Abstract:
There may be situations in which a ship at sea is lost and GPS is not available due to jamming, and neither a position fix nor GPS is available, or the heading and attitude sensors are degraded. A system and method allow estimating a ship&#39;s heading and pitch using radar range measurements, multiple antennas and satellite ephemeris data.

Description:
BACKGROUND 
     It is important for at least navigational purposes for a moving or movable vehicle to know its heading and/or pitch. In the case of a ship at sea, there may be situations in which the ship is lost and there are no landmarks in sight. In a battle zone, there may be situations in which a ship at sea has damaged heading and/or attitude sensors, or in which the sensors are severely degraded due to man-made or natural anomalies. For example, it is well known that the earth&#39;s electromagnetic magnetic field has an extreme effect on magnetic sensors as a ship approaches the earth&#39;s poles. 
     Improved or alternative arrangements are desired for heading and/or pitch determinations. 
     SUMMARY 
     A method is for determining at least one of heading and pitch of a movable platform such as a ship. This method comprises the step of operating a radar system mounted at forward and aft positions on the movable platform, where the forward and aft positions define a baseline, to measure the ranges of at more than two satellites. Ephemerides for the satellites are obtained. Using a computer and the satellite ephemerides, delta-range equations are set up expressing a baseline vector in terms of the range measurements. A least squares solution is obtained for the heading and pitch using the delta-range equations and the range measurements from the forward and aft locations. 
     A system for determining at least one of heading and pitch of a movable platform comprises a radar arrangement mounted at forward and aft positions on the movable platform, where the forward and aft positions define a baseline. The radar arrangement is operable to measure the ranges of at least two satellites from the forward and aft positions. Sources are provided of ephemerides for the satellites and estimates of the heading and/or pitch. A computer is coupled to the radar arrangement, and to the sources of satellite ephemerides and estimates, for setting up delta-range equations expressing a baseline vector in terms of the range measurements, and for obtaining a least squares solution for the heading and pitch using the delta-range equations and the range measurements from the forward and aft locations. 
     A method for determining at least one of heading and pitch of a movable platform, where the movable platform carries a first radar arrangement including an antenna located at one of forward-located and aft-located positions and also carries a second radar arrangement including an antenna located at the other one of the forward-located and aft-located positions. The method comprises the step of measuring the range to at least two Earth satellites from the first and second radar arrangements, to form measured ranges. The method also comprises the steps of obtaining ephemeris data for the satellites and obtaining an initial estimate of heading and pitch. In a computer process, the method calculates an estimated vector representing a baseline extending between the forward- and aft-located positions, and calculates estimated range differences between the satellites and the forward and aft locations. A geometry matrix is formed from the satellite-ship geometry. A Jacobian matrix is formed representing variation in the baseline vector as a function of the heading and pitch. An incremental solution for heading and pitch is obtained using the geometry matrix, Jacobian matrix and range measurements. The incremental position is solved, and the estimate of heading and pitch is updated. 
     A system for determining heading and pitch of a movable platform comprises a radar system located on the movable platform at one of a forward-located and an aft-located position. The movable platform also carries a radar receiver located at the other one of the forward-located and aft-located positions. The radar system and receiver measure the ranges to at least two Earth satellites from the radar and receiver, respectively, to form measured ranges. Sources are provided of satellite ephemeris data and of estimate of heading and pitch. The system further includes a computer or processor for, in a computer process, calculating the estimated baseline vector extending between the forward and aft positions, and estimated range differences between the satellites and the forward and aft locations. In the computer process, a geometry matrix is formed from satellite-ship geometry. A Jacobian matrix is formed representing variation in the baseline vector as a function of the estimated range. In the computer process, an incremental solution for heading and pitch is formed, and the estimated heading and pitch are updated by the incremental solution. 
     A method for determining at least one of heading and pitch of a movable platform carrying a radar system located at one of a forward and aft position on the platform and a radar signal receiver located at the other of the forward and aft position. The method comprises the step of, using both the forward- and aft-located systems, determining the range and range rate of at least two Earth-orbiting satellites. Satellite ephemerides are obtained. Using a computer, range and range rate equations are generated from data originating from the forward-located and aft-located radar and receiver systems, to thereby generate forward and aft range equations and forward and aft range rate equations. The difference is taken between the forward and aft range equations to thereby form a difference range equation. The difference is taken between the forward and aft rate equations to thereby form a difference rate equation. The forward and aft range and range rate equations are solved simultaneously to determine the heading and pitch. In a particular mode of this method, the step of simultaneously solving includes the steps of applying a least-squares simultaneous solution. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
         FIGS. 1A and 1B  together constitute a simplified representation of a ship at sea illuminating a plurality of satellites with forward-located radar system and receiving radar signals from the satellites with an aft-located radar receiver, and including a computer, for ship heading and pitch) determination according to an aspect of the disclosure; 
         FIG. 2  is a geometry representation relating to the scenario of  FIGS. 1A and 1B ; 
         FIG. 3  is a simplified control or logic flow chart or diagram illustrating system steps according to an aspect of the disclosure; 
         FIG. 4  is a simplified overall logic flow chart or diagram illustrating steps according to an aspect of the disclosure; 
         FIG. 5  is a simplified block diagram of a system according to an aspect of the disclosure; and 
         FIG. 6  is a simplified functional block diagram illustrating a computer which may be used for calculations aboard a vehicle. 
     
