Abstract:
Planar vibratory gyroscope structures are provided which are inherently symmetric, which facilitate the use of simple monolithic fabrication processes and which enable the use of sensitive control and sense systems. A planar vibratory member of these structures has a hollow frame, a plate that has a plate perimeter and is positioned within the frame and a plurality of elongate beams which couple the plate to the frame wherein each of the beams is proximate to and substantially parallel to a respective portion of the plate perimeter. The exterior rim of the planar member can be supported by a substrate which provides room for easy access to the plate with mode control and sense systems.

Description:
BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     The present invention relates generally to gyroscopes and more particularly to vibratory gyroscopes. 
     2. Description of the Related Art 
     Gyroscopes are devices which can sense angular rotation and/or rotation rate. Accordingly, they are useful in situations (e.g., satellite attitude control) where observation of other inertial indicators (e.g., cosmic bodies such as the sun) is temporarily obscured. 
     A variety of gyroscope concepts have been developed. For example, gyroscopes based upon gimballed spinning wheels and gyroscopes based upon laser rings have been shown to be highly accurate. Although these devices find use in numerous applications (e.g., inertial navigation), their high expense and large size discourage a wider use. 
     In contrast to these gyroscopes, the concept of vibratory gyroscopes is based on rotation-induced energy exchange between modes of vibrating members. This concept is exemplified by an analysis of ringing wine glasses that was performed in 1890 by G. H. Bryan. In a flexural mode, the lip of a wine glass vibrates in elliptical-shaped modes that have two nodal diameters. When the wine glass is rotated, Bryan found that the node lines lag behind (precess) the rotation of the wine glass (e.g., during a 90° rotation, the node lines were observed to precess by ˜27°). This nodal lag is, therefore, an indication of angular rotation. 
     Although highly accurate hemispherical resonator gyroscopes have been built using the wine glass example (e.g., see Wright, David, et al., “The HRG Applied to a Satellite Attitude Reference System”, from  Guidance and Control  by Culp, R. D., et al., American Astronautical Society, 1994, volume 86, pp. 57—63), their nonplanar form is difficult to miniaturize and requires complicated, expensive fabrication processes. 
     Other nonplanar vibratory gyroscope structures have been investigated (e.g., see Putty, Michael W., et al., “A Micromachined Vibrating Ring Gyroscope”,  Solid-State Sensor and Actuators Workshop , Jun. 13-16, 1994, pp. 213-220). For example, cantilevered beams have been used to form vibratory gyroscopes. Experience with these devices has shown them to be difficult to mount and to be sensitive to temperature and spurious vibrations. To overcome the difficulties of cantilevered beams, tuning fork gyroscopes have been developed. These are balanced devices which are easier to mount and less sensitive to linear vibrations. However, fabrication and temperature drift problems limit the matching of input and output mode frequencies which, in turn, degrades the gyroscope&#39;s sensitivity. Misalignment of mass centers can also produce an undesirable vibration response which causes bias errors. 
     In contrast to these vibratory gyroscope types, the cost and size of planar vibratory gyroscopes is relatively low because they are mechanically simple (e.g., there is an absence of rotating parts) and their design typically facilitates miniaturization and batch fabrication with micromachining techniques. In addition, the precision of micromachining has enabled many vibratory gyroscopes to achieve impressive accuracy. 
     One conventional planar vibratory gyroscope employs a vibrating ring as its sensing element (e.g., see Johnson, Jack D., et al., “Surface Micromachined Angular Rate Sensor”, 1995  SAE Conference Paper  950538, pp 77-83). This ring element can be considered to be a slice out of Bryan&#39;s wine glass. In a controlled resonance, the ring assumes an elliptical pattern in which four nodes on the ring have no deflection and four antinodes on the ring are each located between a pair of nodes and exhibit maximal radial deflection. In response to rotation, the angular position of the nodes lags the angular position to which the gyroscope is rotated. 
     Another planar vibratory gyroscope is typically referred to as a clover-leaf gyroscope (e.g., see Tang, Tony K., et al., “Silicon Bulk Micromachined Vibratory Gyroscope”, 1996  Solid-State Sensor and Actuator Workshop,  Hilton Head, S.C., June 2-6) because it has a planar member whose outline resembles a four leaf clover. This member is suspended by four thin wires or beams from a housing and a metal post is coupled to the center of the member with an orientation orthogonal to the member&#39;s plane. The thin clover leaves provide large areas for electrostatic driving and capacitive sensing. 
