Optimal Guidance Law Research Articles

Nonlinear Optimal Guidance for Intercepting a Stationary Target

The problem of optimally guiding an interceptor to a stationary target is studied in a nonlinear setting. First of all, it is shown that a global solution does not exist for the typical free-time minimum-effort nonlinear optimal intercept problem. This leads to consideration of the linear combination of the control effort and engagement duration as the objective function. The necessary conditions for the optimal intercept problem with the new objective function are found to be parameterized by a scalar, reducing the problem of deriving the optimal guidance law to the problem of finding the zeros of a real-valued function. Moreover, a semianalytical form for the real-valued function is devised, and the interval for its zeros is restricted, allowing the use of a brute-force search to efficiently find all the zeros. As a result, the nonlinear optimal guidance law can be efficiently established. Finally, the characteristics of the guidance law are exemplified and studied through simulations, showing that the nonlinear optimal guidance law performs better than the conventional proportional navigation, especially for cases with large initial heading errors.

Optimal Impact Angle Guidance for Exoatmospheric Interception Utilizing Gravitational Effect

This paper aims to develop a new optimal intercept angle guidance law for exoatmospheric interception by utilizing gravity. A finite-time optimal regulation problem is formulated by considering the instantaneous zero-effort-miss and the intercept angle error as the system states. The analytical guidance command is then derived based on Schwarz's inequality approach and Lagrange multiplier concept. Capturability analysis using instantaneous linear time-invariant system concept is also presented to provide better insights of the proposed guidance law. Theoretical analysis reveals that the proposed optimal guidance law encompasses previously suggested optimal impact angle constrained guidance laws. Numerical simulations with some comparisons clearly demonstrate the effectiveness of the proposed guidance law.

Optimal Cooperative Guidance Laws in a Multiagent Target–Missile–Defender Engagement

A cooperative interception scenario between a target, its defenders, and missiles attacking said target is considered. Optimal cooperative guidance laws are developed for two distinct scenarios: 1) a scenario with a missile using a known maneuver, yielding an optimal strategy for the target and its defenders; and 2) a scenario with a missile using an unknown maneuver, which is treated as a zero-sum differential game, between the team consisting of a target and its defenders and a team of the attacking missiles. From the general solutions of the two scenarios, analytical solutions are synthesized and analyzed for two cases: 1) a single target with a single defender, as well as a single attacking missile; and 2) a single target with two defenders, as well as two attacking missiles.

Generalized explicit guidance with optimal time-to-go and realistic final velocity

This paper presents an approach to compute the optimal time-to-go and final velocity magnitude in the Generalized Explicit (GENEX) guidance. Time-to-go and final velocity magnitude are two critical input parameters in GENEX guidance implementation. Optimal time-to-go selects that optimal solution which yields less cost compared to the costs yielded by other optimal solutions. In addition to it, the input of realistic final velocity lowers the cost further. These developments relax the existing limitations of GENEX, thereby making this optimal guidance law more optimal, effective and generic.

Evasion-Pursuit Strategy against Defended Aircraft Based on Differential Game Theory

The active defense scenario in which the attacker evades from the defender and pursues the target is investigated. In this scenario, the target evades from the attacker, and the defender intercepts the attacker by using the optimal strategies. The evasion and the pursuit boundaries are investigated for the attacker when the three players use the one-to-one optimal guidance laws, which are derived based on differential game theory. It is difficult for the attacker to accomplish the task by using the one-to-one optimal guidance law; thus, a new guidance law is derived. Unlike other papers, in this paper, the accelerations of the target and the defender are unknown to the attacker. The new strategy is derived by linearizing the model along the initial line of sight, and it is obtained based on the open-loop solution form as the closed-loop problem is hard to solve. The results of the guidance performance for the derived guidance law are presented by numerical simulations, and it shows that the attacker can evade the defender and intercept the target successfully by using the proposed strategy.

Adaptive Estimation and Cooperative Guidance for Active Aircraft Defense in Stochastic Scenario.

