Coordinate Measuring Machines and Machine Tools Selfcalibration and Error Correction

Coordinate Measuring Machines and Machine Tools Selfcalibration and Error Correction G. Belforte, B. Bona, E. Canuto, F. Donati, F. Ferraris, 1. Gorini, S. Morei, M. Peisino, S. Sartori Submitted by R. Levi ( l ) , IMGC, Politecnico, CSS, Torino/ltaly Received on February 27,1987 - Accepted by the Editorial Committee A procedure, by which it is possible to detect the geometrical errors of a coordinate measuring machine and a ABSTRACT: machine tool and to obtain the parameters necessary for software correction of these errors, has been studied, put into operation and tested. This procedure, called "selfcalibration", uses the results of the measurements automatically made by the machine on a three-dimensional standard placed at different positions in the working machine volume. The dimensions of the standard need not be known with high accuracy: it is sufficient to guarantee their stability during procedure operations and to measure three scale factors separately. For procedure adjusting and testing, nine machines have been employed, seven of which operating in industrial environment (two of these being working stations). The results proved that machine performance can be improved by a factor ranging between three and ten. KEY UORDS: Dimensional metrology, calibration, measuring machines, machine tools, automatic error correction, identification, simulation. 1) FOREUORD position transducers placed along the three coordinate measurement axes, which determine an arbitrary reference frame M The research concerns the study, design, adjustment and testing usually different from R. of software methods for the correction of geometrical errors of measuring machines and machine tools in their normal operation. -T=[Xt,Vt,Zt] is an array expressing the coordinates of the tool end in a reference integral with the head. The correction algorithms refer to a machine model identified by represent,respectively, head trans1 a-E=[ex, ey ,e,] and Q=[m,e.@] means of an automatic calibration procedure (selfcalibration) tion and rotation errors to be corrected, due to systematic carried out by the machines themselves. The purpose of the procomponents. ject is to define a method with which: -E represents the error due to non-repeatable or unknown compo-costs of machine manufacturing can be reduced partly by avoiding nents whose correction is unnecessary or not possible. the procedures of mechanical adjustments on the guides; -machine accuracy can be increased to levels close to their -Pk and Uk represent a rototranslation, unknown but constant, during piece machining or measurement, whlch takes account of resolution; the arbitrary character of the reference frame. The knowledge of -a machine can easily be re-calibrated after a period of operaPk and Uk is irrelevant. tion or maintenance service; Consequently, E(P,) and Q(Pm) are error components to be -machine error in the whole working volume can be determined. corrected by means of a mathematical error model which will This method must be applicable by machine operators in inhereafter be called the "model", whenever no ambiguity arises. dustrial environment, automatic, and suitable for as many types of machines as possible. The basic idea is to use the measuring and computing system of Table 1 - Machines used in the research. the machine, automatically controlled to measure a three-dimensional standard,in order to identify the behaviour of the machine 11 length [m] as regards systematic errors due to geometrical irregularities. Type Model Neither the standard's dimensions, nor their long-term stability turer need to be accurately known. On the other hand, it is necessary M1 bridge D.E.A. JOTA 1,32 that dimension stability, during automatic calibration which does measuring Torino 2203 not exceed some hours, should be about ten times better than the machine 0'97 0'97 accuracy required on the machine . Not more than three scale M2 bridge D.E.A. JOTA 1.32 factors remain unknown,and must be determined separately. measuring Torino 2203 machine The method must evidence: 0'97 0'97 -machine errors, supplied either in table form with the possibility of point interpolation, or in polynomial form; -the well-known 18 error functions of the machine, whether calculated without corrections, or remaining after corrections; -the estimate, with corrections in progress, of residual errors of the machine when measuring the distance between two points.

