1. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 3, March (2014), pp. 20-30, © IAEME
20
ADVANCEMENT AND STIMULATION OF FIVE DEGREE OF FREEDOM
ROBOT LEVER ARM
Saifuldeen Abed Jebur1
, Prabhat Kumar Sinha2
, Ishan Om Bhargava3
1
Automation & Systems Research Center /Industrial Development & Research Directorate / Ministry
of Science and Technology of republic Iraq.+ Department of Mechanical Engineering (SSET), Sam
Higginbottom Institute of Agriculture Technology and Sciences, Allahabad U.P (INDIA)
2
Department of Mechanical Engineering (SSET), Sam Higginbottom Institute of Agriculture
Technology and Sciences, Allahabad U.P (INDIA)
3
Department of Mechanical Engineering (SSET), Sam Higginbottom Institute of Agriculture
Technology and Sciences, Allahabad U.P (INDIA)
ABSTRACT
In this article the development of virtual software package, where a Robot Lever arm has been taken
as a case study. MATLAB will be used for testing motional kinematics. It has adopted the design
methodology as a tool for analyzing characteristics of the Robot lever arm. Moreover, the model
analysis is carried in order to analyze through kinematics and testing the virtual arm by comparing
between the approaches applied to the arm in terms of kinematics. The development of this robot
lever arm is used as an guide tool in enhancing the applied experimental research opportunities and
improving it’s application.
KEYWORDS: Modeling, MATLAB, Robot Lever Arm, Kinematics.
INTRODUCTION
Over the last two decades, artificial intelligence has been based on mobile robotics and
promoted the development of lever arm. The goal of this research work is to design and develop a
five degree of freedom robot lever arm for determining it’s motional characteristics using
MATLAB. The robot lever arm is chosen as a case study in this research. MATLAB will be used
for testing motional characteristics of the arm. A complete study and mathematical analysis for the
kinematics, is presented and implemented. This is implemented and applied to the robot lever arm
and it’s physical characteristics. A comparison between the kinematic solutions of the robot arm’s
physical motional behavior is discussed. Many industrial robot arms are built with simple
INTERNATIONAL JOURNAL OF MECHANICAL ENGINEERING
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ISSN 0976 – 6340 (Print)
ISSN 0976 – 6359 (Online)
Volume 5, Issue 3, March (2014), pp. 20-30
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2. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 3, March (2014), pp. 20-30, © IAEME
21
geometries such as intersecting or parallel joint axes to simplify the associated kinematics
computations [MAN 01]. Papers that developed software for modeling 2D and 3D robots arm such
as [MAN 01, KOY 02 and GUR 03], Kinematics is analyzed by [PAS 04] using V-Realm Builder
2.0 for virtual reality prototyping and testing the of designs before the implementation phase of the
robot . Martin and Arya in [ROH 05, WIR 06], developed Robot Simulation Software for forward
and inverse kinematic using VRML and MATLAB. [JAM 07] reported the development of the
software for robot. [KOL 08] has presented on educational purposes in Education Conferences. The
work has adopted the virtual reality interface design methodology utilizing MATLAB. It started
from the kinematics of the robot arm taking into consideration the position and orientation of robot
joint. This research is on the kinematical and mathematical analysis of the robot lever arm. The
focus of this paper is on developing components related to MATLAB.
KINEMATIC MODELLING
For the robot lever arm analysis is done, its purpose is to carry the analysis of the movements
of each part of the lever arm mechanism. The kinematics analysis is divided into forward and inverse
kinematic analysis. The forward kinematics consists of finding the position of the arm in the space
knowing the movements of its joints as F (θ1,θ2 ,…,θn ) = [ x, y, z, R] , and the inverse kinematics
consists of determining the joint variables corresponding to a given position and orientation as
F ( x, y, z, R)= θ1,θ2 ,…,θn . Figure shown below shows a simplified block diagram of kinematic
modeling.
