Receiver Clock Modeling

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In this approach, the receiver clock error is augmented to the navigation state and estimated The augmented model is x = [n, e, , RL , RR , c tr , c tr ] x = [ n, e, , RL , RR , c tr , c tr ] According to eqn (849), the clock estimate is propagated through time as c tr (k) = c tr (k 1) + c tr (k 1) Tk (925)

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The pseudorange and carrier phase measurements are de ned in eqns (86) and (825) Based on those equations, the pseudorange and carrier phase measurements can be predicted from the navigation state as i i x R( , pi ) + c tr + i i x = R( , p ) + c tr + i + N i = (926) (927)

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where the measurements are assumed to be simultaneous and the time arguments have been dropped to simplify the notation Note that prediction of the carrier phase measurements assumes that estimates of the carrier phase integer ambiguities N i are available The discussion below assumes that the integer estimates are correct and available If the integer ambiguities are incorrect, the phase measurements would be biased by an integer number

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CHAPTER 9

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GPS AIDED ENCODER-BASED DR

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of wavelengths If the integer ambiguities are not available, then the phase measurements are not used Alternatively, the phase measurements could be used either to smooth the pseudoranges or through temporal di erencing to estimate the velocity In this approach, the measurement residuals for the i-th satellite are computed as i (t) i (t) i i (t) i (t) (t) = i (t) = The error models for the measurement residuals are i (t) (t)

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(928) (929)

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= hi x + i g = hi x g +

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(930) (931)

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2 2 where i N (0, ) and i N (0, ) In the above expression, the i i errors and have been absorbed into i and i , respectively The measurement matrix for the i-th satellite is

hi = [ht , ht , 0, 0, 0, 1, 0] g 1 2

(932)

where ht = he Re , ht and ht denote the rst two components of ht , and t 1 2 p (pe i ) he = pe i The de nition of ht is related to the discussion in eqn p 2 (818) with the a-frame selected as the tangent frame The augmented continuous-time error state model has F(t) = 0 0 0 0 0 0 0 0 0 0 0 0 0 0 sin( ) u u cos( ) 0 0 0 0 0

uL cos( ) 2RL uL sin( ) 2RL 1 uL L RL uR 2RR uR 2RR

cos( ) 0 0 sin( )

uR RR

0 0 0 0 0

1 L

R 0 0 0

0 R 0 0

0 0 0 , 0 1

(933)

= I7 , and = [ n , e , , L , R , , f ] The PSD of and f are S and Sf , respectively, as discussed in Section 8432

Measurement Di erencing

In this approach, the measurement from one satellite is subtracted from the measurements of all other satellites to eliminate the receiver clock bias

95 GPS AIDING Therefore, the state and error state remain x = [n, e, , RL , RR ] x = [ n, e, , RL , RR ]

and F(t), , and remain as de ned in eqn (922) Assuming that m satellites are available and that measurements from the rst satellite are subtracted from all the others, the di erenced pseudorange and carrier phase measurements can be predicted from the navigation state as i i = = x x R( , pi ) + i R( , p1 ) + 1 (934)

x x R( , pi ) + i R( , p1 ) + 1 + N i N 1 (935)

for i = 2, , m The same assumptions and comments apply as were discussed following eqns (926-927) In this approach, the measurement residuals for the i-th satellite would be computed as i (t) i (t) = = i (t) 1 (t) i (t) i (t) 1 (t) i (t) (936) (937)

For i = 2, , m, the error models for the measurement residuals are i (t) (t)