Moody Revisited: Least-Squares Solutions of the Union Jack Surface Plate Measurement Method

Abstract

For the calibration of surface plate, the classical Moody method is still commonly used. In this method the straightness of a number of lines over a surface plate in a union-jack configuration is measured and combined into a flatness measurement. The measurement of the two center lines is used to determine so-called closure errors. A shortcoming of this method is that it gives an ambiguous value for the central height and that the measurements of the central lines are not involved in the evaluation. This research shows how the lines can be incorporated in the measurement evaluation in a least-squares sense. This gives a measurement redundancy leading to an 18% reduction in the uncertainty. Also, it is shown that a further reduction in the uncertainty is possible when using the gravity vector as a common reference, as can be done when using electronic levels. A least-squares evaluation of measurements taken in this way gives an even further redundancy, leading to a reduction in the uncertainty of 29% relative to the traditional evaluation according to the Moody method. This is illustrated with an actual measurement example and additional Monte Carlo simulations.

1. Introduction

The measurement of flatness, and in particular the flatness measurements of surface plates, is an established measurement problem in dimensional metrology. For flatness measurements of surface plates, traditional methods are based on either autocollimators, electronic levels, or laser interferometer systems operated in angular measurement mode, where the variation in the gradient between two sides of a base is used to calculate the surface flatness [1]. The measurement can take place on a rectangular grid, where it is recognized that diagonals must be measured if autocollimators or laser interferometer systems with angular optics (on a base) are used [2]. The configuration where measurements are taken for two diagonals and three lines in both the x and y direction is called the Union Jack configuration because of its resemblance to the flag of the United Kingdom. The measuring method using this configuration is also called the Moody method after the author that first explicitly gave an evaluation method [3] based on an earlier indication in a classical textbook [4].

The Union Jack configuration is well scalable, so that also for large surfaces measurements can be taken within a reasonable amount of time, and the calculation after Moody is reasonably straightforward. This evaluation method is currently in widespread use and implemented in the software of most manufacturers of autocollimators, electronic levels, and displacement laser interferometer systems and is prescribed in ASME B89.3.7-2013 [5]. The evaluation and uncertainty of rectangular grid configurations has been studied rather extensively [6,7,8], the sources of uncertainty in the evaluation according to the Moody method were evaluated by Drescher [9], taking the established method as the basis. A comparison of autocollimator and laser-interferometer-based measurements, also compared to CMM-measurements, using the Moody method was carried out by Zahwi et al. [10]. A comparison between electronic level measurements, autocollimator and laser interferometer measuring using the Moody method was made by Espinoza et al. [11], who concluded that there is no essential difference in accuracy between the methods. A method based on gauge pins in a Union Jack configuration, using the Moody evaluation method, was proposed by Garzón [12].

In this paper the original Moody evaluation is reviewed as a solution of the more general measurement problem that is presented as a matrix calculation. Based on this presentation, the measurements of the central lines that are omitted in the calculation of the node points according to Moody are integrated in a least-squares sense. For measurement using electronic levels, the gravity vector can be used as a common reference for measurement lines having a same direction. This calculation, that potentially gives a further error reduction, is shown and evaluated as well. An appealing feature of the Moody evaluation is that the reference plane, defined by equaling the endpoints of the respective diagonals, is integrated in the calculation. However, this reference plane may be different to the minimum-zone plane that should be the basis for a flatness evaluation.

The uncertainty of these three evaluation methods with an additional minimum-zone evaluation regarding the reference plane, is illustrated for a measurement example, supplemented with Monte Carlo simulations.

2. Measurement Configuration and Basic Evaluation

In the basic measurement configuration, the straightness of three lines is measured in both the x and y direction, and two additional diagonal lines. This is illustrated in Figure 1, where the rectangular horizontal surface to be measured is viewed from above.

Indicated are the measured lines L_1..8 and the node points A..I. The indication and orientation of these lines and points may vary between various publications and software implementations. The heights of the node points are denoted by A..I. The number of measured points taken at each line may vary depending on the measurement base and the size of the object. In the example sketched in Figure 1, N_x = 8 height differences are measured along the lines in the x direction, N_y = 6 height differences along the lines in the y direction and N_d =10 height differences are measured along the diagonal lines. The individual measurements m_k,i indicate the height difference in the measured points along line L_k. These measurements will have a bias depending on the setting of the measuring device, and a bias may be subtracted from the measurements of each line as the variations in the gradients are relevant rather than the gradients themselves. Also, the topography may be rotated so that the line starts and ends in defined points. For example, if points X and Y have a defined height X and Y, the heights h_k,j, j = 0..N along a line L_k between these points are given by:

$$h_{k,j}=X+{\sum }_{i=1}^j{m}_{k,i}-{\displaystyle \frac{j}{N}}{\sum }_{i=1}^N{m}_{k,i}+(Y-X){\displaystyle \frac{j}{N}}.$$

(1)

This is illustrated in Figure 2 for N = 6.

