Computer aided error analysis for a threedimensional precision surface mapping system

Computer aided error analysis for a threedimensional precision surface mapping system M. Hill, J.W. McBride, D. Zhang & J. Loh Mechanical Engineering, U~riversity of Southampton, UK Abstract This paper describes the development of a three-dimensional (3-D) non-contact surface mapping system. Both systematic and random sources of error are discussed, including thermal, mechanical and electrical errors. Simulation has been used to model some of these errors. Measurement of calibrated spheres has been adopted for the calibration of the 3-D coordinate measuring system. Several sphere fitting parameters, including radius, sphericity and residual error, are employed as references to judge the errors of 3-D measurement system. Uncertainty in the radius is inspected to evaluate the random error. Deviation of mean radius, sphericity and residual errors are used to estimate the orthogonality between the axes. The technique provides a simple, low-cost method for estimating the errors present in such a measurement system. 1 Introduction Three-dimensional metrology has brought about a significant change in dimensional measurement. It provides more comprehensive and detailed information about geometry than 2-D measurement methods. On the down side, 3- D measurement systems are more complicated in their mechanical structure, error sources and data processing. This paper describes an analysis of the performance of a coordinate measurement system (CMS) in terms of error analysis and calibration. The accuracy of a measuring instrument is determined by many factors, such as the geometric error of the structure, the deformation caused by the movement of the carriage, temperature variations and sensor errors etc. Thus, the quality of the performance of a CMS depends on the assembly and calibration processes. During assembly, key factors include the perpendicularity of each pair of axes and the displacements of driving shafts [l]. Many different methods for the calibration of

280 Laser \tetrolog arid Uacl7m Performonce such an instrument have been adopted by researchers, including interferometry and the use of grid plates [2][3]. The 3-D precision non-contact surface mapping system (PSMS) currently being developed at the University of Southampton adopts a laser confocal sensor, which possesses high accuracy, high stability and the ability to measure transparent film [4]. As with any coordinate measurement system, a large number of factors make a significant contribution to the measurement results. An approach has been developed for initial calibration, which involves numerical simulation based on a mathematical model of the system behaviour, referred to as computer aided error analysis and calibration. The aim is to verify the system accuracy and guide the assembly of the system. The approach avoids the need for additional high precision instrumentation. 2 Error sources within the PSMS The PSMS consists of three precision stages, providing movement of a measured sample in the X and Y directions, and the movement of the laser sensor in the vertical axis. The X-axis stage is mounted above the Y-axis stage, which in turn is mounted on the base breadboard. These two stages provide the horizontal motion with encoders having a resolution of up to 0.02pm. A confocal laser sensor is used to measure the Z coordinate of the measured sample over a two-dimensional (2-D) X-Y grid. Based on the structure of this system, the factors affecting the geometric accuracy can be summarised as follow [5]: (i) Errors due to mechanical wear, which introduce unwarranted degrees of freedom, such as lack of straightness and squareness of guides, etc. (ii) Deformation caused by the weight of the structure. (iii) Errors in measuring system, which can be mechanical, thermal or electrical in nature, such as the sensor accuracy, drive system accuracy, stage resolution. (iv) Thermally induced structural changes. Considering the four error sources above, item (i) could lead to orthogonality errors between X- and Y-axis and position errors, item (ii) could induce a tilt error in the Z-axis relative to the XY-plane, whereas items (iii) and (iv) could result in random errors. The observed errors at the probing points of the measurement system can be classified as systematic and random. It is assumed that all errors are random, except for the orthogonality error between the axes which is the most significant systematic error. 3 Theoretical analysis of error calibration Calibrated spheres are widely applied to the calibration of 3-D co-ordinate measurement systems and their probes [6]. In this paper, computer aided error analysis and calibration has been conducted by measuring a calibrated sphere. In the calibration, four parameters are examined to characterise the errors of the measurement system:

