Abstract
Accurate and rapid prediction of reentry trajectory and landing point is the basis to ensure the reentry vehicle recovery and rescue, but it has high requirements for the continuity and stability of real-time monitoring and positioning data and the fidelity of the reentry prediction model. In order to solve the above contradiction, based on the theory of relative entropy and closeness in fuzzy learning, research on real-time evaluation of reentry reachability is presented in this article. With the Monte Carlo analysis data during the design and evaluation of the reentry vehicle control system, the reentry trajectory feature information base is designed. With the matching identification decision strategy between the identified trajectory and trajectory feature base, the reachability of the reentry vehicle, reachable trajectory, and landing point can be predicted. The simulation results show that by reasonably selecting the time window and using the evaluation method designed in this article, making statistics of the trajectory sequence number and frequency identified based on relative entropy and closeness method, the reachability evaluation results can be given stably, which is suitable for the real-time task evaluation of TT&C system.
1 Introduction
For the real reentry vehicle and reentry environment, the mathematical simulation model and its corresponding real system (reentry vehicle aerodynamic model, sensor model, actuator model, atmospheric environment model, aerodynamic ablation model, etc.) generally have high nonlinearity and strong coupling, it also has many uncertain influencing factors, and then, the reentry trajectory envelope is large (Hale et al. 2002, Phillips 2003, Richie 1999, Vinh et al. 1980, Peña-Asensio et al. 2021). At the same time, restricted by the tracking ability of TT&C equipment, complete and continuous telemetry data of the reentry vehicle with preset trajectory cannot be obtained in the real reentry process, and then based on the state of its attitude, navigation and positioning, guidance and control systems to predict and judge the reachability of reentry vehicle will be difficult (Vinh 1981, Tang et al. 2019). However, for the TT&C equipment, search and rescue system, it is necessary to realize the guidance of tracking equipment, recover and search the reentry vehicles by evaluating the impact point dispersion and predicting reachability of the reentry vehicle, according to the real-time measured positioning data, high fidelity reentry trajectory prediction model and guidance algorithm (Ono et al. 2020, Du and Liu 2017, Jiang et al. 2020, Dong et al. 2022, Mehta et al. 2017).
The above contradiction has a strong dependence on the stability of real-time measurement data, the accuracy of reentry trajectory high fidelity model and the robustness of the guidance module. The real-time trajectory data of high-speed reentry vehicles can be smoothed by filtering algorithm (Wei et al. 2022, Huang et al. 2020, Wang et al. 2014), but the high-fidelity model of reentry trajectory prediction and guidance calculation module is generally a “black box” for the TT&C equipment, search and rescue system. High computer performance and quantitative evaluation results should be required if using the Monte Carlo analysis method to predict the impact distribution in real time. Then, the Monte Carlo analysis method can only be used as the impact analysis of uncertain factors for the control system, but not suitable for the real-time reachability capability evaluation of reentry vehicles for TT&C dependent itself.
The research on the reachability of reentry vehicle mainly focuses on such as the optimal design of reentry orbit (De Grossi et al. 2021, Fieee et al. 2005, Chen et al. 2021, Meng et al. 2015, Wang et al. 2019, Roh et al. 2020), reentry trajectory and TT&C station determination method (Mansell and Grant 2018, Haitao et al. 2021), through error transfer analysis (Wu et al. 2021, Wang and Grant 2017), aerodynamic shape (Li et al. 2020) and reentry trajectory comprehensive optimization method (Taheri et al. 2021, Vivani and Pezzella 2015, Graves and Harpold 1972), guidance method comparison (Terui et al. 2020, Gamble et al. 1988, Lu 2008, Rea and Putnam 2007, Bairstow 2007, 2006, Putnam et al. 2008, Wang et al. 2021, Guo et al. 2021, You et al. 2021, Succa et al. 2016, Fang et al. 2018). Considering constraints of specific models and guidance algorithms such as no-fly zone, heating and overload, it consumes a lot of computing time with analysis reachable domain, and the real-time performance is not good. Some scholars hope to build a model through the neural network artificial intelligence theory to realize reentry trajectory prediction (You et al. 2020; Sánchez-Sánchez and Izzo 2018, Ma et al. 2021, Yang and Wang 2020), but it also needs to consume a lot of training time and calculation energy consumption in advance, so the flexibility is limited.
