Proportional Navigation Guidance Research Articles

Time-Critical Unified Rendezvous Guidance for an Unmanned Autonomous Vehicle

Abstract This paper addresses the time-critical rendezvous problem for a pursuing autonomous unmanned vehicle, e.g., an unmanned aerial vehicle (UAV), guided using the concept of true-proportional navigation guidance, which is a variant of proportional-navigation guidance. In existing vehicle routing and flight time-constrained guidance techniques, specific rendezvous guidance commands are designed based on the specific motion of the target. In contrast to that, we propose a unified guidance command for a UAV that guarantees a time-critical rendezvous with a target that moves arbitrarily. We explore the purview of true-proportional navigation guidance and posit that a guidance law thus designed may be a potential candidate for designing time-critical rendezvous strategies against various target motions, even when the pursuer does not necessarily have a speed advantage over the target. We first derive a closed-form expression for the flight duration until rendezvous, over which we exercise control to make the pursuing vehicle rendezvous with the target at any feasible time prescribed a priori. Next, we ensure that the necessary flight time-based error variable converges to zero with an optimal convergence pattern with respect to a suitable cost function. We finally validate the efficacy of the proposed unified guidance command via numerical simulations.

Data‐driven observability analysis and enhancement for radome slope estimation with constraints on impact angle and terminal acceleration

AbstractThe authors address the problem of observability analysis and enhancement for missiles equipped with strapdown seekers, especially considering the line‐of‐sight angle‐only measurements and the constraints on impact angle and terminal acceleration. First, based on the generalised polynomial chaos expansion, a novel data‐driven observability analysis method is proposed to quantitatively analyse the observability of the radome aberration estimation system in real time, including the observability condition, the observability degree of each state and the effect of stochastic noises. The results show that under conventional proportional navigation guidance, the radome aberration estimation system is weakly observable with line‐of‐sight angle‐only measurements. To address this problem, a novel manoeuvre algorithm, the observability‐enhanced guidance law, is introduced to enhance the system observability within specified constraints, thereby improving the estimation performance. This guidance law incorporates a bias term to meet terminal constraints, and optimises the navigation gain using the proposed observability indices. Simulations are conducted to verify the performance of the proposed guidance law in conjunction with a target state estimator.

Data-Driven-Method-Based Guidance Law for Impact Time and Angle Constraints

To increase the hit efficiency and lethality of a flight vehicle, it is necessary to consider the vehicle’s guidance law concerning both impact time and angle constraints. In this study, a novel and straightforward impact time and angle control guidance law that is independent of time-to-go and small angle approximations is proposed with two stages using a data-driven method and proportional navigation guidance. First, a proportional navigation guidance-based impact angle control guidance law is designed for the second stage. Second, from various initial conditions on the impact angle control guidance simulation with various initial conditions, the input and output datasets are obtained to build a mapping network. Using the neural network technique, a mapping network model that can output the ideal flight path angle in flight is constructed for impact time control in the first stage. The proposed impact time and angle control guidance law reduces to the proportional navigation guidance law when the flight path angle error converges to zero. The simulation results show that the proposed guidance law delivers excellent performance under various conditions (including cooperative attack) and features better acceleration performance and less control energy than does the comparative impact time and angle control guidance law. The results of this research are expected to supplement those exploring various paradigms to solve the impact time and angle control guidance problem, as concluded in the current literature.

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

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

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 piecewise cooperative guidance with smooth switching topology

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

Multiple-stage spatial–temporal cooperative guidance without time-to-go estimation

This paper investigates the spatial–temporal cooperative guidance problem for multiple flight vehicles without relying on time-to-go information. First, a two-stage cooperative guidance strategy, namely the cooperative guidance and the Proportional Navigation Guidance (PNG) stage strategy, is developed to realize the spatial–temporal constraints in two dimensions. At the former stage, two controllers are designed and superimposed to satisfy both impact time consensus and impact angle constraints. Once the convergent conditions are satisfied, the flight vehicles will switch to the PNG stage to ensure zero miss distance. To further extend the results to three dimensions, a planar pursuit guidance stage is additionally imposed at the beginning of guidance. Due to the independence of time-to-go estimation, the proposed guidance strategy possesses great performance in satisfying complex spatial–temporal constraints even under flight speed variation. Finally, several numerical simulations are implemented to verify the effectiveness and advantages of the proposed results under different scenarios.

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.

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

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

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

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

Analytical Prescribed Performance Guidance with Field-of-View and Impact-Angle Constraints

Aiming at the impact-angle-control guidance with field-of-view constraint in the three-dimensional space, a novel analytical prescribed performance guidance law is proposed. First, a new prescribed performance function is designed in the relative range domain, and the common drawbacks of most existing prescribed performance functions, such as large overshoot, time dependency, and infinite-time convergence, are avoided. Second, the three-dimensional biased pure proportional navigation guidance frame is selected. The biased term is derived based on the prescribed performance control theory and the optimal error dynamic method, for realizing the terminal impact-angle constraint. After that, the analytical solution of missile lead angle is derived under some linearization approximations, and two methods are respectively developed for obtaining the range of the guidance gain and updating its value to guarantee the satisfaction of the field-of-view constraint during the guidance process. Moreover, for the proposed guidance law and methods, the tracking error could be ensured to remain in the predefined performance boundaries. Finally, numerical simulations are conducted to verify the effectiveness of the new theoretical findings.

