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Uncertainty estimation in form errorevaluation of freeform surfaces for precision

metrology

ARTICLE in PROCEEDINGS OF SPIE - THE INTERNATIONAL SOCIETY FOR OPTICAL ENGINEERING AUGUST 2015

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Xiangchao Zhang

Fudan University

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Hao Zhang

Ordnance Engineering College

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Xiaoying He

Fudan University

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Available from: Xiangchao Zhang

Retrieved on: 29 December 2015

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Uncertainty estimation in form error evaluation of freeform surfaces

for precision metrology

Xiangchao Zhang1*, Hong Xiao2, Hao Zhang1, Xiaoying He1and Min Xu1

1. Shanghai Engineering Research Centre of Ultra-Precision Optical Manufacturing,

Fudan University, Shanghai, 200438, P. R. China

2. Laboratory of Precision Manufacturing Technology, China Academy of Engineering Physics,

Mianyang, 621900, P.R. China

ABSTRACT

Freeform surfaces are widely applied in precision components to realize novel functionalities. In order to evaluate the

form qualities of the manufactured freeform parts, matching/fitting is required. The uncertainty of the obtained form

error measures needs to be estimated so as to assess the reliability of the evaluating process. The conventional GUM

approach is not suited for complex nonlinear models. In this paper we develop a Monte-Carlo method to estimate the

uncertainty of the fitted position, shape and form error parameters. Based on the correlation analysis, the effects of some

major factors, including objective functions, noise amplitudes, surface shapes etc can be determined, and subsequently

the significant factors influencing the evaluated form errors can be specified. By appropriate planning of the measuring

and matching procedures, the uncertainty of the evaluation results can be effectively reduced, and thereby improving the

reliability of freeform surface characterization.

Keywords: Precision metrology, freeform surface, uncertainty, Monte-Carlo method

1. INTRODUCTION

With the rapid development of advanced design and manufacturing technologies, freeform components are increasingly

applied in modern opto-mechanical systems, because of their compact sizes, small weights, flexibility in

design/utilization and some attractive capabilities of system integration, realizing various novel functions and remedying

the drawbacks of traditional components. These freeform components realize the intended functionalities by their forms;thereby the evaluation of the form errors with respect to their nominal shapes has become a central task of freeform

surface metrology.

In the ISO standards and commercial precision instruments, various fitting objectives and algorithms have been

developed. In ISO 5459, the objective functions of least squares, minimum zone, one-sided Chebyschev, maximuminscribed, minimum circumscribed and the Lp-norm are defined [1, 2]. As for the freeform surfaces, the L1norm, the

least squares and the minimum zone fitting are commonly adopted in accordance with the specific properties and

applications of the measured data,

1

1

2

1

norm min

least squares min

minimum zone min max

N

i

iN

i

i

ii

L d

d

d

=

=

(1)

In the equation, didenotes the signed deviation of data pointpiwith respect to the nominal shape andNis the number of

data points. Among these three criteria, the least squares approach is most widely used due to its ease of computation and

unbiasedness for the normally distributed noise, but bias will be caused in the obtained form error metrics. The solutions

of the L1norm and the minimum zone fitting are unbiased for long-tailed and uniformly distributed errors, respectively.

In addition, the minimum zone approach can obtain the smallest parameter of form deviation, complying with the

*Corresponding author. Tel.: +86-21-51630347. Fax.: +86-21-65641344. Email: zxchao@fudan.edu.cn.

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definitions in the ISO 1101 [3]. However, these two objectives are non-differentiable, and thus very difficult to be solved,

especially for the freeform surfaces with complex mathematical representations. Recently some algorithms have been

proposed to solve the L1norm and minimum zone fitting/matching problem [4-7]. Researchers mainly focused on theform error parameters obtained, but the reliability of the solutions, i.e. the rotational angles, translation vectors, axis

orientations, shape parameters etc were not examined rigorously.

In order to verify the reliability and stability of the fitting algorithms, the uncertainty of the solutions have to be

estimated. The approaches to the uncertainty estimation have been investigated by many groups and scholars. Some of

the typical principles or methods are provided by Evaluation of measurement data-Guide to the expression of

uncertainty in measurement (GUM) [8] and Monte Carlo (MC) method [9]. The GUM technique can be employed to

assess simple processes, but it is impractical or impossible for the nonlinear minimum zone fitting of freeform surfaces.

Some researchers estimated the uncertainty of straightness and flatness calculation using MC method [10, 11]. The MC

method is a sampling technique that provides rich information by propagating the distributions for the input quantities.

