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Rubeo_ThesisDefense_rev3
- 1. A Feasibility Study on the Application of Laser Tracking Interferometers for Machine Tool Error Mapping by Mark Rubeo at Charlotte Faculty Advisor: Dr. Robert J. Hocken Committee Members: Dr. Edward Morse Dr. Tony Schmitz 1
- 2. Build an Error Model of the Machine Tool Measure the Machine Tool’s 21 Parametric Errors Calculate the 3- Dimensional Error Map Design and Fabricate a Mounting Fixture Measure Volumetric Errors Calculate the 3- Dimensional Error Map Compare the Results Compare the error map produced by the traditional quasi- rigid body kinematic error model to the error map generated through laser tracker based measurements. Project Objectives 2
- 3. Background Information Quasi-Rigid Body Error Mapping Laser Tracker Error Mapping ● Machine Tool Errors ● Measurement Methods ● Laser Trackers ● Mathematical Model ● Parametric Errors ● Error Map ● Fixture Design ● Measurement Correction ● Error Map 3
- 4. 4 Errors Affecting Machine Work-Zone Accuracy Quasi-Static Errors: • Geometric • Load-Induced • Thermal Dynamic Errors: • Kinematic • Vibration-Induced • Spindle Error Motions
- 5. Y Z X εYX εZX εXX δXX δYXδZX εYθ εZθ εXθ δXθ δYθ δZθ Y Z Xθ Error Motions of a Linear Axis Scale Error (Linear Displacement Error) 𝛿 𝑋𝑋(𝑋) Y Straightness of the X Axis 𝛿 𝑌𝑋(𝑋) Z Straightness of the X Axis 𝛿 𝑍𝑋(𝑋) Roll of the X Axis 𝜀 𝑋𝑋(𝑋) Pitch of the X Axis 𝜀 𝑌𝑋(𝑋) Yaw of the X Axis 𝜀 𝑍𝑋(𝑋) Error Motions of a Rotary Axis Radial Translation Error 𝛿 𝑋𝜃(𝜃) Radial Translation Error 𝛿 𝑌𝜃(𝜃) Axial Translation Error 𝛿 𝑍𝜃(𝜃) Tilt Error about the X Axis 𝜀 𝑋𝜃(𝜃) Tilt Error about the Y Axis 𝜀 𝑌𝜃(𝜃) Positioning Error (Displacement Error) 𝜀 𝑍𝜃(𝜃) Geometric Errors 5
- 6. Kinematic Errors 6 Squareness Measurement Techniques: • Diagonal Displacement • Optical Square • Mechanical Square • 2 Dimensional Contour
- 7. Direct Measurement Methods: • Material-based • Optical • Gravity-based Indirect Measurement Methods: • Artifact Standards • Machined Test Parts • Contour Measurements • Displacement Measurements • Requires Identification Strategies Direct and Indirect Measurement Methods 7
- 8. • API Tracker 3 Absolute Accuracy of a 3D Coordinate o ± 5 ppm (2 Sigma) Laser Accuracy o IFM: > ± 0.5 ppm o ADM: > ± 15 µm or 1.5 ppm Angular Accuracy o 3.5 µm∙m-1 • Active Target Centering Accuracy o ± 3 µm Z Y X φ θ r (r,φ,θ) Laser Tracking Interferometers 8
- 9. • Considers the motion of the machine structural elements without considering the forces which act upon them • Assumptions: Principle of Superposition Determinism Quasi-rigid structural elements Positioning error is a function of nominal machine position Errors associated with an axis are a function of the position of that axis alone • 1988 Monarch VMC45 (Three-Axis Machining Center) Classification: XYFZ (Part-to-Tool) Quasi-Rigid Body Kinematic Error Model 9
- 10. 𝑋 PX = ZF + TZ − XY − YF PX = R X R Y R Z −1 TZ + ZF − YF − XY Y X Y' αXY Squareness Sign Convention Mathematical Model 10
- 11. 𝑋 X = − XC + δXX X δYX X δZX X Y = δXY Y − YC ∙ αXY − YC + δYY Y δZY Y Z = δXZ Z + ZC ∙ αXZ δYZ Z + ZC ∙ αYZ ZC + δZZ Z T = XT YT −ZT PX = R X R Y R Z −1 TZ + ZF − YF − XY R X = 1 εZX X −εYX X −εZX X 1 εXX X εYX X −εXX X 1 R Z −1 = 1 −εZZ Z εYZ Z εZZ Z 1 −εXZ Z −εYZ Z εXZ Z 1 Mathematical Model 11
- 12. PX = X Y Z = XC + ∆X YC + ∆Y ZC + ∆Z ∆X ∆Y ∆Z = X − XC Y − YCl Z − ZCl ∆X = δXX X − δXY Y + δXZ Z + Y ∙ αXY + Y εZX X + εZY Y + Z ∙ αXZ − Z εYX X + εYY Y + XT + YT εZX X + εZY Y − εZZ Z + ZT εYX X + εYY Y − εYZ Z ∆Y = δYY Y − δYX X + δYZ Z − X ∙ εZX X + Z ∙ αYZ + Z εXX X + εXY Y + XT −εZX X − εZY Y + εZZ Z + YT + ZT −εXX X − εXY Y + εXZ Z ∆Z = δZZ Z − δZY Y − δZX X + X ∙ εYX X − Y εXX X + εXY Y + XT εYX X + εYY Y − εYZ Z − YT εXX X + εXY Y − εXZ Z − ZT PX = R X R Y R Z −1 TZ + ZF − YF − XY Mathematical Model 12 Commanded Machine Position Positioning Error
