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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 Formula 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 Formula${\mbi {\cal Y}}=[{\mbi y}_{1}, {\mbi Y}]$ the random matrix of GNSS array observables and Formula${\mbi {\cal B}}=[{\mbi b}_{1}, {\mbi {RF}}]$, Formula${\mbi {\cal A}}=[{\mbi a}_{1}, {\mbi Z}]$, and Formula${{\mbi {\cal C}}}=[{\mbi d}_{1}, {\bf 0}]$ the matrices containing the deterministic parameters that need to be estimated under the attitude orthonormality and ambiguity integer constraints Formula TeX Source $${\mbi R} \in \BBO^{3 \times q} \quad {\rm and} \quad {\mbi {Z}} \in \BBZ^{fs \times r} \eqno{\hbox{(2)}}$$Formula${\mbi {\cal B}}$ is the matrix of antenna positions, Formula${\mbi {\cal A}}$ the matrix of carrier phase ambiguities and Formula${{\mbi {\cal C}}}$ 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 Formula$\otimes$ and the Formula${\rm vec}$-operator. For their properties, see, e.g., [29], [30]. The expectation and covariance matrix of a random vector Formula${\mbi x}$ are denoted as Formula${\rm E}({\mbi x})$ and Formula${\rm Cov}({\mbi x})$, respectively. For the covariance matrix of a random matrix Formula${\mbi X}$, we often write Formula${\mbi Q}_{{\mmb {XX}}}$ instead of Formula${\rm Cov}({\rm vec}({\mbi X}))$. For the weighted squared norm, the notation Formula$\Vert . \Vert_{\mmb Q}^{2}=(.)^{T} {\mbi Q}^{-1}(.)$ 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

POSITIONING AND 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 Formula$r$ tracking satellite Formula$s$ on frequency Formula${f_{j}=c}/ {\lambda_{j}}$ (Formula$c$ is speed of light; Formula$\lambda_{j}$ is Formula$j$th wavelength) are denoted as Formula$\phi_{r,j}^{s}$ and Formula$p_{r,j}^{s}$, respectively. When two satellites, Formula$s$ and Formula$t$, 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] Formula 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, Formula$c_{\phi ,r}^{st} = {\tt t}_{r}^{st}-\delta {\tt s}_{,j}^{st}- {\tt o}_{r}^{st}$ and Formula$c_{p,r}^{st} = {\tt t}_{r}^{st}-d {\tt s}_{,j}^{st}- {\tt o}_{r}^{st}$, assumed known. The unknown deterministic parameters in (3) are the receiver position coordinates in vector Formula${\mbi b}_{r}$, the ionospheric delay Formula$i_{r}^{st}$ on frequency Formula$f_{1} \left({\mu_{j}=\lambda_{j}^{2} /\lambda_{1}^{2}}\right)$ and the carrier-phase ambiguity Formula$a_{r,j}^{st}$. The row-vector Formula$({\mbi g}_{r}^{st})^{T}$ contains the difference of the unit-direction vectors to satellites Formula$s$ and Formula$t$. 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, Formula$a_{r,j}^{st} = -\varphi_{,j}^{st}(t_{0}) + z_{r,j}^{st}$. The ambiguity is constant in time as long as the receiver keeps lock.

The PPP corrections Formula$c_{\phi ,r}^{st}$ and Formula$c_{p,r}^{st}$ consist of the tropospheric delay Formula${\tt t}_{r}^{st}$, the satellite phase and code clock delays, Formula$\delta {\tt s}_{,j}^{st}$ and Formula$d {\tt s}_{,j}^{st}$, and the receiver relevant orbital information Formula${\tt o}_{r}^{st}$ 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 Formula${\tt t}_{r}$, 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 Formula${\tt t}_{r}^{z}$ as unknown parameter in (3). Then Formula${\tt t}_{r} = {\tt t}_{r}^{\prime}+l_{r} {\tt t}_{r}^{z}$, with Formula${\tt t}_{r}^{\prime}$ provided by the a priori model, Formula$l_{r}$ the satellite elevation dependent mapping function [37] and Formula${\tt t}_{r}^{z}$ the unknown to be estimated tropospheric zenith delay.

To write (3) in vector-matrix form, it is assumed that receiver Formula$r$ tracks Formula$s+1$ satellites on Formula$f$ frequencies. Defining the Formula$fs \times 1$ SD phase and code observation vectors as Formula${\mmb{\phi}}_{r}=[{\mmb{\phi}}_{r,1}^{T}, \ldots, {\mmb{\phi}}_{r,f}^{T}]^{T}$ and Formula${\mbi p}_{r}=[{\mbi p}_{r,1}^{T}, \ldots, {\mbi p}_{r,f}^{T}]^{T}$, where Formula${\mmb{\phi}}_{r,j}=[\phi_{r,j}^{12}, \ldots, \phi_{r,j}^{1(s+1)}]^{T}$, Formula${\mbi p}_{r,j}=[p_{r,j}^{12}, \ldots, p_{r,j}^{1(s+1)}]^{T}$, Formula$j=1, \ldots, f$, with a likewise definition for the atmospheric delays, the ambiguities and the corrections, the system of Formula$2fs$ SD observation equations of receiver Formula$r$ follows as: Formula 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 Formula${\mbi e}_{f}=(1, \ldots, 1)^{T}$, Formula${\mbi G}_{r}=[({\mbi g}_{r}^{12})^{T}, \ldots, ({\mbi g}_{r}^{1(s+1)})^{T}]^{T}$, Formula${\mmb{\mu}}=(\mu_{1}, \ldots, \mu_{f})^{T}$, Formula${\mmb{\Lambda}} = {\rm diag}[\lambda_{1}, \ldots, \lambda_{f}]$, Formula${\mbi C}_{\phi;r} = {\mbi e}_{f} \otimes ({\mbi t}_{r} - {\mbi o}_{r}) - {{\mbi \delta}} {\mbi{s}}$ and Formula${\mbi C}_{p;r} = {\mbi e}_{f} \otimes ({\mbi t}_{r} - {\mbi o}_{r}) - {\mbi ds}$. 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, Formula${\mbi i}_{r}$ of (4) becomes part of Formula${\mbi C}_{\phi;r}$ and Formula${\mbi C}_{p;r}$, 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 Formula$r+1$ antennas/receivers, all tracking the same Formula$s+1$ GNSS satellites on the same Formula$f$ 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, Formula$q$ and Formula$r$, tracking the same Formula$s+1$ satellites on the same Formula$f$ frequencies, the DDs are defined as Formula${\mmb{\phi}}_{qr} = {\mmb{\phi}}_{r} - {\mmb{\phi}}_{q}$ and Formula${\mbi p}_{qr} = {\mbi p}_{r} - {\mbi p}_{q}$. 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 Formula${\mbi a}_{qr} = {\mbi a}_{r}- {\mbi a}_{q}$ 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 Formula${\mbi a}_{qr}$, we write Formula${\mbi z}_{qr} = {\mbi a}_{qr}$.

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, Formula${\mbi C}_{\phi;r}$ and Formula${\mbi C}_{p;r}$, 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 Formula${\mbi G} = {\mbi G}_{q} = {\mbi G}_{r}$. For two nearby antennas, Formula$q$ and Formula$r$, the vectorial DD observation equations follow therefore from (4) as Formula 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 Formula${\mbi b}_{qr} = {\mbi b}_{r}- {\mbi b}_{q}$ 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 Formula$fs \times r$ phase and code observation matrices as Formula${\mmb {\Phi}} = [{\mmb{\phi}}_{12}, \ldots, {\mmb{\phi}}_{1(r+1)}]$ and Formula${\mbi {\cal P}}=[{\mbi p}_{12}, \ldots, {\mbi p}_{1(r+1)}]$, the Formula$3 \times r$ baseline matrix as Formula${\mbi B}=[{\mbi b}_{12}, \ldots, {\mbi b}_{1(r+1)}]$, and the Formula$fs \times r$ DD integer ambiguity matrix as Formula${\mbi Z}=[{\mbi z}_{12}, \ldots, {\mbi z}_{1(r+1)}]$. The multivariate equivalent to the DD single-baseline model (5) follows then as Formula 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 Formula${\mbi B}$ and Formula${\mbi Z}$. The matrix Formula${\mbi B} \in \BBR^{3 \times r}$ consists of the Formula$r$ unknown baseline vectors and the matrix Formula${\mbi Z} \in \BBZ^{fs\times r}$ consists of the Formula$fsr$ unknown DD integer ambiguities.

The array geometry is described by the baseline matrix Formula${\mbi B}$. Once Formula${\mbi B}$ 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 Formula$q=1,2,3$ 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 Formula$q \times r$ matrix Formula${\mbi F}$. Then Formula${\mbi B}$ and Formula${\mbi F}$ are related as Formula TeX Source $${\mbi B} = {\mbi R}{\mbi F} \eqno{\hbox{(7)}}$$ in which the Formula$q \leq r$ column vectors of Formula${\mbi R}$ are orthonormal, i.e., Formula${\mbi R}^{T} {\mbi R} = {\mbi I}_{q}$ or Formula${\mbi R} \in \BBO^{3 \times q}$. From this matrix equation, one can solve the attitude matrix Formula${\mbi R}$ in a least-squares sense, once an estimate of Formula${\mbi B}$ 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 Formula${\mbi y}_{1}=[{\mmb{\phi}}_{1}^{T}, {\mbi p}_{1}^{T}]^{T}$, Formula${\mmb{\gamma}}_{1} = [{\mbi C}_{\phi ;1}^{T}, {\mbi C}_{p;1}^{T}]^{T}$, Formula${\mbi Y}=[{\mmb {\Phi}}^{T}, {\mbi {\cal P}}^{T}]^{T}$, Formula${\mbi H}=[{\mmb{\Lambda}}^{T}, {\bf 0}^{T}]^{T}$, Formula${\mbi h}=[- {\mmb{\mu}}^{T}, + {\mmb{\mu}}^{T}]^{T}$, the models (4) and (6) can be written in the compact form Formula 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 Formula${\mbi M}=({\mbi e}_{2f} \otimes {\mbi G})$, Formula${\mbi N}=({\mbi H} \otimes {\mbi I}_{s})$, and Formula${\mbi d}_{1}=({\mbi h} \otimes {\mbi I}_{s}) {\mbi i}_{1}+ {\mmb{\gamma}}_{1}$. 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 Formula${\mbi b}_{1}$ from Formula${\mbi y}_{1}$, while the second set is used to determine its attitude, i.e., to determine the rotation matrix Formula${\mbi R}$ from Formula${\mbi Y}$ 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 Formula$[{\mbi y}_{1}, {\mbi Y}]$.

