Patent Publication Number: US-7711476-B2

Title: Aided INS/GPS/SAR navigation with other platforms

Description:
This application is a continuation in part of U.S. Patent and Trademark Office application Ser. No. 11/159,475, filed Jun. 23, 2005 now U.S. Pat. No. 7,395,156. 
    
    
     BACKGROUND OF THE INVENTION 
     1. Field of the Invention 
     This invention is in the field of improved sensor motion (position, velocity, acceleration) accuracy during GPS unavailability using multiple sensors locating a particular geo-target in a common time frame. 
     2. Description of the Related Art 
     Synthetic Aperture Radar (SAR) is used for ground mapping as well as target identification. The general principle behind SAR is to coherently combine the amplitude and phase information of radar returns from a plurality of sequentially transmitted pulses. These pulses are from a relatively small antenna on a moving platform. As the platform moves, the information contained in the pulses is coherently combined to arrive at a high resolution SAR image. 
     The plurality of returns creating a SAR image generated by the transmitted pulses along a presumed known path of the platform make up an array. Theoretically, during the array, amplitude as well as phase information returned from each of the pulses, for each of many range bins, is preserved. The SAR image is formed from the coherent combination of the amplitude and phase of return(s) within each range bin, motion compensated for spatial displacement of the moving platform during the acquisition of the returns for the duration of the array. 
     The clarity of a SAR image is in many respects dependent on the quality of the motion compensation applied to each radar return contributing to the coherent SAR image computation. Motion compensation shifts the phase of each radar sample (typically an I+jQ complex quantity derived from an analog to digital converter) in accordance with the motion in space of the moving platform with respect to a reference point. The SAR imaging process depends on the coherent, phase accurate summing of all radar returns expected within an array. These principles are detailed by W. G. Carrara, R. S. Goodman and R. M. Majewski in  Spotlight Synthetic Radar , Boston, Artech House, 1995, incorporated herein in its entirety by reference. 
     Thus, the accuracy of the motion compensation for phase coherence applied to each radar A/D sample is critical to SAR imaging. Typically, an inertial navigation system (INS) using accelerometers and gyroscopes derives velocity, acceleration and position information for use in radar motion compensation. The INS is updated from various sources, such as satellite based Global Positioning Systems (GPS) or pre-stored geo registered features of the terrain in the vicinity of the path of the moving radar. Using an INS aided by GPS may subject the GPS to jamming, corrupting the GPS signal. Consequently, the GPS jamming may induce a position error that may manifest itself in certain applications as a blurring of SAR images, reducing SAR utility. 
     Using a plurality of pre-stored geo registered features (position references) instead of GPS updates of the INS requires added memory and interfacing within the radar, increasing parts count and reducing reliability. The concept of using pre-stored geo registered features for increased position accuracy is described in U.S. Pat. No. 5,485,384, titled On Board Navigation System For An Aerial Craft Including a Synthetic Aperture Sideways Looking Radar issued Jan. 16, 1996 to B. Falconnet and U.S. Pat. No. 5,432,520 titled SAR/GPS Inertial Method of Range Measurement, issued Jul. 11, 1995 to Schneider et al., both incorporated herein in their entirety by reference. An alternative to using pre-stored geo registered features is thus desirable to avoid generating an accurate geo registered feature database, updating it, and interfacing it to a sensor, such as a radar system. 
     SUMMARY OF THE INVENTION 
     The need for geo-registered features is avoided and above limitations reduced by a system for estimating a motion of a first sensor during a time interval, said system comprising a second sensor, said first sensor and said second sensor sensing the same geo-location during said time interval. The first sensor computes a first target location error from sensing the geo-location. The second sensor also computing a second target location error from sensing the same geo-location. A data link interconnects the first sensor and the second sensor, the data link transmitting the second target location error computed by the second sensor to the first sensor during the time interval. A processor at said first sensor combines the first target location error and the second target location error in a first sensor observation model, where the sensor observation model is descriptive of the motion of the first sensor. The observation model is used with a Kalman filter to update the position of the first sensor. 
     The geo-location is not pre-registered, nor known in advance. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWING 
       In the Drawing: 
         FIG. 1  is a sensor location system of the prior art where the absence of GPS positioning information is compensated for by determining sensor location with respect to geo registered feature(s) using a SAR, or FLIR sensor of the geo-registered feature; 
         FIG. 1A  is a Kalman filter configuration of the prior art using the structure of  FIG. 1  to update the position of a sensor; 
         FIG. 2  shows a target location error (TLE) ellipsoid typical of the output from a Kalman filter using a single sensor; 
         FIG. 3  shows the bidirectional data link between sensors (platforms) enabling transmission of data between sensors; 
         FIG. 4  shows position updating of sensor  0  with TLEs from a plurality of platforms  1 , . . . i . . . (n−1) used within the Kalman filter of sensor  0 ; 
         FIG. 5  shows the combination of two TLEs, from sensor  1  (E 1 ) and TLE from sensor  0  (E 0 ); and 
         FIG. 6  shows a typical flow diagram of the method used in this disclosure. 
     
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     The present disclosure describes a method and apparatus for geo-locating the position of a sensor when Global Positioning Satellite (GPS) transmission is jammed, or unavailable. Instead of using GPS, the position of the sensor is computed by combining a plurality of TLEs derived from sensing a geo-location by a plurality of sensors. The TLEs thus derived are transmitted using a data link from independent sensors motion measurements, where the sensors may be separated in space and not co-located. The geo-location is not known in advance, nor is it pre-registered. 
     As shown in  FIG. 1 , in the prior art, sensor motion is initially measured by an inertial measurement unit  110 . The output from inertial measurement unit  110  is used by inertial navigation system processor  106  to update a Kalman filter  102  in conjunction with GPS signal from processor  108  within INS/GPS integrator  100 . If however, GPS jamming detector  104  determines that the GPS signal is jammed or unavailable, then guidance process  118  directs a SAR/FLIR sensor  116  to perform location measurements on a geo-registered feature  120 . The output from SAR/FLIR sensor  116  is examined in image processor  114  and compared with contents of Catalog of Registered Features  112 . Position of SAR/FLIR sensor  116  is extracted from data provided from image processor  114  in Extract Position  118 . This yields the position of sensor  116  with respect to the geo-registered feature  120 . The position of SAR/FLIR sensor  116  thus derived from image processor  114  and Extract Position  118  is used within Kalman filter  102  to augment inertial navigation information, instead of the unavailable GPS information. The limitation of this prior art approach is that a catalog of geo registered features  112  has to exist for SAR/FLIR processor sensor  116  for its current geographical position. 
     The motion of sensor  116  (position, velocity and acceleration) is updated and projected by a motion navigator. An example of a motion navigator used with this disclosure is the classical Kalman filter, (Kalman and Bucy, 1960) well known in the art, and described by A. Gelb in  Applied Optimal Estimation,  1974, The Analytic Science Corporation, The MIT Press, incorporated herein in its entirety by reference. The function of the motion navigator in this disclosure, exemplified by the Kalman filter, is to (optimally) predict the position, velocity and acceleration of a platform containing a sensor at known time intervals between inputs from a position sensor, such as, for example, a Synthetic Aperture Radar (SAR), Forward Looking Infrared (FLIR), and/or SONAR. 
     As shown in  FIG. 1A , further detailing  FIG. 1 , the estimation of motion is made from a position sensor generating input data  101  at time t=1. SAR/FLIR position sensor  116  measures position during an update time interval, and presents results at the end of the time interval. The Kalman filter evaluates changes in position measured by the position sensor over many previous intervals to estimate future motion, or trajectory of the sensor during the next measuring interval. 
     Continuing with  FIG. 1A , in computing the estimation of sensor motion within a Kalman filter, an error covariance matrix  103  is computed along with an observation model  105 , typically made up of the Observation Matrix H k  and Measurement Noise Matrix R k . Based on the observation model, optimal gains for particular parameters are computed in Compute Optimal Gains K k    107  and used by Update Sensor Position and Filter Parameters  111 . The cycle is repeated during the next time interval, t=2. 
     The input data is presented to the Kalman filter, described by the Gelb reference above, and described as follows:
 
 {right arrow over (x)}   k =Φ k−1   {right arrow over (x)}   k−1   +{right arrow over (w)}   k−1   , {right arrow over (w)}   k   ˜N ({right arrow over (0)} , Q   k )
 
 {right arrow over (z)}   k   =H   k   {right arrow over (x)}   k   +{right arrow over (v)}   k   , {right arrow over (v)}   k   ˜N ({right arrow over (0)} , R   k )
 
 E[{right arrow over (x)} (0)]= {right arrow over (x)}   0   , E[{right arrow over (x)}   0   {right arrow over (x)}   0   T   ]=P   0   , E[{right arrow over (w)}   j   {right arrow over (v)}   k   T ]={right arrow over (0)} ∀j,k  
 
 P   k   − =Φ k−1   P   k−1   + Φ k−1   T   +Q   k−1  
 
 P   k   + =( I−K   k   H   k ) P   k   − 
 
 K   k   =P   k   −   H   k   T   [H   k   P   k   −   H   k   T   +R   k ] −1    Equation 1
 
where Φ k  is the transition matrix for interval k.
 
