# A Theoretical and Experimental Identification with Featured Structures for Crucial Position-Independent Geometric Errors in Ultra-Precision Machining

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Volumetric Error Modelling for PIGEs

_{c}denotes the rotating angle of the C-axis; and I

_{4×4}denotes a fourth-order identity matrix.

_{x}, E

_{y}, and E

_{z}, respectively. The volumetric error is written as:

## 3. Recognition of the Crucial PIGEs with the Featured Structures

#### 3.1. Specific Squareness Errors S_{cx} and S_{cy} under End-Face Turning

_{z}is related with two squareness errors, S

_{cx}and S

_{cy}

_{,}of the C-axis, and the moving distances x of the X-axis, and y of the Y-axis. To recognize only the specific squareness error S

_{cx}, a featured structure is proposed for end-face turning along the X-direction, where the Y-axis moving distance is fixed at 0, as shown in Figure 2. Resultantly, Equation (7) is further simplified as:

_{z}is its resulting machining form error, which would be obtained through measuring the machined surface, i.e., a taper surface, at a taper angle α

_{1}, as shown in Figure 2. Resultantly, the squareness error S

_{cx}is expressed as:

_{cy}, the same featured structure is designed for end-face turning along the Y-direction, where the X-axis moving distance is fixed at 0. Resultantly, Equation (7) is further simplified as:

_{z}would be obtained via measuring the machined taper surface at a taper angle α

_{2}, as shown in Figure 3. Resultantly, the squareness error S

_{cy}is expressed as:

#### 3.2. Specific Squareness Error S_{xy} under End-Square Milling

_{x}and E

_{y}are related with five squareness errors S

_{xy}, S

_{yz}, S

_{cy}, S

_{xz}, and S

_{cx}, the moving distances x of the X-axis and z of the Z-axis, and the rotating angle θ

_{c}of the C-axis. To recognize the specific squareness error S

_{xy}, a featured structure is proposed for end-square milling in the X–Y plane, where the C-axis is fixed at the rotating angle of 0, and the Z-axis moving distance is the cutting depth, hence regarded as the infinitesimal, as shown in Figure 4. Resultantly, the volumetric error E

_{x}in Equation (5) can be ignored, and Equation (6) is further simplified as:

_{y}is its resulting machining form error, which would be obtained via measuring the form of the machined square, i.e., a rhombus, at two diagonal lengths L

_{MN}and L

_{PQ}(the purple dashed line in Figure 4). The lengths are obtained from the position information of four vertices, which are recorded via the intersections of the residues of the two toolpaths (the yellow part of Figure 4). Resultantly, the squareness error S

_{cx}is expressed as:

#### 3.3. Specific Squareness Error S_{yz} under Lateral-Square Milling

_{y}and E

_{z}are related with five squareness errors S

_{xy}, S

_{yz}, S

_{cy}, S

_{xz}, and S

_{cx}, the moving distances x, y, and z of the X-, Y-, and Z-axes, and the rotating angle θ

_{c}of the C-axis. To recognize the specific squareness error S

_{yz}, a featured structure is proposed for lateral-square milling in the Y–Z plane, where the C-axis is fixed at the rotating angle of 0, the X-axis moving distance is the cutting depth, hence regarded as the infinitesimal, and the Y- and Z-axis moving distances are equal, as shown in Figure 5. Resultantly, Equations (6) and (7) are, respectively, simplified as:

_{cy}and S

_{yz}, and the Z-axis moving distance. The squareness error S

_{cy}only makes the designed featured structure rotate around the machined square at a certain angle without any geometric changes, as shown in Figure 5b, while the squareness error S

_{yz}changes it into a rhombus, as shown in Figure 5c. Accordingly, only considering its form accuracy, the volumetric error E

_{z}in Equation (15) could be ignored, so that Equation (14) is further simplified as:

_{y}would be obtained via measuring the form of the machined square at two diagonal lengths L

_{UV}and L

_{RW}, as shown in Figure 5c. Resultantly, the squareness error S

_{yz}is expressed as:

#### 3.4. Specific Squareness Error S_{xz} under Cylinder Turning

_{xz}, a featured structure is proposed for cylinder turning along the Z-direction, where the X-axis moving distance is the cutting depth, hence regarded as the infinitesimal, as shown in Figure 6. Besides, the volumetric errors E

_{x}and E

_{y}of Equations (5) and (6) in the Cartesian coordinate system are transformed into the volumetric error E

_{r}along the radial direction in the cylindrical coordinate system, via the expression E

_{r}= E

_{x}·cosθ

_{c}+ E

_{y}·sinθ

_{c}[30]. Resultantly, the volumetric error E

_{r}is expressed as:

_{c}changes rapidly, while the Z-axis moves relatively slowly. Accordingly, the volumetric error E

_{r}is further maximized as:

_{r}is its resulting machining form error, which would be obtained via measuring the machined surface, i.e., a cone, at a taper angle β, as shown in Figure 6. Resultantly, the squareness error S

_{co}is expressed as:

_{xz}is expressed as:

## 4. Experimental Setup

## 5. Results and Discussion

_{cx}and S

_{cy}, the machining form errors under end-face turning along the X- or Y-directions under the machining conditions of Table 3 and Table 4 were measured and extracted, as shown in Figure 8. Whether turning along the X-direction or Y-direction, the surface profiles are not the lines, but the tapered curves, and their taper heights are about 6 µm at the feed distance of 105 mm. Fitted via the least-square method, the taper angles α

_{1}and α

_{2}were obtained at 179.9896° and 179.9858° (Table 5), respectively. According to Equations (9) and (11), the specific squareness errors S

_{cx}and S

_{cy}were 18.72″ and −25.56″, respectively, as shown in Table 6. The results verify that the featured structures under end-face turning could be greatly amplified to recognize the specific squareness errors S

_{cx}and S

_{cy}.

_{xy}, the machining form error under end-square milling in the X–Y plane at the machining conditions of Table 3 and Table 4 was measured and extracted, as shown in Figure 9. The two diagonal lengths, L

_{MN}and L

_{PQ}, are 98,996.126 µm and 99,003.545 µm (Table 5) at the noticeable difference of 7.419 µm, which indicates that the machined square was deformed into a rhombus. According to Equation (13), the specific squareness error S

_{xy}is 15.46″, as shown in Table 6. The results prove that the featured structures under end-square milling would be significantly designed to recognize the specific squareness error S

_{xy}.

_{yz}, the machining form error under lateral-square milling in the Y–Z plane at the machining conditions of Table 3 and Table 4 was measured and extracted, as shown in Figure 10. The two diagonal lengths, L

_{UV}and L

_{RW}, are 99,000.582 µm and 99,008.822 µm (Table 5) at the significant difference of 8.240 µm, which identifies that the machined square was deformed into a rhombus. According to Equation (17), the specific squareness error S

_{yz}is 17.17″, as presented in Table 6. The results confirm that the featured structures under lateral-square milling would be dramatically designed to recognize the specific squareness error S

_{xy}.

_{xz}, the machining form error under cylinder turning along the Z-direction at the machining conditions of Table 3 and Table 4 was measured and extracted, as shown in Figure 11. After fitting, the taper angle β was 19.80″ (Table 5), which means that the machined cylinder was deformed into a cone. According to Equation (21), the specific squareness error S

_{xz}was 23.98″, as listed in Table 6. The results support the theory that the featured structures under cylinder turning could be prominently designed to recognize the specific squareness error S

_{xz}.

## 6. Conclusions

- (1)
- A volumetric error model has been proposed for PIGEs, to significantly reveal the relationship between the five squareness errors and their resulting machining form errors in UPM. The volumetric error is coupled with other squareness errors, and changes with the motion position along each axis.
- (2)
- Moreover, the featured structures have been designed, machined, and measured to efficiently decouple the specific squareness errors from their form errors in UPM, and to successfully recognize crucial PIGEs. The values of the five specific squareness errors identified are between 15″ and 26″.
- (3)
- Further, it is a potential means to improve the form accuracy of UPM, through the identification of crucial PIGEs with compensation.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

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**Figure 1.**The system of the employed four-axis UPM machine: (

**a**) its schematic diagram, (

**b**) its motion chains.

**Figure 2.**End-face turning along the X-direction for the featured structure: (

**a**) S

_{cx}= 0, (

**b**) S

_{cx}≠ 0.

