Linear-quadratic and norm-bounded combined differential game guidance scheme with obstacle avoidance for attacking defended aircraft in three-player engagement

Abstract

A differential game guidance scheme with obstacle avoidance, based on the formulation of a combined linear quadratic and norm-bounded differential game, is designed for a three-player engagement scenario, which includes a pursuer, an interceptor, and an evader. The confrontation between the players is divided into four phases (P1-P4) by introducing the switching time, and proposing different guidance strategies according to the phase where the static obstacle is located: the linear quadratic game method is employed to devise the guidance scheme for the energy optimization when the obstacle is located in the P1 and P3 stages; the norm-bounded differential game guidance strategy is presented to satisfy the acceleration constraint under the circumstance that the obstacle is located in the P2 and P4 phases. Furthermore, the radii of the static obstacle and the interceptor are taken as the design parameters to derive the combined guidance strategy through the dead-zone function, which guarantees that the pursuer avoids the static obstacle, and the interceptor, and attacks the evader. Finally, the nonlinear numerical simulations verify the performance of the game guidance strategy.

Headnote

ABSTRACT

A differential game guidance scheme with obstacle avoidance, based on the formulation of a combined linear quadratic and norm-bounded differential game, is designed for a three-player engagement scenario, which includes a pursuer, an interceptor, and an evader. The confrontation between the players is divided into four phases (P1-P4) by introducing the switching time, and proposing different guidance strategies according to the phase where the static obstacle is located: the linear quadratic game method is employed to devise the guidance scheme for the energy optimization when the obstacle is located in the P1 and P3 stages; the norm-bounded differential game guidance strategy is presented to satisfy the acceleration constraint under the circumstance that the obstacle is located in the P2 and P4 phases. Furthermore, the radii of the static obstacle and the interceptor are taken as the design parameters to derive the combined guidance strategy through the dead-zone function, which guarantees that the pursuer avoids the static obstacle, and the interceptor, and attacks the evader. Finally, the nonlinear numerical simulations verify the performance of the game guidance strategy.

Keywords:

Active defense aircraft

Differential game theory

Three-player confrontation

Energy optimization

Acceleration constraint

Obstacle avoidance

1. Introduction

Differential Game (DG) theory was established by Isaacs [1] in 1965, which referred to address the dynamic confrontation, competition, or cooperation problem of two-player and multipleplayer. Based on the selection of the cost function, two classes of DG theories are obtained: Linear Quadratic Differential Game (LQDG) [2] and Norm-bounded Differential Game (NDG) [3]. The formulation of LQDG is the miss distance and the energy square integral term of the pursuer and the evader. In NDG, the terminal miss distance of the players is introduced to the performance index, and the maximum maneuverability of the game participant is considered. In practical combat, an air defense detection region is established by the defensive system to probe the presence of the enemy's aircraft in the game engagement. Therefore, the aircraft must evade the spatial region to avoid the detection before the game engagement in order to complete the combat mission. The engagement problem becomes the game conflict with the static obstacle avoidance since the air defensive detection region is abstracted as a static obstacle.

The DG theory was first utilized by Rusnak of Israel Rafael Company to cope with the Pursuer- Interceptor-Evader (PIE) [4] issue, which was described that the aircraft (evader) released an intercepting vehicle (interceptor) to hit an incoming enemy's vehicle (pursuer). Rusnak [5,6] discussed the Lady-BanditBodyguard game concept: the bandit attacked the lady, the lady avoided the bandit, and the bodyguard intercepted the bandit and protected the lady. On this basis, the conception of Lady-BanditBodyguard was expanded by Rusnak [7] into the PIE framework and devised the cooperative defense guidance scheme of the interceptor-evader team.

Recently, the PIE game guidance issue utilizing the DG method has become a research hotspot for the improvement of the cooperative characteristic between the interceptor and the evader. A new LQDG guidance scheme was proposed by Liang et al. [8,9] to study the game scenario of attacking an active defense aircraft. Ye et al. [10,11] designed a proximate interception vector guidance strategy to investigate the PIE problem by the LQDG method. On this basis, the LQDG switching guidance strategy was presented in Ref. [12], which guaranteed that the evader switched the multiple guidance schemes to avoid the interceptor under the assumption of incomplete information. Based on the PIE engagement geometry relationship, Liang et al. [13] employed the LQDG theory to handle the PIE game problem for the multi-agent system, and propose the definitions of pure policy, roundabout policy, rendezvous policy, and cooperative evasion policy. A novel role switch scheme was designed in Ref. [14]. for the PIE confrontation scenario, which considered the influence of unknown parameters and capture radius. Ref. [15]. analyzed the PIE engagement model and devised a cooperative prediction guidance policy in three-dimensional geometry. Cooperative LQDG guidance schemes with the constraint of interception angle were proposed by Liu et al. [16] for the game scenario of multiple pursuers intercepting an active defense evader based on the framework of PIE. Combining deep learning and LQDG approaches, a guidance strategy was developed by Shalumov [17,18] to study the online launch-time selection of the interceptor in the PIE engagement. Ma et al. [19] presented a fault-tolerant control scheme for a second-order nonlinear system via LQDG. The problem of the interceptor-evader team intercepting the pursuer was considered by Shima [20] to design the guidance law by the NDG approach. Sun et al. [21] designed the NDG guidance scheme to study the PIE issue, and identified the boundary conditions for the interceptor to complete the interception task. However, the guidance strategy required an accurate guidance model, and the design process was too complicated to apply in engineering. The NDG theory was employed by Hayoun [22] to devise the guidance strategy from the perspective of hit-to-kill in a pursuit and evasion scenario. A cooperative incomplete game strategy was proposed by Ren et al. [23] for multiple Unmanned Aerial Vehicles(UAV). Li et al. [24] designed a novel constraint game scheme to address the complex air combat environment of multiple UAVs.

