Patent Publication Number: US-2018031603-A1

Title: Systems and methods for detecting inertial parameters using a vibratory accelerometer with multiple degrees of freedom

Description:
CROSS REFERENCE TO RELATED APPLICATION 
     This application claims the benefit of copending, commonly-assigned U.S. Provisional Patent Application No. 62/367,626 filed Jul. 27, 2016, which is hereby incorporated by reference herein in its entirety. 
    
    
     FIELD OF THE INVENTION 
     This invention generally relates to systems and methods for detecting and measuring inertial parameters, such as acceleration. In particular, the systems and methods relate to multiple degrees of freedom inertial sensors with reduced common mode error. 
     BACKGROUND 
     Vibratory inertial sensors typically oscillate a sense structure at a known actuation frequency and can monitor perturbations of the sense structure to obtain measurements of inertial parameters or forces. Common mode error, a form of coherent interference resulting from package deformations, temperature gradients, parasitic capacitance, or other electrical noise, may affect the sensitivity of the inertial sensor. This may be particularly pronounced in sensors with multiple sensing signals, where common mode error in both signals becomes combined to produce an even greater error source. 
     SUMMARY 
     Accordingly, systems and methods are described herein for determining an inertial parameter with an inertial device having multiple degrees of freedom. A device comprises a first mass with a first degree of freedom and a second sense mass mechanically coupled to the first sense mass and with a second degree of freedom. A first time domain switch can be coupled to the first sense mass, and a second time domain switch can be coupled to the second sense mass. A drive structure can be configured to oscillate the first sense mass and the second sense mass in a differential frequency mode. The first time domain switch and the second time domain switch can each produce an electrical signal in response to oscillations of the first sense mass and the second sense mass. A processor in signal communication with the first time domain switch and the second time domain switch can be configured to determine an inertial parameter based in part on time intervals produced by the electrical signal. 
     In some examples, the first sense mass and the second sense mass of the inertial device can oscillate in the differential frequency mode, and the first time domain switch and the second time domain switch can produce a differential signal. In some examples, the inertial device can further comprise coupling springs mechanically coupled to the first sense mass to the second sense mass, and anchoring springs independently mechanically coupled to each of the first sense mass and the second sense mass and a central anchoring structure. The central anchoring structure can be rigidly coupled to a support structure. In some examples, the inertial parameter can be determined using a spring constant of the respective anchoring springs and a spring constant of the coupling springs to reduce the frequency of the differential frequency mode. 
     In some examples, the common mode frequency component of the electrical signal produced by the first time domain switch and the second time domain switch can be substantially eliminated from the differential signal. 
     In some examples, the first degree of freedom and the second degree of freedom can be in a vertical dimension. In some examples, the inertial parameter can be acceleration in the vertical dimension. 
     In some examples, the first time domain switch can further comprise a first electrode at a first radial distance of the first sense mass and a second electrode at a second radial distance of the first sense mass. As the first sense mass and the second sense mass oscillate at the differential frequency mode, the processor can be configured to detect a differential in capacitance of the first electrode and the second electrode. In some examples, the time intervals can be based in part on the times at which the differential in capacitance is equal to zero. In some examples, the first sense mass and the second sense mass raise and lower in the vertical dimension above the support structure. In some examples, the first sense mass and the second sense mass can oscillate in vertical torsional rotation about the central anchoring structure. 
     In some examples, the first degree of freedom and the second degree of freedom can be in a horizontal dimension. In some examples, the inertial parameter can be acceleration in the horizontal dimension. In some examples, the first sense mass can be mechanically coupled to the second sense mass with a frame, and the frame can oscillate in differential motion with the first sense mass and the second sense mass in-plane with the horizontal dimension. 
     In some examples, the first time domain switch can comprise a first set of capacitive teeth that can produce a first capacitive current, and the second time domain switch can comprise a second set of capacitive teeth that can produce a second capacitive current. The first capacitive current can be out of phase with the second capacitive current. In some examples, the differential signal can be a linear combination of the first capacitive current and the second capacitive current. 
     Another example described herein in a method for determining an inertial parameter using multiple degrees of freedom by oscillating a first sense mass in a first degree of freedom, oscillating a second sense mass mechanically coupled to the first sense mass in a second degree of freedom, coupling a first time domain switch to the first sense mass, and a second time domain switch to the second sense mass, producing an electrical signal in response to oscillations of the first sense mass and the second sense mass from each of the first time domain switch and the second time domain switch, and wherein a drive structure oscillates the first sense mass and the second sense mass at a differential frequency mode, and determining an inertial parameter based in part on time intervals produced by the electrical signal. 
     In some examples, the method can include producing a differential signal from the first sense mass and the second sense mass as the first sense mass and the second sense mass oscillate in the differential frequency mode. In some examples, the method can include mechanically coupling the first sense mass to the second sense mass with coupling springs, and mechanically coupling each of the first sense mass and the second sense mass to a central anchoring structure with anchoring springs. The central anchoring structure can be rigidly coupled to a support structure. In some examples, the method can include determining the inertial parameter using a spring constant of the respective anchoring springs and a spring constant of the coupling springs and reducing the frequency of the differential frequency mode. In some examples, the method can include eliminating a common mode frequency component of the electrical signal produced by the first time domain switch and the second time domain switch from the differential signal. 
     In some examples, oscillating the first sense mass in the first degree of freedom and oscillating the second sense mass mechanically coupled to the first sense mass in the second degree of freedom can include wherein the first degree of freedom and the second degree of freedom are in a vertical dimension. In some examples, determining the inertial parameter based in part on time intervals produced by the electrical signal can include wherein the inertial parameter is acceleration in the vertical dimension. In some examples, producing the electrical signal in response to oscillations of the first sense mass from the first time domain switch can include generating a capacitance from a first electrode at a first radial distance of the first sense mass, generating a capacitance from a second electrode at a second radial distance of the first sense mass, and as the first sense mass and the second sense mass oscillate at the differential frequency mode, detecting a differential in capacitance of the first electrode and the second electrode. 
     In some examples, determining the inertial parameter based in part on time intervals produced by the electrical signal can include wherein the time intervals are based in part on a plurality of times at which the differential in capacitance is equal to zero. In some examples, oscillating the first sense mass in the first degree of freedom and oscillating the second sense mass mechanically coupled to the first sense mass in the second degree of freedom can include raising and lowering the first sense mass and the second sense mass in the vertical dimension above the support structure. In some examples, oscillating the first sense mass in the first degree of freedom and oscillating the second sense mass mechanically coupled to the first sense mass in the second degree of freedom can include oscillating in vertical torsional rotation about the central anchoring structure. 
     In some examples, oscillating the first sense mass in the first degree of freedom and oscillating the second sense mass mechanically coupled to the first sense mass in the second degree of freedom can include wherein the first degree of freedom and the second degree of freedom are in a horizontal dimension. In some examples, the method can include determining the inertial parameter based in part on time intervals produced by the electrical signal can include wherein the inertial parameter is acceleration in the horizontal dimension. In some examples, the method can include producing a first capacitive current from the first time domain switch comprising a first set of capacitive teeth, and producing a second capacitive current from the second time domain switch comprising a second set of capacitive teeth, and wherein the first capacitive current can be out of phase with the second capacitive current. In some examples, determining the inertial parameter based in part on time intervals produced by the electrical signal can include determining a linear combination of the first capacitive current and the second capacitive current. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Further features of the subject matter of this disclosure, its nature and various advantages, will be apparent upon consideration of the following detailed description, taken in conjunction with the accompanying drawings, in which like reference characters refer to like parts throughout, and in which: 
         FIG. 1  depicts a conceptual model of a multiple degrees of freedom inertial sensor, according to an illustrative implementation; 
         FIG. 2  is a graph showing an example of a frequency response of a multiple degrees of freedom inertial sensor, according to an illustrative implementation; 
         FIG. 3  depicts a multiple degrees of freedom inertial sensor configured to oscillate in a vertical direction, according to an illustrative implementation; 
         FIG. 4  depicts the differential mode vertical movement of a multiple degrees of freedom inertial sensor, according to an illustrative implementation; 
         FIG. 5  depicts the common mode vertical movement of a multiple degrees of freedom inertial sensor, according to an illustrative implementation; 
         FIG. 6  depicts a multiple degrees of freedom inertial sensor configured for torsional oscillation in a vertical direction, according to an illustrative implementation; 
         FIG. 7  depicts the differential mode torsional movement of a multiple degrees of freedom inertial sensor, according to an illustrative implementation; 
         FIG. 8  depicts the common mode torsional rotational movement of a multiple degrees of freedom inertial sensor, according to an illustrative implementation; 
         FIG. 9  depicts two views of an inertial sensor with recessed moveable beams used for measuring perturbations and oscillations in a vertical direction, according to an illustrative implementation; 
         FIG. 10  depicts two views of an inertial sensor with recessed fixed beams used for measuring perturbations in a vertical direction, according to an illustrative implementation; 
         FIG. 11  depicts eight configurations of fixed and moveable beams which may be used in a multiple degrees of freedom inertial sensor to measure perturbations in a vertical direction, according to an illustrative implementation; 
         FIG. 12  depicts three cross views of the movement of one sense mass of a multiple degrees of freedom inertial sensor and electrodes for measuring perturbations in a vertical direction, according to an illustrative implementation; 
         FIG. 13  depicts three cross views of the movement of one sense mass of a multiple degrees of freedom inertial sensor and electrodes in a second configuration for measuring perturbation in a vertical direction, according to an illustrative implementation; 
         FIG. 14  depicts differential mode vertical movement of a multiple degrees of freedom inertial sensor with packaging deformations, according to an illustrative implementation; 
         FIG. 15  depicts an overhead view of a multiple degrees of freedom inertial sensor for measuring perturbations in a horizontal plane, according to an illustrative implementation; 
         FIG. 16  depicts three views, each showing a schematic representation of movable and fixed elements of a plurality of time-domain switches used to sense perturbations of a multiple degrees of freedom inertial sensor in a horizontal plane, according to an illustrative implementation; 
         FIG. 17  depicts a process for extracting inertial information from an inertial sensor, according to an illustrative implementation; 
         FIG. 18  depicts a conceptual schematic of a one degree of freedom sense mass oscillation, according to an illustrative implementation; 
         FIG. 19  is a graph showing the in phase and out of phase capacitive response to a sense mass oscillation produced by TDS structures of a multiple degrees of freedom inertial sensor, according to an illustrative implementation; 
         FIG. 20  depicts in phase and out of phase capacitive sense structures for sensing perturbations in a horizontal plane, according to an illustrative implementation; 
         FIG. 21  is a graph representing the relationship between analog signals derived from a multiple degrees of freedom inertial sensor and the displacement of a sense mass of a multiple degrees of freedom inertial sensor, according to an illustrative implementation; 
         FIG. 22  is a graph illustrating a current response to the displacement of a sense mass of a multiple degrees of freedom inertial sensor, according to an illustrative implementation; 
         FIG. 23  is a graph showing a rectangular-wave signal produced from zero-crossing times of the current signal depicted in  FIG. 24 , according to an illustrative implementation; 
         FIG. 24  is a graph showing time intervals produced from non-zero crossing reference levels, according to an illustrative implementation; 
         FIG. 25  is a graph showing the effects of an external perturbation on the output signal of the multiple degrees of freedom inertial sensor, according to an illustrative implementation; 
         FIG. 26  is a graph depicting capacitance as a function of the displacement of a sense mass of a multiple degrees of freedom inertial sensor, according to an illustrative implementation; 
         FIG. 27  is a graph depicting the first spatial derivative of capacitance as a function of the displacement of a sense mass of a multiple degrees of freedom inertial sensor, according to an illustrative implementation; 
         FIG. 28  is a graph depicting the second spatial derivative of capacitance as a function of the displacement of a sense mass of a multiple degrees of freedom inertial sensor, according to an illustrative implementation; 
         FIG. 29  is a graph depicting the time derivative of the capacitive current as a function of displacement of a sense mass of a multiple degrees of freedom inertial sensor, according to an illustrative implementation; 
         FIG. 30  is a graph depicting the displacement offsets of two sense masses as a result of common mode error, according to an illustrative implementation; 
         FIG. 31  is a graph depicting the results of differential sensing on the sensed displacement of a multiple degrees of freedom inertial sensor, according to an illustrative implementation; 
         FIG. 32  is a graph representing position of a sense mass relative to time, according to an illustrative implementation; 
         FIG. 33  is a graph representing velocity of a sense mass relative to time, according to an illustrative implementation; 
         FIG. 34  is a graph representing acceleration of a sense mass relative to time, according to an illustrative implementation; 
         FIG. 35  is a graph representing capacitance relative to angular position, according to an illustrative implementation; 
         FIG. 36  is a graph representing capacitive slope relative to angular position of a sense mass, according to an illustrative implementation; 
         FIG. 37  is a graph representing capacitive curvature relative to angular position of a sense mass, according to an illustrative implementation; 
         FIG. 38  is a graph representing capacitance relative to time and produced in response to oscillations of a sense mass, according to an illustrative implementation; 
         FIG. 39  is a graph representing capacitive slope relative to time and produced in response to oscillations of a sense mass, according to an illustrative implementation; 
         FIG. 40  is a graph representing capacitive curvature relative to time and produced in response to oscillations of a sense mass, according to an illustrative implementation; 
         FIG. 41  is a graph representing differential capacitance relative to time and produced in response to oscillations of a sense mass, according to an illustrative implementation; 
         FIG. 42  is a graph representing capacitive slope relative to time and produced in response to oscillations of a sense mass, according to an illustrative implementation; 
         FIG. 43  is a graph representing capacitive curvature relative to time and produced in response to oscillations of a sense mass, according to an illustrative implementation; 
         FIG. 44  is a graph representing capacitance relative to the vertical position of a sense mass, according to an illustrative implementation; 
         FIG. 45  is a graph representing capacitive slope relative to the vertical position of a sense mass, according to an illustrative implementation; 
         FIG. 46  is a graph representing capacitive curvature relative to the vertical position of a sense mass, according to an illustrative implementation; 
         FIG. 47  is a graph representing capacitance relative to time and produced in response to oscillations of a sense mass, according to an illustrative implementation; 
         FIG. 48  is a graph representing capacitive slope relative to time and produced in response to oscillations of a sense mass, according to an illustrative implementation; 
         FIG. 49  is a graph representing capacitive curvature relative to time and produced in response to oscillations of a sense mass, according to an illustrative implementation; 
         FIG. 50  depicts a flow chart of a method for extracting inertial parameters from a nonlinear periodic signal, according to an illustrative implementation; 
         FIG. 51  depicts a flow chart of a method for determining transition times between two values based on a nonlinear periodic signal, according to an illustrative implementation; and 
         FIG. 52  depicts a flow chart of a method for computing 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 reducing common mode error when detecting and measuring inertial parameters using a vibratory accelerometer. 
     Vibratory accelerometers use the measured perturbations of an oscillating sense mass to determine inertial parameters and forces acting on a sensor. These perturbations may be physical perturbations of the sense mass from a neutral equilibrium, and may be converted to analog electrical signals as a result of the electro-mechanical nature of a sensing system. Any accelerometer may be sensitive to temperature changes, long-term mechanical creep, environmental vibrations, packaging deformations, parasitic capacitance, drift in bias voltages, drift in any internal voltage references, and other environmental or electrical noise sources. In accelerometers, these error sources will affect the accuracy of the sensor, thus reducing its ability to measure inertial parameters and inertial forces such as an input acceleration. 
     One form of error that affects accelerometers is common mode error. Common mode error is a form of interference, for example, coherent interference, where an error exists equally and in phase on multiple signal paths, and is therefore not easily distinguished or isolated from the desired signal information, since combining signal paths together will simply compound or amplify the error. Examples include temperature changes, long-term mechanical creep, environmental vibrations, packaging deformations, parasitic capacitance, drift in bias voltages, drift in any internal voltage references, ground loops, and other environmental or electrical error or noise sources that result in systematic errors. 
     One way to reduce the affects of these error sources is to employ sensing techniques that produce multiple signals as a result of a single motion in such a way that their linear combination will in fact remove or detect the systematic error present in both signals. One of these techniques is “differential sensing,” where computing the difference between two signals results in the elimination of common mode error present in both signals, leaving a scalar multiple of the “true” signal without common mode error. For example, two signals may be generated so that a first signal is phase offset from the second signal by 180°. These “anti-phase” signals may then be subtracted from each other to remove common mode error. In another example, two signals may be generated that are inverses of each other using positive or negative-biased electrodes, and then may be subtracted from each other to remove common mode error. Any other sensing technique that produces a difference or “differential” between two signals may be used to implement differential sensing. 
     Common mode error may also occur in a specific frequency range of a sense mass oscillation in a vibratory accelerometer. While differential sensing techniques may be employed, a downside of a vibratory accelerometer with a single sense mass is that there is only a single motion from which to generate electrical signals in response to perturbations, and there is only a single resonant frequency response of the sense mass. Thus the frequency range at which inertial parameters are measured may in fact also be the frequency range in which common mode error primarily resides. In this case, differential sensing techniques may not be able to fully remove the common mode error signal from the measured output signal. 
     In multiple degrees of freedom inertial sensor, however, more than one sense mass may be coupled together, producing multiple detectable motions in response to a single external perturbation or acceleration. The motion of each sense mass is a degree of freedom of the inertial sensor system. In the context of a vibratory accelerometer in which the sense masses are driven into oscillation, each degree of freedom will correspond to an additional normal mode frequency response of the system. For example, in a two degree of freedom sense structure system with two sense masses that are both actuated at a drive frequency, the system will respond at a range of frequencies that are a function of the drive frequency, the mass of each sense mass, the coupling between the masses, and other structural factors. However, the system will have two “natural frequency modes” which correspond to the eigenvalue solutions of the equations of motion of the system. These natural frequency modes, which are the frequencies at which the system would oscillate in the absence of driving forces, will be resonant frequencies of the two degree of freedom system. Oscillations at these frequencies will amplify the motion of both sense masses, resulting in amplitude peaks in the frequency response of the system. For an N-degree of freedom oscillating system, there will be N corresponding natural modes for each of the N eigenvalue solutions to the system&#39;s equations of motion (where N is any positive integer). 
     These natural modes will correspond to both a characteristic frequency and a characteristic physical motion of the sense masses. Again, in the example of a typical two-degree of freedom system, one natural mode, a “low” natural mode, will generally correspond to in-phase, common mode motion of the two sense masses, wherein both masses move together with the same amplitude in the same direction. In a typical system, this “low” natural mode will be at a lower energy or frequency than a second “high” natural mode. This second “high” natural mode will generally correspond to anti-phase, differential motion of the two sense masses, when both masses will move with the same amplitude in opposite directions, 180° out of phase with each other. A typical N-degree of freedom system will have this same minimum “low” natural mode, where all N masses move in-phase with each other, and a maximum “high” natural mode, where the maximum number of alternating pairs of the N masses move anti-phase with each other. For example, in a typical four degree of freedom system, the “high” natural mode will correspond to motion in which masses  1  and  2  move out of phase with each other, masses  2  and  3  move out of phase with each other, and masses  3  and  4  move out of phase with each other. 
     However, while the natural frequency modes will always correspond to characteristic physical motions of the sense masses, it is possible to introduce structural forces to the system to alter the typical correspondence described above. For example, one may create a system where the differential, anti-phase motion of sense masses actually corresponds to the lower energy, lower frequency natural mode response of the multiple degrees of freedom system. In this case, the common mode, or in phase motion of the sense masses would in fact be at the higher energy, higher frequency natural mode response. 
     The natural frequency modes of a multiple degrees of freedom vibratory accelerometer are useful because they allow for the isolation of common mode error to the in-phase response of the accelerometer, and detection of inertial parameters primarily at a second, anti-phase frequency response in which common mode error can be eliminated via differential movement of the sense masses. Isolating sensing to the differential mode thus allows for the elimination of common mode errors when measuring inertial parameters. The multiple natural mode frequency responses of the system also allow for more flexibility in engineering the system, since it allows one to tune the in-phase frequency response to a frequency range of common mode error, and tune the out-of-phase, measurement frequency response to the desired sensing range, which may in fact be at the first, lower frequency mode response of the system. 
     Thus sensing of acceleration may be done primarily at the differential frequency mode, in which the sense masses of the multiple degrees of freedom accelerometer move anti-phase to each other in the lower natural frequency mode response. In this mode, the common mode error affecting each sense mass will be eliminated from the signal by subtracting or combining the signals from each sense mass. Since the signals will be 180° out of phase with each other, any common mode error present in both signals will be eliminated from the resulting combined output signal, leaving only the desired signal reflecting the sense mass&#39; displacement. 
     In vibratory accelerometers, because the physical movement of the sense mass translates to its output analog signal, the physical frequency of oscillation of the sense mass has a direct relation to the sensitivity of the inertial sensor. For accelerometers, the ratio of the linear displacement of a sense mass to the input acceleration, which describes the ability of a signal (denoted S accel ) produced by the sense mass to detect acceleration has the general relation: 
     
