## 2 System Models of Integrated Navigation  
### Integrated Navigation Dynamics Model  
The heliocentric inertial coordinate system is chosen [43]. The orbit dynamics model [44] of the navigation system during the cruise phase for the SSBE with the Sun as the center and the large planet as the gravitational perturbations [45] is established as follows:  
\[\left\{\begin{array}{l}\dot{\mathbf{r}}=\mathbf{v}\\ \dot{\mathbf{v}}=-\mu_{s}\frac{\mathbf{r}_{ps}}{\mathbf{r}_{ps}^{3}}-\sum\limits_{i}^{N_{p} }\mu_{i}\left[\frac{\mathbf{r}_{pi}}{\mathbf{r}_{pi}^{3}}-\frac{\mathbf{r}_{si}}{\mathbf{r}_{si} ^{3}}\right]+\alpha\end{array}\right. \tag{1}\]  
where \(\mathbf{r}=\begin{bmatrix}x&y&z\end{bmatrix}^{T}\) and \(\mathbf{v}=\begin{bmatrix}\dot{x}&\dot{y}&\dot{z}\end{bmatrix}^{T}\) are the vector of the position and velocity for the SSBE probe in the system of heliocentric inertial coordinate, respectively. \(\mu_{s}\) is the constant of heliocentric attraction. \(\mathbf{r}_{ps}\) is the vector of the relative position between the heliocentric and the SSBE probe. The gravitational constant is denoted as \(\mu_{i}\) for the \(i\)th perturbation planet object. The vector of the relative position between the \(i\)th perturbation planet object and the probe is denoted by \(\mathbf{r}_{pi}\). The vector of the relative position between the \(i\)th perturbation planet and the heliocentric in the inertial system is denoted by \(\mathbf{r}_{si}\), \(\mathbf{r}_{si}=\mathbf{r}_{pi}+r\). \(N_{p}\) represents the number of planetary targets for perturbing planets. \(\alpha\) represents a variety of disturbances in the cruise phase environment that are not modeled and can be treated as process noise.  
The orbit dynamic model for the cruise phase of the Earth-to-Jupiter is expressed as follows [46].  
\[\left\{\begin{array}{l}\dot{x}=v_{x}\\ \dot{y}=v_{y}\\ \dot{z}=v_{z}\\ \dot{v}_{x}=-\mu_{s}\frac{x}{r_{ps}^{3}}-\mu_{j}\left[\frac{x-x_{j}}{r_{ps}^ {3}}+\frac{x_{j}}{r_{s}^{3}}\right]-\mu_{m}\left[\frac{x-x_{m}}{r_{ps}^{3}}+ \frac{x_{m}}{r_{sm}^{3}}\right]-\mu_{e}\left[\frac{x-x_{e}}{r_{ps}^{3}}+\frac {x_{e}}{r_{ps}^{3}}\right]+w_{x}\\ \dot{v}_{y}=-\mu_{s}\frac{y}{r_{ps}^{3}}-\mu_{j}\left[\frac{y-y_{j}}{r_{pi}^{3 }}+\frac{y_{j}}{r_{ss}^{3}}\right]-\mu_{m}\left[\frac{y-y_{m}}{r_{ps}^{3}}+ \frac{y_{m}}{r_{ss}^{3}}\right]-\mu_{e}\left[\frac{y-y_{e}}{r_{ps}^{3}}+\frac {y_{e}}{r_{ps}^{3}}\right]+w_{y}\\ \dot{v}_{z}=-\mu_{s}\frac{z}{r_{ps}^{3}}-\mu_{j}\left[\frac{z-z_{j}}{r_{ps}^{3} }+\frac{z_{j}}{r_{ss}^{3}}\right]-\mu_{m}\left[\frac{z-z_{m}}{r_{ps}^{3}}+ \frac{z_{m}}{r_{sm}^{3}}\right]-\mu_{e}\left[\frac{z-z_{e}}{r_{ps}^{3}}+\frac {z_{m}}{r_{ss}^{3}}\right]+w_{z}\end{array}\right. \tag{2}\]  
Here, \((x_{j},y_{j},z_{j})\), \((x_{m},y_{m},z_{m})\), and \((x_{e},y_{e},z_{e})\) are the position vectors of Jupiter, Mars, and Earth, respectively. The relative positions between the center of mass of Jupiter, Mars, and Earth and the probe are denoted as \(r_{pi}\), \(r_{rm}\), and \(r_{pe}\), respectively. The relative positions between the center of mass of Jupiter, Mars, and Earth and the heliocentric are denoted as \(r_{si}\), \(r_{sm}\), and \(r_{se}\), respectively.  
The state variable \(\mathbf{X}=\begin{bmatrix}\mathbf{r}^{T}&\mathbf{v}^{T}\end{bmatrix}^{T}\) is selected. The above orbital dynamics model is converted into the equation of state of the system as follows:  
\[\mathbf{X}(t)=f[\mathbf{X}(t),t]+\mathbf{\omega}(t) \tag{3}\]  
where \(\mathbf{\omega}(t)\) is the perturbation force that is not modeled considering solar radiation pressure and other planetary bodies, \(\mathbf{\omega}(t)\) = [0, 0, 0, \(\omega_{x}\), \(\omega_{y}\), \(\omega_{z}\)]T. Compared to the perturbations of Sun, Jupiter, Mars, and Earth, \(\omega_{x}\), \(\omega_{y}\), and \(\omega_{z}\) are very small and can be treated as the error noise of the system state in the dynamic model of the cruise phase. It is simplified to the state error covariance \(\hat{Q}_{k}\) with the zero mean white noise, \(\hat{Q}_{k}=\left[\hat{Q}_{rk},\hat{Q}_{pk}\right]^{T}\).