Metrological Apparatus and Method for Adjusting the Attitude of a Rotation-Symmetrical Workpiece

Abstract

A metrological apparatus (15) is disposed for adjustment of an attitude of a workpiece (16) having an arcuate upper surface (17) relative to a rotary axis (C) of the metrological apparatus (15). The workpiece (16) is brought into a first rotary position (c1). A plurality of measured points within a measuring plane on the upper surface (17) is recorded. The workpiece (16) is moved into a further rotary position (c2) about the rotary axis (C), and again measured points in the measuring plane (E) on the upper surface (17) of the workpiece (16) are recorded. Based on these recorded measured points, the actual attitude (Li) of the workpiece (16) deviation from a specified target attitude (Ls) are determined. Adjustment parameters are determined, and an adjustment assembly (24) of the metrological apparatus (15) is activated as a function of the calculated adjustment parameters to adjust the workpiece (16).

Claims

1. Method for adjusting an attitude of a rotation-symmetrical workpiece (16) having an arcuate upper surface (17) in a metrological apparatus (15) that comprises a workpiece support (23) that can be driven about a rotary axis (C), and that—via an adjustment assembly (24)—can be tilted relative to the rotary axis (C) and moved at a right angle relative to the rotary axis (C) in two spatial directions, wherein the metrological apparatus (15) comprises a sensor unit (18) configured to measure measuring points (x.sub.ij, y.sub.ij, z.sub.ij, c.sub.i) in a machine coordinate system (KM) of the metrological apparatus (15) on an outside surface of the workpiece (16), the method comprising: S1: Measuring a first set of measuring points (x.sub.ij, y.sub.ij, z.sub.ij, c.sub.1) in a measuring plane (E) of the machine coordinate system (KM) on the upper surface (17) of the workpiece (16) in a first rotary position (c.sub.1) of the workpiece (16) about the rotary axis (C), S2: Rotating the workpiece support (23) with the workpiece (16) by an angle of rotation (δ) about the rotary axis (C) into a second rotary position (c.sub.2), S3: Measuring of a second set of measuring points (x.sub.2j, y.sub.2j, z.sub.2j, c.sub.2) in the same measuring plane (E) in the machine coordinate system (KM) on the upper surface (17) of the workpiece (16) in the second rotary position (c.sub.2) of the workpiece (16), S4: Determining a tilt angle (γ) and a shift (t) out of an actual attitude (Li) of the workpiece (16) described by the first and second sets of measuring points (x.sub.ij, y.sub.ij, z.sub.ij, c.sub.i) into a target attitude (Ls) of the workpiece (16), S5: Activating the adjustment assembly (24) as a function of the tilt angle (γ) and the shift (t) in order to bring the actual attitude (Li) into coincidence with the target attitude (Ls).

2. Method according to claim 1, wherein the machine coordinate system (KM) has a coordinate axis (ZM) parallel to the rotary axis (C) and two coordinate axes (XM, YM) at a right angle relative to the rotary axis (C).

3. Method according to claim 2, wherein the tilt angle (γ) has a tilt angle component (α, β) about one of the two coordinate axes (XW, YW) of the workpiece coordinate system (KW).

4. Method according to claim 2, wherein the shift (t) has a shift component (t.sub.x, t.sub.y) in a direction of one of the two coordinate axes (XW, YW) of the workpiece coordinate system (KW).

5. Method according to claim 4, wherein the shift (t) has a ZW shift component (t.sub.z) in a direction of a third coordinate axis (ZW) of the workpiece coordinate system (KW), said axis being oriented parallel to the rotary axis (C).

6. Method according to claim 1, further comprising, following step S5, verifying coincidence of the actual attitude (Li) with the target attitude (Ls) in a step S6.

7. Method according to claim 6, further comprising repeating steps S1 to S6 in response to determining in step 6 that the actual attitude (Li) and the target attitude (Ls) do not coincide with a specified accuracy.

8. Method according to claim 1, further comprising converting in step S4 the first and second sets of measuring points (x.sub.ij, y.sub.ij, z.sub.ij, c.sub.i) for determining the tilt angle (γ) and the shift (t) into workpiece coordinate system measurement points (x.sub.i,j.sup.KW, y.sub.i,j.sup.KW, z.sub.i,j.sup.KW) in the workpiece coordinate system (KW) of the workpiece (16).

9. Method according to claim 1, further comprising fitting in step S4 the workpiece coordinate system measurement points (x.sub.i,j.sup.KW, y.sub.i,j.sup.KW, z.sub.i,j.sup.KW) for determining the tilt angle (γ) and the shift (t) into a known target geometry of the upper surface (17) of the workpiece (16) in such a manner that deviation between the geometry described by the workpiece coordinate system measurement points (x.sub.i,j.sup.KW, y.sub.i,j.sup.KW, z.sub.i,j.sup.KW) and the target geometry is minimal.

