Abstract:
An apparatus and/or method that corrects for tuning errors in vibrating structure gyroscopes that are configured to be driven along a plurality of axes without the need for dedicated torque elements. The correction is accomplished by introducing a phase offset in the drive signal of one or more of the drive elements relative to other drive elements to minimize or reduce the quadrature signal. The tuning may be accomplished as a one time “set and forget” adjustment, as a manual adjustment performed at the discretion of the user, or as a closed loop active correction system. The technique improves the tuning of the resonator assembly, rather than merely compensating for the mistuning. Accordingly, for various embodiments of the invention, the destructive interference between the plurality of drive axes is reduced. Conservation of vibrational energy present in the resonator is thus enhanced, with less vibrational energy transferred to the support structure.

Description:
RELATED APPLICATION 
     The present application claims the benefit of U.S. Provisional Patent Application No. 61/105,236, filed on Oct. 14, 2008, the disclosure of which is hereby incorporated by reference in its entirety. 
    
    
     FIELD OF THE INVENTION 
     The present disclosure is directed generally to the field of gyroscopes. More specifically, the present disclosure is directed to a vibrating structural gyroscope having quadrature control. 
     BACKGROUND OF THE INVENTION 
     Vibrating structural gyroscopes (VSGs) have found use in a number of applications involving the detection of rotational rate and position, including attitude sensors and gyrocompasses. Rotational rate and/or position is typically determined by exciting a resonator along one or more drive axes to drive the resonator into an oscillation or vibration pattern and detecting a change in the output signal. The output signal may include a “quadrature” signal or component, herein defined as the component of a complex signal that is 90 degrees out of phase in the time domain with the in-phase component. This quadrature signal is generally considered an unwanted signal that can cause output errors. 
     Determination of quadrature signal from a rotation rate signal is known in the art. The quadrature signal may be obtained by demodulating the rotation rate signal out-of-phase with respect to the drive oscillation. Such determination of the quadrature signal by demodulation is presented in greater detail in a paper by Dr. D. D. Lynch, “Coriolis Vibratory Gyros,” presented at Symposium Gyro Technology, Stuttgart, Germany, 1998 (Lynch), and by U.S. Pat. No. 5,629,472 to Varnham et al. (Varnham). 
     Some VSGs are configured to drive the resonator assembly along a plurality of drive axes, with the drive axes being offset with respect to each other or, alternatively, being substantially coincident (i.e. defining substantially the same axis in three-dimensional space). The resonant frequency of a VSG resonator assembly will typically differ between respective drive axes. For example, a resonator assembly being driven along two in-phase drive axes typically has a different resonant frequency for the first drive axis than for the second drive axis; that is, each drive axis of the resonator assembly is said to have a unique resonant frequency. The resonator assembly may be designed and manufactured so that the resonant frequencies of the respective drive axes are close. Tuning processes may also be practiced to bring the respective resonant frequencies even closer. However, the resonant frequencies may never be perfectly tuned, particularly over a range of temperatures, because temperature can change the characteristics of the materials of the resonator assembly and cause a degradation of the tuning of the resonator assembly. This degradation and the resulting quadrature signal has been reported in the literature Lynch. Therefore, a quadrature signal can appear when both axes of a dual axis system are driven, resulting in an errant indication of rotation rate and/or rotational position. 
     A variety of options are available to the artisan to counter the effects of quadrature signals. Some methods focus on altering the vibration characteristics of the resonator. For example, Varnham discloses correction of the quadrature component by mechanically adjusting the resonator mass or mass distribution to tune the gyroscope. U.S. Pat. No. 4,951,508 to Loper et al. (Loper) discloses correcting the quadrature component by electrically adjusting the spring stiffness to tune the gyroscope. Other U.S. Patents and Published Patent Applications (e.g. U.S. Pat. Nos. 6,481,285 and 6,934,660 and U.S. Patent Application Publication No. 2007/0089510) also disclose this technique. 
     Likewise, U.S. Pat. No. 6,883,361 to Wyse (Wyse) discloses a method and apparatus whereby a DC voltage is introduced near a vibrating ring resonator to incite an oscillating force from the vibration, which coincidentally alters the stiffness of the vibrating element, which can be used to cancel quadrature component. Wyse discloses a “set and forget” system, with no dynamic adjustment for automatic or feedback control. Also, U.S. Pat. No. 6,675,630 to Challoner, et al. (Challoner) discloses a method and apparatus whereby a quadrature signal is applied as a DC bias voltage to affect a phase offset in the drive loop. The methods disclosed by Wyse and Challoner require at least one extra electrode in addition to the drive electrodes to accomplish the stiffening. 
     Other techniques focus on electronically compensating for the quadrature signal. For example, U.S. Pat. No. 7,120,548 to Malvern et al. (Malvern) discloses a technique whereby the quadrature signal is minimized by feeding a quadrature-correcting phased signal to a dedicated “torquing” element, causing a vibration that interacts with the driven oscillation pattern to drive the quadrature signal to a minimum, thereby actively correcting for the mistuning. U.S. Pat. No. 7,240,533 to Fell et al. (Fell I) presents a variation of this technique by including a phase corrector in the torque control loop that drives the quadrature torque energy directly into the sensed quadrature signal to correct for the effects of the quadrature component. Other examples where a quadrature signal is added to the torque signal to correct for quadrature signal include U.S. Pat. Nos. 7,188,522 and 7,216,525. 
     U.S. Pat. No. 7,231,823 to Schroder (Schroder) discloses a system wherein a “disturbance component” of the read signal is measured and a frequency offset is implemented as needed to match the disturbance. U.S. Pat. Nos. 7,249,488 and 7,337,665 disclose systems similar to Schroder. 
     Other systems, such as disclosed in “A Second Generation Silicon Ring Gyroscope” by C. Fell, I. Hopkins and K. Townsend (Fell II), utilize phase locked loops which control the oscillator so that there is either no phase difference or a known phase difference between the drive frequency and the oscillator frequency. Such systems are constantly being adjusted to lock in the phase relationship, and are therefore subject to phase jitter in the phase locked loop. 
     The disclosures of the above-mentioned patents and publications are hereby incorporated by reference herein in their entirety except for explicit definitions contained therein as follows: U.S. Pat. No. 5,629,472 (Varnham), U.S. Pat. No. 4,951,508 (Loper), U.S. Pat. No. 6,883,361 (Wyse), U.S. Pat. No. 6,675,630 (Challoner), U.S. Pat. No. 7,120,548 (Malvern), U.S. Pat. No. 7,240,533 (Fell I), U.S. Pat. No. 7,231,823 (Schroder), and paper by Dr. D. D. Lynch, “Coriolis Vibratory Gyros,” presented at Symposium Gyro Technology, Stuttgart, Germany, 1998 (Lynch). 
