# New Approach of the Variable Fractional-Order Model of a Robot Arm
## 3 The Robot Arm Description
In order to achieve the optimal effect, the following links position and speed functions were considered during numerical simulations:  
\[f(x)=\Big{(}\varphi_{1}-\frac{\pi}{2}\Big{)}^{2}+\Big{(}\varphi_{2}-\frac{\pi}{ 2}\Big{)}^{2}, \tag{25}\]  
\[f(v)=\dot{\varphi}_{1}^{2}+\dot{\varphi}_{2}^{2}, \tag{26}\]  
When a function (25) reaches the minimum value of zero, the vertical position of the links is realized (Figure 2b). On the other hand, the minima of the function (26) indicate the lowest possible velocities of the manipulator links after a given time \(\Delta t\). Figure 3a,b show three-dimensional projections of surfaces representing the functions _f(x)_ and _f(v)_ over the two-dimensional space of parameters \(M_{0}\), and \(M_{1}\), while Figure 4a,b shows (in red) areas of these parameters, where functions (25) and (26) reach minimum values. The calculations were carried out by numerically integrating the equations of motion using the 4 th order Runge-Kutta method. The time values of the drive parameters (Figure 1) are \(\Delta t_{0}\) = 0.05 [s], \(\Delta t_{1}\) = 0.45 [s], \(\Delta t_{2}\) = 0.30 [s], and \(\Delta t_{3}\) = 0.20 [s]. The time intervals of the driving torques were selected based on numerical analysis, which was carried out at the fixed values \(M_{0}\) = 9.0 [Nm], \(M_{1}\) = 48.0 [Nm], and the constant on start-up phase \(\Delta t_{0}\) = 0.05 [s]. However, the parameters \(\Delta t_{1}\) and \(\Delta t_{2}\) changed during  
Figure 4: The red color indicates areas of these parameters where f(x) and f(v) reach minimum values. (**a**) The areas of parameters M\({}_{0}\) and M\({}_{1}\) where f(x) reaches minimum values. (**b**) The areas of parameters M\({}_{0}\) and M\({}_{1}\) where f(v) reaches minimum values. Figure 3: Three-dimensional projections of surfaces representing functions f(x) and f(v) over the two-dimensional parameter space \(M_{0}\), \(M_{1}\). (**a**) Three-dimensional projections of surfaces representing f(x) functions over the two-dimensional parameter space M\({}_{0}\), M\({}_{1}\). (**b**) Three-dimensional projections of surfaces representing f(v) functions over the two-dimensional parameter space M\({}_{0}\), M\({}_{1}\). analysis (\(\Delta t_{3}\) is the result of select the remaining intervals). The results of this numerical experiment are presented in Figure 5a,b, where the quantities f(x) and f(v) are given as a function of the time intervals \(\Delta t_{1}\) and \(\Delta t_{2}\). It can be seen that the possibly low-speed values (green areas in the bottom left of the diagram) in Figure 5b coincide with the range where the manipulator reaches the vertical position (green stripe in Figure 5a). In the last phase of movement, \(\Delta t_{3}\) the drive is switched off. Such control of drive torques allows us to achieve the lowest possible values of velocities and angular accelerations when the manipulator links reach the vertical position. This fact is confirmed by the time courses of displacements, velocities, and accelerations presented in Figure 6. The determined driving torques are \(M_{0}=9.46\) [Nm] and \(M_{1}=49.44\) [Nm]. These moments deviate slightly from the optimal values resulting from Figure 4a,b (areas marked in red). The reason for this slight discrepancy may be a different method ([PERSON]'s algorithm) or the step of integration of the equations of motion in the case of the diagrams shown in Figure 6. It can be seen that at \(\Delta t=1.0\) [s] the cells occupy the required vertical position \(\varphi_{1}=\varphi_{2}=\pi/2\) at relatively low values of velocities and accelerations (Figure 6). Figure 7 shows the results of the numerical analysis analogous to the one shown in Figure 6, but taking into account the trajectory limitation of the links by the presence of the base. Contact with the ground means that the first link cannot occupy positions defined by a negative angle \(\varphi_{1}\). Hence, when integrating equations, an additional condition was introduced in the form \"if \(\varphi_{1}<0\) then \(\varphi_{1}=0\), \(\dot{\varphi}_{1}=0\)\".