As a theory of processing small sample data, fuzzy learning theory is more and more favored by researchers and engineers in the fields of machine learning, artificial intelligence and so on. It is an important basis for analyzing the performance of learning machines and developing new learning algorithms. Fuzzy relative entropy (Lin 1991) and closeness theory (De Luca and Termini 1972) are two important theories in fuzzy clustering evaluation. Fuzzy relative entropy, also known as K–L divergence, describes the degree of similarity between samples with a probability distribution. The smaller the relative entropy is, the more similar the probability distribution is. The closeness degree describes the similarity degree by describing the ratio of overlapping area to non-overlapping area after the transformation of the sample vector membership function. The greater the closeness degree is, the greater the similarity degree is. Therefore, these two theories can be used to measure and identify the coincidence degree between systems. At present, relative entropy theory and closeness theory are mainly used to solve multi-attribute decision-making problems (Bi et al. 2015, Ning et al. 2019, Li et al. 2022), such as emergency decision-making, customer credit evaluation and competence strength and satisfaction evaluation.
Considering the inevitable correlation between the current flight path state of the reentry vehicle and the reachable area (Qiao et al. 2017), based on the fuzzy learning theory, a reachability evaluation strategy for the reentry vehicle is presented in this article. Establishing the fuzzy set by trajectory flight envelope information generated as the basic feature library, the trajectory data are generated based on the high fidelity reentry trajectory prediction model and guidance model with Monte Carlo trajectory analysis. Constructing the trajectory fuzzy vector to be identified based on the real-time measured reentry flight path feature information of the reentry vehicle, the matching identification strategy and reentry reachability evaluation method are designed in this article. The maximum possibility of the reachable path and landing point for the real reentry vehicle are calculated, so as to evaluate the reachability and provide guidance for trajectory and landing point prediction. It provides a new method for real-time reachability evaluation with TT&C equipment tracking and guidance with abnormal tracking conditions.
2 Relative entropy and closeness in fuzzy learning
2.1 Definition of fuzzy relative entropy
Defining the probability distribution as
where n is the number of fuzzy vectors,
The relative entropy of known probability distribution Q to distribution P is defined as:
where p is the Lagrange multiplier, and H(P,Q) reflects the difference between distribution P and Q.
For
As it is similar to relative entropy, fuzzy relative entropy could be defined to reflect and measure the difference between two fuzzy vectors based on probability distribution accordingly.
Assuming that
Therefore, the relative entropy of fuzzy vector
This reflects the difference between two fuzzy vectors. However, the above formula has a disadvantage, that is, when
In fact,
Similarly,
It has complete significance and practicability, which is called fuzzy relative entropy. It could characterize the difference between set A and set B. That is, the smaller the fuzzy relative entropy, the smaller difference between them.
It is easy to prove that
2.2 Closeness of fuzzy set
Fuzzy recognition mainly focuses on which fuzzy set is closest to a known one. The given fuzzy subset is defined as:
Represent
Defining
where
Definition 1
Definition 2
Closeness – A finite universe is defined as
3 Evaluation strategy of reachability for reentry vehicle based on fuzzy learning
3.1 Establishing the fuzzy vector of reentry trajectory
3.1.1 The information feature database of reentry reachable trajectory
Monte Carlo analysis is carried out based on the high fidelity dynamic model and uncertainty error term, and the sub-satellite point trajectories are calculated. Set the sub-satellite point of
The feature reflects the attribute values of fuzzy features, and each feature trajectory is a set of vectors related to time, latitude, longitude and height of sub-satellite points.
Set the latitude, longitude and altitude of the sub-satellite point of the trajectory to be identified as
3.1.2 Standardizing the fuzzy vector
Setting
That is, normalizing each element in the fuzzy vector to (0,1). Recording that the normalized trajectory modulus vector to be identified as
3.2 Calculating the relative entropy and closeness of reentry trajectory
According to formula (7), calculating the fuzzy relative entropy between the trajectory information to be identified and the feature database vector, then generating the relative entropy matrix
According to formulas (8) and (9), calculating the closeness of the specially identified trajectory information as follows and generating the closeness matrix:
where
3.3 Strategy of matching identification
Considering the relative entropy can not only reflect the distribution characteristics of the target feature attributes to be identified, but also obtain more accurate target similarity by weighting, and finally improve the accuracy of target recognition. When calculating the closeness, the accuracy of target recognition will be reduced when the identified vector is not in the feature base or there is noise interference. Therefore, the following recognition decision-making strategy is formulated:
When
When
3.4 The reachability evaluation method for reentry vehicles
The specific steps of evaluating the reentry reachability for reentry vehicles based on fuzzy theory are as follows. Figure 1 is the flowchart of the evaluation method of reachability for reentry vehicles.