       
         
           
             
               
                 
                   
                     S 
                     accel 
                   
                   ∝ 
                   
                     1 
                     
                       f 
                       s 
                       2 
                     
                   
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
     where f S  is the frequency of oscillation of the sense mass. As can be appreciated, in order to increase the sensitivity of the accelerometer, one would ideally minimize the value of f S . Thus by introducing structural forces into the system that make the differential, anti-phase motion of the sense masses correspond to the lower frequency mode response, one may accomplish differential sensing, eliminate common mode noise, and still preserve the sensitivity of the accelerometer. 
       FIG. 1  depicts a conceptual model of a multiple degrees of freedom inertial sensor, according to an illustrative implementation.  FIG. 1  will demonstrate the geometric choices available to reject or eliminate the common mode response of the inertial sensor from the differential mode response which is used to measure inertial parameters. 
     Differential sensing may first be achieved by mechanically driving the two sense masses  110  and  112  in their natural differential frequency mode, meaning in opposite directions at the same amplitude, as indicated by the arrows  126   a  and  126   b  respectively. Springs  106   a ,  106   b , and  108  may be configured such that this differential frequency mode is the first, lower frequency mode response of the system. Springs  106   a ,  106   b , and  108  may be configured such that the common mode motion, in which sense masses move in-phase in the same direction, is the second, higher frequency mode response of the system. The sense masses  110  and  112  may be suspended in the z axis from anchors  102  and  104  by anchoring springs  106   a  and  106   b  above a bottom layer (not shown) of the multiple degrees of freedom inertial sensor. A coupling spring  108  may mechanically couple the two sense masses  110  and  112  together. Springs  106   a ,  106   b  and  108  may be substantially compliant in only the x axis, as shown in  FIG. 1 . Sense structures, shown in  FIG. 1  as time-domain switch (“TDS”) structures  120   a  and  120   b  can convert the oscillations of sense masses  110  and  112  into analog electrical signals derived from the displacement of the sense masses  110  and  112 . The TDS structures  120   a  and  120   b  are each composed of one set of teeth  122  and  118  coupled to the sense masses  110  and  112  respectively, and a second set of teeth  114  and  116  rigidly coupled to the bottom layer of the multiple degrees of freedom inertial sensor. Each mass may be driven anti-phase to each other with independent drive structures (not shown). 
     The drive structures described herein may be capacitive comb drives. The capacitive comb drives may have one stationary set of teeth rigidly coupled to the bottom layer of a multiple degrees of freedom inertial sensor, while a second, interdigitated set is connected to the sense mass, such as sense mass  110  or  112 . The drive structures may also be any device capable of driving the sense masses into oscillation. The electrical signal controlling the drive structures may be a constant electrical signal generated through feedback circuitry to maintain the differential frequency mode of the sense masses  110  and  112 . The feedback circuitry may also adjust a drive voltage to the drive structures until the amplitude of the sense masses  110  and  112  oscillation reaches a desired setpoint. This setpoint may be an amplitude associated with a resonant frequency or natural mode frequency of the multiple degrees of freedom inertial sensor. This setpoint may be an amplitude associated with a differential frequency mode response of the multiple degrees of freedom inertial sensor, which occurs at the first, lower frequency mode response of the system. Another example of a control signal may be a periodic “pinged” signal that is turned on and off, creating a stepped electrostatic force to initiate harmonic oscillation. The “pinged” signal may be coordinated between drive structures on opposite sides of the sense masses  110  and  112  in the x-axis, to create a “push/pull” electrostatic force. The drive structures may be powered on or off in response to a user initiating or closing an application on a mobile device. Start up times of oscillating inertial devices can range from 10 milliseconds to multiple seconds, depending on the quality factor of the sense masses and other design factors. 
     In combination with the differential motion of the sense masses  110  and  112 , the differential sensing of acceleration may also be achieved with in and out of phase TDS structures, as shown at  120   a  and  120   b . The in-phase TDS structure has teeth  122  and  114  that are in alignment in their neutral position, meaning that when the sense mass  110  has a net zero force acting on it, the teeth  122  are at a minimum distance in the y-direction from the teeth  114 , as shown in  FIG. 1 . The out of phase TDS structure has teeth  118  and  116  that are anti-aligned in their neutral position, meaning that when the sense mass  112  has a net zero force acting on it, the teeth  118  are at a maximum distance in the y-direction from teeth  116 . As sense masses  110  and  112  oscillate differentially at 180° out of phase with each other in the directions indicated by arrows  126   a  and  126   b , the aligned, or in-phase TDS structure  120   a  and the out of phase TDS structure  120   b  will each produce signals that are themselves differential and 180° out of phase with each other. The teeth  122 ,  114 , and  118  and  116  may be configured to produce signals any phase shift angle from each other as desired. The resulting analog signals will thus be produced by both differential motion as shown at  126   a  and  126   b , and differential detection. The analog signals from teeth  120   a  and  120   b  may be linearly combined with each other as desired, and will reject common mode error both from the sense masses  110  and  112 &#39;s physical motion, and from the electrical sensing of their displacement. The signals produced by in and out of phase TDS structures are discussed in more detail with reference to  FIGS. 21-22 . These structures as shown in  FIG. 1  may also be replaced by any of the sensing structures described herein, and for example, in reference to  FIGS. 8-12, 15 and 21-22 . 
     Anchoring springs  106   a  and  106   b , as well as coupling spring  108  and any of the springs described herein each have an inherent value called a spring constant. A spring constant is an intrinsic property of a spring, which describes its relative compliance to outside forces. Thus springs with low spring constants expand or comply more to outside forces than springs with high spring constants. The spring constants of springs  106   a ,  106   b  and  108  and any of the springs described herein may each be defined purely by the geometry and material of the springs. The stiffness of the springs  106   a ,  106   b  and  108  and any of the springs described herein can be affected by temperature. Thus, changes in ambient or sensor temperature can result in changes in spring stiffness, resulting in changes in resonant frequency of the structure  100 . Springs  106   a ,  106   b  and  108  may be comprised of a uniform isotropic material, such as doped or undoped silicon. Springs may also have varying widths, segments, segment lengths, and moments of inertia to tailor portions of the springs and achieve the desired spring constants. Springs  106   a ,  106   b  and  108  may be configured to lower the frequency associated with differential motion of the sense masses, such that in a two degree of freedom system the first natural frequency mode response corresponds to differential motion, while the second natural frequency mode response corresponds to common mode, in-phase motion. 
     The natural frequencies of the two degree of freedom system as shown in  FIG. 1  will be dependent on the masses of the two sense masses  110  and  112 , denoted M 1  and M 2 , the spring constants of the anchoring springs  106   a  and  106   b , denoted k 1  and k 2 , and the spring constant of the coupling spring, denoted k C . In a typical example where M 1 =M 2  and k 1 =k 2 , the two natural frequencies of the system shown in  FIG. 1  might be: 
     
       
         
           
             
               
                 
                   
                     ω 
                     D 
                   
                   = 
                   
                     
                       
                         
                           k 
                           1 
                         
                         + 
                         
                           2 
                            
                           
                             k 
                             C 
                           
                         
                       
                       
                         M 
                         1 
                       
                     
                   
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
             
               
                 
                   
                     ω 
                     C 
                   
                   = 
                   
                     
                       
                         k 
                         1 
                       
                       
                         M 
                         1 
                       
                     
                   
