Abstract:
A Micro-Electro-Mechanical Systems (MEMS) inertial sensor systems and methods are operable to determine linear acceleration and rotation. An exemplary embodiment applies a first linear acceleration rebalancing force via a first electrode pair to a first proof mass, applies a second linear acceleration rebalancing force via a second electrode pair to a second proof mass, applies a first Coriolis rebalancing force via a third electrode pair to the first proof mass, applies a second Coriolis rebalancing force via a fourth electrode pair to the second proof mass, determines a linear acceleration corresponding to the applied first and second linear acceleration rebalancing forces, and determines a rotation corresponding to the applied first and second Coriolis rebalancing forces.

Description:
BACKGROUND OF THE INVENTION 
     Micro-Electro-Mechanical Systems (MEMS) inertial measurement units contain three gyroscopes and three accelerometers for detecting changes in attitude and acceleration. Typically, the three gyroscopes and the three accelerometers are mounted on separate orthogonal axis, each with their own set of control and read-out electronics. It is appreciated that there is an inherent cost in the assembly of the MEMS inertial measurement unit in view that the three gyroscopes and the three accelerometers must be precisely installed, in view that a relatively large amount of processing capacity is required to process information form six separate units, and in view of the power source requirements to power the three gyroscopes and the three accelerometers. Many applications require a reduction in size, computational requirements, power requirements, and cost of a MEMS inertial measurement unit. In view of these constraints, it would be advantageous to reduce the number of sensing devices in a MEMS inertial measurement unit. 
     A conventional MEMS gyroscope may be used to determine angular rotation by measuring Coriolis forces exerted on resonating proof masses. A conventional MEMS gyroscope includes two silicon proof masses mechanically coupled to and suspended from a substrate, typically glass, using one or more silicon flexures. A number of recesses etched into the substrate allow selective portions of the silicon structure to move back and forth freely within an interior portion of the device. In certain designs, substrates can be provided above and below the silicon structure to sandwich the proof masses between the two substrates. A pattern of metal traces formed on the substrate(s) can be used to deliver various electrical bias voltages and signal outputs to the device. 
     A drive system for many MEMS gyroscopes typically includes a number of drive elements that cause the proof mass to oscillate back and forth along a drive axis perpendicular to the direction in which Coriolis forces are sensed. In certain designs, for example, the drive elements may include a number of interdigitated vertical comb fingers configured to convert electrical energy into mechanical energy using electrostatic actuation. Such drive elements are described, for example, in U.S. Pat. No. 5,025,346 to Tang et al., entitled “LATERALLY DRIVEN RESONANT MICROSTRUCTURES,” and U.S. Pat. No. 7,036,373 to Johnson et al., entitled “MEMS GYROSCOPE WITH HORIZONTALLY ORIENTED DRIVE ELECTRODES,” both of which are incorporated herein by reference in their entirety. 
     Other types of MEMS devices may be used to measure both linear acceleration and rotation. However, such MEMS devices are operated in an open loop mode wherein the acceleration and rotation (gyro) responses are coupled with and depend on each other. Accordingly, systems that independently measure both linear acceleration and rotational movement require at least two different devices so that the acceleration sensing is decoupled from the rotation sensing, which may result in increased complexity and costs. 
     SUMMARY OF THE INVENTION 
     Systems and methods of determining linear acceleration and rotation using a Micro-Electro-Mechanical Systems (MEMS) inertial sensor are disclosed. An exemplary embodiment has a first proof mass; a second proof mass; a first electrode pair operable to apply a first linear acceleration rebalancing force to the first proof mass; a second electrode pair operable to apply a second linear acceleration rebalancing force to the second proof mass; a third electrode pair operable to apply a first Coriolis rebalancing force to the first proof mass; and a fourth electrode pair operable to apply a second Coriolis rebalancing force to the second proof mass. 
     In accordance with further aspects, an exemplary embodiment applies a first linear acceleration rebalancing force via a first electrode pair to a first proof mass, applies a second linear acceleration rebalancing force via a second electrode pair to a second proof mass, applies a first Coriolis rebalancing force via a third electrode pair to the first proof mass, applies a second Coriolis rebalancing force via a fourth electrode pair to the second proof mass, determines a linear acceleration corresponding to the applied first and second linear acceleration rebalancing forces, and determines a rotation corresponding to the applied first and second Coriolis rebalancing forces. 
     In accordance with further aspects, another exemplary embodiment senses a change in a first capacitance between a first electrode of a first electrode pair and a first proof mass, senses a change in a second capacitance between a second electrode of the first electrode pair and the first proof mass, senses a change in a third capacitance between a first electrode of a second electrode pair and a second proof mass, senses a change in a fourth capacitance between a second electrode of the second electrode pair and the second proof mass, senses a change in a fifth capacitance between a first electrode of a third electrode pair and the first proof mass, senses a change in a sixth capacitance between a second electrode of the third electrode pair and the first proof mass, senses a change in a seventh capacitance between a first electrode of a fourth electrode pair and the second proof mass, and senses a change in an eighth capacitance between a second electrode of the fourth electrode pair and the second proof mass. The embodiment is operable to determine a linear acceleration from the sensed first capacitance, the sensed second capacitance, the sensed third capacitance, and the sensed fourth capacitance. The embodiment is also operable to determine a rotation from the sensed fifth capacitance, the sensed sixth capacitance, the sensed seventh capacitance, and the sensed eighth capacitance. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Preferred and alternative embodiments are described in detail below with reference to the following drawings: 
         FIG. 1  is a conceptual perspective view of electrodes and proof masses for a portion of an embodiment of the inertial sensor; 
         FIG. 2  is a conceptual perspective view of electrodes and proof masses for an alternative embodiment of the inertial sensor; 
         FIG. 3  is a conceptual side view of an embodiment of the inertial sensor; 
         FIG. 4  is a conceptual side view of an embodiment of the inertial sensor with applied initialization rebalancing forces; 
         FIG. 5  is a conceptual side view of an embodiment of the inertial sensor with an applied linear acceleration; 
         FIG. 6  is a conceptual side view of an embodiment of the inertial sensor with an applied rotation; 
         FIGS. 7-9  illustrate applied and sensing voltages for embodiments of the inertial sensor; and 
         FIG. 10  is a block diagram illustrating an exemplary implementation of a digital signal processing system coupled to an embodiment of the inertial sensor. 
     