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

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

Improvements in Classical Proportional Navigation Guidance Using Fuzzy Logic

The performance of classical Proportional Navigation guidance is limited against highly maneuvering targets and in presence of measurements noise resulting into large miss distances and lower hit probability. Also, the advanced Augmented PN guidance technique requires target state estimation. In this paper, fuzzy logic has been used to implement PN guidance and Membership Functions of fuzzy logic has been tuned using genetic algorithm. The proposed Fuzzy Based PN (FBPN) guidance gives superior performance than PN and APN in terms of lower miss distance, higher hit ratio and better ability to handle measurement noise at all target attack aspects. Deterministic simulations have proved the superior performance and robustness of proposed scheme for various conditions such as step target maneuver upto 9g, weaving maneuver and presence of measurement noise. Further, Monte Carlo simulations have proved that proposed scheme enhances interception performance in realistic scenarios by improving the hit ratio upto 48% and rms miss distance upto 41%. Limitations of APN scheme in terms of target state estimation is also mitigated in the proposed guidance scheme. Finally, global asymptotic stability of fuzzy homing guidance loop is proved using circle criterion.

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

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

Visual versus visual-inertial guidance in hawks pursuing terrestrial targets.

The aerial interception behaviour of falcons is well modelled by a guidance law called proportional navigation, which commands steering at a rate proportional to the angular rate of the line-of-sight from predator to prey. Because the line-of-sight rate is defined in an inertial frame of reference, proportional navigation must be implemented using visual-inertial sensor fusion. By contrast, the aerial pursuit behaviour of hawks chasing terrestrial targets is better modelled by a mixed guidance law combining information on the line-of-sight rate with information on the deviation angle between the attacker's velocity and the line-of-sight. Here we ask whether this behaviour may be controlled using visual information alone. We use high-speed motion capture to record n = 228 flights from N = 4 Harris' hawks Parabuteo unicinctus, and show that proportional navigation and mixed guidance both model their trajectories well. The mixed guidance law also models the data closely when visual-inertial information on the line-of-sight rate is replaced by visual information on the motion of the target relative to its background. Although the visual-inertial form of the mixed guidance law provides the closest fit, all three guidance laws provide an adequate phenomenological model of the behavioural data, whilst making different predictions on the physiological pathways involved.

Capurability analysis for arbitrarily high-speed maneuvering targets

The capturability of an arbitrarily maneuvering target featuring speed superiority over an interceptor is analyzed for Augmented Pure Proportional Navigation (APPN) and Retro-Augmented Proportional Navigation (RAPN) guidance. This paper focuses on intercepting arbitrary maneuvers to study more general interception problems. A comparative analysis of the capture region between head-on interception related to APPN and head-pursuit interception related to RAPN is proposed. The results indicate that RAPN performs better than APPN in capturability. It is concluded that increasing the target velocity, which increases the velocity ratio, significantly weakens the capturability of the interceptor, and the average acceleration and relative distance affect the location of the capture region but not its size. The analysis is based on prior knowledge of the target maneuver, which inevitably leads to deviations from actual maneuvers in practical engagement, so a deviation analysis is implemented. The effective capture region shrinks as the absolute value of acceleration deviation increases, and the RAPN has a better deviation fault tolerance compared with the APPN. The results reveal that a larger relative distance can weaken the deviation fault tolerance, and the target velocity has opposite effects on head-on and head-pursuit interception.

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

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

Adaptive Finite Time Intercept Guidance

This paper summarizes several key issues of importance related to intercept of high-speed maneuvering targets under stratospheric altitude conditions, where it is unlikely that the interceptor will have a significant speed and/or maneuverability advantage. It explains why one cannot rely on traditional proportional navigation guidance and presents simulation results for a recently developed finite time method of guidance. A simplified version of finite time intercept guidance is derived, and a modification required to allow it to be practically implemented is described. Simulation results are presented to illustrate the advantages of the simplified version.

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

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

A generalized three-dimensional cooperative guidance law for various communication topologies with field-of-view constraint

In this paper, a generalized three-dimensional (3D) cooperative guidance law with seeker’s field-of-view (FOV) constraints is designed for multiple missiles to achieve salvo attack against a stationary target. The proposed generalized guidance law is composed of two parts: a proportional navigation guidance (PNG) component for target capture and a biased feedback component for simultaneous arrival. Two novel auxiliary functions are integrated into the biased feedback component to guarantee the satisfaction of FOV constraint. Furthermore, a time-varying lower bound that contains the impact time error is utilized to avoid the guidance command singularity. The proposed guidance law can be easily applied to various missile communication topologies, including centralized connected, distributed connected, and no-connected topologies. The convergence of impact time error, FOV constraint satisfaction, and command non-singularity of the proposed guidance law are theoretically analyzed and proved. Extensive numerical simulations with comparative studies are conducted to demonstrate the effectiveness and advantages of the proposed guidance law.