Boost-Phase ballistic missile trajectory estimation with ground based radar

A conditional boost-phase trajectory estimation method based on ballistic missile （BM） information database and classification is developed to estimate and predict boos-phase BM trajectory. The main uncertain factors to describe BM dynamics equation are reduced to the control law of trajectory pitch angle in boost-phase. After the BM mass at the beginning of estimation, the BM attack angle and the modification of engine thrust denoting BM acceleration are modeled reasonably, the boost-phase BM trajectory estimation with ground based radar is well realized. The validity of this estimation method is testified by computer simulation with a typical example.

Stability analyses of the mass abrasive projectile high-speed penetrating into a concrete target Part III: Terminal ballistic trajectory analyses

During the high-speed penetration of projectiles into concrete targets （the impact velocity ranges from 1.0 to 1.5 km/s）, important factors such as the incident oblique and attacking angles, as well as the asymmetric abrasions of the projectile nose induced by the target-projectile interactions, may lead to obvious deviation of the terminal ballistic tra- jectory and reduction of the penetration efficiency. Based on the engineering model for the mass loss and nose-blunting of ogive-nosed projectiles established, by using the Differ- ential Area Force Law （DAFL） method and semi-empirical resistance function, a finite differential approach was pro- grammed （PENTRA2D） for predicting the terminal ballistic trajectory of mass abrasive high-speed projectiles penetrating into concrete targets. It accounts for the free-surface effects on the drag force acting on the projectile, which are attributed to the oblique and attacking angles, as well as the asymmetric nose abrasion of the projectile. Its validation on the prediction of curvilinear trajectories of non-normal high-speed pene- trators into concrete targets is verified by comparison with available test data. Relevant parametric influential analyses show that the most influential factor for the stability of ter- minal ballistic trajectories is the attacking angle, followed by the oblique angle, the discrepancy of asymmetric nose abrasion, and the location of mass center of projectile. The terminal ballistic trajectory deviations are aggravated as the above four parameters increase.

Problem of the reference height of the projectile trajectory as a reduced meteo-ballistic weighting factor

This paper deals with the issue of preparation of the aiming angles with the use of tabular firing tables and needed determination of the ballistic elements μ_B(ballistic wind w_B,w_(xB),w_(ZB),ballistic(virtual) temperature τ_B.ballistic density p_B) from the standardized met messages.The weighting factors are used for the calculation of ballistic elements μ_B that are incorporated into the trajectory calculations characteristics of weapon and ammunition.Two different methodologies practically used in the praxis are analysed and compared.For the comparison of the two methodologies the reference height of trajectory determined from the weighting factor functions is employed.On the basis of the analyses conducted,the potential for further increase in accuracy of these aiming angles preparation methods is pointed out.

Ballistic Trajectory Extrapolation and Correction of Firing Precision for Multiple Launch Rocket System

The research on multiple launch rocket system(MLRS)is now even more demanding in terms of reducing the time for dynamic calculations and improving the firing accuracy,keeping the cost as low as possible.This study employs multibody system transfer matrix method(MSTMM),to model MLRS.The use of this method provides effective and fast calculations of dynamic characteristics,initial disturbance and firing accuracy.Further,a new method of rapid extrapolation of ballistic trajectory of MLRS is proposed by using the position information of radar tests.That extrapolation point is then simulated and compared with the actual results,which demonstrates a good agreement.The closed?loop fire correction method is used to improve the firing accuracy of MLRS at low cost.

Improved theory of projectile trajectory reference heights as characteristics of meteo-ballistic sensitivity functions

Projectile trajectories calculated under non-standard conditions are considered to be perturbed. The tools utilized for the analysis of perturbed trajectories are weighting factor functions(WFFs) which are a special kind of sensitivity functions. WFFs are used for calculation of meteo ballistic elements mB(ballistic wind wB, virtual temperature tB, pressure pB, density rB, speed of sound a) as well. An effect of weapon system parameters can be incorporated into calculations through the reference height of trajectory-RHT.RHT are also calculated from WFFs. Methods based on RHT are far more effective than traditional methods that use weighting factors q.We have found that the existing theory of RHT has several shortcomings due to we created an improved theory of generalized RHT which represent a special sensitivity parameters of dynamical systems. Using this theory will improve methods for designing firing tables, fire control systems algorithms, and meteo message generation algorithms.

