The implementation of simultaneous trilateration to measure dynamic three-dimensional contours using the laser ball bar

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Title:
The implementation of simultaneous trilateration to measure dynamic three-dimensional contours using the laser ball bar
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xii, 197 leaves : ill. ; 29 cm.
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Schmitz, Tony L., 1970-
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Mechanical Engineering thesis, Ph.D   ( lcsh )
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Thesis:
Thesis (Ph.D.)--University of Florida, 1999.
Bibliography:
Includes bibliographical references (leaves 193-196).
Statement of Responsibility:
by Tony L. Schmitz.
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Typescript.
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Vita.

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THE IMPLEMENTATION OF SIMULTANEOUS TRILATERATION TO MEASURE
DYNAMIC THREE-DIMENSIONAL 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 DDM-935138 and DGE-9354980. 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
Two-Dimensional Dynamic Measurement Tools............................. ............ 11
Software for Pre-Machining 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
2-D M easurem ents............................................................................................. 108
Dynamic/Static Repeatability .................................................................. 117
2-D Contouring Accuracy ........................ ................................................. 122



iii








3-D 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

1-1. Laser Ball B ar .............................................................. ... ................... 4

1-2. Trilateration Tetrahedron.................................... ........................................ 4

1-3. Sequential Trilateration........................................................ .......................6

1-4. Sim ultaneous Trilateration................................................... .......................6

2-1. Heidenhain G rid Plate..................................................... .. ..................... 13

2-2. Magnetic Ball Bar............................... ........................ 13

2-3. Ball Bar Contouring Error...................... ....................... 15

2-4. Circular Path Section....................... .......... .. ................... 15

2-5. Circular Test Setup............................................................ ...................... 16

3-1. Plane M irror Reflection ..................... .....................................21

3-2. Polarizing Filter.................................................................. ...................... 24

3-3. Linearly Polarized Light..................................................... ...... ................. 26

3-4. Left Circularly Polarized Light............................. .... ...................26

3-5. Young's Double Slit Experiment ................................................29

3-6. M ichelson Interferom eter..................................................... ..................... 31

3-7. Twymann-Green Interferometer..................... ...... .....................33

3-8. Mach-Zehnder Interferometer.................... ......... ......................33

3-9. Alignment Accuracy .................... ........ ......................36








3-10. Beat Phenom enon............................................................. .......................39

3-11. Heterodyne Phase Relationships ...................... .. ...........................41

3-12. Beat Frequency Shift ................................................ ..................... 44

3-13. Acousto-optic M odulator ................................................... ...................... 46

3-14. Z eem an Split ............................................................................ ......................46

3-15. Linear Displacement Interferometer......................... ...................... 50

3-16. Angular Measurement Setup................. ...... ...........................50

3-17. Fiber Delivery/Collection.....................................................51

3-18. Total Internal Reflection............................. ...................... 51

3-19. Refractive Index Profiles....................... .. ................ .......................54

3-20. Single-m ode Fibers............................................................. ...........................58

4-1. Laser B all Bar ........................................................................... .......... 60

4-2. Simultaneous Trilateration................................................ .......................60

4-3. Socket Diameter Constraints ............................................ ....................... 63

4-4. Tool Socket Spring A id..................................................... .......................65

4-5. Castigliano M odeling........................... ....... ..................... 65

4-6. Spring Aid Range of Motion.............................................................................67

4-7. Magnetic Flux Lines ............................. .....................67

4-8. Static Free Body Diagram .................................................... .....................70

4-9. Friction Test Stand .................................................................... ....................73

4-10. C alibration C ouple.................................................................... .....................73

4-11. Contact Area Geometry...................................................... 76

4-12. Tool Point Bracket....................................................79








4-13. Spring Scale Measurement...................... ........ .......................79

4-14. Optics Package...................... ..................................................... 80

4-15. Optics Package Placement................................... ..... ....................80

4-16. 4-DOF Positioning Mechanism .............................. ..... ......................83

4-17. LBB Unsensed Length........................... ...... .... ..................... 85

4-18. Sphere/Motion Misalignment...................... ........ ......................88

4-19. LBB Interferometer Deadpath......................... ............................88

4-20. Initialization Fixture Offsets............................. ................. .....................92

4-21. Castigliano Model Loading Conditions............................ ......................95

4-22. LBB Moment Calculation Sections..................... .............................95

4-23. Left Sphere Total Angular Rotation..................... ... .......................98

