Patent Publication Number: US-2018031602-A1

Title: Converting rotational motion to linear motion

Description:
BACKGROUND 
     Monolithic inertial sensors can contain proof masses that move in response to inertial perturbations such as accelerations and rotations. Some inertial sensors contain proof masses that are driven in oscillation. A linear drive can drive a proof mass in linear oscillation, and a rotational drive can drive a proof mass in rotational oscillation. For proof masses that are driven in linear oscillation, any component of motion that is not aligned with the primary axis of measurement can reduce the signal-to-noise level of the sensor. 
     SUMMARY 
     Accordingly, systems and methods are described herein for converting rotational motion to linear motion. A system can include a proof mass, a rotational drive configured to rotate about a z axis, and a first structure that connects the rotational drive to the proof mass. The first structure can include a major axis that passes from a first anchor to the proof mass and is aligned with a y axis when the first structure is at rest, the y axis perpendicular to the z axis, and a coupling spring with a stiffness along a minor axis perpendicular to the major axis that is different than a stiffness along the major axis. The system can include a second structure including a drive spring with a stiffness along the y axis that is different than a stiffness along an x axis perpendicular to the y and z axes. The system can also include a second anchor connected to the proof mass by the second structure. 
     The coupling spring and the drive spring can be configured to cause the proof mass to move substantially along the x axis when the rotational drive rotates about the z axis. The coupling spring can be configured to bend when the rotational drive rotates. 
     A center of mass of the proof mass can be radially between a point at which the drive spring is attached to the proof mass and a point at which the coupling spring is attached to the proof mass. The drive spring can exert, on the proof mass, a torque that substantially prevents rotation of the proof mass about the center of mass. 
     The first structure can include an arm. The stiffness of the coupling spring along the minor axis can be substantially greater than the stiffness of the coupling spring along the major axis. The stiffness of the drive spring along the y axis can be substantially greater than the stiffness of the drive spring along the x axis. 
     The system can include a second drive spring connected to the proof mass and a third anchor, the second drive spring with a stiffness along the y axis that is different than a stiffness along an x axis. 
     The drive spring can be configured to expand when the rotational drive rotates about the z axis with a first rotation vector and compress when the rotational drive rotates about the z axis with a second rotation vector opposite to the first rotation vector. 
     The first structure can include a drive frame. The stiffness of the coupling spring along the major axis can be substantially greater than the stiffness of the coupling spring along the minor axis. The stiffness of the drive spring along the y axis can be substantially greater than the stiffness of the drive spring along the x axis. 
     The proof mass can include a sensor configured to characterize the motion of the proof mass along the x axis. The sensor can include a comb and/or a time-domain-switched structure. The sensor can be configured to determine an acceleration of the system along the x axis, and/or a velocity of the proof mass along the x axis. 
     The system can include a second proof mass connected to the rotational drive by a third structure including a second coupling spring and a third anchor connected to the second proof mass by a fourth structure including a second drive spring. The second coupling spring and the second drive spring can be configured to cause the second proof mass to move substantially along the y axis when the rotational drive rotates about the z axis. 
     The coupling spring can include a first coupling joint connected to an end of the arm, first and second flex arms connected to the first coupling joint, and first and second forks connected to the first and second flex arms, respectively. The system can include third and fourth flex arms connected to the first and second forks, respectively, and a second coupling joint connected to the third and fourth flex arms and to the proof mass. 
     The drive spring can include an anchor fork connected to the second anchor, an anchor arm connected to the anchor fork, and a first drive fork connected to the anchor arm. The drive spring can also include a drive arm connected to the first drive fork, and a second drive fork connected to the drive arm and to the proof mass. 
     The second drive spring can include a second anchor fork connected to the third anchor, a second anchor arm connected to the second anchor fork, and a third drive fork connected to the second anchor arm. The second drive spring can also include a second drive arm connected to the third drive fork, and a fourth drive fork connected to the second drive arm and to the proof mass. 
     The coupling spring can include a driving fork connected to the drive frame, first and second driving arms connected to the driving fork, and first and second middle forks connected to the first and second driving arms, respectively. The coupling spring can also include first and second middle arms connected to the first and second middle forks, respectively, and a first driven fork connected to the first and second middle arms. The coupling spring can also include a driven arm connected to the first driven fork and a second driven fork connected to the driven arm and to the proof mass. 
     The coupling spring can include a first coupling joint connected to the drive frame, first and second flex arms connected to the first coupling joint, and first and second forks connected to the first and second flex arms, respectively. The coupling spring can also include third and fourth flex arms connected to the first and second forks, respectively, and a second coupling joint connected to the third and fourth flex arms and to the proof mass. 
     The drive spring can also include an anchor fork connected to the second anchor, an anchor arm connected to the anchor fork, and a first drive fork connected to the anchor arm. The drive spring can also include a drive arm connected to the first drive fork and a second drive fork connected to the drive arm and to the proof mass. 
     The system can also include a second proof mass connected to the rotational drive by a third structure including a second coupling spring and a third anchor connected to the second proof mass by a fourth structure including a second drive spring. The second coupling spring and the second drive spring can be configured to cause the second proof mass to move substantially along the third axis when the rotational drive rotates about the second axis. 
     The system can include a third proof mass connected to the rotational drive by a fifth structure including a third coupling spring and a fourth anchor connected to the third proof mass by a sixth structure including a third drive spring. The third coupling spring and the third drive spring can be configured to cause the third proof mass to move substantially along the first axis when the rotational drive rotates about the second axis. 
     The system can include a fourth proof mass connected to the rotational drive by a seventh structure including a fourth coupling spring and a fifth anchor connected to the fourth proof mass by an eighth structure including a fourth drive spring. The fourth coupling spring and the fourth drive spring can be configured to cause the fourth proof mass to move substantially along the third axis when the rotational drive rotates about the second axis. 
     The system can include a fifth proof mass connected to the rotational drive by a ninth structure including a fifth coupling spring and a sixth anchor connected to the fifth proof mass by a tenth structure including a fifth drive spring. The fifth coupling spring and the fifth drive spring can be configured to cause the fifth proof mass to move substantially along a fourth axis when the rotational drive rotates about the second axis, the fourth axis perpendicular to the second axis. 
     The system can include a sixth proof mass connected to the rotational drive by a eleventh structure including a sixth coupling spring and a seventh anchor connected to the sixth proof mass by a twelfth structure including a sixth drive spring. The sixth coupling spring and the sixth drive spring can be configured to cause the sixth proof mass to move substantially along the fourth axis when the rotational drive rotates about the second axis. 
     The system can include a seventh proof mass connected to the rotational drive by a thirteenth structure including a seventh coupling spring and a eighth anchor connected to the seventh proof mass by a fourteenth structure including a seventh drive spring. The system can also include an eighth proof mass connected to the rotational drive by a fifteenth structure including an eighth coupling spring and a ninth anchor connected to the eighth proof mass by a sixteenth structure including an eighth drive spring. The seventh coupling spring and the seventh drive spring can be configured to cause the seventh proof mass to move substantially along a fifth axis when the rotational drive rotates about the second axis, the fifth axis perpendicular to the second and fourth axes. Furthermore, the eighth coupling spring and the eighth drive spring can be configured to cause the eighth proof mass to move substantially along the fifth axis when the rotational drive rotates about the second axis. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIG. 1  depicts an inertial sensor comprising spring systems that convert rotational motion to linear motion, according to an illustrative implementation; 
         FIG. 2  depicts an enlarged view of an area of interest depicted in  FIG. 1 , with a time-domain-switched subassembly displaced in a clockwise direction from its neutral position, according to an illustrative implementation; 
         FIG. 3  depicts the inertial sensor shown in  FIG. 1  when a drive comb has rotated the arm counterclockwise from its neutral position, according to an illustrative implementation; 
         FIG. 4  depicts an enlarged view of a coupling spring, according to an illustrative implementation; 
         FIG. 5  depicts the coupling spring shown in  FIG. 4  when an arm is rotated clockwise from its neutral position, according to an illustrative implementation; 
         FIG. 6  depicts an inertial sensor with springs that convert rotational motion to linear motion, according to an illustrative implementation; 
         FIG. 7  depicts an enlarged view of an area of interest shown in  FIG. 6 , according to an illustrative implementation; 
         FIG. 8  depicts the inertial sensor shown in  FIG. 6  when drive combs have caused a drive frame to rotate counterclockwise about the z-axis of the inertial sensor, according to an illustrative implementation; 
         FIG. 9  depicts an enlarged view of a drive spring when drive combs have rotated the drive frame counterclockwise about the z-axis, according to an illustrative implementation; 
         FIG. 10  depicts the drive spring shown in  FIG. 9  when the drive combs have rotated the drive frame clockwise about the z-axis from its neutral position, according to an illustrative implementation; 
         FIG. 11  depicts a coupling spring of the inertial sensor shown in  FIG. 6  when the drive combs have rotated the drive frame counterclockwise about the z-axis, according to an illustrative implementation; 
         FIG. 12  depicts the coupling spring shown in  FIG. 11  when the drive combs have rotated the drive frame clockwise about the z-axis from its neutral position, according to an illustrative implementation; 
         FIG. 13  depicts an inertial sensor with springs that convert rotational motion to linear motion, according to an illustrative implementation; 
         FIG. 14  depicts the inertial sensor shown in  FIG. 13  when drive combs have rotated a drive frame counterclockwise about the z-axis of the inertial sensor from its neutral position, according to an illustrative implementation; 
         FIG. 15  depicts an enlarged view of a gyroscope subassembly of the inertial sensor shown in  FIG. 13  when a drive frame is in its neutral position, according to an illustrative implementation; 
         FIG. 16  depicts a view of the gyroscope subassembly shown in  FIG. 15  when the drive combs have rotated the drive frame counterclockwise about the z-axis of the inertial sensor from its neutral position, according to an illustrative implementation; 
         FIG. 17  depicts three views, each showing a schematic representation of parts of a movable element and a fixed element, according to an illustrative implementation; 
         FIG. 18  schematically depicts an exemplary process used to extract inertial information from an inertial sensor with periodic geometry, according to an illustrative implementation; 
         FIG. 19  depicts a graph which represents the association of analog signals derived from an inertial sensor with zero-crossing times and displacements of an inertial sensor, according to an illustrative implementation; 
         FIG. 20  depicts a graph showing effects of an external perturbation on input and output signals of an inertial sensor, according to an illustrative implementation; 
         FIG. 21  depicts a graph that illustrates a response in the form of an electrical current to an oscillator displacement, according to an illustrative implementation; 
         FIG. 22  depicts a graph showing a rectangular waveform and signal representing zero-crossing times of the current signal depicted in  FIG. 21 , according to an illustrative implementation; 
         FIG. 23  is a graph which illustrates additional time intervals of the displacement curve depicted in  FIG. 21 , according to an illustrative implementation; 
         FIG. 24  is a graph that depicts the relationship between capacitance of the inertial sensor depicted in  FIG. 18  and displacement of the movable element depicted in  FIG. 17 , according to an illustrative implementation; 
         FIG. 25  is a graph that depicts the relationship between displacement and the first derivative of capacitance with respect to displacement, according to an illustrative implementation; 
         FIG. 26  is a graph that depicts the relationship between displacement and the second derivative of capacitance with respect to displacement, according to an illustrative implementation; and 
         FIG. 27  is a graph that depicts the relationship between time, the rate of change of capacitive current, and displacement, according to an illustrative implementation. 
