INTELLIGENT INVERSE KINEMATIC CONTROL OF SCORBOT-ER V PLUS ROBOT MANIPULATOR by editor.ijaet

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In this paper, an Adaptive Neuro-Fuzzy Inference System (ANFIS) method based on the Artificial Neural Network (ANN) is applied to design an Inverse Kinematic based controller forthe inverse kinematical control of SCORBOT-ER V Plus. The proposed ANFIS controller combines the advantages of a fuzzy controller as well as the quick response and adaptability nature of an Artificial Neural Network (ANN). The ANFIS structures were trained using the generated database by the fuzzy controller of the SCORBOT-ER V Plus.The performance of the proposed system has been compared with the experimental setup prepared with SCORBOT-ER V Plus robot manipulator. Computer Simulation is conducted to demonstrate accuracyof the proposed controller to generate an appropriate joint angle for reaching desired Cartesian state, without any error. The entire system has been modeled using MATLAB 2011.

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									International Journal of Advances in Engineering & Technology, Nov 2011.
©IJAET                                                             ISSN: 2231-1963



             INTELLIGENT INVERSE KINEMATIC CONTROL OF
               SCORBOT-ER V PLUS ROBOT MANIPULATOR
                            Himanshu Chaudhary and Rajendra Prasad
                     Department of Electrical Engineering, IIT Roorkee, India




ABSTRACT
In this paper, an Adaptive Neuro-Fuzzy Inference System (ANFIS) method based on the Artificial Neural
Network (ANN) is applied to design an Inverse Kinematic based controller forthe inverse kinematical control of
SCORBOT-ER V Plus. The proposed ANFIS controller combines the advantages of a fuzzy controller as well as
the quick response and adaptability nature of an Artificial Neural Network (ANN). The ANFIS structures were
trained using the generated database by the fuzzy controller of the SCORBOT-ER V Plus.The performance of
the proposed system has been compared with the experimental setup prepared with SCORBOT-ER V Plus robot
manipulator. Computer Simulation is conducted to demonstrate accuracyof the proposed controller to generate
an appropriate joint angle for reaching desired Cartesian state, without any error. The entire system has been
modeled using MATLAB 2011.

KEYWORDS: DOF, BPN, ANFIS, ANN, RBF, BP
  I.     INTRODUCTION
Inverse kinematic solution plays an important role in modelling of robotic arm. As DOF (Degree of Freedom) of
robot is increased it becomes a difficult task to find the solution through inverse kinematics.Three traditional
method used for calculating inverse kinematics of any robot manipulator are:geometric[1][2] ,
algebraic[3][4][5] and iterative [6] methods. While algebraic methods cannot guarantee closed form
solutions. Geometric methods must have closed form solutions for the first three joints of the
manipulator geometrically. The iterative methods converge only to a single solution and this solution
depends on the starting point.
The architecture and learning procedure underlying ANFIS, which is a fuzzy inference system
implemented in the framework of adaptive networks was presented in [7]. By using a hybrid learning
procedure, the proposed ANFIS was ableto construct an input-output mapping based on both human
knowledge (in the form of fuzzy if-then rules) and stipulated input-output data pairs.
Neuro-Genetic approach for the inverse kinematics problem solution of robotic manipulators was
proposed in [8]. A multilayer feed-forward networks was applied to inverse kinematic problem of a 3-
degrees-of freedom (DOF) spatial manipulator robot in [9]to get algorithmic solution.
To solve the inverse kinematics problem for three different cases of a 3-degrees-of freedom (DOF)
manipulator in 3D space,a solution was proposed in [10]usingfeed-forward neural networks.This
introduces the fault-tolerant and high-speed advantages of neural networks to the inverse kinematics
problem.
A three-layer partially recurrent neural network was proposed by [11]for trajectory planning and to
solve the inverse kinematics as well as the inverse dynamics problems in a single processing stage for
the PUMA 560 manipulator.
Hierarchical control technique was proposed in[12]for controlling a robotic manipulator.It was based
on the establishment of a non-linear mapping between Cartesian and joint coordinates using fuzzy
logic in order to direct each individual joint. Commercial Microbot with three degrees of freedom was
utilized to evaluate this methodology.
Structured neural networks based solution was suggested in[13] that could be trained quickly. The
proposed method yields multiple and precise solutions and it was suitable for real-time applications.



