Differential game strategy in three-player evasion and pursuit scenarios

Transcription

1 Journal o Systems Engineering and Electronics Vol. 29, No. 2, pril 218, pp Dierential game strategy in three-player evasion and pursuit scenarios SUN Qilong 1,QINaiming 1,*, XIO Longxu 2, and LIN Haiqi 1 1. Department o erospace Engineering, Harbin Institute o Technology, Harbin 151, China; 2. The Rocket Force Equipment cademy, Beijing 196, China bstract: conlict o three players, including an attacker, a deender, and a target with bounded control is discussed based on the dierential game theories in which the target and the deender use an optimal pursuit strategy. The current approach chooses the miss distance as the outcome o the conlict. Dierent optimal guidance laws are investigated, and easible conditions are analyzed or the attacker to accomplish an attacking task. For some given conditions, the attacker cannot intercept the target by only using a one-to-one optimal pursuit guidance law; thus, a guidance law or the attacker to reach a critical sae value is investigated. Speciically, the guidance law is divided into two parts. Beore the engagement time between the deender and the attacker, the attacker uses this derived guidance law to guarantee that the evasion distance rom the deender is sae, and that the zero-eort-miss (ZEM) distance between the attacker and the target is the smallest. ter that engagement time, the attacker uses the optimal one-toone guidance law to accomplish the pursuit task. The advantages and limited conditions o these derived guidance laws are also investigated by using nonlinear simulations. Keywords: three players, dierential game theory, zero-eortmiss (ZEM) distance, guidance law. DOI: /JSEE Introduction In an aerial scenario, it is diicult or a missile to attack an aircrat, because the aircrat always launches a deender against the missile. Thus, it is necessary to develop a easible guidance law or a missile to attack an aircrat. The scenario involves three agents, namely an aircrat (target), an attacking missile (attacker), and a deending missile (deender). The one-on-one engagement has been investigated or a long time and several guidance laws have been presented [1 3]. In recent years, various pursuit-evasion scenarios Manuscript received February 23, 217. *Corresponding author. This work was supported by the National Natural Science Foundation o China ( ). involving multiple agents have been developed. scenario in which two players pursuing a target was described in [4 6]. In those studies, the authors always solved the problem using a dierential theory. In contrast to two attackers pursuing one target, Xiao et al. [7] presented a cooperative guidance law or a multimissile system in which the missiles cooperated with each other to attack the target. Reerences [8,9] described a scenario where one attacker pursued two targets. In the scenario, the two evaders cooperated with each other to evade the pursuer. n active deense system has been more popular recently. Boyell and Shinar [1 12] described a scenario in which a deender and a ixed or slowly moving target constituted the deended system, which used a derived optimal guidance law against the missile. In that study, the authors always assumed that the position and the trajectory o the target were known to the deender, and that the position and the motion state o the attacker were also known to the deender. Rusnak [13,14] presented a dierential game or three persons: a lady, a bandit, and a bodyguard. The bandit pursued the lady and the lady tried to evade the bandit; meanwhile, the bodyguard minimized the distance rom the bandit to protect the lady. The problem was solved by using a dierential theory o multiple players. Later, Rusnak [15] investigated a high-order participant or an active deense system. Ratnoo and Shima [16] proposed an approach to protect the target; the deender used the line-o-sight guidance law to pursue the attacker. Reerences [17,18] presented an optimal cooperative guidance law or a target and a deender against an attacking missile using dierential and linear quadratic theories, respectively. The authors considered the target and the deender as a cooperative system, and they assisted each other to win the game. Shima and Ratnoo [19] analyzed a three-player problem in which an attacker and a deender used dierent guidance laws; they analyzed the conditions or the attacker to win the game. Later, Rubinsky and Gut-

