## 1 Introduction
[11] proposed a weighting method to transform the time-jerk optimization problem into a single-objective problem and obtained the optimal solution through the Sequential Quadratic Programming (SQP) method. This method effectively mitigated the complexity inherent in algorithmic resolution, but it cannot accurately capture the nonlinear or intricate interrelations among multiple objectives. Therefore, certain performance targets cannot achieve the desired optimization outcomes. Furthermore, the fixed weights in the optimization process preclude dynamic adjustments to accommodate varying constraint conditions [12; 13]. There are some disadvantages of using the weighting method to convert multi-objective optimization to single-objective optimization. First, the selection of weights depends on the subjective judgment of the decision maker, and small adjustments of weights can lead to significant changes in the solution. In addition, the weighting method can usually find only one solution, and it is difficult to obtain multiple solutions on the Pareto frontier, especially when there are serious conflicting objectives, and it may not be able to efficiently consider all objectives. In contrast, the multi-objective optimization method can provide a set of non-dominated solutions to help the decision-maker trade-off among multiple objectives and explore more possible solutions through the diversity maintenance strategy, which deals with objective conflicts more comprehensively and reduces the influence of subjective factors. Therefore, despite the simplicity of the weighting method implementation, the multi-objective optimization approach is more advantageous in dealing with complex relationships and providing decision support [14]. In a recent study, [PERSON] et al. [15] introduced an enhanced version of the dingo optimization algorithm based on simulated annealing (SA-DOA). This new approach effectively reduces time and jerk while enhancing the execution efficiency and motion control performance of manipulators by optimizing quintic non-uniform rational B-spline (QNURBS) trajectories. [PERSON] et al. [16] put forth an external archives self-searching multi-objective particle swarm optimization algorithm (EASS-MOPSO), which constructs joint position curves using quartic B-spline curves. It establishes a multi-objective optimization model focusing on time, energy, and jerk, thereby achieving seamless and smooth trajectory optimization without discontinuities. To enhance the operational efficiency and mechanical performance of a six-degree-of-freedom manipulator, [PERSON] et al. [17] proposed an advanced multi-objective particle swarm optimization (IMPSO) algorithm. It used the improved NSGA-II algorithm to sort the particles and optimize the weight allocation of the multi-objective function, enabling comprehensive evaluation of operational time, joint torque, and jerk. It realized a more optimal trajectory in the joint space. [PERSON] et al. [18] put forth a multi-objective trajectory planning model for manipulators based on the improved multi-objective salp swarm algorithm (LMSSA). This model combines logistic-sine chaotic mapping, small-hole imaging learning strategy, and golden-sine development strategy with the objective of optimizing the control nodes of the seventh-order B-spline curve and completing the multi-objective trajectory planning of manipulators with time-jerk-energy. [PERSON] et al. [19] proposed an SDE-NSGA-II algorithm to solve the problems of the traditional NSGA-II algorithm in multi-objective trajectory optimization, such as being easy to fall into the local optimum, slow convergence speed, etc. The SDE-NSGA-II algorithm adopts the initialization of the population of the good point set as well as the improvement of the evolution strategy with the DC operator. The results show that the improved algorithm solves the Pareto solution set with a more uniform distribution and obtains a more continuous and smooth trajectory.