Design of an Imaging Optical System for Large-Sized Stepped Shaft Diameter Detection

: Addressing the prevalent issues of low accuracy, low efficiency, and poor image quality in online diameter measurement of large-sized stepped shafts, this study introduces a novel method based on a symmetrical dual-telecentric optical path utilizing dual CCDs, specifically designed for step shafts with diameters ranging from 600 mm to 800 mm. By developing and optimizing an imaging system grounded in the object-image dual-telecentric optical path principle and employing Zemax software for comprehensive analysis and optimization, this research achieves significant findings. The system’s Airy disk radius is calculated at 3.204 µ m; the modulation transfer function (MTF) remains above 0.6 across various fields of view at a spatial cutoff frequency of 71.4 lp/mm, with smooth MTF curves; the field curvature is confined within 0.1 µ m; and the distortion is maintained below 0.1%, fulfilling high-quality imaging requirements. Additionally, a tolerance analysis is conducted to ensure the system’s stability and reliability. Applied to an experimental setup for measuring the diameter of large-sized step shafts, the system demonstrates an improved measurement precision of 0.02 mm. This research offers a robust technical solution for the high-precision online measurement of large stepped shaft diameters, presenting significant practical implications for enhancing productivity and product quality.


Introduction
Large-sized stepped shafts are commonly used in large-scale mechanical equipment as positioning and installation components [1].The online automatic detection of their diameter parameter has emerged as a crucial technology in measurement equipment research.The imaging optical system plays a vital role in detecting the diameter of these shafts and directly impacts the measurement accuracy.Currently, the widely used projection method for diameter measurement is effective for axis-like components due to its relatively simple technical implementation and the ease of designing collimating optical systems [2].However, this method has limitations when it comes to larger diameter sizes, primarily due to the limited projection area and the small effective size of the linear array CCD used.Hence, there is a need to propose a diameter projection measurement method suitable for larger-sized diameters.Currently, optical imaging methods for measuring diameter using the projection method include parallel light imaging, collimated light imaging, and dual-CCD imaging.The parallel light imaging method is only suitable for measuring diameters in the range of 30 mm to 100 mm and is not applicable for measuring large-sized stepped shafts.The collimated light imaging method is suitable for measuring objects of various shapes, but for measuring the diameter of large-sized stepped shafts, it may require a longer focal length and a larger collimation angle to obtain clear imaging effects.In contrast, the dual-CCD imaging method utilizes two symmetric CCD sensors to capture the object's images and measure the diameter based on these images.It has higher accuracy Appl.Sci.2024, 14, 3423 2 of 14 and real-time performance and allows for adjustment of the detection range by adjusting the distance between the two CCDs.For large-sized stepped shafts, the dual-CCD imaging method can better handle larger objects.However, further improvement is needed in terms of imaging quality.Therefore, this paper proposes the addition of a dual-telecentric optical system based on the dual-CCD imaging method.
At the beginning of the 21st century, researchers utilized projection methods for diameter measurements [3], which were improved and upgraded by subsequent studies.In recent years, many scholars have expanded upon the projection method to explore dual-telecentric optical imaging systems [4,5].For instance, Ye Wenwei designed an optical system with dual fields of view, featuring two magnification factors, which were −0.275 and −0.1375, respectively, primarily addressing the complications and accuracy issues associated with changing lenses of different magnifications [6].However, this system requires 15 spherical lenses, making its structure overly complex compared to standard dual-telecentric systems.Cai Dali focused on the application requirements of machine vision in detecting the positional accuracy of pin needles, designing a dual-telecentric optical system with a depth of field of 0.5 mm, maximum distortion less than 0.1%, and telecentricity controlled within 0.007 • , correcting various aberrations to enhance detection accuracy [7].Yet, at a spatial frequency of 208 lp/mm, the lens contrast is low, with an MTF value just over 0.2, indicating poor imaging quality.Luo Chunhua proposed a dualtelecentric optical system with ultra-wide depth of field and low distortion, comprising seven spherical lenses with distortion less than 0.1% and telecentricity at a maximum of 0.013 • .This system achieves industrial measurement requirements in terms of image quality with a good cost-performance ratio, even with fewer lenses [8].However, its object-side field of view is only 60 mm, making it suitable only for measuring small diameters and not capable of measuring large-sized step shafts.Xianbin Fu introduced a method using laser scanners to measure diameters within the range of 200 mm-900 mm.However, the error margin between 2 mm and 6 mm renders it unsuitable for high-precision measurements [9].
To address these issues, this research employed the dual-telecentric principle to design a symmetrical dual-telecentric optical path dual-CCD system.This approach resolved the problems of low imaging quality and precision encountered when measuring large-sized stepped shafts.It lays the theoretical foundation for the imaging system of the large-sized stepped shaft diameter measurement experimental system and its applications, providing a feasible solution for measuring the diameter of large-sized stepped shafts.
This innovative system resolves problems such as low measurement accuracy and poor image quality caused by the misalignment between the scale position and the optical receiving surface.As a result, it establishes a solid theoretical foundation for developing large-sized stepped shaft diameter gauges.This technology was successfully applied to imaging systems for large-sized stepped shaft diameter gauges, laying the groundwork for further research in the field.