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International Journal of Advances in Engineering & Technology, Nov 2011.
©IJAET                                                             ISSN: 2231-1963
To overcome the discontinuity of the inverse kinematics function,a novel modular neural network
system that consists of a number of expert neural networks was proposed in[14].
Neural network based inverse kinematics solution of a robotic manipulator was suggested in[15]. In
this study, three-joint robotic manipulator simulation software was developed and then a designed
neural network was used to solve the inverse kinematics problem.
An Artificial Neural Network (ANN) using backpropagation algorithm was applied in [16]to solve
inverse kinematics problems of industrial robot manipulator.
The inverse kinematic solution of the MOTOMAN manipulator using Artificial Neural Network was
implemented in [17]. The radial basis function (RBF) networks was used to show the nonlinear
mapping between the joint space and the operation space of the robot manipulator which in turns
illustrated the better computation precision and faster convergence than back propagation (BP)
networks.
Bees Algorithm was used to train multi-layer perceptron neural networks in [18]to model the inverse
kinematics of an articulated robot manipulator arm.
This paper is organized into four sections. In the next section, the kinematicsanalysis (Forward as well
as inverse kinematics) of SCORBOT-ER V Plus has been derived with the help of DH algorithm as
well as conventional techniques such as geometric[1][2], algebraic[3][4][5] and iterative [6] methods.
Basics of ANFIS are introduced in section3. It also explains the wayfor input selection for ANFIS
modeling. Simulation results are discussed in section 4. Section 5 gives concluding remarks.
II.     KINEMATICS OF SCORBOT-ER V PLUS
SCORBOT-ER V Plus [19] is a vertical articulated robot, with five revolute joints. It has a Stationary
base, shoulder, elbow, tool pitch and tool roll. Figure 1.1 identifies the joints and links of the
mechanical arm.




2.1. SCORBOT–ER V PLUS STRUCTURE
All joints are revolute, and with an attached gripper it has six degree of freedom. Each joint is
restricted by the mechanical rotation its limits are shown below.
Joint Limits:
Axis 1: Base Rotation: 310°
Axis 2: Shoulder Rotation: + 130° / – 35°
Axis 3: Elbow Rotation: ± 130°
Axis 4: Wrist Pitch: ± 130°
Axis 5: Wrist Roll Unlimited (electrically 570°)
Maximum Gripper Opening: 75 mm (3") without rubber pads 65 mm (2.6") with rubber pads
The length of the links and the degree of rotation of the joints determine the robot’s work envelope.
Figure 1.2 and 1.3 show the dimensions and reach of the SCORBOT-ER V Plus. The base of the robot
is normally fixed to a stationary work surface. It may, however, be attached to a slide base, resulting
in an extended working range.




      159                                                               Vol. 1, Issue 5, pp. 158-169
International Journal of Advances in Engineering & Technology, Nov 2011.
©IJAET                                                             ISSN: 2231-1963




2.2. FRAME ASSIGNMENT TO SCORBOT–ER V PLUS
For the kinematic model of SCORBOT first we have to assign frame to each link starting from base
(frame {0}) to end-effector (frame {5}). The frame assignment is shown in figure 1.4.




Here in model the frame {3} and frame {4} coincide at same joint, and the frame {5} is end– effector
position in space.


    Joint i                       (   )           (   )                                 Operating range
      1            − /2            16             349                   1               −155° + 155°
      2             0             221               0                   2               −35° + 130°
      3             0             221               0                   3               −130° + 130°
      4              /2             0               0                /2 + 4             −130° + 130°
      5             0               0             145                   5                −570° 570°

2.3. FORWARD KINEMATIC OF SCORBOT–ER V PLUS
Once the DH coordinate system has been established for each link, a homogeneous transformation
matrix can easily be developed considering frame {i-1} and frame {i}. This transformation consists of
four basic transformations.

