Patent Publication Number: US-9417066-B2

Title: Electronic bias compensation for a gyroscope

Description:
BACKGROUND INFORMATION 
     1. Field 
     The present disclosure relates generally to vibratory gyroscopes and, in particular, to a method and apparatus for electronically compensating for bias in vibratory gyroscopes. 
     2. Background 
     Gyroscopes are used for measuring and/or maintaining orientation. As used herein, a “gyroscope” is a sensor configured to detect and measure the angular motion of an object relative to an inertial frame of reference. Further, as used herein, an “inertial frame of reference” may be a coordinate system or set of axes that is non-accelerating. In other words, an inertial frame of reference is a frame of reference in which Newton&#39;s first law of motion is true. Newton&#39;s first law of motion states that the velocity of a body remains constant unless the body is acted upon by an external force. 
     A Coriolis vibratory gyroscope (CVG) is configured to be driven to vibrate along a first axis. Vibration along the first axis while the Coriolis vibratory gyroscope is being rotated about a fixed input axis generates a Coriolis force that induces vibrations along a second axis. These vibrations may be measured and used to determine an angular velocity for the rotation of the Coriolis vibratory gyroscope about the fixed input axis. 
     However, bias may contribute to the measurements of the vibrations. Bias may be the error in the measurements due to factors such as, for example, without limitation, temperature, part inconsistencies, and other suitable factors. Calibration of these gyroscopes during manufacturing of the gyroscopes may be less accurate than desired. 
     For example, calibration of these gyroscopes during manufacturing processes may use test data as compared to substantially real-time data. In particular, these calibration techniques may not take into account the effects of the temperature in the environment in which a gyroscope is being operated and/or inconsistencies that may develop over time from the time at which the gyroscope was manufactured. Further, some currently available systems for compensating for this bias may be unable to reduce the bias from these vibration measurements to within selected tolerances. 
     Therefore, it would be desirable to have a method and apparatus that takes into account one or more of the issues discussed above as well as possibly other issues. 
     SUMMARY 
     In one illustrative embodiment, a method for compensating for bias of a gyroscope is provided. Bias measurements for a plurality of drive angles are generated using the gyroscope. A set of equations for the bias of the gyroscope is identified using a model for motion of the gyroscope. The set of equations includes a set of parameters for the bias of the gyroscope. A set of values for the set of parameters is identified using the bias measurements and the set of equations. 
     In another illustrative embodiment, an apparatus comprises a compensation system. The compensation system is configured to receive bias measurements for a plurality of drive angles from a gyroscope. The compensation system is configured to identify a set of equations for the bias of the gyroscope using a model for motion of the gyroscope. The set of equations includes a set of parameters for the bias of the gyroscope. The compensation system is configured to identify a set of values for the set of parameters using the bias measurements and the set of equations. 
     In yet another illustrative embodiment, a method for electronically compensating for bias of a gyroscope is provided. Bias measurements for a plurality of drive angles are generated using the gyroscope. A set of equations for the bias of the gyroscope is identified using a model for motion of the gyroscope and an assumption that an inertial rate for the gyroscope is substantially zero. The set of equations includes a set of parameters for the bias of the gyroscope. A set of values for the set of parameters is identified using the bias measurements and the set of equations. The bias of the gyroscope at a selected drive angle is estimated using the set of equations with the set of values for the set of parameters to form an estimated bias. The estimated bias is subtracted from a measurement generated by the gyroscope for the selected drive angle to substantially compensate for the bias of the gyroscope. 
     The features, functions, and benefits can be achieved independently in various embodiments of the present disclosure or may be combined in yet other embodiments in which further details can be seen with reference to the following description and drawings. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       The novel features believed characteristic of the illustrative embodiments are set forth in the appended claims. The illustrative embodiments, however, as well as a preferred mode of use, further objectives, and features thereof will best be understood by reference to the following detailed description of an illustrative embodiment of the present disclosure when read in conjunction with the accompanying drawings, wherein: 
         FIG. 1  is an illustration of a bias compensation environment in the form of a block diagram in accordance with an illustrative embodiment; 
         FIG. 2  is an illustration of a functional model for a gyroscope in accordance with an illustrative embodiment; 
         FIG. 3  is an illustration of the orbit of an element for a gyroscope in accordance with an illustrative embodiment; 
         FIG. 4  is an illustration of a process for compensating for bias of a gyroscope in the form of a flowchart in accordance with an illustrative embodiment; 
         FIG. 5  is an illustration of a process for identifying a set of equations for the bias of a gyroscope in the form of a flowchart in accordance with an illustrative embodiment; 
         FIG. 6  is an illustration of a process for identifying a set of values for a set of parameters in a set of equations for the bias of a gyroscope in the form of a flowchart in accordance with an illustrative embodiment; and 
         FIG. 7  is an illustration of a data processing system in accordance with an illustrative embodiment. 
     
    
    
