Modeling and Calculation of Limit Magnitude Detection of Orbital Optoelectric Tracking System

Abstract

In order to evaluate the tracking capability of optoelectric tracking for an orbital target, the limit magnitude detection performance calculation model and its calculation method are studied. Combining the optical signal characteristics of the tracked orbital target, the background, and the CCD noise, the framework of the limit magnitude calculation model of the system for dynamic target detection is constructed. The relationships between the limit magnitude and the signal-to-noise ratio threshold of the optical signal characteristics, the exposure time of the CCD camera, and the dark current of the CCD imaging are studied and analyzed while considering the sunlight illumination condition, so that the calculation function and its change curve are given. The limit magnitude detection capability of the system is verified by the simulated experiment and the synchronized tracking test, and the detection distance maximum error of the model calculation is 3.6 m. The results show that under certain illumination conditions, when the exposure time of the CCD camera is longer and the SNR threshold is lower, the limit magnitude detection performance of the system is better, and the tracking performance of the system is more stable.

1. Introduction

The optoelectric tracking system is complex, integrating optical imaging technology, precision mechanical technology, electronic technology, precision tracking and control technology, image recognition technology, etc. [1,2,3,4]. It is capable of realizing the rapid detection and recognition of the target and completing its tracking and measurement. With the increasing development of optoelectric imaging technology and the highly automated test technology, it has a wide range of applications in the military, aerospace, industrial monitoring, and other fields [5,6,7]. In the shooting range test, it is an important piece of equipment in the testing of parameters for guns and other weapons, and it plays an important role for dynamic target detection and recognition under a complex environment. The optical detection capability constrains its accuracy and the reliability of the tracking, so the evaluation of that is an important component [8,9,10]. The evaluation methods can be categorized into the model method, simulation method, and test method. Researchers have carried out some basic discussions and research on the model method, mainly focusing on optoelectric imaging detectors, but there is relatively little research on the influence of the environment, etc. [11,12]. Based on the basic knowledge of radiance and photometry, this paper focuses on the detection performance of the optoelectric tracking system from the target signal characteristics, background noise, and CCD dark current noise aspect; establishes the limit magnitude calculation model; analyzes the limit magnitude detection capability; and provides the basis for the development of a high-speed and dynamic target tracking system for field operation.

The optoelectric tracking device is commonly used for the measurement of target flight attitude and external ballistic data in the shooting range. Because of its advantages of high measurement accuracy, strong performance stability, good intuition, and good resistance to ground clutter interference, countries around the world have researched optoelectric tracking devices since the 1940s. The world’s first optoelectric tracking device of KTH-41 theodolite was successfully developed at the Peenemuende shooting range in Germany in 1940, and was used for the external ballistic trajectory measurement of missiles. The United States from the 1970s on began the study of optoelectric detection systems, and the White Sands Missile Range developed an intelligent TV tracking system for multi-target tracking in 1974. China developed the first large-scale movies theodolite model 150-1 in 1974, which was successfully tested and approved by the Commission for Science, Technology, and Industry for National Defense. Optoelectric theodolite model 778 was developed in 1985, which is the fourth-generation product of the optical measuring device in the shooting range. Its comprehensive performance has reached a new level of the same kind of instruments in China. This research has provided effective means for the mission of launch, measurement and control, and observation and attitude control in the launch phase of the spacecraft, such as missiles and satellites in the shooting range; at the same time, it can be used for weapon accuracy identification and performance detection in a conventional shooting range.