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
     where ω D  is typically the higher differential mode which would normally correspond to anti phase motion of the sense masses  110  and  112 , and ω C  is typically the lower common mode corresponding to in phase motion of the sense masses  110  and  112 . These are the classic frequency solutions to the two-degree of freedom system shown in  FIG. 1 , and are given here as examples of the dependency of the frequencies on each of the variables in the system shown in  FIG. 1 . However, it is possible to introduce structural forces into the system shown in  FIG. 1  such that the differential motion of the sense masses corresponds to the lower energy, lower frequency natural mode response. The system shown in  FIG. 1  may also have different mass values for sense mass  110  and  112 , and different values for the spring constants of  106   a  and  106   b . The system shown in  FIG. 1  may have more than two sense masses. In all of these cases, the natural frequencies of the system will still depend on the spring constants of the coupling or anchoring springs, and the masses of the sense masses. For any N degree of freedom system, the N natural modes will also depend on the masses of the N sense masses and the spring constants of all of the coupling and anchoring springs. These variables are all values that can be fixed and determined through fabrication of the multiple degrees of freedom inertial sensor, meaning that the natural modes will also be fixed. 
     As shown in Equations (2) and (3), the values of the common mode and differential mode frequencies of oscillation may be determined by selecting the stiffness of the coupling and anchoring springs, as well as the masses of the sense masses  110  and  112 . The differential frequency mode may be between 500 and 20,000 Hz, and is preferably 5,000 Hz. 
       FIG. 2  is a graph showing an example of a frequency response of a multiple degrees of freedom inertial sensor, according to an illustrative implementation. The x axis shows the drive frequency, while the y axis shows the displacement amplitude of the sense masses. In the low frequency region  210 , the sense mass response is approximately linear. As shown in  FIG. 2 , the low frequency differential mode response produces a peak in amplitude at  202 , while the higher frequency common mode response produces a second peak in the amplitude response at  204 . In a two-degree of freedom system, these two amplitude peaks correspond to the two natural mode frequencies of the system. Detection and sensing of acceleration occurs at the lower frequency, differential mode response at  202 . The distance between these two normal mode responses, shown at  212 , may be adjusted by changing the spring constants or mass values of the multiple degrees of freedom inertial sensor. The spring constants and mass values will also define the amplitudes and frequencies of peaks  202  and  204 . To isolate the differential frequency response at  202  from the common mode response at  204 , the distance  212  may be increased. The width of the peaks, shown at  206  and  208 , may be defined by the Q factor of the multiple degrees of freedom inertial sensor. 
       FIG. 2  is an example of the frequency response of a two degree of freedom inertial sensor with two sense masses, however there may be any number of sense masses, where additional degrees of freedom will result in the same number of additional peaks in the amplitude response. Thus an N degree of freedom inertial sensor will have N number of corresponding peaks across the full frequency spectrum in the sense masses amplitude response. For an N degree of freedom inertial sensor, there will be a lowest common mode frequency, and a higher differential mode frequency response. 
       FIG. 3  depicts a multiple degrees of freedom inertial sensor configured to oscillate in a vertical direction, according to an illustrative implementation.  FIG. 3  is an implementation of the conceptual diagram shown in  FIG. 1  for z axis oscillation and sensing of z axis acceleration for a two degree of freedom system.  FIG. 3  includes a central anchor  320  rigidly coupled to a bottom layer  326  of the multiple degrees of freedom inertial sensor, a first sense mass  310  coupled to the anchor  320  with a first pair of anchoring springs  316   a  and  316   b , and a second sense mass  312  coupled to the central anchor  320  via a second pair of anchoring springs  314   a  and  314   b . The sense mass  310  is mechanically coupled to the sense mass  312  via coupling springs  318 . The sense masses  310  and  312  may be suspended in the z axis above the bottom layer  326 . Sense structures  302  and  304  rigidly coupled to the bottom layer  326  may detect the sense masses  310  and  312  motion in the z axis and convert it to an analog electrical signal. 
     The multiple degrees of freedom inertial sensor  300  comprises three layers: a device layer containing the features depicted at  302 ,  304 ,  306 ,  308 ,  310 ,  312 ,  314   a ,  314   b ,  316   a ,  316   b ,  318 ,  320 , and  322 , a bottom layer  326 , and a cap layer (not shown). The bottom layer  326  and the cap layer may be made from different wafers than the device layer. One or more of the features of the device layer may be made from the wafers containing the bottom layer  326  and/or the cap layer. The space between the bottom layer  326  and the cap layer may be at a constant pressure below atmospheric pressure. The space between the bottom layer  326  and the cap layer may be at partial vacuum. A getter material such as titanium or aluminum may be deposited on the interior of the space to maintain reduced pressure over time. 
     The anchoring springs  314   a ,  314   b ,  316   a  and  316   b  are shown in  FIG. 3  as rectangular structures hinging sense masses  310  and  312  to the central anchor  320 . These springs may also contain “u” bends, may be serpentine or in any configuration that allows the sense masses  310  and  312  to rotationally oscillate in the z direction about the central anchor  320 . This motion is described in further detail with reference to  FIG. 4-5 . The coupling springs  318  are shown with “u” bends, but may be serpentine or in any configuration that restricts the motion of the sense masses  310  and  312  in they or x direction so that they oscillate in the z-axis. The springs  314   a ,  314   b ,  316   a ,  316   b  and  318  may also be configured to promote the differential motion of sense masses  310  and  312  over their common mode motion, where an example of the differential motion is shown in  FIG. 4 , and is the first, lower frequency mode, and an example of the common mode motion is shown in  FIG. 5 , and is the second, higher frequency mode. 
     The motion of the sense masses  310  and  312  (as described in  FIG. 4  and  FIG. 5 ) may have both a rotational, torsional component and an out-of-plane bending component of motion in the z axis. The anchoring springs  314   a ,  314   b ,  316   a  and  316   b  may have different stiffness responses to the torsional movement as opposed to the bending movement. The springs  314   a ,  314   b ,  316   a  and  316   b  may have a lower effective spring constant in response to torsional motion, and a higher effective spring constant in response to bending motion. The common mode motion may have a larger bending component of motion than torsional component of motion, whereas the differential motion may have a larger torsional component of motion than bending component of motion. As a result of the variable stiffness of the springs in response to these two components of motion, the differential motion may be associated with a lower energy, lower frequency response of the system, while the common mode, in-phase motion may be associated with a higher energy, higher frequency response of the system. Other structural forces may also increase the stiffness response of the system to the common mode motion, effectively increasing the energy and frequency response associated with common mode motion, while lowering the frequency response associated with differential motion. 
     The sense masses  310  and  312  are shown in  FIG. 3  as rectangles with removed interiors. Sense masses  310  and  312  may also be in any topography that allows for symmetric, differential motion, as described in  FIG. 4 . The location of the removed mass (shown as the removed interior of sense masses  310  and  312 ) may be chosen such that the center of mass of the sense masses  310  and  312  are located away from the anchor  320 . Thus, more mass is left towards the ends of the sense masses  310  and  312  that are shown in  FIG. 3  interfacing with the sense structures  302  and  304  respectively, placing the center of mass  332  and  330  away from the anchor  320 . In addition to removing mass from the sense masses  310  and  312  to place the center of mass  332  and  330 , it is possible to adjust the thickness of the sense masses in the z-dimension. The locations  332  and  330  promote differential motion of the sense masses  310  and  312 , as well as the out-of-plane motion in the z direction. The center of mass of the sense masses  310  and  312  may thus be centered in their y dimension, as shown at  332  and  330  and also offset in their x dimension, as shown at  310  and  312  to encourage the desired oscillation motion. The center of mass may also be placed anywhere that may produce differential motion of the two sense masses  310  and  312 . 
     Sense structures are shown with a first set of teeth at  306  and  308  coupled to the sense masses  310  and  312  respectively. A second set of teeth  304  and  302  are shown as interdigitated, for example at  322 , with teeth  306  and  308 , and rigidly coupled to the bottom layer  326  of the multiple degrees of freedom inertial sensor. The teeth of these sense structures may be configured such that the analog signals produced by one set will be out of phase with the other, thus differentially sensing the oscillations of sense masses  310  and  312 . These sense structures may be any of the TDS structures described herein, for example those described in further detail with reference to  FIGS. 9-13, 16 and 19-20 . These structures may also be any capacitive, optical or general means for producing an electrical signal in response to the sense masses  310  and  312  displacement and oscillation. The sense structures may also be parallel plate capacitors formed between electrodes deposited in the bottom layer  326  below each of the sense masses  310  and  312 , and the bottoms of the sense masses  310  and  312  themselves, such that movement of the sense masses  310  and  312  in the z dimension are translated to changes in capacitance between the sense masses  310  and  312  and the bottom layer  326 . Differential driving of the sense masses and differential sensing of their oscillation will substantially eliminate common mode error from the electrical signals produced by the TDS structures. 
       FIG. 4  depicts the differential mode vertical movement of a multiple degrees of freedom inertial sensor, according to an illustrative implementation. The movement shown in  FIG. 4  is that of the first, lower natural frequency mode of the two-degree of freedom system. A central anchor shown at  408  may connect the sense masses  404  and  406  to an anchoring structure (not shown). The central anchor as shown at  408  may be comprised of the anchoring springs  314   a ,  314   b ,  316   a  and  316   b  and coupling springs  318  as shown in  FIG. 3 . At  400 , the sense masses  404  and  406  have free ends  410  that are at a positive altitude angle as indicated at  402 . At any given moment in their oscillation, the altitude angle of sense mass  404  may be the same as that of sense mass  406 . The free ends  410  of sense masses  404  and  406  may be coupled to TDS structures or any other sense structure to convert their displacement to an analog electrical signal.  400  depicts the maximum positive displacement of the sense masses  404  and  406 . 
     At  420 , the free ends  410  of both sense masses  404  and  406  have moved in the negative z-direction from their positions indicated in  400 , rotating about the central anchor  408  and reducing the altitude angle as shown at  422 . At  440 , the free ends  410  of both sense masses  404  and  406  have moved further in the negative z direction, and are in the horizontal plane as indicated at  442 . In this position  440 , sense masses  404  and  406  will be parallel to a bottom layer of the multiple degrees of freedom inertial sensor (not shown). This may be the neutral position of the sense masses  404  and  406 , meaning that in the absence of drive forces they would be at this position  440 . 
     At  460 , the free ends  410  of both sense masses  404  and  406  have moved still further in the negative z direction, and now form a negative altitude angle as indicated at  462 .  460  represents a minimum displacement of the free ends  410 , meaning that the free ends  410  at their lowest point in the z-axis. 
     The sequence of positions  400 ,  420 ,  440  and  460  represent one half cycle of the sense masses  404  and  406  vertical oscillation. To complete the full cycle, the sense masses  404  and  406  will move in the positive z direction from minimum position  460 , going from position  460 , to  440 , to  420 , and reaching their maximum displacement again at  400 . The positions shown in  FIG. 4  are intended as representative models, and may be exaggerated for clarity. 
       FIG. 5  depicts the common mode vertical movement of a multiple degrees of freedom inertial sensor, according to an illustrative implementation. The movement shown in  FIG. 5  is that of the second, higher natural frequency mode of the two-degree of freedom system. This motion is the disfavored motion of sense masses shown in  FIG. 3 , and may contain common mode error in the resulting output signal produced by this motion. A central anchor shown at  508  may connect the sense masses  502  and  504  to an anchoring structure (not shown). The central anchor as shown at  508  may be comprised of the anchoring springs  314   a ,  314   b ,  316   a  and  316   b  and coupling springs  318  as shown in  FIG. 3 . These springs may be configured to disfavor the oscillation response of sense masses  502  and  504  as shown in  FIG. 5 . At  500 , the sense mass  504  is at a positive altitude angle as shown at  512   a , while sense mass  502  is at a negative altitude angle as shown at  512   b . These angles may be equal and opposite of each other. At any give moment in their oscillation, the magnitude of the altitude angle of sense mass  502  is the same as that of sense mass  504 . The free ends  510  of sense masses  502  and  504  may be coupled to TDS structures or any other sense structure to convert their displacement to an analog electrical signal. 
     At  520 , the free ends  510  of the sense masses  502  and  504  have moved in the positive and negative z directions respectively, and are in the horizontal plane as indicated at  522 . In this position  520 , sense masses  502  and  504  will be parallel to a bottom layer of the multiple degrees of freedom sense (not shown). This may be the neutral position of the sense masses  502  and  504 , meaning that in the absence of drive forces they would be at this position  520 . 
     At  540 , the free end  510  of sense mass  502  has moved in the positive z direction, while the free end  510  of sense mass  504  has moved in the negative z direction. Thus sense mass  502  now makes a positive altitude angle as indicated at  542   b , while sense mass  504  makes a negative altitude angle as indicated at  542   a . Finally, at  560 , after further movement of the free end  510  of sense mass  502  in the positive z direction, and further movement of the free end of  510  of sense mass  504  in the negative z direction, the sense mass  504  forms a negative altitude angle as shown at  562   b , while sense mass  502  forms a positive altitude angle as shown at  562   a.    
     Thus in the common mode motion as shown in  FIG. 5 , both sense masses  502  and  504  move as a single mass as they rotate about the central anchor  508 . There is no differential produced by the motion of the two sense masses  502  and  504 . This is not the preferred motion of the sense masses  502  and  504 , and common mode error may be isolated to the frequency response associated with this common mode motion. 
     The sequence of positions  500 ,  520 ,  540  and  560  represent one half cycle of the sense masses  504  and  506  vertical oscillation. To complete the full cycle, the sense masses  504  and  506  will move in the z direction, going from position  560 , to  540 , to  520 , and back to  500 . The positions shown in  FIG. 5  are intended as representative models, and may be exaggerated for clarity. 
       FIG. 6  depicts a multiple degrees of freedom inertial sensor configured for torsional oscillation in a vertical direction, according to an illustrative implementation.  FIG. 6  is another implementation of the conceptual diagram shown in  FIG. 1  for z axis oscillation and sensing of z axis acceleration for a two degree of freedom system.  FIG. 6  includes two central anchors  622  and  628  mechanically coupled to sense masses  612  and  610  respectively and rigidly coupled to a bottom layer  624  of the multiple degrees of freedom inertial sensor  600 . Anchoring springs  618   a  and  618   b  mechanically connect the sense mass  612  to the central anchor  622 , while a second set of anchoring springs  618   c  and  618   d  mechanically connect the sense mass  610  to the central anchor  628 . A coupling spring  614  mechanically couples sense mass  610  to sense mass  612 . Sense structures  602  and  604  may detect the sense masses  610  and  612  torsional motion in the z axis and convert it to an analog electrical signal. Mechanically coupled may mean a physical connection, such as a spring, between elements of the multiple degrees of freedom inertial sensor such that forces are conveyed between them. 
     The multiple degrees of freedom inertial sensor  600  comprises three layers: a device layer containing the features depicted at  602 ,  604 ,  606 ,  608 ,  610 ,  612 ,  614 ,  616 ,  618   a ,  618   b ,  618   c ,  618   d ,  620 ,  622 , a bottom layer  624 , and a cap layer (not shown). The bottom layer  624  and the cap layer may be made from different wafers than the device layer. One or more of the features of the device layer may be made from wafers containing the bottom layer  624  and/or the cap layer. The space between the bottom layer  624  and the cap layer may be at a constant pressure below atmospheric pressure. The space between the bottom layer  624  and the cap layer may be at partial vacuum. A getter material such as titanium or aluminum may be deposited on the interior of the space to maintain reduced pressure over time. 
     The anchoring springs  618   a ,  618   b ,  618   c  and  618   d  are shown in  FIG. 6  as rectangular structures hinging sense masses  610  and  612  to the central anchors  628  and  622 , respectively. These springs may also contain “u” bends, may be serpentine or in any configuration that allows the sense mass  610  to torsionally oscillate in the z direction about an x axis whose origin is centered at  616 . This motion is described in further detail with reference to  FIG. 7-8 . The coupling spring is shown with a bend at  616 , but may have “u” bends, be serpentine, or in any configuration that restricts the motion of the sense masses  610  and  612  to promote the differential motion of the sense masses  610  and  612  over their common mode motion. An example of the differential motion of sense masses  610  and  612  is shown in  FIG. 7 , and is the first, lower natural frequency mode of the system, while an example of the common mode motion is shown in  FIG. 8 , and is the second, higher natural frequency mode of the system. 
     The motion of the sense masses  610  and  612  (as described in  FIG. 7  and  FIG. 8 ) may have both a rotational, torsional component and an out-of-plane bending component of motion in the z axis. The anchoring springs  618   a ,  618   b ,  618   c  and  618   d  (collectively  618 ) may have different stiffness responses to the torsional movement as opposed to the bending movement. The springs  618  may have a lower effective spring constant in response to torsional motion, and a higher effective spring constant in response to bending motion. The common mode motion may have a larger bending component of motion than torsional component of motion, whereas the differential motion may have a larger torsional component of motion than bending component of motion. As a result of the variable stiffnesses of the springs in response to these two components of motion, the differential motion may be associated with a lower energy, lower frequency response of the system, while the common mode motion may be associated with a higher energy, higher frequency response of the system. The distance between springs  618   a  and  618   b  from springs  618   c  and  618   d  may also increase the stiffness response to the bending component of motion, pushing the common mode, out-of phase motion into the higher natural frequency mode response. Other structural forces may also increase the stiffness response of the system to the common mode motion, effectively increasing the energy and frequency response associated with common mode motion, while lowering the frequency response associated with differential motion. 
     The sense masses  610  and  612  are shown in  FIG. 6  as rectangles with removed interiors. Sense masses  610  and  612  may also be in any topography that allows for symmetric, differential motion at the first, lower natural frequency mode. The center of mass of the sense mass  610  may be located at  630  and  632 . The location of the removed mass (shown as the removed interior of sense masses  610  and  612 ) may be chosen such that the center of mass of the sense masses  610  and  612  are located away from their respective anchors  628  and  622 . Thus, more mass is left towards the negative y direction of sense mass  612 , whereas more mass is left towards the positive y direction of sense mass  610 . In addition to removing mass from the sense masses  610  and  612  to place the center of mass  632  and  630 , it is possible to adjust the thickness of the sense masses in the z-dimension. The locations  632  and  630  promote differential motion of the sense masses  610  and  612 , as well as the out-of-plane torsional rotational motion shown in  FIG. 7 . 
     Sense structures  634  and  636  are shown with a first set of teeth at  606  and  608  coupled to the sense masses  610  and  612  respectively. A second set of teeth  602  and  604  are shown as interdigitated with the first set of teeth  606  and  608 , and rigidly coupled to the bottom layer  624  of the multiple degrees of freedom inertial sensor. The teeth of these sense structures may be configured such that the analog electrical signals produced by one set will be out of phase with the other, thus differentially sensing the oscillations of sense masses  610  and  612 . These sense structures may be TDS structures, as describe in further detail with reference to  FIGS. 9-10 and 16 . The sense structures may also be parallel plate capacitors formed between electrodes deposited in the bottom layer  624  below each of the sense masses  610  and  612 , and the bottoms of the sense masses  610  and  612  themselves, such that movement of the sense masses  610  and  612  in the z dimension are translated to changes in capacitance between the sense masses  610  and  612  and the bottom layer  624 . These structures may also be any capacitive, optical or general means for producing an electrical signal in response to the sense masses  510  and  512  displacement. 
       FIG. 7  depicts the differential mode torsional movement of a multiple degrees of freedom inertial sensor, according to an illustrative implementation. The movement shown in  FIG. 7  is that of the first, lower natural frequency mode of the two-degree of freedom system. A central anchor shown at  702  may couple the sense masses  704  and  706  to an anchoring structure (not shown). The central anchor as shown at  702  may be comprised of anchoring springs  618   a ,  618   b ,  618   c  and  618   d , as well as coupling spring  614  as shown in  FIG. 6 . Both sense masses  704  and  706  will oscillate in vertical torsional rotation about a central anchor  702  and axis  720 . 
     At  700 , the free end  714  of sense mass  706  forms a positive altitude angle as indicated at  716 . The other free end  712  of sense mass  706  makes an equal and opposite altitude angle as indicated at  718 . Thus the sense mass  706  is symmetrically “twisted” or rotated about its central x axis in the vertical or z direction. The sense mass  704  is symmetrically “twisted” about the central axis  720  to mirror the motion of sense mass  706 . Thus the corresponding free end  710  of sense mass  704  to the free end  714  of sense mass  706  makes an equal and opposite altitude angle as indicated at  722 . This angle  722  is the same as angle  718 . The other free end  708  makes a positive altitude angle as indicated at  724 . This angle  724  is the same as angle  716 . Thus, throughout the vertical rotational torsional oscillation of sense masses  704  and  706 , the free end  710  may form the same altitude angle as the free end  712 , while the free end  708  will form the same altitude angle as the free end  714 .  700  represents the maximum displacement of free ends  714  and  708 , and the minimum displacement of free ends  710  and  712 . 
     At  740 , free ends  710  and  712  have moved in the positive z direction, forming altitude angles  746  and  744  respectively. Free ends  708  and  714  have moved in the negative z direction, forming altitude angles  748  and  742  respectively. Thus the angles  742 ,  744 ,  746  and  748  are all smaller in magnitude than the angles  716 ,  718 ,  722  and  724 . The sense masses  706  and  704  rotate about the central axis  720 , forming these indicated angles with the horizontal. 
     At  760 , the free ends  710 ,  708 ,  714  and  712  are all level with the horizontal and with each other. The surface of sense masses  706  and  704  are therefore flat and level with each other.  760  represents the midpoint in the oscillation of sense masses  706  and  704 . This may also be the resting position of sense masses  704  and  706 , such that in the absence of torsional forces or drive forces, the sense masses  704  and  706  would remain in this position. The surface of sense masses  704  and  706  may be, at  760 , parallel to a bottom layer of the multiple degree of freedom inertial sensor (not shown). 
     At  780 , the sense masses  706  and  704  have rotated about the central axis  720 . The free end  710  of sense mass  704  has moved in the positive z direction, while the free end  708  of  704  has moved in the negative z direction. The free end  714  of sense mass  706  has moved in the negative z direction, while free end  712  of sense mass  706  has moved in the positive z direction. Thus the free ends  712  and  710  both make positive altitude angles  784  and  788  with the horizontal, respectively, while free ends  708  and  714  both make negative altitude angles  782  and  786  with the horizontal, respectively. The magnitudes of angles  782 ,  784 ,  786  and  788  may all be the same.  780  represents the maximum displacement for free ends  710  and  712 , and a minimum displacement for free ends  708  and  714 . 
     The sequence of positions  700 ,  740 ,  760  and  780  represent one half cycle of the sense masses  704  and  706  vertical torsional rotational oscillation. To complete the full cycle, the sense masses  704  and  706  will rotate about the axis  720 , going from position  780 , to  760 , to  740 , and back to  700 . The positions shown in  FIG. 7  are intended as representative models, and may be exaggerated for clarity. 
       FIG. 8  depicts the common mode torsional rotational movement of a multiple degrees of freedom inertial sensor, according to an illustrative implementation. The motion shown in  FIG. 8  is that of the higher natural frequency mode of the two-degree-of-freedom system. This motion is the disfavored motion of sense masses  610  and  612  shown in  FIG. 6 , and may contain common mode error in the analog electrical output signal produced by its motion. The sense masses  704  and  706  oscillate in tandem about the central axis of rotation  720  in their common mode. 
     At  800 , the free end  710  of sense mass  704  and the free end  714  of sense mass  706  form positive altitude angles with the horizontal, shown at  816  and  822  respectively. The free end  708  of sense mass  704  and the free and  712  of sense mass  706  form negative altitude angles with the horizontal, shown at  818  and  820 . At any given time in the sense masses  704  and  706  oscillation about the central axis  720  in the common mode motion shown in  FIG. 8 , the free end  710  may form the same altitude angle as the free end  714 , while the free end  708  may form the same altitude angle as the free end  712 . The magnitude of altitude angles  822 ,  816 ,  818  and  820  may be the same. 
