A multi-UAV deployment method for border patrolling based on Stackelberg game

doi:10.23919/JSEE.2023.000022

Journal of Systems Engineering and Electronics ›› 2023, Vol. 34 ›› Issue (1): 99-116.doi: 10.23919/JSEE.2023.000022

• SYSTEMS ENGINEERING • Previous Articles Next Articles

A multi-UAV deployment method for border patrolling based on Stackelberg game

Xing LEI¹^,²(), Xiaoxuan HU¹^,²(), Guoqiang WANG¹^,²^,³(), He LUO¹^,³^,*()

¹ School of Management, Hefei University of Technology, Hefei 230009, China
² Key Laboratory of Process Optimization & Intelligent Decision-making, Ministry of Education, Hefei 230009, China
³ Engineering Research Center for Intelligent Decision-making & Information Systems Technologies, Ministry of Education, Hefei 230009, China

Received:2021-07-14 Online:2023-02-18 Published:2023-03-03
Contact: He LUO E-mail:leixing@mail.hfut.edu.cn;xiaoxuanhu@hfut.edu.cn;gqwang2017@hfut.edu.cn;luohe@hfut.edu.cn
About author:
LEI Xing was born in 1994. She received her B.S. degree from Hefei University of Technology in 2016. She is currently pursuing her Ph.D. degree in the School of Management, Hefei University of Technology. Her current research interests include deploying multi-UAVs for border patrols and security game theory. E-mail: leixing@mail.hfut.edu.cn

HU Xiaoxuan was born in 1978. He received his B.S. and Ph.D. degrees from Hefei University of Technology, in 1999 and 2006, respectively. He is a professor in Hefei University of Technology. His current research interests include satellite mission planning, unmanned system intelligence, and aerospace system management. E-mail: xiaoxuanhu@hfut.edu.cn

WANG Guoqiang was born in 1982. He received his B.S. and M.S. degrees from University of Science and Technology of China, in 2004 and 2007, respectively. He received his Ph.D. degree from Hefei University of Technology in 2016. He is a professor in Hefei University of Technology. His current research interests include management and intelligent decision making of unmanned aerial vehicle formation. E-mail: gqwang2017@hfut.edu.cn

LUO He was born in 1982. He received his B.S. and Ph.D. degrees from Hefei University of Technology, in 2004 and 2009, respectively. He is a professor in Hefei University of Technology. His current research interests include intelligent decision making, multi-agent system, and the applications of unmanned aerial vehicle. E-mail: luohe@hfut.edu.cn
Supported by:
This work was supported by the National Natural Science Foundation of China (71971075; 71871079), the National Key Research and Development Program of China (2019YFE0110300), the Anhui Provincial Natural Science Foundation (1808085MG213), and the Fundamental Research Funds for the Central Universities (PA2019GDPK0082).

Abstract

Abstract:

To strengthen border patrol measures, unmanned aerial vehicles (UAVs) are gradually used in many countries to detect illegal entries on borders. However, how to efficiently deploy limited UAVs to patrol on borders of large areas remains challenging. In this paper, we first model the problem of deploying UAVs for border patrol as a Stackelberg game. Two players are considered in this game: The border patrol agency is the leader, who optimizes the patrol path of UAVs to detect the illegal immigrant. The illegal immigrant is the follower, who selects a certain area of the border to pass through at a certain time after observing the leader’s strategy. Second, a compact linear programming problem is proposed to tackle the exponential growth of the number of leader’s strategies. Third, a method is proposed to reduce the size of the strategy space of the follower. Then, we provide some theoretic results to present the effect of parameters of the model on leader’s utilities. Experimental results demonstrate the positive effect of limited starting and ending areas of UAV’s patrolling conditions and multiple patrolling altitudes on the leader ’s utility, and show that the proposed solution outperforms two conventional patrol strategies and has strong robustness.

