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SECTION I

## INTRODUCTION

PRECISE point positioning (PPP), first described in [1], is a global navigation satellite system (GNSS) positioning method that processes pseudorange and carrier phase measurements from a standalone GNSS receiver to compute positions with a high, decimeter or centimeter, accuracy everywhere on the globe. By using satellite orbits and clocks, as well as other corrections (e.g., for Earth rotation, tides and ocean loading, phase wind-up, etc.), the GNSS receiver position along with other parameters, like atmospheric delays, can be estimated [2], [3], [4]. In recent years, services have been developed which allow high accuracy ephemeris data to be made available in real-time to users [5], [6], [7]. Such availability has created, and will continue to create, a wide range of PPP applications [8], [9]. Also, various forms of PPP are possible, like, e.g., single-frequency PPP using global ionospheric maps (GIMs) [3], [4], dual-frequency PPP using ionosphere-free combinations [2], or integer ambiguity resolution [10], [11], [12], [13] enabled real-time kinematic (RTK) PPP [14], [15], [16], [17]. Next to positioning, PPP is also used in remote sensing, [18], [19], as an ionospheric or tropospheric sensor, [20], [21], or for time-transfer [22], [23], [24].

In this paper, we extend the PPP concept to array-aided PPP (A-PPP). A-PPP is a GNSS measurement concept that uses GNSS data from multiple antennas in known formation to realize real-time precise attitude and improved positioning of a (stationary or moving) platform. It is assumed that the local antenna geometry is known in the body (platform) frame and that each of the antennas in the array collects GNSS pseudorange and carrier phase data. The A-PPP principle can then briefly be described as follows. The known array geometry in the platform frame enables successful integer carrier-phase ambiguity resolution, thereby realizing a two-order of magnitude improvement in the between-antenna GNSS pseudoranges. These very precise pseudoranges are then used to determine the platform's earth-fixed attitude, thus effectively making the platform a 3D direction finder. At the same time, the precision of the absolute pseudoranges and carrier phases are improved by exploiting the correlation that exists between these data and the very precise between-antenna pseudoranges. This improvement enables the improved platform parameter estimation. Also integrity improves, since with the known array geometry, redundancy increases, thus allowing improved error detection and multipath mitigation [25].

This contribution is organized as follows. In Section II, the GNSS models for precise point positioning and array-based attitude determination are presented. Their respective estimation problems are usually treated and solved independently. In Section III it is shown why and how this can be improved. A multivariate constrained formulation of the combined position-attitude model is introduced, which is structured as TeX Source \eqalignno{{\rm E}({\mbi {\cal Y}}) =&\, {\mbi{M}} {\mbi {\cal {B}}} + {\mbi N} {\mbi {\cal {A}}} + {{\mbi {\cal C}}} \cr {\rm Cov}({\rm {vec}}({\mbi {\cal Y}})) =&\, {\mmb{\Sigma}} \otimes {\mbi Q} &{\hbox{(1)}}} with ${\mbi {\cal Y}}=[{\mbi y}_{1}, {\mbi Y}]$ the random matrix of GNSS array observables and ${\mbi {\cal B}}=[{\mbi b}_{1}, {\mbi {RF}}]$, ${\mbi {\cal A}}=[{\mbi a}_{1}, {\mbi Z}]$, and ${{\mbi {\cal C}}}=[{\mbi d}_{1}, {\bf 0}]$ the matrices containing the deterministic parameters that need to be estimated under the attitude orthonormality and ambiguity integer constraints TeX Source $${\mbi R} \in \BBO^{3 \times q} \quad {\rm and} \quad {\mbi {Z}} \in \BBZ^{fs \times r} \eqno{\hbox{(2)}}$$${\mbi {\cal B}}$ is the matrix of antenna positions, ${\mbi {\cal A}}$ the matrix of carrier phase ambiguities and ${{\mbi {\cal C}}}$ the matrix of atmospheric delays and satellite-related terms. By means of a decorrelating transformation it is shown which improvements can be realized and how the PPP concept can be extended to array-aided PPP.