     DETAILED DESCRIPTION 
     The different illustrative embodiments recognize and take into account different considerations. For example, the different illustrative embodiments recognize and take into account that some currently available methods for compensating for bias of a gyroscope may use a mechanical system to estimate the bias of the gyroscope. In some cases, this mechanical system may be used to operate the gyroscope and rotate the gyroscope at an angular velocity. Further, this type of mechanical system may introduce additional contributors to the bias of the gyroscope. Consequently, these types of systems may be unable to estimate the bias of the gyroscope within a desired level of accuracy. 
     Thus, the different illustrative embodiments provide a method and apparatus for electronically compensating for the bias of a gyroscope. In one illustrative embodiment, a method for compensating for bias of a gyroscope is provided. Bias measurements for a plurality of drive angles are generated using the gyroscope. A set of equations for the bias of the gyroscope is identified using a model for motion of the gyroscope. The set of equations includes a set of parameters for the bias of the gyroscope. A set of values for the set of parameters is identified using the bias measurements and the set of equations. 
     With reference now to  FIG. 1 , an illustration of a bias compensation environment in the form of a block diagram is depicted in accordance with an illustrative embodiment. In these illustrative examples, bias compensation environment  100  is an example of an environment in which the different illustrative embodiments may be implemented to compensate for bias  101  of gyroscope  102 . 
     As depicted, gyroscope  102  takes the form of vibrating structure gyroscope (VSG)  104  in these examples. In particular, vibrating structure gyroscope  104  may be referred to as Coriolis vibratory gyroscope (CVG)  106 . In these illustrative examples, gyroscope  102  is a sensor configured to measure angular motion relative to inertial frame of reference  108 . Inertial frame of reference  108  is a coordinate system or set of axes that is non-accelerating. 
     In one illustrative example, gyroscope  102  comprises element  110  and frame  112 . Element  110  may be a structure comprising a number of metallic alloys. Element  110  may be configured to vibrate, or resonate, at number of resonant frequencies  114  for element  110 . In some cases, number of resonant frequencies  114  may be substantially equal to number of natural frequencies  115  for element  110 . A natural frequency in number of natural frequencies  115  may be the frequency at which element  110  vibrates along a particular axis when a substantially continuous external force is not being applied to element  110 . In this illustrative example, element  110  may be referred to as a “proof mass” or a resonator in some illustrative examples. 
     Element  110  may be associated with frame  112 . The association between element  110  and frame  112  is a physical association in these depicted examples. A first component, such as element  110 , may be considered to be associated with a second component, such as frame  112 , by being secured to the second component, bonded to the second component, mounted to the second component, welded to the second component, fastened to the second component, and/or connected to the second component in some other suitable manner. The first component also may be connected to the second component using a third component. The first component also may be considered to be associated with the second component by being formed as part of and/or an extension of the second component. 
     For example, without limitation, element  110  may be associated with frame  112  through spring system  116 . Spring system  116  may comprise first set of springs  118  and second set of springs  119 . As used herein, a “set of” items means one or more items. For example, a set of springs means one or more springs. 
     As one specific example, first set of springs  118  may attach element  110  to frame  112  in a direction along first axis  120 . In particular, first set of springs  118  allows element  110  to move linearly in a direction along first axis  120 . Further, second set of springs  119  may attach element  110  to frame  112  in a direction along second axis  122 . Second set of springs  119  allows element  110  to move linearly in a direction along second axis  122 . In this manner, spring system  116  may constrain movement of element  110  to within plane  124  formed by first axis  120  and second axis  122 . In other words, element  110  may have two degrees of freedom. 
     Frame  112 , with element  110  attached to frame  112  through spring system  116  may, may be configured to move with respect to third axis  126 . In particular, frame  112 , with element  110  attached to frame  112 , may rotate about third axis  126 . Third axis  126  is a fixed axis that is substantially perpendicular to plane  124  formed by first axis  120  and second axis  122 . In this manner, first axis  120 , second axis  122 , and third axis  126  are substantially orthogonal to each other. In some illustrative examples, first axis  120 , second axis  122 , and third axis  126  may be referred to as an x-axis, a y-axis, and a z-axis, respectively. 
     Frame  112  may rotate about third axis  126  with angular velocity  128 . Angular velocity  128  also may be referred to as an inertial rate. In other words, angular velocity  128  for frame  112 , with element  110  attached to frame  112 , is the rate of rotation for frame  112  with respect to inertial frame of reference  108 . Angular velocity  128  may be expressed in radians per second, revolutions per second, degrees per second, or in other suitable units. 
     In these illustrative examples, gyroscope  102  also includes control unit  130 . Control unit  130  is configured to control operation of gyroscope  102 . Control unit  130  may be implemented using at least one of a processor, a circuit, an integrated circuit (IC), a microchip, a microprocessor, a computer, and some other suitable type of controller. Further, gyroscope  102  may include a number of additional components, such as, for example, at least one of an electrode, a capacitor, and other suitable components. 
     As used herein, the phrase “at least one of”, when used with a list of items, means that different combinations of one or more of the listed items may be used and only one of each item in the list may be needed. For example, “at least one of item A, item B, and item C” may include, without limitation, item A, or item A and item B. This example also may include item A, item B, and item C, or item B and item C. In other examples, “at least one of” may be, for example, without limitation, two of item A, one of item B, and ten of item C; four of item B and seven of item C; and other suitable combinations. 
     Control unit  130  controls gyroscope  102  such that element  110  moves linearly in a direction along drive axis  121  in plane  124 . In these illustrative examples, this movement of element  110  is vibration of element  110  along drive axis  121 . The angle of drive axis  121  with respect to first axis  120  may be referred to as a drive angle. Control unit  130  may control gyroscope  102  to operate at a selected drive angle such that the actual drive angle is within selected tolerances. 
     Drive axis  121  may be first axis  120  in some illustrative examples. For example, control unit  130  may drive element  110  to vibrate along first axis  120  at number of resonant frequencies  114  using drive control signal  131 . In one illustrative example, drive control signal  131  may be sent to a number of electrodes attached to element  110  to cause element  110  to vibrate along first axis  120  at number of resonant frequencies  114 . 
     In these illustrative examples, vibration of element  110  in the direction along first axis  120 , while frame  112 , with element  110  attached to frame  112 , is rotating about third axis  126 , generates force  132  in a direction along sense axis  123  in plane  124 . In particular, force  132  is a Coriolis force in these examples. Force  132  acts at substantially right angles to angular velocity  128  for frame  112 . 
     Sense axis  123  is an axis that is substantially orthogonal to drive axis  121 . In other words, sense axis  123  is oriented at about 90 degrees with respect to drive axis  121 . Sense axis  123  is the axis along which linear movement of element  110  is induced in response to force  132 . For example, simultaneous vibration of element  110  in the direction along drive axis  121  and rotation of frame  112  about third axis  126  generates force  132 , which induces vibration of element  110  in the direction along sense axis  123 . Drive axis  121  and sense axis  123  are substantially orthogonal to third axis  126 . When drive axis  121  is first axis  120 , sense axis  123  may be second axis  122 . 
     As depicted, control unit  130  uses force rebalance signal  133  to cause the amplitude of vibrations of element  110  along second axis  122  to be substantially zero. In other words, control unit  130  uses force rebalance signal  133  to substantially nullify the movement of element  110  along second axis  122  based on drive control signal  131 . Control unit  130  generates measurements  134  of force rebalance signal  133 . Measurements  134  may be used to determine angular velocity  128 . 
     Measurements  134  may be less accurate than desired when bias  101  of gyroscope  102  contributes to measurements  134 . Bias  101  is the error for gyroscope  102 . For example, bias  101  is the difference between measurements  134  and the measurements that should actually be generated. Bias  101  may be the contribution to measurements  134  when angular velocity  128  is substantially zero. In this manner, bias  101  may be referred to as a zero-rate bias. 
     Bias  101  may be caused by a number of different factors. These factors may include, for example, without limitation, temperature, inconsistencies in the fabrication of the different components for gyroscope  102 , characteristics of element  110 , characteristics of a sensing system in gyroscope  102 , characteristics of control unit  130 , and other suitable factors. These factors also may include damping and stiffness asymmetry between first axis  120  and second axis  122 , asymmetry between drive axis  121  and sense axis  123  alignment, and/or other suitable types of asymmetry. 
     Compensation system  136  may be used to electronically compensate for bias  101 . Compensation system  136  may be implemented using hardware, software, or a combination of the two. For example, compensation system  136  may be implemented within computer system  138 . Computer system  138  may comprise a number of computers. When computer system  138  comprises more than one computer, these computers may be in communication with each other. 
     Compensation system  136  identifies set of equations  140  for bias  101  of gyroscope  102  using model  142  for the motion of gyroscope  102  with respect to non-inertial frame of reference  144  and assumption  146 . Non-inertial frame of reference  144  is a rotating frame of reference. In this illustrative example, non-inertial frame of reference  144  may be formed by first axis  120  and second axis  122 . In other words, plane  124  formed by first axis  120  and second axis  122  may rotate with respect to inertial frame of reference  108 . 
     Model  142  may be a model for the motion of gyroscope  102  along first axis  120  and second axis  122 . In this illustrative example, model  142  takes the form of a number of equations that describe the motion of gyroscope  102  with respect to first axis  120  and second axis  122 . 
     Assumption  146  is an assumption that angular velocity  128  for gyroscope  102  is substantially zero within selected tolerances. In other words, assumption  146  assumes that frame  112  is not rotating about third axis  126  and has a substantially zero inertial rate. Compensation system  136  modifies model  142  based on assumption  146  to form modified model  148 . In particular, modified model  148  is simplified as compared to model  142 . 
     Compensation system  136  identifies set of equations  140  using modified model  148 . Set of equations  140  may include first equation  150  and second equation  152 . First equation  150  is for the portion of bias  101  that is in-phase bias  154 . In-phase bias  154  may be the portion of bias  101  that is substantially in-phase with the linear velocity of the vibrations of gyroscope  102 . Second equation  152  is for the portion of bias  101  that is in-quadrature bias  156 . In-quadrature bias  156  may be the portion of bias  101  that is substantially in-quadrature with the linear velocity of the vibrations of gyroscope  102 . In-quadrature bias  156  also may be referred to as quadrature bias. 
     As depicted, set of equations  140  includes set of parameters  158 . Set of parameters  158  includes one or more parameters that contribute to bias  101 . These parameters may include, for example, at least one of a drive axis misalignment angle, a sense axis misalignment angle, a damping azimuth angle, a damping asymmetry term, a difference between the natural frequencies of drive axis  121  and sense axis  123 , an azimuth angle of drive axis  121  with respect to first axis  120 , and some other suitable parameters. Values for these parameters may be unknown. 
     Compensation system  136  identifies set of values  160  for set of parameters  158  using set of equations  140 , bias measurements  161 , and iterative algorithm  162 . Compensation system  136  identifies set of values  160  for set of parameters  158  within selected tolerances. In other words, compensation system  136  uses set of equations  140 , bias measurements  161 , and iterative algorithm  162  to estimate set of values  160  for set of parameters  158 . 
     Bias measurements  161  may be measurements generated by control unit  130  at plurality of drive angles  164 . A drive angle in plurality of drive angles  164  is the angle of drive axis  121  with respect to first axis  120  along which control unit  130  drives element  110  to move linearly. For example, the drive angle may be selected from a range between about 0 degrees and about 360 degrees with respect to first axis  120 . In one illustrative example, plurality of drive angles  164  may include about 0 degrees, about 45 degrees, and about 90 degrees with respect to first axis  120 . 
     Iterative algorithm  162  may be, for example, without limitation, a least-squares algorithm. Iterative algorithm  162  may be used to solve at least one value in set of values  160  using set of equations  140  and bias measurements  161 . 
     Set of values  160  may be used to identify a particular drive angle at which in-quadrature bias  156  is substantially zero. In other words, set of values  160  may be used to identify the drive angle that substantially zeros out in-quadrature bias  156  within selected tolerances. Gyroscope  102  may be operated at this particular drive angle to calibrate the output of gyroscope  102  using set of equations  140 . 
     Further, with set of values  160  for set of parameters  158 , compensation system  136  may estimate bias  101  of gyroscope  102  at a selected drive angle using set of equations  140  to form an estimated bias. Compensation system  136  may subtract the estimated bias from measurements generated by gyroscope  102  for the selected drive angle to electronically compensate for bias  101  of gyroscope  102 . 
     The illustration of bias compensation environment in  FIG. 1  is not meant to imply physical or architectural limitations to the manner in which an illustrative embodiment may be implemented. Other components in addition to and/or in place of the ones illustrated may be used. Some components may be unnecessary. Also, the blocks are presented to illustrate some functional components. One or more of these blocks may be combined, divided, or combined and divided into different blocks when implemented in an illustrative embodiment. 
     For example, in some illustrative examples, gyroscope  102  may have elements in addition to and/or in place of element  110 . Further, in some case, element  110  may be attached to a case, a housing, a number of structures, or some other suitable type of structure in addition to and/or in place of frame  112 . 
     With reference now to  FIG. 2 , an illustration of a functional model for a gyroscope is depicted in accordance with an illustrative embodiment. Model  200  is an example of a function model for a gyroscope, such as, for example, gyroscope  102  in  FIG. 1 . In this illustrative example, model  200  includes element  202  and frame  203 . Of course, in other illustrative examples, model  200  may include other components for the gyroscope in addition to the ones described in  FIG. 2 . 
     Element  202  is associated with frame  203  by first set of springs  204  along the direction of x-axis  206 . Element  202  is associated with frame  203  by second set of springs  208  along the direction of y-axis  210 . In this illustrative example, x-axis  206  and y-axis  210  are examples of implementations for first axis  120  and second axis  122 , respectively, in  FIG. 1 . 
     As depicted, x-axis  206  and y-axis  210  form plane  212 . Element  202  may vibrate along x-axis  206  at a first natural frequency. Further, element  202  may vibrate along y-axis at a second natural frequency. The first natural frequency may be the same or different from the second natural frequency, depending on the implementation. Vibration of element  202  along x-axis  206  may be a first mode, while vibration of element  202  along y-axis  210  may be a second mode. The first mode and the second mode may be referred to as, for example, a drive mode and a sense mode, respectively. 
     Element  202  may vibrate along x-axis  206  and/or y-axis  210  independently of movement of frame  203  in this illustrative example. In particular, first set of springs  204  and second set of springs  208  may allow element  202  to move along x-axis  206  and y-axis  210  independently of the movement of frame  203 . 
     Motion of element  202  is constrained to within plane  212  in this illustrative example. In one illustrative example, a control unit, such as control unit  130  in  FIG. 1 , may drive element  202  to vibrate along the direction of x-axis  206 . Frame  203  may be rotated about a z-axis  211  that is substantially perpendicular to plane  212 . Rotation of frame  203  about z-axis  211  while element  202  is moved along the direction of x-axis  206  generates a Coriolis force that causes element  202  to vibrate along the direction of y-axis  210 . 
     For example, if element  202  is moved in the direction of arrow  214  along x-axis  206  while frame  203  is rotated about z-axis  211  in the direction of arrow  216 , element  202  may be moved in the direction of arrow  218  along y-axis  210 . If element  202  is moved in the direction of arrow  220  along x-axis  206  while frame  203  is rotated about z-axis  211  in the direction of arrow  216 , element  202  may be moved in the direction of arrow  222  along y-axis  210 . 
     Similarly, if element  202  is moved in the direction of arrow  214  along x-axis  206  while frame  203  is rotated about z-axis  211  in the direction of arrow  224 , element  202  may be moved in the direction of arrow  222  along y-axis  210 . If element  202  is moved in the direction of arrow  220  along x-axis  206  while frame  203  is rotated about z-axis  211  in the direction of arrow  224 , element  202  may be moved in the direction of arrow  218  along y-axis  210 . 
     With reference now to  FIG. 3 , an illustration of the orbit of an element for a gyroscope is depicted in accordance with an illustrative embodiment. In this illustrative example, orbit  300  of an element, such as element  202  from  FIG. 2 , is depicted with respect to x-axis  302  and y-axis  304 . X-axis  302  is the same as x-axis  206  in  FIG. 2 . Y-axis is the same as y-axis  210  in  FIG. 2 . 
     Element  202  may oscillate about origin  305  at the intersection of x-axis  302  and y-axis  304 . Oscillation of element  202  may follow a pendulum-type behavior. In this manner, orbit  300  may be a pendulum orbit in this illustrative example. 
     Parameters for orbit  300  include pendulum angle  306 , principal amplitude  308 , quadrature amplitude  310 , and phase  312 . As depicted, pendulum angle  306 , θ, is an angle with respect to x-axis  302  and defines an axis relative to x-axis  302  along which element  202  may vibrate. Principal amplitude  308 , A, is the amplitude of vibrations for element  202  along the axis defined by pendulum angle  306 . 
     Quadrature amplitude  310 , q, is the amplitude of vibrations for element  202  along the axis in-quadrature to the axis defined by pendulum angle  306 . In other words, quadrature amplitude  310  is the amplitude of vibrations for element  202  along the axis substantially orthogonal to the axis defined by pendulum angle  306 . Further, phase  312 , φ′, is the phase for orbit  300 . 
     A control unit, such as control unit  130  in  FIG. 1 , may be configured to control external force components applied to element  202  and/or frame  203  in  FIG. 2  to control pendulum angle  306 , principal amplitude  308 , quadrature amplitude  310 , and phase  312 . For example, the control unit may control forces exerted on element  202  such that these forces have a same phase as the phase of the natural frequency for element  202 . 
     Further, the control unit may control forces exerted on element  202  such that principal amplitude  308 , A, remains substantially constant during operation of the gyroscope against any damping forces. The rate at which principal amplitude  308 , A, decreases because of damping is proportional to A/τ, in which τ is a damping time constant. Additionally, the control unit may control the forces exerted on element  202  such that pendulum angle  306  and quadrature amplitude  310  remain substantially zero with selected tolerances during operation of the gyroscope. 
     With reference now to  FIG. 4 , an illustration of a process for compensating for bias of a gyroscope in the form of a flowchart is depicted in accordance with an illustrative embodiment. The process illustrated in  FIG. 4  may be implemented using compensation system  136  to compensate for bias  101  of gyroscope  102  in  FIG. 1 . 
     The process begins by generating bias measurements for a plurality of drive angles using the gyroscope (operation  400 ). In operation  400 , the bias measurements are generated with the angular velocity for the gyroscope substantially equal to zero within selected tolerances. 
     The process then identifies a set of equations for the bias of the gyroscope using a model for motion of the gyroscope with respect to a non-inertial frame of reference and an assumption that the angular velocity for the gyroscope is substantially zero within selected tolerances (operation  402 ). The angular velocity for the gyroscope is with respect to an inertial frame of reference. The set of equations includes a set of parameters. The one or more parameters in the set of parameters have unknown values. In this illustrative example, in operation  402 , the set of equations includes a first equation for in-phase bias of the gyroscope and a second equation for in-quadrature bias of the gyroscope. 
     Thereafter, the process identifies a set of values for the set of parameters using the bias measurements and the set of equations (operation  404 ). Operation  404  may be performed using an iterative algorithm, such as, for example, without limitation, a least-squares algorithm. 
     Next, the process estimates the bias of the gyroscope at a selected drive angle using the set of equations with the set of values identified for the set of parameters to form an estimated bias (operation  406 ). The selected drive angle may be, for example, the current drive angle at which the gyroscope is being operated. In this manner, the bias of the gyroscope may be estimated in substantially real-time during operation of the gyroscope. 
     The process then subtracts the estimated bias from one or more measurements generated by the gyroscope for the selected drive angle to substantially compensate for the bias of the gyroscope (operation  408 ), with the process terminating thereafter. In operation  408 , the estimated bias is subtracted from a measurement to substantially remove the contribution of the bias of the gyroscope from the measurement within selected tolerances. In this manner, the bias of the gyroscope may be electronically compensated in substantially real-time. 
     With reference now to  FIG. 5 , an illustration of a process for identifying a set of equations for the bias of a gyroscope in the form of a flowchart is depicted in accordance with an illustrative embodiment. The process illustrated in  FIG. 5  may be an example of one process that may be used to implement operation  402  in  FIG. 4 . 
     The process begins by identifying a model for the motion of the gyroscope (operation  500 ). The gyroscope comprises an element associated with a frame. In this illustrative example, the model comprises the following equations: 
                   {             x   ¨     -     k   ⁡     (       2   ⁢   Ω   ⁢     y   .       +       Ω   .     ⁢   y       )       +       2   τ     ⁢     x   .       +       Δ   ⁡     (     1   τ     )       ⁢     (         y   .     ⁢   sin   ⁢           ⁢   2   ⁢           ⁢     θ   τ       +       x   .     ⁢   cos   ⁢           ⁢   2   ⁢     θ   τ         )       +       (       ω   2     -       k   ′     ⁢     Ω   2         )     ⁢   x     -                 ωΔ   ⁢           ⁢     ω   ⁡     (       x   ⁢           ⁢   cos   ⁢           ⁢   2   ⁢     θ   ω       +     y   ⁢           ⁢   sin   ⁢           ⁢   2   ⁢           ⁢     θ   ω         )         =       f   x     +       γ   x     ⁢     g   x                       y   ¨     +     k   ⁡     (       2   ⁢   Ω   ⁢     x   .       +       Ω   .     ⁢   x       )       +       2   τ     ⁢     y   .       -       Δ   ⁡     (     1   τ     )       ⁢     (         y   .     ⁢   cos   ⁢           ⁢   2   ⁢           ⁢     θ   τ       -       x   .     ⁢   sin   ⁢           ⁢   2   ⁢     θ   τ         )       +       (       ω   2     -       k   ′     ⁢     Ω   2         )     ⁢   y     +                 ωΔ   ⁢           ⁢     ω   ⁡     (       y   ⁢           ⁢   cos   ⁢           ⁢   2   ⁢     θ   ω       -     x   ⁢           ⁢   sin   ⁢           ⁢   2   ⁢           ⁢     θ   ω         )         =       f   y     +       γ   y     ⁢     g   y                         (   1   )               
where
 
