Missile Trajectory Research Articles

Command-Shaping Guidance Law Based on a Gaussian Weighting Function

In this paper, a new homing guidance law is introduced based on the energy cost weighted by a Gaussian function in order to shape the missile's trajectory and distribute the acceleration command during the engagement. This law can improve guidance performance by alleviating sensitivity with respect to initial heading error and by reducing the possibility of command saturation through distributing acceleration demand properly.

Steepest-Ascent Revisited: Unconstrained Missile Trajectory

A Steepest-Ascent numerical procedure for offline trajectory optimization of a surface-to-surface missile attacking a stationary target is presented. A detailed numerical solution, starting from building the mathematical formulation till generating an offline angle of attack control history, is illustrated. A novel approach for guessing a nominal control program is conducted. The formulation entails nonlinear 2-DOF missile flight dynamics with mixed boundary conditions. The technique of changing of variables is adopted to convert the free-final-time optimization problem to a fixed-final-time one. The influences of the optimization algorithm controlling parameters are investigated. A new form of the weighting matrix is proposed. A novel technique of relaxation factors for eliminating the terminal states violation is described. Finally, a comparison between the obtained control history and the one obtained by a nonlinear optimal control package “GPOPS” is presented. The results indicate that the use of Steepest-Ascent method, to a great extent, is comparable in accuracy and iteration time to the pseudospectral optimization package “GPOPS.” This paper, to a great extent, is a detailed procedure for the method of Steepest-Ascent analyzed and verified by the authors from many undetailed sources to disclosure of the main problems faced by user of Steepest-Ascent.

An Efficient Computation of Effective Ground Range Using an Oblate Earth Model

An effcient method is presented to calculate the ground range of a ballistic missile trajectory on a nonrotating Earth. The spherical Earth model does not provide good approximation of distance between two locations on the surface of Earth. We used oblate spheroid Earth model because it provides better approximations. The effective ground range of a ballistic missile is an arc-length of a planner elliptic (or circle) curve which passes through the launch and target points on the surface of Earth model. A general formulation is presented to calculate the arc-length of an elliptic (or circle) curve which is the intersection of oblate Earth model and a plane. Explicit formulas are developed to calculate the coordinates of center of the ellipse as well as major and minor axes which are necessary ingredients for the calculation of effective ground range.

Design of Midcourse Trajectory for Tactical Ballistic Missile Intercept on the Basis of Zero Effort Miss

Since the maneuverability of the tactical ballistic missile, the midcourse trajectory of interception missile needs design online. Firstly, this paper reasonably simplifies the models and explores the analytical expressions of their relative motion state to obtain a kind of rapid and high-precision computing method of zero effort miss. Secondly, it applies constant thrust for the interception missile, burns out shut-down solid engine, reflects on the index requirement of overloaded interception missile, and designs midcourse trajectory of the interception missile with Pontryagin maximal principle, in accordance with the analytical expressions of zero effort miss. Finally, with the simulating calculation, the prediction methods of zero effort miss and trajectory design method provided by this paper are applied to a situation simulation which is close to the fact as much as possible so as to verify the applicability in the interception of tactical ballistic missile.

Study on the Trajectory Simulation of the Ballistic Missile

Based on the geocentric coordinate system, the object coordinate system as well as the theory of elliptic trajectory, it makes research and analysis of the ballistic-parameter-based trajectory generation method of the ballistic missile and, meanwhile, verifies the effectiveness and rationality of this method. It, under the assumed condition of elliptic trajectory, also researches the solution process of the ballistic parameters in the midcourse of the ballistic missile and, meanwhile, makes the simulation verification on the midcourse trajectory of the ballistic missile according to the radar sights variation relation in the object coordinate system during the motion process.

A Conjugate Gradient Optimization Algorithm for Wavy Trajectory of Missile

Since the traditional parabola trajectory is prone to be head off, wavy trajectory has the flight characteristics of defense penetration ability and increasing range. On the basis of flight characteristics of wavy trajectory missile, a method for flight program angle was designed. A specific formula was given. An optimization model was established. A conjugate gradient optimizing algorithm was used to solve the model and the result is compared with the former trajectory. It shows that optimized wavy trajectory can prove the feasibility of increasing range and valid defense penetration ability.

Numerical Performance Study of Small Size Lifting-Body Reentry Vehicle

The multiphase numerical optimization study has been carried out for a 2-stage boost vehicle and small size X-33 type lifting-body reentry vehicle with heat rate and dynamic pressure constraint. The problem has been modeled as a nonlinear, multiphase optimal control problem with the objective to compute the optimum burn-out conditions as well as the best control deflections that would maximize the cross range performance of the boost-glide vehicle under study. The study has been performed using hp-adaptive Pseudospectral method. Comparative performance of the lifting-body vehicle with conventional ballistic missile trajectory has also been carried out. It has been found that for vehicle under study, and near maximum down range, the optimum burn-out angle is 13.6 degree which results in a cross-range of more than 100 km.

