PATENT DOCUMENT

Abstract:
A sensor system includes a first measurement subsystem including a driver and a measurement component. The measurement component includes a set of sensors for generating complimentary signals representing each of one or more characteristics. The complimentary signals representing a characteristic include related error components. The driver is configured to accept the complimentary signals for a characteristic and to determine a measurement value for the characteristic by combining the complimentary signals to reduce effects of the error components in the complimentary signals.

Full Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
       [0001]    This application claims the benefit of the following U.S. Provisional Applications: No. 60/791,821, filed Apr. 13, 2006, titled “Orientation Sensor”; No. 60/873,495, filed Dec. 7, 2006, titled “Orientation Sensor”; and No. 60/900,537, filed Feb. 9, 2007, titled “Orientation Sensor.” Each of these applications is incorporated herein by reference. 
     
     BACKGROUND 
       [0002]    This invention relates to a sensor system. 
         [0003]    Highly accurate compasses, magnetometers, and orientation sensors have a variety of uses including, but not limited to, vehicle navigation, underwater towed arrays, seismic measurements, and orientation sensors. High accuracy is in part obtained by correcting the compass and magnetometer readings for latitude by sensing both vertical and horizontal components of the earth&#39;s magnetic field using a three-axis magnetometer. The three-axis magnetometer not only generates signals representative of latitude and heading. New advances in accelerometer technology now allow the development of a sensor that has no moving parts. 
       SUMMARY 
       [0004]    In one aspect, in general, a sensor system includes a first measurement subsystem including a driver and a measurement component. The measurement component includes a set of sensors for generating complimentary signals representing each of one or more characteristics. The complimentary signals representing a characteristic include related error components. The driver is configured to accept the complimentary signals for a characteristic and to determine a measurement value for the characteristic by combining the complimentary signals to reduce effects of the error components in the complimentary signals. 
         [0005]    Aspects can include one or more of the following features. 
         [0006]    The sensor system includes an orientation computation module for using the measurement value in computing an orientation. 
         [0007]    The set of sensors includes two accelerometers substantially aligned along a common axis and fixed in opposite orientations. Each of the two accelerometers is configured to generate a different one of the complimentary signals. 
         [0008]    The set of sensors includes two (or more) accelerometers substantially aligned along each of three different axes and fixed in opposite orientations along the axes. Each of the two accelerometers for an axis is configured to generate a different one of the complimentary signals. 
         [0009]    The system includes a second measurement subsystem that has a driver and a measurement component. The measurement component is configurable for generating one of a set of complimentary signals in each of a corresponding set of configurations. 
         [0010]    The measurement component of the second measurement subsystem includes a magnetometer that is configurable to be coupled, to the driver in two opposite polarities. Each of the two polarities is used for generating a different one of the complimentary signals. 
         [0011]    The magnetometer includes a two-terminal fluxgate coil, and a switching circuit that includes two configurations for coupling terminals of the coil to the driver. 
         [0012]    The second measurement subsystem is calibrated in absence of substantial interference. 
         [0013]    The second measurement subsystem is configured to estimate a degree of error in measurement values provided by the subsystem in the presence of an interference. 
         [0014]    The interference includes an iron (e.g., a hard iron or soft iron) interference. 
         [0015]    The degree of error includes a quantity characterizing an RMS error in heading. 
         [0016]    The one or more characteristics include three substantially orthogonal components of orientation, and the measurement component includes a sensor configuration for generating complimentary signals for each of the orientation components. 
         [0017]    The first measurement subsystem is configured to measure orientation, and the sensor system further includes a second measurement subsystem for measuring magnetic characteristics. 
         [0018]    An orientation computation module uses measurement values for orientation from the first measurement subsystem and measurement values from the second measurement subsystem for determining an orientation of the sensor. 
         [0019]    The sensor configuration for generating complimentary signals representing each of one or more characteristics represents a substantially symmetric transfer function between the characteristic and the signal. 
         [0020]    In another aspect, in general, a magnetometer is calibrated in the absence of substantial interference. Later, the magnetometer is provides an estimate of a degree of errors, for example, a degree of RMS error in bearing, in the presence of an interference, such as in the presence of a hard iron or a soft iron interference. 
         [0021]    In another aspect, in general, an orientation sensor includes a three-axis fluxgate device, two three-axis accelerometers, an analog to digital converter, a microprocessor, and various electronics to drive the magnetometer coils and filler and condition the data signals. A serial communication interface is also provided. The orientation sensor can be used in a number of applications including, but not limited to, weapons orientation, attitude sensing, bore holes, and vehicle navigation. The sensor provides high accuracy with low power consumption in a small package that can be used even on or near objects containing large amounts of ferrous material in either the horizontal or vertical position. 
         [0022]    In other aspects, in general, a sensor system includes one or more of the following. 
         [0023]    The sensor system implements an orientation sensor. 
         [0024]    The system includes a three-axis magnetometer and dual three-axis accelerometers. 
         [0025]    The system includes a three-axis magnetometer, dual three-axis accelerometers, and one or more axes of gyros. 
         [0026]    The system has no moving parts. 
         [0027]    The system implements a procedure for self-calibration of the orientation sensor without requiring external fixtures. 
         [0028]    The system implements a calibration of an orientation sensor in the presence of large amounts of ferrous material, such as a weapon or vehicle. 
         [0029]    The system implements a calibration to make the system substantially latitude insensitive. 
         [0030]    The system reverses the direction of the magnetometer coils and combines the results allowing use of the instrument as either a magnetometer or a gradiometer. 
         [0031]    The system includes accelerometers oriented in opposite directions and combines the data from these accelerometers. 
         [0032]    A coil energizing pattern of a square wave with “off” times at the zero crossings is used. 
         [0033]    Infinite gain feedback using double integrators is used. 
         [0034]    Low-power consumption achieved by the implementation of different CPU states and sequencing of electronic component power settling versus reading. 
         [0035]    Aspects can include one or more of the following advantages. 
         [0036]    Real-time measurement of pitch, roll, directional heading is accomplished by utilizing a high through put AtoD converter, a hardware MAC (Multiplier/Accumulator), and a RISC architecture 16 bit microcontroller. 
         [0037]    Accuracy better than 0.10 degree via IIR filtering and 24 Bit storage and processing. 
         [0038]    The sensor system can be mounted and successfully calibrated on a weapon. The sensor can be calibrated even on weapons with large amounts of ferrous material 
         [0039]    Complementary pointing vectors improve accuracy by doubling signal strength and eliminating even order harmonic distortions. 
         [0040]    Long term battery operation as well as high speed, high bandwidth, sampling can exist simultaneously in the same unit through unique computational methodology. 
         [0041]    Auto-zeroing function permits long term stability of the magnetometers while removing electronic offsets associated with the electronics. This technique also improves the temperature stability of offset and scale factors and permits dual operation as both a magnetometer and a gradiometer. 
         [0042]    The systems are not limited to weapons orientation. The sensor works well on ferrous vehicles and applications as diverse as antenna positioning. 
         [0043]    A compass can measure its own error without external equipment. In prior systems, the accuracy of the internal calibration has been verified many times by using a precision “step and repeat” rotary table. Using internal calibration, such rotary table procedures can be avoided. 
         [0044]    Other features and advantages of the invention are apparent from the following description, and from the claims. 
     
     
       DESCRIPTION OF DRAWINGS 
         [0045]      FIG. 1  is a block diagram. 
           [0046]      FIG. 2  is a perspective schematic view. 
           [0047]      FIG. 3  is a signal timing diagram. 
           [0048]      FIG. 4  is a block diagram. 
           [0049]      FIGS. 5A  and B are circuit schematics. 
           [0050]      FIGS. 6 and 7  are flowcharts. 
           [0051]      FIGS. 8 and 9  are tables. 
           [0052]      FIG. 10  is a perspective view of coordinate axes. 
       