For the detection performance estimation of the optoelectric imaging system, a variety of evaluation theories and models have been proposed [13,14,15], which can be mainly classified into three categories. One is based on the Johnson criterion [16], without considering the nature of the target and the defects of the image; the corresponding capabilities of the photoelectric imaging system, such as the detection and recognition of the target, are determined by the resolution method of the target equivalent fringe. In order to make up for the shortcomings of the single frequency of Johnson’s criterion, various evaluation methods and parameters have also been proposed, such as the modulation transfer function area (MTFA), which considers the image quality relating to the integral of the difference between the modulation transfer function (MTF) and the contrast of the human eye; the integral contrast sensitivity (ICS), which considers the image quality to be a function of the integrals of the quotient for the MTF and the contrast transfer function (CTF); and the square of root integral (SORI), which considers that the image quality is related to the integral of the quotient for the MTF and CTF. The second is the target task performance (TTP) criterion [17,18]; it was established to solve the single frequency constraining the Johnson criterion, and replaced the Johnson criterion with the TTP criterion. It takes into account the effects of all spatial frequency information below the threshold frequency on the task requirements. The third is the magnitude [19,20], which is a measure of the luminosity of a celestial body. In order to measure the detection performance of astronomical telescopes, the limit magnitude is used, which is the faintest magnitude that a telescope can see. These criteria and methods give qualitative and quantitative evaluation criteria for the detection and identification capabilities of the optoelectric imaging system.

Generally, the volume of the optoelectric tracking measurement device is large, and it is not easy to track the fast-moving target. The above performance evaluation criteria are mainly from the image signal results, without analyzing the influencing factors in the imaging process. For the stable tracking of fast-moving targets, this paper mainly designs the orbital optoelectric tracking system combining with a rotating mirror to reduce the tracking movement load, and establishes the limit magnitude detection calculation model with the target photometric signal characteristics, background noise, and CCD camera noise based on the magnitude evaluation method, which provides the basis of the research and development of the high-speed and dynamic target tracking system for the shooting range test.

This paper is organized as follows: the orbital optoelectric tracking system is designed in Section 2; then, the limit magnitude detection capability model of it is constructed in Section 3; computational and experimental analyses are carried out in Section 4; and finally, the conclusion is given in Section 5.

2. Orbital Optoelectric Tracking System Design

2.1. Basic Principle of Orbital Optoelectric Tracking System

The orbital optoelectric tracking system is mainly composed of the imaging plane of the CCD camera, the optical lens, the camera rotation mechanism, and the image transmission and processing module. The imaging plane of the CCD camera is mainly used to capture the dynamic target information in the optical field-of-view; the optical lens of different focal lengths is mainly used to detect the target at different distances; and the camera rotating mechanism is mainly used to make the camera obtain the target imaging information at each moment on the trajectory, which is connected with the rotating axis of the CCD camera. The camera is controlled by an external clock for synchronization imaging in the process of the target’s movement on the predetermined track, and achieves real-time tracking. Figure 1 is the principle schematic diagram of the orbital optoelectric tracking system.

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 $AB=s$, $O$ is the position of the optical center of the tracking system at the initial moment; $O^{\prime }$ and $O^{″}$ are the positions of the optical centers at two moments in the tracking process, and under ideal conditions, the optical centers of the three moments are the same. The three different moments represent the relationship between the tracking platform and the orbital target. The point H is the middle position of the tracked track, and $P_1$, $P_2$, and $P_3$ are the camera imaging position relationships between the three states in the tracking process.

The indexes $i$ and $n$ are the serial numbers of the tracked target position. It is assumed that ${\theta }_i$ and ${\theta }_n$ are the uniform control angles of the tracking platform from No. $i$, and $s_i$, $s_{i-1}$, and $s_n$ are the displacement of the target, which is the tracking location of the rotating optical lens.

If the speed of the target movement is $v$, under the premise of the known theoretical trajectory length, the theoretical trajectory is divided into a series of displacements according to equal unit time, i.e., a time-division trajectory is carried out within the orbital distance of $s$. The functional relationship between the displacements $s_i$ and the total length $s$ is $s={\displaystyle \sum _{i=1}^n{s}_i}$. Assuming that during the unit time $\Delta t$, the theoretical rotating angle of the target relative to the observation point of the tracking platform is ${\theta }_i$ for the entire trajectory of the target, the total rotating angle of the optical axis is $\theta ={\displaystyle \sum _{i=1}^n{\theta }_i}$. Based on the above analysis, in order to meet the target tracking conditions, it is only necessary to make the tracking system rotate to the corresponding theoretical angle in each unit time; the ballistic target can be tracked with time-division, the corresponding moment image of the target on the orbital can be monitored, and the corresponding parameter information can be provided by image processing.

