Direct Kinematic modeling of 6R Robot using Robotics Toolbox
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The document discusses the direct kinematic modeling of a 6-degree-of-freedom robotic arm using Peter Corke's robotics toolbox in MATLAB. It details the approach to represent the kinematics and dynamics of the robotic manipulator, employing Denavit-Hartenberg parameters for calculation. The paper also presents simulation results, including both the end-effector trajectory and joint movements over time.
![Prashant Badoni Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 6, Issue 1, (Part - 3) January 2016, pp.151-154
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Direct Kinematic modeling of 6R Robot using Robotics Toolbox
Prashant Badoni*
*(Department of Mechanical Engineering, Graphic Era University, Dehradun-248002)
ABSTRACT
The traditional approaches are insufficient to solve the complex kinematics problems of the redundant robotic
manipulators. To overcome such intricacy, Peter Corke’s Robotics Toolbox [1] is utilized in the present study.
This paper aims to model the direct kinematics of a 6 degree of freedom (DOF) Robotic arm. The Toolbox uses
the Denavit-Hartenberg (DH) Methodology [2] to compute the kinematic model of the robot.
Keywords - Direct Kinematics Solution, MATLAB, Robotic Toolbox, Robot manipulator.
I. INTRODUCTION
Kinematic modeling of a robot is concerned with
its motion without consideration of forces. The
kinematic modeling of a robot is usually categorized
into forward or direct kinematics and inverse
kinematics. In the direct kinematics problem, the
location of end effector in the work space i.e. position
and orientation, is determined based on the joint
variables [3] [4]. The inverse kinematics problem
refers to finding the values of the joint variables that
allows the manipulator to reach the given location.
The relationship between direct and inverse
kinematics is illustrated in Fig.1.
Fig.1. Relationship between direct and inverse kinematics
The present paper describes the modeling and
analysis of a 6R robot with the application of
Robotics Toolbox [1] in MATLAB. The Toolbox is
based on a general method of representing kinematics
and dynamics of serial link manipulators by
description matrices. In this study, the toolbox is used
for direct kinematic solution of a six link robot
manipulator. The toolbox is also capable of plotting
3D graphical robot and allows users to drive the robot
model. A general serial 6R robot is shown in Fig.2.
Fig.2. Schematic of a general serial 6R robot manipulator [5]
II. DIRECT KINEMATIC MODEL
Direct kinematic analysis of 6R robot is done by
using Robotics Toolbox in MATLAB. Various
functions are used to complete the analysis
conveniently. First, create the six links and set the D-
H parameters as given in Table 1.
Table 1: D-H parameters of the manipulator
Links ai αi di θi Range
1 0 -90 1 θ1 0 to 360
2 0 90 4 θ2 -27 to 189
3 10 0 3 θ3 -241 to 82
4 10 0 0 θ4 0 to 360
5 0 -90 3 θ5 -100 to 100
6 0 0 1 θ6 0 to 360
Here θi is joint angle, di is joint offset, ai is link
length and αi is the twist angle. The limits of each
joint angle is given in the above table and utilized in
the m-file.
The joint variable is set to be zero at the starting.
A few workspace variables are created to define the
useful joint angles: qz for all starting position, qr for
ready position or reach position. The trajectory
between any of these two joint angle vectors can be
generated with respect to time. After that, the forward
kinematic can be computed easily by using fkine
function, which returns a homogeneous
transformation for the last link of the robot. Here is
the source code in MATLAB.