    
    
     DESCRIPTION 
     1 Abbreviations and Definitions 
       [0053]    ADC Analog to Digital Converter 
         [0054]    DAS Digital Attitude Sensor 
         [0055]    DMX Digital Magnetometer with Accelerometers 
         [0056]    RMS Root Mean Square 
         [0057]    WOM Weapons Orientation Sensor. 
       2 Orientation Sensor 
       [0058]    Referring to  FIG. 1 , in an example of a sensor system, an orientation sensor  100  provides orientation information based on a number of sensors. One example of use of the orientation sensor is as a Weapons Orientation Sensor (WOM) to measure the orientation of a weapon, such as of an M16 rifle. Other examples of sensor systems that make use of similar or the same features include Digital Attitude Sensors (DAS). 
         [0059]    “The orientation sensor  100  includes a magnetometer subsystem  110 , which includes a three-axis fluxgate device  112 , and an accelerometer subsystem  120  that includes two three-axis accelerometers  122 ,  124 . The magnetometer subsystems includes an analog to digital converter and various electronics to drive the magnetometer coils and filter and condition the data signals, which are collectively illustrated as a driver  116 . As is discussed further below, the magnetometer subsystem  110  also includes a switch  114 , which can reverse the polarity of the connections between the driver  116  and the fluxgate devices  112 . The accelerometer subsystem  120  also includes a driver  126 , which drives both accelerometers  122 ,  124 . These accelerometers are mounted in substantially opposite orientations (see  FIG. 2 ). A control subsystem  140  includes a microprocessor  142  as well as associated program storage  144  and data storage  146 , and also includes an interface  148 , which in some examples uses a serial communication link  150  for interchanging information with a host system. The system optionally includes a gyroscope subsystem, which includes one or more gyros  132  and an associated, driver  134 , which can provide angular rate of change data. Angular rate of change alternatively can be measured in heading, roll, and pitch by simply sampling the degree changes on those three axes per unit of time, typically the sample period. 
         [0060]    The magnetometer  110  includes a 3-axis fluxgate device  112  with axes designated as (x,y,z). Referring to  FIG. 2 , the x-axis is nominally aligned with the sensor case and points in the direction toward the electronic connector which is also generally in the direction of travel. The y-axis nominally points athwart ship to port at zero roll angle. The z-axis points vertically downward (toward the center of the Earth) at zero roll and zero pitch angles. 
         [0061]    The accelerometer subsystem  120  includes two 3-axis accelerometer devices  122 ,  124 . Continuing to refer to  FIG. 2 , the coordinate systems for the accelerometer devices are defined with the following convention. For accelerometer # 1 , the coordinate system is left-handed and is the same as the magnetometer coordinate system with a 180° rotation about the z-axis. For accelerometer # 2 , the coordinate system is right-handed with the x-axis nominally aligned with the sensor case pointing toward the electronic connector, the y-axis nominally points athwart ship to port at zero roll angle, and the z-axis points vertically upward (away from the center of the Earth) at zero roll and zero pitch angles. 
         [0062]    The system electronics include an analog board, a digital board, two accelerometer boards, a nose board, and a coil holder board. The analog board contains the magnetometer and accelerometer analog signal control and conditioning circuits and a temperature circuit. The digital board contains the microcontroller and power supply regulation circuits, reference voltage circuits, external communications interface circuits, and a high-resolution analog to digital converter. There are two types of accelerometer boards. The first mounts the accelerometer package such that the x-axis is normal to the analog board plane (pointing away from it) and the y-axis is parallel to the analog board plane. The second mounts the accelerometer package such that the x-axis is parallel to the analog board plane and the y-axis is normal to the analog plane (pointing into it). Two of each board type are used for a total of four accelerometer boards and accelerometers. The coil holder board provides a connection backplane for the three fluxgate coils. The nose board provides a connection backplane to connect the digital board to the mechanical nose piece in which the input/output connector is mounted. 
       3 Calibration 
       [0063]    A procedure for self-calibration is described in U.S. Pat. No. 4,698,912, titled “Magnetic Compass Calibration,” issued to Fowler et. al. in October, 1987, which is incorporated herein by reference. Embodiments of the present approach make use of techniques described in that patent as well a number of new calibration features. 
         [0064]    Examples of the system use a calibration approach that relates to the use of the difference of Factory “Air” (without ferrous sources of deviation) calibrated values and the values calibrated, by the “KAL” method (auto-calibrated in the presence of ferrous material or permanent magnets) to quantitatively determine the type (hard of soft iron) and magnitude of the external sources of deviation. The embedded firmware (e.g., in program storage  144 ) then gives a precise value for the “B &amp; C” (hard iron terms) and the “D &amp; E” (soft iron terms), and even computes its own root mean square (RMS) error. Having reported its own error terms, the system can then automatically update its registers (e.g., in data storage  146 ) and mitigate errors related to those terms in the measurements provided to the host system. 
         [0065]    The calibration process involves the following steps. As part of the calibration process, the sensor rapidly samples the x and y magnetometer axes (i.e., the cosine and sine of heading). The system stores the maximum and minimum readings of each of these uncalibrated cosine and sine signals (waveforms as a function of time). In the absence of offset errors, the sum of the maximum and minimum would be zero. The average of the (signed) maximum and minimum of each of these signals yields the offset (the “B” term for the x-axis and “C” term for the y-axis) (e.g., x-bias and y-bias) after conversion to degrees. For example, the bias is due to an effect of permanent magnetism on the weapon. 
         [0066]    In addition to the bias terms, a y/x ratio is also computed as the average differences of y-peak-to-peak/x-peak-to-peak=(y-max−y-min)/(x-max−x-min). This becomes the “D” when converted to degrees. This term is due, for example, to soft iron along the fore and aft line of symmetry. This is called the “lubber line” on a ship, but is along the axis of the barrel for a rifle. 
         [0067]    The “E” term is determined by measuring the phase angle between the two waveforms at the Intercardinal points (045, 135, 225, and 315 degrees) using waveform analysis techniques. This error is caused by asymmetrically positioned soft iron on either side of the sensor. 
         [0068]    The RMS error is computed as 
         [0000]        RMS =( B   2   +C   2   +D   2   +E   2 ) 1/2    
       Since there are no more deviational influences in the physical model, this result is extremely close to the RMS value if computed with a precision indexer (rotary table). 
       [0069]    This calibration forms is an iterative process with the RMS error being transmitted to the host system and every time a calibration revolution is made, the processor updates the B, C, D and E terms. At some point the RMS error will fall below the specified target and the operator can be informed that calibration can cease. 
         [0070]    In a test of 4 WOM sensors removed after having been calibrated on the M16, they all showed about a 3° “D” term variation when removed from the weapon and run in “Air”. This variation is due to the large amount of soft iron present in the barrel and the receiver of the M16. Had the experiment been run the other way in going from an “Air” calibrated sensor to being mounted on the gun, a −3° “D” term would have been generated. Typically, even large error terms can be reduced to near zero in only one “KAL” revolution. If a small residual term remains, it can be eliminated with a second pass. 
         [0071]    Latitude insensitively is very important if the weapon or vehicle traverses large changes in latitude. Most magnetic auto-calibration algorithms only compensate for X and Y bias errors at one particular latitude. By separating out actual X and Y bias errors from changes in X and Y biases due to contamination from the vertical field, the sensor system is effectively insensitive to vertical field changes and therefore to latitude. 
       4 High Accuracy 
     4.1 Multiple Accelerometers 
       [0072]    In the embodiment of the sensor system illustrated in.  FIGS. 1-2 , two accelerometers are pointed in opposite directions to improve the pitch and roll accuracy. Using multiple accelerometers can improve accuracy. For example, assuming measurement error is statistically independent for each of the sensors, using N sensors would yield a reduction of RMS error by a factor of N 1/2 . If four accelerometers were installed, then a factor of two improvement could be expected. 
         [0073]    Further reduction in error can be achieved by selecting the orientations of the accelerometers to take advantage of certain statistical dependence of the measurement error in the different accelerometers. In some examples, the axes of micromachined accelerometers are configured in opposite directions, as shown in  FIG. 2 . Such configurations can provide additional benefits contributing to overall accuracy and long term stability are achieved. Without being limited to by theory or motivation for the configuration, reasons that such configurations may be beneficial are as follows. 
         [0074]    Since the sensors are MEMs devices, they are often well matched in certain errors, such as nonlinearity. By employing sensors in complementary positions, some of these nonlinear errors cancel, if the difference of the two axes is taken. For example, if the sensor output can be characterized by 
         [0000]        Gx   1   =b   1   +m   1   X+a   1   X   2 , and 
         [0000]    
       
