Positioning and Navigation Using the Russian Satellite System
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The carrier phase measurement in units of length, Φ S R = λ S ϕ S R , is now substituted by the long expression in order to demonstrate the frequency-dependency of that term. Different GLONASS satellites S and r transmit their signals using different carrier frequencies f S and f r . Therefore, the single difference receiver clock error does not cancel in the cycles notation of the double difference observation equation, Eq. (8.2.15). It neither can be ignored. For a single difference receiver clock error of 1 ms, which is not unusual, and adjacent GLONASS carrier frequencies on L 1 , this term evaluates to 562.5 cycles, or more than 100 m. The single difference receiver clock error does cancel in the notation using units of length, Eq. (8.2.16), where the coefficient in the single difference observation equation was the satellite-independent speed of light c instead of the carrier frequency. In that case, however, the single difference integer ambiguities remain in the equation with different factors. Due to their being scaled with the wavelengths of the satellite signal, they cannot be contracted to one double difference term without losing their integer nature. On the other hand, the single difference inter-channel biases can be combined to a double difference term in the equation using units of length Eq. (8.2.16), but remain as single difference expressions in the equation using units of cycles Eq. (8.2.15). Thus, with the single difference receiver clock offset not cancelling, Eq. (8.2.15) contains one more unknown (2n + 3 with respect to 2n + 2 for a set of n satellites including the reference satellite) than Eq. (8.2.16). This number of unknowns would be decreased further, if a way was found to either drop the clock error term from Eq. (8.2.15) or to combine the ambiguity terms in (8.2.16) without losing their integer character. A proposed solution to this latter problem is introduced in Section 8.4. As with the system of single difference observation equations, these are too many unknowns to solve this system, even when code pseudorange observations to the satellites are added. Again, the inter- channel biases must be neglected or determined in some other way to make the system of double difference carrier phase and code pseudorange observations to n GLONASS satellites determined for at least four GLONASS satellites using Eq. (8.2.16) or five satellites in the case of Eq. (8.2.15). 8.2 Carrier Phase Measurements 109 Analogously to the pseudorange observation equations, for the double differenced carrier phase ob- servation equations in a combined GPS/GLONASS scenario with a GPS satellite i and a GLONASS satellite j three different cases must be distinguished, depending on the reference satellite: 1. The reference satellite r GP S is a GPS satellite. The double difference observation equations can be written as ∆ϕ ir GP S U R = 1 λ GP S ∆ ir GP S U R + ∆N ir GP S U R + f GP S · ∆δt ir GP S ,T rop U R + ∆ε ir GP S U R (8.2.17) ∆ϕ jr GP S U R = 1 λ j ∆ j U R − 1 λ GP S ∆ r GP S U R + ∆N jr GP S U R + f j −f GP S · (∆δt U R +∆L U R,GP S )+ f j · ∆δt U R,HW +f j · ∆δt j,T rop U R −f GP S · ∆δt r GP S ,T rop U R + (8.2.18) f j · ∆δt j U R,ICB + ∆ε j U R in units of cycles or λ GP S ∆ϕ ir GP S U R = ∆ ir GP S U R +λ GP S ∆N ir GP S U R +c· ∆δt ir GP S ,T rop U R +λ GP S ∆ε ir GP S U R (8.2.19) λ j ∆ϕ j U R − λ GP S ∆ϕ r GP S U R = ∆ jr GP S U R + λ j ∆N j U R − λ GP S ∆N r GP S U R + c · ∆δt U R,HW + c · ∆δt jr GP S ,T rop U R + c · ∆δt j U R,ICB + λ j ∆ε j U R − (8.2.20) λ GP S ∆ε r GP S U R in units of length, respectively. Of course, the GPS/GPS double difference reads like what one would expect from GPS only pro- cessing. The single difference receiver clock offset, common to both satellites, cancels, and all other single difference terms can be combined to double difference expressions. For the GLONASS/GPS double difference, as already observed in the GLONASS