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Abstract

Robots with complex motion paths are very rarely designed. The main obstacle is the lack of the necessary mathematical apparatus, despite the fact that the theory was proposed by Newton. We managed to formulate a new method for obtaining linear transformation parameters. It can be used for ellipse, hyperbola, as well as for other complex flat differentiable curves. A theorem for receiving the values of a transformed curve is formulated in the general case. The theoretical algorithm and the results of experimental studies using the geometric modeling method are presented. The proposed method works for all test curves, but this does not mean that it can always be used. Separation of the characteristic equation imposes additional complexity. Additional research is necessary, but it can be applied to many mechatronic frameworks now.

Keywords

Trajectory robot complex plane curve nonorthogonal basis symmetries

Introduction

Kinematics pairs with simple trajectories of motion are the basis for the design of robots, including at the present time. However, even Newton proposed to design mechatronic systems with complex trajectories of motion. Designing a robot with a complex trajectory of motion is possible due to approximation methods, but the accuracy of movement is lost. Engineering calculations are based on the movement of the point. If the motion is simple, the developer gets a simple dependence on the functions (hyperbolic) sine and cosine of one argument. The trajectory of motion is an ellipse (hyperbola). Both curves belong to the class of centrally symmetric conic sections. The solution of the resulting dependence was found in 17^th centure.¹ To be fair, it should be noted that the final calculation methods were completed in the middle of the last century.²

The ideal status is to achieve gait moving of robot will be similar to gait pattern of human or others living organism.^3,4 Gait pattern is a unique dynamic stereotype represented by gait parameters. Description of modeling and control of robot with the shape of two link snake with using of geometric mechanics at modeling mixed mechanical locomotion system is described by Tlach et al.⁵

If the resultant system depends on a double parameter or includes the product (partial) of trigonometric functions, then an analytical solution has not been found so far. Approximating methods are used for calculations. They make it possible to find a solution, but technical meanings are lost. Various tests of product parameters are required for the final design. Obviously, if the engineer received an analytical solution in this case, the time of manufacture and production is decreased significantly.

The algorithms and methods of approximation are studding in the present for robots moving calculations. New approaches are proposed.^6

–10 The primary character of all methods is that they have an error. It is obvious that the design will mostly satisfy the necessary conditions if the accuracy of the calculations is ideal and the product precession will depend on manufacturing admittance only.

Let us try to find an analytical solution for complex motions using a new geometric theory.¹¹

Nonclassic method of linear transformations

The task of developing a mechatronic system by a complex motion trajectory was formulated by Newton.⁴ He defined the concept of a canonical equation. For example, for conical sections, it has the form $A x^{2} + 2 B x y + C y^{2} + D x + E y + F = 0$ . A solution for centrally symmetric sections was obtained, but the calculation of the parameters for the parabola required considerable time. The Klein invariant allowed this to be done.^2,12,13 The more high class of curves is designed by the equation $A x^{3} + 3 B x^{2} y + 3 C x y^{2} + D y^{3} + 3 E x^{2} + 6 F x y + 3 G y^{2} + 3 H x + 3 K y + L = 0$ . A finding algorithm for the parameters of the curve belonging to this class has not been found so far. But Newton classification has higher degrees. Algebraic and differential geometry is currently studying curves. The curve determined by the parametric system of equations is accepted by these geometries. We combine this representation with the Dieudonne theory of symmetries as a property of the Euclidean plane.¹¹

Let there be an trajectory by figure $Φ$ —planar differentiable curve in the Euclidean plane $R \times R$ . Curve is defined in a Cartesian coordinate linear system by the parametric equations: $\{\begin{matrix} x = k_{x} f_{x} (t) \\ y = k_{y} f_{y} (t) \end{matrix}$ , where $x, y, t, k_{x}, k_{y} \in R$ . A parametric system of equations $\{\begin{matrix} x = k_{x} t \\ y = k_{y} t \end{matrix}$ cannot be used. She defines a line, thus, functions $f_{x} (t) \neq t$ and $f_{y} (t) \neq t$ at the same time. We can compose a system of parametric equations $\{\begin{matrix} x = t \\ y = k_{y} f_{y} (t) \end{matrix}$ or $\{\begin{matrix} x = k_{x} f_{x} (t) \\ y = t \end{matrix}$ , that is, solve any canonical equation from Newton’s classification. We carry out any transformation of $Φ$ by the matrix $(\begin{matrix} a & h \\ g & b \end{matrix})$ , where $a, b, h, g \in R$ . It is necessary to obtain the parameters of the transformed figure. In others words, we must calculate of the characteristic equation $T \vec{v} = λ \vec{v}$ , where $T$ is the transformation matrix, $λ$ refers scalars, $\vec{v}$ is a vector.

