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Calibration of a 3D-ball plate
Precision Engineering 33 (2009) 1–6
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Precision Engineering journal homepage: www.elsevier.com/locate/precision
Calibration of a 3D-ball plate T. Liebrich a,∗ , B. Bringmann b , W. Knapp b a
Inspire/ETH Zurich, Tannenstrasse 3, 8092 Zurich, Switzerland Institute of Machine Tools and Manufacturing/ETH Zurich, Tannenstrasse 3, 8092 Zurich, Switzerland b
a r t i c l e
i n f o
Article history: Received 9 July 2007 Received in revised form 25 February 2008 Accepted 27 February 2008 Available online 7 March 2008 Keywords: Machine tools Calibration CMM Ball plate Reversal method Uncertainty estimation
a b s t r a c t The presented 3D-ball plate is used for testing machine tools with a workspace of 500 mm × 500 mm × 320 mm. The artefact consists of a 2D-ball plate which is either located by a kinematic correct coupling on a base plate or on a spacer. The spacers are placed between the base plate and the ball plate and are also kinematic coupled to the other elements of the artefact. The kinematic couplings provide a high repeatability of the measurement setup. Because of the specific application the known calibration procedures for 2D-ball plates are not applicable. A calibration method for the pseudo-3D-artefact on a coordinate measuring machine (CMM) is presented, with the aim to minimise the influence of geometric CMM errors. Therefore a computer simulation is used to analyse the effects of these disturbing errors on the calibration of the ball plate and the spacers. Using a reversal method, the plate is measured at four different horizontal positions after rotating the ball plate around its vertical axis. A couple of the CMM errors, e.g., a squareness error C0Y between the Xand Y-axis of the CMM, can be eliminated by that method—others have to be determined with additional measurements, e.g., the positioning errors EXX or EYY of the X- and Y-axis, respectively. The paper also contains a measurement uncertainty estimation for the calibration by use of experiments, tolerances and Monte Carlo-simulations. The achieved uncertainty for ball positions in the working volume is less than 2.1 m (coverage factor k = 2). © 2008 Elsevier Inc. All rights reserved.
1. Introduction The geometric errors of machine tools influence the accuracy of the produced parts. It is therefore recommended to check the geometric accuracy of machine tools regularly. The presented 3Dartefact enables testing of machining centres with a vertical spindle and a workspace of 500 mm × 500 mm × 320 mm (see Bringmann [1,2]). The 3D-artefact is based on a 2D-ball plate, a standard tool in calibrating CMMs. To create a 3D-artefact the ball plate has to be repositioned in different known locations. Therefore spacers with different heights are inserted between a base plate and the ball plate (see Fig. 1). The resulting translation and rotation of the ball plate with reference to the ball plate on the base plate has to be well known. With this build-up a pseudo-3D-artefact is created defining a grid of points (embodied by spheres), which positions are measured by the machine tool to be checked [2,3]. With a calibrated artefact the relative errors of machine tools at the nominal positions of a grid can be determined by measuring the individual
positions of the spheres. A model of the machine tool with its possible geometric errors is necessary to calculate the real deviations of the machine tool based on the measured grid with an optimization algorithm. Advantages of this measurement procedure are reduced measuring times, reduced measurement uncertainties as well as volumetric measurement and compensation of geometric errors [2]. Typically, the calibration of a ball plate is done by a coordinate measuring machine (CMM). In DKD [4] a method for calibrating 2D-ball plates is presented: the suggested proceeding is a reversal method for rotating the ball plate around its X-, Y- and Z-axis. Because of the specific application of the ball plate in the presented 3D-artefact – it is only used in a horizontal alignment – the known calibration procedures for 2D-ball plates are not applicable. A method for calibrating the 3D-artefact is developed. Its backgrounds are simulations, which investigate the influence of geometric errors of the CMM on the calibration. 2. Artefact and measurement device
∗ Corresponding author. Tel.: +41 44 632 46 76; fax: +41 44 632 11 25. E-mail address: [email protected] (T. Liebrich). 0141-6359/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.precisioneng.2008.02.003
The artefact is made up of a 2D-ball plate which is either located directly on a base plate or on a spacer.
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T. Liebrich et al. / Precision Engineering 33 (2009) 1–6
Fig. 1. Schematic measurement setup for calibrating machine tools with a 3Dartefact.
The spacers with heights of 80 mm up to 320 mm can be included between the base plate and the ball plate and thus generate a 3D-grid of measuring points (see Fig. 2). With a kinematic coupling between ball plate, spacer and base plate a high repeatability of the measuring setup is obtained [5,6]. The ball plate, a commercially available product and typically used for CMM calibration, consists of 36 spheres creating a quadratic grid with a mesh size of 100 mm. The coordinate system of the ball plate is defined through three spheres in the corners, see CENAM [7]. The coordinate origin is represented by sphere 1, the center points of sphere 1 and sphere 6 define the X-axis. The XY-plane is defined by spheres 1, 6 and 31 and Y is square to the X-axis (see Fig. 3). The measuring device used for calibrating the artefact is a Leitz PMM 864, a CMM in portal design (see ISO [8]). The movement of the table defines the X-axis of the machine coordinate system, the horizontal slide moves in Y-direction and the centre sleeve in Zdirection. Some deviations of the CMM influence the accuracy of the calibration of the artefact. In the following section these deviations are derived.
Fig. 2. Exemplary measurements results (magnification 3000×).
