonstrate the rationale for combining multiple factors in path planning. Hence, the simplest possible combining equation that is intuitive is chosen (remember the weights assigned to each factor must total to 1). Future research can concentrate on applying sturdier procedures such as the analytic hierarchy 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⫽(100⫺77.7)∗0.8⫹(100⫺27.8)∗(1⫺0.8)⫽32.3 Table 4 shows the sampling efficiency computed as the result of the trade-off between the accuracy of flatness and the length of CMM probe path according to a priority coefficient. Evaluating the sampling efficiency in terms of the sample size and the priority coefficient, it was observed that 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 average accuracy of flatness and the average length of CMM probe path for all sample sizes. Fig. 4(a–f) plots the sampling efficiency as a function of the trade-off between accuracy of flatness 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 ranges of the priority coefficient. On the other hand, the efficiency of the aligned systematic sampling method at a sample size of 4 was lowest at a higher range of the priority coefficient and highest at lower ranges of the priority coefficient. Considering the mid-range of the priority coefficient from 0.4 to 0.6, the Halton–Zaremba sequence and the systematic random sampling methods at a sample size of 4 exhibit the highest efficiency. Similar inferences may be made at other sample sizes as shown in Table 4 and Fig. 4(b–e). Fig. 4(f) shows the combined data for all sample sizes. For higher ranges of the priority coefficient, the Halton–Zaremba sequence sampling and the systematic random sampling method provided the highest efficiency, and the aligned systematic sampling method had the lowest

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

4

8

16

32

64

Total mean

Sampling method Priority coefficient

HM HZ AS SR HM HZ AS SR HM HZ AS SR HM HZ AS SR HM HZ AS SR HM HZ AS SR

0

0.2

0.4

0.6

0.8

1

72.2 72.2 75.2 68.5 62.7 59.0 69.0 69.3 47.8 51.4 50.3 55.9 28.1 29.7 41.0 32.8 0 3.8 0.6 7.9 42.2 43.2 47.2 46.9

62.2 65.2 61.0 63.1 58.6 56.7 60.5 61.5 49.5 53.6 48.6 55.6 35.2 36.9 43.4 41.5 14.2 17.4 12.7 21.1 44.0 45.9 45.2 48.6

52.2 58.2 46.8 57.6 54.5 54.4 52.0 53.7 51.1 55.8 46.9 55.4 42.4 44.2 45.8 50.1 28.3 30.9 24.8 34.3 45.8 48.7 43.3 50.3

42.3 51.1 32.6 52.2 50.5 52.2 43.6 46.0 52.8 58.0 45.2 55.1 49.5 51.4 48.3 58.8 42.5 44.5 36.8 47.6 47.5 51.4 41.3 51.9

32.3 44.1 18.4 46.7 46.4 49.9 35.1 38.2 54.4 60.2 43.5 54.9 56.7 58.7 50.7 67.4 56.7 58.0 48.9 60.8 49.3 54.2 39.3 53.6

22.3 37.1 4.2 41.3 42.3 47.6 26.6 30.4 56.1 62.4 41.8 54.6 63.8 65.9 53.1 76.1 70.8 71.6 61.0 74.0 51.1 56.9 37.3 55.3

a HM, Hammersley sequence sampling; HZ, Halton–Zaremba sequence sampling; AS, aligned systematic sampling; SR, systematic random sampling.

efficiency. At lower ranges of the priority coefficient, the sampling method that had the highest efficiency was the aligned systematic and systematic random sampling method, while the Hammersley and the Halton–Zaremba sequence sampling method had the lowest efficiency. Considering that the mid-range of the priority coefficient was from 0.4 to 0.6, the systematic random and the Halton–Zaremba sequence sampling method resulted in the highest efficiency. The selection of a path in CMM inspection must consider both accuracy and time. The sampling path changes based on the method used and the sample size. This paper has thus shown that significant interaction exists between each of the methods and sample sizes. Combining the two objectives and setting a priority for each indicates that different methods are efficient at different sizes and different priority coefficients. This paper represents the first attempt in the literature to combine the interaction between the two factors. It is hence the recommendation to measurement personnel to explore all methods, sample sizes, and their priorities while selecting a path during inspection 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 the sample 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 the sample 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 the sample 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 decreased like a unimodal function through three kinds of sampling methods: the Hammersley sequence sampling, the Halton–Zaremba sequence sampling, and the aligned systematic sampling. The systematic random sampling method behaved in a somewhat irregular function. The irregularity in the 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 Halton–Zaremba sequence sampling method was the most accurate at a mean discrepancy rate of 43.1. The systematic random sampling method, with a mean discrepancy rate of 44.7, was also very accurate. On the other hand, the aligned systematic sampling method was the worst in terms of accuracy with a mean discrepancy rate 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 discrepancy rate of 26.0 at the sample size of 64. The worst accuracy was seen with the aligned systematic sampling 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 at the sample sizes of 16, 32 and 64. In general, the aligned systematic and the systematic random sampling method yielded paths shorter than the Hammersley and the Halton–Zaremba sequence sampling methods. The most accurate sampling method may not always be the most efficient sampling method during inspection of products. For example, in the case of mass-produced products that require only a reasonable accuracy, it will be important to reduce time. Therefore, in this work a new approach is suggested to find the efficient sampling method through a trade-off between the accuracy 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 that integrates the accuracy and path length considerations. The most efficient sampling method was varied according to the priority coefficient and the sample size. The details are summarised in Table 5. Note that this table is derived purely based on the observations during analysis. Considering the total sampling efficiency, the Halton–Zaremba sequence and the systematic random 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 highest efficiency at the low range of the priority coefficient. At the mid-range of the priority coefficient, the Halton–Zaremba sequence and the systematic random sampling method possessed the highest efficiency. This experiment was conducted using rectangular plates. So, the response variable was selected as the accuracy of flatness measurement of the plate surface. In future studies, it is recommended to use other shapes of samples such as spheres, cylinders, and cones and determine relevant Table 5 Efficient sampling methodsa Sample size

4 8 16 32 64 Total a

Priority coefficient Low accuracy priority

Mid accuracy priority

High accuracy priority

AS AS or SR SR AS SR AS or SR

HZ HZ HZ SR SR HZ

SR HZ HZ SR SR HZ or SR

or SR or HM or SR

or SR

HM, Hammersley sequence sampling; HZ, Halton–Zaremba 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 on the 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 size beyond 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. A more challenging task would be to design an integrated program which can detect the efficient method and sampling size automatically and drive the CMM according to the requirement of the accuracy specified and the time constraints of measurement. Appendix A Table A1

Table A1 Accuracy of flatness for thirty sample plates Sample sizea

Sampling method

Hammersley sequence sampling Halton-Zaremba sequence sampling Aligned systematic sampling Unaligned systematic sampling a

Mean Std. dev Mean Std. dev Mean Std. dev Mean Std. dev

4

8

16

32

64

0.000382 0.000162 0.000646 0.000167 0.000074 0.000054 0.000711 0.000246

0.000722 0.000184 0.000820 0.000175 0.000455 0.000200 0.000517 0.000116

0.000970 0.000228 0.001078 0.000227 0.000719 0.000156 0.000934 0.000201

0.001098 0.000206 0.001134 0.000212 0.000911 0.000182 0.001309 0.000256

0.001215 0.000215 0.001230 0.000222 0.001048 0.000201 0.001270 0.000222

Unit: inch.

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