MEMS Tuning-Fork Gyroscope Final Report

MEMS Tuning-Fork Gyroscope Final Report Amanda Bristow Travis Barton Stephen Nary

Abstract MEMS based gyroscopes have gained in popularity for use as rotation rate sensors in commercial products like cars and game consoles because of their cheap cost and small sized compared to traditional gyroscopes. The MEMS gyroscope consists of the basic mechanical structure an electronic transducer to excite the system as well as an electronic sensor to detect the change in the mechanical structures modal shape. The goal of this project was to aid in Dr. Julie Hao's ongoing research by attempting to refine the accuracy and detail given by the MEMS gyroscope to military grade standards enabling the expansion of their use, while maintaining the attraction of their low cost. To achieve this, the structure of the mechanical portion was analyzed to generate the two modal shapes as close as possible within the 15-30kHz range so as to be differentiable from background noise yet still require minimal power to be detected. Then a MATLAB script was used to calculate the theoretical specifications of the transducers and sensors. Finally, the completed design was drawn using AutoCAD for use in creating a mask for the manufacture of the device. Although the project scope did not include the fabrication of the new design because of time and equipment constraints, the project group was able to study the manufacturing and testing process for MEMS devices as well as work on the connection of printed circuit boards to be used in the future for testing gyroscopes.

2

Table of Contents Abstract.......................................................................................................................................2 Introduction and Purpose............................................................................................................5 Background Theory.....................................................................................................................6 Gyroscope Mechanical Structure............................................................................................6 Coriolis Effect..........................................................................................................................8 Electrical Theory: Drive Mode.................................................................................................9 Monitoring Drive Mode.....................................................................................................12 Electrical Theory: Sense Mode.............................................................................................14 MEMS Device Fabrication....................................................................................................16 Work Breakdown.......................................................................................................................19 Mechanical Structure............................................................................................................19 Electrical Calculations...........................................................................................................20 Mask Design.........................................................................................................................20 Gyroscope Testing - PCB.....................................................................................................22 Conclusion.................................................................................................................................22 Appendix A: MATLAB Code for Electrical Calculations............................................................23 Appendix B: Gantt Chart...........................................................................................................24

3

List of Figures Figure 1: Gyroscope Mechanical Structure................................................................................6 Figure 2: Drive Mode Frequency................................................................................................7 Figure 3: Sense Mode Vibration..................................................................................................7 Figure 4: Sense Mode Metal Plates............................................................................................8 Figure 5: Coriolis Acceleration....................................................................................................8 Figure 6: Gyroscope with Comb Drive Transducer.....................................................................9 Figure 7: Comb Drive Finger Detail..........................................................................................10 Figure 8: Voltage Sources.........................................................................................................11 Figure 9: Sense Mode Parallel Plate Capacitors......................................................................14 Figure 10: Mask for a MEMS Device........................................................................................16 Figure 11: Gyroscope Pattern Repetition..................................................................................16 Figure 12: Preparation of Photo Resist Layer...........................................................................17 Figure 13: Wafer Development.................................................................................................17 Figure 14: SOI Wafer................................................................................................................18 Figure 15: SOI Wafer After Etching with HF Acid.....................................................................18 Figure 16: Critical Dimensions..................................................................................................19 Figure 17: Gyroscope Drawing in AutoCAD.............................................................................20 Figure 18: 3D Model from AutoCAD Drawing...........................................................................20 Figure 19: Gyroscope Detail Drawing.......................................................................................21 Figure 20: Gyroscope Pattern on Mask Area...........................................................................21

