=Paper=
{{Paper
|id=Vol-3732/paper05
|storemode=property
|title=2D chaotic path planning for autonomous vehicles
|pdfUrl=https://ceur-ws.org/Vol-3732/paper05.pdf
|volume=Vol-3732
|authors=Roman Voliansky,Nina Volianska,Arief Bramanto Wicaksono Putra
|dblpUrl=https://dblp.org/rec/conf/cmse/VolianskyVP24
}}
==2D chaotic path planning for autonomous vehicles==
https://ceur-ws.org/Vol-3732/paper05.pdf
2D chaotic path planning for autonomous vehicles
Roman Voliansky1,*,†, Nina Volianska1,† and Arief Bramanto Wicaksono Putra2,†
1 National Technical University of Ukraine “Igor Sikorsky Kyiv Polytechnic Institute”, Polytechnichna Str., 37, Kyiv,
03056, Ukraine
2 Politeknik Negeri Samarinda, Str. Jl. Cipto Mangun Kusumo, 75242, Samarinda, Indonesia
Abstract
Our paper is devoted to solution of path-planning problem for various autonomous vehicles. We
offer to solve this problem by assuming that state variables of a dynamical system, which is
considered as trajectory generator, are defined in some coordinate system. This coordinate
system is interconnected with the target coordinate system where the path planning problem is
solved. The use of an intermediate coordinate system makes it possible to form the desired path
by defining transformations from one coordinate system to another. Since a number of such
transformations can be very huge, an enormous number of dynamical systems can be obtained
from one initial one. In our paper, we use a polar-like coordinate system as an intermediate one
and we use trigonometrical functions to define interrelations between polar and orthogonal
coordinate systems. All used coordinate systems are the simplest two-dimensional ones and we
show the methodology of dynamical system transformation from one coordinate system to
another. At the same time, one can easily increase the number of the initial state variables and
extend our approach from plane to space. We prove the benefits of our approach by considering
the Duffing pendulum as a dynamical system that is defined in polar coordinates and then we use
various transformation routines to design a chaotic dynamical system in the orthogonal
coordinate system. These routines are based on the use of various affine transformations of
pendulum state variables. As a result, various chaotic attractors are being formed to use as the
desired paths for unmanned vehicles.
Keywords
chaotic system, chaotic path planning, coordinate transform, autonomous vehicle
1. Introduction
Chaotic path planning is an innovative approach to trajectory generation [1, 2, 3] that
leverages the principles of chaos theory [4] to create non-linear, unpredictable paths [5, 6].
Unlike conventional deterministic [7, 8] or random path planning methods [9, 10], chaotic
path planning [11] introduces complex dynamics that result in extensive coverage and
adaptability, making it particularly useful for applications in robotics, autonomous vehicles,
and exploration.
CMSE’24: International Workshop on Computational Methods in Systems Engineering, June 17, 2024, Kyiv,
Ukraine
∗ Corresponding author.
† These authors contributed equally.
voliansky@ua.fm (R. Voliansky); ninanin@i.ua (N.Volianska); ariefbram@gmail.com (A.B.W. Putra)
0000-0001-5674-7646 (R. Voliansky); 0000-0001-5996-2341 (N. Volianska); 0000-0003-1187-5040
(A.B.W. Putra)
© 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Attribution 4.0 International (CC BY 4.0).
CEUR
ceur-ws.org
Workshop ISSN 1613-0073
Proceedings
Chaos theory deals with systems that exhibit sensitive dependence on initial conditions
[12, 13, 14], leading to behavior that appears random despite being deterministic. This
characteristic is harnessed in chaotic path planning to generate paths that are both
unpredictable and thorough. Commonly used chaotic systems include the logistic map [15],
the Henon map [16], the Lorenz system [17], and the Chua circuit [18, 19, 20] each providing
distinct patterns of chaos that can be tailored to specific application needs. Last decades lots
of papers have appeared in the scientific press to show studies [21, 22, 23, 24], designs [25,
26, 27], and usage of various chaotic systems [28, 29, 30].