From the principle of orbital optoelectric tracking, it can be seen that in order to stably track the target under each moment on the entire predicted ballistic line, it is necessary that the detection ability of the optoelectric detector should be sufficient. Otherwise, it is easy to lose the tracking at the different tracking distances.

2.2. 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; $D{D}^{\prime }$ 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′ is the field-of-view center of the optical camera, E′M and E′N are the distances from the field-of-view edge points to point E′, thus constituting a sector area. In this case, the motion of point M and point N can be described by the following equations:

$$m=x-AM=x-{\displaystyle \frac{l\sin \alpha }{\sin (2\beta +\alpha )}}·{\displaystyle \frac{h/\sin (2\beta )+l}{l}}$$

(1)

$$n=x+AN=x+{\displaystyle \frac{l\sin \alpha }{\sin (2\beta -\alpha )}}·{\displaystyle \frac{h/\sin (2\beta )+l}{l}}$$

(2)

For the motion equations of the target obtained, their angle can be expressed:

$${\displaystyle \frac{\pi }{2}}-2\beta =\arctan \left({\displaystyle \frac{x}{h}}\right)$$

(3)

The simultaneous differentiation of both sides of Equation (3) yields that

$$-2{\displaystyle \frac{\mathrm{d}\beta }{\mathrm{d}t}}={\displaystyle \frac{1}{1+{\left(\frac{x}{h}\right)}^2}}·{\displaystyle \frac{\mathrm{d}x/\mathrm{d}t}{h}}$$

(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 $\omega$:

$$\omega =-{\displaystyle \frac{1}{1+{\left(\frac{x}{h}\right)}^2}}·{\displaystyle \frac{1}{2h}}·v$$

(5)

Equation (6) is obtained by simultaneous differentiation on both sides of Equation (5):

$${\displaystyle \frac{\mathrm{d}\omega }{\mathrm{d}t}}=-{\displaystyle \frac{h^{2}v+hx{v}^2}{4(h^2+x^2)}}$$

(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 $\theta$. During the target movement imaged into the field-of-view of the optical camera through the reflector, the effective distance $\left|MN\right|$ in the field-of-view can be given by the following relationship:

$$|MN|=2x+\left({\displaystyle \frac{\sin \alpha h}{\sin 2\beta }}+l\sin \alpha \right)\cdot {\displaystyle \frac{\sin (2\beta +\alpha )-\sin (2\beta -\alpha )}{{\sin }^2(2\beta )-{\sin }^2\alpha }}$$

(7)

Based on the geometric relationship of the system and the target, it is known that

$$\theta ={\displaystyle \frac{1}{2}}({\displaystyle \frac{\pi }{2}}-\arctan {\displaystyle \frac{l-x}{h}})$$

(8)

Based on the target velocity, the relationship between the angular velocity of the reflector and the target velocity can be derived.

$$\omega ={\displaystyle \frac{v}{2h}}·{\cos }^2\left[\arctan \left(1-{\displaystyle \frac{x}{h}}\right)\right]$$

(9)

3. The Limit Magnitude Detection Capability of the Orbital Optoelectric Tracking System Modeled

The detection capability of the orbital optoelectric tracking system is mainly affected by the target and noise, and the following gives the optoelectric imaging information characteristic model from these two aspects with the limiting magnitude.

3.1. Optoelectric Signal Characterization of Dynamic Targets

The dynamic target in the orbital optoelectric tracking system does not emit light itself, and the radiant brightness of the target comes from the radiation of the sun, which mainly generates brightness information on the detection system by reflecting the solar radiation. Therefore, during the target signal producing process, it is affected by the attributes of the sunlight, target, optical lens, and CCD sensing surface, so the voltage of the target signal is modeled from these aspects.

