Forward and Inverse Kinematic Analysis of Robotic Manipulators

International Research Journal of Engineering and Technology (IRJET) e-ISSN: 2395 -0056
Volume: 04 Issue: 02 | Feb -2017 www.irjet.net p-ISSN: 2395-0072
© 2017, IRJET | Impact Factor value: 5.181 | ISO 9001:2008 Certified Journal | Page 1459
Forward and Inverse Kinematic Analysis of Robotic Manipulators
Tarun Pratap Singh1, Dr. P. Suresh2, Dr. Swet Chandan3
1 M.TECH Scholar, School Of Mechanical Engineering, GALGOTIAS UNIVERSITY, GREATER NOIDA, U.P. INDIA
2,3Asociate Professor, School Of Mechanical Engineering, GALGOTIAS UNIVERSITY, GREATER NOIDA, U.P. INDIA
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Abstract - Today every small and large industry use the
robotic manipulator to complete the various task like as
picking and placing, welding process, painting and
material handling but to complete these task one of the
most important problem is to get the desire positionand
orientation of the robotic manipulators. There are two
method for analyzing the robotic manipulator one is
forward kinematic analysis and another is inverse
kinematic analysis. This project aim to model the
forward and inverse kinematic of 5 DOF and 6 DOF
robotic manipulator. A movement flow planning is
designed and further evaluate all the DH parameter to
calculate the desire position and orientation of the end
effector. Forward kinematics is simple to design but for
inverse kinematic solution traditionalmethod(iterative,
DH notation, transformation)areused.Andcompare the
result with analytical solution and see there are
acceptable error. A FK and IK solution of aspect robotic
manipulator are successfully modeled.
Key Words: Forward and inverse kinematics, DOF
(degree of freedom), transformation, DH convention,
Robotic Arm.
1. INTRODUCTION
Robot is a machine that collects the information about
the environment using some sensors and makes a
decisionautomatically.Todayrobotareusedinvarious
field like as medical, industry, military operation, in
space and some dangerous place. Where human don’t
want to work. But the controlling of robot manipulator
has been challenges with higher DOF. Position and
orientation analysis of robotic manipulator is an
essential step to design and control. In this paper a
basic introduction of the position and orientation
analysis of a serial manipulator is given. A robot
manipulatorconsistsetoflinksconnectedtogetherthis
either serial or parallel manner. The FK analysis is
simple to analysis of model and calculate the position
using the joint angle. But thechallengeintoanalyzethe
IK solution using the position. Complexity of the IK
increases with increase the DOF.so to analyzeIKinthis
Paper use the DH convention and transformation type
solution.
1.1 Kinematics
Kinematics is the branch of mechanics that deals
with the motion of the bodies and system without
considering the force. And the robot kinematics
applies geometry to the study movement of multi
DOF kinematic chains that form the structure of
robot manipulator [1].

Robot kinematic studies the relationship between
the linkages of robot with the position, orientation
and acceleration. And the robotic kinematic analysis
are divided into two types
1.1.1 Forward kinematic
This specified the joint parameter and kinematic
equation is used to compute the position of end
effector from specified value for the each joint
parameter. Or Calculation of the position and
orientation of the robotic manipulator in terms of the
joint variable is called forward kinematic.
1.1.2 Inverse kinematic
This is oppose to the FK. And specified the position of
end effectors and kinematics equation is used to
compute the joint angle from specified position of end
effector.
1.2 DH parameter
This was first introduce by JACQUES DENVIT and
RICHARAD S. HARTENBERG.DHconventionisusedfor
selecting frame of reference for the robotic arm. In this
convention, coordinate frame areattachedtothejoints
between two link to describe the location of each link
relative to its previous [2].
Fig2: Two Coordinate frames system [2]
There are four parameter used in D-H parameter
representation.Theseparametersdescribetherelative
rotation and translation between consecutive frames.
Link length (ai): the distance between the axis z0 and
z1, and this distance measure along the x1 axis.
(Trans, x, ai)
Link offset (di): distance from origin O0 to the
intersection of the x1 axis withz0 measuredalongthez0
axis (Trans, z, ai).
Joint angle (Ɵi): angle from x0 and x1 measured in
plane normal to z0 (ROT, z, Ɵi).
Link twist (αi): angle between Z0 and Z1, measured in
plane normal to X1 axis (ROT, x, αi).
2. Literature Review
In robotic kinematic analysis forward kinematic is
simple to obtain but Obtaining the inverse kinematics
solution has been one of the main concerns in robot
kinematics research. The complexity of the solutions
increases with higher DOFdue torobotgeometry,non-
linearequations(i.e.trigonometricequationsoccurring
when transforming between Cartesian and joint
spaces) and singularity problems. Obtaining the

