Abstract:
The inertial sensor of the present invention utilizes a proof mass suspended from spring structures forming a nearly degenerate resonant structure into which a perturbation is introduced, causing a split in frequency of the two modes so that the mode shape become uniquely defined, and to the first order, remains orthogonal. The resonator is provided with a mass or inertia tensor with off-diagonal elements. These off-diagonal elements are large enough to change the mode shape of the two nearly degenerate modes from the original coordinate frame. The spring tensor is then provided with a compensating off-diagonal element, such that the mode shape is again defined in the original coordinate frame. The compensating off-diagonal element in the spring tensor is provided by a biasing voltage that softens certain elements in the spring tensor. Acceleration disturbs the compensation and the mode shape again changes from the original coordinate frame. By measuring the change in the mode shape, the acceleration is measured.

Description:
This application claims the benefit of U.S. Provisional Application No. 60/119,479, filed Feb. 10, 1999, which is incorporated by reference in its entirety into the present disclosure. 
    
    
     STATEMENT AS TO FEDERALLY SPONSORED RESEARCH 
     The invention described herein was made in the performance of work under a NASA contract, and is subject to the provisions of Public Law 96-517 (35 U.S.C. 202) in which the Contractor has elected to retain title. 
    
    
     BACKGROUND OF THE INVENTION 
     Future space exploration missions require high performance inertial measurement systems for navigation, guidance, and attitude control. Micromachined vibratory gyroscopes are promising candidates to replace conventional gyroscopes for future miniature spacecraft control and avionics applications while simultaneously satisfying stringent physical requirements of low mass, volume, power and cost. U.S. Pat. No. 5,894,090 to Tang et al., assigned to the same assignees as the present invention, describes such a micromachined vibratory gyroscope and is incorporated by reference in its entirety into the present disclosure. The techniques described in Tang et al. can also be used in the fabrication of the present invention. At times accelerometers are required to be part of these inertial measurement systems. There is therefore a need for a micromachined inertial sensor combining rotation and acceleration measurement functions to greatly reduce the complexity, mass, volume and power of such inertial measurement systems. It is also desirable for such an inertial sensor to be economical yet accurate and reliable. 
     SUMMARY OF THE INVENTION 
     The inertial sensor and method of use of the present invention provides an accurate and reliable, yet compact, light-weight, and relatively simple accelerometer and gyroscope combination, or alternatively provides a stand-alone accelerometer. 
     A resonator structure with two perfectly degenerate, or same frequency, modes can be made to move in an arbitrary motion that is a linear combination of the two modes. Furthermore, the mode shapes of the two modes are orthogonal, but otherwise arbitrarily defined. The inertial sensor of the present invention utilizes a proof mass suspended from spring structures forming a nearly degenerate resonant structure into which a perturbation is introduced, causing a split in frequency of the two modes so that the mode shape becomes uniquely defined, and to the first order, remains orthogonal. The resonator is provided with a mass or inertia tensor with off-diagonal elements. These off-diagonal elements are large enough to change the mode shape of the two nearly degenerate modes from the original coordinate frame. The spring tensor is then provided with a compensating off-diagonal element, such that the mode shape is again defined in the original coordinate frame. The compensating off-diagonal element in the spring tensor is provided by a biasing voltage that softens certain elements in the spring tensor. Acceleration disturbs the compensation and the mode shape again changes from the original coordinate frame. By measuring the change in the mode shape, the acceleration is measured. 
     One embodiment of the inertial sensor of the present invention measures acceleration or acceleration and rotation by using a proof mass having a defined center and a mass imbalance such that the center of mass of the structure is spaced from the defined center; the proof mass is suspended from a frame by spring structures; drive circuitry rocks the proof mass about a rocking axis passing through the defined center; bias circuitry supplies a voltage to compensate for the mass imbalance; sensing circuitry measures acceleration by detecting the change of a mode shape of the proof mass; and output circuitry generates a signal indicating the acceleration. 
     In one embodiment the method of the present invention is performed by rocking a proof mass about a first rocking axis passing through a defined center; supplying a bias voltage to compensate for a mass imbalance of the proof mass; measuring acceleration by detecting change of a mode shape of the proof mass; and outputting a signal indicating the acceleration. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     In the drawings, which constitute part of this specification, embodiments demonstrating various features of the invention are set forth as follows: 
     FIG. 1 is a perspective view of the proof mass, spring structures, rim and post of the inertial sensor of the present invention; 
     FIG. 2 is an exploded perspective view of the inertial sensor of FIG. 1, also showing a base plate; 
     FIG. 3 is a diagrammatic view of the proof mass, spring structures and rim of the inertial sensor of FIG. 1; 
