## 6 Simulation Results
### Estimation Phase Results
Both NDO-based methods have significantly lower errors than the ID-based method. Moreover, they have a lower sensitivity to sensor noise due to the non-dependence of double derivatives. Finally, the IID and INDO work for a simulated LLRR despite the BSIPs uncertainties. Figure 5: Simulated squat sequence. Table 3 reports the performance indices for the four human-robot interaction torque estimation methods during a squat exercise. In this table, we can denote that the INDO excels over the other three approaches. We can see that the INDO obtains the lowest Mean Absolute Error (MAE) and Mean Absolute Percentage Error (MAPE) for the hip and knee joints. Moreover, it has the lowest Root Mean Squared Error (RMS) and Root Mean Squared Percentage Error (RMSPE) for the same joints and the second lower value for all four indices in the ankle joint. Moreover, it obtains a value closer to one for the coefficient of correlation (R\({}^{2}\)) index, i.e., the desired value. Therefore, the overall performance of the INDO is superior in comparison to the other three methods in cases with BSIPs uncertainties. Figure 8 depicts the leg press exercise sequence, which presents similar movements to a squat exercise but in a supine position, i.e., with the subject facing up and their lower limbs elevated. The leg press exercise was simulated by using an initial position with \(\mathbf{q}=(40,0,90)^{\circ}\) and \(B_{r}=30^{\circ}\). Subsequently, the subject bends the knees to approach their chest, i.e., the subject begins the downward movement by slowly bending their hips and knees until their legs form a \(90^{\circ}\) angle. Finally, the subject extends their knee. Figure 6: LLRR dynamics during a squat exercise. Figure 7: Comparison of (**a**) IID and [PERSON] et al. [12] ID method, and (**b**) INDO and [PERSON] et al. [12] NDO method, during a squat exercise with \(\mathrm{SNR}=40\,\mathrm{dB}\) and \(20\%\) BSIPs uncertainties. Figure 9 presents the \(\mathbf{q}_{d}\) for the hip, knee, and ankle. These movements are equivalent to the leg press exercise. Moreover, we can ensure that \(\mathbf{\dot{q}}_{d}\) was not more significant than the one a human joint can execute. In addition, we can denote \(\boldsymbol{\tau}_{dyn}\), \(\boldsymbol{\tau}_{int}\), \(\boldsymbol{\tau}_{fr}\), and \(\boldsymbol{\tau}_{d}\) during the estimation phase. It is essential to note that the \(\boldsymbol{\tau}_{int}\) was simulated with a constant value since we have used an isotonic exercise. The subject was relaxed during the first \(5\,\mathrm{s}\) and a constant torque of \(\boldsymbol{\tau}_{int}=(9.8,9.8,0)\,\mathrm{N}\,\mathrm{m}\) was executed by the subject from \(5\,\mathrm{s}\) to \(20\,\mathrm{s}\). These torque magnitudes are equivalent to the requirements presented in Section 2 suggested by the physiotherapists. Finally, at \(20\,\mathrm{s}\), the subject stopped performing the interaction torque. Figure 10 reports the results for the simulation during a leg press exercise. This simulation presents the results of the four estimation methods with \(\mathrm{SNR}=40\,\mathrm{dB}\) and \(20\%\) BSIPs uncertainties. Figure 10a compares the results for the IID and the [PERSON] et al. [12] ID method. The IID has significantly lower errors. The proposed method can accurately estimate the human-robot interaction torque despite BSIPs uncertainties. Both methods have a similar sensitivity to sensor noise. Figure 10b contrasts the simulations for the INDO and the [PERSON] et al. [12] NDO method. This figure depicts that the result obtained by the proposed method is superior to the approaches because it extends it by reducing the sensitivity to BSIPSs uncertainties. Moreover, both methods present an estimation with  
Figure 8: Simulated leg press sequence.