=Paper=
{{Paper
|id=Vol-2859/paper9
|storemode=property
|title=Visualization of Pursuit Differential Game on a Plane
|pdfUrl=https://ceur-ws.org/Vol-2859/paper9.pdf
|volume=Vol-2859
|authors=Lesia Baranovska,Dmytro Hyriavets,Kateryna Dovzhanytsia,Vadym Mukhin
|dblpUrl=https://dblp.org/rec/conf/its2/BaranovskaHDM20
}}
==Visualization of Pursuit Differential Game on a Plane==
https://ceur-ws.org/Vol-2859/paper9.pdf
99
Visualization of pursuit differential game on a plane
© Lesia Baranovska1[0000-0003-0024-8180], © Dmytro Hyriavets 1[0000-0003-1310-0943],
© Kateryna Dovzhanytsia 1[0000-0001-8529-1553], and © Vadym Mukhin1[0000-0002-1206-9131]
1 National Technical University of Ukraine "Igor Sikorsky Kyiv Polytechnic Institute", Kyiv,
Ukraine
lesia@baranovsky.org
Abstract. This paper is dedicated to differential pursuit games. In the theory of
dynamic games, a number of methods have been developed that provide a guar-
anteed result. The scheme of the Method of Resolving Functions is applied in the
work. Sufficient conditions to end of the game have been found. For the task of
simple pursuit, the visualization of the trajectory of the movements of the group
of pursuers and the fugitive on the plane is realized. To do this, a software product
was created in the Python programming language. This software product is a pro-
totype of a "pursuer-evader" simulation system that can be used to select controls
in pursuit tasks. Two cases of fugitive control selection were considered in the
development. In the first one, the control of the fugitive is based on the following
algorithm: the fugitive determines the nearest pursuer in terms of the Euclidean
norm; the fugitive builds his control on the beam, which comes from the position
of the nearest pursuer and crosses the position of the fugitive, in the direction
opposite to the pursuer with maximum speed. In the second case, the control of
the fugitive is set by the user. The visualization of the trajectory of one pursuer
and one evader on the plane was realized for the Pontryagin’s checking example
of a game problem with simple motions. The results have a graphical presenta-
tion. The result showed the coincidence of the estimated time and the actual end
time of the game.
Keywords: dynamic games, conflict-controlled processes, differential games,
pursuit games, group pursuit games.
1 Statement of the problem. Scheme of the method
In the theory of dynamic games (conflict-controlled processes and differential games),
along with Isaacs ideology [1], a number of methods have been developed that provide
a guaranteed result. Such methods include, in particular, the first direct method of L.S.
Pontryagin [2, 3], the method of extreme aiming of M.M. Krasovskii [4] and the method
of resolving functions of A.O. Chikrii [5–15]. In this paper, the last method will be
used, which gives a theoretical justification for the classical rule of parallel pursuit and
the method of convergence by the beam. The method scheme for differential-difference
games is developed in works [16–19]. In [20–22], the problem of rapprochement for a
group of pursuers and a single evader is considered. The scheme of the method of re-
solving functions for differential-difference systems of neutral type was developed in
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons
License Attribution 4.0 International (CC BY 4.0).
100
[23]. A modification of the method is proposed for objects with different inertia in [24].
Game problems of convergence in case of failure of control devices are considered in
works [25-27].
Consider the motion of a controlled object, which is described by a system of
differential equations:
𝑧 𝐴𝑧 𝜑 𝑢 ,𝑣 , 𝑧 ∈ 𝑅 , 𝑙 1, 𝜗, 𝑢 ∈ 𝑈 , 𝑣 ∈ 𝑉 (1)
where 𝐴 are square constant matrices of order 𝑛 , 𝑛 ⋯𝑛 𝑛 , 𝑈 and 𝑉 are
nonempty compacts sets, 𝜑 𝑢 , 𝑣 : 𝑈 𝑉 → 𝑅 , are jointly continuous in their vari-
ables.
Let 𝑧 0 𝑧 ° be the initial condition of the process (1). The terminal set has cylin-
drical form. i.e.
𝑀∗ ⋃ ,..., 𝑀° 𝑀 =⋃ ,..., 𝑀∗ (2)
where 𝑀° are linear spaces in 𝑅 and 𝑀 are compact sets from the orthogonal com-
plement 𝐿 of 𝑀∙ in 𝑅 .