- 13. • 50 mm Target Position Interval • 1 mm Overrun Travel • Feedrate: 1500 mm/min • 5 Bi-Directional Measurement Runs • Federal Electronic Levels EGH-13W1 • Agilent 5519A Laser Interferometer Y Z X Origin of Table CS Target Positions Overrun Positions Parametric Error Measurement 13
- 14. X Axis Parametric Error Measurements 14
- 15. Y Axis Parametric Error Measurements 15
- 16. Z Axis Axis Squareness α [arcseconds] XY 3.0 XZ 16.1 YZ -3.6 Parametric Error Measurements 16
- 17. Maximum Volumetric Error = 19 µm Quasi-Rigid Body Kinematic Error Map 17
- 18. Design Goals • Maximize Rigidity • Minimize Weight Brunson Thread Adapter Bolt Hole Circle (BHC) 1018 C-Channel Structural Steel 1018 Cold-Rolled Steel (CRS) Plate Active Target Retroreflector Laser Tracking Interferometer Laser Tracker Fixture Design 18
- 19. 10.5 kg 0.179 m 10.5 kg Analytical Model FEA Model ωn 2 = k M + 0.2357m k = 3EI L3 ωn = 467 Hz ωn = 464 Hz Laser Tracker Natural Frequency 19
- 20. A B C Kinematic Mounts (4) Zerodur Base Position 1 Position 2 Position 3 Position 4 Calibrated Lengths • A = 149.9979 mm • B = 300.0002 mm • C = 99.9905 mm Measurement Parameters • 120 samples per point • 5 measurements of each point • ½ inch Spherically Mounted Retroreflector (SMR) Calibrated Length Measurements 20
- 21. Position 1 Position 2 Position 3 Position 4 Calibrated Length Measurements 21
- 22. Position 1 Position 2 Position 3 Position 4 Calibrated Length Measurements 22
- 23. δφ α Zenith Axis θ Measurement Number Azimuth Variation [Degrees] 1 -0.000543 2 -0.000005 3 -0.000426 4 -0.000071 5 -0.000415 6 -0.000193 7 -0.000676 8 -0.000242 9 -0.000396 10 -0.000219 Average -0.000319 Standard Deviation 0.000210 Two Face System Test 23
- 24. Y Z X • 50 mm Spatial Grid • 960 Target Points • 120 Samples/pt. • Feedrate: 1500 mm/min • 1 Bi-Directional Run Volumetric Error Measurement 24
- 25. Coordinate Transformation Methods Laser Tracker Coordinate System Table Coordinate System 1. Origin of Table CS 2. Least Squares Fit of X Axis Data 3. Least Squares Fit of Y Axis Data 4. Vector Normal to Plane Spatial Analyzer • Point, Line, Plane • Ambiguous mathematical process Matlab 1. Define the Origin of the Table Coordinate System 2. Define the X Axis 3. Find the Y Axis Average Line 4. Normal vector defines Z Axis 5. Cross Product of X and Z defines Y Axis Laser Tracker measurements must be transformed to the table coordinate system. 25
- 26. X Axis Linear Translation Errors Coordinate transformations via Matlab yield better agreement with the quasi-rigid body model. 26
- 27. 100 W Work Light 250 mm Radiation Shielded Temperature Sensors hP = 1.58L2 × 10−8 h 𝑇 = 46.4βL2 × 10−8 hMAX = hP + hT = 0.3 𝜇𝑚 Parameter Value 𝛽 0.8 𝐾 ∙ 𝑚−1 𝐿 0.898 𝑚 The effects of atmospheric refraction are sufficiently small that they cannot be effectively corrected. 27
- 28. X Y Z Nominal Machine Position Measured Machine Position Measurements performed using IFM show better agreement with the quasi-rigid body model. 28
- 29. • Location of volumetric error maxima differs • Magnitude of volumetric error differs IFM and ADM measurements yield significantly different error maps. 29
- 30. Magnitude of volumetric error differs by up to 66 µm. The quasi-rigid body model and IFM measurements predict similar volumetric error maxima locations. 30
- 31. Laser Tracker Position 1 Laser Tracker Position 2 Exceptions: • Table Bending • Range-dependent measurement uncertainty The machine tool’s error map should be independent of the laser tracker’s mounting position relative to the work volume. 31
- 32. Scatter is within the laser tracker’s measurement uncertainty. Laser tracker measurements from position 1 and position 2 show good agreement. 32
- 33. Error maps predict similar: • volumetric error maxima location • volumetric error magnitude Laser tracker measurements from position 1 and position 2 exhibit strong positive correlation. 33
- 34. • Largest Source of Measurement Error → Horizontal Angle Encoder • Effects of atmospheric refraction are minimal and cannot be effectively corrected. • Horizontal angle deviations due to height of standards error cannot be easily corrected. • IFM measurements yield results that are in better agreement with the quasi- rigid body kinematic model. • The error maps produced with the laser tracker on opposite sides of the work volume yield similar results. 34
