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Adv Syst Sci Appl 2022; 02; 1-10 
Published online at https://ijassa.ipu.ru. 

 

Modeling the Detection of Moving Objects by Means of a 
Spatially Distributed Continuous Monitoring System with 

a Dynamic Structure 
Vladimir I. Goncharenko1,2, Azret A. Kochkarov3*, Danil V. Yatskin4, 

Boris V. Rumiantsev5 
1) Moscow Aviation Institute (National Research University), Moscow, Russia   
2) V.A. Trapeznikov Institute of Control Sciences, Russian Academy of Sciences, Moscow, 
Russia. E-mail: goncharenkovi@mai.ru 
3) Financial University under the Government of the Russian Federation, Moscow, 
Russia. E-mail: akochkar@gmail.com 
4) Moscow Institute of Physics and Technology, Moscow, Russia. E-mail: danil@frtk.ru 
5) Moscow State University, Moscow, Russia. E-mail: rymbor17@gmail.com 

Abstract: The class of monitoring systems, which is equipped with mobile means of detection, is 
considered. Unmanned aerial vehicles are used as detection means. The area of responsibility of 
the system is a geographic space with counteraction for both detection equipment and objects of 
observation. An original approach to the development of a simulation model for detecting moving 
objects of observation in the area of a spatially distributed continuous monitoring system with a 
structure is proposed. The movement of sensors is displayed along trajectories, which are 
Hamiltonian cycles on the terrain graph. The new approach is to use the approach used to ensure 
the flexibility of the resulting solutions to the problem of monitoring space and the ability to 
respond quickly to various factors and other conditions of use based on self-organizing mechanisms. 
At the same time, both the system itself, designed to solve the tasks of continuous monitoring and 
the solutions found for specific monitoring tasks and spatially distributed systems, provide 
continuous monitoring and are resistant to destructive influences of various kinds. 
The constructed simulation model and the experiments performed using the principles of dynamic 
detection of detection means in the area of a spatially distributed continuous monitoring system. 
At each moment of time, the optimal configuration of the parameters is determined, which is used 
as the most effective solution of the problem from the point of time of detecting an object in the 
monitoring area of vision. The model does not depend on the type of sensors used in the network 
and the implementation of the ideological principles embodied in the concept of dynamic systems. 
Based on the results of the experiments carried out, conclusions are drawn about the characteristics 
of continuous monitoring, which provides a basis for further work to optimize these indicators. 

Keywords: continuous monitoring system, unmanned aerial vehicle, spatially distributed system, 
monitoring object, mobile detection facility, Hamiltonian cycle 

1. INTRODUCTION 
Temporarily distributed continuous monitoring systems demonstrate modern service and 
support functions of the airborne dynamic detection system, which ensure the solution of 
problems of maintaining the space-probabilistic-temporal functions (SPTF) at the required 
level [1,2]. SPTF related to monitoring of the probability of detecting objects in a given period 
of time. 

 
* Corresponding author: akochkar@gmail.com 



2                           V.I. GONCHARENKO, A.A. KOCHKAROV, D.V. YATSKIN, B.V. RUMIANTSEV 

Copyright ©2022 ASSA.                                                                                    Adv. in Systems Science and Appl. (2022) 

The main tasks of the systemic continuous monitoring of the class are prompt and high-
precision detection of objects, their further support and timely delivery of the object to the end 
user [3,4]. 

The classical approach to solving the problem of monitoring the space of continuous 
observation over the maximum possible monitoring area. Often this problem is solved using 
stationary systems that are not distributed in nature [5, 6]. The problems of a negative character 
are presented when solving a general character. To provide more flexibility, network 
monitoring systems are used [7-10]. 

However, more attentively, approaches to the organization of the probe, as a rule, are 
focused on the stationary arrangement of the probe between the devices [11-13]. In this case, 
a one-time section of the monitoring zone occurs between the subsequent monitoring of each 
network node of the corresponding section of the zone. This approach is optimal for solving a 
large class of problems. 

