1. Introduction
Optical freeform surfaces are widely used in space optics, projection optical systems [
1,
2,
3], medical endoscope systems [
4,
5,
6], and other fields because of their complex surfaces and multiple degrees of freedom. With the increasing demand for freeform surfaces in the aerospace industry, national defense and the military, precision instruments, and other modern cutting-edge technology fields, a variety of freeform surfaces with a high degree of curvature and steepness have been proposed, and the accuracy of freeform surface manufacturing in various fields has become higher [
7,
8]. However, due to the complex geometric characteristics of highly curved freeform surfaces, the traditional measurement methods are limited by the measurement range and numerical aperture (NA, the product of the half-angle of the objective’s collection cone and the index of refraction of the immersion medium) [
9] of the probe, which cannot precisely measure the full surface topography of a highly curved freeform surface. As a result, profile measurement has become a key technical enabler in the manufacturing and application of such components [
10,
11].
Compared with the traditional contact coordinate measuring machines or similar equipment, the noncontact coordinate point scanning measurement method is more efficient and does not scratch the surface. The sensor in a noncontact rotary coordinate point scanning measurement system is equipped with a rotating axis, which makes the relative position of the optical sensor and the workpiece more flexible. This flexibility effectively solves the problem of the sensors being unable to measure beyond the angular tolerance. However, the extra motion associated with flexibility brings about some challenges to the measurement accuracy of the system, and some measures need to be taken to ensure accuracy. Typical commercial measuring instruments for noncontact rotary coordinate point scanning include Nanomefos from TNO (Deft, The Netherlands) [
12,
13], LuphoScan from Taylor Hobson [
14], and UltraSurf from OptiPro [
15,
16]. The accuracy of Nanomefos can reach 30 nm, which is sufficient to realize the measurement of a large aspheric surface area. However, these systems are too complex and expensive to use in production lines. LuphoScan and UltraSurf adopt the relative rotation scanning mode for the probe and the workpiece. However, LuphoScan realizes the real-time decoupling of the error by measuring the error of the motion system with the help of multiple laser interferometers. UltraSurf does not provide real-time compensation for motion error, and the measurement accuracy is limited by the accuracy of the moving platform. In these measuring devices, the probe has only one degree of rotational freedom, so most of these devices can only measure symmetrical, rotating structural parts. All of the abovementioned measuring instruments rely on a high-precision shafting motion or feedback positioning with the help of a laser interferometer. These systems are complex and expensive, and the requirements for the external environment are very strict.
The core problem with rotary scanning measurement is that the measurement system’s error is greater, the rotary axis system’s calibration is difficult, and the motion accuracy of the system’s displacement table is poor, all of which result in the rotary scanning coordinate measurement system having low efficiency and low measurement accuracy. The error modeling of and compensation in scanning systems with multiple degrees of freedom have become the primary bases of this kind of measurement equipment. In recent years, based on the choice of a high-precision motion platform, researchers have done a lot of work on error modeling and calibration of rotary scanning measurement systems. In terms of system error modeling, Rahman uses the homogeneous transfer matrix to establish the comprehensive spatial error model of the machine tool, which considers the geometric error, thermal error, rotary axis error, and elastic deformation error of the machine tool [
17,
18]. To date, this method is relatively advanced for geometric error modeling of a machine in the published work [
19]. The main methods for calibrating a rotation axis are the reverse method [
20], the three-point method [
21], and the multi-point method [
22]. In the error modeling of a measurement system with multiple degrees of freedom, Du proposed a self-calibration technique for a five-axis, laser-optical measurement system based on a ball bar [
23]. He used the parameter estimation method to obtain the system model’s parameters through the ball bar. The calibration technology enables the system to achieve a measurement accuracy of better than 5 microns. Zhang set up a noncontact scanning measurement system for a four-axis blade profile [
24]. He established a multi-body mathematical model to calculate the measurement space coordinate transformation matrix and used a three-beam interferometer and a standard gauge block to verify the geometric error of the system. The measurement results were improved to some extent, but some measurement errors remained. In summary, the existing rotational scanning measurement systems have been shown to be able to carry out 3D scanning measurement with rotational symmetry, such as measurement of a spherical surface. However, a method for the measurement of a nonrotationally symmetric freeform surface has yet to be developed. The existing rotary scanning measurement method is still limited by key technologies such as system error calibration and compensation.