     The resonator is electrostatically excited in a control mode to rotate about a first axis of the planar member which causes the post to move in a second axis of the planar member that is orthogonal to the first axis. In response to a rotation about a third axis that is orthogonal to the member&#39;s plane, the motion of the oscillating post is displaced into movement along the first axis. This post displacement translates into a sense mode rotation of the planar member about the second axis. Essentially, the post couples energy between the control and sense modes. 
     Although the planar vibratory gyroscopes described above can be miniaturized and can be generally realized with low-cost micromachining techniques, they suffer from various operational defects. For example, the ring gyroscope is planar and symmetric but the sensitivity of its control and sense electrodes is degraded because of the small electrode size required to couple to the ring&#39;s flexing perimeter. In addition, the ring gyroscope&#39;s circular form degrades the precision with which it can be defined in bulk crystalline material by photographic masks. As a second example, the orthogonally mounted post of the clover-leaf gyroscope detracts from its otherwise planar configuration. The post requires a manual assembly procedure which typically degrades the gyroscope&#39;s symmetry. In addition, this gyroscope&#39;s narrow beam supports are a source of high stress and nonlinearity. 
     SUMMARY OF THE INVENTION 
     The present invention is directed to a planar vibratory gyroscope structure which is inherently symmetric, facilitates the use of simple monolithic fabrication processes and enables the use of sensitive control and sense systems. 
     These goals are realized with a planar gyroscopic member which has a frame, a plate that is positioned within the frame and has a plate perimeter and a plurality of elongate beams which are oriented to substantially surround the plate perimeter and arranged to be everywhere substantially equidistant from the plate perimeter. 
     In particular, the planar member forms a system of slots and each of the slots is at least partially interleaved between adjacent ones of the slots so to define beams which each have a first end coupled to the frame and a second end coupled to the plate and proximate to the first end of an adjacent beam. 
     In a four-beam embodiment of the planar member, the plate is particularly suited for vibration modes about second ends of nonadjacent beams. Because of its structural symmetry, these modes are substantially uncoupled, have substantially equal natural frequencies and the natural frequencies substantially track each other over temperature. The equal natural frequencies enhance the planar member&#39;s sensitivity to rotation and the lack of coupling reduces its sensitivity to spurious vibrations. 
     With the exterior rim of the planar member supported by a substrate, the plate is easily accessed with mode control and sense systems. The large area of the plate enhances the size of control and sense electrodes that are positioned proximate to the plate for excitation of controlled vibration modes and sensing of rotation-induced vibration modes. Other conventional position-sensing systems (e.g., tunneling tips) can also be positioned proximate to the plate to sense rotation-induced vibration modes. The structure of the planar member provides robust support beams which facilitate a low torsional-stress design. 
     The simple structure of the planar member facilitates its definition with precise photolithographic processes and subsequent low-cost fabrication (e.g., from crystalline materials such as silicon). Although a rectilinear embodiment of the planar member is especially suited for easy definition and fabrication, the teachings of the invention can be extended to other spatial realizations, e.g., a circular embodiment. 
     Gyroscopes formed with planar members of the invention are suited for various operational modes. In a “whole angle” mode, drive signals are applied to control electrodes to initiate a vibration about an initial axis. Rotation of the gyroscope induces, via Coriolis coupling, a small vibration about an axis that is orthogonal to the initial axis; it is therefore, a small precession of the driven vibration. Sense electrodes generate signals that are indicative of the rotation-induced precession. In a “force to rebalance” mode, the signals from the sense electrodes are fed back to the control electrodes to substantially cancel the rotation-induced precession. In this operational mode, the feedback signal is a measure of the instantaneous rotation rate. In an “open loop” mode, the rotation-induced vibration amplitude about an axis orthogonal to the drive direction is sensed as a measure of rotation rate. 
    