     U.S. Pat. No. 7,526,957 and U.S. Patent Application Publication No. 2007/0256495 to Watson (collectively “Watson”), both assigned to the assignee of the instant application and hereby incorporated by reference herein in their entirety except for explicit definitions contained therein, disclose drive axes that are rotationally skewed relative to the antinode axes of the vibration pattern when the VSG is rotationally at rest. The skewed axes enable the drive elements of a multiple drive axis system to affect a torquing function in addition to sustaining the oscillation pattern, thus eliminating the need for separate dedicated torque elements. Elimination of dedicated torque elements simplifies the resonator assembly and can provide a mirrored symmetry about a plurality of drive axes for more uniform propagation of vibration between the various nodes and antinodes of the system. Watson further discloses a method for minimizing or reducing the signals at the nodes of the oscillation pattern by changing the relative amplitudes of the drive signals along respective skewed drive axes, thus shifting the position of the node on the resonator. 
     The above disclosed techniques and systems that focus on electronically compensating for the quadrature signal do not improve the tuning of the gyroscope. That is, each of the disclosures imposes a force (e.g., the separate torque elements of Fell I or the differing amplitudes of Watson) or simply establishes the error as a known quantity (e.g., the phase-lock system of Fell II). None of these systems or techniques improve the tuning of the resonator assembly. The quadrature signal itself is indicative that energy introduced in the plurality of drive axes is dissipated in destructive interference. Systems that introduce additional forces to accomplish the compensation introduce still more energy that is also dissipated in destructive interference. Such dissipated energy is in many instances transferred to the structure supporting the resonator assembly, and can be reflected back to the resonator, causing additional signals of arbitrary phase that results in further biasing error in the rotational rate signal. 
     A vibrating structural gyroscope system that electronically improves the tuning of the VSG with respect to the inherent tuning error represented by the quadrature signal, rather than merely attempting to compensate for the inherent tuning error of the VSG, would be welcome. 
     SUMMARY OF THE INVENTION 
     Various embodiments of the invention include an apparatus and/or method that corrects for tuning errors in vibrating structure gyroscopes that are configured to be driven along a plurality of axes by correcting for the quadrature signal directly along the input of the drive axes and without use or need for dedicated torque elements. The correction is accomplished by introducing a phase offset in the drive signal of one or more of the drive elements relative to other drive elements to minimize or reduce the quadrature signal. The tuning may be accomplished as a one time “set and forget” adjustment, as a manual adjustment performed at the discretion of the user, or as a closed loop active correction system. The technique improves the tuning of the resonator assembly, rather than merely compensating for inherent tuning errors. Accordingly, for various embodiments of the invention, the destructive interference between the plurality of drive axes is reduced. Conservation of vibrational energy present in the resonator is thus enhanced, with less vibrational energy transferred to the support structure. 
     A resonator assembly typically comprises a resonator element with drive and sense elements. Each of these components can introduce non-uniformities to the resonator assembly. For example, the resonator element may have a thickness or density that varies over the vibrating portion. Also, the drive and/or sense elements may also be of varying thickness or density and/or of non-uniform size. Machining errors and mask misalignment can also contribute to non-uniformity. Furthermore, the various appurtenances that interface with the resonator assembly (e.g., wires, contact tabs) and that hold the assembly together (e.g., solder, adhesives for the elements) can also be of differing mass between the elements, thereby introducing further non-uniformities to the resonator assembly. Moreover, thermal gradients (transient or steady state) that are present on the resonator assembly can introduce further non-uniformities. Accordingly, a given portion of the resonator assembly may have a resonant frequency that differs from the counterpart portions of the resonator and/or the resonator as a whole. 
     The non-uniformities of the resonator assembly can cause a quadrature signal to appear at the sense element. A phase offset in the time domain between the resonator as a whole and a portion of the resonator can manifest itself as a distortion of the sense signal in an out-of-phase (i.e. quadrature) vibration pattern. The out-of-phase vibrational pattern thus imposes a vibrational amplitude at the location of the node that results in a quadrature signal. 
     Without limiting the present application to a particular theory, the aforementioned non-uniformities are believed to cause different resonant frequencies when the resonator assembly is driven along different axes. Consider a resonator assembly that is driven into resonance along two independent drive axes. The resonant frequency of the resonator assembly is often different when driven along the first drive axis than when driven along the second drive axis, and again different when driven along both drive axes. 
     Embodiments of the present invention take advantage of the variation between the resonant frequencies of a resonator assembly generated along two (or more) drive axes. The phase difference between the vibration characteristics can be used to correct the input drive signals of the drive elements along the respective axes so that phase difference and the attendant quadrature signal is reduced, minimized or substantively eliminated. 
     Structurally, various embodiments of the invention comprise a resonator assembly that is driven along at least two drive axes, such as axisymmetric resonators (e.g., ring or cup and tuning fork configurations). The axes may be coincident, or they may be offset with respect to each other. Herein, an “axisymmetric resonator” is one that defines a central axis and wherein the mass of the resonator is distributed substantially equally on both sides of any plane that includes the central axis. 
     In one embodiment, a method for controlling the quadrature of a vibrating structure gyroscope is implemented that involves providing a resonator assembly including an axisymmetric resonator and a plurality of drive elements operatively coupled with the resonator, the drive elements adapted to sustain a resonant oscillation pattern on the axisymmetric resonator. In this embodiment, a first of the drive elements is adapted to be driven along a first drive axis and a second of the plurality of drive elements adapted to be driven along a second drive axis. The first and second drive axes may be coincident or may be offset with respect to each other. At least one sense element is operatively coupled with the axisymmetric resonator and adapted to detect a rotation rate. In some embodiments, the at least one sense element can be adapted to sense a driving oscillation of the resonator assembly as well. The method further comprises driving the resonator assembly along the first drive axis with the first of the drive elements in accordance with a first drive signal, and driving the resonator assembly along the second drive axis with the second of the drive elements in accordance with a second drive signal. A rotation rate signal is measured while the resonator assembly is driven along the first and second drive axes and quadrature signal is inferred from the rotation rate signal. The method further comprises imposing a combined phase offset between the first and second drive signals so that the quadrature signal is maintained at a desired level. The combined phase offset may be accomplished by imposing a first phase offset on the first drive signal and a second phase offset on the second drive signal, the second phase offset being opposite of and substantially equal to the first phase offset. The quadrature signal can be maintained at a minimum magnitude, or maintained at a known but tolerable level for detection of demodulation phase errors. 
