Improving Orbit Prediction of the Two-Line Element with Orbit Determination Using a Hybrid Algorithm of the Simplex Method and Genetic Algorithm

Abstract

With the rapidly increasing number of satellites and orbital debris, collision avoidance and reentry prediction are very important for space situational awareness. A precise orbital prediction through orbit determination is crucial to enhance the space safety. The two-line element (TLE) data sets are publicly available to users worldwide. However, the data sets have uneven qualities and biases, resulting in exponential growth of orbital prediction errors in the along-track direction. A hybrid algorithm of the simplex method and genetic algorithm is proposed to improve orbit determination accuracy using TLEs. The parameters of the algorithm are tuned to achieve the best performance of orbital prediction. Six satellites with consolidated prediction format (CPF) ephemeris and four satellites with precise orbit ephemerides (PODs) are chosen to test the performance of the algorithm. Compared with the results of the least-squares method and simplex method based on Monte Carlo simulation, the new algorithm demonstrated its superiorities in orbital prediction. The algorithm exhibits an accuracy improvement as high as 40.25% for 10 days of orbital prediction compared to that using the single last two-line element. In addition, six satellites are used to evaluate the time efficiency, and the experiments prove that the hybrid algorithm is robust and has computational efficiency.

1. Introduction

The number of space objects has increased rapidly in recent decades because of the reduced cost of manufacturing and launching satellites into orbit [1]. It is estimated that more than 40,500 space debris objects larger than 10 cm and 1,100,000 space debris objects in sizes between 1 cm to 10 cm exist in Earth’s orbit, causing many threats to space missions and other satellites [2,3]. The United Sates Space Surveillance Network (SSN) tracks over 30,000 objects, and its measurements are used by the North American Aerospace Defense Command (NORAD) to generate TLEs, which are the only publicly available data in the world at https://www.space-track.org or https://celestrak.org [4]. The TLE data have been widely used for various space-related applications due to its extensive coverage and accessibility, such as thermospheric density estimation, maneuver detection, orbit prediction and reentry prediction [5,6,7,8]. These applications are related to the orbit determination (OD) processing.

Since the collision between Iridium 33 and Cosmos 2251 in 2009 that produced thousands of pieces of space debris, debris OD has been more crucial to precisely predict the orbit of a space object for space operations and accurately compute conjunctions and collision probabilities [1,9,10]. The TLE is a compact means to obtain modestly accurate and fast calculations, but the limited quality and lack of uncertainty information make it difficult to obtain high-quality ephemerides [11,12]. Since 2013, the way TLEs are generated has changed significantly, and the pattern of spacecraft positioning errors has been more consistent after 2015 [13,14]. TLEs can be used as pseudo-observations to generate an improved TLE for operational use. The TLE OD, in which multiple TLEs are used in least-squares processing or SGP4 orbit determination, could provide more accurate orbit estimates, as well as uncertainty information.

Several methods of orbit determination with the SGP4 model and TLE data are proposed to obtain a more precise orbit of an object at a desired moment [15]. One of these methods is differential correction based on the least-squares method (LST), proposed by Vallado et al. (2008) [12], to improve the quality of the initial TLE, and the orbit prediction error is significantly reduced. Levit (2011) [16] employed the batch least-squares differential correction and high-precision numerical propagator to curve the increase of the predicted error, achieving 10-fold improvement. To deal with outliers existing in the TLE and the ballistic coefficient soaking up errors from various sources, Águeda et al. (2013) [17] applied a simple method of outlier detection and ballistic coefficient estimation before TLE OD, which assures the convergence of the orbit determination process. Batch least-squares is used to perform OD and debias the two-line element sets [18,19]. Another OD method is the extended Kalman filter (EKF). Goh et al. (2018) [20] proposed a finite difference-based EKF to estimate the TLE, and the TLE estimation accuracy is at least two magnitudes higher than other optimization algorithms such as genetic algorithm (GA), and the average computational time is the lowest. However, Jamali et al. (2023) [21] pointed out that the GA outperformed the EKF in estimating the TLE. In fact, when performing nonlinear TLE OD, several problems arise, such as data sparseness and computational costs. In addition, the OD methods mentioned above require a more extensive set of parameters and prior satellite knowledge. To overcome the shortcomings of the OD method based on LST, Liu et al. (2023) [22] proposed a simplex method based on the Monte Carlo technique (SMMC), which avoids the calculation of the Jacobian matrix and enhances the quality of the TLE but is time-consuming.

