# Kinematic Analysis and Verification of a New 5-DOF Parallel Mechanism
## 2 Design and Verification of the Scheme
### Verification of the Primary Scheme
#### 2.2.2 Verification of Degree of Freedom by Screw Theory  
Traditional Grubler-Kutzbach method can only be calculated as five degrees of freedom of the mechanism, which is not clear about the specific situation of the degree of freedom. Therefore, after a long-term development, the idea of solving the degree of freedom of parallel mechanism by screw theory emerged. The degree of freedom of 5-PUS-UPU parallel mechanism is further analyzed by using screw theory. In screw theory, the screw is generally expressed as \((S;\ S_{0})\) or plucker coordinates \((a\ b\ c;\ l\ m\ n)\)[32]. \(S\) is the original part of the rotation and \(S_{0}\) is the dual part of the rotation. The reciprocal product of two screws \((\$_{1}=(S_{1};\ S_{1}^{0}),\ \$_{2}=(S_{2};\ S_{2}^{0}))\) is defined as the sum of the dot products after the exchange of the original and dual part  
\[\$_{1}\circ\$_{2}{=}S_{1}\cdot S_{2}^{0}+S_{2}\cdot S_{1}^{0} \tag{3}\]  
When the reciprocal product formed by \(\$_{1}\) and \(\$_{2}\) is 0:  
\[\$_{1}\circ\$_{2}{=}0 \tag{4}\]  
They are reciprocal, and the screw \(\$_{2}\) is called the reciprocal screw of \(\$_{1}\). Therefore, the reciprocal screw can be obtained based on the fact that the reciprocal product of the two screws is 0. The parallel mechanism designed in this paper uses five identical PUS limbs as the active limbs and one UPU limb as the constrained limb. Now the center point of the Hook joint is set as the origin \(o\), the directions of the two axes are the \(x\)-axis and the \(y\)-axis, respectively, and the \(z\)-axis is perpendicular to the \(xoy\) plane to establish a coordinate system, as shown in Figure 2.  
The kinematic pairs on the active limbs are sequentially decomposed into single-degree-of-freedom motions, and the achievable motion of active limb 1 is expressed in a screw form [33]:  
\[ \begin{array}{l}\$_{1}=(0\;0\;0;\;l_{1}\;m_{1}\;n_{1})\\ \$_{2}=(1\;0\;0;\;0\;0\;0)\\ \$_{3}=(0\;1\;0;\;0\;0\;0)\\ \$_{4}=(1\;0\;0;\;0\;m_{4}\;0)\\ \$_{5}=(0\;1\;0;\;l_{5}\;0\;0)\\ \$_{6}=(0\;0\;1;\;0\;0\;0)\end{array} \]  
where \(l_{1}\), \(m_{1}\), \(n_{1}\), \(m_{4}\), \(l_{5}\) represent coordinate parameters that are not zero, and \(\$_{1}\) is the screw representation of moving pair \(P_{1}\); \(\$_{2}\) is the spiral representation of degree of freedom of rotation of Hook joint \(U_{1}\) around \(x\)-axis; \(\$_{3}\) is the screw representation of degree of freedom of rotation of Hook joint \(U_{1}\) around \(y\)-axis; \(\$_{4}\) is the screw representation of degree of freedom of rotation of spherical hinge \(S_{1}\) around \(x\)-axis; \(\$_{5}\) is the screw representation of the degree of rotation of spherical hinge \(S_{1}\) about the \(y\)-axis, and \(\$_{6}\) is the spiral representation of the degree of rotation of spherical hinge \(S_{1}\) about the \(z\)-axis. The screw system consisting of \(\$_{1}\), \(\$_{2}\), \(\$_{3}\), \(\$_{4}\), \(\$_{5}\), and \(\$_{6}\) has been simplified to:  
\[\left(\begin{array}{cccccc}1&0&0\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\;\; \;\;\;\( \begin{array}{l}\$_{1}=(1\;0\;0;\;0\;0\;0)\\ \$_{2}=(0\;1\;0;\;0\;0\;0)\\ \$_{3}=(0\;0\;0;\;l_{3}\;m_{3}\;n_{3})\\ \$_{4}=(1\;0\;0;\;0\;m_{4}\;0)\\ \$_{5}=(0\;1\;0;\;l_{5}\;0\;0)\end{array} \)  
where \(l_{3}\), \(m_{3}\), \(n_{3}\), \(m_{4}\), \(l_{5}\) represent coordinate parameters and are not 0. The screw system consisting of \(\$_{1}\), \(\$_{2}\), \(\$_{3}\), \(\$_{4}\), and \(\$_{5}\) has been simplified to:  
\[\left( \begin{array}{ccccc}1&0&0&\;\;0&\;0&0\\ 0&1&0&\;\;0&0&0\\ 0&0&0&\;\;l_{5}&0&0\\ 0&0&0&\;\;0&m_{4}&0\\ 0&0&0&\;\;0&0&n_{3}\end{array} \right) \tag{6}\]  
Finally, the reciprocal screw of the constrained limb is: \(\$_{r}=(0\;0\;0;\;0\;0\;1)\). From the physical meaning of the screw and the reciprocal screw, when the screw represents motion, the reciprocal screw represents constraints. That means the constraint limb gives the moving platform a constraint moment rotating around the \(z\)-axis, while the remaining degrees of freedom are unrestricted.