Separation of form from orientation in 3D measurements of aspheric surfaces with no datum

International Journal of Machine Tools & Manufacture 42 (2002) 457–466

Separation of form from orientation in 3D measurements of aspheric surfaces with no datum M. Hill *, M. Jung, J.W. McBride University of Southampton, School of Engineering Sciences, Highfield, Southampton SO17 1BJ, UK Received 28 February 2001; received in revised form 20 September 2001; accepted 3 October 2001

Abstract Measurement and characterisation of 3D form to maintain manufacturing quality has particular problems in cases such as lenses which do not generally have a clear measurement datum. A 3D-form measurement includes information about the form, the orientation and the position of the surface under test. Orientation and position can be design parameters or may result from misalignment of the test specimen on a measurement table. In either case, it is necessary to separate form from orientation and position if the data-set is to be fitted and compared with an “ideal” surface. In this paper two pre-processing algorithms are presented and examples given of the separation of form from orientation and position. The algorithm is applied to simulated data-sets consisting of up to 26,000 discrete points on a square grid, simulating the measurement of an aspheric lens in 3D. The rotationally symmetric dataset is translated for a distance x0, y0 and z0 and rotated about two axes, x and y, to simulate misalignment. To simulate inaccuracies from a manufacturing process, normally distributed random noise is superimposed on the ideal surface. An application of preprocessing using a real data-set is also shown. Furthermore, form fitting is addressed and the interpretation of form by decomposition of the data into error types is discussed.  2002 Elsevier Science Ltd. All rights reserved. Keywords: Pre-processing; Form characterisation; Data decomposition; Error types; Aspheric surface analysis; Aspheric lens

1. Introduction The precision measurement and investigation of aspheric surfaces in three dimensions (3D) is of importance to the manufacture and quality assurance of a number of products, including lenses, contact lenses and electrical contacts. The surface area under investigation is typically not bigger than 50 mm×50 mm. Previous research has focused on the 3D-form measurement and characterisation of spherical surfaces [1,2] and the volumetric analysis of erosion of electrical contact surfaces [3]. Recent research has extended the investigations to cover aspheric surfaces. This has occurred because of developments in lens technology where aspheric lens surfaces have been shown to have significant performance advantages over conventional spherical lenses. The design of aspheric lenses is well understood, but difficulties arise in the measurement and form characterisation of asph-

* Corresponding author. Tel.: +44-23-8059-3075. E-mail address: [email protected] (M. Hill).

eric surfaces after manufacture. Optical surfaces are particularly challenging because of the required precision and because of the lack of an easily identifiable surface feature which could act as a datum. The analysis of aspheric lens surfaces is the main application for the methods proposed in this paper. In this work, the data are defined to a resolution of 10 nm in the vertical (z) axis and are based on a grid in the xy-plane with a grid spacing of typically between 20 µm and 250 µm. Form characterisation of aspheric lens surfaces has been discussed in a previous paper [4], where the idea of separation of geometry from orientation and position in 3D-form measurements, the pre-processing of the data, was introduced. The raw data hold information about form and surface irregularity as well as orientation and position. The parameters that specify the orientation and position are determined first by pre-processing. The form characterisation can then be undertaken. Finally, the measured surface can be decomposed into error types for analysis. In many instances of form measurement (those often

0890-6955/02/$ - see front matter  2002 Elsevier Science Ltd. All rights reserved. PII: S 0 8 9 0 - 6 9 5 5 ( 0 1 ) 0 0 1 4 2 - 0