- 35. • Measure and correct for the scale error of the horizontal angle encoder. • Measure the 3 dimensional grid in the instruments front and back face to average out the height of standards error. • Employ a best fit identification strategy to determine the geometric errors based on the laser tracker measurements. 35
- 36. • Dr. Robert Hocken • Committee Members: • Dr. Edward Morse • Dr. Tony Schmitz • Dr. Jimmie Miller 36
- 37. 37
Editor's Notes
- My name is Mark Rubeo. I’ve been working under the direction of Dr. Hocken on machine tool error mapping using laser interferometers.
- Machine tool error mapping aims to quantify the 3 dimensional positioning accuracy of a machine tool; typically for the purposes of software correction. Traditionally this is done by building a mathematical error model describing the machine tool’s kinematics. The parametric errors of then measured and the mathematical model is used to calculate the error map throughout the machine’s entire work volume. To generate the error map based on laser tracker measurements, the tracker must be mounted on the machine table. To achieve this a mounting fixture was built. Then the 3 dimensional coordinates of the tool with respect to the laser tracker at multiple positions is measured, and the error map is calculated. Finally, the resulting error maps are compared.
- I’ll begin by detailing some relevant background information including machine tool errors, measurement methods, and the basics of laser tracking interferometers. This will be followed by the procedure of machine tool error mapping based on the quasi-rigid body kinematic model. This includes the derivation of the mathematical model, measurement of the parametric errors, and the resulting error map. Finally, I’ll discuss the error mapping based on laser tracker measurements including the mounting fixture design, measurement corrections, and the resulting error map.
- Machine tool accuracy is affected by errors that can be classified into two groups: Quasi-Static and Dynamic Errors. Quasi-static errors include any mechanical error which varies at a rate sufficiently slow as to enable the use of corrective motions of the machine elements including geometric, load-induced, and thermal errors. Additionally, dynamic errors such as kinematic, vibration-induced, and spindle error motions affect machine accuracy. I’ll be going into more detail with geometric and kinematic errors because they’re included in the quasi-rigid kinematic error model.
- Geometric errors are the result of imperfections in the form, position, and orientation of individual positioning elements. Ideally, the positioning elements of a machine tool are designed to have single degree of freedom motion, but due to geometry errors this is not the case. Typically each positioning element, such as a linear or rotary axis, will exhibit errors motions in all six degrees of freedom. The six error motions include three linear translation errors and three angular errors. For a linear axis, the translation error occurring in the direction of motion is called scale error or linear displacement error, and the linear translation errors occurring perpendicular to the direction of motion are called straightness. The angular errors about the X, Y, and Z axes are called roll, pitch, and yaw, respectively. The error motions of a rotary axis are similar but the nomenclature is slightly different.
- Kinematic errors are deviations from the ideal, coordinated motion of multiple axes which includes parallelism and perpendicularity. They are particularly important in machining operations such as contour milling, threading, and gear hobbing. There are multiple techniques for measuring squreness such as diagonal displacement methods, optical squares, mechanical squares, and 2 dimensional contouring measurements.
