Assessment of the volumetric accuracy of a machine with geometric compensation A.P. Longstaff, S.R. Postlethwaite & D.G. Ford Precision Engineering Centre, University of HuddersJield, England Abstract In a production environment it is important to be able to quantify the accuracy of all machines in order to estimate the production capability of the company as a whole. A measure of machine production capability is volumetric accuracy, which is effectively a combination of all sources of geometric error. The requirement for ever tighter tolerances on manufactured parts has led to the development of several systems capable of compensating for the effects of geometric errors inherent in machine tools and CMMs. This is usually acheved by making linear correction actuations to the orthogonal axes at each point throughout the working volume, dependent upon the magnitude of the sum of the errors in that axis direction. This paper discusses the problems involved in assessing the volumetric performance of a machine tool operating with such a volumetric error compensation system. It explains the need for measurement of the effect of angular errors, rather than the error source themselves when quantifying the performance of a compensated machine. It further considers the additional requirement for measuring the angular error sources directly if tool offsets of different lengths are to be considered. This is of particular relevance when long tool or probe offsets are commonly used, thus machining at a distance from the position of measurement when calibrating the machine. The need for a uniform error compensation grid when using any indirect volumetric assessment technique is also noted. It has been found that such a regimen is not necessarily enforced, particularly in the CMM field.
1 Introduction Machining capability is a crucial factor in a modern flexible workshop. With tightening of tolerances to which a machine can perform comes a greater diversity of jobs it is possible to perform on that machine. It is therefore necessary to be able to assess accurately the capability of all machines in a workshop and, where possible, improve performance. This becomes particularly important when in-process probing of machined components is performed on the same machine as that used for component manufacturing. Only by having an accurate machine can the probing results be meaningful. Assessment of machine tool performance is carried out by a variety of methods Broadly, these fall into two categories, direct and indirect measurement. An important consideration is the availability of measurement equipment. Techniques requiring specialist tools, such as tracking laser interferometers, can prove prohibitively expensive. Therefore, techniques using standard laser interferometers or artefacts to provide information for the whole machine have been investigated and applied. ASME B5.54 [21, the American standard for CNC performance evaluation, uses a diagonal test to assess volumetric performance. Although this can allow the rapid evaluation of a machine over the diagonals of the machine, it does not provide information over the entire volume. The localised information produced is also only partially useful in discerning the source of the errors. By using laser interferometry and artefact probing the twenty-one geometric error sources can be measured directly. Using this information it is possible to predict the performance of a machine over the entire working volume [41. Since this technique requires information on each of the error sources, it is possible to analyse the contribution of each of the errors and produce recommendations for methods of improving machine performance by mechanical adjustment or refurbishment. The geometric error information can also be used in compensation systems capable of volumetric error correction on machine tools[5261 and CMMs. Once the compensation has been applied, it remains necessary to evaluate the machine performance to determine the amount of residual error in the machine. This paper discusses the methods required to evaluate the residual error in a machine with volumetric compensation. It also describes the use of additional information when considering the performance of machines operating with a tool or probe that is offset from the end of the ram of the machine. 2 Error synthesis A rigid body model for three-axis Cartesian machines can be used to determine the effect of each of the geometric errors of a machine at any given point in the working volume. The effect of the translation errors, namely linear position and straightness errors, is easily understood. The effect of angular errors is more complex since it is a function of the magnitude and direction of the error, the
axis configuration and the distance over which the error is amplified by movement of a second axis. The magnitude of the angular errors in an axis varies with the position of the axis as it translates along its length. However, the effects of these errors are only experienced when a second, amplifier axis, moves. The resultant positional error is proportional to the amount of movement in this amplifier axis. Because of this, a small angular error will produce a large positional error when amplified by a large axis movement. Conversely, a relatively large angular error may result in a small positioning error if the amplifier axis has only a short stroke. It is possible to determine the linear effect of each angular error at each point in the working volume by using the correct model for the particular machine [31. In addition, analysis of the single-valued squareness errors between each of the axes is performed in the same way. The effect of the angular errors can be combined with the translation errors to evaluate the linear effect of the errors in each of the axis directions. This gives the total geometric displacement error for each axis, as proposed below. This concept is fundamental to the application of volumetric compensation and is the basis for the assessment of the volumetric accuracy of a machine. 2.1 Definitions For an n-axis machine with defmed Cartesian coordinate reference system, assuming rigid body model. At any point, the total geometric displacement error for each of the reference axes can be defmed as the sum of the linear effects, in that direction, resulting from all the geometric errors of all n axes. The volumetric error at any point in the working volume is defined as the vector sum of the total geometric displacement error of the three reference axes at that point. The volumetric accuracy of the machine is defined as the maximum of the volumetric errors in the working volume of the machine. The figure for volumetric accuracy is relative to a specified datum position and relates to a defined working volume. In this way it is possible to assess the performance of a machine for each required component size as well as for the entire machine. 3 Volumetric error compensation By using the synthesis techniques described in the previous section, it is possible to determine the total geometric displacement error for each of the axes at any point in the working volume of a machine. By using a volumetric compensation system [5261 these errors can be reduced by applying a linear correction actuation.
