Surface form measurement

[0002] This invention relates to a method of surface form measurement and is particularly applicable to the measurement of the radial or cylindrical form of an object such as a workpiece. Radial form is related to the roundness of an object and gives an indication of its deviation from circularity. Cylindrical form provides a measure of how close in shape the surface is to a right cylinder.

[0003] In known apparatus for measuring cylindrical form such as the applicants' Talyrond TR 200 (trade mark) measurement apparatus, a generally cylindrical workpiece to be measured is mounted on a rotatable support or turntable. The rotatable support is centred and levelled, for example in the manner described in the applicants' EP-A-0240150 the contents of which are incorporated herein by reference, so that the cylindrical axis of the workpiece is coincident with the spindle of the turntable which defines a rotational axis datum. An axial straight datum is defined by a support column located offset from but parallel to the rotational axis datum. The support column carries a probe arm which is moveable along and transversely of the support column to allow a stylus carried by the probe arm to contact the surface of the workpiece.

[0004] The radial or cylindrical form of the workpiece is measured by rotating the turntable and measuring the displacement of the stylus relative to the axial straight datum. As the axis of the cylinder is aligned with the rotational axis datum, the displacement of the stylus relative to the axial straight datum provides a measurement of the radius at each angular measurement position during rotation of the rotatable support. The radial form or deviation of the cross-section of the workpiece from a circular cross-section can thus be determined. In addition, the cylindrical form of the workpiece may be determined by making measurements at different heights along the workpiece by moving the probe arm along the axial straight datum. Movement of the probe arm transversely and longitudinally of the axial straight datum is measured by appropriate gauges, for example optical or linear variable differential transducer (LVDT) gauges.

[0005] The accuracy of measurements made using such apparatus depends on the accuracy of the rotational axis datum and, especially for cylindrical form, the axial straight datum. It is possible to define the spindle or rotational axis datum to, typically, within ±25 nm (nanometers) so, generally, the mechanical performance of the axial straight datum is the limiting source of error for radial and cylindrical form. Mechanically related errors in the axial straight datum may arise from, for example, long term mechanical instability, for example changes with temperature or other environmental factors, inaccuracies in the actual straightness of the axial straight datum or its parallelism to the rotational axis datum and variability of the interface between the support column defining the axial straight datum and the carriage carrying the probe arm on the support column.

[0006] As described in the manual for the Talyrond TR 200 at section 16.8, the straightness of the traverse, that is effectively the straightness of the axial straight datum, may be checked by mounting a cylinder slightly offset from the rotational axis datum of the turntable normally of the axial straight datum. The stylus is then caused to move axially along the surface of the workpiece by moving the probe arm longitudinally of the support column and the variation in the displacement of the stylus from the axial straight datum with height is plotted to form a first straightness graph. The above-mentioned procedure is then repeated after the turntable has been rotated through 180° to form a second straightness graph. The line bisecting the first and second straightness graphs should be a straight line and any departure from straightness of the bisector represents an error in the straightness of the column support or axial straight datum. The slope of the bisector is a function of any residual relative tilt of the column support and the cylindrical workpiece and is not related to the straightness of the column support.

[0007] The above-described straightness check is intended to be carried out from time to time before making a measurement on a workpiece so as to ensure that the straightness of the axial straight datum lies within acceptable limits and to enable calibration of the apparatus.

[0008] An embodiment of the present invention provides a method of measuring the form of a surface which is nominally symmetric about an axis, for example a surface having a cylindrical form, which method enables the effects of any deviations in the measurement direction in a reference datum to be eliminated or at least reduced during the measurement process.

[0009] In an embodiment, sensing means are used to obtain at least two sets of data for measurement points P_i(h_y) on a surface with the points P_i(h_y) being at different angular positions (with respect to an axis about which relative rotation of the surface and the sensing means is effected) for each measurement and the at least two sets of data are combined to compensate for any deviations in, for example, straightness or parallelism in a datum to which the measurements made by the sensing means are referenced. For example, the two sets of data may be combined by taking the mean of two values measured for the same location on the surface measured with two different orientations of the sensing means to the reference datum to give an indication of the radius (that is the distance from the said axis to the location) at that location. The difference of the same two measurements may be taken to give an indication of the value of any deviation or error in the reference datum. This enables measurements of the form at a given height along a surface to be made which are not dependent on errors in the reference datum in the measurement direction. By making such measurements at different heights along the surface of the object, the form of the object can be determined without being affected by any deviation or error in the reference datum in the measurement direction.

[0010] In an embodiment, two sensing means are used to obtain data for two diametrically opposed points P_i(h_y) and P_i+π(h_y) on a surface so allowing two sets of data to be obtained at the same time. In this example a diameter determined by the separation of the two sensing means may be compared with a diameter obtained by combining the polar data which is obtained individually from the two sensing means and which is referenced to a reference datum. This enables any deviation in the measurement direction in the reference datum to be determined. These measurements may be repeated for other points at the same height and for corresponding points at different heights to determine the overall form of the surface compensated for any deviation in the measurement direction of the reference datum.

[0011] In another embodiment, the sensing means comprises a single sensing element biassable in at least two different directions and means for adjusting the biasing of the sensing element to change the direction in which the sensing element is biased.

[0012] Relative rotation of the surface and the sensing means may be effected by mounting the surface on a rotatable support. As another possibility, relative rotation of the surface and the sensing means may be effected by rotatably mounting the sensing means to a support and rotating the sensing means about the surface.

[0013] In another aspect, the present invention provides apparatus for determining the shape, form, texture or roughness of a surface using a sensing means having adjustable biassing means to enable the sensing means to be biased in different directions.