Validation of the NATO Armaments Ballistic Kernel for use in small-arms fire control systems

In support for the development of a new small-arm ballistic computer based on the NATO Armaments Ballistic Kernel(NABK) for the Canadian snipers, DRDC Valcartier Research Centre was asked to carry out high-fidelity 6 degree-of-freedom(6-DOF) trajectory simulations for a set of relevant vignettes for the snipers, and to compare the direct fire 6-DOF simulation results with those obtained with the 4-DOF NATO Armaments Ballistic Kernel(NABK) adapted to simulate small-arm ammunition trajectories. To conduct this study, DRDC Valcartier Research Centre used BALCO v1.0b. This paper presents(1) the process and the methodology employed to carry out the sniper direct fire solution study,(2) the modeling and the simulation of the sniper projectile, the approach used in calculating the firing solutions, and the results of direct fire simulations for the sniper vignettes, and(3) an analysis of firing solutions obtained with the BALCO engine versus those of NABK. The work presented in this paper serves to validate the use of NABK for the new sniper ballistic computer.

Improved theory of generalized meteo-ballistic weighting factor functions and their use

It follows from the analysis of artillery fire errors that approximately two-thirds of the inaccuracy of indirect artillery fire is caused by inaccuracies in the determination of the meteo parameters included in fire error budget model.Trajectories calculated under non-standard conditions are considered to be perturbed.The tools utilized for the analysis of perturbed trajectories are weighting factor functions(WFFs)which are a special kind of sensitivity functions.WFFs are used for calculation of meteo ballistic elements B(ballistic wind w B,densityρB,virtual temperatureτB,pressure p B)as well.We have found that the existing theory of WFF calculation has several significant shortcomings.The aim of the article is to present a new,improved theory of generalized WFFs that eliminates the deficiencies found.Using this theory will improve methods for designing firing tables,fire control systems algorithms,and meteo message generation algorithms.

Conceptual Design Architecture Modeling and Simulation for Boost Phase Ballistic Missile Defense

The design review, simulation and validation of a Conceptual Design Architecture (CDA) for Ballistic Missile Defense (BMD) are presented. An intercept system that contains a Ground Based Interceptor (GBI) and its guidance sensors (both radar and infrared) are simulated. 3D model using MATLAB is developed for a multistage target with ascent phase acceleration profile that depends on total mass, propellant mass and the specific impulse in the gravity field. The radar cross section (RCS) and infrared radiation (IR) of the target structure is estimated as a function of the flight profile. The Kill Vehicle (KV) design is examined as a function of the KV mass, acceleration capability, aimpoint offset and impact energy to destroy the target. The aim of the CDA is to: detect the launch of a threat ballistic missile, determine whether the detected object is a threat,define the characteristics of the threat ballistic missile, develop a firing solution to negate the threat ballistic missile, engage the threat ballistic missile, and assess the effectiveness for ballistic missile intercept. The architecture is modeled in Matlab.

Impact Point Prediction and Lateral Correction Analysis of Two-dimensional Trajectory Correction Projectiles

Compared with the one-dimensional trajectory correction technology which adjusts longitudinal range, not only does the two-dimensional trajectory correction technology adjust the force in velocity direction, but also need to modulate the lateral force or trajectory (perpendicular to the vertical plane of fire direction). Therefore, the structure of control cabin of two-dimensional trajectory correction projectile (TDTCP) is more complicated than that of one-dimensional trajectory correction projectile (ODTCP). To simplify the structure of control cabin of TDTCP and reduce the cost, a scheme of adding a damping disk to the control cabin of ODTCP has been developed recently. The damping disk is unfolded at the right moment during its flight to change the ballistic drift of spin stabilized projectile. For this technical scheme of TDTCP, a fast and accurate impact point prediction method based on extended Kalman filter is presented. An approximate formula for predicting the ballistic drift and trajectory correction quantity is deduced. And the lateral correction capability for different fire angles and its influencing factors are analyzed. All the work is valuable for further research.

QPSO-based algorithm of CSO joint infrared super-resolution and trajectory estimation

The midcourse ballistic closely spaced objects（CSO） create blur pixel-cluster on the space-based infrared focal plane,making the super-resolution of CSO quite necessary.A novel algorithm of CSO joint super-resolution and trajectory estimation is presented.The algorithm combines the focal plane CSO dynamics and radiation models,proposes a novel least square objective function from the space and time information,where CSO radiant intensity is excluded and initial dynamics（position and velocity） are chosen as the model parameters.Subsequently,the quantum-behaved particle swarm optimization（QPSO） is adopted to optimize the objective function to estimate model parameters,and then CSO focal plane trajectories and radiant intensities are computed.Meanwhile,the estimated CSO focal plane trajectories from multiple space-based infrared focal planes are associated and filtered to estimate the CSO stereo ballistic trajectories.Finally,the performance（CSO estimation precision of the focal plane coordinates,radiant intensities,and stereo ballistic trajectories,together with the computation load） of the algorithm is tested,and the results show that the algorithm is effective and feasible.

