Patent Publication Number: US-11644578-B2

Title: System for determining high-integrity navigation solutions via optimal partial fixing of floating-point integer ambiguities

Description:
CROSS-REFERENCE TO RELATED APPLICATIONS 
     For purposes of the USPTO extra-statutory requirements, the present application constitutes a divisional application of U.S. patent application Ser. No. 16/369,567 entitled HIGH INTEGRITY PARTIAL ALMOST-FIX SOLUTION, filed Mar. 29, 2019, which application issued Jan. 11, 2022 as U.S. Pat. No. 11,221,419. 
     For purposes of the USPTO extra-statutory requirements, the present application constitutes a continuation of the U.S. patent application Ser. No. 15/065,260 entitled HIGH INTEGRITY PARTIAL ALMOST-FIX SOLUTION, filed Mar. 9, 2016, which application issued Apr. 30, 2019 as U.S. Pat. No. 10,274,606. 
     Said U.S. patent applications Ser. Nos. 16/369,567 and 15/065,260 are herein incorporated by reference in their entirety. 
    
    
     GOVERNMENT LICENSE RIGHTS 
     This invention was made with government support under Government Contract No. N00019-08-C-0034, JOINT PRECISION APPROACH AND LANDING SYSTEM DEVELOPMENT, awarded by the Department of the Navy. The government has certain rights in the invention. 
    
    
     BACKGROUND 
     In selected differential Global Positioning System (GPS) based applications, such as the Joint Precision Approach and Landing System (JPALS), relative navigation functions must meet requirements for high accuracy, high integrity, and high availability. To ensure sub-meter accuracy (e.g., within 20 cm), navigation solutions may be formed using GPS carrier phase measurements. GPS carrier phase integer ambiguities may then be fixed via a Real Time Kinematic (RTK) technique such as the Least-squares AMBiguity Decorrelation Algorithm (LAMBDA) or an integer bootstrapping algorithm. To ensure high integrity, protection levels can be computed to bound errors in the computed navigation solution to a desired degree of confidence. Protection levels depend, among other factors, on the success rate of the carrier phase integer ambiguity fixing process. Generally, if the probability of correct fix (PCF) is sufficiently high, both accuracy and protection levels may be improved with fixed integer ambiguities (as opposed to float integer ambiguities). If the PCF is not high enough, however, improved protection levels may not result from estimated fixed integer ambiguities (relative to estimated float integer ambiguities), even though nominal accuracy may improve. Consequently, both availability and accuracy may suffer, as a fixed solution may not be available after protection levels exceed alert limits. 
     One technique for maintaining the PCF (i.e., ensuring a reasonably high success rate) while maintaining high accuracy may involve partial ambiguity fixing: fixing only a subset of integer ambiguities that can be estimated with higher confidence while using float integer ambiguity estimates for the remainder, trading accuracy for lower protection levels. However, in the original carrier phase measurement ambiguity domain, using integer bootstrapping alone may lead to lower PCF and higher protection levels. Use of LAMBDA algorithms with integer bootstrapping to decorrelate ambiguities prior to fixing may increase the PCF and lower protection levels; however, the use of LAMBDA may complicate partial fixing. 
     SUMMARY 
     In a first aspect, a system for determining high-integrity navigation solutions is disclosed. In embodiments, the system includes one or more processors configured for receiving multiple sets of satellite-based navigational data from diverse sources using common navigational satellites, e.g., Global Positioning System (GPS) carrier phase measurements from a GPS receiver on board the aircraft in which the system is embodied as well as GPS data from another proximate vehicle (e.g., a carrier on which the aircraft is approaching to land upon). Based on the received GPS data, the system generates a set of GPS floating-point ambiguities. By decorrelating the floating-point ambiguities according to Least-squares AMBiguity Decorrelation Adjustment (LAMBDA) algorithms, the system generates a baseline navigational solution in the LAMBDA domain. From this baseline solution, the system selects decorrelated floating-point ambiguities to be fixed, one at a time. The system determines a protection level associated with a partial almost-fix solution (PAFS) based on the set of floating-point ambiguities and a Probability of Almost Fix (PAF) based on the probability that the PAFS is bounded by the protection level. The system fixes a first decorrelated floating-point ambiguity in the LAMBDA domain and updates the PAFS by substituting the floating-point ambiguity with the corresponding fixed ambiguity. While the PAF remains above a predetermined threshold and the protection level remains below a predetermined alert level, the system selects additional decorrelated floating-point ambiguities to fix and continues to update the PAFS by substituting the appropriate floating-point ambiguities with the corresponding fixed ambiguities. Should the PAF reach the threshold level, or the protection level reach the alert level, the system reverts to and outputs the most recently updated PAFS, replacing the most recently substituted fixed ambiguity with its corresponding floating-point ambiguity from the original set. 
     In a further aspect, a Joint Precision and Approach Landing System (JPALS) is also disclosed. The JPALS is embodied aboard an aircraft and includes a GPS or similar satellite-based navigational receiver for determining position data based on GPS carrier phase signals received from navigational satellites. In embodiments, the JPALS includes processors for receiving the GPS signals used by the navigational receiver to determine ownship position as well as additional GPS carrier phase data from other sources (e.g., a carrier or other vehicle proximate to the aircraft, and upon which the aircraft is approaching to land). Based on the received GPS data, the JPALS generates a set of GPS floating-point ambiguities. By decorrelating the floating-point ambiguities according to LAMBDA algorithms, the JPALS generates a baseline navigational solution in the LAMBDA domain. From this baseline solution, the JPALS selects decorrelated floating-point ambiguities to be fixed, one at a time. The JPALS determines a protection level associated with a partial almost-fix solution (PAFS) based on the set of floating-point ambiguities and a Probability of Almost Fix (PAF) based on the probability that the PAFS is bounded by the protection level. The JPALS fixes a first decorrelated floating-point ambiguity in the LAMBDA domain and updates the PAFS by substituting the floating-point ambiguity of the original set with the corresponding fixed ambiguity. While the PAF remains above a predetermined threshold and the protection level remains below a predetermined alert level, the JPALS selects additional decorrelated floating-point ambiguities to fix and continues to update the PAFS by substituting the appropriate floating-point ambiguities with the corresponding fixed ambiguities. Should the PAF reach the threshold level, or the protection level reach the alert level, the JPALS reverts to and outputs the most recently updated PAFS, replacing the most recently substituted fixed ambiguity with its corresponding floating-point ambiguity from the original set. 
     This Summary is provided solely as an introduction to subject matter that is fully described in the Detailed Description and Drawings. The Summary should not be considered to describe essential features nor be used to determine the scope of the Claims. Moreover, it is to be understood that both the foregoing Summary and the following Detailed Description are example and explanatory only and are not necessarily restrictive of the subject matter claimed. 
    
