Abstract:
The accuracy of relative position estimates between two vehicles, generated using a satellite-based navigation system, is improved by eliminating at an early stage in the calculation process, errors affecting simultaneously both receivers. The standard absolute navigation equations are modified to solve directly for relative position, while avoiding a final step differentiation, thereby providing increased precision.

Description:
CROSS-REFERENCES TO RELATED APPLICATIONS 
     Not applicable. 
     STATEMENT REGARDING FEDERALLY-SPONSORED RESEARCH OR DEVELOPMENT 
     Not applicable. 
     REFERENCE TO A MICROFICHE APPENDIX 
     Not applicable. 
     BACKGROUND OF THE INVENTION 
     This invention relates generally to the field of navigation systems which use a constellation of earth-orbiting satellites to determine the relative position between two users located on or near the earth&#39;s surface. More specifically, the invention depicts a method and apparatus for improving the accuracy of relative position estimates in such a satellite-based navigation system. It also relates to the field of collision avoidance as the knowledge of such accuracy is a key factor in preventing mid-air and ground collisions. Several satellite-based navigation systems are currently used by the worldwide community. For ease of discussion, the following disclosure will specifically focus on the NAVSTAR Global Positioning System (GPS) operated and maintained by the Department of Defense. This system, which is used for navigation, position determination and time-transfer applications, consists of a 24-satellite constellation. Each of them radiates precisely timed signals coded so that a receiver on or near the earth&#39;s surface can determine both the transmission time delay (or equivalently, distance) from the satellite to the receiver and the precise satellite position. By simultaneously receiving such signals from at least four satellites, the receiver can determine its position and time. 
     However, several sources or errors adversely affect the accuracy of such determination. These errors are primarily due to: 
     satellite ephemeris uncertainties (lack of precision in the satellite position on its pre-determined orbit) 
     selective availability (deliberate degradation of the signals by the Department of Defense for non military users) 
     ionospheric and tropospheric propagation delays 
     satellite and receiver clock drifts 
     multipath (multiple reflexion and scattering of a signal) 
     receiver noise. 
     Although the differentiation of two nearby receivers&#39; absolute positions (i.e., calculated independently one from the other) enables one to partly eliminate some of these errors, this type of post-processing, abundantly shown in the prior art, is by essence unable to attain the accuracy of a direct relative positioning method, cancelling out at the source errors common to both receivers. Accordingly, what is needed is a new method solving directly for the relative position of two users so that the accuracy of such position is improved. 
     BRIEF SUMMARY OF THE INVENTION 
     The invention is an apparatus and method for use with a satellite-based navigation system. Its purpose is to improve the accuracy of relative position estimates by accounting for errors common to both receivers. In an embodiment featuring the Global Positioning System (GPS), all satellites in view by both receivers are used to calculate the relative position vector. The fundamental GPS equation system is modified in such a fashion so as to solve directly for the relative position of the two receivers in an earth-centered, earth-fixed (ECEF) system of coordinates. 
    
    
     BRIEF DESCRIPTION OF THE DRAWINGS 
     FIG. 1 is a diagram depicting a typical geometry featuring two nearby aircraft separated by an unknown distance and the four satellites that both of them have in view. 
     FIG. 2 is a block diagram describing the operations that need to be performed during the post-processing in order to calculate the relative position of said two aircraft. 
     FIG. 3 is a graph illustrating relative versus absolute navigation performance. 
    
    
     DETAILED DESCRIPTION OF THE INVENTION 
     The present invention is now described with reference to the figures where like reference numbers denote like elements/steps. 
     In the preferred embodiment, the NAVSTAR Global Positioning System is used. As previously mentioned, the system includes twenty-one operational and three spare satellites which orbit the earth in six orbits. The invention is described in the environment of two proximate aircraft 150 and 170 separated by an unknown distance 160, as shown in FIG. 1. A representative GPS constellation includes four GPS satellites 110, 120, 130 and 140 for transmitting GPS data. This configuration was adopted for the sake of clarity. In reality, both aircraft can have &#34;in view&#34;, i.e., receive signals from up to nine satellites. Communication link channel 180 represents the communications link between aircraft 150 and 170. In the preferred embodiment, communication channel 180 has an Automatic Dependent Surveillance-Broadcast (ADS-B) capability. ADS-B has been described in numerous instances in the prior art. Communication channel 180 is used to transfer data, specifically position and time between aircraft 150 and aircraft 170. The GPS constellation comprising satellites 110, 120, 130 and 140 may optionally include one or more pseudolites. A pseudolite or &#34;pseudo-satellite&#34; is a transmitting system located on the earth&#39;s surface which emulates a GPS satellite. Because a pseudolite has a fixed, known position, it can greatly enhance the relative position estimates derived from GPS. For ease of discussion herein, only GPS satellites will be referenced. It should be understood however, that where position data from a satellite is required, pseudolite data may be substituted. 
     The equipment used in the preferred embodiment to record GPS data includes two 12 parallel channel receivers (GPS 25 XL), mounted on aircraft 150 and 170, as well as two antenna kits (GA 27A) available from Garmin, each of which is linked to one of the receivers. 
     GPS signals, broadcast every second, contain ephemeris data, i.e. several sets of parameters characterizing each of the predetermined orbits satellites are constrained to move on, GPS time by means of which precise satellite positions can be determined. By noting the time at which signals are received, GPS receivers can compute the propagation time delay or equivalently (by multiplying by the speed of light) the estimated distances between satellites and receivers also denoted as raw pseudoranges. This terminology reflects the fact that those measurements are only an approximation of the true distances, the integrity of which is corrupted by different sources of errors as described in the section devoted to the background of this invention. 
     Ephemeris data, GPS time and raw (i.e. uncorrected) pseudoranges are the key elements to determine relative position 160 between aircraft 150 and 170. The post-processing operations on which this invention is founded are illustrated in a flow chart shown in FIG. 2. The first step consists in calculating the satellite position in the ECEF (earth-centered, earth-fixed) reference frame. A preexisting algorithm can be found in the second edition of the Global Positioning System Standard Positioning Service Signal Specification dated June 1995. This algorithm however superfluously introduces the functions cos -1  and tan -1 , the use of which may be critical for values at the boundary of the definition domain. A modified version of this algorithm represented by step 210 overcomes this issue by using only the sine and the cosine of both the true anomaly, ν k , and argument of latitude, φ k  defined as follows: ##EQU1## where k ε{1, . . . ,6}, e denotes the eccentricity of the elliptical orbits, E k  and ω, the eccentric anomaly and argument of perigee respectively. By further using the relations: ##EQU2## the second harmonic perturbations (corrections made to orbit parameters such as the argument of latitude, radius and inclination) are thus computed while avoiding any problem of definition in the mathematical sense. 
     At step 220, the satellite clock correction function is performed. At step 230, an external position device such as an inertial reference unit is used to calculate an initial estimate of the unit vectors (which characterize the direction) between aircraft and satellites. The matrices featured at step 240 represent the quintessence of this invention. Their calculation is presented in what follows. 
     The true distance between aircraft 150 and 170 and the satellites they have in view can be expressed as: 
     