A comparison between the LQDG and NDG theories is performed to elaborate that the above DG methods have drawbacks in dealing with the PIE game issue. The constraint of control input isn't imposed for the LQDG theory, therefore the limitation of the aircraft's maximum maneuverability is exceeded to failure of the game engagement. On the contrary, although the overload constraint of the aircraft is taken into consideration in NDG, the fuel cost of the aircraft is ignored to cause excessive energy consumption. To overcome the shortcomings of LQDG and NDG, the Combined Differential Game (CDG) theory was founded by many scholars. For the long fight duration, the LQDG method prefers to design the guidance scheme, which greatly reduces the loss of fuel. In short-time engagement, the NDG guidance strategy is perfectly acceptable since the maximum maneuver is employed to increase the success rate of the combat. A novel CDG guidance scheme was proposed in Refs. [25-27]. to deal with the problem of a one-onone pursuit-evasion game by combining the advantage of LQDG and NDG, which gave the game solution properties. Yan et al. [28] designed a CDG evasion guidance scheme for the scenario of a oneto-one game, which guaranteed that a hypersonic vehicle evaded the interception of the pursuer with unknown dynamics. The switching time was introduced by Chao et al. [29] to study the issue of the PIE game and present the CDG guidance strategy.

It is worthy to notice that the research on the avoidance of the static obstacle has attracted the attention of many scholars in recent years. The problem of the virtual ellipsoid avoiding the static obstacle was studied by Guo et al. [30] based on the HamiltonJacobi-Bellman equation. A cooperative pursuit-evasion strategy was proposed by Sun et al. [31] to avoid the static and dynamic obstacles. The issue of path planning and path tracking was considered in Ref. [32]. utilizing adaptive dual-horizon-parameters to evade the static obstacle. A novel predictive trajectory tracking control algorithm was presented by Huo et al. [33] to guarantee that the unmanned aerial vehicle avoids the static and dynamic obstacles. A class of energy-optimal guidance scheme was designed in Ref. [34]. based on a proximity approach with obstacle avoidance.

Optimal-control-based guidance strategies were proposed to address the confrontation issue with collision avoidance, which guaranteed that the pursuer detected and avoided the static obstacle under the circumstance of energy optimization. Combining the optimal control theory and dead-zone function, a novel guidance strategy with collision avoidance was presented by Weiss and Shima [35] to study the one-to-one game problem that the pursuer avoided a static obstacle and captured the evader. On this basis, a pursuit and evasion game scenario with multipleobstacle was considered by Kumar et al. [36] according to the optimal control method, which guaranteed that the pursuer could evade multiple-obstacle and intercept the evader with the desired terminal angle. Ref. [37]. designed a cooperative optimal guidance scheme for an n-on-n engagement scenario where multiple pursuers avoided collision between the pursuers and intercepted the evaders. The guidance strategies proposed in the abovementioned literatures had to make assumptions about the maneuvering form of the evader and failed to consider the overload limit of the aircraft.

The CDG theory is employed to propose the guidance scheme for obstacle avoidance by combining LQDG and NDG in this paper. There are four phases which are referred to as P1-P4 based on the switching time during the engagement. When the obstacle is in the P1 and P3 stages, the PIE issue with collision avoidance is taken into consideration to present the LQDG guidance strategy in the sense that the energy consumption of the pursuer is optimized; the maximum maneuver is utilized to engage with the interceptorevader team by virtue of the NDG theory for the obstacle located in P2 and P4.

The main contributions of this paper are proposed as follows.

1) When the obstacle is located in the P1 and P3 stages, the objective of the CDG guidance scheme is to guarantee that the pursuer avoids the static obstacle with a lower energy consumption based on the LQDG theory. And the guidance scheme doesn't produce a chattering phenomenon in the phases.

2) The requirement of bounded control input is satisfied in the CDG scheme for the obstacle located in the P2 and P4 phases. Therefore, it improves the success rate of the combat mission that the CDG guidance scheme adopts the maximum maneuver to fight against the interceptor and the evader due to the function of NDG.

3) The radii of the static obstacle and the interceptor are design parameters in the dead-zone functions to ensure that the pursuer avoids the static obstacle with the different obstacle radii and the interceptor with the different killing radii during the game process.

The organization of this paper is listed as follows. In Section 2, the engagement geometric relationship including nonlinear and linear kinematics is introduced. The PIE game issue is decomposed as LQDG/NDG and the ZEM is given in Section 3. Section 4 elaborates on the design process of the CDG guidance strategy with obstacle avoidance. Section 5 performs the nonlinear numerical simulations to validate the performance of the CDG guidance strategy with collision avoidance.

2. Preliminaries

2.1. Nonlinear kinematics

As shown in Fig. 1, the planar engagement scenario consists of a static Obstacle (О) and three players, a Pursuer (P), an Evader (Е), and an Interceptor (I). The Line-of-Sight (LOS) angles and the relative distances between the adversaries are represented as J; and R;, je (PE, PI, PO). The velocities, flight path angles and accelerations of each player are described as V;, y;, and a;,ie f£P, 1, Е}, respectively. And Ro is the static obstacle radius.

The following nonlinear differential equations are utilized to express the nonlinear relationship among the pursuer, the interceptor, the evader, and the obstacle.

The closing velocities are

... (1)

... (2)

... (3)

The LOS angle speeds are given as

... (4)

... (5)

The flight control dynamics of the adversaries are expressed by the following linear equations.

