Berthing Trajectory Prediction for Azimuth Stern Drive Tugs Based on Transformer and Mixture Density Network

Objectives ：To enhance the trajectory prediction accuracy and uncertainty characterization capability of azimuth stern drive (ASD) tugs under complex working conditions, a novel ship trajectory prediction method integrating Transformer with Mixture Density Network (MDN) is proposed.Methods Taking harbor tugs as the research object, the proposed approach designates position, heading, and speed as state variables, while selecting port/starboard rudder angles and port/starboard propeller rotational speeds as control variables. The global attention mechanism within the Transformer encoder is exploited to deeply mine long-term temporal dependencies in time-series data. Results Concurrently, an MDN is introduced to construct a parametric probability density function, thereby realizing probabilistic forecasting for future multi-step motion states. Comparative results against the conventional Long Short-Term Memory (LSTM) network and the standard Transformer model demonstrate that the proposed method yields prominent advantages in both prediction accuracy and robustness. Specifically, compared with the LSTM network, the average displacement error (ADE) of single-step prediction is reduced by 57.7%, and the overall accuracy in 5-step prediction is elevated by over 10.9%, with the root mean square error (RMSE) exhibiting a significantly gentler growth trend as the prediction horizon extends. Furthermore, compared with the standard Transformer model, the prediction precision regarding position, heading, and speed is comprehensively enhanced. The Z-score calibration test verifies the model's capacity for rational characterization of prediction uncertainty, revealing that the normalized residual distribution is highly consistent with the standard normal distribution.Conclusions This methodology not only improves ship trajectory prediction accuracy but also achieves effective quantification of motion uncertainties under complex conditions, providing a reliable technical instrument for tug motion modeling and state forecasting.