44 Lute, \ti~tr.oloq~ ci11d tlric 11i11e l'er./oi~iirtrrrc r Figure 1 (a) and (b) show that, because of the rotation of the Z-axis about the X-axis, for different positions of the Z-axis the end of the ram is offset from the nominal position in the Y-axis. This is the linear effect of the geometric error. Figure 1 (c) then shows how this error can be corrected by moving the Y-axis by an amount equal and opposite to this error. Correction (b) (c> Figure 1 : Linear correction for effect of angular error A more simple approach to error correction is used for co-ordinate measuring machines (CMMs). Because CMMs are used to record the position of the machine when probing, it is not necessary to correct the physical position of the axes, but rather to adjust the recorded or displayed position by software manipulation. These correction methods do not remove the angular error itself, but adjust the axis coordinates by the total geometric displacement error. This creates a problem when attempting to evaluate the volumetric accuracy of the compensated machine. It is no longer valid to use an electronic level or laser interferometer with angular optics to measure the magnitude of the angular error since only the effect of the error has been reduced. A strategy of calculating the magnitude of the residual angular error from measurement of its effect is required. This is achieved by measuring the linear positioning errors of an axis down two separate measurement lines that are a known distance apart. The same target positions are used for each measurement. The angular error, e, is then calculated for each target position by:
where ei is the error at target position i ai is the linear error at target position i for the datum measurement bi is the linear error at target position i for the second measurement d is the displacement between the two measurement runs. The sign of the error is determined by the sign of the linear errors and the sign of the displacement. -- - - - Measured - Forward, - Measured - Reverse l - Calculated - Folward Calculated - Reverse -1400-1200 -1000-800 600-400 -200 Axis position (mm) Figure 2: Comparison of angular measurement techniques Figure 2 shows the X-axis rotation about the Y-axis error (X-axis pitch) from a typical machine. The measured data was taken from a bi-directional laser measurement using angular optics. The calculated angle was derived from two linear measurements at a displacement of 800mrn in 2. It can be seen that there is good correlation between the data acquisition methods. The use of linear measurements to determine angular error can be applied to any machine, whether compensated or not. The advantage is that the measurement is of the unwanted effect of the error, rather than the source of the error, which cannot be compensated by electronic means. However, the
disadvantage is that the laser must be set-up at two distinct positions for each measurement of angular error. The method can also be extended to angular error of an axis about itself, more commonly described as the roll error. In this situation it is often necessary to use two straightness measurements in order to assess the improvement achieved by applying compensation. 4 Effect of Tool Offset The above discussion concerns three-axis volumetric compensation. For such a system, the correction is applied for errors at a point, usually the end of the ram, where tooling would normally be attached. This generally corresponds to the position where measurement optics are mounted during calibration. The prevalence of in-process probing and the potential use of long tools introduces the probability of work being performed offset from this position. To analyse the errors at such a working point requires additional data and hrther geometric error terms in the model. 4.1 Three-axis model Initially consider a machine without volumetric compensation. The general three-axis rigid-body model that is used for volumetric assessment considers only those geometric error sources that contribute to an error at the end of the ram. For example, for a gantry machine (Figure 3) rotation of the Z-axis about itself (Z-axis roll) does not produce a geometric error at the end of the Z-axis. Therefore, this error source is not included in the error calculations. Figure 3: Gantry Machine For a similar machine with a tool offset, additional error components must be measured. Since the positional error resulting from angular error is a function of the length of the amplifier axis, any increase in this distance due to tool length will also result in an increase in the magnitude of the error.
