# Optimization Method of the Clamping Force for Large Cabin Parts
## 2 The Clamping Contact Model and the Basic Equation
### Clamping Matrix  
After determining the contact model, it is also necessary to determine the effect of the clamping force exerted by the clamping mechanism on the object and the mapping relationship between the two coordinates. Figure 5a shows the schematic diagram of the clamping mechanism clamping a large-sized axial object. The object coordinate system B-X\({}_{0}\)Y\({}_{0}\)Z\({}_{0}\) is located at the object's geometric center. In order to later simplify the calculation process of the force optimization, the contact model between the surface of the clamping mechanism and the object is chosen as the point contact model with friction, which is shown in Figure 5b. The contact point is B\({}_{i}\) (\(i=1,\ldots,\) N). The contact coordinate system B\({}_{i}\)-X\({}_{i}\)Y\({}_{i}\)Z\({}_{i}\) is located on the contact surface between each clamping mechanism and the object. It is assumed that the position vector of the \(i\)-th contact point in the object coordinate system is described as \(\mathbf{r}_{i}=\left[r_{ix}\ \ r_{iy}\ \ r_{iz}\right]^{\mathrm{T}}\). The corresponding unit direction vector of each coordinate axis in each contact coordinate system is denoted as \(\mathbf{z}_{i}\), \(\mathbf{x}_{i}\), \(\mathbf{y}_{i}\). Then, the force exerted by the  
Figure 4: Contact model: (**a**) Frictionless contact; (**b**) point contact with friction; (**c**) soft contact.  
\(i\)-th clamping mechanism on the object which is transformed from the contact coordinate system to the object coordinate system is shown as follows:  
\[\mathbf{f}_{Bi}=\mathbf{G}\mathbf{f}_{i} \tag{2}\]  
in which:  
\[\mathbf{G}_{i}=\begin{bmatrix}\mathbf{z}_{i}&\mathbf{x}_{i}&\mathbf{y}_{i}\\ \mathbf{r}_{i}\times\mathbf{z}_{i}&\mathbf{r}_{i}\times\mathbf{x}_{i}&\mathbf{r}_{i}\times\mathbf{y}_ {i}\end{bmatrix} \tag{3}\]  
Let \(\mathbf{G}=\begin{bmatrix}\mathbf{G}_{1}&\mathbf{G}_{2}&\cdots&\mathbf{G}_{N}\end{bmatrix}\in R ^{6\times 3N}\). Then, \(\mathbf{G}\) is called the clamping matrix. After the clamping mechanisms have clamped the object, the object might deviate from its equilibrium state because of different external disturbances, such as impact, roll and  
Figure 5: The schematic of the clamping mechanism clamping an object: (**a**) The distribution of the clamping mechanism; (**b**) simplified point contact model of the clamping mechanism and object’s surface.  
vibration during transportation. At the same time, the external disturbance also causes the clamping force to change to balance the external disturbance. Finally, the clamped object will return to its original equilibrium state or change to a new equilibrium state to prevent its instability, which will cause serious consequences. In order to study the resistance ability to external disturbance of the clamping system composed of the clamping mechanism and object, assuming that the external force screw received by the object is \(\mathbf{W}_{\text{ext}}\in R^{6}\), then the equilibrium equation between the clamping force generated by all clamping mechanisms and external force can be expressed as follows:  
\[\mathbf{Gf}=-\mathbf{W}_{\text{ext}} \tag{4}\]  
According to Equations (1) and (4), the optimization of the clamping force can be expressed as:  
\[\left\{\begin{array}{ll}\min&\mathbf{f}\\ \text{S.T}&\mathbf{Gf}=-\mathbf{W}_{\text{ext}}\\ &\mathbf{f}_{i}\in\mathbf{FC}\end{array}\right. \tag{5}\]