An Approximate Calculation Method for Lateral Trajectory Correction

The two-dimensional trajectory correction needs to adjust not only the force in velocity direction,but also the lateral force or lateral trajectory (normal to the perpendicular plane of fire direction) . Therefore,its structure of control cabin is more complicated than that of one-dimensional trajectory correction projectiles (ODTCP). In order to simplify the structure and reduce the cost,a scheme of adding a damping disc to the control cabin of ODTCP has been developed recently. The damping disc will unfold at the right moment during its flight to change the ballistic drift of rotary projectiles. Aimed at this technical scheme,a mathematical model of two-dimensional trajectory corrections was discussed according to the theory of exterior ballistics. An approximate formula for predicting the ballistic drift and trajectory correction was deduced. The capability of lateral trajectory correction and the flight stability of TDTCP were also analyzed. All the work is valuable for further research.

An Algorithm for Determination of Projectile Attitude Angles in Projectile Trajectory Prediction

A fast and accurate algorithm is established in this paper to increase the precision of ballistic trajectory prediction.The algorithm is based on the six-degree-of-freedom(6 DOF)trajectory equations,to estimate the projectile attitude angles in every measuring time.Hereby,the algorithm utilizes the Davidon-Fletcher-Powell(DFP)method to solve nonlinear equations and Doppler radar trajectory test information containing only position coordinates of the projectile to reconstruct the angular information.The″position coordinates by the test″and″angular displacements by reconstruction″at the end phase of the radar measurement are used as an initial value for the trajectory computation to extrapolate the trajectory impact point.The numerical simulations validate the proposed method and demonstrate that the estimated impact point agrees very well with the real one.Morover,other artillery trajectory can be predicted by the algorithm,and other trajectory models,such as 4 DOF and 5 DOF models,can also be incorporated into the proposed algorithm.

滑翔增程制导炮弹弹道特性分析与设计

滑翔增程制导炮弹与普通的舰炮相比,精度高,射程远,成本低,能够在对岸进行火力支援等方面发挥重要作用,是国家军队急需发展以应对现代化作战的重要制导武器。针对滑翔增程制导炮弹初始发射条件的选取和发动机点火时刻的设计,采用遍历性计算方法进行了分析,得到了最优发射条件和发动机点火时刻。以此为基础,采用最大最大升阻比法设计了增程弹道,仿真结果表明,以最大升阻比为原则的方案弹道结合选取适当的初始发射条件和发动机点火时刻,能够有效提高滑翔增程制导炮弹的射程,对于指导工程实践和作战运用具有重要意义。

拓展高原环境的航空炸弹弹道快速解算方法

为快速、精确地求解高原环境下的航空炸弹弹道,在经典弹道模型的基础上,面向高原环境提出一种基于改进Runge-Kutta的航空无控炸弹弹道解算(improved Runge-Kutta based uncontrolled bombs trajectory calculation,IRK-UBTC)方法。首先,针对炸弹下落过程中时刻变化的环境参数,提出了基于起飞点的空气比重函数估算方法。然后,引入战斗机所处环境的温度、湿度、纬度、科尔奥利力等信息对经典炸弹质心运动微分方程组进行重构。最后,在三阶Runge-Kutta算法的基础上,提出了带误差控制机制的微分方程组实时解算方法。实验结果表明,该方法在1.4 GHz主频的嵌入式计算机中的解算周期小于5 ms,且最大射程误差小于0.2%,能够被有效应用于高原地区航弹弹道的实时解算。

双层三阶多项式拟合的弹道导弹发射点估计方法

发射点估计是炮位侦察校射雷达的重要任务之一。在仅利用雷达测量数据的情况下,先通过三阶函数拟合弹道导弹主动段时间-高度曲线估计导弹的绝对发射时刻,从而求得各个弹道轨迹点相对于绝对发射时刻的相对发射时间,然后再次利用三阶函数拟合主动段相对时间-水平位移,进而得到弹道导弹发射点估计结果。通过弹道导弹运动模型仿真具有一二级助推的弹道数据,并增加雷达观测误差,分析验证了所提方法的有效性和精确性。经测试,所提方法求解过程简单,对主动段弹道拟合准确,发射点估计精度达到0.6%以内,定位误差较传统方法降低了50%以上。