Ballistic Measurement and Parameter Estimation of Ballistic Missile with Early Warning Satellite

Space infrared early warning satellite is an important platform for detecting ballistic missile launch and trajectory. This paper established observation satellite motion model, and the motion state of the satellite at any time was gotten based on its initial information by numerical integration methods. According the theory of double satellites location and satellite observation data of ballistic missile, the observation trajectory of ballistic missile was fitted. The trajectory was used to build the motion model of ballistic missile, and the position, speed and other parameters of the ballistic missile can be inferred. Finally, simulation experiment was conducted.

Design of a Scene Federate in the Proving Ground Simulation Platform

In order to reduce the risk of the range missile experiment and decrease the cost of the range missile development, we design and implement a scene federate in the proving ground simulation platform based on HLA, and analyze and solve the key problems of entity behavior modeling, multi-channel display. By integrating the under tested radar measuring system, optical measuring system with the computer simulation system, utilizes the feature information of the proving ground and weapon systems, and the missile trajectory measurement parameters to vividly exhibit a three-dimensional scene for the whole process of the missile proving ground test. The system can aid the experimentalist for observation, command, and decision-making.

Embedded Systems - Missile Detection/Interception

Missile defense systems are often related to major military resources aimed at shielding a specific region from incoming attacks. They are intended to detect, track, intercept, and destruct incoming enemy missiles. These systems vary in cost, efficiency, dependability, and technology. In present times, the possession of these types of systems is associated with large capacity military countries. Demonstrated here are the mathematical techniques behind missile systems which calculate trajectories of incoming missiles and potential intercept positions after initial missile detection. This procedure involved the use of vector-valued functions, systems of equations, and knowledge of projectile motion concepts.

Fast Intercept Trajectory Optimization for Multi-stage Air Defense Missile Using Hybrid Algorithm

A hybridalgorithmconsists of particle swarm optimization (PSO) and sequential quadratic programming (SQP) is proposed, and appliesto solve the optimal trajectory with maximizing the terminal velocity at predict impact point (PIP) for multi-stage air defense missile. This hybrid optimization approach combines the advantages of PSO as a global optimizer and complemented with SQP to find the accurate local optima. A simple plane motion equation of multi-stage air defense is establishedwith respect to trajectory referenceframe firstly, and restrictions on path parameters and terminal conditions are modeled.For task with given PIP, intercept trajectory optimization problem is a classical continuous optimal control problem. Then control of angle of attack (AOA) is parameterized according to empirical equations so that converting the continuousoptimal control problem into a parameters optimization problem.Finally, the hybrid algorithm is employed to solve this parameters optimization problem effectively with high accuracy. Several simulations and comparative cases are carried out, simulation results illustrate the hybrid method is feasible and it can fast converges to the optimal solution. Comparison results with the conventional optimization algorithm confirm that the proposed algorithm is more accurate and effective and more suited for missile trajectory profile optimization.

Consecutive tracking for ballistic missile based on bearings-only during boost phase

This paper proposes a modified centralized shifted Rayleigh filter (MCSRF) algorithm for tracking boost phase of ballistic missile (BM) trajectory with a highly nonlinear dynamical model based on bearings-only. This paper contributes three folds. Firstly, the mathematical model of an MCSRF for multiple passive sensors is derived. Then, minimum entropy based one-dimensional optimization search to adaptively adjust the probability of the different filters for real time state estimation is deployed. Finally, the unscented transform (UT) is introduced to resolve the asymmetric state estimation problem. Simulation results show that the proposed algorithm can consecutively track the BM precisely during the boost phase. In comparison with the unscented Kalman filter (UKF) algorithm, the proposed algorithm effectively reduces the tracking position and velocity root mean square (RMS) errors, which will make more sense for early precision interception.

Accuracy test and analysis for infrared search and track system

Infrared search and track (IRST) attracts many attentions nowadays. It is important to evaluate the performance of the detection and tracking performance for the development of IRST. This paper presents a tracking accuracy indoor test method for IRST. The proposal approach compares the simulated target trajectories with the tracks given by the IRST and calculates the dynamic tracking accuracy of the tested system. An infrared point target track simulator was built to simulate the actual target and missile trajectories. The movement of the simulator was driven by a high-precision stepper motor. A blackbody was used to simulate the infrared radiation of targets. The simulator was controlled by computer via RS-232 serial port. The simulator moving speed, direction and the operation angle value was managed in a software interface written in Visual Basic. The moving point target's coordinates, azimuth and elevation angle trajectories were reconstructed through the algorithm of coordinate translation. A comparison between the simulated target track and IRST recorded trajectory was performed to get the tracking accuracy of tested system. The paper describes the tracking accuracy test system design and implementation in detail. The experimental test and result analysis are provided to verify the effectiveness.