    
     The novel features of the invention are set forth with particularity in the appended claims. The invention will be best understood from the following description when read in conjunction with the accompanying drawings. 
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a plan view of a vibratory gyroscope embodiment of the present invention; 
     FIG. 2 is a view along the plane  2 — 2  of FIG. 1; 
     FIG. 3 is a view of the gyroscope of FIG. 1 taken along the plane  3 — 3  of FIG. 2, the figure shows a control electrode system and a sense electrode system; 
     FIG. 4 is an enlarged view of structure within the curved line  4  of FIG. 2; 
     FIG. 5 is a view similar to FIG. 1 which illustrates control and sense mode axes; 
     FIG. 6 is a schematized view of a gyroscope system which includes elements of the gyroscope of FIG. 1; 
     FIG. 7 is a plan view of another vibratory gyroscope embodiment of the present invention; 
     FIG. 8 is a view along the plane  8 — 8  of FIG. 7; 
     FIG. 9 is a view of the gyroscope of FIG. 7 taken along the plane  9 — 9  of FIG. 8, the figure shows another sense electrode embodiment; 
     FIG. 10 is a plan view of another vibratory planar member of the present invention; 
     FIGS. 11A and 11B are elevation views of other position-sensing systems which can be substituted for the sense electrode systems of FIGS. 3 and 9; 
     FIGS. 12A and 12B are schematics of a control electrode system and a sense electrode system in the vibratory gyroscope of FIG. 4; and 
     FIG. 13 is a table which illustrates exemplary design parameters in gyroscope embodiments of the present invention. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     A planar vibratory gyroscope  20  is illustrated in FIGS. 1-4. The gyroscope features a planar vibratory member  22  which is inherently symmetric, which facilitates the use of simple monolithic fabrication processes and which enables the use of sensitive control and sense systems. 
     In addition to the planar vibratory member  22 , the gyroscope  20  includes a substrate  24  which carries the planar member  22 . As particularly shown in FIG. 1, the planar member  22  has a frame  26 , a plate  28  which extends laterally to a plate perimeter  30  and four elongate beams  32  which couple the plate  28  to the frame  26 . 
     The beams  32  are oriented to substantially surround the plate  28  with each of the beams  32  arranged proximate to and everywhere substantially equidistant from the plate perimeter  30  (e.g., the beam  32 A lies equidistant from the perimeter portion  30 A). The substrate  24  has a raised rim  34  which surrounds a face  36 . To support the planar member  22 , its frame  26  abuts and is preferably bonded to the rim  34 . 
     As particularly seen in FIG. 3, the gyroscope  20  has a control electrode system  40  which includes four coplanar control electrodes  41 . The gyroscope also has a vibration sensing system in the form of a sense electrode system  42  which includes four coplanar sense electrodes  43 . The control electrode system  40  and the sense electrode system  42  are arranged in a coplanar relationship and positioned between the plate  28  and the substrate  24 . 
     Each of the control electrodes  41  and the sense electrodes  43  have a triangular shape. The control electrodes  41  are grouped together so that their outer edges  46  define a square shape. The sense electrodes  43  are grouped about the square shape of the control electrodes  41  so that their outer edges  48  define another and larger square shape. The outer edges  48  preferably lie directly below the plate perimeter  30 . The enlarged view of FIG. 4 shows that a conductive sheet  50  covers the underside of the plate  28  so that this conductive sheet  50  is spaced directly above the control and sense electrode systems  40  and  42 . 
     The planar member  22  can be economically fabricated because it forms a system  59  of slots which are arranged to define the frame  26 , the plate  28  and the elongate beams  32 . Each of the slots is partially interleaved with two adjacent slots. For example, the slot  60 B is positioned partially outside of adjacent slot  60 A and partially inside of adjacent slot  60 C. Each of the elongate beams  32  are configured to have a first end  62  that is coupled to the frame  26  and a second end  64  that is coupled to the plate  28 . The elongate beams  32  are oriented so that the second end  64  of each beam is proximate to the first end  62  of an adjacent beam  32 . 
     The plate perimeter  30  is substantially formed by inner portions  60 I of each slot. Imaginary extension lines  66  (shown in broken lines in FIG. 1) of these inner portions indicate the plate perimeter  30  in the areas where the beam second ends  64  couple to the plate  28 . 
     In an exemplary fabrication process, the planar member  22  and the substrate  24  are both formed of silicon. The locations of the slots  60  and the face  36  can be defined with conventional photolithographic techniques and formed by conventional etching techniques. 
