## 2 Neural Network-Based Orbit Uncertainty Propagation Orbit Determination Framework
### Orbit Determination Method
\begin{array}{l}\mathbf{x}_{k+1}=\phi(t_{k+1};\mathbf{x}_{k},t_{k})+\mathbf{ \omega}_{k+1}\\ \mathbf{z}_{k+1}=\mathbf{h}(\mathbf{x}_{k+1})+\mathbf{\varepsilon}_{k+1}\end{array} \right. \tag{11}\]  
where \(\mathbf{x}_{k}=\mathbf{x}(t_{k})\) and \(\mathbf{x}_{k+1}=\mathbf{x}(t_{k+1})\) represent the orbital state at epochs \(t_{k}\) and \(t_{k+1}\), respectively, \(\mathbf{z}_{k+1}\) is the measurement vector at epoch \(t_{k+1}\), \(\mathbf{h}(\cdot)\) denotes the measurement model, \(\mathbf{\omega}_{k+1}\) is the process noise, and \(\mathbf{\varepsilon}_{k+1}\) is the measurement noise. We assume that the process and measurement noises are zero-mean Gaussian errors, with their covariance being \(\mathbf{Q}_{k+1}\) and \(\mathbf{R}_{k+1}\), respectively, i.e., \(\mathbb{E}(\mathbf{\omega}_{k+1}\mathbf{\omega}_{k+1}^{T})=\mathbf{Q}_{k+1}\) and \(\mathbb{E}(\mathbf{\theta}_{k+1}\mathbf{\theta}_{k+1}^{T})=\mathbf{R}_{k+1}\). Note that Equation (11) is a discrete format of the OD system, which should be distinguished from Equations (1)-(10). Based on the MANN-UT presented in Equations (9) and (10), the process of the MANN-UKF is provided as follows:  
A. Time Update  
* Given the estimated state \(\mathbf{\hat{x}}_{k}\) and the associated covariance \(\mathbf{P}_{k}\) at epoch \(t_{k}\), calculate the predicted state \(\mathbf{\hat{x}}_{k+1|k}\) and the associated covariance \(\mathbf{P}_{k+1|k}\): \[\mathbf{\hat{x}}_{k+1|k}\approx\phi(t_{k+1};\mathbf{\hat{x}}_{k},t_{k})+\sum_{i=0}^{2 n}w_{i}^{(m)}(\delta\mathbf{\chi}_{i}^{\rm 2 BP}+\mathbf{\hat{e}}_{\rm MANN,i})\] (12) \[ \begin{split}&\mathbf{P}_{k+1|k}=\sum_{i=0}^{2n}w_{i}^{(c)} \Big{[}\delta\mathbf{\chi}_{i}^{\rm 2 BP}+\mathbf{\hat{e}}_{\rm MANN,i}-\sum_{j=0}^{2n}w_{j}^{(m)}(\delta\mathbf{\chi}_{j} ^{\rm 2 BP}+\mathbf{\hat{e}}_{\rm MANN,j})\Big{]}\\ &\times\Big{[}\delta\mathbf{\chi}_{i}^{\rm 2 BP}+\mathbf{\hat{e}}_{\rm MANN,i}-\sum_{j=0}^{2n}w_{j}^{(m)}(\delta\mathbf{\chi}_{j}^{\rm 2 BP}+\mathbf{\hat{e}}_{\rm MANN,j})\Big{]}^{T}+\mathbf{Q}_{k+1}.\end{split} \] (13) B. Measurement update
* Calculate the Cholesky decomposition of \(\mathbf{P}_{k+1|k}\) and generate sigma points \(\mathbf{\chi}_{i,k+1|k}\). * Calculate the predicted measurement \(\mathbf{\hat{z}}_{k+1|k}\) and the associated covariance \(\mathbf{P}_{\mathbf{xz},k+1|k}\) and \(\mathbf{P}_{\mathbf{zz},k+1|k}\): \[\mathbf{z}_{i,k+1|k}^{*}=\mathbf{h}(\mathbf{\chi}_{i,k+1|k})\] (14) \[\mathbf{\hat{z}}_{k+1|k}=\sum_{i=0}^{2n}w_{i}^{(m)}\mathbf{z}_{i,k+1|k}^{*}\] (15) \[\mathbf{P}_{\mathbf{xz},k+1|k}=\sum_{i=0}^{2n}w_{i}^{(c)}(\mathbf{\chi}_{i,k+1|k}^{*}- \mathbf{\hat{x}}_{k+1|k})(\mathbf{z}_{i,k+1|k}^{*}-\mathbf{\hat{z}}_{k+1|k})^{T}\] (16)\[\mathbf{P}_{\mathbf{zz},k+1|k}=\sum_{i=0}^{2n}w_{i}^{(c)}(\mathbf{z}_{i,k+1|k}^{*}-\mathbf{\hat{z} }_{k+1|k})(\mathbf{z}_{i,k+1|k}^{*}-\mathbf{\hat{z}}_{k+1|k})^{T}+\mathbf{R}_{k+1}. \tag{17}\]
* When the measurement \(\mathbf{z}_{k+1}\) has been collected, calculate the estimated state \(\mathbf{\hat{x}}_{k+1}\) and the associated covariance \(\mathbf{P}_{k+1}\) at epoch \(t_{k}\): \[\mathbf{K}_{k+1}=\mathbf{P}_{\mathbf{xz},k+1|k}\mathbf{P}_{\mathbf{zz},k+1|k}^{-1}\] (18) \[\mathbf{\hat{x}}_{k+1}=\mathbf{x}_{k+1|k}+\mathbf{K}_{k+1}(\mathbf{z}_{k+1}-\mathbf{ \hat{z}}_{k+1|k})\] (19) \[\mathbf{P}_{k+1}=\mathbf{P}_{k+1|k}-\mathbf{K}_{k+1}\mathbf{P}_{\mathbf{zz},k+1|k}\mathbf{K}_{k+1}^ {T}.\] (20)  
Compared with traditional UKF, the advantage of using the proposed MANN-UKF is that it actually only needs to integrate one sigma sample point to calculate the prediction state \(\mathbf{\hat{x}}_{k+1|k}\) and covariance matrix \(\mathbf{P}_{k+1}\) instead of integrating all thirteen sigma sample points, as the remaining twelve sigma sample points can be predicted by MANN. The estimation accuracy of MANN-UKF may be worse than that of UKF because of the inevitable prediction error when using MANN. This section introduces a neural network-assisted Earth-Jupiter transfer OD system, wherein MANN construction and training are shown in the next section.