Geometrical error compensation of machines with significant random errors

ISA TRANSACTIONS® ISA Transactions 44 共2005兲 43–53

Geometrical error compensation of machines with significant random errors Kok Kiong Tan, Sunan Huang Department of Electrical and Computer Engineering, National University of Singapore, Singapore, Singapore

共Received 6 January 2003; accepted 17 May 2004兲

Abstract In this paper, we present a new statistical approach towards soft geometrical error compensation of machines with significant errors. The approach, based on an analysis of the probability of the random error recurring, can reduce the adverse influence of random errors on the compensation of systematic errors. The proposed methodology is made up of three steps. First, error classes are defined from the error bands obtained from calibration. Second, the probability of the magnitude of random error belonging to each of these classes is computed based on the density of the data set within the class. Based on these probabilities, the most probable systematic part of the error measurement can be statistically deduced. Finally, the geometrical error compensation is carried out based on this value. Experimental results are provided for the linear error compensation of a single-axis piezo-ceramic motion system. © 2005 ISA—The Instrumentation, Systems, and Automation Society. Keywords: Compensation; Geometrical error; Laser interferometer; Position error

1. Introduction In automated positioning machines such as coordinate measuring machines 共CMM’s兲 and machine tools, the relative position errors between the end effector of the machine and the workpiece directly affect the quality of the final product or the process concerned. These positioning inaccuracies arise from various sources, including static/ quasistatic sources such as geometrical errors from the structural elements, tooling and fixturing errors, thermally induced and load induced errors, and also dynamic ones due to the kinematics of the machine 关1–5兴. These errors may be generally classified under two main categories: systematic errors which are completely repeatible and reproducible and random errors which vary under apparently similar operating conditions. Several CMM manufacturers reported that geometric error compensation by software techniques has pro-

duced a reduction of production costs, which they estimate to be between 5% and 50% 关6兴. While widespread incorporation of error compensation in machine tools remains to be seen, the application in CMM’s is tremendous and today it is difficult to find a CMM manufacturer who does not use error compensation in one form or another 关7,8兴. Compensation for geometrical errors in machines is not new. The development in error compensation is well documented by Evans 关9兴. Early compensation methods mainly utilized mechanical correctors, in the form of leadscrew correctors, cams, reference straightedges, etc. Compensation via mechanical correction, however, inevitably increases the complexity of the physical machine. Furthermore, mechanical corrections rapidly cease to be effective due to mechanical wear and tear. The evolution of control systems from mechanical and pneumatic-based subsystems to microprocessor-based systems has opened up a

0019-0578/2005/$ - see front matter © 2005 ISA—The Instrumentation, Systems, and Automation Society.

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Kok Kiong Tan, Sunan Huang / ISA Transactions 44 (2005) 43–53

Fig. 1. Illustration of setup.

wide range of new and exciting possibilities. Software-based error compensation schemes thus blossomed in the 1970s. The first implementation was on a Moore N.5 CMM, a pioneering piece of work which earned Hocken the CIRP Taylor Medal in 1977. Since then, there has been an explosion of interests in soft compensation of machine errors with new methods developed and implemented 共see Refs. 关7,10–16兴兲. The effectiveness of any soft error compensation scheme highly relies on the reliability of the geometrical error information obtained from the prescribed calibration methodology. By error compensation, we are referring to the systematic component of the geometrical error. Errors of a random nature cannot be compensated. Unfortunately, the error measurements obtained during the machine calibration will inevitably contain both systematic and random components. It is essential to effectively separate the two components in order to correctly compensate for the systematic error. Most of the existing error compensation methods do not adequately address the influence of random errors on the compensation of the systematic errors. The geometrical error to be compensated is usually based on the mean value of the error samples calibrated. This mean value is essentially assumed to be the equivalent systematic error component which can be compensated, since the random ones are assumed to be filtered after the averaging process. Unfortunately, this is not true in general and compensation based on the mean value can lead to a grossly inadequate compensation, especially when there exists significant random errors during the calibration process. Invalid samples or outliers arising due to short and momentary disturbances or noise during the calibration exercise can distort the mean to deviate far from the actual systematic error present. Consequently, using the error mean can lead to insufficient or excessive compensation. Thus, a weighted approach taking into account the

