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THE IMPLEMENTATION OF SIMULTANEOUS TRILATERATION TO MEASURE DYNAMIC THREEDIMENSIONAL CONTOURS USING THE LASER BALL BAR By TONY L. SCHMITZ A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT OF THE REQUIREMENTS FOR THE DEGREE OF DOCTOR OF PHILOSOPHY UNIVERSITY OF FLORIDA ACKNOWLEDGMENTS The author would like to thank his advisor, Dr. John Ziegert, for his ideas, patience, and affable administrative style. The author would also like to thank the renowned Dr. Jiri Tlusty, Dr. John Schueller, Dr. Ramakant Srivastava, and Dr. Ali Seireg for serving on his supervisory committee. At the top of the list for "others to thank" are Dr. Chris Mize for his Laser Ball Bar design expertise and David Smith for his continual tutelage in the finer points of frequency analysis and signal processing. The author would also like to express his appreciation to Rick Glos of the Heidenhain Corporation for the loan of the KGM 101 Grid Encoder used for the STLBB verification tests. This work was supported in part by the National Science Foundation under grant numbers DDM935138 and DGE9354980. Other financial support was supplied by the Department of Energy/National Academy of Engineering 1998 "Integrated Manufacturing Predoctoral Fellowship." TABLE OF CONTENTS page ACKNOW LEDGM ENTS.............. ..........................................................................ii LIST OF FIGURES ................. .. ........................................................v ABSTRACT ................................... .................. ......................xi CHAPTERS 1 INTRODUCTION ................... ... ............. ...................... 1 2 LITERATURE REVIEW.............................. ......................9 Dynamic M measurements .................. ................................................................9 TwoDimensional Dynamic Measurement Tools............................. ............ 11 Software for PreMachining Contour Evaluation ....................... ........................ 18 3 DISPLACEMENT MEASURING INTERFEROMETRY...................................... 19 Introduction ................. .. ......................................................... 19 Polarization.................. ................................................................... 23 Interference ........................................................................... .................. 28 Heterodyne Interferometry .......................................................... ....................... 37 Fiber Optics in Interferometry ....................................................... .....................49 4 STLBB DESIGN ...................................................................................................59 Tool Socket Joint.................. .. ........................................................61 Tool Point Bracket...........................................................77 Optics Package................................. ............... ............................................. 78 Error Budget............................. ............................................84 5 DYNAMIC MEASUREMENT RESULTS ...................................................... 106 Initialization Repeatability ................. ...................................................... 106 2D M easurem ents............................................................................................. 108 Dynamic/Static Repeatability .................................................................. 117 2D Contouring Accuracy ........................ ................................................. 122 iii 3D Measurements............................................. 130 6 MACHINING VERIFICATION ................................... ....................... 140 Introduction ................................................. ................................................ 140 STLBB Results ............................................... 141 Experim mental M ethod....................... .............................. ........................... ...... 146 7 CONCLUSIONS ............................. .................152 Com pleted W ork........ ................ ............................. .............................. 152 Future W ork ........................................................................... ..................... 155 APPENDICES A PROCESS CAPABILITY.............................................. 158 B TRANSFORMATION OF BALL BAR COORDINATES INTO MACHINE COORDINATES .............................................. 160 C SURFACE LOCATION ERROR IN MACHINING ............................................ 162 LIST OF REFERENCES ............................................................... 193 BIOGRAPHICAL SKETCH.................................................... 197 iv LIST OF FIGURES Figure page 11. Laser Ball B ar .............................................................. ... ................... 4 12. Trilateration Tetrahedron.................................... ........................................ 4 13. Sequential Trilateration........................................................ .......................6 14. Sim ultaneous Trilateration................................................... .......................6 21. Heidenhain G rid Plate..................................................... .. ..................... 13 22. Magnetic Ball Bar............................... ........................ 13 23. Ball Bar Contouring Error...................... ....................... 15 24. Circular Path Section....................... .......... .. ................... 15 25. Circular Test Setup............................................................ ...................... 16 31. Plane M irror Reflection ..................... .....................................21 32. Polarizing Filter.................................................................. ...................... 24 33. Linearly Polarized Light..................................................... ...... ................. 26 34. Left Circularly Polarized Light............................. .... ...................26 35. Young's Double Slit Experiment ................................................29 36. M ichelson Interferom eter..................................................... ..................... 31 37. TwymannGreen Interferometer..................... ...... .....................33 38. MachZehnder Interferometer.................... ......... ......................33 39. Alignment Accuracy .................... ........ ......................36 310. Beat Phenom enon............................................................. .......................39 311. Heterodyne Phase Relationships ...................... .. ...........................41 312. Beat Frequency Shift ................................................ ..................... 44 313. Acoustooptic M odulator ................................................... ...................... 46 314. Z eem an Split ............................................................................ ......................46 315. Linear Displacement Interferometer......................... ...................... 50 316. Angular Measurement Setup................. ...... ...........................50 317. Fiber Delivery/Collection.....................................................51 318. Total Internal Reflection............................. ...................... 51 319. Refractive Index Profiles....................... .. ................ .......................54 320. Singlem ode Fibers............................................................. ...........................58 41. Laser B all Bar ........................................................................... .......... 60 42. Simultaneous Trilateration................................................ .......................60 43. Socket Diameter Constraints ............................................ ....................... 63 44. Tool Socket Spring A id..................................................... .......................65 45. Castigliano M odeling........................... ....... ..................... 65 46. Spring Aid Range of Motion.............................................................................67 47. Magnetic Flux Lines ............................. .....................67 48. Static Free Body Diagram .................................................... .....................70 49. Friction Test Stand .................................................................... ....................73 410. C alibration C ouple.................................................................... .....................73 411. Contact Area Geometry...................................................... 76 412. Tool Point Bracket....................................................79 413. Spring Scale Measurement...................... ........ .......................79 414. Optics Package...................... ..................................................... 80 415. Optics Package Placement................................... ..... ....................80 416. 4DOF Positioning Mechanism .............................. ..... ......................83 417. LBB Unsensed Length........................... ...... .... ..................... 85 418. Sphere/Motion Misalignment...................... ........ ......................88 419. LBB Interferometer Deadpath......................... ............................88 420. Initialization Fixture Offsets............................. ................. .....................92 421. Castigliano Model Loading Conditions............................ ......................95 422. LBB Moment Calculation Sections..................... .............................95 423. Left Sphere Total Angular Rotation..................... ... .......................98 424. Cosine Error Due to Interferometer Rotation...........................................98 425. M oving Retroreflector Rotation................................... ........... ... 100 51. LBB Initialization Fixture................................. .... ..................... 107 52. LBB Initialization Repeatability ........................... ............... 107 53. Heidenhain Grid Plate Setup.................................................. 110 54. ST L B B Setup ....................................................................... ...................... 110 55. STLBB Verification Contours .......................................... ................... 112 56. Angle Path Comparison (35 ipm, 0.1g)....................... ...... ......... ........... 114 57. Angle Path Comparison (70 ipm, 0.5g).................... ..................... 114 58. Step Path Comparison (70 ipm, 0.lg)..................... ........................... 115 59. Step Path Comparison (70 imp, 0.5g) ......................................................... 115 510. Sultan Path Comparison (35 ipm, 0.lg) .......................... .................... 116 511. Sultan Path Comparison (70 ipm, 0.lg) ..................... ..................... 116 512. Square Path Comparison (70 ipm, 0.5g) .......................................... 118 513. Triangle Path Comparison (35 ipm, 0.lg) ....................... ..................... 118 514. Circular Path Comparison (70 ipm).......................... .................... ... 119 515. Heidenhain Dynamic Repeatability Test.................................................... 119 516. STLBB Dynamic Repeatability Test ................... .......................... ... 121 517. Capacitance Probe Setup............................. ... ...................... 121 518. STLBB Angle Path (70 ipm, 0.1g)....................... ................... 123 519. STLBB Angle Path (70 ipm, 0.5g)................................ ...................... 123 520. STLBB Square Path (70 ipm, 0.5g)................ ........... ........ .............. 126 521. STLBB Square Path Reversal (70 ipm, 0.5g).............................................. 126 522. STLBB Sultan Path (70 ipm)............................... ... ..................... 128 523. STLBB Sultan Path Reversal (70 ipm)..................... .... .............. 128 524. STLBB Circular Path (35 ipm)...................... .... ........... ........... 129 525. STLBB CounterClockwise Circular Path (70 ipm)........................................... 129 526. STLBB Clockwise Circular Path (70 ipm, 0.lg)........................................ 131 527. STLBB Corkscrew 3D Path and STLBB Measurement................................... 131 528. STLBB Hemisphere 3D Path and STLBB Measurement ................................. 133 529. STLBB Oblique Rectangle 3D Path...................... .................... 133 530. STLBB Oblique Circle 3D Path.................................. .............. 134 531. STLBB Corkscrew 3D Path Section..................................... ..................... 134 532. STLBB Hemisphere 3D Path Section.................................. ...................... 135 533. STLBB Oblique Rectangle 3D Path Section.............................................. 135 534. STLBB Oblique Rectangle XY Plane .................................. ...................... 137 535. STLBB Oblique Rectangle XZ Plane........................ ......... ............. 137 536. STLBB Oblique RectangleYZ Plane......................... .................... .. 139 61. Diamond Tool Path ............................................................. 142 62. X Reversal Error............................................................ ...................... 142 63. Y Reversal Error............................................................ ...................... 143 64. XY Gain Mismatch Error..................................................... 143 65. Diamond Path Gain Mismatch Effect (70 ipm)................................................ 145 66. Determination of Ad (35 ipm Test) ...................................... ....... ............. 145 67. Frequency Response Function (FRF) for 0.5" Diameter Endmill ..................... 147 C Up/D own M killing ............................................................ ..................... 163 C2. Undercut/Overcut in Milling ............................................................ 163 C3. Cutting Forces in Down M illing................ ................ ........................ 167 C4. YForce Components (r < 1) .......................................... 167 C5. System Frequency Response............................................. ... 169 C6. Phasor Diagram (r < 1)............................................................ 169 C7. YDisplacement Components (r < 1) ............................................ 171 C8. YForce Components (r = 1) .................................... ....................... 172 C9. Phasor Diagram (r = 1)........................................................... 175 C10. Phasor D iagram (r > 1)................................................... ............................ 175 Cl Undercut/Overcut/Undercut Transition..................... .. ........ ........... .. 179 C12. Simulation Flow Diagram ............................... .......................... 179 C13. Part G eom etry..................................................................... ................ 182 ix C14a. X Direction Direct Transfer Function........................ ........... ........... 182 C14b. Y Direction Direct Transfer Function........................................................... 183 C15. PTP Force Diagram ................................................ ......................................... 186 C 16. N C Path ......................................................................... ...... ............ 186 C17. Radial Asynchronous Error..................................... ............... 190 C18. D I/D2 Surface Location Error .................................................................... ....190 C19. Full Range Simulation (0.5" diameter tool, 3.25" overhang) .......................... 192 C20. Full Range Simulation (0.75" diameter tool, 1.5" overhang) .......................... 192 Abstract of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy THE IMPLEMENTATION OF SIMULTANEOUS TRILATERATION TO MEASURE DYNAMIC THREEDIMENSIONAL CONTOURS USING THE LASER BALL BAR By TONY L. SCHMITZ May 1999 Chairman: Dr. John Ziegert Major Department: Mechanical Engineering Computer numericallycontrolled (CNC) multiaxis machine tools are an integral part of modern manufacturing. These machines operate in an overall openloop mode (i.e., although the positions of the individual axes are servocontrolled, the actual spatial coordinates of the tool or endeffector are unknown). Since the machining process is open loop, the ability to monitor the cutting tool/workpiece positional relationship and predict the final part dimensions is limited. At this time there are no preprocess measurements which can be performed to accurately predict the final dimensions of a machined part. This produces a fundamental gap in the ability to model the machining process. Although static measurements may be performed on a given machine tool to characterize its (static positioning) accuracy, the final part dimensions are a function of the machine tool's dynamic spatial positioning accuracy. A tool which could dynamically measure the tool position along three dimensional contours to micrometer accuracy would close this gap and permit rapid verification of CNC part programs. This research describes the design and construction of a sensor, the simultaneous trilateration laser ball bar system (STLBB system), which has the capability of measuring these 3D contours to micrometerlevel accuracy. The data obtained from these 3D measurements is useful for the evaluation of the controller performance during contouring and the measurement of the relative contributions of both quasistatic (geometric) positioning errors and controller errors to the part dimensional errors. The ability to verify the CNC part program without machining a series of test parts is another benefit of this sensor. This reduces scrap and increases the manufacturing process efficiency, especially in situations where machining time is high and material is expensive (e.g., the aerospace industry). In order to completely close the open loop machining process, the effects of forced vibrations caused by the steadystate cutting forces on the final workpiece dimensions are also investigated. A simulation is included which computes the surface location errors introduced during the cutting process. These errors may then be applied as a type of post processing filter to the STLBB dynamic path measurements to accurately predict the final machined part dimensions. CHAPTER 1 INTRODUCTION Computer numericallycontrolled (CNC) multipleaxis machine tools are an integral part of modem manufacturing. These machines operate in an overall openloop mode (i.e., although the positions of the individual axes are servocontrolled, the actual spatial coordinates of the tool or endeffector are unknown). In continuouspath NC systems, there is contact between the cutting tool and workpiece while up to five axes are in motion. Therefore, the final workpiece dimensions are directly related to the positional relationship between the tool and workpiece. Since the machining process is essentially open loop (the tool/workpiece contact may loosely close the positional loop and affect the overall machine tool stiffness), the ability to monitor this relationship and predict the final part dimensions is limited. At this time there are no preprocess measurements which can be performed to accurately predict the final dimensions of a machined part. This produces a fundamental gap in the ability to model the machining process. Although static measurements may be performed on a given machine tool to characterize its (static positioning) accuracy, the final part dimensions are a function of the machine tool's dynamic spatial positioning accuracy. A tool which could dynamically measure the tool position along three dimensional contours to micrometer accuracy would close this gap and permit rapid verification of CNC part programs. This research describes the design and construction of a sensor, the simultaneous trilateration laser ball bar system (STLBB system), which has the capability of measuring these 3D contours. The data obtained from the 3D measurements is useful for the evaluation of the controller performance during contouring and the measurement of the relative contributions of both quasistatic positioning errors and controller errors to the part dimensional errors. The ability to verify the CNC part program without machining a series of test parts is another benefit of this sensor. This reduces scrap and increases the manufacturing process efficiency, especially in situations where machining time is high and material is expensive (e.g., the aerospace industry). The ability to characterize the accuracy of 3D contours prior to machining could also have farreaching effects in the field of process capability. In modem industry, there is a "growing interest in quantifying the ability of a process to satisfy customer requirements" [1]. A process capability study provides information on the existing process performance (e.g., producing dimensionally correct parts by machining) with respect to the predefined requirements (e.g., the engineering drawings). It can also suggest possible process improvements and allow design engineers to select an appropriate process to meet the accuracy requirements. A brief overview of the statisticallybased process capability theory is given in Appendix A. A process capability study of a given machine tool, for example, may be completed to isolate the amount of process variation contributed solely by the machine. Such a study would attempt to remove other known variation sources (i.e., setup, operator influence, workpiece material, maintenance, pallet errors) and therefore, concentrate on the variations inherent to the machining process (i.e., thermal fluctuations, geometric errors, spindle errors, cutting force errors). In a typical study, a given number of parts (at least 50 recommended) are machined over some thermal duty cycle under normal machining conditions [1]. These parts are then measured, a process stability analysis performed, and one or more process capability indices (PCIs) utilized to predict future performance [2]. The necessity of machining and measuring such a large number of parts makes this process costly and time consuming. The STLBB system could be used to dynamically measure the corresponding contours in real time over the same thermal duty cycle. This would eliminate the cost of material and tooling and halve the total process time. With the inclusion of possible spindle and cutting force errors in the STLBB analysis, the process capability of a machine tool could be evaluated without cutting a single test part. The STLBB system is based on the laser ball bar (LBB), a precision linear displacement measuring device developed at the University of Florida by Ziegert and Mize [3]. It consists of a twostage 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 (see Figure 11). The LBB has been shown to be accurate to submicrometer levels during static measurements [4]. Once initialized, the LBB uses trilateration to measure the spatial coordinates of points along a CNC part path. The six edges of a tetrahedron formed by three base sockets (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 (LBI, LB2, LB3) shown in Figure 12 are measured once and remain fixed during the motion of the tool socket. The three basetotool socket lengths (denoted Li, L2, L3 in Figure 12) are measured during execution of the applicable CNC part program. Once the coordinates Inner Telescoping Tube Optics Package (InterFeroneter) Figure 11: Laser Ball Bar Z x Base Socket #1 4 S Base Socket #3 LB 2 Base Socket #2 Figure 12: Trilateration Tetrahedron in the LBB frame have been determined, they may be transformed into machine coordinates using the homogeneous transformation matrix (HTM) between the LBB and machine coordinate systems. A brief outline of HTMs and the method used for the determination of the transformation between the LBB and machine coordinates are given in Appendix B. Previous research using the LBB has focused on sequential trilateration and quasi static measurements (i.e., motion is stopped to take the measurements). In sequential trilateration, the same part path is traversed three times, measuring the lengths of one basetotool socket leg at a number of static points during each repetition. The setup for sequential trilateration is shown in Figure 13. This method requires a spatially repeatable measurement trigger since the tool socket must be in exactly the same position (for a given point) for each of the three measurements. When quasistatic measurements are performed, the machine repeatability governs the accuracy of the measurement trigger and, therefore, the measured coordinates. For most machine tools, the short term static repeatability is substantially better than the absolute positioning accuracy, and this process yields satisfactory results. However, for dynamic path measurements, the spatially repeatable measurement trigger is difficult to implement. The parametric error map for a 3axis milling machine was constructed by Kulkari using both the quasistatic sequential trilateration LBB technique and the methods described in the ASME B5.54 Standard, "Methods for Performance Evaluation of Computer Numerically Controlled Machining Centers" [5]. The agreement between the two results verified the LBB technique [6]. A second sequential trilateration study was completed by Srinivasa which measured the positioning errors on a 2axis turning Tool Socket Spindle y x z Machine Table Figure 13: Sequential Trilateration Tool Socket Spindle Y x z Machine Table Figure 14: Simultaneous Trilateration center over a thermal duty cycle and correlated these errors with the temperature gradients within the machine tool using a neural network. A PCbased error compensation system was then used to compensate for the errors predicted by the neural network [7]. The above methods give a rapid, efficient way to measure the quasistatic errors of a machine tool and thus evaluate its static positioning accuracy. However, these methods do not characterize the dynamic positioning accuracy of the machine tool. This research focuses on simultaneous trilateration, where three LBBs ride on a single sphere at the tool point to completely define all three basetotool socket lengths with one execution of the CNC program. Since all three leg lengths are captured at once, the measurement trigger need not be spatially repeatable, and this method is better suited to dynamic measurements (see Figure 14). In previous work, the feasibility of using the LBB as a dynamic measuring device was demonstrated [8]. Sequential trilateration was implemented to measure the coordinates of a 2D circular contour using one LBB and the encoder feedback signal as a measurement trigger. Measurements obtained using an independent device verified the LBB results. The work completed in this research includes the implementation of simultaneous trilateration to measure 3D dynamic part paths to micrometer level accuracies. Other subtasks which were completed are the novel design of a tool socket/joint to support the 3 LBBs, the optics/fiberoptics design and component selection for the linear displacement interferometers inside each LBB, a redesign of the individual LBBs with respect to previous models (e.g., extended and retracted size, optics configuration, optics positioning system), 2D verification of the STLBB results, 3D contour measurements using the STLBB, cutting force experiments to quantify the surface location error which results 8 from forced vibrations of the cutting tool, machining verification tests and software development including NC programming. CHAPTER 2 LITERATURE REVIEW Although there are no other systems currently available to measure three dimensional dynamic contours to micrometerlevel accuracy, the usefulness of dynamic measurements and tools which permit the accurate, dynamic measurement of two dimensional