                                 0
                                  T5 = 0T1 * 1T2 * 2T3 * 3T4 * 4T5              (1)
                               C1 0 − S1 a1 * C1 
                                                 
                          0T =  S1 0 C1 a1 * S1                              (2)
                            1  0 −1 0      d1 
                                                  
                               
                               0 0    0     1 
                                                 




    160                                                                       Vol. 1, Issue 5, pp. 158-169
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©IJAET                                                             ISSN: 2231-1963
                                                      C2             − S2                   0 a2 * C2 
                                                      S              C2                     0 a2 * S 2 
                                                1
                                                 T2 =  2                                                                                                                (3)
                                                      0                      0              1    0 
                                                                                                       
                                                      0                      0              0    1 
                                                      C3             − S3                   0 a3 * C3 
                                                      S              C3                     0 a3 * S3 
                                                2
                                                 T3 =  3                                                                                                                (4)
                                                      0                      0              1    0 
                                                                                                       
                                                      0                      0              0    1 
                                                    − S 4 0                                 C 4 0
                                                                                            S 4 0
                                               3T =  C 4 0                                                                                                              (5)
                                                 4  0     1                                  0 0
                                                                                                 
                                                       0  0                                  0 1
                                                    C 5 − S 5                                0 0
                                                                                             0 0
                                               4T =  S 5 C 5                                                                                                            (6)
                                                 5 0      0                                  1 d 5
                                                                                                  
                                                    0     0                                  0 1

Finally, the transformation matrix is as follow: -

             −S S − C C S
                  1   5
                             −C S + C S S
                                  1       5
                                             CC234
                                                   C (a + a C + a C + d C )
                                                         5   1            1       5    234      1   234   1       1        2       2        3        23       5     234

             CS −SCS         CC +SS S       SC     S (a + a C + a C + d C ) 
      T = T =                                                               
          0       1   5       1       5       234    1       5        1       5       234       1   234   1       1        2       2        3        23       5     234
                                                                                                (7)
              −C C                                  (d − a S − a S − d S ) 
              5

                          5
                                 SC
                                  234
                                             −S                  5   234                            234       1        2       2        3       23        5       234

                                                                            
                   0              0          0                 1            
Where, = ( ), = ( )               = ( + + ),            = ( + + ).
The T is all over transformation matrix of kinematic model of SCORBOT-ER V Plus, from this we
have to extract position and orientation of end –effector with respect to base is done in the following
section.
2.4. OBTAINING POSITION IN CARTESIAN SPACE
The value of , , is found from last column of transformation matrix as: -

                                                                         (8)
                                                X = C1 (a1 + a2 C2 + a3C23 + d5C234 )
                           Y = S1 (a1 + a2 C2 + a3C23 − d5C234 )         (9)
                          Z = ( d1 − a2 S 2 − a3 S23 − d5 S 234 )       (10)
For Orientation of end-effector frame {5} and frame {1} should be coincide with same axis but in our
model it is not coincide so we have to take rotation of −90° of frame {5} over y5 axis, so the overall
rotation matrix is multiplied with −90° as follow: -
                                                                           cos( −90o ) 0 sin(−90o ) 
                                                                                                     
                                                                     Ry =       0       1     0      
                                                                                                     
                                                                           − sin(−90o ) 0 cos(−90o ) 
                                                     0 0 −1
                                               Ry = 0 1 0 
                                                                                                                                                                        (11)
                                                    1 0 0 
                                                           
The Rotation matrix is: -
                                  0 0 −1  − S1S5 − C1C5 S234                                                       −C5 S1 + C1 S5 S234                                 C1C234 
                              R = 0 1 0  ×  C1 S5 − S1C5 S 234
                                                                                                                   C1C5 + S1S5 S234                                    S5C234 
                                                                                                                                                                                 
                                  1 0 0  
                                                −C5C234                                                                             S5C234                             − S234 
                                                                                                                                                                                 



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International Journal of Advances in Engineering & Technology, Nov 2011.
©IJAET                                                             ISSN: 2231-1963
                        C5C234                 − S5C234                S 234 
              R =  C1S5 − S1C5 S 234
                                            C1C5 + S1S5 S234          S5C234 
                                                                              
                   − S1S5 − C1C5 S 234
                                           −C5 S1 + C1 S5 S 234       C1C234 
                                                                                                                         (12)

Pitch: Pitch is the angle of rotation about y5 axis of end-effector




                          pitchβ = θ 2 + θ 3 + θ 4 = θ 234                                                    (13)
                                                      2        2
                          θ 234 = a tan 2( r13, ± r 23 + r 33 )              (14)
Here we use atan2 because its range is [− , ], where the range of atan is [− /2, /2].
Roll: The      = 5 is derived as follow: -
                   θ5 = a tan 2(r12 / C234 , r11 / C234 )                    (15)
Yaw: Here for SCORBOT yaw is not free and bounded by 1.

