Random Error Analysis of Launch and Injection Positions for Distributed Multi Micro-nano Satellite System

Accurate satellite orbit calculation has always been a focus in the aerospace field. Especially, the error distribution of the orbit involving various internal and external factors in the satellite launch process has an important influence on the effectiveness of the satellite cluster. To qualitatively and quantitatively analyze the cooperative efficiency of multi-satellite clusters involving various random errors, this paper analyzes the random error sources in the launch process of micro-nano satellites and studies the distribution law of these satellite orbit position errors. Based on the parameter estimation and the central limit theorem of probability theory, the expression of the error function for the satellite’s orbit position is also constructed. In addition, the probability distribution model of the satellites in a certain range is designed, and the relationship between the number of satellites and the overall errors is proposed. Finally, the result is verified by the simulation experiment, which provides the rationale behind the position and altitude adjustment and constellation optimization of micro-nano satellites.


Introduction
The distributed satellite system is a kind of cluster made up of multiple satellites that are physically disconnected but can achieve the same space missions [1].Micro-nano satellites generally refer to small satellites that are less than 100 kg and have practical functions [2].Due to their small size, low cost, short production time, fast technology update, and other unique advantages, in recent years, micro-nano satellites have increasingly formed distributed satellite systems in the form of clusters, constellations, and formation flying for tasks such as communications, earth observation, science and technology, security, scientific exploration, and space logistics in military and civilian fields [3].Distributed micronano satellite systems also play an indispensable role in the development of the Internet.Distributed micro-nano satellite clusters have advantages that a single small satellite cannot match.With the requirements of information development in various countries, small satellite projects are booming.The launch of small satellites also has a rapid growth period.In 2022, the global number of spacecraft launched was 2505, with small satellites accounting for 93.6% [4,5].Since 2010, with the continuous improvement of small satellite business application capabilities, commercial capital has flooded into the aerospace field, sparking a global "orbital revolution".Satellite orbit precision control has always been a key point of interest for many experts and scholars in the field of space.In 1983, the Chinese rocket scientist Ru [6] analyzed the deviation of the synchronous satellite launch orbit and estimated the deviation of each orbital element before a satellite was sent into orbit.By combining the high-precision position requirements of navigation satellites, Peng et al. [7] introduce a way of controlling the thrust size and controlling the time point that the orbit chooses to achieve orbit insertion within the nominal range.[8] presents a scheme that uses a vernier engine to control payload position and altitude to launch orbit precisely.[9] puts forward a method to meet the active terminal attitude requirements of the payload on the premise of ensuring accurate orbit entry into orbit.While studying small satellites, [10] and [11] investigate the relative orbit maneuver control technology of micro-nano satellites.[12] analyzes the Beidou navigation satellite system in differences in positioning accuracy based on satellite distribution probability.The current research focuses mainly on the precise control of the orbit position for satellites to launch into their orbits and the calculation of effectiveness accuracy.Small satellites have been rapidly developed and applied in recent years, and they differ from medium and high-orbit satellites in their manufacturing costs, launching mode, functions, and so on.By analyzing the distribution probability of the orbit position of micro-nano satellites based on probability theory and then discussing the influence of the error of the integrated orbit position of the satellite cluster on its overall efficiency, we can determine if there is a need to adjust the orbit position of the satellite, which is important for further improving the satellite launch efficiency and cost reduction.

Analysis of launching the micro-nano satellite into orbit
The way of launching a micro-nano satellite is different from medium and high orbit satellites and the random errors are different from each other methods.
Due to the influence of different launch methods, technology, the precision of relevant instruments, external conditions, and human factors, there are inevitably random errors in the orbital position of every satellite of the cluster.
It is assumed that the error caused by launching technical capacity is 1 X , the impact of ground system reliability is 2 X , the impact of wind speed and other climatic factors is 3 X , the impact of space radiation environment is 4 X , the impact of engine ignition time accuracy is 5 X , the error about fuel device is 6 X , the impact of the rocket's flight when it releases the satellite is 7 X , the error in satellite release time interval is 8 X , the error in satellite layout is 9 X , manufacturing errors in satellite shape and size is 10 X , and so on.So the error of orbit position of the i-th satellite can be expressed as the following equation:

Error distribution function
As is well known, if a random variable is influenced by many factors and none of them has a decisive influence on it, then the random variable generally will obey normal distribution [13].Because the position error of a satellite when it enters orbit is caused by tiny errors resulting from various factors, and these tiny errors are mutually independent, every single one of them has little effect on the sum, so the error of orbit position obeys normal distribution, which is and the orbit position also obeys normal distribution, we can express the position as 2 0 , and 0 P is the expected position of the satellite in orbit.

Determination of distribution function parameters
The mean P and standard deviation V of the distribution function of orbital position error can be estimated using parameter estimation methods based on sampling in engineering practice.Parameter estimation includes point estimation and interval estimation, and in this case, point estimation is used to determine the parameters of the function.Common point estimation methods include moment estimation, maximum likelihood estimation, least squares estimation, and Bayesian estimation.In statistical problems, maximum likelihood estimation is often used first, and when it is no longer effective, moment estimation is used [14].
If there is one error sample from the constellation orbital errors x1, x2, …, xn, set the position error is Yi and it obeys the Gaussian distribution with Yi aN P ,V2 , then the probability density of Yi can be present as the following equation: The likelihood function of the error can be expressed as Equation ( 3): After taking the logarithm of both sides, the following equation can be obtained: Further rearrangement yields the following likelihood equations: Combining Equations (4) and (5), we can obtain the following equation.
The maximum likelihood estimates of P and 2 V can be obtained by solving Equation ( 6):

The probability distribution of the in-orbit position for the micro-nano cluster
When a large number of satellites are launched into orbit, their positions in orbit show a normal distribution because of various random factors.To study the performance of the constellation, the percentage of satellites within a certain range can be estimated using the Chebyshev inequality and the distribution function.On the basis of the determined distribution function of the satellite positions, the overall error of the constellation can be researched using the central limit theorem.

By the Chebyshev inequality estimation
The Chebyshev inequality: For a random variable X with mean P and variance 2 V , for any positive number e, it can hold that From the expression of the Chebyshev inequality, the estimation is meaningless when V H because this expression always holds true.In this case, the generalized inequality of Chebyshev can be used to estimate a more accurate result. where During the cluster launch process, the initial orbital position of the satellite named i is denoted by ai.Due to the effect of random errors during the constellation launch process, the expected value of the actual orbital position E [i is as ai, and its variance D [i is as V2 .For ease of analysis, the position of each satellite is standardized by setting Și as i i a [ V After standardization, the position of the satellite named i can be denoted by Și.
According to the generalized inequality of Chebyshev, in the process of launching a constellation, the probability of the satellite position error being within the error range 2 H can be expressed as follows.
For each satellite's actual position in the constellation, it can be restored using the normalization method, while its error remains unchanged.Based on the generalization of Chebyshev's inequality, the number or proportion of satellites within a certain range of deviation in the constellation can be estimated.

Estimation of the distribution probability based on the distribution function
If the function of the satellite's initial position is the Gaussian distribution, the mean and variance are P and2 V , respectively.Then, the probability of the satellite's initial orbital position between [a, b] can be calculated using the following equation.
where X is the actual position, and ' X is the standardization of X .
If the probability of the satellite being within the error range is greater than t , we can find a value 0 x and let it meet 0 Therefore, 0 1 2 Looking up the table of standard normal distribution, we can determine the value of 0 x V and obtain the value of 0 x .