1.1 """~

II

2) OEFINITION OF THE M C H I N E IWDEL AND RELEVANT HYPOTHESES

All the machines used in the present research are described in table 1. Machines M1, M4 and M6 have been used for model study. The model definition, valid for all the kinds of machines used in the research and easily extendable to other kinds of machines, refers to a set of working hypotheses and definitions appearing in the Appendix. The nominal attitude of the head being assumed not to vary during machining or measuring processes, the following vectorial relation is obtained: P = Pm - E(Pm) - Q(Pm) n T + Pk + Uk n Pm + E, c11 where: -The symbol n stands for the external product operation. shows the "true" position of a point in the measure-P=[X,Y,Z] ment volume, as defined by the three "true" coordlnates X,Y,Z, which are determined with respect to reference frame R of orthogonal axes integral with the base and arbitrary. -P,=[Xm,Ym,Zm] is the reading made by the machine with its own

Annals of the CIRP Vol. 36/1/1987

1 I I I I I

I I

manufact. INNSE Bresci center

CPl/P

mobile column Geneve c.m.m.

560M

SIP -

longitudi MOORE nal corm. U.S.A.

3.2

0.27

N.3A

The model adopted is expressed by a combination of the following 18 known functions, each depending upon one variable: -nine translatory functions called Sij, where the first index denotes the moving axis which is the variable the function depends on, while the second indicates the direction along which the error occurs; for example, SXy represents the rectilinear error along direction y o f the movement along x and is a function of x alone. -nine rotatory functions called Rij, having the same meaning. The adoption of this model assumes machine rigidity, which has been experimentally verified, as will be seen later. The E and Q expressions for bridge machines, which can easily

be extended to other types, are given by vectorial relations: E(P,) = S, + Sy + S, + R, fl Pyz + Ry fl (Pz Po) C2l a(P,) = R, + Ry + Rz where: -Si (with i=x,y,z) is an array of functions: Si=[Six.Siy.Siz], -Ri (with i=x,y,z) is an array of functions: Ri=[Rix.Riy.Riz], -Pyz=[O,Ym,ZmI s -Pz =[O.0 PZmI. -Po = [ O , O ,Zo], with Zo being the height of the bridge. The adequacy of the model for these machines was first checked by computing beforehand, on the basis of some of the functions measured, other functions Sij (and Rij) in the working volume at different positions, and by subsequently comparing them with the functions measured at the same positions, not yet used for calculation. The results obtained in this test confirm that the model sufficiently corresponds to the behaviour o f the machine.

-

3) REALIUTION AN0 TESTIW O f THE MACHINE

'St

SIMULATOR Fig. 2

A software package, prepared to simulate, on the basis of eqs. [l] and [2], the behaviour of a machine, was tested on machines M1, lu14 and 07 (table 1). Fig. 1 shows the block diagram of the whole set of programs. The core is the SIMAC program, which does not refer to a particular type of machine. Input data are given with the interface programs typical of each machine, which translate sign conventions, the names of strokes, etc. into univocal format. Data concerning the 18 functions necessary for SIMAC operation are provided in standard format by the interface programs FUN.XXX. The 18 functions in the SIMAC program are described in tables containing the values of the function sampled at fixed and constant intervals of the variable. The values within the sampling interval are generated by linear interpolation. The simulator can work in three ways, briefly described below. Reconstruction mode. It provides at the output any o f the 18 functions i n a zone o f the volume other than that in which they have been supplied. The comparison between the reconstructed

-

Checks of the rigid-body model with the simulator working in reconstruction mode. X and Z are the coordinates related to the axes not involved in measured functions. The functions used in calculations are measured in the axes origin.

