Measurement Technology for Precision Machines

• Shuming Yang
• Changsheng Li
• Guofeng Zhang
Reference work entry
Part of the Precision Manufacturing book series (PRECISION)

Abstract

wThe precision of machine tools is the foundation of machining to achieve high precision. Therefore, an investigation on the measurement technology for precision machines is extremely important for precision/ultraprecision machining. This chapter provides an overall introduction to the measurement of precision machines, including the precision along linear axes and rotary axes, e.g., the precision of length, precision of angle, straightness, squareness, parallelism, flatness, and runout. The basic principles, achievable accuracy, application, and characteristics of a wide variety of measurement methods and instruments with precision in the nanoscale or below are discussed. It is found that digitization, automation, and flexibility are the development trends of modern instruments used for the measurement of precision machine tools. Though a number of measurement methods have been developed during these years, the existed methods face challenges to comprehensively characterize the performance of machine tools, especially for ultraprecision machine tools with a precision better than 100 nm. In addition, online measurement still remains to be investigated to improve the performance of precision machining.

Keywords

Measurement Metrology Online measurement Machine tool Precision machining

Introduction

There is standard nomenclature for defining the possible errors in a mechanical system. A rigid body has 6 degrees of freedom, i.e., three in translation, x, y, and z (including linear position, horizontal straightness and vertical straightness), and three in angular motion or “tilt” commonly referred to as “roll,” “pitch,” and “yaw,” as shown in Fig. 1a. In addition, each axis must be perpendicular to one another (i.e., totaling three squareness relationships). Therefore, an arrangement of three-axis orthogonal machine tool produces potential uncertainties in motion of 21 degrees of freedom. For a nominal rotational movement, the six component errors are two radial error motions, one axial error motion, the angular position error, and two tilt error motions. Figure 1b shows these component errors for a C movement.

Principles of Measurement for Machine Tool Metrology

Nakazawa (1994) gives four principles of measurement that point out some of the basis of good measurement or metrology. They are summarized here:
1. 1.

When measuring the dimensions of an object, one must know its temperature. The temperature differences in the structure will introduce errors due to linear expansion of machine elements, and the magnitude depends on the temperature difference, dimension, and coefficients of expansion.

2. 2.

Avoid the application of force. Noncontact methods of measurement are preferable to contact methods. Contact stresses generated by the contact of two bodies under load will generate deformation due to the elasticity of the bodies, and the magnitude depends on the contact force, the equivalent modulus of elasticity of the two bodies, and the equivalent radius of contact, as revealed by Hertz.

3. 3.

The precision of the measuring instrument must be five or ten times higher than the expected precision of the measured object. This is because the precision of the measuring instrument is one key component of the measurement uncertainty.

4. 4.

Be aware of the behavior of the instrument. It is important for machinist or metrologists to understand the factors that influence the performance of the instrument to use it properly.

Abbé’s principle is one of the basic rules for the metrology of machine tools. Abbé observed that “If errors in parallax are to be avoided, the measuring system must be placed co-axially with the axis along which displacement is to be measured on the workpiece” (Bryan 1979). Figure 2 shows a one-dimensional measurement system, which violates the Abbé’s principle. Abbé arm S exists between the standard scale 1 and measured line 2. If the measuring frame 3 generates an inclination angle ϕ for the straightness deviation of the lead rail movement, it will produce a measurement error of
$$\Delta L=S\;\tan \phi = S\phi$$

Measurement of Dimension

In practical metrological measurements for the inspection and calibration work undertaken on machine tools, the use of equipment such as gauge blocks, length bars, angle gauges, straightedges, and dial gauges together with dial test indicators forms the foundation for machine verification.

Gauge Block

The individual gauge block is a metal or ceramic block that has been ground and lapped to a specific thickness. Gauge blocks come in sets of blocks with a range of standard lengths, as shown in Fig. 3a. The blocks are joined by the combined action of sliding and a pressure process called wringing, which causes their ultraflat surfaces to adhere together. By utilizing blocks from, say, a large set of 105 blocks, one may create any of the required lengths up to its dimensional additive capacity – in 0.001 mm incremental steps.

Length Bars

When it becomes impractical to utilize gauge blocks for longer length measurements, it is the normal practice to use length bars instead. These length bars are normally produced from high-carbon high-chromium steel with a round section of Φ30 mm for greater stability and ease of handling, as shown in Fig. 4. Both of their ends are threaded and recessed and precision lapped to meet with the stipulated standards requirements of finish, flatness, parallelism, and gauge length. They have a threaded hole in the center which allows the bars to be safely and accurately joined together in various combinations. Airy points (Airy 1846) are utilized for accurate and precision metrology. The length standard is supported in such a way as to lessen its anticipated bending, or droop, while allowing the ends of the length bar to become parallel or square (Fig. 5a). These Airy points are symmetrically arranged around the center of the length bar and are separated by a distance.

The Bessel support points (i.e., after Friedrich William Bessel) are the support positions for an evenly loaded bar, where the droop for both the middle and ends of the bar are at a minimum (i.e., termed its neutral axis). These supporting points (Fig. 5b, bottom), often known as the points of minimum deflection, are not valid for end standards such as length bars but can be utilized more effectively for the support points for flat metrology tables or similar level items.

Dial Gauges, Dial Test Indicators, and Inductive Gauges

Dial gauges are probably the most universal metrology instruments utilized in basic inspection/verification/calibration activities. The ubiquitous dial gauge is usually available in two distinct types: (i) plunger-type (Fig. 6a) with the digital variety shown in Fig. 6b and (ii) lever-type (Fig. 6c) often referred to as a dial test indictor. Dial indicators typically measure ranges from 0.25 to 300 mm (0.015–12.0 in), with graduations of 0.001–0.01 mm. The smallest graduation of dial gauges is normally 1 μm. It can be replaced by the inductive gauge if high precision is required. For example, the precision of TESA GT21 inductive gauge (Fig. 6d) can be lower than 0.1 μm.
When mounting the dial gauge and then inspecting with a test mandrel in, say, a turning center’s headstock as shown in Fig. 7, the dial gauge/test mandrel alignment procedure should be carefully performed so that the dial gauge’s sensor axis is both perpendicular and aligned to the axis to be inspected, thus ensuring that both sine and cosine errors are effectively minimized. The actual difference in shaft center lines will be half the total indicator runout (TIR).

Straightness Calibration with Straightedges

Straightness measurement systems consist of a straightness reference and a displacement indicator. In order to measure the straightness errors with a material reference, the straightedge reference is placed in direction of the machine axis. As a straightness reference, a calibrated ruler or a stretched wire (for long axes) can be used. The axis is then moved, while a distance sensor (capacitance gauges, electronic gauges, or dial gauges) measures the lateral displacement.

