Field-of-view limited guidance with impact angle constraint and feasibility analysis

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Abstract

A look angle shaping guidance law with impact angle and seeker's field-of-view (FOV) constraints considering the time-varying speed is proposed. A look angle profile with respect to the relative range is first constructed as a cubic polynomial. The desired impact angle and the maximum look angle are expressed as functions of a parameter in the look angle profile. The impact angle constraint under limited FOV is achieved by solving this parameter with boundary conditions. Moreover, the feasible region of the desired impact angle is derived to ensure seeker's lock-on condition. By combining the first order look angle dynamics and the reference look angle profile, the impact angle control guidance (IACG) law with FOV constraint is obtained. The proposed guidance law is derived without relying on the linearized engagement kinematics, switching logic, constant speed assumption, and the LOS rate information during the guidance process. Moreover, the proposed guidance technique is helpful for user/designers to select feasible impact angle constraints in advance without tuning parameters. Finally, numerical simulations under a realistic model are performed to validate the effectiveness of the proposed guidance law.

Introduction

The zero miss distance is the primary guidance objective for conventional missile guidance laws, such as the well-known proportional navigation (PN) guidance law [1], [2]. However, with the enhanced defense capability of targets, the conventional guidance laws that only meet zero miss distance may lead to the reduction in the survivability and kill probability. For instance, the recently developed tanks and warships are equipped with hard armor and close-in weapon system that can greatly compromise the attack efficiency of incoming missiles. To address this issue, the homing guidance law considering impact angle constraint can provide an effective strategy. Because a specified impact angle is able to maximize the warhead lethality of the tactical missile [3], [4] by attacking a weak spot on the armored target or to increase the survivability [5] by avoiding the defense area of the anti-air systems.

In recent years, plenty of advanced guidance laws have been proposed to address the impact angle constraint. One early work investigating the impact angle control guidance (IACG) law was presented in [6], where the guidance problem was approximated by a linear system and then solved by linear quadratic optimization techniques. Song et al. [7] developed an optimal impact angle control law against moving targets by minimizing a quadratic performance index with final state constraints. In [8], a generalized form of energy minimization guidance law with impact angle constraint for arbitrary order missile dynamics was derived, and the guidance command is presented as a linear combination of the step and the ramp acceleration responses of the missile. To attack stationary targets from a specified direction, a PN-based guidance law for hypersonic gliding vehicles with adaptive guidance parameters was proposed by Lu et al. [9]. Then, a two-phased PN guidance approach was developed in [10]. Therein, an orientation guidance strategy is adopted with the navigation gain smaller than 2 in the first phase and then the guidance gain switches to greater than 2 to fulfill the desired impact angle. Similar to [10], another two-phased PN-based impact angle control law was proposed in [11] by adding a constant bias term to the pure PN guidance commands during the first phase. Refs. [12], [13], [14] proposed the IACG laws for maneuvering targets with the sliding mode control technique due to its efficiency and robustness.

In practice, the missile may lose the target and even fail the guidance mission while generating undesired maneuvers for impact angle control, especially for seekers with a narrow field-of-view (FOV). Therefore, the missile's FOV limit should be considered to ensure the seeker's lock-on condition during the homing process.

The existing IACG laws considering the seeker's FOV constraint can be roughly divided into two categories: linear and nonlinear methods. The linear approaches are based on the linearized engagement kinematics. One early optimal IACG law with the FOV constraint was presented in [15]. Therein, the FOV limit is considered as an inequality constraint. As an extension of [15], [16] minimizes the performance index with the positive weighting function of the range-to-go to achieve the impact angle and FOV constraints simultaneously. In [17], [18], the guidance laws were expressed as functions of time-to-go or range-to-go. In [17], with the linearized engagement kinematics, the guidance gains can be adjusted in advance to restrict the maximum missile look angle within the FOV constraint. To enhance target observability, the IACG law [18] was derived from the guidance law [17] by adding a term proportional to the cross range.

