Multi-GNSS PPP/INS tightly coupled integration with atmospheric augmentation and its application in urban vehicle navigation

Abstract

Precise point positioning (PPP) is receiving increasing interest due to its cost-effectiveness, global coverage and high accuracy. However, its application in the urban environment is still full of challenges due to the satellite tracking sky-view. Thus, we presented a comprehensive positioning model by fusing the multi-GNSS (global navigation satellite system) combination, GNSS/INS (inertial navigation system) tightly coupled integration as well as the ionospheric and tropospheric augmentation in the undifferenced and uncombined PPP. The performance of this model in dual-frequency and single-frequency positioning was assessed with two experiments that denoted as T019 and T023, respectively, and both the experiments were carried out in a real urban environment. Particularly, the experiment T023 was carried out in the Second Ring Road of Wuhan city, which can be regarded as a typical downtown environment. Concerning the regional atmospheric augmentation, observations from 5 reference stations with an inter-station distance of about 40 km were also collected during the experimental time. The comparison between reference stations suggested that the regional tropospheric model had a precision of better than 0.6 cm in terms of zenith tropospheric delay, while the regional ionospheric model had a precision of around 0.5 total electron content unit in terms of Vertical Total Electron Content. It can be concluded that the GPS-only PPP can be improved significantly for urban vehicle navigation with these techniques, i.e., the multi-GNSS, INS tightly coupled integration and the atmospheric augmentation, through the positioning analysis, while INS tightly coupled integration makes the most contributions under the downtown environment, and the improvement of the regional atmospheric augmentation in single-frequency PPP is more significant since that single frequency is more sensitive to the ionospheric delay. In addition, it is proved that the regional atmospheric augmentation accelerates positioning convergence. The 3D positioning root-mean-square (RMS) with the comprehensive positioning model for dual frequency are 0.22 m and 0.77 m for T019 and T023, respectively. Concerning single-frequency PPP, the 3D RMS is 0.45 m and 1.17 m for T019 and T023, respectively. Moreover, taking the lane-level navigation under the downtown environment of T023 into consideration, we further presented the cumulative frequency of the horizontal positioning error less than 1 m, i.e., $P\left( {\sqrt {{\text{d}}N^{2} + {\text{d}}E^{2} } < 1\;{\text{m}}} \right)$, and the best solution can be found with PPP by fusing all the techniques, in which $P\left( {\sqrt {{\text{d}}N^{2} + {\text{d}}E^{2} } < 1\;{\text{m}}} \right)$ is 99.0% and 93.2% for dual frequency and single frequency, respectively.

Appendix

Here, we presented the details concerning the PPP/INS observation model. First, a few symbols and notions are defined for future reference: $\otimes$ and $\circ$ are the Kronecker product and Schur product, respectively (Rao 1973; Davis et al. 1962); $\times$ demotes the skew-symmetric matrix of the vector (Shin 2005); and the notions are defined as:

$$ {\varvec{z}}_{{\varvec{s}}} = \left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 & 0 \\ \end{array} } & {\begin{array}{*{20}c} \ldots & 0 \\ \end{array} } \\ \end{array} } \right)^{{\text{T}}} $$

(19)

$$ {\varvec{u}}_{{\varvec{s}}} = \left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} 1 & 1 \\ \end{array} } & {\begin{array}{*{20}c} \ldots & 1 \\ \end{array} } \\ \end{array} } \right)^{{\text{T}}} $$

(20)

$$ {\varvec{Z}}_{{\varvec{s}}} = \left( {\begin{array}{*{20}c} 0 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 0 \\ \end{array} } \right) $$

(21)

$$ {\varvec{U}}_{{\varvec{s}}} = \left( {\begin{array}{*{20}c} 1 & \cdots & 0 \\ \vdots & \ddots & \vdots \\ 0 & \cdots & 1 \\ \end{array} } \right) $$

(22)

$$ {\text{diag}}\left( {\varvec{a}} \right) = \left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} {a_{1} } \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} \vdots \\ 0 \\ \end{array} } \\ \end{array} } & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ {a_{2} } \\ \end{array} } \\ {\begin{array}{*{20}c} \vdots \\ 0 \\ \end{array} } \\ \end{array} } \\ \end{array} } & {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\begin{array}{*{20}c} \cdots \\ \cdots \\ \end{array} } \\ {\begin{array}{*{20}c} \ddots \\ \cdots \\ \end{array} } \\ \end{array} } & {\begin{array}{*{20}c} {\begin{array}{*{20}c} 0 \\ 0 \\ \end{array} } \\ {\begin{array}{*{20}c} \vdots \\ {a_{n} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right) $$

