Patent Publication Number: US-10317210-B2

Title: Whole angle MEMS gyroscope on hexagonal crystal substrate

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     The presented application claims priority under 35 U.S.C. § 119(e) to U.S. Provisional Application Ser. No. 62/302,474, filed on Mar. 2, 2016 entitled WHOLE ANGLE MEMS GYROSCOPE and U.S. Provisional Application Ser. No. 62/165,365, filed on May 22, 2015, entitled WHOLE ANGLE MEMS GYROSCOPE, both of which are hereby incorporated herein by reference. 
    
    
     GOVERNMENT LICENSE RIGHTS 
     This invention was made with government support under N66001-16-C-4020 awarded by SPAWAR SYSTEMS CENTER PACIFIC on behalf of DARPA. The government has certain rights in the invention. 
    
    
     BACKGROUND OF THE INVENTION 
     The present disclosure generally relates to vibrating structure gyroscopes and more specifically to Microelectromechanical System (MEMS) based vibrating structure gyroscopes. Vibrating structure gyroscopes utilize solid-state resonators, of different shapes, to measure orientation or rotation rate based on the principle that a vibrating object tends to continue vibrating (i.e., oscillate) in a fixed orientation in space as its support rotates, and any vibrational deviation of the object can be used to derive a change in direction. Vibrating structure gyroscopes may be manufactured with MEMS based technology. For example, vibrating structure gyroscopes may be fabricated on silicon or glass wafers using a sequence of steps including photolithography, etching and deposition, or any other MEMS based technology. 
     Vibrational deviations in a resonator of a MEMS based gyroscope may be caused by a Coriolis force. For example, a mass moving at a given velocity will experience Coriolis acceleration when the mass is also rotated with an angular velocity. The Coriolis acceleration is perpendicular to the velocity and the angular velocity. The Coriolis acceleration vector is given by a c =−2(v×Ω), where v is the velocity vector and Ω is the angular velocity vector. Coriolis acceleration is thus indicative of the angular velocity of rotation. 
     Many MEMS based gyroscopes are configured to operate in a rate mode. In a rate mode of operation, vibration of one axis (i.e., a drive axis) of a MEMS based gyroscope is driven at a fixed amplitude in a closed loop while Coriolis-induced motion is read out on the other axis (i.e., a sense axis). In such a rate mode, the amplitude of the Coriolis-induced motion read out on the sense axis is indicative of a rate of angular movement of the gyroscope. Rate mode operated gyroscopes are limited in that the Coriolis-induced motion measurements are limited by the dynamic range of the open-loop sense axis. For fast movements of the gyroscope, the open-loop sense axis may not be able to “keep up” with the movement of the gyroscope. In addition, spring non-linearities at high rates of rotation may cause errors. Some MEMS based gyroscopes operated in rate mode attempt to avoid these problems by also operating the sense axis in a closed loop and monitoring the level of force required to maintain the amplitude of the sense axis at a fixed level. However, such gyroscopes are limited by the closed sense loop bandwidth and the maximum force capable of being exerted by the rebalance. 
     One example of a MEMS based gyroscope configured to operate in a rate mode is a Tuning Fork (TF) gyroscope. A Tuning Fork gyroscope includes a pair of relatively large lumped-element proof masses that are driven to oscillate, in an in-plane axis, with equal amplitude but in opposite directions. When a TF gyroscope is rotated, the Coriolis force creates an orthogonal vibration (i.e., an out-of-plane vibration) in the proof masses that can be sensed by a variety of mechanisms. By monitoring out-of-plane vibrations of the proof masses, the rate of rotation of the TF gyroscope can be determined. 
     Another mode of operation for MEMS based gyroscopes is a whole angle mode (otherwise known as an integrating or rate integrating mode). In a whole angle mode of operation, two axes, having identical frequency and damping, are coupled by Coriolis motion. The axes are driven such that the total vibrational amplitude of the two axes is sustained, but the distribution of energy between the two axes is allowed to change freely. Accordingly, a Coriolis force causes energy to be transferred from one axis to the other as the gyroscope rotates. By measuring the distribution of energy between the axes, an angle of rotation (with respect to a starting angle) can be read out. As energy can freely transfer from one axis to the other in a MEMS based gyroscope operating in whole angle mode, there is no limit on the rate at which the axes can transfer motion. As such, whole angle operating gyroscopes avoid the dynamic range issues discussed above with regard to rate mode operating gyroscopes and typically provide a higher level of performance and higher bias stability. 
     One important requirement of a whole angle operating MEMS based gyroscope is that the two modes be identical (i.e., degenerate) with regard to frequency and damping. If the frequencies differ substantially, a Coriolis force caused by rotation of the gyroscope will not be sufficient to transfer energy from one mode to the other and the vibration will stay “locked” to a single axis. This will interfere with the free transfer of motion between modes and the free precession of the mode shape of the gyroscope. A whole angle operating MEMS based gyroscopes must therefore be designed and fabricated with exceptional symmetry and with mode structures that are insensitive to expected fabrication variations. In addition, it is also typically desired for whole angle operating MEMS based gyroscopes to provide low damping (i.e., long ring down time) and matched damping for principle axes. This is because low overall damping correlates to low damping differences between the two axes and on-axis damping may result in gyroscope bias when drive forcers are misaligned. A mismatch (or mismatch drift) may result in a bias (or bias drift). 
     Traditional whole angle MEMS based gyroscopes include an axially or cylindrically symmetric and continuous structure that is driven to excite two vibratory modes of the structure (i.e., an n=2 vibratory mode where two points of the ring are moving away from the center of the ring while two other points of the ring are moving toward the center of the ring). Rotation of the gyroscope results in a Coriolis force that causes movement (i.e., either inward or outward motion) of other points of the symmetric structure. By monitoring the movement of the symmetric structure in two in-plane axes, the angle of rotation of the gyroscope can be determined. 
     One common example of a whole angle operating gyroscope is a Hemispherical Resonator Gyroscope (HRG) (otherwise known as a wine-glass gyroscope). An HRG includes a thin hemispherical shell, anchored by a stem. The shell is driven to a flexural resonance and a gyroscopic effect is obtained from the inertial property of resulting flexural standing waves. An HRG is typically reliable and accurate; however, they are also typically large and costly. 
     Another example of a whole angle operating gyroscope is a ring gyroscope. Ring gyroscopes include axially symmetric and continuous rings that are driven in an n=2 vibratory mode, as discussed above. The movement of the ring is monitored to determine an angle of rotation of the ring gyroscope. The performance of such ring gyroscopes is limited in that due to the limited mass of the rings, the sensitivity of the gyroscope is relatively low and the bias instability is relatively high. 
     SUMMARY 
     A new MEMS based gyroscope design is provided that combines the best features of a lumped-element TF gyroscope and a rotationally symmetric gyroscope. Certain embodiments efficiently use relatively large masses (e.g., like a TF gyroscope) supported by relatively weak flexures to provide low damping and hence high sensitivity while maintaining an eight-fold symmetry conducive to the n=2 vibratory mode used in most whole angle based gyroscopes to provide high dynamic range. As discussed in more detail below, certain embodiments are capable of operating in both rate and whole angle mode, are low cost, and are easily fabricated. 
     One aspect of the present disclosure is directed to a gyroscope comprising an axially symmetric structure, and a plurality of transducers, each configured to perform at least one of driving and sensing motion of the axially symmetric structure, wherein the plurality of transducers is configured to drive the axially symmetric structure in at least a first vibratory mode and a second vibratory mode, and wherein the gyroscope is implemented on a hexagonal crystal-based substrate. 