Dieudonne considered three types of symmetries on the plane: the unitary matrix $(\begin{matrix} 1 & 0 \\ 0 & 1 \end{matrix})$ , the mirror matrixes $(\begin{matrix} - 1 & 0 \\ 0 & 1 \end{matrix})$ and $(\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix})$ , permutations $(\begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix})$ . The unitary matrix does not change the Euclidean plane. Mirror symmetries determine the direction of the numerical axes. The permutation matrix has not been studied in detail previously. Let us look at a classic orthogonal basis (Figure 1). Line AB defines the action of permutation symmetry. Then four non-orthogonal basis $0 e_{1} {e^{'}}_{1}$ , $0 e_{1} {e^{'}}_{2}$ , $0 e_{2} {e^{'}}_{1}$ , and $0 e_{2} {e^{'}}_{2}$ exist for an orthogonal basis $0 e_{1} e_{2}$ . The basis $0 e_{1} {e^{'}}_{1}$ is the most important, since it defines the essence of the permutation symmetry vividly.¹¹

Figure 1.

Nonorthogonal basis.

This basis determined the new direct method

\{\begin{matrix} c f_{x} (t + α) = a f_{x} (t + β) + h f_{y} (t + β) \\ d f_{y} (t + α) = g f_{x} (t + β) + b f_{y} (t + β) \end{matrix},

where $β$ is the angle of permutation symmetry and α is its own angle. The solution method for centrally symmetric conic sections was published earlier in this journal,^14,15,16 so we will not dwell on it.

Linear transformations of complex curves

Theoretical studies of the existence region of the angle $β$ made it possible to find four types of special linear transformations¹⁵: $\{\begin{array}{l} x = m f_{x} + n f_{y} \\ y = m f_{x} - n f_{y} \end{array}$ , $\{\begin{array}{l} x = - m f_{x} + n f_{y} \\ y = m f_{x} + n f_{y} \end{array}$ , $\{\begin{array}{l} x = m f_{x} + k n f_{y} \\ y = - n f_{x} + k m f_{y} \end{array}$ , and $\{\begin{array}{l} x = k m f_{x} + n f_{y} \\ y = - k n f_{x} + m f_{y} \end{array}$ . Computer modeling that revealed formulas for conic sections can be used for any Jordan curves. The angle is $β \in {0, \pm π / 2, \pm π}$ for this system.

The characteristic equation $T \vec{v} = λ \vec{v}$ is used in many fields of science, such as physics, economics, mechanics, cryptography, and so on. The result of the conversion depends on the permutation and mirror symmetries and vector may be others by type after transformation as write Coolidge in the literature.¹⁶

Theorem

A transformation $T = (\begin{matrix} a & h \\ g & b \end{matrix})$ , where $det T \neq 0$ , $a, b, g, h \in R$ , can modify the characteristic equation $T (\begin{matrix} f_{x} (t) \\ f_{y} (t) \end{matrix}) = (\begin{matrix} λ_{x} f_{x} (t) \\ λ_{y} f_{y} (t) \end{matrix})$ to equation $T (\begin{matrix} f_{x} (t) \\ f_{y} (t) \end{matrix}) = (\begin{matrix} λ_{x} f_{y} (t) \\ λ_{y} f_{x} (t) \end{matrix})$ .

Proof

Let us prove the reverse. Let us suppose a transformation $T$ does not exist. Let us consider a curve with parametrical equations system $\{\begin{matrix} x = f_{x} (t) \\ y = f_{y} (t) \end{matrix}$ and employ a linear transformation $(\begin{matrix} 0 & m \\ n & 0 \end{matrix})$ to it, where $n, m \neq 0$ , then the resulting trivial solution of the characteristic equation will be equal $\{\begin{matrix} x^{'} = m f_{y} (t) \\ y^{'} = n f_{x} (t) \end{matrix}$ , $det T = - n k \neq 0$ . The vector has changed its type.