Optimal impact angle control guidance law considering the seeker’s field-of-view limits

A new optimal guidance problem with impact angle constraint and seeker’s field-of-view limits is investigated for a missile with a strapdown seeker. Impact angle control to satisfy the terminal flight path angle constraint causes missile trajectories to be very curved. Since a strapdown seeker has a narrower field-of-view than a gimbaled seeker, a missile with a strapdown seeker using the impact angle control guidance law can miss its target within the field-of-view during the homing phase, leading to the failure of the mission. Therefore, considering the seeker’s field-of-view limits is a key issue for missiles with strapdown seekers when implementing the impact angle control guidance law. To handle the seeker’s field-of-view limits in the homing guidance problem, a look angle, which is defined as the angle between the velocity vector and the line-of-sight, is considered as an inequality constraint. Based on the optimal control theory with a state variable inequality constraint, the optimal impact angle control guidance law with the seeker’s field-of-view limits is designed. The proposed guidance law is made up of three types of optimal acceleration commands: the first command is to reach the maximum look angle of the seeker, the second is to keep the seeker look angle constant on the constraint boundary, and the third is to intercept the target with the desired impact angle. Nonlinear simulations are performed to validate the proposed approach. In addition, comparisons with other guidance laws considering the limitation of the seeker look angle are carried out via nonlinear simulations, and the results show that the proposed guidance law is more efficient in terms of the control energy.

Research of Trajectory Tracking Control of Cruise Missile Based on Observer

To achieve cruise missile accurate trajectory tracking control, an observer-based tracking control method is designed. An observer is developed to estimate the states and control signals of desired trajectory as the inputs of the tracking controller. The linear quadratic optimal control is used to realize full-state feedback control for trajectory tracking. A certain cruise missile is used for the tracking simulation and the result shows satisfactory performance, the control method is simple and suitable for engineering applications.

The Simulation of Submarine-Launched Missile Out-Water Movement Based on MATLAB/Simulink

According to the characteristic of out-water movement of submarine-launched missile, a mathematical model was developed. Then a simulation model was established in Matlab/simulink environment based on the mathematical model, and the simulation was made. The result of the simulation shows that the simulation model can describe the submarine missile out-water movement perfectly, and the study may provide powerful base to the submarine missile trajectory design.

INS Alignment Using Onboard Computer Software for Missile Trajectory

INS Alignment Using Onboard Computer Software for Missile Trajectory

Design and Implementation of Digital Missile Trajectory Controller

Design and Implementation of Digital Missile Trajectory Controller

Design of impact angle control guidance laws via high-performance sliding mode control

In this article, a new impact angle control guidance law for a stationary or slowly moving target is developed using the high-performance sliding mode control methodology. The law can produce various missile trajectories by changing the damping ratio of the guidance loop motion during interception in order to achieve not only the satisfaction of the terminal impact angle constraint but also the additional guidance objectives. Since the proposed guidance law is given using the general function term, different forms of guidance laws can be designed by choosing specific functions. It can also lead to zero magnitude of the guidance command at the terminal time. The first-order lag system and acceleration disturbances such as measurement noise are considered in the guidance law design. These properties can provide the advantages for the guidance operation, whereas other impact angle control guidance laws may not easily allow this. Numerical simulations are performed to investigate the characteristics and performance of the proposed guidance law. Some application results of the proposed guidance law are also shown.

How bullet trajectory affects outcomes of civilian gunshot injury to the spine

We analysed retrospectively the effect of missile trajectory on outcomes from civilian gunshot injury to the spine (GSIS) between 1994 and 2008. Most of the 98 patients were male (88.8%). A minority (8%) of patients had multiple column injuries and a bone or bullet fragment in the spinal canal (14%). Neurologic injury was seen in 33%; and external bracing was applied to 30% of patients. The odds of bracing among patients with multiple levels of damage were 3.4 times than for patients with a single vertebral level of damage. The odds of paralysis among black patients were 6.33 times the odds among non-black patients. The odds of paralysis among patients with a fragment in the spinal canal were 12.99 times those without. We conclude that the supero-inferior trajectory affects the number of vertebral levels involved and consequently the need for bracing. The lateral trajectory affects neurological outcomes.