# Optimization Algorithm for 3D Smooth Path of Robotic Arm in Dynamic Obstacle Environments
## 1 Introduction  
With the continuous expansion of industrial applications for robotic arms, utilizing robotic manipulators to accomplish flexible tasks in more open environments poses a new challenge for current industrial robotic arms. Therefore, under such demands, the path planning of more flexible and intelligent robotic arms has become a core research task in intelligent control, representing a critical technology that must be addressed for robotic arms to complete complex tasks. Since the 1950s, researchers have proposed numerous algorithms to seek optimal paths for mobile robots. Based on their characteristics, these algorithms are generally classified into three categories [1]: classical algorithms, bionic algorithms, and artificial intelligence algorithms. Classical algorithms include cellular decomposition (CD), sampling-basedmethods [2], graph search algorithms [3], artificial potential field methods [4], dynamic window approaches, and so on. Bionic algorithms encompass genetic algorithms [5], ant colony optimization [6], particle swarm algorithms [7], firefly algorithms [8], bacterial foraging algorithms [9], artificial bee colony algorithms [10], and so on. Artificial intelligence algorithms primarily include fuzzy control algorithms [11] and neural network algorithms. Cellular decomposition (CD) is further divided into regular cellular decomposition (RCD), approximate cellular decomposition (ACD), and exact cellular decomposition (ECD) based on networking strategies. [PERSON] and [PERSON] [12] applied ACD in the field of complete coverage path planning for mobile robots. [PERSON]; [PERSON]; et al. [13] proposed recursive reward-modified approximate cellular decomposition (RRMACD), which constructs small cubic cells in the three-dimensional environment where unmanned aerial vehicles (UAVs) operate, taking into account various constraints such as UAV mobility, geometry, and static and dynamic environmental obstacles, without invoking other algorithms to reduce computational load and complexity. Sampling-based methods (SBM), specifically the rapidly exploring random tree (RRT) algorithm and its optimized versions, are motion planning algorithms based on sampling. The RRT algorithm proposed by [PERSON] [14] is one of the most classic sampling algorithms, suitable for solving path planning problems with different motion constraints and capable of handling problems with multiple degrees of freedom. The RRT algorithm [14], as a path planning algorithm with relatively weak environmental perception, cannot guarantee the optimality of the path. When the environment contains a large number of obstacles or narrow passages, the algorithm's convergence speed slows down, and its efficiency drops sharply. Therefore, many classic optimization solutions have been proposed for the RRT algorithm. The RRT-Connect algorithm [15] addresses the shortcomings of the RRT algorithm, such as the blindness of its search space and insufficient memory for node expansion links, to improve the search speed in space. There are also many optimized variants, such as RRT-Smart [16], SMRS-RRT [17], and Fast-RRT [18]. Researchers have been dedicated to optimizing the RRT algorithm, mainly focusing on three aspects: random sampling methods [14], branch expansion strategies [19], and integration with robot kinematics [20]. These methods have the advantages of quickly searching free space and real-time collision detection. Graph search algorithms (GSA), with [PERSON]'s algorithm as a classic example, are graph search algorithms designed for finding optimal paths from a single source point. The A* algorithm, a heuristic search algorithm for finding the shortest path in static environments, and its optimized variants are also included in GSA. The A* algorithm improves upon [PERSON]'s algorithm by introducing a heuristic function for node selection, which significantly reduces the total number of expanded nodes and improves the efficiency of trajectory planning.