Baseline length constraint approaches for enhancing GNSS ambiguity resolution: comparative study

Abstract

Reliable and correct carrier phase ambiguity resolution is the key to global navigation satellite system (GNSS) high-precision navigation and positioning applications. For kinematic situations, such as moving-baseline-based positioning and attitude determination, the baseline length between two antennas is constant; such a priori information could contribute to integer ambiguity resolution, especially when only a few satellites are viewed. In this research, three different approaches using baseline information—the linearized joint adjustment method, validation method, and constrained LAMBDA (CLAMBDA) method—are comprehensively evaluated through theoretical and experimental analyses. The performance of each method is assessed in terms of the ambiguity success rate and baseline solution accuracy with static and kinematic GPS/BDS datasets. The additional baseline length constraint improves the precision of the float solution and the ambiguity fixed success rate (compared to the standard LAMBDA method), but there are differences in the performance of the three methods. Specifically, the performance of the linearized joint adjustment method primarily depends on baseline length and improves as the baseline length increases; however, caution should be exercised for short baselines. The validation method and CLAMBDA method both consider the quadratic form of ambiguity residuals and baseline length constraint for selecting the ambiguity solution. However, the validation method directly judges the ambiguity to obtain a locally optimal solution, whereas the CLAMBDA method constructs a rigorous mathematical formula to obtain a globally optimal solution. Moreover, because the linearized joint adjustment method and CLAMBDA method primarily contribute to the float and fixed solution, respectively, we fused the two methods to improve the ambiguity resolution success rate. The results confirm that the combined algorithm achieves better performance that exceeds that of either individual method for a baseline length of tens of meters.

Appendix: Proof of (14)

For ${\varvec{A}} = \left[ {\begin{array}{*{20}c} {\lambda {\varvec{I}}_{n} } & {{\varvec{0}}_{n} } \\ \end{array} } \right]^{{\text{T}}}$, ${\varvec{B}} = \left[ {\begin{array}{*{20}c} {{\varvec{G}}_{n}^{{\text{T}}} } & {{\varvec{G}}_{n}^{{\text{T}}} } \\ \end{array} } \right]^{{\text{T}}}$, ${\varvec{C}} = {\varvec{C}}_{1}$, and ${\varvec{P}}_{{{\varvec{yycc}}}} = {\text{diag}}\left( {{{{\varvec{P}}_{n} } \mathord{\left/ {\vphantom {{{\varvec{P}}_{n} } {\delta_{\phi }^{2} }}} \right. \kern-\nulldelimiterspace} {\delta_{\phi }^{2} }},{{{\varvec{P}}_{n} } \mathord{\left/ {\vphantom {{{\varvec{P}}_{n} } {\delta_{{\text{P}}}^{2} }}} \right. \kern-\nulldelimiterspace} {\delta_{{\text{P}}}^{2} }},{{{\varvec{P}}_{1} } \mathord{\left/ {\vphantom {{{\varvec{P}}_{1} } {\delta_{c}^{2} }}} \right. \kern-\nulldelimiterspace} {\delta_{c}^{2} }}} \right)$, $\delta_{c}^{2}$ is the variance of the observed baseline length constraint and ${\varvec{P}}_{1}$ is the unit weight of order 1 in

$$\left\{ {\begin{array}{*{20}c} {{\varvec{A}}^{{\text{T}}} {\varvec{P}}_{{{\varvec{yy}}}} {\varvec{A}} = \frac{{\lambda^{2} }}{{\delta_{\phi }^{2} }}{\varvec{P}}_{n} ;\;{\varvec{B}}^{{\text{T}}} {\varvec{P}}_{{{\varvec{yy}}}} {\varvec{B}} = \frac{{\delta_{\phi }^{2} + \delta_{{\text{p}}}^{2} }}{{\delta_{\phi }^{2} \delta_{{\text{p}}}^{2} }}{\varvec{G}}_{n}^{{\text{T}}} {\varvec{P}}_{n} {\varvec{G}}_{n} } \\ {{\varvec{A}}^{{\text{T}}} {\varvec{P}}_{{{\varvec{yy}}}} {\varvec{B}} = \frac{\lambda }{{\delta_{\phi }^{2} }}{\varvec{P}}_{n} {\varvec{G}}_{n} ;\;{\varvec{B}}^{{\text{T}}} {\varvec{P}}_{{{\varvec{yy}}}} {\varvec{A}} = \frac{\lambda }{{\delta_{\phi }^{2} }}{\varvec{G}}_{n}^{{\text{T}}} {\varvec{P}}_{n} } \\ {{\varvec{A}}^{{\text{T}}} {\varvec{P}}_{{{\varvec{yy}}}} {\varvec{y}} = \frac{\lambda }{{\delta_{\phi }^{2} }}{\varvec{P}}_{n} {\varvec{l}}_{\phi } ;\;{\varvec{B}}^{{\text{T}}} {\varvec{P}}_{{{\varvec{yy}}}} {\varvec{y}} = \frac{1}{{\delta_{\phi }^{2} }}{\varvec{G}}_{n}^{{\text{T}}} {\varvec{P}}_{n} {\varvec{l}}_{\phi } + \frac{1}{{\delta_{{\text{p}}}^{2} }}{\varvec{G}}_{n}^{{\text{T}}} {\varvec{P}}_{n} {\varvec{l}}_{{\text{p}}} } \\ {{\varvec{C}}^{{\text{T}}} {\varvec{P}}_{{{\varvec{cc}}}} {\varvec{C}} = \frac{1}{{n\delta_{C}^{2} }}{\varvec{C}}^{{\text{T}}} {\varvec{P}}_{n} {\varvec{C}};\;{\varvec{C}}^{{\text{T}}} {\varvec{P}}_{{{\varvec{cc}}}} {\varvec{y}}_{{\varvec{c}}} = \frac{1}{{\delta_{C}^{2} }}{\varvec{C}}^{{\text{T}}} {\varvec{P}}_{1} {\varvec{y}}_{{\varvec{c}}} } \\ \end{array} } \right.$$