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

Definition 1 (SD Covariance Matrix)

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

The structure of the covariance matrix Formula${\rm Cov}({\rm {vec}}({\mmb{\Upsilon}}))$ 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 Formula${\mbi Q}_{r}$ and Formula${\mbi Q}_{f}$, while the cofactor matrices Formula${\mbi Q}_{\phi}$ and Formula${\mbi Q}_{p}$ 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 Formula$[{\mbi y}_{1}, {\mbi Y}]$ follows, with (9), from applying the variance-covariance propagation law (i.e., propagation of second order (central) moments) to Formula$[{\mbi y}_{1}, {\mbi Y}] = {\mmb{\Upsilon}} [{\mbi C}_{1}, {\mbi D}_{r}]$, with Formula${\mbi D}_{r}$ the differencing matrix and Formula${\mbi C}_{1}=[1, 0, \ldots 0]^{T}$. 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 Formula 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 Formula 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 Formula${\mbi B} = {\mbi RF}$, Formula${\mbi R} \in \BBO^{3 \times q}$, Formula${\mbi Z} \in \BBZ^{fs \times r}$.

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

B. A Decorrelating Transformation

Although the equations of Formula${\mbi y}_{1}$ and Formula${\mbi Y}$ [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 Formula${\mbi {\cal D}}: \BBR^{2f(r+1)s} \ura{} \BBR^{2f(r+1)s}$ be given as Formula 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 Formula${\rm {vec}}([{\bar {\mbi y}}, {\mbi Y}]) = {\mbi {\cal D}} {\rm {vec}}([{\mbi y}_{1}, \mbi {Y}])$. Then Formula 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 Formula 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 Formula${\mbi B} = {\mbi {RF}}$, Formula${\mbi R} \in \BBO^{3 \times q}$, Formula${\mbi Z} \in \BBZ^{fs \times r}$, where Formula${\mbi e}_{r}=(1, \ldots, 1)^{T}$, Formula${\rm {vec}}([{\bar {\mbi {b}}}, {\mbi B}]) = {\mbi {\cal D}} {\rm {vec}}([{\mbi b}_{1}, \mbi {B}])$, Formula${\rm {vec}}([{\bar {\mbi a}}, {\mbi Z}]) = {\mbi {\cal D}} {\rm {vec}}([{\mbi a}_{1}, \mbi {Z}])$.

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 Formula${\mbi S}$ is blockdiagonal, while Formula${\mmb{\Sigma}}$ is not, it follows that the observation equations of the decorrelated Formula${\bar {\mbi y}}$ and Formula${\mbi Y}$ 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 Formula${\mbi y}_{1}$ and Formula${\mbi Y}$ is taken into account. The model for Formula${\bar {\mbi y}}$ 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 Formula 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 Formula${\bar {\mbi y}} = {\mbi y}_{1}- {\mbi Y} {\mbi D}_{r}^{+} {\mbi C}_{1}$ and Formula${\mbi D}_{r}^{+}=({\mbi D}_{r}^{T} {\mbi Q}_{r} {\mbi D}_{r})^{-1} {\mbi D}_{r}^{T} \mbi {Q}_{r}$.

The precision of Formula${\bar {\mbi y}}$ is always better than that of Formula${\mbi y}_{1}$. This can be shown as follows. Since Formula$1=({\mbi C}_{1}^{T} {\mbi e}_{r})^{2}=({\mbi C}_{1}^{T} {\mbi Q}_{r}^{1 \over 2}. {\mbi Q}_{r}^{-1 \over 2} {\mbi {e}}_{r})^{2}$Formula$=({\mbi C}_{1}^{T} {\mbi Q}_{r} {\mbi C}_{1}) ({\mbi {e}}_{r}^{T} {\mbi Q}_{r}^{-1} {\mbi e}_{r}) \cos^{2}(\alpha)$ (cosine rule) and Formula$\alpha \ne 0$, since Formula${\mbi C}_{1} \ne {\mbi e}_{r}$ for Formula$r>1$, the strict inequality Formula$({\mbi e}_{r}^{T} {\mbi Q}_{r}^{-1} {\mbi e}_{r})^{-1} < ({\mbi C}_{1}^{T} \mbi {Q}_{r} {\mbi C}_{1})$ holds, and therefore, from (10) and (13), the matrix inequality Formula TeX Source $${\rm Cov}({\bar {\mbi y}}) < {\rm Cov}({\mbi y}_{1}) \eqno{\hbox{(16)}}$$ follows. Hence, any linear function of Formula${\bar {\mbi y}}$ will always have a smaller variance than the same function of Formula${\mbi y}_{1}$. As an example, consider an array with Formula$n$ receivers that all are of the same quality. Then Formula${\mbi Q}_{r}$ is a unit matrix and Formula${\rm Cov}({\bar {\mbi y}}_{1}) = {{1} \over {n}} {\rm Cov}({\mbi y}_{1})$. This '1 over Formula$n$' rule improvement propagates then also into A-PPP's parameter estimation, thus resulting in improved results.

The A-PPP model can be applied in different ways. Although A-PPP, like PPP, can be used for other applications than positioning, e.g., remote sensing or time-transfer, attention will be restricted here to positioning. Three different positioning modes are considered: platform positioning, on-platform positioning and between-platform positioning. Each require different information from the array.

1) Platform Positioning (Without Ambiguity Resolution)

This is the simplest A-PPP variant, as it can be solved in exactly the same way as any of the current PPP-variants. Formula${\mbi Y}$ is the only array information that is required to construct Formula${\bar {\mbi y}}$. Since the baseline matrix Formula${\mbi B}$ and the ambiguity matrix Formula${\mbi Z}$ do not need to be known, the solution of (15) can do without solving the attitude observation equations of model (13).

To interpret the platform positioning vector, recall from (13) that Formula${\bar {\mbi {b}}} = {\mbi b}_{1}- {\mbi B} {\mbi D}_{r}^{+} {\mbi C}_{1}$. Since Formula${\mbi B} {\mbi D}_{r}^{+}=[{\mbi b}_{1}, \ldots, {\mbi b}_{r}] {\mbi P_{D_{r}}}$ and Formula${\mbi P_{D_{r}}} = {\mbi I}_{r}- {\mbi Q}_{r}^{-1} {\mbi e}_{r}({\mbi e}_{r}^{T} \mbi {Q}_{r}^{-1} {\mbi e}_{r})^{-1} {\mbi e}_{r}^{T}$, it follows that Formula${\bar {\mbi {b}}} = [{\mbi b}_{1}, \ldots, {\mbi b}_{r}] {\mbi Q}_{r}^{-1} {\mbi e}_{r}(\mbi {e}_{r}^{T} {\mbi Q}_{r}^{-1} {\mbi e}_{r})^{-1}$. Hence, Formula${\bar {\mbi {b}}}$ is the weighted least-squares combination of the Formula$r+1$ antenna positions. For a diagonal weight matrix Formula${\mbi Q}_{r}^{-1}={\rm diag}[w_{1}, \ldots, w_{r+1}]$, for instance, the position vector Formula${\bar {\mbi {b}}}$ is equal to the weighted average Formula TeX Source $${\bar {\mbi b}} = {{\sum\limits_{i=1}^{r+1} w_{i} {\mbi b}_{i}} \over {\sum\limits_{i=1}^{r+1} w_{i}}}. \eqno{\hbox{(17)}}$$ Thus A-PPP, based on (15), determines the position of the “center of gravity” of the antenna configuration rather than that of a single antenna position. If needed, these two positions can be made to coincide by using a suitable symmetry in the array geometry. That is, Formula${\bar {\mbi {b}}} = {\mbi b}_{1}$ if Formula$\sum_{i=1}^{r+1} w_{i} {\mbi b}_{1i} = {\bf 0}$.

Table 1
TABLE I SINGLE-FREQUENCY, KINEMATIC (MOVING PLATFORM) N-E-U ONE-SIGMA POSITIONING PRECISION OF PPP AND FOUR-ANTENNA A-PPP
Table I illustrates the single-frequency platform positioning performance of PPP and A-PPP. The experiment took place in Perth, Australia, on 30 July 2010 (05:24:00–07:03:59 UTC). The platform consisted of four Sokkia GSR2700 ISX antenna/receivers, three of which were placed in a triangle, 2 meters apart, with the fourth one exactly in the middle of the triangle. The 1-Hz single-frequency L1 GPS phase and code data were collected with a zero degree cut-off elevation angle. To allow for low velocity (pedestrian) platform movement the data was processed in kinematic mode, using as a priori PPP corrections, final IGS orbits, final IGS 30-second clock corrections, and final GIM maps. Table I shows the empirically determined North-East-Up (N-E-U) standard deviations (in millimeters) for kinematic PPP and A-PPP. The improvements are clearly visible, although horizontal positioning benefits more than vertical positioning. The height improvement is less, because the PPP and A-PPP common a-priori corrections uncertainty impacts the vertical component most.

2) Platform Positioning (With Ambiguity Resolution)

PPP with integer ambiguity resolution is possible by means of externally provided corrections that transform the PPP ambiguities to integers [14], [15], [16], [17]. The advantage of this PPP-RTK method over standard PPP is the considerable strengthening the integer constraints bring to the model. The question is now whether one can still take advantage of this in the A-PPP setup. Afterall, with A-PPP, the ambiguity vector Formula${\bar {\mbi a}}$ of (15) remains noninteger even after the original SD ambiguities have been corrected to integers. The weighted average of integers is namely generally noninteger.

