A novel pose calibration method for laser displacement sensor in the five-axis on-machine measurement system

The five-axis on-machine measurement system, equipped with a laser displacement sensor, offers an effective approach for conducting on-site inspections of freeform surface parts. It is crucial to address the calibration error pertaining to the sensor’s pose, encompassing both the beam direction and the zero position, as it directly impacts the measurement accuracy. This paper presents a new approach for the calibration of a laser displacement sensor’s pose, which is mounted on the spindle of a five-axis gantry machine tool. The proposed method utilizes measurements taken from multiple angles of a standard ball. Through the utilization of spherical constraints and coordinate translation transformation, the Levenberg-Marquardt iteration method is employed to derive the sensor’s beam direction, while the least square method is used to determine the sensor’s zero position. Applying this proposed method, the maximum error in the distance between the centers of the standard double ball is below 0.008 mm, which can meet the precision requirements of on-site inspection of freeform surface parts.


Introduction
Freeform surface parts play a crucial role in the aerospace industry, such as aero-engine blades, impellers, and aircraft skins.The quality of the contour of free-form surface parts has a direct impact on the aerodynamic performance of aero engines and is crucial for ensuring the safety of aircraft flight.[1].However, the manufacturing technology for freeform surface parts is intricate, involving numerous processes, and the inspection tasks between each step are substantial [2].Using a particular turbine engine as an example, it contains more than 1000 blades, with each blade involving 23 different steps in its process.Typically, coordinate measuring machines (CMMs) are employed to measure freeform surface parts.Nevertheless, this method is time-consuming and can introduce re-clamping errors.Onmachine measurement (OMM) offers a solution to these challenges and is widely adopted due to its ability to compensate for machining based on measurement results [3]- [5].
The use of an OMM system with a laser displacement sensor (LDS) offers an effective approach for conducting on-site inspections of freeform surface parts [6].In this measurement system, the LDS is installed in a fixed position, which is initially unknown until calibration.The precision in calibrating the sensor's beam direction and initial zero position significantly influences the contour detection accuracy of freeform surface components.Therefore, calibrating the pose of the LDS is crucial for improving the measurement precision of the system.
Bi et al. [7] developed a laser OMM system that combined the LDS with a 3-axis CMM.Calibration of the beam direction was achieved through the use of a standard sphere, resulting in a measurement accuracy of 0.03 mm.Sun and Li have developed an innovative model to correct inaccuracies in LDS, considering the errors caused by the angle of incidence [8].Yang et al. [9] introduced an inner diameter measuring device that utilized three laser displacement sensors fixed in a single plane for measurement.The beam direction of the sensors was calibrated by scanning a ring gauge.Kou et al. [10] proposed a calibration method for the beam direction of the LDS by using a standard ball.After calibration, the measurement accuracy of the 3-axis on-machine measurement (OMM) system reached 0.02 mm.However, these calibration methods are applicable only to 3-axis moving platforms.Due to the constraints of the freedom of movement offered by the 3-axis platform, calibration of the zero position of the sensor is not feasible.
Ibaraki et al. [11] implemented the LDS on the spindle of a five-axis cradle machine tool and introduced an asynchronous calibration method for determining the beam direction and zero position of the sensor.The influence of the position and direction errors of the workpiece relative to the machine's rotation axis and the geometric error of the rotation axis on the measurement results were analyzed [12].However, this method is not applicable to five-axis gantry machine tools.Lu et al. [13] employed sine gauge calibration blocks for calibrating the beam direction of the LDS on a four-axis measurement platform.They considered the effects of rotation and deflection angles, introducing a calibration technique for the laser beam that is effective in various positions and orientations.It should be noted that the aforementioned pose calibration methods for the five-axis on-machine measurement system are specific to five-axis cradle machine tools and may impose strict requirements on raw measurement data.
A novel pose calibration method for the laser on-machine measurement system using a five-axis gantry machine tool is presented, which is intended to address the issues mentioned earlier.Based on the spherical constraints, the beam direction of LDS is preliminarily calibrated.Subsequently, by measuring the standard sphere from multiple poses, the pose of LDS, encompassing both the beam direction and zero position, is further accurately calibrated.
The subsequent parts of this paper are organized as follows.Section 2 establishes the mathematical model of the five-axis laser measurement system.Section 3 details the calibration method for the beam direction and zero position.In Section 4, several experiments are conducted to evaluate the effectiveness of the proposed method and the measurement accuracy of the calibrated five-axis laser OMM.Finally, Section 5 concludes the findings of this research.