     The error covariance propagation matrix P k   +   103 , Kalman filter gain K k    107 , measurement model  105  (observation matrix H k , and observation noise matrix R k ) are shown in  FIG. 1A . 
     A typical target location error (TLE) ellipsoid  202  derived from computations in  FIG. 1  and  FIG. 1A  is shown in  FIG. 2 . The TLE is due to uncertainty in the position of the sensor (or its associated platform), with respect to a coordinate system. Thus, the TLE represents an error independent of sensor capabilities or accuracy in detecting or tracking a target. There are three perpendicular quantities that define the extent of a TLE: vertical track  204 , cross track  206  and down-track  208  error. 
     Equation 1 depends on only one sensor (or platform) to derive target location and update the trajectory estimation of the target from data at t=1 to be used for the time interval k until the next update at t=2. 
     In contrast, this disclosure derives an improved (measurement) noise matrix R k  and an observation matrix H k  for use within a sensor navigator, e.g. Kalman filter. This disclosure combines a plurality of independent common geo-location measuring sensors (independent sources of a target location), each having its own independent TLE for overall measurement. The combination of TLEs provide a more accurate estimate of target position as compared to any one of the contributing sensors. That is, a plurality of measurements from individual, independent sensors on separate, or the same platform(s), are combined to obtain a lower, new TLE. The new TLE is of higher accuracy, lesser extent, lesser volume, as compared to the TLE from any one of the sensors used to form the combined, new TLE. In some applications, the more accurate TLE obtained from combining observation models from SAR and/or FLIR sensors avoids the need for the inclusion of geo-registered features in target location computations. The same concept may be applied to sonar in an ocean environment. 
     Reducing Target Location Errors (TLE). 
     Errors in computed target geo-location using a single platform can be primarily attributed to three major sources: 
     sensor position errors 
     sensor bearing errors and 
     Digital Terrain Elevation Data (DTED) errors. 
     For this disclosure, above three errors are being treated as being statistically independent, i.e. fully de-correlated, and have a zero mean Gaussian distribution. Other errors, not cited, are assumed insignificantly small compared to the above three errors, and thus are not considered. It is further assumed that the target to be tracked is acquired, that is, its position is detected at time intervals used to compute its new position during the next time interval. DTED noise is treated as part of the measurement noise in the Z direction using the Earth Center Earth Fixed (ECEF) coordinate frame. 
     Errors in a target position sensor directly transform to the slant coordinate frame. These errors are converted into the platform coordinate system using the angle ψ described in the parent application. 
     As detailed in the parent application, the slant plane consists of vectors {right arrow over (V)} D  and {right arrow over (R)}. The slant coordinate frame is defined by: 
     x s  is along the vehicle velocity axis {right arrow over (V)}; 
     z s  is perpendicular to the slant plane; 
     y s  forms a right hand coordinate system with x s  and z s . 
     The platform coordinate frame (sensor body coordinate frame) is defined by: 
     x p  is sensor (vehicle) velocity direction; 
     y p  is sensor (vehicle) right wing; 
     z p  forms a right hand coordinate system with x p  and y p . 
     The relationship between the slant coordinate frame and the platform coordinate system is given by: 
               dx   p     =       Δ   ⁢           ⁢       r   -&gt;     ·     (         V   -&gt;     V     +           R   -&gt;     R     ·   cos     ⁢           ⁢   ϕ       )         +           Δ   ⁢           ⁢     V   -&gt;       V     ·     R   -&gt;     ·   cos     ⁢           ⁢   ϕ                     dy   p     =           (         Δ   ⁢           ⁢     r   -&gt;       R     +       Δ   ⁢           ⁢     V   -&gt;       V       )     ·     R   -&gt;     ·   sin     ⁢           ⁢     ϕ   ·   cos     ⁢           ⁢   ψ     -     Δ   ⁢           ⁢       r   -&gt;     ·         V   -&gt;     ×     R   -&gt;                V   -&gt;     ×     R   -&gt;              ·   sin     ⁢           ⁢   ψ                     dz   p     =           (         Δ   ⁢           ⁢     r   -&gt;       R     +       Δ   ⁢           ⁢     V   -&gt;       V       )     ·     R   -&gt;     ·   sin     ⁢           ⁢     ϕ   ·   sin     ⁢           ⁢   ψ     -     Δ   ⁢           ⁢       r   -&gt;     ·         V   -&gt;     ×     R   -&gt;                V   -&gt;     ×     R   -&gt;              ·   cos     ⁢           ⁢   ψ                 where             ψ   =       tan     -   1       ⁢               ⁢     R             ⁢   z                   ⁢               ⁢       R             ⁢   x               ⁢   2       ⁢           +           ⁢     R             ⁢   y               ⁢   2                                   R   -&gt;     R     =     (       R   x     ,     R   y     ,     R   z       )           
and Δ{right arrow over (r)} and Δ{right arrow over (V)} are the vectors of the vehicle navigation position and velocity errors, respectively, in the body coordinate frame.
 
     {right arrow over (V)} is the velocity vector of the sensor platform (vehicle); 
     {right arrow over (R)} is the range vector between the vehicle position and a geo-location; and 
     φ is the angle between the sensor platform (vehicle) flight path and the line of sight between the sensor platform (vehicle) and the geo-location sensed by a plurality of sensors. 
     {right arrow over (r)} and Δ{right arrow over (V)} are the first six Kalman filter error states defined as
 
Δ {right arrow over (r)} =(δ r   x   ,δr   y   ,δr   z ) T  
 
and
 
Δ {right arrow over (V)} =(δ V   x   ,δV   y   ,δV   z ) T  
 
     in the platform coordinate frame and subscript T denotes the transpose of the vector. 
     If Δ{right arrow over (r)} and Δ{right arrow over (V)} are defined in the ECEF coordinate frame, then these error states are transformed into the platform reference frame where the error equations are defined. The Kalman filter error states {right arrow over (x)} k , observation matrix H k  and measurement noise matrix R k  are defined as: 
     
       
         
           
             
               
                 
                   x 
                   -&gt; 
                 
                 k 
               
               = 
               
                 
                   ( 
                   
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         r 
                         x 
                       
                     
                     , 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         r 
                         y 
                       
                     
                     , 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         r 
                         z 
                       
                     
                     , 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         V 
                         x 
                       
                     
                     , 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         V 
                         y 
                       
                     
                     , 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         V 
                         z 
                       
                     
                     , 
                     0 
                     , 
                     … 
                     ⁢ 
                     
                         
                     
                     , 
                     0 
                   
                   ) 
                 
                 T 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 H 
                 k 
               
               = 
               
                 ( 
                 
                   
                     
                       
                         
                           h 
                           00 
                         
                         , 
                         
                           h 
                           01 
                         
                         , 
                         
                           h 
                           02 
                         
                         , 
                         
                           h 
                           03 
                         
                         , 
                         
                           h 
                           04 
                         
                         , 
                         
                           h 
                           05 
                         
                         , 
                         0 
                         , 
                         … 
                         ⁢ 
                         
                             
                         
                         , 
                         0 
                       
                     
                   
                   
                     
                       
                         
                           h 
                           10 
                         
                         , 
                         
                           h 
                           11 
                         
                         , 
                         
                           h 
                           12 
                         
                         , 
                         
                           h 
                           13 
                         
                         , 
                         
                           h 
                           14 
                         
                         , 
                         
                           h 
                           15 
                         
                         , 
                         0 
                         , 
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                         , 
                         0 
                       
