## 3 Experimental Validation: FO-PI Design for a UR10 Robot
Thus, the transfer function of a motor drivetrain attached to a robot joint can be written as:  
\[\frac{\Omega(s)}{V_{a}(s)}=\frac{K_{m}K_{a}}{J_{m}s+B} \tag{16}\]  
However, for the purpose of designing the fractional order PI controller using the FO-KC auto-tuning principle, there is no need to determine the parameters of the actual model in (16), but only its frequency response at a selected gain crossover frequency, as well as the corresponding frequency response slope. The gain crossover frequency selected for the UR10 robot is \(\omega_{c}=15.7\) rad/s. A sine signal is applied on the robot reference as indicated next and shown in Figure 3:  
\[w(t)=5\sin(\omega_{c}t) \tag{17}\]  
It is important to note that an open loop response is impossible to obtain, therefore a closed loop test is performed with a simple P controller, with a proportional gain \(k_{p}=15\). Using this information, along with some filtering techniques [19; 22], it is possible to obtain the required robot frequency response and its corresponding frequency response slope:  
\[P(j\omega_{c})=-0.001143-0.02714j \tag{18}\]  
\[\frac{\mathrm{d}P(j\omega)}{\mathrm{d}\omega}\bigg{|}_{\omega=\omega_{c}}=-0. 0006172-0.001826j \tag{19}\]  
\begin{table}
\begin{tabular}{c c c c c} \hline
**Joint** & \(\theta\) & \(d\) & \(a\) & \(\kappa\) \\ \hline
1 & \(\theta_{1}\) & \(0.12\) & \(0\) & \(\pi/2\) \\
2 & \(\theta_{2}\) & \(0\) & \(-0.61\) & \(0\) \\
3 & \(\theta_{3}\) & \(0\) & \(-0.57\) & \(0\) \\
4 & \(\theta_{4}\) & \(0.16\) & \(0\) & \(\pi/2\)The imposed phase margin to design the FO-PI controller in (1) is selected as _PM_ = 60\({}^{\circ}\). The forbidden region center \(C\), and its radius, \(R\), are determined using (2):  
\[C=\
icefrac{{1}}{{cos(PM)}}=2,\ R=\sqrt{C^{2}-1}=1.73 \tag{20}\]  
The slope of the forbidden region border is next computed according to (6):  
Figure 2:0  
where \(\alpha=90^{{}^{\circ}}-PM=30^{{}^{\circ}}\). The controller frequency response can be computed considering (5):  
Figure 2:1  
where the loop frequency response is \(L(j\omega_{c})=M_{L}e^{j\varphi_{L}}=-0.5-0.866j\). Performing now the last step of the FO-KC auto-tuning method implies taking small increments for the fractional order \(\lambda\) and computing the parameters of the FO-PI, as well as the slope of the loop frequency response. Then, for each fractional order \(\lambda\), it is possible to evaluate the difference in (3) and select the minimum value. This then results in the optimal FO-PI controller:  
Figure 2:2  
To implement the controller in (23), a novel direct discretization method is used [23]. The approximation of the FO-PI controller to a discrete-time equivalent of order 4 is done using a sampling period of 0.1 s. The frequency response of the FO-PI controller in (23) and its discrete-time approximation is given in Figure 2:3  
The performance of the fractional order controller in (23), designed using the FO-KC auto-tuning method, has been evaluated in a real time application using a UR10 robot considering a pick and place movement. Since joints 2 and 3 have larger variation in comparison with other joints, the designed controller was implemented and applied to these joints. Multiple experiments were conducted in order to validate and test the robustness of the controller. For the pick and place task, the first experiment was conducted using the next configuration [\(-60\), \(-25\), \(-50\), \(-30\), \(170\), \(260\)] and the obtained results are given in Figures 5 and 6. Figure 2:4 Frequency response of the FO-PI controller and its discrete-time approximation. Figure 2:5 Signals used to determine the robot frequency response and the corresponding frequency response slope at the gain crossover frequency.