In recent years, a data-driven orbit prediction method has been investigated to accurately predict trajectories of cataloged objects. Li et al. (2021) [23] employed a gradient boosting decision tree and convolutional neural networks to model and correct orbit error patterns mined from TLEs. Bizalion et al. (2023) [24] used neural networks to correct the bias of TLE automatically. Subramanian et al. (2023) [25] showed that training a physics informed neural network ordinary differential equation over a few measurements can reduce the propagation error. Thammawichai et al. (2024) [1] proposed a nonlinear programming model to estimate orbital elements, which has proven to be simple and computationally effective. Although the data-driven based orbit propagation promises to provide quick and accurate orbit predictions, it is likely to suffer in cases where the system dynamics differ significantly from the training data. For example, a recurrent neural network is trained on objects with different ballistic coefficients, and the average relative error could increase by almost 36% [26]. Therefore, dynamics-based OD methods are likely to be more effective when training data under similar conditions are limited [27].

In fact, the essence of the OD process is to find a global optimum in the convergence radius of the true TLE. The LST method and simplex method (SM) are easy to get stuck in local solutions. Kazemi et al. (2024) [15] suggested that artificial intelligence, such as the GA, has the potential to find the global optimum solution and should play a more prominent role in this domain than ever before. However, one of the main obstacles in applying GAs to complex problems has been the high computational cost due to their slow convergence rate. Yen et al. (1998) [28] applied the probabilistic SM method and elite-based GA (SMGA) to solve the problems mentioned above, and the hybrid algorithm contributes significantly to the performance improvement. In this hybrid algorithm, the probabilistic SM is used as an additional operator to generate chromosomes of the next generation for the GA, which is described in Section 2. The partition-based hybrid of the probabilistic simplex and GA is particularly suitable for complex optimization problems in which the sensitivity of variables varies widely [28]. As a matter of fact, there are several orders of sensitivity differences between orbital elements in the TLE. Hence, the SMGA method is introduced into the process of the TLE OD to improve the accuracy of orbit prediction and reduce the processing time.

The structure of this paper is organized as follows. In Section 2, the SMGA method is described, In Section 3, the influences of the SMGA parameters on the OD performance are discussed, In Section 4, the performance of the SMGA is analyzed and compared with those of the LST and SMMC methods. In Section 5, the conclusions and future work are presented.

2. SMGA Method

2.1. Genetic Algorithm

GA is a global search and optimization technique, inspired by the biological evolution of nature. The important concepts of GA are chromosome, population and fitness function. Chromosome often refers to a candidate solution or an individual, and a set of chromosomes form a population. GA evolves a population of individuals to a population of high-quality individuals, according to the fitness function [29]. During each generation, three basic genetic operators of selection, crossover and mutation are sequentially applied to chromosomes with certain probabilities. In this paper, rank selection will be employed to make sure that all the chromosomes have a chance to be selected, differential crossover to enhance the capability of population detection and non-uniform mutation to maintain the diversity of the population in the initial stage of search and improve the local fine-tuning ability of GA in the later stage of the search [30].

2.2. Simplex Method

The simplex method developed by Nelder and Mead is proposed to solve the problem of minimizing an objective function in N-dimensional space. This part is mainly from the work of Yen et al. (1998). For an optimization problem involving N variables, the simplex method searches for an optimal solution by N + 1 points, denoted as $x_1$, $x_2$, $\dots$, $x_{N+1}$. The method continually forms new simplices by replacing the worst point in the simplex, denoted as $x_w$, with a new point $x_r$ generated by reflecting $x_w$ over the centroid $\overline{x}$ of the remaining points:

$$x_r=\overline{x}+(\overline{x}-x_w)$$

(1)

where $\overline{x}$ is the average of N + 1 points. If $x_r$ is a good point, a new point farther along the reflection direction is generated using the equation

$$x_e=\overline{x}+\alpha (\overline{x}-x_w)$$

(2)

where $\alpha$ is the expansion coefficient ($1<\alpha$). If $x_r$ is worse than $x_w$ in the original simplex, a new point close to the centroid on the same side of $x_w$ is generated using the equation

$$x_c=\overline{x}-\beta (\overline{x}-x_w)$$

(3)

where $\beta$ is the contraction coefficient ($0<\beta <1$). If $x_r$ is not worse than $x_w$ but is worse than the second worst point in the original simplex, a new point close to the centroid on the opposite side of $x_w$ is generated using the contraction coefficient $\beta$:

$$x_c=\overline{x}+\beta (\overline{x}-x_w)$$

(4)

