Kinematic analysis of a novel 3-DOF actuation redundant parallel manipulator using artificial intelligence approach

Robotics and Computer-Integrated Manufacturing 27 (2011) 157–163

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Robotics and Computer-Integrated Manufacturing journal homepage: www.elsevier.com/locate/rcim

Kinematic analysis of a novel 3-DOF actuation redundant parallel manipulator using artificial intelligence approach Dan Zhang a,b,n, Jianhe Lei a a b

Faculty of Engineering and Applied Science, University of Ontario Institute of Technology, Ontario, Canada L1H 7K4 College of Mechanical Engineering, DongHua University, Shanghai 200051, PR China

a r t i c l e in fo

abstract

Article history: Received 27 November 2009 Received in revised form 1 July 2010 Accepted 12 July 2010

Kinematic analysis is one of the key issues in the research domain of parallel kinematic manipulators. It includes inverse kinematics and forward kinematics. Contrary to a serial manipulator, the inverse kinematics of a parallel manipulator is usually simple and straightforward. However, forward kinematic mapping of a parallel manipulator involves highly coupled nonlinear equations. Therefore, it is more difficult to solve the forward kinematics problem of parallel robots. In this paper, a novel three degreesof-freedom (DOFs) actuation redundant parallel manipulator is introduced. Different intelligent approaches, which include the Multilayer Perceptron (MLP) neural network, Radial Basis Functions (RBF) neural network, and Support Vector Machine (SVM), are applied to investigate the forward kinematic problem of the robot. Simulation is conducted and the accuracy of the models set up by the different methods is compared in detail. The advantages and the disadvantages of each method are analyzed. It is concluded that n-SVM with a linear kernel function has the best performance to estimate the forward kinematic mapping of a parallel manipulator. & 2010 Elsevier Ltd. All rights reserved.

Keywords: Parallel kinematic manipulator Support vector machine Artificial neural networks Forward kinematic problem

1. Introduction A parallel kinematic manipulator is a closed-loop mechanism, in which the end-effector is connected to the base by at least two independent kinematic chains. Generally, it comprises two platforms, which are connected by joints and legs acting in parallel [1]. Nowadays, many applications of parallel robots can be found in various industrial fields, such as manufacturing production configurations [2], micro-motion parallel robot for medical applications [3], assembly robot in automotive applications [4], deep sea exploration [5], and so on. More recently, they have been used in the development of high precision machine tools [6]. Redundancy in parallel manipulators is divided into kinematic redundancy and actuation redundancy [7]. This paper introduces a novel 3-DOF parallel manipulator with actuation redundancy, which can be used as a micro-motion platform. Because the design and analysis of a parallel manipulator with actuation redundancy is very complex, therefore, only the kinematic analysis is given, the issues regarding dynamic, stiffness analysis, and workspace optimization will be presented in other sources. Kinematic analysis includes inverse kinematics and forward kinematics. Forward kinematic problem (FKP) is to compute the

n Corresponding author at: Faculty of Engineering and Applied Science, University of Ontario Institute of Technology, Oshawa, Ontario, Canada L1H 7K4. Tel.: +1 905 721 8668x2965; fax: +1 905 721 3370. E-mail address: [email protected] (D. Zhang).

0736-5845/$ - see front matter & 2010 Elsevier Ltd. All rights reserved. doi:10.1016/j.rcim.2010.07.003

