Improving ambiguity resolution success rate in the joint solution of GNSS-based attitude determination and relative positioning with multivariate constraints

Abstract

Usually, the global navigation satellite system (GNSS)-based attitude determination and the GNSS-based relative positioning are treated as separate problems. In this contribution, the two problems are resolved as a combined solution of a joint model with multivariate constraints, in which the known geometry of the antenna array on the rover platform is used as constraints, thereby improving the parameter estimation in not only the attitude determination but also the relative positioning. The objective function of the joint model is decomposed and then reduced so that only the terms concerning the constraints are retained, thus transforming the objective function into the exact form of that of the GNSS-based multivariate constrained attitude determination model. Then the multivariate constrained least-squares ambiguity decorrelation adjustment method together with the efficient shrinkage or expansion with bounds strategies, which are shown to work well in the attitude determination, can be extended to the joint model. In experimental validation, the joint model is tested with both simulated and real GPS/BDS datasets in terms of ambiguity resolution success rate and the precision of solutions. The results confirm that the joint model ensures the highest ambiguity resolution success rate among the existing unaided GNSS-based relative positioning methods, and the improvement is more obvious when the observation model is weaker, for example, when the number of available satellites is fewer or when the precision of observations is poorer, thus showing an overall best performance in both reliability and precision aspects.

Appendix: Proof of (12)

With ${\varvec{Q}}_{{\hat{\overline{\varvec{b}}}\left( {\overline{\varvec{a}}} \right)\hat{\overline{\varvec{b}}}\left( {\overline{\varvec{a}}} \right)}}^{ - 1} = \overline{\varvec{B}}^{\rm T} {\varvec{Q}}_{{\overline{\varvec{y}}\overline{\varvec{y}}}}^{ - 1} \overline{\varvec{B}}$, $\overline{\varvec{B}} = \left[ {\begin{array}{*{20}c} {{\varvec{F}}^{\rm T} } & 0 \\ 0 & 1 \\ \end{array} } \right] \otimes {\varvec{G}}$, and ${\varvec{Q}}_{{\overline{\varvec{y}}\overline{\varvec{y}}}} = {\varvec{P}}_{n} \otimes {\varvec{Q}}_{{{\varvec{yy}}}}$, the second term of (11) equals