     To enhance a description of an exemplary operation process of the gyroscope, it is helpful in FIG. 5 to assign reference numbers  41 A,  41 B,  41 C and  41 D to the control electrodes and  43 A,  43 B,  43 C and  43 D to the sense electrodes. It is also helpful to define an orthogonal coordinate system  70  which has orthogonal axes  71 ,  72  and  73  (axes  71  and  72  are coplanar with the planar member  22  and the axis  73  is orthogonal to the planar member  22 ). 
     The spatial arrangement of the plate  28  and the beams  32  causes the plate  28  to be suited for a first vibration about the axis  71  (i.e., about second ends  64 A and  64 C of nonadjacent beams  32 ) and a second vibration about the axis  72  (i.e., about second ends  64 B and  64 D of nonadjacent beams  32 ). Because the planar member  22  is spatially symmetric, these vibration modes are substantially uncoupled, i.e., if the plate  28  is excited into a vibration about the axis  71  and the gyroscope is not physically rotated, substantially none of the vibration energy will be diverted into a vibration about the axis  72 . This uncoupled feature of the gyroscope  20  reduces its sensitivity to spurious inputs (e.g., external vibration). 
     The symmetry of the planar member  22  also causes the natural vibration frequencies about the first and second axes  71  and  72  to be substantially matched. If a gyroscope&#39;s modes have different vibration frequencies, energy exchange is discouraged which means the gyroscope&#39;s rotation sensitivity is degraded. Accordingly, the high degree of matching of natural vibration frequencies found in the gyroscope  20  enhances its sensitivity. 
     The conductive sheet ( 50  in FIG. 4) on the bottom of the plate  28  forms a capacitor with each of the sense electrodes  43 A-D. The capacitance of these capacitors changes as the plate vibrates about the axes  71  and  72  and this capacitance change can be sensed in any conventional capacitance-sensitive circuit (e.g., a resonant circuit). Thus, vibration of the plate  28  about the axis  71  is sensed as capacitance changes that involve the electrodes  43 A and  43 C. Similarly, vibration of the plate  28  about the axis  72  is sensed as capacitance changes that involve the electrodes  43 B and  43 D. Although the positions of the control and sense electrode systems may be interchanged, positioning the sense electrodes  43  near the plate&#39;s perimeter ( 30  in FIG. 1) enhances the sensing sensitivity. 
     By changing voltage potentials and polarities between the control electrodes  41 A,  41 B,  41 C and  41 D and the plate&#39;s conductive sheet ( 50  in FIG.  4 ), the plate  28  can be attracted to and repelled from each electrode in a controlled manner (to effect this action, the conductive sheet can be coupled through a sheet extension to a potential such as ground). For example, application of alternating voltages to the control electrodes  41 A and  41 C will excite a vibration of the plate  28  about the axis  71 . In the absence of physical rotation of the gyroscope  20 , essentially none of this vibration energy will be coupled into vibration about the axis  72  because of the uncoupled nature of the plate  28 . 
     If the plate  28  is freely vibrating about the axis  71  and the gyroscope  20  is rotated about the axis  73 , some of the energy will be converted by precession into a vibration about the axis  72 . The altered vibration amplitude about the axis  71  is sensed through sense electrodes  43 A and  43 C and the altered vibration amplitude about the axis  72  is sensed through sense electrodes  43 B and  43 D. A combination of these sensed amplitudes is indicative of the precession angle and, hence, of the rotation angle through which the gyroscope was rotated. The above-described operational process of the gyroscope  20  is conventionally referred to as a “whole angle” operational mode. In this mode, vibration amplitude may be sustained, without disturbing orientation of the vibrating pattern (i.e., without inducing erroneous precession), by application of drive voltage to all electrodes at twice the vibration frequency. 
     FIG. 6 is a schematized view of a gyroscope system  80  in which elements of the gyroscope  20  of FIGS. 1-5 are separated to enhance a further understanding of the gyroscope&#39;s operation. In particular the drive electrode system  40  and the sense electrode system  42  have been spaced on opposite sides of the planar member  22  to facilitate descriptions of gyroscopic operational processes. 
     The signal of a voltage generator  82  is applied differentially (indicated by inverter  83 ) to opposed electrodes of the control electrode system  40  to thereby control vibration of the plate  28  about a control axis  71 . In a negative feedback loop  94 , the output from a switch  90  is applied differentially (indicated by inverter  85 ) to opposed electrodes of the control electrode system so as to control vibration of the plate  28  about a sense axis  72 . 
     In a first open loop process, a switch  90  is placed in an open position to deactivate the feedback loop  94 . In response to the signal of the voltage generator  82 , a vibration is induced in the plate  28  about the control axis  71 . When the gyroscope is rotated about the axis  73  of FIG. 5, energy is transferred to a rotation-induced vibration whose rotation-induced axis is orthogonal to the controlled vibration axis. 