     FIG. 4 is a diagrammatic view of the eight electrode structure of the inertial sensor of FIG. 1; 
     FIG. 5 is a diagrammatic view of the proof mass positioned adjacent to the electrode structure; 
     FIG. 6 is a block diagram of the circuitry for driving and sensing signals from the inertial sensor of FIG. 1; 
     FIG. 7 is a block diagram of the circuitry for driving and sensing signals from a four electrode version of the inertial sensor of FIG. 1; 
     FIG. 8 is a graph of rotation test results showing the response of rotation output and acceleration output to rotation; 
     FIG. 9 is a graph of acceleration test results showing the response of rotation output and acceleration output as the sensor is rotated in Earth&#39;s gravity; 
     FIG. 10 is a perspective view illustrating an embodiment of a proof mass without a post passing though the center; and 
     FIG. 11 is a diagrammatic view showing one embodiment of a sense electrode which uses alternating conductors biased by voltages to produce a fringing electric field. 
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENTS 
     Although detailed illustrative embodiments are disclosed herein, other suitable structures and machines for practicing the invention may be employed and will be apparent to persons of ordinary skill in the art. Consequently, specific structural and functional details disclosed herein are representative only; they merely describe exemplary embodiments of the invention. 
     FIG. 1 shows a simplified perspective view and FIG. 2 shows a simplified exploded perspective view of a portion of one embodiment of an inertial sensor according to the present invention. The inertial sensor  10  has an inertial sensing structure or proof mass  12  suspended from a rim  22  by four spring structures  24 ,  26 ,  28 ,  30 . The combined proof mass  12  and four spring structures  24 ,  26 ,  28 ,  30  form a resonator structure  13 . In one embodiment, the proof mass  12  is a substantially planar, two fold symmetric cloverleaf structure having four square leaves  14 ,  16 ,  18 ,  20 . Other resonating structures, symmetrical or non-symmetrical can be substituted for the four leaf clover structure  12 . Also, the proof mass, spring structures and frame can all be made of a monolithic micromachined body of silicon or other suitable semi-conductor material. 
     The spring structures  26 ,  30  lie along a spring or drive axis x, while the spring structures  24 ,  28  lie along a spring or sense axis y. The rim  22  is attached to a base plate  32 . The base plate  32  can be quartz or any other suitable material. A post  34  is rigidly attached through a defined center or a geometrical center  36  of the proof mass  12  such that the center of mass  38  of the post  34  is aligned with the cloverleaf geometrical center  36 . At the geometrical center  36  can be an opening through which the post  34  is received. The post  34  also passes through an opening at a center  39  of the base plate  32  between electrodes  42 ,  44 ,  48 ,  50 ,  52 ,  56 ,  58 ,  60  patterned on the base plate  32 . The electrodes  42 ,  44  are drive electrodes which receive an AC voltage from an electrical circuit  46  (FIG.  6 ), thereby exerting a force on the corresponding leaves  14 ,  16  to drive the proof mass  12  to oscillate. Sensing electrodes  48 ,  50 ,  52  are also connected to the electrical circuit  46  to capacitively detect the motions of the proof mass  12 . 
     The proof mass  12  has a small mass imbalance  54  which moves the center of mass of the proof mass  12  away from the geometrical center  36 . The mass imbalance  54  can be created by adding or removing material from one or more of the four silicon leaves  14 ,  16 ,  18 ,  20 . The mass imbalance  52  can also be created by asymmetries occurring during fabrication. In a system with a perfectly balanced proof mass suspended from four spring structures  24 ,  26 ,  28 ,  30  having degenerate spring constants, the proof mass will rock about the drive and sense axes x, y aligned with the four spring structures as illustrated in FIG.  3 . However, due to the mass asymmetry, the drive electrodes  42 ,  44  cause the resonator structure  13  to have a mode shape rocking about nodes aligned with rocking axes x′ and y′, also illustrated in FIG.  3 . The nodes are the lines about which the proof mass  12  rocks. The electrical circuit  46  compensates electrostatically for the mass imbalance by applying different electrostatic biases to the electrodes  58 ,  60  to rotate the rocking axes x′ and y′, and thus the mode shape, into a position of alignment with the drive axis x, aligned with the length of the spring structures  26 ,  30 , and the sense axis y, aligned with the length of the spring structures  24 ,  28 . 
     Applicants have found that subjecting the electrostatically compensated inertial sensor  10  to a z-axis directed component of acceleration also changes the mode shape of the resonator structure  13 . Measuring this change in mode shape provides an accurate measurement of acceleration. Examining the resonator structure  13  mathematically provides valuable insight into the reasons why electrostatic compensation and acceleration effect the mode shape and into ways to optimize the inertial sensor  10  to measure acceleration. 
     In Lagrangian mechanics the equations of motion are written as                      ∂     ∂   t              ∂   L       ∂       θ   .     i           -       ∂   L       ∂     θ   i           =   0     ,           [   1   ]                                
     where L is the Lagrangian in coordinates (θ i ,{dot over (θ)} i ), θ i  is the rocking angle of a structure about an i-axis, and {dot over (θ)} i  is the time derivative of θ i , representing the angular rocking rate about the j-axis. 
     The Lagrangian, L, is defined as: 
     