                       ω   =       K   m         ,     
     ⁢       ω   2     =         ω   1   2     +     ω   2   2       2       ,     
     ⁢   and     ⁢     
     ⁢         ω   ⁢           ⁢   Δω     =         ω   1   2     -     ω   2   2       2       ,             (   2   )               
where
 
                       1   τ     =       1   2     ⁢     (       1     τ   1       +     1     τ   2         )         ⁢     
     ⁢   and   ⁢     
     ⁢         Δ   ⁡     (     1   τ     )       =     (       1     τ   1       -     1     τ   2         )       ,     
     ⁢   and             (   3   )               
where x is a position of the element in the gyroscope with respect to an x-axis, {dot over (x)} is the first derivative of x with respect to time, {umlaut over (x)} is the second derivative of x with respect to time, y is a position of the element in the gyroscope with respect to a y-axis, {dot over (y)} is the first derivative of y with respect to time, ÿ is the second derivative of y with respect to time, Ω is the angular velocity of the gyroscope with respect to an inertial frame of reference, k is a gain, τ is a damping time constant, 1 indicates a drive axis for the element, 2 indicates a sense axis for the element, τ 1  is the damping time constant with respect to the drive axis, τ 2  is the damping time constant with respect to the sense axis, ω is the natural frequency of the element in the gyroscope, ω 1  is the natural frequency for the drive axis, ω 2  is the natural frequency for the sense axis, Δ indicates a difference, θ τ  is the azimuth of the τ 1  damping axis with respect to the x-axis and also may be referred to as the damping azimuth angle, k′ is a gain, θ ω  is the azimuth of the drive axis with respect to the x-axis, f x  is an external force component exerted on the element along the x-axis, f y  is an external force component exerted on the element along the y-axis for a force-rebalance signal, γ x  is a gain with respect to the x-axis, γ y  is a gain with respect to the y-axis, g x  is a linear acceleration component for the frame with respect to the x-axis, g y  is a linear acceleration component for the frame with respect to the y-axis, K is the spring constant for the spring connecting the element to the frame of the gyroscope along the x-axis, and m is the mass of the element in the gyroscope. The term,
 
               Δ   ⁡     (     1   τ     )       ,         
may be referred to as a damping asymmetry term.
 
     The process then modifies the model for the motion of the gyroscope based on an assumption that the angular velocity for the gyroscope is substantially zero within selected tolerances to form a modified model (operation  502 ). In operation  502 , an assumption is also made that the contribution of the linear acceleration components for the frame are negligible. The modified model comprises the following equations: 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             x 
                             ¨ 
                           
                           + 
                           
                             
                               2 
                               τ 
                             
                             ⁢ 
                             
                               x 
                               . 
                             