If the points X and Y have the same height and the height of the point in the center is defined as zero, the heights along a line L_k are given by:

$$h_{k,j}={\sum }_{i=1}^j{m}_{k,i}-{\displaystyle \frac{j}{N}}{\sum }_{i=1}^N{m}_{k,i}-{\sum }_{i=1}^{{\displaystyle \frac{N}{2}}}{m}_{k,i}+{\displaystyle \frac{1}{2}}{\sum }_{i=1}^N{m}_{k,i}.$$

(2)

The basic measurements used for the calculations of the heights of the nodes A..I are the height differences measured between these points, denoted by M_1..16 and indicated in Figure 3, for example:

$$M_6-M_5=A+C-2B={\sum }_{i={\displaystyle \frac{{1+N}_x}{2}}}^{N_x}{m}_{1,i}-{\sum }_{i=1}^{N_x/2}{m}_{1,i}.$$

(3)

Assuming that the reference directions of the lines are not related—as is commonly the case when the measurements are taken using an autocollimator or a laser interferometer, with a mirror or angular optics respectively on a measurement base—the measurement situation for calculating the heights A..I can be summarized in matrix-notation as given in Equation (4):

$$\left(\begin{array}{ccccccccc}1& 0& 0& 0& 0& 0& 0& 1& -2\\ 0& 0& 1& 0& 0& 0& 1& 0& -2\\ 1& -2& 1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& -2& 1& 1& 0\\ 1& 0& 0& -2& 0& 0& 1& 0& 0\\ 0& 0& 1& 0& -2& 0& 0& 1& 0\\ 0& 0& 0& 1& 1& 0& 0& 0& -2\\ 0& 1& 0& 0& 0& 1& 0& 0& -2\end{array}\right)\left(\begin{array}{c}A\\ B\\ C\\ D\\ E\\ F\\ G\\ H\\ I\end{array}\right)=\left(\begin{array}{c}{M}_2-M_1\\ M_4-M_3\\ M_6-M_5\\ M_8-M_7\\ M_{10}-M_9\\ M_{12}-M_{11}\\ M_{14}-M_{13}\\ M_{16}-M_{15}\end{array}\right)$$

(4)

For example: M₅ = B − A and M₆ = C − B, giving M₆ − M₅ = A − 2B + C; this is the third row in the matrix in Equation (4). This set of equations cannot be solved as the height and orientation of the reference plane is not defined. In the next sections it is shown how this is solved using the Moody-method and alternatives are investigated.

3. Measurement Evaluation

3.1. Evaluation According to Moody

In the original Moody method, measurements M₁₃..M₁₆ are not used in the calculation of the heights of points A..I, however at the end of the evaluations the line L₇ is fitted between the calculated points B and E, and L₈ is fitted between the calculated points D and F, using Equation (1). In the evaluation, the point I is defined as having a zero height (I = 0), and the endpoints of the diagonals are defined to have a same value; this means that H = A and G = C. Substituting this in (4) gives the characteristic matrix in Equation (5):

$$\left(\begin{array}{cccccc}2& 0& 0& 0& 0& 0\\ 0& 0& 2& 0& 0& 0\\ 1& -2& 1& 0& 0& 0\\ 1& 0& 1& 0& 0& -2\\ 1& 0& 1& -2& 0& 0\\ 1& 0& 1& 0& -2& 0\end{array}\right)\left(\begin{array}{c}A\\ B\\ C\\ D\\ E\\ F\end{array}\right)=\left(\begin{array}{c}{M}_2-M_1\\ M_4-M_3\\ M_6-M_5\\ M_8-M_7\\ M_{10}-M_9\\ M_{12}-M_{11}\end{array}\right)$$

(5)

This equation involves a square matrix that can readily be inverted, giving the solution given in Equation (6):

$$\left(\begin{array}{c}A\\ B\\ C\\ D\\ E\\ F\end{array}\right)=\left(\begin{array}{rrrrrr}0.5& 0& 0& 0& 0& 0\\ 0.25& 0.25& -0.5& 0& 0& 0\\ 0& 0.5& 0& 0& 0& 0\\ 0.25& 0.25& 0& 0& -0.5& 0\\ 0.25& 0.25& 0& 0& 0& -0.5\\ 0.25& 0.25& 0& -0.5& 0& 0\end{array}\right)\left(\begin{array}{c}{M}_2-M_1\\ M_4-M_3\\ M_6-M_5\\ M_8-M_7\\ M_{10}-M_9\\ M_{12}-M_{11}\end{array}\right)$$

(6)

For example: A = H = 0.5(M₂ − M₁) and B = 0.5(M₅ − M₆) + 0.25· (M₂ + M₄ − M₁ − M₃). After these heights are determined, the points in-between are calculated using Equation (1). Then the height of the lowest point is subtracted from all measured points and the surface is plotted using these values. The height of the highest point is the flatness deviation FLTt(EDPL) with the diagonal endpoints (ED) defining the reference plane (PL).

This is equivalent to the original sequence of calculations given by Moody:

• The diagonals A-H and G-C are evaluated using Equation (2), giving A, C, G and H and all other points on the lines.
• The perimeter lines A-C, C-H, A-G and G-H are evaluated using Equation (1), giving B, D, E and F and all other points on the line.
• The inner lines L₇ (between B and E) and L₈ (between D and F) are evaluated using Equation (1) giving all points on the line including additional values for I that are called the closure errors.
• The height of the lowest point is subtracted from all measured points, and the surface is plotted using these values. The height of the highest point is the flatness deviation FLTt(EDPL).

Equation (6) can be used to evaluate the propagation of uncertainties. For all measurement methods, such as electronic levels, autocollimators and laser interferometers, the measurements along the different lines can be considered as independent. Assuming an equal uncertainty u(M_i) in all measurements M_i, we find for the uncertainties in the corners A, C, G and H: $u\left(A,C,G,H\right)=u\left(M_i\right)/\sqrt{2}$ and for the node points along the edges $u\left(A,C,G,H\right)=u\left(M_i\right)\sqrt{3/4}$.

This result can be refined considering the amount of measurements taken along each line. Assuming the uncertainties in the individual measurements $u\left(m\right)$ are equal and independent, assuming a same uncertainty in the measured angle and a same length of the measurement base, it can be derived from Equation (3) that $u\left(M_2-M_1\right)=u\left(M_4-M_3\right)=u\left(m\right)\sqrt{N_d}; u\left(M_6-M_5\right)=u\left(M_8-M_7\right)=u\left(M_{14}-M_{13}\right)=u\left(m\right)\sqrt{N_x} \mathrm{a}\mathrm{n}\mathrm{d} u\left(M_{12}-M_{11}\right)=u\left(M_{10}-M_9\right)=u\left(M_{16}-M_{15}\right)=u\left(m\right)\sqrt{N_y}$.

