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9SCIENTIFIC PROGRESS VOLUME 3 I ISSUE 4 I 2022 _ISSN: 2181-1601
Scientific Journal Impact Factor (SJIF 2022=5.016) Passport: http://sjifactor.com/passport.php?id=22257
SIMULATION OF A TWO-LINK MANIPULATOR
Lyudmila Petrovna Varlamova Farukh N. Xo'jaqulov
National University of Uzbekistan National University of Uzbekistan
ABSTRACT
The paper considers the solution of direct and inverse problems of kinematics for a two-link manipulator. The object of research is a two-link two-degree mechanism. The article considered theoretical and practical knowledge about the features of solving the inverse problem of kinematics for manipulators, the study of automation tools for these tasks, as well as building a model of the considered mechanism. The calculation of the two-link two-degree mechanism chosen for the study was made.
Key words: inverse problem of kinematics (OZK), manipulator, two-link mechanism.
Introduction
When creating robotic systems, the direct and inverse problems of kinematics are calculated.
Let's consider these tasks on a standard example of a manipulator.
The direct problem involves calculating the position (X, Y, Z) of the manipulator working body according to its kinematic scheme and the given orientation (Ai, A2... An) of its links (n is the number of degrees of freedom of the manipulator, A are the rotation angles).
The inverse problem is the calculation of the angles (A1, A2 ... An) according to the given position (X, Y, Z) of the working body and, again, the known scheme of its kinematics (Fig. 1)
B/ / 02 / >'
j^ß Llij -"A
)01 J ->
Fig. 1. A two-link manipulator Uzbekistan www.scientificprogress.uz Page 1107
SCIENTIFIC PROGRESS VOLUME 3 I ISSUE 4 I 2022 _ISSN: 2181-1601
Scientific Journal Impact Factor (SJIF 2022=5.016) Passport: http://sjifactor.com/passport.php?id=22257
Solution of direct and inverse problems of kinematics
Thus, the solution of the direct problem tells where the working body of the manipulator will be located, at given angles of its joints, and the inverse problem, on the contrary, says: how the manipulator needs to "turn out" so that its working body is in a given position.
Consider an example of a direct kinematics problem. We have a manipulator that can work only in one plane and has two links. The first link L1 is fixed on the base and rotated by an angle 0b the second link L2 is attached to the end of the first joint and rotated relative to it by an angle 02.
The working body of the manipulator is located at the 01end of the second link. The direct task of kinematics is to find the coordinates of the working body (x, y) according to the given L1, L2, 0b 02.
L1 and L2 are, respectively, the lengths of the manipulator links; determined by the design of the manipulator. Solution:
here, we have two reference systems - the first, associated with the attachment point of the link L1 - O, and the second - with the origin at the attachment point - A.
Let's find the displacement of the second system relative to the first one (the coordinates of point A in the reference system O): XA = L1*cos(01) YA = L1*sin(01)
Coordinates (x, y) in the reference frame of the link L2 x'' = L2*cos(02) y'' = L2*sin(02)
По рисунку видно, что в системе O, звено L2 повёрнуто относительно плеча на 01+ 02:
x' = L2*cos(01+ 02) y' = L2*sin(01+ 02) т.о.:
x = XA + x' = L1*cos(01) + L2*cos(01+ 02) y = YA + y' = L1*sin(01) + L2*sin(01+ 02)
Now, consider an example of an inverse kinematics problem. The same figure, but now you need to find such angles 01 and 01+ 02 that will allow the manipulator with the shoulder L1 and elbow L2 to place the working body at a given point (x, y).
Draw a line B connecting the origin O with a given point (x, y).
Scientific Journal Impact Factor (SJIF 2022—5.016) Passport: http://sjifactor.com/passport.php?id—22257
B2 = x2 + y2 x = B*cos(P) y = B*sin(P)
в is the angle between axis OX and line B a is the angle between line B and shoulder Lj отсюда: 0i = в - a
в = arccos( x/B ) or в = arctg( y/x )
and ais found using the cosine theorem, which says:
For a flat triangle with sides a,b,c and angle alpha opposite side a, the relation is true:
a2 = b2 + c2 - 2*b*c*cos(alpha)
in our case, according to the cosine theorem:
L22 = B2 + Li2 - 2*B*Li*cos(a)
=> a = arccos( Li2 - L22 + B2 / 2*B*Li )
01 = в - a = arccos( x/B ) - arccos( LiA2 - L2A2 + BA2 / 2*B*Li ) using the same cosine theorem, we find the angle 02: as you can see from the figure, the angle 02 is equal to i80 - the angle OAx Obviously, the hand can be positioned differently: (Fig2)
Fig 2.
the formulas for 0! and 02 will not change, but the signs of the angles will change:
Scientific Journal Impact Factor (SJIF 2022—5.016) Passport: http://sjifactor.com/passport.php?id—22257
01 = ß + a (Fig.3)
Fig. 3.
Then, by modeling with the help of the ANSYS program, a model of the movement of a two-link manipulator was built (Fig.4).
Inverse kinematics problem L(s) = sqrt(x(s)A2+y(s)A2);
alpha2(s) = acos((L(s)A2 - M1A2 - M2A2)/(2*M1*M2)); psi(s) = atan(y(s)/x(s));
phi(s) = acos((L(s)A2 + M1A2 - M2A2)/(2*L(s)*M1)); alphal(s) = psi(s) - phi(s); %% Links
Link1x(s,:) = [0:0.1 :M1]*cos(alpha1(s)); Link1y(s,:) = [0:0.1 :M1]*sin(alpha1(s)); Link2x(s,:) =
[0:0.1:M2]*cos(alpha1(s)+alpha2(s))+M1*cos(alpha1(s)); Link2y(s,:) =
[0:0.1 :M2]*sin(alpha1(s)+alpha2(s))+M1*sin(alpha1(s)); %% Drawing
plot(Link1x(s,:),Link1y(s,:),'linewidth',2); hold on;
plot(Link2x(s,:),Link2y(s,:),'r',,linewidth,,2) hold on;
Scientific Journal Impact Factor (SJIF 2022—5.016) Passport: http://sjifactor.com/passport.php?id—22257
plot(x(1:s),y(1:s),'k,,,linewidth,,3) drawnow; pause(0.1); axis equal end
a - ü • si - a n b ■ a
Fig.4. The movement of a two-link manipulator
Conclusion
Modern technologies do not stand still and are in constant modernization, new ways of controlling robots and manipulators are opening up. Automation of the inverse problem of kinematics is necessary and will significantly simplify the control of simple manipulator mechanisms.
REFERENCES
1. Kolyubin S.A. Dynamics of robotic systems. - St. Petersburg: ITMO University. 2017, -117p.
2. Robotics, Fu K., Gonzalez R., Lee K. 1989.- 614p.
3. Mechanics of industrial robots: textbook. manual for technical colleges: in 3 books / ed. K. V. Frolova, E. I. Vorobieva. Book. 1: Kinematics and Dynamics / E. I. Vorobyov, S. A. Popov, G. I. Sheveleva. M.: Higher school, 1988.- 304 p.
