Calibration Result

The AACMM adopted in this paper is Romer 2028, with the working diameter 2800 mm, repeatability 0.025 mm (3 (upsigma)), and length accuracy 0.066 mm (3 (upsigma)). The laser line sensor is self-made with the working depth 80 mm, and line length 65 mm. The repeatability is 0.008 mm, the measuring accuracy is 0.045 mm, and the standard working distance is 90 mm.

According to the proposed calibration method, 40 lines on the sphere are captured from different directions. Using obtained 40 conjugate pairs, the parameters in Eq. (2) are computed as:

$$left[begin{array}{cc}begin{array}{cc}{l}_{y} & {l}_{z}end{array}& {t}_{x}\ begin{array}{cc}begin{array}{c}{m}_{y}\ {n}_{y}\ 0end{array}& begin{array}{c}{m}_{z}\ {n}_{z}\ 0end{array}end{array}& begin{array}{c}{t}_{y}\ {t}_{z}\ 1end{array}end{array}right]=left[begin{array}{ccc}-0.997642& 0.0314107& 1.079\ 0.0596228& -0.043697& -3.465\ begin{array}{c}-0.033985\ 0end{array}& begin{array}{c}-0.998555\ 0end{array}& begin{array}{c}-211.168\ 1end{array}end{array}right].$$

(9)

The 2D data points on 40 measured lines are then transformed into 3D data points by substituting Eq. (9) into Eq. (4), and the 3D data points are employed to fit a sphere. Figure 6 demonstrates the distance between each point and the fitted surface. The biggest distance from the points outside the sphere and inside the sphere to the fitted surface are 0.483 mm and 0.576 mm, respectively.

Figure 6
figure 6

Error of the sphere fitted using the 3D data points transformed by the calibration result before searching optimization

To improve the accuracy of the parameters in ({{varvec{T}}}_{6}^{{varvec{L}}}), the searching algorithm was executed four times. The cycle index of each loop is always fixed at 11, while the incremental steps for each time are listed in Table 1.

The optimized result of current iteration will be taken as a reference value for the next iteration. The final optimized parameters are

$$left[begin{array}{cc}begin{array}{cc}{l}_{y} & {l}_{z}end{array}& {t}_{x}\ begin{array}{cc}begin{array}{c}{m}_{y}\ {n}_{y}\ 0end{array}& begin{array}{c}{m}_{z}\ {n}_{z}\ 0end{array}end{array}& begin{array}{c}{t}_{y}\ {t}_{z}\ 1end{array}end{array}right]=left[begin{array}{ccc}-0.997193& 0.0348994& 1.465\ 0.0644936& -0.048820& -3.170\ begin{array}{c}-0.038018\ 0end{array}& begin{array}{c}-0.998197\ 0end{array}& begin{array}{c}-211.604\ 1end{array}end{array}right].$$

(10)

The 2D data points on the 40 lines are also transformed into 3D data points by substituting Eq. (10) into Eq. (4), and the 3D data points are employed to fit a sphere. In Figure 7, the biggest distance from the point outside the sphere and inside the sphere to the fitted surface are 0.175 mm and 0.142 mm, respectively. It can be noticed that the accuracy the parameters in ({{varvec{T}}}_{6}^{{varvec{L}}}) is greatly improved after the searching optimization.

Figure 7
figure 7

Error of the sphere fitted using the 3D data points transformed by the calibration result after searching optimization

Error Analysis

The spatial accuracy of the AACMM is 0.066 mm (3 (upsigma)), and the accuracy of the laser line sensor is 0.045 mm, which is usually considered as the error of a single scanned point. In this study, we test this error on a Hexagon bridge CMM. After the sensor is mounted on the CMM, the extrinsic parameters are calibrated. A sphere with the nominal radius 19.875 mm is scanned using the sensor. The scanned 3D data points are then applied to fit sphere with a fitted radius at 19.870 mm. The error between the nominal radius and the fitted radius is 0.005 mm, and the fitted radius is the result of all scanning point operations. We define this error as the systematic error of the sensor. While the biggest distance from the 3D points to the fitted sphere surface is 0.045 mm, it is regarded as the random error of the sensor.

