Manipulability and Kinematics Analysis of Three-Linked Robotic Arm Using Mathematical Program

 

Yokesh K. S.*, Nandakumar N

Department of Mechanical Engineering, Government College of Technology, Coimbatore, Tamil Nadu, India.

*Corresponding Author E-mail: ksyokesh98@gmail.com

 

ABSTRACT:

The Kinematics study deals with the motion of the bodies and the mechanism without consideration of force. Geometry is applied by robot kinematics to the research movement of multi-degree freedom kinematic links that shape the robot manipulator structure. The most critical problem is locating the end effector and angle of the robot manipulator. Used in automation industries to perform various tasks. To overcome this problem, Singular Value Decomposition (SVD) of the Jacobin matrix has been derived using length and angle for robotic manipulator links. In this work, a mathematical program for solving kinematics of the three linked robotic arms has been developed. End effector position x, y and joint angles θ1, θ2, θ3 for the robotic arm have been simulated and values are obtained by using MATLAB software. From the developed mathematical program, the required progression positions of the robotic arm over time have been achieved.

 

KEYWORDS: Kinematics, Manipulability, Workspace, Robot arm, Joint angle

 

 

INTRODUCTION:

Robot kinematics examines the motion of multi DOF connections and joints used to build robotic structures using morphology. The value of mechanics indicates that the robot attachments are stable relations and that its joints are capable of pure translation or rotating. It investigates the relationship between these parameters and the connection of links and joints, as well as the direction, velocity, and acceleration of each link in the mechanical manipulator, without taking into account the forces that influence this motion. The relationship between movement and the forces and torques that go with it is explored in robot mechanics. Flexibility, collision avoidance, and singularity avoidance are all topics covered by Robot Kinematics. Each component of the robot is responsible for the linear motion that is used to determine a reference point, resulting in a robot made up of many components, each with its relation. The robot's single manipulator arm is a source of concern for us in terms of performance. The manipulator's immovable foundation part is named 0, and the first connection linked to it is named 1, then 2, and so on until the last nth connection is named n. Each relation is given a number that is individually assigned. A manipulator is recognized as a robot arm. It consists of a number of joints divided in space by the connections of the arm. The joints are where the movement happens in the arm. The basic components of the robot arm consist of a base, joints, links, and a gripper. The base is the fundamental component connected to the arm, which can either be fixed or dynamic. The joint is adjustable and is connected by two distinct links. The link is fixed and serves the gripper. The gripper would be the last component of the robot arm and can be used for holding and shifting objects. The goal of the robot arm research is to conduct an analysis of the motions of each component of the interactive content. Kinematic analysis1,2 is a division of classical mechanics that explains how structures and entities move beyond consideration of the cause of movement. Fig. 1 shows a simple block diagram of a kinematic model. There are two different problems to address in the kinematic evaluation of the manipulator location: forward kinematics and inverse kinematics3-5. Forward kinematics required the solution of a forward translation formula in order to obtain the position of the robotic link in the form of displacements and angles between connections. Inverse kinematics encompasses solving the backward translation formula to predict the association between the connections and the arm's location in the workspace. Sergei Evgenievich Inanov et al. presented an analytical approach for studying statistical models of three degrees of freedom of robot dynamics6. These approaches provide a solution to the system's nonlinear components. They created and implemented a software for evaluating multi-link robot manipulators. The software kit interfaces to the mysql database infrastructure, which uses the Network database engine to control the design. To monitor the manipulator's gripping direction, a graphical module interface was developed. Tahseen F. Abaas et al. presented a kinematics modeling of a 5 DOF robot arm used for pick and place operations7,8. Forward, inverse, and Velocity kinematics are derived and calculated using the Jacobian matrix. They adopted a modelling method to simulate the location and angle of the gripper. The modelling method has been developed based on the DH parameters and solved using MATLAB software and the movement of the robotic arm is controlled using a microcontroller. The errors in position results are validated. Shravan Anand Komakula et al. derived an analytical method which determines the spatial configuration of the robot arm9-11. This method establishes an approach to analyze the workspace of a 3 link chain motion at joints. Development of spatial control algorithm in the computation of conventional inverse kinematics. The program has been developed in MATLAB software as the tool to achieve the working space of a robotic arm from the geometric aspects of the work. Ruthber Rodriguez Serrezuela et al. designed and implemented a kinematic model for a robot arm manipulator12-15. In this robot, performance is checked mathematically by using the proposed matrices of the DH method. Direct kinematics and inverse was developed with a toolbox implemented in MATLAB; Arduino using a card like interface to determine and control the position of the robotic arm. Ali Roshanianfard et al. investigated a new robotic arm with 5 DOF work for harvesting heavy crops, and the robotic arm was designed and analyzed using Solid Works software16-18. The forward and reverse kinematics were solved using the D-H process. With the right adjustments, the adaptive controller could provide improved results even without the expense of traditional benefit selection. The findings demonstrated that the optimization under investigation was successful.