The primary motivation behind chaotic path planning is to achieve superior area
coverage and unpredictability. In autonomous robotic systems, this translates to more
efficient exploration, improved obstacle avoidance, and enhanced security against pattern
recognition by adversaries [31, 32]. For instance, robotic vacuum cleaners can utilize
chaotic paths to ensure thorough cleaning, while surveillance drones can unpredictably
patrol areas to prevent detection [33, 34].
In practice, chaotic path planning involves the following steps:
• Initialization: Selection of a chaotic system and definition of initial conditions
[35, 36, 37].
• Sequence Generation: Iteration of the chaotic map or solving differential
equations to produce a sequence of points [38, 39].
• Path Construction: The connection of these points to form a continuous path,
adjusted in real-time for obstacle avoidance and environmental changes [40, 41].
It is clearly understood that the solution of the third task depends on coordinate systems
where a chaotic system is defined. Many authors consider the system motion in the
conventional orthogonal coordinate system [42]. Nevertheless, the huge number of such
systems' attractors are known and that is why the vehicle that is moving along them can be
intercepted.
We offer to avoid this drawback of chaotic path planning routine by using some
coordinate transformation to define system dynamics in various coordinates.
Our paper is organized as follows: firstly, we show the possibility of attractor changing
by studying system dynamics in various coordinate systems. Then we design the
generalized chaotic system by considering transformation from polar to cartesian
coordinates. At third, we study the peculiarities of chaotic systems implementation by using
discrete-time devices. At fourth, we consider an example of a chaotic path planning system's
design and study.
2. Method
2.1. Second order dynamical system motions in polar coordinates
Let us consider the simplest second order dynamical system which motion is described as
follows
𝑦̇1 = 𝑦2 ; 𝑦̇ 2 = −𝑦1 . (1)
If one starts to study (1) with phase plane method he obtains very trivial linear
oscillations (Figure 1a) and phase trajectories (Figure 1b).
a) System time response b) System phase portrait
Figure 1: System simulation results.
Due to the simplicity of the above-given oscillations one can use (1) in various
applications including planning of circle motions for any devices.
It is necessary to say that in the classical control theory studying of all dynamical systems
are performed by using the orthogonal cartesian coordinates and assuming that system
state variables make some orthogonal basis in the system state space.
It is clear that the above-mentioned coordinates are not the only possible ones and
studying of system motions depend on how to interpret the system state variables and
which coordinate basis they form.
Generally speaking, one can use any coordinates to describe system motions in it. It is
clearly understood that such changing of the system of coordinate cause occurring of phase
portraits which differ from known existing one. That is why we offer to perform such
coordinate transformations to solve path planning problems and design novel motion
trajectories which can be used as paths for some autonomous vehicles.
In our paper we consider system state variables as coordinates which are given in a polar
system of coordinates and their generalizations. Thus, one of system state variable is
considered as a linear position of some body. This position can be considered as distance ρ
from some origin 0 to a representing point A and another represent angular position φ of
the considered body (Figure 2). In the Figure 2 we assume that linear position is
represented by y1 and angular one is y2, it is necessary to say that fore the system (1) it does
not matter how to interpret each state variable because they have similar values and forms
but differ with initial phase. For more complex system it can produce different phase
portraits.
It is clear that polar and orthogonal cartesian coordinates interrelates each other with
well-known dependencies, which can be written down for considered case in such a way
𝑥1 = 𝑦1 cos(𝑦2 ) ; 𝑥2 = 𝑦1 𝑠𝑖𝑛(𝑦2 ). (2)
Figure 2: State variables interpretation in polar coordinates.
Results of numerical solution of (1) and (2) are shown in Figure 3.
a) System time response b) System phase portrait
Figure 3: Output variables of dynamical system (1) and (2) with initial condition y1(0)=π.
It is clear that shown in Figure 3 simulation results more complex than given in Figure 1
and allow to form more complex motion trajectories. Moreover, because the use of
trigonometric functions oscillations become nonlinear and depend values of 𝑦2 state
variable. System output variables have different form and amplitude.
