Flabio Dario Mirelez Delgado, Elena Elsa Bricio Barrios, Santiago Arceo
Díaz
buoys anchored at strategic points in the breeding tank and scaffolding with buckets that
release the food throughout the structure [11].
Having a mobile vehicle offers the advantage of programming a route that allows food to
be dispersed in as extensive an area as possible, reducing problems such as aggressive
competition between specimens to appropriate areas where excess food accumulates at
the bottom of the pond. Since tilapias tend to consume food until it settles to the bottom,
this strategy helps to avoid problems such as poisoning of specimens and the formation
of harmful compounds associated with the decomposition of uneaten food, reducing the
appearance of diseases.
This approach can reduce the investment cost of the microprocessor, sensor and
microcontroller required to carry out a predefined route, thus addressing the problem
from an analytical perspective. Therefore, this work proposes the design of power routes
based on the Fourier series to calculate the speed and acceleration of the motors,
minimizing the cost of developing these autonomous systems. Unlike other numerical
strategies, Fourier series have been characterized by their ability to: i) decompose the
trajectory into a series of sine and cosine functions, ii) create a smooth and safe trajectory
if a control point is decided, and iii) allow calculating the speed and acceleration of the
vehicle in real time for the incorporation of control schemes [12].
Materials and methods
Case study:
This proposal was developed in two stages: the identification of the needs of the
aquaculturists at the study site and the computational development of a control algorithm
that could be implemented in an autonomous aquatic vehicle.
In the first stage, data were collected corresponding to the tilapia breeding tanks in the
“San Buenaventura” aquaculture farm, located in the Municipality of Armería, Colima.
The identification of needs was developed through unstructured interviews with
aquaculturists and aquaculture farm operators, where emphasis was placed on the need
for an automated feeding system for tilapia in their juvenile and adult stages. These
specimens are raised in artificial ponds in an area of 50 by 100 meters and a depth of
1.6 meters [13]. In these ponds it is common for support personnel to enter the body of
water and disperse the feed by spraying the feed as it moves along the pond. However,
the social dynamics of tilapia are based on the survival of the fittest, which is reflected
in their aggressive behavior when feeding, preventing other tilapia from accessing food
through attacks on the fins, causing the premature death of others. specimens and, in
some cases, by not accessing food, the tilapia report weights below the average [8].
To improve the degree of food dispersion, a scheme was designed in which the vehicle
moves to specific points and, after stopping, the food is released (see Figure 1). This
proposal was reported by Rodríguez in [11] who proposed the design of an autonomous
land vehicle that moved between beams that crossed the rearing pond.
Fourier series in the planning and optimization of trajectories of autonomous aquatic vehicles
The second stage, was based on the definition and delimitation of the case study, was
carried out in the facilities of the Tecnológico Nacional campus Colima and in the facilities
of the National Polytechnic Institute. All computational development was coded in the
MATLAB 2023b programming language.
The creation of an autonomous vehicle requires the application of the principles of
mobile robotics. A classic example that can be used as a basis is the creation of a twowheeled differential robot with a center of mass at the center of its axle and a passive
wheel has a Cartesian configuration given by (see Figure 2), where are the generalized
Cartesian coordinates, and are the coordinates of the center of mass of the robot and is
the orientation angle of the robot, so its kinematic model can be expressed as shown in
equation (1) reported [14].ξ=[x y θ]ξxyθ
Flabio Dario Mirelez Delgado, Elena Elsa Bricio Barrios, Santiago Arceo
Díaz
where is the linear speed of the mobile robot and ω is the angular speed, and the nonholonomic constraint (−) is also considered. The definition for both speeds is given by
equations (2) and (3):Vx cos ( ̇ θ)y cos ( ̇ θ)=0
where and are the angular velocities of the left and right wheel respectively, is the wheel
radius and corresponds to the distance between wheels through the axle that joins them.
With the previous equations, a more complete kinematic model can be obtained, as
shown in equation (4):ψ ̇_1 ψ ̇_2 r_ω 2R
This representation is useful for a boat-type water vehicle, since the linear and angular
velocities can be used as inputs for a control system or the angular velocities if the vehicle
had wheels with blades on its sides. In any scenario it is feasible to modify the position of
the device within a body of water using two control inputs.
Parametric function:
This design was reformulated by Hernández in [15], who proposed a homogeneous
distribution of feeding at specific points in the breeding pond, taking into consideration the
existence of aeration systems located near the center of the pond that fulfill the function
of oxygenating the body of the fish. water, promoting suitable conditions for the growth of
tilapia [16]. Although Hernández in [15] reports that the tracers are capable of describing
a feeding route, the first and second derivative of the proposed parametric function
diverges, that is, this system requires redefinition if this proposal is to be implemented in
an aquatic vehicle. This work addressed this problem using Fourier series.
Trajectory planning by Fourier series:
In general, any closed trajectory can be represented mathematically by a periodic function,
that is, a function that repeats itself at regular intervals of its independent variable, called
“periods”. However, in many cases the periodic functions that describe these trajectories
may have characteristics that make the calculation of the first and second derivative difficult [17].