Carrying out Monte Carlo analysis, based on the high fidelity dynamic model, the identified system uncertainty factors and error interference terms; calculating characteristic information of sub-satellite points to generate the reentry trajectory characteristic information database.
Filtering and smoothing the real-time positioning data in the process of real reentry flight as the trajectory to be identified, and calculating sub-satellite points to generate the characteristic information of the trajectory to be identified.
Processing the trajectory feature information calculated in steps (1) and (2) by fuzzy vector standardization.
Calculating and sorting the calculation results of fuzzy relative entropy and closeness, which of the trajectory to be identified relative to the reentry trajectory feature database, according to the specified real-time data accumulation evaluation time window
Finding the most matching trajectory in the current real-time data window and trajectory feature database according to the matching identification standard. Setting the trajectory matching serial number as K, taking the landing point and landing time corresponding to this trajectory as the currently identified reachability evaluation result.
Repeating steps (2) to (5), counting the trajectory matching sequence number as
Flowchart of reentry reachability evaluation method for reentry vehicles.
4 Simulation and result
Taking the Skip Entry with the characteristics of secondary reentry trajectory as the simulation verification object, the feasibility of the reentry reachability evaluation method based on fuzzy learning is verified.
4.1 Simulation settings
Considering the separation point of 5,000 km module and vehicle (Huang et al. 2020, Li et al. 2020), set the initial deviation term according to the random normal distribution, as shown in Table 1. Generate 300 groups of deviation trajectories based on the Monte Carlo method as the reentry trajectory characteristic information base of the reentry vehicle.
Error source setting (for module separation point)
| Serial number | Category | Error term | Range |
|---|---|---|---|
| 1 | Initial condition deviation | Height (km) | ±5% |
| 2 | Speed (M/s) | ±1% | |
| 3 | Reentry angle | ±0.2 | |
| 4 | Longitude (°) | ±0.2 | |
| 5 | Latitude (°) | ±0.2 | |
| 6 | Velocity Azimuth (°) | 0.2 | |
| 7 | Mass (kg) | ±10.0 | |
| 8 | Deviation of atmospheric density, sound velocity and dynamic model | Atmospheric density, sound velocity | ±20% |
| 9 | Lift coefficient, Cl | ±20% | |
| 10 | Drag coefficient, Cd | ±20% |
Figures 2 and 3 show the altitude and sub-satellite point dispersion diagram of 300 groups of trajectories in the trajectory feature information database. The GNC system guidance capability of the simulation object reentry vehicles can ensure that the altitude dispersion range of first skipping out is about 40 km and the landing point dispersion range is ±50 km.
Taking the first of the 300 characteristic trajectories generated by the Monte Carlo analysis method as the reference nominal trajectory, and the total time of the trajectory is 2,000 s.
The height dispersion diagram in trajectory feature information base.
The distribution diagram of sub-satellite points in trajectory feature information base.
Considering the distance
In order to test the calculated energy consumption, the sliding recognition time window is set in two ways, once per second and once every accumulated 10 s data, so as to evaluate the recognition window setting and results in this article. For the skip reentry spacecraft, the recognition window is set reasonably, and the recognition results of the two windows are evaluated at the same time.
The trajectory number in the trajectory feature database starts from serial number 1.
Programing language and computer configuration: Visual Studio C++ 2010, Intel Core i5-4570pentium (R), CPU 3, 20 GHz, 4-GB memory.
Error source setting of trajectory to be identified
| Serial number | Category | Error term | Range |
|---|---|---|---|
| 1 | Trajectory deviation | Longitude (°) | ±0.04 |
| 2 | Latitude (°) | ±0.04 | |
| 3 | Height (m) | ±50 |
4.2 Simulation results
4.2.1 Identifying once per second
In the current test environment, the total calculation takes 10.72 s to complete 2,000 evaluations. Table 3 gives the calculation results of fuzzy relative entropy and closeness calculated with 300 groups of data in the feature database every 1 s with the trajectory to be identified.
Data matching results per second
| Serial number | 1st second | 2nd second | 3rd second | 2,000th second |
|---|---|---|---|---|
| Fuzzy relative entropy | 0.57743 × 10−6 | 0.01847 × 10−6 | 0.16495 × 10−6 | 0.00846 × 10−6 |
| Match result | Group 1 | Group 1 | Group 1 | Group 1 |
| Maximum and minimum closeness | 0.99992 | 0.99997 | 0.99994 | 0.999828 |
| Match result | Group 1 | Group 1 | Group 1 | Group 4 |
Figure 4 shows the recognition result based on the relative entropy between the data per second and the feature database, that is, the matching sequence number. Figure 5 makes frequency statistics of the recognized sequence number. As can be seen from Figures 4 and 5, the four serial numbers that appear more frequently are as follows: 389 times for serial number 1,357 times for serial number 106, 284 times for serial number 36 and 187 times for serial number 141.
