neural robot

8/18/2019 neural robot

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Implementation of Artificial Neural

Network applied for the solution of inverse

kinematics of 2-link serial chainmanipulator.

Satish Kumar 1*, Kashif Irshad

2

1*Department of Mechanical Engineering IILM, Greater Noida INDIA2 Department of Mechanical Engineering, Aligarh Muslim University, INDIA

*Corresponding Author: e-mail: [email protected] Tel +91-8287823829,9555597343

Abstract

In this study, a method of artificial neural network applied for the solution of inverse kinematics of 2-linkserial chain manipulator. The method is multilayer perceptrons neural network has applied. This unsupervised

method learns the functional relationship between input (Cartesian) space and output (joint) space based on alocalized adaptation of the mapping, by using the manipulator itself under joint control and adapting the solution

based on a comparison between the resulting locations of the manipulator's end effectors in Cartesian space withthe desired location. Even when a manipulator is not available; the approach is still valid if the forwardkinematic equations are used as a model of the manipulator. The forward kinematic equations always have aunique solution, and the resulting Neural net can be used as a starting point for further refinement when themanipulator does become available. Artificial neural network especially MLP are used to learn the forward andthe inverse kinematic equations of two degrees freedom robot arm.

A set of some data sets were first generated as per the formula equation for this the input parameter X and Ycoordinates in inches. Using these data sets was basis for the training and evaluation or testing the MLP model.Out of the sets data points, maximum were used as training data and some were used for testing for MLP. Back-

propagation algorithm was used for training the network and for updating the desired weights. In this workepoch based training method was applied.

Keywords: ANN, MLP, Robot Manipulator, Inverse Kinematics

1. INTRODUCTION

The term robot has been applied to a wide variety of mechanical devices, from children's toys to guidedmissiles. An important class of robots is the manipulator arms, such as the PUMA robot. These manipulators are

used primarily in materials handling, welding, assembly, spray painting, grinding, deburring etc.The robot manipulator is created from a sequence of link and joint combinations. The links are the rigid

members connecting the joints, or axes. The axes are the movable components of the robot that cause relativemotion between adjoining links. The mechanical joints used to construct the manipulator consist of five

principal types. Two of the joints are linear, in which the relative motion between adjacent links is non-rotational, and three are rotary types, in which the relative motion involves rotation between links.A revolute joint rotates about a motion axis and a prismatic joint slide along a motion axis [Rao D. H. andGupta M. M; 1994]. Each robot joint location is usually defined relative to neighboring joint. The relation

between successive joints is described by 4X4 homogeneous transformation matrices that have orientation and position data of robots. The number of those transformation matrices determines the degrees of freedom ofrobots. The product of these transformation matrices produces final orientation and position data of an n degreesof freedom robot manipulator. Robot control actions are executed in the joint coordinates while robot motionsare specified in the Cartesian coordinates. Conversion of the position and orientation of a robot manipulator end-effectors from Cartesian space to joint space, called as inverse kinematics problem, which is of fundamental

importance in calculating desired joint angles for robot manipulator design and control. In most roboticapplications the desired positions and orientations of the end effectors are specified by the user in Cartesiancoordinates. The corresponding joint values must be computed at high speed by the inverse kinematics

transformation.

Satish Kumar et al. / International Journal of Engineering Science and Technology (IJEST)

ISSN : 0975-5462 Vol. 4 No.09 September 2012 4012


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The different techniques used for solving inverse kinematics can be classified as algebraic [Alavandar S. and Nigam M. J. 2008], geometric [Morris A. S. And A. Mansor; 1997] and iterative [Ahmad Z. and Guez; 1990].[Karliket al. 1999]developed an improved approach to the solution of inverse kinematics problems for robotmanipulators. A structured artificial neural-network (ANN) approach has been proposed here to control the

motion of a robot manipulator. Work has been undertaken to find the best ANN configurations for this problem.Both the placement and orientation angles of a robot manipulator are used to fin the inverse kinematics

solutions.[Xia and Wang; 2001 ] developed a Dual Neural Network for Kinematic control of Redundant RobotManipulators the inverse kinematics problem in robotics can be formulated as a time-varying quadraticoptimization problem. The proposed dual network is also shown to be capable of asymptotic tracking for the

motion control of kinematic ally redundant manipulator.[Patino et al. 2002] demonstrate neural networks for advanced control of robot manipulators. Presents an