10. Method according to claim 9, further comprising specifying the target geometry of the workpiece (16) in a way to also describe the target attitude (Ls) concurrently.

11. Method according to claim 10, further comprising allocating each of the workpiece coordinate system measurement points (x.sub.i,j.sup.KW, y.sub.i,j.sup.KW, z.sub.i,j.sup.KW) to a target point (P.sub.ijx, P.sub.ijv, P.sub.ijz) of the target geometry and, depending on the vectors (V.sub.ij) of each of the workpiece coordinate system measurement points (x.sub.i,j.sup.KW, y.sub.i,j.sup.KW, z.sub.i,j.sup.KW) to the allocated target point (P.sub.ijx, P.sub.ijv, P.sub.ijz), determining the tilt angle (γ) and the shift (t) in such a manner that deviation between the measured geometry and the target geometry is minimal.

12. Method according to claim 9, wherein the coordinate origin of the workpiece coordinate system (KW) is located on an axis of symmetry (A) of the workpiece (16) and on the upper surface (17).

13. Method according to claim 1, further comprising performing the steps S2 and S3 N times, wherein N is a natural number unequal to zero, so that the measuring points (x.sub.ij, y.sub.ij, z.sub.ij, c.sub.i) are measured in at least two rotary positions (c.sub.1, c.sub.2), wherein a target angle of rotation (C.sub.i+1-C.sub.i) is equal to 180° divided by N+1.

14. Method according to claim 13, wherein the performing the steps S2 and S3 N times brings an adjusted angle of rotation (δ) to within 10° of the target angle of rotation (C.sub.i+1-C.sub.i).

15. Method according to claim 13, further comprising performing the steps 51 through S5 for a specified number of iterations M, wherein performing the steps S2 and S3 for the N times of rotary positions (c.sub.i) achieves an accuracy of the axis of symmetry (A) of the workpiece (16) along the rotary axis (C) greater than performing the steps S2 and S3 for less than N times.

16. Method according to claim 14, further comprising performing the steps S2 and S3 for the number N of rotary positions (c.sub.i) for which a measuring value sequence of measuring points (x.sub.ij, y.sub.ij, z.sub.ij, c.sub.i) is measured, wherein the steps S1 through S5 are performed for fewer iterations M to a specified accuracy of adjustment of the axis of symmetry (A) of the workpiece (16) along the rotary axis (C).

17. Method according to claim 1 wherein, whenever a position of adjustment axes (25, 26) of the adjustment assembly (24) is not known with sufficient accuracy, step S5 further comprises: S5a: Activating of the adjustment assembly (24) for tilting the workpiece (16) as a function of the tilt angle (γ), S5b: Rotating of the workpiece support (23) with the workpiece (16) into a first adjusted rotary position (c.sub.i) and measuring a first adjusted set of measuring points (x.sub.ij, y.sub.ij, z.sub.ij, c.sub.i) in the measuring plane (E) of the machine coordinate system (KM) on the upper surface (17) of the workpiece (16) in the first rotary position (c.sub.i), S5c: Rotating of the workpiece support (23) with the workpiece (16) into a second adjusted rotary position (c.sub.i+1) and measuring a second adjusted set of measuring points x.sub.i+1,j, y.sub.i+1,j, z.sub.i+1,j, c.sub.i+1) in the measuring plane (E) of the machine coordinate system (KM) on the upper surface (17) of the workpiece (16) in the second rotary position (c.sub.i+1), S5d: Determining an actual value of the tilt angle (γ) and the shift (t) out of the actual attitude (Li) of the workpiece (16) described by the measured first adjusted set of measuring points ((x.sub.ij, y.sub.ij, z.sub.ij, c.sub.i)) and the second adjusted set of measuring points (x.sub.i+1,j, y.sub.i+1,j, z.sub.i+1,j, c.sub.i+1) into the target attitude (Ls) of the workpiece (16), S5e: Activating the adjustment assembly (24) as a function of the shift (t) from step S5d in order to bring the actual attitude (Li) in coincidence with the target attitude (Ls).

