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Abstract
A single field of view (FOV) celestial navigation system (SFCNS) is a navigation system that uses only one camera to observe stars. This system not only requires nonrefracted stars for attitude measurement but also refracted stars via the atmosphere for position measurement. Different from traditional star sensors, the ability of SFCNS to observe a sufficient number of nonrefracted stars and refracted stars is not only related to the camera parameters used by the system but also to the observation attitude of the system. Moreover, its measurement accuracy depends on these parameters. For the SFCNS, both a reasonable distribution of the area ratio of the refraction and nonrefraction regions in the image and a proper design of camera parameters are important. Thus, a single-FOV-double-region global parameter optimization model is established to describe the relationship between the observation attitude, camera parameters, union star detection probability, and measurement error. The smallest measurement error and the maximum union star detection probability are set as the optimization objects. Through optimization, an optimal design scheme is provided. Eventually, a simulation experiment is conducted, proving that the proposed optimal design scheme has a union detection probability of 61%, an attitude measurement error of 0.57′′, and an orbit height measurement error below 100 m.
© 2020 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
To reduce the dependence of spacecraft on the ground segment and improve the survivability of their system, spacecraft autonomous navigation is an important development direction in the field of aerospace in the future. In recent years, a celestial navigation method, which uses a star sensor to measure its attitude and starlight refraction to measure its orbit altitude indirectly [1–3], is proposed. Compared with traditional inertial navigation devices [4], the measurement error of the celestial navigation system using this method does not accumulate over time.
Celestial navigation systems, which measure the attitude on the basis of star identification and the position on the basis of the principle of starlight refraction, can be divided into two categories according to the number of cameras in the system. The first type of design scheme uses two or more cameras to observe refracted stars and non-refracted stars [1,5], respectively, while the second type uses one camera to observe refracted and non-refracted stars simultaneously [6]. In the first multi-FOV scheme, cameras can be designed as different dedicated cameras according to their observation objects. However, the system volume and weight are usually large. Its application scenarios are also limited. In the second single-FOV-double-region resolution, two regions are in one field of view–the refraction region and the nonrefraction region. The camera must be designed to be capable of performing refracted star observation and nonrefracted star observation tasks. Therefore, the design process is complicated. However, its system volume and weight, cost, and power consumption are greatly reduced, thus extending its application range. By modeling and optimizing this camera, the single-FOV-double-region celestial navigation system (SFCNS) can have high system performance, small volume and weight, and low manufacturing cost.
Some achievements have been made in the research of the SFCNS, but no detailed modeling and analysis of its system parameters have been carried out. Qian et al. proposed a design scheme for a satellite celestial navigation system [6]. In this scheme, a single circular FOV was used to observe both refracted and nonrefracted stars. However, its system parameters were not modeled or optimized. Ning et al. proposed a navigation algorithm that uses starlight refraction to calculate the orbital altitude [7,8]. Its study of celestial navigation was only in the algorithm level. Li et al. also studied the navigation algorithm [9]. The optimization of system parameters was also ignored.
Several system parameter optimization methods for star sensors that are similar to the SFCNS have been proposed in recent years. Wei et al. gave the best exposure time for high dynamic star sensors according to the star imaging model and the centroid positioning algorithm [10]. Yan et al. modeled and analyzed the star imaging model and provided the optimal parameters for star sensors [11]. For multiexposure intensified star sensors, Yu et al. modeled and analyzed its system and obtained the optimal parameters [12]. For intensified star sensors, Yan et al. modeled the image intensifier on the basis of the star point imaging model and calculated the optimal system parameters [13]. The system parameters in the above methods were all optimized in several separated parts. Their optimal parameters were optimal in their own parts, but the whole optimal scheme might not be globally optimal.