         FIG. 28  depicts a flow chart of a method used to extract inertial parameters from a nonlinear periodic signal, according to an illustrative implementation; 
         FIG. 29  depicts a method for determining times of transition between two values based on nonlinear periodic signals, according to an illustrative implementation; and 
         FIG. 30  depicts a method to compute inertial parameters from time intervals, according to an illustrative implementation. 
     
    
    
     DETAILED DESCRIPTION 
     To provide an overall understanding of the disclosure, certain illustrative implementations will now be described, including systems and methods for converting rotational motion to linear motion. 
     When a vertically-oriented lever is rotated about a pivot point, the end of the lever distal from the pivot point traces an arc: it moves in a circumferential direction. As the distal end of the lever traces the arc, the distal end moves horizontally and also in the vertical direction. The spring mechanisms described herein substantially remove this vertical component of motion, converting rotational motion to linear motion. 
     Some types of sensors, such as vibratory accelerometers and Coriolis force vibrating gyroscopes, require a proof mass to be oscillated linearly along an axis. Inertial parameters such as accelerations and rotations can affect the oscillating proof mass. In some examples, such as vibratory accelerometers, the oscillations become offset from the neutral point due to an acceleration. To sense inertial parameters acting along multiple axes, an inertial sensing apparatus requires proof masses that oscillate along multiple axes. The systems and methods described herein integrate multiple sensors with proof masses oscillating along different axes into a single multi-axis device driven by a single rotational drive. This allows the motion of each of the proof masses to be synchronized in frequency, phase, and amplitude. 
     The systems and methods described herein may integrate multiple sensors with proof masses into a single multi-axis device by converting rotational motion to linear motion, allowing inertial sensors requiring linear proof mass motion to be driven by a rotational drive. The frequency and phase of the inertial sensors are synchronized because the same drive system actuates each of the inertial sensors. 
     By placing the inertial sensors at appropriate azimuthal positions on the rotational drive, sensors moving in orthogonally linear directions can be realized. The amplitude of each of the inertial sensors is controlled by its distance from the pivot point of the rotational drive. Because all of the inertial sensors are driven by the same drive, any drifting in the drive electronics will affect the frequency, phase, and amplitudes of the inertial sensors in the same manner. Likewise, drift due to other factors such as temperature, mechanical stress, or external forces will also affect all of the inertial sensors in the same manner. Because the inertial sensors are located relatively close to each other on the same drive frame, mechanical stresses such as packaging stress which deform the overall package of the inertial sensor, will tend to cause little relative motion between various parts of the inertial sensor. Thus, the ratio of the drive amplitude of one inertial sensor to the drive amplitude of another inertial sensor is determined by the geometry of the inertial device as fabricated and are not typically changed by any other factors. This results in an inertial device with sensors that have very stable amplitude ratios, and essentially the same frequency and phase. Thus, the inertial sensors of the inertial device are mechanically synchronized in frequency, phase, and amplitude ratio. 
     The power consumed by drive electronics is often the largest fraction of total power consumed by an oscillating inertial device. The energy required to power the drive electronics is often significantly more than the kinetic energy required to oscillate the resonators. Thus, driving multiple inertial sensors with a single oscillating drive reduces the overall power consumption by reducing the number of systems of drive electronics. Furthermore, oscillating inertial sensors often do not oscillate continuously, but only oscillate when their output is required. This may occur when, for example, a user begins using a navigation or virtual reality application of a mobile device that requires inertial sensing. Thus, oscillating resonators are required to start and stop frequently. Starting an oscillating resonator requires adjusting a drive voltage of the resonator in a closed-loop fashion until the amplitude of the oscillations increases to a desired setpoint. Start up times of oscillating inertial devices can range from 10 milliseconds to multiple seconds, depending on the quality factor of the resonators and on other factors. When multiple sensors are driven by a single rotational drive, they can be started and stopped in unison. 
     Springs in the inertial devices may have certain configurations. In some examples, the tailored stiffness and compliance of springs described herein is achieved purely by the geometry of the springs. In some examples, the springs comprise a uniform isotropic material, such as doped or undoped silicon. In other examples, the material properties of the springs are tailored in various portions of the spring to achieve desired variations in stiffness and compliance. 
     Driving a proof mass with a rotational drive can result in more nonlinearity in the motion of the proof mass due to the rotation. The spring systems described herein can substantially linearize the motion of the proof mass of the inertial sensors by controlling and minimizing off-axis motion. The spring systems can achieve this goal by including springs with higher stiffness in off-axis directions, and/or by counterbalancing with springs that convert parasitic off-axis motion into motion in on-axis directions. In some examples, the remaining off-axis (rotational) component of the motion of the proof mass is 100 PPM of the on-axis (linear) component. In some examples, the off-axis (rotational) component is as low as 10 PPM or as high as 1000 PPM of the on-axis (linear) component. Thus, for a proof mass on a vertically-oriented arm and rotating about the origin and having an oscillation in the x direction of 1 micron, the proof mass only moves in the y direction by 1 nanometer (corresponding to 1000 PPM), 0.1 nanometers (corresponding to 100 PPM) or as little as 0.01 nanometers (corresponding to 10 PPM). 
       FIG. 1  depicts an inertial sensor  100  comprising spring systems that convert rotational motion to linear motion. The inertial sensor  100  includes a central anchor  102  and a drive comb  104 . The drive comb  104  is an example of a rotational drive.  FIG. 1  only depicts movable portions of the drive comb  104 , but the drive comb  104  also includes fixed portions that are not shown. The inertial sensor  100  also includes six gyroscope subassemblies  106 ,  110 ,  112 ,  114 ,  118 , and  120 . In addition, the inertial sensor  100  includes time-domain-switched (TDS) subassemblies  108  and  116 .  FIG. 1  also depicts a coordinate system  122  with an x-y-z coordinate system sharing a z-axis and an origin with a u-v-z coordinate system. While the coordinate system  122  is depicted offset from the inertial sensor  100  for clarity, the origin of the coordinate system  122  is located at the center of the central anchor  102 . The x- and y-axes are orthogonal to each other. The u- and v-axes are orthogonal to each other and are rotated by −45 degrees from the x- and y-axes, respectively.  FIG. 1  also depicts an area of interest  101 . 
     The inertial sensor  100  comprises three layers, a device layer containing the features depicted in  FIG. 1 , a bottom layer (not shown), and a cap layer (not shown). In some examples, the bottom layer and cap layer are made from different wafers than the device layer. In some examples, one or more features of the device layer can be made from the wafers containing the bottom layer and/or the cap layer. The region between the bottom and cap layers can be at a pressure below atmospheric pressure. In some examples, a gettering material such as titanium or aluminum is deposited to maintain the reduced pressure for an extended period of time after manufacturing the inertial sensor. 
     The central anchor  102  is anchored to one or both of the bottom and cap layers and is the central pivot point of the inertial sensor  100 . The drive comb  104  causes the respective subassemblies to rotationally oscillate about the central anchor  102 . This oscillation causes the gyroscope subassemblies  106 ,  110 ,  114 , and  118  to move at a drive velocity. When the inertial sensor  100  is rotated, a Coriolis force proportional to the rotation rate causes proof masses of the gyroscope subassemblies  106 ,  110 ,  114 , and  118  to deflect. 
     The gyroscope subassemblies  106 ,  110 ,  114 , and  118  provide differential sensing for rotation about the x- and y-axes. Here, and throughout, rotations, accelerations, displacements, and other parameters described with reference to the x- and y-axes can be mathematically transformed to reference the u- and v-axes instead, and vice versa, by a simple rotation of the relevant coordinate system. This transformation can be performed by signal processing circuitry. Capacitor electrodes (not shown) are located either above or below a respective proof mass of each of the gyroscope subassemblies  106 ,  110 ,  114 , and  118 . The capacitor electrodes can be located in a cap layer and/or a bottom layer. These capacitor electrodes detect motion of the respective proof masses in the z direction in response to a rotation about the x-axis, the y-axis, or another axis in the x-y plane. The gyroscope subassemblies  112  and  120  contain proof masses that deflect radially in response to rotations about the z-axis. 
     The TDS subassemblies  108  and  116  can be used to measure drive velocity, acceleration along the u-axis, or both. For either measurement, accuracy is improved if the subassembly  108  and/or  116  oscillates purely along the u-axis. The systems and methods described herein convert the rotational motion imparted by the drive comb  104  into linear motion along the u-axis. 
     In some examples, the inertial sensor  100  does not contain a TDS structure, such as the TDS structure  235  of the TDS subassembly  108  which is further described with reference to  FIG. 2 , but instead uses one or more drive sense combs for both velocity measurement and drive comb regulation. In some examples, the inertial sensor  100  does not include drive sense combs and uses the TDS structure (e.g.,  235 ) for both velocity measurement and drive comb regulation. In some examples, the inertial sensor  100  contains both TDS structures (e.g.,  235 ) and drive sense combs and uses the TDS structure (e.g.,  235 ) for drive comb regulation and the drive sense combs for velocity measurement. In some examples, the inertial sensor  100  uses the TDS structure (e.g.,  235 ) for velocity measurement and the drive sense combs for drive comb regulation. 