position and orientation of the end-effector of the manipulator based on a set of joint angles. In most cases, joint angles can be computed independently. However, the forward kinematics problem of parallel manipulators is generally complicated. Different efforts have been made in solving FKP either in general cases or in special cases. Generally, there are four different methods to solve this problem: analytical approaches; use of additional sensors or transducers; numerical methods; and neural network based approaches [8]. Some scholars applied Newton– Raphson method [9–11] or other analytical approaches for solving the FKP [12–17]. However, most of these analytical approaches were devised for special configurations of parallel manipulators [8]. To address this issue of generalization, numerical approaches, e.g., Newton–Raphson method, have been proposed [18–22]. Recently, artificial neural network methodology has received considerable attention for solving the FKP of parallel robots, and some meaningful results have been achieved [8,23,24]. Although artificial neural network (ANN) is a useful tool for solving the FKP of parallel robot, it still has some intrinsic disadvantages: (1) the multi-layer neural networks cannot be adapted for on-line application, while the control system of a parallel robot requires real-time processing. (2) The ability of the artificial neural network to map the input and output relation is completely dependent on the accurate training of the system, the sample size is large and the training method is crucial. A very large amount of data is required in order to ensure that the results are statistically accurate. (3) It has the problems such as slow convergence speed, local minima, and poor generalization.

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D. Zhang, J. Lei / Robotics and Computer-Integrated Manufacturing 27 (2011) 157–163

Therefore, it is necessary to investigate other methods in order to solve the FKP of parallel robots. Unlike empirical risk minimization used in other methods, Support Vector Machine (SVM) introduced by Vapnik [26], is based on the principle of structural risk minimization. It is originated in modern statistical learning theory, and has found a wide range of real-world applications recently. In good generalization, the absence of local minima and the sparse representation of solution are the advantages of considering SVM as one of the powerful tools for classification and regression. Zha [27] used SVM for solving the FKP of a 6-SPS Stewart platform, some basic results were obtained. In this paper, we plan to investigate an SVM regression for solving the FKP of the novel 3-DOF parallel manipulator with actuation redundancy. The results show that SVM is more powerful for FKP of robots compared with other intelligent methods, such as MLP and RBF neural networks. In the following sections, first, geometric modeling and kinematic analysis of the parallel robot are developed, and then an overview of an SVM is presented. The results from the applications of the SVM and neural networks to solve the FKP of the parallel manipulator are presented and the results are compared. Finally, the conclusion and future work are given.

Fig. 2. Schematic representation of the spatial 3-DOF actuation redundant parallel manipulator.

2. Geometric modeling and kinematic analysis 2.1. Geometric modeling The proposed 3-DOF parallel manipulator with actuation redundancy and a passive leg is shown in Figs. 1 and 2. This manipulator is composed of a moving plate, a fixed base, four (4) limbs with identical kinematic structure and one (1) passive limb. The four (4) limbs connect the fixed base by a universal joint followed by a prismatic joint and a spherical joint attached to the moving plate. A linear actuator drives each prismatic joint. The 5th leg is fixed on the base platform, followed by a prismatic joint, and then connected to the moving plate by a universal joint. The degree of freedom of the 5th leg is equal to three; therefore, the overall structure has 3-DOF [28]. Fig. 3. A fixed reference frame O xyz is attached to the base of the mechanism and a moving coordinate frame Ouxuyuzu is connected to the moving platform. In Fig. 2, the points of attachment of the actuated legs to the base are represented with Ai and the points of Fig. 3. The passive leg.

attachment of legs to the moving platform are represented with Bi, with i¼1, y, 4, while point O0 is located at the center of the moving platform with the coordinate of P (x, y ,z). 2.2. Inverse kinematics The position vector of points Ai and Bi with respect to the coordinate frames O and O0 , respectively, can be written as O

a i ¼ ½ra cos bi , ra sin bi , 0T , i ¼ 1, 2, 3, 4

O0

b i ¼ ½rb cos bi , rb sin bi , 0T , i ¼ 1, 2, 3, 4

ð1Þ

where ra and rb are the lengths of the OAi and OuBi , respectively, angle bi is measured from the x-axis to the line OAi and is equal to the x0 axis to the line O0 Bi. To facilitate the analysis, a position vector,A p, is used to define the position of the moving platform Fig. 1. CAD model of the spatial 3-DOF actuation redundant parallel manipulator.

