# Optimization of 3D Tolerance Design Based on Cost–Quality–Sensitivity Analysis to the Deviation Domain

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## Abstract

**:**

## 1. Introduction

## 2. Parametric Modeling and Representation of 3D Tolerance Zone Based on New GPS

#### 2.1. Small Displacement Torsor Theory and Homogeneous Transformation Matrix (HTM) Principle

_{x}, then rotating around the y-axis by angle θ

_{y}, and then rotating around the z-axis by angle θ

_{z}, the new coordinate system is a homogeneous rotation transformation relative to the original coordinate system The matrix is:

#### 2.2. 3D Tolerance Modeling

#### 2.2.1. Create a Coordinate System

#### 2.2.2. Kinematic Model of Three-Dimensional Tolerance Accumulation Based on HTM

#### 2.2.3. The Basic Properties of the Tolerance Zone and Its Torsor Interval Description

## 3. Mathematical Modeling of 3D Tolerance Allocation

#### 3.1. Establishment of 3D Tolerance Mathematical Model

- By studying the SDT torsor constraint relationship of different functional feature surfaces, the tolerance zone and clearance domain between parts are mapped to the deviation domain, and the geometric variation is represented in six directions.
- Based on the mathematical model of cost-tolerance in the classic two-dimensional dimensional chain, the single independent variable is replaced with six deviation components through torsor parameters of the deviation domain. Because the deviations in the six directions are orthogonal and linearly independent, the deviation cost can be represented by the product of each direction.
- Next, we establish the relationship between the deviation domain and the cost function. According to Table 1, it can be found that each torsor has a constraint boundary. After combining the objective function and constraint conditions, a three-dimensional tolerance mathematical model is established.
- In order to verify the validity and correctness of the mathematical model, a three-dimensional function graph is made under the given correlation coefficient to judge whether the model is in line with actual production and use.
- In addition to the geometric variation constraints of the SDT torsor, it is also necessary to consider the functional requirements constraints and assembly tolerance constraints, and they should be expressed in the deviation domain.
- Combining the objective function and constraint conditions based on the deviation domain, the three-dimensional tolerance mathematical model is built.

#### 3.2. Processing Cost–Tolerance Model Based on HTM-SDT

_{1}, T

_{2}], the coordinates at the four vertices are expressed as (a,b,0), (a,−b,0), (−a,−b,0), and (−a,b,0), the constraint model of torsor deviation in the Z-axis direction at the four vertices can be obtained by Equations (7) and (8):

#### 3.3. Quality Loss–Tolerance Model Based on HTM-SDT

#### 3.4. Sensitivity–Tolerance Model Based on HTM-SDT

#### 3.5. Constraints

#### 3.5.1. Constraints of Machining Capacity and Geometric Function

#### 3.5.2. Constraints of Assembly Tolerance

## 4. Steps of Optimal Allocation for 3D Tolerance Based on Modified Bat Algorithm

#### 4.1. Basic Concepts of Functional Structural Tolerance

#### 4.2. Construction of Assembly Tolerance Loop Based on Functional Surface

- Determine the geometric constraints and degrees of freedom of the two functional surfaces for functional requirements.
- Select the part where one of the functional surfaces is located, and filter out the surface of the part that participates in the constraint degree of freedom.
- Remove irrelevant part feature surfaces.
- Continue the above steps to find the next part and its surface that has geometric constraints with the functional surface of the part until it returns to the original functional surface.

#### 4.3. Optimal Allocation for 3D Tolerance Based on Modified Bat Algorithm

## 5. Case Study

#### 5.1. Set up a Coordinate System

#### 5.2. Generation of Assembly Tolerance Loop

_{1}is the deviation clearance between the functional surface 1 of the part 1 and the functional surface 1 of the part 2, and so on.

_{2}in Equation (53), and so on.