Object-Space Telecentric Optical Path Design and Optimization
Telecentric optical paths can be classified into object-side telecentric, image-side telecentric, and dual-telecentric paths.The object-side telecentric path features a simple structure and a larger working distance range.However, compared to the dual-telecentric path, it exhibits greater distortion and aberration in the edge field of view.On the other hand, the image-side telecentric path provides high image quality with fewer secondary errors.Nevertheless, relative to the dual-telecentric path, it typically has a smaller working distance range and requires higher lens quality and manufacturing precision.Dual-telecentric optical systems combine the advantages of both object-side and image-side telecentric paths while avoiding the errors introduced by these paths during detection [10].The schematic diagram is shown in Figure 1.
paths while avoiding the errors introduced by these paths during detection [10].The sch matic diagram is shown in Figure 1.Dual-telecentric optical systems typically adopt a symmetric structure, with the a erture stop located at the center of the system, ensuring that the chief ray is incident an emergent parallel to the optical axis.The incident of parallel rays guarantees a sufficie depth of field in the system.The emergence of parallel rays ensures that the magnificatio remains constant when the working distance changes within the depth of field range, an the position of the chief ray remains unchanged.Therefore, the system has no paralla [11][12][13][14].

Symmetrical Double-Telecentric Optical Path for Large-Sized Stepped Shaft Detection
Given the measurement range of diameters from 600 mm to 800 mm, the projectio method is unsuitable for measurement.Therefore, an improved method based on a dua field-of-view optical path with dual CCDs is proposed [15], as illustrated in Figure 2. Th system comprises five main components: illumination, the object under measurement, r ception, signal processing, and actuation.The working principle of the system is as fo lows: Light emitted from the source is transformed into parallel beams through a collim tion and beam expansion system.After error compensation by a plane mirror, the light projected onto the surface of the object under measurement, forming shadows due to th object's obstruction.These shadows are captured by a dual-field-of-view optical imagin system and received by two CCD sensors.Since the size of the object under measureme exceeds the sensing area of the CCD sensors, the two CCDs can only capture the edges the object.After edge data acquisition, the actual size of the object under measurement determined through the signal processing module in conjunction with previously mea ured dimensions of standard parts.For the diameter measurement of large-sized ste shafts, a wide parallel light source and a dual-field-of-view optical imaging system a required.Thus, a collimation and beam expansion system is added between the lig source and the object under measurement.A plane mirror, added between the collimatio and beam expansion system and the object under measurement [16], compensates for a gular errors due to installation.The light source utilizes a LASER525nm-100nW with laser wavelength of 525 nm.The CCD employed is a TCD1501D, with a pixel size of 7 µ × 7 µm.Dual-telecentric optical systems typically adopt a symmetric structure, with the aperture stop located at the center of the system, ensuring that the chief ray is incident and emergent parallel to the optical axis.The incident of parallel rays guarantees a sufficient depth of field in the system.The emergence of parallel rays ensures that the magnification remains constant when the working distance changes within the depth of field range, and the position of the chief ray remains unchanged.Therefore, the system has no parallax [11][12][13][14].

