On-the-fly ambiguity resolution method for pseudolite/INS integration based on double-difference square observations

Abstract

Resolving carrier phase ambiguities is the prerequisite for achieving high-precision positioning applications based on pseudolite positioning systems. To improve the probability of resolving ambiguities, pseudolite observations are usually integrated with measurements from inertial navigation systems (INS) in an “on-the-fly” ambiguity resolution (OTF-AR) process. Traditional OTF-AR for pseudolite/INS integration is realized by approximately linearizing a nonlinear problem based on the initial states. However, due to the close distance between base stations and the receiver, the measurement model of a pseudolite positioning system is always highly nonlinear. If the initial position is not sufficiently accurate, significant truncation errors may occur, leading to convergence difficulties or even AR failure in traditional OTF-AR. We present an improved OTF-AR method for pseudolite/INS integration. In the proposed method, pseudolite measurements and inertial measurements are combined in a tight integration mode. In addition, double-difference square observations are utilized to reduce the effect of truncation errors, and the reliability of the resolved ambiguities is improved significantly. To validate the proposed method, simulations and real-world experiments are designed and demonstrated. Simulations and experimental results show that, compared to existing methods, the proposed method converges more rapidly and own stronger robustness. It can achieve accurate ambiguity resolution, even when severe initial state errors are present.

Appendix: Computation method of the INS error matrix

When calculating (4), the INS error matrix ${\mathbf{F}}_{{{\text{ie}}}}$ is an essential part. The INS error matrix ${\mathbf{F}}_{{{\text{ie}}}}$ is calculated through the following equalities (Goshen-Meskin and BarItzhack 1992):

$${\mathbf{F}}_{{{\text{ie}}}} = \left[ {\begin{array}{*{20}c} {{\mathbf{M}}_{{{\text{aa}}}} } & {{\mathbf{M}}_{{{\text{av}}}} } & {{\mathbf{M}}_{{{\text{ap}}}} } \\ {{\mathbf{M}}_{{{\text{va}}}} } & {{\mathbf{M}}_{{{\text{vv}}}} } & {{\mathbf{M}}_{{{\text{vp}}}} } \\ {{\mathbf{0}}_{3 \times 3} } & {{\mathbf{M}}_{{{\text{pv}}}} } & {{\mathbf{M}}_{{{\text{pp}}}} } \\ \end{array} } \right]$$

where

$${\mathbf{M}}_{{{\text{aa}}}} = \left[ {\begin{array}{*{20}c} 0 & {\Omega \sin L + \frac{{v_{{\text{E}}} \tan L}}{{R_{{\text{N}}} + h}}} & { - \Omega \cos L - \frac{{v_{{\text{E}}} }}{{R_{{\text{N}}} + h}}} \\ { - \Omega \sin L - \frac{{v_{{\text{E}}} \tan L}}{{R_{{\text{N}}} + h}}} & 0 & { - \frac{{v_{{\text{N}}} }}{{R_{{\text{M}}} + h}}} \\ {\Omega \cos L + \frac{{v_{{\text{E}}} }}{{R_{{\text{N}}} + h}}} & {\frac{{v_{{\text{N}}} }}{{R_{{\text{M}}} + h}}} & 0 \\ \end{array} } \right]$$

$${\mathbf{M}}_{{{\text{av}}}} = \left[ {\begin{array}{*{20}c} 0 & { - \frac{1}{{R_{{\text{M}}} + h}}} & 0 \\ {\frac{1}{{R_{{\text{N}}} + h}}} & 0 & 0 \\ {\frac{\tan L}{{R_{{\text{N}}} + h}}} & 0 & 0 \\ \end{array} } \right]$$

$${\mathbf{M}}_{{{\text{ap}}}} = \left[ {\begin{array}{*{20}c} 0 & 0 & {\frac{{v_{{\text{N}}} }}{{\left( {R_{{\text{M}}} + h} \right)^{2} }}} \\ { - \Omega \sin L} & 0 & { - \frac{{v_{{\text{E}}} }}{{\left( {R_{{\text{N}}} + h} \right)^{2} }}} \\ {\Omega \cos L + \frac{{v_{{\text{E}}} \sec^{2} L}}{{R_{{\text{N}}} + h}}} & 0 & { - \frac{{v_{{\text{E}}} \tan L}}{{\left( {R_{{\text{N}}} + h} \right)^{2} }}} \\ \end{array} } \right]$$