(i) Deviation of radius: the difference between mean of fitting radius and nonllnal one. (ii) Uncertainty of radius: an index to express the reliability of estimated radius. It can be expressed by the estimated standard deviation. (iii) Sphericity: the radial difference between two concentric spheres that circumscribe and inscribe the form error. It is related to tolerance zone [7]. (iv) Residual error: error after the best-fit sphere is removed. 3.1 Orthogonality error between X and Y axes One potential source of inaccuracy is the orthogonality of the X-Y stage motion. If the X and Y stages are misaligned, the actual measured grid will be skewed as shown in Figure 1. When the measured points from the skewed grid are mapped onto an ideally orthogonal grid, the sphere data will be distorted. In order to investigate the effect of this error, computer simulation is carried out. The simulation assumes that the error of the angle between the X- and Y-axis (referred to as the orthogonality error) is 0 and the calibrated sphere is measured with ths measurement system. Residual error caused by orthogonality error (Q = l") is shown in Figure 2. From this figure, it can be shown that the error is symmetrical about the sphere centre, and that with the increase of scan area (and subsequently the measured segment angle), the amplitude of the residual errors increases. Hence, the following simulations were carried out over various segment angles in order to judge the influence as the measured area increased. Skewed gnd and ------Square gnd and JIOZ-- " 5 2 actual sphmcal data mapped data 15-~2 Figure 1 : The effect of mapping a Figure 2: Residual error caused by sphere from a skewed grid orthogonality error (B = 1 ") Obviously, thls geometric error causes a deformation of the measured data of a spherical surface. Computer simulated results in Figure 3(a) show that this error causes the deviation of fitting radius from the nominal value. Two more factors: the sphericity and residual error were investigated along with the deviation in the fitting radius. In Figure 3(b) and (c), an almost linear relationship in both sphericity and residual errors can be observed with increasing errors in the orthogonality of the axes and the measured segment angle. The knowledge of the

orthogonality error on these three factors will enable prediction of the measurement system behaviour based on experimental results. -+ 0 5 degree --c 1 degree -A- I 5 degree 10 20 30 40 50 10 20 30 40 50 (a) Deviation of radius (b) sphericity 10 20 3 0 4 0 5 0 (c) residual error Figure 3: Simulation results with different orthogonality error 3.2 Tilt of Z-axis to XY-plane The error caused by the tilt of the Z-axis to the XY-plane (referred to as tilt error) has influences on form measurement of severely curved surfaces, which is much more significant than it is on flat or nearly flat surfaces. Assume that a calibrated sphere is measured under a 3-D CMS with a tilt error of p. Figure 4 illustrates a cross-section of a sphere, which includes the ideal and actual probe direction. When the probe moves from position A to B, the actual displacement should be from B to K. However, with a tilt error of 9, displacement is now taken from B to C. The error induced by the tilt error is given as S z-il it = BC-BK. S can be derived from the geometric relationship in Figure 4, as shown in the following equations. From equation (l), SL-fll, can be shown as a function of angle p and probe position on spherical surface and is illustrated in Figure 5.

Figure 4: geometry relationship of tilt Figure 5 : illustration of error on a sphere surface (p = 1 O) y =90 -y, OH = Rtg p + AB OH OC R sin a sin y cos p OH sin a = - cos y, R Rtg y, + AB a = arcsin( cos P R /? =90m-(180 "-a-y)=a-p EC = R sin /? = R sin( a - p) DC EC-AB Rsin(a-p)-AB BC =-- - - sin p sin p sin p BK = AF = R - J R ~ - A B ~ R sin( a - p ) - AB 6,_,,,, = BC - BK = - R - J R ~ - A B ~ (1) sin p The influence of the tilt error on sphere measurement has been investigated using simulation. The results in Figure 6 show that the tilt error mainly affects the deviation of radius from its nominal value. It can be seen that as the tilt error increases, the fitting radius deviates more significantly. With the sphericity and residual error shown in Figure 6(b) and (c), the influence of the tilt error is not as apparent as that due to the orthogonality error. The magnitudes of the sphericity and residual error are much smaller when compared to results shown in Figure 3(b) and (c). Similarly, the knowledge gained here will once again be useful in the prediction of the measurement system behaviour.