18. Metrological apparatus (15) disposed for aligning a rotation-symmetrical workpiece (16) having an arcuate upper surface (17), the apparatus comprising: a workpiece support (23) that can be driven about a rotary axis (C), said workpiece support configured to be tilted, via an adjustment assembly (24), relative to the rotary axis (C) and configured to be movable at a right angle relative to the rotary axis (C) in two spatial directions, a sensor unit (18) configured to measure measuring points (x.sub.ij, y.sub.ij, z.sub.ij, c.sub.i) in a machine coordinate system (KM) of the metrological apparatus (15) on an outside surface of the workpiece (16), and a control device (21) that is configured to: S1: measure a first set of measuring points (x.sub.ij, y.sub.ij, z.sub.ij, c.sub.1) in a measuring plane (E) of the machine coordinate system (KM) on the upper surface (17) of the workpiece (16) in a first rotary position (c.sub.1) of the workpiece (16) about the rotary axis (C), S2: rotate Rotating the workpiece support (23) with the workpiece (16) by an angle of rotation (6) about the rotary axis (C) into a second rotary position (c.sub.2), S3: measure a second set of Measuring of several measuring points (x.sub.2j, y.sub.2j, z.sub.2j, c.sub.2) in the same measuring plane (E) in the machine coordinate system (KM) on the upper surface (17) of the workpiece (16) in the second rotary position (c.sub.2) of the workpiece (16), S4: determine a tilt angle (γ) and a shift (t) out of an actual attitude (Li) of the workpiece (16) described by the first and second sets of measuring points (x.sub.ij, y.sub.ij, z.sub.ij, c.sub.i) into a target attitude (Ls) of the workpiece (16), S5: activate Activating the adjustment assembly (24) as a function of the tilt angle (γ) and the shift in order to bring the actual attitude (Li) into coincidence with the target attitude (Ls).

Description

BRIEF DESCRIPTION OF THE DRAWINGS

[0032] Advantageous embodiments of the invention can be inferred from the dependent claims, the description and the drawings. Hereinafter, preferred exemplary embodiments are explained in detail with reference to the appended drawings. They show in

[0033] FIG. 1 a schematic representation of the principle of an exemplary embodiment with a workpiece that is to be measured,

[0034] FIG. 2 a schematic representation of the principle of the determined shift of the axis of symmetry of the workpiece relative to an rotary axis of the metrological apparatus,

[0035] FIGS. 3a to 3c a representation of the principle of the adjustment of the workpiece, wherein the axis of symmetry of said workpiece is oriented along the rotary axis of the metrological apparatus,

[0036] FIG. 4 a representation of the principle of a first measurement of an upper surface of the workpiece in a measuring plane in a first rotary position,

[0037] FIG. 5 a representation of the principle of a first measurement of an upper surface of the workpiece in a measuring plane in a second rotary position,

[0038] FIG. 6 a qualitative representation of an exemplary measurement in the first rotary position,

[0039] FIG. 7 a qualitative representation of an exemplary measurement in the second rotary position,

[0040] FIG. 8 a representation of the principle for adapting the measured points on the surface of the workpiece in a target geometry of the surface, and

[0041] FIG. 9 a representation of the principle of a further procedure for fitting the measured values in a known target geometry of the upper surface of the workpiece. FIG. 1 schematically shows an exemplary embodiment of a metrological apparatus 15 in the manner of a block diagram. The metrological apparatus 15 may be a coordinate measuring device or a shape measuring device and can be universally used for several measuring tasks. It is disposed for measuring a rotation-symmetrical workpiece 16 with an arcuate upper surface 17. To do so, measured points on the upper surface 17 can be recorded. In so doing, each measuring point is defined by a point in a machine coordinate system KM. In accordance with the example, the machine coordinate system is embodied as a Cartesian coordinate system with three coordinate axes, namely an X-coordinate axis XM in an X-direction, a Y-coordinate axis YM in Y-direction and a Z-coordinate axis ZM in Z-direction.

DETAILED DESCRIPTION

[0042] A workpiece coordinate system KW has a Z-coordinate axis ZW that is oriented along an axis of symmetry A of the rotation-symmetrical workpiece 16. The coordinate origin is preferably located at the point at which the axis of symmetry A intersects the arcuate upper surface 17. In the exemplary embodiment, the upper surface 17 is curved in a convex manner and has its zenith in the coordinate origin on the axis of symmetry A. The workpiece 16, for example, is a workpiece having an aspherical upper surface 17. Furthermore, the workpiece coordinate system KW has an X-coordinate axis XW and a Y-coordinate axis YW, each being oriented at a right angle relative to each other and at a right angle relative to the Z-coordinate axis ZW, whereby they form a Cartesian coordinate system KW.

[0043] The metrological apparatus 15 comprises a sensor unit 18 for recording measured points on the outside surface of the workpiece 16. In so doing, the sensor unit 18 can be moved relative to the workpiece 16 via a machine axle assembly 19. In the exemplary embodiment described here, the machine axle assembly 19 comprises three translatory machine axles that can be used to move the sensor unit 18 respectively in the direction of one of the coordinate axes XM, YM, ZM of the machine coordinate system KM. In addition or as an alternative to the translatory machine axles of the machine axle assembly 19, there could also be provided one or more rotative machine axles.