#### 5.3. Establish a Mathematical Model of Deviation Optimization

#### 5.3.1. Objective Function

#### 5.3.2. Constraints

#### 5.3.3. Mathematical Model of Optimal Deviation Allocation

#### 5.4. Solve the Model with Genetic Bat Algorithm

## 6. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## References

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Tolerance Zone | Describe the Area | Homogeneous Transformation Matrix | Constraint Inequality |
---|---|---|---|

Between two parallel lines | $\mathrm{T}=\left[\begin{array}{cccc}1& -\Delta \mathsf{\gamma}& 0& 0\\ \Delta \mathsf{\gamma}& 1& 0& \Delta \mathrm{v}\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$ | $-\frac{\mathrm{t}}{\mathrm{L}}\le \Delta \mathsf{\gamma}\le \frac{\mathrm{t}}{\mathrm{L}}$ $-\frac{\mathrm{t}}{2}\le \Delta \mathrm{v}\le \frac{\mathrm{t}}{2}$ $\left|\Delta \mathrm{v}\right|+\left|\frac{\mathrm{t}\Delta \mathsf{\gamma}}{2\mathrm{L}}\right|\le \frac{\mathrm{t}}{2}$ | |

Between two parallel curves | $\mathrm{T}=\left[\begin{array}{cccc}1& -\Delta \mathsf{\gamma}& 0& \Delta \mathrm{u}\\ \Delta \mathsf{\gamma}& 1& 0& \Delta \mathrm{v}\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$ | $-\frac{\mathrm{t}}{{\mathrm{l}}_{\mathrm{xy}}}\le \Delta \mathsf{\gamma}\le \frac{\mathrm{t}}{{\mathrm{l}}_{\mathrm{xy}}}$ $-\frac{\mathrm{t}}{2}{\mathrm{n}}_{\mathrm{x}}\le \Delta \mathrm{u}\le \frac{\mathrm{t}}{2}{\mathrm{n}}_{\mathrm{x}}$ $-\frac{\mathrm{t}}{2}{\mathrm{n}}_{\mathrm{y}}\le \Delta \mathrm{v}\le \frac{\mathrm{t}}{2}{\mathrm{n}}_{\mathrm{y}}$ | |

Between two parallel planes | $\mathrm{T}=\left[\begin{array}{cccc}1& 0& \Delta \mathsf{\beta}& 0\\ 0& 1& -\Delta \mathsf{\alpha}& 0\\ -\Delta \mathsf{\beta}& \Delta \mathsf{\alpha}& 1& \Delta \mathrm{w}\\ 0& 0& 0& 1\end{array}\right]$ | $-\frac{\mathrm{t}}{{\mathrm{L}}_{1}}\le \Delta \mathsf{\alpha}\le \frac{\mathrm{t}}{{\mathrm{L}}_{1}}$ $-\frac{\mathrm{t}}{{\mathrm{L}}_{2}}\le \Delta \mathsf{\beta}\le \frac{\mathrm{t}}{{\mathrm{L}}_{2}}$ $-\frac{\mathrm{t}}{2}\le \Delta \mathrm{w}\le \frac{\mathrm{t}}{2}$ $\left|\Delta \mathrm{w}\right|+\left|\frac{{\mathrm{L}}_{2}\Delta \mathsf{\alpha}}{2}\right|+\left|\frac{{\mathrm{L}}_{1}\Delta \mathsf{\beta}}{2}\right|\le \frac{\mathrm{t}}{2}$ | |

Between two parallel surfaces | $\mathrm{T}=\left[\begin{array}{cccc}1& -\Delta \mathsf{\gamma}& \Delta \mathsf{\beta}& \Delta \mathrm{u}\\ \Delta \mathsf{\gamma}& 1& -\Delta \mathsf{\alpha}& \Delta \mathrm{v}\\ -\Delta \mathsf{\beta}& \Delta \mathsf{\alpha}& 1& \Delta \mathrm{w}\\ 0& 0& 0& 1\end{array}\right]$ | $-\frac{\mathrm{t}}{{\mathrm{l}}_{\mathrm{yz}}}\le \Delta \mathsf{\alpha}\le \frac{\mathrm{t}}{{\mathrm{l}}_{\mathrm{yz}}}$ $-\frac{\mathrm{t}}{{\mathrm{l}}_{\mathrm{xz}}}\le \Delta \mathsf{\beta}\le \frac{\mathrm{t}}{{\mathrm{l}}_{\mathrm{xz}}}$ $-\frac{\mathrm{t}}{{\mathrm{l}}_{\mathrm{xy}}}\le \Delta \mathsf{\gamma}\le \frac{\mathrm{t}}{{\mathrm{l}}_{\mathrm{xy}}}$ $-\frac{\mathrm{t}}{2}{\mathrm{n}}_{\mathrm{x}}\le \Delta \mathrm{u}\le \frac{\mathrm{t}}{2}{\mathrm{n}}_{\mathrm{x}}$ $-\frac{\mathrm{t}}{2}{\mathrm{n}}_{\mathrm{y}}\le \Delta \mathrm{v}\le \frac{\mathrm{t}}{2}{\mathrm{n}}_{\mathrm{y}}$ $-\frac{\mathrm{t}}{2}{\mathrm{n}}_{\mathrm{z}}\le \Delta \mathrm{w}\le \frac{\mathrm{t}}{2}{\mathrm{n}}_{\mathrm{z}}$ | |