    
    
     DETAILED DESCRIPTION 
     As mentioned, there may be situations when a ship at sea has damaged heading and attitude sensors, or where the sensors are severely degraded due to man-made or natural anomalies. For example, it is well known that the earth&#39;s electromagnetic magnetic field has an extreme effect on magnetic sensors as a ship approaches the earth&#39;s poles. A system to determine ship heading and pitch according to an aspect of the disclosure uses the ship&#39;s radar range measurements, satellite ephemeris data and a remote passive radar sensor (antenna) tuned to the same radar frequency as the ship&#39;s active radar (or a second active radar system) to determine a ship&#39;s heading and pitch α with the aid of a computer and algorithms. The system can be self-contained, in that it may be independent of external (off-ship) sensors or transmissions, and also independent of the earth&#39;s magnetic field.  FIGS. 1A and 1B  together constitute a simplified representation of a ship  112  at sea  110 . Ship  112  has an active radar system, the antenna of which is at a location designated  114   a . For simplicity, the radar antenna at location  114   a  is given the designation “ 114   a .” Radar antenna  114   a  is capable of transmitting radar signals, and bouncing them from satellites within range of the radar to measure ranges. Ship  112  also has another sensor or radar antenna, located at a point  114   b , which is aft of point  114   a ; the sensor or antenna is designated as “ 114   b .” Antenna  114   b  is also capable of receiving the radar returns bouncing from the satellite. Point  114   b  is located at a fixed distance L from the active radar antenna  114   a , and a line  109  extending therebetween is termed a “baseline”. The ship&#39;s heading is depicted as ψ, pitch as θ, and roll as φ. The ship&#39;s heading and pitch can be determined, according to an aspect of the disclosure, using radar information together with satellite ephemeris data. 
       FIG. 1A  illustrates three satellites denominated k, l, and m. The range or distance from radar location  114   a  is designated by a subscript a, and the range from radar location  114   b  is designated by a subscript b. Thus, the distance or range between satellite k and radar at location  114   a  is designated r a   k , the distance or range between satellite k and radar receiver location  114   b  is designated r b   k , the distance or range between satellite l and radar location  114   a  is given as r a   l , the distance or range between radar receive location  114   b  is designated r b   l , the distance or range between satellite m and radar location  114   a  is designated r a   m , and the distance or range between satellite m and receive location  614   b  is designated r b   m . 
       FIG. 2  illustrates geometry associated with the determination of the ship heading and pitch reference according to aspects of the disclosure. In  FIG. 2 , the gravitational center of the Earth is designated EC. For a ship with antennas at forward and aft locations  114   a  and  114   b  of  FIG. 1A  or  1 B, the baseline  109  vector {right arrow over (x)} ab  connecting the two antennas as in  FIG. 2  can be solved for by using radar range measurements from the two antennas. Baseline vector {right arrow over (x)} ab  of  FIG. 2  is known in the vehicle body frame coordinates, which body frame coordinates are defined by unit vectors {circumflex over (b)} x , {circumflex over (b)} y  and {circumflex over (b)} z  of  FIG. 1B . Obtaining this vector in the vehicle navigation frame (coordinates North, East, Up) allows the heading to be determined. Calculation of this baseline vector in the vehicle navigation frame is accomplished by utilizing the difference in range measurements from the two antennas. The geometry between a satellite and the baseline orientation is shown in  FIG. 2 . The range difference Δr ab   (k)  is the projection of the negative of the line of sight unit vector lôs (k)  onto the baseline vector {circumflex over (x)} ab . The range difference is
 
Δ r   ab   (k)   =r   a   (k)   =r   b   (k)   =−lô   (k)   ·{circumflex over (x)}   ab   (k)   Eq 1
 
Because the range of the satellite is much greater than the baseline length, the line-of-sight direction from each antenna to the satellite being measured is assumed to be the same.
 