Mathematical model of five-axis laser measurement system
The LDS is attached to the end of the spindle of the gantry machine tool, resulting in a five-axis measurement system.To calibrate the pose of the LDS, a standard ball is utilized.Figure 1 illustrates the system measurement process, which involves defining four coordinate systems to facilitate the description of the system's functioning.(1) Machine coordinate system {} M .The origin of the machine tool coordinate system is defined at the point, where the encoder values of the X-axis, Y-axis, and Z-axis are all zero, and the direction of the coordinate axis is along the movement direction of the machine.
(2) Reference coordinate system {} T .The origin of the reference coordinate system is located at the spindle datum point of the machine tool.The direction of this coordinate system changes corresponding to the A-axis and C-axis angles.In the initial state, when the angles of the A and C axes are both zero, it aligns with {} M .(3) Sensor coordinate system {S}.The origin is located at the point where the measurement value of the sensor is 0 mm and is expressed as M Si p in {} M .The directions of the three coordinate axes are the same as those of the reference coordinate system {} T .(4) Spherical coordinate system {} B .The origin T ,, x y z  =  p is located at the center of the standard sphere.The direction of the reference coordinate system is also aligned with {} M .i d is defined as the measurement value of the sensor.The zero position in {} T is defined as , and the beam direction in {} T is defined as It is assumed that the rotation angle of the A-axis is denoted as  and that of the C-axis is denoted as  .The rotation matrices about the A-axis and C-axis are expressed as follows respectively.
The rotation matrix of {} T relative to {} M is expressed as: cos sin cos sin sin = sin cos cos cos sin 0 sin cos The intersection M i p can be calculated by Equation (4): ( )

Methodology for calibrating the pose of the laser displacement sensor
The beam direction t and zero position l of the LDS directly determine the coordinates of the measurement point.Precise calibration of the sensor's pose is essential for maintaining the accuracy of measurements on freeform surface components

Calibration of beam direction
When the machine tool spindle is in the initial pose, the A-axis and C-axis angles are both zero, and the rotation matrix p is calculated by using Equation (5): The intersection B i p can be calculated by Equation (6): (7) where r represents the standard sphere radius.8) can be constructed: For the resolution of the nonlinear equations, an objective function is established as follows: Next, the L-M iteration method is employed to solve the objective function, and the beam direction t can be initially calibrated.

Calibration of zero position
Calibrating the zero position is achieved through the measurement of a standard ball in various poses.A set of spherical points is measured when the pre-set measurement value is 1 d , and the machine tool spindle remains at the first pose.Then, the spindle datum points M Ti p and measurement values of these spherical points are obtained.According to Equation (4), a spherical point in {} M can be expressed as: ( ) where 1 d is the pre-set measurement value, ( ) is the actual measurement value, 1i d is the tiny difference between the pre-set measurement value and the actual measurement value, CA RR is the rotation matrix of the reference coordinate system {} T relative to the machine coordinate system {} M corresponding to the first pose.Equation ( 10) can be obtained by transforming Equation ( 11): where ( ) t + l is a fixed value for the first set of spherical points, thus the geometric significance of the left-hand side of Equation ( 11) can be described as a fixed translation of all these spherical points relative to their absolute coordinates M i p .Subsequently, a sphere with a diameter approximating the standard ball's theoretical diameter r can be obtained by the least square fitting of the left-hand side of the equations corresponding to these spherical points.
The standard ball's center is p , and we let ' M B p as the spherical center after the translation.Equation ( 12) can be obtained by the least square fitting of both sides of Equation ( 11): ( ) where p is obtained by the least square fitting of ( ) t is the beam direction obtained by nonlinear solution in Section 3.1, while t and l in ( ) are unknown values to be obtained.Equation ( 12) can be arranged into the form of linear equations: where I is the 3×3 unit matrix, M B p , t and l are 3×1 unknown vectors.There are 9 unknown values and 3 equations.The standard ball is measured at least 3 times under different poses and pre-set measurement values.Then the unknown vectors can be solved:

System configuration
The five-axis OMM system, illustrated in Figure 3, comprises a five-axis gantry machine tool, an LDS, a computer, and a wireless communication module.The machine tool, Changzheng Machine Tool Group's GMC1600H/2, has three translation axes X, Y, and Z, and two rotation axes A and C. The motion ranges for axes X, Y, and Z are 3000 mm, 1600 mm, and 1000 mm, respectively.The positioning accuracies for all three translational axes are 0.020 mm, while the repeated positioning accuracies are 0.010 mm.A and C axes have rotation angles of ±100° and ±190°, respectively.The positioning accuracies for the A and C axes are 8", with repeated positioning accuracies of 5".The sensor utilized in the system is a high-performance LDS called optoNCDT ILD2300-50, developed by Micro-Epsilon.
The sensor has a measuring range of 45 mm to 95 mm, the optimal incident angle range of the sensor is within 25°, and the measurement accuracy is within 0.010 mm.

