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Abstract
With the development of large low earth orbit (LEO) communication constellations, the efficiency of laser inter-satellite link (ISL) establishing become the bottleneck for subsequent large-scale launch and rapid networking applications of LEO communication constellations. Hence, we establish the pointing jitter error structure of LEO communication experiment satellites (LCES) system. The error structure can be used to trace the source of errors and evaluate the in-orbit jitter. And we derive an analytical expression of the acquisition probability density function (PDF) which comprehensively considering the influence of the scanning region, the pointing jitter error, the overlap factor and the in-orbit jitter error. The multi-parameter influenced acquisition model is validated by Monte Carlo (MC) simulations and semi-physical tests. The results reveals that the multi-parameter influenced acquisition model can be used to guide the in-orbit ISL establishing.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
In recent years, the infrastructure construction of space broadband internet based on large LEO communication satellite constellations have reached a new climax. Starlink, Kuiper, OneWeb, China Satellite Network system and other large LEO communication satellite constellations projects have been launched to satisfy the business needs of global coverage and low-latency communication across continents [1–3]. The free space optical communication have become the key technology to solve the bottleneck of microwave communication and construct space-based internet [4–8]. At present, lots of demonstration missions have achieved the great success for inter-satellite laser communication [9–14]. However, the above mentioned demonstration missions have costed months of in-orbit debugging before laser ISL is established the first time. For reference, ARTEMIS and OICETS had been launched in July 2001 and August 2005, respectively, and the inter-satellite laser link was first established in December 2005 [10]. ANIFRE and TerraSAR-X satellites had been launched in April and June 2007, respectively, and the first time for inter-satellite laser link establishment was February 2008 [11]. Alphasat and Sentinel-1A had been launched in July 2013 and April 2014, respectively, and the first time for inter-satellite laser link establishment was October 2014 [12]. Long period in-orbit debugging can be tolerated for demonstration and verification tasks, but for large LEO communication satellite constellations, it will cost a lot of manpower, time and financial resources, which will bring great uncertainties for subsequent large-scale launch and rapid networking applications of corresponding systems. Hence, it is necessary to establish an acquisition model which can guide the in-orbit ISL establishing progress.
During the acquisition process, the target satellite will appear in the field of uncertainty (FOU). As illustrated in Fig. 1, the transmitter uses beacon beam to scan a certain region with the polar range set to $\theta _u$ until the receiver detects the signal [15]. There are two situations for the acquisition failure. Firstly, the target satellite appears in the scanning region, but due to the influence of in-orbit jitter, the beacon beam failed to ’hit’ the target satellite and the acquisition fails. Secondly, the variance of FOU is too large and the scanning region $\theta _u$ is set too small, leading to the target satellite appearing outside the scanning range and unable to be acquired. Therefore, it is very important to establish an accurate acquisition model under the influence of in-orbit jitter to improve the efficiency of inter-satellite laser link establishment. Toyoshima established the relationship between bit error rate and the ratio of laser divergence angle to in-orbit jitter, and deduced the mathematical model of the optimal divergence angle for inter-satellite laser link [16]. Friederichs built a beaconless scanning model under the influence of micro-vibration [17]. Ma deduced the influence of in-orbit jitter on acquisition probability by calculating the scanning lost region [18]. Hechenblaikner further improved the acquisition probability model, pointing out that even if the acquisition fails in the track $n$ under the influence of in-orbit jitter, there still has a chance to complete acquisition in the track $n+1$, and deduced the mathematical model of the acquisition probability under the influence of in-orbit jitter [19]. The above studies have accumulated a good theoretical and technical foundation for solving the problem of efficient ISL establishment affected by in-orbit jitter, but there are still some aspects need to be improved for practical engineering applications. On the one hand, the existing models are not modeled with the probability density of target satellite position. Previous models assume that target satellite is uniformly distributed between track $n$ and $n+1$ , however, the actual target satellite position is related to the pointing jitter error, and the acquisition probability ignores the situation that target satellite falls outside the scanning area. On the other hand, the divergence angle parameter has become an un-adjustable parameter of the system after the satellite is launched in orbit. Considering the pointing jitter error and in-orbit jitter error are quite different in different satellites, it is very meaningful to build a acquisition probability model which comprehensively considers the influence of the scanning region, the pointing jitter error, the overlap factor and in-orbit jitter error.