KINEMATIC MODELLING BLOCK DIAGRAM
A commonly used convention for selecting frames of reference in robotic applications is the
Denavit Hartenberg. In this convention each homogenous transformation Ti is represented as a
product of four basic transformations. The kinematics analysis is divided into two solutions, the first
one is the solution of Forward kinematics, and the second one is the inverse kinematics solution.
Forward kinematics has been determining the position of robot lever arm if all joints are known.
Where as the inverse kinematics is being used to calculate what each joint variable must be if the
desired position and orientation of end point is determined. Hence, Forward kinematics is defined as
transformation from joint space to Cartesian space where as Inverse kinematics is defined as
transformation from Cartesian space to joint space.
Forward
Kinematic
Geometric
Parameters
Inverse
Kinematics
Position and Orientation of
the end-Effector
Joints Movements

3. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 3, March (2014), pp. 20-30, © IAEME
22
A commonly used convention for selecting frames of reference in robotic applications is the
Denavit Hartenberg frame. In this convention each homogenous transformation ܶ௜ is represented as
a product of four basic transformations.
Ti = ܴ‫ݐ݋‬ሺ‫,ݖ‬ ߠ௜ሻ ܶ‫ݏ݊ܽݎ‬ሺ‫,ݖ‬ ݀௜ሻܶ‫ݏ݊ܽݎ‬ሺ‫,ݔ‬ ߙ௜ሻܴ‫ݐ݋‬ሺ‫,ݔ‬ ߙ௜ሻ
OORDINATE FRAMEC
Where the notation Rot (x, α୧) stands for rotation about x୧ axis by α୧, Trans(x, α୧) is
translation along xi axis by a distance αi, Rot(z, θi) stands for rotation about ezi axis by θi, and
Trans(z, di) is the translation along zi axis by a distance di.
ܶ ൌ ൦
ܿఏ೔
െ‫ݏ‬ఏ೔
0 0
‫ݏ‬ఏ೔
0
ܿఏ೔
0
0 1
0
0
0 0 0 1
൪ ൦
1 0 0 0
0
0
1 0
0 1
0
݀௜
0 0 0 1
൪ ൦
1 0 0 ߙ௜
0
0
1 0
0 1
0
0
0 0 0 1
൪ ൦
1 0 0 0
0
0
ܿఈ೔
െ‫ݏ‬ఈ೔
‫ݏ‬ఈ೔
ܿఈ೔
0
0
0 0 0 1
൪
൦
ܿఏ೔
െ‫ݏ‬ఏ ೔
ܿఈ೔
‫ݏ‬ఏ೔
‫ݏ‬ఈ೔ ܽ௜ܿఏ೔
‫ݏ‬ఏ೔
0
ܿఏ೔
ܿఈ೔
െܿఏ೔
‫ݏ‬ఈ೔
‫ݏ‬ఈ೔
ܿఈ೔
ܽ௜‫ݏ‬ఈ೔
݀௜
0 0 0 1
൪=
Where the four quantities θi, ai, di, αi are the parameters of link i and joint i. The description
below illustrates the link frames attached so that frame {i} is attached rigidly to link i. The various
parameters in previous equation are given the following are
ܽ௜(Length) is the distance from ‫ݖ‬௜to‫ݖ‬௜ାଵ, measured along ‫ݖ‬௜;
ߙ௜ (Twist), is the angle between ‫ݖ‬௜and ‫ݖ‬௜ାଵ, measured about ‫ݔ‬௜;
݀௜ (Offset), is the distance from ‫ݔ‬௜to‫ݔ‬௜ାଵmeasured along ‫ݖ‬௜; and
ߠ௜ (Angle), is the angle between‫ݔ‬௜ ܽ݊݀‫ݔ‬௜ାଵmeasured about ‫ݖ‬௜;
In the usual case of a revolute joint, the joint variable with the other three quantities are fixed.
