## 5 Simulations  
### Simulation Setting  
The sensors used in SINS are assumed to have mid-precision. The constant bias and random noise of gyroscopes and accelerometers are illustrated in Table 1. The AUV sails along the \(y\)-axis with a velocity of 5 m/s. Driven by interference of the current, the AUV sails with a swinging motion, and the swinging motion has the function \(A\)\(\sin(2\pi f\cdot t+\eta_{0})+\theta_{0}\), in which \(A\) and \(f\) are the amplitude and frequency of the swinging and \(\eta_{0}\) and \(\theta_{0}\) are the initial phase and swinging center. The swinging parameters are shown in Table 2. The initial longitude, latitude, and height of the AUV are set as \(118^{\circ}\) E, \(32^{\circ}\) N, and \(-20\) m.  
Based on the ideal motion, the theoretical sensor outputs can be obtained by back-stepping of the navigation solution. When the sensor errors are added into the theoretical outputs, the real sensor data can be simulated. To assess the navigation accuracy, the ideal motion can be used as a reference. During the simulation, the update frequency of the sensor data and navigation solution are both 100 Hz. In the simulation, without considering the constant error, the random noise in DVL, MCP, and PS is assumed as white noise with variances of 0.5 m/s, 2\({}^{\circ}\), and 0.5 m. When these errors are added to the ideal motion, the real outputs of the DVL, MCP, and PS can be simulated. During simulation, the update frequency of the DVL, MPC, and PS is set to 10 Hz. Straight line trajectory is chosen in the simulation, and the velocity is set as above because this trajectory has no canceling effect on error growth [5; 6]. To simulate the sudden velocity change in DVL, velocities of 2 m/s, 5 m/s, and 1 m/s are added to the real output of the DVL along three directions during the period 2000-2010 s.  
When the initial parameters for SINS are set, the integrated navigation, including the initial alignment, is executed. Among those initial parameters, the initial velocity and position are set without any error, whereas the attitude has misalignment angles of 0.2\({}^{\circ}\), 0.2\({}^{\circ}\), and 1.0\({}^{\circ}\) in the pitch, roll and yaw. The parameters for the Kalman filter in Section 2 are set as:  
\[\mathbf{X}_{\text{SINS/DVL0}}=\left[ \begin{array}{cccccccccccc}0&0&0&0&0&0&0&0&0&0 &0&0&0&0&0&0\\ \end{array} \right]^{\text{T}},\]  
\[\mathbf{P}_{\text{SINS/DVL0}}=\text{diag}\left[ \begin{array}{cccccccccccc}(0. 5\text{m/s})^{2}&(0.5\text{m/s})^{2}&(0.5\text{m/s})^{2}&(5^{\circ})^{2}&(5^{ \circ})^{2}&(15^{\circ})^{2}&(30/R_{e})^{2}&(30/R_{e})^{2}&(1\text{m})^{2}\\ &&(500\text{uq})^{2}&(500\text{uq})^{2}&(500\text{uq})^{2}&(500\text{uq})^{2}&( 5^{\circ}/\text{h})^{2}&(5^{\circ}/\text{h})^{2}&(5^{\circ}/\text{h})^{2}\\ \end{array} \right],\]  
\[\mathbf{Q}_{\text{SINS/DVL}}=\text{diag}\left[ \begin{array}{cccccccccccc}(50 \text{uq})^{2}&(50\text{uq})^{2}&(50\text{uq})^{2}&(0.01^{\circ})^{2}&(0.01^{ \circ})^{2}&(0.01^{\circ})^{2}&(30/R_{e})^{2}&(30/R_{e})^{2}&(0.1\text{m})^{2} &0&0&0&0&0&0\\ \end{array} \right],\]  
\[\mathbf{X}_{\text{SINS/DVL0}}=\mathbf{X}_{\text{SINS/MCP}}=\mathbf{X}_{\text{SINS/PS0}},\mathbf{P}_{ \text{SINS/DVL0}}=\mathbf{P}_{\text{SINS/MCP}}=\mathbf{P}_{\text{SINS/PS0}},\mathbf{Q}_{ \text{SINS/DVL}}=\mathbf{Q}_{\text{SINS/MCP}}=\mathbf{Q}_{\text{SINS/MCP}}\]  
\[\mathbf{R}_{\text{SINS/DVL}}=\text{diag}\left[ \begin{array}{cc}(0.5\text{m/s})^{ 2}&(0.5\text{m/s})^{2}&(0.5\text{m/s})^{2}\\ \end{array} \right],\mathbf{R}_{\text{SINS/MCP}}=(0.5^{\circ})^{2}\text{ and }\mathbf{R}_{\text{SINS/PS}}=(0.5m)^{2}\]  
Two schemes are compared. In Scheme 1, the data fusion method introduced in Section 2 is simulated. In Scheme 2, the data fusion method with the velocity tracing method in Section 3, with the detecting threshold set at 50, is adopted. In Scheme 2, the parameters for the velocity tracing filter are set as \(\mathbf{X}_{by0}=\left[ \begin{array}{cc}V_{by}&0\\ \end{array} \right]^{\text{T}},\mathbf{P}_{by0}=\left[ \begin{array}{cc}1000&1000 \\ \end{array}