Direct
Kinematics
Inverse
Kinematics
Link Parameters
Joint
angles
Position and
Orientation of
end-effector
Link Parameters
Joint
angles
RESEARCH ARTICLE OPEN ACCESS](https://image.slidesharecdn.com/u61031511541-160727051148/85/Direct-Kinematic-modeling-of-6R-Robot-using-Robotics-Toolbox-1-320.jpg)
![Prashant Badoni Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 6, Issue 1, (Part - 3) January 2016, pp.151-154
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% create links using D-H parameters
% L = Link([theta, d, a, alpha]
L(1) = Link([0 1 0 –pi/2]);
L(2) = Link([0 4 0 pi/2]);
L(3) = Link([0 3 10 0]);
L(4) = Link([0 0 10 0]);
L(5) = Link([0 3 0 –pi/2]);
L(6) = Link([0 1 0 0]);
% Joint limits
L(1).qlim = pi/180*[0 360];
L(2).qlim = pi/180*[-27 189];
L(3).qlim = pi/180*[-241 82];
L(4).qlim = pi/180*[0 360];
L(5).qlim = pi/180*[-100 100];
L(6).qlim = pi/180*[0 360];
% create the robot model
Robot = SerialLink(L);
Robot.name = ‘Robot’
% starting position
qz = [0 0 0 0 0 0];
% ready position
qr = [pi/4 0 pi/4 0 –pi/2 pi];
% generate a time vector
t = [0:0.056:2];
% computes the joint coordinate trajectory
q = jtraj(qz, qr, t);
% direct kinematics for each joint co-ordinate
T = Robot.fkine(q)
Joint space trajectory can be animated by using the
following command:
>>Robot.plot(q)
Starting and ready position of the robot manipulator
from this simulation are given below in Fig.3 and
Fig.4:
Fig.3. Zero Pose of Robot
Fig.4. Ready Pose of Robot
3D Trajectory of the end-effector of the robot can be
plotted by:
>>figure,plot3(squeeze(T(1,4,:)),squeeze(T(2,4,:)),
squeeze(T(3,4,:)));
xlabel('X(inch)'),ylabel('Y(inch)'),zlabel('Z(inch)'),
title(‘End-effector 3D Trajectory’),grid;
Fig.5. Robot direct kinematics end-effector trajectory
Cartesian coordinates x, y, z of end-effector can be
plotted against time by:
>>figure,plot(t,squeeze(T(1,4,:)),
'-',t,squeeze(T(2,4,:)),'--',t,squeeze(T(3,4,:)),'-.');
xlabel('Time(s)'), ylabel('Coordinate(inch)'),grid,
legend('X','Y','Z'), title('End-effector Position');
Fig.6. Robot direct kinematics end-effector X, Y, Z coordinates](https://image.slidesharecdn.com/u61031511541-160727051148/85/Direct-Kinematic-modeling-of-6R-Robot-using-Robotics-Toolbox-2-320.jpg)
![Prashant Badoni Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 6, Issue 1, (Part - 3) January 2016, pp.151-154
www.ijera.com 153 | P a g e
Generalized coordinates for each joint of the robot
can be plotted against time by:
>>subplot(6,1,1); plot(t,q(:,1));
xlabel('Time(s)'); ylabel('Joint1(rad)');
subplot(6,1,2); plot(t,q(:,2));
xlabel('Time(s)'); ylabel('Joint2(rad)');
subplot(6,1,3); plot(t,q(:,3));
xlabel('Time(s)'); ylabel('Joint3(rad)');
subplot(6,1,4); plot(t,q(:,4));
xlabel('Time(s)'); ylabel('Joint4(rad)');
subplot(6,1,5); plot(t,q(:,5));
xlabel('Time(s)'); ylabel('Joint5(rad)');
subplot(6,1,6);plot(t,q(:,6));
xlabel('Time(s)'); ylabel('Joint6(rad)');
Fig.7. Robot direct kinematics joint angles against time
Velocity and acceleration profile can be obtained by:
>> [q,qd,qdd] = jtraj(qz, qr, t)
Velocity profile of joints of the robot manipulator is
given by:
>>subplot(6,1,1); plot(t,q(:,1));title(‘Velocity’);
xlabel('Time(s)'); ylabel('Joint1(rad/s)');
subplot(6,1,2); plot(t,q(:,2));
xlabel('Time(s)'); ylabel('Joint2(rad/s)');
subplot(6,1,3); plot(t,q(:,3));
xlabel('Time(s)'); ylabel('Joint3(rad/s)');
subplot(6,1,4); plot(t,q(:,4));
xlabel('Time(s)'); ylabel('Joint4(rad/s)');
subplot(6,1,5); plot(t,q(:,5));
xlabel('Time(s)'); ylabel('Joint5(rad/s)');
