Proportional Navigation Guidance Law Research Articles

Minimum-Data-Driven Guidance for Impact Angle Control

This paper investigates the impact-angle-control guidance problem for varying-speed flight vehicles with constrained acceleration. A learning-based bias proportional navigation guidance (L-BPN) law is proposed to achieve impact-angle-constrained impact by constructing a deep neural network (DNN) for nonlinear mapping between the impact angle and the bias term. During the process of dataset establishment, the impact of state variables is evaluated by sensitivity analysis to minimize the quantity of training data. This approach also effectively accelerates sample generation and improves the training efficiency. The simulation results verify the effectiveness of the proposed L-BPN law and demonstrate its advantages over the existing algorithms.

Dynamic Encircling Cooperative Guidance for Intercepting Superior Target with Overload, Impact Angle and Simultaneous Time Constraints

This paper proposes a dynamic encircling cooperative guidance (DECG) law to enable multiple interceptors to cooperatively intercept a superior target, considering low velocity, limited overload, impact angle and simultaneous arrival constraints. First, the feasible escaping area of the target is analyzed and a dynamic encircling strategy for the target is established. This strategy efficiently provides virtual escaping points, allowing interceptors to dynamically encircle the target without excessive energy consumption, ultimately leading to a successful interception. Second, to enhance the physical feasibility of the kinematic equations governing the interaction between interceptors and target at the virtual escaping points, the independent variable is substituted and the kinematic equations are remodeled. Convex optimization is employed to address the multi-constraint optimal guidance problem for each interceptor, thereby facilitating simultaneous interception. Compared with the existing guidance laws, DECG has a more practical and feasible cooperative strategy, is able to handle more constraints including the interceptor’s own constraints and cooperative constraints, and does not rely on the precise calculation of explicit remaining flight time in the guidance law implementation. Lastly, the effectiveness, superiority and robustness of the DECG law are evaluated through a series of numerical simulations, and its performance is compared with that of the cooperative proportional navigation guidance law (CPNG).

Three-dimensional piecewise cooperative guidance with smooth switching topology

To investigate the issue of large-scale multi-missile cooperative attacks on a same target, this paper presents a three-dimensional piecewise cooperative guidance law based on smooth switching topology, realizing time consensus and impact angle constraints. Firstly, to implement the piecewise cooperative guidance law design, we establish the attack surface of the arc profile according to the expected LOS angle and the communication switching point, and obtain the virtual impact point (VIP). Secondly, a switching function is designed to ensure the smooth transition of the communication topology, mitigating the chattering issue in existing communication topology switching, while a time consensus protocol control input is implemented to facilitate the time cooperative attack of the missile in the LOS direction. Meanwhile, a fixed-time terminal sliding mode guidance law is established to guarantee the satisfaction of the impact angle constraint at the VIP. Additionally, an adaptive three-dimensional proportional navigation guidance (PNG) law is designed to guarantee zero miss distance at the real impact point (RIP). Finally, simulations involving four missiles attacking stationary targets simultaneously validate the effectiveness and superiority of the piecewise cooperative guidance law.

Reinforcement learning-based adaptive spiral-diving Manoeuver guidance method for reentry vehicles subject to unknown disturbances

Abstract This paper proposed a reinforcement learning-based adaptive guidance method for a class of spiral-diving manoeuver guidance problems of reentry vehicles subject to unknown disturbances. First, the desired proportional navigation guidance law is designed for the vehicle based on the initial conditions, terminal constraints and the curve involute principle. Then, the first-order multivariable nonlinear guidance command tracking model considering unknown disturbances is established. And the controller design problem caused by the coupling of control variables is overcome by introducing the coordinate transformation technique. Moreover, the actor-critic networks and corresponding adaptive weight update laws are designed to cope with unknown disturbances. With the help of Lyapunov direct method, the stability of the system is proved. Subsequently, the range values of the guidance parameters are analysed. Finally, the validity as well as superiority of the proposed method are verified by numerical simulations.