[0014] Embodiments of the present invention will now be described, by way of example, with reference to the accompanying drawings in which:

[0015] FIG. 1 is a simplified schematic perspective view one example of metrological apparatus suitable for use in a method in accordance with the invention;

[0016] FIGS. 2aand 2bare diagrammatic side and plan views, respectively, of part of the apparatus shown in FIG. 1 for illustrating a first step in a first embodiment of a method in accordance with the invention;

[0017] FIGS. 2cand 2dare views similar to FIGS. 2aand 2bfor illustrating a second step in a first embodiment of a method in accordance with the present invention;

[0018] FIGS. 3ato 3dare diagrams for explaining a method embodying the invention;

[0019] FIG. 4ais a schematic perspective view of one example of apparatus suitable for use in a second embodiment of a method in accordance with the present invention;

[0020] FIG. 4bis a diagrammatic side view of the apparatus shown in FIG. 4a;

[0021] FIGS. 5aand 5bare diagrams for explaining in more detail a method embodying the invention using apparatus such as that shown in FIGS. 4aand 4b;

[0022] FIG. 6ais a diagrammatic side view of a second example of apparatus suitable for use in a second embodiment of a method in accordance with the present invention;

[0023] FIG. 6bis a top plan view of part of the apparatus shown in FIG. 6a;

[0024] FIG. 7 is a simplified diagrammatic perspective view for illustrating the principles of another embodiment of apparatus suitable for use in a method in accordance with the invention;

[0025] FIG. 8 is a simplified diagrammatic side elevational view of a further embodiment of apparatus suitable for use in a method embodying the invention;

[0026] FIG. 9 is a diagrammatic side view similar to FIG. 2aof apparatus similar to that shown in FIG. 1 but having a different gauge for carrying out a third embodiment of a method in accordance with the invention;

[0027] FIG. 10 is an enlarged view of part of the gauge shown in FIG. 9 for explaining the third embodiment of a method in accordance with the invention;

[0028] FIG. 11 is a diagrammatic view similar to FIG. 10 for illustrating a modified form of the gauge shown in FIG. 10;

[0029] FIG. 12 is a part cross-sectional view of part of the gauge of the apparatus shown in FIGS. 9 to 11;

[0030] FIGS. 13a, band care diagrammatic views for illustrating a variation of a method in accordance with the present invention;

[0031] FIG. 14 shows a diagrammatic side view of a modified coordinate measuring machine suitable for use in a method in accordance with the present invention; and

[0032] FIG. 15 is a simplified block diagram of a control system for the apparatus shown in the preceding Figures.

[0033] Referring now to FIG. 1, a first example of a metrological apparatus 1 suitable for use in a method in accordance with the invention comprises a work bench 2 (only part of which is shown) which carries a rotatable support or turntable 4 for receiving a workpiece (not shown) to be measured. The turntable 4 has a rotational axis shown by the dashed line A in FIG. 1. A support column 5 extends perpendicularly of the work bench 2 and has a longitudinal axis shown by the dashed line B. The rotational axis A of the turntable 4 defines a rotational axis datum while the support column 5 defines an axial straight datum in the form of a plane passing through the axis B parallel to the rotational axis datum A.

[0034] A carriage 6 is moveable along the support column 5 parallel to the axial straight datum B by means, for example, of a motor (not shown) or manually. The manner in which the carriage 6 is moved along the support column 5 is conventional and will not be described further. Measurement of the actual location of the carriage 6 along the support column 5 is effected by a suitable conventional linear transducer, for example an optical transducer arrangement such as that described in U.S. Pat. No. 4,807,152, the contents of which are incorporated herein by reference.

[0035] A probe arm 7 is mounted to the carriage 6 so as to be moveable transversely of the axial straight datum B and radially of the rotational axis datum A. Again, any suitable form of conventional mechanism such as a lead screw may be used for moving the probe arm 7 relative to the carriage 6. A stylus 9 is mounted to the probe arm 7 by means of a pivotable arm 8. Measurement of the position of the probe arm 7 (and thus the stylus 9) is made using a suitable conventional transducer (not shown).

[0036] In use of the apparatus, as shown diagrammatically in FIG. 2a,a workpiece 10 of a generally cylindrical external form 10ais mounted to the turntable 4 so that its axis of rotation is coincident with the rotational axis datum A. This may be achieved by, for example, using the centring and levelling arrangement disclosed in EP-A-0240150 or International Patent Application Publication No. WO96/12162, or any other suitable known centring and levelling technique.

[0037] The carriage 6 is adjusted to a desired measurement starting height, for example a height h above the base of generally cylindrical workpiece 10. The moveable probe arm 7 is then driven transversely of the carriage 6 until the stylus 9 contacts the surface 10aof the workpiece. Contact between the stylus 9 and the surface 10aof the workpiece may be determined by any suitable mechanism, for example that described in EP-A-0240151 so that the probe arm 7 is driven so as to maintain the deflection of the stylus 9 within the range of operation of an associated transducer. The pivot arm 8 is maintained at a fixed orientation. The distance between the rotational axis datum A and the axial straight datum B minus the displacement of the stylus 9 relative to the axial straight datum B provides a measure R of the radius of the workpiece 10 at the measurement point P at the height h on the workpiece.

[0038] As the workpiece 10 is rotated about the rotational datum axis A, a set of radius values P_i(h₁) is obtained for each angular orientation, i, of the workpiece at a height h.

[0039] Measurements at a different height h₂along the workpiece can be made by driving the carriage 6 along the column 5 and then proceeding to measure a further set of radius values P_i(h₂).

[0040] After sufficient sets of data at different desired heights h_yhave been obtained, the probe can be driven using the extending arm 7 and the pivotal arm 8 so as to address a second measurement position p′ diametrically opposed to the original position P.

[0041] If necessary the carriage 6 may be lifted in order to avoid interference between the workpiece and the stylus 9.

[0042] The stylus 9 can then be driven to the height h₁once more and another set of radius values obtained P¹_(i+π)(h₁). The carriage 6 can then be driven to all the previous height h_yto obtain corresponding sets of radius data P_i+π(h_y) for each height.

[0043] The obtained radial or polar data thus gives two measurements of the radius at each point P_i(h_y) on the surface 10aof the workpiece 10, with the workpiece having been rotated through 180° between the two measurements. By taking the mean of the two measurements at each point P_i(h_y), errors in the axial straight datum B can be compensated for. As indicated above, these errors may arise from mechanical inaccuracies in the straightness of the support column 5 defining the axial straight datum, changes with time in the support column 5 and, for example, variations in the interface between the carriage 6 and the support column 5. Typically, such deviations may be of the same order as deviations from roundness or circularity of the cross-section of a nominally circular cross-section workpiece.

[0044] FIG. 3ashows a cross-section 10a′ of the workpiece 10 at a measurement height h. The dashed line circle C represents the circle having the radius r which is the closest fit to the cross-sectional shape or form of the workpiece 10. δr represents the “out of roundness” or deviation from circularity of the cross-section of the workpiece 10 at the measurement point P at the height h.

[0045] As indicated above, the axial straight datum B may deviate from straightness for various reasons. The deviation of the axial straight datum B from the ideal B₁is shown as δc in FIG. 3a.

[0046] Consider the measurement of a radius at a particular point on the surface at orientation θ of the full data set P_i(h). The radius at this point measured closest to the axial straight datum B (that is the position shown in FIGS. 2aand 2b) is given by:

[0047] Pθ(h)=r−δr+δc  1)

[0048] The radius P_j(h) is thus dependent on any error δc in the measurement direction in the straightness of the axial straight datum B.

