International Journal of Aerospace Engineering

Volume 2019, Article ID 8502870, 17 pages

https://doi.org/10.1155/2019/8502870

## Optimal Midcourse Guidance Algorithm for Exoatmospheric Interception Using Analytical Gradients

School of Astronautics, Beihang University, Xueyuan Road No. 37, Haidian District, Beijing, China

Correspondence should be addressed to Liang Yang; moc.liamtoh@aaub.gnailgnay

Received 2 June 2019; Accepted 23 August 2019; Published 16 September 2019

Academic Editor: Maj D. Mirmirani

Copyright © 2019 Wenhao Du 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

This paper is aimed at providing a semianalytical method to solve the optimal exoatmospheric interception problem with the minimum fuel consumption. A nonlinear programming (NLP) problem with the minimum velocity increment, which involves Lambert’s problem with unspecified time-of-flight, is firstly formulated. Then, a set of Karush-Kuhn-Tucker conditions and the Jacobian matrix corresponding to those conditions are derived in an analytical manner, even though the derivatives are mathematically complicated and computationally onerous. Therefore, the Newton-Raphson method can be used to efficiently solve this problem. To further decrease computational cost, a near-optimal initialization method reducing the dimension of the search space is presented to provide a better initial guess. The performance of the proposed method is assessed by numerical experiments and comparison with other methods. The results show that this method is not only of high computational efficiency and accuracy but also applicable to onboard guidance.

#### 1. Introduction

Exoatmospheric missile defense system is a generic term conveying an interceptor designed to destroy any ballistic targets delivering nuclear, chemical, biological, or conventional warheads outside the atmosphere. It is of great significance for national security, and therefore, the majority of powerful countries such as the United States, Russia, and China have developed their own missile defense systems [1–3]. In the midcourse, both interceptor and target follow Keplerian orbits and are in high-speed interception engagement with the relative velocity greater than 10 km/s. Thus, “hit-to-kill” is the only valid way of completely destroying invading targets, which brings forward high requirements on the performance of guidance system [4–6]. Actually, exoatmospheric midcourse guidance is a process to calculate an initial velocity vector to ensure a successful impact with zero miss, which theoretically can be categorized into a type of Lambert’ problem [7–9]. Our purpose in this paper is to develop a guidance algorithm from the point of solving Lambert’s problem efficiently.

In 1809, Gauss [10] developed the first iterative process to solve Lambert’s problem, but it suffers from the singularity problem. Battin et al. [11, 12] improved Gauss’s method and proposed an elegant algorithm that efficaciously removed the singularity. Their method is based on a new transformation and a new iteration function to achieve fast convergence. It is mathematically elegant and practically implementable, but its derivation is founded on the complicated geometric properties of conic sections. Avanzini [13] presented an intuitive and simplified version of Battin’s method by parametrizing admissible orbits in terms of the transverse eccentricity component. An iterative method based on Householder’s root solver was designed in [14], where the first and second derivatives of the time-of-flight (TOF) equation are analytically derived to increase the rate of convergence. In [15], Levi-Civita regularization was applied to Lambert’s problem, and a Newton-Raphson method in combination with safety checks was adopted to achieve both speed and robustness. These works have successfully promoted the application of Lambert’s problem in the space science community. Additionally, Nelson and Zarchan’s efficient method [16, 17] was popular in defense applications [18]. This method regarded the flight path angle as an iterative variable and employed a secant method combined with boundary values to speed up the routine. Obviously, the aforementioned researchers focused on numerical procedures to deal with Lambert’s problem by searching a variable iteratively, because it is impossible to derive an analytical solution due to transcendental equations. Therefore, it is still of great significance to speed up the convergence of iterative algorithms, especially for onboard autonomous scenarios requiring computation [19–22].