Design of Convergent and Accurate Guidance Law with Finite Time in Complex Adversarial Scenarios

The target can deceive the flight vehicle by releasing an infrared decoy to make the line-of-sight (LOS) angle rate deflect greatly, thus causing the flight vehicle to miss the target. Therefore, in order to accurately strike the target in complex adversarial scenarios, this paper proposes a finite-time convergence guidance law (FTCG) combined with a finite-time disturbance observer (FTDO). The complex adversarial scenario is established by combining the relative motion model between the flight vehicle and the target and the motion model of the infrared decoy. Based on this, considering the dynamic characteristic of the flight vehicle’s autopilot, a guidance model is obtained. Utilizing sliding mode control theory and finite-time control theory, an FTCG of the LOS angle rate is designed. Then, the finite-time convergence of the guidance law is proved and the total convergence time is derived. Finally, for the target maneuvering that is difficult to measure in the guidance law, an FTDO is used to estimate and compensate for the target maneuvering in the guidance law. Simulation results show that the FTCG can make the LOS angle rate quickly converge and accurately strike the target in different scenarios, with a good guidance accuracy and robustness. Compared with the sliding mode guidance law (SMGL) and the adaptive sliding mode guidance law (ASMGL) based on an extended state observer (ESO), the advantages of the designed guidance law are illustrated. Finally, FTCG is extended to be three dimensional and compared with the proportional navigation guidance law (PNG) to further illustrate its effectiveness in a three-dimensional coordinate system.

Improved Fixed-Time Convergence Guidance Scheme Design and Its Application to Missile

This study presents a fixed-time convergence guidance scheme for impact time and angle control. First, two improved fixed-time stable systems are presented, which have smaller initial control command and better terminal convergence.16 An improved fixed-time extended state observer is proposed to provide accurate estimation of system states and disturbance, which effectively solves peaking value problem. Furthermore, an improved fixed-time sliding mode controller is derived, which avoids the singular problem and achieves faster convergence rate with smaller initial control command. Second, a new guidance scheme with impact angle and impact time constraints is proposed for intercepting a stationary target. By introducing a virtual target, the guidance process is divided into two stages. The proposed fixed-time controller is employed in the first stage. The method with a virtual leader ensures that the missile intercept the virtual target with desired line-of-sight angles at a specific time. By using the proportional navigation guidance law, the missile keeps travelling with desired flight-path angles to hit the real target in the second stage,17 so as to achieve the impact time and angle control. Finally, the feasibility and effectiveness of the proposed guidance scheme in different engagement scenarios are verified by numerical simulations with comparisons.

Optimal Penetration Guidance Law for High-Speed Vehicles against an Interceptor with Modified Proportional Navigation Guidance

Aiming at the penetration problem of high-speed vehicles against a modified proportional guidance interceptor, a three-dimensional mathematical model of attack–defense confrontation between the high-speed vehicle and the interceptor is established in this paper. The modified proportional navigation guidance law of the interceptor is included in the model, and control constraints, pitch angle velocity constraints, and dynamic delay are introduced. Then, the performance index of the optimal penetration of high-speed vehicles is established. Under the condition of considering the 180-degree BTT control, the analytical solutions of the optimal speed roll angle and the optimal overload of high-speed vehicles are obtained according to symmetric Hamilton principle. The simulation results show that the overload switching times of high-speed vehicles to achieve optimal penetration are N − 1, where N is the modified proportional guidance coefficient of the interceptor. When the maximum speed roll angle velocity is [60, 90] degrees per second, the penetration effect of high-speed vehicles is good. Finally, the optimal penetration guidance law proposed in this paper can achieve a miss distance of more than 5 m when the overload capacity ratio is 0.33.

Three-dimensional impact-time-constrained proportional navigation guidance using range-varying gain

This paper presents a novel feedback guidance law to address the impact time control problem of homing missiles in three-dimensional (3D) space. First, a range-varying gain is incorporated into the conventional proportional navigation guidance (PNG) law, which leads to globally precise time-to-go solution and achievable impact time region. Next, an impact time control guidance (ITCG) law is developed by augmenting the proposed varying-gain PNG law with a feedback biased term. Unlike most existing studies, the proposed ITCG law does not involve any linearized approximations or numerical iterative schemes. Furthermore, it can achieve the desired impact time and high-precision interception for a wide launch envelop with arbitrary initial leading angles. Finally, extensive numerical simulations, including comparison studies and Monte Carlo assessments, are carried out to validate the theoretical findings.

