The proportional navigation dilemma-pure or true?

Two generic classes of proportional navigation (PN) laws are compared in detail. One class consists of pursuer-velocity-referenced systems, which include pure proportional navigation (PPN) and its variants; the second category consists of line-of-sight- (LOS-) referenced systems such as true proportional navigation (TPN), generalized true proportional navigation (GTPN), and generalized guidance laws. The existing closed-form solutions are discussed in detail, and the classical linear and quasilinear analytical solutions are summarized. A critical comparison is then made with regard to the definition, implementation, and analytical aspects of the guidance laws, including the method, the nature of solution, and an appraisal of the behavior of the pursuer motion resulting from the laws. It is established that, in spite of some restricted advantages in the solvability of the equations of motion, the LOS-referenced PN schemes suffer from serious limitations in terms of implementation and trajectory behavior. Among the major drawbacks are forward velocity variation requirement, relatively large control effort requirement, restrictions on initial engagement conditions to ensure intercept, lack of robustness, and possibility of unbounded acceleration. It is concluded that PPN is a better guidance law in a practical sense than TPN and its generalizations     

On the generalization of true proportional navigation

A simple framework to define generalized true proportional navigation (GTPN) guidance laws is presented. It is shown that this framework subsumes many of the generalizations presented in the earlier literature. The capture regions of a number of GTPN guidance laws are obtained through a rigorous qualitative analysis. The method of analysis is simpler and lends itself directly to an easy geometrical interpretation. A considerable amount of misinterpretation in the previous results, arising out of certain basic misconceptions, are corrected here. It is shown that a logical application of the guidance philosophy, through a minor modification of GTPN to take into account the direction of rotation of the line-of-sight (LOS), contributes substantially to the expansion of the capture region in the relative velocity space. In particular, it is shown that the capture region also extends to the negative closing velocity region, thus making the modified GTPN almost comparable, so far as the domain of capturability of guidance laws is concerned, to the pure proportional navigation (PPN) guidance law. A number of new results on the exact bounds on the capture region are derived and illustrated through examples     

An analysis of modern versus classical homing guidance

Three terminal guidance laws incorporated in a generic interceptor are compared: first- and second-order modern guidance laws (MGLs), that are matched to the second-order autopilot of the interceptor, and proportional navigation (PN). The sensitivity of miss distance (MD) between the interceptor and a target to autopilot time constant, damping ratio, and update rate is shown. For the chosen interceptor-target scenario, the second-order MGL achieves significantly improved MD over the other two guidance laws     

Three-dimensional Large Landing Angle Guidance Based on Two-dimensional Guidance Laws

 Both the design process and form of the three-dimensional (3D) suboptimal guidance law (3DSGL) are very complex. Therefore, we propose the use of two-dimensional (2D) guidance laws to meet the guidance requirements of 3D space. By analyzing the relationship between the flight-path angle and its projections on OXY and OXZ planes, we obtain the ideal design requirements of the guidance laws. Based on the requirements, we design a 2D suboptimal guidance law used in the horizontal plane; combining the 2D vertical suboptimal guidance law, we create a whole ballistic simulation of six degree-of-freedom. The results are compared with those using other three guidance modes in the case of large windage of the initial azimuth angle. When the proportional navigation guidance (PNG) law is used in the horizontal planes, the landing angle will obviously decrease. With the proposed guidance mode, the large landing angle can be realized and meet the guidance precision requirements. Moreover, the required overload can decrease to meet the control requirement. The effects of the proposed guidance mode are close to that of 3DSGL despite its very simple form.     

Unified approach to missile guidance laws: a 3D extension

Since the proportional navigation guidance law was first introduced, many of the researchers had proposed different methodologies to investigate the corresponding performances of all the existing guidance laws. Even though a unified approach was proposed a few years ago, other authors found that under the proposed framework, all the existing guidance laws, namely ideal proportional navigation (IPN), true proportional navigation (TPN), and pure proportional navigation (PPN), were indeed special cases of the mentioned general guidance law. However, the results were restricted to two-dimensional space. In this paper, the author not only extends the results to three-dimensional space, but also to general IPN (GIPN), general TPN (GTPN), and PPN. Unlike conventional researchers, a modified polar coordinate (MPC) is adopted. It is shown that with the property of this MPC, for the line of sight (LOS) based guidance laws (GIPN and GTPN) the number of differential equations required to fully describe the relative dynamics can be reduced from six to three, however, for the missile's velocity-based guidance law, i.e., PPN, five differential equations are required. All the terms of differential equations involve only products and additions of variables. For all the mentioned guidance laws in this paper, only two transformed variables are required to describe the capture region, while the third variable is required to provide the condition of finite turn rate.     

Three-dimensional cooperative guidance laws against stationary and maneuvering targets

This paper presents the cooperative strategies for salvo attack of multiple missiles based on the classical proportional navigation(PN) algorithm.The three-dimensional(3-D) guidance laws are developed in a quite simple formulation that consists of a PN component for target capture and a coordination component for simultaneous arrival.The centralized algorithms come into effect when the global information of time-to-go estimation is obtained, whereas the decentralized algorithms have better performance when each missile can only collect information from neighbors.Numerical simulations demonstrate that the proposed coordination algorithms are feasible to perform the cooperative engagement of multiple missiles against both stationary and maneuvering targets.The effectiveness of the 3-D guidance laws is also discussed.     