One of the most common straightness measurement systems is the combination of a straightedge and a dial gauge. Straightedges can be produced from a variety of materials which include steel, cast iron, granite, Zerodur™, etc. They are usually provided with feet-invariably set at the Bessel points for minimum deflection. A Zerodur™ straightedge shown in Fig. 8 is an axis-calibrating nanocenter lathe. This type of straightedge material is produced in Germany by Schott AG. It is made from lithium aluminosilicate glass ceramic. Its low coefficient of thermal expansion, high 3D homogeneity, and good chemical stability make it a very good solution for high-precision metrology.
The accuracy and precision of straightness of the working faces for straightedges are tested by direct comparison with say, a master surface table, as shown in Fig. 9. Two calibration-grade and equal-sized gauge blocks are situated on the surface table just below the Bessel support points, with the straightedge on its working face being situated over these paired gauge blocks. And the gap (i.e., vertical distance) between the lower working face of the straightedge and the datumed surface table is measured at various points along its length by lightly and gently sliding in gauge blocks of appropriate size, which gives an indication of the parallelism degree along the length of straightedge.
By employing reversal methods (rotating the standard), the calibration errors and the reference errors can be separated. However, gravitational deformation will always point in the same direction. For the reversal method, the separation of reference errors is customarily achieved by reversing the position of a reference object along with the sensitive direction of the instrument and then simply repeating a sequence of measurements as a function of linear position or angle. In the first orientation, the machine error will be added to the result, while in the second orientation, the machine error will be subtracted from the result. For example, in an experimental setup shown in Fig. 10, an indicator is mounted on the travelling carriage in such a way that it is aligned in the direction of interest (i.e., usually orthogonal to the axis), touching a straightedge mounted in the desired vicinity. Assuming that the machine slide straightness is given by a function “M(x)” and the departure of the straightedge is given by a function “S(x),” it is then possible to calculate the indicator output for this position by adding these two functions of ‘M(x)’ and ‘S(x)’ together and then calling it “I1(x).”

Once satisfactorily completed, it simply reverses the straightedge by rotating along with its long axis and remounting the indicator so that it is touching the straightedge, but it now has had its direction or sign reversed. This action is important to note, because now when one calculates the indicator output “I2(x),” it is possible to see an apparent reversal of the machine axis; the ostensible lack of change in the straightedge output being despite the fact that this straightedge has been reversed.

Spindle calibration with Cylindrical Precision Mandrels

Cylindrical precision mandrels often have a self-holding taper and are also known as inspection mandrels, which are utilized for calibrating both the alignment of machine tool’s spindle axis and its runout. Typically, cylindrical taper mandrels (shown Fig. 11, left) are usually made of hardened and stabilized alloy steel. Standard sizes and accuracy of these straight and taper mandrels are typically manufactured to ISO 2063:1962.
Squareness Calibration with Precision Squares
Precision squares are commonly made from cast iron or granite (Fig. 12), which can be used for the calibration of the squareness of machine tool axes with the aid of dial gauges. The reversal method can be also used for the squareness calibration. The disadvantages are that the requirement for testing environment is relatively high and the measuring range is limited, generally no more than 1 m. An appropriate indicator is used on the slideway(s) of the machine to derive angular information (In), as shown in Fig. 13 (Evans et al. 1996). α is the out of squareness of the machine axes and β the test square:
$$\alpha =\frac{l_1+{l}_2}{2}$$
(1)
$$\beta =\frac{l_2-{l}_1}{2}$$
(2)

Measurement of Angle

Sine bar, angle gauges, polygonal mirrors, serrated tooth circle dividers, and rotary encoders are used for establishing motion through an angle or measuring angles.

Sine Bar

Firstly, we will mention here the traditional method of using a sine bar to establish an angle. This method is comprised of a sine bar, a block (like a gauge block – with a precise length), and an auxiliary flat surface (Fig. 14) from Nakazawa (1994). The sine bar is designed to act as a gage or angle reference or to make angular measurements of mechanical parts mounted on it. Usually, a stack of gauge blocks is set in the correct combination to form length h. The corners of the right triangle, A and B, are hardened gage pins. The inclined surface is hardened, ground, and lapped to a high degree of flatness. The device is fairly reliable at low angles, less than 15°, but becomes increasingly inaccurate as the angle increases.

Angle Gauges

A combination set of angle gauges normally consists of a specific range of separate gauges as shown in Fig. 15, which may be utilized in conjunction with one square block and one parallel straightedge. Like linear gauge blocks, angle gauge blocks can also be wrung together to build up a desired angle. In addition, they can also be subtracted to form a smaller angle as a difference of two larger angles. These angle gauges are normally made of hardened alloy steel, and the actual measuring faces are previously lapped and polished to a high degree of accuracy and flatness, similarly to that of gauge blocks. Flatness of each lapped face is 0.2 μm or less, and the faces are square to the ground sides within 2 μm. The contents of 27-piece, 16-piece, and 15-piece sets have been calculated so that it is possible to wring together gauges to form 32,400 different angles from 0° 0′ 0″ to 90° 0′ 0″ in 10-second increments. Additionally, the 27-piece set is also able to produce obtuse angles. The deviation from the marked angle for the laboratory master grade angle gauges is smaller than ±0.25 sec.

Precision Polygons

Optical polygons (Fig. 16a) are designed for use with autocollimators (Fig. 17) in the precise measurement of angle spacing and calibration of the circular-divided scales (with circles for optical-dividing equipment, such as rotary tables). Many of the precision polygons of today are produced from fused silica, with a special aluminum protective coating. The largest metallic precision polygon usually has 72 faces at an included angular interval of 5°. However, a typical precision polygon might have 12 sides at an interval of 30°, which is suitable for most types of inspection/verification/calibration work. These polygons commonly have equiangular faces, but for certain special purposes, unequal angles can also be provided by the optical manufacturers. In practical use, the polygon might be mounted on the indexing plate to be verified/calibrated, or a circular scale to be divided, as shown in Fig. 16b. The autocollimator is set to receive a reflection from any one of the faces of the polygon, and the reading is noted down accordingly. Then the polygon is rotated until the same reading is observed by the autocollimator, and thus obviously, the angle of the rotation will be equal to the angle between the polygon faces. It is possible to divide the circle as well as undertake any required recalibration.

Davidson Optical Polygons are available in standard configurations of 8, 9, 12, 18, 24, and 36 facets and may be ordered with any number of facets, up to 72. The flatness of facets is λ/10 and accuracy of polygons is ±5 arc seconds. As a result, its calibration of 1/10 arc second can be achieved.