The active aircraft defense problem is investigated for the stochastic scenario wherein a defending missile (or a defender) is employed to protect a target aircraft from an attacking missile whose pursuit guidance strategy is unknown. For the purpose of identifying the guidance strategy, the static multiple model estimator (sMME) based on the square-root cubature Kalman filter is proposed, and each model represents a potential attacking missile guidance strategy. Furthermore, an estimation enhancement approach is provided by using pseudo-measurement. For each model in the sMME, the model-matched cooperative guidance laws for the target and defender are derived by formulating the active defense problem as a constrained linear quadratic problem, where an accurate defensive interception and the minimum evasion miss distance are both considered. The proposed adaptive cooperative guidance laws are the result of mixing the model-matched optimal cooperative guidance laws in the criterion of maximum a posteriori probability in the framework of the sMME. By adopting the adaptive cooperative guidance laws, the target can facilitate the defender’s interception with the attacking missile with less control effort. Also, simulation results show that the proposed guidance laws increase the probability of successful target protection in the stochastic scenario compared with other defensive guidance laws.

Linear Quadratic Optimal Control-Based Missile Guidance Law With Obstacle Avoidance

Two missile guidance algorithms are proposed for the intercept and for the rendezvous of a maneuvering target, while avoiding a static obstacle by a specified avoidance distance. The guidance algorithms are derived as solutions of a linear quadratic optimal control problem and they are expressed as formulas reminiscent of the classical (augmented) proportional navigation type guidance laws. These formulas include correction terms that are responsible for obstacle avoidance and are parameterized by the minimum avoidance distance. Simulation results are used to demonstrate the performance of the two algorithms.

Impact time and angle control guidance independent of time-to-go prediction

A novel cruise guidance method that can meet the terminal latitude and longitude, time and angle constraints independent of time-to-go prediction is studied in this paper. The impact time and angle control guidance (ITACG) goal is achieved through two steps. The first step is the construction of the complete motion model, the constant altitude and velocity guidance laws, and the optimal proportional guidance law which can satisfy the terminal position constraint based on the in-plane relative motion equation. The impact time control is the second step, which can be achieved by the integrated utilization of optimal guidance and maneuvering flight. Additional maneuvering trajectories used to control the impact time are designed in both cases with and without angle constraint, and the maneuvering parameters are directly solved online without time-to-go prediction. Simulation results verify the effectiveness of the guidance method, as well as the adaptability to different initial conditions and terminal constraints.

Guidance and Control Method for Vehicle with Strap-down Seeker

Guidance and control method was presented in this paper for flight vehicle with a strap-down seeker. Firstly, an extended Kalman filter was constructed in LOS (line of sight) coordinate system to estimate the guidance signal based on the information measured by strap-down seeker and inertial measurement unit. Secondly, an optimal guidance law which could reduce the target maneuvering and guidance system dynamics was proposed. Thirdly, three-loop acceleration autopilot with pseudo-angle of attack feedback was presented to control the flight vehicle. Finally, the proposed methods were simulated under typical condition. The simulation results demonstrated that the proposed methods were valid, and good performance could be achieved by the guidance and control method with the guidance signal estimated by the extended Kalman filter.

Field-of-view constrained two-stage guidance law design for three-dimensional salvo attack of multiple missiles via an optimal control approach

This study considers the cooperative guidance problem for multiple missiles against a stationary target with the field-of-view constraint. A two-stage guidance scheme is proposed to realize salvo attack of multiple missiles without using any information on time-to-go or its estimation. In the first guidance phase, this study integrates the cooperative guidance problem, state tracking problem and field-of-view constraint into a unified optimal control framework. A non-quadratic field-of-view constraint cost is innovatively constructed via an inverse optimal control approach, which leads to an analytical, distributed, and optimal guidance law to provide favored initial conditions for the latter phase. The stability and optimality of the closed-loop system is also proven. With respect to the second guidance phase, missiles disconnect from each other and are governed by the typical pure proportional navigation guidance law to eventually realize the salvo attack mission. Numerical simulations are performed to validate the theoretical results.

