Proportional Navigation Research Articles

Survival of Unmanned Aerial Vehicles by Luring Pursuers in Angle Constrained Interception

In this paper, a sliding mode control-based cooperative guidance strategy for a team of two unmanned aerial vehicles (aircraft), which ensures their survival against pursuers, is proposed. The pursuing interceptors are assumed to employ augmented proportional navigation guidance. The cooperating aircraft maneuver in a way to lure their respective pursuers to an impact angle constrained collision. The problem of controlling impact angle is first transformed to that of tracking line-of-sight angle and its rate, and then the same is achieved by enforcing sliding mode on appropriate chosen switching surface. Furthermore, the allocation of aircraft’s lateral accelerations is performed in a way to minimize the instantaneous aerodynamic drag acting on both unmanned aerial vehicles. Numerical simulations are presented to validate the effectiveness of the proposed guidance strategy under different engagement geometries.

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.

Linear Covariance Analysis for Head-On Target Engagements

A linear covariance simulation framework is developed and validated for a two-dimensional target engagement scenario. The scenario comprises a single interceptor equipped with an inertial navigation system aided by absolute position measurements, as well as range, angle, and range rate measurements relative to the incoming target. The interceptor uses a proportional navigation guidance law to engage the target, modeled as nominally constant velocity perturbed by Singer motion accelerations. The linear covariance framework is developed by linearizing the differential and measurement equations about the nominal trajectory, and forming an augmented system comprising truth and navigation state dispersions. In contrast with sample-based methods, the developed linear covariance framework can calculate the truth state dispersion covariance and the estimation error covariance throughout the engagement in a single run. It also provides several advantages over analysis methods such as the adjoint technique or traditional covariance analysis. The linear covariance computational efficiency is exploited to rapidly analyze a head-on target engagement problem.

Rope-Hook Recovery Controller Designed for a Flying-Wing UAV

Due to the complexity of landing environments, precision guidance and high-precision control technology have become key to the rope-hook recovery of shipborne unmanned aerial vehicles (UAVs). The recovery process was divided into three stages and a reasonable guidance strategy had been designed for them, respectively. This study separated the guidance and control issues into an outer guidance loop and an inner control loop. The inner loop (attitude control loop) controled the UAV to follow the acceleration commands generated by the outer loop (trajectory tracking loop). The inner loop of the longitudinal controller and the lateral controller were designed based on active disturbance rejection control (ADRC), which has strong anti-interference ability. In the last phase, the outer loop of the longitudinal controller switched from a total energy control system (TECS), which greatly decoupled the altitude channel and speed channel, to the proportional navigation (PN) guidance law, while the outer loop of lateral controller switches from the proportional control law based on the L1 guidance law, which can reduce the tracking error and deviation, to the PN guidance law, which considerably enhances the tracking precision. Finally, the simulation data and flight test data show that the controller has strong robustness and good tracking precision, which ensures safe rope-hook recovery.

Optimal Guidance With Active Observability Enhancement for Scale Factor Error Estimation of Strapdown Seeker

This article investigates the problem of scale factor error estimation for missiles with a strapdown seeker. We first theoretically prove that the seeker scale factor error will introduce a parasitic loop in the guidance loop and, hence, result in instability of the guidance loop. This fact reveals that it is necessary to extract the required guidance information and estimate the scale factor error simultaneously in real time. To this end, the observability of scale factor error estimation is analyzed, and the theoretical results show that this information is weakly or poorly observable under conventional proportional navigation guidance. To address this problem, a new optimal guidance law that can be utilized to improve the estimation performance is proposed by optimizing terminal miss distance, control energy, and observability in an integrated manner. The characteristics of the proposed optimal guidance law are also theoretically analyzed to provide better insights of the proposed approach. Extensive numerical simulations are conducted to support the proposed algorithm.

Quick identification of guidance law for an incoming missile using multiple-model mechanism

A guidance law parameter identification model based on Gated Recurrent Unit (GRU) neural network is established. The scenario of the model is that an incoming missile (called missile) attacks a target aircraft (called aircraft) using Proportional Navigation (PN) guidance law. The parameter identification is viewed as a regression problem in this paper rather than a classification problem, which means the assumption that the parameter is in a finite set of possible results is discarded. To increase the training speed of the neural network and obtain the nonlinear mapping relationship between kinematic information and the guidance law parameter of the incoming missile, an output processing method called Multiple-Model Mechanism (MMM) is proposed. Compared with a conventional GRU neural network, the model established in this paper can deal with data of any length through an encoding layer in front of the input layer. The effectiveness of the proposed Multiple-Model Mechanism and the performance of the guidance law parameter identification model are demonstrated using numerical simulation.