2 SUN Qilong et al.: Dierential game strategy in three-player evasion and pursuit scenarios 353 man [2,21] presented a three-player scenario in which a missile evaded a deender and continued to pursue the target. In this scenario, the target and deender were independent. Garcia et al. [22 24] presented a deended system that contains a target and a deender. The target and the deender send inormation to each other and seek an optimal cooperative strategy against the homing missile. They investigated the guidance law by using nonlinear dynamics. Kumar and Shima [25] derived an optimal cooperative guidance law by using nonlinear dynamics, assuming that the attacking missile used a linear proportional guidance law. Particularly, in that study, they used a zero-eort velocity to ensure zero terminal relative lateral velocity, and doing so reduced the sensitivity o the guidance law to erroneous time-to-go estimates. In this study, we derive the optimal guidance laws or the attacker to intercept the target assuming that the deender and the target use the optimal one-to-one guidance laws. This study also chooses the miss distance as the outcome o the conlict; however, it is dierent rom the guidance law presented by Rubinsky and Gutman. For some cases, the attacker can accomplish the attacking task by using the one-to-one optimal guidance law provided that a reasonable pursuit acceleration within the maximal acceleration or the attacker is chosen. Using the optimal one-toone guidance law damages the collision triangle slightly and the acceleration o the attacker does not have to be much greater than the target; thus, as shown in [22], it is not necessary to evade the deender irst. For some situations, it is diicult to accomplish the task i the attacker is only assigned to a pursuit acceleration within the maximal acceleration. Thus, we derive a guidance law or the attacker to evade the deender to a critical sae value, and to guarantee that the zero-eort-miss (ZEM) distance between the attacker and the target is the smallest. Then the attacker uses the optimal one-to-one guidance law to accomplish the task. 2. Problem ormulation 2.1 Nonlinear dynamics The problem consists o three players: an attacker (), a target (T) and a deender (D). It is assumed that the deender is launched rom the target to intercept the attacker. In the problem, the target and the deender s positions, velocities, and accelerations are known to the attacker. The endgame scenario is described in Fig. 1. The range between the players is deined by R; the velocity o the player is deined by V ; the light path angle is denoted by γ; the line o sight is denoted by LOS; λ presents the angle between line o sight and the X axis, and a the lateral acceleration o the player. The subscripts, T, and D correspond to the attacker, the target and the deender. The subscripts T and D present the corresponding parameters between the attacker and the target, and the attacker and the deender, respectively. Fig. 1 The range rate is given by Engagement geometry Ṙ T = V cos(γ λ T )+V T cos(γ T + λ T ) (1) Ṙ D = V cos(γ λ D )+V D cos(γ D + λ D ). (2) The LOS rate is given by λ T = V T sin(γ T + λ T ) V sin(γ λ T ) R T (3) λ D = V D sin(γ D + λ D ) V sin(γ λ D ). (4) R D The dynamics o each player is obtained by ẋ i = i x i + b i u i, i = {, T, D} (5) a i = C i x i + d i u i, i = {, T, D} (6) where x i is the state vector o the internal state variables o each agent with dim (x i )=n i ; u i represents its controller and u i u i max,i = {, T, D}; i, b i, C i,andd i are the player s dynamics state-space model matrices. The path angle is in the ollowing orm: 2.2 Linearized kinematics γ i = a i V i, i = {, T, D}. (7) It is assumed that the problem occurs in the endgame phase and the light near the collision triangle can be linearized around LOS. The subscript represents the initial state. The relative displacement between and T normal to LOS T is denoted as y T and the relative displacement between and D normal to LOS D is denoted as y D. The accelerations o and T normal to LOS T and

3 354 Journal o Systems Engineering and Electronics Vol. 29, No. 2, pril 218 the acceleration o D normal to LOS D are denoted by u il {i =, T, D}, and they satisy u L = a cos(γ λ T )= C x cos(γ λ T )+d u u TL = a T cos(γ T + λ T )= C T x T cos(γ T + λ T )+d T u T u DL = a D cos(γ D + λ D )= C D x D cos(γ D + λ D )+d D u D where u,u T,andu D are the controllers normal to the corresponding LOS and they satisy the ollowing orm: u = u cos(γ λ T ) u T = u T cos(γ T + λ T ). (9) u D = u D cos(γ D + λ D ) The state vector o the linearized engagement is expressed as ollows: (8) x =[y T ẏ T x T T y D ẏ D x T D x T ] T (1) where the dimension o x is 4+n + n T + n D. It is assumed that, T and D obey ideal dynamics, and LOS T and LOS D are coincident. The equations o motion corresponding to (1) are given by ẏ T = x 2 ÿ T = u T u ẋ T = ẋ = ẏ D = x nt+4. (11) ÿ D = u D u ẋ D = ẋ = The equations can be written in the ollowing orm: where ẋ = x + B[u T u D ] T + Cu (12) 1 = 1, 1 1 B =, C =. 1 1 By linearizing the kinematics around the collision triangles, the intercept times are considered to be ixed and can be given by t T = R T Ṙ T, t D = R D Ṙ D (13) where R T and R D are the initial ranges between and T, and between and D, respectively. ṘT and ṘD are the initial relative velocities. In order to deend the target, the deender must intercept the attacker prior to the intercept time between the attacker and the target. It is considered that ater t D, the deender will disappear. We deine anewparametert i go in the ollowing orm: t i go = t i t, i = {T, D} (14) where t T go is the time-to-go o and T, and td go time-to-go o and D. 3. Dierential game description 3.1 Order reduction is the The ZEM distance between the attacker and the target can be obtained by using (15) below. Z T (t) =D T Φ(t T,t)x. (15) Similarly, the ZEM distance between the attacker and the deender can be obtained by Z D (t) =D D Φ(t D,t)x (16) where Φ(t T,t) and Φ(tD,t) are the transition matrices corresponding to (12) and they are expressed as { Φ(t D,t)= Φ(t D,t), Φ(t D,t D )=I Φ(t T. (17),t)= Φ(tT,t), Φ(tT,tT )=I where I represents the identity matrix. D D and D T are given by { DD =[ 1 nt 1 1 nd 1 n ] D T =[1 1 nt 1 nd 1 n ]. Equations (15) and (16) can be rewritten in the ollowing orm: { ZD (t) =y D +ẏ D (t D t) Z T (t) =y T +ẏ T (t T. (18) t) In order to ind the dynamics o Z T (t) and Z D (t),we dierentiate them with respect to time and combine them with (17). Their dynamics are given as { ŻD (t) =(t D t)( u + u D ) Ż T (t) =(t T t)( u + u T ). (19)