    
     
       BRIEF DESCRIPTION OF THE DRAWINGS 
       Implementations of the inventive concepts disclosed herein may be better understood when consideration is given to the following detailed description thereof. Such description makes reference to the included drawings, which are not necessarily to scale, and in which some features may be exaggerated and some features may be omitted or may be represented schematically in the interest of clarity. Like reference numerals in the drawings may represent and refer to the same or similar element, feature, or function. In the drawings: 
         FIG.  1 A  illustrates exemplary embodiments of a system according to the inventive concepts disclosed herein; 
         FIG.  1 B  is a block diagram illustrating an exemplary embodiment of a system according to the inventive concepts disclosed herein; 
         FIG.  2 A  is a block diagram illustrating operations of the system of  FIG.  1 B ; 
         FIG.  2 B  is a block diagram illustrating alternative operations of the system of  FIG.  1 B ; 
         FIGS.  3 A- 3 B  are process flow diagrams illustrating a method according to embodiments of the inventive concepts disclosed herein; and 
         FIGS.  4 A- 4 C  are process flow diagrams illustrating a method according to embodiments of the inventive concepts disclosed herein. 
     
    
    
     DETAILED DESCRIPTION OF EXEMPLARY EMBODIMENTS 
     Before explaining at least one embodiment of the inventive concepts disclosed herein in detail, it is to be understood that the inventive concepts are not limited in their application to the details of construction and the arrangement of the components or steps or methodologies set forth in the following description or illustrated in the drawings. In the following detailed description of embodiments of the instant inventive concepts, numerous specific details are set forth in order to provide a more thorough understanding of the inventive concepts. However, it will be apparent to one of ordinary skill in the art having the benefit of the instant disclosure that the inventive concepts disclosed herein may be practiced without these specific details. In other instances, well-known features may not be described in detail to avoid unnecessarily complicating the instant disclosure. The inventive concepts disclosed herein are capable of other embodiments or of being practiced or carried out in various ways. Also, it is to be understood that the phraseology and terminology employed herein is for the purpose of description and should not be regarded as limiting. 
     As used herein a letter following a reference numeral is intended to reference an embodiment of the feature or element that may be similar, but not necessarily identical, to a previously described element or feature bearing the same reference numeral (e.g.,  1 ,  1   a ,  1   b ). Such shorthand notations are used for purposes of convenience only, and should not be construed to limit the inventive concepts disclosed herein in any way unless expressly stated to the contrary. 
     Further, unless expressly stated to the contrary, “or” refers to an inclusive or and not to an exclusive or. For example, a condition A or B is satisfied by anyone of the following: A is true (or present) and B is false (or not present), A is false (or not present) and B is true (or present), and both A and B are true (or present). 
     In addition, use of the “a” or “an” are employed to describe elements and components of embodiments of the instant inventive concepts. This is done merely for convenience and to give a general sense of the inventive concepts, and “a” and “an” are intended to include one or at least one and the singular also includes the plural unless it is obvious that it is meant otherwise. 
     Finally, as used herein any reference to “one embodiment,” or “some embodiments” means that a particular element, feature, structure, or characteristic described in connection with the embodiment is included in at least one embodiment of the inventive concepts disclosed herein. The appearances of the phrase “in some embodiments” in various places in the specification are not necessarily all referring to the same embodiment, and embodiments of the inventive concepts disclosed may include one or more of the features expressly described or inherently present herein, or any combination of sub-combination of two or more such features, along with any other features which may not necessarily be expressly described or inherently present in the instant disclosure. 
     Broadly, embodiments of the inventive concepts disclosed herein are directed to a system for determining precision navigation solutions based on GPS carrier phase measurements received from one or more GPS receivers. The system may increase the confidence level of navigation solutions by decorrelating GPS carrier phase integer ambiguities via Least-squares AMBiguity Decorrelation Adjustment (LAMBDA) and partially fixing the integer ambiguities in the LAMBDA domain. System integrity may be enhanced via Partial Almost Fixing, or determining the probability that floating-point navigation solutions are included within correct integer ambiguities or within a subset of incorrect integer ambiguities with single-cycle or small multiple-cycle errors. 
     Referring now to  FIG.  1 A , an exemplary embodiment of a system  100  according to the inventive concepts disclosed herein may be embodied in a Joint Precision Approach and Landing System (JPALS) aboard an aircraft  102  operating proximate to an aircraft carrier or similar ship  104 ; the aircraft  102  may be based aboard the ship  104 . Both the aircraft  102  and the ship  104  may continually generate position information based on GPS signals received from one or more common GPS satellites  106 . For example, the aircraft  102 , not performing a precision landing operation to the ship  104 , may determine its own position based on signals received from GPS satellites (which need not be common in the case of a single aircraft determining a standalone position). The aircraft  102  may also receive coarse position and speed data ( 108 ) associated with the ship  104  for the purposes of coarse relative navigation. The aircraft  102   a , however, executing a precision approach operation to the ship  104  (e.g., within 10 NM of the ship  104 ) may receive from the ship  104  approach data and GPS measurements ( 108   a ) received in turn by the ship  104  from the common GPS satellites  106 . As the aircraft  102   a  approaches the ship  104  (e.g., toward a landing aboard the ship  104 ), the ship  104  may provide increasingly sophisticated approach data  108   a  so that the aircraft  102   a  may determine a precise navigational vector for precision approach to the ship  104 . 
     Referring now to  FIG.  1 B , an exemplary embodiment of a system  100  for determining precise navigation solutions may be embodied in aircraft-based JPALS equipment  100   a  aboard the aircraft  102 . The system may receive GPS measurements from the common GPS satellites  106  (via an onboard GPS receiver  110 ) and the ship  104 , including GPS carrier signals. Based on the received GPS measurements, the system  100  may generate floating-point baseline navigation solutions  114  comprising sets of GPS floating-point carrier phase ambiguities  112 . For example, the floating-point baseline navigation solution  114  may include a set of n floating-point carrier phase ambiguities  112  (where n is a positive integer). The determined GPS floating-point carrier phase ambiguities  112  may be decorrelated via LAMBDA decorrelation to obtain LAMBDA-domain floating-point ambiguities  116 . While existing Geometric Extra Redundancy Almost Fix Solution (GERAFS) functionality in a JPALS  100   a  may convert the LAMBDA-domain floating-point ambiguities  116  back to the original carrier-phase measurement ambiguity domain (e.g., via the inverse of a LAMBDA transformation matrix), the system  100  may generate LAMBDA-domain fixed integer ambiguities  118  by fixing the LAMBDA-domain floating-point ambiguities  116  in the LAMBDA domain. 
     For example, â may be a vector of floating-point carrier-phase ambiguities  112  in the original carrier-phase measurement ambiguity domain, {hacek over (a)} may be a vector of fixed integer ambiguities (not shown) in the original carrier-phase measurement ambiguity domain, {circumflex over (b)} may be a vector of an estimated floating-point baseline navigation solution  114 , {hacek over (b)} may be a vector of a more precise fixed-ambiguity (fully fixed) baseline navigation solution  120 , Q {circumflex over (b)}â  may be the error covariance between the floating-point baseline navigation solution  114  and the floating-point carrier-phase ambiguities  112 , and Q â  may be the error covariance of the floating-point carrier-phase ambiguities  112 . Therefore, according to Real Time Kinematic (RTK) theory 
                     b   ⋓     =       b   ^     -       Q       b   ^     ⁢     a   ^         ⁢         Q     a   ^         -   1       ⁡     (       a   ^     -     a   ⋓       )                   (   1   )               
and the floating-point baseline navigation solution {circumflex over (b)} ( 114 ) may be updated with a list ( 118   a ) of partially fixed integer ambiguities {hacek over (a)} to obtain a partial almost-fix navigation solution  124  which may approximate the fixed-ambiguity baseline navigation solution {hacek over (b)} ( 120 ). The correction e in the floating-point baseline navigation solution {circumflex over (b)} ( 114 ) may be defined as:
 