approach and a systematic design methodology to adaptive motion control based on neural networks (NNs) forhigh-performance robot manipulators, for which stability conditions and performance evaluation are given.Simulation results showing the practical feasibility and performance of the proposed approach to robotics aregiven.[Mayorga and Pronnapa Sanongboon;2002]developed Inverse kinematics and geometrically boundedsingularities prevention of redundant manipulators an Artificial Neural Network approach. This article presents

an Artificial Neural Network (ANN) approach for fast inverse kinematics computation and effective

geometrically bounded singularities prevention of redundant manipulators.[Manocha and Canny; 2007 ] presented an algorithm and implementation for efficient inverse kinematics for ageneral 6R manipulator. When stated mathematically, the problem reduces to solving a system of multivariateequations. They make use of the algebraic properties of the system and the symbolic formulation used forreducing the problem to solving univariate polynomial. However, the polynomial is expressed as a matrix

determinant and its roots are computed by reducing to an eigen value problem.[Alavandar and Nigam; 2008] developed Neuro-Fuzzy based approach for Inverse Kinematics Solution ofIndustrial Robot Manipulators. In this paper, using the ability of ANFIS (Adaptive Neuro-Fuzzy inferenceSystem) to learn from training data, it is possible to create ANFIS, an implementation of a representative fuzzyinference system using a BP neural network-like structure, with limited mathematical representation of thesystem. Computer simulations conducted on 2 DOF and 3DOF robot manipulator shows the effectiveness of the

approach.[H. Sadjadian and H.D. Taghirad; 2008] developed Neural Networks Approaches for Computing the Forward

Kinematics of a Redundant Parallel Manipulator. In this paper, different approaches to solve the forwardkinematics of a three DOF actuator redundant hydraulic parallel manipulator are presented. It is concluded thatANFIS presents the best performance compared to MLP, RBF and PNN networks in this particular application.[Gallaf ; 2008 ]developed Neural Networks for Multi-Finger Robot Hand Control. This paper investigates theemployment of Artificial Neural Networks (ANN) for a multi-finger robot hand manipulation in which theobject motion is defined in task-space with respect to six Cartesian based coordinates. The paper demonstrates

the proposed algorithm for a four fingered robot hand, where inverse hand Jacobian plays an important role inrobot hand dynamic control.We used MLP (multiple layer perceptrons) method and comparison with MIMO system which uses a Widrow-Hoff type error correction rule. This unsupervised method learns the functional relationship between input(Cartesian) space and output (joint)space based on a localized adaptation of the mapping, by using themanipulator itself under joint control and adapting the solution based on a comparison between the resulting

locations of the manipulator's end effectors in Cartesian space with the desired location. Even when a

manipulator is not available; the approach is still valid if the forward kinematic equations are used as a model ofthe manipulator. The forward kinematic equations always have a unique solution, and the resulting Neural netcan be used as a starting point for further refinement when the manipulator does become available. Artificialneural network especially MLP are used to learn the forward and the inverse kinematic equations of two degreesfreedom robot arm. The technique is independent of arm configuration, including the number of degrees of

freedom and the link geometry.[ Jenhwa Guo and Vladimir Cherkassky; 1999].

1.1 BackgroundIn this report, a method of artificial neural network applied for the solution of inverse kinematics of 2-link serialchain manipulator. The method is multilayer perceptrons neural network has applied. The main objective of is to predict the values of joint angles (inverse kinematics), as we know that there is no unique solution for theinverse kinematics even mathematical formulae are complex and time taking so it is better to find out solutionthrough neural network.




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1.2 ObjectiveThe main objective of the report is to find out the solution for inverse kinematics of manipulator with the help ofneural network method. Validation of the NN methods ensures future selection of the correct method of NN.From the literature it is well described that there is no unique solution for inverse kinematics. This is why it is

significant to apply artificial neural networks models. Here work has been undertaken to find the best ANNconfiguration for the problem.

1.3 MethodologyOn the basis of Literature Survey we have proposed one method for the solution of inverse kinematics ofmanipulator, the proposed methodis multilayer perceptrons to validate the performance of MLP for inverse

kinematics problem, simulation studies will be carried out by using MATLAB [Youshen Xia and Jun Wang2001]. Many researchers have followed MLP, PPN, RBF and FLANN with MISO (multi input single output)

system. Here in this research we have applied MLP with MIMO (multi input multi output) system. A set ofsome data sets were first generated as per the formula equation for this the input parameter X and Y coordinatesin inches. Using these data sets was basis for the training and evaluation or testing the MLP model. Out of thesets data points, maximum were used as training data and some were used for testing for MLP. Back- propagation algorithm was used for training the network and for updating the desired weights. In this workepoch based training method was applied.