                           
                           + 
                           
                             
                               Δ 
                               ⁡ 
                               
                                 ( 
                                 
                                   1 
                                   τ 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   
                                     y 
                                     . 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   
                                     θ 
                                     τ 
                                   
                                 
                                 + 
                                 
                                   
                                     x 
                                     . 
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     θ 
                                     τ 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               ω 
                               2 
                             
                             ⁢ 
                             x 
                           
                           - 
                           
                             ω 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ω 
                               ( 
                               
                                 
                                   x 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     θ 
                                     ω 
                                   
                                 
                                 + 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               y 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 ω 
                               
                             
                             ) 
                           
                           = 
                           
                             f 
                             x 
                           
                         
                       
                     
                     
                       
                         
                           
                             y 
                             ¨ 
                           
                           + 
                           
                             
                               2 
                               τ 
                             
                             ⁢ 
                             
                               y 
                               . 
                             
                           
                           - 
                           
                             
                               Δ 
                               ⁡ 
                               
                                 ( 
                                 
                                   1 
                                   τ 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               ( 
                               
                                 
                                   
                                     y 
                                     . 
                                   
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   
                                     θ 
                                     τ 
                                   
                                 
                                 - 
                                 
                                   
                                     x 
                                     . 
                                   
                                   ⁢ 
                                   sin 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     θ 
                                     τ 
                                   
                                 
                               
                               ) 
                             
                           
                           + 
                           
                             
                               ω 
                               2 
                             
                             ⁢ 
                             y 
                           
                           + 
                           
                             ω 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ω 
                               ( 
                               
                                 
                                   y 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   cos 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   2 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   
                                     θ 
                                     ω 
                                   
                                 
                                 - 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               x 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               sin 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               
                                 θ 
                                 ω 
                               
                             
                             ) 
                           
                           = 
                           
                             f 
                             y 
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   4 
                   ) 
                 
               
             
           
         
       
     
     Thereafter, the process identifies a misalignment model for the misalignment of the drive axis (operation  504 ). The misalignment of the drive axis occurs when the axis along which the element of the gyroscope is driven to vibrate is different from the drive axis indicated by the drive control signal generated by the control unit for the gyroscope. 
     In other words, the control unit may generate a drive control signal to control the element to move along the drive axis. However, the actual movement of the element may be along an axis offset from the drive axis by angle a. This angle, a, may be referred to as a drive axis misalignment angle. Further, with this type of misalignment, the axis along which the element is induced to vibrate in response to a Coriolis force may be offset from the sense axis by angle b. This angle, b, may be referred to as a sense axis misalignment angle. In this illustrative example, these angles, a and b, may be in radians (rads). 
     The misalignment model may be defined by the following equations: 
                       [           f     x   ,   actual                 f     y   ,   actual             ]     =           [         1       a           b       1         ]       ︸     E   ⁡     (     a   ,   b     )           ⁡     [           f   x               f   y           ]       =       E   ⁡     (     a   ,   b     )       ⁡     [           f   x               f   y           ]           ,           (   5   )               
where f x,actual  is the actual force exerted along the x-axis and f y,actual  is the actual force exerted along the y-axis.
 
     The process then identifies a number of transformations for transforming the model for the motion of the gyroscope to take into account a drive axis between the x-axis and the y-axis (operation  506 ). For example, the gyroscope may be driven to vibrate along an axis between the x-axis and the y-axis. The angle between this drive axis and the x-axis is the drive angle. The drive axis and the sense axis that is substantially orthogonal to the drive axis form a rotated frame of reference with respect to the frame of reference formed by the x-axis and the y-axis. 
     A first transformation is defined as follows: 
                       [           x   s               y   s           ]     =       [           cos   ⁢           ⁢   φ           sin   ⁢           ⁢   φ                 -   sin     ⁢           ⁢   φ           cos   ⁢           ⁢   φ           ]     ⁡     [         x           y         ]         ,           (   6   )               
where φ is the drive angle, x s  is the position of the element with respect to the drive axis, y s  is the position of the element with respect to the sense axis, cos is the cosine function, and sin is the sine function.
 
     A second transformation needed for deriving the model for the motion of the gyroscope with respect to the rotated frame of reference is defined as follows: 
                       [         x           y         ]     =         [           cos   ⁢           ⁢   φ             -   sin     ⁢           ⁢   φ               sin   ⁢           ⁢   φ           cos   ⁢           ⁢   φ           ]     ⁡     [           x   s               y   s           ]       =         M   ⁡     (   φ   )       ⁡     [           x   s               y   s           ]       .     
     ⁢   Further         ,           (   7   )                   [           x   .               y   .           ]     =           ⅆ     M   ⁡     (   φ   )           ⅆ   t       ⁡     [           x   s               y   s           ]       +       M   ⁡     (   φ   )       ⁡     [             x   .     s                 y   .     s           ]           ,           (   8   )                   [           x   ¨               y   ¨           ]     =             ⅆ   2     ⁢     M   ⁡     (   φ   )           ⅆ     t     2   ⁢                   ⁡     [           x   s               y   s           ]       +     2   ⁢         ⅆ     M   ⁡     (   φ   )           ⅆ   t       ⁡     [             x   .     s                 y   .     s           ]         +       M   ⁡     (   φ   )       ⁡     [             x   ¨     s                 y   ¨     s           ]           ,   and           (   9   )                   [           f     x   ,   actual                 f     y   ,   actual             ]     =           [         1       a           b       1         ]       ︸     E   ⁡     (     a   ,   b     )           ⁢       M   ⁡     (   φ   )       ⁡     [           f     x   s                 f     y   s             ]         =       EM   ⁡     (   φ   )       ⁡     [           f     x   s                 f     y   s             ]           ,           (   10   )               
where {dot over (x)} s  is the first derivative of x s  with respect to time, {umlaut over (x)} s  is the second derivative of x s  with respect to time, {dot over (y)} s  is the first derivative of y s  with respect to time, ÿ s  is the second derivative of y s  with respect to time, f x     s    is the external force component exerted on the element in the direction of the drive axis through x s , and f y     s    is the external force component exerted on the element in the direction of the sense axis through y s .
 
     The process uses the modified model for the motion of the gyroscope and the number of transformations identified to form a transformed model for the motion of the gyroscope (operation  508 ). In operation  508 , an assumption is made that the angular velocity for the gyroscope is at a lower rate than the rate at which the vibrations of the sense axis are being measured. 
     Consequently, the first and second derivatives in equations (8) and (9) may be negligible. Equations (8) and (9) may be simplified as follows: 
                     [           x   .               y   .           ]     =       M   ⁡     (   φ   )       ⁡     [             x   .     s                 y   .     s           ]               (   11   )                 [           x   ¨               y   ¨           ]     =         M   ⁡     (   φ   )       ⁡     [             x   ¨     s                 y   ¨     s           ]       .             (   12   )               
The transformed model may be defined as follows:
 
                       [             x   ¨     s                 y   ¨     s           ]     +         [             2   τ     +       Δ   ⁡     (     1   τ     )       ⁢   cos   ⁢           ⁢   2   ⁢     θ   τ                 Δ   ⁡     (     1   τ     )       ⁢   sin   ⁢           ⁢   2   ⁢     θ   τ                   Δ   ⁡     (     1   τ     )       ⁢   sin   ⁢           ⁢   2   ⁢     θ   τ               2   τ     -       Δ   ⁡     (     1   τ     )       ⁢   cos   ⁢           ⁢   2   ⁢     θ   τ               ]       ︸     D   ⁡     (     τ   ,     Δ   ⁡     (     1   τ     )       ,     θ   τ       )           ⁡     [           x   .               y   .           ]       +         [             ω   2     -     ω   ⁢           ⁢   Δ   ⁢           ⁢   ω   ⁢           ⁢   cos   ⁢           ⁢   2   ⁢     θ   ω                 -   ω     ⁢           ⁢   Δ   ⁢           ⁢   ω   ⁢           ⁢   sin   ⁢           ⁢   2   ⁢     θ   ω                   -   ω     ⁢           ⁢   Δ   ⁢           ⁢   ω   ⁢           ⁢   sin   ⁢           ⁢   2   ⁢     θ   ω               ω   2     +     ω   ⁢           ⁢   Δ   ⁢           ⁢   ω   ⁢           ⁢   cos   ⁢           ⁢   2   ⁢     θ   ω               ]       ︸     S   ⁡     (     ω   ,     Δ   ⁢           ⁢   ω     ,     θ   ω       )           ⁡     [         x           y         ]         =       E   ⁡     (     a   ,   b     )       ⁢       M   ⁡     (   φ   )       ⁡     [           f     x   s                 f     y   s             ]                 (   13   )               
with the transformed model being rewritten as follows:
 
     
       
         
           
             
               
                 
                   
                     
                       [ 
                       
                         
                           
                             
                               
                                 x 
                                 ¨ 
                               
                               s 
                             
                           
                         
                         
                           
                             
                               
                                 y 
                                 ¨ 
                               
                               s 
                             
                           
                         
                       
                       ] 
                     
                     + 
                     
                       
                         [ 
                         
                           
                             
                               
                                 ω 
                                 x 
                                 2 
                               
                             
                             
                               0 
                             
                           
                           
                             
                               0 
                             
                             
                               0 
                             
                           
                         
                         ] 
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 x 
                                 s 
                               
                             
                           
                           
                             
                               
                                 y 
                                 s 
                               
                             
                           
                         
                         ] 
                       
                     
                     + 
                     
                       
                         
                           
                             
                               M 
                               
                                 - 
                                 1 
                               
                             
                             ⁡ 
                             
                               ( 
                               φ 
                               ) 
                             
                           
                           ⁢ 
                           DM 
                           ⁢ 
                           
                             ( 
                             φ 
                             ) 
                           
                         
                         
                           ︸ 
                           H 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 
                                   x 
                                   . 
                                 
                                 s 
                               
                             
                           
                           
                             
                               
                                 
                                   y 
                                   . 
                                 
