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8th Vienna International Conference on Mathematical Modelling 8th Conference on Modelling 8th Vienna Vienna18International International Conference on Mathematical Mathematical Modelling February - 20, 2015. Vienna University of Technology, Vienna, 8th Vienna International Conference on Mathematical Modelling February 18 -- 20, 2015. Vienna University of Technology, Available online at Vienna, www.sciencedirect.com February 18 20, 2015. Vienna University of Technology, Vienna, Austria February 18 - 20, 2015. Vienna University of Technology, Vienna, Austria Austria Austria

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IFAC-PapersOnLine 48-1 (2015) 655–656

Modeling of Elastic Robotic Arm using Modeling Modeling of of Elastic Elastic Robotic Robotic Arm Arm using using Soft-Computing Algorithm Soft-Computing Algorithm Algorithm Soft-Computing

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Hammam Tamimi ∗∗∗ Dirk S¨ offker ∗∗∗ Hammam o Hammam Tamimi Tamimi ∗ Dirk Dirk S¨ S¨ offker ffker ∗ Hammam Tamimi Dirk S¨ o ffker ∗ Chair of Dynamics and Control, University of Duisburg-Essen, ∗ ∗ of Dynamics and Control, University of Duisburg-Essen, ∗ Chair Chair of Control, University Chair of Dynamics Dynamics and Control, University of of Duisburg-Essen, Duisburg-Essen, Duisburg, Germanyand (e-mail: [email protected]) Duisburg, Germany (e-mail: [email protected]) Duisburg, Germany (e-mail: [email protected]) Duisburg, Germany (e-mail: [email protected]) Abstract: This paper proposes the use of the least square support vector machine (LS-SVM) Abstract: This paper proposes the use of the least square support vector machine (LS-SVM) Abstract: paper the of least support vector (LS-SVM) Abstract:toThis This paper proposes the use use arm. of the theDynamic least square square support vectorismachine machine (LS-SVM) algorithm model anproposes elastic robotic system modeling important as the algorithm to model an elastic robotic arm. Dynamic system modeling is important as algorithm to obtaining model an ana elastic elastic robotic arm. for Dynamic systemAcquiring modelinganis is accurate important as the the algorithm to model robotic arm. Dynamic system modeling important as the first step in suitable controller any system. model of first step in obtaining a suitable controller for any system. Acquiring an accurate model of first step step in obtaining obtaining ainput-output suitable controller controller for any any using system. Acquiring an accurate accurate modelless of first in suitable for system. an model of elastic robotic based ona measurements theAcquiring LS-SVM algorithm requires elastic robotic based on input-output measurements using the LS-SVM algorithm requires less elastic robotic based on input-output measurements using the LS-SVM algorithm requires less elastic robotic based on input-output measurements thealgorithm LS-SVM algorithm requires less knowledge about the physical-laws of the system. The using LS-SVM achieves global, unique knowledge about the physical-laws of the system. The LS-SVM algorithm achieves global, unique knowledge about the the system. LS-SVM algorithm achieves global, knowledgeand about the physical-laws physical-laws of thecompared system. The The LS-SVM algorithm achieves global, unique unique solution, requires less training of time with other soft computing algorithms. In this solution, and requires less training time compared with other soft computing algorithms. In solution, and requires requires less training timealgorithm comparedtowith with other soft computing algorithms. In this this solution, and training time compared other computing algorithms. In this paper, a successful useless of the LS-SVM model thesoft elastic robotic arm as multi-input paper, a successful use of the LS-SVM algorithm to model the elastic robotic arm as multi-input paper, a a successful successful useisof ofdemonstrated. the LS-SVM LS-SVM algorithm algorithm to model model the elastic elastic robotic arm as as multi-input multi-input paper, use the to the robotic arm multi-output system The simulation results illustrate the efficiency and high multi-output system is demonstrated. The simulation results illustrate the efficiency and high multi-output system is multi-output of system is demonstrated. demonstrated. The simulation simulation results results illustrate illustrate the the efficiency efficiency and and high