     
       
         
           
             
               
                 
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     To increase the success rate of the fixing process, floating-point carrier-phase ambiguities â ( 112 ) in the original carrier-phase measurement ambiguity domain may be transformed into decorrelated floating-point ambiguities z ( 116 ) in the LAMBDA domain using a LAMBDA transformation Z, such that 
                   z   =       Z   T     ⁢     a   .               (   3   )               
The error covariance between the floating-point baseline navigation solution {circumflex over (b)} ( 114 ) and the LAMBDA-domain floating-point ambiguities z ( 116 ) may be Q {circumflex over (b)}{circumflex over (z)} , and the error covariance of the LAMBDA-domain floating-point ambiguities z may be Q {circumflex over (z)} , such that
 
     
       
         
           
             
               
                 
                   
                     
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     In addition, Q {circumflex over (z)} =Z T Q â Z and therefore 
     
       
         
           
             
               
                 
                   
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     It may follow from equations (2), (4), and (5) that 
             e   =         Q       b   ^     ⁢     a   ^         ⁢         Q     a   ^         -   1       ⁡     (       a   ^     -     a   ⋓       )         =         Q       b   ^     ⁢     z   ^         ⁢     Z     -   1       ⁢       ZQ     z   ^         -   1       ⁢       Z   T     ⁡     (       a   ^     -     a   ⋓       )         =       Q       b   ^     ⁢     z   ^         ⁢         Q     z   ^         -   1       ⁡     (         Z   T     ⁢     a   ^       -       Z   T     ⁢     a   ⋓         )                   
and from equation (3) that e=Q {circumflex over (b)}{circumflex over (z)} Q {circumflex over (z)}   −1 ({circumflex over (z)}−ž); therefore, equation (1) may become
 
                       b   ⋓     =       b   ^     -       Q       b   ^     ⁢     z   ^         ⁢         Q     z   ^         -   1       ⁡     (       z   ^     -     z   ⋓       )             ,           (   6   )               
showing that the correction in the floating-point baseline navigation solution {circumflex over (b)} may be equivalently defined and corrected in the LAMBDA domain. Accordingly, both partial fixing (e.g., fixing only a subset  122  of decorrelated floating-point carrier-phase ambiguities  116  to generate a partial almost-fix navigation solution  124 ) and partial correction updates (e.g., removing only a subset of associated errors in the floating-point baseline navigation solution  114 ) may be accomplished via fixing decorrelated floating-point ambiguities  116  in the LAMBDA domain. For example, any unfixed floating-point ambiguities  116  in the LAMBDA domain may be equivalent to floating-point ambiguities z, and therefore corresponding elements of the vector ({circumflex over (z)}−ž) may be zero.
 
