The calculations of Jordan curves trajectory of the robot movement

Abstract

Calculation of the moving mechanism on a trajectory described by a planar differentiable curve is challenging. The difficulty arises in compliance with high-precision movement. The new method of calculation based on the intrinsic properties of the plane is proposed. The method is based on mathematical linguistics and relational algebra. First, these disciplines are applied to analyze the intrinsic properties of the Euclidean plane. Calculation of classic and new methods for moving point for the flat rod is shown for comparison. Analytical formulas for Jordan curves can be obtained in some cases. The experiments in the areas of geometric modeling and control of a robot are listed briefly. The analytic solution was obtained by nontraditional methods not only for centrally symmetric conic sections but also for Jordan curves.

Introduction

Trajectories of the mechatronic system were calculated based on the principles of Archimedean mechanism. The design is constructed of linear and circular movements, as it exists in nature. At the same time, a set of movements is a much more complex path, such as the epicycloid, hypocycloid, and so on. The accuracy of the calculation mechanism depends on the methods of geometrical control. ¹ There are only approximate methods for complex differentiable curves. Let us try to provide accurate calculation methods on the basis of amendments to the analytic geometry. ²

The new branches of geometry appear by transferring the results of other mathematical sections, such as set theory, algebra, and so on, generally. Mathematics is the language with which God has written the universe. The key words in this Galileo aphorism will assume the words “language” and “write.” Let’s remove the emphasis from the word “mathematics” as has been done yet. Proof of Galileo thesis and output of similarity method by Leibniz are identical to a large extent. Hilbert formulating axioms for Euclidean plane suggested that the construction of linguistic rules must be considered. The main results in mathematical linguistics are prepared using semiotic and structural analysis. They are base solutions in the natural and formal language theory ³ and used in artificial intelligence theory sometimes. Symmetry as the basis of the similarity method and their analysis using semiotic method was applied in the research.

Convex geometry is greatly close to analytic geometry. Important practical results are obtained from its use. It is formulated by adding Zermelo-Fraenkel (ZF) axioms of set theory. The main objective of the reported studies is to get sensible results in mechatronic systems and Computer Aided Design/Computer Aided Manufacturing (CAD/CAM). Codd’s relational algebra as a practical part of ZF theory has been applied additionally. Proofs by Galileo and Leibniz are based on geometric constructions. We used mathematical linguistics to improve the accuracy of geometrical modeling in CAD/CAM, mechatronics, and so on. We obtained the Euclidean plane as text.

The table of symmetries

Hilbert formulating axioms for Euclidean plane suggested that the construction of linguistic rules must be considered. We obtained the Euclidean plane as text by analogy with semiotic analysis of drawings according Leibniz’s method of similarity. Levels of study of the text are internal relations in plane. The basic postulates used were permutation, mirror and unitary matrix symmetries by Dieudonne ⁴ ; table automorphisms and transfer symmetry by Weyl; definition of symmetry by Born; and binary automorphisms by Bachman.

Main attention of research was devoted to permutation symmetry. ⁵ The Euclidean plane is a relation table. The proof is easy, because relation algebra may be working with finite and indefinite table. The conduct of symmetry was considered by relation algebra and semiotic analysis. Application of the method to the Dieudonne automorphisms shows that binary symmetry belongs to two mathematical disciplines: set theory and universal algebra.

Extended table of Dieudonne symmetries was built on the basis of knowledge symmetry and relational algebra. The current presentation of this table is:

Existing of set (A≠Ø Zermelo).

Existing of relation (a₁Ra₂ Codd).

Membership element of set (a∈A Fraenkel).

Universal relation (f : Ω → Ω' implication).

Linguistic description of the set (Descartes).

Linguistic presentation of the relation (Descartes).

Saving cardinality (m(A) = const Lagrange).

Saving power relations (n = const in $C_1{x}^n+C_2{y}^n+C_3{x}^{n-1}{y}^{n-1}+\cdots +C_{k-1}x+C_{k}y+C_0$ Klein).

Linguistic order ( $\overrightarrow{v}=xi+yj+zk+w$ Hamilton).

Mathematical order (a_i ≺ a_i+1, where a_i,a_i+1 ∈ R Kantor).

Permutation ( a_i ↔ a_j).