Let the spectral irradiance of the sun be $E(\lambda )$ and $\lambda$ be the wavelength, and then the solar radiation received by the dynamic target above the element $ds$ is

$$d{F}_1={{\displaystyle \int }}_{\lambda }E(\lambda )ds\cos \varphi d\lambda$$

(10)

where $\varphi$ is the angle between the sunlight direction and the normal direction of the dynamic target surface element $ds$. If the diffuse reflection coefficient of the dynamic target surface element $ds$ is $c(\lambda )$, then the reflecting light luminous flux of the dynamic target surface element $ds$ is

$$d{F}_2={\displaystyle {\int }_{\lambda }c}(\lambda )E(\lambda )ds\cos \varphi d\lambda$$

(11)

Considering the dynamic target surface element $ds$ as a fully extended surface, the luminous flux of the CCD receiving surface element $d{s}^{\prime }$ at an angle $\theta$ in the normal direction and at a distance $R$ is

$$dF={\displaystyle \frac{\cos \theta }{\pi R^2}}{\displaystyle {\int }_{\lambda }c}(\lambda )E(\lambda )ds\cos \varphi d\lambda d{s}^{\prime }$$

(12)

Integrating over the entire irradiated receiving surface element $d{s}^{\prime }$ for a dynamic target, the detection surface illuminance corresponding to the target’s spectral properties is obtained as

$$E_m={\displaystyle \sum _i\frac{c(\lambda )E_0}{\pi R^2}{\displaystyle {\int }_s\cos }\theta \cos \varphi ds}$$

(13)

where $E_0$ is the illuminance of the sun reaching the detection surface and $i$ is the serial number of a plane, sphere, column, cone, etc.

Based on the optical detection principle and the illumination of the target, the spectral radiance obtained by the orbital optoelectric tracking system is

$$E_1={\displaystyle {\int }_{{\lambda }_1}^{{\lambda }_2}{E}_m}(\lambda )d\lambda$$

(14)

In Equation (14), ${\lambda }_1$ and ${\lambda }_2$ are the spectral wavelength ranges to which the imaging CCD detector responds. According to the effective aperture of the optical lens, if $\mathsf{\Phi }$ is the total luminous flux received in the orbital optoelectric tracking system, it can be expressed by Formula (15).

$$\mathsf{\Phi }={\tau }_0\cdot {\displaystyle {\int }_{{\lambda }_1}^{{\lambda }_2}\frac{1}{4}}\cdot \pi D^2\cdot E_b(\lambda )d\lambda$$

(15)

where $D$ is the effective aperture diameter of the optical lens if the optical transmittance of the optical lens is ${\tau }_0$.

Assuming that the CCD exposure time is $t_0$, the total radiant energy acquired on the light-sensitive surface of the CCD detection element is

$$Q=t_0\cdot {\tau }_0\cdot {\displaystyle {\int }_{{\lambda }_1}^{{\lambda }_2}\frac{1}{4}}\cdot \pi D^2\cdot E_b(\lambda )d\lambda$$

(16)

According to the principle of CCD-photocoupled detection, assuming that the spectral quantum efficiency of the CCD detector is $\eta (\lambda )$, the number of spectral photogenerated electrons on the CCD sensing surface is

$$N(\lambda )={\displaystyle \frac{1}{4M}}\cdot \pi D^2\cdot t_0\cdot {\tau }_0\cdot {\displaystyle {\int }_{{\lambda }_1}^{{\lambda }_2}\eta }(\lambda )\cdot {\displaystyle \frac{\lambda }{hc}}{E}_b(\lambda )d\lambda$$

(17)

where $M$ is the amount of pixels of the target imaging coverage energy of the CCD, and the voltage output from the CCD photoreceptor is

$$V_o={\displaystyle \frac{1}{4M}}\cdot \pi D^2\cdot {\displaystyle \frac{e\cdot {\delta }_t\cdot t_0\cdot {\tau }_0}{A_V\cdot C_r}}\cdot {\displaystyle {\int }_{{\lambda }_1}^{{\lambda }_2}\eta }(\lambda )\cdot {\displaystyle \frac{\lambda }{hc}}{E}_b(\lambda )d\lambda$$

(18)

In Equation (18), ${\delta }_t$ is the total charge transfer efficiency, $A_V$ is the amplifier gain, $C_r$ is the readout equivalent capacitance, and $e$ is the electron charge. The voltage amplitude is the output from the optoelectric detection CCD element under different illumination levels with the relevant parameters of the orbital optoelectric tracking system.

