## 7 Conclusions  
We proposed the Identified ID-based method (IID) and the Identified NDO-based method (INDO). Both methods estimate the human-robot interaction torque for a simulated Lower Limb Rehabilitation Robot (LLRR), which is usable for isotonic exercises despite the uncertainties of the Body Segment Inertial Parameter (BSIPs). Both methods have two phases. (1) Phase one is the calibration phase, which allows an online parameter estimation to reduce BSIPs uncertainties. (2) Phase two is the estimation phase, where the human-robot interaction torque uses the estimated parameters under the assumption that the BSIPs uncertainties have been reduced. The first method combines a transformed Inverse Dynamics (ID), a friction model, and an online parameter estimation method. The second method uses the conjunction of a transformed Nonlinear Disturbance Observer (NDO) and a friction model fed by the output of an online parameter estimation method. Two sets of simulations were conducted using white Gaussian noise (WGN) with a signal-to-noise ratio \(\text{SNR}=40\,\text{dB}\) and \(20\%\) BSIPs uncertainties. The two proposed methods' performance was compared to two of the best methods reported in the literature. Both proposed methods obtained the best Coefficient of Correlation, Mean Absolute Error, and Root Mean Squared Error compared to the benchmarks. Moreover, the IID and INDO met the vast majority of the requirements, i.e., accomplishing 72.2% and 88.9%, respectively. In contrast, both methods reported in the literature only accomplish 27.8% and 33.3% of the requirements when using simulations with under noise and BSIPs uncertainties. In addition, the INDO is superior to the IID since it reduces the sensitivity to sensor noise. In conclusion, this paper proposes two human-robot interaction torque estimation methods, which extend two methods reported in the literature and copes with BSIPs uncertainties without using additional sensors. Future work includes a practical realization of proposed observers within a real environment with the actual LLRR Nukawa and conducting a study with healthy subjects and patients. This work requires additional approval from the ethics committee. Moreover, the validation would ideally involve hardware using a torque sensor on each joint to measure the actual interaction torque experimentally. Thus, we would be able to compare this measure with the torque estimation made via the two proposed methods. A challenge for implementation in the physical robot is the existence of static friction, which the methods do not consider. Therefore, the estimated torques may be different from the real ones during the start and stop of the robot. Conceptualization, J.C.Y., S.R., M.O., V.Z.P., J.A.M., A.A.-J. and M.J.B. ; methodology, J.C.Y., S.R., M.O., V.Z.P., J.A.M., A.A.-J. and M.J.B. ; validation, J.C.Y., S.R., M.O., V.Z.P., J.A.M., A.A.-J. and M.J.B. ; formal analysis, J.C.Y., S.R., M.O., V.Z.P., J.A.M. and M.J.B. ; investigation, J.C.Y., S.R., M.O., V.Z.P., J.A.M., A.A.-J. and M.J.B. ; writing--original draft preparation, J.C.Y. and S.R. ; writing--review and editing, J.C.Y., S.R., M.O., V.Z.P., J.A.M., A.A.-J. and M.J.B. ; software, J.C.Y., S.R., M.O., V.Z.P., J.A.M. and M.J.B. ; funding acquisition, J.C.Y., S.R., V.Z.P., J.A.M. and M.J.B. All authors have read and agreed to the published version of the manuscript. **Funding:** This paper was framed under the project \"KINA: Virtual Reality System for Lower Limb Rehabilitation of APM or IED Victims\" funded by the call 808 of 2018 for projects from the _Ministerio de Ciencia, Tecnologia e Innovacion de Colombia_ (Minicncias), project 121080864383. Moreover, the Ph.D. studies of [PERSON] were funded by Micinciencias and the Universidad Pontificia Bolivariana (UPB) under call 727 of 2015. In addition, the Australian Academy of Science (AAS) and Micnicnicas granted the funds for his Ph.D. internship within the Australia-Americas Ph.D. Research Internships Program 2019. Additionally, this work was supported by UNAM-PAPIIT IN 102121.