CC BY
15Impact Factor: SJIF 2019 - 5.11 ТСХШЩЖМ НАУКИ
2020 - 5.497
2021 - 5.81
УДК: 004.896:621.865
DIRECT AND INVERSE KINEMATICS OF MANIPULATOR ROBOTS
YEGANA NOVRUZ KiZi ALiYEVA AGiL TOFiK OGLU MAMMADLi
Azerbaijan State Oil and Industry University
Annotation: Currently in various fields industry, industrial handling robots are used, used to perform a wide range of technological tasks. The main type of manipulation systems of robots are mechanical manipulators. n modern industry, robotics has found wide application as an effective means of automating production, replacing human labor in difficult or dangerous conditions. Solving the direct and inverse problems of the kinematics of the robotic arm, as well as deriving the equations of the dynamics of the robotic arm, you can proceed to conduct research on a mathematical model.
The article is devoted to the study of the characteristics of kinematics and dynamics of the manipulator, and its mathematical model is also formed.
Key words: robot, manipulator, control system, kinematics, trajectory, Denavit-Hartenberg.
Ключевые слова: робот, манипулятор, система контроля, кинематика, Денавита-Хартенберга, траектория.
One of the reasons for the close attention to mechatronics at the present time is the constant increase in requirements for automation of production and the associated need to create production machines and complexes of them - mechanical systems with fundamentally new properties. Manipulation robot is a multi-stage, multifunctional manipulator designed to reproduce some of the working functions of human hands in order to perform various tasks. [1].
According to its structure, the manipulator is a multi-link machine, between the individual elements of which there are mechanical connections. Depending on the application, various schemes for constructing the mechanical part of the manipulator can be used, but still the main design is a sequence of links interconnected by rotational/translational pairs [2].
Industrial manipulator robots are currently used in various industrial sectors to perform a wide range of technological tasks. The main type of manipulator system for robots is mechanical manipulators. They are complex electromechanical objects with a number of unique features. Firstly, manipulator robots have a complex kinematic structure, containing many independent or interconnected links. Secondly, changing the position of these links in space affects the physical forces acting on the manipulator. Thirdly, there is a need for synchronous control of a large number of motors.
Bodies that are rigid and form part of the mechanical system of a manipulator are called links. Two links in a movable connection that are in contact with each other are called a kinematic pair. The classification of kinematic pairs can be based on the number of connections or the number of degrees of freedom of the mechanism components. The connection of links that form kinematic pairs with each other is classified as a kinematic chain. The study of manipulator kinematics involves all geometric and time-related properties of motion. The relationships between motion and the forces/moments that generate it constitute the subject of dynamics, which will also be considered. The kinematic and dynamic parameters of complex mechatronic objects, such as manipulator robots, have a significant influence on the characteristics of control systems. Therefore, it is necessary to provide a detailed mathematical description of these parameters, which is convenient for use in implementing control systems.
To study the processes in a manipulator, it is necessary first of all to create its kinematic model, i.e., a model that relates the movement of its links to the position of the end effector in space.
In three-dimensional space, to indicate the location of a point in space, it is sufficient to define its coordinates in an inertial coordinate system. When describing the position of a rigid body, its own
ОФ "Международный научно-исследовательский центр "Endless Light in Science"
Impact Factor: SJIF 2019 - 5.11 TE^HHK^ ^YKH
2020 - 5.497
2021 - 5.81
coordinate system is associated with it. Three parameters defining the orientation of the axes of the associated coordinate system relative to the inertial one (Euler angles) and three coordinates of the origin of the associated coordinate system are used. Thus, the main task of describing the kinematic characteristics of a manipulator is the transformation between its own and the selected working coordinate systems.
The method based on the use of Denavit-Hartenberg coordinate systems [4] has become widely used. The manipulator we are studying consists of two links: a link that rotates around a vertical axis with a generalized coordinate q1 and a link that rotates along a horizontal axis with a generalized coordinate q2. The movement of these coordinates determines the position of the manipulator. To plan the trajectory of the manipulator's movement and determine its position in space, we will solve the direct (FK) and inverse (IK) kinematics problems.