     At  840 , the free ends  710  and  714  have moved in the negative z direction, while the free ends  708  and  712  have moved in the positive z direction. The free ends  710  and  714  form positive altitude angles  842  and  848  respectively. The free ends  708  and  712  form negative altitude angles  844  and  846  respectively. The magnitude of altitude angles  842 ,  844 ,  846  and  848  may be the same. 
     At  860 , the free ends  710  and  714  have moved further in the negative z direction, while free ends  708  and  712  have moved further in the positive z direction. The free ends  708 ,  710 ,  712 , and  714  are level with the horizontal, and therefore do not form any altitude angles with the horizontal.  860  may be the resting position of the sense masses  704  and  706 , meaning that in the absence of drive forces or outside perturbations they would return to this position. At  860 , the sense masses  704  and  706  may be parallel to a bottom layer of the multiple degrees of freedom inertial sensor (not shown). 
     At  880 , the free ends  708  and  712  have moved in the positive z direction, while the free ends  710  and  714  have moved in the negative z direction. Free ends  708  and  712  therefore form positive altitude angles with the horizontal, shown at  888  and  884 , respectively. The magnitude of altitude angles  882 ,  884 ,  886  and  888  may be the same. At  880 , the free ends  710  and  714  may be at their minimum displacement, while free ends  708  may be at their maximum displacement.  880  may be the halfway point in the period of oscillation of sense masses  702  and  706 . To complete a full cycle, the free ends may move from position  880  to  860 , to  840  and return to  800 . 
       FIG. 9  depicts two views of an inertial sensor with recessed moveable beams used for measuring perturbations and oscillations in a vertical direction, according to an illustrative implementation.  FIG. 9  depicts a fixed element  904  and a moveable element  902 . The fixed element  904  includes beams  906   a ,  906   b , and  906   c  (collectively, beams  906 ). The moveable element  902  includes beams  908   a ,  908   b ,  908   c , and  908   d  (collectively, beams  908 ). The fixed beams  906  are the same height as the fixed beam  904 , and the moveable beams  908  are shorter than the fixed beams  906  and the moveable element  902  by a distance  920 . The beams shown in  FIG. 9  form a TDS structure capable of measuring time intervals. 
       FIG. 10  depicts two views of an inertial sensor with recessed fixed beams used for measuring perturbations in a vertical direction, according to an illustrative implementation.  FIG. 10  depicts a moveable element  1002  and a fixed element  1004 . The moveable element  1002  has moveable beams  1008   a ,  1008   b ,  1008   c , and  1008   d  (collectively, beams  1008 ). The fixed beam  1004  includes fixed beams  1006   a ,  1006   b , and  1006   c  (collectively, beams  1006 ). The fixed beams  1006  are recessed by a distance  1020  such that the top surface of the fixed beams  1006  is lower than the top surface of the fixed element  1004  and the top surface of the moveable beams  1008 . The structures depicted in  FIGS. 9 and 10  can be used to implement any of the structures depicted in  FIG. 11 . The beams shown in  FIG. 10  form a TDS structure capable of measuring time intervals. 
       FIG. 11  depicts eight configurations of fixed and moveable beams which may be used in a multiple degrees of freedom inertial sensor to measure perturbations in a vertical direction, according to an illustrative implementation.  FIG. 11  includes views  1100 ,  1102 ,  1104 ,  1106 ,  1108 ,  1110 ,  1112 , and  1114 . The view  1100  includes a fixed beam  1116  and a moveable beam  1118  that is shorter than the fixed beam  1116 . At rest, the lower surface of the moveable beam  1118  is aligned with the lower surface of the fixed beam  1116 . As the moveable beam is displaced upward by one-half the difference in height between the two beams, the capacitors between the two beams is at a maximum. When the capacitance is at a maximum, the capacitive current is zero and can be detected using a zero-crossing detector as described herein. 
     The view  1102  includes a moveable beam  1120  and a fixed beam  1122 . The moveable beam  1120  is taller than the fixed beam  1122 , and the lower surfaces of the moveable fixed beams are aligned in the rest position. As the moveable beam is displaced downward by a distance equal to one-half the distance in height of the two beams, capacitance between the two beams is at a maximum. 
     The view  1104  includes a fixed beam  1124  and a moveable beam  1126  that is shorter than the fixed beam  1124 . The center of the moveable beam is aligned with the center of the fixed beam such that in the rest position, the capacitance is at a maximum. 
     The view  1106  includes a fixed beam  1130  and a moveable beam  1128  that is taller than the fixed beam  1130 . At rest, the center of the moveable beam  1128  is aligned with the center of the fixed beam  1130  and capacitance between the two beams is at a maximum. 
     The view  1108  includes a fixed beam  1132  and a moveable beam  1134  that is the same height as the fixed beam  1132 . At rest, the lower surface of the fixed beam  1132  is above the lower surface of the moveable  1134  by an offset distance. As the moveable beam  1134  moves upward by a distance equal to the offset distance, capacitance between the two beams is at a maximum because the overlap area is at a maximum. 
     The view  1110  includes a fixed beam  1138  and a moveable beam  1136  that is the same height as fixed beam  1138 . In the rest position, the lower surface of the moveable beam  1136  is above the lower surface of the fixed beam  1138  by an offset distance. As the moveable beam travels downward by a distance equal to the offset distance, the overlap between the two beams is at a maximum and thus capacitance between the two beams is at a maximum. 
     The view  1112  includes a fixed beam  1140  and a moveable beam  1142  that is shorter than the fixed beam  1140 . In the rest position, the lower surfaces of the two beams are aligned. As the moveable beam  1142  moves upwards by a distance equal to one-half the difference in height between the two beams, overlap between the two beams is at a maximum and thus capacitance is at a maximum. 
     The view  1114  includes a fixed beam  1146  and a moveable beam  1144  that is taller than the fixed beam  1146 . In the rest position, the lower surface of the moveable beam  1144  is below the lower surface of the fixed beam by an arbitrary offset distance. As the moveable beam  1144  moves downwards such that the center of the moveable beam  1144  is aligned with the center of the fixed beam  1146 , the overlap area reaches a maximum and thus capacitance between the two beams reaches a maximum. For each of the configurations depicted in  FIG. 11 , a monotonic motion of the moveable beam produces a non-monotonic change in capacitance resulting in an extremum in capacitance. For all of the configurations depicted in  FIG. 11 , when capacitance between the two beams is at a maximum, the capacitive current is zero. The beams shown in  FIG. 11  may be used to measure time intervals between zero-crossings. These zero-crossings may be used to determine inertial parameters. 
       FIG. 12  depicts three cross views of the movement of one sense mass of a multiple degrees of freedom inertial sensor and electrodes for measuring perturbations in a vertical direction, according to an illustrative implementation.  FIG. 12  shows the bottom layer  1202  of the multiple degrees of freedom inertial sensor, a sense mass comprised of connected segments  1208   a ,  1208   b  and  1208   c  (collectively  1208 ), and sense electrodes  1204   a  and  1204   b . The sense mass  1208  has a pivot point  1206 , around which it rotates in a vertical direction as shown at  1220  and  1240 . At  1200 , the sense mass  1208  may be at equilibrium, meaning that in the absence of drive forces or external perturbations, it would remain at this position. At  1200 , the sense mass may be parallel to the bottom layer  1202 . 
     The central anchor depicted at  1206  may include coupling springs and drive springs to mechanically connect the sense mass  1208  to a second sense mass (not shown) of a multiple degrees of freedom inertial sensor. The central anchor depicted at  1206  may be rigidly coupled to the bottom layer  1202 . The sense mass may be driven by drive structures (not shown) positioned below the sense mass  1208  on the bottom layer  1202 , or in any other configuration capable of producing the oscillation shown at  1200 ,  1220  and  1240 . The electrodes  1204   a  and  1204   b  are spaced at a radius  1212  and  1210 , respectively, from a rotational pivot point  1206  of the proof mass. Radius  1210  is smaller than radius  1204   a . Additionally, as shown, the electrode  1204   b  has a smaller area than the electrode  1204   a , and thus  1204   b  has a smaller nominal capacitance than 1212. The electrodes  1204   a  and  1204   b  may be rigidly coupled to the bottom layer  1202 . They are shown as separated by the segment of the sense mass  1208   b.    
     The inner walls of the sense mass, shown at  1214 , interface with the sense electrodes  1204   a  and  1204   b , and may contain electrodes or capacitive plates, meaning that the sense electrodes and sense masses may form parallel plate capacitors between each other, producing capacitive current as the result of their relative movement and change in capacitance. Additionally, as shown, the first electrode  1204   b  has a smaller area than the second electrode  1204   a , and thus the first electrode has a smaller nominal capacitance than the second electrode. 
     At  1220 , the sense mass  1208  has reached its maximum vertical displacement, forming a positive altitude angle  1222  as a result of the movement of its free end as indicated by arrow  1224 . At  1240 , the sense mass  1208  has reached its minimum vertical displacement, forming a negative altitude angle  1242  as a result of the movement of its free end as indicated by arrow  1244 . Angle  1222  may have the same magnitude as angle  1242 . 
     As the proof mass rotates in the directions indicated at  1224  and  1244 , both the capacitance of the first electrode  1204   b  and second electrode  1204   a  will decrease from the maximum capacitance shown at position  1200 . Since the second electrode  1204   a  is positioned at a larger radius  1212 , the electrode has an offset relative to the tilting proof mass that increases faster than that of the first electrode  1204   b . This also means that the second electrode  1204   a &#39;s capacitance decreases faster than that of the first electrode  1204   b . As such, during a rotation of the proof mass  1208 , the second electrode  1204   a &#39;s capacitance decreases from a magnitude greater than to a magnitude less than that of the first electrode  1204   b &#39;s capacitance. Thus, at some particular altitude angle ±φ, the capacitance of the first electrode  1204   b  and the second electrode  1204   a  will be equal, giving a differential capacitance of zero at angle ±φ. This capacitance relation between the first electrode  1204   b  and the second electrode  1204   a  is shown in further detail with reference to  FIGS. 35-43 . An algorithm, such as the Cosine algorithm, or any of the algorithms as described with reference to  FIG. 24 , is able to use these points of zero differential capacitance to determine acceleration and other inertial parameters. 
       FIG. 13  depicts three cross views of the movement of one sense mass of a multiple degrees of freedom inertial sensor and electrodes in a second configuration for measuring perturbation in a vertical direction, according to an illustrative implementation.  FIG. 13  shows the bottom layer  1302  of the multiple degrees of freedom inertial sensor, a sense mass comprised of connected segments  1312   a ,  1312   b  and  1312   c  (collectively,  1312 ), and sense electrodes  1306   a  and  1306   b . The sense mass  1312  may have a pivot point (not shown) located at the left-most end of sense mass segment  1312   a  as shown in  FIG. 13 , which allows it to oscillate in the vertical direction as shown at  1320  and  1340 . At  1300 , the sense mass  1312  may be at equilibrium, meaning that in the absence of drive forces or external perturbations, it would remain at this position. At  1300 ,  1320  and  1340 , the sense mass may be parallel to the bottom layer  1302 . 
     The pivot point may include coupling springs and drive springs to mechanically connect the sense mass  1312  to a second mass (not shown) of a multiple degrees of freedom inertial sensor. The pivot point may be rigidly coupled to the bottom layer  1302 . The sense mass  1312  may be driven by drive structures (not shown) positioned below the sense mass  1312  on the bottom layer  1302 , or in any other configuration capable of producing the oscillation shown at  1320  and  1340 . Electrode  1306   a  has the same area as electrode  1306   b , and electrodes  1306   a  and  1306   b  may be rigidly coupled to the bottom layer  1302 . 
     In the equilibrium position  1300 , the first electrode  1306   a  is vertically offset upward relative to the proof mass segment  1312   a , and the second electrode  1306   b  is vertically offset downward to the proof mass segment  1312   c . Segment  1312   b  is offset downward to the first electrode  1306   a  on the left side, and offset upwards to the second electrode  1306   b  on the right side. As shown in  FIG. 13 , this is achieved by aligning the bottoms of the proof masses and the bottoms of the electrodes  1306   a  and  1306   b , and etching a gap of distance  1210  shown at  1304 . This gap may be approximately 4 μm deep. 
     At  1320 , the proof mass  1312  has moved in the vertical z direction as indicated by the arrow  1322 . At  1320 , the proof mass  1312  may have reached its maximum positive displacement in the z direction. At  1340 , the proof mass  1312  has moved in the negative z direction as indicated by the arrow  1342 . At  1340 , the proof mass  1312  may have reached its minimum negative z displacement. As the proof mass  1312  oscillates in the z direction, it may move from position  1320 , to position  1300 , to position  1340 , and then back to  1300  and  1320  to complete a full oscillation cycle. 
     As the proof mass moves in the directions indicated at  1322  and  1342 , one electrode&#39;s capacitance will increase and the other electrode&#39;s capacitance will decrease. For example, as proof mass  1312  lowers, the second electrode  1306   b  that has a downward offset will approach a maximum capacitance when the second electrode  1306   b  and the proof mass  1312  are aligned. The first electrode  1306   a , which has an upward offset, will have a decrease capacitance as the electrode&#39;s vertical separation from the proof mass  1312  increases. The converse is true as the proof mass  1312  moves in the positive z direction. As a specific upward position, the first electrode  1306   a &#39;s capacitance will have a maximum, and at a specific downward vertical position, the second electrode  1306   b  will have a maximum. At each of these maxima, the slope of the capacitance with respect to time will be zero as the proof mass translates in the z direction. Because these zero-slope points correspond to fixed proof mass positions, an algorithm, such as the Cosine algorithm, as discussed with reference to  FIG. 12 , is able to use these points to determine acceleration. 
       FIG. 14  depicts differential mode vertical movement of a multiple degrees of freedom inertial sensor with packaging deformations, according to an illustrative implementation. The motion shown in  FIG. 14  is that of the first, lower natural frequency mode of the two-degree-of-freedom system.  FIG. 14  shows a central anchor  1406  which is rigidly coupled to the bottom layer  1408  of the multiple degrees of freedom inertial sensor  1400 . A first sense mass  1402  and a second sense mass  1404  may be mechanically coupled to the central anchor  1406  with springs (not shown). A package deformation  1420  may cause a tilt in the bottom layer  1408  of the multiple degrees of freedom inertial sensor, shown by the angle  1418 . Sense electrodes  1410   a  and  1410   b  may sense the oscillations and perturbations of the sense masses  1402  and  1404  in the z direction, respectively, by detecting changes in capacitance between electrodes  1410   a  and  1410   b , and electrodes located on the undersides of the sense masses  1402  and  1404 . The sense masses  1402  and  1404  may be mechanically driven with drive combs (not shown) for example, as discussed in more detail with reference to  FIG. 1 . As each sense mass  1402  and  1404  oscillates, it moves up and down in the z direction as shown at  1412 , going from minimum distances  1414  and  1418  from the sense electrode  1410   a  and  1410   b , respectively, to maximum distances  1416  and  1420  from the sense electrode  1410   a  and  1410   b , respectively, in one half cycle. The sense electrodes  1410   a  and  1410   b  may be any TDS structure, and may be one of the TDS structures described in  FIGS. 1-13 , capable of sensing oscillations and perturbations in the vertical direction. As shown in  FIG. 14 , the package deformation may cause a tilt  1418  in the anchoring structure  1406 , which may result in uneven oscillation of the sense mass  1402  from  1404 . As shown in  FIG. 14 , this may result in sense mass  1402  having a larger minimum distance  1414  from the sense electrode  1410   a  than the sense mass  1404 &#39;s minimum distance  1418  from the sense electrode  1410   b . The tilt angle  1418  may also lead to the sense mass  1402  having a larger maximum distance  1416  from the sense electrode  1410   a  than the sense mass  1404 &#39;s maximum distance  1420  from the sense electrode  1410   b . The difference in these minimum and maximum distances between sense masses  1402  and  1404  may result in common mode error in the electrical signals produced by the sense electrodes  1410   a  and  1410   b  as a result of the oscillations of sense masses  1402  and  1404 . 
     The common mode error that results from tilt  1418  may be removed as a result of the differential motion of sense masses  1402  and  1404 , as shown in  FIG. 14 . Examples of the removal of package deformation or other common mode error from the differential motion of the two degree of freedom inertial sensor  1400  are discussed in more detail with reference to  FIGS. 30 and 31 . 
       FIG. 15  depicts an overhead view of a multiple degrees of freedom inertial sensor for measuring perturbations in a horizontal plane, according to an illustrative implementation. The multiple degree of freedom inertial sensor  1500  is shown with two sense masses  1502  and  1504 , which are each mechanically coupled to a frame  1506  and  1508  with coupling springs  1516   a ,  1516   b ,  1512   a  and  1512   b  respectively. The frame  1506  and  1508  is mechanically coupled to a central anchor  1510  with anchoring springs  1514   a  and  1514   b . The central anchor  1510  may be rigidly coupled to a bottom layer (not shown) of the multiple degrees of freedom inertial sensor  1500 . The sense masses  1502  and  1504  may oscillate in a differential mode as indicated by arrows  1520  and  1518 , where, for example, the sense mass  1502  may move in a negative y direction at the same time that the sense mass  1518  moves in a positive y direction. The differential motion shown by arrows  1518  and  1520  is that of the first, lower natural mode frequency response of the two-degree-of-freedom system. Common mode motion, where the sense masses  1502  and  1504  move in the same direction, will be at the second, higher natural frequency mode of the system. The sense masses  1502  and  1504  may be driven with comb drives or any other drive structure capable of producing the oscillating motion as indicated by arrows  1520  and  1518 . TDS sensors, described in further detail with reference to  FIGS. 9-13 and 16  may convert the oscillation of sense masses  1520  and  1504  to an electrical signal capable of sensing perturbations such as acceleration of the multiple degrees of freedom inertial sensor  1500  in the horizontal plane. 
     The springs  1516   a ,  1516   b ,  1512   a , and  1512   b  will each have a spring constant that, together with the mass of sense masses  1520  and  1504 , and the mass of the frame  1506  and  1508 , will define the resonant frequency of sense mass  1520  and  1504 . The spring constant of springs  1512   a ,  1512   b ,  1516   a  and  1516   b  may all be the same. The spring constant of springs  1512   a ,  1512   b ,  1516   a  and  1516   b  may be lower than the spring constant of springs  1514   a  and  1514   b . The spring constants and masses of the multiple degrees of freedom inertial sensor  1500  may be adjusted to lower the differential frequency mode of sense masses  1502  and  1504 , as well as to favor the differential motion indicated by arrows  1520  and  1518 . The springs  1512   a  and  1512   b ,  1514   a ,  1514   b ,  1516   a ,  1516   b , may have a lower effective spring constant in response to the differential, out-of-phase motion of sense masses  1502  and  1504  than to the common mode, in-phase motion of sense masses  1502  and  1504 . The lower, natural frequency mode response of the system shown in  FIG. 15  may thus be associated with differential motion of the sense masses, while the second, higher natural frequency mode response of the system may be associated with common mode motion of the sense masses. 
     One end of the frame  1522  may move differentially with respect to the other end of the frame  1524 , so that as the sense masses  1502  and  1504  oscillate differentially as indicated by the arrows  1520  and  1518 , the frame  1506  and  1408  will oscillate with the same differential motion. Thus as the sense mass  1504  moves in the positive y direction, the end  1522  will also move in the positive y direction. As the sense mass  1502  moves in the negative y direction, the end  1524  will also move in the negative y direction. The differential motion of the sense masses  1502  and  1504  may be differentially sensed with in and out of phase TDS structures as described in further detail with reference to  FIG. 16 . The frame  1506 ,  1508 , and sense masses  1502  and  1504  may be driven with a drive structure (not shown) as discussed in further detail with reference to  FIG. 1 . 
     The multiple degrees of freedom inertial sensor  1500  allows for differential motion of two sense masses in the horizontal plane. The frame  1522  as shown, allows for the coupling of sense masses necessary to produce a system with multiple resonant frequencies, while still allowing for differential motion of the sense masses in the horizontal plane. 
       FIG. 16  depicts three views, each showing a schematic representation of movable and fixed elements of a plurality of time-domain switches used to sense perturbations of a multiple degrees of freedom inertial sensor in a horizontal plane, according to an illustrative implementation. The sense mass of a multiple degrees of freedom inertial sensor can be coupled to the movable element  1602 , while the fixed element  1604  may be rigidly coupled to the bottom layer of the multiple degrees of freedom inertial sensor. The movable element  1602  and the fixed element  1604  each include a plurality of interdigitated, equally spaced beams. In  FIG. 16 , the fixed element  1604  includes beams  1606   a ,  1606   b  and  1606   c  (collectively, beams  1606 ). The movable element  1602  includes beams  1608   a  and  1608   b , and is separated from the fixed element  1604  in the x direction by a distance  1622 . The distance  1622  will increase and decrease as the movable element  1602  oscillates with respect to the fixed element  1604  in the x direction. The distance  1622  is selected to minimize parasitic capacitance when the movable element  1602  is in the rest position, while also taking into consideration the ease of manufacturing the structure  1600 . The view  1640  depicts an area of interest noted by the rectangle  1624  of view  1620 .  1620  is an overhead view of the perspective view shown at  1600 . 
     Each of the beams  1606  and  1608  includes multiple sub-structures, or teeth, protruding in a perpendicular axis to the long axis of the beams (shown in  FIG. 16  as they and x axis, respectively). The beam  1606   b  includes teeth  1648   a ,  1648   b , and  1648   c  (collectively, teeth  1648 ). The beam  1608   b  includes teeth,  1650   a ,  1650   b  and  1650   c  (collectively, teeth  1650 ). Adjacent teeth on a beam are equally spaced according to a pitch  1642 . Each of the teeth  1648  and  1650  has a width defined by the line width  1646  and a depth defined by a corrugation depth  1652 . Opposing teeth are separated by a tooth gap  1654 . As the movable beam  1606   b  oscillates along the axis  1610  with respect to the fixed beam  1606   b , the tooth gap  1644  remains unchanged. 
     A capacitance may exist between the fixed beam  1606   b  and the movable beam  1608   b  coupled to the sensing mass. As the movable beam  1608   b  oscillates along the axis  1610  with respect to the fixed beam  1606   b , this capacitance will change. As the teeth  1650   a ,  1650   b  and  1650   c  align with opposing teeth  1648   a ,  1648   b  and  1648   c  respectively, the capacitance will increase. The capacitance will then decrease as these opposing sets of teeth become less aligned with each other as they move in either direction along the x-axis. At the position shown in view  1640 , the capacitance is at a maximum as the teeth  1650  are aligned with teeth  1648 . As the moveable beam  1602  moves monotonically along the axis  1610 , the capacitance will first gradually decrease and then gradually increase as the Nth moving tooth becomes less aligned with the Nth fixed tooth, and then aligned with the (N±i)th fixed tooth, where i=1, 2, 3, 4 . . . i max  This process is repeated for the full range of motion for the Nth tooth, where the minimum of the sense mass&#39;s displacement occurs at the (N−i max )th fixed tooth, and the maximum of the sense mass&#39;s displacement occurs at the (N+i max )th fixed tooth. 
     The capacitance may be degenerate, meaning that the same value of capacitance occurs at multiple displacements of the moveable beam  1608   b . For example, the capacitance value when the Nth moving tooth is aligned with the (N+1)th fixed tooth may be the same when the Nth moving tooth is aligned with the (N+2)th fixed tooth. Thus when the moveable beam  1608   b  has moved from its rest position by a distance equal to the pitch  1642 , the capacitance is the same as when the moveable beam  1608   b  is in the rest position. 
       FIG. 17  depicts a process for extracting inertial information from an inertial sensor, according to an illustrative implementation  FIG. 17  includes a representative inertial sensor  1700  which experiences an external perturbation  1701 . This inertial sensor may be an accelerometer, a gyroscope, a multiple degrees of freedom inertial sensor, or any other sensor capable of producing the signals shown in  FIG. 17  and able to detect an inertial parameter. A drive signal  1710  causes a moveable portion of the multiple degrees of freedom inertial sensor  1700  to oscillate. This moveable portion of the multiple degrees of freedom inertial sensor  1700  may be the sense mass. An analog frontend (AFE) electrically connected to a moveable element and a fixed element of a TDS structure of the inertial sensor measures the capacitance between them and outputs a signal based on this capacitance. The AFE may measure 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  1700  are generated at  1702  and  1704  and combined at  1706  into a combined signal. A signal processing module  1708  processes the combined analog signal to determine inertial information. One or more processes can convert the combined analog signal into a rectangular waveform  1712 . This may be done using a comparator, by amplifying the analog signal to the rails, or by other methods. 
     The rectangular waveform  1712  has high and low values, with no substantial time spent transitioning between them. 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  1718  of the sense mass crosses reference levels  1714  and  1716 . The reference levels  1714  and  1716  correspond to physical locations along the path of motion of the sense mass. 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. 18  depicts a conceptual schematic of a one degree of freedom sense mass&#39; oscillation, according to an illustrative implementation. A sense mass  1818  is attached to springs  1820  and  1822 , which may be coupled to a drive mass, and which each compress or extend as the sense mass  1818  oscillates in the axis of displacement  1824 . The spring constants of springs  1820  and  1822  will determine the force extension relationship of the proof mass. This can be modeled by Hooke&#39;s law, whereby the force F applied to the sense mass results in displacement Δx according to the relation: 
         F=kΔx   (4)
 