3.2. Optical Signal Characterization of Noise

Suppose the target radiated photon noise is $n_1$, the background radiated photon noise is $n_2$, and CCD sensing surface dark current noise is $n_3$. The target radiation photon noise is the incident photon flow caused by the random undulation of the detection circuit when the target comes into the optical detection system, and $n_1={(N_s)}^{1/2}$; the equivalent photon number of the background radiation noise is $n_2={\left(N_b\right)}^{1/2}$, wherein $N_b$ is the number of photogenerated electrons by the background; and the dark current noise is a random process signal caused by the carrier thermal effect that is a kind of white noise, and its electron number $n_3$ is equal to the square root of the number of the dark current electrons $N_d$, expressed as $n_3={(N_d)}^{1/2}$. The equivalent noise voltage output from the orbital optoelectric tracking system is

$$V_n={\displaystyle \frac{1}{C_r}}e{\delta }_t{A}_V\sqrt{N_s+N_d+N_b}$$

(19)

3.3. Calculation Model and Analysis for Limit Magnitude of Orbital Optoelectric Tracking System

3.3.1. Limit Magnitude Model under Unillumination Conditions

The detection capability of the orbital optoelectric tracking system can be measured by the signal-to-noise ratio:

$$SNR={\displaystyle \frac{V_o}{V_n}}={\displaystyle \frac{\pi D^2\cdot t_0\cdot {\tau }_0\cdot {\displaystyle {\int }_{{\lambda }_1}^{{\lambda }_2}\eta }(\lambda )\cdot \frac{\lambda }{hc}{E}_b(\lambda )d\lambda }{4{MA}_V^2\sqrt{N_s+N_d+N_b}}}$$

(20)

From the signal-to-noise ratio Equation (20), it can be seen that if the $SNR$ is larger, the detection performance and the detection capability of the optical detection system are stronger. But for the orbital optoelectric tracking system, in order to stably obtain the target image information at each moment on the full ballistic trajectory, it mainly relies on the detection capability of the system, in addition to the stability of the rotary mechanism. Assuming that the minimum detectable signal-to-noise ratio threshold of the system is $K_{\min }$, the inequality (21) can be established from the Equation (20).

$$N(\lambda )\ge (K_{{\min }^2}+\sqrt{K_{{\min }^2}+4(N_d+N_b)K_{{\min }^2}})/2$$

(21)

From the relationship between the magnitude and the luminosity, the radiant illuminance, the luminosity $E_{\mathrm{m}}$ of the target on the incident surface of the optical system, is obtained as

$$E_m=E_0\cdot {2.512}^{-m}lx$$

(22)

where $E_0$ is the luminosity of the 0 limit magnitude and $E_0=2.65\times {10}^{-6}lx$. The luminosity $E_{\mathrm{m}}$ should satisfy the condition of Formula (23)

$$E_m=E_0\cdot {2.512}^{-m}Lx\ge {\displaystyle \frac{2Mhc}{{\lambda }^{*}}}\cdot {\displaystyle \frac{K_{{\min }^2}+\sqrt{K_{{\min }^4}+4(N_d+N_b)K_{{\min }^2}}}{t_0\cdot {\tau }_0\cdot \eta \cdot \pi D^2}}{\mathrm{W}/\mathrm{m}}^2$$

(23)

where ${\lambda }^{*}$ is the average value of the detection wavelength of the system (usually ${\lambda }^{*}=580\mathrm{nm}$), and $\eta$ is the spectral quantum efficiency of the CCD detector at the average value of the detection wavelength.

With the above analysis, the limit magnitude $m$ of the system is

$$m\le 1.0857\cdot [-10.392-\lg ({\displaystyle \frac{2Mhc}{\lambda }}\cdot {\displaystyle \frac{K_{{\min }^2}+\sqrt{K_{{\min }^4}+4(N_d+N_b)K_{{\min }^2}}}{t_0\cdot {\tau }_0\cdot \eta \cdot \pi D^2}})]$$

(24)

From Equation (24), it can be seen that the limit magnitude of the system is related to the signal-to-noise ratio threshold, the dark current, the background noise, and the optical lens parameters, wherein $h$ is Planck’s constant and $c$ is the speed of light.

