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12ANALYSIS OF THE DIRECT AND INVERSE KINEMATICS IN ROBOTICS
Malika Shukhrat qizi Keldiyararova Abdumalik Asror o'g'li Ziyatov
Karshi engineering economics institute
ABSTRACT
The main target of the article is to learn classification of robotic systems and analyze the trajectory planning of computer controlled industrial manipulators. During the research many useful information about the role of robots in industry and new generation of them were gathered.
Keywords. Industry, manipulator, forward kinematics, inverse kinematics, degree of freedom, position and orientation, trajectory.
Kinematics describes the movements of bodies without considerations of the cause. The relations are fundamental for all types of robot control and when computing robot trajectories. More advanced robot control involves for example moments of inertias and their effects on the acceleration of the single robot joints and movement of the tool.
In kinematic models, position, velocity, acceleration and higher derivatives of the position variables of the robot tool are studied. Robot kinematics especially studies how various links move with respect to each other and with time. This implies that the kinematic description is a geometric one.
Using coordinate frames attached to each joint, shown in Figure 2.1, the position p and orientation ^ of the robot tool can be defined in the Cartesian coordinates x,y,z with respect to the base frame 0 of the robot by successive coordinate transformations. This results in the relation
P0=fl0>n + d0 (1.1)
where p0 and pn are the position of the tool frame expressed in frame 0 and tool frame n, respectively. The rotation matrix R0n describes the rotation of the frame n with respect to the base frame 0, and gives the orientation The vector d0n describes the translation of the origin of frame n relative to the origin of frame 0.
The rigid motion (1.1) can be expressed using homogeneous transformations H as
in
Po = (?) Pn = (?)
Pc = HSPn = (R° f)P„ (1.2)
It must be mentioned that there is a variety of formulations of the kinematics, based on vectors, homogeneous coordinates, screw calculus and tensor analysis. The efficiency of any of these methods is very sensitive to the details of the analytical
formulation and its numerical implementation. The efficiency also varies with the intended use of the kinematic models.
There are two sides of the same coin describing the kinematics: forward kinematics and inverse kinematics. The concepts are illustrated in Figure 1.1 and they are shortly described in the following sections.
Figure 1.1
Figure 1.1: In forward kinematics the joint variables ql,...,q6 are known and the position and orientation of the robot tool are sought. Inverse kinematics means to compute the joint configuration ql,...,q6 from a given position and orientation of the tool.
In Part I, the focus is on modelling the forward kinematics. The main interest is the principal structure, and issues regarding efficiently implementation have not been considered.
The work is based on homogeneous transformations using the Denavit-Hartenberg (D-H) representation, which gives coordinate frames adapted to the robot structure. In this section the essential kinematic relations are introduced.
Position and orientation
The aim of forward kinematics (also called configuration or direct kinematics) is to compute the position and orientation of the robot tool as a function of the joint variables ql,..., qn, as is illustrated in Figure 1.1. The position p and orientation <p of the tool frame n relative to the base frame 0 of the robot are denoted
x = (l) = f(q) (13)
Where q = (ql,..., qn) is the vector ofjoint variables. The forward kinematics for a robot having n degrees of freedom is given by the homogeneous transformation as
T£(q) = (1.4)
The orientation ^ of the tool frame n relative the base frame 0 is given by the rotation matrix Ron, and the position p of the origin of the tool frame with respect to the base frame is given by the translational part d0n. Note that both R0n and d0n are functions of the joint variables.
Velocity and acceleration
Starting from the direct kinematics relation (1.4), it is possible to derive a relationship between the joint velocities 'q and the linear velocity v and angular velocity ® of the tool. This relation is expressed by the Jacobian J(q),which in general is a function of the joint variables q. To sum up, the velocity is expressed as a function of q and q by the relation
x = {i) = 0=J(i)4 (15)
Differentiating this expression gives the acceleration of the tool expressed in joint variables q and the derivatives q, q as
* = {{} = Q = Tt (J(q= J(q)« + Tt (J(q)q) (16)
Inverse kinematics means computing the joint configuration q1,...,qn of the robot from the position p and orientation 9 of the robot tool, as is shown in Figure 1.1 Solving the inverse kinematics problem is important when transforming the motion specifications of the tool into the corresponding joint angle motions to be able to execute the motion. It is however in general a much more difficult problem to solve for a serial robot than the forward kinematics problem discussed above. For parallel kinematic robots as Figure 1.1, it is easier to calculate the inverse kinematics.