$$\begin{aligned} \left\| {\hat{\overline{\varvec{b}}}\left( {\overline{\varvec{a}}} \right) - \overline{\varvec{b}}} \right\|_{{{\varvec{Q}}_{{\hat{\overline{\varvec{b}}}\left( {\overline{\varvec{a}}} \right)\hat{\overline{\varvec{b}}}\left( {\overline{\varvec{a}}} \right)}} }}^{2} & = \left[ {\begin{array}{*{20}c} {\hat{\overline{\varvec{b}}}_{1} \left( {\overline{\varvec{a}}} \right) - {\varvec{X}}_{{\varvec{C}}} } \\ {\hat{\overline{\varvec{b}}}_{2} \left( {\overline{\varvec{a}}} \right) - {\varvec{b}}_{{\text{m}_{1} \text{r}}} } \\ \end{array} } \right]^{\rm T} {\varvec{Q}}_{{\hat{\overline{\varvec{b}}}\left( {\overline{\varvec{a}}} \right)\hat{\overline{\varvec{b}}}\left( {\overline{\varvec{a}}} \right)}}^{ - 1} \left[ {\begin{array}{*{20}c} {\hat{\overline{\varvec{b}}}_{1} \left( {\overline{\varvec{a}}} \right) - {\varvec{X}}_{{\varvec{C}}} } \\ {\hat{\overline{\varvec{b}}}_{2} \left( {\overline{\varvec{a}}} \right) - {\varvec{b}}_{{\text{m}_{1} \text{r}}} } \\ \end{array} } \right] \\ & = \left[ {\begin{array}{*{20}c} {\hat{\overline{\varvec{b}}}_{1} \left( {\overline{\varvec{a}}} \right) - {\varvec{X}}_{{\varvec{C}}} } \\ {\hat{\overline{\varvec{b}}}_{2} \left( {\overline{\varvec{a}}} \right) - {\varvec{b}}_{{\text{m}_{1} \text{r}}} } \\ \end{array} } \right]^{\rm T} \left( {\left[ {\begin{array}{*{20}c} {{\varvec{F}}^{\rm T} } & 0 \\ 0 & 1 \\ \end{array} } \right] \otimes {\varvec{G}}} \right)^{\rm T} \left( {{\varvec{P}}_{n} \otimes {\varvec{Q}}_{{{\varvec{yy}}}} } \right)^{ - 1} \left( {\left[ {\begin{array}{*{20}c} {{\varvec{F}}^{\rm T} } & 0 \\ 0 & 1 \\ \end{array} } \right] \otimes {\varvec{G}}} \right)\left[ {\begin{array}{*{20}c} {\hat{\overline{\varvec{b}}}_{1} \left( {\overline{\varvec{a}}} \right) - {\varvec{X}}_{{\varvec{C}}} } \\ {\hat{\overline{\varvec{b}}}_{2} \left( {\overline{\varvec{a}}} \right) - {\varvec{b}}_{{\text{m}_{1} \text{r}}} } \\ \end{array} } \right] \\ & = {\mathop {\left[ {\begin{array}{*{20}c} {\hat{\overline{\varvec{b}}}_{1} \left( {\overline{\varvec{a}}} \right) - {\varvec{X}}_{{\varvec{C}}} } \\ {\hat{\overline{\varvec{b}}}_{2} \left( {\overline{\varvec{a}}} \right) - {\varvec{b}}_{{\text{m}_{1} \text{r}}} } \\ \end{array} } \right]}\limits_{{1 \times 3\left( {q + 1} \right)}}}^{\rm T} \left( {\mathop {\left[ {\begin{array}{*{20}c} {{\varvec{F}}^{\rm T} } & 0 \\ 0 & 1 \\ \end{array} } \right]}\limits_{{n \times \left( {q + 1} \right)}} \otimes {\varvec{I}}_{3} } \right)^{\rm T} \left( {{\varvec{P}}_{n}^{ - 1} \otimes \mathop {{\varvec{G}}^{\rm T} }\limits_{3 \times 2s} \mathop {{\varvec{Q}}_{{{\varvec{yy}}}}^{ - 1} }\limits_{2s \times 2s} \mathop {\varvec{G}}\limits_{2s \times 3} } \right)\left( {\mathop {\left[ {\begin{array}{*{20}c} {{\varvec{F}}^{\rm T} } & 0 \\ 0 & 1 \\ \end{array} } \right]}\limits_{{n \times \left( {q + 1} \right)}} \otimes {\varvec{I}}_{3} } \right)\mathop {\left[ {\begin{array}{*{20}c} {\hat{\overline{\varvec{b}}}_{1} \left( {\overline{\varvec{a}}} \right) - {\varvec{X}}_{{\varvec{C}}} } \\ {\hat{\overline{\varvec{b}}}_{2} \left( {\overline{\varvec{a}}} \right) - {\varvec{b}}_{{\text{m}_{1} \text{r}}} } \\ \end{array} } \right]}\limits_{{3\left( {q + 1} \right) \times 1}} \\ \end{aligned}$$

Let the LDL^T decomposition of ${\varvec{P}}_{n}^{ - 1} { = }\left( {\frac{1}{2}{\varvec{D}}_{{\text{A}}} {\varvec{D}}_{{\text{A}}}^{\rm T} } \right)^{ - 1}$ be given as ${\varvec{P}}_{n}^{ - 1} = {\varvec{L}}_{n}^{{ - {\rm T}}} {\varvec{D}}_{n}^{ - 1} {\varvec{L}}_{n}^{ - 1}$, where ${\varvec{L}}_{n}^{ - 1} = \left[ {\begin{array}{*{20}c} 1 & {} & {} & {} & {} \\ \vdots & \ddots & {} & {} & {} \\ { - \frac{1}{i}} & { - \frac{1}{i}} & 1 & {} & {} \\ \vdots & \vdots & \vdots & \ddots & {} \\ { - \frac{1}{n}} & { - \frac{1}{n}} & { - \frac{1}{n}} & \cdots & 1 \\ \end{array} } \right]$ with i being the row number, and the diagonal entries of ${\varvec{D}}_{n}^{ - 1}$ equal to $\frac{i + 1}{{2i}}$, $i = 1,\;\ldots ,\;n$. If we set ${\varvec{M}} = - \frac{1}{n}{\varvec{e}}_{n - 1}^{\rm T}$, ${\varvec{P}}_{n}^{{{ - }1}}$ is rewritten as ${\varvec{P}}_{n}^{{{ - }1}} = \left[ {\begin{array}{*{20}c} {{\varvec{L}}_{n - 1}^{ - 1} } & {} \\ {{\varvec{M}}^{\rm T} } & 1 \\ \end{array} } \right]^{\rm T} \left[ {\begin{array}{*{20}c} {{\varvec{D}}_{n - 1}^{ - 1} } & {} \\ {} & {\frac{2n}{{n + 1}}} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {{\varvec{L}}_{n - 1}^{ - 1} } & {} \\ {{\varvec{M}}^{\rm T} } & 1 \\ \end{array} } \right]$. Then it follows that