      
       Gx 
       2 
       =b 
       2 
       −m 
       2 
       X+a 
       2 
       X 
       2  
      
     
         [0000]    recalling that the acceleration X opposite for the second accelerometer, the difference of the two complementary outputs for an axis will read 
         [0000]        Gx   1   −Gx   2 =( b   1   −b   2 )+ X ( m   1   +m   2 )+( a   1   −a   2 ) X   2    
         [0000]    In other words the offsets and the even order nonlinearity terms tend to cancel, and the odd order terms are generally doubled. Since the residual b terms (offsets) can be measured and canceled, as well as the m terms (gain), we are left with Gx=2 m X, where m is the average of the two m terms. 
         [0075]    Referring to  FIG. 5B , accelerometers  522  and  524 , each representing one axis of 3-axis accelerometers  122  and  124  (see  FIG. 1 ), respectively. Each accelerometer signal is passed to a double-throw switch  536  while permits an AtoD to read the acceleration reading from the two accelerometers in succession. The switch  536  can be synchronized with the analog-to-digital conversion. In some embodiments, the switch (or equivalent circuitry) forms an AC signal from the alternations between the accelerometers, and this AC signal can be AC coupled to circuitry for processing the signal prior to digitization. In some examples, rather than switching between the two accelerometers, and analog summation of the two signals if performed prior to digitization. 
       4.2 Coil Flipping 
       [0076]    In some examples, the magnetometer subsystem  110  employs what is referred to herein as a “coil flipping” feature. Viewed generally, both the use of complementary accelerometers and electrically flipping the orientation of excitation of the fluxgate cores are examples of a general technique of using complementary configurations with complementary devices (e.g., two accelerometers) or complementary uses or excitations of a physical device (e.g., two polarities of excitation of fluxgate coils). Using complementary configurations can have a benefit of cancelling dependent measurement errors from measurements using the complementary configurations. 
         [0077]    The coil flipping feature gives the magnetometer subsystem improved offset and high temperature stability. Referring to  FIG. 1 , the 3-axis fluxgate  112  is coupled to the driver  116  through a switch  114 . The switch is controllable to drive each of the two-terminal fluxgate coils (one for each axis) in each of two polarities. That is, the switch  114  implements a Double Pole Double Throw (DPDT) switch allowing the polarity to be reversed. 
         [0078]    Referring to  FIG. 4 , which illustrates elements of associated with one axis of the magnetometer subsystem  110 , a fluxgate  112 A includes a two terminal fluxgate coil  312 . A driver  116 A includes a signal generator  334 , which drives the coil  312 , and an integrator  332 , which senses the coil  312 . The switch  114 A includes an arrangement of switches  322  such that the polarity of the drive to the coil can be inverted, and an arrangement of switches  324 , such that together the switches  322 ,  324  pass the coil signal to the integrator  322  with a consistent polarity. 
         [0079]    Referring to  FIGS. 3 and 5 , the coil/coil-bar (CCB) signal drives the state of the switches  322 , and the DETECT signal drives the state of the switches  324 . In the absence of the CCB signal flipping the orientation of the coil, the output v 0  is a DC signal proportional to the measured field. The CCB signal flips the orientation periodically, for example, every 32 cycles of the IdriveAC signal. This results in the output v 0  being an AC signal that has a peak-to-peak amplitude equal to twice the amplitude of the (DC) signal generated by the coil, e.g., 60,000 nT for the DC signal is converted to ±60,000 nT for the AC signal. In some examples, use of such an AC signal is used in an AC-coupling arrangement such that the AC signal is processed further before digitization. For example, such an AC signal may have advantages in certain applications, such as with very low field strengths. 
         [0080]    If there are offsets associated with the electronics (e.g., in the integrator  332 , signal generator  334 , and feedback loop and analog-to-digital converters) that is not illustrated) then the effect of such offset can effectively be eliminated through the use of the switch  114 A. For example, assume the fluxgate is aligned with a positive field B. Then with the switches  322  in the plus direction, the magnetometer output will read Hout 1 =H+off. Now if the switches  322  reverse the coil polarity, then the output will read Hout 2 =−H+off. If these two readings are then subtracted we get Hout 1 −Hout 2 =2H+(off−off), canceling all of the offsets in the electronics, including the A-to-D converter itself. 
       4.3 Fluxgate Control 
       [0081]    As introduced above, three fluxgate coil drive and signal processing circuit channels are controlled by a small microcontroller ( 142 , see  FIG. 1 ) that produces a four cycle alternating waveform as shown in  FIG. 3 . In some examples, the three channels are entirely or effectively identical. In step  1 , the coil is driven with a constant current (Idrive) in the positive direction until the core is fully saturated (Ifb+). In step  2  the current is turned off, and the energy in the coil and core is allowed to “flyback”. Steps  3  (Ifb−) and  4  are a repeat of  1  and  2  but with the coil current in the negative direction. An integrator is switched at twice the drive frequency (DETECT) and captures the integral of the positive and negative excursions (Ifb±), as well as the “flyback energy” of the coil voltage in response to the constant current AC drive. 
         [0082]    A schematic of a single axis of the magnetometer subsystem is shown in  FIG. 5A . The Idrive AC is an AC constant current fluxgate drive. Ifb is DC Fluxgate current, which is equal to H*L/N, where H is the Magnetic field, L is the fluxgate length, and N is the number of turns of coil wire. The output Vo is equal to R*H*L/N. Note that Vo drives the current Ifb through R into a virtual ground. Neither the resistance of the coil nor any changes due to temperature affects Vo. 
         [0083]    The output Vo of the integrator reflects the flux balance in the core as shown in  FIG. 5A . The voltage produced by the coil in response to the constant current excitation is 
         [0000]    
       
         
           
             
               E 
               = 
               
                 N 
                  
                 
                   ( 
                   
                     
                        
                       Φ 
                     
                     
                        
                       t 
                     
                   
                   ) 
                 
               
             
             , 
           
         
       