paths have been widely discussed in literature. Methods of dynamic data capture, data evaluation, and the currently available tools which provide 2D measurements will be outlined in the following paragraphs. Additionally, recent releases of computer software which focus on premachining CNC contour evaluation will be discussed briefly. Dynamic Measurements Geometric calibration of machine tools using the procedures outlined in national and international standards can be costly and time consuming, especially for large machines requiring the measurement of all possible geometric errors (21 for a 3axis machine). However, the importance of calibration to machine tool builders and users cannot be overlooked. For builders and users alike, calibration procedures attempt to provide a means of comparing the actual machine tool performance with the product specifications. Users can also use calibration to check and maintain the required performance of the machine tool during daily production. Typical calibration standards call for static measurements (commonly using a laser interferometer) along a single axis with a positionbased measurement trigger (i.e., the machine axis is stopped at regular or random intervals during measurements). The time required to stop, settle, make several measurements (to average out vibrations), and start again is considerable when several static measurements are to be made. In order to minimize this time, the number of measurement points taken is limited. For such coarse measurement intervals, cyclic errors with a high spatial frequency (such as errors associated with leadscrew pitch) may be overlooked. If the data is to be used for compensation, this can be a major issue. In research described in [9] and [10], a timebased measurement trigger was imposed on a linear displacement interferometer system to collect calibration data dynamically along a single axis. The dynamic data capture reduced the overall measurement time and the ability to measure at high temporal frequencies yielded high spatial resolution. In this work, a RenishawTM laser interferometer system was used to measure axis position, straightness, or angular (pitch and yaw) errors at sampling rates up to 5 kHz. Unlike static measurements, however, the raw dynamic data required extensive processing to obtain useful information. For position errors, a line was fit to the dynamic data which represented the nominal axis positions for a constant velocity axis motion. The error was the difference between the measured positions and constant velocity positions at appropriate time steps. To calculate the straightness errors, a line was again fit to the data to remove any misalignment between the laser and optics. This line was then subtracted from the original data to give the straightness errors. Using the dynamic procedure, the measurements were found to be more "noisy" than the results from static tests. This is a direct consequence of the absence of averaging in the dynamic tests (i.e., in static tests, several measurements are taken at each position and the results averaged). The papers suggest that the dynamic data should be processed to remove the "random" high frequency components (although it may be argued that there are no random components, only unexplained ones). Two postprocess software methods were used to massage the data: averaging and lowpass filtering. The filter implemented was a digital representation of a 3rd order Butterworth lowpass filter. The cutoff frequency was userselected in the software. The averaging was implemented using the "boxcar" method. In this technique, the data was split into a series of blocks and the mean of each block computed. The user was prompted for the time interval of the individual data blocks for this method. Dynamic positioning tests at various sampling frequencies and axis velocities were compared to static measurements. Close agreement between the two suggested that dynamic measurements are a valid way to complete machine tool calibration. TwoDimensional Dynamic Measurement Tools A product capable of 2D dynamic measurements has been commercialized by the Heidenhain Corporation. The Heidenhain grid plate system is composed of an optical grid plate with a waffletype grating of closely spaced lines (4 Itm signal period) and a noncontact scanning head which can measure translations in two directions. The grid plate is attached to a mounting base. This base is mounted in the plane to be measured (on an XY table, for instance) and the scanning head is fixed perpendicular to the plate (on the Z axis attached to the spindle, for example). This system measures the relative planar motion of the two bodies for any type of curvilinear path in the plane of the mounting base and within the area of the optical grid plate (see Figure 21). The recorded motions allow the user to observe the dynamic effects of the machine tool's performance on the CNC tool path. Common singleaxis measurements (e.g., positioning, straightness, reversal), as well as the circular test, were performed in [11] using the Heidenhain grid plate. The straightness of a short path (100 mm) was measured by moving in a single direction (at 4 m/min) and observing the lateral deviations of the scanning head. For the machine tool used in this research, a total straightness error of 0.6 im was found. The reversal error of a single axis was also measured for a single traverse and over a three hour duty cycle to test thermal effects. The circular test (circle commanded in NC controller and actual path measured) was performed to see the deviation of the recorded path from a true circle (10 mm diameter circle with a feed rate of 6 m/min). An average error in the path diameter as well as high frequency radial oscillations were detected. The cornering test was also completed to check possible overshoot or undershoot. An overshoot of 0.02 mm was seen at 6 m/min for the machine tool in question. The results of all these tests showed that the grid plate encoder is a valuable tool for examining the dynamic performance of a machine tool. In a related paper [12], the Heidenhain grid plate encoder was used, along with appropriate models, to optimize the dynamic performance of fast feed drives on high speed machine tools. A model was developed to represent a particular threeaxis machine tool. The machine tool was modeled as serial connections of the motor, ballscrew, table Grid Plate _ Machine Table Figure 21: Heidenhain Grid Plate Circular Test Path Magnetic Ball Bar Socket Fixed to  Machine TabLe Figure 22: Magnetic Ball Bar Scanning Head Spindle " Free Socket LVDT Readout and structure in two axes (X, Y), while the Z axis dynamics were neglected. Cornering motions in the XY plane were then simulated in software and measured (10 m/min, 1 m/s2) using the grid plate system. The axis gains were then adjusted according to the simulation and measurement results. The conclusion was that the limiting factor was the flexibility of the machine tool structure, rather than motor tuning issues. Another tool commonly used to perform the circular test is the magnetic ball bar. The ball bar is a device which measures small changes in its lengths using a precision transducer (such as an LVDT) (see Figure 22). As part of a complete calibration of a threeaxis vertical spindle machine tool, the 2D contouring errors were measured by the circular test using a magnetic ball bar in [13]. A 150 mm radius circular path (4 m/min tangential velocity) was programmed into the controller and the deviations from this path recorded. The two most common forms of contouring error were recognized: path radius error and reversal spikes at a change in quadrant (see Figure 23). Typical motion for a section of this circular path is shown in Figure 24. The causes of the errors shown in Figure 23 were attributed to the servo lag in the controller and the time delay of the acceleration/deceleration control. By reducing the controller time constant from 50 to 8 ms and adding feed forward velocity control and nonlinear friction compensation, the path errors were minimized. This offline servo tuning improved the 2D contouring performance at this local area in the work volume significantly (approximately 20 times improvement at 2 m/min). Another body of work [14] also included attempts to identify the major motion error sources on CNC machines using the 2D circular test. The test setup is shown in Figure 25. In this research, the motion error signal (proportional to the change in path Commanded Figure 23: Ball Bar Contouring Error Path End  Path Center Error Path Start Figure 24: Circular Path Section tar Test Path Gap Sensor SEncoder Figure 25: 2D Circular Test Setup radius from ideal) were taken to be linearly related to the motion error sources. These motion error signals were divided into those which are directional dependent and those which are nondirectional dependent. Directional error patterns are said to be a result of such errors as backlash and loop gain discrepancy. Nondirectional errors result from such sources as squareness, straightness, and uniform positioning errors. By performing two circular tests in opposite directions, nondirectional dependency was ascertained by averaging the two signals. The directional errors were found by subtracting the averaged record from the original signals. A frequency analysis was then used to find the individual nondirectional errors such as uniform positioning, straightness and squareness errors. A weighted residual method was implemented to identify the directional errors such as backlash and loop gain mismatch. Aside from machine tool calibration, there are other applications which require dynamic measurements. The laser interferometer provides an attractive method of completing vibration measurements because the measurements are noncontact and have a high resolution. If these measurements are to be performed on large bandwidth systems, highfrequency dynamic data capture is necessary. An example of this highfrequency, dynamic measurement requirement is the calibration of piezoelectric transducers (PZT). PZTs are typically used in the control systems of lithography and diamond turning to correct rapidly changing errors or provide fine position resolution. In these cases, the frequency response and bandwidth of the transducer must be well defined. Dynamic measurements can provide this information. 18 Software for PreMachining Contour Evaluation The attractiveness of verifying CNC code prior to machining a test part has not gone unnoticed in the manufacturing community. Several computeraided manufacturing (CAM) software programs which boast of this capability have been released. Although such programs add collision protection and other NC error checks, as well as tool path optimization, to the manufacturing process, they cannot predict part dimensional errors due to such sources as thermal gradients, geometric positioning errors, machine tool dynamics, and/or cutting force effects. A sensor such as the STLBB system is required to account for these error sources. CHAPTER 3 DISPLACEMENT MEASURING INTERFEROMETRY Introduction A logical starting point in a discussion of displacement measuring interferometry is a brief review of the fundamentals of light theory, polarization, and interference. Light is a transverse electromagnetic wave with wellestablished relationships among the velocity, wavelength, and frequency. The velocity of light in a given medium, v, is related to the speed of light in vacuum, c, by the index of refraction, n. This relationship is given in Equation 31. A similar relationship can be seen between the wavelength of light in a given medium, X, and the wavelength in vacuum, i0. See Equation 32. At standard temperature and pressure (temperature = 20C, pressure = 760 mm Hg, relative humidity = 50%), the index of refraction of air is 1.000271296 [15]. The velocity of light in air is, therefore, slightly less than the velocity of light in vacuum. v=c/n, where c = 2.99792458e8 m/s (31) X= / n (32) Additionally, the velocity is related to the wavelength and frequency, f, of light as shown in Equation 33 [16]. An examination of the electromagnetic spectrum reveals a range of frequencies from 3 Hz (AC power radio waves) to 3e24 Hz (gamma rays) with corresponding wavelengths of le8 m to le16 m. The visible spectrum is generally characterized by wavelengths between 380 nm (violet) and 770 nm (red) [17]. v=Xf (33) The relationships given in Equations 32 and 33 allude to the wave nature of light (i.e., when speaking of the wavelength of light). However, the wave model cannot always predict the behavior of light. Light behavior is better described by a theory which incorporates both wave and particle models, generally referred to as the waveparticle duality. In some situations, the wave theory is useful to describe the behavior of light (e.g., interference) and in others the particle nature is more applicable (e.g., photoelectric effect) [17]. The theory of quantum electrodynamics (QED) incorporates the dual nature of light and successfully predicts all types of light behavior (e.g. diffraction, reflection, refraction, partial reflection, etc.) using probability theory. As an example, consider the simple case of reflection off a plane mirror. Let there be a very low intensity, single frequency (monochromatic) source that emits only one particle of light, or photon, at a time. This light is incident on a plane mirror with a photomultiplier positioned as shown in Figure 31 to detect any reflected photons. Only photons which actually reflect off the mirror surface will be considered (i.e., any photons which travel directly from the source to the detector will be neglected). The photons are free to travel in any path from the source to the detector, and they actually do. A separate 'probability arrow' can be drawn to represent each possible path that a photon may take from the detector to the source. The length of the 'arrow' for 21 Surface Normal Source Detector Mirror Figure 31: Plane Mirror Reflection each path represents the probability that a photon would take this particular path. The angular orientation