# Automatic Estimation of Dynamic Lever Arms for a Position and Orientation System
## 3 Proposed SINS/GPS Integrated Navigation Model
### Error Distribution Analysis
\begin{array}{c}\overset{\wedge}{\mathbf{K}}_{k}\\ \mathbf{0}_{18\times 3}\end{array} \right]. \tag{29}\]  
Since the gain matrix \(\mathbf{K}_{k}^{*}\) of full-order filter model is known, the covariance calculation loop is updated as follows:  
\[\mathbf{P}_{k/k-1}^{*}=\mathbf{\Phi}_{k/k-1}^{*}\mathbf{P}_{k-1}^{*}\mathbf{\Phi}_{k/k-1}^{* \mathsf{T}}+\mathbf{\Gamma}_{k-1}^{*}\mathbf{\Omega}_{k-1}^{*}\mathbf{\Gamma}_{k-1}^{* \mathsf{T}}, \tag{30}\]  
\[\mathbf{P}_{k}^{*}=(\mathbf{I}-\mathbf{K}_{k}^{*}\mathbf{H}_{k}^{*})\mathbf{P}_{k/k-1}^{*}( \mathbf{I}-\mathbf{K}_{k}^{*}\mathbf{H}_{k}^{*})^{\mathsf{T}}+\mathbf{K}_{k}^{*}\mathbf{R}_{k} ^{*}\mathbf{K}_{k}^{*\mathsf{T}}. \tag{31}\]  
Firstly, based on linear propagation characteristics of associated errors, and setting \(\mathbf{Q}^{*}=\mathbf{0}\) and \(\mathbf{R}^{*}=\mathbf{0}\) in Equations (30) and (31), the mean square deviation propagation matrix of initial errors can be calculated as follows:  
\[\mathbf{S}_{k}^{*}=(\mathbf{I}-\mathbf{K}_{k}^{*}\mathbf{H}_{k}^{*})\mathbf{\Phi}_{k/k-1}^{*} \mathbf{S}_{k-1}^{*}, \tag{32}\]  
with the initial value \(\mathbf{S}_{0}^{*}=\mathbf{I}\). Secondly, setting \(\mathbf{P}_{0}^{*}=\mathbf{0}\) and \(\mathbf{R}^{*}=\mathbf{0}\) in Equations (30) and (31), the propagation matrix of the \(j\)th process noise \(W_{j}\) is obtained by:  
\[\mathbf{A}_{k/k-1}^{j}=\mathbf{\Phi}_{k/k-1}^{*}\mathbf{A}_{k-1}^{j}\mathbf{\Phi}_{k/k-1}^{* \mathsf{T}}+\mathbf{\Gamma}_{k-1}^{*}\mathbf{e}_{j,k-1}\mathbf{\Gamma}_{k-1}^{*\mathsf{T}}, \tag{33}\]  
\[\mathbf{A}_{k}^{j}=(\mathbf{I}-\mathbf{K}_{k}^{*}\mathbf{H}_{k}^{*})\mathbf{A}_{k/k-1}^{j}( \mathbf{I}-\mathbf{K}_{k}^{*}\mathbf{H}_{k}^{*})^{\mathsf{T}}, \tag{34}\]  
with the initial value \(\mathbf{A}_{0}^{j}=\mathbf{0}\), where \(\mathbf{e}_{jj}\) denotes the 39 \(\times\) 39 matrix with \(\mathbf{e}(j,j)=1\) and all else zeros. If there are \(l\)-dimensional nonzero process noises, Equations (33) and (34) should be executed \(l\) times with different \(\mathbf{e}_{jj}\) to obtain all of the propagation matrices. Thirdly, setting \(\mathbf{P}_{0}^{*}=\mathbf{0}\) and \(\mathbf{Q}^{*}=\mathbf{0}\) in Equations (30) and (31), the propagation matrix of the \(j\)th measurement noise \(V_{j}\) is obtained by:  
\[\mathbf{B}_{k/k-1}^{j}=\mathbf{\Phi}_{k/k-1}^{*}\mathbf{B}_{k-1}^{j}\mathbf{\Phi}_{k/k-1}^{* \mathsf{T}}, \tag{35}\]  
\[\mathbf{B}_{k}^{j}=(\mathbf{I}-\mathbf{K}_{k}^{*}\mathbf{H}_{k}^{*})\mathbf{B}_{k/k-1}^{j}( \mathbf{I}-\mathbf{K}_{k}^{*}\mathbf{H}_{k}^{*})^\[\sigma_{xi}=\sqrt{\sum_{j=1}^{n}S_{j,i}^{2}\sigma_{x_{j0}^{2}}^{2}+\sum_{j=1}^{l}A _{ii}^{j}Q_{jj}+\sum_{j=1}^{m}B_{ii}^{j}R_{jj}}, \tag{37}\]  
where \(\sigma_{x_{j0}^{*}}^{2}\), \(Q_{jj}\), and \(R_{jj}\) are the deviations of the \(j\)th initial state error, \(j\)th process noise, and \(j\)th measurement noise, respectively, while \(A_{ii}^{j}\) and \(B_{ii}^{j}\) are the \(i\)th diagonal elements of covariance matrices \(\mathbf{A}_{k}^{j}\) and \(\mathbf{B}_{k}^{j}\), respectively. Based on the linear error model, the mean square deviation \(\sigma_{xi,xj}\) of linear combination of \(x_{i}^{*}\) and \(x_{j}^{*}\) can be expressed as \(\sqrt{\sigma_{xi}^{2}+\sigma_{xj}^{2}}\), and so on. Therefore, according to the above covariance analyzing method, the error distribution budgets of the full order filter or real-world model can be constructed from the calculation of the reduced order filter. The complexity and computational burden of simulation are greatly reduced.