Proportional Navigation Guidance Research Articles

Three-Dimensional Nonsingular Cooperative Guidance Law with Different Field-of-View Constraints

In this paper, a distributed three-dimensional (3D) nonsingular cooperative guidance law is designed for multiple missiles with different seeker’s field-of-view (FOV) constraints to achieve salvo attack against a stationary target. The guidance law is derived from an extended biased proportional navigation guidance (PNG). The PNG is augmented by a cooperative guidance term to ensure that times-to-go of the missile group reach consensus in a fixed time before interception. Two auxiliary functions are included into the biased guidance term to guarantee nonsingularity of the guidance law and meet the FOV constraints. The proposed design does not require guidance law switching that is used in some existing biased or multistage guidance designs, which allows for a smooth guidance command. A feasible impact time set is given for the cooperative mission when applying the proposed guidance law. The nonsingularity, FOV constraint satisfaction, and closed-loop stability of the proposed guidance law are theoretically analyzed and proved. Moreover, the criteria for selecting all guidance parameters are provided to facilitate the guidance design. Finally, numerical simulations with comparative studies are conducted to verify the effectiveness of the proposed guidance law and demonstrate the advantages over existing cooperative guidance laws.

Modified three‐dimensional true proportional navigation and its inverse optimal form

AbstractA novel three‐dimensional modified true proportional navigation (MTPN) guidance and its inverse optimal form are proposed for the interception of the nonmaneuvering and maneuvering targets, and the nonlinear dynamic characteristics of pursuit situations and target maneuvers are taken into full account. The proposed approach is feasible without knowing any prior information or estimate of the target maneuvers, which are regarded as the unknown disturbance input. Therefore, the TPN‐type guidance is extended to intercept the maneuvering targets with bounded unknown accelerations. Moreover, the explicit physical significance is given for each component of the navigation gain expression of MTPN. The inverse optimal control approach and the input‐to‐state stability are introduced to the MTPN design. Using the MTPN, the homing missile can conduct the terminal homing guidance phase in any particular direction with respect to their nonmaneuvering or maneuvering targets through choosing an appropriate inertial reference coordinate system in practice. Using the proposed schemes, the line‐of‐sight (LOS) rate is globally exponentially stable for nonmaneuvering targets, and is input‐to‐state stability (ISS) for maneuvering targets with bounded unknown accelerations. Performed simulation results have confirmed the effectiveness of the proposed scheme.

A Combined Linear–Quadratic/Bounded Control Differential Game Guidance Law

A new guidance law, based on the solution of a combined linear–quadratic and norm-bounded differential game, is proposed. It hinges on the insight that the hard constraints on the lateral acceleration are critical in the last part of the flight, and in contrast, at the early stages of the flight, it is important to avoid excessive speed loss due to induced drag. This yields a linear–quadratic differential game formulation for the initial stage and a norm-bounded differential game formulation for the last stage of the engagement. In an important particular case of players with ideal dynamics, the proposed guidance law is shown to be biased proportional navigation guidance with a deadzone. Due to its simple structure, the proposed guidance law is easy to implement in practice. The performance of the guidance law is illustrated using numerical simulations, and it is shown to perform better than comparable classical guidance laws.

Typical Fault Estimation and Dynamic Analysis of a Leader-Follower Unmanned Aerial Vehicle Formation

In this paper, the typical fault estimation and dynamic analysis are presented for a leader-follower unmanned aerial vehicle (UAV) formation system with external disturbances. Firstly, a dynamic model with proportional navigation guidance (PNG) control of the UAV formation is built. Then, an intermediate observer design method is adopted to estimate the system states and faults simultaneously. Based on the graph theory, the topology relationship between each node in the UAV formation has been also analyzed. The estimator and the system error have been created. Moreover, the typical faults, including the components failure, airframe damage, communication failure, formation collision, and environmental impact, are also discussed for the UAV system. Based on the fault-tolerant strategy, five familiar fault models are proposed from the perspectives of fault estimation, dynamical disturbances, and formation cooperative control. With an analysis of the results of states and faults estimation, the actuator faults can be estimated precisely with component failure and wind disturbances. Furthermore, the basic dynamic characteristics of the UAV formation are discussed. Besides, a comparison of two cases related to the wind disturbance has been accomplished to verify the performance of the fault estimator and controller. The results illustrate the credibility and applicability of the fault estimation and dynamic control strategies for the UAV system which are proposed in this paper. Finally, an extension about the UAV formation prognostic health management system is expounded from the point of view of the fault-tolerant control, dynamic modeling, and multifault estimation.

