Design And Analysis Of Three-Dimensional Guidance Laws Based On Nonlinear Formulation

The design of guidance law plays a key role in enhancing the capacity of modern precision guided weapons.With the increasing requirements on the precision strike ability of guided weapons in modern warfare environment,new requirements are proposed in designing guidance laws for intercepting maneuvering targets in large off-boresight launching conditions and controlling terminal impact angle and/or impact time,so as to improve the capture zone and capability of air-to-air missiles,increase combat effectiveness of air-to-surface guided weapons,and enhance the abilities of defense penetration and cooperative attack of anti-ship missiles.Based on nonlinear relative kinematics,some traditional guidance laws are analyzed and improved in the paper,and several new guidance laws are designed for different mission requirements.The specific work of this paper is as follows:The concept of plane angle has been adopted in describing the three-dimensional(3D)relative kinematics.And the zero-miss-distance property of the 3D pure proportional navigation(PPN)guidance law has been studied.It has been proven that if the missile is launched toward the target and the missile speed is greater than the target speed,the 3D PPN guidance law would guarantee the intercept of maneuvering targets as long as the navigation gain is greater than 1;in addition,the capture region and the guidance command of the 3D PPN guidance law are analyzed based on the dynamics of relative heading error angle(RHEA),obtaining new capture region in large off-boresight launching conditions and new sufficient conditions to guarantee the boundedness of commanded acceleration.The traditional proportional navigation(PN)guidance requires large,often excessively large,control command in the final phase to capture the maneuvering target.To alleviate this difficulty,the augmented proportional navigation(APN)guidance law has been introduced using the target acceleration information.The traditional APN is the augmentation of true PN(TPN),of which the commanded acceleration is directed normal to the line-of-sight,and thus it is difficult to be applied to aerodynamically controlled weapon systems.Because PPN guidance law is more realistic than TPN for an aerodynamically controlled interceptor,from the point of view of implementation,augmentation of PPN guidance law to account for target maneuvers would be a more realistic proposition.However,to the best of authors’ knowledge,the capturability performance of APPN has been only studied in planar pursuit situations.In this paper,using the concept of RHEA,a strict mathematical analysis on the capturablity of 3D APPN is performed.Compared with the analysis in the previous work,it is shown that the RHEA based analysis relaxes conditions on speed ratio,navigation gain,and augmentation parameter for capturability and boundedness of commanded acceleration.Optimal control method is one of the most commonly used approaches when designing new guidance laws.In order to obtain an explicit form of optimal guidance law,the most common strategy is to linearize the equation of kinematics firstly,and then utilize the linear quadratic optimal control method to design an optimal guidance law.This kind of strategy is only applicable for the case when the missile-target relative geometrics maintains near the collision course during the entire engagement and the target maneuvers according to a specific mode.What’s more,most of such guidance laws are designed in planar engagements with the assumption of a nonmaneuvering or static target.Therefore,this kind of optimal guidance law has limited scope of application.In this paper,unlike many previous studies on guidance problems for maneuvering targets,the homing problem is formulated in a relative virtual frame in which the origin is attached to the target.By compensating the target velocity and acceleration in this manner,the original problem is transformed to the one with a velocity-varying interceptor against a stationary target.By selecting the range as the independent variable and by introducing the notion of pseudocontrol,the dynamics of the relative geometry is then captured by a set of linear equations.By solving the transformed optimal control problem in which the optimality criterion is the integrated pseudocontrol energy weighted by the range function,two optimal guidance law are obtained analytically in a simple function of the current states of the target.When the traditional dual-plane decomposition method is utilized to design 3D guidance laws with impact angle constraint,coupling items brought by dual-plane decomposition have usually been ignored.In order to overcome the design defect,a new decoupling method is proposed in this paper.Based on such decoupling method,a new three-dimensional guidance law with impact angle constraint is derived.Theoretical analysis of the boundedness of commanded acceleration and capture region of the guidance law are conducted,using non-linear kinematics.For the non-linear guidance problem with simultaneous restriction of the impact time and impact angle in the case with large initial launch angle and/or designated impact angle,theoretical solution for time-to-go of the guidance law with impact angle constraint proposed by the paper has been analyzed,and a simple,valid and high-precision method for time-to-go estimation is proposed based on the linear interpolation method.Using the dynamics of time-to-go,a three-dimensional guidance law with simultaneous restriction of impact time and impact angle is proposed,along with a limited time convergence property.