of the 'arrow' symbolizes the time it takes for the photon to reach the detector. For example, a longer path will be represented by an 'arrow' which is rotated clockwise by several degrees (i.e., the second hand on a stopwatch after several seconds from vertical), while the 'arrow' for a very short path would be nearly vertical (i.e., the second hand is near vertical when the count in seconds is near zero). Since the photons have an equal probability of following any path, all arrows have equal lengths. The orientations of the individual arrows will vary, however, since the path lengths are not equal. A photon which travels straight down to the mirror and then to the detector has a longer path length than one which reflects off the mirror center. If the arrow orientations represent the time of flight, the path taken will determine the arrow direction. The implementation of vector addition to sum the individual arrows from all possible paths gives a large resultant arrow. The length of the arrow is the probability amplitude and a long arrow implies an event is likely to occur. Therefore, QED predicts light will indeed reflect off the mirror. Furthermore, the magnitude of the resultant arrow is most strongly influenced by the arrows which represent paths near the middle of the mirror. Since paths near the center of the mirror take approximately the same time and, therefore, have basically the same orientation, it is these paths of least distance (and time) that most affect the final arrow length and high probability of reflection. Therefore, the assumption that light travels in paths of least time (i.e., the law of reflection which states that the incident angle equals reflected angle) is an acceptable approximation, although not an exact picture of light behavior [18]. Polarization Light can be described as a transverse electromagnetic wave with fluctuating electric (E) and magnetic (B) fields which are mutually perpendicular to one another, as well as the direction of propagation of the wave. The polarization of the wave is defined, by convention, as the direction of the electric field vector (since this is the more easily measurable quantity). If this vector lies in a plane for all temporal and spatial positions, the light is said to be linearly polarized. Randomly polarized (or unpolarized) light, on the other hand, has an E vector which does not lie in a single plane and does not vary spatially in a repeatable manner. This unpolarized light, such as sunlight or the light emitted by a hot filament, may be linearly polarized by passing the light through a PolaroidTM, or polarizing filter. This dichroic material, originally developed by E. H. Land, allows one polarization component (e.g., the vertical) to pass through and absorbs all others. The transmission axis (the vertical axis in this example) is defined as the direction in the material which suppresses vibrations and, therefore, absorbs little of the light energy. The direction orthogonal to the transmission axis blocks the light polarized in this direction by allowing the material electrons to vibrate when excited by the light, thereby absorbing the light energy. Figure 32 represents the case of absorbing horizontal polarizations and transmitting the vertical. Other methods of producing polarized light include reflection, scattering and birefringence [17]. The reader may be aware that quality sunglasses have lenses made of dichroic polarizing material. Light reflected from a horizontal surface, such as the highway, has a 24 Transmitting Axis Unpolarized Light ti / Direction SoF Propagation Absorbing 1 Linearly Polarized Axis Light Figure 32: Polarizing Filter polarization state which is mostly linearly polarized in the horizontal plane. The lenses, with a vertical polarization axis, almost completely absorb the horizontal orientation and substantially reduce glare. Next, consider two, equalmagnitude waves with perpendicular polarizations (electric field vectors E1 and E2 perpendicular) and identical directions of propagation. If these two waves are in phase, the resultant polarization, found by the vector addition of E1 and Ez, will lie in a plane oriented at 450 with respect to each of the original E fields. This is also termed linearly polarized light and is shown in Figure 33. If these two, equalmagnitude waves are now shifted in phase by 900 relative to one another, the resultant vector no longer lies in a single plane. Instead, it traces out a helical path along the direction of propagation. This situation is defined as circular polarization. The electromagnetic wave is termed right circularly polarized if the helical path is clockwise as one looks back along the direction of propagation and left circularly polarized if the direction is counterclockwise. Figure 34 shows an example of left circular polarization. Viewed headon, the polarization vector traces out the Lissajous figure of resultant electromagnetic field vibration [17]. In this case, the figure is a circle. In the more general case, when the phase shift is neither 00 or 900 (or an integer multiple) or the magnitudes of the two electric field magnitudes are not equal, the polarization is defined as elliptical. In this instance, the polarization vector traces an elliptical path along the direction of propagation (i.e., the Lissajous figure is an ellipse). Depending on the phase difference, the tip of the resultant electric field vector may trace clockwise or counterclockwise paths. U I^^ im^ Direction of Propagation Figure 33: Linearly Polarized Light E, E Left Circular I Potarization I I I I, I" Direction of I, Propagation Figure 34: Left Circularly Polarized Light In many situations, it is desired to change the polarization of light by shifting one of the two waves with orthogonal electric field vectors relative to the other, as described previously. A birefringent material is particularly useful in this instance. Birefringent materials exhibit a dependence of the index of refraction on the direction of polarization of the incident electromagnetic wave. Notice from Equation 31 that a higher refractive index yields a slower velocity of propagation in the given material. Therefore, when the two waves with perpendicular polarization enter the material (with the optical axis parallel to one polarization), one wave will travel slower than the other (due to a higher n) and introduce some phase shift between the two on exit from the material. In order to convert linearly polarized light to circular polarized (or vice versa), a 900 phase shift must be introduced in one axis relative to the other. The name for the birefringent optic which completes this conversion, a quarterwave plate (or quarterwave phase retarder), is derived from the '/4 cycle delay which results from this 900 phase shift. The quarterwave plate has defined two optical axes, which are mutually orthogonal. The slow axis introduces a 900 phase shift (due to a higher index of refraction) in the light (of appropriate wavelength) which travels along this polarization relative to light which propagates along the fast axis polarization. As a final word in this brief outline of polarization, it should be noted that an elegant representation of the polarization of electromagnetic waves is given by the Jones vector notation. The Jones vector is a 2x 1 column vector which denotes the polarization state of a wave using real and imaginary values. Also, the polarizing elements described previously may be modeled by 2x2 matrices. The matrix premultiplication of the Jones vector by the matrices of the polarizing elements which act on the wave gives the final polarization state of the original wave. The reader is referred to [17] for more information on this topic. Interference The basis for interferometry is the interference of two or more electromagnetic waves in space. The magnitude of the resultant electric field is defined by the principle of linear superposition [16]. When two coherent (constant initial phase), nonorthogonal waves (for simplicity, assume identical polarizations) arrive at the same point, their phase relationship determines the final magnitude. Constructive interference occurs when the waves are in phase and gives a total magnitude equal to the sum of the individual magnitudes. Destructive interference takes place when the waves are 1800 out of phase. The resultant amplitude is the difference between the two individual amplitudes. For equal initial amplitudes, total cancellation occurs. A popular example of this twosource interference is shown in Figure 35. In this experiment, first performed by Thomas Young in 1802, a coherent monochromatic source is incident on two narrow slits in a screen. This source is typically taken to be a point source which has wavefronts, or loci of points with the same phase, which are ideally spherical in shape. The two light paths which show through the two narrow (ideally point source) slits also exhibit coherent spherical wave fronts. A screen placed in the path of these two waves shows alternating bright and dark bands. This is an interference pattern which demonstrates both constructive (bright bands) and destructive (dark bands) interference between the light waves emitted from the two slits. 29 Monochromatic Source Stit Screen Figure 35: Young's Double Slit Experiment The monochromatic wavelength of the source may then be calculated, according to Equation 34, by measuring the distance between the slits (d), the distance between the viewing screen and the slits (R), and the distance between adjacent bright band centers (5). For completeness, it should be noted that the effects of diffraction (also an interference phenomenon), or bending of the light waves around the slit of finite width in the screen, have been neglected in this simple analysis. ?=d /R (34) The first displacement measuring interferometer was developed by Albert Michelson in 1881 [17]. In this experiment, light was incident on a partially silvered mirror or beam splitter (glass with a thin silver coating). This beam splitter allowed some of the light to pass through and reflected the rest. The reflected portion of the light traveled to a moveable mirror mounted on a fine pitch micrometer. The transmitted light passed through a second glass plate with the same thickness as the beam splitter to a fixed, reference mirror. The second plate, or compensator plate, was necessary to ensure that both the reflected and transmitted light traveled through an equal distance in glass, referred to as the optical path length (OPL). When the mirror mounted on the micrometer was translated relative to the fixed mirror (without changing its angular orientation), light to dark (uniform intensity) transitions due to constructive and destructive interference were seen once the beams had recombined at the beam splitter, as shown in Figure 36. A relative displacement of onehalf the wavelength of the source light (?\2) produced a light darklight transition, or fringe, at the viewing position. A lightdarklight transition 31 Moving Mirror Beam Splitter Diffuse Light Silver Compensator Coating P te Fixed Viewing Mirror Point Figure 36: Michelson Interferometer occurred for a 3600 (one wavelength) phase shift between the electric fields of the two waves [15]. This corresponded to the halfwave displacement of the mirror since the reflected wave must travel both to and from the mirror and therefore has an overall motion of one wavelength. Two common variations of the Michelson interferometer are the TwymannGreen and MachZehnder interferometers. The TwymannGreen, shown in Figure 37, uses a coherent point source rather than the extended source of the Michelson. The coherent source allows for unequal path lengths in the two legs of the interferometer. For this reason, the TwymannGreen interferometer is also called a Laser Unequal Path Interferometer (LUPI) [19]. Collimating lenses are used to convert the diverging rays of the point source to (essentially) parallel rays within the interferometer. The rest of the interferometer operation is analogous. The TwymannGreen interferometer is typically used to measure imperfections within other optics, provided the optics within the interferometer are high quality, by placing the optic to be measured within one of the measurement paths. The optic distorts the normally planar wavefronts by slowing the light which passes through. Interference fringes can be seen at the interferometer output due to variations in the index of refraction across the optic face. The shape of these fringes indicates where the index of refraction is constant. Therefore, the fringe shapes can be thought of as a contour map which shows the variation of the light path length through the optic and may be used to polish the glass to a particular shape [20]. The MachZehnder, shown in Figure 38, has two beam splitters and two mirrors arranged in a rectangular pattern. Typically, one path is disturbed while the other remains fixed to measure the relative change in path length. Moving Mirror Fixed Mirror Viewing Point Figure 37: TwymannGreen Interferometer Beam Mirror Spitter Recombined Beam', Reft oec ted ar / I r a r BRe B re Spl t Ler" Figure 38: MachZehnder Interferometer Smittc 'am Mirror The lightdark transitions in the Michelson interferometer discussed previously form the foundation for homodyne (DC), or single frequency, interferometry. After the two beams travel the measurement and reference paths, respectively, they must interfere. As noted, interference only occurs for beams with the same polarization state. If initially the two beams each have planar polarization perpendicular to one another, a polarizing filter may be placed in the path, oriented at 450 to each. A portion of each wave is absorbed by the polarizing filter and a portion passes through, each with the same polarization and now interfered. To find the total relative displacement of the moving mirror, the recombined beams are first incident on a photodetector. The photodetector then produces a signal proportional to the light intensity. For a X\4 motion of the moving mirror, the signal will vary from the maximum value to ideally zero (onehalf fringe), assuming the fixed and moving mirrors initially had an equal path length. (The resolution may be increased by adding electronic circuitry to interpolate between the light and dark conditions.) The displacement is calculated by multiplying the number of fringes by V2 (the scale factor for a singlepass Michelson interferometer). The measurement signal in a DC interferometer is therefore a function of the photodetector output amplitude. Several problems may arise from this reality. First, the system is susceptible to variations in the source intensity and ambient light since the measurement amplitude will vary for no motion. Second, a loss of the beam from either leg (or both) cannot be distinguished from total destructive interference. Finally, if motion is stopped at a maximum, there is no directional knowledge when motion starts again since either direction will give the same reduction in photodetector output. To obtain directional knowledge, part of the interfered signal can be split off, directed through a quarter wave plate to shift the signal by 900 and compared to the original signal. This is analogous to the quadrature detection used in rotary encoders. To this point, the interference of the two waves recombinedd at the beam splitter) which provides uniform illumination at the photodetector has been taken for granted. If the two beams are not precisely parallel (mirrors are not perfectly orthogonal), fringes of equal spacing will be seen at the photodetector rather than the desired uniform intensity [17]. These fringes appear as dark bands across the otherwise bright output. As one mirror moves relative to the other, the fringes translate, but do not change the total intensity incident on the photodetector. Therefore, no motion is observed. A simple proof of this concept is to superimpose two transparent plastic sheets, each with finely spaced lines printed on one side. If the lines are not perfectly parallel when the sheets are placed over one another, fringes will be seen which move across the sheets during relative motion. If the lines on the two sheets are perfectly parallel, however, uniform intensity (bright and dark) will be seen for relative motion (i.e., for vertical translation with essentially horizontal lines). In reality, the alignment of the mirrors in the Michelson interferometer is challenging and timeconsuming. It requires precise, highresolution angular adjustments of the mirrors and a stable optical bench to maintain the alignment. The minimum required alignment accuracy can be quantified according to Figure 39. As an example, consider two 3 mm diameter beams of green (coherent) laser light with perfectly planar wavefronts and equal wavelengths of 510 nm. As shown in Figure 39, the maximum allowable misalignment occurs when the edge of wavefront #2a overlaps with wavefront Wave #1 7 Wave #2 C  0 _________   e i! oc a GX(^ Figure 39: Alignment Accuracy lb. For the typical values given, this predicts a maximum misalignment, a, of 35.07 arc sec (see Equation 35). a = tan' (X/ D) 3600 arcsec, where = 510e9 m (35) D = 3e3 m Heterodyne Interferometrv Modem users of linear displacement measuring interferometers rely mainly on heterodyne (AC), or twofrequency, interferometry. The light source is, in most instances, a frequencystabilized HeliumNeon (HeNe) laser tube with some method of generating a second frequency (within the bandwidth of modem electronics) from the natural HeNe center frequency (roughly 474e3 GHz) within the laser head. Several longitudinal modes may actually be available within the single TEMoo transverse mode of light emitted from the laser head, depending on the length of the lasing cavity. However, the frequency split between modes is normally hundreds of megahertz and beyond the bandwidth of most modern electronics. These adjacent longitudinal modes each satisfy the resonant condition for lasing, which is dependent on both the laser cavity length and the wavelength of light, at different frequencies under the HeNe gain curve (see Equation 36). Furthermore, the physics of the process requires that adjacent modes be orthogonally polarized. The frequency difference between two longitudinal modes may be calculated according to Equation 37 [26]. The TEMoo transverse mode has a Gaussian crosssectional intensity profile. The variation in irradiance across the beam is given in Equation 38 [17]. 2L = mX, where L = laser cavity length (36) m = integer longitudinal mode number X = light wavelength c f,, f, where fi = light frequency (37) 2Ln c = speed of light n = refractive index of medium 8y2 I= e D where I= beam intensity (38) y = transverse beam direction D = beam width at given position In heterodyne systems, by definition, the frequencies of the measurement and reference beams differ slightly. When these two beams of different frequency are recombined at the beam splitter (after traveling different paths) and passed through a polarizing filter, rather than pure constructive or destructive interference, linear superposition yields a wave with a periodically varying amplitude and some phase as shown in Figure 310. If the two beam frequencies are considered as phasors (or vectors) rotating with unequal angular velocities, the phasor with the higher frequency will periodically overtake the slower phasor and the two will have a relative angular velocity of 2nt (f, f2) radians/second. The time (in seconds) it takes for one phasor to overtake the other is (f,  f2)l. If the two phasors are equal in amplitude (A) and have a small difference in frequency, the interference resultant, y, is described by Equation 39. This is essentially a sine wave with a slowly varying, or modulated, amplitude (given by the cosine term). The frequency of the varying amplitude envelope, or beat frequency, is equal to the difference between the two individual frequencies of the measurement and reference beams (ft f2) 39 Ampitude Time f21 Figure 310: Beat Phenomenon Figure 310: Beat Phenomenon in Hz. The frequency of the actual waveform is the average of the two original signal frequencies, 0.5 (fl + f2) [21][22]. y = A sin (%t + A sin Oct (39) = 2A cos ((i o2)/2)t sin ((w + Q~)/2)t This heterodyne technique carries the displacement information in the phase of the interfered measurement and reference signals rather than the amplitude. In the frequency domain, a motion of the moving retroreflector causes a change in frequency of the beat signal due to the Doppler shift. For a perfectly sinusoidal beat signal, the single Fourier transform spike at the beat frequency can be viewed sliding left or right (depending on the displacement direction) using a spectrum analyzer. In this way, the frequency/phase relationship can be considered analogous to the velocity/position relationship, where the frequency is simply the circular velocity. As noted, the measurement and reference beams can be represented by phasors which are rotating with angular velocities equal to the respective optical frequencies of the two beams. For no retroreflector motion (and no subsequent Doppler shift), one phasor periodically overtakes the other (as described previously). The combination of these two vectors, therefore, provides a resting beat frequency equal to the difference between the two optical frequencies. During motion, the phase of the measurement vector with respect to the reference vector either grows larger or smaller (i.e. the beat frequency is either up or downshifted) depending on the direction of the moving mirror. Unlike homodyne systems, which have a zero Hertz beat frequency at rest and can therefore only exhibit an upshift in frequency regardless of the motion direction, heterodyne systems Frequency Position  Velocity Phose Frequency/Phase Velocity/Position IMaginary Measurement Phasor Reference Phasor Resa Phasor Diagram Figure 311: Heterodyne Phase Relationships provide directional sensitivity by the increase or decrease in relative phase. Using phase measuring electronics, the instantaneous phase between the two phasors is recorded and converted to displacement (see Figure 311). In the time domain, the beat frequency sinusoid can be seen translating in either direction on an oscilloscope. The amount of shift is the phase change, which carries the displacement information. The Doppler shift, which is responsible for the phase change in the interfered signals, can be explained in several ways. One can characterize the shift by the time of flight, such as the change in pitch observed when a car with the horn blowing passes by or Doppler radar. It is also possible to think of the light as being stretched by the velocity of the moving reflector [21]. An intuitive explanation, provided by John Beckwith of Lawrence Livermore National Laboratory in Livermore, CA, is to consider the light path as a pipeline [23]. The wavelengths, X, can then be thought of as filling up the pipeline. For a wavelength of 632.8 nm (HeNe light), this gives a wavenumber or Vnumber, X', of 1580278.1 m'. The wavenumber represents the number of ks in one meter of this imaginary pipeline. If the pipeline now begins lengthening at I m/s, there will be 1580278.1 more ks entering the pipeline than exiting. This is analogous to moving the mirror away from the beam splitter in the Michelson interferometer, except that this number is doubled because the light must travel both to and from the mirror (this represents a fold factor of 2 or scale factor of 1/2). Therefore, a negative frequency shift in the beat frequency of 2*1580278.1 Hz/m/s will be seen for this motion direction because there is a deficit of ?s. To further clarify the Doppler shift concept, the author would also like to add the notion of a 'bag of light,' or ks, attached to the end of the pipeline. As the pipeline is lengthening, the 'bag of ,s' is emptying and the deficit describes the negative frequency shift previously noted. Now consider the mirror moving toward the beam splitter. In this case, the pipeline will be shrinking at 1 m/s and there will be 1580278.1 more ks exiting the pipeline than entering (i.e., the 'bag of ks' is filling). Now the frequency shift will be positive 2*1580278.1 Hz/m/s for the interferometer. This number gives a general rule of thumb for the frequency shift of red HeNe laser light in a singlepass Michelson interferometer. The shift for a velocity of 1 ft/s is roughly 1 MHz (or 1 MHz/ft/s); the actual value is 0.9633375 MHz/ft/s. In modern heterodyne interferometry, however, the situation is a little more complex. The Doppler shifted frequency is the difference between the optical frequency (474e12 Hz for HeNe), designated fo, and the second frequency, fo f,, normally generated within the laser head by either acoustooptic modulation or Zeeman split. This second frequency can be upshifted or downshifted by the amount fs. In this case, consider an upshifted second frequency of fo + fs. For no retroreflector motion, the two optical frequencies remain fixed in the frequency spectrum. The beat frequency is, therefore, the difference between the two, fs. Either the optical or shifted frequency signal could function as the reference beam, so either may be Doppler shifted (by retroreflector motion). For instance, if the fo mirror was moving away from the beam splitter, this signal would be Doppler shifted down by an amount fd. According to Figure 312, this would produce an increase in the beat frequency and a positive phase (and therefore displacement) since the total difference Frequency (Hz) Original Beat Frequency Shifted Beat Frequency Figure 312: Beat Frequency Shift Magnitude I foFd fo i between the two signals is increasing. Moving the fo + f, mirror in the same direction will have the opposite effect. A brief description of two common methods to generate the second, shifted frequency for use in heterodyne systems will now be provided (recall that the naturally available adjacent longitudinal modes are separated by hundreds of megahertz and the frequency split is generally outside the electronics' bandwidth). In the acoustooptic technique, a single longitudinal mode (of the TEMoo transverse wave) is passed through an acoustooptic modulator. This device is, in its simplest form, a piece of glass with a PZT attached to one edge. This transducer is driven at the shifting frequency by a stable quartz oscillator and produces a traveling acoustic (sound) wave in the glass. This traveling sound wave produces successive compressions and rarefractions in the medium which changes the index of refraction periodically along its length. The periodically varying refractive index produces a moving diffraction grating. When linearly polarized, monochromatic light is incident on this phase diffraction grating at the Bragg angle, O9, a large portion of the beam is diffracted and frequency shifted (for certain crystals the polarization vector is also rotated by 900), while the rest is transmitted (with no frequency shift) (see Figure 313). The diffracted and transmitted waves are diverging at twice the Bragg angle and are, therefore, spatially separated. The Bragg angle, which is proportional to the driving frequency, facoustc, is given in Equation 310. The two frequencies may then be recombined into a single heterodyne beam. 2 OB = X \ acoustic, where ,acous, = Vacoustic / acoustic Vacoutic = acoustic velocity (310) OptC iStorder opticc + facost, I Transmitted ptic acoustic Othorder Transducer SLocal Osciltator Figure 313: Acoustooptic Modulator Magnetic Field f + f''r t" A F He Nop e ube / H( Ne laser TubW ' Figure 314: Zeeman Split The Zeeman frequency split is accomplished by placing a magnetic field around the HeNe laser tube. At low magnetic power, there exists one (preferential) linearly polarized, longitudinal mode in the laser tube (provided the laser cavity is "short"). At some higher power level, this single mode snaps into two circularly polarized modes (one left, one right) of different frequencies. The actual frequency difference between the two modes is dependent on the magnetic field strength. The HeNe medium is able to support the two modes of different wavelengths (which are not harmonics of one another) in one