Direct Intercept Guidance using Differential Geometry Concepts

This paper examines the application of differential geometry to the engagement of both nonmanoeuvring and manoeuvring targets. The kinematics of the engagement for both manoeuvring and nonmanoeuvring target are developed and expressed in differential geometric terms. Two-dimensional geometry is then used to determine the intercept conditions for a straight line target and a constant manoeuvre target. The intercept conditions for both targets are developed for the case when the interceptor missile guides onto a straight line interception. These two cases are shown to have a common set of core conditions such that it enables a unified guidance law to be developed. The guidance law is shown to be globally stable using Lyapunov theory, so that guidance capture is assured for any initial condition. The analysis and guidance law design does not rely on local linearisation and can be shown to produce guidance trajectories that mirror proportional navigation for the straight line interception of a nonmanoeuvring target for which proportional navigation was originally developed. The paper finishes with simulation in two dimensions, illustrating the convergence and solution properties of the approach.     

现实真比例导引拦截任意机动目标捕获区域

主动机动突防技术的发展给大气层外动能拦截带来了严重困难。假设目标机动加速度方向垂直于弹目视线、形式任意且有界，分析了用于大气层外拦截制导的现实真比例导引律（RTPN）对该任意机动目标的捕获区域。首先在考虑拦截器存在机动过载饱和限制的情况下，推导了比例导引系数的选择范围，可使RTPN的制导指令加速度在拦截过程中不超过拦截器饱和过载；其次基于不等式分析方法，在不考虑拦截器过载限制时，推导了RTPN对任意机动目标的捕获区域；然后结合拦截器机动过载限制，推导了RTPN对任意机动目标的更加实用的捕获区域；最后通过数值仿真算例，验证了所提出的理论。     

一种无人机定距盘旋跟踪制导律及稳定性证明

对地面目标的定距盘旋跟踪问题是无人机（UAVs）在任务应用阶段需要面临的重要问题之一,如何在传感器信息受限的情况下完成跟踪任务是目前的研究热点。首先针对地面固定目标设计了一种仅依赖测距传感器的制导律,不再需要传统的视线角信号以及目标和无人机自身的定位信息;其次,设计了李雅普诺夫函数对该制导律的稳定性进行了严格数学证明;最后,将该制导律推广到对地面匀速和变速移动目标的跟踪制导。相比于现有制导律,所提出的制导律结构更为简洁,仅有一个设计参数,并且制导策略更为合理。仿真结果表明所提出的制导律能够实现无人机对地面固定和移动目标的稳定跟踪。     

Biased PNG law for impact with angular constraint

A new homing guidance law is proposed to impact a target with a desired attitude angle. It is a variation of the conventional proportional navigation guidance (PNG) law which includes a supplementary time-varying bias. The proposed guidance law does not require a time-to-go estimation and has a simpler form. Analytic conditions for fulfilling the guidance goal are also provided. Simulation results demonstrate that the proposed guidance law has wider launch envelopes than the previous one and shows a good performance even against a maneuvering target     

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     

基于纯比例导引的拦截碰撞角约束制导策略

拦截碰撞角约束制导是当前导弹制导研究的关键问题之一。首先基于理想比例导引（IPN）律拦截非机动目标的解析解,推导了纯比例导引律（PPN）拦截固定目标的解析解,得到了弹目相对距离、制导指令加速度和导弹前置角的显示表达式,并进一步得到了拦截碰撞角与弹目相对运动状态和比例导引系数之间的解析表达式。其次,基于该解析表达式,提出了基于PPN的拦截碰撞角约束制导策略（PPNIACG）,并探讨了在铅垂面内进行落角约束打击和水平面内进行拦截碰撞角约束打击的2种实现方式。最后,以弹道成型制导律（TSG）和最优碰撞角约束制导律（OIACG）为参考,通过数值仿真算例,对PPNIAC的拦截性能进行了对比分析,验证了所提出制导策略的有效性和正确性。     

