Kinematic Relative GPS Positioning Using State-Space Models: Computational Aspects

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1 Kinematic Relative GPS Positioning Using State-Space Moels: Computational Aspects Xiao-Wen Chang, Mengjun Huang, McGill University, Canaa BIOGRAPHIES Dr. Chang is Associate Professor of Computer Science at McGill University. He hols a B.Sc an an M.Sc in Computational Mathematics from Nanjing University, China, an a Ph.D in Computer Science from McGill University. His main research area is scientific computing with particular emphasis on matrix computations an applications. He has been involve in GNSS research since 1998 an his current research interests inclue various estimation techniques for GNSS positioning an fast algorithms for ambiguity etermination. His publications can be foun at chang. Mr. Huang has just grauate from McGill University with an M.Sc in Computer Science. He hols a B.Sc in Computer Science from East China University of Science an Technology. He is intereste in numerical algorithms an software esign for linear estimation an their applications in GNSS positioning. ABSTRACT For inematic relative GPS positioning the often use approach is to establish a iscrete linearize statespace moel, an then apply a stanar Kalman filtering technique to estimate the positions. However, there are a few shortcomings with this approach. Since the integer ambiguities are constant (suppose there are no cycle slips), the state-space moel for positioning is special. In orer to apply a stanar Kalman filtering technique, this approach enlarges the timevariant state vectors (e.g., position-velocity vectors) to inclue the integer ambiguities. This not only maes the estimation problem larger leaing to higher computational cost, but also results in singular covariance matrices for the process noise vectors in the process equations since the covariance matrix corresponing to the integer ambiguity vector is a zero matrix. Then some stanar Kalman filtering techniques, such as the square root information filter cannot be applie. The typical metho people have use to hanle this singularity problem is to artificially assign a small covariance matrix to the ambiguity vector. But this is an approximation an oes not loo elegant in theory. In orer to avoi the above problems, for short baseline inematic relative positioning, we present a computationally efficient an numerically reliable approach. Our approach can be regare as an extension of the stanar information square root filtering technique. The basic iea is to write all available measurement equations (we use both coe an carrier phase measurements base on L1 signals) an process equations together to form a large linearize moel, which has special structures. Then base on this moel, we evelop a recursive algorithm to compute the least squares estimates of positions an velocities. We obtain not only the estimate of the current position (this is calle filtering), but also the estimates of previous positions (this is calle smoothing). Our algorithm is computationally efficient an numerically reliable. One thing which maes the estimation problem much more complicate an also more interesting is that the integer ambiguity vector may change ue to cycle slips, loss of signals, an satellite rising/setting. In this paper we show how to hanle this problem in our algorithm. Unlie the typical literature in GNSS, we give etails about the algorithm so that people can implement it without ifficulties. 1 INTRODUCTION The often use approach to inematic relative GPS positioning is to employ ifferencing techniques to establish linearize state-space moels an then apply the Kalman filtering techniques to estimate the receiver positions (an velocities et al), see, e.g., 8, 9, 11, 12, 13, 14, 17, 19. The state-space moel June 25, Cambrige, MA 937

2 can be written as: y = A x + B z + v, v N(,Σ v ), (1) x +1 = F x + u +1, u +1 N(,Σ u +1 ), (2) for = 1, 2,..., where (1) an (2) are referre to as the measurement (or observation) equation an the process (or state) equation, respectively, x is the vector incluing position, velocity et al, z is the integer ambiguity vector, A, B an F are nown matrices, an v an u +1 are the ranom measurement noise vector an process noise vector, respectively, following normal istributions. In orer to apply a stanar Kalman filtering technique, a typical approach is to stac x an z as a state vector, so the above moel is transforme to: y = x A B + v z, v N(,Σ v ), (3) x+1 F = x u+1 +, (4) z u+1 w +1 I N (, z Σ u +1 w +1 ). (5) This approach not only (unnecessarily) maes the estimation problem larger so that the computational cost will be higher, but also results in a singular