    
    
     DETAILED DESCRIPTION OF THE PREFERRED EMBODIMENT 
     Embodiments of the inertial sensor  100  decouple acceleration sensing and rotation sensing so that rotation and acceleration are independently determinable.  FIG. 1  is a block diagram of a portion of an embodiment of an inertial sensor  100 . The exemplary portion of the inertial sensor  100  is operable to sense either linear acceleration or rotation. Other portions of the inertial sensor  100  that sense rotation are described and illustrated below. 
     The illustrated portion of the inertial sensor  100  comprises a first proof mass  102  (interchangeably referred to herein as the left proof mass  102 ) and a second proof mass  104  (interchangeably referred to herein as the right proof mass  104 ). The left proof mass  102  is between an upper sense electrode  106  (interchangeably referred to herein as the upper left sense (ULS) electrode  106 ) and a lower sense electrode  108  (interchangeably referred to herein as the lower left sense (LLS) electrode  108 ). The right proof mass  104  is between an upper sense electrode  110  (interchangeably referred to herein as the upper right sense (URS) electrode  110 ) and a lower sense electrode  112  (interchangeably referred to herein as the lower right sense (LRS) electrode  112 ). 
     The left proof mass  102  is separated from the ULS electrode  106  by a gap (G ULS ) which defines a capacitance that is dependent upon the separation distance between the left proof mass  102  and the ULS electrode  106 . Similarly, the left proof mass  102  is separated from the LLS electrode  108  by a gap (G LLS ) which defines a capacitance that is dependent upon the separation distance between the left proof mass  102  and the LLS electrode  108 . Changes in the capacitances associated with the gaps G ULS  and G LLS , caused by linear acceleration or rotation is detectable. 
     The right proof mass  104  is separated from the URS electrode  110  by a gap (G URS ) which defines a capacitance that is dependent upon the separation distance between the right proof mass  104  and the URS electrode  110 . Similarly, the right proof mass  104  is separated from the LRS electrode  112  by a gap (G LRS ) which defines a capacitance that is dependent upon the separation distance between the right proof mass  104  and the LRS electrode  112 . Changes in the capacitances associated with the gaps G URS  and G LRS , caused by linear acceleration or rotational movement, is detectable. 
     The proof masses  102 ,  104  are capacitively coupled to drive electrodes (not shown) which impart a “back-and-forth” motion to the proof masses  102 ,  104  as an alternating current (AC) voltage is applied to the drive electrodes. The drive electrodes cause the proof masses  102 ,  104  to oscillate back and forth in resonance along a drive axis (the illustrated x axis). The drive axis and the y axis define an in-plane motion of the proof masses  102 ,  104 . The relative direction of motion of the left proof mass  102 , as denoted by the direction vector  114 , is opposite from the direction of motion of the right proof mass  104 , as denoted by the direction vector  116 , during one half cycle of the resonant motion. Thus, the proof masses  102 ,  104  are illustrated as moving away from each other in  FIG. 1 . During the next half cycle of the resonant motion, the proof masses  102 ,  104  move towards each other. It is appreciated that embodiments of the inertial sensor  100  may be implemented in MEMS based devices having various configurations of drive electrodes. 
       FIG. 2  is a conceptual perspective view of electrodes and proof masses for an alternative embodiment of the inertial sensor  100 . Electrode groups  202 ,  204  are oriented above and below the proof masses  102 ,  104 , respectively. Electrodes  206 ,  208 ,  210  and  212  are oriented above and substantially the same distance from the left proof mass  102  as illustrated, and define the gap G ULS . Electrodes  214 ,  216 ,  218  and  220  are oriented below and substantially the same distance from the left proof mass  102  as illustrated, and define the gap G LLS . Electrodes  222 ,  224 ,  226  and  228  are oriented above and substantially the same distance from the right proof mass  104  as illustrated, and define the gap G URS . Electrodes  230 ,  232 ,  234  and  236  are oriented below and substantially the same distance from the right proof mass  104  as illustrated, and define the gap G LRS . In alternative embodiments, the electrodes may be positioned at different distances from their respective proof masses. In other embodiments, arrays of electrodes or a plurality of electrodes may be used to apply one or more of the rebalancing forces described herein. 