     Thus as an inertial force is applied to the sense mass, it will respond with a displacement Δx that may be measured by a change in capacitance or any other electrical signal relating the physical displacement to a measurable output. The k value or spring constant of a multiple degrees of freedom inertial sensor is determined by the geometry of the springs. The geometric and fabrication considerations for determining this spring constant are discussed in more detail with reference to  FIG. 1-8 . 
       FIG. 19  is a graph showing the in phase and out of phase capacitive response to a sense mass oscillation produced by TDS structures of a multiple degrees of freedom inertial sensor, according to an illustrative implementation.  FIG. 19  demonstrates the translation of the linear displacement of a sense mass into a non-linear electrical signal. An in-phase signal  1904  may be generated by TDS geometry that maximizes capacitance at a sense mass&#39; resting position. An out of phase signal  1902  may be generated by TDS geometry that minimizes capacitance at a sense mass&#39; resting position. An in-phase and an out of phase signal may be separated by a phase difference of 90° as is shown at  FIG. 19 , or any other phase difference desired. The in phase  1904  and out of phase  1902  signals may result from the displacement of the same sense mass, such that the moveable components of the TDS structures that generate signals  1904  and  1902  are both coupled to the same sense mass. The in phase  1904  and out of phase  1902  signals may be subtracted, averaged, or otherwise combined to produce a single measurement reflective of a proof mass displacement. This measurement may be based on time intervals produced by zero-crossings of an analog electrical signal output by the TDS structures shown. The period  1906  of an in phase signal  1904  may be determined entirely by the geometry of the TDS teeth. The in phase  1904  and out of phase  1902  signals may have the same zero crossings as shown at  1908 ,  1912  and  1914 . 
       FIG. 20  depicts in phase and out of phase TDS structures for sensing perturbations in a horizontal plane, according to an illustrative implementation. Both moveable elements  2026  and  2030  are shown in at their resting equilibrium without inertial forces or drive forces acting on either of them. The pitch or distance between teeth  2032  defines the distance between peaks of capacitance, or the phase of the resulting nonlinear capacitive signal. A voltage may be applied between fixed element  2024  and moveable element  2026 , as well as between fixed element  2028  and moveable element  2030 . The distance  2036  between fixed element  2024  and moveable element  2026  defines a minimum distance between teeth corresponding to a maximum of capacitance. The distance  2034  between fixed element  2028  and moveable element  2030  defines a maximum distance between teeth corresponding to a minimum of capacitance. As moveable elements  2034  and  2036  oscillates linearly in the axis  2038 , the capacitance between teeth will oscillate between the minimum “aligned” state where the distance between teeth is  2036 , and the non aligned state where the distance between teeth is  2034 . This will in turn produce an electrical signal as discussed in detail with reference to  FIG. 19 . The moveable elements  2026  and  2030  may be coupled to the same sense mass, such that the electrical signal produced between elements  2024  and  2026 , and  2028  and  2030  will correspond to the same physical displacement. The fixed elements  2028  and  2024  may be rigidly coupled to a support structure or other anchoring architecture of the composite mass inertial sensor. 
     Signals generated from in phase structures  2024  and  2026 , and out of phase structures  2028  and  2030  may be linearly combined to produce differential signals. Differential signals may be produced by subtracting a signal produced by  2024  and  2026  from a signal produced by  2028  and  2030 . This differential signal may eliminate common mode error produced by parasitic capacitance, temperature variations, packaging deformations, ground loops, drifts in voltage bias, or any other sources of electrical error that may affect both signals. 
       FIG. 21  is a graph representing the relationship between analog signals derived from a multiple degrees of freedom inertial sensor and the displacement of a sense mass of a multiple degrees of freedom inertial sensor, according to an illustrative implementation. The graph  2100  represents signals derived from an oscillator in which opposing teeth are aligned at the rest position, as described in further detail with reference to  FIG. 20 . This oscillator may be the sense mass of a multiple degrees of freedom inertial sensor coupled to a TDS structure. The graph  2100  includes curves  2102 ,  2104 , and  2106 . The curve  2102  represents an output of an AFE such as a transimpedence amplifier (TIA). Since a TIA outputs a signal proportional to its input current, the curve  2102  represents a capacitive current measured between moveable and fixed elements of a multiple degrees of freedom inertial sensor. The curve  2106  represents an input acceleration applied to the accelerometer. The input acceleration represented by curve  1206  is shown as a 15 g acceleration at 20 Hz, but may be any outside perturbation, force or acceleration. The curve  2104  represents displacement of the sense mass of a composite mass inertial sensor. 
       FIG. 21  includes square symbols indicating points at which the curve  2102  crosses zero. Since capacitive current  2102  is proportional to the first derivative of capacitance, these zero-crossings in the current represent local maxima or minima (extrema) of capacitance between a moveable element and a fixed element of the multiple degrees of freedom inertial sensor.  FIG. 21  includes circular symbols indicating points on the curve  2104  corresponding to times at which curve  2102  crosses zero. The circular symbols indicate the correlation between the physical position of a moveable element of the multiple degrees of freedom inertial sensor and zero-crossing times of the outputs of the signal  2102 . 
     At the time  2118 , the curve  2102  crosses zero because the displacement  2104  of the moveable element of the sense mass is at a maximum and the oscillator is instantaneously at rest. Here, capacitance reaches a local extremum because the moveable element has a velocity of zero, not necessarily because teeth or beams of the oscillator are aligned with opposing teeth or beams. At time  2120 , the TIA output curve  2102  crosses zero because the oscillator displacement reaches the +d 0  location  2108 . The +d 0  location  2108  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  2122 , the TIA output curve  2102  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 are in an aligned position with the centers of gaps between teeth of the fixed element, resulting in a minimum in capacitance. This minimum in capacitance occurs at a location of +d 0 /2  1210 , corresponding to a displacement of one-half the pitch distance in the positive direction. 
     At time  2124 , the TIA output curve  2102  crosses zero because teeth of the movable element are aligned with teeth of the fixed element, producing a maximum in capacitance. The time  2124  corresponds to a time at which the movable element is at the rest position, indicated by the zero displacement  2112  on the curve  2104 . At time  2126 , the TIA output  2102  crosses zero because teeth of the movable element are once again anti-aligned with teeth of the fixed element, producing a local minimum in capacitance. This anti-alignment occurs at a displacement of −d 0 /2  2114 , corresponding to a displacement of one-half the pitch distance in the negative direction. 
     At time  2128 , the TIA output  2102  crosses zero because the teeth of the movable element are in an aligned position with respect to the teeth of the fixed element, creating a local maximum in capacitance. This local maximum in capacitance occurs at a displacement of −d 0    2116 , corresponding to a displacement of the pitch distance in the negative direction. At time  2130 , the TIA output curve  2102  crosses zero because the movable element has an instantaneous velocity of zero as it reverses direction. This reversal of direction is illustrated by the displacement curve  2104 . As at time  2118 , when the movable element has a velocity of zero, capacitance does not change with time and thus the current and TIA output (which are proportional to the first derivative of capacitance) are zero. 
       FIG. 22  is a graph illustrating a current response to the displacement of a sense mass of a multiple degrees of freedom inertial sensor, according to an illustrative implementation. The graph  2200  includes a current curve  2202  and a displacement curve  2204 . The current curve  2202  represents an input signal for a TIA and may be produced by TDS structures coupled to a sense mass of a multiple degrees of freedom inertial sensor. The TIA may produce an output signal such as the TIA output curves  2102  as shown in  FIG. 21  in response to displacement of the sense mass of a multiple degrees of freedom inertial sensor. The current curve  2202  is a capacitive current generated between fixed and movable elements of the multiple degrees of freedom inertial sensor in response to displacement  2204 . The current curve  2202  crosses zero at numerous times, including times  2224 ,  2226 ,  2228 , and  2230 . At the times  2224  and  2230 , the movable element has a displacement of −d 0 , where d 0  may correspond to the pitch distance between teeth of a TDS structure. At the times  2226  and  2228 , the movable element has a displacement of +d 0 . 
     The graph  2200  includes two time intervals T 43    2232  and T 61   2234 . The time interval T 43    2232  corresponds to the difference in time between time  2226  and time  2228 . The time interval T 61    2234  corresponds to the time difference between times  2224  and  2230 . Thus, time interval T 61    2234  corresponds to the time between subsequent crossings of the −d 0    2216  location, and the time interval T 43    2232  corresponds to the time interval between subsequent crossings of the +d 0    2208  location. The methods used to determine the time intervals T 43    2232  and T 61    2234  can be used to determine other time intervals, such as between a crossings of the +d 0    2208  and the next subsequent crossing of the −d 0    2216  level, between a time interval between a crossing of the −d 0    2416  level and the next crossing of the +d 0    2208  level, between the time  2230  and the next crossing of the +d 0    2208  level, between crossings of the zero  2212  level, between zero-crossings due to a maximum or minimum of displacement, or between any other combination of zero-crossings of the current curve  2202  or a TIA output signal corresponding to the current curve  2202 . 
       FIG. 23  is a graph showing a rectangular-wave signal produced from zero-crossing times of the current signal depicted in  FIG. 22 , according to an illustrative implementation. The graph  2300  includes a rectangular waveform curve  2336 . The rectangular waveform curve  2336  has substantially two values: a high value and a low value. While the rectangular waveform curve  2336  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  2336  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  2336  from the current curve  2202  shown in  FIG. 22  is to use a comparator to detect zero-crossings of the current curve  2202 . When the current curve  2202  has a value greater than a reference level (such as zero), the comparator outputs a high value, and when the current curve  2202  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  2202  transitions from a negative value to a positive value, and the comparator&#39;s output transitions from high to low when the current curve  2202  transitions from a positive value to a negative value. Thus, times of rising edges of the rectangular waveform curve  2336  correspond to times of negative-to-positive zero-crossings of the current curve  2304 , and falling edges of the rectangular waveform curve  2336  correspond to positive-to-negative zero-crossings of the current curve  2202 . This can be seen at time  2324 , where the rectangular waveform curve  2336  transitions from a negative to positive value, corresponding to a zero crossing at  2224  in  FIG. 22 . The same can be seen at time  2328  corresponding to zero crossing  2228 . The rectangular waveform curve  2336  transitions from a positive value to a negative value at times  2326  and  2330 , corresponding to a zero crossing at  2226  and  2230  in  FIG. 22 , respectively. 
     The rectangular waveform curve  2336  includes the same time intervals  2232  and  2234  as the current curve  2202 . One benefit of converting the current curve  2202  to a rectangular waveform signal such as the rectangular waveform curve  2336  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. 24  is a graph showing time intervals produced from non-zero crossing reference levels, according to an illustrative implementation. The graph  2400  includes times  2436  and  2438 . The graph  2400  includes the time interval T 94    2440  and the time interval T 76    2442 , which represent crossing times of the displacement curve  2404  of reference levels  2408  and  2416  respectively. The time interval T 94    2440  corresponds to the time interval between times  2428  and  2438 . The time interval T 76    2442  corresponds to the time interval between times  2430  and  2436 . The graph  2400  also includes time interval T 43    2432  and T 61   2434 , corresponding to a time interval between times  2426  and  2428 , and  2424  and  2430 , respectively. The reference levels, shown at  2408 ,  2412 , and  2416  may be any value within the displacement range of the sense mass. The reference levels  2408 ,  2412  and  2416  may be predetermined, and may correspond to the physical geometry of a TDS structure, such as the pitch distance between teeth. 
     As can be seen with reference to  FIG. 25 , the sense mass displacement as shown by the displacement curve  2504  experiences an offset that is correlated with input acceleration as indicated by the acceleration curve  2506 . Thus, one way to detect shifts of the displacement of a sense mass and thus input acceleration is to compare relative positions of zero-crossing times of a displacement curve produced by the sense mass. As shown in  FIG. 24 , a sum of the time intervals T 43    2432  and T 94    2440  represents a period of oscillation as does a sum of the periods T 61    2434  and T 36    2442 . In comparing a subset of the period, such as comparing the time interval T 43    2432  with the sum of T 43    2432  and T 94    2440  represents the proportion of time that the sense mass spends at a displacement greater than +d 0    2408 . 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 a sense mass can be determined from the time intervals depicted in  FIG. 24  using equations (5), (6), and (7). 
     
