# New Approach of the Variable Fractional-Order Model of a Robot Arm
## 2 Mathematical Preliminaries
### Grinuwald-Letnikov Definition  
For a given function \(f(t)\)eR of a real variable \(t\), the first-order left derivative has the form  
\[f^{\prime}(t)=\frac{df(t)}{dt}=\underset{h\to 0}{lim}\frac{f(t)-f(t-h)}{h}, \tag{3}\]  
The second-order left derivative is as follows\[f^{\prime\prime}\left(t\right)=\frac{d^{2}f(t)}{dt^{2}}=\underset{h \to 0}{lim}\frac{f^{\prime}(t)-f^{\prime}(t-h)}{h}\] \[=\underset{h\to 0}{lim}\frac{\frac{f(t)-f(t-h)}{h}-\frac{f(t-h)-f(t-2h) }{h}}{h}\] \[=\underset{h\to 0}{lim}\frac{f(t)-2f(t-h)+f(t-2h)}{h^{2}} \tag{4}\] \[=\underset{h\to 0}{lim}\frac{1}{h^{2}}[1\ -2\ 1]\begin{bmatrix}f(t)\\ f(t-h)\\ f(t-2h)\end{bmatrix},\]  
For any order \(n\), one has  
\[f^{(n)}(t)=\frac{d^{n}f(t)}{dt^{n}}=\underset{h\to 0}{lim}\frac{1}{h^{n}} \Big{[}a_{0}^{(n)}\quad a_{1}^{(n)}\quad\cdots\quad a_{n}^{(n)}\Big{]}\ \begin{bmatrix}f(t)\\ f(t-h)\\ \vdots\\ f(t-nh)\end{bmatrix}, \tag{5}\]  
or  
\[f^{(n)}(t)=\frac{d^{n}f(t)}{dt^{n}}=\underset{h\to 0}{lim}\frac{1}{h^{n}} \sum_{i=0}^{n}a_{i}^{(n)}f(t-hi), \tag{6}\]  
where the coefficients \(a_{i}^{(n)}\) are defined as follows  
\[a_{i}^{(n)}=\Bigg{\{}(-1)^{i}\frac{1}{\underset{h\to 0}{lim}}\quad for \quad i=0 \tag{7}\]  
or  
\[a_{i}^{(n)}=a_{i-1}^{(n)}\bigg{(}1-\frac{n+1}{i}\bigg{)}, \tag{8}\]  
The above formulas are valid for any order including fractional ones. Then, one has  
\[{}_{t_{0}}D_{t}^{-n}f(t)=\underset{h\to 0}{lim}h^{-n}\sum_{i=0}^{\frac{t-t_{0}}{ \tau}}a_{i}^{(-n)}f(t-hi)=\frac{1}{(n-1)!}\ \int_{t_{0}}^{t}(t-\tau)^{n-1}f(\tau)d\tau, \tag{9}\]  
The variable fractional-order backward difference (VFOBD) of a bounded discrete function \(f_{k}\) is defined as follows:  
\[{}_{0}^{GL}\Delta_{k}^{(v_{k})}f_{k}=a_{k}^{(v_{k})}*f_{k}=\sum_{i=0}^{k}a_{i }^{(v_{k})}f_{k-i}, \tag{10}\]  
where  
\[a_{k}^{(v_{k})}=\left\{\begin{array}{ccc}0&\text{\ \ \ \ for \ \ }k<0\\ 1&\text{\ \ \ \ for \ \ }k=0\\ (-1)^{k}\binom{v_{k}}{k}&\text{\ \ \ for \ \ }k\geq 1\end{array},\right. \tag{11}\]  
\[\binom{v_{k}}{k}=\frac{v_{k}(v_{k}-1)(v_{k}-2)\cdots(v_{k}-k+1)}{k!}, \tag{12}\]  
\(v_{k}-order\)\(of\)\(the\)\(fractional\)\(difference\)\(in\)\(the\)\(step\)\(k\).  
On the basis of the above equations, one derives a general formula  
\[\begin{bmatrix}a_{0}^{(n)}&a_{1}^{(n)}&\cdots&a_{n-1}^{(n)}&a_{n}^{(n)} \end{bmatrix}=\begin{bmatrix}a_{0}^{(n-m)}&a_{1}^{(n-m)}&\cdots&a_{n-1}^{(n-m)}&a_ {n}^{(n-m)}\end{bmatrix}\\ \begin{bmatrix}a_{0}^{(m)}&a_{1}^{(m)}&a_{2}^{(m)}&a_{3}^{(m)}&\cdots&0&0&0\\ 0&a_{0}^{(m)}&a_{1}^{(m)}&a_{2}^{(m)}&\cdots&0&0&0\\ 0&0&a_{0}^{(m)}&a_{1}^{(m)}&\cdots&0&0&0\\ 0&0&0&a_{0}^{(m)}&\cdots&0&0&0\\ \vdots&\vdots&\vdots&\vdots&&\vdots&\vdots&\vdots\\ 0&0&0&0&\cdots&a_{0}^{(m)}&a_{1}^{(m)}\\ 0&0&0&0&\cdots&0&0&a_{0}^{(m)}\end{bmatrix}, \tag{13}\]  
Equation (13) denotes a discrete convolution and \(\
u_{k}>0\) is the differentiation order function. For negative order functions, Equation (10) defines variable fractional-order integration (discrete summation). One should realize that for integer order \(\mathbb{Z}_{+}\
i\
u_{k}=n>0\), one immediately obtains a classical backward difference.  
\[\begin{split}\partial_{0}^{GL}\Delta_{k}^{(n)}f_{k}=a_{k}^{(n)}*f _{k}=\sum_{i=0}^{k}a_{i}^{(n)}f_{k-i}=\Delta^{(n)}f_{k},\end{split} \tag{14}\]  
The above difference may serve as an approximation of a left derivative.  
\[\begin{split}\frac{d^{n}f(t)}{dt^{n}}=\left.\frac{d^{n}f(t)}{dt^{ n}}\right|_{t=tk}\approx\frac{\partial_{0}^{GL}\Delta_{k}^{(n)}f_{kh}}{h^{n}}= \frac{1}{h^{n}}\sum_{i=0}^{k}a_{i}^{(n)}f_{kh-ih}=\frac{\Delta^{(n)}f_{k}}{h^{n }},\end{split} \tag{15}\]