Mirror (a_i ·−1 = −a_i ).

Since the connection between automorphisms 10 and 12 was first opened by Gilbert, the symmetry of knowledge may be named in his honor. Symmetry with numbers 1 and 2 is combined as $\left(\begin{array}{cc}1& 0\\ 0& 1\end{array}\right)$ Gilbert opened the structural properties of symmetry, and Euler’s formula e^iπ = −1 to determine the relationship between them in universal algebra. The main feature of the table is a hierarchy. Symmetry with a minor number determines of action of the following automorphisms. Symmetries are joined of the two sciences, so there are two methods for solving the trajectories calculation.

The linear transformations for central symmetric conic sections

Elliptic and hyperbolic trajectory can be calculated using nonprojective transformations by the classical method ⁶ or direct analytical method. ^2,5,7 Nonprojective solutions for Jordan curves are absent.

Let there be an arbitrary figure Φ —Jordan curve in the Euclidean plane R ² in a Cartesian coordinate system defined by the parametric equation:

$\{\begin{array}{c}x=f_x(t)\\ y=f_y(t)\end{array}$

1

where functions f_x  and f_y (t) are piecewise continuous. If the equation’s figure is defined, it is always possible to write. Class of shapes defined only by the implicit function F(x, y) = 0 is not considered in this research. We carry out any transformation of the figure Φ defined by the matrix, where it is necessary to obtain the parameters of the transformed figure (to solve the characteristic equation).

The basis of the direct method of linear transformations is a manifestation of the adjustable symmetry, namely $\{\begin{array}{c}c{f}_x(t+\alpha )=a{f}_x(t+\beta )+h{f}_y(t+\beta )\\ d{f}_y(t+\alpha )=g{f}_x(t+\beta )+b{f}_y(t+\beta )\end{array}$ , where β —angle of permutation symmetry. Nonorthogonal basis ⁷ is a form from the corners α and β (Figure 1). Let us consider the solution of the characteristic equation for the centrally symmetric conic sections. Only equation of own angle α take of the classic method ⁶ :

Figure 1.

Nonorthogonal basis.

$\text{tan}2\alpha =\frac{2\left(bh+ag\right)}{\left(a^2+h^2\right)-\left(b^2+g^2\right)}$

2

Parameters semiaxes are considered difficult in the classical method, since they represent a radical dependence.

A new direct analytical method for the linear transformation was proposed earlier. Free from radicals, it is more simple and accessible for further mathematical derivations. The method is based on the permutation symmetry and other symmetries. ⁷

We calculate the angle β which is symmetrical to own corner α for the system:

$\text{tan}2\beta =\frac{2(gb+ha)}{a^2-h^2-b^2+g^2}$

3

in the first step. The angle α is determined using the two equations:

$\text{tan}{\alpha }_1=\frac{a\text{tan}\beta -h}{b-g\text{tan}\beta }$

4

and

$\text{tan}{\alpha }_2=\frac{b\text{tan}\beta +g}{a+h\text{tan}\beta }$

5

Angles are equal if the calculation and transformation is correct. The coefficients λ ₁ = c and λ ₂ = d are equal.

$c_1=\frac{a\text{cos}\beta +h\text{sin}\beta }{\text{cos}\alpha }$

6

and

$c_2=\frac{b\text{sin}\beta +g\text{cos}\beta }{\text{sin}\alpha }$

7

$d_1=\frac{a\text{sin}\beta -h\text{cos}\beta }{\text{sin}\alpha }$

8

and

$d_2=\frac{b\text{cos}\beta -g\text{sin}\beta }{\text{cos}\alpha }$

9

The initial system will be $\{\begin{array}{l}x=c_1\text{cos}(t+\alpha )\hfill \\ y=d_1\text{sin}(t+\alpha )\hfill \end{array}$ and $\{\begin{array}{l}x=c_2\text{cos}(t+\alpha )\hfill \\ y=d_2\text{sin}(t+\alpha )\hfill \end{array}$ Both solutions are valid and the choice is not necessary. Semiaxes are equivalent c ₁ = c ₂ and d ₁ = d ₂. The decision does not have square roots in contrast to the classical, so it can easily be used in further calculations.

