# Optimization Method of the Clamping Force for Large Cabin Parts
## 3 Calculation and Optimization Method of the Clamping Force  
### The Friction Cone Constraint  
In the clamping process, in order to avoid excessive output clamping force causing damage to the clamping mechanism and object, it is required that the clamping force should be as small as possible which should satisfy each constraint and realize stable clamping. Among each constraint, the constraint of the friction cone is nonlinear. [PERSON] et al. point out in their research that the constraint of the friction cone can be described as a symmetric matrix of a specific structure, so Equation (1) can be expressed as:  
\[\mathbf{P}_{i}^{3\times 3}=\begin{bmatrix}\mu f_{\text{zz}}&0&f_{\text{iy}}\\ 0&\mu f_{\text{zz}}&f_{\text{ix}}\\ f_{\text{ix}}&f_{\text{iy}}&\mu f_{\text{zz}}\end{bmatrix}\succ 0 \tag{6}\]  
Let the block diagonal matrix \(\mathbf{P}=\text{diag}\big{(}\mathbf{P}_{1}\quad\mathbf{P}_{2}\quad\cdots\quad\mathbf{P}_{N} \big{)}\in R^{3N\times 3N}\). Then, the matrix \(\mathbf{P}\) can be called the description matrix. Because matrix \(\mathbf{P}_{i}\) is a symmetric positive definite matrix, its specific structure can be expressed using a linear homogeneous equation:  
\[\mathbf{A}_{fi}\cdot\text{vec}(\mathbf{P}_{i})=0 \tag{7}\]  
in which:  
\[\mathbf{A}_{fi}=\begin{bmatrix}1&0&0&0&-1&0&0&0&0\\ 1&0&0&0&0&0&0&0&-1\\ 0&0&1&0&0&0&-1&0&0\\ 0&0&0&0&0&1&0&-1&0\\ 0&1&0&0&0&0&0&0&0\\ 0&0&0&0&1&0&0&0&0\end{bmatrix} \tag{8}\]  
vec() is defined as the vectorization operator of the matrix. From the description matrix \(\mathbf{P}\) and Equation (7), it can be known that:  
\[\mathbf{A}_{f}\cdot\text{vec}(\mathbf{P})=0 \tag{9}\]  
Equation (9) shows that the nonlinear constraint of the friction cone is transformed into a linear constraint by the specific structure of the matrix, where the matrix \(\mathbf{A}_{f}=\text{diag}\big{(}\mathbf{A}_{f1}\quad\mathbf{A}_{f2}\quad\cdots\quad\mathbf{A} _{fN}\big{)}\in R^{6N\times 9N^{2}}\) is related to Equation (8). Equation (9) represents the constraint of the friction cone that includes all the clamping mechanisms.  
Similarly, according to Equation (9), Equation (4) can also be rewritten into the linear constraint form, which can be expressed as:  
\[\mathbf{A}_{e}\cdot\text{vec}(\mathbf{P})=-\mathbf{W}_{\text{ext}} \tag{10}\]  
in which:  
\[\mathbf{A}_{ei}=\left[\begin{array}{cc}\frac{1}{\mu}\cdot\mathbf{z}_{i}\\ \frac{1}{\mu}\cdot(\mathbf{r}_{i}\times\mathbf{z}_{i})\end{array}\right.\quad 0\in R^{6 \times 1}\quad\begin{array}{cc}\mathbf{x}_{i}\\ \mathbf{r}_{i}\times\mathbf{x}_{i}\end{array}\quad 0\in R^{6\times 2}\quad \begin{array}{cc}\mathbf{y}_{i}\\ \mathbf{r}_{i}\times\mathbf{y}_{i}\end{array}\quad 0\in R^{6\times 3}\right] \tag{11}\]  
In Equation (10), \(\mathbf{A}_{e}=\left(\mathbf{A}_{e1}\quad\mathbf{A}_{e2}\quad\cdots\quad\mathbf{A}_{eN}\right) \in R^{6\times 9N^{2}}\) is related to the clamping matrix \(\mathbf{G}\) and it is also affected by the clamping configuration and the number of contact points. By combining Equations (9) and (10), it can be expressed as:  
\[\mathbf{A}\cdot\text{vec}(\mathbf{P})=\begin{bmatrix}\mathbf{A}_{f}\\ \mathbf{A}_{e}\end{bmatrix}\cdot\text{vec}(\mathbf{P})=\begin{bmatrix}0\in R^{6N}\\ -\mathbf{W}_{\text{ext}}\end{bmatrix}=\mathbf{q} \tag{12}\]