## 2 Configuration of the Remote Center Compliance (RCC) Mechanism
### Rotational Precision Analysis
\Big{]}^{\mathrm{T}} \tag{21}\]  
The pitch \(h\) and the rotation axis \(l\) of the twist \(\hat{\xi}_{1}\) can be calculated as  
\[ \begin{array}{l}h_{1}=\omega_{1}{}^{\mathrm{T}}\
u_{1}=0\ ;\\ l_{1}=\left\{\omega_{1}\times\
u_{1}+\lambda\omega_{1}:\ \lambda\in\mathbf{R}\right\}\\ =\Big{\{}\big{[}0,\ \lambda,-C_{X,F_{X}}\big{/}C_{\theta_{Y},F_{X}} \big{]}^{\mathrm{T}}:\ \lambda\in\mathbf{R}\Big{\}}\end{array}  \tag{22}\]The twist coordinates corresponding to the other five column vectors of the compliance matrix can be obtained in the same way:  
\[ \begin{array}{l}\xi_{2}{=}\Big{[}\begin{array}{cc}\
u_{2}&\omega_{2}\\ \end{array}\Big{]}^{\mathrm{T}}=\Big{[}\begin{array}{cc}0&C_{Y,F_{Y}}/C_{ \theta_{X},F_{Y}}&0&1&0&0\\ \end{array}\Big{]}^{\mathrm{T}}\\ \xi_{3}{=}\Big{[}\begin{array}{cc}\
u_{3}&\omega_{3}\\ \end{array}\Big{]}^{\mathrm{T}}=\Big{[}\begin{array}{cc}0&0&1&0&0&0\\ \end{array}\Big{]}^{\mathrm{T}}\\ \xi_{4}{=}\Big{[}\begin{array}{cc}\
u_{4}&\omega_{4}\\ \end{array}\Big{]}^{\mathrm{T}}=\Big{[}\begin{array}{cc}0&C_{Y,M_{X}}/C_{ \theta_{X},M_{X}}&0&1&0&0\\ \end{array}\Big{]}^{\mathrm{T}}\\ \xi_{5}{=}\Big{[}\begin{array}{cc}\
u_{5}&\omega_{5}\\ \end{array}\Big{]}^{\mathrm{T}}=\Big{[}\begin{array}{cc}C_{X,M_{Y}}/C_{ \theta_{Y},M_{Y}}&0&0&0&1&0\\ \end{array}\Big{]}^{\mathrm{T}}\\ \xi_{6}{=}\Big{[}\begin{array}{cc}\
u_{6}&\omega_{6}\\ \end{array}\Big{]}^{\mathrm{T}}=\Big{[}\begin{array}{cccccc}0&0&0&0&0&1\\ \end{array}\Big{]}^{\mathrm{T}}\end{array}  \tag{23}\]  
All the other pitches can be obtained as  
\[ \begin{array}{l}h_{2}={\omega_{2}}^{\mathrm{T}}\
u_{2}=0\ ;\ h_{3}={\omega_{3}}^{ \mathrm{T}}\
u_{3}=0\ ;\\ h_{4}={\omega_{4}}^{\mathrm{T}}\
u_{4}=0\ ;\ h_{5}={\omega_{5}}^{\mathrm{T}}\
u_{5}=0\ ;\ h_{6}={ \omega_{6}}^{\mathrm{T}}\
u_{6}=0\ ;\end{array}  \tag{24}\]  
All the axes for the other twists can be further obtained as  
\[ \begin{array}{l}l_{2}=\Big{\{}\big{[}\lambda,\ 0,C_{Y,F_{Y}}/C_{\theta_{X},F_{Y} }\big{]}^{\mathrm{T}}:\ \lambda\in\mathbf{R}\Big{\}}\\ l_{3}=l_{6}=\Big{\{}\big{[}0,\ 0,\lambda\big{]}^{\mathrm{T}}:\ \lambda\in \mathbf{R}\Big{\}}\\ l_{4}=\Big{\{}\big{[}\lambda,\ 0,C_{Y,M_{X}}/C_{\theta_{X},M_{X}}\big{]}^{ \mathrm{T}}:\ \lambda\in\mathbf{R}\Big{\}}\\ l_{5}=\Big{\{}\big{[}0,\ \lambda,-C_{X,M_{Y}}/C_{\theta_{Y},M_{Y}}\big{]}^{ \mathrm{T}}:\ \lambda\in\mathbf{R}\Big{\}}\end{array}  \tag{25}\]  
In order to validate that the actual rotation center is very close to Point \(O_{RC}\), let \(l=12.5\) mm, \(D=1.5\) mm, \(\theta=45^{\circ}\), and \(|OO_{1}|=10\) mm. The material for the mechanism is chosen as nylon PA 2200 with a Young's Modulus of 1646 MPa and a Poisson's ratio of 0.33. From Equations (1)-(11), we obtain the factors of the compliance matrix:  
\[ \begin{array}{l}C_{X,F_{X}}=C_{Y,F_{Y}}{=}3.1929\times 10^{-5}\ \ (\mathrm{m}/ \mathrm{N})\\ C_{X,M_{Y}}=C_{Y,M_{X}}{=}2.956\times 10^{-3}\ \ (\mathrm{m}/\mathrm{N} \mathrm{m})\\ C_{\theta_{X},F_{Y}}=C_{\theta_{Y},F_{X}}{=}-2.956\times 10^{-3}\ \ (\mathrm{rad}/ \mathrm{N})\\ C_{\theta_{X},M_{X}}=C_{\theta_{Y},M_{Y}}{=}3.1459\times 10^{-1}\ \ (\mathrm{rad}/ \mathrm{N}\mathrm{m})\end{array}  \tag{26}\]  
Because all the pitches of the six twists are zero, the motions of the end-effector can be regarded as pure rotations about the axes \(l_{1}-l_{6}\) of the six twists. The real twist axes \(l_{1}-l_{6}\) and the ideal twist axes are illustrated in Figure 6. The distance between the real twist axes and ideal twist axes is  
\[ \begin{array}{l}\big{|}2.956\times 10^{-3}/3.1459\times 10^{-1}-10\times 10^{-3 }\big{|}=0.6\times 10^{-3}\ \mathrm{m}\\ \big{|}3.1929\times 10^{-5}/2.956\times 10^{-3}-10\times 10^{-3}\big{|}=0.8 \times 10^{-3}\ \mathrm{m}\end{array}  \tag{27}\]  
which is defined as the center shift or the parasitic error of the remote center of the RCC mechanism [15; 17]. It can be seen from Figure 6 that the real twist axis does not overlap with the ideal twist axis, which goes through point \(O_{RC}\), and the distance between the two axes is 0.6 mm (or 0.8 mm), which cannot be neglected in precision-required applications.