This study aims to analyze four different telescope pointing correction models to verify the highest accuracy of the laser ranging telescope corrected by the back propagation (BP) neural network model optimized by the proposed genetic algorithm and Levenberg–Marquardt. In this process, first, the observation data of 95 stars are used to solve the coefficients of the four models, and then the pointing accuracy of the telescope corrected by those four models is verified by the detection results of 22 stars. The results indicate that the pointing accuracy of the telescope corrected by the three traditional pointing correction models, the mount model, the spherical harmonic function model, and the basic parameter model, reaches approximately 15 in. in the azimuth and ∼10 in. in the pitch; however, the BP neural network model optimized by the genetic algorithm and Levenberg–Marquardt has a pointing accuracy of 3.42 in. in the azimuth and 2.44 in. in the pitch. Finally, different space debris is detected by the telescope corrected by this model. The results show that the pointing accuracy of the telescope corrected by this model probably increases to nine times in the azimuth and three times in the pitch. The results of this study prove that the BP neural network model optimized by the genetic algorithm and Levenberg–Marquardt greatly increases the pointing accuracy of the telescope and thus significantly improves the success rate of space debris detection.

## I. INTRODUCTION

Space debris is the space junk generated by human beings in space during space activities, and it mainly comes from an invalid spacecraft, the last stage arrow of a carrier rocket, and the debris of spacecraft disintegrating in the orbit. According to incomplete statistics, there are ∼17 000 space debris with a radar cross section of >0.1 m^{2} in space and hundreds of millions of space debris with a radar cross section of <0.1 m^{2}. The space debris mainly distributes in the geostationary earth orbit and low earth orbit. Their existence seriously affects the safety of a spacecraft in orbit.^{1,2} Therefore, debris monitoring has attracted significant attention worldwide. Even though geodetic technologies have the highest measurement accuracy, laser ranging technology is widely used to detect space debris. At present, there are >50 global laser observation stations and a global satellite laser ranging measurement and tracking network that have been formed, and both of which are of great importance to the monitoring of space debris.

Laser ranging technology is one of the most accurate space technologies in space target tracking technology, and single ranging accuracy of cooperative targets such as in-orbit spacecraft and the noncooperative targets such as space debris can reach centimeter level and it is developing into the millimeter level.^{3,4} However, in the process of space debris detection by laser ranging technology, because of the small size, poor prediction accuracy, and no reflection prism on the surface of space debris, it belongs to a noncooperative target,^{5} making the pointing position of the telescope deviate from the actual position of the space debris, which prevents laser ranging systems from accurately detecting the space debris. Therefore, the precise pointing and guidance of the laser ranging system become the key to the accurate detection of space debris.

In view of the low pointing accuracy of the telescope of the laser ranging system at present, the method of correcting the telescope pointing error of the laser ranging system was proposed by Scholars to improve the pointing accuracy of the telescope. One is to improve the pointing accuracy of the telescope by improving the processing and assembly accuracy of the hardware equipment. The second is to establish a pointing error correction model to improve the pointing accuracy of the telescope.^{6,7} Since the first method is time-consuming and laborious and the accuracy improvement is limited, the second method is widely adopted to improve the pointing accuracy of the telescope in the laser ranging observation stations. The traditional telescope pointing error correction models include the spherical harmonic function model, the basic parameter model, and the mount model; among them, the spherical harmonic function model is simple in form and can be used to fit various errors; however, the correlation among the model coefficients is relatively large, the parameters have no actual physical meaning, and the model is not stable enough. In contrast, the basic parameter model and the mount model are relatively stable, and these two types of models are built on the basis of determining the interrelationship among the physical factors causing the pointing errors. In addition, these two types of model functions are clear in form, the convergence is fast in the process of solving function, and each parameter has a definite physical meaning. However, it is difficult to comprehensively consider the factors affecting the pointing accuracy and accurately describe all types of error laws. In addition, the overall accuracy of the model is not very high.^{8,9} On this basis, different laser ranging telescope pointing error correction models have been proposed. Li *et al.*^{10} and Wang^{11} proposed the use of image processing to establish the telescope pointing error correction model; Zhu^{12} proposed the use of the back propagation (BP) neural network model to correct the telescope pointing error. Both methods were tested and applied in the mobile laser ranging station; Xu and Zhue^{13} proposed the use of radial basis function neural network for modeling; Prestage and Coulson^{14} and Lewis *et al.*^{15} added several corrected empirical terms based on the description of the main sources of error and predicted the trajectory of space debris in the airspace; Zhao^{16} used the quaternion method to model the pointing error of the radio telescope and analyzed and summarized the error items affecting the pointing accuracy of the radio telescope; Ukita *et al*.,^{17} Kanzawa *et al*.,^{18} Kong *et al*.,^{19} Gawronski *et al.*,^{20} Gu,^{21} and others investigated the perspective of the effect of the azimuth rail irregularity error on the pointing accuracy of the telescope and proposed the look-up table method to improve the pointing accuracy of the telescope. The accuracy of the telescope pointing correction model proposed can reach about 10 in.