The proposed SFCNS is different from the traditional star sensor. The traditional star sensor only needs to observe nonrefracted stars, whereas the SFCNS in this study needs to observe both refracted and nonrefracted stars in one FOV. Changes in the observation attitude of the navigation system directly affect the area ratio of the refraction region and the nonrefraction region in the image. Although using a larger FOV can simultaneously expand the region field of view (RFOV) of the refraction and nonrefraction regions, the FOV cannot be expanded indefinitely because an excessively large FOV cannot position the centroids of stars accurately. Without expanding the FOV, the area of the two regions cannot expand simultaneously because they share the same FOV. Therefore, setting the FOV angle and observation attitude reasonably and allocating the ratio of the refraction star area and the nonrefraction star regions in the FOV can make both regions obtain not only a high star detection probability (SDP) but also a high measurement accuracy. Evidently, the optimization methods for the system parameters of the traditional star sensor cannot be applied to the SFCNS in this study because the effects of the refraction region and the observation attitude are not considered. In addition, the optimal solution obtained by the partial optimization methods adopted by the traditional model may not be the global optimal solution. To solve the above problems, a single-FOV-double-region global optimization model is established in this study. Given that the changes of the FOV and the observation attitude directly affect the area ratio of the refraction region and the non-refraction region in the image, this model includes not only various traditional system parameters but also the observation attitude. After establishing and solving the optimization model of the union star detection probability (USDP) and the measurement error, a global optimal design scheme of the smallest measurement error of the maximum USDP is obtained.
The structure of this paper is as follows. Chapter 2 describes the mathematical model and its optimization established in this study. Chapter 3 gives the optimal solution obtained by the optimization calculation of the model in this study. Then, the experimental verification is carried out. Finally, Chapter 4 provides the conclusion.
2. Methodology
The principle of starlight refraction for positioning is shown in Fig. 1. When the starlight passes through the atmosphere, it causes refraction. The refraction angle is related to the height of the starlight from the ground. Thus, the position of the carrier can be calculated after measuring the refraction angle of the starlight. As the height of the starlight increases, the refraction angle continues to diminish and eventually becomes indistinguishable. Therefore, according to the height of the light, the region where the starlight has a detectable refraction angle is defined as the refraction region. The region far from the earth without refraction is defined as the nonrefraction region in this study.
The proposed SFCNS is different from the traditional star sensor. The traditional star sensor only needs to observe nonrefracted stars. However, the SFCNS in this study needs to observe both refracted and non-refracted stars in one FOV. Figure 2(a) shows the simulation image of an SFCNS under a certain FOV angle and observation attitude. In the blue refraction region in the figure, a number of refracted stars can be observed to calculate the position of the carrier, while in the yellow nonrefraction region, nonrefracted stars can be observed to calculate the attitude of the carrier. The refraction stars also appear in the gray region in the figure. However, the refracted stars appearing in this region cannot be used for the position measurement because of their indistinguishable refraction angle. Thus, they should not be considered. The detailed region definition will be given later.
As shown in Fig. 2(B), if the observation attitude of the navigation system changes, the shape and area ratio of the two regions in the image change accordingly. As shown in Fig. 2(C), if the FOV angle of the camera changes, the shape and area ratio of the two regions and the star centroid accuracy change. Evidently, for the two regions, the greater the proportion of a region or the FOV angle of the system, the greater the probability of stars appearing in this region. Although using a larger FOV can simultaneously expand the region field of view (RFOV) of the refraction and nonrefraction regions, the FOV cannot be expanded indefinitely because an excessively large FOV cannot position the centroids of stars accurately. Without expanding the FOV, the area of the two regions cannot expand simultaneously because they share the same FOV. Therefore, setting the FOV angle and observation attitude reasonably and allocating the ratio of the refraction star area and the nonrefraction star regions in the FOV can make both regions obtain not only a high star detection probability but also a high measurement accuracy. Evidently, the optimization methods for the system parameters of the traditional star sensor cannot be applied to the SFCNS in this study because the effects of the refraction region and the observation attitude are not considered.
2.1 Single-FOV-double-region global parameter optimization model
To set the field angle, lens aperture, and exposure time in the most optimal way and to allocate the ratio of the two regions in the FOV reasonably, a single-FOV-double-region global parameter optimization model is established in this study, as shown in Fig. 3. This model includes not only various traditional system parameters but also the observation attitude. In this model, traditional system parameters are calculated, such as signal-to-noise ratio (SNR), limit magnitude, and star centroid error. Then, the observation attitude and the two regions are defined. Finally, the relationship between the USDP and measurement error of parameters is obtained, and the global optimization model is derived.
This section is divided into three parts to describe the model, including navigation measurement error, observation attitude, and USDP.
2.1.1 Navigation measurement error modeling
Navigation measurement information consists of attitude information and position information. For the proposed SFCNS, the attitude is calculated after observing nonrefracted stars, whose principle is the same as that of star sensors. The position is calculated on the basis of the principle of starlight refraction after observing refracted stars. The calculation of attitude and position needs to undergo the star identification algorithm first. Afterward, the attitude can be directly calculated from the information of identified stars, and the position can be obtained by the calculated attitude.