1.4 Scope of the Present Work In this study the MLP has been proposed for the solution of inverse kinematics problem of robot manipulator.However, it has some limitations. There are several types of soft computing methods are available which can beused for finding the solution, but this is beyond the scope of this thesis but this technique can be used for the

future scope of the thesis. These methods are followed:Application of fuzzy inference system (FIS)Functional link artificial neural network (FLANN)Evaluation computation

2. RESULT AND DISCUSSION

To analyse the Manipulator position in joint space and also to validate the performance of MLP for inversekinematics problem, simulation studies are carried out by using MATLAB.

2.1 Data Generation:

Let us take the 2-dimensional input space with a two-joint robotic arm and given the desired co-ordinate, the problem reduces to finding the two angles involved. Let θ1 be the angle between the first arm and the base. Let

θ2 be the angle between the second arm and the first arm.Let the length of the first arm be L1= 12 and that of the second arm be L2= 8.Let us assume that the first jointhas limited freedom to rotate and it can rotate between 0 and 180 degrees. Similarly assume that the second jointhas limited freedom to rotate and can rotate between 0 and 180 degrees.

Hence, 0< =θ1=< pi and 0< =θ2=< pi. Now for every combination of θ1and θ2 values the X and Y co-ordinates are deduced using Forward kinematics

formulae.

X = l1cos θ1+ l2cos(θ1+θ2) (2.1)

Y = l1sin θ1+ l2 sin(θ1+θ2) (2.2)

With the help of MATLAB programming, the data is generated for all combination of θ1 and θ2 values andsaved into a matrix to be used as training data. Plotting of points shows all the X-Y data points generated bycycling through different combinations of θ1 and θ2 and deducing x and y co-ordinates for each.

2.2 Calculation of Desired Values of (Θ1& Θ2)

The θ1 and θ2 values are deduced mathematically from the x and y coordinates using inverse kinematicsformulae given in 4.10.The MATLAB programming is used to calculate mathematically the desired values ofθ1& θ2.Let THETA1D and THETA2D are the variables that hold the values of θ1 and θ2 deduced using the

inverse kinematics formulae.




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0 20 40 60 80 100 120 140 160

0.65

0.7

0.75

0.8

0.85

NUMBER OF SAMPLES (INPUTS)

V A L U E S O F D E S

I R E D V A L U E S O F T H E T A 1

0 20 40 60 80 100 120 140 160

2.32

2.34

2.36

2.38

2.4

2.42

2.44


V A L U E S

O F

D E S I R E D V A L U E

O F

T H E T A

2

Fig-1 Graph shows the desired values of θ1

Fig-2 Graph shows the desired values of θ2

Above Fig 1 and 2 shows the line graph representing all possible values of θ1& θ2 for 160 sample input data points. In the graph it shows that the values are in alternate order it is due to the fact that the each sample ischosen from its higher point to lower point but it is not necessary all time.

2.3 Calculation and Testing to Predict the Values Through Ann

MATLAB programming is used to calculate the predicted values let it is THETA1P & THETA2P. The newffcommand is used to create the back propagation neural network. In this report the 50 data points are used tocreate and train the back propagation multilayer neural network in which there are 10 hidden layers. After thetraining of network the simulation is done by using ‘sim’ command. There are total 160 sample inputs data points are simulate through the network in 8 epochs separately for θ1 and θ2 values. ‘trainsig’ command is usedfor the sigmoidal transfer function. Other commands are used in default condition as described earlier in

previous chapter. 160 sample points are first used to evaluate the values of θ1. After this the same data pointsare used to evaluate the values of θ2.Performance of the network and the predicted values of θ1 and θ2by neural network are shown in below Fig- 3and 4 and Table 1.