Nevertheless, in recent years, decentralized networks have become more relevant, one of 
the most important advantages of which is the ability to dynamically change the configuration 
and adapt the solution to the monitoring problem to any changes in the conditions of the 
problem, provoked by factors of different nature. Thus, it is necessary to develop methods 
associated with finding solutions to similar problems [14,15]. 

It is in this ideology that models for the dynamic allocation of means of a spatially 
distributed continuous monitoring system are developed. 

An original approach to the development of a simulation model for detecting moving 
objects of observation in a spatially distributed continuous monitoring system is proposed. The 
developed simulation model and the conducted experiments use the principles of dynamic 
detection of detection means in the area of a spatially distributed continuous monitoring 
system. 

2. MATHEMATICAL MODEL OF THE DYNAMIC PLACEMENT 
As the area of responsibility of the continuous monitoring system in this paper is considered 
the monitoring area which is bounded by a given square on the plane. Obstacles are placed in 
the monitoring area, represented in the form of parts of the plane bounded by closed curves. 
Neither monitoring objects nor monitoring tools can be placed or moved in these areas. Fig. 1 
shows the layout of the monitoring objects in a given area of responsibility of the monitoring 
system. 

In the general case, mountains, hills, insurmountable for ground and air purposes, water 
bodies, as well as elements of infrastructure in an urbanized area can be considered as 
obstacles. In the area of responsibility of the system (monitoring area), monitoring objects can 
freely move either along random or along specified trajectories, taking into account their 
technical capabilities. 

Monitoring is carried out by means on the basis of Sensors, which at any given time can 
be located at a certain point in the monitoring area. Each monitoring tool has a limited field of 
view, which in planar projection onto the earth's surface is a circle of radius R. The radius of 
the field of view of each tool depends on the weather conditions in the monitoring area. Also, 
like the radius R, the speed of the Sensor in the area of responsibility of the entire system also 
depends on the weather conditions. In this work, the failure of monitoring tools is also taken 
into account. 

 
 
 
 
 



MODELING THE DETECTION OF MOVING OBJECTS BY MEANS OF A SPATIALLY DISTRIBUTED  3 

Copyright ©2022 ASSA.                                                                                    Adv. in Systems Science and Appl. (2022) 
 

 
Fig. 1. General layout of monitoring tools in the area of responsibility of the system 

To formalize the problem of placing detection equipment, the following designations are 
used (Fig. 1): 

 
Ω!" – a set of points of the system's area of responsibility, 
𝑆!" – the total value of the area of the system's area of responsibility, 
Ω#$% – a set of points of obstacle areas in the area of responsibility of the system, 
Ω& – a set of visibility zone points, 
𝑛' – 𝑙-th vertex of the graph by dividing the area of responsibility of the system. 
Assuming that the number of targets detected during the monitoring 𝑡 ∈ [𝑡(; 𝑡)], inversely 

proportional to mean time < 𝑡 >[!!;	!"]  detection of one target at a time 𝑡 ∈ [𝑡(; 𝑡)] , the 
following statement can be obtained: 

< 𝑡 >[!!;	!"]	= 	 ∫ 𝑑𝑡. ⋅ 𝑡.𝑝((𝑡.)
!"
!!

, ((1) 

where 𝑝((𝑡) – probability density of detecting a monitored object at a point in time 𝑡.  
Consider the case when the monitoring object is randomly located in the system's area of 

responsibility at the moment 𝑡( and does not move in the considered period of time 𝑡 ∈ [𝑡(; 𝑡)]. 
The randomness of the location of the monitoring object in this case, in the absence of a priori 
information, should be regarded as uniform. The assumption about the immobility of the 
monitoring object is valid provided that the speed of its movement is significantly less than 
the speed of movement of the sensors. (𝑣ОМ ≪ 𝑣СО). Under the condition of immobility of the 
monitoring object, normalization ∫ 𝑑𝑡. ⋅ 𝑝((𝑡.)