covariance matrix for each process noise vector (note that the noise covariance matrix of w +1 is a zero matrix in (5)) so that some stanar Kalman filter techniques, such as the square root information filter (see, e.g., 1) cannot be applie. To overcome this singularity problem, the typical metho (see, e.g., 12, 14) is to artificially assigne a scale ientity matrix δi (δ is a small positive scalar) as the covariance matrix to w +1. But this approximation oes not conform with the physical characteristic of the integer ambiguity vector an may cause a numerical problem if δ is too small or lea to a loss of accuracy of the position estimates if it is not small enough. In this paper, for short baseline inematic positioning we propose a new approach to avoi the above ifficulties. We combine all the measurement equations (for L1 signals) an process equations to form a linearize state-space moel. Base on the moel, we esign a recursive approach to compute the least squares (LS) estimates of the positions. Computing the corresponing error covariance matrices of the estimates are also consiere. We exploit the structure of the moel to mae the algorithm computationally efficient an use orthogonal transformations to mae it numerically reliable, an also tae the storage efficiency into consieration. This approach avois the mentione rawbacs of the current approach an is an extension of the recursive least squares (RLS) metho presente in 4, which use only the coe an carrier measurement equations for positioning, an an extension of the square root information filter given in 16. The rest of this paper is organize as follows. In Section 2 we present the state-space moel. In Section 3 we use orthogonal transformations which mae full use of the structures of the moel to compute the LS estimates of the positions an velocities, an the corresponing error covariance matrices, an show how to hanle satellites rising an setting. Finally a summary is given in Section 4. Throughout this paper bol upper case letters are use to enote matrices an bol lower case letters are use to enote vectors. The unit matrix is enote by I an its ith column by e i, while e 1, 1,...,1 T (we use to mean is efine to be ). The 2-norm of a vector or a matrix is enote by 2. The jth column of a matrix A is enote by A(:, j), the (i, j) entry of A by A(i, j), an the ith entry of a vector a by a(i). The mean an covariance are enote by E{ } an cov{ }, respectively. For a ranom vector x which is normally istribute with mean u an covariance Σ, we enote it by x N(u,Σ). 2 THE MATHEMATICAL MODEL In this section, we establish a state-space moel for short baseline inematic relative positioning base on the measurement equations an process equations. In inematic relative positioning, one receiver is set up at a surveye site whose exact position is nown, an the other receiver is roving an its positions are to be estimate. We assume that the carrier phase an coe measurements of the reference receiver are available at the roving receiver. We also assume the baseline is short such that the ionospheric reflection an tropospheric reflection can (almost) be cancele after applying the between-receiver single-ifference technique. We consier only the L1 carrier since many receivers can only receive the L1 signal. But it is easy to exten our approach to the ual frequency case. For short baseline relative positioning, in 4, we erive the following single ifference carrier phase an coe measurement equations corresponing to satellite i at epoch φ i = λ 1 (ω i e i ) T b + N i + β φ + νi, (6) ρ i = λ 1 (ω i ei )T b + β ρ + µi, (7) where the units of each of the terms in above equations are number of wavelengths, φ i an ρi are the single ifference carrier phase measurement an coe measurement, respectively; λ is the wavelength of the June 25, Cambrige, MA 938

3 L1 carrier; b is the baseline vector pointing from the reference receiver to the roving receiver; ω i satisfies ω i = 2h i r b 2 /( h i r 2 + h i r b 2 ), (8) with h i r being the vector pointing from the reference receiver to satellite i (note that ω i is close to 1); ei is the unit vector pointing from the mipoint of the baseline to satellite i; N i is the single ifference integer ambiguity; β φ is the single ifference receiver cloc error (incluing harware elay an initial phase) for the carrier phase measurement; β ρ is the single ifference receiver cloc error (incluing harware elay) for the coe measurement; an ν i an µi are the single ifference noises (incluing multipath errors) for the carrier phase measurement an the coe measurement, respectively. We have an important assumption (see 19, Sec 3.4) for (6) an (7) that all the carrier phase noises ν i (or coe noises µj l ) for