A Numerical Method For Constructing Geo-Location Isograms

Transcription

1 A Numerial Method For Construting Geo-Loation Isograms Mike Grabbe The Johns Hopkins University Applied Physis Laboratory Laurel, MD Memo Number GVW--U- June 9, 2 Introdution Geo-loation is often performed using measurements of salar quantities suh as range, Time Differene of Arrival (TDOA), range rate, Frequeny Differene of Arrival (FDOA), Angle of Arrival (AOA). Eah of these an be represented geometrially as a line on the earth s surfae along whih the measured value is onstant. In general, a line or path along whih a speified parameter is onstant is referred to as an isogram. For a TDOA measurement, this line is often alled an isohrone. The ability to generate points along a measurement isogram is neessary if the isogram is to be plotted for visualization of the geo-loation senario. Also, points along the isogram assoiated with one measurement may be treated as idate target loations used to determine the validity of suessive measurements. The purpose of this paper is to present a numerial method that an be used to generate points along any isogram on the surfae of a WGS84 earth model. The funtional representations of the isograms for TDOA FDOA measurements will be given as examples. 2 Coordinate Frames The Earth Centered Earth Fixed (ECEF) frame has its x axis through the intersetion of the equator the Greenwih Meridian, its z axis through the North Pole, its y axis oriented to reate a right hed oordinate system. The World Geodeti System 984 (WGS84) [ models the earth s surfae as an oblate spheroid (ellipsoid), whih allows Cartesian ECEF position on the earth s surfae to be represented using the angles longitude geodeti latitude. The WGS84 was developed by the National Imagery Mapping Ageny, now the National Geospatial-Intelligene Ageny, has been aepted as a stard for use in geodesy navigation.

2 3 Position On An Isogram Position on the WGS84 ellipsoid is defined in the ECEF frame by [2 p = r E os (ψ) os (θ) sin (ψ) os (θ) ( ε 2 ) sin (θ) () where ψ is longitude, θ is geodeti latitude, ε is the earth s eentriity. earth s transverse radius of urvature defined by r E = r eq ε 2 sin 2 (θ) The term r E is the (2) where r eq is the earth s equatorial radius. We will define the angle α suh that Squaring both sides of the above gives tan (α) = ε 2 tan (θ) (3) tan 2 (α) = ( ε 2 ) sin 2 (θ) os 2 (θ) = ε2 sin 2 (θ) os 2 (θ) (4) From this we have therefore that + tan 2 (α) = ε2 sin 2 (θ) os 2 (θ) os (α) = os (θ) ε 2 sin 2 (θ) (5) (6) Combining (2) (6) gives r E os (θ) = r eq os (α) (7) r E ( ε 2 ) sin (θ) = r eq ε 2 sin (α) (8) Substituting (7) (8) into () gives an alternate representation of ECEF position p = r eq os (ψ) os (α) sin (ψ) os (α) ε 2 sin (α) (9) The expression given in (9) is a simpler parameterization of position than the one given in (), sine r eq is a onstant r E is a funtion of latitude. We will use (9) to represent a point on an isogram. 2

3 The derivatives of this position vetor with respet to the angles ψ α will be needed later are sin (ψ) os (α) p ψ = r eq os (ψ) os (α) () p α = r eq os (ψ) sin (α) sin (ψ) sin (α) ε 2 os (α) () 4 Isogram Funtional Representation Let p be the ECEF position vetor (9) of a point on an isogram let m be the value of a geo-loation measurement. We will represent the isogram assoiated with this measurement as the set S = {p: h (p) = m} (2) where h is a funtion unique to eah measurement type. We will show the form of h of the derivative h p, whih will be needed later, for measurements of TDOA FDOA. To do so, the following quantities are defined. Let p p 2 be the ECEF positions of sensors 2 respetively. Then the ranges from a point p on an isogram to these sensors are Unit vetors along these diretions are ( u = r = p p (3) r 2 = p 2 p (4) r ( u 2 = Let v v 2 be the ECEF veloities of sensors 2. r 2 ) (p p) (5) ) (p 2 p) (6) Then v = ṗ (7) v 2 = ṗ 2 (8) If we assume that the position p on the isogram is stationary, then (3)-(8) give that the range rates are ṙ = r ṗ p = r ( ) v = (p p p) T v = u T v (9) r ṙ 2 = u T 2 v 2 (2) 3

