1 Dr. Guoqing Zhou 8. Mathematical Model CET 318 Book: p. 183-202 1. Point Positioning 1.1 Point Positioning with Code Ranges 1. Code Range Observation Equation Measured Distance: (t)δc(t)ρ)( j i ?+= j i j i tR )(tR j i (t)δ j i ? (t)ρ j i c )(tR j i i j Epoch at t (t)ρ j i (t)δ j i ? )Z-(t)(Z)Y-(t)(Y)X-(t)(X(t)ρ 2 i j i 2 i j i 2 i j i ++= j i (t)δc)Z-(t)(Z)Y-(t)(Y)X-(t)(X)( 1 i 2 i 12 i 12 i 11 ?+++= iiii tR (t)δc)Z-(t)(Z)Y-(t)(Y)X-(t)(X)( 2 i 2 i 22 i 22 i 22 ?+++= iiii tR (t)δc)Z-(t)(Z)Y-(t)(Y)X-(t)(X)( 3 i 2 i 32 i 32 i 33 ?+++= iiii tR (t)δc)Z-(t)(Z)Y-(t)(Y)X-(t)(X)( 4 i 2 i 42 i 42 i 44 ?+++= iiii tR Additional Satellites (Simultaneously): An additional epoch, new satellite clock biases must be modeled due to clock drift. Fortunately, the satellite clock information is transmitted via the broadcast navigation message in the form of three polynomial coefficients a 0 , a 1 , a 2 with a reference time t c . Additional Epoch: 2 210 j )()( (t)δ cc ttattaa ?+?+=? (t)δ(t)δ (t)δ i jj i +=? Combined Bias: Combined UnknownKnown (t)δc(t)ρ (t)δc)( i j ??=?? j i j i tR Observation Equation: UnknownKnown 2 1.2 Point Positioning with Carrier Phase 1. Phase Range Observation Equation (t)δfN(t)ρ)( j i j iλ 1 ?++=Φ jj i j i t )(t j i Φ Measured carrier phase expressed in cycles. N j i Phase ambiguity integer number, integer ambiguity, or integer unknown. j f Wavelengthλ Frequency of the satellite signal (t)δfN(t)ρ (t)δf)( i j iλ 1 j ??+=?Φ jj i jj i t Observation Equation: UnknownKnown 1.3 Point Positioning with Doppler Data (t)δ(t)ρ)( j i D D ?+= ctD j i j i )(tD j i Observed Doppler shift scaled to range rate (t)ρ j i D Instantaneous radial velocity between the satellite and the receiver (t)δ j i D ? Time derivative of the combined clock 2. Differential Positioning (DGPS) DGPS calculates pseudorange corrections (PRC) and range rate corrections (RRC) (located at A) which are transmitted to the remote receiver (located at B) in near real time. 2.1 DGPS with Code Ranges )(cδ-)(cδ )(δ)(ρ )( 0A0 j 0 j A00 tttttR j A j A +?+= )(δ 0 j A t? Radial orbital error The code range correction for satellite j at reference epoch t 0 is )(cδ)(cδ -)(δ- )(δ)( )(PRC 0A0 j 0 j A 0 j A00 ttt ttRt j A j +?= +?= Station A From a time series of range corrections, the range rate correction RRC j (t 0 ) can be evaluated by numerical differentiation. )-)(t(PRC)(PRC )(PRC 000 tttt jjj += The code range at station B at epoch t can be modeled )(cδ-)(cδ )(δ)(ρ )( B jj B tttttR j B j B +?+= ))(δ-)(δ())(δ-)(δ()(δ )(PRC)()( AB j A j BB corr ttcttt ttRtR jj B j B ??+= += Neglecting the difference of the radial orbital errors )(δc)(ρ)( AB j Bcorr tttR j B ??= Combined error of receiver clocks Station B 2.2 DGPS with Carrier Phases The pseudorange derived from carrier phases at station A at epoch t 0 )(tcδ-)(tcδλN )(tρ)(tρ)(λΦ 00 j A0 j A00 j A A jj A t ++?+= The phase range correction at reference epoch t 0 is )(tcδ-)(tcδλN )(tρ- )(tρ)(tλ)(PRC 00 j A0 j A 0 j A0 j A0 j A j t ???= +Φ?= 3 )-)(t(PRC)(PRC )(PRC 000 tttt jjj += The phase range correction at any epoch t is Following the same procedure as before )(δcN)(ρ)(Φ AB j AB j Bcorr j B ttt ???+=λ Combined integer ambiguity 3. Relative Positioning Basic Principle: The objective of relative positioning is to determine the coordinates of an unknown point with respect to a known point. AB AB Relative positioning can be performed with code ranges, or with phase ranges. Subsequently, only phase ranges are explicitly considered. Linear combinations of station A and B leading to 1. Single-differences, 2. Double differences, and 3. Triple-differences. 3.1 Phase Differences 1. Single-Differences: Two points (A, B) and one satellite (j) are involved. (t)δfN(t)ρ (t)δf)( A j Aλ 1 j jj A jj A t ?+=?Φ (t)δfN(t)ρ (t)δf)( B j Bλ 1 j jj B jj B t ?+=?Φ Difference of the two equation is (t)]δ(t)[δfNN(t)]ρ-(t)[ρ)()( AB j A j Bλ 1 ???+=Φ?Φ jj A j B j A j B tt (t)δfN(t)ρ)( AB j ABλ 1 jj AB j AB t ?