To resolve the problem of the nonintegerness of Formula${\bar {\mbi a}}$, use is made of the relation Formula TeX Source $${\bar {\mbi a}} = {\mbi a}_{1}- {\mbi Z}{\mbi D}_{r}^{+} {\mbi {C}}_{1} \eqno{\hbox{(18)}}$$ which shows that one can undo the effect of averaging and express Formula${\bar {\mbi a}}$ in Formula${\mbi a}_{1}$, provided the integer matrix Formula${\mbi Z}$ is known. Hence, the A-PPP RTK observation equations become Formula TeX Source $${\rm E}(\buildrel = \over{\mbi y}) = {\mbi M} {\bar {\mbi {b}}} + {\mbi N} {\mbi a}_{1} + {\mbi d}_{1} \eqno{\hbox{(19)}}$$ with the Formula${\mbi Z}$-corrected observation vector given as Formula TeX Source $${{\buildrel = \over{\mbi y}}} = {\bar {\mbi y}}+ {\mbi N} {\mbi Z} {\mbi D}_{r}^{+} {\mbi C}_{1}. \eqno{\hbox{(20)}}$$ Since (19) has the same structure as the original PPP RTK equations, it can be solved in the same way. As to the required array information, now both Formula${\mbi Y}$ and Formula${\mbi Z}$ are needed. Formula${\mbi Y}$ is needed to obtain Formula${\bar {\mbi y}}$ from Formula${\mbi y}_{1}$, and Formula${\mbi Z}$ is needed to obtain Formula$\buildrel = \over{\mbi y}$ from Formula${\bar {\mbi y}}$. The A-PPP system (19) can therefore only be solved, after Formula${\mbi Z}$ has been solved from the attitude equations of model (13).

Critical in the application of (19) is how fast and how well the integer matrix Formula${\mbi Z}$ can be estimated. Preferably this should be on a single-epoch (instantaneous) basis, with a sufficiently high probability of correct integer estimation, i.e., Formula${\rm Prob}[{\mathcheck {\mbi Z}} = {\mbi Z}] \approx 1$, where Formula${\mathcheck {\mbi Z}}$ is the integer estimator of Formula${\mbi Z}$ [11], [48]. Only if this probability, also referred to as the ambiguity success rate, is sufficiently close to one, can one neglect the uncertainty in the integer estimator Formula${\mathcheck {\mbi Z}}$ of Formula${\mbi Z}$ and does Formula${\rm Cov}({\buildrel = \over{\mbi y} }) \approx {\rm Cov}({\bar {\mbi y}}) < {\rm Cov}({\mbi y}_{1})$ hold, meaning that one can take advantage of the improved precision of Formula${\bar {\mbi y}}$ over Formula${\mbi y}_{1}$. Section IV shows that such array integer ambiguity resolution is indeed possible with our method of integer Formula${\mbi Z}$-estimation.

To illustrate the potential of A-PPP RTK, the GPS experiment of Table I was repeated but now with ionospheric- and satellite clock corrections provided by a regional dual-frequency CORS network [17]. The results showed cm-level positioning accuracy (Formula$\sigma_{N} = 10, \sigma_{E} = 11, \sigma_{U} = 21 [mm]$) and a one-minute time-to-fix, twice faster for A-PPP RTK than PPP RTK.

3) On-Platform Positioning

Next to determining the position of the platform, it is often also of importance to be able to determine the position of an arbitrary point on the platform. In many applications, for instance, the platform will be equipped with additional (remote sensing) sensors. The sensor positions are then needed so as to be able to colocate the remote sensing data with an earth-fixed frame.

Let Formula${\mbi b_{\tt S}}$ and Formula${\mbi f_{\tt S}}$ be the position vector of the sensor in the earth-fixed frame and in the platform-fixed frame, respectively. Then Formula TeX Source $${\mbi b}_{\tt S}={\bar {\mbi b}} + {\mbi R}({\mbi f}_{\tt S}-{\bar {\mbi f}}) \eqno{\hbox{(21)}}$$ with Formula${\bar {\mbi f}}=- {\mbi F} {\mbi D}_{r}^{+} {\mbi C}_{1}$ the counterpart of Formula${\bar {\mbi {b}}}$ in the platform-fixed frame. Hence, since Formula${\mbi f_{\tt S}}$ and Formula${\bar {\mbi f}}$ are assumed known, Formula${\bar {\mbi {b}}}$ and Formula${\mbi R}$ are needed to determine Formula${\mbi b_{\tt S}}$, the sensor position in the earth-fixed frame. Thus next to positioning, now also an attitude solution is needed.

As with Formula${\bar {\mbi {b}}}$, the rotation matrix Formula${\mbi R}$ can be determined with or without integer ambiguity resolution. But, as is shown in Section IV, the quality of Formula${\mbi R}$ is rather poor for small sized arrays, when solved without the integer ambiguity constraints. Therefore the integer ambiguity resolved rotation matrix has preference and, as is shown in the next section, it can be determined with a high success rate with our method of integer Formula${\mbi Z}$-estimation.

4) Between-Platform Positioning

The A-PPP concept can also be applied to the important field of relative navigation and formation flying. Examples of applications that can benefit from multiplatform A-PPP include land (robotics and cars [49], [50]), air (uninhabited air vehicles [41], [51]), and space (spacecraft formations and attitude [42], [52]) systems.

Consider two A-PPP equipped platforms, Formula${\tt P}$ and Formula${\tt Q}$, each having a system of observation equations like (19). By taking the between-platform difference, one gets Formula TeX Source $${\rm E}({{\bar {\mbi y}}}_{\tt PQ}) = {\mbi M} {\bar {\mbi {b}}}_{\tt PQ} + {\mbi N} {\mbi z}_{\tt PQ}+ {\mbi {d}}_{1{\tt PQ}} \eqno{\hbox{(22)}}$$ with Formula${\bar {\mbi y}}_{\tt PQ} = {\bar {\mbi y}}_{\tt Q}- {{\bar {\mbi y}}}_{\tt P}$ and a likewise definition for Formula${\bar {\mbi {b}}}_{\tt PQ}$, Formula${\mbi z}_{\tt PQ}$ and Formula${\mbi d}_{1{\tt PQ}}$.

To solve (22), the Formula${\mbi Y}{\rm s}$ and Formula${\mbi Z}{\rm s}$ of both platforms are needed, but not their attitude. The rotation matrices of the two platforms, Formula${\mbi R}_{\tt P}$ and Formula${\mbi R}_{\tt Q}$, would be needed though, if next to the relative position, also the between-platform relative attitude, Formula${\mbi R}_{{\tt P} {\tt Q}} = {\mbi R}_{\tt Q} {\mbi R}_{\tt P}^{T}$, is required.

Note, importantly, that the DD ambiguity vector Formula${\mbi z}_{{\tt P} {\tt Q}}$ in (22) is integer. The following example illustrates how its success-rate can be improved by A-PPP. The success rates are given in Table II for the case the ionospheric delays are assumed absent in the model (ionosphere fixed), as for the case they are estimated as unknown parameters (ionosphere float). They are based on 1 Hz GPS phase and code tracking, with zero degree cut-off elevation angle, from two identical Sokkia-receiver equipped platforms, hundred meter apart, having the same configuration as in the experiment of Table I.

Table 2
TABLE II SINGLE-FREQUENCY, SINGLE-EPOCH, BETWEEN-PLATFORM AMBIGUITY SUCCESS RATES Formula$[\%]$ FOR TWO SINGLE-ANTENNA AND TWO QUADRUPLE-ANTENNA PLATFORMS, SEPARATED BY SHORT (IONOSPHERE-FIXED) AND LONG (IONOSPHERE-FLOAT) DISTANCES (SF: SINGLE-FREQUENCY, DF: DUAL-FREQUENCY, XYZ-FIXED: STATION COORDINATES KNOWN, WL: WIDELANE)

Table II shows the single-epoch succes rate improvement when going from a one-antenna equipped pair (1-1) to a quadruple-antennas equipped platform pair (4-4). The second column of Table II shows a significant improvement of the single-frequency (SF), ionosphere-fixed success rate, thus enabling faster single-frequency precise baseline positioning. Such improvement is not seen for the dual-frequency (DF) case. However, when compared with the SF results, we do see that the SF, multiantennas platform has a close to standard dual-frequency receiver performance.

The results of the fourth column indicate that A-PPP equipped CORS stations, having known coordinates [1], [17], can also benefit significantly. Finally, the last column of Table II indicates the A-PPP improvement of widelane (WL) ambiguity resolution. When positioning under ionosphere-float, full ambiguity resolution is often replaced by partial ambiguity resolution using the widelane [53].

We remark that the success rates of Table II are unconditional, since they are not conditioned on assuming the integer ambiguity matrices of both platforms, Formula${\mbi Z}_{\tt P}$ and Formula${\mbi Z}_{\tt Q}$, known [cf. (20)]. Hence, Table II's success rates give the probabilities of correctly estimating the between-platform integer ambiguities, irrespective of whether the integer array ambiguities of both platforms were estimated correctly or not. The method used for integer estimating the array ambiguities is described in the next section. There the improved performance of the method is also compared with the standard method of integer ambiguity resolution.

SECTION IV

CONSTRAINED ARRAY ESTIMATION

In the previous section it was shown that different A-PPP versions require different array information. For platform positioning without ambiguity resolution, it suffices to know Formula${\mbi Y}$, (cf. 15). On-platform positioning, however, requires both Formula${\mbi Y}$ and Formula${\mbi R}$, [cf. (21)], while any version that includes integer ambiguity resolution needs Formula${\mbi Z}$ as well. In order to make these A-PPP applications possible, it is shown in this section how to best estimate Formula${\mbi Z}$ and Formula${\mbi R}$.

