Global Navigation Satellite Systems, Inertial Navigation, and Integration. Mohinder S. Grewal

case. Whenever the effectivity time of the leap second event, as indicated by the WNLFS and DN values, is in the past relative to the user's current GPS time, the expression given for tUTC in the first case earlier is valid except that the value of ΔtLFS is used instead of ΔtLS. The GPS control segment coordinates the update of UTC parameters at a future upload in order to maintain a proper continuity of the tUTC timescale.

2.5 Example: User Position Calculations with No Errors

      2.5.1 User Position Calculations

      This section demonstrates how to go about calculating the user position, given ranges (pseudoranges) to satellites, the known positions of the satellites, and ignoring the effects of clock errors, receiver errors, propagation errors, and so on.

      Then, the pseudoranges will be used to calculate the user's antenna location.

      2.5.1.1 Position Calculations

      Neglecting clock errors, let us first determine the position calculation with no errors:

ρ r	=	pseudorange (known)
x, y, z	=	satellite position coordinates (known), in ECEF
X, Y, Z	=	user position coordinates (unknown)

      where x, y, z, X, Y, Z are in the ECEF coordinate system. (It can be converted to ENU.)

Position calculation with no errors is

      (2.36)

      Squaring both sides yields

      (2.37)

(2.38)

where r equals the radius of the Earth and C_b is the clock bias correction. The four unknowns are (X, Y, Z, C_b). Satellite position (x, y, z) is calculated from ephemeris data. For four satellites, Eq. (2.38) becomes

      (2.39)

      with unknown 4 × 1 state vector

      We can rewrite the four equations in matrix form as

      or

(2.40)

      where

Y	=	vector (known)
M	=	matrix (known)
X ρ	=	vector (unknown)

Then, we premultiply both sides of Eq. (2.40) by M⁻¹:

      If the rank of M, the number of linear independent columns of the matrix M, is less than 4, then M will not be invertible.

      2.5.2 User Velocity Calculations

Differentiate Eq. (2.21) with respect to time without C_b.

(2.41)

      Differentiate Eq. (2.41) with respect to

(2.42)

      where .

      In classical navigation geometry, the components (3 × 3) of this unit vector are often called direction cosine. It is interesting to note that these components are the same as the position linearization shown in Eqs. (2.26a) and (2.26b).

Equations (2.42) and (2.26b) will be used in GPS/INS tightly coupled implementation as measurement equations for pseudoranges and/or delta pseudoranges in chapters 11 and 12 in the extended Kalman filters. Equation (2.27) will be used in integrity determination of GNSS satellites in Chapter 9 and from Eq. (2.41),

(2.43)

      where

	=	range rate (known)
ρ r	=	range (known)
(x, y, z)	=	satellite positions (known)
(, , )	=	satellite rates