Key words: border patrol, unmanned aerial vehicle (UAV), Stackelberg game, compact linear programming, dominated strategy elimination

Xing LEI, Xiaoxuan HU, Guoqiang WANG, He LUO. A multi-UAV deployment method for border patrolling based on Stackelberg game[J]. Journal of Systems Engineering and Electronics, 2023, 34(1): 99-116.

Figures/Tables 14

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References 41

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Set	UAV	Percentage difference	$ {R_d} = R,\left\| H \right\| = 1 $		$ \left\| {{R_d}} \right\| = \left\lceil {{r \mathord{\left/ {\vphantom {r 3}} \right. } 3}} \right\rceil ,\left\| H \right\| = 1 $		$ {R_d} = R,\left\| H \right\| = 3 $		$ \left\| {{R_d}} \right\| = \left\lceil {{r \mathord{\left/ {\vphantom {r 3}} \right. } 3}} \right\rceil ,\left\| H \right\| = 3 $
Set	UAV	Percentage difference	PD-ACS	PD-PCS	PD-ACS	PD-PCS	PD-ACS	PD-PCS	PD-ACS	PD-PCS
Set A	20	Aver	1.62	1.69	−0.05	0.02	5.08	5.14	4.88	4.95
		Max	3.75	3.70	0.08	0.20	11.07	10.99	10.73	10.75
		Min	0.70	0.74	−0.35	−0.43	2.28	2.32	2.17	2.22
	All	Aver	1.02	1.06	−0.03	0.01	3.20	3.24	3.08	3.12
		Max	3.75	3.70	0.08	0.20	11.07	10.99	10.73	10.75
		Min	0.18	0.19	−0.35	−0.43	0.58	0.58	0.55	0.56
Set B	20	Aver	2.63	3.03	−0.31	0.10	6.48	6.85	6.23	6.61
		Max	5.84	7.10	−0.11	0.68	13.83	14.98	13.32	14.42
		Min	1.14	1.30	−1.11	−0.42	2.86	3.07	2.80	2.90
	All	Aver	1.65	1.90	−0.19	0.10	4.10	4.34	3.94	4.18
		Max	5.84	7.10	−0.03	0.68	13.83	14.98	13.32	14.42
		Min	0.29	0.33	−1.11	−0.42	0.72	0.78	0.71	0.73
Set C	20	Aver	6.94	10.43	1.14	4.93	7.60	11.06	6.65	10.15
		Max	16.29	23.54	3.20	11.25	17.21	24.38	14.38	21.75
		Min	3.09	4.38	0.29	1.86	3.39	4.67	2.96	4.33
	All	Aver	4.42	6.67	0.71	3.11	4.82	7.05	4.20	6.46
		Max	16.29	23.54	3.20	11.25	17.21	24.38	14.38	21.75
		Min	0.78	1.11	0.07	0.47	0.86	1.19	0.75	1.10

