=Paper=
{{Paper
|id=Vol-2711/paper37
|storemode=property
|title=Mathematical Modeling of Motion of Iron Bird Target Node of Security Data Management System Sensors
|pdfUrl=https://ceur-ws.org/Vol-2711/paper37.pdf
|volume=Vol-2711
|authors=Olena Tachinina,Oleksandr Lysenko,Iryna Alekseeva,Valeriy Novikov
|dblpUrl=https://dblp.org/rec/conf/icst2/TachininaLAN20
}}
==Mathematical Modeling of Motion of Iron Bird Target Node of Security Data Management System Sensors==
https://ceur-ws.org/Vol-2711/paper37.pdf
Mathematical Modeling of Motion of Iron Bird Target
Node of Security Data Management System Sensors
Olena Tachinina1[0000-0001-7081-0576], Oleksandr Lysenko2[0000-0002-7276-9279],
Iryna Alekseeva2 [0000-0002-2878-6514], Valeriy Novikov2[0000-0003-4199-9968]
1 National Aviation University, Kyiv, Ukraine
tachinina5@gmail.com
2 National Technical University of Ukraine «Igor Sikorsky Kyiv Polytechnic Institute», Kyiv,
Ukraine
lysenko.a.i.1952@gmail.com, alexir1@ukr.net, novikov1967@ukr.net
Abstract. The article is devoted to presentation of the results of mathemati-
cal modeling of motion of the target node, which imitates a mini-drone and is
mounted on the carriage of iron bird (test bench) for testing the sensors of secu-
rity data management system. It is demonstrated that mathematical modeling
made it possible to predict the possibility of self-oscillations during modeling of
protective maneuvers of mini-drones and to confirm effectiveness of the algo-
rithmic method of test bench upgrading. The results of computer simulations of
the bench operation show that self-oscillations and an unstable mode of opera-
tion of a hypothetical typical biaxial iron bird (test bench) may occur during
iron bird modeling of both deterministic and stochastic motion of the mini-
drone. These self-oscillations did not occur earlier when modeling the motion
of inertial objects, because the cross-links between the channels when iron bird
modeling with low accelerations were not significant. The simulation results
showed the existence of a scientific and technical problem that arises when try-
ing to use the available iron birds for modeling the motion of maneuverable tar-
gets. To solve the scientific and technical problem, it is proposed to perform al-
gorithmic adaptation of the servo drive control system.
Keywords: Mathematical Modeling of Motion, Target Node, Iron Bird, Sensor
Network, Flying Robots, Drones, Security Data Management System.
1 Introduction
Currently the availability of anti-drone defense is one of the most important condi-
tions for solving the problem of ensuring the security (sustainable functioning) of
national critical infrastructure [1–4, 10].
The most topical task is protecting against unauthorized access into the protected
zone of individual mini-drones, its groups and a "mosquito" raid of mini-drones. One
way to solve this problem is to use a security data management system (SDMS).
Source information about the target in such systems comes from stationary, mobile
or quasi-mobile sensor networks. This information is used in SDMS for subsequent
Copyright © 2020 for this paper by its authors. Use permitted under Creative Commons
License Attribution 4.0 International (CC BY 4.0). ICST-2020
identification, inspection, localization (interception) and, if necessary, destruction of a
mini-drone (for example, by a narrow electromagnetic pulse) [1].
The critical elements of SDMS sensor networks are target sensors (TS). The quali-
ty of information that is used to make a decision in the SDMS depends on the effi-
ciency of the TS. TSs can operate on one physical principle (homogeneous TSs) or
different ones (multi-sensors of a target or heterogeneous TSs). The most important
element of the TS is the target tracking unit (TTU). At the stage of validation of made
design decisions, TSs together with TTUs are sent for iron bird testing.
The test benches of modeling flat (two-dimensional) or spatial (three-dimensional)
target motion are used for testing.
The target’s maneuvers are modeled by the motion of bench’s target node (Target
Motion Simulators (TMS)) using tracking drives that operate along two or three mu-
tually perpendicular axes.