In this paper, we propose a noncontact dual-axis rotary coordinate scanning measurement method based on the confocal to solve the problem of highly curved optical freeform surface measurement. In order to ensure surface information was always captured within the sensing range, Cheng, F. et al. developed an adaptive surface tracing algorithm [
25]. However, this method may not meet the angular tolerance of the probe at some measuring points. The innovation of the method in this paper is that two orthogonal rotating axes are equipped with sensors for space-relative pose scanning measurement. Compared with the existing rotary scanning method, the proposed method has more degrees of freedom, and theoretically can adapt to the changes in curvature in any direction on a highly curved freeform surface. In addition, the proposed method provides a system error calibration technique for the rotation axis errors. The key system errors that affect the measurement accuracy are first analyzed through establishing an error model. Then, the real error values are obtained by the optimal calculation in the calibration process. The advantages of this method are that it avoids the use of additional high-precision equipment to calibrate the system and solves system error problems conveniently and cheaply. Finally, the feasibility of the proposed measurement method is verified by measuring typical devices and comparing the results with those obtained using advanced commercial equipment.
4. Calibration Model for the Measurement System
In
Section 3 of this paper, the key errors of the system are determined and their effects on the final measurement results are analyzed, and
Section 4 provides a method to quantify the system key error values determined in
Section 3. From the analysis of system error in
Section 3, we found that the installation errors
f(
B0) and
f(
A0) of the rotating axes and the XZ perpendicularity error had the greatest influence on the final surface shape measurement. Therefore, these errors were considered in the calibration model first. In this section, we establish a calibration model for the system and use this model to calibrate the actual installation pose of the rotating axis and the relationship between the X-axis and the Z-axis. We calculate the deviation between the true value and the ideal value, and compensate for it in the actual measurement process to accurately control the rotation and translation of the probe and achieve the goal of accurate measurement of the measured point.
First, we discuss the installation position and pose of the rotating axis. We take a single rotating axis as an example, and the results can be compounded when there are two axes. The calibration model is shown in
Figure 7. The direction and position of the central axis of the rotating axis are expressed by six unknown variables: The coordinates of any point of the central axis in the global coordinate system (
a,
b,
c) and the vector of its direction (
l,
m,
n). The reference zero position of the probe at the initial position is the origin of the global coordinate system:
C0 (0, 0, 0). The indication of the probe
h0 is the relative distance between the measured point and the reference zero point; thus, the coordinate of the measured point is
P0 (0, 0,
h0). The probe was rotated clockwise by θ degrees and translated by (
x1,
y1,
z1) to Position 1. We established the measurement coordinate system with the reference zero position of the probe after rotation and translation as the origin O’M. The coordinates of the measured point in the measurement coordinate system are (0, 0,
h′). The rotation transformation between the global coordinate system and the measurement coordinate system is represented by a matrix
M. The coordinate of the measured point at Position 1 in the global coordinate system is:
The rotation transformation matrix for any point P0 that rotates θ degrees around the rotation axis determined by (a, b, c, l, m, n) is:
In this study, the standard ball was used as the calibration part, and the rotation axis was rotated by multiple angles sequentially to obtain the position
P1,
P2…
Pn of the corresponding rotation matrix
M1,
M2…
Mn in the global coordinate system at multiple angles. Using the distribution of the points to be measured on the standard sphere as a constraint condition, the model was optimized to obtain the actual pose and direction of the rotation axis and the relationship between the X-axis and the Z-axis. The optimization constraint function is shown in Equation (10), where
Cr is the center of the standard sphere, and
R is the radius of the standard ball.