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found in the use of co-ordinate measurement machines for instance) alignment of the measurements to remove orientation errors can be achieved by reference to a measurement datum, or by searching for easily identifiable features within the data set. The requirements of aspheric lenses dictates that they rarely include an identifiable feature which can provide a datum and it is to such cases that this work is addressed. Before data interpretation is outlined in more depth the manufacture and form measurement of aspheric lens surfaces is briefly described. 1.1. Manufacturing methods for aspheric lens surfaces The manufacture of aspheric lens surfaces by diamond turning or grinding is most economical for low quantity production. Replication methods such as moulding, on the other hand, are more suited for mass production because the difficult and expensive task of generating high precision aspheric surfaces is reduced to the manufacture of a few master surfaces and the cost can be spread over many replicas. The aspheric lens analysed in this paper was manufactured by single point diamond turning of PMMA (polymethyl methacrylate). 1.2. Surface measurement methods for aspheric surfaces Methods for the measurement of form and surface irregularity are classified into two groups, contact and non-contact methods. Contact methods, such as stylus instruments, measure the mechanical geometry and therefore the surface directly. Non-contact methods obtain the values indirectly, by measuring the optical path length or optical path difference and converting it into height values. Due to the different principles, fine surface details might vary on the same surface depending on the measurement method. It has been suggested that optical methods tend to enhance noise whereas stylus instruments tend to reduce noise, because fine details of the geometry are integrated (or even damaged) by the stylus tip [5]. Some years ago, surface measurement was exclusively two-dimensional (2D). However fine surface details (e.g. Ra-values) vary on the same surface from place to place and with the direction in which the stylus or probe is tracked over the surface. Also precision characterisation of form is difficult due to the small amount of information about form in a 2D measurement. More information may be obtained by taking several closely spaced, parallel, 2D measurements and combining them to create a 3D profile of the surface. In a 3D surface map (Fig. 1), a trace refers to a collection of data-points along the x-axis and hence, a single trace is equivalent to a 2D measurement of the surface at a certain position.

Fig. 1. Graphical definition of terms used in 3D-form measurement and data analysis.

2. Data interpretation The problems of data interpretation of surfaces that have no clearly identifiable datum become obvious when considering a relatively simple surface. The simplest data to interpret are measurements of planes and spheres. The standard equation for a plane, Eq. (1), already includes parameters to specify orientation (mx and my) and position (c) and the parameters can be easily derived by a linear least squares method. Hence no attention needs to be given to the problem of separation of geometry from orientation and position. z⫽mxx⫹myy⫹c

(1)

The standard equation of a sphere, Eq. (2), has similar benefits. Parameters to specify the centre position of a sphere (x0, y0 and z0) are easily incorporated and parameters for orientation can be neglected due to symmetry. Many different algorithms for sphere fitting exist [1,6– 8]. z⫽z0⫹(r2⫺(x⫺x0)2⫺(y⫺y0)2)1/2

(2)

Other more complicated geometries such as cylinders, cones and other aspheric surfaces are different to planes and spheres in that they require a separation of geometry from orientation and position at some point during data interpretation. 2.1. Data interpretation without pre-processing In the method for the form characterisation of cylinders and cones outlined by Forbes [6] additional parameters are used to take orientation and position of such surfaces into account. Forbes published two iterative algorithms based on the Gauss–Newton strategy, in order

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to evaluate the parameters for geometry, orientation and position simultaneously. The algorithms are specific to the particular geometry (cylinder or cone) and cannot easily be transferred to data interpretation of other surface shapes. Sourlier [9] developed an algorithm that is independent of the surface geometry. He also identified the problem of separation of geometry from orientation and position in 3D-form measurements. However, similarly to Forbes, parameters for geometry, orientation and position are derived simultaneously. This can be disadvantageous, as it sets limitations on the choice of algorithms for parameter evaluation as well as data analysis. In this paper an approach is suggested in which the separation of geometry from position and orientation takes place initially.

MSD⫽MS⫺DS

(3)

Finally, data interpretation after pre-processing allows the use of simplified surface equations for form characterisation. Simplified equations are defined as equations without parameters to describe orientation or position. They exist in a local co-ordinate system (LCS) that has the characteristics that the vertex of the surface is in the origin and the vertical axis is a symmetry axis. In BSISO 10110-12 [11] Eq. (4) is used to describe rotationally symmetric aspheric surfaces in a LCS: z⫽

冉

1 R

·r2

1 1+ 1−(1+k)· 2·r2 R

冊

⫹A4r4⫹A6r6⫹A8r8

1/2

(4)