tube due to the presence of a circular birefringence [23] (see Figure 314 [15]). In other words, the two modes see effectively different laser cavity lengths, due to the different indices of refraction, and satisfy the condition for lasing (Equation 36) at two different wavelengths (frequencies). The mathematics behind the heterodyne interference will now be discussed. The math is not complex and relies simply on the trigonometric relationship 2 cos(A) cos (B) = cos (A+B) + cos (AB). The electric field (E) of the resultant beam after interference of the measurement (moving mirror) and reference beams (fixed mirror) is expressed in Equation 311. Again, the interference of the two beams with initially perpendicular planar polarizations is accomplished using a 450 polarizing filter. The subscript m refers to the measurement beam and the subscript r to the reference beam. E = Em cos ((0. + ()t + Oo + Od) + Er cos (at + Oo), (311) where E,, E, = magnitude (V/m) oa = optical frequency (rad/s) c% = shift frequency (rad/s) o = initial phase (rad) Od = Doppler phase shift (rad) When the interfered beam is incident on a photodetector, the output is proportional to the field strength squared (i ~ E2). The quantity E2 and resultant simplifications are shown in Equation 312. The final simplification step is the detector output because all frequencies except o% are outside the detector's bandwidth (i.e. the detector acts like a low pass filter) and the DC component is not used in the dynamic, heterodyne measurement. E2 = E2 cos2 ((r, + OL)t + 0o + Od) + Er2 cos2 ( obt + 0) (312) + 2 Em E, cos ((aO + %)t + 0o + 0d) cos (ot + 0o) Expand to obtain: = 0.5 E,2 (cos (2(". + %o)t + 2(o + 20d) + cos (0)) + 0.5 E,2 (cos (2cot + 20o) + cos (0)) + 2 E, E, (0.5 (cos (2bt + Oct + 20o + d) + cos (cOt + 0d))) Ignore all terms which contain the optical frequency (outside detector bandwidth): = 0.5E2 + 0.5E2 + E Er cos (Mt + d) Neglect the DC components of the signal: = E E, cos (Mt + Id) A typical linear interferometer consists minimally of a twofrequency HeNe laser head with the two frequencies occupying perpendicular polarizations (polarization coded), a polarization beam splitter, two retroreflectors, and a heterodyne receiver with the necessary phase measuring electronics. Variations in the optical setup allow measurement of linear and angular displacement, straightness of travel, flatness, squareness, and parallelism, as well as changes in the refractive index of air (although all are derived from a change in displacement) [15]. Simple linear displacement is measured as shown in Figure 315. At the polarization beam splitter it can be seen that the polarization parallel to the plane of incidence is transmitted, while the vertical polarization is reflected. The polarization parallel to the plane of incidence (the plane of the page) is normally referred to as the transverse magnetic (TM) mode, or 'p' mode. The polarization perpendicular to the plane of incidence is called the transverse electric (TE) mode, or 's' mode [17]. Either path can act as the reference or measurement leg of the Michelson interferometer. The beams are recombined (for interference) when they return from their respective retroreflectors. The retroreflectors, glass optics with three silvered faces containing a solid right angle, simplify the alignment procedure greatly by returning a beam parallel to the incident beam (after three reflections) regardless of the incident angle [20]. This nearly guarantees interference provided there are no significant changes in the wavefront shape of the transmitted and reflected beams. Figure 316 shows a method of measuring angular deviations using only a polarization beam splitter (PBS), a 900 prism, and two retroreflectors. Fiber Optics in Interferometrv A discussion of modern heterodyne interferometry would be remiss if a brief description of fiber optics and their application to heterodyne interferometry were not included. Although the main use of fiber optic technology is in the field of communications, metrology applications, such as fiber optic sensors and fiber delivery/collection of heterodyne measurement signals, are becoming quite common. Heterodyne fiber delivery/collection will be discussed here. An example heterodyne fiber delivery/collection system is shown in Figure 317. The different fiber types shown in the figure, singlemode polarization maintaining (SMPM) and multimode (MM), will be discussed after a brief review of total internal Fixed Retro fo Moving Retro aser Head * 's' polarization 'p' polarization Figure 315: Linear Displacement Interferometer Prism *tros Input Output PBS _ sin c c change in path length Ii Figure 316: Angular Measurement Setup SMPM Loe Fiber L MM Fiber Detector Figure 317: Fiber Delivery/Collection 2 Surface Normal S 90 > n 3nb 4 4 S Critical Anle for TIR 4 Figure 318: Total Internal Reflection reflection, launching conditions, fiber numerical aperture (NA), and mode distribution in fibers. When light travels from one medium to a medium with a different refractive index, it obeys Snell's law of refraction. See Equation 313. In the case of light entering a medium with a lower index of refraction, nb, the refracted angle, (b, is greater than the incident angle, 0a (i.e., the light bends away from the surface normal) (see Figure 318). As the incident angle is increased, the refracted angle in the second medium increases until it reaches 900. At this point, the interface between the two mediums begins to act as a perfect mirror and the light is reflected back into the first medium according to Snell's law (incident angle equals reflected angle). This situation is called total internal reflection (TIR). The critical angle of incidence for TIR may be derived from Snell's law by substituting 900 for the refracted angle. The critical angle, OQfcai, is given in Equation 314. If the first medium is a transparent rod, or fiber optic, and the surrounding medium is air (or any medium with a lower refractive index), light input at the critical angle will be trapped within the rod by TIR [16]. n, sin a, = nb sin b (313) sin Ocnicrl = nb / na, where nb < n, (314) The light trapped by TIR in the fiber will only remain in the fiber provided it does not come into contact with another surface. If the fiber touches a surface with a higher index of refraction, TIR will no longer occur. To circumvent this problem, optical fibers consist of a core, cladding and protective jacket. The core carries the light and has a higher index of refraction (no,,) than the cladding (rnlading) which surrounds the core. Two typical crosssectional profiles for the core index of refraction are shown in Figure 319. The step index profile has an abrupt discontinuity at the cladding interface, while the gradedindex profile has a smoothly changing index of refraction, typically parabolic in shape. The advantage of gradedindex fiber is seen in communications. For a step index, the path length for a ray entering at the maximum angle for TIR (cone angle) is longer than the path length for one which enters along the fiber axis. This different path time (due to a constant refractive index) will smear, or broaden, a pulse (bit) of information by modal distortion and limit the data transfer rate (bandwidth). A graded index fiber with a parabolic profile significantly reduces this problem [24]. Additionally, a reduction in the number of modes propagating in the fiber lessens the effect of modal distortion. Material dispersion, inherent to all fiber optics, also limits the system bandwidth by pulse broadening. In any given material there is a dependence of the refractive index on the wavelength of light traveling in the medium. Since there are no completely monochromatic sources, the input light encompasses a range of wavelengths, referred to as the free spectral range (at full width half maximum). As this range of wavelengths propagates through the fiber, the information carried in the longer wavelengths travels faster than the information in the shorter wavelengths (i.e., the refractive index decreases with wavelength for normal dispersion). Therefore, the input pulse spreads in time and, in the extreme case, overlaps with the next pulse. Adjacent pulses can then no longer be distinguished and the data transfer rate is limited [17]. 54 8'e Core Cladding Step Groded Index Index Jacket Profile Profile Figure 319: Refractive Index Profiles The cone of light which may be accepted by a fiber is a function of the core and cladding indices of refraction. The halfangle (0c) of the acceptance cone may be calculated according to Equation 315 for a step index fiber. n, sin 0, = (nco2 ncladding2) 0, where ni = 1 for light from air (315) A fiber's numerical aperture (NA), a measure of the lightgathering capability of the fiber, is also a function of the core and cladding refractive indices. The NA for a step index fiber with light incident on the fiber from air is given in Equation 316. It should be noted that this particular figure of merit is independent of the fiber geometry. NA = (nc~2 ncladdmg2).5 (316) A fiber's NA is an important consideration in the launch conditions of light into a fiber. For example, if light is introduced into the fiber with a cone of diverging rays greater than the maximum cone angle of the fiber, some of the source light cannot be propagated by the fiber. This situation is referred to as overfillingg" the fiber and causes transmission losses. If the source cone of light is less than the maximum cone angle the fiber can accept, the fiber is "underfilled," but results in less attenuation than in the overfilled case. If light is being coupled into the fiber from a collimated source (i.e., the source emits light in only one direction), a converging (double convex) lens may be used to focus the light on the fiber core (a challenging task which is mainly a function of the angle at which the light strikes the fiber from the lens). The NA of the beam may now be found according to Equation 317. This NA must then be compared to the fiber NA to avoid overfilling. NAam = n sin (ro / f), where ro = initial beam waist radius (317) f = lens focal length n = air refractive index (= 1) Once light is coupled into the fiber, it is next important to understand how it propagates in the fiber. By combining Maxwell's equations, a wave equation may be obtained which can be solved for the distribution of the electromagnetic field across the fiber face, or the guided modes. A solution of the wave equation for these modes depends on the fiber geometry and index profile of the core and cladding. To determine the number of modes which will be supported by a given fiber, the normalized wavenumber, or Vnumber, may be calculated (see Equation 318 [24]). An alternate analysis, developed in [17], suggests that the maximum number of supported modes (mn.x) may be calculated according to Equation 319. V = ko (a) NA, where ko = free space wavenumber = 27t / X (318) a = fiber core radius lm d 2 m 2= NA) where d = fiber core diameter (319) 2 A,, It can be seen that a small core diameter, small NA, or a large free space wavelength, Ao, will decrease the Vnumber and reduce the number of modes supported by the fiber. For V < 2.405, only a single mode (HEn) will be supported. Fibers which support just this single mode are known as single mode fibers. As the Vnumber is increased, more modes may be carried by the fiber. These fibers are referred to as multimode fibers. Typical multimode communications fibers may have Vnumbers from 50 to 150. In many cases, it is desirable to maintain the polarization state of the input light through the length of a single mode fiber. This is the case for the fiber feed from the laser head to the interferometer (e.g., the single modepolarization maintaining or SMPM fiber shown in Figure 317). For the fiber to maintain the input polarization state, it is stressed along a single axis to produce a birefringence in the fiber (see Figure 320 [25]). Light linearly polarized along the stressed axis travels at a slower rate than light orthogonally polarized to this axis. If the two orthogonal frequency components from the laser head are aligned with these birefringent (fast and slow) axes when launching into the fiber, sensitivity to environmental effects is reduced and the input polarizations are maintained. For an ideal fiber and perfect launch conditions, the fiber input and output light could be described by the same wave equation. In reality, there is a small amount of polarization "leakage" between the fast and slow axes and small relative phase shifts introduced due to mechanical and environmental cable perturbations. Again referring to the heterodyne system shown in Figure 317, once the two orthogonal polarizations have traversed the Michelson interferometer (to introduce a relative phase shift between reference and measurement signals) and interfered at the polarizing filter, there is no longer any need to maintain the polarization state because the displacement information is now carried in the phase of the optical signal. Therefore, the light may now be carried on a high NA multimode (MM) fiber to the phase measuring electronics. Multimode fibers generally have a much larger core diameter and are easier to couple into than singlemode fibers. Elliptical stressed cladding Bow tie Core/cladding Stress applying part Circular Slow axis Figure 320: Singlemode Fibers CHAPTER 4 STLBB DESIGN As noted in Chapter 1, the Laser Ball Bar (LBB) is a precision linear displacement measuring device. It consists of a twostage telescoping tube with a precision sphere mounted at each end. A heterodyne displacement measuring interferometer is aligned inside the telescoping tube and measures the relative displacement between the two spheres (see Figure 41). Once initialized, the LBB uses trilateration to measure the spatial coordinates of points along a CNC part path. The six sides of the tetrahedron formed by three base sockets (attached to the machine table) and a tool socket (mounted in the spindle) are measured, and, by geometry, the coordinates of the tool position (end effector) can be calculated. In simultaneous