An essential component of A-PPP processing is solving the constrained array estimation problem. This novel multivariate, orthonormality-constrained, mixed integer least-squares (ILS) problem is solved in Section IV. In contrast to the existing constrained ILS problems, as box-constrained ILS [26] and ellipsoid-constrained ILS [27], our problem is a mixed real/integer least-squares problem, of the multivariate type, with orthonormality constraints on the real-valued parameters. As is shown, the two type of constraints play a distinct role. The integer matrix constraint is necessary to obtain the most precise instantaneous attitude and position solution, whereas the inclusion of the orthonormality constraint in the ambiguity objective function is essential to achieve a high probability of correction integer estimation [28].

In the following, a frequent use is made of the Kronecker product $\otimes$ and the ${\rm vec}$-operator. For their properties, see, e.g., [29], [30]. The expectation and covariance matrix of a random vector ${\mbi x}$ are denoted as ${\rm E}({\mbi x})$ and ${\rm Cov}({\mbi x})$, respectively. For the covariance matrix of a random matrix ${\mbi X}$, we often write ${\mbi Q}_{{\mmb {XX}}}$ instead of ${\rm Cov}({\rm vec}({\mbi X}))$. For the weighted squared norm, the notation $\Vert . \Vert_{\mmb Q}^{2}=(.)^{T} {\mbi Q}^{-1}(.)$ is used. Although the terminology of weighted least-squares estimation is used throughout, the given least-squares (LS) estimators are also maximum likelihood estimators in the Gaussian case and best linear unbiased estimators (BLUEs) in the linear model case, since the inverse covariance matrix of the GNSS observables is used as weight matrix.

SECTION II

## POSITIONINGAND ATTITUDE

In this section we present the PPP observation equations for positioning and the array observation equations for attitude determination. Although these models are currently restricted to the usage of single- or dual-frequency GPS data, we formulate them for the general multifrequency case, thus enabling next generation GNSS application as well.

### A. Precise Point Positioning

The undifferenced carrier-phase and pseudorange (code) observables of a GNSS receiver $r$ tracking satellite $s$ on frequency ${f_{j}=c}/ {\lambda_{j}}$ ($c$ is speed of light; $\lambda_{j}$ is $j$th wavelength) are denoted as $\phi_{r,j}^{s}$ and $p_{r,j}^{s}$, respectively. When two satellites, $s$ and $t$, are tracked, one can form the between-satellite, single-differenced (SD) phase, and code observables, of which the linear(ized) observation equations are given as [31], [32], [33], [34], [35] TeX Source \eqalignno{{\rm E}(\phi_{r,j}^{st}) =&\, ({\mbi {g}}_{r}^{st})^{T} {\mbi b}_{r}-\mu_{j} i_{r}^{st}+\lambda_{j} a_{r,j}^{st}+c_{\phi, r}^{st} \cr {\rm E}(p_{r,j}^{st}) =&\, ({\mbi g}_{r}^{st})^{T} {\mbi b}_{r}+\mu _{j} i_{r}^{st}+c_{p, r}^{st} &{\hbox{(3)}}} with the PPP correction terms, $c_{\phi ,r}^{st} = {\tt t}_{r}^{st}-\delta {\tt s}_{,j}^{st}- {\tt o}_{r}^{st}$ and $c_{p,r}^{st} = {\tt t}_{r}^{st}-d {\tt s}_{,j}^{st}- {\tt o}_{r}^{st}$, assumed known. The unknown deterministic parameters in (3) are the receiver position coordinates in vector ${\mbi b}_{r}$, the ionospheric delay $i_{r}^{st}$ on frequency $f_{1} \left({\mu_{j}=\lambda_{j}^{2} /\lambda_{1}^{2}}\right)$ and the carrier-phase ambiguity $a_{r,j}^{st}$. The row-vector $({\mbi g}_{r}^{st})^{T}$ contains the difference of the unit-direction vectors to satellites $s$ and $t$. The between-satellite differencing has eliminated the receiver phase and the receiver code clock offsets. Likewise, the initial receiver phases are absent in the SD ambiguity, as it only contains the satellite initial phases and integer ambiguity, $a_{r,j}^{st} = -\varphi_{,j}^{st}(t_{0}) + z_{r,j}^{st}$. The ambiguity is constant in time as long as the receiver keeps lock.

