2.1. Overview of CAT System

This study used a bridge-type CMM (Innovalia Metrology, Spark gage 06.05.05) as the positioning mechanism and a line laser probe (Datapixel, Optiscan 1040-L) as the optical probe. The CAT system comprised a path generation system and measurement simulator for shape measurements. Kodatuno, a geometric calculation library for path generation systems ¹⁴, and a PC configured with an Intel Core i7-8700 CPU, Nvidia GeForce RTX 2060 GPU, and 16 GB of memory were used for simulations.

2.2. Bridge-Type CMM

The CMM has five axes: three orthogonal axes and two rotational axes. The operating range of the three orthogonal axes is 600 \(\times\) 500 \(\times\) 500 mm. In addition, the CMM has an inclined axis and a rotational axis for posture control. The operating range of the inclined axis is \(-180°\) to 180° and that of the rotational axis is 0° to 105°. The inclination and rotational angles were set at intervals of 7.5°. In addition, the measurement accuracy was independent of the temperature changes.

2.3. Line Laser Probe

The line laser probe shown in Fig. 4 uses a diffuse-reflection-based detection method and a triangulation-based measurement principle. The characteristics of the used probe are a maximum measurement width of 40 mm and a measurement distance of \(93\pm 20\) mm from the laser emission unit. The measurable area (field of view) was defined as the rectangular area obtained based on the measurement width and distance. In addition, the measurement accuracy is \(\pm 10\) μm.

3.1. Overview of Constraints

In a path generation system, the scanning paths are regarded as repetitions of the relative positioning of the probe against the target surface. This positioning is determined by the relative distance and angle and is closely related to the quality of measured data. In general, the reference measurement distance is ideal for the relative distance, and the posture perpendicular to the measured surface is ideal for the relative angle ¹⁸. Furthermore, constraints were imposed on the path-generation system by the equipment.

3.2. Constraints by Probe

Constraints (1)–(3) must be considered based on the measurement principles of the probe. In this context, the triangle connected to the scanning line and the position of laser incidence is defined as the laser plane, as shown in Fig. 4.

1. Probe positioning such that the measured surface falls within the field of view (FOV).

2. Probe positioning that prevents occlusion.

3. Scanning direction set along the normal vector to the laser plane.

In (1), the range can be detected when the laser light is limited by the probe specifications. Therefore, the probe must be positioned such that the measured surface falls within the defined FOV. In (2), the reflected laser light must be detected by light-receiving devices, as shown in Fig. 5. Therefore, occlusion must not occur in either the incident or reflected direction. In addition, it is desirable to consider the scanning direction, as described in (3).

3.3. Constraints by Position Mechanism

During the measurement, the positioning mechanism controls the position and posture of the probe. Therefore, it is necessary to consider constraints (4) and (5).

1. 4.

   The posture of the probe cannot be changed during the measurement.

2. 5.

   The probe posture cannot be controlled around the posture vector D\(_{\textrm{k}}\), which represents the direction of the laser emission.

Regarding (4), a change in posture should be avoided to prevent deterioration of the measurement accuracy of the CMM. Regarding (5), perfect control of the posture of the line laser probe requires three degrees of freedom; however, the system controls probe posture using only two rotational axes of the CMM in this study.

4.1. Overview of Path Generation

This system automatically generates a scanning path by loading the shape data of the measurement target and selecting the surface for measurement. This information included occlusion occurrence, and measurement accuracy was evaluated for each surface. Subsequently, a measurement position was generated for each surface based on this information. Additionally, a measurement position was generated for each posture. This method can be applied to free-form surfaces that fit within an FOV. Even if the surface is not fit within the FOV, the method can be applied by changing the scanning posture and dividing into multiple scanning paths.

4.2. Determination of Measurement Posture

As shown in Fig. 6, the measurement posture can be expressed by the posture vector D\(_{\textrm{k}}\) based on the rotation and inclination angles. Because each angle can be set at intervals of 7.5°, D\(_{\textrm{k}}\) was determined from 49 rotation-angle candidates \(\times\) 15 inclination candidates. In this system, areas that cannot be measured based on the occurrence of occlusions were defined on the target surface. An appropriate posture was selected for each measurable surface. First, judgment points were generated on the target surface, as shown in Fig. 7. The judgment vectors N\(_{\textrm{k}}\) were generated in the direction normal to the target surface from the judgment point. The distance between the judgment points was determined to be smaller than the point distance per irradiation, which was determined based on the measurement period and scanning speed. Then, one judgment point was selected, and intersections with the straight line in the N\(_{\textrm{k}}\) direction and each surface were checked. When an intersection is detected, interference between the surface and the rest of the straight line in the D\(_{\textrm{k}}\) direction was carried out.

