Complexity

Volume 2018, Article ID 7470823, 19 pages

https://doi.org/10.1155/2018/7470823

## Integrated Estimation/Guidance Law against Exoatmospheric Maneuvering Targets

School of Automation, National Key Laboratory of Science and Technology on Multispectral Information Processing, Huazhong University of Science and Technology, Wuhan, China

Correspondence should be addressed to Lei Liu; nc.ude.tsuh@ieluil

Received 5 June 2018; Revised 10 November 2018; Accepted 21 November 2018; Published 10 December 2018

Guest Editor: Zhaojie Ju

Copyright © 2018 Mao Su 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

An integrated guidance integrated estimation/guidance law is designed for exoatmospheric interceptors equipped with divert thrusters and optical seekers to intercept maneuvering targets. This paper considers an angles-only guidance problem for exoatmospheric maneuvering targets. A bounded differential game-based guidance law is derived against maneuvering targets using zero-effort-miss (ZEM). Estimators based the extended Kalman filter (EKF) and the unscented Kalman filter (UKF) are designed to estimate LOS rates that are contaminated by noise and target maneuver. Furthermore, to improve the observability of the range, an observability enhancement differential game guidance law is derived. The guidance law and the estimator are integrated together in the guidance loop. The proposed integrated estimation/guidance law has been tested in several three-dimensional nonlinear interception scenarios. Numerical simulations on a set of Monte-Carlo simulations prove the validity and superiority of the proposed guidance law in hit-to-kill interception.

#### 1. Introduction

The problem of exoatmospheric interception has been studied for decades. Clearly, the interceptor must hit and kill the target at the end of the terminal phase. Both the midcourse and terminal guidance phases take place when the interceptor is outside the atmosphere [1, 2]. The major difference between these two phases is that the guidance law in the midcourse phase is executed using a ground-based long-range radar (LRR) station, and the interceptor does not lock-on the target, while in the terminal phase the interceptor uses an onboard seeker to guide itself after locking on to the target. That is to say, the patterns of getting information from the target are different, so the guidance laws are different.

Is there a single guidance law that can be used during both phases? In [3], the authors discussed this problem. In the terminal phase, the relative velocity is large, while the acceleration is relatively small. This generates fragility of the interception with respect to the LOS rates. For this problem, the midcourse law directed from the ground station must reduce LOS rates as much as possible within a required range.

An exoatmospheric interceptor is usually equipped with a strapdown optical seeker that can measure only LOS angles [4]; information relating to relative range and velocity is not measured. Even more concerning, the optical seeker’s measurements of LOS angles are nearly always corrupted by noise, resulting in inaccuracies in the LOS rate information. Using data from the ground-based station during the midcourse phase, the interceptor can get information on the position, velocity and even acceleration of the target. The more information needed, the more advanced the guidance laws should be that are used. For the reasons mentioned above, the designs of the guidance laws during the two phases are different. In [3], the interceptor uses a thrust vector control (TVC) system to adjust its flight path during the midcourse phase. The TVC system can provide axial acceleration for the interceptor. Because of the burning time limit, however, the TVC system can work only during a certain period of time, after which axial acceleration is no longer available, and the interceptor can use only a Divert Control System (DCS) [5] to steer itself in order to hit the target. Unlike TVC, DCS does not provide axial acceleration. The difference between TVC and DCS results in corresponding differences in the guidance laws applied. The “terminal” part of the terminal phase is the so-called “endgame” phase [6], and in this phase the interceptor must rely on the information from the onboard seeker for guidance to hit the target.

Differential game theory is a natural setup to discuss pursuit-evasion problems [7]. The most common pursuit-and-evasion game, called a “zero-sum differential game,” deals with two entities in relation to terminal cost function. In generating guidance laws, it is common practice to linearize with respect to a collision course, which implies linearized kinematics. There are two versions of the game [8]: the first is the “linear quadratic differential game” (LQDG), and the second is the “norm differential game” (NDG). In LQDG, the controls are unbounded and the cost function is the weighted sum of three quadratic terms: the square of the miss distance and two penalty terms that represent the integrals of the respective control energy of the players. The optimal solution of this formulation is linear. In NDG, on the other hand, the controls are hard bound and the cost is purely terminal to account for imposed on the miss distance. Contrary to LQDG, the optimal strategies in NDG are nonlinear; at a certain time before termination, the guidance law becomes pure bang-bang. In exoatmospheric interception, the divert thrusters are “on-off” devices, so the guidance commands from NDG are suitable for this kind of actuator. The guaranteed miss distance [9] for an interceptor must be very small for hit-to-kill, especially for evasive maneuvers. NDG uses the miss distance at terminal time as its cost function, another problem about time-to-go that has been studied by Shaul and Sergey [3, 10, 11]. They present an exoatmospheric guidance law based on a bounded differential game and a fourth-order time-to-go calculated while both the interceptor and the target perform the optimal strategies in the game, the guidance commands are not chatting and the directions of the optimal acceleration are constant. However, this kind of guidance is not suitable for an interceptor equipped with lateral thrust, and a first-pass phenomenon will occur if the initial condition is not perfect.