[0049] The same radius measured at the same point on the surface (orientation θ) of the workpiece when the gauge is in the second position P′ shown in FIGS. 2cand 2dis:

[0050] Pθ′(h)=r−δr−δc  2)

[0051] Taking the mean of equations 1) and 2) gives:

[0052] ½(Pθ(h)+Pθ′(h))=r−δr  3)

[0053] which is the true radius of the component at orientation θ independent of any error δc in the direction of the measurement in the axial straight datum B.

[0054] Thus taking the mean of the two radius measurements Pθ(h) and Pθ′(h) for all values of θ around the circumference of the workpiece at height h will give a complete picture of the cross section and shape (and thus of the radial form of workpieces 10) at the height h, independent of any error in the measurement direction of the axial straight datum B.

[0055] The method described above also prevents any lack of straightness in the axial straight datum affecting the measurement of cylindrical form. To explain this, reference will now be made to FIGS. 3bto 3dwhich are schematic diagrams showing the use of a method embodying the present invention to measure the cylindrical form of a perfect cylinder 100 when the axial straight datum B is bowed so as to be concave when viewed from the workpiece 100. It will, of course, be appreciated that the degree of lack of straightness of the axial straight datum B has been considerably exaggerated in FIGS. 3band 3c.

[0056] As shown in FIG. 3b,when measurements of the radius of points P around the circumference of the workpiece 100 are made at different heights h along the workpiece 100 with the stylus 9 in the position shown in FIGS. 2aand 2b,the resulting cylindrical form data set represents the workpiece 100 as being barrel-shaped because of the curvature of the axial straight datum B. In contrast, when measurements of the cylindrical form are made with the stylus 9 displaced by 180° from the position shown in FIG. 3bthen, as shown in FIG. 3c,the cylindrical form data represents the workpiece 100 as having an inwardly-bowed surface.

[0057] By combining a measurement made of the radius at a point P_i(h_y) using the arrangement shown in FIG. 3bwith a measurement of the same point P_i(h_y) with the arrangement shown in FIG. 3c,a set of cylindrical form data is obtained from which the influence of the lack of straightness of the axial straight datum B has been removed so that the workpiece 100 is correctly represented as a perfect cylinder as shown in FIG. 3d.

[0058] Thus, the cylindrical form of a workpiece can be determined independently of any errors in the measurement direction in the axial straight datum, so avoiding or at least reducing the need to effect calibration to determine the straightness and/or parallelism of the axial straight datum B before carrying out a measurement.

[0059] Although FIGS. 2ato 3dshow the measurement of the cylindrical form of an external surface of a workpiece, it will, of course, be appreciated that the above-described method can be applied to measurement of the cylindrical form of an accessible interior surface of a workpiece with, as will be appreciated by those skilled in the art, appropriate modification of the probe arm and stylus.

[0060] FIG. 4ashows a schematic perspective view of apparatus 1asuitable for use in a second embodiment of a method in accordance with the invention. FIG. 4bis a schematic side view of the apparatus 1ashowing a workpiece 12 mounted on the turntable 4.

[0061] The apparatus shown in FIG. 4adiffers from that shown in FIG. 1 mainly in that the carriage 6 carries two probe arms 7aand 7beach of which carries a respective stylus 9aand 9bto form a twin gauge 90. The probe arms 7aand 7bare received in respective mounting members 11aand 11bwhich engage a guide rail 6aof the carriage 6 so as to be moveable along the guide rail 6a.Any suitable drive arrangement may be provided for driving the mounting members 11a,11balong the guide rail 6a,for example, a rack and pinion type arrangement may be provided. The mounting members 11aand 11bmay be arranged to be driven manually or by a motor (not shown) so that rotation of the motor in one direction causes the carriages 10aand 10bto move in opposite directions. Of course, the mounting members 11aand 11bmay be independently driven to provide greater flexibility.

[0062] As shown more clearly in the side view of FIG. 4b,the carriage 6 and mounting members 11aand 11bare arranged such that the styli 9aand 9blie along a diameter of the turntable 4, in the example shown along a diameter perpendicular to a line between the rotational axis datum A and the axial straight datum B defined by the support column 5.

[0063] The probe arms 7aand 7bmay be mounted so as to be longitudinally moveable by any suitable drive means within the mounting members 11aand 11bto effect minor adjustments in the position of the styli 9aand 9bin a direction parallel to the rotational axis datum A.

[0064] The styli 9aand 9bshown in FIG. 4aare elongate members extending parallel to the rotational axis datum. Each stylus 9a,9bhas a tip portion 9c,9dprojecting outwardly (that is away from the other tip portion) at, for example, a right angle to the stylus. The styli 9aand 9bshown in FIGS. 4aand 4bare thus suited for measuring the cylindrical form of an internal surface of a hollow body or workpiece 12 as indicated diagrammatically in FIG. 4b.

[0065] The provision of twin styli means that the measurements of each point P_i(h_y) and of each point P′_i(h_y) diametrically opposite to P_i(h_y) can be made without having to re-position the probe or stylus. This means that any errors due to re-positioning of the stylus in the arrangement described above with reference to FIGS. 2 and 3 can be avoided. Also, the use of a pair of styli enables two sets of data to be obtained at the same time, one for the point P_i(h_y) and the other for the point P_i+π(h_y) 180° displaced around the circumference of the workpiece from the point P_i(h_y). This reduces the amount of time required to obtain a full set of data for the cylindrical form so that temporal and thermal changes should be less significant.

[0066] In use of the apparatus shown in FIGS. 4aand 4b,the hollow workpiece 12 is centered and levelled on the turntable 4 in the manner described above so that its nominal axis is coincident with the rotational axis datum A. The carriage 6 is then operated to cause the styli 9aand 9bto move into the hollow workpiece 12 until they reach a desired measurement height h. The mounting members 11aand 11bare then driven so as to move the styli 9aand 9bapart until the styli contact respective diametrically opposed points P_i(h_y) and P_i+π(h_y) on the internal surface of the workpiece 12.

[0067] The position of each stylus 9aand 9bis determined using any suitable conventional measurement transducer, (for example an optical or electromagnetic transducer arrangement) associated with the drive mechanisms of the mounting members 11aand 11b.The measurement obtained from one of the styli 9aand 9bthus provides an indication of the radius at point P_i(h_y) while the measurement of the position of the other one of the styli 9aand 9bprovides an indication of the radius at the point P_i+π(h_y). The separation of the styli 9aand 9bmay also be determined to give a measurement of the diameter d_cbetween the points P_i(h_y) and P_i+π(h_y). This diameter d_cis independent of the rotational axis datum A and the axial straight datum B and thus does not involve any errors resulting from inaccuracies in the axial straight datum B along the diameter d_c.