       FIG. 2  depicts an enlarged view of the area of interest  101  from  FIG. 1 , with a proof mass  246  of the TDS subassembly  108  displaced in the clockwise direction from its neutral position. The proof mass  246  has a center of mass  248 . The center of mass  248  is the point at which the mass-weighted position vectors of each portion of the proof mass  246  sum to zero. The center of mass of an object is not necessarily located on or within the object, and in  FIG. 2  the center of mass  248  is indeed not located within the proof mass  246 . 
       FIG. 2  also depicts a rotational spring  224  and an arm  226 . The rotational spring  224  comprises a plurality of proximal ends and a plurality of distal ends. The proximal ends are connected to the anchor point and the distal ends are connected to a circular frame  229 . The arm  226 , as well as a plurality of other arms, comprises a proximal end and a distal end. The arm  226  has a major axis running along its length and a minor axis that is perpendicular to the major axis and in the u-v plane. When the arm is at rest, the major axis is aligned with the v axis and the minor axis is aligned with the u axis. The u axis is perpendicular to the z and v axes. The proximal end of the arm  226  is connected to the circular frame  229 . The rotational spring  224  allows the circular frame and the arms to rotate about the z-axis, located at the center of the central anchor  102 . As the arm  226  rotates about the z-axis, the distal end of the arm  226  travels in an arc. Without any of the spring systems described herein, the proof mass  246  would also travel in an arc and thus would have both u and v components of motion. However, one or more of the spring systems described herein substantially eliminate the v component of motion, resulting in proof mass  246  moving almost entirely along the u-axis in response to rotation caused by the drive comb  104 . 
     The distal end of the arm  226  is connected to a coupling spring  228 . The coupling spring  228  transmits circumferential motion (that is, motion perpendicular to the long axis of the arm  226 ) to the proof mass  246  through a coupling joint  462 . Because the coupling spring  228  has an open center, the coupling spring  228  is compliant in the radial direction (that is, the direction parallel to the long axis of the arm  226 ). Because the coupling spring  228  is rigid in the circumferential direction but compliant in the radial direction, the proof mass  246  moves with the arm  226 , but the gap between the proof mass  246  and the distal end of the arm  226  can vary. 
     The coupling spring  228  works in tandem with a pair of drive springs  225  and  227  to convert rotational motion to linear motion of the proof mass  246 . The drive spring  225  comprises an anchor fork  211 , an anchor arm  209 , a drive fork  207 , a drive arm  205 , and a drive fork  203 . The anchor arm  209  is connected to an anchor  213  at the anchor fork  211 . The anchor  213  is anchored to the bottom layer and/or the cap layer and is not moved by the drive comb  104 . The drive arm  205  is connected to the anchor arm  209  at the drive fork  207 . The drive arm  205  is connected to the proof mass  246  at the drive fork  203 . The anchor arm  209  and the drive arm  205  are compliant in the u direction but rigid in the v direction. Thus, while the distance along the u-axis between the anchor fork  211  and the drive fork  203  can vary, the distance along the v-axis between the two forks does not vary. 
     The structure of the drive spring  227  is a mirror image of the structure of the drive spring  225  and comprises a drive fork  215 , a drive arm  217 , an drive fork  219 , an anchor arm  221 , and an anchor fork  223 . The drive spring  227  is compliant in the u direction but rigid in the v direction. Thus, the drive fork  215  and the anchor fork  223  can move relative to each other in the u direction but cannot do so in the v direction. The drive arms  205  and  217  and the anchor arms  209  and  221  are compliant in the u direction but stiff in the v and z directions because their dimensions in u are much smaller than their dimensions in v and z. Because the drive springs  225  and  227  are not perfect springs, they are not perfectly rigid and thus have finite stiffnesses. Thus, the drive springs  225  and  227  do allow some motion of the proof mass  246  in the u direction. However, although the drive springs  225  and  227  are compliant in the u direction, they are stiff in the v direction such that motion of the proof mass  246  in the v direction is small. Thus, the coupling spring  228  and the drive springs  225  and  227  convert rotational motion about the z axis into linear motion of the proof mass  246  substantially along the u axis. 
     The spring systems described herein (e.g., the drive springs  225  and  227  and the coupling spring  228 ) can also convert rotational motion to linear motion through dynamic effects. The dynamic effects occur because the center of mass  248  is located at a different radius from the central anchor  102  than the points at which the drive springs are connected to the proof mass  246 . For the TDS subassembly  108 , the drive springs  225  and  227  are attached to the proof mass  246  at the drive forks  203  and  215 . The drive comb  104  exerts, on the arm  226  and coupling spring  228 , a torque about the central anchor  102 . The coupling spring  228  then exerts a force on the proof mass  246  that is in the +u direction and acting through the coupling joint  462 . This force can be resolved into a resolved torque about the center of mass  248  and a resolved force acting through the center of mass  248 . Thus, if the drive comb  104  is exerting a clockwise torque, and the arm  226  is rotating clockwise about the central anchor  102 , the resolved torque be counterclockwise and will tend to rotate the proof mass  246  counterclockwise about the center of mass  248 . The radius of the center of mass  248  is greater than the radius of the coupling joint  462  and less than the respective radii of the drive forks  203  and  215  (where the radii are measured with respect to the central anchor  102 ). However, because the center of mass  248  is radially between the coupling joint  462  and the drive forks  203  and  215 , the drive forks  203  and  215  exert a counter-torque that is clockwise about the center of mass  248 . 
     This counter-torque tends to rotate the proof mass  246  clockwise about the center of mass  248 , thus counteracting the tendency of the resolved torque to rotate the proof mass  246  counterclockwise about the center of mass  248 . The directions of the resolved torque and the counter-torque would be reversed for counterclockwise torques exerted on the arm  226  by the drive comb  104 . The properties of the TDS subassembly  108  can affect the magnitudes of the resolved torque and the counter-torque. Some properties that affect these magnitudes include the mass of the proof mass  246 , the location of the center of mass  248  (especially the radial distance from the central anchor  102 ), the locations of the drive forks  203  and  215  (especially the radial distances from the central anchor  102 ), the stiffnesses of the drive springs  225  and  227  and the coupling spring  228 , and the location of the coupling spring  228 . By choosing these and other properties such that the counter-torque mostly or fully counteracts the resolved torque, the counter-torque substantially prevents rotation of the proof mass  246  about the center of mass  248 . Thus, rotational motion about the z axis is converted into motion of the proof mass  246  substantially along the u axis. 
       FIG. 2  depicts the area of interest  101  ( FIG. 1 ) of inertial sensor  100  ( FIG. 1 ) when the drive comb  104  has rotated the arm  226  in the counterclockwise direction from its neutral position. The coupling spring  228  has transmitted the u component of this rotation to the proof mass  246 . The drive springs  225  and  227  have allowed the proof mass  246  to move in the +u direction while preventing it from moving in the v direction. Because the drive springs  225  and  227  have prevented the proof mass  246  from moving in the v direction, the distance between the proof mass  246  and the distal end of the arm  226  has increased. Because the coupling spring  228  is compliant in the v direction, the distance between the proof mass  246  and the distal end of the arm  226  can change while motion in the u direction is still transmitted. Thus, the coupling spring  228  and the drive springs  225  and  227  have converted the rotational motion of the arm  226  into linear motion of the proof mass  246 . 
       FIG. 2  also depicts anchors  230  and  231  and comb sensors  232  and  234 . The anchors  230  and  231  are anchored to the bottom layer and/or the cap layer and do not move relative to the central anchor  102 . The comb sensors  232  and  234  experience a change in capacitance when the proof mass  246  moves in the u direction. The comb sensors  232  and  234  can characterize the motion of the proof mass  246  along the u axis. In some examples, the outputs from the comb sensors  232  and  234  are used to determine the velocity of the proof mass  246  in the u direction. In other examples, the outputs from the comb sensors  232  and  234  are used to regulate the velocity at which the arm  226  is oscillated by the drive comb  104 . In other examples, the output of one of the comb sensors  232  and  234  is used to regulate the drive comb  104  in closed-loop feedback and the output of the other of the comb sensors  232  and  234  is used to determine the velocity in the u direction of the proof mass  246 . 
     The TDS subassembly  108  includes a TDS structure  235  configured to characterize motion of the proof mass  246  in the u direction. The TDS structure  235  includes a movable beam  236  comprising a plurality of equally spaced teeth  238 . The TDS structure  235  also includes a fixed element  244  comprising a fixed beam  242 , itself comprising a plurality of teeth  240 . The fixed element  244  is anchored to the bottom layer and/or the cap layer and does not move relative to the central anchor  102 . The TDS structure  235  can produce nonlinear capacitive signals for determining velocity in the u direction of the proof mass  246 , an offset in oscillations along the u direction of the proof mass  246 , or both. The systems and methods described with reference to  FIGS. 17-30  can be used to determine this velocity and offset. The offset in the oscillations is proportional to an acceleration acting on the inertial sensor  100  in the u direction. 
       FIG. 3  depicts the area of interest  101  ( FIG. 1 ) of inertial sensor  100  ( FIG. 1 ) when the drive comb  104  has rotated the arm  226  counterclockwise from its neutral position. The coupling spring  228  has transmitted motion in the −u direction to the proof mass  246 . The drive springs  225  and  227  have allowed the proof mass  246  to move in the −u direction while preventing it from moving in the v direction. The drive spring  225  compresses slightly while the drive spring  227  expands slightly. Because the proof mass  246  does not move in the v direction, the coupling spring  228  expands slightly in the v direction to allow the v distance between the proof mass  246  and the distal end of the arm  226  to vary. Thus, the coupling spring  228  and the drive springs  225  and  227  have converted the rotational motion of the arm  226  into linear motion of the proof mass  246 .  FIG. 3  also depicts an area of interest  350 . 
       FIG. 4  depicts an enlarged view of the area of interest  248  ( FIG. 3 ), showing the coupling spring  228  in detail. The coupling spring  228  comprises a coupling joint  448 , flex arms  450 ,  452 ,  458 , and  460 , forks  454  and  456 , and a coupling joint  462 . The coupling spring  228  is connected to the distal end of the arm  226  at the coupling joint  448 . The coupling joint  448  is connected to the flex arms  450  and  452 . The flex arm  458  is connected to the flex arm  450  at the fork  454 . The flex arm  460  is connected to the flex arm  452  at the fork  456 . The flex arms  458  and  460  are connected to the proof mass  246  at the coupling joint  462 .  FIG. 4  depicts the coupling spring  228  when the arm  226  is at its neutral position. The coupling spring  228  is compliant along the major axis (aligned with the v axis when at rest) and stiff along the minor axis (aligned with the u axis when at rest). 