If the pixel error of the star centroid positioning is $\Delta x$, then the star positioning error $\Delta \theta$ after the star identification can be calculated as [14]:
where $W$ is the image width in pixels and $\theta _{fov}$ is the FOV angle of the imaging system. Furthermore, when using $N$ stars for attitude calculation, the attitude measurement error isThe calculation of the position relies on the principle of starlight refraction as shown in Fig. 1. Given that the atmosphere is a nonuniform medium, the optical path of the starlight bends when it travels in the earth’s atmosphere. The bending degree is related to the atmospheric state, and the atmospheric refractive index changes with the change of atmospheric density. Given that the atmospheric density changes with its altitude, the atmospheric refractive index also changes with its altitude. When an observer observes a star through the atmosphere, the star position deviates slightly from the original position because the starlight is bent.
According to the starlight refraction model of Lillestrand et al. [15], combined with the addition of Wang et al. [16] and White et al. [17], the starlight refraction model can be expressed as [7]
where $h_a$ is the height of the observation target point in kilometers, $\theta _R$ is the refraction angle in radians, and the values of the remaining parameters are shown in Table 1.
Table 1. Parameter value of star refraction formula
When using a refracted star to measure the orbit height of the carrier, the direct measurement is the refraction angle $\theta _R$ of the refracted star. Then, the height of the starlight $h_a$ can be calculated using Eq. (3). Finally, the orbit height can be calculated according to Fig. 1 as
where the angle $\alpha$ between the line of starlight and the line from the center of the earth to the camera center can be calculated by the star centroid positioning algorithm. $R_e$ is the earth’s radius.By derivation of Eq. (4), the error of orbit height measurement is
where is derived from Eq. (3).The angle $\alpha$ is calculated from the star centroid positioning, so its error $\Delta \alpha =\Delta \theta$. Then, the error of orbit height is
For refraction stars with low starlight, the measurement accuracy of the orbit height is high. However, given the earth’s troposphere, extremely low starlight is affected by clouds and atmospheric glow. The troposphere is approximately 10-12 km at low latitudes, and 18 km at mid-high latitudes. Therefore, the refraction region has a lower limit, which is defined as $h_L=20\ \mathrm {km}$.
Although the star refraction phenomenon is used for positioning, it is a harmful factor to the attitude measurement. When the height of the starlight exceeds 87 km, its refraction angle is less than $0.1^{\prime \prime }$, which can be ignored. Therefore, the lower limit to the nonrefraction region is defined as $h_N=87\ \mathrm {km}$.
2.1.2 Observation attitude modeling
To measure the attitude and position, both refracted and nonrefracted stars are required in the image. They can appear simultaneously in the FOV only when the navigation system is in a certain observation attitude. To describe the definition of the observation attitude of the proposed SFCNS, a geographic coordinate system is established first. This coordinate system takes the optical center of the camera as the origin, the $z_e$ axis points to the zenith, and the $x_e$ and $y_e$ axis points to the south and east, respectively. Given that the earth can be considered a sphere, the rotation of the SFCNS around the $z_e$ axis can be ignored according to its symmetry. Therefore, the observation attitude of the SFCNS relative to the earth can be defined as having two rotations: the pitch rotation about the $y_e$ axis and the roll rotation about the camera’s visual axis $z_c$.
Figure 5 shows that the first rotation angle is defined as the angle between the visual axis $z_c$ and the $z_e$ axis, which is called the observation zenith angle (OZA), denoted as $\theta _{oza}$. The second one is defined as the rotation angle of the system around the $z_c$ axis, which is called the roll angle, denoted as $\theta _{roll}$. When the center of the earth is directly below the image, the roll angle is defined as 0.
The relationship between the area ratio of the refraction and nonrefraction region and the observation attitude $(\theta _{oza},\ \theta _{roll})$ is modeled and calculated in detail below. For ease of calculation, the earth is regarded as a regular sphere, and the atmospheres at different altitudes are regarded as regular spheres with different radii. Then, the image of the earth and the atmosphere at a certain altitude in the camera can be regarded as the projection of the sphere in the camera. According to the pinhole camera model [18], the edge of a sphere in the image is a conical curve, which can be calculated as
The parameters in the equation areThus, the interval between atmosphere at any two altitudes in the image can be written as
Though the area of $a$ can be calculated by some approximate function [19], the approximate function has a large error. Thus, the area is calculated by the computer in this study. Evidently, this area is related to the two altitudes, namely, the observation attitude and the FOV angle of the imaging system. Therefore, the area of $a$ can be denoted as $A_{h_1,h_2}\left (\theta _{oza},\theta _{roll},\theta _{fov}\right )$. Thus, the area of refraction region is denoted as $A_r=S_{h_L,h_H}$ and that of the nonrefraction region as $A_n=A_{h_N,\infty }$.