     In some embodiments, the method further includes providing at least one drive sensor adapted to sense a driving oscillation of the resonator assembly, providing a phase-locked loop driving system having an input operatively coupled with the at least one driving sensor and having outputs operatively coupled to the first and the second of the plurality of drive elements and causing the phase-locked loop driving system to output the first and the second drive signals, the first and the second drive signals having a frequency that substantially matches the frequency of the drive oscillation. 
     In certain embodiments, the drive elements of the resonator assembly are adapted to sustain the oscillation pattern to include a plurality of anti-node pairs that define a plurality of reference axes when the resonator assembly is rotationally at rest, each of the anti-node pairs being diametrically opposed about the central axis, each of the plurality of reference axes passing through a corresponding one of the plurality of anti-node pairs. In these embodiments, the first drive axis of the resonator assembly is offset by a first rotational offset relative to a first of the plurality of reference axes, the first drive axis being other than coincident with any of the plurality of reference axes. Also, the second drive axis of the resonator assembly provided in the step of providing is offset by a second rotational offset relative to a second of the plurality of reference axes, the second drive axis being other than coincident with any of the plurality of reference axes, the second rotational offset being in a direction opposite from the first rotational offset. 
     A quadrature controlled vibrating structure gyroscope (QCVSG) is disclosed in an embodiment of the invention. The QCVSG comprises a resonator assembly including an axisymmetric resonator defining a central axis. The resonator assembly may include (but is not limited to) a resonator selected from the group consisting of a cup resonator, a hemispherical resonator, a ring resonator, a two-tine fork resonator and a four-tine fork resonator. The QCVSG may include a plurality of drive elements operatively coupled with the resonator and adapted to sustain a resonant oscillation pattern on the axisymmetric resonator, a first of the drive elements adapted to be driven along a first drive axis, a second of the plurality of drive elements adapted to be driven along a second drive axis. The first and second drive axes may be substantially coincident or offset with respect to each other. The offset can be a rotational offset. At least one sense element is operatively coupled with the axisymmetric resonator, the at least one sense element being adapted to detect a rotation rate signal. The QCVSG further includes a control system operatively coupled with the plurality of drive elements and adapted to sustain an oscillation pattern on the axisymmetric resonator and to infer a quadrature component from said rotation rate signal, the control system further being adapted to control a combined phase offset between the first and second drive elements for control of the magnitude of the quadrature component. 
     In some embodiments, the QCVSG is configured so that the control system and the drive elements are adapted to sustain the oscillation pattern to include a plurality of anti-node pairs that define a plurality of reference axes when the resonator assembly is rotationally at rest, each of the anti-node pairs being diametrically opposed about the central axis, each of the plurality of reference axes passing through a corresponding one of the plurality of anti-node pairs. The first drive axis can be offset by a first rotational offset relative to a first of the plurality of reference axes, the first drive axis being other than coincident with any of the plurality of reference axes. The second drive axis can be offset by a second rotational offset relative to a second of the plurality of reference axes, the second drive axis being other than coincident with any of the plurality of reference axes, the second rotational offset being in a direction opposite from the first rotational offset. In some embodiments, the control system of the QCVSG is operatively coupled with the at least one sense element, the control system being a closed loop control system that utilizes the quadrature signal as a feedback signal. The QCVSG may further include an automatic gain control adapted to provide the amplitude of the drive signal, a quadrature phase adjustment source adapted to provide the phase offset, a sine wave reference adapted to provide a SIN(ωt) function, and a cosine wave reference adapted to provide the COS(ωt) function. 
     In one embodiment, the QCSVG is adapted for control by a central microprocessor. The central microprocessor is operatively coupled to the axisymmetric resonator and a computer-readable medium. In this embodiment, the computer-readable medium includes instructions for control of the quadrature controlled vibrating structure gyroscope. The instructions include: driving the resonator assembly along the first drive axis in accordance with a first drive signal; driving the resonator assembly along the second drive axis in accordance with a second drive signal; measuring a rotation rate signal while the resonator assembly is driven along the first and second drive axes; inferring a quadrature signal from the rotation rate signal; and imposing a combined phase offset between the first and second drive signals so that the quadrature signal is maintained at a desired level. The desired level may be of minimum magnitude. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  depicts a cup resonator assembly that is driven along two drive axes; 
         FIG. 2  depicts the vibration amplitude and phase vs. frequency for an ideal VSG having an overall resonant frequency of ω N ; 
         FIG. 3  depicts the vibration amplitude and phase vs. frequency for a VSG driven along a first drive axis that produces a resonant frequency ω N1  that is at a lower frequency than the overall resonant frequency ω N  of the VSG in an embodiment of the invention; 
         FIG. 4  depicts the vibration amplitude and phase vs. frequency for the VSG characterized in  FIG. 3  driven along a second drive axis that produces a resonant frequency ω N2  that is at a higher frequency than the overall resonant frequency ω N  of the VSG in an embodiment of the invention; 
         FIG. 5  depicts the phase deviations of the vibration amplitude and phase vs. a composite frequency of the VSG characterized in  FIGS. 3 and 4  driven along the first and second drive axes in an embodiment of the invention; 
         FIG. 6  is a schematic of a quadrature control drive circuit operatively coupled to a cup gyroscope in an embodiment of the invention; 
         FIG. 7  depicts a 4-tine fork resonator assembly that is driven along two diagonal axes; 
         FIG. 7A  is a plan view of the 4-tine fork resonator and drive elements of  FIG. 7 ; 
         FIG. 8  is a schematic of a quadrature control drive circuit operatively coupled to the drive elements of a 4-tine fork gyroscope in an embodiment of the invention; 
         FIG. 9  is a schematic of a cup gyroscope having drive elements arranged along skewed drive axes and a quadrature control drive circuit utilizing a phase locked loop driving system in an embodiment of the invention; 
         FIG. 10  is a perspective view of a cup gyroscope having a sense element located at the same radial location as a drive element for use in an embodiment of the invention; 
         FIG. 11  is a perspective view of a 2-tine fork resonator assembly; and 
         FIG. 12  is a schematic of a quadrature-controlled vibrating structure gyroscope implementing a 2-tine quadrature control system in an embodiment of the invention. 
     