programs provide the readings the machine would give, should it actually perform the measurements on the standard in the established position (input and output data are respectively provided by interface programs STAND and RES, shown in fig. 1, which build the tables). Correction mode. With input data represented by the coordinates of any point read by the machine, the simulator output provides the three corrections t o be introduced in the coordinates and the three attitude angles of the measuring head, calculated by means of the 18 functions (MEAS and POS programs). I n order to test the simulator in the last two ways of operation, the following procedure was applied: -some standards having spheres at the vertices o f a parallelepiped were constructed. The sphericity error of these spheres is guaranteed not to exceed 0,5 om (fig. 3); -the sphere center coordinates in a reference system integral with the standard were determined with an uncertainty of 0,8 urn; -the standards were measured by the manufacturers on machines M1,M and 07 in at least 8 different locations in the working volume and at each position with two different probes (the last condition is essential in order to obtain information on angle attitude of the measuring head); -the collected data were introduced into the simulator and measurement results were compared with those predicted by the simulator. In table 2 the most significant results concerning two kinds of machines show satisfactory agreement between simulated and experimental data. 4) THEORETICAL STUDY

FOR THE IDENTIFICATION OF THE CORRECTION

FUIICTIOMS As regards both measuring machines and machine tools, the theoretical study for the identification procedure was based upon the measurement of the point coordinates in a standard placed at various positions in the measurement volume and on a set of

L Fig. 1

-

Block diagram of the set of programs making up the machine simulator.

functions and those measured in the same zone of the volume is a way to verify the adequacy of the model used by the simulator. Fig. 2 records,as an example,the results obtained with one machine. The difference between reconstructed and measured values always keeps within 2 or 3 times the machine resolution. Simulation mode. With the size of a three-dimensional standard (in terms of distances between characteristic points) and the standard position in the volume as input data, the simulator

360

298

Fig. 3

-

A three-dimensional standard made to test the simulator i n simulation and correction modes. millimeters.

Dimensions

in

EXPERIMENTAL. VALUES Axis y Axis z Axis x

-

-10,2 -18.7 -18.3 - 9.2 17.4 9,8 10,9 18.3

-

MACHINE M1 - ERRORS

-

1.8 0,3 0,3 0.4 1.7 0.9

-

1,2

I

18,7

I

MACH 1,s

2.5

I

9,2

MACHINE M1

I

3.0 -12,4 19,9 12.3 -20,3 - 7,s 15,O

-24,9 -21,o 39,O 33,7 -34.3 -27.7 21.1 13,5

3,4 11,5 13,6

-12,o 2,3 3.8 -13.3

3.0

14.7

RESIDUALS AFTER SIMULATION Axis x Axis y Axis z

13.6

-

10,8

-17,4

4,s 8,3 2.0

0.0 7.1 3,9 0,9 3.4

-

-

-

0,6 1.4 2.1 0,6 0.9 0.3 0,2 0,8

- 0,5

-

-

1 1.1 I MAXIMIM VALUES 1 2.1 I

-

--

3,2 1,7 2,s 1,2 4,O 1,s 3,8 2.3

I

2,O

I

0.6

-

0.3 2.9

-

2.3 0.5

5.1

0,2

-

0.3 0,7 0.1 0.2

2.0

1,3

- 2,O - 1,4

I

-

- 0,l - 0,8

1,9 0,7 0.5 1.6 1.2 0.4 2.0

-

appropriately weighted squares of the term

2,O o'8

I

- 3,6 1,8 - 2,2 3,2 - 3.0 0.71 - 0,l 3.1

28.0

I

2.5

39-0

I

>,6

hypotheses, which are listed in the Appendix. Only the assumptions that are most important for the application of the method are given. As regards identification in particular, the following main hypotheses have been put: -Rigidity of tridimensional standards during the selfcalibration process. -Rigidity of the probe, which makes it possible to neglect errors in the measurement equation. The array elements can be identified at each probe change with a qualification procedure. -Regularity of the 18 functions, in the sense that they do not show local discontinuities; the function can, therefore, be analytically expressed with a finite number of parameters, provided that "model" errors are small with respect to desired machine accuracy. -The model can be expressed as a linear combination of the various error terms. The higher-order terms are negligible. This means that errors have to be small enough to allow error components to be added and the relevant relations to be linearized. The model, as regards the part to be corrected and then identified (terms E and Q of eq. [l)),has been described by representing each Sij and R i j function with a reduced-order Legendre polynomial, with the u coefficients unknown. The model will hereafter be synthetically indicated by terms E(P,,p) and Q(P,,p). The model, for the part that cannot be corrected (term c of eq. [l]), has been described in two different ways, according to two a1 ternative identification phi 1 osophi es: PROCEDURE A) Error e can be expressed by a white noise, with zero mean value, stationary and Gaussian. PROCEDURE 8) Error is unknown but bounded. The equation used for identification becomes, for each measurement point:

-

-

Ti t Pk t Ok n Pm t C = 0, n(Pm,p) Pm - P E(Pm,p) [3] where the terms have the meaning as in eq. [l]; in detail: -P are the point coordinates of the standard; the distances between points are determined following calibration performed using sufficiently accurate instruments (the possibility of not performing such a calibration i s considered below): P is, therefore, known, to within a rototranslation, which is taken into account in Pk and in Uk; -p are the unknown coefficients whose values must be identified; -Pk and ak are unknowns whose values are irrelevant; -subscript k indicates the positions of the standard in the measurement volume; -subscript i is the ordinal number of the probe employed.

c

of the measurement

functional minimization. Procedure B refers to techniques of a deterministic kind developed in the last few years(8) and only recently tested in some cases of actual application. Identification provides the estimate that minimizes the uncertainty of the u coefficients. In statistical terms, the method is equivalent to a maximal likelihood estimate assuming a uniform distribution of E errors. Linear programming algorithms are used for the estimation. As regards procedure A, the conditions sufficient for identifiability have been proved. These are expressed as prerequisites which must be satisfied by the p parameters: -in the general case where E and Q are described in the form of a polynomial expansion in the three-dimensional variable P,; -in the particular case where they are represented by the 18 functions of only one variable, each expressed by polynomial expansion. The demonstration, here omitted for the sake of brevity, also points out the separability condition of the E and Q functions; with them it is obligatory to use m n y probes (at least two or, better, three if it is advantageous to identify also the n CWOnent along the line joining the ends of the two first probes) to measure points of the standard at each position. Identifiability eventually defines the characteristics of the standard, which will be described in the following paragraph. Procedures A and B have been studied in order to make selfcalibration, that is identification, possible even if the same standard has been calibrated without any particular precision. With these procedures identification is possible even with a totally unknown standard. In the following, reference i s made only to the selfcalibration procedure with an unknown standard. In this case, also P appears among the unknowns in eq. [3]: there are, therefore, more unknowns than equations and the problem cannot be worked out unless it is assumed that the distances between the measurement points of the standard remain constant and enough movements of the standard in the measurement volume are carried out, to obtain the necessary number of equations. Identifiability has been demonstrated also in this case, by establishing conditions relating the number of standard's positions to the number and position of measurement points on the standard; in particular, to the established hypotheses there is added a condition of non-colinearity between the p parameters of the model and the positions of the measured points. Identification can be carried out without one scale factor (three factors, with certain kinds of standards), which must be determined separately by other methods. This last condition means, for the polynomial model with the 18 functions, that not more than three coefficients of the first-order terms of the polynomial are non-identifiable. 5)

DEFINITION OF THE STANDARD

Account being taken of the different manufacturing and operational requirements,three-dimensional standards,with some spheres mounted on them, proved to be the most suitable ones: they can be positioned horizontally in the measurement volume. The masurement points are represented by the sphere centers, estimated by the interpolation algorithms present in all the machines used for the research. Such algorithms use information obtained through more coordinate measurements of points on the sphere surface. Requirements as to the number and accessibility of the measurement points suggested the construction of standards with a roughly pyramidal shape, with spheres placed on several planes. Tests were first carried out with the standard equipped with 24 spheres on five planes, as shown in fig. 4. The diameter of spheres is 17 nan and the maximal sphericity error is 0.5 pm.