4

Introduction and Purpose Gyroscopes are becoming more desirable in many different applications because of their ability to sense and report rotation. There are many industries using gyroscopes in their products. Increasingly precise gyroscopes are in demand, and research is being done to meet that demand. The main considerations involved in designing and fabricating a gyroscope are how it works, the design attributes which need to be added, and the cost considerations involved. Gyroscopes are designed to sense rotation, which they do by taking advantage of the Coriolis effect. The Coriolis effect is the name given to the acceleration a moving body within a rotating reference frame seems to experience. The Coriolis effect can be demonstrated by having a person stand stationary on the x axis and a particle move along the positive y axis. Once a rotation is added about the z axis, the particle appears to move towards the observer standing in the rotating reference frame. When used in a gyroscope, the Coriolis effect allows the detection of angular velocity. Two modes of vibrations are used to do this. The first mode of vibration is the drive mode. This mode is intentionally introduced to the system to take advantage of the Coriolis effect by having two proof masses moving horizontally in opposite directions from one another. The second mode of vibration is caused by introducing an angular velocity to the already moving device. The Coriolis effect causes the proof masses to move in a different direction than the first mode. The proof masses then move vertically in directions opposite each other. Sense electrodes are placed above the proof masses and can sense the distance that the proof mass is moving. These sense electrodes are able to calculate capacitance which is a function of distance. This capacitance is directly proportional to the angular velocity, thus giving an accurate measurement of angular velocity in terms of capacitance. The design of a gyroscope is very important for its application. The most common gyroscope is the Micro-Electro Mechanical System (MEMS) tuning for gyroscope. This gyroscope is designed with a symmetrical tuning fork structure. This design uses two proof masses. The structure is composed of long thin beams and anchored in the center, leaving the proof masses free to move. It is essential to adjust all of the dimensions so that the desired frequency range is attained, allowing the data to be collected and interpreted without excess interference from background noise. Gyroscopes are used in many different industries, some new and some old. They are used for gaming in systems such as the Wii Motion Plus attachment, and telecommunications in devices such as the iPhone 4. Gyroscopes are also used in the automotive industry for many different purposes as well as for space exploration and ship building. Gyroscopes are desirable because of their small size, low power use, and low cost when mass produced. Most gyroscopes are now made from layered silicon wafers. These wafers are micro machined such that the gyroscope’s form is exposed from the middle of the wafer. The purpose of this project is to assist Dr. Julie Hao with her continuing research in MEMS technology by refining an existing gyroscope design to try to improve the sensitivity of the device. Although fabrication of the completed design is beyond the scope of this project (because of the time consuming nature of the MEMS manufacturing process), the project group will learn about the manufacture and testing of MEMS devices and gain some firsthand experience in Dr. Hao's lab by working with an existing gyroscope design. As time permits, the group will also attempt to connect the printed circuit boards needed to test a MEMS gyroscope so that Dr. Hao and her graduate students may use them in their research.

5

Background Theory Gyroscope Mechanical Structure MEMS tuning-fork style gyroscopes are small silicon devices that detect rotation. Although they are currently not as sensitive as traditional rotating ring gyroscopes, because of their small size and low cost, they are used in many different electronic devices, including the iPhone 4 and the Wii Motion Plus accessory. The basic mechanical structure of a MEMS gyroscope is shown in Figure 1 below.

Anchors

Proof Masses

Figure 1: Gyroscope Mechanical Structure The structure is fixed at the anchors shown above, while the rest of the structure, including the large proof masses on either side, is free to move. Since the gyroscope uses the Coriolis effect to detect rotation, the device must be in constant motion to function as the Coriolis force only acts on moving bodies (for more about the Coriolis Effect, click here). This motion is accomplished by vibrating the structure at one of its natural frequencies to achieve the motion shown in Figure 2. This vibration is referred to as the Drive Mode.

6

Figure 2: Drive Mode Frequency When the structure begins to rotate, the Coriolis force acting on the moving proof masses changes the direction of the vibration from horizontal to vertical as seen in Figure 3. This vertical vibration corresponds to a higher natural frequency of the structure than the horizontal Drive Mode vibration and is referred to as Sense Mode.

Figure 3: Sense Mode Vibration Metal plates are placed above the proof mass as shown in Figure 4. Together with the proof mass, these plates form a capacitor. As the proof mass vibrates in drive mode, the distance between the proof mass and the plates remains constant. Since capacitance for a parallel plate capacitor such as this one is a function of the distance between the two plates, in Drive Mode the capacitance also remains constant. Once the structure begins to rotate and enters Sense Mode, as the proof mass moves vertically, the distance between it and the plates changes, which changes the capacitance. This chance in capacitance can be detected by electronic equipment 7

and converted to indicate the corresponding rotation.

Figure 4: Sense Mode Metal Plates

Coriolis Effect The Coriolis Effect is the name given to the acceleration experienced by a moving point in a rotating reference frame. This Coriolis acceleration is defined as

 V a coriolis =2 × 

(1)

 is the angular velocity of reference frame and V is the velocity of the particle within this reference   is clockwise about the z-axis). frame, as shown in Figure 5 below 1 (note that the direction of angular velocity  where

z z 

v d

Rotation rate

Velocity

 z ×v d a =2  Coriolis Acceleration

x y Figure 5: Coriolis Acceleration

1

Figure 5 used with the permission of Dr. Julie Hao.