                     
                   
                   
                     
                       
                         
                           h 
                           20 
                         
                         , 
                         
                           h 
                           21 
                         
                         , 
                         
                           h 
                           22 
                         
                         , 
                         
                           h 
                           23 
                         
                         , 
                         
                           h 
                           24 
                         
                         , 
                         
                           h 
                           25 
                         
                         , 
                         0 
                         , 
                         … 
                         ⁢ 
                         
                             
                         
                         , 
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                 ) 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 and 
                 ⁢ 
                 
                   
                       
                   
                   ⁢ 
                   
                       
                   
                 
                 ⁢ 
                 
                   R 
                   k 
                 
               
               = 
               
                 ( 
                 
                   
                     
                       
                         
                           r 
                           00 
                         
                         , 
                       
                     
                     
                       
                         0 
                         , 
                       
                     
                     
                       0 
                     
                   
                   
                     
                       
                         0 
                         , 
                       
                     
                     
                       
                         
                           r 
                           11 
                         
                         , 
                       
                     
                     
                       0 
                     
                   
                   
                     
                       
                         0 
                         , 
                       
                     
                     
                       
                         0 
                         , 
                       
                     
                     
                       
                         r 
                         22 
                       
                     
                   
                 
                 ) 
               
             
           
         
       
     
     Assume that the vectors 
     
       
         
           
             
               
                 
                   R 
                   -&gt; 
                 
                 R 
               
               = 
               
                 ( 
                 
                   
                     R 
                     x 
                   
                   , 
                   
                     R 
                     y 
                   
                   , 
                   
                     R 
                     z 
                   
                 
                 ) 
               
             
             , 
             
               V 
               = 
               
                  
                 
                   V 
                   -&gt; 
                 
                  
               
             
           
         
       
       
         
           
             
               R 
               = 
               
                  
                 
                   R 
                   -&gt; 
                 
                  
               
             
             , 
             
               
                 
                   V 
                   -&gt; 
                 
                 V 
               
               = 
               
                 ( 
                 
                   
                     p 
                     x 
                   
                   , 
                   
                     p 
                     y 
                   
                   , 
                   
                     p 
                     z 
                   
                 
                 ) 
               
             
           
         
       
       
         
           and 
         
       
       
         
           
             
               
                 
                   V 
                   -&gt; 
                 
                 V 
               
               × 
               
                 
                   R 
                   -&gt; 
                 
                 R 
               
             
             = 
             
               ( 
               
                 
                   q 
                   x 
                 
                 , 
                 
                   q 
                   y 
                 
                 , 
                 
                   q 
                   z 
                 
               
               ) 
             
           
         
       
     
     Therefore, the target location error (TLE), as derived in the parent application can be written in the following form: 
     
       
         
           
             
               dx 
               p 
             
             = 
             
               
                 
                   
                     ( 
                     
                       
                         p 
                         x 
                       
                       + 
                       
                         
                           
                             R 
                             x 
                           
                           · 
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                         ⁢ 
                         
                             
                         
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                   ⁢ 
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                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     r 
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                 + 
                 
                   
                     ( 
                     
                       
                         p 
                         y 
                       
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                           · 
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                   ⁢ 
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                   ⁢ 
                   
                     r 
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                         p 
                         z 
                       
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                           · 
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                         ⁢ 
                         
                             
                         
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                     ) 
                   
                   ⁢ 
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                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     r 
                     z 
                   
                 
                 + 
                 
                   
                     [ 
                     
                         
                     
                     ⁢ 
                     
                       
                         
                           R 
                           x 
                         
                         · 
                         cos 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         ϕ 
                         · 
                         
                           R 
                           / 
                           V 
                         
                       
                     
                     ] 
                   
                   ⁢ 
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                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     V 
                     x 
                   
                 
                 + 
                 
                   
                     [ 
                     
                       
                         
                           R 
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                       ⁢ 
                       
                           
                       
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                   ⁢ 
                   
                     V 
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                         sin 
                       
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
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                           R 
                           / 
                           υ 
                         
                       
                     
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                   ⁢ 
                   δ 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     V 
                     z 
                   
                   ⁢ 
                   
                     
 
                   
                   ⁢ 
                   
                     dy 
                     p 
                   
                 
               
               = 
               
                 
                   
                     
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                             · 
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                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ϕ 
                             · 
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                           ⁢ 
                           
                               
                           
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                           ⁢ 
                           
                               
                           
                           ⁢ 
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                           ⁢ 
                           
                               
                           
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                           ⁢ 
                           
                               
                           
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                           ⁢ 
                           
                               
                           
                           ⁢ 
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                           ⁢ 
                           
                               
                           
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                           ⁢ 
                           
                               
                           
                           ⁢ 
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                       ) 
                     
                     ⁢ 
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       r 
                       z 
                     
                   
                   + 
                   
                     
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                             R 
                             x 
                           
                           · 
                           sin 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ϕ 
                           · 
                           cos 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ψ 
                           · 
                           
                             R 
                             / 
                             V 
                           
                         
                       
                       ] 
                     
                     ⁢ 
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       V 
                       x 
                     
                   
                   + 
                   
                     
                       [ 
                       
                           
                       
                       ⁢ 
                       
                         
                           
                             R 
                             y 
                           
                           · 
                           sin 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ϕ 
                           · 
                           cos 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ψ 
                           · 
                           
                             R 
                             / 
                             V 
                           
                         
                       
                       ] 
                     
                     ⁢ 
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       V 
                       y 
                     
                   
                   + 
                   
                     
                       [ 
                       
                         
                           
                             R 
                             z 
                           
                           · 
                           sin 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ϕ 
                           · 
                           cos 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ψ 
                           · 
                           
                             R 
                             / 
                             V 
                           
                         
                       
                       ] 
                     
                     ⁢ 
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       V 
                       z 
                     
                     ⁢ 
                     
                       
 
                     
                     ⁢ 
                     
                       dz 
                       p 
                     
                   
                 
                 = 
                 
                   
                     
                       ( 
                       
                         
                           
                             
                               R 
                               x 
                             
                             · 
                             sin 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ϕ 
                             · 
                             sin 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ψ 
                         
                         - 
                         
                           
                             
                               q 
                               x 
                             
                             · 
                             cos 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ψ 
                         
                       
                       ) 
                     
                     ⁢ 
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       r 
                       x 
                     
                   
                   + 
                   
                     
                       ( 
                       
                         
                           
                             
                               R 
                               y 
                             
                             · 
                             sin 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ϕ 
                             · 
                             sin 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ψ 
                         
                         - 
                         
                           
                             
                               q 
                               y 
                             
                             · 
                             cos 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ψ 
                         
                       
                       ) 
                     
                     ⁢ 
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       r 
                       y 
                     
                   
                   + 
                   
                     
                       ( 
                       
                         
                           
                             
                               R 
                               z 
                             
                             · 
                             sin 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           
                             ϕ 
                             · 
                             sin 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ψ 
                         
                         + 
                         
                           
                             
                               q 
                               z 
                             
                             · 
                             cos 
                           
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           ψ 
                         
                       
                       ) 
                     
                     ⁢ 
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       r 
                       z 
                     
                   
                   + 
                   
                     
                       [ 
                       
                         
                           
                             R 
                             x 
                           
                           · 
                           sin 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ϕ 
                           · 
                           sin 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ψ 
                           · 
                           
                             R 
                             / 
                             V 
                           
                         
                       
                       ] 
                     
                     ⁢ 
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       V 
                       x 
                     
                   
                   + 
                   
                     
                       [ 
                       
                         
                           
                             R 
                             y 
                           
                           · 
                           sin 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ϕ 
                           · 
                           sin 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ψ 
                           · 
                           
                             R 
                             / 
                             V 
                           
                         
                       
                       ] 
                     
                     ⁢ 
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       V 
                       y 
                     
                   
                   + 
                   
                     
                       [ 
                       
                         
                           
                             R 
                             z 
                           
                           · 
                           sin 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ϕ 
                           · 
                           sin 
                         
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         
                           ψ 
                           · 
                           
                             R 
                             / 
                             V 
                           
                         
                       
                       ] 
                     
                     ⁢ 
                     δ 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       V 
                       z 
                     
                   
                 
               
             
           
         
       
     