Symmetrical Double-Telecentric Optical Path for Large-Sized Stepped Shaft Detection
Given the measurement range of diameters from 600 mm to 800 mm, the projection method is unsuitable for measurement.Therefore, an improved method based on a dualfield-of-view optical path with dual CCDs is proposed [15], as illustrated in Figure 2. The system comprises five main components: illumination, the object under measurement, reception, signal processing, and actuation.The working principle of the system is as follows: Light emitted from the source is transformed into parallel beams through a collimation and beam expansion system.After error compensation by a plane mirror, the light is projected onto the surface of the object under measurement, forming shadows due to the object's obstruction.These shadows are captured by a dual-field-of-view optical imaging system and received by two CCD sensors.Since the size of the object under measurement exceeds the sensing area of the CCD sensors, the two CCDs can only capture the edges of the object.After edge data acquisition, the actual size of the object under measurement is determined through the signal processing module in conjunction with previously measured dimensions of standard parts.For the diameter measurement of large-sized step shafts, a wide parallel light source and a dual-field-of-view optical imaging system are required.Thus, a collimation and beam expansion system is added between the light source and the object under measurement.A plane mirror, added between the collimation and beam expansion system and the object under measurement [16], compensates for angular errors due to installation.The light source utilizes a LASER525 nm-100 nW with a laser wavelength of 525 nm.The CCD employed is a TCD1501D, with a pixel size of 7 µm × 7 µm.

Parameter Computation
Based on the symmetric dual-telecentric optical path detection method used in the system and considering the detection range of the large stepped shaft diameter detector (600 mm-800 mm), the maximum detection range has been increased to 820 mm in order to improve fault tolerance.The calculation of the object-side linear field of view can be expressed as follows: 2y = (820 The TCD1501D CCD is selected, with a photosensitive receiving area of 35 mm.Additionally, the image-side linear field of view is as follows:

Parameter Computation
Based on the symmetric dual-telecentric optical path detection method used in system and considering the detection range of the large stepped shaft diameter dete (600 mm-800 mm), the maximum detection range has been increased to 820 mm in o to improve fault tolerance.The calculation of the object-side linear field of view ca expressed as follows:

2
(820 600) / 2 110 mm y = − = The TCD1501D CCD is selected, with a photosensitive receiving area of 35 mm.ditionally, the image-side linear field of view is as follows: 35 mm y ′ = Therefore, the magnification ratio [17] is the following: The TCD1501D CCD has a pixel size of 7 µm × 7 µm.Based on the Nyquist freque [18] theory, a clear image display requires a minimum of two sampling points for spatial frequency.Consequently, the CCD's maximum resolution is at least twice the p size.Therefore, the spatial frequency is defined as follows: 1000 spatial frequency lp / mm 2 pixel = × In the equation, "pixel" represents the size of each pixel.From the above equatio Therefore, the magnification ratio [17] is the following: The TCD1501D CCD has a pixel size of 7 µm × 7 µm.Based on the Nyquist frequency [18] theory, a clear image display requires a minimum of two sampling points for each spatial frequency.Consequently, the CCD's maximum resolution is at least twice the pixel size.Therefore, the spatial frequency is defined as follows: In the equation, "pixel" represents the size of each pixel.From the above equation, it can be deduced that the maximum resolution spatial frequency of this type of CCD is 71.4 lp/mm.In order to be distinguishable by the photosensitive device, the resolution power of the designed double-telecentric system should be greater than 0.3 at 71.4 lp/mm.
The formula for calculating the Airy disk size [19] is as follows: where λ 0 represents the wavelength of the laser and F is the F-number.In order to fully utilize the CCD's resolving power, it is required that the Airy disk size be smaller than the CCD pixel size.Therefore, F can be set to 10.
In a telecentric optical system, the depth of field determines the imaging clarity of the system [20] and can be represented by the object-space depth.The formula is as follows: where ∆ represents the depth of field, β is the magnification factor, and F is the working F-number of the system.The parameter k is a specific value (commonly taken as (0.015-0.008)).Based on calculations, the depth of field for the system is determined to be 8.2 mm.Therefore, the main technical specifications of the object-image telecentric system can be found in Table 1.In the design process of optical systems, selecting an appropriate initial structure is a critical step.Currently, there are generally two methods for determining the initial structure of a system: The first method is based on the system's relevant design specifications, utilizing the Seidel aberration theory to establish aberration balancing equations for design purposes or employing the PW method directly to design the initial structure of the system [21].Although the method of using aberration balancing equations has notable advantages for simple system structures, it becomes computationally complex and redundant and requires a considerable workload for complex optical systems.It is difficult to achieve an efficient and high-quality system design using this method.The second method combines patent databases or existing references to determine an initial structure that meets the system's specifications for subsequent design.This approach utilizes patented structures found in optical patent databases or references as the initial structure.Regarding the complex imaging system for large-scale stepped shaft diameter detection, the second method in this article is employed to determine the initial structure of the system [22].Firstly, a specific initial structure is chosen based on the optical design manual and the design specifications provided in Table 1.Then, scaling is performed according to the size of the image plane.Following that, an optimization function is set up to control the far focus of the object side, using the operand RANG to control the pupil distance and object distance.
Using the operand property TTHI allows for controlling the air gap using the operand properties MNCA/MXCA/MNEA, and controlling the glass thickness is achieved using the operand properties MNCG/MXCG/MNEG.Finally, Zemax software is utilized to optimize the initial structure and generate the structure of the object-side far-focus system.The parameters are presented in Table 2, the system layout is illustrated in Figure 3a, and the spot diagram in Figure 3b.rameters are presented in Table 2, the system layout is illustrated in Figure 3a, and the s diagram in Figure 3b.From Figure 3b, it can be observed that this is an object-space telecentric system with an Airy disk radius of 9.605 µm.The optical system is divided into five fields of view, each representing the image height when the rays converge at the image plane after passing through the optical system.The upper-left field corresponds to an image height of 0 mm with an RMS spot radius of 1.814 µm.The upper-right field corresponds to an image height of 16.244 mm with an RMS spot radius of 5.093 µm.The middle field corresponds to an image height of 26.370 mm with an RMS spot radius of 6.278 µm.The lower-left field corresponds to an image height of 35.603 mm with an RMS spot radius of 5.007 µm.The lower-right field corresponds to an image height of 47.551 mm with an RMS spot radius of 3.378 µm.All of these values are smaller than the Airy disk radius, indicating that the spots where the rays converge for each field of view are within the black circles shown in the diagram.These black circles represent the Airy disk radius, demonstrating that the rays converge through the system into a point rather than a diffuse spot, indicating the excellent imaging capabilities of the optical system.

