## 4 Workspace Simulation  
According to the self-insulating bionic joint design described above and the overall mechanical design of the mechanical arm, a simulation environment of the mechanical arm end workspace is built to verify the working adaptability of the bionic self-insulating mechanical arm in the hidden space environment of the cable tunnel. As shown in Figure 9, an equivalent simulation model of the mechanical arm is built. The main dimensions of the robot arm used in the simulation are set to \(l=65\) cm, \(h=15\) cm. The joint rotation angle \(\alpha\) is limited to between \(110^{\circ}\) and \(-110^{\circ}\). The angle \(\theta_{m2}\) is limited to between \(-60^{\circ}\) and \(60^{\circ}\). The angle \(\theta_{i2}\) is limited to between \(-60^{\circ}\) and \(60^{\circ}\). The angle \(\theta_{b2}\) is limited to between \(-60^{\circ}\) and \(60^{\circ}\). The Monte Carlo method is used in MATLAB R2024b to simulate its workspace. The Monte Carlo method is used to generate joint working angle groups. Its flow chart is shown in Figure 10.  
Figure 8: Rope length extension curve. Figure 7: Schematic diagram of end load force simulation. In the process of randomly generating angle groups within the joint working angle range, the mechanical arm self-collision judgment condition is added to discard the unusable angle groups. The generated usable angle groups are used to generate the space coordinate points of the mechanical arm end through the forward kinematics solution process established in the previous article. The generation of space coordinate points stops when a certain density requirement is reached, and finally a workspace point set is formed. During the generation process, the end working coordinates are calibrated and recorded. The generation process is shown in Figure 11a,b. Figure 10: Monte Carlo method workspace generation. Figure 9: Equivalent simulation model of the mechanical arm workspace. The mechanical arm's three-dimensional workspace exhibits a hemispheroidal configuration, as illustrated by the generated point cloud in Figure 11b. At the same time, this paper generates uniformly distributed joint angle combinations within the range of the robot joint angle limit. The position of the end effector is calculated  
Figure 11: Simulation image of the end-effector workspace. (**a**) Mechanical arm working trajectory; (**b**) workspace point cloud; (**c**) operability assessment results; (**d**) singularity evaluation results; (**e**) cable tunnel hidden space mechanical arm end working trajectory. for each set of joint angles (positive kinematics), and the Yoshikawa operability index is calculated based on the position Jacobian matrix \(J_{pose}\):  
\[\omega=\sqrt{\det(J_{pose})_{pose}^{T}} \tag{27}\]  
This indicator reflects the flexibility of the end effector at the current position. The larger the value, the stronger the operability. The results are presented in the form of a heat map, as shown in Figure 11c. The high operability area is mainly distributed in the middle workspace of the robot arm, mainly corresponding to the situation when the elbow joint is at an intermediate angle and the Jacobian matrix condition number is small. The low operability area appears at the edge of the workspace, especially when each joint is close to the extreme angle, and the robot arm is close to kinematic singularity. Regarding the singularity of the robot arm, this paper performs singular value decomposition (SVD) on the Jacobian matrix and extracts the minimum singular value \(\sigma_{min}\). The threshold \(\sigma_{min}\) < 0.06 is defined as the singular region, and the robot arm is judged to be in a singular configuration, and the rest is the normal region. The analysis results are shown in Figure 11d. Singular region (red point): It appears in two typical configurations, one is the elbow joint straight, and the other is the upper and lower shoulder joints and wrist joints are aligned, that is, when the axis coincides. In the safe area, the robot arm has high motion stability.