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APPLICATION OF FUZZY KNOWLEDGE BASE FOR CORRECTED MEASURED POINT DETERMINATION IN COORDINATE METROLOGY, A. Wozniak , R. Mayer, M. Balazinski, In proceeding of: Fuzzy Information Processing Society, 2007. NAFIPS '07. Annual Meeting of the North American

Application of fuzzy knowledge base for corrected measured point determination in coordinate metrology* A. Wozniak

M. Balazinski

Institute of Metrology and Measuring Systems Warsaw University of Technology Sw.A. Boboli 8 Street, 02-525 Warsaw, Poland [email protected]

Department of Mechanical Engineering École Polytechnique de Montréal C.P. 6079, Succ. Centre-Ville, Montréal, H3C 3A7, Canada [email protected]

R. Mayer Department of Mechanical Engineering École Polytechnique de Montréal C.P. 6079, Succ. Centre-Ville, Montréal, H3C 3A7, Canada [email protected] Abstract - This paper describes an application of fuzzy logic for corrected measured point determination in coordinate metrology. The correction method works on a series of indicated points obtained by contact scanning of the measured surface with a spherical tip probe. The outline of the probe ball defines an arc for each measured point, each such arc being delimited by the points of intersection with the preceeding and the following arcs. As a first approximation the corrected measured point is estimated as the mid-point of the arc. The refinement to the method consists in determining an angular compensation to be applied to the mid-point estimation and calculating the associated indicated measured point coordinate values. To determine an angular compensation a rule-based approach to decision making using fuzzy logic techniques is proposed. In this approach, we consider imprecise vague knowledge as a set of rules linking a finite number of conditions with a finite number of conclusions. The representation of such imprecise knowledge by means of fuzzy linguistic terms makes it possible to carry out quantitative processing in the course of inference based on the compositional rule of inference that is used for handling uncertain (imprecise) knowledge, often called approximate reasoning or fuzzy reasoning. Such knowledge can be collected and delivered by a human expert (e.g., decision maker, designer, process planner, machine operator, etc.). For our case, this knowledge is expressed by a finite number of heuristic fuzzy rules of the Multiple Input Single Output type (MISO). The paper will include a brief discussion of a new compensation procedure of probe radius tip and related test results of the same probe head carried out on a fixed bridge Mitutoyo Legex 910 CMM equipped with a MPP300 scanning probe in the continuous scanning mode.

I. INTRODUCTION

Coordinate measuring machines (CMMs) are instruments used for the inspection of part geometry and size in laboratories as well as in industry. Modern CMMs are equipped with scanning probe allowing high point density and automated form error measurements, graphic visualization of results, numerical data archiving in electronic media and integration with CAD/CAM systems. Therefore, these machines are utilized more frequently where reliability, speed and precision of measurements are required. The specified accuracy of these CMMs with scanning modes is in the sub-micrometer range. However, the probing system accuracy depends on the design parameters of the transducer, probe configuration, measuring strategy and the algorithms of probe radius compensation [1-3]. The usual built-in real-time CMM software [4-6] for processing scanned data points results in some smoothing of the real shape of the surface. Most of these are based on NURBS or other surface or 2D contour models to estimate the direction to apply the effective tip radius correction. The direction is the normal vector to the fitted indicated measured points surface. As a result, the information about the real shape of the measured surface can be distorted. In order to accurately measure on coordinate measuring machine we propose a new method for corrected measured point determination in a CMM measuring process. In this study, we discuss the application of fuzzy logic for a new method of corrected measured point determination (or probe radius compensation) in coordinate metrology. II. PRINCIPLE OF THE METHOD OF CORRECTED MEASURED POINT DETERMINATION IN COORDINATE METROLOGY

*

This work and Adam Woźniak was supported by the HOMING Program and Supporting Grant of Foundation for Polish Science. This work was also funded in part by NSERC Discovery Grant RGPIN-155677-02 with equipment funded by the Canadian Foundation for Innovation grant FCI-3000