The error of the AACMM and the laser sensor can cause errors in the process of calibration. Even if a fixed point is exactly measured, the errors in ({{varvec{T}}}_{0}^{6}) and (({y}_{mathrm{L}},{z}_{mathrm{L}})) can also cause errors in ({{varvec{T}}}_{6}^{{varvec{L}}}). Since (({y}_{mathrm{L}},{z}_{mathrm{L}})) is achieved by circle fitting, the error of (({y}_{mathrm{L}},{z}_{mathrm{L}})) is much smaller than the sensor error, which suggests that the error of the AACMM is the main error source of calibration error. In summary, there exist four types of errors in this system, which are defined as follows:

({err}_{A}): Error of the AACMM,

({err}_{SS}): The systematic error of the laser sensor,

({err}_{SR}): Random error of the laser sensor,

({err}_{C}): Calibration error of the system.

The maximum total error of the system is

$$begin{aligned} {err}_{Total}={err}_{A}+{err}_{SS}+{err}_{SR}+{err}_{C} &{=0.066+0.005+0.045+err}_{C}. end{aligned}$$

(11)

Since the magnitude of ({err}_{C}) is unknown, ({err}_{Total}) cannot be directly calculated.

Accuracy Test

The measuring accuracy of the system is tested by scanning a plate and a reference sphere. The plate was scanned five times from different directions. For each scanning, only one or two joints of the AACMM were rotated, while other joints were kept fixed as far as possible. Firstly, the points on each scanned data patch are used to fit a plane, and the biggest distance from the points on each data patch to the corresponding fitted plane is defined as ({D}_{PPi}) (i = 1, 2, 3, 4, 5), listed in Table 2. Then, the points on all five data patches are together utilized to fit a plane, and the biggest distance from the points on five data patches to the fitted plane is 0.193 mm.

Table 2 Biggest distance from the points on each data patch to the fitted plane

The measurement error mainly depends on the error of the arm and the sensor, and the error of arm comes from the position errors of the six axes. When scanning the plate from one direction, only one or two joints of the AACMM were rotated with other joints kept fixed. Thus, the biggest distance from the points to the fitted plane is majorly affected by the error of the one or two joints and the sensor. On the other hand, the position error of the entire data patch is mainly caused by the position errors of other axes. If all five data patches are applied to fit a plane, the fitted error could be attributed to the position errors of the six axes and the sensor. As a result, fitted error of five data patches (around 0.193 mm) is significantly greater than errors in single data patch ({D}_{PPi}).

The sphere was also scanned from five directions. At each direction, only one or two joints of the AACMM were rotated. The points on the edge of each data patch have greater errors, thereby removed from the data patch using the software Surfacer V10.0. Five spheres were fitted using the remaining points on each data patch, defined as ({SPH}_{i}) (i = 1, 2, 3, 4, 5). In Table 3, ({R}_{i}) represents the fitted radius, and ({D}_{PSi}) (i = 1, 2, 3, 4, 5) gives the biggest distance from the points on each data patch to the corresponding fitted sphere. Then the points on five data patches are utilized to fit a sphere, defined as ({SPH}_{ALL}). The biggest distance from the points to the fitted sphere is 0.219 mm, which is greater than ({D}_{PSi}) in Table 3. The distances between the center of ({SPH}_{ALL}) and the centers of ({SPH}_{i}) are also given in Table 3.

Table 3 Biggest distance from the points on each data patch to the fitted sphere (mm)

It can be seen from the above test:

  1. (1)

    According to Eq. (11), total errors of the system are 0.193 mm and 0.219 mm, corresponding to scanning a plate and a sphere from five directions, respectively.

  2. (2)

    In Table 3(left|{SPH}_{ALL}-{SPH}_{i}right|) can represent the error between two data patches. Apparently, this error is caused by the position errors of different joints of the AACMM. Hence, ({err}_{A}) is its main error source.

  3. (3)

    The ({R}_{i}) in Table 3 is close to the nominal radius of the sphere, since ({R}_{i}) is the radius of the fitted sphere by the data scanned from one direction. In this case, ({err}_{A}) is much smaller than 0.066 mm, and ({err}_{SR}) is greatly reduced after sphere fitting. Therefore, ({err}_{C}) and ({err}_{SS}) are the main error sources in ({R}_{i}).

Typically, when measuring a complicated part, all joints are required to be rotated. In the above test, the total error of measuring a sphere is 0.219 mm. This error can be regarded as the measuring error of this system, since when scanning the sphere from five directions, all of the joints of the AACMM are rotated.

It can be found that the error of the AACMM is the main error source. It is worth noting that the calibration error is also an error source, but its magnitude cannot be found out.

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