Fractional Calculus Guidance Algorithm in a Hypersonic Pursuit-Evasion Game

T HE development of aircraft guidance, navigation, and control systems has been a long-standing research area. Numerous methods relating to the enhancement of aircraft performance under various mission parameters have been developed in response to a need for more reliable and robust guidance systems. Current guidance systems applied to commercial, civilian, and unmanned aircraft rely on the knowledge of a flight path, specified bywaypoints located in inertial space. Most missions are considered successful when the vehicle reaches the designatedwaypoint at which new commands are issued to the vehicle to proceed to the next waypoint. Two common types of conventional aircraft guidance are the direct-to-waypoint (DTW) and track-to-waypoint (TTW) methods in relation to pathfollowing between designated waypoints. The DTWmethod simply issues heading commands to the vehicle based on the angular difference between thewaypoint and vehicle.When thevehicle reaches the waypoint, the control system issues a new command to guide the aircraft to the next waypoint. The TTW method aims to follow the track betweenwaypoints. In this guidancemethod the control system aims to minimize the lateral offset between the prescribed flight path and the aircraft’s position, issuing heading commands that return the vehicle to the nominal flight path. The track method therefore places the additional constraint on a flight path that the vehicle must follow in order to reach the waypoint, rather than simply reaching the waypoint. However, both methods are far from optimal. This is evident in how the aircraft transitions between flight paths after reaching awaypoint. During theseflight-path transitions, the aircraft will often overshoot the desiredflight path to correct its track, particularlywhen the flight-path transition angle is acute. Various control strategies have been investigated to alleviate or minimize flight-path deviations. Such strategies include applyingmodern control methods such as receding-horizon control [1,2] and model predictive control [3] to anticipate flight-path changes and take control action before reaching a goal while maintaining adequate vehicle flight performance. Missile guidance and control systems operate on similar principles to commercial, civilian, and unmanned aircraft guidance and control algorithms. The primary mission for missile systems is to intercept a moving target using information about the relative position and velocity between the pursuer and target. One of thefirstmethods used in missile guidance was pursuit guidance (PG) [4–7]. The method operates by forcing the angular displacement error between a pursuer and its target to zero. Control commands scaled by a proportional factor of the current error are then issued to direct the pursuer along the line of sight (LOS) between the pursuer and target. PG solutions, however, do not consider the path taken or the levels of system performance required by the pursuer in reaching the target, resulting in a far-from-optimal solution. To address this problem of suboptimality, additional parameters have been introduced to enhance missile performance. One method includes taking into account the motion of the commanded line of sight between the pursuer and target [4,7–10], issuing lateral acceleration commands based on tracking error and tracking error rate to the target. This approach has been shown to improve overall interceptor performance compared with conventional PG [4,7]. Another suchmethod aims tomodify the level of control the guidance algorithm possesses over the vehicle by adjusting the level of proportional gain. This is achieved by gain scheduling [11] to select gain values based on current interceptor states. In addition to modifying internal missile guidance and control parameters such as variable gains and LOS rate estimation, mission performance can be enhanced by manipulating the trajectory taken by the pursuer to the targets. A good example of such a method is discussed in [12], in which a missile aims to exploit the aerodynamic benefits of high-altitude flight by tracking a virtual target at some initially high altitude that is not necessarily along the trajectory to the true target. This Note discusses the development of a guidance law fusing the virtual-target concepts with those of pursuit guidance for implementation into an aircraft guidance system. This Note develops a path-following aircraft guidance algorithm that pursues synthetic waypoints using only a small set of guidance parameters, extending the virtual-target concept to complete aircraft guidance. The path is defined by the track between a minimal set of waypoints at specified locations, removing the need for a smooth path to be defined or the need for complicated path-switching logic or trajectory planning when awaypoint is reached. The synthetic waypoint travels along the path between waypoints, with the trailing aircraft traveling a smooth path generated through its own dynamics in following the synthetic waypoint. The guidance law is tested by varying guidance parameters, thus assessing vehicle sensitivity to and overall system performance of parameter variations. The following sections discuss the basic concepts of missile and aircraft guidance and provide a detailed description of the structure of the synthetic-waypoint guidance algorithm. A discussion on the implementation of the algorithm into the underlying aircraft control system will also be presented, followed by an analysis of the performance of the guidance algorithm in nonlinear simulation. Received 30 June 2009; revision received 3 November 2009; accepted for publication 9November 2009.Copyright©2009 by theAmerican Institute of Aeronautics and Astronautics, Inc. All rights reserved. Copies of this paper may be made for personal or internal use, on condition that the copier pay the $10.00 per-copy fee to the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923; include the code 0731-5090/10 and $10.00 in correspondence with the CCC. ∗Graduate Research Student, School of Aerospace, Mechanical and Mechatronic Engineering; e.medagoda@aeromech.usyd.edu.au. Senior Lecturer, School of Aerospace, Mechanical and Mechatronic Engineering; pwg@aeromech.usyd.edu.au. JOURNAL OF GUIDANCE, CONTROL, AND DYNAMICS Vol. 33, No. 2, March–April 2010