... (8)

Where, x; is the state vector; uj is the guidance command of each player, and satisfies ui < |u';["·, |u';|"·· is the overload limitation of each player; A;, b;, C;, and d; are dynamic model matrices.

The flight path angle rates of the pursuer, the interceptor, and the evader are given as follows.

... (9)

2.2. Linear kinematics

Egs. (1)-(6) are able to be linearized based on the assumption that the scenario is located near the collision triangles, the speeds Vi ie(P, LE) are constant and A;,j€{PE,PI,PO} are small. As shown in Fig. 2, define the separation between the players and the initial line-of-sight LOS;o, ¡E (PI, PO, PE} as y, je (P,1,E,O). у; = у; - yp, ¡E (PE, PI, PO), jefE, 1,0) is acquired according to the above definition.

The guidance commands of each player normal to LOS, ie (PI, PO, PE} are respectively ид, je {P,1,E}. The initial state of the players is represented by subscript O. x;, ie {P, I, E} given in Eq. (7) is the state of each adversary, and the relationship between x; and a; is represented in Eq. (8).

Based on the above definitions, it can be obtained

... (10)

Eq. (10) is rewritten as

... (11)

where, a = cos(yp, - Арво), b = COS(YE, + Apg,), and с = cos(y,, + Ари).

Select the following state vector of the linearized model to acquire

... (12)

The dimension of x is np + ng + Np + 6. Take the derivative of the state variable x to obtain the following equations.

... (13)

The state space equation is

... (14)

Since Egs. (1)-(6) are linearized around the collision triangles, ре, бурь and бро among the pursuer, the interceptor, the evader, and the obstacle are fixed and calculated by

... (15)

3. Game framework

It can be known from Ref. [29] that four phases, P1: (0, ty], P2: ЕЕ (ts, р], РЗ: ге (бр, 152], and P4: (2, trpe], in the engagement process are obtained according to the prescribed switching time ty; and ts.

For t€ (0, ур], the game is comprised of the pursuer, the interceptor, and the evader, respectively. In P1, the guidance strategies are designed by virtue of the LQDG method for each adversary; the NDG/LQDG guidance schemes are employed by the adversaries to

fight each other in P2. The game confrontation consists of the pursuer and the evader in the P3 and P4 phases because the engagement of the pursuer-interceptor terminates after the tp; moment. In P3, the pursuer and the evader use the LQDG guidance schemes; the NDG method is employed to acquire the guidance schemes for the players in P4.

It is worthy to point out that the issue of collision avoidance is ignored in the above approach which designs the CDG guidance strategy. Therefore, the pursuer may be detected by the defensive positions to cause the failure of the pursuit task. To overcome this drawback, a novel CDG guidance scheme with obstacle avoidance is proposed to fight the interceptor-evader team and the static obstacle. Since the framework of the CDG game issue is formed with the P1-P4 stages, the method for designing the guidance scheme is determined according to the phases in which the static obstacle is located. According to the above analysis, the LQDGmethod-based guidance strategy is derived for the adversaries when the obstacle is located in P1 and P3; the players adopt the NDG guidance scheme to fight each other in the case of the obstacle located in P2 and P4.

Remark 1. t. and t., are prescribed as switching time based on the final time between pursuer-interceptor and pursuer-evader. The energy cost of the pursuer is reduced by increasing the switching time ts] and t, in P1 and РЗ. Contrarily, the fuel consumption of the pursuer increases for the P2 and P4 stages due to decreasing the switching time t;1 and ts.

3.1. Order reduction

ZEM [38] is applied to lower the issue complexity through the reduction of the system order, which means the miss distance if none of the adversaries in engagement apply any control from the current time onward. Zpg, Zp;, and Zpo are respectively ZEMs between the pursuer and the evader, between the pursuer and the interceptor, between the pursuer and the obstacle.

Based on the above definitions and the terminal projection transformation [39], it can be obtained:

... (16)

where, ..., ..., Dpo = [O1x(m-ne+m+4) 1 O | and ф is a state transition matrix.

Differentiating Eq. (16), we yield the following formulas.

...

...

...

(17)

т т

where, hp, (t) = Drro (tros, E) [o Brg 01. (n 14 | App, (©) =

РОрЕФ (бра, DC.

Similarity,

... (18)

... (19)

where, ..., ..., ..., ...

Remark 2. The function of ZEMs given in Eq. (16) is to reduce the order of Eq. (14). It can be known from the above analysis that the order of the system is changed from np + ng +n; + 6 to 1, which lowers the difficulty of solving the PIE problem with collision avoidance.

3.2. Game decomposition and cost function design

According to the principle of the CDG guidance scheme with obstacle avoidance, the following cost functions are obtained by the ZEM and square integral term of control input.

The LQDG theory is utilized to devise the following cost function through trpo = ts; when the static obstacle is located in P1.

... (20)

For the interceptor-evader team, the equation is yielded as follows.

... (21)

when the obstacle is located in P2, the following formula is given according to the NDG approach.

... (22)

where,

...

К, is considered as the killing radius of the interceptor, à is a positive value to ensure Zpı(İşpı) > Ri.

The following cost function is presented for the evader according to the LQDG method.

... (23)

For the situation where the static obstacle is located in P3, the engagement becomes a one-to-one game between the pursuer and the evader since the interceptor ceases to exist. Therefore, the cost

function is given:

... (24)

where, ...

where, Zp is defined as the boundary value of the game space in P4, ap,, and œ are the weight parameters of the pursuer and the evader.

When the obstacle is located in P4, the cost function between

the adversaries is obtained by virtue of the NDG theory.