     Signals indicative of the rotation-induced vibration are generated by the sense electrode system  42  and coupled through a signal processor  83  to an output port  86 . The signal processor  83  performs necessary signal operations (e.g., capacitance to voltage conversion and vector combination of signals from different pairs of electrodes of the sense electrode system  42 ) to generate a desired signal form at the output port  86 . This output signal is the open loop indication of the gyroscope&#39;s rotation angle. 
     In a second “force to rebalance” operational process, the switch  90  is closed and the output of the signal processor  83  is coupled to the drive electrode system  40 . The amplitude of the processor signal may be adjusted, e.g., through an amplifier  84 . If the gain of the feedback loop  94  is sufficiently high, precession in the planar member  22  of rotation-induced vibration is substantially canceled and the output signal at the output port  86  is therefore indicative of the instantaneous rotation rate. In FIG. 6, therefore, the switch  90  is used to select between open loop and force to rebalance operational processes. 
     In the gyroscope system  80 , the signals of the voltage generator  82  and the feedback loop  94  are applied to opposite electrodes to effect vibration modes about the axes  71  and  72 . In other system embodiments, the voltage generator&#39;s signal can be applied to one adjacent pair of electrodes and the feedback loop&#39;s signal applied to the other adjacent pair. This will effect a 45° rotation of the axes  71  and  72 . 
     FIGS. 7-9 illustrate another vibratory gyroscope embodiment  100  of the present invention. These views are similar to FIGS. 1-3 with like elements represented by like reference numbers. Instead of the triangularly-shaped control electrodes  41  of the gyroscope  20  (of FIG.  1 ), the gyroscope  100  has a control electrode system  102  that includes control electrodes  103 . The latter control electrodes have a square shape, are positioned proximate to one side of the planar member  22  and are arranged so that their outer edges  104  lie directly below the plate perimeter ( 30  in FIG.  1 ). 
     Similarly, the gyroscope  100  has a sense electrode system  106  in which sense electrodes  107  have a square shape, are positioned proximate to an opposite side of the planar member  22  and are arranged so that their outer edges lie directly above the plate perimeter. 
     As in the gyroscope  20 , the planar member&#39;s frame  26  abuts the rim  34  of a substrate  24 . In addition, a second substrate  110  extends over and protects the sense electrode system  106 . The second substrate  110  is similar to the substrate  24  and a rim  112  of the second substrate abuts and is preferably bonded to another side of the planar member&#39;s frame  26 . 
     Because of the larger size of the control electrodes  103  and sense electrodes  107 , the gyroscope  100  is more sensitive than the gyroscope  20  but the additional sensitivity is gained at the cost of increased size and complexity. 
     The vibratory planar member  22  of FIGS. 1 and 2 represents a rectilinear embodiment of the invention&#39;s teachings. These teachings may be extended to various other embodiments. For example, a circular embodiment  122  is illustrated in FIG.  10 . In particular, the planar member  122  has a hollow frame  126 , a plate  128  which extends laterally to a plate perimeter  130  and four elongate beams  132  which couple the plate  128  to the frame  126 . The beams  132  are oriented to substantially surround the plate  128  and each beam  132  is arranged to be everywhere substantially equidistant from the plate perimeter  130  (e.g., the beam  132 A lies equidistant from the perimeter portion  130 A). 
     Similar to the planar member  22 , the planar member  122  can be fabricated with the aid of a system  139  of slots  140  which are arranged to define the frame  126 , the plate  128  and the elongate beams  132  which each have a first end  142  and a second end  144 . Each of the slots  140  is partially interleaved with two adjacent slots and is configured so that the second end  144  of each beam is proximate to the first end  142  of an adjacent beam  132 . The plate perimeter  130  is substantially formed by inner portions  1401  of each slot. 
     In the gyroscope  20  of FIGS. 1-3, a position-sensing system in the form of a sense electrode system  42  was used for sensing rotation-induced vibration modes. The gyroscope  100  of FIGS. 7-9 used a different sense electrode system  106 . Other conventional position-sensing systems can be substituted to form still other gyroscope embodiments. 