Optimal pure proportional navigation for maneuvering targets

The optimal pure proportional navigation (PPN) guidance law with time-varying navigation gains is considered. Unlike the conventional optimal PPN approach where linearized model was assumed in the optimization process, this work exploits the exact nonlinear formulation of PPN to derive analytically the optimal time trajectory of the navigation gain to minimize a performance index which is a weighted sum of the final time and the integral of the squared acceleration. It is verified that the PPN scheme with constant navigation gain is not only optimal in the vicinity of the interception point, but also optimal for the whole trajectory, if the navigation constant is designed by the methodology proposed here. Based on the optimization results for nonmaneuvering targets, a recursive optimal PPN scheme is proposed for maneuvering targets, wherein the optimal navigation gain and time-to-go are predicted recursively during the interception, and trajectory and performance of the interceptor guided by optimal recursive PPN scheme are evaluated analytically.     

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     

Performance analysis of PNG laws for randomly maneuvering targets

The performance of the conventional proportional navigation guidance (PNG) law for a randomly maneuvering target is considered. By means of the Lyapunov method, it is proved that an ideal missile guided by the conventional PNG law can always intercept a target which maneuvers with time-varying normal acceleration provided that the navigation constant is sufficiently large. The authors also propose a modified PNG (MPNG) law which seems to improve the performance of the conventional PNG law at the final phase of pursuit. Simulation results demonstrate that the MPNG law demands less missile acceleration at the final phase of pursuit than the conventional PNG law     

带有攻击角度和攻击时间控制的三维制导

 在三维空间导引动力学与运动学模型的基础上,假设目标静止,而导弹本身以恒速运动,根据实际的攻击角度与设定的攻击角度误差,分析和设计了期望的视线（LOS）角运动学,基于李雅普诺夫稳定性理论设计了带有攻击角度控制的三维导弹导引律。为了对攻击时间进行预测与控制,假设导弹本身以恒速或者匀加/减速运动,先将导弹导引到预定的攻击角度上,根据待飞直线距离对待飞时间进行估算,再根据预测时间误差,确定导弹按照特定的圆弧轨迹机动飞行的指令和机动飞行的时间,通过机动飞行来对时间误差进行补偿,最后,再利用所设计的导引律攻击目标。给出了仿真结果。     

三维制导的几何方法与鲁棒控制方法

 在考虑导弹速度和目标速度均为时变的情况下,将微分几何方法与李雅普诺夫稳定理论结合起来,提出了一种导弹三维导引规律设计新方法。对机动目标,PPNG方法的主要问题是在导引末段,它要求导弹在俯仰和偏航通道上具有较大的加速度,为了解决这一问题,本文应用基于李雅普诺夫稳定理论的鲁棒控制方法,提出了一种三维鲁棒制导算法。这种方法在导引末段不需要过大的加速度命令,不需要知道目标精确的加速度和速度方位信息。     

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.     

径向基神经网络在优化导引律中的应用

应用神经网络的非线性的特点,在纯比例导引律的导引下,分析战机的运动轨迹,通过采样取得的一系列载机的控制输入数据,根据追踪导引控制的内在要求(目标视线角速度的绝对值减小并趋于零),设计载机导引控制器的相应的离散输出值,离线训练一个径向基神经网络模块,嵌入到载机控制回路中,作为战机导引的控制输入,实现载机在纯比例导引下的性能优化;仿真结果表明,在节能及省时方面,能达优化纯比例导引律的目的。     

基于Leader-Follower编队的无人机协同跟踪地面目标制导律设计

对地面目标的自动跟踪是无人机在任务应用阶段需要解决的重要问题之一,多无人机协同跟踪能够提高对目标运动状态的估计精度并降低目标丢失的概率,因而具有重要研究意义。本文提出了一种基于Leader-Follower编队的无人机协同跟踪制导方法,解决了传统Standoff跟踪模式对地面目标的速度范围限制问题。首先,通过控制无人机的航向不断趋近于地面目标牵连跟踪圆切线方向的方法设计了Leader无人机自动跟踪地面目标的制导律并完成了稳定性证明;其次,通过控制Follower无人机的速度和航向角逐渐趋近于Leader无人机速度和航向的协同跟踪策略,分别设计了Follower无人机自动跟踪Leader无人机的制导律和编队相位协同制导律并完成了稳定性证明;最后,分别针对静止目标、匀速直线运动目标和变速运动目标的跟踪问题进行了仿真验证,结果表明所提出的制导方法能够实现对不同运动状态地面目标的自动协同跟踪,并且跟踪性能优于基于李雅普诺夫向量法的制导方法。     