Cooperative optimal guidance of hypersonic glide vehicles by real-time optimization and deep learning

This paper investigates the cooperative optimal guidance of hypersonic glide vehicles (HGVs) in the terminal phase with consideration on time-varying velocity, aerodynamic forces, and practical constraints. The cooperative optimal guidance problem is formulated as an optimal control problem with time as the independent variable, which brings great convenience in controlling the impact time. We propose proper convexification techniques to convexify this problem and apply successive convex optimization to get the solution of the original problem. To achieve cooperative guidance of multiple HGVs with time coordination constraint, a deep learning–based approach is proposed to find an optimal common impact time assigned to all the HGVs. The training samples required by deep learning are obtained by convex optimization. An algorithm is then presented to summarize the cooperative optimal guidance strategy. In each guidance loop, the common impact time is updated according to mission conditions, and the guidance commands are generated by the successive solution procedure in real time. Numerical examples will be provided to demonstrate that the proposed cooperative optimal guidance algorithm is effective and efficient, and it can achieve better performance than a popular cooperative proportional navigation guidance law.

Investigation of the Performance Characteristics of the Angle-based Proportional Navigation Guidance Law

Guidance commands generated by the proportional navigation guidance (PNG) law which constitutes the most popular one among the guidance laws applied on the guidance munition fired against predetermined targets are in the form of linear acceleration or angular speed as dictated by the relevant engagement geometry. As a result of the studies in which notable linear acceleration- and angle-based guidance laws are compared, it is seen the lateral acceleration values obtained with the PNG law occur in a lower level than the results of the angle-based guidance laws. However, the angle-based guidance laws lead to lower final miss distances. In this study, the PNG law is so adapted that it yields angle-based guidance commands and then it is applied upon a short-range air-to-surface missile against a maneuvering surface target as well as the velocity pursuit guidance law that is nothing but a version of PNG law, linear homing guidance law, and body pursuit guidance law. After the computer simulations, it is observed that the angle-based PNG law produces smaller final miss distances compared to its original form. The resulting lateral accelerations are in admissible levels. Also, the engagement duration values with the target happen to be almost the same.

Parameter Identification of a PN-Guided Incoming Missile Using an Improved Multiple-Model Mechanism

An active defense against an incoming missile requires information of it, including a guidance law parameter and a first-order lateral time constant. To this end, assuming that a missile with a proportional navigation (PN) guidance law attempts to attack an aerial target with bang-bang evasive maneuvers, a parameter identification model based on the gated recurrent unit (GRU) neural network is built in this paper. The analytic identification solutions for the guidance law parameter and the first-order lateral time constant are derived. The inputs of the identification model are available kinematic information between the aircraft and the missile, while the outputs contain the regression results of missile parameters. To increase the training speed and the identification accuracy of the Model, an output processing method called improved multiplemodel mechanism (IMMM) is proposed in this paper. The effectiveness of IMMM and the performance of the established model are demonstrated through numerical simulations under various engagement scenarios.

Three-Dimensional Impact Angle and Time Control Guidance Law Based on Two-Stage Strategy

In this article, a 3-D nonlinear guidance law is developed for attacking a stationary target with desired impact angle and time. The guidance law is divided into two stages—impact angle control guidance (IACG) and proportional navigation guidance (PNG). In the IACG stage, a sliding-mode guidance law is established to guarantee the satisfaction of the impact angle constraint, and in the second stage, the 3-D PNG law is applied to ensure zero miss distance. The switch between the IACG and the PNG stages is determined by a guidance parameter related to the impact time constraint. By the virtue of Newton iteration method, an appropriate guidance parameter is obtained by solving an implicit nonlinear equality after a small number of iterations. Several numerical simulations are conducted to verify the feasibility and effectiveness of the proposed guidance law under different scenarios.