Inst	Zone	UAV	$ {R_d} = R,\left\| H \right\| = 1 $		$ \left\| {{R_d}} \right\| = \left\lceil {{r \mathord{\left/ {\vphantom {r 3}} \right. } 3}} \right\rceil ,\left\| H \right\| = 1 $		$ {R_d} = R,\left\| H \right\| = 3 $		$ \left\| {{R_d}} \right\| = \left\lceil {{r \mathord{\left/ {\vphantom {r 3}} \right. } 3}} \right\rceil ,\left\| H \right\| = 3 $
Inst	Zone	UAV	PD-ACS	PD-PCS	PD-ACS	PD-PCS	PD-ACS	PD-PCS	PD-ACS	PD-PCS
A01	200	5	0.95	0.90	−0.06	−0.11	2.89	2.84	2.72	2.67
		10	1.89	1.80	−0.12	−0.21	5.70	5.60	5.36	5.27
		15	2.83	2.69	−0.18	−0.32	8.42	8.29	7.93	7.79
		20	3.75	3.56	−0.23	−0.43	11.07	10.90	10.43	10.25
A07	400	5	0.48	0.48	−0.01	−0.02	1.46	1.46	1.38	1.38
		10	0.96	0.95	−0.03	−0.04	2.91	2.90	2.75	2.74
		15	1.44	1.42	−0.04	−0.06	4.33	4.31	4.09	4.08
		20	1.92	1.89	−0.06	−0.08	5.73	5.71	5.42	5.40
A13	600	5	0.32	0.33	0.00	0.00	0.98	0.98	0.92	0.93
		10	0.65	0.65	0.00	0.00	1.94	1.95	1.84	1.85
		15	0.97	0.98	0.00	0.01	2.90	2.91	2.75	2.76
		20	1.29	1.30	0.00	0.01	3.85	3.86	3.65	3.66
A19	800	5	0.24	0.24	−0.01	−0.02	0.73	0.73	0.68	0.68
		10	0.47	0.47	−0.03	−0.03	1.45	1.45	1.36	1.36
		15	0.71	0.71	−0.04	−0.05	2.17	2.16	2.03	2.03
		20	0.94	0.94	−0.06	−0.06	2.88	2.87	2.70	2.69
A25	1000	5	0.19	0.20	−0.01	0.00	0.58	0.59	0.55	0.56
		10	0.38	0.39	−0.01	0.00	1.17	1.18	1.11	1.12
		15	0.57	0.59	−0.02	0.00	1.74	1.76	1.65	1.67
		20	0.76	0.79	−0.03	0.00	2.32	2.34	2.20	2.22

Inst	TP	UAV	$ {R_d} = R,\left\| H \right\| = 1 $		$ \left\| {{R_d}} \right\| = \left\lceil {{r \mathord{\left/ {\vphantom {r 3}} \right. } 3}} \right\rceil ,\left\| H \right\| = 1 $		$ {R_d} = R,\left\| H \right\| = 3 $		$ \left\| {{R_d}} \right\| = \left\lceil {{r \mathord{\left/ {\vphantom {r 3}} \right. } 3}} \right\rceil ,\left\| H \right\| = 3 $
Inst	TP	UAV	PD-ACS	PD-PCS	PD-ACS	PD-PCS	PD-ACS	PD-PCS	PD-ACS	PD-PCS
C01	6	5	4.35	6.49	0.72	2.94	4.61	6.75	3.71	5.86
		10	8.50	12.56	1.43	5.79	9.01	13.04	7.28	11.38
		15	12.48	18.23	2.13	8.56	13.20	18.91	10.71	16.58
		20	16.29	23.54	2.83	11.25	17.21	24.38	14.02	21.47
C02	12	5	3.87	5.70	0.54	2.43	4.26	6.08	3.75	5.58
		10	7.60	11.08	1.09	4.81	8.34	11.79	7.36	10.85
		15	11.18	16.15	1.63	7.13	12.24	17.15	10.84	15.82
		20	14.62	20.93	2.16	9.39	15.98	22.18	14.18	20.52
C03	18	5	3.87	5.58	0.39	2.16	4.25	5.96	3.54	5.26
		10	7.59	10.85	0.77	4.28	8.33	11.57	6.96	10.25
		15	11.16	15.83	1.15	6.35	12.23	16.84	10.26	14.97
		20	14.60	20.53	1.53	8.37	15.96	21.80	13.44	19.45
C04	24	5	3.90	5.80	0.81	2.77	4.27	6.16	3.81	5.71
		10	7.65	11.26	1.61	5.46	8.35	11.94	7.47	11.09
		15	11.25	16.41	2.41	8.08	12.26	17.36	10.99	16.17
		20	14.72	21.26	3.20	10.63	16.01	22.45	14.38	20.95
C05	30	5	3.96	6.17	0.56	2.84	4.28	6.48	3.73	5.95
		10	7.76	11.95	1.11	5.60	8.37	12.54	7.33	11.54
		15	11.41	17.38	1.66	8.29	12.29	18.20	10.79	16.80
		20	14.92	22.48	2.21	10.89	16.05	23.50	14.12	21.75
C06	36	5	3.75	5.59	0.61	2.51	4.13	5.96	3.70	5.54
		10	7.35	10.86	1.21	4.96	8.08	11.56	7.27	10.78
		15	10.82	15.84	1.82	7.34	11.87	16.83	10.70	15.73
		20	14.16	20.54	2.41	9.67	15.51	21.79	14.01	20.40