... (25)

The specific form of the dead-zone function Wpo[Zro(trpo)] given in the cost Eqs. (20)-(25) is

0

...

Remark 3. ap,, ay and ap,, & are the prescribed parameters of control input for the players. The objective of ap, and ap, is to reduce the energy consumption of the pursuer during the engagement. The parameters ay and ap are active for increasing the fuel loss of the interceptor-evader team. As shown in Figs. 3 and 4, Wi[ -]is the dead-zone functions [19] of state variables Z;(t;), ¡E (PO, PE), je {fPO,s2}, which guarantee that the pursuer avoids the static obstacle and satisfies the constraints of the game space. Wp[-] is a piecewise function of the state variable Zp;(trp;) in order to evade the interceptor at the final time.

4. Solutions of game

The solutions of the game that the obstacle is located in the P1-P4 phases, utilizing the DG theory, are acquired in this section according to the corresponding cost function Egs. (20)-(25). Note that the Hamiltonian functions are obtained to address the PIE game issue and design the CDG guidance scheme with obstacle avoidance.

1) The following Hamiltonian functions are devised for the obstacle located in P1. The engagement of the P1 is shown in Fig. 5.

... (26)

... (27)

The adjoint equations and transversality conditions are written as

... (28)

Substitute Üpo ро (©), Ед. (19) апа Apo = 2Wpo [Zpo (trpo)] into Eq. (26) to derive the following equation.

... (29)

For the pursuer, the following formula is acquired.

... (30)

The open-loop game strategy of the pursuer is

... (31)

The following equation is obtained by substituting Eq. (31) into Eq. (19), and integrating from t to tro.

... (32)

where,

... (33)

Bringing Eq. (32) into Ypo[Zro (trro)], we get

... (34)

Substituting Eq. (34) into Eq. (31), the closed-loop game scheme of the pursuer is obtained as follows.

... (35)

Similarly, bringing Egs. (17) and (18) into Hamiltonian Eq. (27) obtains

... (36)

For the evader and the interceptor, the optimal guidance strategies are acquired by the following formulas.

... (37)

The open-loop guidance schemes of the interceptor-evader are given based on the above derivation.

... (38)

Substituting Eqs. (31) and (38) into Eqs. (17) and (18), we obtain

... (39)

... (40)

The following formulas are given by bringing Ape = 2Zpe(trpo) and Ap; = 22 (typo) into Egs. (39) and (40), and integrating from t to trpo- After some lengthy, but straightforward manipulation, we get

... (41)

... (42)

where,

Substituting Eqs. (41) and (42) into Eq. (38), the following closed-loop optimal guidance strategies are acquired.

... (43)

... (44)

2) For the obstacle located in ?2, the engagement relationship is illustrated in Fig. 6. The Hamiltonian functions are designed as

... (45)

... (46)

The adjoint equations and transversality conditions are obtained as

... (47)

Substituting Apo = Pro [Zpo(tfpo)]IT and App = Wp (Zp (tre) into Eq. (45), we obtain

... (48)

After straightforward manipulation, bringing ging · ... Eqs. (18) and (19) into Eq. (48), we obtain

... (49)

The optimal guidance strategy of the pursuer can be obtained.

Bringing Egs. (50) and (53) into Ед. (17), and integrating from t

... (50)

to typ, we derive the closed-loop optimal guidance strategy of the evader.

The following pure pursuit strategy given in Ref. [21] is utilized by the interceptor to attack the pursuer.

... (51)

Substituting Eq. (17) and Apg = 2Zpg (tsp) into the Hamiltonian function (46), the LQDG theory is adopted to design the game scheme of the evader.

... (52)

The open-loop optimal guidance strategy of the evader is obtained by minimizing Eq. (52).

... (53)

... (54)

Remark 4. Substituting Egs. (50) and (51) into Eq. (18), and integrating from t to бр yield the following equation for [Za (t)| > (Ry + д).

...

... (55)

...

It can be obtained from Eq. (55) that

... (56)

Therefore,

... for - ...

3) When the static obstacle is located in P3, as shown in Fig. 7, the participants in the game are the pursuer and the evader since the engagement between the pursuer and the interceptor is completed after the moment гр. Based on the cost Eq. (24), the following Hamiltonian function is presented to handle the PIE problem.

... (57)

The adjoint equations and transversality conditions are

... (58)

Substituting W;[Z;(t)] and à;, ie (PO, PE}, Egs. (17) and (19) into Eq. (57), we get

... (59)

The following equations are given to design the optimal game schemes of the pursuer and the evader.

... (60)

The open-loop и; and up are derived according to Egs. (59) and (60).

... (61)

Ед. (62) is obtained by substituting up and Wpo Zro(trro)] into Eq. (19), and integrating from t to trpo.

... (62)

where, ...

Similarly, substituting Eq. (61) and Wpg[Zpg(tsy)] into Eq. (17), and integrating from t to t,>, we get

... (63)

where, ...

Based on Refs. [21,27,40], Zp is calculated as

... (64)

Remark 5. From Fig. 8 it is concluded that Zp divides the ZEM between the pursuer and the evader into singular and regular regions. Zpe(t) converges to zero at troy when it is located in the singular region. Therefore, |Zpg(ts;)| < Zp is satisfied to complete the pursuit mission that the pursuer attacks the evader at tsp.

Substituting Eqs. (62) and (63) into WrolZro(trro)] and Woe [Zp (552), we obtain

... (65)

... (66)

Substituting Eqs. (65) and (66) into Eq. (61), the closed-loop optimal guidance strategies for the pursuer and the evader are obtained.

... (67)

... (68)

4) When the obstacle is located in P4, the following equation is given according to the combat intention in Fig. 9.

