## 1 Introduction  
Efficient robotic trajectory planning is essential for the real-time control and operation of robots. It involves the generation of a robot's trajectory in space, taking into account the initial and final motion states, such as position, velocity, acceleration, and jerk, while satisfying predetermined boundary conditions and constraints on velocity, acceleration, and jerk. There are different methods for trajectory planning, including Cartesian space and joint (driving limb) space [(1)]. In this paper, we focus on trajectory planning in joint (driving limb) space to avoid coupling problems that exist in Cartesian space trajectory planning. Trajectory planning can also be regarded as an optimization problem involving establishing a weighted objective function that considers motion energy consumption, actuator torque, jerk, and acceleration, and then employing corresponding optimization algorithms to solve it [(1; 2; 3; 4; 5; 6; 7; 8; 9)]. Many researchers have looked into the multi-objective hybrid criterion and proposed various multi-objective optimization methods for planning robot paths [(10; 11; 12)]. Typically, research on trajectory planning for robots involves optimizing a performance indicator that satisfies predefined constraints, such as time-optimal planning [(13; 14; 15)] and minimum jerk trajectory planning [(16; 17; 18; 19)]. The common optimization method used is to construct a continuous acceleration change curve and optimize the selection of curve parameters to ensure boundary conditions and constraints on acceleration, velocity, and motion curve are met, and the optimal performance indicator is achieved. Some of the commonly used acceleration curve selections include a seven-segment step curve under zero initial and final acceleration condition, concatenated homogeneous polynomials, construction of conditional tree acceleration curve selection, cubic spline curve, and quintic B-spline curve [(20; 21)]. However, the aforementioned optimization algorithms are limited by issues such as computing efficiency, which makes it impossible to achieve real-time planning of robot trajectories. As a result, there is a need for a highly efficient and accurate optimization algorithm. Quantum particle swarm optimization (QPSO) is an intelligent optimization algorithm that utilizes the principles of particle swarm optimization (PSO) and quantum computing theory [22]. The PSO algorithm has been applied to robot motion controller parameter optimization [23], and although it is highly effective in solving complex optimization problems, QPSO, like any other algorithm, has certain limitations. For instance, parameter selection, high algorithmic complexity, sensitivity to initial values, and the challenge of locally optimal solutions can be significant challenges. Consequently, many researchers have attempted to improve QPSO to overcome these shortcomings and develop more efficient and effective optimization algorithms that can meet the needs of various applications. As a result, many scholars have made corresponding improvements to QPSO in order to address these issues. These improvements have led to the development of more efficient and effective optimization algorithms that can better meet the needs of various applications. For example, QPSO, which dynamically adjusts the number of quantum bits and quantum rotation angles has been proposed to improve the algorithm's convergence speed and global search capabilities [24]. [PERSON] proposed a teamwork evolutionary particle swarm optimization algorithm (TEQPSO) that balances global search and local search. The TEQPSO algorithm can utilize the group information more effectively and improves the algorithm's performance [25]. [PERSON] and [PERSON] proposed the quantum-behaved particle swarm optimization (QPSO) algorithm with the local attractor point subject to a Gaussian probability distribution (GAQPSO). The results show that the GAQPSO algorithm is an effective method that can greatly improve the performance of QPSO [26].