     For N range measurements, where N is greater than three (N&gt;=3), Equation 1 can be written in vector form as equation 2 
                     Δ   ⁢           ⁢       r   →     ab       =       [           Δ   ⁢           ⁢       r   →     ab     (   1   )                   Δ   ⁢       r   →     ab     (   2   )                 ⋮             Δ   ⁢       r   →     ab     (   N   )               ]     =       [             -   l     ⁢     o   ^     ⁢     s       (   1   )     T                     -   l     ⁢     o   ^     ⁢     s       (   2   )     T                 ⋮               -   l     ⁢     o   ^     ⁢     s       (   N   )     T               ]     ·         x   →     ab                           Eq   ⁢           ⁢   2               
The matrix on the right side of Equation 2 is the geometry matrix G. Equation 2 can be re-written as
 
Δ {circumflex over (r)}   ab   =G·{circumflex over (x)}   ab   Eq 3
 
The matrix G can be determined using the ship&#39;s position together with satellite ephemeris data. The matrix G expresses the line-of-sight vectors pointing from the ship position to each satellite. Eq. 3 is a linear equation with three unknowns, namely the three components of three-dimensional baseline vector {circumflex over (x)} ab , [{circumflex over (x)} ab   (1)  {circumflex over (x)} ab   (2)  {circumflex over (x)} ab   (3) ]. In the case in which measurements are available from three or more satellites, {circumflex over (x)} ab  can be obtained using a least squares solution.
 
 {circumflex over (x)}   ab =( G   T   W   −1   G ) −1   G   T   W   −1   Δ{circumflex over (r)}   ab   Eq 4
 
where W is a weight matrix used in case more accurate measurements are to be favored over less accurate measurements. Weight W can be selected to be the covariance of the delta-range measurements
 
 W =diag└σ Δr1   2 τ Δr2   2  . . . σ Δr3   2 ┘  Eq. 5
 
where σ Δrt   2  represents the uncertainty in the delta-range measurement for satellite i.
 
     Equation 4 solves for the vector {right arrow over (x)} ab  in the navigation reference frame. Since this vector is also known in the vehicle body frame, the heading of the ship can be determined. If the antennas are located on the roll axis (φ), the heading Ψ and pitch θ can be determined as follows, referring to  FIGS. 1A and 1B . 
                   ψ   =       tan     -   1       ⁡     (           x   →     ab     ⁡     (   1   )             x   →     ab     ⁡     (   2   )         )               Eq   ⁢           ⁢   6               θ=sin −1 ( {right arrow over (x)}   ab (3))  Eq 7
 
     where the parenthetical numbers 1, 2, and 3 refer to the components of the baseline vector. If the antennas are located off the roll axis, pitch and roll will be coupled and only heading will be observable. 
     The above discussion relates to determination of heading and pitch for the case of N range measurements, where N is greater than three (N≧3), thus requiring the presence of at least three satellites on which to make measurements. The determination of ship heading and pitch with only N=2 satellites is described next. In the case that only two range measurements are available in the context of the scenario of  FIGS. 1A and 1B , ship heading and pitch can still be determined, even though the full ship attitude cannot.  FIG. 1B  further defines the ENU axes and unit coordinate vectors {circumflex over (b)} x , {circumflex over (b)} y , {circumflex over (b)} z  attached to the ship&#39;s body. Referring to  FIG. 1B , if the two antennas  114   a  and  114   b  are located on the vehicle roll axis φ, the baseline vector in the East-North-Up (ENU) frame can be expressed as shown in Equation 8, where L is the length of the baseline. 
                       x   →     ab     =     L   ⁡     [           sin   ⁢           ⁢   ψ   ⁢           ⁢   cos   ⁢           ⁢   θ               cos   ⁢           ⁢   ψ   ⁢           ⁢   cos   ⁢           ⁢   θ               sin   ⁢           ⁢   θ           ]               Eq   ⁢           ⁢   8               
The unit vectors {circumflex over (b)} x {circumflex over (b)} y {circumflex over (b)}{circumflex over (b z )} of  FIG. 1B  represent a coordinate frame attached to the ship body. Rotation about {circumflex over (b)} x  is the pitch θ, and rotation about {circumflex over (b)} y  is the roll φ. In the case of a ship at sea, the heading can be calculated as the angle between North and {circumflex over (b)} y . If the antennas are located off the roll axis, then the angle θ will include components of pitch and roll, and only the heading can be determined. To solve for heading ψ and pitch θ, Equation 8 can be linearized and the solution obtained iteratively. The variation of {right arrow over (x)} ab  will have the form
 
