# Overall Profile Measurements of Tiny Parts with Complicated Features with the Cradle-Type Five-Axis System

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Measurement Scheme

## 3. Coordinate System Construction and Path Planning

_{0}.

**P**of the exit pupil are (0, 0, d + r + h

_{ep_}_{0}_{0}), and the initial coordinates

**P**of the reference point are (0, 0, r + h

_{ref_}_{0}_{0}). The measurement data transmitted from the measurement system to the computer at a time includes six terms (x

_{i}, y

_{i}, z

_{i}, α, β, and h

_{i}), which represent the real-time position information of the X, Y, Z, A, and C axes, and the measurement data of the chromatic confocal probe.

_{0}) to (x

_{1}, y

_{1}, z

_{1}+ d + r + h

_{0}), and the reference point will move to (x

_{1}, y

_{1}, z

_{1}+ r + h

_{0}). In three-dimensional space, each rotary axis can be positioned by two points on it, which is named c

_{1}(a, b, c) and c

_{2}. According to the rotation transformation principle of rigid body in three dimensions, the vector of the axis

**p**and rotary matrix

**R**can be written as follows:

**P**

_{ep_}**and reference point**

_{1}**P**are expressed as Equations (3) and (4).

_{ref_}_{1}**m**can indicate the spatial orientation of the probe, and can be calculated according to the current positions of the exit pupil of the probe and reference point:

_{1}, the spatial coordinates of the first measuring point in the global measurement coordinate system

**P**can be calculated as Equation (6).

_{point_}_{1}_{n}, y

_{n}, z

_{n}, α

_{n}, and β

_{n}) at the nth measuring point and the data (x

_{n+}

_{1}, y

_{n+}

_{1}, z

_{n+}

_{1}, α

_{n+}

_{1}, β

_{n+}

_{1}, and h

_{n+}

_{1}) obtained at the n + 1th measuring point, the process of calculating the coordinates P

_{point_n}

_{+1}of the n + 1th point was as follows:

_{add}, y

_{add}, z

_{add}, α

_{add}, and β

_{add}of the X, Y, Z, A, and C axes, respectively. Due to the cradle-type structure, rotations around the C axis change the directions of linear motions by X, Y, and Z stages, while rotations around the A axis change the directions of X, Y, and Z stages and the vector of the C axis. Each motion is recorded, and these vectors were calculated. The translation matrix

**T**

_{i}is expressed as Equation (7), where

**axis**

_{x_i},

**axis**

_{y_i}, and

**axis**

_{z_i}are the vectors of X, Y, and Z stages, and

**O**is a 3 × 3 matrix of zeros.

**P**_

_{ep}

_{n+}**and reference point**

_{1}**P**_

_{ref}

_{n+}**during the n + 1th measurement.**

_{1}**P**and

_{ep}__{n}**P**represent the coordinates of the exit pupil and reference point during the nth measurement.

_{ref}__{n}_{n+}

_{1}. In summary, based on the iterative theory, the spatial coordinates of all measured points can be derived. All points were on the scanning paths and coordinates were calculated in the same measurement coordinate system recursively. Point cloud registration was not needed because the relative position of these points was consistent with the actual situation.

## 4. Error Correction Theory of the Cradle-Type Measurement System

#### 4.1. Error Identification

_{1}, δθ

_{2}, Δx, Δy, and Δz), dynamic errors of the linear stage (Δd), and dynamic errors of the rotary stage (δβ). The first two belong to static system errors, which stem from the inaccurate clamping and is fixed when the system is built, and the latter two belong to dynamic system errors, which stem from motions of the linear or rotary axes and are random. The clamping errors of measured workpieces are divided into tilt errors (δβ

_{w}

_{1}and δβ

_{w}

_{2}) and centrifugal errors (Δx

_{w}and Δz

_{w}), and both are static errors. The schematic diagram of clamping errors is shown in Figure 4a, with a cylinder as the workpiece. Cylinders have central axes, which can indicate the degrees in which workpieces are usually tilted. The axis of the workpiece may not be perpendicular to the rotary stage surface, so there is a certain angle between this axis and the axis of the stage. Tilt errors are the angles between the projection of the axis of the measured workpiece and coordinate axes in the plane, which is perpendicular to the C axis. Although the axis of the workpiece is parallel to the C axis, they are not coincident. The distance between two parallel axes represents the degree of deviation of two axes, and the two orthogonal components of this distance in the plane perpendicular to the C axis are called centrifugal errors.