2.1. Machine errors of the CMM The geometric errors of CMMs (and also of machine tools) can be classified in location errors and component errors. Location errors describe the position and orientation between two different axis motions, e.g., squareness or parallelism deviations. Component errors describe errors of the moving components themselves, e.g., positioning or straightness deviations. Every component error CE can be described mathematically by a Fourier series: CE(ω) =
∞
Ai cos(iω − ϕi )
i=1
For an exact description of one component error an infinite number of parameters are needed. In the simulation of the geometric behaviour of the CMM (Section 3), the component errors are classified in linear and harmonic errors. A linear error means that the component error increases linearly with a proceeded length. A harmonic error has the form of the corresponding term of the Fourier series. In the following this is marked with (lin), respectively (harm).
Fig. 3. Ball plate with its coordinate system (left) and measurement setup for calibrating a spacer (right).
T. Liebrich et al. / Precision Engineering 33 (2009) 1–6
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Fig. 4. Component errors (left) and orientation errors (right) of a linear axis X.
Table 1 List of errors of a CMM with three linear axes Direction
Component errors X-axis
Orientation errors Y-axis
Z-axis
X-axis a
X Y Z A
EXX positioning EYX horizontal straightness EZX vertical straightness EAX roll
EXY horizontal straightness EYY positioning EZY vertical straightness EAY pitch
EXZ straightness in X EYZ straightness in Y EZZ positioning (EAZ)b tilt around X
X0X zero position – – –
B
EBX pitch
EBY roll
(EBZ)b tilt around Y
B0Xa squareness of X to Z
C
ECX yaw
ECY yaw
(ECZ)b roll
C0Xa squareness of X to Y
a b
Y-axis
Z-axis
– Y0Ya zero position – A0Ya squareness of Y to Z –
– – Z0Za zero position A0Z squareness of Z to Y B0Z squareness of Z to X –
C0Y squareness of Y to X
Set to zero as this defines an element of the coordinate system of the CMM. Included for the applied calibration method in EXZ, EYZ and EZZ.
The coordinate system of the CMM is defined by the three linear axes X, Y and Z. A linear axis has six component errors and three orientation errors (see Fig. 4, ISO [9]). The notation of these errors is according to ISO [9]. From the nine orientation errors of a three-axes CMM six have to be cancelled because they define the coordinate system of the CMM: X0X, Y0Y and Z0Z define the origin of the CMM coordinate system; B0X and C0X define the X-axis of the CMM coordinate system; A0Y defines the Y-direction of the CMM coordinate system (see ISO [10]). From this it follows that totally 21 geometric deviations of the CMM exist, which are summarized in Table 1. The tilt and roll movements of the Z-axis (EAZ, EBZ and ECZ) are included in EXZ, EYZ and EZZ and cannot be differed from them because the length of the stylus is constant and not changed during calibration.
the X-axis of the CMM. According to following sections, the CMM errors can be separated by the reversal method in compensable and non-compensable errors, see Tables 2 and 3. 3.1.1. Linear straightness error of Y: EXY(lin) A linear increasing straightness error over the traverse path of 5 m per length of the ball plate has the same effect like a squareness error C0Y between the X- and Y-axis. Fig. 6a shows the resulting
3. Simulation The calibration procedure is derived from the suggested procedure in DKD [4] and is composed of four different alignments of the artefact on the CMM table. Because the ball plate is used in this application only in a horizontal alignment the calibration method also consists only of horizontal measurements, rotated 90◦ with respect to each other (see Fig. 5). This reversal method (see Evans et al. [11]) is reproduced in the simulation which contains a kinematic model of the CMM, evaluating the influence of individual CMM errors on measuring a grid that represents the ball plate. 3.1. Discussion of the influence of CMM errors The different courses of CMM errors and their effects on calibrating the ball plate are discussed by means of a horizontal straightness error EXY. Therefore we assume that for an orientation 0◦ of the artefact the X-axis of the ball plate is parallel to
Fig. 5. Reversal method for calibrating the artefact (pictures A–D): orientation 0◦ , 90◦ , 180◦ and 270◦ ; X, Y: coordinate system of ball plate; XCMM , YCMM : coordinate system of CMM.
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T. Liebrich et al. / Precision Engineering 33 (2009) 1–6
Fig. 6. Simulation of CMM errors (left to right): EXY(lin), EXY(1harm) and EXY(2harm): magnification 5000×; amplitude 5 m. Table 2 Eliminated errors of CMM PMM 864 by reversal method
X-axis Y-axis Z-axis
Component errors
Orientation errors
EYX(lin), EYX(1harm), EAX(lin), EAX(harm), EBX(harm), ECX(lin) EXY(lin), EXY(1harm), EAY(harm), EBY(lin), ECY(harm) EXZ, EYZ, EAZa , EBZa , ECZb
– C0Y A0Z, B0Z
a
EAZ and EBZ cannot be differed from EXZ, EYZ, EZZ because the length of the stylus is not changed. The influence of EAZ and EBZ on EZZ is a second-order effect and therefore they are listed up with compensable CMM errors. b ECZ has no influence on the calibration (only centre probe used) and is therefore listed up with the compensable CMM errors.
grids for orientations of 0◦ and 90◦ of the artefact on the CMM table. Because of the change in sign (symmetry in Y-direction) this error is compensated with a perfect reversal method by rotations of the artefact around 90◦ . CMM errors with this characteristic can be eliminated because of having a difference between the mean value of an arbitrary sphere during the reversal measurements and its nominal position which consists of high order terms and can therefore be neglected. That is the case if the Taylor series, which approximates this difference, does not contain a linear term. Fig. 7 shows that CMM error C0Y, squareness of Y to X, is of second order for the Y-distance between spheres if C0Y is small: L = Ynom (1 − cos(C0Y)). Figs. 7 and 8 show that CMM error C0Y, squareness of Y to X, is eliminated by the reversal method for orientations of the ball plate of 0◦ and 90◦ . For example, sphere 31 has for orientation 0◦ the coordinates 31 r0◦ = (−x,500 − L,0) and for 90◦ orientation 31 r90◦ = (x,500 − L,0). The mean value addicts 31 rmean = (0,500 − L,0). Because L is of second-order C0Y is eliminated (analogous for orientation 180◦ and 270◦ ).