       
           L=T−U,   [3] 
       
     
     where T is the structure&#39;s kinetic energy and U is the structure&#39;s potential energy. For the proof mass suspended from the spring structures:                U   =       1   2          K   ij          θ   i          θ   j              
        and           [   4   ]                 T   =       1   2          I   ij            θ   .     i            θ   .     j         ,           [   5   ]                                
     where K ij  is the spring tensor of the spring structures, I ij  is the inertial tensor of the inertial sensing structure, θ j  is the rocking angle of the inertial sensing structure relative to a j-axis perpendicular to the i-axis about which the structure rocks, and {dot over (θ)} j  is the time derivative of θ j , representing the angular rocking rate about the j-axis. Substituting EQUATIONS [3]-[5] into EQUATION [1], the equations of motion become:                    1   2          I   ij            θ   ¨     j       +       1   2          K   ij          θ   j         =   0.           [   6   ]                                
     The equations of motion in matrix form are then                    (           I   11           I   12               I   21           I   22           )          (             θ   ¨     1                 θ   ¨     2           )       +       (           K   11           K   12               K   21           K   22           )          (           θ   1               θ   2           )         =       (         0           0         )     .             [   7   ]                                
     For the inertial sensor  10  of FIGS. 1 and 2, the equations of motion describing the proof mass  12  and spring structures  24 ,  26 ,  28 ,  30  of the resonator structure  13  can be written as                    (             I   xx     +       y   2        m             -   xym               -   xym             I   yy     +       x   2        m             )          (             θ   ¨     x                 θ   ¨     y           )       +       (             K   xx     -     γ   1     -     γ   2             -     γ   1                 -     γ   1               K   yy     -     γ   1             )          (           θ   x               θ   y           )         =       (         0           0         )     .             [   8   ]                                
     The inertial tensor has been modified by terms accounting for the mass imbalance  54  having a magnitude of m and located at position (x, y) as illustrated in FIG.  3 . θ x  is the rocking angle of the proof mass  12  relative to the x-axis and θ y  is the rocking angle of the proof mass  12  relative to the γ-axis as shown in FIG.  1 . The terms {dot over (θ)} x  and {dot over (θ)} y  are the time derivatives of θ x  and θ y , respectively, representing the angular rocking rates. The spring tensor is modified by the electrostatic spring softening terms γ 1  and γ 2  to compensate for the mass imbalance  54 . 
     The term γ 1  represents the spring softening provided by the diagonal biasing electrode  58  and the term γ 2  represents the electrostatic spring softening provided by the spring-axis biasing electrode  60 . FIG. 4 illustrates the electrodes  42 ,  44 ,  48 ,  50 ,  52 ,  56  and FIG. 5 illustrates the proof mass  12  positioned adjacent to the electrodes  42 ,  44 ,  48 ,  50 ,  52 ,  56 ,  58 ,  60 . The terms γ 1  and γ 2  can be written as:                  γ   1     =         r   1   2          C   a          V   1   2         2        d   3                
        and           [   9   ]                 γ   2     =         r   2   2          C   a          V   2   2         d   3               [   10   ]                                
     where r 1  and r 2  are the distances from the center  39  of the base plate  32  to the diagonal biasing electrode  58  and the spring-axis biasing electrode  60 , respectively. The terms V 1  and V 2  represent the compensating biasing voltage applied to the diagonal biasing electrode  58  and the spring-axis biasing electrode  60 , respectively. The term d represents the average distance between the electrodes and the proof mass  12  as illustrated in FIG.  5  and the term C a *d represents the average capacitance between the biasing electrodes  58 ,  60  and the proof mass  12 . C a  is equivalent to the product of the permitivity (ε) between the leaf structures and the electrodes, and the area of the electrodes (A). 
     To rotate the resonant modes from the x′-y′ coordinate system back to the x-y coordinate system, the cross terms in EQUATION [8] are forced to cancel individually yielding: 
     
