International Journal of Aerospace Engineering

Volume 2019, Article ID 5316396, 23 pages

https://doi.org/10.1155/2019/5316396

## An Analytical Method for Calculating Short-Term Space Interception Probability

Correspondence should be addressed to Wenzhe Ding; moc.liamg@321ehznewgnid

Received 9 July 2019; Revised 24 August 2019; Accepted 5 September 2019; Published 3 November 2019

Academic Editor: Paolo Gasbarri

Copyright © 2019 Wenzhe Ding et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

For small unguided space interceptors, the interception probability is an important index to evaluate their strike capability. However, the Monte Carlo (MC) based simulation method not only consumes a lot of computation time but also theoretically lacks interpretability. In this regard, this paper proposes an analytical method that can quickly and accurately calculate the short-term space interception probability. Firstly, by considering the effect of perturbation force on spacecraft as the effect of external force acceleration, the analytical calculation formula of the short-term state error covariance propagation of space target and interceptor is deduced. Next, by projecting the state error and rotating the coordinate system, the joint error distribution of the target and the interceptor in the calculation coordinate system at the time of closest approach (TCA) is obtained. Thereby convert the calculation of the space interception probability into the integral of the 2-dimensional probability density function in the circular domain. Then, the Laplace transform and Taylor expansion are used to obtain the exact power series expression and the maximum truncation error of the integral calculation, and the analytical calculation of the short-term space interception probability is realized. Finally, the effectiveness of the proposed method is verified by a simulation example. The proposed method can directly calculate the space interception probability according to the initial state error distribution of the target and interceptor, and the whole calculation process does not contain double integral operation. The proposed method has high computational efficiency, is suitable for on-orbit calculation, and provides effective support for the rapid evaluation of the strike capability of the space interceptor.

#### 1. Introduction

The interceptor can be mounted on a satellite platform as a payload and to perform orbital interceptor missions such as space debris removal. Small nonguided interceptors (such as projectiles) have the advantages of lightweight and can be fired continuously, but due to the lack of guidance, it is necessary to make a reasonable assessment of their strike capability. As an important index to evaluate the striking ability of small unguided interceptors, interception probability plays an important role in the design, optimization, and performance evaluation of interceptors. Literature [1] systematically compares the calculation methods of circular error probability (CEP). Literature [2] applied the bootstrap method to resample small samples and calculated the cyclic error caused by repeated sampling, so as to analyze the influence of various error sources on the shooting accuracy of electromagnetic railgun (EMRG). In literature [3], the ground strike accuracy of EMRG is analyzed with Latin hypercube design (LHD). However, all the above researches are based on ground tests. Due to the unique motion of the interceptor in space, the above research methods are no longer applicable.

When calculating the probability of intercept in space, the initial error of interceptor should be analyzed first. The source of the interceptor’s initial error can be classified into two categories. One is the position error of interceptor caused by the position error of the satellite platform. The other is the interceptor velocity error caused by satellite velocity error, attitude adjustment error, launch velocity error, and other factors. Therefore, the initial error of the interceptor can be simplified into the state error of the interceptor after launching in orbit. Different from the ground strike test, the initial state of the target will have some initial errors due to the limited ground observation ability. Due to the influence of gravity and various kinds of perturbation, the initial states of the target and the interceptor will change continuously throughout the interception process. Therefore, it is necessary to study the distribution of state errors at TCA according to the initial state error distribution of the target and the interceptor. Then, when calculating the space interception probability, we first need to study the uncertainty propagation problem in orbit mechanics.

As for the uncertainty propagation problem in orbital mechanics, there is much relevant research literature [4–6]. For the linear propagation process of error, the literature [7] used linear covariance analysis to study the orbit error propagation. When the error propagation time is too long and the system has strong nonlinearity, the literature [8, 9] uses Monte Carlo to study the error propagation problem. In order to reduce the computational cost of MC, some related techniques for error propagation have been proposed, such as unscented transformation (UT) [10], polynomial chaos (PC) [11, 12], differential algebraic (DA) [13, 14], and Gaussian mixture model (GMM) [15–17]. For the short-term space interception task, the propagation time of the state errors of the target and the interceptor is short during the whole interception process. After the propagation, the orbital deviations of both are small, so the Tschauner-Hempel (TH) equation can be used to describe the relative motion between the “mean orbit” and the “deviation orbit” of the target and the interceptor, so as to obtain the analytical solution of the target and interceptor state error propagation.