$${\mathbf{M}}_{{{\text{va}}}} = \left[ {\begin{array}{*{20}c} 0 & { - f_{{\text{U}}} } & {f_{{\text{N}}} } \\ {f_{{\text{U}}} } & 0 & { - f_{{\text{E}}} } \\ { - f_{{\text{N}}} } & {f_{{\text{E}}} } & 0 \\ \end{array} } \right]$$

$$\begin{aligned}{\mathbf{M}}_{{{\text{vv}}}} = &\left[ {\begin{array}{*{20}c} 0 & { - v_{{\text{U}}} } & {v_{{\text{N}}} } \\ {v_{{\text{U}}} } & 0 & { - v_{{\text{E}}} } \\ { - v_{{\text{N}}} } & {v_{{\text{E}}} } & 0 \\ \end{array} } \right]{\mathbf{M}}_{{{\text{av}}}} \\- &\left[ {\begin{array}{*{20}c} 0 & { - \Omega \sin L - \frac{{v_{{\text{E}}} \tan L}}{{R_{{\text{N}}} + h}}} & {\Omega \cos L + \frac{{v_{{\text{E}}} }}{{R_{{\text{N}}} + h}}} \\ {\Omega \sin L + \frac{{v_{{\text{E}}} \tan L}}{{R_{{\text{N}}} + h}}} & 0 & {\frac{{v_{{\text{N}}} }}{{R_{{\text{M}}} + h}}} \\ { - \Omega \cos L - \frac{{v_{{\text{E}}} }}{{R_{{\text{N}}} + h}}} & { - \frac{{v_{{\text{N}}} }}{{R_{{\text{M}}} + h}}} & 0 \\ \end{array} } \right]\end{aligned}$$

$${\mathbf{M}}_{{{\text{vp}}}} = \left[ {\begin{array}{*{20}c} 0 & { - v_{{\text{U}}} } & {v_{{\text{N}}} } \\ {v_{{\text{U}}} } & 0 & { - v_{{\text{E}}} } \\ { - v_{{\text{N}}} } & {v_{{\text{E}}} } & 0 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} 0 & 0 & {\frac{{v_{{\text{N}}} }}{{\left( {R_{{\text{M}}} + h} \right)^{2} }}} \\ { - 2\Omega \sin L} & 0 & { - \frac{{v_{{\text{E}}} }}{{\left( {R_{{\text{N}}} + h} \right)^{2} }}} \\ {2\Omega \cos L + \frac{{v_{{\text{E}}} \sec^{2} L}}{{R_{{\text{N}}} + h}}} & 0 & { - \frac{{v_{{\text{E}}} \tan L}}{{\left( {R_{{\text{N}}} + h} \right)^{2} }}} \\ \end{array} } \right]$$

$${\mathbf{M}}_{{{\text{pv}}}} = \left[ {\begin{array}{*{20}c} 0 & {\frac{1}{{R_{{\text{M}}} + h}}} & 0 \\ {\frac{\sec L}{{R_{{\text{N}}} + h}}} & 0 & 0 \\ 0 & 0 & 1 \\ \end{array} } \right]$$

$${\mathbf{M}}_{{{\text{pp}}}} = \left[ {\begin{array}{*{20}c} 0 & 0 & { - \frac{{v_{{\text{N}}} }}{{\left( {R_{{\text{M}}} + h} \right)^{2} }}} \\ {\frac{{v_{{\text{E}}} \sec L\tan L}}{{R_{{\text{N}}} + h}}} & 0 & { - \frac{{v_{{\text{E}}} \sec L}}{{\left( {R_{{\text{N}}} + h} \right)^{2} }}} \\ 0 & 0 & 0 \\ \end{array} } \right]$$

where $\Omega$ is the earth rotation rate, $L$, $\lambda$ and $h$ are latitude, longitude and altitude values, $f_{{\text{E}}}$, $f_{{\text{N}}}$ and $f_{{\text{U}}}$ are components of the accelerometer specific force vector ${\mathbf{f}}^{{\text{n}}}$ in east, north and up directions.