CPREDICTION OF INVERSE KINEMATICS SOLUTION OF A REDUNDANT MANIPULATOR USING ANFIS

18
International Journal of Research and Innovation (IJRI)
CPREDICTION OF INVERSE KINEMATICS SOLUTION
OF A REDUNDANT MANIPULATOR USING ANFIS
Manish.B.J 1
, Dilip Maha2
, K.rajanikanth3
,
1 Research Scholar, Department Of Mechanical Engineering, Aurora's Scientific Technological & Research Academy, Hyderabad, India
2 Professor , Department Of Mechanical Engineering, Aurora's Scientific Technological & Research Academy, Hyderabad, India
3 Associate professor,Department Of Mechanical Engineering, Aurora's Scientific Technological & Research Academy, Hyderabad, India
*Corresponding Author:
Manish.B.J ,
Research Scholar, Department Of Mechanical Engineering,
Aurora's Scientific Technological & Research Academy,
Hyderabad, India
Published: July 28, 2014
Review Type: peer reviewed
Volume: I, Issue : II
Citation: Manish.B.J, Research Scholar (2014)
CPREDICTION OF INVERSE KINEMATICS SOLUTION OF
A REDUNDANT MANIPULATOR USING ANFIS
INTRODUCTION
Introduction to Robotics Word robot was coined by
a Czech novelist Karel Capek in 1920. The term ro-
bot derives from the Czech word robota, meaning
forced work or compulsory service. A robot is repro-
grammable, multifunctional manipulator designed
to move material, parts, tools, or specialized devices
through various programmed motions for the per-
formance of a variety of tasks [1]. A simpler version
it can be define as, an automatic device that per-
forms functions normally ascribed to humans or a
machine in the form of a human.
History of Robotics The first industrial robot named
UNIMATE; it is the first programmable robot de-
signed by George Devol in1954, who coined the
term Universal Automation. The first UNIMATE was
installed at a General Motors plant to work with
heated die-casting machines
The first industrial robot: UNIMATE
In 1978, the Puma (Programmable Universal Ma-
chine for Assembly) robot is developed by Victor
Scheinman at pioneering robot company Unimation
with a General Motors design support. These robots
are widely used in various organisations such as
Nokia Corporation, NASA, Robotics and Welding or-
ganization.
Abstract
In this thesis, a method for forward and inverse kinematics analysis of a 5-DOF and a 7- DOF Redundant manipulator
is proposed. Obtaining the trajectory and computing the required joint angles for a higher DOF robot manipulator is one
of the important concerns in robot kinematics and control. The difficulties in solving the inverse kinematics equations
of these redundant robot manipulator arises due to the presence of uncertain, time varying and non-linear nature of
equations having transcendental functions. In this thesis, the ability of ANFIS is used to the generated data for solving
inverse kinematics problem. A single- output Sugeno-type FIS using grid partitioning has been modeled in this work.
The forward kinematics and inverse kinematics for a 5-DOF and 7-DOF manipulator are analyzed systemically. The Ef-
ficiency of ANFIS can be concluded by observing the surface plot, residual plot and normal probability plot. This current
study in using different nonlinear models for the prediction of the IKs of a 5-DOF and 7-DOF Redundant manipulator
will give a valuable source of information for other modellers.
Keywords: 5-DOF and 7-DOF Redundant Robot Manipulator; Inverse kinematics; ANFIS; Denavit-Harbenterg (D-H)
notation.
1401-1402

19
Puma Robotic Arm
Then the robot industries enters a phase of rapid
growth to till date, as various type of robot are being
developed with various new technology, which are
being used in various industries for various work.
Few of these milestones in the history of robotics
are given below.
Components and Structure of Robots
The basic components of an industrial robot are:
The manipulator
The End-Effector (Which is a part of the manipula-
tor)
The Power supply
The controller the manipulator
The End-Effector (Which is a part of the manipula-
tor)
The Power supply
The controller
Forward Kinematics And Inverse Kinematics
In this section of the thesis the forward kinematics
and the inverse kinematics of the 5- dof and 7-dof
redundant manipulator is discussed. The denavit-
harbenterg (d-h) notation for these two manipula-
tors is discussed with steps used for deriving the
forward kinematics is presented. Then this chapter
is concluded with the solution of inverse kinematics
for the 5-dof redundant manipulator is given. The
forward kinematics is concerned with the relation-
ship between the individual joints of the robot ma-
nipulator and the position (x,y, and z) and orienta-
tion of the end-effector. Stated more formally, the
forward kinematics is to determine the position and
orientation of the end-effector, given the values for
the joint variables of the robot. The joint variables
are the angles between the links in the case of revo-
lute or rotational joints, and the link extension in
the case of prismatic or sliding joints. The forward
kinematics is to be contrasted with the inverse kin-
ematics, which will be studied in the next section
of this chapter, and which is concerned with deter-
mining values for the joint variables that achieve a
desired position and orientation for the end-effector
of the robot. The above mention theory is explained
diagrammatically in figure.

