## 1 Introduction  
Recent years have seen the rise of manipulators as critical components in numerous industries, as advancements in industrial technology have led to their growing importance. Consequently, the demand for high-performance manipulators has also increased significantly. Research on manipulators primarily includes aspects such as structural design [1], trajectory planning, and trajectory tracking [2; 3]. Among these, trajectory planning is crucial in the design process of manipulators. Many experts and scholars have conducted extensive research on various optimization problems related to manipulators, such as time-optimal [4], energy-optimal [5], impact-optimal [6], and multi-objective optimization [7], and have proposed a variety of solutions. In recent years, research and discussion have increasingly focused on developing efficient trajectory planning methods for manipulators, with a particular emphasis on minimizing the time required to complete a task. When planning a time-optimal trajectory for manipulators, the goal is to generate a path that satisfies the manipulator's kinematic constraints while also minimizing the time required to complete the task. One approach that was proposed in paper [8] is to use multiple quadratic radial basis functions to generate smooth motion trajectories for the manipulator. This method uses two objective functions, one based on the total running time and the other on the square of the jerk, which are minimized in order to achieve a manipulator's optimal trajectory in time. Experimental results show that this method can generate highly smooth trajectories within a relatively short running time. Paper [9] proposed a smooth and efficient trajectory planning method by combining an adaptive elitit genetic algorithm with singularity avoidance mechanism and quintic polynomials. The algorithm includes an elite population and an adaptive adjustment mechanism, which improves the recognition ability of the optimal solution. Furthermore, the singularity avoidance mechanism is introduced to prevent singular points in the generated trajectory. The combination of the adaptive elitist genetic algorithm with quintic polynomials generates smooth and efficient motion trajectories. In paper [10], an enhanced teaching-based optimization algorithm is proposed, which incorporates variable neighborhood search to keep the algorithm out of local optima. The algorithm is further combined with the quintic B-spline function to generate time-optimal motion trajectories. The particle swarm optimization algorithm was first improved by introducing population exchange operations and a tail elimination principle to enhance the algorithm's search ability [11]. Then, fuzzy reward-penalty theory was introduced to plan manipulator trajectory accurately while effectively reducing computational complexity for the manipulator, which leads to improved optimization efficiency by minimizing redundant decoupling operations. An improved adaptive cuckoo search algorithm was presented in paper [12] based on the continuous quintic B-spline curve algorithm for motion trajectory planning of manipulators, which can effectively generate time-optimal joint space smooth trajectories. Although the papers mentioned above have made positive contributions to time-optimal trajectory planning for manipulators, they did not address the issue of the motion path of manipulators. Specifically, they only achieved time-optimal trajectory planning on a fixed path without considering whether that path was the optimal solution. In contrast to the previous approaches, paper [13] introduced a method that not only plans the time-optimal trajectory for a manipulator but also searches for the optimal path, considering both aspects simultaneously. Given the start and final points, the intermediate path points and the running time are treated as variables, and the time-optimal trajectory is planned by means of an improved genetic algorithm.