This gives for the uncertainty in the heights of the corner points u(A) = u(C) = u(G) = u(H) = 0.5·$\sqrt{N_d}$·u(m), for the uncertainty in the heights of points B and F along the perimeter $u\left(B\right)=u\left(F\right)=u\left(m\right)\sqrt{0.125·N_d+0.25·N_x}$ and for the uncertainty in the heights of points D and E along the edges $u\left(D\right)=u\left(E\right)=u\left(m\right)\sqrt{0.125·N_d+0.25·N_y}$. For example: for N_d = 10, N_x = 8 and N_y = 6 we find u(A,C,G,H) = 1.58·u(m), u(B,F) = 1.80·u(m) and u(D,E) = 1.66·u(m).

The lines L₇ and L₈ may not give a zero value in point I. In the original Moody calculation method, these discrepancies are taken as a measure for the measurement error and are called ‘closure errors’. The closure errors can be denoted as E_BF and E_DE and can be written as the measured height of I along the lines L₇ and L₈ respectively as following:

$$E_{BF}=0.5·\left(B+F+M_{16}-M_{15}\right)=0.25\left(\left(M_1-M_2\right)+\left(M_3-M_4\right)+{(M}_6-M_5\right)+\left(M_8-M_7\right))+0.5(M_{16}-M_{15})$$

(7)

and

$$E_{DE}=0.5·\left(D+E+M_{14}-M_{13}\right)=0.25\left(\left(M_1-M_2\right)+\left(M_3-M_4\right)+{(M}_{10}-M_9\right)+\left( M_{12}-M_{11}\right))+0.5(M_{14}-M_{13})$$

(8)

For perfect measurements with zero uncertainty, $u\left(m_i\right)=0$, these closure errors are zero. The expected deviations from zero of the closure errors for real measurements can be expressed as the uncertainty in their value as:

$$\sqrt{{〈E_{BF}〉}^2}=u\left(m\right)·\sqrt{0.125·N_d+0.125·N_x+0.25·N_y}$$

(9)

and

$$\sqrt{{〈E_{DE}〉}^2}=u\left(m\right)·\sqrt{0.125·N_d+0.125·N_y+0.25·N_x}$$

(10)

For example: for N_d = 10, N_x = 8 and N_y = 6, applying Equations (9) and (10), we find for the expected closure error in the y direction $\sqrt{{〈E_{BF}〉}^2}=1.93·u\left(m\right)$and for the expected closure error in the x direction $\sqrt{{〈E_{DE}〉}^2}=2·u\left(m\right)$. Comparing this with the expected errors in the points A..H this means that the closure errors give a reasonable estimate of the errors in the points A..H.

From Equations (7) and (8) an estimate of the uncertainty can be made; however both equations are dependent. Using the Gram–Schmidt orthogonalization principle the closure error E_DE can be modified to give a measure of the uncertainty $u\left(m\right)$ that is independent of E_BF, giving the following equation:

$$\begin{gathered} E_{DEM}=E_{DE}-0.25\ast E_{BF}={\displaystyle \frac{3}{16}}\left(\left(M_1-M_2\right)+\left(M_3-M_4\right)\right)+{\displaystyle \frac{1}{16}}\left(\left(M_5-M_6\right)+\left(M_7-M_8\right)\right)+ \\ {\displaystyle \frac{1}{4}}\left(\left(M_{10}-M_9\right)+\left(M_{12}-M_{11}\right)\right)+0.5\left(M_{14}-M_{13}\right)+{\displaystyle \frac{1}{8}}(M_{15}-M_{16}) \end{gathered}$$

(11)

The expected value can be expressed as:

$$\sqrt{{〈E_{DEM}〉}^2}=u\left(m\right)·\sqrt{0.07·N_d+0.258·N_x+0.141·N_y}$$

(12)

The uncertainty $u\left(m\right)$can be estimated from the root mean square of the $u\left(m\right)$ values derived from Equations (9) and (12):

$$u\left(m\right)=\sqrt{{\displaystyle \frac{E_{DEM}^2}{2(0.07·N_d+0.258·N_x+0.141·N_y)}}+{\displaystyle \frac{E_{BF}^2}{2(0.125·N_d+0.125·N_x+0.25·N_y)}}}$$

(13)

This means that the standard uncertainty $u\left(m\right)$ can be estimated from the measurements with two degrees of freedom.

3.2. Least-Squares Evaluation Based on Straightness Measurements

Measurements M_13..16 can be involved in the evaluation rather than giving closure errors in point I. This extends the matrix given in Equation (5) to:

$$\left(\begin{array}{c}\begin{array}{cccccc}2& 0& 0& 0& 0& 0\\ 0& 0& 2& 0& 0& 0\\ 1& -2& 1& 0& 0& 0\\ 1& 0& 1& 0& 0& -2\\ 1& 0& 1& -2& 0& 0\\ 1& 0& 1& 0& -2& 0\\ 0& 0& 0& 1& 1& 0\\ 0& 1& 0& 0& 0& 1\end{array}\end{array}\right)\left(\begin{array}{c}A\\ B\\ C\\ D\\ E\\ F\end{array}\right)=\left(\begin{array}{c}{M}_2-M_1\\ M_4-M_3\\ M_6-M_5\\ M_8-M_7\\ M_{10}-M_9\\ M_{12}-M_{11}\\ M_{14}-M_{13}\\ M_{16}-M_{15}\end{array}\right)$$

(14)