The LCES system built by Shanghai Engineering Center for Microsatellites (SECM) consists of two satellites, which was launched in 2021 and has completed in-orbit tests recently. This paper focuses on the influence of in-orbit jitter on the acquisition of ISL based on LCES system in-orbit tests. In Section 2, the pointing jitter error structure is established to trace the source of errors, and to evaluate the in-orbit jitters. In Section 3, the mathematical model of acquisition probability is established with comprehensively considering the influence of the scanning region, the pointing jitter error, the overlap factor and in-orbit jitter error. Section 4. presents the verificationa of the mathematical model of acquisition probability through the MC simulation and the LCES system semi-physical tests.
2. Pointing jitter error structure
Before establishing ISL, it is necessary to evaluate the size of uncertain region to determine the setting of scanning range. In the scanning process, the laser terminal read the broadcast information, which includes local satellite attitude, local satellite position and target satellite position, and calculate the center point of the target satellite in real time. Then, the spiral scanning is performed on the uncertain region around the calculated center point to complete the acquisition. Firstly, satellite broadcast data are used to calculate the center point of the target satellite, and these three broadcast data all contain errors, so it will cause errors in the calculation of the center point of the target satellite. Secondly, other errors which does not involve in the calculation process such as micro-vibration error, laser terminal actuator error, thermal deformation error, point-ahead error, will also affect the target satellite position. This paper takes the LCES system as an example, the pointing jitter error structure is carried out from the calculation process of pointing to the central point.
The ISL building strategy of the LCES system is "scanning-staring". The "scanning-staring" is that two satellites point to each other based on predicted positions calculated by the known ephemeris. Satellite-B staring at the center point, and Satellite-A scans around the center point until the laser signal of Satellite-A enters the receiver of Satellite-B, and then Satellite-B points to Satellite-A according to the incident direction of Satellite-A to complete bidirectional acquisition. The system parameters of laser terminal are shown in Table 1. The installation diagram of the LCES system and laser terminal is shown in the Fig. 2. The laser terminal installed on the $+Y$ plane of the satellite is consistent with the satellite coordinate system $o'-x'y'z'$, and the orbit coordinate system is J-2000 system o-xyz. Attitude data represent the rotation from the orbit coordinate system $o-xyz$ to the satellite coordinate system $o'-x'y'z'$ in the order of yaw-roll-pitch (rotation around Z axis first, then X axis, and finally Y axis). The initial image plane of the laser terminal tracking camera is $o'-y'z'$, and the initial normal vector of the camera image plane is the unit vector $\overrightarrow {x'}$ along the $x'$ axis, and $\overrightarrow {x'}$ is also the pointing direction of the laser terminal optical axis. The laser terminal is capable of 2-dimension rotation. The rotation angle from $x'$ axis in the $o'-x'z'$ plane is defined as the azimuth angle of the laser terminal $\theta _{Ax}$, which is positive clockwise and initial pointing is $0 rad$. The angle formed between the direction of the optical axis of the laser terminal and the $o'-x'z'$ plane is defined as the zenith angle $\theta _{Az}$, which is positive towards $+y'$ direction and negative towards $-y'$ direction.
Table 1. The parameters of laser terminal on LCES system
Taking Satellite-A as an example, the laser terminal tracks the positions of Satellite-B by calculating the local satellite (Satellite-A) position $(x_A; y_A; z_A)$, target satellite (Satellite-B) position $(x_B; y_B; z_B)$ and local satellite attitude $(\varphi _{Ax}; \varphi _{Ay}; \varphi _{Az})$. Diagram of theoretical pointing is shown in Fig. 3. The laser terminal loads the roll angle $\varphi _{Ax}$, pitch angle $\varphi _{Ay}$ and yaw angle $\varphi _{Az}$ of local satellite into the rotation matrix $T_{A}$ as shown in Eq. (1), and calculates the unit vector expression of laser terminal coordinate system ($\overrightarrow {x'}=(1; 0; 0)$, $\overrightarrow {y'}=(0; 1; 0)$, $\overrightarrow {z'}=(0; 0; 1)$) in the orbit coordinate system ($\overrightarrow {x''}$, $\overrightarrow {y''}$, $\overrightarrow {z''}$) as shown in Eq. (2). The azimuth angle $\theta _{Ax}$ and the zenith angle $\theta _{Az}$ can be denoted as Eq. (4) and Eq. (5), respectively, where $\overrightarrow {P_A}$ is the the vector from Satellite-A to Satellite-B as shown in Eq. (3). The broadcast frequency is $1Hz$, and the laser terminal updates the pointing angle center point ($\theta _{Ax}$, $\theta _{Az}$) in real time according to the above calculation process.