     The system  100  may additionally determine an error covariance Q {circumflex over (b)}  of the partial almost-fix navigation solution  124  approximating the fixed-ambiguity baseline navigation solution {hacek over (b)} ( 120 ). For example, if M is a matrix defining the decorrelated floating-point ambiguities z ( 116 ) being fixed in the LAMBDA domain (resulting in LAMBDA-domain fixed integer ambiguities  118 ) for the partial almost-fix navigation solution  124 , M may be a diagonal matrix corresponding to the partial almost-fix navigation solution such that 
               M   =     [           m   11         0       0       0           0         m   22         0       0           0       0       …       0           0       0       0         m   nn           ]       ;                 m   ii     =     {             1   ⁢     (   fixed   )       ,             i   =   1     ,   …   ,     n   f                   0   ⁢     (float/unfixed)       ,             i   =       n   f     +   1       ,   …   ,   n                   
where n f  may be the total number of LAMBDA-domain fixed ambiguities  118  with (n-n f ) ambiguities remaining floating-point. A fully fixed navigation solution (i.e., a fixed-ambiguity navigation solution  120 ) may be indicated where M is set to the identity matrix, while a floating-point baseline navigation solution  114  may be indicated by a zero matrix. The partial update equation (6) may become
 
                       b   ⋓     =       b   ^     -       Q       b   ^     ⁢     z   ^         ⁢       Q     z   ^         -   1       ⁢     M   ⁡     (       z   ^     -     z   ⋓       )             ,           (   7   )               
and the error covariance
 
                     Q     b   ⋓       =       Q     b   ^       -         Q       b   ^     ⁢     z   ^         ⁡     (         MQ     z   ^         -   1       +       Q     z   ^         -   1       +   M   -         Q     z   ^         -   1       ⁢     MQ     z   ^       ⁢       MQ     z   ^         -   1           )       ⁢     Q       z   ^     ⁢     b   ^                     (   8   )               
of the partial almost-fix navigation solution  124  may be useful in error covariance analysis.
 
     Generally, the system  100  may improve the PCF and the reliability of the fixing process through generation of a partial almost-fix navigation solution  124 , e.g., by partial fixing of floating-point ambiguities  116  in the LAMBDA domain as described above. However, to ensure sufficiently high system integrity for incorporation in JPALS equipment  100   a  ( FIG.  1 A ), partial fixing in the LAMBDA domain may be combined with other techniques. For example, Geometry Extra Redundant Almost Fix Solution (GERAFS) functionality of the JPALS  100   a  meets integrity requirements via the Probability of Almost Fix (PAF). The system  100   a  may improve on GERAFS functionality by fixing decorrelated floating-point ambiguities  116  in the LAMBDA domain rather than transforming the floating-point ambiguities back to their original domain for fixing. However, GERAFS may define the PAF as the probability that a partial almost-fix navigation solution  124  is bounded by protection levels defined or predetermined by the JPALS equipment  100   a , which bounds errors in the partial almost-fix navigation solution  124  to the desired confidence level. 
     Similarly, if a fully fixed solution, i.e., a fixed-ambiguity baseline navigation solution  120 , cannot be achieved without protection levels exceeding alert limits, the system  100  may generate the partial almost-fix solution  124  by fixing only a partial list  118   a  of floating-point ambiguities  116  to generate fixed integer ambiguities  118 . The system  100  may compute an integrity bound (e.g., protection level) based on the GERAFS PAF, or the probability that the floating-point baseline navigation solution  114  is derived from floating-point ambiguities  112  within either Pull-in Regions (PIR) containing correctly fixed integer ambiguities ( 118 ) or a selected set of Enlarged Pull-in Regions (EPIR) containing incorrectly fixed floating-point ambiguities ( 116 ) with errors of no more than a single cycle or a small-multiple cycle. The system  100  may determine the probability that the partial almost-fix navigation solution is within the computed integrity bound based on a partial list of fixed integer ambiguities  118 . The EPIR may provide additional system integrity at the expense of large protection levels even when only single-cycle errors are included. 
     Broadly speaking, the system  100  may provide (e.g., via airborne integrity processing) protection levels to bound errors in the partial almost-fix navigation solution  124  and/or the fixed-ambiguity baseline navigation solution {hacek over (b)} ( 120 ), the protection levels based on data and estimates from airborne and/or ship system components aboard the ship  104  as well as the aircraft  102 . Protection level components may vary depending on the operational mode of the system  100  (e.g., autonomous navigation/surveillance (airborne-state only); precision approach/surveillance (relative states determined by precision relative navigation (RelNav) processing and incorporating the most complex array of error sources as described below, as well as the most stringent integrity levels; ship relative navigation (e.g., in the absence of precision RelNav)). Broadly speaking, protection levels may include vertical and lateral components (vertical protection level (VPL), lateral protection level (LPL)) respectively bounding vertical and lateral position errors. 
     For example, ship system data and estimates may pertain to error/confidence levels of predicted ship states and/or geometric relations between various shipboard and aircraft locations, e.g., local-level (LL) positions of GPS antennas and/or inertial measurement units (IMU) aboard the ship  104  and aircraft  102 ; ship reference points (SRP); stabilized touchdown points (TDP); aircraft guidance control points (GCP); baseline vectors between shipboard elements in the ship reference frame (or from the SRP to the stabilized TDP in the LL frame); relative position solutions from the SRP and stabilized TRP to the aircraft GCP in the LL frame; and errors associated with the extrapolation of prior relative position solutions according to ship and aircraft motion, due to the need for solutions based on latent measurements, e.g., wherein airborne position measurements may be stored until synchronized ship data arrives via uplink. Error components may broadly include differential GPS (DGPS) errors; ship antenna-to-TDP survey errors; ship/aircraft lever arm errors; relative velocity errors; ship/aircraft acceleration/propagation errors; and path stabilization errors. 
     With respect to the system  100 , protection levels for bounding error in a user relative state may be reduced to contributions from multiple error sources, e.g., double-differenced GPS, ship/TDP state. airborne state, airborne lever arm, and flight path guidance, represented in an approach reference frame. For example, with respect to GERAFS protection level calculation, GPS error may be based in a minimum probability that the set of fixed integer ambiguities ( 118 ,  118   a ) corresponding to the fixed-ambiguity solution  120  or the partial almost-fix solution  124  is either in the conventional PIR (e.g., minimum Probability of Correct Fix, PCF min ) or in the EPIR (e.g., minimum Probability of Almost Fix, PAF min ) relative to the floating-point solution  114 . For example, with respect to the HO unfaulted case, the PCF min  and PAF min  thresholds may have nominal values set by the system (e.g., 0.998 and 1-4e −7  respectively). 
     With respect to the floating-point solution  114 , error sources may be assumed from non-faulted measurement sources and therefore characterized by single-sigma terms in each dimension, e.g., σ DGPS_Float , σ Rel_State_Prop , σ Ship_LevArm_Att , σ Ship_Survey , σ Air_LevArm_Att , σ Air_Survey , σ Ship_State_Prop . Some of these terms may be common for protection-level calculation purposes across float and fixed solutions and may be consolidated into a common term σ common   2 . Additional combined terms, e.g., σ′ Float , may be derived from the covariance matrices of the floating-point solution set. 
     Still with respect to the floating-point solution  114 , protection level calculation may follow from the sigma levels of error sources described above. For example, 1-sigma bounds σ vert_Float  and σ lat_Float  may be based on these common and combined terms with respect to the vertical and lateral differenced line-of-sight unit vectors respectively. Vertical and lateral protection levels may then be calculated: 
               VPL   Float     =       K     ffmd   ⁢   _   ⁢   float       ⁢     σ     vert   ⁢   _   ⁢   Float                       LPL   Float     =       K     ffmd   ⁢   _   ⁢   float       ⁢     σ     lat   ⁢   _   ⁢   Float               
where, e.g., the K-factor
 