     According to one embodiment, each transducer is located at a periphery of the axially symmetric structure. In one embodiment, at least one transducer is configured to electrostatically drive motion of the axially symmetric structure. In another embodiment, at least one transducer is configured to magnetically drive motion of the axially symmetric structure. In one embodiment, at least one transducer is configured to optically drive motion of the axially symmetric structure. In another embodiment, at least one transducer is configured to piezoelectrically drive motion of the axially symmetric structure. In one embodiment, at least one transducer is configured to thermally drive motion of the axially symmetric structure. 
     According to another embodiment, the plurality of transducers is further configured to drive the axially symmetric structure in an n=2 vibratory mode. In one embodiment, the first vibratory mode and the second vibratory mode are 45° apart. In another embodiment, the plurality of transducers is further configured to drive motion of the axially symmetric structure at a fixed amplitude in the first vibratory mode and to sense motion of the axially symmetric structure in the second vibratory mode. In one embodiment, the gyroscope further comprises a controller coupled to the plurality of transducers, wherein the plurality of transducers is further configured to provide signals to the controller based on the sensed motion of the axially symmetric structure in the second vibratory mode, and wherein the controller is configured to calculate a rate of rotation of the gyroscope based on the signals. 
     According to one embodiment, the plurality of transducers is further configured to drive motion of the axially symmetric structure such that a total vibrational energy is maintained across the first vibratory mode and the second vibratory mode and to sense a distribution of energy between the first vibratory mode and the second vibratory mode. In one embodiment, the gyroscope further comprises a controller coupled to the plurality of transducers, wherein the plurality of transducers is further configured to provide signals to the controller based on the sensed distribution of motion between the first vibratory mode and the second vibratory mode, and wherein the controller is configured to calculate an angle of rotation of the gyroscope based on the signals. 
     According to another embodiment, the axially symmetric structure comprises a central anchor, a plurality of internal flexures, a plurality of masses, each mass coupled to the central anchor via at least one of the plurality of internal flexures and configured to translate in a plane of the gyroscope, and a plurality of mass-to-mass couplers, each mass-to-mass coupler coupled between two adjacent masses of the plurality of masses, wherein the plurality of transducers is configured to drive the plurality of masses in at least the first vibratory mode and the second vibratory mode. In one embodiment, each mass-to-mass coupler includes a bar coupled to each adjacent mass via a flexural hinge, wherein the bar is configured to operate such that circumferential motion of one of the two adjacent masses of the plurality of masses to which it is coupled depends on radial motion of the other one of the two adjacent masses. In one embodiment, the plurality of masses includes a plurality of wedge-shaped masses. 
     According to one embodiment, the gyroscope further comprises a plurality of outside anchors, a plurality of outside shuttles, each located at a periphery of a corresponding one of the plurality of masses, and a plurality of outside flexures, wherein each mass of the plurality of masses is suspended between the central anchor and the plurality of outside anchors via the plurality of internal flexures and the plurality of outside flexures, and wherein each one of the plurality of outside shuttles is configured to restrict rotation of its corresponding one of the plurality of masses. In one embodiment, each one of the plurality of outside shuttles is further configured to decouple x- and y-motion of its corresponding one of the plurality of masses. In another embodiment, each one of the plurality of outside shuttles is further configured to prevent force from being applied circumferentially to its corresponding one of the plurality of masses. 
     According to another embodiment, the gyroscope further comprises a plurality of internal shuttles, each one of the plurality of internal shuttles coupled between the central anchor and a corresponding one of the plurality of masses and configured to restrict rotation of its corresponding one of the plurality of masses. In one embodiment, each one of the plurality of internal shuttles is further configured to decouple x- and y-motion of its corresponding one of the plurality of masses. 
     According to one embodiment, the gyroscope further comprises a plurality of angled electrodes, each angled electrode coupled to a corresponding one of the plurality of masses and configured to trim the cross spring term of the corresponding one of the plurality of masses. In one embodiment, in trimming the cross spring term of its corresponding one of the plurality of masses, each angled electrode is configured to generate a radial force component in the second vibratory mode in response to a circumferential motion of its corresponding one of the plurality of masses in the first vibratory mode, and generate a circumferential force component in the second vibratory mode in response to a radial motion of its corresponding one of the plurality of masses in the first vibratory mode, wherein the radial force component and the circumferential force component are configured to either assist or oppose the vibration of the plurality of masses in the second vibratory mode to trim the cross spring term. 
     According to another embodiment, the hexagonal crystal-based substrate is a Silicon Carbide (SiC) based substrate. In one embodiment, the SiC based substrate is a 4H—SiC based substrate. In another embodiment, the hexagonal crystal-based substrate is oriented in the (0001) plane of a hexagonal crystal. 
     According to one embodiment, the gyroscope has a thickness, and wherein the plurality of internal flexures includes flexures having a width that is at least five times narrower than the thickness of the gyroscope. In one embodiment, the gyroscope is a Microelectromechanical System (MEMS) based gyroscope. In another embodiment, the axially symmetric structure is one of a ring and a disk. 
     Another aspect of the present disclosure is directed to a gyroscope comprising an axially symmetric structure, means for driving the axially symmetric structure in an n=2 vibratory mode of a rotationally symmetric gyroscope, and means for operating the gyroscope in one of a rate mode of operation and a whole angle mode of operation, wherein the gyroscope is implemented on a hexagonal crystal-based substrate. 
     According to one embodiment, the gyroscope further comprises means for decoupling radial and circumferential motion of each one of the plurality of masses. In another embodiment, the gyroscope further comprises means for trimming a cross spring term of the gyroscope. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Various aspects of at least one embodiment are discussed below with reference to the accompanying figures, which are not intended to be drawn to scale. The figures are included to provide illustration and a further understanding of the various aspects and embodiments, and are incorporated in and constitute a part of this specification, but are not intended as a definition of the limits of the invention. In the figures, each identical or nearly identical component that is illustrated in various figures is represented by a like numeral. For purposes of clarity, not every component may be labeled in every figure. In the figures: 
         FIG. 1  is a diagram illustrating one embodiment of a MEMS based gyroscope according to aspects of the present invention; 
         FIG. 2  is a diagram of an internal flexure according to aspects of the present invention; 
         FIG. 3  is a diagram of an outer flexure according to aspects of the present invention; 
         FIG. 4  is a diagram illustrating two degenerate vibratory modes of a MEMS based gyroscope according to aspects of the present invention; 
         FIG. 5A  is a diagram illustrating one embodiment of a K xy  trim electrode and a mass-to-mass coupler according to aspects of the present invention; 
         FIG. 5B  is a diagram illustrating directional planes of a cubic crystal structure according to aspects of the present invention; 
         FIG. 5C  is a diagram illustrating one embodiment of the orientation of a MEMS based gyroscope relative to the crystallographic orientation of a cubic crystal structure according to aspects of the present invention; 
         FIG. 5D  is a diagram illustrating another embodiment of the orientation of a MEMS based gyroscope relative to the crystallographic orientation of a cubic crystal structure according to aspects of the present invention; 
         FIG. 5E  is a diagram illustrating directional planes of a hexagonal crystal structure according to aspects of the present invention according to aspects of the present invention; 
         FIG. 5F  is a diagram illustrating one embodiment of the orientation of a MEMS based gyroscope relative to the crystallographic orientation of a hexagonal crystal structure according to aspects of the present invention; 
         FIG. 5G  is a graph illustrating the variation in Young&#39;s modulus between Silicon-Carbide (4H—SiC) and Silicon (Si).  FIG. 6A  is a diagram of another embodiment of a MEMS based gyroscope according to aspects of the present invention; 
         FIG. 6A  is a diagram of another embodiment of a MEMS based gyroscope according to aspects of the present invention; 
         FIG. 6B  is a diagram of another embodiment of a MEMS based gyroscope according to aspects of the present invention; 
         FIG. 6C  is a diagram illustrating further detail of the MEMS based gyroscope shown in  FIG. 6B  according to aspects of the present invention; 
         FIG. 6D  is a diagram of another embodiment of a MEMS based gyroscope according to aspects of the present invention; 
         FIG. 6E  is a diagram illustrating further detail of the MEMS based gyroscope shown in  FIG. 6D  according to aspects of the present invention; 
         FIG. 6F  is a diagram of another embodiment of a MEMS based gyroscope according to aspects of the present invention; 
         FIG. 7A  is a diagram of another embodiment of a MEMS based gyroscope according to aspects of the present invention; 
         FIG. 7B  is a diagram of another embodiment of a MEMS based gyroscope according to aspects of the present invention; 
         FIG. 8  is a diagram of another embodiment of a MEMS based gyroscope according to aspects of the present invention; 
         FIG. 9  is a diagram illustrating further detail of the MEMS based gyroscope shown in  FIG. 8  according to aspects of the present invention; 
         FIG. 10  is a diagram illustrating further detail of the MEMS based gyroscope shown in  FIG. 8  according to aspects of the present invention; 
         FIG. 11  is a diagram of another embodiment of a MEMS based gyroscope according to aspects of the present invention; 
         FIG. 12  is a diagram of another embodiment of a MEMS based gyroscope according to aspects of the present invention; 
         FIG. 13  is a diagram of another embodiment of a MEMS based gyroscope according to aspects of the present invention; 
         FIG. 14  is a diagram of another embodiment of a MEMS based gyroscope according to aspects of the present invention; 
         FIG. 15  is a diagram of another embodiment of a MEMS based gyroscope according to aspects of the present invention; 
         FIG. 16  is a diagram illustrating further detail of the MEMS based gyroscope shown in  FIG. 15  according to aspects of the present invention; 
         FIG. 17A  is a diagram of another embodiment of a MEMS based gyroscope according to aspects of the present invention; 
         FIG. 17B  is a diagram illustrating further detail of the MEMS based gyroscope shown in  FIG. 17A  according to aspects of the present invention; 
         FIG. 17C  is a diagram illustrating further detail of the MEMS based gyroscope shown in  FIG. 17A  according to aspects of the present invention; 
         FIG. 18  is a diagram of another embodiment of a MEMS based gyroscope according to aspects of the present invention; and 
         FIG. 19  is a diagram illustrating further detail of the MEMS based gyroscope shown in  FIG. 18  according to aspects of the present invention. 
