## 3 Scara Robot
### Linearization Method  
Now, the linearization method of [24; 25; 26; 27; 28; 29; 30; 31; 32] is applied to the scara robot. We write the scara robot as (1), and we detail it as:  
\[ \begin{gathered} F(z_{1},z_{2})=\tau,y=Dz,\\ F(z_{1},z_{2})=\left[\begin{array}{c}z_{2}\\ W(z_{1})z_{2}^{\bullet}+V(z_{1},z_{2})z_{2}+X(z_{1})\end{array}\right],\\ F(z_{1},z_{2})=\left[\begin{array}{cc}F_{1}&F_{2}&\cdots&F_{6}\end{array} \right]^{T},\\ z=\left[\begin{array}{cc}z_{1}&z_{2}\end{array}\right]^{T},D=\left[\begin{array} []{cc}I&0\end{array}\right],\tau=\left[\begin{array}{cc}0&u\end{array} \right]^{T}\end{gathered}  \tag{9}\]  
Figure 1: Scara robot. where \(F(z_{1},z_{2})\in\Re^{6\times 1}\) is a nonlinear function vector containing all the robot terms, \(F_{1},F_{2},\cdots,F_{6}\), which are described as follows:  
\[ \begin{array}{c}F_{1}=z_{21},\,F_{2}=z_{22},\,F_{3}=z_{23},\\ F_{4}=-\frac{\left[\left(m_{1}^{2}c_{2}^{2}+m_{3}t_{2}^{2}\right)+\frac{1}{1}C_ {2}\left(m_{1}c_{2}+m_{3}t_{2}\right)\right]}{\left[I_{3}+m_{1}t_{2}^{2}+m_{ 2}\left(t_{1}^{2}+t_{2}^{2}\right)+m_{3}\left(t_{1}^{2}+t_{2}^{2}\right)+2I_{ 1}C_{2}\left(m_{1}c_{2}+m_{3}t_{2}\right)\right]}z_{22}\\ +\frac{\left[2I_{3}C_{2}\left(m_{2}c_{2}+m_{3}t_{2}\right)\right]}{\left[I_{3} +m_{1}t_{2}^{2}+m_{2}\left(t_{1}^{2}+m_{3}t_{2}^{2}\right)\right]}z_{22}z_{21} \\ +\frac{\left[I_{3}C_{2}\left(m_{2}c_{2}+m_{3}t_{2}\right)\right]}{\left[I_{3} +m_{1}t_{2}^{2}+m_{2}\left(t_{1}^{2}+t_{2}^{2}\right)+m_{3}\left(t_{1}^{2}+t_ {2}^{2}\right)+2I_{1}C_{2}\left(m_{2}c_{2}+m_{3}t_{2}\right)\right]}z_{22}z_{2 2}\\ +\frac{\left[I_{3}C_{2}\left(m_{2}c_{2}+m_{3}t_{2}\right)\right]}{\left[I_{3} +m_{1}t_{2}^{2}+m_{2}\left(t_{1}^{2}+t_{2}^{2}\right)+m_{3}\left(t_{1}^{2}+t _{2}^{2}\right)+2I_{1}C_{2}\left(m_{2}c_{2}+m_{3}t_{2}\right)\right]}u_{1},\\ F_{5}=-\frac{\left[\left(m_{1}t_{2}^{2}+m_{3}t_{2}\right)\right]+I_{1}C_{2} \left(m_{2}c_{2}+m_{3}t_{2}\right)\right]}{\left[I_{3}+\left(m_{2}c_{2}+m_{3} t_{2}\right)\right]}z_{21}^{\bullet}\\ -\frac{\left[2I_{3}C_{2}\left(m_{1}c_{2}+m_{3}t_{2}\right)\right]}{\left[I_{3} +\left(m_{2}t_{2}^{2}+m_{3}t_{2}\right)\right]}z_{21}z_{21}+\frac{\left[I_{3} +\left(m_{2}t_{2}^{2}+m_{3}t_{2}^{2}\right)\right]}{\left[I_{3}+\left(m_{2}t_ {2}^{2}+m_{3}t_{2}^{2}\right)\right]}u_{2},\\ F_{6}=\left[g\right]+\frac{1}{\left[m_{3}\right]}u_{3},\end{array}  \tag{10}\]  
\(m_{1}\), \(m_{2}\), \(m_{3}\) are the masses of the links one, two, three, \(z_{11}=\theta_{1}\), \(z_{12}=\theta_{2}\), are the angles of the joints one, two in rad, \(z_{13}=l_{c3}\) is the length of the link three, in m, and \(g\) is the acceleration gravity constant. \(l_{1}=l_{2}=0.3\) m, \(l_{c1}=l_{1}/2\), \(l_{c2}=l_{2}/2\), \(m_{1}=m_{2}=m_{3}=0.3\) kg, \(l_{13}=l_{1}+l_{2}+l_{3}\), \(l_{1}=0.0208\) kgm\({}^{2}\), \(l_{2}=J_{3}=0.0127\) kgm\({}^{2}\), \(g=9.81\) m/s\({}^{2}\). By using the linearization method described in (2), the linear model is:  
\[ \begin{array}{c}\overset{\bullet}{z}=Az+Bu,\,y=Dz,\\ A=\left[\begin{array}{cccccc}0&0&0&1&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1\\ 0&0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ 0&0&0&0&0\\ \end{array}\right],\\ B=\left[\begin{array}{cccc}0&0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ 0&0&0&0\\ \end{array}\right],\\ D=\left[\begin{array}{cccc}1&0&0&0&0&0\\ 0&1&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&1&0&0&0\\ \end{array}\right],\end{array}  \tag{11}\]  
By using the linearization method described in (3), the controllability matrix is:  
\[ \begin{array}{c}C=\left[\begin{array}{cccccc}0&0&0&c_{14}&0&0\\ 0&0&0&0&c_{25}&0\\ 0&0&0&0&0&c_{36}\\ c_{41}&0&0&0&0&0\\ 0&c_{52}&0&0&0&0\\ 0&0&c_{63}&0&0&0\\ \end{array}\right],\end{array}  \tag{12}\]  
\(c_{14}=c_{41}=4.1331\), \(c_{25}=c_{52}=21.529\), \(c_{36}=c_{63}=3.3333\). Since the rank of the controllability matrix is \(6\), the scara robot is controllable. By using the linearization method described in (4), the observability matrix is:  
\[O=\left[ \begin{array}{cccccc}1&0&0&0&0&0\\ 0&1&0&0&0&0\\ 0&0&1&0&0&0\\ 0&0&0&1&0&0\\ 0&0&0&0&1&0\\ 0&0&0&0&0&1\end{array}