Game (1), (2) is considered complete if for some 𝑙 1, 𝜗 the condition 𝑧 ∈ 𝑀∙ ful-
filled. The pursuers use quasi-strategies, and the evader uses software control..
Let 𝜋 be the orthogonal projection mapping from 𝑅 onto the subspace 𝐿 . Con-
sider the multivalued mappings
𝑊 𝑡, 𝑣 𝜋𝑒 𝜑 𝑈 ,𝑣 ,
𝑊 𝑡 𝑊 𝑡, 𝑣 , 𝑡 0, 𝑣 ∈ 𝑉.
∈
Pontryagin condition. The mappings 𝑊 𝑡 ∅ for all 𝑙 1, 𝜗, 𝑡 0.
There exists at least one Borelian selector 𝛾 𝑡 ∈ 𝑊 𝑡 [28-32]. Let’s fix it and set
𝜉 𝑡, 𝑧 , 𝛾 ∙ 𝜋𝑒 𝑧 𝛾 𝜏 𝑑𝜏.
Let’s assign a resolving function to each pursuer:
𝛼 𝑡, 𝜏, 𝑧 , 𝑣, 𝛾 ∙
sup 𝛼 0: 𝑊 𝑡 𝜏, 𝑣 𝛾 𝑡 𝜏 ⋂𝛼 𝑀 𝜉 𝑡, 𝑧 , 𝛾 ∙ ∅ (3)
If 𝜉 𝑡, 𝑧, 𝛾 ∙ ∈ 𝑀 , then put 𝛼 𝑡, 𝜏, 𝑧, 𝑣, 𝛾 ∙ ∞, 0 𝜏 𝑡, 𝑣 ∈ 𝑉. In other
cases the function 𝛼 𝑡, 𝜏, 𝑧, 𝑣, 𝛾 ∙ assumes finite values for each 𝜏 ∈ 0, 𝑡 , 𝑣 ∈ 𝑉.
Let’s take 𝛾 ∙ 𝑐𝑜𝑙𝑢𝑚𝑛 𝛾 ∙ , … , 𝛾 ∙ and let
Г 𝛾 ∙ :𝛾 𝑡 ∈ 𝑊 𝑡 ,𝑡 0, 𝑙 1, 𝜗 .
101
Introduce the function
𝑇 𝑧, 𝛾 ∙ min 𝑡 0: inf max 𝛼 𝑡, 𝜏, 𝑧 , 𝑣 𝜏 , 𝛾 ∙ 𝑑𝜏 1.
∙ ∈ ,…,
Theorem 1 [5, 33]. Assume that the Pontryagin condition holds for the conflict-control
process (1), (2), 𝑀 𝑐𝑜𝑀 , 𝑙 1, … , 𝜗, for the initial state 𝑧 ° and a certain selector
𝛾 𝑡 ∈ Г we have 𝑇 𝑧 ° , 𝛾 ° ∙
°
∞.
Then at least for one l, the corresponding trajectory of process (1) can be steered
from the initial state 𝑧 ° to the set 𝑀∗ at the moment 𝑇 𝑧 ° , 𝛾 ° ∙ .
Theorem 2 [5]. Assume that the control process (1), (2) is linear (i.e., 𝜑 𝑢 , 𝑣 𝑢
𝑣 , the Pontryagin condition holds, the multivalued mappings 𝜋 𝑒 𝑈 𝑟 𝑡 𝑆 , and
the sets 𝑀 𝜖 𝑆 , 𝑙 1, 𝜗, where 𝑟 𝑡 are continuous nonnegative numerical func-
tions, 𝜖 𝑐𝑜𝑛𝑠𝑡 0, and 𝑆 is the unit ball centered at zero in the space 𝐿 .
Then the resolving functions 𝛼 𝑡, 𝜏, 𝑧 , 𝑣 𝜏 , 𝛾 ∙ for 𝜉 𝑡, 𝑧 , 𝛾 ∙ ∉ 𝑀 are large
roots of the quadratic equations
𝜋𝑒 𝑣 𝛾 𝑡 𝜏 𝛼 𝜉 𝑡, 𝑧 , 𝛾 ∙ 𝑟 𝑡 𝜏 𝛼𝜖,
0 𝜏 𝑡, 𝑙 1, 𝜗, 𝑣 ∈ 𝑉, 𝛾 𝑡 ∈ 𝑊 𝑡
with respect to 𝛼 , 𝛼 0.