     
    
    
     DETAILED DESCRIPTION 
     Aspects and embodiments described herein provide a MEMS based gyroscope design that combines the best features of a TF gyroscope and a rotationally symmetric gyroscope. Certain embodiments efficiently use relatively large masses (e.g., similar to a TF gyroscope) to provide high sensitivity while maintaining an eight-fold symmetry conducive to the n=2 vibratory mode used in most whole angle based gyroscopes to provide high dynamic range. Certain embodiments are capable of operating in both rate and whole angle mode, may be low cost, and may be easily fabricated. 
     It is to be appreciated that embodiments of the methods and apparatuses discussed herein are not limited in application to the details of construction and the arrangement of components set forth in the following description or illustrated in the accompanying drawings. The methods and apparatuses are capable of implementation in other embodiments and of being practiced or of being carried out in various ways. Examples of specific implementations are provided herein for illustrative purposes only and are not intended to be limiting. Also, the phraseology and terminology used herein is for the purpose of description and should not be regarded as limiting. The use herein of “including,” “comprising,” “having,” “containing,” “involving,” and variations thereof is meant to encompass the items listed thereafter and equivalents thereof as well as additional items. References to “or” may be construed as inclusive so that any terms described using “or” may indicate any of a single, more than one, and all of the described terms. 
     Referring to  FIG. 1 , there is illustrated one embodiment of a MEMS based gyroscope  100  configured according to aspects of the present disclosure. The gyroscope  100  includes eight wedge-shaped masses  102 , internal flexures  104 , outer flexures  106 , a central anchor  110 , outer anchors  114 , outer shuttles  112 , inner shuttles  113 , and mass-to-mass couplers  108 . 
     Each mass  102  is suspended between an inner shuttle  113  and an outer shuttle  112  via internal flexures  104  and outer flexures  106  and is configured to translate in the plane of the gyroscope  100  on the flexures  104 ,  106 . An internal flexure  104  is shown in greater detail in  FIG. 2  and an outer flexure  106  is shown in greater detail in  FIG. 3 . According to one embodiment (shown in  FIGS. 2 and 3 ) where the gyroscope  100  is fabricated on a silicon wafer, the internal flexures  104  and outer flexures  106  include areas of silicon  105  defined by slots  107  (i.e., areas of empty space) in the silicon that are configured to allow a coupled mass  102  to vibrate (i.e., translate) in the plane of the gyroscope  100 . In other embodiments, the flexures  104 ,  106  may be constructed differently. The central anchor  110  and the outer anchors  114  are configured to support the structure. The mass-to-mass couplers  108  are coupled between adjacent masses  102 . 
     Each internal shuttle  113  and outer shuttle  112  is configured to enforce radial forcing on its corresponding mass  102  and restrict rotation of its corresponding mass  102  by decoupling x- (i.e., radial) and y- (i.e., circumferential) motion. For example, each outer shuttle  112  is configured to only move radially (i.e., in a direction  115  in towards the anchor or out away from the anchor  110 ) and each inner shuttle  113  is configured to move only circumferentially (i.e., in direction  119 ) around the anchor  110 . Accordingly, each mass  102  may only translate in the plane of the gyroscope  100  (i.e. move radially and/or circumferentially in the plane of the gyroscope  100 ). By decoupling the x- and y-motion, the stiffness of each direction may be designed independently. Also, out-of-plane motion may be suppressed by utilizing flexures of high aspect ratio (i.e. whose width is at least five times narrower than the thickness of the flexure for the whole planar gyroscope  100 ). For example, a gyroscope that is 100 microns in thickness may utilize flexures which are five times narrower (e.g., 20 microns wide), ten times narrower (e.g., 10 microns wide), or twenty times narrower (5 microns wide). In addition, each outer shuttle  112  may also reduce errors caused by misaligned drive/sense transducer  116  electrodes by being stiff in the direction of the misaligned force component (i.e., by only moving radially in and out, the outer shuttle  112  reduces any erroneous circumferential forces on the mass  102  from the drive). 
     The gyroscope  100  also includes drive/sense transducers  116 . For the ease of illustration, only one drive/sense transducer  116  is shown in  FIG. 1 ; however, a drive/sense transducer  116  is located at the periphery of each mass  102  in the gyroscope  100 . Each drive/sense transducer  116  is capable of driving motion of its corresponding mass  102  and of sensing motion of its corresponding mass  102 . For example, in at least one embodiment, each drive/sense transducer  116  is an electrostatic transducer (e.g., a variable capacitor) that includes an electrode positioned at the periphery of a corresponding mass  102  and an electrode positioned on a corresponding outer shuttle  112 . When a voltage is applied to the electrode at the periphery of the mass  102 , motion of the mass is electrostatically driven. Each drive/sense transducer  116  is also configured to sense motion of its corresponding mass and provide a signal indicative of the motion to an external controller/processor  117 . According to one embodiment, the drive/sense transducers  116  are clapper or comb drives; however, in other embodiments, any other appropriate type of circuit capable of driving and sensing motion of a mass  102  may be utilized. For example, in other embodiments, magnetic, piezoelectric, thermal, or optical based transducers may be utilized. 
     As discussed above, the drive-sense transducers  116  are located at the periphery of each mass  102 ; however, in other embodiments, a transducer  116  may be located at any other position adjacent a mass  102 . In addition, according to at least one embodiment, each drive-sense transducer  116  is located internal to a mass  102 . 