2.5. HOME POSITION IN MODELING
At home position all angle are zero so in equation (1.7) put                      1   = 0,    2   = 0,    3   = 0,   4   = 0,    5   =0
So the transformation matrix reduced to:-
                   0     0 1 a1 + a2 + a3 + d5   0              0 1 603
                   0     1 0         0          0               1 0 0 
          THome   =                            =                                                                            (16)
                    −1   0 0         d1          −1             0 0 349 
                                                                        
                   0     0 0         1          0               0 0 1 
The home position transformation matrix gives the orientation and position of end-effector frame.
From the 3×3 matrix orientation is describe as follow, the frame {5} is rotated relative to frame {0}
such that 5 axis is parallel and in same direction to 0 axis of base frame; 5is parallel and in same
direction to 0 axis of base frame; and 5axis is parallel to 0but in opposite direction. The position is
                                                                T
given by the 3 × 1 displacement matrix a1 + a2 + a3 + d5 0 d1  .
                                                              

2.6. INVERSE KINEMATICS OF SCORBOT-ER V PLUS
For SCORBOT we have five parameter in Cartesian space is x, y, z, roll ( ), pitch ( ).For joint
parameter evaluation we have to construct transformation matrix from five parameters in Cartesian
coordinate space. For that rotation matrix is generated which is depends on only roll, pitch and yaw of
robotic arm. For SCORBOT there is no yaw but it is the rotation of first joint 1.
So the calculation of yaw is as follow: -
                         α = θ1 = a tan 2( x, y )                          (17)
Now for rotation matrix rotate frame {5} at an angle – about its x axis then rotate the new frame {5′
} by an angle with its own principal axes ′ , finally rotate the new frame {5′′} by an angle with
its own principal axes ''.
                                                  = (− )∗ ( )∗ ( )
                                   1   0         0   Cγ         0 Sγ  Cα                − Sα    0
                                                                     
                                 =  0 Cβ        Sβ  ×  0        1 0  ×  Sα
                                                                                            Cα      0
                                                                                                      
                                   0 − Sβ       Cβ   − Sγ       0 Cγ   0                 0      1
                                                                                                 




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©IJAET                                                             ISSN: 2231-1963
                            Cα Cγ                −Cγ Sα                Sγ       
                                                                                
                  =  Cβ Sα − Cα S β Sγ     Cβ Cα + S β Sγ Sα          Cγ S β    
                     − S β Sα − Cα Cβ Sγ − S β Cα + Sα Cβ Sγ          Cβ Cγ     
                                                                                                 (18)
Now rotate matrix by 90° about y axis: -
                                                    COS (90o )        0 SIN (90o ) 
                                                                                   
                                     Ry ( −90o ) =       0            1     0      
                                                             o                  o 
                                                    − SIN (90 )       0 COS (90 ) 
                                   0 0 1
                    R y (−90o ) =  0 1 0 
                                                                                                 (19)
                                   −1 0 0 
                                          
After pre multiplying the equation 19 with equation 18, one will get following rotation matrix: -
                       − S β Sα − Cα Cβ Sγ    − S β Cα + Sα Cβ Sγ     Cβ Cγ 
                                                                             
                    =  Cβ Sα − Cα S β Sγ      Cβ Cα + S β Sγ Sα       Cγ S β 
                              −Cα Cγ               Cγ Sα               − Sγ 
                                                                                                 (20)
So, the total transformation matrix is as follows: -
                        − S β Sα − Cα Cβ Sγ     − S β Cα + Sα Cβ Sγ       Cβ Cγ     X
                        C S −C S S              Cβ Cα + S β Sγ Sα         Cγ S β    Y
                    T = β α         α β γ                                            
                               −Cα Cγ                  Cγ Sα               − Sγ     Z
                                                                                     