Autocollimator

Although interferometry was applied to the establishment of standards of measure and eventually replaced the autocollimator as the preferred means for measurement of angular errors of form in mechanisms and workpieces, the autocollimator is still in use and serves as a practical instrument or a component of many measurements. If the autocollimator is used in conjunction with a reflecting mirror, it can accurately measure very small deviations from a datum angle. A visual autocollimator operates by projecting an image onto a target mirror/reflector as shown in Fig. 19, which is then used to measure the deflection of the returned image against a graduated scale, either visually (Fig. 18(a) or by means of an electronic detector – such as in the digital autocollimator/digital autocollimator (Fig. 18(b)). With the optical autocollimator, it can quantify angles as small as 0.5 arc seconds or even be ≥100 times more accurate for electronic/digital autocollimator.

Autocollimators can be used to measure angle (indexing head accuracy), straightness (machine tool slides in two axes), squareness (column to slideways), parallelism (slideways), etc.

Straightness, Squareness, and Parallelism Calibration

Figure 19a shows the schematic of measuring straightness with an autocollimator. Using a reflector (on a carriage) and a dual axis autocollimator, up to 200 measuring steps can be taken for straightness checking by moving the reflector carriage along the slideway in equidistant steps. Any out-of-straightness in either of the two slideway surfaces X and Y (side and top of the slide) will cause the reflector carriage to change angle with respect to the autocollimator, and it is these changes which are measured and computed to determine the errors in straightness. Measurement of parallelism of two machine tool rails can be achieved using an autocollimator with an optical square and mirror, as shown in Fig. 19c.

Autocollimation can also measure the squareness using a pentaprism that can deflect the light path with 90° (Jia et al. 2015). As shown in Fig. 20, straightness of plane A was firstly measured by the autocollimation and reflector, and then straightness measurement of plane B was carried out by the pentaprism. This method is particularly suitable for large objects, such as machine tool guide, large workpiece, and so on.

Flatness Calibration

The autocollimator can also be used for the calibration of flatness of a granite or cast iron table, as shown in Fig. 21a. The table should be labeled according to the Moody method or grid method. The Moody method (Espinosa et al. 2008; Meijer and Heuvelman 1990) was founded by JC Moody in 1955 and has been widely accepted to provide a quick calibration method for round, rectangular, and square surface tables. It is also referred to as the “Union Jack” method. It provides a map where all measurements must be conducted to grade the surface table. It is consisting of eight measuring lines in which each line can be divided into a multiple of measuring stations. For a rectangular plate, the Moody map is consist of three long lines, three short lines, and two diagonal lines as shown in Fig. 21a. For a square surface plate, we will have six equal lines and two diagonal lines. The same can also be assumed for a round surface table as it was seen in Fig. 21a. The number of measuring stations is determined by the length of the line and the agreed increment on measurement which is called the foot spacing. Figure 22 shows the procedures to measure the flatness of a granite table using the Moody method. This method does have one disadvantage – all of the points on all of the eight lines must be measured and plotted. This can cause problems in defining a foot spacing which will meet this requirement, particularly on slotted tables/surfaces where the position of one or more slots may coincide with the required position of one of the feet of the flatness base. With the grid method (Kim and Raman 2000), any number of lines may be taken in two orthogonal directions across the surface as shown in Fig. 21b. While the incremental nature of the measurement technique requires that all points on a given line are measured, it is not necessary to take measurements on all lines. This allows the measurement lines to be configured to avoid “obstructions” or to provide details in a given area. A disadvantage of both grid methods is that they require a reference plane. An alternative technique such as the Moody method is needed to define the reference plane.

Angular Calibration of a Precision Polygon

The autocollimator with the multiangular prism angle measurement principle has long been considered as the most precise means of angular positional determination. This technique of angle determination remains the national angle reference in many countries. Consequently, the calibration of such features of the angular measurement, namely, by the multiangular prism/polygon, can be considered as extremely important. A precision polygon can be calibrated by utilizing two autocollimators based upon the known fact that a circle is a continuous function, namely, whatever the values of the individual polygon’s angles, they exactly total to 360°; the setup which is typically configured is depicted in Fig. 23. For example, if an eight-sided polygon requires calibration prior to use, then the two autocollimators are positioned at an included angle of 45° with respect to each other. So, if “R1” and “R2” are the readings taken on the autocollimators and identified as “1” and “2,” respectively, with “S  ” being the angle between the normals of faces “A” and “B” and, “T  ” being the angle between these autocollimators, then by simple geometry we get
$$S+{R}_1=T+{R}_2$$
(3)
In order to complete the polygon’s geometrical faces for all values of “S  ,” “T  ,” and (“R2R1”)
$$\sum S=\sum T+\sum \left({R}_2-{R}_1\right)$$
(4)
“ΣS = 360°,” so
$$360=\sum T+\sum \left({R}_2-{R}_1\right)$$
(5)
By dividing through by the number of sides of the polygon, n, i.e.,
$$360/n=\sum T/n+\sum \left({R}_2-{R}_1\right)/n$$
(6)

A complete set of readings “Σ(R2R1)” can be established, enabling the value of “T  ” to be determined. “T  ” can be substituted back into equation, for each face, enabling the angle for each of these faces to be simply determined.

Calibration of a Rotary Table

Figure 24 shows the setup to calibrate the rotary table (Smith 2016). To ensure proper alignment, the polygon is mounted on the rotary table using an inside diameter as a reference. The inside diameter center line is parallel to the faces and square to the base. After alignment, one of the mirror faces on the polygon is rotated toward the autocollimator and zeroed; then the rotary table readout is zeroed. During inspection the table is rotated until its readout is the nominal angle of the polygon (45° increments for an eight-sided polygon). The next face should be aligned to the autocollimator. If not, the error can be read on the autocollimator. An alternative to the polygon is the ultraprecision index table. The typical angular accuracy of an index table is 0.25 arc second. A 360-position index table yields 1° resolution. To use an index table, a plane mirror is placed on the center of rotation and parallel to the axis of rotation. The index table is aligned in the same manner as a polygon. During inspection, the rotary table is rotated to 23°, for example, and the index table is counter-rotated 23°. If not again, the error can be read on the autocollimator.

Micro-alignment Telescope

The alignment telescope decomposes the autocollimator into two optical tools, the collimator and the telescope. Similar to the autocollimator, micro-alignment telescope is used for checking and setting, for example, alignment (series of bores or bearings); squareness (column to a base); parallelism (series of rollers); level/flatness (machine bed foundation); and straightness (rails or guideways).