In the latter category, some nonlinear IACG laws considering the FOV constraint have been developed based on the pure proportional navigation (PPN) or biased pure proportional navigation guidance. In [19], [20], the FOV and impact angle constraints were achieved by switching the bias terms of the IACG law. A switched-gain guidance scheme based on the PPN was investigated in [21] to handle the FOV constraint by numerically solving the navigation gains. In [22], a two-stage PPN guidance law provided a closed-form solution for the choice of navigation gains while considering the FOV and impact angle constraints. In [23], a composite guidance scheme was proposed. The first phase employs a modified deviated pure pursuit with the error feedback loop of the look angle to maintain the constant look angle of the seeker, and the second phase adopts PN to intercept the moving target at a desired impact angle. To improve the guidance robustness, e.g., against the nonlinear flight dynamics that are necessary to be considered in practice [24], the sliding mode control technique and the trajectory shaping method are utilized to design the nonlinear IACG laws. The FOV constrained IACG law was investigated in [25] by utilizing the sliding mode control and integral barrier Lyapunov methods. By applying the backstepping technique, the constrained guidance task was transformed into a nonlinear tracking control problem with partial state constraint [26]. An FOV constrained IACG law that only requires the bearing angles was derived in [27] via constructing a sigmoid function based sliding surface. More recently, a trajectory shaping guidance law with impact angle and FOV constraints was proposed in [28] via developing reference line-of-sight profiles.

Although the FOV constrained IACG laws have been well studied, this problem is still open due to the following challenges. 1) The multi-phase guidance structure and switching logic [19], [20], [21], [22], [23] may lead to discontinuous and even abrupt-jumping guidance commands during the flight. 2) Approximations such as model linearization [15], [16], [17], [18] or constant speed assumption [25], [26], [27] are required. 3) Another practical issue is that the achievable impact angle set is not provided for FOV consideration in most IACG law designs.

In this work, a look angle shaping IACG law considering seeker' FOV limit and time-varying speed is developed by constructing a look angle profile with respect to the relative range in a cubic polynomial form. The advantages of the proposed technique are in order:

1) The desired impact angle and FOV constraints are achieved under the varying-speed nonlinear guidance model. Moreover, the zero-miss distance and bounded terminal acceleration are ensured.

2) The guidance technique does not require the switching scheme and LOS rate information during the guidance process. The guidance law only involves one single parameter that can be easily determined and does not need trial-and-error tuning.

3) The achievable impact angle set under limited FOV is provided for feasibility analysis such that users can avoid blind selection of desired impact angles in advance.

The remainder of this paper is organized as follows: The nonlinear guidance model is presented in Section 2. Section 3 describes the derivation of the proposed guidance law and the achievable impact angle set. In Section 4, numerical simulations are conducted to evaluate the performance of the guidance law. Finally, concluding remarks are provided in Section 5.

Section snippets

Problem statement

The guidance mission considered in this work is to intercept a stationary target with impact angle and FOV constraints under varying speed. The two-dimensional engagement geometry is shown in Fig. 1, where the missile and the target are denoted as M and T, respectively. XIOIYI is the inertial reference frame. R and λ represent the relative range and line-of-sight (LOS) angle of the missile with respect to the target, respectively. VM denotes the speed of the missile, which is perpendicular to

IACG law with FOV constraint

To achieve the guidance objectives given by (6), an FOV constrained IACG law considering the time-varying speed is developed. For this goal, a look angle profile with respect to the relative range in a cubic polynomial form is constructed. The maximum look angle and the impact angle can be expressed as functions of a single unknown parameter designed in the look angle profile. To meet the FOV constraint, the safe range of this unknown parameter is analytically obtained. Meanwhile, the

Numerical simulations

In this section, a series of numerical simulations are conducted to evaluate the performance of the proposed guidance law (15). For all the simulation cases, if not specifically given, the engagement scenario is set as follows. The initial position of the missile is (XM0,YM0)=(0,0)Km. The target is located at (XT,YT)=(10,0)Km. A realistic missile model utilized in [33] is taken into account. The missile velocity is time-varying, which is governed by (5) with an initial speed VM0=250m/s. The

Conclusion

In this paper, an impact angle control guidance law is proposed under the missile speed variation and seeker's FOV limit for homing missiles. The guidance law is developed by designing a reference look angle profile with respect to the relative range. The achievable impact angle set under limited FOV is provided by feasibility analysis, which can be calculated in advance and only depends on initial guidance conditions. Simulation results under different engagement scenarios show that the

Declaration of Competing Interest

There is no conflict of interest to declare in the article.

Acknowledgements

This work was supported partially by the National Natural Science Foundation of China (61960206011 and 61633003), and Beijing Natural Science Foundation (JQ19017). The authors greatly appreciate the above financial support. The authors would also like to thank the associate editor and reviewers for their valuable comments and constructive suggestions that helped to improve the paper significantly.

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