After error propagation, target and interceptor reach the closest position at the TCA. At this point, there is a high relative speed between the target and the interceptor and the overall contact time between them is extremely short during the encounter. Therefore, short-term encounter model can be used to define the encounter process between the target and the interceptor [18]. The motion trajectories between the target and the interceptor in the encounter process are assumed to be straight lines, and the uncertainty of the velocities of the target and the interceptor in the encounter process is ignored. So, the probability calculation can be converted to solve the integral of the two-dimensional probability density function in the encounter plane. For this double integral, the literature [19] uses the Monte Carlo method for calculation. In order to speed up the calculation efficiency, Foster and Estes [20], Patera [21], and Alfano [22] proposed different numerical integration models, respectively. Besides, Chan [23] proposed an analytical method to approximate the integral formula. The method approximates the integral domain. When the probability density function (PDF) is close to the circular distribution, it can well approximate the actual probability value. However, when the ratio of the semiminor axis length of PDF elliptic distribution is large, the probability calculation will be biased. Compared with Chan’s method, the analytical calculation method based on the Laplace transform [24] avoids the approximation of the integral domain and is more suitable for the calculation of interception probability.

In summary, the calculation of space interception probability is a complex multistage problem. In this issue, uncertain propagation calculation of orbit and collision probability calculation is involved. Although an exact solution to the space interception probability can be obtained by a large number of Monte Carlo simulations, it often requires a large amount of computational cost and time cost. In this regard, this paper combines the characteristics of short-term space interception task to define “short-term” from two aspects: The first is that the overall time of the interception process is short. The second is that the overall contact time between the target and the interceptor is short. Based on the above definition, this paper presents an analytical method to calculate the short-time space interception probability. This method divides the space interception task into two stages: error propagation and instantaneous collision, and realizes the analytical calculation of the short-term space interception probability by modeling the two phases separately. In the error propagation phase, we establish the state error uncertainty propagation model for the target and interceptor based on the TH equation. By considering the perturbation effect as the external force acceleration, the analytical calculation formula of the uncertainty propagation is obtained. By formula, we can directly calculate the error covariance matrices of the state of the target and interceptor at the TCA. In the transient collision phase, we unify the state vectors and state errors of the target and interceptor at the TCA into the calculation coordinate system, thereby transforming the calculation of the interception probability from the 3-dimensional problem to the 2-dimensional problem. The Laplace transform and Taylor expansion are used to obtain the expression of the power series of the space interception probability, which realizes the accurate analytical calculation of the space interception probability. The main innovation of this paper is to provide a precise method to solve short-term space interception probability in a pure analytic way. The advantage of this method is that it combines the uncertainty propagation calculation of orbit with the space collision probability calculation. Through the multistage modeling, an accurate analytical method for calculating the short-term space interception probability is given. Using this method, the short-term space interception probability can be calculated directly according to the initial state distribution of the target and the interceptor. The Laplace transform method used in the modeling avoids the double integral operation required in the final probability calculation, which makes the method more suitable for on-orbit calculation and provides the effective support for the rapid evaluation of the interception capability of the space interceptor.

#### 2. State Transition Matrix and Analytical Solution for Uncertainty Propagation Calculation

For the space interception probability problem, the error uncertainty propagation of target orbit and interceptor orbit should be considered first. In this section, the relative motion theory is used to study the error propagation of the two orbits. The mean orbit and deviation orbit of the spacecraft are regarded as two close space objects of “real” and “virtual,” respectively, so the propagation rule of orbital error can be described by relative motion theory.

##### 2.1. Relative Motion Model and State Transition Matrix under the Elliptical Orbit

In this section, Local Vertical Local Horizontal (LVLH) coordinate system of spacecraft mean orbit is used to establish the relative motion equation. The origin of the coordinate system lies at the center of mass of the “real object.” The -axis points the center of the mass of the “real object” along the center of the earth. The -axis points to the direction of movement of the “real object” and is perpendicular to the -axis in the mean orbital plane of the spacecraft. The -axis satisfies the right-hand rule. Figure 1 shows the LVLH coordinate system of the spacecraft mean orbit.