- Direct measurement methods, otherwise known as parametric measurement methods, aim to isolate and quantify a single geometric error of an axis without the involvement of the other axes [6]. Direct measurement methods are well established and widely used in the machine building industry [13]. Material-based methods use calibrated artifacts such as ball plates, step gauges, and line scales; optical methods such as laser interferometry use the stabilized wavelength of light; and gravity-based methods such as differential levels use the earth’s gravitational field. Indirect measurement methods involve the coordinated motion of multiple axes and results in an error quantity that is a convolution of multiple, superimposed geometric errors [6]. In order to detect the individual error terms some type of identification strategy, such as best fit methods, must be employed. Indirect measurement techniques may use artifact standards, machined test parts, contour measurements, and displacement measurements.
- The operating concept for most laser trackers is the same. Although, there are some design variations amongst the different manufacturers and models. A laser beam is generated either in the tracker’s base or in the rotating head and the beam is passed through a polarizing beam splitter. The beam is divided and a portion of it is diverted to a position sensing detector (PSD) while the other portion is reflected to a rotating mirror. The rotating mirror is mounted on a two-axis gimbal that is able to steer the beam by rotating about two orthogonal axes. These rotations, called the azimuthal and elevation angles, are measured by precision angle encoders. The distance, or range, measurement is performed through interferometry (IFM) or absolute distance measurement (ADM) which uses the time of flight measuring principle. Laser tracker measurements are reported in spherical coordinates (r, φ, θ) as shown in FIGURE 2.1. In order to for the laser tracker to maintain a beam lock on the retroreflector throughout the entire work volume, a retroreflector with a large acceptance angle is required. For this reason the active target was used. The active target operates in a similar fashion to the laser tracker. The active target has two orthogonal rotation axes which allow it to position the retroflector such that the beam is always locked.
- The quasi-rigid body kinematic error model makes various assumptions. The principle of superposition asserts that the various individual geometric error sources may be isolated and measured individually and then combined mathematically to yield an error map. The assumption of determinism asserts that machine tools position systematically and through proper calibration the accuracy may be improved. Additionally, it is assumed that the positioning errors of an individual axis are a function of the nominal position of that axis alone. In fact, the positioning error is a function of the direction of approach, velocity, acceleration, and the position of other axes. Because some of these assumptions are known to be in error, the quasi-rigid body kinematic error model is imperfect but has been successfully implemented for decades and will serve as the standard for comparison in this study.
- The first step to building the machine tool’s mathematical model is to assign coordinate systems to the machine’s structural components. In this case, the X, Y, and Z axis as well as the machine’s frame. Next, we define vectors pointing from one coordinate system to another around the machine’s structural loop in manner which reflects the machine’s kinematics. The goal is to derive an expression for the part vector which defines the position of the tool with respect to the workpiece, or X, coordinate system. In order to calculate the part vector in the X coordinate system, the X, Y, Z, and tool vector must be rotated into the X coordinate system through the use of rotation matrices.
- The matrix equation for calculating the part vector has been derived. Next, the X, Y, Z, and tool vector must be defined in a manner such that the sign convention is consistent with our vector model and includes the linear translation and squareness errors of that axis. For example, the X vector points in the negative X direction. Additionally, notice the inclusion of axis squareness, denoted as alpha, in the definition of the Y and Z vectors. The three dimensional, infinitesimal rotation matrices include the angular error terms of the X, Y, and Z axes.
- After expanding the matrix equation, the resulting expressions include the commanded machine position as well as the cumulative effect of the parametric errors. In order to extract the positioning errors, delta x delta y and delta z, the commanded machine position is simply removed. The resulting expressions allow for the calculation of the x, y, and z positioning error at any location within the work volume. Second order and higher terms have been removed from this expression because their overall affect is minimal.
- To measure the parametric errors, a table coordinate system was defined and the six parametrics errors of each axis were measured relative to this coordinate system. The target positions were separated by 50 mm intervals with 1 mm overrun position at each end in an attempt to eliminate reversal error.
- This figure shows the results of the 5 bi-directional measurement runs for each X axis parametric error. Forward runs are denoted by solid lines with circular markers and reverse runs by dotted lines and asterisk markers. The errors are consistent with what is expected of a machine such as this. Most of the geometric errors show a high degree of repeatability, but the pitch error shows drift. I believe this is most likely due to setup hysteresis.
- The Y axis geometric errors display the same high degree of repeatability, and in most cases are less than 5 micrometers for the translation errors and 5 arcseconds for the angular errors. The exception is the scale error of the Y axis which reaches a maximum of about 11 micrometers.