Any singular transformation obeys a permutation symmetry as well as a non-singular conversion, since the resultant parametric system has the form $\{\begin{matrix} x^{'} = k f^{'} (t) \\ y^{'} = k f^{'} (t) \end{matrix}$ .¹⁴

The trivial solution of the characteristic equation exists for scaling, axial compression, and rotation. The above systems did not differ much from simple transformations. The orthogonal basis does not change after all the listed conversions.

Various hypotheses have been put forward about changing the system (1) for complex curves, including:

The appearance of an independent angle $ξ$ of the rearrangement symmetry when the parameter $n t$ and $n = 2$ ;

Change the record of the view function $G : f (n t) \to f_{1}^{n} (t)$ ;

If there is $f_{x} (n t)$ , $f_{y} (k t)$ , $n < k$ , that is $f_{x} (n t) \times f_{x} (l τ)$ , where $l = k - n$ .

All hypotheses are refuted theoretically.¹¹

The transformation of shear $(\begin{matrix} 1 & h \\ 0 & 1 \end{matrix})$ or $(\begin{matrix} 1 & 0 \\ g & 1 \end{matrix})$ introduces asymmetry into the transformed image. The number of Weyl symmetries decreases. Further studies are based on this type of linear transformation.

Computational experiments were carried out using the method of geometric modeling. The program algorithm is quite simple.

The requested transformation parameters.

We output curve(s) will transform every part of the line of it.

Output is black.

Forward transformation parameters.

Output a new curve(s) with parameters obtained in yellow.

The experiments were carried out on plane curves with at least two symmetries C_i or D_i (Weyl symmetries), in particular, the Jeromeau lemniscate, Pascal’s cochlea, various epicycloids and hypocycloids, and so on.

To solve this problem it was assumed that for the Jordan curves, in addition to the method of finding the parameters of a linear transformation, it is necessary to consider the inverse transformation from a local nonorthogonal linear coordinate system to an orthogonal one. Let us consider that the angles of the vectors of the basis have the values α and $β$ , then for the return of the point to the rectangular linear coordinate system it is necessary to make the transformation $\frac{P}{det P} = \frac{1}{cos α sin β + sin α cos β} (\begin{matrix} cos α & sin α \\ - sin β & cos β \end{matrix})$ .¹⁷ This conversion reduces the number of symmetries in the resulting image, so this hypothesis is preferable for obtaining analytical dependencies.

The large number of alternatives that needs to be considered is the biggest problem for subsequent research.

The first condition for the multiplicity of solutions is the location of the basis. The classical method can have its own angle α of inclination in the range $[- π / 2, π / 2]$ and the proposed method of the whole plane. Therefore, the exact direction of the vectors of the frame is also unknown and can be different.

The second manifold is determined by the form of the characteristic equation. The form of the characteristic equation has four variants. The first and second conditions contain the same situations, therefore the number of actual variants does not exceed 10. The angle $β$ does not belong to the set $\{0, \pm π, \pm π / 2\}$ for the considered variants, therefore it is impossible to apply the results obtained earlier.

We assume that the central angle of a nonorthogonal basis can be different and depends on the type of transformation shear by X $(\begin{matrix} 1 & 0 \\ g & 1 \end{matrix})$ or shear by Y $(\begin{matrix} 1 & h \\ 0 & 1 \end{matrix})$ . The shape of the curve is determined by the value of the parameter $g (h)$ , so it is necessary to consider the following cases: transformations $g, h \geq 1$ , $g, h \leq - 1$ , $g, h < 1$ , and $g, h > - 1$ . The shear transformation is not orthogonal, so it is suitable for an experiment for an arbitrary transformation.

A total of 256 options were considered. The results of the experiments and two small trivial theorems have made it possible to reduce the number of variants to 16. Computational experiments were carried out. They made it possible to find a non-projective solution for extracting the analytic formula of the transformed plane differentiable curve.

General solution algorithm

Lemma

Each arbitrary nonsingular matrix $T = (\begin{matrix} a & h \\ g & b \end{matrix})$ can be represented by the product of a symmetry and compression matrix $T = (\begin{matrix} 1 & h / b \\ g / a & 1 \end{matrix}) (\begin{matrix} a & 0 \\ 0 & b \end{matrix})$ .