A

p ¼ OOu ¼ ½px , py , pz T

ð2Þ

D. Zhang, J. Lei / Robotics and Computer-Integrated Manufacturing 27 (2011) 157–163

And a rotation matrix,A R B , is used to define the orientation of the moving platform with respect to the fixed base A

R B ¼ Rz ðfÞRY ðyÞRx ðcÞ 2 cos f cos y cos f sin y sin csin f cos c 6 ¼ 4 sin f cos y sin f sin y sin ccos f cos c sin y cos y sin c

cos f sin y cos csin f sin c

3

sin f sin y cos ccos f sin c 7 5 cos y cos c

159

Table 2 Optimal kinematic parameters of the 3-DOF PKM. ra rb

21.75 mm 59.50 mm {  151, 151} {  151, 151} 130–180 mm

y1 y2 d5

ð3Þ where c, y, f denote the three successive rotations of the moving platform about the fixed x, y, and z axes. Combining Eqs. (2) and (3), we obtain a 4  4 transformation matrix A T B " # A RB Ap A TB ¼ ð4Þ 0 1

q24 ¼ d25 þ ra2 þ rb2 þ 2ra rb cos y2 þ 2rb d5 sin y1 sin y2

ð13Þ

Hence, the length of the four (4) actuated legs can be calculated by the square root of the above four (4) equations. The optimal kinematic parameters are shown in Table 2 [25].

3. An overview of support vector machine

T

Hence the six variables, denoted as x ¼ ½px , py , pz , c, f, y , completely define the position and orientation (pose) of the moving platform. Table 1. We note that the selection of the three independent variables may depend on the constraints of the mechanism. Since the passive leg is a 3-DOF open-loop chain, and its parameters can be determined by d5, y1, and y2. Based on the Denavit–Hartenberg (DH) table of the passive link, the transformation from the moving platform to the fixed base can be written as A

T B ¼ A T 1 ðd5 Þ1 T 2 ðy1 Þ2 T 3 ðy2 Þ

ð5Þ

i

where T j denotes 3  3D-H transformation matrix. The 0th link frame is attached on the fixed base with the z0-axis pointing along the first joint axis of the passive leg 2 3 cy1 cy2 cy1 sy2 sy1 6 sy A cy2 0 7 RB ¼ 4 ð6Þ 5 2 sy1 cy2 sy1 sy2 cy1 and  p¼ 0

0

d5

T

ð7Þ

For inverse kinematics, the three independent parameters are given and the problem is to find the actuated leg length. With Fig. 2, a vector-loop equation can be written for each actuated leg as below B

qi ¼ qi si ¼ p þ A R B b i ai

f ðxÞ ¼ ðw FðxÞÞ þ b

ð14Þ

n

where wCR , bCR, and F denotes a non-linear transformation from Rn to high dimensional space. The quality of estimation is measured by the loss function and the e-insensitive loss function by Vapnik, which is very popular and has the form [26] ( 9f ðxÞy9e, for 9f ðxÞy9 Z e Gðf ðxÞyÞ ¼ ð15Þ 0 otherwise In the next section, we will investigate the most effective SVR methods for solving the FKP of the proposed 3-DOF parallel manipulator with an actuation redundancy.

4. Forward kinematic solution using SVM

ð8Þ

where qi is the length of the ith actuated leg and si is a unit vector pointing along the direction of the ith actuated leg. Hence, qi can be computed by dot-multiplying qi with itself to yield. B

Suppose that there is a training data set fðx1 ,y1 Þ, ::::, ðxm , ym Þg, each xi CRm represents the input space of the sample and has a corresponding target value yi CR for i¼1, y , m, where m corresponds to the size of the training data [26,29]. The purpose of support vector machine regression is to construct a hyperplane that can approximate a nonlinear input–output mapping accurately to determine a specific nonlinear function with the black box method. The generic support vector regression (SVR) estimating function is [30–32]