[0068] It will, of course, be appreciated that measurements of the radial polar data can P_i(h_y) be obtained using just one of the two styli 9aand 9b.However, as mentioned above, using both styli to obtain simultaneous measurements for diametrically opposed points reduces considerably the total time required for measurement of the cylindrical form and the effects of errors which are temporal in nature (for example thermal effects) are minimised.

[0069] As in the example described above with reference to FIGS. 2 and 3, the cylindrical form data P_i^K(h_y) may be obtained by measurement at each height h_yin turn sequentially for each measurement position P & P′, that is in order, P_i(h₁), P_i(h₂), P_i(h₃) . . . P_i(h_n) and then P_i′(h₁), P_i′(h₂), P_i′(h₃) . . . P_i′(h_n).

[0070] In practice the full data required to describe the cylindrical form P_i^K(h₁) can be determined by changing i, y and K in any order.

[0071] In the second embodiment illustrated in FIG. 3 it is preferable to obtain both data sets P_i(h_y) and P_i′(h_y) whilst the carriage remains at height h_yto avoid errors in re-positioning the carriage 6 to the height h_ywhich might arise if both the data sets are not taken at this time.

[0072] As will be appreciated, the radius R measured by one of the styli 9aand 9bat a point on the surface at an orientation θ at height h will again be given by:

[0073] P_θ(h)=r−δr+δc  4)

[0074] while the radius measured by the other stylus of the same point on the surface of the workpiece will be:

[0075] P_θ′(h)=r−δr−δc  5)

[0076] where δr is again the out of roundness of the workpiece at the point being measured and δc is the deviation from straightness in the measurement direction in the axial straight datum B.

[0077] The radial polar data obtained from each of the two styli 9aand 9bmay thus be combined, as described above with reference to FIGS. 2ato 3d,for each measurement point P_i(h_y) so as to obtain a set of data for the cylindrical form of the workpiece which is independent of any error in the straightness of the axial straight datum in the measurement direction.

[0078] As well as permitting assessment of surfaces independent of the axial straight datum (and therefore of any error δc), the apparatus shown in FIGS. 4aand 4balso enables an indication of the actual lack of straightness δc in the axial straight datum at a given height to be determined. Thus, subtracting equations 4) and 5) gives:

[0079] P_δ(h)−P_θ¹(h)=2δc  6)

[0080] Hence, taking the difference between any two corresponding data points and dividing by 2 gives δc. Assessment of δc can thus be made at a nominated test orientation θ or by a suitable averaging process over a number of values of θ.

[0081] In the above explanation, it has been assumed that the rotational axis of the cylindrical surface being measured is precisely positioned upon the spindle axis. It has also been assumed that the stylus or styli are exactly positioned on a diameter (that is on crest). However, in practice, the rotational axis C of the workpiece cylindrical surface may be somewhat offset from the spindle axis A.

[0082] FIGS. 5aand 5billustrate this diagrammatically. In FIG. 5a,the radius at the point P_iis being measured by one stylus 9aof a pair of styli 9aand 9b.In FIG. 5b,the radius at the point P_iis being measured by the other stylus 9bwhen the workpiece 12 has rotated through 180°. FIGS. 5aand 5bshow the rotational axis C of the workpiece cylindrical surface as being offset from the spindle axis A by a distance e (the eccentricity) at an angle α to the diameter between the styli 9aand 9b.

[0083] It is well known in the art that the surface traced out by a point on a circle which is rotated off-axis is approximated by a limicon in polar coordinates. Thus, the radius R_P1at point P_iin FIG. 5ais given by:

[0084] R_P1=F+g+ecos(α)   7)

[0085] where 2F is the distance between the null positions of the two gauges or styli 9aand 9b(manually set to the component diameter) and g is the offset of the centre point O_Fbetween the two gauges from the spindle axis A in the direction of the gauge reading. The offset g will therefore include any error δc in the straightness of the axial straight datum in the measurement direction.

[0086] When the workpiece has been rotated through 180° to the position shown in FIG. 5bthen the equation of the point P_iin polar coordinates is given by:

[0087] R_P2=F+g+ecos(α+π)   8)

[0088] As indicated above, however, the styli may be slightly off-crest by a cresting distance H. For the first position of the point P_ion the surface of the workpiece shown in FIG. 5athis is given by: $\begin{array}{cc}\sin {\theta }_1=\frac{H}{R_{\mathrm{P1}}\left(x+{\theta }_1\right)}& 9)\end{array}$

[0089] where θ₁is the angle between the desired measurement position P_iand the actual measurement position (the cresting angle) and R_P1(x+θ₁) represents the measurement at the point on the circumference of the workpiece 12 offset from the point P_iby the cresting angle θ₁.

[0090] The above equation can be solved for θ by using an appropriate mathematical computing package or software program, for example “Maple 5” which is a symbolic mathematics package produced by Maple Software of 450 Phillips Street, Waterloo, Ontario, Canada.

[0091] Using the above-mentioned mathematics package to solve for the cresting angle θ₁as a series in terms of the cresting error distance H gives: $\begin{array}{cc}{\theta }_1=\frac{H}{(F+g+e\cos \left(\alpha \right)}+\frac{H^{2}e\sin \left(\alpha \right)}{{\left(F+g+e\cos \left(\alpha \right)\right)}^3}+O\left(H^3\right)& 10)\end{array}$

[0092] where the series is truncated at terms of the order of H³.