trilateration, three LBBs ride on a single sphere at the tool point to completely define the tetrahedron with one execution of the CNC program (the three lengths between the base sockets are measured prior to executing the part program and remain fixed throughout the measurement). Figure 42 illustrates this method. The physical implementation of simultaneous trilateration required the design of a joint at the tool socket to support all three LBBs and a bracket to mount the joint at the tool point on the spindle. A redesign of the past generation LBB was also completed. The tool socket joint, tool point bracket, and new LBB mechanical and optical design will now be outlined. Optlcs Package (nterfero eter) cison Sphere "I Precision Sphere Inner Telescoping Middle Telescopng Tube  ^l/. Figure 41: Laser Ball Bar Tool Socket Sp ndle x ~z ' Base Socket iVi Muc hine Tabe Figure 42: Simultaneous Trilateration Tool Socket Joint Simultaneous trilateration requires that the axes of the three LBBs meet at a single point which coincides with the tool point (to minimize Abbe offset errors). During the execution of a CNC part program, the coordinates of the tool point are measured at finite intervals along the path to define the contouring accuracy. As the spatial coordinates of the tool point vary, the lengths of the individual LBBs change as well as the angles between the LBBs. This calls for a joint which provides three independent angular degrees of freedom while prohibiting relative translations between the endpoints of each of the three LBB axes (e.g., a spherical joint). This joint provided a challenging design problem when combined with the minimum space, low friction, and ease of assembly/disassembly requirements of the STLBB system. For the first generation STLBB system, the initial concept was the use of a single precision sphere at the tool socket with three magnetic sockets utilized to attach the LBBs to the sphere. In a first analysis, the main concern with this design appeared to be the coefficient of friction between the sphere and sockets. However, one design change in the LBBs used in this research was a 50% increase in length. This length increase expanded the work volume for the LBB, but also added weight. Therefore, adequate magnetic attractive force to secure the LBBs to the sphere was also an important consideration. Additionally, it was required to minimize the crosssectional area of the sockets to reduce interference between the three sockets riding on the single sphere. Socket Interference A minimum angle between adjacent LBB sockets of 450 was selected to maximize the measurement volume, while still allowing adequate space for the magnets (a larger magnet crosssectional area provides more attractive force). This angle, combined with the sphere diameter (D) constrains the allowable LBB socket diameter (see Figure 43). do, = Dsin 4 (41) From Equation 41, the maximum LBB socket diameter can be calculated. For a sphere diameter of 38.1 mm (1.5"), the maximum allowable LBB socket diameter is 14.58 mm (0.574"). A 12.7 mm (0.5") diameter by 12.7 mm long cylindrical neodyminum magnet was then chosen to fit inside the LBB socket and attach the socket to the tool sphere. Initial tests using the LBBs and these magnets provided less than desirable results. Although near vertical static positions could be held, horizontal positions or motions with high accelerations caused the sockets to tip off the tool sphere. To remedy these problems, a spring aid was designed to provide additional support for the LBBs at the tool sphere. Magnetic/Spring Force The requirement for this spring aid was to provide a mechanical force along a line approximately parallel to the magnetic force (or the LBB axis), while minimizing the motion limitations placed on the measurement volume. From experimental Precision Sphere Figure 43: Socket Diameter Constraints measurements, it was determined that an additional 8.9 N (2 lbf) would provide adequate holding power, but not introduce an excessive normal force which would increase the friction force between the sphere and LBB socket. The spring aid is shown in Figure 44. It is composed of a tool socket and sphere, upper bearing with pin connection, spring assembly, and lower collar with pin connection (attached to the LBB socket). The pin connections allow the spring assembly to move as the LBB socket rotates about the sphere center and the upper sintered brass bearing allows rotation about the vertical axis. Two LBBs require this spring aid. The other is rigidly fixed to the tool sphere using a threaded connection. The radius and crosssectional dimensions of the spring steel assembly which provides the holding force were calculated using Castigliano's Theorem on Deflections [33]. The spring was modeled as a curved beam with the required force applied at the free end (see Figure 45). By selecting representative values for the force (F), radius (R), Young's modulus (E), deflection (qF), and width (w), the required thickness (t) could be calculated. M dM qF = El  RdO (42) In Equation 42, the formula for the deflection of the spring end in the direction of the force is given (deflections due to axial and shear loads have been neglected). The solution of this definite integral and substitution of wt3/12 for the 2nd area moment of inertia (I) provides an equation for the necessary thickness. Using the dimensions shown Upper Bearing Pin Connectior Tool Socket T ool Sphere Pin Connection BB Socket Figure 44: Tool Socket Spring Aid H Figure 45: Castigliano Modeling in Equation 43, the spring aid was found to provide a range of motion of 22.5" to 380 from the vertical axis (see Figure 46). The minimum angle of 22.5" results from interference between adjacent LBBs on the tool sphere. The maximum angle, 760 between adjacent LBBs, was imposed by the spring aid design. Because the two spring assembly upper bearings were stacked upon one another, there was a constraint placed on the maximum angular deviation of each spring assembly. In the final spring aid design, the maximum range of motion was chosen to satisfy this constraint. t= 67FRE 0.8mm (43) Ewq, where F= 8.9N q = 3.15 mm R = 0.03 m w = 12.7 mm E = 200e9 N/m2 The tool socket holds the tool sphere using a magnetic connection. In this case, the size requirements were relaxed so a magnet of adequate flux could be selected. In fact, it was necessary to limit the magnetic force so that unnecessary friction forces would not be developed between the sphere and the tool socket during rotations of the LBB which is rigidly connected to the sphere. The magnetic flux lines for the tool socket geometry are shown in Figure 47a. The 440 stainless steel (SST) socket completes the magnetic circuit through the sphere. The neodymium magnetic is encased in a brass (nonmagnetic) sleeve to force the flux lines out the plane of the magnet. It was found that the magnetic attractive force could be reduced by decreasing the height of the socket. The new predicted flux lines for a short Figure 46: Spring Aid Range of Motion 378 N Sinutoted Flux Lines Magnet / . Figure 47: Magnetic Flux Lines 25.6 N ?5.6 N socket are shown in Figure 47b. For a 31.75 mm (1.25") tool socket, the magnetic force was found to be 37.8 N (8.5 lbf), which is adequate to support the three LBBs. Maximum Allowable Coefficient of Friction Another important consideration for the tool socket joint is the coefficient of friction at the sphere/LBB socket interface. If the coefficient of friction between the LBB socket and the tool sphere is large, the socket will not slide around the sphere surface. Instead, the socket would tend to tip off the sphere and cause measurement errors [35]. A static analysis of the LBB socket/sphere interface was therefore performed to find the maximum allowable coefficient of friction to prevent tipping. The planar free body diagram for the static case is shown in Figure 48. This figure shows the normal and friction forces at the socket/sphere interface (Ni and (IN, (i = 1 to 2)), the spring aid force (S), the magnetic force (M), the LBB weight (W), and a force applied to the LBB end (P). The static equations of equilibrium are shown in Equation 44. The simultaneous solution of these equations was found using the symbolic mathematics software Maple V Release 4.0TM. This solution yields an expression for the sliding force (Pslide) in terms of the unknown (l) and the knowns (0, r, S, 4, W, d, LI, L2, M and p) (see Equation 45). FX =0 0 = P cose + S sin o + M sin N, sin(0 1) LN, cos(6 f)  N2 sin(0 + 1) pN2 cos( + 3) 1 F, =o (44) 0 = P sin 0 + S cosl + M cos6 N, cos(0 p) + N, sin( fl)  N2 cos(6 + f) + pN2 sin( + ) W IMo =0 0 = N, r UN2r + PL, Sd +WL, sin Pslide = (sin(0)2 p2 r2 S sin()) + sin(0) R2 r2 cos(0) W S d 12 LI cos(0) (45) sin(0) 12 r2 M + sin(6) L1 S d mu sin(0)2 L1 W L2 1i W L2 sin(0)2 I3 L, cos(0) L1 S d + S d '3 LI sin(0) g3 r2 sin(0)2 W + W L2 sin(0) 12 Li cos(9) + cos(9) Li W L2 sin(9) r3 r sin(0) S d cos(P) + I3 r2 sin(0)2 S cos(o) 2 sin(9) 2 r cos(0) cos(p) Li W + cos(P) L1 g r W sin(8) pg r S d cos(p) cos(0) cos(p) Ll 1( r M sin(o)2 R r cos(P) Li W + i r sin(0) sin(P) Li cos(0) W p3 r cos(0) sin(P) L1 sin(6) W + sin(p) L1 W g2 r + W L2 sin(0) sin(P) L1 cos(P3) g cos(0) + W L2 sin(8) sin(p) R3 cos(P) L1 cos(8) S d sin(P) LI cos(3) 4 cos(O) S d sin(P) 4' cos(1P) L1 cos(0) + W L2 sin(0)2 sin(P) LI cos(p) sin(8) S d sin(p) Li cos(1) sin(O) S d sin(3) g2 cos(p) L, + W L2 sin(0)2 sin(p) 2 cos(P) L, + 3 r W L2 sin(8)2 cos(P) + p3 r sin(0)2 cos(p) L1 W + sin(9) R2 r cos(P) L1 M + sin(0)2 ir r W L2 cos(P) cos(0) L, W L2 sin(O) sin(p)2 2 sin(0)2 g2 r sin(P) L1 W + W L2 sin()2 sin(P)2 g3 L1 S d sin(p)2 LI 4 sin(0) S d sin(P)2 .' L1 sin(0) + p2 cos(0) sin(P)2 L, S d M cos(8) LI sin(P) I2 r + cos(O) Li S d sin(P1)2 I cos(0) sin(p)2 LI W L2 sin(0) + W L2 sin(0)2 sin(p)2 L, gt L r sin(O) sin(p) L, M sin(O) 12 r2 cos(O) S cos(o) + 2 sin(0) g2 r cos(8) cos(P) L, S cos(() cos(() L, p r S cos(4) + 3 r cos(0) sin(P) Li sin(O) S cos()) 1r r sin(0) sin(P) L1 cos(6) S cos(o) sin(P) L1 S cos(o) I2 r 13 r sin(0)2 cos(P) L, S cos(4) + sin(0)21 gr cos(p) LI S cos(Q) + 2 p2 r sin(0)2 sin(P) L, S cos( ) p2 r cos(P) L1 S sin(g) I3 r sin(0) S sin(4) cos(8) + g3 r sin(O) cos(P) Li S sin(o) cos(9) sin(0) l r S sin(o) LI cos(8) cos(P) + p1 r sin(P) LI S sin(o) sin(o)2 2 r sin(8) sin(P) L1 S sin(() cos(O) p2 r sin(P) L, S sin(4) + 2 sin(o0)22 r S sin(4) LI cos(P) g r sin(0)2 sin(P) L1 S sin(4)) / ((L1 o ) c os(P) 2 C() L, + W2 r) (sin(0) g r + cos(0) cos(3) L, + sin(0) sin(p) Ll ji sin(0) cos(P) L + gi cos(O) sin(p3) Li)) 70  N N M NFi Figure 48: Static Free Body Diagram At the limit between sliding of the socket over the sphere and tipping of the socket off the sphere, the normal force (N2) becomes zero [35]. An expression for the tipping force is then obtained by setting N2 equal to zero in Equation 44. The tipping force (Pip) is given in Equation 46. (46) Pip = (S sin()) g r M sin(0) r sin( p) S d + sin(9 p) W L2 sin(0) 4 cos(0 p) S d + g cos(0 p) W L2 sin(0)) / (cos(0) g r + sin(O p) Li + t cos(0 P) L, ) The maximum allowable static coefficient of friction may then be obtained by equating the expressions for Pslide and Ptip and solving for the unknown, p.. In order to simplify this calculation, the values for the known variables were substituted into this equality. The known values for the limiting case (maximum tendency towards tipping) used in this calculation are given in Table 41. Table 41: Maximum Coefficient of Friction Calculation Constants Variable Value 0 380 0 230 d 1.62 mm S 8.9 N M 11.1N L2 304.8 mm L, 609.6 mm r 19.05 mm 13 22.750 W 8.9 N The value for p is then obtained by solving the fourth order polynomial for the four roots and selecting the appropriate value. The equality (Pside = Pip) with the variables replaced by the values in Table 41 is shown in Equation 47. From the solution set, it was determined that the maximum allowable static coefficient of friction is 0.479. At higher values, the LBB socket will tend to tip off rather than slide around the tool sphere [35]. The next step, therefore, was to measure the coefficient of friction between commercially available coatings and the 440 SST tool sphere and select a coating which met the friction requirement at the lowest cost. Pide= Pip 75.91/3 + 312.94/2 128.64 + 312.80 12.39p 3.85 (47) (21.3882 22.13)(5.85/ + 23.15) 2256pu +6.31 3.958 1.280 = 0.479 0 Coefficient of Friction Measurements Two commercially available coatings (S334B Teflon and amorphous diamond) were applied to several LBB sockets and the resulting static coefficient of friction measured using a straingagebased friction test stand [36]. The straingagebased friction test stand is shown in Figure 49. The main components are the base, support arms, straingage bridge mounted on a torsion member, the precision sphere and the socket. The first step in the measurement procedure was to calibrate the test stand. This was accomplished by measuring the strain induced in the torsion member by a known applied couple. The couple was placed on the torsion member by loading a shaft through the tube center by two equal and opposite forces Support Arm Torsion Member StrainGoge Bridge Precision Sphere LBB Socket S Base Figure 49: Friction Test Stand To Pulley  Shaft 2 625' Torsion f5"y/ Member  o Pulley Sop View I. ow Fric tion Pulley Side View F my F my Figure 410: Calibration Couple perpendicular to the shaft (see Figure 410). The moment (M) produced by this couple is equal to the magnitude of the force multiplied by the distance (d) between the two forces. M = F d (48) The strain in the fullbridge wheatstone circuit was measured for a range of applied torques using a Vishay V/E20A Strain Gage Indicator. This unit contains a DC power supply and fixed gain differential amplifier which converts the change in resistance across the wheatstone bridge into microstrain and displays it on a digital readout. A linear relationship between the applied torque and resulting strain was determined and the slope of a least squares bestfit line through the data points recorded. Once this relationship was obtained, the next step was to begin the actual friction measurements. The test procedure follows: 1) The shaft through the torsion member was removed and the 38.1 mm (1.5") diameter tooling ball attached with structural adhesive to the lower end. 2) A coated socket was placed on the socket support below the ball. 3) A known mass was placed directly above the ball to produce a normal force. 