                     δ   ⁢           ⁢       x   →     ab       =       [             ∂         x   →     ab     ⁡     (   1   )             ∂           ⁢   ψ     ⁢                       ∂         x   →     ab     ⁡     (   1   )             ∂           ⁢   θ     ⁢                           ∂         x   →     ab     ⁡     (   2   )             ∂           ⁢   ψ     ⁢                       ∂         x   →     ab     ⁡     (   2   )             ∂           ⁢   θ     ⁢                           ∂         x   →     ab     ⁡     (   3   )             ∂           ⁢   ψ     ⁢                       ∂         x   →     ab     ⁡     (   3   )             ∂           ⁢   θ     ⁢                     ]     ⁡     [           δ   ⁢           ⁢   ψ               δ   ⁢           ⁢   θ           ]               Eq   ⁢           ⁢   9               
Applying this to Equation 8 results in
 
                     δ   ⁢           ⁢       x   →     ab       =       [           cos   ⁢           ⁢   θ   ⁢           ⁢   cos   ⁢           ⁢   ψ             -   sin     ⁢           ⁢   ψ   ⁢           ⁢   sin   ⁢           ⁢   θ                 -   cos     ⁢           ⁢   θ   ⁢           ⁢   sin   ⁢           ⁢   ψ             -   cos     ⁢           ⁢   ψ   ⁢           ⁢   sin   ⁢           ⁢   θ             0         cos   ⁢           ⁢   θ           ]     ⁡     [           δ   ⁢           ⁢   ψ               δ   ⁢           ⁢   θ           ]               Eq   ⁢           ⁢   10               
or
 
                     δ   ⁢           ⁢       x   →     ab       =     J   ⁡     [           δ   ⁢           ⁢   ψ               δ   ⁢           ⁢   θ           ]               Eq   ⁢           ⁢   11               
Defining the difference between the measured and predicted delta ranges δΔ{right arrow over (r)} ab ≡Δ{right arrow over (r)} ab , the delta range equation (Equation 3) can be expressed in differential form
 
δΔ {right arrow over (r)}   ab   =G·δ{right arrow over (x)}   ab   Eq 12
 
Substituting for δ{right arrow over (x)} ab  from Equation 11 results in
 
                     δ   ⁢           ⁢   Δ   ⁢           ⁢       r   →     ab       =     GJ   ⁡     [           δ   ⁢           ⁢   ψ               δ   ⁢           ⁢   θ           ]               Eq   ⁢           ⁢   13               
A solution is obtained iteratively. The steps are set forth below in conjunction with the logic flow chart or diagram  300  of  FIG. 3 . In  FIG. 3 , the logic begins at a START block  310 , and flows to a block  311 . Block  311  represents the measuring of the range to the two satellites using the forward and aft antennas at locations  614   a  and  614   b . From block  311 , the logic  300  flows to a block  312 . Block  312  represents the importation of satellite ephemeris data. The logic  300  flows to a block  314 , which represents the importing or generation of an initial guess as to the values for ψ and θ, namely heading {circumflex over (ψ)} and pitch {circumflex over (θ)}. From block  314 , the logic  300  flows to a block  316 . Block  316  represents calculation of the estimated vector {circumflex over (x)} ab  extending between antennas  114   a  and  114   b 
 
                       x   ^     ab     =     L   ⁡     [           sin   ⁢           ⁢     ψ   ^     ⁢           ⁢   cos   ⁢           ⁢     θ   ^                 cos   ⁢           ⁢     ψ   ^     ⁢   cos   ⁢           ⁢     θ   ^                 sin   ⁢           ⁢     θ   ^             ]               Eq   ⁢           ⁢   14               
From block  316 , the logic  300  flows to a block  318 . Block  318  represents calculation of estimated delta-ranges, which is the difference in range between antennas  114   a  and  114   b  to each satellite
 
                     Δ   ⁢           ⁢       r   ^     ab       =     [                      x   ^     a     -       x   →       (   1   )              -              x   ^     b     -       x   →       (   1   )                                     x   ^     a     -       x   →       (   2   )              -              x   ^     b     -       x   →       (   2   )                      ]             Eq   ⁢           ⁢   15               
Block  320  represents determination or formation of the Geometry matrix, as shown in Equation 3.
 