     
         [0000]    where Φ is the flux in the coil core, N is the number of turns on the coil, and t is the time during which the flux change takes place. The integrator (Detector balance amp) captures this voltage change as E=N ∫Φdt. In the absence of an external field, the flux excursion in response to a no net DC constant current excitation has a zero integral. However, an external magnetic field causes an imbalance in the BH loop, causing the integrator to move toward a positive or negative voltage limit. 
         [0084]    The integrator output is fed into a constant current source (Current source amp) that is summed with the AC constant current fluxgate drive (Idrive AC). This results in a negative feedback that re-balances the fluxgate core and produces a zero average flux within the core (null-balance). The integrator now stabilizes at a point that produces a constant current that exactly cancels any external field and results in a zero bias on the fluxgate core. This zero bias on all three magnetometers results in each magnetometer behaving as if it were non-magnetic, canceling any influence that one might have on a close neighbor. This permits a non-symmetric arrangement of magnetometers to act as if all three axes were mutually orthogonally but intersecting, a physical impossibility. 
         [0085]    Note that in  FIG. 5A , both integrators have only a capacitor across their feedback terminals (output connected to non-inverting input). This arrangement may have advantages related to sensitivity to temperature variation. 
       5 Low Power Operation 
       [0086]    Examples of the sensor system  100  can perform data acquisition of magnetometer and accelerometer data, filter it with an IIR filter, and store the filtered resulting vectors as raw values. As a mechanism for reducing power consumption, only when data is requested does it calibrate the raw data, transform the vectors to the horizontal and vertical planes and transmit the results. This technique combined with turning elements on and off reduces power while preserving bandwidth. 
       6 Signal Processing 
       [0087]    A high level flow diagram of the sensor data processing is shown in  FIG. 6 . Each of the major blocks in the flow diagram are described in detail below. 
       6.1 Pre-Transform Calibration 
       [0088]    After analog-to-digital conversion of the raw sensor signals (step  610 ), the digitized signals are processed in a pre-transform calibration step (step  612 ). This processing on the raw output of the sensor output includes a number of steps as shown in  FIG. 7 . The individual sensor signals from each axis are corrected for common mode errors, random noise, and sensor and electronics corrections related to gain, offset, and axes alignment. This section first defines the coordinate system used in computations and then describes in detail the require correction computations. 
       6.2 Coordinate Systems 
       [0089]    An individual sensor output from the magnetometers and the accelerometers is taken from an analog-to-digital converter (ADC) for each channel of the multi-axis sensors. These digitized sensor readings are defined herein as “raw” readings and are the input for the subsequent processing. 
         [0090]    As introduced above, the magnetometer subsystem  110  uses a 3-axis fluxgate device  112  (see  FIG. 1 ) with axes designated as (x,y,z). As illustrated in  FIG. 2 , the x-axis is nominally aligned with the case of the system and points in the direction toward the electronic connector (which is defined as “North” pointing for the system), the y-axis nominally points athwart ship to port at zero roll angle. The z-axis points vertically downward (toward the center of the Earth) at zero roll and zero pitch angles. 
         [0091]    The accelerometer subsystem  120  includes two 3-axis accelerometers  122 ,  124 . As illustrated in  FIG. 2 , the coordinate systems for the accelerometer devices are defined with the following convention. For accelerometer # 1  ( 122 ), the coordinate system is left-handed and is the same as the magnetometer coordinate system with a 180° rotation about the z-axis. For accelerometer # 2  ( 124 ), the coordinate system is right-handed with the x-axis nominally aligned with the case of the system pointing toward the electronic connector, the y-axis nominally points athwart ship to port at zero roll angle, and the z-axis points vertically upward (away from the center of the Earth) at zero roll and zero pitch angles. 
         [0092]    The accelerometers are used to estimate the sensor body roll and pitch attitude with respect to the local gravity vector. This information is used to align the magnetometer axes with the plane of the Earth&#39;s local tangent and thence to estimate heading using the geomagnetic field. 
       6.2.1 Preprocessing 
       [0093]    There are nine individual sensor outputs from the ADC, three magnetometer axes and six accelerometer axes. For each of these axes two preprocessing steps are used to compensate for common mode DC errors and to reduce random noise using an infinite impulse response (IIR) digital filter. 
       6.2.1.1 Common Mode Rejection 
       [0094]    For each of the axes, a common mode rejection (CMR) computation of the following form is performed: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       Q 
                       CMR 
                     
                     = 
                     
                       
                         
                           Q 
                           + 
                         
                         - 
                         
                           Q 
                           - 
                         
                       
                       2 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   1 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where Q is a placeholder variable name with Q + , representing the “positive” polarity axis and Q − , representing the “negative” polarity axis. To make this concrete, consider the accelerometer x-axis CMR computation 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       G 
                       xCMR 
                     
                     = 
                     
                       
                         
                           G 
                            
                           
                               
                           
                            
                           
                             1 
                             x 
                           
                         
                         - 
                         
                           G 
                            
                           
                               
                           
                            
                           
                             2 
                             x 
                           
                         
                       
                       2 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   2 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where G1 x  represents the output of the x-axis for accelerometer # 1  and G2 y  represents the output of the x-axis for accelerometer # 2 . 
         [0095]    In some examples of the system, as illustrated in  FIGS. 1-2 , there is only one set of physical sensing axes, as illustrated for the fluxgate  112  axes in  FIG. 2 . These axes correspond to the “positive” magnetometer axes. To create the corresponding “negative” axes the polarity of the sensor drive current, voltage sense, and feedback current is reversed electronically by using the switch  113 , which uses a DPDT (Double Pole Double Throw) MUX (implemented using switches  322 , see  FIG. 4 ) to reverse the two terminals of the fluxgate itself. The result for each magnetometer axis is defined to be positive with current in the direction corresponding to a “coil” signal (C) and is defined to be negative with current of the opposite polarity corresponding to a “coil-bar” signal (CB). Hence, the form of the CMR computation for the magnetometer is, for example, 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       H 
                       xCMR 
                     
                     = 
                     
                       
                         
                           H 
                           xC 
                         
                         - 
                         
                           H 
                           xCR 
                         
                       
                       2 
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   3 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where H x  is the output of the magnetometer x-axis with coil (C) or coil-bar (CB) as indicated. 
         [0096]    Using the above conventions, the table shown in  FIG. 8  shows the six computations that must be performed on the accelerometer and magnetometer axes to determine the CMR signals. 
       6.2.1.2 IIR Filter 
       [0097]    Each of the six CMR signals is next subjected to a low-pass smoothing digital filter (step  720 ) to improve signal-to-noise ratio. The form of the filter is a simple IIR exponential averaging filter. 
         [0098]    The filter is implemented on a sample-by-sample basis as new data are available from the ADC (after CMR correction). The filter equation is defined as 
         [0000]        Q   IIR   [n ]=(1−β) Q   IIR   [n− 1 ]+βQ   CMR   [n− 1],   (4) 
         [0000]    where [n] represents the current value of the digital sample stream, [n−1] represents the previous value of the digital sample stream, Q IIR  represents the output of the IIR filter, Q CMR  represents the output of the common-mode-rejection computation, and β=2 −m  is a weighting factor with m=0, 1, 2, 3, . . . Q is a surrogate variable and for the computations takes on the value of H x,y,z  or G x,y,z , as appropriate. 
         [0099]    The frequency response corresponding to the IIR filter of Equation (4) may be written as 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         
                            
                           
                             
                               Y 
                                
                               
                                 ( 
                                 φ 
                                 ) 
                               
                             
                             
                               X 
                                
                               
                                 ( 
                                 φ 
                                 ) 
                               
                             
                           
                            
                         
                         2 
                       
                       ≡ 
                       
                         
                            
                           
                             H 
                              
                             
                               ( 
                               φ 
                               ) 
                             
                           
                            
                         
                         2 
                       
                     
                     = 
                     
                       
                         β 
                         2 
                       
                       
                         
                           2 
                            
                           
                             ( 
                             
                               1 
                               - 
                               β 
                             
                             ) 
                           
                            
                           
                             ( 
                             
                               1 
                               - 
                               
                                 cos 
                                  
                                 
                                     
                                 
                                  
                                 φ 
                               
                             
                             ) 
                           
                         
                         + 
                         
                           β 
                           2 
                         
                       
                     
                   
                   , 
                 
               
               
                 
                   ( 
                   5 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where β is the filter weight that prescribes the low-pass corner frequency, φ=2πfτ is dimensionless phase, f is frequency in Hz, and τ is the fixed time interval in seconds between successive digital samples in the sample stream. By solving Equation (5) for β as a function of the low-pass corner frequency, f L , we can select β to achieve certain performance features. Similarly, for a given β we can compute the corresponding low-pass corner frequency, f L . The equations for these computations are as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     β 
                     = 
                     
                       
                         ( 
                         
                           
                             cos 
                              
                             
                                 
                             
                              
                             
                               φ 
                               L 
                             
                           
                           - 
                           1 
                         
                         ) 
                       
                       + 
                       
                         
                           
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   
                                     φ 
                                     L 
                                   
                                 
                               
                               ) 
                             
                             2 
                           
                           + 
                           
                             2 
                              
                             
                               ( 
                               
                                 1 
                                 - 
                                 
                                   cos 
                                    
                                   
                                       
                                   
                                    
                                   
                                     φ 
                                     L 
                                   
                                 
                               
                               ) 
                             
                           
                         
                       
                     
                   
                   , 
                   
                       
                   
                    
                   
                     
 
                   
                    
                   
                     
                       φ 
                       L 
                     
                     = 
                     
                       2 
                        
                       
                           
                       
                        
                       π 
                        
                       
                           
                       
                        
                       
                         f 
                         L 
                       
                        
                       τ 
                     
                   
                   , 
                   
                       
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   6 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       f 
                       L 
                     
                     = 
                     
                       
                         1 
                         
                           2 
                            
                           
                               
                           
                            
                           π 
                            
                           
                               
                           
                            
                           τ 
                         
                       
                        
                       
                         
                           cos 
                           
                             - 
                             1 
                           
                         
                          
                         
                           ( 
                           
                             1 
                             - 
                             
                               
                                 β 
                                 2 
                               
                               
                                 2 
                                  
                                 
                                   ( 
                                   
                                     1 
                                     - 
                                     β 
                                   
                                   ) 
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                   
                   , 
                   
                       
                   
                    
                   
                     
                       where 
                        
                       
                           
                       
                        
                       
                         f 
                         L 
                       
                     
                     = 
                     
                       
                         
                           φ 
                           L 
                         
                         
                           2 
                            
                           
                               
                           
                            
                           π 
                            
                           
                               
                           
                            
                           τ 
                         
                       
                       . 
                     