The matrix notation used is homogeneous .Here H represents a rotation by angle α about the current
x-axis followed by a translation of units along the x-axis, followed by a translation of d units along
the z-axis, followed by a rotation by angle θ about the z-axis, which is

4. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 3, March (2014), pp. 20-30, © IAEME
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H= ܴ‫ݐ݋‬௫,ఈ െ ܶ‫ݏ݊ܽݎ‬௫,ఈܶ‫ݏ݊ܽݎ‬௭,ௗܴ‫ݐ݋‬௭,ఏ
൦
ܿఏ െ ‫ݏ‬ఏ 0 ܽ
ܿఈ‫ݏ‬ఏ
‫ݏ‬ఏ‫ݏ‬ఈ
ܿఏ೔
െ‫ݏ‬ఈ
‫ݏ‬ఈܿఏ ܿఈ
െ݀‫ݏ‬ఈ
݀ܿఈ
0 0 0 1
൪=
The most general homogeneous transformation is of the form
൤
ܴ‫݊݋݅ݐܽݐ݋‬ ܶ‫݊݋݅ݕ݈ܽݏ݊ܽݎ‬
ܲ݁‫݁ݒ݅ݐܿ݁݌ݏݎ‬ ‫݈݁ܽܿݏ‬ ݂ܽܿ‫ݎ݋ݐ‬
൨=‫ܪ‬ ൌ ൤
ܴଷ‫כ‬ଷ ݀ଷ‫כ‬ଵ
݂ଵ‫כ‬ଷ ‫ݏ‬ଵ‫כ‬ଵ
൨
= ቎
݊௫ ‫݋‬௫ ܽ௫ ‫݌‬௫
݊ଵ
݊௭
‫݋‬௬ ܽ௬
‫݋‬௭ ܽ௭
‫݌‬௬
‫݌‬௭
0 0 0 1
቏
Where the three by three augmented matrix, [R3x3
, represents the rotation, the three by one
augmented matrixes, d3x1
, represent the translation; the f1x3
represents the perspective transformation
and S1x1
is the factor. The direct kinematics made from the composition of homogeneous
transformation matrices, where each rotation corresponds to one four by four augmented matrix:
ܶ௜
௝
ൌ ܶ௜ାଵ
௝
… ܶ௜
௜ିଵ
MODELING THROUGH MATLAB
Our main motive is to investigate and to develop the robot lever arm using MATLAB. The
Hartenberg analysis is the most approximate, method for using the direct kinematics using relevant
parameters that have been used. In this, analysis, there is a defined coordinate transformation
between two frames, where the position and orientation are fixed one with respect to the other and it
is possible to work with homogeneous matrix transformations. .
.
THE KINEMATIC LINK DIGRAM

5. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 3, March (2014), pp. 20-30, © IAEME
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ܶ௜
௜ିଵ
is a homogenous matrix which is defined to transform the coordinates of a point from
frame i to frame i-1. The matrixܶ௜
௜ିଵ
is not constant, but varies as the configuration of the robot lever
arm is changed. However, the assumption that all joints are either revolute or prismatic means that
ܶ௜
௜ିଵ
is a function of only a single joint variable, namely qi.
ܶ௜
௜ିଵ
ൌ ܶ௜
௜ିଵ
ሺ‫ݍ‬ଵሻ
The homogenous matrix that transforms the coordinates of a point from frame i to frame j is
denoted by ܶ௜
௝
(i > j). Denoting the position and orientation of the end joint with respect to the inertial
or the base frame by a three dimensional vector݀௡
଴
and a 3x3 rotation matrixܴ௡
଴
, respectively, we
define the homogenous matrix as
ܶ௡
଴
ൌ ൤ܴ௡
଴
݀௡
଴
0 1
൨
Then the position and orientation of the end joint in the inertial frame has been mentioned as
ܶ௡
଴
ൌ ሺ‫ݍ‬ଵ, ‫ݍ‬ଶ , … … . , ‫ݍ‬௡ሻ ൌ ܶଵ
଴ሺ‫ݍ‬ଵሻܶଶ
ଵሺ‫ݍ‬ଵሻ … . ܶ௡
௡ିଵ
ሺ‫ݍ‬ଵሻ
Each homogenous transformation ܶ௜
௜ିଵ
is of the form of
ܶ௜
௜ିଵ
ൌ ൤ܴ௜
௜ିଵ
݀௜
௜ିଵ
0 1
൨
Hence
ܶ௜
௝
ൌ ܶ௝ାଵ
௝
… ܶ௜
௜ିଵ
ൌ ൤ܴ௜
௝
݀௜
௝
0 1
൨
The matrix ܴ௜
௝
expresses the orientation of frame i relative to frame j (i > j) and is given by the
rotational parts of the ܶ௜
௝
-matrices (i > j) as
ܴ௜
௝
ൌ ܴ௜ାଵ
௝
… . ܴ௜
௝ିଵ
The vectors ݀௜
௝
(i> j) are given recursively by the formula
݀௜
௝
ൌ ݀௝ାଵ
௝
൅ ܴ௜ିଵ
௝
݀௜
௜ିଵ
THE COORDINATE FRAME
Robot lever arm has five rotational joints and a moving grip. Joint 1 represents the shoulder
and its axis of motion is z1. This joint provides a angular ߠଵ motion around z1 axis in x 11 plane. Joint
2 is identified as the Upper Arm and its axis is perpendicular to Joint 1 axis. It provides a rotational
ߠଶangular motion around z2 axis in x2y2 plane. Z3axes of Joint 3 and Joint 4 are parallel to Joint 2.
