## Appendix D Transformed Model  
The introduction of dynamic coefficients \(\boldsymbol{\chi}_{i}\) is a convenient regrouping of the dynamic parameters, i.e., a linear parametrization of dynamics. The robot dynamics depend in a nonlinear way on some of these parameters. Therefore, we assumed \[\chi_{1} =Ic_{1,zz}+Ic_{2,zz}+Ic_{3,zz}+{L_{1}}^{2}\,m_{2}+{L_{1}}^{2}\,m_{3}+ {L_{2}}^{2}\,m_{3}+{b_{1}}^{2}\,m_{1}+{b_{2}}^{2}\,m_{2}+{b_{3}}^{2}\,m_{3}\] \[\chi_{2} =L_{1}\,g_{0}\,m_{2}+L_{1}\,g_{0}\,m_{3}+b_{1}\,g_{0}\,m_{1}\] \[\chi_{3} =m_{3}\,{L_{2}}^{2}+m_{2}\,{b_{2}}^{2}+m_{3}\,{b_{3}}^{2}+Ic_{2,zz} +Ic_{3,zz}\] \[\chi_{4} =L_{1}\,L_{2}\,m_{3}+L_{1}\,b_{2}\,m_{2}\] \[\chi_{5} =L_{2}\,g_{0}\,m_{3}+b_{2}\,g_{0}\,m_{2} \tag{117}\] \[\chi_{6} =m_{3}\,{b_{3}}^{2}+Ic_{3,zz}\] \[\chi_{7} =L_{2}\,b_{3}\,m_{3}\] \[\chi_{8} =L_{1}\,b_{3}\,m_{3}\] \[\chi_{9} =b_{3}\,g_{0}\,m_{3}\]  
Therefore, we obtained a transformed \(\mathbf{M}(\chi,\mathbf{q})\) defined as  
\[\mathbf{M}(\chi,\mathbf{q})= \begin{bmatrix}m_{11}(\chi,\mathbf{q})&m_{12}(\chi,\mathbf{q})&m_{13}(\chi,\mathbf{q})\\ m_{21}(\chi,\mathbf{q})&m_{22}(\chi,\mathbf{q})&m_{23}(\chi,\mathbf{q})\\ m_{31}(\chi,\mathbf{q})&m_{32}(\chi,\mathbf{q})&m_{33}(\chi,\mathbf{q})\end{bmatrix}  \tag{118}\]  
with  
\[m_{11}(\chi,\mathbf{q})=\chi_{1}+2\chi_{4}c_{2}+2\chi_{7}c_{3}+2\chi_{8}c_{23} \tag{119}\]  
\[m_{22}(\chi,\mathbf{q})=\chi_{3}+2\,\chi_{7}c_{3} \tag{120}\]  
\[m_{33}(\chi,\mathbf{q})=\chi_{6} \tag{121}\]  
\[m_{12}(\chi,\mathbf{q})=m_{21}(\chi,\mathbf{q})=\chi_{3}+\chi_{4}c_{2}+2\chi_ {7}c_{3}+\chi_{8}c_{23} \tag{122}\]  
\[m_{13}(\chi,\mathbf{q})=m_{31}(\chi,\mathbf{q})=\chi_{6}+\chi_{7}c_{3}+\chi_ {8}c_{23} \tag{123}\]  
\[m_{23}(\chi,\mathbf{q})=m_{32}(\chi,\mathbf{q})=\chi_{6}+\chi_{7}c_{3} \tag{124}\]  
The Lagrangian \(\mathcal{L}(\mathbf{q},\dot{\mathbf{q}})\) of the mechanical system can be defined as  
\[\mathcal{L}(\mathbf{q},\dot{\mathbf{q}})=\mathcal{T}(\mathbf{q},\dot{\mathbf{ q}})-\mathcal{U}(\mathbf{q})=\frac{1}{2}\sum_{ij}m_{ij}(\mathbf{q})\dot{ \mathbf{q}}_{i}\dot{\mathbf{q}}_{j}-\mathcal{U}(\mathbf{q}) \tag{125}\]  
where \(\mathcal{T}\) and \(\mathcal{U}\) denote the total KE and PE of the system, respectively. The Euler-Lagrange equations can be defined as  
\[\frac{d}{dt}\frac{\partial\mathcal{L}}{\partial\dot{\mathbf{q}}_{k}}-\frac{ \partial\mathcal{L}}{\partial\dot{\mathbf{q}}_{k}}=\mathbf{u}_{k}\quad\mathbf{k}=\bm {1},\mathbf{2},\mathbf{3} \tag{126}\]  
where \(\mathbf{u}_{k}\) is the generalized force associated with the generalized coordinate \(\mathbf{q}_{i}\). Therefore, we rewrite the dynamic equations as  
\[\underbrace{\sum_{j}m_{kj}(\mathbf{q})\ddot{\mathbf{q}}_{j}}_{\text{Inertial terms}}+\underbrace{\sum_{ij}c_{kij}(q)\dot{\mathbf{q}}_{i}\dot{\mathbf{q}}_{j}}_{ \text{Centrifugal ($i$=$) and Coriolis ($i$\
eq$) terms}}+\underbrace{\frac{\partial\mathbf{U}}{\partial\dot{ \mathbf{q}}_{k}}}_{\text{Gravity terms $\mathbf{g}_{k}(\mathbf{q})$}}=\mathbf{u}_{k} \tag{127}\]  
where \(c_{kij}=c_{kji}\) are the Christoffel symbols of the first kind. Subsequently, we computed the Centrifugal and Coriolis term \(c(\mathbf{q},\dot{\mathbf{q}})\) taking into account that the \(k\)th component of vector \(\mathbf{C}\) is defined as  
\[c_{k}(\dot{\mathbf{q}})=\dot{\mathbf{q}}^{T}C_{k}(\mathbf{q})\dot{\mathbf{q}} \tag{128}\]  
where  
\[m_{22}(\chi,\mathbf{q})=\chi_{3}+2\,\chi_{7}c_{3} \tag{120}\]0and \(M_{k}\) is the \(k\)-th column of matrix \(M(q)\). Subsequently, we obtained a transformed \(\mathbf{C(\chi,q,\dot{q})}\) as  
\[\mathbf{C(\chi,q,\dot{q})}= \begin{bmatrix}C_{1}(\chi,q,\dot{q})\\ C_{2}(\chi,q,\dot{q})\\ C_{3}(\chi,q,\dot{q})\end{bmatrix}