If the orbital optoelectric tracking system detects under unillumination conditions, the background noise can be neglected, which is 0, and Equation (24) can be transformed into Equation (25).

$$m\le 1.0857\cdot [-10.392-\lg ({\displaystyle \frac{2Mhc}{\lambda }}\cdot {\displaystyle \frac{K_{{\min }^2}+\sqrt{K_{{\min }^4}+4N_d{K}_{{\min }^2}}}{t_0\cdot {\tau }_0\cdot \eta \cdot \pi D^2}})]$$

(25)

3.3.2. Extreme Magnitude Detection Equivalent Model in Illuminated Conditions

Under illuminated conditions, the photon number of the background noise equivalent is related to the sunlight incident direction. When the angle between the optical axis of the orbital optoelectric tracking system and the sunlight incident direction is greater than 90 degrees, the system works with a backlight. Then, there is no influence of the background light on the noise, and the limit magnitude detection of the system is the same as that of the model under unilluminated conditions.

Under illumination conditions, when the angle between the optical axis of the system and the sunlight incident direction is less than 90 degrees, the system is unavoidably affected by the background light. The background noise cannot be ignored, and it also needs to consider the angle $\theta$ between the optical axis of the system and the sunlight incident direction. Therefore, it is necessary to study the relationship between the limit magnitude of the system and the exposure time, the signal-to-noise ratio threshold, and the sunlight incident direction under illumination conditions. The illuminance of the solar spectral radiation received by the system at the distance $L$ from the center of the sun can be expressed as follows:

$$E_b(\lambda )=R^2{C}_1/({\lambda }^5{L}^2(e^{(C_2/\lambda T)}-1))\cos \theta$$

(26)

where $C_1=3.742\times {10}^{-4}\mathrm{W}\cdot {\mathrm{um}}^2$ and $C_2=1.4388\times {10}^4\mathrm{um}\cdot \mathrm{K}$ are the first and second radiation constants. $T=5900\mathrm{K}$ is the temperature of the equivalent radiation blackbody of the sun. $R=6.9599\times {10}^8\mathrm{m}$ is the radius of the sun, and $\theta$ is the angle between the normal of the sensing surface of the CCD optodetector and the incident light from the sun.

Since the background radiation is a surface light source, take $M_b=N_x\times N_y$. Then, the background radiation noise equivalent photon number for the solar spectral radiance illuminance is expressed as follows:

$$n_b^{\prime }={(s\eta (\lambda /hc)E_b{\tau }_0{t}_0/M_b)}^{{\displaystyle \frac{1}{2}}}$$

(27)

Then, $N_b$ is transformed to $N_b^{\prime }$.

$$N_b^{\prime }={\displaystyle \frac{1}{4M}}\cdot \pi D^2\cdot \eta \cdot {\displaystyle \frac{{\lambda }^{*}}{hc}}\cdot E_b\cdot {\tau }_0\cdot t_0$$

(28)

The limit magnitude of the system under illumination conditions must meet Formula (29), which is a limit magnitude calculating model for the system under illumination conditions.

$$m\le 1.0857\cdot [-10.392-\lg ({\displaystyle \frac{2Mhc}{\lambda }}\cdot {\displaystyle \frac{K_{{\min }^2}+\sqrt{K_{{\min }^4}+4(N_d+N_b^{\prime })K_{{\min }^2}}}{t_0\cdot {\tau }_0\cdot \eta \cdot \pi D^2}})]$$

(29)

