International Journal of Aerospace Engineering

Volume 2018 (2018), Article ID 7403639, 13 pages

https://doi.org/10.1155/2018/7403639

## A Two-Phased Guidance Law for Impact Angle Control with Seeker’s Field-of-View Limit

Correspondence should be addressed to Jie Guo; nc.ude.tib@1891eijoug

Received 25 July 2017; Revised 27 October 2017; Accepted 12 November 2017; Published 25 March 2018

Academic Editor: Hikmat Asadov

Copyright © 2018 Haoqiang Zhang 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

A two-phased guidance problem with terminal impact angle constraints and seeker’s field-of-view limit is addressed in this paper for a missile against a nonmaneuvering incoming target. From the conventional PN guidance without any constraints, it is found that satisfying the impact angle constraint causes a more curved missile trajectory requiring a large look angle. To avoid the look angle exceeding the seeker’s physical limit, a two-phased look angle control guidance scheme with the terminal constraint is introduced. The PN-typed guidance law is designed for each guidance phase with a specific switching condition of line-of-sight. The proposed guidance law is comprised of two types of acceleration commands: the one in the initial phase which aims at controlling the missile’s look angle to reach the limit and the other for final phase which is produced by switching the navigation gain. The monotonicity of the line-of-sight angle and look angle is analyzed and proved to support the proposed method. To evaluate the specific navigation gains for both initial and final phases, the scaling coefficient between them is discussed by solving a quadratic equation with respect to the initial navigation gain. To avoid a great abrupt acceleration change at the switching instant, a minimum coefficient is chosen. Extensive simulations are performed to validate the efficiency of the proposed approach.

#### 1. Introduction

The impact angle control guidance laws (IACGL) have been widely studied for several years [1], for both stationary and moving targets. Practically, the engagement cannot be implemented successfully for a homing missile without the target information from a seeker. The seeker’s field-of-view (FOV) is a critical limit for missile engagement, especially when the missile utilizes a seeker with narrow FOV and intercepts a high-speed target with terminal angle constraints [2, 3]. In such a strict condition, the line-of-sight (LOS) violates the FOV limit more easily. Meanwhile, the missile’s detection process might be within a short period of time for moving targets.

A considerable number of previous works focusing on the impact angle guidance used the optimal control theory or the proportional navigation (PN) guidance method. The optimal control theory is mainly used to design those guidance laws in the assumption of linear engagement model. This method is typically applied with a class of cost functions involving the quadratic term of the control input which needs to be minimized [4]. By using the Schwartz inequality, the guidance laws which considered the terminal impact angle constraints (TIAC) were developed with the estimated time-to-go [5, 6], especially for a type of linear quadratic optimal control problem. The optimal guidance laws (OGL) for impact angle control which were proposed in [7, 8] also paid attention to the other performance of the shaping trajectory, such as the target observability and the acceleration constraints. Because of its efficiency and ease of implementation, PN guidance law is more popular and widely used [9]. There are also many modified PN-typed laws related to the impact angle control problem. A biased PN (BPN) guidance law was adopted in [10] to control the missile to impact a target with terminal angular constraints. This guidance law is a variation of the conventional PN guidance law, which is combined with a supplementary time-varying bias to control the impact angle. Lu et al. [11] introduced a three-dimensional guidance law, which was also based on PN with adaptive guidance parameters, to control a hypersonic vehicle in its terminal phase to impact a stationary target. Ratnoo and Ghose [12, 13] proposed a two-phased variable navigation gain PN guidance law for intercepting stationary and moving (nonmaneuvering) targets with all possible impact angles, but not considering the FOV limitation. On the other hand, sliding mode control (SMC) theory becomes more and more popular to be applied to design IACGL. A finite-time convergent sliding mode guidance law with terminal impact angle constraint was presented in [14], which was mainly addressed by the finite-time convergence stability theory. Zhao et al. [15] designed a SMC-based guidance law for an unpowered lifting reentry vehicle against a stationary target, to satisfy the TIAC with high guidance accuracy. Recently, the interception against maneuvering target was proposed based on several new SMC methods, for example, the nonsingular fast sliding mode (NFSM), with consideration of TIAC [16, 17].