Because of the use of nonlinear function, the considered system depends on initial
conditions. In Figure 4, we show simulation results for (1) with twice reduced initial value
of 𝑦1 variable.
a) System time response b) System phase portrait
Figure 4: Output variables of dynamical system (1) and (2) with initial condition y1(0)=π/2.
Analysis of simulation results from Figure 3 and Figure 4 allows us to claim that
considering of system state variables in polar coordinates allows to design nonlinear
dynamical system which can be used to produce motions path which are tremendously
differ from those which are produced buy interpreting state variables in orthogonal
cartesian coordinates.
2.2. The Generalized Motion in the Orthogonal Coordinate Basis
Analysis of curves which are studied by using polar coordinate system shows that in the
most general case distance ρ is considered as function of body angular position φ. We claim
that the opposite is also true, so angular position φ can depend on linear one ρ. This
assumption gives us the possibility to generalize (2) as follows
𝑥1 = 𝑓1 (𝑦1 , 𝑦2 ) cos(𝑓2 (𝑦1 , 𝑦2 )) ; 𝑥2 = 𝑓1 (𝑦1 , 𝑦2 ) sin(𝑓2 (𝑦1 , 𝑦2 )). (4)
From control theory viewpoint (1) and (3) form dual channel dynamical system which
motion is defined by following operator equations
𝑥1 = 𝑓1 (𝑦1 , 𝑦2 ) 𝑐𝑜𝑠(𝑓2 (𝑦1 , 𝑦2 )) ; 𝑥2 = 𝑓1 (𝑦1 , 𝑦2 ) 𝑠𝑖𝑛(𝑓2 (𝑦1 , 𝑦2 )) ; (5)
𝑠𝑦1 = 𝑠𝑦1 (0) + 𝑦2 ; 𝑠𝑦2 = 𝑠𝑦2 (0) − 𝑦1 ,
where 𝑠 is a Laplace operator and 𝑦1 (0), 𝑦2 (0) are initial values of system state variable.
One can find block-diagram of such system in Figure 5.
Contrary to the initial dynamical system (1) which is considered as core system and that
block-diagram is shown in dashed rectangular, the dynamical system (5) can be considered
as a dual channel system. The parallel channels of this system are designed by using
trigonometric expressions with nonlinear generalized arguments which depend on initial
dynamical system state variables.
Figure 5: Scheme of the generalized dynamical system.
Since the dynamical system which is shown in Figure 5 is described with differo-
algebraic equations it causes some inconveniences to use control theory methods to study
systems of such type.
We offer to avoid this fact and rewrite (5) by using differential equations only. To do this
we differentiate (4) by time
𝜕𝑓1 (𝑦1 , 𝑦2 ) 𝜕𝑓1 (𝑦1 , 𝑦2 ) (6)
𝑥̇ 1 = ( 𝑦2 − 𝑦1 ) cos(𝑓2 (𝑦1 , 𝑦2 )) +
𝜕𝑦1 𝜕𝑦2
𝜕𝑓2 (𝑦1 , 𝑦2 ) 𝜕𝑓2 (𝑦1 , 𝑦2 )
+𝑓1 (𝑦1 , 𝑦2 ) ( 𝑦1 − 𝑦2 ) sin(𝑓2 (𝑦1 , 𝑦2 )) ;
𝜕𝑦2 𝜕𝑦1
𝜕𝑓1 (𝑦1 , 𝑦2 ) 𝜕𝑓1 (𝑦1 , 𝑦2 )
𝑥̇ 2 = ( 𝑦2 − 𝑦1 ) 𝑠𝑖𝑛( 𝑓2 (𝑦1 , 𝑦2 )) +
𝜕𝑦1 𝜕𝑦2
𝜕𝑓2 (𝑦1 , 𝑦2 ) 𝜕𝑓2 (𝑦1 , 𝑦2 )
+𝑓1 (𝑦1 , 𝑦2 ) ( 𝑦1 − 𝑦2 ) cos(𝑓2 (𝑦1 , 𝑦2 )).
𝜕𝑦2 𝜕𝑦1
It is clear that the use of (6) requires to have input signals y1 and y2 which are produced
by dynamical system (1). Thus, one can consider (1) as the generator of harmonic signals
for nonlinear dual channel transformation system (6). The main drawback of such system
is interrelations between channels by input signals. This drawback does not allow to
perform trajectory calculations by using (6) in parallel way for each channel because it is
necessary to obtain common inputs.