                                 s 
                               
                             
                           
                         
                         ] 
                       
                     
                     + 
                     
                       
                         
                           ( 
                           
                             
                               
                                 
                                   M 
                                   
                                     - 
                                     1 
                                   
                                 
                                 ⁡ 
                                 
                                   ( 
                                   φ 
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 SM 
                                 ⁡ 
                                 
                                   ( 
                                   φ 
                                   ) 
                                 
                               
                             
                             - 
                             
                               [ 
                               
                                 
                                   
                                     
                                       ω 
                                       x 
                                       2 
                                     
                                   
                                   
                                     0 
                                   
                                 
                                 
                                   
                                     0 
                                   
                                   
                                     0 
                                   
                                 
                               
                               ] 
                             
                           
                           ) 
                         
                         
                           ︸ 
                           G 
                         
                       
                       ⁡ 
                       
                         [ 
                         
                           
                             
                               
                                 x 
                                 s 
                               
                             
                           
                           
                             
                               
                                 y 
                                 s 
                               
                             
                           
                         
                         ] 
                       
                     
                   
                   = 
                   
                     
                       
                         
                           
                             M 
                             
                               - 
                               1 
                             
                           
                           ⁡ 
                           
                             ( 
                             φ 
                             ) 
                           
                         
                         ⁢ 
                         
                           E 
                           ⁡ 
                           
                             ( 
                             
                               a 
                               , 
                               b 
                             
                             ) 
                           
                         
                         ⁢ 
                         M 
                         ⁢ 
                         
                           ( 
                           φ 
                           ) 
                         
                       
                       
                         ︸ 
                         K 
                       
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               f 
                               
                                 x 
                                 s 
                               
                             
                           
                         
                         
                           
                             
                               f 
                               
                                 y 
                                 s 
                               
                             
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     The new matrices may be defined as follows: 
                   {           H   =           M     -   1       ⁡     (   φ   )       ⁢     DM   ⁡     (   φ   )         =     [           h   11           h   12               h   21           h     22   ⁢                     ]                   G   =             M     -   1       ⁡     (   φ   )       ⁢     SM   ⁡     (   φ   )         -     [           ω   x   2         0           0       0         ]       =     [           g   11           g   12               g   21           g   22           ]                   K   =           M     -   1       ⁡     (   φ   )       ⁢     E   ⁡     (     a   ,   b     )       ⁢     M   ⁡     (   φ   )         =     [           k   11           k   12               k   21           k   22           ]                       (   15   )               
and the transformed model for the motion of the gyroscope further simplified as follows:
 
     
       
         
           
             
               
                 
                   
                     [ 
                     
                       
                         
                           
                             
                               x 
                               ¨ 
                             
                             s 
                           
                         
                       
                       
                         
                           
                             
                               y 
                               ¨ 
                             
                             s 
                           
                         
                       
                     
                     ] 
                   
                   + 
                   
                     
                       [ 
                       
                         
                           
                             
                               ω 
                               x 
                               2 
                             
                           
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                           
                             0 
                           
                         
                       
                       ] 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               x 
                               s 
                             
                           
                         
                         
                           
                             
                               y 
                               s 
                             
                           
                         
                       
                       ] 
                     
                   
                   + 
                   
                       
                     
                       
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     
                                       h 
                                       11 
                                     
                                     ⁢ 
                                     
                                       
                                         x 
                                         . 
                                       
                                       s 
                                     
                                   
                                   + 
                                   
                                     
                                       h 
                                       12 
                                     
                                     ⁢ 
                                     
                                       
                                         y 
                                         . 
                                       
                                       s 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       h 
                                       21 
                                     
                                     ⁢ 
                                     
                                       
                                         x 
                                         . 
                                       
                                       s 
                                     
                                   
                                   + 
                                   
                                     
                                       h 
                                       22 
                                     
                                     ⁢ 
                                     
                                       
                                         y 
                                         . 
                                       
                                       s 
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                         + 
                         
                           [ 
                           
                             
                               
                                 
                                   
                                     
                                       g 
                                       11 
                                     
                                     ⁢ 
                                     
                                       x 
                                       s 
                                     
                                   
                                   + 
                                   
                                     
                                       g 
                                       12 
                                     
                                     ⁢ 
                                     
                                       y 
                                       s 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     
                                       g 
                                       21 
                                     
                                     ⁢ 
                                     
                                       x 
                                       s 
                                     
                                   
                                   + 
                                   
                                     
                                       g 
                                       22 
                                     
                                     ⁢ 
                                     
                                       y 
                                       s 
                                     
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                       = 
                       
                         K 
                         ⁡ 
                         
                           [ 
                           
                             
                               
                                 
                                   f 
                                   
                                     x 
                                     s 
                                   
                                 
                               
                             
                             
                               
                                 
                                   f 
                                   
                                     y 
                                     s 
                                   
                                 
                               
                             
                           
                           ] 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
     Thereafter, the process identifies a force-rebalance model for the force-rebalance signal that substantially nullifies the vibrations with respect to the sense axis (operation  510 ). For example, when the vibrations with respect to the sense axis are substantially nullified, 
                   {                 y   ¨     s     =   0                   y   .     s     =   0                     y   .     s     =   0     ,           ⁢     
     ⁢   and             (   17   )                   [             x   ¨     s                 y   ¨     s           ]     +       [           ω   x   2         0           0       0         ]     ⁡     [           x   s               y   s           ]         =       [         0           0         ]     .             (   18   )               
Consequently, the force-rebalance signal may be solved for using the following equation:
 
                         [             h   11     ⁢       x   .     s                   h   21     ⁢       x   .     s             ]     +     [             g   11     ⁢     x   s                   g   21     ⁢     x   s             ]       =     K   ⁡     [           f     x   s                 f     y   s             ]         ,           (   19   )               
which may be simplified to
 
                     [               h   11     ⁢       x   .     s       +       g   11     ⁢     x   s                       h   21     ⁢       x   .     s       +       g   21     ⁢     x   s               ]     =         [           g   11           h   11               g   21           h   21           ]     ⁡     [           x   s               x   s           ]       =       K   ⁡     [           f     x   s                 f     y   s             ]       .               (   20   )               
Solving for the force-rebalance signal gives the following equation:
 
                       [           f     x   s                 f     y   s             ]     =         K     -   1       ⁡     [           g   11           h   11               g   21           h   21           ]       ⁡     [           x   s                 x   .     s           ]         ,           (   21   )               
which may be expanded to
 
     
       
         
           
             
               
                 
                   
                     f 
                     
                       y 
                       s 
                     
                   
                   = 
                   
                     
                       
                         Δ 
                         ⁡ 
                         
                           ( 
                           
                             1 
                             τ 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         sin 
                         ( 
                         
                           
                             2 
                             ⁢ 
                             φ 
                           
                           - 
                           
                             2 
                             ⁢ 
                             
                               θ 
                               τ 
                             
                           
                         
                         ⁢ 
                         
                             
                         
                         ) 
                       
                       ⁢ 
                       
                         
                           x 
                           . 
                         
                         s 
                       
                     
                     - 
                     
                       
                         a 
                         2 
                       
                       ⁢ 
                       
                         Δ 
                         ⁡ 
                         
                           ( 
                           
                             1 
                             τ 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             
                               2 
                               ⁢ 
                               φ 
                             
                             - 
                             
                               2 
                               ⁢ 
                               
                                 θ 
                                 τ 
                               
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           x 
                           . 
                         
                         s 
                       
                     
                     + 
                     
                       
                         b 
                         2 
                       
                       ⁢ 
                       
                         Δ 
                         ⁡ 
                         
                           ( 
                           
                             1 
                             τ 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             
                               2 
                               ⁢ 
                               φ 
                             
                             - 
                             
                               2 
                               ⁢ 
                               
                                 θ 
                                 τ 
                               
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           x 
                           . 
                         
                         s 
                       
                     
                     + 
                     
                       
                         
                           3 
                           ⁢ 
                           a 
                         
                         4 
                       
                       ⁢ 
                       ω 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ω 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             
                               2 
                               ⁢ 
                               φ 
                             
                             - 
                             
                               2 
                               ⁢ 
                               
                                 θ 
                                 ω 
                               
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         x 
                         s 
                       
                     
                     + 
                     
                       
                         a 
                         2 
                       
                       ⁢ 
                       
                         Δ 
                         ⁡ 
                         
                           ( 
                           
                             1 
                             τ 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             2 
                             ⁢ 
                             
                               θ 
                               τ 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           x 
                           . 
                         
                         s 
                       
                     
                     + 
                     
                       
                         b 
                         2 
                       
                       ⁢ 
                       
                         Δ 
                         ⁡ 
                         
                           ( 
                           
                             1 
                             τ 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             2 
                             ⁢ 
                             
                               θ 
                               τ 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           x 
                           . 
                         
                         s 
                       
                     
                     - 
                     
                       
                         1 
                         
                           τ 
                           ⁡ 
                           
                             ( 
                             
                               1 
                               - 
                               ab 
                             
                             ) 
                           
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             b 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 x 
                                 . 
                               
                               s 
                             
                           
                           - 
                           
                             a 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               
                                 x 
                                 . 
                               
                               s 
                             
                           
                           + 
                           
                             a 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   φ 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 x 
                                 . 
                               
                               s 
                             
                           
                           + 
                           
                             b 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               cos 
                               ⁡ 
                               
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   φ 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               
                                 x 
                                 . 
                               
                               s 
                             
                           
                         
                         ) 
                       
                     
                     - 
                     
                       ω 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ω 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             
                               2 
                               ⁢ 
                               φ 
                             
                             - 
                             
                               2 
                               ⁢ 
                               
                                 θ 
                                 ω 
                               
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         x 
                         s 
                       
                     
                     + 
                     
                       
                         a 
                         4 
                       
                       ⁢ 
                       ω 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ω 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             
                               2 
                               ⁢ 
                               φ 
                             
                             + 
                             
                               2 
                               ⁢ 
                               
                                 θ 
                                 ω 
                               
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         x 
                         s 
                       
                     
                     - 
                     
                       
                         a 
                         4 
                       
                       ⁢ 
                       ω 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ω 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             
                               2 
                               ⁢ 
                               φ 
                             
                             - 
                             
                               2 
                               ⁢ 
                               
                                 θ 
                                 ω 
                               
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         x 
                         s 
                       
                     
                     + 
                     
                       
                         b 
                         4 
                       
                       ⁢ 
                       ω 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ω 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       cos 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             2 
                             ⁢ 
                             φ 
                           
                           + 
                           
                             2 
                             ⁢ 
                             
                               θ 
                               ω 
                             
                           
                         
                         ) 
                       
                       ⁢ 
                       
                         x 
                         s 
                       
                     
                     - 
                     
                       
                         a 
                         
                           ab 
                           - 
                           1 
                         
                       
                       ⁢ 
                       ω 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ω 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             2 
                             ⁢ 
                             
                               θ 
                               ω 
                             
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         
                           x 
                           s 
                         
                         . 
                       