Fig. 7. Difference in Y-direction is a second-order term if C0Y 3◦ .
is chosen 5 m, the phase is 0. For rotations of the artefact around 180◦ the ‘bow’ has a change in sign, therefore EXY(1harm) of the CMM is also eliminated with the suggested reversal method. If the phase is not equal to 0 or , respectively, the elimination is not perfect and a little error emerges which has to be regarded in the uncertainty of the calibration (see Section 3.3). 3.1.3. Second harmonic of straightness error of Y: EXY(2harm) The amplitude of EXY(2harm) is chosen to 5 m and the phase is /2. The resulting course is shown in Fig. 6c. For these parameters the grid has no change in sign and consequently an elimination is not possible by rotating the ball plate around its vertical axis. CMM errors which show no symmetric change in sign cannot be elimi-
3.1.2. First harmonic of straightness error of Y: EXY(1harm) The first harmonic of EXY is described with a half wavelength of a sinus over the length of the ball plate, see Fig. 6b. The amplitude Table 3 Non-compensable errors of the CMM PMM 864
X-axis Y-axis Z-axis
Component errors
Orientation errors
EXX, EYX(2harm), EZX, EBX(lin), ECX(harm) EYY, EXY(2harm), EZY, EAY(lin), EBY(harm), ECY(lin) EZZ
– – –
Fig. 8. Compensable CMM error C0Y (pictures A and B): orientation 0◦ and 90◦ .
T. Liebrich et al. / Precision Engineering 33 (2009) 1–6
Fig. 9. Result of additional measurement EYY.
nated by this reversal method. Again, the influence of the phase has to be considered in the calibration uncertainty (Table 4). 3.2. Additional measurements To have an upper limit of the amplitude of non-compensable CMM errors these values can be taken from the machine specifications or have to be acquired by additional measurements with auxiliary measurement devices. These measurements are exemplified by positioning accuracy and repeatability of Y-axis (EYY). Therefore a comparator system is used which allows also the measurement of horizontal and vertical straightness. The resulting bidirectional accuracy A (see ISO [12]) is determined to 2.6 m (see Fig. 9). So it is traceable to choose the amplitude of geometric CMM errors in the Monte Carlo-simulation arbitrarily within this range. If the measured CMM errors cause a non-negligible influence on the calibration and therefore enlarge the calibration uncertainty, the CMM has to be compensated. That minimises the contribution to the calibration uncertainty due to geometric CMM errors. 3.3. Calibration uncertainty The positions of the spheres are afflicted with an uncertainty due to several perturbing influences during the calibration. These can be classified in contributors coming from the CMM (Table 4) and in contributors coming from the artefact (Table 5). The uncertainty is estimated as recommended in ISO [13]. Thus repeated measurements or Monte Carlo-simulations can be used to calculate the standard uncertainty of each contributor. The evaluation of the standard uncertainty is then based on assumptions which depend on the limit of variation and the type of distribution.
5
Fig. 10. Calculation of the standard uncertainty for probing the sphere with a Monte Carlo-simulation (1000 iterations, assumed probing error P = 0.5 m).
The main contributors are drift of the CMM, the repeatability of the positioning of the ball plate on the base plate, respectively, on a spacer and the non-perfect reversal method. A perfect reversal method cannot be realised because it is not possible to rotate the artefact exactly in steps of 90◦ . So the equalisation of the CMM errors succeeds not completely and therefore an uncertainty contributor is caused. Moreover, the non-compensable CMM errors cause a further contributor to the calibration uncertainty (see Section 3.1.3). To investigate the standard uncertainty of a non-perfect reversal method combined with the influence of non-compensable CMM errors a Monte Carlo-simulation is done. This represents the behaviour of the CMM with its possible geometric errors, which are assumed to be random values (rectangular distributed) within a known range. In Fig. 10 the result of such a simulation is shown. For all Monte Carlo-simulations the parameters are chosen arbitrarily within the specified range in 1000 runs. The resulting measurement uncertainties are shown in Table 6. The traceability of the calibration result is provided by the use of calibrated measurement devices and therefore it is included in the calibration uncertainty. 4. Calibration procedure The calibration procedure is classified in calibrating the ball plate and in calibrating the spacers. 4.1. Calibration of the ball plate During calibration the ball plate lies on the base plate which is fixed on the table of the CMM. First the coordinate system of the
Table 4 Standard uncertainty of CMM Contributor
Uncertainty of drift (measurement) Uncertainty of calculating the center of a sphere (Monte Carlo-simulation with uncertainty of the probing system: P = 0.5 m) Uncertainty of a non-perfect reversal method and non-compensable CMM errors (Monte Carlo-simulation with accuracy of orientation of artefact of ±1◦ and geometric CMM errors of ±1.5 m, respectively, ±1.5 m/m) CMM standard uncertainty
Standard uncertainty X (m)
Y (m)
Z (m)
0.29 0.10
0.23 0.10
0.29 0.16
0.85
0.81
0.48
0.90
0.85
0.58
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T. Liebrich et al. / Precision Engineering 33 (2009) 1–6
Table 5 Standard uncertainty of artefact Contributor
Standard uncertainty (m) X
Y
Z
Uncertainty of repeatability of setup (measurement)
0.28 (1) 0.25 (2) 0.37 (3)
0.26 (1) 0.24 (2) 0.24 (3)
0.06 (1) 0.06 (2) 0.25 (3)
Uncertainty of form deviation of spheres (Monte Carlo-simulation with F = 1 m)
0.20
0.21
0.32
Uncertainty of temperature equivalence between CMM and artefact: T = ±0.3 ◦ C, ˛(artefact) = 11.6 m/(m K), ˛(CMM) = 8 m/(m K) (glass scales)
0.31
0.31
0 (1) 0.05 (2) 0.20 (3)
Uncertainty of thermal expansion coefficients ˛ = ±2 m/(m K) at temperature deviation of (T = 20 ◦ C) < 0.3 ◦ C
0.17
0.17
0 (1) 0.03 (2) 0.11 (3)
Artefact standard uncertainty
0.49 (1) 0.47 (2) 0.55 (3)
0.49 (1) 0.48 (2) 0.48 (3)
0.32 (1) 0.33 (2) 0.47 (3)
Index (1) characterizes the calibration of the ball plate on the base plate, the indices (2) and (3) stand for spacers with nominal height of 80 mm and 320 mm, respectively.
