Closed-form guidance law for velocity maximization with impact angle constraint

Final velocity and impact angle are critical to missile guidance.Computationally efficient guidance law with compre-hensive consideration of the two performance merits is challeng-ing yet remains less addressed.Therefore,this paper seeks to solve a type of optimal control problem that maximizes final velocity subject to equality point constraint of impact angle con-straint.It is proved that the crude problem of maximizing final velocity is equivalent to minimizing a quadratic-form cost of cur-vature.The closed-form guidance law is henceforth derived using optimal control theory.The derived analytical guidance law coincides with the widely-used optimal guidance law with impact angle constraint(OGL-IAC)with a set of navigation parameters of two and six.On this basis,the optimal emission angle is determined to further increase the final velocity.The derived optimal value depends solely on the initial line-of-sight angle and impact angle constraint,and thus practical for real-world appli-cations.The proposed guidance law is validated by numerical simulation.The results show that the OGL-IAC is superior to the benchmark guidance laws both in terms of final velocity and missing distance.

Design and Analysis of Terminal Guidance Law for Kinetic Interceptors based on Pulse Engine

A new terminal guidance law is proposed based on a solid propellant pulse engine and an improved proportional navigation method to address the terminal guidance issue for kinetic interceptors.On this basis,the start-stop curve of the pulse motor during the terminal guidance process is designed,along with its start-up logic.The effectiveness of the proposed guidance strategy is verified through simulation.

Computational intelligence interception guidance law using online off-policy integral reinforcement learning

Missile interception problem can be regarded as a two-person zero-sum differential games problem,which depends on the solution of Hamilton-Jacobi-Isaacs(HJI)equa-tion.It has been proved impossible to obtain a closed-form solu-tion due to the nonlinearity of HJI equation,and many iterative algorithms are proposed to solve the HJI equation.Simultane-ous policy updating algorithm(SPUA)is an effective algorithm for solving HJI equation,but it is an on-policy integral reinforce-ment learning(IRL).For online implementation of SPUA,the dis-turbance signals need to be adjustable,which is unrealistic.In this paper,an off-policy IRL algorithm based on SPUA is pro-posed without making use of any knowledge of the systems dynamics.Then,a neural-network based online adaptive critic implementation scheme of the off-policy IRL algorithm is pre-sented.Based on the online off-policy IRL method,a computa-tional intelligence interception guidance(CIIG)law is developed for intercepting high-maneuvering target.As a model-free method,intercepting targets can be achieved through measur-ing system data online.The effectiveness of the CIIG is verified through two missile and target engagement scenarios.

EVOLUTIONARY FUZZY GUIDANCE LAW WITH SELF-ADAPTIVE REGION

Effective guidance is one of the most important tasks to the performance of air-to-air missile. The fuzzy logic controller is able to perform effectively even in situations where the information about the plant is inaccurate and the operating conditions are uncertain. Based on the proportional navigation, the fuzzy logic and the genetic algorithm are combined to develop an evolutionary fuzzy navigation law with self-adapt region for the air-to-air missile guidance. The line of sight (LOS) rate and the closing speed between the missile and the target are inputs of the fuzzy controller. The output of the fuzzy controller is the commanded acceleration. Then a nonlinear function based on the conventional fuzzy logic control is imported to change the region. This nonlinear function can be changed with the input variables. So the dynamic change of the fuzzy variable region is achieved. The guidance law is optimized by the genetic algorithm. Simulation results of air-to-air missile attack using MATLAB show that the method needs less acceleration and shorter flying time, and its realization is simple.[KH*3/4D]

Head Pursuit Variable Structure Guidance Law for Three-dimensional Space Interception

This article aims to develop a head pursuit （HP） guidance law for three-dimensional hypervelocity interception, so that the effect of the perturbation induced by seeker detection can be reduced. On the basis of a novel HP three-dimensional guidance model, a nonlinear variable structure guidance law is presented by using Lyapunov stability theory. The guidance law positions the interceptor ahead of the target on its tlight trajectory, and the speed of the interceptor is required to be lower than that of the target, A numerical example of maneuvering ballistic target interception verifies the rightness of the guidance model and the effectiveness of the proposed method.