(23)

thus, ${\varvec{z}}_{{\varvec{s}}}$ is a $s$ by $1$ vector with zero entries and ${\varvec{u}}_{{\varvec{s}}}$ is a $s$ by $1$ vector with one entries, while ${\varvec{Z}}_{{\varvec{s}}}$ is a $s$ by $s$ matrix with zero entries and ${\varvec{U}}_{{\varvec{s}}}$ is a $s$ by $s$ identity matrix, and ${\text{diag}}\left( {\varvec{a}} \right)$ denotes the diagonal matrix with the elements of vector ${\varvec{a}} = \left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} {a_{1} } & {a_{2} } \\ \end{array} } & {\begin{array}{*{20}c} \ldots & {a_{n} } \\ \end{array} } \\ \end{array} } \right)^{{\text{T}}}$ on the main diagonal. The dimensions and lengths of such vectors will generally be obvious from context, and the symbols are exactly the same as that of Sect. 2.

According to Eqs. (19)–(23), and denotes the frequency number as $k$, then the design matrix ${\varvec{H}}_{{{\varvec{PPP}}}}$ is expressed as:

$$ {\varvec{H}}_{{{{\varvec{\updelta}}}{\varvec{x}}_{{\varvec{r}}} }} = \left( {\begin{array}{*{20}c} {{\varvec{u}}_{{2 \cdot {\varvec{k}}}} } \\ 0 \\ \end{array} } \right) \otimes \left( {\begin{array}{*{20}c} {h_{r}^{1} } \\ \vdots \\ {h_{r}^{j} } \\ \end{array} } \right) $$

(24)

$$ {\varvec{H}}_{{t_{r} }} = \left( {\begin{array}{*{20}c} {{\varvec{u}}_{{2 \cdot {\varvec{j}} \cdot {\varvec{k}}}} } \\ {{\varvec{z}}_{{\varvec{j}}} } \\ \end{array} } \right) $$

(25)

$$ {\varvec{H}}_{{{{\varvec{\updelta}}}{\varvec{T}}_{{\varvec{w}}} }} = \left( {\begin{array}{*{20}c} {{\varvec{u}}_{{2 \cdot {\varvec{k}}}} } \\ 0 \\ \end{array} } \right) \otimes \left( {\begin{array}{*{20}c} {\alpha_{r}^{1} } \\ \vdots \\ {\alpha_{r}^{j} } \\ \end{array} } \right) $$

(26)

$$ {\varvec{H}}_{{{\varvec{b}}_{{\varvec{r}}} }} = \left( {\begin{array}{*{20}c} {{\varvec{z}}_{{\varvec{j}}}^{{\mathbf{T}}} } & {{\varvec{u}}_{{\varvec{j}}}^{{\mathbf{T}}} } & {\begin{array}{*{20}c} {{\varvec{z}}_{{\varvec{j}}}^{{\mathbf{T}}} } & {{\varvec{z}}_{{\varvec{j}}}^{{\mathbf{T}}} } & {{\varvec{z}}_{{\varvec{j}}}^{{\mathbf{T}}} } \\ \end{array} } \\ \end{array} } \right)^{{\mathbf{T}}} $$

(27)

$$ {\varvec{H}}_{{{\varvec{N}}_{{\varvec{r}}} }} = \left( {\begin{array}{*{20}c} {{\varvec{Z}}_{j \cdot k} } \\ {{\varvec{U}}_{j \cdot k} } \\ {{\varvec{z}}_{{\varvec{k}}}^{{\varvec{T}}} \otimes {\varvec{Z}}_{{\varvec{j}}} } \\ \end{array} } \right) $$

