## 4 Discussion
\begin{table}
\begin{tabular}{c c c c c c c c c c c} \hline \hline
**Profile** & **H\({}_{\text{P}}\) (Steps)** & **Q** & **R\({}_{\text{u}}\)** & **R\({}_{\text{Au}}\)** & **CF** & **Ref** & **TI (s)** & **TID (\%)** & **FI (Ns)** & **FID (\%)** & **FI CV * 100** \\ \hline P1 & 10 & 3.8 & 0 & 10 & 1.14 & R1 & 0.256 & \(-\)2.4 & 8.35 & 33.17 & 7.14 \\ (cyan) & & & & & 1.45 & R2 & 0.077 & \(-\)2.7 & 3.08 & 17.92 & 2.66 \\ P4 & 5 & 3.8 & 0 & 10 & 1 & R1 & 0.252 & \(-\)0.8 & 10.59 & 15.29 & 0.46 \\ (black) & & & & & 1.45 & R2 & 0.077 & \(-\)2.7 & 3.83 & \(-\)2.19 & 0.50 \\ P5 & 5 & 1.5 & 0 & 10 & 1.23 & R1 & 0.254 & \(-\)1.6 & 9.36 & 25.10 & 0.34 \\ (orange) & & & & & 2.29 & R2 & 0.078 & \(-\)4 & 2.93 & 21.87 & 0.63 \\ P6 & 5 & 6 & 0 & 10 & 1 & R1 & 0.255 & \(-\)2 & 11.01 & 11.93 & 0.88 \\ (violet) & & & & & 1 & R2 & 0.077 & \(-\)2.7 & 3.03 & 19.12 & 1.47 \\ P7 & 5 & 3.8 & 1 & 10 & 1 & R1 & 0.252 & \(-\)0.8 & 6.69 & 46.47 & 0.55 \\ (blue) & & & & & 1.45 & R2 & 0.076 & \(-\)1.3 & 2.72 & 27.35 & 1.25 \\ \hline \hline \end{tabular}
\end{table} 
Table 5: HIL MPC tuned parameters and the resulting performance indices. The colors refer to the corresponding curves presented in Figures 5 and 6.  
the control input to be instantly issued to the plant and, thus, acts accordingly. Moreover, in a real human-machine interaction, it could be even more difficult to retrieve unknown data related to the physical characteristics of the subject, which could be highly specific and impractical to measure. However, the outcomes of this study signal that, in order to achieve a better matching between the MIL and HIL results, a refinement of the analytical model is necessary to improve the performance of the predictive algorithm. For example, detailed modeling of the actuator's dynamics could be necessary to overcome part of the limitations observed in the current architecture of the control system, especially for short-lasting perturbations. In addition, a more refined model of the plant could help in the identification of the optimal control parameters, that have been obtained by iterative trial and error approach in the current study. With respect to the model simulations, experimental trials showed lower tracking accuracy, as confirmed by the significantly higher force impulse deviations collected in Table 5. Among the several selections of control parameters, the one corresponding to the P4 profile (see Figures 5 and 6) showed the best results in terms of impulse accuracy. In such configuration, the FID observed for the short-lasting R2 reference profile was even lower than the one measured for the R1 profile, however, it was not possible to assess the same result regarding the tracking accuracy. As shown in Figure 6, the oscillations in the force profile were particularly substantial for the 75 ms reference profile. A similar result was highlighted for the other selections of control parameters (see Figure 5 and Table 5). A large overshoot in the contact force profile could be potentially dangerous for the subject, therefore it must be avoided in a final embodiment of the perturbation system. The CF was tuned to achieve a reasonable compromise between the accuracy of the force impulse and the steadiness of the control but, still, it was not possible to remove the undesired overshoot without affecting the general performance of the perturbation device.