Table 6 Measurement uncertainties U(k = 2) for calibrating the 3D-artefact
U(k = 2) base plate U(k = 2) spacer 80 mm U(k = 2) spacer 320 mm
X (m)
Y (m)
Z (m)
2.0 2.0 2.1
2.0 2.0 2.0
1.3 1.3 1.5
ball plate is defined (see Section 2) and afterwards all 36 spheres are measured by probing each sphere with five points. This is done for all four orientations of the artefact on the CMM (see Section 3). The positions of the spheres are calculated by averaging these four measurements. This procedure allows an elimination of the compensable errors of the CMM. 4.2. Calibration of the spacers The procedure for calibrating the spacers is similar to that of the ball plate: first the corner spheres of the ball plate positioned on the base plate are measured and the coordinate system is defined. Then the spacer is inserted and the corner spheres are measured again with the presented reversal method by rotating the artefact around its vertical axis in steps of 90◦ . This allows a compensation of the straightness and squareness errors of the Z-axis. By measuring the four corner spheres, the translatory and rotary shift of the ball plate when resting on a spacer with respect to the ball plate resting on the base plate are measured redundantly. The displacement of the ball plate on a spacer with respect to the pose on the base plate can be determined from the averaged vectors for each corner sphere by a best-fit algorithm. 5. Summary and conclusions A measurement procedure for calibrating the 3D-artefact has been introduced. This method is based on a simulation, which
investigates the influence of geometric errors of the CMM on the calibration. A reversal method with four different orientations of the artefact on the CMM is used. This allows the elimination of some CMM errors which do not influence the calibration of the artefact. A measurement uncertainty is estimated: the achieved uncertainty of ball positions in the working volume is 2.1 m in X- and Y-direction and 1.5 m in the Z-direction. References ¨ [1] Bringmann B, Kung A, Knapp W. A measuring artefact for true 3D machine testing and calibrating. Ann CIRP 2005;54:471–4. [2] Bringmann B, Improving geometric calibration methods of multi-axes machining centers by examining error interdependency effects. PhD thesis, ETH Zurich; 2007. [3] Bringmann B, Knapp W. Machine tool calibration: geometric test uncertainty depends on machine tool performance. In: Proceedings for Lamdamap. 2007. p. 211–20. [4] Deutscher Kalibrierdienst (DKD), Guideline for the DKD-calibration of test plates in the form of ball-plate and bore plate, PTB, PTB draft, 1992. [5] Schellenkens P, Rosielle N, Vermeulen H, Vermeulen M, Wetzels S, Pril W. Design for precision: current status and trends. Ann CIRP 1998;47(2):557–86. [6] Hale LC, Slocum AH. Optimal design techniques for kinematic couplings. Precis Eng 2000;25(2001):114–27. [7] Centro Nacional de Metrolog´ıa (CENAM). Calibration of coordinate measuring machine two-dimensional artifacts. CENAM; 2005. [8] ISO 10360-1:2000. Acceptance and reverification tests for coordinate measuring machines (CMM). Part 1. Vocabulary. ISO; 2000. [9] ISO 230-1:1999. Test code for machine tools. Part 1. Geometric accuracy of machines operating under no-load or finishing conditions. ISO; 1999. [10] ISO 5459:1981. Technical drawings – Geometrical tolerancing – Datums and datum-systems for geometrical tolerances. ISO; 1981. [11] Evans C, Hocken RJ, Estler WT. Self-calibration: reversal, redundancy, error separation and ‘absolute testing’. Ann CIRP 1996;45(2):617–34. [12] ISO/TR 230-2:2006. Test code for machine tools. Part 2. Determination of accuracy and repeatability of positioning numerically controlled axes. ISO; 2006. [13] ISO/TR 230-9:2005. Test code for machine tools. Part 9. Estimation of measurement uncertainty for machine tool tests according to series ISO 230 basic equations. ISO; 2005.
