## 6 Simulation Results
### Estimation Phase Results  
Based on the parameters from Table 2 for the simulation, we can see that  
\[\|\dot{\mathbf{M}}(\mathbf{q})\|\leq\zeta=30 \tag{32}\]  
Finally, with the identity matrix \(\mathbf{I}_{3}\in\mathbb{R}^{3\times 3}\) and (25) would turn into  
\[\left[ \begin{array}{cc}\mathbf{Y}+\mathbf{Y}^{T}-30\mathbf{I}_{3}&\mathbf{Y }^{T}\\ \mathbf{Y}&\mathbf{\Gamma}^{-1}\end{array} \right]\geq 0. \tag{33}\]  
Subsequently, we used the MATLAB(r) Robot Control Toolbox(tm) to solve this LMI-based analysis, obtaining a design for the NDO. The solution with \(\zeta=30\) for (33) was \(\mathbf{Y}=\mathbf{X}=0.0028\mathbf{I}\) and \(\mathbf{\Gamma}=0.9542\times 10^{-3}\mathbf{I}\). Hence, satisfying the LMI. Therefore, we successfully designed the NDO for the human-robot interaction torque estimation phase despite BSIPs uncertainties for our LLRR. For the estimation phase, we used two exercises. The first simulated exercise was the squat, based on the requirements defined by the physiotherapists, see Section 2. The second exercise was a leg press. This exercise presents similar movements to a squat exercise, but in a supine position, i.e., with the subject facing up and their lower limbs elevated. These two exercises were simulated, and a physiotherapist validated the simulation trajectories. Figure 5 presents a simulation of the squat exercise sequence, using an standing initial position with \(\mathbf{q}=(-90,0,90)^{\circ}\) and \(B_{r}=90^{\circ}\). Subsequently, the subject bends the knees to  
Figure 4: LLRR vector parameters error \(\tilde{\mathbf{\chi}}\) during the estimation phase. descend the body toward the floor with the heels on the floor, and the upper body remains aligned in the vertical plane [45]. It is important to note that our LLRR cannot let the subject descend their body toward the floor with their heels on the floor. Therefore, we simulated an \"inverted\" squat exercise, i.e., the subject does not lower their body towards the ground, but their feet rise towards their chest. Then, finally, the subject extends their knees to return to the standing position. Figure 6 presents the \(\mathbf{q}_{d}\) for the hip, knee, and ankle. These movements are equivalent to the squat exercise. Moreover, we ensured that \(\dot{\mathbf{q}}_{d}\) was not more significant than the one a human joint can execute. In addition, we can observe \(\boldsymbol{\tau}_{dym}\), \(\boldsymbol{\tau}_{int}\), \(\boldsymbol{\tau}_{fri}\), and \(\boldsymbol{\tau}_{a}\) during the estimation phase. It is important to note the \(\boldsymbol{\tau}_{int}\) had constant value since we simulated an isotonic exercise. During this simulation, the subject was relaxed during the first 5 s. Subsequently, we simulated a constant torque of \(\boldsymbol{\tau}_{int}=(9.8,9.8,0)\) N m, which is equivalent to the requirements presented in Section 2 suggested by the physiotherapists. The simulation evaluated the response to sensor noise, i.e., the simulation used white Gaussian noise (WGN) with a signal-to-noise ratio \(\text{SNR}=40\) dB and 20% BSIPs uncertainties. We selected a WGN with \(\text{SNR}=40\) dB since it is the one used in the paper reported by [PERSON] et al. [36]. Figure 7 reports the results for the simulation during the squat exercise. This simulation presents the outcome of the four estimation methods with \(\text{SNR}=40\) dB and 20 % BSIPs uncertainties. Figure 7a compares the results for the IID and the [PERSON] et al. ID method. In this figure, we can observe that the IID works when there are BSIPs uncertainties. As expected, sensor noise affects both methods due to the computation of double derivatives to calculate velocity and acceleration. Therefore, our IID extends the technique reported in the literature by [PERSON] et al. [12]. Figure 7b contrasts the simulations for the INDO and the [PERSON] et al. NDO method. In this figure, we can denote that the INDO notably reduces the sensitivity to BSIPs uncertainties. Therefore, our INDO extends the technique reported in the literature by [PERSON] et al. [12].