## 2 Mathematical Modeling Strategy
### Development of Mathematical Model 1
 \tag{9}\]  
where \(T^{r}_{m^{a}_{j,2}}\) corresponds to  
\[T^{r}_{m^{a}_{j,2}}=\frac{1}{2}m^{a}_{j,2}\bigg{(}\Big{(}\dot{x}^{a,r}_{j,2} \Big{)}^{2}+\Big{(}\dot{y}^{a,r}_{j,2}\Big{)}^{2}\bigg{)} \tag{10}\]  
Correspondingly, the formal expression of the potential energy is defined as follows  
\[V=\sum_{k=1}^{3}V_{m^{f}_{k}}(\theta_{1,2},\theta_{2,2},\theta_{3,2})+\sum_{ k=1}^{3}V_{m^{a}_{k,1}}(\gamma_{1},\gamma_{2},\gamma_{3})+\sum_{k=1}^{3}V_{m^{a}_ {k,2}}(\gamma_{1},\gamma_{2},\gamma_{3}) \tag{11}\]  
The first section of individual potential energies corresponds to the masses of the fixed mechanical sections of the BRD (solid links), i.e., \(V_{m^{f}_{k}}\), respectively, for the three sections of the device:  
\[V_{m^{f}_{k}}=m^{f}_{k}gy^{f}_{k} \tag{12}\]  
The second section of individual potential energies corresponds to the fixed mechanical sections of each linear actuator in the BRD, that is, \(V_{m^{a}_{j,1}},j=1,2,3\), respectively, for the three actuators of the device:  
\[V_{m^{a}_{j,1}}=m^{a}_{j,1}gy^{a}_{j\mu} \tag{13}\]  
The third section of individual potential energies corresponds to the mobile mechanical sections of each linear actuator in the BRD, that is, \(V_{m^{a}_{j,2}},j=1,2,3\), respectively, for the three actuators of the device:  
\[V_{m^{a}_{j,2}}=m^{a}_{j,2}gy^{a}_{j\mu} \tag{14}\]  
The calculus of both the potential and kinetic energies in the BRD uses the estimation of the centers of masses of the fixed sections in the robot \(x^{f}_{j}\) and \(y^{f}_{j}\) that correspond to  
\[ \begin{split}& x^{f}_{1}=\frac{1}{2}L^{f}_{1}cos(\theta_{1,2})\\ & y^{f}_{1}=\frac{1}{2}L^{f}_{1}sin(\theta_{1,2})\end{split}  \tag{15}\]  
\[ \begin{split}& x^{f}_{2}=L^{f}_{1}cos(\theta_{1,2})-\frac{1}{2}L^{ f}_{2}cos(\theta_{2,2})\\ & y^{f}_{2}=L^{f}_{1}sin(\theta_{1,2})-\frac{1}{2}L^{f}_{2}sin( \theta_{2,2})\end{split}  \tag{16}\]  
\[ \begin{split}& x^{f}_{3}=L^{f}_{1}cos(\theta_{1,2})-L^{f}_{2}cos( \theta_{2,2})+\frac{1}{2}L^{f}_{3}cos(\theta_{3,2})\\ & y^{f}_{3}=L^{f}_{1}sin(\theta_{1,2})-L^{f}_{2}sin(\theta_{2,2})+ \frac{1}{2}L^{f}_{3}sin(\theta_{3,2})\end{split}  \tag{17}\]The configuration of the linear actuators in the form shown in Figure 1 implies the necessity of connecting the active length of the actuators with respect to the angular motion obtained at each articulation in the legs of the BRD. These relationships are the following:  
\[ \begin{array}{c}\left(L_{1,1}^{f}\right)^{2}=\left(L_{1}^{0}\right)^{2}+\left( L_{1}^{a}\right)^{2}-2L^{0}L_{1}^{a}cos\big{(}\gamma_{1}^{c}\big{)}\\ \left(L_{2,1}^{f}\right)^{2}=\left(L_{1,3}^{f}\right)^{2}+\left(L_{2}^{a} \right)^{2}-2L_{1,3}^{f}L_{2}^{a}cos(\gamma_{2}^{c})\\ \left(L_{3,1}^{f}\right)^{2}=\left(L_{2,3}^{f}\right)^{2}+\left(L_{3}^{a} \right)^{2}-2L_{2,3}^{f}L_{3}^{a}cos(\gamma_{3}^{c})\end{array}  \tag{18}\]  
Complementarily, the angles \(\gamma_{j}=\pi/2-\gamma_{j}^{c}\) with \(j=1,2,3\). The corresponding coordinates for the fixed sections of the linear actuators are \(\left(x_{ja}^{a},y_{ja}^{a}\right)\) while the coordinates for the mobile section are defined by \(\left(x_{jb}^{a},y_{jb}^{a}\right)\), which can be expressed as follows  
\[ \begin{array}{c}x_{ja}^{a}=x_{j,1}^{a,r}+\frac{1}{2}L_{j,1}^{a}sin\big{(} \gamma_{j}\big{)}\\ x_{jb}^{a}=x_{j,1}^{a,r}+L_{j,1}^{a}sin\big{(}\gamma_{j}\big{)}+\frac{1}{2}L_ {j,2}^{a}sin(\gamma_{j})\\ y_{ja}^{a}=y_{j,1}^{a,r}+\frac{1}{2}L_{j,1}^{a}cos(\gamma_{j})\\ y_{jb}^{a}=y_{j,1}^{a,r}+L_{j,1}^{a}cos(\gamma_{j})+\frac{1}{2}L_{j,2}^{a}cos (\gamma_{j})\end{array}  \tag{19}\]  
where \(x_{1,1}^{a,r}=x_{0^{\prime}}^{\prime}\), \(y_{1,1}^{a,r}=y_{0}^{\prime}\); \(x_{2,1}^{a,r}=x_{b}\), \(y_{2,1}^{a,r}=y_{b}\) and \(x_{3,1}^{a,r}=x_{d}\), \(y_{3,1}^{a,r}=y_{d}\). The relationships for the coordinates of the fixed and mobile sections require the analysis of the connections between the lengths of the actuators and the angular motions, which are  
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