Zero-effort-miss Research Articles

Homing Guidance Law Design against Maneuvering Targets Based on DDPG

A novel homing guidance law against maneuvering targets based on the deep deterministic policy gradient (DDPG) is proposed. The proposed guidance law directly maps the engagement state information to the acceleration of the interceptor, which is an end-to-end guidance policy. Firstly, the kinematic model of the interception process is described as a Markov decision process (MDP) that is applied to the deep reinforcement learning (DRL) algorithm. Then, an environment of training, state, action, and network structure is reasonably designed. Only the measurements of line-of-sight (LOS) angles and LOS rotational rates are used as state inputs, which can greatly simplify the problem of state estimation. Then, considering the LOS rotational rate and zero-effort-miss (ZEM), the Gaussian reward and terminal reward are designed to build a complete training and testing simulation environment. DDPG is used to deal with the RL problem to obtain a guidance law. Finally, the proposed RL guidance law’s performance has been validated using numerical simulation examples. The proposed RL guidance law demonstrated improved performance compared to the classical true proportional navigation (TPN) method and the RL guidance policy using deep-Q-network (DQN).

Inverse Optimal Zero Effort Miss Guidance Based on Disturbance Observer

To intercept a maneuvering target in a two-dimensional plane, the inverse optimal guidance law based on zero effort miss (ZEM) and disturbance observer (DO) is studied in this paper. Firstly, the relative kinematics equation is simplified to obtain the missile-target ZEM and its dynamics. In order to enhance the robustness of the inverse optimal guidance law, the integral of the ZEM is introduced as a new state to form an augmented system with the original system based on the idea of proportional integral (PI) control. Then, the target maneuver acceleration is assumed as the unknown external disturbance of the guidance augmented system, which is estimated by the DO. Based on the estimated value of DO and the backstepping method, the inverse optimal guidance law is designed to reduce the adverse effect of the disturbance on the guidance system. Finally, simulations are designed to verify the effectiveness of the inverse optimal guidance method based on DO.

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.

Optimal predictive sliding-mode guidance law for intercepting near-space hypersonic maneuvering target

The design of optimal guidance law for intercepting a near-space hypersonic maneuvering target with bounded inputs is considered. Firstly, a maneuvering model for near-space hypersonic aircraft is given. Then, the aircraft acceleration prediction can be obtained using this model with two neural networks. By using the target acceleration prediction, which is taken into account when calculating the Zero Effort Miss (ZEM), an optimal sliding-mode guidance law is proposed to fulfill the guidance task. An adaptive sliding-mode switch term is designed to deal with actuator saturation and prediction errors. Finally, numerical simulations show that the proposed guidance law can reduce the energy consumption and the terminal acceleration command of the interceptor effectively.

A Stochastic Characterization of the Capture Zone in Pursuit-Evasion Games

Pursuit-evasion games are used to define guidance strategies for multi-agent planning problems. Although optimal strategies exist for deterministic scenarios, in the case when information about the opponent players is imperfect, it is important to evaluate the effect of uncertainties on the estimated variables. This paper proposes a method to characterize the game space of a pursuit-evasion game under a stochastic perspective. The Mahalanobis distance is used as a metric to determine the levels of confidence in the estimation of the Zero Effort Miss across the capture zone. This information can be used to gain an insight into the guidance strategy. A simulation is carried out to provide numerical results.

PN Modified Differential Game Strategy for Two-on-one Cooperative Interception

In order to deal with the Chaff, Infrared Decoy, Electronic Jamming and the escaping maneuvering of the target, two-on-one cooperative interception strategy is discussed in this paper. The planar engagement of two air-to-air missiles cooperative intercepting one target by the alliance of differential game (DG) strategy and proportional navigation (PN) guidance law is investigated. The equations of motion for two-on-one cooperative intercepting are derived first, and zero-effort-miss (ZEM) representation technique is employed. Then a linear quadratic differential game cooperative zero-sum cost function is phrased, and the cooperative control variables of two-on-one cooperative guidance law under pure DG strategy are derivate. Finally, the proportional navigation guidance law is introduced to alleviate the acceleration command oscillation. The simulation results in the ideal dynamic scenarios show that, the combination of DG and PN guidance law is effective to the two-on-one cooperative pursuit-evasion problem, especially for the high maneuvering target at the cost of much more control effort.

Robust intercept guidance law with predesigned zero-effort miss distance convergence for capturing maneuvering targets

In this paper, a homing guidance in a two-dimensional target-interceptor scenario is studied with an unknown target maneuver strategy. In order to capture the target with acceptable accuracy, a robust guidance law with a predesigned zero-effort miss (ZEM) distance convergence for the interceptor is proposed. In order to guarantee the ZEM distance to converge into a small region around the predesigned terminal value within a finite time, a novel function of ZEM distance is proposed. The ZEM distance convergence profile can be reasonably controlled according to the interceptor's initial state. Based on the designed profile, an initial delay control and prescribed performance control are utilized to achieve a high terminal miss distance accuracy and account for uncertainties caused by the target's unknown maneuverability. Four maneuvering strategies of the target are tested to verify the performance of the proposed guidance law.

