## 1 Introduction
### Description of Example Manipulators  
For the example manipulators used in this paper (Figures 1-3), the fixed and moving platforms are described by the frames \(\{O\}\) and \(\{O^{\prime}\}\), respectively. The orientation of the moving plate with respect to the base plate is represented by:  
\[\mathbf{R}=\mathbf{R}_{y}(\theta)\mathbf{R}_{x}(\psi)\mathbf{R}_{z}(\phi), \tag{1}\]  
whereas the position of the moving plate with respect to the base is described by a vector \(\mathbf{p}\). Considering the moving plates as a rigid plane, the position of spherical joints is described by the vector \(\mathbf{a}_{i}\), which is directed from \(O^{\prime}\) to \(A_{i}\) and written as follows:  
\[\mathbf{a}_{i}^{\prime}=\mathbf{R}_{z}(\alpha_{i})\mathbf{T}_{x}(r_{p}). \tag{2}\]  
Variable \(r_{p}\) in Equation (2) implies the radius of the moving platform's circumscribed circle and \(\mathbf{T}_{x}(\cdot)\) represents a translation vector in the x- direction by the specified magnitude, in this case \(r_{p}\). The angle \(\alpha_{i}\) defines the location of three spherical joints measured from the positive x-axis parallel to the moving platform and is given as:  
\[\alpha_{i}=(i-1)\frac{2\pi}{3},\quad\text{ for }\;i=1,2,3. \tag{3}\]  
whereas the revolute joints located on the fixed platform can be measured by the angle \(\xi_{i}\) from the positive x-axis and parallel to the fixed platform as in Equation (4).  
\[\xi_{i}=(i-1)\frac{2\pi}{3},\quad\text{ for }\;i=1,2,3. \tag{4}\]  
The vector from \(\{O\}\) to \(\{B_{i}\}\) can be denoted by \(\mathbf{b}_{i}\). So, it can be described by the following relation:  
\[\mathbf{b}_{i}=\mathbf{R}_{z}(\xi_{i})\mathbf{T}_{x}(b_{i}). \tag{5}\]  
Note that \(r_{p}\) and \(b_{i}\) of Figures 1 and 2 are initially the same and can be equally represented in Equation (5). In solving the inverse kinematics, \(b_{i}\) is the varying length associated to the prismatic joint. Accordingly, for the manipulators shown in Figures 1 and 2, the kinematic transformation from \(\{O\}\) to \(\{O^{\prime}\}\) can be represented by the following relation:  
\[\begin{split}\mathbf{HR}z(\alpha_{i})\mathbf{HT}x(r_{p})&=\bm {HR}z(\xi_{i})\mathbf{HT}x(b_{i})\mathbf{HR}z(\theta_{2i})\mathbf{HT}z(l_{i})\mathbf{HR}x(\theta _{3i})\times\\ &\mathbf{HR}y(\theta_{4i})\mathbf{HR}z(\theta_{5i}),\end{split} \tag{6}\]  
where \(\mathbf{H}\in SE(3)=\{(\mathbf{p},\mathbf{R}):\mathbf{p}\in\mathbb{R}^{3},\mathbf{R}\in\SO(3)\}\), which is a Euclidean space group. Note that the radius of the fixed platform is not constant for the 3-PhRS manipulator because of the actuator layout. Thus, vector \(\mathbf{b}_{i}\) in Equation (5) is also not constant for this manipulator.  
Similarly, the transformation from \(\{O\}\) to \(\{O^{\prime}\}\) for the manipulator shown in Figure 3 can be given by:  
\[\begin{split}\mathbf{HR}z(\alpha_{i})\mathbf{HT}x(r_{p})&= \mathbf{HR}z(\xi_{i})\mathbf{HT}x(b_{i})\mathbf{HT}z(d_{i})\mathbf{HR}z(\theta_{2i})\mathbf{HT}z(l _{i})\mathbf{HR}x(\theta_{3i})\times\\ &\mathbf{HR}y(\theta_{4i})\mathbf{HR}z(\theta_{5i}).\end{split} \tag{7}\]  
Using Equations (6) and (7), the position relation can be derived. However, the detail is left for the readers since it is not complicated, and not the purpose of this paper.  
Figure 1: Kinematic architecture of the 3-RPS manipulator.  
Figure 2: Kinematic architecture of the 3-PhRS manipulator.