20
Work space for the 5-DOF Redundant manipulator.
Considering all the D-H parameters, the x, y and
z coordinates are calculated for 5-DOF Redundant
manipulator End-effector using forward kinematics
equation shown in equations 4-15. For solving the
forward kinematics equations, the angles of rotation
of the joints are taken as tabulated in Table 1. Fig-
ure 4 shows the workspace for 5-DOF Redundant
manipulator
Work space for the 7-DOF Redundant manipulator.
Considering all the D-H parameters, the x, y and z
coordinates (i.e. End-effector
Coordinates) are calculated for 7-DOF Redundant
manipulator using forward kinematics equation as
shown in equations 17-28. For solving the forward
kinematics equations, the angles of rotation of the
joints are taken as tabulated in. Figure shows the
workspace for this manipulator.
ANFIS Architecture
ANFIS stands for adaptive Neuro-fuzzy inference
system developed by Roger Jang [57]. It is a feed for-
ward adaptive neural network which implies a fuzzy
inference system through its structure and neu-
rons. He reported that the ANFIS architecture can
be employed to model nonlinear functions, identify
nonlinear components on-line in a control system,
and predict a chaotic time series. It is a hybrid Neu-
ro-fuzzy technique that brings learning capabilities
of neural networks to fuzzy inference systems. It is
a part of the fuzzy logic toolbox in MATLAB R2008a
software of Math Work Inc. [58]. The fuzzy inference
system (FIS) is a popular computing frame work
based on the concepts of fuzzy set theory, fuzzy if-
then rule, and fuzzy reasoning. It has found suc-
cessful application in a wide variety of fields, such
as automatic control, data classification, decision
analysis, expert system, time series prediction, ro-
botics, and pattern recognition. The basic structure
of a FIS consists of 3 conceptual components: a rule
base, which contains a selection of fuzzy rules: a
database, which defines the membership function
used in fuzzy rules; a reasoning mechanism, which
performs the inference procedure upon the rules
and given facts to derive a reasonable output or con-
clusion. The basic FIS can take either fuzzy input
or crisp inputs, but outputs it produces are almost
always fuzzy sets. Sometime it is necessary to have
a crisp output, especially in a situation where a FIS
is used as a controller. Therefore, method of defuzzi-
fication is needed to extract a crisp value that best
represent a fuzzy set. For solving the IK of 5-DOF
and 7-DOF redundant manipulator used in this
work Sugeno fuzzy inference system is used, to ob-
tain the fuzzy model, The Sugeno FIS was proposed
by Takagi, Sugeno, and Kang [59, 60] in an effort
to develop a systematic approach to generate fuzzy
rules from a given input and output data set. The
typical fuzzy rule in a Sugeno fuzzy model for three
inputs used in this work for both the manipulator
has the form: If x is A, y is B and z is C, then z (x,
y, z), where A, B, C are fuzzy sets in the antecedent,
while (x, y, z) is a crisp function in the consequent.
Usually, f (x, y, z) is a polynomial in the input vari-
ables x, y, and z but it can be any function as long
as it can appropriately describe the output of the
model with the fuzzy region specified by antecedent
of the rule. When f (x, y, and z) is a first order poly-
nomial, the resulting FIS is called first order Sugeno
fuzzy model. When the fuzzy rule is generated, fuzzy
reasoning procedure for the fuzzy model is followed
as shown in Figure 3. Since each rule has a crisp
output, the overall output is obtained via weighted
average, thus avoiding the time consuming process
of defuzzification required in Mamdani model [61].
In practice, the weighted average operator is some-
time replaced with weighted sum operator to reduce
computation further, especially in the training of
FIS.