2 Example of problem for a process with simple matrices
Consider the problem of group pursuit with 𝜗 pursuers and one evader [5, 34]:
𝑧 𝑎𝑧 𝑢 𝑣, 𝑎 0, 𝑧 ∈ 𝑅 , || 𝑢 || 1, || 𝑣|| 1 ,𝑙 1, 𝜗 (4)
The game is considered over if for some 𝑙 𝑥 𝑦.
The terminal set
𝑀∗ ⋃ ,..., 𝑀° =⋃ ,..., 𝑧 :𝑧 0.
Thus we have 𝑊 𝑡 0 , 𝑙 1, 𝜗. The selectors of the multivalued mappings 𝑊 𝑡
are clearly defined as 𝛾 𝑡 0. The functions 𝜉 𝑡, 𝑧 , 𝛾 ∙ 𝑒 𝑧 , 𝑙 1, 𝜗. The
resolving functions of the pursuers
𝛼 𝑡, 𝜏, 𝑧 , 𝑣, 0 max 𝛼 0: 𝛼𝑒 𝑧 ∈𝑒 𝑆 𝑣 𝑒 𝛼 𝑧 ,𝑣 ,
where 𝛼 𝑧 , 𝑣 has the form
102
, , ‖ ‖ ‖ ‖ /
𝛼 𝑧 ,𝑣 ,𝑙 1, 𝜗. (5)
|| ||
Hence we obtain the time to the end the group pursuit:
𝑇 𝑧 min 𝑡 0: inf max 𝑒 𝛼 𝑧 ,𝑣 𝜏 𝑑𝜏 1 .
∙ ∈ ,…,
Letting 𝛿 𝑧 min max 𝛼 𝑧 , 𝑣 . Finally, we deduce the upper bound
|| || ,…,
∗
𝑇 𝑧 ln 1 , (6)
∗
where 𝑎∗ max 𝑎 [1].
,…,
Theorem 3 [5]. Let 𝑧 ° be the initial state of the process (4). Then if 0 ∈ 𝑖𝑛𝑡 𝑐𝑜 𝑧 °
then the problem of group pursuit is solvable at finite time 𝑇 𝑧 ° for which the estimate
(6) is true.
If 𝑡∗ 𝑡∗ 𝑣 ∙ , 𝑡∗ 𝑇 𝑧 ° , is the instant of switching, at which the test function
1 max 𝑒 𝛼 𝑧 ,𝑣 𝜏 𝑑𝜏
,…,
vanishes, then the controls of the pursuers, ensuring of the game at the time 𝑇 𝑧 ° , on
the interval 0, 𝑡∗ have the form
𝑢 𝜏 𝑣 𝜏 𝛼 𝑧 °, 𝑣 𝜏 𝑧 °, 𝑙 1, 𝜗,
and on the interval 𝑡∗ , 𝑇 𝑧 °
𝑢 𝜏 𝑣 𝜏
for the indices 𝑙, satisfying the equality
𝑒 𝛼 𝑧 °, 𝑣 𝜏 𝑑𝜏 max 𝑒 𝛼 𝑧 °, 𝑣 𝜏 𝑑𝜏,
,…,
the form 𝑢 𝜏 𝑣 𝜏 allowing the controls of the remaining pursuers to be arbitrary.
If 0 ∉ 𝑖𝑛𝑡 𝑐𝑜 𝑧 ° then the problem is solvable by means of any constant evader’s
control which furnished minimum to function max 𝛼 𝑧 ° , 𝑣 .
,…,
103
3 Visualization of a simple pursuit on the plane
There are ϑ pursuers and one evader. The positions of the participants are geometric
points, that is we do not take into account their size. Motion of pursuers have the form
𝑥 𝑢 , 𝑥 ∈ 𝑅 ,𝑘 1, || 𝑢 || 𝑎 ,𝑙 1, 𝜗.
The law of motion of the evader 𝑦 𝑣, 𝑦 ∈ 𝑅 , || 𝑣|| 𝑏. Players move at limited
speeds. The numbers al and b show the maximum velocity of the players. the game is
over if for some 𝑙 1, 𝜗, 𝑥 𝑦.