Zone	Goal	Inst	Minimum_UAV
Zone	Goal	Inst	$ {R_d} = R,\left\| H \right\| = 1 $	$ \left\| {{R_d}} \right\| = \left\lceil {{r \mathord{\left/ {\vphantom {r 3}} \right. } 3}} \right\rceil ,\left\| H \right\| = 1 $	$ {R_d} = R,\left\| H \right\| = 3 $	$ \left\| {{R_d}} \right\| = \left\lceil {{r \mathord{\left/ {\vphantom {r 3}} \right. } 3}} \right\rceil ,\left\| H \right\| = 3 $
200	0.2	A01-A06	232	317	148	151
		B01-B06	133	183	97	99
		C01-C06	67	93	65	68
	0.8	A01-A06	33	45	21	21
		B01-B06	19	26	14	14
		C01-C06	10	13	9	10
400	0.2	A07-A12	464	630	295	301
		B07-B12	266	362	194	198
		C07-C12	133	185	130	135
	0.8	A07-A12	65	88	42	42
		B07-B12	37	51	27	28
		C07-C12	19	26	18	19
600	0.2	A13-A18	696	951	443	452
		B13-B18	399	543	290	295
		C13-C18	200	260	194	204
	0.8	A13-A18	97	132	62	63
		B13-B18	56	76	41	41
		C13-C18	28	39	28	29
800	0.2	A19-A24	929	1270	590	603
		B19-B24	531	726	387	394
		C19-C24	266	369	258	270
	0.8	A19-A24	129	176	82	84
		B19-B24	74	101	54	55
		C19-C24	37	51	36	38
1000	0.2	A25-A30	1160	1584	737	754
		B25-B30	663	906	484	492
		C25-C30	332	460	323	338
	0.8	A25-A30	161	220	103	105
		B25-B30	93	126	68	69
		C25-C30	47	64	45	47

Inst	Zone	Goal	Minimum_UAVs
Inst	Zone	Goal	$ {R_d} = R,\left\| H \right\| = 1 $	$ \left\| {{R_d}} \right\| = \left\lceil {{r \mathord{\left/ {\vphantom {r 3}} \right. } 3}} \right\rceil ,\left\| H \right\| = 1 $	$ {R_d} = R,\left\| H \right\| = 3 $	$ \left\| {{R_d}} \right\| = \left\lceil {{r \mathord{\left/ {\vphantom {r 3}} \right. } 3}} \right\rceil ,\left\| H \right\| = 3 $
B01-B06	200	0.2	133	183	97	99
		0.4	76	104	56	57
		0.6	43	58	31	32
		0.8	19	26	14	14
B07-B12	400	0.2	266	362	194	198
		0.4	152	207	111	113
		0.6	85	115	62	63
		0.8	37	51	27	28
B13-B18	600	0.2	399	543	290	295
		0.4	227	309	166	169
		0.6	127	173	93	94
		0.8	56	76	41	41
B19-B24	800	0.2	531	726	387	394
		0.4	303	413	220	225
		0.6	169	231	123	125
		0.8	74	101	54	55
B25-B30	1000	0.2	663	906	484	492
		0.4	378	516	276	281
		0.6	211	288	154	156
		0.8	93	126	68	69