Fig-3Graph shows the Predicted values of θ1




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0 20 40 60 80 100 120 140 1602.37

2.38

2.39

2.4

2.41

2.42

2.43

2.44


P R E D I C T E D

V A L U E

O F

T H E T A

2

Fig-4 Graph shows the Predicted values of θ2

Table 1. Comparison of Desired values and ANN predicted values

S.NO.DESIRED VALUE OF

θ1ANN PREDICTED VALUE

OF θ1DESIRED VALUE OF

θ2ANN PREDICTED VALUE

OF θ2

1 0.846585675 0.808673263 2.406270352 2.430646548

2 0.834232113 0.824876339 2.406192714 2.431057806

3 0.82186534 0.839148244 2.405959843 2.430548103

4 0.809489252 0.852528773 2.405571856 2.42525326

5 0.797107829 0.875174309 2.405028954 2.4163884

6 0.78472512 0.92649984 2.404331414 2.4124183

7 0.772345243 0.872208086 2.403479592 2.40898237

8 0.759972367 0.752438434 2.402473921 2.402945497

9 0.747610713 0.714238745 2.401314908 2.397962404

10 0.735264538 0.683660666 2.400003135 2.392771978

11 0.722938129 0.662703074 2.398539255 2.387627089

12 0.710635792 0.64038021 2.396923992 2.384057849

13 0.698361847 0.620972799 2.395158135 2.382364467

14 0.686120613 0.60984029 2.393242541 2.381823383

15 0.673916402 0.604698645 2.391178129 2.381736335

16 0.661753511 0.602885724 2.388965876 2.381748829

17 0.649636211 0.603291253 2.386606818 2.38174338

18 0.637568739 0.606167458 2.384102045 2.381745454

19 0.625555288 0.613220417 2.381452695 2.381946989

20 0.613600004 0.62747883 2.378659958 2.382831026

21 0.601706969 0.649233113 2.375725067 2.385187253

22 0.845300805 0.670286591 2.393701923 2.389486937

23 0.833099024 0.694324039 2.393625344 2.394838325

24 0.820886388 0.72686979 2.393395643 2.399796051

25 0.808666649 0.783833121 2.393012936 2.40530937

26 0.796443632 0.918734542 2.39247741 2.410524339

27 0.784221231 0.906043681 2.39178933 2.413707396

28 0.7720034 0.863707772 2.390949032 2.419153348




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29 0.759794145 0.847255294 2.389956928 2.428302272

30 0.747597514 0.833882915 2.3888135 2.430959343

31 0.735417591 0.818916526 2.387519299 2.43097375

32 0.723258488 0.801447466 2.386074946 2.430232461

33 0.711124335 0.83990566 2.384481129 2.40487116334 0.699019271 0.834469229 2.382738598 2.403078069

35 0.686947437 0.819641053 2.380848166 2.402466907

36 0.674912965 0.790813032 2.378810708 2.403820667

37 0.662919973 0.772939482 2.376627154 2.40538019

38 0.650972556 0.753103891 2.374298489 2.40483224

39 0.639074774 0.7330505 2.37182575 2.402894812

40 0.627230649 0.733614207 2.369210023 2.400890525

41 0.615444155 0.731228941 2.366452439 2.39761761

42 0.60371921 0.715977128 2.363554174 2.398713533

43 0.844199806 0.700457108 2.381150383 2.400730634

44 0.832146106 0.694027895 2.38107482 2.395232807

45 0.820083764 0.683390025 2.380848166 2.39075224

46 0.808016393 0.670247657 2.380470531 2.389361315

47 0.795947675 0.6605278 2.379942092 2.388902954

48 0.783881355 0.653820838 2.379263101 2.388107008

49 0.771821235 0.647249387 2.378433878 2.385951898

50 0.759771161 0.636769224 2.377454813 2.381703139

51 0.747735022 0.620501494 2.376326364 2.377732515

52 0.735716739 0.618423719 2.375049056 2.380112889

53 0.723720258 0.685739282 2.373623478 2.388047646

54 0.711749541 0.782232059 2.372050283 2.393095553

55 0.699808558 0.791334447 2.370330187 2.394616645

56 0.687901281 0.795880957 2.368463962 2.394339203

57 0.676031675 0.858491118 2.366452439 2.39278124

58 0.664203688 0.886942685 2.364296505 2.389662718

59 0.652421247 0.813601487 2.361997097 2.38742547

60 0.640688249 0.761657084 2.359555203 2.38962238

61 0.629008551 0.742765694 2.356971859 2.39446378

62 0.617385966 0.732380469 2.354248142 2.395931563

63 0.605824257 0.724885847 2.351385174 2.395902272

64 0.843275501 0.71925425 2.368613128 2.395743476

65 0.831366315 0.838434365 2.368538542 2.40420198

66 0.819450568 0.830733583 2.368314818 2.402957465

67 0.807531741 0.810216794 2.367942058 2.403048091

68 0.79561338 0.78233209 2.367420433 2.40494311

69 0.783699089 0.771954346 2.366750181 2.405698926

70 0.771792523 0.748450873 2.365931607 2.404484898

71 0.759897382 0.736032642 2.364965082 2.402468924

72 0.748017402 0.735578339 2.363851039 2.400018424




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73 0.73615635 0.728707101 2.36258998 2.397278411