[0044] The sensor unit 18 comprises a sensor 20, for example a sensor 20 that takes measurements by optical or tactile means. The sensor unit 18 transmits a sensor signal to a control device 21. Based on the sensor signal and the rotary position of the workpiece support 23 around the rotary axis C, it is possible to allocate a coordinate quadruple x.sub.ij, y.sub.ij, z.sub.ij, c.sub.i in the machine coordinate system KM to each measured point on the outside surface—in accordance with the example, on the upper surface 17 of the workpiece 16. The control device 21 also controls the machine axle assembly 19 with the translatory and/or rotatory machine axles.

[0045] Furthermore, the metrological apparatus 15 comprises a workpiece support 23 that can be driven about a rotary axis C by means of a rotative machine axle 22. The workpiece support 23 is disposed to hold the workpiece 16. To do so, the workpiece support 23 may comprise a workpiece holder device suitable therefor, so that the workpiece cannot move relative to the workpiece support 23 during a measurement. The rotative machine axle 22 is activated by the control device 21.

[0046] Furthermore, the metrological apparatus 15 comprises an adjustment assembly 24. Via the adjustment assembly 24, it is possible to move the workpiece support 23 with the workpiece 16 arranged thereon into a desired attitude for measurement. To do so, the adjustment assembly 24 has at least one—according to the example several—adjustment axles that may be configured as translatory axles or pivot axles or tilt axles. In the illustrated example, the adjustment assembly 24 comprises a tilt table 25, by means of which the workpiece support 23 and the workpiece 16 arranged thereon, respectively, can be pivoted or tilted about two axes that are oriented at a right angle relative to each other.

[0047] The tilt table 25 can tilt the workpiece support 23 with the workpiece 16 about a first tilt axis that is oriented parallel to the X-coordinate axis XW of the workpiece coordinate system KW and about a second tilt axis that is oriented parallel to the Y-coordinate axis YW of the workpiece coordinate system KW.

[0048] Furthermore, the adjustment assembly 24 comprises two translatory adjustment axles 26, in which case the one translatory adjustment axle can shift the workpiece support 23 parallel to the X-coordinate axis of the workpiece coordinate system KW, and the respectively other translatory adjustment axle 26 can shift the workpiece support 23 parallel to the Y-coordinate axis of the workpiece coordinate system KW.

[0049] The adjustment assembly 24 is activated by the control device 21. The control device 21 specifies the rotary position c.sub.i about the rotary axis C, as well as the positions for the machine axle assembly 19. The respective attitude or position values relative to the machine coordinate system KM are thus known as target values in the control device 21. It is possible to detect the respective positions of the axles by sensory means and to transmit these to the control device 21, so that actual values are also present in the control device 21, for example in order to perform an adjustment.

[0050] Via the adjustment assembly 24, it is possible to tilt the axis of symmetry A of the workpiece 16 or a workpiece support axis relative to the rotary axis C and to move it relative to the rotary axis C in X-direction and Y-direction of the workpiece coordinate system KW via the adjustment assembly 24.

[0051] FIG. 1 schematically illustrates that the workpiece 16 arranged on the workpiece support 23 is oriented—relative to the rotary axis C—not in a desired target attitude Ls. The axis of symmetry A is inclined relative to the rotary axis C or the Z-coordinate axis ZM of the machine coordinate system KM and/or shifted in the X-Y-plane of the machine coordinate system KM. Therefore, the attitude of the axis of symmetry A can generally be stated by a tilt angle γ and a shift t in the X-Y-plane of the workpiece coordinate system KW (FIGS. 2 and 3). It is also possible, to define the shift t not only in one plane but also in space (shift t* in FIG. 2), so that the shift t* displays not only an X-component t.sub.x, a Y-component t.sub.y, but—in addition—a Z-component t.sub.x. Consequently, it is possible to bring the coordinate origin of the workpiece coordinate system KW and the coordinate origin of the machine coordinate system KM to coincide with each other. In this case, the adjustment assembly 24 comprises a translatory adjustment axle for moving the workpiece support 23 in Z-direction. At this point it should be pointed out that—to avoid confusion—the machine coordinate system KM in FIG. 1 is drawn under the workpiece support 23. The origin of the machine coordinate system KM in the last-mentioned case would have to be defined above the workpiece support 23. Said origin can be specified at any point along the rotary axis C by appropriate initialization, depending on the workpiece 16 to be measured.

[0052] Generally, the tilt angle γ can be described by two tilt angle components α, β. Therefore, the tilt angle component about the X-coordinate axis XW is referred to as the tilt angle component a, and the tilt angle component about the Y-coordinate axis of the workpiece coordinate system KW is referred to as the tilt angle component β. Both tilt angle components α, β together result in the tilt angle γ between the orientation of the axis of symmetry A and the line parallel to the rotary axis C or to the Z-coordinate axis ZM of the machine coordinate system KM.