     As also discussed above, a single transducer  116  is associated with each mass  102 ; however, in other embodiments, each mass  102  may be associated with more than one transducer  116 . For example, in at least one embodiment, each mass  102  is associated with a first transducer that drives motion of the mass  102  and a second transducer that senses motion of the mass  102 . In another embodiment, each mass  102  also includes a third transducer that is utilized for tuning the mass  102  (e.g., tuning the radial spring constant of the mass  102 ). In other embodiments, the transducer(s)  116  associated with each mass  102  may be configured in any appropriate way to drive motion of the mass  102 , sense motion of the mass  102 , and/or tune the mass  102 . 
     As the masses  102  in the gyroscope  100  vibrate, the mass-to-mass couplers  108  couple the motion of the masses  102  together, resulting in an n=2 vibratory mode resembling that of a rotationally symmetric gyroscope (e.g., a ring, disc, or hemispherical gyroscope).  FIG. 4  is a diagram illustrating two degenerate vibratory modes (a first vibratory mode  400  and a second vibratory mode  420 ) of the gyroscope  100 . Considering the first vibratory mode  400 , if the gyroscope  100  is not rotating, two of the masses  405  are translating inwardly and two of the masses  403  are translating outwardly. As shown in  FIG. 4 , the first n=2 vibratory mode shape  400  of the gyroscope is an ellipse  402   a . Due to the mass-to-mass couplers  108 , the other four masses  407 ,  409  are translating circumferentially. Resulting velocity vectors  404  are also shown for each mass. Considering the second vibratory mode  420 , if the gyroscope  100  is not rotating, two of the masses  407  are translating inwardly and two of the masses  409  are translating outwardly. As shown in  FIG. 4 , the second n=2 vibratory mode shape  420  of the gyroscope is an ellipse  402   b  that is rotated 45° in relation to ellipse  402   a  of the first vibratory mode  400 . Due to the mass-to-mass couplers  108 , the other four masses  403 ,  405  are translating circumferentially. Resulting velocity vectors  404  are also shown for each mass. 
     As the gyroscope  100  is rotated (e.g., due to rotation of the system to which the gyroscope  100  is attached) about axis (Z) which is perpendicular to the plane of the gyroscope  100 , the vibratory modes  400 ,  420  exhibit coupling via Coriolis forces. For example, as shown in  FIG. 4 , the Coriolis forces  406  (F c =−2mΩ×v) in the first vibratory mode  400  arising from the velocity vectors  404  force two masses  407  inward and two masses  409  outward, thus exciting the other vibratory mode  420 . 
     The Coriolis forces  406  of the first vibratory mode  400  remove energy from the first vibratory mode  400  and excite motion in the second vibratory mode  420 . For example, as shown in  FIG. 4 , the Coriolis force  406  applied to each mass in the first vibratory mode  400  correspond to the displacement of the mass in the second vibratory mode  420 . Therefore, the Coriolis forces  406  resulting from the rotation of the gyroscope  100  result in the transfer of energy from the first vibratory mode  400  to the second vibratory mode  420 . Similarly, Coriolis forces of the second vibratory mode  420  reduce motion in the second vibratory mode  420  and excite motion in the first vibratory mode  400 . Therefore, the Coriolis forces resulting from the rotation of the gyroscope  100  also result in the transfer of energy from the second vibratory mode  420  to the first vibratory mode  400 . 
     More specifically, the vibratory motion in each mode  400 ,  420  is sinusoidal and as shown in the first vibratory mode  400  of  FIG. 4 , the masses  403  and  405  are at one extreme of their displacement (i.e., the outwardly moving masses  403  are at their largest radial distance and the inwardly moving masses  405  are at their smallest radial distance). At a time T/2=1/(2*f), where f is the frequency of the first vibratory mode  400  (T=1/f is the period and w=2πf is the frequency in radians per second), the masses  403  are closer to the center and the masses  405  are farther from the center. Accordingly, the position of each mass at any time is related to the phase of the oscillation at that time. In other words, if r 403  is the radial position of masses  403  and r 405  is the radial position of masses  405 , then r 403 =r 0 *sin (wt) and r 405 =r 0 *sin (wt+π). 
     The velocity of each mass is out of phase with the displacement by n/2. Therefore, when a mass  403  is at the outward extreme (as shown in  FIG. 4 ), its velocity is actually zero. The velocity vector  404  for each mass shown in  FIG. 4  is actually the velocity at a time t=(n/2w)=1/(4*f)=(T/4) seconds ago. As such, the velocity of a mass  403  can be expressed as v 403 =v 0 *sin(wt+n/2)=r 0   w *cos (wt), and the velocity of a mass  405  can be expressed as v 405 =v 0 *sin(wt+3n/2)=r 0   w *cos (wt+π). These velocity equations show that in addition to the displacement vectors, the velocity vectors also reverse direction sinusoidally. For example, at each half cycle, the velocity vectors reverse direction. The corresponding Coriolis force vectors also reverse direction at each half cycle. This is necessary to excite motion in the other mode, as the other mode also has sinusoidal motion which requires sinusoidal forcing. In at least one embodiment, it is desired to set the frequency of the force sinusoid equal to the resonance frequency of the mode being excited. By setting these frequencies equal, stronger coupling of the modes may be achieved. 
     With reference to the first vibratory mode  400  of  FIG. 4 , the positions of each mass  403 ,  405 ,  407 ,  409  are shown when the masses  403  and  405  are at one extreme of their displacement and the velocity  404  and Coriolis vectors  406  are shown at a time T/4 seconds earlier. The time at which the Coriolis force vectors  406  are shown in the first vibratory mode  400  is the same time at which the velocity vectors  404  are shown in the second vibratory mode  420  (the force is in phase with the velocity of the vibratory motion which is increasing, i.e., the force counteracts damping, which is proportional to velocity), which is again T/4 seconds earlier than the time corresponding to the position of the masses  403 ,  405 ,  407 ,  409  in the second vibratory mode  420 . In other words the velocity vectors  404  and force vector  406  shown in each vibratory mode  400 ,  420  are shown at the same time. 
     As the Coriolis forces  406  in the first vibratory mode  400  excite the masses in the second vibratory mode  420  (i.e., assist the velocity vectors  404  in the second vibratory mode  420 ), the Coriolis forces  406  in the second vibratory mode  420  reduce the motion in the first vibratory mode  400  (i.e., are opposed to the velocity vectors  404  in the first vibratory mode  400 ). As such, energy is transferred from the first vibratory mode  400  to the second vibratory mode  420 . At some point, all of the energy will have transferred and the motion in the first vibratory mode  400  will be zero. 
     At this point, the Coriolis forces  406  in the second vibratory mode  420  remain the same, but are now working with zero velocity in the first vibratory mode  400 . The Coriolis forces  406  excite the motion shown in the first vibratory mode  400 , but with a phase difference of π. Accordingly, the time at which the velocity vectors  404  are as shown in the first vibratory mode  400  of  FIG. 4  is T/2 different than as previously discussed. This means that the Coriolis force vectors  406  in the first vibratory mode  400  also reverse direction, thus reducing motion in the second vibratory mode  420 . As such, energy is transferred back to the first vibratory mode  400  from the second vibratory mode  400 . It is to be appreciated that one complete cycle of energy transfer (i.e., the first vibratory mode  400  to the second vibratory mode  420  and back to the first vibratory mode  400 ) results in a phase shift of π in the vibratory oscillation. The next half cycle (i.e., the first vibratory mode  400  to the second vibratory mode  420 ) will likewise result in motion in the second vibratory mode  420  that is π different than before. 