A stochastic variant of the simplex method is introduced for cost-effectively searching for the optimal solution. The distance between the centroid and the reflected point is to be determined probabilistically. In this variant, $\alpha$ is a random variable within the interval [1, 2] and $\beta$ within the interval [0, 1], and both the coefficients follow the triangular probability density function. This flexibility allows the probabilistic simplex to explore the search space with more freedom and facilitates the fine-tuning of solutions around an optimum.

2.3. SMGA

Both the SM and GA algorithms only need to be sure that the initial state (TLE) is within the convergence radius of the TLE (Liu et al., 2023; Kazemi et al., 2024). The state vector of the TLE to be estimated is defined as $X$:

$$X={\left[\begin{array}{ccc}{B}^{*}& e& \begin{array}{ccc}i& \mathsf{\Omega }& \begin{array}{ccc}\omega & M& n\end{array}\end{array}\end{array}\right]}^T$$

(5)

where $B^{*}$ is the drag-related term in the TLE, $e$ is the orbit eccentricity, $i$ is the orbit inclination, $\mathsf{\Omega }$ is the right ascension of the ascending node (RAAN), $\omega$ is the argument of perigee, $M$ is the mean anomaly, and $n$ is the mean motion. The state $X$ can be used to compute the position and velocity at any epoch using the SGP4 algorithm.

In the view of the GA, a single sample of $X$ is treated as a chromosome, and the parent population is generated randomly in the feasible region of the state vector $X$. The size of the parent population is denoted as $P$. The reproduction of TLE data in the elite-based SMGA is shown in Figure 1. Since $X$ contains 7 parameters, the SM needs 8 points for an initial simplex. The 8 points or chromosomes from the ranked population with the best fitness values are copied to the next generation, and the centroid of the 8 points is computed to implement the simplex search. Since the simplex method is a local optimal search algorithm, $S$ chromosomes can be obtained with the same centroid and probabilistic simplex method, and the $S$ − 8 chromosomes are copied to the next generation. Then, the $P$ − $S$ chromosomes of the next generation are obtained through selection, mutation and crossover operations. To proceed from the parent generation to the next generation, the process of the SMGA algorithm is repeated until a termination condition is met, which is defined by the user.

The pseudo-code of the concise SMGA is as follows (Algorithm 1):

Algorithm 1: SMGA

input: $X_0$, $S$

output: the best chromosome

1   Initialize a random population of size $P$. 2   While (a terminal condition is not met) 3    { 4     Evaluate the fitness of each chromosome; 5     Rank population based on the fitness results; 6     Copy 8 best chromosomes to the next generation; 7     ******Simplex Part****** 8     Apply simplex operator to the top 8 chromosomes and generate $S$ − 8 chromosomes; 9   Copy the generated $S$ − 8 chromosomes to the next generation; 10    ****** GA Operator Part****** 11    Select $P$ − $S$ chromosomes based on ranking, and copy to the next generation; 12    Apply mutation with the mutation probability to the $P$ − $S$ chromosomes; 13    Apply two parents crossover with the crossover probability to the $P$ − $S$ chromosomes; 14   }

It is assumed $k+1$ TLEs of a space object are given, with their reference epochs denoted as $t_i,i=0,1,\dots ,k$. For the $i$-th TLE, the position $r_i$ and velocity $v_i$ at $t_i$ form $y_i=[r_i,v_i]$. Now, one of the $k+1$ TLEs is set as the one to be optimized. Assume that the TLE at $t_0$, denoted as $X_0$, is to be optimized and its initial state is the same as the original TLE. This TLE is then used to compute the position $r_{0i}$ and velocity $v_{0i}$ at reference epochs $t_i$ of the $i$-th TLE, and we obtain $y_{0i}=[r_{0i},v_{0i}]$. To evaluate the fitness of $X_0$, the difference between $y_i$ and $y_{0i}$ is computed and assigned to its corresponding chromosome. Consequently, the fitness function is as follows:

$$\mathrm{Fitness}={\sum }_{i=0}^k{(y_{0i}-y_i)}^2$$

(6)