Through computer calculation, the relationship between the area of the two regions and the parameters is shown in Fig. 6. The two subfigures show the area proportion of the two regions in the image change with the observation attitude. In each subfigure, the upper graph indicates the FOV angle of $8^\circ$, and the lower graph indicates $5^\circ$.
Fig. 6. The relationship between area proportion of the two regions in the image with the observation attitude and FOV angle.
2.1.3 Union star detection probability modeling
To calculate the star detection probability, the darkest detectable star magnitude of the imaging system, namely, the limit magnitude $m_{lmt}$, must be calculated first. The detection of star points in the image requires a high SNR. Dark stars with a low SNR are not able to participate in the calculation or even be extracted from the image because of its large error of star positioning. The SNR of the star point can be calculated as follows
where $S$ is the energy of the star on a single pixel, $B$ is the atmospheric radiation background noise, which can be calculated by an atmospheric radiation simulation software, $n_{add}^2$ is the additive noise of the imaging system, and $V_{th}$ is the low SNR threshold for the star positioning algorithm. The energy of the star point on a single pixel and its magnitude $m$ can be obtained by the equation of magnitude definition, as follows where $m_0$ and $E_0$ are the magnitude and the illuminance near the earth of the reference star, respectively, $\mathcal {K}$ is the percentage of the energy received in a single pixel to the total received energy, and $D$ and $t$ represent the lens aperture and exposure time, respectively. According to Eq. (12) and (13), limit magnitude can be calculated as followsThe SFCNS works depending on the stars in the FOV. A successful measurement can only be completed when enough stars appear in the FOV.
For the measurement in the refraction region, only one star is required. Thus, it is only concerned about the probability of at least one star appearing in the refraction region in a random observation. The refraction star detection probability (RSDP) is defined as $P_r=P(N_r\geq 1)$, where the random variable $N_r$ represents the number of refraction stars in an observation.
Similarly, for the measurement in the nonrefraction region, at least three stars are required by the star identification algorithm to perform star matching [20]. Thus, the nonrefraction star detection probability (NSDP) is defined as $P_n=P(N_n\geq 3)$, where the random variable $N_n$ represents the number of nonrefraction stars in an observation.
In this study, the union star detection probability (USDP) of the SFCNS is defined as the probability of successful measurement, which expresses the probability that enough stars simultaneously appear in the two regions in the FOV in an observation. The definition equation is as follows
In this section, the star detection probability in the refraction region and that in the nonrefraction region are calculated first. Then, the union star detection probability is approximately calculated.
Given that the probability of these events cannot be expressed analytically, the method of multiple experiments is used to obtain the approximate solution of the SDP. The NSDP $P_n$ and the RSDP $P_r$ are related to the limit magnitude and the RFOV of the two regions. Thus, a black box function is established as follows
The numerical solution can be solved by a computer. In the equation, $m_{lmt}$ is the limit magnitude, and $\Omega _e$ is the RFOV solid angle of the refraction region or the nonrefraction region, which can be calculated as
where $\Omega _{fov}=2\pi \left (1-\cos {\frac {\theta _{fov}}{2}}\right )$ is the FOV solid angle and $A$ is the area of the region in pixels.At present, the celestial coordinate system commonly used in the field of astronomy is the equatorial coordinate system. However, the stars in the celestial sphere are mainly in the Milky Way. The angle between the galactic plane and the equatorial plane is approximately $\theta _g\approx 63^\circ$, so the distribution of stars throughout the celestial sphere is a cosine curve, as shown in Fig. 7(A).
To sample the entire $4\pi$ three-dimensional space more scientifically, the galaxy coordinate system [21] is used in this study instead of the equatorial coordinate system commonly used in star catalogs. In this coordinate system, stars are distributed as in Fig. 7(B). The number of stars gradually decreases from low to high latitude, while the number of stars does not change with the change of longitude.