The investigation on telescope pointing correction has reached a certain level. However, since the satellite prediction accuracy is generally in the decimeter level, and the space debris prediction accuracy is in the kilometer level, the current pointing accuracy of the model is still difficult to meet the high-accuracy pointing requirements of the telescopes required for space debris detection. Therefore, on the basis of previous studies, the requirements of the telescope pointing correction with fast and real-time performance are taken into consideration, and the BP neural network model optimized by the genetic algorithm (GA) and Levenberg–Marquardt (LM) is proposed to establish the telescope pointing error correction model, in which GA can determine the optimal initial weight matrix of the BP neural network to avoid the deficiency that the BP neural network is too sensitive to the initial parameters; LM accurately trains the BP neural network in the local solution space to ensure that the global optimal solution can be searched. Based on the 60 cm aperture laser ranging system of the Beijing Fangshan station, the experimental research and application of the real-time pointing correction method for the laser ranging system telescope were carried out. The results show that the application of the BP neural network model optimized by the GA and LM into the modeling of the telescope pointing error in the laser ranging system can greatly improve the pointing accuracy of the telescope and meet the requirements of the telescope pointing accuracy for detecting space debris in the laser ranging system.

## II. TELESCOPE POINTING ERROR CORRECTION MODEL

### A. Spherical harmonic function model

The physical parameters of the spherical harmonic function model have no actual significance. If the zonal harmonic terms are taken to the fourth order and the tesseral harmonic terms are taken to the first order, then the model expression is

where *A* and *H* represent the azimuth and pitch of the telescope, respectively, and $\Delta A$ and $\Delta H$ represent the azimuth and pitch deviations of the telescope, respectively.

### B. Basic parameter model

By analyzing the vertical axis error, horizontal axis error, and collimation axis error of the satellite laser ranging system, the expression of the basic parameter model can be obtained, and each coefficient of the model has actual physical meaning. The expression of the model is

### C. Mount model

The mount model was obtained by further analysis of the frame error of the laser ranging system based on the basic parameter model in Sec. II B and is actually an extension of the basic parameter model. The model contains 23 model coefficients, and each of which has actual physical meaning. The model expression is

### D. Optimized BP neural network model

The BP neural network is a multilayer feed forward neural network that propagates backward according to error and includes an input layer, an implicit layer, and an output layer. In this study, the three-layer BP neural network is used to model, i.e., including only one implicit layer. Although the traditional BP neural network can fit arbitrary nonlinear data,^{22–24} in this study, the model is applied to the pointing correction of the telescope of the laser ranging system oriented to space debris. Because of small size, poor prediction accuracy, and high speed of space debris, achieving the required accuracy for the traditional BP neural network model is difficult. Therefore, the traditional BP neural network was modified by the GA and LM, which can not only globally improve the traditional BP neural network, but also optimize the weights and thresholds of the network, in order to improve the accuracy of the model. The specific optimization process of the model is as follows:

First, GA is used to optimize the initialization parameters of the traditional BP neural network,^{25,26} characterized by strong adaptability to the searching space and can solve any nonlinear, discrete, high-dimensional, multipeak, or noisy data. This algorithm uses the information of the target function value to perform multipoint and multipath optimization. Therefore, GA is a global optimization method suitable for dealing with complex problems. Overall, in this study, this algorithm was applied to the traditional BP neural network model to optimize the weights and thresholds of the BP neural network. The GA optimization procedure for the traditional BP neural network mainly includes five steps:

Encoding

The BP neural network has more connection weights from the input layer to the implicit layer and from the implicit layer to the output layer. Therefore, it is reasonable to use the real number to be encoded. In this study, a three-layer BP neural network was used, and the encoding formula is as follows:

where *s* is the real number encoding and $th$ and $to$ are the thresholds of the implicit layer and the output layer, respectively.

Fitness function

For subsequent operations, GA usually uses the fitness function value as the genetic basis in the global searching process. For the BP neural network, the individual fitness function of the GA can be expressed as

where *F* is the individual fitness and $erri$ is the mean square error.

Selection operation

According to the fitness of each individual, some are selected from the optimal individuals for the next generation of inheritance. The probability that No. *i* individual is selected is

Interlace operation

Randomly select two individuals $a1$ and $a2$ from the first generation. According to formula (10), the two individuals are randomly conducted by interlace operation, and then two new individuals $a\u20321$ and $a\u20322$ are generated, wherein *b* is the random number taken between 0 and 1,

Mutation operation

The new individual is performed with the mutation operation according to the mutation probability, assuming that two new individuals are $a1\u2032\u2032$ and $a2\u2032\u2032$; therefore, the mutation operation is performed according to

Repeat steps (2)–(5) until it is not obvious for the continuous multigeneration changes of the required maximum evolutionary algebra or the fitness value of the optimal individual, then terminate the algorithm, followed by decoding the most adaptive individual solved, leading to the optimal initialization parameters of the BP neural network.

On the basis of optimizing the initialization parameters of the traditional BP neural network by GA, LM (Ref. 27) is used to optimize the shortcomings of local minimization and slow convergence of the traditional BP neural network. The process is as follows:

First, suppose $x=[WihthWhoto]$, then the adjustment amount calculation formula of the weights and thresholds is as follows:

where $\Delta x$ represents the adjustment amount of weights and thresholds and $\mu $ represents the damping factor. Through the adaptive adjustment of the progress factor *I*, when the algorithm starts, a small positive value of $\mu $ is taken, and the value of the progress factor ranges from 0 to 10. Besides,

Then, the adjustment formula of weights and thresholds is as follows:

where $xt$ is the weights and thresholds after *t* iteration and $xt+1$ is the new weights and thresholds after adjustment. The adjusted new weights and thresholds are used to return to formula (13) to continue the calculation until the error $ei$ is less than the allowable value of the training error. Here, the allowable value of the training error is the tolerance, which is taken as 0.001. Then, the data can be tested with the trained weights and thresholds. Figure 1 shows the overall data processing flow of the BP neural network model optimized by the GA and LM.

## III. ANALYSIS OF THE EXPERIMENTAL RESULTS

### A. Star data acquisition

The accuracy of the telescope pointing error correction model of the laser ranging system is directly related to the number of observed stars and the distribution of stars. Therefore, the distribution of the observed stars should be as even as possible in the hemispherical sky area of the observation station. The star data observed in this study were obtained through continuous observation by the laser ranging system in the Beijing Fangshan observation station. In the hemispherical sky above the observation station, a certain number of stars are selected from the Fifth Fundamental Catalog (FK5) for observation according to a certain distribution. During the observation process, if there is no star at the selected node or the stars cannot be observed due to building occlusion, the range can be enlarged by 5° in the azimuth and 2° in the pitch based on the original node, in order to search for the stars closest to the node in this area for observation. The star’s magnitude is in the range of 1–10, the pitch is changed from 20° to 80° at the intervals of 10°, and the azimuth is changed from 0° to 360° at the intervals of 20°. The detailed selection method is listed in Table I.