!"
!!

= 1 . To minimize the value< 𝑡 >[!!;	!"] 
дresidually minimize the standard deviation σ of the distribution 𝑝((𝑡). The value 𝑝((𝑡) can 
be represented as 

𝑝((𝑡) =
2
2!
7∬ 𝑑𝑥.𝑑𝑦. ⋅ 𝑝((𝑥., 𝑦.)3#$(!)

<, (2) 

 



4                           V.I. GONCHARENKO, A.A. KOCHKAROV, D.V. YATSKIN, B.V. RUMIANTSEV 

Copyright ©2022 ASSA.                                                                                    Adv. in Systems Science and Appl. (2022) 

where Ω%6(𝑡) – area of responsibility of the system, "swept" by means of detection at the 
moment in time 𝑡 ∈ [𝑡(; 𝑡)], 𝑝((𝑥, 𝑦) – density of the probability distribution of the initial 
location of the monitoring object at the point (𝑥, 𝑦). In the case of a uniform distribution of 
the initial location of the monitoring object 𝑝((𝑥, 𝑦) = 1/𝑆!" , where 𝑆!"  – system area of 
responsibility. That is why expression (2) can be rewritten as: 

𝑝((𝑡) =
27#$(!)
2!

⋅ (
7#$

. (3) 

Substituting expression (3) into expression (1), obtain: 

< 𝑡 >[!!;	!"]	=
(
7%&
⋅ ∫ 𝑑𝑡. ⋅ 𝑡. 27#$(!.)

2!.
!"
!!

. (4) 

Due to the fact that "sweeping" can occur only as long as 𝑆%6 ≤ 𝑆!" (since at 𝑆%6=𝑆!" the 
monitoring object will be guaranteed to be detected), the task of minimizing the average time 
of object detection has been reduced to the task of maximizing the "sweeping" speed (shown 
at Fig. 2). 

 
 

 
Fig. 2. Schematic representation of the effect of the "sweeping" speed on the length of the integration 

region in expression (4) 

In the case when the monitoring object moves in the area of responsibility of the system 
for a period of time 𝑡 ∈ [𝑡(; 𝑡)] (that is, when the condition 𝑣ОМ ≪ 𝑣СО is not observed), one 
pass of the detectors may not be enough to detect the monitored objects. In this case, it is 
necessary to launch an iterative patrol of the system's area of responsibility, in which at each 
iteration (or, in other words, in a cycle), the system's area of responsibility is swept along the 
routes (in both directions) that were used above when approaching a stationary target. 
Moreover, the less the period 𝑇8 (in time units) of traversing each point (graph node) of the 
target zone by monitoring tools, the faster the monitoring object will be detected. This 
statement clearly demonstrates the limiting case in which the period of the round trip of each 



MODELING THE DETECTION OF MOVING OBJECTS BY MEANS OF A SPATIALLY DISTRIBUTED  5 

Copyright ©2022 ASSA.                                                                                    Adv. in Systems Science and Appl. (2022) 
 

point 𝑇8 → 0. It should be noted that in view of the fact that the location of the monitoring 
object in the area of responsibility of the system is modeled by a random process with a 
uniform distribution over the area of the area of responsibility, none of the points is “selected”. 
Therefore, it is advisable to choose the same period of traversing each point, so 𝑇8 = 𝑐𝑜𝑛𝑠𝑡 
for every 𝑘. Such a patrol mode can be carried out by a set of N detectors following each other 
along the constructed route at the same distance from each other. When the weather conditions 
change and / or the sensors fail, their location should be reconfigured at each subsequent 
moment in time, taking into account the new route and / or the new number of detection means 
of the monitoring system. 

3. COMPUTER MODELING  

Computer simulation is carried out in accordance with the following conditions: 

1. The area of responsibility of the system is covered by nodes with a radius of the field 
of view under current weather conditions. Such a division is necessary to speed up the 
operation of the algorithm and is valid provided that during the time between two iterations in 
time, each monitoring tool has time to move to the next node along the route. At the nodes, a 
route is built along the terrain graph, which is a Hamiltonian cycle along which the detection 
means will move during the monitoring process. 