             K   =       2     ⁢       erf     -   1       ⁡     (     1   -     P   bound       )               
may be based on inverting an error function erf (x) describing the double-sided tails of a Gaussian distribution, and where P bound  is the probability with which the protection level must bound navigation errors. K-factors generally may be derived from design parameters, e.g., an allocated Probability of Not Almost Fix (PNAF; e.g., 4e −7 ), PCF min , and the allocation of integrity risk between PIR and EPIR (hence, K ffmd_float , K ffmd_PIR , K ffmd_EPIR ).
 
     With respect to the fixed-ambiguity solution  120  (based on whether the current set of fixed ambiguities  118  is in the PIR or (rarely) the EPIR), protection level calculation is similar: 
     
       
         
           
             
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               LPL 
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                   ⁢ 
                   EPIR 
                 
               
             
           
         
       
     
     (Lateral protection levels LPL Fixed ,LPL EPIR  may be calculated along similar lines.) Protection levels with respect to the EPIR (e.g., VPL EPIR ,LPL EPIR ) may incorporate wrong-fix biases associated with the EPIRs and added to the 1-sigma contributions described above. With respect to the partial almost-fix solution  124  (and each corresponding set of fixed ambiguities  118   a  as n f  is decremented, as described in greater detail below), vertical and lateral protection levels may be calculated as a maximum of PIR and EPIR-associated terms: 
     
       
         
           
             
               VPL 
               AFS 
             
             = 
             
               max 
               ⁡ 
               
                 ( 
                 
                   
                     VPL 
                     Fixed 
                   
                   , 
                   
                     VPL 
                     EPIR 
                   
                 
                 ) 
               
             
           
         
       
       
         
           
             
               LPL 
               AFS 
             
             = 
             
               max 
               ⁡ 
               
                 ( 
                 
                   
                     LPL 
                     Fixed 
                   
                   , 
                   
                     LPL 
                     EPIR 
                   
                 
                 ) 
               
             
           
         
       
