## 3 Experiment and Result
### Singular Point Avoidance Simulation Experiment  
To validate the effectiveness of the improved FMT* algorithm in avoiding singular points, experiments were conducted using a 6-degree-of-freedom (6-DOF) robotic arm model in MATLAB (version R2023b). Firstly, the initial and final poses were set in the workspace of the robotic arm, and a straight-line trajectory was planned. The start and end points were (0.5, 0.5, 1.2) and (\(-\)0.5, \(-\)0.5, 1.2), respectively. This trajectory passes through a singular point at the shoulder of the robotic arm, with coordinates (0.18, 0.18, 1.2). The MATLAB simulation system framework is shown in Figure 9.  
As shown in Figure 10a, it can be observed that without singular point avoidance, joint 1, joint 4, joint 5, and joint 6 undergo significant changes in joint angles after passing through \begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Algorithm** & **Improved FMT* & **Standard FMT* & **RRT* Smart** & **Standard RRT* & **Standard RRT-Connect** & **Informed RRT* \\ \hline \multirow{8}{*}{Parameters} & Initial sample & & & & & \\  & point: 1000 & & & & & \\  & Secondary samp & Sample point: & Sample point & 0.3 (m) & 0.3 (m) & 1000 \\  & point: 500 & 1000 & 1000 & Maximum & Maximum & Connection \\  & radius: 0.3 (m) & radius: 0.3 (m) & radius: 0.3 (m) & iteration count: & iteration count: & radius: 0.3 (m) \\  & Secondary & & & & 4000 & 4000 \\  & connection radius: & & & & & \\  & 0.15 (m) & & & & & \\ \hline \multirow{2}{*}{\begin{tabular}{c} Average time \\ consumption \\ \end{tabular} } & 1.67 (s) & 1.58 (s) & 1.72 (s) & 2.21 (s) & 0.98 (s) & 1.64 (s) \\ \hline \multirow{2}{*}{\begin{tabular}{c} Time variance \\ \end{tabular} } & 0.16 (s\({}^{2}\)) & 0.11 (s\({}^{2}\)) & 0.13 (s) & 0.38 (s\({}^{2}\)) & 0.09 (s\({}^{2}\)) & 0.21 (s\({}^{2}\)) \\ \hline \multirow{2}{*}{\
\begin{tabular}{c} Average path length \\ \end{tabular} } & 18.8 (m) & 21.7 (m) & 22.1 (m) & 23.6 (m) & 25.2 (m) & 20.8 (m) \\ \hline \multirow{2}{*}{
\begin{tabular}{c} Path length variance \\ \end{tabular} } & 1.12 (m\({}^{2}\)) & 1.20 (m\({}^{2}\)) & 1.18 (m\({}^{2}\)) & 1.14 (m\({}^{2}\)) & 1.39 (m\({}^{2}\)) & 1.15 (m\({}^{2}\)) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: Experimental results of obstacle avoidance path planning under different algorithms. Figure 9: Simulation experiment Simulink system diagram. the singular point. As illustrated in Figure 9(b), by using the singular point avoidance trajectory planning algorithm proposed in this paper, the end effector of the robotic arm successfully avoids the singular point. This ensures that the joint configuration of the robotic arm remains closer to the initial configuration, avoiding drastic changes in joint angles and ensuring the stability and safety of the robotic arm's operation. Applying the proposed algorithm for singular point avoidance, the trajectory planning results for the given singular point are depicted in Figure 11. The red dashed line represents experimental group 1, which did not utilize the singular point avoidance algorithm, while the blue solid line represents experimental group 2, which employed the singular point avoidance algorithm. Since the cost function of this algorithm depends on the singularity and path length, the planned trajectory using the algorithm avoids the singular point and maintains the shortest path in the non-singular region. The variation curves of joint angles and angular velocities for the robotic arm in the simulation experiment are shown in Figure 12. The red dashed line represents experimental group 1, while the blue solid line represents experimental group 2. In Figure 11(a), the joint angle curves can be observed, and it is evident that the joint curves in experimental group 2 are smoother compared to those in experimental group 1, especially in joint 1. In experimental group 1, joint 1 rotates by 3.1 rad, which is equivalent to 177\({}^{\circ}\), while in experimental group 2, joint 1 rotates by 0.3 rad, which is equivalent to 17.2\({}^{\circ}\).