The PPP corrections $c_{\phi ,r}^{st}$ and $c_{p,r}^{st}$ consist of the tropospheric delay ${\tt t}_{r}^{st}$, the satellite phase and code clock delays, $\delta {\tt s}_{,j}^{st}$ and $d {\tt s}_{,j}^{st}$, and the receiver relevant orbital information ${\tt o}_{r}^{st}$ of the two satellites. The satellite ephemerides (orbit and clocks) is publicly available information that can be obtained from global tracking networks [5], [6].

For the tropospheric delay ${\tt t}_{r}$, one usually uses an a priori model, such as, e.g., the model of [36]. In case such modelling is not considered accurate enough, one may compensate by including the residual tropospheric zenith delay ${\tt t}_{r}^{z}$ as unknown parameter in (3). Then ${\tt t}_{r} = {\tt t}_{r}^{\prime}+l_{r} {\tt t}_{r}^{z}$, with ${\tt t}_{r}^{\prime}$ provided by the a priori model, $l_{r}$ the satellite elevation dependent mapping function [37] and ${\tt t}_{r}^{z}$ the unknown to be estimated tropospheric zenith delay.

To write (3) in vector-matrix form, it is assumed that receiver $r$ tracks $s+1$ satellites on $f$ frequencies. Defining the $fs \times 1$ SD phase and code observation vectors as ${\mmb{\phi}}_{r}=[{\mmb{\phi}}_{r,1}^{T}, \ldots, {\mmb{\phi}}_{r,f}^{T}]^{T}$ and ${\mbi p}_{r}=[{\mbi p}_{r,1}^{T}, \ldots, {\mbi p}_{r,f}^{T}]^{T}$, where ${\mmb{\phi}}_{r,j}=[\phi_{r,j}^{12}, \ldots, \phi_{r,j}^{1(s+1)}]^{T}$, ${\mbi p}_{r,j}=[p_{r,j}^{12}, \ldots, p_{r,j}^{1(s+1)}]^{T}$, $j=1, \ldots, f$, with a likewise definition for the atmospheric delays, the ambiguities and the corrections, the system of $2fs$ SD observation equations of receiver $r$ follows as: TeX Source \eqalignno{{\rm E}({\mmb{\phi}}_{r}) =&\, ({\mbi e}_{f} \otimes {\mbi G}_{r}) {\mbi b}_{r} - ({\mmb{\mu}} \otimes {\mbi I}_{s}) {\mbi i}_{r} \cr + &\, ({\mmb{\Lambda}} \otimes {\mbi I}_{s}) {\mbi a}_{r}+ {\mbi C}_{\phi;r} \cr {\rm E}({\mbi p}_{r}) =&\, ({\mbi e}_{f} \otimes {\mbi {G}}_{r}) {\mbi b}_{r} + ({\mmb{\mu}} \otimes {\mbi I}_{s}) {\mbi i}_{r}+ {\mbi C}_{p;r} &{\hbox{(4)}}} with ${\mbi e}_{f}=(1, \ldots, 1)^{T}$, ${\mbi G}_{r}=[({\mbi g}_{r}^{12})^{T}, \ldots, ({\mbi g}_{r}^{1(s+1)})^{T}]^{T}$, ${\mmb{\mu}}=(\mu_{1}, \ldots, \mu_{f})^{T}$, ${\mmb{\Lambda}} = {\rm diag}[\lambda_{1}, \ldots, \lambda_{f}]$, ${\mbi C}_{\phi;r} = {\mbi e}_{f} \otimes ({\mbi t}_{r} - {\mbi o}_{r}) - {{\mbi \delta}} {\mbi{s}}$ and ${\mbi C}_{p;r} = {\mbi e}_{f} \otimes ({\mbi t}_{r} - {\mbi o}_{r}) - {\mbi ds}$. Note that in (4), the first satellite is used as reference (pivot) in defining the SD. This choice is not essential as any satellite can be chosen as pivot.