[0053] The shift t results from a vector addition of the shift components in the different spatial directions, for example from a shift component t.sub.x along the X-coordinate axis XW and a shift component t.sub.y along the Y-coordinate axis YW of the workpiece coordinate system KW.

[0054] FIGS. 3a to 3c show the principle of adjustment in a schematic manner. The workpiece 16 that initially assumes a random actual attitude Li is tilted via the adjustment assembly 24—in accordance with the example, the tilt table 25—by the tilt angle γ, so that the axis of symmetry A is oriented parallel to the rotary axis C (FIG. 3b). Subsequently, the workpiece 16 is moved by the shift t, so that the axis of symmetry A is oriented along the rotary axis C, and the workpiece 16 assumes the target attitude Ls (FIG. 3c). In order to be able to perform the adjustment, the tilt angle γ and the shift t must be determined. The method for determining these parameters will be explained hereinafter with reference to FIGS. 4 to 9.

[0055] FIGS. 4 and 5 are greatly simplified plan views of the workpiece support 23 and the workpiece 16 arranged thereon. The coordinate origin UW of the workpiece coordinate system KW is marked. The coordinate origin UW corresponds to the zenith of the workpiece 16 or the puncture point of the axis of symmetry A through the upper surface 17. The rotary axis C extends perpendicularly to the plane of projection.

[0056] In a first step S1, the workpiece support 23 and the workpiece 16, respectively, are moved into a first rotary position or the actual rotary position is defined as the first rotary position c.sub.1. In this first rotary position c.sub.1 in the first step S1, several measured values in a measuring plane E are recorded. In accordance with the example, the measuring plane E is defined by the machine coordinate axes XM, ZM and thus oriented at a right angle relative to the Y-coordinate axis YM of the machine coordinate system KM. The rotary axis C thus extends inside the measuring plane E. As an alternative thereto, the measuring plane E may also be arranged parallel to and at a distance from the rotary axis C. In this first rotary position c.sub.1, the sensor unit 15 records the coordinates of several measured points x.sub.1j, y.sub.1j, z.sub.1j, c.sub.1 on the upper surface 17.

[0057] In a second step S2, the workpiece support 23 with the workpiece 16 is rotated by an angle of rotation 5 about the rotary axis C into a second rotary position c.sub.2. In the exemplary embodiment, the angle of rotation δ is approximately 90° and may deviate from the target angle of rotation by preferably at most by 5°. In this second rotary position c.sub.2, the measuring points x.sub.2j, y.sub.2j, z.sub.2j, c.sub.2 are measured in the same measuring plane E relative to the machine coordinate system KM (step S3).

[0058] At least two measurements are recorded in two different rotary positions c.sub.1, c.sub.2 with respectively several measuring points x.sub.ij, y.sub.ij, z.sub.ij, c.sub.i. The index i=1, 2, 3, . . . thus indicates the number of the measured value sequence during the measurement of several measuring points in one of the rotary positions. It is also possible to perform two measurements in more than two different rotary positions. The index j=1, 2, 3, . . . indicates the number of the measuring point in a measured value sequence. The number of measuring points in the measuring plane E along the upper surface 17 is selected as a function of the required accuracy.

[0059] Generally, the target angle of rotation δ.sub.soll between two consecutive rotary positions c.sub.i and c.sub.i+1 is calculated as follows:

[00001]$\begin{array}{cc}{\delta }_{\mathrm{soll}}=\frac{180.Math.°}{M}& \left(1\right)\end{array}$

where M is the number of measured value sequences i=1, 2, 3, . . . , M in various rotary positions. In the event of measurements in two different rotary positions c.sub.i, c.sub.2, the target angle of rotation δ.sub.soll thus is equal to 90°. The more rotary positions are used for respectively one measured value sequence, the fewer iterations are required for achieving a specified accuracy of the adjustment of the axis of symmetry A of the workpiece 16 along the rotary axis C. Correspondingly, with a specified number of iterations due to a greater number of rotary positions, it is possible to achieve an improved accuracy of the adjustment of the axis of symmetry of a workpiece 16 along the rotary axis C.

[0060] The recorded measured points x.sub.ij, y.sub.ij, z.sub.ij, c.sub.i describe the actual attitude Li of the workpiece 16. The target geometry of the workpiece surface 17 of the measured workpiece 16, for example an aspherical surface, is known. By fitting the measured points x.sub.ij, y.sub.ij, z.sub.ij, c.sub.i in the known target geometry, it is thus possible to determine the actual attitude Li. The deviation of the actual attitude Li from the desired target attitude Ls of the workpiece 16 can be described by the shift t and the tilt angle y. If the target geometry for the upper surface 17 or the workpiece 16, at the same time, also defines the target attitude, it is possible by fitting the measuring points in the target attitude Ls, also at the same time, to determine the shift t and the tilt angle γ.