3. Settings of the SMGA Parameters

To evaluate the effectiveness of the SMGA algorithm, the settings of the SMGA parameters are discussed in this section. The parameters include the convergence radius of the TLE estimation, the total population size $P$ and the chromosome percentage by SM. The experiments are made with space objects with given TLE data sets, which are used to optimize the TLE, and CPF ephemeris from using satellite laser ranging (SLR) data. The differences between the two positions, one from TLE and the other from CPF, are obtained and used to assess the prediction performance of the optimized TLE. The timespan of TLEs used for this experiment is from day 37 to day 58 of 2018, and the orbit is predicted for 10 days, in which the position is computed every 10 min.

3.1. The Estimation of the Convergence Radius

GA is a global optimization method and computationally intensive, while SM is a local optimization method and computationally compact. Abay et al. (2021) [31] proposed a machine learning approach to predict the initial estimates with the convergence radius of the actual TLEs to decrease the effort and time in estimating TLEs. Hence, the estimation of the convergence radius is very important for reducing the time cost of the GA global search and improving the capability of the SM search.

The LST polynomial orbit element fitting can obtain the variation trend of the TLE, which is applied to detect outliers of TLEs [32]. To determine the convergence radius of the TLE, the LST polynomial fitting is employed to each orbital element in the state vector $X$. The convergence radius is set to 2 standard deviations ($\sigma$) of a fitted orbital element for a confidence of 95%. If a real orbital element in a TLE is out of the 2$\sigma$ range of the fitted orbital element, the real orbital element is treated as an outlier, and the convergence radius is set to 3$\sigma$ for a confidence 99.7%. In order to reveal the influences of the convergence radius on the optimal solution search, a satellite with NORAD ID 37781 is chosen to run the SMGA algorithm. The optimal solution constrained by the convergence radius is denoted as CST for short, and the optimal solution unconstrained is denoted as UNC. The initial population size $P$ is set to 120. The number of chromosomes, $S$, generated by the simplex method is set to 24.

Figure 2 gives the distributions of the UNC and CST solutions of the 7 estimated orbital elements and fitness values for satellite 37781. It is obvious to see that the distributions of UNC and CST are centered around different values and have almost no overlapping area.

Table 1. Original TLEs, UNC and CST solutions at the epoch 58.2361, satellite 37781.

Epoch	B*/ER	$\mathit{e}$	$\mathit{i}$/°	$\mathbf{\Omega }$/°	$\mathit{\omega }$/°	$\mathit{M}$/°	$\mathit{n}$/rev/day
57.0750	1.8160 × 10−4	1.9360 × 10−4	99.3282	69.0729	75.8652	284.2805	13.78718758
57.3653	−4.8704 × 10−4	1.9080 × 10−4	99.3283	69.3581	77.2623	282.8759	13.78718135
57.5830	1.3400 × 10−4	1.9120 × 10−4	99.3284	69.5728	76.2074	283.9263	13.78717890
57.9458	4.0391 × 10−4	1.9420 × 10−4	99.3283	69.9290	68.8161	291.3207	13.78717853
58.0910	7.7336 × 10−4	1.9000 × 10−4	99.3280	70.0726	70.3828	289.7571	13.78719342
58.2361	8.3012 × 10−5	1.8800 × 10−4	99.3277	70.2150	69.1368	290.9974	13.78718417
UNC	2.8012 × 10−5	6.5100 × 10−4	99.3304	70.2150	68.2390	291.8796	13.78718383
CST	2.6804 × 10−4	1.8200 × 10−4	99.3277	70.2110	64.9200	295.2117	13.78718425

lists the optimal solutions of UNC and CST at the epoch 58.2361 and 6 TLEs. The standard deviations of the orbit prediction errors for UNC and CST are 2.252 km and 1.172 km, respectively. The standard deviation of CST is 47.94% smaller than that of UNC. However, the fitness values of UNC are centered around 5.303, while those of CST are around 5.513. It is a contradiction that the larger fitness value of CST has a better performance in the orbit prediction. This phenomenon can be explained by the variation trend of B* and $e$. At the epoch 58.2361, B* is one order of magnitude smaller than those of the other TLEs and thus treated as an outlier according to the convergence radius. The optimal B* of CST can only be obtained in the convergence radius, while the optimal B* of UNC is out of the range. According to the variation trend of B*, the optimal B* of CST is more in line with the trend. Although the variation trend of $e$ is stable, the optimal $e$ of UNC converges to an unexpected value with one order of magnitude smaller.