                                  0                      0                  0       1
                                                                                                   (21)
After comparing the transformation matrix in equation (7) with matrix in equation (21), one can
deduce: -
                                               1 = ,
                                              234 = ,
                                               5= ,
Now, we have 1 and 5 directly but 2, 3           4 are merged in 234 so we have separate them, to
separate them we have used geometric solution method as shown in Figure 1.6




Here for finding     2,   3,   4,   we have X, Y, Z in Cartesian coordinate space from that we can take:-
                               X 1 = ( X 2 + Y 2 ) andY1 = Z          (22)
We have pitch of end-effector 234 = , from that we can find point 2, 2 is calculated as follows: -
                        X 2 = X 1 − d5 cos θ 234
                                                                      (23)
                        Y2 = Y1 + d5 sin θ 234




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International Journal of Advances in Engineering & Technology, Nov 2011.
©IJAET                                                             ISSN: 2231-1963
Now the distance        3and     3can   be found: -
                                                              X 3 = X 2 − a1
                                                              Y3 = Y2
From the low of cosines applied to triangle ABC, we have: -
                                                                  2           2   2
                                                              ( X 3 + Y32 − a2 − a3 )
                                                  cos θ 3 =
                                                                      2 a2 a3

                           θ3 = a tan 2(± 1 − cos 2 θ3 , cos θ3 )                                    (24)
From figure 1.6       2 = −∅ −     or
                         θ 2 = −a tan 2(Y3 , X 3 ) − a tan 2(a3 sin θ3 , a2 + cos θ3 )               (25)
Finally we will get: -
                      θ 4 = θ 234 − θ 2 − θ3                                                         (26)

III.     INVERSE KINEMATICS OF SCORBOT-ER V PLUS USING ADAPTIVE
         NEURO FUZZY INFERENCE SYSTEM (ANFIS)
The proposed ANFIS[7][20][21] controller is based on Sugeno-type Fuzzy Inference System (FIS)
controller.The parameters of the FIS are governed by the neural-network back propagation method.
The ANFIS controller is designed by taking the Cartesian coordinates plus pitch as the inputs, and the
joint angles of the manipulator to reach a particular coordinate in 3 dimensional spaces as the output.
The output stabilizing signals, i.e., joint angles are computed using the fuzzy membership functions
depending on the input variables. The effectiveness of the proposed approach to the modeling is
implemented with the help of a program specially written for this in MATLAB. The information
related to data used to train is given inTable 1.2.


  Sr.        Manipulator       No. of       No. of Parameters                  Total No. of   No. of        No. of       No. of
  No.        Angles            Nodes                                           Parameters     Training      Checking     Fuzzy
                                            Linear        Nonlinear                           Data Pairs    Data Pairs   Rules

  01.        Theta1            193          405           36                   441            4500          4500         81

  02.        Theta2            193          405           36                   441            4500          4500         81

  03.        Theta3            193          405           36                   441            4500          4500         81

  04.        Theta4            193          405           36                   441            4500          4500         81


The procedure executed to train ANFIS is as follows:
(1) Data generation: To design the ANFIS controller, the training data have been generated by using
an experimental setup with the help of SCORBOT-ER V Plus. A MATLAB program is written to
govern the manipulator to get the input –output data set. 9000 samples were recorded through the
execution of the program for the input variables i.e., Cartesian coordinates as well as Pitch. Cartesian
coordinates combination for all thetas are given in Fig.1.7




       164                                                                                    Vol. 1, Issue 5, pp. 158-169
International Journal of Advances in Engineering & Technology, Nov 2011.
©IJAET                                                             ISSN: 2231-1963
(2) Rule extraction and membership functions: After generating the data, the next step is to estimate
the initial rules. A hybrid learning algorithm is used for training to modify the above parameters after
obtaining the Fuzzy inference system from subtracting clustering. This algorithm iteratively learns the
parameter of the premise membership functions and optimizes them with the help of back propagation
and least-squares estimation. The training is continued until the error minimization..The input as well
as output member function used was triangular shaped member function.The final fuzzy inference
system chosen was the one associated with the minimum checking error, as shown in figure 1.8.it
shown the final membership function for the thetas after training.