B

qTi qi ¼ ½p þ A R B b i ai T ½p þ A R B b i ai 

ð9Þ

Eq. (9) can be rewritten as q21 ¼ d25 þra2 þrb2 þ2rb d5 sin y1 cos y2 2ra rb cos y1 cos y2

ð10Þ

q22 ¼ d25 þra2 þrb2 þ2ra rb cos y2 2rb d5 sin y1 sin y2

ð11Þ

q23 ¼ d25 þra2 þrb2 2rb d5 sin y1 cos y2 2ra rb cos y1 cos y2

ð12Þ

Table 1 DH parameters for the passive constraining leg with rigid links. Link

ai

bi

ai

yi

A 1 2 3

0 0 0 0

0 d5 0 0

0 90 90 0

0 0

y1 y2

4.1. Workspace analysis and training data set Based on the kinematic model developed, the structural parameters can be optimized in the sense that the motion range of the end-effector can be maximized. As shown in Table 2, the motion range of the z-translation is 130–180 mm. The motion ranges of x-and y-rotations are about 7151. To get the training data set, we use the closed-form solution of the inverse kinematic position within the workspace of the robot, i.e., four actuator displacements were from solution of the inverse kinematics model, and 1000 data pairs are obtained. This data set includes 1000 input–output pairs for every 11 joint angle. The 1000 data points were selected randomly to form the training data set. Input vector consists of four elements and the output vector, or also called as the target vector, includes x-and y-rotation angles and the z-translation d5. We then invert these data points, so that it can be used as an input–output data set for the forward kinematic position. This data set is partitioned into three disjoint sets: 60% of the vectors are used to train the network, 20% of the vectors are used to validate quality of the network, and the last 20% of the vectors provide an independent test of network (ANN and SVM) generalization to data that the network has never seen [33].

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Table 3 FKP solution model of Z-translation d5. SVM model

Kernel function

SVM parameters C

e-SVR n-SVR

N

Linear Sigmoid

354.283 60.998

Linear Sigmoid

54.454 84.926

0.01 0.152

Gamma

Coefficient

P

0.081

0.000

0.095 1.596

0.046141

0.000

Time (s)

Performance (MSE)

6.15 11.85

1.1e-5 5.1 e-5

1.95 7.53

1.1 e-5 4.7 e-5

Table 4 FKP solution model of the x-rotation angle y1 . SVM model

Kernel function

SVM parameters

n

C

e-SVR n-SVR

Linear Sigmoid

9159.448 4.0629

Linear Sigmoid

975.616 180.329

0.007 0.050

Gamma

Coefficient

P

0.283

0.000

2.432 0.204

0.141

0.000

Time (s)

Performance (MSE)

46.10 36.79

1.13e-5 2.33e-5

1.38 3.31

1. 12e-5 3.81e-5

Table 5 FKP solution model of the y-rotation angle y2 . SVM model

Kernel function

SVM parameters

n

C

e-SVR n-SVR

Linear Sigmoid

262.652 422.225

Linear Sigmoid

54.454 118072.66

0.01 0.165

Gamma

Coefficient

P

0.129

2.512

0.435 0.0001

0.140

1.000

4.2. FKP solution using an SVM When SVM is used for solving regression problems, two models can be applied, i.e., e-SVR and n-SVR regression, they are different because their error functions are not the same, for e-SVR, we have the error function XN XN   1 ð16Þ xi þ xi J ¼ wT wþ C i i 2 which we can minimize it subject to 

wT fðxi Þ þ bi yi r e þ xi

Time (s)

Performance (MSE)

3.36 39.65

2.03e-5 3.17e-5

1.95 37.24

1. 83e-5 4.21e-5

SVM estimators. In our experiment, different parameter sets are investigated. Results show that if the parameters are not set up appropriately, the training time could be much longer, for example, in the last row of Table 5, the training time is over 13 min. The training stop criterion is when the mean square error (MSE) reaches the set values. The training results are shown in Tables 3–5 and will be compared with the ANN method, which was used widely for solving an FKP of parallel robots in the literature. In Tables 3–5, Gamma and P are the parameters of eSVR and Coefficient is the parameter b in the sigmoid kernel function.

