# Autonomous Underwater Vehicle Localization Using Sound Measurements of Passing Ships
## 2 Preliminary Information
### Rpekf  
The RPEKF in [37; 38] is utilized to decrease the range estimation error. In the RPEKF, we utilize \(M\) EKF tracks to decrease the range estimation error. Every EKF track of the RPEKF runs the EKF in Section 3.2 independently, such that every EKF has a different initial range estimate. We assume that the AUV has been moving in underwater environments using the IMU-only localization before time step 0. Suppose that at time step 0, the IMU-only localization estimates the range between the ship and the AUV as \(\Delta\). Also, suppose that the range estimation error at time step 0 is bounded above by \(\delta\). Doing underwater experiments, we can derive the relationship between the operation time of the AUV (under the IMU-only localization) and the increase in localization error. Then, the bound for localization error can be set considering the operation time of the AUV in underwater environments. This range estimation error is unavoidable, since the AUV has been moving in underwater environments using the IMU-only localization. A feasible range interval (\(r_{min}-r_{max}\)) is set considering the localization error of the AUV at time step 0. Here, \(r_{min}\) is set as  
\[r_{min}=\Delta-\delta. \tag{11}\]  
Also, we use  
\[r_{max}=\Delta+\delta. \tag{12}\]  
Moreover, there may be a case where the long-term drift of the AUV is too severe. Assume that the AUV can access the approximate SL of a passing ship. Based on the SL, we can utilize an acoustic propagation model, which simulates the sound emanated from the ship. Suppose that the AUV measures the sound of the ship. Then, using the RAM in Section 2.2, we can derive the feasible region where the AUV can exist. For instance, suppose that the maximum sensing range, say \(mS\), is derived by searching for a target range satisfying Equation (10). In this case, we use  
\[r_{min}=0, \tag{13}\]  
instead of Equation (11). Also, we use  
\[r_{max}=mS, \tag{14}\]  
instead of Equation (12). To maintain a comparable performance of every EKF track, the range interval (\(r_{min}-r_{max}\)) is divided such that the \(j\)-th sub-interval has its range interval as (\(rn(j)-rn(j+1)\)) for \(j\in\{1,\ldots,M\}\). We utilize \(rn(1)=r_{min}\). For \(i\in\{2,\ldots,M+1\}\), we utilize  
\[rn(i)=rn(i-1)*\rho. \tag{15}\]  
Here,  
\[\rho=(r_{max}/r_{min})^{(1/M)}. \tag{16}\]  
Thus, we have \(rn(M+1)=r_{max}\). In total, \(M\) EKF tracks are utilized, and we utilize the likelihood \(p(\mathbf{z}_{k}|j)\) (\(j\in\{1,2,\ldots,M\}\)) from the \(j\)-th EKF track. In \(p(\mathbf{z}_{k}|j)\), \(\mathbf{z}_{k}\) is in Equation (2). Let \(\hat{\mathbf{X}}_{k|k}^{(j)}\) denote the state vector of the \(j\)-th EKF track, utilizing all measurements up to sample-index \(k\). In addition, \(\mathbf{P}_{k|k}^{(j)}\) denotes the error covariance of \(\hat{\mathbf{X}}_{k|k}^{(j)}\). Let \(\hat{\mathbf{X}}_{k|k-1}^{(j)}\) denote the state vector of the \(j\)-th EKF track, utilizing all measurements up to sample-index \(k-1\). Let \(\mathbf{D}_{k}=\{\mathbf{z}_{1},\mathbf{z}_{2},\ldots,\mathbf{z}_{k}\}\) for convenience. The weight of the \(j\)-th EKF track (\(j\in\{1,2,\ldots,M\}\)) is  
\[w_{k}^{(j)}=p(j|\mathbf{D}_{k})\propto p(\mathbf{z}_{k}|j)p(j| \mathbf{D}_{k-1}). \tag{17}\]  
In Equation (17), the likelihood \(p(\mathbf{z}_{k}|j)\) is derived as  
\[p(\mathbf{z}_{k}|j)=\frac{1}{\sqrt{det(\mathbf{S}_{k}^{(j)})}} exp(-0.5\mathbf{r}_{s}^{T}(\mathbf{S}_{k}^{(j)})^{-1}\mathbf{r}_{s}). \tag{18}\]  
Here, \(\mathbf{r}_{s}=\mathbf{z}_{k}-h(\hat{\mathbf{X}}_{k|k-1}^{(j)})\), \(\mathbf{S}_{k}^{(j)}=\mathbf{H}_{k}^{(j)}\mathbf{P}_{k|k-1}^{(j)}(\mathbf{H} _{k}^{(j)})^{T}+\mathbf{R}_{k}\), and \(\mathbf{H}_{k}^{(j)}=\frac{\Delta h(\mathbf{X})}{\partial\mathbf{X}}\big{|}_ {\mathbf{X}=\hat{\mathbf{X}}_{k|k-1}^{(j)}}\).