We offer to avoid this drawback and improve system performance by redefining motion
for each channel with two differential equations. That is why we offer to differentiate 𝑥1 and
𝑥2 output variable for two times.
𝑦2 (7)
𝑥̇ 1 = (𝛻𝑓1 (𝑦1 , 𝑦2 )cos(𝑓2 (𝑦1 , 𝑦2 )) − 𝛻𝑓2 (𝑦1 , 𝑦2 )𝑓1 (𝑦1 , 𝑦2 )sin(𝑓2 (𝑦1 , 𝑦2 ))) (−𝑦 ) ;
1
−𝑦1
𝑥̈ 1 = (𝛻𝑓1 (𝑦1 , 𝑦2 )cos(𝑓2 (𝑦1 , 𝑦2 )) − 𝛻𝑓2 (𝑦1 , 𝑦2 )𝑓1 (𝑦1 , 𝑦2 )sin(𝑓2 (𝑦1 , 𝑦2 ))) (−𝑦 ) +
2
(𝛻 2 𝑓1 (𝑦1 , 𝑦2 )cos(𝑓2 (𝑦1 , 𝑦2 )) − 2𝛻𝑓1 (𝑦1 , 𝑦2 )𝛻𝑓2 (𝑦1 , 𝑦2 )sin(𝑓2 (𝑦1 , 𝑦2 )) −
2 𝑦2
−𝑓1 (𝑦1 , 𝑦2 ) (𝛻 2 𝑓2 (𝑦1 , 𝑦2 )sin(𝑓2 (𝑦1 , 𝑦2 )) + (𝛻𝑓2 (𝑦1 , 𝑦2 )) 𝑐𝑜𝑠(𝑓2 (𝑦1 , 𝑦2 )))) (−𝑦 ).
1
𝑦2 (8)
𝑥̇ 2 = (𝛻𝑓1 (𝑦1 , 𝑦2 )sin(𝑓2 (𝑦1 , 𝑦2 )) + 𝛻𝑓2 (𝑦1 , 𝑦2 )𝑓1 (𝑦1 , 𝑦2 )cos(𝑓2 (𝑦1 , 𝑦2 ))) (−𝑦 ) ;
1
−𝑦1
𝑥̈ 2 = (𝛻𝑓1 (𝑦1 , 𝑦2 )sin(𝑓2 (𝑦1 , 𝑦2 )) + 𝛻𝑓2 (𝑦1 , 𝑦2 )𝑓1 (𝑦1 , 𝑦2 )cos(𝑓2 (𝑦1 , 𝑦2 ))) (−𝑦 ) +
2
(𝛻 2 𝑓1 (𝑦1 , 𝑦2 )sin(𝑓2 (𝑦1 , 𝑦2 )) + 2𝛻𝑓1 (𝑦1 , 𝑦2 )𝛻𝑓2 (𝑦1 , 𝑦2 )cos(𝑓2 (𝑦1 , 𝑦2 )) +
2 𝑦2
+𝑓1 (𝑦1 , 𝑦2 ) (𝛻 2 𝑓2 (𝑦1 , 𝑦2 )cos(𝑓2 (𝑦1 , 𝑦2 )) − (𝛻𝑓2 (𝑦1 , 𝑦2 )) 𝑠𝑖𝑛(𝑓2 (𝑦1 , 𝑦2 )))) (−𝑦 ).
1
Equations (6) and (7) define motion of the considered system in the orthogonal cartesian
coordinates by assuming that 𝑦1 and 𝑦2 are given polar coordinates. Nevertheless, one can
use the first equation in (7) and (8) as well as (4) to solve inverse problem for the
considered dual channel dynamical system and define 𝑦𝑖 as function of 𝑥𝑖 .