8

Electrical Theory: Drive Mode The force necessary to keep the proof masses in constant motion in drive mode is provided by a comb drive transducer placed next to one of the proof masses, forming a capacitor as shown below 2:

or s

Proof mass

An ch

Dr ele ive ctr od e

Sense electrodes

Dr ele ive ctr od e

Capacitor

Proof mass

Tuning electrodes

Figure 6: Gyroscope with Comb Drive Transducer The fingers on the comb drive and the side of the proof mass increase the surface area of the capacitor, which increases the capacitance and allows changes in the system to be detected more easily. Note that only one of the drive electrodes above is used to apply a force to the proof masses; the other is used to monitor the drive mode vibration of the proof masses. The capacitance

C for one finger of the proof mass is given as:

C=20

l0 h g

(2)

where l 0 is the length of the initial overlap between the finger on the proof mass and the finger on the comb drive, h is the height of the gyroscope, g is the gap between the fingers on the comb drive and the finger on the proof mass, and  0 is the permittivity of free space, a constant.

2

Parts of Figure 6 used with the permission of Dr. Julie Hao.

9

Figure 7: Comb Drive Finger Detail To find the total capacitance of the Comb Drive – Proof Mass capacitor, simply multiply the capacity for one finger by the number of fingers, n .

The energy stored in the capacitor,

E is given as: 1 E= C V 2 2

where

(3)

V is the voltage drop across the capacitor. Substituting the value for the total capacitance, C yields:

E=n 0

As the proof mass moves a small distance

l0 h 2 V g

(4)

x in drive mode, the overlap between fingers changes from l 0 to

l 0x and the equation for energy becomes

E=n 0

To find the force

l 0 x h 2 V g

(5)

F acting on the proof mass, take the derivative of E with respect to x :

F=

0 h 2 dE =n V dx g

10

(6)

In order to keep the proof masses in constant motion, an AC voltage source must be used so that the capacitor will not reach a steady state. An AC voltage source is applied to the comb drive transducer as shown below 3. Note that a DC voltage source is applied to the anchor of the gyroscope for use in sense mode.

VDC

~ vAC

Figure 8: Voltage Sources With these two voltage sources applied, the equation for force becomes:

F =n

0 h V DC v AC 2 g

(7)

This can be expanded as:

0 h 2 V DC 2V DC v AC v 2AC  g 0 h 2  h  h =n V DC n 0 2 V DC v AC n 0 v 2AC g g g

F =n

The term

n

0 h 2 is a constant and will only offset the proof mass initially by a small amount. Because it will V g DC

not contribute to the oscillation of the proof mass, it can be disregarded.

3

(8)

Parts of Figure 8 used with permission of Dr. Julie Hao

11

v AC is time dependent, it can be written as v AC =v 0 sin t  where v 0 is the amplitude of the AC voltage and  is the frequency of the AC voltage, which must be equal to the drive mode natural frequency of 0 h 2 0 h 2 2 the gyroscope. Substituting this expression into the term n v AC yields n v 0 sin t  . However, g g 1−cos 2 t  2 since sin t= , this term becomes a function of two times the natural frequency of the 2 Since

system and can be disregarded, leaving:

F =2 n

0 h V v sin  t g DC 0

(9)

Applying Newton's Second Law to the gyroscope mechanical system yields

m

where

d2 x dx D kx= F 2 dt dt

(10)

 k is the spring constant of the system and D is a damping coefficient given as D=

km where Q is Q

a large number, typically greater than 10,000.

Because the displacement of the proof mass is also time dependent, it can be written as

x=q0 sin t , where

q 0 is the amplitude of the displacement. The differential equation above can be solved for

q 0 , producing

F Q k 0 h V DC v 0 =2 n Q gk

q 0=

(11)

This equation gives the amplitude of the proof mass vibration in terms of the amplitude of the AC voltage, which can be used to design the comb drive transducer to achieve a specific amplitude of proof mass vibration.