4 SUN Qilong et al.: Dierential game strategy in three-player evasion and pursuit scenarios 355 The optimal guidance law corresponding to (15) can be obtained as ollows: { u = sign(z D(t))u max u D = sign(z D(t))u max. (2) D The optimal guidance law corresponding to (16) can be obtained as ollows: { u =sign(z T(t))u max u T =sign(z T(t))u max. (21) T 3.2 Optimal evasion trajectories or attacker We investigate the optimal trajectories or to evade rom D. It is assumed that u max >u max D. Ż D (t) is given by Ż D (t) =t D go [sign(z D (t))u max Thus, Z D (t) is obtained by t D (t D Z D (t) =Z D (t =)+ t)[sign(z D (t))u max sign(z D (t))u max D ]. (22) sign(z D (t))u max D ]dt. (23) It is assumed that the kill radius o the deender is smaller than R; thus, i Z D (t = t D ) R, the deender cannot intercept the attacker. Deining Z D (t = t D )= ±R, we can obtain the two border trajectories in the ollowing orm: ZD Θ = R 1 2 (umax umax D )(td go ) 2 ZD Θ = R + 1. (24) 2 (umax umax D )(td go ) 2 The optimal trajectories are shown in Fig. 2, and the two border trajectories are described with thicker lines. In the engagement, and D use the optimal strategy; i Z D is located within the zone between the two border trajectories, the attacker will be intercepted by the deender. Conversely, i Z D is located outside the zone between the two border trajectories, the attacker will win the game. 3.3 Optimal pursuit trajectories or attacker We then solve or the optimal trajectories or to pursue T. It is assumed that u max >u max T. Ż T (t) is given by Ż T (t) = t T go [ sign(z T (t))u max +sign(z T (t))u max T ]. (25) Thus, Z T (t) is obtained by t T Z T (t) =Z T (t =)+ (t T t) [ sign(z T (t T ))u max +sign(z T (t T ))u max T ]dt. (26) It is assumed that the kill radius o the attacker is R 1 ; thus, i Z T (t = t T ) R 1, the attacker can accomplish the attacking task. The two border trajectories are obtained by deining Z T (t = t T )=±R 1: ZT Θ = R ( umax Z Θ T = R ( umax + umax T )(tt go ) 2. (27) + umax T )(tt go ) 2 Fig. 3 shows the optimal trajectories, and the two border trajectories are described with thick lines. In the engagement the attacker uses the optimal pursuit strategy and the target uses the optimal evasion strategy. I Z T is located within the zone between the two border trajectories, the attacker can intercept the target. Conversely, i Z T is located outside the zone between the two border trajectories, the target can evade the attacker. The maximum permitted Z max T by (t =) or the attacker to win the game is given Z max T (t =) = R ( umax + umax T )(tt )2. (28) I the attacker wants to win the game, Z T (t = t D ) should not be greater than a value deined by N, where N can be obtained rom (29). Fig. 2 Optimal trajectories o Z D N = R ( umax + umax T )(tt t D ) 2 (29)

