On Feb 23, 12:49 pm, koobee.wub...@gmail.com wrote:
> On Feb 20, 4:33 am, "Jon" <jon8...@peoplepc.com> wrote:
>
>
>
> > Supposing three satellites transmit signals with time stamps that are picked
> > up by a receiver.  If the clocks on the satellites are closely synchronized,
> > then
>
> > |AD|=|AB|+t_AB
> > |BD|=|BC|+t_BC
> > |CD|=|CA|+t_CA
>
> > Where
> > A,B,C=position vectors of three satellites
> > AB,BC,CA=distances between satellites
> > t_AB,t_BC,t_CA=time differences between signals at D.
> > D=position vector of receiver.
>
> > Once |AD|,|BD|,|CD| are found, the coordinates of D can be derived.
>
> > Is this how it is done?
>
> > summarized at
>
> >http://mypeoplepc.com/members/jon8338/math/id8.html
>
> You need four satellites where each satellite should know its time and
> position.  Time can just be a counter with all the satellites
> synchronized to within a count of each other.  Each satellite then
> just broadcast its time and position information as almanac signal to
> whoever wants to receive it.
>
> Since the distance is traveled by light with a known speed, you can
> then easily form a set of four equations with four unknowns.  The four
> unknowns are your time (relative to the counter in each satellite) and
> position.
>
> **  c^2 (t1 – t)^2 = (x1 – x)^2 + (y1 – y)^2 + (z1 – z)^2
> **  c^2 (t2 – t)^2 = (x2 – x)^2 + (y2 – y)^2 + (z2 – z)^2
> **  c^2 (t3 – t)^2 = (x3 – x)^2 + (y3 – y)^2 + (z3 – z)^2
> **  c^2 (t4 – t)^2 = (x4 – x)^2 + (y4 – y)^2 + (z4 – z)^2
>
> Where
>
> **  (t1, x1, y1, z1) = Time and position of satellite 1
> **  (t2, x2, y2, z2) = Time and position of satellite 2
> **  (t3, x3, y3, z3) = Time and position of satellite 3
> **  (t4, x4, y4, z4) = Time and position of satellite 4
> **  (t, x, y, z) = Time and position of the receiver
>
> Notice that relativistic effect is never needed.

Except that t1, t2, t3, and t4 are Earth-referenced times, not the
native time of the clocks on the satellites, which are t1', t2', t3',
t4'. To get from t1', t2', t3', t4' to t1, t2, t3, t4, the satellites
have programmed in a correction that comes from understanding
relativistic effects.

PD