# New Approach of the Variable Fractional-Order Model of a Robot Arm
## 6 The VFOM and CM Parameters Selection  
In this Section, both models' parameter selection methods are described. The main fitting criterion is a minimum of the square error defined as a difference between measured real system responses \(y_{m,k}\) and simulated \(y_{s,k}\) ones  
\[I_{CM}=\min_{K,\left\lvert 0\right\rvert\,k}\left\lvert y_{m,k}-y_{s,k}(K) \right\rvert^{2} \tag{41}\]  
\[I_{VFOM}=\min_{B_{0},\lambda_{2},\left\lvert 0\right\rvert\,k}\left\lvert y_{m,k}-y _{s,k}(B_{0},\
u_{k})\right\rvert^{2} \tag{42}\]  
First, the coefficient \(B_{0}\) in formula (36) is evaluated. Here, one initially assumes that \(\
u_{k}=1\). Then, for \(u_{k}=u_{0}1_{k}\) (\(1_{k}\) denotes the unit discrete step function) one gets  
\[y_{j,k}=B_{0}\sum_{i=0}^{k}u_{i}=B_{0}u_{j,0}kh\ for\ u_{j,0}\in\mathbb{U} \tag{43}\]  
Figure 14: Robot arm photo.  
Figure 15: Robot arm measured step responses.  
where \(h\) denotes a sampling time. For the chosen time instants (for which the responses achieve for the first time its maximal value) from (43) one gets coefficients  
\[B_{j,0}(u_{j,0})=\frac{y_{k_{max}}}{u_{j,0}k_{max}h}\ for\ u_{j,0}\in\mathbb{U} \tag{44}\]  
Plots of \(B_{j,0}(u_{j,0})y_{j,k}\) are presented in Figure 16. The coefficients (44) are plotted in Figure 17.  
This enables us to evaluate the static characteristic of the model. It is presented in Figure 18. To check the response shapes, the time coefficients are added. The coefficients \(w_{j}(u_{j,0})\) are selected in order to get the same transient times. Comparison of all such responses revealed its changes due to the input signal value. In Figure 18 all mentioned responses are presented.  
Figure 16: Robot arm measured step responses with coefficients (44).  
Figure 17: Plot of coefficients \(B_{j,0}\big{(}u_{j,0}\big{)}\).  
Figure 18: Responses of the robot arm with transformed time.  
As the order functions, one takes  
\[\
u_{k}=-1+c(u_{k})e^{dk} \tag{45}\]  
with \(0<c(u_{k})<1,d<0\). For \(c(u_{k})\) and \(d\) parameters, selection of an ISE (integrated square error) criterion was applied.  
\[I[c(u_{k}),d]=\sum_{i=0}^{k_{\text{max}}}e_{i}^{2} \tag{46}\]  
where  
\[e_{k}=y_{k}-y_{s,k}=y_{k}-B_{0}(u_{k})_{0}^{GL}\Delta_{k}^{\left[-1+c(u_{k})e^ {dk}\right]}u_{k} \tag{47}\]  
For \(0<c(u_{k})<1\), a plot of coefficients \(c(u_{k})\) vs. \(u_{k}\) is presented in Figure 19.  
The responses that were measured and simulated for \(d=-0.001\) are presented in Figure 20. Figure 20 also shows the response obtained by simulation of the classical model where \(\frac{g(m_{1}l_{1}+m_{2}l_{1})}{l_{1}}=0.6\) and \(K=0.1787\) are added (in blue).  
Figure 19: Coefficient \(c(u_{k})\) vs. \(u_{k}\).  
Figure 20: Measured (black) and simulated fractional (red) and classical (blue) robot arm mathematical model step responses.  
In Figure 21, similar responses for different input signals are compared.  
To compare the simulation results, two performance criteria were used: IAE and ISE. The first error is defined by formula (45) and the second is defined as \(e_{k}=y_{k}-q_{1}(\text{kh})\) where \(h=0.0215\) denotes the simulation step. The obtained IAE and ISE criteria values are given in Figure 22.