## 4 Results and Discussion  
### End-Effector Force Estimation  
The first experiment was performed in order to train neural networks and directly compare the performance of the proposed system using strain gauges with the performance of the system when a 6-axis force sensor is used.  
It seems that the performance of the two is similar, due to the fact that the reported validation and test set metrics (root-mean-squared-error--RMSE) reported similar values. The values are summarised in Table 2 and show that the system with strain gauges performs marginally better than the one with the force sensor. This is somewhat surprising, given the difference in price/accuracy between the 6-axis force-torque sensor and our custom-built platform. However, the results might be explained by the earlier note about the mounting configuration. Nevertheless, it demonstrates the validity of the proposed strain gauge-based approach. Showcase examples of the obtained force estimates using both methods are shown in Figure 4 for randomly chosen measurement runs. From the figures, it can be concluded that the performance is also visually similar, i.e., both approaches follow the reference force reasonably well.  
\begin{table}
\begin{tabular}{l l} \hline
**Hyperparameter** & **Possible Values** \\ \hline Activation function & ReLU, ELU, Tanh \\ Number of LSTM layers & 1–3 \\ Number of cells per LSTM layer & 8–64, step 8 \\ Number of FC layers & 2–4 \\ Number of neurons per FC layer & 8–64, step 8 \\ \hline \end{tabular}
\end{table}
Table 1: Hyperparameters search space.  
The LSTM networks that were obtained with hyperparameter optimisation were architecturally similar: both had a single LSTM layer and two fully connected layers but differed in the number of units per layer. Therefore, the network trained with the 6D force-torque sensor data had an LSTM layer with just 8 processing units, followed by two fully connected layers with 32 units each, as shown in Figure 5). On the other hand, the network trained with the strain gauge data had an LSTM layer with 24 units, followed by two fully connected layers with 16 units each, as shown in Figure 6. This suggests that, in shear numbers (and interpreted naively), the neural network for strain gauge-based measurements had 8 units less than its force-based counterpart. Both networks used the exponential linear unit (ELU) as an activation function.  
\begin{table}
\begin{tabular}{l l l l} \hline \hline \# & **System** & **Validation RMSE** & **Test RMSE** \\ \hline
1. & Force sensor & 2.1590 N & 2.0043 N \\
2. & Strain gauges & **1.9226** N & **1.9447** N \\ \hline \hline \end{tabular}
\end{table}
Table 2: Trained neural networks performance for end-effector force estimation.  
Figure 4: Examples of the end-effector force estimation performance of the two systems using a real-world UR5e robotic manipulator.  
It might be concluded from the obtained results in this experiment that the usage of a custom-made force platform based on strain gauges does not negatively impact the performance of the system. Contrary to that, it even slightly improves the performance (with a smaller network size), given that the price of the strain gauges is negligible compared to the price of the 3-axis force sensors. Also, when compared to the force sensing accuracy of 2.1 N RMSE obtained during human-robot interaction in [42], our results demonstrate the practical applicability of the approach.