There is a certain sampling of time. Players choose their motion every time of sam-
pling and move to the directions indicated by them, considering their speed limits. The
evader knows the position of the pursuers now, and the pursuers know the position of
the evader and the motion he has chosen now. Strategy of pursuers is determined by the
following algorithm:
1. Calculate 𝛼 𝑧 , 𝑣 according to the formula
‖𝑧 ‖ 𝑎 ‖𝑣‖ /
𝑧 ,𝑣 𝑧 ,𝑣
𝛼 𝑧 ,𝑣 , 𝑙 1, 𝜗
|| 𝑧 ||
2. Check 𝛼 𝑧 , 𝑣 1, 𝑙 1, 𝜗. If it is true, the motion is calculated by for-
mula 𝑢 𝑣 𝛼 𝑧 , 𝑣 𝑧 , 𝑙 1, 𝜗.
3. If some 𝛼 𝑧 , 𝑣 1, 𝑙 1, 𝜗, then we find the pursuer with the greatest
value 𝛼 max 𝛼 𝑧 , 𝑣 and determine his motion 𝑢 𝑣 𝑧 ,
and the motion of others is set to zero.
To represent the trajectories of the pursuers and the evader in the game for the plane
case, a software product was written in the Python programming language. For the pro-
gram, the input data is the following information: the number of pursuers, initial coor-
dinates of all participants; maximum speed values for all participants. For the algorithm
to work, you need to choose the evader motion. Two cases of evader movement were
used in the development. In the first, the motion of the evader is based on the following
algorithm: 1) the evader determines the nearest pursuer in terms of the Euclidean norm;
2) the evader builds his control on the beam, which comes from the position of the
nearest pursuer and crosses the position of the evader, in the direction opposite to the
pursuer with maximum velocity. In the second case, the motion of the evader is set by
the user.
In the figures, the initial positions of the pursuers are marked with stars, the initial
position of the evader is marked with a rhombus, and the point of capture is marked
with a pentagon.
In the Fig. 1-8, we can see several examples of pursuit game. In the Fig. 1-4, the
evader uses first algorithm. The evader does not choose his movement optimally be-
cause he pays attention only to the nearest pursuer. Such actions can be seen in real life,
when of all the threats, the object notices only the nearest and does not pay attention to
others. In the Fig. 5-8, the evader`s motion is set by user.
104
Fig.1. Two pursuers.
Fig. 2. Two pursuers and surrounded evader.
105
Fig. 3. Four pursuers and surrounded evader.
Fig. 4. Four pursuers with equal velocities.
106
Fig. 5. Four pursuers with equal velocities and user`s movement.
Fig. 6. Four pursuers, surrounded evader and user`s movement.
107
Fig. 7. Three pursuers and user`s movement.
Fig. 8. Two pursuers, surrounded evader and user`s movement.
108
4 Pontryagin’s checking example
The motions of the one pursuer and the one evader are described by the equations [5]:
𝑥 𝑥 2𝑢, 𝑥 ∈ 𝑅, |𝑢| 1, 𝛼, 𝜌 0,
(7)
𝑦 2𝑦 𝑣, 𝑦 ∈ 𝑅, |𝑣| 1, 𝛽, 𝜎 0.
Letting
𝑥 0 6, 𝑥 0 1,
𝛼 1, 𝛽 2, 𝜌 2, 𝜎 1,
𝑦 0 2, 𝑦 0 1.
Substitute the values of the parameters in (7):
𝑥 𝑥 2𝑢, 𝑥 ∈ 𝑅, |𝑢| 1, 𝛼, 𝜌 0,
𝑦 2𝑦 𝑣, 𝑦 ∈ 𝑅, |𝑣| 1, 𝛽, 𝜎 0.
The pursuit is completed when 𝑥 𝑦. Let's move on to the system of first-order
equations. To do this, we’ll use new variables
𝑧 , 𝑧 , 𝑧 , 𝑐𝑜𝑙𝑢𝑚𝑛 𝑧 , 𝑧 , 𝑧 𝑧,
𝑧 𝑥 𝑦, 𝑧 𝑥, 𝑧 𝑦 8
Differentiating over the time (8) , we obtain an equivalent system
𝑧 𝑧 𝑧 ,
𝑧 𝛼𝑧 𝜌𝑢,
𝑧 𝛽𝑧 𝜎𝑣.