⫹A10r10; r⫽(x2⫹y2)1/2

2.2. Data interpretation with pre-processing The pre-processing is used prior to and independent of form characterisation. This approach has some significant advantages, which are detailed below. The same pre-processing algorithm may be used with a wide range of geometries and is independent of the measurement method. Furthermore, using a pre-processing algorithm the measurement set-up can be simplified. It is not required to align a test specimen to high accuracy on a measuring table before a measurement can be taken. Instead the orientation and position of a measured surface are determined by the pre-processing algorithm and alignment takes place thereafter. After pre-processing and alignment a variety of standard data regression methods for precision form characterisation can be used. Previous solutions by Forbes and Sourlier make exclusive use of the L2 norm (based on minimising the sum of the squares) for minimisation of the sum of errors and the iterative Gauss–Newton algorithm for derivation of the best-fit parameters. Pre-processing enables the use of any norm, such as the L⬁ norm that is based on the maximum error to any data point and hence is directly related to standardised tolerancing systems (see for example BS 308-3 [10]). Also, iterative and non-iterative data regression algorithms can be used, the latter of which can be advantageous for the analysis of large data-sets due to their increased computational efficiency. A further advantage becomes apparent in view of data analysis. In BS-ISO 10110-12 [11] recommendations are made for data analysis by decomposition of the measured surface deviation (MSD) into error types. The MSD is obtained by subtracting the desired surface (DS) from the measured surface (MS) as shown in (3). A requirement is that the surfaces are aligned nominally parallel. Pre-processing enables the derivation of orientation and position parameters and hence the alignment of the surface.

459

In this equation: R represents the radius of curvature of the underlying sphere; k, the conic constant determines the nature of the basic (second order) deviation from sphericity (for example a sphere is defined when k=0, a paraboloid when k=⫺1 and a hyperboloid when k⬍⫺1); and coefficients Ai govern the magnitude of any higher order deviations from sphericity. 2.3. Relationship between a local and global coordinate system In surface metrology the global co-ordinate system (GCS) is the measurement machine’s co-ordinate system. The relation of the local to the global frame is represented by six degrees of freedom: three rotational and three translational. Pre-processing identifies the six degrees of freedom. Co-ordinate system alignment by pre-processing may be achieved through a simultaneous rotation [12] and translation of the discrete data in the GCS.→ For the rotation it is necessary to introduce an axis k of unit length and a rotation angle q (see Fig. 2). The rotation axis and the rotation angle are characterised through the orientation parameters, angles a, b and g. In this paper only rotationally symmetric surfaces are considered, hence the rotation about the z-axis (g) can be neglected → and k and q are defined through a and b only, as shown in Eqs. (5) and (6).

冤 冥冤 冥 1

冤冥 kx

→

k ⫽ ky ⫽ kz

冪

1/sin a−1 1+ 1/sin2b−1

±kx· 0

1

2

冪

1/sin2a−1 1/sin2b−1

⫽

冪

1+

±kx· 0

cosec2a−1 cosec2b−1

冪

cosec2a−1 cosec2b−1

(5)

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The algorithms are designed for pre-processing of surfaces, which are concave relative to the LCS (as shown in Fig. 4). This is not a constraint because convex surfaces can be made concave by simple data manipulation. In this work, it is assumed that the surface measurement includes the vertex (the point at which the axis of rotational symmetry meets the surface), which is used as a reference point. 3.1. Pre-processing by contour line fit (CLF)

Fig. 2. The relationship between the local and the global co-ordinate system.

q⫽arcsin

1

冪

(6)

1 1 + −1 sin2a sin2b

The matrix for rotational alignment is given in Eq. (7). The notation q⬘=90°⫺q is used and the simplified symbols v(q⬘)=1⫺cos(q⬘), s(q⬘)=sin(q⬘) and c(q⬘)=cos(q⬘) are applied. After rotational alignment of the co-ordinate systems, positional alignment is carried out through Eq. (8). Note, in the equations below discrete data in global co-ordinates are denoted xi,yi,zi. The position parameters, x0,y0,z0, locate the origin of the LCS within the GCS; the measured data after rotational alignment is denoted x˜ i,y˜ i,z˜ i; and a discrete point in local co-ordinates has the notation xi⬘,yi⬘,zi⬘.