     
     Calculated protection level values (e.g., expressed in terms of meters) may be compared against alert limits. For example, alert limits may indicate the amount of error with respect to lateral or vertical deviation (e.g., also in meters) that would be considered hazardously misleading information (HMI) for a particular application, e.g., a lateral alert limit (LAL) defined as a limit on lateral deviations from a flight path and a vertical alert limit (VAL) defined as a limit on lateral deviations from a flight path. Alert limits may be dependent on a particular application or operating mode; for example, precision surveillance accuracy requirements are less stringent than precision approach accuracy requirements, so precision surveillance alert limits may be greater than precision approach alert limits. If either a vertical or lateral protection level is found to meet or exceed a corresponding vertical or lateral alert limit, the corresponding fixed-ambiguity solution  120  or partial almost-fix solution  124  may be flagged as invalid. 
     Referring to  FIG.  2 A , an implementation of the system  100  of  FIG.  1 B  may generate a floating-point baseline navigation solution  114  including a set of n floating-point carrier-phase ambiguities  112 . The system  100  may decorrelate the set of n floating-point carrier-phase ambiguities  112  in the LAMBDA domain to generate a set of n decorrelated floating-point ambiguities  116  corresponding to the set of n floating-point carrier-phase ambiguities  112 . The system may select a subset  122  of n f  decorrelated floating-point ambiguities to be fixed (n f ≤n). The system may determine protection levels and PAF; while protection levels remain below alert limits and PAF remains above a threshold (e.g., a threshold predetermined or preset by the system  100  or the JPALS equipment  100   a ), the system  100  may generate a set  126  of LAMBDA-domain fixed ambiguities  118  by fixing the subset  122  of n f  decorrelated floating-point ambiguities  116  in the LAMBDA domain. The system may generate a partial almost-fix navigation solution  124  by replacing each floating-point carrier-phase ambiguity  112  of the floating-point baseline navigation solution  114  with the corresponding LAMBDA-domain fixed ambiguity  118  (i.e., the LAMBDA-domain fixed ambiguity corresponding to the decorrelated floating-point ambiguity  116  based on the particular floating-point carrier-phase ambiguity). In the case where the system  100  selects a subset  122  wherein n f =n, and protection levels remain below alert limits while PAF remains above the threshold, the result may be a full fixed-ambiguity navigation solution  120  wherein every floating-point carrier-phase ambiguity  112  is replaced by a LAMBDA-domain fixed ambiguity  118 . The system may output the full fixed-ambiguity navigation solution  120  when this solution is available. If protection levels rise above alert limits or PAF drops below the threshold for a partial almost-fix navigation solution  124  including a subset  122  of size n f , the system  100  may continue the iterative process by decrementing of to (n f −1) by removing a decorrelated floating-point ambiguity  116  from the subset  122 , resulting in a subset  122   a  including (n f −1) decorrelated floating-point ambiguities  116 . The system  100  may generate a set  126   a  of LAMBDA-domain fixed ambiguities  118  by fixing the subset  122   a  of (n f −1) decorrelated floating-point ambiguities  116  in the LAMBDA domain, and generate a partial almost-fix navigation solution  124   a  by replacing the corresponding floating-point carrier-phase ambiguities  112  of the floating-point baseline navigation solution  114  with the (n f −1) LAMBDA-domain fixed ambiguities  118  of the subset  122   a , and output the partial almost-fix navigation solution  124   a  if protection levels remain below alert limits and the PAF remains above the threshold. If the system  100  continues to decrement n f  such that n f  drops to zero, the system may output the original floating-point baseline navigation solution  114 . 
     Referring to  FIG.  2 B , an alternative implementation of the system  100  of  FIG.  1 B  may generate a partial almost-fix navigation solution  124  by fixing floating-point ambiguities  116  in the LAMBDA domain individually, or sequentially, until protection levels reach the alert limits. For example, the system  100  may generate a floating-point baseline navigation solution  114  including an ordered set of corresponding LAMBDA-domain decorrelated floating-point ambiguities  116  based on each floating-point carrier-phase ambiguity  112  of the floating-point baseline navigation solution as in  FIG.  2 A  above. The system  100  may order the LAMBDA-domain decorrelated floating-point ambiguities  116  of the floating-point baseline navigation solution  114 , e.g., in descending order of PCF, the first decorrelated floating-point ambiguity  116   a  to be fixed having the highest PCF, the second decorrelated floating-point ambiguity  116   b  having the next highest PCF, etc. The system  100  may select the first LAMBDA-domain decorrelated floating-point ambiguity  116   a  to be fixed and add the selected first decorrelated floating-point ambiguity to a subset of n f  decorrelated floating-point ambiguities (wherein the value of n f  is initially set to zero). After protection levels and PAF are determined by the system  100 , the system may fix the first selected decorrelated floating-point ambiguity  116   a  in the LAMBDA domain to the value of the corresponding LAMBDA-domain fixed ambiguity  118   b . The system may generate a partial almost-fix navigation solution  124  by updating the first floating-point carrier-phase ambiguity  112   a  of the floating-point baseline navigation solution  114  (and corresponding to the selected first decorrelated floating-point ambiguity  116   a ) with the generated LAMBDA-domain fixed ambiguity  118   b.    