       
         − xym{umlaut over (θ)}   x −γ 1 θ x =0  [11] 
       
     
     
       
         − xym{umlaut over (θ)}   y −γ 1 θ y =0  [12] 
       
     
     
       
         ( I   xx   +y   2   m ){umlaut over (θ)} x +( K   xx −γ 1 −γ 2 )θ x =0  [13] 
       
     
     
       
         ( I   yy   +x   2   m ){umlaut over (θ)} y +( K   yy −γ 1 )θ y =0.  [14] 
       
     
     Solving for γ 1  and γ 2  yields:                γ   1     =       xymK   yy       xym   +     I   yy     +       x   2        m                 [   15   ]                 γ   2     =       K   xx     -         K   yy          (         I   xx     +       y   2        m     +   xym         I   yy     +       x   2        m     +   xym       )       .               [   16   ]                                
     Assuming that y 2 m+xym is small compared to I xx , γ 1  and γ 2  can be rewritten in simpler form as: 
     
       
         γ 1   =xymω   y   2   [17] 
       
     
     
       
         γ 2   =I   xx (ω x   2 −ω y   2 )  [18] 
       
     
     where                ω   x     =         K   xx         I   xx     +   xym   +       y   2        m                   [   19   ]                 ω   y     =           K   yy         I   yy     +   xym   +       x   2        m           .             [   20   ]                                
     EQUATION [18] shows how γ 2  compensates for the frequency split partially caused by the mass imbalance. 
     Equating EQUATIONS [9] and [10] with EQUATIONS [17] and [18], respectively, yields the compensating biasing voltage V 1  to be applied to the biasing electrodes  58  and the biasing voltage V 2  to be applied to the biasing electrode  60  in order to compensate for the mass imbalance:                  V   1              2                   ω   y   2          d   3        mxy         r   1   2          A   1        ɛ                
        and           [   21   ]                 V   2     =           (       ω   x   2     -     ω   y   2       )          d   3          I   xx           r   2   2          A   2        ɛ                 [   22   ]                                
     where ε represents the permitivity between the leaf structures and the electrodes  58  and the electrode  60 , respectively. A 1  and A 2  represent the area of the electrodes  58  and the electrode  60 , respectively. 
     Acceleration along the z-axis results in a modified distance d between the biasing electrodes  58 ,  60  and the proof mass  12  as illustrated in FIG.  5 . The distance d is modified by acceleration a according to:                d   ′     =       -     a     ω   z   2         +   d             [   23   ]                                
     where the term ω z  represents the oscillation frequency of the motion of the proof mass  12  along the z-axis. From EQUATION [23] it can be seen that acceleration along the positive z-axis reduces the distance d by a factor of          a     ω   z   2       .                          
     Substituting EQUATION [23] into EQUATIONS [21] and [22] and assuming that          a       ω   z   2        d            &lt;&lt;   1                            
     results in the approximate change in biasing voltages due to acceleration along the z-axis:                  Δ                   V   1       ≈     a       ω   z   2        d              
        and           [   24   ]                 Δ                   V   2       ≈     a       ω   z   2        d               [   25   ]                                
     To optimize the inertial sensor  10  for sensing acceleration, ΔV 1  and ΔV 2  should be large. EQUATIONS [24] and [25] reveal that smaller values of acceleration can be detected by making ΔV 1  and ΔV 2  relatively large. In order to make ΔV 1  and ΔV 2  large, the frequency ω z  and the distance d should be relatively small. 
