## 4 Construction and Operation of Haptic Equipment
### Design of the Haptic Equipment  
#### 4.1.1 Mechanism Design  
The degree of freedom (DOF) of the haptic device is the number of directions of the allowed motion. The proposed haptic device must respond to the user's action by providing the necessary force and vibration feedback as in the process of sawing and, at the same time, to guide the handle of the device on the required trajectory. Usually the force-feedback can be accomplished with admittance and impedance control, depending on how it is reacting to displacement or force. In this case, the two types of feedback (force and trajectory) have to be coupled: the device must respond to the user's pulling and pushing movements with the proper force and trajectory. The solution is a haptic device with a kinematic structure with two DOF and with two actuators. This structure will couple the feedback from two directions. Other requirements are a small dimension of the parts to decrease the inertia, a reduced number of mechanical joins to reduce the friction (rotation joins are preferred), and a high structural stiffness to enable quick movements. The trajectory of the handle has to be close to linear, and the position and trajectory of the handle has to maximize the range of motions for the wrist and elbow. A close chain mechanism with two DOF can be designed with five pairs with one degree of freedom for each. The solution with only revolute joints is presented in Figure 9. The pairs can be replaced one by one with translational joins, but this solution does not respect the requirements of low friction, low inertia, and high stiffness for quick response. Figure 8: Distribution of frequency: (**a**) on x axis-longitudinal, (**b**) on y axis–transversal, (**c**) on z axis–vertical. The proposed solution for the haptic device has five revolute joins (A-E) and four links (Figure 9). Two motors electric motors are present in the haptic device, one in revolute G (Motor I) and one in E (Motor II). In addition, one rotational sensor is present for the measurement of the rotation in the joint G. Motor I is responsible for providing the force feedback. The transmission angle is an indication of the efficiency of a power transmission. This is defined by the acute angle between the directions of the velocity-difference vector of the driving point and the absolute-velocity vector of the driven point. The maximum efficiency in the actuators' torque transmission is in the position when link one is perpendicular to link two, and link four to link two. Due to the long stroke of the handle, these perpendicularity conditions were chosen to be fulfilled only at the middle of the handle's path. The horizontal displacement of the handle (\(x_{A}\)) is imposed by the user. The task of Motor II is to guide the handle in the vertical direction by imposing the \(y_{A}\) coordinate of the handle and to generate the vibrations in the vertical direction. The angle \(\theta_{1}\) is measured by the rotational sensor, so the coordinates of revolute joint B (Equation (1)) can be easily found:  
\[x_{B}=x_{G}+l_{BG}\cos\theta_{1} \tag{1}\]  
\[y_{B}=y_{G}+l_{BG}\sin\theta_{1}\]  
where \(l_{BG}\) is the length of the link one, the distance between the two revolute joints' axis of rotation. By imposing the vertical position \(y_{A}\) of the handle, the coordinates of the revolute joint C will be as follows (Equation (2)):  
\[x_{C}=x_{B}+l_{BC}\cos(\mathrm{asin}(y_{B}-y_{A})/l_{BA}) \tag{2}\]  
\[y_{C}=y_{B}+l_{BC}\sin(\mathrm{asin}(y_{B}-y_{A})/l_{BA})\]  
The angle \(\theta_{2}\) which is imposed by the second motor and will generate the required trajectory can be computed as follows (Equation (3)):  
\[\theta_{2}=2\tan^{-1}\left(-K-\sqrt{K^{2}+L^{2}-M^{2}}\right)/(L-M) \tag{3}\]  
where  
\[K=2l_{DE}(y_{C}-y_{E}) \tag{4}\]  
\[L=2l_{DE}(x_{C}-x_{E}) \tag{5}\]  
\[M=l_{DE}{}^{2}+\left(x_{C}-x_{E}\right)^{2}+\left(y_{C}-y_{E}\right)^{2}-l_{CD} {}^{2} \tag{6}\]  
and \(l_{DE}\) is the length of link four and \(l_{CD}\) is the length of link three.