## 2 Two Methods to Determine the Controllability and Observability of Robots
### Linearization Method to Determine the Controllability and Observability of Robots  
The robot model considered is detailed. The robot model is important because it is the starting point in determining the controllability and observability of robots with this method. For the robots with \(n\) degrees of freedom, considering the contact force and constraints, the robot model is:  
\[ \begin{gathered} F(z_{1},z_{2})=\tau,y=Dz,\\ F(z_{1},z_{2})=\Bigg{[}\begin{array}{c}z_{2}\\ W(z_{1})z_{2}^{\bullet}+V(z_{1},z_{2})z_{2}+X(z_{1})\end{array}\Bigg{]},\\ F(z_{1},z_{2})=\big{[}\begin{array}{cc}F_{1}&F_{2}&\cdots&F_{2n}\end{array} \big{]}^{T},\\ z=\big{[}\begin{array}{cc}z_{1}&z_{2}\end{array}\big{]}^{T},D=\big{[} \begin{array}{cc}I&0\end{array}\big{]},\tau=\big{[}\begin{array}{cc}0&u \end{array}\end{bmatrix}^{T}\end{gathered}  \tag{1}\]  
\(I\in\Re^{n\times n}\) 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 velocity of 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, \(F(z_{1},z_{2})\in\Re^{2n\times 1}\) is a nonlinear function vector containing all the robot terms, \(D\in\Re^{2n\times n}\), and \(y\in\Re^{n\times 1}\) is the output. Now, the linearization method of [24; 25; 26; 27; 28; 2  
\(A\in\Re^{2n\times 2n}\), \(B\in\Re^{2n\times n}\) Since the linear robot model has been obtained in (2), now the known methods for linear systems [31; 32] are used to determine the controllability and observability of robots. Using the linear robot model (2) of the linearization method, the controllability matrix is:  
\[C=\left[ \begin{array}{cccc}B&AB&A^{2}B&\cdots&A^{2n-1}B\end{array} \right], \tag{3}\]  
If the rank of the controllability matrix \(C\) is equal to \(2n\), the robot model of the linearization method is controllable around the origin. Using the linear robot model (2) of the linearization method, the observability matrix is:  
\[O=\left[ \begin{array}{cccc}D&DA&DA^{2}&\cdots&DA^{2n-1}\end{array} \right]^{T}, \tag{4}\]  
If the rank of the observability matrix \(O\) is equal to \(2n\), the robot model of the linearization method is observable around the origin. Transformed Structural Properties Method to Determine the Controllability and Observability of Robots  
The linear robot model with gravity terms considered is detailed. The linear robot model with gravity terms is important because it is the starting point in determining the controllability and observability of robots with this method. By using the fact that the inverse of the robot inertia matrix \(W(z_{1})\in\Re^{n\times n}\) is well defined, (1) is represented as the following linear robot model with gravity terms:  
\[ \begin{array}{l}\overset{\bullet}{z}=A_{1}z+B_{1}u+G(z_{1}),\,y=Dz,\\ A_{1}=\left[\begin{array}{cccc}0&I\\ 0&-W^{-1}(z_{1})V(z_{1},z_{2})\end{array}\right],\\ B_{1}=\left[\begin{array}{cccc}0\\ W^{-1}(z_{1})\end{array}\right],\\ G(z_{1})=\left[\begin{array}{cccc}0\\ -W^{-1}(z_{1})X(z_{1})\end{array}\right],\\ G(z_{1})=\left[\begin{array}{cccc}G_{1}&G_{2}&\cdots&G_{2n}\end{array}\right]^ {T},\\ z=\left[\begin{array}{cccc}z_{1}&z_{2}\end{array}\right]^{T},D=\left[\begin{array} []{cccc}I&0\end{array}\right],\end{array}