Contents lists available at ScienceDirect
Precision Engineering journal homepage: www.elsevier.com/locate/precision
Calibration of a 3D-ball plate T. Liebrich a,∗ , B. Bringmann b , W. Knapp b a
Inspire/ETH Zurich, Tannenstrasse 3, 8092 Zurich, Switzerland Institute of Machine Tools and Manufacturing/ETH Zurich, Tannenstrasse 3, 8092 Zurich, Switzerland b
a r t i c l e
i n f o
Article history: Received 9 July 2007 Received in revised form 25 February 2008 Accepted 27 February 2008 Available online 7 March 2008 Keywords: Machine tools Calibration CMM Ball plate Reversal method Uncertainty estimation
a b s t r a c t The presented 3D-ball plate is used for testing machine tools with a workspace of 500 mm × 500 mm × 320 mm. The artefact consists of a 2D-ball plate which is either located by a kinematic correct coupling on a base plate or on a spacer. The spacers are placed between the base plate and the ball plate and are also kinematic coupled to the other elements of the artefact. The kinematic couplings provide a high repeatability of the measurement setup. Because of the specific application the known calibration procedures for 2D-ball plates are not applicable. A calibration method for the pseudo-3D-artefact on a coordinate measuring machine (CMM) is presented, with the aim to minimise the influence of geometric CMM errors. Therefore a computer simulation is used to analyse the effects of these disturbing errors on the calibration of the ball plate and the spacers. Using a reversal method, the plate is measured at four different horizontal positions after rotating the ball plate around its vertical axis. A couple of the CMM errors, e.g., a squareness error C0Y between the Xand Y-axis of the CMM, can be eliminated by that method—others have to be determined with additional measurements, e.g., the positioning errors EXX or EYY of the X- and Y-axis, respectively. The paper also contains a measurement uncertainty estimation for the calibration by use of experiments, tolerances and Monte Carlo-simulations. The achieved uncertainty for ball positions in the working volume is less than 2.1 m (coverage factor k = 2). © 2008 Elsevier Inc. All rights reserved.
1. Introduction The geometric errors of machine tools influence the accuracy of the produced parts. It is therefore recommended to check the geometric accuracy of machine tools regularly. The presented 3Dartefact enables testing of machining centres with a vertical spindle and a workspace of 500 mm × 500 mm × 320 mm (see Bringmann [1,2]). The 3D-artefact is based on a 2D-ball plate, a standard tool in calibrating CMMs. To create a 3D-artefact the ball plate has to be repositioned in different known locations. Therefore spacers with different heights are inserted between a base plate and the ball plate (see Fig. 1). The resulting translation and rotation of the ball plate with reference to the ball plate on the base plate has to be well known. With this build-up a pseudo-3D-artefact is created defining a grid of points (embodied by spheres), which positions are measured by the machine tool to be checked [2,3]. With a calibrated artefact the relative errors of machine tools at the nominal positions of a grid can be determined by measuring the individual
positions of the spheres. A model of the machine tool with its possible geometric errors is necessary to calculate the real deviations of the machine tool based on the measured grid with an optimization algorithm. Advantages of this measurement procedure are reduced measuring times, reduced measurement uncertainties as well as volumetric measurement and compensation of geometric errors [2]. Typically, the calibration of a ball plate is done by a coordinate measuring machine (CMM). In DKD [4] a method for calibrating 2D-ball plates is presented: the suggested proceeding is a reversal method for rotating the ball plate around its X-, Y- and Z-axis. Because of the specific application of the ball plate in the presented 3D-artefact – it is only used in a horizontal alignment – the known calibration procedures for 2D-ball plates are not applicable. A method for calibrating the 3D-artefact is developed. Its backgrounds are simulations, which investigate the influence of geometric errors of the CMM on the calibration. 2. Artefact and measurement device
∗ Corresponding author. Tel.: +41 44 632 46 76; fax: +41 44 632 11 25. E-mail address: [email protected] (T. Liebrich). 0141-6359/$ – see front matter © 2008 Elsevier Inc. All rights reserved. doi:10.1016/j.precisioneng.2008.02.003
The artefact is made up of a 2D-ball plate which is either located directly on a base plate or on a spacer.
2
T. Liebrich et al. / Precision Engineering 33 (2009) 1–6
Fig. 1. Schematic measurement setup for calibrating machine tools with a 3Dartefact.
The spacers with heights of 80 mm up to 320 mm can be included between the base plate and the ball plate and thus generate a 3D-grid of measuring points (see Fig. 2). With a kinematic coupling between ball plate, spacer and base plate a high repeatability of the measuring setup is obtained [5,6]. The ball plate, a commercially available product and typically used for CMM calibration, consists of 36 spheres creating a quadratic grid with a mesh size of 100 mm. The coordinate system of the ball plate is defined through three spheres in the corners, see CENAM [7]. The coordinate origin is represented by sphere 1, the center points of sphere 1 and sphere 6 define the X-axis. The XY-plane is defined by spheres 1, 6 and 31 and Y is square to the X-axis (see Fig. 3). The measuring device used for calibrating the artefact is a Leitz PMM 864, a CMM in portal design (see ISO [8]). The movement of the table defines the X-axis of the machine coordinate system, the horizontal slide moves in Y-direction and the centre sleeve in Zdirection. Some deviations of the CMM influence the accuracy of the calibration of the artefact. In the following section these deviations are derived.
Fig. 2. Exemplary measurements results (magnification 3000×).