... (69)

The adjoint equations and transversality conditions are obtained as

... (70)

Substitute Egs. (17), (19) and (70) and Üpo [Zpo(t)] into Eq. (69) to yield

... (71)

ug and up are obtained by minimizing the Hamiltonian function (71).

... (72)

... (73)

Remark 6. The guidance strategies with the obstacle avoidance

hold if

... (74)

... (75)

According to the definitions of Сро (©) and Gj (г),

... (76)

It can be concluded from the monotonical increase of Egs. (65) and (66) that ... Ro for ... for ... for ...

... (77)

Similarly, ...

Remark 7. In order to alleviate the chattering phenomenon in P2 and P4, the following saturation function sat(e) is utilized to replace the signum function sign( e).

...

Where, · is a positive constant.

5. Simulations

The CDG guidance strategy with obstacle avoidance is verified through nonlinear numerical simulations. Suppose the engagement scenario where the pursuer evades the obstacle, the interceptor, and attacks the evader in order to complete the pursuit task. The assumption that the information required in the CDG guidance strategy can be measured by sensors onboard is imposed in this section. The CDG guidance scheme with obstacle avoidance given by Egs. (82)-(91) is elaborated in the Appendix for the players with ideal proper dynamic.

When the obstacle is located in P1, the pursuer employs the LQDG guidance scheme Eq. (82) to evade the static obstacle and save fuel, and the evader-interceptor team utilizes the LQDG guidance strategies Eqs. (83) and (84); When the obstacle is located in P2, the NDG guidance scheme Eq. (85) is obtained to avoid the obstacle and the interceptor, the pure pursuit strategy Eq. (86) is used by the interceptor to fight with the pursuer, and the LQDG guidance strategy of the evader is acquired in Eq. (87); The framework of the engagement becomes pursuit and evasion game since the interceptor ceases to exist for the case that the obstacle is located in P3. The pursuer adopts the LQDG method to obtain the guidance strategy Eq. (88), and the evader performs the LQDG maneuver scheme Eq. (89); For the obstacle located in P4, the pursuer utilizes the NDG guidance strategy Eq. (90) to evade the static obstacle and attack the evader, and the evader employs the NDG guidance strategy Eq. (91).

According to the above analysis, four simulation scenarios are established to verify the effectiveness of the CDG guidance strategy proposed in this paper, which respectively illustrates that the pursuer evades the static obstacle, the interceptor, and attacks the evader when the obstacle is located in the P1-P4 stages. The simulations for three different values of the obstacle avoidance distance Ко: О m (no avoidance), 400 т, and 600 т are performed to demonstrate the robustness of the game strategy.

Case 1. The simulation parameters are given as follows.

The position of the static obstacle О is selected as хо = 29.6720 km, Уо = 212.0300 km; the killing radius of the interceptor is 40 т, and à is 50 m; the constant { is 3; the weight parameters are ap, 0.05, ay = 3.8, ap, = 5, and ag = 8, respectively. Based on Table 1 and the position of the obstacle, the final time are calculated as бро = 10.1798 5, tp; = 17.7729 5, and tp = 28.5259 s. And the switching time are ts] = trpo = 10.1798 s and ty; = 23 s, respectively. The obstacle avoidance distance of P2-P4 is 0 m.

Fig. 10 shows the flight trajectory of the game scheme for different obstacle radii when the obstacle is located in P1, which indicates that the pursuer evades the static obstacle, the interceptor, and intercepts the evader.

In Figs. 11 and 12, the variation tendencies of Zp; and Zp, for time are demonstrated under the situation of the different obstacle radii. It can be seen from Fig. 11 that, for the different Ro, the game strategy guarantees that Zp; converges and remains (Rı + 6) = 90 т at the terminal time tp, which expresses that the pursuer successfully accomplishes the task. The ZEM between the pursuer and the evader satisfies the game space |Zpg| <Zp = 149.6231 m at the switching time ty; and converges to О m at the terminal time trpg according to Fig. 10.

Fig. 13 shows the time evolutions of Zpo for the different obstacle radii. Based on the position of the static obstacle and Eq. (92), the initial value of |Zpo| is calculated as 1.3388 x 10 т > Ro. The initial value of Zpo is determined as О m in order to be better to illustrate the performance of the CDG guidance strategy. The simulation result demonstrates that Zpg increases from O m to the minimum required distance Ro (O т, 400 т, 600 т) and maintains the value at the final time trpo. From the above analysis, it is known that the pursuer evades the static obstacles for the different radii to verify the plots in Fig. 10.

The variation tendencies of Zp; and Zpg with respect to time are represented in Figs. 14 and 15 for the different killing radii of the interceptor when the obstacle radius Ro is 400 т. In Fig. 14, Zp; can converge to (R +6) at the terminal time tep, which coincides with the engagement trajectories in Fig. 10. For the different Ri, according to Fig. 15, Zpg converging to О m at the terminal time trpg represents that the pursuer intercepts the evader.

Fig. 16 is presented to show the time evolution of up for the different obstacle radii Ro. It can be seen from Fig. 16 that the initial commanded acceleration of the pursuer increases and converges to 0 m/s? when the minimum avoidance distance is increased. The overload constraint of the pursuer is satisfied due to |up| < up.

Case 2. The following simulation parameters are given in Table 2 to demonstrate the performance of the guidance strategies Egs. (50), (51) and (54) for the situation that the static obstacle is located in P2.