4. Calculation and Experiment Analysis

4.1. Computation Analysis

According to the detection and tracking principle of the orbital optoelectric tracking system and the calculation model of the limit magnitude, under different lighting conditions, the relationship between the limit magnitude and the exposure time of the CCD camera, the relationship between the limit magnitude and the signal-to-noise ratio threshold of the system, and the relationship between the limit magnitude and the dark current of the system are calculated. According to the geometric relationships of the system in Figure 2, assuming that the average wavelength of the CCD detector sensing is 580 nm, the average imaging pixel number of the target is $M=6$ during its entire movement on the trajectory; the transmittance rate of the optics is ${\tau }_0=0.88$; the aperture diameter of the optical lens is $D=200\mathrm{mm}$; the electron number in the dark current is $N_d=150e^{-}$; and the spectral quantum efficiency of the CCD detector in the average detection wavelength is $\eta =0.65$. If the minimum detectable signal-to-noise ratio threshold of the system is $K_{\min }=3.5$, Planck’s constant is $h=6.6260693\times {10}^{-34}$ J s, the light velocity is $c=3.0\times {10}^8\mathrm{m}/\mathrm{s}$, the exposure time of the CCD camera is $t_0=0.3\mathrm{s}$, and the background illuminance is 2 × 10⁴ cd/m², then according to Equation (25), the detectable limit magnitude can be calculated to be 10.4 magnitudes.

If the CCD camera exposure time, the system’s signal-to-noise ratio threshold, and the dark current are changed, their corresponding change curves can be obtained according to Equations (25) and (29). Figure 3a shows the relationship between the limit magnitude detection and the SNR threshold $K_{\min }$, Figure 3b shows the relationship between the limit magnitude detection and the exposure time of the CCD camera, Figure 3c shows the relationship between the limit magnitude detection and the dark current of the CCD camera, and Figure 3d shows the relationship between the limit magnitude detection and the background lumination.

From Figure 3a, it can be seen that lowering the signal-to-noise ratio threshold can improve the system’s limit magnitude detection, which is conducive to the tracking of the target. Especially in the background environment of the relatively strong light conditions, the higher the signal-to-noise ratio, the better the limit magnitude detection performance and the more stable the system tracking target. The lower signal-to-noise ratio threshold can be selected under the conditions of the higher signal-to-noise ratio.

From Figure 3b, it can be seen that the longer the exposure time of the CCD sensor, the higher the limit magnitude detection of the system, which indicates that in the tracking system, choosing a suitable exposure time can also improve the limit magnitude detection capability.

From Figure 3c, it can be seen that reducing the dark current in the system can increase the limit magnitude detection, and the detection capability is improved. But from the curve change rate, the influence of the dark current change on the limit magnitude detection is small. The change in the limit magnitude detection of the system is only 0.2 magnitude when the dark current is changed from $100e^{-}$ to $160e^{-}$. Therefore, in order to effectively improve the limit magnitude detection, the exposure time should be increased as much as possible, and the signal-to-noise ratio threshold should be lowered. At the same time, the effective aperture of the optical lens should be increased.

Figure 3d gives the relationship between the limit magnitude detection of the system and the exposure time of the CCD camera, as well as the relationship with the signal-to-noise ratio threshold under illuminated conditions. As can be seen from Figure 3d, the limit magnitude detection decreases significantly when the background light is enhanced. Therefore, under strong light conditions, a light filter is needed to attenuate the influence of the strong background light on the limit magnitude detection in order to improve the tracking performance.

For Equation (29), taking $N_x=N_y=256$ and $L=1.49597892\times {10}^8\mathrm{km}$, the changing curve between the limit magnitude detection and the sunlight incident direction under illuminated conditions is obtained, as shown in Figure 4:

As can be seen from Figure 4, since the sunlight is relatively strong, just a little sunlight incident greatly reduces the limit magnitude detection of the system. Therefore, the system basically can only adopt the backlight working mode or work in the ground shadow. Especially when the background light is enhanced, the limit magnitude detection decreases significantly. Therefore, the system, when under strong illumination conditions, needs to take measures to increase the light filter to weaken the influence of the strong background light on the limit magnitude detection in order to improve the tracking performance.