Such a solution depends on functions 𝑓𝑖 (𝑦1 , 𝑦2 ) and in the most general case can be
defined as follows
𝑦1 = 𝑔11 (𝑥1 , 𝑥̇ 1 ) = 𝑔21 (𝑥2 , 𝑥̇ 2 ); 𝑦2 = 𝑔12 (𝑥1 , 𝑥̇ 1 ) = 𝑔22 (𝑥2 , 𝑥̇ 2 ), (9)
where 𝑔𝑖𝑗 (∙) are nonlinear functions which are inverse to functions 𝑓1 (𝑦1 , 𝑦2 ) and 𝑓2 (𝑦1 , 𝑦2 ).
If one substitutes (9) into the last equations of (7) and (8), he can rewrite these equations
in such a way
𝑥̈ 1 = 𝑞11 (𝑥1 , 𝑥̇ 1 )cos(𝑞12 (𝑥1 , 𝑥̇ 1 )) + 𝑞13 (𝑥1 , 𝑥̇ 1 )𝑐𝑜𝑠(𝑞12 (𝑥1 , 𝑥̇ 1 )); (10)
𝑥̇ 2 = 𝑞21 (𝑥2 , 𝑥̇ 2 )cos(𝑞22 (𝑥2 , 𝑥̇ 2 )) + 𝑞23 (𝑥2 , 𝑥̇ 2 )𝑐𝑜𝑠(𝑞12 (𝑥2 , 𝑥̇ 2 )), (11)
where 𝑞𝑖𝑗 (∙) are some nonlinear functions which are obtained after substituting (9) into (7)
and (8) and performing algebraic simplifications.
The proposed approach allows us to define two independent nonlinear 2nd order
differential equations which allows to define components of acceleration of moved body
which motion is defined in the orthogonal coordinate basis.
It is clearly understood that these components define projections of moved body
acceleration vector and its length can be found in a trivial way
(12)
𝑎 = √𝑥̈ 12 + 𝑥̈ 22 .
One can use (12) to check if the planned motion can be physically implemented and
suitable for the considered body, body speed and position can be found by integrating (10)
and (11).
3. Results and Discussion
3.1. Duffing Pendulum Modeling and Simulating
It is clear that one can use both regular and chaotic dynamical system equation to define
their state variables as polar coordinates in some space plane. Moreover, in case of the use
chaotic systems it becomes possible to design dynamical system with predefined chaotic
attractor.
Let us show the benefits of the usage of our approach by considering well-known Duffing
equation
𝑦1̇ = 𝑦2 ; 𝑦2̇ = −𝑎11 𝑦2 − 𝑎01 𝑦1 − 𝑎03 𝑦13 + 𝑏1 𝑐𝑜𝑠(𝑏2 𝑡), (13)
where 𝑦1 and 𝑦2 are pendulum position and speed, 𝑎𝑖𝑗 are pendulum factors, 𝑏1 and 𝑏2 are
external harmonic signal parameters.
In this paper we study Duffing pendulum with following parameters 𝑎11 = 0.02, 𝑎01 =
1, 𝑎03 = 5, 𝑏1 = 8, 𝑏2 = 0.5. Also, we assume zero initial conditions for the considered
motion. Numerical solution of the Duffing equation (13) is shown in Figure 6.
a) System time response b) System phase portrait
Figure 6: Numerical solution of (13).
Analysis of Figure 6 proves that under some conditions Duffing pendulum can be
considered as chaotic dynamical system with external harmonic excitation. Figure 6 makes
the initial base to compare with novel results. Due to the features of attractor in Figure 6b
we call it as infinity-like attractor. It necessary to say that the Duffing pendulum is
considered because of this is a one of simplest second order chaotic systems. Nevertheless,
one can use any chaotic system as the core to perform the above-given transformations.
3.2. Duffing Pendulum Dynamic in Polar Coordinates
3.2.1. The Simplest Polar to Cartesian Coordinate Transformations
At first, we consider the simplest case of the proposed approach use. In this case we assume
that core dynamical system is defined as (13) and transformation expressions given by (2).