 G=G ( {circumflex over (x)}   a   ,ŷ   a   ,{circumflex over (z)}   a )  Eq 16
 
The Jacobian matrix is formed in logic block  322 , as shown in Equation 10.
 
 J=J ({circumflex over (ψ)},{circumflex over (θ)})  Eq 17
 
Since the product of G and J is square (2×2) and is full rank, the incremental solution can be obtained by taking the inverse of the product
 
                     [           δ   ⁢           ⁢   ψ               δ   ⁢           ⁢   θ           ]     =         (   GJ   )       -   1       ⁢   δ   ⁢           ⁢   Δ   ⁢           ⁢       r   →     ab               Eq   ⁢           ⁢   18               
This incremental solution for heading ψ and pitch θ is calculated in block or step  324  of  FIG. 3 . From block  324 , the logic flows to a block  326 , representing the updating of the estimates
 
{circumflex over (ψ)} + ={circumflex over (ψ)} − +δψ
 
{circumflex over (θ)} + ={circumflex over (θ)} − +δθ  Eq 19
 
The logic  300  of  FIG. 3  flows from block  326  by way of a logic path  328  back to step  316 . The logic flows around the logic loop using the updated estimates at each iteration. The iteration may be ended in known manners upon convergence.
 
       FIG. 4  is a simplified overall control or logic flow chart or diagram illustrating steps in the determination of ship heading and pitch. In  FIG. 4 , the logic  400  begins at a START block  410 , and flows to a block  412 , which represents measuring the range to at least two satellites using at least the two antennas  614   a  and  614   b  of  FIGS. 1A and 1B  on board the platform or ship  12 , the heading and pitch of which are to be determined. From block  412 , the logic  400  flows to a block  414 , representing the loading of satellite ephemerides information. From block  414 , the logic flows to a decision block illustrated as parts or portions  416   a  and  416   b . Decision block portion  416   a  routes the logic to a block  418  if the number of satellites for which range measurements have been made exceeds two, while decision block portion  416   b  routes the logic to block  430  if the number of satellites is two. 
     Block  418  in the 3-satellite portion (&gt;2) of the logic flow  400  of  FIG. 4  represents formation of delta-range equations (equation 2) from at least the two antennas. From block  418 , logic  400  flows to a block  420 . Block  420  represents the use of geometry equations (equation 4) to solve for the platform baseline vector(s) in the navigation reference frame. Block  422  represents determination of the heading ψ and pitch α from the platform baseline vector(s) (Equations 6 and 7). 
     In the two-satellite portion (=2) of the flow chart  400  of  FIG. 4 , the logic proceeds from decision block portion  416   b  to a block  430 . Block  430  of logic  400  of  FIG. 4  represents the formation of an initial estimate or guess as to platform pitch {circumflex over (θ)} and heading {circumflex over (ψ)}, and corresponds to block  314  of  FIG. 3 . Block  432  of  FIG. 4  represents calculation of the platform baseline vector using equation 14, and corresponds to block  316  of  FIG. 3 . Block  434  of  FIG. 4  represents formation of the delta-range equations 15, corresponding to block  318  of  FIG. 3 . At this point in the processing, it is assumed that position is known, and therefore G is also known. Block  436  of  FIG. 4  represents the formation of the Jacobian matrix (Equation 17) representing the variation in the baseline vector, and corresponds to block  322  of  FIG. 3 . Block  438  of  FIG. 4  represents the obtainance of an incremental least-squares solution for platform heading and pitch (equation 18), and the updating of the current estimates, corresponding to blocks  324  and  326  of  FIG. 3 . From block  438  of  FIG. 4 , the logic  400  flows to a decision block  440 . Block  440  determines if the number of iterations has reached a limit value. If the number of iterations has not reached a limit value, the logic  400  leaves decision block  440  by the NO output and returns by a logic path  441  to block  432 , to begin another iteration. Eventually, the number of iterations will have reached the limit value, and the logic will leave decision block  440  by the YES output. From either block  422  or  440  the logic will reach block  424 . Block  424  represents the end state of the process, namely with platform heading and pitch. The logic  400  ends at an END block  426 . 