4-24. Cosine Error Due to Interferometer Rotation...........................................98

4-25. M oving Retroreflector Rotation................................... ........... ... 100

5-1. LBB Initialization Fixture................................. .... ..................... 107

5-2. LBB Initialization Repeatability ........................... ............... 107

5-3. Heidenhain Grid Plate Setup.................................................. 110

5-4. ST L B B Setup ....................................................................... ...................... 110

5-5. STLBB Verification Contours .......................................... ................... 112

5-6. Angle Path Comparison (35 ipm, 0.1g)....................... ...... ......... ........... 114

5-7. Angle Path Comparison (70 ipm, 0.5g).................... ..................... 114

5-8. Step Path Comparison (70 ipm, 0.lg)..................... ........................... 115

5-9. Step Path Comparison (70 imp, 0.5g) ......................................................... 115

5-10. Sultan Path Comparison (35 ipm, 0.lg) .......................... .................... 116








5-11. Sultan Path Comparison (70 ipm, 0.lg) ..................... ..................... 116

5-12. Square Path Comparison (70 ipm, 0.5g) .......................................... 118

5-13. Triangle Path Comparison (35 ipm, 0.lg) ....................... ..................... 118

5-14. Circular Path Comparison (70 ipm).......................... .................... ... 119

5-15. Heidenhain Dynamic Repeatability Test.................................................... 119

5-16. STLBB Dynamic Repeatability Test ................... .......................... ... 121

5-17. Capacitance Probe Setup............................. ... ...................... 121

5-18. STLBB Angle Path (70 ipm, 0.1g)....................... ................... 123

5-19. STLBB Angle Path (70 ipm, 0.5g)................................ ...................... 123

5-20. STLBB Square Path (70 ipm, 0.5g)................ ........... ........ .............. 126

5-21. STLBB Square Path Reversal (70 ipm, 0.5g).............................................. 126

5-22. STLBB Sultan Path (70 ipm)............................... ... ..................... 128

5-23. STLBB Sultan Path Reversal (70 ipm)..................... .... .............. 128

5-24. STLBB Circular Path (35 ipm)...................... .... ........... ........... 129

5-25. STLBB Counter-Clockwise Circular Path (70 ipm)........................................... 129

5-26. STLBB Clockwise Circular Path (70 ipm, 0.lg)........................................ 131

5-27. STLBB Corkscrew 3-D Path and STLBB Measurement................................... 131

5-28. STLBB Hemisphere 3-D Path and STLBB Measurement ................................. 133

5-29. STLBB Oblique Rectangle 3-D Path...................... .................... 133

5-30. STLBB Oblique Circle 3-D Path.................................. .............. 134

5-31. STLBB Corkscrew 3-D Path Section..................................... ..................... 134

5-32. STLBB Hemisphere 3-D Path Section.................................. ...................... 135

5-33. STLBB Oblique Rectangle 3-D Path Section.............................................. 135








5-34. STLBB Oblique Rectangle X-Y Plane .................................. ...................... 137

5-35. STLBB Oblique Rectangle X-Z Plane........................ ......... ............. 137

5-36. STLBB Oblique RectangleY-Z Plane......................... .................... .. 139

6-1. Diamond Tool Path ............................................................. 142

6-2. X Reversal Error............................................................ ...................... 142

6-3. Y Reversal Error............................................................ ...................... 143

6-4. X-Y Gain Mismatch Error..................................................... 143

6-5. Diamond Path Gain Mismatch Effect (70 ipm)................................................ 145

6-6. Determination of Ad (35 ipm Test) ...................................... ....... ............. 145

6-7. Frequency Response Function (FRF) for 0.5" Diameter Endmill ..................... 147

C- Up/D own M killing ............................................................ ..................... 163

C-2. Undercut/Overcut in Milling ............................................................ 163

C-3. Cutting Forces in Down M illing................ ................ ........................ 167

C-4. Y-Force Components (r < 1) .......................................... 167

C-5. System Frequency Response............................................. ... 169

C-6. Phasor Diagram (r < 1)............................................................ 169

C-7. Y-Displacement Components (r < 1) ............................................ 171

C-8. Y-Force Components (r = 1) .................................... ....................... 172

C-9. Phasor Diagram (r = 1)........................................................... 175

C-10. Phasor D iagram (r > 1)................................................... ............................ 175

C-l Undercut/Overcut/Undercut Transition..................... .. ........ ........... .. 179