Ring | $\mathrm{T}=\left[\begin{array}{cccc}1& 0& 0& \Delta \mathrm{u}\\ 0& 1& 0& \Delta \mathrm{v}\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$ | $-\frac{\mathrm{t}}{2}\le \Delta \mathrm{u}\le \frac{\mathrm{t}}{2}$ $-\frac{\mathrm{t}}{2}\le \Delta \mathrm{v}\le \frac{\mathrm{t}}{2}$ $\Delta {\mathrm{u}}^{2}+\Delta {\mathrm{v}}^{2}\le \frac{{\mathrm{t}}^{2}}{4}$ | |

Circle | $\mathrm{T}=\left[\begin{array}{cccc}1& 0& 0& \Delta \mathrm{u}\\ 0& 1& 0& \Delta \mathrm{v}\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$ | $-\frac{\mathrm{t}}{2}\le \Delta \mathrm{u}\le \frac{\mathrm{t}}{2}$ $-\frac{\mathrm{t}}{2}\le \Delta \mathrm{v}\le \frac{\mathrm{t}}{2}$ $\Delta {\mathrm{u}}^{2}+\Delta {\mathrm{v}}^{2}\le \frac{{\mathrm{t}}^{2}}{4}$ | |

Sphere | $\mathrm{T}=\left[\begin{array}{cccc}1& 0& 0& \Delta \mathrm{u}\\ 0& 1& 0& \Delta \mathrm{v}\\ 0& 0& 1& \Delta \mathrm{w}\\ 0& 0& 0& 1\end{array}\right]$ | $-\frac{\mathrm{t}}{2}\le \Delta \mathrm{u}\le \frac{\mathrm{t}}{2}$ $-\frac{\mathrm{t}}{2}\le \Delta \mathrm{v}\le \frac{\mathrm{t}}{2}$ $-\frac{\mathrm{t}}{2}\le \Delta \mathrm{w}\le \frac{\mathrm{t}}{2}$ $\Delta {\mathrm{u}}^{2}+\Delta {\mathrm{v}}^{2}+\Delta {\mathrm{w}}^{2}\le \frac{{\mathrm{t}}^{2}}{4}$ | |

Cylinder | $\mathrm{T}=\left[\begin{array}{cccc}1& 0& \Delta \mathsf{\beta}& \Delta \mathrm{u}\\ 0& 1& -\Delta \mathsf{\alpha}& \Delta \mathrm{v}\\ -\Delta \mathsf{\beta}& \Delta \mathsf{\alpha}& 1& 0\\ 0& 0& 0& 1\end{array}\right]$ | $-\frac{\mathrm{t}}{\mathrm{L}}\le \Delta \mathsf{\alpha}\le \frac{\mathrm{t}}{\mathrm{L}}$ $-\frac{\mathrm{t}}{\mathrm{L}}\le \Delta \mathsf{\beta}\le \frac{\mathrm{t}}{\mathrm{L}}$ $-\frac{\mathrm{t}}{2}\le \Delta \mathrm{u}\le \frac{\mathrm{t}}{2}$ $-\frac{\mathrm{t}}{2}\le \Delta \mathrm{v}\le \frac{\mathrm{t}}{2}$ ${\left(\mathrm{u}+\frac{\mathrm{L}\Delta \mathsf{\alpha}}{2}\right)}^{2}+{\left(\mathrm{v}+\frac{\mathrm{L}\Delta \mathsf{\beta}}{2}\right)}^{2}\le \frac{{\mathrm{t}}^{2}}{4}$ | |