    
    
     DETAILED DESCRIPTION 
     Referring to  FIG. 1 , an example of a resonator assembly  18  that can be driven along two independent drive axes  20  and  22  is depicted to establish nomenclature. The resonator assembly  18  includes a resonator element  24  (depicted as a cup resonator), a first pair of drive elements  26 ,  28 , a second pair of drive elements  30 ,  32 , and various sense element pairs  34   a ,  34   b  and  35   a ,  35   b . Both drive element pairs  26 ,  28  and  30 ,  32  are operatively coupled with the resonator element  24 , the resonator element  24  defining a central axis  36  about which rotation of the resonator assembly  18  is sensed. 
     The drive elements  26 ,  28  and  30 ,  32  of each drive element pair are positioned in diametric opposition to each other about the central axis  36 . In this embodiment, the first and second drive element pairs  26 ,  28  and  30 ,  32  define the first drive axis  20  and the second drive axis  22 , respectively. Herein, a “drive axis” defines a vector along which a forcing function is imposed to sustain an oscillation pattern on a given resonator assembly. Accordingly, for the resonator assembly  18  depicted in  FIG. 1 , each of the drive axes  20  and  22  passes substantially through the center of the drive elements of the respective drive element pair  26 ,  28  and  30 ,  32 . It is noted that for other configurations drive axes do not pass through the centers of drive elements (see, e.g.,  FIGS. 7 ,  7 A and  8  and attendant discussion). 
     The drive axes  20  and  22  of the resonator assembly  18  are oriented at substantially 90° with respect to each other. More generally, the second drive axis  22  of the  FIG. 1  embodiment is located at an antinode location of a vibration pattern generated by the first drive axis  20 , and vice-versa. The sense element pairs  34   a ,  34   b  and  35   a ,  35   b  may be located proximate the node location of the vibration pattern and each adapted to generate respective output signals  39   a ,  39   b ,  41   a  and  41   b  indicative of the local oscillation (amplitude and frequency). The sense element pairs  34   a ,  34   b  and  35   a ,  35   b  may also define a pair of sense element axes  40  and  42 , respectively. 
     Referring to  FIGS. 2 through 5 , various amplitude and phase characteristics of a resonator assembly (e.g., resonator assembly  18 ) as a function of the drive frequency are described. Each of the  FIGS. 2 through 5  present an amplitude ratio ordinate  46  (logarithmic) and a phase ordinate  48  (linear), both referenced against a frequency ratio abscissa  50  (logarithmic). The amplitude ratio ordinate  46  represents the ratio of the vibration amplitude of the resonator assembly to the amplitude of the drive element. The frequency ratio abscissa  50  represents the ratio of the drive frequency ω to the resonant or natural frequency ω N  of the resonator assembly. An amplitude curve or function  52  (or  52   a ,  52   b  or  52   c ) is plotted against the amplitude ordinate and a phase curve or function  54  (or  54   a ,  54   b  or  54   c ) is plotted against the phase ordinate  48 . 
       FIG. 2  depicts ideal amplitude and phase characteristics  56 , i.e. the characteristics that an ideal resonator. Note that the amplitude function  52  is at unity with and in phase with the drive amplitude at low frequency ratios (e.g., the amplitude ratio of 1 at the frequency ratio of 0.1 in  FIG. 2 ). At high frequency ratios, the amplitude function  52  is very low relative to the drive amplitude, and lags in phase by 180 degrees as represented by the phase function  54  (e.g., the negligible amplitude ratio at the frequency ratio of 10 in  FIG. 2 ). When the drive frequency is at the resonant frequency ω N  of the resonator assembly (frequency ratio of unity), the amplitude function  52  are much higher than the drive amplitude, and the phase function  54  depicts that the lag of the vibrations of the resonator lag the drive amplitude by −90 degrees. 
     Consider now a non-ideal resonator that is driven along two or more drive axes, and having amplitude and phase characteristics as depicted in  FIGS. 3 and 4 , respectively. While the resonator may have an overall or system resonant frequency of ω N  when driven along all the drive axes, the resonant frequency of the resonator may be different from the system resonant frequency ω N  when driven only along a first of the drive axes. That is, the resonator may have a first axis-driven resonant frequency ω N1  ( FIG. 3 ) and a second axis-driven resonant frequency ω N1  ( FIG. 4 ) that differ substantially from each other and from the system resonant frequency ω N . 
     Here, the amplitude function  52   a  of the first axis-driven frequency characteristics  58  of the resonator assembly ( FIG. 3 ) are depicted as having a decreased magnitude at the system resonant frequency ω N  relative to the magnitude of the amplitude function  52   a  at ω N1 . The phase function  54   a  of the first axis-driven frequency characteristics  58  is said to have a phase offset α at the system resonant frequency ω N  such that the phase function  54   a  is at −(90+α) (that is, −90−α) at ω=ω N . 
     Likewise, the amplitude function  52   b  of the second axis-driven frequency characteristics  60  of the resonator assembly ( FIG. 4 ) is also depicted as having a decreased magnitude at the system resonant frequency ω N  relative to the magnitude of the amplitude function  52   b  at ω N2 , but with the phase function  54   b  at −(90−α) (that is, −90+α) at the system resonant frequency ω N . 
     For dual drive axis systems such as depicted in  FIG. 1 , the resonant frequency ω N  of the resonator assembly  18  when driven along both drive axes  20  and  22  may be roughly the average of the first and second axis-driven resonant frequencies ω N1  and ω N2 . The phase functions  54   a  and  54   b  may be said to have a combined phase offset  64  relative to each other and having magnitude of 2α, with the 90° phase lag being arbitrarily chosen at the midspan of the 2α interval ( FIG. 5 ). 
     A method of the various embodiments of the invention is to drive each of the drive axes (e.g.,  20  and  22 ) at an appropriate phase relative to each other or relative to the aggregate characteristics of the resonator assembly to compensate for the differences between the respective axis-driven characteristics  58  and  60 . By establishing the proper phase difference between the drive functions imposed on the respective drive axes, a tuned amplitude and phase characteristic  62  having an amplitude function  52   c  and a phase function  54   c  results that more closely approximates the ideal amplitude and phase characteristics  56 . 
     In some embodiments, the phase compensation takes the form of a preset signal established to minimize the quadrature signal at a nominal operating condition. In other embodiments, the phase compensation may be manually set for an existing operating condition to reduce the quadrature signal. In still other embodiments, the phase compensation is provided as a closed loop active correction system for dynamic minimization or reduction of the quadrature signal across a range of operating conditions. 
     A phase compensation technique for resonator assemblies that are driven along two axes is now described mathematically. To affect correction of the quadrature signal in accordance with certain embodiments of the invention, it is desired that a total phase difference equal to the combined phase offset  64  of 2α be sustained between the first and second drive axes. Accordingly, desired drive signals D 1  and D 2  for the first and second drive axes, respectively, are expressed as:
 