Assuming the measurements M_1..16 have equal uncertainties, Equation (14) can be written as $\mathit{\alpha }\overline{a}=\overline{y}$, with as the least-squares solution $\overline{a}={\left({\mathit{\alpha }}^{\prime }·\mathit{\alpha }\right)}^{-1}\mathit{\alpha } \overline{y}$, where the ‘-sign denotes the transposed matrix and ${\left({\mathit{\alpha }}^{\prime }·\mathit{\alpha }\right)}^{-1}\mathit{\alpha }$ is the pseudo-inverse. This gives as a solution:

$$\left(\begin{array}{c}A\\ B\\ C\\ D\\ E\\ F\end{array}\right)={\displaystyle \frac{1}{60}}\left(\begin{array}{rrrrrrrr}24& -6& 3& 3& 3& 3& 6& 6\\ 6& 6& -23& 7& 2& 2& 4& 14\\ -6& 24& 3& 3& 3& 3& 6& 6\\ 6& 6& 2& 2& -23& 7& 14& 4\\ 6& 6& 2& 2& 7& -23& 14& 4\\ 6& 6& 7& -23& 2& 2& 4& 14\end{array}\right)\left(\begin{array}{c}{M}_2-M_1\\ M_4-M_3\\ M_6-M_5\\ M_8-M_7\\ M_{10}-M_9\\ M_{12}-M_{11}\\ M_{14}-M_{13}\\ M_{16}-M_{15}\end{array}\right)$$

(15)

This gives I = 0 for all lines through the center. The uncertainty in the corner points is given by u(A,C,G,H) = $u\left(m\right)·\sqrt{{0.17·N}_d+0.015·N_x+0.015·N_y}$, for the uncertainty in the points B and F along the perimeter $u\left(B,F\right)=u\left(m\right)\sqrt{0.02·N_d+0.165·N_x+0.0567·N_y}$and for the uncertainty in the points D and E along the perimeter $u\left(D,E\right)=u(\mathrm{m})\sqrt{0.02·N_d+0.0567·N_x+0.165·N_y}$. For example: for N_d = 10, N_x = 8 and N_y = 6 we find u(A,C,G,H) = 1.382·u(m), u(B,F) = 1.364·u(m) and u(D) = 1.282·u(m). Obviously, the uncertainty in the points along the perimeters benefits most from incorporating the lines L₇ and L₈ in the evaluation. The overall improvement of the uncertainty can be estimated by comparing the root mean square of the uncertainties in the node points: $Q=\sqrt{{\displaystyle \frac{4·{1.382}^2+2·{1.36}^2+2·{1.364}^2}{4·{1.58}^2+2·{1.8}^2+2·{1.66}^2 }}}$ = 0.82, this means that overall, the uncertainty in the height of the nodes is reduced by 18% by applying the least-squares solution instead of the traditional ‘Moody’ evaluation, and the height of the center point I is fixed unequivocally.

The solution can be refined by weighing the measurements by the inverse square of their uncertainty, replacing Equation (14) by Equation (16):

$$\left(\begin{array}{c}\begin{array}{rrrrrr}2/{\mathrm{N}}_{\mathrm{d}}& 0& 0& 0& 0& 0\\ 0& 0& 2/{\mathrm{N}}_{\mathrm{d}}& 0& 0& 0\\ 1/{\mathrm{N}}_{\mathrm{x}}& -2/{\mathrm{N}}_{\mathrm{x}}& 1/{\mathrm{N}}_{\mathrm{x}}& 0& 0& 0\\ 1/{\mathrm{N}}_{\mathrm{x}}& 0& 1/{\mathrm{N}}_{\mathrm{x}}& 0& 0& -2/{\mathrm{N}}_{\mathrm{x}}\\ 1/{\mathrm{N}}_{\mathrm{y}}& 0& 1/{\mathrm{N}}_{\mathrm{y}}& -2/{\mathrm{N}}_{\mathrm{y}}& 0& 0\\ 1& 0& 1/{\mathrm{N}}_{\mathrm{y}}& 0& -2/{\mathrm{N}}_{\mathrm{y}}& 0\\ 0& 0& 0& 1/{\mathrm{N}}_{\mathrm{x}}& 1/{\mathrm{N}}_{\mathrm{x}}& 0\\ 0& 1/{\mathrm{N}}_{\mathrm{y}}& 0& 0& 0& 1/{\mathrm{N}}_{\mathrm{y}}\end{array}\end{array}\right)\left(\begin{array}{c}A\\ B\\ C\\ D\\ E\\ F\end{array}\right)=\left(\begin{array}{c}\left(M_2-M_1\right)/N_d\\ \left(M_4-M_3\right)/N_d\\ \left(M_6-M_5\right)/N_x\\ \left(M_8-M_7\right)/N_x\\ \left(M_{10}-M_9\right)/N_y\\ \left(M_{12}-M_{11}\right)/N_y\\ \left(M_{14}-M_{13}\right)/N_x\\ \left(M_{16}-M_{15}\right)/N_y\end{array}\right)$$

(16)

and proceed as above. However we found no significant further decrease in uncertainties for this and other grid sizes so we conclude that Equation (15) can well be applied generally for reasonably square measurement configurations with no major differences in the number of measured points in the lines.

The heights of the points in-between the nodes may be calculated using Equation (1). However, the adjustment of the points means that in fact the straightness measurements along lines L₁..L₈ are adjusted halfway the lines and a kink in a line may appear in a measurement that gave an apparent straight line. A more smooth adjustment can be obtained by making a parabolic correction to the measured line, for example for line L₃:

$$h_{3,j}=A+{\sum }_{i=1}^j{m}_{1,i}-(1+C-A){\displaystyle \frac{j}{N_x}}{\sum }_{i=1}^{N_x}{m}_{1,i}+4·\left(A+C-2B-M_6+M_5\right)·\left(\left({\displaystyle \frac{j}{N_x}}\right)-{\left({\displaystyle \frac{j}{N_x}}\right)}^2\right).$$

(17)

This may especially be useful if many points are taken along the lines, for example by a profilometer. In our implementation, Equation (17) is used for the calculation of the points between the nodes.