high performance the proposed approach.The performance of the proposed approach. performance performance of of the the proposed proposed approach. approach. © 2015, IFAC (International Federation of Automatic Control) Hosting by Elsevier Ltd. All rights reserved. Keywords: System identification, multi-input multi-output system, soft-computing algorithm, Keywords: System identification, multi-input multi-output system, soft-computing algorithm, Keywords: System identification, multi-input multi-output Keywords: System identification, multi-input multi-output system, system, soft-computing soft-computing algorithm, algorithm, support vector machine, multi-step ahead perdition. support vector machine, multi-step ahead perdition. support support vector vector machine, machine, multi-step multi-step ahead ahead perdition. perdition. 1. INTRODUCTION u ∈ Rm and output vector y ∈ Rrrr is done by estimating m m 1. INTRODUCTION u ∈ R and output vector yy ∈ Rr is done by estimating m 1. INTRODUCTION u ∈ R 1. INTRODUCTION u R and and foutput output vector ∈R R is is done done by by estimating estimating the∈ function () givenvector as y ∈ the function f () given as the function function ff() () given given as as This paper focuses on the modeling of elastic robotic arm. the This focuses on the elastic robotic arm.  This paper paper focuses on manipulators the modeling modeling of of elastic robotic since arm. This paper focuses on the modeling of robotic arm. Building light weight is elastic advantageous yˆ(k − 1), yˆ(k − 2), ..., yˆ(k − p), yˆ(k) = f  Building light weight manipulators is advantageous since yyˆˆ(k) = ff yyˆˆ(k − 1), yˆ(k − 2), ..., yˆ(k − p), Buildingactuators light weight weight manipulators is advantageous advantageous since (k) = Building light manipulators is smaller can be used for driving the joints ofsince the  y ˆ (k) = f yˆ(k (k − − 1), 1), yyˆˆ(k (k − − 2), 2), ..., ..., yyˆˆ(k (k − − p), p), smaller actuators can be used for driving the joints of the  smaller actuators can be used for driving the joints of the , (1) smaller actuators can be usedthis fortranslates driving thetojoints of the manipulator. Consequently, less energy  u(k), u(k − 1), ..., u(k − p) manipulator. Consequently, this translates to less energy u(k), u(k − 1), ..., u(k − p) ,, (1) manipulator. Consequently, this translates to less energy u(k), u(k − 1), ..., u(k − p) manipulator. translates less energy consumptions.Consequently, Additionally, this damage to thetomanipulator u(k), u(k − 1), ..., u(k − p) , (1) (1) consumptions. Additionally, damage to the manipulator consumptions. Additionally, damage to the manipulator consumptions. Additionally, damage to the manipulator system due to accidental collisions can be avoided by where yˆ is the estimated output, k is the time step, p the system due collisions can be by yˆ is estimated output, system flexibility. due to to accidental accidental collisions canmanipulators be avoided avoided exby where where is the the estimated output, k k is is the the time time step, step, p p the the system due to accidental can be avoided by adding Further, collisions light-weight where is the estimated k is the time step, p the numberyyˆˆ of previous steps.output, adding flexibility. Further, light-weight manipulators exnumber of previous steps. adding flexibility. Further, light-weight manipulators exnumber of of previous previous steps. steps. adding flexibility. Further, light-weight manipulators ex- number hibit higher speed manipulators compared to conventional hibit higher to Suykens and Vandewalle (1999), the dynamic hibit manipulators. higher speed speed manipulators manipulators compared compared to to conventional conventional According hibit higher speed manipulators compared to conventional rigid According to Suykens Vandewalle (1999), the dynamic According to unknown Suykens and and Vandewalle (1999), the dynamic rigid manipulators. According to Suykens and Vandewalle (1999), dynamic model of the system estimated usingthe LS-SVM is rigid manipulators. manipulators. rigid model of the unknown system estimated using LS-SVM modelas of the the unknown unknown system system estimated estimated using using LS-SVM LS-SVM is is Due to the flexibility of the elastic robotic arm varies model of is given Due flexibility of the the elastic elastic robotic robotic arm arm varies varies given as Due to to the the flexibility of as Due to the of the elastic model roboticis