## 1 Introduction  
The utilization of robotic manipulators enhances the efficiency and automation of industrial manufacturing processes. In the 3c industry (computer, communication, consumer electronics), rapidly updated products and complex production patterns require highly flexible robotic arms to complete tasks. In the process of performing the task, the robotic arm may collide with obstacles, so a real-time obstacle avoidance algorithm is needed to control the robotic arm [1; 2; 3; 4; 5; 6]. The obstacle avoidance algorithm is widely studied in the field of robot motion planning, and it is a computational method to find the path from the starting point to the target point without colliding with obstacles when the robotic manipulator is working in an environment with obstacles. At present, researchers have proposed various obstacle-avoidance algorithms, such as genetic algorithm, rapidly exploring random tree (RRT), and probabilistic roadmap (PRM). However, they all have limitations in practical applications [7]. The RRT algorithm begins by selecting the starting point as the root node and proceeds to generate a randomly expanded tree structure by iteratively sampling and adding leaf nodes, and the shortest path from the starting point along the leaf node to the target position is the obstacle avoidance path. Nevertheless, the calculation speed of RRT is slow, and the smoothness of the trajectory obtained by this algorithm is unsatisfactory, which makes the manipulator shake easily [8]. The genetic algorithm finds an intermediate point with position and velocity information, and connects the starting position to the intermediate point and the intermediate point to the target position through two polynomial paths. The two paths are concatenated to obtain an obstacle avoidance path. Although the genetic algorithm can obtain a smooth obstacle avoidance trajectory, the computational efficiency is very low [9; 10]. The PRM algorithm establishes a complete undirected graph by sampling and collision detection, and then obtains a feasible obstacle avoidance path through a graph search algorithm (such as the A* algorithm). However, the process of establishing the map using the PRM algorithm is complex and time-consuming. Moreover, the connection between two nodes in the undirected graph does not take into account the robot's motion model [11; 12]. Although the above algorithms can obtain obstacle avoidance trajectories, their real-time performance is weaker than the artificial potential field method. This paper only discusses the artificial potential field method. The artificial potential field (APF) method is a widely recognized motion planning strategy in the field of robotics. In 1986, [PERSON] first proposed the artificial potential field method (APF) for the motion planning of robots [13]. The APF method extensively borrows the concept of potential fields from physics and transforms the path planning problem of robots or moving objects into a problem of searching for the optimal path in a virtual potential field. The core idea is to consider the target location as an attractive source and obstacles or infeasible areas as repulsive sources. High potential fields are assigned around obstacles, while low potential fields are assigned around the target location. By calculating the forces and directions, the APF guides the robot or object to move along the direction of decreasing potential energy, thereby achieving the goal of avoiding obstacles and reaching the target location. The APF method is characterized by its simplicity, intuitiveness, ease of implementation, and real-time capabilities. As a result, it has been widely applied in the industrial field. For example, in automated warehousing systems, robots utilize the APF method to navigate around obstacles and find the shortest path based on the layout and obstacle positions within the warehouse. This potential field guidance enables them to efficiently and swiftly accomplish tasks such as cargo handling [14; 15].