Biased Proportional Navigation Guidance Research Articles

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.

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.

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.

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 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).

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.

Cooperative Guidance of Seeker-Less Missiles for Precise Hit

A novel cooperative guidance scenario is proposed that implements fire-and-forget attacks for seeker-less missiles with a cheap finder for stationary targets and without requiring real-time communication among missiles or precise position information. Within the proposed cooperative scenario, the classic leader-follower framework is utilized, and a two-stage cooperative guidance law is derived for the seeker-less missile. Linear-quadratic optimal control and biased proportional navigation guidance (PNG) are employed to develop this two-stage cooperative guidance law to minimize the control cost in the first stage and to reduce the maximum acceleration command in the second stage when the acceleration command is continuous. Simulations and comparisons are conducted that demonstrate the effectiveness and advantages of the proposed guidance law.

Coverage-based cooperative guidance strategy against highly maneuvering target

Considering highly maneuvering target, a novel cooperative guidance strategy, which aims to make multiple missiles' joint reachable sets cover the target evasion region, is proposed. The assumed engagement scenario is that multiple missiles separate with each other at the beginning of homing phase and then cooperatively intercept the highly maneuvering target which cannot be intercepted by single missile due to insufficient maneuverability. Firstly, for biasing the reachable sets of different missiles to cover different subintervals of target maneuvering range as expected, a biased proportional navigation guidance law (BPN) is designed by introducing the virtual aiming point. Secondly, for the missile-team without communication capability, the minimum relative maneuverability superiority and minimum team size for covering the whole target maneuvering range are derived based on the BPN respectively, and one open-loop cooperative guidance strategy is proposed with an acceptable assumption about the handover errors of midcourse guidance. While for the missile-team with communication capability, a novel coordinating variable is designed and the corresponding closed-loop guidance strategy is proposed. Numerical simulations with different target maneuver modes are presented to verify the guidance performance of the proposed cooperative guidance strategy.

Generalized optimal impact-angle-control guidance with terminal acceleration response constraint

To study the optimal impact-angle-control guidance problem with multiple terminal constraints, a generalized optimal impact-angle-control guidance law with terminal acceleration response constraint (GOIACGL-TARC) is proposed. In the deriving, a time-to-go − nth power weighted object function is adopted to derived the GOIACGL-TARC and a general expression of GOIACGL-TARC is presented. Based on the general expression of GOIACGL-TARC, three guidance laws, GOIACGL-TARC1/TACC0/TACC1 are proposed and the inheritance relationship between GOIACGL-TACC0/TACC1/TARC1 and the conventional optimal guidance law with impact angle constraint is demonstrated. Performance analysis of the proposed guidance laws shows that in the case of GOIACGL-TACC0, the terminal acceleration is not zero at n = 0 and only as n > 0, the terminal acceleration converges to zero; in the case of GOIACGL-TACC1 and GOIACGL-TARC1, GOIACGL-TARC1 can guarantee the acceleration response to reach the exactly zero value but GOIACGL-TACC1 cannot, which can only guarantee the acceleration command to reach the exactly zero value. It is pointed out that compared with the biased proportional navigation guidance law, GOIACGL-TARC1 has an outstanding guidance performance in acceleration response, miss distance, and terminal impact angle error.

Observer-Based Guidance Law With Impact Angle Constraint

Accounting for the autopilot as second-order dynamics, an observer-based guidance law with terminal impact angle constraint is designed using the dynamic surface control method. Some first-order low-pass filters are introduced into the designing process to avoid the occurrence of high-order derivatives of the line of sight (LOS) angle in the expression of the guidance law such that the guidance law can be implemented in practical applications. The proposed guidance law is effective in compensating for the second-order autopilot lag. In simulation of intercepting targets with sinusoidal acceleration, the guidance law is compared with the biased proportional navigation guidance (BPNG) law in the presence of missile autopilot lag. Simulation results show that the proposed observer-based guidance law with terminal impact angle constraint is able to guide a missile with large autopilot lag to impact a target with a desired angle and achieve a small miss distance, even if the target escapes in a great and fast maneuver.

Biased PNG With Terminal-Angle Constraint for Intercepting Nonmaneuvering Targets Under Physical Constraints

A biased proportional navigation guidance law, considering the seeker's look angle and acceleration capability limits, for impact-angle-control against nonmaneuvering targets is proposed. The proposed law consists of two time-varying biases: One is to achieve the desired impact angle within the command limit; the other is to keep the look angle constant for not exceeding the boundary value. Under physical constraints, the maximum achievable impact angle is obtained for practical implementation. The performance of the proposed law is demonstrated through nonlinear simulations.

Optimal Guidance Around Circular Trajectories for Impact-Angle Interception

A new linear optimal guidance law for impact-angle interception is presented. The linearization of the nonlinear equations of motion is performed around a nominal circular trajectory, thus allowing the linearization to be valid far from the initial line of sight. The proposed law can be viewed as a generalization of the optimal rendezvous problem. Closed-form expressions for the miss distance, the intercept angle error, and the control effort are derived for the linear model. The transformation from the relative position and the relative velocity (with respect to the circular trajectory) to the relative inscribed angle and the relative inscribed angle rate is presented. Using this transformation, the implementation of the optimal impact-angle guidance law based on the inscribed angle is derived. The application in various scenarios enforcing different impact angles is studied via nonlinear simulations, and it is shown that, for the investigated parameters, the proposed law performs better when compared to biased proportional navigation guidance. The nonlinear simulations also show that, for investigated parameters, although the optimality is guaranteed only around the nominal circular trajectory, the proposed guidance law enables imposing of the desired impact angles even when the deviations from the nominal trajectory are not small.

Midcourse Guidance Law Based on High Target Acquisition Probability Considering Angular Constraint and Line-of-Sight Angle Rate Control

Random disturbance factors would lead to the variation of target acquisition point during the long distance flight. To acquire a high target acquisition probability and improve the impact precision, missiles should be guided to an appropriate target acquisition position with certain attitude angles and line-of-sight (LOS) angle rate. This paper has presented a new midcourse guidance law considering the influences of random disturbances, detection distance restraint, and target acquisition probability with Monte Carlo simulation. Detailed analyses of the impact points on the ground and the random distribution of the target acquisition position in the 3D space are given to get the appropriate attitude angles and the end position for the midcourse guidance. Then, a new formulation biased proportional navigation (BPN) guidance law with angular constraint and LOS angle rate control has been derived to ensure the tracking ability when attacking the maneuvering target. Numerical simulations demonstrates that, compared with the proportional navigation guidance (PNG) law and the near-optimal spatial midcourse guidance (NSMG) law, BPN guidance law demonstrates satisfactory performances and can meet both the midcourse terminal angular constraint and the LOS angle rate requirement.

A novel biased proportional navigation guidance law for close approach phase

A novel biased proportional navigation guidance (BPNG) law is proposed for the close approach phase, which aims to make the spacecraft rendezvous with the target in specific relative range and direction. Firstly, in order to describe the special guidance requirements, the concept of zero effort miss vector is proposed and the dangerous area where there exists collision risk for safety consideration is defined. Secondly, the BPNG, which decouples the range control and direction control, is designed in the line-of-sight (LOS) rotation coordinate system. The theoretical analysis proves that BPNG meets guidance requirements quite well. Thirdly, for the consideration of fuel consumption, the optimal biased proportional navigation guidance (OBPNG) law is derived by solving the Schwartz inequality. Finally, simulation results show that BPNG is effective for the close approach with the ability of evading the dangerous area and OBPNG consumes less fuel compared with BPNG.