## 1 Introduction
This method effectively reduces the driving force of the trolley while ensuring rapid and accurate positioning, thereby suppressing the swinging of the payload sway angle. [PERSON] et al. [PERSON] et al. (2020) devised a sliding mode control (SMC) method augmented with an extended state observer (ESO) for the positioning and two-stage swing suppression of a three-dimensional double-pendulum crane. Experimental findings demonstrate the method's capability to achieve rapid and accurate positioning while effectively mitigating swing oscillations. [PERSON] et al. [PERSON] et al. (2020) proposed a disturbance observer for estimating external disturbances to address the adaptive fuzzy finite-time control problem based on disturbance observers for strict-feedback nonlinear systems. The effectiveness of this approach was verified through simulation studies on a single-link manipulator. [PERSON] et al. [PERSON] et al. (2020) proposed an observer-based differential evolutionary constraint control method for the safe reference tracking of robots. [PERSON] et al. [PERSON] et al. (2020) proposed an enhanced anti-sway tracking control strategy based on adaptive neural networks for offshore crane systems, and the effectiveness of this method was validated through experiments. The PID control is a commonly used linear control technique in crane systems, where its three key parameters significantly influence the system's stability and response characteristics. Therefore, selecting suitable parameters has become an important issue. In recent years, numerous scholars have devoted themselves to researching various intelligent optimization methods to achieve the optimization of PID parameters. [PERSON] et al. [PERSON] et al. (2020) proposed a backpropagation neural network PID (BP-PID) control method for DC motor speed regulation. They compared its performance to the fuzzy PID (FPID) and traditional PID control methods, demonstrating its superiority. [PERSON] et al. [PERSON] et al. (2020) proposed a fuzzy PID control strategy optimized using the bee colony algorithm. The improved PID controller achieved excellent control precision in the torque control of magnetorheological (MR) clutches. [PERSON] et al. [PERSON] et al. (2020) proposed a combined Cuckoo Search Optimization with an Improved Grey Wolf Optimization (CSO-IGWO) algorithm for tuning PID controller parameters. This algorithm utilizes the tent chaotic mapping method for population initialization and incorporates Levy flight strategy to update position equations, enabling grey wolf individuals to make large jumps to expand the search space without easily becoming trapped in local optima. [PERSON] et al. (2020) combined the wandering lion optimization algorithm (LSO) with fuzzy PID control, introducing the LSO algorithm and incorporating a scattering operator, effectively enhancing its global search performance. This method reduces the probability of the algorithm becoming stuck in local optima and employs a nonlinear adaptive method to update inertia weights and learning coefficients. [PERSON] (2019) proposed an improved bat control algorithm to enhance the trajectory tracking accuracy of robots. Experimental results indicate that the improved bat control algorithm achieved the best increase in trajectory tracking accuracy, and the proposed control method can be applied to other systems. [PERSON] et al. (2020) proposed an opposition-based learning jumping spider optimization algorithm to adjust the parameters of PID controllers, aiming to address the intelligent control problems related to the inlet flow rate, pressure, and temperature of heat exchangers in gas turbine cooling systems. Since the sparrow search algorithm (SSA) proposal in 2020 (2020), the SSA algorithm has been applied in many fields (24; 25). It mainly simulates the foraging behavior of sparrows to solve global optimization problems.