For example, let us consider the motion of a point located outside the axis of the flat rod. The system of equations describing the motion of the third point is $\{\begin{array}{l}x=p\text{cos}t+s\text{sin}t\hfill \\ y=s\text{cos}t+(l-p)\text{sin}t\hfill \end{array}$ , where l —crank length, p,s —coordinates of the point (x, y) in the coordinate system of the flat rod. Solution proposed method is $\{\begin{array}{l}x=(l-p+s\text{cot}\alpha )\text{cos}(t+\alpha )\hfill \\ y=(p-s\text{cot}\alpha )\text{sin}(t+\alpha )\hfill \end{array}$ and $\{\begin{array}{l}x=(l+s\text{tan}\alpha )\text{cos}(t+\alpha )\hfill \\ y=(l-p-s\text{tan}\alpha )\text{sin}(t+\alpha )\hfill \end{array}$ , where $\text{tan}2\alpha =\frac{2s}{2l-p}$ .

Solution classical method for the transformation matrix $T=\left(\begin{array}{cc}p& s\\ s& l-p\end{array}\right)$ by the inverse matrix $T^{-1}=\frac{1}{detT}\left(\begin{array}{cc}l-p& -s\\ -s& p\end{array}\right)$  is $\{\begin{array}{c}x={\lambda }_1\text{cos}(t+\alpha )\\ y={\lambda }_2\text{sin}(t+\alpha )\end{array}$  or $\{\begin{array}{c}x={\lambda }_2\text{cos}(t+\alpha )\\ y={\lambda }_1\text{sin}(t+\alpha )\end{array}$ ,

Where ${\lambda }_{1,2}=\frac{l\pm \sqrt{{\mathit{\text{l}}}^2-4((\mathit{\text{l}}-\mathit{\text{p}})\mathit{\text{p}}-{\mathit{\text{s}}}^2)}}{2\left((l-p)p-s^2\right)}$ . Angle α is not change. As can be seen from the results, the proposed method allows to be used in subsequent calculations easily. For example, if you can solve differential equations.

Application for planar differentiable curves

Unfortunately, general solution for Jordan curve could not be obtained. If the result of the calculations obtained for the systems:

$\{\begin{array}{l}x=m{f}_x+n{f}_y\hfill \\ y=m{f}_x-n{f}_y\hfill \end{array}$

10

$\{\begin{array}{l}x=-m{f}_x+n{f}_y\hfill \\ y=m{f}_x+n{f}_y\hfill \end{array}$

11

$\{\begin{array}{l}x=m{f}_x+kn{f}_y\hfill \\ y=-n{f}_x+km{f}_y\hfill \end{array}$

12

$\{\begin{array}{l}x=km{f}_x+n{f}_y\hfill \\ y=-kn{f}_x+m{f}_y\hfill \end{array}$

13

then the solution may be found.

For example, the system $\{\begin{array}{l}x=\text{cos}t-2\text{sin}2t\hfill \\ y=\text{cos}t+2\text{sin}2t\hfill \end{array}$  for the sake of m = 2 and n = 2 in equation can be reduced to the form $\{\begin{array}{l}x=c\text{cos}(t+\alpha )\hfill \\ y=d\text{sin}2(t+\alpha )\hfill \end{array}$ .

Parameters for converting epicycloid modulo 2 with parametric system of equations $\{\begin{array}{c}x=3R\text{cos}t-R\text{cos}3t\\ y=3R\text{sin}t-R\text{sin}3t\end{array}$ (Figure 2) for the geometric modeling unfortunately led to incorrect results. The same parameters for the astroid (hypocycloid modulo 4) and ellipse matched exactly. Astroid is described by two identical systems of parametric equations $\{\begin{array}{c}x=\frac{3}{4}R\text{cos}\frac{t}{4}+\frac{1}{4}R\text{cos}\frac{3t}{4}\\ y=\frac{3}{4}R\text{sin}\frac{t}{4}-\frac{1}{4}R\text{sin}\frac{3t}{4}\end{array}$  and $\{\begin{array}{c}x=Rco{s}^3\frac{t}{4}\\ y=Rsi{n}^3\frac{t}{4}\end{array}$ . The test was performed for transforming $\left(\begin{array}{cc}1& 1.2\\ 1& -1.2\end{array}\right)$ .

Figure 2.

Error in transformation of epicycloid.