The Eq. (4) and Eq. (5) illustrates the influence of local satellite position error $\sigma _{pos\_local}$, attitude measurement error $\sigma _{att\_mea}$ and target satellite position error $\sigma _{ephemeris}$ on the calculation of pointing center point. The center point calculation error $\sigma _{p\_cen}=\sqrt {{\Delta \theta _{Ax}}^2+{\Delta \theta _{Az}}^2}$ can be obtained by substituting the above errors into Eq. (4) and Eq. (5).
Except for the error $\sigma _{p\_cen}$ introduced by the pointing center calculation, other errors are not involved in the calculation process, but they are accompanied by the whole acquisition process, including: point-ahead error $\sigma _{p\_ahead}$, thermal deformation error $\sigma _{thermal}$, optical coaxiality calibration error $\sigma _{cali}$, local satellite attitude stability error $\sigma _{att\_sta}$, micro-vibration error $\sigma _{microvib}$, laser terminal actuator error $\sigma _{actuator}$. Among the above errors, $\sigma _{p\_cen}$, $\sigma _{p\_ahead}$, $\sigma _{thermal}$, $\sigma _{cali}$ are bias errors, and $\sigma _{att\_sta}$, $\sigma _{microvib}$, $\sigma _{actuator}$ are random errors, which can cause in-orbit jitter error $\varepsilon$ during scanning. The total pointing jitter error $\sigma$ is given as the linear sum of those by the following Eq. (6) and Eq. (7) [20].
3. Mathematical model of acquisition probability
According to the analysis of Section.2, the target satellite usually would not appear at the pointing center ($\theta _{Ax}$, $\theta _{Az}$), but within an uncertain area. For each acquisition process, the actual target satellite position is related to the pointing jitter error $\sigma$. The laser terminal scans spirally around the central point and covers the whole scanning region $\theta _{u}$. The spiral scanning is analysed in this paper, and other scanning strategies could also reach the same conclusion through the same derivation process.
The expression of spiral scanning in polar coordinates is shown in Eq. (8) [21], where $\theta$ is the polar angle and $\rho (\theta )$ is the polar diameter. The step length is defined as $I_\theta$ which is related to beacon beam divergence angle $\theta _b$, and can be expressed as $I_\theta =\theta _b(1-\alpha )$. Where $\alpha$ is the overlap factor, considering the overlap between the illumination areas in order to avoid the leaky scanning. The azimuth angle $\theta _{Lx}$ and the zenith angle $\theta _{Lz}$ of the laser terminal are expressed as Eq. (9).
The azimuth and the zenith pointing error follow the identical distribution [22], which are zero-mean Gaussian variable with variance of $\sigma ^2$ ($\sigma$ can be obtained from Eq. (6), and independent with each other. Then the radial deviation error is Rayleigh distributed and the probability density function (PDF) is shown in Eq. (10) [23].
In realistic acquisition process, the scanning beam is jittering around the ideal trajectory. We assume that the camera integration time has been set to the optimal, and the link budget is enough, that is, the acquisition will succeed when the beacon beam waist ‘hit‘ the target satellite. As shown in Fig. 4, the target satellite is assumed to be located somewhere in between spiral track n and spiral track n+1. The jitter of the scanning beam is described by a random variable $x_n$, which is normally distributed with zero mean and a variance $\varepsilon ^2$ ($\varepsilon$ can be obtained from Eq. (7) and a PDF given by a Gaussian function $h(x_n)$. There are two cases for the acquisition failure. In case 1, the target satellite appears in the scanning region, but due to the influence of in-orbit jitter, the beacon beam fails to ’hit’ the target satellite and the acquisition fails. In case 2, the variance of pointing jitter error $\sigma ^2$ is too large and the scanning region $\theta _u$ is set too small, leading to the target satellite appearing outside the scanning range and unable to be acquired. Therefore, the probability of acquisition failure $P_{lost}$ can be expressed by the Eq. (11), where, $p_1$ represents the probability of case 1 and $p_2$ represents probability of case 2.
The case 1 occurs if the acquisition fails on the track n, and then fails again on the track n+1 due to in-orbit jitter. Let the target satellite appear between the track n and track n+1, and the polar diameter of the target satellite at this time is $\rho _B$. At this time, the polar diameter difference between the target satellite and the track n of the scanning curve is denoted as $x$, $0\leq x <I_\theta$. For any target satellite location $\rho _B$, the offset polar distance $x$ can be described as $x=\rho _B-\rho _n$. Where $\rho _n$ denotes the polar diameter for track n, $\rho _{n+1}$ denotes the polar diameter for track n+1, and $nI_\theta \leq \rho _B<(n+1)I_\theta$.