Principle

The alignment telescope consists of two separate elements, a collimating unit and a focusing telescope, as shown in Fig. 25; the body of each of which is cylindrically ground to a truly accurate and precise known outside diameter. The collimator is an optical instrument consisting of a well-corrected objective lens with an illuminated reticle at its focal plane. The emerging beam is parallel (collimated beam), so that the image of the reticle is projected at infinity. It is customary for the collimating unit to contain the light source and condensers, in front of which is positioned a reticle in the focal plane of the collimating lens.
The telescope (Fig. 26) is an optical tool that images an object at a far distance (usually preset to infinity) into the image plane of the objective lens. The image is then magnified and visually inspected by an eyepiece. For measurement purposes usually a graduated reticle is located in the image plane. The magnification of the telescope is given by the ratio of the objective focal length and the eyepiece focal length. Two micrometer drums are situated within the telescope unit with the thimble being graduated with intervals of 0.02 mm, situated either side of the zero position and being click stop adjustable. These two micrometre drums are set at 180° apart and when the alignment telescope is mounted. The focusing knob is normally positioned at 45° to the vertical, as shown in Fig. 25. In this telescope orientation, the left-hand drum controls the movement of the line-of-sight in a horizontal plane, while the right-hand drum controls the movement in the vertical plane.

Applications

With its optical and mechanical axes aligned to within 3 arc seconds, a typical accuracy of 50–70 μm at 30 m is achievable with the micro-alignment telescope. Closer distances produce better results with a best accuracy of around 5–10 μm. The micro-alignment telescope generates a straight line-of-sight (LOS) from zero to infinity, as shown in Fig. 27. This forms the basic reference from which all measurements are taken. To measure squareness or parallelism, a pentaprism is used to deviate the straight line through exactly 90°. A similar rotating pentaprism (sweep optical square) is used to generate a plane for flatness measurement, as shown in Fig. 28(a).

The setting or checking of squareness (shown in Fig. 28b) utilizes a lamphouse to illuminate the crosslines, with the telescope being focused to infinity. Accordingly, from this telescope the rays of light are collimated and are successively reflected back along their own path from a mirror set square to this line-of-sight, forming a reverse image of the crosslines, on the actual original crosslines. A specific feature of the crossline patterns is two pairs of short heavy lines which produce a bold reverse image for precise setting. In consequence, if the mirror is slightly tilted, this crossline image will be displaced. It is worth to emphasize the point that when the telescope is focused at infinity, the micrometers are then ineffective; they cannot be employed in measurement process; no graticule or target other than the crosslines are utilized for autocollimation. A representative micro-alignment telescope system offers the following advantages when either inspecting or calibrating machines: optical and mechanical axes alignment to ≤3 arc seconds and concentric within 6 μm, an achievable accuracy and precision to ≤0.05 mm at 30 m, and an extensive field of view of which ranges from 50 mm at 2 m to 600 mm at 30 m. In addition to the performance of the autocollimators and alignment telescope themselves, the earth curvature and atmospheric refraction will also influence their calibration precision. The effective radius of the earth curvature becomes more noticeable over much longer linear distances, especially when a line-of-sight configuration with the telescope is compared with a level surface setup by a gravity-controlled instrument. The line-of-sight deviates from a straight line owing to refraction within the same atmosphere due to the density gradient in the atmosphere resulting from differential heating/cooling of the layers of air at varying heights above the earth surface.

Clinometers

The Taylor Hobson Talyvel electronic levels and a range of precision clinometers are used for level and flatness measurement (surface plates and granite tables), straightness and twist (roll) measurement (machine slides), squareness measurement (of machine columns), angle measurement (remote monitoring of movement of structures), parallelism measurement, etc.

Principle

The traditional type of clinometer utilizes the spirit-level principle (Fig. 29). In this metrological application, the clinometer incorporates a spirit level mounted onto a rotary assembly member, which is then carried within an instrument’s housing. One face of the housing forms the base of the instrument. Situated within this housing is a circular scale that is incorporated with the angle of inclination of its rotary member carrying the level being relative to the datum base. The angle inspected can then be established by the in situ circular scale.
Recently the compact digital inclinometers without a bubble level are widely used, as shown in Fig. 30. Incorporating a highly stable pendulum transducer in the level unit and a rechargeable battery, the Talyvel 6 is simple to calibrate and operate, with a fast measurement response time and exceptional stability. Talyvel 6 provides rapid and simple reading of angle of tilt and measurement relative to gravity; it can also function as a comparator to detect departures from a preset attitude, which may not necessarily be level. As shown in Fig. 31, a pendulum is suspended from a low stiffness torsion spring. Pendulum rotation can be induced by a linear acceleration along its sensitive axis (acceleration mode) or by a component of the gravity force along that axis caused by tilt (tilt mode). The pendulum displacement is sensed by a photoelectric position sensor, the output of which is amplified and fed to the coil of an electromagnetic torque attached to the pendulum. The polarity of the connection is chosen to produce a reaction torque which tends to return the pendulum to its zero position (negative feedback). Since the input and reaction torque’s exactly balance each other, the current producing the reaction torque is a very accurate and linear function of the acceleration and is used as output. In the tilt measuring mode, the output is proportional to the sine of the angle of tilt. This fact limits the effectivity of the sensor for tilt angles approaching ±90°.

Straightness and Parallelism Calibration

A clinometer can be used to measure the straightness of a slideway by moving the clinometer with a constant interval. For some metrological inspection applications, e.g., parallelism, two of these level units may be required – denoted here as “A” and “B” (see Fig. 32) which can then be controlled from the single display unit. A clinometer provides a differential system for measuring the actual difference in the inclination of two surfaces, as well as their departure from absolute level. This metrological application of differential levelling is of particular value in specific applications such as measuring the relative deflections, while machines are in actual production and in the accurate assembly of precision machinery, furthermore for the monitoring twist, or deflection on exceptionally slowly tilting surfaces.

Flatness Measurement

One of its most common uses of Talyvel electronic level is checking the flatness of granite and cast iron tables. Flatness can be measured and displayed by employing either the union jack (i.e., Moody) method or by the grid method. Figure 33 shows the Talyvel electronic level that was used to conduct the flatness measurement by participant A to grade the UUT (unit under test). The electronic level was firstly validated with the aid of three gauge blocks. It was then zeroed on the first measuring station. Then it moved to the next measuring station and noted down the angle reading on the dial. This was repeated for all measuring lines to capture the data on all measuring stations. The data was then fed to a Taylor and Hobson flatness program to compute the results. The uncertainly budget was also calculated by taking all the contributors into consideration.
To operate the electronic level, the level is first adjusted, so the meter reads zero at the starting position (at the beginning of each line for evaluation). The electronic level is then translated along a straightedge, which acts as guide, as shown in Fig. 33. Measurements are taken at evenly spaced distances along the line to be evaluated. The actual flatness of the measurement line will be the algebraic sum of the readings, as shown in Fig. 34. Finally, the flatness map can be drawn according to the error of all the measuring points, as shown in Fig. 35. Two electronic levels can be used to eliminate the influence of level weight. The first level remains stationary and the second is translated along the straightedge. The difference between the two measurements is recorded. If the surface plate tilts due to the weight of the translating level, the differential measurement will not be effected, while an absolute measurement made with a single level would be effected.