subplot(6,1,6);plot(t,q(:,6));
xlabel('Time(s)'); ylabel('Joint6(rad/s)');
Fig.8. Joint velocity against time
Acceleration profile is given by:
>>subplot(6,1,1); plot(t,q(:,1));title(‘Acceleration’);
xlabel('Time(s)'); ylabel('Joint1(rad/s2)');
subplot(6,1,2); plot(t,q(:,2));
xlabel('Time(s)'); ylabel('Joint2(rad/s2)');
subplot(6,1,3); plot(t,q(:,3));
xlabel('Time(s)'); ylabel('Joint3(rad/s2)');
subplot(6,1,4); plot(t,q(:,4));
xlabel('Time(s)'); ylabel('Joint4(rad/s2)');
subplot(6,1,5); plot(t,q(:,5));
xlabel('Time(s)'); ylabel('Joint5(rad/s2)');
subplot(6,1,6);plot(t,q(:,6));
xlabel('Time(s)'); ylabel('Joint6(rad/s2)');
Fig.9. Joint acceleration against time
III. CONCLUSION
In this paper, the direct kinematic model of a 6R
robot is presented by utilizing the principal features
of the Robotic Toolbox in MATLAB. The Toolbox
provides several essential tools for the modeling and
analysis of the robotic manipulator, as well as
analyzing the experimental results.](https://image.slidesharecdn.com/u61031511541-160727051148/85/Direct-Kinematic-modeling-of-6R-Robot-using-Robotics-Toolbox-3-320.jpg)
![Prashant Badoni Int. Journal of Engineering Research and Applications www.ijera.com
ISSN: 2248-9622, Vol. 6, Issue 1, (Part - 3) January 2016, pp.151-154
www.ijera.com 154 | P a g e
REFERENCES
[1] P.I. Corke, A Robotics Toolbox for
MATLAB, IEEE Robotics and Automation
Magazine, Vol.3, No.1, pp.24-32, March
1996.
[2] J. Denavit, R.S. Hartenberg, A Kinematic
Notation for Lower-Pair Mechanisms Based
on Matricies, Trans. of ASME J.App. Mech.
vol. 77, pp.215-221, June 1955.
[3] J.J. Crage, Introduction to Robotics
Mechanics and Control, 3rd Edition,
Prentice Hall, 2005.
[4] M.W. Spong, S. Hutchinson and M.
Vidyasagar, Robot Modeling and Control,
1st
Edition, Jon Wiley & Sons Inc, 2005.
[5] Zhongtao Fu, Wenyu Yang and Zhen Yang,
Solution of Inverse Kinematics for 6R Robot
Manipulators With Offset Wrist Based on
Geometric Algebra, Journal of Mechanism
and Robotics, August 2013, Vol. 5.](https://image.slidesharecdn.com/u61031511541-160727051148/85/Direct-Kinematic-modeling-of-6R-Robot-using-Robotics-Toolbox-4-320.jpg)
Direct Kinematic modeling of 6R Robot using Robotics Toolbox
- 1. Prashant Badoni Int. Journal of Engineering Research and Applications www.ijera.com ISSN: 2248-9622, Vol. 6, Issue 1, (Part - 3) January 2016, pp.151-154 www.ijera.com 151 | P a g e Direct Kinematic modeling of 6R Robot using Robotics Toolbox Prashant Badoni* *(Department of Mechanical Engineering, Graphic Era University, Dehradun-248002) ABSTRACT The traditional approaches are insufficient to solve the complex kinematics problems of the redundant robotic manipulators. To overcome such intricacy, Peter Corke’s Robotics Toolbox [1] is utilized in the present study. This paper aims to model the direct kinematics of a 6 degree of freedom (DOF) Robotic arm. The Toolbox uses the Denavit-Hartenberg (DH) Methodology [2] to compute the kinematic model of the robot. Keywords - Direct Kinematics Solution, MATLAB, Robotic Toolbox, Robot manipulator. I. INTRODUCTION Kinematic modeling of a robot is concerned with its motion without consideration of forces. The kinematic modeling of a robot is usually categorized into forward or direct kinematics and inverse kinematics. In the direct kinematics problem, the location of end effector in the work space i.e. position and orientation, is determined based on the joint variables [3] [4]. The inverse kinematics problem refers to finding the values of the joint variables that