       FIG. 5  depicts an enlarged area of interest  248  ( FIG. 3 ) and in particular, the coupling spring  228  when the arm  226  is rotated clockwise from its neutral position. The coupling spring  228  has transmitted the u component of this rotation to the proof mass  246  while preventing the proof mass  246  from moving in the v direction. The coupling spring  228  allows the v component of distance between the distal end of the arm  226  and the proof mass  246  to increase by deforming in the v direction. This deformation of the coupling spring  228  causes the flex arms  450 ,  452 ,  458 , and  460  to bend. This deformation of the coupling spring  228  also causes the fork  454  to move closer to the proof mass  246  while the fork  456  moves further away. The geometry of the coupling spring  228  is deflected to result in this bending. This bending behavior provides the combination of compliance in the v direction and rigidity in the u direction. Accordingly, the geometry of the coupling spring allows the proof mass  246  to move substantially only in the u direction when the arm  226  is rotated about the z axis. 
       FIG. 6  depicts an inertial sensor  600  with springs that convert rotational motion to linear motion.  FIG. 6  also depicts an area of interest  601 . The inertial sensor  600  includes a central anchor  602  and a rotational spring  604 . The inertial sensor  600  also includes a rotational drive comprising thirty-two drive combs, eight of which are labeled in  FIG. 6  as drive combs  616 ,  618 ,  620 ,  624 ,  626 ,  628 ,  630 , and  632 . The inertial sensor  600  includes twelve drive sense combs, four of which are labeled in  FIG. 6  as drive sense combs  634 ,  636 ,  638 , and  640 . The inertial sensor  600  includes a drive frame  605 , which is connected to the central anchor  602  by the rotational spring  604 .  FIG. 6  also depicts a coordinate system  622  with an x-y-z coordinate system sharing a z-axis and an origin with a u-v-z coordinate system. While the coordinate system  622  is depicted offset from the inertial sensor  600  for clarity, the origin of the coordinate system  622  is located at the center of the central anchor  602 . The x- and y-axes are orthogonal to each other. The u and v axes are orthogonal to each other and are rotated by −45 degrees from the x- and y-axes, respectively. 
     The drive combs (e.g.,  616 ,  618 ,  620 ,  624 ,  626 ,  628 ,  630 , and  632 ) cause the drive frame  605  to rotate about the z-axis. The drive sense combs (e.g.,  634 ,  636 ,  638 , and  640 ) provide output signals that can be used for closed-loop control of the drive combs (e.g.,  616 ,  618 ,  620 ,  624 ,  626 ,  628 ,  630 , and  632 ), measurement of the velocity of the drive frame  605 , or both. In some examples, some of the drive sense combs (e.g.,  634 ,  636 ,  638 , and  640 ) are used for closed-loop control and some are used for measuring the velocity of the drive frame  605 . The inertial sensor  600  also includes TDS structure  614 . The TDS structure  614  produces a nonlinear capacitive signal used to measure drive velocity of the drive frame  605 . The drive velocity of the drive frame  605  can be determined using the systems and methods described with reference to  FIGS. 17-30 . 
     The inertial sensor  600  includes gyro subassemblies  606 ,  608 ,  610 , and  612 . The gyro subassemblies  606  and  610  include proof masses  966  and  611 , respectively, both configured to deflect in the y and z directions due to Coriolis forces when the inertial sensor  600  is rotated about the z- and y-axes, respectively. The gyroscope subassemblies  608  and  612  contain proof masses  609  and  613 , respectively, both configured to deflect in the x and z directions due to Coriolis forces when the inertial sensor  600  is rotated about the z- and y-axes, respectively. 
     In some examples, the inertial sensor  600  does not contain a TDS structure  614  or other TDS structure and uses the drive sense combs (e.g.,  634 ,  636 ,  638 , and  640 ) for both velocity measurement and drive comb regulation. In some examples, the inertial sensor  600  does not include drive sense combs (e.g.,  634 ,  636 ,  638 , and  640 ) and uses the TDS structure, e.g.,  614  for both velocity measurement and drive comb regulation. In some examples, the inertial sensor  600  contains both TDS structures  614  and/or others and drive sense combs (e.g.,  634 ,  636 ,  638 , and  640 ) and uses the TDS structure  614  and/or other TDS structures for drive comb regulation and the drive sense combs (e.g.,  634 ,  636 ,  638 , and  640 ) for velocity measurement. In some examples, the inertial sensor  600  uses the TDS structure  614  and/or others for velocity measurement and the drive sense combs (e.g.,  634 ,  636 ,  638 , and  640 ) for drive comb regulation. 
     In some examples, the inertial sensor  600  does not have a central anchor  602 . In these examples, the drive frame  605  is anchored to the bottom layer and/or the cap layer at an outer location. 
       FIG. 7  depicts an enlarged view of the area of interest  601  ( FIG. 6 ). At the center of  FIG. 7  is the gyroscope subassembly  606 . The gyroscope subassembly  606  is connected to the drive frame  605  by coupling springs  742  and  744  and by drive springs  746 ,  748 ,  750 , and  752 . The drive springs and coupling springs depicted in  FIG. 7  operate in a similar manner as the drive springs (e.g.,  225  and  227 ) and coupling springs (e.g.,  228 ) depicted in  FIGS. 1-5 , but have different geometry. In contrast to the coupling spring  228  ( FIG. 2 ), which is located radially inward from the proof mass  246  ( FIG. 2 ), the coupling springs  742  and  744  of the inertial sensor  600  are located circumferentially adjacent to the gyroscope subassembly  606 . The coupling springs  742  and  744  are rigid in the x direction but are compliant in the y direction. Thus, the coupling springs  742  and  744  transfer motion in the x direction from the drive frame  605  to the gyroscope subassembly  606  while allowing relative motion between the drive frame  605  and the gyroscope subassembly  606  in the y direction. The drive springs  746 ,  748 ,  750 , and  752  are rigid in the y direction but are compliant in the x direction. Because the drive springs  746 ,  748 ,  750 , and  752  are not perfect springs, they are not perfectly rigid and thus have finite stiffnesses. Thus, the drive springs  746 ,  748 ,  750 , and  752  do allow some motion of the gyroscope subassembly  606  in the y direction. However, the drive springs  746 ,  748 ,  750 , and  752  have high stiffnesses in the y direction such that motion of the gyroscope subassembly  606  in the y direction is small. Thus, the drive springs  746 ,  748 ,  750 , and  752  allow the gyroscope subassembly  606  to move in the x direction but substantially prevent it from moving in the y direction. Accordingly, the combination of the coupling springs  740  and  744  and the drive springs  746 ,  748 ,  750 , and  752 , with appropriately tailored geometry, stiffness and compliance, convert rotational motion of the drive frame  605  about the z-axis into linear motion of the gyroscope subassembly  606  substantially along the x-axis. 
       FIG. 7  also depicts details of the TDS structure  614 . The TDS structure  614  includes movable teeth  758 , fixed teeth  756 , and an anchor  754 . The anchor  754  is anchored to the bottom layer and/or the cap layer and does not move relative to the central anchor  602 . Thus, the fixed teeth  756  also do not move relative to the central anchor  602 . The movable teeth  758  are connected to the drive frame  605  and rotate with it. As the movable teeth  758  rotate about the z-axis, the capacitance between the fixed teeth  756  and the movable teeth  758  varies nonlinearly. The velocity of the drive frame  605  can be determined using the systems and methods described with reference to  FIGS. 17-30 . The velocity of the drive frame  605  is then used for determining rates of rotation acting upon the inertial sensor  600 . 
       FIG. 8  depicts an enlarged view of area of interest  601  ( FIG. 6 ) when the drive combs have caused the drive frame  605  to rotate counterclockwise about the z-axis. The coupling springs  742  and  744  have transmitted motion in the x direction to the gyroscope subassembly  606  while allowing relative motion in the y direction between the gyroscope subassembly  606  and the drive frame  605 . The drive springs  746 ,  748 ,  750 , and  752  have prevented any relative motion in the y direction between the gyroscope subassembly  606  and the drive frame  605  while allowing relative motion in the x direction. The drive springs  746  and  748  have closed slightly while the drive springs  750  and  752  have opened slightly. The point at which the coupling springs  742  attaches to the drive frame  605  is offset in the −y direction from the point at which the coupling spring  742  attaches to the gyroscope subassembly  606 . Likewise, the point at which the coupling spring  744  attaches to the drive frame  605  is offset in the +y direction from the point at which the coupling spring  744  attaches to the gyroscope subassembly  606 . Because the coupling springs  742  and  744  allow this offset, they allow relative motion in the y direction. Because the coupling springs  742  and  744  and drive springs  746 ,  748 ,  750 , and  752  are symmetric, they behave symmetrically when the drive frame  605  rotates clockwise. 
     The gyroscope subassembly  606  contains a proof mass  966  that is deflected by a Coriolis force in response to rotations of the inertial sensor  600 . When the inertial sensor  600  is rotated about the y-axis, a Coriolis force causes the proof mass  966  to deflect in the z direction. When the inertial sensor  600  is rotated about the z-axis, a Coriolis force causes the proof mass  966  to deflect in the y direction. 
       FIG. 9  depicts an enlarged view of part of the gyroscope subassembly  606 , and in particular the drive spring  746  when the drive combs (e.g.,  616 ,  618 ,  620 ,  624 ,  626 ,  628 ,  630 , and  632 ) have rotated the drive frame  605  counterclockwise about the z-axis.  FIG. 9  depicts anchors  954  and  970  which are anchored to the bottom layer and/or the cap layer and do not move relative to the central anchor  602  ( FIG. 6 ). The drive spring  746  includes an anchor fork  956 , an anchor arm  958 , a middle fork  960 , a drive arm  962 , and a drive fork  965 . The anchor  954  is connected to the proximal end of the anchor arm  958  by the anchor fork  956 . The distal end of the anchor arm  958  is connected to the distal end of the drive arm  962  by the middle fork  960 . The proximal end of the drive arm  962  is connected to a drive frame  964  of the gyroscope subassembly  606  by the drive fork  965 . The forks  956 ,  960 , and  964  flex to allow the drive frame  964  to move in the −x direction, but the arms  958  and  962  are rigid, substantially preventing the drive frame  964  from moving in the y direction. 