Variable structure guidance law for attacking surface maneuver targets

The characteristics of surface maneuver targets are analyzed and a 3-D relative motion model for missiles and targets is established. A variable structure guidance law is designed considering the characteristics of targets. In the guidance law, the distance between missiles and targets as well as the missile-target relative velocity are all substituted by estimation values. The estimation errors, the target＇s velocity, and the maneuver acceleration are all treated as bounded disturbance. The guidance law proposed can be implemented conveniently in engineering with little target information. The performance of the guidance system is analyzed theoretically and the numerical simulation result shows the effectiveness of the guidance law.

Extended optimal guidance law with impact angle and acceleration constriants

The extended optima straints of miss distance and Schwartz inequality. To reduce guidance law with terminal conmpact angle is derived by the terminal acceleration and eliminate gravity disturbance absolutely, the object function, which designs the weight of control command to be the power function of time-to-go＇s reciprocal, is given. And the gravity is considered when building the state equation. Based on the parsing express of the guidance command change with varying time and adjoint system analysis method, the command characteristics and the non-dimensional miss distance of the guidance law are analyzed, a design principle of guidance order coefficients is discussed. Finally, based on the requirement of engineering, the method to calculate the guidance condition and maximal required acceleration of the guidance law is given. The simulation demonstrates that not only the guidance law can satisfy the terminal position and impact angle constraints, but also the terminal acceleration can be converged toward zero, which will support a good situation for the terminal angle of attacking control.

Impact angle control over composite guidance law based on feedback linearization and finite time control

The impact angle control over guidance(IACG) law against stationary targets is proposed by using feedback linearization control(FLC) and finite time control(FTC). First, this paper transforms the kinematics equation of guidance systems into the feedbackable linearization model, in which the guidance law is obtained without considering the impact angle via FLC. For the purpose of the line of sight(LOS) angle and its rate converging to the desired values, the second-order LOS angle is considered as a double-integral system. Then, this paper utilizes FTC to design a controller which can guarantee the states of the double-integral system converging to the desired values. Numerical simulation illustrates the performance of the IACG, in contrast to the existing guidance law.

Three-dimensional impact angle constrained distributed cooperative guidance law for anti-ship missiles

This paper investigates the problem of distributed cooperative guidance law design for multiple anti-ship missiles in the three-dimensional(3-D)space hitting simultaneously the same target with considering the desired terminal impact angle constraint.To address this issue,the problem formulation including 3-D nonlinear mathematical model description,and communication topology are built firstly.Then the consensus variable is constructed using the available information and can reach consensus under the proposed acceleration command along the line-of-sight(LOS)which satisfies the impact time constraint.However,the normal accelerations are designed to guarantee the convergence of the LOS angular rate.Furthermore,consider the terminal impact angle constraints,a nonsingular terminal sliding mode(NTSM)control is introduced,and a finite time convergent control law of normal acceleration is proposed.The convergence of the proposed guidance law is proved by using the second Lyapunov stability method,and numerical simulations are also conducted to verify its effectiveness.The results indicate that the proposed cooperative guidance law can regulate the impact time error and impact angle error in finite time if the connecting time of the communication topology is longer than the required convergent time.

Adaptive weighting impact angle optimal guidance law considering seeker's FOV angle constraints

In this paper, a new adaptive optimal guidance law with impact angle and seeker’s field-of-view(FOV) angle constraints is proposed. To this end, the generalized optimal guidance law is derived first. A changeable impact angle weighting(IAW) coefficient is introduced and used to modify the guidance law to make it adaptive for all guidance constraints. After integrating the closed-form solution of the guidance command with linearized engagement kinematics, the analytic predictive models of impact angle and FOV angle are built, and the available range of IAW corresponding to constraints is certain. Next, a calculation scheme is presented to acquire the real-time value of IAW during the entire guidance process. When applying the proposed guidance law, the IAW will keep small to avoid a trajectory climbing up to limit FOV angle at an initial time but will increase with the closing target to improve impact position and angle accuracy, thereby ensuring that the guidance law can juggle orders of guidance accuracy and constraints control.

