International Journal of Aerospace Engineering

Volume 2019, Article ID 6827982, 14 pages

https://doi.org/10.1155/2019/6827982

## Research on Minimum Time Interception Problem with a Tangent Impulse under Relative Motion Models

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

Received 15 January 2019; Revised 30 March 2019; Accepted 17 April 2019; Published 5 August 2019

Academic Editor: Joseph Morlier

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

The minimum time interception problem with a tangent impulse whose direction is the same as the satellite’s velocity direction is studied based on the relative motion equations of elliptical orbits by the combination of analytical, numerical, and optimization methods. Firstly, the feasible domain of the true anomaly of the target under the fixed impulse point is given, and the interception solution is transformed into a univariate function only with respect to the target true anomaly by using the relative motion equation. On the basis of the above, the numerical solution of the function is obtained by the combination of incremental search and the false position method. Secondly, considering the initial drift when the impulse point is freely selected, the genetic algorithm-sequential quadratic programming (GA-SQP) combination optimization method is used to obtain the minimum time interception solution under the tangent impulse in a target motion cycle. Thirdly, under the high-precision orbit prediction (HPOP) model, the Nelder-Mead simplex method is used to optimize the impulse velocity and transfer time to obtain the accurate interception solution. Lastly, the effectiveness of the proposed method is verified by simulation examples.

#### 1. Introduction

For the orbital interception problem under the two-body model, the Lambert method can be used to solve it when the initial orbital elements of a satellite and a target are known [1]. When the impulse point and the interception point are given, the orbital transfer time can be obtained by the Kepler equation, and the initial velocity required for the orbital transfer can be solved by expressing the transfer time as a univariate function of other parameters [2, 3]. When the relative distance between the satellite and the target is small, the state transition matrix can be constructed to solve the initial velocity required for the orbital transfer. On the relative motion problem, if the target is running on a circular orbit, the Clohessy-Wiltshire (CW) equation can be used to describe the relative motion [4]. For the case where the target is running in an elliptical orbit, the Tschauner-Hempel (TH) equation can be used to describe the relative motion [5]. In 2002, Yamanaka and Ankersen obtained the state transition matrix described by the true anomaly by solving the TH equation [6]. Therefore, for the orbital interception problem in relative motion, the state transition matrix can be utilized to solve the impulse velocity required by the interceptor at the impulse moment.

For orbital interception tasks such as space debris removal, the tangent impulse has a simpler impulse direction, which makes the attitude adjustment of the satellite at the impulse point more convenient. So tangent impulse interception is a better interception method. For the tangent problem of coplanar elliptical orbits, Adamyan et al. solved the cotangent transfer orbit by the geometric method and obtained the analytical expression between the orbital parameters and the velocity vector [7]. Thompson et al. studied three types of tangent problems by using the Hodograph theory [8]. However, to ensure that the interceptor and the target have the same flight time, the above research does not apply to the problem of orbital interception. In 2012, Zhang et al. obtained the conditions for the existence of transfer solutions for three types of tangent orbits by using the relationship between the orbital semilatus rectum and the flight direction angle [9]. At the same time, the tangent impulse intercept problem with the minimum time when the target orbit is an ellipse is also studied [10]. Wang et al. studied this when the orbit was hyperbolic [11]. However, the above research contents are all based on absolute motion. For the unguided close-range interception problem requiring shorter interception time, it is necessary to study the tangent impulse interception with the minimum time under the condition of relative motion.

In addition to the above studies, there are many related studies on the tangent impulse orbit maneuver problem in recent years [12–15]. But for these studies, the models used are simplified models. Therefore, in order to reduce the impact of the perturbation on the intercepting orbit, it is necessary to further optimize the interception orbit. For the Lambert problem considering J2 perturbation, under the premise of setting the terminal precision, the state transition-sensitive matrix of the two-body model is often used to iteratively obtain the required initial velocity by the shooting method [16]. When the number of flight laps is vast, the homotopy method can be used to divide the entire time interval into small intervals so that the initial velocity can gradually converge [17]. However, for the high-precision extrapolation model adopted in this paper, the above shooting method is no longer applicable due to the lack of useful gradient information. At the same time, tangent interception limits the direction of impulse velocity, which makes it impossible to solve the interception problem by fixing the terminal position.

The minimum time interception problem can be classified as an optimization problem. Therefore, the corresponding optimization model can be established by combining different optimization indicators and solved by a direct method or indirect method. The direct method transforms the optimization problem of intercepting orbit into a nonlinear programming (NLP) problem and uses the optimization algorithm to solve it. In recent years, some scholars have used hybrid optimization algorithms to solve the single-impulse interception orbit optimization problem [18, 19]. Among them, GA, as a global optimization algorithm, is insensitive to initial values and has strong robustness, which can exhibit strong global search ability. The disadvantage is that the local search ability is weak and the result precision is low. Therefore, GA often provides initial values for gradient information-based optimization methods [20]. The SQP method is sensitive to the initial value, has a small convergence radius, and is easy to fall into the local optimum. But for the NLP problem, it can quickly converge to get a high-precision solution [21]. Therefore, combining GA and SQP is an effective method for solving orbit optimization problems.

Based on the above analysis, the paper studies the minimum time interception problem when the tangent impulse is used. First of all, the relative motion model based on an elliptic orbit is used to transform the interception solution into a univariate function only about the true anomaly of the target. Next, all the solutions in the feasible region are obtained by numerical iteration. Then, considering the initial drift segment, the GA-SQP combination optimization method is used to obtain the minimum time interception solution under the tangent impulse in a target motion cycle. Finally, in the absence of effective gradient information, the Nelder-Mead simplex method [22] is adopted to optimize the impulse velocity increment and the transfer time to obtain an accurate interception solution under the high-precision extrapolation model. The main innovation of this paper is to provide an accurate method to solve the interception solution of the minimum time tangent impulse under relative motion by combining analytical, numerical, and optimization methods. The advantages of this method are as follows: (1) Combine the GA and SQP method to optimize the impulse position, and then obtain the accurate minimum time tangent impulse interception solution without providing the initial value. (2) The Nelder-Mead simplex method adopted in the optimization process avoids the dependence on the gradient information. Under the condition that the target is relatively close to the satellite, the accurate solution under the high-precision HPOP model can be obtained just by the initial solution provided by the linear TH equation.

#### 2. Existence Condition of Tangent Impulse Interception Solution

For the tangent impulse interception problem of space targets, it is necessary to judge the existence of the interception solution of the satellite firstly at the impulse moment and give the feasible domain of the solution. In this regard, a detailed proof has been given in the literature [9, 10]. According to the conclusion in the literature [10], for the shooting point in Figure 1, there is a solution only when the interception point is located at .