Inverse position and orientation
The general inverse kinematics problem is given the homogeneous transformation H = (R(q) d(q)) , (17)
find a solution, or possibly multiple solutions, of the relation
= (q±.....qn)= A1(q1).....An(qn) = H (1.8)
H represents the desired position and orientation of the tool, and the joint variables q1,...,qn are to be found so that the relation Tq = (q1 ,...,qn) = H is fulfilled. The relation (1.8) gives 12 nontrivial, nonlinear equations in n unknown variables, written as
Tij = (qi.....qn), i = 1,2,3, j = 1.....4. (1.9)
The forward kinematics always has a unique solution for serial robots. The inverse kinematics problem can however have a solution, not necessarily unique, or no solution. Even if a solution may exist, the relation (1.8) generally gives complicated nonlinear functions of the joint variables, which makes it more complicated.
When solving the inverse kinematics problem, a solution of the form
Qk = fkihi,-,^), k = l,...,n, (1.10)
is the most useful. One example of an application is when tracking a welding seam and the desired location originates from a vision system. A closed form of the kinematic equations is an advantage, since it needs less computing power compared to an iterative solution. The inverse kinematic equations generally have multiple solutions, and closed form solutions then make it easier to develop rules for choosing a particular solution.
Inverse velocity and acceleration
The Jacobian J(q) relates the joint velocities q and the Cartesian velocity X = (u ()T of the tool. Finding the inverse velocity and acceleration is actually easier than finding the inverse position p and orientation under the assumption that the joint variables q are known. When the Jacobian is square and nonsingular ( detj(q) ^ 0 ), the relation 1.5 gives
q = (J(q))-1x (1.11)
If the robot does not have six joints, the non-square Jacobian cannot be inverted. Then there is a solution to (1.5) if and only if X lies in the range space of the Jacobian, that is
rank J(q) = rank(J(q)) X). (112)
When the robot has n > 6 joints, the relation can be solved for q using the pseudoinverse J f, as in
q= JfX + (I — JfJq) b, (1.13)
where b is an arbitrary vector. The pseudoinverse can be constructed using singular value decomposition; J = UEVT and J f= VI f UT. The first term J f X in (1.13) is related to minimum-norm joint velocities, which can be seen from the formulation of the inverse velocity problem as a constrained linear optimization problem .The second term, named the homogeneous solution, means that all vectors of the form (I — J f J) b, lie in the nullspace of J. If X = 0, it is possible to generate internal motions, described by (I — J f J) b, that reconfigure the robot structure without changing the position and orientation of the tool.
The acceleration q can be determined under the assumption that the Jacobian J(q) is non singular and that the joint variables q and the derivatives qare known. The acceleration is then computed using (1.6), giving
q = (j(q)Y\x — Tr(J(l))i)
Conclusion. Today we find most robots working for people in industries, factories, warehouses, and laboratories. Robots are useful in many ways. For instance, it boosts economy because businesses need to be efficient to keep up with the industry
competition. Therefore, having robots helps business owners to be competitive, because robots can do jobs better and faster than humans can, e.g. robot can built, assemble a car. Yet robots cannot perform every job; today robots roles include assisting research and industry. Finally, as the technology improves, there will be new ways to use robots which will bring new hopes and new potentials.
Efficiency is always one of the key aspects of industrial world and world class manufacturing. Good and services we are utilizing now is the result of new technologies, know-how's and innovations. In order to get best possible circumstances from whole bunch of scenarios, one should use highly effective information technology system and robots.
In this article, described the performance analysis of robots which is important in many fields of industrial facility, by working in high accuracy, and get possible performance. It begins with the definition of the robotics and continues with the robot kinematics.
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