## 5 Results
### Example 2  
Subsequently, we focus on the estimation of the longitudinal behavior of the fully submerged hydrofoil craft (FSHC). The FSHC's nonlinear model, encompassing both translational and rotational movements, is constructed on the basis of rigid body dynamics and momentum theorem [33,34]:  
\[(Z_{\dot{w}}-m)\dot{w}+Z_{w}w+Z_{z}z+Z_{\dot{q}}\dot{q}+(Z_{q}+U_{e}m)q+Z_{\theta }\theta=-Z_{\delta_{e}}\delta_{e}-Z_{\delta f}\delta_{f}-Z_{S} \tag{63}\]  
\[M_{\dot{w}}\dot{w}+M_{w}w+M_{z}z+(M_{\dot{q}}-I_{y})\dot{q}+M_{q}q+M_{\theta} \theta=-M_{\delta_{e}}\delta_{e}-M_{\delta f}\delta_{f}-M_{S} \tag{64}\]  
\[ \begin{bmatrix}\dot{z}\\ \dot{\theta}\end{bmatrix} = \begin{bmatrix}\cos\theta&0\\ 0&1\end{bmatrix}\begin{bmatrix}w\\ q\end{bmatrix} + \begin{bmatrix}-U_{e}\sin\theta\\ 0\end{bmatrix}  \tag{65}\]  
where \( \begin{bmatrix}\dot{z}&\dot{\theta}\end{bmatrix} \) refers to the velocity of the vertical movement and the velocity of the pitch angle in an earth-fixed frame, \( \begin{bmatrix}w&q\end{bmatrix} \) holds an equivalent interpretation in the body-fixed coordinate, and \(U_{e}\) is the steady speed of the FSHC. The means of the variable of (63), (64) can refer to the [31,32]. The nonlinear second-order state space function of the FSHC is designed as:  
\[\dot{x}_{1}=x_{2} \tag{66}\]  
\[\dot{x}_{2}=f(x_{1},x_{2})+Bu+DW \tag{67}\]  
Equations (66) and (67) are discretized through the application of the fourth-order Runge-Kutta method, resulting in a nonlinear discrete model for the longitudinal behavior of the FSHC, together with a linear measurement function. \[x_{k}=f(x_{k-1},u_{k-1})+d(t)+w_{k-1} \tag{68}\]  
\[z_{k}=H_{k}x_{k}+v_{k} \tag{69}\]  
where \(x_{k}= \begin{bmatrix}z&\dot{z}&\theta&\dot{\theta}\end{bmatrix} ^{T}\in R^{4}\) are the state of the system, \(z_{k}= \begin{bmatrix}z&\theta\end{bmatrix} ^{T}\in R^{2}\) represents the measurement state, \(u_{k}\in R^{2}\) means the control item, and \(H_{k}\) denotes the measurement matrix. \(w_{k}\sim N(0,Q_{k}),\;v_{k}\sim N(0,R_{k})\) are the system noises and measurement noises. The system's simulation duration is defined as \(t=100\) s with a sampling interval of \(T=0.1\) s. A total of \(\Upsilon=50\) Monte Carlo iterations are conducted. The initial states and covariance matrix are specified as \(X_{0}= \begin{bmatrix}0.01&0.01&0.01\end{bmatrix} ^{T}\), \(P_{0}=diag( \begin{bmatrix}100&10&100&10\end{bmatrix} )\). The measurement matrix is \(H_{k}= \begin{bmatrix}1&0&0&0\\ 0&0&1&0\end{bmatrix} \). The measurement noise covariance is \(R=diag( \begin{bmatrix}0.06&0.25\end{bmatrix} )\) and the system noise covariance set as \(Q=diag( \begin{bmatrix}1&1&1&1\end{bmatrix} )\). The parameters of the FSHC can refer to [33,34]. In this case, \(\delta\) is selected as \(\delta=0.4-0.5\). The fuzzy rule is as follows:  
\[\text{If}\;\Big{\|}z_{k}-\hat{z}_{k|k-1}\Big{\|}>0.1\text{, then decrease the }\rho,\;\rho=0.01 \tag{70}\]  
\[\text{If}\;\Big{\|}z_{k}-\hat{z}_{k|k-1}\Big{\|}<0.1\text{, then decrease the }\rho,\;\rho=400 \tag{71}\]  
We assume the fact that the FSHC maintains a constant speed of \(45\) km with a significant wave height of \(1.5\) m and the encounter angles of ocean waves are set at \(30^{\circ}\). We further assume that the measurement noise adheres to a non-Gaussian noise distribution as:  
\[r_{1}(k) \sim 0.9N(0,0.06)+0.1N(0,6) \tag{72}\] \[r_{2}(k) \sim 0.9N(0,0.25)+0.1N(0,25) \tag{73}\]In order to validate the competence of the MKMC-CSVSF, the CKF, CSVSF and the MCSVSF are compared in the experiment for the longitudinal behavior estimation of the FSHC. The estimation performance of the CKF, CSVSF and proposed MKMC-CSVSF for the heave motion of the FSHC is shown in Figure 2 while the corresponding estimation errors are depicted in Figure 3. The results of the estimate and discrepancies for heave velocity are exhibited in Figures 4 and 5, respectively. The estimation results and errors for the pitch angle of the FSHC are displayed in Figures 6 and 7.