       
         
           
             
               
                 
                   d 
                   = 
                   
                     
                       
                         2 
                          
                         
                           d 
                           0 
                         
                          
                         
                           cos 
                            
                           
                             ( 
                             
                               π 
                                
                               
                                 
                                   T 
                                   61 
                                 
                                 
                                   P 
                                   
                                     m 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                       
                         
                           cos 
                            
                           
                             ( 
                             
                               π 
                                
                               
                                 
                                   T 
                                   61 
                                 
                                 
                                   P 
                                   
                                     m 
                                      
                                     
                                         
                                     
                                      
                                     1 
                                   
                                 
                               
                             
                             ) 
                           
                         
                         - 
                         
                           cos 
                            
                           
                             ( 
                             
                               π 
                                
                               
                                 
                                   T 
                                   43 
                                 
                                 
                                   P 
                                   
                                     m 
                                      
                                     
                                         
                                     
                                      
                                     2 
                                   
                                 
                               
                             
                             ) 
                           
                         
                       
                     
                     - 
                     
                       d 
                       0 
                     
                   
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
             
               
                 
                   
                     P 
                     
                       m 
                        
                       
                           
                       
                        
                       1 
                     
                   
                   = 
                   
                     
                       T 
                       61 
                     
                     + 
                     
                       T 
                       76 
                     
                   
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
             
               
                 
                   
                     P 
                     
                       m 
                        
                       
                           
                       
                        
                       2 
                     
                   
                   = 
                   
                     
                       T 
                       43 
                     
                     + 
                     
                       T 
                       94 
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
     Displacement of the sense mass can be converted to an acceleration using Hooke&#39;s Law (shown in equation (4)). Displacement of the sense mass can be calculated recursively for each half cycle of the sense mass. Using this information, the displacement of the sense mass can be recorded as a function of time. This allows the calculation of external perturbations with zero drift and lower broadband error. 
     In some examples, the out-of-plane sensor includes periodic capacitive sensors, in which the capacitance between the sense mass and a fixed portion of the sensor varies non-monotonically as a function of z(t), which represents the out-of-plane displacement of the sense mass. This non-linear capacitive variation may be known, repeatable, and periodic. The non-linear capacitance produced by a single electrode may be modeled by a trigonometric or otherwise periodic function. The non-linear capacitance may be shown as: 
     
       
         
           
             
               
                 
                   
                     
                       
                         
                           
                             S 
                             MAP 
                           
                            
                           
                             ( 
                             t 
                             ) 
                           
                         
                         = 
                           
                          
                         
                           
                             C 
                             0 
                           
                           + 
                           
                             
                               C 
                               1 
                             
                             · 
                             
                               sin 
                                
                               
                                 [ 
                                 
                                   
                                     
                                       2 
                                        
                                       π 
                                     
                                     P 
                                   
                                   · 
                                   
                                     x 
                                      
                                     
                                       ( 
                                       t 
                                       ) 
                                     
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                   
                   
                     
                       
                         = 
                           
                          
                         
                           
                             C 
                             0 
                           
                           + 
                           
                             
                               C 
                               1 
                             
                             · 
                             
                               sin 
                                
                               
                                 [ 
                                 
                                   
                                     
                                       2 
                                        
                                       π 
                                     
                                     P 
                                   
                                   · 
                                   
                                     ( 
                                     
                                       
                                         A 
                                          
                                         
                                             
                                         
                                          
                                         
                                           sin 
                                            
                                           
                                             ( 
                                             
                                               
                                                 ω 
                                                 d 
                                               
                                                
                                               t 
                                             
                                             ) 
                                           
                                         
                                       
                                       + 
                                       Δ 
                                     
                                     ) 
                                   
                                 
                                 ] 
                               
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   8 
                   ) 
                 
               
             
           
         
       
     
     Where C 0  and C 1  are constants that may be defined by the geometry of the sense electrodes, P is a period such as those give by equations (6) and (7), and ω d  is a frequency of oscillation in the out-of-plane direction. Using equation (5), one may utilize the relationship between capacitance and displacement to model the displacement by a periodic function, such as the following: 
         z ( t )= A  sin(ω d   t )+Δ  (9)
 
     Measurements of capacitance, given in equation (5), may thus allow one to solve for the variables in equation (6), such as frequency ω d , offset Δ, amplitude A and displacement z(t). By repeatedly solving for these variables, the amplitude, frequency and offset of the motion of the sense mass can be determined with respect to time. The offset may be proportional to the external acceleration or other perturbing forces of measurement interest. 
     To obtain these parameters, the times at which the out-of-plane sensor has predetermined values of capacitance are measured. At these times, the sense mass is known to be at a position that is given by equation (10), where n is a positive integer. 
     
       
         
           
             
               
                 
                   
                     
                       
                         2 
                          
                         π 
                       
                       P 
                     
                     · 
                     
                       z 
                        
                       
                         ( 
                         t 
                         ) 
                       
                     
                   
                   = 
                   
                     n 
                     · 
                     π 
                   
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
           
         
       
     
     The oscillator is known to be at a displacement that is a multiple of P/2, where P is a period that may be given, for example, by equations (6) or (7), by tracking the number of times at which the capacitance equals the predetermined capacitance. The number of times at which the oscillator crosses displacements of P/2 can be tracked to overcome issues of degeneracy in capacitance. In particular, successive times at which the oscillator displacement equals +P/2 and −P/2 (δt and δt−, respectively) are measured and used to solve for A, ω d , and Δ. Equation (11) shows the calculation of ω d  as a function of the time intervals. 
     
       
         
           
             
               
                 
                   
                     ω 
                     d 
                   
                   = 
                   
                     
                       
                         2 
                          
                         π 
                       
                       Period 
                     
                     = 
                     
                       2 
                        
                       π 
                        
                       
                         2 
                         
                           ( 
                           
                             
                               δ 
                                
                               
                                   
                               
                                
                               
                                 t 
                                 1 
                                 + 
                               
                             
                             + 
                             
                               δ 
                                
                               
                                   
                               
                                
                               
                                 t 
                                 2 
                                 + 
                               
                             
                             + 
                             
                               δ 
                                
                               
                                   
                               
                                
                               
                                 t 
                                 1 
                                 - 
                               
                             
                             + 
                             
                               δ 
                                
                               
                                   
                               
                                
                               
                                 t 
                                 2 
                                 - 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
     Exploiting the similarity of the measured time intervals combined with the fact that all time measurements were taken at points at which the capacitance equaled known values of capacitance and the oscillator displacement equaled integral multiples of P/2, the system of equations (12) and (13) can be obtained. 
     