**Figure 3.**End-face turning along the Y-direction for the featured structure: (

**a**) S

_{cy}= 0, (

**b**) S

_{cy}≠ 0.

**Figure 4.**End-square milling in the X–Y plane for the featured structure: (

**a**) S

_{xy}= 0; (

**b**) S

_{xy}≠ 0.

**Figure 5.**Lateral-square milling in the Y–Z plane as the featured structure: (

**a**) S

_{yz}= 0 and S

_{cy}= 0, (

**b**) S

_{yz}= 0 but S

_{cy}≠ 0, (

**c**) S

_{cy}= 0 but S

_{yz}≠ 0.

**Figure 6.**Cylinder turning along the Z-direction for the featured structure: (

**a**) S

_{cz}= 0, (

**b**) S

_{cz}≠ 0.

**Figure 7.**UPM experiments for the featured structures under: (

**a**) end-face turning for S

_{cx}or S

_{cy}, (

**b**) end-square milling for S

_{xy}, (

**c**) lateral-square milling for S

_{yz}, (

**d**) cylinder turning for S

_{xz}.

**Figure 8.**The measured taper angles under end-face turning for S

_{cx}and S

_{cy}: (

**a**) α

_{1}along the X-direction, (

**b**) α

_{2}along the Y-direction.

**Figure 9.**The measured diagonal lengths under end-face square milling for S

_{xy}: (

**a**) M for L

_{MN}, (

**b**) Q for L

_{PQ}, (

**c**) P for L

_{PQ}, and (

**d**) N for L

_{MN}.

**Figure 10.**The measured diagonal lengths under lateral-square milling for S

_{yz}: (

**a**) U for L

_{UV}, (

**b**) W for L

_{RW}, (

**c**) R for L

_{RW}, and (

**d**) V for L

_{UV}.

**Figure 11.**The measured taper angle β under cylinder turning for S

_{xz}: (

**a**) the measured topography, and (

**b**) the fitted topography.

Number | Symbol | Description |
---|---|---|

1 | S_{xy} | Squareness error between X-axis and Y-axis |

2 | S_{xz} | Squareness error between X-axis and Z-axis |

3 | S_{yz} | Squareness error between Y-axis and Z-axis |

4 | S_{cx} | Squareness error between C-axis and X-axis |

5 | S_{cy} | Squareness error between C-axis and Y-axis |

Adjacent Bodies | $\mathbf{Error}\mathbf{Transformation}\mathbf{Matrix}{}_{\mathit{i}}{}^{\mathit{j}}\mathit{T}{}_{\mathit{e}}$ | $\mathbf{Motion}\mathbf{Transformation}\mathbf{Matrix}{}_{\mathit{i}}{}^{\mathit{j}}\mathit{T}{}_{\mathit{m}}$ |
---|---|---|

0–1 (X-axis) | ${}_{0}{}^{1}T_{e}={I}_{4\times 4}$ | ${}_{0}{}^{1}T_{m}=\left[\begin{array}{cccc}1& 0& 0& x\\ 0& 1& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 0\end{array}\right]$ |

1–2 (Y-axis) | ${}_{1}{}^{2}T_{e}=\left[\begin{array}{cccc}1& -{S}_{xy}& 0& 0\\ {S}_{xy}& 1& -{S}_{yz}& 0\\ 0& -{S}_{yz}& 1& 0\\ 0& 0& 0& 1\end{array}\right]$ | ${}_{1}{}^{2}T_{m}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& y\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$ |

