# Angle-Only Cooperative Orbit Determination Considering Attitude Uncertainty
## 2 Attitude and Orbit Determination Model
### Review of Fifth-Order CKF  
In this section, the fifth-order CKF algorithm is briefly summarized. First, consider a discrete nonlinear system as:  
\[ \begin{split}\mathbf{x}_{k}&=\mathbf{f}(\mathbf{x}_{k-1})+\mathbf{w }_{k-1}\\ \mathbf{z}_{k}&=\mathbf{h}(\mathbf{x}_{k})+\mathbf{v}_{k}\end{split}  \tag{25}\]  
where \(\mathbf{x}_{k}\in\mathbb{R}^{n}\) is the state vector at time epoch \(k\) and \(\mathbf{z}_{k}\in\mathbb{R}^{m}\) is the measurement. \(\mathbf{w}_{k-1}\in\mathbb{R}^{n}\) and \(\mathbf{v}_{k}\in\mathbb{R}^{m}\) denote the independent system and measurement noise, respectively, and are both considered independent and white Gaussian distributions with covariances \(\mathbf{Q}_{k-1}\) and \(\mathbf{R}_{k}\), respectively. The optimal Bayesian filters contains two steps: the prediction step and the update step. Both of the two steps require us to calculate the Gaussian weighted integration \(\int_{\mathbb{R}^{n}}g(\mathbf{x})\mathbf{N}(\mathbf{x};\bar{\mathbf{x}},\mathbf{P})dx\), where \(g(\mathbf{x})\) is a nonlinear function. The integral with respect to the general Gaussian distribution \(\mathbf{N}(\mathbf{x};\bar{\mathbf{x}},\mathbf{P})\) can be further approximated by:  
\[ \begin{split}\int_{\mathbb{R}^{n}}g(\mathbf{x})\mathbf{N}(\mathbf{x};\bar{ \mathbf{x}},\mathbf{P})dx&=\int_{\mathbb{R}^{n}}g(\mathbf{\mathrm{A}}\mathbf{x}+ \bar{\mathbf{x}})\mathbf{N}(\mathbf{x};\bar{\mathbf{x}},\mathbf{P})dx\\ &\approx\sum_{i=1}^{N_{P}}W_{i}g(A\gamma_{i}+\bar{\mathbf{x}})\end{split}  \tag{26}\]  
where \(\mathbf{P}=\mathbf{S}\mathbf{S}^{\mathrm{T}}\), \(N_{P}\) is the total number of points, and \(\mathbf{\gamma}_{i}\) and \(W_{i}\) are the quadrature points and weights, respectively, corresponding to the Gaussian distribution \(\mathbf{N}(\mathbf{x};\bar{\mathbf{x}},\mathbf{P})\). Specifically, the integral with respect to \(\mathbf{N}(\mathbf{x};\mathbf{0},\mathbf{I})\) can be approximated by the following quadrature rule:  
\[\int_{\mathbb{R}^{n}}g(\mathbf{x})N(\mathbf{x};\mathbf{0},\mathbf{I})dx\approx\sum_{i=1}^{N_ {P}}W_{i}g(\mathbf{\gamma}_{i}) \tag{27}\]  
According to the cubature rule and Equation (26), in fifth-order CKF, the cubature points \(\xi_{i}\) are given by [21]:  
\[ \begin{cases}\mathbf{\zeta}_{0}=\mathbf{0}_{n\times 1}\\ \mathbf{\zeta}_{i}=\mathbf{\mathrm{A}}_{k}\,\sqrt{n+2}\mathbf{e}_{i}\\ \mathbf{\zeta}_{i+n}=-\mathbf{\mathrm{A}}_{k}\,\sqrt{n+2}\mathbf{e}_{i}\\ \mathbf{\zeta}_{j+2n}=\mathbf{\mathrm{A}}_{k}\,\sqrt{n+2}\mathbf{\mathrm{s}}_{j}^{+}\\ \mathbf{\zeta}_{j+2n+n(n-1)/2}\,=-\mathbf{\mathrm{A}}_{k}\,\sqrt{n+2}\mathbf{s}_{j}^{+}\\ \mathbf{\zeta}_{j+2n+n(n-1)}=\mathbf{\mathrm{A}}_{k}\,\sqrt{n+2}\mathbf{s}_{j}^{-}\\ \mathbf{\zeta}_{j+2n+3n(n-1)/2}\,=-\mathbf{\mathrm{A}}_{k}\,\sqrt{n+2}\mathbf{s}_{j}^{-}\\ i=1,2,\ldots,n\,\,;\,\,j=1,2,\ldots,n(n-1)/2\end{cases}  \tag{28}\]where \(n\) is the state dimension of the system to be estimated, \(\mathbf{A}_{k}\) is the Cholesky decomposition of covariance matrix \(\mathbf{P}_{k}\) at epoch \(t_{k}\) and \(P_{k}=A_{k}\Lambda_{k}^{T}\), and \(\mathbf{e}_{i}\) is the \(i\)-th column of the \(n\)-th identity matrix \(\mathbf{I}_{n}\). The point sets \(\mathbf{s}_{j}^{+}\) and \(\mathbf{s}_{j}^{-}\) are given as follows. \[ \begin{split}\{\mathbf{s}_{j}^{+}\}&=\left\{(\mathbf{e}_{p }+\mathbf{e}_{q})/\sqrt{2}\ |\ p<q,p,q=1,2,\ldots,n\right\}\\ \{\mathbf{s}_{j}^{-}\}&=\left\{(\mathbf{e}_{p}-\mathbf{e}_{q})/ \sqrt{2}\ |\ p<q,p,q=1,2,\ldots,n\right\}\end{split}  \tag{29}\]  
The corresponding weight \(w_{i}\) of each Cubature point \(\zeta_{i}\) is given by  
\[w_{i}=\left\{ \begin{array}{ll}\frac{2}{n+2}&i=0\\ \frac{(n+2)}{(n+2)^{2}}&i=1,\ldots,2n(n-1)\\ \frac{4-n}{(n+2)^{2}}&i=2n(n-1)+1,\ldots,2n^{2}\end{array} \right. \tag{30}\]  
Then, the fifth-order CKF algorithm is summarized as follows:  
i. Time updating:  
\[