CET 318 The Eighth Lecture 8, Mathematical Model 1. Point Positioning Book:p.183-202 Dr Guoding Zhou 1.1 Point Positioning with Code ranges Epoch at t Additional Satellites(Simultaneously): L Code Range Observation Equatio )2 R/()=pf(t)+c△6() R2()=√x(p2+(Y2(NY2+20)12)+cB p (t) p!(1) R()=x0四+(Y:0+2(区+c8 △6{(U) R(=x-四3+(Y:1x3+:02+0 Measured distance: p{(t)=√(x()-x,)2+(Y(t)-Y)2+(z{()-z)2 Additional Epoch: Combined bias lock biases must be △6;(t)=6(1)+6,(t) tation message in the form of thre Combined Known Unknown coefficients ao, a, a y with a reference △6(t)=a0+a1(1-1)+a2(1-1) Observation Equation R/()-c△8(t)=p(t)-c△6,(t nknown
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: ( ) ρ (t) c δ (t) j = + ∆ i j i j i R t R (t) j i δ (t) j ∆ i ρ (t) j i c R (t) j i i j Epoch at t ρ (t) j i δ (t) j ∆ i ρ (t) (X (t) - X ) (Y (t) - Y ) (Z (t) - Z ) 2 i j i 2 i j i 2 i j = i + + j i ( ) (X (t) - X ) (Y (t) - Y ) (Z (t) - Z ) c δ (t) 1 i 2 i 2 1 i 2 1 i 1 1 Ri t = i + i + i + ∆ ( ) (X (t) - X ) (Y (t) - Y ) (Z (t) - Z ) c δ (t) 2 i 2 i 2 2 i 2 2 i 2 2 Ri t = i + i + i + ∆ ( ) (X (t) - X ) (Y (t) - Y ) (Z (t) - Z ) c δ (t) 3 i 2 i 2 3 i 2 3 i 3 3 Ri t = i + i + i + ∆ ( ) (X (t) - X ) (Y (t) - Y ) (Z (t) - Z ) c δ (t) 4 i 2 i 2 4 i 2 4 i 4 4 Ri t = i + i + i + ∆ 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 a0, a1, a2 with a reference time tc. Additional Epoch: 2 0 1 2 j δ (t) ( ) ( ) c c ∆ = a + a t − t + a t − t δ (t) δ (t) δ (t) i j j ∆ i = + Combined Bias: Combined Known Unknown ( ) c δ (t) ρ (t) c δ (t) i j − ∆ = − ∆ j i j i R t Observation Equation: Known Unknown
1. 2 Point Positioning with Carrier Phase II. Phase Range Observation Equation 1.3 Point Positioning with Doppler data Φ!(1)=p()+N!+f△6:(t) D(t)=p;(t)+c△6:(t) p/(r)Measured carrier phase expressed in cycles integer unknown D (r)Observed Doppler shift scaled to range rate f Frequency of the satellite signal stantaneous radial velocity between the Observation Equation Ad (t) Time derivative of the combined clock Φ(1)-f6(t)=kp(t)+N/-f△6(t) Unknown 2.1 DGPS with Code Ranges 2. Differential Positioning (DGPS) Station A DGPS calculates pseudorange corrections(PRC)and R4(0)=p4(0)+△8(0)+c6(t0)-c6A(0) nge rate corrections(RRC)(located at A)which are △8(0) Radial orbital er transmitted to the remote receiver(located af B)in near real time The code range correction for satellite j at reference epoch to is PRC(t0)=-R(t0)+8(t0) (t0)-c6(t0)+c6A(0) rom a time series of range corrections, the range rate correction RRC(to) can be evaluated by numerical PRC()=PRC (o)+ PRC (o)(t-t) 2.2 DGPS with Carrier Phases Station B The pseudorange derived from carrier phases at station The code range at station B at epoch t can be modeled R(n)=p()+△6(n)+c6(n)-c6a(1) Ao(o)=P(to)+ApA(to)+NA+c8(to)-c8, (t.) RI(=R()+PRC(0) The phase range correction at reference epoch to is 6g(r)+(△8()-64()-c(6(1)6A(t) PRC()=-(t0)+p(t) Neglecting the difference of the radial orbital errors =-△p(tn)-N-c6/(t0)-c6(t。) R()am=p()-c△6a() Combined error of receiver clocks
2 1.2 Point Positioning with Carrier Phase 1. Phase Range Observation Equation ( ) ρ (t) N f δ (t) j i j λ i 1 Φ = + + ∆ j j i j i t (t) j Φi Measured carrier phase expressed in cycles. N j i Phase ambiguity integer number, integer ambiguity, or integer unknown. j f λ Wavelength Frequency of the satellite signal ( ) f δ (t) ρ (t) N f δ (t) i j λ i j 1 Φ − = + − ∆ j j i j j i t Observation Equation: Known Unknown 1.3 Point Positioning with Doppler Data ( ) ρ (t) δ (t) j i D D t = D + c∆ j i j i D (t) j i Observed Doppler shift scaled to range rate ρ (t) j i D Instantaneous radial velocity between the satellite and the receiver δ (t) 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 ( ) ρ ( ) δ ( ) cδ ( ) - cδ ( ) 0 A 0 j 0 j 0 0 A R t t t t t j A j A = + ∆ + δ ( ) 0 j A ∆ t Radial orbital error The code range correction for satellite j at reference epoch t0 is - δ ( ) - cδ ( ) cδ ( ) PRC ( ) ( ) δ ( ) 0 A 0 j 0 j A 0 j 0 0 A t t t t R t t j A j = ∆ + = − + Station A From a time series of range corrections, the range rate correction RRCj (t0) can be evaluated by numerical differentiation. PRC ( ) PRC ( ) PRC ( )(t - ) 0 0 0 t t t t j j j = + The code range at station B at epoch t can be modeled ( ) ρ ( ) δ ( ) cδ ( ) - cδ ( ) B j j B R t t t t t j B j B = + ∆ + δ ( ) ( δ ( ) - δ ( )) (δ ( ) - δ ( )) ( ) ( ) PRC ( ) B A j A j B B corr t t t c t t R t R t t j j B j B = + ∆ − = + Neglecting the difference of the radial orbital errors ( ) ρ ( ) c δ ( ) AB j corr B R t t t 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 t0 λΦ ( ) ρ (t ) ρ (t ) λN cδ (t ) - cδ (t ) 0 0 j 0 A j 0 0 A j A A j j A t = + ∆ + + The phase range correction at reference epoch t0 is - ρ (t ) λN cδ (t ) - cδ (t ) PRC ( ) λ (t ) ρ (t ) 0 0 j 0 A j A 0 j 0 A j 0 A j A j t = ∆ − − = − Φ +
The phase range correction at any epoch t is RC/(n)=PRC(o)+PRC(oXt-to Following the same procedure as before AdB(Ocor=PB(1)+ANAB-CA8AB() Combined integer ambigui 3. Relative Positioning Basic Principle: 3.1 Phase Differences The objective of relative positioning is to determine the 1. Single- Differences coordinates of an unknown point with respect to a Two points(A, B)and one satellite() are involved. known point. ()-f6(t)=÷p么(t)+N-f6A(t) Linear combinations of station a and b leading to ag(0)-f8(1=tPd(t)+NH-f8B(t) 2. Double differences and Difference of the two equation is 3. Triple-differences -f[6B()-6A( Final form of the single-difference equation Relative positioning can be performed with code anges, or with phase ranges. Subsequently, only da()=pa()+N-f6( phase ranges are explicitly considered. Satellite clock bias has been cancelled 2. Double-Differences: 2. Triple-Differences Assuming the two points A, B, and the two satellites j, k, Two double-differences between two epochs t t2 are two single differences are d/ r(o=+pB(t+NAB-f8aB(t) ΦA(2)=p(2)+NA dB(1)=P4B(t)+NAB-f8aB(t) Differencing two double-differences dA(1)-l()=Hpt(t)-pl(t)+NAB-NAB Φ(2)-①(1)=p4(t2)-p(t1月 Final form of the double-difference equation Final form of the triple-difference equation a AR(1)=+/As 1B(t)+NAB f=f ()-p5()=+ptn) Receiver clock bias has been cancelled if simultaneous Effect for the ambiguities has been cancelled, thus the observations and equal frequencies of satellite signals munity from changes in the ambiguities
3 PRC ( ) PRC ( ) PRC ( )(t - ) 0 0 0 t t t t j j j = + The phase range correction at any epoch t is Following the same procedure as before Φ ( ) ρ ( ) N c δ ( ) AB j AB j corr B j B λ t = t + ∆ − ∆ t 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. A B 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. ( ) f δ (t) ρ (t) N f δ (t) A j λ A j 1 j j A j j A Φ t − = + − ( ) f δ (t) ρ (t) N f δ (t) B j λ B j 1 j j B j j B Φ t − = + − Difference of the two equation is ( ) ( ) [ρ (t) - ρ (t)] N N f [δ (t) δ (t)] B A j A j λ B 1 Φ − Φ = + − − − j j A j B j A j B t t ( ) ρ (t) N f δ (t) AB j λ AB 1 j j AB j AB Φ t = + − 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 ( ) ρ (t) N f δ (t) AB j λ AB 1 j j AB j AB Φ t = + − ( ) ρ (t) N f δ (t) AB k λ AB 1 k k AB k AB Φ t = + − ( ) ρ (t) N jk λ AB 1 Φ = + jk AB jk AB t ( ) ( ) [ρ (t) - ρ (t)] N N j AB k λ AB 1 Φ − Φ = + − j AB k AB j AB k AB t t f f k j = 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 t1, t2 are jk λ 1 AB 1 Φ ( 1 ) = ρ (t ) + N 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 λ 2 AB 1 Φ ( 2 ) = ρ (t ) + N jk AB jk AB t Differencing two double-differences ( ) ( ) ρ (t ) λ 12 1 2 1 jk AB jk AB jk AB Φ t − Φ t = ( ) ( ) [ρ (t ) - ρ (t )] λ 2 1 1 2 1 jk AB jk AB jk AB jk AB Φ t − Φ t =