$$\begin{aligned} \left\| {\hat{\bar{\varvec{b}}}\left( {\bar{\varvec{a}}} \right) - \bar{\varvec{b}}} \right\|_{{\varvec{Q}_{{\hat{\bar{\varvec{b}}}\left( {\bar{\varvec{a}}} \right)\hat{\bar{\varvec{b}}}\left( {\bar{\varvec{a}}} \right)}} }}^{2} & = \left[ {\begin{array}{*{20}c} {\hat{\bar{\varvec{b}}}_{1} \left( {\bar{\varvec{a}}} \right) - \varvec{X}_{\varvec{C}} } \\ {\hat{\bar{\varvec{b}}}_{2} \left( {\bar{\varvec{a}}} \right) - \varvec{b}_{{\text{m}_{1} \text{r}}} } \\ \end{array} } \right]^{{\rm T}} \left( {\left[ {\begin{array}{*{20}c} {\varvec{F}^{{\rm T}} } & {\mathbf{0}} \\ {\mathbf{0}} & 1 \\ \end{array} } \right] \otimes \varvec{I}_{3} } \right)^{{\rm T}} \left( {\left[ {\begin{array}{*{20}c} {\varvec{L}_{{n - 1}}^{{ - 1}} } & {} \\ {\varvec{M}^{{\rm T}} } & 1 \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\varvec{D}_{{m - 1}}^{{ - 1}} } & {} \\ {} & {\frac{{2n}}{{n + 1}}} \\ \end{array} } \right]\left[ {\begin{array}{*{20}c} {\varvec{L}_{{n - 1}}^{{ - 1}} } & {} \\ {\varvec{M}^{{\rm T}} } & 1 \\ \end{array} } \right] \otimes \varvec{Q}_{{\hat{\varvec{b}}\left( \varvec{a} \right)\hat{\varvec{b}}\left( \varvec{a} \right)}}^{{ - 1}} } \right) \\ & \quad \times \left( {\left[ {\begin{array}{*{20}c} {\varvec{F}^{{\rm T}} } & {\mathbf{0}} \\ {\mathbf{0}} & 1 \\ \end{array} } \right] \otimes \varvec{I}_{3} } \right)\left[ {\begin{array}{*{20}c} {\hat{\bar{\varvec{b}}}_{1} \left( {\bar{\varvec{a}}} \right) - \varvec{X}_{\varvec{C}} } \\ {\hat{\bar{\varvec{b}}}_{2} \left( {\bar{\varvec{a}}} \right) - \varvec{b}_{{\text{m}_{1} \text{r}}} } \\ \end{array} } \right] \\ & = \left\{ {\left( {\left[ {\begin{array}{*{20}c} {\varvec{L}_{{n - 1}}^{{ - 1}} } & {} \\ {\varvec{M}^{{\rm T}} } & 1 \\ \end{array} } \right] \otimes \varvec{I}_{3} } \right)\left( {\left[ {\begin{array}{*{20}c} {\varvec{F}^{{\rm T}} } & {\mathbf{0}} \\ {\mathbf{0}} & 1 \\ \end{array} } \right] \otimes \varvec{I}_{3} } \right)\left[ {\begin{array}{*{20}c} {\hat{\bar{\varvec{b}}}_{1} \left( {\bar{\varvec{a}}} \right) - \varvec{X}_{\varvec{C}} } \\ {\hat{\bar{\varvec{b}}}_{2} \left( {\bar{\varvec{a}}} \right) - \varvec{b}_{{\text{m}_{1} \text{r}}} } \\ \end{array} } \right]} \right\}^{{\rm T}} \left\{ {\left[ {\begin{array}{*{20}c} {\varvec{D}_{{m - 1}}^{{ - 1}} } & {} \\ {} & {\frac{{2n}}{{n + 1}}} \\ \end{array} } \right] \otimes \varvec{Q}_{{\hat{\varvec{b}}\left( \varvec{a} \right)\hat{\varvec{b}}\left( \varvec{a} \right)}}^{{ - 1}} } \right\} \\ & \quad \times \left\{ {\left( {\left[ {\begin{array}{*{20}c} {\varvec{L}_{{n - 1}}^{{ - 1}} } & {} \\ {\varvec{M}^{{\rm T}} } & 1 \\ \end{array} } \right] \otimes \varvec{I}_{3} } \right)\left( {\left[ {\begin{array}{*{20}c} {\varvec{F}^{{\rm T}} } & {\mathbf{0}} \\ {\mathbf{0}} & 1 \\ \end{array} } \right] \otimes \varvec{I}_{3} } \right)\left[ {\begin{array}{*{20}c} {\hat{\bar{\varvec{b}}}_{1} \left( {\bar{\varvec{a}}} \right) - \varvec{X}_{\varvec{C}} } \\ {\hat{\bar{\varvec{b}}}_{2} \left( {\bar{\varvec{a}}} \right) - \varvec{b}_{{\text{m}_{1} \text{r}}} } \\ \end{array} } \right]} \right\} \\ & = \left[ {\begin{array}{*{20}c} {\left( {\varvec{L}_{{n - 1}}^{{ - 1}} \varvec{F}^{{\rm T}} \otimes \varvec{I}_{3} } \right)\left( {\hat{\bar{\varvec{b}}}_{1} \left( {\bar{\varvec{a}}} \right) - \varvec{X}_{\varvec{C}} } \right)} \\ {\left( {\varvec{M}^{{\rm T}} \varvec{F}^{{\rm T}} \otimes \varvec{I}_{3} } \right)\left( {\hat{\bar{\varvec{b}}}_{1} \left( {\bar{\varvec{a}}} \right) - \varvec{X}_{\varvec{C}} } \right) + \left( {\hat{\bar{\varvec{b}}}_{2} \left( {\bar{\varvec{a}}} \right) - \varvec{b}_{{\text{m}_{1} \text{r}}} } \right)} \\ \end{array} } \right]^{{\rm T}} \left[ {\begin{array}{*{20}c} {\varvec{D}_{{n - 1}}^{{ - 1}} \otimes \varvec{Q}_{{\hat{\varvec{b}}\left( \varvec{a} \right)\hat{\varvec{b}}\left( \varvec{a} \right)}}^{{ - 1}} } & {} \\ {} & {\frac{{2n}}{{n + 1}}\varvec{Q}_{{\hat{\varvec{b}}\left( \varvec{a} \right)\hat{\varvec{b}}\left( \varvec{a} \right)}}^{{ - 1}} } \\ \end{array} } \right] \\ & \quad \times \left[ {\begin{array}{*{20}c} {\left( {\varvec{L}_{{n - 1}}^{{ - 1}} \varvec{F}^{{\rm T}} \otimes \varvec{I}_{3} } \right)\left( {\hat{\bar{\varvec{b}}}_{1} \left( {\bar{\varvec{a}}} \right) - \varvec{X}_{\varvec{C}} } \right)} \\ {\left( {\varvec{M}^{{\rm T}} \varvec{F}^{{\rm T}} \otimes \varvec{I}_{3} } \right)\left( {\hat{\bar{\varvec{b}}}_{1} \left( {\bar{\varvec{a}}} \right) - \varvec{X}_{\varvec{C}} } \right) + \left( {\hat{\bar{\varvec{b}}}_{2} \left( {\bar{\varvec{a}}} \right) - \varvec{b}_{{\text{m}_{1} \text{r}}} } \right)} \\ \end{array} } \right] \\ & = \left( {\hat{\bar{\varvec{b}}}_{1} \left( {\bar{\varvec{a}}} \right) - \varvec{X}_{\varvec{C}} } \right)^{{\rm T}} \left( {\varvec{FP}_{{n - 1}}^{{ - 1}} \varvec{F}^{{\rm T}} \otimes \varvec{Q}_{{\hat{\varvec{b}}\left( \varvec{a} \right)\hat{\varvec{b}}\left( \varvec{a} \right)}}^{{ - 1}} } \right)\left( {\hat{\bar{\varvec{b}}}_{1} \left( {\bar{\varvec{a}}} \right) - \varvec{X}_{\varvec{C}} } \right) + \left[ {\hat{\bar{\varvec{b}}}_{2} \left( {\bar{\varvec{a}},X_{\varvec{C}} } \right) - \varvec{b}_{{\text{m}_{1} \text{r}}} } \right]^{{\rm T}} \varvec{Q}_{{\hat{\bar{\varvec{b}}}_{2} \left( {\bar{\varvec{a}},X_{\varvec{C}} } \right)\hat{\bar{\varvec{b}}}_{2} \left( {\bar{\varvec{a}},X_{\varvec{C}} } \right)}}^{{ - 1}} \left[ {\hat{\bar{\varvec{b}}}_{2} \left( {\bar{\varvec{a}},X_{\varvec{C}} } \right) - \varvec{b}_{{\text{m}_{1} \text{r}}} } \right] \\ \end{aligned}$$