I degree -A- l 5 degree -7 284 Luser Zletr,ology und Ifuch~ne Performance 10 20 30 40 50 10 20 30 40 50 (a) Deviation of fitting radius (b) Sphericity 10 20 3 0 4 0 50 (c) Residual error Figure 6: Simulation results with different tilt error 3.3 Analysis of Random Error 3.3.1 Investigation of random error Figure 7: Residual error Figure 8: Histogram of residual e&r Random errors exist in any measurement system and can be produced by many possible sources of error, as mentioned in section 2. Residual error, after the best fitting sphere has been removed, from the measurements of a 22.00064mm calibrated sphere (100x 100 grid points) are presented in Figure 7. The histogram

Laser \ Ietroiop und \/achrne Per/ormnnt e 285 in Figure 8 shows that the residual error approximates to a normal distribution with a standard deviation of 0.54pm. On this basis, the random error of the system are assumed to be close to a normal distribution 3.3.2 Measurement uncertainty Random errors result in an uncertainty in the measurement results. The uncertainty is a strong function of sampling strategy [S]. The sampling strategy can be broken into three components: the size; mode; and the number of points [g]. For sphere measurement, the sampling size corresponds to the measured segment angle. All measurements have been taken using a uniform grid-sampling mode. To investigate the uncertainty of measured radius, simulations were carried out using different segment angles and number of sampling points. Two reference spheres of radii 7.00008rnm and 22.00064rnm were used. Results obtained from the two spheres were identical, which suggested that the measurement uncertainty is independent of the radius. The simulations were repeated over 100 times with different measured segment angles and numbers of sampling points. Assuming the standard deviation of the instrumental random error and the fitting radius are o and o~ respectively, the ratio of or/ois known as the error propagation factor [6]. A mathematical description of the relation between o~ and o can be expressed as Equation (2) and referred to f(n, ryl as error propagation factor function: where OR =f(n, 0 (2) n - the line number of sampling grid. W- segment angle of measured sphere. 0 20 40 60 80 100 30 50 70 90 segment angle (degree) segment angle 230 (degree) Figure 9: Error propagation coefficient The error propagation coefficients obtained from simulation results are presented in Figure 9, which is approximated by following exponential function: f(n,y)= k,.vk2 (3) where k, and k2 depend on the number of sampling points (nm), which are presented in Figure 10 and approximated by following equations.

286 Laser. Iletrolop und \luchine Pe~:for.mance 0 20 40 60 80 0 20 40 60 80 Sampling grid line Sampling grid line Figure 10: Presentation of coefficient kl and k2 From the above analysis, the error propagation coefficient can be used for: Characterisation of the random error of the measurement system By repeatedly measuring a calibration sphere with a fixed segment angle and number of sampling points, calculating the standard deviation of fitting radius, the standard deviation of the random error of the measurement system can be derived by the error propagation function. Determination of the error limit for single measured result If a sphere were measured using a measurement system with known random and negligible systematic error, the standard deviation or error limit of the fitting radius could be determined by the error propagation coefficient function. 4. Evaluation of measurement system error Measurements of a calibrated sphere (r=7.00008mm) have been taken on the PSMS. These results were deemed to have included the systematic and random error of the measurement system. Using the results, comparisons can then be made with simulated measurements with assumed errors. The measurements were repeated 50 times, scanning a square area of 3mm and 15x15 data points. The results were presented in Table 1. From the experimental results in Table 1, the mean and standard deviation of the radius estimates indicate that random and systematic errors exist in the measurement system. Assuming that there are only random errors in the measurement system, the standard deviation of random error can be derived from Equation (2) and o = o, l(k,tyk2) = 0.3prn.