inverse kinematics solution requires the solution of
nonlinear equations having transcendental functions.
To solving the IK equation many researcher used
method like algebraic [3], geometric [4], and iterative
[5] for complex manipulators, these methods are time
consuming and produce highly complex mathematical
formulation. Whichcan’tsolveeasily.Calderonetal.[6]
proposed a hybrid approach to inverse kinematicsand
control and a resolve motion rate control method are
experimented to evaluate their performances in terms
of accuracy and time response in trajectory tracking.
Xu et al. [7] proposed an analyticalsolutionfora5-DOF
manipulator to follow a given trajectory while keeping
the orientation of one axis in the end-effector frame by
consideringthesingularpositionproblem.Ganetal.[8]
derived a complete analytical inverse kinematics (IK)
model, which is able to control the P2Arm to any given
position and orientation, in its reachable space, so that
the P2Arm gripper mounted on a mobile robot can be
controlled to move to any reachable position in an
unknown environment.HereinthisprojectusetheD-H
convention used to solve forward kinematic equation
than by using these equations find the value of IK.
3. Kinematic Modelling of 5 DOF and 6 6DOF
robotic manipulators
Take the simple 5r robotic manipulator which have
revolute joint only and similar way for 6DOF use 6r
robotic manipulator
3.1 Modelling of 5 DOF
Fig 4: Coordinate frame for the 5-DOF Redundant
manipulator
Coordinate frame assignment for 5 degree of freedom
shown above. They are establish using the D-H
conversation for each joint coordinate. And frame 5 is
auxiliary frame attach to end effector and the frame 6
have the same direction as frame 5 and show no
rotation for end effectors. The D-H parameter for 5
DOF is given
Table 1: D-H convention for 5 degree of
freedom
fram
e
Joint
offse
t in
m
Theta
(degree
)
Link
length in
m
Twist
angle
(Degre
e)
Initial
joint
value
In m
or
deg.
Final
joint
value
In m or
degree
0-1 D1=
0.13
variabl
e
A1 = o.7 -90 -365 365
1-2 0 variabl
e
A2=1.6 0 -150 30
2-3 0 variabl
e
0 -90 60 -210
3-4 D4=
0.14
variabl
e
0 90 0 360
4-5 0 variabl
e
0 -90 60 -90
5-
EE
D6=
0.16
variabl
e
0 0 0 0

In above table initial joint value and the final joint
value denote the limit of the value of theta for
orientation. These are maximum value at that
manipulator rotate.
3.2 Modelling of 6 DOF
In the 6 DOF one joint added at the end effector and all
the configuration is same as 5 DOF because adding of
one joint one DOF increase. Here all the joint are
revolute type and have two type of joint one is that
which have the limited rotation and another is which
have rotation 360 or many more like base and end
effector joint both are second type joint which have no
limitation of angle.
Fig 5: Puma5606DOFrobotwithframeassignment
Table 2: D-H convention for 6 degree of freedom
fram
e
Joint
offse
t in
m
Theta
(degree
)
Link
length in
m
Twist
angle
(Degre
e)
Initial
joint
value
In m
or
deg.
Final
joint
value
In m or
degree
0-1 0 variabl
e
0 -90 -185 185
1-2 D2=
0.5
variabl
e
A2= 0.7 0 -155 35
2-3 D3=
.094
8
variabl
e
A3 =
0.948
90 -130 154
3-4 D4=
0.68
variabl
e
0 -90 -350 350
4-5 0 variabl
e
0 90 -130 130
5-
EE
D6=
0.85
3
variabl
e
0 0 -350 350
4. Kinematic analysis of manipulators
By using the D-H conversation easily calculate the
homogeneous transformation matrix and by usingthis
FK equation found. The D-H conventions uses a
product of four basic transformation to represent the
homogeneous transformation and denoted by Ai.
The A matrix is a homogenous 4x4 transformation
matrix which describe the position of a point on an
object and the orientation of the object in a three
dimensional space. The homogeneous transformation
matrix from one frame tothenextframecanbederived
by the using D-H parameters.
In D-H convention, each homogeneous transformation
matrix Ai is represented as a product of four basic
transformations as follows.
Ai = (ROT, z, Ɵi)(Trans, z, ai)(Trans, x, ai)(ROT, x, αi).

