# Optimization Method of the Clamping Force for Large Cabin Parts

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. The Clamping Contact Model and the Basic Equation

#### 2.1. The Contact Model

**f**

_{i}= [f

_{i}

_{z}, f

_{i}

_{y}, f

_{i}

_{x}]

^{T}is the clamping force vector of the i-th clamping mechanism in the contact coordinate system, μ is the friction coefficient, f

_{i}

_{x}and f

_{i}

_{y}represent the component force of the i-th clamping mechanism in two tangential directions in the contact coordinate system, and f

_{i}

_{z}is the normal positive force of the i-th clamping mechanism in the contact coordinate system (i = 1…n). In order to ensure the friction force at the contact point always exists, each normal positive force should meet f

_{i}

_{z}≥ 0, which is the unidirectional force constraint of the point contact model with friction.

#### 2.2. Clamping Matrix

_{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, …, 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 ${\mathit{r}}_{i}={\left[{r}_{i\mathrm{x}}\hspace{1em}\hspace{1em}{r}_{i\mathrm{y}}\hspace{1em}\hspace{1em}{r}_{i\mathrm{z}}\right]}^{\mathrm{T}}$. The corresponding unit direction vector of each coordinate axis in each contact coordinate system is denoted as

**z**

_{i},

**x**

_{i},

**y**

_{i}. Then, the force exerted by the i-th clamping mechanism on the object which is transformed from the contact coordinate system to the object coordinate system is shown as follows:

**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 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 ${\mathit{W}}_{\mathrm{ext}}\in {R}^{6}$, then the equilibrium equation between the clamping force generated by all clamping mechanisms and external force can be expressed as follows:

## 3. Calculation and Optimization Method of the Clamping Force

#### 3.1. The Friction Cone Constraint

**P**can be called the description matrix. Because matrix

**P**

_{i}is a symmetric positive definite matrix, its specific structure can be expressed using a linear homogeneous equation:

**P**and Equation (7), it can be known that:

**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:

#### 3.2. Gradient Flow Optimization Method for the Clamping Force

**P**is a symmetric positive definite matrix, Equation (5) can be transformed into a convex optimization problem which describes matrix

**P**with affine constraints in a smooth Riemannian manifold. According to Equations (6) and (12), the convex optimization problem can be expressed as below:

_{p}and W

_{i}are weight coefficients. The former is used to control the magnitude of the normal clamping force, and the latter is used to control the distance between the clamping force vector and the edge of the friction cone.

**I**is the identity matrix and tr() is defined as the trace operation of the matrix. A linear constrained gradient flow using the Riemann metric can be expressed as:

**A**. In order to start the iterative calculation, it is also necessary to discretize the linear constrained gradient flow, which can be expressed as:

## 4. The Illustrative Example and Simulation

_{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, …, 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 μ = 0.4.

**W**

_{ext}= [10, 50, −147, 2, −8, 5]

^{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. The principle of the penalty function algorithm is to define a penalty function outside the feasible domain and gradually approach the extreme point from the outside of the feasible domain. This method is suitable for optimization problems with equality constraints or inequality constraints. The principle of the Lagrange multiplier algorithm is to introduce the Lagrange multiplier and slack variable according to the KKT condition to combine the equality constraint, inequality constraint and objective function during optimization. The principle of the sequential quadratic programming algorithm is to simplify the nonlinear programming problem to a quadratic programming problem at the approximate solution of the objective function and find its optimal solution via quadratic programming. The principle of the genetic algorithm is to start from any initial population, generate a group of individuals more suitable for the environment via random selection, crossover and mutation operations and, finally, converge on a group of individuals most suitable for the environment to obtain the optimal solution to the problem. The application, calculation process and parameter settings of the above algorithms can be queried by analyzing and combining with the MATLAB 2018b help document [32]. In order to be expressed clearly, the calculated initial value vectors of each clamping mechanism are combined into a vector

**F**

_{j}= [

**f**

_{1},

**f**

_{2}, …,

**f**

_{8}]

^{T}, j = 1, 2, 3, 4; below, the

**F**

_{optj}is combined in the same way. The numbering sequence of each clamping mechanism is the same as that in Figure 7. According to the penalty function algorithm, Lagrange multiplier algorithm, sequential quadratic programming algorithm and genetic algorithm, the initial value of the clamping force optimization is calculated successively. The initial value of the clamping force optimization

**F**

_{j}obtained by calculation is shown in Table 1.

_{p}and W

_{i}are finally set as 2 and 0.1, respectively, and the convergence error is set as 0.001. The target function of the gradient flow optimization method and the changes in the normal contact force at the contact point between the surface of the clamping mechanism and the object are obtained, which are shown in Figure 8, Figure 9, Figure 10 and Figure 11.

**F**

_{optj}are shown in Table 2.

