On 06/23/2013 01:52 AM, Jim Lux wrote:
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)
Indeed.
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.
Hmm. You could tabulate it even. It would be quite a bit of core-memory,
but achieveable.
Oh, and it isn't full 43200 s, it's only about 11 hours and 58 min.
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.
Oh yes.
With a RAIM algorithm you could use extra channels to overcome
deficiencies in the crudeness of the calculations.
Would be neat if there would be a PLL steering of the revolving calender
to maintain with minimum error. The T error would be a natural detector
to use. Extra grade if individual birds got adjusted.
Cheers,
Magnus
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