z-axis provides ߠଷand ߠସangular motions in x3y3 and x4y4 planes respectively. Joint five are
identified as the grip rotation. And it’s z5 axis is vertical to z4 axis and it provides ߠହangular motions
in x5y5 plane. A graphical view of all the joints has been presented below.

6. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
ISSN 0976 – 6359(Online), Volume 5, Issue 3, March (2014), pp. 20-30, © IAEME
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COORDINATE FRAME OF ROBOT LEVER ARM
THE PARAMETERS
The Denavit-Hartenberg analysis is preferred and is the most approximate method .The direct
kinematics is determined from some parameters that have been defined, for each mechanism. The
homogeneous transformations and matrix has been used for analysis, for coordinate transformation
between two frames, where the position and orientation are fixed one with respect to the other it is
possible to work with elementary homogeneous transformation operations. D-H parameters for AL5B
defined for the assigned frames.
i ݀௜ ߠ௜
1 0 0 dଵ ߠଵ
‫כ‬
2 90 0 0
3 0 aଷ 0
4 0 aସ 0 ( ߠସ-90) *
5 -90 0 dହ ߠହ
‫כ‬
6 0 0 0 Gripper
By substituting the parameters from into coordinate equation the transformation matrices T1
to T6 can be obtained.
ܶଵ
଴
ൌ ൦
ܿఏభ
െ‫ݏ‬ఏభ
0 0
‫ݏ‬ఏభ
0
ܿఏభ
0
1 0
0
݀ଵ
0 0 0 1
൪
ܶଶ
ଵ
ൌ
‫ۏ‬
‫ێ‬
‫ێ‬
‫ۍ‬
ܿఏమ
െ‫ݏ‬ఏమ
0 0
0
‫ݏ‬ఏమ
0 െ1
ܿఏమ
0
0
0
0 0 0 1‫ے‬
‫ۑ‬
‫ۑ‬
‫ې‬
ܶଷ
ଶ
ൌ ൦
ܿఏయ
െ‫ݏ‬ఏయ
0 ߙଷ
‫ݏ‬ఏయ
0
ܿఏయ
0
0 1
0
0
0 0 0 1
൪

7. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
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ܶସ
ଷ
ൌ ൦
ܿఏర
െ‫ݏ‬ఏర
0 ߙସ
‫ݏ‬ఏర
0
ܿఏర
0
0 1
0
0
0 0 0 1
൪
ܶହ
ସ
ൌ
‫ۏ‬
‫ێ‬
‫ێ‬
‫ۍ‬
ܿఏఱ
െ‫ݏ‬ఏఱ
0 0
0
‫ݏ‬ఏఱ
0 0
ܿఏఱ
0
݀ହ
0
0 0 0 1 ‫ے‬
‫ۑ‬
‫ۑ‬
‫ې‬
ܶீ௥௜௣௣௘௥ ൌ ൦
ܿఏఱ
െ‫ݏ‬ఏఱ
0 0
‫ݏ‬ఏఱ
0
ܿఏఱ
0
0 1
0
0
0 0 0 1
