## 4 Results and Discussion
### Joint-Side Torque Estimation  
The goal of this experiment was to train a neural network for joint-side torque estimation and to assess its performance. Training this network may be thought of as learning of the inverse dynamic model of the robot from collected data about robot kinematics and dynamics in the presence of external forces acting on the robot.  
In this experiment, similarly to the previous one, two networks were optimised: one with 6D force-torque sensor measurement, and one with the proposed force-measuring platform. Both of the optimised architectures have the same number of LSTM layers (1 layer), the same number of fully-connected layers (2 layers), and the same activation function (ELU). These were all the same as in the previous experiment. However, this time, the two networks ended equally also in the number of neurons in each layer (LSTM: 32 neurons; fully-connected layers: 8 and 16 neurons), as shown in Figure 7. Thus, the only difference between them is that the one that uses strain gauge readings as input has one more input, since force-torque sensors produce three readings in each time instance, but strain gauges produce four (since there are four of them installed on the platform). The validation and test set RMSE metrics for both networks are reported in Table 3.  
\begin{table}
\begin{tabular}{c c c c} \hline \hline \# & **System** & **Validation RMSE** & **Test RMSE** \\ \hline
1. & Force sensor & 2.9294 Nm & 3.0430 Nm \\
2. & Strain gauges & **2.9137** Nm & **3.0063** Nm \\ \hline \hline \end{tabular}
\end{table}
Table 3: Trained neural networks performance for joint torques estimation  
Figure 5: Neural network for end-effector force estimation based on force sensor.  
Figure 6: Neural network for end-effector force estimation based on multiple strain gauge sensors.  
The obtained metrics again demonstrate marginally better performance when using the newly-proposed system (and, thus, prompting the construction-related note mentioned earlier). Showcase examples may be examined in Figure 8, from which is evident that ground truth data are somewhat noisy, especially the lower joints data. Neural networks mostly managed to filter out the noise and produce fair estimates using the network that used force-torque sensor data as input, and also the one that used strain gauges data as input, but to a lesser degree, especially in joint 4. This might be due to the sensitivity of the 24-bit A/D converter used to acquire data from strain gauges, as strain gauge readings have been fed to the neural networks in their raw form, i.e., they have not been preprocessed in any way (i.e., filtering or calibration).  
Figure 8: _Cont._  
Figure 7: Neural network for joint torque estimation (the same network was obtained in both analysed cases).  
Another issue that should be taken into account when developing neural network-based models, both for the torque and for the force estimation, is the effect of the longtime operation of the mobile manipulator. It is safe to assume that during the robot's lifetime, there will be changes in robot dynamic parameters (due to normal wear-and-tear and possibly by changing defective parts) which will have effects on the accuracy of the learned models or any other model, including the analytical one (thus, highlighting the fact this is not an issue that is inherent only to our approach but present in all other model-based approaches). However, this issue can, in our view, be dealt with by the occasional retraining of the model (something like transfer learning): an approach that requires significantly fewer data points than original neural network training and one that is much simpler than performing full manipulator identification, as needed in other methods/approaches.