An Improvement in Three-Dimensional Pure Proportional Navigation Guidance

This article proposes an improved version of 3-D pure proportional navigation (PPN) against a manoeuvring target. The main research hypothesis is that the performance of 3-D PPN can be improved by properly selecting the direction of the guidance command as there exists an infinite number of potential directions complying with the PPN concept in 3-D space. Analysis on the relative motion confirms the validity of the hypothesis and leads to the development of a new guidance algorithm. Unlike traditional 3-D PPN, the guidance algorithm developed adapts the direction, but maintains the magnitude of the commanded acceleration proportional to only the line-of-sight rate. The validity and performance of the proposed guidance algorithm are investigated through theoretical analysis and numerical simulations.

Adaptive True Proportional Navigation Guidance Based On Heuristic Optimization Algorithms

The PN-guidance (Proportional Guidance) still continues to be improved, because it is the simplest, cheapest and most reliable algorithm. One of the most important techniques to improve PN-guidance is to adapt the navigation constant depending on time. In this study, first, the entire adaptation methods for PN-guidance are classified, then the adaptation process is online achieved by using heuristic optimization during guiding the missile. The novelty of this study is that the heuristic optimization approach is used at the first time to update the navigation constant of PN-guidance. It is considered that having short program code, fast convergence speed and just simple algebraic computations without derivative are vital advantages of heuristic algorithms using into missile systems. In this scope, an Adaptive True-PN (ATPN) guidance algorithm is designed by optimizing navigation constants varying according to the target behavior. The results show that while the acceleration gap of the pitch axis decreases 21.8%, the acceleration gap of yaw axis reduces 39.68%. These reductions mean that while the missile guided by ATPN is maneuvering, it is exposed to less acceleration and less strain.

Attack behaviour in naive gyrfalcons is modelled by the same guidance law as in peregrine falcons, but at a lower guidance gain.

ABSTRACTThe aerial hunting behaviours of birds are strongly influenced by flight morphology and ecology, but little is known of how this relates to the behavioural algorithms guiding flight. Here, we used GPS loggers to record the attack trajectories of captive-bred gyrfalcons (Falco rusticolus) during their maiden flights against robotic aerial targets, which we compared with existing flight data from peregrine falcons (Falco peregrinus). The attack trajectories of both species were well modelled by a proportional navigation (PN) guidance law, which commands turning in proportion to the angular rate of the line-of-sight to target, at a guidance gain N. However, naive gyrfalcons operate at significantly lower values of N than peregrine falcons, producing slower turning and a longer path to intercept. Gyrfalcons are less manoeuvrable than peregrine falcons, but physical constraint is insufficient to explain the lower values of N we found, which may reflect either the inexperience of the individual birds or ecological adaptation at the species level. For example, low values of N promote the tail-chasing behaviour that is typical of wild gyrfalcons and which apparently serves to tire their prey in a prolonged high-speed pursuit. Likewise, during close pursuit of typical fast evasive prey, PN will be less prone to being thrown off by erratic target manoeuvres at low guidance gain. The fact that low-gain PN successfully models the maiden attack flights of gyrfalcons suggests that this behavioural algorithm is embedded in a guidance pathway ancestral to the clade containing gyrfalcons and peregrine falcons, though perhaps with much deeper evolutionary origins.

Quaternion Based Three-Dimensional Impact Angle Control Guidance Law

This article deals with a guidance law for impact angle control (IAC) in the 3-D space. Extended from the 2-D IAC law, the proposed guidance law is derived using a quaternion. The general form of the IAC law comprises a biased proportional navigation guidance (PNG) law, wherein a component to control the impact angle is added to the PNG law. Since the biased term consists of a rotation angle and Euler axis obtained by a quaternion coordinate transformation between the missile's velocity vector and the desired impact angle, the magnitude of the guidance command depends on the rotation angle multiplied by the biased term. Therefore, from attempting to minimize the control energy, the optimal impact roll angle causing the rotation angle to be zero is mathematically derived. The Euler axis that satisfies this optimal impact roll angle is composed of two velocity vectors; the angle between the two velocity vectors is the minimum rotation angle. Numerical simulations are performed for stationary and moving targets according to the desired impact angle.

$${H}_{\infty }$$ Approach to Performance Analysis of Missile Control Systems with Proportional Navigation Guidance Laws

This paper presents a performance analysis of missile control systems considering nonlinear pursuit dynamics and proportional navigation guidance (PNG) laws. In order to analyze the system stability, missile dynamics are simplified as a linearized model which is often considered in missile autopilot design and the well-known $${{{H}}}_{\boldsymbol{\infty }}$$ approach is employed. A sufficient condition guaranteeing miss distance and the closed loop stability is proposed. The provided condition of missile system stability can be numerically easily verified by a linear matrix inequality (LMI).