     Note that the dx p  (down-track), dy p  (cross-track), and dz p  vertical track components form an ellipsoid of TLE. 
     The following observation matrix is applicable, 
     Equation 2
           h   00   =p   x   +R   x ·cos φ
   h   01   =p   y   +R   y ·cos φ   h   02   =p   z   +R   z ·cos φ   h   03   =R   x ·cos φ· R/V      h   04   =R   y ·cos φ· R/V      h   05   =R   z ·cos φ· R/V      h   10   =R   x ·sin φ·cos ψ− q   x ·sin ψ   h   11   =R   y ·sin φ·cos ψ− q   y ·sin ψ   h   12   =R   z ·sin φ·cos ψ− q   z ·sin ψ   h   13   =R   x ·sin φ·cos ψ· R/V      h   14   =R   y ·sin φ·cos ψ· R/V      h   15   =R   z ·sin φ·cos ψ· R/V      h   20   =R   x ·sin φ·sin ψ+ q   x ·cos ψ   h   21   =R   y ·sin φ·sin ψ+ q   y ·cos ψ   h   22   =R   z ·sin φ·sin ψ+ q   z ·cos ψ   h   23   =R   x ·sin φ·sin ψ· R/V      h   24   =R   y ·sin φ·sin ψ· R/V      h   25   =R   z ·sin φ·sin ψ· R/V    Equation 2       

     Now the measurement noise is derived. The measurement noises are determined by the TLE of each platform. An example of the measurement noise matrix is described using a two dimensional observation model and two sensors, followed by more general scenarios. 
     It is assumed that all the sensors (platforms) in each scenario are in a common time frame, observing the same geo-location (target(s)), and sharing navigation information instantaneously via a data link system. The scenarios considered are examples of the teachings herein, and are as follows. 
     Case I. Platform  0  is Updated from Sensor Platforms i Where i=1, 2, . . . n−1 
     a. Platform  0  is updated from platform  1  with two dimensional measurement noise matrix. 
     b. Platform  0  is updated from Platform  1  with three dimensional noise matrix. 
     c. Platform  0  is updated from all other platforms i where i=1, 2 . . . n−1 with three dimensional measurement noise matrix. 
     Case II. Platform i is Updated from Other Platforms j,(≠i), Where i, j=0, 1 . . . n−1 
     Case Ia Embodiment 
     Platform  0  is updated from Platform  1  with two dimensional measurement noise matrix. 
     In this case Platform  1  has higher navigation accuracy than that of Platform 0. The navigation accuracy of Platform  0  is updated with independent observations (target measurements), separate and distinct from those obtained in Platform  1  via the data link system. The error state vector, the observation matrix and the measurement noise matrices of the Kalman filter for Platform  0  are: 
     
       
         
           
             
               
                 
                   x 
                   -&gt; 
                 
                 k 
               
               = 
               
                 
                   ( 
                   
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         r 
                         x 
                       
                     
                     , 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         r 
                         y 
                       
                     
                     , 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         r 
                         z 
                       
                     
                     , 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         V 
                         x 
                       
                     
                     , 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         V 
                         y 
                       
                     
                     , 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         V 
                         z 
                       
                     
                     , 
                     0 
                     , 
                     … 
                     ⁢ 
                     
                         
                     
                     , 
                     0 
                   
                   ) 
                 
                 T 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 H 
                 k 
               
               = 
               
                 
                   ( 
                   
                     
                       
                         
                           
                             h 
                             00 
                           
                           , 
                           
                             h 
                             01 
                           
                           , 
                           
                             h 
                             02 
                           
                           , 
                           
                             h 
                             03 
                           
                           , 
                           
                             h 
                             04 
                           
                           , 
                           
                             h 
                             05 
                           
                           , 
                           0 
                           , 
                           … 
                           ⁢ 
                           
                               
                           
                           , 
                           0 
                         
                       
                     
                     
                       
                         
                           
                             h 
                             10 
                           
                           , 
                           
                             h 
                             11 
                           
                           , 
                           
                             h 
                             12 
                           
                           , 
                           
                             h 
                             13 
                           
                           , 
                           
                             h 
                             14 
                           
                           , 
                           
                             h 
                             15 
                           
                           , 
                           0 
                           , 
                           … 
                           ⁢ 
                           
                               
                           
                           , 
                           0 
                         
                       
                     
                   
                   ) 
                 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 and 
               
             
           
         
       
       
         
           
             
               R 
               k 
             
             = 
             
               ( 
               
                 
                   
                     
                       r 
                       00 
                     
                   
                   
                     0 
                   
                 
                 
                   
                     
                       0 
                       , 
                     
                   
                   
                     
                       r 
                       11 
                     
                   
                 
               
               ) 
             
           
         
       
     
     The derivation of r 00  and r 11  is shown in  FIG. 5 . In  FIG. 5 , the lengths OP 0  (denoted as a 0 ) is the down track error while OQ 0  (denoted as b 0 ) is the cross track error of the TLE obtained from platform  0 . The lengths OP 1  (denoted as a 1 ) is the down track error while OQ 1  (denoted as b 1 ) is the cross track error of the TLE obtained from platform  1 . The TLE for platform  0  (namely E 0 ) is an ellipsoid which is formed by the semi-major axis a 0  and semi minor axis b 0  Similarly, the TLE for platform  1  (namely E 1 ) is an ellipsoid which is formed by the semi-major axis a 1  and semi minor axis b 1 . 
     Since platform  0  can obtain the TLE information from platform  1 , the intersection of these two ellipsoids, E 0  ∩ E 1 , becomes a more accurate TLE for platform  0 . Therefore, for platform  1 , the down track error a 1  will be reduced to r DR1 , the length of OA 1 . Similarly, for platform  0 , the down track error a 0  will be reduced to r DR0 , the length of OA 0 . This information is used to update and to bind the navigation errors for platform  0 . The minimum of r DR0 and a 1  is the standard deviation of the measurement noise for the down-track. Similarly, the minimum of r CR0  and b 1  is the standard deviation of the measurement noise for the down-track. The two dimensional measurement noise matrix of the Kalman filter for platform  0  is as follows: 
     
       
         
           
             
               
                 
                   R 
                   k 
                 
                 = 
                 
                   ( 
                   
                     
                       
                         
                           r 
                           00 
                         
                       
                       
                         
                           r 
                           01 
                         
                       
                     
                     
                       
                         
                           r 
                           10 
                         
                       
                       
                         
                           r 
                           11 
                         
                       
                     
                   
                   ) 
                 
               
               ; 
               
                 
                   r 
                   01 
                 
                 = 
                 
                   
                     r 
                     10 
                   
                   = 
                   0 
                 
               
             
             , 
             
               
                 r 
                 00 
               
               = 
               
                 r 
                 
                   D 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   01 
                 
               
             
             , 
             
               
                 r 
                 11 
               
               = 
               
                 r 
                 
                   C 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   01 
                 
               
             
           
         
       
       
         
           
             
               where 
               ⁢ 
               
                   
               
               ⁢ 
               
                 r 
                 
                   D 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   01 
                 
               
             
             ⁢ 
             
               = 
               denote 
             
             ⁢ 
             
               
                 min 
                 ⁢ 
                 
                   { 
                   
                     
                       a 
                       0 
                     
                     , 
                     
                       r 
                       
                         DR 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                     
                   
                   } 
                 
                 ⁢ 
                 and 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   r 
                   
                     C 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     01 
                   
                 
               
               ⁢ 
               
                 = 
                 denote 
               
               ⁢ 
               
                 min 
                 ⁢ 
                 
                   
                     { 
                     
                       
                         b 
                         0 
                       
                       , 
                       
                         r 
                         
                           CR 
                           ⁢ 
                           
                               
                           
                           ⁢ 
                           1 
                         
                       
                     
                     } 
                   
                   . 
                 