12

Monitoring Drive Mode As mentioned above, only one of the comb drive transducers is required to apply a force to the proof mass to keep it in motion. This leaves the other proof mass free to be used to monitor the vibration of the proof mass in drive mode. By measuring the current i from this transducer, the displacement of the proof mass can be found through the following relationship. Since Q , the electric charge, can be expressed as C V , and current is merely the derivative of charge with respect to time, an expression for electric current can be obtained by evaluating

dQ d = C V DC  dt dt

(12)

By applying the chain rule, this becomes

i=

however, since

dV DC dC V DC C dt dt

V DC is a constant, the term C

i=

As mentioned in the previous section,

C=

dV DC reduces to zero leaving dt

dC V dt DC

(14)

2 n0 h l 0 x , so taking the derivative of this function leaves g

dC 2 n 0 h dx = dt g dt Since

(13)

(15)

x=q0 sin t  can also be expressed as x=q0 e i t , dx =A i e i t =i  x dt

13

(16)

This value can then be substituted into the equation for

dC which can then be inserted into the equation for dt

current, producing the relationship between current and displacement which will allow the vibration of the proof mass to be monitored in drive mode:

i=V DC

2n h i  x g

(17)

Electrical Theory: Sense Mode Just as in drive mode a comb drive transducer is used to indirectly measure the proof mass vibration, in sense mode, capacitance is also used to track the vibration of the proof mass. However, sense mode does not use a single comb drive transducer, but rather a pair of metal plates (seen below) which form a pair of parallel plate capacitors with the proof mass. Both plates over each proof mass are connected to the detection equipment, which helps to amplify their signal in order to aid detection.

Figure 9: Sense Mode Parallel Plate Capacitors

The initial capacitance for a parallel plate capacitor is given as and height of the plate, respectively, and can be expressed as

C 0=

0 w h , where w and h are the width d0

d 0 is the gap between the plates. Given that the sense-mode current

14

i s=

and that

C 0 dy V d 0 dt DC

dy = s y 0 , where  s is the sense mode angular frequency and dt

(18)

y 0 is the amplitude of the

sense mode vibration. The amplitude of sense mode vibration is a function of the force applied to the proof mass:

y 0=

F Qs  k −k elec 

(19)

Q s is a coefficient relating to the damping of the system, just as in drive mode, k is the stiffness constant of the system, and k elec is the Electrostatic Stiffness of the system, which is defined as V2 k elec =C 0 DC . Since the force acting on the proof masses in sense mode is a result of the Coriolis d 20 where

acceleration, it can be expressed as:

m 2 z×v d 

(20)

where m is the mass of the gyroscope, z is the rotation signal input to the system and v d is the drive mode velocity of the proof masses, which can be expressed as the product of the drive mode vibration amplitude and the drive mode angular velocity, q 0  d . As a scalar, the force can be expressed as

2 mz q0 d

(21)

Substituting this force into the sense mode vibration amplitude yields

y 0=

2 mz q0 d Q s k −k elec

(22)

This amplitude can then be substituted into the current equation, which gives

i=

C 0 V DC  s 2 m z q0  d Q s d 0  k −k elec 

15

(23)

The sensitivity of the gyroscope,

i z

then becomes

C 0 V DC  s 2 mq 0 d Q s i = d 0  k −k elec  z

(24)

MEMS Device Fabrication In order to make a MEMS gyroscope one must transfer the design from a CAD file to a Mask and finally to a silicon wafer. To do this a CAD program is used to design the mask. The design is patterned many times on the mask so as to utilize the space on the wafer as efficiently as possible. The program most used to fabricate the mask is CADENCE. This program allows a auxiliary device to laser etch chrome away from glass to reveal the desired design. Below is an example of a mask. Note that light has the ability to pass through the mask.

Figure 10: Mask for a MEMS Device Below is an example of the gyroscope design pattered to form a 1cm by 1cm silicon chip. This would be one of many squares on a mask.

16

Figure 11: Gyroscope Pattern Repetition The silicon wafer must then be prepared by adding a photo resist layer. A round wafer is loaded onto a spin coater, photo resist is applied and the wafer is allowed to spin which adds a thin uniform layer of photo resist.

Figure 12: Preparation of Photo Resist Layer After preparing the wafer, the mask is aligned above the wafer. It is exposed to UV light which changes the properties of the photo resist that is exposed. After this exposure is complete the wafer is developed. In a positive photo resist situation the areas that were exposed to light are etched away, however the opposite is true if a negative photo resist is used.

