The Development of a MEMS Gyroscope forAbsolute Angle Measurement

Damrongrit Piyabongkarn and Rajesh RajamaniDepartment of Mechanical Engineering

University of MinnesotaMinneapolis, MN 55455

AbstractMEMS gyroscopes are typically designed to measure angular

rate of rotation. A measurement of the angle itself is useful in manyapplications but cannot be obtained by integrating the angular ratedue to the presence of bias errors which cause a drift. This paperpresents an innovative design for a vibrating gyroscope that candirectly measure both angle and angular rate. The design is basedon the principle of measuring the angle of free vibration of asuspended mass with respect to the casing of the gyroscope. Severalcritical challenges have to be handled before the theoretical sensingconcept can be converted into a reliable practical sensor. Theseinclude compensating for the presence of dissipative forces,mismatched springs, cross-axis stiffness and transmission of rotarytorque. These challenges are addressed by the development of acomposite nonlinear feedback control system that compensates foreach of the above effects and ensures that the mass continues tobehave as a freely vibrating structure. Theoretical analysis andsimulation results presented in the paper show that the gyroscopecan accurately measure both angle and angular rate for lowbandwidth applications.

Keywords: MEMS, gyroscope, angle measurement, angular rate

1. INTRODUCTION

Gyroscopes have played an important role in aviation, spaceexploration and military applications (Lawrence, 1998, [1]). Untilrecently, high cost and large size made their use in automobiles andother consumer products prohibitive. With the advent of Micro-Electro-Mechanical Systems (MEMS), gyroscopes and otherinertial measurement devices can now be produced cheaply and invery small packages in the micro domain. An example of this arethe MEMS accelerometers now used in some automobiles to detectcollisions for air bag deployment (Eddy, et. al., 1998 [2]).

In order to estimate the absolute angle, θ , with a traditionalMEMS rate-gyroscope, one would have to integrate the angular ratesignal Ω with respect to time. The problem with this method is thatbias errors in the angular rate signal from the gyroscope willinevitably cause the integrated angle value to drift over time, sinceall gyroscopes have at least a small amount of bias error in theirangular rate signal. This paper develops a sensor design to directlymeasure absolute angle. The design can also be combined withtraditional angular rate measurement to provide a sensor in anintegrated package that measures both angle and angular rate.

There are a large number of applications where a gyroscopethat can measure angle would be useful. A common application ismeasurement of the heading or orientation of a highway vehicle.The measurement of orientation is useful in computer-controlledsteering of the vehicle as well as in differential braking systemsbeing developed by automotive manufacturers for vehicle skidcontrol (Rajamani, et. al., 2000, [3], Tseng, at. al., 1999, [4]).

An important additional benefit of the proposed design isthat it would also contribute towards improving the accuracy ofthe regular rate gyroscopes (see section 5). The proposed designis novel in that it breaks new ground by introducing sophisticatedcontrol systems into the MEMS domain. It is the use ofadvanced control techniques that leads to a new sensor makingthe measurement of a new variable (absolute angle) possible.

A number of researchers and research groups have workedon MEMS vibratory-rate gyroscopes. A good description of theworking of a basic angular rate gyroscope can be found in Clark,et.al. [5]. Here the gyroscope consisted of a single massoscillating longitudinally with rotation induced lateral deflectionsbeing sensed capacitively. A summary of the basic workingprinciple of an angular rate gyroscope is also provided below insection 2.

The bulk of literature on MEMS vibratory gyroscopes dealswith different embodiments of the above basic sensing conceptdescribed in [5]. There are varying designs and implementationusing a number of different fabrication processes. Bernstein,et.al.[6] demonstrated a tuning fork gyroscope using two massesthat oscillate laterally. An external oscillation induced oscillationout of plane of the device. Boeing is now developing acommercial version of this device. Juneau, et. al. [7]demonstrated a dual-axis gyroscope which could simultaneouslymeasure two rotational rates. It consisted of a rotating disk inwhich deflection induced by rotations about two axes weremeasured out of plane of the device. A gyroscope based on avibrating ring was demonstrated by Putty, et. al. [8]. The workby Kranz, et. al., [9] focused on a new one-mass-inside-anotherdesign that reduced measurement errors due to mechanical cross-talk.

All of the above results relate to gyroscopes that measureangular rate. None of the above results have developed agyroscope that measures absolute angle.