Optimal guidance for acceleration constrained missile andmaneuvering target

Explicit formulas of optimal guidance laws for an acceleration-constrained, arbitrary-order missile and maneuvering target are derived. These formulas are given in terms of the transfer function and acceleration constraint of the missile and the transfer function/shaping filter of the target. Optimal full-state feedback guidance law is synthesized against a target performing a barrel roll maneuver, and compared with the performance of proportional navigation (PN) for minimum and nonminimum phase missile. Simulation of a third-order missile shows the relative gain from using the full-order guidance law     

考虑运动耦合的BTT导弹三维制导律设计

针对倾斜转弯(BTT)导弹制导过程中的双通道耦合问题,设计了一种新型三维(3D)制导律。首先,通过矢量描述的方式,将导弹运动学模型分解成俯冲、转弯和耦合3项,俯冲和转弯两项在笛卡儿坐标系下描述,耦合项则在SO(3)空间中描述。然后,针对无终端约束和有终端约束情况,分别采用比例控制和SO(3)群上的广义比例-微分(PD)控制方式进行了相应的制导律设计。所得制导律既克服了李群方法的结构复杂性,又避免了解耦方法的信息损失。仿真表明,所设计的制导律能够满足BTT导弹精确制导的要求。     

Generalized guidance law for homing missiles

The concept of a generalized guidance law is presented, and the closed-form solution for a homing missile pursuing a maneuvering target according to generalized guidance laws is given. It is shown that the guidance laws appearing in the literature are merely special cases of the one proposed by the authors. The derived generalized forms of capture area, missile acceleration, and homing time duration that are derived provide insight into the performance of the guidance laws being considered and lead to the discovery of new ones. The problem of finding a nonlinear optimal guidance law for a homing missile with controlled acceleration, applied so as to capture a maneuvering target with a predetermined trajectory while minimizing a weighted linear combination of time of capture and energy expenditures, is solved in closed form. The derived optimal control law is equal to the LOS (line of sight) rate multiplied by a trigonometric function of the aspect angle. Numerical simulation shows that the resulting guidance law appears to yield a significant advantage over true proportional navigation     

带落角约束的圆弧比例导引律和偏置比例导引律的研究

为了实现导弹以一定落角命中目标,研究了两种带有落角约束的圆弧比例导引律和偏置比例导引律。首先建立弹目相对运动数学模型和框架动力学数学模型,然后阐述了两种导引律的基本原理,接着对导弹在三维空间内攻击静止与运动目标进行数学仿真试验,仿真实验证明两种导引律都能以较小的脱靶量和期望落角命中目标,最后从原理上和仿真实验上对两种导引律进行了对比研究。     

Launch Envelope Optimization of Virtual Sliding Target Guidance Scheme

This paper presents an optimization of the performance of a recently proposed virtual sliding target (VST) guidance scheme in terms of maximization of its launch envelope for three-dimensional (3-D) engagements. The objective is to obtain the launch envelope of the missile using the VST guidance scheme for different lateral launch angles with respect to the line of sight (LOS) and demonstrate its superiority over kinematics-based guidance laws like proportional navigation (PN). The VST scheme uses PN as its basic guidance scheme and exploits the relation between the atmospheric properties, missile aerodynamic characteristics, and the optimal trajectory of the missile. The missile trajectory is shaped by controlling the instantaneous position and the speed of a virtual target which the missile pursues during the midcourse phase. In the proposed method it is shown that an appropriate value of initial position for the virtual target in 3-D, combined with optimized virtual target parameters, can significantly improve the launch envelope performance. The paper presents the formulation of the optimization problem, obtains the approximate models used to make the optimization problem more tractable, and finally presents the optimized performance of the missile in terms of launch envelope and shows significant improvement over kinematic-based guidance laws. The paper also proposes modification to the basic VST scheme. Some simulations using the full-fledged six degrees-of-freedom (6-DOF) models are also presented to validate the models and technique used.     

空间拦截修正比例导引律研究

基于追踪导引律和比例导引律,提出了一种新的空间拦截末制导律。通过调整两种导引法的系数决定其在导引法中的权重。考虑空间末段拦截的具体情况,在相对运动方程进行线性化处理的基础上可以得出运动方程的解析解。将非线性方程得出的数值解和这种导引律的解析解相比,仿真表明这种方法的解析解与数值解一致,且避免了最优制导复杂的解析结果。     

Empirical virtual sliding target guidance law design: an aerodynamic approach

We present a novel empirical virtual sliding target (VST) guidance law for the midcourse phase of a long range surface-to-air missile that uses the simplicity of the conventional proportional navigation (PN) guidance law while exploiting the aerodynamic characteristics of a missile's flight through the atmosphere to enable the missile to achieve superior performance than that achieved by conventional PN guidance laws. The missile trajectory emulates the trajectory of an optimal control based guidance law formulated on a realistic aerodynamic model of the missile-target engagement. The trajectory of the missile is controlled by controlling the speed of a virtual target that slides towards a predicted intercept point during the midcourse phase. Several sliding schemes, both linear and nonlinear, are proposed and the effect of the variation of the sliding parameters, which control the sliding speed of the virtual target, on the missile performance, are examined through extensive simulations that take into account the atmospheric characteristics as well as limitations on the missile in terms of the energy available and lateral acceleration limits. Launch envelopes for these sliding schemes for approaching and receding targets are also obtained. These results amply demonstrate the superiority of the proposed guidance law over the conventional PN law.     