Set the position of the static obstacle as хо = 32.5550 km and Уо = 212.7600 km. The killing radius of the interceptor is given as 40 т, and ó is 50 m; the constant { is 3; and the parameters are chosen as ap, = 13.4, 4 = 2.8; ap, = 3, ag = 8. The final time trpo, trp,, and trp are respectively calculated as 13.6707 5, 18.0030 s and 30.8746 s by virtue of the parameters given in Table 2 and the coordinates of the obstacle. And the switching time t;1 and ts, are selected as 6 s and 23 s, respectively. The relationship between the terminal time and the switching time is ts; < бро < ру to validate that the static obstacle is located in P2. The obstacle avoidance distance of P1, РЗ and P4 is О т.

The engagement trajectories among the pursuer, the static obstacle, the interceptor and the evader are illustrated for the different Ко in Fig. 17. The simulation result represents that, for the different obstacle radii, the pursuer is able to evade the static obstacle and the interceptor, hit the evader when the obstacle is located in P2.

The variation trend of Zp; and Zpg with respect to time are shown for the different Ко by Figs. 18 and 19, respectively. From Fig. 18 it can be seen that Zp; converges to (К, +6) = 90 т at tp, which means that the pursuer successfully evades the interceptor. In Fig. 19, the condition of game space is met according to |Zpg(ts2)| < Zp = 303.8423 т for the different Ro. It is indicated that the pursuer attacks the evader through the fact that Zpg converges to О mat the final time tpg.

Fig. 20 shows the time evolutions of Zpo with respect to the different obstacle radii. Based on Eq. (92), the initial value of Zpo is calculated as |Zpo| = 1.5569 x 10 т > Ко. In order to demonstrate the effectiveness of the game strategy, similar to Case 1, the initial value of Zpo is given as O m and Zpg converges from O т to the minimum required miss distance in Fig. 20, which represents that the pursuer evades the static obstacle with the different radii at 'гро

Zp; and Zpg With respect to time are shown in Figs. 21 and 22 for the different killing radii, when the obstacle radius Ко is 400 т. From Fig. 21, the game scheme ensures that Zp; converges to (R +6) at the terminal time tp; in case of the different killing radii.

According to the above analysis, the pursuer evades the interceptor with the different killing radii. It can be seen from Fig. 22 that Zpg converges to 0 m, which represents that the pursuer hits the evader at the terminal time trpr.

The variation tendency of the acceleration up with time for the different obstacle radii is shown in Fig. 23. Based on Fig. 23, the acceleration of the pursuer is smaller than |uM·| in case of the different obstacle radii.

Case 3. When the static obstacle is located in P3, the players in the game become the pursuer and the evader since the engagement between the pursuer and the interceptor is completed after the tp; moment. The simulation parameters are listed in Table 3.

Choose the coordinate of the static obstacle as хо = 67.8920 km, Yo = 222.0800 km. The killing radius of the interceptor R¡ and ó are given as 40 m and 50 m; the constant 6 is 3. The weight parameters are ap, = 12.4, aj = 3.8, ap, = ag = 0.01. typ}, trpo, and Гуру are calculated as 15.5793 s, 21.7131 s and 29.4049 s according to Table 3 and the position of the obstacle. The switching time are selected as ts; = 8 s and t, = 25 5, respectively. The relationship between the terminal time and the switching time is бр < бро <İs2 to demonstrate that the static obstacle is located in P3. The obstacle avoidance distance of P1, P2 and P4 is 0 m.

The engagement trajectories among the pursuer, the interceptor, and the evader for the different obstacle radii are shown in Fig. 24, which represents that the pursuer evades the interceptor, the static obstacle, and hits the evader when the obstacle is located in P3.

Note that Figs. 25 and 26 show the evolutions of Zp; and Zpg with time for the different obstacle radii, respectively. It can be observed from Fig. 25 that Zp converges to (К, +6) = 90 т at tgp, which consists of the plots in Fig. 24. The engagement between the pursuer and the interceptor terminates in P3 to illustrate that the static obstacle will not affect Zp; during the combat. The CDG guidance scheme guarantees that Zpg converges to О m at the final time type and satisfies the game constraint of game space based on |Zpe (ts2)| = 5.3023 т <7р = 95.0741 т in Fig. 26.

Zpo with time for the different obstacle radii is illustrated in Fig. 27. Similarly, the initial value of Zpo is calculated as 2.63 x 10% m> Ro by virtue of (92). The initial value of |Zpo| is given as 0 m to validate the effectiveness of the game strategy. In Fig. 27, Zpo converges to the minimum required miss distance when the obstacle is located in P3, which expresses that the pursuer is able to avoid the obstacle with the different radii Ко.

The variation tendencies of Zp; and Zpg with respect to time for the different R; are illustrated by Figs. 28 and 29, respectively. Note that Zp; converges to (R +6) at the terminal time tp; in Fig. 28, which shows that the pursuer avoids the interceptor with different killing radii. From Fig. 29 it can be seen that Zpg reduces to O т at tepe, Which demonstrates that the pursuer is able to hit the evader and complete the attack mission.

The time evolutions of Zpg and up for the different Zpg (trp1) and Ro are demonstrated by Figs. 30 and 31, respectively. According to Fig. 30, Zpg converges to Zp = 95.0741 т at the final time tpg and О m at type for the different Zpg (Гуру) when the obstacle radius Ro is 200 m. The game space constraint is satisfied to accomplish the pursuit task through the above analysis in Fig. 30. As shown in Fig. 31, up in P3 grows gradually with the increase of the obstacle radii Ro and meets the overload limitation in the whole game stage.

Case 4. When the obstacle is located in P4, the pursuer employs the maximum maneuver to evade the static obstacle and attack the evader according to the NDG method. The parameters in Table 4 are given to demonstrate the performance of Eqs. (90) and (91).

