# A Noncontact Method for Calibrating the Angle and Position of the Composite Module Array
## 1 Introduction  
With the rapid development of computer and manufacturing technology, freeform surfaces are playing an increasingly important role in the fields of bionics, optics, precision machining, and measurement [1; 2; 3; 4]. [PERSON] et al. [5] developed a construction iteration optimization method to design a double freeform surface lens for point sources, as well as extended sources to achieve uniform illuminance. [PERSON] et al. [6] utilized a double-sinusoidal freeform surface to achieve 2-D displacement and angle measurement of machine tools. [PERSON] et al. [7] achieved 2-D microdisplacement measurement on the elliptical paraboloid. [PERSON] et al. [8] designed a 3-D displacement measurement system based on an optical freeform surface for a multiaxis system. [PERSON] et al. [9] employed sinusoidal-groove linear reflective grating to measure the rotation angle, linear displacement, and vertical straightness. In view of the limitations of the application of single freeform surfaces in large range scenarios, some scholars used arrays to meet the application. [PERSON] et al. [10] proposed a 2-D displacement measurement method based on the rotating parabolic array, which extended the measurement range of the system in a discrete manner. [PERSON] et al. [11] formed freeform surfaces into an L-shaped array and established an error identification model to identify 21 geometric errors of the machine tool. [PERSON] et al. [12] used an LED array and photoelectric detection sensor array to measure dust concentrations in differentareas. [PERSON] et al. [13] designed a kind of light emitting diode (LED) freeform lens array based on optical freeform surfaces and adjustable Cartesian candela distributions to achieve uniform illumination. The calibration of the array is a means to enhance accuracy. [PERSON] et al. [14] proposed a novel method for adaptive calibration of the electrode array shifts by improving the robustness of control systems. [PERSON] et al. [15] developed a refined coordinates mapping method to achieve camera array intrinsic and extrinsic calibration, which can improve measurement accuracy. [PERSON] et al. [16] proposed a self-calibration method for the vertex distance of an elliptical parabolic array. The system error of the displacement measurement after calibration was reduced from 100 to 3 \(\upmu\)m. [PERSON] et al. [17] presented a calibration method for the camera array in a spherical arrangement, which can solve the problem that the camera with opposite features was invisible at opposite positions. [PERSON] et al. [18] described a self-calibration method for a ball plate that can be extended to calibrate not only the pitch position of the balls, but also the radial and height position on the circular ball plate. The composite module studied in this paper is composed of a plane and a rotating paraboloid. [PERSON] and [PERSON] have introduced the application of the composite module array in displacement measurement and 21 geometric errors of the machine tool detection, showing the broad prospect of the composite module array [10; 11]. Owing to the existence of installation error, the relative angle and position of each composite module deviate from the theoretical value, which creates a huge error in the application of composite module arrays in measurement and other fields. Designing an appropriate calibration method is a necessary prerequisite for measurement. Therefore, a reliable method is urgently needed to calibrate the angle and position of the composite module array to ensure the high precision of the measurement. At the same time, in order to meet the high-precision measurement requirements of the array, each composite module adopts a higher processing accuracy, while the contact measurement makes it easy to destroy its surface characteristics. Moreover, the special geometric features of the rotating paraboloid make it difficult for contact measurement to adopt a uniform measurement standard for all composite modules.