ifferent satellites an ifferent epochs are unbiase inepenently istribute with the same normal istribution, an ν i an µ j l are also assume to be inepenent. Suppose there are m visible satellites an efine y φ φ1, y ρ φ m ρ1, E λ 1 (ω1 e1 )T, ρ m (ω mem )T (9) a N1, v φ N m ν1, v ρ ν m µ1.. µ m We can combine m single ifference carrier-phase measurement equations (6) an m single ifference coe measurement equations (7) together to give y φ = E b + a + β φ e + vφ, vφ N(, σ2 φi m ), (1) y ρ = E b + β ρ e + vρ, vρ N(, σ2 ρ I m), (11) where we assume that the stanar eviations σ φ an σ ρ are nown. Note that in E, ω i ei epens on the baseline b. In other wors, (1) an (11) are actually nonlinear equations in terms of b. We write E E(b ). (12) This E is nown once b is nown. Since E(b ) is not very sensitive to changes in b, with receiver s ynamic available, it is usually sufficient to use the preicte estimate b 1 of b (see the en of Section 3.1) to compute E(b 1 ) an use it to replace E in (1) an (11). Then, when y ρ an yφ are available, we can obtain a better estimate of b by the metho to be given later. We coul use the better estimate to re-compute an approximation to E an estimate b again if necessary. Thus from now we just assume that the E in (1) an (11) are nown. In the single ifference measurement equations (1) an (11), there are still the unnown errors β ρ an β φ. A typical way to remove these errors is by using the ouble ifference technique. However, ouble ifference maes the measurements correlate. Thus, we just follow 3 an 4 to use orthogonal transformations to eliminate these errors, leaving the measurement equations still uncorrelate. We first eliminate β φ in the carrier-phase measurement equation (1). Let P R mm be a Householer transformation (see, e.g., 6, p29) such that Pe = me 1, P I 2uuT u T u, u = e 1 e. (13) m p T Partition P, where P P = e, I m 1 eet m m. m Multiplying both sies of (1) by P from the left, we obtain pt y φ pt E Py φ = p T mβ φ pt v b φ P E + a+ + P Pv φ. (14) Since only the first equation in (14) inclues the term β φ, it can be roppe off for position estimation to give: P y φ = P E b + Pa + Pv φ, (15) where Pv φ N(, σ2 φ I m 1). We have use the simple Householer transformation instea of the often use ouble ifference technique to eliminate the unnown error β φ, leaving the transforme measurements still uncorrelate. Since P is (m 1) m, it oes not have full column ran an we will not be able to get a unique estimate of a. If we set P a as a new vector, we woul lose the integer nature of a. So following 3 an 4 we introuce the ouble ifference integer ambiguity (DDIA) vector z: z = N 2 N 1, N 3 N 1,..., N m N 1 T, (16) where without loss of generality we choose satellite 1 as the reference satellite. Note that z is still a vector of integers. Define G I m 1 eet m m, J e, I m 1. (17) It is easy to verify from (13) that G is nonsingular an P = GJ, P a = GJa = Gz. (18) June 25, Cambrige, MA 939

4 Thus (15) becomes Py φ = PE b + Gz + Pv φ. (19) Similarly, we can eliminate β ρ from the coe measurement equation (11) an obtain where Pv ρ N(, σ2 ρ I m 1). Py ρ = PE b + Pv ρ, (2) In orer to combine (19) an (2), we efine σ 1 φ, H y P y φ σρ 1 Py ρ σ 1 B φ G, v σ 1 φ σ 1 ρ σ 1 φ σ 1 ρ P v φ Pv ρ Thus we can rewrite (19) an (2) as PE, (21) PE, y = H b + Bz + v, v N(, I 2(m 1) ). (22) In this measurement equation, we use the geocentric Cartesian coorinate system. However, we woul lie to wor with the local geoetic coorinate system later. This can easily be one by pre-multiplying b by the orthogonal transformation matrix which transforms the coorinates in the geocentric Cartesian coorinate system to the coorinates in the local geoetic coorinate system (N,E,U) with the reference receiver as the origin, an by post-multiplying H by the transpose of the orthogonal transformation matrix in (22). For etails on the transformations of coorinates in ifferent coorinate systems, see, e.g., 12, Secs 6.2, 7.1. To simplify the notation, we just assume that the components of b x, y, z T in (22) are alreay the coorinates in the local geoetic coorinate system. Now we woul lie to introuce the process equations. There are ifferent types of process equations to moel the ynamics of receivers, see, e.g., 1, 2. In this paper, we aopt the following process equations which are suitable for short baseline relative positioning with low ynamics (see, e.g., 2, Chap 6, 12, Sec 1.3.3, 14, Appenix L) x +1 = Fx + ū +1, ū +1 N(,Σū), (23) for = 1, 2,..., where x = x, ẋ, y, ẏ, z, ż T with ẋ, ẏ, ż being the velocities in each irection, the transition matrix F satisfies F = C C 1 T, C =, 1 C with T being the time interval between two consecutive epochs, an the noise covariance