                                                                                                                                                                                                                                                                                                                       D e g r e e o f m e m b e r s h ip D e g r e e o f m e m b e r s h ip




                                                                                                                                                                                                                                                                                                                                                                                                                                                       D e g r e e o f m e m b e r s h ip D e g r e e o f m e m b e r s h ip
        D e g re e o f m e m b e rs h ip D e g re e o f m e m b e rs h ip




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                                                                             in11 m f1           i n 1 m f2            i n 1 m f3                                                                                                   in12 m f1                         i n 2 m f2                     i n 2 m f3                                                                                in1 m f1
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                                                                                                                                                                                                                                                                                                                                                                                                  -0 . 5               0                  0 .5                                                                                          -0 . 4   -0 . 2        0       0 .2   0 .4
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                                                                                                                                                                                                                                                                                                                                                                                                in13 m f1          in 3 m f2              i n 3 m f3                                                                              4
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                                                                             i n1 m f1           i n 3 m f2             i n 3 m f3                                                                                                   4
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                                                                              1
                                                                            in1 m f1            i n 1 m f2            i n 1 m f3                                                                                                      2
                                                                                                                                                                                                                                 i n 1 m f1                        i n 2 m f2                   in 2 m f3                                                                                          1
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                                                                            0 .5                                                                                                                                                 0 .5                                                                                                                                                          0 .5                                                                                                                            0 .5
                                                                                                                                   o f m e m b e r sDh ei pg r e e
     o f m e m b e r sDh e pg r e e




                                                                                                                                                                                                                                                                                                                       o f m e m b e r s Dh e pg r e e




                                                                                                                                                                                                                                                                                                                                                                                                                                                       o f m e m b e r s Dh e pg r e e
                          i




                                                                                                                                                                                                                                                                                                                                            i




                                                                                                                                                                                                                                                                                                                                                                                                                                                                            i
                                                                               0                                                                                                                                                   0                                                                                                                                                             0                                                                                                                                0
                                                                               -0 .5                0                 0 .5                                                                                                                  -0 .4      -0 . 2           0          0 .2       0 .4                                                                                                -0 .5                0                  0 .5                                                                                          -0 . 4   -0 .2         0       0 .2   0 .4
                                                                                                in p u t1                                                                                                                                                          in p u t 2                                                                                                                                      in p u t 1                                                                                                                             in p u t 2

                                                                             i n13 m f1         i n 3 m f2            i n 3 m f3                                                                                                      4
                                                                                                                                                                                                                                 i n 1 m f1                        i n 4 m f2                   in 4 m f3                                                                                       i n13 m f1          i n 3 m f2            in 3 m f3                                                                                 4
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                               i n 1 m f1                 in 4 m f2             i n 4 m f3




                                                                            0 .5                                                                                                                                                 0 .5                                                                                                                                                          0 .5                                                                                                                            0 .5
                                                                                                                                   D e g re e
     D e g re e




                                                                                                                                                                                                                                                                                                                       D e g re e




                                                                                                                                                                                                                                                                                                                                                                                                                                                       D e g re e
                                                                               0                                                                                                                                                   0                                                                          θ3                                                                                 0                                                                                                                                0
                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                                             θ4
                                                                              -0 .2       0   0 .2      0 .4   0 .6    0 .8                                                                                                                  -4          -2             0            2          4                                                                                                -0 . 2      0   0 .2       0 .4   0 .6    0 .8                                                                                          -4        -2          0         2      4
                                                                                                 in p u t3                                                                                                                                                         in p u t 4                                                                                                                                       in p u t 3                                                                                                                            in p u t 4




(3) Results: The ANFIS learning was tested on a variety of linear and nonlinear processes. The
ANFIS was trained initially for 2 membership functions for 9000 data samples for each input as well
as output. Later on, it was increased to 3 membership functions for each input. To demonstrate the
effectiveness of the proposed combination, the results are reported for a system with81 rules and a
system with an optimized rule base. After reducingthe rules the computation becomes fast and it also
consumes less memory. The ANFIS architecture for θ1 is shownin Fig. 1.9.