       FIG. 5  is a simplified block diagram illustrating a system according to an aspect of the disclosure for determining heading and pitch. In  FIG. 5 , active radar  14  is coupled to antenna  114   a  and ancillary radar receiver  514  is coupled to antenna  114   b . Radar  14  and receiver  514  are coupled to computer  14   c  for making available satellite range information. Computer  14   c  is coupled to a source  504  of satellite ephemerides information and to a source  506  of heading and pitch estimates. Computer  14   c  is preprogrammed to use the information from the radar and receiver, and from the ephemeris and estimate sources, to determine at least one of the heading and pitch according to aspects of the disclosure. 
       FIG. 6  is a simplified diagram in block and schematic form illustrating a representative computer which may be used as  14   c . In  FIG. 6 , computer  1100  includes a processor or board  1110  and outboard elements such as a monitor  1112 , user controls such as a keyboard and/or mouse, illustrated as a block  1114 , local area network (LAN)  1116 , additional buses  1118  such as PCI and/or USB, and read-only memory (ROM)  1120 , which is ordinarily a hard drive, and additional ROM  1122 , which may be, for example, a flash memory stick or capacitance disk (CD). The main portion of the computer processor or board  1110  includes a central processing unit (CPU)  1134 , which communicates with a cache dynamic memory  1138 . At initial turn-on of the computer  1100 , a power-on reset illustrated as a block  1154  enables a preloaded basic input/output system (BIOS) flash memory, which loads cache  1138  with information that initializes the booting sequence by the CPU. When booted, CPU  1134  may communicate with a coprocessor illustrated as  1136 , and also communicates with main dynamic memory (DRAM)  1132  and a local bus  1158 . Local bus  1158  provides communication between the CPU and other elements of the computer, as for example the video processor  1140  and video random-access memory  1142  for driving a monitor. Local bus  1158  also communicates by way of a bridge  1144  to external ROM  1120  and to user controls  1118 . Local bus  1158  further communicates by way of a port  1148  with other ROM  1122  if desired, by way of a USB or PCI bridge or port  1150  with external buses, and/or by way of a local area network (LAN) port  1146  with a LAN  1116 . Those skilled in the art will understand how to use one or more computers to perform the processing required by elements of the disclosure. 
     Other embodiments will be apparent to those skilled in the art. For example, while the system of  FIGS. 1A and 1B  has been described as having an active radar system with an antenna at location  114   a  and a radar receiver antenna at location  114   b , the active radar antenna could be at location  114   b  and the radar receiver at location  114   a . Similarly, since it is only necessary to know the ranges from the forward and aft locations to the satellites, both the forward and aft antenna locations  114   a  and  114   b  could be associated with independent active radar systems; this is disadvantageous because of cost and because of possible interference between the transmissions of the two active radars. The “location” or position of the radar or receiver is deemed to be the position of the associated antenna. 
     A method according to an aspect of the disclosure is for determining at least one of heading and pitch of a movable platform  112 ) carrying a radar system ( 114   c ) located at one of a forward ( 114   a ) and aft ( 114   b ) position on the platform ( 112 ) and a radar signal receiver located at the other of the forward ( 114   a ) and aft ( 114   b ) position. The method comprises the step ( 311 ) of operating a radar system mounted in a movable platform ( 112 ), using both the forward- ( 114   a ) and aft-located ( 114   b ) systems (antennas), and determining the range and range rate of at least two Earth-orbiting satellites. Satellite ephemerides are obtained. Using a computer ( 14   c ), range and range rate equations ( 300 ) are generated from data originating from the forward-located and aft-located radar and receiver systems, to thereby generate forward and aft range equations and forward and aft range rate equations. The difference is taken between the forward and aft range equations to thereby form a difference range equation (equation  53 ). The forward and aft range equations are solved simultaneously to determine the heading and pitch. In a particular mode of this method, the step of simultaneously solving includes the steps of applying a least-squares simultaneous solution. 