On motion camouflage as proportional navigation.

Motion camouflage is a stealth behaviour by which an insect can appear stationary at a fixed point while approaching or escaping another moving insect. Although several approaches have been proposed to generate motion camouflage in simulated and real agents, the exact mechanisms insects use to perform this complex behaviour are not well understood, especially considering their limited perceptual and computational resources. This paper sheds light on the possible underlying control mechanisms insect might use to generate motion camouflage, by training and analysing a series of motion camouflage controllers using reinforcement learning. We first investigate through simulations the most relevant information available to the insect that can be used to perform motion camouflage and analyse the learnt controllers. The results of this analysis drove us to hypothesise two simpler control mechanisms which, we show, can also generate motion camouflage. The proposed controllers are an extension of proportional navigation, another interception technique found in nature, and therefore, both animal behaviours seem to be connected. Motion camouflage can lead, among others, to novel approaches to closely observe animals in the wild, record sports events or gather information in military operations without being noticed.

Guidance law of interceptors against a high-speed maneuvering target based on deep Q-Network

This paper proposes a novel guidance law for intercepting a high-speed maneuvering target based on deep reinforcement learning, which mainly includes the interceptor–target relative motion model and value function approximation model based on deep Q-Network (DQN) with prioritized experience replay. First, a method called prioritized experience replay is applied to extract more efficient samples and reduce the training time. Second, to cope with the discrete action space of DQN, a normal acceleration is introduced to the state space, and the normal acceleration rate is chosen as the action. Then, the continuous normal acceleration command is obtained using numerical integral method. Third, to make the line-of-sight (LOS) rate converge rapidly, the reward function whose absolute value tends to zero has been constructed. Finally, compared with proportional navigation guidance (PNG) and the Q-Learning-based guidance law (QLG), the simulation experiments are implemented to intercept high-speed maneuvering targets at different acceleration policies. Simulation results demonstrate that the proposed DQN-based guidance law (DQNG) can obtain continuous acceleration command, make the LOS rate converge to zero rapidly, and hit the maneuvering targets using only the LOS rate. It also confirms that DQNG can realize the parallel-like approach and improve the interception performance of the interceptor to high-speed maneuvering targets. The proposed DQNG also has the advantages of avoiding the complicated formula derivation of traditional guidance law and eliminates the acceleration buffeting.

Optimization of the PNG Law for a Dual-Spin Mortar with Fixed Canards

The limited instantaneous overload available and the curved trajectory lead to adaptivity problems for the proportional navigation guidance (PNG) of a guided mortar with a fixed-canard trajectory correction fuze. In this paper, the optimization of a PNG law with gravity compensation is established. Instead of using the traditional empirical method, the selection of the proportional navigation constants is formulated as an optimization problem, which is solved using an intelligent optimization algorithm. Two optimization schemes are proposed for constructing corresponding optimization models. In schemes 1 and 2, the sum squared error between the impact point and target and the circular error probability (CEP), respectively, are taken as the objective function. Monte Carlo simulations are conducted to verify the effectiveness of the two optimization schemes, and their guidance performance is compared through trajectory simulations. The simulation results show that the impact point dispersion can be efficiently reduced under both proposed schemes. Scheme 2 achieves a lower CEP, which is approximately 2.9 m and 2.4 times smaller than that achieved by scheme 1. Moreover, the mean impact point is closer to the target.

Variable Coefficient Terminal-Phase Guidance Design for Deep Space High-Speed Impact

Asteroid impact poses a major threat to human survival. High-speed direct impact is a feasible and effective means to change the orbit of an asteroid. This paper takes Bennu, a potentially hazardous asteroid , as object to design the terminal guidance law of deep space high-speed impact. In view of the characteristics of high relative speed and high impact accuracy, the acceleration command is derived based on the Augmented Proportional Navigation Guidance law (APNG), considering the acceleration of the target. Moreover, the proportional coefficient is designed as a function of the relative distance to meet both the requirements of reducing fuel consumption and improving impact accuracy. An ignition strategy is designed to convert the continuous command into impulse command, without the need to give maneuvering time in advance. The simulation results show the effectiveness of the proposed algorithm. The miss distance constraint is satisfied, and the ignition times and fuel consumption are less than those of the constant proportional coefficient algorithm.