Given the limit magnitude $m_{lmt}$ and the solid angle of the RFOV $\Omega _e$, a circular FOV with the same solid angle of $\Omega _e$ is used to approximate the irregularly shaped RFOV. The sample direction of FOV changes at equal intervals in the galactic longitude and latitude. Then, the probability of the events can be calculated. As shown in Fig. 8, $k$ points from $0\sim 360^\circ$ at regular intervals in the longitude direction are denoted as $\{\psi _1,\psi _2,\ldots ,\psi _k\}$. $(2l+1)$ points from $-90\sim 90^\circ$ at regular intervals in the latitude direction are denoted as $\{-\phi _l,\ldots ,-\phi _1,\phi _0,\phi _1,\ldots ,\phi _l\}$. The sample value can be recorded as
The sampling points with the same latitude but different in longitudes are evenly distributed on the sphere. Thus, the weight of each point is the same. The probability of the event $P$ appearing in the sampling group is regarded as the probability of the event occurring at this latitude line. Its statistical value is
Given that the sampling points in different latitudes are not evenly distributed on the spherical surface, the weight of each sampling point can be set to the circumference of the latitude line. The probability of the occurrence of event $P$ in the entire celestial sphere can be calculated as
Finally, the USDP can be calculated. The accurate solution of the USDP is calculated by Eq. (15). However, the event of $\left (N_r\geq 1\land N_n\geq 3\right )$ is difficult to test because the RFOV solid angle of these two regions changes in two dimensions. Considering that the star density depends on the visual direction and the limit magnitude, the two events of $(N_r\geq 1)$ and $(N_n\geq 3)$ are highly related. Thus, the USDP can be approximately calculated as
2.2 System parameter optimization method
2.2.1 Dimensional reduction of optimization parameters
The five-dimensional optimization model of USDP and measurement error with observation zenith angle, roll angle, FOV angle, exposure time, and lens aperture was established above. Two of these parameters can be calculated before the global optimization.
First , the lens aperture can be optimized alone. It only affects the limit magnitude of the imaging system by affecting the imaging energy of the star point. Meanwhile, it has no effect on other parameters. Evidently, the lens aperture can be set to the maximum achievable value to make the imaging system achieve the maximum limit magnitude. However, considering the lens designation, the number of aperture cannot be infinitely small. Thus, in this study, the minimum number of aperture $F_{min}^\#=1.1$ is used for the largest and optimal lens aperture. If the image size is $W\times W$ in pixels and the pixel size is $\omega \times \omega$, then the optimal lens aperture can be represented as a function of FOV angle $\theta _{fov}$ as
Similarly, the exposure time also plays a similar role to the lens aperture. It only affects the limit magnitude. Given that the energy of star point accumulates faster than noise energy, increasing the exposure time can effectively increase the limit magnitude. However, considering the rotation of the navigation system, the star tailing phenomenon is caused by too long exposure time. The energy of the star point cannot be concentrated in one pixel. Thus, the exposure time has an upper limit. The optimal exposure time is set to the value with the maximum SNR [11] as
where $\omega _s=0.3rad/s$ is the angular velocity of the carrier.According to the equation above, the lens aperture and exposure time are converted into a function of the FOV angle. Thus, the parameters of the proposed optimization model degenerate into three parameters, namely, the observation zenith angle, roll angle and FOV angle. The optimization parameter can be denoted as a vector as $\vec {x}=\left (\theta _{fov},\theta _{oza},\theta _{roll}\right )^T$. The reasonable definition domain of each parameter can be defined as the follows
2.2.2 Description of optimization problem
To achieve the high stability operation and high precision measurement of the SFCNS, two objectives are set to the optimization in this study: minimizing the measurement error and maximizing the USDP.
The measurement error can be divided into attitude measurement error and position measurement error. These two errors are linearly related to the star positioning error, which is proportional to the FOV angle. Thus, the FOV angle can be used as the equivalent measurement error instead of the original actual measurement error for the optimization calculation. The equivalent measurement error can be denoted as
According to the above model, the USDP can be denoted as a function with the FOV angle and the observation attitude as $P_u\left (\vec {x}\right )$. Then, the objective function of the multiobjective optimization can be written as
3. Result and analysis
3.1 Optimal parameters
The proposed optimization model has different optimal results if the image detectors or the orbit running heights are different. According to actual requirements, the optimization condition is set as follows. The existing CMV4000-E12 CMOS image sensor [22] is chosen as the detector of the SFCNS, whose parameters can be seen in Table 2. The Hipparcos catalog [23] is used as the origin data to calculate the USDP in which the stellar instrument magnitudes of the CMV4000-E12 detector are estimated by a method in infinite-dimensional space [24]. The carrier orbit height is set to 400 km. The atmospheric radiation under the moonlight conditions is calculated in the ModTRAN.