[0093] By analogy, the cresting error H at the position 2 shown in FIG. 5bis given by: $\begin{array}{cc}\sin {\theta }_2=\frac{H}{R_{\mathrm{P2}}\left(x+{\theta }_2\right)}& 11)\end{array}$

[0094] Using the same mathematics package to solve for θ₂in terms of a series in H gives: $\begin{array}{cc}{\theta }_2=\frac{H}{\left(R-g-e\cos \left(\alpha \right)\right)}+\frac{H^{2}e\sin \left(\alpha \right)}{{\left(F-g-e\cos \left(\alpha \right)\right)}^3}+O\left(H^3\right)& 12)\end{array}$

[0095] The signal S₁from the stylus 9ain FIG. 5awith the cresting error given in equation 10) is:

[0096] S₁=R_P1(x+θ₁)cos(θ₁)   13)

[0097] The signal from stylus 9bat position 2 in FIG. 5b(that is when the component has been rotated through 180°) with the cresting error given in equation 12) is:

[0098] S₂=R_P2(x+π−θ₂)cos(θ₂)   14)

[0099] Solving for S₁as a Taylor series in the cresting error H gives: $\begin{array}{cc}{S}_1=\left(F+g+e\cos \left(\alpha \right)\right)-\frac{\mathrm{He}\sin \left(\alpha \right)}{\left(F+g+e\cos \left(\alpha \right)\right)}+O\left(H^2\right)& 15)\end{array}$

[0100] Similarly, solving for S₂as a Taylor series in terms of the cresting error H gives: $\begin{array}{cc}{S}_2=\left(F-g+e\cos \left(\alpha \right)\right)+\frac{\mathrm{He}\sin \left(\alpha \right)}{F-g-e\cos \left(\alpha \right)}+O\left(H^2\right)& 16)\end{array}$

[0101] Taking the mean of the two signals S₁and S₂and expanding this as a Taylor series in the cresting error H gives: $\begin{array}{cc}{S}_{\mathrm{mean}}=\left(F+e\cos \left(\alpha \right)\right)+\left(-\frac{1}{2}\left(\frac{e\sin \left(\alpha \right)}{F+g+e\cos \left(\alpha \right)}\right)+\frac{1}{2}\left(\frac{e\sin \left(\alpha \right)}{F-g-e\cos \left(\alpha \right))}\right)\right)H+O\left(H^2\right)& 17)\end{array}$

[0102] where, as in equations 10) and 12), O indicates terms of the order of the indicated power. Converting the series of equation 17) into a polynomial in the cresting error or distance H gives: $\begin{array}{cc}\mathrm{PS}=\left(F+e\cos \left(\alpha \right)\right)+\left(-\frac{1}{2}\left(\frac{e\sin \left(\alpha \right)}{F+g+e\cos \left(\alpha \right)}\right)+\frac{1}{2}\left(\frac{e\sin \left(\alpha \right)}{F-g-e\cos \left(\alpha \right))}\right)\right)H& 18)\end{array}$

[0103] Typically, F=1.25 mm, g=−0.005 mm, e=0.001 mm and H=0.001 mm.

[0104] Because F is very much larger than g and e, the terms in sinα in equation 18) will be very similar to one another and will effectively cancel out. Thus, to first order, the measure of the radius obtained by taking the mean of the signals from the stylus 9ain FIG. 5aand the stylus 9bin FIG. 5bis not affected by the cresting error H or the offset g.

[0105] In the examples described above, the workpiece is itself rotated. However, other arrangements can be envisaged wherein the measuring arrangement (the stylus or styli) is rotated rather than the workpiece.

[0106] FIG. 6aillustrates, very schematically, such a type of apparatus 1bsuitable for use in a method in accordance with the invention. The apparatus shown in FIG. 6ais, like the apparatus shown in FIGS. 4aand 4b,a twin gauge system. However, in the apparatus shown in FIG. 6a,the turntable 4 is omitted and replaced by a rotatable spindle 40 (which may have any suitable conventional form) mounted to an arm 6afixed to the support column 5.

[0107] The spindle 40 carries a frame 60 having downwardly depending arms 61 to which is slidably mounted a cross-bar 62 carrying respective mounting chambers 63 for the probe arms 7aand 7b.The cross-bar 62 is movable along the arms 61 in a manner similar to the manner in which the carriage 6 is movable along the support column 5 in FIG. 1. In this example, the arms 61 define the axial straight or reference datum B. The position of the cross-bar 62 with respect to the arms 61 may be determined in a manner similar to that used to determine the position of the carriage 6 in FIG. 1.

[0108] FIG. 6bis a top plan view of part of the apparatus of FIG. 6ashowing more clearly the arrangement of the cross-bar 62 and probe arms 7aand 7b.This arrangement enables the styli 9aand 9bto be moved along the height of the workpiece 13 to enable measurements to be taken at different heights.

[0109] In the example shown in FIG. 6a,the styli 9aand 9bare configured with inwardly pointing tip portions 9cand 9dfor measuring an external surface. However, the styli 9aand 9bshown in FIGS. 6aand 6bcould be replaced by the styli shown in FIG. 4 to enable an internal surface to be measured, with appropriate dimensioning of the components of the apparatus to enable the styli to be accommodated within the workpiece being measured.

[0110] The apparatus 1bshown in FIG. 6 is used in a similar manner to the apparatus shown in FIGS. 4aand 4bexcept, in this case, the workpiece 13 is mounted to the work bench 2 so as to be aligned with the spindle axis A′ and the spindle 40 rather than the turntable 4 is rotated to obtained the polar data.

[0111] In the examples described above, the cylindrical form is determined by making measurements of the radial form at different heights along the workpiece by raising or lowering the stylus or styli. However, another way of measuring cylindrical form is to provide two or more styli or pairs of styli spaced at calibrated fixed distances from one another in a direction defining the axial straight datum.

[0112] FIG. 7 shows schematically part of an apparatus to illustrate the principle of providing a number of twin gauges 90a.In the example shown in FIG. 7, three twin gauges 90 are mounted to a common support 61, causing the sets of twin gauges to be nominally aligned to the component axis which defines the equivalent to the axial straight datum B. Of course, if it is desired only to make measurements at three distinct heights along a workpiece, then the support carrying the twin gauges may be fixed to the support column 5.

[0113] As shown very diagrammatically in FIG. 7, each of the twin gauges 90ais in the form of a fixed U-shaped member 91 with the arms 91aand 91bextending perpendicularly of the support or carriage 6′. The free end of each arm 91aand 91bsupports a stylus 9c(shown simply by an arrow in FIG. 7) in any suitable conventional manner.

[0114] In the arrangement shown in FIG. 7, the arms 91aand 91bof the U-shaped members 91 are fixed in position. It may, however, be possible to mount the arms 91aand 91bon guide rails in a manner similar to that discussed above with reference to FIG. 4aso as to enable the nominal separation of the styli 9cof each twin gauge 90ato be adjusted to allow differently sized work pieces to be measured using the same apparatus.

[0115] In operation of the apparatus illustrated by FIG. 7, the workpiece 14 is mounted and centred in conventional manner on the turntable (not shown in FIG. 7) so as to be positioned between the respective styli of the gauges 90a.The styli are then driven in conventional manner to bring them into contact with the surface of the workpiece. Measurements of the radial data sets P_θ(h) can then be made in the manner described above simultaneously at each of three heights h1, h2 and h3 along the workpiece 14. Because, as described above, deviations from straightness of the axial straight datum are compensated for using a method embodying the invention, it is not necessary for the twin gauges 90ato be calibrated with respect to one another. Of course, each individual gauge should be calibrated in a conventional manner. In addition, in order to ensure compatibility of the twin gauges 9a,calibration of the gauge separation is also required.