Interferometry

Interferometry is most widely used for displacement/length measurement today as well as some form measurement. It can also be used for measurement of positioning accuracy and straightness of the moving axis, positional accuracy of a rotary axis, surface form of a surface plate, perpendicularity of two nominally orthogonal axes, etc. It is a measure of relative movement (measurement from an initial position) rather than an absolute measurement (measurement of a specific position). Different selections of optics pass the laser beam through different paths, allowing a variety of measurement modes (e.g., linear, angular, straightness) to be taken from a single laser unit. Without reliable and accurate wavelength compensation, linear measurement errors of 20 ppm (parts per million) would be common in typical conditions. These errors can be reduced to ±0.5 ppm by applying precise environmental compensation, which is better than many other methods including tracker laser system or ballbar testing.

Renishaw XL-80 laser interferometer (Beno et al. 2013; Józwik and Czwarnowski 2015; Parkinson et al. 2012) is one of the most widely used high-precision interferometers for machine tool calibration. Key XL system components are a compact laser head (XL-80), an independent environmental compensator system (XC-80), and a comprehensive and powerful software suite. Together with the measurement optics, they form a highly accurate measurement and analysis system. The achievable specifications of XL-80 are shown in Table 1.
Table 1

Achievable specifications of XL-80 laser measurement system. (Courtesy of Renishaw plc)

Measurement

Range

Resolution

Accuracy

Linear

0–80 m

0.001 μm

±0.5 ppm

Angular

±10°

0.01 arc sec

±0.0002A ±0.1 ±0.007F arc sec

Straightness

0.1–4.0 m (short range)

0.01 μm (short range)

±0.005A ±0.5 ±0.15 M2 μm (short range)

1–30 m (long range)

0.1 μm (long range)

±0.025A ±5 ± 0.015 M2 μm (long range)

Rotary

Up to 25 revolutions

0.01 μm

±5 μm/m

Flatness

±1.5 mm

0.01 μm

±0.002A ±0.02 M2 μm

Squareness

±3/M mm/m

0.01 μm

±0.005A ±2.5 ±0.8 M μrad (short range)

±0.025A ±2.5 ±0.08 M μrad (long range)

M = measurement distance in meters of the longest axis

Linear Measurement

The setup measures linear positioning accuracy of an axis (Fig. 36) by comparing the movement displayed on the machine’s controller with that measured by the laser. The setup provides an accuracy of ±0.5 ppm (parts per million) with a resolution of 1 nm. During linear measurement the laser system measures the change in relative distance between a reference and measurement optical path. Either optic of the linear beam-splitter and the retro-reflector can be moving, providing the other optic remains stationary.

Angular Measurement

Pitch and yaw angular errors are among the largest contributors to machine tool and CMM positioning errors. Even a small error at the spindle can cause a significant effect at the tool tip. The setup shown in Fig. 37 can measure maximum angular deflections of up to ±10° with a resolution of 0.01 arc secs. Angular measurements are made by monitoring the change in optical path generated by the movement of the angular reflector. The angular interferometer is best mounted in a fixed position on a machine. The angular reflector is then mounted to the moving part of the machine.

Straightness Measurement

Straightness measurements (Fig. 38) record errors in the horizontal and vertical planes perpendicular to an axis movement. Straightness errors will have a direct effect on the positioning and contouring accuracy of a machine. The components used in this measurement comprise a straightness beam splitter (Wollaston prism) and a straightness reflector. The straightness reflector is mounted to a fixed position on the table even if it moves. The straightness beam splitter should then be mounted in the spindle. Straightness measurements are made by monitoring the change in optical path generated by the lateral displacement of the straightness reflector or straightness beam splitter. A combination of two straightness measurements makes it possible to assess the parallelism of independent axes.

Rotary Axis Measurement

Recent international standards state a rotary axis should be calibrated in a number of ways, which include 0.1° increments through 5°; 3° intervals through 360°; and 0°, 90°, 180°, and 270° positions and nine further random angular positions through 360°. It is extremely difficult to complete these measurements using autocollimators and optical polygons. Automated testing with rotary axis calibrator enables rotary axes to be checked at any angular position and far more quickly than with any other methods with precision down to 1 arcsec. The rotary setup uses an XL-80 laser, an XR20-W rotary axis calibrator, and an angular interferometer (Józwik and Czwarnowski 2015). A typical rotary axis calibrator setup is shown in the machining center configuration in Fig. 39a. An angular reflector is mounted on top of the rotary axis calibrator, which in turn is also mounted on top of the machine tool’s rotary table axis. In this case, it is an integral rotary table which is presently being calibrated. As the machining center’s axis under test is rotated from one target position to the next, the rotary axis calibrator is driven in the opposite direction in order to maintain alignment with that of the angular interferometer. When the axis under test stops at each target position, the positioning error is calculated by comparing this target position, with the arithmetic sum of the angular readings from the laser interferometer and from that of the rotary axis calibrator. This specific rotary action allows the calibration of the axis over a full 360° or even over multiple revolutions.

Flatness Measurement

Flatness measurement analyzes the form of a surface. This enables a 3D picture to be built up and documents the deviations from a perfectly flat surface. The flatness kit contains two flatness mirrors and three flatness bases to suit the size of the surface (Fig. 39b). The flatness mirrors not only rotate horizontally but also tilt vertically. This allows horizontal and vertical adjustment of the laser beam. In addition, angular measurement optics are required for flatness measurements. Two standard methods of conducting flatness measurements are supported by the laser software, i.e., the Moody method in which measurement is restricted to eight predefined lines and the grid method in which any number of lines may be taken in two orthogonal directions across the surface.

Squareness Measurement

The measurement of the squareness (Józwik and Czwarnowski 2015) is the extension of the straightness measurement in the two-dimensional direction. The setup and principle of squareness are shown in Figs. 40 and 41. The squareness measurement is to measure the straightness of two nominal orthogonal axes according the same reference. Then compare the straightness of the two axes to get the squareness of the two axes. The reference usually refers to the optical alignment axis of the mirror during two measurements. The mirror is neither moved nor adjusted during two measurements to maintain the reference line unchanged. The optical square is used for at least one measurement, allowing the laser beam to be aligned with the former straight line without moving the straightness of the mirror. Laser interferometer measurement is one of the most reliable ways to measure squareness. However, the Abbe’s offset combined with angular errors during the motion of an axis causes Abbe’s error. In addition, difficulty in the optical square setup causes restrictions on other optics and limitations of the measurable range. Recently, Lee et al. present mathematical approaches that can be used to eliminate Abbe’s error and to estimate squareness over the full range by using the best fit of straightness data without an optical square (Józwik and Czwarnowski 2015).

