# Kinematic and Workspace Analysis of RRU-3 RSS: A Novel 2T2R Parallel Manipulator
## 3 Kinematic Analysis
### Position Problem Equations  
The loop closure equations of each KC are used to set up the position problem equations. The loop closure equation for the \(i\)-th KC is given by Equation (1). \[\mathbf{A}_{i}=\mathbf{C}_{i}+\mathbf{C}_{i}\mathbf{B}_{i}+\mathbf{B}_{i}\mathbf{A}_{i}\ \ \ \ \ i=1,\ldots,\,4 \tag{1}\]  
As shown in Figure 1, \(\mathbf{C}_{i}\) corresponds to the coordinates of the actuated kinematic joints of the mechanism. Their values are defined in Equation (2). \[\mathbf{C}_{1}= \begin{bmatrix}-F&0&0\end{bmatrix} ^{T}\ \ \ \ \ \mathbf{C}_{2}= \begin{bmatrix}0&-N&0\end{bmatrix} ^{T}\ \ \ \ \ \mathbf{C}_{3}= \begin{bmatrix}F&0&0\end{bmatrix} ^{T}\ \ \ \ \ \mathbf{C}_{4}= \begin{bmatrix}0&N&0\end{bmatrix} ^{T} \tag{2}\]  
\(\mathbf{B}_{i}\) points refer to the coordinates of the intermediate kinematic joints. These points always remain in planes parallel to \(XZ\) plane, and their coordinates depend on the values of the actuators in each position (\(\beta_{i}\)), as shown in Equation (3). Hereon, for brevity _sine_ and _cosine_ will be represented by \(s\) and \(c\), respectively. \[\mathbf{B}_{1}= \begin{bmatrix}-F-R\!\cdot\!c\beta_{1}&0&R\!\cdot\!s\beta_{1}\end{bmatrix}  ^{T}\ \ \ \ \mathbf{B}_{2}= \begin{bmatrix}-R\!\cdot\!c\beta_{2}&-N&R\!\cdot\!s\beta_{2}\end{bmatrix}  ^{T} \tag{3}\]  
Finally, the coordinates of the kinematic joints attached to the mobile platform, \(\mathbf{A}_{i}\), can be obtained using Equation (4)  
\[\mathbf{A}_{i}=\mathbf{p}+\textbf{Rot}\!\cdot\!\mathbf{a}_{i}^{T}\,i=1,\ldots,\,4 \tag{4}\]  
where \(\mathbf{p}= \begin{bmatrix}x&0&z\end{bmatrix} ^{T}\) is the position vector of \(\mathbf{P}\) in the fixed reference frame. \(\mathbf{Rot}\) is the rotation matrix that relates the mobile reference frame with the fixed one, whose expression (Equation (5)) can be obtained by applying a rotation \(\varphi\) about the \(Y\)-axis and a rotation \(\theta\) about the \(u\)-axis. Vectors \(\mathbf{a}_{i}\), given by Equation (6), define the position of the \(\mathbf{A}_{i}\) points in the mobile reference frame. \[\mathbf{Rot}= \begin{bmatrix}c\varphi&s\varphi\!\cdot\!s\theta&s\varphi\!\cdot\!c \theta\\ 0&c\theta&-s\theta\\ -s\varphi&c\varphi\!\cdot\!s\theta&c\varphi\!\cdot\!c\theta\end{bmatrix}  \tag{5}\]  
\[\mathbf{a}_{1}= \begin{bmatrix}-F&0&0\end{bmatrix} ^{T}\ \ \ \ \ \mathbf{a}_{2}= \begin{bmatrix}0&-N&0\end{bmatrix} ^{T}\ \ \ \ \mathbf{a}_{3}= \begin{bmatrix}F&0&0\end{bmatrix} ^{T}\ \ \ \ \ \mathbf{a}_{4}= \begin{bmatrix}0&N&0\end{bmatrix} ^{T} \tag{6}\]  
By substituting Equations (5) and (6) for each KC in Equation (4), the expression of the \(\mathbf{A}_{i}\) points can be obtained, as shown in Equation (7). \[\mathbf{A}_{1}= \begin{bmatrix}x-F\!\cdot\!c\varphi\\ 0\\ z+F\!\cdot\!s\varphi\end{bmatrix} \ \ \ \ \mathbf{A}_{2}= \begin{bmatrix}x-N\!\cdot\!s \varphi\!\cdot\!s\theta\\ -N\!\cdot\!c\theta\\ z-N\!\cdot\!c\varphi\!\cdot\!s\theta\end{bmatrix} \ \ \ \ \mathbf{A}_{3}= \begin{bmatrix}x+F\!\cdot\!c\varphi\\ 0\\ z-F\!\cdot\!s\varphi\end{bmatrix} \ \ \ \ \mathbf{A}_{4}= \begin{bmatrix}x+N\!\cdot\!s \varphi\!\cdot\!s\theta\\ N\!\cdot\!c\theta\\ z+N\!\cdot\!c\varphi\!\cdot\!s\theta\end{bmatrix}  \tag{7}\]  
Isolating the term \(\mathbf{B}_{i}\mathbf{A}_{i}\) of the loop closure equation and taking into account that the module of \(\mathbf{B}_{i}\mathbf{A}_{i}\) is the length of the upper links (\(L\)) of each KC, the Equation (1) can be