冤冥 冤 x˜ i

kxkxv(q⬘)+c(q⬘)

y˜ i ⫽ kxkyv(q⬘)+kzs(q⬘) kykyv(q⬘)+c(q⬘) z˜ i

冥冤 冥

kxkyv(q⬘)−kzs(q⬘) kxkzv(q⬘)+kys(q⬘)

xi

kykzv(q⬘)−kxs(q⬘) · yi

kxkzv(q⬘)−kys(q⬘) kykzv(q⬘)+kxs(q⬘) kzkzv(q⬘)+c(q⬘)

(7)

zi

[xi⬘,yi⬘,zi⬘]⫽[x˜ i,y˜ i,z˜i]⫺[x0,y0,z0]

(8)

Pre-processing by contour line fit (CLF) is an iterative pre-processing algorithm for rotationally symmetric surfaces, where the analysis of contour lines is used to determine the position and the orientation of a surface. A rotationally symmetric surface has circular contour lines, which are concentric to the surface vertex. Hence, circle fitting of the contour lines can be used to identify the vertex and therefore the position parameters x0, y0 and z0. If the surface is subject to a rotation, a=b⫽90°, the contour lines are distorted. Analysis of the distortion is used to estimate the rotation of the surface. One iteration of the algorithm is outlined below. For circle fitting of the contour lines a stable algorithm is required and circle fitting with the minimum zone method is used. Nearly flat surfaces can cause the algorithm to diverge, if it is not damped to ensure that the orientation of the surface is corrected gradually. This is effected by operating on the data set by a fixed proportion of the calculated correction each iteration. start iteration Search for the lowest point in the current data set and move it to the origin of the global co-ordinate system (GCS). Slice the discrete surface n-times parallel to the xy-plane to obtain contour lines. Fit a circle through each contour and keep a record of the centre co-ordinates of the contour, together with its position on the vertical axis, the height of the contour line hCi. for i=1 to n contour(i)=slice at height(i)

3. Algorithms for the pre-processing of discrete data Two pre-processing algorithms are outlined in this paper. The algorithms can be used for the evaluation of the position parameters (x0, y0, z0) and the orientation parameters (a, b) of a rotationally symmetric surface.

[x0i,y0i]=best fit circle through contour end

冧

(9)

Fit a least squares line through [hCi, x0i] and through [hCi, y0i]. The slope of the two best-fit lines is used to define

M. Hill et al. / International Journal of Machine Tools & Manufacture 42 (2002) 457–466

the orientation parameters, α and β. The position parameters x0 and y0 are calculated by averaging all x0i and y0i respectively. Values of x0 and y0 so calculated will converge on the true values as the orientations α and β are removed from the realigned data.

冘 n

冘 n

1 1 x0= · x0i ; y0= · y0i n i⫽1 n i⫽1

Calculate the radius of a sphere to intersect the measured surface. The sphere radius is defined through the vertex (P1) and a randomly chosen point (P3). r⫽冑⌬x2+⌬y2+⌬z2

(11)

⫽冑(x1−x3)2+(y1−y3)2+(z1−z3)2

冧

tan(a)=slope of x centres relative to height tan(b)=slope of y centres relative to height

461

(10) End

Calculate a point P4=[x4,y4,z4] on the intersection line of the sphere with the measured surface that is point symmetric to P3. xdummy⫽x1⫾冑r2−(y1−y3)2−(z1−zi)2 ; dxi

(12)

⫽ABS(xi⫺xdummy)

3.2. Pre-processing by local axis search (LAS) Pre-processing by local axis search (LAS) is an iterative pre-processing algorithm for rotationally symmetric and plane symmetric surfaces. The aim is to locate the vertex and the symmetry axis of a surface, which enables calculation of the position parameters and the orientation parameters. The position parameters are determined through the vertex. The initial estimate of the vertex is identified as the point that is closest to the origin of the global co-ordinate system (GCS). This point need not be close to the vertex but is simply an arbitrary starting point. The orientation parameters are estimated from the direction of the symmetry axis. Two points are necessary to uniquely identify the symmetry axis of a rotationally symmetric surface, for example the vertex and a second point. The estimated vertex is known from above and a second point is derived as follows. The measured surface is intersected by a sphere with radius r, around the vertex. The intersection line, located in a plane perpendicular to the symmetry axis is then analysed. For rotationally symmetric surfaces the intersection line is a circle and for plane symmetric surfaces it is an ellipse. The centre of the circle or ellipse is the required second point. One iteration of the algorithm is outlined below. The vertex has the notation P1=[x1,y1,z1]. The second point to specify the symmetry axis is P2=[x2,y2,z2]. A random point chosen from the data set P3=[x3,y3,z3] is used to define r, the radius of the intersecting sphere. The discrete data from the surface measurement has the notation [xj ,yj ,zj ]. Fig. 3 shows the points used in this process. start iteration

ydummy⫽y1⫾冑r2−(x1−x3)2−(z1−zi)2 ; dyi⫽ABS(yi

(13)