4.2. Experiment Analysis

In order to verify the calculation model of the limit magnitude detection performance, a certain type tracking platform is used. The tracking platform parameters are as follows: the optical lens focal length is 85 mm–120 mm, the transmittance is ${\tau }_0=0.85$, the aperture is $D=240\mathrm{mm}$, the electron number in the dark current is $N_d=150e^{-}$, the spectral quantum efficiency of the average detecting wavelength by the CCD detector is $\eta =0.65$, the CCD camera exposure time is $t_0=0.2\mathrm{s}$, and the vertical distance of the tracking platform optical center to the predefined orbital is 58 m.

The angles required for the predetermined tracking and actual control of the tracking platform are given, the unit time of the rotation angle control is set to 0.02 s, and the tracking effect can be seen from the target velocity measurement error. According to the optical spatial geometric relationships in Figure 2 and Equations (20) and (25), the limit magnitude and the corresponding distance are calculated and compared with the actual detection distance of the tracking platform.

Table 1. Experimental data of detection distance model calculation and actual test.

No.	Angle of Predetermined Tracking/°	Control Angle of Tracking Platform/°	Error between Theoretical and Measured of Target Velocity/%	Detection Distance Calculated with Model/m	Detection Distance from Actual Test of Tracking Platform/m
1	5.409	5.401	0.62	77.9	75.2
2	10.356	10.349	0.13	72.9	70.5
3	20.748	20.755	0.41	65.8	64.6
4	30.061	30.075	0.05	62.1	61.3
5	35.944	35.932	0.54	60.8	60.1
6	40.475	40.481	0.17	60.2	59.4
7	44.102	44.089	0.22	60.0	59.3
8	49.355	49.363	0.35	60.2	59.5
9	54.331	54.342	0.04	60.8	60.2
10	59.622	59.610	0.36	62.0	61.1
11	64.516	64.524	0.28	63.7	62.6
12	69.245	69.238	0.19	65.8	64.3
13	74.913	74.907	0.31	69.2	67.8
14	80.854	80.846	0.46	74.0	72.1
15	86.227	86.239	0.53	79.8	76.2

is the experimental data obtained in the test.

From the results of the data in Table 1, when the distance of the tracking platform is close to $OH$, the detection distance model calculating data error is small; at the A and B ends of the ballistic, the detection distance model calculating data error is large, and the maximum error is 3.6 m. It is mainly reflected in the two ports of the ballistic; the distance between the target and the tracking platform is larger than the detection distance $OH$, which makes the detection ability insufficient, and a large deviation in the tracking platform. In order to improve this condition, the effective aperture of the optical lens can be enlarged to improve the detection ability. At the same time, the model calculation detection distance is slightly larger than the distance of the actual test. That is mainly because in the actual test environment, there are many influence factors that are non-quantifiable, such as the atmospheric attenuation, the energy reflection of the target surface, the stability of the tracking rotary platform, and so on.

5. Conclusions

This paper carries out a study on the limit magnitude model and an analysis of the orbital optoelectric tracking system based on its principle. It deduces the calculation model of the limit magnitude; analyzes the relationships between the limit magnitude detection and the exposure time of the CCD camera, the threshold value of the signal-to-noise ratio, and the dark current in different lighting conditions; and gives the corresponding change curves. At the same time, the variation of the tracking error under different detection distances is verified by the experiment of the synchronous tracking. The results show that under certain illumination conditions, a longer exposure time of the CCD and a smaller dark current of the tracking system are conducive to the improvement of the detection capability.

The contributions of this paper are as follows: Based on an orbital optoelectric tracking system with a rotating reflector, we describe a limit magnitude modeling method for detection performance calculation to solve the dynamic target detection in orbit. We construct an analysis framework for the limit magnitude model, which combines the optical information of the dynamic target, background, and noise while considering the sunlight illumination condition.

In this paper, only the effects of the optoelectric information characteristics of the target, uniform background, and CCD camera noise on the detection performance of the orbiting optoelectric tracking system are considered, while the actual detection performance will be affected by changes of the environment and the vibration characteristics of the tracking platform, etc. The theoretical model and calculation methods of the paper provide a good theoretical design basis for the subsequent development of tracking systems for long-distance and large dynamic targets, such as the forward-looking infrared (FLIR) system, infrared search and track (IRST) system, electro-optical targeting system (EOTS), etc.