Varying-Gain Proportional Navigation Guidance for Precise Impact Time Control

This paper proposes a varying-gain proportional navigation guidance (PNG) law that can serve as the baseline for precise control of impact time. First, a varying-gain PNG law is developed by an inverse design strategy from the explicit time-to-go estimation of the conventional PNG law, which leads to high control efficiency and accurate time-to-go prediction. Next, to show its potential significance, the varying-gain PNG law is augmented with a biased feedback command to obtain an impact time control guidance (ITCG) law. As a step forward in more practical applications considering missile speed variation induced by thrust, drag, and gravity, another ITCG law is derived from an efficient time-to-go prediction method by taking the exact trajectory-to-go as the independent variable. Further, the varying-gain PNG law and the planar ITCG laws are extended to the three-dimensional guidance scenario by steering the missile to fly on an engagement plane. Compared with similar existing results, the proposed ITCG laws based on the varying-gain PNG law do not involve any linearized approximations in the derivation, and thus possess higher impact precision, wider feasible region, and less energy consumption. Finally, several numerical simulations are performed to support the analytical findings.

Three-dimensional coverage-based cooperative guidance law with overload constraints to intercept a hypersonic vehicle

This paper proposes a coverage-based cooperative guidance (CBCG) law to enable low-velocity and constrained-manoeuvrability interceptors to intercept a hypersonic vehicle, considering overload constraints. First, the reachable, feasible, and escape areas are considered to establish the guidance law under a nonlinear interception model. Second, the problem of intercepting the hypersonic vehicle is transformed into a coverage problem by using the coverage strategy. A guidance command including basis and bias terms is generated considering the standard trajectory. The basis term generates the guidance command for nonstandard trajectories to nullify the line-of-sight angular rate at the time of interception to ensure successful interception, and the bias term generates the guidance command for the standard trajectory to enhance the coverage ability of the interceptors. The optimal number and location constraints of the interceptors are theoretically analysed. The effectiveness, applicability, superiority, sensitivity, and robustness of the CBCG law are evaluated through a series of numerical simulations, and its performance is compared with that of the proportional navigation guidance law. The results demonstrate that the CBCG law can ensure that at least one interceptor successfully intercepts the hypersonic vehicle, with high robustness to errors and noise.

An Exoatmospheric Homing Guidance Law Based on Deep Q Network

A homing guidance law for exoatmospheric interceptor based on the Deep Q Network (DQN) algorithm is proposed in this paper. Aiming at the exoatmospheric interception problem, the guidance agent is built with the help of the deep reinforcement learning theory, and the action command is given according to the measurement information of the exoatmospheric interceptor for the accurate interception of the target. The homing guidance problem is first transformed into a Markov decision process, and a three-dimensional (3D) interception scenario is established. Then, the reward function considering the line-of-sight (LOS) rate and the final zero-effort-miss (ZEM) is designed, and the homing guidance problem is transferred to the reinforcement learning framework. After that, DQN is utilized to solve the exoatmospheric interception problem, and the guidance agent is obtained through a large amount of training. Finally, the guidance performance of DQN homing guidance law is verified by numerical simulation examples and compared with the classical true proportional navigation (TPN) guidance law. The results show that the guidance performance of the homing guidance law is better than that of TPN.

Dynamic interception point guidance algorithm based on particle swarm optimization

The engagement of target-interceptor is an extremely complicated and nonlinear problem. Most literatures of developed guidance algorithms are hard to work in real-time missile guidance systems because of the complicated design of controllers, restriction in specific condition or excess computing loading. In this paper, the proposed guidance algorithm computes the predicted interception point of target-interceptor and applies particle swarm optimization to optimize the lateral acceleration control commands of missile where the definition of fitness function can guide the missile toward the predicted interception point when the computed fitness value is the minimum. According to the results of simulation experiments, the proposed method has the satisfied target-kill performance to the superior aircraft with high agility. The missile can greatly revise the flight route toward the computed collision course at the initial pursuit stage and the course curve of missile is flatter than the other two guidance laws. Besides, the proposed method can reduce the occurrence of big lateral acceleration control commands acting on the missile to avoid unlocking the evasive target at the terminal stage. As a result, the proposed guidance algorithm based on particle swarm optimization is very effective without using the complicated nonlinear control methods and excess storage burden of computer. It is a simple and feasible missile guidance algorithm due to the advantages of simplicity and effectiveness just like the proportional navigation guidance law but the performance of proposed guidance algorithm is better than proportional navigation guidance law and the other guidance algorithm designed by particle swarm optimization.