Equations of the direct kinematics problem. Solving the direct kinematics problem is an important first step in using a new robot, a kind of introduction to it. The direct kinematics problem of manipulators is formulated as follows: the kinematic scheme of the manipulator is given and at a certain moment in time, the values of the generalized coordinates that determine the position of all links of the manipulator relative to each other are known [4]. It is necessary to determine the position and orientation of the gripper of the last link of the manipulator in the coordinate system associated with the base.
Figure 1. Two-link, two-degree mechanics
The study two-degree mechanism (Figure 1) is a two-link mechanical manipulator with two rotational pairs.
Each link of the manipulator is an absolutely rigid homogeneous rod of length 1. The first link, 1, goes from the fixed base and is located at an angle q1 to it. The second link, 1, is attached to the end of the first link and is located at an angle q2 to it. The manipulator's gripper is located at the end of the second link. Thus, we have a coordinate system associated with the origin at the attachment point of the arm O-1 and a coordinate system at the elbow attachment point. First, let's determine how the second coordinate system is displaced relative to the first, that is, the coordinates of point A in the O coordinate system (formula (1)):
xA = l* cos^qi) ;yA = l* sin(qi) (1)
Then, the coordinates (x, y) in the first link's coordinate system are represented by formula
(2):
x =l*cos(q2 );y = I * sin(q2) (2)
From Figure 1, we see that in the O coordinate system, the second link is rotated with respect to the first link by (qx + q2):
x' = I * cos(q1 + q2 );y' = I* sin(q1 + q2) (3)
thus, we obtain the solution presented by formula (4):
x = xA + x' = I * cos(q1) + I * cos(q1 + q2)
ОФ "Международный научно-исследовательский центр "Endless Light in Science"
Impact Factor: SJIF 2019 - 5.11 TE^HHK^ ^YKH
2020 - 5.497
2021 - 5.81
y = yA+y' = 1* sm(qi) + I * sin(qi + q^) (4)
Let's consider the second method for calculating the direct kinematics problem. We will use the Denavit-Hartenberg notation to describe the rotational connections between the links. Denavit and Hartenberg proposed a matrix method for sequentially constructing coordinate systems associated with each link of a kinematic chain [4]. The essence of the Denavit-Hartenberg notation is to form a homogeneous transformation matrix with a dimension of 4x4, which describes the position of the coordinate system of each link relative to the coordinate system of the previous link (the upper left submatrix of 3x3 represents the rotation matrix; the upper right submatrix of size 3x1 represents the position vector of the origin of the rotated coordinate system relative to the absolute coordinate system; the lower left submatrix of size 1x3 defines the perspective transformation; the fourth diagonal element is the global scaling factor). This makes it possible to sequentially transform the coordinates of the manipulator end-effector from the coordinate system associated with the last link to the base coordinate system, which is the inertial coordinate system for the considered dynamic system.
Figure 2. Coordinate systems attachment of a two-link manipulator.
For the considered robot shown in Figure 2, the joint variables are q1 and q2, and the link parameters have values of a1=a2=0, d1=d2=0, a1=a2=l.
We write the transformation matrices (5) for each link of the investigated manipulator. Then, for the matrix -1T(i = 1,2) we have:
cos(q1) sin(q1) 0 I * cos^)'
1T =
-sin^i) cos^) 0 I * cos(q2)
0
0 0
1 0
0 1
1T =
Now, we form the matrix \T,
cos(q2) -sin(q2) 0 I * cos(q2)
sinfa) 0
cosfa) 0
0 1
I * sin(q2) 0
0 0 0 1
by multiplying the matrices listed above
cos(q1) sin(1) 0 l*c1 + l* c12
\T =
(5)
(6)
—sin(q1) cos(q1) 0 I * s1 + I * s12 0 0 1 0 0 0 0 1 Where c1=cos(q1), s1=sin(q1), c12=cos(q1+q2), s12=sin(q1+q2). The equations represented by formula (6) describe the kinematics of a two-degree-of-freedom manipulator. They allow calculating the position and orientation of the coordinate system {2} relative to the coordinate system {0} of the robot manipulator. Thus, the results obtained by solving the direct kinematics problem using the
0
Impact Factor: SJIF 2019 - 5.11 TE^HHK^ ^YKH
2020 - 5.497
2021 - 5.81
geometric approach and the Denavit-Hartenberg method coincide. DH representation is often used for manipulators with six or more degrees of freedom [3].