Multiconstraint adaptive three-dimensional guidance law using convex optimization

The traditional guidance law only guarantees the accuracy of attacking a target. However, the look angle and acceleration constraints are indispensable in applications. A new adaptive three-dimensional proportional navigation(PN) guidance law is proposed based on convex optimization. Decomposition of the three-dimensional space is carried out to establish threedimensional kinematic engagements. The constraints and the performance index are disposed by using the convex optimization method. PN guidance gains can be obtained by solving the optimization problem. This solution is more rapid and programmatic than the traditional method and provides a foundation for future online guidance methods, which is of great value for engineering applications.

Sliding mode control and Lyapunov based guidance law with impact time constraints

This paper analyses the issue of impact time control of super-cavitation weapons impact fixed targets which mainly refer to the ships or submarines who lost power, but still have combat capability. Control over impact time constraints of guidance law(ITCG) is derived by using sliding mode control(SMC) and Lyapunov stability theorem. The expected impact time is realized by using the notion of attack process and estimated time-to-go to design sliding mode surface(SMS). ITCG contains equivalent and discontinuous guidance laws, once state variables arrive at SMS,the equivalent guidance law keeps the state variables on SMS,then the discontinuous guidance law enforces state variables to move and reach SMS. The singularity problem of ITCG is also analyzed. Theoretical analysis and numerical simulation results are given to test the effectiveness of ITCG designed in this paper.

Observer-based Guidance Law Accounting for Second-order Dynamics of Missile Autopilots

Accounting for the missile autopilot as second-order dynamics, an observer-based guidance law is designed based on the dynamic surface control method. Some first-order low-pass filters are introduced into the design process to avoid the occurrence of high-order derivatives of the line of sight angle in the expression of guidance law such that it can be implemented in practical applications. The proposed guidance law is effective in compensating the bad influence of the autopilot lag on guidance accuracy. In the simulations of intercepting non maneuvering targets, targets with step acceleration, and targets with sinusoidal acceleration respectively, the guidance law is compared with the adaptive sliding mode guidance law in the presence of missile autopilot lag. The simulation results show that the proposed guidance law is able to guide a missile to accurately intercept a maneuvering target, even if it escapes in a great and fast maneuver and the autopilot has a relatively large lag.

Extended differential geometric guidance law for intercepting maneuvering targets

Without assumptions made on motion states of missile and target, an extended differential geometric guidance law is derived. Through introducing a line of sight rotation coordinate system, the derivation is simplified and has more explicit physical significances. The extended law is theoretically applicable to any engagement scenarios. Then, on basis of the extended law, a modified one is designed without the requirement of target acceleration and an approach is proposed to determining the applied direction of commanded missile acceleration. Qualitative analysis is carried out to study the capture performance and a criterion for capture is given. Simulation results indicate the two laws are effective and make up the deficiency that pure proportional navigation suitable for endoatmospheric interceptions cannot deal with high-speed maneuvering targets. Furthermore, the correctness of the criterion is validated.

Proportional Navigation Guidance Law with Variable Coefficient Gravity Compensation

Based upon a discussion on the merits and limitations of proportional navigation（PN）guidance law in which constant gravity compensation is included as a part,a counterpart having varying compensations,which changes with pitching angle and line-of-sight angle,is substituted.Flight trajectory simulation over a submissile model is conducted,resulting in increased impact angle,shorter miss distance,smaller maximum normal overload and narrower terminal angle of attack.

