## 4 Simulation Experiments
### Friction Torque Simulation Experiment  
To suppress vibration-inducing factors during the modeling stage, a friction disturbance residual modeling experiment was designed as part of the parameter identification process. The goal was to further extract and characterize the dynamic contribution of nonlinear friction to the total joint torque. Since vibrations in actual robotic manipulators are largely caused by friction and structural nonlinearities, random noise terms were introduced in the simulation to approximate external disturbances, thereby reflecting the dynamic effects of vibration on the manipulator. Although this setup cannot fully replicate physical vibrations, it effectively tests the stability and accuracy of the identification model. Three curves were analyzed in the experiment:  
The black curve represents the true friction torque, serving as the benchmark of the ideal model and used to evaluate the accuracy of BLS predictions. The red curve represents simulated  
\begin{table}
\begin{tabular}{c c c c c c c} \hline \hline
**Joint Parameters** & **Joint 1** & **Joint 2** & **Joint 3** & **Joint 3** & **Joint 5** & **Joint 6** \\ \hline \(q_{i,0}\) & \(-0.1738\) & \(-0.0922\) & \(0.0631\) & \(-0.6137\) & \(0.4425\) & \(-0.0799\) \\ \(a_{i,1}\) & \(-0.7341\) & \(-0.0259\)  
sensor measurements, obtained by adding high-frequency vibration and white noise to the true friction torque, which reflects the fluctuations commonly observed in real-world sensor signals. The blue curve represents the friction torque estimated by the BLS from noisy observations, aiming to approximate the true model through nonlinear feature mapping. To better illustrate the prediction performance of the BLS model under noise interference, a local segment of the signal was magnified in Figure 6. The enlarged view clearly shows that, despite the presence of measurement noise, the BLS-predicted signal (blue curve) closely follows the true friction curve (black curve), demonstrating strong denoising capability and accurate fitting in transient regions. The experimental results are shown in Figure 6. For each joint of the six-degree-of-freedom manipulator, the true friction torque and the predicted friction disturbance  
Figure 6: Friction torque prediction for robotic manipulator joints. (**a**) Joint 1; (**b**) Joint 2; (**c**) Joint 3; (**d**) Joint 4; (**e**) Joint 5; (**f**) Joint 6. Each subfigure shows the comparison between the true friction torque, noisy measurements, and BLS model predictions. output by the BLS network were extracted and compared. This analysis verifies whether the BLS network can effectively filter out high-frequency noise and extract meaningful friction torque under actual control conditions, while also evaluating whether the prediction accuracy of the friction torque meets the requirements of the control system. Figure 7a illustrates the torque prediction error for Joint 1. The error remains within \(\pm\)0.008 N\(\cdot\)m, with slight asymmetry. A peak deviation of +0.05 N\(\cdot\)m occurs at the acceleration transition (t \(\approx\) 2.1 s), while a negative peak of -0.06 N\(\cdot\)m appears near the velocity reversal (t \(\approx\) 8.7 s). During the steady motion phase (t = 4-7 s), the error stabilizes. These results confirm the BLS model's ability to maintain continuity and dynamic responsiveness at motion inflection points. Figure 7: Friction torque error. (**a**) Joint 1; (**b**) Joint 2; (**c**) Joint 3; (**d**) Joint 4; (**e**) Joint 5; (**f**) Joint 6. Each subfigure shows the prediction error of friction torque for the corresponding joint. Figure 7b illustrates the error pattern for Joint 2, showing periodic oscillations of \(\pm\)0.018 N-m around t \(\approx\) 3.5 s. The frequency corresponds to the second-order bending mode of the manipulator, indicating flexible coupling interference. Despite this, the BLS network constrains residual vibration energy effectively, demonstrating its suitability for flexible joint modeling and dynamic decoupling.