                   
                 
               
               
                 
                   ( 
                   7 
                   ) 
                 
               
             
           
         
       
     
         [0100]    Since β takes on discrete values of m=0, 1, 2, . . . , we can compute the discrete values of the dimensionless frequency parameter, φ L  or, given a sampling interval, τ, we can compute the discrete values of the corner frequency, f L . The table in  FIG. 9  shows low-pass filter parameters for discrete values of m and for τ=0.01, i.e., 100 Hz sampling rate. Also shown in  FIG. 9  is the reciprocal of the corner frequency, T L , which gives an estimate of the IIR filter settling time needed for the cited case. 
         [0101]    With suitably selected values β, Equation (4) is applied to each of the six CMR corrected signals that are computed as shown in the table in  FIG. 8 . The result is a set of six filtered signals; namely for the magnetometer the signals are H xIIR , H yIIR , and H zIIR  and for the accelerometer the signals are G xIIR , G yIIR , and G zIIR . 
       6.2.2 Gain, Offset, Alignment Corrections 
       [0102]    For each of the sensor axes a correction for misalignment of the axes and adjustments for differences in physical and electronic gain and offset is computed. The correction is based, in part, on a so-called mx+b calculation. All of the preprocessing steps defined above are applied to the input, data stream from the ADC regardless of the sensor situation, i.e., whether the sensor is being calibrated or being used to sense operational signals. 
         [0103]    In the pre-transform calibration, described in this section, the sensor is subjected to controlled conditions. Two constraints are placed on the sensor:
       Sensor is successively placed in a V-block at zero roil and at 180° roll and   Sensor is allowed to remain stationary for sufficiently long that the IIR filter output has reached steady state.       
 
         [0106]    In each of the V-block positions, after the IIR filter has stabilized, a recording is made of the magnetometer and accelerometer measurements. Specifically, the values used are
       H x0 , H y0 , H z0 , G x0 , G y0 , G z0 , and   H x180 , H y180 , H z180 , G x180 , G y180 , G z180 .       
 
         [0109]    These values are used to compute the coefficients of the correction equation, which may be expressed as 
         [0000]        Q   xcal   =K   xgain   Q   xIIR   K   xoffset   Q   xIIR   +K   dXdZ   Q   zIIR   +K   dXdY   Q   yIIR ,   (8) 
         [0000]    where the Q ()IIR  are the output of the IIR preprocessing filter, and the K (▪)  are coefficients computed as shown below using the values determined in the V-block test. 
         [0110]    The first two terms on the right-hand side of Equation (8) are adjustments for the gain and offset of the sensor and the associated electronics. The “mx+b” name for Equation (8) derives from these two terms, where the “m” is identified as K xgain  and the “b” is identified as K xoffset . While the form of this equation appears to resemble mx+b, this is really only a single point correction equation. There is no “slope” and “intercept” per se; these are simply correction coefficients related to the particular sensor and its construction. The last two terms in Equation (8) are correction, for misalignment of the sensor axes, i.e., corrections for deviations away from the axes being precisely orthogonal. 
       7 Mathematical Aspects of Data Calibration and Measurement 
       [0111]    Mathematical aspects that are used in the design of examples of the system are discussed in this section. Note that the derivations set forth are not necessarily the only ways of configuring examples of the system. 
       7.1 General Set-Up 
     7.1.1 Definition of Axes 
       [0112]    Referring to  FIG. 10 , the axes of the sensor (either an accelerometer or a magnetometer) are S 1 , S 2  and S 3 . In general, these are non-orthogonal and are not aligned with the projectile axes. They are related to an orthogonal coordinate set G 1 , G 2 , G 3  via Gram-Schmidt orthogonalization. The orthogonal projectile coordinate system is denoted by P 1 , P 2  and P 3 . Finally, the orthogonal earth system is denoted by E 1 , E 2  and E 3  in which the E 1  axis is aligned with the magnetic field (H) and the E 3  axis is aligned with the gravitational field (g). 
         [0113]    In  FIG. 10 , the sensor, Gram-Schmidt, projectile, and earth axes (denoted by S, G, P, and E respectively). The Gram-Schmidt and projectile axes are related via rotations through three angles α 1 , α 2 , α 3  analogous to Euler angles. The projector and earth axes are related by rotations through Euler angles φ, θ, ψ. 
       7.1.2 Relationship Between Sensor Frame {S 1 ,S 2 ,S 3 } &amp; Projectile Frame {P 1 ,P 2 ,P 3 } 
       [0114]    The sensor frame is related to the projectile frame by orthogonalization and by rotations. The orthogonalization transformation can be represented by a Gram-Schmidt matrix. We take the S 1  and P 1  axes to be identical, and the matrix to transform from the sensor axes to the Gram-Schmidt is as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         T 
                         = 
                       
                       S 
                       G 
                     
                     = 
                     
                       ( 
                       
                         
                           
                             1 
                           
                           
                             0 
                           
                           
                             0 
                           
                         
                         
                           
                             
                               
                                 - 
                                 
                                   λ 
                                   2 
                                   
                                     - 
                                     1 
                                   
                                 
                               
                                
                               
                                 
                                   
                                     e 
                                     _ 
                                   
                                   2 
                                   S 
                                 
                                 · 
                                 
                                   
                                     e 
                                     _ 
                                   
                                   1 
                                   G 
                                 
                               
                             
                           
                           
                             
                               λ 
                               2 
                               
                                 - 
                                 1 
                               
                             
                           
                           
                             0 
                           
                         
                         
                           
                             
                               
                                 - 
                                 
                                   λ 
                                   3 
                                   
                                     - 
                                     1 
                                   
                                 
                               
                                
                               
                                 
                                   
                                     e 
                                     _ 
                                   
                                   3 
                                   S 
                                 
                                 · 
                                 
                                   
                                     e 
                                     _ 
                                   
                                   1 
                                   G 
                                 
                               
                             
                           
                           
                             
                               
                                 - 
                                 
                                   λ 
                                   3 
                                   
                                     - 
                                     1 
                                   
                                 
                               
                                
                               
                                 
                                   
                                     e 
                                     _ 
                                   
                                   3 
                                   S 
                                 
                                 · 
                                 
                                   
                                     e 
                                     _ 
                                   
                                   2 
                                   G 
                                 
                               
                             
                           
                           
                             
                               λ 
                               3 
                               
                                 - 
                                 1 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                   , 
                   
                     
 
                   
                    
                   
                     
                       λ 
                       i 
                     
                     = 
                     
                        
                       
                         
                           
                             e 
                             _ 
                           
                           i 
                           G 
                         
                         - 
                         
                           
                             ∑ 
                             
                               j 
                               = 
                               1 
                             
                             i 
                           
                            
                           
                             
                               ( 
                               
                                 
                                   
                                     e 
                                     _ 
                                   
                                   i 
                                   G 
                                 
                                 · 
                                 
                                   
                                     e 
                                     _ 
                                   
                                   j 
                                   S 
                                 
                               
                               ) 
                             
                              
                             
                               
                                 e 
                                 _ 
                               
                               j 
                               S 
                             
                           
                         