4 4. TDOA Measurement If the signal is transmitted at time t =, then the times when the signal is reeived by the sensors are t = r (2) t 2 = r 2 (22) where is the speed of light. The sensor measurement is of the quantity ( ) tdoa = t 2 t = (r 2 r ) (23) whih gives r 2 r = tdoa (24) If we treat m = tdoa as our measurement, then the measurement funtion in (2) is h (p) = r 2 r = p 2 p p p (25) therefore h p = ut 2 u T (26) 4.2 FDOA Measurement If f is the frequeny of the transmitted signal, then the frequenies reeived by the sensors are [3 ( ) f = ṙ f (27) The sensor measurement is of the quantity whih gives If we treat m = ( f f 2 = fdoa = f 2 f = ( ) ṙ2 f (28) ( ) f (ṙ 2 ṙ ) (29) ( ) ṙ 2 ṙ = fdoa (3) f ) fdoa as our measurement, then the measurement funtion in (2) is h (p) = ṙ 2 ṙ = u T 2 v 2 u T v (3) therefore ( ) ( ) h p = v T [ 2 I u2 u T 2 + v T [ I u u T r 2 r (32) 4

5 5 Isogram Constrution The representation of position given in (9) of an isogram in (2) define an impliit funtional relationship between ψ α. If we treat α as a funtion of ψ, then an isogram an be onsidered to be a plot of pairs (ψ, α) as shown in Figure below. Figure : Isogram Unit Tangent Vetor Also shown in Figure is a unit vetor tangent to the isogram. If we let s represent the path length (in radians) along the isogram from a speified initial position (ψ, α ), then we an express this unit vetor as [4 u T = (33) [ dψ ds dα ds If s is the path length from the initial position (ψ, α ) to some point (ψ, α ), then therefore [ ψ α ψ = ψ + α = α + = s s [ ψ α ( ) dψ ds (34) ds ( ) dα ds (35) ds + s u T ds (36) An isogram will be onstruted by numerially integrating the unit tangent vetor (33) with respet to ar length from a speified initial position. This gives a sequene of n points (ψ i, α i ) for i =,,..., n. The geodeti latitude θ i ECEF position vetor p i assoiated with eah point an be omputed using (3) (9) respetively. The method for omputing the unit tangent vetor will now be given. 5

6 A vetor tangent to the isogram at the point (ψ, α) is given by v T = [ dα dψ (37) To ompute the derivative dα dψ, treat position p as a funtion of ψ α differentiate the expression for an isogram given in (2) d {h (p)} = d {m} = h h dψ + ψ α dα = dα dψ = h/ ψ h/ α (38) Now substitute (38) into (37) v T = [ h/ ψ h/ α = h α [ h α h ψ (39) This shows that the following vetor is also tangent to the isogram at the point (ψ, α) [ h [ h p d T = α h p α = ψ h p p ψ (4) The unit tangent vetor is then u T = ± d T d T (4) where the sign an be seleted to determine the diretion taken along the isogram from the initial position. The quantities needed to ompute this vetor are p ψ, p h α, p. The first two were given in () () are the same for eah measurement type. The quantity h p is unique to eah measurement type is given in (26) for a TDOA measurement in (32) for an FDOA measurement. Note that using (4) instead of (37) to onstrut the unit vetor avoids a singularity is infinite. if dα dψ 6

7 6 Simulation Example Matlab R was used to simulate the numerial method developed in this paper for the following geoloation senario. An airraft a UAV are reeiving a ommuniations signal being transmitted by a stationary ground target. The reeived signal is proessed by the two platforms to generate a measurement of TDOA of FDOA. The values of the senario parameters are given in the table below. Senario Parameter Units Value target longitude degrees 7 target latitude degrees 36 target altitude feet target signal frequeny megahertz 5 airraft speed knots 4 airraft heading degrees 9 airraft altitude feet 3, range from target to airraft nautial miles 7 bearing from target to airraft degrees UAV speed knots 2 UAV heading degrees UAV altitude feet, range from target to UAV nautial miles 4 bearing from target to UAV degrees This senario produed the following parameter values that effet the shape of the TDOA FDOA isograms. Parameter Units Value range differene nautial miles 3.2 TDOA miro seonds 86.4 range rate differene meters/seond 24.9 FDOA hertz 2.5 7

8 The airraft, UAV, target, isograms are shown in Figure 2 below. ds used for numerial integration was. degrees. The path length inrement airraft TDOA FDOA 36.8 latitude (deg) target UAV longitude (deg) Figure 2: TDOA FDOA Isograms 7 Referenes [ NIMA Tehnial Report TR835.2, Department of Defense World Geodeti System 984, Its Definition Relationships With Loal Geodeti Systems, 3 rd Ed., January 2. [2 Farrell, J.A. M. Barth, The Global Positioning System Inertial Navigation, MGraw- Hill, 999. [3 Serway, R.A. J. Jewett, Physis for Sientists Engineers, Brooks/Cole, 24. [4 Kreyszig, E., Advaned Engineering Mathematis, 8 th Ed., John Wiley & Sons,