 D 1 =K ·SIN(ω t +α)  Eqn. (1)
 
 D 2 =−K ·SIN(ω t −α)  Eqn. (2)
 
where K is the amplitude of the drive signal (e.g., in volts), ω is the drive frequency and t is a time parameter. The trigonometric identity for the sine of the sum of two angles is:
 
SIN(ω t +α)=COS(α)SIN(ω t )+SIN(α)COS(ω t )  Eqn. (3)
 
For small angles α, Eqn. (3) simplifies to
 
SIN(ω t +α)=SIN(ω t )+α·COS(ω t )  Eqn. (4)
 
Therefore, the drive signals D 1  and D 2  may be expressed as
 
                     D   ⁢           ⁢   1     =       K   ·     SIN   ⁡     (     ω   ⁢           ⁢   t     )         +     K   ⁢           ⁢     α   ·     COS   ⁡     (     ω   ⁢           ⁢   t     )                     Eqn   .           ⁢     (   5   )                         D   ⁢           ⁢   2     =         -   K     ·     SIN   ⁡     (     ω   ⁢           ⁢   t     )         +     K   ⁢           ⁢     α   ·     COS   ⁡     (     ω   ⁢           ⁢   t     )                         =         -   K     ·     SIN   ⁡     (     ω   ⁢           ⁢   t     )         +     P   ·     COS   ⁡     (     ω   ⁢           ⁢   t     )                         Eqn   .           ⁢     (   6   )                 
The expression Kα·COS(ωt) is herein referred to as the phase adjustment component P·COS(ωt), where P=Kα and is the amplitude of the quadrature phase adjustment signal (e.g., numerical reference  90  at  FIG. 6 ).
 