74 0.724318013 0.70999317 2.361182464 2.400249966

75 0.712506197 0.698760749 2.359629112 2.399183707

76 0.700724713 0.691932261 2.357930605 2.393165896

77 0.688977373 0.679907548 2.356087679 2.39006473578 0.677267982 0.66761673 2.354101125 2.389191378

79 0.665600331 0.658988352 2.351971786 2.388687151

80 0.653978187 0.652548663 2.349700558 2.387503051

81 0.642405291 0.645253972 2.34728838 2.384661265

82 0.630885348 0.633293996 2.344736239 2.38025341

83 0.619422019 0.620708763 2.342045165 2.378476115

84 0.608018917 0.644073828 2.339216228 2.383889137

85 0.842521156 0.740410627 2.356087679 2.390947962

86 0.830753046 0.795712056 2.356014032 2.393979036

87 0.818980332 0.787803502 2.355793126 2.394519208

88 0.807206372 0.819491649 2.355425056 2.39367291

89 0.795434581 0.884801548 2.354909986 2.391416546

90 0.78366843 0.860514743 2.354248142 2.38836534

91 0.771911436 0.782474624 2.353439813 2.387763719

92 0.760167158 0.750855586 2.352485352 2.39208585

93 0.748439187 0.736242685 2.351385174 2.395442185

94 0.736731145 0.728017398 2.350139752 2.395900997

95 0.725046672 0.721379842 2.348749621 2.395714875

96 0.713389423 0.718563393 2.347215373 2.395801623

97 0.701763059 0.836474119 2.345537653 2.4038922

98 0.690171242 0.825501656 2.343717164 2.403159302

99 0.678617626 0.799973859 2.34175466 2.404139689

100 0.667105851 0.778368701 2.339650944 2.405862626

101 0.655639537 0.772248403 2.337406869 2.4058515

102 0.644222276 0.74892074 2.335023332 2.404188357

103 0.632857627 0.739633298 2.332501274 2.402081684

104 0.621549109 0.73677302 2.329841677 2.399075227

105 0.610300194 0.724596691 2.327045563 2.398022056

106 0.841930446 0.705837461 2.343571665 2.400856859

107 0.830300095 0.697806442 2.343498923 2.397002702

108 0.818666986 0.689812713 2.343280728 2.391730131

109 0.807034355 0.677365103 2.342917173 2.389677247

110 0.795405495 0.666122379 2.342408412 2.389037481

111 0.783783749 0.658229472 2.34175466 2.388361646

112 0.772172505 0.651840504 2.340956194 2.386719748

113 0.760575186 0.643890177 2.340013348 2.38338621

114 0.74899525 0.63251202 2.338926518 2.379775732

115 0.737436178 0.632535527 2.337696154 2.381046103

116 0.725901471 0.692063272 2.336322766 2.387874755




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117 0.714394643 0.782941833 2.334806917 2.392811849

118 0.702919212 0.793268667 2.333149222 2.394296663

119 0.691478698 0.795868827 2.331350352 2.394038103

120 0.680076612 0.854130536 2.329411023 2.392621591

121 0.668716453 0.888555778 2.327332004 2.389831732122 0.657401699 0.820963608 2.325114105 2.387631254