(24)

According to (13), we obtain:

$$\begin{aligned} {\varvec{Q^{\prime}}}_{{{\varvec{\hat{a}\hat{a}}}}} { = }{\varvec{N}}_{{1{|}2}}^{{ - 1^{\prime } }} = & \;\left[ {{\varvec{A}}^{{\text{T}}} {\varvec{P}}_{{{\varvec{yy}}}} {\varvec{A}} - {\varvec{A}}^{{\text{T}}} {\varvec{P}}_{{{\varvec{yy}}}} {\varvec{B}}\left( {{\varvec{B}}^{{\text{T}}} {\varvec{P}}_{{{\varvec{yy}}}} {\varvec{B}} + {\varvec{C}}^{{\text{T}}} {\varvec{P}}_{{{\varvec{cc}}}} {\varvec{C}}} \right)^{ - 1} {\varvec{B}}^{{\text{T}}} {\varvec{P}}_{{{\varvec{yy}}}} {\varvec{A}}} \right]^{ - 1} \\ = & \;\left[ {\frac{{\lambda^{2} }}{{\delta_{\phi }^{2} }}{\varvec{P}}_{n} - \frac{\lambda }{{\delta_{\phi }^{2} }}{\varvec{P}}_{n} {\varvec{G}}_{n} \times \left( {\frac{{\delta_{\phi }^{2} + \delta_{{\text{p}}}^{2} }}{{\delta_{\phi }^{2} \delta_{{\text{p}}}^{2} }}{\varvec{G}}_{n}^{{\text{T}}} {\varvec{P}}_{n} {\varvec{G}}_{n} + \frac{1}{{n\delta_{C}^{2} }}{\varvec{C}}^{{\text{T}}} {\varvec{P}}_{n} {\varvec{C}}} \right)^{ - 1} \times \frac{\lambda }{{\delta_{\phi }^{2} }}{\varvec{G}}_{n}^{{\text{T}}} {\varvec{P}}_{n} } \right]^{ - 1} \\ \approx & \;\left[ {\frac{{\lambda^{2} }}{{\delta_{\phi }^{2} }}{\varvec{P}}_{n} - \frac{\lambda }{{\delta_{\phi }^{2} }}{\varvec{P}}_{n} {\varvec{G}}_{n} \times \frac{{n\delta_{C}^{2} \left( {\delta_{{\text{p}}}^{2} + \delta_{\phi }^{2} } \right)}}{{n\delta_{C}^{2} \left( {\delta_{{\text{p}}}^{2} + \delta_{\phi }^{2} } \right) + \delta_{\phi }^{2} \delta_{{\text{p}}}^{2} }}\frac{{\delta_{\phi }^{2} \delta_{{\text{p}}}^{2} }}{{\delta_{\phi }^{2} + \delta_{{\text{p}}}^{2} }}\left( {{\varvec{G}}_{n}^{{\text{T}}} {\varvec{P}}_{n} {\varvec{G}}_{n} } \right)^{ - 1} \times \frac{\lambda }{{\delta_{\phi }^{2} }}{\varvec{G}}_{n}^{{\text{T}}} {\varvec{P}}_{n} } \right]^{ - 1} \\ \end{aligned}$$

(25)

Using the matrix reversing formula $\left( {{\varvec{D}}{ + }{\varvec{ACB}}} \right)^{ - 1} = {\varvec{D}}^{ - 1} - {\varvec{D}}^{ - 1} {\varvec{A}}\left( {{\varvec{C}}^{ - 1} + {\varvec{BD}}^{ - 1} {\varvec{A}}} \right)^{ - 1} {\varvec{BD}}^{ - 1}$, Eq. (25) can be expressed as:

$$\begin{aligned} {\varvec{Q^{\prime}}}_{{{\varvec{\hat{a}\hat{a}}}}} { = } & \;\frac{{\delta_{\phi }^{2} }}{{\lambda^{2} }}\left[ {{\varvec{I}}_{n} { + }\frac{{n\delta_{C}^{2} \left( {\delta_{{\text{p}}}^{2} + \delta_{\phi }^{2} } \right)}}{{n\delta_{C}^{2} (\delta_{{\text{p}}}^{2} + \delta_{\phi }^{2} ) + \delta_{\phi }^{2} \delta_{{\text{p}}}^{2} }}\frac{{\delta_{{\text{p}}}^{2} }}{{\delta_{\phi }^{2} + \delta_{{\text{p}}}^{2} }}{\varvec{G}}_{n} \left( {{\varvec{G}}_{n}^{{\text{T}}} {\varvec{P}}_{n} {\varvec{G}}_{n} - \frac{{n\delta_{C}^{2} \left( {\delta_{{\text{p}}}^{2} + \delta_{\phi }^{2} } \right)}}{{n\delta_{C}^{2} (\delta_{{\text{p}}}^{2} + \delta_{\phi }^{2} ) + \delta_{\phi }^{2} \delta_{{\text{p}}}^{2} }}\frac{{\delta_{p}^{2} }}{{\delta_{\phi }^{2} + \delta_{{\text{p}}}^{2} }}{\varvec{G}}_{n}^{{\text{T}}} {\varvec{P}}_{n} {\varvec{G}}_{n} } \right)^{ - 1} {\varvec{G}}_{n}^{{\text{T}}} {\varvec{P}}_{n} } \right] \\ \approx & \;\frac{{\delta_{\phi }^{2} }}{{\lambda^{2} }}\left[ {{\varvec{Q}} + \frac{{n\delta_{C}^{2} }}{{\delta_{\phi }^{2} }}{\varvec{G}}_{n} \left( {{\varvec{G}}_{n}^{{\text{T}}} {\varvec{P}}_{n} {\varvec{G}}_{n} } \right)^{ - 1} {\varvec{G}}_{n}^{{\text{T}}} } \right] \\ { = } & \;\frac{{n\delta_{C}^{2} }}{{\lambda^{2} }}\left[ {\frac{{\delta_{\phi }^{2} }}{{n\delta_{C}^{2} }}{\varvec{Q}} + {\varvec{G}}_{n} \left( {{\varvec{G}}_{n}^{{\text{T}}} {\varvec{P}}_{n} {\varvec{G}}_{n} } \right)^{ - 1} {\varvec{G}}_{n}^{{\text{T}}} } \right] \\ \approx & \;\frac{{n\delta_{C}^{2} }}{{\lambda^{2} }}{\varvec{G}}_{n} \left( {{\varvec{G}}_{n}^{{\text{T}}} {\varvec{P}}_{n} {\varvec{G}}_{n} } \right)^{ - 1} {\varvec{G}}_{n}^{{\text{T}}} \\ \end{aligned}$$

(26)

Equation (26) presents the precision of the float ambiguity. Similarly, we obtain:

$$\begin{aligned} {\varvec{Q^{\prime}}}_{{{\varvec{\hat{b}\hat{b}}}}} = {\varvec{N}}_{{2|1}}^{{ - 1^{\prime } }} = & \;\left[ {\left( {{\varvec{B}}^{{\text{T}}} {\varvec{P}}_{{{\varvec{yy}}}} {\varvec{B}} + {\varvec{C}}^{{\text{T}}} {\varvec{P}}_{{{\varvec{cc}}}} {\varvec{C}}} \right) - {\varvec{B}}^{{\text{T}}} {\varvec{P}}_{{{\varvec{yy}}}} {\varvec{A}}\left( {{\varvec{A}}^{{\text{T}}} {\varvec{P}}_{{{\varvec{yy}}}} {\varvec{A}}} \right)^{ - 1} {\varvec{A}}^{{\text{T}}} {\varvec{P}}_{{{\varvec{yy}}}} {\varvec{B}}} \right]^{ - 1} \\ = & \;\left[ {\left( {\frac{{\delta_{\phi }^{2} + \delta_{{\text{p}}}^{2} }}{{\delta_{\phi }^{2} \delta_{{\text{p}}}^{2} }}{\varvec{G}}_{n}^{{\text{T}}} {\varvec{P}}_{n} {\varvec{G}}_{n} + \frac{1}{{n\delta_{C}^{2} }}{\varvec{C}}^{{\text{T}}} {\varvec{P}}_{n} {\varvec{C}}} \right) - \frac{\lambda }{{\delta_{\phi }^{2} }}{\varvec{G}}_{n}^{{\text{T}}} {\varvec{P}}_{n} \left( {\frac{{\lambda^{2} }}{{\delta_{\phi }^{2} }}{\varvec{P}}} \right)_{n}^{ - 1} \frac{\lambda }{{\delta_{\phi }^{2} }}{\varvec{P}}_{n} {\varvec{G}}_{n} } \right]^{ - 1} \\ { = } & \;\frac{{n\delta_{C}^{2} }}{{n\delta_{C}^{2} + \delta_{{\text{p}}}^{2} }}\delta_{{\text{p}}}^{2} \left( {{\varvec{G}}_{n}^{{\text{T}}} {\varvec{P}}_{n} {\varvec{G}}_{n} } \right)^{ - 1} \\ \end{aligned}$$

(27)

Equation (27) presents the precision of the float baseline.