Fourier series in the planning and optimization of trajectories of autonomous aquatic vehicles
In this aspect, Fourier series play a crucial role in approximating periodic functions,
since they allow a function to be efficiently decomposed into an infinite combination
of sines and cosines. This representation significantly simplifies the calculation of the
first and second derivative, by expressing the periodic function in terms of known
trigonometric functions, whose derivatives consist only of sums of sines and cosines,
facilitating mathematical analysis and optimizing calculations. By using Fourier series, a
more manageable representation of the original function is achieved, which simplifies the
differentiation process and improves analysis efficiency. The structure of a Fourier series
is described below [17].
A function is considered periodic if there exists a positive number, such that for each
value of the independent variable, , in the domain of it holds that:f(t)Ttf(t)
f(t)=f(t+nT), (5)
where is any non-zero integer and the number corresponds to the period of the function
and represents the length of one complete repetition cycle.nT
Fourier series [18] propose that any periodic function can be represented as an infinite
series:
where the sinusoidal functions represent harmonic oscillations with frequencies integer
multiples of the fundamental frequency (proportional to). The coefficients1/T
Flabio Dario Mirelez Delgado, Elena Elsa Bricio Barrios, Santiago Arceo
Díaz
determine the size of the contribution that each sinusoidal function has in the series and
the fundamental frequency is defined as.ω_0=2π/T
Results and discussion
Creation of tracking trajectories based on parametric equations:
Once we have the routes or trajectories that the device or vehicle must travel (see section
“parametric function”), it was found that the proposed trajectory can be approximated by
a Fourier series of five terms, as well as its transitive dynamics in the Figure 3.
After obtaining the parameterized trajectories as well as their derivatives, the next step
is to select a control law that can be applied to the mobile robot to precisely follow the
change of positions in the work area, that is, that the robot can stick to the route. To
achieve the above, desired linear (12) and angular (13) speeds are established and, in this
way, establish an error to be minimized between the real speeds of the prototype and the
desired ones according to the trajectory to be fulfilled [19].
From the desired speeds that are a function of the desired positions, the control speeds
or the control law can be established that allows position errors to be minimized and
therefore follow the established routes [20], as can be seen in (14) and (15).
En general, los docentes manifiestan que este tipo de prácticas pedagógicas despiertan
motivación en los estudiantes hacia el aprendizaje de las matemáticas y su utilización en
otras disciplinas. Refieren a las capacidades de sus estudiantes como a ese conjunto de
conocimientos, habilidades, actitudes y competencias que desarrollan dentro del proceso
de trabajo cooperativo con el propósito de la toma de decisiones en la primera actividad
y la generación y valoración de oportunidades para encontrar soluciones en la segunda.
La experiencia permitió visualizar el compromiso con el que abordan los estudiantes
las actividades propuestas y la generación de un ambiente de aula con implementación
de aprendizaje cooperativo que favorece la relación entre docentes y estudiantes
generando alternativas de trabajo en búsqueda de lograr un aprendizaje significativo de
las matemáticas.
The methodology used to simulate the behavior of a surface aquatic robot on a body of
water, including the desired trajectories, the kinematic model and the control law, was built
in Simulink. Figure 4 shows the general diagram in Simulink as well as the connections
between the main blocks (path generator, control and kinematic model).
Following the methodology proposed by Chin in [2], a virtual environment generated in
Simulink was built, an aquatic mobile robot was placed with initial conditions (10 m x 5 m)
inside a simulated pond of 50 m x 50 m represented in Figure 5.
Figure 6 reports the performance of the trajectory tracking in the x and y direction, as
well as its errors in the orientation of the vehicle (). At first the magnitude of the errors is
considerable but asymptotically they are minimized, although the error is not cancelled.
This remains within a range considered acceptable.θ
Conclusions
This work reports, from a numerical simulation perspective, the use of mathematical
strategies to define, carry out and optimize food distribution routes inside a tilapia
breeding tank, significantly reducing the cost of electronic components of an autonomous
unmanned vehicle.
It is important to note that, for the selected parametric function (lenmiscata), it was
necessary to couple a Fourier series to the tracers that would allow obtaining expressions
of the first and second derivatives that are input variables in the kinematic and correction
action model. The main advantage of using the Fourier series is that it can represent any
function, therefore, any trajectory that you want to follow in the plane can have its first
and second derivative and be applied to this proposal.
As future work, it is proposed to add natural disturbances such as air currents, irregular
air flows generated by aeration systems and even the presence of native fauna in the area.
Likewise, it is proposed to extend this proposal by allowing the vehicle to be programmed
based on the growth stage of the specimen, which will allow better nutrition and thus
gain in weight and size.
Acknowledgments:
The authors thank the staff of the Department of Mathematics and Statistics and for
the Master’s Degree in Mathematics Education of the Francisco de Paula Santander
University, which allowed us to share the conference given at the XIX International Meeting
of Applied Mathematics and XIV Meeting of Statistics, based in Cucuta, Colombia.
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