To study the effect of interpreting system state variables we consider both cases
𝜌 = 𝑦1 ; 𝜑 = 𝑦2 (14)
and
𝜌 = 𝑦2 ; 𝜑 = 𝑦1 . (15)
Simulation results for the dynamical system (13) with transformation (2) and variables
(14) are shown in Figure 7 and Figure 8 illustrates results of numerical solution of (13) and
(2) with variables (15).
a) System time response b) System phase portrait
Figure 7: Numerical solution of (13) with transformations (2) and variables (14).
a) System time response b) System phase portrait
Figure 8: Numerical solution of (13) with transformations (4) and variables (15).
Analysis of given in Figure 7 and Figure 8 simulation results allows us to claim that
interpretation of Duffing pendulum state variables as variables in polar coordinates with
following transformation into orthogonal coordinates allows us to dramatically change
Duffing pendulum attractor and form of its oscillations. Moreover, the use 𝑦1 as pendulum
linear position and 𝑦2 as its angular position gives us the possibility to increase the
nonlinearity of oscillations and make them more complex. At the same time, the use of 𝑦2 as
pendulum linear position and y1 as its linear one makes pendulum oscillation less nonlinear.
In both cases pendulum attractors differ from the initial one. Furthermore, the assuming
that pendulum variables are defined in polar coordinates avoid occurring of dual scroll
attractors and produce novel attractors for this dynamical system which we call as eye-like
for given in Figure 7b attractor and clover-like attractor for attractor which is given in
Figure 8b.
3.2.2. The Affine Polar to Cartesian Coordinate Transformations
Now we make transformations (14) and (15) more complex and assume that the pendulum
linear position in polar coordinates is defined as linear combination of Duffing pendulum
state variables and its angular position is unchanged
𝜌 = 𝑦1 + 𝑦2 ; 𝜑 = 𝑦2 (16)
and
𝜌 = 𝑦1 + 𝑦2 ; 𝜑 = 𝑦1 . (17)
One can find simulation results for dynamical systems which are described by (13), (16)
and (13), (17) in Figure 9 and Figure 10.
a) System time response b) System phase portrait
Figure 9: Numerical solution of (13) with transformations (2) and variables (16).
a) System time response b) System phase portrait
Figure 10: Numerical solution of (13) with transformations (4) and variables (17).
In Figure 11 and Figure 12 we show simulation results for the case when linear
pendulum position is defined by only one pendulum state variable and its angular position
is sum of pendulum state variables. In other words, following transformations are used
𝜌 = 𝑦1 ; 𝜑 = 𝑦1 + 𝑦2 (18)
and
𝜌 = 𝑦2 ; 𝜑 = 𝑦1 + 𝑦2 . (19)
a) System time response b) System phase portrait
Figure 11: Numerical solution of (13) with transformations (4) and variables (18).
a) System time response b) System phase portrait
Figure 12: Numerical solution of (13) with transformations (4) and variables (19).
Analysis of given in Figure 9 – Figure 12 simulation results shows that if one defines
linear combinations of Duffing pendulum state variables and then use the obtained
combinations as initial signals for transformation (2) it becomes possible to change system
attractors in tremendous ways.
It is necessary to say that usage of linear combinations of Duffing pendulum state
variables also allows us to form hidden attractors. One of such attractors (Figure 13) can be
obtained if one uses following transformation which is the combination of the two previous
ones
𝜌 = 𝑦1 + 𝑦2 ; 𝜑 = 𝑦1 + 𝑦2 . (20)
The main feature of the shown in Figure 13 system that in polar coordinates its linear
and angular positions equal each other.
a) System time response b) System phase portrait
Figure 13: Numerical solution of (13) with transformations (4) and variables (20).
Such an attractor is formed because of the quite stable amplitude of chaotic oscillations
in the considered system. Contrary to the above-considered systems the last one forms
motion trajectories which quite close each other but motions along these trajectories
becomes according to unpredictive laws.
Similar results can be obtained in special cases when one of pendulum positions is
assumed as constant (Figure 14 and Figure 15)
𝜌 = 𝑟; 𝜑 = 𝑦1 + 𝑦2 ; (21)
𝜌 = 𝑦1 + 𝑦2 ; 𝜑 = 𝑓 . (22)
a) System time response b) System phase portrait
Figure 14: Numerical solution of (13) with transformations (4) and variables (21).
a) System time response b) System phase portrait
Figure 15: Numerical solution of (13) with transformations (4) and variables (22).