Ai =
4.1 Forward kinematic analysis:
By using equation 1st calculate the transformation
matrix for each joint and this equation have 3 fix
component and one is variable that’s one is theta and
have one 3*3 rotational matrix that show the
orientation of the end effector and have one 1*3 type
matrix which show the position of the end effector.
At the base of the robot,
it can be started with the first joint and then transform
to the second joint, then to the third until to the arm-
end of the robot, and eventually to the end effectors.
The total transformation betweenthebaseoftherobot
and the hand is
By multiplying these equation we can get the final
equation for forward kinematics and compare the
value with equation one and finally get the kinematic
equation

4.2 Inverse kinematic analysis:
Inverse kinematic analysis is the opposite of the
forward kinematic analysis. The corresponding
variables of each joint could found with the given
location requirement of the end of the manipulator in
the given references coordinates system.
Inverse kinematic analysis is done by multiplying each
inverse matrix of T matrices on the left side of above
equation and then equalizing the corresponding
elements of the equal matrices of both ends [9].
To solve the angle use equation 1st to calculate the
value of each angle by using the end effector position
On comparing equation (20) and equation (1)
By the value of S1 and a2 C2, calculate the value of C1
Now similarly get the values of all theta in terms of
position and the orientation.
5. Results
5.1 Result for 5 degree of freedom
By using the above equation solve the position and
orientation for 20 iteration by using the given angle if
the position of the end effector are given than use the

inverse kinematic equation to get the angle of each
joint
Table 3: all possible angle for 5 DOF
Above table shows the all possible joint values for 5
DOF robotic manipulator. And used to calculate the
value of position and orientation of end effector using
forward kinematic. Similarly for the inverse kinematic
analysis in which the value of position of the end
effector is known and using these value calculate the
joint value for each link. The all possible joint values of
end effector are given in above table.
Fig 6: plot for joint value
The below table shows the all possiblepositionofEEat
the given joint valueandcalculatebyusingFKequation
and this value is further used for IK.
End X End Y End Z
71.1641 -6.22605 210
71.43058 -5.49949 210.536
73.07748 -0.3708 214.2102
75.71005 14.04008 223.7318
73.0788 43.05237 240.8419
47.10427 86.09795 265.6854
-27.6663 115.2092 296.2026
-134.304 58.38066 327.8614
-140.923 -112.938 354.2492
65.55203 -206.678 368.853
248.9949
-5.97E-
15 367.559
81.94976 258.3784 350.5228
-218.21 174.8769 322.1171
-252.509 -109.763 288.9284
-61.2086 -254.887 257.1425
117.7797 -215.28 230.8243
198.0047 -116.649 211.6126
214.6015 -39.7969 199.3397
211.5727 1.073525 192.8173
208.3327 16.03968 190.3564
Table 4: Position value for End effector of 5DOF
Fig7: plot for link 1 position
Fig 8: plot for link 2 position
jointValue1 jointValue2 jointValue3 jointValue4 jointValue5
-365 -150 60 0 60
-364.403 -149.853 59.77903 0.29463 59.87724
-360.291 -148.839 58.25821 2.322386 59.03234
-349.494 -146.177 54.26494 7.646741 56.81386
-329.497 -141.246 46.86864 17.50848 52.7048
-298.683 -133.648 35.47183 32.70422 46.37324
-256.497 -123.246 19.86864 53.50848 37.7048
-203.494 -110.177 0.264944 79.64674 26.81386
-141.291 -94.8388 -22.7418 110.3224 14.03234
-72.4026 -77.8527 -48.221 144.2946 -0.12276
0 -60 -75 180 -15
72.40255 -42.1473 -101.779 215.7054 -29.8772
141.2907 -25.1612 -127.258 249.6776 -44.0323
203.4941 -9.82337 -150.265 280.3533 -56.8139
256.4967 3.245762 -169.869 306.4915 -67.7048
298.6831 13.64789 -185.472 327.2958 -76.3732
329.4967 21.24576 -196.869 342.4915 -82.7048
349.4941 26.17663 -204.265 352.3533 -86.8139
360.2907 28.83881 -208.258 357.6776 -89.0323
364.4026 29.85268 -209.779 359.7054 -89.8772