The uncertainty $u\left(m\right)$ may still be estimated from Equations (7), (8), (11) and (13). An alternative is the calculation of the Chi-square value given by:

$$\begin{gathered} {\chi }^2={\left(2A-M_2+M_1\right)}^2+{\left(2C-M_4+M_3\right)}^2+{\left(A-2B+C-M_6+M_5\right)}^2+{\left(A+C-2F-M_8+M_7\right)}^2+{\left(A+C-2D-M_{10}+M_9\right)}^2 \\ +{\left(A+C-2E-M_{12}+M_{11}\right)}^2+{\left(D+E-M_{14}+M_{13}\right)}^2+{\left(B+F-M_{16}+M_{15}\right)}^2. \end{gathered}$$

(18)

Due to the evaluation method, the terms in Equation (18) are very correlated because there are only two degrees of freedom (six heights are calculated from eight independent measurements). For example, the first and the second term in Equation (18) are equal, as are the third and fourth term. From Equation (18) the measurement uncertainty $u\left(m\right)$ can be estimated as:

$$u\left(m\right)=\sqrt{{\displaystyle \frac{{\chi }^2}{0.5·N_d+0.75·N_x+0.75·N_y}}}$$

(19)

In practice this gives the same result as Equation (13).

3.3. Least-Squares Evaluation Based on Lines Having a Common Reference

When using electronic levels for the gradient measurements, the parallelism between the lines can be measured as well by keeping the same reference direction, for example by keeping the reference level at the same position on the object during the measurements in the x and y directions respectively. In this case, instead of the second derivative of each line (M_i+1-M_i), the height differences between the node points M_i can be considered directly. The measured height differences m_i,j of lines L₃, L₄ and L₇ have a common offset (tilt) in the x direction and the measured height differences in L₅, L₆ and L₈ have a common tilt in the y direction. The diagonal lines L₁ and L₂ have no common reference with other lines, therefore for these lines the second derivative is considered to be M₂-M₁ for line L₁ and M₄-M₃ for line L₂, as in Equation (4). The reference height is defined again by defining I = 0.

This gives as equations:

$$\left(\begin{array}{cccccccc}1& 0& 0& 0& 0& 0& 0& 1\\ 0& 0& 1& 0& 0& 0& 1& 0\\ -1& 1& 0& 0& 0& 0& 0& 0\\ 0& -1& 1& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& -1& 0\\ 0& 0& 0& 0& 0& -1& 0& 1\\ -1& 0& 0& 1& 0& 0& 0& 0\\ 0& 0& 0& -1& 0& 0& 1& 0\\ 0& 0& -1& 0& 1& 0& 0& 0\\ 0& 0& 0& 0& -1& 0& 0& 1\\ 0& 0& 0& -1& 0& 0& 0& 0\\ 0& 0& 0& 0& 1& 0& 0& 0\\ 0& -1& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 0& 1& 0& 0\end{array}\right)\left(\begin{array}{c}A\\ B\\ C\\ D\\ E\\ F\\ G\\ H\end{array}\right)=\left(\begin{array}{c}{M}_2-M_1\\ M_4-M_3\\ M_5-\alpha \\ M_6-\alpha \\ M_7-\alpha \\ M_8-\alpha \\ M_9-\beta \\ M_{10}-\beta \\ M_{11}-\beta \\ M_{12}-\beta \\ M_{13}-\alpha \\ M_{14}-\alpha \\ M_{15}-\beta \\ M_{16}-\beta \end{array}\right)$$

(20)

Here the constants α and β represent the tilt of the reference plane, expressed as the height difference between the points measured by M_i in the x and y direction respectively. The least-squares solution of Equation (20) is:

$$\left(\begin{array}{c}A\\ B\\ C\\ D\\ E\\ F\\ G\\ H\end{array}\right)={\displaystyle \frac{1}{96}}\left(\begin{array}{cccccccccccccc}30& 6& -33& -15& -9& -15& -33& -15& -9& -15& -18& -6& -18& -6\\ 12& 12& 26& -26& 2& -2& -14& -10& -14& -10& -4& 4& -44& -4\\ 6& 30& 15& 33& 15& 9& -9& -15& -33& -15& 6& 18& -18& -6\\ 12& 12& -14& -10& -14& -10& 26& -26& 2& -2& -44& -4& -4& 4\\ 12& 12& 10& 14& 10& 14& 2& -2& 26& -26& 4& 44& -4& 4\\ 12& 12& 2& -2& 26& -26& 10& 14& 10& 14& -4& 4& 4& 44\\ 6& 30& -9& -15& -33& -15& 15& 33& 15& 9& -18& -6& 6& 18\\ 30& 6& 15& 9& 15& 33& 15& 9& 15& 33& 6& 18& 6& 18\end{array}\right)\left(\begin{array}{c}{M}_2-M_1\\ M_4-M_3\\ M_5-\alpha \\ M_6-\alpha \\ M_7-\alpha \\ M_8-\alpha \\ M_9-\beta \\ M_{10}-\beta \\ M_{11}-\beta \\ M_{12}-\beta \\ M_{13}-\alpha \\ M_{14}-\alpha \\ M_{15}-\beta \\ M_{16}-\beta \end{array}\right)$$

(21)