Pursuer’s Control Strategy for Orbital Pursuit-Evasion-Defense Game with Continuous Low Thrust Propulsion

This paper studies the orbital pursuit-evasion-defense problem with the continuous low thrust propulsion. A control strategy for the pursuer is proposed based on the fuzzy comprehensive evaluation and the differential game. First, the system is described by the Lawden’s equations, and simplified by introducing the relative state variables and the zero effort miss (ZEM) variables. Then, the objective function of the pursuer is designed based on the fuzzy comprehensive evaluation, and the analytical necessary conditions for the optimal control strategy are presented. Finally, a hybrid method combining the multi-objective genetic algorithm and the multiple shooting method is proposed to obtain the solution of the orbital pursuit-evasion-defense problem. The simulation results show that the proposed control strategy can handle the orbital pursuit-evasion-defense problem effectively.

Error distribution of zero-effort miss distance under mode mismatch

ABSTRACT Zero-effort miss (ZEM) distance is a critical parameter for many kinds of guidance laws. For highly manoeuvring target interception, ZEM becomes a random variable with an unknown prior due to noise-corrupted measurements and unknown bounded target manoeuvres. Currently, the distribution of ZEM is usually evaluated by the method of Monte Carlo simulation. In this paper, an analytical approach is proposed under the configuration of a linear estimator with a separate mode decision-maker. The resulting ZEM estimation error is subject to a biased Gaussian distribution when taking a fixed mode decision delay of the mode decision-maker into consideration. Finally, the analytical distribution is validated by comparison with Monte Carlo simulation and some meaningful use cases are discussed. The results in this paper give an insight to the design, analysis, and performance evaluation of the guidance system.

Integrated Estimation/Guidance Law against Exoatmospheric Maneuvering Targets

An integrated guidance integrated estimation/guidance law is designed for exoatmospheric interceptors equipped with divert thrusters and optical seekers to intercept maneuvering targets. This paper considers an angles-only guidance problem for exoatmospheric maneuvering targets. A bounded differential game-based guidance law is derived against maneuvering targets using zero-effort-miss (ZEM). Estimators based the extended Kalman filter (EKF) and the unscented Kalman filter (UKF) are designed to estimate LOS rates that are contaminated by noise and target maneuver. Furthermore, to improve the observability of the range, an observability enhancement differential game guidance law is derived. The guidance law and the estimator are integrated together in the guidance loop. The proposed integrated estimation/guidance law has been tested in several three-dimensional nonlinear interception scenarios. Numerical simulations on a set of Monte-Carlo simulations prove the validity and superiority of the proposed guidance law in hit-to-kill interception.

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.

Differential game strategy in three-player evasion and pursuit scenarios

A conflict of three players, including an attacker, a defender, and a target with bounded control is discussed based on the differential game theories in which the target and the defender use an optimal pursuit strategy. The current approach chooses the miss distance as the outcome of the conflict. Different optimal guidance laws are investigated, and feasible conditions are analyzed for the attacker to accomplish an attacking task. For some given conditions, the attacker cannot intercept the target by only using a one-to-one optimal pursuit guidance law; thus, a guidance law for the attacker to reach a critical safe value is investigated. Specifically, the guidance law is divided into two parts. Before the engagement time between the defender and the attacker, the attacker uses this derived guidance law to guarantee that the evasion distance from the defender is safe, and that the zero-effort-miss (ZEM) distance between the attacker and the target is the smallest. After that engagement time, the attacker uses the optimal one-toone guidance law to accomplish the pursuit task. The advantages and limited conditions of these derived guidance laws are also investigated by using nonlinear simulations.

Zero-Effort-Miss Shaping Guidance Laws

This paper suggests a new approach in designing homing guidance laws to enable direct shaping of the pattern of zero-effort-miss (ZEM) as desired. The proposed approach uses the concept of weighted ZEM and its specific desired error dynamics: the former is to provide an additional degree of freedom in shaping actual ZEM and the latter is to guarantee a finite-time convergence. Utilization of these two concepts allows simple determination of the guidance law that can achieve the desired pattern of ZEM. The resultant guidance law is shown a type of proportional navigation guidance law with the specific form of time-varying gain not revealed in previous studies. It provides unique information on how the time-varying gain should be shaped to obtain the desired pattern of ZEM. Accordingly, the resultant guidance laws can cope with various operational objectives in a more direct way compared with the previously existing approaches. This paper also performs theoretical analysis to investigate the properties of designed guidance laws including the closed-loop solutions of ZEM and acceleration command. Also, we determine the feasible set of desired ZEM patterns that can be achieved in the proposed framework. Two illustrative examples are considered to show how to design guidance laws using the proposed approach. Moreover, the characteristics of the guidance laws designed are validated and demonstrated via numerical simulations.

