## 2 Two Methods to Determine the Controllability and Observability of Robots
### Linearization Method to Determine the Controllability and Observability of Robots
 \tag{5}\]  
\(I\in\Re^{n\times n}\) is the identity matrix, \(z_{1}\in\Re^{n\times 1}\) is the joint angle or link displacement of the robot, \(z_{2}\in\Re^{n\times 1}\) is the velocityof the robot, \(z\in\Re^{2n\times 1}\) is the state, \(W(z_{1})\in\Re^{n\times n}\) is a matrix with the inertia terms, which is symmetric and positive definite, \(V(z_{1},z_{2})\in\Re^{n\times n}\) is a matrix with the centripetal and Coriolis terms, \(X(z_{1})\in\Re^{n\times 1}\) is a vector with the gravity terms, \(u\in\Re^{n\times 1}\) is the input, \(G(z_{1})\in\Re^{2n\times 1}\) is a nonlinear function vector containing only the gravity terms of \(F(z_{1},z_{2})\in\Re^{2n\times 1}\), and \(y\in\Re^{n\times 1}\) is the output. \(A_{1}\in\Re^{2n\times 2n}\), \(B_{1}\in\Re^{2n\times n}\), \(D\in\Re^{2n\times n}\). Now the transformed structural method of this study is presented to determine the controllability and observability of robots, which is the combination of the linearization method of [24; 25; 26; 27; 28; 29; 30; 31; 32] and the structural properties method of [33; 34; 35; 36; 37; 38; 39; 40]. The linearization method is an approximation of the robot model around the origin, and the structural properties method is a different representation of the nonlinear model based on transformations. The combination of the linearization and structural properties methods contributes to evade the problems of both methods by evading the structural conditions requirement and decreasing the approximation error. The robot model (1) was transformed to obtain a linear robot model with the gravity terms (5), and the gravity terms are linearized around the origin by the linearization method to obtain the following linear robot model:  
\[\begin{array}{c}\stackrel{{\bullet}}{{=}}Az+Bu,\,y=Dz,\\ A=A_{1}+A_{2},\,B=B_{1}+B_{2},\\ A_{1}=\left[ \begin{array}{cc}0&I\\ 0&-W^{-1}(z_{1})V(z_{1},z_{2})\\ \end{array} \right],\\ B_{1}=\left[ \begin{array}{c}0\\ W^{-1}(z_{1})\\ \end{array} \right],\\ A_{2}=\frac{\partial C(z_{1})}{\partial z}\Big{|}_{z=0,\mu=0},\,B_{2}=\frac{ \partial C(z_{1})}{\partial u}\Big{|}_{z=0,\mu=0},\\ A_{2}=\left[ \begin{array}{c}\frac{\partial C_{1}}{\partial z_{11}}\frac{ \partial C_{1}}{\partial z_{12}}\dots\frac{\partial C_{1}}{\partial z_{m}}\\ \frac{\partial C_{2}}{\partial z_{11}}\frac{\partial C_{2}}{\partial z_{12}} \dots\frac{\partial C_{2}}{\partial z_{m3}}\\ \vdots\vdots\vdots\vdots\vdots\\ \frac{\partial C_{2n}}{\partial z_{11}}\frac{\partial C_{2n}}{\partial z_{12}} \dots\frac{\partial C_{2n}}{\partial z_{m4}}\\ \vdots=0,\mu=0\\ B_{2}=\left[\begin{array}{c}\frac{\partial C_{1}}{\partial u_{1}}\frac{ \partial C_{1}}{\partial u_{1}}\dots\frac{\partial C_{1}}{\partial u_{m}}\\ \frac{\partial C_{2}}{\partial u_{1}}\frac{\partial C_{2}}{\partial u_{2}} \dots\frac{\partial C_{2}}{\partial u_{n}}\\ \vdots\vdots\vdots\vdots\vdots\\ \frac{\partial C_{2n}}{\partial u_{1}}\frac{\partial C_{2n}}{\partial u_{2}} \dots\frac{\partial C_{2n}}{\partial u_{n}}\\ \end{array}\right]_{z=0,\mu=0}\\ z=\left[\begin{array}{cc}z_{1}&z_{2}\\ \end{array}\right]^{T},\,D=\left[\begin{array}{cc}I&0\\ \end{array}\right],\end{array}  \tag{6}\]  
\(I\in\Re^{n\times n}\) is the identity matrix and \(z=[z_{1},z_{2}]^{T}\in\Re^{2n\times 1},\,z_{1}\in\Re^{n\times 1},\,z_{2}\in\Re^{n\times 1}\), and \(u\in\Re^{n\times 1}\) are described in (5), \(A\in\Re^{2n\times 2n},\,B\in\Re^{2n\times n}\). Since the linear robot model has been obtained in (6), now the known methods for linear systems [31; 32] are used to determine the controllability and observability of robots. Using the linear robot model (6) of the transformed structural properties method, the controllability matrix is:  
\[C=\left[ \begin{array}{cccccc}B&AB&A^{2}B&\dots&A^{2n-1}B\\ \end{array} \right], \tag{7}\]  
If the rank of the controllability matrix \(C\) is equal to \(2n\), the robot model of the transformed structural properties method is controllable around the origin. Using the linear robot model (6) of the transformed structural properties method, the observability matrix is:  
\[O=\left[