               
             
           
         
       
     
     Now, we will derive r DR0  and r CR . The length of OA 0  and OA 1  can be computed as follows. 
     Let us denote
 
a 0 =DR 0 =length of  OP 0   
 
b 0 =CR 0 =length of  OQ 0   
 
a 1 =DR 1 =length of  OP 1   
 
b 1 =CR 1 =length of  OQ 1   
 
     The general two-dimensional elliptic equation is 
     
       
         
           
             
               r 
               2 
             
             = 
             
               
                 
                   a 
                   2 
                 
                 ⁢ 
                 
                   b 
                   2 
                 
               
               
                 
                   
                     b 
                     2 
                   
                   ⁢ 
                   
                     cos 
                     2 
                   
                   ⁢ 
                   θ 
                 
                 + 
                 
                   
                     a 
                     2 
                   
                   ⁢ 
                   
                     sin 
                     2 
                   
                   ⁢ 
                   θ 
                 
               
             
           
         
       
       
         
           
             Thus 
             , 
             
               
 
             
             ⁢ 
             
               
                 r 
                 
                   DR 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   1 
                 
               
               = 
               
                 
                   Length 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   of 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   
                     
                       OA 
                       0 
                     
                     _ 
                   
                 
                 = 
                 
                   
                     
                       
                         a 
                         1 
                         2 
                       
                       ⁢ 
                       
                         b 
                         1 
                         2 
                       
                     
                     
                       
                         
                           b 
                           1 
                           2 
                         
                         ⁢ 
                         
                           cos 
                           2 
                         
                         ⁢ 
                         θ 
                       
                       + 
                       
                         
                           a 
                           1 
                           2 
                         
                         ⁢ 
                         
                           sin 
                           2 
                         
                         ⁢ 
                         θ 
                       
                     
                   
                 
               
             
           
         
       
       
         
           
             
               r 
               
                 CR 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 1 
               
             
             = 
             
               
                 Length 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 of 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     OB 
                     0 
                   
                   _ 
                 
               
               = 
               
                 
                   
                     
                       a 
                       1 
                       2 
                     
                     ⁢ 
                     
                       b 
                       1 
                       2 
                     
                   
                   
                     
                       
                         b 
                         1 
                         2 
                       
                       ⁢ 
                       
                         sin 
                         2 
                       
                       ⁢ 
                       θ 
                     
                     + 
                     
                       
                         a 
                         1 
                         2 
                       
                       ⁢ 
                       
                         cos 
                         2 
                       
                       ⁢ 
                       θ 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               r 
               
                 DR 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 0 
               
             
             = 
             
               
                 Length 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 of 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     OA 
                     1 
                   
                   _ 
                 
               
               = 
               
                 
                   
                     
                       a 
                       0 
                       2 
                     
                     ⁢ 
                     
                       b 
                       0 
                       2 
                     
                   
                   
                     
                       
                         b 
                         0 
                         2 
                       
                       ⁢ 
                       
                         cos 
                         2 
                       
                       ⁢ 
                       θ 
                     
                     + 
                     
                       
                         a 
                         0 
                         2 
                       
                       ⁢ 
                       
                         sin 
                         2 
                       
                       ⁢ 
                       θ 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               r 
               
                 CR 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 0 
               
             
             = 
             
               
                 Length 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 of 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     OB 
                     1 
                   
                   _ 
                 
               
               = 
               
                 
                   
                     
                       a 
                       0 
                       2 
                     
                     ⁢ 
                     
                       b 
                       0 
                       2 
                     
                   
                   
                     
                       
                         b 
                         0 
                         2 
                       
                       ⁢ 
                       
                         sin 
                         2 
                       
                       ⁢ 
                       θ 
                     
                     + 
                     
                       
                         a 
                         0 
                         2 
                       
                       ⁢ 
                       
                         cos 
                         2 
                       
                       ⁢ 
                       θ 
                     
                   
                 
               
             
           
         
       
       
         
           where 
         
       
       
         
           
             
               cos 
               ⁢ 
               
                   
               
               ⁢ 
               θ 
             
             = 
             
               
                 
                   
                     V 
                     
                       H 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       0 
                     
                   
                   ⟶ 
                 
                 
                   V 
                   
                     H 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
               
               · 
               
                 
                   
                     V 
                     
                       H 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       1 
                     
                   
                   ⟶ 
                 
                 
                   V 
                   
                     H 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   sin 
                   2 
                 
                 ⁢ 
                 θ 
               
               = 
               
                 1 
                 - 
                 
                   
                     cos 
                     2 
                   
                   ⁢ 
                   θ 
                 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 
                   V 
                   
                     H 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 ⟶ 
               
               = 
               
                 ( 
                 
                   
                     V 
                     
                       x 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       0 
                     
                   
                   , 
                   
                     V 
                     
                       y 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       0 
                     
                   
                   , 
                   0 
                 
                 ) 
               
             
             , 
             
               
 
             
             ⁢ 
             and 
           
         
       
       
         
           
             
               
                 V 
                 
                   H 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   1 
                 
               
               ⟶ 
             
             = 
             
               ( 
               
                 
                   V 
                   
                     x 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                   
                 
                 , 
                 
                   V 
                   
                     y 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                   
                 
                 , 
                 0 
               
               ) 
             
           
         
       
     
     For platform  0  and platform i, the general forms for r DRi  and r CRi  are denoted as 
     
       
         
           
             
               r 
               
                 DR 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 i 
               
             
             = 
             
               
                 Length 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 of 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     OA 
                     0 
                   
                   _ 
                 
               
               = 
               
                 
                   
                     
                       a 
                       i 
                       2 
                     
                     ⁢ 
                     
                       b 
                       i 
                       2 
                     
                   
                   
                     
                       
                         b 
                         i 
                         2 
                       
                       ⁢ 
                       
                         cos 
                         2 
                       
                       ⁢ 
                       θ 
                     
                     + 
                     
                       
                         a 
                         i 
                         2 
                       
                       ⁢ 
                       
                         sin 
                         2 
                       
                       ⁢ 
                       θ 
                     
                   
                 
               
             
           
         
       
       
         
           
             
               r 
               
                 CR 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 i 
               
             
             = 
             
               
                 Length 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 of 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 
                   
                     OB 
                     0 
                   
                   _ 
                 
               
               = 
               
                 
                   
                     
                       a 
                       i 
                       2 
                     
                     ⁢ 
                     
                       b 
                       i 
                       2 
                     
                   
                   
                     
                       
                         b 
                         i 
                         2 
                       
                       ⁢ 
                       
                         sin 
                         2 
                       
                       ⁢ 
                       θ 
                     
                     + 
                     
                       
                         a 
                         i 
                         2 
                       
                       ⁢ 
                       
                         cos 
                         2 
                       
                       ⁢ 
                       θ 
                     
                   
                 
               
             
           
         
       
       
         
           where 
         
       
       
         
           
             
               cos 
               ⁢ 
               
                   
               
               ⁢ 
               θ 
             
             = 
             
               
                 
                   
                     V 
                     
                       H 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       0 
                     
                   
                   ⟶ 
                 
                 
                   V 
                   
                     H 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
               
               · 
               
                 
                   
                     V 
                     
                       H 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       i 
                     
                   
                   ⟶ 
                 
                 
                   V 
                   
                     H 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     i 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 
                   sin 
                   2 
                 
                 ⁢ 
                 θ 
               
               = 
               
                 1 
                 - 
                 
                   
                     cos 
                     2 
                   
                   ⁢ 
                   θ 
                 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 
                   V 
                   
                     H 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 ⟶ 
               
               = 
               
                 ( 
                 
                   
                     V 
                     
                       x 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       0 
                     
                   
                   , 
                   
                     V 
                     
                       y 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       0 
                     
                   
                   , 
                   0 
                 
                 ) 
               
             
             , 
             
               
 
             
             ⁢ 
             and 
           
         
       
       
         
           
             
               
                 V 
                 
                   H 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   i 
                 
               
               ⟶ 
             
             = 
             
               ( 
               
                 
                   V 
                   
                     x 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                   
                 
                 , 
                 
                   V 
                   
                     y 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                   
                 
                 , 
                 0 
               
               ) 
             
           
         
       
     
     V x0  and V y0  are the x and y component of the vehicle velocity for platform  0 . V xi  and V yi  are the x and y component of the vehicle velocity for platform i. 
     V H0  and V Hi  are the magnitudes of vectors {right arrow over (V H0  )} and {right arrow over (V Hi )} respectively. θ is the angle between the semi-major axes of platform  0  and i. 
     Case Ib 
     We will choose the down-track and vertical-track of the TLE as two dimensional elliptic axes. Thus
         b 1 =cross-track   c 1 =vertical-track       

     The general elliptic equation for two dimension is 
     
       
         