The program algorithm is quite simple:

The requested transformation parameters;

Asteroid, epicycloids and ellipse are output after transformation by every part of the line of it and the output is black;

Forward transformation parameters;

The new obtained curves are output in red.

With minimal appeals to the mathematical theory, the research method of geometric modeling will be carried out. Consider the three curves, astroid, epicycloid modulo 2, and the ellipse. The last curve is not a contradictory method of calculation. We will investigate the direct analytical method as a black box, and the results are compared with the mathematical theory.

Solution of the characteristic parametrical systems (10–13) can be different. Vector $\overrightarrow{v}=\left(\begin{array}{c}x\\ y\end{array}\right)$  does not change after the transformation of the ellipse. Vector belongs to the set $\overrightarrow{v}\in \left\{\left(\begin{array}{c}x\\ y\end{array}\right),\left(\begin{array}{c}y\\ x\end{array}\right),\left(\begin{array}{c}-y\\ -x\end{array}\right)\right\}$ after the transformation of the Jordan curve. Transformations with symmetry can simplify the system of equations for the complex movements. Own angle determines the position of the curve on the plane. It is customary, but decision depends on the angle for Jordan curves. The results of research are shown in Table 1. The number in the last column indicates the number of vector representation. The vector representation is shown sequentially in the table and is related to its determination as determined by the last two vectors υ. As can be seen from the results, the proposed method allows to be used in subsequent calculations easily. For example, if you can solve differential equations, for example, in control theory. ⁸ Main distinctions between methods are as follows: Euler’s method does not work for the transformation of the scaling and rotation, singular different conversion, and no solution for the parabola. Principal differences are finding in singularities, but they are not important for technical applications. Some of the results obtained for plane differentiable curves may be relevant to physics. Results for arbitrary linearly independent transformations of planar differentiable curves are not prepared unfortunately.

Table 1.

Results of experiments.

Matrix	Angle β°	Angle α°	Angle α 1°	Angle α 2°	Semiaxis c	Semiaxis d	Vector
$\left(\begin{array}{cc}1& 0.6\\ 1& -0.6\end{array}\right)$	0.0	45.0	−135.0	45.0	28.2843	−16.9706	1
$\left(\begin{array}{cc}-1& 0.6\\ -1& -0.6\end{array}\right)$	0.0	45.0	−135.0	−135.0	28.2843	−16.9706	1
$\left(\begin{array}{cc}-1& -0.6\\ -1& 0.6\end{array}\right)$	0.0	45.0	45.0	−135.0	−28.2843	16.9706	1
$\left(\begin{array}{cc}1& -0.6\\ 1& 0.6\end{array}\right)$	0.0	45.0	45.0	45.0	28.2843	16.9706	1
$\left(\begin{array}{cc}1& 1.2\\ 1& -1.2\end{array}\right)$	90.0	−45.0	135.0	−45.0	−33.9411	28.2843	2
$\left(\begin{array}{cc}-1& 1.2\\ -1& -1.2\end{array}\right)$ (Figure 3a)	90.0	−45.0	−45.0	−45.0	33.9411	28.2834	2
$\left(\begin{array}{cc}-1& -1.2\\ -1& 1.2\end{array}\right)$	90.0	−45.0	−45.0	135.0	−33.9411	28.2834	2
$\left(\begin{array}{cc}1& -1.2\\ 1& 1.2\end{array}\right)$	90.0	−45.0	135.0	135.0	33.9411	28.2834	2
$\left(\begin{array}{cc}-2.1& 3.72\\ -3.1& -2.52\end{array}\right)$ (Figure 3b)	−90.0	−34.1145	145.88	145.88	−89.8639	−74.886	3
$\left(\begin{array}{cc}2.1& 3.72\\ -3.1& 2.52\end{array}\right)$	90.0	34.1145	34.1145	34.1145	89.8639	74.886	2
$\left(\begin{array}{cc}-2.1& -3.72\\ -3.1& 2.52\end{array}\right)$	90.0	−34.1145	−34.1145	145.88	−89.8639	74.886	2
$\left(\begin{array}{cc}2.1& -3.72\\ -3.1& -2.52\end{array}\right)$	−90.0	34.1145	−145.886	34.1145	89.8639	−74.886	3
$\left(\begin{array}{cc}-2.1& -3.72\\ 3.1& -2.52\end{array}\right)$	90.0	34.1145	−145.886	−145.886	89.8639	74.886	2
$\left(\begin{array}{cc}2.1& -3.72\\ 3.1& 2.52\end{array}\right)$	−90.0	−34.1145	−34.1145	−34.1145	89.8639	74.886	3
$\left(\begin{array}{cc}-2.1& 3.72\\ 3.1& 2.52\end{array}\right)$	−90.0	34.1145	34.1145	−145.886	−89.8639	74.886	3
$\left(\begin{array}{cc}2.1& 3.72\\ 3.1& -2.52\end{array}\right)$	90.0	−34.1145	145.886	−34.1145	−89.8639	74.886	2

