## 3 Results
### B. Results with the 3-DOF Simulated Model with a Reduction Ratio  
Figure 4B presents the learning curves for the two architectures with a reduction ratio R of 0.01. This time, RMSE is under 0.05 radians--3 degrees--for all three motors. As observed in Figure 4A, the INN method reaches its optimal performance with 100 trajectories vs. 250 for the GNN method. Figure 5 presents the average relative performance of both methods, with the reference PD controller as a function of the ratio between the inertia \(\mathbf{I}_{m}\) of the motor rotor and the mass matrix of the system (\(\mathbf{RMR}\)). The results indicate similar performances for both methods for a ratio above 10 with a relative RMSE of 0.15 compared to the PD controller and a progressive deterioration of performances of the INN method when the ratio decreases. For a ratio of 0.2--obtained when there is no reduction ratio in the motor--the GNN method demonstrates a relative RMSE of 0.22 compared to 0.56 for the PD controller, i.e., 2.5 times less. Figure 6 shows the relative performances of both methods compared to the reference PD controller depending on its accuracy--or parameters--for two ratios between the frequency of the dynamic integration and the frequency of the feedforward NN compensation to construct an optimal dataset. When the step time corresponds to the interval between two commands from the neural network, the RMSE of the NN is nearly proportional to that of the PD controller, showing a reduction of 90% of its tracking errors. When the frequency of the dynamic integration is ten times higher than the frequency of the NN, the reduction in tracking errors compared to the PD controller is between 80% and 85%, resulting in RMSE values that are two times more important than when the timestamps are the same. Moreover, when the reference PD is more aggressive--i.e., manages to track the trajectories with an RMSE less than 3 degrees--we can see that the performances of the NN are no longer proportional to that of the PD with an RMSE remaining around 0.5 degrees when the two frequencies differ. Figure 6: Relative performances of the NN-based control structure compared to the reference PD controller as a function of the accuracy of the PD controller and the step time of the direct dynamic integration. To design an optimal neural network, the median relative RMSE was compared for different sizes of NNs for both INN and GNN methods tested under various PID parameters. The results are presented in Table 3. All tested architectures, which have at least as many neurons as the number of associated motors on each layer, achieved a median relative accuracy between 0.935 and 1.063 compared to the smallest neural network, representing less than a 10% difference in accuracy. On the contrary, when the number of neurons per layer was fewer than the number of associated motors, the NN failed to converge. This is true for the NN with only 1 neuron in the GNN method, which had an RMSE 24.26 times greater than other NN configurations. Figure 7 shows the relative performances of both methods compared to the reference PD controller depending on the noise present in the training data. In the absence of noise, both INN and GNN methods achieved a 90% improvement in trajectory tracking over the reference PD controller, with an RMS error of 0.86 degrees. A gradual decrease in the accuracy of the system is observed for noise between 0 and 0.5 degrees and is also observed for both INN and GNN methods. Moreover, when the noise exceeds 0.5 degrees, the RMSE becomes 1.8 times larger than in the absence of noise in the data. However, the RMSE of the NN remains relatively stable--ranging between 1.40 and 1.65 degrees--which is still five times less than the RMSE of the PD controller. Note that the noise in this figure is quite high; for instance, the noise in the real systems in this study is estimated to be 0.2 degrees (one encoder step).