## 3 Intelligent Integrated Navigation Q-Learning FEKF Algorithm
### Design of the \(Q\)-Learning-Based FKF for Intelligent Integrated Navigation
The QLFEKF information update includes time update and measurement update, in which the update of time is updated autonomously in the parallel sub-filter and primary filter, the specific update process is as follows:Since there is not measurement information in the global main filter, the measurement information is located in the parallel sub-filter and is only carried out in the state estimation learning filter in the sub-filter. The specific update process is described as follows:

\[\left\{ \begin{array}{l}K_{k}^{\mathcal{S}e(n)}=\mathbf{P}_{psk}^{\mathcal{S}e(n)} \left(H_{ps,k}^{\mathcal{S}e(n)}\right)^{T}\!\!\left[H_{ps,k}^{\mathcal{S}e(n) }\right]\mathbf{P}_{ps,k}^{\mathcal{S}e(n)}\!\!\left[H_{ps,k}^{\mathcal{S}e(n)} \right)^{T}\!+\hat{\mathbf{R}}_{pk}^{n}\right]^{-1}\\ \hat{\mathbf{X}}_{ps,k}^{\mathcal{S}e(n)}=\hat{\mathbf{X}}_{ps,k|k-1}^{\mathcal{S}e(n)}+ K_{k}^{\mathcal{S}e(n)}\mathbf{y}_{ps,nk}\\ \mathbf{P}_{ps,k}^{\mathcal{S}e(n)}=\left[I-K_{k}^{\mathcal{S}e(n)}H_{ps,k}^{ \mathcal{S}e(n)}\right]\mathbf{P}_{ps,k|k-1}^{\mathcal{S}e(n)}\left[I-K_{k}^{ \mathcal{S}e(n)}H_{ps,k}^{\mathcal{S}e(n)}\right]^{T}+K_{k}^{\mathcal{S}e(n)} \hat{\mathbf{R}}_{pk}^{n}\!\left(K_{k}^{\mathcal{S}e(n)}\right)^{T}\end{array} \right. \tag{25}\]

The federated EKF fuses the information of the state estimates \(\mathbf{\hat{X}}_{ps,k}^{\mathcal{S}e(p)}\) and \(\mathbf{\hat{X}}_{ps,k}^{\mathcal{S}e(d)}\) as well as the error covariance matrices \(\mathbf{P}_{ps,k}^{\mathcal{S}e(p)}\) and \(\mathbf{P}_{ps,k}^{\mathcal{S}e(d)}\) of the two sub-filters and the main filter. The optimal global state estimation is obtained as follows:

\[\left\{ \begin{array}{l}\hat{\mathbf{X}}_{ps,k}^{\mathcal{S}}=\left[\sum\limits_{ n=p,d}^{n}\!\left(\mathbf{P}_{ps,k}^{\mathcal{S}(n)}\right)^{-1}+\left(\mathbf{P}_{ps,k}^{ m}\right)^{-1}\right]^{-1}\\ \mathbf{P}_{ps,k}^{\mathcal{S}}=\mathbf{P}_{ps,k}^{\mathcal{S}}\left[\sum\limits_{n=p, d}^{n}\!\left[\left(\mathbf{P}_{ps,k}^{\mathcal{S}(n)}\right)^{-1}\hat{\mathbf{X}}_{ps,k}^{ \mathcal{S}e(n)}\right]+\left(\mathbf{P}_{ps,k}^{m}\right)^{-1}\hat{\mathbf{X}}_{ps,k}^ {m}\right]\end{array} \right. \tag{26}\]

The output result of the QLFEF algorithm is the global state estimate, including the PVSE \(\hat{\mathbf{X}}_{ps,k}\) and corresponding state error estimate covariance \(\mathbf{P}_{ps,k}\). Figure 5: Structure diagram of the \(Q\)-learning-based FEKF intelligent integrated navigation. \[\left\{ \begin{array}{l}\hat{\mathbf{X}}_{ps,k}=\hat{\mathbf{X}}_{ps,k}^{g}\\ \mathbf{P}_{ps,k}=\mathbf{P}_{ps,k}^{g}\end{array} \right. \tag{27}\]

## 4 Simulation and Results Analysis

The high-accuracy, effectiveness, and feasibility of the autonomous intelligent integrated navigation method is an important index to improve the autonomous operation ability of the cruise phase for the SSB probe. In order to further analyze the performance for this presented method, which is compared with the integrated navigation FEKF algorithm to verify the high accuracy of the state estimation of the X-ray pulsar/solar and target planetary Doppler velocity-measuring intelligent integrated navigation algorithm based on the QLEFKF. ### Simulation Initial Conditions

This section presents the initial position state and velocity sate parameters of the probe cruise phase for the solution of the orbital dynamics model at the beginning. The initial simulation conditions are shown in Table 1 as follows. The simulation time is from 1 September 2026 12:00:00:00.000 UTCG to 7 September 2026 12:00:00:00.000 UTCG. Thus, the total simulation time is 6 days. Three X-ray pulsars are considered as one of the measurement models for intelligent integrated navigation, namely PSR B1937+21, PSR B0531+21, and PSR J1821-24. Their specific parameters are shown in Table 2. The X-ray pulsar detector has an effective area of 1 m\({}^{2}\). The vector of the position for the SSB relative to the solar center of mass is calculated by the ephemeris DE441 from the Jet Propulsion Laboratory (JPL).