This gives as a solution the heights A..H in a tilted plane, that can be leveled separately to the diagonal endpoints, the least-squares plane or the minimum-zone plane. In order to enable a comparison of the uncertainties to the solutions given in Section 3.1 and Section 3.2, a leveling to equal the diagonal endpoints must be integrated into the solution. This is achieved as follows:

Taking A − H = 0 from Equation (14) gives an equation for α + β; and taking C − G = 0 gives an equation for α − β. Solving these two equations gives:

$$\alpha ={\displaystyle \frac{3}{16}}\left(M_5+M_6+M_7+M_8\right)+{\displaystyle \frac{1}{16}}\left(M_9-M_{10}-M_{11}+M_{12}\right)+{\displaystyle \frac{1}{8}}\left(M_{13}+M_{14}\right),$$

(22)

and

$$\beta ={\displaystyle \frac{1}{16}}\left(M_5-M_6-M_7+M_8\right)+{\displaystyle \frac{3}{16}}\left(M_9+M_{10}+M_{11}+M_{12}\right)+{\displaystyle \frac{1}{8}}(M_{15}+M_{16})$$

(23)

Using Equations (22) and (23), the variables in the right-hand side of Equation (21) can be expressed in the measurements M_i by the following transformation:

$$\left(\begin{array}{c}{M}_2-M_1\\ M_4-M_3\\ M_5-\alpha \\ M_6-\alpha \\ M_7-\alpha \\ M_8-\alpha \\ M_9-\beta \\ M_{10}-\beta \\ M_{11}-\beta \\ M_{12}-\beta \\ M_{13}-\alpha \\ M_{14}-\alpha \\ M_{15}-\beta \\ M_{16}-\beta \end{array}\right)={\displaystyle \frac{1}{16}}\left(\begin{array}{rrrrrrrrrrrrrrrr}-16& 16& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& -16& 16& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0& 0\\ 0& 0& 0& 0& 13& -3& -3& -3& -1& 1& 1& -1& -2& -2& 0& 0\\ 0& 0& 0& 0& -3& 13& -3& -3& -1& 1& 1& -1& -2& -2& 0& 0\\ 0& 0& 0& 0& -3& -3& 13& -3& -1& 1& 1& -1& -2& -2& 0& 0\\ 0& 0& 0& 0& -3& -3& -3& 13& -1& 1& 1& -1& -2& -2& 0& 0\\ 0& 0& 0& 0& -1& 1& 1& -1& 13& -3& -3& -3& 0& 0& -2& -2\\ 0& 0& 0& 0& -1& 1& 1& -1& -3& 13& -3& -3& 0& 0& -2& -2\\ 0& 0& 0& 0& -1& 1& 1& -1& -3& -3& 13& -3& 0& 0& -2& -2\\ 0& 0& 0& 0& -1& 1& 1& -1& -3& -3& -3& 13& 0& 0& -2& -2\\ 0& 0& 0& 0& -3& -3& -3& -3& -1& 1& 1& -1& 14& -2& 0& 0\\ 0& 0& 0& 0& -3& -3& -3& -3& -1& 1& 1& -1& -2& 14& 0& 0\\ 0& 0& 0& 0& -1& 1& 1& -1& -3& -3& -3& -3& 0& 0& 14& -2\\ 0& 0& 0& 0& -1& 1& 1& -1& -3& -3& -3& -3& 0& 0& -2& 14\end{array}\right)\left(\begin{array}{c}{M}_1\\ M_2\\ M_3\\ M_4\\ M_5\\ M_6\\ M_7\\ M_8\\ M_9\\ M_{10}\\ M_{11}\\ M_{12}\\ M_{13}\\ M_{14}\\ M_{15}\\ M_{16}\end{array}\right)$$

(24)

Combining Equations (21) and (24) gives:

$$\left(\begin{array}{c}A\\ B\\ C\\ D\\ E\\ F\end{array}\right)={\displaystyle \frac{1}{96}}\left(\begin{array}{rrrrrrrrrrrrrrrr}-30& 30& -6& 6& -9& -3& 3& 9& -9& -3& 3& 9& -6& 6& -6& 6\\ -12& 12& -12& 12& 32& -32& -4& 4& 4& 8& 4& 8& -4& 4& -32& 8\\ -6& 6& -30& 30& 3& 9& -9& -3& 3& 9& -9& -3& -6& 6& -6& 6\\ -12& 12& -12& 12& 4& 8& 4& 8& 32& -32& -4& 4& -32& 8& -4& 4\\ -12& 12& -12& 12& -8& -4& -8& -4& -4& 4& 32& -32& -8& 32& -4& 4\\ -12& 12& -12& 12& -4& 4& 32& -32& -8& -4& -8& -4& -4& 4& -8& 32\end{array}\right)\left(\begin{array}{c}{M}_1\\ M_2\\ M_3\\ M_4\\ M_5\\ M_6\\ M_7\\ M_8\\ M_9\\ M_{10}\\ M_{11}\\ M_{12}\\ M_{13}\\ M_{14}\\ M_{15}\\ M_{16}\end{array}\right)$$

(25)