 

Figure 1: Block diagram for kinematic model

 

METHODOLOGY:

The methodology refers to the mathematical program that was used in the design and analysis of three-linked Robotic arm.

a.     Design of three-linked robotic arm

b.     Analysis of Joint parameters

c.     Analysis of Manipulability of the ellipse

d.     Analysis of Workspace

 

Design of three-linked Robotic arm:

The basic 2D diagram of a three-link robotic arm with linkage location is shown in Fig. 2. As seen in Fig. 2, the length of the link is l1, l2, and l3. The first link's angle of reference to the x-axis is determined counterclockwise. In comparison to the previous link, the angles of corresponding links are determined counterclockwise. The equations for the three-link robotic arm were derived using this approach, and the program was written in MATLAB software19,20.

 

Figure 2: Three-link robotic arm in XY plane.

 

Analysis of Joint parameters:

The forward direction of the manipulator is defined by the term in the equations. As a result, angular position and velocities are used to estimate the orientation of end effector locations. The complicated issue of inverse direction, which involves determining how and when to move the angular position to produce a better end-effector motion, is frequently associated with robotic manufacturing. The issue could be interpreted as a minimum of two equations with 3 unknown variables. In particular, there are several solutions to these issues, and the Jacobian's Singular Value Decomposition will benefit in identifying one. The lengths of the links L1 = 2cm, L2 = 3cm, and L3 = 1cm are used in the computation. The location of the end effector in the X-Y graph is determined using the Kinematics equation and considering constants as θ1 = 45°, θ2 = 60° and θ3 = -25°. The kinematics end effector movement of the robotic arm with links and joints are described in the following equation.

 

X3 = L1*cos (θ1) + L2*cos (θ1 + θ2) + L3*cos (θ1 + θ2 + θ3)

Y3 = L1*sin (θ1) + L2*sin (θ1 + θ2) + L3*sin (θ1 + θ2 + θ3)

 

X3 and Y3 are the robotic end effector location. The velocity of X3 and Y3 are rearranging in matrix form as shown in equation 3.

 

 

The 2 X 3 matrix in equation 3 is recognized as the Jacobian matrix and symbolized by J (θ). In Jacobian matrix (+) function is defined as pseudoinverse.

 

To get the orientation at the following control interval, the orientations are updated based on the velocities. The updated orientation is then applied to equation in the results next phase.

 

Analysis of Manipulability of an ellipse:

Manipulation ellipses were created using the following equations x and y. The very first segment of a program for the manipulability ellipsoid of the robotic arm is similar to the analysis of joint parameters where the joint locations, joint angles, and Jacobian matrix were described, followed by the SVD of a Jacobian. The principal axis' distances, commonly represented through a and b. To make the ellipsoid suit in the diagram frame, a scale value of 0.3 is assigned to the sizes. Whenever the robotic arm seems to be in a predetermined position, a manipulability ellipsoid can be used to imagine in which areas the end effector can travel. The ellipsoid could be used to describe this concept by combining analysis of joint parameters. The SVD of the Jacobian matrix can be used to evaluate the scale and form of an ellipsoid. The ellipse's axis distances singular values. The ellipsoid contracts in a direction opposite to the manipulator also at singularities. Most of a program is used to perform certain minimal yet crucial activities, including establishing parameters, modifying values, and displaying the output. The equation used to generate ellipse manipulability and to generate link position and angle are given below