Three-dimensional midcourse guidance using neural networks forinterception of ballistic targets

A suboptimal midcourse guidance law is obtained for interception of free-fall targets in the three-dimensional (3D) space. Neural networks are used to approximate the optimal feedback strategy suitable for real-time implementation. The fact that the optimal trajectory in the 3D space does not deviate much from a vertical plane justifies the use of the two-dimensional (2D) neural network method previously studied. To regulate the lateral errors in the missile motion produced by the prediction error of the intercept point, the method of feedback linearization is employed. Computer simulations confirm the superiority of the proposed scheme over linear quadratic regulator guidance and proportional navigation guidance as well as its approximating capability of the optimal trajectory in the 3D space     

True proportional navigation with maneuvering target

We analytically obtain the capture regions for the true proportional navigation (TPN) missile guidance law against an intelligently maneuvering target. Two versions of TPN are considered. The first is the original TPN which assumes the commanded lateral acceleration to be directly proportional to the line-of-sight (LOS) rate only, the proportionality factor being an arbitrary constant or dependent only on the initial closing velocity. The other, known as RTPN (realistic TPN), assumes the commanded lateral acceleration to be directly proportional to the LOS rate and also the current closing velocity. The target is assumed to maneuver in such a way as to increase the LOS rate and thus directly oppose the proportional navigation (PN) philosophy of annulling the LOS rate. A necessary and sufficient condition for capture is derived for the original TPN, and using it, the exact capture region is obtained. A sufficient condition for capture is derived for RTPN and is used to obtain its capture region partially. Some necessary conditions for capture are also derived for RTPN and are used to obtain an upper bound on its complete capture region. Using these conditions some important results on the existence of capture regions and a comparative study of capturability of TPN laws are also presented     

Optimization of biased proportional navigation [guided projectiles]

An analytical treatment of the biased proportional navigation (BPN) is carried out with the aim of optimizing the bias parameter. It is shown that optimum biasing can lead to significantly more control-effort-efficient PN guidance in a wide variety of engagement situations, especially those involving higher target maneuvers. The performance of the BPN is compared with the standard (unbiased) PN system for the general case of a maneuvering target, and performance of the BPN is maximised to obtain the optimum bias value. The optimum bias is expressed through a simple algebraic equation, which can be readily solved. For the special (and very useful) case of the effective navigation constant being equal to three, the equation reduces to a quadratic, leading to an explicit expression for the optimum bias. Specific examples are provided to show the benefits of the BPN law. The higher control efficiency of the law is especially useful in extra-atmospheric interception, where the savings in control effort directly translates to a saving of propellent which forms part of the payload     

Short-time stability of proportional navigation guidance loop

Stability characteristics of proportional navigation (PN) guidance are analyzed by using the short-time stability criterion which is extended here to accommodate time-varying state weights and time-varying bounds of the state norm. As short-time stability is defined over a specified time interval, its application to the stability analysis of a homing guidance loop that operates up to a finite time gives more accurate results than previous studies. Furthermore, within the framework of short-time stability, zero effort miss and acceleration command, which are the most important variables determining guidance performance, can be directly related with guidance loop stability. An application to a PN guidance loop with a 1st-order missile/autopilot time lag shows that the stability condition based on short-time stability is less conservative than the previous results based on hyperstability and Popov stability     

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     

飞行器比例导引综述

基于导弹制导和机器人控制等领域，对追踪／截获动目标——导引问题的研究，综观几十年来大量国内外研究成果，系统分析与比较了各种比例导引律的含义、特点与性能。指出纯比例导引及其改进是非常有效的导引方式，可以有效地用于机动目标追踪／截获导引，能为相关领域机动目标导引问题的研究，包括战斗机追踪导引设计提供参考。     

A novel biased proportional navigation guidance law for close approach phase

A novel biased proportional navigation guidance (BPNG) law is proposed for the close approach phase, which aims to make the spacecraft rendezvous with the target in specific relative range and direction. Firstly, in order to describe the special guidance requirements, the concept of zero effort miss vector is proposed and the dangerous area where there exists collision risk for safety consideration is defined. Secondly, the BPNG, which decouples the range control and direc-tion control, is designed in the line-of-sight (LOS) rotation coordinate system. The theoretical anal-ysis proves that BPNG meets guidance requirements quite well. Thirdly, for the consideration of fuel consumption, the optimal biased proportional navigation guidance (OBPNG) law is derived by solving the Schwartz inequality. Finally, simulation results show that BPNG is effective for the close approach with the ability of evading the dangerous area and OBPNG consumes less fuel compared with BPNG.