Between two concentric cylindrical surfaces | $\mathrm{T}=\left[\begin{array}{cccc}1& 0& \Delta \mathsf{\beta}& \Delta \mathrm{u}\\ 0& 1& -\Delta \mathsf{\alpha}& \Delta \mathrm{v}\\ -\Delta \mathsf{\beta}& \Delta \mathsf{\alpha}& 1& 0\\ 0& 0& 0& 1\end{array}\right]$ | $-\frac{\mathrm{t}}{\mathrm{L}}\le \Delta \mathsf{\alpha}\le \frac{\mathrm{t}}{\mathrm{L}}$ $-\frac{\mathrm{t}}{\mathrm{L}}\le \Delta \mathsf{\beta}\le \frac{\mathrm{t}}{\mathrm{L}}$ $-\frac{\mathrm{t}}{2}\le \Delta \mathrm{u}\le \frac{\mathrm{t}}{2}$ $-\frac{\mathrm{t}}{2}\le \Delta \mathrm{v}\le \frac{\mathrm{t}}{2}$ ${\left(\mathrm{u}+\frac{\mathrm{L}\Delta \mathsf{\alpha}}{2}\right)}^{2}+{\left(\mathrm{v}+\frac{\mathrm{L}\Delta \mathsf{\beta}}{2}\right)}^{2}\le \frac{{\mathrm{t}}^{2}}{4}$ |

Tolerance Types | Function Expression |
---|---|

Reciprocal model | $\mathrm{C}\left(\mathrm{t}\right)={\mathrm{a}}_{0}+{\mathrm{a}}_{1}{\mathrm{t}}^{-1}$ |

Exponential model | $\mathrm{C}\left(\mathrm{t}\right)={\mathrm{a}}_{0}+{\mathrm{a}}_{1}{\mathrm{e}}^{-{\mathrm{a}}_{2}\mathrm{t}}$ |

Power-exponential model | $\mathrm{C}\left(\mathrm{t}\right)={\mathrm{a}}_{0}+{\mathrm{a}}_{1}{\mathrm{t}}^{-{\mathrm{a}}_{2}}$ |

Negative square model | $\mathrm{C}\left(\mathrm{t}\right)={\mathrm{a}}_{0}+{\mathrm{a}}_{1}{\mathrm{t}}^{-2}$ |

Composite power-exponential and exponential model | $\mathrm{C}\left(\mathrm{t}\right)={\mathrm{a}}_{0}+{\mathrm{a}}_{1}{\mathrm{e}}^{-{\mathrm{a}}_{2}\mathrm{t}}+{\mathrm{a}}_{3}{\mathrm{t}}^{-{\mathrm{a}}_{4}}$ |

Composite linear and exponential model | $\mathrm{C}\left(\mathrm{t}\right)={\mathrm{a}}_{0}+{\mathrm{a}}_{1}\mathrm{t}+{\mathrm{a}}_{2}{\mathrm{e}}^{-{\mathrm{a}}_{3}\mathrm{t}}$ |

Cubic model | $\mathrm{C}\left(\mathrm{t}\right)={\mathrm{a}}_{0}+{\mathrm{a}}_{1}\mathrm{t}+{\mathrm{a}}_{2}{\mathrm{t}}^{2}+{\mathrm{a}}_{3}{\mathrm{t}}^{3}$ |

Polynomial model | $\mathrm{C}\left(\mathrm{t}\right)={{\displaystyle \sum}}_{\mathrm{i}=0}^{\mathrm{n}}\left({\mathrm{c}}_{\mathrm{i}}{\mathrm{t}}^{\mathrm{i}}\right)$ |

Functional Surfaces | Sphere | Plane | Cylinder | Helical | Rotational | Prismatic | General |
---|---|---|---|---|---|---|---|

MGDE | point | plane | line | Point and line | Point and line | Line and plane | Point and line and plane |

Degrees of invariances | 3 rotations | 1 rotation 2 translation | 1 rotation 1 translation | 1 helical displacement | 1 rotation | 1 translation | None |

Reference element | sphere center | plane | cylinder axis | Rotation axis and points on the surface | Rotation axis and points on the surface | Straight line in translation direction and plane in determined direction | Any combinations of surface elements |

Type | p1,1 | p1,1/2,1 | p2,1 | p2,3 | p2,3/3,1 | p3,1 | p3,3 | p3,3/1,3 | p1,3 |
---|---|---|---|---|---|---|---|---|---|

∆α1 | −0.0626 | 0.0686 | 0.1339 | 0.0666 | 0.0750 | −0.0711 | 0.2132 | −0.2062 | 0.3365 |

∆α2 | −0.1070 | −0.1661 | 0.0687 | 0.0328 | 0.0769 | 0.3695 | 0.0554 | −0.0226 | 0.0033 |

∆α3 | −0.0083 | 0.0885 | −0.0164 | 0.1955 | −0.0300 | 0.2892 | −0.2052 | −0.2146 | 0.0073 |

∆α4 | −0.0598 | −0.0036 | 0.2237 | 0.1418 | 0.3842 | 0.0350 | 0.2310 | −0.1199 | 0.3371 |