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
     The process then uses the force-rebalance model to identify the set of equations for the bias of the gyroscope (operation  512 ), with the process terminating thereafter. In operation  512 , the following assumption is made: 
                   {             x   s     =       c   0     ⁢     cos   ⁡     (       ω   x     ⁢   t     )                           x   .     s     =       -     c   0       ⁢     ω   x     ⁢     sin   ⁡     (       ω   x     ⁢   t     )           ,                   (   23   )               
where c 0  is the amplitude component in phase with the drive axis, and ω x  is the natural frequency of the element with respect to the x-axis.
 
     The set of equations identified in operation  512  includes a first equation for an in-phase bias and a second equation for an in-quadrature bias. The in-phase bias may be represented by the following equation: 
                       B       -     c   0       ⁢     ω   x         =         Δ   ⁡     (     1   τ     )       ⁢     sin   ⁡     (       2   ⁢   φ     -     2   ⁢     θ   τ         )         +         a   +   b     2     ⁢     Δ   ⁡     (     1   τ     )       ⁢     (       cos   ⁡     (     2   ⁢     θ   τ       )       -     cos   ⁡     (       2   ⁢   φ     -     2   ⁢     θ   τ         )         )       -         b   ⁡     (     1   +     cos   ⁡     (     2   ⁢   φ     )         )       -     a   ⁡     (     1   -     cos   ⁡     (     2   ⁢   φ     )         )           τ   ⁡     (     1   -   ab     )             ,           (   24   )               
where B is the in-phase bias. In equation (24), the in-phase bias, B, is scaled by the term, c 0 ω x . Equation (24) may be further simplified as follows:
 
     
       
         
           
             
               
                 
                   
                     B 
                     
                       
                         - 
                         
                           c 
                           0 
                         
                       
                       ⁢ 
                       
                         ω 
                         x 
                       
                     
                   
                   = 
                   
                     
                       
                         Δ 
                         ⁡ 
                         
                           ( 
                           
                             1 
                             τ 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             
                               2 
                               ⁢ 
                               φ 
                             
                             - 
                             
                               2 
                               ⁢ 
                               
                                 θ 
                                 τ 
                               
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         
                           a 
                           + 
                           b 
                         
                         2 
                       
                       ⁢ 
                       
                         Δ 
                         ⁡ 
                         
                           ( 
                           
                             1 
                             τ 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   τ 
                                 
                               
                               ) 
                             
                           
                           - 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 
                                   2 
                                   ⁢ 
                                   φ 
                                 
                                 - 
                                 
                                   2 
                                   ⁢ 
                                   
                                     θ 
                                     τ 
                                   
                                 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         
                           b 
                           ⁡ 
                           
                             ( 
                             
                               1 
                               + 
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     2 
                                     ⁢ 
                                     φ 
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                         - 
                         
                           a 
                           ⁡ 
                           
                             ( 
                             
                               1 
                               - 
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     2 
                                     ⁢ 
                                     φ 
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                       
                         τ 
                         ⁡ 
                         
                           ( 
                           
                             1 
                             - 
                             ab 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   25 
                   ) 
                 
               
             
           
         
       
     
     The in-quadrature bias may be represented by the following equation: 
     
       
         
           
             
               
                 
                   
                     quad 
                     
                       c 
                       0 
                     
                   
                   = 
                   
                     
                       
                         - 
                         
                           ( 
                           
                             1 
                             + 
                             a 
                           
                           ) 
                         
                       
                       ⁢ 
                       ω 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ω 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             
                               2 
                               ⁢ 
                               φ 
                             
                             - 
                             
                               2 
                               ⁢ 
                               
                                 θ 
                                 ω 
                               
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         
                           a 
                           + 
                           b 
                         
                         4 
                       
                       ⁢ 
                       ω 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ωcos 
                         ⁡ 
                         
                           ( 
                           
                             
                               2 
                               ⁢ 
                               φ 
                             
                             + 
                             
                               2 
                               ⁢ 
                               
                                 θ 
                                 ω 
                               
                             
                           
                           ) 
                         
                       
                     
                     - 
                     
                       
                         a 
                         
                           ab 
                           - 
                           1 
                         
                       
                       ⁢ 
                       ω 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ωcos 
                         ⁡ 
                         
                           ( 
                           
                             2 
                             ⁢ 
                             
                               θ 
                               ω 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   26 
                   ) 
                 
               
             
           
         
       
     
     In this manner, the process illustrated in  FIG. 5  provides a method for identifying a set of equations, such as equation (25) and equation (26) for the bias of the gyroscope. The unknown parameters in these equations include a, b, 
               Δ   ⁡     (     1   τ     )       ,         
θ τ , Δω, and θ ω . In one illustrative example, these equations may be used in operation  404  in  FIG. 4  to identify values for these parameters.
 
     With reference now to  FIG. 6 , an illustration of a process for identifying a set of values for a set of parameters in a set of equations for the bias of a gyroscope in the form of a flowchart is depicted in accordance with an illustrative embodiment. The process illustrated in  FIG. 6  may be an example of one process that may be used to implement operation  404  in  FIG. 4 . 
     The process begins by making a group of assumptions (operation  600 ). In operation  600 , the group of assumptions includes assumptions with respect to the in-phase bias of the gyroscope. These assumptions are as follows: 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           ab 
                           ≈ 
                           0 
                         
                       
                     
                     
                       
                         
                           
                             
                               Δ 
                               ⁡ 
                               
                                 ( 
                                 
                                   1 
                                   τ 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             a 
                           
                           ≈ 
                           0 
                         
                       
                     
                     
                       
                         
                           
                             
                               Δ 
                               ⁡ 
                               
                                 ( 
                                 
                                   1 
                                   
                                     τ 
                                     ⁢ 
                                     
                                         
                                     
                                   
                                 
                                 ) 
                               
                             
                             ⁢ 
                             b 
                           
                           ≈ 
                           0 
                         
                       
                     
                     
                       
                         
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             
                               θ 
                               τ 
                             
                           
                           ≈ 
                           1 
                         
                       
                     
                     
                       
                         
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             
                               θ 
                               τ 
                             
                           
                           ≈ 
                           
                             2 
                             ⁢ 
                             
                               θ 
                               τ 
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               Δ 
                               ⁡ 
                               
                                 ( 
                                 
                                   1 
                                   τ 
                                 
                                 ) 
                               
                             
                             ⁢ 
                             
                               ( 
                               
                                 2 
                                 ⁢ 
                                 
                                   θ 
                                   
                                     τ 
                                     ⁢ 
                                     
                                         
                                     
                                   
                                 
                               
                               ) 
                             
                           
                           ≈ 
                           0 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   27 
                   ) 
                 
               
             
           
         
       
     
     Further, the group of assumptions also includes assumptions with respect to the in-quadrature bias of the gyroscope. These assumptions are as follows: 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             θ 
                             ω 
                           
                           = 
                           
                             90 
                             + 
                             
                               θ 
                               
                                 ω 
                                 ⁢ 
                                 
                                     
                                 
                                 ⁢ 
                                 0 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             sin 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             
                               θ 
                               ω 
                             
                           
                           = 
                           
                             
                               
                                 - 
                                 sin 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                               ⁢ 
                               
                                 θ 
                                 
                                   ω 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   0 
                                 
                               
                             
                             ≈ 
                             
                               
                                 - 
                                 2 
                               
                               ⁢ 
                               
                                 θ 
                                 
                                   ω 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   0 
                                 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             cos 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             2 
                             ⁢ 
                             
                               θ 
                               ω 
                             
                           
                           = 
                           
                             
                               
                                 - 
                                 cos 
                               
                               ⁢ 
                               
                                   
                               
                               ⁢ 
                               2 
                               ⁢ 
                               
                                 θ 
                                 
                                   ω 
                                   ⁢ 
                                   
                                       
                                   
                                   ⁢ 
                                   0 
                                 
                               
                             
                             ≈ 
                             1 
                           
                         
                       
                     
                     
                       
                         
                           
                             a 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ω 
                           
                           ≈ 
                           0 
                         
                       
                     
                     
                       
                         
                           
                             b 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             ω 
                           
                           ≈ 
                           0 
                         
                       
                     
                     
                       
                         
                           ab 
                           ≈ 
                           0 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   28 
                   ) 
                 
               
             
           
         
       
     
     The process then uses the set of assumptions to modify the set of equations to form a modified set of equations (operation  602 ). The modified set of equations includes a first modified equation for the in-phase bias and a second modified equation for the in-quadrature bias. The first modified equation for the in-phase bias is defined as follows: 
     
       
         
           
             
               
                 
                   
                     B 
                     
                       
                         - 
                         
                           c 
                           0 
                         
                       
                       ⁢ 
                       
                         ω 
                         
                           x 
                           ⁢ 
                           
                               
                           
                         
                       
                     
                   
                   = 
                   
                     
                       [ 
                       
                         
                           
                             
                               sin 
                               ⁡ 
                               
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   φ 
                                 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 1 
                                 - 
                                 
                                   cos 
                                   ⁡ 
                                   
                                     ( 
                                     
                                       2 
                                       ⁢ 
                                       φ 
                                     
                                     ) 
                                   
                                 
                               
                               τ 
                             
                           
                           
                             
                               - 
                               
                                 
                                   1 
                                   + 
                                   
                                     cos 
                                     ⁡ 
                                     
                                       ( 
                                       
                                         2 
                                         ⁢ 
                                         φ 
                                       
                                       ) 
                                     
                                   
                                 
                                 τ 
                               
                             
                           
                         
                       