 

x = a cost

y = b sint

 

Analysis of Workspace:

The robot arm was mathematically modelled using the Jacobian approach and kinematics and then evaluated utilizing MATLAB to determine the workspace. The x and y coordinates are determined utilizing kinematics equations for each variation of θ1, θ2, and θ3 parameters. Data is created for all combinations of θ1, θ2, and θ3 parameters and stored into a database for use as labelled data in the corresponding source file.

 

[THETA1, THETA2, THETA3] = meshgrid (θ1, θ2, θ3);

 

Mesh grid generates two-dimensional network locations based on positions found in variables x and y. This programming is used to construct a mesh network. X is a variable in which each row is a duplicate of x, and Y is a variable in which each column is a duplicate of y. The X and Y positions display a graph of distance (y) rows and distance (x) columns. Assume that the very first joint link has minimal rotational mobility, with a range of 0° to 90°. Then the second link joint has minimal rotational flexibility and could only rotate a range of 0° and 180°, and then the third joint has minimal rotational control and could only rotate within 0° and 90°.

 

RESULT AND DISCUSSION:

Results of Joint parameters:

Numerous outcomes of considering input variables of link length of l1= 2m, l2= 3m, and l3=1m, and angles of a joint as 45°, 60°, and -20° respectively, end effector location and marginal increase in joint angle of link are calculated. Its main and progressive motion were estimated at 1s cycle and visualised in the XY graph. The graph resulting from program is seen in fig. 4. A speed of 5 m/s has been used in the analysis, so motions of the robotic manipulator were performed. From fig. 4a, the progressive forward trajectory is determined with a positive velocity on the X-axis and a negative velocity on the Y-axis. That robotic manipulator gradually decreases and right at a 45° from the present condition, and thus the arm travels as appropriate. Joint angles and End effector positions of robotic arm determined at the velocity of Ẋ= 5 and Ẏ= -5 are shown in table 1. The same program, with the same boundary position but a distinct possible end-effector motion, yielded a significant outcome. From fig. 4b, the progressive inverse trajectory is estimated with a positive velocity on the X-axis and Y-axis. Initially, the arm going as intended, with the end effector going up and right at a 45°. The robotic manipulator moves as expected initially, with the effector position going up and right at 45°. This depicts the arm link attempting to reach a distant location, but it begins to appeal up. The robot arm is nearly absolutely extended out to θ2 = 0 and θ3 = 0 and closer to its computation singularity, which describes such conduct. The robot arm can't travel outward very rapidly as it would in many locations throughout this design. However, the unusual and intriguing occurs in the final iteration. Table 2 shows the Joint angles and end effector location of the robotic arm estimated at the velocity of Ẋ = 5 and Ẏ = 5. Joint parameters are illustrated in fig 3.

 

Figure 3: Plot for Joint parameters for a)  Ẋ = 5 and  Ẏ = -5 b)  Ẋ = 5 and  Ẏ = 5

 

Table 1: Measured link parameters for  Ẋ = 5 and  Ẏ = -5

Positions

Joint angle

End Effector location

θ1

θ2

θ3

x3

y3

0

45

60

-20

0.805

5.298

1

41.25

63.71

-20.5

0.922

5.196

2

37.16

67.99

-20.99

1.026

5.096

3

33.59

71.18

-21.33

1.138

4.996

4

30.07

74.24

-21.44

1.244

4.896

5

26.63

77.24

-21.55

1.356

4.796

6

23.2

80.12

-22.27

1.469

4.694

7

19.95

82.63

-22.31

1.575

4.586

8

16.54

85.1

-23.17

1.678

4.486

9

13.35

87.34

-23.17

1.791

4.386

10

10.28

89.46

-23.58

1.910

4.286

 

