## 5 Performance Evaluation of Optimal Manipulator Designs
#### Jacobian of the Limb with Decoupled Joint  
For the manipulators shown in Figure 10, the motion of the limb with the decoupled joint is generated by a \(4\xi_{0}-15_{\infty}\) screws. Taking the center of the second revolute joint as a reference point, the direction vector of the restriction screw for the limb shown in Figure 10 can be obtained from a five system as:  
\[ \begin{split} s_{\text{c12}_{\parallel}}&=-[s_{52_ {\parallel}}\cdot(s_{32_{\parallel}}\times s_{12_{\parallel}})]\bigg{(} \boldsymbol{r}_{54}\times\boldsymbol{s}_{42_{\parallel}}\bigg{)}\times s_{2 2_{\parallel}}\\ &\quad+[\boldsymbol{s}_{52_{\parallel}}\cdot(\boldsymbol{s}_{42 _{\parallel}}\times s_{12_{\parallel}})]\bigg{(}\boldsymbol{r}_{53}\times s _{32_{\parallel}}\bigg{)}\times s_{22_{\parallel}}\\ &\quad-[\boldsymbol{s}_{52_{\parallel}}\cdot(\boldsymbol{s}_{42 _{\parallel}}\times s_{32_{\parallel}})]\bigg{(}\boldsymbol{r}_{52}\times s _{12_{\parallel}}\bigg{)}\times s_{22_{\parallel}}.\end{split}  \tag{41}\]  
Figure 9: Manipulability of PhRS Figure 5c where parasitic motion along x-axis is removed. Because \(\mathbf{r}_{54}=0\), the direction vector in Equation (41) can be simplified to:  
\[ \begin{split}\mathbf{s}_{c12_{\parallel}}&=+[\mathbf{s}_{52_{ \parallel}}\cdot(s_{42_{\parallel}}\times\mathbf{s}_{12_{\parallel}})]\bigg{(} \mathbf{r}_{53}\times s_{32_{\parallel}}\bigg{)}\times\mathbf{s}_{22_{\parallel}}\\ &\qquad-[\mathbf{s}_{52_{\parallel}}\cdot(\mathbf{s}_{42_{\parallel}} \times\mathbf{s}_{32_{\parallel}})]\bigg{(}\mathbf{r}_{52}\times\mathbf{s}_{12_{\parallel }}\bigg{)}\times\mathbf{s}_{22_{\parallel}}.\end{split}  \tag{42}\]  
With the condition of linear independence of \(\mathbf{s}_{32\parallel},\mathbf{s}_{42\parallel},\mathbf{s}_{52\parallel}\), the moment vector is zero. Then, the second limb restriction screw becomes:  
\[\mathbf{\xi}_{c12}= \begin{bmatrix}\mathbf{0}&\mathbf{s}_{c12\parallel}\end{bmatrix} ^{T}, \tag{43}\]  
where \(\mathbf{s}_{c12\parallel}=\mathbf{s}_{32\parallel}(\mathbf{r}_{53}\cdot\mathbf{s}_{22 \parallel})+\mathbf{s}_{12\parallel}(\mathbf{r}_{52}\cdot\mathbf{s}_{22\parallel})\). Note that \(\mathbf{r}_{53}=I_{22}\) and \(\mathbf{r}_{52}=I_{21}+I_{22}\). From further simplification of the restriction screw, we can see vector \(\mathbf{s}_{c12\parallel}\) is parallel to \(\mathbf{s}_{12\parallel}\). By transforming the reference point and taking the constraint wrench of all three legs, we obtain the constraint matrix. Then, with the extended Jacobian, we obtain the first revolute joint wrench can be obtained from:  
\[ \begin{bmatrix}\mathbf{s}_{52\parallel}^{T}&(\mathbf{s}_{52\parallel}\times I_{22})^ {T}\\ \mathbf{s}_{42\parallel}^{T}&(\mathbf{s}_{42\parallel}\times I_{22})^{T}\\ \mathbf{s}_{32\parallel}&\mathbf{0}\\ \mathbf{0}^{T}&\mathbf{s}_{22\parallel}^{T}\\ \mathbf{0}&\mathbf{s}_{c12\parallel}^{T}\end{bmatrix}\begin{bmatrix}\mathbf{w}_{p12 \perp}\\ \mathbf{w}_{p12\parallel}\end{bmatrix} =\mathbf{0}:3\mathbf{\hat{s}}_{0}-2\mathbf{\hat{s}}_{ \infty}. \tag{44}\]  
For Equation (44), the direction vector is simultaneously orthogonal to \(\mathbf{s}_{21}\) and \(\mathbf{s}_{c12\parallel}\). Therefore, \(\mathbf{w}_{p12\parallel}=\mathbf{s}_{22}\times\mathbf{s}_{c12\parallel}\). The moment vector is obtained from Equation (44), and thus \(\mathbf{w}_{p12\perp}=\mathbf{M}_{1}^{\dagger}\mathbf{M}_{2}\mathbf{w}_{p12\parallel}\), where \(\mathbf{M}_{1}\) and \(\mathbf{M}_{2}\) are matrices spanned by the set of direction  
Figure 10: Non-Parasitic motion enhanced performance manipulators (**left**). Isolated limb (**right**). and moment vectors of Equation (44), respectively. The active (prismatic) joint wrench is also obtained from the following constraint equation:  
\[