3.2 Correlations of the phase combinations Covariance Matrix 1. Single- Differences Assuming the phase random error is following a normal distribution with mean, 0 and variance, 8. The measured phases Two single-differences can be expressed by are linearly independent or uncorrelated. The covariance matri d d Φ)=62 dte()100-114(n) dR(n) 2. Double-Differences Three satellites j, k, I withj as reference, A, B two point at 3. Triple-Differences epoch t, d()=A()-ΦAB() The covariance of a single triple-difference is computed Φ()=Φl()-(t by applying the covariance propagation law Matrix Form: Two triple-differences with the same epochs sharing one satellite are considered The first difference using the satellites j, k; the second difference using j, I From Error Propagation Law: ov( DD)=28CC/=282 This shows that phases of double-differences are correlated The single-, double-, and triple-differencing will be investigate with respect to the number of observation equ It is assumed that the two sites a and B are able to observe the same 3.3 Static Relative Positioning 1. The Undifferenced Phase unknown) between point A The two data sets could separately, which would be △A equivalent to point positioning
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, δ2. The measured phases are linearly independent or uncorrelated. The covariance matrix for the phases is cov( ) δ 2 Φ = I 1. Single-Differences: Φ Φ Φ Φ − − = Φ Φ ( ) ( ) ( ) ( ) 0 0 1 1 1 1 0 0 ( ) ( ) 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 Φ = Φ − Φ AB k AB jk AB t ( ) (t) (t) j Φ = Φ − ΦAB l AB jl AB t Φ Φ Φ − − = Φ Φ ( ) ( ) ( ) 1 0 1 1 1 0 ( ) ( ) t t t t t l AB k AB j AB jl AB jk AB DD C SD From Error Propagation Law: = = 1 2 2 1 cov( ) 2δ 2δ 2 T 2 DD CC 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 tripledifference using the satellites j,k; the second tripledifference 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 R (t) j i j
4. Triple-Differences 2. A Single-Difference may be expressed for each satellite and each epoch. The number of unknowns is written below: e()=p(n)n1≥+ Φ2(1)=p(t)+NA+f△6AB(t) int coordinates. For a single triple-d re necessary 3. Double-Differences ΦA()=p(t)+NA nm=4,n1≥2, epoch=2 3. 4 Kinematic Relative Positioning In kinematic relative positioning, the receiver on the Assignment 8 nown point A remains fixed. The second receiver moves, 1. Write the observation equations of point positioning with nd its position is determined at arbitrary epochs code ranges, carrier phases, and Doppler (25 points). 2. Write the observation equations of Differential positioning Considering point B and satellite j, the geometric distance 3. Write the observation equations of Relative positioning (0)-n()2+(Y(a2+(z(12(2 with single-, double-, and triple- phase differences(25 Three coordinates are unknown at each epoch. Thus 4. How do we initialize the static and kinematic vectors of the total number of unknown site coordinates is 3 n for epochs for single, double-, and triple-difference
5 2. A Single-Difference may be expressed for each satellite and each epoch. The number of unknowns is written below: ( ) ρ (t) N f δ (t) AB j λ AB 1 Φ = + + ∆ j j AB j AB t 3. Double-Differences: ( ) ρ (t) N jk λ AB 1 Φ = + jk AB jk AB t 4. Triple-Differences: ( ) ρ (t ) λ 12 1 12 jk AB jk AB Φ t = (n 1) (n 2) t j j n − + ≥ (1) 2, n t 4 min j n = ≥ The triple-difference model includes only the three unknown point coordinates. For a single triple-difference, two epochs are necessary. (2) 4, n t 2, epoch min 2 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. ρ ( ) (X (t) - X ( )) (Y (t) - Y ( )) (Z (t) - Z ( )) 2 i 2 B 2 B t t t t j j j j 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 3 nt 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)?
