# Single Cylinder-Type Piezoelectric Actuator with Two Active Kinematic Pairs
## 3 Mathematical Model of the Contact Force Control  
A mathematical model for sphere positioning in space through contacting force control was developed. One active kinematic pair of the actuator was used for sphere positioning on the plane. This kind of positioning is performed by moving the piezoelectric cylinder on the plane. Another active kinematic pair is used to rotate the sphere about three axes. All previously studied positioning systems had only one kinematic pair, i.e., the active cylinder was moved on the plane or the sphere was rotated about different axes. Combining two kinematic pairs and applying the brakes, as shown in Figure 3, we obtain one active cylinder with two kinematic pairs (Figure 4a). The configuration of electrodes should be as follows (Figure 4b):  
* the sphere rotates on the cylinder and the cylinder does not move, when electrodes 3, 4, 5 are excited
* the cylinder moves on the plane and the sphere does not rotate, when electrodes 6, 7, 8 are excited  
Different electrode switching and trajectory planning algorithms of the cylinder movement on the plane and sphere rotation are developed [16; 17]. However, there is no research performed and trajectory planning algorithm developed, when two active kinematic pairs are used. Therefore, a new algorithm must be created. Based on the previous investigation of trajectory planning algorithms, it can be concluded that only a fine trajectory planning algorithm is suitable for this task [14; 15]. The fine trajectory planning algorithm allows exciting two or more electrodes of the cylinder at the same time for obtaining a linear motion of the cylinder or rotation of the sphere. This method has the advantage of achieving the maximum available speed of movement, when certain contact forces are used. Therefore, the main task of the trajectory planning algorithm is to calculate the function of the contacting forces that can be obtained by exciting a particular electrode section of the piezoelectric cylinder. The cylinder moves on the plane by the trajectory described as function \(C(t)\) (Figure 8a), the sphere rotates around its central axis by the trajectory described as function \(S(\tau)=\overrightarrow{P}\left(\left|\overrightarrow{R}\right|,\theta(\tau ),\phi(\tau)\right)\) (Figure 8b), where \(\theta(\tau)\in[0,\pi]\) is an angle between positive \(z\) axis and \(\overrightarrow{r};\phi(\tau)\in[0,2\pi]\) is an angle between positive \(x\) axis and projection in \(xy\) plane of \(\overrightarrow{r}\) (Figure 8); \(t\) and \(\tau\) are parameters of the trajectories. Each kinematic pair is controlled at different value intervals:  
\[ \begin{split}\left|\overrightarrow{f}_{I}\right|& \in\left[f^{t}_{min^{\prime}},f^{t}_{max}\right]\\ \left|\overrightarrow{f}_{I\tau}\right|&\in\left[f^{ \tau}_{min^{\prime}},f^{t}_{max}\right]\end{split}  \tag{1}\]  
Figure 7: Trajectories of the contact point. The orientation of contact forces \(J=3,4,5\) of the sphere does not change throughout the movement time:  
\[\overrightarrow{f_{J}}^{\circ}=(\cos(\alpha_{\gamma t}),\cos(\beta_{\gamma t}), \cos(\gamma_{\gamma t})) \tag{2}\]  
where \(\alpha_{I},\beta_{I},\gamma_{I}\) is an angle between \(x,y,z\) axis and forces \(\overrightarrow{f}_{\gamma t}\). The rotation direction of the sphere is the same as the direction of the unit tangent vector at point \(\tau\) of function \(S(\tau)\). The unit tangent vector also shows the direction, in which the total force must be active. The unit tangent vector \(\overrightarrow{T}(\tau)=\langle T_{x}|T_{y}|T_{z}\rangle\) is calculated by the formula [14]:  
\[\overrightarrow{T}(\tau)=\frac{\overrightarrow{\tau}^{\prime}(\tau)}{\left| \overrightarrow{\tau}^{\prime}(\tau)\right|} \tag{3}\]  
The distance from the center of the sphere to the plane, where the contacts of the cylinder are located, are equal to \(d=\sqrt{R^{2}-D^{2}}\), where \(R\) is the radius of the sphere; \(D\) is the radius of the cylinder and \(d<R\).