Homing Guidance Law with Falling Angle and Flying Time Control

In this paper,a new homing guidance method is used to control the flying time and falling angle for guided missiles. Through this approach,it finds the approximate solution to the quadratic equation of time-togo,which is used for the formula derivation of the flying time control command. In this guidance law design,the acceleration rate control command is adopted. The guidance law is composed of a PN guidance command and a flying time control command. Firstly,it obtains a desired falling angle with accurate guidance. Secondly,it introduces to satisfy the constraint of flying time. The flying time control requires an assumption on the future evolution of missile,which is called time-to-go. To cope with the time-varying speed of missiles,a method of compensating the estimation of time-to-go is presented. The new guidance law is evaluated by using a simulation of typical terminal guidance for rocket-propelled torpedo. The simulation results show that the guidance achieves excellent control performance and exhibits insensitivity to initial trajectory parameter over a widen flight envelope.

Three-dimensional cooperative guidance law for multiple missiles with impact angle constraint

This paper proposes a cooperative guidance law for attacking a ground target with the impact angle constraint based on the motion camouflage strategy in the line-of-sight(LOS)frame.A dynamic model with the impact angle constraint is established according to the relative motion between multiple missiles and the target.The process of cooperative guidance law design is divided into two stages.Firstly,based on the undirected graph theory,a new finite-time consensus protocol on the LOS direction is derived to guarantee relative distances reach consensus.And the value of acceleration command is positive,which is beneficial for engineering realization.Secondly,the acceleration command on the normal direction of the LOS is designed based on motion camouflage and finite-time convergence,which can ensure the missiles reach the target with the desired angle and satisfy the motion camouflage state.The finitetime stability analysis is proved by the Lyapunov theory.Numerical simulations for stationary and maneuver targets have demonstrated the effectiveness of the cooperative guidance law proposed.

A trajectory shaping guidance law with field-of-view angle constraint and terminal limits

In this paper, a trajectory shaping guidance law,which considers constraints of field-of-view(FOV) angle, impact angle, and terminal lateral acceleration, is proposed for a constant speed missile against a stationary target. First, to decouple constraints of the FOV angle and the terminal lateral acceleration, the third-order polynomial with respect to the line-ofsight(LOS) angle is introduced. Based on an analysis of the relationship between the looking angle and the guidance coefficient,the boundary of the coefficient that satisfies the FOV constraint is obtained. The terminal guidance law coefficient is used to guarantee the convergence of the terminal conditions. Furthermore, the proposed law can be implemented under bearingsonly information, as the guidance command does not involve the relative range and the LOS angle rate. Finally, numerical simulations are performed based on a kinematic vehicle model to verify the effectiveness of the guidance law. Overall, the work offers an easily implementable guidance law with closed-form guidance gains, which is suitable for engineering applications.

New three-dimensional guidance law for BTT missiles based on differential geometry and Lie-group

A new non-decoupling three-dimensional guidance law is proposed for bank-to-turn （BTT） missiles with the motion coupling problem. In this method, the different geometry is taken for theoretically modeling on B-IT missiles＇ motion within the threedimensional style without information loss, and meanwhile, Liegroup is utilized to describe the line-of-sight （LOS） azimuth when the terminal angular constraints are considered. Under these cir- cumstances, a guidance kinematics model is established based on differential geometry. Then, corresponding to no terminal angular constraint and terminal angular constraints, guidance laws are re- spectively designed by using proportional control and generalized proportional-derivative （PD） control in SO（3） group. Eventually, simulation results validate that this developed method can effectively avoid the complexity of pure Lie-group method and the information loss of the traditional decoupling method as well.

Fuzzy sliding mode control guidance law with terminal impact angle and acceleration constraints

In this paper, a novel fuzzy sliding mode control（FSMC） guidance law with terminal constraints of miss distance, impact angle and acceleration is presented for a constant speed missile against the stationary or slowly moving target. The proposed guidance law combines the sliding mode control algorithm with a fuzzy logic control scheme for the lag-free system and the first-order lag system. Through using Lyapunov stability theory, we prove the sliding surface converges to zero in finite time. Furthermore, considering the uncertain information and system disturbances, the guidance gains are on-line optimized by fuzzy logic technique. Numerical simulations are performed to demonstrate the performance of the FSMC guidance law and the results illustrate the validity and effectiveness of the proposed guidance law.