     The system  100  may iterate through successive partial almost-fix navigation solutions  124   b - d  while protection levels remain below alert limits and PAF remains above the threshold level by selecting successive decorrelated floating-point ambiguities to fix (e.g., the decorrelated floating-point ambiguity  116  having the highest remaining PCF), one at a time, until alert limits are exceeded or PAF drops below the threshold. For example, the system  100  may continue the iterative process by selecting the second decorrelated floating-point ambiguity  116   b  (having the highest remaining PCF) to be fixed. The system  100  may fix the second decorrelated floating-point ambiguity  116   b  in the LAMBDA domain (to the value of the LAMBDA-domain fixed ambiguity  118   c ) and generate an updated partial almost-fix navigation solution  124   b  wherein the second floating-point carrier phase ambiguity  112   b  of the floating-point baseline navigation solution  114  (and the prior partial almost-fix navigation solution  124 ) is replaced with the second LAMBDA-domain fixed ambiguity  118   c  derived from the second decorrelated floating-point ambiguity  116   b  corresponding to the second floating-point carrier phase ambiguity  112   b . As each successive floating-point ambiguity  116  is fixed to the value of a fixed integer ambiguity  118 , the PCF may be lowered and more EPIR may be included to meet the PAF requirement. If the protection levels associated with a partial almost-fix navigation solution  124   c  generated by the system  100  exceed the alert limits (or if the PAF drops below the threshold), the system  100  may remove the most recently added LAMBDA-domain fixed ambiguity  118   c  in favor of the floating-point carrier phase ambiguity  112   c  it replaced, reverting to the immediately prior partial almost-fix navigation solution ( 124   d ) wherein protection levels remained below alert limits and PAF remained above the threshold value. For example, to achieve the partial almost-fix navigation solution  124   c , the system  100  may have most recently replaced the n f -th floating-point carrier-phase ambiguity  112   c  (of the set  114  of n floating-point carrier-phase ambiguities) with the LAMBDA-domain fixed ambiguity  118   d  derived from fixing the n f -th decorrelated floating-point ambiguity  116   c  corresponding to the n f -th floating-point carrier-phase ambiguity  112   c . The system  100  may replace the most recently updated LAMBDA-domain fixed ambiguity  118   d  with the n f -th floating-point carrier-phase ambiguity  112   c  to restore and output the immediately prior partial almost-fix navigation solution  124   d . As the result of this process, the system  100  may output the original floating-point baseline navigation solution  114  (if n f =0) or a full fixed-ambiguity navigation solution  120  (if n f =n), as shown by  FIG.  2 A . 
     Referring now to  FIG.  3 A , an exemplary embodiment of a method  200  for determining a precision navigation solution according to the inventive concepts disclosed herein may include one or more of the following steps. At a step  202 , the system  100  may receive GPS information from at least two GPS sources, the GPS sources corresponding to one or more common GPS satellites  106 . For example, the system  100  may be embodied in aircraft-based JPALS equipment  100   a  which receives GPS information directly from common GPS satellites  106  via a GPS receiver  112  onboard the aircraft  102  in which the system  100  is embodied as well as GPS information  108 ,  108   a  from a ship  104  or other vehicle proximal to the aircraft  102 . The system  100  may receive GPS carrier waves associated with GPS signals. 
     At a step  204 , the system  100  generates a floating-point baseline navigation solution  114  including a set of n floating-point carrier-phase ambiguities  112  based on the received GPS information. The floating-point baseline navigation solution  114  and set of n floating-point carrier-phase ambiguities  112  may be associated with the original domain (e.g., carrier-phase measurement ambiguity domain). 
     At a step  206 , the system  100  generates a set of n Least-squares AMBiguity Decorrelation Adjustment (LAMBDA)-domain floating-point ambiguities  116  corresponding to the set of n floating-point carrier phase ambiguities  112  by decorrelating the set of n floating-point carrier-phase ambiguities via LAMBDA algorithms. The system  100  may generate error covariances associated with the LAMBDA-domain floating-point ambiguities  116 . 
     At a step  208 , the system  100  defines a subset  122  of the set of n decorrelated LAMBDA-domain floating-point ambiguities  116 , the subset  122  including n f  decorrelated LAMBDA-domain floating-point ambiguities to be fixed in the LAMBDA domain. 
     At a step  210 , the system  100  determines protection levels associated with a partial almost-fix navigation solution  124  based on the probability that the partial almost-fix navigation solution  124  is derived from within a PIR of correctly fixed LAMBDA-domain fixed integer ambiguities  118  (e.g., having zero errors) or an EPIR of incorrect (e.g., unfixed, incorrectly fixed) LAMBDA-domain floating-point integer ambiguities  116  wherein the errors associated with the LAMBDA-domain floating-point integer ambiguities are no more than a single cycle (or, in the alternative, multiple cycles). The EPIR may be narrowed to incorrectly fixed LAMBDA-domain floating-point integer ambiguities  116  involving errors of a single cycle. 
     At a step  212 , the system  100  determines a PAF based on the probability that the partial almost-fix navigation solution  124  is bounded by the determined protection levels. 
     Referring to  FIG.  3 B , at a step  214 , the system  100  determines whether the PAF is higher than a predetermined threshold and the determined protection levels are under predetermined alert limits. 
     While the PAF is above the threshold value and protection levels are under alert limits, at a step  216 , the system  100  generates a set  126  of n f  LAMBDA-domain fixed integer ambiguities  118  by fixing the subset  122  of n f  decorrelated LAMBDA-domain floating-point ambiguities  116  in the LAMBDA domain. 
     At a step  218 , the system  100  generates the partial almost-fix navigation solution  124  by updating the n f  floating-point carrier-phase ambiguities  112  (of the floating-point baseline navigation solution  114 ) corresponding to the subset  122  of n f  decorrelated LAMBDA-domain floating-point ambiguities  116  with the set  126  of n f  LAMBDA-domain fixed integer ambiguities  118 . 
     At a step  220 , the system  100  outputs the updated partial almost-fix navigation solution  124 . The output solution may be a full fixed ambiguity solution  120 . 
     If either the PAF drops below the threshold or the protection levels exceed the alert limits, at a step  222  the system  100  decrements n f  by removing a decorrelated LAMBDA-domain floating point ambiguity  116  from the subset  122  of n f  decorrelated LAMBDA-domain floating-point ambiguities, thereby setting n f  to n f −1. At a step  224 , the system  100  determines whether of is zero (e.g., the subset  122  of n f  decorrelated LAMBDA-domain floating-point ambiguities  116  is empty). If of is non-zero, the method  200  resumes at step  214  with the decremented n f . If of is zero, at a step  226  the system  100  outputs the floating-point baseline navigation solution  114 . 
     Referring now to  FIG.  4 A , an exemplary embodiment of a method  300  for determining a precision navigation solution according to the inventive concepts disclosed herein may be implemented similarly to the method  200  of  FIGS.  3 A- 3 B , in that the steps  302  and  304  are implemented similarly to the steps  202  and  204  of the method  200 . 
