## 6 Simulation Results
### Requirements Evaluation  
Table 5 presents an evaluation of the performance of the two proposed human-robot interaction torque estimation methods according to the requirements defined in Section 2. These requirements were defined to evaluate the performance of the human-robot torque estimation methods, to be used as biofeedback to favor isotonicity, despite the BSIP's uncertainties, through a simulation. In this table, we can observe that both proposed methods meet the vast majority of the requirements, i.e., accomplishing 16 (88.9%) and 13 (72.2%) of them. In contrast, both methods reported in the literature only accomplish 27.8% and 33.3% of the requirements when using simulations under noise and BSIP's uncertainties. Therefore, our proposed methods may be employed as a human-robot interaction torque estimation method for a simulated lower limb rehabilitation robot. Moreover, it is usable for isotonic exercises, as biofeedback, despite the BSIP's uncertainties. In addition, the INDO is superior to the ID-based method since it reduces the sensitivity to sensor noise. This result is reasonable because of the nature of the observer and because the NDO works without double derivates to compute \(\hat{\mathbf{q}}\). It is noticeable that the proposed IID and INDO behaved appropriately in the area where the methods were working. The good behavior of both methods was intensively verified using stimuli and simulations in the region in which the LLRR works. Let us perform a stability analysis of suggested observers, considering that the IID and INDO are methods comprised of two phases. Both methods have the same phase one, i.e., a calibration phase to reduce BSIP's uncertainties. The calibration phase for the IID and INDO inherits the stability from the online parameter estimation reported by [PERSON] et al. [36]. We have shown that the calibration phase complies with the input trajectory design and the LMI design, e.g., the Genetic Algorithm optimization for the condition number ensured a persistent excitation trajectory a gave us an adequate input for the online parameter estimation. Thus, we obtained a trajectory whose points gave a well-conditioned observation matrix. Therefore, the calibration phase can use the same trajectory regardless of the patient's BSIP's, and ensure that this estimator can converge during the calibration phase. The second phase for the IID is based on a transformed ID. This method and transformation turn into a dynamical equation rather than an observer. Moreover, the stability properties are invariant under base changes. Therefore, this phase does not require a stability analysis. The second phase for the INDO focuses on the convergence of the transformed NDO, i.e., the methods are used one after the other, and no feedback is used within them. Convergence was shown for the simulations, evidencing that it is very likely to work in the region of operation. This condition was preserved by ensuring the requirements for the transformed NDO design by using the LMI stated in Theorems 1 and 2. To sum up, the IID and INDO stability is based on the fact that the convergence of each estimator separately is ensured. Moreover, they are executed one after the other. Therefore, the stability of each phase is maintained. This work's contribution consists of two methods that do not require the physiotherapist to make an exact measurement of the BSIPs of the patient's limbs, but allow the use of approximate values. This contribution is possible since we based our methods on separating the parameter identification phase from the torque estimation phase and both from the controller itself. To do this, the physiotherapist may ask the patient to be in a state