In this work, design of three-dimensional (3D) proportional navigation guidance law based on input-output linearization (IOL) for tactical missiles is proposed. For the three-dimensional nonlinear missile-target engagement dynamics, the line-of-sight (LOS) rates in pitch and yaw planes are taken as outputs and the IOL theory is used to derive the guidance laws without ignoring the cross-coupling terms. Simulations are carried out by considering maneuvering target, and the results are presented to showcase the efficacy of the proposed design. Further, comparison of the performance of the proposed guidance law with augmented proportional navigation guidance(APNG) laws is carried out and the related results are presented. Lastly, implementation issues of the proposed guidance law are discussed.

The proportional guidance method has been widely used in various missiles. However, with the continuous improvement of target maneuverability and the increasingly ugly combat environment, the classical proportional guidance method can no longer meet people's requirements for ballistic parameters, so it is improved and perfected. The proportional guidance law is extremely necessary. In this paper, the establishment of the proportional guidance process based on the guidance principle is studied, and the trajectory established by the classical proportional guidance method and the proportional guidance trajectory based on the guidance principle are designed and simulated, and the simulation results are analyzed. The simulation accuracy analysis is carried out with the example, and the result proves that the proposed method effectively reduces the method error in the calculation of the guidance ballistics.

Impact Angle Constrained Suboptimal Terminal Guidance of Interceptors for Air Defence

The main problem in bringing autonomy to any vehicle lies in the design of a suitable guidance law. For truly autonomous operation, the vehicle needs to have a reliable Navigation, Guidance and Control (NGC) system of which the guidance system is the key element that generates suitable trajectories to be followed. In this review paper, various guidance laws found in the literature and their relevance to autonomous underwater vehicles (AUVs) are discussed. Since existing guidance laws for underwater vehicles have emulated from tactical airborne missile systems, a number of approaches for the missile guidance systems are considered. Finally, potential guidance strategies for AUVs are proposed.

In this paper realistic Six Degrees of Freedom simulation of a pursuer evader engagement during terminal guidance has been carried out using a class of nonlinear guidance law based on Feedback Linearization. This guidance law has been developed as a regulator problem based on nonlinear control theory where line of sight rate derivative along both yaw and pitch plane are made zero by taking line of sight angle and line of sight rate as output feedback. The line of sight angles and rates have been computed from noisy seeker measurements through a nonlinear state observer. The available seeker measurements are pursuer-evader relative range, relative rage rate, gimbal angles and line of sight rates along yaw and pitch plane in seeker inner gimbal frame. The present guidance law has been validated in presence of realistic non Gaussian and correlated seeker measurement noise, estimator, autopilot lag and highly time varying pursuer velocity during engagement. It is demonstrated through realistic Six Degrees of Freedom pursuer-evader engagement simulation in the presence of above constraints in a practical environment that the proposed guidance law is a viable option for intercepting maneuvering target with high heading error over the well known proportional navigation guidance law. This specific type of guidance law being not available in published literature, is claimed as quiet novel.

A heuristic predictive line-of-sight (LOS) guidance law is introduced to intercept a high-speed maneuvering target. A combination of continuous ant colony system and tabu search optimization algorithms is proposed to generate the optimal predictive commands of LOS guidance law. Prediction is driven by the previous positions of the target to estimate the next positions of it. Thus, the guidance system is continually solving a dynamic optimization problem in order to determine the acceleration commands by minimizing a cost function subject to actuators saturation. This innovation distinguishes the proposed guidance law from the classic LOS guidance, described by a simple relation between the deviation from the LOS and the guidance commands and use only the current measurement. The performance of the proposed guidance law is verified by carrying out several simulations of three dimensional scenarios. By still requiring simple and low-cost hardware in the new guidance law, simulation results show that the proposed heuristic predictive line-of-sight guidance law can prepare comparable performance in the whole case studies which show the robustness and adaptability of the proposed scheme.