The system of SD observation (4) forms the basis for multifrequency PPP. In case of single-frequency PPP, ${\mbi i}_{r}$ of (4) becomes part of ${\mbi C}_{\phi;r}$ and ${\mbi C}_{p;r}$, as the ionospheric delays are then provided externally by GIMs [3], [4], [8]. As demonstrated in [38] and [39], the single- and dual-frequency PPP convergence times depend significantly on the precision of the code and ionosphere-free observables. The variance reduction achieved by A-PPP (cf. Section III-C) will therefore reduce their convergence times.

### B. The GNSS Array Model

Now consider a platform-fixed array of $r+1$ antennas/receivers, all tracking the same $s+1$ GNSS satellites on the same $f$ frequencies. With two or more antennas, one can formulate the so-called double-differences (DD), which are between-antenna differences of between-satellite differences. For two antennas, $q$ and $r$, tracking the same $s+1$ satellites on the same $f$ frequencies, the DDs are defined as ${\mmb{\phi}}_{qr} = {\mmb{\phi}}_{r} - {\mmb{\phi}}_{q}$ and ${\mbi p}_{qr} = {\mbi p}_{r} - {\mbi p}_{q}$. In the DDs, both the receiver clock offsets and the satellite clock offsets get eliminated. Moreover, since double differencing eliminates all initial phases, the DD ambiguity vector ${\mbi a}_{qr} = {\mbi a}_{r}- {\mbi a}_{q}$ is an integer vector. This is an important property. Inclusion of integer constraints into the model, strengthens the parameter estimation process and allows one to determine the noninteger parameters with a significantly improved accuracy [11], [40]. To emphasize the integerness of the DD ambiguity vector ${\mbi a}_{qr}$, we write ${\mbi z}_{qr} = {\mbi a}_{qr}$.

The array size is assumed such that also the between-antenna differential contributions of orbital pertubations, troposphere, and ionosphere are small enough to be neglected. Hence, the two correction terms, ${\mbi C}_{\phi;r}$ and ${\mbi C}_{p;r}$, that are present in the between-satellite SD model (4), can be considered absent in the DD array model [5], [31], [32], [33]. Also, since the unit-direction vectors of two nearby antennas to the same satellite are the same for all practical purposes, we have ${\mbi G} = {\mbi G}_{q} = {\mbi G}_{r}$. For two nearby antennas, $q$ and $r$, the vectorial DD observation equations follow therefore from (4) as TeX Source \eqalignno{{\rm E}({\mmb{\phi}}_{qr}) =&\, ({\mbi e}_{f} \otimes {\mbi G}) {\mbi b}_{qr}+ ({\mmb{\Lambda}} \otimes {\mbi I}_{s}) {\mbi z}_{qr} \cr {\rm E}({\mbi p}_{qr}) =&\, ({\mbi e}_{f} \otimes {\mbi G}) {\mbi b}_{qr} &{\hbox{(5)}}} in which ${\mbi b}_{qr} = {\mbi b}_{r}- {\mbi b}_{q}$ is the baseline vector between the two antennas.