C-12. Simulation Flow Diagram ............................... .......................... 179

C-13. Part G eom etry..................................................................... ................ 182



ix








C-14a. X Direction Direct Transfer Function........................ ........... ........... 182

C-14b. Y Direction Direct Transfer Function........................................................... 183

C-15. PTP Force Diagram ................................................ ......................................... 186

C -16. N C Path ......................................................................... ...... ............ 186

C-17. Radial Asynchronous Error..................................... ............... 190

C-18. D I/D2 Surface Location Error .................................................................... ....190

C-19. Full Range Simulation (0.5" diameter tool, 3.25" overhang) .......................... 192

C-20. 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 THREE-DIMENSIONAL CONTOURS USING THE LASER BALL BAR

By

TONY L. SCHMITZ

May 1999

Chairman: Dr. John Ziegert
Major Department: Mechanical Engineering

Computer numerically-controlled (CNC) multi-axis machine tools are an integral

part of modern manufacturing. These machines operate in an overall open-loop mode

(i.e., although the positions of the individual axes are servo-controlled, the actual spatial

coordinates of the tool or end-effector 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 pre-process 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 3-D contours to micrometer-level accuracy. The data obtained from these 3-D

measurements is useful for the evaluation of the controller performance during contouring

and the measurement of the relative contributions of both quasi-static (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 steady-state 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 numerically-controlled (CNC) multiple-axis machine tools are an

integral part of modem manufacturing. These machines operate in an overall open-loop

mode (i.e., although the positions of the individual axes are servo-controlled, the actual

spatial coordinates of the tool or end-effector are unknown). In continuous-path 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 pre-process 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 3-D contours. The data obtained from the 3-D

measurements is useful for the evaluation of the controller performance during contouring

and the measurement of the relative contributions of both quasi-static 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 3-D contours prior to machining could

also have far-reaching 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 pre-defined 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

statistically-based 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 two-stage telescoping tube with a precision sphere mounted at each

end. A heterodyne displacement measuring interferometer is aligned inside the tube and

measures the relative displacement between the two sphere centers (see Figure 1-1). The

LBB has been shown to be accurate to sub-micrometer 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 1-2 are measured once and remain fixed during the motion of

the tool socket. The three base-to-tool socket lengths (denoted Li, L2, L3 in Figure 1-2)

are measured during execution of the applicable CNC part program. Once the coordinates














Inner Telescoping Tube-


-Optics Package (InterFeroneter)


Figure 1-1: Laser Ball Bar


Z


x



Base Socket #1 4


S Base Socket #3





LB 2




Base Socket #2


Figure 1-2: 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

base-to-tool socket leg at a number of static points during each repetition. The setup for

sequential trilateration is shown in Figure 1-3. 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 quasi-static 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 3-axis milling machine was constructed by

Kulkari using both the quasi-static 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 2-axis turning













Tool Socket


Spindle

y

x z


Machine Table


Figure 1-3: Sequential Trilateration


Tool Socket


Spindle
Y

x z


Machine Table


Figure 1-4: 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 PC-based 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 quasi-static 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 base-to-tool 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 1-4). 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 2-D 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 3-D dynamic part paths to micrometer level accuracies. Other

sub-tasks which were completed are the novel design of a tool socket/joint to support the

3 LBBs, the optics/fiber-optics 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), 2-D verification of the STLBB results, 3-D 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 micrometer-level 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 2-D

measurements will be outlined in the following paragraphs. Additionally, recent releases

of computer software which focus on pre-machining 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 3-axis

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 position-based 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 time-based 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 post-process 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 user-selected 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.

Two-Dimensional Dynamic Measurement Tools


A product capable of 2-D dynamic measurements has been commercialized by the

Heidenhain Corporation. The Heidenhain grid plate system is composed of an optical

grid plate with a waffle-type grating of closely spaced lines (4 Itm signal period) and a

non-contact 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 X-Y 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 2-1). The recorded

motions allow the user to observe the dynamic effects of the machine tool's performance

on the CNC tool path.

