The kinematics miniature are established for a 4 DOF robotic arm. Denavit-Hartenberg (DH) convention and the product of exponential formula are used for solving kinematic problem based on screw theory. For acquiring simple matrix for inverse kinematics a new simple method is derived by solving problems like robot base movement, actuator restoration. Simulations are done by using MATlab programming for the kinematics exemplary.

1. B. Siva Kumar et al. Int. Journal of Engineering Research and Applications www.ijera.com
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A New Method For Solving Kinematics Model Of An RA-02
B. Siva Kumar1
*, J.Sushma2
And G.Srikanth3
Assistant Professor*1, 2
, PG Scholar3
, Department of Mechanical Engineering
VNR Vignan Jyothi Institute of Engineering & Technology1,2
,GITAM University3
, Hyderabad, India
ABSTRACT
The kinematics miniature are established for a 4 DOF robotic arm. Denavit-Hartenberg (DH) convention and the
product of exponential formula are used for solving kinematic problem based on screw theory. For acquiring
simple matrix for inverse kinematics a new simple method is derived by solving problems like robot base
movement, actuator restoration. Simulations are done by using MATlab programming for the kinematics
exemplary.
Keywords - Robotics; DH convention; product of exponentials; kinematics; simulations.
I. INTRODUCTION
Nowadays, robotics (Robot kinematics) are an
affluent area, which is the foundation of robot control
contained two kinds of problems: the one is
calculating gesture and position of end actuators from
the known angle of each joint, which is called
forward kinematics; the other one is finding out angle
of each joint from a known gesture and position of
end actuators, which is named inverse kinematics.
Between these two problems, the latter one, inverse
kinematics, is more complicated and usually has
multiple solutions due to its nonlinear condition.
Jasjit Kaur et al [1] Using soft computing techniques
like genetic algorithm analysis and simulation of
robotic arm having three links-manipulator is done.
In [2,3] an approach to solving specific joint angles
for positioning of the robot arm is presented.
The inverse kinematics complication was
disintegrating into sub problems was proposed and
interactive simulations are done for robot
manipulators is given in [4], where the forward and
inverse kinematics of a robotic arm called
Katana450.Mainly robot simulator is introduced by
Lodes [5]. The algorithm in [6-8] only need once
inverse matrix calculation, nevertheless robot
modeling is a little more sophisticated which leading
to a more complicated calculation. Considering
modular modeling for achieving Multi-robot
configuration and interchangeability, reference [9]
has a remarkable amelioration.
Modeling and Simulation of 4 DOF robotic arm
are presented in this paper. Section II below presents
introduction according to literature survey, Section
III below is kinematics manipulator based on the
algebraic method, Section IV below is the
manipulator simulation, Section V concludes results
and conclusion.
II. MANIPULATOR KINEMATICS
Kinematics study of robot manipulators is done
by using DH convention. Analysis is done on RA-2, a
four-joint spatial manipulator. In equivalent way of
PUMA 560 analysis is done by removing the wrist
and adding one more in-surge joint.
A. DH Convention:
Denavit-Hartenberg (DH) convention is
frequently used in the scrutiny of the kinematic
manipulator. DH parameters for individual links and
different parameters are used for formulating a DH
table.
Table 1:DH Parameters
Frame (i) 𝜶𝒊−𝟏 𝒂𝒊−𝟏 𝒅𝒊 𝜽𝒊
1 0 0 0 𝜃1
2 -900
0 0 𝜃2
3 0 𝑎2 𝑑3 𝜃3
4 -900
0 𝑑4 𝜃4
Finally, a revolution matrix enclosed with
coordinating frames is achieved and derivation of the
relationship among joints and positions is to be done,
which is presented in Fig.1 and DH criterion in Fig.
2.
Fig.1The Robot Arm
RESEARCH ARTICLE OPEN ACCESS

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Fig.2 DH Parameters
B. Forward Kinematics
The equalized frame connected to individual link
for asset its shape in the nearby frames using the
solid motion formula in order to do we need a DH
table as follows as per Fig 2.
Fig.3 Robot Arm with 5 independent movements
By implementing the DH method [10] for the
joint correlate, the DH-table can be formulated as
listed above in Table I. The link dimensions as shown
in Fig. 2 are l1=11. 5cm, l2=12cm, and l3, 4=9cm
Forward Transformation Matrices: By using DH
table,the revolution matrix can we easily evaluate
with frame ranging Bi to frame Bi-1
The DH parameters are akin to the configuration
of the robot. Moreover, kinematics equations of the
manipulator based on the DH convention provide
some singularity making the equations difficult to
solve or unsolvable in some cases.