These statements allowed finding a method of linear transformations for complex curves.

Let us split the transformation matrix into the product of two matrices: $(\begin{matrix} a & 0 \\ 0 & b \end{matrix})$ and $(\begin{matrix} 1 & h / b \\ g / a & 1 \end{matrix})$ .

Let us find the transformation $(\begin{matrix} 1 & h / b \\ g / a & 1 \end{matrix}) (\begin{matrix} f_{x} (t) \\ f_{y} (t) \end{matrix})$ parameters as a centrally symmetric conic section: $β$ is angle of permutation symmetry and α is its own angle.

Let us consider a new curve description system: $\{\begin{matrix} x = a k_{x} f_{x} (t) \\ y = b k_{y} f_{y} (t) \end{matrix}$ .

Let us perform an additional transformation $P$ depending on the angle $β$ . If the angle $β$ is negative, then the transformation is $P = (\begin{matrix} cos β & - sin β \\ cos α & sin α \end{matrix})$ or $P = (\begin{matrix} cos β & sin β \\ cos α & - sin α \end{matrix})$ ; if positive, then $P = (\begin{matrix} - cos β & sin β \\ cos α & sin α \end{matrix})$ or $P = (\begin{matrix} cos β & sin β \\ - cos α & sin α \end{matrix})$ . This is a transition from a nonorthogonal basis to an orthogonal basis.

Let us transform $(\begin{matrix} - 1 & 0 \\ 0 & 1 \end{matrix})$ for the parametric system of equations of the trajectory which has the form $\{\begin{matrix} x = k_{x} f_{x} (t) \\ y = k_{y} t \end{matrix}$ and transformation $(\begin{matrix} 1 & 0 \\ 0 & - 1 \end{matrix})$ if the system has an alternative description. This step of the algorithm is not complete, as it is necessary to take into account the characteristic equation separation. Additional research works are needed. However, the algorithm should be correct for most of the developed robots.

Let us rotate the curve by its own angle α and multiply it by the scalars ${k^{'}}_{x}$ and ${k^{'}}_{y}$ from step 2.

The analytical formula of the transformed complex curve is found as $\{\begin{matrix} x = {k^{'}}_{x} f_{x} (t + α) \\ y = - {k^{'}}_{y} f_{y} (t + α) \end{matrix}$ .

The lemma does not hold for transformation groups $T_{1} = (\begin{matrix} 0 & h \\ g & b \end{matrix})$ and $T_{2} = (\begin{matrix} a & h \\ g & 0 \end{matrix})$ . Let us transform the matrix $T_{1}$ and $T_{2}$ using permutation symmetry ${T^{'}}_{1} = (\begin{matrix} h & 0 \\ b & g \end{matrix})$ and ${T^{'}}_{2} = (\begin{matrix} h & a \\ 0 & g \end{matrix})$ , then the solution will be found, but the resulting system will take the form $\{\begin{matrix} x = - {k^{'}}_{y} f_{y} (t + α) \\ y = {k^{'}}_{x} f_{x} (t + α) \end{matrix}$ .

Let’s consider an example. Let the Jeromeau lemniscate (Figure 2) be transformed with a matrix $(\begin{matrix} 1 & h \\ 0 & 1 \end{matrix})$ , where $h > 0$ , then the analytic representation of the system of equations will be $\{\begin{matrix} x = c^{'} (- cos β cos (t + α) + sin β cos (t + α) sin (t + α)) \\ y = d^{'} (cos α cos (t + α) + sin α cos (t + α) sin (t + α)) \end{matrix}$ , where $c^{'} = c / (1 - h)$ , $d^{'} = d / (1 - h)$ .

Figure 2.
Jeromeau lemniscate.

The matrices of the transition from the basis to the basis coincide with the classical matrix, but the rows in the matrix are permutated. In addition, there are two matrices.

Let us look at the results of geometric modeling in detail.

Experiments

The research work was carried out over the Jeromeau lemniscate to simplify, which is defined by the parametric system $\{\begin{matrix} x = cos t \\ y = cos t sin t \end{matrix}$ . The results of the geometric modeling of the Jeromeau lemniscate are presented in Table 1.

Table 1.
Computer experiments.