If interference is detected in all straight lines in the D\(_{\textrm{k}}\) direction, this judgment point is regarded as a point that cannot be measured due to occlusion occurrence and is deleted.

The target shape when the occlusion occurs is shown in Fig. 8. The measurable vectors are shown in blue, and the unmeasurable vectors are shown in red. The angles between N\(_{\textrm{k}}\) and D\(_{\textrm{k}}\) are then calculated at the remaining judgment points. If the calculated angle is less than or equal to 45°, the judgment point is regarded as a measurable point at this D\(_{k}\). This process was applied to all D\(_{\textrm{k}}\), and the D\(_{\textrm{k}}\) with the most measurable points was selected as the measurement posture at the surface. In the case where multiple measurement postures have the same number of measurable points, the one with the smaller angle between D\(_{\textrm{k}}\) and the average of N\(_{\textrm{k}}\) was selected as the measurement posture. This process was repeated for all the judgment points on the target surfaces. Even when occlusion occurs in a certain posture, the target can be measured by changing the posture. An example of this is shown in Fig. 9.

4.3. Division of Measurement Surface

The target surfaces were divided for each selected posture in the measurement domain as shown in Fig. 10. An offset point was generated. The distance between this point and the measurable judgment point was equal to the reference measurement distance in the direction of the posture vector. A plane was generated with its normal vector aligned with D\(_{\textrm{k}}\) of the measurement posture. The distance between the generated plane and other measurable points was then calculated. If this distance was less than the reference measurement distance of the probe, then the other measurable point belonged to the same measurement domain as the other measurable point. Otherwise, a new plane was generated for the measurable point belonging to the new measurement domain.

4.4. Generation of Scanning Path

Scanning paths were generated to comprehensively cover the measurable points in each generated measurement domain. In this context, considering constraint (3), the scanning paths were set as straight lines. Furthermore, when multiple target surfaces or multiple measurement domains exist, the intermediate path along the \(Z\)-axis and the connection path connecting to the nearest measurement scanning path were used to construct a single-path stroke. When using the shape shown in Fig. 7, a scanning path can be generated within 3 min.

5.1. Overview of Overlap

When using a 3D measurement system, duplication of the measurement area occurs because of various factors. Examples include the wide measurement area of the probe and division of the scanning path into several paths. In the study, duplication of the measurement area, defined as “overlap,” is shown in Fig. 11. The overlap helps avoid areas lacking point clouds. However, various problems are caused by this overlap, as mentioned below. Accordingly, it is desirable to reduce the overlapping parts and limit the point-cloud density to satisfy the requirements.

5.2. Problems

Problems caused by the occurrence of overlap are variations in the point cloud density. This leads to an increase in unnecessary data volume and errors in geometric shape extraction. However, the processing method for preventing overlap depends on the operator and has not been explicitly defined. Therefore, reducing overlap is an important component of product inspection and machining after measurement.

6.1. Overview of Simulator

Measurement simulators have been developed for the building and manufacturing industries. However, they primarily focus on the prediction of the measured point cloud, and the overlap has not received significant attention. Therefore, this measurement simulation aims to detect overlapping parts through calculations using an automatically generated scanning path and measurement conditions. The results are output as a colored mesh and an area of overlapping parts. Consequently, we can add information to the measurement point cloud, indicating whether each point is near an overlapping part. Based on this result, point-cloud reduction can be performed.

6.2. Import of Simulation Data

First, the triangular mesh model data generated from the CAD data of the target object were imported. The two mesh models have different mesh sizes, as shown in Fig. 12. We defined the mesh model with a smaller mesh size as “detailed mesh,” and the one with a larger mesh size as “simplified mesh.” These were used to limit the calculation range and simplify the calculations. The size of the detailed mesh was set to be the same as the point distance per irradiation, which was determined by the measurement period and scanning speed. In addition, the detailed and simplified meshes corresponded geometrically, and several triangles of the detailed mesh were contained within each triangle of the simplified mesh. Second, the characteristics of the laser probe and measurement conditions were imported for use in calculations.