                       ] 
                     
                     ⁡ 
                     
                       [ 
                       
                         
                           
                             
                               Δ 
                               ⁡ 
                               
                                 ( 
                                 
                                   1 
                                   τ 
                                 
                                 ) 
                               
                             
                           
                         
                         
                           
                             a 
                           
                         
                         
                           
                             b 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   29 
                   ) 
                 
               
             
           
         
       
     
     The second modified equation for the in-quadrature bias is defined as follows: 
     
       
         
           
             
               
                 
                   
                     quad 
                     
                       c 
                       0 
                     
                   
                   = 
                   
                     
                       
                         - 
                         ω 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ω 
                         ⁡ 
                         
                           ( 
                           
                             
                               
                                 sin 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     2 
                                     ⁢ 
                                     φ 
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     2 
                                     ⁢ 
                                     
                                       θ 
                                       
                                         ω 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         0 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                             - 
                             
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     2 
                                     ⁢ 
                                     φ 
                                   
                                   ) 
                                 
                               
                               ⁢ 
                               
                                 sin 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     2 
                                     ⁢ 
                                     
                                       θ 
                                       
                                         ω 
                                         ⁢ 
                                         
                                             
                                         
                                         ⁢ 
                                         0 
                                       
                                     
                                   
                                   ) 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                     ≈ 
                     
                       
                         - 
                         ω 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       Δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       ω 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             2 
                             ⁢ 
                             φ 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   30 
                   ) 
                 
               
             
           
         
       
     
     The process then identifies a set of values for the set of parameters in the modified set of equations using the bias measurements for the plurality of drive angles and an iterative algorithm (operation  604 ), with the process terminating thereafter. The bias measurements in operation  604  may be the bias measurements generated by the gyroscope for the plurality of drive angles in operation  400  in  FIG. 4 . 
     As one illustrative example, the plurality of drive angles may include angles of about 0 degrees, about 45 degrees, and about 90 degrees with respect to the x-axis. Of course, in other illustrative examples, any angles between about 0 degrees and about 90 degrees with respect to the x-axis may be selected. 
     For the angles about 0 degrees, about 45 degrees, and about 90 degrees with respect to the x-axis, equation (29) may be rewritten as follows: 
                       1       -     c   0       ⁢     ω   x         ⁡     [           B   ⁡     (     φ   =   0     )                 B   ⁡     (     φ   =   45     )                 B   ⁡     (     φ   =   90     )             ]       =         [         0       0         -     2   τ               1         1   τ           -     1   τ               0         2   τ         0         ]     ⁡     [           Δ   ⁡     (     1   τ     )               a           b         ]       =     C   ⁡     [           Δ   ⁡     (     1   τ     )               a           b         ]                 (   31   )                 with   ⁢           ⁢     det   ⁡     (   C   )         =       4     τ     2   ⁢                 ≠   0             (   32   )               
The solution for the parameters, a, b, and
 
               Δ   ⁡     (     1   τ     )       ,         
is given by the following:
 
                     [           Δ   ⁡     (     1   τ     )               a           b         ]     =         C     -   1       ⁡     (       1       -     c   0       ⁢     ω   x         ⁡     [           B   ⁡     (     φ   =   0     )                 B   ⁡     (     φ   =   45     )                 B   ⁡     (     φ   =   90     )             ]       )       =         1       -     c   0       ⁢     ω   x         ⁡     [           -     1   2           1         -     1   2               0       0         τ   2               -     τ   2           0       0         ]       ⁡     [           B   0               B   45               B   90           ]                 (   33   )               
where B 0  is the bias measurement for the drive angle of about 0 degrees with respect to the x-axis, B 45  is the bias measurement for the drive angle of about 45 degrees with respect to the x-axis, and B 90  is the bias measurement for the drive angle of about 90 degrees with respect to the x-axis.
 
     Further, in operation  604 , equation (25) may be rewritten as follows: 
                       (         B   new         -     c   0       ⁢     ω   x         +         b   ⁡     (     1   +     cos   ⁡     (     2   ⁢   φ     )         )       -     a   ⁡     (     1   -     cos   ⁡     (     2   ⁢   φ     )         )           τ   ⁡     (     1   -   ab     )           )       Δ   ⁡     (     1   τ     )         =       sin   ⁡     (       2   ⁢     φ   ^       -     2   ⁢     θ   τ         )       +         a   +   b     2     ⁢     (       cos   ⁡     (     2   ⁢     θ   τ       )       -     cos   ⁡     (       2   ⁢     φ   ^       -     2   ⁢     θ   τ         )         )                 (   34   )               
where the damping azimuth angle, θ t , is the only unknown parameter. The iterative algorithm may be used to solve for θ τ .
 
     Further, in solving for the parameters in the second modified equation for the in-quadrature bias, equation (30), bias measurements for the angles of about 0 degrees and about 45 degrees with respect to the x-axis may be selected. Consequently, equation (30) may be rewritten as follows: 
     
       
         
           
             
               
                 
                   { 
                   
                     
                       
                         
                           
                             
                               quad 
                               ⁡ 
                               
                                 ( 
                                 
                                   φ 
                                   = 
                                   45 
                                 
                                 ) 
                               
                             
                             
                               c 
                               0 
                             
                           
                           = 
                           
                             
                               - 
                               ω 
                             
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ωcos 
                               ⁡ 
                               
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   
                                     θ 
                                     
                                       ω 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       0 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                     
                       
                         
                           
                             
                               quad 
                               ⁡ 
                               
                                 ( 
                                 
                                   φ 
                                   = 
                                   0 
                                 
                                 ) 
                               
                             
                             
                               c 
                               0 
                             
                           
                           = 
                           
                             ω 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             Δ 
                             ⁢ 
                             
                                 
                             
                             ⁢ 
                             
                               ωsin 
                               ⁡ 
                               
                                 ( 
                                 
                                   2 
                                   ⁢ 
                                   
                                     θ 
                                     
                                       ω 
                                       ⁢ 
                                       
                                           
                                       
                                       ⁢ 
                                       0 
                                     
                                   
                                 
                                 ) 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   35 
                   ) 
                 
               
             
           
         
       
     
     If the natural frequencies in the drive axis and the sense axis are asymmetrical, Δω≠0, then the azimuth angle, θ ω0 , may be estimated as follows: 
                         θ   ^       ω   ⁢           ⁢   0       =       1   2     ⁢     arctan   ⁡     (       q   0       -     q   45         )           ,           (   36   )               
where {circumflex over (θ)} ω0  is the estimated azimuth angle and q 0  and q 45  are the in-quadrature biases at the drive angles of about 0 degrees and about 45 degrees, respectively, with respect to the x-axis. The natural frequency asymmetry for any drive angle may be estimated as follows:
 
                       Δ   ⁢           ⁢     ω   ^       =       quad   ⁡     (   φ   )           -     c   0       ⁢   ω   ⁢           ⁢     sin   ⁡     (       2   ⁢   φ     -     2   ⁢       θ   ^       ω   ⁢           ⁢   0           )             ,           (   37   )               
where Δ{circumflex over (ω)} is the estimated difference between the natural frequencies for the drive axis and the sense axis.
 
     The set of values identified in operation  604  for the set of parameters in the set of equations for the bias of the gyroscope may be used to estimate the bias of the gyroscope for any drive angle. In particular, the in-phase bias and in-quadrature bias may be estimated for any drive angle. 
     Further, in these illustrative examples, the estimated azimuth angle, {circumflex over (θ)} ω0 , may be used to identify the particular drive angle identified in operation  406  in  FIG. 4 . For example, the particular drive angle may be identified as follows: 
                   φ   =       90   +       θ   ^       ω   ⁢           ⁢   0         =     90   +       1   2     ⁢     arctan   ⁡     (       q   0       -     q   45         )                     (   38   )               
The values estimated for a and b may be used to solve the following equation for the drive angle:
 
     
       
         
           
             
               
                 
                   
                     
                       
                         - 
                         
                           ( 
                           
                             1 
                             + 
                             a 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             
                               2 
                               ⁢ 
                               φ 
                             
                             - 
                             
                               2 
                               ⁢ 
                               
                                 θ 
                                 ω 
                               
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         
                           a 
                           + 
                           b 
                         
                         4 
                       
                       ⁢ 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             
                               2 
                               ⁢ 
                               φ 
                             
                             + 
                             
                               2 
                               ⁢ 
                               
                                 θ 
                                 ω 
                               
                             
                           
                           ) 
                         
                       
                     
                     - 
                     
                       
                         a 
                         
                           ab 
                           - 
                           1 
                         
                       
                       ⁢ 
                       
                         cos 
                         ⁡ 
                         
                           ( 
                           
                             2 
                             ⁢ 
                             
                               θ 
                               ω 
                             
                           
                           ) 
                         
                       
                     
                   
                   = 
                   0 
                 
               
               
                 
                   ( 
                   39 
                   ) 
                 
               
             
           
         
       
     
     Still further, the output of the gyroscope may be calibrated during operation of the gyroscope in operation  408  in  FIG. 4  using the following equation: 
     
       
         
           
             
               
                 
                   
                     B 
                     
                       
                         - 
                         
                           c 
                           0 
                         
                       
                       ⁢ 
                       
                         ω 
                         x 
                       
                     
                   
                   = 
                   
                     
                       
                         Δ 
                         ⁡ 
                         
                           ( 
                           
                             1 
                             τ 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         sin 
                         ⁡ 
                         
                           ( 
                           
                             2 
                             ⁢ 
                             
                               φ 
                               ^ 
                             
                           
                           ) 
                         
                       
                     
                     + 
                     
                       
                         
                           a 
                           + 
                           b 
                         
                         2 
                       
                       ⁢ 
                       
                         Δ 
                         ⁡ 
                         
                           ( 
                           
                             1 
                             τ 
                           
                           ) 
                         
                       
                       ⁢ 
                       
                         ( 
                         
                           1 
                           - 
                           
                             cos 
                             ⁡ 
                             
                               ( 
                               
                                 2 
                                 ⁢ 
                                 φ 
                               
                               ) 
                             