Optimal control based guidance law to control both impact time and impact angle

A new guidance law with both impact time and impact angle constraints is proposed in this paper. We first simplify missile dynamics under small heading error approximation, and derive an optimal guidance law with impact angle constraint against a stationary target. By adding a feedback controller to the obtained optimal guidance law, both impact time and angle requirements are achieved and the singularity problem is tackled. The proposed guidance law can be used to perform a simultaneous formation cooperative attack. Numerical simulations are demonstrated to illustrate the effectiveness of the proposed method.

Entry trajectory planning with terminal full states constraints and multiple geographic constraints

This paper proposes an online entry trajectory planning algorithm satisfying terminal full states constraints, path constraints and multiple geographic constraints for lifting-body entry vehicles. The vehicle is considered as a 3DOF point mass. The entry trajectory is divided into the initial descent phase, gliding phase and terminal guidance phase. In the gliding phase, a piecewise polynomial in the altitude-versus-velocity plane is used to plan the longitudinal trajectory and a new heading angle corridor based bank angle reversal logic is designed to satisfy all geographic constraints simultaneously. In the terminal guidance phase, an optimal guidance law with terminal angle constraints is adopted to generate the trajectory. Finally, the terminal full states constraints including terminal velocity direction constraints are implemented by iterating over the altitude in the gliding phase. Then the highly constrained entry trajectory planning problem is converted into a one-parameter search problem. The key of this planning algorithm is the use of terminal guidance phase, through which the terminal velocity direction is constrained strictly rather than kept around the line-of-sight angle and the range error caused by great arc assumption is also avoided. Simulation results with the CAV-H model show that this algorithm can generate entry trajectories satisfying complex constraints rapidly and is suitable for various missions.

Practical generalized optimal guidance law with impact angle constraint

In this paper, we provide a practical solution to the generalized optimal guidance problem with an impact angle constraint. The optimal guidance problem with arbitrary weighting functions is extended to explicitly consider a missile dynamic lag effect as well as a missile velocity variation. Therefore, compared to existing results, the proposed result can prevent performance degradation due to the dynamic lag effect and the velocity variation, which is an essential issue in practice. Besides, since the proposed guidance law is formulated from the generalized optimal control framework, it can directly inherit a vital feature of the framework: providing an additional degree of freedom in shaping a guidance command for achieving a specific guidance operational goal. An illustrative example is provided in order to validate this property. In this study, the proposed solution is also compared with the existing solutions. The comparison results indicate that the proposed result is a more general and practical solution. Finally, numerical simulations are also conducted to demonstrate the practical significance of the proposed method.

Cooperative guidance for simultaneous attack: a fully distributed, adaptive, and optimal approach

ABSTRACT This paper provides a fully distributed, adaptive, and optimal approach to address the problem of simultaneous attack against a maneuvering target with multiple missiles. For the case that the target acceleration is bounded, the relative distances and velocities between missiles and target are used as consensus variables. The communication graph and missile-target dynamics are decoupled by using neighborhood information and adaptive coupling gain coefficients. Without knowing the global information, the simultaneous hitting of the maneuvering target can be achieved by the designed fully distributed adaptive guidance law. On that basis, a cooperative guidance law with optimal performance is designed, which achieves simultaneous attack and ensures minimum attack error with minimum control and communication cost. The fully distributed adaptive and optimal cooperative guidance law can be combined to apply to a variety of attack scenarios. Numerical examples are provided to validate the effectiveness of the proposed control methods.

Novel guidance model and its application for optimal re-entry guidance

ABSTRACTAiming at three-dimensional (3D) terminal guidance problem, a novel guidance model is established in this paper, in which line-of-sight (LOS) range is treated as an independent variable, describing the relative motion between the vehicle and the target. The guidance model includes two differential equations that describe LOS’s pitch and yaw motions in which the pitch motion is separately decoupled. This model avoids the inaccuracy of simplified two-dimensional (2D) guidance model and the complexity of 3D coupled guidance model, which not only maintains the accuracy but also simplifies the guidance law design. The application of this guidance model is studied for optimal re-entry guidance law with impact angle constraint, which is presented in the form of normal overload. Compared with optimal guidance laws based on traditional guidance model, the proposed one based on novel guidance model is implemented with the LOS range instead of time-to-go, which avoids the problem of the time-to-go estimation of traditional optimal guidance laws. Finally, the correctness and validity of the guidance model and guidance law are verified by numerical simulation. The guidance model and guidance law proposed in this paper provide a new way for the design of terminal guidance.