The probability of acquisition failure in the track n and n+1 can be expressed as Eq. (12) and Eq. (13) [19], respectively. Where $erfc(x)$ is the complementary error function.
According to Eq. (10) that the probability density function of $x$ is the piecewise sum of the $\rho _B$ probability density function, as shown in Fig. 5, and the probability density function for $x$ can be expressed as Eq. (14).
Combining Eq. (14) with Eq. (12) and Eq. (13), the probability for case 1 $p_1$ can be expressed as Eq. (15).
In practice, the upper limit of integration in Eq. (15) can not reach to infinity, because the scanning will not stop until the polar angle reaches the scanning region. Therefore, the maximum number of scanning tracks is determined as Eq. (16). Where $|x|$ is floor integer operation. Eq. (15) in practical application should be expressed as Eq. (17). If the target satellite appears outside the scanning area, it must miss scanning, so $p_2$ can be expressed as Eq. (18). At this point, we have an expression that can evaluate the acquisition probability $p_{acq}$ under the influence of in-orbit jitter as shown in Eq. (19), where, $p_{acq}=1-p_{lost}$ and $p_{lost}$ is the non-detection probability.
4. Discussion of results
In Section 3, we obtain an analytical expression that can evaluate the acquisition probability with in-orbit jitter. It expresses the relationship between the overlap factor, in-orbit jitter, the ratio of scanning region to pointing jitter error and the non-detection probability. It should be noticed that If $\alpha >0.5$, there will be more than two spiral tracks for potential detection of the beam, which make Eq. (19) not suitable. Herein, we will discuss the Eq. (19) in $\alpha \leq 0.5$. This section will first analyze the numerical and MC simulation results. The MC simulation process is presented in Fig. 6. Then, the accuracy of the model is verified by the semi-physical tests with in-orbit scanning pattern of the LCES system.
4.1 Numerical and MC simulation results
Figure 7 plots the various probabilities $p_1, p_2, p_{lost}$ against changes of the overlap factor $\alpha$ for a given $\sigma /\theta _b=1.7, \varepsilon /\theta _b=0.12, \theta _u/\theta _b=6$ ($\theta _b=0.8mrad, \sigma =1.36mrad, \varepsilon =0.096mrad, \theta _u=4.8mrad$) in Fig. 7-(a), against changes of the in-orbit jitter $\varepsilon$ for a given $\sigma /\theta _b=1.7, \theta _u/\theta _b=6, \alpha =0.45$ ($\theta _b=0.8mrad, \sigma =1.36mrad, \theta _u=4.8mrad, I_\theta =0.44mrad$) in Fig. 7-(b), and against changes of the ratio of scanning region to pointing jitter error $\theta _u/\sigma$ for a given $\sigma /\theta _b=1.7, \varepsilon /\theta _b=0.12, \alpha =0.45$ ($\theta _b=0.8mrad, \sigma =1.36mrad, \varepsilon =0.096mrad, I_\theta =0.44mrad$) in Fig. 7-(c).
Fig. 7. The probabilities $p_1, p_2, p_{lost}$ against changes of $\alpha$ in (a), against changes of $\varepsilon$ in (b) and against changes of $\theta _u/\sigma$.
It can be seen from Fig. 7-(a) that with the increase of the overlap factor $\alpha$, the probability of $p_1$ can be reduced exponentially. However, for $p_{lost}$, when the overlap factor is large enough, it will have little relationship with $p_1$ and it is limited by the influence of $p_2$. In this case, $p_1=3.8E-4, p_2=1.97E-3$, and $p_{lost}=2.35E-3$ when $\alpha =0.45$. $p_{lost}$ is mainly determined by the size of the scanning region. Thus, it is meaningless to increase the overlap factor in this case, which guides us to increase the scanning region in subsequent debugging to reduce the probability of $p_2$.
As can be seen from Fig. 7-(b), when in-orbit jitter $\varepsilon$ exceeds a certain value, $p_1$ probability should be reduced preferably by increasing overlap factor $\alpha$. When $\sigma /\theta _b$ is greater than 0.15, the overlap factor $\alpha$ should be preferentially increased to reduce the probability of $p_1$ rather than expanding the scanning region.