Image-Space Telecentric Optical Path Design and Optimization
By scaling the object-space telecentric system based on the image plane size, using the same operands for control, and setting up an optimization function, the optical system is optimized to achieve a telecentric optical system on the image side that meets the requirements.The system parameters are shown in Table 3.The system layout diagram is shown in Figure 4a, and the spot diagram of the image-space telecentric system is shown in Figure 4b.From Figure 3b, it can be observed that this is an object-space telecentric system with an Airy disk radius of 9.605 µm.The optical system is divided into five fields of view, eac representing the image height when the rays converge at the image plane after passin through the optical system.The upper-left field corresponds to an image height of 0 mm with an RMS spot radius of 1.814 µm.The upper-right field corresponds to an imag height of 16.244 mm with an RMS spot radius of 5.093 µm.The middle field correspond to an image height of 26.370 mm with an RMS spot radius of 6.278 µm.The lower-left field corresponds to an image height of 35.603 mm with an RMS spot radius of 5.007 µm.Th lower-right field corresponds to an image height of 47.551 mm with an RMS spot radiu of 3.378 µm.All of these values are smaller than the Airy disk radius, indicating that th spots where the rays converge for each field of view are within the black circles shown i the diagram.These black circles represent the Airy disk radius, demonstrating that th rays converge through the system into a point rather than a diffuse spot, indicating th excellent imaging capabilities of the optical system.

Image-Space Telecentric Optical Path Design and Optimization
By scaling the object-space telecentric system based on the image plane size, using th same operands for control, and setting up an optimization function, the optical system is op timized to achieve a telecentric optical system on the image side that meets the requirements The system parameters are shown in Table 3.The system layout diagram is shown in Figur 4a, and the spot diagram of the image-space telecentric system is shown in Figure 4b.From Figure 4b, it can be observed that this is an image-space telecentric system with an Airy disk radius of 6.405 µm.The optical system is divided into five fields of view, each representing the image height when the rays converge at the image plane after passing through the optical system.The upper-left field corresponds to an image height of 0 mm with an RMS spot radius of 1.266 µm.The upper-right field corresponds to an image height of 5.257 mm with an RMS spot radius of 1.392 µm.The middle field corresponds to an image height of 8.694 mm with an RMS spot radius of 1.567 µm.The lower-left field corresponds to an image height of 12.035 mm with an RMS spot radius of 1.753 µm.The lower-right field corresponds to an image height of 16.798 mm with an RMS spot radius of 1.961 µm.All of these values are smaller than the Airy disk radius.In the spot diagrams of each field of view, the RMS spot radius is smaller than the Airy disk radius, indicating the optical system possesses high imaging capability.

Double-Telecentric Optical Path Design and Optimization
The initial structure of a double-telecentric system is obtained.After analyzing parameters such as telecentricity, MTF, and Airy disk radius, it is determined that the system fails to meet the requirements for high-precision detection of large-sized stepped shafts.Therefore, optimization of the system is necessary.
After repeated optimization design of the system, the planning of various indicators is carried out.The system structural parameters are shown in Table 4, the system layout diagram is shown in Figure 5a, the system MTF is shown in Figure 5b, the system spot diagram is shown in Figure 5c, and the system distortion/field curve is shown in Figure 5d.From Figure 4b, it can be observed that this is an image-space telecentric system with an Airy disk radius of 6.405 µm.The optical system is divided into five fields of view, each representing the image height when the rays converge at the image plane after passing through the optical system.The upper-left field corresponds to an image height of 0 mm with an RMS spot radius of 1.266 µm.The upper-right field corresponds to an image height of 5.257 mm with an RMS spot radius of 1.392 µm.The middle field corresponds to an image height of 8.694 mm with an RMS spot radius of 1.567 µm.The lower-left field corresponds to an image height of 12.035 mm with an RMS spot radius of 1.753 µm.The lower-right field corresponds to an image height of 16.798 mm with an RMS spot radius of 1.961 µm.All of these values are smaller than the Airy disk radius.In the spot diagrams of each field of view, the RMS spot radius is smaller than the Airy disk radius, indicating the optical system possesses high imaging capability.