Identification results of trajectory matching sequence number based on fuzzy relative entropy.
Identification results of matching sequence number frequency statistics based on fuzzy relative entropy.
Figure 6 shows the recognition result based on the closeness between the data and the feature database per second, that is, the matching sequence number. Figure 7 makes frequency statistics of the recognized sequence number. It can be seen from Figures 6 and 7 that the four sequence numbers that appear more frequently are 309 times for serial number 262, 248 times for serial number 36, 206 times for serial number 171 and 198 times for serial number 1.
Identification results of trajectory matching sequence number based on fuzzy closeness.
Identification results of matching sequence number frequency statistics based on fuzzy closeness.
Based on the aforementioned calculation results, taking the separation point of 5,000 km capsule as the time zero, it is matched with the feature database every second. A total of about 2,000 calculations can be seen:
When t <1,200 s, the reentry vehicle is located before the first reentry phase, and the “matching” recognition results are relatively concentrated, and the recognition frequency is about 200 times. This shows that the reentry vehicle is less affected by the atmosphere at this stage, and the trajectory to be identified is a high similarity to the trajectory feature database. The evaluation results of reachability are mainly concentrated in five trajectories.
When 1,200 s < t <1,300 s, the reentry vehicle is located before the first skip and after the first reentry phase and the “matching” recognition result is divergent. This indicates that the reentry vehicle is seriously affected by the atmosphere and the GNC system needs to start guidance planning, the probability of following any trajectory increases. At this time, it is difficult to evaluate the reachability.
When 1,300 s < t < 1,700 s, the reentry vehicle is before the second reentry and after the first skip phase, although the “matching” recognition result diverges, the number 1 appears the most times, and the reachability of the reentry vehicle evaluation result can converge in the only one trajectory. The divergence of the recognition result indicates that the GNC system of the reentry vehicle needs to conduct guidance planning again. Although the probability of following any trajectory increases, it does not affect the position of the final landing point.
When t < 1,700 s, the reentry vehicle is in the second reentry phase, and the “matching” recognition result is number 1. The known trajectory to be identified is generated by adding deviation disturbance to the data of No. 1. The reentry reachability evaluation algorithm designed in this article identifies itself with disturbance, and the correctness of the algorithm has been verified.
Determining the strategy of matching identification. Identifying once every second, mainly according to the second strategy of Section 2.3.
The evaluation results of reentry vehicle reachability are as follows: before the first reentry phase, take the trajectory number 36 as the prediction result, and take the last point of the trajectory as the landing time and landing point prediction; after the first reentry phase, take the trajectory number 1 as the prediction result, and take the last point of the trajectory as the landing time and landing point of the landing point prediction.
4.2.2 Identifying once every accumulated 10 s data
In the current test environment, the total calculation takes 22.36 s and is evaluated 200 times. Table 4 gives the calculation results of fuzzy relative entropy and closeness calculated with 300 groups of data in the feature database every 10 s of the trajectory to be identified.
Results of recognition once every 10 s
| Serial number | 1–10 s | 1–20 s | 1–30 s | 1–2,000 s |
|---|---|---|---|---|
| Fuzzy relative entropy | 0.04306 × 10−4 | 0.10183 × 10−4 | 0.40296 × 10−4 | 0.79562 × 10−4 |
| Match result | Group 180 | Group 180 | Group 36 | Group 1 |
| Maximum and minimum closeness | 0.99993 | 0.99993 | 0.99992 | 0.99978 |
| Match Result | Group 153 | Group 15 | Group 153 | Group 1 |
Figure 8 shows the recognition result based on the relative entropy between the data per 10 s and the feature database, that is, the matching sequence number. Figure 9 makes frequency statistics of the recognized sequence number. As shown in Figures 8 and 9, the two serial numbers that appear more frequently are 117 times for serial number 36 and 73 times for serial number 1.
Identification results of trajectory matching sequence number based on fuzzy relative entropy.
Identification results of matching sequence number frequency statistics based on fuzzy relative entropy.
Figure 10 shows the recognition result based on the closeness between the data and the feature database per 10 s, that is, the matching sequence number. Figure 11 makes frequency statistics of the recognized sequence number. Figures 10 and 11 show the two sequence numbers that appear more frequently are 73 times for serial number 1 and 71 times for serial number 36.