     Opposing electrodes form electrode pairs. For example, electrodes  206  and  214  form an electrode pair. Electrode pairs may be operated in relation to each other, as described in greater detail below. Other electrode pairs include electrodes  208  and  216 , electrodes  210  and  218 , electrodes  212  and  220 , electrodes  222  and  230 , electrodes  224  and  232 , electrodes  226  and  234 , and electrodes  228  and  236 . A pair of selected electrodes for each of the proof masses  102 ,  104  corresponds to the above-described electrodes  106  and  108 , or electrodes  110  and  112 . The gaps between the proof mass and each of the electrodes of an electrode pair, when a voltage is applied there across, results in a detectable capacitance. For example, the gap G ULS  between the electrode  208  and the proof mass  102  results in a first capacitance. Similarly, the gap G LLS  between the proof mass  102  and the electrode  216  results in a second capacitance. When the proof mass  102  moves, the above described first and second capacitances change. The capacitance changes may be determined by sensing the change in a current from an amplifier (not shown) coupled to the electrode  208  and/or the electrode  216 . 
       FIG. 3  is a conceptual side view of an embodiment of the inertial sensor  100 . Here, the proof masses  102 ,  104  are illustrated as aligned with each other along the x axis. A flexure  302  supports the left proof mass  102  between the gaps G ULS  and G LLS . A flexure  304  supports the right proof mass  104  between the gaps G URS  and G LRS . The flexures  302  and  304  are attached to anchor  306 . In this exemplary embodiment, anchor  306  is attached to the lower substrate  308 , although the anchor  306  may be attached to the upper substrate  310 , or may be attached to both substrates  308 ,  310 , in alternative embodiments. The flexures  302 ,  304  are flexible members that have spring-like characteristics such that when the proof masses  102 ,  104  are driven by the drive electrodes (not shown), the proof masses  102 ,  104  will resonate. 
     In other embodiments, the anchor  306  may be attached to the upper substrate  310 . Some embodiments may employ a plurality of flexures to couple the proof masses  102 ,  104  to various anchor points in the MEMS device. In some embodiments, the flexures  302 ,  304  may be connected to different anchors. 
     In the exemplary embodiment of the inertial sensor  100 , the proof masses  102 ,  104  are suspended such that the gaps G ULS  and G LLS , and the gaps G URS  and G LRS , are equal to each other. Accordingly, the upper and lower capacitances associated with the proof masses  102 ,  104  and the illustrated electrodes are substantially equal (with respect to each other). For example, assuming that the surface areas and other characteristics of the electrodes  206 ,  214 ,  228 , and  236  are substantially the same, the capacitance between the electrode  206  and the left proof mass  102 , the capacitance between the electrode  214  and the left proof mass  102 , the capacitance between the electrode  228  and the right proof mass  104 , and the capacitance between the electrode  236  and the right proof mass  104 , are substantially the same. In alternative embodiments, the capacitances may be different from each other. 
     A linear acceleration in a direction along the illustrated z axis causes the proof masses  102 ,  104  to move together in the same direction and at substantially the same rate and/or distance. This movement is referred to herein as movement in a “common mode.” The common mode movement of the proof masses  102 ,  104  causes substantially the same change in the electrode-to-proof mass capacitance of electrode pairs across the gaps G ULS  and G URS , and substantially the same change in the electrode-to-proof mass capacitance of the electrode pairs across gaps G LLS  and G LRS . That is, assuming that the upper and lower gaps (G URS , G LRS , G ULS , and G URS ) are the same (i.e.: balanced), the magnitudes of the changed capacitance of electrode pairs across the gaps G ULS  and G URS , and the magnitudes of the changed capacitance of the electrode pairs across gaps G LLS  and G LRS , are substantially the same. If the gaps G ULS , G LLS , G URS , and G LRS , are unbalanced, the upper capacitances vary substantially the same amount, and the lower capacitances vary substantially the same amount, since the forces which move the proof masses  102 ,  104  that result in the change of these capacitances are nearly equal. Linear acceleration can be determined from the sensed common mode changes in capacitance. 