∆β1 | 0.2322 | −0.0184 | 0.2598 | −0.2113 | 0.3229 | −0.0298 | −0.0993 | −0.0328 | −0.0641 |

∆β2 | −0.1472 | 0.3395 | −0.1493 | −0.1989 | 0.2057 | −0.1731 | −0.0340 | −0.1542 | 0.1300 |

∆β3 | −0.1926 | 0.0816 | 0.1306 | 0.2561 | −0.0248 | 0.1936 | 0.3226 | −0.0024 | 0.3058 |

∆β4 | −0.0019 | 0.3776 | −0.1030 | 0.1998 | 0.2418 | 0.13816 | 0.1633 | −0.1097 | 0.1772 |

∆w1 | 0.0558 | −0.1005 | 0.0327 | 0.1723 | 0.1229 | 0.1767 | −0.1343 | −0.1327 | 0.0079 |

∆w2 | 0.3817 | 0.0896 | 0.1699 | −0.1589 | 0.3161 | −0.1238 | −0.1077 | −0.1144 | 0.3893 |

∆w3 | 0.0312 | 0.0115 | −0.1237 | 0.1247 | 0.3588 | −0.2559 | 0.1361 | −0.0632 | 0.0810 |

∆w4 | −0.0037 | −0.0810 | 0.2691 | 0.3744 | −0.1031 | 0.1071 | −0.0366 | −0.0150 | −0.0700 |

Functional Surface | T_{min} | T_{max} |
---|---|---|

p1,1 | −0.0037 | 0.3817 |

p1,1/2,1 | −0.1005 | 0.0896 |

p2,1 | −0.1237 | 0.2691 |

p2,3 | −0.1589 | 0.3744 |

p2,3/3,1 | −0.1031 | 0.3588 |

p3,1 | −0.2559 | 0.1767 |

p3,3 | −0.1343 | 0.1361 |

p3,3/1,3 | −0.1327 | −0.0150 |

p1,3 | −0.0700 | 0.3893 |

Functional Surface | T_{min} | T_{max} |
---|---|---|

p1,1 | −0.3724 | 0.3378 |

p1,1/2,1 | −0.3043 | 0.2805 |

p2,1 | −0.3889 | 0.2604 |

p2,3 | −0.2918 | 0.1297 |

p2,3/3,1 | −0.2882 | 0.2972 |

p3,1 | −0.2517 | 0.3030 |

p3,3 | −0.1318 | 0.3349 |

p3,3/1,3 | −0.3451 | −0.1162 |

p1,3 | −0.3870 | 0.2681 |

p1,1 | p1,1/2,1 | C(T) | L(T) | ∆C(T) | Total Cost | |
---|---|---|---|---|---|---|

Result of the single objective function | T_{1,2} | 1.3653 | 1.8829 | 111.8426 | 56.6813 | 493.6342 |

T_{2,2} | 1.0708 | 2.6002 | 68.7968 | 92.6071 | ||

T_{3,2} | 1.0214 | 2.7176 | 62.5955 | 93.9102 | ||

Result of the multi objective function | T_{1,2} | 1.0096 | 2.7503 | 61.1575 | 94.5597 | 477.9916 |

T_{2,2} | 1.2587 | 2.0571 | 95.0595 | 58.1012 | ||

T_{3,2} | 0.7582 | 3.2887 | 34.4920 | 126.5256 |

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**MDPI and ACS Style**

Yang, K.; Gan, Y.; Cao, Y.; Yang, J.; Wu, Z.
Optimization of 3D Tolerance Design Based on Cost–Quality–Sensitivity Analysis to the Deviation Domain. *Automation* **2023**, *4*, 123-150.
https://doi.org/10.3390/automation4020009

**AMA Style**

Yang K, Gan Y, Cao Y, Yang J, Wu Z.
Optimization of 3D Tolerance Design Based on Cost–Quality–Sensitivity Analysis to the Deviation Domain. *Automation*. 2023; 4(2):123-150.
https://doi.org/10.3390/automation4020009

**Chicago/Turabian Style**

Yang, Kaili, Yi Gan, Yanlong Cao, Jiangxin Yang, and Zijian Wu.
2023. "Optimization of 3D Tolerance Design Based on Cost–Quality–Sensitivity Analysis to the Deviation Domain" *Automation* 4, no. 2: 123-150.
https://doi.org/10.3390/automation4020009