- The Z axis geometric errors of the machine aren’t as smooth as the X and Y axes particularly above the Z = 0 postion. This axis is driven by rack and pinion as opposed to ballscrew driven like the X and Y axes. Additionally, it features a milling clamp feature wherein the spindle is locked to provide additional stiffness. During measurement it was observed that the geometric errors varies depending on whether the milling clamp was on or off. For the duration of these measurements the milling clamp was turned on. It is noteable that the roll of the Z axis was not measured. There are two reasons for this. One, this error term doesn’t contribute to the positioning error in a meaningful way because this is the direction of the rotational spindle axis. And Two, there is no readily available method for measuring this error. This table shows the axis squareness as measured using the diagonal displacement method. Notice that the XZ squareness is significantly higher than the other two squareness terms, and it varied considerably with engagement of the milling clamp feature.
- Once the parametric errors have been measured, the error map of the machine tool may be calculated. The figure on the left displays a color map of the volumetric error magnitude on the planar faces of the work volume. The top figure is the top and front two planes, and the bottom figure show the rear two and bottom planes. The figure on the left is a colored vector map of the volumetric errors throughout the work volume. It is observed that the maximum volumetric error occurs here at a value of 19 micrometers. The is noticeable oscillation of the error magnitude in the Z direction within the work volume which is likely due to the behavior of the Z axis.
- Now, I’ll begin to discuss the error mapping procedure based on laser tracker measurements. In order to obtain the position vector of the tool in the workpiece coordinate system, the laser tracker must be rigidly mounted to the machine table. To this end a mounting fixture was designed with the goal of maximizing rigidity and minimizing weight.
- An analytical and FEA model were used to predict the fundamental frequency of the fixture. The setup was modeled as a cantilever beam with a mass at the free end. Using these equations, the natural frequency was calculated as 467 Hz. This was corroborated by the FEA model, using the same boundary conditions. It should be noted that these boundary conditions do not accurately reflect the real boundary conditions and it is likely that the natural frequency of the setup is significantly lower than these models predict.
- Some basic performance tests were conducted with the laser tracker. The goal of the performance test is gain some understanding of what measurement component, range, horizontal angle or vertical angle, contributes the most error to the measurement. An artifact composed of zerodur with four kinematic mounts and three calibrated lengths, detailed here, was mounted in four positions in the work volume. Each calibrated length was measured 5 times in both IFM and ADM mode and in all positions.
- This figure shows the deviation of the length measured by the laser tracker from the calibrated length at each artifact position. Notice that position 1 and position 2 which exercise a large horizontal angular range display a high degree of measurement scatter, and position 3 which is almost entirely range-based contains the least scatter.
- The figure on the left shows average length deviation of a single calibrated length. The error bars represent one standard deviation. In the treatment of measurement data, it is often the case wherein data points which vary significantly from the mean are considered as outliers and disregarded. This technique is typically bolstered by statistical methods, but ignoring measurement data is inherently in contradiction of the scientific method. The figure on the right shows the average length deviation and standard deviation of the calibrated length measured with the outliers removed. It is notable that the deviations fall within plus or minus 10 micrometers.
- During the two face test, the discrete position of a retroreflector is measured in the arbitrarily named “front face” of the instrument. Then, the laser tracker head is rotated 180°, nominally, about the azimuthal and elevation axes and re-locks to the retroreflector. The discrete position is now measured in the “back face” of the instrument. Nominally, the difference in the measured horizontal angle from the “front” and “back” face should be 180°, but due to the height of standards error, this is not the case. It was believed that the systematic deviation of the angular measurement due to the height of standards error could be compensated for in the measurement results, but as you see the standard deviation of the angular measurement is of the same magnitude as the average which indicates that this error cannot be effectively corrected.
- For the machine calibration using the laser tracker, the work volume was divided into a three dimensional spatial grid with target positions at 50 mm intervals for a total of 960 target points. Each target point was sample 120 times and averaged to produce the final measurement result.
- In order to calculate the error map, the measurement data which is collected in the laser tracker’s coordinate frame must be transformed to the table coordinate system that was established for the quasi-rigid body kinematic error model. The software package called Spatial Analyzer, which interfaces with the laser tracker and allows for control over the measurement procedure may be used to perform the coordinate transformation, but the mathematical process is largely unknown. It was decided that performing the coordinate transformation using Matlab offered a degree of control over the process. In order to do the transformation a coordinate system based on the measured data was constructed.
- In most cases, laser-based measurement techniques rely on the propagation of light rays through the atmosphere. This is commonly referred to as the optical line of sight. When this optical line of sight passes through open air it cannot be assumed to have a straight trajectory because of light refraction due to index gradients, variations in the speed of light, and turbulence caused by time-dependent variations in the index of refraction [16]. The equations shown here allow for the calculation of the vertical deviation of the laser beam due to vertical pressure and temperature gradients. I will discuss this in more detail later.