123 0.646135804 0.762233195 2.322758184 2.389376747

124 0.634922188 0.741869671 2.320265139 2.394177413

125 0.623764235 0.730654127 2.317635908 2.395746007

126 0.612665285 0.723824005 2.314871466 2.395698801

127 0.841497428 0.719028263 2.331062818 2.395591852

128 0.830001638 0.720118759 2.330990947 2.39614558

129 0.818504827 0.833782777 2.330775363 2.403875571

130 0.807010119 0.818491256 2.330416155 2.403800475

131 0.79552069 0.790744711 2.32991347 2.4054508

132 0.784039762 0.779769598 2.329267511 2.406514295

133 0.772570597 0.77527519 2.328478543 2.405921686

134 0.761116495 0.752178915 2.327546885 2.403984566

135 0.749680781 0.742928805 2.326472913 2.401636591

136 0.738266808 0.736801801 2.325257056 2.398437983

137 0.726877942 0.719764153 2.323899799 2.399309796

138 0.715517564 0.703448737 2.32240168 2.400237897

139 0.704189059 0.696997313 2.320763285 2.394897572

140 0.69289581 0.688064387 2.318985253 2.390813412

141 0.681641194 0.675949958 2.317068267 2.389429037

142 0.670428575 0.665700296 2.315013061 2.388838723

143 0.6592613 0.658341515 2.312820408 2.387920556

144 0.648142688 0.652071807 2.310491127 2.385877945

145 0.637076031 0.644452486 2.308026075 2.382577985

146 0.626064583 0.638722015 2.305426148 2.380877047

147 0.61511156 0.664305098 2.302692278 2.384933498

148 0.841216512 0.751046763 2.318558961 2.391057679

149 0.829852193 0.799021864 2.318487929 2.393773131

150 0.818488493 0.790608423 2.318274861 2.394105638

151 0.807128428 0.821521851 2.317919841 2.393196584

152 0.795775062 0.883675835 2.317423008 2.391130202

153 0.784431501 0.864446336 2.316784559 2.388450194

154 0.773100891 0.784884894 2.316004743 2.387853039

155 0.76178641 0.749469016 2.315083865 2.391907471

156 0.750491262 0.734209468 2.314022284 2.395271758

157 0.739218674 0.725865541 2.312820408 2.395645159

158 0.727971888 0.720429304 2.311478701 2.395468317

159 0.716754157 0.718776365 2.309997674 2.395694916

160 0.705568737 0.724711187 2.308377887 2.396934771




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0 20 40 60 80 100 120 140 160

0.65

0.7

0.75

0.8

0.85

0.9

0.95

NUMBER OF SAMP LE (INPUTS)

V A L U E

O

F

T H E T A

1

THETA1 DESIRED

THETA1 ANN PREDICTED

0 20 40 60 80 100 120 140 160

2.3

2.35

2.4

.


V A L U E

O

F

T H E T A

2

THETA 2 DESIRED

THETA 2 ANN PREDICTED

Fig 5 Graph shows the comparison of desired and predicted values of θ1

Fig-6 Graph Shows a comparison of Desired and Predicted values for θ2

The above Fig 5 and 6 represents the comparison of desired values and the ANN predicted values of 160 input

data points for both θ1 and θ2. From the graph it is clear that the Network results obtained are approximatelyequal to the desired values and also within the acceptable range.

2.4 Calculation for Mean Square Error (MSE)The calculated values and predicted values are evaluated to find the mean square error with the MATLAB programming which shows the following results. Mean square error is observed by the difference of desiredvalues and predicted values. Findings of the errors are in the 1e-3 range which is a fairly good number for the

application it is being used in.After the evaluation of 160 input data points for the prediction of θ1 values the result occurred by the network is

in the form of MSE. The above fig 6.8 to 6.11 shows the mean square error, best fit values and validation checksfor different stages like training, testing and validation for θ1 values. Mean square error is in the range of -0.22

to 0.18 x 10-2 which is under considerable range of 0.01. the best fit result is obtained at the epoch 2 and thevalue is 0.0060. From the graph of regression it is clear that the most of the data input points give the betteroutput result and approximately equal to the target value. Neural network calculate the gradient of the slope andalso the validity at each epoch the best gradient value is obtained on the epoch 8 and is equal to 4.695e-0.10. and

the total no. of validation check is 6.So the above results obtained for θ1 values are satisfy the desired values.




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Fig- 7 Performance Graph representing the different results for θ1

Fig- 8 Regression Graph θ1 representing the best fit data for different stages.