36 1

Oigital computer, since they were not meant to be directly used on the machines at the present stage of the research work. These programs can solve u p to

J/0

c Fig. 4

0

8 0 I.M.GW7

N = 3 * Nb * Nk * Ni c41 scalar equations, where -Nb represents the number of spheres in the standard, -Nk the number o f positions of the standard, -Ni the number of probes employed. Relation [4] is valid if it is assumed that all the spheres are measured and all the probes are used for each position; the programs are modular, in order to make identification possible also when, in the different positions, not all spheres are measured and not all the probes are used.

0

300

- Cast

1

iron standard with 24 spheres.Dimensions in millimeters.

Later, the truncated-pyramid solution shown in fig. 5 was chosen for reasons of easier transport and the short time during which the stability of the standard is needed. The figure shows a standard with 36 spheres, made up of welded sections of A37 iron. The spheres were of the same type as in the previous standard. After construction, the structure underwent the usual stabilizing thermal treatments. Spheres were mounted in the standard in conical seats, with brass brackets, in such a way as to obtain a kinematic constraint without the use of adhesives. As regards the position of the standard in the measurement volume, the identification procedure requires that: -The standard should be placed at different positions on different planes: on one of these (plane o ) movements must be such as to cover the whole measurement plane. On the other planes a reduced number of positions is sufficient. -At least on plane a, the positions should be such as to permit partial over1 appi ng . -The standard should be turned to some of the positions by a sizable angle (approximatly 45". go', 180' are suitable). In actual performance,more positions than those strictly needed for identification were adopted.The positions in excess served as comparison terms for preliminary checks of the algorithms. Prior to construction, tests were made within the identification program described in the following paragraph by simulating the proposed standard and the procedure for point coordinate measurement by means of the simulator used in "simulation mode".

I

tL?:m procedurecr

I1

in the machine

STANDARD mowments

PROCESS meaarrement of points of the standard

riwi rnParmatla.

a

PROCESS

p k e points measurements

PROCESS

oOIcl0t.d m m w w m d s of @nts

The unknowns are: -n coefficients v o f the polynomials; -3*k parameters related to Pk translations; -3*k parameters related to Ok rotations; -3(Nb-1) coordinates of sphere centres. The number n depends on the order of each polynomial and on the values of some coefficients possibly known. The programs are structured in such a way as to allow a choice to be made among

-

Selfcalibration procedure. Sununary data for the calibration of two tested machines.

Machine Standard used Number of standard spheres Number of standard positions Number of probes Number of measurement points Number of identified functions Highest order of functions i Number of identified coefficienta Number of unknowns

I? - Truncated

M2 IMGC/5 22 43

M4 IMGC/4 17

1

2 1700

50

946 14 4 48

4 54

946

850

17

370

pyramid structured standard in section and 36 spheres. Dimensions in millimeters.

iron

The comparison between the results obtained from identification and the subsequent correction algorithms (with the simulator used in "correction mode") and the "exact" values of the points simulated in the whole working volume, made it possible to verify the adequacy of the standards and of the methods for position choice during the identification procedure. 6) IMWTIFICATION PROCEDURE

- FIRST RESULTS

The identification and correction programs are s h w in the block diagram of fig. 6. They were written in FORTRAN for VAX

362

of the pi.Q.

