Objective To address the cooperative control problem in multi-tug towing systems, which are characterized by multiple constraints and strong physical coupling, a multi-objective particle swarm optimization (MOPSO) method integrated with a multi-criterion switching strategy is proposed.

Method First, a comprehensive mathematical model of the motion dynamics of the multi-tug towing system is developed. This model fully captures the 3-DOF nonlinear dynamics of both the towed ship and each individual tug. This highly coupled dynamic system is then reformulated as a multi-objective optimization problem. The key conflicting objectives are carefully defined as follows: (1) Minimizing the tracking errors of the towed ship's position and velocity, (2) Minimizing the tracking errors of the tugs' position and velocity. Specifically, this involves reducing the variance in inter-tug distances and relative orientations to achieve a safe, compact, and coordinated tug formation under diverse conditions. Next, tailored velocity and position update rules for the particle swarm are developed to construct a multi-objective particle swarm optimization (MOPSO) control framework. The velocity update mechanism incorporates an adaptive inertia weight, which helps balance global exploration during the early stages and fine local exploitation in later stages. Building on this MOPSO foundation, a multi-criterion switching strategy is introduced as a core innovation. To prevent the particle swarm from becoming trapped in local optima during the search process, this strategy adopts a fixed-interval switching mechanism combined with random selection. A predefined switching interval is set, and whenever the iteration count reaches a multiple of this interval, a new evaluation criterion is randomly selected from a predefined set of criteria to guide the subsequent iterations. The predefined criteria for selecting the global best particle from the external archive are as follows: (1) Total Loss Minimization Criterion; (2) Towed Ship Loss Minimization Criterion; (3) Maximum Loss Minimization Criterion; (4) Tug Loss Balancing Criterion. By periodically and randomly switching among these complementary criteria at fixed intervals, the strategy dynamically adjusts the search direction. This approach effectively avoids local optima, enhances diversity in the Pareto archive, and ultimately yields a more uniform, well-converged, and globally optimal Pareto front that is tailored to the complex, coupled dynamics of the multi-tug towing system.

Results Simulation results demonstrate that, compared with the traditional optimal control method, the proposed approach improves the convergence speed by 21.88%, while reducing the towed ship's longitudinal and lateral errors by 43.39% and 48.99%, respectively. The tracking errors of all four tugs are also reduced, resulting in smoother and more compact formation control. Furthermore, compared with the single-criterion method, the proposed multi-criterion switching strategy achieves a more uniform distribution of the Pareto front, with the solution set being closer to the ideal optimal region. The convergence speed is improved by 7.41%, and the longitudinal and lateral errors of the towed ship are reduced by 11.47% and 10.27%, respectively, thereby achieving a better balance among multiple objectives.

Conclusion The proposed method enhances the control accuracy of the towed ship while achieving coordinated control among the tugs and optimizing the overall system performance.