## 5 Proposed Methods
### Transforming the Model  
The objective of this section is to present the model (1) transformed as a linear in the parameters system that will be used for the identification of the parameters. Consider the LLRR reported in Section 3. Each link of the LLRR is characterized by ten dynamic parameters expressed as  
\[m_{i}\,,\quad\mathbf{r}_{Ci}=\left( \begin{array}{c}r_{xi}\\ r_{yi}\\ r_{zi}\end{array} \right)\,,\quad\mathbf{I}_{Ci}=\left( \begin{array}{c}I_{Ci, xx}\\ I_{Ci,xy}\\ I_{Ci,yz}\\ I_{Ci,xz}\end{array} \right)\,. \tag{9}\]  
However, the LLRR's dynamics depend in a nonlinear way on some of these parameters, e.g., through the combination \(I_{ci,zz}+mi_{x}^{2}\). Many standard parameters do not play a role in the dynamic model of the given LLRR. Therefore, one can isolate \(p<10\) independent groups of parameters. These grouped parameters are called dynamic coefficients. These parameters are the only ones that affect robot dynamics. The base inertial parameters or identifiable parameters are the minimum inertial parameter set used to completely characterize and obtain a dynamic model of a robot. These parameters can be obtained from the standard dynamic parameters by eliminating all the parameters that have no effect in the dynamic model and grouping some of them in linear combinations as reported by [PERSON] et al. [37]. The introduction of base inertial parameters \(\mathbf{\chi}_{i}\in\mathbb{R}^{p\times 1}\) is a convenient regrouping of the dynamic parameters, i.e., a linear parametrization of LLRR's dynamics. Therefore, we should obtain a vector \(\mathbf{\chi}\) that contains the base inertial parameters. Thus, the dynamic model (1) can be written such that the parameters enter in a linear fashion as  
\[\mathbf{\tau}=\mathbf{W}(\mathbf{q},\dot{\mathbf{q}},\ddot{\mathbf{q}})\mathbf{\chi} \tag{10}\]  
where \(\mathbf{W}(\mathbf{q},\dot{\mathbf{q}},\ddot{\mathbf{q}})\in\mathbb{R}^{N\times p}\) is the regressor matrix. It is important to note that it will be denoted as \(\mathbf{W}\) in the rest of this report to simplify the notation. Next, we will present the adjustment for the mathematical model of the LLRR (1) to the form (10) indicated by the formulation reported by [PERSON] et al. [36]. Therefore, we assumed a set of parameters \(\mathbf{\chi}_{i}\) as  
\[ \begin{split}\chi_{1}&=Ic_{1,zz}+Ic_{2,zz}+Ic_{3,zz }+{L_{1}}^{2}\,m_{2}+{L_{1}}^{2}\,m_{3}+{L_{2}}^{2}\,m_{3}+{b_{1}}^{2}\,m_{1}+{ b_{2}}^{2}\,m_{2}+{b_{3}}^{2}\,m_{3}\\ \chi_{2}&=L_{1}\,g_{0}\,m_{2}+L_{1}\,g_{0}\,m_{3}+b_ {1}\,g_{0}\,m_{1}\\ \chi_{3}&=m_{3}\,{L_{2}}^{2}+m_{2}\,{b_{2}}^{2}+m_{3 }\,{b_{3}}^{2}+Ic_{2,zz}+Ic_{3,zz}\\ \chi_{4}&=L_{1}\,L_{2}\,m_{3}+L_{1}\,b_{2}\,m_{2}\\ \chi_{5}&=L_{2}\,g_{0}\,m_{3}+b_{2}\,g_{0}\,m_{2}\\ \chi_{6}&=m_{3}\,{b_{3}}^{2}+Ic_{3,zz}\\ \chi_{7}&=L_{2}\,b_{3}\,m_{3}\\ \chi_{8}&=L_{1}\,b_{3}\,m_{3}\\ \chi_{9}&=b_{3}\,g_{0}\,m_{3}\end{split}  \tag{11}\]  
See Table 2 for the definition of all these parameters. Subsequently, we obtained the transformed matrices \(\mathbf{M}(\mathbf{\chi},\mathbf{q})\), \(\mathbf{C}(\mathbf{\chi},\mathbf{q},\dot{\mathbf{q}})\), and \(\mathbf{G}(\mathbf{\chi},\mathbf{q})\) defined as  
\[\mathbf{M}(\mathbf{\chi},\mathbf{q})= \begin{bmatrix}m_{11}(\mathbf{\chi},\mathbf{q})&m_{12}(\mathbf{ \chi},\mathbf{q})&m_{13}(\mathbf{\chi},\mathbf{q})\\ m_{21}(\mathbf{\chi},\mathbf{q})&m_{22}(\mathbf{\chi},\mathbf{q})&m_{23}(\mathbf{\chi},\mathbf{q})\\ m_{31}(\mathbf{\chi},\mathbf{q})&m_{32}(\mathbf{\chi},\mathbf{q})&m_{33}(\mathbf{\chi},\mathbf{q}) \end{bmatrix}  \tag{12}\]  
\[\mathbf{C}(\mathbf{\chi},\mathbf{q},\dot{\mathbf{q}})= \begin{bmatrix}C_{1}(\mathbf{\chi},\mathbf{q}, \dot{\mathbf{q}})\\ C_{2}(\mathbf{\chi},\mathbf{q},\dot{\mathbf{q}})\\ C_{3}(\mathbf{\chi},\mathbf{q},\dot{\mathbf{q}})\end{bmatrix}  \tag{13}\]  
and  
\[\mathbf{G}(\mathbf{\chi},\mathbf{q})= \begin{bmatrix}g_{1}(\mathbf{\chi},\mathbf{q})\\ g_{2}(\mathbf{\chi},\mathbf{q})\\ g_{3}(\mathbf{\chi},\mathbf{q})\end{bmatrix}  \tag{14}\]  
by using the same process implemented in Section 3.