Table 1 : Example comparisons of measured and simulated results I I Mean I STD of ( Mean / Residual ( Results Experiment r 4 Simulation 1 Radius (mm) 7.0024 Radius (ar) (pm) 7.1263 Sphericity (pm) 4.5270 Error (W) 0.6739 7.0000 7.0267 1.5373 0.2368 6.9999 8.1 309 4.4633 0.6938 Simulation 3 7.0024 8.0149 4.4733 0.6953 Simulations were carried out with assumed values of error after detailed analysis of possible error angles. Three example simulations are shown in Table 1. In Simulation 1, the measurement system was simulated with random errors having a standard deviation of 0. 3 but ~ no ~ systematic errors at all. The results obtained were then compared to the actual experimental data. It can be seen in Table 1 that only standard deviation of radius from both experiment and simulation coincides. The three other factors were found to possess significant differences. These differences indicated that systematic errors are likely to be present in the measurement system. Simulation 2 was performed with assumed systematic errors. The orthogonality error was set at 1" with no tilt error in the system. Results from Simulation 2 showed that the standard deviation, sphericity and residual error were found to be similar to the experimental results. However, the mean of radius was still slightly different compared to the experiment. In Section 3.2, it was found that as the tilt error gets larger, the fitting radius deviates more significantly. In both Simulation 1 and 2, no tilt error was included and the mean radius obtained were found to be quite identical. The tilt and orthogonality errors of 1.5" and 1" respectively were thus included in the simulated measurement system. The results obtained from Simulation 3 were found to be close to the measurements taken from the experiment. These results showed that random and systematic errors do exist in the measurement system. On the basis of the simulations the errors are estimated to be as follows: Random error (a) under 0.3 pm Orthogonality error (8) : under 1' Tilt error (p) under 1.5' Work is currently under way to improve the measurement set-up and to verify these error estimates.

288 Laser Zletroloe and 2/achrne Performance 5 Conclusion This paper has discussed how the magnitude and nature of errors in an instrumentation system can be estimated from the measurement of a calibration sphere. Using results from simulations, the random and systematic errors in an actual measurement system can be analysed and predicted. This method is simple and easy to realised without any attached instruments. It also provides a basis for the numerical simulation of a CMS based on a mathematical model of system behaviour. References [l] Matsuda, J., Nakaya, T., The Assembly/ Adjustment of coordinate measuring machine and the analysis oferror, Int. J. Japan Soc. Proc. Eng., Vol. 25 No.2 pplll-117. 1991. [2] Zhang, G., Error compensation of coordinate measurement machines, Annals of the CIRP, Vol. 34, pp445-448, 1985. [3] Zhang, G., Fu, J., A method for optical CMM calibration using a grid plate, Annals of the CIRP Vol49, pp399-402, 2000. [4] McBride, J., Hill, M., Loh, J., Sensitivity ofa 3-0 surfice mapping system to environmental perturbation, To be published in LAMDAMAP, 2001. [S] Burdekin, M., Voutsadopoulos, C., Computer aided calibration of the geometric errors of multi-axis coordinate measuring machines, Proc Instn, Mech Engrs Vol 195, pp23 1-239, 1981 [6] Bourdet, P., Lartigue, C., and Leveaux, F, effects ofdata point distribution and mathematical model on finding the best-fit sphere to data, Precision Engineering, Vol. 15, No.3, pp150-157, July, 1993. [7] Kanada,T., Evaluation of spherical form errors-comptctation of sphericity by means of minimum zone method and some examinations with using simulated data, Precision Engineering, Vol. 17, pp28 l-289,1995. [8] Phillips, S.D., Borchardt, B. Estler W.T., and Buttress, J., The estimation of measurement uncertainty of small circular features measured by coordinate measuring machines, Precision engineering, Vol., 22, pp87-97,1998. [9] Choi, W., Kerfess, T., Jonathan C., Sampling uncertainty in coordinate measuring data analysis, Precision Engineering, Vol. 22, pp. 153-163, 1998.