Investigation of the Performance Characteristics of the Angle-based Proportional Navigation Guidance Law

Guidance commands generated by the proportional navigation guidance (PNG) law which constitutes the most popular one among the guidance laws applied on the guidance munition fired against predetermined targets are in the form of linear acceleration or angular speed as dictated by the relevant engagement geometry. As a result of the studies in which notable linear acceleration- and angle-based guidance laws are compared, it is seen the lateral acceleration values obtained with the PNG law occur in a lower level than the results of the angle-based guidance laws. However, the angle-based guidance laws lead to lower final miss distances. In this study, the PNG law is so adapted that it yields angle-based guidance commands and then it is applied upon a short range air-to-surface missile against a maneuvering surface target as well as the velocity pursuit guidance law that is nothing but a version of PNG law, linear homing guidance law, and body pursuit guidance law. After the computer simulations, it is observed that the angle-based PNG law produces smaller final miss distances compared to its original form. The resulting lateral accelerations are in admissible levels. Also, the engagement duration values with the target happen to be almost the same.

Capturability of 3D RTPN guidance law against true-arbitrarily maneuvering target with maneuverability limitation

The capturability of the Three-Dimensional (3D) Realistic True Proportional Navigation (RTPN) guidance law is thoroughly analyzed. The true-arbitrarily maneuvering target is considered, which maneuvers along an arbitrary direction in 3D space with an arbitrary but upper-bounded acceleration. The whole nonlinear relative kinematics between the interceptor and target is taken into account. First, the upper-bound of commanded acceleration of 3D RTPN is deduced, using a novel Lyapunov-like approach. Second, the reasonable selection range of navigation gain of 3D RTPN is analyzed, when the maneuver limitation of interceptor is considered. After that, a more realistic definition of capture is adopted, i.e., the relative range is smaller than an acceptable miss-distance while the approaching speed is larger than a required impact speed. Unlike previous researches which present Two-Dimensional (2D) capture regions, the inequality analysis technique is utilized to obtain the 3D capture region, where the three coordinates are the closing speed, transversal relative speed, and relative range. The obtained capture region could be taken as a sufficient-but-unnecessary condition of capture. The new theoretical findings are all given in explicit expressions and are more general than previous results.

Learning prediction-correction guidance for impact time control

This paper investigates the problem of impact-time-control and proposes a learning-based computational guidance algorithm to solve this problem. The proposed guidance algorithm is developed based on a general prediction-correction concept: the exact time-to-go under proportional navigation guidance with realistic aerodynamic characteristics is estimated by a deep neural network and a biased command to nullify the impact time error is developed by utilizing the emerging reinforcement learning techniques. To deal with the problem of insufficient training data, a transfer-ensemble learning approach is proposed to train the deep neural network. The deep neural network is augmented into the reinforcement learning block to resolve the issue of sparse reward that has been observed in typical reinforcement learning formulation. Extensive numerical simulations are conducted to support the proposed algorithm.

Pursuer Navigation Based on Proportional Navigation and Optimal Information Fusion

Pursuer navigation is proposed based on the three-dimensional proportional navigation law, and this method presents a family of navigation laws resulting in a rich behavior for different parameters. Firstly, the kinematics model for the pursuer and the target is established. Secondly, the proportional navigation law is deduced through the kinematics model. Based on point-to-point navigation, obstacle avoidance is implemented by adjusting the control parameters, and the combination can enrich the application range of obstacle avoidance and guidance laws. Thirdly, information fusion weighted by diagonal matrices is used for decreasing the tracking precision. Finally, simulations are conducted in the MATLAB environment. Simulation results verify the availability of the proposed navigation law.