𝑻 𝟏
𝟎
=
𝒄 𝟏 −𝒔 𝟏 𝟎 𝟎
𝒔 𝟏 𝒄 𝟏 𝟎 𝟎
𝟎
𝟎
𝟎
𝟎
𝟏
𝟎
𝟎
𝟏
(𝟏)
𝑻 𝟐
𝟏
=
𝒄 𝟐 −𝒔 𝟐 𝟎 𝟎
𝟎 𝟎 𝟏 𝟎
−𝒔 𝟐
𝟎
−𝒄 𝟐
𝟎
𝟎
𝟎
𝟎
𝟏
(𝟐)
𝑻 𝟑
𝟐
=
𝒄 𝟑 −𝒔 𝟑 𝟎 𝒂 𝟐
𝒔 𝟑 𝒄 𝟑 𝟎 𝟎
𝟎
𝟎
𝟎
𝟎
𝟎
𝟎
𝒅 𝟑
𝟏
(𝟑)
𝑻 𝟒
𝟑
=
𝒄 𝟒 −𝒔 𝟒 𝟎 𝟎
𝟎 𝟎 𝟏 𝒅 𝟑
−𝒔 𝟒
𝟎
−𝒄 𝟒
𝟎
𝟎
𝟎
𝟎
𝟏
(𝟒)
𝑻 𝟒
𝟎
= 𝑻 𝟏
𝟎
𝑻 𝟐
𝟏
𝑻 𝟑
𝟐
𝑻 𝟒
𝟑
(𝟓)
𝑻 𝟒
𝟎
=
𝒓 𝟏𝟏 𝒓 𝟏𝟐 𝒓 𝟏𝟑 𝒑 𝒙
𝒓 𝟐𝟏 𝒓 𝟐𝟐
𝒓 𝟐𝟑 𝒑 𝒚
𝒓 𝟑𝟏
𝟎
𝒓 𝟑𝟐
𝟎
𝒓 𝟑𝟑
𝟎
𝒑 𝒛
𝟏
(𝟔)
Where:
𝒓 𝟏𝟏 = 𝒄 𝟒 𝒄 𝟏 𝒄 𝟐𝟑 + 𝒔 𝟏 𝒔 𝟒 (𝟕)
𝒓 𝟐𝟏 = 𝒄 𝟒 𝒔 𝟏 𝒄 𝟐𝟑 − 𝒄 𝟏 𝒔 𝟒 (𝟖)
𝒓 𝟑𝟏 = −𝒄 𝟒 𝒔 𝟐𝟑 (𝟗)
𝒓 𝟏𝟐 = 𝒄 𝟒 𝒔 𝟏 − 𝒔 𝟒 𝒄 𝟏 𝒄 𝟐𝟑 (𝟏𝟎)
𝒓 𝟐𝟐 = −𝒄 𝟏 𝒄 𝟒−𝒔 𝟒 𝒔 𝟏 𝒄 𝟐𝟑 𝟏𝟏
𝒓 𝟑𝟐 = 𝒔 𝟒 𝒔 𝟐𝟑 𝟏𝟐
𝒓 𝟏𝟑 = −𝒄 𝟏 𝒔 𝟐𝟑 (𝟏𝟑)
𝒓 𝟐𝟑 = −𝒔 𝟏 𝒔 𝟐𝟑 𝟏𝟒
𝒓 𝟑𝟑 = −𝒄 𝟐𝟑 (𝟏𝟓)
𝒑 𝒙 = −𝒅 𝟑 𝒔 𝟏−𝒅 𝟒 𝒄 𝟏 𝒔 𝟐𝟑 + 𝒂 𝟐 𝒄 𝟏 𝒄 𝟐 𝟏𝟔
𝒑 𝒚 = 𝒄 𝟏 𝒅 𝟑+𝒅 𝟒 𝒔 𝟏 𝒔 𝟐𝟑 + 𝒂 𝟐 𝒄 𝟐 𝒔 𝟏 (𝟏𝟕)
𝒑 𝒛 = −𝒂 𝟐 𝒔 𝟐−𝒅 𝟒 𝒄 𝟐𝟑 + 𝑳 𝟏 (𝟏𝟖)
In addition, in the DH convention, the common
normal is not defined properly when axes of the two
joints are parallel. In this case, the DH method has a
singularity, where a little change in the spatial
coordinates of the parallel joint axes can create a
huge misconfiguration in representation of the DH
coordinates of their relative position.
C. Inverse Kinematics
Solving these equations algebraically, known as
the inverse kinematics, requires that we need to know
the joint variables θ1, θ2, θ3 and θ4 for a given end
effector position [px, py, pz ] and orientation ϕ . We
get from equations (16) to (18), by dividing,
squaring, adding and using some trigonometric
formulas:
𝜃1 = tan−1
𝑑𝑦
𝑑𝑥
𝜃1 = tan−1
𝑐, ± 𝑟2 − 𝑐2 − tan−1
𝑎, 𝑏
𝜃3 = cos−1
𝐴2
+ 𝐵2
+ 𝐶2
− 𝑙2
2
− 𝑙3
2
2 𝑙2 𝑙3
Where
a= 𝑙3 sin 𝜃3, b= 𝑙2 + 𝑙3 cos 𝜃3,c= dZ- 𝑙1 −
𝑙4 sin 𝜙 , 𝑎𝑛𝑑 𝑟 = 𝑎2 + 𝑏2.