A multi-UAV deployment method for border patrolling based on Stackelberg game

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Set	Case	Percentage difference	$ {R_d} = R,\left\| H \right\| = 1 $	$ \left\| {{R_d}} \right\| = \left\lceil {{r \mathord{\left/ {\vphantom {r 3}} \right. } 3}} \right\rceil ,\left\| H \right\| = 1 $	$ {R_d} = R,\left\| H \right\| = 3 $	$ \left\| {{R_d}} \right\| = \left\lceil {{r \mathord{\left/ {\vphantom {r 3}} \right. } 3}} \right\rceil ,\left\| H \right\| = 3 $
Set A	Prop	PDP-I	−0.59	−0.43	0.02	0.06
		PDP-ACS	0.44	−0.46	3.22	3.14
		PDP-PCS	0.48	−0.42	3.26	3.18
	Cover	PDC-I	−0.40	−0.29	0.02	−0.04
		PDC-ACS	0.63	−0.32	3.22	3.04
		PDC-PCS	0.67	−0.28	3.26	3.08
Set B	Prop	PDP-I	−1.38	−0.97	0.00	0.21
		PDP-ACS	0.31	−1.16	4.10	4.13
		PDP-PCS	0.56	−0.90	4.34	4.37
	Cover	PDC-I	−0.70	−0.51	−0.02	0.21
		PDC-ACS	0.97	−0.70	4.08	4.14
		PDC-PCS	1.23	−0.44	4.32	4.38
Set C	Prop	PDP-I	−2.59	−1.33	−0.02	0.24
		PDP-ACS	1.99	−0.60	4.80	4.43
		PDP-PCS	4.32	1.84	7.04	6.68
	Cover	PDC-I	−1.38	−0.92	0.01	0.20
		PDC-ACS	3.14	−0.20	4.83	4.39
		PDC-PCS	5.44	2.23	7.06	6.65

Set	UAV	IR-CPU/%	$ {R_d} = R,\left\| H \right\| = 1 $	$ \left\| {{R_d}} \right\| = \left\lceil {{r \mathord{\left/ {\vphantom {r 3}} \right. } 3}} \right\rceil ,\left\| H \right\| = 1 $	$ {R_d} = R,\left\| H \right\| = 3 $	$ \left\| {{R_d}} \right\| = \left\lceil {{r \mathord{\left/ {\vphantom {r 3}} \right. } 3}} \right\rceil ,\left\| H \right\| = 3 $
Set A	20	Aver	72.32	60.05	82.26	84.50
		Max	96.40	89.07	98.06	98.89
		Min	37.33	9.76	22.49	12.26
	All	Aver	73.33	61.38	81.56	84.99
		Max	97.13	90.74	98.06	98.95
		Min	17.84	9.76	18.67	2.16
Set B	20	Aver	76.24	57.83	82.83	84.79
		Max	96.56	87.21	97.15	98.69
		Min	27.98	5.66	21.49	35.57
	All	Aver	74.73	59.59	84.64	84.14
		Max	96.89	91.22	97.52	98.82
		Min	6.57	5.66	18.86	17.70
Set C	20	Aver	80.33	72.75	81.23	82.37
		Max	98.26	90.55	97.16	98.66
		Min	45.19	45.87	18.71	1.81
	All	Aver	79.42	70.53	80.20	83.23
		Max	98.26	90.73	97.16	98.86
		Min	35.63	24.81	9.41	1.81

Inst	Zone	IR-CPU
Inst	Zone	$ {R_d} = R,\left\| H \right\| = 1 $	$ \left\| {{R_d}} \right\| = \left\lceil {{r \mathord{\left/ {\vphantom {r 3}} \right. } 3}} \right\rceil ,\left\| H \right\| = 1 $	$ {R_d} = R,\left\| H \right\| = 3 $	$ \left\| {{R_d}} \right\| = \left\lceil {{r \mathord{\left/ {\vphantom {r 3}} \right. } 3}} \right\rceil ,\left\| H \right\| = 3 $
A04	200	90.50	86.86	92.61	93.10
A10	400	91.54	85.78	97.75	96.87
A16	600	66.13	86.45	96.41	96.25
A22	800	61.65	84.55	96.59	95.96
A28	1 000	86.80	88.62	95.36	95.71

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