## 4 Original ID-Based and NDO-Based Methods
[34], the NDO gain matrix \(\mathbf{L}(\mathbf{q},\dot{\mathbf{q}})\) can be computed taking into account that it depends only on \(\mathbf{q}\) as  
\[\mathbf{L}(\mathbf{q})=\mathbf{X}^{-1}\hat{\mathbf{M}}^{-1}(\mathbf{q}) \tag{7}\]  
where \(\mathbf{X}\) is an \(n\times n\) invertible and constant matrix to be determined. Additionally, remember that the robot's inertia matrix is symmetric and positive definite and, thus, invertible. Then, by substituting (7) into (6), we obtain \(\mathbf{p}\) in terms of \(\dot{\mathbf{q}}\), as shown below in the disturbance observer auxiliary vector:  
\[\mathbf{p}(\dot{\mathbf{q}})=\mathbf{X}^{-1}\dot{\mathbf{q}}. \tag{8}\]  
[PERSON] et al. [12] evaluated five methods for human-robot interaction torque estimation. These methods ranged from relying only on commanded actuation torques, i.e., motor currents, to techniques that use the whole dynamics of the system, such as the ID-based method and the NDO-based method. The ID-based method obtained 20 to 22% average error, while the NDO-based method obtained 12 to 18% average error when using accurate model parameters. Both methods obtained better results than the other three methods evaluated, which got more than 25% average error. Moreover, they analyzed the sensibility of the ID and NDO methods to inaccuracies in model parameter estimations. This analysis shows considerable sensitivity of both approaches to model parameter variations. Moreover, they evidenced the behavior of both estimation methods with sensor noise. The noise in the estimations with the NDO method was smaller than the one obtained with the ID method. Therefore, one of the main differences between both methods is noise reduction. In addition, the NDO method is a filtered version since the nature of the observer can work without double derivates to compute the acceleration. To sum up, [PERSON] et al. [12] demonstrated how these systems work and state that their method could adequately estimate human-robot interaction. However, they also report that both methods require an accurate system model. Therefore, we wanted to take advantage of the fact that there was a benchmark to compare their study with our extended methods. Consequently, we based our approaches on their methods and used them for lower limbs. Moreover, we used more degrees of freedom and reduced the sensitivity to BSIPs uncertainties. In addition, they reported only the effects of constant offsets added to the elements of model matrices dynamic matrices, not the BSIPs by themselves, which is a more realistic approach. Finally, they only tested the sensitivity of the methods to variations in the parameter mass of the second limb, not all BSIPs with uncertainties at once. For those above, we selected these two approaches, i.e., the ID and NDO method reported by et al. [12]. [PERSON] et al. [12] evaluated the behavior of their ID-based method (2) and NDO-based method (6) in the presence of sensor noise and uncertainties. They reported that both methods are suitable to estimate the human-robot interaction torque. However, both approaches failed to estimate the human-robot interaction torque \(\mathbf{\hat{r}}_{int}\) when there are uncertainties. Therefore, this paper extends the ID-based and NDO-based methods to cope with BSIPs uncertainties.