     For example, FIG. 11A illustrates a position-sensing system in the form of a tunneling tip  160  whose probe tip  162  is carried by a translation driver in the form of a piezoelectric transducer  164 . The transducer  164  moves the probe tip  164  so as to maintain it in close proximity with the plate  28  to generate a measurable tip-to-plate interaction. An electrical potential is imposed across a tip-to-plate gap  165  and this potential causes tip and plate electrons to form a tunneling current  166  whose magnitude is extremely sensitive to the dimension of the gap  165 . Typically, a control loop responds to the tunneling current  166  by applying a control signal  168  to the transducer  164 . In response to the control signal, the transducer vertically translates the probe tip  162  to maintain a constant tunneling current  166 . The control signal  168  is, therefore, an accurate indicator of the position of the plate  28 . 
     There are numerous variations of the tunneling tip  160  of FIG.  11 A. One is the tip structure  170  shown in FIG.  11 B. In this position-sensing system, a tip probe  172  is coupled by a resilient cantilever  174  to a translation driver  175 . A laser  176  emits a laser beam  177  which is reflected from the cantilever  174  and received by a detector  178 . The output signal of the detector  178  is, therefore, a function of the cantilever&#39;s deflection. Typically, a control loop applies a control signal  179  to the translation driver  175  to maintain a constant detector signal and, hence, a constant cantilever deflection. The control signal  179  is, therefore, an accurate indicator of the position of the plate  28 . 
     An understanding of the operation of gyroscopes of the present invention can be further enhanced by a mathematical investigation of motion in terms of the moment of inertia J and torque T in the vibratory plate  28  of FIG.  5 . Accordingly, FIGS. 12A and 12B are schematics of the control and sense electrode systems of FIG. 5 taken along an exemplary x axis (axis  71  in FIG.  5 ). FIG. 12A shows the sense electrodes  43 A and  43 C that sense rotation about the x axis and FIG. 12B shows the control electrodes  41 A and  41 C that control rotation about the x axis. Electrode voltages along the y axis that are equivalent to those of FIG. 12A are V 3s  and V 4s  and equivalent voltages to those of FIG. 12B are V 3c  and V 4c . The electrodes have an area A, the centroids of the control electrodes are radially offset by r c  and the centroids of the sense electrodes are radially offset by r s . 
     Sense voltages that are generated in FIG. 12A are approximately proportional to plate rotation  x  about the x and y axes (axes  71  and  72  in FIG. 5) and the axial translation z. The sense voltages are expressed as: 
     V xs =V 2s −V 1s =K   ,  x  V ys ≡V 4s −V 3s ≡K     y , and 
     V zs =V 1s +V 2s +V 3s +V 4s ≡K z z in which            K   ϑ     =     2          r   s       d   o                C   o          V   o           C   s     +     C   o             ,       K   z     =       4     d   o                C   o          V   o           C   s     +     C   o             ,       C   o     =         ɛ   o        A       d   o                                
     and C s =input stray capcitance. 
     The equivalent sensor noise angle due to front end electronics noise ε e  is  ne +ε ne /K   . The torques about the x and y axes are proportional respectively to V xc  and V yc ., i.e., 
     V 1c +V xc +V yc +V b , V 2c =−V xc =V b , V 3c =V b , V 4c =−V yc +V b , T x ≡K T V xc  and T y ≡K T V yc            in                 which                   K   T       =     2        r   c                C   o          V   b         d   o       .                              
     Given the rotational variables, 
     ω=[ω x , ω y , Ω]=inertial rate of the substrate  24 , and 
     =[ x ,  y , 0]=small rotations of the plate  28  with respect to the substrate  24 , 
     the relationships 
     {umlaut over ()} x &gt;&gt;{dot over (ω)} x , {umlaut over ()} y &gt;&gt;{dot over (ω)} y , {dot over ()} x &gt;&gt;ω x , {dot over ()} y &gt;&gt;ω y  and Ω&lt;&lt; x  and  y , 
     and with the axes  71 ,  72  and  73  of FIG. 5 represented respectively as x, y and z, simplified equations of motion of the vibratory plate  28  have been derived as: 
     J x {umlaut over ()} x +(J z −J y −J x ){dot over ()} y Ω+k x   x =T x , 
     J y {umlaut over ()} y −(J z −J y −J x ){dot over ()} x Ω+k y   y =T y , and            J   x     =       J   y     =       ρ        (         w   2          h   3       +       w   4        h       )       12         ,       J   z     =       ρ                   w   4        h     6       ,         J   z     -     (       J   x     +     J   y       )       =     -       ρ                   w   2          h   3       6         ,                          
     wherein w and h=width and thickness of the plate  28 , t=width of the beams  32 , E=elastic modulus of the planar member  22  and ρ=density of the planar member  22  and plate inertias and natural frequencies are given by            ω   x     =       ω   y     =             k     ϑ                 y         J   y         ≅           4        Eh   3        t       12      w            12     ρ        (         w   2          h   3       +       w   4        h       )               =         4        Eh   2        t       ρ                     w   3          (       h   2     +     w   2       )                   ,                  ω   z     =             k   z     m       ≅           16        Eh   3        t         w   3        12            1     ρ                   w   2        h             =         4        Eh   3        t       ρ                   w   5        3             ,                  k     ϑ                 y       ≅       4      EI     w       ,         k   z     ≅         16      EI       w   3                     and                 I       =           h   3        t     12     .                              
