Please note this is a comparison between Version 2 by Rita Xu and Version 1 by Youyu Liu.
通过引入高概率目标偏差策略可以提高快速探索随机树（The search efficiency of a rapidly exploring random tree (RRT）的搜索效率。在存在多个复杂障碍的情况下，具有固定步长的高概率目标偏差策略将落入局部最优，从而降低搜索效率。) can be improved by introducing a high-probability goal bias strategy. In the case of multiple complex obstacles, the high-probability goal bias strategy with a fixed step size will fall into a local optimum, which reduces search efficiency.
Path planning is an essential component of robot motion planning and is a research hotspot in the field of robotics and other related intelligent fields 
. Among them, the manipulator, as an important industrial robot, has autonomy and intelligence levels that are crucial for improving production efficiency and quality. Path planning can help the manipulator automatically plan the optimal path, reduce dependence on staff, and improve the autonomy of the manipulator. Path planning can be flexibly adjusted and optimized according to different task characteristics, thereby improving the motion accuracy and speed of the manipulator, which directly affects production efficiency and quality. Compared with a single manipulator, the form of a dual manipulator collaborative operation can meet the needs of complex, intelligent, and compliant modern industrial systems, and dual manipulators have more advantages in efficiency and performance [2,3]
and are gradually gaining attention in the industry. Path planning for dual manipulators is an important work of collaborative operations 
. To enhance the adaptability and flexibility of dual-manipulator systems, it is necessary to flexibly adjust and optimize paths based on various production environments and task requirements. The equipment transformation and upgrading of production lines are of great significance, but there are high requirements for efficiency, real-time performance, and safety in collaborative operations 
. To address these challenges, researchers are continuously developing and improving various path planning methods for dual manipulators to enhance their efficiency and precision and meet different demands in industrial production environments. Therefore, different path planning algorithms need to be designed or selected for various fields to achieve the goal of fast matching and application.
2. Rapidly Exploring Random Tree for a Dual Manipulator
The classic 3D path planning algorithms for robots can be roughly divided into three categories. The first type of path planning algorithm is based on searches, such as Dijkstra and A* algorithms [6,7]
. This algorithm is based on a graph structure in which each node represents the robot’s location, and each edge represents its movement path. By searching through the graph structure and calculating a heuristic function for each node to evaluate the distance to the endpoint, the optimal path is found. Qing et al. 
proposed an improved Dijkstra algorithm that saves all equidistant shortest paths during the path search process, although it can solve several shortest path planning problems; in some cases, it may be difficult to obtain the complete graph structure, and there are issues such as large search space, high computational complexity, and poor real-time performance.
The second type of path planning algorithm is based on rules, such as the artificial potential field method [9,10]
. The main idea is to design an artificial potential field to simulate the perception and decision-making process of the robot during movement and achieve path planning. The artificial potential field method has the advantages of simple algorithm implementation, easy understanding and use, rapid calculation of robot movement paths, and high real-time performance. Therefore, Xia et al. 
proposed an improved velocity potential field (IVPF) algorithm based on the artificial potential field method to address the inherent drawbacks of traditional algorithms. However, utilizing tangential velocity to avoid local minimum problems leads to poor path quality. The artificial potential field method only considers the relationship between the robot and obstacles, ignoring the constraints among robots themselves, which may lead to locally optimal solutions in some cases.
To address this issue, the third type of path planning algorithm based on sampling is widely applied in various fields, such as the rapidly exploring random tree (RRT) 
and probabilistic roadmap (PRM) 
. The main idea is to search for the optimal feasible path through random sampling in the environment. Sampling-based algorithms are not limited by the type of environment and can be applied to path planning problems in various complex environments, with high robustness and reliability. Li et al. 
improved the PRM algorithm by using a pseudorandom sampling strategy with the spatial principal axis as a reference axis and optimized the path using Bezier curves. However, the roadmap construction rate is unstable in three-dimensional environments. Liu et al. 
proposed a grid-local PRM method, which has high efficiency and real-time performance. However, this type of algorithm has weak scalability and a low roadmap reuse rate. To address this issue, the RRT algorithm and its variations have been proposed. The RRT algorithm has wide applicability, high efficiency, strong scalability, good determinism, and real-time computation, which effectively solves the path planning problem with high-dimensional space constraints. As a result, the RRT algorithm has become one of the most commonly used and effective algorithms in path planning.
On this basis, Kuffner et al.  proposed an RRT-connect double tree algorithm by randomly expanding paths at the same time at the start and goal nodes. It is superior to the RRT algorithm in terms of search performance. However, it is difficult to find the optimal path due to randomness. To solve this problem, scholars made some improvements to the RRT-connect algorithm. For example, based on the triangle inequality, Kang et al.  proposed an RRT-connect algorithm based on the triangle inequality principle by re-wiring path nodes, which has outstanding performance in terms of path length. However, there may be problems, such as non-differentiable linear sections with sharp corners and constraints with the kinematics of the manipulator. Based on the idea of dynamic step size , Li et al.  proposed a variable step size RRT (VT-RRT) by transforming the search space of random nodes in the RRT algorithm and adaptively adjusting the search step size according to the goal and the position of obstacles in the current point. This algorithm effectively reduces path planning time and optimizes sampling direction. However, it generates too many path nodes, resulting in longer paths. To improve the adverse effects of variable step size, Zhang et al.  proposed a path planning method for a manipulator based on the artificial potential field and bidirectional rapidly exploring random tree (BiRRT-APF) algorithm, aiming to solve the problem of low search efficiency and high randomness. However, its goal orientation is poor. Shao et al.  proposed a motion planning method based on the goal bias RRT algorithm (G-RRT), which reduces invalid searches by guiding the direction of random sampling. However, the one-way search is less efficient, and the resulting path is not optimal. Liu et al.  proposed a goal bias bidirectional rapidly exploring random tree (GBI-RRT) algorithm, which improves the success rate of node expansion. However, in complex and high-dimensional environments, this algorithm generates redundant nodes, resulting in overly complex paths. The types of path planning algorithms are shown in Table 1
below, as well as the advantages and disadvantages of each algorithm. Due to the existence of overlapping workspaces, the path planning of dual manipulators should deal with the interference of static and dynamic obstacles at the same time. In response to the above content, a sampling-based RRT path planning algorithm is adopted to improve and optimize the shortcomings of the algorithm and is deployed on a dual manipulator.
The types of path planning algorithms.
Regarding the aforementioned issues, this research proposes a bidirectional potential field probabilistic step-size RRT algorithm for the path planning of dual manipulators by angle selection. The main contributions of this research are as follows:
基于 Based on the
RRT连接算法和双向搜索特性，引入高目标概率偏差策略，使随机点沿目标方向进行采样。-connect algorithm and the characteristics of bidirectional searches, the high goal probability bias strategy is introduced to enable the random points to be sampled along the goal direction.
角度选择用于限制双树搜索的方向，避免对周围区域的冗余采样。 Angle selection is used to limit the direction of dual-tree searches and avoid redundant sampling to the surrounding area.
基于动态步长思想，创新性地将随机值作为步长参数，并通过随机性的动态变化自适应调整搜索步长以应对环境。引入人工势场法应对多障碍环境。Based on the idea of dynamic step size, random values are innovatively used as step size parameters, and the search step size is adaptively adjusted by the dynamic changes of randomness to cope with the environment. The artificial potential field method is introduced to deal with multi-obstacle environments.
贪婪算法用于路径优化，去除路径上的冗余节点并找到最短路径。 A greedy algorithm is used for path optimization, removing redundant nodes on the path and finding the shortest path.