# Design and Impedance Control of a Traveling Wave Ultrasonic Motor with a Spherical Rotor
## 5 The Problem of Direct Torque Control  
In order to test the effects of the force feedback based on direct torque control of the TWUM, the proportional-integral-derivative (PID)-based control scheme whose task it was to make the motor behave as a linear spring with ideal stiffness of 0.5 mNm/\({}^{\circ}\) was formulated, as shown in Figure 7. When the operator operates the handle to rotate output shaft and the brake is set free, the desired torque \(\tau_{d}\) is calculated through multiplying the actual angular displacement (\(\theta_{0}-\theta_{r}\)) using the virtual stiffness \(K_{d}\), where \(\theta_{0}\) is the initial angle of the handle and \(\theta_{r}\) is the actual angle of the handle. The torque error (\(e_{\tau}=\tau_{d}-\tau_{r}\)) is the input into PID force controller to calculate the control variable, where the \(\tau_{r}\) is the actual interactive torque imposed by the operator. It is noted that the signal collected by the dynamic torque transducer often contained various noises, which could not accurately reflect the output state of the motor and was not suitable for close-loop control system. Therefore, a low-pass filter circuit and a Kalman filter were used to denoise the original signal [21]. The phase difference was modulated using a limiting discrete PID controller according to:  
\[\varphi(k)=\left\{\begin{array}{c}\pi/2,\text{ while }\varphi(k)\geq\pi/2\\ K_{P}e_{\tau}(k)+K_{I}\sum\limits_{i=0}^{k}e_{iI}+K_{D}[e_{\tau}(k)-e_{\tau}(k -1)]\\ -\pi/2,\text{ while }\varphi(k)\leq-\pi/2\end{array}\right.. \tag{5}\]  
\begin{table}
\begin{tabular}{c c c c c c} \hline \hline
**Driving** & **Driving Voltage** & **Preload** & **Holding** & **Stalling** & **Maximum** \\
**Frequency** & **(peak-peak)** & **Proj** & **Torque** & **Torque** & **Speed** \\ \hline
44.5 kHz & 300 V & 3.5 kgf & 0.18 Nm & 0.12 Nm & 210 rpm \\ \hline \hline \end{tabular}
\end{table}
Table 2: Properties of the designed TWUM with a spherical rotor.  
Figure 6: Steady-state speed versus phase difference under various external torques.  
In this experiment, the operator rotates the handle backwards and forwards. When the handle is in the initial position, the operator should not feel any resistance. When pushing the handle to rotate, the resistance felt by the operator should be proportional to the angular displacement.  
The results given in Figure 8 show that there was a significant difference between the measured and desired torque with large oscillations of the force feedback. In order to quantify the force feedback effect, the actual stiffness shown in Figure 8 was calculated from the measured data obtained during the experiment.  
When the handle position was close to the initial position, the stiffness value changed greatly. The enlarged view of data plot in Figure 8b shows that the stiffness value had a large jump when  
Figure 8: Force feedback based on the direct torque control: (**a**) comparison of the measured and desired torque, and (**b**) the calculated stiffness with focused detail.  
Figure 7: Direct torque control scheme.  
the direction changed. According to the statistics of the stiffness data, the mean value was about 0.483 mNm/\({}^{\circ}\) which was close to the desired value, but the standard deviation was about 1.955 mNm/\({}^{\circ}\), which is very large and meant that the operator felt very unreal. The reason seems to be the disadvantage of about \(\pm 20^{\circ}\) of control dead zone. In addition, Equation (3) shows that the parameters affecting the torque were not easy to determine.