APPLICATION OF FUZZY KNOWLEDGE BASE FOR CORRECTED MEASURED POINT DETERMINATION IN COORDINATE METROLOGY, A. Wozniak , R. Mayer, M. Balazinski, In proceeding of: Fuzzy Information Processing Society, 2007. NAFIPS '07. Annual Meeting of the North American

The proposed compensation method involves the following steps: 1) performing a series of high density measurements on the geometric characteristic to be measured with a probe ball; 2) the outline of the probe ball defines an arc for each measurement point; 3) calculating for each of the arcs the points of intersection with the preceding and the following arcs; 4) for each arc, estimating the point of contact with the characteristic as being the mid-point of the arc and 5) determining an angular compensation using fuzzy knowledge base and applying respectively a further compensation based on the corresponding angular adjustment. Consider a series of indicated measured positions of probe ball (centres) temporary referred to some XY coordinate system obtained while scanning a freeform surface as shown in Fig. 1a). a)

b) Oi+1

Pi+1

Oi Oi-1

Ai+1

R R

Pi

additional preceding point, Oi-1 and one following, Oi+1. Considering the stylus tip external envelope at point Oi, it can be stated that since the stylus tip is always in contact with the part surface and because no part matter can be located within the stylus tip boundary, the real point of contact of the stylus tip with the measuring surface lies somewhere on arc AiAi+1. Points Ai and Ai+1 are points of intersection of the three circles with centres Oi, Oi-1 and Oi+1 respectively. All three circles have radius R equal to the stylus tip radius as calculated during the preliminary CMM’s own procedure of probing system qualification. All points on the arc AiAi+1 are possible candidates as corrected measured point associated with the indicated measured point Oi; however, as a first approximation, the real point of contact of the stylus tip with the measuring surface Si is evaluated at the mid-point on the arc AiAi+1. However, and especially in the case of measurement features with internal discontinuities, some adjustment of the corrected measured point may be needed. As a first guess, a preliminary point Si was selected at the mid-point on the arc AiAi+1. Next, taking into consideration the mutual position of the points Oi and neighbors Oi-m,… Oi-1 and Oi+1,… Oi+n (where m and n are a number of preceding and following points respectively) an angular adjustment Δαi is obtained to improve on the initial mid-point and so complete the correction. The calculation of Δαi maybe performed using a variety of known rule-based or other AI techniques such as neural networks. In the experimental implementation of the stylus tip envelop method (STEM) we have opted for a calculation of Δαi using a fuzzy logic algorithm. The fuzzy input is an array with two components: Δzi and Δki. The first one, Δzi is the distance of point Oi to line zi crossing points Oi-1 and Oi+1 while the second component, Δki is the distance from point Oi to the line ki crossing points Oi-2 and Oi-1. Thus, the following are possible linguistic values defined for each input variables and the conclusion.

Δαi

Si Ai Pi-1

Fig. 1 Analysis of geometry of scanning path for corrected measured point determination in a CMM scanning process.

Let’s join together the arcs limited by the points of intersection of the outline of the probe tip (ball). The so obtained envelop (ticker line shown in Fig. 1a) circumscribes very well the shape of the real surface expect for the geometric filtering of the real geometric features of the surface which occurs by virtue of the mechanical contact interface, i.e. the stylus tip non zero sphere diameter. Consider point Oi as one of the data points describing the position of the centre of the spherical stylus tip as recorded by the CMM (shown in Fig. 1b). Let’s take into consideration one

TABLE I POSSIBLE LINGUISTIC VALUES

APPLICATION OF FUZZY KNOWLEDGE BASE FOR CORRECTED MEASURED POINT DETERMINATION IN COORDINATE METROLOGY, A. Wozniak , R. Mayer, M. Balazinski, In proceeding of: Fuzzy Information Processing Society, 2007. NAFIPS '07. Annual Meeting of the North American delta z

minus

b.minus

Oi

delta k Oi

delta a

the universe of discourse U. If a knowledge base of MISO system is used, the compositional rule of inference (CRI) may be written symbolically as:

Oi+1

Oi-2 Ai+1

Oi-2

Ai

Oi-1

Pi

Oi-2

Ai

Oi+1

Ai+1

Oi

Oi

Oi-2

Oi-1

U ' = C 'o... o (B 'o( A'o Rl )) .