Table 2. Parameters of CMV4000-E12 detector
Considering that the objective function of the proposed optimization problem is a black box function but the optimization parameters are bounded, the response surface method [25] is used to find its global optimal solution in the parameter space. Using the calculations, the objective function and the optimal point are shown in Fig. 9. The value of the objective function is shown by the size and color of the marked points in the figure, and its optimal point is shown by the red arrow in the figure. Figure 10 shows an image under the optimal parameters. The yellow region is the nonrefraction region, the blue region is the refraction region, and the black one is the earth. Some stars are still present in the white region. However, the stars in this region either are blocked by clouds or have an excessively small refraction angle, which is unsuitable for measurement. The optimal system parameters corresponding to the optimal point are shown in Table 3.
Table 3. Optimal system parameters
3.2 Simulation validation experiment
To verify the optimization results of the proposed single-FOV-double-region global optimization model, a simulation verification experiment is designed. The simulation experiment mainly includes two parts: star image generation module and celestial navigation module. The star image generation module uses the star catalog data and the testing optimal system parameters as the original data. According to the input reference attitude and position, the physical process of imaging and atmospheric refraction, the background light interference and the various noises in the imaging process, a simulation star image is eventually generated. The celestial navigation module calculates the attitude and position of the carrier on the basis of the input star image. First, the star points in the image are extracted by the star centroid algorithm. Then, they are identified by the triangle star identification algorithm [20]. Afterward, according to the identified nonrefraction stars, the attitude can be solved. Finally, the position can be solved by the attitude and the identified refracted stars.
In this experiment, 10,000 simulation images of random attitudes and positions are generated, thus satisfying the testing optimal observation attitude. The simulation attitudes and positions are considered the true values, while the attitudes and positions calculated from the simulation images are considered measurement values. The result is shown in Fig. 11.
In these 10,000 simulation tests, the detection the NSDP is 97%, and the RSDP is 62%. The USDP is 61%. The proportion of attitude measurement error less than 1 arc second was 89%, and the proportion of orbit height measurement error less than 100 m was 99%. The experimental results are consistent with the expectations calculated under the optimal system parameters of the model, which proves that the proposed single-FOV-double-region global optimization model is correct.
3.3 Simulation comparison experiment
To prove that the proposed design scheme is optimal, an experiment using the control variable method is conducted. On the basis of the optimal design scheme, the exposure time, lens aperture, observation zenith angle, roll angle, and FOV angle are changed to test their union star detection probabilities and measurement errors. For each change of parameters, 10,000 random simulation tests, which are the same as those in the first experiment, are conducted. In the experiment of the FOV angle, the lens aperture and exposure time change simultaneously with the FOV according to Eq. (21) and (22). The experimental results are shown in Fig. 12.
The figures show that the USDP decreases regardless of which parameter is changed. Although the change of parameters can improve the measurement accuracy in some conditions, the improvement of the measurement accuracy is very limited. Considering the significant decrease in the USDP, this change is evidently not worth the gains. Therefore, the proposed design scheme is the optimal design scheme.
4. Conclusion
In this paper, a global optimization model of the system parameters is proposed for the single-FOV-double-region navigation system. Considering that the changes of the FOV and the observation attitude directly affect the area ratio of the refraction region and the non-refraction region in the image, this model includes not only various traditional system parameters but also the observation attitude. After establishing and solving the optimization model of the union star detection probability and the measurement error, a global optimal design scheme of the smallest measurement error and the maximum union star detection probability is obtained. A simulation verification experiment is also conducted in this study, which proves that the proposed single-FOV-double-region global optimization model has a high accuracy in describing the operation of the SFCNS. In addition, the optimal design scheme is compared with other schemes to prove that the proposed optimal scheme has the largest USDP and almost the smallest attitude and position measurement errors. In conclusion, the single-FOV-double-region global optimization model describes the working process of the SFCNS with high accuracy, and can be applied to its optimization task.
Funding
National Natural Science Foundation of China; National Key Research and Development Program of China.
Disclosures
The authors declare no conflicts of interest.
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