(28)

$$ {\varvec{H}}_{{{\varvec{a}}_{{\varvec{r}}} }} = \left( {\begin{array}{*{20}c} {\left( {\begin{array}{*{20}c} 1 \\ { - 1} \\ \end{array} } \right) \otimes \left( {\begin{array}{*{20}c} {\frac{40.3}{{f_{1}^{2} }}} \\ {\frac{40.3}{{f_{2}^{2} }}} \\ \end{array} } \right)} \\ 0 \\ \end{array} } \right) \otimes \left( {{\varvec{u}}_{5}^{{\varvec{T}}} \otimes \left( {\begin{array}{*{20}c} {{\varvec{\gamma}}_{r}^{1} } \\ \vdots \\ {{\varvec{\gamma}}_{r}^{j} } \\ \end{array} } \right) \circ \left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} 1 \\ \vdots \\ 1 \\ \end{array} } & {\begin{array}{*{20}c} {\hbox{d}L_{r}^{1} } \\ \vdots \\ {\hbox{d}L_{r}^{j} } \\ \end{array} } & {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\hbox{d}B_{r}^{1} } \\ \vdots \\ {\hbox{d}B_{r}^{j} } \\ \end{array} } & {\begin{array}{*{20}c} {\hbox{d}L_{r}^{12} } \\ \vdots \\ {\hbox{d}L_{r}^{j2} } \\ \end{array} } & {\begin{array}{*{20}c} {\hbox{d}B_{r}^{12} } \\ \vdots \\ {\hbox{d}B_{r}^{j2} } \\ \end{array} } \\ \end{array} } \\ \end{array} } \right)} \right) $$

(29)

$$ {\varvec{H}}_{{{\varvec{r}}_{{\varvec{r}}} }} = \left( {\begin{array}{*{20}c} {\left( {\begin{array}{*{20}c} 1 \\ { - 1} \\ \end{array} } \right) \otimes \left( {\begin{array}{*{20}c} {\frac{40.3}{{f_{1}^{2} }}} \\ {\frac{40.3}{{f_{1}^{2} }}} \\ \end{array} } \right)} \\ 0 \\ \end{array} } \right) \otimes \left( {{\varvec{u}}_{5}^{{\varvec{T}}} \otimes \left( {\begin{array}{*{20}c} {{\varvec{\gamma}}_{r}^{1} } \\ \vdots \\ {{\varvec{\gamma}}_{r}^{j} } \\ \end{array} } \right) \otimes {\varvec{U}}_{{\varvec{j}}} } \right) $$

(30)

By the way, the Eq. (15) in our previous study Gu et al. (2015a) should be corrected to Eqs. (29) and (30) of this study.

Concerning ${\varvec{N}}^{{\varvec{e}}}$ in Eq. (12) we have:

$$ {\varvec{N}}^{{\varvec{e}}} = \frac{kM}{{r^{3} }}\left( {\begin{array}{*{20}c} { - 1 + \frac{{3x^{2} }}{{r^{2} }}} & {\frac{3xy}{{r^{2} }}} & {\frac{3xz}{{r^{2} }}} \\ {\frac{3xy}{{r^{2} }}} & { - 1 + \frac{{3y^{2} }}{{r^{2} }}} & {\frac{3yz}{{r^{2} }}} \\ {\frac{3xz}{{r^{2} }}} & {\frac{3yz}{{r^{2} }}} & { - 1 + \frac{{3z^{2} }}{{r^{2} }}} \\ \end{array} } \right) + \left( {\begin{array}{*{20}c} {\omega_{e} } & 0 & 0 \\ 0 & {\omega_{e} } & 0 \\ 0 & 0 & 0 \\ \end{array} } \right) $$

(31)

where $kM$ is the product of the gravitational constant and the mass of the earth; ${\varvec{x}}_{{{\varvec{INS}}}}^{{\varvec{e}}} = \left( {\begin{array}{*{20}c} x & y & z \\ \end{array} } \right)^{{\text{T}}}$; $r = \sqrt {x^{2} + y^{2} + z^{2} }$; and $\omega_{e}$ is the earth rotation rate.