[0116] The arrangement shown in FIG. 7 may be modified to enable simultaneous measurement of radial form at a number of different heights of an internal surface of a workpiece.

[0117] FIG. 8 illustrates very diagrammatically a modified form of the apparatus shown in FIG. 6a.In the example shown in FIG. 8, the frame 60 and depending arms 61 are replaced by a rigid rod 65 to which are fixedly attached pairs of opposed radially extending arms 66a,b,ceach of which carries a respective stylus 9e.The apparatus shown in FIG. 8 also differs from that shown in FIG. 6ain that a movable carriage 6 is supported on the support column 5 in a manner similar to that shown in FIG. 1. However any suitable arrangement for nominally aligning the rigid rod 65 with the component axis may be used. For example, the spindle 40 and rigid rod 65 may be carried by a frame which is designed to be portable and to be secured directly to a component to be measured. As in the apparatus of FIG. 7, the reference datum B is defined by the alignment of the twin gauges 9e.

[0118] In this example, a workpiece 15 is first mounted on the work bench 2 so as to be aligned with the rotation axis A′ of the spindle 40. The carriage 6 is then lowered so as to insert the rigid rod 65 carrying the probe arm 66 and styli 9cinto the hollow workpiece. Once the rod 65 has been correctly inserted within the hollow workpiece 15, the probe arms 66care driven so as to bring the styli 9cinto contact with the interior surface of the workpiece 15. Measurements of radial form at each of three different heights are then taken in the manner described above by rotating the spindle 40.

[0119] The apparatus shown in FIG. 8 has the advantage that, as in the apparatus shown in FIG. 6, the alignment of the twin gauges 9e(as determined by the rigid rod 65 in this case is not critical because any deviations or errors are compensated for in the manner described above.

[0120] Of course, the apparatus shown in FIG. 8 could be modified by providing a turntable 4 on the workbench 2 and fixing the rod 65 against rotation to the carriage 6.

[0121] The apparatus shown in FIGS. 7 and 8 may, of course, by modified to provide any desired number of pairs of twin gauges. It may also be possible to mount the twin gauges of FIG. 7 on a movable carriage similar to the carriage 6 shown in FIG. 1 to enable three simultaneous sets of polar data to be obtained at two or more different sets of three heights along the workpiece. Similarly, the carriage 6 shown in FIG. 8 may be moved to enable measurements to be made at more than one set of three heights.

[0122] FIG. 9 illustrates, by way of a view similar to FIG. 2a,another way of carrying out a method embodying the invention. The apparatus shown in FIG. 9 is similar to that shown in FIGS. 1 and 2aexcept for the structure of the gauge G carried by the moveable probe arm 7. Only those aspects of the apparatus shown in FIG. 9 which differ from that described above with reference to FIGS. 1 and 2awill now be described.

[0123] The stylus of the apparatus shown in FIG. 9 comprises a stylus arm 92 carrying a stylus 92awhich may be, for example, in the form of a ruby or sapphire ball. The stylus arm 92 is arranged to be pivotable in a direction extending radially of the rotational axis so as to allow the stylus point 92ato contact opposite ends P_iand P_i+θof a diameter of an internal surface 100aas shown more clearly in FIG. 10.

[0124] In order to measure the cylindricity of the surface 100a,the carriage 6 is given to a desired first measurement height h₁with the probe arm 7 positioned so that the probe arm 8′ is aligned with the rotational axis datum A. This may be achieved by using a fixed length probe arm or by driving a moveable probe arm 7 to bring the arm 8′ into alignment with the rotational axis datum A.

[0125] The stylus arm 92 is then pivoted by biassing means as will be described below to bring it into contact with the point P_i(h₁). The turntable 4 carrying the workpiece 100 is then rotated about the rotational axis A and a set of radius values P_i(h₁) is obtained for each angular orientation i, of the workpiece at the height h₁.

[0126] The stylus arm 92 is then pivoted using the biassing means to be described below to bring the stylus point 92ainto contact with the point P_i+πdiametrically opposed to the point P_ion the internal surface 100aof the workpiece. A further set of radius values P_i′π(h₁) is then obtained for each angular orientation of the workpiece at the height h₁.

[0127] Measurements at different heights h_yalong the workpiece are made by driving the carriage 6 along the column 5 to new heights h_yand repeating the radius value measurements for P_i(h_y) and P_i+π(h_y) at each height h_y.

[0128] Alternatively, the sets of radial values P_i(h_y) may first be obtained at the different heights h_yand then the biassing reversed and the sets of radial values P_i+π(h_y) obtained for the different heights h_y.

[0129] The obtained radial or polar data P_i(h_y) and P_i+π(h_y) are then processed as described above to obtain data representing the cylindricity of the internal surface 100a.

[0130] Although FIGS. 9 and 10 are schematic, it will be appreciated that pivoting of the stylus arm 92 causes arcuate motion which produces uncertainties in the height at which contact is made. Such uncertainties can be minimised by using a stylus tip or point 92awith a width which is as close as possible to the bore diameter of the workpiece 100 being measured so that the arcuate motion is kept to a minimum. Such a method is particularly useful for measuring the cylindricity of balls with diameters in the region of 5 to 10 mm.

[0131] Where it is desired to measure the cylindricity of larger diameter balls, a modified form of gauge G′ may be used as shown in FIG. 11 where the free end of the stylus arm 92 carries a cross-bar 92′ having at each end a stylus tip or point 92a′. This modified form of gauge G′ can be used to measure the cylindricity of internal balls having diameters significantly larger than the diameter of the stylus tips 92a′ while still limiting the arcuate motion of the stylus arm 92. If desired, the length of the cross-piece 92′ may be made adjustable by, for example, means of a micrometer adjustment or the like so as to enable the gauge G′ to be optimised to fit various different diameters of bore.

[0132] FIG. 12 is a cross-sectional view of part of a probe arm 8′ to show the manner in which the stylus arm 92 may be biassed between the two diametrically opposed positions. For simplicity, electrical connections and other conventional components, for example the rear plug of the probe arm 80′ which enables it to be connected to the probe arm 7, are omitted from FIG. 12.

[0133] As shown in FIG. 12, the probe arm 8′ has a housing 80 having an opening 80afrom which an end 81aof a stylus support arm 81 extends. As shown in phantom lines an end 92′ of the stylus arm 92 is held in place within a recess of the end 81aof the stylus support arm 81 by means of a spring clip 82.