2. BACKGROUND ON THE WORKING OF ANGULAR-RATE MEMS GYROSCOPES

The traditional suspended mass vibrating gyroscope consistsof a mass suspended by elastic members that allow it to travel inboth the x and y directions as shown in Fig. 1a. The MEMSembodiment of such a suspended mass gyroscope is shown inFig. 1b [10]. The equations of motion for such a system can berepresented by

mxF

yyxxmxc

xxx ++++−−= DDDDDDDD θθθω 222 (1)

myF

xxyymyc

yyy +−−+−−= DDDDDDDD θθθω 222 (2)

2

where mxk

x =2ω , m

yky =2ω , xxmxc ωξ2= , yymyc ωξ2= and θ

is the rotation of the gyroscope casing and xF and yF are

controlled forces applied to the mass in the x and y directionsrespectively. In order to measure the angular rate with thisgyroscope, the x mode is driven sinusoidally using xF at an

amplitude of Fd and a frequency of dω . The frequency dω is

usually significantly higher than the specified bandwidth of thegyroscope. At this high frequency, the terms 2θDx , 2θDy , θDDy and

θDDx are small and can often be neglected [5]. If necessary theseterms can also be appropriately compensated for by using the forces

xF and yF . The equations of motion are then given by

mxF

yxmxc

xxx ++−−= DDDDD θω 22 (1a)

m

yFxy

m

ycyyy +−−−= DDDDD θω 22 (2a)

mass

2/xk 2/xk

2/xc 2/xc

2/yc2/yk

x

y

2/yk 2/yc

xF

yF

(a)

(b)Fig. 1 Suspended mass vibrating gyroscope

(a) Schematic diagram of a vibratory gyroscope(b) Vibratory-rate gyroscope fabricated in

the MCNC MUMPs process [10]

When the gyroscope experiences an external angularvelocity, Ω=θD , the xDΩ2 term in equation (2a) causes the y modeto vibrate at the driven frequency dω with an amplitude that is

proportional to the angular rate, Ω. By demodulating the y outputsignal at the driven frequency, the value of the angular rate canbe obtained.

Note that y axis motion can also be caused by several othersources, including external forces and initial conditions. Bymeasuring the y axis motion at only the driven frequency dω ,

we isolate the effect of the angular rotation Ω on y axis motion.Thus, angular rate can be measured accurately in spite of otherexternal disturbances.

3. ANGLE MEASUREMENT

The proposed design for angle measurement is based on theprinciple of measuring the angle of free vibration of a suspendedmass with respect to the casing of the gyroscope. The idea behindthe methodology is as follows. If the mass is given an initialcondition so that it vibrates in a known direction, the angle, θ ,of the gyroscope in the global frame can be determined bykeeping track of the angle, α , of the direction of vibration of themass in the local frame (see Fig. 2). When the gyroscopeexperiences a rotation in the global frame the mass continues tovibrate in the same direction with respect to the global frame.

Fig. 2. Mass motion with initial x-axis oscillation

Since this method relies on the free vibration of the mass,the presence of any damping will drive the motion of the mass tozero. To overcome the affects of damping, energy needs to beadded to the system. The energy will be added by usingactuators to apply small forces to the x and y modes proportionalto the x and y velocities respectively. Such forces couldcounteract the damping in the system. If it were possible toselect forces that would counteract the damping exactly (i.e.addition of negative damping), the mass would sustain freevibration indefinitely. However it is not possible to select suchforces because of the changing damping of the system,manufacturing imperfections, temperature fluctuations and thepresence of other energy loss mechanisms in the system. If thenegative damping were chosen too large, the amplitude of motionwill become unbounded, and if it were chosen too small, theamplitude of motion will still decay. In order to overcome thissituation, a control loop is used to keep the total energy of thesystem constant. The current energy of the system is comparedto the desired energy of the system and an energy error term iscalculated. The forces counteracting the damping of the systemare continuously adjusted using the energy error in order to keepthe energy of the system constant:

3

xym

yyk

xm

xxkExF DDD

−−−−= 2

22

22

22

2α (3)

yym

yyk

xm

xxkEyF DDD

−−−−= 2

22

22

22

2α (4)

where E is the energy set point and

+++ 2

2

2

2

2

2

2

2y

my

ykx

mxxk

DD represents the instantaneous

energy of the system.In this section, we first show that this control system will

ensure that the mass vibrates with constant energy approximatlyequal to E. Equation (1) and (2) are modified to include thespecific forces xF and yF applied by the control system:

yyxxmxc

xxx DDDDDDDD θθθω 222 +++−−=

xym

yykx

mxxk

Em

DDD

−−−−+ 2

22

22

22

2

α (5)

xxyym

ycyyy DDDDDDDD θθθω 222 −−+−−=

yym

yykx

mxxk

Em

DDD

−−−−+ 2

22

22

22

2

α (6)

where α is a constant positive gain. Consider the following energyfunction for this system:

22

122

122

122

1yykxxkymxmV +++= DD (7)

The time derivative of the energy function is:

][][] yykymyxxkxmxyyykxxxkyymxxmV +++=+++= DDDDDDDDDDDDDDD (8)

Substituting equations (5) and (6) into equation (8) yields:

]2222

2[ 22222 xymyk

xmxkEymmymxxcxV yxx DDDDDDDDDDD

−−−−++++−= αθθθ

]2222

2[ 22222 yymyk

xmxk

Exmmxmyycy yxy DDDDDDDDDD

−−−−+−−+−+ αθθθ

(9)Typical parameter values in MEMS gyroscopes are nω = 40k

rad/s and amplitudes of x and y of about 5 µ m. If the maximumangular rate is 60 rad/s and maximum angular acceleration is 6000rad/s2, it can be seen from equation (9) that xxc D and yyc D are

much greater than θθθ DDDD mymymx ,2,2 and θDDmx . Hence theequation (9) can be approximated by

( ) ( )yVEyycyxVExxcxV DDDDDDD −+−+−+−= αα [][ (10)

This can further be re-written as

EyxycyxcxVyxV )22(]22[)22( DDDDDDD +++−=++ αα (11)

Due to the fact that )22( yx DD +α is always greater than zero, theenergy function, V, has stable dynamics. The steady state of theenergy function, ssV , is

)22(

)22(

yx

ycyxcxEssV

DD

DD

+

+−=

α (12)

By selecting an appropriate positive α , the steady state ofthe energy function can be made close to the desired energy.Thus the energy of the x and y modes will always remainbounded implying that yxyx DD,,, will also remain bounded.

Figure 3 below shows the performance of the anglemeasurement gyroscope. In this simulation, the gyroscope frameexperiences a ramp angular motion from zero to 45 degrees afterwhich the angle remains at 45 degrees as shown in Fig. 3a. InFig. 3c, the line of oscillation of the gyroscope moves from 0degrees to 45 degrees in the local gyroscope frame. From theangle of this line of oscillation, the angular motion of the casingcan be inferred. The line of oscillation in the global frame (asshown in Fig. 3d) always remains at 0 degrees. The physicalparameters used are typical values for MEMS gyroscopes [9].The suspended mass is taken to be 0.9425e-6 g. The springstiffness for both x and y springs is 4180.8 and the dampingcoefficient is 0.2.

(a) (b)

(c) (d)Fig. 3 Angle measurement (a) angular motion from 0 to 45degrees (b) x and y motion (c) motion in gyroscope frame

(d) motion in global frame

4. ANGLE MEASUREMENT IN THE PRESENCE OFIMPERFECTION

A real MEMS gyroscope fabricated through manymicrofabrication steps will not be the ideal 2-dof spring-masssystem shown in Fig. 1. Common imperfections in fabricationare reflected as asymmetry in spring stiffness and damping. Also,cross-coupling forces occur along the two local axes when themass orientation is not the same as that of the gyroscope frame.

4

As shown in the following sub-sections, the presence of even smallimperfections can have a huge impact on the performance of theabove angle-measurement system. These imperfections must behandled before this theoretical sensing concept can be convertedinto a reliable practical sensor.

4.1 Mismatched spring stiffness compensationIf the spring stiffness is the same in both axes, x and y, the

system is said to be isotropic. When the isotropic oscillator isallowed to freely oscillate, the precession of the straight line ofoscillation provides a measure of the angle of rotation, as we haveseen in the previous section. However, if the spring stiffness is notthe same in both directions, the mass no longer oscillates in astraight line even for a constant angle input. Instead the line ofoscillation keeps rotating around the frame with constant x and yamplitudes. Figure 4 shows the simulation results in both the localgyroscope frame and the global frame when there is a small 1%difference in the values of xk and yk . The angle input for the

gyroscope casing is the same motion from 0 to 45 degrees asdiscussed in the previous section.

(a) (b)Fig. 4 Anisotropic oscillating

(a) motion in gyroscope frame(b) motion in global frame

Thus even a small 1% difference between xk and yk causes

the angle of free vibration to keep changing and never reach steadystate.

There will always be some difference between xk and yk due

to manufacturing imperfections. One can attempt to compensatefor the asymmetry actively using the xF and yF actuators.

However, a direct compensation of the type

xxkrefkxF )( +−= (13)

yykrefkyF )( +−= (14)

cannot solve the problem due to noise and bias in the calculatedperiodic stiffness signals, xk and yk , as well as in the force

generation of xF and yF .

The following PI control law is proposed instead.

