## 6 Simulation Results
### Estimation Phase Results
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline \multirow{2}{*}{**Method**} & \multicolumn{5}{c}{**Error (Hip, Knee, Ankle)**} \\ \cline{2-6}  & **MAE** & **MAPE (\%)** & **RMSE** & **RMSF (\%)** & \(\mathbf{R}^{2}\) \\ \hline [PERSON] et al. [12] ID & (73.0, 22.2, 6.67) & (1192, 362, \(\sim\)) & (312, 128, 6.75) & (5105, 2095, \(\sim\)) & (6.64 \(\times\) 10\({}^{-3}\), 0.002, \(\sim\)) \\ IID & (7.9, 159, **0.468**) & (618, 260, \(\sim\)) & (243, 100, **0.691**) & (3973, 1634, \(\sim\)) & (3.74 \(\times\) 10\({}^{-4}\), 0.002, \(\sim\)) \\ [PERSON] et al. [12] NDO & (31.3, 4.99, 6.65) & (512, 814, \(\sim\)) & (33.7, 547, 6.77) & (550, 89.3, \(\sim\)) & (0.047, 0.855, \(\sim\)) \\ INDO & (**0.04, 953**), 0.814 & (**16.9, 15.5, \(\sim\))** & (**1.34, 122**), 1.02) & (**21.8, 19.3, \(\sim\)**) & (**0.931, 0.939, \(\sim\)**) \\ \hline \hline \end{tabular}
\end{table} 
Table 3: Human–robot interaction torque estimation methods performance indices during a squat exercise with \(\mathrm{SNR}=40\,\mathrm{dB}\) and \(20\%\) BSIPs uncertainties. Bold numbers highlight the best metric value. Figure 9: LLRR dynamics during a leg press exercise. the lowest noise since their non-dependence on double derivatives. As a result, the errors obtained with the IID and INDO are significantly lower. Therefore, we can state that our proposed methods stand out for a proper torque estimation of the human-robot interaction torque even with sensor noise and BSIPs uncertainties. Thus, extending the work reported by [PERSON] et al. [12]. Table 4 reports the performance indices for the four human-robot interaction torque estimation methods during a leg press exercise. In this table, we can observe that the INDO excels over the other three approaches. We can see that the INDO obtains the lowest MAE, MAPE, RMS, and RMSPE for all three joints. Moreover, it obtained the closest value to one for the R\({}^{2}\) index, i.e., the desired value. Therefore, the INDO's general performance outweighs in comparison of the other three methods. As mentioned by [PERSON] et al. [12], their ID method is suitable for estimating the interaction torque when there is no noise in the sensors and there are no uncertainties in the parameters. It is expected to be noticeably affected by noise since it uses double derivatives to calculate velocity and acceleration. Additionally, it does not have a correction term like the NDO and only depends on the system's dynamics. In contrast, in Figures 7a and 10a, it can be seen that the IID notably reduces the sensitivity to BSIPs uncertainties. Therefore, the estimation with the IID outperforms when there are BSIPs uncertainties. However, the IID is affected by noisy sensors since the dependence of the doubles derivatives is not removed. As demonstrated by [PERSON] et al. [12], their NDO method is good at estimating the interaction torque when there are no uncertainties in the parameters. Moreover, it is less sensitive to noise in the sensors than the ID. The [PERSON] et al. NDO method is a filtered version of the [PERSON] et al. ID method due to the nature of the observer and its non-dependence on double derivatives, only the first-order one. However, the [PERSON] et al. NDO method is just as sensitive to parameter uncertainties as the [PERSON] et al. ID method [12]. The INDO method notably reduces the sensitivity to BSIPs uncertainties, see Figures 7b and 10b. In addition, the INDO inherits the lower sensitivity to noise from