High Accuracy Coordinate Conversion Method for Air Traffic Control Applications

DESCRIPTION OF THE FIGURES

These and other objects, features and advantages of the invention will be more fully appreciated with reference to the accompanying figures.

FIG. 1 is a diagram of the prior art technique for the transformation of air traffic control information from multiple radar observations into a common system plane.

FIG. 2 is a schematic diagram of the method of the invention.

FIG. 3 is a flow diagram of the method of the invention.

FIG. 4 is a system block diagram of the data processing system for carrying out the method of the invention.

FIG. 5 is a three-dimensional diagram illustrating the geometric aspects of the stereographic projection in accordance with the invention.

FIG. 6 illustrates a relationship between geodetic and geocentric latitude.

FIG. 7 illustrates the angular relationship between the target and the radar vectors.

FIG. 8 illustrates radar coordinate system rotation.

FIG. 9 illustrates spherical triangle identities.

FIG. 10 illustrates the deviation from the normal for the configuration of the cross-section of the Earth's ellipsoid and the target.

FIG. 11 illustrates the radial distance to the ellipsoid surface for a geocentric latitude.

DISCUSSION OF THE PREFERRED EMBODIMENT

As previously mentioned, the shape of the Earth is that of an oblate ellipsoid. Thus, a plane tangent to any point of the surface of the Earth has a normal which passing through the point of tangency does not necessarily pass through the center of the Earth. Hence, although values of longitude for a position on the surface of the Earth do not vary in magnitude between a geocentric approximation of a spherical surface for the Earth and a geodetic longitude for an ellipsoidal Earth, the geocentric latitude for a spherical representation of the Earth does have a different value from the true geodetic latitude. The geodetic latitude is the angle which the normal to the ellipsoid at a station makes with the plane of the geodetic equator, in an ellipsoidal representation of the Earth. Geodetic longitude is the angle between the plane of the geodetic meridian at a station and the plane of the geodetic meridian at Greenwich, for an ellipsoidal representation of the Earth. The geocentric latitude is the angle at the center of the ellipsoid used to represent the Earth, between the plane of its Equator and a straight line radius vector to a point on the surface of the ellipsoid. This differs from the geodetic latitude because the Earth is an ellipsoid rather than a sphere, and the meridians are ellipses. Since the parallels of latitude are considered to be circles, the geodetic longitude is geocentric, and a separate expression is not required. The difference between the geocentric and the geodetic latitudes has a maximum of about 11.6 minutes at a latitude of 45.degree.. Because of the oblate ellipsoidal shape of the Earth, the length of a degree of geodetic latitude is not everywhere the same, increasing from about 59.7 nautical miles at the Equator, to about 60.3 nautical miles at the poles.

FIG. 2 is a schematic diagram of the method of the invention. Multiple radar observations of aircraft are converted to a single coordinate system in a new transformation technique. In the left-hand column labeled "Observations," a plurality of N radar stations sense aircraft in the air traffic control system. At the bottom of this column is an example of a radar station 21 observing aircraft A and a radar station 22 observing aircraft B. In the next column labeled "Nearly Exact Geometric Calculation," the first eleven steps of the method as shown in FIG. 3, are carried out. The method is carried out in a data processing system such as is shown in the system diagram of FIG. 4, wherein the system central processing unit CPU 50 performs the method steps depicted in FIG. 3. The CPU 50 is a data processor which executes a computer program embodying the flow diagram of FIG. 3. It can accept digital data streams from several radar stations A.sub.1, A.sub.2, etc., and can perform the method of FIG. 3 to yield the system plane coordinates x and y for many aircraft detected by the radar stations. The output values of x and y from the CPU 50 are then displayed on the flat, cathode ray tube (CRT) display 60 for viewing by the air traffic controller. The method of FIG. 3 has its first eleven steps transforming the radar measurements of the aircraft into worldwide geographic (geodetic) latitude and longitude coordinates in an iterative procedure making use of nearly exact geometric construction. A transformed system is indicated in the third column of FIG. 2, labeled "Earth Coordinates Latitude and Longitude." The method's output at this point is an improvement over the prior art transformation techniques, because the worldwide geographic coordinates for the aircraft are directly calculated for an ellipsoidal shaped Earth. The last two columns of FIG. 2 labeled "Exact Transformation" represent the remainder of the steps 12 and 13 of the method of FIG. 3. The final transformation is exact and converts the geographic latitude and longitude of the aircraft to coordinates on a common stereographic system plane 40' used by the air traffic controller. The location of the detached aircraft in the system plane 40' is displayed on the system plane display 60. Other functions such as the computation of collision paths, which use the coordinates of the detected aircraft in the common stereographic system plane 40', can be carried out in the CPU 50 or in an attached processor unit 70, as shown in FIG. 4. By using the computer method in accordance with the invention, more air traffic control information is produced and the quality and accuracy of the information is better than that which has been available in prior art systems.