The convergence radius of the TLE is very important for obtaining an optimal solution. Its determination is influenced by the size or number of TLEs used in the LST polynomial orbit element fitting. To see this influence, satellites with NORAD IDs 27386, 36508 and 37781 are chosen to investigate the relationship between the size of the TLEs and the best convergence radius.

Table 2. The orbit prediction errors (km) and fitness values with different sizes of the TLEs.

NORAD ID		Size
		3	5	7	9
27386	Error	8.124	8.121	8.128	8.125
	Fitness	2.392	2.400	2.500	2.400
36508	Error	14.159	11.193	11.198	11.188
	Fitness	1.344	1.213	1.220	1.216
37781	Error	6.129	6.021	6.003	5.948
	Fitness	1.064	1.029	1.098	1.055

lists the satellite orbit prediction errors and fitness values with different sizes. Figure 3 gives insight into the differences in orbit prediction errors and relative position errors of different sizes with a size = 5. For 27386, it seems that the size of the TLEs has the least impact on the orbit prediction error, but a size = 5 has a significant advantage over others, as shown in Figure 3; for 36508 and 37781, the overall trend of orbit prediction errors is that the errors decrease with the increasing size of the TLEs; the differences in the prediction error and fitness values between size = 4 and size = 5 are the largest and then become smaller when the size of the TLEs is larger than 5. It can be seen from Figure 3 that the orbit prediction errors at size = 5 are smallest, and thus, the number of TLEs is set to 5 when running the SMGA algorithm.

3.2. Population Size and Hybrid Percentage

The population size and hybrid percentage of SM and GA also influence the fitness value and time efficiency, and they are investigated with NORAD IDs 32711, 35752, 39741, 36111, 37867 and 40315. As shown by Figure 4, the fitness values fall sharply and then keep stable around a constant value when the population size is beyond 120. The time cost of the SMGA algorithm decreases linearly with the population size increase in the beginning and then linearly increases after the population size is 80. The most time-effective population size is about 70–90, while the fitness values still decrease. Therefore, we chose a population size = 120 to balance the fitness value and time efficiency.

Figure 5 shows the relationship between the fitness value and the hybrid percentage of the SM and GA. It is visible that the minimum fitness value is achieved when the number of chromosomes generated by the SM algorithm accounts for 20% of the total population. For 36111 and 40135, the fitness values do not vary, while the percentage increases. When the percentage is 0% or 100%, the SMGA becomes the GA or the probabilistic SM. In the following experiments, the hybrid percentage of the SM and GA is set to 20%.

4. Performance of the SMGA

In this section, 16 satellites are chosen to evaluate the performance of the SMGA in orbit prediction accuracy and computation time efficiency. The LST and SMMC orbit determination methods are employed to compare with the performance of the SMGA method. The NORAD IDs of the satellites used for evaluation of the prediction accuracy are 27944, 36508, 37781, 39086, 41579 and 43215, which have CPF orbits, and 39634, 41335, 41456 and 43437, which are four Sentinel satellites with precise orbit ephemerides. The satellites used for evaluation of the time efficiency are 32711, 35752, 36111, 37867, 39741 and 40315.

Table 3. Orbital parameters of the test satellites.

Norad ID	Satellite Name	Apogee/km	Perigee/km	$\mathbf{Inclination}/°$
27944	Larers	689	673	98.00
32711	NavStar 62	20,706	19,659	54.97
35752	NavStar 64	20,329	20,038	54.28
36111	SL-4 Deb	251	243	64.18
36508	CryoSat 2	721	718	92.02
37781	Haiyang 2A	969	967	99.32
37867	Cosmos 2476	19,192	19,068	64.48
39086	Saral	785	781	98.53
39634	Sentinel 1A	697	695	98.18
39741	NavStar 70	20,277	20,085	55.58
40315	Cosmos 2501	19,182	19,077	64.27
41335	Sentinel 3A	803	802	98.62
41456	Sentinel 1B	492	567	98.18
41579	Cosmos 2517	944	942	99.27
43215	PAZ	510	507	97.44
43437	Sentinel 3B	804	801	98.62

shows the specific information about these satellites. Their orbits are almost circular, with inclinations ranging from 54.28$°$ to 99.32$°$. The TLEs used for the orbital prediction are given in Appendix A.

