Hangju Cho

Optimal Guidance Laws with Terminal Impact Angle Constraint

Optimal guidance laws providing the specified impact angle as well as zero terminal miss distance are generalized for arbitrary missile dynamics. The optimal guidance command is represented by a linear combination of the ramp and the step responses of the missile’s lateral acceleration. Optimal guidance laws in the form of the state feedback for the lag-free and the first-order lag system are derived, and their characteristics are investigated. Practical timeto-go calculation methods, which are important for the implementation of the optimal guidance laws, are proposed to consider the path curvature. Nonlinear and adjoint simulations are performed to investigate the performance of the proposed laws.

Time-to-go weighted optimal guidance with impact angle constraints

In this paper, the optimal guidance law with terminal constraints of miss distance and impact angle is presented for a constant speed missile against the stationary target. The proposed guidance law is obtained as the solution of a linear quadratic optimal control problem with the energy cost weighted by a power of the time-to-go. Systematic selection of guidance gains and trajectory shaping are possible by adjusting the exponent of the weighting function. A new time-to-go calculation method taking account of the trajectory curve is also proposed for implementation of the proposed law. Nonlinear and adjoint simulations are performed to investigate the performance of the proposed guidance law and time-to-go calculation method.

Recursive time-to-go estimation for homing guidance missiles

This paper addresses the problem of computing accurate time-to-go estimates, which is an important issue in implementing various optimal guidance laws developed for missiles of time-varying velocity. A recursive time-to-go computation method which updates the time-to-go in a noniterative way is presented. The recursive method includes an error compensation feature which explicitly computes the time-to-go error produced by nonzero initial heading errors. The proposed method is simple and straightforward to implement for any missile velocity profiles. Various numerical examples show that the proposed method works effectively for optimal guidance laws as well as proportional navigation and augmented proportional navigation.

Closed-form solutions of optimal guidance with terminal impact angle constraint

Optimal guidance laws and their closed-form solutions for controlling the impact angle as well as terminal miss are investigated in this paper. Generalized formulation of optimal guidance laws for constant speed missiles with a higher order system dynamics can be represented by a linear combination of a step and a ramp acceleration response of the missile. In the case of lag-free system, optimal guidance command is linear time varying and, therefore, the entire missile states can be expressed by polynomial functions in terms of time-to-go. We also proposed practical and precise time-to-go calculation methods in consideration of curved missile trajectory, which can be applied for various impact angle controllers. Basic properties including error analyses for the optimal guidance law for lag-free system are shown by numerical examples.

Optimal Impact Angle Control Guidance Law Based on Linearization About Collision Triangle

C ONTROL of missile-target-relative geometry is one of the desired features of guidance in many modern applications. A typical example is to impact a ground target in a direction perpendicular to the tangent plane of the terrain with very high precision both in miss distance and impact angle [1]. Various needs for the maximum warhead effectiveness, and sometimes enhancement of survivability of the missile launch vehicle, in naval applications call for guidance laws that can achieve a specified final direction of approach to the target as well [2]. Also, ensuring a small angle of the missile body relative to the target during the whole engagement process is critical in the case of missiles with strapdown seekers [3]. This necessity of control of terminal engagement geometry has been amajor thrust for much of the researchwork in the area of guidance law design with impact angle constraints. In this Note, a novel method of optimal impact angle control guidance law development based on linear quadratic optimal framework [4] against an arbitrary maneuvering target is presented. Throughout the Note, the missile velocity profile is assumed to be arbitrary. The equation of motion of a missile is often written in terms of the angular variables associated with velocity vectors; in this case, the missile acceleration is computed as the angular rate of its velocity vector multiplied by the magnitude of the velocity, which is directly realizable for aerodynamically controlled missiles. The main problem here is the fact that the kinematics is now nonlinear, defying closed-form solutions of many optimal guidance problems of interest. Thus, it has been common practice to linearize the kinematics (e.g., [5]) or approximate by linear equations [6] to come up with a nice linear quadratic optimal guidance problem. A natural question to follow is then how we linearize the kinematics in a right manner. This question, however, has not been addressed adequately in the literature and linearization has been performed in many cases with the usual assumption of small values of angular variables involved. As a result, the guidance laws often yield poor performance when the associated angular variables get larger. Usually, the collision triangle is defined to be the triangle formed by the initial positions of target and missile, and the intercept point at which the missile hits the target when flown by a straight (with zero effort) line. When no specific requirement on final engagement geometry is posed, the linearization about the usual collision triangle works reasonably well (e.g., [7]; also see [8]). If a specific impact angle between the missile and target velocity vectors is required, however, the usual collision triangle no longer serves as zero-effort collision geometry because the missile trajectory may largely deviate from the collision triangle to satisfy the specific impact angle requirement. This is why some papers just assume before linearization that the end game is initiated with a collision triangle satisfying closely the impact angle requirement [9]. No attempt, however, has been made yet to address specific questions such as what collision trianglewe should be looking for and howwe compute and use it for the linear optimal guidance problem formulation. In this Note, we introduce and use, as the basis of linearization, the perfect (or zero effort) collision triangle for the impact angle control problem, which varies depending on the value of the prescribed impact angle, and solve a linear optimal guidance problem.

Time-To-Go Estimation Using Guidance Command History

Abstract In this paper, we address the time-to-go estimation problem which is one of major parameters determining the performance of various optimal guidance laws. The strong curvature of the trajectory is a main factor that increases the time-to-go estimation error and the curvature is determined by the given guidance command history over the flight. Therefore a time-to-go estimation algorithm is proposed using guidance command histories. Since the proposed method involves trigonometric integrands in estimating the time-to-go, we approximate the sinusoidal functions by polynomial functions of the time of flight through the Talyor series expansion. The performance of the proposed time-to-go estimation algorithm is investigated by various numerical examples, and the results show that it works effectively.

Optimal Guidance Solution for Impact Angle Control Based on Linearization about Collision Triangle

Abstract In this paper, an optimal guidance problem is investigated where specific missile-target geometry at the time of the intercept of, or a terminal impact angle with respect to, a maneuvering target is prescribed. The kinematics is linearised to obtain a closed-form optimal solution and the proposed approach addresses the question of how to accomplish the linearization in a right manner. We use, as the basis of linearization, the perfect (or zero effort) collision triangle for the impact angle control problem which varies depending on the value of the prescribed impact angle, and solve a linear optimal guidance problem. The performance of the resultant guidance law is illustrated by performing some numerical simulations and the selection of a guidance variable that effectively capture the distance between the missile and the zero effort collision triangle is discussed.

Trajectory Modulation Guidance Law for Anti-ship Missiles

In this paper, a new optimal homing guidance law with terminal angle constraints is developed to improve the survivability of anti-ship missiles. Since it is di cult for a pursuing missile to hit missiles with weave maneouvres and impact angle control, the proposed guidance generates stable and robust weave manoeuvres while guaranteeing homing along with a desired terminal angle. A new virtual state variable is introduced to provide one more degree of freedom resulting in weave manoeuvres on the missile trajectory. The physical meaning and role of the virtual state variable is also investigated. We analyse the e ect of the design parameters on weave patterns, such as the weighting factor in the optimal guidance problem and initial value of the virtual state variable.

The Closed-Form Solution and its Approximation of the Optimal Guidance Law

Abstract In this paper, the optimal homing guidance problem is investigated for the general missile/target models described in the state-space. The closed-form solution of the optimal guidance law is derived, and its asymptotic properties are studied as the time-to-go goes to infinity or zero. Futhermore, several polynomial approximations of the optimal guidance law are suggested for real-time applications