Biased Proportional Navigation Research Articles

Enhanced Computational Biased Proportional Navigation with Neural Networks for Impact Time Control

Advanced computational methods are being applied to address traditional guidance problems, yet research is still ongoing regarding how to utilize them effectively and scientifically. A numerical root-finding method was proposed to determine the bias in biased proportional navigation to achieve the impact time control without time-to-go estimation. However, the root-finding algorithm in the original method might experience efficiency and convergence issues. This paper introduces an enhanced method based on neural networks, where the bias is directly output by the neural networks, significantly improving computational efficiency and addressing convergence issues. The novelty of this method lies in the development of a reasonable structure that appropriately integrates off-the-shelf machine learning techniques to effectively enhance the original iteration-based methods. In addition to demonstrating its effectiveness and performance of its own, two comparative scenarios are presented: (a) Evaluate the time consumption when both the proposed and the original methods operate at the same update frequency. (b) Compare the achievable update frequencies of both methods under the condition of equal real-world time usage.

Three-dimensional free-will arbitrary time cooperative guidance law against moving target

A novel three-dimensional free-will arbitrary time cooperative guidance law based on biased proportional navigation is proposed for intercepting a high-value moving target. In contrast to existing research, the presented method does not require controlling the interceptors' axial speed or prediction of the interception point. First, the engagement plane is constructed so that a three-dimensional guidance problem can be formulated as a planar problem. Second, a new accurate time-to-go estimation algorithm against the moving target is deduced based on the formulated problem model. Third, a free-will arbitrary time cooperative guidance law is designed utilizing the leader-follower architecture. Multiple missiles can intercept a moving target simultaneously and the intercept time of each missile will converge within a free-will arbitrary time. Fourth, the proposed cooperative guidance law under different communication circumstances is analyzed. When the communication resources are abundant, missiles can reach dynamic average consensus. Meanwhile, the system is robust against the communication imperfection. Finally, numerical simulations are presented to verify the effectiveness of the time-to-go estimation algorithm and proposed cooperative guidance law.

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.

Three‐dimensional cooperative guidance for simultaneous attack with specified impact time based on fixed‐time convergence

AbstractIn this paper, distributed three‐dimensional (3D) cooperative guidance laws are designed for leader–follower multiple missiles to achieve simultaneous attack at a specified time. The guidance laws are derived from the biased proportional navigation guidance (PNG) law. Two fixed‐time distributed algorithms with time‐varying coefficients are proposed to track the state of the leader under undirected and directed topologies, respectively. The upper bounds of convergence times for the tracking algorithms are obtained. The biased guidance terms of followers are designed to track the leader with the proposed algorithms and the biased guidance term of the leader is designed to achieve specified impact time. Then, an auxiliary function is included into the biased guidance terms to meet the field‐of‐view (FOV) constraints. The leading angles' properties and convergence times under the proposed guidance laws are further analyzed. Numerical simulations are given to verify the effectiveness of the proposed cooperative guidance laws.

Dynamic Line of Sight Shaping for Impact Time and Angle Control Guidance

This paper presents a novel guidance law enabling to control together of the impact time and angle for planar engagement scenarios. The developed guidance command has a form of biased proportional navigation, and the bias term employs the angle and time constraints at the terminal phase. Specifically, the bias term with nonlinear expression contains the second derivative of the line of sight angle. Thus, shaping a Bézier curve with the C2 continuity guarantee allows us to meet final constraints. Besides, one extra design variable is needed to remove the impact time error related to angle error arising from nonlinearity. Control point locations of resulting 4th degree Bézier curve are determined by boundary conditions, except one being determined using an offline Newton iteration method to eliminate the error. Simulation studies, including a comparison study, are conducted to show the effectiveness of the proposed ITACG law.