Therefore
𝑧 𝑧 𝑧 ,
𝑧 𝑧 2𝑢,
𝑧 2𝑧 𝑣
The terminal set 𝑀∙ 𝑧: 𝑧 0 , and 𝑀° 𝑧: 𝑧 0, 𝑀 𝑧: 𝑧 𝑧 𝑧 0.
Then 𝐿 𝑧: 𝑧 𝑧 0 𝑅, 0, 0 . The operator of orthogonal projecrion
𝜋: 𝑅 → 𝐿 is given by the matrix
1 0 0
𝜋 0 0 0 .
0 0 0
Here matrix A and the control domains are
0 0
𝑈 2𝑢 : |𝑢| 1 ,𝑉 0 : |𝑣| 1.
0 𝑣
The fundamental matrix of a homogeneous system is
109
1 𝑒 1 𝑒
1
𝑒 1 2 ,
0 𝑒 0
0 0 𝑒
𝑊 𝑡 2𝑆 ∗ 𝑆 2 𝑆 𝑤 𝑡 𝑆.
4
We set 𝛾 𝑡 ≡ 0, 𝑧 0 1 . Then ξ 𝑡, 𝑧, 0 4 1 1.
1
By Theorem 2, we obtain the resolving function 𝛼 𝑡, 𝜏, 𝑧, 𝑣, 0 as a large positive
root of the quadratic equation
1 𝑒 1 𝑒
𝑣 𝛼ξ 𝑡, 𝑧 0 , 0 2.
2 1
Next, by virtue of the capture time 𝑇 𝑧 0 ,0 is defined as min 𝑡
0: ||
𝑑𝜏 1 , we have 𝑇 5.
|| , ,
Let’s find the control of the pursuer. Since the motion occurs on the plane and the
set M consists of one point, we conclude that
, , °, , , °,
𝑢 𝑣 𝜏 ,𝑇 5,
and therefore
, , °, , , °,
𝑢1 𝑣 𝜏 ,𝑢 𝑙𝑒𝑥 min 𝑢1.
5 Visualisation of Pontryagin’s checking example
To represent the trajectories of the pursuers and fugitives in the game described in the
previous paragraph and for the case of the plane, was created a software in the Python
programming language. This is a prototype of a modeling system "fugitive-pursuers",
which can be used to select control in control tasks.
The inputing data for the program is the following information:
1. The values of the parameters α, β, σ, ρ;
2. Initial coordinates of all participants.
Consider the algorithm by which the software works:
1. Check the conditions 𝜌 𝜎,
2. Сalculate the fundamental matrix 𝑒 .
3. Check the condition 𝑊 𝑡 ∅, and find 𝑤 𝑡 .
110
4. Сalculate ξ 𝑡, 𝑧, 0 𝑧 𝑧 𝑧
5. Сalculate fugitive delay time as min 𝑡 0: ||
𝑑𝜏 1
|| , ,
6. Construct the pursuer control as
𝛼 𝑇, 𝜏, 𝑧 ° , 𝑣 𝜏 , 0 𝜉 𝑇, 𝑧 ° , 0
𝑢 𝑣 𝜏
𝜋𝑒
, , ,
𝛼 𝑡, 𝜏, 𝑧, 𝑣, 0 +
| , , |
/
1 𝑒 1 𝑒 1 𝑒
𝜎 𝑣, ξ 𝑡, 𝑧, 0 |ξ 𝑡, 𝑧, 0 | 𝜌 𝜎 ||𝑣||
𝛽 𝛼 𝛽
||ξ 𝑡, 𝑧, 0 ||
7. Using the Runge-Kutta method, we construct the solutions of
differential equations at time t.
Let's use this control example to check the program.
Letting
𝑥 0 6, 𝑥 0 1
𝛼 1, 𝛽 2, 𝜌 2, 𝜎 1, .
𝑦 0 2, 𝑦 0 1
We get the capture time T that coincides with the time obtained in our calculations (see
Fig. 9).
111
Fig. 9. The results of the program.
Letting
𝑥 0 10, 𝑥 0 1
𝛼 1, 𝛽 2, 𝜌 2, 𝜎 1,
𝑦 0 3, 𝑦 0 1
We obtain the capture time T (see Fig. 10).
Fig. 10. The results of the program
112
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