4) A torque was placed on the socket and the maximum strain just before movement was recorded. 5) Next, the value of the moment (M) corresponding to the measured strain was calculated using the graph of the calibration data. 6) The externally applied friction force was then calculated assuming that a couple was again applied. The friction force was then equal to the moment (M), divided by the diameter of the socket (docket). 7) Finally, the static coefficient of friction (SCOF) was found from the ratio of the friction force to the normal force. SCOF = Ffricon / Ncontact It is important to note that the normal force produced by putting a known weight above the ball is not the true normal used in the friction calculations since the ball is resting on a threepoint contact socket. See Figure 411 for the actual geometry of the contact area. The equation for the normal used in the friction equation is given in Equation 410. The results of the friction measurements for S334B Teflon and amorphous diamond coated sockets on the 440 stainless steel tooling ball are shown in Table 42. Ncontc cos() = Napplied (410) Table 42: SCOF Experimental Results S334B Teflon Amorphous Diamond Nap, i.g) Strain (xle6) SCOF N.pplnd (i) Strain (.xle6) SCOF 500 4 .077 500 3 .114 700 5 .069 700 3 .082 900 6 .064 900 4 .084 1295 7 .052 1295 5 .073 1453.5 8 .053 1453.5 6 .078 1748.5 9.5 .053 1748.5 7 .076 2202 10.5 .046 2202 8 .069 Precision Sphere 9=190 Contact 3 Figure 411: Contact Area Geometry N Contact N contact 3 A statistical analysis of the SCOF data yielded a mean of 0.059 with a standard deviation (95% confidence level) of 0.011 for the Teflon data and a mean of 0.082 with a standard deviation of 0.015 for the amorphous diamond data. Both tested coatings showed SCOFs well below the calculated maximum value of 0.479. Further considerations included the wear characteristics or durability of each coating, the cost per part, and the coating thickness uniformity. Because the Teflon coating was nearly an order of magnitude less expensive than the amorphous diamond, the determining factor was cost and the LBB sockets were coated with S334B Teflon. Tool Point Bracket The bracket which supports the tool socket joint is shown in Figure 412. The sphere center of the joint is located on the spindle centerline with an offset (in the Z direction) of 104.8 mm (4.125") from the face of a standard 50 taper tool holder. The design constraint imposed on this item was a maximum deflection of 1.5 gm at the cantilevered bracket end for a 35.6 N (8 lbf) load, P. The experimental determination of this load is shown in Figure 413. The spring scale reading was recorded under static conditions (4 lbf) and under a 0.5 g vertical acceleration (6 lbf). The 8 lbf value for the load was chosen as a worst case scenario. The bracket crosssectional dimensions were then calculated according to Equation 411. The material selected for the bracket was 301 SST due to its high elastic modulus and good corrosion resistance. For a square crosssectional dimension (b) of 34.16 mm (1.345"), the cantilever deflection at the angled bracket end (qp) was calculated to be 1.51 gtm, which was at the limit of the design constraint. k I ( d,) q= + l + J(F +wx2 +wddx (411) q GA El 2 GA (o 1 (L'd) Wd,2 WX, , + Fx, + wdx + 2 + 2 (d, +x EI 2 I1 + L Fd, + L,dy+ (wd, +wL +F)dy where F = 35.6 N LI = 0.0928 m L2 = 0.0921 m di = 0.0222 m d2 = 0.0342 m I = b4/12 = 1.135e7 m4 G = Shear Modulus = 75.98e9 N/m2 A = b2 = 1.167e3 m2 E = Young's Modulus = 193e9 N/m2 w = Distributed Weight = pgA = pgb2 = 90.68 N/m k = Correction Coefficient for Shear Strain = 1.2 [33] xt, x2, and y = integration variables along length of bracket Experimental verification of this bracket deflection analysis was also performed. A known force (37.8 N) was applied to the end of the bracket and the resulting deflection measured with a capacitance probe fixed to ground. The deflection predicted by the analysis for this force was 1.59 ptm, while the measured deflection was 1.4 gtm. The measured deflection was less than the prescribed design constraint and the design was, therefore, acceptable. Optics Package The optical configuration of the Michelson interferometer (aligned between the spheres at the ends of the LBB) for the first generation STLBB system will now be described. There were three main design considerations which defined the final optical layout. First, because the heterodyne laser signal is introduced into the interferometer though a singlemode, polarization maintaining fiber optic, it is necessary to generate a Tool Point Bracket Tool Holder' Figure 412: Tool Point Bracket Spring Scale Spindle Quill Spindle Centerline Spindle Magnetic Sockets Figure 413: Spring Scale Measurement Poltrizer Reference Retroreflectar QWP PBS Pris7 2n Local Re.f OWP Beam Outp 0  Ouput Mea Beam Pris 3   Input 2F Laser Be NPBS T 4 mm NPBS Pris 11 I Input 2F S" ..... L..aser Be To Moving Retroreftector erence ut isurement requency Ian frequency an Figure 414: Optics Package COutput Port For Local Reference Beon Output Port for Meosurement Beam 7 I requreny i.ospr Bean art For Figure 415: Optics Package Placement I '. L 'o local reference signal. This signal accounts for the unavoidable relative phase shifts between the two frequencies introduced by cable deformations. Second, aesthetically it was desired to place both the input and output fiber optic cables on the same side of the optics package. Third, as an overall design concern, the size of each component in the LBB must be minimized. The required optics and their configuration are shown in Figure 414. The position of the optics package within the LBB is shown in Figure 415. Figure 414 shows the four millimeter diameter input laser beam at the lower right hand side of the top view. This beam is split into two components by the nonpolarization beam splitter (NPBS). The transmitted portion (approximately 15%) is routed around the setup (via prisms 1 and 2) through a linear polarizer oriented at 450 to the two orthogonal polarizations in the heterodyne signal. The two frequency components then interfere at the polarizing filter. The interference signal is carried to a photodetector by a multimode (MM) fiber optic where the electronic local reference signal is generated. The reflected portion from the NPBS travels to the Michelson interferometer. At the polarization beam splitter (PBS), the vertical (s) polarization is reflected and the horizontal (p) is transmitted. The linearly polarized p light then passes through a quarter wave plate (QWP), orientated at 450 to the light linear polarization, which converts the polarization state to circular. After three internal reflections within the retroreflector, the light returns parallel to itself and passes once again through the QWP. The circular polarization state is now transformed into linear, but now the polarization vector is rotated 900 with respect to the original orientation (a 1800 relative phase shift between the electric and magnetic fields has occurred). The light returning from the reference retroreflector is therefore functionally s light and is reflected at the PBS. The s light originally reflected by the PBS follows an analogous path, except for the 900 turn imposed by total internal reflection in prism 3 (attached to the corresponding QWP). When the two light beams (with some relative Doppler phase shift) recombine at the PBS and pass through the polarizing filter (oriented at 450 to either linear polarization), they also interfere and are carried to the measurement photodetector (again on a MM fiber). Any phase change in the local reference signal is then subtracted from the measurement signal by the system electronics to obtain the final measurement signal which represents the actual retroreflector motion. In order to secure an interference signal (in the Michelson interferometer portion of the optical layout), the return beam from the moving retroreflector must overlap the beam from the reference retroreflector. This requires a positioning system which provides both translational and angular adjustments of the optics package relative to the moving retroreflector. The four degreeoffreedom device which provides these adjustments is shown in Figure 416. The top two adjustment screws provide pitch and vertical translations, while the two side screws allow yaw and horizontal motions. The fifth screw oriented at 1350 to the other screws serves as a locking mechanism. Three important design considerations to note are: 1) the dimple under the screw marked 'A' prevents axial motions of the optics package within the LBB, 2) the fine pitch of all screws (80 threads per inch) allows the necessary precise adjustments, and 3) the locking screw is springloaded so constant adjustment of this screw while aligning the interferometer is unnecessary. Screw A Vertical Pitch Horizontal Yow I[ Locking Screw Screw A  Locking Screw Screw A Figure 416: 4DOF Positioning Mechanism Ol Error Budget In order to predict the final accuracy of the STLBB system, it was necessary to identify each error source and tabulate these errors in an error budget. Error budgets are typically used in the design process to select components or processes which will meet the required system accuracy. In this case, Mize had previously outlined an error budget for the first generation laser ball bar, so the primary error sources had already been characterized [27]. Mize's work has been modified for the STLBB error budget. Twelve key error sources are described and quantified in the following paragraphs. In the error calculations, the error has been calculated as the difference between the measured and actual values and individual errors have been assumed to be independent of the others. 1) Thermal Error Due to Unsensed Length An interferometer measures the relative path difference between the reference and measurement paths. In the LBB, the reference path is fixed inside the optics package, while the measurement path changes with the motion of the moving retroreflector mounted at the end of the telescoping tubes. Ideally, the measurement path would include the entire LBB length (sphere center to sphere center). In reality, there are portions of the LBB total length outside the measurement path. The portion of the LBB outside this path represents the unsensed length (see Figure 417). Changes in this length due to thermal variations over the course of a measurement introduces an error. For example, if the spheres at the end of the LBB were fixed in two, thermally stable magnetic sockets and the unsensed length expanded due to a temperature increase, the interferometer would 85 PBS Center 13350 A un nun Unsensed Length 22.685 Stoin ess Steel Unsensed Length Figure 417: LBB Unsensed Length Stalntess Steel Unsensed Length show a decrease in displacement although the overall length had not changed. Alternately, if the LBB was unconstrained and the unsensed length increased, the interferometer would not show a change in displacement even though the overall length was now greater. This error can be calculated according Equation 412. Figure 417 shows an unsensed length of 57.035 mm in 304 stainless steel and 13.350 in 6061T6 aluminum. The coefficients of thermal expansion, a, are 17.8 ppm/OC for 304 stainless steel and 24.3 ppm/C for the 6061T6 aluminum [32]. For a temperature change of 0.250C, these values give a total error of 0.335 pm. erroruns~ = [(a L)ss + (a L)AI] AT, where L = unsensed length (412) AT = temperature change 2) Cosine Error This error is inherent to linear displacement interferometers. If the laser beam is not exactly parallel with the direction of motion, a proportional error (i.e., one which increases with the measurement distance) will be established which is a function of the angle of misalignment [28]. The cosine error may be calculated according to Equation 4 13. The angle of misalignment (0) is found by measuring the lateral offset of the return beam from the moving retroreflector. The angle is related to the return offset according to Equation 414. Because HeNe laser beams have a Gaussian crosssectional intensity profile, it is difficult to visually resolve the edge of the beam to find the exact return offset. The best case resolution of this lateral displacement was assumed to be 0.5 mm. This gives a cosine error of 0 to 0.066 um for an angular misalignment of 0.030. errorcom = L (cos 0 1), where L = range of motion = 470.5 mm (413) 0 = tan' (offset / 2 L) (414) 3) Sphere/Motion Misalignment This error is present when the center of one of the spheres at the end of the LBB does not lie on the line defining the direction of tube extension. This situation is shown in Figure 418. For no offset, the change in displacement of the LBB (AL) is equal to the difference between final and initial lengths (L' and L, respectively). However, a lateral offset of one sphere from the line of displacement due to incorrect assembly will result in a difference between the sphere center displacement and the recorded displacement. This error is equal to the difference between the change in displacement recorded by the interferometer and the actual change in displacement between sphere centers (see Equation 415). For a measured sphere center offset of 0.25 mm, a maximum change in displacement of 470.5 mm and an initialization length (L) of 425.7 mm, the total error is 0 to 0.039 im. error,,j,, = AL (L' L) (415) = AL [L2 + 2AL (L2 offset2)5 + AL] 0.5 + L 4) Deadpath Error When the measurement and reference path lengths are not equal at initialization (the zero displacement point), deadpath error, or DPE, is introduced for any AL AL L'L Figure 418: Sphere/Motion Misalignment Fixed Retroreftector LF L RLd Figure 419: LBB Interferometer Deadpath Zero Position 
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