Evasion and pursuit guidance law against defended target

This paper investigates a new approach for a scenario in which an Attacker attempts to intercept a defended aerial Target. The problem is formulated as a game among three players, an Attacker, a Defender, and a Target, with bounded controls. In the considered pursuit–evasion problem, the Target uses an optimal evasion strategy and the Defender uses an optimal pursuit strategy. The proposed approach focuses on the miss distance as the outcome of the conflict. The infeasible region for the initial Zero-Effort-Miss (ZEM) distance between the Attacker and the Defender, for a scenario in which the Attacker evades the Defender, is analyzed, assuming that the Attacker uses a control effort chosen from the permitted control region. The sufficient conditions are investigated under which, for ideal players, the Attacker can pursue the Target while evading the Defender launched by the Target. The guidance provided on how the Attacker can accomplish the task is divided into two parts. During the final time between the Attacker and the Defender, the Attacker chooses the control effort that guarantees the miss distance, and then uses the optimal pursuit strategy to accomplish the task. The derived guidance law is verified by nonlinear simulation.

Collision avoidance ZEM/ZEV optimal feedback guidance for powered descent phase of landing on Mars

A novel zero-effort-miss (ZEM)/zero-effort-velocity (ZEV) optimal feedback guidance is proposed in order to rule out the possibility of Martian surface collision caused by the classical ZEM/ZEV optimal feedback guidance. The main approach is to add a collision avoidance term, which has self-adjustment capacity to ensure the near fuel optimality. Its main improvement is that it can not only successfully avoid collisions with the thruster constraint but also guarantee the near fuel optimality, and both of them are pivotal performances in Mars landing missions. Simulations are made to show the effectiveness of the proposed guidance and the parameters effects are simulated as well to analyze the properties of the proposed guidance.

One active debris removal control system design and error analysis

The increasing expansion of debris presents a significant challenge to space safety and sustainability. To address it, active debris removal, usually involving a chaser performing autonomous rendezvous with targeted debris to be removed is a feasible solution. In this paper, we explore a mid-range autonomous rendezvous control system based on augmented proportional navigation (APN), establishing a three-dimensional kinematic equation set constructed in a rotating coordinate system. In APN, feedback control is applied in the direction of line of sight (LOS), thus analytical solutions of LOS rate and relative motion are expectedly obtained.To evaluate the effectiveness of the control system, we adopt Zero-Effort-Miss (ZEM) in this research as the index, the uncertainty of which is directly determined by that of LOS rate. Accordingly, we apply covariance analysis (CA) method to analyze the propagation of LOS rate uncertainty.Consequently, we find that the accuracy of the control system can be verified even with uncertainty and the CA method is drastically more computationally efficient compared with nonlinear Monte-Carlo method. Additionally, to justify the superiority of the system, we further discuss more simulation cases to show the robustness and feasibility of APN proposed in the paper.

3D-nonlinear vector guidance and exo-atmospheric interception

This paper presents a nonlinear 3D-vector guidance law. It is based on kinematic vector equations, where the commanded thrust is bounded, and the interceptor does not know the target evasive strategy. The resulting game theoretic guidance law with miss-distance as the cost, is nonlinear, aligned along the zero-effort-miss (ZEM) direction, and the time-to-go is a solution of a quartic polynomial equation in feedback form. This time-to-go is the actual one provided both players play optimal. If the evader does not play optimal, capture occurs earlier. Moreover, optimal strategies in line-of-sight (LOS) coordinates are also presented. It is found that these strategies are functions of the range, range rate, and the LOS rate. Simulations demonstrate the possibility of a miss at the first pass, but eventual capture.

Improved ZEM/ZEV feedback guidance for Mars powered descent phase

An improved zero-effort-miss (ZEM) and zero-effort-velocity (ZEV) optimal feedback guidance algorithm is proposed in this paper for Mars powered descent phase, which has taken into account the no-subsurface constraints when designing the guidance law. Detailed descriptions have been given to this algorithm. The main advantage of this optimal guidance law is that it can successfully avoid collision to the Martian surface, at the same time preserving high landing precision and fuel optimality. For future Mars landing mission, the thrust-limited engine is most likely to be used. We propose a simulation method to study and determine the optimal final time which can minimize the fuel usage while ensuring precision landing and no-subsurface constraints. Monte Carlos are run to test the robustness and reliability of the improved and original guidance algorithms which show the superiority of the improved guidance law in preventing collision to the surface over the original one.

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.

Linear Quadratic Differential Game Strategies with Two-pursuit Versus Single-evader

In order to intercept the future targets that are characterized by high maneuverability, multiple interceptors may be launched and aimed at single target. The scenario of two missiles P and Q intercepting a single target is modeled as a two-pursuit single-evader non-zero-sum linear quadratic differential game. The intercept space is decomposed into three subspaces which are mutually disjoint and their union covers the entire intercept space. The effect of adding the second interceptor arises in the intercept space of both P and Q (PQ-intercept space). A guidance law is derived from the Nash equilibrium strategy set (NESS) of the game. Simulation studies are focused on the PQ-intercept space. It is indicated that 1) increasing the target's maneuverability will enlarge PQ-intercept space; 2) the handover conditions will be released if the initial zero-effort-miss (ZEM) of both interceptors has opposite sign; 3) overvaluation of the target's maneuverability by choosing a small weight coefficient will generate robust performance with respect to the target maneuvering command switch time and decrease the fuel requirement; and 4) cooperation between interceptors increases the interception probability.