     Accordingly, the trigonometric functions of the drive signals D 1  and D 2  of Eqns. (5) and (6) are expressed solely in terms of the drive frequency-time product ωt, so that the various components of the drive signals D 1  and D 2  are readily implemented by electronic circuitry and/or computer control. Example embodiments that implement Eqns. (5) and (6) are presented below. 
     Referring to  FIG. 6 , a quadrature-controlled vibrating structure gyroscope (QCVSG)  70  is depicted in an embodiment of the invention. The QCVSG  70  includes the resonator assembly  18  operatively coupled to a quadrature control system  72 . In the depicted embodiment, the quadrature control system  72  includes: a sine wave reference  76  for outputting a sine wave reference signal  78 ; a cosine wave reference  80  for outputting a cosine wave reference signal  82 ; an automatic gain control (AGC)  84  for outputting an AGC signal  86  that affects the amplitude of the drive signal K; and a quadrature phase adjustment source  88  for outputting a quadrature phase adjustment signal  90  that affects the amplitude P of the phase adjustment component P·COS(ωt). The sine wave reference signal  78  and the AGC signal  86  may be routed through a first signal multiplier  94  to produce a variable gain sine wave signal  96 . The cosine wave reference signal  82  and the quadrature phase adjustment signal  90  may be routed through a second multiplier  98  to produce a phase adjustment cosine signal  100 . 
     In this embodiment, the phase adjustment source  88  can comprise a demodulator that accepts and demodulates the output signal  39   a  indicative of rotation rate in an out-of-phase manner with respect to the drive oscillation to provide the quadrature signal. The phase adjustment source  88  may also include a filter system to provide a conditioned quadrature phase adjustment signal  90 . While only one rotation rate signal is depicted as being utilized for the phase adjustment source  88 , those skilled in the art will recognize that any or all of the output signals  39   b ,  41   a  and  41   b  may optionally be utilized as well. 
     In certain embodiments, the variable gain sine wave signal  96  and the phase adjustment cosine signal  100  are routed in parallel to first and second adders  101  and  102 , respectively, to produce the first and second drive signals D 1  and D 2 , respectively, with the variable gain sine wave signal  96  being routed through an inverter  103  to affect the subtraction operation of Eqn. (6). The first and second drive signals D 1  and D 2  are routed to the first and second drive element pairs  26 ,  28  and  30 ,  32 , respectively. 
     The various components of the QCVSG  70  may comprise hardware electronics that execute the various functions of the QCVSG  70  without need for a central processor. Alternatively, the QCVSG  70  may be adapted for control by a central microprocessor μ that controls the QCVSG  70  pursuant instructions (e.g., software or firmware) stored on a computer-readable medium  99 . 
     In operation, the quadrature control system  72  executes the phase compensation described at Eqns. (5) and (6). The various control components of the quadrature control system  72  can relate to the various variables and functions of Eqns. (5) and (6) as follows: The amplitude of the drive signal K may be affected by the AGC  84 ; the phase offset α may be affected by the quadrature phase adjustment source  88 ; the SIN(ωt) function may be affected by the sine wave reference  76 ; and the COS(ωt) function may be affected by the cosine wave reference  80 . 
     The foregoing focuses on the manifestation of Eqns. (5) and (6). It is understood, however, that other circuitry and/or computer controlled embodiments may be implemented to affect any of the Eqns. (1) through (6) or their equivalents, and still be within the scope of the invention. 
     While the resonator assembly in the embodiment of  FIG. 6  is depicted as having drive element pairs and sense element pairs, it is understood that single elements may be utilized in place of any of the drive or sense element pairs. The respective drive and/or sense element axes in such an arrangement is defined as substantially normal to the single respective element and may pass near or through the central axis. 
     Referring to  FIG. 7 , a 4-tine fork resonator assembly  104  having four tines  105   a ,  105   b ,  105   c  and  105   d  and defining a central axis  106  is depicted. The 4-tine fork resonator assembly  104  depicted in  FIG. 7  is the QUAPASON gyroscope, commercially available from Sagem of Paris, France. A gyroscope implementing a 4-tine fork resonator assembly is also depicted in U.S. Pat. No. 5,597,955 to Leger, et al., which is hereby incorporated by reference in its entirety except for explicit definitions contained therein. 
     It is noted that the 4-tine fork resonator assembly  104  is an “axisymmetric” resonator because the mass of the 4-tine fork resonator assembly  104  is equally distributed on both sides of any plane that includes the central axis  106 . 
     The 4-tine fork resonator assembly  104  includes four drive element pairs  107 ,  108 ,  109  and  110 , and four sense element pairs  111 ,  112 ,  113  and  114 . Each member of each pair in  FIG. 7  is identified with “a” and “b” designations (e.g.,  108   a  and  108   b ). The elements located on the aft side of the perspective view of  FIG. 7  are depicted in phantom. Note that the pairing of the drive elements are on the corners of a respective tine (e.g., drive elements  108   a  and  108   b  on the outside corner of tine  105   b ), while the pairing of the sense elements is on the same face of the 4-tine fork resonator assembly  104  (e.g., sense elements  112   a  and  112   b  on the face of the 4-tine fork resonator assembly  104  defined by tines  105   b  and  105   c ). The drive element pairs  107 ,  108 ,  109  and  110  are centered substantially on a first plane that is normal to the central axis  106 . Likewise, the sense element pairs  111 ,  112 ,  113  and  114  are centered substantially on a second plane that is normal to the central axis  106  (i.e. parallel to the first plane). 
     Referring to  FIG. 7A , the 4-tine fork resonator assembly  104  is depicted as having a vibration pattern that causes the tines  105   a ,  105   b ,  105   c  and  105   d  to deflect toward the central axis  106 . To produce this pattern, the drive element pairs that are diametrically or diagonally opposed (e.g., drive element pairs  107  and  109 ) are energized simultaneously to drive the diagonal tines toward the central axis  106 . Also, activation of the diagonal pairs may be alternated so that when a first of the diagonal pairs (e.g., drive element pairs  107  and  109 ) is driven inward, a second of the diagonal pairs (e.g., drive element pairs  108  and  110 ) is driven outward, but with both diagonal pairs being driven at the same frequency. In this way, the tines  105   a  and  105   c  are driven diagonally along a drive axis  115  that passes substantially through the central axis  106 . Likewise, the tines  105   b  and  105   d  are driven diagonally along a drive axis  116  and in complimentary oscillation (out of phase) with respect to tines  105   a  and  105   c , as depicted in  FIG. 7A . 
     The sense element pairs  111 ,  112 ,  113  and  114  of the 4-tine fork resonator assembly  104  are axially offset along the tines  105   a ,  105   b ,  105   c  and  105   d  relative to the location of the drive element pairs  107 ,  108 ,  109  and  110 . The sense element pairs  111 ,  112 ,  113  and  114  as depicted in  FIGS. 7 and 7A  will each detect a component of the drive oscillation but in opposing phases so that addition of each member of sense element pairs will cancel the drive component. 
     Also, the drive components along each drive axis  115  and  116  can be inferred by adding the signals from the sense elements that are located on the opposing corners (e.g., the addition of signals from sense elements  111   b ,  112   a ,  113   b  and  114   a  to infer the drive component along drive axis  116 ). 
     Referring to  FIG. 8 , a QCVSG  117  is depicted utilizing the 4-tine fork resonator assembly  104  as the resonator assembly in an embodiment of the invention. The same quadrature control system  72  as depicted for the QCVSG  70  of  FIG. 6  may be utilized, by virtue of the dual axis drive. For the 4-tine fork resonator assembly  104 , the drive axis  115  is defined as passing equidistant between and on the same plane as the centroids of the diagonally opposed drive element pairs  107  and  109 . Likewise, the drive axis  116  is defined as passing equidistant between and on the same plane as the centroids of the diagonally opposed drive element pairs  108  and  110 . Also, a pair of sense element axes  118  and  119  may be defined as passing equidistant between and on the same plane as the centroids of the diagonally opposed sense element pairs  111 ,  113  and  112 ,  114 . 
     Referring to  FIG. 9 , a skewed-drive QCVSG  120  is depicted in an embodiment of the invention. The skewed-drive QCVSG  120  includes a skewed-drive resonator assembly  122  and a torque-and-quadrature control system  124 . The skewed-drive resonator assembly includes first and second drive axes  126  and  128  that pass substantially through the centers of the drive elements  26 ,  28  and  30 ,  32 , respectively, and are “skewed” relative to a rotationally uniform distribution. That is, the first and second drive axes  126  and  128  are intentionally rotationally offset at angles of −Θ and +Θ, respectively, relative to a 90° orientation. The skewed-drive resonator assembly  122  and the torque-and-quadrature control system  124  include many of the same components and aspects as the resonator assembly  18  and quadrature control system  72 , identified by the same numerical references. 
     An advantage of the skewed-drive QCVSG  120  is that the drive elements  26 ,  28 , and  32 , in addition to sustaining the oscillation pattern on the skewed-drive resonator assembly  122 , can torque the oscillation pattern, thus eliminating the need for an additional torque element or elements. Accordingly, in one embodiment, the drive signals D 1  and D 2  of Eqns. (5) and (6) are tailored to accomplish the torque function by adding a torque component:
 
