So this is a pretty specific question, and one that has been puzzling me for a while. Either I'm missing something very basic…or. Anyway, here goes :
There's a flight, that has a dish on it. The flight's dish has reported to me some data : Latititude, Longitude of the flight, Azimuth angle (of satellite relative to flight body), and Elevation angle (of satellite relative to flight body ). I also have satellite longitude (and I know that the satellite is geosynchronous).
From the above I do two things.
1) Using Lat, Lon, Satellite Longitude, I assume level flight and nose of flight pointed north and calculate (x1,y1,z1) of the satellite. ( A unit vector really, pointing in the satellite's direction )
2) Using the Az El reported by the flight, I calculate with the (x2,y2,z2) really is according to the orientation of the flight (shifted co-ordinate frame).
If I calculate the rotation matrix between these two points, should I be able to solve for heading, bank, and attitude of the flight?
At first glance it looks like a variation of the 'lookat' problem on this page:
I assume that you know the height of the aeroplane and the satellite relative to the earth? (if not I don't know how you would do it).
If so you know the position of the aeroplane and the satellite relative to the earth. Then using the stuff on the web page would give what angle you would have to rotate to point to the centre of the earth. So you know the orientation relative to the earth so you know the heading etc.
I don't know if the above is correct, but even it it is I would not like to do the calculations, it seems horribly messy especially using Euler angles.
Hmm. I'm a bit confused. Let me start here :
1) Why do I need the altitude of the satellite? I have AZ / EL in earth frame and AZ/EL angles in body frame. Which means I can calculate x1,y1,z1 and x2,y2,z2, the cartesian co-ordinates in each of these frames. AZ/ EL are the spherical co-ordinates' angle components…so the only thing missing here is 'r' which is range to the satellite ( r, Az, El ) giving the complete spherical co-ordinate set. However, when normalizaing the x1,y1,z1 and x2,y2,z2 to unit vectors, the r term naturally drops out… which made me thing I don't need it to start with. Here's what I've done :
x1 = r sin ( elevation )
y1 = r cos( elevation ) cos ( azimuth )
z1 = r cos (elevation ) sin ( azimuth ) ;
L2 norm = sqrt ( x1^2 + y1^2 + z1^2 ) = r
so if i use the normalized version, the exact position of the satellite becomes unnecessary right?
I will look into the lookat method. I actually did solve for the rotation matrix and the euler's angles, but what puzzled me is that I was getting unrealistic values for heading, bank and attitude. I felt like I was doing something wrong…… :-/
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