only scenario, due to the different fre- quencies of the participating satellites either the single difference receiver clock offset, combined with the GPS hardware delay, ∆δt U R + ∆L U R,GP S does not cancel (in the cycles notation) or the two single difference integer ambiguities cannot be contracted to one double difference term (in the length notation). In addition, the single difference receiver inter-system hardware delay ∆δt U R,HW remains in the mixed GLONASS/GPS double difference, as already observed in the pseudorange double difference processing, as well as the single difference receiver inter-channel biases. For a scenario with m GPS and n GLONASS satellites, the set of observation equations contains m + 2n + 4 unknowns. 2. The reference satellite r GLO is a GLONASS satellite. The double difference observation equations can in that case be written as ∆ϕ ir GLO U R = 1 λ GP S ∆ i U R − 1 λ r GLO ∆ r GLO U R + ∆N ir GLO U R + f GP S − f r GLO · (∆δt U R + ∆L U R,GP S ) − f r GLO · ∆δt U R,HW + (8.2.21) f GP S · ∆δt i,T rop U R − f r GLO · ∆δt r GLO ,T rop U R − f r GLO · ∆δt r GLO U R,ICB + ∆ε i U R ∆ϕ jr GLO U R = 1 λ j ∆ j U R − 1 λ r GLO ∆ r GLO U R + ∆N jr GLO U R + f j − f r GLO · (∆δt U R + ∆L U R,GP S ) + (8.2.22) f j − f r GLO · ∆δt U R,HW + f j · ∆δt j,T rop U R − f r GLO · ∆δt r GLO ,T rop U R + f j · ∆δt j U R,ICB − f r GLO · ∆δt r GLO U R,ICB + ∆ε j U R 110 8 OBSERVATIONS AND POSITION DETERMINATION in units of cycles or λ GP S ∆ϕ i U R − λ r GLO ∆ϕ r GLO U R = ∆ ir GLO U R + λ GP S ∆N i U R − λ r GLO ∆N r GLO U R − c · ∆δt U R,HW + c · ∆δt ir GLO ,T rop U R − (8.2.23) c · ∆δt r GLO U R,ICB + λ GP S ∆ε i U R − λ r GLO ∆ε r GLO U R λ j ∆ϕ j U R − λ r GLO ∆ϕ r GLO U R = ∆ jr GLO U R + λ j ∆N j U R − λ r GLO ∆N r GLO U R + c · ∆δt jr GLO ,T rop U R + c · ∆δt jr GLO U R,ICB + (8.2.24) λ j ∆ε j U R − λ r GLO ∆ε r GLO U R in units of length, respectively The GPS/GLONASS double difference is equivalent to the GLONASS/GPS double difference from above, but with inverse signs. For the GLONASS/GLONASS double difference, again due to the different frequencies of the participating satellites the single difference receiver clock offset ∆δt U R + ∆L U R,GP S does not cancel in the cycles notation, nor does the single difference receiver inter-system hardware delay ∆δt U R,HW . They both cancel in the notation using units of length. But here again, both single difference integer ambiguities remain in the equation, with different factors. This turns the other way around for the single difference inter-channel biases. They can be combined to one double difference term in the equation using units of length, but remain as single difference terms with different factors in the equation using units of cycles. For a scenario with m GPS and n GLONASS satellites, the set of observation equations contains m + 2n + 4 unknowns. 3. Separate reference satellites r GP S for GPS and r GLO for GLONASS In that case the double difference observation equations become ∆ϕ ir GP S U R = 1 λ GP S ∆ ir GP S U R + ∆N ir GP S U R + f GP S · ∆δt ir GP S ,T rop U R + ∆ε ir GP S U R (8.2.25) ∆ϕ jr GLO U R = 1 λ j ∆ j U R − 1 λ r GLO ∆ r GLO U R + ∆N jr GLO U R + f j −f r GLO · (∆δt U R + ∆L U R,GP S ) + f j −f r GLO · ∆δt U R,HW + f j · ∆δt j,T rop U R − f r GLO · ∆δt r GLO ,T rop U R + (8.2.26) f j · ∆δt j U R,ICB − f r GLO · ∆δt r GLO U R,ICB + ∆ε j U R in units of cycles or λ GP S ∆ϕ ir GP S U R = ∆ ir GP S U R +λ GP S ∆N ir GP S U R +c· ∆δt ir GP S ,T rop U R +λ GP S ∆ε ir GP S U R (8.2.27) λ j ∆ϕ j U R − λ r GLO ∆ϕ r GLO U R = ∆ jr GLO U R + λ j ∆N j U R − λ r GLO ∆N r GLO U R + c · ∆δt jr GLO ,T rop U R + c · ∆δt jr GLO U R,ICB + (8.2.28) λ j ∆ε j U R − λ r GLO ∆ε r GLO U R in units of length, respectively. This is the combination of the GPS/GPS and GLONASS/GLONASS double differences of the previ- ous two cases. In the GPS/GPS double difference, the single difference receiver clock offset, common Download 5.01 Kb. Do'stlaringiz bilan baham: 
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