# A Novel Virtual Sensor for Estimating Robot Joint Total Friction Based on Total Momentum
## 3 Simulation and Analysis
### Payload Experiment  
#### 4.3.1 Motor Low-Velocity Experiment  
The desired position trajectory was 24 \(\pi\)sin (0.6\(\pi\)t) in motor low-velocity experiments with payload, and its first derivative was the desired velocity. The experiment time was extended to 20 s to see whether accumulative errors of the observer exist. The \(K_{\mu}\) was tuned to 80 by using the tuning method described in Section 4.1. The quality of the link payload as shown in Figure 13 is m = 0.55 kg, the distance of link COM is \(r_{c}\) = 0.18 m, and the moment of inertia of the link relative to the Z-axis at the COM is I = \(6.1\times 10^{-3}\) kg\(\cdot\)m\({}^{2}\), which was measured in SolidWorks.  
The observed friction torque (red curve) is shown in Figure 14a. It can be seen that the changes of observed friction torque are stable and there is no obvious accumulative errors with the increase of time. However, larger model uncertainties and more measurement noise might result in more obvious accumulative errors because of the integral item in the proposed virtual sensor as time goes by. In this condition, more accurate dynamics model and filtering processing are needed. In addition, it should be explained here that all the initial peak errors occurring from 0 s to 1 s in the figures below are caused by dynamic tuning and tracking of the controller, and it is more reasonable to only consider the static errors after 1 s. After the controller compensates for the observed friction torque, the RMSE of motor trajectory tracking static errors are obtained. These values are listed in Table 6 and shown in Figure 14b,c. The figures and table suggest that the position tracking performance of the controller  
Figure 12: Motor high-velocity without payload experiment results. (**a**) Observed total friction torque mapped to the motor side at high velocities. (**b**) Normalized actual motor velocity (NV) and normalized friction torque (NF). (**c**) Motor position tracking errors at high velocities. (**d**) Motor velocity tracking errors at high velocities.  
with observed friction compensation did not obviously improve compared with the case without observed friction compensation. However, the velocity tracking performance of the controller with observed friction compensation considerably improved compared with the case without observed friction compensation, and the RMSE of static velocity tracking error decreased by 9.8%. The reason for this is that the controller position tracking performance was not significantly improved, but the controller velocity tracking performance was notably improved, after compensating for the observed friction torque, as has been explained in Section 4.2.1.  
Figure 14: Motor low-velocity with payload experiment results. (**a**) Observed total friction torque (red curve) obtained by the proposed method mapped to the motor-side and estimated motor-side friction torque by the Coulomb and viscous friction model at low velocities. (**b**) Motor position tracking errors at low velocities. (**c**) Motor velocity tracking errors at low velocities.  
Figure 13: The motor with link payload after the gearbox.