## 4 Discussion  
The first main result is highlighted by the simulated model without reduction ratio in Figure 4A. The INN method is effective solely when the joint dynamics are not coupled with the other joints' dynamics--i.e., the impact of the joint movements on the dynamics of the other joints is negligible. In fact, the INN method performs well for the last motor, with a 91% reduction in tracking errors. This is because the third motor at the end of the arm is not affected by the movements of the first two motors. Contrarily, the INN method does not outperform the PD controller for the first two motors, with an RMSE of 0.2 radians similar to that of the PD controller. The second main result is highlighted in Figure 4B: the INN method outperforms the GNN method when the dynamics of each joint are not coupled with the other ones and when the training dataset comprises fewer than 250 trajectories. This can be explained by the lower number of parameters required for the INN method. Moreover, both methods present a similar accuracy when more training data are available, with RMSE values ranging from 0.034 and 0.048 for the INN method and from 0.035 and 0.049 for the GNN method. These first two results align with those in [4], where the GNN method was applied to a flexible Baxter robot because of the dependence on the joint dynamics and the statement that it would be possible to use the INN method on an ABB industrial robot because of the decoupled dynamics [4,5]. The decoupling of the joint dynamics occurs when the term RMR in Equation (2) becomes negligible compared to the inertia of the servomotor \(\mathbf{J}_{m}\). This means that the influence of the mass matrix on the system is negligible compared to the inertia of the servomotor due to the reduction ratio. This result is consistent with the equations presented in [29]. Specifically, Figure 5 shows that this decoupling occurs when the inertia of the rotor of the motor \(\mathbf{J}_{m}\) is more than 10 times larger than the term _RMR_ in Equation (2). In this case, the INN method requires less data than the GNN method to complete the learning process: 100 trajectories vs. 250 trajectories for a 3-DOF robot arm. Moreover, the more complex the controlled robot arm, the more trajectories required for the training of the NN in the GNN method. To our knowledge, no study has explicitly demonstrated this result. Figure 3 also shows that the more trajectories available for training, the more accurate will be the control structure. Notably, the amount of data required for the training is 50 times less than the 500 trajectories of 25 s used in [4]. These ones likely used more trajectories than necessary to ensure an optimal training process of their RNN. no more than 10 times larger than _RMR_ in Equation (2). In this case, the GNN method provides better accuracy. For example, when the reduction ratio is 1, the RMSE is 2.5 times smaller than the INN method. In fact, the global NN in the GNN method can learn the dependencies between the different joints and maintain accuracy close to that achieved with decoupled dynamics. To our knowledge, no study has explicitly compared the INN and GNN approaches in systems with coupled dynamics. To summarize, these first results provide guidelines for both designing the neural network system and its training: if the ratio is greater than 10, a small neural network per servomotor can be used effectively. If the ratio is less than 10, a larger neural network for the entire system will yield better accuracy. Moreover, when the \(\mathbf{J_{m}}/(\mathbf{RMR})\) ratio is between 10 and 100, the accuracy of the GNN method might be slightly better than the accuracy of the INN method. Thus, if the system accuracy is crucial, it may be beneficial to use both methods to see if the improvement in accuracy justifies the use of the GNN method. Further, Figure 6 demonstrates that to achieve the best possible accuracy for both the INN and GNN methods, the step time of the training data must match the PD frequency used during the dataset construction.