## 2 Methods  
The trajectory planning and singularity avoidance algorithm for robotic arm obstacle avoidance based on an improved FMT is shown in Figure 1. The system receives inputs \begin{table}
\begin{tabular}{c c c c} \hline \hline
**Reference** & **Authors** & **Problem Addressed** & **Methods Used** \\ \hline
[4] & [PERSON] et al. & Merging RRT\({}^{\star}\) and PRM for obstacle avoidance planning. & FMT\({}^{\star}\) \\ \hline
[5] & [PERSON]. & Enhancing FMT\({}^{\star}\) for dual manipulator obstacle avoidance. & BIT\({}^{\star}\) \\ \hline
[6] & [PERSON]. & Applying bidirectional RRT algorithm to plan collision-free trajectory of robotic arms. & BI-RRT\({}^{\star}\) \\ \hline
[7] & [PERSON] et al. & Apply bidirectional FMT\({}^{\star}\) to handle spacecraft power failure and arm deployment. & BI-FMT\({}^{\star}\) \\ \hline
[8] & [PERSON] et al. & Applying CPSO to solve the problem of singular points in the trajectory of robotic arms. & CPSO with DLS \\ \hline
[9] & [PERSON] et al. & Adjusting robotic arm trajectories based on minimum singular values. & Dynamic DLS \\ \hline
[10] & [PERSON] et al. & Trajectory planning of gantry system using \(B\)-spline and least squares method. & Least squares with \(B\)-splines \\ \hline
[11] & [PERSON] et al. & Triangular collision planes and cost functions for obstacle avoidance planning. & Geometric-based predictable obstacle avoidance \\ \hline
[12] & [PERSON] et al. & Real-time trajectory smoothing via shortcutting for manipulators. & Learning-based approaches \\ \hline
[13] & [PERSON] et al. & Model-free deep learning for UR5 robotic arm path planning. & Deep reinforcement learning \\ \hline
[15] & [PERSON] et al. & Genetic algorithm with singularity avoidance for trajectory optimization. & AEGA-SA \\ \hline
[16] & [PERSON] et al. & Differential vector optimization for smooth, singularity-free trajectory planning. & High-order joint smooth trajectory planning method \\ \hline
[17] & [PERSON] et al. & Custom potential functions for singular configuration avoidance. & Improved artificial potential field method \\ \hline
[18] & [PERSON] et al. & Learning-based control strategy for robotic arm task constraints. & A purely reactive method for maximizing manipulability \\ \hline
[19] & [PERSON] [PERSON] et al. & Planning with maximized manipulability without known constraints. & Learning-based methods combine manipulability index \\ \hline
[20] & [PERSON] & Addressing singularities with dexterity index and trajectory replanning. & Optimization methods using dexterity index \\ \hline
[21] & [PERSON] & Optimizing humanoid robot vision and operation through torso positioning. & Trajectory planning method combined with a dexterity map \\ \hline
[22; 23] & [PERSON] et al. & Introducing geometric perception singularity index for trajectory planning. & Geometry-aware Singularity Avoidance costs \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Problems addressed and methods used by different scholars in literature reviews. of obstacle information in Cartesian space, along with the start and end coordinates of the robotic arm end effector. First, path points are sampled in the robotic arm workspace environment. After initial sampling, initial path generation, and secondary sampling, the sampling phase ends. The next phase is the path expansion phase, where the path points obtained from secondary sampling are connected. Starting from the root node, which is the start point of the robotic arm end effector, an improved cost function that includes singularity avoidance is used to expand the nodes. At each expansion, the path point with the minimum cost within the connection radius is searched. When there are no unconnected path points in the space, the path optimization phase is entered. Each complete path is checked for collisions to prevent collisions between the robotic arm links. Then, the cost of collision-free paths is calculated, and the path with the minimum cost is obtained. Finally, the trajectory is smoothed using \(B\)-spline curves to achieve a smoother motion.