# Improved Design of Imaging System for Online Detection of Large-Sized Step-Shaft Runout Errors
## 2 Principle and Simulation Analysis  
### The Working Principle of the Runout Detection System  
The runout detection system is an integral part of photoelectric detection in large-sized, stepped-shaft automatic measuring equipment, whose systems include an optical runout detection system, an optical diameter detection system, a base, brackets, etc. A large-sized, stepped-shaft photoelectric detection system is shown in Figure 1.  
A runout detection optical system consists of a laser, a collimated focusing optical system, an optical imaging system, a photodetector, a motor drive, and a signal processor, as shown in Figure 2. The system uses direct laser triangulation for runout detection, which has the advantages of concentrated light intensity, a simple structure, and a low error rate. Direct laser triangulation can be used to determine a measured object's displacement because a point of displacement can be an imaging spot, and this point can be used as the measurement position for a one-to-one correspondence, which is more suitable for runout detection along a stepped axis. Figure 1: Large-sized stepped-shaft photoelectric detection system. The measurement principle of direct laser triangulation is shown in Figure 3, where the beam emitted by the laser diode is vertically incident on the reference plane through the collimated focusing optical system and then converges on the photosensitive surface of the photodetector through the lens of the optical imaging system after reflection occurs. When the step axis in the rotation process produces a jump, the measured point from the reference plane moves along the direction of the incident optical axis from point \(A\) to point \(B\), resulting in a displacement of y. With the change in the image point in the photodetector from point \(A^{\prime}\) to point \(B^{\prime}\) and with the displacement of \(y^{\prime}\), the detector at the spot position can detect the amount of change in order to calculate the runout error along the stepped axis [10; 11]. Figure 3: Schematic diagram of direct laser triangulation. Figure 2: Component diagram of optical system for runout detection. This may be calculated according to optical imaging theorem and the direct laser triangulation method:  
\[\frac{B^{\prime}C^{\prime}}{BC}=\frac{OC^{\prime}}{OC}=\frac{OB^{\prime}}{OB} \tag{1}\]  
\[\frac{B^{\prime}C^{\prime}}{BC}=\frac{OC^{\prime}}{OC}=\frac{L^{\prime}-A^{ \prime}C^{\prime}}{L+AC} \tag{2}\]  
where \(B^{\prime}C^{\prime}=y^{\prime}\times sin\beta\); \(A^{\prime}C^{\prime}=y^{\prime}\times cos\beta\); \(C=y\times sin\alpha\); and \(AC=y\times sin\alpha\). From the imaging conditions of Gaussian paraxial optical system, we can see the following:  
\[\frac{1}{l}+\frac{1}{l^{\prime}}=\frac{1}{f} \tag{3}\]  
where \(l\) is object distance, \(l=L\), \(l^{\prime}\) is the image distance, and \(l^{\prime}=L^{\prime}\). Let the focal length of the imaging optical system be \(f\). When the laser beam is incident on the surface of the measured object, Equation (2) can be substituted to obtain the following formula:  
\[L^{\prime}=\frac{f\times L}{L-f} \tag{4}\]  
Substitute \(l^{\prime}\) into Equation (1) to find the displacement \(L\) of the measured object. It can be shown that when the actual position is under the reference position, the runout \(y\) is as follows:  
\[y=\frac{y^{\prime}\times(L-f)\times\sin\beta}{f\times\sin\alpha-y^{\prime} \times\left(1-\frac{f}{L}\right)\times\sin(\alpha+\beta)} \tag{5}\]  
When the actual position is at the reference position, the runout \(y\) is as follows:  
Figure 2.0  
The distance between the intersection point \(A\) of the principal optical axis of the imaging system, the incident beam, and the center O is \(L\); the distance between the intersection point \(A^{\prime}\) of the principal optical axis of the imaging system and the photodetector and the center O is \(L^{\prime}\); the angle between the principal optical axis and the incident beam is \(\alpha\); and the angle between the photodetector and the principal optical axis is \(\beta\).