Common single-axis 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 three-axis machine

tool. The machine tool was modeled as serial connections of the motor, ballscrew, table

















Grid Plate _







Machine Table



Figure 2-1: Heidenhain Grid Plate


Circular Test Path

Magnetic Ball Bar


Socket Fixed to -
Machine TabLe




Figure 2-2: 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 X-Y 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 2-2). As part of a complete calibration of a

three-axis vertical spindle machine tool, the 2-D 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 2-3). Typical motion for a

section of this circular path is shown in Figure 2-4. The causes of the errors shown in

Figure 2-3 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 off-line servo tuning improved the 2-D 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 2-D circular test. The test setup is shown in

Figure 2-5. In this research, the motion error signal (proportional to the change in path










Commanded


Figure 2-3: Ball Bar Contouring Error


Path
End -














Path
Center


Error


Path
Start


Figure 2-4: Circular Path Section















tar Test Path

Gap Sensor


SEncoder


Figure 2-5: 2-D 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 non-contact and have a

high resolution. If these measurements are to be performed on large bandwidth systems,

high-frequency dynamic data capture is necessary.

An example of this high-frequency, 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 Pre-Machining Contour Evaluation


The attractiveness of verifying CNC code prior to machining a test part has not

gone unnoticed in the manufacturing community. Several computer-aided 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 well-established 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 3-1. A similar relationship can be seen between the wavelength of light in a

given medium, X, and the wavelength in vacuum, i0. See Equation 3-2. 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 (3-1)

X= / n (3-2)



Additionally, the velocity is related to the wavelength and frequency, f, of light as

shown in Equation 3-3 [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 le-16 m. The visible spectrum is generally

characterized by wavelengths between 380 nm (violet) and 770 nm (red) [17].



v=Xf (3-3)



The relationships given in Equations 3-2 and 3-3 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 wave-particle

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 3-1 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 3-1: 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

3-2 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 3-2: 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, equal-magnitude 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 3-3.

If these two, equal-magnitude 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 counter-clockwise. Figure 3-4 shows an example of left

circular polarization. Viewed head-on, 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

counter-clockwise paths.






















U I^^ im^ Direction of
Propagation



Figure 3-3: Linearly Polarized Light



E,
E

Left Circular
I Potarization


I I
I I,



I-" Direction of
I, Propagation


Figure 3-4: 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 3-1 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 quarter-wave plate (or quarter-wave

phase retarder), is derived from the '/4 cycle delay which results from this 900 phase shift.

The quarter-wave 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 pre-multiplication 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), non-orthogonal

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 two-source interference is shown in Figure 3-5. 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 3-5: Young's Double Slit Experiment








The monochromatic wavelength of the source may then be calculated, according to

Equation 3-4, 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 (3-4)



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 3-6. A

relative displacement of one-half the wavelength of the source light (?\2) produced a light-

dark-light transition, or fringe, at the viewing position. A light-dark-light transition





31




Moving Mirror


Beam
Splitter

Diffuse
Light





Silver Compensator
Coating P te
Fixed
Viewing Mirror
Point

Figure 3-6: Michelson Interferometer








occurred for a 3600 (one wavelength) phase shift between the electric fields of the two

waves [15]. This corresponded to the half-wave 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 Twymann-Green

and Mach-Zehnder interferometers. The Twymann-Green, shown in Figure 3-7, 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 Twymann-Green 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 Twymann-Green 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 Mach-Zehnder, shown in Figure 3-8, 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 3-7: Twymann-Green Interferometer

Beam
Mirror Spitter





Recombined
Beam',


Reft oec ted
ar





/ I r a r
BRe
B re
Spl t Ler"

Figure 3-8: Mach-Zehnder Interferometer


Smittc
'am


Mirror








The light-dark 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 (one-half 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 single-pass 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 time-consuming. It requires precise, high-resolution 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 3-9. 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 3-9, 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 3-9: Alignment Accuracy








lb. For the typical values given, this predicts a maximum misalignment, a, of 35.07 arc-

sec (see Equation 3-5).


a = tan' (X/ D) 3600 arc-sec, where = 510e-9 m (3-5)
D = 3e-3 m

Heterodyne Interferometrv


Modem users of linear displacement measuring interferometers rely mainly on

heterodyne (AC), or two-frequency, interferometry. The light source is, in most instances,

a frequency-stabilized Helium-Neon (He-Ne) laser tube with some method of generating a

second frequency (within the bandwidth of modem electronics) from the natural He-Ne

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 He-Ne gain curve

(see Equation 3-6). 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 3-7 [26]. The TEMoo transverse mode has a Gaussian

cross-sectional intensity profile. The variation in irradiance across the beam is given in

Equation 3-8 [17].