2. ESTIMATION PREEDURE OF FITTING UNCERTAINTY

The purpose of uncertainty estimation for surface fitting is to identify the significant factors influencing the fitting

quality, and assess the reliability of the fitted results, so that the performance of the utilized fitting algorithms can be

validated.

As for the fitted results, the obtained rotation angles, translation components and shape parameters are stored. Threeform error parameters are used to evaluate the relative deviations between the fitted surface and the input noisy data: the

arithmetic average (AA), the root-mean-squares (RMS) and the peak-to-valley (PV),

1

2

1

1AA

1RMS

PV max min

N

i

iN

i

i

i iii

dN

dNd d

=

=

(2)

It is known that the objective function of surface fitting should be consistent with the form error parameter to be

calculated. In surface metrology the form deviations are usually assessed along the z direction only. It is known the

solutions of such algebraic fitting are unstable and sensitive to measurement noise [12]. Moreover, the obtained error

parameters are prone to be biased and over-estimated when the surfaces are highly curved [13]. As a result we adoptorthogonal distance fitting here, i.e. the form deviations are calculated along the normal vectors of the nominal surfaces.

To assess the influence of the different configurations, including the objective functions, noise amplitudes and surface

shapes, the fitting program is run repetitively forMtimes, with random noise added onto the sampled data. To make the

calculation results faithfully reflect the properties of the fitting algorithms, it is important to ensure that no bias is

introduced by the added noise, i.e. the mean shapes and positions of the noisy data should always be identical with the

reference datums.

1

1

1

1

shape preserving 0

rest in direction 0

rest in direction 0

rest in direction 0

N

i

iN

i

iN

i

iN

i

i

d

x x

y y

z z

=

=

=

=

=

=

=

=

(3)

Hereix , i and i are thex,yandzcomponents of the form deviation di, respectively.

In practice, outliers and defects are usually removed in advance, hence they are not considered here.

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Then the distribution of the parameters of interest is analyzed. The mean, standard deviation, skewness and kurtosis are

calculated respectively to assess the bias, dispersion, symmetry and peakedness of the distribution of each calculated

parameter,

1

2

1

3

31

4

41

1mean

1standard deviation ( )1

1skewness ( )

( 1)1

kurtosis ( )( 1)

N

i

iN

i

iN

i

iN

i

i

ME xN

SD x ME N

SK x ME N SD

KU x MEN SD

=

=

=

=

=

=

=

=

(4)

The implementing procedure of Monte Carlo estimation of the fitting uncertainty is presented in Figure 1.

Figure 1. Procedure of Monte-Carlo estimation of fitting uncertainty

3. UNCERTAINTY ESTIMATION OF BICONIC SURFACE FITTING

The representations of freeform surfaces are very complex. To make the estimation results typical and representative, a

biconic surface is adopted [4], as shown in Figure 2.

2 2

2 2

2 22 2 2 2

2 2

/ /

1 1 (1 ) / (1 ) /

M Mx y i j

i j

i jx x y y

x R y Rz a x b y

k x R k y R = =

+

= + +

+ + +

(5)

In the equation, Rx=50 mm and Ry=60 mm are the radii of curvature and cx= -0.3 and cy=0.5are conic factors. The

coefficients of polynomial terms are set as a4=3.5e-6, b4=-2.8e-6, a6=-1.9e-10 and b6=6.3e-9. A set of 200 200 points

are taken as measured data.

generate

surface data

add random

noise

surface fitting

change

configuration

analyze

distribution

M times

establish

correlation

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Figure 2. Biconic surface

3.1 Effect of fitting objectives

Three fitting criteria the L1 norm, least squares (LS) and minimum zone (MZ) are compared. A surrogate function

approach based on the majorize-minimize theory [4], the fast Levenberg-Marquardt algorithm [14] and the differentialevolution method [7] are adopted to solve these three optimization problems, respectively. It is worth noting that the

obtained shape parameters and form errors of the orthogonal distance fitting are irrelevant to the surface orientation;

therefore no translation/rotation is implemented on the data set before fitting.

Random Gaussian noise with standard deviation of 0.6 m is superimposed on the sampled data points, so as to make the

data uncertainty consistent with the Carl Zeiss Prismo Ultra CMM. The programs are implemented repetitively for 300

times, and the distributions of the obtained rotation angles, translation components and shape parameters are presented.

It can be seen that the L1norm and LS fitting can always obtain correct positions. The relative rotation angles are within

10-3

. The lateral and vertical translation errors are less than 0.9m and 80 nm, respectively. This is due to the symmetry

of the data sets and the added random noise, thus no misalignment will be caused in surface fitting. But the MZ fitting

results depend on several feature points only, henceforth the variations of these extreme points can make the fitted

surface shifted or tilt. The SD of the translation components Tx, Tyand Tzand rotation angles x, yand zare 66.9m,

25.0m, 8.0m, 0.033, 0.088 and 0.043, respectively.