F
Table 3

Fig. 5

1

2 1, 0

k

Qc

-40

-80

-I20

Fig.7-Trend diagram for one of the 18 functions.Comparison between the identified (solid line) and measured (broken line) values.

the unknowns to be identified. Identification methods require a number of equations higher by at least one order of magnitude than the number of unknowns. The programs were tested by means of the simulator, following the procedure mentioned in the preceding paragraph. I n the light o f the computing time needed and of the results obtained, it was decided to use the identification procedure A for field testing. The results mentioned below thus concern only this procedure. The 18 functions were approximated with polynomials of the fourth order at most. The number of positions, of measurement points, of the identified coefficients, and the total number of unknowns are sumnarized in table 3. Figure 7 shows the results obtained by identification for the estimate of one of the 18 functions, compared with the values actually measured with traditional techniques. Table 4 shows the values of some coefficients of the identified polynomials. These were associated with the mean square value estimate, which is an index of the confidence level of the value obtained. Table 4-Results of selfcalibration. Values of some coefficients of the identified polynomials and associated confidence index. lOrde+ of coefficients1

1

I

2

1

4

3

-

P, = P,,, E - Q n T where Pc is the array representing the corrected point, while the other terms bear their usual meaning. The correction is valid to within a rototranslation. Correction can be performed by polynomial interpolation on the basis of the 18 functions. About one hundred multiplications are necessary for each point, but the memory storage is limited to the recording of polynomial coefficients, which, as mentioned in the previous paragraph, are about fifty. Only the M9 machine was systematically checked, by using four linear standards 77,5 mn, 1 5 5 , O m, 232,5 min and 310,O m long. Each was placed at 26 positions in the machine volume and its length was calculated with and without corrections. Consequently, checking involved 104 length measurements; the maximal dispersion in not-corrected length measurements was 2.6 pm; with corrections using selfcalibration results was 2,O urn. This result, which shows high machine accuracy even without software corrections, was obtained with the selfcalibration process carried out by means of a non-optimized standard. These first results are sumnarized in table 6, which evidences the highest residual error. 8 ) CONCLUSIWS

FUNCTION -2.8

Value Index Value Index

25,5 1.0

I

t

Rxx

Value Index

I

-3.5

-0,4

0.9

I

I

-28.0

-2,2

2.4

-3.5

2.6

1,3

0,9

0,8

Table 5 shows the indexes denoting the adequacy of the identification performed and refer to the estimate additionally carried out by the program of the "true" positions of the spheres in the standard;this estimate is based upon the readings taken by the machine with the standard at the various positions. The table shows the square norm (with respect to the estimated "true" positions of the spheres in the standard) of the error calculated urior to correction and the norm estimated after correction.

7) REALIZATION AND TESTIN6 OF THE PROGRAn FOR ERROR CORRECTION The program installed on the machines used for the present research provides: -correction of the coordinates of the measured points during measurement operations, -indication of the correction of position and attitude of the head during piece machining. Table 5

-

Results of selfcalibration.Origina1 (before corrections) and residual (after corrections) norm of the errors. The norm is the normalized square root of the s m of coordinate dispersion squares. Values in micrometers.

Y

z

Original norm

18,O

17.6

4,7

14,7

Residual norm

2.2

2,3

2,o

2,2

MACHINE COORDINATES

Standard's length

[ml

X

Measured length dispersion IMAX - MINI IWI

HEAN

Corrected length dispersion IMAX - MINI

[PI

155.0 232,s 310.0 The correction equations are derived from eq. regards a measuring machine, are:

[l]