probabilities associated with different magnitudes of the random error component in the error sample can extract a more accurate and reliable data set of systematic errors for compensation. Such a statistical approach requires a large set of samples to be calibrated, in order to have a sufficiently large data density to operate on. This can be easily facilitated by modern automated and DSP-based data logging and analysis systems. In this paper, we explore the use of a statistical approach to reduce the adverse influence of random errors, as an inevitable part of the calibrated data set, on the compensation of systematic errors. The basic idea is to deduce and isolate the most likely systematic geometrical error components from a data set which is infiltrated with random ones, by appropriately analyzing the probability of the magnitude of random errors. The approach is simple and directly amenable to practical applications. It consists of three main steps. First, from the geometrical error information collected, the error band is split into small consecutive classes. Second, the probability of the random error falling into each class is calculated based on the density of data clusters falling within the class. These probabilities are used in a statistical analysis to deduce the most probable systematic error from the data set. Finally, compensation is done based on this statistically deduced error. The proposed statistical approach has the advantage that the influence of random errors, including those arising due to intermittent faults, noise and disturbances, can be largely removed and isolated from adversely affecting the compensation effort. This ad-

Fig. 2. Experimental setup.

Kok Kiong Tan, Sunan Huang / ISA Transactions 44 (2005) 43–53

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Fig. 3. Raw data set for linear errors.

vantage becomes significant when the precision requirements of the machine get correspondingly higher. 2. Proposed methodology We define the random error ␦ ( x ) as the deviation of the actual error measurement from the mean computed over the entire data set. Mathematically, this can be expressed as

␦ 共 x 兲 ⫽⌬y 共 x 兲 ⫺⌬y¯ 共 x 兲 ,

共1兲

where ⌬y is the actual error measurement, and ⌬y ¯ is the aforementioned mean computed from the data set, i.e., n

1 ⌬y ¯ ⫽ 兺 ⌬y (k) . n k⫽0

共2兲

n is the total number of measurements available. In general, the expectation E 关 ␦ 兴 ⫽0. It is well-known that only systematic errors can be compensated. Random errors which can possibly occur during the calibration process should be minimized ideally by a proper system design,

since they cannot be compensated. The main difficulty, which is being addressed in this paper, is not to attempt to compensate for random errors, but to reduce the influence of the random errors from adversely affecting the compensation of the systematic errors, since both the systematic and the random component co-exist in the same error measurement. The challenge is how to effectively distillate the two components so that compensation can be based purely on the systematic error only. ¯ to make the final comMost approaches use ⌬y pensation. In these approaches, ⌬y ¯ is assumed to be free of the influence of random errors and thus, it is taken to be equivalent to the systematic component which can be compensated. Unfortunately, these methods will yield a grossly inadequate compensation when a significant amount of random errors is present in the error measurements. Invalid samples or outliers arising due to short and momentary disturbances or noise during the calibration exercise can distort the mean to deviate far from the actual systematic error present. Consequently, using just the sample mean can lead to insufficient or excessive compensation. Thus a

Table 1 Mean errors. Sample points 共mm兲 Error mean ⌬y ¯ 共␮m兲

5 1.1 1.1

10

15

20

25

30

35

40

45

50

2.2

3.4

4.9

6.0

7.8

8.9

10.7

12.7

14.0

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Fig. 4. Error probability analysis 共5–20 mm兲.

weighted approach to filter the influence of random errors using a statistical analysis may be used to extract a more accurate and reliable data set of systematic errors for machine compensation. To this end, we will first split the error band into subclasses. Consider a one-dimensional space that consist of 2k⫹1 error classes s j , j⫽0, ⫾1, ⫾2, . . . , ⫾k. The random error ␦ defined in Eq. 共1兲 is assumed to fall into the s j class if

P共 s j 兲⫽

共 2 j⫺1 兲 L⬍ ␦ ⭐ 共 2 j⫹1 兲 L,

j⫽0,⫾1,⫾2, . . . ,⫾k,

Thus the actual class size used should depend on the resources available in the acquisition of the raw data set. The probability of a random error with a magnitude falling into each of these classes can be computed from the number of error samples within the class. If the class s j contains n j random error samples, then

共3兲

where L is the class size. While we would like the class size to be as small as possible to maximize the resolution of errors across the classes, a small L will correspondingly requires a large data set.

nj . n

共4兲

This definition of probability clearly satisfies the axioms of the axiomatic theory when n→⬁. Ideally, if ⌬y ¯ is an adequate representation of the systematic component of the geometrical error, we expect most random errors to be contained in the set s 0 , i.e., the random error class containing

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Fig. 5. Error probability analysis 共25– 40 mm兲.

␦ ⫽0 should be registered with the highest probability. Otherwise, any bias phenomenon in the random error should be used to offset the error mean for a more accurate reconstruction of the

systematic error component. This mean offset ¯␦ can be obtained from the probabilities associated with the various error classes 关 P ( s j ) , j⫽0,⫾1, ⫾2, . . . ,⫾k] as

Fig. 6. Error probability analysis 共45–50 mm兲.