17

Figure 13: Wafer Development For a MEMS gyroscope to work, most of the mechanical structure must be able to freely move, with the exception of anchors, which hold the whole device in place. In order to accomplish this, a special type of wafer is used. This wafer is composed of several layers. A top layer of silicon is used for the mechanical structure which is 30 microns thick, Silicon dioxide which will release the mechanical structure, and more silicon which creates the bulk substrate. This type of wafer, called a Silicon-On-Insulator (SOI) wafer, is pictured below:

Figure 14: SOI Wafer To release the mechanical structure a powerful acid, such as hydrofluoric acid is used, resulting in a structure like the one in the following image:

18

Figure 15: SOI Wafer After Etching with HF Acid The silicon dioxide does not completely etch away. The mechanical structure of the MEMS gyroscope has two large proof masses, which would not be able to be released from the substrate. To account for this, tiny holes are added into the proof masses which then allow the chemicals to etch enough silicon dioxide to release the structure.

19

Work Breakdown Though all team members were involved in each stage of the design process, each group member accepted primary responsibility for one area of the project, according to the following table. Group Member

Responsibility

Travis Barton

Modelling of Mechanical Structure

Stephen Nary

Theoretical Calculations for Electrical Components

Amanda Bristow

Coupling of Mechanical and Electrical Structures – Mask Design Table 1: Primary Project Responsibilities

Mechanical Structure After receiving the basic gyroscope design from Dr. Hao, the group began modelling the mechanical structure in COMSOL so that a modal analysis could be performed. In order to get the drive and sense modes to be within 50-100 hertz of each other, while still being between 15-30 kilohertz, the group began modifying the lengths of the two critical beams that correspond to the these frequencies. These critical lengths are shown in the following figure. L2

L1

Figure 16: Critical Dimensions Keeping L2 constant at 620 µm, the group initially varied L1 by 10 µm lengths above and below the initial 520 µm value and began with to see how the change in length affected the drive frequency. After finding where the drive frequency crossed below the sense frequency, the group proceeded to vary L1 by 1 µm at a time to see how close to the desired range the design could get by just changing the first variable. Once these results had been collected and tabulated, the group moved onto varying L2 while keeping L1 constant to see how this would affect the sense frequency. After collecting all these values and recording them in excel, it was possible to compare and contrast them to see approximately how the frequencies would change when adjusting both of the variables at the same time. From these results, the group searched for all the points where the drive and sense frequencies were within the 50-100 Hz range of each other, and both above 15 kHz. The drive and sense mode only were within the 50-100 Hz range of each other four times within the data sets, two of which had to be disregarded because the drive mode for them was below the 15 kHz required. This narrowed the selection rather quickly, and made it simple to select which numbers to use. After selecting the optimal values, the group proceeded to re-run the COMSOL analysis to verify that the selected values matched the predicted frequencies and ranges. With the finite element analysis completed, the group finalized the design, and used COMSOL to determine the strain energy and vibration amplitude of the drive mode to use in the electrical calculations.

20

Electrical Calculations Once the physical dimensions of the gyroscope had been finalized, the natural frequencies of interest (15872.9 Hz for drive mode and 15951.4 Hz for sense mode), as well as the maximum displacement of the proof masses, were used as input for MATLAB code to calculate the amplitude of the necessary AC voltage for the drive mode transducer. The code also calculated the sensitivity of the gyroscope according to formula 24 in the theory section. The results of the calculations are summarized in the table below. Please see Appendix A for a copy of the completed code.

Drive Mode Frequency

15872.9 Hz

Sense Mode Frequency

15951.4 Hz

Vibration Amplitude

3.636x10-6 m

AC Voltage Amplitude

0.4810 V

Drive Mode Current

6.4214x10 -8 A

Sensitivity

1.1554x10 -6

Table 2: Results of Electrical Calculations

Mask Design Once all of the design work was completed, the group had all of the necessary dimensions to make a complete model of the Micro Electrical Mechanical System (MEMS) gyroscope. Below is an image of an AutoCAD drawing of a complete MEMS Gyroscope and a solid model based on the drawing.