[0061] The Z-contour lines described by the measured points x.sub.ij, y.sub.ij, z.sub.ij, c.sub.i along the X-coordinate axis XM of the machine coordinate system KM are illustrated as a qualitative example in FIGS. 6 and 7. Each measurement in a rotary position c.sub.i describes such a contour line.

[0062] Each measured point of a measurement can be described by its coordinates x.sub.ij, y.sub.ij, z.sub.ij in the space of the machine coordinate system KM, as well as the rotary position c.sub.i, so that a coordinate quadruple x.sub.ij, y.sub.ij, z.sub.ij, c.sub.i is the result. As explained, the index i refers to the number of the measured value sequence, and the index j refers to the number of the point of a measured value sequence. In accordance with the example, the measured points are first transformed by the machine coordinate system KM into the workpiece coordinate system KW:

[00002]$\begin{array}{cc}\left(\begin{array}{c}{x}_{i,j}^{\mathrm{KW}}\\ y_{i,j}^{\mathrm{KW}}\\ z_{i,j}^{\mathrm{KW}}\end{array}\right)=\left(\begin{array}{ccc}\cos \left(c_i\right)& -\sin \left(c_i\right)& 0\\ \sin \left(c_i\right)& \cos \left(c_i\right)& 0\\ 0& 0& 1\end{array}\right)*\left(\begin{array}{c}{x}_{i,j}\\ y_{i,j}\\ z_{i,j}\end{array}\right)& \left(2\right)\end{array}$

[0063] The obtained measured points x.sub.i,j.sup.KW, y.sub.i,j.sup.KW, z.sub.i,j.sup.KW are fitted into the known target geometry while minimizing the deviation. Two basic options of such a fitting are shown in a highly schematic manner by FIGS. 8 and 9. Each of the individual measured points x.sub.i,j.sup.KW, y.sub.i,j.sup.KW, z.sub.i,j.sup.KW can be allocated a target point P.sub.ij on the upper surface 17 of the workpiece 16 in the target attitude Ls—the target geometry of the upper surface 17 being known. For example, the allocation for all measured points x.sub.i,j.sup.KW, y.sub.i,j.sup.KW, z.sub.i,j.sup.KW may take place along the same direction (FIG. 8), for example, parallel to the Z-coordinate axis ZM or, alternatively, in the direction of the normal (FIG. 9) relative to the upper surface 17 of the workpiece 16 that is in the target attitude Ls. From each measured point x.sub.i,j.sup.KW, y.sub.i,j.sup.KW, z.sub.i,j.sup.KW, a vector V.sub.ij points to an allocated target point P.sub.ij on the upper surface 17 of the workpiece 16. Each of the vectors V.sub.ij represents the deviation of the actual attitude Li from the target attitude Ls, and the sough adjustment parameters, i.e., the tilt angle γ and the shift t, can be determined based on these. In so doing, the adjustment parameters (tilt angle y, shift t) are determined in such a manner that the deviation during the fitting of the measured points x.sub.i,j.sup.KW, y.sub.i,j.sup.KW, z.sub.i,j.sup.KW into the target geometry of the upper surface 17 is as small as possible. For this purpose, a suitable measure of quality may be determined. For example, the method of the smallest error squares or another known mathematical procedure may be employed in order to minimize the determined measure of quality or the deviation.

[0064] After the determination of the shift t and the tilt angle γ in the fourth step S4, the adjustment assembly 24 is activated by the control device 21. In so doing, the tilt axles of the tilt table 25 are first activated in order to perform the appropriate tilting of the workpiece 16 in order to orient the axis of symmetry A parallel to the rotary axis C. Subsequently, the translatory adjustment axles 26 are activated in order to shift the axis of symmetry A of the workpiece 16 toward the rotary axis C. The activation of the adjustment assembly 24 aims to bring the actual attitude Li of the workpiece 16 to coincide with the target attitude Ls.

[0065] In a sixth step S6 a repeated measurement may be performed in one or more rotary positions, and the deviation of the actual attitude Li from the target attitude Ls can be evaluated. If a desired accuracy has not been reached yet, actual values for the tilt angle γ and the shift t can be calculated and the adjustment assembly 24 can be activated consistent with the calculated parameters. This iterative process can be repeated several times until the specified accuracy is reached or no substantial improvement of accuracy can be achieved by additional iterations.

[0066] The method explained hereinabove in general will be illustrated with reference to an example hereinafter. The mean squared error is minimized here as the measure of quality in order to fit the transformed measured points x.sub.i,j.sup.KW, y.sub.i,j.sup.KW, z.sub.i,j.sup.KW in the target geometry or the target attitude Ls.