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Fig. 7. Probability analysis after compensation with mean value 共5–20 mm兲.

¯␦ ⫽ 兺 ˜␦ j P 共 s j 兲 , j

共5兲

where ˜␦ j ⫽2 jL refers to the center of the s j class, i.e., „( 2 j⫺1 ) L, ( 2 j⫹1 ) L…. The ‘‘most likely’’ systematic error, in a statistical sense, can be thus ¯ ⫹¯␦ . computed to be ⌬y The procedure to implement the proposed method is described below: 1. Split the error band into subclasses, 2. Calculate the random error probability of each subclass according to Eq. 共4兲, 3. Compute the center of the ith subclass according to Eq. 共5兲, ¯ ⫹¯␦ . 4. Set the error compensation value ⌬y Finally, the geometrical error of machine can be compensated based on this statistically deduced systematic error.

3. Experiments In this section, experimental results are provided to illustrate the effectiveness of the proposed method when applied to linear error compensation of a linear piezo-ceramic motion system. To do this experiment, it first needs to define a zero point. Usually, the index is defined as zero point. Second, the laser interferometer is placed on the probe or payload. Third, the reflector is placed closely to the end of the linear motor as shown in Fig. 1. The controller is required to follow the step by step based on a constant interval. Fig. 2 shows the photo of the experimental setup. The piezo-ceramic motor used is a single axis servo motor manufactured by Nanomotion Ltd., which has an optical encoder with an effective resolution of 0.1 ␮m. The main sources of random errors arising in this application include the vary-

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Fig. 8. Probability analysis after compensation with mean value 共25– 40 mm兲.

ing contact friction conditions, the stick-slip effect, clearances in shift mechanisms, deformations of bearing elements, transients from the sensor electronics, changes in ambient temperature, and even minor structural vibrations.

DSPACE control development and rapid prototyping system, in particular the all-in-one DS1102 board, is used for both calibration and compensation purposes. DSPACE integrated the entire development cycle seamlessly into a single environ-

Fig. 9. Probability analysis after compensation with mean value 共45–50 mm兲.

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Fig. 10. Probability analysis after proposed compensation 共5–20 mm兲.

sured. Fig. 3 shows the raw data set of linear errors collected from 40 complete bi-directional travel, i.e., n⫽40. The average values at each of these points from the 40 samples are computed and tabulated in Table 1. The combined error data band is split into subclasses with L⫽2 ␮ m and the probability of the random error component falling within each class computed accordingly. Figs. 4 – 6 show a clear bias phenomenon in the random error from zero. The probability of random error falling in class s 0 is lower than that in class s 1 in eight out of the ten

ment, so that individual development stages between simulation and test can be run and rerun, without frequent re-adjustment. MATLAB and SIMULINK can be integrated directly with DSPACE to constitute a powerful real-time data acquisition and analysis system. The error compensation is carried out in the form of a look-up table embedded into a SIMULINK control block. Calibration is carried out using a laser interferometer at specific points located at 5-mm intervals along the 50-mm travel. Hence the linear errors at ten points along the linear piezo motor are mea-

Table 2 Mean offsets. Sample points 共mm兲 Mean offset value ¯␦ 共␮m兲

5 2

10 2

15 0

20 2

25 1

30 0

35 2

40 2

45 1

50 2

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Fig. 11. Probability analysis after proposed compensation 共25– 40 mm兲.

calibration positions. This indicates that the sample mean does not adequately reflect the actual systematic error present. Compensation is then done based on these mean values. Re-calibration is carried out after compen-

sation at the same positions, and the experimental results are shown in Figs. 7–9. As expected, from these figures, it is evident that after compensation, the ‘‘likely’’ random error class continues to exhibit a strong bias from class s 0 .

Fig. 12. Probability analysis after proposed compensation 共45–50 mm兲.