     At a step  306 , the system  100  generates a set of n LAMBDA-domain floating-point ambiguities  116  corresponding to the set of n floating-point carrier phase ambiguities  112  by decorrelating the set of n floating-point carrier-phase ambiguities via LAMBDA algorithms. The system  100  may generate error covariances associated with the decorrelated LAMBDA-domain floating-point ambiguities  116 . The system  100  may order then decorrelated floating-point ambiguities  116 , e.g., by descending PCF (the first decorrelated floating-point ambiguity  116   a  having the highest PCF, the second decorrelated floating-point ambiguity  116   b  having the next highest PCF, and so forth). 
     At a step  308 , the system  100  selects from the set of n decorrelated LAMBDA-domain floating-point ambiguities  116  a single decorrelated LAMBDA-domain floating-point ambiguity  116   a  to be fixed. The system  100  may select the first decorrelated LAMBDA-domain floating-point ambiguity  116   a  having the highest PCF. 
     At a step  310 , the system  100  adds the selected decorrelated LAMBDA-domain floating-point ambiguity  116   a  to a subset  122  of n f  decorrelated LAMBDA-domain floating-point ambiguities. 
     Referring to  FIG.  4 B , at a step  312 , the system  100  determines protection levels associated with a partial almost-fix navigation solution  124  based on the probability that the partial almost-fix navigation solution  124  is derived from within a PIR of correctly fixed LAMBDA-domain fixed integer ambiguities  118  (e.g., having zero errors) or an EPIR of incorrect (e.g., unfixed, incorrectly fixed) LAMBDA-domain floating-point integer ambiguities  116  wherein the errors associated with the LAMBDA-domain floating-point integer ambiguities  116  are no more than a single cycle (or, in the alternative, multiple cycles). The EPIR may be narrowed to incorrectly fixed LAMBDA-domain floating-point integer ambiguities  116  involving errors of a single cycle or less. 
     At a step  314 , the system  100  determines a PAF based on the probability that the partial almost-fix navigation solution  124  is bounded by the determined protection levels. 
     At a step  316 , the system  100  generates a LAMBDA-domain fixed ambiguity  118   b  by fixing the selected decorrelated LAMBDA-domain floating-point ambiguity  116   a  in the LAMBDA domain. 
     At a step  318 , the system  100  generates the partial almost-fix navigation solution  124  by updating the floating-point carrier-phase ambiguity  112   a  of the floating-point baseline navigation solution  114  and corresponding to the selected decorrelated LAMBDA-domain floating-point ambiguity  116   a  with the generated LAMBDA-domain fixed ambiguity  118   b.    
     Referring to  FIG.  4 C , at a step  320 , the system  100  determines whether the PAF is higher than a predetermined threshold and the determined protection levels are under predetermined alert limits. 
     While the PAF is above the threshold value and protection levels are under alert limits, at a step  322 , the system  100  increments n f  by selecting an additional decorrelated LAMBDA-domain floating-point ambiguity  116   b  to be fixed. For example, the system  100  may select the first unfixed decorrelated LAMBDA-domain floating-point ambiguity  116   b  (e.g., the unfixed decorrelated LAMBDA-domain floating-point ambiguity having the highest remaining PCF, or the (n f +1) th  decorrelated LAMBDA-domain floating-point ambiguity). 
     At a step  324 , the system  100  generates an additional LAMBDA-domain fixed ambiguity  118   c  by fixing the selected decorrelated LAMBDA-domain floating-point ambiguity  116   b  in the LAMBDA domain. 
     At a step  326 , the system  100  updates the partial almost-fix navigation solution ( 124   b ) by updating the floating-point carrier-phase ambiguity  112   b  of the partial almost-fix navigation solution and corresponding to the selected decorrelated LAMBDA-domain floating-point ambiguity  116   b  with the generated LAMBDA-domain fixed ambiguity  118   c  based on the selected decorrelated LAMBDA-domain floating-point ambiguity  116   b.    
     If either the PAF drops below the threshold or the protection levels reach the alert limits, at a step  328  the system  100  updates the most recently added LAMBDA-domain fixed ambiguity  118   d  of the partial almost-fix navigation solution  124   c  with the corresponding floating-point carrier-phase ambiguity  112   b  that the most recently added fixed ambiguity  118   d  replaced. 
     At a step  330 , the system  100  outputs the updated partial almost-fix navigation solution  124   d . The output solution may be a full fixed ambiguity solution  120  (e.g., if n f =n); the output solution may be the original floating-point baseline navigation solution  114  (e.g., if n f =0). 
     As will be appreciated from the above, systems and methods according to embodiments of the inventive concepts disclosed herein may simplify the partial fixing process. As both the fixing of decorrelated floating-point ambiguities  116  and the corresponding update of the floating-point baseline navigation solution  114  with fixed ambiguities  118  are accomplished in the LAMBDA domain, the partial list of fixed integer ambiguities used to update the floating-point baseline navigation solution is straightforward. Similarly, as only the partial list of fixed integer ambiguities  118  is used to compute the PAF of the partial almost-fix navigation solution  124 , high system integrity can be preserved within tighter protection levels compared to conventional GERAFS functionality. 
     It is to be understood that embodiments of the methods according to the inventive concepts disclosed herein may include one or more of the steps described herein. Further, such steps may be carried out in any desired order and two or more of the steps may be carried out simultaneously with one another. Two or more of the steps disclosed herein may be combined in a single step, and in some embodiments, one or more of the steps may be carried out as two or more sub-steps. Further, other steps or sub-steps may be carried in addition to, or as substitutes to one or more of the steps disclosed herein. 
     From the above description, it is clear that the inventive concepts disclosed herein are well adapted to carry out the objects and to attain the advantages mentioned herein as well as those inherent in the inventive concepts disclosed herein. While presently preferred embodiments of the inventive concepts disclosed herein have been described for purposes of this disclosure, it will be understood that numerous changes may be made which will readily suggest themselves to those skilled in the art and which are accomplished within the broad scope and coverage of the inventive concepts disclosed and claimed herein.