3D/6D Laser Interferometer

A 3D interferometer design is usually based on that of the 1D interferometer with added optoelectronic circuits (e.g., CCD cameras, PSDs, four-field detectors) to enable the simultaneous measurement of the lateral motions of the reflector. The accuracy of measurement along the given axis is equal to that of the interferometer, while in the other two axes, it is usually lower. An example of such a design is the laser interferometer with a four-field detector, shown in Fig. 42 (Kwaśny et al. 2011). It is a bifrequency interferometer exploiting the heterodyne method. In the case of bifrequency lasers, laser beam stabilization is of major importance, which consists in measuring the frequency resulting from the Doppler effect. A major drawback of this solution is the limitation of the measuring arm travel speed in one direction to about 0.3 m/s. If a proper splitter is used, it is possible to measure the other axis, but then the laser beam power drops by 50%. This may make it difficult or impossible to perform measurements on larger machine tools.
By adding more optoelectronic circuits, 6D lasers were built. By using the latter, the measuring time can be reduced up to 80%. Besides measuring the positioning error, horizontal and vertical straightness, and angular errors (pitch and yaw), a 6D laser system can be used to measure the roll error. This means that the components of a 6D laser measuring system need to be set and calibrated only once for each of the measured machine tool axes. The 6D XD™ Laser from Automated Precision Inc. (API) is a multidimensional laser measurement system that simultaneously measures linear, angular, straightness, and roll errors for rapid machine tool error assessment. Table 2 shows the measurement accuracies of the API XD laser system compared to 1D interferometer and 3D interferometer.
Table 2

Measuring accuracy of API XD laser systems compared to 1D interferometer and 3D interferometer (Kwaśny et al. 2011)

The main sources of error/uncertainty arising from the use of laser calibration systems are well known and documented and must be minimized if valid readings are to be obtained from the machine or equipment under test. The so-called weather station automatically compensates for the linear displacement readings from the laser on behalf of any variations in both the air refractive index and material temperature. So just some significant values of uncertainty are incurred when setting up and utilizing laser systems on machines, including:
• Cosine error. This will be present when the machine tool’s travel is not identical to that of the interferometer’s beam.

• Abbé offset error. This is when the position of the laser measurement is shifted away (i.e., offset) from the position of the actual axis cutter travel.

• Dead path errors. These are influenced by ambient conditions in the same manner as the displacement part of the beam. So, the interferometer reasons that the path length is the distance moved by the mirror, but in actuality this path length is affected by ambient conditions, the distance from the interferometer block to the matched mirror.

The system does not perceive that there is additional air in the measurement arm and as a result will not compensate for changes in the wavelength of the laser in that portion of the beam. As a consequence, the general equation for the air dead path error (EADP) is provided as follows:
$${E}_{\mathrm{ADP}}=D\left({\lambda}_{\mathrm{air}}-{\lambda}_0\right)/{\lambda}_0$$
(7)
where “EADP” is the air dead path error, “D” is the separation between the optics at datum (i.e., the dead path), “λair” is the current laser wavelength, and “λ0” was the laser wavelength – when the system was datumed. The general equation for material dead path error (EMDP) is
$${E}_{\mathrm{MDP}}= D\alpha T$$
(8)
where EMDP is the material dead path error, ‘α’ is the linear coefficient of expansion of the material in the dead path, and “T  ” is the change in temperature of the material since the system was first datumed.

Normally, material dead path errors are potentially much more significant than its associated air dead path errors. The optimum methodology to lessen these undesirable conditions is to apply good metrology practices.

The material dead path – this should be made by situating the setup of the laser as closely and directly as possible to the point of metrological interest.

Any changes in material temperature during measurement – this should be made by initially stabilizing the temperature and/or promptly completing the measurements.

The optics separation (when the system is datumed) – this should be made either by utilizing a preset reading or by employing a beam splitter as the moving optic.

Tracking Laser

The original laser tracker was patented in the USA in 1987. These now very popular trackers can be considered as polar coordinate measuring systems, being capable of high-accuracy measurements over quite long linear distances. For example, in one of the systems now being utilized by NPL, it can measure at distances of up to 40 m to within discrete accuracies of measurement of ±60 μm. The current laser trackers can make measurements to ~10 μm accuracy without any special geometry or data processing. By choosing advantageous geometry, by calibrating repeating errors, and by averaging random errors, the instrument’s tracking allows it to measure to even tighter levels of <1 μm.

Principle

Laser trackers take measurements using a spherical coordinate system as shown in Fig. 43a, which means that any given point is specified by three numbers: (a) the radial distance of the point from the tracker; (b) the polar or zenith angle, which measures elevation; (c) the azimuth angle, measured on a reference plane that passes through the tracker and is orthogonal to the zenith. The numbers for the zenith and azimuth angles are determined by means of two angular encoders which measure the orientation in the tracker’s gimbal along its mechanical axes. The radial distance of the point from the tracker is determined using an interferometer (IFM), an absolute distance meter (ADM) or a combination of the two.

Laser trackers equipped with IFMs split their lasers in two: one beam travels directly to the interferometer, while the other travels to a target device in contact with the component being measured. The two lasers interfere with one another inside the interferometer (hence the name), resulting in a cyclic change equal to one quarter of the laser’s wavelength – approximately 0.0158 μm. Each time the target device changes its distance from the tracker. By counting these cyclic changes, what’s known as fringe counting, the laser tracker can determine the distance the laser has travelled. ADMs use infrared light from a semiconductor laser which bounces off the target device and reenters the tracker. The infrared light is then converted into an electrical signal for time-of-flight analysis.

Applications

The basic laser tracker setup consists of the tracker itself (Fig. 43a), a target device (Fig. 43b), and a computer running the tracker’s application software. The most common target device is a retroreflector, which reflects the laser beam back in the direction from which it came. The most common retroreflector design is the spherically mounted retroreflector (SMR) as shown in Fig. 43b, which has the advantage of keeping the center of the retroreflector at a constant distance from the surface being measured. To take measurements with the tracker, the inspector firstly sets up a laser tracker on a tripod, with an unobstructed view of the object to be measured (see Fig. 44). Then the inspector removes a target from the base of the tracker and carries it to the object, for certain regions of its features to be measured, moving smoothly to allow the laser tracker to follow these movements of the target. During this procedure, the inspector individually places this target against the object, triggering measurements to be taken at preselected points.
PTB and NPL have jointly developed a method for error mapping of machine tools (Schwenke et al. 2005). The concept is based on displacement measurements between reference points that are fixed to the base and offset points fixed to the machine head. These measurements are realized by a tracking interferometer that is mounted on the workpiece table and a retroreflector that is attached to the machine head. Figure 45 shows a setup in principle. For each of the three laser positions, the machine tool performs spindle motions maximally filling its workspace. In each point of the space grid, the axis motions are stopped, and the displacement of the reflector relative to the stationary reference ball is recorded. Machine tool errors are defined as the differences between the programmed displacements and the measured ones.

