# Improved Design of Imaging System for Online Detection of Large-Sized Step-Shaft Runout Errors
## 2 Principle and Simulation Analysis
### Imaging Optical System Design
#### 2.2.1 Parameter Calculation of Imaging Optical System  
The effective photosensitive surface of the PSD-1315 detector used in the imaging optical system is 1.3 \(\times\) 15 mm\({}^{2}\). In order to eliminate the detection error caused by the nonlinearity of the PSD, only the middle part is used, that is, the spot is imprinted within the range of \(\pm\)3.5 mm, being placed as close to the center of the detector as possible. Due to the runout error detection of the step axis in the range of \(\pm\)1.5 mm, it can be known that the linear field of view of the object is 2\(y\) = 3 mm. The imaging range of the PSD position detector is \(\pm\)3.5 mm, that is, the line field of view of the image square is 2\(y^{\prime}\) = 7 mm. The magnification of the imaging optical lens can be obtained as follows:  
\[k=\frac{2y^{\prime}}{2y}=2.333 \tag{8}\]  
Combined with the image position, determined using Formulas (5) and (6) in the above direct laser triangulation method, the relationship between magnification and the angle of the imaging optical system can be obtained using the following formula:  
\[k=\frac{dy}{dy^{\prime}}=\frac{f(L-f)\sin\beta\sin\alpha}{\left[f\sin\alpha\mp y ^{\prime}\Big{(}1-\frac{f}{L}\Big{)}\sin(\alpha+\beta)\right]^{2}} \tag{9}\]  
The position of the PSD position detector is determined by its angle \(\mu\) from the horizontal line, where  
\[\mu=90^{\circ}-\alpha-\beta \tag{10}\]  
From the working principles of the imaging optical system, it can be seen that the runout error of the step axis is measured based on the vertical incidence of the laser beam on the surface of the measured object. When the incident beam is not perpendicular to the surface of the measured object, the tilt of the surface changes the spatial distribution of the scattered light to be equivalent to that of the imaging optical system, resulting in a change in the position of the scattered light spot in the photodetector relative to its vertical incidence. If this is calculated according to the position at this time, there is bound to be an error, namely the inclination error, which is the root cause of measurement errors when assessing the surface of a measured object. It can be seen that when the surface of the measured object is tilted, the position of the light-energy centroid of the image spot on the position detector is shifted to a certain extent relative to its geometric center, and so the light-energy centroid is no longer the geometric center of the image spot. Under ideal conditions, the measured object surface can be approximately regarded as a diffuse reflective surface, and the influence of the object surface tilt is analyzed based on [PERSON]'s law [11]. The schematic diagram of Lambert's Law is shown in Figure 4, and the spatial distribution of the scattered field can be expressed as follows:  
\[I(\alpha)=I_{0}\cos\alpha \tag{11}\]The strip-surface element dS, which is perpendicular to the receiving surface, is used on the lens of the imaging system, as shown in Figure 5. Since the overall thickness of the lens is small relative to the entire system, it can be ignored. Based on [PERSON]'s formula, the light energy received by the lens is calculated using the following Formula [13]:  
\[E=\pi I_{0}\frac{R^{2}}{f}\left(1+2\frac{y}{f}\cos\alpha\right)\cos(\alpha-\theta) \tag{12}\]  
where \(f\) is the measured distance, \(R\) is the radius of the imaging system, \(\alpha\) is the angle between the incident laser and the main optical axis of the imaging system lens, \(\theta\) is the angle between the horizontal plane and the reference plane, that is, the object plane inclination angle, and \(I_{0}\) is the scattered light intensity generated by the laser beam, which is incident on the side under ideal conditions. According to formula (12), it can be seen that the intensity of scattered light energy on the measured object surface is affected by the slope angle \(\theta\). The displacement measurement error \(\Delta X\) can be obtained using Formulas (4)-(6), shown above, and the calculation formula the gross error.