The position of the static obstacle is selected as хо = 79.9110 km, yo = 225.2700 km; the killing radius of the interceptor R is 40 m and à is 50 m; the constant 6 is 3; the weight parameters are set as ap, = 11.4, a = 3.8; ap, = 3 and az = 8. The terminal time tfpı = 13.0751 5, trpo = 21.7131 s and tpg = 28.3538 s are obtained by virtue of the parameters in Table 4 and the coordinate of the obstacle. The switching time are chosen as tı = 8 sand ty = 18.5 5, respectively. The obstacle is located in P4 according to the relationship of the final time t;> < ро < ре. The obstacle avoidance distance of P1-P3 is O т.

The engagement trajectories of the players for the different obstacle radii are shown in Fig. 32 when the obstacle is located in P4. Based on Fig. 32, the pursuer avoids the interceptor and the static obstacle, and attacks the evader.

Figs. 33 and 34 describe the variation tendencies of Zp; and Zpg with respect to time for the different obstacle radii Ro, respectively. From Fig. 33 it is seen that Zp; converges to (К; +6) = 90 т at tgp; and isn't influenced by the obstacle radii Ro. In Fig. 34, Zpg reduces to О m for the different Ro at the final time t;pg.From Figs. 33 and 34 it can be concluded that the pursuit task is completed according to the convergences of Zp; and Zp at the terminal time.

The time evolution of Zpo for the different obstacle radii is illustrated in Fig. 35 under the circumstance that the obstacle is located in P4. Similarly, the initial value of Zpo is given as O min order to validate the CDG guidance strategy. According to Fig. 35, the guidance strategy guarantees Zro(trro)| > Ro to represent that the pursuer is able to evade the static obstacle for Ro.

The simulation results of Zp; and Zpg with the time for the different R are presented in Figs. 36 and 37 when the obstacle radius is 400 т. Zp; and Zpg converge to (К, +06) and O m for the different R, at the final time, which show that the pursuer is able to complete the attack mission under the circumstance of the different killing radii.

The variation tendency of up with regard to time for the different Ко is shown in Fig. 38. It can be known from Fig. 38 that the acceleration of the pursuer satisfies the limitation of overload in the whole phase.

6. Conclusions

A PIE issue with collision avoidance is considered to propose the combined guidance scheme in view of LQDG and NDG theories, which consists of the P1-P4 phases. The corresponding guidance strategies based on the stage in which the static obstacle is located are designed to complete the attacking task. The following conclusions are drawn.

1) When the obstacle is located in the P1 and P3 stages, the CDG guidance strategy is presented to lower the energy cost by employing the LQDG theory and the dead-zone function. There is no chattering phenomenon since the guidance strategy in P1 and P3 does not include the sign function.

2) The NDG guidance scheme is designed to enable that the pursuer evades the static obstacle, the interceptor and hits the evader for the case that the obstacle is located in P2 and P4. Therefore, the maximum guidance command is employed to increase the success rate of completing the pursuit mission.

3) The static obstacle radius and the killing radius of the interceptor are considered in the combined game scheme with obstacle avoidance, which ensures that the pursuer evades the static obstacle with the different obstacle radii and the interceptor with the different killing radii.

Declaration of competing interest The authors declare that they have no known competing

financial interests or personal relationships that could have appeared to influence the work reported in this paper.

Acknowledgements

This work is supported by National Natural Science Foundation (NNSF) of China under (Grant No. 62273119).

Appendix

The specific form of the CDG guidance scheme with obstacle avoidance is given for the players having ideal dynamics in the P1-P4 phases. The parameters of Eqs. (7) and (8) are respectively A; = O, b; = 0, С; = 0, а; = 1, ie {P,E,1}. By defining the following formulas as

... (78)

... (79)

Based on the above definitions,

... (80)

It is obtained from Egs. (17)-(19) that

... (81)

when the obstacle is located in P1,

... (82)

... (83)

... (84)

... (85)

... (86)

... (87)

when the obstacle is located in P3,

... (88)

... (89)

When the obstacle is located in P4,

... (90)

... (91)

ZEM adopts the nonlinear form given in Ref [41]. for the sake of the accuracy of the simulations.

... (92)

Sidebar

ARTICLE INFO

Article history:

Received 11 March 2024

Received in revised form

2 May 2024

Accepted 28 June 2024

Available online 2 July 2024

Footnote

* Corresponding author.

E-mail address: [email protected] (T. Chao).

Peer review under responsibility of China Ordnance Society

References

References

Isaacs R. Differential games: a mathematical theory with applications to warfare and pursuit, control and optimization. 1965. New York, NY, USA.

Shima T, Golan OM. Linear quadratic differential games guidance law for dual controlled missiles. IEEE Trans Aero Electron Syst 2007;43(3):834-42.

Turetsky М, Shinar J. Missile guidance laws based on pursuit-evasion game formulations. Automatica 2003;39(4):607-18.

Farugi FA. Differential game theory with applications to missiles and autonomous systems guidance. 2017. р. 102-3.

Rusnak I. The lady, the bandit and the body-guard game[C]. The 44™ Israel annual conference on aerospace science. 2004.

Rusnak I. The lady, the bandit and the body-guards-a two team dynamic game. IFAC Proc Vol 2005;38(1):441-6.

Rusnak 1. Guidance laws in defense against missile attack[C]. In: IEEE 25% convention of electrical and electronics engineers in Israel; 2008. p. 90-4.

Liang HZ, Wang JY, Wang YH, et al. Optimal guidance against active defense ballistic missiles via differential game strategies. Chin J Aeronaut 2020;33(3): 978-89.

