# Kinematic and Workspace Analysis of RRU-3 RSS: A Novel 2T2R Parallel Manipulator
## 3 Kinematic Analysis
### Velocity Problem  
The loop closure equations for the mechanism can be rewritten as Equation (12). By differentiating it with respect to time, we obtain the velocity problem equations shown in Equation (13). \[\mathbf{P}=\mathbf{C}_{i}+\mathbf{C}_{i}\mathbf{B}_{i}+\mathbf{B}_{i}\mathbf{A}_{i}+\mathbf{A}_{i}\mathbf{P} \tag{12}\]  
\[\mathbf{v}_{P}=\mathbf{\Omega}_{i1}\times\mathbf{C}_{i}\mathbf{B}_{i}+\mathbf{\Omega}_{i2}\times\bm {B}_{i}\mathbf{A}_{i}+\mathbf{\Omega}_{P}\times\mathbf{A}_{i}\mathbf{P} \tag{13}\]  
where \(\mathbf{v}_{P}\) and \(\mathbf{\Omega}_{P}\) are the linear and angular velocity of the mobile platform and, for each \(i\)-th KC, \(\mathbf{\Omega}_{i1}\) is the angular velocity of the lower links and \(\mathbf{\Omega}_{i2}\) is the angular velocity of the upper link. In order to cancel the terms corresponding to passive velocities, we dot-multiply Equation (13) by the unit vectors \(\mathbf{s}_{i}\), defined in the direction of the upper links (Equation (14)), and we obtain Equation (15). \[\mathbf{s}_{i}=\frac{\mathbf{B}_{i}\mathbf{A}_{i}}{\left\|\mathbf{B}_{i}\mathbf{A}_{i}\right\|} \tag{14}\]  
Figure 3: Example of the symbolic notation used for the inverse kinematic problem solutions (working modes). Figure 2: Equivalence between circumference-sphere intersection and two circumferences intersection in RSS KCs. \[\mathbf{s}_{i}\mathbf{T}\cdot\mathbf{v}_{P}=\mathbf{s}_{i}\mathbf{T}\cdot(\mathbf{\Omega}_{i1}\times C_{i} \mathbf{B}_{i})+\mathbf{s}_{i}\mathbf{T}\cdot(\mathbf{\Omega}_{P}\times\mathbf{A}_{i}\mathbf{P}) \tag{15}\]  
Considering the existing constraints, we can express the linear velocity of \(\mathbf{P}\) as Equation (16), the angular velocity of the mobile platform as a combination of the aforementioned two rotations (Equation (17)), and the angular velocity of the lower links as the angular rotation of the rotatory actuators, as shown in Equation (18). \[\mathbf{v}_{P}=\dot{\mathbf{x}}\cdot\mathbf{i}+\dot{z}\cdot\mathbf{k} \tag{16}\]  
\[\mathbf{\Omega}_{P}=\dot{\varphi}\cdot\mathbf{j}+\dot{\theta}\cdot\mathbf{u}\quad\mathbf{u}=c \varphi\cdot\mathbf{i}-s\varphi\cdot\mathbf{k} \tag{17}\]  
\[\mathbf{\Omega}_{i1}=\dot{\beta}_{i}\cdot\mathbf{j} \tag{18}\]  
We obtain Equation (19) by isolating all the input (\(\dot{\beta}_{i}\)) and output (\(\mathbf{v}_{P}\) and \(\mathbf{\Omega}_{P}\)) terms in Equation (15) on different sides of the equation. \[\mathbf{s}_{i}\mathbf{T}\cdot\mathbf{v}_{P}+(\mathbf{PA}_{i}\times\mathbf{s}_{i})^{T}\cdot\mathbf{ \Omega}_{P}=(\mathbf{C}_{i}\mathbf{B}_{i}\times\mathbf{s}_{i})^{T}\cdot\dot{\beta}_{i} \cdot\mathbf{j} \tag{19}\]  
We write Equation (19) in matrix notation, and we obtain Equation (20), which represents the input-output velocity equation of the mechanism,  
\[\mathbf{v}_{P}=\mathbf{\Omega}_{i1}\times\mathbf{C}_{i}\mathbf{B}_{i}+\mathbf{\Omega}_{i2}\times\bm {B}_{i}\mathbf{A}_{i}+\mathbf{\Omega}_{P}\times\mathbf{A}_{i}\mathbf{P} \tag{13}\]0  
where \(\mathbf{J}_{x}\) is the Jacobian matrix of the direct kinematic problem (DKP), \(\mathbf{J}_{q}\) is the Jacobian matrix of the inverse kinematic problem (IKP), \(\dot{\mathbf{x}}\) is the input velocity vector and \(\dot{\mathbf{q}}\) is the output velocity vector. Their expressions are shown in the following equations:  
\[\mathbf{J}_{x}=\left[