SCIENCE CHINA Information Sciences, Volume 61, Issue 11: 112206(2018) https://doi.org/10.1007/s11432-017-9408-6

## Evasion strategies of a three-player lifeline game

• AcceptedMar 5, 2018
• PublishedOct 15, 2018
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### Abstract

This study examines a multi-player pursuit-evasion game, more specifically, a three-player lifeline game in a planar environment, where a single evader is tasked with reaching a lifeline prior to capture. A decomposition method based on an explicit policy is proposed to address the game qualitatively from two main aspects: (1) the evader's position distribution to guarantee winning the game (i.e., the escape zone), which is based on the premise of knowing the pursuers' positions initially, and (2) evasion strategies in the escape zone. First, this study decomposes the three-player lifeline game into two two-player sub-games and obtains an analytic expression of the escape zone by constructing a barrier, which is an integration of the solutions of two sub-games. This study then explicitly partitions the escape zone into several regions and derives an evasion strategy for each region. In particular, this study provides a resultant force method for the evader to balance the active goal of reaching the lifeline and the passive goal of avoiding capture. Finally, some examples from a lifeline game involving more than one pursuer are used to verify the effectiveness and scalability of the evasion strategies.

### Acknowledgment

This work was supported by Innovative Research Groups of National Natural Science Foundation of China (Grant No. 61621063) and Key Program of National Natural Science Foundation of China (Grant No. 1613225).

### References

[1] Yuksek B, Ure N K. Optimization of allocation and launch conditions of multiple missiles for three-dimensional collaborative interception of ballistic target. Int J Aerospace Eng, 2016, 2016: 9582816. Google Scholar

[2] Liu Y F, Li R F, Wang S Q. Orbital three-player differential game using semi-direct collocation with nonlinear programming. In: Proceedings of the 2nd International Conference on Control Science and Systems Engineering, Singapore, 2016. 217--222. Google Scholar

[3] Shen D, Pham K, Blasch E, et al. Pursuit-evasion orbital game for satellite interception and collision avoidance. Proc SPIE, 2011, 8044: 284--287. Google Scholar

[4] Casbeer D W, Garcia E, Pachter M. The target differential game with two defenders. In: Proceedings of 2016 International Conference on Unmanned Aircraft Systems, Arlington, 2016. 202--210. Google Scholar

[5] Li Y, Mu Y F, Yuan S. The game theoretical approach for multi-phase complex systems in chemical engineering. J Syst Sci Complex, 2017, 30: 4-19 CrossRef Google Scholar

[6] Wei N, Zhang Z. Competitive access in multi-RAT systems with regulated interference constraints. Sci China Inf Sci, 2017, 60: 022306 CrossRef Google Scholar

[7] Zhu B, Xie L H, Han D. A survey on recent progress in control of swarm systems. Sci China Inf Sci, 2017, 60: 070201 CrossRef Google Scholar

[8] Chen P, Sastry S. Pursuit controller performance guarantees for a lifeline pursuit-evasion game over a wireless sensor network. In: Proceedings of the 45th IEEE Conference on Decision and Control, San Diego, 2016. 691--696. Google Scholar

[9] Mu Y F, Guo L. How cooperation arises from rational players?. Sci China Inf Sci, 2013, 56: 112201 CrossRef Google Scholar

[10] Gao H W, Petrosyan L, Qiao H. Cooperation in two-stage games on undirected networks. J Syst Sci Complex, 2017, 30: 680-693 CrossRef Google Scholar

[11] Isaacs R. Differential Games: A Mathematical Theory with Applications to Warfare and Pursuit, Control and Optimization. Hoboken: John Wiley and Sons, 1965. Google Scholar

[12] Chen J, Zha W Z, Peng Z H. Multi-player pursuit-evasion games with one superior evader. Automatica, 2016, 71: 24-32 CrossRef Google Scholar

[13] Zha W Z, Chen J, Peng Z H. Construction of barrier in a fishing game with point capture. IEEE Trans Cybern, 2017, 47: 1409-1422 CrossRef PubMed Google Scholar

[14] Pan S, Huang H M, Ding J, et al. Pursuit, evasion and defense in the plane. In: Proceedings of American Control Conference, Montreal, 2012. 4167--4173. Google Scholar

[15] Chen M, Zhou Z Y, Tomlin C J. Multiplayer reach-avoid games via low dimensional solutions and maximum matching. In: Proceedings of American Control Conference, Portland, 2014. 1444--1449. Google Scholar

[16] Ibragimov G, Karapanan P, Alias I A. Pursuit differential game of two pursuers and one evader in R2 with coordinate-wise integral constraints. In: Proceedings of 2015 International Conference on Research and Education in Mathematics, Kuala Lumpur, 2015. 223--226. Google Scholar