     To lower the natural frequencies of a thick plate, its associated beams can be thinned by etching from one or both sides of the plate. 
     For a specific quality factor Q m  and a temperature T emp , Brownian motion causes a sensed angular rotation of,          α     y   B     2     =       4        k   B          T   emp          ω   y           J   y          Q   m                                
     in which k B  is Boltzman&#39;s constant. 
     For ideal control and torque rebalance, 
     {dot over ()} x = 0 ω x  cos(ω x t) and  y ≡{dot over ()} y ≡0, 
     respectively, which produces                T   y     =       2          J   t          (     1   -       J   z       2        J   t           )            Ωϑ   o          ω   x          cos        (       ω   x        t     )         =     2        J   t        k                 Ω                   ϑ   o          ω   x          cos        (       ω   x        t     )                       =         K   r        Ω                   cos        (       ω   x        t     )         =         ρ                   w   2          h   3       6        Ω                   ϑ   o          cos        (       ω   x        t     )                             in                 which                   J   t       =       J   y     =     J   x         ,                  the                 angular                 gain                 k     =       1   -       ρ                   w   4        h       ρ        (         w   2          h   3       +       w   4        h       )           =       γ   2         γ   2     +   1           ,                γ   =     h   w       ,       and                   K   r       =     2      k                   ϑ   o          ω   x            J   t     .                                
     It is noted that the angular gain approaches 0 for very thin plates and is ½ for a plate in the shape of a cube. 
     With ideal torque rebalance, the demodulated control voltage is,                      V   yc          (     2      cos                   ω   x        y     )       _     ≅                      T   y          (     2      cos                   ω   x        t     )       _       K   T         =       2        J   t        k                 Ω                   ϑ   o          ω   x         2        r   c              C   o          V   b         d   o                       =                        J   t        k                   ϑ   o          d   o          ω   x           r   c          C   o          V   b            Ω     =       K   sc          Ω   .                                      
     wherein the overbar indicates a baseband signal. 
     Alternatively, with ideal open loop operation T y =0, the demodulated sense voltage is:                    V   ys          (     2                   cos        (         ω   x        t     +     φ   o       )         )       _     ≅                    K   ϑ            ϑ   y          (     2        cos        (         ω   x        t     +     φ   o       )         )         _                 =                  K   ϑ          Q   t            2      k                   ϑ   o          ω   x         ω   y   2          Ω                 =                2          r   s       d   o              C   o         C   s     +     C   o              Q   t            2      k                   ϑ   o          ω   x         ω   y   2          Ω                   =                  K   so        Ω       ,                                
     in which φ O  is the sense mode phase shift at the drive frequency. 
     Q t  is the tuning gain          Q   t     =     1           (         (       ω   x       ω   y       )     2     -   1     )     2     +       (     Q   m     )       -   2                                    
     between the control and sense modes, the equivalent Brownian noise rate noise Ω nB  is          Ω   nB     =           J   t          α   nB         K   r       =         α   nB       2      k                   ϑ   o          ω   x                (     rad   /   sec     )     /     Hz                                  
     and the equivalent electronic rate noise Ω ne  is          Ω   ne     =         ϑ   ne       Q   t       =         ɛ   ne         Q   t          K   ϑ                (     rad   /   sec     )     /       Hz     .                                  
     An exemplary relationship for Q t  is            Q   t     ≥   100     ,       when                            ω   x     -     ω   y              ω   y         ≤     0.5      %                 and                   Q   m       &gt;   1000     ,                          
     and exemplary design parameters include the following: 
     