Double-Telecentric Optical Path Design and Optimization
The initial structure of a double-telecentric system is obtained.After analyzing parameters such as telecentricity, MTF, and Airy disk radius, it is determined that the system fails to meet the requirements for high-precision detection of large-sized stepped shafts.Therefore, optimization of the system is necessary.
After repeated optimization design of the system, the planning of various indicators is carried out.The system structural parameters are shown in Table 4, the system layout diagram is shown in Figure 5a, the system MTF is shown in Figure 5b, the system spot diagram is shown in Figure 5c, and the system distortion/field curve is shown in Figure 5d.
According to Figure 5b, it can be concluded that at a spatial cutoff frequency of 71.4 lp/mm, the MTF of each field of view is higher than 0.6, meeting the design requirements.According to Figure 5c, the Airy disk radius is 3.204 µm and the RMS radius is 2.660 µm.The RMS radius is smaller than the Airy disk radius, indicating that the imaging quality of the dual-telecentric system meets the requirements.The field curvature reflects the degree of bending of the image plane, and the distortion affects the accuracy of imaging [23].According to Figure 5d, the field curvature correction is within a range of 0.1 mm, meeting the design requirements.The system distortion is less than 0.1%, also meeting the design requirements.According to Figure 5b, it can be concluded that at a spatial cutoff frequency of 71.4 lp/mm, the MTF of each field of view is higher than 0.6, meeting the design requirements.According to Figure 5c, the Airy disk radius is 3.204 µm and the RMS radius is 2.660 µm.The RMS radius is smaller than the Airy disk radius, indicating that the imaging quality

Discussion
During the actual optical manufacturing process, there may be various errors that can affect the performance of the system.Therefore, it is necessary to use software simulations to analyze the impact of these errors on the system's performance [24].
To ensure that the CCD sensor has a high resolution during the detection process, tolerance analyses of the imaging optical system are required to ensure that the MTF is not lower than 0.3.In this study, sensitivity analysis was used for the tolerance analyses of the imaging system, with a test wavelength of 525 nm and 500 Monte Carlo samples.The geometric mean MTF of the optical system at 71.4 lp/mm was taken as the standard.The first set of tolerance settings and analysis results are shown in Table 5, and the worst deviation types are shown in Figure 6.

Discussion
During the actual optical manufacturing process, there may be various errors th can affect the performance of the system.Therefore, it is necessary to use software sim lations to analyze the impact of these errors on the system's performance [24].
To ensure that the CCD sensor has a high resolution during the detection proces tolerance analyses of the imaging optical system are required to ensure that the MTF not lower than 0.3.In this study, sensitivity analysis was used for the tolerance analys of the imaging system, with a test wavelength of 525 nm and 500 Monte Carlo sample The geometric mean MTF of the optical system at 71.4 lp/mm was taken as the standar The first set of tolerance settings and analysis results are shown in Table 5, and the wor deviation types are shown in Figure 6.According to the first tolerance analysis results, it can be observed that the thickne deviation of the fifth lens element (H-ZLAF50D) has the greatest impact on the MTF the imaging system, while the influence of other parameters is relatively small.Therefor after readjusting the tolerance settings, a subsequent analysis was conducted.The analys results are shown in Figure 7.According to the first tolerance analysis results, it can be observed that the thickness deviation of the fifth lens element (H-ZLAF50D) has the greatest impact on the MTF of the imaging system, while the influence of other parameters is relatively small.Therefore, after readjusting the tolerance settings, a subsequent analysis was conducted.The analysis results are shown in Figure 7.These analysis results indicate that over 98% of the samples have an MTF decrea less than 0.1, achieving satisfactory tolerance analysis results that meet practical req ments.

Experiment
The dual-telecentric system designed in this study has been applied to a large-s stepped shaft diameter measurement experimental system.This measurement syste composed of a symmetrical dual-telecentric dual-CCD system, a CCD signal proces system, and an upper-computer display system.The configuration of the large-s stepped shaft diameter measurement system, incorporating the dual-telecentric sys is shown in Figure 8.The system was tested using standard parts with diameters of 652 mm and 750 The measurements were conducted using the large-sized stepped shaft diameter m urement experimental system without our imaging system installed, the same sy with our imaging system installed, and a Keyence measuring instrument for compar The measurement results are presented in Figure 9.These analysis results indicate that over 98% of the samples have an MTF decrease of less than 0.1, achieving satisfactory tolerance analysis results that meet practical requirements.