       FIG. 9  also depicts a proof mass  966  and a sense comb  968 . The sense comb  968  is configured for detecting motion of the proof mass  966  in they direction. 
       FIG. 10  depicts an enlarged view of part of the gyroscope subassembly  606 , and in particular the drive spring  746 , when the drive combs (e.g.,  616 ,  618 ,  620 ,  624 ,  626 ,  628 ,  630 , and  632 ) have rotated the drive frame  605  clockwise about the z-axis from its neutral position. The forks  956 ,  960 , and  964  have flexed, allowing the drive spring  746  to expand slightly. This opening of the drive spring  746  allows the drive frame  964  to move in the x direction. Because the arms  958  and  962  are rigid, the drive spring  746  prevents the drive frame  964  from moving in the y direction. Accordingly, the drive spring  746  allows the gyroscope subassembly  606  to move in the x direction but substantially prevents it from moving in the y direction. 
       FIG. 11  depicts an enlarged view of part of the gyroscope subassembly  606 , and in particular the coupling spring  742 , when the drive combs have rotated the drive frame  964  counterclockwise about the z-axis. The coupling spring  742  includes a driving fork  1172 , driving arms  1174  and  1176 , middle forks  1178  and  1180 , middle arms  1182  and  1184 , driven fork  1186 , driven arm  1188 , and driven fork  1190 . The proximal ends of the driving arms  1174  and  1176  are connected to the drive frame  605  by the driving fork  1172 . The distal end of the middle arm  1182  is connected to the distal end of the driving arm  1174  by the middle fork  1178 . The distal end of the driving arm  1176  is connected to the distal end of the middle arm  1184  by the middle fork  1180 . The proximal ends of the middle arms  1182  and  1184  are connected to each other and to the proximal end of the driven arm  1188  by the driven fork  1186 . The distal end of the driven arm  1188  is connected to the drive frame  964  by the driven fork  1190 . As the drive frame  605  rotates about the z-axis, the forks  1172 ,  1178 ,  1180 ,  1186 , and  1190  flex, allowing the drive frame  605  to move in the y direction relative to the drive frame  964 . The arms  1174 ,  1176 ,  1182 ,  1184 , and  1188  are rigid in the x direction, thus transmitting the x component of the rotation of the drive frame  605  to the drive frame  964 . Because relative motion between the drive frames is allowed in the y direction, the coupling spring  742  is compliant in the y direction. Because the coupling spring  742 , which connects the gyroscope subassembly  606  to the drive frame  605 , is compliant in the y direction but rigid in the x direction, the coupling spring  742  transmits only the x component of the rotation of the drive frame  605  to the gyroscope subassembly  606 . The coupling springs  742  and  744  ( FIGS. 7-8 ) have symmetric geometry. 
       FIG. 12  depicts an enlarged view of part of the gyroscope subassembly  606 , and in particular the coupling spring  742 , when the drive combs (e.g.,  616 ,  618 ,  620 ,  624 ,  626 ,  628 ,  630 , and  632 ) have rotated the drive frame  605  clockwise about the z-axis from its neutral position. The forks  1172 ,  1178 ,  1180 ,  1186 , and  1190  have flexed, allowing the driving fork  1172  to move in the +y direction relative to the driven fork  1190 . The driven fork  1190  does not move in the y direction, while the position of the driving fork  1172  moves in an arc centered on the z-axis as the drive frame  605  rotates. The coupling spring  742  transmits only the x component of the motion along this arc to the drive frame  964  and the gyroscope subassembly  606 . Accordingly, the coupling spring  742 , in conjunction with the coupling spring  744  and the drive springs  746 ,  748 ,  750 , and  752 , converts rotational motion of the drive frame  605  about the z-axis into linear motion of the gyroscope subassembly  606  along the x-axis. 
       FIG. 13  depicts an inertial sensor  1300  with springs that convert rotational motion to linear motion. The inertial sensor  1300  includes a central anchor  1302 , which is anchored to the bottom layer (not shown) and/or the cap layer (not shown) below a device layer of the inertial sensor  1300  depicted in  FIG. 13 . The inertial sensor  1300  includes a drive frame  1305  connected to the central anchor  1302  by a rotational spring  1304 . The inertial sensor  1300  includes a rotational drive comprising a plurality of drive combs (not shown) which cause the drive frame  1305  to rotationally oscillate about the z-axis.  FIG. 13  also depicts a coordinate system  1322  with an x-y-z coordinate system sharing a z-axis and an origin with a u-v-z coordinate system. While the coordinate system  1322  is depicted offset from the inertial sensor  1300  for clarity, the origin of the coordinate system  1322  is located at the center of the central anchor  1302 . The x- and y-axes are orthogonal to each other. The u- and v-axes are orthogonal to each other and are rotated by −45 degrees from the x- and y-axes, respectively. The inertial sensor  1300  includes TDS structures  1314  (only part of which are shown) and drive sense combs (not shown) to measure the velocity of the drive frame  1305  and to regulate the drive combs in closed-loop control. The velocity and amplitude of motion of the drive frame  1305  can be determined using the systems and methods described with reference to  FIGS. 17-30 . 
     In some examples, the inertial sensor  1300  does not contain a TDS structure and uses the drive sense combs for both velocity measurement and drive comb regulation. In some examples, the inertial sensor  1300  does not include drive sense combs and uses the TDS structure for both velocity measurement and drive comb regulation. In some examples, the inertial sensor  1300  contains both TDS structures and drive sense combs and uses the TDS structure for drive comb regulation and the drive sense combs for velocity measurement. In some examples, the inertial sensor  1300  uses the TDS structure for velocity measurement and the drive sense combs for drive comb regulation. 
     In some examples, the inertial sensor  1300  does not have a central anchor  1302 . In these examples, the drive frame is anchored to the bottom layer and/or the cap layer at an outer location. 
     The inertial sensor  1300  includes gyroscope subassemblies  1306 ,  1308 ,  1310 , and  1312 . When the inertial sensor  1300  is rotated about the x-axis, a Coriolis force causes proof masses of the gyroscope subassemblies  1308  and  1312  to deflect in the z direction. When the inertial sensor  1300  is rotated about the z-axis, a Coriolis force causes the proof masses of the gyroscope subassemblies  1306  and  1310  to deflect in they direction and the proof masses of the gyroscope subassemblies  1308  and  1312  to deflect the x direction. When the inertial sensor  1300  is rotated about the y-axis, a Coriolis force causes proof masses of the gyroscope subassemblies  1306  and  1310  to deflect in the z direction. Electrodes (not shown) mounted either above or below the device layer depicted in  FIG. 13  detect the deflection in the z direction in the proof masses of the gyroscope subassemblies  1306 ,  1308 ,  1310 , and  1312 . These electrodes measure the respective deflections by measuring a change in capacitance. Electrodes (not shown) anchored to the bottom layer and/or the cap layer but extending into the device layer measure the deflection of the proof masses of the gyroscope subassemblies  1306 ,  1308 ,  1310 , and  1312  in the x-y plane by measuring a change in capacitance. The inertial sensor  1300  also includes TDS structures (not shown) configured to measure motion of the proof masses of the gyroscope subassemblies  1308  and  1312  along the y-axis. The motion measured by the TDS structures can be used to calculate velocity of the drive frame  1305 , acceleration of the inertial sensor  1300  in the y direction, or both. 
     The inertial sensor  1300  includes four coupling springs, one of which is a coupling spring  1318 . In contrast to the coupling spring  228  of the inertial sensor  100  and the coupling springs  742  and  744  of the inertial sensor  600 , the coupling spring  1318  is located radially outward from the gyroscope subassembly  1306 . The inertial sensor  1300  also includes eight drive springs, two of which are drive springs  1314  and  1316 . 
       FIG. 14  depicts the inertial sensor  1300  when the drive combs have rotated the drive frame  1305  counterclockwise about the z-axis from its neutral position. The drive springs and couplings springs have converted this rotational motion of the drive frame  1305  into linear motion in the −x direction for the gyroscope subassembly  1306 , linear motion in the +x direction for the gyroscope subassembly  1310 , linear motion in the +y direction for the gyroscope subassembly  1308 , and linear motion in the −y direction for the gyroscope subassembly  1312 . 
       FIG. 15  depicts an enlarged view of the gyroscope subassembly  1306  when the drive frame  1305  is in its neutral position.  FIG. 15  depicts an anchor  1528  that is anchored to the bottom layer (not shown) and/or the cap layer (not shown) and does not move relative to the central anchor  1302 . The anchor  1528  is connected to the drive springs  1314  and  1316 . The drive springs  1314  and  1316  have a similar geometry to, and function in a similar manner as, the drive springs  225  ( FIG. 2 ),  227  ( FIG. 2 ),  746  ( FIG. 7 ),  748  ( FIG. 7 ),  750  ( FIG. 7 ), and  752  ( FIG. 7 ). The coupling spring  1318  is connected to an outer rim  1307  of the drive frame  1305 . The outer rim  1307  is rigidly connected to the drive frame  1305  and rotates with it. The coupling spring  1318  has a similar geometry and functions in a similar manner as the coupling spring  228  ( FIG. 2 ).  FIG. 15  also depicts the springs  1524  and  1526  which allow a proof mass of the gyroscope subassembly  1306  to deflect in the z direction. 
       FIG. 16  depicts an enlarged view of the gyroscope subassembly  1306  when the drive combs have rotated the drive frame  1305  counterclockwise from its neutral position.  FIG. 15  also depicts a drive frame  1520  of the gyroscope subassembly  1306 . The drive frame  1520  receives the motion in the x direction transmitted by the coupling spring  1318  and transmits that x motion to a proof mass of the gyroscope subassembly  1306 . The coupling spring  1318  includes coupling links  1630  and  1644 , flex arms  1632 ,  1634 ,  1640 , and  1642 , and forks  1636  and  1638 . The distal end of the coupling link  1630  is connected to the outer rim  1307  of the drive frame  1305 . The proximal end of the coupling link  1630  is connected to the flex arms  1632  and  1634 . The left ends of the flex arms  1632  and  1640  are connected by the fork  1636 , and the right ends of the flex arms  1634  and  1642  are connected by the fork  1638 . The right end of the flex arm  1640  and the left end of the flex arm  1642  are connected to the drive frame  1520  by the coupling link  1644 . Because the drive frame  1305  is rotated from its neutral position, the flex arms  1632 ,  1634 ,  1640 , and  1642  bend slightly to allow relative motion in the y direction between the coupling links  1630  and  1644  while transmitting the x component of the rotation to the drive frame  1520  by the coupling link  1644 . 