Inverse Kinematics Equations.Unlike direct kinematics, there are significantly more approaches to solving the inverse kinematics problem. The inverse kinematics problem consists of calculating the joint parameters of a manipulator knowing the coordinates of its end effector. To solve the inverse kinematics problem, we will use the kinematic diagram presented in Figure 3.
Figure 3. Kinematic diagram of a two-degree-of-freedom mechanism
Inverse kinematics allows us to find the generalized coordinates q1 and q2 for known Cartesian coordinates (x, y) of the end effector of the second link. Let's make additional constructions and introduce an additional angle a, for which we obtain (7) from the law of cosines:
d2 = I2 + (x2 + y2) — 2x1 x jx2 +y2 x cos(k) Then we will express a and obtain (2.8):
l2+(x2+y2)-d2
<x= arccos-
2xlxjx2+y2
From Figure 5, we have: tg(q1+a)=y/x. Thus, we obtain (2.9):
* У
q, = arctg--a
(7)
(8)
x
From the triangle containing the angle q2, taking into account the simplifications made, we get (10):
tg(q± + q2) =
From here, we obtain (11):
y-l*sin(q!) x-l*cos(q1)
q? = arctq-^^ — q->
42 a x-l*cos(q1) 41
(10)
(11)
Thus, the solution of the inverse kinematics problem is written in the form (12):
41
+ y
arctq — x
arccos■
I2 + (x2 +y2) — d
2*1* jx2 + y2
q2 = arctg
x2
y-l*sin(q!) x-l*cos(q1)
41
(12)
The obtained expressions allow us to precisely determine the required position of the manipulator links based on their desired absolute coordinates and spatial configuration. The first equation allows us to find the generalized coordinate q1 based on the known coordinates (x,y), and the second equation finds the angle q2. Thus, they are the solution to the inverse kinematics problem. An important feature of IK is the specificity of trigonometric function calculations. As a result of the calculations, there may be intervals where solutions will not exist, particularly when calculating the arccosine function. Another feature is that if there are three or more links, the problem will have an unlimited number of solutions, excluding special cases. When controlling the robot, it is necessary to choose the solution that is closest to the current position of the links. There is no universal solution
2
Impact Factor: SJIF 2019 - 5.11 TE^HHK^ ^YKH
2020 - 5.497
2021 - 5.81
for IK. Different approaches to solving the problem are required for various robot kinematic schemes, but the problem always boils down to simple school geometry. When solving IK, attention should be paid to instability when the links are fully extended. The joint position angles are obtained by calculating the arccosine function, and in this case, the argument of the arccosine becomes numbers close to 1 and -1. Even due to small errors in the calculation, the number under the arccosine can go slightly above 1 or slightly below -1. However, as we know, there is no arccosine for these numbers. Therefore, it is necessary to limit the argument under the arccosine in the range from -0.999 to 0.999. If the argument goes beyond this interval, make it equal to the boundary value of the interval.
REFERENCES
1.Зенкевич С. Л., Ющенко А. С. Основы управления манипуляционнымироботами. М.: Изд-во МГТУ им. Н. Э. Баумана- 2004. - 480 с
2. Юревич Е.И. Управление роботами и робототехническими системами.СПб. - 2001. - 168 с.
3.Craig J.J. Introduction to robotics:Mechanics and control. 3rd ed. Reading, MA: Addison -Wesley, 1986. P-19-256. 77
4. С.Ф. Бурдаков А., В.А. Дьяченко. Проектирование манипуляторов промышленных роботов и роботизированных комплексов. М.: Высшая школа, 1986. - 264 с.