Experiment
The dual-telecentric system designed in this study has been applied to a large-sized stepped shaft diameter measurement experimental system.This measurement system is composed of a symmetrical dual-telecentric dual-CCD system, a CCD signal processing system, and an upper-computer display system.The configuration of the large-sized stepped shaft diameter measurement system, incorporating the dual-telecentric system, is shown in Figure 8.These analysis results indicate that over 98% of the samples have an MTF decrea less than 0.1, achieving satisfactory tolerance analysis results that meet practical requ ments.

Experiment
The dual-telecentric system designed in this study has been applied to a large-s stepped shaft diameter measurement experimental system.This measurement syste composed of a symmetrical dual-telecentric dual-CCD system, a CCD signal proces system, and an upper-computer display system.The configuration of the large-s stepped shaft diameter measurement system, incorporating the dual-telecentric sys is shown in Figure 8.The system was tested using standard parts with diameters of 652 mm and 750 The measurements were conducted using the large-sized stepped shaft diameter m urement experimental system without our imaging system installed, the same sys with our imaging system installed, and a Keyence measuring instrument for compari The measurement results are presented in Figure 9.The system was tested using standard parts with diameters of 652 mm and 750 mm.The measurements were conducted using the large-sized stepped shaft diameter measurement experimental system without our imaging system installed, the same system with our imaging system installed, and a Keyence measuring instrument for comparison.The measurement results are presented in Figure 9.In Figure 9, 'A' represents the large-sized stepped shaft diameter measu perimental system without the installation of our imaging system, 'B' repr Keyence measuring instrument, and 'C' represents the large-sized stepped sha measurement experimental system with our imaging system installed.From F can be observed that in the measurement of a standard component with a diam mm, the maximum diameter measured by the large-sized stepped shaft diam urement experimental system without our imaging system is 652.055mm, and mum diameter is 652.047mm.The Keyence measuring instrument measures a diameter of 652.041 mm and a minimum diameter of 652.30 mm.The maximu the large-sized stepped shaft diameter measurement experimental system wi tem installed is 652.032mm, and the minimum error is 652.024mm.From Figur be observed that in the measurement of a standard component with a diameter the maximum diameter measured by the large-sized stepped shaft diameter me experimental system without our imaging system is 750.055mm, and the mi ameter is 750.050mm.The maximum diameter measured by the Keyence me strument is 750.043mm, and the minimum diameter is 750.044mm.The maxi of the large-sized stepped shaft diameter measurement experimental system wi tem installed is 750.036mm, and the minimum error is 750.028mm.Through co it can be seen that the installation of our imaging system significantly reduces in the measurement of the stepped shaft diameter in the experimental system to the system without our imaging system installed.Furthermore, the errors a duced when compared to the Keyence measuring instrument.