T

yi w fðxi Þbi r e þ xi xi , xi Z 0, i ¼ 1, :::, N for n-SVM, we have the error function   1 1 XN  ðxi þ xi Þ J ¼ wT wC ne þ i 2 N

5. A comparison of SVM and ANN methods

ð17Þ

which we can also minimize it subject to ðwT fðxi Þ þ bi Þyi r e þ xi  yi ðwT fðxi Þ þbi Þ r e þ xi  xi , xi Z 0, i ¼ 1, :::, N In order to set up a regression model, we have experimented

e- and n-SVR models with the linear and sigmoid kernel functions. Proper setting of e (insensitive zone) and C (regularization) parameters is crucial to an SVM regression. Recently, some researchers proposed analytical selection of SVM parameters [34]. Although these formulas could give some hints on the determinations of the SVM parameters, it is still a timeconsuming task to find the suitable parameter values for the

Neural networks are used for solving FKP in the sense that they are able to create internal representations through training example sets. Due to their learning capabilities, they are often applied as adaptive function estimators to estimate the input– output relation of a system. Therefore, they are utilized to represent the mapping from the joint space to the work space. Forward neural network with back propagation algorithm and radial basis function (RBF), which is the most widely used neural network, will be investigated here for the FKP solution. 5.1. Multilayer perceptron (MLP) The most common neural networks used in solving the FKP are the multilayer perceptron (MLP), therefore, it is used as a comparable method to compare with SVM estimator in the first

D. Zhang, J. Lei / Robotics and Computer-Integrated Manufacturing 27 (2011) 157–163

step. In our case, the nonlinear unknown function has been approximated as F : R4 -R3

ð18Þ

A graphical representation of an MLP for the solution of FKP is shown in Fig. 4. The input of the ANN is the vector q¼ [q1, q2, q3, q4]0 corresponding to the length of actuated legs. On the other hand, the targets are two rotation angles y1 , y2 and one translation in the z-axis d5, which represent the orientation of the end-effector task space. Since there is no theoretical method to determine the numbers of the layers and the neurons in each layer, therefore, the multiple neural networks (one hidden layer and two hidden layers) with different neurons have been tested. The neurons in the hidden layers have sigmoid activation functions. The output layer has linear neurons. In this procedure, a termination criterion was set as an MSE and all the weighting coefficients were initially assigned randomly. Then input vectors from the test data set are presented to the trained backpropagation network. The responses of the network, i.e., two rotation angles and one translation in the z-axis d5, are compared with the targets in the test data. Network performances with different network structures are compared in Table 6. Aiming at finding a best model for FKP of robot, over forty feed forward networks, which have one or two hidden layers with variable neuron numbers, were trained. Using the stated criteria, six networks with best performance were selected as shown in Table 6. It can be seen that networks with two hidden layers (N1 ¼25 and N2 ¼25) have better performance in general and would be selected as the model of an FKP. For training the networks, different performance indices could be used, such as

F : R4

the sum of square output errors, mean square or mean absolute error, etc. Here, we also adopted an MSE, which is used widely by researchers, as the performance index. Just as the training of SVM in Section 4, all the simulations were done using MATLAB. Fig. 5 shows the training process of the neural network, which has two hidden layers with 25 neurons for each hidden layer. Fig. 6 is the testing of the trained ANN.

