## 5 Results and Discussion
### Verification of the Improved GA's Ability to Adjust Inter-Unit Distances and Overall Speed
Figure 6 shows the thrust allocation and inter-unit distance variations for a 5-unit MATC system, comparing scenarios with and without the inclusion of the penalty function. Figure 6a shows the thrust allocation results with the penalty function applied, demonstrating that all thrust values remain within 3.5 N. In contrast, Figure 6b presents the thrust allocation results without the penalty function, where thrust values exceed 3.5 N (e.g., u5). Figure 6c displays the inter-unit distance variations with the penalty function, revealing significant oscillations due to the constrained thrust values, which limit rapid adjustments. Conversely, Figure 6d illustrates the inter-unit distance variations without the penalty function, showing smaller oscillations as the thrust values are unconstrained, enabling faster and more flexible adjustments. Despite this, the improved GA offers a significant improvement in managing thrust constraints, which is critical for the stability and efficiency of MATC systems. Except for Unit 5, which is affected by ocean current disturbances, the thrusts of the other units gradually stabilize. The introduction of the penalty function effectively constrains the optimization process, ensuring that each unit's thrust stays within the predefined range, thereby maintaining the system's stability and controllability. Additionally, these oscillations are primarily caused by the periodic nature of ocean current disturbances acting on Unit 5. In the early stages of system operation, the MATC system rapidly adjusts the thrust allocation, allowing the system to quickly return to the ideal state. However, because of the continuous fluctuations in ocean current, slight oscillations occur before the units achieve the desired inter-unit distance. Over time, these  
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline \(t\) (s) & **0** & **1.500** & **3.000** & **4.500** & **6.000** & **7.500** & **9.000** & **10.500** & **12.000** & **13.500** & **15.000** \\ \hline \(u_{1}\) (N) & 3.985 & 2.272 & 2.577 & 2.610 & 2.360 & 2.162 & 2.125 & 2.079 & 2.058 & 2.191 & 2.386 \\ \(u_{2}\) (N) & 3.985 & 2.524 & 2.799 & 2.809 & 2.598 & 2.419 & 2.307 & 2.247 & 2.271 & 2.385 & 2.565 \\ \(u_{3}\) (N) & 4.038 & 2.615 & 2.740 & 2.717 & 2.595 & 2.44  
Verification of the Thrust Allocation Ability of the HHO+BP When the Number of AUV Units in MATC System Is Large  
If the offline training process is excluded and the thrust optimization is directly performed using the improved GA, the inter-unit distance variation for the 30-unit MATC system is shown in Figure 7a. The inter-unit distance variation for the thrust distribution after offline training (HHO+BP) is presented in Figure 7b. As shown in Figure 7a, due to the long computation time for thrust allocation, the MATC system is unable to execute the optimal thrust allocation in time, leading to noticeable oscillations in the allocated thrust and preventing the inter-unit distance from converging to the ideal value. At 10.5 s, the average inter-unit distance is 0.1217 m, the maximum is 0.1653 m (with a relative deviation of 37.75%), and the minimum is 0.1121 m (with a relative deviation of 6.58%). Therefore, directly using online optimization methods for thrust allocation in cases with a large number of AUV units shows significant drawbacks. Figure 7: The variation of inter-unit distance with different thrust allocation methods. (**a**) Directly using the improved GA. (**b**) Using the offline training (HHO+BP). In contrast, as shown in Figure 7b, the inclusion of the offline training process significantly reduces the online optimization time, enabling the system to quickly execute the thrust allocation results. Consequently, the thrust allocation stabilizes rapidly, and the inter-unit distance converges to the ideal value. At 10.5 s, the average inter-unit distance is 0.1200 m, the maximum is 0.1201 m (with a relative deviation of 0.08%), and the minimum is 0.1199 m (with a relative deviation of 0.08%).