[ ])ˆˆ()ˆˆ( ykxkpkykxkIkxxF DDD −+−= (15)

[ ])ˆˆ()ˆˆ( ykxkpkykxkIkyyF DDD −+−−= (16)

where xk and yk are the measured stiffness values calculated

from x and y measurement in real time.As shown in Fig. 5, the signal frequencies can be calculated

by measuring the time interval of each half cycle:

iti

πω = , ,..3,2,1=i (17)

Then, the stiffness values can be calculated from the frequenciesas

2ˆimik ω= (18)

where k is a periodic signal. Thus, this measuring scheme givesus the zero bias signal, or the real stiffness value will be equal tothe mean value of the calculated signal as follow,

N

N

i ik

realk∑== 1

ˆ (19)

Fig. 5 Time interval measuring for stiffness value

The forces tune both stiffness values to be equal to eachother by integration of the difference between xk and yk .

Figure 6 shows the result of using the above control law inthe case where there is a 10% difference between xk and yk . It

can be seen that the control law ensures that the mass vibratesalong a steady state straight line. The angle of rotation can beinferred from the angle of oscillation.

(a) (b)Fig. 6. Stiffness tuning control, ky=1.1*kx

(a) motion in gyroscope frame(b) motion in global frame

4.2 Addressing of cross-coupling spring stiffness

5

The spring stiffness along the two axes can change due torotation of the suspended mass with respect to the gyroscope frame(Fig. 7). The ability of the springs to transmit forces perpendicularto their axis results in cross-coupling terms. The third equation ofmotion describing the rotation of the mass with respect to thecasing is the following [9],

θφφφ IkI −−= (20)

Fig. 7 Schematic representation of a simple mass spring systemwith two orthogonal modes[9]

When the mass rotates with respect to the frame the springstiffness in both the x and y axes is changed. For the isotropicsystem, the equations of motion can be written as

yyymxyk

xxx DDDDDDDD θθθωξω 2)22(2 ++−=−++ (21)

xxxmxyk

yyy DDDDDDDD θθθωξω 2)22(2 −−−=−++ (22)

where φγccxyk = , ( ) 222xykmxk −= ω and

m

k=ω

The cross-coupling forces result in ellipticity of the nominalstraight-line motion and in frequency change, as can be seen in Fig.8.

Fig. 8 Cross-coupling stiffness effect

We can use the ability to simultaneously measure angular-rateand the observer of Section 6 to estimate the measured angleaccurately in the presence of the above imperfection. The observer(discussed in section 6) ensures that the angle can be accuratelyestimated in spite of the elliptical motion.

5. COMBINED ANGLE AND ANGULAR-RATEMEASUREMENT

It is possible to combine the angular rate use of a gyroscopewith its ability to also measure angle. In this case the x-axis forceis chosen to be a sum of the feedback force defined by equation(3) at steady state and an additional sinusoidal excitation at adriven frequency dω :

)sin(22

22

ˆ2

22

2

ˆtddFxy

myyk

xm

xxkExF ωα +−−−−=

DDD

(23)

The y-axis force remains the same as the feedback forceused for angle measurement:

yym

yykx

mxxk

EyF DDD

−−−−= 2

22

2

ˆ2

22

2

ˆα (24)

In the presence of an angular rate θD , the sinusoidal x-axisexcitation term in equation (23) causes vibrations along the y-axis at the excitation frequency dω . The value of the angular

rate can be obtained by demodulating the y axis motion at thedriving frequency.

In implementing the above combined system, the x and ysignals coming from the gyroscope need to go through a separateband pass filter at the natural frequency in order to determine theangle value. When running the gyroscope at both its naturalfrequency and the driving frequency the relative amplitude ofoscillation at the two frequencies becomes important. It isnecessary to ensure that the driving frequency, dω , is

sufficiently far away from the natural frequency of the system.Roughly it was found that the free vibration signal provided agood measurement of the angular motion of the gyroscope aslong as the driving frequency remained about an order ofmagnitude different from the natural frequency.

Fig. 9 Angular rate signal.

Fig. 10 Angle signal.

6

The results of running the gyroscope with both angular ratefunctionality and angle functionality are shown in Figures 9 and 10(the input signal was a sine wave with an amplitude of 1 radian anda frequency of 100 rad/sec).

6. OBSERVER ESTIMATION

Since angle and angular rate are both measured by thegyroscope, a filter can be used to exploit the angular ratemeasurement so as to improve the absolute angle estimates.Consider the following observer.