In order not to affect the investigation of the observation station, the sighting error and zero point error in the telescope axis error are not observed before the tracking of the stars; therefore, the errors of azimuth and elevation of the observation results are relatively large, especially the azimuth error increases greatly with increasing elevation angle. In addition, when the observed elevation angle is relatively low and the azimuth angle is around 0°, the telescope will be obscured by the surrounding buildings and cannot be used; therefore, the observation points should be encrypted by the nearby nodes. Finally, 95 stars were observed. The distributions of the stars and the deviation of the azimuth and pitch are shown in Figs. 2 and 3, respectively.

### B. Solving model coefficients

The original observation data of the obtained 95 stars are modeled by the spherical harmonic function model, the mount model, the basic parameter model, and the BP neural network model optimized by the GA and LM proposed in this study, and the coefficients of different models are solved. Among them, the coefficients of the spherical harmonic function model, the mount model, and the basic parameter model are all solved by the least squares method, and the coefficients of the BP neural network model optimized by the GA and LM are obtained through the continuous training of the network to obtain the optimal coefficient of the model. The number of nodes in the input layer, implicit layer, and output layer of the BP neural network is determined as 2, 10, and 2, respectively, and all the model coefficients are listed in Table II.

### C. Accuracy analysis

After solving the coefficients of all the models, 22 stars were observed again the next day besides the observed 95 stars, which were used to test the accuracy of all the models, i.e., the four models established were used to correct the pointing errors of 22 stars. The error in the distribution results of the azimuth and pitch of the stars corrected by the four models is shown in Figs. 4–7. The figures indicate that the corrected error distribution of those three traditional pointing correction models, the spherical harmonic function model, the mount model, and the basic parameter model, is still scattered, while that of the proposed BP neural network model optimized by the GA and LM is basically concentrated near zero, indicating that the accuracy of the BP neural network model optimized by the GA and LM is higher than those of the traditional models.

In order to compare the accuracy of the four models, the absolute errors of the telescope pointing toward 22 stars in the azimuth and pitch after correction of the four models were calculated. The results are listed in Table III.

Table III shows that the absolute error of the telescope pointing after correction by the spherical harmonic function model, the mount model, and basic parameter model is approximately 10 in., while that of the telescope pointing after the BP neural network model modification optimized by the GA and LM is not >10 in. In order to further compare the accuracy of the four models, statistical analysis was conducted on the mean error of the azimuth and pitch toward 22 stars after the four model modifications. The results are shown in Table IV, indicating that the accuracy of the telescope pointing correction models of the three traditional models, the spherical harmonic function model, the mount model, and the basic parameter model, reaches ∼15 in. in the azimuth, and ∼10 in. in the pitch; however, that of the BP neural network model optimized by the GA and LM in this study reaches 3.42 in. in the azimuth and 2.44 in. in the pitch. This indicates that the BP neural network model optimized by the proposed GA and LM further improves the pointing accuracy of the telescope, and it also overcomes the shortcomings of the laser ranging system in the detecting space debris because of small size, poor prediction accuracy, and high speed of space debris, improving the success rate of the space debris detection.

In order to compare the complexity of different algorithms, the calculation time of the spherical harmonic function model, the mount model, the basic parameter model, and the BP neural network model optimized by the GA and LM is counted. The computer platform used is Core i5 Dell personal computer, central processing unit is 3.20 GHz, random access memory is 16.0 GB, 100 times of calculations were conducted on each algorithm, and the average time of each algorithm is shown in Table V.