Analyzing the overall error using the Central Limit Theorem
In the actual working process of a constellation, the efficiency depends on the joint effect of satellites in the constellation.According to the central limit theorem, the average result of a large number of random phenomena generally tends to be stable, and the behavior of a single random phenomenon has almost no impact on the total average result produced by a large number of random phenomena.This means that although the orbital position of an individual satellite inevitably has random deviation and can affect the efficiency, under a large number of random orbit errors of satellites, these random errors may cancel out and compensate for each other, resulting in a stable overall average result [15].The central limit theorem is present as follows.
ˈ ˈ ˈ ˈ are mutually independent and identically distributed random variables, and Then for any real number x the distribution function n F x The theorem states that when n is large enough, n Y approximately follows a standard normal distribution: . 1 -0,1 Thus: The probability is that the total position errors of all satellites in the constellation within > @ The probability that the average position error of each satellite in the constellation within > @ W W , can be expressed as From Equation (21), it can be analyzed that the probability of overall error will decrease as n increases.In other words, if the number of satellites in the constellation reaches a certain large number, the total deviation or the effects caused by deviations can be estimated.On the basis of engineering requirements, it can be determined whether adjustments are needed for the orbital position and attitude of satellites.

Problem description
A constellation with 42 micro-nano satellites will be launched.The constellation is designed as a circular orbit sun-synchronous with an altitude of 526 km.The satellites are arranged as a Walker constellation to perform Earth observation tasks.The proportion of the number of satellites within a certain range of height error during the launch process will be analyzed, and the influence of the number of satellites on the overall error of the distributed satellite constellation will be discussed to determine the relationship between the positioning error and the system efficiency of the constellation.

Problem analysis
According to the characteristics of the constellation, each satellite has the same altitude, eccentricity, and argument of perigee.Because they are all sun-synchronous satellites, the inclination of the orbit at this altitude is 97.5054°.Based on the launch conditions and experimental data, it is estimated that the expected value of the semi-axis of the orbit is 6, 904.14 km (assuming the radius of the Earth is 6, 378.14 km), and the variance is 3045 2 .Under these conditions, the satellites of the constellation are launched into orbit, and the proportion of satellites whose altitude error is within a certain range of 1 km, 2 km, 3 km, 4 km, and 4.5 km will be analyzed, respectively.In addition, the relationship between the number of satellites and the altitude error will also be discussed when it meets specific performance requirements.

Calculate the proportion of satellite altitude error within a certain range
When the number of satellites is relatively large, it can be roughly estimated based on the Chebyshev Equation ( 9).When the Chebyshev inequality estimation is invalid, its generalization of Equation ( 10) can be used to estimate.In this example, the minimum error 2 H is taken as 1 km, 2 km, 3 km, 4 km, and 4.5 km, and the maximum error 1 H is taken as 10 km.
Combining Equations ( 10) and ( 22) to obtain the estimated probability within a certain error range: Under the condition that the in-orbit position of the satellite obeys normal distribution, the expected value and variance can also be estimated according to the distribution function Equation (12).According to the Chebyshev inequality, the generalization of the Chebyshev inequality and the estimation results of the distribution function are shown in Table 1 From the data in Table 1, it can be seen that the results calculated using the distribution function are relatively accurate.

Relationship between the number of satellites and overall errors based on the central limit theorem
According to known conditions and Equation (20), it is obtained: where , n is the satellite number in the constellation, P is 6904.14km, and V is 3.045 km.
If an error range is given, the relationship curve between the number of satellites n and the error H can be calculated from Equation ( 22) by looking up the standard normal distribution probability table.
In this example, 90% of the satellites are required to be within the nominal range.The relationship expression is obtained as follows: 1.65 3045 We use MATLAB to draw the relationship curve between the number of satellites in the constellation and the mean error, which is shown in Figure 1.From the curve, it can be seen that if 90% of the satellites are required to be within the specified error range, the deviation range will decrease as the satellite number of the constellation increases.Similarly, the relationship curve between the number of satellites and the deviation can be analyzed at any other ratio.
Assuming that the satellite number of the constellation is 1000, fitting the probability curve of the deviation range and the satellite proportion can be obtained from Figure 2. In Figure 2, it can be seen that as probability increases, the deviation range will increase.In Figure 2, it can be seen that as the probability increases, the deviation range of the satellites will increase.Approximately 50% of the satellite error range is within 65 m.If 95% of the satellites in the constellation are required to be within the specified error range, the attitude error range will reach 140 m.