Analysis of Figure 13 – Figure 15 proves the possibility of design chaotic systems with
hidden attractors by using Duffing equations.
3.3. Duffing Pendulum-based Chaotic Dynamical System Design and Implementation
We design all above-considered chaotic systems by taking into account the given in previous
section approach which is based on the taking into account (13) and differentiate (2) for
two times for each linear and angular pendulum position in the polar coordinates.
Now let us show the system design which is based on (13), (2).
We show such a design by differentiating first expression in (2)
𝑥̇ 1 = 𝑦2 cos(𝑦2 ) − 𝑦1 (𝑏1 cos(𝑏2 𝑡) − 𝑎11 𝑦2 − 𝑎01 𝑦1 − 𝑎03 𝑦13 ) sin(𝑦2 ). (23)
It is clear that both of expressions (2) and (23) are nonlinear for state variables 𝑦𝑖 . That
is why, we use numerical methods to solve them
𝑥1 (24)
𝑦1 = ;
𝑐𝑜𝑠(𝑦2 )
𝑥1 𝑥3
𝑥̇ 1 −𝑦2 cos (𝑦2 )+𝑥1 (𝑏1 cos(𝑏2 𝑡)−𝑎11 𝑦2𝑖−1 −𝑎01 −𝑎03 31(𝑦 )) 𝑡𝑎𝑛(𝑦2 )
𝑐𝑜𝑠(𝑦2 ) 𝑐𝑜𝑠 2
𝑦2 = 𝑦2 − 3 .
𝑎01 𝑥1 𝑠𝑖𝑛(𝑦2 ) 3𝑎03 𝑥1 𝑠𝑖𝑛(𝑦2 )
𝑦2 𝑠𝑖𝑛(𝑦2 )−𝑐𝑜𝑠(𝑦2 )−𝑥1 (𝑎11 + + ) 𝑡𝑎𝑛(𝑦2 )−
𝑐𝑜𝑠2 (𝑦2 ) 𝑐𝑜𝑠4(𝑦2 )
𝑎 𝑥 𝑎 𝑥3
−𝑥1 (𝑎11 𝑦2 + 01 1)− 03 1 )(1+𝑡𝑎𝑛2 (𝑦 ))
2
𝑐𝑜𝑠(𝑦2 𝑐𝑜𝑠3 (𝑦2 )
Then we differentiate (23) one more time
..
𝑥1 = −𝑦1 cos (𝑏2 𝑡)2 cos (𝑦2 )𝑏12 + 𝑏1 ((2𝑎03 𝑦14 + 2𝑎01 𝑦12 + 2𝑎11 𝑦1 𝑦2 + 1)cos (𝑦2 + (25)
+sin (𝑦2 )(𝑎11 𝑦1 − 2𝑦2 ))cos (𝑏2 𝑡) + 𝑦1 sin (𝑦2 )sin (𝑏2 𝑡)𝑏1 𝑏2 −
−(𝑎03 𝑦14 + 𝑎01 𝑦12 + 𝑎11 𝑦1 𝑦2 + 1)(𝑎03 𝑦13 + 𝑎01 𝑦1 + 𝑎11 𝑦2 ) cos(𝑦2 ) − sin(𝑦2 ) ×
× (𝑎03 𝑎11 𝑦14 − 5𝑎03 𝑦2 𝑦13 + 𝑎01 𝑎11 𝑦12 + 𝑎11
2
𝑦1 𝑦2 − 3𝑎01 𝑦1 − 2𝑎11 𝑦22 ).
and substitute into the second derivative (25) values of 𝑦1 and 𝑦2 from (24).
Other equations are defined in the similar way. We offer to use numerical approximation
of derivative operator with feedback differences
𝑑 1−𝑧 −1 (26)
𝑑𝑡
≈ 𝑧 −1 𝑇
,
−1
where 𝑧 is a shift operator and 𝑇 is a discretization time to implement (25) and similar
equations in the digital devices.