     The transfer of energy between modes can also appear similar to a rotation of the modal shape (i.e., the second vibratory mode  420  appears as a 45° rotated version of the first vibratory mode  400 ). It is to be appreciated that, from a frame of reference affixed to the gyroscope  100 , the gyroscope structure  100  does not rotate; rather, the orientation of the overall vibratory mode (superposition of modes  400  and  420  in varying proportions) appears to rotate in that frame as that frame of reference is rotated. At overall vibratory mode orientations between 45°, some combination of the two modes  400  and  420  will appear, i.e. all masses will be moving both radially and circumferentially, in proportion to the angular distance between the current orientation and the starting orientation. For rate mode operation, the control commands exert rebalancing forces to null (maintain at zero) the radial motion of either masses  407 ,  409  or masses  403 ,  405 . For whole angle mode operation, the exchange of energy (or apparent rotation of the overall vibratory mode) is allowed to occur without interference. Because of damping, forces must be continually applied along the direction of largest motion (or the direction of the overall vibratory mode), which is accomplished by applying the appropriate proportion of force (determined by the vector components of the orientation) via both the transducers  116  located at the masses  407 ,  409  and the masses  403 ,  405 . 
     By monitoring the motion of the masses  102  in the gyroscope  100  the rotation of the gyroscope  100  can be determined. For example, in a rate mode of operation of the gyroscope  100 , one of the modes (e.g., the first vibratory mode  400  or the second vibratory mode  420 ) is continually driven at a fixed amplitude by the drive/sense transducers  116 , and the motion of the other mode is monitored, by the drive/sense transducers  116 . Signals from the drive/sense transducers  116  based on the sensed motion are provided to the controller  117  (coupled to the drive/sense transducers  116 ) and the controller  117  can determine a rate of rotation of the gyroscope  100  based on the signals. For example, in one embodiment, the motion of masses  102  in one mode (e.g., the first vibratory mode  400  or the second vibratory mode  420 ) is driven at a fixed amplitude (by the drive/sense transducers  116 ) while the other mode is measured (by the drive/sense transducers  116 ) in an open loop. In such an embodiment, the amplitude of motion sensed by the drive/sense transducers  116  is proportional to the rate of rotation of the gyroscope. In another embodiment, the motion of masses  102  in one mode (e.g., the first vibratory mode  400  or the second vibratory mode  420 ) is driven at a fixed amplitude (by the drive/sense transducers  116 ) while the motion of masses  102  in the other mode is fixed at zero, by the drive/sense transducers  116 , in a closed-loop. The required feedback force necessary to maintain the motion of the masses  102  at zero is proportional to the rate of rotation of the gyroscope  100 . 
     In a whole-angle mode of operation, the motion of masses  102  in both modes  400 ,  420  is driven by the drive/sense transducers  116  such that the total vibrational amplitude of the masses  102  across both modes  400 ,  420  is sustained, but the distribution of energy between the two modes  400 ,  420  is allowed to change freely. By measuring the distribution of motion between the modes  400 ,  420  with the drive/sense transducers  116  and providing signals based on the distribution of motion to the controller  117 , an angle of rotation (with respect to a starting angle) can be read out by the controller  117 . According to one embodiment, the control scheme discussed in U.S. Pat. No. 7,318,347, titled “HEMISPHERICAL RESONATOR GYRO CONTROL”, filed on May 9, 2005, which is herein incorporated by reference in its entirety, is utilized to maintain the total vibrational amplitude constant across the modes  400 ,  420  while allowing the distribution of energy to transition freely between the modes  400 ,  420  subject to Coriolis forces. 
     The arrangement of the masses and flexures in the MEMS based gyroscope discussed above result in a gyroscope that combines the best features of a lumped element TF gyroscope and a rotationally symmetric gyroscope. The use of relatively large masses on relatively weak flexures enables low damping, high momentum, and high sensitivity which may result in low Brownian motion noise (the dominant resolution limit in MEMS based symmetric angular rate gyroscopes). The symmetrical eight mass configuration enables the gyroscope to behave like a continuous rotationally symmetric vibratory gyroscope (e.g., such as a hemisphere, ring, or disc gyroscope). The mass-to-mass couplers cause the eight masses to move in the n=2 vibratory mode, which is characterized by two opposite masses  403  moving out (i.e., away from center), the two opposite masses  405  (90° from the masses moving out) moving in (i.e., in towards the center), and the other four masses  407 ,  409  moving circumferentially. This vibratory mode emulates the vibratory mode of a rotationally symmetric gyroscope and is necessary so that the Coriolis forces, due to rotation about the Z axis, couple the two modes together, allowing the rate of rotation or the angle of rotation to be sensed based on the vibrations of the masses. 
     The particular arrangement of the link between adjacent masses (i.e., the mass-to-mass couplers  108  shown in  FIG. 1 ) is important to ensuring the correct modal structure of the gyroscope. For example, Applicant has appreciated that in order to couple the two modes together, each mass-to-mass coupler  108  must be stiff in the circumferential direction, yet compliant in the radial direction. For example,  FIG. 5A  is a diagram of one embodiment of a mass-to-mass coupler according to aspects of the present invention. The mass-to-mass coupler  108  includes a stiff “bar”  111  that has a flexural hinge  109  on either side. The stiff “bar”  111  operates such that the circumferential motion of a mass  102  to which it is coupled depends on the radial motion of the masses adjacent partner. Accordingly, as the masses  102  vibrate, the stiff “bar”  111  results in the gyroscope  100  operating in the n=2 vibratory mode. In other embodiments, any other type of mass-to-mass coupler can be used that couples masses together such that the radial and circumferential motions of adjacent masses  102  act in such a way as to result in the n=2 vibratory mode where two opposite masses  102  are moving radially outward, two opposite masses  102  are moving radially inward, and the masses  102  located in between move circumferentially. For example, according to some embodiments, the structure of the mass-to-mass coupler may be configured differently (e.g., as discussed below with regard to  FIG. 10 ), the gyroscope may include multiple mass-to-mass couplers coupled between each adjacent mass (e.g., as discussed below with regard to  FIG. 12 ), and in addition to being coupled between adjacent masses, the mass-to-mass coupler may also be coupled to a central anchor via a flexural hinge (e.g., as discussed below with regard to  FIG. 16 ). According to some other embodiments, a mass-to-mass coupler includes a ring (or rings) that are configured to couple all masses together (e.g., as discussed below with regard to  FIGS. 17-19 ). 
     According to at least one embodiment, the gyroscope  100  also includes angled electrodes  118  which enable the cross-spring term of the masses  102  to be trimmed. The cross spring term in the gyroscopes  100  equations of motion describes the mechanical coupling between modes  400 ,  420 . More specifically, the cross spring term is a spring constant quantifying the amount of force applied in a direction that increases one mode proportionally in response to the motion of the other mode. It is advantageous to reduce or eliminate this term as it is desirable to have each mode only be excited by rotation of the gyroscope  100 . Typically, this coupling is intentionally minimized by the design of the suspension (i.e., the springs or flexures). However, fabrication imperfections may cause the cross spring term to be non-zero. In a rate mode of operation, this may result in a quadrature error when the demodulation phase is also imperfect. As a result, standard TF gyroscopes typically tune out the cross-spring term by compensating it with a variable bias applied to the driver transducers (e.g., the drive combs). 
     In a whole-angle operating gyroscope, the cross-spring term may also lead to errors. For example, the performance of a whole angle operating gyroscope depends on the frequencies of two vibratory modes being equal. Reducing the frequency split between modes to zero is not possible unless the cross spring term is also reduced to zero. Some existing whole-angle-capable gyroscopes (e.g., such as a quad mass gyroscope) provide a capability of tuning only the on-axis spring term, and hence cannot achieve perfect mode matching. Such gyroscopes rely on the mechanical cross-spring term being small by design. Other vibratory MEMS based gyroscopes (e.g., such as ring gyroscopes) use electrostatic forcers located at specific locations to provide a tunable spring force that compensates for the mechanical cross spring term. None of the rate or whole angle based methods of compensating for the cross spring term are applicable to the gyroscope  100  discussed above. Accordingly, the gyroscope  100  includes the angled electrodes  118  which are configured to trim the cross-spring term of the masses  102 . 