൪
Now the link transformations can be calculated to find the single transformation that is
relating frames
ܶହ
଴
ൌ ܶଵ
଴
ܶହ
଴
ܶହ
଴
ܶହ
଴
ܶହ
଴
ൌ ቎
݊௫ ‫݋‬௫ ܽ௫ ‫݌‬௫
݊௬
݊௭
‫݋‬௬ ܽ௬
‫݋‬௭ ܽ௭
‫݌‬௬
‫݌‬௭
0 0 0 1
቏
The transformation given here is a function of all five variables. From the robots joint
position, the coordinate position and orientation of the last link is computed .The orientation and
position of are calculated using:
݊௫ ൌ ሺሺܿଵ ܿଶ ܿଷିܿଵ ‫ݏ‬ଶ ‫ݏ‬ଷ ሻܿସ ൅ ൫‫ܿـ‬ଵ ܿଶ ‫ݏ‬ଷି ܿଵ ‫ݏ‬ଶ ܿଷሻ ‫ݏ‬ସ൯ܿହ ൅ ‫ݏ‬ଵ ‫ݏ‬ହ
n୷ ൌ ቀሺsଵ cଶ cଷି sଵ sଶ sଷ ሻcସ ൅ ൫‫ـ‬sଵ cଶ sଷ െ sଵ sଶ cଷ ൯sସ ቁ cହ െ cଵ sହ
݊௭ ൌ ሺሺ‫ݏ‬ଶ ܿଷା ܿଶ ‫ݏ‬ଷ ሻܿସ ൅ ൫‫ݏـ‬ଶ ‫ݏ‬ଷ ൅ ܿଶ ܿଷ ൯‫ݏ‬ସሻܿହ
‫݋‬௫ ൌ ‫ـ‬ሺሺܿଵ ܿଶ ܿଷ _ܿଵ ‫ݏ‬ଶ ‫ݏ‬ଷ ሻܿସ+(‫ـ‬ܿଵ ܿଶ ‫ݏ‬ଷ _ܿଵ ‫ݏ‬ଶ ܿଷ ሻ‫ݏ‬ସ ሻ‫ݏ‬ହ ൅ ‫ݏ‬ଵ ܿହ
‫݋‬௬ ൌ ‫ـ‬ ቀ൫‫ݏ‬ଵ ܿଶ ܿଷ ‫ݏـ‬ଵ ‫ݏ‬ଶ ‫ݏ‬ଷ ൯ܿସ ൅ ൫‫ݏـ‬ଶ ܿଶ ‫ݏ‬ଷ ‫ݏـ‬ଵ ‫ݏ‬ଶ ܿଷ ൯‫ݏ‬ସ ቁ ‫ݏ‬ହ ‫ܿـ‬ଵܿହ
‫݋‬௭ ൌ ሺܿଶܿଷ‫ݏـ‬ଶ‫ݏ‬ଷሻ‫ݏ‬ସሻ‫ݏ‬ହିሺሺ‫ݏ‬ଶܿଷ ൅ ܿଶ‫ݏ‬ଷሻܿସ
ܽ௫ ൌ ‫ـ‬൫ܿଵܿଶܿଷ‫ܿـ‬ଵ‫ݏ‬ଶ‫ݏ‬ଷ൯‫ݏ‬ସ ൅ ሺ‫ܿـ‬ଵܿଶ‫ݏ‬ଷ‫ܿـ‬ଵ‫ݏ‬ଶܿଷሻܿସ
ܽ௬ ൌ ‫ـ‬൫‫ݏ‬ଵܿଶܿଷ‫ݏـ‬ଵ‫ݏ‬ଶ‫ݏ‬ଷ൯‫ݏ‬ସ ൅ ሺ‫ݏـ‬ଵܿଶ‫ݏ‬ଷ‫ݏـ‬ଵ‫ݏ‬ଶܿଷሻܿସ
ܽ௭ୀሺܿଶܿଷ‫ݏـ‬ଶ‫ݏ‬ଷሻܿସ_ሺ‫ݏ‬ଶܿଷ ൅ ܿଶ‫ݏ‬ଷሻ‫ݏ‬ସ
݀௫ ൌ ൫‫ـ‬൫ܿଵܿଶܿଷ‫ܿـ‬ଵ‫ݏ‬ଶ‫ݏ‬ଷ൯‫ݏ‬ସ ൅ ൫‫ܿـ‬ଵܿଶܿଷ‫ܿـ‬ଵ‫ݏ‬ଶܿଷ൯ܿସ൯݀ହ ൅ ൫ܿଵܿଶܿଷ‫ܿـ‬ଵ‫ݏ‬ଶ‫ݏ‬ଷ൯ܽସ ൅ ܿଵܿଶܽଷ
݀௬ =ሺെ൫ܿଵܿଶܿଷ‫ݏـ‬ଵ‫ݏ‬ଶ‫ݏ‬ଷ൯‫ݏ‬ସ ൅ ൫‫ݏـ‬ଵܿଶ‫ݏ‬ଷ െ ‫ݏ‬ଵ‫ݏ‬ଶܿଷ൯ܿସሻ݀ହ ൅ ൫‫ݏ‬ଵܿଶܿଷ‫ݏـ‬ଵ‫ݏ‬ଶ‫ݏ‬ଷ൯ܽସ +‫ݏ‬ଵܿଶܽଷ
݀௭ ൌ ሺെሺ‫ݏ‬ଶܿଷ ൅ ܿଶ‫ݏ‬ଷሻ‫ݏ‬ସ ൅ ሺെ‫ݏ‬ଶ‫ݏ‬ଷ ൅ ܿଶܿଷሻ݀ହ ൅ ሺ‫ݏ‬ଶܿଷ ൅ ܿଶ‫ݏ‬ଷሻܽସ ൅ ܿଶܽଷ ൅ ݀ଵ
KINEMATIC ANALYSIS THROUGH MATLAB
In the kinematic window displayed below the main purpose is to compute the position of
end joint by entering the angle values .This is the done through the use of MATLAB module. This
gives the resulting matrices.

8. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
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LEVER ARM COORDINATE ARRANGEMENT