5.2. Radial basis function (RBF) Generally, RBF networks are applied successfully for many applications such as function approximation, time series prediction, system modeling, and control. Radial Basis Function (RBF) neural network can also provide an alternative to back propagation networks, while maintaining a high level of accuracy. Therefore, as the second choice for the FKP model, we made some investigations during the experiment. Typically, RBF networks have an input layer, a hidden layer with a nonlinear RBF activation function and a linear output layer. The activation function of the hidden layer takes the form yi ðtÞ ¼ wi0 þ

nh X

wij yj ðxÞ

i ¼ 1, :::, m

j¼1

R3

q1 1

q2 2

q3

d5

q4

Input Layer

Hidden Layer

Output Layer

Fig. 4. Structure of an ANN model for the solution of FKP.

Table 6 MLP models and network performances. Network structure

No. of neurons

Network performance Training time (min)

MSE

MLP (one hidden layer)

N1 ¼ 25 N1 ¼ 30 N1 ¼ 35

4.133 3.317 5.333

9.5e-5 9.3e-5 9.0e-5

MLP (two hidden layers)

N1 ¼ 20 N2 ¼ 20 N1 ¼ 25 N2 ¼ 25 N1 ¼ 30 N2 ¼ 30

3.167

7.2e-5

4.983

1.25e-5

8.317

9.73e-5

161

Fig. 5. Training process of an ANN model.

ð19Þ

162

D. Zhang, J. Lei / Robotics and Computer-Integrated Manufacturing 27 (2011) 157–163

Fig. 7. Simulation result of the x-rotation angle y1, using an SVM regression.

Fig. 6. Testing of the trained ANN.

Table 7 RBF models and network performances. Network structure

Training time (min)

MSE

RBF1 RBF2

11.167 9.817

6.2 e-5 6.7 e-5

where

yj ðxÞ ¼ exp

JXmj J2

!

2s2j

ð20Þ

X is a d-dimensional input vector with elements xi and, mj is the vector determining the center of the strictly positive radially symmetric function yj. Input and output data sets are the same as that of training MLP network. Same network evaluation indices were used as that of the MLP network. Over twenty different network structures with different parameters were trained and compared. Two best networks were selected and the performances of the networks are shown in Table 7. From Tables 6 and 7, we can see, the MLP with two hidden layers provides the better performance. From the experiments, we found that RBF network needs more samples than MLP in order to achieve the same accuracy as a backpropagation network. Meanwhile, the performance is dependent on RBF units or cluster centres. The correct choice of centres is analogous to choosing the number of nodes in hidden layers of back propagation networks. The performance of the developed models is assessed using a new data set. We assume that the rotation angle y2 ¼0 and the z-translation d5 ¼130 mm, and the y1 of the end-effector starts from  151 and ends at 151 according to sinðotÞ, 100 newly obtained data pairs are used for the model validation. Mean absolute error (MAE) is applied here for the validation MAE ¼

n 1X absðyi di Þ n i

yi is the output of the model and di is the target value y1. With the equation, by calculating the performances MAE for the best MLP, RBF, and SVR models with the 100 samples, we obtain MAE(MLP) ¼0.195, MAE(RBF) ¼0.09, and MAE(SVR) ¼0.01. From the results, it has been shown that the performance of the MLP is the worst and the performance of the SVR is the best. This is because ANN is based on Empirical Risk Minimisation (ERM) approach and has the problems such as local minimal point, over fitting, and poor generalization. If the ANN based models have been trained with a known sample set, they can possess very good output prediction ability, otherwise the performance of the models could be degraded greatly. The SVM is based on Structural Risk Minimization (SRM) principle; therefore, it has good generalization. Figs. 7 and 8 are the comparison between the best MLP model and SVR with the linear kernel function for estimating the x-rotation angle y1. In Figs. 7 and 8, the units of y1 is degree. It shows that an SVM is an effective method for the solution of parallel robots.

