# Design, Simulation, and Comparison of Advanced Control Strategies for a 3-Degree-of-Freedom Robot
## 3 Description of the System Under Study
### Dynamic Model of the Scrobot Robot  
The dynamic model of the Scrobot robot is obtained through the Euler-Lagrange equations, using (15) and (16) to form (17) and applying (18) to form (19). \[K_{i}\left(q,\dot{q}\right)=\frac{1}{2}m_{i}v^{2}+\frac{1}{2}l_{i}\dot{q}^{2} \tag{15}\]  
\[U_{i}(q)=m_{i}gh \tag{16}\]  
\[L\left(q,\dot{q}\right)=K_{i}\left(q,\dot{q}\right)-U_{i}(q) \tag{17}\]  
\[\frac{d}{dt}\left(\frac{\partial L}{\partial\dot{q}_{i}}\right)-\frac{\partial L }{\partial\dot{q}_{i}}=\tau \tag{18}\]  
By adding the terms \(F(q)+\tau_{d}\), Equation (19) is derived. \[\tau=M(q)\ddot{q}+C\left(q,\dot{q}\right)\dot{q}+G(q)+F(q)+\tau_{d} \tag{19}\]  
The Scrobot robot shown in Figure 3 employs three degrees of freedom, allowing it to determine its position in space (x, y, z) [24]. Figure 4 presents the diagram of the robot under study. To model its kinematics and dynamics, the coordinate axes and centroid systems consider \(q_{1}\), \(q_{2}\), \(q_{3}\) and \(l_{1}\), \(l_{2}\), \(l_{3}\) as generalized coordinates and lengths of the first, second, and third links, respectively. Furthermore, \(l_{c1}\), \(l_{c2}\), \(l_{c3}\) represent the lengths of the first, second, and third links from the origin to the centroid [24]. This results in the matrix M (20) and the vectors C (30), G (34), and F (38). \[M=\left[ \begin{array}{ccc}\mathrm{M}_{11}&\mathrm{M}_{12}&\mathrm{M}_{13}\\ \mathrm{M}_{21}&\mathrm{M}_{22}&\mathrm{M}_{23}\\ \mathrm{M}_{31}&\mathrm{M}_{32}&\mathrm{M}_{33}\end{array} \right] \tag{20}\]  
\[\mathrm{M}_{11}=(\mathrm{l}_{\mathrm{c}3}\mathrm{cos}(\theta_{2}+\theta_{3})+ \mathrm{l}_{2}\mathrm{cos}(\theta_{2}))^{2}\mathrm{m}_{3}+\mathrm{I}_{1 zz}+ \mathrm{I}_{2 zz}+\mathrm{I}_{3 zz}+\left(\frac{\left(\mathrm{l}_{\mathrm{c}2} ^{2}\mathrm{cos}(2\theta_{2})\right)}{2}+\frac{\mathrm{l}_{\mathrm{c}2}^{2}}{2 }\right)\mathrm{m}_{2} \tag{21}\]  
\[\mathrm{M}_{12}=\mathrm{I}_{2 zz}+\mathrm{I}_{3 zz} \tag{22}\]  
\[\mathrm{M}_{13}=\mathrm{I}_{3 zz} \tag{23}\]  
\[\mathrm{M}_{21}=\mathrm{I}_{2 zz}+\mathrm{I}_{3 zz} \tag{24}\]  
\[\mathrm{M}_{22}=\left(\mathrm{l}_{2}{}^{2}+2\mathrm{cos}(\theta_{3})\mathrm{l }_{2}\mathrm{l}_{3}+\mathrm{l}_{3}{}^{2}\right)\mathrm{m}_{3}+\mathrm{m}_{2} \mathrm{l}_{2}{}^{2}+\mathrm{I}_{2 zz}+\mathrm{I}_{3 zz} \tag{25}\]  
\[\mathrm{M}_{23}=\left(\mathrm{l}_{\mathrm{c}3}{}^{2}+\mathrm{l}_{2}\mathrm{cos }(\theta_{3})\mathrm{l}_{\mathrm{c}3}\right)\mathrm{m}_{3}+\mathrm{I}_{3 zz} \tag{26}\]  
\[\mathrm{M}_{31}=\mathrm{I}_{3 zz} \tag{27}\]  
\[\mathrm{M}_{32}=\left(\mathrm{l}_{\mathrm{c}3}{}^{2}+\mathrm{l}_{2}\mathrm{ cos}(\theta_{3})\mathrm{l}_{\mathrm{c}3}\right)\mathrm{m}_{3}+\mathrm{I}_{3 zz} \tag{28}\]  
\[\mathrm{M}_{33}=\mathrm{l}_{\mathrm{c}3}{}^{2}\mathrm{m}_{3}+\mathrm{I}_{3 zz} \tag{29}\]  
Figure 4: Diagram of the SCORBOT-ER V plus robot in two positions, considering the coordinate axes and centroid systems. (**a**) Elbow down. (**b**) Elbow up. \[\mathrm{M}_{12}=\mathrm{I}_{2 zz}+\mathrm{I}_{3 zz} \tag{22}\]0 Robot SCORBOT-ER V plus and its diagram. \[C=\left[ \begin{array}{c}C_{11}\\ C_{21}\\ C_{31}\end{array} \right] \tag{30}\]  
\[ \begin{array}{c}C_{11}=-\mathrm{d}\mathrm{d}_{1}\mathrm{d}\mathrm{d}_{2} \Big{(}\mathrm{m}_{2}\mathrm{sin}(2\uptheta_{2})\mathrm{l}_{\mathrm{c}2}{}^{2} +2\mathrm{m}_{3}(\mathrm{l}_{\mathrm{c}3}\mathrm{cos}(\uptheta_{2}+\uptheta_{3 })+\mathrm{l}_{2}\mathrm{cos}(\uptheta_{2}))(\mathrm{l}_{\mathrm{c}3}\mathrm{ sin}(\uptheta_{2}+\uptheta_{3})+\mathrm{l}_{2}\mathrm{sin}(\uptheta_{2})) \Big{)}\\ \qquad\qquad-2\mathrm{d}\mathrm{q}_{1}\mathrm{d}\mathrm{d}\mathrm{q}_{3}\mathrm{ l}_{\mathrm{c}3}\mathrm{m}\mathrm{sin}(\uptheta_{2}+\uptheta_{3})(\mathrm{l}_{ \mathrm{c}3}\mathrm{cos}(\uptheta_{2}+\uptheta_{3})+\mathrm{l}_{2}\mathrm{cos }(\uptheta_{2}))\end{array}  \tag{31}\]  
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