Pi

Oi-1 Ai

Oi+1

Oi

-

Oi-1

Oi Oi-2

Oi-1

Having evaluated the angular deviation Δαi for each indicated measured point, the corrected measured point coordinates Pi may be estimated.

To solve the specific problem of correction of the stylus tip radius a rule-based approach to decision making using fuzzy logic techniques is proposed [7]. In this approach we can consider imprecise vague knowledge as a set of rules linking a finite number of conditions with a finite number of conclusions. The representation of such imprecise knowledge by means of fuzzy linguistic terms makes it possible to carry out quantitative processing in the course of inference based on the compositional rule of inference that is used for handling uncertain (imprecise) knowledge, often called approximate reasoning or fuzzy reasoning [8]. Such knowledge can be collected and delivered by a human expert (e.g., decision maker, designer, process planner, machine operator, etc.). This knowledge, expressed by a finite number (r=1,2,..,Rn) of heuristic fuzzy rules of the Multiple Input Single Output type (MISO) for our case, may be written in the form: (r ) R MISO : if a is A ( r ) and b is B ( r ) and ... and c is C ( r ) then u is U ( r )

(1) where A

, B

(r )

,..., C

(r )

denote values of linguistic

variables a, b,...,c (conditions) defined in the following universes of discourse: A, B, ..., C and U k(r ) , stands for the value of the independent linguistic variable u (conclusion) in

(4)

Taking into account sup-min (sup-prod) operations as compositional operators, min (prod) for implication, min (prod) for sentence connective 'and' and max (sum) for sentence connective 'also', respective inference results are obtained. To obtain a correction for the stylus tip radius it is proposed to apply a sum-sumprod-prod-prod inference engine The compositional rule of inference CRI [8] can be written as below:

III. FUZZY LOGIC APPLICATION

(r )

(3)

In the last formula, Rl aggregates all the rules, i.e. where sentence connective "also" denotes any t- or s-norm (e.g. min, max operators) or averages. The symbol o stands for the compositional rule of inference operators (e.g. sup-min, supprod etc.). The compositional rule of inference applied to Equation (3) may be written in the form:

Ai+1

Oi Oi-1

b.plus

Pi

Oi

Oi+1

zero

Rl = also i ( Rl (r ) )

Oi-1

Oi-1

(2)

The global relation Rl will be expressed as:

Oi

Oi+1

Oi

plus

U ' = ( A' , B' , C ') o Rl .

Oi-1

Oi-1

U ' (u ) =

∑ sup[(C ' (c) ⋅ ... ⋅ B' (b) ⋅ A

(r )

r a∈A b∈B M c∈C

(a ))⋅ B ( r ) (b ) ⋅ ...C (r ) (c ) ⋅ U ( r ) (u )] (5)

The center of gravity is used for deffuzification: Rn

∑U ' (u ) ⋅ u j

u' =

j =1 Rn

∑U ' (u )

j

.

(6)

j

j =1

For the purpose of stylus tip radius correction the fuzzy input (fuzzy observation) is a set of two parameters normalized to the range [0,1]: Δzi and Δki. The angular deviation Δαi also normalized to the range [0,1] is an output parameter (conclusion). For these inputs and output parameters and as formulated in Equation (1) the heuristic fuzzy rules of the MISO type may be written in the form: (r ) RMISO : if ∆zi is Z ( r ) and ∆ki is K ( r ) then ∆α is Α( r )

(7)