                       
                        
                     
                   
                 
               
               
                 
                   ( 
                   9 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where e 1   A  is a unit vector pointing in the i-direction in the frame denoted by superscript A. Note that since the sensor coordinate system is not orthogonal, the transformation matrix  T   S   G  is not orthogonal, and therefore its inverse does not equal its adjoint. 
         [0115]    We define rotations, analogous to the Euler angle rotations, to transform between the projectile frame and the frame determined via Gram-Schmidt orthogonalization. Specifically, we assume that rotations through α 3  about the P 3 -axis, α 2  about the P 2 -axis and α 1  about the P 1 -axis (in that order) transform the projectile axes to the Gram-Schmidt axes. Then the rotation matrix from the Gram-Schmidt vectors to the projectile axes is 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         T 
                         = 
                       
                       G 
                       P 
                     
                     = 
                     
                       ( 
                       
                         
                           
                             
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     α 
                                     2 
                                   
                                   ) 
                                 
                               
                                
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     α 
                                     3 
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       
                                         sin 
                                          
                                         
                                           ( 
                                           
                                             α 
                                             2 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         sin 
                                          
                                         
                                           ( 
                                           
                                             α 
                                             1 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           
                                             α 
                                             3 
                                           
                                           ) 
                                         
                                       
                                     
                                     - 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       cos 
                                        
                                       
                                         ( 
                                         
                                           α 
                                           1 
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       sin 
                                        
                                       
                                         ( 
                                         
                                           α 
                                           3 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       
                                         sin 
                                          
                                         
                                           ( 
                                           
                                             α 
                                             2 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           
                                             α 
                                             1 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           
                                             α 
                                             3 
                                           
                                           ) 
                                         
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       sin 
                                        
                                       
                                         ( 
                                         
                                           α 
                                           1 
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       sin 
                                        
                                       
                                         ( 
                                         
                                           α 
                                           3 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     α 
                                     2 
                                   
                                   ) 
                                 
                               
                                
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     α 
                                     3 
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       
                                         sin 
                                          
                                         
                                           ( 
                                           
                                             α 
                                             2 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         sin 
                                          
                                         
                                           ( 
                                           
                                             α 
                                             1 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         sin 
                                          
                                         
                                           ( 
                                           
                                             α 
                                             3 
                                           
                                           ) 
                                         
                                       
                                     
                                     + 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       cos 
                                        
                                       
                                         ( 
                                         
                                           α 
                                           1 
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       cos 
                                        
                                       
                                         ( 
                                         
                                           α 
                                           3 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                           
                           
                             
                               
                                 
                                   
                                     
                                       
                                         sin 
                                          
                                         
                                           ( 
                                           
                                             α 
                                             2 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         cos 
                                          
                                         
                                           ( 
                                           
                                             α 
                                             1 
                                           
                                           ) 
                                         
                                       
                                        
                                       
                                         sin 
                                          
                                         
                                           ( 
                                           
                                             α 
                                             3 
                                           
                                           ) 
                                         
                                       
                                     
                                     - 
                                   
                                 
                               
                               
                                 
                                   
                                     
                                       sin 
                                        
                                       
                                         ( 
                                         
                                           α 
                                           1 
                                         
                                         ) 
                                       
                                     
                                      
                                     
                                       cos 
                                        
                                       
                                         ( 
                                         
                                           α 
                                           3 
                                         
                                         ) 
                                       
                                     
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     α 
                                     3 
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     α 
                                     2 
                                   
                                   ) 
                                 
                               
                                
                               
                                 sin 
                                  
                                 
                                   ( 
                                   
                                     α 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     α 
                                     2 
                                   
                                   ) 
                                 
                               
                                
                               
                                 cos 
                                  
                                 
                                   ( 
                                   
                                     α 
                                     1 
                                   
                                   ) 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                    
                   
                     
 
                   
                    
                   and 
                 
               
               
                 
                   ( 
                   10 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       T 
                       = 
                     
                     S 
                     P 
                   
                   = 
                   
                     
                       
                         T 
                         = 
                       
                       G 
                       P 
                     
                     · 
                     
                       
                         T 
                         = 
                       
                       S 
                       G 
                     
                   
                 
               
               
                 
                   ( 
                   11 
                   ) 
                 
               
             
           
         
       
     
         [0000]    is the transformation matrix from the sensor to the projectile frame. Then vectors measured in the sensor frame (S) can be converted to the corresponding vectors in the projectile frame (P) using the relationship 
         [0000]          v     P   = T     S   P   · v     S    (12) 
       7.1.3 Relationship Between Projectile Frame {P 1 ,P 2 ,P 3 } &amp; Earth Frame {E 1 ,E 2 ,E 3 } 
       [0116]    The projectile coordinate frame is related to the earth coordinate frame by rotations through the Enter angles (pitch—θ, roll—φ and heading—ψ), which give rise to the following transformation matrix from the projectile axes to the earth axes: 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       T 
                       = 
                     
                     P 
                     E 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             
                               cos 
                                
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                              
                             
                               cos 
                                
                               
                                 ( 
                                 ψ 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   
                                     
                                       sin 
                                        
                                       
                                         ( 
                                         θ 
                                         ) 
                                       
                                     
                                      
                                     
                                       sin 
                                        
                                       
                                         ( 
                                         ϕ 
                                         ) 
                                       
                                     
                                      
                                     
                                       cos 
                                        
                                       
                                         ( 
                                         ψ 
                                         ) 
                                       
                                     
                                   
                                   - 
                                 
                               
                             
                             
                               
                                 
                                   
                                     cos 
                                      
                                     
                                       ( 
                                       ϕ 
                                       ) 
                                     
                                   
                                    
                                   
                                     sin 
                                      
                                     
                                       ( 
                                       ψ 
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   
                                     
                                       sin 
                                        
                                       
                                         ( 
                                         θ 
                                         ) 
                                       
                                     
                                      
                                     
                                       cos 
                                        
                                       
                                         ( 
                                         ϕ 
                                         ) 
                                       
                                     
                                      
                                     
                                       cos 
                                        
                                       
                                         ( 
                                         ψ 
                                         ) 
                                       
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     sin 
                                      
                                     
                                       ( 
                                       ϕ 
                                       ) 
                                     
                                   
                                    
                                   
                                     sin 
                                      
                                     
                                       ( 
                                       ψ 
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               cos 
                                
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                              
                             
                               sin 
                                
                               
                                 ( 
                                 ψ 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   
                                     
                                       sin 
                                        
                                       
                                         ( 
                                         θ 
                                         ) 
                                       
                                     
                                      
                                     
                                       sin 
                                        
                                       
                                         ( 
                                         ϕ 
                                         ) 
                                       
                                     
                                      
                                     
                                       sin 
                                        
                                       
                                         ( 
                                         ψ 
                                         ) 
                                       
                                     
                                   
                                   + 
                                 
                               
                             
                             
                               
                                 
                                   
                                     cos 
                                      
                                     
                                       ( 
                                       ϕ 
                                       ) 
                                     
                                   
                                    
                                   
                                     cos 
                                      
                                     
                                       ( 
                                       ψ 
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                         
                           
                             
                               
                                 
                                   
                                     
                                       sin 
                                        
                                       
                                         ( 
                                         θ 
                                         ) 
                                       
                                     
                                      
                                     
                                       cos 
                                        
                                       
                                         ( 
                                         ϕ 
                                         ) 
                                       
                                     
                                      
                                     
                                       sin 
                                        
                                       
                                         ( 
                                         ψ 
                                         ) 
                                       
                                     
                                   
                                   - 
                                 
                               
                             
                             
                               
                                 
                                   
                                     sin 
                                      
                                     
                                       ( 
                                       ϕ 
                                       ) 
                                     
                                   
                                    
                                   
                                     cos 
                                      
                                     
                                       ( 
                                       ψ 
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                         
                       
                       
                         
                           
                             - 
                             
                               sin 
                                
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               cos 
                                
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                              
                             
                               sin 
                                
                               
                                 ( 
                                 ϕ 
                                 ) 
                               