(19)

where $\hat{\overline{\varvec{b}}}_{2} \left( {\overline{\varvec{a}},X_{{\varvec{C}}} } \right) = \hat{\overline{\varvec{b}}}_{2} \left( {\overline{\varvec{a}}} \right) + \left( {{\varvec{M}}^{\rm T} {\varvec{F}}^{\rm T} \otimes {\varvec{I}}_{3} } \right)\left( {\hat{\overline{\varvec{b}}}_{1} \left( {\overline{\varvec{a}}} \right) - {\varvec{X}}_{{\varvec{C}}} } \right)$ and ${\varvec{Q}}_{{\hat{\overline{\varvec{b}}}_{2} \left( {\overline{\varvec{a}},X_{{\varvec{C}}} } \right)\hat{\overline{\varvec{b}}}_{2} \left( {\overline{\varvec{a}},X_{{\varvec{C}}} } \right)}} = \frac{n + 1}{{2n}}{\varvec{Q}}_{{\hat{\varvec{b}}\left( {\varvec{a}} \right)\hat{\varvec{b}}\left( {\varvec{a}} \right)}}$ denote the conditional baseline solution of ${\varvec{b}}_{{{\text{m}}_{1} {\text{r}}}}$ (conditioned on $\overline{\varvec{a}}$ and ${\varvec{X}}_{{\varvec{C}}}$) and its v–c matrix, respectively. Since

$$\begin{aligned} {\varvec{Q}}_{{\hat{\overline{\varvec{b}}}\left( {\overline{\varvec{a}}} \right)\hat{\overline{\varvec{b}}}\left( {\overline{\varvec{a}}} \right)}} & = \left[ {\left( {\left[ {\begin{array}{*{20}c} {{\varvec{F}}^{\rm T} } & 0 \\ 0 & 1 \\ \end{array} } \right] \otimes {\varvec{G}}} \right)^{\rm T} \left( {{\varvec{P}}_{n} \otimes {\varvec{Q}}_{{{\varvec{yy}}}} } \right)^{ - 1} \left( {\left[ {\begin{array}{*{20}c} {{\varvec{F}}^{\rm T} } & 0 \\ 0 & 1 \\ \end{array} } \right] \otimes {\varvec{G}}} \right)} \right]^{ - 1} \\ & = \left[ {\begin{array}{*{20}c} {{\varvec{FP}}_{n - 1}^{ - 1} {\varvec{F}}^{\rm T} + \frac{2n}{{n + 1}}{\varvec{FMM}}^{\rm T} {\varvec{F}}^{\rm T} } & {\frac{2n}{{n + 1}}{\varvec{FM}}} \\ {\frac{2n}{{n + 1}}{\varvec{M}}^{\rm T} {\varvec{F}}^{\rm T} } & {\frac{2n}{{n + 1}}} \\ \end{array} } \right]^{ - 1} \otimes {\varvec{Q}}_{{\hat{\varvec{b}}\left( {\varvec{a}} \right)\hat{\varvec{b}}\left( {\varvec{a}} \right)}} \\ \end{aligned}$$