According to Fig. 7-(c), when the scanning region is small, $p_2$ is the main cause of acquisition failure. When the scanning region increases to a certain extent, the probability that the target satellite does not appear in the scanning area is very low. And for this situation, the main factor of acquisition failure is induced by the in-orbit jitters. In this case, when $\theta _u/\sigma >4.5$, $p_1$ has become the main constraint factor, and it should give priority to increasing the overlap factor $\alpha$ rather than continuing to expand the scanning region.
We performed a total of $10^6$ Monte Carlo simulations. In the MC simulations, the beacon beam is moving along an Archimedean spiral accompanied by the in-orbit jitter error $\varepsilon$, and the target satellite position was sampled according to the pointing jitter error $\sigma$. When target satellite falls within the beacon beam’s coverage, we mark this simulation as an acquisition success, otherwise, we mark this simulation as an acquisition failure. The results have been plotted in Fig. 7 and can be found to be in good agreement with the analytical predictions.
4.2 Semi-physical tests
The acquisition probability result requires a large number of repeated tests under the same parameter conditions. The acquisition probability statistics can be completed by counting the number of successful acquisition events, which is difficult to complete in the in-orbit tests since the chance of in-orbit testing is extremely limited. For this reason, we carry out the semi-physical in-orbit test that Satellite-A execute spiral scanning while Satellite-B does not cooperate with it. Affected by the in-orbit jitter $\varepsilon$, the scanning pattern of Satellite-A under the influence of in-orbit jitter is obtained through in-orbit scanning tests. Under the scanning pattern, the position of Satellite-B is sampled according to pointing jitter error $\sigma$. The Satellite-B appearing in the area covered by the scanning pattern is marked as an acquisition success. Otherwise, the event is marked as an acquisition failure. Monte Carlo method is used to calculate the acquisition probability, which realizes the semi-physical test.
We first evaluate the pointing jitter error $\sigma$ of the LCES system. According to Section 2, the evaluation results combined with in-orbit data are as follows.
The attitude measurement accuracy $\sigma _{att\_mea}$ is determined by the star sensor performance. In the LCES system, $\sigma _{att\_mea}=0.081 mrad$. The real-time orbit determination accuracy of the LCES system is 100m. Hence, $\sigma _{pos\_local}=0.05mrad$ for the inter-satellite distance of 2000km. The ephemeris error $\sigma _{ephemeris}$ is determined by the extrapolation accuracy. For the LCES system, the time interval of dense orbit is 20 minutes. The in-orbit measurement results of ephemeris error are shown in the Fig. 8, and the extrapolation points are updated every 1200s, thus ensuring that the single-axis ephemeris error is better than 350m and the three-axis ephemeris error is better than 400m. Compared with the traditional extrapolation method, the dense orbit method significantly reduces the ephemeris error [24]. For the distance between satellites of 2000km, $\sigma _{ephemeris}=0.2mrad$. According to Eq. (4) and Eq. (5), we can obtain that $\sigma _{p\_cen}=0.22mrad$.
The optical coaxiality calibration error $\sigma _{cali}$ is determined by the laser terminal performance, $\sigma _{cali}=0.01 mrad$ in the LCES system. The thermal deformation error is determined by the variation of in-orbit temperature, $\sigma _{ther} =0.58mrad$ in the LCES system. And the point-ahead error is $\sigma _{p\_ahead}=0.017 mrad$ in the LCES system. Substituting the above results into Eq. (7), we can get $\gamma =0.62mrad$.
The micro-vibration spectral density of the LCES system is simulated based on the finite element model, which takes the ground measured data of momentum wheel as an input. The vibration power spectral density is shown in Fig. 9, where the root-mean-square (RMS) value of approximately $\sigma _{microvib}=14.24\mu rad$. This result is close to RMS of the inter satellite link experiment (SILEX) vibration spectral density measured by European Space Agency (ESA) [20,25], which is $16\mu rad$.
The attitude stability error $\sigma _{att\_sta}$ is shown in Fig. 10. During the inter-satellite link establishing mission, the satellite will perform the geocentric pointing attitude, and the angular velocity of the orbit system should be $0^{\circ }/s$ theoretically. At this time, the angular velocity error of the orbit system collected by the attitude control system is the attitude stability error. The root mean square of x, y and z axis data is calculated respectively, and the mean square of the three is the attitude stability error. Combining the measurement results with the broadcast data at the frequency of $1Hz$, it could be concluded that $\sigma _{att\_sta}=0.087mrad$. And the actuator error is $\sigma _{actuator}=0.073mrad$. Substituting the above results into Eq. (7), we can get $\varepsilon =0.11mrad$. Finally, we get the pointing jitter error of the LCES system $\sigma =0.73mrad$.