           
             
               r 
               2 
             
             = 
             
               
                 
                   b 
                   2 
                 
                 ⁢ 
                 
                   c 
                   2 
                 
               
               
                 
                   
                     c 
                     2 
                   
                   ⁢ 
                   
                     cos 
                     2 
                   
                   ⁢ 
                   θ 
                 
                 + 
                 
                   
                     b 
                     2 
                   
                   ⁢ 
                   
                     sin 
                     2 
                   
                   ⁢ 
                   θ 
                 
               
             
           
         
       
       
         
           
             
               r 
               
                 VR 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 1 
               
             
             = 
             
               
                 
                   
                     b 
                     1 
                     2 
                   
                   ⁢ 
                   
                     c 
                     1 
                     2 
                   
                 
                 
                   
                     
                       c 
                       1 
                       2 
                     
                     ⁢ 
                     
                       cos 
                       2 
                     
                     ⁢ 
                     θ 
                   
                   + 
                   
                     
                       b 
                       1 
                       2 
                     
                     ⁢ 
                     
                       sin 
                       2 
                     
                     ⁢ 
                     θ 
                   
                 
               
             
           
         
       
       
         
           where 
         
       
       
         
           
             
               cos 
               ⁢ 
               
                   
               
               ⁢ 
               θ 
             
             = 
             
               
                 
                   
                     V 
                     
                       H 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       0 
                     
                   
                   ⟶ 
                 
                 
                   V 
                   
                     H 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
               
               · 
               
                 
                   
                     V 
                     
                       H 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       i 
                     
                   
                   ⟶ 
                 
                 
                   V 
                   
                     H 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     i 
                   
                 
               
             
           
         
       
       
         
           
             
               
                 sin 
                 2 
               
               ⁢ 
               θ 
             
             = 
             
               1 
               - 
               
                 
                   cos 
                   2 
                 
                 ⁢ 
                 θ 
               
             
           
         
       
       
         
           
             
               
                 V 
                 
                   H 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   0 
                 
               
               ⟶ 
             
             = 
             
               ( 
               
                 0 
                 , 
                 
                   V 
                   
                     y 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
                 , 
                 
                   V 
                   
                     z 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     0 
                   
                 
               
               ) 
             
           
         
       
       
         
           
             
               
                 V 
                 
                   H 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   i 
                 
               
               ⟶ 
             
             = 
             
               ( 
               
                 0 
                 , 
                 
                   V 
                   
                     y 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                   
                 
                 , 
                 
                   V 
                   
                     z 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     1 
                   
                 
               
               ) 
             
           
         
       
     
     Note that we also can obtain r CR1  from the above equations. However, this r CR1  is the same as r CR1  that was derived in Case Ia. 
     Note that the r VR1  can also be derived from the vertical-track and cross-track of the TLE ellipsoid by using the above method. 
     The h ij (i=0,1,2 j=0,1,3,4,5) are defined in Equation 2 which are derived from the TLE for platform  0 . 
     Case Ic—Platform  0  is Updated from All Other Platforms i (i=1, 2, . . . n−1) 
     In one embodiment, as shown in  FIG. 3 , Sensor  0   301  is connected to all other sensors, sensor  1   303 , sensor i  307  and sensor (n−1)  305  using bidirectional data link  309 . Sensor  0 , an example of a typical sensor, is also described in  FIG. 4 , as sensor  0 ,  400 . The structure of sensor  0  is applicable to sensor  1   303 , sensor i  307  and sensor (n−1)  305 . Thus TLEs are exchanged among all platforms for a particular geo-location (target). 
     For each pair, ( 0 ,i), which is platforms  0  and i, compute r CR     (i-1)   , r DR     (i-1)   , and r VR     (i-1)    from Case Ib above. Thus the H and R matrices can be derived as follows, 
     
       
         
           
             
               
                 
                   x 
                   -&gt; 
                 
                 k 
               
               = 
               
                 
                   ( 
                   
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         r 
                         x 
                       
                     
                     , 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         r 
                         y 
                       
                     
                     , 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         r 
                         z 
                       
                     
                     , 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         V 
                         x 
                       
                     
                     , 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         V 
                         y 
                       
                     
                     , 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         V 
                         z 
                       
                     
                     , 
                     0 
                     , 
                     … 
                     ⁢ 
                     
                         
                     
                     , 
                     0 
                   
                   ) 
                 
                 T 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 H 
                 k 
               
               = 
               
                 ( 
                 
                   
                     
                       
                         
                           h 
                           00 
                         
                         , 
                         
                           h 
                           01 
                         
                         , 
                         
                           h 
                           02 
                         
                         , 
                         
                           h 
                           03 
                         
                         , 
                         
                           h 
                           04 
                         
                         , 
                         
                           h 
                           05 
                         
                         , 
                         0 
                         , 
                         … 
                         ⁢ 
                         
                             
                         
                         , 
                         0 
                       
                     
                   
                   
                     
                       
                         
                           h 
                           10 
                         
                         , 
                         
                           h 
                           11 
                         
                         , 
                         
                           h 
                           12 
                         
                         , 
                         
                           h 
                           13 
                         
                         , 
                         
                           h 
                           14 
                         
                         , 
                         
                           h 
                           15 
                         
                         , 
                         0 
                         , 
                         … 
                         ⁢ 
                         
                             
                         
                         , 
                         0 
                       
                     
                   
                   
                     
                       
                         
                           h 
                           20 
                         
                         , 
                         
                           h 
                           21 
                         
                         , 
                         
                           h 
                           22 
                         
                         , 
                         
                           h 
                           23 
                         
                         , 
                         
                           h 
                           24 
                         
                         , 
                         
                           h 
                           25 
                         
                         , 
                         0 
                         , 
                         … 
                         ⁢ 
                         
                             
                         
                         , 
                         0 
                       
                     
                   
                 
                 ) 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 and 
                 ⁢ 
                 
                   
                       
                   
                   ⁢ 
                   
                       
                   
                 
                 ⁢ 
                 
                   R 
                   k 
                 
               
               = 
               
                 ( 
                 
                   
                     
                       
                         
                           r 
                           00 
                         
                         , 
                       
                     
                     
                       
                         0 
                         , 
                       
                     
                     
                       0 
                     
                   
                   
                     
                       
                         0 
                         , 
                       
                     
                     
                       
                         
                           r 
                           11 
                         
                         , 
                       
                     
                     
                       0 
                     
                   
                   
                     
                       
                         0 
                         , 
                       
                     
                     
                       
                         0 
                         , 
                       
                     
                     
                       
                         r 
                         22 
                       
                     
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             
               
                 r 
                 01 
               
               = 
               
                 
                   r 
                   02 
                 
                 = 
                 
                   
                     r 
                     10 
                   
                   = 
                   
                     
                       r 
                       12 
                     
                     = 
                     
                       
                         r 
                         20 
                       
                       = 
                       
                         
                           r 
                           21 
                         
                         = 
                         0 
                       
                     
                   
                 
               
             
             , 
             
                 
             
           
         
       
       
         
           
             
               
                 r 
                 00 
               
               = 
               
                 r 
                 
                   D 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   01 
                 
               
             
             , 
             
               
                 r 
                 11 
               
               = 
               
                 r 
                 
                   C 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   01 
                 
               
             
             , 
             
               
                 r 
                 22 
               
               = 
               
                 
                   r 
                   
                     V 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     01 
                   
                 
                 ⁢ 
                 
                     
                 
               
             
           
         
       
       
         
           where 
         
       
       
         
           
             
               
                 r 
                 
                   D 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   01 
                 
               
               ⁢ 
               
                 = 
                 denote 
               
               ⁢ 
               
                 min 
                 ⁢ 
                 
                   { 
                   
                     
                       a 
                       0 
                     
                     , 
                     
                       r 
                       
                         DR 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                     
                     , 
                     
                       r 
                       
                         DR 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                       
                     
                     , 
                     … 
                     , 
                     
                       r 
                       
                         DR 
                         ⁡ 
                         
                           ( 
                           
                             n 
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                   
                   } 
                 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 r 
                 
                   C 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   01 
                 
               
               ⁢ 
               
                 = 
                 denote 
               
               ⁢ 
               
                 min 
                 ⁢ 
                 
                   { 
                   
                     
                       b 
                       0 
                     
                     , 
                     
                       r 
                       
                         CR 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                     
                     , 
                     
                       r 
                       
                         CR 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                       
                     
                     , 
                     … 
                     , 
                     
                       r 
                       
                         CR 
                         ⁡ 
                         
                           ( 
                           
                             n 
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                   
                   } 
                 