Experimental Comparison of Nonlinear Guidance Laws for Unmanned Aircraft

A PD nonlinear dynamic inverse guidance law finite time convergence is proposed for the problem of UAV's collision avoidance in this paper. First, a proportional-derivative nonlinear dynamic inverse guidance law, which can be applied to multiple initial angles and speeds, is designed to ensure the security of UAV's collision avoidance. In addition, the stability of guidance law is proved by the circle criterion, and stability conditions are used to design coefficients of guidance law. In order to analysis the performance of this guidance law, a simulation system based on 6-DOF UAV is established. The results conclusively demonstrate that the maneuvering range of collision avoidance is smaller than the collision avoidance method based on traditional nonlinear dynamic inverse guidance.

This paper investigates a new concept of “overhead-attack”. Firstly, a guidance law with the terminal angle restraint proposed, which is given by establishing the right Hamiltonian. Furthermore the practical application in Engineering is also given in this paper. The guidance law provides a trajectory scheme with terminal angle restraint, which is efficient for penetration bombs or missiles. Simulations show that the guidance law is effective for overhead-attack, which requires higher overload for the bomb as the increase of the terminal angle. Secondly, this paper investigates four important factors for penetration bombs: the guidance system, the homing head, the range, and the maneuverability. The analyses of those four factors are also given. It has been proven that this guidance law works effectively for a bomb that needs to attack an “overhead” target. It confirms that the method works in practical application.

A nonlinear optimal guidance law with constraint on impact angle is proposed in this paper for interception of incoming high-speed ballistic missile targets. The guidance law is developed based on the recently developed linear gauss pseudospectral model predictive control (LGPMPC) method which is to solve the nonlinear optimal control problem with quadratic performance index and hard terminal constraints. The LGPMPC method combines the philosophies of linear quadrature optimal control, the Model Predictive Control and Gauss Pseudospectral Method. A design of engaging an incoming high-speed ballistic missile target with a specified impact angle in two dimensions is carried out to evaluate the performance of the proposed guidance law. In this work, the initial guess is produced by biased PN guidance law in the two dimensions. Simulation results show that the proposed guidance law has high computational efficiency in achieving ideal impact angles with high accuracy and the near-zero miss distance. And the generated lateral acceleration is small and smooth as expected. Moreover, performance of the proposed guidance law is superior to other guidance laws by comparative studies.

A new adaptive finite time nonlinear guidance law to intercept maneuvering targets

A novel nonlinear impact angle guidance law for stationary targets, based on Lyapunov stability theory, is proposed in this work. In the proposed guidance law, nonlinear line-ofsight dynamics of the engagement geometry between the pursuer and stationary target is studied and the state equations for the line-of-sight angle and its rate are derived. The lineof-sight angle is aimed to be controlled to satisfy impact angle constraints. Then a sliding manifold which drives the line-of-sight angle and its rate to exponential stability is constituted. Robust Control Lyapunov Function is introduced to the system to control the line-of-sight dynamics to head it towards defined sliding manifold. The necessary commands are analytically calculated. Finally the comparison with a different impact angle control guidance law is performed to summarize the proposed guidance law performance.

This paper presents a nonlinear guidance law for controlling the impact time of a missile-target engagement in three dimensions. The guidance law is initially formulated for a two-dimensional encounter before being extended to a three-dimensional scenario. The guidance law is constructed such that nonlinear kinematics are neither linearized nor approximated. The guidance law offers a precise closed-form expression for impact time, which is the principal contribution of this work. Choosing an integrable heading error profile is the central concept of this research for determining an accurate impact time. A two-step strategy is utilized to accomplish an interception at a predetermined time. Because the control parameter is related to the desired impact time with an exact expression, a simple and accurate control of impact time is attained. The method provides equations for calculating the minimum and maximum impact times, enabling the determination of the range of impact times that can be achieved. The proposed guidance law can ensure that the seeker field of view restriction is not violated, If the initial heading error is within this limit. The developed method is utilized in salvo attacks, in which multiple missiles simultaneously strike an enemy target, increasing the missiles’ effectiveness against the target’s self-defense mechanism. To validate the effectiveness of the proposed strategy, MATLAB-based numerical simulations of 2D and 3D engagements, including salvo attacks, are conducted.

In this paper, a new guidance law considering control loop dynamics is developed in order to intercept highly maneuvering targets. Since the great advantage of using fractional-order calculation at integer order control is to improve robustness and performance properties, a fractional terminal sliding mode guidance law is designed based on Lyapunov stability theory. As the autopilot dynamic has a significant contribution to guidance loop stability, it has been considered as a first order transfer function. Finally, the proposed guidance law is compared with the finite time proportional guidance law and the effectiveness and robustness of the proposed law against target maneuvers is demonstrated by means of simulation.