The sensor under test is placed in a special rotary support, which is spatially locat-
ed relative to the target node, taking into account the scale of modeling. TTU of TS is
checked for stability of the target capture and quality of tracking. At the time of writ-
ing, the corresponding specialized companies have available two- and three-axis
modeling test benches (iron birds) (see Fig. 1 [5]).
Fig. 1. Example of combination of two-axis hypothetic test benches on one support [5]:
1– rotary support for TS; 2 – TMS of first two-axis test bench; 3 – TMS of first two-axis sec-
ond test bench.
Existing test benches are designed for iron bird simulation of the motion of objects
that are less maneuverable compared to mini-drones. This means that the dynamics of
test benches themselves when simulating the target motion can be neglected. If you
simulate the motion of super maneuverable mini-drones using same test benches, then
significant dynamic errors arise. There are two ways to solve the problem of modeling
the motion of a target, which is a maneuverable mini-drone. The first way is the crea-
tion of specialized test benches with improved dynamic characteristics and the second
– is the algorithmic modernization of existing test benches, which is economically
feasible. The article is devoted to the description of method for algorithmic moderni-
zation of existing biaxial test benches.
2 Problem statement
The biaxial bench (iron bird) of simulation of the target motion is considered. The
drives of the iron bird allow moving the carriage with the mounted target assembly.
The carriage (target node) movements imitate the maneuvers of the target in relation
to SDMS target sensor, which tracks this movement. The equipment mounted on the
carriage imitates the target’s identification features (for example, unmasking acoustic
or heat radiation, reflective properties of a mini-drone when it is illuminated with
radio-frequency radiation or laser locator light waves). We assume the easiest option
for fixing SDMS target sensor - the sensor is fixed stationary. The plane of observa-
tion for the target motion is located at a certain distance from SDMS target sensor,
taking into account the scale and angle of modeling. The position of this plane is set
by the axes of movement of the iron bird target node.
Typical movements of the mini-drone are assigned to the target node, then the tar-
get tracking device, which is part of SDMS target sensor, is experimentally checked
for functioning.
The iron bird target node should adequately simulate the motion of the mini-drone
relative to SDMS target sensor. The iron bird should not introduce significant errors
in the form of trajectory of the mini-drone and the rate of movement along it. This
means that the dynamic properties of the tracking drives of target unit should not
significantly affect the typical nature of the movement that is being modeled. If the
target is inertial, then the dynamic properties of tracking drives of the target node can
be neglected.
However, the modern mini-drones have small mass and main moments of inertia
and are high-powered. This allows them to perform fast and complex maneuvers.
Such maneuvers were not available for those types for which the existing iron birds
were designed.
We will check the possibilities of the existing biaxial iron birds for the adequate
execution of protective maneuvers of mini-drones on the computer mathematical
model of target node motion. All calculations and construction of the computer math-
ematical models are performed using MATLAB + Simulink computer mathematics
system.
A computer mathematical model of the typical dynamics of mini-drones motion
relative to the target sensor along ОХ and ОY axes of the observation plane of target
motion is set by Transfer Fon2 and Transfer Fon6 blocks (ref. Fig. 2) [6]. ОХ and ОY
axes are located in the plane of observation of target movement: ОY axis is the line
perpendicular to the horizon plane and passing through the target node, ОХ axis is the
line passing through the target node and parallel to the line of intersection of the ob-
servation plane of the target motion with the horizon plane.
Fig. 2. Computer mathematical model of movement of the target node of a hypothetical typical
biaxial iron bird, which is designed to test the sensors of a security data management system.
On XYGraph1 and XYGraph2 plotters, we can observe the trajectories of reference
motion of the mini-drone, and result of modeling this motion using a typical hypothet-
ical biaxial iron bird with tracking DC drives.
The reference motion of mini-drone is a test movement when the target sensor will
be checked (tested). The mathematical model of drives that operate along OX and OY
axes is assumed to be the same and is specified in the simulation scheme (Fig. 2) by
State Spase1 and State Spase2 blocks (the model parameters are shown in the dia-
gram).
