# Fast Measurement and Reconstruction of Large Workpieces with Freeform Surfaces by Combining Local Scanning and Global Position Data

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Description of the Combined System

#### 2.1. System Design

#### 2.2. Multi-Sensor Data Fusion

_{W}Y

_{W}Z

_{W}and the coordinate system of MAXscan is the local coordinate system O-X

_{L}Y

_{L}Z

_{L}. With the movement of MAXscan, the ith local coordinate system is expressed by O-X

_{Li}Y

_{Li}Z

_{Li}(i = 1, 2, 3 …, n).

**Figure 4.**(

**a**) TBR for the laser tracker mounted on a nest; (

**b**) RRTs pasted on the object; (

**c**) RRT pasted on the center of a 1.5-inch tip on the nest to assess the same point with the laser tracker and MAXscan.

_{W}Y

_{W}Z

_{W}is noted by ξ

_{W}= [X

_{W}Y

_{W}Z

_{W}] and that in O-X

_{Li}Y

_{Li}Z

_{Li}is ξ

_{L}= [X

_{L}Y

_{L}Z

_{L}]. The spatial relationship of all the local point clouds relative to the global coordinate system can be achieved by the scale parameter k, the rotation matrix R and the translation vector T. The coordinate transformation equation is expressed by:

_{T}y

_{T}z

_{T}]

^{T}.

_{T}y

_{T}z

_{T}ω φ к k]

^{T}.

^{2}[Var(V)]

^{−1}, σ

^{2}is the variance factor of unit weight.

_{T0}y

_{T0}z

_{T0}ω

_{0}φ

_{0}к

_{0}k

_{0}]

^{T}is expressed by:

^{−5}.

_{i}(i = 1, …, n) is the 3-D coordinate from the ith sensor, and n is the number of the sensors. It usually can be expressed by means of a function of θ

_{i}:

_{i}and neglecting the quadratic term:

_{i}is the Jacobian matrix with respect to θ:

_{i}:

_{i}for the ith sensor is determined by the covariance matrix V [M

_{i}]:

^{−1}> V[Mi]

^{−1}. Thus, V[$\widehat{{M}_{i}}$] < V[M

_{i}]. This proves that after the multi-sensor data fusion, the data accuracy is improved.

## 3. Experiments

#### 3.1. Standard Establishment

Object | Standard Value (mm) |
---|---|

AB | 127.7948 |

AC | 329.9191 |

AD | 487.4985 |

AE | 1015.1636 |

AF | 2211.4268 |

**Figure 5.**The standard values of the distances between the points measured by the laser tracker. (

**a**) The measurement of AB, AC and AD; (

**b**) The measurement of AE and AF.

#### 3.2. Small Size

_{W}Y

_{W}Z

_{W}and O-X

_{Li}Y

_{Li}Z

_{Li}. Then, all the measured volumes are separated to two scanning stations. As shown in Figure 7, points A, 1, 2, 3 and 4 are scanned at station 1, and points D, 1, 2, 3 and 4 are scanned at station 2. Meanwhile, points 1, 2, 3 and 4 are measured by the laser tracker as the reference global position points. Based on the principle of the multi-sensor data fusion, the points from both scanning stations can be transferred into the global coordinate system. Finally, we can calculate the length of AD in the global coordinate system. This process is repeated 10 times, and the results are compared with the standard. The deviations are shown in Figure 8. From Figure 9, through the error comparison by two methods for AC and AD, it is clearly observed that the accuracy is improved by the combined method, demonstrating that the combined method is effective to decrease the accumulated error.

**Figure 9.**The error comparison by two methods: (

**a**) The measured object is AC; (

**b**) The measured object is AD.

#### 3.3. Large Scale

**Figure 10.**(

**a**) Measurement error of AE and AF using the scanner; (

**b**) Measurement error of AE and AF using the combined method.

## 4. Practical Application

_{ij}(0 ≤ i ≤ n, 0 ≤ j ≤ m) is the control points, ω

_{ij}is corresponding weight of P

_{ij}, N

_{ik}(u) and N

_{jl}(v) are the normalized B-spline base functions of order k and l, respectively, defined over vector U = {u

_{0}, u

_{1}, …, u

_{n}, …, u

_{n+k}} and V = {v

_{0}, v

_{1}, …, v

_{m}, …, v

_{m+k}}.

**Figure 14.**The fitting result comparison: (

**a**) The point data; (

**b**) The conventional method; (

**c**) The proposed method.

Mean Error | Max Error | Standard Deviation | |
---|---|---|---|

Lateral deviation | 0.0000 | 0.0001 | 0.0000 |

Negative normal deviation | −0.0821 | −0.4300 | 0.0582 |

Positive normal deviation | 0.0810 | 0.3477 | 0.0704 |

Euclidean error | 0.0816 | 0.4300 | 0.0647 |

## 5. Error Analysis and Synthesis

_{W}, the measurement error of the handheld laser scanning sensor ∆

_{S}, data processing error Δp, the transformation error ∆

_{T}and the surface reconstruction error ∆

_{R}. Assuming all of the errors meet a normal distribution, the total error may be considered as:

_{R}of the gear rim is 0.0816 mm. Thus, according to the Equation (16), the total error is 0.0963 mm.

## 6. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## References

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**MDPI and ACS Style**

Chen, Z.; Zhang, F.; Qu, X.; Liang, B.
Fast Measurement and Reconstruction of Large Workpieces with Freeform Surfaces by Combining Local Scanning and Global Position Data. *Sensors* **2015**, *15*, 14328-14344.
https://doi.org/10.3390/s150614328

**AMA Style**

Chen Z, Zhang F, Qu X, Liang B.
Fast Measurement and Reconstruction of Large Workpieces with Freeform Surfaces by Combining Local Scanning and Global Position Data. *Sensors*. 2015; 15(6):14328-14344.
https://doi.org/10.3390/s150614328

**Chicago/Turabian Style**

Chen, Zhe, Fumin Zhang, Xinghua Qu, and Baoqiu Liang.
2015. "Fast Measurement and Reconstruction of Large Workpieces with Freeform Surfaces by Combining Local Scanning and Global Position Data" *Sensors* 15, no. 6: 14328-14344.
https://doi.org/10.3390/s150614328