In case of more than two antennas, the single-baseline model (5) can be generalized to a multibaseline array-model. Since the size of the array is assumed small, the model can be formulated in multivariate form, thus having the same design matrix as that of the single-baseline model (5). For the multivariate formulation, we take antenna 1 as the reference antenna (i.e., the master) and we define the $fs \times r$ phase and code observation matrices as ${\mmb {\Phi}} = [{\mmb{\phi}}_{12}, \ldots, {\mmb{\phi}}_{1(r+1)}]$ and ${\mbi {\cal P}}=[{\mbi p}_{12}, \ldots, {\mbi p}_{1(r+1)}]$, the $3 \times r$ baseline matrix as ${\mbi B}=[{\mbi b}_{12}, \ldots, {\mbi b}_{1(r+1)}]$, and the $fs \times r$ DD integer ambiguity matrix as ${\mbi Z}=[{\mbi z}_{12}, \ldots, {\mbi z}_{1(r+1)}]$. The multivariate equivalent to the DD single-baseline model (5) follows then as TeX Source $${\rm E} \left[ \matrix{{\mmb {\Phi}} \cr {\mbi {\cal P}}} \right]= \left[ \matrix{{\mbi e}_{f} \otimes {\mbi G} & {\mmb{\Lambda}} \otimes {\mbi I}_{s} \cr {\mbi e}_{f} \otimes {\mbi G} & {\bf 0}} \right] \left[ \matrix{{\mbi B} \cr {\mbi {Z}}} \right]. \eqno{\hbox{(6)}}$$ The unknowns in this model are the matrices ${\mbi B}$ and ${\mbi Z}$. The matrix ${\mbi B} \in \BBR^{3 \times r}$ consists of the $r$ unknown baseline vectors and the matrix ${\mbi Z} \in \BBZ^{fs\times r}$ consists of the $fsr$ unknown DD integer ambiguities.

The array geometry is described by the baseline matrix ${\mbi B}$. Once ${\mbi B}$ has been determined, the attitude of the platform can be determined if use is made of the known array geometry in the platform-fixed frame. Let $q=1,2,3$ be the dimension of the antenna array (linear, planar or three dimensional) and let the coordinates in the platform-fixed frame of the known array geometry be given by the column vectors of the $q \times r$ matrix ${\mbi F}$. Then ${\mbi B}$ and ${\mbi F}$ are related as TeX Source $${\mbi B} = {\mbi R}{\mbi F} \eqno{\hbox{(7)}}$$ in which the $q \leq r$ column vectors of ${\mbi R}$ are orthonormal, i.e., ${\mbi R}^{T} {\mbi R} = {\mbi I}_{q}$ or ${\mbi R} \in \BBO^{3 \times q}$. From this matrix equation, one can solve the attitude matrix ${\mbi R}$ in a least-squares sense, once an estimate of ${\mbi B}$ is available from solving (6) [41], [42].

SECTION III

## ARRAY-AIDED POSITIONING

In this section, array-aided PPP is introduced as generalization of the PPP concept. Its various positioning applications are described together with the improvements that can be realized.

### A. A Combined Position-Attitude Model

Usually the point positioning model (4) is processed independently from the attitude determination model (6). Moreover, in current GNSS attitude determination methods, also the integer estimation problem is treated separately from the attitude estimation process. Existing approaches either first resolve the integer ambiguities and then use the precise baseline estimates for attitude determination [43], [44], [45] or they use the baseline length constraints only for validation purposes [46], [47]. In this section, however, we combine the two models, (4) and (6), and show the improvement that a combined processing brings.

If we define ${\mbi y}_{1}=[{\mmb{\phi}}_{1}^{T}, {\mbi p}_{1}^{T}]^{T}$, ${\mmb{\gamma}}_{1} = [{\mbi C}_{\phi ;1}^{T}, {\mbi C}_{p;1}^{T}]^{T}$, ${\mbi Y}=[{\mmb {\Phi}}^{T}, {\mbi {\cal P}}^{T}]^{T}$, ${\mbi H}=[{\mmb{\Lambda}}^{T}, {\bf 0}^{T}]^{T}$, ${\mbi h}=[- {\mmb{\mu}}^{T}, + {\mmb{\mu}}^{T}]^{T}$, the models (4) and (6) can be written in the compact form TeX Source \eqalignno{{\rm E}({\mbi y}_{1}) = &\, {\mbi Mb}_{1} + {\mbi {Na}}_{1} + {\mbi d}_{1} \cr {\rm E}({\mbi Y}) = &\, {\mbi {MB}} + {\mbi NZ} &{\hbox{(8)}}} where ${\mbi M}=({\mbi e}_{2f} \otimes {\mbi G})$, ${\mbi N}=({\mbi H} \otimes {\mbi I}_{s})$, and ${\mbi d}_{1}=({\mbi h} \otimes {\mbi I}_{s}) {\mbi i}_{1}+ {\mmb{\gamma}}_{1}$. Note that the two sets of observation equations have no parameters in common. This is the reason why one has treated the two equation sets of (8) separately. The first set is then used to determine the position of the array, i.e., to determine ${\mbi b}_{1}$ from ${\mbi y}_{1}$, while the second set is used to determine its attitude, i.e., to determine the rotation matrix ${\mbi R}$ from ${\mbi Y}$ via (7). However, despite the lack of common parameters in (8), the data of the two sets are correlated and therefore not independent. Thus in order to be able to solve the system (8) rigorously, one needs to take this correlation into account. This is possible if we know the complete covariance matrix of $[{\mbi y}_{1}, {\mbi Y}]$.