基于Bahdanau注意力机制的大口径火炮双向GRU轨迹预测

针对大口径火炮弹丸轨迹预测问题,提出一种基于Bahdanau注意力机制与双向门控循环单元网络的序列到序列轨迹预测模型(S2S ATT-BiGRU)。通过不同条件下六自由度弹丸运动模型仿真,得到大量弹丸轨迹数据样本,并利用滑动窗口法和差分法构造仿真数据集;将所提轨迹预测模型与GRU模型在仿真数据集下进行实验。研究结果表明,S2S ATT-BiGRU模型在输入时间为0.5 s预测未来时间为5 s的情况下,其MSE为13.89 m^(2),MAE为1.84 m, MAPE为0.29%,其预测精度远高于GRU模型。在其余情况下S2S ATT-BiGRU模型的预测结果均优于GRU模型。这表明S2S ATT-BiGRU模型具有更强的存储输入轨迹序列信息和自适应关注重要输入轨迹信息的能力,为弹丸轨迹预测研究提供有利参考。

Trajectory prediction of ballistic missiles using Gaussian process error model

Ballistic Missile Trajectory Prediction(BMTP)is critical to air defense systems.Most Trajectory Prediction(TP)methods focus on the coast and reentry phases,in which the Ballistic Missile(BM)trajectories are modeled as ellipses or the state components are propagated by the dynamic integral equations on time scales.In contrast,the boost-phase TP is more challenging because there are many unknown forces acting on the BM in this phase.To tackle this difficult problem,a novel BMTP method by using Gaussian Processes(GPs)is proposed in this paper.In particular,the GP is employed to train the prediction error model of the boost-phase trajectory database,in which the error refers to the difference between the true BM state at the prediction moment and the integral extrapolation of the BM state.And the final BMTP is a combination of the dynamic equation based numerical integration and the GP-based prediction error.Since the trained GP aims to capture the relationship between the numerical integration and the unknown error,the modified BM state prediction is closer to the true one compared with the original TP.Furthermore,the GP is able to output the uncertainty information of the TP,which is of great significance for determining the warning range centered on the predicted BM state.Simulation results show that the proposed method effectively improves the BMTP accuracy during the boost phase and provides reliable uncertainty estimation boundaries.

火炮武器高原射击时的弹道特性研究

在对高原条件分析的基础上,对火炮武器弹丸在高原上的陀螺飞行稳定性、动态飞行稳定性、转速衰减规律、章动规律、动力平衡角、射程、偏流以及地面密集度等弹道特性进行了理论研究,通过六自由度弹道计算,得到了各主要弹道参数随海拔高度的变化量.结果表明,随着海拔高度的增大,弹丸的陀螺稳定性和动态稳定性增强,动力平衡角和偏流明显增大,地面密集度有所提高.

基于推力加速度模板的主动段弹道跟踪方法

对弹道导弹的主动段跟踪是导弹防御中重要的环节。针对此问题提出了一种基于推力加速度模板的主动段跟踪算法,该算法需要主动段推力加速度信息支持。给出了一种简捷的推力加速度模板构建方法。利用近常速运动模型描述了导弹运动特性,解决了对导弹发射时间的估计问题和模板概率的递推计算问题,并将该方法应用到空间预警系统中,通过仿真验证了该方法具有较好的识别效果,位置和速度估计误差都接近于CRLB限。

弹道导弹中段突防弹道设计与验证

针对中段多脉冲弹道规划及评价问题,提出了一种新的弹道设计及评价方法。借鉴轨迹规划中节点搜索及扩展的思想,将多脉冲弹道的脉冲点火点视为节点,建立了多脉冲机动变轨模型。基于该模型,对突防弹道进行了设计及优化。运用＂博弈论＂的思想,在考虑敌方防御系统的探测机理、性能及部署基础上将敌方预测弹道与实际弹道的差值作为评价函数,以尽量降低敌方预测弹道的精度。最后,设计了导弹突防效能地面验证平台以验证所设计弹道的突防效率与弹道评价方法的有效性。选取了一个例子进行分析及验证,结果表明：相比最省能量弹道,设计的导弹防御系统对突防弹道的预测误差提高了5到15倍,达到了100~300km,飞行时间缩短了8.53%,而且规划时间不到140s。实验显示本文提出的弹道设计方法,能够在很短时间内规划出一条突防概率很高的弹道。