with the property of the inverse matrix $\left[ {\begin{array}{*{20}c} {\varvec{A}} & {\varvec{U}} \\ {\varvec{V}} & {\varvec{D}} \\ \end{array} } \right]^{ - 1} = \left[ {\begin{array}{*{20}c} {\left( {{\varvec{A}} - {\varvec{UD}}^{ - 1} {\varvec{V}}} \right)^{ - 1} } & { - \left( {{\varvec{A}} - {\varvec{UD}}^{ - 1} {\varvec{V}}} \right)^{ - 1} {\varvec{UD}}^{ - 1} } \\ { - \left( {{\varvec{D}} - {\varvec{VA}}^{ - 1} {\varvec{U}}} \right)^{ - 1} {\varvec{VA}}^{ - 1} } & {\left( {{\varvec{D}} - {\varvec{VA}}^{ - 1} {\varvec{U}}} \right)^{ - 1} } \\ \end{array} } \right]$, this v–c matrix is reformulated as ${\varvec{Q}}_{{\hat{\overline{\varvec{b}}}\left( {\overline{\varvec{a}}} \right)\hat{\overline{\varvec{b}}}\left( {\overline{\varvec{a}}} \right)}} = \left[ {\begin{array}{*{20}c} {\left( {{\varvec{FP}}_{n - 1}^{ - 1} {\varvec{F}}^{\rm T} } \right)^{ - 1} } & { - \left( {{\varvec{FP}}_{n - 1}^{ - 1} {\varvec{F}}^{\rm T} } \right)^{ - 1} {\varvec{FM}}} \\ { - {\varvec{M}}^{\rm T} {\varvec{F}}^{\rm T} \left( {{\varvec{FP}}_{n - 1}^{ - 1} {\varvec{F}}^{\rm T} } \right)^{ - 1} } & {\frac{n + 1}{{2n}} + {\varvec{M}}^{\rm T} {\varvec{F}}^{\rm T} \left( {{\varvec{FP}}_{n - 1}^{ - 1} {\varvec{F}}^{\rm T} } \right)^{ - 1} {\varvec{FM}}} \\ \end{array} } \right] \otimes {\varvec{Q}}_{{\hat{\varvec{b}}\left( {\varvec{a}} \right)\hat{\varvec{b}}\left( {\varvec{a}} \right)}}$.

Thus, we get

$$\begin{aligned} {\varvec{Q}}_{{\hat{\overline{\varvec{b}}}_{2} \left( {\overline{\varvec{a}}} \right)\hat{\overline{\varvec{b}}}_{1} \left( {\overline{\varvec{a}}} \right)}} & = - {\varvec{M}}^{\rm T} {\varvec{F}}^{\rm T} \left( {{\varvec{FP}}_{n - 1}^{ - 1} {\varvec{F}}^{\rm T} } \right)^{ - 1} \otimes {\varvec{Q}}_{{\hat{\varvec{b}}\left( {\varvec{a}} \right)\hat{\varvec{b}}\left( {\varvec{a}} \right)}} \\ {\varvec{Q}}_{{\hat{\overline{\varvec{b}}}_{1} \left( {\overline{\varvec{a}}} \right)\hat{\overline{\varvec{b}}}_{1} \left( {\overline{\varvec{a}}} \right)}} & = \left( {{\varvec{FP}}_{n - 1}^{ - 1} {\varvec{F}}^{\rm T} } \right)^{ - 1} \otimes {\varvec{Q}}_{{\hat{\varvec{b}}\left( {\varvec{a}} \right)\hat{\varvec{b}}\left( {\varvec{a}} \right)}} \\ \end{aligned}$$

(20)