and, as

An overall test of the method i s being carried out and will still require a long time. I n fact, only its extensive application to several machines will provide quantitative parameters for a comparison, as regards both gain in testing time and improvement in machine performance. In order to carry out general testing, two conditions must be satisfied: -Machine users must be trained to use machines no longer almost geometrically perfect,but having the needed precision provided by the correction programs contained in the computer control system. -Machine manufacturers must reorganise the system for checking the quality of their production. Training and reorganisation will take a long time. The present results are, therefore, only partial, since they have not even been obtained on all the machines used for the research and since these machines do not represent all the existing types. Account being also taken that the standards used were at times optimized for a machine and unsuitable for another, the results so far obtained have neverthless confirmed the validity of the method applied, so that this procedure can be said to yeld undisputable advantages and to reconmend itself as a very good candidate for CCN-machine testing and correction. The following synoptical conclusions are added to the reported analytical results: 1)If the machine building process is interrupted at the m m e n t when the desired repeatability has been obtained, (without particularly improving the rectilinearity and orthogonality of movements), a saving of 200 to 500 working hours of inspectors and highly specialized workers is obtained. 2)Once the relevant necessary organization has been made, selfcalibration requires from 10 to 30 machine hours, with the assistance of an operator whose only task is to move the standard and launch the measurement programs. 3)The performances of a machine,when it is corrected by the selfcalibration results,are ten to a hundred times better than those achievable by the machine in the condition already described in point 1) and two to five times better than those obtainable after rectilinearity and orthogonality have been improved. 4)The same performance described above can be obtained by imasuring on the machine (still in the conditions described in point 1) the eighteen error functions by traditional techniques and by computing corrections with a simulation program like the one described in paragraph 3. This operation, however, requires between 30 and 60 hours of an experienced inspector, very expensive equipment and the relevant maintenance, and a thorough knowledge of several data format conventions. With respect to the direct measurement of the eighteen functions, the selfcalibration procedure has a still m r e significant advantage, than can be inferred from a mere comparison of the skill and experience required by the working procedure. Up to

now, a f t e r e r r o r i d e n t i f i c a t i o n , the correction functions were computed by adopting a rigid-body model t o describe the behaviour o f the machine. This choice i s compulsory, when a d i r e c t measurement of the eighteen functions i s adopted, as they are the only ones experimentally accessible, but i s purely a u x i l i a r y i n the s e l f c a l i b r a t i o n procedure; a model, i n fact, may n o t consider the physical structure o f a machine, but only the conditions necessary f o r a good performance. For each machine type can therefore be adopted a s u i t a b l e e r r o r model t h a t leaves out any i n t u i t i v e modes o f e r r o r combination i n the machine volume.This i s a f i e l d s t i l l open t o investigation. 9) ACKNOULEDGMEIITS The authors are p a r t i c u l a r l y indebted to: -the experts o f DEA, PRIMA INDUSTRIE and INNSE, f o r t h e i r essent i a l c o n t r i b u t i o n t o the d e f i n i t i o n and development o f the research, w i t h f r u i t f u l discussions and the coordination o f the measurements c a r r i e d out. P.C. Cresto o f IMGC f o r the programs he s p e c i a l l y prepared -dr. for standard measurement analysis applied f o r simulator testing.

3 -The t o o l i s a r i g i d body. 4 -A reference frame, r , i s connected w i t h the head, i n respect o f which the t o o l remains fixed. 5 -The t o o l i s f i x e d t o the head i n such a way t h a t v a r i a t i o n s i n i t s attitude,during the whole measurement o r piece machining process,produce n e g l i g i b l e errors.The p o s i t i o n o f the t o o l end i n the reference frame r i s glven by the array T=[Xt.Yt.Zt]. 6 -It is possible t o i d e n t i f y the t o o l model ( t h a t i s , terms X t , Yt.Zt) by a q u a l i f i c a t i o n procedure performed when mounting the t o o l on the head. 7 -The t o o l has a c y l i n d r i c a l symnetry. 8 -The symnetry o f the t o o l axis coincides w i t h one o f the axes o f reference frame r o f the head. 9 -There must e x i s t a reference frame M,with respect t o which the machine c a r r i e s out i t s own readings. Readings are given by the array Pm=[Xm,Ym,Zm], referred t o reference frame M. 10-The p o s i t i o n o f the head does not depend on the movements necessary t o reach the p o s i t i o n i t s e l f . 11-During the whole measurement procedure the nominal a t t i t u d e o f the head keeps constant. 12-The E errors i n the working volume are small enough t o leave room f o r the sum o f the various e r r o r comoonents. B)

(7)

(9)