⫺ydummy) Note that some dummy points need to be defined through δxj and δyj: Pa and Pb are the points for which δxj is smallest and Pc and Pd are the points for which δyj is smallest respectively, hence δxa→Pa, δxb→Pb, δyc→Pc, δyd→Pd. The dummy points are required for linear interpolation of P4

冋 冋

x4=xa+ dxa· y4=yc+ dyc·

册 册

xa−xb dxa+dxb

yc−yd dyc+dyd

z4=z1+冑ABS(r2−x4−x1)2−(y4−y1)2)

冧

(14)

Calculate the second point to identify the local axis. x3+x4 y3+y4 z3+z4 ; y 2⫽ ; z2⫽ x 2⫽ 2 2 2

(15)

The parameters for position (x0, y0 and z0) and orientation (α and β) are defined through P1 and P2 as follows: a⫽arccos

x2−x1

冑(x −x ) +(z −z ) 2

1

2

2

1

2

;

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M. Hill et al. / International Journal of Machine Tools & Manufacture 42 (2002) 457–466

Fig. 3. The key points (+, P1–P4) and the dummy points (쐌, Pa–Pd) used in each iteration of the LAS algorithm: (a) shown looking “down into” a paraboloic surface and (b) shown enlarged omitting the surface.

b⫽arccos

y2−y1

冑(y −y ) +(z −z ) 2

冋 冋

1

2

2

册 册

x0= x1−

z1·(x2−x1) ·sina z2−z1

y0= y1−

z1·(y2−y1) ·sinb z2−z1

1

(16)

2

z0=冑x21+y21+z21+x0·cota+y0·cotb

冧

(17)

End

4. Algorithm testing The pre-processing algorithms have been tested on a rotationally symmetric, aspheric surface of second order, which was generated using Eq. (18). The conic constant is set to k=⫺1, making the test surface a paraboloid. The surface is simulated on a square base of 6 mm in the xy-plane, with the vertex in the centre. The position and the orientation of the test surface are: x0=10 mm, y0=⫺4 mm, z0=1 mm, a=90.3°, and b=89.5°. The parameters are chosen arbitrarily within a range that reflects typical misalignment of a surface in this research. Three typical test surfaces are shown in Fig. 4. f(x,y)⫽

1 R

冪

1+

·(x2+y2)

1 1−(1+k)· 2·(x2+y2) R

(18)

used with: k⫽⫺1 In the algorithm testing, the form of the parabolic test

surface is altered by changing the radius of curvature. Changing the radius of curvature (R) causes the parabolic test surface to become flatter or steeper over the measured area. To test the sensitivity of the algorithms to noise, a normally distributed “white” surface irregularity with a standard deviation s is superimposed on the test surface. A normally distributed surface irregularity is characteristic of many common machining processes [13]. In addition to “white” noise, band limited noise with an appropriate 2D autocorrelation function was applied. The band-limited noise was generated by applying a 2D FIR filter to 2D white noise. Fig. 5 shows a typical 1D autocorrelation function along either a row or column of the band-limited noise and Fig. 6 shows example noise surfaces for both white and band limited noise cases respectively. Finally, the robustness of the algorithms to different grid spacings (dx and dy) was tested. It should be noted that a change in grid spacing automatically results in a change in the number of discrete points in the data set as the base (area of measurement) remains constant. The simulated results discussed below represent summaries of 60 simulations for each condition, with different noise data and different positioning of the vertex of the data set relative to the measurement grid. To provide a realistic test, it is important not to assume that a measurement point falls exactly on the vertex itself. Each simulation generated a 61×61 grid with an x and y spacing of 100 µm (i.e. covering a projected area of 6×6 mm. Examples of the simulated results are shown in Figs. 7–10. Each graph shows the mean distance of the estimated vertex position from the true vertex position (solid lines, left-hand y scale) and the mean difference between the true angular orientation and the angular orientation estimate (dotted lines, right-hand y scale). Fig. 7 shows the variation of these mean errors with form radius in the presence of 100 nm Gaussian white noise. Radii chosen for simulation were 1 mm, 3 mm and 10

M. Hill et al. / International Journal of Machine Tools & Manufacture 42 (2002) 457–466

Fig. 4.