A New United Proportional Navigation Guidance for Impact Angle Constraint without Measurement Distance between Vehicle and Target

This paper proposed a united proportional navigation guidance (UPNG) method to alleviate the guidance command saltation with an impact angle constraint under the condition of no real-time distance between the vehicle and the target (line-of-sight (LOS) distance). Firstly, based on the biased proportional navigation guidance (BPNG), a smooth-biased proportional navigation guidance (SBPNG) method was proposed, whose bias term was designed as a trigonometric function. In SBPNG method, due to the continuous smooth change of the bias term, the guidance command would not saltus anymore, and the impact angle was controlled by the bias integral component. Secondly, biased on SBPNG method, the united proportional navigation guidance (UPNG) method combining SBPNG and variable coefficient proportional navigation guidance (VCPNG) was established. In UPNG method, because there was no LOS distance, the guidance coefficient was designed as a function of the difference between the expected impact angle and the estimated impact angle, so the closed-loop control of impact angle was realized. Finally, a lot of simulation experiments on different guidance laws were carried out without real-time LOS distance. The results verify that the UPNG method proposed in this paper solves the problem of guidance command saltation effectively and has better robustness in impact angle control.

Rocket landing guidance using convex optimization and proportional navigation considering performance-limited engine

In recent years, convex optimization has been widely applied in aircraft trajectory optimization and guidance. For rocket vertical landing, the dynamics are highly nonlinear, which easily causes convergence issues in convex optimization. Moreover, for the existing convex optimization methods, it is difficult to implement landing guidance with a performance-limited engine in which the thrust cannot be continuously and instantly regulated. To address these two issues, a novel rocket vertical landing guidance method is developed, in which the guidance in the normal and tangential planes with respect to the rocket velocity is separated to enhance the convergence property of convex optimization. Normal guidance adopts biased proportional navigation to easily satisfy the pinpoint landing with the required attitude angles. For tangential guidance, convex optimization and model predictive control are employed to optimize the thrust command that satisfies the landing velocity limit. Based on this, an impulse equivalent transformation method is designed to derive thrust commands that satisfy the engine limits. Simulation results indicate that the proposed guidance method can achieve precise landing and satisfy various constraints while being highly robust to uncertainties. Moreover, the efficiency, convergence property, and control smoothness are significantly improved.

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.

Nonlinear modified bias proportional navigation guidance law against maneuvering targets

To realize the terminal acceleration constraint for a bias proportional navigation guidance law without usage of switching logics, this paper proposes a modified bias term and presents a terminal acceleration constrained bias proportional navigation guidance law against maneuvering targets. First, a so-called virtual planar coordinates whose origin is attached to the point mass of the target is built, so that the original maneuvering target is transformed to a virtual stationary target. On this basis, the common structure of bias proportional navigation guidance law is presented. To realize the terminal acceleration constraint, a modified bias term related to the relative distance between target and missile is used to improve the bias proportional navigation guidance law. With the virtual look angle and the line-of-sight angle constrained, the proposed modified bias proportional navigation guidance law can intercept the maneuvering targets in a desired attack angle. Comparisons with the optimal guidance law in the linear system are carried out, and the proposed law is proved to be near-optimal. The numerical simulation results demonstrate the all-aspect interception capability of the proposed law against maneuvering targets.

Three-Dimensional Biased Proportional Navigation Guidance Based on Spatial Rotation of Predicted Final Velocity

This study presents the design of three-dimensional biased proportional navigation guidance laws for arrival at a stationary target along a desired direction based on spatial rotation of predicted final velocity vector. The focus is on full constructive derivation using vector-form expressions without introducing local representation of rotation such as Euler angles or quaternions. The proposed approach synthesises the bias command in the form of an angular velocity vector through realisation of the predictive control design philosophy, the direction which has been unexplored in a three-dimensional setting. The proposed approach avoids heuristic choices and approximations in the design process and hence overcomes the limitation of earlier studies. The vector-form design approach provides theoretical and practical advantages including rigour in derivation, clear geometric understandings about the problem provided by identification of the most effective direction for rotation of final velocity, independence from selection of a fixed coordinate system, avoidance of singularities in local representations, more direct trajectory shaping, and simple implementation.

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.

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.

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.

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.