21
From the above theory, it can be conclude that the
antecedent of a fuzzy rule define a local fuzzy region
while the consequent describe the behavior within
the region via various constituent. The consequent
constitute of a first order Sugeno model is a linear
equation. Different consequent constitute results in
different FIS, but their antecedent are always same.
ANFIS model structure used for 5-DOF Redundant
manipulator
ANFIS Architecture used for 7-DOF Redundant
manipulator. For solving the inverse kinematics
equation of 7-DOF Redundant manipulator, in this
work, the grid partitioning option in the ANFIS tool-
box is used. For each input, 7 membership func-
tion (Gaussian membership) are used along with
343(=7^3) fuzzy rules are applied for all three in-
puts. For the Neuro-fuzzy model used, 2187 data
points are analytically obtained from MATLAB, of
which 1640 are used for training and the remaining
547 are used for testing (validating).
RESULT AND DISCUSSION
In this section of the thesis the surface plots, the
residual plots and the normal probability plots for
the 5-DOF and 7-DOF redundant manipulator is
carried out. The surface plots obtained for this type
of manipulators explains the efficiency of the ANFIS
methodology. The residual plots obtained by com-
paring the predicted data from the ANFIS and the
analytical data show that, the data predicted using
ANFIS methodology deviate very less from the ana-
lytical data. The last section of this chapter is con-
cluded with obtaining the normal probability plots.
The details of the plots are explains in the following
section.
3-D Surface viewers Analysis In this section the 3-D
surface plots, obtained for the 5-DOF and 7-DOF
Redundant manipulator is discussed. The surface
plot displays both the connecting lines and faces of
the surface in color. The surf command in MATLAB
tool is use to create the 3-D surface plots of the ma-
trix data. The surface plot explains the relation be-
tween the output and two inputs.
3-D Surface plots obtained for all joint angles of
5-DOF Redundant manipulator Figures show sur-
face plot of five ANFIS networks relating inputs with
joint angles of 5-DOF Redundant manipulator. Fig-
ure 15 indicates the surface plot between Cartesian
coordinates y and z for
It shows that when the values of y and z moving
in a positive direction, there is a marginal increase
followed by a decrease in1 values. The inputs-out-
put surface plot of 2 is shown in Figure. The Figure
depicts that the value of2 increases linearly when
moving in the positive direction of y coordinate to
some values of y and then there is a sudden in-
crease of 2 values.
No significant change in the value of 2 is observed
with change in values of z coordinate. By moving
from negative direction to the positive direction of x
and y coordinates, the 3 value decreases first then
followed by slightly increase, can be easily conclude
from figure. Similarly the surface plot of5 with input
variables x and z coordinate is depicted in figure. It
shows that the value of inputs has significant effect
in determining the value of
It concludes from the surface plot that the contribu-
tion of interdependent parameters toward obtaining
the output can easily provide through the ANFIS
algorithm and can be hardly obtained otherwise
without employing massive computations. All the
surface viewer plots show that the total surface is
covered by the rule base

22
3-D Surface plots obtained for all joint angles of
7-DOF Redundant manipulator The following Figure
19-25 shows the three dimensional surface plot of
ANFIS network relating to the joint angle of 7-DOF
Redundant manipulator. Figure indicates the sur-
face plot between Cartesian coordinates y and z for
When the value of z decreases, there is a sudden in-
crease in 1 value followed by decrease at the middle
range of z value and there is no significant change in
1 value for y coordinate. The inputs-output surface
plot of 2 is shown in Figure. The Figure depicts that
the value of 2 decrease first followed by increase, for
the increase in the value of z. No significant change
in the value of 2 is observed with change in values
of y coordinate. When y changes from positive value
to negative value, there is a marginal increase in the
value of 3 as well as there is no significant change
with the value of z, as clearly noticed from Figure
With the increase in y value, at its middle range,
the value of 5 decrease first then increase, where
as there is no significant change for values of z, as
depicted in Figure 23. Similarly, the 3 dimensional
surface viewers for 4, 6, and 7 can be explain. All
the surface plots obtained from ANFIS, are continu-
ous, smooth and the total surface is covered by the
rule base.
Residual Plot Analysis
Residuals are the difference between the
predicted output from the model (ANFIS)