Liang HZ, Li Z, Wu JZ, et al. Optimal guidance laws for a hypersonic multiplayer pursuit-evasion game based on a differential game strategy. Aerospace 2022;9(2):1-17.

Ye D, Shi MM, Sun ZW. Satellite proximate pursuit-evasion game with different thrust configurations. Aero Sci Technol 2020;99:1-10.

Ye D, Shi MM, Sun ZW. Satellite proximate interception vector guidance based on differential games. Chin J Aeronaut 2018;31(6):1352-61.

Tang X, Ye D, Huang L, et al. Pursuit-evasion game switching strategies for spacecraft with incomplete-information. Aero Sci Technol 2021;119:1-20.

Liang L, Deng F, Peng ZH, et al. A differential game for cooperative target defense. Automatica 2019;102:58-71.

Liang L, Deng F, Liu MB, et al. Analysis of role switch for cooperative target defense differential game. IEEE Trans Automat Control 2021;66(2):902-9.

Shi H, Zhu JH, Kuang MC, et al. Cooperative prediction guidance law in targetattacker-defender scenario. Sci China Inf Sci 2021;64(No.4):149201.

Liu F, Dong XW, Li QD, et al. Cooperative differential games guidance strategies for multiple attackers against an active defense target. Chin ] Aeronaut 2022;35(5):374-89.

Shalumov М. Cooperative online Guide-Launch-Guide policy in a targetmissile-defender engagement using deep reinforcement learning. Aero Sci Technol 2020;104:1-12.

Shalumov V. Online launch-time selection using deep learning in a target-missile-defender engagement. J Aero Inf Syst 2019;16(6):224-36.

Ma Y], Meng ОУ, Jiang В, et al. Fault-tolerant control for second-order nonlinear systems with actuator faults via zero-sum differential game. Eng Appl Artif Intell 2023;123:106342.

Shima T. Optimal cooperative pursuit and evasion strategies against a homing missile. J Guid Control Dynam 2011;34(2):414-25.

Sun QL, Zhang CF, Liu N, et al. Guidance laws for attacking defended target. Chin J Aeronaut 2019;32(10):2337-53.

Hayoun SY, Shima T. Necessary conditions for "hit-to-kill" in missile interception engagements. J Guid Control Dynam 2018;41(4):916-28.

Z. Ren, D. Zhang, S. Tang, et al., "Cooperative maneuver decision making for multi-UAV air combat based on incomplete information dynamic gamel]]", Defence Technology, Vol. 27, pp. 308-317.

Li SY, Chen М, Wang YH, et al. Air combat decision-making of multiple UCAVs based on constraint strategy games. Defence Technology 2022;18:368-83.

Turetsky V, Weiss M, Shima T. A combined linear-quadratic/bounded control differential game guidance law. IEEE Trans Aero Electron Syst 2021;57(5): 3452-62.

Hayoun SY, Weiss M, Shima T. A mixed L2Loodifferential game approach to pursuit-evasion guidance. IEEE Trans Aero Electron Syst 2016;52(6):2775-88.

Turetsky V, Hayoun SY, Shima T, et al. On the value of differential game with asymmetric control constraints[C]. IFAC-PapersOnLine 2018;51(32):799-804.

Yan T, Cai YL, Xu B. Evasion guidance for air-breathing hypersonic vehicles against unknown pursuer dynamics. Neural Comput Appl 2021;37(7): 5213-24.

Chao T, Wang XT, Wang SY, et al. Linear-quadratic and norm-bounded differential game combined guidance strategy against active defense aircraft in three-player engagement. Chin J Aeronaut 2023;36(8):331-50.

Guo SJ, Jing LJ, Dai ZP, et al. Collision avoidance problem of ellipsoid motion. Mathematics 2022;10:3478.

Sun ZY, Sun HB, Li P, et al. Cooperative strategy for pursuit-evasion problem in the presence of static and dynamic obstacles. Ocean Eng 2023;279:114476.

Li YH, Fan JK, Liu Y, et al. Path planning and path tracking for autonomous vehicle based on MPC with adaptive dual-horizon-parameters. Int J Automot Technol 2022;23(5):1239-53.

Huo D, Li D, Chai RQ, et al. Collision-free model predictive trajectory tracking control for UAVs in obstacle environment. IEEE Trans Aero Electron Syst 2023;59(3):2920-32.

Ye D, Sun JY, Xiao Y, et al. Energy optimal guidance for proximity approach with obstacle avoidance. Aero Sci Technol 2022;130:107949.

Weiss M, Shima T. Linear quadratic optimal control-based missile guidance law with obstacle avoidance. IEEE Trans Aero Electron Syst 2019;55(1): 205-14.

Kumar S, Weiss M, Shima T. Minimum-effort intercept angle guidance with multiple-obstacle avoidance. J Guid Control Dynam 2018;41(6):1355-69.

Jha B, Tsalik R, Weiss M, et al. Cooperative guidance and collision avoidance for multiple pursuers. J Guid Control Dynam 2019;42(7):1506-18.

Shima T, Idan M, Golan OM. Sliding-mode control for integrated missile autopilot guidance. J Guid Control Dynam 2006;29(2):250-60.

Bryson E, Ho CY. Applied optimal control[M]. Blaisdell Waltham; 1969. 154155, 282-289.

Glizer VY, Turetsky V. Complete solution of a differential game with linear dynamics and bounded controls. Appl Math Res eXpress 2007:1-49.

Shinar J, Glizer VY, Turetsky V. The effect of pursuer dynamics on the value of linear pursuit-evasion games with bounded controls in Advances in Dynamic Games-Theory, vol. 13. Switzerland: Birkhauser; 2013. р. 313-50.