A. The Array and its Constraints

To determine Formula${\mbi R}$ and Formula${\mbi Z}$, the array-part of model (13) needs to be solved. Would one only need Formula${\mbi R}$, the simplest approach would be to solve model (13) with (7) in a least-squares sense while disregarding the integerness of Formula${\mbi Z}$. In case of GNSS, however, this approach suffers from the drawback that a disregard of the integerness of Formula${\mbi Z}$, implies that the baseline solution, and therefore the solution of Formula${\mbi R}$ as well, is driven by the relatively poor code data.

Alternatively therefore, one could solve the array-part of model (13) for Formula${\mbi B}$ in a least-squares sense, but now with the integerness of Formula${\mbi Z}$ enforced, and then use this baseline solution to solve for Formula${\mbi R}$. This second approach is an improvement over the first. Still, however, it can be further improved upon, since the determination of the integer matrix Formula${\mbi Z}$ will then not have benefitted from the orthonormality of Formula${\mbi R}$. As will be shown, this improvement turns out to be very significant indeed.

The above discussion makes clear that both constraints, the orthonormality constraint of Formula${\mbi R}$ in Formula${\mbi B} = {\mbi R}{\mbi F}$ and the integer constraint on Formula${\mbi Z}$, need to be enforced from the beginning. The aim of this section is therefore to show how the following, orthonormality-constrained, multivariate (mixed) integer model can be solved in a weighted least-squares sense.

Definition 4 (Constrained Array Model)

The Formula$2fs \times r$ matrix observation equation and covariance matrix of the constrained array model are given as Formula TeX Source $${\rm E}({\mbi Y}) = {\mbi M}{\mbi R}{\mbi F}+ {\mbi N}{\mbi Z}, {\rm Cov}({\rm {vec}}({\mbi Y})) = {\mbi P} \otimes {\mbi Q} \eqno{\hbox{(23)}}$$ with the two sets of constraints Formula TeX Source $${\mbi R} \in \BBO^{3 \times q} {\rm and} {\mbi Z} \in \BBZ^{fs\times r} \eqno{\hbox{(24)}}$$ and where Formula${\mbi P} = {\mbi D}_{r}^{T} {\mbi Q}_{r} {\mbi D}_{r}$.

Thus the unknown parameters in this array model are the matrices Formula${\mbi R}$ and Formula${\mbi Z}$, constrained by (24).

B. The Role of Integer Ambiguity Resolution

To get a better understanding of the role played by integer ambiguity resolution in determining Formula${\mbi R}$, let us for the moment disregard the first constraint, Formula${\mbi R} \in \BBO^{3 \times q}$, and consider, instead of the second constraint, the two extremes cases: Formula${\mbi Z}$ is known or Formula${\mbi Z}$ is completely unknown.

Lemma 1 (Formula${\mbi Z}$ Known, Formula${\mbi R}$ Unknown)

Let the Formula${\mbi Z}$-constrained LS-estimator of Formula${\mbi R}$ in model (23) be defined as Formula TeX Source $${\mathhat {\mbi{R}}}({\mbi Z}) = \arg \min_{{\mmb R} \in \BBR^{3 \times q}}\Vert {\rm {vec}}({\mbi Y}- {\mbi M}{\mbi R}{\mbi F}- \mbi {NZ})\Vert_{{\mmb Q_{\mmb {YY}}}}^{2} \eqno{\hbox{(25)}}$$ with Formula${\mbi Q_{YY}} = {\rm Cov}({\rm {vec}}({\mbi Y}))$. Then Formula${\mathhat {\mbi{R}}}({\mbi Z})$ and its covariance matrix are given as Formula TeX Source $$\eqalignno{{\mathhat {\mbi R}}({\mbi Z}) =&\, {\mbi {M}}^{+}({\mbi Y}- {\mbi N}{\mbi Z}) {\mbi F}^{+} \cr \mbi {Q_{{\mathhat {R}}(Z) {\mathhat {R}}(Z)}} =&\, ({\mbi F} \mbi {P}^{-1} {\mbi F}^{T})^{-1} \otimes ({\mbi M}^{T} {\mbi {Q}}^{-1} {\mbi M})^{-1} &{\hbox{(26)}}}$$ with the LS-inverses Formula${\mbi M}^{+}=({\mbi M}^{T} {\mbi Q}^{-1} {\mbi M})^{-1} {\mbi M}^{T} {\mbi Q}^{-1}$ and Formula${\mbi F}^{+} = {\mbi P}^{-1} {\mbi F}^{T}({\mbi F} {\mbi P}^{-1} {\mbi F}^{T})^{-1}$.

Proof

The proof is given in the Appendix.

This LS estimator of Formula${\mbi R}$ is denoted as Formula${\mathhat {\mbi{R}}}({\mbi Z})$ to emphasize its dependence on the value taken for Formula${\mbi Z}$.

Through the presence of the matrices Formula${\mbi M}$ and Formula${\mbi F}$, we clearly recognize the contributions of both the receiver-satellite geometry, via Formula${\mbi M}$, and the antenna-array geometry, via Formula${\mbi F}$. Without the antenna-array geometry (i.e., Formula${\mbi F} = {\mbi I}$ or Formula${\mbi R} = {\mbi B}$), the solution would read Formula${\mathhat {\mbi B}}({\mbi Z}) = {\mbi M}^{+}({\mbi Y}- {\mbi N}{\mbi Z})$. But with the antenna-array geometry included, a further least-squares mapping takes place, from Formula${\mathhat {\mbi B}}({\mbi Z})$ to Formula${\mathhat {\mbi R}}({\mbi Z}) = {\mathhat {\mbi B}}({\mbi Z}) {\mbi F}^{+}$.

Now the other extreme, that of a completely unconstrained Formula${\mbi Z}$-matrix, is considered.

Lemma 2 (Formula${\mbi Z}$ Unknown, Formula${\mbi R}$ Unknown)

Let the unconstrained LS-estimators of Formula${\mbi R}$ and Formula${\mbi Z}$ in model (23) be defined as Formula TeX Source $$\{{\mathhat {\mbi R}}, {\mathhat {\mbi Z}}\} = \arg \min_{{\mmb R} \in \BBR ^{3 \times q}, {\mmb Z} \in \BBR^{fs \times r}} \Vert {\rm {vec}}({\mbi Y}- {\mbi M}{\mbi R}{\mbi F}- {\mbi N}{\mbi Z})\Vert_{\mmb {Q_{YY}}}^{2}. \eqno{\hbox{(27)}}$$ Then Formula${\mathhat {\mbi{R}}}$, Formula${\mathhat {\mbi Z}}$ and their covariance matrices are given as Formula TeX Source $$\eqalignno{{\mathhat {\mbi{R}}} =&\, {\bar {\mbi M}}^{+}({\mbi Y}) {\mbi F}^{+} \cr {\mathhat {\mbi Z}} =&\, {\mbi N}^{+}({\mbi Y}- {{\mbi M} {\mathhat {\mbi R}}{\mbi F}}) &{\hbox{(28)}}}$$ and Formula TeX Source $$\eqalignno{{\mbi Q_{{\mathhat {R}} {\mathhat {R}}}} =&\, ({\mbi F} {\mbi P}^{-1} {\mbi F}^{T})^{-1} \otimes ({\bar {\mbi M}}^{T} {\mbi Q}^{-1} {\bar {\mbi M}})^{-1} \cr {\mbi Q_{{\mathhat {Z}} {\mathhat {Z}}}} =&\, ({\mbi P}^{-1} \otimes {\bar {\mbi N}}^{T} {\mbi Q}^{-1} {\bar {\mbi N}} \cr & \quad + {\bar {\mbi P}}^{-1} \otimes {\mbi N}^{T} {\mbi Q}^{-1} {\mbi P_{\mmb M}{\mbi N}})^{-1} &{\hbox{(29)}}}$$ with Formula${\bar {\mbi M}}^{+}=({\bar {\mbi M}}^{T} {\mbi Q}^{-1} {\bar {\mbi M}})^{-1} {\bar {\mbi M}}^{T} {\mbi Q}^{-1}$, Formula${\mbi N}^{+}=({\mbi N}^{T} {\mbi Q}^{-1} {\mbi N})^{-1} {\mbi N}^{T} {\mbi Q}^{-1}$, Formula${\bar {\mbi M}} = {\mbi P_{\mmb N}^{\mmb \perp}} {\mbi M}$, Formula${\mbi P_{\mmb N}^{\mmb \perp}} = {\mbi I}- {\mbi N}{\mbi N}^{+}$, Formula${\bar {\mbi N}} = {\mbi P_{\mmb M}^{\mmb \perp}} {\mbi N}$, Formula${\mbi P}_{\mmb M}^{\mmb \perp} = {\mbi I}- {\mbi M}{\mbi M}^{+}$ and Formula${\bar {\mbi P}}^{-1}=({\mbi I}- {\mbi F}^{+} {\mbi F}) {\mbi P}^{-1}$.

Proof

The proof is given in the Appendix.

Since Formula${\mbi Z}$ is assumed unknown in (27), the precision of Formula${\mathhat {\mbi{R}}}$ is, of course, poorer than that of Formula${\mathhat {\mbi{R}}}({\mbi Z})$. Importantly, in case of GNSS, this difference is very significant. In case of GNSS, the precision of Formula${\mathhat {\mbi{R}}}({\mbi Z})$ is driven by the very precise carrier-phase measurements, while the precision of Formula${\mathhat {\mbi{R}}}$ is driven by the relatively low precision code measurements. Denoting the phase variance as Formula$\sigma_{\phi}^{2}$ and the code variance as Formula$\sigma_{p}^{2}$, the covariance matrices of the two attitude estimators can shown to be related as Formula TeX Source $${\mbi Q_{{\mathhat {\mmb R}}({\mmb Z}) {\mathhat {\mmb R}}({\mmb Z})}} \approx {{\sigma_{\phi}^{2}} \over {\sigma_{p}^{2}}} \quad \mbi {Q_{{\mathhat {R}} {\mathhat {R}}}} \eqno{\hbox{(30)}}$$ where, in case of current GPS, Formula${\sigma_{\phi}^{2} \over \sigma_{p}^{2}} \approx 10^{-4}$ [40]. This shows that a very large precision improvement in the determination of the attitude matrix Formula${\mbi R}$ can be realized if one would be able to integer estimate Formula${\mbi Z}$ with negligible uncertainty, i.e., with a success rate Formula${\rm {Prob}}[{\mathcheck {\mbi Z}} = {\mbi Z}] \approx 1$. Achieving the latter, is the goal of integer ambiguity resolution [28].

