Page 175 - Intro to Space Sciences Spacecraft Applications

162 Introduction to Space Sciences and Spacecraft Applications

and receiver clocks may not be in perfect synchronization. This clock off-
set time is shown in Figure 7-5 sketch as the tc time difference between
the satellite signal and the receiver’s replica code. If it is assumed that the
satellite is the absolute time reference, then this clock offset time would
produce an error in range if not allowed for. The method for determining
the value of the clock offset time is described as part of the ranging
process described next.

Range Determination

The range equation (eq. 7-6) would now look like:

R = C, (to f t,) (7-8)
In terms of an X, Y, Z coordinate system, this equation would be of the form:

R1 = I(X,(t,) - XI2 + (YI(t1) - YI2 + (Zl(t1) - z)211/2 cdtc)
f
where the X1(,,,, Y1(,,), Zl(tl) terms represent the position of the transmit-
ter (satellite) at the time of transmission (tl). X, Y, and Z represent the
position of the receiver computed from the range information derived
from the correlation offset time c, to, and c,,, tc represents the range
error due to the satellitekeceiver clock offset time.
Assuming that the position of the satellite is known precisely at all
times (via a navigation message as described earlier), this equation has
four unknowns involving the X, Y, Z position coordinates of the receiver
and the receiver clock offset time tc. To solve for these unknowns, at least
four range signals must be available at the same time to solve four simi-
lar equations simultaneously:

-
R1 = [(X,,,,) - XI2 + (y~(q) Y12 + (Zl(tl) - Z)211” * Cm(tcI
R2 = [(X2(t2) - XI2 + (Y2(t2, - Y12 + (G(t.7) - Z)21’” cm(tc>

R3 = [(X3(t3) - XI2 + (Y3(t3) - V2 + (Z3(t3) - Z)21’n * cm(tc)

R4 = [(X4(,, - X)’ + (Y4(t4) - Y)’ + (Z4(t4, - Z)21’’2 * Cm(Q
Note that the same receiver clock offset time tc appears for each of the
received signals. This assumes that each of the received signals comes from
satellite transmitters synchronized exactly with absolute time. If this were