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

Abstract

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.