Figure 4: Robotic arm progressive position for a)  Ẋ = 5 and  Ẏ = -5 b)  Ẋ = 5 and  Ẏ = 5

 

Table 2: Measured link parameters for  Ẋ = 5 and  Ẏ = 5

Positions

Joint angles

End effector location

θ1

θ2

θ3

x3

y3

0

45

60

-20

0.729

5.31

1

47.5

54.69

-19.95

0.829

5.41

2

51.89

48.82

-19.45

0.922

5.504

3

54.56

42.19

-19.27

1.024

5.588

4

58.93

34.1

-17.54

1.052

5.682

5

64.23

24.47

-14.79

1.22

5.768

6

72

10.97

-10.97

1.302

5.83

7

85.04

-15.01

-8.81

1.384

5.806

Results of Manipulability ellipse:

The SVD of the Jacobian robotic arm mathematical program is created to obtain manipulability ellipses, and the outcomes are shown in fig. 5. From fig. 5a it shows the orientation of the robotic arms end effector location and manipulability ellipse with angular values of θ1 = 15°, θ2 = 100°, and θ3 = 150° provided to the created program. From a starting position, the arm can move on both axes, and the y-axis requires more flexibility than the x-axis. Fig. 5b shows the orientation of the manipulator end-effector location and ellipse with an angular position of 1 = 35°, 2 = 30°, and 3 = 15° assigned to a designed program. The robotic arm seems too far extended out during this design, as well as the ellipse is far more curved, illustrating the reality that perhaps the end effector can't travel as far radially stretched because it is parallel to the X-axis.

 

Figure 5: Position of Ellipse when (a) θ1 = 15°, θ2 = 100°, and θ3 = 150° (b) θ1 = 35°, θ2 = 30°, and θ3 = 15°

 

Results of Workspace:

The mesh grid of the 2-D scattered plot of the coordinates found in variables x and y is illustrated in Fig. 5. The scatter resolution could be seen spreading across the area within the zone ringed by end-effector points in a loop with such a distance of 9 cm from the origin, which would be the robot's workspace volume. The program was written in a looping method graph in which the three joint angles are expected to rise in 5 intervals for bound joint angles and 10 intervals for unrestrained joints, resulting in 50,653 end-effector positions in the workspace and depicting the positions as well as joint variables in the appropriate graphs. The scatter plot can be further compiled to measure the mobility of the arms.

 

Figure 6: End effectors work positions

 

CONCLUSION:

In this project work, the Kinematic analysis of robotic arms using the Mathematical Program has been developed and performed successfully. The following conclusions have been made from the results obtained using the MATLAB program.

·       For given positive X direction velocity (X = 5 m/s) and negative Y direction Velocity ( Y = -5 m/s) the complete progressive position of robotic arm have been achieved

·       For the given positive X direction velocity (X = 5 m/s) and positive Y direction velocity (Y = 5 m/s), the progressive position is achieved only up to seventh iteration. The arm is almost entirely extended over (θ2 = 0° and θ3 = 0°), which induces this conduct. It tends to retrench.

·       Outstretched position has been identified after θ1= 85.04, θ2= -15.01, and θ3= -8.81 corresponding end effector positions of X3 = 0.692, Y3 = 2.903.

·       By using different combinations of Joint angles, the cyclic end-effector positions have been generated using the MATLAB program.

 

CONFLICT OF INTEREST:

The authors have no conflicts of interest regarding this investigation.

 

ACKNOWLEDGMENTS:

I thank the Government College of Technology, Coimbatore, Tamil Nadu, India, which gave full support in conducting the research work in this paper.

 

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Received on 07.05.2021            Accepted on 27.09.2021

©A&V Publications all right reserved

Research J. Engineering and Tech. 2021;12(3):59-65.

DOI: 10.52711/2321-581X.2021.00010