4.1. Performance Evaluation by CPF Data

The fitting span of the TLE data is from the 33rd to the 58th day of 2018, except for satellite 43215, which is from the 278th to the 303rd day. The CPF data are taken as true values, and the true prediction errors are calculated with positions computed from the original TLE, LST-TLE, SMMC-TLE and SMGA-TLE.

Table 4. Comparison of the orbit prediction error RMSs with different TLE data for the satellites with CPF ephemeris.

NORAD ID	Original TLE/km	LST/km	SMMC/km	SMGA/km	Against Error RMSs with Original TLE in %
					LST	SMMC	SMGA
27944	1.237	2.065	1.710	1.202	66.94 $\downarrow$	38.24 $\downarrow$	2.83 $\uparrow$
36508	1.562	3.335	5.643	1.411	113.51 $\downarrow$	261.27 $\downarrow$	9.67 $\uparrow$
37781	1.816	1.389	6.297	1.085	23.51 $\uparrow$	246.75 $\downarrow$	40.25 $\uparrow$
39086	0.786	2.167	1.435	0.784	175.70 $\downarrow$	82.57 $\downarrow$	0.25 $\uparrow$
41579	1.134	2.655	1.402	0.910	134.13 $\downarrow$	23.63 $\downarrow$	19.75 $\uparrow$
43215	10.979	27.093	25.852	10.885	146.77 $\downarrow$	135.47 $\downarrow$	0.85 $\uparrow$

$\uparrow$ represents the accuracy improvement; $\downarrow$ represents a decrease in accuracy.

shows the comparisons of the orbit prediction error root mean square (RMS), and the RMS is obtained through all coordinate differences of all the processed epochs over the test period. Relative to the error RMS with the original TLE, the change is calculated, respectively, with error RMSs from using the LST-TLE, SMMC-TLE and SMGA-TLE. The symbol $\uparrow$ represents accuracy improvement, while symbol $\downarrow$ represents a decrease in accuracy. With the LST-TLE, there is only one error RMS lower than that of the original TLE (23.51% for satellite 37781). The error RMSs of the SMMC-TLEs are all larger than those of the original TLEs, but there are five error RMSs smaller than those of the LST-TLEs. The error RMSs with the SMGA-TLEs are better than those with the original TLEs: the maximum improvement is 40.25%, and the minimum improvement is 0.85%. It is noted that the error RMSs with the SMGA-TLEs for 39086 and 43215 are 0.974 km and 10.885 km and are only 0.002 km and 0.094 km larger than those with the original TLEs, respectively. Overall, the SMGA can improve the orbit prediction accuracy when the quality of the original TLE is ordinary or maintain the orbit prediction accuracy when the quality of the original TLE is good. In other words, the SMGA method is robust in finding an optimal TLE solution.

The variations of the orbital prediction errors over time are shown in Figure 6 and Figure 7. The orbital prediction errors of the LST-TLEs are the worst overall, except for 37781. The orbital prediction errors of the LST-TLEs in the W-direction grow exponentially over time, while the orbital prediction errors of the SMMC-TLEs in the S-direction linearly increase over time. As usual, the orbital prediction errors of the SMGA-TLEs are mainly in the S-direction.

4.2. Performance Evaluation by POD

The fitting span of the TLE data is from the 262nd to the 272nd day of 2021. The PODs of four Sentinel satellites are used to calculate the orbital prediction errors with positions provided by the original TLE, LST-TLE, SMMC-TLE and SMGA-TLE. It lists the error RMSs of the orbit prediction with different TLE data in

Table 5. Comparison of the orbit prediction error RMSs with different methods for the Sentinel satellites.

