4.1 Linearization of Observation Equation Approximate point position (Xo, Yo, Za) 4. Adjustment of Mathematical p()=√x(0)-xa)+(Y1()-Y2+(210)-z2 GPS Model ()=pl(n) p!(n)=pa(t)-a△x4-al2△y-a2△Z 4.2 Linearization model for point The satellite clock bias is assumed to be known. this Positioning with Code Ranges sumption makes sense because satellite clock correctors can be received from the navigation message The elementary model for point positioning with code ranges Four Satellit R()=p()+c6(t)-co1(t) =ax△X1+ay△Y+a2△Z-c6,(t) In the model, only the clocks are modeled. The ionosphere F2=a3△X+a3△Y4+a24Z·c6() troposphere, and other minor effects are neglected 13= R(1)-p()-c6l(t)=ax△X+ay△Y4+a2△Z1·c6,(t) Observation Equation =ax△X1+a△Y4+a2△Z1-c6,(t) l=Ax =?a=?ay=?a2=? 4.3 Linearization model for point Observation Equation Positioning with Carrier Phases Above fact reflects The procedure is the same as code range ineaized Equation: 1. Point positioning with phases in this form cannot be olved epoch by epoch 入()-P(0)-c6(0)=a△X+ayAY+a2,AZ,+N nt positioning with code ranges, the nu 0 knowns by a new clock term. of unknowns is now increased by the ambiguities 3. For two epochs there are eight eq and Four Satellites again unknowns(still an underdetermined problem ). For three epochs there are 12 equations and 10 unknowns, thus, slightly overdetermined problem. The coefficients of the coordinate increments are supplemented with the time parameter t. Obviously, the four equations do not I=Ax2 4. Adjustment of Mathematical GPS Model 4.1 Linearization of Observation Equation Taylor series with respect to the approximate point ρ ( ) (X (t) - X ) (Y (t) - Y ) (Z (t) - Z ) 2 i 2 i 2 i j i j j j t = + + f(X , Y , Z ) ρ ( ) (X (t) - X ) (Y (t) - Y ) (Z (t) - Z ) i0 i0 i0 2 i0 2 i0 2 i0 j i0 = = + + j j j t (X , Y , Z ) Approximate point position i0 i0 i0 j i i i j j i i i j j i i i j Z p t Z t Z Y p t Y t Y X p t X t X t t ∆ − − ∆ − ∆ − − = − ( ) ( ) ( ) ( ) ( ) ( ) ρ ( ) ρ ( ) 0 0 0 0 0 j 0 i0 j i Xi i Yi i Zi Zi t = t − a ∆X − a ∆Y a ∆ j 1 1 1 i0 j i ρ ( ) ρ ( ) - 4.2 Linearization Model for Point Positioning with Code Ranges The elementary model for point positioning with code ranges is given by R ( ) p ( ) δ (t) - δ (t) i j j i j i t = t + c c In the model, only the clocks are modeled. The ionosphere, troposphere, and other minor effects are neglected. Lineaized Equation: R ( ) p ( ) δ (t) a X a Y a Z - δ (t) i i 1 i Z 1 i Y 1 X j j i j i i i i t − t − c = ∆ + ∆ + ∆ c a X a Y a Z - δ (t) i i 1 i Z 1 i Y 1 X 1 i i i l = ∆ + ∆ + ∆ c ? a ? a ? a ? 1 Z 1 Y 1 X 1 i i i l = = = = The satellite clock bias is assumed to be known. This assumption makes sense because satellite clock correctors can be received from the navigation message. Four Satellites: a X a Y a Z - δ (t) i i 1 i Z 1 i Y 1 X 1 i i i l = ∆ + ∆ + ∆ c a X a Y a Z - δ (t) i i 2 i Z 2 i Y 2 X 2 i i i l = ∆ + ∆ + ∆ c !!!!! 3 l = !!!!! 4 l = Observation Equation: l = Ax 4.3 Linearization Model for Point Positioning with Carrier Phases λΦ ( ) ( ) δ (t) a X a Y a Z λN - δ (t) i j i i 1 i Z 1 i Y 1 X j 0 j i i i t p t c c j i − i − = ∆ + ∆ + ∆ + The procedure is the same as code range Compared to point positioning with code ranges, the number of unknowns is now increased by the ambiguities. Lineaized Equation: Four Satellites again: l = Ax The coefficients of the coordinate increments are supplemented with the time parameter t. Obviously, the four equations do not solve for the eight unknowns. Observation Equation: Above fact reflects 1. Point positioning with phases in this form cannot be solved epoch by epoch. 2. Each additional epoch increases the number of unknowns by a new clock term. 3. For two epochs there are eight equations and nine unknowns (still an underdetermined problem). For three epochs there are 12 equations and 10 unknowns, thus, a slightly overdetermined problem. l = Ax