Let us simulate the movement of mini-drone in the plane of observation of the tar-
get movement. We assume that along each axis of mobility of the hypothetical iron
bird there is a follow-up digital drive with a digital proportional integral differential
controller.
The transmission coefficients of the proportional, integral, and differential signals
of digital controller, which was used to simulate operation of the prototype test bench,
were assumed to be the same for both channels and equal 1, 0.08, and 1.47 (the pa-
rameters of digital proportional integral differential controller were calculated by
Ziegler-Nichols method [7]).
Three types of the mini-drone reference motion were considered: “oval”; "Lissa-
jous figure"; "Random walk" [3, 4]. The result of modeling these reference motions is
shown in Fig. 3 (a, c), Fig. 4 (a, c), Fig. 5 (a, c). The dimension along ОХ and ОY
axes is meters.
а) b)
Fig. 3. Graphic image of the reference trajectory of oval mini-drone (a) and result of reproduc-
ing this movement with a hypothetical typical biaxial iron bird (b): position of P1 and P2
switches (Fig. 2) «down» and «up».
а) b)
Fig. 4. Graphic image of the reference trajectory of « Lissajous figure» mini-drone (a) and
result of the reproduction of this movement by a hypothetical typical biaxial iron bird (b): posi-
tion of P1 and P2 switches (Fig. 2) is «up» and «down», and P3 and P4 – «down» and «up».
The results of computer simulations show that during iron bird modeling of both
deterministic and stochastic movements of the mini-drone, self-oscillations and an
unstable mode of operation of a hypothetical typical biaxial iron bird may occur.
These self-oscillations did not occur earlier when modeling the motion of inertial
objects, because the cross-links between the channels in the iron bird modeling with
low accelerations were not significant.
а) b)
Fig. 5. Graphic image of the reference trajectory of «random walk» mini-drone (a) and result
of reproducing this movement with a hypothetical typical biaxial iron bird (b): position of P1,
P3 switches (Fig. 2) and P2, P4, «up» and «down»
The simulation results showed the existence of a scientific and technical problem
that arises when trying to use the available iron birds to simulate the movement for
which these test benches were not designed.
The easiest way to solve a scientific and technical problem is to perform an algo-
rithmic adaptation of the drive control system. We show how this can be done.
3 Problem solving
To eliminate self-oscillations of the target node that arise when modeling the pro-
tective maneuvers of mini-drones, the algorithmic adaptation of servo drives operat-
ing along the axes of mobility of a hypothetical iron bird was performed. The algo-
rithmic adaptation consisted in application of the modal control algorithm to correct
dynamic properties of the servo drives. The modal controller was synthesized accord-
ing to the following procedure:
1. A continuous mathematical model of the target unit drive of a hypothetical typi-
cal biaxial iron bird in the form of a MIMO LTI model was identified. Identification
performed using specifications of DC motors [8].
Based on the identification results, it was assumed that the MIMO LTI (Multiple
Input Multiple Output) drive models on the OX and OY channels practically coincide
and are described by equations in the two-dimensional state space [9]
dX/dt=A·X(t)+B·U(t)· (1)
Y(t)=C·X(t)+D·U(t) (2)
where X(t)=[x1(t); x2(t)], x1(t)=i(t) – current in the armature circuit of a DC motor,
x2(t)=w(t) – angular velocity of the armature rotation; U(t)=[u1(t); u2(t)], u1(t) – con-
trol signal, which is fed to the independent excitation winding, u2(t) – signal simulat-
ing the effect of cross-coupling between ОХ and ОY channels;
A=[-25 -7.5;7.5 0], B=[5 0;0 -5], C=[1 0;0 1], D=[0 0;0 0].
2. Parameters of the drive’s discrete mathematical model are analytically calculat-
ed. The functions MATLAB+Simulink «ss» and «c2d» of a computer mathematics
system were used for calculations.
The period of quantization in time Т0 was calculated based on the following rea-
soning.
It is accepted that at the input of analog to digital converters, which are part of the
digital meters of the armature circuit current and angular velocity of rotation of the
armature of DC motors of OX and OY drive channels, the low-pass filters operate.