2. At each iteration, the detector moves along the constructed route by exactly one node. 
A monitored object is considered detected (registered) if it falls within the scope of at least one 
detection tool. 

For the computer experiment, the MATLAB software package was used. 
It is proposed to carry out computer modeling of the deployment of mobile detection 

devices based on SENSORs in the area of responsibility of the system in the form of four 
main stages. 

At the first stage, the input parameters of the task are initialized (the size of the monitoring 
area, the location of obstacles, the number of detectors, the size of the visibility area of one 
detector, the number of monitored objects, the speed of the monitored objects). The speed of 
movement of the detectors is determined by the size of their field of view, since in one iteration 
the sensor is shifted by a distance of the order of 2𝑅, where 𝑅 – the radius of the visibility area 
of the detector, which is a circle centered at the sensor location. Thus, the average speed of 
each detection tool over the monitoring time is < 𝑣%9:% >!	≈ 	2𝑅	𝑝𝑖𝑥/𝑖𝑡𝑒𝑟𝑎𝑡𝑖𝑜𝑛 (2𝑅 pixel per 
iteration).  

At the second stage, a graph of the movement of detectors in the area of responsibility of 
the system is built. For this, the area of responsibility of the system is covered by the visibility 
areas of the detection means. The location of the nodes is carried out by means of the gradient 
descent method, the function of which is the area of the covered area of responsibility of the 
system. The center of each of these scopes represents the top of the graph. Two vertices are 
connected by edges if the distance between them is < 2𝑅 + 0.2 ⋅ 2𝑅. An example of building 
a graph is shown in Fig. 3. 



6                           V.I. GONCHARENKO, A.A. KOCHKAROV, D.V. YATSKIN, B.V. RUMIANTSEV 

Copyright ©2022 ASSA.                                                                                    Adv. in Systems Science and Appl. (2022) 

 
Fig. 3. Left: Filling the target area with sensor scopes. Right: building a graph of the movement of 

detectors based on this filling. The red line marks the Hamiltonian cycle 
At the third stage, a Hamiltonian cycle of traversing all the vertices of the constructed 

graph is constructed. On the constructed cycle, detectors are placed with the same distance 
between two neighboring ones (if the distance is counted along the Hamiltonian cycle, see Fig. 
4). Such an arrangement provides an equal traversal period for each vertex of the graph during 
the monitoring process. 

 
Fig. 4. Left: the location of the sensors on the Hamiltonian cycle.  

Right: Hamiltonian cycle on the plotted graph 
The fourth stage is directly devoted to the computer experiment. The number of detected 

(registered) monitoring objects is calculated for N iterations. At each iteration, the detectors 
are shifted along the Hamiltonian cycle by one vertex. The detected monitoring object is 
considered registered and cannot be re-registered in the future. The ratio of the number of 
registered objects of monitoring at the end of monitoring to their initial number determines the 
probability of object detection (registration) during the monitoring period. Monitored objects 
move in the system's area of responsibility in a random manner with equal probability at each 
iteration, the offset distance per iteration is initialized by the user and describes the speed of 
the monitored objects. A frame of an iteration of a computer experiment at a fixed time is 
shown in Fig. 5. 



MODELING THE DETECTION OF MOVING OBJECTS BY MEANS OF A SPATIALLY DISTRIBUTED  7 

Copyright ©2022 ASSA.                                                                                    Adv. in Systems Science and Appl. (2022) 
 

 
Fig. 5. A frame of an iteration of a computer experiment 

4. RESULTS AND DICCUSSION 

As a result of the experiment, was built the dependence ;'()
;*

(𝑣, 𝑁%9:%) = 𝑃(𝑣, 𝑁%9:%), which 
is the dependence of the share of registered monitoring objects on their speed and the number 
of detection tools. The share of registered objects of monitoring can be interpreted as the 
probability of registering one object of monitoring during the monitoring period. The 
experimentally obtained dependence 𝑃(𝑣, 𝑁%9:%) with an increase in the number of detectors 
is illustrated in Fig. 6 and 7. 