2–3 (C-axis) | ${}_{2}{}^{3}T_{e}=\left[\begin{array}{cccc}1& 0& {S}_{cx}& 0\\ 0& 1& -{S}_{cy}& 0\\ -{S}_{cx}& {S}_{cy}& 1& 0\\ 0& 0& 0& 1\end{array}\right]$ | ${}_{2}{}^{3}T_{m}=\left[\begin{array}{cccc}\mathrm{cos}{\theta}_{c}& -\mathrm{sin}{\theta}_{c}& 0& 0\\ \mathrm{sin}{\theta}_{c}& \mathrm{cos}{\theta}_{c}& 0& 0\\ 0& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$ |

3–4 (workpiece) | ${}_{3}{}^{4}T_{e}={I}_{4\times 4}$ | ${}_{3}{}^{4}T_{m}={I}_{4\times 4}$ |

0–5 (Z-axis) | ${}_{0}{}^{5}T_{e}=\left[\begin{array}{cccc}1& 0& {S}_{xz}& 0\\ 0& 1& 0& 0\\ -{S}_{xz}& 0& 1& 0\\ 0& 0& 0& 1\end{array}\right]$ | ${}_{0}{}^{5}T_{m}=\left[\begin{array}{cccc}1& 0& 0& 0\\ 0& 1& 0& 0\\ 0& 0& 1& z\\ 0& 0& 0& 1\end{array}\right]$ |

5–6 (tool) | ${}_{5}{}^{6}T_{e}={I}_{4\times 4}$ | ${}_{5}{}^{6}T_{m}={I}_{4\times 4}$ |

Squareness Error | Featured Structure | Feed Direction | Feed Rate (mm min ^{−1}) | Spindle Speed (rpm) | Feed Distance (mm) | Cutting Depth (μm) |
---|---|---|---|---|---|---|

S_{cx} | End-face turning in the X-direction | X | 10 | 1000 | 105 | 2 |

S_{cy} | End-face turning in the Y-direction | Y | 10 | 1000 | 105 | 2 |

S_{xy} | End-square milling in the X–Y plane | X and Y | 10 | 20,000 | 70 | 5 |

S_{yz} | Lateral-square milling in the Y–Z plane | Y and Z | 10 | 20,000 | 70 | 5 |

S_{xz} | Cylinder turning in the Z-direction | Z | 10 | 1000 | 70 | 2 |

Tool | Tool Nose Radius (mm) | Tool Rake Angle (°) | Front Clearance Angle (°) |
---|---|---|---|

Turning tool | 0.3258 | 0 | 12 |

Milling tool | 0.3700 | 0 | 7 |

Item | Result |
---|---|

The taper angle α_{1} (°) | 179.9896 |

The taper angle α_{2} (°) | 179.9858 |

The lengths L_{MN} and L_{PQ} (μm) | 98,996.126 and 99,003.545 |

The lengths L_{UV} and L_{RW} (μm) | 99,000.582 and 99,008.822 |

The taper angle β (″) | 19.80 |

Squareness Error | Result (″) |
---|---|

Squareness error S_{cx} | 18.72 |

Squareness error S_{cy} | −25.56 |

Squareness error S_{xy} | 15.46 |

Squareness error S_{yz} | 17.17 |

Squareness error S_{xz} | 23.98 |

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## Share and Cite

**MDPI and ACS Style**

Zhang, L.; Zhang, S.
A Theoretical and Experimental Identification with Featured Structures for Crucial Position-Independent Geometric Errors in Ultra-Precision Machining. *Machines* **2023**, *11*, 909.
https://doi.org/10.3390/machines11090909

**AMA Style**

Zhang L, Zhang S.
A Theoretical and Experimental Identification with Featured Structures for Crucial Position-Independent Geometric Errors in Ultra-Precision Machining. *Machines*. 2023; 11(9):909.
https://doi.org/10.3390/machines11090909

**Chicago/Turabian Style**

Zhang, Li, and Shaojian Zhang.
2023. "A Theoretical and Experimental Identification with Featured Structures for Crucial Position-Independent Geometric Errors in Ultra-Precision Machining" *Machines* 11, no. 9: 909.
https://doi.org/10.3390/machines11090909