## 4 Original ID-Based and NDO-Based Methods  
The objective of this section is to present the original ID-based and NDO-based methods and their limitations. Let us introduce the formulation of the ID-based method reported by [PERSON] et al. [12]:  
\[ \begin{array}{l}\hat{\mathbf{\tau}}_{d}=\hat{\mathbf{M}}(\mathbf{q})\ddot{ \mathbf{q}}+\hat{\mathbf{C}}(\mathbf{q},\dot{\mathbf{q}})+\hat{\mathbf{G}}( \mathbf{q})-\mathbf{\tau}_{a}\\ \hat{\mathbf{\tau}}_{fri}=F_{v}\dot{\mathbf{q}}\\ \hat{\mathbf{\tau}}_{int}=\hat{\mathbf{\tau}}_{d}-\mathbf{\tau}_{a}-\hat{\mathbf{\tau}}_{fri} \end{array}  \tag{2}\] \begin{table}
\begin{tabular}{c c c c c} \hline \hline \(\mathbf{i}\) & \(\mathbf{\pi}_{i}\) & \(\mathbf{a}_{i}\) & \(\mathbf{d}_{i}\) & \(\mathbf{q}_{i}\) \\ \hline
1 & 0 & \(\ell_{1}\) & 0 & \(q_{1}\) \\
2 & 0 & \(\ell_{2}\) & 0 & \(q_{2}\) \\
3 & 0 & \(\ell_{3}\) & 0 & \(q_{3}\) \\ \hline \hline \end{tabular}
\end{table} 
Table 1: D-H Parameters for our 3-DOF LLRR. Figure 1: LLRR configuration. It is important to note that this ID-based method uses double derivatives to compute \(\bar{\mathbf{q}}\), which significantly increases the sensitivity of the model to sensor noise. Moreover, this method uses the estimates \(\bar{\mathbf{M}}(\mathbf{q})\), \(\hat{\mathbf{C}}(\mathbf{q},\dot{\mathbf{q}})\), and \(\hat{\mathbf{G}}(\mathbf{q})\) of the actual \(\mathbf{M}(\mathbf{q})\), \(\mathbf{C}(\mathbf{q},\dot{\mathbf{q}})\), and \(\mathbf{G}(\mathbf{q})\) dynamic matrices, respectively. Taking into account that  
\[\mathbf{M}(\mathbf{q})=\hat{\mathbf{M}}(\mathbf{q})+\Delta\mathbf{M}(\mathbf{ q}), \tag{3}\]  
\[\mathbf{C}(\mathbf{q},\dot{\mathbf{q}})=\hat{\mathbf{C}}(\mathbf{q},\dot{ \mathbf{q}})+\Delta\mathbf{C}(\mathbf{q},\dot{\mathbf{q}}), \tag{4}\]  
and  
\[\mathbf{G}(\mathbf{q})=\hat{\mathbf{G}}(\mathbf{q})+\Delta\mathbf{G}(\mathbf{ q}) \tag{5}\]  
where \(\Delta\mathbf{M}(\mathbf{q})\), \(\Delta\mathbf{C}(\mathbf{q},\dot{\mathbf{q}})\), and \(\Delta\mathbf{G}(\mathbf{q})\) are deviations of the dynamic matrices due to BSIP's uncertainties, these estimates may turn into sensibility of the method to BSIP's uncertainties. These dynamic matrices \(\hat{\mathbf{M}}(\mathbf{q})\), \(\hat{\mathbf{C}}(\mathbf{q},\dot{\mathbf{q}})\), and \(\hat{\mathbf{G}}(\mathbf{q})\) may be computed using an initial estimate of the parameters of the LLRR. However, they may have uncertainties when an amputee subject is using the LLRR, e.g., in the masses, CoM, or Inertias. Now, let us introduce the formulation of the NDO-based method reported by [PERSON] et al. [12; 34; 35]. This NDO has the following formulation:  
\[ \begin{array}{l}\dot{\mathbf{z}}=-\mathbf{L}(\mathbf{q},\dot{\mathbf{q}}) \mathbf{z}+\mathbf{L}(\mathbf{q},\dot{\mathbf{q}})\{\hat{\mathbf{C}}(\mathbf{ q},\dot{\mathbf{q}})+\hat{\mathbf{G}}(\mathbf{q})-\boldsymbol{\tau}_{a}- \mathbf{p}(\mathbf{q},\dot{\mathbf{q}})\}\\ \dot{\mathbf{\tau}}_{d}=\mathbf{z}+\mathbf{p}(\mathbf{q},\dot{\mathbf{q}})\\ \frac{d}{d!}\mathbf{p}(\mathbf{q},\dot{\mathbf{q}})=\mathbf{L}(\mathbf{q}, \dot{\mathbf{q}})\hat{\mathbf{M}}(\mathbf{q})\bar{\mathbf{q}}\end{array}  \tag{6}\]  
where \(\mathbf{L}(\mathbf{q},\dot{\mathbf{q}})\) is the observer gain matrix, \(\hat{\mathbf{\tau}}_{d}\) is an estimation of the disturbances in a lumped term, \(\hat{\mathbf{M}}(\mathbf{q})\), \(\hat{\mathbf{C}}(\mathbf{q},\dot{\mathbf{q}})\), and \(\hat{\mathbf{G}}(\mathbf{q})\) are initial estimates of the dynamic matrices, \(\mathbf{z}\) and \(\mathbf{p}(\mathbf{q},\dot{\mathbf{q}})\) are an auxiliary variable and vector to avoid the differentiation of position measurements more than once, since it may add problems as mentioned before. In order to design the observer, the challenge is to determine \(\mathbf{L}(\mathbf{q},\dot{\mathbf{q}})\) and \(\mathbf{p}(\mathbf{q},\dot{\mathbf{q}})\). According to [PERSON] et al.