     Each angled electrode  118  is configured such that the electrode generates a radial force component in response to circumferential motion of a corresponding mass  102  and the electrode generates a circumferential force component in response to radial motion of the corresponding mass  102 . For example, as shown in  FIG. 5A , in response to circumferential motion  127  of a mass  102  (Mass 1), the electrode  118  applies a radial force component  125  to the same mass  102  (Mass 1) and in response to radial motion  125  of the mass  102  (Mass 1), the electrode  118  applies a circumferential force component  127  to the same mass  102  (Mass 1). Similarly, in response to circumferential motion  123  of the mass  102  (Mass 2), the electrode  118  applies a radial force component  121  to the same mass  102  (Mass 2) and in response to radial motion  121  of the mass  102  (Mass 2), the electrode  118  applies a circumferential force component  123  to the same mass  102  (Mass 2). 
     The magnitude of the circumferential and radial force components applied by an electrode  118  depends on the voltage applied to the electrode  118 . Each electrode  118  is angled such that circumferential or radial motion in one vibratory mode (e.g., vibratory mode  400  or  420 ) results in a corresponding circumferential or radial force in the other mode. The resulting force, due to circumferential or radial motion, will either assist or oppose motion in the vibratory mode depending on which electrodes  118  are used and the voltages applied to them. Therefore, the cross spring term can be cancelled regardless of the polarity of the cross spring term. 
     According to one embodiment, the MEMS based gyroscope  100  is implemented on a substrate (e.g., substrate  430  shown in  FIG. 4 ). In at least one embodiment, the substrate  430  is a cubic crystal structure. As illustrated in  FIG. 5B , a cubic crystal structure (e.g., single crystal silicon (Si)) exhibits anisotropic elastic properties such that flexures oriented along the (100) direction plane behave differently from flexures having the same geometry but oriented along the (110) direction plane (the (100) direction plane and the (110) direction plane being 45° apart). This causes errors in whole angle operation because the natural frequencies of the two vibratory modes will differ. 
     Historically the difference in moduli between the (100) direction and the (110) direction has been compensated for by adjusting flexure width; however, in at least one embodiment, these modulus differences are compensated for by rotating the entire device by 22.5° relative to the (100) direction plane. For example, as shown in  FIG. 5C , when constructing the device, the geometry of the device is oriented relative to the crystallographic orientation of silicon such that the (100) plane falls along direction  433  and the (110) plane falls along direction  435 . This places the axis of the corresponding flexures (e.g., flexures A and B) along a line  502  that is halfway between the (100) direction plane  433  and the (110) direction plane  435  (i.e., 22.5° between the (100) and (110) direction planes), rather than exactly on either the (100) direction plane  433  or the (110) direction plane  435 . The result is that all flexures having the same geometry have the same modulus, and will thus behave in the same way. This may save design time and also reduce errors that arise from imperfect width-based compensation, which is often attempted using simplified analytical models that imperfectly predict the stiffness of complicated flexures. As discussed above, the MEMS based gyroscope  100  is implemented on a silicon based substrate; however, in other embodiments, the MEMS based gyroscope  100  can be implemented on any type of cubic crystal structure. 
     In another embodiment, planar isoelasticity in a cubic crystal based substrate (e.g., substrate  430 ) is achieved by utilizing a wafer with a (111) crystal orientation. For example, as shown in  FIG. 5D , when the device is implemented on a cubic crystal based substrate  430  with a (111) crystal orientation, the in-plane directions are (110) and/or (112), all of which have the same modulus. The result is that all flexures having the same geometry will thus behave in the same way without requiring compensation (e.g., as described above with respect to the (100) crystal orientation). 
     According to another embodiment, the MEMS based gyroscope  100  is implemented on a hexagonal crystal based substrate  430  (e.g., a Silicon Carbide (4H—SiC) based substrate). As illustrated in  FIGS. 5E-5F , for wafers (i.e., substrates) oriented in the (0001) plane of a hexagonal crystal, the in-plane directions ((1 1 00) and (11 2 0)) resemble those of a (111) plane oriented cubic crystal in that they have the same modulus. The result is that all flexures having the same geometry will thus behave in the same way without requiring compensation (e.g., as described above with respect to the (100) cubic crystal orientation). As illustrated in  FIGS. 5D and 5F , whereas the cubic crystal axis repeats every 60°, the hexagonal crystal axes repeat every 120°. As discussed above, the MEMS based gyroscope  100  is implemented on a Silicon-Carbide based substrate; however, in other embodiments, the MEMS based gyroscope  100  can be implemented on any type of hexagonal crystal structure. 
     Applicant has appreciated that in addition to not requiring compensation (e.g., rotation) to account for differences in stiffness (as discussed above), the use of a hexagonal crystal based substrate (e.g., substrate  430 ) is also advantageous because stiffness variations remain low even if the angle between the device plane and the crystallographic plane is imperfect. For example,  FIG. 5G  is a graph illustrating how Young&#39;s modulus, which quantifies the material contribution to stiffness, varies in silicon-carbide (4H—SiC) compared to silicon (Si).  FIG. 5G  assumes 0.01 radian planar misalignment. As shown in  FIG. 5G , silicon-carbide is less sensitive to misalignment then silicon (by two orders of magnitude). Such low-sensitivity for misalignment holds for any hexagonal crystal whose stiffness along the principal direction is of the form: 
     
       
         
           
             
               
                 
                   
                       
                   
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     A single-crystal 4H—SiC substrate (or material layer), for example substrate  430  shown in  FIG. 5F , may have an approximately on-axis orientation of (0001) with the first surface (top surface) of the 4H—SiC substrate (or material layer) being a silicon face and the second surface (bottom surface) of the 4H—SiC material being a carbon face having an approximate on-axis orientation of (000 1 ). In some embodiments, a single-crystal 4H—SiC material or substrate may have an on-axis orientation in the range of 0 degree+/−0.1 degree. In other embodiments, a single-crystal 4H—SiC material or substrate may have an on-axis orientation in the range of 0 degree+/−0.5 degree. Electrostatic coupling to a 4H—SiC beam may require that the beam have an electrical conductivity and thus the on-axis 4H—SiC material may be doped N-type or P-type. According to at least one embodiment, the on-axis 4H—SiC substrate (or material) is N-type doped with a doping concentration in the range of 5×10 18  cm 3  to 1×10 19  cm −3 ; however, in other embodiments, a different doping concentration may be utilized. Suspended beams (i.e., flexures) in silicon carbide (4H—SiC) can be made from bulk 4H—SiC substrates or from epitaxial layers grown on bulk 4H—SiC or silicon substrates. 
     A single-crystal 4H—SiC substrate (or material layer), for example substrate  430  shown in  FIG. 5F , may have an off-axis orientation with respect to the (0001) c-axis, with the first surface (top surface) of the 4H—SiC substrate (or material layer) being a silicon face and the second surface (bottom surface) of the 4H—SiC material being a carbon face having off-axis orientation with respect to (000 1 ) c-axis. The off-axis orientation can be in the range of 0.5 to 10 degrees with respect to the (0001) axis. The accuracy of the off-axis orientation can be as low as +/−0.1 degree. The off-axis 4H—SiC substrate (or material) may be doped N-type of P-type. According to one embodiment, the off-axis 4H—SiC substrate (or material) is N-type doped using nitrogen dopant atoms; however, in other embodiments, another type of dopant atoms may be utilized. According to at least one embodiment, an N-type SiC substrate has a 4 degree or 8 degree off-axis orientation with respect to the (0001) axis. In one embodiment, the substrate  430  is an N-type off-axis or on-axis SiC substrate manufactured by Cree, Inc. of Durham, N.C. or II-VI Incorporated of Saxonburg, Pa. 