               
             
           
         
       
       
         
           and 
         
       
       
         
           
             
               r 
               
                 V 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 01 
               
             
             ⁢ 
             
               = 
               denote 
             
             ⁢ 
             
               min 
               ⁢ 
               
                 { 
                 
                   
                     c 
                     0 
                   
                   , 
                   
                     r 
                     
                       VR 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       1 
                     
                   
                   , 
                   
                     r 
                     
                       VR 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       2 
                     
                   
                   , 
                   … 
                   , 
                   
                     r 
                     
                       VR 
                       ⁡ 
                       
                         ( 
                         
                           n 
                           - 
                           1 
                         
                         ) 
                       
                     
                   
                 
                 } 
               
             
           
         
       
     
     Case II 
     Sensor i is updated from other sensors j (≠i) at the same time where j=0, 1, . . . , n-1 for a fixed i where I=0, 1, . . . , n-1. All sensors, including sensor  0 , are sensing the same geo-location A, typically a conveniently discernable target. 
     This case is illustrated in  FIG. 4 . Here, sensor  0   400  has a GPS receiver  402  for receiving navigation signals from a constellation of GPS satellites. GPS Jamming Detector  414  detects a jammed or unavailable GPS signal. If the GPS navigation signal is valid, the GPS location information is passed to Kalman filter  412 , which in turn supplies position information to Update Guidance Position, Velocity and Attitude  416  within processor  420 . 
     Conversely, if GPS Jamming Detector  414  determines that the GPS signal is unusable, that is, jammed or corrupted, an alternative source of navigation information is activated. Inertial Navigation System SAR, FLIR Platform motion  404  determines the location of a sensor (on a platform) and geo-location A, typically using a SAR or FLIR sensor such as sensor  116  in  FIG. 1 . Note that geo-location A is not cataloged, nor pre-stored or geo-registered in a data base. Returning to  FIG. 4 , sensor  400  uses Generates TLE equations  406  to compute its own TLE. These TLE from sensor  0   400  are sent to other sensor (platforms) using two way link  418 . 
     In receiving TLE from other sensors using two way link  418 , a coordinate transformation  408  is performed on incoming TLE from sensors  1 ,  2  . . . (n-1) to match their TLE to the current position of sensor  0   400 . This and the data from sensor  0 , is input into observation model for sensor  0   400  (Compute Observation Matrix H and Measurement Noise Matrix  410 ). 
     The observation matrix H and Measurement Noise Matrix are part of the input into Kalman Filter  412  to substitute for the missing GPS information. 
     This case is an extension of the Case Ic from sensor (platform)  0  to any sensor (platform) i. In this case, the observation model is as follows. 
     
       
         
           
             
               
                 
                   x 
                   → 
                 
                 k 
               
               = 
               
                 
                   ( 
                   
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         r 
                         x 
                       
                     
                     , 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         r 
                         y 
                       
                     
                     , 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         r 
                         z 
                       
                     
                     , 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         V 
                         x 
                       
                     
                     , 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         V 
                         y 
                       
                     
                     , 
                     
                       δ 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         V 
                         z 
                       
                     
                     , 
                     0 
                     , 
                     … 
                     ⁢ 
                     
                         
                     
                     , 
                     0 
                   
                   ) 
                 
                 T 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 H 
                 k 
               
               = 
               
                 ( 
                 
                   
                     
                       
                         
                           h 
                           00 
                         
                         , 
                         
                           h 
                           01 
                         
                         , 
                         
                           h 
                           02 
                         
                         , 
                         
                           h 
                           03 
                         
                         , 
                         
                           h 
                           04 
                         
                         , 
                         
                           h 
                           05 
                         
                         , 
                         0 
                         , 
                         … 
                         ⁢ 
                         
                             
                         
                         , 
                         0 
                       
                     
                   
                   
                     
                       
                         
                           h 
                           10 
                         
                         , 
                         
                           h 
                           11 
                         
                         , 
                         
                           h 
                           12 
                         
                         , 
                         
                           h 
                           13 
                         
                         , 
                         
                           h 
                           14 
                         
                         , 
                         
                           h 
                           15 
                         
                         , 
                         0 
                         , 
                         … 
                         ⁢ 
                         
                             
                         
                         , 
                         0 
                       
                     
                   
                   
                     
                       
                         
                           h 
                           20 
                         
                         , 
                         
                           h 
                           21 
                         
                         , 
                         
                           h 
                           22 
                         
                         , 
                         
                           h 
                           23 
                         
                         , 
                         
                           h 
                           24 
                         
                         , 
                         
                           h 
                           25 
                         
                         , 
                         0 
                         , 
                         … 
                         ⁢ 
                         
                             
                         
                         , 
                         0 
                       
                     
                   
                 
                 ) 
               
             
             , 
             
                 
             
             ⁢ 
             
               
 
             
             ⁢ 
             and 
           
         
       
       
         
           
             
               R 
               k 
             
             = 
             
               ( 
               
                 
                   
                     
                       
                         r 
                         00 
                       
                       , 
                     
                   
                   
                     
                       0 
                       , 
                     
                   
                   
                     0 
                   
                 
                 
                   
                     
                       0 
                       , 
                     
                   
                   
                     
                       
                         r 
                         11 
                       
                       , 
                     
                   
                   
                     0 
                   
                 
                 
                   
                     
                       0 
                       , 
                     
                   
                   
                     
                       0 
                       , 
                     
                   
                   
                     
                       r 
                       22 
                     
                   
                 
               
               ) 
             
           
         
       
       
         
           
             
               
                 r 
                 01 
               
               = 
               
                 
                   r 
                   02 
                 
                 = 
                 
                   
                     r 
                     10 
                   
                   = 
                   
                     
                       r 
                       12 
                     
                     = 
                     
                       
                         r 
                         20 
                       
                       = 
                       
                         
                           r 
                           21 
                         
                         = 
                         0 
                       
                     
                   
                 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 r 
                 00 
               
               = 
               
                 r 
                 
                   Di 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   1 
                 
               
             
             , 
             
               
                 r 
                 11 
               
               = 
               
                 r 
                 
                   Ci 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   1 
                 
               
             
             , 
             
               
                 r 
                 22 
               
               = 
               
                 r 
                 
                   Vi 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   1 
                 
               
             
           
         
       
       
         
           where 
         
       
       
         
           
             
               
                 r 
                 
                   Di 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   1 
                 
               
               ⁢ 
               
                 = 
                 denote 
               
               ⁢ 
               
                 min 
                 ⁢ 
                 
                   { 
                   
                     
                       a 
                       i 
                     
                     , 
                     
                       r 
                       
                         DR 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         0 
                       
                     
                     , 
                     
                       r 
                       
                         DR 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                     
                     , 
                     … 
                     ⁢ 
                     
                         
                     
                     , 
                     
                       r 
                       
                         DR 
                         ⁡ 
                         
                           ( 
                           
                             i 
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                     , 
                     
                       r 
                       
                         DR 
                         ⁡ 
                         
                           ( 
                           
                             i 
                             + 
                             1 
                           
                           ) 
                         
                       
                     
                     , 
                     
                       … 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         r 
                         
                           DR 
                           ⁡ 
                           
                             ( 
                             
                               n 
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                     
                   
                   } 
                 
               
             
             , 
             
               
 
             
             ⁢ 
             
               
                 r 
                 
                   Ci 
                   ⁢ 
                   
                       
                   
                   ⁢ 
                   1 
                 
               
               ⁢ 
               
                 = 
                 denote 
               
               ⁢ 
               
                 min 
                 ⁢ 
                 
                   { 
                   
                     
                       a 
                       i 
                     
                     , 
                     
                       r 
                       
                         CR 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         1 
                       
                     
                     , 
                     
                       r 
                       
                         CR 
                         ⁢ 
                         
                             
                         
                         ⁢ 
                         2 
                       
                     
                     , 
                     … 
                     ⁢ 
                     
                         
                     
                     , 
                     
                       r 
                       
                         CR 
                         ⁡ 
                         
                           ( 
                           
                             i 
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                     , 
                     
                       r 
                       
                         CR 
                         ⁡ 
                         
                           ( 
                           
                             i 
                             + 
                             1 
                           
                           ) 
                         