C. Ambiguity Resolution Without Orthonormality Constraint

Although matrix Formula${\mbi Z}$ is not known, we know that its entries are all integers. Hence, if one could estimate these entries with a probability of correct integer estimation that is sufficiently close to one, one could treat the integer estimated Formula${\mbi Z}$ for all practical purposes as known and therefore indeed compute a very precise attitude matrix.

With the integer constraints included, the LS problem turns into a (mixed) integer least-squares (ILS) problem [13]. To determine its solution, we first write its objective function as a sum-of-squares.

Lemma 3 (Multivariate Orthogonal Decomposition)

Let Formula${\mathhat {\mbi{R}}}({\mbi Z})$, Formula${\mathhat {\mbi{R}}}$, Formula${\mathhat {\mbi Z}}$ and their covariance matrices be given as in (26), (28), and (29) and let Formula${\mathhat {\mbi E}} = {\mbi Y}- {\mbi M} {\mathhat {\mbi{R}}} {\mbi F}- {\mbi N} {\mathhat {\mbi Z}}$. Then Formula TeX Source $$\displaylines{\Vert {\rm vec} ({\mbi Y}- {\mbi M}{\mbi R}{\mbi F}- {\mbi N}{\mbi Z})\Vert_{{{\mmb Q}_{\mmb{YY}}}}^{2} = \Vert {\rm vec}({\mathhat {\mbi E}})\Vert_{{{\mmb Q}_{\mmb {YY}}}}^{2} \hfill \cr \hfill +\Vert {\rm vec}({\mathhat {\mbi Z}}- {\mbi Z})\Vert_{{{\mmb Q}_{{\mathhat {\mmb Z}} {\mathhat {\mmb Z}}}}}^{2}+\Vert {\rm vec}({\mathhat {\mbi R}}({\mbi Z})- {\mbi R})\Vert_{{\mmb Q_{{\mathhat {\mmb R}}({\mmb Z}) {\mathhat {\mmb R}}({\mmb Z})}}}^{2}. \quad{\hbox{(31)}}}$$

Proof

The proof is given in the Appendix.

With this orthogonal decomposition the following result can be proven.

Lemma 4 (Formula${\mbi Z}$ Integer, Formula${\mbi R}$ Unknown)

Let the Formula${\mbi R}$-unconstrained (mixed) integer LS-estimators of Formula${\mbi R}$ and Formula${\mbi Z}$ in model (23) be defined as Formula TeX Source $$\displaylines{\{{\mathhat {\mbi R}}_{\tt U},{\mathhat {\mbi Z}}_{\tt U}\} = \arg \min_{{\mmb R} \in \BBR^{3 \times q}, {\mmb Z} \in \BBZ^{fs \times r}} \hfill \cr \hfill \Vert {\rm vec}({\mbi Y}- {\mbi M}{\mbi R}{\mbi F}- {\mbi N}{\mbi Z})\Vert_{{\mmb Q}_{\mmb {YY}}}^{2}.\quad{\hbox{(32)}}}$$ Then Formula${\mathhat {\mbi{R}}}_{\tt U}$ and Formula${\mathcheck {\mbi Z}}_{\tt U}$ are given as Formula TeX Source $$\eqalignno{{\mathhat {\mbi{R}}}_{\tt U} =&\, {\mathhat {\mbi{R}}}({\mathcheck{\mbi Z}}_{\tt U}) \cr {\mathcheck{{\mbi {Z}}}}_{\tt U} =&\, \arg {\mathop{\min}\limits_{{\mmb Z} \in \BBZ^{fs \times r}}}\Vert {\rm vec}({\mathhat {\mbi Z}}- {\mbi Z})\Vert_{{\mmb Q_{{\mathhat {\mmb Z}} {\mathhat {\mmb Z}}}}}^{2}. &{\hbox{(33)}}}$$

Proof

Since the first term on the right-hand side (RHS) of (31) is a constant, while the third term on the RHS can be made zero for any Formula${\mbi Z}$, i.e., by setting Formula${\mbi R} = {\mathhat {\mbi{R}}}({\mbi Z})$, the integer ambiguity matrix solution, Formula${\mathcheck {\mbi Z}}_{\tt U}$, follows from integer minimizing the second term on the RHS of (31). Substitution of this integer solution into Formula${\mathhat {\mbi{R}}}({\mbi Z})$ gives the attitude matrix solution of (33). Formula$\hfill\square$

The estimators of (33) are given the suffix Formula$(.)_{\tt U}$ to emphasize that this solution is still Formula${\mbi R}$-orthonormality unconstrained.

If the probability mass function of Formula${\mathcheck {\mbi Z}}_{\tt U}$ is sufficiently peaked at the true but unknown value Formula${\mbi Z}$, i.e., Formula${\rm Prob}[{\mathcheck {\mbi Z}}_{\tt U} = {\mbi Z}] \approx 1$, then the uncertainty in Formula${\mathcheck {\mbi Z}}_{\tt U}$ can be neglected for all practical purposes and the covariance matrix of Formula${\mathhat {\mbi{R}}}_{\tt U}$ can be approximated by Formula${\mbi Q_{{\mathhat {\mmb R}}({\mmb Z}) {\mathhat {\mmb R}}({\mmb Z})}}$, being the covariance matrix of the very precise estimator Formula${\mathhat {\mbi{R}}}({\mbi Z})$, cf. (30). Hence, if a precise orthonormal attitude matrix is asked for, one can use as estimator Formula${\mathcheck {\mbi R}}_{\tt U} = {\mathcheck {\mbi R}}({\mathcheck {\mbi Z}}_{\tt U})$, with Formula${\mathcheck {\mbi R}}({\mbi Z})$ defined as the orthonormality-constrained least-squares solution Formula TeX Source $${\mbi{\mathcheck{R}}}({\mbi Z}) = \arg \min_{{\mmb R} \in \BBO^{3 \times q}}\Vert {\rm vec}({\mathhat {\mbi{R}}}({\mbi {Z}})- {\mbi R})\Vert_{{\mmb Q_{{\mathhat {\mmb R}}({\mmb Z}) {\mathhat {\mmb R}}({\mmb Z})}}}^{2} . \eqno{\hbox{(34)}}$$ This problem reduces to Wahba's problem [54], [55], [56], also known as the “orthogonal Procrustes problem” [57], [58], in case the covariance matrix of Formula${\mathhat {\mbi{R}}}({\mbi Z})$ would have the special structure Formula${\mbi Q_{{\mathhat {\mmb R}}({\mmb Z}) {\mathhat {\mmb R}}({\mmb Z})}}=({\mbi F} {\mbi P}^{-1} {\mbi F}^{T})^{-1} \otimes {\mbi I}$, with Formula${\mbi P}$ diagonal. The many solution methods of Wahba's problem have been reviewed in [59]. One of the simplest is based on the singular value decomposition (SVD) of the baseline matrix. It allows a direct computation of the attitude matrix.

For our GNSS array, Formula${\mbi Q_{{\mathhat {\mmb R}}({\mmb Z}) {\mathhat {\mmb R}}({\mmb Z})}}$ is fully populated, and therefore no such direct solution, as with Wahba's problem, can be used to solve (34). This nonlinear least-squares problem is therefore iteratively solved by means of one of the gradient descent methods, like the Gauss-Newton method, having a local linear rate of convergence, or the Newton method, having a local quadratic rate of convergence. These gradient descent methods work well when initialized by a good starting value. In our case, the solution of Wahba's problem provides a good starting value. For a numerical-statistical analysis of nonlinear least-squares procedures as used in positioning, we refer to, e.g., [60], [61], [62].

D. Ambiguity Resolution With Orthonormality Constraint

As is shown in [45], the required high probability of correct integer estimation of Formula${\mathcheck {\mbi Z}}_{\tt U}$ is feasible in the GNSS multifrequency case Formula$(f > 1)$, but generally problematic in the single-frequency case. This shows that the single-frequency array model needs a further strengthening.

To increase the strength of the array model, we now include the orthonormality constraint Formula${\mbi R} \in \BBO^{3 \times q}$ from the start. With this constraint rigorously incorporated into the integer estimation process, a higher probability of correct integer estimation can be achieved. The inclusion of the constraint Formula${\mbi R} \in \BBO^{3 \times q}$ is thus not so much for the purpose of forcing the solution of Formula${\mbi R}$ to be orthonormal per se, but rather to aid the integer ambiguity resolution process.

With both the integer constraint and the orthonormality constraint included, the minimization problem becomes a constrained (mixed) ILS problem.

Theorem 2 (Formula${\mbi Z}$ Integer, Formula${\mbi R}$ Orthonormal)

Let the orthonormality-constrained (mixed) integer LS-estimators of Formula${\mbi R}$ and Formula${\mbi Z}$ in model (23) be defined as Formula TeX Source $$\{{\mathcheck {\mbi R}}, {\mathcheck {{\mbi {Z}}}}\} = \arg \min_{{\mmb R} \in \BBO^{3 \times q}, {\mmb Z} \in \BBZ^{fs \times r}} \Vert {\rm vec}({\mbi {Y}}- {\mbi M}{\mbi R}{\mbi F}- {\mbi N}{\mbi Z})\Vert_{{\mmb Q_{\mmb {YY}}}}^{2} .\eqno{\hbox{(35)}}$$ Then Formula${\mathcheck {\mbi R}}$ and Formula${\mathcheck {\mbi Z}}$ are given as Formula TeX Source $${\mathcheck {\mbi R}} = {\mathcheck {\mbi R}}({\mathcheck {\mbi Z}}) {\rm and} {\mathcheck {\mbi Z}} = \arg {\mathop{\min}\limits_{{\mmb Z} \in \BBZ ^{fs \times r}}} J({\mbi Z}) \eqno{\hbox{(36)}}$$ with the ambiguity objective function given as Formula TeX Source $$J({\mbi Z}) = \Vert {\rm vec}({\mathhat {\mbi Z}}- {\mbi Z})\Vert_{{\mmb Q_{{\mathhat {\mmb Z}} {\mathhat {\mmb Z}}}}}^{2} + \Vert {\rm vec}({\mathhat {\mbi{R}}}({\mbi Z})- {\mathcheck {\mbi R}}({\mbi {Z}}))\Vert_{{\mmb Q_{{\mathhat {\mmb R}}({\mmb Z}) {\mathhat {\mmb R}}({\mmb Z})}}}^{2}. \eqno{\hbox{(37)}}$$

Proof

The proof is given in the Appendix.