     Thus, a method according to an aspect of the disclosure as illustrated on the greater-than-two-satellite portion of  FIG. 4  is for determining at least one of heading and pitch of a movable platform ( 12 ) such as a ship. This method comprises the step of operating a radar system ( 14 ,  14   a ) mounted at forward and aft positions ( 114   a ,  114   b ) on the movable platform ( 12 ), where the forward and aft positions define a baseline, to measure the ranges ( 412 ) of at more than two (&gt;2) satellites ( 18 ). Ephemerides for the satellites ( 18 ) are obtained ( 414 ). Using a computer ( 14   c ) and the satellite ephemerides, delta-range equations are set up ( 418 ) expressing a baseline vector in terms of the range measurements. A least squares solution is obtained ( 420 ) for the heading and pitch ( 422 ) using the delta-range equations and the range measurements from the forward and aft locations. 
     A system according to another aspect of the disclosure relates to  FIG. 3  and the satellites-equal-two portion of the flow chart of  FIG. 4 . This system is for determining at least one of heading and pitch of a movable platform ( 12 ). The system comprises a radar arrangement ( 14 ,  14   a ) mounted at forward and aft positions ( 114   a ,  114   b ) on the movable platform ( 12 ), where the forward and aft positions define a baseline, the radar arrangement being operable to measure the ranges ( 412 ) of at least two satellites ( 18 ) from the forward and aft positions. Sources are provided of ephemerides ( 414 ) for the satellites ( 18 ) and estimates of the heading and/or pitch. A computer ( 14   c ) is coupled to the radar arrangement, to the sources of satellite ephemerides and estimates, for setting up delta-range equations ( 418 ) expressing a baseline vector in terms of the range measurements, and for obtaining ( 420 ) a least squares solution for the heading and pitch ( 422 ) using the delta-range equations and the range measurements from the forward and aft locations. 
     A method according to another aspect of the disclosure relates to  FIG. 3  and to the two-satellite (=2) portion of  FIG. 4  is for determining at least one of heading and pitch of a movable platform ( 12 ). The movable platform ( 12 ) carries a first radar arrangement ( 14 ) including an antenna located at one of forward-located ( 114   a ) and aft-located ( 114   b ) positions and also carries a second radar arrangement including an antenna located at the other one of the forward-located and aft-located positions. The method comprises the step of measuring the range ( 311 ,  412 ) to at least two Earth satellites ( 18 ) from the first ( 114   a ) and second ( 114   b ) radar arrangements, to form measured ranges. The method also comprises the steps of obtaining ephemeris data ( 312 ,  414 ) for the satellites and obtaining ( 314 ,  430 ) an initial estimate of heading and pitch. In a computer process, the method calculates ( 316 ,  432 ) an estimated vector representing a baseline extending between the forward- and aft-located positions, and calculates ( 318 ,  434 ) estimated range differences between the satellites and the forward and aft locations. A geometry matrix is formed ( 320 ] from the satellite-ship geometry. A Jacobian matrix is formed ( 322 ,  436 ) representing variation in the baseline vector as a function of heading and pitch. An incremental solution for heading and pitch is obtained from the using the geometry matrix, Jacobian matrix and range measurements. The incremental position is solved, and the estimate of heading and pitch is updated. 
     A system for determining heading and pitch of a movable platform ( 12 ) according to another aspect of the disclosure comprises a radar system ( 14 ) located on the movable platform ( 12 ), with an antenna ( 114   a ) at one of a forward-located ( 114   a ) and an aft-located position ( 114   b ). The movable platform also carries a radar receiver ( 514 ) with an antenna ( 114   b ) located at the other one of the forward-located ( 114   a ) and aft-located ( 114   b ) positions. The radar system ( 14 ) and receiver ( 514 ) measure the ranges ( 311 ,  412 ) to at least two Earth satellites ( 18 ) from the radar ( 14 ) and receiver ( 514 ), respectively, to form measured ranges. Sources are provided of satellite ephemeris data ( 564 ) and of estimate of heading and pitch ( 566 ). The system further includes a computer or processor ( 14   c ) for, in a computer ( 14   c ) process, calculating ( 316 ,  432 ) the estimated baseline vector extending between the forward and aft positions, and estimated range differences between the satellites and the forward and aft locations ( 318 ,  434 ). In the computer process, a geometry matrix is formed ( 320 ) from the ship-satellite geometry. A Jacobian matrix is formed ( 322 ,  436 ) representing variation in the baseline vector as a function of the estimated range. In the computer process, an incremental solution ( 322 ,  436 ) for heading and pitch is formed, and the estimated heading and pitch are updated by the incremental solution.