2.1. Machine errors of the CMM The geometric errors of CMMs (and also of machine tools) can be classified in location errors and component errors. Location errors describe the position and orientation between two different axis motions, e.g., squareness or parallelism deviations. Component errors describe errors of the moving components themselves, e.g., positioning or straightness deviations. Every component error CE can be described mathematically by a Fourier series: CE(ω) =
∞
Ai cos(iω − ϕi )
i=1
For an exact description of one component error an infinite number of parameters are needed. In the simulation of the geometric behaviour of the CMM (Section 3), the component errors are classified in linear and harmonic errors. A linear error means that the component error increases linearly with a proceeded length. A harmonic error has the form of the corresponding term of the Fourier series. In the following this is marked with (lin), respectively (harm).
Fig. 3. Ball plate with its coordinate system (left) and measurement setup for calibrating a spacer (right).
T. Liebrich et al. / Precision Engineering 33 (2009) 1–6
3
Fig. 4. Component errors (left) and orientation errors (right) of a linear axis X.
Table 1 List of errors of a CMM with three linear axes Direction
Component errors X-axis
Orientation errors Y-axis
Z-axis
X-axis a
X Y Z A
EXX positioning EYX horizontal straightness EZX vertical straightness EAX roll
EXY horizontal straightness EYY positioning EZY vertical straightness EAY pitch
EXZ straightness in X EYZ straightness in Y EZZ positioning (EAZ)b tilt around X
X0X zero position – – –
B
EBX pitch
EBY roll
(EBZ)b tilt around Y
B0Xa squareness of X to Z
C
ECX yaw
ECY yaw
(ECZ)b roll
C0Xa squareness of X to Y
a b
Y-axis
Z-axis
– Y0Ya zero position – A0Ya squareness of Y to Z –
– – Z0Za zero position A0Z squareness of Z to Y B0Z squareness of Z to X –
C0Y squareness of Y to X
Set to zero as this defines an element of the coordinate system of the CMM. Included for the applied calibration method in EXZ, EYZ and EZZ.
The coordinate system of the CMM is defined by the three linear axes X, Y and Z. A linear axis has six component errors and three orientation errors (see Fig. 4, ISO [9]). The notation of these errors is according to ISO [9]. From the nine orientation errors of a three-axes CMM six have to be cancelled because they define the coordinate system of the CMM: X0X, Y0Y and Z0Z define the origin of the CMM coordinate system; B0X and C0X define the X-axis of the CMM coordinate system; A0Y defines the Y-direction of the CMM coordinate system (see ISO [10]). From this it follows that totally 21 geometric deviations of the CMM exist, which are summarized in Table 1. The tilt and roll movements of the Z-axis (EAZ, EBZ and ECZ) are included in EXZ, EYZ and EZZ and cannot be differed from them because the length of the stylus is constant and not changed during calibration.
the X-axis of the CMM. According to following sections, the CMM errors can be separated by the reversal method in compensable and non-compensable errors, see Tables 2 and 3. 3.1.1. Linear straightness error of Y: EXY(lin) A linear increasing straightness error over the traverse path of 5 m per length of the ball plate has the same effect like a squareness error C0Y between the X- and Y-axis. Fig. 6a shows the resulting
3. Simulation The calibration procedure is derived from the suggested procedure in DKD [4] and is composed of four different alignments of the artefact on the CMM table. Because the ball plate is used in this application only in a horizontal alignment the calibration method also consists only of horizontal measurements, rotated 90◦ with respect to each other (see Fig. 5). This reversal method (see Evans et al. [11]) is reproduced in the simulation which contains a kinematic model of the CMM, evaluating the influence of individual CMM errors on measuring a grid that represents the ball plate. 3.1. Discussion of the influence of CMM errors The different courses of CMM errors and their effects on calibrating the ball plate are discussed by means of a horizontal straightness error EXY. Therefore we assume that for an orientation 0◦ of the artefact the X-axis of the ball plate is parallel to
Fig. 5. Reversal method for calibrating the artefact (pictures A–D): orientation 0◦ , 90◦ , 180◦ and 270◦ ; X, Y: coordinate system of ball plate; XCMM , YCMM : coordinate system of CMM.
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T. Liebrich et al. / Precision Engineering 33 (2009) 1–6
Fig. 6. Simulation of CMM errors (left to right): EXY(lin), EXY(1harm) and EXY(2harm): magnification 5000×; amplitude 5 m. Table 2 Eliminated errors of CMM PMM 864 by reversal method
X-axis Y-axis Z-axis
Component errors
Orientation errors
EYX(lin), EYX(1harm), EAX(lin), EAX(harm), EBX(harm), ECX(lin) EXY(lin), EXY(1harm), EAY(harm), EBY(lin), ECY(harm) EXZ, EYZ, EAZa , EBZa , ECZb
– C0Y A0Z, B0Z
a
EAZ and EBZ cannot be differed from EXZ, EYZ, EZZ because the length of the stylus is not changed. The influence of EAZ and EBZ on EZZ is a second-order effect and therefore they are listed up with compensable CMM errors. b ECZ has no influence on the calibration (only centre probe used) and is therefore listed up with the compensable CMM errors.
grids for orientations of 0◦ and 90◦ of the artefact on the CMM table. Because of the change in sign (symmetry in Y-direction) this error is compensated with a perfect reversal method by rotations of the artefact around 90◦ . CMM errors with this characteristic can be eliminated because of having a difference between the mean value of an arbitrary sphere during the reversal measurements and its nominal position which consists of high order terms and can therefore be neglected. That is the case if the Taylor series, which approximates this difference, does not contain a linear term. Fig. 7 shows that CMM error C0Y, squareness of Y to X, is of second order for the Y-distance between spheres if C0Y is small: L = Ynom (1 − cos(C0Y)). Figs. 7 and 8 show that CMM error C0Y, squareness of Y to X, is eliminated by the reversal method for orientations of the ball plate of 0◦ and 90◦ . For example, sphere 31 has for orientation 0◦ the coordinates 31 r0◦ = (−x,500 − L,0) and for 90◦ orientation 31 r90◦ = (x,500 − L,0). The mean value addicts 31 rmean = (0,500 − L,0). Because L is of second-order C0Y is eliminated (analogous for orientation 180◦ and 270◦ ).