Conclusions
A highly advanced and low-distortion dual-telecentric imaging system wa for the purpose of large-sized stepped shaft diameter detection.The system e cellent imaging quality, with a detection range of 110 mm, a working distance and an Airy disk radius of 3.204 µm.At a spatial cutoff frequency of 71.4 lp/mm transfer functions across various fields of view are all above 0.6, while the field remains within 0.1mm and distortion is controlled within 0.1%.
When applied to the experimental system for measuring the diameter of a stepped shaft using standard components, it was found that the introduction aging system significantly reduced measurement errors.
Furthermore, besides its high-precision detection capabilities for large-siz shafts, this system holds promising applications in fields such as projectors an tical paths.In Figure 9, 'A' represents the large-sized stepped shaft diameter measurement experimental system without the installation of our imaging system, 'B' represents the Keyence measuring instrument, and 'C' represents the large-sized stepped shaft diameter measurement experimental system with our imaging system installed.From Figure 9a, it can be observed that in the measurement of a standard component with a diameter of 652 mm, the maximum diameter measured by the large-sized stepped shaft diameter measurement experimental system without our imaging system is 652.055mm, and the minimum diameter is 652.047mm.The Keyence measuring instrument measures a maximum diameter of 652.041 mm and a minimum diameter of 652.30 mm.The maximum error of the large-sized stepped shaft diameter measurement experimental system with our system installed is 652.032mm, and the minimum error is 652.024mm.From Figure 9b, it can be observed that in the measurement of a standard component with a diameter of 750 mm, the maximum diameter measured by the large-sized stepped shaft diameter measurement experimental system without our imaging system is 750.055mm, and the minimum diameter is 750.050mm.The maximum diameter measured by the Keyence measuring instrument is 750.043mm, and the minimum diameter is 750.044mm.The maximum error of the large-sized stepped shaft diameter measurement experimental system with our system installed is 750.036mm, and the minimum error is 750.028mm.Through comparison, it can be seen that the installation of our imaging system significantly reduces the errors in the measurement of the stepped shaft diameter in the experimental system compared to the system without our imaging system installed.Furthermore, the errors are also reduced when compared to the Keyence measuring instrument.

Conclusions
A highly advanced and low-distortion dual-telecentric imaging system was designed for the purpose of large-sized stepped shaft diameter detection.The system exhibits excellent imaging quality, with a detection range of 110 mm, a working distance of 60 mm, and an Airy disk radius of 3.204 µm.At a spatial cutoff frequency of 71.4 lp/mm, the MTF transfer functions across various fields of view are all above 0.6, while the field curvature remains within 0.1 mm and distortion is controlled within 0.1%.
When applied to the experimental system for measuring the diameter of a large-sized stepped shaft using standard components, it was found that the introduction of this imaging system significantly reduced measurement errors.
Furthermore, besides its high-precision detection capabilities for large-sized stepped shafts, this system holds promising applications in fields such as projectors and laser optical paths.

Figure 2 .
Figure 2. Schematic diagram of the diameter detection system.

Figure 2 .
Figure 2. Schematic diagram of the diameter detection system.

Figure 6 .
Figure 6.Worst deviation types for the first tolerance analysis.

Figure 6 .
Figure 6.Worst deviation types for the first tolerance analysis.

Figure 7 .
Figure 7. Results of the second tolerance analysis.

Figure 8 .
Figure 8. Experimental system for measuring the diameter of large-sized stepped shafts.

Figure 7 .
Figure 7. Results of the second tolerance analysis.

Figure 7 .
Figure 7. Results of the second tolerance analysis.

Figure 8 .
Figure 8. Experimental system for measuring the diameter of large-sized stepped shafts.

Figure 8 .
Figure 8. Experimental system for measuring the diameter of large-sized stepped shafts.

Figure 9 .
Figure 9. Measurement results.(a) Measurement of the 652 mm standard part; (b) mea the 750 mm standard part.

Figure 9 .
Figure 9. Measurement results.(a) Measurement of the 652 mm standard part; (b) measurement of the 750 mm standard part.

Table 1 .
Main technical indexes of double-telecentric optical system.
3. Design and Optimization of Double-Telecentric Optical System3.1.Object-Space Telecentric Optical Path Design and Optimization

Table 2 .
Structural parameters of the optimized object-space telecentric system.

Table 2 .
Structural parameters of the optimized object-space telecentric system.

Table 3 .
Structural parameters of the optimized image-space telecentric system.

Table 3 .
Structural parameters of the optimized image-space telecentric system

Table 4 .
Structural parameters of double-telecentric system.

Table 4 .
Structural parameters of double-telecentric system.

Table 5 .
Results of the first tolerance analysis.

Table 5 .
Results of the first tolerance analysis.