Sat Jun 22 19:52:28 EDT 2013

```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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