To determine the covariance matrix of $[{\mbi y}_{1}, {\mbi Y}]$, we first have to define the covariance matrix of the SD phase and code observables.

#### Definition 1 (SD Covariance Matrix)

Let ${\mmb{\Upsilon}}=[{\mbi y}_{1}, \ldots, {\mbi y}_{r+1}]$, with ${\mbi y}_{i}=[(({\mbi I}_{f} \otimes {\mbi D}_{s}^{T}) {\mmb{\varphi}}_{i})^{T}, (({\mbi {I}}_{f} \otimes {\mbi D}_{s}^{T}) {\mmb{\rho}}_{i})^{T}]^{T}$ and ${\mbi D}_{s}^{T}=[- {\mbi e}_{s}, {\mbi I}_{s}]$, where the undifferenced phase and code data vectors of antenna $i$ are given as ${\mmb{\varphi}}_{i}=[{\mmb{\varphi}}_{i,1}^{T}, \ldots, {\mmb{\varphi}}_{i,f}^{T}]^{T}$, ${\mmb{\varphi}}_{i, j}=[\phi_{i,j}^{1}, \ldots, \phi_{i,j}^{s+1}]^{T}$ and ${\mmb{\rho}}_{i}=[{\mmb{\rho}}_{i,1}^{T}, \ldots, {\mmb{\rho}}_{i,f}^{T}]^{T}$, ${\mmb{\rho}}_{i,j}=[p_{i,j}^{1}, \ldots, p_{i,j}^{s+1}]^{T}$, $j=1, \ldots, f$. Then the covariance matrix of ${\rm vec}({\mmb{\Upsilon}})$ is given as TeX Source $${\rm Cov}({\rm {vec}}({\mmb{\Upsilon}})) = {\mbi Q}_{r} \otimes {\mbi Q} \quad {\rm with} \quad {\mbi Q} = {\mbi {Q}}_{f} \otimes {\mbi Q}_{s} \eqno{\hbox{(9)}}$$ where ${\mbi Q}_{s} = {\rm blockdiag}[{\mbi D}_{s}^{T} {\mbi Q}_{\phi} {\mbi D}_{s}, {\mbi {D}}_{s}^{T} {\mbi Q}_{p} {\mbi D}_{s}]$ and ${\mbi Q}_{r}$, ${\mbi Q}_{f}$, ${\mbi Q}_{\phi}$ and ${\mbi Q}_{p}$ are positive definite cofactor matrices.

The structure of the covariance matrix ${\rm Cov}({\rm {vec}}({\mmb{\Upsilon}}))$ has been defined such that it accomodates differences in the phase precision, differences in the code precision, frequency dependent tracking precision, satellite elevation dependency and differences in quality of the antenna/receivers in the array. The precision contribution of antenna/receivers and frequency can be specified through ${\mbi Q}_{r}$ and ${\mbi Q}_{f}$, while the cofactor matrices ${\mbi Q}_{\phi}$ and ${\mbi Q}_{p}$ identify the relative precision contribution of phase and code, including the satellite elevation dependency. The covariance between the phase observables and the code observables is assumed zero.