     Further, a rotation in a direction around the illustrated y axis causes the proof masses  102 ,  104  to move in opposite directions and at substantially the same rate and/or distance in the z direction. This movement is referred to herein as movement in a “differential mode.” The differential mode movement of the proof masses  102 ,  104  is caused by Coriolis forces. This differential mode movement of the proof masses  102 ,  104  (movement in opposite directions) causes substantially the same magnitudes of change in the electrode-to-proof mass capacitance of electrode pairs across the gaps G ULS  and G LRS , and substantially the same magnitudes of change in the electrode-to-proof mass capacitance of the electrode pairs across gaps G LLS  and G URS . Rotation can be determined from the sensed differential mode changes in capacitance. 
     As noted above, embodiments of the inertial sensor  100  provide decoupling between acceleration sensing and rotation sensing so that rotation and acceleration are independently sensed and determined. In the preferred embodiment, the quadrature forces, which are ninety degrees out-of-phase from the Coriolis forces, are also decoupled from the acceleration and Coriolis forces. Accordingly, rebalancing forces for linear acceleration, Coriolis, and/or quadrature forces are separately applied to electrode pairs to maintain the position of the proof masses  102 ,  104  in a fixed position such that the capacitances associated with the respective electrode pairs across gaps G ULS , G LLS , G URS , and G LRS , are substantially matched. Thus, when an unbalance between the positions of the proof masses  102 ,  104  occurs (detectable from the changes in the electrode-to-proof mass capacitances of the electrode pairs across the gaps G ULS , G LLS , G URS , and G LRS ), rebalancing forces operate to self center the proof masses  102 ,  104 . 
     A Coriolis rebalancing force is applied to proof mass  102  by a selected electrode pair. A Coriolis rebalancing force is also applied to proof mass  104  by another selected electrode pair. The applied Coriolis rebalancing force self centers the proof masses  102 ,  104  during a rotation of the inertial sensor  100 . The magnitude of the required Coriolis rebalancing force corresponds to the amount of rotation. Similarly, an applied linear acceleration rebalancing forces self center the proof masses  102 ,  104  during a linear acceleration of the inertial sensor  100 . The magnitude of the required linear acceleration rebalancing force corresponds to the amount of linear acceleration. Since the linear acceleration rebalancing force is provided by a direct current (DC) voltage applied to selected electrode pairs, the linear acceleration rebalancing force can be differentiated from the Coriolis rebalancing force. That is, because a linear acceleration (which induces a time varying acceleration force in the z-axis) is different from a rotation (which induces a force that is modulated at the drive frequency of the proof masses  102 ,  104 ), the linear acceleration rebalancing force and the Coriolis rebalancing force can be separately determined. 
       FIG. 4  is a conceptual side view of an embodiment of the inertial sensor  100  with applied initialization rebalancing forces  402 , illustrated as vectors  402 . Selected electrodes may be operated to exert an initialization rebalancing force to its respective proof mass  102 ,  104 . Accordingly, the gaps G ULS , G LLS , G URS , and G LRS , may be set to be equal to each other, or set to a desired value. 