Their upper cutoff frequency coincides with the upper cutoff frequency of the use-
ful signal and is equal to 51.4 rad/s for a typical drive. The time sampling period is
found T0 Ј p / 51.4 = 0.0611 s by the quantization theorem. It was accepted that
T0= 0.06 s.
The discrete mathematical model of the drives of ОХ and ОY channels has the fol-
lowing form
X(n+1)=Ad·X(n)+ Bd·U(n) (3)
Y(n)=C·X(n)+D·U(n) (4)
where Ad=[0.1841 –0.2256; 0.2256 0.9359], Bd=[0.1504 0.04274; 0.04274 –0.2928].
3. The modal controller for correction of the dynamic properties of the drives of
OX and OY channels was calculated [7, 9].
The calculation results to the values of matrix amplification factors T, K,
inv(T*Bd) for:
1) modal analyzers Gain 7, Gain 10
T=[-1.1860 -0.3955;-0.3955 -1.1860] (fig.6).
Each modal analyzer converts the output signals of digital sensors of the drive motor
armature current and the angular velocity of this armature into virtual coordinates in
two-dimensional space;
2) modal regulators Gain 8, Gain 11
K = [ -0.2407 0; 0 0.3607] (fig.6).
Each modal controller calculates a virtual control vector.
3) modal synthesizers Gain 9, Gain 12
inv(T*Bd)=[ -5.7616 1.1356;-1.9215 3.4057] (fig.6).
Each modal synthesizer computes a real control vector.
After connecting the modal controller, the digital proportional integral differential
controller was reconfigured in the servo system of ОХ and ОY channels according to
Ziegler–Nichols method [7].
The computer mathematical model and simulation results of functioning of the
modernized hypothetical typical biaxial iron bird are presented in Fig. 6, 7 (a, b), 8 (a,
b) and 9 (a, b).
Fig. 6. Computer mathematical model of motion of the target node of a hypothetical typical
biaxial iron bird, which is designed to test the sensors of security data management system.
а) b)
Fig. 7. Graphic image of the reference trajectory of the oval mini-drone (a) and result of repro-
duction this motion by a hypothetical typical biaxial iron bird (b): positions of P1 and P2
switches (Fig. 6) are “down” and “up”.
а) b)
Fig. 8. Graphic image of the reference trajectory of “Lissajous figure” mini-drone (a) and
result of reproduction of this motion by a hypothetical typical biaxial iron bird (b): positions of
P1 and P2 switches (Fig. 6) are “up” and “down”, and P3 and P4 – down and up.
а) b)
Fig. 9. Graphic image of the reference trajectory of “random walk” mini-drone (a) and result of
reproduction this movement by a hypothetical typical biaxial iron bird (b): positions of switches
(Fig. 6) P1, P3 and P2, P4 are “up” and “down”.
4 Conclusions
1. Mathematical modeling of the target assembly motion, which was virtually
driven by a hypothetical biaxial bench for testing the sensors of security data man-
agement system, allowed: to predict the possibility of self-oscillations in the simula-
tion of protective maneuvers of mini-drones; confirm the effectiveness of algorithmic
method of upgrading the test bench.
To implement the algorithmic method of upgrading the test bench it is necessary to
identify: a mathematical model of movement of the carriage (target node) of iron bird,
taking into account cross-links between the motion simulation channels; mathematical
model of extreme maneuvers of mini-drones.
2. The complex algorithm of servo drives operation proposed in the article consists
of two algorithms: the digital modal controller algorithm and the digital proportional
integral differential controller algorithm. The first one corrects the dynamic properties
of electric motor, and the second – provides a quasi-adaptive control with respect to
the disturbing effect of cross-links.
3. The iron birds, modernized by the algorithmic method described in the article,
can be used to test the tracking nodes for individual mini-drones or clusters of mini-
drones (distributed drones). These tracking nodes can be used in data management
systems with a human operator or artificial intelligence in remote control loop of in-
spection and destruction means of security data management systems.
4. The direction of further research should be considered construction of a mathe-
matical model of the motion of target node, which will be installed on the carriage of
three-axis iron bird for testing the sensors of security data management system.
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