# Design and Testing of Two Haptic Devices Based on Reconfigurable 2R Joints
## 2 Reconfigurable Joint Designs  
In this paper, two reconfigurable joint designs are considered: an active 2R spherical mechanism-based joint (see Figure 1a) and a differential gear-based design (see Figure 1b). The complete kinematic analysis, workspace characterization and finite element analysis (FEA) of the active 2R spherical mechanism-based joint can be found in previous works [3,28]. The kinematic analysis and quasi-static analysis of the differential gear-based joint can be found in a previous study [29]. Subsequently, we indicate the main features and analyses of those previous works, in order to understand the functioning of the joints and hence to be able to follow the current developments. In Figure 1a it can be seen that the active 2R spherical mechanism-based joint consists of several main parts: the round base 1 with the top ring and the central sphere 2. The  
Figure 1: The reconfigurable joint designs: (**a**) an active 2R spherical mechanism-based joint; (**b**) differential gear-based joint. central sphere has two slots with rectangular profiles through which it is actuated with two sets of cranks. The red cranks 3 control one DOF of the joint (angle \(\psi\)) directly. The other DOF (angle \(\lambda\)) is controlled with the blue crank 4 through the connection link 5. The differential gear based joint (see Figure 1b) consists of the base 1, central sphere 2 and the red input part 3. The red input part is designed to be actuated with the belt drive and controls the output angle \(\psi\). The output angle \(\lambda\) is controlled with two blue shafts 4 through the bevel gear 5. The blue shafts can be directly connected to the actuators. In order to understand the operation of the active 2R spherical mechanism-based joint better, information on its kinematics should be provided. The position problem can be formulated as  
\[\left\{ \begin{array}{c}f_{1}:\psi+\alpha=0\\ f_{2}:\left(r^{2}sin\beta^{2}+(rcosacos\beta-lsin\alpha)^{2}\right)^{0.5}sin \lambda+rsin\beta=0\end{array} \right. \tag{1}\]  
where \(\alpha\) and \(\beta\) are input variables, \(\psi\) and \(\lambda\) are output variables, and \(r\) and \(l\) are geometrical parameters of the joint. More information can be found in [28]. The active 2R spherical mechanism-based joint Jacobian can be written as follows:  
\[\left[ \begin{array}{c}1\\ 0\end{array} \right.\left(r^{2}sin\beta^{2}+(rcosacos\beta-lsin\alpha)^{2} \right)^{0.5}\cos\lambda\left.\right]\left\{ \begin{array}{c}\dot{\psi}\\ \dot{\lambda}\end{array} \right\}=\left[ \begin{array}{cc}-1&0\\ \frac{-\partial f_{2}}{\partial\alpha}&\frac{-\partial f_{2}}{\partial\beta} \end{array} \right]\left\{ \begin{array}{c}\dot{\alpha}\\ \dot{\beta}\end{array} \right\} \tag{2}\]  
where  
\[\frac{\partial f_{2}}{\partial\alpha}=\frac{\sin\lambda(l\cos\alpha+r\cos \beta\sin\alpha)(l\sin\alpha-r\cos\beta\cos\alpha)}{\left(r^{2}(\sin\beta)^{2} +(l\sin\alpha-r\cos\beta\cos\alpha)\right)^{0.5}} \tag{3}\]  
\[\frac{\partial f_{2}}{\partial\beta}=r\cos\beta+\frac{\sin\lambda\left(2r^{2} \cos\beta\sin\beta+2r\cos\alpha\sin\beta(l\sin\alpha-r\cos\beta\cos\alpha) \right)}{2\left(r^{2}(\sin\beta)^{2}+(l\sin\alpha-r\cos\beta\cos\alpha) \right)^{0.5}} \tag{4}\]  
From Equation (2), it can be found that for this joint design, two direct singular positions exist within the practical workspace, satisfying the following equations:  
\[sin\beta=0\rightarrow\beta=0\pm\pi \tag{5}\]  
\[rcosacos\beta-lsin\alpha=0\rightarrow\beta=0\to tan\alpha=\frac{r}{l} \tag{6}\]  
These singular positions exist for each of the blue actuators; however, they cannot be reached simultaneously. For the current design parameters for one of the actuators the singular position can be reached when the input angles satisfy the range \(\alpha=25^{\circ},\ \beta=\pm\pi\); for the other actuator the range is \(\alpha=-25^{\circ},\ \beta=\pm\pi\). Thus, one of the actuators will always keep the joint under control. A detailed study of this topic can be found in [28].