Substitution of (20) into the last equation of (19) gives

$\begin{aligned} \left\| {\hat{\overline{\varvec{b}}}\left( {\overline{\varvec{a}}} \right) - \overline{\varvec{b}}} \right\|_{{{\varvec{Q}}_{{\hat{\overline{\varvec{b}}}\left( {\overline{\varvec{a}}} \right)\hat{\overline{\varvec{b}}}\left( {\overline{\varvec{a}}} \right)}} }}^{2} & = \left( {\hat{\overline{\varvec{b}}}_{1} \left( {\overline{\varvec{a}}} \right) - {\varvec{X}}_{{\varvec{C}}} } \right)^{\rm T} {\varvec{Q}}_{{\hat{\overline{\varvec{b}}}_{1} \left( {\overline{\varvec{a}}} \right)\hat{\overline{\varvec{b}}}_{1} \left( {\overline{\varvec{a}}} \right)}}^{ - 1} \left( {\hat{\overline{\varvec{b}}}_{1} \left( {\overline{\varvec{a}}} \right) - {\varvec{X}}_{{\varvec{C}}} } \right) + \left[ {\hat{\overline{\varvec{b}}}_{2} \left( {\overline{\varvec{a}},X_{{\varvec{C}}} } \right) - {\varvec{b}}_{{\text{m}_{1} \text{r}}} } \right]^{\rm T} {\varvec{Q}}_{{\hat{\overline{\varvec{b}}}_{2} \left( {\overline{\varvec{a}},X_{{\varvec{C}}} } \right)\hat{\overline{\varvec{b}}}_{2} \left( {\overline{\varvec{a}},X_{{\varvec{C}}} } \right)}}^{ - 1} \left[ {\hat{\overline{\varvec{b}}}_{2} \left( {\overline{\varvec{a}},X_{{\varvec{C}}} } \right) - {\varvec{b}}_{{\text{m}_{1} \text{r}}} } \right] \\ & = \left\| {\hat{\overline{\varvec{b}}}_{1} \left( {\overline{\varvec{a}}} \right) - {\varvec{X}}_{{\varvec{C}}} } \right\|_{{{\varvec{Q}}_{{\hat{\overline{\varvec{b}}}_{1} \left( {\overline{\varvec{a}}} \right)\hat{\overline{\varvec{b}}}_{1} \left( {\overline{\varvec{a}}} \right)}} }}^{2} + \left\| {\hat{\overline{\varvec{b}}}_{2} \left( {\overline{\varvec{a}},{\varvec{X}}_{{\varvec{C}}} } \right) - {\varvec{b}}_{{\text{m}_{1} \text{r}}} } \right\|_{{{\varvec{Q}}_{{\hat{\overline{\varvec{b}}}_{2} \left( {\overline{\varvec{a}},{\varvec{X}}_{{\varvec{C}}} } \right)\hat{\overline{\varvec{b}}}_{2} \left( {\overline{\varvec{a}},{\varvec{X}}_{{\varvec{C}}} } \right)}} }}^{2} \\ \end{aligned}$ in which $\hat{\overline{\varvec{b}}}_{2} \left( {\overline{\varvec{a}},X_{{\varvec{C}}} } \right) = \hat{\overline{\varvec{b}}}_{2} \left( {\overline{\varvec{a}}} \right) + \left( {{\varvec{M}}^{\rm T} {\varvec{F}}^{\rm T} \otimes {\varvec{I}}_{3} } \right)\left( {\hat{\overline{\varvec{b}}}_{1} \left( {\overline{\varvec{a}}} \right) - {\varvec{X}}_{{\varvec{C}}} } \right) = \hat{\overline{\varvec{b}}}_{2} \left( {\overline{\varvec{a}}} \right) - {\varvec{Q}}_{{\hat{\overline{\varvec{b}}}_{2} \left( {\overline{\varvec{a}}} \right)\hat{\overline{\varvec{b}}}_{1} \left( {\overline{\varvec{a}}} \right)}} {\varvec{Q}}_{{\hat{\overline{\varvec{b}}}_{1} \left( {\overline{\varvec{a}}} \right)\hat{\overline{\varvec{b}}}_{1} \left( {\overline{\varvec{a}}} \right)}}^{ - 1} \left( {\hat{\overline{\varvec{b}}}_{1} \left( {\overline{\varvec{a}}} \right) - {\varvec{X}}_{{\varvec{C}}} } \right)$.

□