Field-of-view-constrained impact angle control guidance with error convergence before interception considering speed changes

An impact angle control guidance law is proposed for stationary target interception considering missile's field-of-view limit and speed changes. The proposed impact angle control guidance law is structured as a biased proportional navigation with a time-varying bias. The proposed guidance law does not involve any switching logic for maintaining lock-on; hence, the guidance command is continuous during the entire engagement. Unlike the most existing studies, the proposed method guarantees that the impact angle error converges to zero before interception without the constant-speed assumption. To realize these desirable properties, the positive invariance of the bounded look angle interval and the change of independent variable are utilized. Numerical simulations are conducted to demonstrate the performance of the proposed guidance law.

Optimal three-dimensional impact time guidance with seeker’s field-of-view constraint

This paper proposes a new three-dimensional optimal guidance law for impact time control with seeker’s Field-of-View (FOV) constraint to intercept a stationary target. The proposed guidance law is devised in conjunction with the concept of biased Proportional Navigation Guidance (PNG). The guidance law developed leverages a nonlinear function to ensure the boundedness of velocity lead angle to cater to the seeker’s FOV limit. It is proven that the impact time error is nullified in a finite-time under the proposed method. Additionally, the optimality of the biased command is theoretically analyzed. Numerical simulations confirm the superiority of the proposed method and validate the analytic findings.

Distributed target-encirclement guidance law for cooperative attack of multiple missiles

The target-encirclement guidance problem for many-to-one missile-target engagement scenario is studied, where the missiles evenly distribute on a target-centered circle during the homing guidance. The proposed distributed target-encirclement guidance law can achieve simultaneous attack of multiple missiles in different line-of-sight directions. Firstly, the decentralization protocols of desired line-of-sight angles are constructed based on the information of neighboring missiles. Secondly, a biased proportional navigation guidance law that can arbitrarily designate the impact angle is cited. The missiles can achieve all-aspect attack on the target in an encirclement manner by combining the biased proportional navigation guidance law and dynamic virtual targets strategy. Thirdly, the consensus protocol of simultaneous attack is designed, which can guarantee that all missiles’ time-to-go estimates achieve consensus asymptotically, and the convergence of the closed-loop system is proved strictly via the Lyapunov stability theory. Finally, numerical simulation results demonstrate the performance and feasibility of the proposed distributed target-encirclement guidance law in different engagement situations.

Impact time control using biased proportional navigation for missiles with varying velocity

A feasible guidance scheme with impact time constraint is proposed for attacking a stationary target by missiles with time-varying velocity. The main idea is to replace the constant velocity with the future mean velocity; therefore, the existing time-to-go estimation algorithm of the proportional navigation guidance law can be improved to adapt to varying conditions. In order to obtain the prediction of the velocity profile, the velocity differential equation to the downrange is derived, which can be numerically integrated between the current downrange and the target position by the on-board computer. Then, a third-order polynomial is introduced to fit the velocity profile in order to calculate the future mean velocity. At the beginning of each guidance loop, the future mean velocity is predicted and the time-to-go information is updated, based on which a novel biased proportional navigation guidance law is established to achieve the impact time constraint. Finally, numerical simulation results verified the effectiveness of the time-to-go estimation algorithm and the proposed law.

Intercepting Higher-Speed Targets Using Generalized Relative Biased Proportional Navigation

To intercept a higher-speed target in the terminal guidance phase, this paper proposes a generalized relative biased proportional navigation (BPN) law. In order to enlarge the capture domain of the classical proportional navigation (PN) law and make full use of the maneuverability of a missile, the paper designs time-varying navigation coefficients; thus the modified PN guidance law integrates the advantages of the PN guidance law with those of the retro-PN guidance law. In order to intercept high-speed targets with impact angle constraints, the relative BPN law is introduced, and the impact angle is achieved by controlling the relative flight-path angle. In order to improve the performance of the guidance law for intercepting higher-speed maneuvering targets, some compensation measures are designed for guidance commands. Extensive simulations are conducted to verify the design features of the proportional navigation law.