Identification results of trajectory matching sequence number based on fuzzy closeness.
Identification results of matching sequence number frequency statistics based on fuzzy closeness.
Based on the aforementioned calculation results, taking the separation point of 5,000 km capsule as the time zero, the data accumulated every 10 s are matched with the feature database, and a total of about 200 calculations are made. It can be seen that:
When t < 1,200 s, the reentry vehicle is located before the first reentry phase, and the “match” recognition result is serial number 36.
When t > 1,200 s, the reentry vehicle is located after the first reentry phase, and the “match” recognition result is serial number 1.
Determining the strategy of matching identification. Identify once every 10 s, mainly according to the first strategy of Section 2.3.
The reachability evaluation result of the reentry vehicle is the same as that in Section 3.2.1. The result shows that when there are enough members in the special identification data set, it can better reflect its essential characteristics. According to the demand, the identification data window can be reasonably selected to reduce the calculation energy consumption and improve the identification probability and evaluation rate at the same time.
4.2.3 Trajectory deviation between to be identified and identification results
Figures 12 and 13 show the trajectory deviation relationship between four groups of data of serial numbers 1, 36, 106, and the trajectory to be identified. It also shows that the matching degree value between the reentry vehicle and number 36 trajectory is high before the first reentry phase, and the matching degree value between the reentry vehicle and number 1 trajectory is high after the first reentry phase. The simulation recognition result is correct.
Height deviation.
Sub-satellite point deviation.
5 Conclusion
The real-time reachability evaluation method of reentry vehicles based on fuzzy learning is presented in this article. Compared with the traditional real-time trajectory and landing point prediction method of reentry vehicles based on high fidelity reentry and guidance model, this method only needs the sample trajectory data, which is generated by the Monte Carlo method, and based on the calculation of fuzzy relative entropy and closeness, then it can realize the real-time reachability evaluation of reentry vehicle. The accuracy and evaluation speed can adapt to the needs for the rapid response to real-time tasks. It provides a new idea and way for reentry trajectory prediction, landing point prediction and reachability evaluation of non-cooperative reentry vehicles. It is suitable for the TT&C equipment tracking and guidance with abnormal tracking conditions.
In order to test the calculation of energy consumption and the accuracy of analysis and evaluation, the sliding recognition time window is set in two ways, once per second and once every accumulated 10 s data in this article. For the skip reentry spacecraft, the recognition window is set reasonably, and the recognition results of the two windows are evaluated. After analysis, the recognition of the two time windows requires 10.72 s (2,000 evaluations in total) and 22.36 s (200 evaluations in total), respectively. Obviously, the calculation speed is faster with evaluating once per second. However, through the comparison between Figures 7 and 9, it can be found that there are more similar tracks identified in the tracking database with the evaluation result of once per second. Accumulating 10 s data, with the increase of feature information, that is, when there are many members in the data set to be identified, the most similar tracks can be better identified, and the accuracy is also improved. Of course, the recognition window can also be longer, but we will be anxious to know which reentry track the vehicle will take in real time. Increasing the measurement data accumulation time will not improve the recognition accuracy qualitatively. However, the accuracy and calculation energy consumption of 10 s recognition can suit the requirements of real-time tasks. Therefore, it is suggested to select it reasonably according to the display requirements of real-time monitoring system in practical application.
Moreover, the entry phase when the reentry vehicle enters the black barrier and cannot obtain the measurement data is not simulated during the simulation, but when the recognition window is selected once for 10 s, Figures 8 and 10 show that the recognition and evaluation result has already converged to one track, which does not affect the presentation of the evaluation result.
Of course, in order to improve the accuracy of this method for reachability evaluation and landing point prediction, we need to rely on a large number of trajectory feature samples as the basic identification database. Subsequently, the intelligent machine learning method can be used to analyze and learn the trajectory information of Monte Carlo analysis in time domain and frequency domain, so as to improve its trajectory and landing point prediction ability.
Acknowledgments
The author would like to thank Ke Xu of the State Key Laboratory of Astronautic Dynamics and Wei Zhang of the State Key Laboratory of Spacecraft In-Orbit Fault Diagnosis and Maintenance for their support and interest in this research.
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Funding information: This research was funded by the National Natural Science Foundation of China (Grant No. 11772356, No. U21B2050).
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Author contributions: All authors have accepted responsibility for the entire content of this manuscript and approved its submission.
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Conflict of interest: Authors state no conflict of interest.
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