                             
                           
                         
                         
                           
                             
                               cos 
                                
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                              
                             
                               cos 
                                
                               
                                 ( 
                                 ϕ 
                                 ) 
                               
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   13 
                   ) 
                 
               
             
           
         
       
     
       Then vectors can be transformed from the sensor to the earth frame using the relationship 
       [0117]    
       
         
           
             
               
                 
                   
                       
                   
                    
                   
                     
                       
                         
                           v 
                           _ 
                         
                         E 
                       
                       = 
                       
                         
                           
                             
                               T 
                               = 
                             
                             P 
                             E 
                           
                           * 
                           
                             
                               T 
                               = 
                             
                             S 
                             P 
                           
                           * 
                           
                             
                               v 
                               _ 
                             
                             S 
                           
                         
                         = 
                         
                           
                             
                               T 
                               = 
                             
                             P 
                             E 
                           
                           * 
                           
                             
                               v 
                               _ 
                             
                             P 
                           
                         
                       
                     
                     , 
                     
                       
 
                     
                      
                     
                       
                         v 
                         i 
                         E 
                       
                       = 
                       
                         
                           
                             ∑ 
                             
                               j 
                               , 
                               
                                 k 
                                 = 
                                 1 
                               
                             
                             3 
                           
                            
                           
                             
                               
                                 ( 
                                 
                                   T 
                                   P 
                                   E 
                                 
                                 ) 
                               
                               ij 
                             
                              
                             
                               
                                 ( 
                                 
                                   T 
                                   S 
                                   P 
                                 
                                 ) 
                               
                               jk 
                             
                              
                             
                               
                                 ( 
                                 
                                   v 
                                   S 
                                 
                                 ) 
                               
                               k 
                             
                           
                         
                         = 
                         
                           
                             ∑ 
                             
                               j 
                               = 
                               1 
                             
                             3 
                           
                            
                           
                             
                               
                                 ( 
                                 
                                   T 
                                   P 
                                   E 
                                 
                                 ) 
                               
                               ij 
                             
                              
                             
                               
                                 ( 
                                 
                                   v 
                                   P 
                                 
                                 ) 
                               
                               j 
                             
                           
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   14 
                   ) 
                 
               
             
           
         
       
     
       7.2 Relating these Transformations to Calibrations 
       [0118]    First, all accelerometer and magnetometer readings are corrected for offset and gains. Then the calibration matrix is determined by measuring the sensor reading on the j-axis (in the sensor frame) when the system is accelerated in the i-direction (in the projectile frame) to obtain the i-j entry. 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           T 
                           = 
                         
                         S 
                         P 
                       
                       ) 
                     
                     ij 
                   
                   = 
                   
                     
                       
                         
                           
                             e 
                             _ 
                           
                           i 
                           p 
                         
                         · 
                         
                           
                             e 
                             _ 
                           
                           j 
                           s 
                         
                       
                       ⇒ 
                       
                         
                           e 
                           _ 
                         
                         i 
                         p 
                       
                     
                     = 
                     
                       
                         ∑ 
                         
                           j 
                           = 
                           1 
                         
                         3 
                       
                        
                       
                         
                           
                             ( 
                             
                               
                                 T 
                                 = 
                               
                               S 
                               P 
                             
                             ) 
                           
                           ij 
                         
                          
                         
                           
                             e 
                             _ 
                           
                           j 
                           s 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   15 
                   ) 
                 
               
             
           
         
       
     
       The (small) off-diagonal terms are the so-called “cross-product” terms (“dXdY”, dYdZ” and “dZdX”). 
       [0119]    We demonstrate how the calibration matrix is calculated via an example. The term ( T   S   P ) 31  is determined by taking 100 readings of G 1   S  and G 2   S  when the accelerometer is in the gravitational field. The projectile is then rotated through a roll angle of 180 degrees, and the measurements are repeated. Then 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       ( 
                       
                         
                           T 
                           = 
                         
                         s 
                         p 
                       
                       ) 
                     
                     31 
                   
                   = 
                   
                     
                       
                         
                           G 
                           x 
                           p 
                         
                          
                         
                           ( 
                           
                             ϕ 
                             = 
                             0 
                           
                           ) 
                         
                       
                       - 
                       
                         
                           G 
                           x 
                           p 
                         
                          
                         
                           ( 
                           
                             ϕ 
                             = 
                             180 
                           
                           ) 
                         
                       
                     
                     
                       
                         
                           G 
                           z 
                           p 
                         
                          
                         
                           ( 
                           
                             ϕ 
                             = 
                             0 
                           
                           ) 
                         
                       
                       + 
                       
                         
                           G 
                           z 
                           p 
                         
                          
                         
                           ( 
                           
                             ϕ 
                             = 
                             180 
                           
                           ) 
                         
                       
                     
                   
                 
               
               
                 
                   ( 
                   16 
                   ) 
                 
               
             
           
         
       
     
         [0120]    Note that ( T   S   P ) 31 ≠( T   S   P ) 12 , indicating that “dx dy≠dy dx”. 
       7.3 Relating these Transformations to Measurements 
     7.3.1 Determining Roll and Pitch from Accelerometer Measurements 
       [0121]    When measuring the earth&#39;s gravitational field which is aligned with the z-coordinate in the earth frame, Equation (14) reduces to the form 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         ( 
                         
                           
                             T 
                             = 
                           
                           p 
                           E 
                         
                         ) 
                       
                       y 
                     
                     · 
                     
                       ( 
                       
                         
                           
                             0 
                           
                         
                         
                           
                             0 
                           
                         
                         
                           
                             
                               - 
                               g 
                             
                           
                         
                       
                       ) 
                     
                   
                   = 
                   
                     
                       - 
                       
                         g 
                          
                         
                           ( 
                           
                             
                               
                                 
                                   - 
                                   
                                     sin 
                                      
                                     
                                       ( 
                                       θ 
                                       ) 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     cos 
                                      
                                     
                                       ( 
                                       θ 
                                       ) 
                                     
                                   
                                    
                                   
                                     sin 
                                      
                                     
                                       ( 
                                       ϕ 
                                       ) 
                                     
                                   
                                 
                               
                             
                             
                               
                                 
                                   
                                     cos 
                                      
                                     
                                       ( 
                                       θ 
                                       ) 
                                     
                                   
                                    
                                   
                                     cos 
                                      
                                     
                                       ( 
                                       ϕ 
                                       ) 
                                     
                                   
                                 
                               
                             
                           
                           ) 
                         
                       
                     
                     = 
                     
                       
                         
                           v 
                           _ 
                         
                         p 
                       
                       = 
                       
                         ( 
                         
                           
                             
                               
                                 G 
                                 x 
                                 p 
                               
                             
                           
                           
                             
                               
                                 G 
                                 y 
                                 p 
                               
                             
                           
                           
                             
                               
                                 G 
                                 z 
                                 p 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   17 
                   ) 
                 
               
             
           
         
       
     
         [0000]    where we have used the result (T P   E ) −1 =(T P   E ) P . Therefore roll and pitch can be determined as follows: 
         [0000]    
       
         
           
             
               
                 
                   
                     ϕ 
                     = 
                     
                       
                         tan 
                         
                           - 
                           1 
                         
                       
                        
                       
                         ( 
                         
                           
                             G 
                             y 
                             p 
                           
                           
                             G 
                             z 
                             p 
                           
                         
                         ) 
                       
                     
                   
                   , 
                   
                       
                   
                    
                   
                     θ 
                     = 
                     
                       - 
                       
                         
                           tan 
                           
                             - 
                             1 
                           
                         
                         ( 
                         
                           
                             G 
                             x 
                             p 
                           
                           
                             
                               
                                 
                                   ( 
                                   
                                     G 
                                     y 
                                     p 
                                   
                                   ) 
                                 
                                 2 
                               
                               + 
                               
                                 
                                   ( 
                                   
                                     G 
                                     z 
                                     p 
                                   
                                   ) 
                                 
                                 2 
                               
                             
                           
                         
                         ) 
                       
                     
                   