23
and the actual values of joint angles. The
residual plot is a graph that shows the re-
siduals in the vertical axis and the inde-
pendent variables in the horizontal axis. If
the points in the residual plot are randomly
dispersed around the horizontal axis, the
prediction model is considered to be appro-
priate for the data i.e. there is no drift in
the data. In this section the residual plots
are obtained for training and testing data
of all joint angles of 5-DOF Redundant ma-
nipulator. It depicts the distribution of re-
siduals of all joint angles are in the positive
and negative axis of the plot. The residual
plots for 5-DOF and 7-DOF are shown in
following section.
The Residual plot of Training data for all
joint angle of 5-DOF Redundant manipula-
tor. The residual plots of training data for
of 5-DOF Redundant robot.
manipulator are depicted in Figures 27-31
respectively. The residual plot shows a fair-
ly random pattern as some of the residuals
are in positive and some are lies in the neg-
ative side of the horizontal axis. Figure 27
shows a random pattern indicating a good
fit for training data of 1. As a very large
number of residuals lie close to the hori-
zontal axis shown in Figure 28, it indicates
a reasonably good fit for 2. The Figures 29-
30 indicates a decent fit to the model of 3
and 4 as most of the residuals lie between
-0.01 to 0.01. The Figure 31 explains the
residual plot for training data of 5. It indi-
cates a few of the residuals of 5 lies beyond
the range -0.1 to 0.1 and does not alter the
prediction model of the data. The average
absolute error (actual minus and predicted
values) for the training data are found to
be 0.0700, 0.0011, 0.0330, 0.0850, and
0.0240 for the joint coordinates are found
to be 0.06, 0.03, 0.09, 0.10, and 0.11 re-
spectively.
The Residual plot of testing data for all joint
angle of 5-DOF Redundant manipulator.
The residual plots of testing data for and
5 of 5-DOF Redundant robot manipulator
are studied. The residual plot shows a fair-
ly random pattern as some of the residuals
are in positive axis and some are lies in the
negative axis of the of the graph. Figure
shows a random pattern indicating a good
fit for training data
1. As a very large number of residuals lie
close to the horizontal axis shown in Figure
33, it indicates a reasonably good fit for
2. The residuals for 3 lie between -0.2 to
0.2 and distributed over both sides of the
mean line. It indicates that the prediction
model is well suited for the study Figure
The Figures 35-36 indicates a decent fit to
the model of4 and 5 as most of the residu-
als lie between -0.03 to 0.03.
Normal probability plot analysis of Training
data for all joint angle of 5-DOF Redundant
manipulator

24
CONCLUSION
In this study, the inverse kinematics solution using
ANFIS for a 5-DOF and 7-DOF Redundant manipu-
lator is presented. The difference in joint angle de-
duced and predicted with ANFIS model for a 5-DOF
and 7-DOF Redundant manipulator clearly depicts
that the proposed method results with an acceptable
error. The modeling efficiency of this technique was
obtained by taking three end-effector coordinates as
input parameters and five and seven joint positions
for a 5-DOF and 7-DOF Redundant manipulator
respectively as output parameters in training and
testing data of NF models. Also, the ANFIS model
used with a smaller number of iteration steps with
the hybrid learning algorithm. Hence, the trained
ANFIS model can be utilized to solve complex, non-
linear and discontinuous kinematics equation com-
plex robot manipulator; thereby, making ANFIS an
alternative approach to deal with inverse kinemat-
ics. The analytical inverse kinematics model derived
always provide correct joint angles for moving the
arm end-effector to any given reachable positions
and orientations. As the ANFIS approach provides a
general frame work for combination of NN and fuzzy
logic. The efficiency of ANFIS for predicting the IK
of Redundant manipulator can be concluded by ob-
serving the 3-D surface viewer, residual and normal
probability graphs. The normal probability plots of
the model are also plotted. The normal probability
plot of residuals of training and testing data ob-
tained from ANFIS shows that the data set of ANFIS
are approximately normally distributed. The meth-
ods used for deriving the inverse kinematics model
for the these manipulators could be applied to other
types of robotic arms, such as the EduBots devel-
oped by the Robotica Ltd, Pioneer 2 robotic arm
(P2Arm), 5-DOF Lynx 6 Educational Robot arm. It
can be concluded that the solution developed in this
paper will make the PArm more useful in applica-
tion with unpredicted trajectory movement in un-
known environment.
Future Work
In this work a hybrid neuro-fuuzy technology is
used for the study of inverse kinematics of redun-
dant robot manipulator. ANFIS is adopted for solv-
ing the IK of higher DOF robot manipulator. Due to
its compactness and adaptive nature this technol-
ogy is highly efficiency in predicting the IK of higher
DOF robot manipulator. So this technology can use
in different robot in different field to know the joint
angles, orientations, and the robot working space to
avoid osstacles. The robotics industry has reached
one plateau with the successful introduction of ro-
bots into automotive manufacturing for spot weld-
ing and painting, are two areas where robotic usage
is almost universal. There are several other areas
where the usage of robotics is in its infancy and this
chapter is dedicated to brief descriptions of some of
these fields along with a quick assessment of their
current status. A 20 meters long and 6-DOF remote
robot manipulator is commonly used in space for
repairing satellites and other coordinated activities
on self-propelled platform. So ANFIS can be used to
this robot for its free positioning and to determine
its path. Apart from this, the Neuro fuzzy technique
can be used in various fields to determine the posi-
tions 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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Author
Manish.B.J 1
,
Research Scholar, Department Of Mechanical
Engineering, Aurora's Scientific Technological &
Research Academy, Hyderabad, India
Dilip Maha2
,
Professor , Department Of Mechanical Engineering,
Aurora's Scientific Technological & Research
Academy, Hyderabad, India
K.rajanikanth3
,
Associate professor,Department Of Mechanical
Engineering, Aurora's Scientific Technological &
Research Academy, Hyderabad, India

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