[0134] The stylus support arm 81 is mounted to a mounting block 83 by means of a flexure spring 84 which defines a pivot axis of the stylus support arm 81. A U-shaped spring 86 is mounted to a pin 85 extending transversely of the stylus support arm 81. The free ends of the U-shaped spring 86 captured therebetween a pin 87 mounted eccentrically on a drum 88. Drum 88 is rotatable either manually or by means of a motor (not shown) for example a DC gear motor.

[0135] The spring 86 forms the biassing means of the stylus arm 92. The spring 86 is shown in a neutral or central position in FIG. 12 in which the stylus arm 92 is biassed neither to left or right in FIGS. 9 to 11.

[0136] When the drum 88 is rotated to a certain angle in an anti-clockwise direction, the pin 87 moves through that angle causing the spring 86 to apply a bias force to the stylus support arm 81 causing the stylus arm 92 to pivot downwardly in FIG. 12. In contrast, when the drum 88 is rotated in a clockwise direction, the movement of the pin 87 applies a biassing force to cause the stylus arm 92 to move upwardly in FIG. 12.

[0137] Rotation of the drum 88 may be sensed using any suitable form of shaft encoder, for example a Hall effect arrangement. Different degrees in biassing in both the clockwise and anticlockwise directions may be applied by allowing the drum to be rotated to different degrees. For example, the drum may have two sets of positions in both the clockwise and anti-clockwise directions to give moderate and heavy biassing. The use of a suitable shaft encoder and a motorised drive for the drum 88 enables adjustment of the biassing to be carried out under computer control with the shaft encoder providing signals to the computer processing means to indicate the position of the drum 88.

[0138] Pivoting of the stylus support arm 81 is detected in a conventional manner using a linear variable differential transducer (LVDT) 200 comprising a pair of coils 201 and 202 mounted to the mounting block 83 and a ferrite core 203 secured to a shaft 204 carried by the stylus support arm.

[0139] The pivotal movement of the stylus support arm 81 is limited by means of a metal plate 300 secured at one end to a mounting block 89 within the housing 80. The free end of the plate 300 is received in a recess 81bprovided at the rear end of the stylus support arm. A set screw 301 determines the initial position of the metal plate 300. The set screw may be adjusted by, for example, means of a screwdriver through an aperture 80′ in the housing. A limit screw 302 limits the upward movement of the metal plate 300 and thus of the end of the stylus support arm 81.

[0140] In the operation of the gauge shown in FIG. 12, when the stylus tip 92ahas been located within the internal bore of the workpiece 100 at the desired height by movement of the carriage 6 and adjustment of length of the movable probed arm 7 (if appropriate), the rotatable drum 88 is rotated either manually or by means of an electric motor to move the pin 87 off-centre so as to bias the stylus arm 92 towards one of the two opposed positions so that the stylus arm 92 pivots under the biassing force until the stylus tip 92 contacts the inner surface 100aof the workpiece.

[0141] As will be appreciated by those skilled in the art, roundness measuring apparatus such as that being described is designed to measure movements of the order of micrometers whereas the distance moved by the stylus tip 92aunder the biassing force before it reaches the workpiece surface 100awill be of the order of millimeters. Thus, although the LVDT will be designed to detect the position of the stylus support arm (and thus of the stylus tip 92a) throughout the possible range of movement under the biassing force, the measurements obtained by the apparatus during the pivotal movement under the influence of the biassing force will be out of the range of the expected measurement. Contact with the surface 100aof the workpiece is determined to be that point at which the measurements resulting from the signal supplied by the LVDT fall within the expected range. It is not necessary to know this position absolutely because it is not the absolute position of the point on the workpiece surface which is being measured but rather the polar variation or out of roundness of the workpiece surface 100aat that particular height.

[0142] Although the gauge described above with reference to FIG. 12 is primarily intended for measuring internal bores, such a reverse-biassing gauge may also be used in other circumstances where it is desirable to measure two opposed surfaces. For example, such a reverse-biassed gauge may be used in the apparatus shown in FIGS. 2ato 2dwith the stylus being biassed towards the left in FIG. 2aand towards the right in FIG. 2c. Such a reverse-biassing gauge may also be used where relative measurements of opposed surfaces are required, for example to enable a lower surface to be measured from beneath and an upper surface to be measured from above to determine any variations in thickness of an object.

[0143] In the embodiments described above, the two sets of polar data at each height are averaged to compensate for variations in the reference datum B. Another method of processing the polar data sets to compensate for errors in the vertical datum B will now be described with reference to FIGS. 13ato 13c. This method may be used in any of the embodiments described above.

[0144] FIG. 13ashows at A1 examples of polar data obtained at three heights h₁, h₂and h₃during the first measurement, that is with the measurement points P_i. FIG. 13bshows at A2 the corresponding polar data obtained for the heights h₁, h₂and h₃during the second set of measurements, that is with the measurements taken at an angular separation from the measurements shown in FIG. 13a, generally with the measurements taken at point P_i+π.

[0145] In this method, instead of averaging the two sets of data P_i(h_y) and P_i+π(h_y) for each value of y, each set of polar data P_i(h_y) and P_i+π(h_y) is fitted to a circle. As described in the book entitled “Lets Talk Roundness” by H Dagnell M.A., there are four different reference circles specified in the British, US and Japanese standards, namely, the least squares circle, the minimum zone reference circles, maximum inscribed reference circle and minimum circumscribed reference circle. In the present case, the least squares circle method is used where the mathematical definition of the circle is that for which the sum of the squares of a sufficient number of equally spaced radial points measured from the circle to the profile has minimum value. This fitting procedure may be carried out using any appropriate conventional least squares fitting algorithms. The least square circle (C_i(h_l), C_i(h₂), and C_i(h₃) obtained are shown at C1 in FIG. 13a.

[0146] The same procedure is carried out with the second set of data taken at the orientation P_i+πto derive the least squares circles C_i+π(h_l), C_i+π(h₂) and C_i+π(h₃) shown in FIG. 13b.

[0147] The average of the reference circles C₁(h_y) and C_i+π(h_y) at each height h_yare obtained to provide the data {overscore (C)} shown in FIG. 13. The least squares circle {overscore (C)}_i(h_y) for a height h_yis then subtracted from the average circle C(_y) for that height and added to the raw polar data P_i(h_y) for that height. The resulting data P_e(h_y) represents the actual shape of the surface 100aat the height h_ycorrected for any errors in the vertical axis datum B.