Fig 12: surface plot for position of end effector
5.2 Result for 6 degree of freedom
By using the forward kinematic analysis get all the
possible position of end effectors. And these value are
further used for calculation of inverse kinematic in
which calculate the angle using the position of end
effectors. Take all possible joint value as input for
forward kinematic analysis and get all the possible
position of the end effector. And for the inverse
kinematic position values of the endeffectoraretaken
as input and by using this calculate the joint value for
each joint to get the desire orientation.
jointValue1 jointValue2 jointValue3 jointValue4 jointValue5 jointValue6
0 0 0 0 0 0
0.049105 0.049105 -0.04911 0.049105 0.049105 0.049105
0.387064 0.387064 -0.38706 0.387064 0.387064 0.387064
1.274457 1.274457 -1.27446 1.274457 1.274457 1.274457
2.918079 2.918079 -2.91808 2.918079 2.918079 2.918079
5.450703 5.450703 -5.4507 5.450703 5.450703 5.450703
8.918079 8.918079 -8.91808 8.918079 8.918079 8.918079
13.27446 13.27446 -13.2745 13.27446 13.27446 13.27446
18.38706 18.38706 -18.3871 18.38706 18.38706 18.38706
24.04911 24.04911 -24.0491 24.04911 24.04911 24.04911
30 30 -30 30 30 30
35.95089 35.95089 -35.9509 35.95089 35.95089 35.95089
41.61294 41.61294 -41.6129 41.61294 41.61294 41.61294
46.72554 46.72554 -46.7255 46.72554 46.72554 46.72554
51.08192 51.08192 -51.0819 51.08192 51.08192 51.08192
54.5493 54.5493 -54.5493 54.5493 54.5493 54.5493
57.08192 57.08192 -57.0819 57.08192 57.08192 57.08192
58.72554 58.72554 -58.7255 58.72554 58.72554 58.72554
59.61294 59.61294 -59.6129 59.61294 59.61294 59.61294
59.95089 59.95089 -59.9509 59.95089 59.95089 59.95089

Fig 13: surface plot for joint value
End X End Y End Z
526.628 139.692 1100.599
527.0173 140.1437 1100.13
529.6739 143.2708 1096.88
536.4599 151.6324 1088.162
548.2687 167.6693 1071.309
564.3856 193.6511 1043.588
581.9591 231.3008 1002.315
595.9011 280.9468 945.3523
599.6258 340.4005 871.9466
586.8507 404.2318 783.6389
554.0557 464.3222 684.7641
502.4461 512.0784 582.0885
438.1163 541.5359 483.4645
370.0438 551.6382 395.9441
306.9984 546.2927 324.1966
255.0976 532.2926 269.9224
216.9997 516.4974 232.3015
192.4576 503.7352 208.942
179.3259 496.0306 196.7058
174.3527 492.9474 192.118

6. Conclusion and future work
To conclude, this paper proposed mathematical
approach for solving the forward and inverse
kinematic for 5 DOF and 6 DOF (PUMA 560) robotic
manipulators. The experimental result is obtained by
using robot-analyzer and mat-lab software and
compare the result with the result calculate by using
the D-H transformation.
This technique can be used in various field to
determine the positions and orientations. It can be
used for:
 Under water manipulator
 Nuclear, toxic waste disposal and mining
robot
 Firefighting, construction and agricultural
robot
 Medical application
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Forward and Inverse Kinematic Analysis of Robotic Manipulators

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