In Equation (25) the solutions for G and H are omitted, as they are equal to C and A respectively. Assuming a same uncertainty u(m) in every measured height difference m_ij, it can be derived from Equation (3) that $u\left(M_{1..}{M}_4\right)=u\left(m\right)\sqrt{N_d/2}; u\left(M_{5..8}\right)=u\left(M_{\mathrm{13,14}}\right)=u\left(m\right)\sqrt{N_x/2} \mathrm{a}\mathrm{n}\mathrm{d} u\left(M_{9..12}\right)=u\left(M_{\mathrm{15,16}}\right)=u\left(m\right)\sqrt{N_{y/2}}$. This gives for the uncertainty in the corner points u(A,C,G,H) = $\sqrt{{0.1·N}_d+0.014·N_x+0.014·N_y}·u\left(m\right)$, for the uncertainty in the points B and F along the perimeter $u\left(B,F\right)=\sqrt{0.031·N_d+0.115·N_x+0.065·N_y}·u\left(m\right)$and for the uncertainty in the points D and E along the edges $u\left(D,E\right)=\sqrt{0.031·N_d+0.065·N_x+0.115·N_y}·u\left(m\right)$. For example: for N_d = 10, N_x = 8 and N_y = 6 we find u(A,C,G,H) = 1.09·u(m), u(B,F) = 1.27·u(m) and u(D,E) = 1.23·u(m). Comparing this to the uncertainty of the traditional Moody method, similarly as in Section 3.2, this gives an overall error reduction of 29% compared to the traditional Moody method given in Section 3.1, and a 13% reduction compared to the least-squares solution given in Section 3.2. This improved uncertainty of level measurements using a constant reference to gravity, for example by a differential measurement using a reference level on a fixed position, must be weighed against the additional care needed to keeping this reference constant.

A measure for the measurement uncertainty can be obtained by defining the Chi-square function based on Equation (20) using the values of α and β calculated in Equations (22) and (23):

$$\begin{gathered} {\chi }^2={\left(2A-M_2+M_1\right)}^2+{\left(2C-M_4+M_3\right)}^2+{\left(B-A-M_5+\alpha \right)}^2+{\left(C-B-M_6+\alpha \right)}^2+{\left(F-G-M_7+\alpha \right)}^2 \\ +{\left(H-F-M_8+\alpha \right)}^2+{\left(D-A-M_9+\beta \right)}^2+{\left(G-D-M_{10}+\beta \right)}^2+{\left(E-C-M_{11}+\beta \right)}^2 \\ +{\left(H-E-M_{12}+\beta \right)}^2+{\left(\alpha -D-M_{13}\right)}^2+{\left(\alpha +E-M_{14}\right)}^2+{\left(\beta -B-M_{15}\right)}^2+{\left(\beta +F-M_{16}\right)}^2. \end{gathered}$$

(26)

From Equation (26) the measurement uncertainty $u\left(m\right)$ can be estimated as:

$$u\left(m\right)=\sqrt{{\displaystyle \frac{{\chi }^2}{0.875·N_d+1.3125·N_x+1.3125·N_y}}}$$

(27)

This uncertainty is calculated with 10 degrees of freedom because 16 independent measurements are taken from which six heights are determined.

4. Reference Plane Definition

The definition of the reference plane by initially putting I = 0 and equaling the diagonal endpoints has its advantages as it is uniquely defined and, as it is shown in this paper, can be directly integrated in the measurement evaluation and uncertainty evaluation. When considering the node points only, the least-squares plane can be used as well, giving as boundary conditions A + B + C = G + F + H; A + D + G = C + E + H and A + B + C + D + E + F + G + H + I = 0. This can be incorporated in Equation (5) instead of putting A = H, C = G and I = 0. However for determining a proper least-square plane all measured heights must be taken into account, including the points between the nodes, requiring an additional calculation. In form deviations in the ISO Geometric Product Specification system, it is common to take the minimum-zone planes as the reference, according to which the flatness is defined as the minimum distance between two parallel planes within which the entire surface resides. This can be calculated by rotating the reference plane around the x- and y-axis in such a way that the difference between the minimum and maximum height is minimal. Writing the measured coordinates as h_ij(x_i,y_i) this gives:

$$h_{ij}^{\prime }(x_{ij},y_{ij})={min}_{\alpha ,\beta }\left(range\right(\left(h_{ij}\left(x_{ij},y_{ij}\right)-\alpha ·x_{ij}-\beta ·y_{ij}\right))$$

(28)

where α is the rotation around the y-axis with a positive angle, affecting the x-coordinates and β is the rotation around the x-axis with a negative angle, affecting the y-coordinates. This can be evaluated by standard mathematical routines for minimizing a function depending on two variables: the two angles α and β. An alternative is an exchange algorithm that determines the critical points that define the minimum zone plane [13]. After this operation, the z-value of the lowest point can be subtracted from all measured points. Note that at the time the Moody method was designed (1955) this calculation was virtually impossible, computers being hardy available. The orientation of the minimum-zone plane depends on the actually measured topography and is task-specific. Therefore, general expressions for the uncertainties cannot be made and Monte Carlo simulations must be carried out to evaluate the propagation of uncertainties.

5. Monte Carlo Simulations

In Section 3 the uncertainties in the node points could be analyzed; however, for the uncertainties in the points in between an analytical calculation is virtually impossible. This is also the case for the flatness deviation FLTt(EDPL), where it depends on the actual topography which points determine the flatness deviations. Therefore, in order to evaluate the uncertainty for actual measurements using both the minimum-zone plane as a reference as well as equaling the diagonal endpoints, Monte Carlo simulations were carried out using the following steps:

• The measurements are evaluated according to one of the methods given in Section 3.1, Section 3.2 and Section 3.3, giving the heights h_i,j that are used as a reference in the simulations.
• The flatness deviation FLTt is evaluated for both the diagonal endpoints reference, giving FLTt(EDPL), and the minimum zone plane, giving FLTt(MZPL). These values are taken as a reference in the simulations.
• The measurement uncertainty u(m) is estimated from the measurements using the appropriate method given in Section 3.1, Section 3.2 and Section 3.3. It may be adjusted for unrealistic values or for research purposes.
• A set of simulated measurements m_ij is generated using the calculated values of h_ij and u(m) as following:

$$m_{ij}=h_{i,j+1}-h_{i,j}+r·u\left(m\right)$$

(29)

where r is a Gaussian distributed random number with zero mean and a standard deviation of unity. The application of Equation (29) simulates measurements of the calculated topography h_i,j where the measuring instrument introduces errors with a standard uncertainty u(m) that follows a Gaussian distribution.