Optimal Proportional-Integral Guidance With Reduced Sensitivity to Target Maneuvers

This paper proposes a new optimal guidance law based on the proportional-integral (PI) concept to reduce the sensitivity to unknown target maneuvers. Compared to the existing PI guidance laws, the proposed guidance command is derived in the optimal control framework while guarantees finite-time convergence. The kinematics equation with respect to the zero-effort-miss (ZEM) is utilized and the integral ZEM is augmented as a new system state. The proposed guidance law is derived through the Schwarz's inequality method. The closed-form solution of the proposed guidance law is presented to provide better insight of its properties. Additionally, the working principle of the integral command is investigated to show why the proposed guidance law is robust against unknown target accelerations. The analytical results reveal that the proposed optimal guidance law is exactly the same as an instantaneous direct model reference adaptive guidance law with a prespecified reference model. The potential significance of the obtained results is that it can provide a point of connection between PI guidance laws and adaptive guidance laws. Therefore, it allows us to have better understanding of the physical meaning of both guidance laws and provides the possibility in designing a new guidance law that takes advantages of both approaches. Finally, the performance of the guidance law developed is demonstrated by nonlinear numerical simulations with extensive comparisons.

Impact Time Control Based on Time-to-Go Prediction for Sea-Skimming Antiship Missiles

This paper proposes a novel approach for guidance law design to satisfy the impact-time constraints for a certain class of homing missiles. The proposed guidance law provides proper lateral acceleration commands that make the impact time error converge to zero by the time of impact. This scheme can be applied to any existing guidance law for which a formula of predicted time to go is available. Convergence of time-to-go errors is supported by Lyapunov stability. The optimal guidance law and the impact angle control guidance law are extended by the proposed method for impact-time-control guidance and impact-time-and-angle-control guidance, respectively. The performance of the extended guidance laws is demonstrated by numerical simulation.

English

English

Impact Angle Control Law with Sinusoidal Evasive Maneuver for Survivability Enhancement

In this paper, a new guidance law to enhance the survivability of anti-ship missiles against anti-air threats is presented. Sinusoidal maneuver acceleration is considered as a feed-forward input to a first-order lag system dynamics on the plane. Subject to this system dynamics a planar optimal guidance law for a stationary target is then derived to achieve zero miss distance and a specified impact angle while minimizing the control effort. By independent application of this planar law for homing guidance on the vertical and the horizontal plane, three-dimensional evasive guidance is easily synthesized. The phase difference as well as the magnitude and frequency of the sinusoidal acceleration inputs for the vertical and the horizontal guidance is the design parameter to shape the three-dimensional spiral or barrel-roll evasive maneuver pattern. Enhanced survivability with the proposed law against anti-air missiles is verified via numerical simulations.

Hyperbolic tangent function weighted optimal intercept angle guidance law

In this paper, a new optimal guidance law with terminal constraints on miss distance and intercept angle is proposed for a missile with time-varying velocity against a maneuvering target. The proposed optimal guidance law is obtained by solving a linear quadratic optimal control problem. Miss distance and intercept angle costs are weighted by variants of hyperbolic tangent function and energy cost is weighted by a power of time-to-go. The rule for determining a key parameter in the variant of hyperbolic tangent function is studied using the theory of second order differentiation, and its validity is verified by numerical simulation. In the purpose of easy implementation, an analytical solution of the proposed optimal guidance law is derived using assumption of constant velocity when calculating integral of missile's velocity from current time to final time. The significant contribution of this paper lies in that it is the first time a variant of hyperbolic tangent function is developed and employed as weighting coefficients of constraints on miss distance and intercept angle. Therefore, the proposed optimal guidance law can reduce acceleration command at the initial phase and increase missile's terminal velocity. Nonlinear numerical simulations clearly demonstrate effectiveness of the proposed optimal guidance law.