     FIG. 6 illustrates a circuit  46  for measuring the change in mode shape in order to measure acceleration. Additionally, this circuit allows for simultaneous and independent measurement of the acceleration applied to the inertial sensor  10  along the z-axis as well as the rotation of the inertial sensor about the z-axis so that the inertial sensor  10  serves as both an accelerometer and gyroscope. First, the proof mass  12  is driven by the drive electrodes  42 ,  44  to rock about the rocking axes x′ and y′, as illustrated in FIG.  3 . The compensating electrostatic spring-softening biasing voltages V 1 , V 2  of EQUATIONS [21] and [22] are applied to the diagonal biasing electrode  58  and the spring-axis biasing electrode  60 , respectively, to compensate for the mass imbalance and rotate the nodes of the mode shape back into alignment with the spring axes x, y. The inertial sensor  10  is subjected to a z-axis directed component of acceleration. As explained above, the z-axis directed component of acceleration rotates the nodes of the mode shape out of alignment with the spring axes x, y. The circuit  46  provides torque or force rebalancing and drive amplitude control to the resonator structure  13  as follows. 
     An output  62  of the sensing electrode  52  is provided to a preamplifier  64 . The output  66  of the preamplifier  64  is connected to an automatic gain control amplifier (“AGC amplifier”)  68 . The output  66  of the preamplifier is also provided to a voltage controlled amplifier  70  which is controlled by the voltage output of the AGC amplifier  68 . The AGC amplifier also receives a reference voltage V REF  as input. The mixer  70  outputs a drive signal  71  which is split into two signals. The drive signals are fed back to drive electrodes  42 ,  44  respectively. The drive signals are in phase and thus provide torque about the drive or spring axis x and provide drive amplitude control. 
     The outputs  76 ,  78  of the spring axis sensing electrodes  48 ,  50  are provided to preamplifiers  80 ,  82 , respectively. The outputs  84 ,  86  of the preamplifiers  80 ,  82  are then fed to the inputs  88 ,  90  of a differential amplifier  92 . The differential amplifier  92  outputs a sense signal  93  which is split into two signals. One of the signals is fed back in phase to the drive electrode  42  and the other signal is fed into an inverter prior to being fed back to the electrode  44 . The sense signals fed to the drive electrodes  42 ,  44  are thus out of phase in order provide force or torque rebalancing about the sense or spring axis y. 
     The force or torque rebalancing substantially eliminates rocking motion of the proof mass about the spring or sense axis y caused by the rotation mass imbalance or acceleration. Also, the applicants have found that the output acceleration and rotation signals are in quadrature, or at 90°, with each other. This allows the present invention to use the same method as in U.S. Pat. No. 5,894,090 to Tang, cited above, to measure the rotation about z-axis, while simultaneously using the present method to measure acceleration. 
     It can be shown that the acceleration signal is in quadrature with both the rotation signal and the drive signal as follows: 
     
       
           T   D =τ D  cos ωt  [32] 
       
     
     
       
           T   S =τ D  cos θ sin θ cos ω t −τ D  sin θ cos θ cos(ωt+φ)  [33] 
       
     
     where T D  drive torque functions and T s  is the sense torque function representing the torque required to prevent rocking about the sense axis caused by the drive torque T D , τ D  is the torque amplitude about the drive axis x, t is time, ω is the rocking frequency about the drive axis x and θ is the angle between the drive axis x and the rotated axis x′ as illustrated in FIG.  3 . The sense torque function T s  is in phase with the acceleration signal. Making several simplifying assumptions results in: 
     
       
           T   S ≈τ D θφ sin ωt  [34] 
       