     For example, as conceptually illustrated in  FIG. 4 , during fabrication of an inertial sensor  100 , the left proof mass  102  may not be in its designed ideal position  404  between the electrodes. Here, the left proof mass  102  is illustrated in a non-ideal position  406  such that the gaps G ULS  and G LLS  are not substantially equal. The non-ideal position  406  of the left proof mass  102 , even though acceptable from a fabrication perspective, may be sufficiently different from the ideal position  404  as a result of design and/or fabrication tolerances so as to impart inaccuracies in the detection of linear accelerations and/or rotational movement. Initialization rebalancing forces, illustrated as vectors  402 , may be applied by one or more selected electrodes to reposition the left proof mass  102  to, or very near to, its designed ideal position  404 . The initialization rebalancing forces may be equal, or may be unique, depending upon the amount of initialization rebalancing required to position a proof mass into its ideal position. Preferably, the initialization rebalancing forces result from DC biases applied to the selected electrodes. The initialization rebalancing forces may be determined prior to use of the inertial sensor  100 , such as by bench testing after fabrication. 
       FIG. 5  is a conceptual side view of an embodiment of the inertial sensor  100  with an applied linear acceleration, denoted by the acceleration vectors  502  (corresponding to a movement in the negative z axis direction). Inertial forces (illustrated as vectors  504 ) are exerted on the proof masses  102 ,  104 . Accordingly, the proof masses  102 ,  104  are moved towards the upper substrate  310  during the period of acceleration. The flexures  302 ,  304  will operate to return the proof masses  102 ,  104  to their initial positions (see  FIG. 3 ) when the acceleration ceases. 
     The above-described common mode movement of the proof masses  102 ,  104  causes substantially the same change in the electrode-to-proof mass capacitance of electrode pairs across the gaps G ULS  and G LLS , and the electrode pairs across the gaps G URS  and G LRS , respectively. That is, the magnitude of the changed electrode-to-proof mass capacitance of electrode pairs across the gaps G ULS  and G URS , and the magnitude of the changed electrode-to-proof mass capacitance of electrode pairs across the gaps G LLS  and G LRS , are substantially the same. In response to the movement of the proof masses  102 ,  104 , a linear acceleration rebalancing force may be applied via selected electrode pairs to reposition the proof masses  102 ,  104  back to their original position. Linear acceleration can be determined from the amount of the applied linear acceleration rebalancing force and/or from the sensed common mode changes in capacitance. 
       FIG. 6  is a conceptual side view of an embodiment of the inertial sensor  100  with an applied rotation, denoted by the rotation vector  602 , (corresponding to a rotation movement around the y axis). Inertial forces, illustrated as vectors  604  and  606 , are exerted on the proof masses  102 ,  104 , respectively. Accordingly, the proof mass  102  is moved towards the upper substrate  310  during the period of rotation and the proof mass  104  is moved towards the lower substrate  308  during the period of rotation. The flexures  302 ,  304  will operate to return the proof masses  102 ,  104  to their initial positions (see  FIG. 3 ) when the rotation ceases. 
     The above-described differential mode movement of the proof masses  102 ,  104  causes a detectable change in the electrode-to-proof mass capacitance of electrode pairs across the gaps G ULS , G LLS , G URS , and G LRS . The magnitude of the changed electrode-to-proof mass capacitance of electrode pairs across the gaps G ULS  and G LRS , and the magnitude of the changed electrode-to-proof mass capacitance of electrode pairs across the gaps G LLS  and G URS , are substantially the same (assuming initial balancing of the gaps G URS , G LRS , G ULS , and G URS ). In response to the movement of the proof masses  102 ,  104 , a Coriolis rebalancing force may be applied via selected electrode pairs to reposition the proof masses  102 ,  104  back to their original position. Rotation can be determined from the applied Coriolis rebalancing force and/or from the sensed differential mode changes in capacitance. 
       FIG. 7  illustrates applied and sensing voltages for a portion of an embodiment of the inertial sensor  100  illustrated in  FIG. 1 . The voltages V ULS , V LLS , applied by the electrode pair  106  and  108 , and the voltages V URS , and V LRS , applied by the electrode pair  110  and  112 , correspond in part to the linear acceleration rebalancing force. 
     The applied voltages have three components that provide three functions, linear acceleration rebalancing, rotation sense biasing, and acceleration sense pickoff. The applied upper left sense plate voltage (V ULS ) may be defined by equation (1) below:
 
