# Modeling and Calculation of Limit Magnitude Detection of Orbital Optoelectric Tracking System
## 2 Orbital Optoelectric Tracking System Design
### Orbital Optoelectric Tracking System Based on Rotating Reflector  
When tracking a fast-moving target frequently in a short period of time with the orbital optoelectric tracking system above, the camera needs to rotate, and this easily leads to wear and damage. So, a rotating reflector is introduced on the basis of the system above. The reflector is controlled by a motor, which makes the tracking system much simpler. It utilizes an optical camera to capture the dynamic target image reflected in the reflector instead of directly capturing the dynamic target itself. The schematic diagram of the orbital optoelectric tracking system based on the rotating reflector shown in Figure 2. It adopts the CCD camera, optical lens, and reflector as the core components. In order to make the rotation angle and velocity of the reflector accurately match the target's motion velocity and the high-speed optical camera clearly collect the target's motion image, the tracking system adopts the principal optical axis tracking mode, that is, the principal optical axis as the horizontal line, to ensure that the center of the reflector always intersects with the optical camera principal optical axis in the target tracking process. Assuming that point A is the starting position of the target's movement, point B is the stopping position of the target's movement, and \(AB=s\), The indexes \(i\) and \(n\) are the serial numbers of the tracked target position. \(s_{i}\), \(s_{i-1}\), and \(s_{n}\) are the displacement of the target, which is the tracking location of the reflector; \(DD^{\prime}\) is the reflector, and \(OH=h\) is the perpendicular distance between the reflector and the orbital line. The field-of-view angle of the optical camera is \(\alpha\), and the angle between the reflector and the principal optical axis of the optical camera is \(\beta\). Based on the principle of light reflection, the field-of-view angle of the optical camera is reflected by the reflector and intersected at the two points \(M\) and \(N\) of the orbit. The target A moves along the orbit. Let the mass center coordinate of the target be \(x\); the length of \(OE\) is \(l\). Based on the geometric relationships, the motion range of the field-of-view edge points of the optical camera in the motion direction of the target can be determined as follows: assuming that point \(E^{\prime}\) is the field-of-view center of the optical camera, \(E^{\prime}M\) and \(E^{\prime}N\) are the distances from the field-of-view edge points to point \(E^{\prime}\), thus constituting a sector area. In this case, the motion of point \(M\) and point \(N\) can be described by the following equations:  
\[m=x-AM=x-\frac{l\sin\alpha}{\sin(2\beta+\alpha)}\cdot\frac{h/\sin(2\beta)+l}{l} \tag{1}\]  
Figure 2: Schematic diagram of orbital optoelectric tracking system with rotating reflector. \[n=x+AN=x+\frac{l\sin\alpha}{\sin(2\beta-\alpha)}.\frac{h/\sin(2\beta)+l}{l} \tag{2}\]  
For the motion equations of the target obtained, their angle can be expressed:  
\[\frac{\pi}{2}-2\beta=\arctan\Bigl{(}\frac{x}{h}\Bigr{)} \tag{3}\]  
The simultaneous differentiation of both sides of Equation (3) yields that  
\[-2\frac{\mathrm{d}\beta}{\mathrm{d}t}=\frac{1}{1+\bigl{(}\frac{x}{h}\bigr{)}^{ 2}}.\frac{\mathrm{d}x/\mathrm{d}t}{h} \tag{4}\]  
wherein \(\mathrm{d}x/dt\) is the motion velocity \(v\) of target \(S_{i}\), and \(d\beta/dt\) indicates the change in the angle \(\beta\) with time, i.e., the angular velocity of the reflector \(\omega\):  
\[\omega=-\frac{1}{1+\bigl{(}\frac{x}{h}\bigr{)}^{2}}.\frac{1}{2h}.v \tag{5}\]  
Equation (6) is obtained by simultaneous differentiation on both sides of Equation (5):  
\[\frac{\mathrm{d}\omega}{\mathrm{d}t}=-\frac{h^{2}v+hxv^{2}}{4(h^{2}+x^{2})} \tag{6}\]  
According to the principle of optical reflection and the laws of kinematics, suppose the angular velocity of the reflector is \(\omega\) and the angle of rotation is \(\theta\).