matrix Σū = q x W T 3 /3 T q y W, W = 2 /2 T q z W 2 /2 T with q x, q y an q z being the power spectral ensities of the process noise. In orer to mae the state vector b in the measurement equation (22) the same as the state vector x in the process equation (23), we moify the matrix H to inclue three zero columns: A H (:, 1),, H (:, 2),, H (:, 3),, thus (22) can be rewritten as y = A x + Bz + v, v N(, I 2(m 1) ). (24) In orer to esign a recursive algorithm for positionvelocity estimation, we nee to have an estimate an its error covariance matrix for the initial state vector x 1. The usual assumption is that we have E{x 1 } = x 1, cov{x 1 x 1 } = Σ 1, which can be rewritten as x 1 = x 1 ū 1, E{ū 1 } =, cov{ū 1 } = Σ 1. (25) In practice, we can choose x 1 an Σ 1 by the following strategy. At the beginning ( = 1), we may have some iea about the location of the roving receiver, so can give an initial estimate of b 1. If we o not have any goo information about the location of the roving receiver, we can tae a zero vector as an initial estimate of b 1. Let the initial estimate of b 1 be enote by x 1, y 1, z 1 T. For each velocity component, we tae zero as an initial estimate. Thus the initial estimate of x 1 is x 1 = x 1,, y 1,, z 1, T. Following 1, we tae the error covariance matrix Σ 1 of x 1 to be a iagonal matrix. Due to the large uncertainty of x 1, the iagonal elements of Σ 1 shoul usually tae large values. The choice of specific values epens on a specific application. The equations (23) (25) give the mathematical moel for our positioning algorithm. 3 A RECURSIVE LS ALGORITHM If a goo process moel for the roving receiver is not available, 4 has alreay shown how to use only measurement equations (see (22)) to esign an efficient an numerically reliable recursive LS algorithm for positioning. Now we woul lie to exten this approach June 25, Cambrige, MA 94

5 to the case where the process moel (23) which is assume to be an accurate moel is available. In the extension, we will use the ieas given in 16, which presente a square-root information filter (a variant of the stanar Kalman filter) to compute the LS estimates for a stanar state-space moel (forme by equation (1) with no the B z term an equation (2)). 3.1 Position Estimation Following 16, we first transform (23) an (25) so that the process noise vectors will be ientically istribute with zero mean an unit covariance matrices as the measurement noise vectors in (24). Suppose we have the Cholesy factorizations Σ 1 1 = U T 1 U 1, (Σū) 1 = U T U, (26) where U 1 an U are upper triangular. Notice that the time-invariant process noise covariance matrix Σū is in a bloc-iagonal form, so we can easily obtain U W / q x U = U W / q y U W /, q z where U W = 6/(T 3T) 3/ T 1/. T Multiplying both sies of (25) an (23) by U 1 an U from the left, respectively, leaing to c U 1 x 1 = U 1 x 1 U 1 ū 1 U 1 x 1 + u 1, (27) Ux +1 = U Fx + Uū +1 Fx u +1, (28) for = 1, 2,..., where u 1 N(, I 6 ), u +1 N(, I 6 ). At epoch, we combine all available process equations (27) an (28 an measurement equations (24) to form the following linear moel c U 1 u 1 B or y 1 y 2 y A 1 F U x 1 = A 2 B x 2 F x + U z A B v 1 u 2 v 2 u v (29) y () = A () x () + v (), v () N(, I). (3) Notice that since U 1 an U are nonsingular, A () has full column ran. For numerical stability, we use the QR factorization metho to fin the LS estimate of x () in (3) (see 6, Sec 5.3). Suppose we fin an orthogonal matrix Q such that (Q () ) T A () = (Q () 1 )T (Q () A () = 2 )T R (), (31) where R () is nonsingular upper triangular. Then the linear moel (3) can be transforme to (Q () 1 )T y () (Q () = 1 )T y () R () (Q () x () + 1 )T v () (Q (), 1 )T v () where transforme noise vector still has zero mean an (Q () unit covariance matrix, i.e., 1 )T v () (Q () N(, I). 1 )T v () Then the LS estimate ˆx () satisfies R ()ˆx () = (Q () 1 )T y (), an its error covariance matrix is cov{x () ˆx () } = (R () ) T R () 1. (32) Now we give etails for computing (31). We will follow 4 an 16 to mae full use of special structure of A (). First we set ˆR 1 U 1, Ĉ 1. For each j we fin an orthogonal matrix Q T j to zero A j as follows: ˆRj Q T Ĉ j j = B A j Rj Cj, (33) M j where R j is nonsingular upper triangular an Q j is orthogonal. Consiering the structures of ˆR j R 66, A j R (2m 2)6 an B R (2m 2)(m 1) in (33), we use Givens rotations to eliminate the non-zero elements of A j columnwise from the left to the right an for each column from the bottom to the top. For each non-zero element in the ith row (2 i 2m 2) of A j, we combine it with the (i 1)th element in the same column to construct the rotation. For each nonzero element in the first row of A j, we combine it with the corresponing iagonal entry of ˆRj in the same column to construct the rotation. We give part of the process for m = 6 schematically in Figure 1, where the letters inicate the orer of the zero elements becoming non-zeros in A by the Givens rotations an these elements will be eliminate later; the numbers in squares inicate the orer of these zero elements in B becoming non-zeros after the Givens rotations. Note June 25, Cambrige, MA 941