    165                                                                                                                                                                                                                                                                                                                                                                                                                                   Vol. 1, Issue 5, pp. 158-169
International Journal of Advances in Engineering & Technology, Nov 2011.
©IJAET                                                             ISSN: 2231-1963
        Five angles have considered for the representation of robotic arm. But as the 5 is independent
of other angles so only remaining four angles was considered to calculate forward kinematics. Now,
for every combination of 1, θ2, θ3 andθ4 values the x and y as well as z coordinates are deduced using
forward kinematics formulas.
IV.                                                                     SIMULATION RESULTS AND DISCUSSION
The plots displaying the root-mean-square error are shown in figure 1.10. The plot in blue represents
error1, the error for training data. The plot in green represents error2, the error for checking data.
From the figure one can easily predict thatthere is almost null difference between the training error as
well as checking error after the completion of training of ANFIS.
                                                                                                 E r r o r C u rve s                                                                                                                                                  E r r o r C u rve s
                                                                 0 .9                                                                                                                                                              0.34
              R M S E (R oo t M ea n S q u a r e d E rro r )




                                                                                                                                                                                R M S E (R oo t M ea n S q u a r e d E rro r )
                                                                0.85
                                                                                                                                                                                                                                   0.32


                                                                 0 .8
                                                                                                                                                                                                                                     0 .3

                                                                0.75
                                                                                                                                                                                                                                   0.28
                                                                 0 .7

                                                                                                                                                                                                                                   0.26
                                                                0.65


                                                                                                                                                                                                                                   0.24
                                                                 0 .6


                                                                0.55
                                                                        0   2   4        6   8         10              12    14   16   18         20
                                                                                                                                                            θ1                                                                     0.22
                                                                                                                                                                                                                                            0   2   4   6         8         10              12   14    16        18       20
                                                                                                                                                                                                                                                                                                                                θ2
                                                                                                    E poc hs                                                                                                                                                             E poc hs




                                                                                                                                                                              R M S E (R o o t M e a n S q u a r e d E r r o r )
                                                                                                 E r r o r C u rve s                                                                                                                                                  E r r o r C u rve s
              R M S E (R o o t M e a n S q u a r e d E rro r)




                                                                 0 .7                                                                                                                                                              0 .44



                                                                                                                                                                                                                                   0 .42
                                                                0 .65

                                                                                                                                                                                                                                     0 .4

                                                                 0 .6
                                                                                                                                                                                                                                   0 .38


                                                                0 .55
                                                                                                                                                                                                                                   0 .36



                                                                 0 .5                                                                                                                                                              0 .34




                                                                0 .45
                                                                                                                                                            θ3                                                                     0 .32
                                                                                                                                                                                                                                            0   2   4   6         8         10              12   14    16        18       20    θ4
                                                                        0   2   4        6   8         10              12    14   16   18         20                                                                                                                     E pochs
                                                                                                    E poc hs




In addition to above error plots, the plot showing the ANFIS Thetas versus the actual Thetasare given
in figures1.11,1.12,1.13 and 1.14 respectively. The difference between the original thetas values and
the values estimated using ANFIS is very small.
                                                                                                                                             Theta1 and ANFIS Prediction theta1
3


2


1


0


-1


-2


-3
                                                                                                                                                                                                                                                              Experimental Theta1
                                                                                                                                                                                                                                                              ANFIS Predicted Theta1
-4
     0                                                                          50                                100                       150                                                                                    200                      250                                  300                                 350
                                                                                                                                                            Time (sec)




                                                                                                                                                  Theta2 and ANFIS Prediction theta2
     2
                                                                                                                                                                                                                                                                                                            Experimental Theta2
                                                                                                                                                                                                                                                                                                            ANFIS Predicted Theta2
 1.5



     1



 0.5



     0



-0.5



     -1
          0                                                                         50                                 100                   150                                                                                   200                      250                                  300                                 350
                                                                                                                                                             Time (sec)




                                  166                                                                                                                                                                                                                   Vol. 1, Issue 5, pp. 158-169
International Journal of Advances in Engineering & Technology, Nov 2011.
©IJAET                                                             ISSN: 2231-1963
                                                Theta3 and ANFIS Prediction Theta3
3


