Читать книгу Global Navigation Satellite Systems, Inertial Navigation, and Integration - Mohinder S. Grewal - Страница 68

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.