[17] Alias I A, Ibragimov G, Rakhmanov A. Evasion Differential Game of Infinitely Many Evaders from Infinitely Many Pursuers in Hilbert Space. Dyn Games Appl, 2017, 7: 347-359 CrossRef Google Scholar

[18] Kuchkarov A, Ibragimov G, Ferrara M. Simple motion pursuit and evasion differential games with many pursuers on manifolds with euclidean metric. Discrete Dyn Nat Soc, 2016, 2016: 1386242. Google Scholar

[19] Zhou Y, Li J X, Wang D L. Target tracking in wireless sensor networks using adaptive measurement quantization. Sci China Inf Sci, 2012, 55: 827-838 CrossRef Google Scholar

[20] Selvakumar J, Bakolas E. Evasion from a group of pursuers with a prescribed target set for the evader. In: Proceedings of 2016 American Control Conference, Boston, 2016. 155--160. Google Scholar

[21] Li W. Formulation of a cooperative-confinement-escape problem of multiple cooperative defenders against an evader escaping from a circular region. Commun Nonlinear Sci Num Simul, 2016, 39: 442-457 CrossRef ADS Google Scholar

[22] Yao Y, Zhang P, Liu H. Optimal switching target-assignment based on the integral performance in cooperative tracking. Sci China Inf Sci, 2013, 56: 012203 CrossRef Google Scholar

[23] Garcia E, Casbeer D W, Pachter M. Active target defense using first order missile models. Automatica, 2017, 78: 139-143 CrossRef Google Scholar

[24] Merz A. Homicidal chauffeur: a differential game. Dissertation for Ph.D. Degree. Palo Alto: Stanford University, 1971. Google Scholar

[25] Bopardikar S D, Bullo F, Hespanha J P. A cooperative homicidal chauffeur game. Automatica, 2009, 45: 1771-1777 CrossRef Google Scholar

[26] Merz A W. The game of two identical cars. J Opt Theory Appl, 1972, 9: 324-343 CrossRef Google Scholar

[27] Shankaran S, Stipanovic D M, Tomlin C J. Collision avoidance strategies for a three-player game. In: Proceedings of International Society of Dynamic Games, Wroclaw, 2008. 253--271. Google Scholar

[28] Averboukh Y, Baklanov A. Stackelberg solutions of differential games in the class of nonanticipative strategies. Dyn Games Appl, 2014, 4: 1-9 CrossRef Google Scholar

[29] Exarchos I, Tsiotras P, Pachter M. UAV collision avoidance based on the solution of the suicidal pedestrian differential game. In: Proceedings of AIAA Guidance, Navigation, and Control Conference, San Diego, 2016. Google Scholar

• Figure 1

(Color online) Three-player lifeline game in the plane, where $\varphi$, $\phi_1$, and $\phi_2$ are the moving directions of players E, P$_1$, and P$_2$ in the realistic game space, respectively.

• Figure 2

(Color online) Barrier of the two-player lifeline game with $v_p=1.5$, $v_e=1$, and $l=2$.

• Figure 3

(Color online) Barrier of the three-player lifeline game with $u=10$, $h_1=5$, $h_2=10$, $l=2$, and $w=1.5$.

• Figure 4

(Color online) Force analysis of the evader, where $\alpha$ and $\beta$ are the directions of the forces $F_1$ and $F_2$, respectively, and $\varphi$ is the direction of the resultant force of the evader E.

• Figure 5

(Color online) Three-player lifeline game with two different evasion strategies. (a) The evader moves vertically down the lifeline; (b) the evader uses the resultant force strategy (28). The initial positions of the players are E $=(3,15)$, P$_1=(-10,5)$, P$_2=(10,20)$. $w=1.2$ and $l=2$. The black curves are the barriers of the game, the blue, green and red curves are the trajectories of the pursuers P$_1$, P$_2$, and the evader E, respectively.

• Figure 6

(Color online) Partition of the escape zone for the three-player lifeline game, where the blue curves are the barriers, and the red curves are described by (25). The initial positions of P$_1$ and P$_2$ are $(-20,5)$ and $(20,20)$, respectively. $w=1.2$ and $l=2$.

• Figure 7

(Color online) Lifeline game with more than two pursuers. (a) The evader escapes from three pursuers and reaches the lifeline; (b) the evader escapes from four pursuers and reaches the lifeline. The initial positions of the players are E $=(50,180)$, P$_1=(-200,200)$, P$_2=(200,100)$, P$_3=(300,150)$, and P$_4=(-150,50)$. $w=4/3$ and $l=2$.

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