Fig- 9 Graph representing the gradient, validation check at different epochs for θ1

After the evaluation of 160 input data points for the prediction of θ1 values the result occurred by the network isin the form of MSE. The above Fig 7 to 9 shows the mean square error, best fit values and validation checks fordifferent stages like training, testing and validation for θ1 values. Mean square error is in the range of -0.22 to0.18 x 10-2 which is under considerable range of 0.01. the best fit result is obtained at the epoch 2 and the value

is 0.0060. From the graph of regression it is clear that the most of the data input points give the better outputresult and approximately equal to the target value. Neural network calculate the gradient of the slope and alsothe validity at each epoch the best gradient value is obtained on the epoch 8 and is equal to 4.695e-0.10. and thetotal no. of validation check is 6.So the above results obtained for θ1 values are satisfy the desired values.

Similarly the evaluation of 160 input data points for the prediction ofθ2 values the result occurred by thenetwork is in the form of MSE.




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0 20 40 60 80 100 120 140 160-0.08

-0.06

-0.04

-0.02

0

0.02

0.04

D E S I R E D -

P R E D I C T E D

MEAN SQURE ERROR

NUMBER OF INPUTS FOR THETA 2

Fig- 10. Graph representing the Mean square error (MSE) for θ2

Fig- 11 Performance Graph representing the different results for θ2

Fig- 12 Regression Graph for θ2 representing the best fit data for different stages.




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Fig- 13 Graph representing the gradient, validation check at different epochs for θ2

The above Fig 10 to 13 shows the mean square error, best fit values and validation checks for different stageslike training, testing and validation for θ2values. Mean square error is in the range of -0.07 to 0.02 x 10-2 which

is under considerable range of 0.01. the best fit result is obtained at the epoch 2 and the value is 1.2e-.005. Fromthe graph of regression it is clear that the most of the data input points give the better output result andapproximately equal to the target value. Neural network calculate the gradient of the slope and also the validityat each epoch the best gradient value is obtained on the epoch 8 and is equal to 1.224e-0.005. and the total no. of

validation check is 6. So the above results obtained for θ2values are satisfy the desired values.

3. CONCLUSION AND SCOPE FOR FUTURE WORK

Mathematical models relay on assuming the structure of the model in advanced, which may be sub-optimal.Consequently, many mathematical models fail to simulate the complex behavior of inverse kinematics problem.In contrast, ANN (artificial neural networks) is based on the data input/output data pairs to determine thestructure and parameters of the model. Moreover, ANN’s can always be updated to obtain better results by presenting new training examples as new data become available. This artificial neural network based joint

angles prediction model can be useful tool for the production engineer’s to estimate the motion of themanipulator accurately.

3.1 Future ScopeIn this study the MLP has been proposed for the solution of inverse kinematics problem of robot manipulator.However, it has some limitations. There are several types of soft computing methods are available which can beused for finding the solution, but this is beyond the scope of this thesis but this technique can be used for the

future scope of the thesis. These methods are followed:Application of fuzzy inference system (FIS)

Functional link artificial neural network (FLANN)Evaluation computation

NomenclatureANN Artificial Neural Network

MLP Multiple Layer PerceptronANFIS Adaptive Neuro-Fuzzy inference System

AcknowledgementI wish to express my profound gratitude, respect and honour to my venerable supervisor Dr K.P. Roy, Professor,Department of Mechanical Engineering, IILM, Greater Noida, INDIA for his illuminative and preciousguidance, constant supervision, critical opinion and timely suggestion, constant useful encouragement and

technical tips which has always been a source of inspiration during the preparation of the project.I would also like to thank my all classmates and friends for their good and cordial company, healthy discussion

and helpful attitude during the study.I will be failing in my duties if I miss to express my profound and deepest sense of gratitude to my father Mr.Chandrika Prasad, and other members of family for their keen interest in my studies, manifold assistance,immense support and encouragement, without which it was impossible to complete this dissertation.

Finally before concluding, I remember once again the “GOD” who gave me power, energy, & enthusiasm toaccomplish this work.




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Biographical notes

Satish Kumar received B. Tech. from Shivaji University, India in 2008 and M.Tech from IIT Delhi, India in2012, respectively. He is the Head of Department (HOD) in Mechanical Engineering, IILM, Greater NoidaIndia. His area of research is Robotics, Mechatronics and Non-conventional resource of energy.

Kashif Irshad received B. Tech. and M.Tech from Aligarh Muslim University, India in 2008 and 2011,respectively. He is a Assistant Professor in the Department of Mechanical Engineering, IILM, Greater Noida

India. His area of research was Ergonomics, Robotics and Mechatronics.

Received xx 20xxAccepted xx 20xx

Final acceptance in revised form xx 20xx



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