Figure 18: 3D Model from AutoCAD Drawing

Figure 17: Gyroscope Drawing in AutoCAD

Pictured below is an image of the drawing of a MEMS gyroscope depicting all the important dimensions in microns.

21

Figure 19: Gyroscope Detail Drawing Once the gyroscope had been drawn once in AutoCAD, the design was copied many times, in order to use as much of the mask surface as possible, depicted as a blue circle in the following image.

Figure 20: Gyroscope Pattern on Mask Area

22

Gyroscope Testing - PCB With a very limited knowledge of electrical systems, the group was given an opportunity to learn more about how electrical circuits like our MEMS device work on a larger scale by working on the printed circuit boards (PCBs) used to test the device. Not only did the PCB work give a concrete example of how connections would be made, it also would serve to benefit the continuing research by enabling the graduate students to have access to an appropriate interface to allow for testing of the MEMS Tuning Fork Gyroscope optimization. Despite initial assumptions that assembling the PCB's would be a fairly simple process, the group soon found out just how delicate the chips and resistors required for it to function are, and how difficult it is to make a solid solder connection for them. Although it was a steep learning curve as none of the group members had every done any kind of solder work before, it was very informative and helpful in understanding just what goes on in the manufacturing of electrical devices, and why precision soldering is so important. With a significant time investment, the group finished one PCB, and though there was insufficient time to use it to test a MEMS Tuning Fork Gyroscope, the group was able to confirm through preliminary testing that it is indeed a working board and ready for the graduate students to use in their experiments.

Conclusion A gyroscope’s purpose is to detect rotation. Although there are currently gyroscopes in production, this project has been formed to increase the accuracy of existing gyroscopes. Throughout this semester group members have been exposed to all aspects of MEMS manufacturing as well as gyroscope theory and design. They have designed a gyroscope with a difference in frequency of 78.5Hz, a vibration amplitude of 3.636x10 -6 m, an AC voltage amplitude of 0.4810 V, and drive mode current of 6.4214x10 -8 A and a sensitivity of 1.1554x10 -6.All of these results are well within the desired goal range decided upon at the beginning of the project. Various design constraints were faced in the duration of the project which were all overcome. Along with the purpose of increasing the accuracy of an existing gyroscope, this project was also meant to be a continuation of a faculty member’s research, which will carry on in later semesters. The work performed this semester will be a great stepping stone for another group in the future to hopefully study and fabricate to compare to the theoretical data collected.

23

Appendix A: MATLAB Code for Electrical Calculations % Mechanical Structure Dimensions and physical properties l=400*10^-6; %meters w=400*10^-6; %%meters h=30*10^-6; %meters freq1=15872.9; freq2=15951.4; omega1=2*pi*freq1; omega2=2*pi*freq2; q0=3.636*10^-6; viben = 1548.3364*10^-12; k = (2*viben)/(q0)^2 m= k/(2*pi*freq1)^2 % Drive Mode settings n=25; g=3*10^-6;

%drive mode frequency, Hz %sense mode frequency, Hz %meters %Strain Energy of mech structure, Joules %spring constant of system %mass of proof mass

%number of fingers on proof mass %gap between proof mass fingers and %comb drive fingers; meters %permutivity of free space %volts DC Voltage

epsilon=8.854*10^-12; V=40; Q=10000; v0=(q0*k*g)/(2*n*epsilon*h*V*Q) %AC Voltage Amplitude % Drive Mode confirmation I1=V*(2*n*epsilon*h)*omega1*q0/(g)%drive mode current

% Sense Mode d0= 3*10^-6; %gap between proof mass and transducer, meters C0=epsilon*omega2*h/d0 kelec=(C0*(V^2))/(d0^2) Qs=9000; sensitivity = -(C0*V*omega2*2*m*q0*omega1*Qs)/(d0*(k-kelec))

24

Appendix B: Gantt Chart

25

Works Referenced Hao, Dr. Julie. Case Study – Gyroscope. Powerpoint presentation for ME695/895: Design and Modelling of MEMS Devices. 2010. Sharma, A., Zaman, F., Amini, B., & Ayazi, F. A High-Q In-Plane SOI Tuning Fork Gyroscope. Atalanta, Georgia: Georgia Institute of Technology, 2004 Yazdi, N., Ayazi, F., & Najafi, K. Micromachined Inertial Sensors. Ann Arbor, Michigan: National Science Foundation, 1998.

26