Hocken, R., Simpson, J.A., Borchardt, B., Lazar, J., Reeve, C., Stein, P., 1983, Three dimensional Metrology, Annals o f CIRP, Vol. 32, 403:408. Knapp, W., 1983, Test o f the three-dimensional uncertainty of machine t o o l s and measuring machines and i t s r e l a t i o n t o the machine errors, Annals o f CIRP, Vol. 32, 459:464. Busch, K . , Kunanann, H., Waldele, F., 1984, Numerical e r r o r correction o f a coordinate measuring machine, Proc. I n t . Symp. on Metrologie f o r Qual. Contr. i n Prod., Tokyo, 218 :282. Busch, K., Kunzmann, H., WYldele, F., 1985, Calibration o f coordinate measuring machines, Report t o BCR. Duffie, N.A., Yang, S.M., 1985, Generation o f parametrickinematic error-correction functions from volumetric e r r o r measurements, Annals o f CIRP, Vol. 34, 435:438. Veale, R., Charlton, T., Borchardt, B., Hocken, Zhang, G., R., 1985, Error compensation o f coordinate measuring machines, Annals o f CIRP, Vol. 34, 445:448. Jerdrzejewski, J., Kwasny, W., M i l e j s k i , D., 1985, Selected diagnostic methods f o r machine t o o l s acceptance tests, Annals o f CIRP, Vol. 34, 343:346. Milanese, M., Tempo, R . , 1985, Optimal algorithms theory. f o r robust estimation and prediction, IEEE Transactions on Automatic Control, Vol. AC-30, 730:738. Donati, F, Canuto, E., 1986, S e l f - c a l i b r a t i o n o f coordinate measuring machines: theory and f i r s t r e s u l t s , Proc. 5th I n t . Symp. MIC, Innsbruck. 1986.

APPENDIX

- HYPOTHESES

A l l the hypotheses needed f o r the application o f the correction method described i n the a r t i c l e are l i s t e d i n the present Appendix. As already pointed out, most o f these hypotheses have been expressly checked during the research. A l l o f them proved compat i b l e w i t h the machines chosen f o r t e s t i n g the method, as i s demonstrated by the correction r e s u l t s obtained. For the convenience o f the reader the d e f i n i t i o n s r e l a t i n g t o the various hypotheses are here stated. A)

GENERAL HYWTHESES NEEDED FOR EQUATION [l].

1 -During a l l the operations o f the machine the piece f i x e d t o i t remains r i g i d . 2 -There i s a reference frame R f i x e d t o the base, w i t h respect t o which the piece remains fixed. The absolute p o s i t i o n o f a p o i n t w i t h respect t o the reference frame R i s given by the array P=[X,Y,Z].

364

HYPOTHESES NEEDED FOR EQUATIMS [ Z ]

AND [3].

13-The E e r r o r s are random, stationary and have a Gaussian d i s t r i b u t i o n with zero mean value (Procedure A) or, a l t e r n a t i v e l y , are not known but bounded (Procedure 6). 14-The machine has no l o c a l d i s c o n t i n u i t i e s e i t h e r i n i t s s t r u c t u r e o r i n p o s i t i o n transducers.Possible d i s c o n t i n u i t i e s are pre-solved with algorithms o r correction tables (e.g. f o r " o p t i c a l l i n e s " ) not considered i n the present research. 15-The structure o f the machine i s r i g i d enough t o be modelized w i t h the 18 functions. C)

HYPOTHESES CMIDITIONING STANDARD'S C O N S l W J C T I ~ AND MEASUREMENT PERFORlUWCE WRING SELFCALIBRATION.

16-The number of probes and t h e i r r e l a t i v e p o s i t i o n s are such as t o permit e r r o r components E and Q t o be separated. 17-The shape o f the standard and the p o s i t i o n s i t takes i n the measurement volume during s e l f c a l i b r a t i o n are such as t o allow the i d e n t i f i c a t i o n equation system t o be solved.