## 4 Integrated Positioning
### Attitude Bootstrapping  
The attitude bootstrapping method [18, 19] uses a decorrelation technique to decouple the combined system in Equation (26), such that the subsystems still yield the optimal solution. Using decorrelation matrix:  
\[\mathcal{D} = \left[\begin{array}{ccc}I_{n_{1}}&0&0\\ 0&I_{n_{2}}&0\\ -\frac{1}{2}e_{n_{1}}^{T}P_{n_{1}}^{-1}&\frac{1}{2}e_{n_{2}}^{T}P_{n_{2}}^{-1 }&1\end{array}\right]\otimes I_{2 fm} \tag{29}\]  
The decorrelated system reads:  
\[\mathrm{E}\left(\mathcal{Y}^{\prime}\right) = AZ^{\prime}+G\mathcal{R}^{\prime}B_{0} \tag{30}\] \[\mathrm{D}\left(\mathrm{vec}(\mathcal{Y}^{\prime})\right) = P^{\prime}\otimes Q_{yy} \tag{31}\]where \(\mathcal{Y}^{\prime}=\begin{bmatrix}Y^{1}&Y^{2}&y^{{}^{\prime}12}\end{bmatrix}\) is the decorrelated observation matrix, \(\mathcal{R}^{\prime}=\begin{bmatrix}R^{1}&R^{2}&b^{{}^{\prime}12}\end{bmatrix} \in\mathbb{R}^{3\times(q_{1}+q_{2}+1)}\) is the combined rotation matrices and between-platform baseline after decorrelation, \(\mathcal{Z}^{\prime}=\begin{bmatrix}Z^{1}&Z^{2}&z^{{}^{\prime}12}\end{bmatrix}\) is the combined ambiguity matrix after decorrelation and:  
\[P^{\prime} = \mathrm{blockdiag}\left(P_{n_{1}},\;P_{n_{2}},\;\eta\right) \tag{32}\]  
with:  
\[y^{{}^{\prime}12} = y^{12}-\frac{1}{n_{1}+1}\sum_{r=1}^{n_{1}}y_{r}^{1}+\frac{1}{n_ {2}+1}\sum_{r=1}^{n_{2}}y_{r}^{2} \tag{33}\] \[z^{{}^{\prime}12} = z^{12}-\frac{1}{n_{1}+1}\sum_{r=1}^{n_{1}}z_{r}^{1}+\frac{1}{n_ {2}+1}\sum_{r=1}^{n_{2}}z_{r}^{2}\] (34) \[b^{{}^{\prime}12} = b^{12}-\frac{1}{n_{1}+1}\sum_{r=1}^{n_{1}}R^{1}b_{0,r}^{1}+\frac {1}{n_{2}+1}\sum_{r=1}^{n_{2}}R^{2}b_{0,r}^{2}\] (35) \[\eta = \frac{n_{t}+1}{2(n_{1}+1)(n_{2}+1)} \tag{36}\]  
This decorrelation keeps the platform processing intact as in Equation (16) and only alters the between-platform model. As a result of decorrelation, the ambiguities in Equation (34) may not be an integer. However, once platform ambiguities are determined reliably using MC-LAMBDA with decoupled platform models in Equation (30), the model for the between-platform baseline can be rearranged as:  
\[\mathrm{E}(y^{{}^{\prime\prime}12}) = Az^{12}+Gb^{{}^{\prime}12}\qquad z^{12}\in\mathbb{Z}^{fm} \tag{37}\] \[\mathrm{D}(y^{{}^{\prime\prime}12}) = \eta Q_{yy} \tag{38}\]  
where:  
\[y^{{}^{\prime\prime}12} = y^{{}^{\prime}12}+\frac{1}{n_{1}+1}\sum_{r=1}^{n_{1}}Az_{r}^{1} -\frac{1}{n_{2}+1}\sum_{r=1}^{n_{2}}Az_{r}^{2} \tag{39}\]  
Due to the reduction of variance-covariance by a factor of \(\eta\), this model yields improved ambiguity resolution and baseline estimation compared to the standard positioning model in Equation (24). That is, the use of array-aided positioning reduces the variance-covariance matrices of the float ambiguities and ambiguity-fixed baseline estimators by a factor of \(\eta\). For the rotating frame with two antennas and a single antenna at the reference station in Figure 1, the variance reduction factor is \(\eta=\frac{3}{4}\). Note that the between-platform baseline estimate in Equation (35) corresponds to between-centroids of antenna arrays, naturally yielding the parameter of interest for the geo-referencing system in Figure 1. The unconstrained mixed-integer least-squares problem defined in Equations (37) and (38) can be solved efficiently using the LAMBDA method [25] providing ambiguity-fixed baseline estimate \(\breve{b}^{{}^{\prime}12}(\breve{\mathcal{Z}}^{\prime})\) and associated variance-covariance matrix \(Q_{\breve{b}^{{}^{\prime}12}(\breve{\mathcal{Z}}^{\prime})\breve{b}^{{}^{ \prime}12}(\breve{\mathcal{Z}}^{\prime})}\).