 V   ULS   =−V   SB   −V   A   +V   p  sin(ω p   t )  (1)
 
where V SB  is the applied voltage of the sense bias (a DC bias voltage), where V A  is the voltage of the applied linear acceleration rebalancing force, where V p  is an applied AC pick off voltage, and where ω p  is the frequency of the applied AC pick off voltage V p . The current i SPO  results from imbalances in the position of the proof masses  102 ,  104 .
 
     The applied lower left sense plate voltage (V LLS ), the applied upper right sense plate voltage (V URS ), and the applied lower right sense plate voltage (V LRS ), may be defined by equations (2), (3), and (4), respectively, below:
 
 V   LLS   =V   SB   −V   A   −V   p  sin(ω p   t )  (2)
 
 V   URS   =V   SB   +V   A   +V   p  sin(ω p   t )  (3)
 
 V   LRS   =−V   SB   +V   A   −V   p  sin(ω p   t )  (4)
 
     An amplifier system  702  is communicatively coupled to detect voltages and/or currents from the proof masses  102 ,  104 . The output of the amplifier system  702  corresponds to the sensed pick off voltage, V SPO . V SPO  may be defined by equation (5) below.
 
 V   SPO   =[V   Ω ·cos(ω m   t )]+[ V   Q ·sin(ω m   t )]+[ V   CM ·sin(ω p   t )]  (5)
 
where V Ω  is the portion of V SPO  that is proportional to the rotation motion, where V Q  is the quadrature component of V Ω , where V CM  is the portion of V SPO  that is proportional to the common mode motion (caused by the linear acceleration), and where ω m  is the applied motor frequency.
 
       FIG. 8  illustrates applied and sensing voltages for the embodiment of the inertial sensor  100 . Included are the above-described applied voltages V ULS , V LLS , V URS , and V LRS , corresponding to the linear acceleration rebalancing force, which are applied by the electrode pair  208 ,  216  to proof mass  102 , and by the electrode pair  226 ,  234  to proof mass  104 . Other embodiments may apply the linear acceleration rebalancing force using different selected electrodes. In some embodiments, the electrodes  208 ,  216 ,  226 , and  234  may be used to inject currents (or voltages) used for sensing common mode movement and/or differential mode movement of the proof masses  102 ,  104 . 
     The electrode pair  210 ,  218  provides a Coriolis rebalancing force to the proof mass  102 . Similarly, the electrode pair  224 ,  232  applies a Coriolis rebalancing force to the proof mass  104 . Preferably, the Coriolis rebalancing force applied to the proof mass  102  is opposite in direction and of equal magnitude to the Coriolis rebalancing force applied to the proof mass  104 . Other embodiments may apply the Coriolis rebalancing force using different selected electrodes. 
     The Coriolis rebalancing force, corresponding to V CUL , applied by electrode  210  may be defined by equation (6) below:
 
 V   CUL   =V   COR  sin(ω m   t /2)  (6)
 
where V COR  is a Coriolis voltage, and where ω m t/2 is the one half of the frequency of the motor frequency of proof masses  102 ,  104 .
 