Note, importantly, that the ambiguity objective function (37) differs from that of Formula${\mathcheck {\mbi Z}}_{\tt U}$ [cf. (33)] by the presence of the Formula${\mbi R}$-dependent second term. The presence of both constraints is therefore felt when evaluating this objective function. In integer minimizing Formula$J({\mbi Z})$, not only the weighted distance between Formula${\mbi Z}$ and Formula${\mathhat {\mbi Z}}$ counts [as is the case in (33)], but also the weighted distance between Formula${\mathhat {\mbi{R}}}({\mbi Z})$ and its closest orthonormal matrix Formula${\mathcheck {\mbi R}}({\mbi Z})$. The weights are determined by the inverses of Formula${{\mbi Q}_{{\mathhat {\mmb Z}} {\mathhat {\mmb Z}}}}$ and Formula${\mbi Q_{{\mathhat {\mmb R}}({\mmb Z}) {\mathhat {\mmb R}}({\mmb Z})}}$, respectively. And as remarked earlier, in case of GNSS, the covariance matrix Formula${\mbi Q_{{\mathhat {\mmb R}}({\mmb Z}) {\mathhat {\mmb R}}({\mmb Z})}}$ is driven by the very precise carrier-phase data [cf. (30)]. Thus the second term in the ambiguity objective function (37) receives a relatively large weight and contributes significantly to the improved success rate performance of Formula${\mathcheck {\mbi Z}}$ over Formula${\mathcheck {\mbi Z}}_{\tt U}$. This method is therefore also our method of choice for realizing array-aided PPP.

The following experiment illustrates the very high success rates that can be achieved when working with the ambiguity objective function (37), instead of with the standard quadratic ambiguity objective function of (33). Located at a stationary point in Limburg, the Netherlands, a fixed array of three antennas (a Trimble Zephyr Geodetic L1/L2, the Master, and two Trimble Geodetic W Groundplane, the auxiliaries), connected to three Trimble receivers (a Trimble R7 and two Trimble SSi), was used to collect (10:44–13:29 UTC) and process 1 Hz data, with a zero cut-off elevation angle. The two baselines formed by the three antennas have lengths 2.214 m and 1.742 m, with a 66.4–degree relative orientation.

The single-frequency, single-epoch success rates are given in Table III as function of the number of tracked satellites and the number of antennas used. For each configuration, the unconstrained (U) and constrained (C) success rates are given, based on using (33) and (37), respectively. The number of tracked satellites was artificially reduced to show the robustness against constellation availability. Also, different baselines have been included in the model: for the two single-baseline (dual-antennas) cases only the baseline length is used as a priori constraint, whereas for the two-baseline (triple-antennas) case the complete geometry is used to construct the Formula${\mbi F}$ matrix of (7).

Table 3
TABLE III SINGLE-FREQUENCY, SINGLE-EPOCH, UNCONSTRAINED- (u) AND CONSTRAINED-(C) AMBIGUITY SUCCESS RATES Formula$[\%]$ FOR TWO DUAL-ANTENNA PLATFORMS AND ONE TRIPLE-ANTENNA PLATFORM. THE SUCCESS RATES ARE GIVEN FOR A VARYING NUMBER OF TRACKED SATELLITES (5–9)

As the results show (compare the U- and C-columns), the success rates improve dramatically when the constraints are exploited using (37). For the worst scenario, with only five satellites in view, the inclusion of the dual-antennas length constraint is sufficient to increase the success rate from about 9% to about 75%–79%. The constrained success rate increases even further to 99.6% when the full sets of constraints for the three antennas is exploited. Also note that the constrained success rates are far more robust against variability in number of tracked satellites than the unconstrained success rates are. The results of Table III are typical for the performance of the ambiguity objective function (37). The high success rates show that real-time A-PPP platform ambiguity resolution is possible and that reinitialization, in case of a complete loss of lock, only requires a few epochs at most.

The estimation theory and method presented is not restricted to a minimum or maximum antenna separation. But the implicit assumption is of course that the antennas do not interfere with one another. As to the suitability of antenna spacing: smaller spacing makes the integer estimation process simpler (if less than 0.5 wavelength the simplest integer rounding techniques get closer to optimal performance in terms of maximizing the success rate). Larger spacing, however, improves platform's attitude resolution performance [cf. (26) and (29)].

E. The Integer Ambiguity Search

An integer search is needed to solve for the integer minimizer of Formula$J({\mbi Z})$ [cf. (37)]. Such search is conceptually simple in principle. The search can be confined to any nonempty discrete set of the type Formula TeX Source $$\Omega (\chi^{2}) = \{{\mbi Z} \in \BBZ^{fs \times r} \vert J({\mbi Z}) \leq \chi^{2}\} \eqno{\hbox{(38)}}$$ where Formula$\chi^{2}$ is a user-defined positive constant; it controls the size of the search space. If Formula$\Omega (\chi^{2})$ is nonempty, then the integer minimizer is, by definition, contained in it. It is then found by first collecting all integer matrices inside Formula$\Omega (\chi^{2})$, followed by selecting the one that returns the smallest function value Formula$J({\mbi Z})$.

Although conceptually simple, the actual search in our A-PPP case turns out to be somewhat more complex. To appreciate this complexity, several issues need to be addressed. First, consider the shape of the search space Formula$\Omega (\chi^{2})$. The search space would be ellipsoidal, in case the second, Formula${\mbi R}$-dependent, term in Formula$J({\mbi Z})$ would be absent [cf. (37)]. The presence of this attitude-dependent term, however, turns Formula$\Omega (\chi^{2})$ into a nonellipsoidal, nonconvex search space. This effect is emphasized the more so, since the covariance matrix Formula${\mbi Q_{{\mathhat {\mmb Z}} {\mathhat {\mmb Z}}}}$ in the first term of Formula$J({\mbi Z})$ is driven by the relatively poor code precision, while the covariance matrix Formula${\mbi Q_{{\mathhat {\mmb R}}({\mmb Z}) {\mathhat {\mmb R}}({\mmb Z})}}$ in the second term of Formula$J({\mbi Z})$ is driven by the very precise carrier phase precision.

Second, consider the user-defined positive constant Formula$\chi^{2}$ in (38). It determines the size of the search space. Any choice Formula$\chi^{2}=J({\mbi Z}_{0})$, with Formula${\mbi Z}_{0}$ an integer matrix, guarantees that the search space is nonempty. At the same time, however, one would like Formula$\chi^{2}$ to be small enough, so as to have a search space with not too many integer candidates. Therefore some care needs to be excersized in choosing Formula${\mbi Z}_{0}$. Any Formula${\mbi Z}_{0}$ which is too far from the actual integer minimizer can be expected to result in a too large Formula$\chi^{2}$, especially due to the amplifying effect of the second term in Formula$J({\mbi Z})$. We therefore choose Formula${\mbi Z}_{0} = {\mathhat {\left\lceil {\mbi Z}^{\ast} \right\rfloor}}$, where Formula${\mathhat {\mbi Z}}^{\ast}$ is the real-valued minimizer of Formula$J({\mbi Z})$ and Formula$\left\lceil . \right\rfloor$ denotes rounding to nearest integer. It is our experience that this choice works very well. Note that Formula${\mathhat {\mbi Z}}^{\ast}$ is the orthonormality-constrained least-squares solution of the ambiguity matrix Formula${\mbi Z}$.

Third, consider the actual evaluation of Formula$J({\mbi Z})$ (cf. 37). Any such evaluation also requires the evaluation of Formula${\mathcheck {\mbi R}}({\mbi Z})$ and therefore, for any candidate Formula${\mbi Z}$, the solution of a nonlinear constrained least-squares problem like (34). Since such minimization for every candidate in Formula$\Omega (\chi^{2})$ is a computational burden on the search, the search efficiency can be improved if one would be able to work with an easier-to-evaluate function of which the level set would still be a good approximation to Formula$\Omega (\chi^{2})$. We therefore work with an easy-to-evaluate, sharp upper bounding function Formula$J^{\prime}({\mbi Z})$, having as level set Formula$\Omega ^{\prime}(\chi^{2})=\{{\mbi Z} \in \BBZ^{fs \times r} \quad \vert \quad J^{\prime}({\mbi Z}) \leq \chi^{2}\}$. Then Formula TeX Source $$J({\mbi Z}) \leq J^{\prime}({\mbi Z}) {\rm and} \Omega ^{\prime}(\chi^{2}) \subset \Omega (\chi ^{2}) . \eqno{\hbox{(39)}}$$ Note that for any orthonormal matrix Formula${\mbi R}_{0} \in \BBO^{3 \times q}$, the function Formula TeX Source $$J^{\prime}({\mbi Z}) = \Vert {\rm vec}(\mbi {\mathhat {Z}}- {\mbi Z})\Vert_{{\mmb Q_{{\mathhat {\mmb Z}} {\mathhat {\mmb Z}}}}}^{2} + \Vert {\rm vec}({\mathhat {\mbi{R}}}({\mbi Z})- {\mbi R}_{0})\Vert_{\mmb {Q_{{\mathhat {R}}(Z) {\mathhat {R}}(Z)}}}^{2} \eqno{\hbox{(40)}}$$ is an upper bound for Formula$J({\mbi Z})$. Thus Formula$J^{\prime}({\mbi Z})$ can expected to be a good approximation to Formula$J({\mbi Z})$, if Formula${\mbi R}_{0}$ is a good approximation to Formula${\mathcheck {\mbi R}}({\mbi Z})$. This suggests that we take the closed form solution of Wahba's problem as our choice for Formula${\mbi R}_{0}$. And indeed, it is our experience that this choice results in a sharp upper bound. The drawback of this choice is, however, that it requires the solution of an SVD for every candidate Formula${\mbi Z}$. Our method of choice is therefore to use, instead of the SVD, the (weighted) Gramm-Schmidt orthogonalization of Formula${\mathhat {\mbi{R}}}({\mbi Z})$. This is faster to execute than the SVD and still results in a sharp enough upper bound.