2L = mX, where L = laser cavity length (3-6)
m = integer longitudinal mode number
X = light wavelength

c
f,, f, where fi = light frequency (3-7)
2Ln
c = speed of light
n = refractive index of medium

-8y2
I= e D where I= beam intensity (3-8)
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 3-10.

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 3-9. 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 3-10: Beat Phenomenon







Figure 3-10: 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 (3-9)
= 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 down-shifted) 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 up-shift 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 3-11: 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 3-11). 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 (He-Ne light), this gives a wavenumber or V-number, 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 He-Ne laser light in a single-pass 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 He-Ne), designated fo, and the second frequency, fo f,, normally

generated within the laser head by either acousto-optic 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 3-12, 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 3-12: Beat Frequency Shift


Magnitude
I


fo-Fd 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 acousto-optic

technique, a single longitudinal mode (of the TEMoo transverse wave) is passed through an

acousto-optic 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 3-13). 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 3-10. The two

frequencies may then be recombined into a single heterodyne beam.


2 OB = X \ acoustic, where ,acous, = Vacoustic / acoustic
Vacoutic = acoustic velocity


(3-10)
















OptC -iStorder opticc + facost,






I Transmitted ptic
acoustic Othorder

Transducer


SLocal Osciltator



Figure 3-13: Acousto-optic Modulator







Magnetic Field

f +







f''r -t" A F
He Nop e ube /

H( Ne laser- TubW '


Figure 3-14: Zeeman Split








The Zeeman frequency split is accomplished by placing a magnetic field around the

He-Ne 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 He-Ne 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 3-14 [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 3-6) 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 (A-B). The electric field (E) of the resultant beam after interference of

the measurement (moving mirror) and reference beams (fixed mirror) is expressed in

Equation 3-11. 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), (3-11)

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 3-12. 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) (3-12)
+ 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 two-frequency He-Ne 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 3-15. 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 3-16 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 3-17.

The different fiber types shown in the figure, single-mode 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 3-15: Linear Displacement Interferometer


Prism


*tros


Input


Output


PBS _


sin c c change in path length
Ii


Figure 3-16: Angular Measurement Setup















SMPM --Loe
Fiber L


MM Fiber


Detector


Figure 3-17: Fiber Delivery/Collection





2

Surface
Normal

S 90 > n

3nb





4 4

S- Critical Anle for TIR 4


Figure 3-18: 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 3-13. 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 3-18).

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

3-14. 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 (3-13)

sin Ocnicrl = nb / na, where nb < n, (3-14)



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 cross-sectional profiles for the core index of refraction are shown in

Figure 3-19. The step index profile has an abrupt discontinuity at the cladding interface,

while the graded-index profile has a smoothly changing index of refraction, typically

parabolic in shape. The advantage of graded-index 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 3-19: 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 half-angle (0c) of the acceptance cone may be

calculated according to Equation 3-15 for a step index fiber.



n, sin 0, = (nco2 ncladding2) 0, where ni = 1 for light from air (3-15)



A fiber's numerical aperture (NA), a measure of the light-gathering 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 3-16. It should be

noted that this particular figure of merit is independent of the fiber geometry.



NA = (nc~2 ncladdmg2).5 (3-16)



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 3-17. This NA must then be compared to the fiber NA to avoid

overfilling.



NAam = n sin (ro / f), where ro = initial beam waist radius (3-17)
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 V-number, may be calculated (see Equation 3-18 [24]). An alternate analysis,

developed in [17], suggests that the maximum number of supported modes (mn.x) may be

calculated according to Equation 3-19.



V = ko (a) NA, where ko = free space wavenumber = 27t / X (3-18)
a = fiber core radius

lm d 2
m- 2= NA) where d = fiber core diameter (3-19)
2 A,,








It can be seen that a small core diameter, small NA, or a large free space

wavelength, Ao, will decrease the V-number 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 V-number is

increased, more modes may be carried by the fiber. These fibers are referred to as

multimode fibers. Typical multimode communications fibers may have V-numbers 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 mode-polarization maintaining or SMPM fiber

shown in Figure 3-17). 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 3-20 [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 3-17, 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 single-mode fibers.


Elliptical stressed
cladding Bow tie









Core/cladding
Stress applying part


Circular











Slow axis


Figure 3-20: Single-mode 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 two-stage 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 4-1).