arm varies given problem risesflexibility such as the dynamic considered given as problem rises as model is M problemnonlinear, rises such suchstructural as the the dynamic dynamic model istheconsidered considered  problem rises such as the dynamic model considered M highly vibrations, and is accurate  M highly nonlinear, structural vibrations, and the accurate M αj K(q(k), qj ) + b,  y ˆ (k) = (2)  highly nonlinear, nonlinear, structural vibrations, and the the accurate accurate highly positioning of the structural end effectorvibrations, is reduced.and y ˆ (k) = α qqjj )) + b, (2) jj K(q(k), y ˆ (k) = α K(q(k), + b, (2) positioning of the end effector is reduced. y ˆ (k) = α K(q(k), q ) + b, (2) j=1 j j positioning of the end effector is reduced. positioning of the end effector is reduced. j=1 j=1 This paper concerns with the use of least squares support j=1  This concerns with of squares support This paper paper concerns with the thetouse use of least least squares support q(k) = yˆ(k − 1), yˆ(k − 2), ..., yˆ(k − p), u(k), u(k − This paper concerns with the use of least vector machine (LS-SVM) model an squares elastic support robotic where where q(k) =  yyˆˆ(k − 1), yyˆˆ(k − 2), ..., yˆ(k − p), u(k), u(k − vector machine (LS-SVM) to model an elastic robotic where q(k) (k ..., p), u(k), − vector machine (LS-SVM) to model an elastic robotic where q(k)−= = (kj − − 1), yˆ(k (k − − 2), 2), ..., yyˆˆ(k (k − −M p), is u(k), u(k − vector machine (LS-SVM) to model elastic arm. The implemented method has twoangoals, therobotic result 1), is 1), Lagrange multiplier, theu(k num..., u(k p),yˆα arm. The implemented method has two goals, the result is Lagrange multiplier, M is the num1), ..., u(k − p) , α j arm. The implemented method has two goals, the result is Lagrange multiplier, M is the num1), ..., u(k − p) , α model should be able to make accurate multi-step ahead 1), j is Lagrange multiplier, M is the num..., u(k − p) , α ber of nonzero Lagrange multipliers, q is called support j j model be able to accurate multi-step ahead ber of nonzero Lagrange multipliers, qqjj is called support model should should be the ablemethod to make makeshould accurate multi-step ahead vector, ber Lagrange is model should be able to make accurate multi-step ahead prediction, and allow the modeling ber of of nonzero nonzero Lagrange multipliers, is called called support kernel function K(,multipliers, ), and b is qaj bias term.support prediction, and the method should allow the modeling vector, kernel function K(, ), and b is a bias prediction, androbotic the method method should allow the the modeling vector, vector, kernel kernel function function K(, K(, ), ), and and bb is is aa bias bias term. term. prediction, and the allow modeling of the elastic arm asshould multi-input multi-output term. of the elastic robotic arm as multi-input multi-output graphical representation of recurrent LS-SVM model of the the elastic elastic robotic robotic arm arm as as multi-input multi-input multi-output multi-output The of system. The graphical representation of recurrent LS-SVM model The graphical representation of model system. The graphical representation of recurrent recurrent LS-SVM model is shown in figure 1. It should be noted thatLS-SVM the multi-step system. system. is shown in figure 1. It should be noted that the multi-step is shown in figure 1. It should be noted that the multi-step is shown in figure 1. It should be noted that the multi-step ahead prediction is achieved by feeding back the output to 2. SYSTEM IDENTIFICATION USING SUPPORT ahead prediction is ahead prediction is achieved achieved by by feeding feeding back back the the output output to to 2. USING ahead prediction is achieved by feeding back the output to the model input vector. 2. SYSTEM SYSTEM IDENTIFICATION IDENTIFICATION USING SUPPORT SUPPORT 2. SYSTEM IDENTIFICATION USING SUPPORT VECTOR MACHINE the model input vector. the model input vector. VECTOR MACHINE the model input vector. VECTOR MACHINE MACHINE VECTOR As outlined by Yan et al. (2003), the SVM has less tuning As by (2003), SVM less As outlined outlined by Yan Yan et ettoal. al. (2003), the the SVM has has(MLP) less tuning tuning outlined by Yan et al. (2003), the SVM has less tuning The Support Vector Machine (SVM) algorithm was pro- As effort in comparison multilayer perceptron netThe Support Vector Machine (SVM) algorithm was proeffort in comparison to multilayer perceptron (MLP) netThe