## 3 Literature Classification and Analysis
### Mathematical Models  
The trade-off between model accuracy and its complexity should also be taken into consideration. The classical mathematical model (2) of the robot arm is inspired by the well-known Newton-Euler formalism. Such a dynamic system can be described by two classical differential second-order equations. This approach is based on a non-deformable suspension and a fixed mobile platform. As a discrete time, classical mathematical model, one takes an approximation of the derivative by a difference [6].  
The variable fractional-order difference equation model approach is based on the general form of a linear, time-invariant variable fractional-order difference equation:  
\[\sum_{i=0}^{n}A_{i0}^{GL}\Delta_{k}^{(\
u_{k})}y_{k}=\sum_{j=0}^{m}B_{j0}^{GL} \Delta_{k}^{(\mu_{k})}u_{k}, \tag{27}\]  
where \(A_{i},B_{j}\) are constant coefficients with \(A_{n}=1;\
u_{n,k}>v_{n-1,k}>\ldots>v_{1,k}>v_{0,k}=0\) are order functions; \(\mu_{m,k}>\mu_{m-1,k}>\ldots>\mu_{1,k}>\mu_{0,k}\geq 0\) are order functions; \(u_{k},y_{k}\) are the robot arm input and output functions.  
The robot arm possesses the integration element properties. Hence, if a classical backward difference is considered, then from (27) one obtains the equation:  
\[\begin{array}{c}{}_{0}^{GL}\Delta_{k}^{(\
u_{k})}y_{k}=B_{0}u_{k},\end{array} \tag{28}\]  
Describing a variable fractional-order integrator, two parameters to be selected: a coefficient \(B_{0}\) and an order function \(v_{k}\). Note that this is a generalization of the fractional-order discrete integrator, which in turn is a generalization of the classical (first-order) integrator. It was shown that (28) can be transformed to an equivalent form.  
\[y_{k}=B_{0}^{GL}\Delta_{k}^{(\mu_{k})}u_{k}, \tag{29}\]  
where \(\mu_{k}=-v_{k}\). The non-linear behavior may be expected, so it can be taken as \(B_{0}(u_{k})\).  
The final model is represented by Equation [6]:  
\[y_{k}=B_{0}(u_{k})_{0}^{GL}\Delta_{k}^{[\mu_{k}(u_{k})]}u_{k}, \tag{30}\]  
where \(\mathbf{y}_{k}\) represents the output or response of the system at discrete time \(k\), \(B_{0}(u_{k})\) represents the coefficient or function associated with the input \(u_{k}\) at time \(k\), and \(\Delta_{k}^{[\mathbf{y}_{k}(u_{k})]}\) represents the variable fractional-order backward difference of the response \(y_{k}\) at time \(k\) with respect to the order \(\mu_{k}\).  
Comparing these models with the observed values of the robot arm, the model based on the variable fractional-order difference equation yields better results. Further research is needed in two areas, according to the article: one is the effect of platform movement on the tested system, and the other is the selection of the appropriate order function with respect to the recorded real-time data [6].  
The variable fractional-order difference equations model performed better than the classical Newton-Euler formalism model, as presented by the final simulation results. Integral Squared Error and Integral Absolute Error were the two performance criteria that were utilized for judgment. It was found that the variable fractional-order model, with its higher number of parameters than the classical model, more accurately depicts the behavior of the robot arm in practice. The open problem of the choice of a suitable order function that can be effectively applied to real-time system signals is one of the prominent problems of the research. It is suggested that future work investigate the impact of the mobility of the movable platform on the robot arm's performance. In addition, the robustness of the suggested model was verified through comparisons of simulated outcomes with experimental results. Overall, the results indicate that this new direction has a very high chance of increasing the accuracy in modeling transitory behaviors and creating sophisticated control schemes for use in robotics [7].