5 356 Journal o Systems Engineering and Electronics Vol. 29, No. 2, pril Case 2: the signs o Z T (t=) and Z D (t =) are the same In this case, we can obtain signz D (t) = signz T (t); thus, the attacker s optimal guidance law o (2) is contrary to the optimal guidance law o (21), which means that i the attacker uses the optimal pursuit guidance law to pursue the target, it will simultaneously perorm the worst in evading the deender. Fig. 5 shows the ZEM distance or Case 2 as a unction o time. We only analyze the positive values o Z T (t =)and Z D (t =), because the analysis is the same or negative values. Fig. 3 Optimal trajectories o Z T 4. Guidance law o the attacker In order or the attacker to win the game, we need to analyze the required conditions or the attacker to accomplish the evasion and pursuit task and then design a easible guidance law or the attacker. We divide the engagement into two cases. 4.1 Case 1: the signs o Z T (t=) and Z D (t=) are opposite In this case, we can obtain signz D (t) =signz T (t); thus, the optimal evasion and pursuit control eorts or the attacker are the same and the attacker can accomplish the optimal evasion and pursuit action simultaneously in the engagement. Fig. 4 shows the time evolution o ZEM distance or Case 1. In this igure, Z D represents the ZEM between the deender and the attacker or Case 1 as a unction o time. Z T represents the time evolution o ZEM between the target and the attacker. The meanings o the two lines are the same or the subsequent igures. Fig. 5 ZEM distance or case 2 as a unction o time When the attacker only uses the optimal pursuit guidance law to intercept the target, the required conditions or the attacker to avoid the deender can be divided into two cases. The irst required condition is that Z D (t =)is a relatively large positive value. The schematic igure or the irst condition is shown in Fig. 6. In the igure, ZD Θ is the border trajectory above the zero axis. It is noted that Z D (t D ) = R is a critical value. For this case, Z D (t =)satisies the ollowing orm: Z D (t = t D )=Z D (t =)+ t D ( sign(z T (t))u max umax D )td go The critical Z D (t =)can be obtained by Z D (t =)= dt = R. (3) Fig. 4 ZEM distance or case 1 as a unction o time R [sign(z T(t))u max + umax D ](td ) 2. (31) I Z D (t = ) R + [sign(z T (t))u max + u max D ](td ) 2 /2, the attacker can avoid the deender. Combining (28) and (31), we can obtain the initial conditions o Z D (t =)and Z T (t =)or the attacker to accom-

6 SUN Qilong et al.: Dierential game strategy in three-player evasion and pursuit scenarios 357 plish the task, and these conditions are in the ollowing orm: Z D (t =) R [sign(z T(t))u max + u max D ](td ) 2. (32) Z T (t =) R ( umax +umax T )(tt )2 Speciically, by assuming that Z T (t) beore t D, we can write (32) in the ollowing orm: 1 Z D (t =) R (umax 2 ( umax 1 2 ( umax + u max T )(t T + umax T R ( umax + umax D )(td ) 2 t D ) 2 )(tt)2 Z T (t =) + u max T )(t T ) 2. (33) Similarly, it is assumed that when Z T (t D ), the critical Z D (t =)can be obtained by I 1 2 (umax + u max D ) Z D (t =)= [ (t D ) 2 Z D (t =) (u max +u max D ) 1 2 [ u max (t D 2R umax D ) 2 ]. (36) 2R u max umax D the attacker can avoid the deender. Under the condition that Z T (t D ), by combining (36) and (28), we obtain the initial conditions o Z D (t =)and Z T (t =) or the attacker to accomplish the task, and the conditions are in the ollowing orm: Z D (t =) 1 2 (umax + u max D ) [ ] (t D ) 2 2R 1 2 ( umax 1 2 ( umax (u max + umax T )(tt + umax T umax D ) )(tt t D ) 2 )2 Z T (t =) R ( umax + u max T )(t T ) 2 ],. (37) Fig. 6 ZEM distance or the irst condition as a unction o time The second required condition is that Z D (t = ) is a relatively small positive value. The schematic igure or the second condition is shown in Fig. 7. Here, ZD Θ is the border trajectory below the zero axis. It is noted that Z D (t D ) = R is a critical value. For this case Z D (t =)satisies the ollowing orm: Z D (t =)+ [ sign(z T (t))u max t D (t D t) sign(z D(t))u max D ]dt = R (34) and t Z D (t =)+ (t D t)( u max t D t (t D t)( signz T (t)u max umax D )dt+ + u max D )dt = R. (35) Fig. 7 ZEM distance or the second condition as a unction o time I Z D (t =)is not a relatively large positive value or a relatively small positive value, or the attacker to accomplish the attacking task, we need to design a guidance that guarantees the evasion and pursuit distance, which are Z D (t D ) R and Z T (t T ) R 1. We only analyze the positive values o Z T (t =)and Z D (t =).Itis