463

Simulated parabolic test surfaces with radius of curvature: (a) r=10, (b) r=3, (c) r=1. All dimensions and axes in mm.

Fig. 7. Means of error magnitudes of the estimated position of the vertex (marked P Err) and the estimated angular orientation (marked A Err) for the two algorithms. Errors are plotted against increasing form radius, and Gaussian white noise with a standard deviation of 100 nm is superimposed on the form.

Fig. 5. Typical autocorrelation plot for a row or column of the band limited 2D noise.

mm. The CLF algorithm provides acceptable results in all cases, but its accuracy decreases significantly as the form radius increases. This is due to its reliance on fitting through circular sections through the parabola at a distance from the nominal vertex. The reliability of this process reduces as the data profile becomes flatter — that

Fig. 6.

is the radius increases while the measurement area stays constant, or the measurement area increases while the radius is held constant. Conversely, the LAS algorithm is unstable for very high curvatures, but approaches the accuracy of the CLF algorithm at higher radii (note that for low radii, the LAS errors are off the scale of the graph). Fig. 8 shows the sensitivity of the algorithms to Gaussian white noise, and it can be seen that (at a radius of 10 mm) the LAS algorithm shows an improved performance over the CLF algorithm at higher noise levels.

White random noise and filtered band limited noise surfaces.

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Fig. 8. Means of error magnitudes of the estimated position of the vertex (marked P Err) and the estimated angular orientation (marked A Err) for the two algorithms. Errors are plotted against increasing magnitudes of Gaussian white noise superimposed on a form of nominal radius 10 mm.

a small radius of curvature. The deviation of the derived position and orientation parameters increases as the radius of curvature increases. On nearly flat surfaces the algorithm may diverge. Repetition of the tests with the same base area but a finer grid spacing has little influence under most conditions, but does improve estimates when pre-processing flatter surfaces. Pre-processing by local axis search is unstable for low radius forms (relative to the base area and the grid spacing), but may provide better estimates than the CLF algorithm for relatively flat surfaces. Tests with finer grid spacing improved estimates for small radius forms significantly. The LAS algorithm shows greater stability with respect to high magnitudes of surface irregularity.

5. Data interpretation of an aspheric lens surface

Fig. 9. Means of error magnitudes of the estimated position of the vertex (marked P Err) and the estimated angular orientation (marked A Err) for the two algorithms. Errors are plotted against increasing form radius, and band limited Gaussian noise with a standard deviation of 100 nm is superimposed on the form.

Figs. 9 and 10 use the same parameters as Figs. 7 and 8, but in these cases the noise is band-limited (correlated). The trends seen in the figures are the same, but it can be seen that the CLF algorithm is rather more sensitive to band limited noise than is the LAS algorithm. In general, pre-processing by contour line fit is suitable for steep aspheric surfaces and hence surfaces with

Fig. 10. Means of error magnitudes of the estimated position of the vertex (marked P Err) and the estimated angular orientation (marked A Err) for the two algorithms. Errors are plotted against increasing magnitudes of band limited Gaussian noise, superimposed on a form of nominal radius 10 mm.

After testing of the proposed pre-processing algorithms on simulated data, an example is now given of their application, enabling an aspheric lens surface to be analysed and decomposed into different error types. A surface measurement of an aspheric lens has been taken on a stylus based Form Talysurf. The grid spacing in x is dx=200 µm and the spacing in y is dy=130 µm. A graph of the full measurement (164×164 points) is shown in Fig. 11. The surface has the same shape as the simulated data used in the previous section with a 10 mm radius, but a larger base area, making it suitable for analysis with the CLF algorithm. For data interpretation a reduced data-set (100×154 points) is used in which unwanted sections such as the flat areas at the beginning and the end of each trace are cut off as shown in Fig. 11. The analysed area is approximately 20×20 mm.

Fig. 11. Surface measurement of an aspheric lens surface. The graph shows the surface as obtained from the measurement system, before pre-processing. Only the marked area is analysed.