Sliding mode lateral stand-off tracking control of finless airship

The paper addresses the stand-off tracking control problem for an underactuated finless airship having uncertainties and wind disturbances. The outer loop design sliding surface is based on the line of sight angle rate and its distance from a target, for the inner control loop, generates the reference heading angle commands, the proposed backstepping sliding-mode controller based on the yaw error angle generating commands by actuating the yaw-gyro propulsion. Theoretical results show that the closed loop system is globally asymptotically stable. With the assumption of target movement as disturbances, this framework is also applied to linear and variable direction moving targets. The numerical simulations are conducted for stand-off tracking control for a stationary and a moving target or targets, having servos dynamics and sensor models with measurement noises and external disturbances, and the results verify the effectiveness of the proposed framework for stand-off target tracking. The results obtained by comparing traditional proportional navigation and LQR design demonstrate the capability of the airship to execute a stand-off trajectory tracking with uncertainties and wind disturbances.

The object-oriented dynamic task assignment for unmanned surface vessels

This paper investigates the task assignment and guidance issues of unmanned surface vessels (USVs) interception. When the USVs formation is invaded by some moving objects during its escort, it is necessary for the unmanned systems to assign defenders to prevent attackers approaching the vulnerable target in antagonistic scenarios. This action requires efficient guidance and task assignment strategies. With this in mind, this paper presents the Integral Proportional Navigation Guidance (IPNG) with Tabu Dynamic Consensus-Based Auction Algorithm (TDCBAA) in marine interception scenario. First, IPNG is introduced in the interception game considering the USV kinematic model, which can effectively reduce the individual interception time. Second, a new bidding function is designed for moving objects interception with the consideration of the attackers’ types, positions and interception time. Finally, a TDCBAA is designed to solve the task assignment subproblem, resulting in a shorter overall interception time and a higher interception success rate. Simulations demonstrate that the proposed algorithm can optimize the allocation of defenders in real-time and intercept the attackers more quickly compared with other classical algorithms, which is more suitable in situations where attackers are approaching from all directions.

FOV constrained guidance law for nonstationary nonmaneuvering target interception with any impact angle

This paper presents the nonstationary nonmaneuvering target interception with all possible desired impact angles in a two-dimensional (2D) aerial engagement scenario, where the target can move in any direction. The paper also considers the field-of-view (FOV) constraint for designing the guidance law so that the target is always visible while following the missile trajectory in the entire engagement time, which makes it feasible for real world applications. The guidance law is based on the pure proportional navigation (PPN) to achieve any impact angle of the entire angular spectrum. The proposed guidance law is then simulated for intercepting a nonstationary nonmaneuvering target using a kinematic model of a missile to demonstrate the efficacy of the presented scheme. A comparison with the related work existing in the literature has also been added to establish the superiority of the present work.

Guidance Law with Desired Impact Time and FOV Constrained for Antiship Missiles Based on Equivalent Sliding Mode Control

Impact time control guidance (ITCG) is an important approach to achieve saturation attack on targets. With the increasing complexity of warfare requirements for missiles, an ITCG with field-of-view (FOV) constrained for antiship missiles is proposed based on equivalent sliding mode control. Firstly, in view of the accuracy of the calculation of remaining impact time for guidance law, the large initial lead angle is taken into consideration in the estimation of remaining flying time in which there is no need for the assumption of small angle approximation. Besides, for the sake of promoting the practical application value of the proposed guidance law, FOV is considered so that it can satisfy the actual working performance of the seeker. Then, combined with the concept of predicted interception point (PIP), the proposed guidance law is applied to attack a moving target. Numerical analysis is carried out for different initial lead angles, various impact time, different methods of estimating remaining flying time, and cooperative attack conditions. Compared with proportion navigation guidance (PNG), the feasibility and effectiveness of the guidance law are verified. Simulation results demonstrate that the proposed guidance law can guarantee the constraints of both impact time and FOV effectively.

Distributed cooperative guidance law for multiple missiles with input delay and topology switching

This paper considers the simultaneous attack of a stationary target by multiple missiles. A novel fixed-time distributed guidance law based on the proportional navigation (PN) guidance law is designed by integrating a consistent control technique into the guidance strategy. This guarantees that the time-to-go of the missile becomes consistent. The guidance law adopts a discrete design, and a compensation item driven by normal acceleration is added to tangential acceleration. This eliminates the potential singularity problem when the heading angle is zero before the consistency is obtained, and thus the multiple missile system still converges in fixed time. In addition, the proposed guidance law can be applied to both undirected and directed graphs. Furthermore, two improved guidance laws are proposed to improve the robustness of the system against adverse effects caused by input delays and topology switching failures and to provide a theoretical proof. Finally, a simulation is used to verify the performance of the distributed guidance law and its robustness against the above failures.

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.