7 358 Journal o Systems Engineering and Electronics Vol. 29, No. 2, pril 218 assumed that during the inal time t D, the attacker employs a consistent control eort deined by u.bydeining Z D (t D ) R [21], and substituting u into (19) and integrating it, we obtain t D Z D (t =)+ (t D t)( u max D u )dt R. (38) Thus, the control range or the attacker to evade the deender is obtained by u u max D + 2Z D(t =) 2R (t D ) 2. (39) Substituting u into (19) and integrating it, we obtain Z T (t D )= Z T (t =)+ 1 2 ( u +u max T )[2t T t D (t D ) 2 ]. (4) It is noted that Z T (t D ) is a monotonically decreasing unction o u. Within the control range or the attacker to evade the deender, the largest value o the attacker s acceleration satisies u = u max D + 2Z D(t =) 2R (t D ) 2. (41) Thus, u = u max D +[2Z D(t =) 2R]/(t D ) 2 is a border value that can guarantee Z T (t D ) is the smallest value within the permitted control range or the attacker to reach the sae area at t = t D. Thus, during the inal time t D, we select u = u max D +2(Z D(t =) R)/(t D ) 2 as the control eort o the attacker; ater t D, the attacker uses the optimal guidance law o (21) to accomplish the pursuit task. It is known that or the attacker to accomplish the task, based on (29), Z T (t D ) should satisy (42). [2t T t D (t D ) 2 ] R ( umax +u max T )(t T t D ) 2. (44) Thus, combining (44) and Z T (t =), we can obtain the permitted range or Z T (t =)in the ollowing orm: Z T (t =) R ( umax {[ 1 u max D 2 +umax T 2Z D(t =) 2R (t D ) 2 )(tt ] + u max T t D ) 2 } [2t T td (t D ) 2 ]. (45) It is known that i we use the derived guidance law, and i Z T (t =)satisies (45), the attacker can accomplish the evasion and pursuit task. The derived guidance law has the ollowing orm: u = u max D + 2Z D(t=) 2R (t D, t t D ) 2. (46) sign(z T (t))u max, td t t T Fig. 8 shows the ZEM distance or the derived guidance law as a unction o time. It is noted that the attacker uses the designed guidance law to ensure that the ZEM distance between the attacker and the deender reaches the bounded value R at t = t D, and to guarantee that Z T (t D ) is the smallest value. ter t = t D, the attacker uses the optimal one-to-one guidance law as outlined in (21) to accomplish the pursuit task. Z T (t D ) R ( umax + umax T )(tt Substituting (4) into (42), we can obtain t D ) 2 (42) Z T (t =)+ 1 2 ( u + u max T R ( umax )[2tT td (t D ) 2 ] + u max T )(t T t D ) 2. (43) Then, substituting (41) into (43), we can obtain {[ ] } Z T (t =)+ 1 u max D 2Z D(t=) 2R 2 (t D + u max ) 2 T Fig. 8 time ZEM distance or the derived guidance law as a unction o 5. Simulation In this section, the dierent guidance laws are veriied and the required conditions are analyzed. The initial parameters are given in Table 1. R and R 1 are assumed to be 8 m and 2 m, respectively.

8 SUN Qilong et al.: Dierential game strategy in three-player evasion and pursuit scenarios 359 Table 1 Initial parameters Parameter T D Initial position (X,Y )/km (5,.2) (, ) (5,.2) Initial speed V i /(m/s) Simulation o Case 1 The initial conditions are given in Table 2. The initial Z T (t=) and Z D (t=) are m and m, respectively. Table 2 Initial parameters or Case 1 Parameter T D Initial course γ i /( ) Maximal acceleration u i max /(m/s 2 ) For this case, i the attacker pursues the target by using the one-to-one optimal guidance law, it can simultaneously evade the deender. We use nonlinear simulation to analyze this situation. Fig. 9 shows the trajectories o the three players. It is noted that the miss distance between the attacker and the target, and the miss distance between the attacker and the deender are.428 m and m, respectively. Thus, the attacker can avoid the deender and intercept the target. The inal time, t T respectively. and t D, are s and s, Fig. 1 shows the time evolution o the time-to-go. It can be observed that t D go is almost linear in orm; however, at t T go, there exists a turning point where Z T is a very small value close to zero. ter the turning point, the accelerations o the attacker and the target display the bang-bang phenomenon. Fig. 11 shows the trajectories o the three players when the maximal acceleration o the deender becomes 8 m/s 2. Here, the miss distance between the attacker and the deender decreases to m and the attacker is intercepted. For this case, i the attacker wants to accomplish the task, it must have a greater maximal acceleration. It is assumed that the attacker s maximal acceleration becomes 1 m/s 2. Thus, the maximal accelerations o the attacker and the deender are 1 m/s 2 and8m/s 2, respectively. Fig. 12 and Fig. 13 show the trajectories o the three players and the time evolution o the time-to-go, respectively. Fig. 1 Time-to-go with parameters shown in Table 2 Fig. 11 Trajectories with parameters shown in Table 2 except u D max =8m/s 2 Fig. 9 Trajectories with parameters shown in Table 2 Fig. 12 Trajectories with parameters shown in Table 2 except u D max =8m/s 2 and u max = 1 m/s 2