In accordance with the invention, the transformation from radar observables of the target slant range, azimuth, and altitude, to target position coordinates in the stereographic display system plane, requires a coordinate conversion process followed by a stereographic projection process. The measured target azimuth is converted to geocentric azimuth (as if the radar's axis were cited along the geocentric vertical), solving for target geocentric latitude and longitude. This geocentric latitude and longitude are then converted to geodetic coordinates. Then the geodetic coordinates of the target position are transformed to the conformal sphere. Finally the target position from the conformal sphere is projected onto the stereographic display system plane. This iterative technique is necessary because the Earth centered radial distance to the target is initially unknown.

The method for performing the transformation in a data processing system, in accordance with the invention, is described in the flow diagram of FIG. 3. The details of each step in the flow diagram of FIG. 3, are given in the Table II, and an explanation of the terms in Table II is provided in Table I.

Two commonly used terms and their definitions are that of "radial distance," which is the distance from the center of the Earth to a point in space, and secondly, the term, "sub-", is a prefix which denotes the geodetic latitude of a point of interest.

The method in accordance with the invention takes the target data for detected aircraft, which is measured relative to a fixed radar station's location, and calculates the coordinates for the detected aircraft in a stereographic plane, which is a two-dimensional representation of the original three-dimensional configuration for the aircraft flying above the surface of the Earth. The method of the invention is executed in a computer system such as that shown in FIG. 4, as an iterative procedure. In performing the iterative processing, the iterative steps are steps 2 through step 9, inclusive, in Table II. The geocentric coordinates of the aircraft, the geocentric coordinates of the subtarget point, that is, the geodetic latitude of the target, and the geodetic coordinates of the target are calculated. The geodetic coordinates of the target and the subtarget point are the same. The required output of the iteration which is required to continue the method, namely steps 11 through 13, uses only the geodetic coordinates of the target. The geodetic coordinates in the final steps are converted to coordinates on a conformal sphere and then the coordinates on the conformal sphere are transformed to the two-dimensional coordinates of a stereographic plane.

In step 1, the method begins with the assignment of estimates of parameters required in the iterative steps 2 through 9. FIG. 5 and FIG. 10 present the geometry to derive the relationships. Equation 1 approximates the subtarget radial distance by the subradar radial distance, and Equation 2 approximates the target radial distance by the sum of the subtarget radial distance and the target altitude. Equation 3 converts the radar measurements into units that can be operated upon, namely radians.

In step 2A, the angular separation is calculated between the line from the Earth's center to the target and from the Earth's center to the radar. FIGS. 5 and 7 describe the equations in this step. Equation 4 saves computation by calculating the interim value "A" which is used more than once. Equation 5 is the result of the Law of Cosines, used on the triangle in FIG. 7. Equation 6 yields the angle of interest.

Step 2B provides the elevation angle of the target above the plane tangent to a sphere having a radius equal to the radar radial distance. Refer to FIG. 7. Equation 7 is derived from the Law of Sines applied to the triangle in FIG. 7. Equation 8 is the angle of interest calculated from the arcsine of the quantity calculated in Equation 7.

Step 3 converts the radar azimuth from the geographic coordinate system to the geocentric system. This is necessary in order to get all of the parameters expressed in one coordinate system. FIG. 8 and Table III illustrate the coordinate rotation which transforms the azimuth from the radar's coordinate system (that is, the geodetic or geographic) to the geocentric coordinate system. Equations 9, 10 and 11 describe the processing for the special cases where the geodetic azimuth is 90.degree., 180.degree. or 270.degree., respectively.

Steps 3A and 3B calculate the geocentric azimuth in another iterative procedure. Equation 12 approximates the geocentric azimuth by the geodetic azimuth. Equation 13 solves for the geocentric azimuth, more exactly than does Equation 12.

Step 3C corrects the calcuated geocentric azimuth for ambiguity, which arises since the arctangent of an argument has multiple solutions.