Moreover, for some situations where the issue of convergence time is critical, two types of methods are commonly used to improve convergence rate. The first one focuses on analytical gradients [23]. Bate et al. [24] presented an effective algorithm named as the -iteration method, where the slope of the TOF with respect to the semilatus rectum could be expressed in an analytical manner. Ahn and Lee [25] analytically derived the gradient of the flight path angle with respect to the TOF from the conditions of the two-point orbital boundary value problem, and their method was fully verified by comprehensive numerical experiments. Furthermore, another practical way to accelerate the convergence of iteration process is improving the initial guess. In 1990, Gooding [26] made pioneering efforts in this field and used Halley’s iterations to set the initial guess, thereby achieving fast convergence properties. Then, Izzo [27] improved Gooding’s method by inverting the linear approximation to the TOF curves, which significantly reduced the computational complexity. Arora and Russell [28] proposed a fast and robust algorithm to solve multiple revolution Lambert’s problem, in which a new geometry parameter was introduced to simplify the TOF equation, and therefore, an accurate initial guess can be provided for rapid root-solving. In [29], Ahn and Lee also developed a method using a two-dimensional table involving the geometric characteristic and the normalized TOF to interpolate an initial guess, whose efficiency was validated by various numerical experiments. It should be noted that the aforementioned methods assume that the TOF is required to be a given constant.

The optimal exoatmospheric interception problem with the minimum fuel consumption is to find the optimal velocity increment to eliminate interception error and retain enough energy for a successful collision. This kind of problem is abbreviated as the minimum velocity increment problem (MVIP) in this paper. It is worth noting that the aforementioned algorithms cannot be directly applied to the MVIP, because the TOF should be regarded as an unspecified parameter in the optimization process [30–32]. Commonly, an additional single-variable algorithm needs to be developed to repeatedly solve Lambert’s problem so as to determine the optimal TOF. However, this type of method has high computational cost because of its overreliance on a two-level iterative process. For onboard autonomous scenarios, the computation time for generating commands is very critical. Therefore, it is necessary to develop a new algorithm for solving the MVIP in an inexpensive manner.

The core objective of this paper is to provide a semianalytical method for solving the MVIP using analytical gradients, and it has potential advantage in onboard guidance. The main contributions of this paper are stated as follows. Firstly, a typically nonlinear programming problem (NLP) with the minimum velocity increment is formulated, in which two transcendental equations characterizing Lambert’s problem and Keplerian motion are regarded as equality constraints in order to reduce computational difficulties. Secondly, a set of Karush-Kuhn-Tucker (KKT) conditions for this NLP problem is derived to determine the minimum velocity increment. To speed up convergence, the Jacobian matrix is successfully derived on each parameter in an analytical way, even though the expressions are mathematically complicated and computationally onerous. Accordingly, root-finding algorithms such as the Newton-Raphson method can be effectively used to find the optimal solution with a high accuracy. Thirdly, it is found that the variable, semilatus rectum , is insensitive to the variation of TOF. A near-optimal initialization method, which considers as a constant and reduces the dimension of the search space, is developed to provide a better initial guess. That further accelerates the rate of convergence. In a word, the proposed method is attractive from the point of view of low computational cost, high accuracy, and the fact that it has potential to be used as a baseline algorithm for onboard guidance. Finally, the performance of this method is sufficiently verified by comprehensive experiments and comparison with the existing methods.

The rest of the paper is organized as follows. Section 2 provides a detailed description of the MVIP, which involves orbital motion and Lambert’s problem, respectively. Section 3 introduces the method used to solve the problem in this paper, which is composed of the KKT conditions, analytical gradients, initialization method, and the procedure of implementation. Section 4 presents the numerical results to demonstrate the performance of the method. Finally, conclusions are given in Section 5.

#### 2. Problem Formulation

The engagement geometry of the MVIP is illustrated in Figure 1(a). At the moment , the target performs an orbital maneuver and its trajectory is transferred from the Keplerian orbit orb1 to a new orbit orb2. The exoatmospheric kill vehicle (EKV), which is a representative interceptor designed for exoatmospheric interception, needs to change its velocity vector from to so as to score a direct hit on the predicted intercept point (PIP) PIP_{2}. Commonly, a PIP can be uniquely determined by orbital equations if the TOF is given. Then, the midcourse guidance problem can be formulated as the determination of an orbit having a specified transfer time and connecting two position vectors, which is known as Lambert’s problem. Actually, there must be an optimal TOF corresponding to the minimum velocity increment , where the symbol “” denotes the Euclidean norm of vector. To demonstrate the MVIP more clearly, a schematic representation for the correlation between Keplerian motion and Lambert’s problem is presented in Figure 1(b).