[0148] The method described above with reference to FIG. 13 has advantages in that angular phase errors which may be produced if the angular separation of the two measurement positions is not precisely known are significantly reduced because such errors can have only a minor effect on the position or radius of the reference circle. This should avoid false averaging problems which may otherwise occur where the wavelengths of variations of the surface 100aare short.

[0149] It will, of course, be appreciated that the correction shown in FIG. 13 can be applied to either the set of data P_i(h_y) or the data P_i+π(h_y) by subtracting the appropriate set of reference circles C_i(h_y) or C_i+π(h_y).

[0150] A method embodying the present invention may also applied to a modified coordinate measuring machine in which, as described in, for example, GB-A-1137238 or GB-A-2160975 (the contents of which are herein incorporated by reference) measurements are made on an x, y, z rectangular coordinate system.

[0151] FIG. 14 illustrates very schematically an example of coordinate measuring machine modified to be used in a method in accordance with the present invention.

[0152] As shown in FIG. 14, the coordinate measuring machine comprises a work bench 20 carrying two guide rails 21aand 21bwhich extend spaced-apart and parallel to one another in a direction perpendicular to the plane of the paper (the x direction). A U-shaped frame 22 is supported so as to be drivable along the guide rails 21aand 21bby any suitable drive means and using any suitable form of bearing arrangement. The cross-bar 22aof the U-shaped frame 22 carries a movable carriage 23 which supports a support member 24. The support member 24 is moveable relative to the carriage 23 in the z direction by any appropriate drive mechanism. Suitable measurement transducers are associated with the x, y and z direction drive mechanisms, for example optical or electromagnetic transducers as mentioned above, to enable to precise location in x, y and z coordinates of a probe carried by the support 24 to be determined.

[0153] The components of the coordinate measuring machine described above are conventional.

[0154] As shown in FIG. 14 the work bench 20 is provided with a rotatable support or turntable 4bsimilar to that shown in FIG. 1. In this case, the z axis support 24 carries a fixed probe arm 7acarrying a single stylus 9a. The coordinate measuring machine shown in FIG. 14 may be used in the manner described above with reference to FIGS. 2ato 2dand 3aand 3dby causing the carriage 23 to move along the cross-bar 22ato enable a first measurement of the radius at a point P on the circumference at height h of the workpiece 15 to be measured with the carriage 23 at a first position y1 and a second measurement of the radius at the point P to be measured when the turntable has been rotated through 180° and the carriage 23 moved to a position y2 shown in phantom lines in FIG. 14. The radial and cylindrical form can then be determined as described above with reference to FIGS. 2ato 2dand 3ato 3d. As another possibility, the apparatus shown in FIG. 14 may be modified by replacing the single stylus with one, or possibly more, calliper gauges to enable the methods described above with reference to FIG. 4 to be performed.

[0155] In each of the examples described above, the radial and cylindrical form data are generally calculated by a suitable programmed computer processing system which receives input signals from the transducers of the apparatus. FIG. 15 shows a simplified block diagram of a suitable processing system comprising a central processing unit 50 associated with a memory comprising ROM and RAM for storing operational programs and data, respectively.

[0156] During the measurement of the cylindrical form of a workpiece, data from the transducers is logged by the computer processing system and then used to determine the radial and cylindrical form of the workpiece in the manner described above. FIG. 15 shows the central processing unit 50 as being coupled to three transducers T1, T2 and T3 providing signals representing the rotation of the turntable, the height of the carriage along the column support 5 and the position of the probe arm 7. Of course, the number of transducers will depend upon the particular apparatus, as described above. Thus, where a motorised reversible-biassed gauge such as that shown in FIG. 12 is used, a further transducer T4 in the form of a suitable shaft encoder will be provided to supply signals indicative of the biassing of the gauge. The central processing unit 50 will also normally be connected, via appropriate conventional control interface circuits (not shown), to motors M1, M2 and M3 for controlling the rotation of the turntable, the height of the carriage and the position of the extending arm 7 respectively. Again, where a motorised reversible bias gauge is used, the central processing unit 50 will normally be connected, via appropriate conventional control interface circuits (not shown) to control operation of a motor M4 of the reversible-bias gauge.

[0157] The processing system may be provided with an input device 52 in the form of a keyboard which may be associated with a mouse 52aor other appropriate pointing device to enable a user to input information concerning, for example, recent calibration of the apparatus or machine, or instructions regarding the particular method to be used. In addition, a display unit such as a cathode ray tube 53 may be associated with the control processing unit 50 to enable, using appropriate conventional software, visual displays of the measurements made with the apparatus to be provided. As an alternative or addition to the visual display unit, an output device such as a printer or plotter may be provided to enable a hard copy to be provided of the measured data in the form of, for example, tabulated radial data and/or a graphical representation of the measured data.

[0158] Although in the examples described above, two measurements of the polar data at a single position at a given height on a workpiece are made, if desired three or more such measurements may be made by, for example, providing three or more measurements stylised equally angularly spaced around the periphery of the workpiece.

[0159] Also, although in the examples described above a set of polar data is first obtained at a given height, it may be possible or desirable in some circumstances to move the stylus or stylii axially of the workpiece or component to obtain data at different heights h and then to effect relative rotation between the workpiece and the sensing means.

[0160] If desired, where a twin gauge is provided, a first measurement of diameter may be made by determining the relative separation of the two gauges. This first measurement is independent of any errors in the axial straight datum B and may be compared with a second measurement of diameter obtained by combining the radii measured at the two opposed points by the stylii to give an indication of the actual deviation or error in the axial straight datum at that diameter.

[0161] Although the above examples describe ways of determining the cylindrical form of a workpiece surface, the present invention may be used to determine the form of any suitable surface, for example any surface which is nominally symmetric about an axis of rotation. For example, the present invention may be used to determine the form of a surface of constant polygonal, ellipsoidal or irregular cross-section or a tapered surface such as a frustoconical or conical surface or a surface of non-constant cross-section.

[0162] The present invention may also be used to determine the form of any surface that is produced by a machine tool such as a lathe or grinder. For example, the apparatus may be provided as an attachment to a lathe or grinder with for example a twin gauge pair being attached to the tool post and the spindle of the lathe or grinder being used to effect rotation to obtain data while the workpiece or component is still in the chuck or jaw of the machine tool. Such an arrangement may allow measurements of the form of a surface to be made at an intermediate stage in the working of the component by the machine tool to check that the machining is proceeding correctly.

[0163] Although specific examples of the present invention have been described above, various modifications and alternatives will be apparent to those skilled in the art.