[0067] The nominal design (target geometry) of the workpiece 16 or the upper surface 17 is known. In accordance with the example, this is an asphere. The height value or z-value z.sub.asp of the asphere may be expressed, for example, by the following asphere formula:

[00003]$\begin{array}{cc}{z}_{\mathrm{asp}}\left(x^{\mathrm{KW}},y^{\mathrm{KW}}\right)=z_{\mathrm{asp}}\left(r^{\mathrm{KW}}\right)=\frac{{\left(r^{\mathrm{KW}}\right)}^2/R_0}{1+\sqrt{1-\left(1+k\right).Math.{\left(\frac{r^{\mathrm{KW}}}{R_0}\right)}^2}}+\underset{n=2}{\overset{N}{.Math.}}.Math..Math.{A_{2.Math..Math.n}\left(r^{\mathrm{KW}}\right)}^{2.Math..Math.n}.Math..Math.\mathrm{with}& \left(3\right)\\ .Math.r^{\mathrm{KW}}=\sqrt{{\left(x^{\mathrm{KW}}\right)}^2+{\left(y^{\mathrm{KW}}\right)}^2}& \left(4\right)\end{array}$

[0068] where R.sub.0 is the base radius of the asphere, k is the conical constant, and A.sub.2n are the aspheric coefficients. In equation (3), the origin of the workpiece coordinate system KW is in the zenith of the asphere.

[0069] The measured points transformed into the workpiece coordinate system KW are thus fitted in the target geometry (nominal aspheric design) given by equation (3) in such a manner that the mean squared error between the z-coordinate of the fitted in points and the z-value z.sub.asp of the target geometry at the allocated locations x, y of the fitted-in measured points is minimal. To do so, according to the example, two rotation or tilt parameters (tilt angle components α, β) and two or three translation parameters (shift components t.sub.x, t.sub.y and optionally t.sub.z) are determined so that the following expression is minimal:

Σ.sub.i=1.sup.IΣ.sub.j=1.sup.J.sup.i({tilde over (z)}.sub.i,j.sup.KW−z.sub.asp({tilde over (x)}.sub.i,j.sup.KW, {tilde over (y)}.sub.i,j.sup.KW)).sup.2  (5)

In so doing, {tilde over (x)}.sub.i,j.sup.KW, {tilde over (y)}.sub.i,j.sup.KW und {tilde over (z)}.sub.i,j.sup.KW, are calculated by means of

[00004]$\begin{array}{cc}\left(\begin{array}{c}{\tilde{x}}_{i,j}^{\mathrm{KW}}\\ {\tilde{y}}_{i,j}^{\mathrm{KW}}\\ {\tilde{z}}_{i,j}^{\mathrm{KW}}\end{array}\right)=\left(\begin{array}{ccc}1& 0& 0\\ 0& \cos \left(\alpha \right)& -\sin \left(\alpha \right)\\ 0& \sin \left(\alpha \right)& \cos \left(\alpha \right)\end{array}\right)*\left(\begin{array}{ccc}\cos \left(\beta \right)& 0& \sin \left(\beta \right)\\ 0& 1& 0\\ -\sin \left(\beta \right)& 0& \cos \left(\beta \right)\end{array}\right).Math.\left(\begin{array}{c}{x}_{i,j}^{\mathrm{KW}}\\ y_{i,j}^{\mathrm{KW}}\\ z_{i,j}^{\mathrm{KW}}\end{array}\right)+\left(\begin{array}{c}{t}_x\\ t_y\\ t_z\end{array}\right)& \left(6\right)\end{array}$

[0070] Inasmuch as, in accordance with the example, the target geometry has the coordinate origin of the workpiece coordinate system KW at the zenith of the asphere and the rotary axis C in the machine coordinate system KM extends through the coordinate origin of the machine coordinate system KM, the shift components t.sub.x in X-direction of the workpiece coordinate system KW and t.sub.y in Y-direction of the workpiece coordinate system KW indicate the shift of the axis of symmetry A of the workpiece 16 at a right angle relative to the rotary axis C, and the tilt angle components α about the X-direction of the workpiece coordinate system KW and β about the Y-direction of the workpiece coordinate system KW form the tilt angle γ between the axis of symmetry A and the rotary axis C.

[0071] After determining the shift t and the tilt angle γ, the adjustment assembly 24 can be activated accordingly in order to bring the actual attitude Li into coincidence with the target attitude Ls. As explained, the method can be repeated iteratively in order to increase the accuracy.