Derived from Eqs. (8) to (10), the matrixes in the state Eq. (11) are presented as:

$$ {\mathbf{F}}\varvec{ = }\left( {\begin{array}{*{20}l} 0 \hfill & {\varvec{U}} \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ {{\varvec{N}}^{{\varvec{e}}} } \hfill & { - 2{\varvec{\omega}}_{{{\varvec{ie}}}}^{{\varvec{e}}} \times } \hfill & {{\varvec{C}}_{{\varvec{b}}}^{{\varvec{e}}} {\varvec{f}}_{{\varvec{b}}} \times } \hfill & {{\varvec{C}}_{{\varvec{b}}}^{{\varvec{e}}} } \hfill & 0 \hfill & {{\varvec{C}}_{{\varvec{b}}}^{{\varvec{e}}} {\varvec{diag}}\left( {{\varvec{f}}^{{\varvec{b}}} } \right)} \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & { - {\varvec{\omega}}_{{{\varvec{ie}}}}^{{\varvec{e}}} \times } \hfill & 0 \hfill & { - {\varvec{C}}_{{\varvec{b}}}^{{\varvec{e}}} } \hfill & 0 \hfill & { - {\varvec{C}}_{{\varvec{b}}}^{{\varvec{e}}} {\varvec{diag}}\left( {{\varvec{\omega}}_{{{\varvec{ib}}}}^{{\varvec{b}}} } \right)} \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & { - 1/{\varvec{\tau}}_{{{\varvec{ba}}}} } \hfill & 0 \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & { - 1/{\varvec{\tau}}_{{{\varvec{bg}}}} } \hfill & 0 \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & { - 1/{\varvec{\tau}}_{{{\varvec{sa}}}} } \hfill & 0 \hfill \\ 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & 0 \hfill & { - 1/{\varvec{\tau}}_{{{\varvec{sg}}}} } \hfill \\ \end{array} } \right) $$

(32)

$$ {\varvec{G}} = {\text{diag}}\left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\varvec{U}} & {{\varvec{C}}_{{\varvec{b}}}^{{\varvec{e}}} } & { - {\varvec{C}}_{{\varvec{b}}}^{{\varvec{e}}} } \\ \end{array} } & {\begin{array}{*{20}c} {\varvec{U}} & {\varvec{U}} \\ \end{array} } & {\begin{array}{*{20}c} {\varvec{U}} & {\varvec{U}} \\ \end{array} } \\ \end{array} } \right) $$

(33)

$$ {\varvec{w}} = {\text{diag}}\left( {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\varvec{Z}} & {{\varvec{w}}_{{\varvec{v}}} } & {{\varvec{w}}_{\phi } } \\ \end{array} } & {\begin{array}{*{20}c} {{\varvec{w}}_{{{\varvec{ba}}}} } & {{\varvec{w}}_{{{\varvec{bg}}}} } \\ \end{array} } & {\begin{array}{*{20}c} {{\varvec{w}}_{{{\varvec{sa}}}} } & {{\varvec{w}}_{{{\varvec{sg}}}} } \\ \end{array} } \\ \end{array} } \right) $$

(34)

The design matrixes corresponding to the INS state parameters in Eq. (18) can be expressed as

$$ \varvec{ H}_{\phi } = \left( {\begin{array}{*{20}c} {{\varvec{u}}_{{2 \cdot {\varvec{k}}}} } \\ 0 \\ \end{array} } \right) \otimes \left( {\begin{array}{*{20}c} {h_{r}^{1} } \\ \vdots \\ {h_{r}^{j} } \\ \end{array} } \right) \otimes \left( {\tilde{\varvec{C}}_{{\varvec{b}}}^{{\varvec{e}}} {\varvec{l}}^{{\varvec{b}}} \times } \right) $$

(35)

$$ {\varvec{H}}_{{{{\varvec{\updelta}}}{\varvec{v}}_{{\varvec{r}}} }} = {\varvec{z}}_{3}^{{\text{T}}} \otimes {\varvec{z}}_{{2 \cdot {\varvec{j}} \cdot {\varvec{k}} + {\varvec{j}}}} $$

(36)

$$ \varvec{ H}_{{\varvec{B}}} = \varvec{ H}_{{\varvec{S}}} = {\varvec{z}}_{6}^{{\text{T}}} \otimes {\varvec{z}}_{{2 \cdot {\varvec{j}} \cdot {\varvec{k}} + {\varvec{j}}}} $$

(37)

$\varvec{ H}_{\phi }$ indicates that the GNSS observations are sensitive to the attitude through the lever-arm correction vector in Eq. (15), while the zero entries in ${\varvec{H}}_{{{{\varvec{\updelta}}}{\varvec{v}}_{{\varvec{r}}} }}$, ${ }{\varvec{H}}_{{\varvec{B}}}$ and $\varvec{ H}_{{\varvec{S}}}$ indicate that the GNSS observations have no direct relation with the velocity and sensor errors of INS.