       
         
           
             
               
                 
                   
                     z 
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       + 
                       
                         P 
                         2 
                       
                     
                     = 
                     
                       
                         A 
                         · 
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 ω 
                                 d 
                               
                                
                               
                                 
                                   δ 
                                    
                                   
                                       
                                   
                                    
                                   
                                     t 
                                     1 
                                     + 
                                   
                                 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       Δ 
                     
                   
                 
               
               
                 
                   ( 
                   12 
                   ) 
                 
               
             
             
               
                 
                   
                     z 
                      
                     
                       ( 
                       t 
                       ) 
                     
                   
                   = 
                   
                     
                       - 
                       
                         P 
                         2 
                       
                     
                     = 
                     
                       
                         A 
                         · 
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 ω 
                                 d 
                               
                                
                               
                                 
                                   δ 
                                    
                                   
                                       
                                   
                                    
                                   
                                     t 
                                     1 
                                     - 
                                   
                                 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                       + 
                       Δ 
                     
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
     The difference of equations (6) and (7) allows the amplitude A to be determined as in equation (14). 
     
       
         
           
             
               
                 
                   A 
                   = 
                   
                     P 
                     
                       
                         cos 
                          
                         
                           ( 
                           
                             
                               ω 
                               d 
                             
                              
                             
                               
                                 δ 
                                  
                                 
                                     
                                 
                                  
                                 
                                   t 
                                   1 
                                   + 
                                 
                               
                               2 
                             
                           
                           ) 
                         
                       
                       - 
                       
                         cos 
                          
                         
                           ( 
                           
                             
                               ω 
                               d 
                             
                              
                             
                               
                                 δ 
                                  
                                 
                                     
                                 
                                  
                                 
                                   t 
                                   1 
                                   - 
                                 
                               
                               2 
                             
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
     The sum of the equations (6) and (7) allows the offset Δ to be determined as in equation (15). 
     
       
         
           
             
               
                 
                   Δ 
                   = 
                   
                     
                       - 
                       
                         A 
                         2 
                       
                     
                     · 
                     
                       [ 
                       
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 ω 
                                 d 
                               
                                
                               
                                 
                                   δ 
                                    
                                   
                                       
                                   
                                    
                                   
                                     t 
                                     1 
                                     + 
                                   
                                 
                                 2 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           cos 
                            
                           
                             ( 
                             
                               
                                 ω 
                                 d 
                               
                                
                               
                                 
                                   δ 
                                    
                                   
                                       
                                   
                                    
                                   
                                     t 
                                     1 
                                     - 
                                   
                                 
                                 2 
                               
                             
                             ) 
                           
                         
                       
                       ] 
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
     In some examples, an excitation field itself is varied with time. For example one or more of the components is attached to a compliant structure but is not actively driven into oscillation. Instead, the time varying signal is generated by varying, for example, voltage between the components. External perturbations will act on the compliant component, causing modulation of the time-varying nonlinear signal produced by the component. 
     Nonlinear, non-monotonic, time varying signals can be generated with a fixed set of electrically decoupled structures with which a nonlinear time-varying force of variable phase is generated. The time-varying force may be caused by the application of voltages of equal magnitude and different phase to each of the set of structures. This generates signals at phases determined by the phase difference of the applied voltages. 
     Sets of nonlinear signals with identical or differing phases can be combined to form mathematical transforms between measured output signals and system variables such as amplitude, offset, temperature, and frequency. Combinations of nonlinear signals with identical or differing phases can be included to minimize or eliminate a time varying force imparted on a physical system that results from measurement of the nonlinear signal. For example, two separate signals can be included within the system at 0° and 180° of phase, such that each signal is the inverse of the other. An example set of signals of this nature are the signals +A*sin(ωt) and −A*sin(ωt) for phases of 0° and 180° respectively. 
     Mathematical relationships between the periodic nonlinear signals and external perturbations can be applied to extract inertial information. For example, mathematical relationships can be applied in a continuous fashion based on bandwidth and data rates of the system. In some examples, mathematical relationships can be applied in a periodic sampled fashion. Mathematical relationships can be applied in the time or the frequency domains. Harmonics generated by the sensor can be utilized mathematically to shift frequency content to enable filtering and removal of lower frequency, drift-inducing noise. Harmonics can also be used to render the sensor insensitive or immune to these drift-inducing noise sources by applying one or more mathematical relationships to decouple the inertial signal from other system variables. 
     In some implementations, assist structures uniquely identify when external perturbations cause an offset in the physical structure of the device. Offsets can be integral or non-integral multiples of a pitch of tooth spacing. These assist structures are electrically isolated from one another and from the main nonlinear periodic signal. 
     To sense external perturbations in the z axis, normal to the plane of the wafer, corrugations may be formed on one or more surface of the sensor. In some examples, corrugated comb fingers are formed with height differences. In some examples, vertically corrugated teeth are formed in a self-aligned in-plane structure used for x or y axis sensing. In some examples, vertical corrugations are added to one or more plates of a capacitor. 
     In some examples, materials used to form the device may be varied spatially to result in a time-varying component of capacitance resulting from device motion. For example, oxides, other dielectrics, metals, and other semiconductors can be deposited or patterned with spatial variations. These spatial variations in dielectric constant will result in time variations of capacitance when components of the sensor are moved relative to each other. In some examples, both top and bottom surfaces of silicon used to form a proof mass include vertical corrugations. In some examples, both top and bottom cap wafers surrounding the device layer of silicon include vertical corrugations. In some examples, one or more of spatial variations in material, corrugation of the top of the device layer of silicon, corrugation of the bottom device layer of silicon, corrugation of the top cap wafer, and corrugation of the bottom cap wafer are used to form the sensor. In some examples, a vernier capacitor structure is used to form the sensor. 
     Signals output by the systems and methods described herein can include acceleration forces, rotational forces, rotational accelerations, changes in pressure, changes in system temperature, and magnetic forces. In some examples, the output signal is a measure of the variation or stability of the amplitude of a periodic signal, such as the oscillator displacement. In some examples, the output signal is a measurement in the variation or stability of the frequency of the periodic signal. In some examples, the output is a measurement of the variation or stability of the phase of the periodic signal. In some examples, the output signal includes a measurement of time derivatives of acceleration, such as jerk, snap, crackle, and pop, which are the first, second, third, and fourth time derivatives of acceleration, respectively. 
     In addition to measuring the inertial parameters from time intervals, in some examples, periodicity in physical structures is utilized to detect relative translation of one of the structures by tracking rising and falling edges caused by local extrema of capacitance, these local extrema of capacitance corresponding to translation of multiples of one half-pitch of the structure periodicity. The number of edges counted can be translated into an external acceleration. In some examples, an oscillation is applied to the physical structure, and in other examples, no oscillation force is applied to the physical structure. 
     A nonlinear least-squares curve fit, such as the Levenburg Marquardt curve fit, can be used to fit the periodic signal to a periodic equation such as equation (16). 
         A  sin( Bt++Dt+E   (16)
 
     In equation (10), A represents amplitude, B represents frequency, C represents phase, E represents the offset of an external acceleration force, and D represents the first derivative of the external acceleration force, or the time-varying component of acceleration of the measurement. The measurement period is one-half of the oscillation cycle. Additionally, higher-order polynomial terms can be included for the acceleration as shown in equation (17). 
         A  sin( Bt+C )+ Dt   3   +Et   2   +Ft+G+ . . .   (17)
 
     In some examples, the input perturbing acceleration force can be modeled as a cosine function as shown in equation (18), in which D and E represent the amplitude and frequency of the perturbing acceleration force, respectably. 
         A  sin( Bt+C )+ D  cos( Et )  (18)
 
