## 6 Conclusions  
In this paper, we propose the EM-MOGCPF algorithm, an improved state estimation approach based on the CPF framework, for underwater robots operating in environments with nonlinear and non-Gaussian noise distributions. Our algorithm leverages a hybrid Gaussian model to approximate the significant type density function, and employs the EM algorithm instead of traditional resampling to mitigate the particle degradation problem.  
Figure 11: RMSE of sea state degree 0.  
Figure 12: (**a**) RMSE of sea state degree 1–2, (**b**) RMSE of sea state degree 3.  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Simulation Case** & **Attitude** & **PF** & **CKF** & **CPF** & **EM-MOGCPF** \\ \hline \multirow{4}{*}{Sea state degree: 0} & \(X\) & 1.8577 & 1.0618 & 0.8291 & 0.5633 \\  & \(Y\) & 1.2232 & 0.9602 & 0.7297 & 0.5290 \\  & \(Z\) & 3.8294 & 0.8827 & 0.6871 & 0.5140 \\  & \(\psi\) & 2.5201 & 1.5164 & 1.5018 & 0.9196 \\ \hline \multirow{4}{*}{Sea state degree: 1–2} & \(X\) & 0.8289 & 1.1010 & 0.6611 & 0.4392 \\  & \(Y\) & 2.6940 & 1.4307 & 0.8125 & 0.3738 \\  & \(Z\) & 6.9908 & 3.9050 & 1.0363 & 0.4771 \\  & \(\psi\) & 1.6541 & 1.7259 & 0.8849 & 0.4083 \\ \hline \multirow{4}{*}{Sea state degree: 3} & \(X\) & 2.1690 & 3.5037 & 1.2855 & 0.5956 \\  & \(Y\) & 7.5129 & 5.0190 & 1.6607 & 0.3094 \\ \cline{1-1}  & \(Z\) & 5.9652 & 10.9400 & 3.3708 & 0.6318 \\ \cline{1-1}  & \(\psi\) & 8.7070 & 14.6010 & 6.7736 & 2.1357 \\ \hline \hline \end{tabular}
\end{table}
Table 3: RMSE of different algorithms in three sea states.  
We also use a modified algorithm for underwater anchor chain cleaning robot observation, and validate the performance of the algorithm under three different sea state conditions. Our experimental results demonstrate that the proposed algorithm can accurately estimate the robot state and achieve strong convergence even in the presence of different noise distributions, while effectively resisting the influence of noise. The algorithm exhibits good environmental adaptability and can address the problem of robot state estimation in underwater nonlinear and non-Gaussian environments. Moreover, the proposed algorithm can be applied to other state estimation fields.  
Future research includes developing a simulation platform for underwater robot state observation, exploring underwater robot models with more degrees of freedom and parameter estimation, and conducting additional underwater experiments to further evaluate the real-time performance and accuracy of the algorithm.  
Conceptualization, C.C. and B.W.; methodology, C.C. and B.W.; software, C.C. and Y.Z.; validation, B.W. and Z.J.; formal analysis, B.W. and C.C.; writing--original draft preparation, B.W. and C.C.; writing--review and editing, B.W. and C.C.; visualization, C.C.; supervision, Z.J. and B.W. All authors have read and agreed to the published version of the manuscript.  
This research was funded by the Shanghai Science and Technology Commission Innovation Action Plan Project [20 dz1206500].  
Not applicable.  
The authors are grateful for the support of the Shanghai Engineering Research Center of Hadal Science and Technology, Shanghai Ocean University, Shanghai, China.  
The authors declare no conflicts of interest.