The required covariance matrix of $[{\mbi y}_{1}, {\mbi Y}]$ follows, with (9), from applying the variance-covariance propagation law (i.e., propagation of second order (central) moments) to $[{\mbi y}_{1}, {\mbi Y}] = {\mmb{\Upsilon}} [{\mbi C}_{1}, {\mbi D}_{r}]$, with ${\mbi D}_{r}$ the differencing matrix and ${\mbi C}_{1}=[1, 0, \ldots 0]^{T}$. The complete structure of the combined positioning-attitude model can therefore be summarized as follows.

#### Definition 2 (Combined Position-Attitude Model)

The multivariate observation equations and covariance matrix of the combined position-attitude model are given as TeX Source \eqalignno{{\rm E}([{\mbi y}_{1}, {\mbi Y}]) =&\, {\mbi {M}}[{\mbi b}_{1}, {\mbi B}]+ {\mbi N}[{\mbi a}_{1}, {\mbi {Z}}]+[{\mbi d}_{1}, {\bf 0}] \cr {\rm Cov}({\rm {vec}}([{\mbi {y}}_{1}, {\mbi Y}])) =&\, {\mmb{\Sigma}} \otimes {\mbi Q} &{\hbox{(10)}}} with cofactor matrices TeX Source $${\mmb{\Sigma}} = [{\mbi C}_{1}, {\mbi D}_{r}]^{T} {\mbi {Q}}_{r}[{\mbi C}_{1}, {\mbi D}_{r}] \quad {\rm and} \quad {\mbi {Q}} = {\mbi Q}_{f} \otimes {\mbi Q}_{s} \eqno{\hbox{(11)}}$$ and with the constraints ${\mbi B} = {\mbi RF}$, ${\mbi R} \in \BBO^{3 \times q}$, ${\mbi Z} \in \BBZ^{fs \times r}$.

The nonzero correlation between ${\mbi y}_{1}$ and ${\mbi Y}$ is due to the nonzero term ${\mbi C}_{1}^{T} {\mbi Q}_{r} {\mbi D}_{r} \ne 0$ in (11).

### B. A Decorrelating Transformation

Although the equations of ${\mbi y}_{1}$ and ${\mbi Y}$ [cf. (10)] have no parameters in common, their nonzero correlation implies that treating the positioning problem independently from the attitude determination problem is suboptimal. To properly take the nonzero correlation into account, the two sets of observation equations need to be considered in an integral manner.

We now show how the nonzero correlation can be taken into account, while still being able to work with a system of observation equations that has the same structure as the original one (10). The idea is the following. We first decorrelate the two sets of data with an appropriate decorrelating transformation [cf. (12)]. Then we use the decorrelating transformation to reparametrize the parameters such that the positioning-parameters and the array-parameters are decoupled again. Thus a transformed system of decorrelated equations is obtained with the same structure as the orginal system and that therefore can be solved as such.