     
     From EQUATIONS [32] and [34] it can be seen that T D  and T s  are in quadrature to each other and thus the acceleration signal is in quadrature with both the rotation signal and the drive signal. 
     The output  71  from the voltage controlled amplifier  70  and the output  93  from the differential amplifier  92  are applied to inputs  102 ,  100  of the demodulator  94  where they are demodulated in phase to produce an output signal indicating the amount of rotation experienced by the inertial sensor  10 . 
     The output  71  from the voltage controlled amplifier  70  is provided to a 90° phase shifter  98  so that it can be demodulated by the demodulator  96  in quadrature with the output  93  of the differential amplifier  92 . The demodulator  96  then outputs a signal indicating the amount of acceleration experienced by the inertial sensor  10 . 
     A four electrode embodiment of a circuit  107  for measuring rotation and acceleration is illustrated in FIG.  7 . The circuit  107  functions similarly to circuit  46  of FIG. 6, but does not have the spring axis bias electrode. With only a single bias electrode  114 , the circuit  107  is unable to completely compensate for the mass imbalance caused mode rotation. 
     FIGS. 8 and 9 show outputs of the four electrode dual mode embodiment of the inertial sensor  10 . FIG. 8 is a graph of inertial sensor  10  output, measured in millivolts, as a function of the inertial sensor  10  rotation rate, measured in degrees/second, about the z-axis. The graph of FIG. 8 was produced by positioning the inertial sensor  10  with the post  34  extending vertically and rotating the inertial sensor  10  about the post  34  along the z-axis at varying rotation rates Ω z  as illustrated in FIG.  1 . The inertial sensor  10  output can be seen to be a linear function of the inertial sensor  10  rotation rate. Due to the imperfect spring softening compensation of the mass imbalance, there is some acceleration output even though there is no acceleration. FIG. 9 is a graph of inertial sensor  10  output, measured in millivolts, as a function of time as the inertial sensor  10  is rotated in Earth&#39;s gravity. At time=0 seconds the post  34  is approximately vertical and pointing upwards and thus the z-directed component of acceleration is at a maximum. At time=10 seconds the post is approximately horizontal and thus the z-directed component of acceleration zero. At time=19 seconds the post is approximately vertical and pointing downwards and thus the z-directed component is at a minimum. There is some rotation output even though there is no rotation of the inertial sensor  10 . 
     FIG. 10 illustrates an embodiment of a proof mass  13  without a post passing though the center. As explained in U.S. Pat. No. 5,894,090 to Tang, cited above, a post serves an important function in rotation measurements. However, when only measuring acceleration, the proof mass  131  of FIG. 10, without a post, can be used. 
     FIG. 11 illustrates one embodiment of a sense electrode  105  which uses alternating conductors biased by voltages  107 ,  109  to produce fringing electric field  121 . As a proof mass resonates near the sense electrode  105 , it disturbs the electric field resulting in an output to sensing circuitry connected to the sense electrode  105 . This embodiment can be used for detecting the motion of a proof mass of any material capable of disturbing the electric field  121 . 
     There are many other possible circuits for measuring the change in mode shape in order to provide an output indicative of acceleration. For example, the bias V 1 , V 2  can be increased or decreased in response to feedback indicating that acceleration has caused rotation of the mode shape. The acceleration, according EQUATIONS [24] and [25] is directly proportional to the change in electrostatic spring softening bias voltage required to compensate for the node shape rotation. 
     The z-axis directed component of acceleration can alternatively be determined using an open-loop embodiment of the electrical circuit  46 . In this embodiment, sensing electrodes  48 ,  50 ,  52  are used to detect the amount of rotation of the resonant modes away from the spring axes x, y due to the z-directed component of acceleration. The acceleration is determined from the rotational displacement of the resonant modes or mode shape. The rotational displacement of the resonant modes or mode shape can be determined by using, for example, the Modal ID of the structure. 
     While the above description contains many specific features of the invention, these should not be construed as limitations on the scope of the invention, but rather as an example of one preferred embodiment thereof. Many other variations are possible. Accordingly, the scope of the invention should be determined not by the embodiments illustrated, but by the appended claims and their legal equivalents.