# Nonlinear Analysis and Reliability Analysis of Multilink Mechanism Considering Mixed Clearance
## 2 Creation of Kinematic Pair Clearance Model
### Establishment of a Kinematic Model for the Clearance in Prismatic Joints  
The clearance model for prismatic joints, as illustrated in Figure 2. The length, width, and thickness of the slider are denoted as \(L\), \(W_{T}\), and \(T\), respectively, while the distance between the two sides of the guideway is \(H_{T}\). The variable \(a\) represents half perimeter of rectangular contact surface of slider, and is given by \(a=T+L\). The clearance value for the prismatic joint is denoted as \(c_{T}\), and its expression is as follows:  
\[c_{T}=\frac{H_{T}-W_{T}}{2} \tag{21}\]  
Positional relationship between the slider and guideway is illustrated in Figure 3.  
Figure 2: Clearance model of prismatic joint: (**a**) main view; (**b**) side view.  
The analytical model for the clearance in prismatic joints is illustrated in Figure 4. The slider is denoted as component \(m\), and the guideway as component \(n\). \(o_{m}\) and \(o_{n}\) represent the centroids of the slider and guideway, respectively. \(o_{m}-x_{m}y_{m}\) and \(o_{n}-x_{n}y_{n}\) denote the local coordinate system of the slider and the guideway respectively. The system coordinate system is represented by \(O\)-\(XY\).  
Suppose that \(P\), \(Q\), \(R\), and \(T\) are geometrically constrained points on rail that may come into contact with slider and suppose that \(A_{m},\ B_{m},\ C_{m}\), and \(D_{m}\) represent four corners of the slider. Similarly, \(A_{n},\ B_{n},\ C_{n}\), and \(D_{n}\) denote points on the guideway that are closest to the corners of the slider. Due to the similarity in the positional equations for each corner point on the slider, take point \(A\) as an illustration. The vector of positions of any point \(H\) on the \(i\)-component can be shown as:  
\[\mathbf{r}_{i}^{H}=\mathbf{r}_{i}+\mathbf{A}_{i}\mathbf{s}_{i}^{H}(i=m,n) \tag{22}\]  
where \(\mathbf{r}_{i}\) represents the position vector of centroid of component \(i\) in fixed coordinate system, \(\mathbf{A}_{i}\) is matrix of transformations, and \(\mathbf{s}_{i}^{H}\) indicates the vector of positions of any point \(H\) in local coordinate system.  
Position vector connecting point \(A_{m}\) on the slider and point \(A_{n}\) on the guideway is:  
\[\mathbf{\delta}_{T}=\mathbf{r}_{n}^{A}-\mathbf{r}_{m}^{A} \tag{23}\]  
Figure 4: Analysis model for clearance in prismatic joint.  
Figure 3: Positional relationship between slider and guideway.  
The normal vector on the surface of the guide rail is:  
\[\mathbf{n}_{T}=\left(\mathbf{t}_{T}^{y}-\mathbf{t}_{T}^{X}\right)^{T} \tag{24}\]  
where \(\mathbf{t}_{T}^{X}\) and \(\mathbf{t}_{T}^{y}\) are components of the tangential vector \(\mathbf{t}_{T}\) in \(X\) and \(Y\) directions, respectively.  
When a collision occurs between slider and guide rail, vector \(\delta_{T}\) and normal vector \(\mathbf{n}_{T}\) of the guideway surface are collinear and oriented in opposing direction.  
Consequently, the collisions of the slider with the rail are conditional on [44].  
\[\mathbf{n}_{T}^{\mathrm{T}}\delta_{T}<0 \tag{25}\]  
The embedment depth of guide and slider is:  
\[\delta_{T}=\sqrt{\delta_{T}^{\mathrm{T}}\delta_{T}} \tag{26}\]