Variations

Having in mind higher precision in determining the position of a point in space, research on the realization of the multilateration procedure in accordance with the GPS (global positioning system) principle, i.e., through the use of not one but several electronically interconnected transmitters, is underway (Schwenke et al. 2005; Wang et al. 2011). The MultiTrace system made by Etalon (Fig. 46) has been designed for this purpose. Such a solution may substantially reduce test time.
As part of the project entitled “Volumetric Accuracy for Large Machine Tool,” carried out jointly by Boeing, Siemens, Mag Cincinnati, and Automated Precision Inc. (API), methodology enabling the precise calibration of large multiaxis machine tools in a very short time (in a matter of hours) has been developed (Kwaśny et al. 2011). A special tracking laser, called API T3, using the company’s volumetric error compensation (VEC) technology (Bangert 2009), working in tandem with the patented active target (Fig. 47) is used. Active target devices are new-generation retroreflectors equipped with two drives; hereby they can automatically position and track the laser beam, preventing it from being interrupted. The angle of laser beam incidence on the retroreflector can change to a rate of 50°/s. Since the retroreflector can rotate by a large angle, only one tracking laser position is required for measuring volumetric error.

Telescoping Ballbars

Component dimensional and finish defects may result from bad tooling, worn spindles, or workpiece clamping, but the major causes of defects can usually be attributed to positioning errors in the machine tool itself. In the past the workpiece accuracy may have been achieved by machining test or “master” parts and then inspecting them. However, this was time-consuming and gave limited confidence when machining parts with geometries different to the master part. Ballbar testing first developed by Bryan provides the quickest (typically 10 min), easiest, and most effective way to monitor machine tool condition and quickly diagnose problems that may require maintenance and the error sources that produce them.

Principle

The basis of the Renishaw QC20-W ballbar is shown in Fig. 48a. Any of the errors of a machine tool will cause the radius of the circle to deviate from the programmed circle. If you could accurately measure the actual circular path and compare it with the programmed path, you would have a measure of the machine’s performance. The heart of the system is the ballbar itself, a very high accuracy, telescoping linear sensor with precision balls at each end. In use the balls are kinematically located between precision magnetic cups, one attached to the machine table and the other to the machine spindle or spindle housing. This arrangement enables the ballbar to measure minute variations in radius as the machine follows a programmed circular path. The resolution and accuracy of the QC20-W ballbar system are 0.1 μm and ± (0.7 + 0.3% L) μm. The error of the distance transducer alone does not exceed ±0.5 mm. The Bluetooth device enables data transmission for a distance of up to 10 m. Figure 48b shows the schematic diagram of ballbar testing.

In an archetypal ballbar test, the ballbar might be assembled to sweep, say, a 150 mm radius, but for other tests the ballbar’s radii can be changed, ranging from 50 mm to that a larger radius of >1000 mm becomes possible. A longer radial ballbar length could be employed to increase sensitivity to geometry errors such as squareness; conversely a very small ballbar radius might be utilized to highlight any dynamic errors such as servo mismatch.

A Closer Examination of Machine Tool Inaccuracies

Diagnostic testing with a telescoping ballbar for machine tool verification can reveal a number of machine-related problems. According to the ballbar troubleshooting guide, the problems of the machine tools can be identified, and corresponding measures can be taken. Take the squareness, for example. This error geometry is shown as an oval ballbar plot (Smith 2016) (Fig. 49), usually occurring when orthogonal axes are no longer moving at 90°, relative to one another. This type of error may possibly be due to a bent axis or some other form of axis misalignment. This squareness error ovality on the plot will tilt it at 45° with respect to the two axes, remaining in the same position regardless of the direction of travel of the ballbar.

Since it was designed for only simple three-axis milling centers and three-axis lathe centers with ball screws and feed drives, the expert system aiding inference about the causes of errors is a series limitation of the above method. Consequently, only simple circular interpolation tests can be carried out for other machines, both conventional ones and with parallel kinematics. Another drawback of the ballbar device is that measurements are taken only in a selected part of the machine tool workspace. Despite the above drawbacks, the ballbar enables machine tool diagnostics and does not necessitate long breaks in the manufacturing process, mainly owing to measurement simplicity and speed.

Variations of Ballbar

Similar to the magnetic ballbar, the laser ballbar (LBB) (Schmitz et al. 2001; Ziegert and Mize 1994) uses the telescoping assembly to maintain beam alignment with the moving retroreflector and features a much longer range of motion than does Bryan’s with relative displacements measured via laser interferometry. It consists of a two-stage telescoping tube with a precision sphere mounted at each end. A heterodyne displacement measuring interferometer is aligned inside the tube and measures the relative displacement between the two sphere centers.

Once initialized, the LBB uses trilateration to measure the spatial coordinates of points along a CNC part path (Schmitz and Ziegert 2000). The six edges of a tetrahedron formed by three base sockets (rigidly attached to the machine table) and a tool socket (mounted in the spindle) are measured, and, by geometry, the spatial coordinates of the tool position in the LBB coordinate system are calculated. The three lengths between the three base sockets (denoted LB1, LB2, LB3) shown in Fig. 50 are measured once and are assumed to remain fixed during the motion of the tool socket. The three base-to-tool socket lengths (denoted L1, L2, L3 in Fig. 50) are measured simultaneously during a single execution of the applicable CNC part program. Then the coordinates of the tool point are calculated from the ballbar readings to define the contouring accuracy.
Another kind of laser ballbar is laser vector measurement technique (Janeczko et al. 2000; Wang and Liotto 2002) for the determination and compensation of volumetric position errors. A laser noncontact measurement of static positioning and dynamic contouring accuracy of a CNC machine tool (Figs. 51 and 52) is a relatively recent machine tool calibration technique, its operation being based upon a single-aperture Laser Doppler Displacement Meter (LDDM) using the laser vector method. Briefly, when a laser beam is reflected from a target, the Doppler frequency shift is proportional to the velocity. Since the frequency shift is the change of the phase and the velocity is the change of the position, after the integration with respect to time, the Doppler phase shift is proportional to the position. Once the phase is measured, the position can be determined. The foremost features are that the measurement is noncontact where the circular path radii can be varied continuously, ranging from 1 to 150 mm; the laser/ballbar uses a laser displacement meter for the measurement, and the accuracy is typically 1 ppm and traceable to NIST. However, two sets of measurements with two setups are needed for the laser/ballbar to generate the circular path compare to telescoping ballbar, only one setup and one set of measurement.
Although ballbar and the latest laser ballbar (LBB) are capable of two-axis error measurements, they are still sensitive to one dimension only. A novel design that integrates the merits of a LBB and laser tracking systems (LTSs) has been proposed for the three-dimensional measurement of moving objects in real time. The system, called the 3D Laser Ballbar (3D LBB) (Fan et al. 2004), is based on the spherical coordinate principle containing only one precision laser linear measurement device and two precision laser rotary encoders in the gimbal base with an extendable ballbar (Fig. 53a). Three sensors simultaneously record the ball positions and transform into the Cartesian coordinate in real time. The LaserTRACER-MT (MT, mechanical tracking) system made by ETALON (Fig. 53b) is an exemplary commercial implementation of the above idea. The system can be used to calibrate small- and medium-sized machine tools with their assemblies shifted by up to 1.5 m.