STUDY OF OPTIMAL SLIDING-MODE GUIDANCE LAW

An optimal guidance law based on missile-target line-of-sight (LOS) angular rate is presented for intercepting a nonmaneuvering target. It is then integrated with sliding-mode control theory by using reaching-law of sliding-mode, in order to derive an optimal sliding-mode guidance law for intercepting a maneuvering target. The new guidance method's robustness against target maneuvers and good miss distance performance are proved by the second method of Lyapunov and simulation results. The presented guidance law is simple to implement in practical applications.     

Three-dimensional guidance law based on adaptive integral sliding mode control

For the terminal guidance problem of missiles intercepting maneuvering targets in the three-dimensional space, the design of guidance laws for non-decoupling three-dimensional engage-ment geometry is studied. Firstly, by introducing a finite time integral sliding mode manifold, a novel guidance law based on the integral sliding mode control is presented with the target acceler-ation as a known bounded external disturbance. Then, an improved adaptive guidance law based on the integral sliding mode control without the information of the upper bound on the target accel-eration is developed, where the upper bound of the target acceleration is estimated online by a designed adaptive law. The both presented guidance laws can make sure that the elevation angular rate of the line-of-sight and the azimuth angular rate of the line-of-sight converge to zero in finite time. In the end, the results of the guidance performance for the proposed guidance laws are pre-sented by numerical simulations. Although the designed guidance laws are developed for the con-stant speed missiles, the simulation results for the time-varying speed missiles are also shown to further confirm the designed guidance laws.     

Three-dimensional time-varying sliding mode guidance law against maneuvering targets with terminal angle constraint

This paper deals with the problem of intercepting maneuvering targets with terminal angle constraints for missiles subjected to three-dimensional non-decoupling engagement geometry.To achieve the finite-time interception and satisfactory overload characteristics, a time varying sliding mode control methodology is developed based on a time base generator function. The main feature of the proposed guidance law guarantees the Line-of-Sight(LOS) angles to converge to small neighborhoods of the desir...     

An adaptive fast fixed-time guidance law with an impact angle constraint for intercepting maneuvering targets

Sliding mode guidance laws based on a conventional terminal sliding mode guarantees only finite-time convergence, which verifies that the settling time is required to be estimated by selecting appropriate initial launched conditions. However, rapid convergence to a desired impact angle within a uniform bounded finite time is important in most practical guidance applications. A uniformly finite-time/fixed-time convergent guidance law means that the convergence (settling) time is predefined independently on initial conditions, that is, a closed-loop convergence time can be estimated a priori by guidance parameters. In this paper, a novel adaptive fast fixed-time sliding mode guidance law to intercept maneuver targets at a desired impact angle from any initial heading angle, with no problems of singularity and chattering, is designed. The proposed guidance law achieves system stabilization within bounded settling time independent on initial conditions and achieves more rapid convergence than those of fixed-time stable control methods by accelerating the convergence rate when the system is close to the origin. The achieved acceleration-magnitude constraints are rigorously enforced, and the chattering-free property is guaranteed by adaptive switching gains. Extensive numerical simulations are presented to validate the efficiency and superiority of the proposed guidance law for different initial engagement geometries and impact angles.     

机动目标三维空间导引运动建模及导引律研究

给出了导弹—目标追逃运动的空间矢量方程，在此基础上推导了三维导引运动模型。对机动目标应用李雅普诺夫稳定性分析方法提出了一种自适应变结构制导算法。六自由度数学仿真结果表明了模型的正确性和制导算法的有效性。     

A unified formulation of optimal guidance-to-collision law for accelerating and decelerating targets

This paper provides a unified formulation of optimal guidance-to-collision law for a target with an arbitrary acceleration or deceleration. The collision course for general target acceleration or deceleration is first determined from the engagement geometry in conjunction with the nonlinear engagement kinematics in the proposed approach. The heading error defined in the collision course is then adopted as a variable to be nullified for accomplishing the intercept condition.The proposed guidance ...