allows the manipulator to reach the given location. The relationship between direct and inverse kinematics is illustrated in Fig.1. Fig.1. Relationship between direct and inverse kinematics The present paper describes the modeling and analysis of a 6R robot with the application of Robotics Toolbox [1] in MATLAB. The Toolbox is based on a general method of representing kinematics and dynamics of serial link manipulators by description matrices. In this study, the toolbox is used for direct kinematic solution of a six link robot manipulator. The toolbox is also capable of plotting 3D graphical robot and allows users to drive the robot model. A general serial 6R robot is shown in Fig.2. Fig.2. Schematic of a general serial 6R robot manipulator [5] II. DIRECT KINEMATIC MODEL Direct kinematic analysis of 6R robot is done by using Robotics Toolbox in MATLAB. Various functions are used to complete the analysis conveniently. First, create the six links and set the D- H parameters as given in Table 1. Table 1: D-H parameters of the manipulator Links ai αi di θi Range 1 0 -90 1 θ1 0 to 360 2 0 90 4 θ2 -27 to 189 3 10 0 3 θ3 -241 to 82 4 10 0 0 θ4 0 to 360 5 0 -90 3 θ5 -100 to 100 6 0 0 1 θ6 0 to 360 Here θi is joint angle, di is joint offset, ai is link length and αi is the twist angle. The limits of each joint angle is given in the above table and utilized in the m-file. The joint variable is set to be zero at the starting. A few workspace variables are created to define the useful joint angles: qz for all starting position, qr for ready position or reach position. The trajectory between any of these two joint angle vectors can be generated with respect to time. After that, the forward kinematic can be computed easily by using fkine function, which returns a homogeneous transformation for the last link of the robot. Here is the source code in MATLAB. Direct Kinematics Inverse Kinematics Link Parameters Joint angles Position and Orientation of end-effector Link Parameters Joint angles RESEARCH ARTICLE OPEN ACCESS
- 2. Prashant Badoni Int. Journal of Engineering Research and Applications www.ijera.com ISSN: 2248-9622, Vol. 6, Issue 1, (Part - 3) January 2016, pp.151-154 www.ijera.com 152 | P a g e % create links using D-H parameters % L = Link([theta, d, a, alpha] L(1) = Link([0 1 0 –pi/2]); L(2) = Link([0 4 0 pi/2]); L(3) = Link([0 3 10 0]); L(4) = Link([0 0 10 0]); L(5) = Link([0 3 0 –pi/2]); L(6) = Link([0 1 0 0]); % Joint limits L(1).qlim = pi/180*[0 360]; L(2).qlim = pi/180*[-27 189]; L(3).qlim = pi/180*[-241 82]; L(4).qlim = pi/180*[0 360]; L(5).qlim = pi/180*[-100 100]; L(6).qlim = pi/180*[0 360]; % create the robot model Robot = SerialLink(L); Robot.name = ‘Robot’ % starting position qz = [0 0 0 0 0 0]; % ready position qr = [pi/4 0 pi/4 0 –pi/2 pi]; % generate a time vector t = [0:0.056:2]; % computes the joint coordinate trajectory q = jtraj(qz, qr, t); % direct kinematics for each joint co-ordinate T = Robot.fkine(q) Joint space trajectory can be animated by using the following command: >>Robot.plot(q) Starting and ready position of the robot manipulator from this simulation are given below in Fig.3 and Fig.4: Fig.3. Zero Pose of Robot Fig.4. Ready Pose of Robot 3D Trajectory of the end-effector of the robot can be plotted by: >>figure,plot3(squeeze(T(1,4,:)),squeeze(T(2,4,:)), squeeze(T(3,4,:))); xlabel('X(inch)'),ylabel('Y(inch)'),zlabel('Z(inch)'), title(‘End-effector 3D Trajectory’),grid; Fig.5. Robot direct kinematics end-effector trajectory Cartesian coordinates x, y, z of end-effector can be plotted against time by: >>figure,plot(t,squeeze(T(1,4,:)), '-',t,squeeze(T(2,4,:)),'--',t,squeeze(T(3,4,:)),'-.'); xlabel('Time(s)'), ylabel('Coordinate(inch)'),grid, legend('X','Y','Z'), title('End-effector Position'); Fig.6. Robot direct kinematics end-effector X, Y, Z coordinates