_{p}and W

_{i}to a certain value. It also confirms the feasibility and effectiveness of the gradient flow optimization method applied to the clamping force optimization process of a large cabin part. However, the gradient flow optimization method of the clamping force may cause the problem that the objective function’s value and the clamping force’s value will get stuck in the local optimal value during the optimization process like in other gradient optimization methods. This problem may also be produced in the initial value calculation process of the clamping force optimization. By only analyzing the above four groups of the clamping force optimization simulations, it cannot be determined whether the iteration is getting stuck in a local optimal value. It is necessary to carry out a clamping force experiment and comprehensively analyze the experimental results and relevant simulation results.

## 5. Experiment

**F**

_{n}= [

**f**

_{n1},

**f**

_{n2}, …,

**f**

_{n8}]

^{T}. This vector can be regarded as the normal clamping force value of each clamping mechanism after a stable clamping, which can be expressed as follows:

**F**

_{ne}calculated by the simulation is obtained as follows:

_{p}, W

_{i}and the convergence error are taken as 2, 0.1 and 0.001, respectively. The friction coefficient μ of each contact point is taken as 0.4. Then, the convergence value of the normal clamping force of each clamping mechanism

**F**

_{nopt}is obtained by using the gradient flow optimization method, which is shown as follows:

## 6. Conclusions

_{p}and W

_{i}, the distribution of the clamping mechanism, the configuration of the clamping mechanism and the pose of the clamped object. These uncertainties will affect the final optimization results. As a gradient optimization method, the gradient flow optimization method of the clamping force may also cause a critical problem of getting stuck in the local optimal value during the optimization process like other gradient optimization methods. This problem may also be produced when calculating the initial optimization value of the clamping force. But this problem is not discussed comprehensively in this paper. Moreover, because the clamped object is a large-mass and large-size axial part, the error of the initial optimized value and the optimal clamping force will be large. How to avoid the problem of the local optimal value which may be produced in the process of the clamping force optimization and how to more reasonably equate the surface contact model to the point contact model between the clamping mechanisms and the object when designing experiments are the factors that are the key points and difficulties during the optimization method of the clamping force that require further research.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**3D model of the ring pneumatic fixture: 1. Body of the ring pneumatic fixture, 2. Felt at the end of the clamping mechanism, 3. Clamping mechanism.

**Figure 4.**Contact model: (

**a**) Frictionless contact; (

**b**) point contact with friction; (

**c**) soft contact.

**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.

**Figure 7.**The example model of 8 modular clamping mechanisms clamping the object: (

**a**) The distribution of 8 clamping mechanisms; (

**b**) simplified point contact model of the clamping mechanism and object’s surface.

**Figure 8.**The iterative process of the gradient flow method in which the initial value is calculated using the penalty function algorithm: (

**a**) Value of the target function; (

**b**) normal contact force of the upper clamping mechanism; (

**c**) normal contact force of the lower clamping mechanism.

**Figure 9.**The iterative process of the gradient flow method in which the initial value is calculated using the Lagrange multiplier algorithm: (

**a**) Value of the target function; (

**b**) normal contact force of the upper clamping mechanism; (

**c**) normal contact force of the lower clamping mechanism.

**Figure 10.**The iterative process of the gradient flow method in which the initial value is calculated using the sequential quadratic programming algorithm: (

**a**) Value of the target function; (

**b**) normal contact force of the upper clamping mechanism; (

**c**) normal contact force of the lower clamping mechanism.

**Figure 11.**The iterative process of the gradient flow method in which the initial value is calculated using the genetic algorithm: (

**a**) Value of the target function; (

**b**) normal contact force of the upper clamping mechanism; (

**c**) normal contact force of the lower clamping mechanism.

**Figure 12.**Experimental equipment: (

**a**) Experimental prototype and its supporting device of the ring pneumatic fixture: 1. Experimental prototype, 2. Clamping mechanism 3. Auxiliary support bracket, 4. Roll support bracket. (

**b**) Experimental force sensor. (

**c**) Four-channel data acquisition card. (

**d**) Force measuring system with the upper computer: 1. Experimental prototype, 2. Force sensor attached to the surface of the clamping mechanism, 3. Data acquisition card, 4. Upper computer.