#### 4.2. Error Calibration and Compensation

#### 4.2.1. Calibration and Compensation of Static Errors of Rotary Stages

**L**

_{1},

**L**

_{2},

**L**

_{3}, and

**L**

_{4}, and the other three axes were generated by the rotation of the first axis

**L**

_{1}. Four axes were symmetrically distributed around the rotary axis of the rotary stage, so that the real vector of it could be calculated by optimization. The coordinates of the two points on the rotary axis were the optimization objects. The first axis

**L**

_{1}will be rotated 90°, 180°, and 270° around the optimized rotary axis and the form

**L**

_{2′},

**L**

_{3′},

**L**

_{4′},

**L**

_{2′},

**L**

_{3′}, and

**L**

_{4′}did not coincide with

**L**

_{2},

**L**

_{3}, and

**L**

_{4}, and the optimization was finished when the total distance between them reached the minimum.

**-X**

^{’}**Y**

^{’}**Z**

^{’}**of the rotary stage did not coincide with the measurement coordinate system O-XYZ. Coordinates of the direction vectors**

^{’}**O**,

^{’}X^{’}**O**,

^{’}Y^{’}**O**and the origin

^{’}Z^{’},**O**in O-XYZ can be expressed according to the static errors of the rotary stage. By using the coordinate space transformation matrices, the coordinates of the measured point

^{’}**P**can be converted to the coordinates

^{’}**P**in the global measurement coordinate system:

**P**x,

**P**y, and

**P**z represent the direction vectors of O’X’, O’Y’, and O’Z’ in the global measurement coordinate system.

_{i}between corresponding points in the virtual point clouds and the optimized point clouds along the Z axis, which can explain the compensation effect and the reliability of the optimized rotary axis:

_{virtual_i}and z

_{optimized_i}are the Z coordinates of the ith point in the virtual and the optimized point clouds.

#### 4.2.2. Calibration and Compensation of Clamping Errors

_{1}(x

_{1}, y

_{1}, and z

_{1}) and N

_{2}(x

_{2}, y

_{2}, and z

_{2}) on the rotary axis were used to determine the position and the orientation of the axis in the optimization, and the cylinder axis can be expressed as Equations (12) and (13):

_{i}between the measured point M

_{i}(x

_{i}, y

_{i}, z

_{i}) and the cylinder axis can be calculated as Equation (14):

## 5. Experiments and Results

#### 5.1. System Construction and Error Calibration

#### 5.2. Measurement Results

_{i}between the measured ith point and the corresponding tiny triangle surface was calculated. Suppose the number of total points is n, then the standard deviation σ is obtained by the calculation process of Equation (16). This standard deviation can reflect the dispersion degree of the deviation between the actual and ideal coordinates, indicating the concentration of the error distribution in a single measurement. The standard deviation in the cylinder measurement dropped from 101.27 to 12.42 μm, as shown in Figure 11a, which confirms the compensation effect. The same area of the standard cylinder was measured four times, and the performance is presented in Figure 11b. The measurement results were expanded along the angle, and the distance from each point to the cylinder axis was calculated. It can be seen from the figure that the error distribution of each measurement was basically the same, which proved that the repeatability was good.

## 6. Conclusions

- (1)
- An optical, cradle-type, non-registration point-scanning measurement method was proposed, which does not need the point cloud registration process and adapts to multiple complicated features of different sizes. This measurement system has strong flexibility.
- (2)
- A process to identify major error terms in measurement systems and apply calibration and compensation on them was proposed. The advantages of this process are that it does not rely on any additional high-precision equipment and promotes the system’s accuracy conveniently and efficiently. This method can also be applied to correct error terms in other measurement systems with rotary axes.
- (3)
- A five-axis experiment setup was built and tiny parts were measured in experiments. The measurement accuracy and capability of overall profile measurements were verified by measuring standard workpieces and complicated tiny parts separately. It was proved that in terms of overall profile measurement, this cradle-type five-axis measurement system had more advantages than some commercial instruments. It is worth noting that, measurement accuracy can be further improved if hardware with higher accuracy is used in the future. With the help of a premeasurement by an external measurement device, a fully automatic measurement is the next research goal.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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**Figure 2.**Schematic diagrams for coordinates recursion: (

