Tactical preference based objectives for solving an evasion problem of fighter in air combat

Evasive maneuvers of fighter only consider the objective of miss distance in traditional air combat decision problems. The amalgamative tactical demands of achieving self-conflicting evasive objectives in actual air combat are taken into account in this paper. A method to generate a strategy of evasive maneuvers based on tactical preference is proposed. Tactical preference is given to higher miss distance, less energy consumption, and higher terminal superiority. The evasion problem in air combat is defined and reformulated into a multi-objective optimization problem, which is solved by a multi-objective evolutionary algorithm. Simulation results show the feasibility and effectiveness of the proposed method, and a set of approximate Pareto-optimal solutions are obtained to satisfy the tactical preference.


Introduction
Fighter aircraft are faced with increasingly threatening high-precision air-to-air missiles (AAM) in the modern air combat mode. How to minimize the lethality of enemy AAM through evasive maneuvers is a necessary skill for the fighter, which is of great significance to improve the survival probability of the fighter. Therefore, the decision of evasive maneuvers of the fighter has received widespread attention.
Imado [1] and Akdag [2] made a comparative study of some features of different evasive maneuvers against a tactical missile through mathematical simulation and analysis. The analytic solution of optimal maneuver has been solved by a zero-sum two-person differential game method [3][4]. Ong [5] treated the problem as an approximated parameter optimization problem, and sequential quadratic programming was used to solve it. A receding horizon control scheme for obtaining nearoptimal controls in a feedback form was addressed in [6]. A similar problem was solved in [7] using nonlinear model predictive control, and the Gauss Pseudospectral method was used to solve the model. The search for an optimal evading solution was performed by using parallel evolutionary programming in [8]. Besides, an evasive maneuver decision was presented based on the artificial neural network in [9] for evading incoming missile.
Evasive maneuvers of fighter only consider the objective of miss distance in traditional air combat decision problems. However, the fighter usually has multiple tactical objectives when confronting incoming missiles in actual air combat [10], such as higher miss distance, less energy consumption [11] and higher terminal superiority. Therefore, this problem that involves several objectives has no unique optimal solution but rather a set of approximate Pareto-optimal solutions, which can exhibit different tactical preferences of the pilot [12]. The evasion problem in air combat is defined and reformulated into a multi-objective optimization problem. Then a multi-objective evolutionary algorithm based on  [13] is proposed to solve the model. An attempt is made to find a nondominant and feasible solution set with tactical preference for the problem.

Problem analysis and formulation
2.1Description of the problem Ensuring tactical superiority and survivability in air combat is crucial for the fighter. In the face of incoming missile, the essential prerequisite of the fighter threatened by AAM is to avoid being hit. On this basis, consider that air warfare is a continuous and multi-round process with missile attacks. Therefore, the fighter needs to consider the whole battle efficiency and tactical superiority, not just successful evasion.
In this research, we will consider multiple tactical objectives of evasion in realistic air combat, including higher miss distance, less energy consumption, and higher terminal superiority. Maximizing the miss distance means increasing the survival probability, decreasing energy consumption means more energy for subsequent multi-round combat after a successful evasion, and maximizing the terminal superiority means a superior situation in the next missile duel.
This research focuses on defining and reformulating the evasion problem in air combat into a multi-objective optimization problem, then finding a non-dominant and feasible solution set with tactical preference for the problem. Firstly, establish dynamical models and constraints of the fighter and missile. Then simulation end conditions of the evasion problem in three-dimensional space are defined. The details of the optimization model of evasive maneuvers strategy are presented, and the objective space is solved by digital simulation. Finally, the MOEA/D is designed to find the approximate Pareto-optimal strategy set of evasive maneuvers.
It should be noted that several assumptions that simplify the problem to a certain extent without losing practicability are listed: Six variables used to describe the motion in (1)   x v The definitions of the motion equations of the missile are similar to that of the fighter, that are, down range m x , altitude m y , cross range m z , velocity m v , flight-path angle m  , and heading angle m  , respectively. The mass of the missile m m and the thrust m T are functions of the missile's flight time t . g is the acceleration of gravity. The drag force m D is given as tabular data. The guidance law adopts the proportional navigation guidance scheme, and be given through first-order lag from command signals [10].

End condition of simulation
The results of the evasion can be determined through simulation if the initial conditions are certain, which include the following possible scenarios: (a) Failed evasion. If the distance between the missile and fighter is the minimum controlled velocity of the missile), that will lead to the self-destruction of the missile as out of control. Besides, if d r r  in the terminal phase, the fighter is also believed to have successfully evaded the missile.

3.1Evasive objectives based on tactical preference
As previously mentioned, the multi-objective of evasive maneuver include: maximizing the miss distance as much as possible to increase the survival probability, decreasing the energy consumption as much as possible for subsequent multi-round combat after a successful evasion, and maximizing the terminal superiority as much as possible for a superior situation in the next missile duel.
Based on this tactical preference in evasion, the optimization model of evasive maneuver strategy for the fighter is defined as Where system model (6) is constituted by (1) and (3), and 0 x is the initial state. The state vector refers to ( , , , , , , , , , , , ) , and the control vector is ( , , ) Equation (7) represents the constraints of the system model, which is constituted by (2)  (a) Miss distance The miss distance is the distance when the closing velocity between the fighter and the missile is zero. The most basic objective of the fighter is to make the miss distance greater than the missile's damage radius, that is, ( ) However, if the fighter is hit, ( ) m J u is set as a large constant for punishment. (b) Energy consumption As can be seen from dynamic equations of the fighter in (1), thrust coefficient  directly reflects the energy consumption of the fighter with a positive correlation relation. Therefore, in this paper, it is designed as where d t  is the decision-making period in simulation, and M refers to the terminal period.
, and max f v is the maximum velocity of the fighter.

3.2Solution algorithm (a)
Normalized MOEA/D According to the MOEA/D [13], the number of the subproblems is denoted by N , then let  Figure 1 shows a set of non-dominated solutions that reflect the different evasive tactical preferences of the fighter while ensuring security. Each solution represents a strategy, that is, a feasible evasive flight path. Therefore, this problem has no unique optimal solution but rather a set of approximate Pareto-optimal solutions, which can exhibit different evasive tactical preferences. Such as higher miss distance, less energy consumption, or higher terminal superiority.

4.2Simulation experiment 2
To visualize the impact of tactical preferences on flight paths, Figure 2 shows an evasive trajectory that satisfies a specific tactical preference. The tactical preference is expressed by weighting the objective functions, that is, 1  The tactical preference in the Figure 2 is characterized by security objective first, followed by energy consumption and terminal superiority. Therefore, the obtained trajectory of the fighter has lower energy consumption and higher terminal superiority than the trajectory with miss distance as the only objective.

Conclusion and future work
This paper studies evasive maneuver strategy of the fighter with multiple tactical objectives when confronting incoming missile. The amalgamative tactical demands of achieving self-conflicting evasive objectives in air combat are taken into account, such as higher miss distance, less energy consumption and higher terminal superiority. The evasion problem is defined and reformulated into a multi-objective optimization problem. Using scenario-based simulations, a set of approximate Paretooptimal solutions (i.e., non-dominated evasive strategies) are obtained, which satisfy different tactical preferences of the fighter while ensuring security. In general, the proposed method is feasible and effective for solving the problem.
Future research directions mainly include considering the uncertainty of information, building more accurate models, enhancing algorithm efficiency, and improving objective functions.