# New Approach of the Variable Fractional-Order Model of a Robot Arm
## 4 The Robot Arm's Mathematical Models
### Classical Mathematical Model (CM) of the Robot Arm  
The theoretical model of a robot arm is presented in Figure 12[6; 10].  
Following the Euler-Lagrange formalism, two classical differential second-order equations describing the dynamic system are as follows  
\[D[q(t)]\ddot{q}(t)+C\big{[}q(t),\dot{q}(t)\big{]}\dot{q}(t)+g[q(t)]=\tau(t) \tag{27}\]  
where  
\[\tau(t)=\begin{bmatrix}\tau_{1}(t)\\ \tau_{2}(t)\end{bmatrix}\text{- drive moments,}\]  
\[q(t)=\begin{bmatrix}q_{1}(t)\\ q_{2}(t)\end{bmatrix}\text{- coordinates of the manipulator links position,}\]  
\[D[q(t)]=D=\begin{bmatrix}I_{1}+I_{2}&I_{2}\\ I_{2}&I_{2}\end{bmatrix}\text{-moments of inertia matrix of the driving system,}\]  
\[C\big{[}q(t),\dot{q}(t)\big{]}=\begin{bmatrix}h(t)\dot{q}_{2}(t)&h(t)\dot{q}_ {2}(t)+h(t)\dot{q}_{1}(t)\\ -h(t)\dot{q}_{1}(t)&0\end{bmatrix}=\]  
\[=m_{2}I_{1}l_{c2}sin[q_{2}(t)]\begin{bmatrix}-\dot{q}_{2}(t)&-\dot{q}_{2}(t)- \dot{q}_{1}(t)\\ \dot{q}_{1}(t)&0\end{bmatrix},\]  
\[h(t)=-m_{2}I_{1}l_{c2}sin[q_{2}(t)],\]  
\[g[q(t)]=g\begin{bmatrix}(m_{1}l_{c1}+m_{2}I_{1})cos[q_{1}(t)]+m_{2}l_{c2}cos[q _{1}(t)+q_{2}(t)]\\ m_{2}l_{c2}cos[q_{1}(t)+q_{2}(t)]\end{bmatrix},\]  
g--acceleration of the Earth,  
\(I_{1},I_{2}\)--lengths of the robot arm,  
\(l_{c1},l_{c2}\)--distance from the precedent joint to the arm centroid.  
Now one assumes that the second arm is immobilized. This means that \(\ddot{q}_{2}(t)=0\)  
Figure 12: The theoretical model of robot arm.  
\(\dot{q}_{2}(t)=0\), and \(q_{2}(t)=q_{2}=\) const, \(\tau_{2}(t)=0\). From Equation (27), one obtains  
\[\begin{bmatrix}I_{1}+I_{2}&I_{2}\\ I_{2}&I_{2}\end{bmatrix}\ddot{q}(t)+\dot{q}_{1}(t)m_{2}I_{1}l_{2}sin[q_{2}] \begin{bmatrix}0&-1\\ 1&0\end{bmatrix}\dot{q}(t) \tag{28}\] \[+g\bigg{[}(m_{1}l_{c1}+m_{2}l_{1})cos[q_{1}(t)]+m_{2}l_{c2}cos[q_{ 1}(t)+q_{2}]\bigg{]}=\begin{bmatrix}\tau_{1}(t)\\ 0\end{bmatrix}\]  
Pre-multiplication of both sides of Equation (28) by a matrix  
\[\begin{bmatrix}I_{1}+I_{2}&I_{2}\\ I_{2}&I_{2}\end{bmatrix}^{-1}=\begin{bmatrix}\frac{1}{I_{1}}&-\frac{1}{I_{1}} \\ -\frac{1}{I_{1}}&\frac{1}{I_{1}}+\frac{1}{I_{2}}\end{bmatrix} \tag{29}\]  
yields  
\[\begin{split}\ddot{q}(t)+\dot{q}_{1}(t)m_{2}l_{1}l_{2}sin[q_{2}] \begin{bmatrix}0&\frac{1}{I_{1}}\\ \frac{1}{I_{2}}&-\frac{1}{I_{1}}\end{bmatrix}\dot{q}(t)\\ +g\begin{bmatrix}\frac{1}{I_{1}}&-\frac{1}{I_{1}}\\ -\frac{1}{I_{1}}&\frac{1}{I_{1}}+\frac{1}{I_{2}}\end{bmatrix}\bigg{[}\begin{bmatrix} m_{1}l_{c1}+m_{2}l_{1}cos[q_{1}(t)]+m_{2}l_{c2}cos[q_{1}(t)+q_{2}]\\ m_{2}l_{c2}cos[q_{1}(t)+q_{2}]\end{bmatrix}\\ =\begin{bmatrix}\tau_{1}(t)\\ 0\end{bmatrix}\end{split} \tag{30}\]  
Further transformations lead to  
\[\begin{split}\begin{bmatrix}\ddot{q}_{1}(t)\\ 0\end{bmatrix}+m_{2}l_{1}l_{2}sin(q_{2})\begin{bmatrix}0\\ \frac{\dot{q}_{2}^{2}(t)}{I_{2}}\end{bmatrix}\\ +g\bigg{[}\frac{(m_{1}l_{c1}+m_{2}l_{1})cos[q_{1}(t)]}{I_{1}}+\frac{m_{2}l_{ c2}cos[q_{1}(t)+q_{2}]}{I_{2}}\bigg{]}=\begin{bmatrix}\tau_{1}(t)\\ 0\end{bmatrix}\end{split} \tag{31}\]  
From the first row, one has  
\[\ddot{q}_{1}(t)+\frac{g(m_{1}l_{c1}+m_{2}l_{1})cos[q_{1}(t)]}{I_{1}}=\tau_{1} (t)\]  
Additionally, the second row must be satisfied  
\[\begin{split}\frac{m_{2}l_{1}(l_{2}sin(q_{2})}{I_{2}}\dot{q}_{1} ^{2}(t)+\frac{g(m_{1}l_{c1}+m_{2}l_{1})cos[q_{1}(t)]}{I_{1}}\\ +\frac{gm_{2}l_{c2}cos[q_{1}(t)+q_{2}]}{I_{2}}=0\end{split} \tag{32}\]  
As a discrete time, mathematical model, one takes an approximation of the derivative by a difference  
\[\begin{split}\frac{q_{1}(kh)-2q_{1}(kh-h)+q_{1}(kh-2h)}{h^{2}}+ \frac{g(m_{1}l_{c1}+m_{2}l_{1})cos[q_{1}(kh)]}{I_{1}}\\ =\tau_{1}(kh)\end{split} \tag{33}\]  
In further considerations to verify adopted assumptions, only the first link was the subject of the analysis. Mechanical friction and backlash are omitted. Now one denotes:  
\[K=\frac{g(m_{1}l_{c1}+m_{2}l_{1})}{I_{1}} \tag{34}\]  
Hence, under the assumptions mentioned above, in the model there is only one parameter to evaluate.