6 that the last m 1 rows of B are all zeros before applying the rotations, thus after the rotations, the last m 4 rows of M are zeros. It is easy to now a total of 6m rotations are neee. a b c e f Figure 1: Scheme of transformation (33) After the transformation (33), another orthogonal matrix Q j is esigne such that R j Cj R j R j,j+1 C j Q T j M j = M j, (34) F U ˆRj+1 Ĉ j+1 where R j an ˆR j+1 are both upper triangular. We also use Givens rotations to zero F R 66 by taing avantage of the upper triangular structure of R j R 66 an U R 66. The transformation on the first two bloc columns of (34) is schematically shown in Figure 2, where the number in a circle inicates the element is eliminate in the ith rotation, while the number in a square inicates this element is generate by the ith rotation. When we apply the above transformations to A (), we also apply them to y () in (29) or (3). The transformations are enote by Q T j where ĉ 1 c. ĉj cj =, QT y j j j c j j = c j j ĉ j+1, (35) Figure 2: Scheme of transformation (34) Therefore, finally we have transforme (29) to c 1 R 1 R 12 C 1 1 M 1 c 2 R 2 R 23 C 2 x 1 2 M 2 x 2 x (36) c R C z M where to simplify the presentation, we have omitte the transforme noise vector, which still has zero mean an a unit covariance matrix since we use orthogonal transformations, an so we use instea of =. Later, we will continue to use this notation when we transform the moel by orthogonal transformations. Reorering the equations in (36), we obtain two submoels: x c 1 R 1 R 12 C 1 1 c 1 x 2 R 1 R 1, C 1 x c R. C z (37) 1 M 1 z. (38) 1 M 1 M Obviously we nee to first estimate z by (38) an then estimate x,...,x 1 by (37). The estimate of the DDIA vector z can be compute in a recursive fashion. Suppose at epoch 1 we have June 25, Cambrige, MA 942

7 compute the following orthogonal transformations M 1 1 T T M 2 1 = S 1, T T 2 1 = f 1, f 1 M 1 1 where T T 1 is orthogonal, an S 1 is nonsingular upper triangular. Then at epoch, with the newly obtaine M an (cf. (33) an (35)) we use Householer transformations to compute T T S 1 M = S, T T f 1 = f f 1, ˆf f. ˆf (39) In the above transformations, for computational efficiency, we have to mae full use of the structures of S 1 (which is upper triangular) an M (whose last m 4 rows are zero). Note that we never form or store T 1 an T. Then (38) is transforme to f f S z. (4) So we can compute the estimate z of z by solving the (m 1) (m 1) upper triangular system S z = f. (41) Then from (37) we can obtain the estimate x of x an estimates x j of x j for j = 1, 2,...,1 by solving triangular linear systems: R x = c C z (42) R j x j = c j R j,j+1 x j+1 C j z. (43) In other wors, at epoch, we can obtain not only the estimate of the current position, which is calle the filtere estimate, but also the estimates of the previous positions, which are calle the smoothe estimates. In real-time applications, smoothe estimates are usually not require, then at epoch, only R, C, c, S an f shoul be temporarily store for position estimation at next epoch. Before we finish this section, we woul lie to mention that we nee to get the preicate estimate x 1 of x before we use the measurement equation (24) to obtain the filtere estimate x. After we obtain the filtere estimate x 1 1 of x 1, from the process equation (23) (with replace by 1), we can immeiately obtain x 1 = Fx 1 1. The first, thir, an fifth elements of x 1 just form the preicate estimate b 1 of the baseline vector b. This b 1 is use to compute an approximation to E to eal with the nonlinearity problem, see the iscussion given in Section Computing Error Covariance Matrices In orer to have some iea about the errors of the estimates, we nee to compute the corresponing error covariance matrices cov{x j x j } for j = 1, 2,..., at epoch. We first consier j =. Combine the th bloc equation of (37) an the top part of (4) to give c R C x. (44) f S z Note that (44) etermines the estimate z of z an the estimate x of x. In orer to obtain cov{x x }, we ecouple x an z in (44) by multiplying an orthogonal Z T to the coefficient matrix from the left to zero the C bloc (see 3, 4): Z T R C R =, (45) S N S where Givens rotations are use to tae avantage of the upper triangular structures of R an S an prouce upper triangular R. We zero C columnwise from