     The drive spring  1314  includes an anchor arm  1656 , a fork  1652 , and a drive arm  1648 . The drive spring  1316  includes an anchor arm  1658 , a fork  1654 , and a drive arm  1650 . The respective proximal ends of the anchor arms  1656  and  1658  are connected to the anchor  1528 . The distal end of the anchor arm  1656  is connected to the distal end of the drive arm  1648  by the fork  1652 . Likewise, the distal end of the anchor arm  1658  is connected to the distal end of the drive arm  1650  by the fork  1654 . The proximal ends of the drive arms  1648  and  1650  are connected to the drive frame  1520  by respective forks. 
     The drive springs  1314  and  1316  are stiff in they direction but compliant in the x direction. Thus, as the coupling spring  1318  transmits the x component of rotation to the drive frame  1520 , the drive springs  1314  and  1316  prevent the drive frame  1520  from moving in the y direction. As the drive frame  1305  rotates counterclockwise, the fork  1652  flexes to allow the drive spring  1314  to close slightly and the fork  1654  flexes to allow the drive spring  1316  to open slightly. This flexing, opening, and closing allows the drive frame  1520  to move in the x direction. Because the drive springs  1314  and  1316  are not perfect springs, they are not perfectly rigid and thus have finite stiffnesses. Thus, the drive springs  1314  and  1316  do allow some motion of the drive frame  1520  in the y direction. However, the drive springs  1314  and  1316  have high stiffnesses in the y direction such that motion of the drive frame  1520  in the y direction is small. Because of the geometry, stiffness, and compliance of the coupling spring  1318  and the drive springs  1314  and  1316 , the inertial sensor  1300  converts the rotational motion of the drive frame  1305  into linear motion of the gyroscope subassembly  1306  substantially along the x-axis. 
       FIG. 17  depicts three views  1700 ,  1730 , and  1760 , each showing a schematic representation of parts of a moveable element  1702  and a fixed element  1704 . The TDS structures described herein can include the moveable element  1702  and the fixed element  1704 . The oscillating mass of the TDS structure can include the moveable element  1702 . The movable element  1702  and the fixed element  1704  depicted in  FIG. 17  each include a plurality of structures, or beams. In particular, the fixed element  1704  includes beams  1706   a ,  1706   b , and  1706   c  (collectively, beams  1706 ). The moveable element  1702  depicted in  FIG. 17  includes beams  1708   a  and  1708   b  (collectively, beams  1708 ). The moveable element  1702  is separated from the fixed element  1704  by a distance WO  1732 . The distance WO  1732  can change as the moveable element  1702  oscillates with respect to the fixed element  1704 . The distance WO  1732  affects parasitic capacitance between the movable element  1702  and the fixed element  1704 . The distance WO  1732  is selected to minimize parasitic capacitance when the movable element  1702  is in the rest position, while maintaining manufacturability of the sensor. The view  1760  depicts an area of interest noted by the rectangle  1740  of view  1730 .  FIG. 17  depicts an example of TDS structures with teeth on parallel beams. In other examples, TDS structures include teeth on other geometrics. However, the same general principles described with reference to  FIGS. 17-30  apply to TDS structures with other geometrics. 
     Each of the beams  1706  and  1708  includes multiple sub-structures, or teeth, protruding perpendicularly to the long axis of the beams. The beam  1706   b  includes teeth  1710   a ,  1710   b , and  1710   c  (collectively, teeth  1710 ). The beam  1708   b  includes teeth  1712   a ,  1712   b  and  1712   c  (collectively, teeth  1712 ). Adjacent teeth on a beam are equally spaced according to a pitch  1762 . Each of the teeth  1710  and  1712  has a width defined by the line width  1766  and a depth defined by a corrugation depth  1768 . Opposing teeth are separated by a tooth gap  1764 . As the moveable beam  1708   b  oscillates along the moving axis  1701  with respect to the fixed beam  1706   b , the tooth gap  1764  remains unchanged. In some examples, manufacturing imperfections cause the tooth spacing to deviate from the pitch  1762 . However, provided that the deviation is negligible compared to the pitch  1762 , the deviation does not significantly impact operation of the sensor and can be neglected for the purposes of this disclosure. 
     A capacitance exists between the fixed beam  1706   b  and the moveable beam  1708   b . As the moveable beam  1708   b  oscillates along the moving axis  1701  with respect to the fixed beam  1706   b , the capacitance changes. The capacitance increases as opposing teeth of the teeth  1710  and  1712  align with each other and decreases as opposing teeth become less aligned with each other. At the position depicted in the view  1760 , the capacitance is at a maximum and the teeth  1710  are in an aligned position with respect to the teeth  1712 . As the moveable beam moves monotonically along the moving axis  1701 , the capacitance changes non-monotonically, since a maximum in capacitance occurs as the teeth  1710  and  1712  are in an aligned position. 
     The capacitance can be degenerate, meaning that the same value of capacitance can occur at different displacements of the moveable beam  1708   b . When the moveable beam  1708   b  has moved from its rest position by a distance equal to the pitch  1762 , the capacitance is the same as when the moveable beam  1708   b  is in the rest position. 
       FIG. 18  schematically depicts an exemplary process used to extract inertial information from an inertial sensor with periodic geometry.  FIG. 18  includes an inertial sensor  1800  which experiences an external perturbation  1801 . The inertial sensor  1800  can include the system  100 , and the external perturbation  1801  can include the input inertial parameter  102 . A drive signal  1810  causes a movable portion of the sensor  1800  to oscillate. The moveable portion can be the moveable element  1702 . An analog frontend (AFE) electrically connected to the moveable element  1702  and to the fixed element  1704  measures the capacitance between them and outputs a signal based on the capacitance. The AFE can do this by measuring a capacitive current or a charge. Zero-crossings of the AFE output signal occur when the AFE output signal momentarily has a magnitude of zero. Zero-crossings of an output signal from the inertial sensor  1800  are generated at  1802  and  1804  and combined at  1806  into a combined signal. A signal processing module  1808  processes the combined analog signal to determine inertial information. One or more processes can convert the combined analog signal into a rectangular waveform  1812 . This can be accomplished using a comparator, by amplifying the analog signal to the rails, or by other methods. 
     The rectangular waveform  1812  comprises a rectangular pulse stream having high and low values, with no substantial time spent transitioning between high and low values. Transitions between high and low values correspond to zero-crossings of the combined analog signal. The transitions between high and low values and zero-crossings occur when a displacement  1818  of the movable element  1702  crosses reference levels  1814  and  1816 . The reference levels  1814  and  1816  correspond to physical locations of movable portions of the sensor  1800 . Because the zero-crossings are associated with specific physical locations, displacement information can be reliably determined independent of drift, creep and other factors which tend to degrade performance of inertial sensors. 
       FIG. 19  depicts a graph  1900  which represents the association of analog signals derived from the inertial sensor  1800  with zero-crossing times and displacements of the inertial sensor. The graph  1900  represents signals derived from an oscillator in which opposing teeth are aligned at the rest position. The graph  1900  includes curves  1902 ,  1904  and  1906 . The curve  1902  represents an output of an AFE such as a transimpedance amplifier (TIA). Since a TIA outputs a signal proportional to its input current, the curve  1902  represents a capacitive current measured between movable and fixed elements of an inertial device such as the inertial device  1800 . The curve  1906  represents an input acceleration that is applied to the inertial device  1800 . The input acceleration represented by the curve  1906  is a 15 g acceleration at 20 Hz. The curve  1904  represents displacement of the movable element of the inertial device  1800  as it oscillates. 
       FIG. 19  includes square symbols indicating points on the curve  1902  at which the curve  1902  crosses the zero level. These zero-crossings in the current represent local maxima or minima (extrema) of capacitance between the movable element and the fixed element of the inertial device, because capacitive current is proportional to the first derivative of capacitance.  FIG. 19  includes circular symbols indicating points on the curve  1904  corresponding to times at which the curve  1902  crosses zero. The circular symbols indicate the correlation between physical position of a movable element of the oscillator and zero-crossing times of the outputs of signal  1902 . 
     At the time  1918 , the curve  1902  crosses zero because the displacement of the movable element of the oscillator is at a maximum and the oscillator is at rest, as indicated by the displacement curve  1904 . Here, capacitance reaches a local extremum because the movable element has a velocity of zero, not necessarily because teeth or beams of the oscillator are aligned with opposing teeth or beams. At time  1920 , the TIA output curve  1902  crosses zero because the oscillator displacement reaches the +d 0  location  1908 . The +d 0  location  1908  corresponds to a displacement in a positive direction equal to the pitch distance and is a point at which opposing teeth or beams are aligned to produce maximum capacitance. At time  1922 , the TIA output curve  1902  crosses zero because the movable element of the oscillator is at a position at which the teeth are anti-aligned. This occurs when the teeth of the movable element  1702  ( FIG. 17 ) are in an aligned position with the centers of gaps between teeth of the fixed element  1704 , resulting in a minimum in capacitance. This minimum in capacitance occurs at a location of +d 0 /2  1910 , corresponding to a displacement to one-half the pitch distance in the positive direction. 
     At time  1924 , the TIA output curve  1902  crosses zero because teeth of the movable element  1702  ( FIG. 17 ) are aligned with teeth of the fixed element  1704  ( FIG. 17 ), producing a maximum in capacitance. The time  1924  corresponds to a time at which the movable element is at the rest position, indicated by the zero displacement  1912  on the curve  1904 . At time  1926 , the TIA output  2002  crosses zero because teeth of the movable element  1702  ( FIG. 17 ) are anti-aligned with teeth of the fixed element  1704  ( FIG. 17 ), producing a local minimum in capacitance. This anti-alignment occurs at a displacement of −d 0 /2  1914 , corresponding to a displacement of one-half the pitch distance in the negative direction. 
