# Optimization Method of the Clamping Force for Large Cabin Parts
## 4 The Illustrative Example and Simulation  
The following is an illustrative example of the clamping mechanism clamping a large cabin part. The gradient flow optimization method is used to optimize the clamping force. All of the following calculations and simulation processes in this section are completed using MATLAB 2018b. As shown in Figure 7a, a large-size and large-mass cylindrical cabin part commonly used in the aerospace field is taken as the clamped object. Eight modular clamping mechanisms are used to clamp the object. In order to meet the operational requirements of the aerospace enterprises and the rolling function of the ring pneumatic fixture, according to the internal technical documents of the aerospace enterprise, two clamping mechanisms with an interval of 180\({}^{\circ}\) on the ring pneumatic fixture are regarded as the same group. The number of the same group of clamping mechanisms is adjacent in order. The number is counterclockwise from the upper side to the lower side of the clamping mechanism. The different numbering order of the clamping mechanism will affect the clamping matrix, description matrix and the numerical order of each clamping force. It will not affect the calculation process and the final optimization results. Suppose the ring pneumatic fixture is in a static state. The radius of the clamped object is \(r=0.7\) m. The mass of the clamped object is \(15\) T; B-X\({}_{0}\)Y\({}_{0}\)Z\({}_{0}\) is the object coordinate system located at the object's center. B\({}_{i}\) is the contact coordinate system (\(i=1,\ldots,8\)). Each contact point is regarded as the point contact model with friction, which is shown in Figure 7b. The contact point between each clamping mechanism and the object is the origin point of each contact coordinate system. The friction coefficient at each contact point has the same value. In order to prevent the rigid contact between the clamping mechanism and the clamped object from further damaging the object's surface, the surface of the clamping mechanism in the ring pneumatic fixture is covered with a layer of \(5\) mm felt. Felt materials usually have a high friction coefficient. For comprehensive consideration, the friction coefficient here is set as \(\mu=0.4\). Figure 6: Calculation process of the gradient flow optimization method. According to the relationship between the coordinate system of the contact point and the object coordinate system, the clamping matrix of the example can be obtained using Equation (3):  
\[\mathbf{G}= \begin{bmatrix}0&1&0&0&1&0&0&1&0\\ 0.8&0&0.6&-0.8&0&0.6&0.3&0&0.95\\ 0.6&0&-0.8&-0.6&0&-0.8&0.95&0&-0.3\\ 0&0&-0.7&0&0&0.7&0&0&-0.6947\\ 0&0.42&0&0&-0.42&0&0&0.665&0\\ 0&-0.56&0&0&0.56&0&0&-0.21&0\\ \end{bmatrix}  \tag{18}\]  
\[ \begin{array}{cccccccc}0&1&0&0&1&0&0&1&0\\ -0.3&0&0.95&-0.3&0&0.95&0.3&0&0.95\\ -0.95&0&-0.3&0.95&0&0.3&-0.95&0&0.3\\ 0&0&0.6947&0&0&-0.6947&0&0&0.6947\\ 0&0.665&0&0&0.665&0&0&-0.665&0\\ 0&-0.21&0&0&0.21&0&0&-0.21&0\\ \end{array} \]  
\[ \begin{array}{cccccccc}0&1&0&0&1&0\\ -0.8&0&0.6&0.8&0&0.6\\ 0.6&0&0.8&-0.6&0&0.8\\ 0&0&-0.7&0&0&0.7\\ 0&0.42&0&0&-0.42&0\\ 0&0.56&0&0&-0.56&0\end{array} \]  
Considering the gravity of the clamped object, suppose that a random external force \(\mathbf{W}_{\text{ext}}=[10,\,50,\,-147,\,2,\,-8,\,5]^{\text{T}}\) is exerted onto the object after the clamping mechanism has clamped the object. In order to obtain the initial optimization value required by the gradient flow optimization method, according to Equation (5), the objective function, number of variables, boundary conditions and constraints of the clamping force optimization are determined. The penalty function algorithm, Lagrange multiplier algorithm, sequential quadratic programming algorithm and genetic algorithm are respectively used to calculate the initial optimized value of the clamping force satisfying both Equations (1) and (4) by using the fmincon function and genetic algorithm toolbox in MATLAB 2018b.