- 3. Prashant Badoni Int. Journal of Engineering Research and Applications www.ijera.com ISSN: 2248-9622, Vol. 6, Issue 1, (Part - 3) January 2016, pp.151-154 www.ijera.com 153 | P a g e Generalized coordinates for each joint of the robot can be plotted against time by: >>subplot(6,1,1); plot(t,q(:,1)); xlabel('Time(s)'); ylabel('Joint1(rad)'); subplot(6,1,2); plot(t,q(:,2)); xlabel('Time(s)'); ylabel('Joint2(rad)'); subplot(6,1,3); plot(t,q(:,3)); xlabel('Time(s)'); ylabel('Joint3(rad)'); subplot(6,1,4); plot(t,q(:,4)); xlabel('Time(s)'); ylabel('Joint4(rad)'); subplot(6,1,5); plot(t,q(:,5)); xlabel('Time(s)'); ylabel('Joint5(rad)'); subplot(6,1,6);plot(t,q(:,6)); xlabel('Time(s)'); ylabel('Joint6(rad)'); Fig.7. Robot direct kinematics joint angles against time Velocity and acceleration profile can be obtained by: >> [q,qd,qdd] = jtraj(qz, qr, t) Velocity profile of joints of the robot manipulator is given by: >>subplot(6,1,1); plot(t,q(:,1));title(‘Velocity’); xlabel('Time(s)'); ylabel('Joint1(rad/s)'); subplot(6,1,2); plot(t,q(:,2)); xlabel('Time(s)'); ylabel('Joint2(rad/s)'); subplot(6,1,3); plot(t,q(:,3)); xlabel('Time(s)'); ylabel('Joint3(rad/s)'); subplot(6,1,4); plot(t,q(:,4)); xlabel('Time(s)'); ylabel('Joint4(rad/s)'); subplot(6,1,5); plot(t,q(:,5)); xlabel('Time(s)'); ylabel('Joint5(rad/s)'); subplot(6,1,6);plot(t,q(:,6)); xlabel('Time(s)'); ylabel('Joint6(rad/s)'); Fig.8. Joint velocity against time Acceleration profile is given by: >>subplot(6,1,1); plot(t,q(:,1));title(‘Acceleration’); xlabel('Time(s)'); ylabel('Joint1(rad/s2)'); subplot(6,1,2); plot(t,q(:,2)); xlabel('Time(s)'); ylabel('Joint2(rad/s2)'); subplot(6,1,3); plot(t,q(:,3)); xlabel('Time(s)'); ylabel('Joint3(rad/s2)'); subplot(6,1,4); plot(t,q(:,4)); xlabel('Time(s)'); ylabel('Joint4(rad/s2)'); subplot(6,1,5); plot(t,q(:,5)); xlabel('Time(s)'); ylabel('Joint5(rad/s2)'); subplot(6,1,6);plot(t,q(:,6)); xlabel('Time(s)'); ylabel('Joint6(rad/s2)'); Fig.9. Joint acceleration against time III. CONCLUSION In this paper, the direct kinematic model of a 6R robot is presented by utilizing the principal features of the Robotic Toolbox in MATLAB. The Toolbox provides several essential tools for the modeling and analysis of the robotic manipulator, as well as analyzing the experimental results.
- 4. Prashant Badoni Int. Journal of Engineering Research and Applications www.ijera.com ISSN: 2248-9622, Vol. 6, Issue 1, (Part - 3) January 2016, pp.151-154 www.ijera.com 154 | P a g e REFERENCES [1] P.I. Corke, A Robotics Toolbox for MATLAB, IEEE Robotics and Automation Magazine, Vol.3, No.1, pp.24-32, March 1996. [2] J. Denavit, R.S. Hartenberg, A Kinematic Notation for Lower-Pair Mechanisms Based on Matricies, Trans. of ASME J.App. Mech. vol. 77, pp.215-221, June 1955. [3] J.J. Crage, Introduction to Robotics Mechanics and Control, 3rd Edition, Prentice Hall, 2005. [4] M.W. Spong, S. Hutchinson and M. Vidyasagar, Robot Modeling and Control, 1st Edition, Jon Wiley & Sons Inc, 2005. [5] Zhongtao Fu, Wenyu Yang and Zhen Yang, Solution of Inverse Kinematics for 6R Robot Manipulators With Offset Wrist Based on Geometric Algebra, Journal of Mechanism and Robotics, August 2013, Vol. 5.