     The Coriolis rebalancing force, corresponding to V CLL , applied by electrode  218 , the Coriolis rebalancing force, corresponding to V CUR , applied by electrode  224 , and the Coriolis rebalancing force, corresponding to V CLR , applied by electrode  232 , may be defined by equations (7), (8), and (9), respectively, below:
 
 V   CLL   =V   COR  cos(ω m   t/ 2)  (7)
 
 V   CUR   =V   COR  cos(ω m   t/ 2)  (8)
 
 V   CLR   =V   COR  sin(ω m   t/ 2)  (9)
 
     Some embodiments may apply optional quadrature rebalancing forces via the optional electrodes  206 ,  214 ,  228 , and  236 . The quadrature rebalancing forces are proportional to the induced motor motion of the proof masses  102 ,  104 . In the exemplary embodiments illustrated in  FIGS. 2-9 , four electrodes are illustrated (at each end of the proof masses  102 ,  104 ) that are used for the application of quadrature rebalancing forces. In alternative embodiments, a single electrode pair for each of the proof masses  102 ,  104  may be used to apply quadrature rebalancing forces. The single pair of quadrature rebalancing electrodes may be placed in any suitable position with respect to its proof masse  102 ,  104 . In alternative embodiments, quadrature rebalancing electrodes are optional or are not used. 
       FIG. 9  illustrates applied voltages and sensing voltages for an alternative embodiment of the inertial sensor  100 . The electrodes  208 ,  216 ,  226 , and  234  are coupled to pick off amplifier systems  902 ,  904 ,  906 , and  908 , respectively, to sense or pick off voltages at their respective electrodes. This embodiment allows compensation of parasitic signals injected into the proof masses  102 ,  104 , which may result in undesirable applied parasitic forces. That is, parasitic coupling effects between the rotational forces and the linear acceleration forces may be mitigated since the frequency of parasitic terms will be higher (ω p +ω m /2). 
     The amplifier system  902  outputs a signal V ULSP . The amplifier systems  904 ,  906 , and  908 , output the signals V LLSP , V URSP , and V LRSP , respectively. Rotational output, V RATE , may be derived from the output of the amplifier systems  902 ,  904 ,  906 , and  908 , in accordance with equation (10), below:
 
 V   RATE   =V   ULSP   +V   LRSP   −V   LLSP   −V   URSP   (10)
 
       FIG. 10  is a block diagram illustrating an exemplary implementation of a processing system  1002  coupled to an embodiment of the inertial sensor  100 . In an exemplary embodiment, the processing system is a digital signal processing (DSP) electronics system. The processing system  1002  may be implemented as an analog, as a digital system, or a combination thereof, and may be implemented as software, hardware, or a combination of hardware and software, depending upon the particular application. 
     The amplifier system  702  provides the sensed pick off voltage, V SPO , to the processing system  1002 . Demodulators  1004 ,  1006  and  1008  demodulate V SPO  by stripping off the AC portions of V SPO . The 90 degree clock applied to demodulator  1004  and the 0 degree clock applied to demodulator  1006  correspond to a multiplied motor signal at different phases (90 degrees and 0 degrees, respectively). 
     The low pass filter  1010  processes the output of the demodulator  1004  and outputs a Coriolis output signal to a proportional-integral-derivative (PID) controller  1012 . A low pass filter  1014  and a PID controller  1016  process the output of the demodulator  1004  and outputs a quadrature output signal. A low pass filter  1018  and a PID controller  1020  process the output of the demodulator  1008  and outputs an acceleration output signal corresponding to the common mode imbalance in capacitance. The output signals are used to generate the outputs V ULS , V LLS , V URS , and V LRS , corresponding to the above-described linear acceleration rebalancing force, and are used to generate the outputs V CUL , V CLL , V CUR , and V CLR , corresponding to the above-described Coriolis rebalancing force. 
     Embodiments of the inertial sensor  100 , operable to sense and determine linear acceleration and rotation, may be incorporated into an inertial measurement unit. Since one inertial sensor  100  sense linear acceleration and rotation, three inertial sensors  100  may be used to construct one inertial measurement unit rather than the three gyroscopes and the three accelerometers used in a conventional inertial measurement unit. Accordingly, costs and/or size may be reduced since fewer components are used. 
     While the preferred embodiment of the invention has been illustrated and described, as noted above, many changes can be made without departing from the spirit and scope of the invention. Accordingly, the scope of the invention is not limited by the disclosure of the preferred embodiment. Instead, the invention should be determined entirely by reference to the claims that follow.