       
         
               
               
             
           
               
                   
               
             
             
               
                 control amplitude υ o  = 0.1° 
                 mechanical quality Q m  = 50,000 
               
               
                   
                 (typical for crystal silicon in 
               
               
                   
                 vacuum) 
               
               
                 applied voltages V o  = V b  = 10v 
                 stray capacitance C s  = 1pf 
               
               
                 electronic noise ε ne  = 100 nv/Hz 
                 tuning gain Q t  = 100. 
               
               
                   
               
             
          
         
       
     
     Performance of planar gyroscopes is degraded by the presence of noise signals, e.g., electronic noise and Brownian (thermal) noise. In the planar gyroscope  20  of FIGS. 1-4, electronic noise is reduced by increasing the width of the plate  28  and Brownian noise is generally reduced by increasing the plate&#39;s thickness. A thick plate also enhances angular gain k and, hence, scale factor sensitivity (e.g., when w=h, k=½). In contrast, a thinner plate will reduce the manufacturing time required to etch the plate&#39;s slot system  59 . These are examples of various considerations which influence design selections of plate width w, plate thickness h, beam width t and the elastic modulus E and density ρ of the planar member  22 . 
     Table 200 of FIG. 13 illustrates selected values for some of these parameters in three exemplary designs of the gyroscope embodiments of the present invention. For low cost stellar inertial navigation, a gyroscopic angle random walk of &lt;0.03°/hour is typically required. Design 1 in table 200, for example, has a thin plate which achieves this requirement. 
     It is estimated that this design can be realized at a cost of ˜100 dollars/wafer with polished silicon wafers having a thickness of ˜400 micrometers. It is further estimated that more than 1000 planar members can be batch fabricated and packaged using three six-inch wafers in ˜2 hours based on an ion etch rate of ˜3 micrometers/hour. Designs 2 and 3 gain improved performance with thicker and wider plates and thicker beams. 
     Lack of mechanical precision is a source of temperature-sensitive biases and drift in vibratory gyroscopes. Accordingly, bias and drift are reduced in the present invention because of its symmetry and its compatibility with photolithographic fabrication processes. 
     Although the planar members of the invention have been shown to have four beams of similar cross section, the teachings of the invention can be extended to configurations that have different numbers of beams and beams that have different cross sections. The beams have been shown and described as being coplanar with the plate (e.g., the beams  32  and the plate  28  of FIG.  2 ). Although this configuration may facilitate fabrication of the invention, nonplanar embodiments may also find utility when practicing the teachings of the invention. The conductive sheet  50  of FIG. 4 can be applied with various conventional materials and processes (e.g., silicon doping). It can be coupled to ground with various conventional structures (e.g., a sheet extension or a thin wire). 
     While several illustrative embodiments of the invention have been shown and described, numerous variations and alternate embodiments will occur to those skilled in the art. Such variations and alternate embodiments are contemplated, and can be made without departing from the spirit and scope of the invention as defined in the appended claims.