**a**) calculate the first point coordinates; (

**b**) calculate the n + 1th point coordinates.

**Figure 3.**Schematic diagrams for different measurement paths: (

**a**) rotary scanning paths; (

**b**) raster paths; (

**c**) free scanning paths.

**Figure 4.**Error identification and simulation: (

**a**) a schematic diagram for clamping errors of workpieces; (

**b**) simulation results on clamping errors.

**Figure 6.**The point cloud distribution of the rotary axis calibration in simulation: (

**a**) θ = 0°; (

**b**) θ = 90°; (

**c**) θ = 180°; (

**d**) θ = 270°.

**Figure 7.**The distance distribution in the XOY plane after rotary-axis compensation in simulation: (

**a**) θ = 0°; (

**b**) θ = 90°; (

**c**) θ = 180°; (

**d**) θ = 270°.

**Figure 8.**The error distribution in XOY plane after clamping-error compensation in simulation: (

**a**) θ = 0°; (

**b**) θ = 90°; (

**c**) θ = 180°; (

**d**) θ = 270°.

**Figure 12.**Application results on two tiny parts: (

**a**) a cross cylinder; (

**b**) a microtriangular pyramid.

**Figure 13.**Comparation of results: (

**a**) the measure process on the CMM; (

**b**) measurement results (50% spherical surface) by the CMM; (

**c**) the measure process on our five-axis system; (

**d**) results of the same measured area (25% spherical surface) by the five-axis system; (

**e**) measurement results of 70% of the spherical surface by the five-axis system.

Error Types | Specific Error Terms | Symbols | Ratios |
---|---|---|---|

System errors | Static errors of linear stages | δx, δy, δz | 2.430% |

Dynamic errors of linear stages | Δd | 0.367% | |

Static errors of rotary stages | δθ_{1}, δθ_{2}, Δx, Δy, Δz | 73.013% | |

Dynamic errors of rotary stages | δβ | 0.083% | |

Clamping errors of workpieces | Tilt errors | δβ_{w}_{1}, δβ_{w}_{2} | 17.247% |

Centrifugal errors | Δx_{w}, Δz_{w} | 6.860% |

Hardware | Travel/Range | Accuracy | Others |
---|---|---|---|

X/Y/Z axes | 200 mm | 1 μm | \ |

A/C axes | 360° | 0.004° | Surface radius of C-axis stage: 30 mm |

Probe | 400 μm | 0.1 μm | NA: ±28° |

Standard cylinder | r: 10 mm | Cylindricity: 14 μm | \ |

Error Source | Error Terms | Value (° or μm) |
---|---|---|

A axis | δθ_{1}, δθ_{2} | 0.0741, 0.3413 |

Δx, Δy, Δz | (0.8424, −0.0012, 0.0177) | |

C axis | δθ_{1}, δθ_{2} | 0.5035, −0.8466 |

Δx, Δy, Δz | (0.2906, 0.5074, 0.6875) | |

Workholding device | δβ_{w}_{1}, δβ_{w}_{2} | −5.6936, 2.2561 |

Δx_{w}, Δz_{w} | (0.0651, −0.2498) |

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## Share and Cite

**MDPI and ACS Style**

Liu, L.; Zhu, L.; Miao, L.; Li, C.; Fang, C.; Zhang, X.
Overall Profile Measurements of Tiny Parts with Complicated Features with the Cradle-Type Five-Axis System. *Sensors* **2021**, *21*, 4609.
https://doi.org/10.3390/s21134609

**AMA Style**

Liu L, Zhu L, Miao L, Li C, Fang C, Zhang X.
Overall Profile Measurements of Tiny Parts with Complicated Features with the Cradle-Type Five-Axis System. *Sensors*. 2021; 21(13):4609.
https://doi.org/10.3390/s21134609

**Chicago/Turabian Style**

Liu, Lei, Linlin Zhu, Li Miao, Chen Li, Changshuai Fang, and Xiaodong Zhang.
2021. "Overall Profile Measurements of Tiny Parts with Complicated Features with the Cradle-Type Five-Axis System" *Sensors* 21, no. 13: 4609.
https://doi.org/10.3390/s21134609