[0072] In the aforementioned exemplary embodiments the arrangement of the adjustment axles of the adjustment assembly 24 was assumed as being known. As is schematically illustrated by FIG. 3a, the workpiece 16 is pivoted by means of the tilt table 25 about a tilt point KS that is located at a distance from the workpiece 16. If this distance is not exactly known, the method described hereinabove can be modified as described hereinafter.

[0073] Following the determination of a first value for the tilt angle γ (in the fourth step S4), the tilt table 25 can first be activated in order to move the workpiece support 23 consistent with the determined tilt angle γ.

[0074] Subsequently, the workpiece support 23, for example, is again moved into the first rotary position c.sub.1, and another measurement of a measuring value sequence in the measuring plane E is performed. Then the workpiece support 23 is rotated, for example into the second rotary position c.sub.2, and again a measurement of a measuring value sequence in the measuring plane E is performed. Following these two measurements of the measuring value sequences, again an actual value for the tilt angle and the shift is calculated based on the actual measurements. Inasmuch as after a tilting of the workpiece support 23 and the repeated measurement it is known how the tilt attitude of the workpiece support 23 or the workpiece 16 has changed, it is now possible—based on the actual measurements and the determined actual tilt angle, as well as the determined actual shift—to activate the adjustment assembly 24 in order to bring the actual attitude Li into coincidence with the target attitude Ls. It is also understood in this method, that a measurement is performed in at least three rotary positions; however, it also possible to perform measurements in more than three rotary positions as has already been explained in conjunction with other exemplary embodiments.

[0075] The invention relates to a method and a metrological apparatus 15 that is disposed for the adjustment of an attitude of a workpiece 16 having an arcuate upper surface 17 relative to a rotary axis C of the metrological apparatus 15. The workpiece 16 is brought by a workpiece support 23 into a first rotary position c.sub.1. A plurality of measured points within a measuring plane on the upper surface 17 is recorded. The workpiece 16 is moved into a further rotary position c.sub.2 about the rotary axis C, and again measured points in the measuring plane E on the upper surface 17 of the workpiece 16 are recorded. Based on these recorded measured points, it is possible to determine the actual attitude Li of the workpiece 16, as well as the deviation from the specified target attitude Ls. In the target attitude Ls, the axis of symmetry A of the workpiece 16 is moved to coincide with the rotary axis C. To do so, adjustment parameters, for example a tilt angle γ and a shift t, are determined, and an adjustment assembly 24 of the metrological apparatus 15 is activated as a function of the calculated adjustment parameters in order to adjust the workpiece 16.

LIST OF REFERENCE SIGNS

[0076] 15 Metrological apparatus [0077] 16 Workpiece [0078] 17 Upper surface of the workpiece [0079] 18 Sensor unit [0080] 19 Machine axle assembly [0081] 20 Sensor [0082] 21 Control device [0083] 22 Rotative machine axle [0084] 23 Workpiece support [0085] 24 Adjustment assembly [0086] 25 Tilt table [0087] 26 Translatory adjustment axle [0088] α Tilt angle component around the x-direction of the workpiece coordinate system [0089] β Tilt angle component around the y-direction of the workpiece coordinate system [0090] γ Tilt angle [0091] δ Angle of rotation [0092] δ.sub.soll Target angle of rotation [0093] A Axis of symmetry of the workpiece [0094] C Rotary axis [0095] c.sub.i Rotary position of the workpiece about the rotary axis in the machine coordinate system KM [0096] KM Machine coordinate system [0097] KW Workpiece coordinate system [0098] KS Tilt point [0099] Li Actual attitude [0100] Ls Target attitude [0101] P.sub.ij Target point [0102] t Shift [0103] t.sub.x Shift component in x-direction of the workpiece coordinate system [0104] t.sub.y Shift component in y-direction of the workpiece coordinate system [0105] t.sub.z Shift component in z-direction of the workpiece coordinate system [0106] UW Coordinate origin of the workpiece coordinate system [0107] V.sub.ij Vector [0108] x.sub.ij Coordinate of a measuring point in the machine coordinate system [0109] x.sub.i,j.sup.KW x-coordinate of a measuring point in the machine coordinate system [0110] XM X-coordinate axis of the machine coordinate system [0111] XW X-coordinate axis of the of the workpiece coordinate system [0112] y.sub.ij y-coordinate of a measuring point in the machine coordinate system [0113] y.sub.i,j.sup.KW y-coordinate of a measuring point in the workpiece coordinate system [0114] YM Y-coordinate axis of the machine coordinate system [0115] YW Y-coordinate axis of the workpiece coordinate system [0116] z.sub.ij z-coordinate of a measuring point in the machine coordinate system [0117] z.sub.i,j.sup.KW z-coordinate of a measuring point in the workpiece coordinate system [0118] ZM Z-coordinate axis of the machine coordinate system [0119] ZW Z-coordinate axis of the workpiece coordinate system