     At time  1928 , the TIA output  1902  crosses zero because the teeth of the movable element  1702  ( FIG. 17 ) are in an aligned position with respect to the teeth of the fixed element  1704  ( FIG. 17 ), creating a local maximum in capacitance. This local maximum in capacitance occurs at a displacement of −d 0    1916 , corresponding to a displacement equal to such distance in the negative direction. At time  1930 , the TIA output curve  1902  crosses zero because the movable element  1702  ( FIG. 17 ) has a velocity of zero as it reverses direction. This reversal of direction is illustrated by the displacement curve  1904 . As at time  1918 , when the movable element has a velocity of zero, capacitance is not changing with time and thus the current and TIA output (which are proportional to the first derivative of capacitance) are zero. 
       FIG. 20  depicts a graph  2000  showing the effect of an external perturbation on input and output signals of any of the inertial sensors described herein. The graph  2000  includes the TIA output curve  2002  which is similar to the TIA output curve  1902 , the displacement curve  2004  which is similar to the displacement curve  1904 , and the input acceleration curve  2006  which is similar to the input acceleration curve  1906 .  FIG. 20  also depicts the location+d 0    2008  which is similar to the location+d 0    1908 , the location+d 0 /2  2010  which is similar to the location +d 0 /2  1910 , the location 0  2012  which is similar to the location −  1912 , the location −d 0 /2  2014  which is similar to the location −d 0 /2  1914 , and the location −d 0    2016  which is similar to the location −d 0    1916 . The graph  2000  depicts the same signals depicted in the graph  1900 , and the only difference is that the graph  2000  represents a longer duration of time than the graph  1900 . With a longer duration of time displayed in the graph  2000 , the periodicity of the input acceleration curve  2006  is more easily discerned. In addition, maximum displacement crossings  2020  and minimum displacement crossings  2022  can be discerned in the graph  2000  to experience a similar periodicity. In contrast to the maximum displacement crossings  2020  and the minimum displacement crossings  2022 , the amplitude of which varies with time, zero-crossings of the TIA output signal  1902  triggered by alignment or anti-alignment of teeth of the fixed and movable elements  1704  ( FIG. 17 ) and  1702  ( FIG. 17 ) at the locations +d 0    2008 , +d 0 /2  2010 , 0  2012 , −d 0 /2  2014 , and −d 0    2016  are stable with time. These reference crossings, the amplitude of which are stable with time, provide stable, drift-independent indications of oscillator displacement and can be used to extract inertial parameters. 
       FIG. 21  depicts a graph  2100  that illustrates the response of a current to an oscillator displacement. The graph  2100  includes a current curve  2102  and a displacement curve  2104 . The current curve  2102  represents an input signal for a TIA. The TIA may produce an output signal such as one or both of the TIA output curves  1902  and  2002  in response. The current curve  2102  is a capacitive current between the fixed beam  1704  ( FIG. 17 ) and the movable beam  1702  ( FIG. 17 ) in response to displacement of the movable beam  1702  ( FIG. 17 ) according to the displacement curve  2104 . The current curve  2102  crosses zero at numerous times, including times  2124 ,  2126 ,  2128 , and  2130 . At the times  2124  and  2130 , the movable element  1702  ( FIG. 17 ) has a displacement of −d 0 , as shown in the graph  2100 . At the times  2126  and  2128 , the movable element  1702  ( FIG. 17 ) has a displacement of +d 0 , shown on the graph  2100 . 
     The graph  2100  includes two time intervals T 43    2132  and T 61   2134 . The time interval T 43    2132  corresponds to the difference in time between time  2126  and time  2128 . The time interval T 61    2134  corresponds to the time difference between times  2124  and  2130 . Thus, time interval T 61    2134  corresponds to the time between subsequent crossings of the −d 0    2116  level, and the time interval T 43    2132  corresponds to the time interval between subsequent crossings of the +d 0    2108  level. The methods used to determine the time intervals T 43    2132  and T 61    2134  can be used to determine other time intervals, such as between a crossings of the +d 0    2108  and the next subsequent crossing of the −d 0    2116  level, between a time interval between a crossing of the −d 0    2116  level and the next crossing of the +d 0    2108  level, between the time  2130  and the next crossing of the +d 0    2108  level, between crossings of the zero  2112  level, between zero-crossings due to a maximum or minimum of displacement, or between any other combination of zero-crossings of the current curve  2702  or a TIA output signal corresponding to the current curve  2102 . 
       FIG. 22  depicts a graph  2200  showing a rectangular waveform signal representing zero-crossing times of the current signal  2102 . The graph  2200  includes a rectangular waveform curve  2236 . The rectangular waveform curve  2236  has substantially two values: a high value and a low value. While the rectangular waveform curve  2236  may have intermediate values as it transitions between the high and low values, the time spent at intermediate values is far less than the combined time spent at the high and low of the values. 
     The rectangular waveform curve  2236  can be produced by a variety of methods, including using a comparator to detect changes in an input signal, by amplifying an input signal to the limits of an amplifier so as to saturate the amplifier (amplifying to the rails), by using an analog-to-digital converter, and the like. One way to produce this rectangular waveform curve  2236  from the current curve  2102  is to use a comparator to detect zero-crossings of the current curve  2102 . When the current curve  2102  has a value greater than a reference level (such as zero), the comparator outputs a high value, and when the current curve  2102  has a value less than the reference level (such as zero), the comparator has a low value. The comparator&#39;s output transitions from low to high when the current curve  2102  transitions from a negative value to a positive value, and the comparator&#39;s output transitions from high to low when the current curve  2102  transitions from a positive value to a negative value. Thus, times of rising edges of the rectangular waveform curve  2236  correspond to times of negative-to-positive zero-crossings of the current curve  2104 , and falling edges of the rectangular waveform curve  2236  correspond to positive-to-negative zero-crossings of the current curve  2102 . 
     The rectangular waveform curve  2236  includes the same time intervals  2132  and  2134  as the current curve  2102 . One benefit of converting the current curve  2102  to a rectangular waveform signal such as the rectangular waveform curve  2236  is that in a rectangular waveform signal, rising and falling edges are steeper. Steep rising and falling edges provide more accurate resolution of the timing of the edges and lower timing uncertainty. Another benefit is that rectangular waveform signals are amenable to digital processing. 
       FIG. 23  depicts a graph  2300  which illustrates additional time intervals of displacement curve  2104 . In addition to the times depicted in the graph  2100 , the graph  2300  includes times  2336  and  2338 . In addition to the time intervals depicted in the graph  2100 , the graph  2300  includes the time interval T 94    2340  and the time interval T 76    2342 . The time interval T 94    2340  corresponds to the time interval between times  2128  and  2338 , both crossings of the d 0    2108  level. The time interval T 76    2342  corresponds to the time interval between times  2130  and  2336 , both crossings of the −d 0    2116  level. 
     As can be seen in  FIG. 19 , the oscillator displacement as shown by the displacement curve  1904  experiences an offset that is correlated with input acceleration as indicated by the acceleration curve  1906 . Thus, one way to detect shifts of the displacement curve  2104  and thus input acceleration is to compare relative positions of zero-crossing times. For example, a sum of the time intervals T 43    2132  and T 94    2340  represents a period of oscillation as does a sum of the periods T 61    2134  and T 36    2342 . In comparing a subset of the period, such as comparing the time interval T 43    2132  with the sum of T 43    2132  and T 94    2340  represents the proportion of time that the oscillator spends at a displacement greater than +d 0    2108 . An increase in this proportion from a reference proportion indicates a greater acceleration in a positive direction than the reference. Likewise, a decrease in this proportion from the reference indicates a greater acceleration in the negative direction. Other time intervals can be used to calculate other proportions and changes in acceleration. 
     In some examples, integrating portions of the rectangular waveform using the systems and methods described herein can be performed to determine relative positions of zero-crossing times and thus acceleration, rotation and/or velocity. In other examples, displacement of an oscillator can be determined from the time intervals depicted in  FIG. 23  using equations 1, 2, and 3. 
     
       
         
           
             
               
                 
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     Displacement of the oscillator can be converted to an acceleration using Hooke&#39;s Law. Displacement of the oscillator can be calculated recursively for each half cycle of the oscillator. Using this information, the displacement of the oscillator can be recorded as a function of time. This allows the calculation of external perturbations with zero drift and lower broadband noise. 
       FIG. 24  depicts a relationship between capacitance of an inertial sensor (e.g., the inertial sensor  1800 ) and displacement of a movable element (e.g., movable element  1702 ).  FIG. 24  includes a capacitance curve  2402  that is periodic and substantially sinusoidal. Thus, monotonic motion of the movable element  1702  ( FIG. 17 ) produces a capacitance that changes non-monotonically with displacement. This non-monotonically is a function of the geometric structure of the sensor  100  and the manner in which the sensor  100  is excited. 
       FIG. 25  depicts a relationship between displacement and the first derivative of capacitance with respect to displacement.  FIG. 25  includes a dC/dx curve  2502  which is periodic and substantially sinusoidal. The dC/dx curve  2502  is the first derivative of the capacitance curve  2402 . As such, the dC/dx curve  2502  crosses zero when the capacitance curve  2402  experiences a local extremum. Capacitive current is proportional to the first derivative of capacitance and thus proportional to, and shares zero-crossings with, the dC/dx curve  2502 . 
       FIG. 26  depicts a relationship between displacement and the second derivative of capacitance with respect to displacement.  FIG. 26  includes a d 2 C/dx 2  curve  2602 . The dC/dx 2  curve  2602  is the first derivative of the dC/dx curve  2502  and as such has a value of zero at local extrema of the dC/dx curve  2502 . The d 2 C/dx 2  curve  2602  indicates the slope of the dC/dx curve  2502  and thus indicates locations at which the current is most rapidly changing. In some implementations, it is desirable to maximize the amplitude of the d 2 C/dx curve  2602  to maximize the steepness of the current curve. This reduces uncertainty in resolving timing of zero-crossings of the current. Reducing uncertainty of the zero-crossing times results in decreased system noise and decreased jitter, as well as, lower gain required of the system. Decreased jitter results in improved resolution of external perturbations. In some implementations, it is desirable to minimize the impact of variable parasitic capacitance, which is parasitic capacitance that varies with oscillator motion. 