Fig. 7. Difference in Y-direction is a second-order term if C0Y 3◦ .
is chosen 5 m, the phase is 0. For rotations of the artefact around 180◦ the ‘bow’ has a change in sign, therefore EXY(1harm) of the CMM is also eliminated with the suggested reversal method. If the phase is not equal to 0 or , respectively, the elimination is not perfect and a little error emerges which has to be regarded in the uncertainty of the calibration (see Section 3.3). 3.1.3. Second harmonic of straightness error of Y: EXY(2harm) The amplitude of EXY(2harm) is chosen to 5 m and the phase is /2. The resulting course is shown in Fig. 6c. For these parameters the grid has no change in sign and consequently an elimination is not possible by rotating the ball plate around its vertical axis. CMM errors which show no symmetric change in sign cannot be elimi-
3.1.2. First harmonic of straightness error of Y: EXY(1harm) The first harmonic of EXY is described with a half wavelength of a sinus over the length of the ball plate, see Fig. 6b. The amplitude Table 3 Non-compensable errors of the CMM PMM 864
X-axis Y-axis Z-axis
Component errors
Orientation errors
EXX, EYX(2harm), EZX, EBX(lin), ECX(harm) EYY, EXY(2harm), EZY, EAY(lin), EBY(harm), ECY(lin) EZZ
– – –
Fig. 8. Compensable CMM error C0Y (pictures A and B): orientation 0◦ and 90◦ .
T. Liebrich et al. / Precision Engineering 33 (2009) 1–6
Fig. 9. Result of additional measurement EYY.
nated by this reversal method. Again, the influence of the phase has to be considered in the calibration uncertainty (Table 4). 3.2. Additional measurements To have an upper limit of the amplitude of non-compensable CMM errors these values can be taken from the machine specifications or have to be acquired by additional measurements with auxiliary measurement devices. These measurements are exemplified by positioning accuracy and repeatability of Y-axis (EYY). Therefore a comparator system is used which allows also the measurement of horizontal and vertical straightness. The resulting bidirectional accuracy A (see ISO [12]) is determined to 2.6 m (see Fig. 9). So it is traceable to choose the amplitude of geometric CMM errors in the Monte Carlo-simulation arbitrarily within this range. If the measured CMM errors cause a non-negligible influence on the calibration and therefore enlarge the calibration uncertainty, the CMM has to be compensated. That minimises the contribution to the calibration uncertainty due to geometric CMM errors. 3.3. Calibration uncertainty The positions of the spheres are afflicted with an uncertainty due to several perturbing influences during the calibration. These can be classified in contributors coming from the CMM (Table 4) and in contributors coming from the artefact (Table 5). The uncertainty is estimated as recommended in ISO [13]. Thus repeated measurements or Monte Carlo-simulations can be used to calculate the standard uncertainty of each contributor. The evaluation of the standard uncertainty is then based on assumptions which depend on the limit of variation and the type of distribution.
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Fig. 10. Calculation of the standard uncertainty for probing the sphere with a Monte Carlo-simulation (1000 iterations, assumed probing error P = 0.5 m).
The main contributors are drift of the CMM, the repeatability of the positioning of the ball plate on the base plate, respectively, on a spacer and the non-perfect reversal method. A perfect reversal method cannot be realised because it is not possible to rotate the artefact exactly in steps of 90◦ . So the equalisation of the CMM errors succeeds not completely and therefore an uncertainty contributor is caused. Moreover, the non-compensable CMM errors cause a further contributor to the calibration uncertainty (see Section 3.1.3). To investigate the standard uncertainty of a non-perfect reversal method combined with the influence of non-compensable CMM errors a Monte Carlo-simulation is done. This represents the behaviour of the CMM with its possible geometric errors, which are assumed to be random values (rectangular distributed) within a known range. In Fig. 10 the result of such a simulation is shown. For all Monte Carlo-simulations the parameters are chosen arbitrarily within the specified range in 1000 runs. The resulting measurement uncertainties are shown in Table 6. The traceability of the calibration result is provided by the use of calibrated measurement devices and therefore it is included in the calibration uncertainty. 4. Calibration procedure The calibration procedure is classified in calibrating the ball plate and in calibrating the spacers. 4.1. Calibration of the ball plate During calibration the ball plate lies on the base plate which is fixed on the table of the CMM. First the coordinate system of the
Table 4 Standard uncertainty of CMM Contributor
Uncertainty of drift (measurement) Uncertainty of calculating the center of a sphere (Monte Carlo-simulation with uncertainty of the probing system: P = 0.5 m) Uncertainty of a non-perfect reversal method and non-compensable CMM errors (Monte Carlo-simulation with accuracy of orientation of artefact of ±1◦ and geometric CMM errors of ±1.5 m, respectively, ±1.5 m/m) CMM standard uncertainty
Standard uncertainty X (m)
Y (m)
Z (m)
0.29 0.10
0.23 0.10
0.29 0.16
0.85
0.81
0.48
0.90
0.85
0.58
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Table 5 Standard uncertainty of artefact Contributor
Standard uncertainty (m) X
Y
Z
Uncertainty of repeatability of setup (measurement)
0.28 (1) 0.25 (2) 0.37 (3)
0.26 (1) 0.24 (2) 0.24 (3)
0.06 (1) 0.06 (2) 0.25 (3)
Uncertainty of form deviation of spheres (Monte Carlo-simulation with F = 1 m)
0.20
0.21
0.32
Uncertainty of temperature equivalence between CMM and artefact: T = ±0.3 ◦ C, ˛(artefact) = 11.6 m/(m K), ˛(CMM) = 8 m/(m K) (glass scales)
0.31
0.31
0 (1) 0.05 (2) 0.20 (3)
Uncertainty of thermal expansion coefficients ˛ = ±2 m/(m K) at temperature deviation of (T = 20 ◦ C) < 0.3 ◦ C
0.17
0.17
0 (1) 0.03 (2) 0.11 (3)
Artefact standard uncertainty
0.49 (1) 0.47 (2) 0.55 (3)
0.49 (1) 0.48 (2) 0.48 (3)
0.32 (1) 0.33 (2) 0.47 (3)
Index (1) characterizes the calibration of the ball plate on the base plate, the indices (2) and (3) stand for spacers with nominal height of 80 mm and 320 mm, respectively.