Support Vector Machine (SVM) algorithm was proeffort in comparison to multilayer perceptron (MLP) netThe Vector Machine algorithm was procomparison to multilayer (MLP) netposedSupport in Vapnik et al. (1996)(SVM) . Although the SVM al- effort work. in Also, the computational timeperceptron required by the SVM posed in Vapnik et al. (1996) .. Although the SVM alwork. Also, the computational time required by the SVM posed in Vapnik et al. (1996) Although the SVM alwork. Also, the computational time required by the SVM posed al. (1996) Although the SVM apal- work. Also,isthe time required by the gorithmin isVapnik widelyetused in the .field of classification algorithm lesscomputational than MLP network; due to the factSVM that gorithm widely used in the apis MLP network; due to the that gorithm is isthe widely used in algorithm the field field of ofin classification classification ap- algorithm algorithm is less less than than MLP network; dueto toobtain the fact fact that gorithm is widely used in the field of classification apalgorithm is less than MLP network; due to the fact plication, use of SVM the field system the SVM algorithm uses linear equations thethat replication, the use of SVM algorithm in the field system the SVM algorithm uses linear equations to obtain the plication, the the isuse use offully SVMexplort algorithm inadaptation the field field system system the SVM SVM algorithm uses linear linear equations to to obtain obtain the rereplication, SVM algorithm the uses equations the reidentification notof . Thein of the the sults whilealgorithm MLP network uses backpropagation algorithm. identification is fully .. The of the while MLP network uses backpropagation algorithm. identification is not not fully explort explort The adaptation adaptation ofprothe sults sults while uses identification is not fully explort . The adaptation the sults while MLP network uses backpropagation backpropagation algorithm. LS-SVM for the modeling of dynamic system wasof Yan et al. MLP (2003)network noted that models obtained algorithm. using the LS-SVM for of system was proYan et al. (2003) noted that models obtained using the LS-SVM for the the modeling modeling of dynamic dynamic system was proal. models using the LS-SVM for the modeling of dynamic system Yan et al. (2003) (2003) noted that models obtained obtained usingones the posed in Suykens and Vandewalle (1999), wherewas the proLS- Yan SVMet algorithm hasnoted betterthat generalization ability than posed in Suykens and Vandewalle (1999), where the LSSVM algorithm has better generalization ability than ones posed algorithm in Suykens Suykenswas andsuccessfully Vandewalle implemented (1999), where wheretothe the LS- SVM SVM algorithm algorithm has better better generalization ability than ones posed in and Vandewalle (1999), LShas generalization ability ones SVM model obtained using MLP network. Additionally, for than the SVM SVM algorithm was successfully implemented to using MLP network. Additionally, for SVM SVM algorithm was successfully implemented to model model obtained obtained using MLP network. Additionally, for the the SVM SVM algorithm successfully implemented to model obtained MLP network. Additionally, for the SVM the double scroll was system. The multi-step ahead prediction there is nousing risk of getting stuck into local minima, thus the the double scroll system. The multi-step ahead prediction there is no risk of getting stuck into local minima, thus the the double scroll system. The multi-step ahead prediction there is no risk of getting stuck into local minima, thus the double scroll system. Thesystem multi-step prediction no risk of getting stuck into minima, thus the the model of unknown dynamic with ahead the input vector there SVM is always results in a unique andlocal global solution. model of unknown dynamic system with the input vector SVM always results in a unique and global solution. model of of unknown unknown dynamic dynamic system system with with the the input input vector vector SVM model SVM always always results results in in aa unique unique and and global global solution. solution. 2405-8963 © Copyright © 2015, 2015, IFAC IFAC (International Federation of Automatic Control) 655Hosting by Elsevier Ltd. All rights reserved. Copyright 2015, IFAC 655 Copyright © 2015, IFAC 655 Peer review© of International Federation of Automatic Copyright ©under 2015,responsibility IFAC 655Control. 10.1016/j.ifacol.2015.05.024