 D 1 =K ·SIN(ω t )+ P ·COS(ω t )+ T ·SIN(ω t )  Eqn. (7)
 
 D 2 =−K ·SIN(ω t )+ P ·COS(ω t )+ T ·SIN(ω t )  Eqn. (8)
 
where P=Kα as discussed in connection with Eqns. (5) and (6) above and T is the amplitude of a torque adjustment signal  134 . The remaining parameters are the same as defined in connection with Eqns. (1) and (2).
 
     It is further noted that once the QCVSG  120  is tuned, the phase offset α may be determined from K and P, that is
 
α= P/K   Eqn. (9)
 
The combined phase offset  64 , then, is given by
 
2α=2 P/K   Eqn. (10)
 
     The skewed-drive resonator assembly  122  includes many of the same components as the quadrature control system  72  of  FIG. 6 , which are identified by the same numerical reference numbers. In addition, the torque-and-quadrature control system  124  includes a torque signal source  136  for outputting the torque adjustment signal  134 , a third signal multiplier  138  to produce a torque adjustment sine wave signal  140 , and a third adder  142  for adding the torque and phase offset components. Also, for the skewed-drive resonator assembly  122 , the quadrature phase adjustment signal  90  can be adjusted to set the quadrature at a desired level. 
     The method does not require the use of phase locked loops, but can be implemented in a phase locked loop driving system  150 , as depicted in  FIG. 9 . The details of one such phase locked loop driving system is disclosed in U.S. Pat. No. 7,411,465, assigned to the assignee of the instant application and hereby incorporated by reference in its entirety except for express definitions included therein. 
     The phase-locked loop driving system  150  can enhance the performance by generating output signals  152 ,  154  that have a fixed relation to the phase of an input signal  156 . A phase-locked loop circuit (not depicted) detects the phase difference between the outputs  152 ,  154  and the input  156  and uses the resulting difference between these signals signal to adjust frequency of an internal oscillator until the outputs  152 ,  154  matches the input  156  in both frequency and phase. Effectively, the phase-locked loop driving system  150  serves as a filter for reducing drive noise and the attendant system noise. When utilized, the phase locked loop driving system  150  is coupled to each of the multipliers  94 ,  98  and  138  depicted in  FIG. 9 . 
     In one embodiment, the input signal  156  to the phase-locked loop driving system  150  comprises a comparison of the drive oscillation of the resonator assembly with the drive signals D 1  and D 2  and thus requires a resonator assembly that is capable of measuring the drive oscillation. Herein, the “drive oscillation” is the resultant oscillation (amplitude and frequency) that is imposed on the resonator assembly by the drive signals D 1  and/or D 2 . The sense elements  34   a ,  34   b  and  35   a ,  35   b  of the skewed-drive resonator assembly  122  depicted in  FIG. 9  are coupled with the resonator element  24  so that the sense element axes  40  and  42  are skewed at angles of −ψ and +ψ, respectively, in relation to nodal axes  160  and  162  that pass through node pairs  164 ,  166  and  168 ,  170 , respectively, when the skewed-drive resonator assembly  122  is in operation and rotationally stationary. The skewed relationship between the sense element axes  40  and  42  and the nodal axes  160  and  162  imposes a component of the drive oscillation on the sense elements  34   a ,  34   b  and  35   a ,  35   b  that can be isolated by adding the output signals  39   a  and  39   b  from the sense element or elements  34   a ,  34   b  on the sense axis  40  with the output signals  41   a  and  41   b  from the sense element or elements  35   a ,  35   b  on the sense axis  42 . A fourth adder  174  is depicted in  FIG. 9  for this purpose. The details of the operation of the skewed sense configuration is described in U.S. Pat. No. 7,526,957, incorporated by reference above. 
     Other axisymmetric resonator assemblies may also be configured to provide a measurement of the drive oscillation. For example, the sense element pairs  111 ,  112 ,  113  and  114  of the 4-tine fork resonator assembly  104  of  FIG. 7  can each detect a component of the drive oscillation such that subtraction of signals produced by elements on adjacent corners will isolate the drive oscillation component. For example, subtraction of the signals generated by sense elements  112   b  and  113   a  can provide an indication of the drive oscillation along the drive axis  115 . Likewise, subtraction of the signals generated by the sense elements  113   b  and  114   a  can provide an indication of the drive oscillation along drive axis  116 . Accordingly, those of skill in the art will recognize that the 4-tine fork resonator assembly  104  can be utilized with the phase-locked loop driving system  150 . 
     Referring to  FIG. 10 , a cup resonator assembly  180  is depicted for use in an embodiment of the invention. The cup resonator assembly  180  includes drive elements  182  and sense elements  184  that are operatively coupled to a resonator element  186  at the same radial location but axially separated. The sense elements  184  thereby provide a measurement of the drive oscillation along the drive axis defined by the drive element  182 , which can be implemented for the input signal  156  to the phase-locked loop driving system  150 . 
     Referring to  FIGS. 11 and 12 , a QCVSG  196  implementing a 2-tine quadrature control system  198  and a 2-tine fork resonator assembly  200  having a tuning fork-like arrangement about a central axis  201  is illustrated in an embodiment of the invention. The 2-tine fork resonator assembly  200  has tines  202   a  and  202   b , each comprising a drive element  204  adapted to be driven along a drive axis  205  and a sense element  206  adapted to sense vibration along a sensing axis  207  that is perpendicular to the major faces of the sense element  206 . The drive elements  204 , when energized, drives the tines  202   a  and  202   b  along the drive axes  205  and causes the tines  202   a  and  202   b  to oscillate in the direction indicated by vector  208 . When the gyroscope  200  is rotated about the sensing axis  201 , the sense elements  206  flex in the direction indicated by vector  210 , generating signals  212  having an amplitude proportional to the rotation of the gyroscope  200  about the rotation sensing axis  201 . 
     For the 2-tine fork resonator assembly  200 , the tines  202   a  and  202   b  vibrate in opposition along substantially coincident drive axes  205 . Because the drive elements  204  are on opposing faces of the 2-tine fork resonator assembly  200 , both are driven with primary drive signals having the same polarity. A phase compensation technique for resonator assemblies that are driven along one axis as for a two-tine tuning fork and its variants is now described mathematically. 
     To affect correction of the quadrature signal in accordance with certain embodiments of the invention, it is desired that a total phase difference equal to the combined phase offset  64  of 2α be sustained between the first and second tines. Accordingly, desired drive signals D 3  and D 4  for the first and second tine drives, respectively, are expressed as:
 