APPLICATION OF FUZZY KNOWLEDGE BASE FOR CORRECTED MEASURED POINT DETERMINATION IN COORDINATE METROLOGY, A. Wozniak , R. Mayer, M. Balazinski, In proceeding of: Fuzzy Information Processing Society, 2007. NAFIPS '07. Annual Meeting of the North American

where Z (r ) and K (r ) denote values of linguistic variables of deviations Δzi and Δki and A (r ) , stands for the value of the conclusion, the angular deviation Δαi. IV. EXPERIMENTAL RESULTS An optical fibre assembly has been used to demonstrate the feasibility of the proposed fuzzy logic algorithm. The optical fibre assembly is a specially prepared shape artefact consisting of an optical fibre glued onto a flat glass surface. It conveniently produces an internal discontinuity with a shape that is easily described by an equation. Tests were carried out on a fixed bridge Mitutoyo Legex 910 CMM equipped with a MPP-300 scanning probe. The maximum permissible scanning probing error of this probe (MPETij) is 0.32 µm, as per standard ISO-10360 [9]. Such fibre assembly artefact has been measured in a section perpendicular to the optical fibre axis. A 2 mm probe stylus tip diameter was used. The measuring velocity and the sampling step were set to 0.2 mm/s and 50 µm respectively. The raw (uncompensated) data have been stored in computer files by the CMM. Next the proposed algorithm and different premises and conclusions have been used for calculation of the corrected measured points set by: 1) CMM software, and 2) our method with the calculation of angular deviation using the fuzzy logic algorithm Fig. 2 shows the results of the CMM built-in compensation algorithm. The dashed line and square symbols represent the corrected measured points using the CMM’s own software. As can be seen, the corrected measured points obtained by the CMM’s own software suggest that some smoothing is performed during the processing of the indicated measured points resulting in a non-existing continuous characteristic. The errors are of the order of several tenths of millimetres.

shows the first trial of premises and conclusions used for corrected points calculation while Fig. 4 shows the compensated results of fiber assembly artifact measurement. The corrected by new method data are represented by black dots. The continuous line represents the real (theoretical) shape of the artefact scanned section. The dashed line and triangular points represent the deviation between corrected measured points and theoretical shape.

Fig. 3 First trial of premises and conclusions and its graphical representation for calculation the angular deviation using fuzzy logic algorithm.

As can be seen, the corrected measured points obtained by the first trial of premises and conclusions configurations (as it is visible in Fig. 3) gives a better estimate of the measuring feature, however the errors of few points are of the order of several tenths of millimeters.

1,3

1,3 1,1

CMM correction Shape of artifact

New correction Shape of artifact dev.(1x)

1,1

0,9

0,9 0,7

0,7 0,5

Y [mm]

Y [mm]

0,5 0,3

0,3

0,1

0,1 -1,5

-1

-0,5

-0,1 0

0,5

1

1,5

-1,5

-1

-0,5

-0,1 0

0,5

1

1,5

-0,3

-0,3 -0,5 X [mm]

-0,5 X [mm]

Fig. 2 Graph of the measurement result of a plane section of the fibre assembly artefact using the CMM built-in method.

The same raw measurement data set has been processed with the proposed new method with arc adjustment. Fig. 3

Fig. 4 Graph of the measurement result of a plane section of the fibre assembly artefact using proposed fuzzy logic correction method and premises, conclusions and rules as per Fig. 3.

APPLICATION OF FUZZY KNOWLEDGE BASE FOR CORRECTED MEASURED POINT DETERMINATION IN COORDINATE METROLOGY, A. Wozniak , R. Mayer, M. Balazinski, In proceeding of: Fuzzy Information Processing Society, 2007. NAFIPS '07. Annual Meeting of the North American

New experiments and analysis keen us to modify premises and conclusions as can be seen in Fig. 5. The corrected measured points obtained by new method (as can be seen on Fig. 6) do not show any smoothing of processing of the scanned indicated measured points. The new configuration of fuzzy algorithm gives a better estimate of the measuring features because it does not show the unreal, in existing measurement points. However, the maximum deviation of the corrected measurement results is approximately equal to 24 microns and this is still too much in case of precise coordinate metrology.

The last modification of premises and conclusions has been presented in Fig. 7. Points processed with the proposed method with arc adjustment, represented by black dots, are all on the expected profile (see Fig. 8). New algorithm of compensation and the last representation of premises, conclusions and rules give the best estimate of the measuring features. Deviations not exceeding 0.3 µm were obtained. This is of a level similar to the CMM machine repeatability itself.