Table V shows that although the calculation time of the BP neural network model optimized by the GA and LM in this study is approximately 0.0004 s longer than those of the other three models, it can be neglected by the current computer. The most important thing is that the BP neural network model optimized by the GA and LM in this study is more accurate than the other three models, showing obvious improvement. Moreover, when the model is used to correct the telescope pointing, it can be calculated directly by using the trained model coefficients. It is easy to operate and has good real-time performance.

### D. Experimental analysis of space debris detection

In order to verify the pointing accuracy of the BP neural network model optimized by the GA and LM oriented to space debris detection in the laser ranging system, the established model was applied to the Beijing Fangshan laser ranging system, in which the space debris is tracked. The Beijing Fangshan laser ranging system uses an event timer to complete the acquisition of the measured distance. Event timer regards the main wave and echo as events. There is a precise clock in the timer, which can accurately record the time of each event. Then, according to the satellite’s predicted orbit, the laser pulse round-trip time intervals can be calculated. In addition, the time of main waves and echos can be identified by computer, the time differences between which is the measured time interval value, and then the observation distance of the satellite can be obtained. This technology exhibited the highest accuracy in all the detection methods of space debris. The technical parameters of the laser generator are listed in Table VI.

At present, the radar cross section of space debris is divided into three grades: (a) The radar cross section of space debris less than 0.1 m^{2} is defined as a small space debris, (b) the radar cross section of space debris between 0.1 and 1 m^{2} is defined as a medium space debris, and (c) the radar cross section of space debris larger than 1 m^{2} is defined as a large space debris. Ten space debris were tracked by the Beijing Fangshan laser ranging system. The information of these 10 space debris is shown in Table VII. Due to space limitations, only the tracking results of the three pieces of spaced debris are listed. The COSPAR IDs of the three space debris are 85009B, 98025E, and 12017E. The results of the pointing accuracy of those three pieces of space debris before and after the model correction are shown in Figs. 8–10, where (a) represents azimuth deviation and (b) represents pitch deviation.

The pointing accuracy of those three pieces of space debris before and after the model correction was statistically analyzed. The results are listed in Table VIII, indicating that the pointing accuracy of the laser ranging system oriented to space debris corrected by the BP neural network model optimized by the GA and LM proposed in this study can reach ∼5 in., which is ∼2 in. lower than the detection result of the star, mainly because the prediction of space debris is not accurate enough. The experimental results show that the pointing accuracy of the corrected model increases approximately nine times in the azimuth and about three times in the pitch, indicating that the BP neural network model optimized by the GA and LM proposed in this study greatly increases the pointing accuracy of the telescope oriented to space debris, thereby significantly improving the success rate of the laser ranging system for space debris detection.

## IV. CONCLUSION

Improving the pointing accuracy of the laser ranging system telescope oriented to space debris is a challenging task, because of small size, poor prediction accuracy, and no reflection prism on the surface of space debris, i.e., it belongs to a noncooperative target. Herein, the BP neural network model optimized by the GA and LM was proposed and successfully developed to correct the pointing error of the telescope, and the accuracy of the model was verified by the laser ranging system in the Beijing Fangshan observation station. The results show that not only the accuracy of this model is superior to that of the three traditional telescope models, but also this model overcomes the shortcomings of slow convergent speed and tending to be trapped in local optimum solution, making the pointing accuracy of the corrected telescope reaches 3.42 in. in the azimuth and 2.44 in. in the pitch. At the same time, the telescope corrected by this model was used to track the space debris, and the pointing accuracy probably increased to nine times in the azimuth and three times in the pitch compared to that before correction. The experimental results of this study prove the effectiveness of the BP neural network model optimized by the GA and LM, which is of great significance for the laser ranging system to detect space debris and provide valuable contribution in improving the success rate of space debris detection.

## ACKNOWLEDGMENT

This study was supported by the National Natural Science Foundation of China (NNSFC) (Grant No. 41774013).