9 36 Journal o Systems Engineering and Electronics Vol. 29, No. 2, pril 218 accomplish the task. For this case, i the attacker wants to intercept the target, it must have a smaller maximal acceleration. It is assumed that the maximal accelerations o the attacker and the deender become 73 m/s 2 and8m/s 2,respectively. Fig. 17 and Fig. 18 show the trajectories o the three players and the time evolution o the time-to-go. Fig. 13 Time-to-go with parameters shown in Table 2 except u D max =8m/s 2 and u max = 1 m/s 2 It can be noted that the miss distance between the attacker and the target and that between the attacker and the deender are.126 m and m, respectively; thus, the attacker can intercept the target. The inal time, t D and t T, are s and s, respectively. Because the maximal acceleration o the attacker increases, the miss distance between the target and the attacker decreases to a smaller value than that shown in Fig. 9, and the turning point occurs earlier than that shown in Fig Simulation o Case 2 Example 1 The initial conditions are given in Table 3. The initial Z T (t =)and Z D (t =)are m and m, respectively. Z D (t =)is a relatively large value or this situation. The attacker can only use the oneto-one optimal guidance law to pursue the target. lthough this case is rare, it does exist and needs to be analyzed. Table 3 Initial parameters or Example 1 Parameter T D Initial course γ i /( ) Maximal acceleration u i max /(m/s 2 ) For this case, i the attacker pursues the target by using the one-to-one optimal guidance law, it makes the worst maneuver to evade the deender. Similar to the previous case, we analyze this situation by using nonlinear simulation. Fig. 14 and Fig. 15 show the trajectories o the three players and the time evolution o the time-to-go. The miss distance between the attacker and the target, and the inal time t T are still.428 m and s, respectively. The miss distance between the attacker and the deender is m, and the attacker can accomplish the task. The inal time t D is s. Fig. 16 shows the trajectories o the three players when the maximal acceleration o the deender is assigned to a greater value, such as 8 m/s 2. Here, the miss distance between the attacker and the deender decreases to m and the attacker cannot Fig. 14 Trajectories with parameters shown in Table 3 Fig. 15 Time-to-go with parameters shown in Table 3 Fig. 16 Trajectories with parameters shown in Table 3 except u D max =8m/s 2

10 SUN Qilong et al.: Dierential game strategy in three-player evasion and pursuit scenarios 361 target, and the inal time t T are still.428 m and s,respectively,whichareshowninFig.19and Fig. 2, respectively. The miss distance between the attacker and the deender is m, and the inal time t D is s. It is assumed that the maximal acceleration o the deender is assigned to a greater value, such as 65 m/s 2. Fig. 17 Trajectories with parameters shown in Table 3 except u D max =8m/s 2 and u max =73m/s 2 Fig. 19 Trajectories with parameters shown in Table 4 Fig. 18 Time-to-go with parameters shown in Table 3 except u D max =8m/s 2 and u max =73m/s 2 It is observed that the miss distance between the attacker and the target and that between the attacker and the deender are.728 m and m, respectively; thus, the attacker can accomplish the task. The inal time t D and t T are s and s, respectively. s the maximal acceleration o the attacker becomes smaller, the miss distance between the target and the attacker rises to a greater value than that shown in Fig. 14 and the turning point occurs later than that shown in Fig. 15. Example 2 The initial conditions are given in Table 4. The initial Z T (t =)and Z D (t =)are m and m, respectively. Z D (t =)is a relatively small value or this situation. The attacker can only use the one-to-one optimal guidance law to pursue the target. This case is common; thus, it is important to analyze it. Table 4 Initial parameters or Example 2 Parameter T D Initial courseγ i /( ) 14 8 Maximal acceleration u i max /(m/s 2 ) We also analyze this situation by using nonlinear simulation. The miss distance between the attacker and the Fig. 2 Time-to-go with parameters shown in Table 4 The trajectories o the three players are shown in Fig. 21, in which the attacker is intercepted by the deender. Through Fig. 7 we can conclude that i we want to increase the miss distance between the attacker and the deender, we can assign a greater value to the maximal acceleration o the attacker. It is assumed that the maximal accelerations o the deender and the attacker are 65 m/s 2 and 1 m/s 2, respectively, and the trajectories or this situation are shown in Fig. 22. It can be noted that although the attacker s maximal acceleration increases, the miss distance between the attacker and the deender cannot increase; this is because the attacker uses the optimal one-to-one guidance law to pursue the target, and the acceleration is restricted by Z T. When the attacker s maximal acceleration becomes greater, Z T reduces to zero more quickly, and