As for the fitted shape parameters, the L1 and LS method behave almost the same. In addition, the shape parameters

associated withxbehaves similarly with those associated withy, e.g.RxandRy, kxand ky, and so forth. As a result only

thex shape parameters are presented in Tables 1 and 2. For the sake of clarity, their units are omitted.

All the fitted shape and form error parameters obey symmetric Gaussian distribution, expect for kxobtained by the MZ

fitting. The uncertainties of the MZ shape parameters are much greater than the LS fitting. However, the obtained form

error parameters have similar uncertainties. The obtained AA and RMS values are equal, but the PV parameters of MZ

fitting is smaller than the LS values by 47%, implying the LS method tends to over-estimate the form deviation if

measured with a zone width.

parameters ME SD SK KU

Rx 50.0000 0.0008 -0.0632 3.2891

kx -0.30000 0.0003 -0.0033 2.7053

a4 3.50e-6 5.96e-10 -0.1692 3.0918

a6 -1.90e-10 4.94e-13 -0.2099 3.4022

AA 4.78e-4 2.84e-6 0.0959 3.0826

RMS 6.00e-4 3.23e-6 0.1067 3.4439

PV 0.0047 2.51e-4 0.6251 3.3067

Table 1. Distributions of the fitted parameters of least squares

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parameters ME SD SK KU

Rx 50.0111 0.2752 -0.0191 3.1276

kx -0.3033 0.0592 0.1645 2.1946

a4 3.51e-6 1.6420e-7 -0.0277 2.8722

a6 -1.95e-10 8.64e-11 0.0786 2.3807

AA 4.79e-4 2.85e-6 0.0826 3.0216

RMS 6.00e-4 3.24e-6 0.1094 3.4112

PV 0.0025 1.80e-4 1.1379 5.4346

Table 2. Distributions of the fitted parameters of minimum zone

3.2 Effect of shapes

The curvedness of the measured surfaces severely influences the reliability of the fitted results [12]. For the sake of

expression clarity, the surface parameters remain unchanged, while the width of the data set in the x direction is

decreased. The ratio between thexandywidths are set to be 0.2, 0.4, 0.6, 0.8 and 1 respectively, and the MZ fitting with

differential evolution is implemented 200 times for each case. Gaussian random noise with standard deviation of 0.6mis added into the data.

It is found that the bias and uncertainties of the motion parameters are not affected by the width ratio. As the data sets are

symmetric about the rotational axis and the added noise has zero mean, hence the program can always obtain the correct

positions for the fitted surfaces. In addition, the obtained form error parameters AA, RMS, and PV are not apparently

affected by the width ratio either.

As for the fitted shape parameters, we define two factors: the relative bias0/ 1 100%s s and relative uncertainty

0( ) / 100%s s . Here s and ( )s are the mean and standard deviation of the fitted shape parameter s, and s0 is its

ideal value.

0.2 0.4 0.6 0.8 110

-4

10-2

100

102

Rx

kx

a4

a6

Ry

ky

b4

b6

0.2 0.4 0.6 0.8 110

-1

100

101

102

103

104

Rxkx

a4

a6

Ryky

b4

b6

(a) bias (b) uncertainty

Figure 3. Relative bias and uncertainty at different width ratios (%)

Thexshape parametersRx, kx, a4and a6are increasingly biased when the data set narrows down in the xdirection. The

uncertainties are getting worse simultaneously. This is straightforward to understand. When the surface is narrowed, the

form is flattened, as a result the data correspond to only a small section on the original surface. Slight variation of data in

thezdirection can cause remarkable change to the fitted shape parameters. This is consistent with the conclusions of [12].Interestingly, theyparametersRy, ky, b4and b6show a different trend. When thexcoordinates are relatively small, the

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11. Calvo, R., Gomez, E. and Domingo, R., Vectorial method of minimum zone tolerance for flatness, straightness,and their uncertainty estimation, International Journal of Precision Engineering and Manufacturing 15(1), 31-44 (2014).

12. Sun, W., McBride, J.W. and Hill, M., A new approach to characterizing aspheric surfaces, Precision Engineering34, 171-179 (2010).

13. Zhang, X., Zhang, H., He, X. and Xu, M., Bias in parameter estimation of form errors, Surface Topography:Metrology and Properties2(3),035006 (2014).

14. Boggs, P.T., Byrd, R.H. and Schabel, R.B., A stable and efficient algorithm for nonlinear orthogonal distanceregression, SIAM J Stat Comput. 8(6), 1052-1078 (1987).