To perform transformation of (25) in discrete-time form, let us rewrite it as follows
𝑥̇ 11 = 𝑥12 ; (27)
2 2 4 2
𝑥̇ 12 = −𝑦1 cos (𝑏2 𝑡) cos (𝑦2 )𝑏1 + 𝑏1 ((2𝑎03 𝑦1 + 2𝑎01 𝑦1 + 2𝑎11 𝑦1 𝑦2 + 1) cos(𝑦2 ))
+sin (𝑦2 )(𝑎11 𝑦1 − 2𝑦2 ))cos (𝑏2 𝑡) + 𝑦1 sin (𝑦2 )sin (𝑏2 𝑡)𝑏1 𝑏2
− (𝑎03 𝑦14 + 𝑎01 𝑦12 + 𝑎11 𝑦1 𝑦2 + 1)(𝑎03 𝑦13 + 𝑎01 𝑦1
+ 𝑎11 𝑦2 ) cos(𝑦2 )
− sin(𝑦2 ) (𝑎03 𝑎11 𝑦14 − 5𝑎03 𝑦2 𝑦13 + 𝑎01 𝑎11 𝑦12 + 3𝑦2 𝑎11
2
𝑦1
2)
− 𝑦2 𝑎01 𝑦1 − 2𝑎11 𝑦2
and apply (26)
𝑥11 = 𝑧 −1 𝑥11 + 𝑇𝑧 −1 𝑥12 ; (28)
𝑥12 = 𝑧 𝑥12 + 𝑧 𝑇(𝑏1 (()2𝑎03 𝑦14 + 2𝑎01 𝑦12 + 2𝑎11 𝑦1 𝑦2 + 1)cos (𝑦2 ) −
−1 −1
𝑦1 cos (𝑏2 𝑡)2 cos (𝑦2 )𝑏12 + +sin (𝑦2 )(𝑎11 𝑦1 − 2𝑦2 ))cos (𝑏2 𝑡) +
𝑦1 sin (𝑦2 )sin (𝑏2 𝑡)𝑏1 𝑏2 − (𝑎03 𝑦14 + 𝑎01 𝑦12 + 𝑎11 𝑦1 𝑦2 + 1)(𝑎03 𝑦13 + 𝑎01 𝑦1 +
𝑎11 𝑦2 )cos (𝑦2 ) − sin (𝑦2 )(𝑎03 𝑎11 𝑦14 − 5𝑎03 𝑦2 𝑦13 + 𝑎01 𝑎11 𝑦12 + 3𝑦2 𝑎11
2
𝑦1 −
2
𝑦2 𝑎01 𝑦1 − 2𝑎11 𝑦2 )).
One can use modern CPU/MCU/FPGA devices to implement the designed chaotic system.
4. Conclusions
The interpreting of dynamical system state variables as coordinates in some coordinate
system which is different from the orthogonal cartesian one gives us the possibility to define
novel dynamical systems with novel oscillations and attractors. One can transform these
attractors into a conventional orthogonal cartesian coordinate system by using various
transformation expressions. The use of these transformations allows us to define a
multichannel dynamical system and bind each channel with corresponding coordinates to
define body position in the plane for each time moment. Combining these moments gives us
the possibility to define the desired trajectory for a considered body. Different
transformation allows us to determine different paths for a vehicle and use them in various
applications.
One can use both regular and chaotic systems as core systems to implement our
approach. In the case of chaotic system usage, one can get various chaotic systems, in which
attractors have features that can be useful in one or another application.
The above-considered systems are designed and studied by using linear combinations of
Duffing pendulum state variables. Even in this case, the transformed system is defined with
quite complex equations due to the trigonometric function use and system nonlinearity.
This fact requires to use of numerical methods to complete the coordinate transformation.
We believe that replacing nonlinear functions with piecewise linear ones makes it possible
to simplify transformation from one coordinate system to another and avoid the use of
numerical methods. We see the check of this hypothesis as one of the future developments
of our work. One more possible problem to solve by using the proposed approach is
studying dynamical systems with nonlinear combinations of their state variables. Also, one
can use the proposed approach to design multidimensional dynamical systems that have
attractors in some N-th dimensional coordinate systems.
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