A New Optimal Guidance Law with Impact Time and Angle Constraints Based on Sequential Convex Programming

This paper proposed an optimal time-varying proportional navigation guidance law based on sequential convex programming. The guidance law can achieve the desired impact angle and impact time with look angle and lateral acceleration constraints. By treating the multiconstraints’ guidance problem as an optimization problem and changing the independent variable to linearize the problem and constraints, the original nonlinear and nonconvex problem is transformed into a series of convex optimization problem so that it can be quickly solved by sequential convex programming. Numerical simulations compared to nonlinear programming and traditional analytical guidance law demonstrate the effectiveness and efficiency of the proposed algorithm. Finally, the proposed guidance law is verified to satisfy different impact time periods and impact angle constraints.

A New Guidance Algorithm Against High-Speed Maneuvering Target

This paper aims to propose a new guidance algorithm for intercepting a high-speed maneuvering target, such as a tactical ballistic missile that flies in a quasi-ballistic trajectory. The motivation of this study lies in the fact that the classical proportional navigation guidance (PNG) undergoes the performance degradation under the above engagement scenario due to its intrinsic property. To deal with the issue, an accurate collision course that reflects the motion characteristics of the ballistic target and missile is first analyzed. The desired look angle that leads to achieving the accurate collision course is then determined with the help of a novel time-to-go calculation method. Finally, the proposed guidance law providing the desired look angle is obtained by leveraging the concept of the biased PNG. Since the proposed method is designed to achieve an accurate collision course, it does not lead to unnecessary maneuvering near the target under the engagement scenario. This property is desirable for reserving the operational margin of maneuverability for reacting to unexpected situations during the engagement. It could improve the capturability of the target. Finally, the performance of the proposed method is verified through numerical simulations in a way to compare with the existing methods such as PNG and the augmented PNG (APNG).

Cooperative Guidance for Multiple Powered Missiles with Constrained Impact and Bounded Speed

In this paper, a cooperative guidance law is proposed for missiles with thrust control to realize impact angle constrained salvo attack, where missile speed limit is also considered. Guidance of a powered missile is decoupled into independent flight direction control with normal acceleration as an input to realize impact angle control/constrained guidance (IACG) and cooperative flight speed control with engine thrust as an input to realize salvo attack. Firstly, an IACG law is derived from a geometrically modified proportional navigation guidance. The trajectory shape and length under this proposed IACG law are proved to be independent of missile speed variation. Then a cooperative missile thrust control (CMTC) law is designed from a constrained consensus algorithm to cooperate remaining trajectory lengths without exceeding missile speed limit. Simplified aerodynamic and engine models of a missile are used in CMTC law design. The lower and upper bounds of speed are considered according to missile maneuverability and limited maximum engine thrust. Multimissile salvo attack with impact angle and time constraints is achieved through the combination of IACG and CMTC. Numerical simulations demonstrate the effectiveness of the proposed cooperative guidance law.

Deep Neural Network-Based Guidance Law Using Supervised Learning

This paper proposes that the deep neural network-based guidance (DNNG) law replace the proportional navigation guidance (PNG) law. This approach is performed by adopting a supervised learning (SL) method using a large amount of simulation data from the missile system with PNG. Then, the proposed DNNG is compared with the PNG, and its performance is evaluated via the hitting rate and the energy function. In addition, the DNN-based only line-of-sight (LOS) rate input guidance (DNNLG) law, in which only the LOS rate is an input variable, is introduced and compared with the PN and DNNG laws. Then, the DNNG and DNNLG laws examine behavior in an initial position other than the training data.

Cooperative guidance law for active aircraft defense with intercept angle constraint

In this paper, the three-body engagement scenario is considered where a target aircraft fires a defender missile to intercept the attacker missile. A cooperative guidance law with the intercept angle constraint for both of the defender missile and target has been presented. With the assumption that the attacker missile uses augmented proportional navigation guidance law, the nonlinear relative motion model of target-attacker-defender engagement is built. Considering the requirement of miss distance and satisfying the intercept angle constraint, the function index is established. The cooperative guidance law is derived based on optimal control theory. Moreover, given initial launch condition, the feasible intercept angle region of defender is analyzed, considering the limited maneuverability of defender and target and the intercept time constraint which means the attacker must be intercepted by the defender prior to hitting the target. Similarly, the feasible launch region of defender is obtained with the given designated intercept angle, variable overload of defender and target, and intercept time constraint. The simulation results further demonstrate that within the feasible region of designated intercept angle and launching condition, the defender can intercept the attacker with designated intercept angle successfully despite of the limited maneuverability. Compared with conventional uncooperative situation, the target-defender cooperation could significantly reduce the maneuverability requirements for the defender.