Furthermore, several works are introduced to design the guidance laws considering the FOV limit as well as the TIAC. Park et al. [18] proposed an optimal impact angle control guidance law with the seeker’s FOV limit for missiles with strapdown seekers. By using the optimal control theory, the look angle which was regarded as a new state for inequality constraint was introduced during the homing phase. In [19], a two-phased scheme was developed in which BPN was used to shape the missile trajectory by making the integral value of the bias to reach a certain value at first and then switched to PN in the second phase. The integral value was calculated from initial engagement conditions and desired impact angle. Based on the same strategy of [19], a few of two-phased BPN (TPBPN) guidance laws were proposed in [20–25] for attacking a stationary target. A TPBPN was proposed in [20] based on the bias-shaping method, which can satisfy both the terminal-angle constraint and the FOV limitation to maintain the seeker’s lock-on condition. The TPBPN in [21] utilized the unbiased and biased pure proportional navigation laws and applied seeker’s FOV maximum value to calculate the bias in the initial phase. A time-varying BPN guidance law was designed by Yang et al. [22] to achieve the angular constraints without violating the look angle. Two time-varying biases were adopted in [22] to keep the seeker’s look angle and to ensure the terminal impact angle, respectively. Along with the thoughts of TPBPN and composite guidance law, Tekin and Erer [23] proposed the PN-gain-switched strategy, which can admit the allowed look angle and acceleration values to meet the demand of impact angle and at the same time, have a relative simple structure for implementation because of the PN form. Based on a two-stage PN guidance law, Ratnoo [24] derived a closed-form solution for the choice of navigation gains to satisfy the look angle and impact angle limit. Wen et al. [25] proposed a new guidance parameter design strategy based on the classical time-to-go weighted impact angle optimal guidance law. A robust guidance law which was based on the switching logic was designed in [26] by an additional term and combined the sliding mode control. This kind of guidance problem was also solved by SMC [27] and handled without any additional switching logic. However, since most of these works dealt with stationary targets, the desired impact angle may not be achieved against moving targets when applying these works to homing missiles. A composite guidance scheme was presented considering the case of a nonmaneuvering moving target, which was composed of modified deviated pure pursuit (DPP) with error feedback loop of look angle for initial guidance phases and PN with for final guidance phases [28]. Park et al. [29] addressed the similar guidance problem and strategy in [28] and extended to consider the command limit.

Based on the studies of previous works, this article draws on the experience of PN-gain-switched strategy from [23] to extend the investigated case to a nonmaneuvering moving target. The algorithm procedures and switching logic are introduced to achieve the guidance law proposed in this work. The main contributions of this work are summarized as follows: (1)We have studied the scaling factor between the different navigation gains of the initial and final guidance phases. The minimum value of this scaling factor is chosen to reduce the abrupt acceleration change at switching instant. Therefore, the specific navigation gain values for both two guidance phases can be calculated according to the scaling factor and the desired impact angle, which is different from [23, 28].(2)Two propositions are presented to prove the monotonicity of the line-of-sight angle and look angle. The switching of guidance phases occurs when a specific LOS angle similar to [28] is satisfied. However, the look angle reaches the FOV limit only at the switching instant, which reduces the required load in initial guidance phase compared with [28].(3)Different from [20–22], the integral biased term is not needed in the proposed method, which indicates that we do not need to estimate the value of time-to-go. It results in a convenient implementation in real missile model.

#### 2. Problem Statement

Consider the planar homing guidance geometry of a missile with narrow FOV against a nonmaneuvering incoming target as shown in Figure 1, where is a Cartesian inertial reference frame. The relative distance between the missile and the target is . The subscripts and denote the missile and the target, respectively. The acceleration vector is applied perpendicularly to the velocity vector . , , and represent the flight-path angle, LOS angle, and look angle. Angles are positive in the counterclockwise direction.