     If the external perturbing acceleration is small in comparison to the internal acceleration of the oscillator itself, a linear approximation may be used to model the perturbing acceleration. In this case, the offset modulation is taken to be small in comparison to the overall amplitude of the generated periodic signal. By doing so, a measurement of a single time period can be taken to be linearly proportional to the external perturbing force. In some examples, multiple time periods may be linearly converted into acceleration and then averaged together to obtain lower noise floors and higher resolution. 
     In some examples, analysis in the frequency domain may be performed based on the periodic nature of the nonlinear signals being generated, as well as their respective phases. Frequency domain analysis can be used to reject common-mode noise. Additionally, the non-zero periodic rate of the signal can be used to filter out low frequency noise or to high-pass or band-pass the signal itself to mitigate low-frequency drift. 
       FIG. 25  is a graph showing the effects of an external perturbation on the output signal of the multiple degrees of freedom inertial sensor, according to an illustrative implementation. The graph  2500  includes the TIA output curve  2502 , a displacement curve  2504 , and an input acceleration curve  2506 .  FIG. 25  also depicts the reference pitch locations +d 0    2508 , +d 0 /2  2510 , 0  2512 ,−d 0 /2  2514 , and −d 0    2516 , where d 0  is the pitch between teeth of a TDS structure, as described in further detail with reference to  FIG. 16 . The graph  2500  depicts the same signals depicted in the graph  2400  of  FIG. 24 , with the x axis of  2500  representing a longer duration of time than is shown in the graph  2400 . The periodicity of the input acceleration curve  2506  is more easily discerned at this time scale. In addition, maximum displacement crossings  2520  and minimum displacement crossings  2522  can be discerned in the graph  2500  to experience a similar periodicity. In contrast to the maximum displacement crossings  2520  and the minimum displacement crossings  2522 , the amplitude of which varies with time, zero-crossings of the TIA output signal  2502  triggered by alignment or anti-alignment of teeth of the fixed and movable elements at the locations +d 0    2508 , +d 0 /2  2510 , 0  2512 ,−d 0 /2  2514 , and −d 0    2516  are time invariant. These reference crossings, the amplitude of which are stable with time, provide stable, drift-independent indications of sense mass displacement and can be used to extract inertial parameters. 
       FIG. 26  is a graph depicting the capacitance as a function of the displacement of a sense mass of a multiple degrees of freedom inertial sensor, according to an illustrative implementation.  FIG. 26  includes a capacitance curve  2602  that is periodic and substantially sinusoidal. Thus, monotonic motion of the movable element, such as described with reference to  FIG. 16 , produces a capacitance that changes non-monotonically with displacement. This non-monotonic change is a function of the geometric structure of the TDS structures shown with reference to  FIG. 16 , and the manner in which the multiple degrees of freedom inertial sensor is excited. 
       FIG. 27  is a graph depicting the first spatial derivative of capacitance as a function of the displacement of a sense mass of a multiple degrees of freedom inertial sensor, according to an illustrative implementation.  FIG. 27  includes a dC/dx curve  2702  which is periodic and substantially sinusoidal. The dC/dx curve  2702  is the first derivative of the capacitance curve  2602 . As such, the dC/dx curve  2702  crosses zero when the capacitance curve  2602  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  2702 . 
       FIG. 28  is a graph depicting the second spatial derivative of capacitance as a function of the displacement of a sense mass of a multiple degrees of freedom inertial sensor, according to an illustrative implementation.  FIG. 28  includes a d 2 C/dx 2  curve  2802 . The dC/dx 2  curve  2802  is the first derivative of the dC/dx curve  2702  and as such has a value of zero at local extrema of the dC/dx curve  2702 . The d 2 C/dx 2  curve  2802  indicates the slope of the dC/dx curve  2702  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  2802  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 error 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 sense mass motion. 
       FIG. 29  is a graph depicting the time derivative of the capacitive current as a function of displacement of a sense mass of a multiple degrees of freedom inertial sensor, according to an illustrative implementation.  FIG. 29  includes a dI/dt curve  2902 . The capacitive current used to determine the dI/dt curve  2902  is obtained by applying a fixed voltage across the capacitor used to produce the capacitive curve  2602 . The dI/dt curve  2902  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 sense mass used to generate the curves shown in  FIGS. 26-29  oscillates about zero displacement and reverses direction at minimum and maximum displacements, the velocity of the sense mass is lowest at its extrema of displacement. At these displacement extrema, the current is also changing less rapidly and thus the dI/dt curve  2902  has a lower magnitude. Using zero-crossings at which the dI/dt curve  2902  has large values results in improved timing resolution and decreased jitter. These zero-crossings occur near the center of the sense mass&#39;s range. 
       FIG. 30  is a graph depicting the displacement offsets of two sense masses as a result of common mode error, according to an illustrative implementation. As shown in graph  3000 , two signals  3002  and  3004  may be produced as a result of the oscillation of two sense masses. Signal  3002  may be produced by a TDS structure coupled to one sense mass, while signal  3004  may be produced by a separate TDS structure coupled to a second sense mass.  FIG. 30  depicts the affect of common mode error on the signals  3002  and  3004  produced from the sense mass oscillation. As shown in  FIG. 30 , the common mode error may result in offsets of the two sense masses in the absence of an inertial or external force. These offsets are shown in  FIG. 30  at  3006  and  3008 , and may correspond to physical offsets of the sense mass oscillations, as shown with reference to  FIG. 14 . These offsets  3006  and  3008  that result from common mode error cause the zero crossing points of each signal  3002  and  3008  to shift as well, where the time interval between zero crossings  3010 ,  3012 ,  3014 ,  3016 ,  3018 , and  3020  becomes either shorter, as shown at  3024 , or longer, as shown at  3022 . The shifted time intervals that result from the offsets  3006  and  3008  may cause the multiple degrees of freedom inertial sensor to detect a non zero inertial parameter even in the absence of any inertial forces or perturbations if only a single signal  3002  or  3004  is used to determine the inertial parameter. For any N degree of freedom sensor, there may be N corresponding signals produced from each of the N oscillating sense masses. The signals  3002  and  3004  may be produced from sense masses  1402  and  1404  with reference to  FIG. 14 , respectively. Thus the offsets may be the result of package deformations of the multiple degrees of freedom inertial sensor. 
       FIG. 31  is a graph depicting the results of differential sensing on the sensed displacement of a multiple degrees of freedom inertial sensor, according to an illustrative implementation. As shown in graph  3100 , the single signal  3102  may result from the linear combination of signals  3002  and  3004  as shown with reference to  FIG. 30 . Signal  3102  may be the result of subtracting signal  3002  from  3004 . As shown in  FIG. 31 , the offsets  3006  and  3008 , which affect each signal path equally, may be removed from the resulting differential signal  3102 , such that the differential signal  3102  oscillates about the x axis  3112  corresponding to zero inertial forces or outside perturbations. As can be seen in  FIG. 31 , the time intervals between zero crossings  3108 ,  3106 ,  3104 , as shown at  3110 , may be regular intervals. The signal  3102  may be produced from the differential sensing of any N degree of freedom sensor, as a result of the linear combination of any N oscillating sense masses. As a result of this differential sensing, the multiple degrees of freedom inertial sensor may correctly detect the absence of outside forces, despite offsets that result in each individual signal produced by each sense mass as a result of package deformations or common mode error. 
       FIG. 32  is a graph representing position of a sense mass relative to time, according to an illustrative implementation. The curve  3202  represents the sinusoidal oscillation of a sense mass about a central anchor. The oscillation shown in  FIG. 32  may be the oscillation of any one of the sense masses described herein. The horizontal axis of  FIG. 32  represents time normalized by period of the sense mass, meaning that  FIG. 32  represents a full period of oscillation of the sense mass. The sense mass shown in  FIG. 32  has a resonant frequency of 2 kHz and thus a period of 500 μs. 
       FIG. 33  is a graph representing velocity of a sense mass relative to time, according to an illustrative implementation. The curve  3302  depicted in  FIG. 33  represents velocity of the sinusoidal oscillation of a sense mass about a central anchor. The oscillation shown in  FIG. 33  may be the oscillation of any one of the sense masses described herein. The curve  3302  is the first time derivative of the curve  3202  as shown in  FIG. 32 . 
       FIG. 34  is a graph representing acceleration of a sense mass relative to time, according to an illustrative implementation. The curve  3402  represents acceleration of the sinusoidal oscillation of a sense mass about a central anchor. The oscillation shown in  FIG. 34  may be the oscillation of any one of the sense masses described herein. The curve  3402  is the first time derivative of the curve  3302  as shown in  FIG. 33 , and the second time derivative of the curve  3202  as shown in  FIG. 32 . 
       FIGS. 32-34  show the relationship between the displacement of a sense mass as shown in  FIG. 32  and the inertial parameters velocity and acceleration as shown in  FIG. 33  and  FIG. 34  respectively. The curves  3202 ,  3302  and  3402  may represent the oscillation of a sense mass in the absence of external perturbations other than the drive force produced by drive structures to actuate the sense mass. As can be seen, local extrema of one signal may translate to a zero-crossing in another. 
       FIG. 35  is a graph representing capacitance relative to angular position of a sense mass, according to an illustrative implementation.  FIG. 35  represents changes in capacitance of a first and second electrode as a sense mass oscillates about a central anchor.  FIG. 35  may represent an output signal produced by the electrodes  1204   a  and  1204   b  as depicted in  FIG. 12 . The signals  3502  and  3504  may be produced in response to the motion depicted in any of the  FIG. 4, 5, 7 or 8 . As shown in  FIG. 12 , because the electrode  1204   a  located at the larger radius  1212  experiences a larger change in position than the electrode  1204   b  located at the smaller radius  1210  for the same angular displacement, the electrode  1204   a  experiences a larger change in capacitance as well. Thus curve  3502  shows the change in capacitance of electrode  1204   a , while curve  3504  shows the change in capacitance of electrode  1204   b . At the angular positions  3508  and  3506 , the capacitance of the two electrodes is equal. As depicted in  FIG. 35 , these angular positions are approximately +/−0.124°. The magnitudes of capacitance curves  3502  and  3506  may vary due to applied bias, rotational mass velocity, temperature, electronic drift, and other such factors, but the physical, angular positions at which the capacitances equal each other are defined by the geometry of the sense mass  1208  and position of the electrodes  1204   a  and  1204   b , and will therefore be invariant under any changes in these outside factors. Thus using differential signal processing, where the curve  3502  and  3504  may be linearly combined and subtracted from each other, the locations  3508  and  3506  will correspond to positions at which the differential in capacitance is equal to zero. As the sense mass oscillates, the differential capacitance can be measured and the times at which the sense mass passes these predetermined angular positions can therefore be determined. 
       FIG. 36  is a graph representing capacitive slope relative to angular position of a sense mass, according to an illustrative implementation. The curves  3602  and  3604  represent changes in the capacitive slope of the capacitance produced by first and second electrodes as a sense mass oscillates about a central anchor. The electrodes may be the electrodes  1204   a  and  1204   b  as depicted with reference to  FIG. 12 . The curve  3602  may correspond to the capacitive slope of electrode  1204   a , while the curve  3604  may correspond to the capacitive slope of electrode  1204   b . The curve  3602  is the first spatial derivative of curve  3502 , and the curve  3604  is the first spatial derivative of curve  3504  as depicted with reference to  FIG. 35 . 
       FIG. 37  is a graph representing capacitive curvature relative to angular position of a sense mass, according to an illustrative implementation. The curves  3702  and  3704  represent changes in the capacitive curvature produced by first and second electrodes as a sense mass oscillates about a central anchor. The electrodes may be electrodes  1204   a  and  1204   b  as depicted with reference to  FIG. 12 . The curve  3702  may correspond to electrode  1204   a , while the curve  3704  may correspond to electrode  1204   b . The curve  3704  is the first spatial derivative of curve  3604 , while the curve  3702  is the first spatial derivative of curve  3602  as depicted with reference to  FIG. 36 . The curve  3702  is the second spatial derivative of the curve  3502 , while the curve  3704  is the second spatial derivative of curve  3504 , as depicted in with reference to  FIG. 35 . 
       FIG. 38  is a graph representing capacitance relative to time and produced in response to oscillations of a sense mass, according to an illustrative implementation. The curves  3802  and  3804  represent changes in the capacitive curvature produced by first and second electrodes as a sense mass oscillates about a central anchor. The electrodes may be electrodes  1204   a  and  1204   b  as depicted with reference to  FIG. 12 . The curve  3802  may be produced by the electrode  1204   a , while the curve  3804  may be produced by the electrode  1204   b . The capacitance can be measured by one or more capacitance-to-voltage (C-to-V) converters. A C-to-V converter can be a charge amplifier, a switch capacitor, a bridge with a general impedance converter (GIC), or another analog front end that produces a voltage corresponding to a measured charge or capacitance. 
       FIG. 39  is a graph representing capacitive slope relative to time and produced in response to oscillations of a sense mass, according to an illustrative implementation. The curves  3902  and  3904  represent changes in the capacitive slope produced by first and second electrodes as a sense mass oscillates about a central anchor. The electrodes may be electrodes  1204   a  and  1204   b  as depicted with reference to  FIG. 12 . The curve  3902  may be produced by the electrode  1204   a , while the curve  3904  may be produced by the electrode  1204   b . The curve  3902  is the first time derivative of curve  3802 , while the curve  3904  is the first time derivative of curve  3804  as shown in  FIG. 38 . The curves  3902  and  3904  can be measured by an analog front end that measures current, such as a transimpedance amplifier (TIA). 
       FIG. 40  is a graph representing capacitive curvature relative to time and produced in response to oscillations of a sense mass, according to an illustrative implementation. The curves  4002  and  4004  represent changes in the capacitive curvature produced by first and second electrodes as a sense mass oscillates about a central anchor. The electrodes may be electrodes  1204   a  and  1204   b  as depicted with reference to  FIG. 12 . The curve  4002  may be produced by the electrode  1204   a , while the curve  4004  may be produced by the electrode  1204   b . The curve  4002  is the first time derivative of curve  3902 , while the curve  4004  is the first time derivative of curve  3904  as shown in  FIG. 39 . The curve  4002  is the second time derivative of curve  3802 , while the curve  4002  is the second time derivative of curve  3802 . As the second time derivatives, curves  4002  and  4004  represent the rates at which the capacitive slopes change. 
       FIG. 41  is a graph representing differential capacitance relative to time and produced in response to oscillations of a sense mass, according to an illustrative implementation. The curve  4102  is the difference of the curves  3802  and  3804  as shown with reference to  FIG. 38 . The curve  4102  can be obtained by measuring the difference of capacitance between the first and second electrodes. The electrodes may be electrodes  1204   a  and  1204   b  as depicted with reference to  FIG. 12 . This may be measured by a differential amplifier, or the capacitance curves  3802  and  3804  can be measured separately and the difference obtained via analog or digital signal processing. The time at which the curve  4102  equals zero are the zero-crossing times shown at  4104 . These zero-crossing times are the times at which the capacitances of the first and second electrodes are equal. These zero-crossing times correspond to the predetermined angular positions at which the two electrodes have the same capacitance. The times shown at  4104  may be detected via analog means and can be converted to a digital signal by a time-to-digital converter (TDC). The digital signal produced by the TDC can be a binary signal that toggles between high and low signals when the zero-crossings  4104  are detected. By measuring the times at which zero-crossings  4104  occur, the time at which the sense mass is at a predetermined angular position may also be determined. 
       FIG. 42  is a graph representing differential capacitive slope relative to time and produced in response to oscillations of a sense mass, according to an illustrative implementation. The curve  4202  represents changes in the slope of differential capacitance between a first and second electrode as a sense mass oscillates about a central anchor. The electrodes may be electrodes  1204   a  and  1204   b  as depicted with reference to  FIG. 12 . The curve  4202  can be obtained by a differential measurement of current from the first and second electrodes. Alternatively, the curve  4202  can be obtained by differentiating the curve  4102  as shown in  FIG. 41  using digital signal processing. The extrema  4204  of the curve  4202  correspond to the zero-crossings of the curve  4102  as shown in  FIG. 41 . Thus, the zero-crossings  4104  can be measured by peak detection of the curve  4202 . This peak detection can be performed via analog or digital means. Furthermore, the magnitude of the capacitive slope curve depicted in  FIG. 4202  corresponds to the steepness of the curve  4102 . A steeper slope at zero-crossing times results in lower timing uncertainty of the zero-crossing time measurements. 
       FIG. 43  is a graph representing differential capacitive curvature relative to time and produced in response to oscillations of a sense mass, according to an illustrative implementation. The curve  4302  represents changes in curvature of differential capacitance between two electrodes as a sense mass oscillates about a central anchor. The electrodes may be electrodes  1204   a  and  1204   b  as depicted with reference to  FIG. 12 . The curve  4302  can be obtained by differentiating the curve  4202  as shown in  FIG. 42  using analog or digital signal processing. The magnitude of the curvature curve  4302  represents the steepness of the slope of the curve  4202 . A higher magnitude of curvature will result in lower timing uncertainty of peak detection measurements of the curve  4302 . 
     The zero-crossing times determined as described with respect to  FIG. 35-43  can be used to determine time periods for use in a cosine method describe with respect to  FIG. 12 . Thus, by using the cosine method and zero-crossing times corresponding to predetermined physical positions of a sense mass, the amplitude, frequency, and offset of the sense mass can be determined. Inertial parameters of the multiple degrees of freedom sense or can be determined from the amplitude, frequency and offset of its sense masses. 
       FIG. 44  is a graph representing capacitance relative to the vertical position of a sense mass, according to an illustrative implementation.  FIG. 44  represents changes in capacitance of a first and second electrode as a sense mass oscillates about a central anchor.  FIG. 44  may represent an output signal produced by the electrodes  1306   a  and  1306   b  as depicted in  FIG. 13 . The oscillation of the sense mass may entail raising and lowering in only the vertical direction as shown in  FIG. 13 . The signals  4402  and  4404  may be produced in response to the motion of a sense mass as shown in  FIG. 13 . Because the electrodes  1306   a  and  1306   b  have different heights (as shown at the gap  1310  in  FIG. 13 ) and are thus aligned with the stationary electrode comprising the sense mass  1312  at different vertical positions, the capacitive curves  4402  and  4404  have local extrema  4406  and  4408  at different vertical positions. The local maximum of each curve corresponds to the vertical position at which the moving electrode positioned on the oscillating sense mass is aligned with the stationary electrode. The vertical position corresponding to the local maximum depends only on the geometry of the stationary electrodes and the moving sense mass. Thus, although the magnitude of capacitance may vary due to bias, sense mass velocity, temperature, electronic drift, or other factors, the vertical position in which the maximum of capacitance occurs for each electrode remains constant. By determining times at which these maxima  4406  and  4408  occur, the times at which the sense mass is in the corresponding vertical position may be determined. 
       FIG. 45  is a graph representing capacitive slope relative to the vertical position of a sense mass, according to an illustrative implementation. The curves  4504  and  4502  represent changes in the capacitive slope of the first and second electrodes as the sense mass oscillates about a central anchor. The electrodes may be  1306   a  and  1306   b  as shown in  FIG. 13 . The curve  4504  may be the first spatial derivative of the curve  4404 , while the curve  4502  may be the first spatial derivative of curve  4402 , as shown in  FIG. 44 . 
       FIG. 46  is a graph representing capacitive curvature relative to the vertical position of a sense mass, according to an illustrative implementation. The curves  4602  and  4604  represent changes in the capacitive curvature of the first and second electrodes as the sense mass oscillates about a central anchor. The electrodes may be  1306   a  and  1306   b  as shown in  FIG. 13 . The curve  4602  may be the first spatial derivative of curve  4502 , while the curve  4604  may be the first spatial derivative of curve  4504 , as shown in  FIG. 45 . 
       FIG. 47  is a graph representing capacitance relative to time and produced in response to oscillations of a sense mass, according to an illustrative implementation. The curves  4702  and  4704  represent changes in capacitance of the first electrode and the second electrode as a sense mass oscillates about a central anchor. The electrodes may be  1306   a  and  1306   b  as shown in  FIG. 13 . The times at which the capacitance experiences a local extremum correspond to either times of zero velocity or times at which the moving sense mass is aligned with the stationary electrode, causing a local maximum in capacitance. 
       FIG. 48  is a graph representing capacitive slope relative to time and produced in response to oscillations of a sense mass, according to an illustrative implementation. The curves  4802  and  4804  represent changes in capacitance of the first electrode and the second electrode as a sense mass oscillates about a central anchor. The electrodes may be  1306   a  and  1306   b  as shown in  FIG. 13 . The curve  4802  is the first time derivative of curve  4702 , while the curve  4804  is the first time derivative of curve  4704 , as shown in  FIG. 47 . Thus the curves  4802  and  4804  represent the rates at which capacitance changes. The capacitive slopes  4802  and  4804  can be measured by an analog front end, such as a TIA, that measures current. The times at which the capacitive slope is equal to zero correspond to times at which the capacitance is at a local extremum or inflection point. These times may correspond to times at which a sense mass is at zero velocity, or times at which the sense mass is aligned with the stationary electrode, causing a local maximum in capacitance. By determining times at which the capacitive slope crosses zero (or zero-crossing times), the corresponding times at which the sense mass is at a predetermined position with respect to the stationary electrode can be determined. 
       FIG. 49  is a graph representing capacitive curvature relative to time and produced in response to oscillations of a sense mass, according to an illustrative implementation. The curves  4902  and  4904  represent changes in the capacitive curvature of the first and second electrodes as a sense mass oscillates about a central anchor. The curve  4902  is the first time derivative of the curve  4802 , while the curve  4904  is the first time derivative of the curve  4804 , as shown in  FIG. 48 . 
       FIG. 50  depicts a flow chart of a method for extracting inertial parameters from a nonlinear periodic signal, according to an illustrative implementation. At  5002 , a first nonlinear periodic signal is received. At  5004 , 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 sensing structures described herein, such as structures depicted in  FIGS. 1-16, 18 and 20 . 
     At  5006 , optionally, the first and second nonlinear periodic signals are combined into a combined signal. This can be accomplished by the element  1706 . If the steps  5004  and  5006  are omitted, the method  5000  proceeds from  5002  directly to  5008 . 
     At  5008 , the signal is converted to a two-valued signal. 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 may be converted by an element such as the element  1706 , and can be one or more of the signals  1712  or  2336 . 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  5010 , 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) such as one or both of the elements  2514  and  3616 . The time intervals determined in this way can be one or more of the intervals  2516 ,  2832 ,  2834 ,  3040 , and  3042 . 
     At  5014 , 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  5016 , 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 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. 51  depicts a flow chart of a method for determining transition times between two values based on a nonlinear periodic signal, according to an illustrative implementation. The method  5100  can be used to perform one or more of the steps  5002 ,  5004 ,  5006 ,  5008 , and  5010  of the method  5000 . 
     At  5102 , a first value of a first nonlinear of a nonlinear periodic signal is received. At  5104 , a second value of a second nonlinear periodic signal is optionally received. 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  5100  can be the same as the first and second nonlinear periodic signals of the method  5000 . 
     At  5106 , the first and second values are optionally combined into a combined value. The values may be combined using the element  1706 . Combining may include summing the values, taking a difference of the values, multiplying the values, or dividing the values. If the optional steps  5104  and  5106  are omitted, the method  5100  proceeds from  5102  directly to  5108 . 
     At  5108 , the first value or the combined value is compared to a threshold. If the value is above the threshold, the method  5100  proceeds to  5110 . 
     At  5110 , a high value is assigned for the current time. If the value is not above the threshold, the method  5100  proceeds to  5112 . At  5112 , a low value is assigned for the current time. The steps  5108 ,  5110  and  5112  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  5100  can be the same as the signal of the method  5000 . 
     At  5114 , 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  5100  proceeds to  5116  where the method  5100  terminates. If the two values are not the same, a transition has occurred and the method proceeds to  5118 . 
     At  5118 , 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  5100  proceeds to  5122 . At  5122 , a falling edge is assigned to the transition. Thus, times having transitions are detected and classified as having either rising or falling edges. At  5124 , 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. 52  depicts a flow chart of a method for computing inertial parameters from time intervals, according to an illustrative implementation. The method  5200  can be used to perform one or more of the steps  5014  and  5016  of the method  5000 . 
     At  5202 , first and second time intervals are received. The first and second time intervals can be determined using the method  5100 . 
     At  5204 , a sum of the first and second time intervals is computed. The sum can be the measured period as described by equations 6 and 7. At  5206 , 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 5. 
     At  5208 , an argument is computed using the ratio. The argument can be one or more of the arguments of the cosine functions of equation 5. 
     At  5210 , a trigonometric function is applied to the argument. The trigonometric function can be any of the trigonometric functions described with respect to step  5004  of the method  5000 . 
     At  5212 , a displacement is computed using one or more geometric parameters and the result of applying the trigonometric function. The displacement can be computed using equation 5. Computing displacement can involve computing more than one trigonometric function, and arguments other than the computed argument of  5208  can be included as arguments of some of the trigonometric functions. 
     At  5214 , one or more inertial parameters are computed using the displacement. The inertial parameters computed can be any of the inertial parameters described with respect to step  5016  of the method  5000 . 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.