                 
               
               
                 
                   ( 
                   18 
                   ) 
                 
               
             
           
         
       
     
       7.3.2 Determining Heading from Magnetometer Measurements 
       [0122]    The heading angle can be determined as follows: consider the decomposition of the Euler transformation matrix into the three separate rotations (notation as in  Harkins,  2003). From Equation (14) we can deduce 
         [0000]    
       
         
           
             
               
                 
                   
                     
                       
                         T 
                         = 
                       
                       p 
                       E 
                     
                     = 
                     
                       
                         
                           
                             
                               E 
                               = 
                             
                             1 
                             y 
                           
                           * 
                           
                             
                               E 
                               = 
                             
                             2 
                             y 
                           
                           * 
                           
                             
                               E 
                               = 
                             
                             3 
                             y 
                           
                         
                         ⇒ 
                         
                           
                             
                               E 
                               = 
                             
                             3 
                           
                           * 
                           
                             
                               v 
                               _ 
                             
                             E 
                           
                         
                       
                       = 
                       
                         
                           
                             E 
                             = 
                           
                           2 
                           y 
                         
                         * 
                         
                           
                             E 
                             = 
                           
                           3 
                           y 
                         
                         * 
                         
                           
                             v 
                             _ 
                           
                           p 
                         
                       
                     
                   
                    
                   
                     
 
                   
                    
                   where 
                 
               
               
                 
                   ( 
                   19 
                   ) 
                 
               
             
             
               
                 
                   
                     
                       
                         E 
                         = 
                       
                       i 
                       
                         - 
                         2 
                       
                     
                     = 
                     
                       
                         E 
                         = 
                       
                       t 
                       p 
                     
                   
                   , 
                   
                       
                   
                    
                   
                     
 
                   
                    
                   
                       
                   
                    
                   
                     i 
                     = 
                     1 
                   
                   , 
                   2 
                   , 
                   3 
                   , 
                   
                     
                       
                         
                           E 
                           = 
                         
                         2 
                         r 
                       
                       * 
                       
                         
                           E 
                           = 
                         
                         3 
                         r 
                       
                     
                     = 
                     
                       ( 
                       
                         
                           
                             
                               cos 
                                
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                           
                           
                             
                               
                                 sin 
                                  
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                                
                               
                                 sin 
                                  
                                 
                                   ( 
                                   ϕ 
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 sin 
                                  
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                                
                               
                                 cos 
                                  
                                 
                                   ( 
                                   ϕ 
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             0 
                           
                           
                             
                               cos 
                                
                               
                                 ( 
                                 ϕ 
                                 ) 
                               
                             
                           
                           
                             
                               - 
                               
                                 sin 
                                  
                                 
                                   ( 
                                   ϕ 
                                   ) 
                                 
                               
                             
                           
                         
                         
                           
                             
                               - 
                               
                                 sin 
                                  
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 cos 
                                  
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                                
                               
                                 sin 
                                  
                                 
                                   ( 
                                   ϕ 
                                   ) 
                                 
                               
                             
                           
                           
                             
                               
                                 cos 
                                  
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                                
                               
                                 cos 
                                  
                                 
                                   ( 
                                   ϕ 
                                   ) 
                                 
                               
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   20 
                   ) 
                 
               
             
           
         
       
     
         [0123]    Since the magnetic field in the earth&#39;s frame points in the x-direction, Equation (19) can be written as 
         [0000]    
       
         
           
             
               
                 
                   
                     ( 
                     
                       
                         
                           
                             
                               cos 
                                
                               
                                 ( 
                                 ψ 
                                 ) 
                               
                             
                              
                             
                               H 
                               s 
                               E 
                             
                           
                         
                       
                       
                         
                           
                             
                               - 
                               
                                 sin 
                                  
                                 
                                   ( 
                                   ψ 
                                   ) 
                                 
                               
                             
                              
                             
                               H 
                               x 
                               E 
                             
                           
                         
                       
                       
                         
                           0 
                         
                       
                     
                     ) 
                   
                   = 
                   
                     ( 
                     
                       
                         
                           
                             
                               
                                 cos 
                                  
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                                
                               
                                 H 
                                 s 
                                 p 
                               
                             
                             + 
                             
                               
                                 sin 
                                  
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                                
                               
                                 sin 
                                  
                                 
                                   ( 
                                   ϕ 
                                   ) 
                                 
                               
                                
                               
                                 H 
                                 y 
                                 p 
                               
                               * 
                               
                                 sin 
                                  
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                                
                               
                                 cos 
                                  
                                 
                                   ( 
                                   ϕ 
                                   ) 
                                 
                               
                                
                               
                                 H 
                                 z 
                                 p 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 cos 
                                  
                                 
                                   ( 
                                   ϕ 
                                   ) 
                                 
                               
                                
                               
                                 H 
                                 y 
                                 p 
                               
                             
                             - 
                             
                               
                                 sin 
                                  
                                 
                                   ( 
                                   ϕ 
                                   ) 
                                 
                               
                                
                               
                                 H 
                                 z 
                                 p 
                               
                             
                           
                         
                       
                       
                         
                           
                             
                               
                                 - 
                                 
                                   sin 
                                    
                                   
                                     ( 
                                     θ 
                                     ) 
                                   
                                 
                               
                                
                               
                                 H 
                                 s 
                                 p 
                               
                             
                             + 
                             
                               
                                 cos 
                                  
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                                
                               
                                 sin 
                                  
                                 
                                   ( 
                                   ϕ 
                                   ) 
                                 
                               
                                
                               
                                 H 
                                 z 
                                 p 
                               
                             
                             + 
                             
                               
                                 cos 
                                  
                                 
                                   ( 
                                   θ 
                                   ) 
                                 
                               
                                
                               
                                 cos 
                                  
                                 
                                   ( 
                                   ϕ 
                                   ) 
                                 
                               
                                
                               
                                 H 
                                 z 
                                 p 
                               
                             
                           
                         
                       
                     
                     ) 
                   
                 
               
               
                 
                   ( 
                   21 
                   ) 
                 
               
             
           
         
       
     
         [0000]    which leads to the result 
         [0000]    
       
         
           
             
               
                 
                   ψ 
                   = 
                   
                     
                       tan 
                       
                         - 
                         1 
                       
                     
                      
                     
                       ( 
                       
                         
                           
                             
                               cos 
                                
                               
                                 ( 
                                 ϕ 
                                 ) 
                               
                             
                              
                             
                               H 
                               s 
                               p 
                             
                           
                           - 
                           
                             
                               sin 
                                
                               
                                 ( 
                                 ϕ 
                                 ) 
                               
                             
                              
                             
                               H 
                               z 
                               p 
                             
                           
                         
                         
                           
                             
                               cos 
                                
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                              
                             
                               H 
                               s 
                               p 
                             
                           
                           + 
                           
                             
                               sin 
                                
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                              
                             
                               sin 
                                
                               
                                 ( 
                                 ϕ 
                                 ) 
                               
                             
                              
                             
                               H 
                               y 
                               p 
                             
                           
                           + 
                           
                             
                               sin 
                                
                               
                                 ( 
                                 θ 
                                 ) 
                               
                             
                              
                             
                               cos 
                                
                               
                                 ( 
                                 ϕ 
                                 ) 
                               
                             
                              
                             
                               H 
                               z 
                               p 
                             
                           
                         
                       
                       ) 
                     
                   
                 
               
               
                 
                   ( 
                   22 
                   ) 
                 
               
             
           
         
       
     
         [0124]    Note that there is an implicit choice of sign for cosine calculated in Equation (18). Note also that the z-equation in Equation (21) is not used and therefore need not be calculated in software. 
         [0125]    Although the above method works in theory, in some examples errors in the calibration lead to a variation in the heading angle as the roll angle is varied. Therefore for some examples of the system, a second “calibration” is performed in which the first mode of variation with roll angle is subtracted out. 
         [0126]    It is to be understood that the foregoing description is intended to illustrate and not to limit the scope of the invention, which is defined by the scope of the appended claims. Other embodiments are within the scope of the following claims.