Under the influence of $\sigma =0.73mrad$, according to Eq. (19), when the scanning region is set to 5mrad, the probability of non-detection is shown in Fig. 11. At this point, the influence of $p_2$ can be ignored, and $p_{lost}=p_1$. This case can be used to study the influence of overlap factor on acquisition probability with in-orbit jitter. We set the scanning region to $\theta _u=5mrad$ with the change of overlap factor $\alpha =0.45$, $\alpha =0.40$ and $\alpha =0.32$, respectively. And an in-orbit test is carried out that Satellite-A execute spiral scanning while Satellite-B does not cooperate with it. The scanning pattern with different overlap factor unders the influence of in-orbit jitter is shown in Fig. 12. The semi-physical test is carried out based on the scanning pattern of Fig. 12. The location of Satellite-B is sampled according to $\sigma =0.73mrad$ for $10^6$ times. We mark the result as acquisition success when Satellite-B falls into the coverage of scanning pattern. The comparison results of $p_{lost}$ in different overlap factor $\alpha$ for the probability of semi-physical test $p_{pattern}$ and acquisition model $p_{model}$ are shown in Table 2. The results show that the acquisition model is in good agreement with the semi-physical tests.
Fig. 11. The non-detection probabilities with different overlap factor $\alpha$ when $\theta _u=5mrad$.
Fig. 12. The scanning pattern of Satellite-A with in-orbit jitter when $\alpha =0.45$ (a), $\alpha =0.40$ (b) and $\alpha =0.32$ (c).
Table 2. The comparison results of $p_{lost}$ in different overlap factor $\alpha$ for semi-physical test $p_{pattern}$ and acquisition model $p_{model}$
According to Eq. (19), the non-detection probabilities with different $\theta _u/\sigma$ are shown in Fig. 13 when the overlap factor is set to 0.5. $p_1$ remains unchanged and $p_2$ will decrease with the increase of scanning region. We carried out the semi-physical test with the overlap factor of 0.5. The scanning pattern under the influence of orbital jitter for $\theta _u/\sigma =3.3(\theta _u=2.4mrad)$, $\theta _u/\sigma =6.5((\theta _u=4.8mrad))$ and $\theta _u/\sigma =7.1(\theta _u=5.2mrad)$ are shown in Fig. 14. Semi-physical tests are peformed based on the scanning pattern of Fig. 14, and statistical results are compared with the model calculation results as shown in Table 3. The results validate that the model had good consistency with the semi-physical tests.
Fig. 14. The scanning pattern of Satellite-A with in-orbit jitter when $\theta _u/\sigma =3.3$ (a), $\theta _u/\sigma =6.5$ (b) and $\theta _u/\sigma =7.1$ (c).
Table 3. The comparison results of $p_{lost}$ in different $\theta _u/\sigma$ for semi-physical test $p_{pattern}$ and acquisition model $p_{model}$
5. Conclusion
We have established the pointing jitter error structure for the LCES system which can be used to trace the source of errors, and evaluated the in-orbit jitters. Based on the pointing jitter error structure, we derive analytical expressions for the probability density functions of acquisition with comprehensively considering the influence of the influence of scanning region, the pointing jitter error, the overlap factor and in-orbit jitter error. The relationship among overlap factor $\alpha$, the in-orbit jitter $\varepsilon$, the ratio of scanning region to pointing jitter error $\theta _u/\sigma$ and non-detection probability $p_{lost}$ have been analysed in numerical results and MC simulation. The semi-physical tests have been carried out and show well agreement with the multi-parameter influenced acquisition model. The results reveal that the multi-parameter influenced acquisition model can be used to guide the in-orbit ISL establishing, thus promoting the networking efficiency of large LEO communication constellations.
Funding
Youth Innovation Promotion Association of the Chinese Academy of Sciences (2022295); Shanghai Sailing Program (22YF1446600).
Acknowledgments
The authors would like to express their thanks to all the members who have been engaged in developing and operating LEO communication experiment satellites and conducting the experiment with their great efforts. The authors especially would like to thank the Shanghai Radio Equipment Research Institute and Shanghai Institute of Optics and Fine Mechanics, for manufacturing the laser terminals on LEO communication experiment satellites and conducting the optical inter-satellite communications experiment for assistance.
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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