6.3. Determination of Scanning Path

The scanning path generated by the CAT system is output as position \((x,y,z)\) in the CAD coordinate system, rotational angle (\(\mathit{rot}\)), and inclination angle (\(\mathit{inc}\)). First, the measurement position was determined based on the start and end points. Therefore, a point cloud was generated by linear interpolation between the two points, considering the scanning speed. These points were saved as “measurement position.” The point clouds of the measurement position from the starting point to the end point were regarded as one scanning path. Subsequently, the generated scanning paths were saved individually as Path 2, Path 3, etc. In addition, the movement vector was defined as a normalized vector connecting the start and end points.

Second, the direction of the laser incidence was determined using \(\mathit{rot}\) and \(\mathit{inc}\). When the angles are defined using Eqs. \(\eqref{eq:1}\) and \(\eqref{eq:2}\), the vector can be represented by normalizing Eq. \(\eqref{eq:3}\) ¹⁹. This was defined as the incident vector.

6.4. Calculation of Intersection

6.5. Judgment of Overlap

The overlap occurs when the measurement areas were duplicated. That is, the same parts were measured using multiple scanning paths. Therefore, we define the criterion for overlap occurrence as the number of times the same parts were measured by different paths. The target was a triangular mesh. By applying this criterion, the number of measurements for a single mesh with different paths can be regarded as the number of overlaps. When the intersection is saved as described in Section 6.4.3, the path number used for the measurement is also saved for each mesh index containing the intersection. As a result, it is possible to judge the overlap occurrence quantitatively, as shown in Fig. 15, based on the number of times different paths are measured.

6.6. Output of Result

If the count saved in Section 6.5 exceeds a certain threshold, the mesh is output as an overlapping occurrence part. The meshes are colored darker red according to the number of measurements. The meshes judged as overlapping parts are colored in blue. The centroid of the triangular meshes judged as an overlapping occurrence were output as the position of overlapped parts. When the generated path and the 1.5 mm detailed mesh were used, the overlap of all surfaces could be detected within 20 s. Through this overlap detection, information can be added to the measurement point cloud, indicating whether each point is near an overlapping part. Using this information, we could efficiently perform point-cloud reduction.

7.1. Results of Measurement and Path Generation

A measurement was conducted using the scanning path generated by the CAT system. The target shape is shown in Fig. 16(a) and the generated scanning path is shown in Fig. 16(b). The point cloud data measured by the CMM are shown in Figs. 17(a) and (d). Consequently, the interference between the target object and the probe did not occur. In addition, no missing point-cloud regions were observed. However, an overlap occurred on part of the top surface. This overlap likely occurred because of excessive division of the scanning path to prevent missing point clouds. It was also found that the occurrence of overlaps caused by the generated path affected the density of the measured point cloud. Therefore, it is considered that the point cloud density can be adjusted by setting an appropriate path interval.

7.2. Results of Overlap Detection

This study conducted a comparative evaluation between the detection results of overlap by the simulator and the measured point cloud data. The simulation path is shown in Fig. 18(a), the detection result of the overlap is shown in Fig. 18(b), and the point cloud data are shown in Fig. 18(c). When the path shown in Fig. 18(a) was used, an overlap was detected within 3 s. An enlarged view of the overlap detection result is shown in Fig. 19(a), and the result of extracting the point cloud within the mesh size from the overlap coordinate values is shown in Fig. 19(b). The average width of the overlap coordinate values in Fig. 19(a) is 12.20 mm, whereas the width of the overlap point cloud in Fig. 19(b) is 13.18 mm. This difference is approximately 1 mm. Because the verification was conducted with a mesh size of 1.5 mm, this difference was unavoidable. Therefore, the detection results were considered sufficient. This error can be reduced by decreasing the mesh size. In addition, when the simulations were performed using the scanning paths generated for the side surfaces and other surfaces, results similar to those shown in Figs. 17(c) and 17(d) were obtained. Consequently, we confirmed that overlap regions were detected using the developed measurement simulator.