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In the Issue of Complex Plane Motions of the Robot
Lozhkin A.G.1,
, Abramov I.V.2, Nikitin Yu.R.2, Arkhipov I.O.1, Sultanov R.O.1
1Departments “Software”, M.T. Kalashnikov Izhevsk State Technical University, Izhevsk, Russia
2Mechatronic Systems department, M.T. Kalashnikov Izhevsk State Technical University, Izhevsk, Russia
Abstract
Two methods (classical and direct) for calculating an elliptical path for the kinematic mechanisms is considered briefly. A new method is proposed for such calculations. It is simpler, convenient for further calculations and can be applied to arbitrary curves in some cases. This method can serve as a basis for building simple robots with complex trajectories.
Keywords: ellipse, calculation of trajectory, kinematic mechanisms, plane differentiable curves
Copyright © 2015 Science and Education Publishing. All Rights Reserved.Cite this article:
- Lozhkin A.G., Abramov I.V., Nikitin Yu.R., Arkhipov I.O., Sultanov R.O.. In the Issue of Complex Plane Motions of the Robot. Journal of Automation and Control. Vol. 3, No. 3, 2015, pp 89-95. https://pubs.sciepub.com/automation/3/3/10
- A.G., Lozhkin, et al. "In the Issue of Complex Plane Motions of the Robot." Journal of Automation and Control 3.3 (2015): 89-95.
- A.G., L. , I.V., A. , Yu.R., N. , I.O., A. , & R.O., S. (2015). In the Issue of Complex Plane Motions of the Robot. Journal of Automation and Control, 3(3), 89-95.
- A.G., Lozhkin, Abramov I.V., Nikitin Yu.R., Arkhipov I.O., and Sultanov R.O.. "In the Issue of Complex Plane Motions of the Robot." Journal of Automation and Control 3, no. 3 (2015): 89-95.
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At a glance: Figures
1. Introduction
Many standards such as the life cycle of a product, company management, etc. assert that the right decisions to be applied in the early stages, which guarantees places without high costs in the following stages. Until now, the first author of the article used the new methods of valuation solely for the design or CAD or CAM [6, 8, 9] but the quality of product design starts much earlier in the development phase of the kinematic mechanisms by team of authors. One of the major tasks of this phase is to calculate the trajectory of motion of the mechanism. Rational calculation of the trajectory of motion actuators automatic machines is important for efficient production. Calculate the trajectories of the kinematic mechanisms studied and necessary mathematical theory in papers [1, 2, 5, 7, 10, 12, 13]. The calculations of the motion trajectory of the kinematic mechanisms are often yields a system of parametric equations of a circle or ellipse
 | (1) |
where 
.
2. Transformation of the Circle
It is proposed to consider the parametric system as a transformation of the circle. Canonical equation of the transformed circle can be converted to determine the inverse transformation of the matrix 
. Matrix 
is equal to 
[Efimov 1967].
Consequently the canonical equation of a circle is as follows:
 |
 |
 | (2) |
It is known from analytic geometry [Efimov 2009], how to get the angle of rotation of the quadratic form of the canonical equation of the second order line. If the general equation written in the form 
, the angle of rotation can be found is 
.
We obtain from (1) the values of the coefficients: 
; 
; 
; 
; 
; 
. The angle of rotation of the resulting curve will be equal to
 |
 | (3) |
We solve the equation (1) from the characteristic equation 
[3], where 
. It is obtain solutions.
 |
 |
 |
The solution
 | (4) |
is obtained, but it is difficult to apply in the coefficients for the other calculations. Engineer must choose one right decisions of the two 
additionally.
It is proposed to use another method [7, 8, 9]. The method is based on the permutation symmetry and other symmetries [8].
3. Direct Analytical Method of Arbitrary Linear Transformations
We calculate the angle 
which is symmetrical own corner 
for system (1)
 | (5) |
in the first step. The angle is determined from the two equations
 | (6) |
and
 | (7) |
Angles are equal if the calculation and transformation is correct. The coefficients 
are equal
 | (8) |
 | (9) |
 | (10) |
 | (11) |
The initial system will be:
 | (12) |
and
 | (13) |
Both solutions are valid and the choice is not necessary.
General solution for any curve could not get unfortunately. If the result of the calculations obtained systems 
, 
, 
, where 
and 
arbitrary trigonometrically functions, the solution can be found similarly. For example, the system 
can be reduced to the form 
.
4. Theoretical Tests
Submitted theorems are very simple and allow the reader to the proof without reference to sources in a foreign language. Let's consider the sufficiency of all the theorems only. Theorem 3 is published the first.
Theorem 1. Angles 
(6) and 
(7) is equivalence.
Proof.
Let’s equating the formula (6) and (7) in value 
: 
, 
or 
. Let's show 
: 
, 
. The value 
is determinate: 
, 
. Results equate to 
, 
, 
, 
, 
, 
. The resulting formula coincides with the formula (5) and hence the angle 
is the same in both formulas.
Remark. Theorem is intended to linearly independent transformations only.
Theorem 2. Parameters 
when 
and 
when 
are equivalence.
Proof.
1. Let’s assume the opposite for coefficient 
. Let’s 
, 
than 
from (8) and (9): 
, 
, 
. We have from (7) and theorem 1 
. Equality holds when 
or 
. Since 
in general not equal 
then 
and parameters are equivalence.
2. Let’s assume the opposite for coefficient 
. Let’s 
, 
than 
from (10) and (11): 
, 
, 
. We have from (6) and theorem 1 
. Equality holds when 
or 
. Since 
in general not equal 
then 
and parameters are equivalence.
Theorem 3. The own angle in classical method and direct analytical method is equivalence.
Proof.
Let's equating the formula (6) and (7) in value 
. Let's deduce the angle 
of the formula (7): 
, 
, 
, 
, 
. Let's deduce the angle 
of the formula (6): 
, 
, 
.
Results equate to 
, 
. We have from theorem 1 
that 
or 
. Let’s consider 
: 
, 
. Value 
is 
, 
. Results equate to 
, 
, 
, 
, 
. We find (2).
The proof of the transformation parameters can be obtained similarly to Theorem 2, but it is cumbersome for publication in the journal.
Let coefficients of matrix are 
, 
. Angle 
is equivalence 
, 
, 
. Since the angle is not define. Let coefficients 
and 
are equality value 
, where 
, than 
, 
, 
и 
. Angle 
is 
, 
or 
, 
. Angle 
is turned out the same for both formulas. Coefficients 
and 
are 
, 
, 
, 
.
Thus the formulas for converting performed in the vicinity of the transformation 
, but do not exist for accurate conversion.
Let coefficients of matrix are 
, 
, 
. Angle 
is equivalence 
, 
. Since the angle is not define also. Let coefficients of matrix are 
, 
, 
, where 
. Angle 
will seek, as 
: 
, 
, 
. Angle 
is 
, 
, 
or 
, 
, 
. Angle 
is turned out the same for both formulas. Coefficients 
and 
: 
, 
, 
, 
. Findings are consistent with the conclusion of the previous conversion.
Method can not be used to rotate 
and scaling (homothety) 
and some other transformations (see theorem 2) [Lozhkin 2012].
5. Geometrical Modeling
Example of geometric modeling the method is presented to calculate the trajectory of the astroid in language AutoLisp. Astroid (hypocycloid modulo 4) is described by two identical systems of parametric equations 
and 
[Gibson 2001].
The program algorithm is quite simple:
1. The requested transformation parameters;
2. We output the desired astroid will transform every part of the line of it. Output is black;
3. Forward transformation parameters;
4. Output a new astroid with parameters obtained in yellow.
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The result of the programs for conversion 
shown in Figure 1. The method works. Parameters are calculated correctly. The result of the programs for conversion 
is shown in Figure 2. Method does not work. The experiment was carried out in AutoCAD 2009.
Two rules to observe when moving algorithm to other languages:
1. Calculate the arc tangent function of two arguments;
2. Remember in which quadrant is a calculation.
It should be noted that the theoretical and modeling researches of Jordan curves is not full.
6. Experiment
Experiments were carried out on the basis of the developed stand with dimensions 1300x1400x1500 mm to check the adequacy of the developed mathematical model. Photo booth is shown in Figure 3.
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The design of the stand was assembled from aluminum profiles, linear modules and fasteners RK Rose + Krieger. Aluminium profiles RK Rose + Krieger Blocan shown in Figure 4.
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Aluminum profiles are profiled rods with longitudinal T-slots. The surface profile is finishing. Linear Module RK Rose + Krieger Blocan have aluminum construction, possess high hardness, large carrying capacity. Assembly is protected from dust and dirt. Linear guides have been installed on a wide aluminum profile and secured the appropriate bracket, Figure 5.
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Two profiles with guides were bonded together using an aluminum sheet on the top side. Four movable slider on the guide are connected to the same sheet of aluminum, 3D model shown in Figure 6.
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Experiments have shown a difference between the theoretical and actual trajectory less than 5%.
Simple and cheap robot to move along the path of the executive body of the ellipse and the circle may develop on the basis of the proposed mathematical model. Photo pneumatic robot is shown in Figure 7.
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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 can with laser burn in different materials is not necessarily circular holes.
7. Plane Differentiable Curves
The authors studied the expression of symmetry Euler not only for conic sections but also more complex forms. Four groups of linear 
, 
, 
, 
, where 
, can be used DAM for any Jordan curve, but the method is not applicable for rotation and dilation for any curve, because they not change the discrete structure of Euclidean plane.
The proposed theory allows the calculation of complex mechatronic systems. Let there be a simple kinematical mechanism built on the basis of three gears: central, moving inside the central wheel and moving outside of the central wheel (Figure 8). Let a point on the first wheel moves according to the law 
. Let the motion of the second and third wheels are defined by the systems of parametric equations 
(asteroid), 
(epicycloids).
It is necessary to design an elliptical gear mechanism to the trajectory of the first wheel to a system of parametric equations 
from second group, where
and 
. The method of calculation can be applied to all wheels. Geometric modeling results are shown in Figure 9.
Solution of the characteristic equation 
can be different. Vector 
does not change after the transformation of the ellipse. Vector belongs to the set 
after the transformation of the planar differentiable curve. Parameters symmetric transformations find possible and to simplify the system of parametric equations for the complex movements as a result. Real coefficients can be used in systems of parametric equations. Thus the analytical solution for calculating opened for the complex trajectories of the mechatronic system. Precision of designing is defined within a computer data storage and precision of machining equipment.
Symmetries has an important influence on the trajectory of this is obvious. Let anticipated changes in trajectory when applying an arbitrary linearly independent transformation. The system of parametric equations is divided into two or four portions of movement:
, 
, 
, 
. This hypothesis is tested in research now.
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8. Summary
If the trajectory of the point of the kinematic mechanism is represented by the parametric equations that can be used a new effective method. Parametric system is used to convert the equations of motion in a circle. The method allows to find the exact characteristics of the mechanism are based on the kinematic trajectory of its motion. The method described by simple algorithms more uncomplicated compared with the classical method.
Acknowledgement
Supported by Minobrnauki of Russian Federation, Grant GZ/TVG 14(01.10).
References
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