NORAD ID	Original TLE/km	LST/km	SMMC/km	SMGA/km	Against Error RMSs with Original TLE in %
					LST	SMMC	SMGA
39634	1.675	7.531	2.441	1.528	349.61 $\downarrow$	45.73 $\downarrow$	8.77 $\uparrow$
41335	1.324	2.110	1.226	0.888	59.37 $\downarrow$	7.40 $\uparrow$	32.93 $\uparrow$
41456	2.140	8.154	2.203	2.073	281.03 $\downarrow$	2.94 $\downarrow$	3.13 $\uparrow$
43437	5.660	5.009	5.257	4.835	11.50 $\uparrow$	7.12 $\uparrow$	14.58 $\uparrow$

$\uparrow$ represents the accuracy improvement; $\downarrow$ represents a decrease in accuracy.

. The LST-TLEs for all four satellites have the poorest accuracies, except for 43437, and the accuracy of 39634 is worsen by 349.61% and that of 41456 by 281.03%. Compared to the LST-TLEs, the SMMC-TLEs are much better, with a maximum accuracy reduction of 45.73%. The prediction accuracy for 41335 and 43437 is improved by 7.4% and 7.12%, respectively, and the prediction accuracy with the SMMC-TLE for 41456 is only worse by 2.94%, which is the smallest of the three prediction errors. Compared to the SMMC-TLEs, the accuracy with the SMGA-TLEs is significantly improved for 41335 and 43437 and that for 39634 is better by 8.77%. For satellite 41456, the improvement is 3.13%, which is small but is still better than that with the SMMC.

Figure 8 and Figure 9 show the orbit prediction errors of the Sentinel satellites with four types of TLEs. The prediction errors with the LST-TLE grow exponentially in the W-direction and S-direction, while the prediction errors with the SMMC-TLE have large errors in the R-direction and S-direction. The position errors of 39634 and 41456 for the SMGA-TLEs are mainly in the R-direction and are very small in the S-direction and W-direction.

4.3. Time Efficiency

The time costs of the SMGA, SMMC and LST methods for another set of satellites are given in

Table 6. Comparison of the time cost for OD with different methods.

NORAD ID		32711	35752	36111	37867	39741	40315
SMGA	Fitness	1.214	1.265	1.113	0.848	0.636	0.365
	Time/s	14.521	17.754	27.272	19.644	13.238	22.272
SMMC	Fitness	1.672	1.586	1.443	1.115	0.923	0.474
	Time/m	24.619	25.773	31.576	26.222	22.041	27.311
LST	Fitness	1.741	1.432	1.521	1.230	1.002	0.530
	Time/m	1.363	1.417	1.833	1.413	1.210	1.658

. It can be seen that the OD runs with the SMGA are the least time-consuming (all in less than 1 min), while the OD runs with the SMMC are very time-consuming (all solutions require more than 20 min), which is unacceptable for real-time applications. Although the time consumption of the LST method is moderate (less than 2 min), the LST method is not suggested due to its large prediction error.

5. Conclusions

The quality of publicly available TLE data is uneven, and thus, the orbit predictions of space objects could have large errors, which introduce potential risks for collision avoidance and reentry prediction. Using multiple TLEs to estimate an optimal TLE is an approach adopted by many researchers. Among the proposed methods, the LST-based or SMMC-based method has the problems of being stuck in the local optimal solutions and are time-consuming. Although the GA has the potential to discover the global optimum solution for complex problems, it often has a high computational cost and a slow convergence. A common strategy for dealing with the GA’s slow convergence problem is to combine a GA with a complementary local search technique. Therefore, the probabilistic SM algorithm is introduced as a cost-effective operator into the GA to perform OD. This results in the SMGA algorithm, a hybrid of the SM and GA algorithms.

The parameter setting for the hybrid algorithm is discussed. For the TLE estimation problem, the number of TLEs used for convergence radius determination is suggested to be five, the population size is to be 120, and the hybrid percentage of SM and GA is set at 20%. Six satellites with CPF orbits and four satellites with PODs are chosen to evaluate the orbit prediction performance of the hybrid algorithm, and the SMGA method seems to be promising. Compared to the orbit prediction error RMSs with the original TLEs, the maximum prediction accuracy improvement is 40.25%, and the minimal improvement is 0.25%. The experiments demonstrate that the SMGA algorithm is robust.

Since we have shown that the minimal fitness value does not mean the best solution, more attention should be paid to the quality of the convergence radius, especially for B*. It is noted that, in some cases, using TLE with negative B* can obtain a higher precision of the orbital prediction. It makes the determination of the convergence radius of B* more complex.