     According to one embodiment, the epitaxial growth of 4H—SiC single-crystal SiC material layers on the silicon face of a hexagonal single-crystal SiC bulk substrate requires the use of off-axis orientation from the (0001) c-axis of more than about 2 degrees. In other embodiments, the epitaxial growth of 4H—SiC single-crystal SiC material layers on the silicon face of a hexagonal single-crystal SiC bulk substrate requires the use of off-axis orientation from the (0001) c-axis of more than 4 degrees. In another embodiment, the epitaxial growth of 4H—SiC single-crystal SiC material layers on the silicon face of a hexagonal single-crystal SiC bulk substrate requires the use of off-axis orientation from the (0001) c-axis of more than about 8 degrees. The 4H—SiC epitaxial layers grown on the silicon face can be doped N-type or P-type to a selected doping level. 
     4H—SiC single-crystal SiC material layers can also be grown on the carbon face of 4H—SiC single-crystal substrates. Similar to the epitaxial growth of 4H—SiC single-crystal SiC material layers on a silicon face as described above, the epitaxial growth of 4H—SiC single-crystal SiC material layers on the carbon face of a hexagonal single-crystal SiC bulk substrate requires the use of off-axis orientation from the (0001) c-axis of more than about 2 degrees in some embodiments, more than 4 degrees in some embodiments, and more than about 8 degrees in some embodiments. The 4H—SiC epitaxial layers grown on the carbon face can be doped N-type or P-type to a selected doping level. According to other embodiments, a hexagonal crystal based substrate  430  (e.g., a Silicon Carbide (4H—SiC) based substrate) may be constructed in another appropriate way. 
     As described above, a MEMS based gyroscope having an axially symmetric structure including relatively large masses (coupled via mass-to-mass couplers) on relatively weak flexures (e.g., the MEMS based gyroscope  100  shown in  FIG. 1 ) can be implemented on a hexagonal crystal based substrate (e.g., substrate  430 ). However, in other embodiments, another type of gyroscope (e.g., a ring gyroscope, disk gyroscope, quad-mass gyroscope, etc.) including another type of axially symmetric structure (e.g., a ring, a disk, other symmetrically coupled masses, etc.) configured to be driven in two vibratory modes of the structure can also be implemented on a hexagonal crystal based substrate. 
       FIG. 6A  is a diagram illustrating another embodiment of a MEMS based gyroscope  600  configured according to aspects of the present disclosure. The gyroscope  600  is the same as gyroscope  100  discussed above with regard to  FIG. 1  except that the gyroscope  600  does not include outside anchors (e.g., such as outside anchors  114  shown in  FIG. 1 ) or outside shuttles (e.g., such as outside shuttles  112  shown in  FIG. 1 ). In addition, the internal shuttles  613  are configured differently. A benefit of such a configuration is that internal stress (which may arise from such causes as thermal expansion coefficient mismatches between the gyroscope material and the substrate material connecting a central anchor  602  and any outside anchors) are avoided. Such thermal mismatches between materials connecting the two anchors may result in the stretching of masses/flexures suspended between the two anchors. By only including one central anchor  602  in the gyroscope  600 , such stretching may be avoided. Additionally, the bias stability of the gyroscope  600  may be improved by reducing the transmission of (unpredictable) external stresses on the gyroscope  600 . However, by only including one central anchor  602 , the force isolation (i.e., decoupling x- and y-motion as discussed above with regard to  FIG. 2 ) provided by the outer shuttles and anchors is eliminated. 
       FIG. 6B  is a diagram illustrating another embodiment of a MEMS based gyroscope  650  configured according to aspects of the present disclosure. The gyroscope  650  is similar to the gyroscope  600  discussed above with regard to  FIG. 6A  except that the gyroscope  650  does not include angled electrodes (e.g., such as the angled electrodes  118  shown in  FIG. 1 ), and the anchor  652 , internal flexures  654 , and internal shuttles  653  are configured differently.  FIG. 6C  is a diagram illustrating further details of the anchor  652 , internal flexures  654 , and internal shuttle  653  of the gyroscope  650 . 
       FIG. 6D  is a diagram illustrating another embodiment of a MEMS bases gyroscope  660  configured according to aspects of the present disclosure. The gyroscope  660  is similar to the gyroscope  650  discussed above with regard to  FIG. 6B  except that the gyroscope  650  includes outer shuttles  662 , outer flexures  664 , and outer anchors  666 .  FIG. 6E  is a diagram illustrating further details of the outer shuttles  662 , outer flexures  664 , and outer anchors  666 . According to one embodiment, the internal shuttles  653  and outer shuttles  662  are identical. This may simplify the design process of the gyroscope  660  and also has the benefit of being more symmetric. The outer shuttles  662  and inner shuttles  653  are both configured to move in a circumferential direction and not in a radial direction (e.g., differently than discussed above with regard to  FIG. 1 ). 
       FIG. 6F  is a diagram illustrating another embodiment of a MEMS based gyroscope  680  configured according to aspects of the present disclosure. The gyroscope  680  is substantially the same as the gyroscope  660  discussed above with regard to  FIG. 6D  except that the outer shuttles, outer flexures and outer anchors are configured differently. For example, as shown in  FIG. 6F , the gyroscope  680  includes outer shuttles  682 , outer flexures  684 , and outer anchors  686 . 
       FIG. 7A  is a diagram illustrating another embodiment of a MEMS based gyroscope  700  configured according to aspects of the present disclosure. The gyroscope  700  is similar to the gyroscope  100  discussed above with regard to  FIG. 1  except that the gyroscope  700  does not include outside anchors or outside shuttles (e.g., as discussed above with regard to  FIG. 6 ), does not include internal shuttles (e.g., such as internal shuttles  113  shown in  FIG. 1 ) and the flexures  702  are configured differently in a serpentine configuration.  FIG. 7B  is a diagram illustrating the MEMS based gyroscope  700  and outer clapper electrodes  704  adjacent each mass  701  of the gyroscope  700 . The outer clapper electrodes  704  are part of adjacent drive/sense transducers (e.g., the drive/sense transducers  116  shown in  FIG. 1 ) utilized to drive and sense motion of the masses  701 . 
       FIG. 8  is a diagram illustrating another embodiment of a MEMS based gyroscope  800  configured according to aspects of the present disclosure. The gyroscope  800  is similar to the gyroscope  100  discussed above with regard to  FIG. 1  except that the gyroscope  800  does not include outside anchors or outside shuttles (e.g., as discussed above with regard to  FIG. 6 ), does not include internal shuttles (e.g., such as internal shuttles  113  shown in  FIG. 1 ) and the flexures  802  are configured differently in a stacked configuration.  FIG. 9  is a diagram illustrating further details of the stacked configuration of flexures  802 . In addition, unlike the mass-to-mass couplers  108  shown in  FIG. 1 , the gyroscope  800  utilizes different mass-to-mass couplers  804  to couple together adjacent masses.  FIG. 10  is a diagram illustrating further details of a mass-to-mass coupler  804 . The mass-to-mass coupler  804  is more stiff in the radial direction and more soft in the circumferential direction. 
       FIG. 11  is a diagram illustrating another embodiment of a MEMS based gyroscope  1100  configured according to aspects of the present disclosure. The gyroscope  1100  is similar to the gyroscope  100  discussed above with regard to  FIG. 1  except that the gyroscope  1100  does not include outside anchors or outside shuttles (e.g., as discussed above with regard to  FIG. 6 ), does not include internal shuttles (e.g., such as internal shuttles  113  shown in  FIG. 1 ) and the flexures  1102  are configured differently in a serpentine configuration. In addition, unlike the mass-to-mass couplers  108  shown in  FIG. 1 , the gyroscope  1100  utilizes mass-to-mass flexures  1104  (e.g., as shown in  FIG. 10 ) to couple together adjacent masses. 