Curve Matrix $\approx β^{\circ}$ $\approx α^{\circ}$ Scalar X Scalar Y P ₁ or P ₂

Lemniscates $(\begin{matrix} 1 & 1 \\ 0.5 & 1 \end{matrix})$ 52 38 1.8 0.3 2 (Figure 3)

Circle $(\begin{matrix} 1.5 & - 1 \\ - 0.5 & 1.2 \end{matrix})$ 52 38 1.6 0.4 2 (Figure 4)

Limacon $(\begin{matrix} 1.5 & - 1 \\ - 0.5 & 1.2 \end{matrix})$ 52 38 1.6 0.4 2 (Figure 5)

Steiner curve $(\begin{matrix} 1.5 & - 1 \\ - 0.5 & 1.2 \end{matrix})$ 52 38 1.6 0.4 2 (Figure 6)

Parabola $(\begin{matrix} 1.5 & - 1 \\ - 0.5 & 1.2 \end{matrix})$ 52 38 1.6 0.4 2 (Figure 7)

Evolute $(\begin{matrix} 1.5 & - 1 \\ - 0.5 & 1.2 \end{matrix})$ 52 38 1.6 0.4 2 (Figure 8)

Evolute $(\begin{matrix} 0.01 & - 1 \\ 1 & 0.7 \end{matrix})$ 1 89 1 0.5 2 (Figure 9)

Evolute $(\begin{matrix} - 0.01 & - 1 \\ 1 & 0.7 \end{matrix})$ −1 -89 1 0.5 1 (Figure 10)

We did not cite all the illustrations for each experiment, but they differ in the form of the curve only. The algorithm may be used.

Additional studies were carried out for limaçon of Pascal by system $\{\begin{matrix} x = (1 + 2 cos t) cos t \\ y = (1 + 2 cos t) sin t \end{matrix}$ , tricuspoid or Steiner curve by system $\{\begin{matrix} x = 2 cos t + cos 2 t \\ y = 2 sin t - sin 2 t \end{matrix}$ , evolute by system $\{\begin{matrix} x = 0.2 (t - sin t) \\ y = 0.2 (1 - cos t) \end{matrix}$ , parabola by system $\{\begin{matrix} x = t \\ y = t^{2} \end{matrix}$ , and circle. The parameter is $t \in [0, 2 π]$ for all curves, except for evolute. Evolution is considered on two periods. The results are shown in Figures 3 to 10. The calculation method is taken from the determination of the transformation parameters of centrally symmetric conic sections.

Figure 3.
Test 1.

Figure 4.
Circle.

Figure 5.
Limacon.

Figure 6.
Steiner curve.

Figure 7.
Parabola.

Figure 8.
Evolute (test 6).

Figure 9.
Evolute (test 7).

Figure 10.
Evolute (test 8).

The algorithm is also checked for a circle (Figure 7). The parameters are the same as for the astroid. The algorithm works, as can be seen from the figure.

The characteristic equation $T \vec{v} = λ {\vec{v}}^{'}$ is divided into four types $\vec{v^{'}} \in \{(\begin{matrix} x \\ y \end{matrix}), (\begin{matrix} - x \\ - y \end{matrix}), (\begin{matrix} y \\ x \end{matrix}), (\begin{matrix} - y \\ - x \end{matrix})\}$ . This follows from the proposed theory and is confirmed by computer simulation. The transformation matrices from tests 7 and 8 differed very little. The correct result was obtained in test 8, and the conjugate curve was received in test 7. So, step 5 of the algorithm is the most difficult. Separation of the characteristic equation affects the parameters and requires additional research.

Practical orientation of research

Kinematic pairs with simple travel paths are the basis for a robot design. Perhaps even Newton has in some way designed mechatronic systems with complex motion trajectories. Designing a robot with a complex trajectory of motion is possible in an approximate manner, in addition it is also possible to follow the designed trajectory of motion during the designing process.

The obtained results can be applied in the field of robotics, mainly due to its accuracy of trajectory calculation and rendering, but also by reverse validation of motion control of monitored robot effectors. The proposed model is applicable for simulation of a two-wheel self-balancing mobile robotic system or other unstable robotic vehicles with parallel wheel alignment (from a physical point of view). Robotic devices belonging to this category of robotic systems require some degree of intelligence (autonomous behavior), described in the literature as a mobile service robot. A convenient addition to the precise regulation and control of mobile service robots is the implementation of inertia navigation, which provides reverse validation and control, for example, for a tilting system implemented on a two-wheel self-balancing mobile robotic system.