Application for the design of robots

The circle and the ellipse are the basis for many of the kinematic mechanisms, such as a flat crank, crank, and so on. Payment mechanism causes the trajectory of its motion. ⁸ Detailed information as a condition of the mentioned calculations are described in detail in the monograph. ⁶ It is also linked to the causes of payment mechanism. The solution of differential equations can be obtained in an analytical form if the trajectory is given using the exact analytical formula.

Calculation of the trajectory of motion actuators and automatic machines with high precision is important for efficient production. ⁹ The calculations of the motion trajectory of the kinematic mechanisms often yield a system of parametric equations of a circle or ellipse $\{\begin{array}{c}x=a\text{cos}t+h\text{sin}t\\ y=g\text{cos}t+b\text{sin}t\end{array}$ , where a, b, h, g, t ∈ R.

Experiments were carried out on the basis of the developed stand to check the adequacy of the developed mathematical model. The design of the stand was assembled from aluminum profiles, linear modules, and fasteners (RK Rose+Krieger, Minden, Germany). Simple and cheap robot (Figure 4) to move along the path of the executive body of the ellipse and the circle may be developed on the basis of the proposed mathematical model. ^{10 –12} Experiments have shown a difference between the theoretical and actual trajectory of less than 5%. This robot can be carved with milling cutter details in the form of an ellipse and a circle made of different materials, such as glass, metal, ceramics, and wood. The robot with laser can burn through in different materials is not necessarily circular holes. Experiments with the Jordan curves are the path of movement carried out in the laboratory plant (Figure 5).

Figure 3.

Results of transformations from Table 1.

Figure 4.

Robot.

Figure 5.

Laboratory stand.

Conclusions

The characteristic parametrical system $T\overrightarrow{v}=\lambda \overrightarrow{v^{\prime }}$ , where T —matrix of transformations, $\overrightarrow{v}$ —vector, λ —set of scalar, is used in many science, such as mechanics, physics, economics, cryptography, and so on. The authors believe the main result of research is the vector belongs $\overrightarrow{v^{\prime }}\in \left\{\left(\begin{array}{c}x\\ y\end{array}\right),\left(\begin{array}{c}y\\ x\end{array}\right),\left(\begin{array}{c}-y\\ -x\end{array}\right)\right\}$ . The result of the conversion depends on the permutation and mirror symmetries. Two other numerical automorphisms influence the analytical results to be identical. This finding affects many of the theoretical and practical applications. Own angle α was studied in great detail. Particular attention should be paid to the theoretical research of angle β in nonorthogonal basis. From this angle, it may receive analytical formulas for converting Jordan curves and complex nonlinear transformations.

The analytical solution for calculating is opened for the complex trajectories of the mechatronic system. Precision of designing is defined within a computer data storage and precision of machining equipment. People design symmetrical mechatronic systems as a rule. Therefore, most of mechatronic systems can be calculated using the method proposed in this article.

Analytical geometry is the simplest of the existing geometries. We have shown the possibility of obtaining exact formulas based on it, the opportunity to build robots on the mathematical theory. The calculation cannot be made for all linear parametric systems. Development of the theory will extend the class of systems, and, in turn, accurate and high-speed robots will enable designing. The stated mathematical calculation is a suitable method for determining the trajectory of mobile robots and industrial robots anywhere. It is necessary to follow the programmed process of the trajectory with extremely high precision. They are mainly engineering technologies, material handling in precise manufacturing process as well as electrical engineering and the fabrication of new products by advanced printing methods.