**Figure 14.**Change in the normal clamping force obtained by the experiment: (

**a**) The normal clamping force change of the upper clamping mechanism; (

**b**) the normal clamping force change of the lower clamping mechanism.

**Figure 15.**The iterative process of the gradient flow method when clamping a 2 t object: (

**a**) Value of the target function; (

**b**) normal contact force of the upper clamping mechanism; (

**c**) normal contact force of the lower clamping mechanism.

Sequence | F_{1} | F_{2} | F_{3} | F_{4} |
---|---|---|---|---|

f_{1z} | 57.74 | 85.4 | 57.69 | 16.1 |

f_{1y} | 1.9 | −2 | 1.89 | 0.21 |

f_{1x} | −6.84 | −12.73 | −7.47 | −1.06 |

f_{2z} | 57.18 | 33.55 | 57.07 | 80.53 |

f_{2y} | −6.82 | −1.9 | −6.81 | −7.09 |

f_{2x} | −4.74 | −4.44 | −3.97 | −5.13 |

f_{3z} | 67.46 | 86.39 | 67.37 | 41.34 |

f_{3y} | 1.6 | 4.78 | 1.59 | 3.35 |

f_{3x} | −8.84 | −10.77 | −9.64 | −1.89 |

f_{4z} | 18.23 | 62.06 | 17.98 | 12.14 |

f_{4y} | 0.93 | 7.1 | 0.97 | 0.83 |

f_{4x} | −2.56 | −9.31 | −1.84 | −2.77 |

f_{5z} | 81.2 | 97.59 | 81.13 | 235.95 |

f_{5y} | 0.34 | −5.59 | 0.33 | 0.07 |

f_{5x} | 7.47 | 10.23 | 6.79 | −0.14 |

f_{6z} | 26.7 | 51.59 | 26.58 | 26.87 |

f_{6y} | −4.27 | −9.66 | −4.27 | −4.13 |

f_{6x} | 0.13 | 1.08 | 0.81 | 0.89 |

f_{7z} | 79.79 | 136.61 | 79.7 | 29.18 |

f_{7y} | −2.34 | −4.4 | −2.34 | −2.39 |

f_{7x} | 3.73 | −7.79 | 2.91 | −0.6 |

f_{8z} | 29.26 | 67.97 | 29.17 | 108.07 |

f_{8y} | −1.3 | 1.677 | −1.35 | −0.81 |

f_{8x} | −0.18 | −11.28 | 0.45 | 0.5 |

Sequence | F_{opt1} | F_{opt2} | F_{opt3} | F_{opt4} |
---|---|---|---|---|

f_{1z} | 23.92 | 16.55 | 25.85 | 11.01 |

f_{1y} | −4.12 | −2.2 | −4.23 | −1.49 |

f_{1x} | −2.01 | −1.08 | −2.6 | −0.9 |

f_{2z} | 41.14 | 24.56 | 41.8 | 37.24 |

f_{2y} | −8.08 | −4.42 | −8.22 | −7.2 |

f_{2x} | −0.87 | −0.45 | −0.77 | −0.55 |

f_{3z} | 59.93 | 56.78 | 63.52 | 26.77 |

f_{3y} | 4.24 | 4.67 | 4.31 | 2.23 |

f_{3x} | −10.68 | −8.35 | −11.45 | −3.69 |

f_{4z} | 0.8 | 3.63 | 0.8 | 1.04 |

f_{4y} | 0.01 | 0.01 | 0.01 | 8.52 |

f_{4x} | −0.01 | −0.02 | −0.03 | −1.87 |

f_{5z} | 74.51 | 82.17 | 73.12 | 154.92 |

f_{5y} | 8.1 | 8.66 | 7.91 | −5.32 |

f_{5x} | 9.71 | 8.74 | 6.32 | 5.05 |

f_{6z} | 18.63 | 18.53 | 8.7 | 8.61 |

f_{6y} | −1.63 | −3.29 | −1.65 | −1.51 |

f_{6x} | 0.04 | 0.1 | 0.06 | 0.08 |

f_{7z} | 47.56 | 64.87 | 44.14 | 10.08 |

f_{7y} | −7.77 | −11.42 | −7.52 | −1.46 |

f_{7x} | 5.08 | 3.38 | 4.19 | 0.49 |

f_{8z} | 4.21 | 8.21 | 3.58 | 20.03 |

f_{8y} | −0.74 | −1.98 | −0.61 | −3.75 |

f_{8x} | −0.07 | 0.21 | 0.07 | 0.48 |

**Table 3.**Experimental average value and the simulated convergence value of the normal clamping force (N).

Number of the Clamping Mechanism | F_{n} | F_{nopt} |
---|---|---|

1 | 1.324 | 1.614 |

2 | 2.893 | 2.542 |

3 | 1.224 | 1.646 |

4 | 3.091 | 1.223 |

5 | 1.013 | 2.027 |

6 | 2.894 | 1.443 |

7 | 1.645 | 3.273 |

8 | 2.873 | 1.394 |

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## Share and Cite

**MDPI and ACS Style**

Yao, S.; Luan, Y.; Ceccarelli, M.; Carbone, G.
Optimization Method of the Clamping Force for Large Cabin Parts. *Appl. Sci.* **2023**, *13*, 12575.
https://doi.org/10.3390/app132312575

**AMA Style**

Yao S, Luan Y, Ceccarelli M, Carbone G.
Optimization Method of the Clamping Force for Large Cabin Parts. *Applied Sciences*. 2023; 13(23):12575.
https://doi.org/10.3390/app132312575

**Chicago/Turabian Style**

Yao, Shuangji, Yijv Luan, Marco Ceccarelli, and Giuseppe Carbone.
2023. "Optimization Method of the Clamping Force for Large Cabin Parts" *Applied Sciences* 13, no. 23: 12575.
https://doi.org/10.3390/app132312575