 D 3 =K ·SIN(ω t +α)  Eqn. (11)
 
 D 4 =K ·SIN(ω t −α)  Eqn. (12)
 
where K is the amplitude of the drive signal (e.g., in volts), ω is the drive frequency and t is a time parameter. The trigonometric identity for the sine of the sum of two angles is:
 
SIN(ω t +α)=COS(α)SIN(ω t )+SIN(α)COS(ω t )  Eqn. (13)
 
For small angles α, Eqn. (13) simplifies to
 
SIN(ω t +α)=SIN(ω t )+α COS(ω t )  Eqn. (14)
 
Substituting Eqn. (14) into Eqns. (11) and (12), the drive signals D 3  and D 4  may be expressed as
 
 D 3 =K ·SIN(ω t )+ K α·COS(ω t )  Eqn. (15)
 
 D 4 =K ·SIN(ω t )− K α·COS(ω t )  Eqn. (16)
 
     Accordingly, the trigonometric functions of the drive signals D 3  and D 4  of Eqns. (15) and (16) are expressed solely in terms of the drive frequency-time product ωt, so that the various components of the D 3  and D 4  functions are readily implemented by electronic circuitry and/or computer control. 
     Other resonator assemblies, such as microelectromechanical system (MEMS) gyroscopes and H-fork gyroscopes, are also driven along a common axes with same polarity primary drive signals. Examples of such gyroscopes are disclosed by U.S. Pat. No. 5,996,410 to Yachi et al.; Zaman, et al., “High Performance Matched-Mode Tuning Fork Gyroscope,” MEMS 2006, Istanbul, Turkey, pp. 22-26, January 2006; and Trusov et al., “Gyroscope Architecture with Structurally Forced Anti-Phase Drive-Mode and Linearly Coupled Anti-Phase Sense-Mode,” IEEE Transducers 2009, Denver Colo., USA, June 2009, the disclosures of which are hereby incorporated by reference in their entirety except for express definitions therein. Eqns. (10)-(15) may be implemented with these devices with the same result. 
     The 2-tine quadrature control system  198  includes many of the same components as the quadrature control system  72  of  FIG. 6 , which are identified by the same numerical reference numbers. A distinction between the 2-tine quadrature control system  198  and the quadrature control system  72  is that the inverter  103  inverts the phase adjustment cosine signal  100  going into the second adder  102  (instead of the variable gain sine wave signal  96 ) to affect the subtraction of the Kα·COS(ωt) component of Eqn. (16). 
     In still another embodiment of the invention, a phase compensation technique is implemented that involves utilizing a “controlled” bias. Consider the bias Bq of the rate signal caused by a demodulated quadrature signal Sq, given by
 
 Bq=K 1 ·Sq·ε   DM   Eqn. (17)
 
where ε DM  is a demodulation phase error of the of the rate signal and K 1  is the gain of the sensing system. The demodulation phase error ε DM  is herein defined as the difference between a desired or targeted demodulation phase and an actual demodulation phase.
 
     In some cases, there may be intrinsic, phase-related phenomena that cause the demodulation phase error ε DM  (e.g., transitions caused by turn on drift or stresses from temperature changes). Such intrinsic causes could be compensated by a controlled bias such as Bq. This controlled bias may involve establishing a tolerable level of quadrature signal to use an intrinsic phase error to generate a specific bias. By tolerating a small amount of quadrature signal Sq, one can control that level of quadrature signal Sq to prevent larger quadrature signals of unknown quantity. In one embodiment, the objective is to use the demodulated quadrature signal Sq to compensate for these other bias error sources that tend to correlate with the demodulation phase error ε DM . Knowing the demodulated quadrature signal Sq, one can determine the magnitude of the demodulation phase error ε DM  and use the information to compensate for false indication of rotation rate. 
     The invention may be embodied in other specific forms without departing from the spirit or essential attributes thereof, and it is, therefore, to be understood that the depicted embodiments are in all respects as illustrative and not restrictive. For example, while the preceding description is directed to cup and four-tine gyroscopes, the methods and apparatuses described herein may be implemented with any gyroscope or resonator that utilizes more than one drive axis, including but not limited to cup resonators, hemispherical resonators, ring resonators and four-tine resonators. 
     The foregoing descriptions present numerous specific details that provide a thorough understanding of various embodiments of the present invention. Each of the figures and methods disclosed herein may be used separately, or in conjunction with other features and methods, to provide improved devices, systems and methods for making and using the same. Therefore, combinations of features and methods disclosed herein may not be necessary to practice the invention in its broadest sense and are instead disclosed merely to particularly describe representative embodiments of the invention. 
     It is to be understood that even though numerous characteristics and advantages of various embodiments are set forth in the foregoing description, together with details of the structure and function of various embodiments, this disclosure is illustrative only. Other embodiments may be constructed that nevertheless employ the principles and spirit of the present invention, which is defined solely by the claims that follow. 
     For purposes of interpreting the claims for the present invention, it is expressly intended that the provisions of Section 112, sixth paragraph of 35 U.S.C. are not to be invoked with respect to a given claim unless the specific terms “means for” or “step for” are recited in that claim.