       FIG. 12  is a diagram illustrating another embodiment of a MEMS based gyroscope  1200  configured according to aspects of the present disclosure. The gyroscope  1200  is similar to the gyroscope  100  discussed above with regard to  FIG. 1  except that the gyroscope  1200  includes two mass-to-mass couplers  1208  coupled between each adjacent mass  1202 , and the flexural hinges on the couplers are straight instead of folded. 
       FIG. 13  is a diagram illustrating another embodiment of a MEMS based gyroscope  1300  configured according to aspects of the present disclosure. The gyroscope  1300  is similar to the gyroscope  100  discussed above with regard to  FIG. 1  except that the gyroscope  1300  does not include outside anchors (e.g., such as outside anchors  114  shown in  FIG. 1 ) or outside shuttles (e.g., such as outside shuttles  112  shown in  FIG. 1 ), and the internal flexures  1314 , internal shuttles  1313 , and anchor  1312  are configured differently. For example, as shown in  FIG. 13 , the internal shuttles  1313  are extended length shuttles. 
       FIG. 14  is a diagram illustrating another embodiment of a MEMS based gyroscope  1400  configured according to aspects of the present disclosure. The gyroscope  1400  is similar to the gyroscope  100  discussed above with regard to  FIG. 1  except that the gyroscope  1400  does not include outside anchors (e.g., such as outside anchors  114  shown in  FIG. 1 ) or outside shuttles (e.g., such as outside shuttles  112  shown in  FIG. 1 ), and the internal flexures  1414 , internal shuttles  1413 , and anchor  1412  are configured differently. For example, as shown in  FIG. 14 , each mass  1402  includes two extended length internal shuttles  1413  and the anchor  1412  extends into each mass  1402 . 
       FIG. 15  is a diagram illustrating another embodiment of a MEMS based gyroscope  1500  configured according to aspects of the present disclosure. The gyroscope  1500  is similar to the gyroscope  100  discussed above with regard to  FIG. 1  except that the gyroscope  1500  does not include outside anchors (e.g., such as outside anchors  114  shown in  FIG. 1 ) or outside shuttles (e.g., such as outside shuttles  112  shown in  FIG. 1 ), and the internal flexures  1514 , internal shuttles  1513 , anchor  1512 , and mass-to-mass couplers  1508  are configured differently.  FIG. 16  is a diagram illustrating further detail regarding the internal flexures  1514 , internal shuttles  1513 , anchor  1512 , and mass-to-mass coupler  1508 . The anchor  1512  extends between each mass  1502 . The mass-to-mass coupler  1508  includes a “stiff” bar  1510  that is coupled between adjacent masses  1502  via flexural hinges  1509 . The bar  1510  is also coupled to the anchor  1512  via a flexural hinge  1511 . 
       FIG. 17A  is a diagram illustrating another embodiment of a MEMS based gyroscope  1700  configured according to aspects of the present disclosure. The gyroscope  1700  is similar to the gyroscope  100  discussed above with regard to  FIG. 1  except that the gyroscope  1700  does not include outside anchors (e.g., such as outside anchors  114  shown in  FIG. 1 ) or outside shuttles (e.g., such as outside shuttles  112  shown in  FIG. 1 ), and the internal flexures  1714 , anchor  1712 , and mass-to-mass couplers are configured differently. As shown in the gyroscope  1700  of  FIG. 17 , the mass-to-mass coupler includes an internal ring  1708  and an external ring  1710  which are configured to couple together adjacent masses  1702 . For example,  FIGS. 17B and 17C  are diagrams illustrating further detail of the internal ring  1708  and the external ring  1710 . According to one embodiment, the internal ring  1708  and external ring  1710  are single continuous rings. In another embodiment, the internal ring  1708  may include multiple independent portions, each portion configured to couple together two adjacent masses  1702 . Similarly, the external ring  1710  may include multiple independent portions, each portion configured to couple together two adjacent masses  1702 . 
       FIG. 18  is a diagram illustrating another embodiment of a MEMS based gyroscope  1800  configured according to aspects of the present disclosure. The gyroscope  1800  includes a plurality of masses  1802 . Each mass  1802  is suspended between two shuttles  1804  via flexures  1806 . Each mass is also coupled to a ring  1808 . The gyroscope  1800  operates in substantially the same way as the gyroscope  100  discussed above with regard to  FIG. 1  except that with the gyroscope  1800 , the ring  1808  acts as the mass-to-mass coupler.  FIG. 19  is a diagram illustrating further details of the mass  1802 , shuttles  1804 , and flexures  1806 . 
     As discussed above, in certain embodiments, the MEMS based gyroscope includes eight masses; however, in other embodiments, the MEMS based gyroscope may include any number of masses. For example, in one embodiment, the MEMS based gyroscope includes any number of masses that is greater than eight and a multiple of four. As also discussed above, the MEMS based gyroscope includes wedge-shaped masses; however, in other embodiments, the masses may be configured in any appropriate shape capable of operating in the n=2 vibratory mode. 
     As discussed above, the MEMS based gyroscope is operated in an n=2 vibratory mode; however, in other embodiments, the MEMS based gyroscope may be operated in some other vibratory mode. For example, in at least one embodiment, the MEMS based gyroscope includes 12 masses and is configured to operate in the n=3 vibratory mode. In other embodiments, the MEMS based gyroscope may be configured to operate in any other vibratory mode (e.g., n=3 vibratory mode, n=4 vibratory mode, n=5 vibratory mode, etc.) and may include an appropriate number of masses. 
     A new MEMS based gyroscope design is provided that combines the best features of a lumped element TF gyroscope and a rotationally symmetric gyroscope. The new design efficiently uses relatively large masses (e.g., like a TF gyroscope) on relatively weak flexures to provide high sensitivity while maintaining an eight-fold symmetry conducive to the n=2 vibratory mode used in most whole angle based gyroscopes to provide high dynamic range. The new design is capable of operating in both rate and whole angle mode, is low cost, and is easily fabricated. 
     According to one embodiment, the MEMS based gyroscope design discussed above may be utilized as a whole angle gyroscope in a miniature system. In another embodiment, the MEMS based gyroscope may be utilized as a whole angle gyroscope in a platform having a high rotation rate that requires a high dynamic range instrument. The MEMS based gyroscope may be utilized in any other whole angle application. 
     The MEMS based gyroscope design could also be used in any application where traditional MEMS gyroscopes are currently used. The combination of large masses on weak springs (providing high momentum and low damping) and matched modes (providing high gain) yields a low Angle Random Walk (ARW), one of the primary performance parameters for gyroscopes. Applicant has appreciated that an ARW on the order of 0.01 deg/rt-hr will be obtained with this gyroscope design and improvement of  10   x  (or more) could be possible by increasing the size of the gyroscope. 
     Various embodiments of systems and methods disclosed herein may have applications in various fields. Applications may encompass the field of precision inertial guidance and navigation, particularly in GPS denied environments. For example, embodiments may be used to guide platforms such as strategic missiles, submarines, Unmanned Underwater Vehicles (UUV), Unmanned Aerial Vehicles (UAV), cruise missiles, aircraft, and tactical munitions. Other examples of applications may include commercial aviation, self-driving vehicles, robotic machinery, personal navigation and consumer electronics such as various computing devices and mobile communication devices. 
     Having described above several aspects of at least one embodiment, it is to be appreciated various alterations, modifications, and improvements will readily occur to those skilled in the art. Such alterations, modifications, and improvements are intended to be part of this disclosure and are intended to be within the scope of the disclosure. Accordingly, the foregoing description and drawings are by way of example only, and the scope of the disclosure should be determined from proper construction of the appended claims, and their equivalents.