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Using the proposed method, the mean offset (¯␦ ) are computed and tabulated in Table 2. These values are used to adjust the mean values in Table 1 for compensation. Calibration is again carried out at the same ten positions after the proposed error compensation. The random errors arising after compensation are shown in Figs. 10–12. Comparing these with Figs. 7–9, the bias phenomenon has been reduced significantly, despite there is a little worse result in x⫽5 mm and x⫽10 mm 共this shows the statistical probability is not perfect in each class; this also implies that furher study should focus on an optimal probability representation兲. The proposed method has achieved a more accurate compensation of systematic errors with respect to the influence of random errors. It is not suprised for this the result. Let us explain why. The conventional mean value method only considers the systematic error appearing in machine. However, the random error always exists in the machine. The proposed method combines both the mean value and the random error together for the error compensation. Thus the result is better than that of the conventional method. For other robust statistics methods, a comparison would be good, but the work to do so is beyond the scope of this project. 4. Conclusion Soft compensation of systematic geometrical errors has the main difficulty that the error measurements contain both systematic and random components. While the random errors cannot be compensated, inadequate treatment of their influence can lead to an inaccurate compensation of the systematic error. A new statistical approach towards soft geometrical error compensation of precision machines has been developed in this paper. The approach, based on an analysis of the probability of the random error recurring, can reduce the adverse influence of random errors on the compensation of systematic errors. The proposed methodology is made up of three steps. First, error classes are defined from the error bands obtained from calibration. Second, the probability of the magnitude of random error belonging to each of these classes is computed based on the density of the data set within the class. Based on these probabilities, the most probable systematic error can be statistically deduced. Finally, the geometrical error

compensation is carried out based on this value. Experimental results are provided on the linear error compensation of a single-axis piezo-ceramic motion system. Consistent with expectations, they show that the compensation error obtained statistically according to the proposed method is quite different from the error mean of conventional methods. The significance of the proposed method is that for a high-precision positioning it is important to compensate the geometrical error enhancing the control performance. This method can also be extended to two-dimensional or threedimensional machine stage. It will be useful to carry out a comparison study 共this is not done in this current work兲 with other robust statistics methods in future research work. References 关1兴 Hayati, S.A., Robot arm geometric link parameter estimation. Proceedings of the 22nd IEEE Conference on Decision and Control, pp. 1477–1483, 1983. 关2兴 Veitschnegger, W.K. and Wu, C.H., Robot accuracy analysis based on kinematics. IEEE J. Rob. Autom. RA-2Õ3, 171–179 共1986兲. 关3兴 Zhang, G., Veale, R., Charlton, T., Hocken, R., and Borchardt, B., Error compensation of coordinate measuring machines. CIRP Ann. 34, 445– 448 共1985兲. 关4兴 Duffe, N.A. and Maimberg, S.J., Error diagnosis and compensation using kinematic models and position error data. CIRP Ann. 36 共1兲, 355–358 共1987兲. 关5兴 Weekers, W.G. and Schellekens, P.H.J., Assessment of dynamic errors of CMMs for fast probing. CIRP Ann. 44 共1兲, 469– 474 共1995兲. 关6兴 Satori, S., Colonnetti, G., and Zhang, G.X., Geometric error measurement and compensation of machines. CIRP Ann. 44 共1兲, 599– 609 共1995兲. 关7兴 Hocken, R. et al., Three dimensional metrology. CIRP Ann. 26 共2兲, 403– 408 共1977兲. 关8兴 Hocken, R., Machine Tool Accuracy, in Report of Technology of Machine Tools 共Vol. 5兲, Lawrence Livermore Laboratory, University of California, 1980. 关9兴 Evans, C.J., Precision Engineering: An Evolutionary View. Cranfield Press, Cranfield, UK, 1989. 关10兴 Love, W.J. and Scarr, A.J., The determination of the volumetric accuracy of multi axis machines. 14th MTDR, pp. 307–315, 1973. 关11兴 Bush, K., Kunzmann, H., and Waldele, F., Numerical Error Correction of Co-Ordinate Measuring Machines. Proc. Int. Symp. on Metrology for Quality Control in Production, Tokyo, pp. 270–282, 1984. 关12兴 Tan, K.K., Lee, T.H., Lim, S.Y., Dou, H.F., and Seet, H.L., Probabilistic approach towards error compensation for precision machines. 3rd International ICSC Symposium on Intelligent Industrial Automation, Genova, Italy, pp. 293–299, 1999. 关13兴 Hornik, K., Stinchcombe, M., and White, H., Multilayer feedforward networks are universal approximators. Neural Networks 2, 359–366 共1989兲.

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关14兴 Fabri, S. and Kadirkamanathan, V., Dynamic structure neural networks for stable adaptive control of nonlinear systems. IEEE Trans. Neural Netw. 7 共5兲, 1151– 1167 共1996兲. 关15兴 Tan, K.K., Lim, S.Y., and Huang, S.N., Two-degreeof-freedom controller incorporating RBF adaptation

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for precision motion control applications. IEEE/ ASME International Conference on Advanced Intelligent Mechatronics 共AIM’99兲, USA, 1999. 关16兴 British Standard: Coordinate Measuring Machines, Part 3. Code of practice, BS 6808, 1989.