KINEMATIC ANALYSIS USING MATLAB MODULE
In this study mathematical and kinematic modeling analysis is done for the lever arm. Robot
arm has been mathematically modeled through Hartenberg method using kinematics and is furthered
analyzed through MATLAB.

9. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
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RESULT
The experimental results of modeling through MATLAB have been analyzed and are used for
the robot lever arm physical characteristic analysis.
THE KINEMATIC MODULE
A initial position angle MATLAB module is given below with zero ߠ ߠ௜ ൌ 0, ݅ ൌ 0,1,2, … ,5
. The transformation matrix is given and this matrix gives the initial position and orientation of the
robot arm.
THE INITIAL POSITION KINEMATIC MODULE
ܶହ
଴
ሺ݂݈݅݊ܽሻ ൌ ቎
0 0 1 347
0
െ1
1 0
0 1
0
70
0 0 0 1
቏
From the above matrix we find that the (x, y, and z) position of the end position is equal to
(347, 0, and 70).

10. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
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INITIAL ORIENTATION OF LEVER ARM
When the following angular are taken for the up position the following matrix is obtained
Matrix, Tହ
଴
which is the orientation of the base of lever arm is shown.
ܶହ
଴
ሺ݂݈݅݊ܽሻ ൌ ቎
െ0.63 െ0.47 0.61 211
0.76
െ0.15
െ0.23 0.61
0.85 0.5
211
205
0 0 0 1
቏
This equation gives the following module which is [x, y, and z] = [211, 211, and 205].
ܶହ
଴
is analyzed through MATLAB and is the final kinematic solution of the robot arm.
INITIAL POSITION OF LEVER ARM

11. International Journal of Mechanical Engineering and Technology (IJMET), ISSN 0976 – 6340(Print),
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REFERENCES
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