Deep reinforcement learning-based impact time control guidance law with constraints on the field-of-view

This paper proposes a novel three-dimensional (3D) intelligent impact time control guidance (ITCG) law with the field-of-view (FOV) strictly constrained based on nonlinear relative motion relationship. This novel guidance law can be utilized to coordinate multiple missiles to attack a target simultaneously. Firstly, the precise time-to-go estimation approach of the 3D coupling proportional navigation guidance (PNG) law is derived, which can improve accuracy of impact time. Secondly, a modified time biased term c⁎ including the guidance gain k1 which can accelerate the convergence rate is incorporated into 3D-PNG by feeding back the FOV error. Therefore, the optimal trajectory can be got to achieve precise impact time, in the meantime, the FOV constraint can be satisfied strictly. Then, the guidance gain k1 is obtained by deep deterministic policy gradient (DDPG) in the reinforcement learning (RL) framework, which makes the proposed guidance law meet smaller time error and has less energy loss. Furthermore, the proposed intelligent law can obtain optimal guidance gains quickly and avoid the tedious manual setting. Ultimately, simulation results demonstrate that missiles can impact the target precisely at desired impact times without violating FOV limit and the optimal guidance gain k1 is obtained adaptively and quickly in various initial conditions.

Multi-missile interception for multi-targets: Dynamic situation assessment, target allocation and cooperative interception in groups

Target allocation is a crucial point in the cooperative interception process. In this paper, a new algorithm of dynamic situation assessment, target allocation and cooperative interception in groups is proposed for multi-target interception with multiple missiles. An effective situation assessment function is proposed, according to which the targets of each intercepting missile are assigned to maximize their efficiency. The new situation assessment function evaluates the flight status and cooperation of missiles, which considers not only the relative position and velocity mentioned in traditional functions, but also relative normal velocity, normal acceleration, maneuverability and cooperation between multiple missiles. During the interception process, the flight status could be evaluated dynamically, thus the allocation of the missiles could be real-time adjusted. To optimize target allocation, a method based on genetic algorithm is applied, which uses the dynamic situation assessment function based proportional navigation guidance law, and we also use proportional navigation guidance law to realize the interception to maneuvering targets with multiple missiles. Simulations for target allocation and target interception validate the effectiveness of the algorithm. Potential adjustments and improvements are proposed according to the simulation results.

Generalized Analysis of Biased Proportional Navigation Guidance with Fractional Power Error Feedback

This study presents a comprehensive analysis of biased proportional navigation guidance laws with variable gain in the bias command that achieve stationary target interception with a desired impact angle under the restriction of permissible look angle. More specifically, this study investigates the bias shaping approach, which takes the bias gain function given by a product of factors in range and look angle. The linear feedback and fractional power feedback of impact-angle error are considered for command structures. For each command form, this study examines the conditions that should be satisfied by the gain functions to guarantee mission accomplishment without violation of the constraints. The analysis particularly focuses on estimating the set of feasible initial or final conditions that ensure compliance with the constraints along the trajectory. Numerical simulation is performed for a range of initial and final flight-path angles to verify the analytical results. The findings of this study provide a general foundation for the design of a nonswitching guidance law for impact-angle control with limited look angle through shaping of gain profiles.

Three-Dimensional Impact Time Control Guidance Considering Field-of-View Constraint and Velocity Variation

The problem of three-dimensional impact time control guidance considering field-of-view constraints and time-varying velocity is investigated in this study. First, considering the effect of gravity and aerodynamic forces on velocity, a simplified numerical estimation algorithm of flight time with a three-dimensional proportional navigation guidance law is derived. Then, based on the structure of the biased proportional navigation guidance law, the effect of the biased term on flight time is analyzed. The biased term is then designed to achieve impact time and field-of-view constraints considering time-varying velocity. Finally, numerical simulations are performed to demonstrate the effectiveness and superiority of the proposed guidance law.