2


1


0


-1


-2
                                                                                                               Experimental Theta3
                                                                                                               ANFIS Predicted Theta3
-3
     0              50         100             150                                   200    250          300                                350
                                                              Time (sec)



                                                Theta4 and ANFIS Prediction Theta4
 2


 1


 0


-1


-2
                                                                                                               Experimental Theta4
                                                                                                               ANFIS Predicted Theta4
-3
     0              50         100             150                                   200    250          300                                350
                                                              Time (sec)




The prediction errors for all thetas appear in the figures 1.15, 1.16, 1.17, 1.18 respectively with a much finer
scale. The ANFIS was trained initially for only 10 epochs. After that the no. of epochs were increased to 20 for
applying more extensive training to get better performance.
                                                     Prediction Errors for THETA 1
3
                                                                                                                  Prediction Error Theta1

2



1



0



-1



-2



-3
     0              50         100             150                                   200    250          300                                350
                                                              Time (sec)



                                                      Prediction Errors for THETA2
     1
                                                                                                                   Prediction Error Theta2

0.5


     0


-0.5


     -1


-1.5
          0          50         100             150                                   200    250         300                                 350
                                                               Time (sec)


                                                      Prediction Errors for THETA3
 2
                                                                                                                   Prediction Error Theta3

 1


 0


-1


-2


-3
     0              50          100            150                                   200    250          300                                 350
                                                              Time (sec)




              167                                                                           Vol. 1, Issue 5, pp. 158-169
International Journal of Advances in Engineering & Technology, Nov 2011.
©IJAET                                                             ISSN: 2231-1963
                                                      Prediction Errors for THETA4
1.5

  1

0.5

  0

-0.5

  -1

-1.5
                                                                                                              Prediction Error Theta4
  -2
       0         50              100            150                                  200   250          300                             350
                                                              Time (sec)




 V.          CONCLUSION
From the experimental work one can see that the accuracy of the output of the ANFIS based inverse
kinematic model is nearly equal to the actual mathematical model output, hence this model can be
used as an internal model for solving trajectory tracking problems of higher degree of freedom (DOF)
robot manipulator. Asingle camera for the reverse mapping from camera coordinates to real world
coordinateshas been used in the present work, if two cameras are used stereo vision can be achieved
andproviding the height of an object as an input parameter would not be required. The methodology
presented herecan be extended to be used for trajectory planning and quite a few tracking applications
with real world disturbances. Thepresent work did not make use of color image processing; making
use of color image processing can helpdifferentiate objects according to their colors along with their
shapes.

ACKNOWLEDGEMENTS
As it is the case in almost all parts of human endeavour so also the development in the field of robotics has been
carried on by engineers and scientists all over the world.It can be regarded as a duty to express the appreciation
for such relevant, interesting and outstanding work to which ample reference is made in this paper.

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©IJAET                                                             ISSN: 2231-1963
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Authors
Himanshu Chaudhary received his B.E. in Electronics and Telecommunication from
Amravati University, Amravati, India in 1996, M.E. in Automatic Controls and Robotics
from M.S. University, Baroda, Gujarat, India in 2000.Presently he is a research scholar in
Electrical Engineering Department, IIT Roorkee, India. His area of interest includes
industrial robotics, computer networks and embedded systems.


Rajendra Prasad received B.Sc. (Hons.) degree from Meerut University, India in 1973. He
received B.E.,M.E. and Ph.D. degree in Electrical Engineering from the University of
Roorkee, India in 1977, 1979 and 1990 respectively. . He also served as an Assistant
Engineer in Madhya Pradesh Electricity Board (MPEB) from 1979- 1983. Currently, he is a
Professor in the Department of Electrical Engineering, Indian Institute of Technology
Roorkee, Roorkee (India).He has more than 32 years of experience of teaching as well as
industry. He has published 176 papers in various Journals/conferences and received eight
awards on his publications in various National/International Journals/Conferences Proceeding papers. He has
guided Seven PhD’s, and presently six PhD’s are under progress. His research interests include Control,
Optimization, System Engineering and Model Order Reduction of Large Scale Systems and industrial robotics.

								
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