       FIG. 27  depicts a relationship between time, the rate of change of capacitive current, and displacement.  FIG. 27  includes a dI/dt curve  2702 . The capacitive current used to determine the dI/dt curve  2702  is obtained by applying a fixed voltage across the capacitor used to produce the capacitive curve  2402 . The dI/dt curve  2702  represents the rate at which the capacitive current is changing with time and thus provides an indicator of the steepness of the current slope. High magnitudes of the dI/dt signal indicate rapidly changing current and high current slopes. Since the oscillator used to generate the curves shown in  FIGS. 24-27  oscillates about zero displacement and reverses direction at displacements of +15 μm and −15 μm, the velocity of the oscillator is lowest at its extrema of displacement. At these displacement extrema, the current is also changing less rapidly and thus the dI/dt curve  2702  has a lower magnitude. Using zero-crossings at which the dI/dt curve  2702  has large values results in improved timing resolution and decreased jitter. These zero-crossings occur near the center of the oscillator&#39;s range. 
       FIG. 28  depicts a flow chart of a method  2800  used to extract inertial parameters from a nonlinear periodic signal. At  2802 , a first nonlinear periodic signal is received. At  2804 , a second nonlinear periodic signal is optionally received. The first nonlinear periodic signal and the optional second nonlinear periodic signal can be generated by any of the TDS structures depicted in  FIGS. 1-16  and received at signal processing circuitry configured to extract an inertial parameter from one or more nonlinear periodic signals. 
     At  2806 , optionally, the first and second nonlinear periodic signals are combined into a combined signal. This can be accomplished by the element  1806 . If the steps  2804  and  2806  are omitted, the method  2800  proceeds from  2802  directly to  2808 . 
     At  2808 , the signal is converted to a two-valued signal by signal processing circuitry that can include a comparator and/or a high-gain amplifier. The two-valued signal can be a signal that has substantially only two values, but may transition quickly between the two values. This two-valued signal can be a digital signal such as that output from a digital circuit element. In some examples, the two-valued signal is produced by amplifying the combined signal or one of the first and second nonlinear signals using a high-gain amplifier. This technique can be referred to as “amplifying to the rails.” The two-valued signal can be the signal  1812 . The two-valued signal can be determined based on a threshold such that if the combined, first, or second signal is above the threshold, the two-valued signal takes on a first value and if below the threshold, the two-valued signal takes on a second value. 
     At  2810 , times of transitions between the two values of the two-valued signal are determined. In some examples, these times can be determined using a time-to-digital converter (TDC) or by an analog to digital converter and digital signal processing. The time intervals determined in this way can be one or more of the intervals  2132 ,  2134 ,  2340 , and  2342 . 
     At  2814 , a trigonometric function is applied to the determined time intervals. The trigonometric function can be a sine function, a cosine function, a tangent function, a cotangent function, a secant function, and a cosecant function. The trigonometric function can also be one or more of the inverse trigonometric functions such as the arcsine, the arccosine, the arctangent, the arccotangent, the arcsecant, and the arccosecant functions. Applying the trigonometric function can include applying a trigonometric function to an argument that is based on the determined time intervals. 
     At  2816 , inertial parameters are extracted from the result of applying the trigonometric function. Extracting the inertial parameters can include curve fitting and computing derivatives of the result. The inertial parameters can be one or more of sensor acceleration, sensor velocity, sensor displacement, sensor rotation rate, sensor rotational acceleration and higher order derivatives of linear or rotational acceleration, such as jerk, snap, crackle, and pop. 
       FIG. 29  depicts a method  2900  for determining times of transition between two values based on nonlinear periodic signals. The method  2900  can be used to perform one or more of the steps  2802 ,  2804 ,  2806 ,  2808 , and  2810  of the method  2800 . 
     At  2902 , a first value of a first nonlinear periodic signal is received at signal processing circuitry that can include a TDC or digital circuitry. At  2904 , a second value of a second nonlinear periodic signal is optionally received at the TDC or digital circuitry. The first and second values are values of the first and second signals at particular moments in time, and can be analog or digital values. The first and second nonlinear periodic signals of the method  2900  can be the same as the first and second nonlinear periodic signals of the method  2800 . 
     At  2906 , the first and second values are optionally combined into a combined value. The values may be combined using the element  1806 , which may include a summing amplifier, a differential amplifier, an analog multiplier, and/or an analog divider. Combining may include summing the values, taking a difference of the values, multiplying the values, or dividing the values. If the optional steps  2904  and  2906  are omitted, the method  2900  proceeds from  2902  directly to  2908 . 
     At  2908 , the first value or the combined value is compared to a threshold. If the value is above the threshold, the method  2900  proceeds to  2910 . 
     At  2910 , a high value is assigned for the current time. If the value is not above the threshold, the method  2900  proceeds to  2912 . At  2912 , a low value is assigned for the current time. The steps  2908 ,  2910  and  2912  can be used to generate a two-valued signal having high and low values from an input signal. The two-valued signal of the method  2900  can be the same as the signal of the method  2800 . 
     At  2914 , the value of the signal for the current time is compared to a value of the signal for an immediately previous time. If the two values are the same, the method  2900  proceeds to  2916  where the method  2900  terminates. If the two values are not the same, a transition has occurred and the method proceeds to  2918 . 
     At  2918 , the sense of the transition (whether the transition is a rising edge or a falling edge) is determined. If the value for the current time is greater than the value for the previous time, a rising edge is assigned to the transition. 
     If the value for the current time is not above the value for the previous time, the method  2900  proceeds to  2922 . At  2922 , a falling edge is assigned to the transition. Thus, times having transitions are detected and classified as having either rising or falling edges. At  2924 , a time interval is determined between the transition and another transition. Time intervals between these transition times can be determined by obtaining a difference in time values between times of transition. 
       FIG. 30  depicts a method  3000  to compute inertial parameters from time intervals. The method  3000  can be used to perform one or more of the steps  2814  and  2816  of the method  2800 . 
     At  3002 , first and second time intervals are received at signal processing circuitry that can include a TDC or digital circuitry. The first and second time intervals can be determined using the method  2900 . 
     At  3004 , a sum of the first and second time intervals is computed using digital signal processing circuitry such as an application specific integrated circuit (ASIC) or a field programmable gate array (FPGA). The sum can be the measured period as described by equations 2 and 3. At  3006 , a ratio of the first time interval to the sum is computed. The ratio can be one or more of the ratios forming part of the arguments of the cosine functions in equation 1. 
     At  3008 , an argument is computed using the ratio by the digital signal processing circuitry. The argument can be one or more of the arguments of the cosine functions of equation 1. 
     At  3010 , a trigonometric function is applied to the argument by the digital signal processing circuitry. The trigonometric function can be any of the trigonometric functions described with reference to step  2904  of the method  2900 . 
     At  3012 , the digital signal processing circuitry computes a displacement using one or more geometric parameters and using the result of applying the trigonometric function. The displacement can be computed using equation 1. Computing displacement can involve computing more than one trigonometric function, and arguments other than the computed argument of  2008  can be included as arguments of some of the trigonometric functions. 
     At  3014 , the digital signal processing circuitry computes one or more inertial parameters using the displacement. The inertial parameters computed can be any of the inertial parameters described with reference to step  2816  of the method  2800 . Inertial parameters can be computed by obtaining one or more derivatives of the displacement with respect to time. Inertial parameters may be extracted using an offset of the computed displacement to determine an external acceleration. In this way, inertial parameters are computed from time intervals. 
     The systems described herein can be fabricated using MEMS and microelectronics fabrication processes such as lithography, deposition, and etching. The features of the MEMS structure are patterned with lithography and selected portions are removed through etching. Such etching can include deep reactive ion etching (DRIE) and wet etching. In some examples, one or more intermediate metal, semiconducting, and/or insulating layers are deposited. The base wafer can be a doped semiconductor such as silicon. In some examples, ion implantation can be used to increase doping levels in regions defined by lithography. The spring systems can be defined in a substrate silicon wafer, which is then bonded to top and bottom cap wafers, also made of silicon. Encasing the spring systems in this manner allows the volume surrounding the mass to be evacuated. In some examples, a getter material such as titanium is deposited within the evacuated volume to maintain a low pressure throughout the lifetime of the device. This low pressure enhances the quality factor of the resonator. From the MEMS structure, conducting traces are deposited using metal deposition techniques such as sputtering or physical vapor deposition (PVD). These conducting traces electrically connect active areas of the MEMS structure to microelectronic circuits. Similar conducting traces can be used to electrically connect the microelectronic circuits to each other. The fabricated MEMS and microelectronic structures can be packaged using semiconductor packaging techniques including wire bonding and flip-chip packaging. 
     As used herein, the term “memory” includes any type of integrated circuit or other storage device adapted for storing digital data including, without limitation, ROM, PROM, EEPROM, DRAM, SDRAM, DDR/2 SDRAM, EDO/FPMS, RLDRAM, SRAM, flash memory (e.g., AND/NOR, NAND), memrister memory, and PSRAM. 
     As used herein, the term “processor” is meant generally to include all types of digital processing devices including, without limitation, digital signal processors (DSPs), reduced instruction set computers (RISC), general-purpose (CISC) processors, microprocessors, gate arrays (e.g., FPGAs), PLDs, reconfigurable compute fabrics (RCFs), array processors, secure microprocessors, and ASICs). Such digital processors may be contained on a single unitary integrated circuit die, or distributed across multiple components. 
     From the above description of the system it is manifest that various techniques may be used for implementing the concepts of the system without departing from its scope. In some examples, any of the circuits described herein may be implemented as a printed circuit with no moving parts. Further, various features of the system may be implemented as software routines or instructions to be executed on a processing device (e.g. a general purpose processor, an ASIC, an FPGA, etc.) The described embodiments are to be considered in all respects as illustrative and not restrictive. It should also be understood that the system is not limited to the particular examples described herein, but can be implemented in other examples without departing from the scope of the claims. 
     Similarly, while operations are depicted in the drawings in a particular order, this should not be understood as requiring that such operations be performed in the particular order shown or in sequential order, or that all illustrated operations be performed, to achieve desirable results. 
     References to axes as x, y, z, u, v, major, and/or minor axes are for the purpose of distinguishing between different axes. A different notation for any given axis, or different axis orientations, can be used without affecting the scope of the disclosure. 
     The terms first, second, third, fourth, fifth, sixth, seventh, eighth, ninth, etc. are used herein to distinguish between elements, components, etc. These terms when used herein do not imply a sequence or order unless clearly indicated by the context.