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 4-2 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 4-1: Laser Ball Bar


Tool Socket


Sp ndle


x ~z


' Base Socket iVi


Muc hine Tabe


Figure 4-2: 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 cross-sectional 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 cross-sectional area provides more attractive force). This angle, combined with

the sphere diameter (D) constrains the allowable LBB socket diameter (see Figure 4-3).




do, = Dsin 4- (4-1)




From Equation 4-1, 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 4-3: 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 4-4. 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 cross-sectional 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 4-5). 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 (4-2)




In Equation 4-2, 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 4-4: Tool Socket Spring Aid


H


Figure 4-5: Castigliano Modeling








in Equation 4-3, the spring aid was found to provide a range of motion of 22.5" to 380

from the vertical axis (see Figure 4-6). 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 (4-3)
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 4-7a.

The 440 stainless steel (SST) socket completes the magnetic circuit through the sphere.

The neodymium magnetic is encased in a brass (non-magnetic) 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 4-6: Spring Aid Range of Motion


378 N


Sinutoted
Flux Lines


Magnet / .





Figure 4-7: Magnetic Flux Lines


25.6 N


?5.6 N








socket are shown in Figure 4-7b. 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 4-8. 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 4-4. 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 4-5).







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
(4-4)
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) (4-5)
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 4-8: 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 4-4. The tipping force (Pip)

is given in Equation 4-6.

(4-6)
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 4-1.



Table 4-1: 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 4-1 is shown in Equation 4-7. 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
(4-7)
(-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 strain-gage-based friction test stand [36].

The strain-gage-based friction test stand is shown in Figure 4-9. The main

components are the base, support arms, strain-gage 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
Strain-Goge
Bridge


Precision
Sphere
LBB
Socket





S Base



Figure 4-9: 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 4-10: Calibration Couple








perpendicular to the shaft (see Figure 4-10). 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 (4-8)




The strain in the full-bridge wheatstone circuit was measured for a range of applied

torques using a Vishay V/E-20A 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 best-fit 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 three-point contact socket. See Figure 4-11 for the actual geometry of the

contact area. The equation for the normal used in the friction equation is given in

Equation 4-10. 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 4-2.


Ncontc cos() = Napplied


(4-10)


Table 4-2: 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 4-11: 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 4-12. 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 4-13. 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 cross-sectional dimensions were then calculated according to

Equation 4-11. The material selected for the bracket was 301 SST due to its high elastic

modulus and good corrosion resistance. For a square cross-sectional 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 (4-11)
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 = 1.135e-7 m4
G = Shear Modulus = 75.98e9 N/m2 A = b2 = 1.167e-3 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 single-mode, polarization maintaining fiber optic, it is necessary to generate a











Tool Point Bracket


Tool Holder-'


Figure 4-12: Tool Point Bracket


Spring
Scale


Spindle Quill






Spindle
Centerline









Spindle










Magnetic Sockets


Figure 4-13: Spring Scale Measurement











Poltrizer
Reference Retroreflectar QWP
PBS







Pris7 2n
Local Re.f
OWP Beam Outp

0 ------ Ouput Mea
Beam

Pris 3 --- -- Input 2-F
Laser Be

NPBS T 4 mm
NPBS
Pris- 11

I Input 2-F
S" ..... L..aser Be









To Moving Retroreftector


erence
ut

isurement

requency
Ian


frequency
an


Figure 4-14: Optics Package


COutput Port For Local Reference Beon

Output Port for Meosurement Beam



7 I


requreny i.ospr Bean


art For


Figure 4-15: 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

4-14. The position of the optics package within the LBB is shown in Figure 4-15.

Figure 4-14 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 non-polarization

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 degree-of-freedom device which provides these adjustments is shown in

Figure 4-16. 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 spring-loaded 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 4-16: 4-DOF 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 4-17). 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 4-17: 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 4-12. Figure 4-17

shows an unsensed length of 57.035 mm in 304 stainless steel and 13.350 in 6061-T6

aluminum. The coefficients of thermal expansion, a, are 17.8 ppm/OC for 304 stainless

steel and 24.3 ppm/C for the 6061-T6 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 (4-12)
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 4-14. Because He-Ne laser beams have a Gaussian cross-sectional 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 (4-13)



0 = tan' (offset / 2 L) (4-14)



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 4-18. 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 4-15). 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) (4-15)
= 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 4-18: Sphere/Motion Misalignment


Fixed
Retroreftector















LF-- ---L RLd





Figure 4-19: LBB Interferometer Deadpath


Zero
Position




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