From Figure 4 (a) it can be seen that applying an extension in the form of a probe or tool will result in an increase in horizontal error. The desired position is Yo but, as a result of angular error, the end of the ram is at Y,. A long probe or tool would further amplify the error to Y2. Figure 4: Tool extension in line with ram Figure 4 (b) shows the case of a compensated machine. The axis has been linearly adjusted by an amount (Yo-YJ to compensate for the angular error, 0. However, a residual error 6Y exists, due to 60, the difference between the angular value used by the compensation system and the true angle. This difference could results f?om uncertainties during measurement of the error, or mechanical changes over time. 6Y is a function of ram extension and the residual angular error, which must be calculated from two linear positioning runs, as described above. However, the length of the tool amplifies the mechanical angular error, 0 resulting in the end of the tool being at a point Y3. The distance Y3 from Yo+GY is calculated from the angle taken using angular optics, or using the linear measurement method with no angular compensation applied. A more complex problem arises when the tool is no longer in line with the axis carrying the tool (Figure 5). In this case, it is essential to have knowledge of additional error components and use a full model of the geometric errors. For the example configuration given in Figure 3, such a probe orientation would require
the angular measurements for the Z-axis and the rotation of the Y-axis about the Z-axis. For the machine whose angular error is presented in Figure 2, in-process probing is undertaken at an extension of 230mm in the X-axis direction. Amplification of the pitch error by the length of the offset would produce an error of up to 18pm in the Z-axis direction. Vertical error Figure 5: Tool extension at an angle to ram As a further example, consider a machine with a vertical spindle in the Z-axis which is used to probe in the Z-axis. Any roll of the z-axis will not produce a measurement error since it has no amplifier axis. However the inclusion of a tool perpendicular to the spindle axis (e.g. nominally the X-axis) will amplify the angular error by the length of the tool in question, producing an error in the third (Y) axis. 5 Analysis Software The Error Simulation Program [41 (ESP), produced at the University of Huddersfield, has been further developed to include the analysis of a machine including tool offsets. A typical parameter selection dialogue box (Figure 6) is split into two sections, clearly separated by the centre horizontal line. In the upper half of the screen the errors affecting the position of the end of the ram (in this case the T-axis) are specified. In the lower half, the additional error files required for analysing the effect of tool offset on the error are input. The tool length and orientation are also specified in this section, allowing prediction of errors during typical usage. For a machine without volumetric compensation, the angular error files affecting the end of the ram will be the same as in those in the upper part of the screen. 5.1 Results A machine with volumetric compensation has been evaluated using the techniques described above and ESP. It was found that 9pm of error in the X- axis resulted fyom the residual angular error after correction. This represented eighty seven percent of the error range for that axis. Thirty six percent of the Z- axis errors derived from the effects of an offset using a typical probe extension employed by the company, mainly due to the large rotational error presented in
Figure 2. Although not causing the machine to be out of tolerance, the ability to identify the source of these errors allows further error avoidance. This could be achieved by modification of the part program by inclusion of offsets depending on tool-length. Figure 6: Error selection screen 6 Uniform Grid While investigating the application of this technique on machines with volumetric compensation several CMMs have been investigated. Correction of errors throughout the volume has been standard practice on such machines for many years. However, it has been found that a practice within the CMM servicing community may produce a spurious analysis. After installation of a CMM, it is necessary to perform regular recalibration of the machine in order to validate any results that are taken from it. This is usually achieved by measuring a small subsection of the working volume, for example taking a linear positioning measurement using a laser at a single position of the axis. If this is found to be outside the desired tolerance by a small amount, adjustment of the compensation grid can be made. However, the error grid calculation is not generally performed in the same way as at first installation and is likely to result in a distorted grid. Obviously, a machine that has such a non-uniform correction map can not be assessed using the rigid body model as described above. The volumetric accuracy will become dependent upon the position of measurement of the errors.
7 Conclusions Volumetric assessment of Cartesian machines using a universal three-axis rigid body error model has been an extremely useful tool. Application of volumetric correction using the same model has allowed machines to produce parts to tighter tolerances. In order to assess the residual error after compensation, a method for calculating angular errors using linear position measurements has been devised. Because of the prevalence of in process probing and machining using long tools, it has become necessary to provide analysis for geometrically induced errors at the end of the tool. The requirement for the measurement of additional error components has been described. Additionally, it has been noted that some angular components must be measured using both the linear positioning method and using standard direct measurement techniques. Software has been developed to perform the additional calculations. It has further been noted that adjusting the error grid of a CMM, as practised by some calibrators, will produce a non-uniform error profile. This is not congruent with the assessment methods described above and will produce uncertainties in the accuracy of the machine throughout the worlung volume. References IS0 230-2, "Determination of accuracy of positioning of numerically controlled machlne tool axes" 1997 ASME B5.54-1992 Standard, "Methods for Performance Evaluation of Computer Numerically Controlled Machining Centres," (reaffirmed in 1998) Postlethwaite, S.R., "Electronic based accuracy enhancement of CNC tools", PhD thesis, University of Huddersfield, 1992 Postlethwaite, S.R. & Ford, D.G., "Geometric error analysis software for CNC machine tools", Laser Metrology and h4achine Performance Ill, pp 305-3 16, 1997 Postlethwaite, S.R. & Ford, D.G., "A practical system for 5 axis volumetric compensation" Laser Metrology and Machine Perfbrmance IV, pp 379-388, 1999 Fletcher, S., Postlethwaite, S.R. & Ford, D.G.. "Volumetric compensation through the machine controller". Submitted for publication in Laser Metrology and Machine Performance V, 2001