11 362 Journal o Systems Engineering and Electronics Vol. 29, No. 2, pril 218 the bang-bang phenomenon occurs earlier. From Fig. 7, it can be concluded that by increasing Z T (t =), we can obtain a greater miss distance between the attacker and the deender. The attacker s initial course is redeined by 2 and the initial Z T (t =)and Z D (t =)are m and m, respectively. The maximal accelerations o the deender and the target are still 65 m/s 2 and 1 m/s 2, respectively, and the nonlinear simulation results are shown in Fig. 23 and Fig. 24. For this case, it is observed that the miss distance between the attacker and the target, and that between the attacker and the deender are.153 m and m, respectively; thus, the attacker can intercept the target. The inal time t D and t T are s and s, respectively. s the maximal acceleration o the attacker becomes greater, the miss distance between the target and the attacker decreases to a smaller value than that shown in Fig. 19 and the turning point occurs earlier than that shown in Fig. 2. Fig. 23 Trajectories with parameters shown in Table 4 except u D max =65m/s 2, u max = 1 m/s 2,andγ = 2 Fig. 24 Time-to-go with parameters shown in Table 4 except u D max =65m/s 2, u max = 1 m/s 2,andγ = 2 Fig. 21 Trajectories with parameters shown in Table 4 except u D max =65m/s 2 Fig. 22 Trajectories with parameters shown in Table 4 except u D max =65m/s 2 and u max = 1 m/s 2 Example 3 The initial conditions are given in Table 5. The attacker uses the guidance law corresponding to (46), which is similarly tested by using nonlinear simulation. Table 5 Initial parameters or Example 3 Parameter T D Initial course γ i /( ) 6 15 Maximal acceleration u i max /(m/s 2 ) Initial position (X,Y )/km (9,.2) (,) (9,.2) Initial speed V i /(m/s) Fig. 25 and Fig. 26 show the trajectories and time-to-go by using the parameters given in Table 5. The miss distance between the attacker and the deender is 8.74 m, and the miss distance between the attacker and the target is.319 m; thus the attacker wins the game. The inal times t T and t D are s and s, respectively. It can be noted that the time evolution o t T go is not linear in orm but inally becomes approximately linear with a decreasing slope. This is because when using this guidance law, the attacker needs to avoid the deender irst and damages the collision triangle heavily; however, a new collision triangle is established during the inal attacking period. Fig. 27 shows the trajectories or the case where the

12 SUN Qilong et al.: Dierential game strategy in three-player evasion and pursuit scenarios 363 maximal acceleration o the deender changes to 1 m/s 2 and the other parameters remain the same. s can be observed, the attacker ails to complete this task. In order to accomplish the task, the attacker s maximal acceleration must be greater than 15 m/s 2.Itisassumed that the maximal accelerations o the deender and the attacker are 1 m/s 2 and 18 m/s 2, while the other parameters are invariant. The new trajectories and timeto-go are shown in Fig. 28 and Fig. 29. The miss distances are m and.429 m, respectively; thus, the attacker wins. The inal time t T and t D are s and s, respectively. It can be noted that using this guidance law, the attacker s maximal acceleration needs to be much greater than that o the target. Fig. 25 Trajectories with parameters shown in Table 5 Fig. 28 Trajectories with parameters shown in Table 5 except u D max = 1 m/s 2 and u max = 18 m/s 2 Fig. 26 Time-to-go with parameters shown in Table 5 Fig. 29 Time-to-go with parameters shown in Table 5 except u D max = 1 m/s 2 and u max = 18 m/s 2 Fig. 27 Trajectories with parameters shown in Table 5 except u D max = 1 m/s 2 Example 4 The initial conditions are given in Table 6. We test the attacker s optimal one-to-one pursuit guidance law and the guidance law identiied in (46) by using nonlinear simulation. Table 6 Initial parameters or Example 4 Parameter T D Initial course γ i /( ) 7 7 Maximal acceleration u i max /(m/s 2 ) Initial position (X,Y )/km (9,.2) (, ) (9,.2) Initial speed V i /(m/s) 3 6 8

13 364 Journal o Systems Engineering and Electronics Vol. 29, No. 2, pril 218 Fig. 3 and Fig. 31 show the trajectories or the two guidance laws, respectively. It can be observed that no matter which guidance law is used, the attacker can intercept the target. s shown in Fig. 31, using the guidance law corresponding to (46) damages the collision triangle between the attacker and the target, and the evasive distance is much smaller than that when using the optimal one-to-one pursuit guidance law as shown in Fig. 3. Under these parameters, using the optimal one-to-one guidance law damages the collision triangle slightly and the attacking miss distance is smaller. maximal acceleration should be much greater than that o the target. Thus, we can choose a reasonable pursuit acceleration within the maximal acceleration or the attacker to win the game, such as the circumstances shown in Fig. 32. I the maximal acceleration o the attacker is 18 m/s 2,we choose 8 m/s 2 as the pursuit acceleration. Hence, we can conclude that i the signs o Z T (t =)and Z D (t =) are the same, or some instances, the attacker, it is not necessary or the attacker to evade the deender irst. Fig. 32 Trajectories by using the optimal pursuit guidance law with parameters given in Table 6 except u max =8m/s 2 Fig. 3 Trajectories by using the optimal pursuit guidance law Fig. 33 Trajectories by using (46) with parameters given in Table 6 except u max =8m/s 2 Fig. 31 Trajectories by using (46) Fig. 32 and Fig. 33 show the trajectories or the two guidance laws, respectively, or the case when the maximal acceleration o the attacker is 8 m/s 2. It can be observed that by decreasing the maximal acceleration o the attacker, the optimal pursuit guidance law still can accomplish the task as shown in Fig. 32. However, the guidance law corresponding to (46) ails; this is because the guidance law corresponding to (46) requires that the attacker s lthough under these parameters, using the optimal pursuit guidance law is better than using the guidance law corresponding to (46), and in some cases using the guidance law rom (46) cannot intercept the target, this guidance law can win the optimal pursuit guidance law in some cases such as the one given in Example 3. Fig. 34 and Fig. 35 show the trajectories o the three players by using the optimal guidance law with the pursuit acceleration being 15 m/s 2 and 18 m/s 2, respectively. It can be observed that the attacker cannot accomplish the attacking task.