The main limitation of the LBB and 3D LBB devices is the minimum length of the telescoping tube, which determines the dead part of the workspace and the maximum size of the workspace in which measurements can be made. The state-of-the-art tracking laser systems have no such limitations.

Measurement of Spindle Error

The performance of precision spindles is a significant contributor to the overall accuracy of a precision machine or instrument. For years, the error motion of the highest precision spindles was essentially too small to reliably quantify because the measurement uncertainty exceeded the measurand (a few nanometers for the best spindles). However, improvements in sensors, instrumentation, data acquisition, signal processing, and structural design are closing this gap and enabling low uncertainty spindle metrology.

Principle and Instrument

A conventional method for the calibration of rotary axis is described in ISO 230-1 (ISO 230-1 1996) which proposes the use of a dial gauge to measure the radial and axial runout deviation at the center hole of the rotary axis. If the gauge cannot be applied at the center hole, it can be used in combination with precise manufactured check gauges which are mounted on the rotary axis. Another possibility is the use of capacitive (Chapman 1985; Marsh et al. 2006) or inductive (Frennberg and Sacconi 1996) sensors, which measure contactless and can be used at much higher rotational speed. Capacitance gauging has several advantages over the alternative Eddy current gauging or inductive sensing, which most notably is that capacitance: insensitive to metal alloy variations, not affected by variations in the surface, allows measuring equipment to be readily calibrated in an approved calibration laboratory, and can be traceable to an international standard.

The spindle error motions (dynamic and thermal) is commonly measured and analyzed by the Spindle Error Analyzer (SEA) system (Grejda et al. 2005; Marsh et al. 2006) as shown in Fig. 54. Master target with a maximum roundness error of 50 nm and its fixture are mounted on the machine tool spindle and will provide the reference surface for measurements taken by this SEA system. A standard setup consists of three sensors mounted in X-, Y-, and Z-direction, respectively. The sensors measure simultaneously the master ball allowing a real-time dynamic radial and axial measurement. With five sensors the tilt of the spindle can also be determined in X- and Y-direction. For five axes of measurement as shown in Figure, a probe is mounted from the bottom to measure movement in the Z-axis. A pair of probes is mounted at right angles (top view) to measure movement in the X and Y axes. A second pair of X and Y probes are mounted to measure a second master ball. The combination of these probe pairs generates tilt measurements. The last degree of freedom is the error of the rotation angle itself, which can be measured with a laser interferometer in combination with a self-centering device and the optical components for an angular measurement.
The most widely used method to present error motion of an axis of rotation is to plot error motion versus the angle of rotation of the spindle in a polar chart (PC) as is shown in Fig. 55 (Shu et al. 2017). The most widely recognized evaluation method is based on least-squares center (LSC) of the polar chart. The LSC is the center of a circle that minimizes the sum of the squares of radial deviations measured from it to the error motion polar plot. The synchronous error motion value is the scaled difference in radius of two concentric circles from LSC just sufficient to contain the synchronous error motion polar plot, while asynchronous error motion value is the maximum scaled width of the asynchronous error motion polar plot, measured along a radial line through the polar chart (PC) center.

Error Separation

Generally, error motion of a spindle is directly measured with a capacitive probe targeting a spherical or cylindrical artifact mounted on the top of the spindle. However, this traditional method no longer applies to error motion measurement for an ultraprecision axis of rotation because of form error of the artifact. As a result, several error separating methods have been developed. The most well-known methods such as Donaldson reversal, multistep, and multiprobe have been demonstrated to have uncertainty of the order of 6–20 nm. The basic concept of Donaldson reversal is the reversed characteristic of artifact form error based on that complete error separation is achieved (Donaldson 1972). As shown in Fig. 56, spindle radial motion is given by R(θ), and the ball roundness is given by B(θ). Assuming that angular information is derived from the spindle position, the indicator outputs, I(θ), for the two positions are
$${I}_1\left(\theta \right)=R\left(\theta \right)+B\left(\theta \right)$$
(9a)
$${I}_2\left(\theta \right)=-R\left(\theta \right)+B\left(\theta \right)$$
(9b)
$$R\left(\theta \right)=\frac{I_1\left(\theta \right)-{I}_2\left(\theta \right)}{2}$$
(9c)
$$B\left(\theta \right)=\frac{I_1\left(\theta \right)+{I}_2\left(\theta \right)}{2}$$
(9d)
Multistep is another well-known error separation method. With sufficient measurement steps and elaborately designed multistep method (Anandan and Ozdoganlar 2016), it is possible to achieve measurement accuracy of nanometer level, although harmonic suppression exists. Figure 57 shows the procedure for multistep method. Record probe readings around several cycles before rotating the artifact by an angle ϕ until N measurements are completed, such that  = 360°. Each measurement should be made at the same starting point, and all the data should be sampled at evenly spaced positions around one cycle of the rotator. Let mi(θ) be the ith measurement, and the radial error motion of spindle axis S(θ) is given by
$$S\left(\theta \right)\approx \frac{1}{N}\sum \limits_{i=1}^N{m}_i\left(\theta \right)$$
(10)

Averaging the multistep measurements separates the error motion of the spindle from the artifact form, except at frequencies that are at integer harmonics of the number of steps. Therefore, caution should be used when interpreting the results. However, when a sufficient number of steps (e.g., 20) are used, the first distorted harmonic occurs at a relatively high frequency. For high quality lapped artifacts, the amplitude of artifact error occurring at these higher harmonics is relatively small (normally a few nanometers) (Marsh et al. 2006).

The multiprobe method, as illustrated in Fig. 58, involves more than two probes spaced at given angles with each other. Compared with multistep methods, multiprobe methods are more suitable for on-machine measurements, because the repeat measurement of the spindle error is not necessary.

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