Calibration experiment
The calibration experiment entails two steps.The first step is to calibrate the beam direction, as shown in Figure 3.The standard ball, with a diameter of 50.7767 mm, is fixed on the workbench of the fiveaxis gantry machine tool.The sensor measures 60 points of the standard single ball with 0° rotational angles of the A-axis and C-axis.Operating on the principle of triangulation, the laser displacement sensor requires that the angle of incidence be kept within its optimal range to ensure the quality of measurements.
The collected measurement data, including the spindle datum points M Ti p and measurement values i d , are obtained and substituted into Equation ( 9).Then, a set of nonlinear equations is formed and resolved by using the L-M iteration method.The calibration result of the beam direction is   T 1 0.0257,0.0004,0.9997 =− t .The second step is the calibration of the zero position, and the process is shown in Figure 4.The sensor measures the standard single ball with 5 groups of rotational angles of the A-axis and C-axis, and the pre-set measurement value.Each group measures 60 points, and all measurement points are still planned with an incident angle within 25°.The raw data of 300 measurement points, including the five-axis coordinates of spindle datum points , , , ,   , and measurement values i d , are obtained.According to Equation (11) and Equation ( 12), the beam direction 1 t obtained in the first step of the calibration experiment, the five-axis coordinates of spindle datum points , , , ,   , and differences of measurement values i d are used to fit 5 spheres.The coordinates of the spherical centers are shown in Table 1.
Table 1.The Coordinates of the 5 Spherical Centers.According to Equation ( 13) and the coordinates of the 5 spherical centers, the overdetermined linear equation is established.Then, the vectors of zero position, beam direction, and the coordinates of the standard ball are obtained by solving these linear equations.The calibration results of beam direction obtained in the first step and the second step are 1 t and 2 t , respectively.2 t is selected as the final calibration result of the beam direction.

Accuracy verification experiment
A ceramic standard double ball with a certified center distance of 60.0200 mm is utilized to verify the measurement accuracy, as illustrated in Figure 5.The measurement errors of the center distance of the standard double ball are presented in Table 3.The maximum error, average error, and standard deviation of errors are 0.0071 mm, 0.0050 mm, and 0.0009 mm, respectively.The measurement errors primarily originate from the positioning error of the five-axis gantry machine tool, the measurement error of the LDS, the calibration error, and the spherical fitting error.The accuracy verification experiment results demonstrate the high accuracy and stability of the five-axis on-machine measurement system, as well as the validity of the proposed calibration method.To verify the effectiveness of the introduced method in practical on-site inspections of freeform surface components, an experimental measurement is carried out on an aero-engine blade.Figure 6 illustrates the blade being clamped twice to measure the suction and pressure surfaces respectively.The measurement point clouds are aligned with the design model to obtain the distribution of profile errors.Comparison is made between the measurement results and PowerScan 2.3M, a commercially available system that utilizes surface-structured light.Figure 7 and Figure 8 illustrate the comparison of profile errors for the suction and pressure surfaces, respectively.The profile error distribution obtained by the calibrated five-axis OMM system presented in this paper corresponds well with that of PowerScan 2.3M.Compared to the design model, the blade is in a semi-finished state with machining allowances and errors.The figures demonstrate that, on the suction surface, the blade profile exhibits a distinct concave shape in the middle position (negative error), whereas it becomes convex at the edge position (positive error).Conversely, on the pressure surface, the blade profile displays a noticeable convex shape in the middle position (positive error), while it becomes concave at the edge position (negative error).The experimental results confirm the practical effectiveness of the five-axis OMM system and the proposed calibration method for on-site inspection of complex surface parts.

Conclusion
To facilitate on-site inspection during the production of blade-like freeform surface components, this paper introduces a novel pose calibration method for the laser displacement sensor in the five-axis OMM.
The mathematical model for the laser triangulation on-machine measurement system, built upon a fiveaxis gantry machine tool, has been formulated.Based on the spherical constraints, the beam direction of LDS is preliminarily calibrated.Subsequently, by measuring the standard sphere from multiple poses, the pose of LDS, encompassing both the beam direction and zero position, is further accurately calibrated.The proposed calibration method is suitable for five-axis gantry machine tools.In the calibrated five-axis OMM system, the measurement error of the standard double center distance is less IOP Publishing doi:10.1088/1742-6596/2761/1/0120209 than 0.008 mm.The system is applied to the measurement of aero-engine blades, which verifies that the proposed method has a good application effect on the rapid on-site inspection of freeform surface parts.

Figure 1 .
Figure 1.The mathematical model of a five-axis laser measurement system.

I
into Equation (4), the intersection pointM i

4 Figure 2
Figure 2 illustrates the calibration process of beam direction.A couple of spherical points are measured by the sensor while maintaining the machine tool spindle at the initial pose.Then spindle datum points M Ti p and the measurement values

Figure 5 .
Figure 5. Measurement of the standard double ball.

Figure 6 .
Figure 6.(a) Blade suction surface measurement and (b) Blade pressure surface measurement.

Figure 7 .
Figure 7.Comparison of suction surface contour error results: (a) Measured by the five-axis OMM, (b) Measured by PowerScan.

Figure 8 .
Figure 8.Comparison of pressure surface contour error results: (a) Measured by the fiveaxis OMM, (b) Measured by PowerScan.

Table 2 .
The Calibration Results.

Table 3 .
The Measurement Errors of the Standard Double Ball Center Distance.