On 6/22/13 4:38 PM, Magnus Danielson wrote:
electromechanical.. like omega receivers. rotary transformers can do
very high quality trig functions, but do you actually need trig
functions assuming you're just solving for X,Y,Z,T.
Oh yes. Check IS-GPS-200F, clause 20.3.3.4.3 User Algorithm for
Ephemeris Determination, found on page 113 and forward. The Table 20-IV
contains the actual formulas. The Kepler's Equation for Eccentric
Anomaly is a bit annoying, since it is not in closed form, so one way or
another of approximation iteration is needed.
Quite a bit of trigonometry goes on just to have each tracked satellites
current position estimated, such that the pseudo-range value taken for
the bird can be diffed out with the position. That process becomes
trivial if the position is known and only time is needed, given that we
cranked out the birds X, Y, Z and T position, which requires trigonometry.
Yes, but that trig can be done VERY slowly, since the cycle time is 12
hours, which is why a resolver/rotary transformer approach seems viable.
(rather, than, say, integrating the satellite state vector)
Are you allowed to externally supply the almanac, in the form of a
electromechanical system. The satellites are in circular orbits and
fairly stable, and with multiple satellites in the same plane.
You could naturally cheat in several interesting ways, but you need
fairly accurate X, Y and Z values for the birds at any given time.
How accurate?? Resolvers are good to about 16 bit accuracy, so I guess
1 part in 60,000. if the orbit circumference is 163 Mm, then a resolver
can determine the position to a few km.
However, I don't know that that is good enough. If you need to know to
1 chip at C/A code rates, 1 microsecond, that's a pretty small fraction
of one 12 hour rev of 43200 seconds. But maybe not.
You'd only need trig to convert X,Y,Z into lat/lon, and for us timenuts
types, do you really need lat/lon? In fact, do you even need to solve
for earth centered coordinates? Why not work in inertial space (whether
your receiver happens to be moving in a circle at 1 rev/24 hrs or flying
in a plane at something else is sort of immaterial)
Once you come to having a X, Y, Z and T, the remaining trig operations
is trivial to what you already have done, so you might as well do them.
I envision something with a common shaft running at 1 rev/12 hours that
drives N rotors (one for each satellite). there's a small motor that
sets the offset of the rotor relative to the shaft to account for small
movements along the orbit plane. That, plus some other transformers
would give you X,Y, and Z for each satellite.
You have a sick mind. What worse is, I understood what you actually meant!
Actually, how bad would your time estimate be if you just assumed
perfect circular orbits with no higher order corrections?
Grabbing a modern set of data, doing the calculations with and without
the proper values would tell you. I would not be surprised if it where
way over the km off. On the other hand, the precision we talk about in
general already throws us off sufficiently, so who cares.
One should realize that we talk about tens of Mm numbers in pseudo-range
distances.
So I think you probably can't get a position fix within 10km, but hey,
what a beast it would be.
Cheers,
Magnus
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