                           
                         
                         ) 
                       
                     
                     - 
                     
                       
                         
                           b 
                           ⁡ 
                           
                             ( 
                             
                               1 
                               + 
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     2 
                                     ⁢ 
                                     
                                       φ 
                                       ^ 
                                     
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                         - 
                         
                           a 
                           ⁡ 
                           
                             ( 
                             
                               1 
                               - 
                               
                                 cos 
                                 ⁡ 
                                 
                                   ( 
                                   
                                     2 
                                     ⁢ 
                                     
                                       φ 
                                       ^ 
                                     
                                   
                                   ) 
                                 
                               
                             
                             ) 
                           
                         
                       
                       
                         τ 
                         ⁡ 
                         
                           ( 
                           
                             1 
                             - 
                             ab 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   40 
                   ) 
                 
               
             
           
         
       
     
     The flowcharts and block diagrams in the different depicted embodiments illustrate the architecture, functionality, and operation of some possible implementations of apparatuses and methods in an illustrative embodiment. In this regard, each block in the flowcharts or block diagrams may represent a module, segment, function, and/or a portion of an operation or step. For example, one or more of the blocks may be implemented as program code, in hardware, or a combination of the program code and hardware. When implemented in hardware, the hardware may, for example, take the form of integrated circuits that are manufactured or configured to perform one or more operations in the flowcharts or block diagrams. 
     In some alternative implementations of an illustrative embodiment, the function or functions noted in the blocks may occur out of the order noted in the figures. For example, in some cases, two blocks shown in succession may be executed substantially concurrently, or the blocks may sometimes be performed in the reverse order, depending upon the functionality involved. Also, other blocks may be added in addition to the illustrated blocks in a flowchart or block diagram. 
     Turning now to  FIG. 7 , an illustration of a data processing system is depicted in accordance with an illustrative embodiment. Data processing system  700  may be used to implement computer system  138  in  FIG. 1 . 
     In this illustrative example, data processing system  700  includes communications framework  702 , which provides communications between processor unit  704 , memory  706 , persistent storage  708 , communications unit  710 , input/output (I/O) unit  712 , and display  714 . 
     Processor unit  704  serves to execute instructions for software that may be loaded into memory  706 . Processor unit  704  may be a number of processors, a multi-processor core, or some other type of processor, depending on the particular implementation. A “number”, as used herein with reference to an item, means one or more items. Further, processor unit  704  may be implemented using a number of heterogeneous processor systems in which a main processor is present with secondary processors on a single chip. As another illustrative example, processor unit  704  may be a symmetric multi-processor system containing multiple processors of the same type. 
     Memory  706  and persistent storage  708  are examples of storage devices  716 . A storage device is any piece of hardware that is capable of storing information, such as, for example, without limitation, data, program code in functional form, and other suitable information either on a temporary basis or a permanent basis. Storage devices  716  also may be referred to as computer readable storage devices in these examples. Memory  706 , in these examples, may be, for example, a random access memory or any other suitable volatile or non-volatile storage device. Persistent storage  708  may take various forms, depending on the particular implementation. 
     For example, persistent storage  708  may contain one or more components or devices. For example, persistent storage  708  may be a hard drive, a flash memory, a rewritable optical disk, a rewritable magnetic tape, or some combination of the above. The media used by persistent storage  708  also may be removable. For example, a removable hard drive may be used for persistent storage  708 . 
     Communications unit  710 , in these examples, provides for communications with other data processing systems or devices. In these examples, communications unit  710  is a network interface card. Communications unit  710  may provide communications through the use of either or both physical and wireless communications links. 
     Input/output unit  712  allows for input and output of data with other devices that may be connected to data processing system  700 . For example, input/output unit  712  may provide a connection for user input through a keyboard, a mouse, and/or some other suitable input device. Further, input/output unit  712  may send output to a printer. Display  714  provides a mechanism to display information to a user. 
     Instructions for the operating system, applications, and/or programs may be located in storage devices  716 , which are in communication with processor unit  704  through communications framework  702 . In these illustrative examples, the instructions are in a functional form on persistent storage  708 . These instructions may be loaded into memory  706  for execution by processor unit  704 . The processes of the different embodiments may be performed by processor unit  704  using computer-implemented instructions, which may be located in a memory, such as memory  706 . 
     These instructions are referred to as program code, computer usable program code, or computer readable program code that may be read and executed by a processor in processor unit  704 . The program code in the different embodiments may be embodied on different physical or computer readable storage media, such as memory  706  or persistent storage  708 . 
     Program code  718  is located in a functional form on computer readable media  720  that is selectively removable and may be loaded onto or transferred to data processing system  700  for execution by processor unit  704 . Program code  718  and computer readable media  720  form computer program product  722  in these examples. In one example, computer readable media  720  may be computer readable storage media  724  or computer readable signal media  726 . 
     Computer readable storage media  724  may include, for example, an optical or magnetic disk that is inserted or placed into a drive or other device that is part of persistent storage  708  for transfer onto a storage device, such as a hard drive, that is part of persistent storage  708 . Computer readable storage media  724  also may take the form of a persistent storage, such as a hard drive, a thumb drive, or a flash memory, that is connected to data processing system  700 . In some instances, computer readable storage media  724  may not be removable from data processing system  700 . 
     In these examples, computer readable storage media  724  is a physical or tangible storage device used to store program code  718  rather than a medium that propagates or transmits program code  718 . Computer readable storage media  724  also is referred to as a computer readable tangible storage device or a computer readable physical storage device. In other words, computer readable storage media  724  is a media that can be touched by a person. 
     Alternatively, program code  718  may be transferred to data processing system  700  using computer readable signal media  726 . Computer readable signal media  726  may be, for example, a propagated data signal containing program code  718 . For example, computer readable signal media  726  may be an electromagnetic signal, an optical signal, and/or any other suitable type of signal. These signals may be transmitted over communications links, such as wireless communications links, optical fiber cable, coaxial cable, a wire, and/or any other suitable type of communications link. In other words, the communications link and/or the connection may be physical or wireless in the illustrative examples. 
     In some illustrative embodiments, program code  718  may be downloaded over a network to persistent storage  708  from another device or data processing system through computer readable signal media  726  for use within data processing system  700 . For instance, program code stored in a computer readable storage medium in a server data processing system may be downloaded over a network from the server to data processing system  700 . The data processing system providing program code  718  may be a server computer, a client computer, or some other device capable of storing and transmitting program code  718 . 
     The different components illustrated for data processing system  700  are not meant to provide physical or architectural limitations to the manner in which different embodiments may be implemented. The different illustrative embodiments may be implemented in a data processing system including components in addition to or in place of those illustrated for data processing system  700 . Other components shown in  FIG. 7  can be varied from the illustrative examples shown. The different embodiments may be implemented using any hardware device or system capable of running program code. As one example, the data processing system may include organic components integrated with inorganic components and/or may be comprised entirely of organic components excluding a human being. For example, a storage device may be comprised of an organic semiconductor. 
     In another illustrative example, processor unit  704  may take the form of a hardware unit that has circuits that are manufactured or configured for a particular use. This type of hardware may perform operations without needing program code to be loaded into a memory from a storage device to be configured to perform the operations. 
     For example, when processor unit  704  takes the form of a hardware unit, processor unit  704  may be a circuit system, an application specific integrated circuit (ASIC), a programmable logic device, or some other suitable type of hardware configured to perform a number of operations. With a programmable logic device, the device is configured to perform the number of operations. The device may be reconfigured at a later time or may be permanently configured to perform the number of operations. Examples of programmable logic devices include, for example, a programmable logic array, a field programmable logic array, a field programmable gate array, and other suitable hardware devices. With this type of implementation, program code  718  may be omitted because the processes for the different embodiments are implemented in a hardware unit. 
     In still another illustrative example, processor unit  704  may be implemented using a combination of processors found in computers and hardware units. Processor unit  704  may have a number of hardware units and a number of processors that are configured to run program code  718 . With this depicted example, some of the processes may be implemented in the number of hardware units, while other processes may be implemented in the number of processors. 
     In another example, a bus system may be used to implement communications framework  702  and may be comprised of one or more buses, such as a system bus or an input/output bus. Of course, the bus system may be implemented using any suitable type of architecture that provides for a transfer of data between different components or devices attached to the bus system. 
     Additionally, a communications unit may include a number of devices that transmit data, receive data, or transmit and receive data. A communications unit may be, for example, a modem or a network adapter, two network adapters, or some combination thereof. Further, a memory may be, for example, memory  706 , or a cache, such as found in an interface and memory controller hub, that may be present in communications framework  702 . 
     Thus, the different illustrative embodiments provide a method and apparatus for electronic bias compensation of a gyroscope. In one illustrative embodiment, a method for electronically compensating for bias of a gyroscope is provided. Bias measurements for a plurality of drive angles are generated using the gyroscope. A set of equations for the bias of the gyroscope is identified using a model for motion of the gyroscope with respect to a non-inertial frame of reference and an assumption that an inertial rate for the gyroscope is substantially zero. The set of equations includes a set of parameters for the bias of the gyroscope. A set of values for the set of parameters is identified using the bias measurements and the set of equations. 
     Further, a drive angle that reduces the in-quadrature bias of the gyroscope to substantially zero is identified using the set of equations. The gyroscope is operated at the drive angle identified. The bias of the gyroscope at a selected drive angle is estimated using the set of equations with the set of values for the set of parameters to form an estimated bias. The estimated bias is subtracted from a measurement generated by the gyroscope for the selected drive angle to substantially compensate for the bias of the gyroscope. 
     The description of the different illustrative embodiments has been presented for purposes of illustration and description and is not intended to be exhaustive or limited to the embodiments in the form disclosed. Many modifications and variations will be apparent to those of ordinary skill in the art. Further, different illustrative embodiments may provide different features as compared to other illustrative embodiments. The embodiment or embodiments selected are chosen and described in order to best explain the principles of the embodiments, the practical application, and to enable others of ordinary skill in the art to understand the disclosure for various embodiments with various modifications as are suited to the particular use contemplated.