5.

The simulated measurements are evaluated and the flatness deviations FLTt(EDPL) and FLTt(MZPL) are noted.

6.

Steps 4 and 5 are repeated some thousand times where the deviation of the mean simulated flatness deviation from the reference is noted as the bias and the standard deviation in the flatness value of the simulated measurements is noted as the standard deviation. This bias is noted as it is recognized that non-repeatable measurements tend to give higher rather than lower form deviation values [14].

As an example we take the measurement of a perfect flat, giving h_ij = 0, measured with a measuring instrument giving a standard deviation of 1 µm, for example an electronic level with a standard deviation of 10 µrad and a measurement base of 100 mm.

Table 1. Result of Monte Carlo simulations of measurements on a perfectly flat surface.

Bias and Standard Deviation in µm u(m) = 1 µm	Classical Moody Evaluation (Section 3.1)	Least Squares Moody Evaluation (Section 3.2)	Least Squares Level Evaluation (Section 3.3)
2 × 2, FLTt(EDPL)	1.99 ± 0.59	1.59 ± 0.54	1.37 ± 0.46
2 × 2, FLTt(MZPL)	1.77 ± 0.56	1.34 ± 0.47	1.11 ± 0.36
8 × 6, FLTt(EDPL)	5.59 ± 1.08	5.08 ± 1.00	4.77 ± 0.87
8 × 6, FLTt(MZPL)	4.89 ± 0.97	4.43 ± 0.87	4.15 ± 0.70

gives the result of the simulations for a 2 × 2 measurement grid (N_d = N_x = N_y = 2) and the 8 × 6-point configuration that was taken as an example: N_d = 10 N_x = 8 N_y = 6.

Table 1 shows that both the least-squares evaluations and taking the minimum zone as the reference plane leads to lower bias and lower uncertainties compared to the classical Moody evaluation. In this case the bias is far larger than the standard deviation (repeatability). This means that both the bias and standard deviations must be considered in an uncertainty budget, especially for surface plates with a flatness deviation in the range of the measurement limit of the used instrument. The bias and standard deviation can be treated as type B and type A uncertainties respectively in further uncertainty evaluations.

6. Measurement Example

A surface plate of 900 mm × 700 mm × 150 mm was measured using electronic levels with a resolution of 1 µrad on a 80 mm measurement base, where an area of 640 mm × 480 mm was measured giving N_d = 10, N_x = 8 and N_y = 6. These numbers of measurements correspond to the examples in Section 3 where the uncertainty was elaborated, however the Equations given in Section 3, especially Equations (6), (15) and (21), are valid for any number of measurements, as was tested for various measurement configurations. The measured topography is not essentially different between the methods and is displayed in Figure 4.

In

Table 2. Results of the measurement evaluations according to the methods described in Section 3.1, Section 3.2 and Section 3.3.

Results in µm	Classical Moody Evaluation (Section 3.1)	Least Squares Moody Evaluation (Section 3.2)	Least Squares Level Evaluation (Section 3.3)
FLTt(EDPL)	3.50	3.46	3.26
FLTt(MZPL)	3.20	3.18	2.98
u(m)	0.15	0.15	0.21
Monte Carlo results; bias and standard deviation for u(m) = 0.21 µm
FLTt(EDPL)	0.23 ± 0.36	0.08 ± 0.34	0.08 ± 0.30
FLTt(MZPL)	0.24 ± 0.31	0.08 ± 0.26	0.09 ± 0.24

the results are summarized regarding the measured topography, the estimated uncertainty in the measurements and the bias and standard deviation based on u(m) = 0.21 µm.

The results in Table 2 confirm the general trends analyzed in Section 3. The higher uncertainty predicted by the least-square level evaluation may be due to the higher degrees of freedom and the involvement of the parallelism of the lines in the evaluation. The significant bias observed in the Monte Carlo simulations of the classical Moody evaluation may be due to the lines L₇ and L₈ that cause the closure error and cross point I which may also be the lowest point defining the FLTt value in the simulations. Most of the values depend on the actual topography and measurements taken, but the general trend is that the least squares evaluations result in lower uncertainties than the classical Moody evaluation.

7. Traceability and Uncertainty Considerations

The traceability to the perfect plane is established by the straight traveling of light through air for autocollimator and laser interferometer measurements, and the gravity vector for level measurements. The effect of the earths’ curvature on level-based measurements is smaller than 0.1 µm deviation for surfaces up to 2 m × 2 m [7]. Temperature and air pressure effects on autocollimator measurements have been investigated extensively by Geckeler et al. [15]. The values of the flatness deviations depend on the length of the measurement base and the calibration of the angular measurement device. Uncertainties in these values are commonly below the percentage level and are mostly not significant for reasonable flat surfaces. Other aspects are the coverage of the surface plate by these measurements, humidity temperature gradients of the surface plate and the size of the measurement feet. These aspects have extensively been treated by Drescher [9] and are not repeated here. This paper is focused on measurement evaluation and its uncertainty.

8. Conclusions

The classical Moody method can be improved by applying a least-squares evaluation using the measurement redundancies. This improves both the repeatability of the results as well as the bias in flatness measurements by typically 20% and it gives a single value for the surface height in the center. Taking the minimum-zone reference plane may give a significantly lower FLTt-value, depending on the topography. Without these improvements, the Moody method tends to give too high FLTt values, increasing the risk of false rejection of the plate. An additionally improved uncertainty of level measurements using a constant reference to gravity, for example by a differential measurement, keeping a reference level on a fixed position, must be weighed against the additional care of keeping this reference constant.