                       
                     
                     , 
                     
                       … 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       
                         r 
                         
                           DR 
                           ⁡ 
                           
                             ( 
                             
                               n 
                               - 
                               1 
                             
                             ) 
                           
                         
                       
                     
                   
                   } 
                 
               
             
           
         
       
       
         
           and 
         
       
       
         
           
             
               r 
               
                 Vi 
                 ⁢ 
                 
                     
                 
                 ⁢ 
                 1 
               
             
             ⁢ 
             
               = 
               denote 
             
             ⁢ 
             
               min 
               ⁢ 
               
                 { 
                 
                   
                     a 
                     i 
                   
                   , 
                   
                     r 
                     
                       VR 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       1 
                     
                   
                   , 
                   
                     r 
                     
                       VR 
                       ⁢ 
                       
                           
                       
                       ⁢ 
                       2 
                     
                   
                   , 
                   … 
                   ⁢ 
                   
                       
                   
                   , 
                   
                     r 
                     
                       VR 
                       ⁡ 
                       
                         ( 
                         
                           i 
                           - 
                           1 
                         
                         ) 
                       
                     
                   
                   , 
                   
                     r 
                     
                       VR 
                       ⁡ 
                       
                         ( 
                         
                           i 
                           + 
                           1 
                         
                         ) 
                       
                     
                   
                   , 
                   
                     … 
                     ⁢ 
                     
                         
                     
                     ⁢ 
                     
                       r 
                       
                         VR 
                         ⁡ 
                         
                           ( 
                           
                             n 
                             - 
                             1 
                           
                           ) 
                         
                       
                     
                   
                 
                 } 
               
             
           
         
       
     
     Note that the classical Kalman filter has a restriction on the cross-correlation between process (matrix Q in equation 1) and measurement noise (matrix R in equation 1). A modified filter is thus used in this disclosure. 
     To preserve uncorrelated relationships between observation model and dynamic model in the Kalman filter for platform i, it is necessary to exclude the down track, cross track and vertical track information obtained from Platform i for the computation of r jj  and R K  matixes. Thus, the observation model is modified as follows. 
                   x   →     k     =       (       δ   ⁢           ⁢     r   x       ,     δ   ⁢           ⁢     r   y       ,     δ   ⁢           ⁢     r   z       ,     δ   ⁢           ⁢     V   x       ,     δ   ⁢           ⁢     V   y       ,     δ   ⁢           ⁢     V   z       ,   0   ,   …   ⁢           ,   0     )     T       ,     
     ⁢       H   k     =     (             h   00     ,     h   01     ,     h   02     ,     h   03     ,     h   04     ,     h   05     ,   0   ,   …   ⁢           ,   0                 h   10     ,     h   11     ,     h   12     ,     h   13     ,     h   14     ,     h   15     ,   0   ,   …   ⁢           ,   0                 h   20     ,     h   21     ,     h   22     ,     h   23     ,     h   24     ,     h   25     ,   0   ,   …   ⁢           ,   0           )       ,           ⁢     
     ⁢   and                 R   k     =     (             r   00     ,           0   ,         0             0   ,             r   11     ,         0             0   ,           0   ,           r   22           )                     r   01     =       r   02     =       r   10     =       r   12     =       r   20     =       r   21     =   0               ,     
     ⁢       r   00     =     r     Di   ⁢           ⁢   1         ,       r   11     =     r     Ci   ⁢           ⁢   1         ,       r   22     =     r     Vi   ⁢           ⁢   1                   where                 r     Di   ⁢           ⁢   1       ⁢     =   denote     ⁢     min   ⁢     {       r     DR   ⁢           ⁢   0       ,     r     DR   ⁢           ⁢   1       ,   …   ⁢           ,     r     DR   ⁡     (     i   -   1     )         ,     r     DR   ⁡     (     i   +   1     )         ,   …   ⁢           ,     r     DR   ⁡     (     n   -   1     )           }         ,     
     ⁢       r     Ci   ⁢           ⁢   1       ⁢     =   denote     ⁢     min   ⁢     {       r     CR   ⁢           ⁢   0       ,     r     CR   ⁢           ⁢   1       ,   …   ⁢           ,     r     CR   ⁡     (     i   -   1     )         ,     r     CR   ⁡     (     i   +   1     )         ,   …   ⁢           ,     r     CR   ⁡     (     n   -   1     )           }                   and               r     Vi   ⁢           ⁢   1       ⁢     =   denote     ⁢     min   ⁢     {       r     VR   ⁢           ⁢   0       ,     r     VR   ⁢           ⁢   1       ,   …   ⁢           ,     r     VR   ⁡     (     i   -   1     )         ,     r     VR   ⁡     (     i   +   1     )         ,   …   ⁢           ,     r     VR   ⁡     (     n   -   1     )           }             
A Typical Method
 
     As shown in  FIG. 6 , above analytical analysis describes a method for estimating the motion of a first sensor, typically on a platform, during a time interval, comprising the following steps: 
     a) sensing a geo location using sense geo-location  602  for the first sensor, sense geo-location  604  for the second sensor, and sense geo-location  606  for the N th  sensor during the time interval; 
     b)computing a first TLE in Compute TLE 1   608  using the first sensor, a second TLE in Compute TLE  2   610  using the second sensor; a third TLE in Compute TLE N  612  using the N th  sensor; 
     c) interconnecting the first sensor, the second sensor and the N th  sensor using a data link, the data link transmitting the second location error TLE  2  from second sensor 2  using Send TLE  2 , and the Nth sensor using Send TLE N to the first sensor during said time interval; 
     e) combining the first target location error (TLE  1 ) associated with the first sensor the second target location error (TLE  2 ) and the N th  target location errors in a first sensor observation model within Combine TLE  1 , and TLE  2  . . . N in Sensor Observation Model 620; 
     f) Using the combined TLE gained from the combination of TLEs from the second and N th  separate, de-correlated measurements within Update Kalman Filter  622  associated with sensor  1 , thereby reducing the TLE for sensor  1 . 
     The first, or second sensor, or both, is a synthetic aperture radar, a FLIR infrared sensor, or a SONAR. The only requirement is that the sensor supplying a TLE observe, or take position measurements with respect to the same reference geo-location. 
     The first sensor and the second sensor sense the geo-location to be used as a reference independently, during the same time interval, or common time frame. It is understood that geo-location sensing can be adjusted in time to insure that reported position observations are synchronized to the exact same time. 
     Application Example 
     Above principles are typically applied for the described cases using a 15 error state Kalman filter defined as having 3 position errors, 3 velocity errors, 3 attitude errors, 3 gyro biases, and 3 accelerometer biases. In this example, a typical 1σ error budget is defined as follows: 
     
       
         
           
               
               
             
               
                   
               
             
            
               
                 Position errors 
                 100 meters 
               
               
                 Velocity errors 
                 5 meters/sec 
               
               
                 Attitude errors 
                 0.5 degrees 
               
               
                 Accelerometer bias 
                 400 × 10 −6  g, g = 9.8 meters/second 2   
               
               
                 Accelerometer white noise 
                 0.05 meter/second/{square root over (hour)} 
               
               
                 Gyroscope bias 
                 0.7 degrees/hour 
               
               
                 Gyroscope white noise 
                 0.05 degrees/{square root over (hour)} 
               
               
                   
               
            
           
         
       
     
     Using above operating figures, the size of the navigation error using the teachings herein is substantially reduced especially over longer time intervals, as compared to the single sensor case. For example, where GPS measurements are lost and updates are received via the two way data link every 100 seconds from a second sensor over a 1000 second interval, the navigation position error is reduced from about 4000 meters to 200 meters and the velocity error is reduced from 4 meter/sec to about 1 meter/sec. 
     All references cited in this document are incorporated herein in their entirety by reference. 
     Although presented in exemplary fashion employing specific embodiments, the disclosed structures are not intended to be so limited. For example, although the target location accuracy improvement herein is described in the context of a radar sensor for target position, the disclosure is also applicable for sonar TLE, or other target locating methods, where a location of a target (or scatterers) is independently determined by a plurality or target measuring platforms, each platform having its own, independent TLE. 
     Those skilled in the art will also appreciate that numerous changes and modifications could be made to the embodiment described herein without departing in any way from the invention.