ECE 6390 Radiolocation Scavenger Hunt 3 (Fall 2006) 

The DeterminAtors 

Some conversions & some calculations were done in MATLAB. Note that radii, cartesian coordinates, and PseudoRanges are in [km]. c is in [m/s]. Lon & Lat will be in [rads], ? in [s]. 

Define some constants.  

Ratl := `+`(6380, .308); 1; Rgps := 20200+Ratl; 1; c := 299860; 1 

PR1 := `*`(0, c); 1; PR2 := -0.3817986e-2*c; 1; PR3 := -0.4077077e-2*c; 1 

X1 := 12229.2419946027; 1; X2 := 1562.1883996874; 1; X3 := -1142.5200543264; 1 

Y1 := -23318.5825600359; 1; Y2 := -18695.1691312511; 1; Y3 := -23040.7222466909; 1 

Z1 := 3631.2184770731; 1; Z2 := 18829.4083432751; 1; Z3 := 13202.8847710001; 1 

Define our equations for Ux, Uy, Uz. 

Ux := Ratl*cos(Lon)*cos(Lat); 1; Uy := Ratl*sin(Lon)*cos(Lat); 1; Uz := Ratl*sin(Lat); 1
Ux := Ratl*cos(Lon)*cos(Lat); 1; Uy := Ratl*sin(Lon)*cos(Lat); 1; Uz := Ratl*sin(Lat); 1 

Define our ranging equations. 

eqn1 := (X1-Ux)^2+(Y1-Uy)^2+(Z1-Uz)^2-(PR1-c*tau)^2; 1 

(12229.2419946027-6380.308*cos(Lon)*cos(Lat))^2+(-23318.5825600359-6380.308*sin(Lon)*cos(Lat))^2+(3631.2184770731-6380.308*sin(Lat))^2-89916019600*tau^2
(12229.2419946027-6380.308*cos(Lon)*cos(Lat))^2+(-23318.5825600359-6380.308*sin(Lon)*cos(Lat))^2+(3631.2184770731-6380.308*sin(Lat))^2-89916019600*tau^2
(12229.2419946027-6380.308*cos(Lon)*cos(Lat))^2+(-23318.5825600359-6380.308*sin(Lon)*cos(Lat))^2+(3631.2184770731-6380.308*sin(Lat))^2-89916019600*tau^2 

eqn2 := (X2-Ux)^2+(Y2-Uy)^2+(Z2-Uz)^2-(PR2-c*tau)^2; 1 

(1562.1883996874-6380.308*cos(Lon)*cos(Lat))^2+(-18695.1691312511-6380.308*sin(Lon)*cos(Lat))^2+(18829.4083432751-6380.308*sin(Lat))^2-(-1144.861282-299860*tau)^2
(1562.1883996874-6380.308*cos(Lon)*cos(Lat))^2+(-18695.1691312511-6380.308*sin(Lon)*cos(Lat))^2+(18829.4083432751-6380.308*sin(Lat))^2-(-1144.861282-299860*tau)^2
(1562.1883996874-6380.308*cos(Lon)*cos(Lat))^2+(-18695.1691312511-6380.308*sin(Lon)*cos(Lat))^2+(18829.4083432751-6380.308*sin(Lat))^2-(-1144.861282-299860*tau)^2 

eqn3 := (X3-Ux)^2+(Y3-Uy)^2+(Z3-Uz)^2-(PR3-c*tau)^2; 1 

(-1142.5200543264-6380.308*cos(Lon)*cos(Lat))^2+(-23040.7222466909-6380.308*sin(Lon)*cos(Lat))^2+(13202.8847710001-6380.308*sin(Lat))^2-(-1222.552309-299860*tau)^2
(-1142.5200543264-6380.308*cos(Lon)*cos(Lat))^2+(-23040.7222466909-6380.308*sin(Lon)*cos(Lat))^2+(13202.8847710001-6380.308*sin(Lat))^2-(-1222.552309-299860*tau)^2
(-1142.5200543264-6380.308*cos(Lon)*cos(Lat))^2+(-23040.7222466909-6380.308*sin(Lon)*cos(Lat))^2+(13202.8847710001-6380.308*sin(Lat))^2-(-1222.552309-299860*tau)^2 

Solve using Maple's solve routine. 

solve({eqn1, eqn2, eqn3}, {Lat, Lon, tau}) 

{Lon = 1.494306032, Lat = -.6234491628, tau = .1057454264}, {tau = -0.7172725977e-1, Lat = .5894240078, Lon = -1.472871755}, {Lat = .3733602757, tau = 0.6841028588e-1, Lon = -.9413211687}, {tau = -.10...
{Lon = 1.494306032, Lat = -.6234491628, tau = .1057454264}, {tau = -0.7172725977e-1, Lat = .5894240078, Lon = -1.472871755}, {Lat = .3733602757, tau = 0.6841028588e-1, Lon = -.9413211687}, {tau = -.10...
{Lon = 1.494306032, Lat = -.6234491628, tau = .1057454264}, {tau = -0.7172725977e-1, Lat = .5894240078, Lon = -1.472871755}, {Lat = .3733602757, tau = 0.6841028588e-1, Lon = -.9413211687}, {tau = -.10...
{Lon = 1.494306032, Lat = -.6234491628, tau = .1057454264}, {tau = -0.7172725977e-1, Lat = .5894240078, Lon = -1.472871755}, {Lat = .3733602757, tau = 0.6841028588e-1, Lon = -.9413211687}, {tau = -.10...
{Lon = 1.494306032, Lat = -.6234491628, tau = .1057454264}, {tau = -0.7172725977e-1, Lat = .5894240078, Lon = -1.472871755}, {Lat = .3733602757, tau = 0.6841028588e-1, Lon = -.9413211687}, {tau = -.10...
{Lon = 1.494306032, Lat = -.6234491628, tau = .1057454264}, {tau = -0.7172725977e-1, Lat = .5894240078, Lon = -1.472871755}, {Lat = .3733602757, tau = 0.6841028588e-1, Lon = -.9413211687}, {tau = -.10...
{Lon = 1.494306032, Lat = -.6234491628, tau = .1057454264}, {tau = -0.7172725977e-1, Lat = .5894240078, Lon = -1.472871755}, {Lat = .3733602757, tau = 0.6841028588e-1, Lon = -.9413211687}, {tau = -.10...
{Lon = 1.494306032, Lat = -.6234491628, tau = .1057454264}, {tau = -0.7172725977e-1, Lat = .5894240078, Lon = -1.472871755}, {Lat = .3733602757, tau = 0.6841028588e-1, Lon = -.9413211687}, {tau = -.10... 

A number of possible solutions exist due to cosine and sine being in our equations (quadrant mapping ambiguity) along with the squared terms. However, we know that we shouldn't have to travel too far to get to the restaraunt, so we can say a good answer will be the one a. in Atlanta b. and if there remains any more ambiguity, the closer solution to Georgia Tech. Van Leer is located at (Lon: -1.4730, Lat: 0.5895) so the solution is easily chosen as the highlighted one above. The final solution in degrees is: 33.771508 N, 84.3893353 W which is The Varsity on North Ave!