The actual search is similar as used in [10] and proceeds as follows. We start the search for an integer candidate in the initial search space Formula$\Omega ^{\prime}(\chi_{0}^{2}) \subset \Omega (\chi_{0}^{2})$, where Formula$\chi_{0}^{2}=J^{\prime}({\mbi Z}_{0})$. Let this candidate be Formula${\mbi Z}_{1}$. Then Formula$J^{\prime}({\mbi Z}_{1})=\chi_{1}^{2} < \chi_{0}^{2}$, which gives the shrunken search space Formula$\Omega ^{\prime}(\chi_{1}^{2})$, in which again an integer candidate is searched, say Formula${\mbi Z}_{2}$. This iterative process of 'search and shrink' is repeated until the integer minimizer of Formula$J^{\prime}({\mbi Z})$, say Formula${\mathcheck {\mbi Z}}^{\prime}$, is found. Since this minimizer need not be the minimizer of Formula$J({\mbi Z})$ (although in practice it usually is), the search space Formula$\Omega (\bar {\chi}^{2}) \supset \Omega ^{\prime}(\bar {\chi}^{2})$, with Formula$\bar {\chi}^{2}=J^{\prime}({\mathcheck {\mbi Z}}^{\prime})$, is searched. The sought-for minimizer Formula${\mathcheck {\mbi Z}}$ is then selected from the candidates in Formula$\Omega (\bar {\chi}^{2})$. In practice, with our choice of bounding function, Formula$\Omega (\bar {\chi}^{2})$ contains only a few candidates and usually even only one. As a result the integer minimizer of Formula$J({\mbi Z})$ can be found efficiently.

SECTION V

SUMMARY AND CONCLUSION

In this paper, the GNSS A-PPP concept was introduced as a generalization of PPP. A-PPP is a GNSS measurement concept that uses GNSS data from multiple antennas in known formation to realize improved GNSS parameter estimation (position, attitude, time, and atmospheric delays).

For its stochastic model a general structure was introduced so as to accomodate differences in phase precision, differences in code precision, frequency dependent tracking precision, satellite elevation dependency and also differences in quality of the antenna/receivers in the array. By means of a decorrelating transformation, applied to a combined positioning and attitude model, it was shown which improvements array-aiding brings to the different forms of positioning. The improvements can be exploited in different ways, e.g., to improve accuracy, to reduce convergence time, to achieve higher success rates or to improve between-platform positioning. The A-PPP improvements were illustrated by means of empirical results obtained from GPS experiments.

To enable fast and accurate A-PPP, a novel orthonormality-constrained multivariate (mixed) integer least-squares problem was introduced and solved. It was shown that its 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 high instantaneous probabilities of correct integer estimation. We also discussed the nonlinear A-PPP ambiguity objective function and presented a search for its integer matrix minimizer.

The A-PPP principle is generally applicable. It applies to single-, dual-, and multifrequency GNSS receivers, as well as to any current and future GNSS (e.g., Europe's Galileo and China's Compass), standalone or in combination. The integer ambiguity resolved array-aiding concept is not restricted to GNSS, as it may apply to, e.g., acoustic phase-based positioning [63] and other interferometric techniques as well.

APPENDIX

Proof of Theorem 1

Application of the one-to-one Formula${\mbi \cal D}$-transformation, Formula${\rm {vec}}([ {\bar {\mbi y}}, {\mbi Y}]) = {\mbi {\cal D}} {\rm {vec}}([{\mbi y}_{1}, \mbi {Y}])$, to the multivariate observation equations of (10), directly gives those of (13).

To derive the covariance matrix Formula${\rm Cov}({\rm {vec}}([{\bar{\mbi y}}, {\mbi Y}]))$, we first substitute Formula${\rm {vec}}([{\mbi y}_{1}, {\mbi Y}]) = {\mmb{\Upsilon}}[{\mbi C}_{1}, {\mbi D}_{r}]$, with Formula${\mmb{\Upsilon}}=[{\mbi y}_{1}, \ldots, {\mbi y}_{r+1}]$, into Formula${\rm {vec}}([{\bar {\mbi y}}, {\mbi Y}]) = {\mbi{\cal D}} {\rm {vec}}([{\mbi y}_{1}, \mbi {Y}])$. This gives Formula TeX Source $${\rm {vec}}([{\bar {\mbi y}}, {\mbi Y}]) = \left([\mbi {P_{D_{r}}^{\perp}} {\mbi C}_{1}, {\mbi D}_{r}]^{T} \otimes {\mbi I}_{2fs} \right) {\rm {vec}}({\mmb{\Upsilon}}) \eqno{\hbox{(A1)}}$$ in which Formula${\mbi P_{D_{r}}^{\perp}}$ denotes the orthogonal projector Formula${\mbi P_{D_{r}}^{\perp}} = {\mbi I}- {\mbi D}_{r}({\mbi D}_{r}^{T} {\mbi Q}_{r} \mbi {D}_{r})^{-1} {\mbi D}_{r}^{T} {\mbi Q}_{r}$. With Formula${\rm Cov}({\rm vec}({\mmb{\Upsilon}}) = {\mbi Q}_{r} \otimes {\mbi Q}$ [cf. (9)], an application of the variance-covariance propagation law to (A1) gives Formula TeX Source $${\rm Cov}({\rm {vec}}([{\bar {\mbi y}}, {\mbi Y}])) = {\mbi S} \otimes \mbi {Q}\eqno{\hbox{(A2)}}$$ with Formula TeX Source $${\mbi S} = [{\mbi P_{D_{r}}^{\perp}} {\mbi C}_{1}, {\mbi {D}}_{r}]^{T} {\mbi Q}_{r}[{\mbi P_{D_{r}}^{\perp}} {\mbi {C}}_{1}, {\mbi D}_{r}] . \eqno{\hbox{(A3)}}$$ Since Formula${\mbi D}_{r}^{T} {\mbi Q}_{r} {\mbi P_{D_{r}}^{\perp}}=0$ and the projector Formula${\mbi P_{D_{r}}^{\perp}}$ can alternatively be expressed as Formula${\mbi P_{D_{r}}^{\perp}} = {\mbi Q}_{r}^{-1} {\mbi e}_{r}({\mbi e}_{r}^{T} \mbi {Q}_{r}^{-1} {\mbi e}_{r})^{-1} {\mbi e}_{r}^{T}$, because Formula${\mbi D}_{r}^{T} {\mbi e}_{r} = {\bf 0}$, we finally obtain Formula TeX Source $$\eqalignno{{\mbi S} = &\, {\rm blockdiag}\left[{\mbi {C}}_{1}^{T} {\mbi Q}_{r} {\mbi P_{D_{r}}^{\perp}} {\mbi {C}}_{1}, ({\mbi D}_{r}^{T} {\mbi Q}_{r} {\mbi {D}}_{r})\right] \cr = &\, {\rm blockdiag}\left[({\mbi {e}}_{r}^{T} {\mbi Q}_{r}^{-1} {\mbi e}_{r})^{-1}, (\mbi {D}_{r}^{T} {\mbi Q}_{r} {\mbi D}_{r})\right]. &{\hbox{(A4)}}}$$ This concludes the proof of the theorem. Formula$\hfill\square$

Proof of Lemma 1

The system of multivariate normal equations of the LS-problem (25) is given as Formula TeX Source $$({\mbi F} {\mbi P}^{-1} {\mbi F}^{T} \otimes {\mbi M}^{T} {\mbi Q}^{-1} {\mbi M}) {\mathhat {\mbi R}}({\mbi Z}) =({\mbi F} {\mbi P}^{-1} \otimes {\mbi M}^{T} {\mbi Q}^{-1}){\rm vec}({\mbi Y}). \eqno{\hbox{(A5)}}$$ With the use of the Kronecker product property Formula$({\mbi A} \otimes {\mbi B})^{-1} = {\mbi A}^{-1} \otimes {\mbi B}^{-1}$ (Formula${\mbi A}$ and Formula${\mbi B}$ invertible matrices), inversion of (A5) gives Formula${\mathhat {\mbi{R}}}({\mbi Z})$ of (26). The covariance matrix Formula${\mbi Q_{{\mathhat {R}}(Z) {\mathhat {R}}(Z)}}$ follows from an application of the variance-covariance propagation law to the expression for Formula${\mathhat {\mbi{R}}}({\mbi Z})$. Formula$\hfill\square$

Proof of Lemma 2

The multivariate normal equations of the LS-problem (35) are given as Formula TeX Source $$\displaylines{\left[\matrix{{\mbi F} {\mbi P}^{-1} {\mbi F}^{T} \otimes {\mbi M}^{T} {\mbi Q}^{-1} {\mbi M} & {\mbi F} {\mbi P}^{-1} \otimes {\mbi M}^{T} {\mbi Q}^{-1} {\mbi N} \cr {\mbi P}^{-1} {\mbi F}^{T} \otimes {\mbi N}^{T} {\mbi Q}^{-1} {\mbi M} & {\mbi P}^{-1} \otimes {\mbi N}^{T} {\mbi Q}^{-1} {\mbi N}} \right] \hfill \cr \hfill \times \left[\matrix{{\rm vec}({\mathhat {\mbi{R}}}) \cr {\rm vec}(\mbi {\mathhat {Z}})} \right] = \left[\matrix{({\mbi F} {\mbi P}^{-1} \otimes {\mbi M}^{T} {\mbi Q}^{-1}){\rm vec}({\mbi Y}) \cr ({\mbi P}^{-1} \otimes {\mbi N}^{T} {\mbi Q}^{-1}){\rm vec}({\mbi Y})} \right]. \quad{\hbox{(A6)}}}$$ After reduction for Formula${\mathhat {\mbi Z}}$, the reduced normal equations are obtained as Formula TeX Source $$[{\mbi F} {\mbi P}^{-1} {\mbi F}^{T} \otimes {\bar {\mbi M}}^{T} {\mbi Q}^{-1} {\bar {\mbi M}}]{\rm vec}({\mathhat {\mbi{R}}}) = [{\mbi F} \mbi {P}^{-1} \otimes {\bar {\mbi M}}^{T} {\mbi Q}^{-1}] {\rm vec}({\mbi Y}) \eqno{\hbox{(A7)}}$$ with Formula${\bar {\mbi M}} = {\mbi P_{N}^{\perp}} {\mbi M}$, Formula${\mbi P_{N}} = {\mbi N}({\mbi N}^{T} {\mbi Q}^{-1} {\mbi N})^{-1} {\mbi N}^{T} \mbi {Q}^{-1}$ and Formula${\mbi P}_{N}^{\perp} = {\mbi I}- {\mbi P}_{N}$. Inversion of (A7) gives Formula${\mathhat {\mbi{R}}}$ of (28).