the left to the right, an for each column we start from the bottom to the top. Only one element of C an one corresponing iagonal entry of S are use to construct one rotation. Since C is 6 (m 1), a total of 6(m 1) rotations are neee. Thus after applying Z T to (44) from the left, we obtain c R x. (46) f N S z Then obviously x, which satisfies R x = c, has the error covariance matrix: cov{x x } = (R T R ) 1. For each j <, we can recursively compute the covariance matrix cov{x j x j }. Suppose the (j +1)th (bloc) equation in (37) was transforme to the following equation when we compute cov{x j+1 x j+1 }. c j+1 R j+1 x j+1, (47) where R j+1 is nonsingular upper triangular. To compute the covariance matrix cov{x j x j }, we stac the jth bloc equation in (37), equation (47), an the top part of (4) together to give: c j c j+1 R j R j,j+1 C j R j+1 x j x j+1. (48) f S z June 25, Cambrige, MA 943

8 Note that equation (48) etermines the estimates z, x j+1 an x j. We just use the same metho as we i for the transformation (45). We first use C j R 6(m 1) an S to zero C j, an then use the transforme R j,j+1 R 66 an R j+1 to zero the former by the Givens rotations. We nee a total of 6(m + 5) rotations. Thus (48) is transforme to c j c j+1 R j R (j+1,j) Rj+1 Cj+1 x j x j+1 f j N j S j z (49) From the first bloc equation in (49) we obtain cov{x j x j } = (R T j R j ) Fixing the Integer Ambiguities In Section 3.1, we have compute the real-value estimate z of the ouble ifference integer ambiguity (DDIA) vector z by (41). However, the elements of z are integers. If we only use its real-value estimate in (42) an (43), we may not be able to obtain high accurate estimates of the positions quicly. To fix the obtaine real-value estimate of z to an integer vector, we can use the original LAMBDA metho 18 or the moifie LAMBDA metho 5, which is more computationally efficient. To use either of these two methos at epoch, we nee two pieces of information. One is the real-value vector z, an the other is the error covariance matrix Σ of z. From the top part of (4), we obtain Σ = cov{z z} = (S T S ) 1. (5) Once z is fixe as a vector of integers an is valiate, we can obtain the corresponing positionvelocity estimates (43). Starting from epoch + 1, we will not nee to estimate z any more. Therefore, to compute the estimate of x i for any i >, we irectly use the integer-value z as the estimate of z at that epoch. In other wors, we solve the following upper triangular systems (cf. (42) an (43)): R i x i i = c i C i z, i > (51) R j x j i = c j R j,j+1 x j+1 i C j z, (52) for j = i 1,...,1 an i >. 3.4 Hanling the Change of Ambiguity Vector So far, we have assume that z has a fixe imension. In GPS-base positioning, this means that we have the same number of satellites in the whole observation span. However in practice, this is not always true. When we have a long observation span, the imension of the DDIA vector z may change ue to satellite rising an setting. Furthermore, when the signal from a satellite is missing for some reason at some epoch, the imension of z changes (this can be regare as satellite setting). While later the signal from this satellite is re-foun, the imension of z changes again (this can be regare as satellite rising). We also have assume that the value of each element of z is constant, but it changes when there is a cycle slip in the corresponing signal (see, e.g., 7, Sec 9.1.2). Detection an repair of cycle slips is an important topic in GPS positioning. We can incorporate a metho for cycle slip etection into our positioning algorithm. We will not use a metho to repair cycle slips since it may not be easy, therefore when a cycle slip is etecte between two epochs, we just assume there were satellite setting an rising between these two epochs, although physically the setting an rising satellites are ientical. We will hanle this problem by following 3, 4 where only measurement equations were use for positioning. The major complicate tas is to upate the estimate of the DDIA vector z from one epoch to the next one. Once z has been estimate at an epoch, the position estimate at this epoch can easily be obtaine, Therefore, the problem is to fin a way to achieve the equivalents of the transformations (33), (34), (35) an (39). Since the DDIA vector may be ifferent for ifferent epochs, we will use z instea of z an B instea of B in the measurement equation (22) at epoch. Note from (16) that every element of the DDIA vector z has the form N n N i, where n correspons to a nonreference satellite an i is the reference satellite. It is important to be aware that we will use the same reference satellite for every element in every DDIA vector at a given epoch j an this reference satellite must be visible at that epoch j, so if it sets between epochs 1 an, for some > j, then (see Case 2 later) we can choose to use a ifferent reference satellite at epoch. When we tal about a DDIA vector at an epoch, a non-reference satellite means any satellite which is not the reference satellite at that epoch. If a satellite sets an rises again, it will be consiere as a new satellite. In the following we list ifferent DDIA vectors we will use: z : (whose elements correspon to) all the nonreference satellites that are visible for at least one epoch from epoch 1 to epoch ; z : all the non-reference satellites that go own between epoch 1 an epoch ; June 25, Cambrige, MA 944