M. Hill et al. / International Journal of Machine Tools & Manufacture 42 (2002) 457–466

465

5.1. Results of data interpretation The application of the CLF pre-processing algorithm to the measured surface shows the orientation to be: a=90.24° and b=90.46°. The position of the vertex is estimated to be at: x0=16.312 mm, y0=10.978 mm and z0=3.422 mm. For form characterisation the best-fit parameters of a 10th order polynomial as shown in Eq. (19) have been identified by use of a linear least squares method. The approach follows the steps as described in McBride et al. [4]. z⫽A2r2⫹A4r4⫹A6r6⫹A8r8⫹A10r10 ; r⫽(x2⫹y2)1/2

(19)

Fig. 12. Symmetrical form error (SFE) of the surface, defined by subtracting a best fit 10th order polynomial from the desired theoretical surface (SFE=DS⫺BFM).

5.2. Decomposition of the data into error types Decomposition into error types is required to identify surface errors. Often it is possible to relate symmetrical form errors to wrong tool settings while a worn cutting tool can cause asymmetrical surface irregularities. Ideally the errors are characterised in a way that will allow the likely cause of the errors to be examined. This can give useful feedback for the manufacture and eventually result in the optimisation of a manufacturing process. One method for decomposition into error types is outlined in BS-ISO 10110-5 [14]. The recommendation is to first calculate the measured surface deviation (MSD) as previously defined in (3). The MSD is then separated into an approximating spherical surface (ASS) and a total irregularity function (TIF), hence: MSD⫽ASS⫹TIF

(20)

The TIF consists of two parts, a rotationally symmetric part and an asymmetric irregularity (AI). The rotationally symmetric part is defined through the approximating aspheric surface (AAS). TIF⫽AAS⫹AI

(21)

Using this decomposition method the symmetrical form error (SFE) of the measured surface can be defined as follows: SFE⫽ASS⫹AAS

Fig. 13. Asymmetric irregularity (AI), defined by subtracting a best fit 10th order polynomial from the measured surface (AI=MS⫺BFM).

The alternative decomposition method has the advantage that only one fitting process is required. The BS-ISO approach requires two fitting processes, one to compute the ASS (see relationship (20)) and one to compute the AAS (see relationship (21)). Also, surface irregularities are more critical in the BS-ISO approach. For computation of ASS or AAS a best-fit through a surface with high magnitude of surface irregularity relative to form is required. Therefore, the fitting algorithms have to be very robust. A comparison of the two decomposition methods is the subject of current research.

(22) 6. Conclusions

An alternative method for decomposition is suggested in this paper. Instead of computing a MSD, the best-fit through the measured surface (BFM) is derived and is used to calculate the SFE (Fig. 12) and AI (Fig. 13) as follows: SFE⫽DS⫺BFM

(23)

AI⫽MS⫺BFM

(24)

Two pre-processing algorithms have been described which are able to estimate the position and the orientation of rotationally symmetric surfaces of second and higher order. Instead of evaluating parameters for geometry, orientation and position simultaneously, preprocessing enables the task to be split into two: first determining the orientation and position second

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determining parameters for a best-fit geometry. This approach has several advantages over previous solutions. The main advantage is the independence of pre-processing from a specific geometry or measurement method. Furthermore, due to the split task, alignment of discrete data is possible before form characterisation takes place, allowing for the use of simplified equations and various data regression methods, iterative or noniterative in nature, which has not been possible before. Pre-processing and alignment of a discrete surface is not only advantageous but it is sometimes necessary before it is possible to proceed with data interpretation. The algorithms have been tested for their proficiency on simulated data. Test data were generated to study the reliability of the algorithms with respect to data representing differing forms, data with varying distributions of points and data with widely differing magnitude of surface irregularity superimposed on the nominal surface geometry. Based on the observations of algorithm testing it is recommended to use pre-processing by contour line fit to derive the orientation and position parameters for surface alignment, whenever possible. For nearly flat surfaces pre-processing by local axis search is the recommended algorithm. Classifying a surface as nearly flat depends mainly on the radius of curvature in relation to the sample area and also on the magnitude of irregularity that is superimposed on the surface. The pre-processing has been applied to the surface measurement of an aspheric lens and data interpretation has been carried further to precision form characterisation and decomposition of the surface into error types. Acknowledgements The authors would like to thank Ocular Sciences, Ltd. for the financial support of the project and Optics and

Vision, Ltd. for providing aspheric lenses for measurement purposes and lens surface details.

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