+=Φ Final form of the single-difference equation Satellite clock bias has been cancelled 2. Double-Differences: Assuming the two points A, B, and the two satellites j, k, two single differences are (t)δfN(t)ρ)( AB j ABλ 1 jj AB j AB t ?+=Φ (t)δfN(t)ρ)( AB k ABλ 1 kk AB k AB t ?+=Φ N(t)ρ)( jk ABλ 1 +=Φ jk AB jk AB t N N(t)]ρ-(t)[ρ)()( j AB k ABλ 1 ?+=Φ?Φ j AB k AB j AB k AB tt ff jk = Final form of the double-difference equation Receiver clock bias has been cancelled if simultaneous observations and equal frequencies of satellite signals 2. Triple-Differences: Two double-differences between two epochs t 1 , t 2 are jk AB1λ 1 1 N)(tρ)( +=Φ jk AB jk AB t Final form of the triple-difference equation Effect for the ambiguities has been cancelled, thus the immunity from changes in the ambiguities. jk AB2λ 1 2 N)(tρ)( +=Φ jk AB jk AB t Differencing two double-differences )(tρ)()( 12λ 1 12 jk AB jk AB jk AB tt =Φ?Φ )](tρ-)(t[ρ)()( 12λ 1 12 jk AB jk AB jk AB jk AB tt =Φ?Φ 4 3.2 Correlations of the Phase Combinations Covariance Matrix: Assuming the phase random error is following a normal distribution with mean, 0 and variance, δ 2 . The measured phases are linearly independent or uncorrelated. The covariance matrix for the phases is δ)cov( 2 I=Φ 1. Single-Differences: ? ? ? ? ? ? ? ? ? ? ? ? ? ? Φ Φ Φ Φ ? ? ? ? ? ? ? ? = ? ? ? ? ? ? Φ Φ )( )( )( )( 1100 0011 )( )( t t t t t t k B k A j B j A k AB j AB Two single-differences can be expressed by SD C Φ 2. Double-Differences: Three satellites j, k, l with j as reference, A, B two point at epoch t, (t)(t))( j AB Φ?Φ=Φ k AB jk AB t (t)(t))( j AB Φ?Φ=Φ l AB jl AB t ? ? ? ? ? ? ? ? ? ? Φ Φ Φ ? ? ? ? ? ? ? ? = ? ? ? ? ? ? Φ Φ )( )( )( 101 011 )( )( t t t t t l AB k AB j AB jl AB jk AB DD C SD From Error Propagation Law: ? ? ? ? ? ? == 21 12 2δ δ2)cov( 22 T CCDD Matrix Form: This shows that phases of double-differences are correlated 3. Triple-Differences: Two triple-differences with the same epochs and sharing one satellite are considered. The first triple- difference using the satellites j,k; the second triple- difference using j,l. The covariance of a single triple-difference is computed by applying the covariance propagation law. 3.3 Static Relative Positioning The single-, double-, and triple-differencing will be investigated with respect to the number of observation equations and unknowns. It is assumed that the two sites A and B are able to observe the same satellites at the same epochs. 1. The Undifferenced Phase: would be no connection (no common unknown) between point A and B. The two data sets could be solved separately, which would be equivalent to point positioning. A B )(tR j i j 5 2. A Single-Difference may be expressed for each satellite and each epoch. The number of unknowns is written below: (t)δfN(t)ρ )( AB j ABλ 1 ?++=Φ jj AB j AB t 3. Double-Differences: N(t)ρ )( jk ABλ 1 +=Φ jk AB jk AB t 4. Triple-Differences: )(tρ )( 12λ 1 12 jk AB jk AB t =Φ )1(n 2)(n t j j n ? + ≥ 4n ,2 )1( t min j ≥=n The triple-difference model includes only the three unknown point coordinates. For a single triple-difference, two epochs are necessary. 2epoch ,2n ,4 )2( mint min j =≥=n 3.4 Kinematic Relative Positioning In kinematic relative positioning, the receiver on the known point A remains fixed. The second receiver moves, and its position is determined at arbitrary epochs. ))(Z-(t)(Z))(Y-(t)(Y))(X-(t)(X)(ρ 2 i 2 B 2 B tttt jjjj B ++= Considering point B and satellite j, the geometric distance Three coordinates are unknown at each epoch. Thus, the total number of unknown site coordinates is 3nt for nt epochs for single-, double-, and triple-difference. Assignment 8 1. Write the observation equations of point positioning with code ranges, carrier phases, and Doppler (25 points). 2. Write the observation equations of Differential positioning with code ranges, carrier phases (25 points). 3. Write the observation equations of Relative positioning with single-, double-, and triple- phase differences (25 points). 4. How do we initialize the static and kinematic vectors of ambiguities (25 points)?