With Formula${\mathhat {\mbi{R}}}$ given, the normal equation for Formula${\mathhat {\mbi Z}}$ follows from (A6) as Formula TeX Source $$\displaylines{({\mbi P}^{-1} \otimes {\mbi N}^{T} {\mbi {Q}}^{-1} {\mbi N}){\rm vec}({\mathhat {\mbi Z}}) \hfill \cr \hfill = ({\mbi P}^{-1} \otimes {\mbi N}^{T} {\mbi {Q}}^{-1}){\rm vec}({\mbi Y}- {\mbi M} {\mathhat {\mbi{R}}} {\mbi F}). \quad{\hbox{(A8)}}}$$ Inversion gives Formula${\mathhat {\mbi Z}}$ of (28). The covariance matrices Formula${\mbi Q_{{\mathhat {R}} {\mathhat {R}}}}$ and Formula${\mbi Q_{{\mathhat {Z}} {\mathhat {Z}}}}$ of (29) follow from an application of the variance-covariance propagation law to the expressions of Formula${\mathhat {\mbi{R}}}$ and Formula${\mathhat {\mbi Z}}$ in (28). Formula$\hfill\square$

Proof of Lemma 3

Denote the objective function Formula$\Vert {\rm vec}({\mbi Y}- {\mbi M}{\mbi R}{\mbi F}- {\mbi N}{\mbi Z})\Vert_{{\mmb Q_{\mmb {YY}}}}^{2}$, which is a quadratic form in Formula${\mbi x}={\rm vec}({\mbi R}, {\mbi Z})$, as Formula${\cal E}({\mbi x})$. Since its gradient vanishes at its minimizer Formula${\mathhat {\mbi x}}={\rm vec}({\mathhat {\mbi{R}}}, {\mathhat {\mbi Z}})$, the quadratic form can be written as the sum of its zero-order and second-order term Formula TeX Source $${\cal E}({\mbi x}) = {\cal E}({\mathhat {\mbi x}})+({\mathhat {\mbi x}}- {\mbi x})^{T} {\cal {\mbi N}}({\mathhat {\mbi x}}- {\mbi x}) \eqno{\hbox{(A9)}}$$ with Formula${\mbi N}$ being the normal matrix of (A6) (it is Formula${{1} \over {2}}$ times the Hessian matrix of Formula${\cal E}({\mbi x})$). Define the blocktriangular transformation Formula TeX Source $${\mbi T} = \left[\matrix{{\mbi I} & {\mbi F}^{+T} \otimes {\mbi M}^{+} {\mbi N} \cr {\bf 0} & {\mbi I}}\right]. \eqno{\hbox{(A10)}}$$ Then Formula TeX Source $${\mbi T}({\mathhat {\mbi x}}- {\mbi x}) = {\rm vec}({\mathhat {\mbi{R}}}({\mbi Z})- {\mbi R}, {\mathhat {\mbi Z}}- {\mbi Z}) \eqno{\hbox{(A11)}}$$ and Formula TeX Source $${\mbi T}^{-T} {\mbi N} {\mbi T}^{-1} = {\rm blockdiagonal}\left[{\mbi Q_{{\mathhat {\mmb R}}({\mmb Z}) {\mathhat {\mmb R}}({\mmb Z})}^{\mmb -1}}, {\mbi Q_{{\mathhat {\mmb Z}} {\mathhat {\mmb Z}}}^{\mmb -1}}\right]. \eqno{\hbox{(A12)}}$$ Hence Formula TeX Source $$\displaylines{({\mathhat {\mbi x}}- {\mbi x})^{T} {\cal {\mbi N}}({\mathhat {\mbi x}}- {\mbi x}) \hfill \cr \hfill = \Vert {\rm vec}({\mathhat {\mbi{R}}}({\mbi Z})- {\mbi R})\Vert_{\mmb {Q_{{\mathhat {R}}(Z) {\mathhat {R}}(Z)}}}^{2}+\Vert {\rm vec}({\mathhat {\mbi Z}}- {\mbi Z})\Vert_{{\mmb Q_{{\mathhat {\mmb Z}} {\mathhat {\mmb Z}}}}}^{2}. \quad{\hbox{(A13)}}}$$ This combined with (A9) concludes the proof. Formula$\hfill\square$

Proof of Theorem 2

Using the orthogonal decomposition (31), we can write Formula TeX Source $$\eqalignno{&{\mathop{\min}\limits_{{\mmb R} \in \BBO^{3 \times q}, {\mmb Z} \in \BBZ^{fs \times r}}}\Vert {\rm vec} ({\mbi {Y}}\!-\! {\mbi M}{\mbi R}{\mbi F}- {\mbi N}{\mbi Z})\Vert_{{{\mmb Q}_{\mmb {YY}}}}^{2} \cr & \quad \!=\! \Vert {\rm vec}({\mathhat {\mbi E}})\Vert_{\mmb {Q_{\mmb {YY}}}}^{2} \cr & \qquad \!+\! {\mathop{\min}\limits_{{\mmb Z} \in \BBZ ^{fs \times r}}} \left(\Vert {\rm vec}({\mathhat {\mbi Z}}\!-\! {\mbi Z})\Vert_{{\mmb Q_{{\mathhat {\mmb Z}} {\mathhat {\mmb Z}}}}}^{2} \right. \cr & \qquad \qquad \!+ \! {\mathop {\min}\limits_{{\mmb R} \in \BBO^{3 \times q}}} \left. \Vert {\rm vec}({\mathhat {\mbi R}}({\mbi Z}) \!-\! {\mbi {R}})\Vert_{{\mmb Q_{{\mathhat {\mmb R}}({\mmb Z}) {\mathhat {\mmb R}}({\mmb Z})}}}^{2} \right). &{\hbox{(A14)}}}$$ Hence, if we define Formula${\mathcheck {\mbi R}}({\mbi Z}) = \arg \min_{{\mmb R} \in \BBO^{3 \times q}}\Vert {\rm vec}({\mathhat {\mbi{R}}}({\mbi Z})- {\mbi R})\Vert_{{\mmb Q_{{\mathhat {\mmb R}}({\mmb Z}) {\mathhat {\mmb R}}({\mmb Z})}}}^{2}$, the integer ambiguity matrix minimizer of (A14) is as follows: Formula TeX Source $$\displaylines{{\mathcheck {\mbi Z}} = \arg {\mathop{\min}\limits_{{\mmb Z} \in \BBZ^{fs \times r}}} \bigg(\Vert {\rm vec}({\mathhat {\mbi Z}}\!-\!{\mbi {Z}})\Vert_{{\mmb Q_{{\mathhat {\mmb Z}} {\mathhat {\mmb Z}}}}}^{2}\hfill \cr \hfill \!+\! \Vert {\rm vec}({\mathhat {\mbi{R}}}({\mbi {Z}})\!-\!{\mathcheck {\mbi R}}({\mbi Z}))\Vert_{\mmb {Q_{{\mathhat {R}}(Z) {\mathhat {R}}(Z)}}}^{2} \bigg) \quad {\hbox{(A15)}}}$$ and the corresponding orthonormal attitude matrix minimizer as Formula${\mathcheck {\mbi R}} = {\mathcheck {\mbi R}}({\mathcheck {\mbi Z}})$. Formula$\hfill\square$

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).

References

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Authors

Peter J. G. Teunissen

Peter J. G. Teunissen

Peter J. G. Teunissen (M'10) received the Ph.D. degree in geodesy (summa cum laude) in 1985, from the Delft University of Technology (TUDelft), Delft, The Netherlands.

Immediately following, he was awarded the prestigious five-year Constantijn en Christiaan Huygens Fellowship (1986–1991) by the Netherlands Organization for the Advancement of Pure Research. In 1988, he attained full Professor of Geodesy and Navigation, TUDelft, where he has held various senior academic positions: Head of Mathematical and Physical Geodesy Department, Faculty of Geodesy (1993–1998); Vice-Dean of Faculty of Civil Engineering and Geosciences (2001–2002); Director of Education (2002–2004), Program Director Delft Research Centre Earth and Atmosphere (2004–2008), Chair of the Netherlands Geodetic Commission (1993–2009), and Head of Earth Observation and Space Systems Department, Faculty of Aerospace Engineering (2003–2006). He is the inventor of the LAMBDA method and has 25 years of research experience in GNSS positioning and navigation. He is currently Federation Fellow of the Australian Research Council working on the theory and modeling for the next generation GNSS.

Dr. Teunissen was elected Fellow of the Royal Netherlands Academy of Sciences in 2000 and Fellow of the International Association of Geodesy (IAG) in 1991. He is a Professor Honoris Causa of Wuhan University (2000) and of Tongji University, Shanghai, China (2010).

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