## 3 An Integrated Tolerance Theory for the Kinematic Accuracy Analysis of Landing  
### Kinematic Accuracy Model Derived from Jacobian-Torsor Model in a Mechanism  
The Jacobian-Torsor model uses FE pairs to represent both the dimensions and variations of an assembly. Based on the results sought in the kinematics of a robot, a generic dispersion in an FE pair can be expressed by a set of six virtual joints and coordinate frames. There are six local coordinate frames in an FE pair. (see Figure 2). The transformation matrix from frame 0 to frame 6 can be deduced to be. \begin{table}
\begin{tabular}{l l} \hline \hline
**Characteristic Type** & \(S/N\) **Ratios** \\ \hline Nominal the best & With scale factor \(\eta=10\text{log}(\mu^{2}/\sigma^{2})\) \\  & With adjustment factor \(\eta=10\text{log}(\sigma^{2})\) \\ Smaller the better & \(\eta=10\text{log}[\frac{1}{n}\sum\limits_{i=1}^{n}y_{i}^{2}]\) \\ Larger the better & \(\eta=10\text{log}[\frac{1}{n}\sum\limits_{i=1}^{n}\frac{1}{y_{i}^{2}}]\) \\ \hline \hline \end{tabular}
\end{table} 
Table 2: \(S/N\) ratios for different characteristic types. Figure 1: Robust Design. \[T_{0}^{6}=\left[ \begin{array}{ccc}S(\omega_{3}^{4})C(\omega_{4}^{5})C(\omega_{5}^ {6})+C(\omega_{3}^{4})S(\omega_{5}^{6})&S(\omega_{3}^{4})S(\omega_{4}^{5})\\ S(\omega_{3}^{4})S(\omega_{5}^{6})-C(\omega_{3}^{4})C(\omega_{5}^{4})C(\omega_{5}^ {6})&C(\omega_{3}^{2})C(\omega_{2}^{3})\\ -S(\omega_{5}^{4})C(\omega_{5}^{6})&C(\omega_{4}^{5})\\ 0&0\\ &S(\omega_{3}^{4})C(\omega_{4}^{5})S(\omega_{5}^{6})-C(\omega_{3}^{4})C(\omega_ {5}^{6})&d_{2}^{3}\\ &-C(\omega_{3}^{4})C(\omega_{4}^{5})S(\omega_{5}^{6})-S(\omega_{3}^{4})C(\omega _{5}^{6})&-d_{1}^{2}\\ &-S(\omega_{4}^{5})S(\omega_{5}^{6})&d_{0}^{1}\\ 0&1\end{array} \right]. \tag{4}\]  
where \(d_{i}^{j}\) is translational variable from the frame i to frame j, \(w_{i}^{j}\) is rotational variable from the frame i to frame j, \(S\) denotes \(sin\), and \(C\) denotes \(cos\). Similarly, transformation matrices between the other frames of FE pairs can be deduced according to this basic transformation matrix and the following Jacobian matrix \(J\) is established based on the transformation matrices between different frames. The Jacobian-Torsor model expressed in DRF is written as  
\[ \begin{bmatrix}\Delta\vec{s}\\ \Delta\vec{\alpha}\end{bmatrix} =\left[ \begin{array}{cc}J_{1}J_{2}J_{3}J_{4}J_ {5}J_{6}&\end{array} \right]_{FE_{1}}\cdots\left[ \begin{array}{cc}J_{6n-5}J_{6 n-4}J_{6n-3}J_{6n-4}J_{6n-5}J_{6n}&\end{array} \right]_{FE_{n}}\cdot \begin{bmatrix} \vec{\delta}_{FE_{i}}\\ \vdots\\ \vec{\delta}_{FE_{n}}\end{bmatrix}  \tag{5}\]  
Figure 2: Virtual joints and coordinate frames in an FE pair. Originally, Jacobian matrix \(J\) can be obtained from the overall transform matrix. For small rotational virtual joints, the ith column of the Jacobian matrix \(J\) is computed as  
\[J_{i}=\left[ \begin{array}{c}\vec{z}_{0}^{i-1}\times(\vec{d}_{0}^{n}-\vec{d}^{ \bar{n}-1})\\ \vec{z}_{0}^{\bar{n}-1}\end{array} \right], \tag{6}\]  
where \(\vec{z}_{0}^{i-1}\) is the third column of \(T_{0}^{i-1}\) and \(\vec{d}^{\bar{n}-1}\) is the last column of \(T_{0}^{i-1}\). However, In a mechanism, as \((\vec{d}_{0}^{n}-\vec{d}_{0}^{\bar{n}-1})\) changes with the movement of the kinematic chain, \((\vec{d}_{0}^{n}-\vec{d}_{0}^{\bar{n}-1})\)can be transformed to \(f_{i}(\eta)\), where \(\eta\) is the input variable of the mechanism. Then,  
\[J_{i}=\left[ \begin{array}{c}\overrightarrow{z}_{0}^{i-1}\times(\overrightarrow {d}_{0}^{n}-\overrightarrow{d}^{\bar{n}-1})\\ \overrightarrow{z}_{0}^{i-1}\end{array} \right]=\left[ \begin{array}{c} \overrightarrow{z}_{0}^{i-1}\times f_{i}(\eta)\\ \overrightarrow{z}_{0}^{i-1}\end{array} \right], \tag{7}\]  
while for small translational virtual joints, there is no contribution to small rotation displacements of the point of interest, and the ith column of the jacobian matrix \(J_{i}\) is computed simply as  
\[J_{i}=\left[ \begin{array}{c}\overrightarrow{z}_{0}^{i-1}\\ 0\end{array}