Simulation by STK
STK is the abbreviation for the Satellite Tool Kit, which supports the entire process of space missions, including design, testing, launch, operation, and mission applications.We simulate and analyze the inorbit position and operational efficiency of the constellation by STK software.A Walker constellation composed of 42 microsatellites for Earth observation.The constellation consists of 7 orbital planes, 6 satellites in each orbital plane, and the altitude of the satellite above the ground is 526 km (expected value).The orbital elements of the satellite are designed as a circular orbit and sunsynchronous satellite.The elements of the first satellite in the first orbital plane are shown in Table 2.The onboard sensor is a rectangular sensor with a horizontal field of view angle of 45.44° and a vertical field of view angle of 1°.The satellite is launched with one arrow and multiple satellites, and random errors are considered in the orbit entry position.The Earth radius is calculated at 6378.14 km and is simulated by STK software to analyze the satellite altitude deviation caused by the error and compare the ground coverage efficiency of the constellation in the case of random error.The simulation time of the experience is from 00:00:00.000on January 1, 2023, to 02:00:00.000on January 1, 2023, and the starting epoch of the satellite's orbit is 00:00:00.000on January 1, 2023.The expected altitude of the satellite's orbit initial position is set to 6904.14 km, with a standard deviation of 3.045 km.We simulate the in-orbit position of the Walker constellation, taking into account the complexity of the satellite's spatial position and practical use of the constellation, and then analyze its altitude error and take the semi-major axis as an example.Among the 42 satellites, the satellite in-orbit altitude distribution proportion of satellites within a certain range are shown in Table 3. Comparing the analysis results in Tables 2 and 3, the average proportion among the 100 simulation times is very close to the distribution function estimation results, that is, when the number of satellites is relatively large, the distribution function can be used for estimation.When there are a few satellites in the constellation, the Chebyshev inequality and its generalization can be used for a rough estimate.

Performance of satellite clusters with injection orbital error.
Due to many effect factors, the satellites of this constellation have random errors when in orbit.After being launched into orbit, they are used to observe the ground of the Earth.According to the requirements of the constellation task, a calculation of the Earth coverage of the grid point was used [16].Through STK simulation, global coverage between 85° north-south latitude was obtained within 2 hours, as shown in Figure 3.The red line represents the cumulative coverage curve of the satellite when there is no error, while the blue lines represent the cumulative coverage curve of the constellation to the ground formed when there is a random error.From the data and the figure, we can find that the impact of orbital altitude error on coverage is very small.On the basis of actual needs, it is necessary to consider whether to adjust the orbital attitude.
In terms of altitude, because there are not too many satellites in the constellation, there is a certain error between its mean and expected value.As the number of satellites increases, the deviation will gradually decrease.

Conclusions
In this paper, according to the relevant knowledge of probability theory, the error distribution of the satellite in-orbit position is analyzed.Based on the law of large numbers, the Central Limit Theorem, and other theories in probability theory, the expression of the error function of the satellite in-orbit position is constructed, and the probability distribution model of constellation comprehensive error under the Gaussian distribution of error is proposed.Taking the satellite orbital semi-major axis deviation as an example, the variation law of its altitude deviation in a certain range is analyzed with the increase in the number of satellites.From the pattern, it can be seen that as the number of satellites increases, the deviation tends to zero.The conclusion was validated by STK simulation.
In the experimental analysis, this article takes the altitude of satellite orbit parameters as an example for analysis and simulation verification.For other orbit parameters, analysis and verification can also be conducted.This study has an important reference value for the launch configuration and application of micro-nano constellation and can be further analyzed for constellations with different functions.

Figure 1 .Figure 2 .
Figure 1.The curve between the number of satellites and deviation under the range of 90% of satellites Figure 2. The curve between satellite probability and deviation in the cluster

1 .
Parameter settings for the STK simulation.

Figure 3 .
Figure 3. Effectiveness diagram of error and error-free orbit entry

Table 1 .
. High Deviations of Satellites

Table 2 .
Elements of the first satellite of the first orbit