Integrated guidance and control framework for the waypoint navigation of a miniature aircraft with highly coupled longitudinal and lateral dynamics

A solution to the waypoint navigation problem for the fixed wing micro air vehicles (MAV) having a severe coupling between longitudinal and lateral dynamics, in the framework of integrated guidance and control (IGC) is addressed in this paper. IGC yields a single step solution to the waypoint navigation problem, unlike conventional multiple loop design. The pure proportional navigation (PPN) guidance law is integrated with the coupled MAV dynamics. A multivariable static output feedback (SOF) controller is designed for the linear state space model formulated in IGC framework. A waypoint navigation algorithm is proposed that handles the minimum turn radius constraint of the MAV and also evaluates the feasibility of reaching a waypoint. Non-linear simulations with and without wind disturbances are performed on a high fidelity 150 mm wingspan MAV model to demonstrate the proposed waypoint navigation algorithm.

Analysis of the Influence of Trace Point Step-Jump Behavior

To explore the influence of the trace point step-jump behavior on a terminal guidance system, an analysis is performed from the line-of-sight rate (LOS rate) and guidance accuracy views for designing an anti-step-jump guidance law. First, the linear terminal guidance model under the trace point jump circumstance is constructed, and then the fundamental reason for the miss distance is investigated by deriving the upper bound of the LOS rate at the initial step-jump moment. Following this, the novel proposed analytical differential adjoint model is established with the adjoint method, and its validity is demonstrated comparing with the numeric derivative model. Based on the adjoint model, the effects of the ratio coefficient, the time constant, and the jump amplitude on the guidance accuracy are explored. Finally, a novel anti-step-jump guidance law is designed to shorten the recovery time of the overload. The simulations have shown that the faster recovery time and higher accuracy are achieved in comparison with the proportional navigation guidance, optimal guidance, and adaptive sliding mode guidance.

Study on Reinforcement Learning-Based Missile Guidance Law

Reinforcement learning is generating considerable interest in terms of building guidance law and solving optimization problems that were previously difficult to solve. Since reinforcement learning-based guidance laws often show better robustness than a previously optimized algorithm, several studies have been carried out on the subject. This paper presents a new approach to training missile guidance law by reinforcement learning and introducing some notable characteristics. The novel missile guidance law shows better robustness to the controller-model compared to the proportional navigation guidance. The neural network in this paper has identical inputs with proportional navigation guidance, which makes the comparison fair, distinguishing it from other research. The proposed guidance law will be compared to the proportional navigation guidance, which is widely known as quasi-optimal of missile guidance law. Our work aims to find effective missile training methods through reinforcement learning, and how better the new method is. Additionally, with the derived policy, we contemplated which is better, and in which circumstances it is better. A novel methodology for the training will be proposed first, and the performance comparison results will be continued therefrom.

A nonlinear robust model predictive differential game guidance algorithm based on the particle swarm optimization

Two-dimensional engagement of a pursuer and a maneuvering target, affected by matched uncertainties, is formulated as a nonlinear differential game. The uncertain guidance problem is converted into a nonlinear model predictive control problem by introducing an appropriate cost function. The objective is to calculate the best guidance commands of the pursuer and the worst possible target maneuvers simultaneously, over a receding horizon. The proposed cost function penalizes the line-of-sight rate, the pursuer acceleration, and the uncertainties. It also rewards the target maneuver. A particle swarm-based dynamic optimization algorithm is developed to solve the nonlinear model predictive differential game, affected by the uncertainties. Performance of the proposed guidance algorithm is evaluated against maneuvering and non-maneuvering targets. The algorithm is also evaluated for the cases when the pursuer has a high initial heading error, and the guidance command is constrained. The statistical performance of the proposed algorithm is evaluated using Monte Carlo simulation. Moreover, a processor in the loop experiment is performed to verify the implementation capability of the proposed algorithm. Finally, a comparison is made between the performance of the suggested algorithm with some other methods including linear- quadratic differential game, state-dependent Riccati equation-differential game, a guidance law on the basis of adaptive dynamic programming, a proportional navigation guidance improved by particle swarm optimization, a guidance algorithm based on the continuous ant colony controller, switched bias proportional navigation, and augmented proportional navigation.

Time-varying Proportional Navigation Guidance using Deep Reinforcement Learning

Time-varying Proportional Navigation Guidance using Deep Reinforcement Learning

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 three-dimensional 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.