MATHMOD 2015 656 February 18 - 20, 2015. Vienna, Austria Hammam Tamimi et al. / IFAC-PapersOnLine 48-1 (2015) 655–656

delays is obtained (p = 53), an accurate LS-SVM model is calculated. In Figure 2 the prediction of the LS-SVM model red and black curves compared to the FEM model curve in blue and green curves while exiting the system with two different pseudo random signals is shown. It can observed that predictions of the LS-SVM model are accurate.

Support vector machine model u(k) K(q(k), q1 ) α1

z −1

α2

K(q(k), q2 )

z −p q(k)

α3

K(q(k), q3 )

yˆ(k)

αn z −1 K(q(k), qn )

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Sim. 1

x x ˆ

x x ˆ

z −2 2

b

1

x1

z −p

0 −1 −2 5

x2

Fig. 1. Recurrent LS-SVM model

0

To ensure accuracy of the dynamic model the following two major practical aspects have to be considered:

10 5 0 −5 −10 20 10

x4

3. PRACTICAL ASPECTS OF USING SOFT-COMPUTING ALGORITHMS FOR MODEL BUILDING

Displacement [mm] x3

−5

0 −10

4. SIMULATION RESULTS In this paper, the elastic robotic arm is considered as cantilever beam. The simulation model is obtained using finite element method (FEM) assuming that the cantilever beam consists of five elements with a control input applied at the tip of the beam. The cantilever beam parameters are given in table 1. The simulation model is excited using two distinctive signals; pseudo random signal and sweep signal. For simplicity, in this paper the displacement signals [x1 to x5 ] are considered. It is important that the excitation signals have wide range of dynamic properties in order to stimulate all the dynamic of the system. In this work the sweep signal is used to train the LS-SVM model and the pseudo random signal is used for the validation of the LS-SVM model. Table 1. Beam parameters Density (ρ) Length(l) Cross section area (A) Modulus of elasticity (E)

2700 kg/m3 1m 3.2 × 10−4 m2 10 × 1010 N/m2

The sum of squared error (SSE) function is considered as a performance measure of the LS-SVM model. The process of selecting p is an iterative process, p is increased until the performance is maximized. Once the number of 656

−20 20 10

x5

• The cross-validation technique is used to overcome the bias-variance dilemma. The most popular method in cross-validation technique is the k-fold crossvalidation. In this work the 2-fold cross-validation, which is the simplest form of k-fold cross-validation, is used. In this method the data are divided into two parts; the first part is used for training, second part is used for validation, then the process is reversed. • For a perfect dynamic model, the input error crosscorrelation should show no correlation and the error autocorrelation should show one nonzero value at zero lag. The dynamic model accuracy can be improved by increasing the number of delay p Beale et al. (2014).

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Fig. 2. LS-SVM model prediction vs. simulation model 5. SUMMARY In this paper, modeling of elastic robotic arm as multipleinput multiple-output system using LS-SVM algorithm is presented. The LS-SVM algorithm has less tuning parameters, is faster to train, also it achieves an unique and global solution. The simulation results show the successful multistep-ahead prediction of LS-SVM model of the elastic robotic arm at different positions. REFERENCES Beale, M.H., Hagan, M.T., and Demuth, H.B. (2014). Neural network toolbox: getting started guide. Mathworks. Suykens, J. and Vandewalle, J. (1999). Least squares support vector machine classifiers. In Neural Processing Letters, volume 9, 293–300. Vapnik, V., Golowich, S., and Smola, A. (1996). Support vector method for function approximation, regression estimation and signal processing. Advances in Neural Information Processing Systems, 9, 281–287. Yan, R., Liu, Y., Jin, R., and Hauptmann, A. (2003). On predicting rare classes with SVM ensembles in scene classification. IEEE International Conference on Acoustics, Speech, and Signal Processing, 3, 21–24.