Fig. 7 Third trial of premises and conclusions and its graphical representation for calculation the angular deviation using fuzzy logic algorithm. 1,3

Fig. 5 Second trial of premises and conclusions and its graphical representation for calculation the angular deviation using fuzzy logic algorithm.

New correction Shape of artifact dev.(1000x)

1,1 0,9 0,7 0,5

Y [mm]

1,3

0,3

New correction Shape of artifact dev.(50x)

1,1 0,9

0,1 -1,5

-1

-0,5

-0,1 0

0,5

1

1,5

0,7 -0,3 Y [mm]

0,5 -0,5 X [mm]

0,3

Fig. 8 Graph of the measurement result of a plane section of the fibre assembly artefact using proposed fuzzy logic correction method as per Fig. 7.

0,1 -1,5

-1

-0,5

-0,1 0

0,5

1

1,5

V. CONCLUSIONS

-0,3 -0,5 X [mm]

Fig. 6 Graph of the measurement result of a plane section of the fibre assembly artefact using proposed fuzzy logic correction method and premises, conclusions and rules as per Fig. 5.

The proposed method for correcting measurements can be applied directly to the data collected during a CMM measuring process. The correction method works on a series of high density indicated points obtained by contact scanning of the measured surface with a spherical tip probe. The feasibility of the new proposed stylus tip envelop method has been verified

APPLICATION OF FUZZY KNOWLEDGE BASE FOR CORRECTED MEASURED POINT DETERMINATION IN COORDINATE METROLOGY, A. Wozniak , R. Mayer, M. Balazinski, In proceeding of: Fuzzy Information Processing Society, 2007. NAFIPS '07. Annual Meeting of the North American

with success on a fixed bridge Mitutoyo Legex 910 CMM equipped with a MPP-300 scanning probe. The new method provides better results then using the CMM built-in method. ACKNOWLEDGMENT The authors would like to thank Mélissa Côté, research associate at École Polytechnique de Montréal for valuable discussion and assistance in some of the tests. REFERENCES [1] A. Woźniak, M. Dobosz, “Methods of testing of static inaccuracy of the CMM scanning probe”, Metrology and Measurement Systems, Vol. X, No. 2 (2003) pp. 191-203. [2] A. Woźniak, “Application of piezotranslator for the dynamic testing of scanning probes in coordinate measuring machines”, Mecatronica, No.1/2004, pp. 93-95 [3] A. Woźniak, R. Mayer, M. Bałaziński, M. Côté, “Investigation into precise measurement of cutting tool edges using coordinate measuring machine”, CIRP 2nd International Conference on High Performance Cutting, Vancouver, Canada, 12-13.06.2006. [4] Y. Zhongwei, Z. Yuping, J. Shouwei, “Methodology of NURBS surface fitting based on off-line software compensation of errors of a CMM”, Precision Engineering 27 /2003, pp. 299-303. [5] C.K. Song, S.W. Kim, “Reverse engineering: autonomus digitization of free-formed surface on a CNC coordinate measuring machine”, “Int. J. of Machine Tools and Manufacture”, 7/1997, pp. 1041-51. [6] N.A. Duffie, S.C. Feng, “Modification of bicubic surface patches using last squares fitting techniques”, Computer in Mechanical Engineering, CIME Research Supplement 3/1985, pp. 57-65. [7] M. Balazinski, M. Bellerose, E. Czogala, “Application of fuzzy logic techniques to the selection of cutting parameters in machining processes”, Fuzzy Sets and Systems, 61/1994, pp. 307-17. [8] L.A. Zedeh, “Outline of a new approach to the analysis of complex systems and decision processes”, IEEE Transactions of Systems, Man and Cybernetics 3/1973, pp. 28-44. [9] ISO 10360-4 2000 Geometrical Product Specifications (GPS) – Acceptance and reverification tests for coordinate measuring machines (CMM) – Part 4: CMMs used in scanning measuring mode, Switzerland, 2000.