## 3 An Integrated Tolerance Theory for the Kinematic Accuracy Analysis of Landing
### Kinematic Accuracy Model Derived from Jacobian-Torsor Model in a Mechanism
\right], \tag{8}\]  
Meanwhile, \([\vec{\delta}_{FE_{1}}\ \cdots\ \vec{\delta}_{FE_{n}}]^{T}\) are the small deviations of FE pairs, and \([\Delta\vec{s}\ \Delta\vec{s}]^{T}\) is the deviation of the FR. They can be represented as small-displacement screws. Letting \(\Delta=[\delta_{1}\ \delta_{2}\ \delta_{3}\ \phi_{1}\ \phi_{2}\ \phi_{3}]\) be a small-displacement screw model, which represents potential variations along and about all three Cartesian axes. Regarding the existence of clearance, when a mechanism moves, FE pairs move, the lengths of FE pairs change, and the small-displacement screw models change as well. For a mechanism, in an FE pair, letting \(\eta\) be the input variable, the projections of lengths and angles, respectively, along and about three axes are  
\[x=f_{x}(d,\eta),y=f_{y}(d,\eta),z=f_{z}(d,\eta), \tag{9}\] \[\theta_{x}=f_{\theta x}(d,\eta),\theta_{y}=f_{\theta y}(d,\eta), \theta_{z}=f_{\theta z}(d,\eta), \tag{10}\]  
and then,  
\[\Delta x=\sum_{i=1}^{n}\frac{\partial f_{x}(d,\eta)}{\partial d_{ i}}\Delta d_{i},\Delta y=\sum_{i=1}^{n}\frac{\partial f_{y}(d,\eta)}{\partial d _{i}}\Delta d_{i},\Delta z=\sum_{i=1}^{n}\frac{\partial f_{y}(d,\eta)}{\partial d _{i}}\Delta d_{i}, \tag{11}\] \[\Delta\theta_{x}=\sum_{i=1}^{n}\frac{\partial f_{\theta x}(d,\eta )}{\partial d_{i}}\Delta d_{i},\Delta\theta_{y}=\sum_{i=1}^{n}\frac{\partial f _{\theta y}(d,\eta)}{\partial d_{i}}\Delta d_{i},\Delta\theta_{z}=\sum_{i=1}^{ n}\frac{\partial f_{\theta z}(d,\eta)}{\partial d_{i}}\Delta d_{i}, \tag{12}\]  
respectively, where \(\Delta x,\Delta y,\Delta z,\Delta\theta_{x},\Delta\theta_{y}\), and \(\Delta\theta_{z}\) are, respectively, small deviations that are functions of the input variables \(\eta\) and \(\Delta d_{i}\). As a consequence, the small-displacement screw model of the ith FE pair in the LRF is  
\[\Delta FE_{i}= \begin{bmatrix}\delta_{1}\\ \delta_{2}\\ \delta_{3}\\ \phi_{1}\\ \theta_{2}\\ \theta_{3}\end{bmatrix} = \begin{bmatrix}\Delta x\\ \Delta y\\ \Delta\theta_{x}\\ \Delta\theta_{y}\\ \Delta\theta_{z}\end{bmatrix} = \begin{bmatrix}\sum\limits_{i=1}^{n}\frac{ \partial f_{x}(d,\eta)}{\partial d_{i}}\Delta d_{i}\\ \sum\limits_{i=1}^{n}\frac{\partial f_{y}(d,\eta)}{\partial d_{i}}\Delta d_{i} \\ \sum\limits_{i=1}^{n}\frac{\partial f_{y}(d,\eta)}{\partial d_{i}}\Delta d_{i} \\ \sum\limits_{i=1}^{n}\frac{\partial f_{\theta y}(d,\eta)}{\partial d_{i}}\Delta d _{i}\\ \sum\limits_{i=1}^{n}\frac{\partial f_{\theta z}(d,\eta)}{\partial d_{i}}\Delta d _{i}\\ \end{bmatrix}  \tag{13}\]\(\Delta\)FE\({}_{i}\) and and the jacobian matrix of a mechanism both change as it moves. For a mechanism which contains n functional elements pairs, kinematic accuracy model in the GRF can be described as  
\[ \begin{bmatrix}\delta_{1}\\ \delta_{2}\\ \delta_{3}\\ \theta_{1}\\ \theta_{2}\\ \theta_{3}\end{bmatrix} =\left[\left[ \begin{array}{cc}J_{1}\,J_{2}\,\cdots \,J_{6}\end{array} \right]_{FE_{1}}\cdots\left[ \begin{array}{cc}J_{6n-5}\,J_ {6n-4}\,\cdots\,,J_{6n}\end{array} \right]_{FE_{n}}\right]. \begin{bmatrix}\Delta \text{FE}_{1}\\ \vdots\\ \Delta\text{FE}_{i}\\ \vdots\\ \Delta\text{FE}_{n}\end{bmatrix}  \tag{14}\]  
here, \([\,\delta_{1}\,\delta_{2}\,\delta_{3}\,\theta_{1}\,\theta_{2}\,\theta_{3}\,]^{T}\) is the small displacement screw model of FR in the GRF. The \(\Delta\)FE\({}_{i}\) and \(J\) are functions of input variables and tolerances. The clearance between parts is taken as a functional pair and modeled as a small-displacement screw model.