## 4 Simulation Experiments
### Joint Total Torque Simulation Experiment  
In vibration control, the dynamic accuracy of joint total torque is directly related to the positional stability of the end-effector. Therefore, to evaluate the ability of the proposed identification model to support control performance at the level of overall dynamic output, a joint total torque comparison simulation experiment was designed.  
Figure 8 presents the comparison between the ideal total torque curves (blue lines) and the BLS-predicted total torque outputs (red lines) for all six joints.  
As shown in Figure 8a, the two curves exhibit a high degree of overall alignment, with near-complete overlap in the main motion region (e.g., from t = 5 s to t = 10 s), indicating that the overall model effectively captures both the inertial and frictional characteristics of the system.  
Minor deviations occur during the start-stop motion phases (around t = 0-2 s and t = 9-10 s), which can be attributed to abrupt coupling effects between system inertia and friction; this behavior falls within the expected scope of a standard identification model.  
Figure 8b shows that the comparison curves for Joint 2 exhibit good overall performance. In particular, during the stable motion phase between 7 and 9 s, the predicted torque closely matches the ideal torque, demonstrating strong predictive capability.  
Near the two prominent velocity reversal points (approximately at t = 4 s and t = 9 s), the predicted torque shows slight lag, with a maximum error of approximately 0.04 N-m. This is caused by the coupling between dynamic inertia variation and the nonstationary nature of friction characteristics. Such behavior is consistent with the inertial delay effects commonly observed in nonlinear dynamic system modeling and falls within a reasonable range.  
In Figure 8c, the total torque comparison for Joint 3 exhibits more complex nonlinear characteristics. The predicted curve (blue line) generally follows the trend of the ideal curve  
Figure 8: Total torque output prediction. (**a**) Joint 1; (**b**) Joint 2; (**c**) Joint 3; (**d**) Joint 4; (**e**) Joint 5; (**f**) Joint 6. Each subfigure shows the predicted total output torque for the corresponding joint.  
(red line), but during several high-frequency motion phases (e.g., t = 7-10 s), noticeable fluctuations appear in the predicted torque.  
These fluctuations are attributed to frequent velocity reversals and load changes in Joint 3, reflecting its complex coupled disturbance dynamics. Despite this, the overall prediction error remains within \(\pm\)0.045 N\(\cdot\)m, highlighting the model's sensitivity to unstructured disturbances.  
In Figure 8d, the torque prediction for Joint 4 appears exceptionally stable. Due to the lower intensity of nonlinear disturbances and the more consistent friction-inertia relationship of this joint, the two curves exhibit a very high degree of alignment. The overall error remains within \(\pm\)0.02 N\(\cdot\)m, with virtually no noticeable fluctuations.  
Figure 8e shows that the total torque comparison for Joint 5 features clear high-frequency disturbance characteristics. The predicted curve generally follows the trend of the ideal curve throughout most of the motion cycle and performs particularly well during high-speed transition phases (e.g., around t = 6 s and t = 9 s). However, slight deviations are observed at certain local points (e.g., t = 3 s), with an amplitude of approximately 0.03 N\(\cdot\)m.  
For Joint 6, as illustrated in Figure 8f, the total torque prediction exhibits frequent disturbances and irregular fluctuations. While the predicted curve shows noticeable stochastic behavior, it closely matches the overall trend of the ideal total torque. The error is evenly distributed, and the maximum deviation remains below 0.02 N\(\cdot\)m, demonstrating strong capability in tracking random disturbances.