Another problem that must be considered when using an onboard optical seeker in the terminal phase is angles-only guidance [4, 5, 12–14]. The implementation of advanced laws such as augmented proportional navigation (APN) and optimal guidance law (OGL) requires that the guidance system be provided with information about the time-to-go () [15]. In the case of angles-only measurements, however, the range and range rate cannot be measured directly; thus, needs to be estimated with noise, which will lead to decreased performance of the advanced laws, making bearings-only or angles-only estimation another problem in exoatmospheric interception [4, 6, 13]. A hybrid Kalman filter has been presented in [4, 6] that bused both Cartesian and spherical states to minimize estimation errors. The relative position and velocity estimations were propagated using Cartesian coordinates, and the measurement updates used spherical coordinates. In [13], a differential game guidance law with bearings-only measurements was derived in two-dimensional interception, an estimator based on an EKF was designed in the guidance loop, and a deterministic and a stochastic option were presented in the paper. An optimal guidance-to-collision law for an accelerating exoatmospheric interceptor as studied in [14], where the guidance law was based on LQDG, and EKF was used as the estimator in the guidance loop.

An integrated estimation/guidance (IEG) algorithm has been presented for maneuvering target interception [16, 17]. These guidance laws are designed by taking into consideration the estimation-delay of the moving target. A ZEM-based integrated estimation and guidance law for the interceptor in the endoatmosphere has been presented [18, 19], in which the ZEM components and time-to-go are estimated in the loop, and the estimation and guidance work in unison. This IEG approach has been found to be very effective in engaging both conventional maneuvering aircraft targets as well as incoming high speed ballistic missiles. In [20, 21], an IEG strategy is proposed that combines an interactive multiple model (IMM) estimator with a differential game guidance law (DGGL) for a realistically modeled seeker-less interceptor. The interceptor is not equipped with a seeker, so target information comes from the ground-based station and guidance commands are transmitted from the station to the interceptor using three-point-guidance.

In this paper, we study a maneuvering target intercepting problem for an exoatmospheric interceptor equipped with an angles-only measurement seeker and lateral thrusters. An angles-only measurements guidance problem is considered for an exoatmospheric maneuvering target that performs a bang-bang evasion maneuver. To solve this problem, an integrated estimation/guidance law which relied on an angles-only nonlinear filter is derived; furthermore, the traditional differential game guidance law is modified to enhance the observability of the range. With passive measurements, nonlinear filters with designs based on EKF and UKF were used to estimate LOS angles, LOS rates, and target maneuver acceleration for both nonmaneuvering and maneuvering targets. The integrated estimation/guidance law combines the estimator and guidance law in one and is effective for exoatmospheric maneuvering target interception. The paper is organized as follows. Section 2 presents the engagement formulation used for guidance law derivation and simulations. A guidance law based on a bounded differential game is discussed in Section 3. In Section 4, an angles-only estimation problem is studied. The system and measurement model in spherical coordinates is derived, and two nonlinear filters, EKF and UKF, are used to design the estimator. The integrated estimation/guidance law is presented in Section 5. Section 6 presents the nonlinear simulation results and is followed by conclusions in Section 7.

#### 2. Problem Formulation

##### 2.1. Vector in Inertial Coordinates

Consider two players, an interceptor* M *and a target* T*, moving in an exoatmospheric space. Let , , , be the position and velocity vectors of the objects in inertial coordinates, respectively. Let and be the interceptor and the target acceleration vectors, respectively, and and be the gravitation vectors of the two players. Then,

Define the relative position and velocity vectors , . Then, assuming , the relative kinematic equations can be written as

In exoatmospheric interception, the acceleration is usually obtained by divert motors, the process that has dynamics. We assume that the dynamics can be represented by arbitrary-order linear equations.

where is the internal state vector of the acceleration in each axis in inertial frame with . is the acceleration command and assumed to be limited in magnitude,

The acceleration vector can be represents as .

Define the state vector where , define command vector , and thenwhere

With the dynamics and constraints above, we associate the terminal cost with the final time* t*_{f}. In particular,where* J* is a terminal cost to be minimized by and maximized by and

##### 2.2. Spherical Coordinates

It is convenient to use spherical coordinates to describe the motion of the two players by using seekers in the endgame. Considering the LOS spherical coordinates , where, is the relative distance, are the elevation and azimuth line-of-sight angles, and are their rates, respectively. In the inertial frame, the relative range is , then

Using the vector derivation method, the relative equations of LOS coordinates can be obtained. Then,where , are, respectively, line-of-sight rates for the LOS coordinates: , , . , , are the actual acceleration of the interceptor and target in the LOS frame, respectively. The engagement geometry and the relationships are shown in Figure 1.