#### Theorem 1 (Decorrelated Positioning-Attitude Model)

Let the invertible transformation ${\mbi {\cal D}}: \BBR^{2f(r+1)s} \ura{} \BBR^{2f(r+1)s}$ be given as TeX Source $${\mbi {\cal D}} = \left[ \matrix{1 & - {\mbi C}_{1}^{T} {\mbi {Q}}_{r} {\mbi D}_{r}({\mbi D}_{r}^{T} {\mbi Q}_{r} {\mbi {D}}_{r})^{-1} \cr {\bf 0} & {\mbi I}_{r}} \right] \otimes {\mbi {I}}_{2fs} \eqno{\hbox{(12)}}$$ and define ${\rm {vec}}([{\bar {\mbi y}}, {\mbi Y}]) = {\mbi {\cal D}} {\rm {vec}}([{\mbi y}_{1}, \mbi {Y}])$. Then TeX Source \eqalignno{{\rm E}([{\bar {\mbi y}}, {\mbi Y}]) =&\, {\mbi M}[{\bar {\mbi {b}}}, {\mbi B}]+ {\mbi N}[{\bar {\mbi a}}, {\mbi Z}]+[{\mbi d}_{1}, {\bf 0}] \cr {\rm Cov}({\rm {vec}}([{\bar {\mbi y}}, {\mbi Y}])) =&\, {\mbi S} \otimes {\mbi Q} &{\hbox{(13)}}} with blockdiagonal cofactor matrix TeX Source $${\mbi S} = {\rm blockdiag}\left[({\mbi e}_{r}^{T} {\mbi {Q}}_{r}^{-1} {\mbi e}_{r})^{-1}, ({\mbi D}_{r}^{T} {\mbi {Q}}_{r} {\mbi D}_{r})\right] \eqno{\hbox{(14)}}$$ and with constraints ${\mbi B} = {\mbi {RF}}$, ${\mbi R} \in \BBO^{3 \times q}$, ${\mbi Z} \in \BBZ^{fs \times r}$, where ${\mbi e}_{r}=(1, \ldots, 1)^{T}$, ${\rm {vec}}([{\bar {\mbi {b}}}, {\mbi B}]) = {\mbi {\cal D}} {\rm {vec}}([{\mbi b}_{1}, \mbi {B}])$, ${\rm {vec}}([{\bar {\mbi a}}, {\mbi Z}]) = {\mbi {\cal D}} {\rm {vec}}([{\mbi a}_{1}, \mbi {Z}])$.

##### Proof

The proof is given in the Appendix.

Compare (13) and (14) to (10) and (11), respectively. The transformed set of (13) has the same structure as the original set (10), but since ${\mbi S}$ is blockdiagonal, while ${\mmb{\Sigma}}$ is not, it follows that the observation equations of the decorrelated ${\bar {\mbi y}}$ and ${\mbi Y}$ can be solved separately. Moreover, the same software packages can be used to solve for the parameters of (13) as has been used hiterto to solve for the parameters of (10). Importantly, however, with (13) the results will then be based on having taken the full covariance matrix into account.

### C. The A-PPP Model and Its Applications

The decorrelating transformation (12) changed the positioning equations, but not those for attitude. Hence, it is the positioning that takes advantage of the array data when the full correlation between ${\mbi y}_{1}$ and ${\mbi Y}$ is taken into account. The model for ${\bar {\mbi y}}$ will be referred to as the array-aided precise point positioning model.

#### Definition 3 (Array-Aided PPP Model)

The observation equations and covariance matrix of the A-PPP model are given as TeX Source \eqalignno{{\rm E}({\bar {\mbi y}}) =&\, {\mbi M} {\bar {\mbi b}}+ {\mbi N} {\bar {\mbi a}}+ {\mbi d}_{1} \cr {\rm Cov}({\bar {\mbi y}}) =&\, \left({\mbi e}_{r}^{T} {\mbi Q}_{r}^{-1} {\mbi e}_{r}\right)^{-1} \otimes {\mbi Q} &{\hbox{(15)}}} with the array-aided data vector ${\bar {\mbi y}} = {\mbi y}_{1}- {\mbi Y} {\mbi D}_{r}^{+} {\mbi C}_{1}$ and ${\mbi D}_{r}^{+}=({\mbi D}_{r}^{T} {\mbi Q}_{r} {\mbi D}_{r})^{-1} {\mbi D}_{r}^{T} \mbi {Q}_{r}$.

### ACKNOWLEDGMENT

The author would like to acknowledge P. Buist and G. Giorgi of the Delft University of Technology, Delft, the Netherlands. They have been instrumental in helping implement and test the multivariate estimation algorithms. L. Huisman and D. Odijk of the Curtin University of Technology, Perth, Australia, executed and analyzed the A-PPP experiments.

## Footnotes

The associate editor coordinating the review of this manuscript and approving it for publication was Prof. Sofia C. Olhede. This work was supported by an Australian Research Council (ARC) Federation Fellowship (project number FF0883188).

The author is with the Faculty of Science and Engineering, Curtin University of Technology, Perth, WA 6845, Australia. He is also with the Faculty of Aerospace Engineering, Delft University of Technology, Delft, The Netherlands (e-mail: p.teunissen@curtin.edu.au).

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