Table 6 Measurement uncertainties U(k = 2) for calibrating the 3D-artefact
U(k = 2) base plate U(k = 2) spacer 80 mm U(k = 2) spacer 320 mm
X (m)
Y (m)
Z (m)
2.0 2.0 2.1
2.0 2.0 2.0
1.3 1.3 1.5
ball plate is defined (see Section 2) and afterwards all 36 spheres are measured by probing each sphere with five points. This is done for all four orientations of the artefact on the CMM (see Section 3). The positions of the spheres are calculated by averaging these four measurements. This procedure allows an elimination of the compensable errors of the CMM. 4.2. Calibration of the spacers The procedure for calibrating the spacers is similar to that of the ball plate: first the corner spheres of the ball plate positioned on the base plate are measured and the coordinate system is defined. Then the spacer is inserted and the corner spheres are measured again with the presented reversal method by rotating the artefact around its vertical axis in steps of 90◦ . This allows a compensation of the straightness and squareness errors of the Z-axis. By measuring the four corner spheres, the translatory and rotary shift of the ball plate when resting on a spacer with respect to the ball plate resting on the base plate are measured redundantly. The displacement of the ball plate on a spacer with respect to the pose on the base plate can be determined from the averaged vectors for each corner sphere by a best-fit algorithm. 5. Summary and conclusions A measurement procedure for calibrating the 3D-artefact has been introduced. This method is based on a simulation, which
investigates the influence of geometric errors of the CMM on the calibration. A reversal method with four different orientations of the artefact on the CMM is used. This allows the elimination of some CMM errors which do not influence the calibration of the artefact. A measurement uncertainty is estimated: the achieved uncertainty of ball positions in the working volume is 2.1 m in X- and Y-direction and 1.5 m in the Z-direction. References ¨ [1] Bringmann B, Kung A, Knapp W. A measuring artefact for true 3D machine testing and calibrating. Ann CIRP 2005;54:471–4. [2] Bringmann B, Improving geometric calibration methods of multi-axes machining centers by examining error interdependency effects. PhD thesis, ETH Zurich; 2007. [3] Bringmann B, Knapp W. Machine tool calibration: geometric test uncertainty depends on machine tool performance. In: Proceedings for Lamdamap. 2007. p. 211–20. [4] Deutscher Kalibrierdienst (DKD), Guideline for the DKD-calibration of test plates in the form of ball-plate and bore plate, PTB, PTB draft, 1992. [5] Schellenkens P, Rosielle N, Vermeulen H, Vermeulen M, Wetzels S, Pril W. Design for precision: current status and trends. Ann CIRP 1998;47(2):557–86. [6] Hale LC, Slocum AH. Optimal design techniques for kinematic couplings. Precis Eng 2000;25(2001):114–27. [7] Centro Nacional de Metrolog´ıa (CENAM). Calibration of coordinate measuring machine two-dimensional artifacts. CENAM; 2005. [8] ISO 10360-1:2000. Acceptance and reverification tests for coordinate measuring machines (CMM). Part 1. Vocabulary. ISO; 2000. [9] ISO 230-1:1999. Test code for machine tools. Part 1. Geometric accuracy of machines operating under no-load or finishing conditions. ISO; 1999. [10] ISO 5459:1981. Technical drawings – Geometrical tolerancing – Datums and datum-systems for geometrical tolerances. ISO; 1981. [11] Evans C, Hocken RJ, Estler WT. Self-calibration: reversal, redundancy, error separation and ‘absolute testing’. Ann CIRP 1996;45(2):617–34. [12] ISO/TR 230-2:2006. Test code for machine tools. Part 2. Determination of accuracy and repeatability of positioning numerically controlled axes. ISO; 2006. [13] ISO/TR 230-9:2005. Test code for machine tools. Part 9. Estimation of measurement uncertainty for machine tool tests according to series ISO 230 basic equations. ISO; 2005.