)ˆ(ˆ θθθθ −+= measKmeas (25)

where measθ is the measured angle, measθC is the measured

angular rate, θ is the estimate of the angle, θ is the estimate of theangular rate, and K is the observer gain. A small value for K willmean that the observer estimate is primarily obtained fromintegrating the angular rate with the angle measurement errorserving mainly to correct the drift. A high value of K will meanthat the observer relies more on the direct angle measurement withthe angular rate measurement playing a role only for very highfrequency measurements.

Based on the ellipticity encountered in the angle measurementscheme and the drift present in the angular rate measurementscheme, the gain K can be chosen in an optimal way so as to obtainthe best possible estimates of angular motion.

Angular estimates obtained using the observer are shown inFig. 11.

Fig. 11. Observer estimation

7. CONCLUSIONS

MEMS gyroscope are typically designed to measure angularrate of rotation. A measurement of the angle itself is useful in manyapplications but cannot be obtained by integrating the angular ratedue to the presence of bias errors which cause a drift. This paperpresented an innovative design for a vibrating gyroscope that candirectly measure both angle and angular rate. The design was basedon the principle of measuring the angle of free vibration of asuspended mass with respect to the casing of the gyroscope.Several critical challenges had to be handled before the theoreticalsensing concept could be converted into a reliable practical sensor.These included compensating for the presence of dissipative forces,mismatched springs, cross-axis stiffness and transmission of rotary

torque. These challenges were addressed by the development ofa composite nonlinear feedback control system that compensatedfor each of the above effects and ensured that the mass continuedto behave as a freely vibrating structure. Theoretical analysis andsimulation results presented in the paper showed that thegyroscope can accurately measure both angle and angular rate forlow bandwidth applications.

References

[1] A. Lawrence, Modern Inertial Technology, Springer-Verlag,New York, 1998.

[2] D.S. Eddy and D.R. Sparks, “Application of MEMStechnology in automotive sensors and actuators,”Proceedings of the IEEE, Vol. 86, No. 8, pp.1747-1755,August 1998.

[3] R. Rajamani, H.S. Tan, B.Law and W.B. Zhang,“Demonstration of Integrated Lateral and LongitudinalControl for the Operation of Automated Vehicles inPlatoons”, IEEE Transactions on Control SystemsTechnology, Vol. 8, No. 4, pp. 695-708, July 2000.

[4] Tseng, E.H., Ashrafi, B., Madau, D., Brown, T.A. andRecker, D., “The Development of Vehicle Stability Control atFord”, IEEE/ASME Transactions on Mechatronics, Vol. 4,No. 3, pp. 223-234, September, 1999.

[5] W.A. Clark, R.T. Howe, et. al., “Micro-machined Z-axisvibratory Rate Gyroscope,” Technical Digest of the SolidState Sensors and Actuators Workshop, pp.283-287, 1996.

[6] J. Bernstein, S. Cho and et. al., “A Micro-Machined Comb-Drive Tuning Fork Rate Gyroscope,” Proceedings of theWorkshop on Microelectromechanical Systems, pp. 143-148,1993.

[7] T. Juneau, A.P. Pisano, et. al., “Dual Axis Operation of aMicromachined Rate Gyroscope”, Tranducers’97, Vol. 2, pp.883-886, 1997.

[8] M. Putty and K. Najafi, “A Micro-Machined Vibrating RingGyroscope”, Technical Digest of the Solid-State Sensors andActuators Workshop, June 13-16, pp. 213-220, 1994.

[9] M. Kranz and G.K. Fedder, “Design, Simulation, andImplementation of Two Novel Micromechanical Vibratory-Rate Gyroscopes”, Report, Department of Electrical andComputer Engineering, Carnegie Mellon University, May,1998.

[10] D.A. Koester, et. al., “Multi-User MEMS Processes(MUMPs) Introduction and Design Rules,” MCNC MEMSTechnology Applications Center, 3021 Cornwallis Road,Research Traingle Park, NC 27709, rev. 3, Oct. 1994, 39pages.

[11] J.W. Gardner, Microsensors: principles and applications,Wiley, NY, 1994.

[12] Yoichi Mochida, Masaya Tamura and Kuniki Ohwada, “Amicromachined vibrating rate gyroscope with independentbeams for the drive and detection modes,” Sensors andActuators, A: Physical 80, pp. 170-178, 2000.

[13] A.M. Shkel, R. Horowitz, A.A. Seshia, S. Park, and R.T.Howe, ”Dynamics and control of micromachinedgyroscopes,” Proceedings of the 1999 American ControlConference, Vol. 3, pp. 2119 –2124, 1999.

[14] J.J. Slotine and W. Li, Applied Nonlinear Control, PrenticeHall, 1991.