## 4 Simulation Experiments
### Total Torque Error Analysis  
To evaluate the accuracy of the identification model under vibratory conditions during motion, the total torque error is analyzed by comparing the true and predicted torques, with a focus on performance under vibration-induced excitation.  
Figure 9 presents the total torque error curves for each joint. Figure 9a shows the prediction error for Joint 1. The error curve exhibits a generally random distribution, with a maximum deviation of less than 1.5 \(\times\) 10\({}^{-12}\) N\(\cdot\)m. Most of the error is concentrated around the start and end phases (0-1 s and 3-4 s), which correspond to dynamic transitions during manipulator start-up and stopping. In the steady-state motion phase (around 8 s), the error amplitude significantly decreases and approaches zero.  
Figure 9b shows the identification error for Joint 2, with a maximum magnitude of approximately 4 \(\times\) 10\({}^{-12}\) N\(\cdot\)m. The error peaks are clearly located near the velocity reversal nodes (around 3 s and 6 s), corresponding to the nonlinear friction-inertia dynamic coupling points previously identified in the analysis. During the intermediate steady motion phase (4-5 s), the error fluctuations are minimal, with absolute values below 1 \(\times\) 10\({}^{-12}\) N\(\cdot\)m.  
The total torque error curve for Joint 3, shown in Figure 9c, exhibits a certain degree of complexity and high-frequency oscillatory characteristics. The overall maximum error is approximately \(-\)2.5 \(\times\) 10\({}^{-12}\) N\(\cdot\)m, primarily concentrated in the time intervals of 3-4 s and 6-7 s.  
These errors are associated with the frequent speed changes and strongly asymmetric frictional coupling characteristics of Joint 3. Nonetheless, the constructed model successfully keeps the error within a reasonable range, with the overall mean error remaining at a low and stable level.  
The error curve for Joint 4 is shown in Figure 9d. The fluctuation amplitude is extremely low, remaining almost entirely within the range of \(\pm\)4 \(\times\) 10\({}^{-12}\) N\(\cdot\)m throughout the motion. This joint is subject to relatively minimal nonlinear disturbances during the system identification process. Such performance is crucial for precision vibration suppression control strategies, as it effectively reduces the negative impact of micro-vibrations on end-effector positioning accuracy.  
Figure 9e presents the total torque error curve for Joint 5, which exhibits typical high-frequency random disturbance characteristics. The maximum error amplitude is approximately 6 \(\times\) 10\({}^{-12}\) N\(\cdot\)m, mainly concentrated in the frequently varying velocity regions between 2-4 s and 6-7 s.  
As discussed earlier, these errors correspond to the high-frequency components of friction torque coupled with nonlinear disturbances from inertial load variation. Overall, the model keeps the error within a relatively small range and demonstrates strong adaptability to high-frequency disturbances and load fluctuations.  
Figure 9: Total torque error. (**a**) Joint 1; (**b**) Joint 2; (**c**) Joint 3; (**d**) Joint 4; (**e**) Joint 5; (**f**) Joint 6. Each subfigure shows the torque prediction error for the corresponding joint.  
Figure 9f shows the error curve for Joint 6. Due to the lower inertia of the end-effector degree of freedom, this joint is more sensitive to frictional noise, resulting in slightly larger errors compared to the previous joints. The maximum error amplitude is around \(\pm 8\times 10^{-12}\) N\(\cdot\)m.  
Nevertheless, the friction modeling network effectively suppresses the influence of high-frequency noise, keeping the overall error stable and within a controlled range. These results further confirm the model's capability for high-precision, small-disturbance capture, making it suitable for robotic manipulator micro-vibration suppression scenarios.