14 SUN Qilong et al.: Dierential game strategy in three-player evasion and pursuit scenarios 365 Meanwhile, even i the attacker s maximal acceleration becomes greater, the miss distance between the attacker and the deender cannot increase to a sae range. This is because when the attacker uses the optimal one-to-one guidance law to pursue the target, the acceleration is restricted by Z T. However, under the parameters given in Example 3, the derived guidance o (46) can accomplish the task. Fig. 34 Trajectories by using the optimal pursuit guidance law with parameters given in Table 5 Fig. 35 Trajectories by using the optimal pursuit guidance law with parameters given in Table 5 except u max = 18 m/s 2 6. Conclusions Guidance laws are investigated or the attacker to win a game involving an attacker, a deender and a target. The game occurs in the endgame; thus, we derive the guidance laws by using linear dynamics and investigate them by using nonlinear simulation. When the signs o Z T (t =) and Z D (t = ) are opposite, the attacker uses the optimal one-to-one guidance law to pursue the target and simultaneously evades the deender. When the signs o Z T (t =)and Z D (t =)are the same, the attacker uses the optimal one-to-one guidance law to pursue the target but simultaneously perorms the worst with respect to the deender. However, the attacker can still accomplish the task under some conditions, speciically when the initial Z D (t =)is a relatively large value or when a relatively small value under Z T (t =)is reasonable. For this case, the collision triangle is slightly damaged and the acceleration o the attacker does not have to be much greater than that o the target. In this scenario, we only need to choose a reasonable pursuit acceleration within the maximal acceleration or the attacker. Thus, it is not necessary or the attacker to evade the deender irst because the guidance law deined in (46) damages the collision triangle heavily, and the acceleration needed by the attacker is much greater than that o the target, as indicated in Example 4. However, or some situations, such as in Example 3, it is diicult to accomplish the task i the attacker is only assigned to a pursuit acceleration within the maximal acceleration. For this case, we need to use the guidance law deined in (46) to accomplish the task, as shown in Fig. 25. Thus, in reality, we need to choose reasonable guidance laws or the attacker to win a game under dierent conditions. The investigation and the derived guidance laws have potential or use in pursuit-evasion problems. Reerences [1] PENG S, PN L, HU T, et al. New three-dimensional guidance law or BTT missiles based on dierential geometry and Liegroup. Journal o Systems Engineering and Electronics, 211, 22(4): [2] SONG J, SONG S. Three-dimensional guidance law based on adaptive integral sliding mode control. Chinese Journal o eronautics, 216, 29(1): [3] LI R, XI Q, WEN Q. Extended optimal guidance law with impact angle and acceleration constriants. Journal o Systems Engineering and Electronics, 214, 25(5): [4] LIU Y, QI N, TNG Z. Linear quadratic dierential game strategies with two-pursuit versus single-evader. Chinese Journal o eronautics, 212, 25(6): [5] KUMKOV S S, MENEC S L, PTSKO V S. Solvability sets in pursuit problem with two pursuers and one evader. IFC Proceedings Volumes, 214, 47(3): [6] KUMKOV S S, MENEC S L, PTSKO V S. Level sets o the value unction in dierential games with two pursuers and one evader, interval analysis interpretation. Mathematics in Computer Science, 214, 8(8): [7] XIO B S, FNG Y W, HU S G, et al. Decision methods or cooperative guidance in multi-aircrat air warare. Systems Engineering and Electronics, 29, 31(3): (in Chinese) [8] FUCHS Z E, KHRGONEKR P P, EVERS J. Cooperative deense within a single-pursuer, two-evader pursuit evasion dierential game. Proc. o the 49th IEEE Conerence on Decision and Control, 211: [9] SCOTT W, LEONRD N E. Pursuit, herding and evasion: a three-agent model o caribou predation. Proc. o the merican Control Conerence, 213:

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