         A.sub.1 S.sub.1,i =ρ.sub.1,i -e.sub.1 -e.sub.1,i       (1) 
    
     
         A.sub.2 S.sub.2,j =ρ.sub.2,j -e.sub.2 -e.sub.2,j       (2) 
    
     In the preferred embodiment i and j were chosen to vary between 1 and 4. It must be understood however that the present study remains valid in the general case when i, resp. j, varies between 1 and n, resp. p, n and p being the total number of satellites each of them has in view. In Eq. (1) and (2), A 1  denotes the position of aircraft 150, S 1 ,i, the position of the i th  satellite it has in view; similarly, A 2  denotes the position of aircraft 170, S 2 ,j, the position of the j th  satellite it has in view. ρ 1 ,i and ρ 2 ,j are pseudoranges reported from the satellites to each aircraft. Lastly, e 1  and e 2  represent the range equivalents of each aircraft clock offsets; e 1 ,i and e 2 ,j, the range equivalents of the satellite clock offsets. 
     Position vectors OA 1   between earth&#39;s center O and aircraft 150 and OA 2   between earth&#39;s center O and aircraft 170 can be decomposed as: ##EQU3## By further introducing unit vectors: ##EQU4## calculated at step 230, the following equations are obtained: ##EQU5## 
     Vectorial Representation 
     The purpose of the next derivation is to express unknown distance 160, noted as A 1  A 2 , as a function of known (i.e., satellite related) parameters. ##EQU6## 
     Using equations (1) to (4), this equation is rewritten as: ##EQU7## 
     By further decomposing OA 2   and OA 1   through S 2  and S 1  respectively, the following equation is obtained: ##EQU8## 
     Separating unknown (i.e., user related) and known variables finally leads to: ##EQU9## 
     The left hand side of this equation features users&#39; relative position and its associated error; the right hand side, observable data. 
     Matrix Representation 
     Let X be a four-dimensioned vector, the first three components of which represent A 1  A 2   coordinates and the fourth, the error component e=e 1  -e 2 . ##EQU10## When i and j vary, the preceding set of equations, exhibited in a generic fashion in Eq. (5), can be expressed as WX=O, where W is seen as a weighted coefficients matrix and O, as an observed parameters vector. W is a np×4 matrix and O, a np×1 matrix, n being the number of satellites in view from aircraft 150 and p, the number of satellites in view from aircraft 170. ##EQU11## Generic elements of arrays W and O, w i ,j and o i ,j, are defined as follows: ##EQU12## 
     In step 250, the system is solved using the pseudo-inverse matrix Ψ=( t  W W) -1  t W. 
     
         X=ΨO                                                   (8) 
    
     This solution represents a least squares approximation of the linear system under consideration. It should be clear nevertheless that other solving algorithms, for instance a total least squares method, could be used instead without departing from what step 250 is aiming at. 
     In FIG. 3, the performance of relative versus absolute navigation techniques is illustrated. Histogram 310 represents a distribution of distances between aircraft 150 and 170 which were computed through standard absolute positioning techniques involving a final step differentiation. Best-fitting gaussian curve 320 is determined using the mean (i.e. average of all distances) and standard deviation (characterizing the dispersion of a distribution) of histogram 310. Similarly, histogram 330 represents a distribution of distances between aircraft 150 and 170 which were computed through relative positioning techniques and curve 340, its associated best-fitting gaussian curve. In this example, an improvement of more than 50% was achieved. ##EQU13## 
     Yet another application of this invention resides in the calculation of differentially corrected relative positions, as is the case within local and wide area augmentation systems (LAAS and WAAS) as well as their European counterpart EGNOS (European Geostationary Navigation Overlay Service). 
     While the invention has been particularly shown and described with reference to a preferred embodiment thereof, it will be understood by those skilled in the art that various changes in form and details may be made therein without betraying the spirit and scope of the invention as defined in the appended claims.