## 3 Analysis of Kinematics  
This paper uses a vector-loop closure equation [21] for each arm to solve the solutions of the inverse kinematic model for the PH-LDR. Figure 3 shows the coordinate frames for the PH-LDR. The three arms of the PH-LDR have an identical kinematic structure. Two coordinate frames are used to analyze the kinematic model of the PH-LDR. There is a fixed Cartesian reference frame \(\{O\}\), for which the origin is located at the point \(O(x_{0},y_{0},z_{0})\) on the fixed base. The frame \(\{O\}\) also serves as the world coordinate. There is also a mobile Cartesian reference frame \(\{P\}\), for which the origin is at the center \(P\big{(}x_{p0},y_{p0},z_{p0}\big{)}\) of the moving platform. \(A_{i}(i=1,2,3)\) are joints that are located at the center of the parallel chain of each arm and there is a pair of universal joints at its two ends and \(B_{i}(i=1,2,3)\) denote the location of joints on the moving platform. Three arms connect the joint \(A_{i}\) and the joint \(B_{i}\) and firmly connect the fixed base to the moving platform. In Figure 3, \(q_{i}(i=1,2,3)\) denote the actuated joint variables that are defined as the length between the point \(O(x_{0},y_{0},z_{0})\) and the points \(A_{i}\). The slider moves when the PH-LDR operates, so \(q_{i}\) are not constants. \(q_{\max}\) is the maximum value for \(q_{i}\) and are equal to the maximum stroke of the rod-less PAs. Table 1 shows the geometry parameters for the PH-LDR. In this study, \(q_{i}\) are defined as the joint variables; and the Cartesian variable vector \({}^{O}P_{P}=\left[x\,y\,z\right]^{\mathrm{T}}\) represents the coordinate of the point \(P\big{(}x_{p},y_{p},z_{p}\big{)}\) with respect to the frame \(\{O\}\). The coordinates of the vector \({}^{O}\overrightarrow{OA_{i}}\) with respect to the frame \(\{O\}\) are:  
\[{}^{O}\overrightarrow{OA_{i}}=q_{i}\overrightarrow{\mathbf{u}}_{i} \tag{1}\]  
where \(\overrightarrow{\mathbf{u}}_{i}=\Big{[}\ \cos\!\Big{(}(i-1)\times 120^{0} \Big{)}\ \ \sin\!\Big{(}(i-1)\times 120^{0}\Big{)}\ \ \ 0\ \Big{]}^{\mathrm{T}}\) is the unit vector along \({}^{O}\overrightarrow{OA_{i}}(i=1,2,3)\). For simplicity, the unit vector of \({}^{P}\overrightarrow{PB_{i}}\) is also defined as the vector \(\overrightarrow{\mathbf{u}}_{i}\) because the orientation of \(\{P\}\) is identical to the orientation of \(\{O\}\) for this design. The coordinate of the vector \({}^{P}\overrightarrow{PB_{i}}\) with respect to the frame \(\{P\}\) is:  
\[{}^{P}\overrightarrow{PB_{i}}=r\overrightarrow{\mathbf{u}}_{i} \tag{2}\]  
where \(r\) is the distance between \(P\big{(}x_{p0},y_{p0},z_{p0}\big{)}\) and \(B_{i}(i=1,2,3)\). In general, the superscript can be ignored if the vector is expressed with respect to the frame \(\{O\}\). In the design of the PH-LDR, the origin of the frame \(\{O\}\) shares the \((x-y)\) plane with the origin of the frame \(\{P\}\), and the orientation of \(\{P\}\) has an orientation that is identical to the orientation of \(\{O\}\), so \({}^{O}\overrightarrow{PB_{i}}\) is equal to \({}^{P}\overrightarrow{PB_{i}}\). For simplicity, therefore, \({}^{O}\overrightarrow{OA_{i}}\) is expressed as \(\overrightarrow{OA_{i}}\), and \({}^{O}\overrightarrow{PB_{i}}=^{P}\overrightarrow{PB_{i}}\) are expressed as \(\overrightarrow{PB_{i}}\).  
Figure 2: Operation at the initial position for the rod-less pneumatic actuators (Pas).