 
Fig. 6. Dependence of the share of registered targets on the speed of targets and the number of sensors 

 



8                           V.I. GONCHARENKO, A.A. KOCHKAROV, D.V. YATSKIN, B.V. RUMIANTSEV 

Copyright ©2022 ASSA.                                                                                    Adv. in Systems Science and Appl. (2022) 

 

 

 
Fig. 7. Dependence of the share of registered objects of monitoring on their speed and the number of 

detection tools 

 

The resulting dependence has characteristic features, among which the following should 
be noted: 

1. There is an area of "stationary" probability, in which for any number of detectors the 
probability ceases to effectively depend on their speed, since addiction 𝑃(𝑣) this region has 
the character of small random deviations from the mean value, which we will call the stationary 
value of the probability 𝑃%!<!. (see Fig. 3, left). 

 
2. The dependence of the stationary probability on the number of detection means is 

approximated with high accuracy by the exponential growth function with saturation (see Figs. 
6 and 8), i.e. 

𝑃%!<!(𝑁%9:%) = 1 − e=>;+(+# (5) 

3. For N?@A? = 1 the dependance 𝑃(𝑣) is monotonically decreasing in the region until the 
stationary probability is reached. Therefore, at N?@A? = 1  the maximum probability of 
registering one monitoring object is achieved at zero target speed 𝑣 = 0.  

4. It is also of interest to study the dependence of the threshold point 𝑣!B transition to 
stationary dependence on the number of detection means. In Fig. 6 for 𝑁%9:% = 	2,3,4 this point 
is shifted to the region of bigger speeds. 
 

 



MODELING THE DETECTION OF MOVING OBJECTS BY MEANS OF A SPATIALLY DISTRIBUTED  9 

Copyright ©2022 ASSA.                                                                                    Adv. in Systems Science and Appl. (2022) 
 

 
Fig. 8. Dependence of the stationary probability of registration of one monitoring object  

on the number of detection means 

5. CONCLUSION  
The proposed approach to optimizing the use of a continuous monitoring system makes it 
possible to efficiently register objects moving randomly in the area of responsibility. systems. 

Within the framework of the budget work, a model that implements the ideology of the 
proposed approach. The model makes it possible to determine the optimal geometric 
arrangement, including the trajectory of movement of the means of a spatially distributed 
continuous monitoring system using the dynamic structure of dynamic targets. For a small 
number of detectors, the model allows you to find the optimal ratio of target speed to sensor 
speed. This, in turn, makes it possible to increase such an important indicator as to the 
probability of registration of one monitoring object. 

The work investigates statistical dependencies, including: 
• stationary probability of registration of one monitoring object from the number of detection 

means; 
• registered objects of monitoring on their speed and the number of means of detection; 
• registration of registered targets from speed and number of sensors. 

The conclusions drawn from the analysis of these dependencies confirm the effectiveness 
of the proposed model. 

In the region of stationary probability, an increase in the speed of sensors does not lead to 
a change in the probability of registering a target object in the singular. This allows us to 
conclude that it is possible to detect monitoring objects moving at a given speed, which makes 
it possible to determine such a threshold value for the detection rate when the efficiency of the 
system (from the point of the probability of object detection) exceeds the maximum values, 
which makes it possible to determine such a threshold value as optimal for solving detection 
tasks. 

 
ACKNOWLEDGEMENTS 

This work has been supported by the grants the Russian Science Foundation,                     
RSF № 21-19-00481. 

 
 



10                           V.I. GONCHARENKO, A.A. KOCHKAROV, D.V. YATSKIN, B.V. RUMIANTSEV 

Copyright ©2022 ASSA.                                                                                    Adv. in Systems Science and Appl. (2022) 

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