In addition
A= (dx-l4cθ1cϕ)
B= (dy-l4 sθ1cϕ),
And
C= (dz-l1-l4 sϕ)
Having determined 𝜃1, 𝜃2 𝑎𝑛𝑑 𝜃3 we can then find
𝜃4 from the end effecter orientation of Φ as follows:
𝜃4 = 𝜙 − 𝜃2 − 𝜃3

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III. PROPOSED METHOD: PE
For the analysis of the robotic manipulator, a
product of exponentials (PE) is another method.
Robot Kinematics considering geometric
representations and different locomotive actions with
joints and angle of rotations can be acquired using
this method. Considering „P‟ is a twist, the forward
kinematics are given
𝑔𝑠𝑡 𝜃 = 𝑒 𝜉1 𝜃1 𝑒 𝜉2 𝜃2 … . 𝑒 𝜉 𝑛 𝜃 𝑛 𝑔𝑠𝑡 0
The above equation is called the product of the
exponential formula for the robot forward kinematics
𝑔𝑠𝑡 𝜃 = Final Configuration of the robot
𝑒 𝜉 𝑛 𝜃 𝑛 = Matrix Exponential
𝑒 𝜉 𝑛 𝜃 𝑛
=
𝑒 𝜃 𝑛 𝜔 𝑛 𝐼 − 𝑒 𝜃 𝑛 𝜔 𝑛 𝜔 𝑛 ∗ 𝑣 𝑛 + 𝜃 𝑛 𝜔 𝑛 𝜔 𝑛
𝑇
𝜃𝑛
0 1
For a pri prismatic joint the twist ξi is given by
𝜉𝑖 =
𝑉𝑖
0
,𝜉𝑖 =
−𝜔𝑖 ∗ 𝑞𝑖
𝜔𝑖
Where ωi Є R3
is a unit vector in the direction of axis
of the twist , 𝑞𝑖 Є R3
is any point of the axis, and 𝑉𝑖 Є
R3 is a
unit vector directing in the translication
direction .In this case ,the twist ξ‟s for different links
of the robot are given by
𝜉1 =
0
0
0
0
0
1
𝜉2 =
𝐿1
0
0
0
−1
0
𝜉3 =
𝐿1
0
−𝐿2
0
−1
0
𝜉4 =
𝐿𝑖
0
−(𝐿2+𝐿3)
0
−1
0
Moreover 𝑔𝑠𝑡 𝜃 =
1 0 0 𝑙2 + 𝑙3 + 𝑙4
0 1 0 0
0
0
0
0
1
0
𝑙1
1
The forward kinematics map of the manipulator has
the form:
𝑔𝑠𝑡 𝜃 = 𝑒 𝜉1 𝜃1 𝑒 𝜉2 𝜃2 … . 𝑒 𝜉 𝑛 𝜃 𝑛 𝑔𝑠𝑡 0
=
𝑅(𝜃) 𝑝(𝜃)
0 1
By expanding terms in the product of exponentials
formula, the above eqn yields
𝑅 𝜃 =
cos(𝜃2+𝜃3+𝜃4) cos 𝜃1 − sin 𝜃1 −sin(𝜃2+𝜃3+𝜃4) cos 𝜃1
cos(𝜃2+𝜃3+𝜃4) sin 𝜃1 cos 𝜃1 −sin(𝜃2+𝜃3+𝜃4) sin 𝜃1
sin(𝜃2+𝜃3+𝜃4) 0 cos(𝜃2+𝜃3+𝜃4)
𝑝 𝜃 =
cos 𝜃1(𝐿3 cos 𝜃2+𝜃3 + 𝐿2 cos 𝜃2 + 𝐿4 cos 𝜃2+𝜃3+𝜃4 )
sin 𝜃1(𝐿3 cos 𝜃2+𝜃3 + 𝐿2 cos 𝜃2 + 𝐿4 cos 𝜃2+𝜃3+𝜃4 )
𝐿1 + 𝐿3 sin 𝜃2+𝜃3 + 𝐿2 sin 𝜃2 + 𝐿4 sin 𝜃2+𝜃3+𝜃4 )
IV. SIMULATION RESULTS
Using the robotics toolbox together with the
Matlab software [11-13], the kinematics of a robotic
arm can be simulated and analyzed based on the DH
convention described before. The toolbox takes a
conventional approach to represent the kinematics
and dynamics of serial-link robot arms.
Figure 4. Home position
Figure 5. Upright position.
Figure 6. Left-down position.
Figure 7. All joints are given angles

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V. CONCLUSION
Kinematics model of a 4 degree-of-freedom
robotic arm is presented using both the DH method
and product of exponential formula. It is proven that
both approaches provide the same solution for the
robot manipulator under study. In addition, the
simulation of the robot manipulator is carried out
using the Matlab software via the robotics toolbox,
through which several positions of the manipulator
are realized based on the DH convention. Although
the results of the product of exponential formula are
not given, they are expected to be same as those of
the DH convention.
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