# A Novel Virtual Sensor for Estimating Robot Joint Total Friction Based on Total Momentum
## 3 Simulation and Analysis
### Velocity Step Experiment  
#### 4.1.1 Motor Low-Velocity Experiment  
The desired low-velocity step sent to the controller was 346 rad/s. The experimental results are shown in Figures 6-8. Figure 6a shows that the velocity step response time required to reach the desired velocity is different for different \(K_{\mu}\). When there is no friction compensation, the step response is the slowest and the time is 0.051 s. When \(K_{\mu}\) is 50, the step response is the fastest (0.041 s) and the step response performance improves by 19.6%. Although the velocity step response performance can also be improved with no friction compensation when \(K_{\mu}\) is 10 or 100, it is not better than that when \(K_{\mu}\)  
Figure 5: The single joint experiment system. The developed driver consisted of a TI DRV8301-HC-EVM board attached to a microcontroller unit TMS320F28035.  
\begin{table}
\begin{tabular}{c c} \hline \hline
**Parameters** & **Values** \\ \hline Torque constant & \(2.76\times 10^{-2}\) N\(\cdot\)m/A \\ Rotor inertia & \(3.33\times 10^{-6}\) kg\(\cdot\)m\({}^{2}\) \\ Gearbox equivalent inertia & \(9.4\times 10^{-7}\) kg\(\cdot\)m\({}^{2}\) \\ Gearbox radio & 81 \\ \hline \hline \end{tabular}
\end{table}
Table 3: Motor and gearbox parameters.  
is 50. Therefore, the system can display a better velocity step response performance when an observed friction disturbance is compensated for by the controller. In addition, the observed friction disturbance is more reasonable when \(K_{\mu}\) is 50 because the controller displays a better velocity step response ability than at other \(K_{\mu}\) values. However, we should not fully determine the appropriate gain coefficient by only considering this one result. Therefore, next, we analyzed the actual motor torque curve and observed the total friction disturbance curve to confirm the choice of \(K_{\mu}\). It should be noted that all the actual step velocities stabilized after reaching the desired velocity. More data are listed in Table 4.  
Figure 8: Observed total friction torque mapped to the motor side at low velocities.  
Figure 6: Motor low- and high-velocity step experiments with friction compensation for different gain coefficients (\(K_{\mu}\)). NC denotes no friction compensation. (**a**) Low-velocity experiment. (**b**) High-velocity experiment.  
Figure 7: Actual motor torque at low velocities.  
As shown in Figure 7, large values of \(K_{\mu}\) result in large motor peak torques. When \(K_{\mu}\) is 100, the motor torque reaches the largest peak torque of 0.1 N\(\cdot\)m at time 0.003 s and \(-\)0.1 N\(\cdot\)m at time 0.206 s, respectively. As shown in Figure 8, large values of \(K_{\mu}\) generate more observed noise and large observed friction torques, which increase the motor peak torque after compensation by the controller. However, the safe torque limit of the motor is set to \(\pm\) 0.1 N\(\cdot\)m, so \(K_{\mu}\) cannot be set to 100. Therefore, we conclude that the best \(K_{\mu}\) is 50 among these three values. It should be noted that we did not find the best \(K_{\mu}\) from 10 to 100 here, but just investigated how to determine the appropriate \(K_{\mu}\) and provided the method for selecting this value. The two observed friction torque mutations in Figure 8 occur because the positive motor acceleration generates a negative friction torque from 0 s to 0.025 s and negative motor deceleration generates a positive friction torque from approximately 0.2 s to 0.22 s. This phenomenon is the same as that illustrated in the actual motor torque curve and is reasonable. It should be noted that when the motor velocity decreases to zero, the observed friction torque does not reach zero and ranges from \(-\)0.0058 to 0.0096 N m when \(K_{\mu}\) is 50 because static friction exists in the single joint system.