Conclusion

Engineering calculations determine the functional capabilities of the mechatronic system. The main method of obtaining is approximation, that is, the inaccuracies were established down in the project before machining. Exact calculation is available for trajectories by conical sections only. Newton proposed a classification for the analytical solution of the problem for complex plane curves, but calculation methods have not yet appeared.

The main disadvantage of the previously obtained types of transformations is the study conducted on the orthogonal basis only. The transformation parameters of a complex curve depend solely on the type of transformation and the properties of space. This is the approach of the article. Initially, a new method for obtaining parameters was found only for centrally symmetric conical sections.^18,19 Further theoretical and experimental studies made it possible to substantiate the applicability of this method to Jordan curves.²⁰ The new method was tested in computer geometric modeling and on the trajectories of robot tests in laboratory conditions. Recent studies have made it possible to find a general calculation algorithm on a nonorthogonal basis. The method combines one type of calculation with any plane curve that describes the trajectory of the robot. The algorithm will help to reveal the meanings and practical solutions for many natural and technical sciences.

The authors believe that the development of a space application method can provide a solution for obtaining the exact trajectory of any robot. Thus, the accuracy of the robot will depend only on the accuracy of the production equipment. Combining the method with Hamiltonian mechanics can become the theoretical basis for development.

Curve	Matrix	$\approx β^{\circ}$	$\approx α^{\circ}$	Scalar X	Scalar Y	P ₁ or P ₂
Lemniscates	$(\begin{matrix} 1 & 1 \\ 0.5 & 1 \end{matrix})$	52	38	1.8	0.3	2 (Figure 3)
Circle	$(\begin{matrix} 1.5 & - 1 \\ - 0.5 & 1.2 \end{matrix})$	52	38	1.6	0.4	2 (Figure 4)
Limacon	$(\begin{matrix} 1.5 & - 1 \\ - 0.5 & 1.2 \end{matrix})$	52	38	1.6	0.4	2 (Figure 5)
Steiner curve	$(\begin{matrix} 1.5 & - 1 \\ - 0.5 & 1.2 \end{matrix})$	52	38	1.6	0.4	2 (Figure 6)
Parabola	$(\begin{matrix} 1.5 & - 1 \\ - 0.5 & 1.2 \end{matrix})$	52	38	1.6	0.4	2 (Figure 7)
Evolute	$(\begin{matrix} 1.5 & - 1 \\ - 0.5 & 1.2 \end{matrix})$	52	38	1.6	0.4	2 (Figure 8)
Evolute	$(\begin{matrix} 0.01 & - 1 \\ 1 & 0.7 \end{matrix})$	1	89	1	0.5	2 (Figure 9)
Evolute	$(\begin{matrix} - 0.01 & - 1 \\ 1 & 0.7 \end{matrix})$	−1	-89	1	0.5	1 (Figure 10)

Footnotes

Declaration of conflicting interests

The author(s) declared no potential conflicts of interest with respect to the research, authorship, and/or publication of this article.

Funding

The author(s) disclosed receipt of the following financial support for the research, authorship, and/or publication of this article: The contribution is sponsored by the project 015STU-4/2018 Specialised laboratory supported by multimedia textbook for subject “Production systems design and operation” for STU Bratislava and by the project 013TUKE-4/2019: Modern educational tools and methods for forming creativity and increasing practical skills and habits for graduates of technical university study programmes. This publication has been written thanks to support of the Operational Program Research and Innovation for the project: “Výskum pokročilých metód inteligentného spracovania informácií”, ITMS code: NFP313010T570 co-financed by the European Regional Development Found.

ORCID iD

Pavol Božek

Alexander Lozhkin

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The precision calculating method of robots moving by the plane trajectories

Abstract

Keywords

Introduction

Nonclassic method of linear transformations

Linear transformations of complex curves

Theorem

Proof

General solution algorithm

Lemma

Experiments

Practical orientation of research

Conclusion

Footnotes

Declaration of conflicting interests

Funding

ORCID iD

References