Step 4 calculates the geocentric latitude of the aircraft because by using relationships of spherical triangles, the geocentric latitude of the aircraft can be calculated. See FIG. 9 and Table IV for a description of this calculation. Equation 14 is the Fortran encoded version of the last equation calculation in Table IV. Equation 15 yields the geocentric latitude of the aircraft by taking the arcsine of the result of Equation 14. Equation 16 calculates the cosine of the target geocentric latitude which is to be used later.

Step 5A and 5B is used to find the difference in the longitudes of the radar and of the aircraft, where the direction westward from Greenwich is taken as a positive value. Equation 17 assumes the value in the special cases where either the radar or the aircraft are above the true North Pole. Equation 18 also comes from FIG. 9. It is derived in the first three Equations in Table IV. The quantity evaluated is the cosine of the longitude difference described above. Equation 19 calculates the longitude difference from the result of Equation 18. Equation 20 corrects the result of Equation 19 for multiple solutions.

Step 6 calculates the difference between the geodetic and the geocentric latitudes of the target. It is depicted in FIG. 10 and in Table V.

Step 6A initializes the value of the latitude difference by setting it equal to the latitude difference at the radar station. Equation 21 performs that approximation.

Step 6B is another iterative process to obtain a more accurate value of the latitude difference. Equation 22 calculates the value used multiple times. This saves some computation. Equation 23 calculates the tangent of the difference of the geocentric latitude and the geodetic latitude of the aircraft. Equation 24 calculates the difference by taking the arctangent of the result of Equation 23.

Step 7 calculates the quantity of interest in the main iterative loop, that is the geodetic latitude of the aircraft (and the subtarget point), as can be seen in FIG. 10.

Step 8 calculates more parameters which will be necessary to obtain a more accurate estimate of the target geodetic latitude in subsequent iterations of steps 2 through 9. See FIG. 10 for these quantities. Equation 26 calculates the difference between the geocentric latitude and the geodetic latitude of the subtarget point. Equation 27 then calculates the geocentric latitude of the subtarget point.

Step 9 calculates more parameters to be used in subsequent iterations of steps 2 through 9. See FIG. 11. Equation 28 calculates the radial distance to the subtarget point. Equation 29 calculates the radial distance to the target more accurately with each iteration.

Step 11 has Equation 30 calculate the longitude of the target from the known radar longitude and the longitude difference calculated in Step 5B.

At this stage, the geodetic latitude and longitude of the target aircraft is established. This information is sufficient for the comparison of all aircraft positions and headings for the entire air traffic control system. However, in order to provide for the display of all target positions and headings on a flat display in a conformal manner, so as to preserve all angles, steps 12 and 13 must be carried out.

Step 12 now projects the geodetic latitude of the aircraft onto the conformal sphere. The property of the conformal projection is that angles measured on the surface of the Earth are preserved on the conformal sphere. Equations 31 and 32 calculate parameters to be used in Equation 33. Equation 33 produces the conformal latitude of the aircraft.

Step 13 transforms points on the conformal sphere into the two-dimensional stereographic plane for display. The inputs are the aircraft conformal latitude, the aircraft longitude, the conformal latitude of the point at which the stereographic plane is tangent to the conformal sphere, the longitude of this point, and the radius of the conformal sphere. Equations 34 through 37 produce parameters which are used to calculate the coordinates in the stereographic plane. Equation 38 calculates the x coordinate of the aircraft in the stereographic plane, and Equation 39 calculates the y coordinate in the stereographic plane.

The resulting x and y coordinates are then available for the display with the position and headings, of all of the aircraft in the air traffic control system. Still further, computations can be made on these x and y coordinates to determine closing velocities and other functions. The display of the system plane is performed on the system plane display 60 in the system block diagram of FIG. 4. The performance of other system plane functions, such as the computation of closing headings, for example, can be performed by the CPU 50 or alternately by the attached data processor 70 shown in FIG. 4.

FIG. 6 illustrates the relationship between geodetic and geocentric latitude. It is seen that because of the ellipsoidal surface of the Earth, if latitude is defined as the normal to a plane tangent to the surface of the Earth, the geodetic latitude will be different from the geocentric latitude.

The resulting method provides the display of aircraft altitudes in the stereographic system plane of 10 nautical miles with an accuracy of about 4 to 5 inches.

The resulting transformation method for air traffic control preserves the inherent accuracy in the input target data sufficient to introduce less than 0.005 nautical miles of error in target positions in the stereographic display system plane coordinates.

Although a specific embodiment of the invention has been disclosed, it will be understood by those having skill in the art, that changes can be made to the specific embodiment without departing from the spirit and the scope of the invention.