9 z : all the non-reference satellites that are visible at epoch ; z r : all the non-reference satellites that are visible at epoch 1 an remain at epoch ; z u : all the non-reference satellites that come up between epoch 1 an epoch ; z : all the non-reference satellites that go own between epoch 1 an epoch. Note that when a satellite rises, the imension of z increases by one, but a setting satellite leaves the imension unchange. In our constant satellite case, z was just z in (16). The following are the relationships between these DDIA vectors which will be use later: z = z 1 z, z = z r z u, z = z = z z 1 z z r z u. (53) Note that z 1 is a rearrangement of the elements of z r an z, so we can fin a permutation matrix Π = Π (1) such that,π(2) Π T z 1 = (Π (1) z )T 1 (Π (2) = z )T 1 z z r. (54) Assume that at the en of epoch 1 we have obtaine the equivalent of the top part of (4) for epoch 1 f 1 S 1 z 1, (55) where S 1 is nonsingular upper triangular. We can partition z 1 as z z 1 = 1. z 1 Then with compatible partitioning, we can rewrite (55) as (1) f 1 S (1) 1 f 1 1 S 1 z 1 z 1 (56) (1) where both S 1 an S 1 are nonsingular upper triangular. Notice that if no satellites rise or set from epoch 1 to epoch 1, then the top part of (56) will isappear an the bottom part is just the top part of (4) with replace with 1. In the following we will combine (56) with the relevant equations at epoch which provie new information about the DDIA vectors in orer to obtain the estimates of the DDIA vectors. We consier two cases separately. Case 1: The reference satellite at epoch 1 remains at epoch. We still use it as the reference satellite at epoch. Suppose at epoch 1, we obtaine the equivalent of the bottom part of (37): c 1 R 1 x 1 + C 1 z 1. (57) At epoch, we first combine (57) with the process equation (28) to give c 1 R 1 C 1 x 1 x. (58) F U z 1 As in (34), we apply the Givens rotations to zero F in the coefficient matrix in (58). Then (58) is transforme to (cf. (34) an (35)) c 1 ĉ R 1 R 1, C 1 ˆR ˆ C x 1 x z 1. (59) Note that we have ecouple x 1 from x an z 1. Once the estimates of x an z 1 are obtaine, the estimate of x 1 can be obtaine immeiately. Now we consier to use the measurement equation (24) at epoch, which, for the new situation, is written as y A x + B z, (6) where a subscript has been ae to B an z an the noise vector has been omitte. Our tas is to combine the bottom part of (59) an (6), both of which involve x an DDIA vectors, to obtain the equivalent of the bottom two bloc equations in (36). In orer to o this, we have to rewrite them. With compatible partitioning (cf. (53)), ˆ C z 1 in (59) can be written as ˆ C ˆ C(1) z z 1 = 1. For ˆ C(2) z 1, use (54) to write z 1 = ˆ C(2) Π Π T z 1 ˆ C(2) = Π(1) Π(2) z 1 z z r. (61) Therefore, we can rewrite the bottom part of (59) as ĉ ˆR ˆ C(1) Π(1) Π(2) x z 1 z z r. (62) Now we rewrite (6). We partition B compatibly with the partitioning of z in (53), so that we have B z = B (1) B (2) z r. z u June 25, Cambrige, MA 945

10 Therefore (6) can be rewritten as y A B (1) B (2) x z r. (63) Combining (62) with (63), we have ĉ ˆR ˆ C(1) y Π(1) Π(2) z u A B (1) B (2) x z 1 z z r (64) In orer to ecouple x from the DDIA vectors, we multiply both sies of (64) by a sequence of Givens rotations from the left to zero A, giving (cf. (33), (35)): c R C(1) M(1) C (2) M (2) C (3) M (3) C (4) M (4) x z u z 1 z z r z u., (65) Note that the bottom bloc equation of (65) involves only DDIA vectors. To compute the estimate of z or give the equivalent of (55) at epoch, we want to combine (56) with the bottom part of (65). In orer to o that, we use (54) S (2) an write 1 z 1 an S 1 z 1 in (56) as follows: 1 z 1 = 1 Π(1) z 1 Π(2) z r, (66) S 1 z 1 = S 1 Π (1) S 1 Π (2) z. (67) Then we stac (56) on the bottom part of (65), an use (66) an (67) to give: S (1) f(1) Π(1) 1 Π(2) z 1 f 1 S 1 Π (1) S 1 Π (2) z z r. M (1) M (2) M (3) z r M (4) (68) Then we perform the following orthogonal transformations to (68): T T = S (1) 1 1Π (1) 1Π (2) S 1 Π (1) S 1 Π (2) M (1) S (1) M (2) S M (3) S, M (4) z u T T f(1) 1 f 1 S (1) = f (1) f ˆf f, ˆf where both an S are nonsingular upper triangular. This has complete the upate an provie the equivalents of (55) an its expane form (56) for epoch : f S z, (69) or (1) f S (1) f S z z, (7) where we have use the relationships among ifference DDIVs given in (53). We now can compute the LS estimates z of z an z of z by solving S z =f, S(1) z =f(1) S (2) z, z = z z Note that if no satellites rise or set between epochs 1 an, z an zu will have no elements, therefore in (54) we can tae Π = Π (2) = I, giving z 1 = z r = z. With the estimate DDIA vectors, we can compute the filtere estimates an smoothe estimates of positionvelocity vectors. Using (53), from the top part of (65) we can obtain x by solving where. R x = c C z, (71) C C(1), C(2), C(3), C(4). (72) Suppose we have compute the estimate x j+1 of x j+1, we will show how to compute the smoothe estimates of x j of x j. From (59), we see the equivalent of its first bloc equation for each epoch j < can be written as c j R j x j + R j,j+1 x j+1 + C j z j. (73) Since the elements of z j are part of z, there exists a matrix P j which is a permutation matrix with possible zero columns ae, such that z j = P j z, so the corresponing estimate z j of z j satisfy z j = P j z. Then we can compute the smoothe estimate x j by solving R j x j = c j R j,j+1 x j+1 C j z j. (74) June 25, Cambrige, MA 946

11 Finally we woul lie to iscuss how to compute the error covariance matrices cov{x j x j } for j = 1, 2,...,. First we consier j =. We combine the top part of (65) with (69) an use (53) to give c f R C S x z, (75) where C is efine by (72). So we can use the same approach of estimating the filtere error covariance matrix cov{x x } presente in Section 3.2 to (75). We fin an orthogonal Z to transform (75) to (cf. (45), (46)): c R x, (76) f N S z where R is nonsingular upper triangular. Therefore we have cov{x x } = (R T R ) 1. Now we consier how to compute cov{x j x j } in the orer of j = 1, 2,...,1. Suppose in the process of fining cov{x j+1 x}, we obtaine the following equation (cf. the top bloc equation in (76)): j+1 = R j+1 x j+1. (77) where R j+1 is nonsingular upper triangular. We combine (73), (77) an (69) together to give: c j c j+1 = R j R j,j+1 Cj P j R j+1 x j x j+1. (78) f S z The above equation has the same structure as (48) in, so we can aopt the same approach to computing cov{x j x j } given there. The Givens rotations can be foun to transform (78) to c j R j c j+1 R (j+1,j) Rj+1 Cj+1 x j x j+1, N j Sj z f j where R j is nonsingular upper triangular. Thus it follows that cov{x j x j } = (R T j R j ) 1, Case 2: The reference satellite (satellite 1, say) of epoch 1 goes own between epochs 1 an. We can choose any satellite visible at epoch 1 an remaining at epoch, without loss of generality assuming that is satellite 2, to be the reference satellite at epoch. Suppose at epoch, the new measurement equation is y A x + B z, (79) where z is the DDIA vector with satellite 2 as the reference satellite. At the en of epoch 1 we have (55), where z 1 R m 1 say, z 1 N 2 N 1, N 3 N 1,..., N m N 1 T. Define the corresponing vector z 1 with satellite 2 as the reference satellite, along with the matrix K z 1 N 1 N 2, N 3 N 2,...,N m N 2 T, 1 K. e I m 2 It is easy to verify that b KK = I, z 1 = K z 1. (8) This implies that we can easily transform a DDIA vector with one satellite as the reference satellite to another DDIA vector with another satellite as the reference satellite. Define S 1 S 1 K, then we can rewrite (55) as f 1 S 1 z 1 = S 1 KK z 1 = S 1 z 1. If we fin an orthogonal matrix from the left to triangularize S 1, we get essentially the same situation as in Case 1, therefore we then can follow the steps in Case 1 to hanle it. 4 Summary A recursive LS approach was presente for short baseline inematic GPS positioning by using combine carrier phase an coe measurements an a process moel. Unlie the approaches using stanar Kalman filter in the literature, our approach gives avantages in numerical stability an efficiency by using orthogonal transformations an taing full avantage of the structure of the problem. We gave full computational etails for computing position-velocity estimates (incluing both filtere an smoothe position-vector estimates) as well as the corresponing error covariance matrices. We also hanle the computation for possible satellite setting an rising. If a process moel in a specific application is not the same as the one use in this paper, our approach can be moifie without ifficulty. Our approach can also be moifie easily to hanle ual frequency signals June 25, Cambrige, MA 947

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