ð21Þ

6. Conclusions In this paper, a novel 3-DOF parallel manipulator with an actuation redundancy is introduced and an SVM based modeling method is applied for FKP solution of the proposed parallel robot. The same as ANN, SVM can learn any highly nonlinear functions when it is used for regression, but adopting a completely different way. The formulation embodies the structural risk minimization principle, as opposed to the empirical risk minimization, which ANN is based on. This feature makes SVM have good generalization, absence of local minimal and sparse representation of solution. In this research, an SVM has been successfully applied to approximate the complex mapping between robot positions, orientations, and robot actuation displacement. The results from an SVM estimation are compared with that of the common neural networks, e.g., MLP and RBF. From the simulation results, we can conclude that SVM solutions have better performance, in terms of convergence speed and better generalization. It is more suitable for the forward kinematic modeling problem of a parallel robot. With the SVM model set up, we can generate the best estimation of orientation and position of a parallel robot. However, it should be pointed out that there is no generic SVM model which can fit for all applications. For the case in the paper, n-SVM, which has a linear kernel function, is the best choice. It was also observed that the parameter selection in the case of SVM has a significant

D. Zhang, J. Lei / Robotics and Computer-Integrated Manufacturing 27 (2011) 157–163

Fig. 8. Simulation result of the x-rotation angle y1, using an MLP.

impact on the performance of the model. With an on-line SVM algorithm, this model can be used for the real-time control of the parallel robot proposed in the paper.

Acknowledgment The authors would like to acknowledge the financial support from the Natural Sciences and Engineering Research Council of Canada (NSERC). The first author gratefully acknowledges the financial support from the Canada Research Chairs program.

References [1] Merlet JP. Parallel robots. Dordrecht: Kluwer Academic Publishers; 2000. [2] Necsulescu DS, Skowronski JM, Shaban-Zanjani H. Low speed motion control of a mechanical system. Dyn Control 1993;3(4):307–22. [3] Merlet J.P., 2001, Micro parallel robot MIPS for medical applications. In IEEE Int. Conf. Emerg Technol Fact Autom, Antibes, 15–18 October 2001. [4] Mintchell GA. Industrial robots: fast, nimble at 30. Control Eng 2002;49(11): 32. [5] Saltaren R. Field and service applications – exploring deep sea by teleoperated robot – an underwater parallel robot with high navigation capabilities. IEEE Robot Autom Mag 2007;14(3):65. [6] Zhang D., Bi Z. Development of reconfigurable parallel kinematic machines using modular design approach, 3rd CDEN/RCCI International Design Conference, Toronto, Canada, p. 63–8, 2006. [7] Lenarcic J, Wenger P. Advances in robot kinematics: analysis and design. Springer; 2008.

163

[8] Parikh PJ, Lam SS. Solving the forward kinematics problem in parallel manipulators using an iterative artificial neural network strategy. Int J Adv Manuf Technol 2009;40(5–6):595–606. [9] Merlet JP. Solving the forward kinematics of a Gough-type parallel manipulator with interval analysis. Int J Robot Res 2004;23(3):221–35. [10] Nanua P, Waldron K, Murthy V. Direct kinematic solution of a Stewart platform. IEEE Trans Robot Autom 1990;6(4):438–44. [11] Waldron K, Raghavan M, Roth B. Kinematics of a hybrid series-parallel manipulation system. J Dyn Syst Meas Control 1989;111(2):211–21. [12] Innocenti C, Parenti-Castelli V. Direct kinematics of the 6–4 fully parallel manipulator with position and orientation decoupled. In: Tzafestas SG, editor. Robotic systems. Amsterdam: Kluwer Academic; 1992. p. 3–10. [13] Ji P, Wu H. A closed-form forward kinematics solution for the 6–6p Stewart platform. IEEE Trans Robot Autom 2001;17(4):522–6. [14] Wampler C. Forward displacement analysis of general six-in-parallel SPS (Stewart) platform manipulators using soma coordinates. Mech Mach Theory 1996;31(3):331–7. [15] Innocenti C. Forward kinematics in polynomial form of the general Stewart platform. ASME J Mech Des 2001;123:254–60. [16] Lee TY, Shim JK. Forward kinematics of the general 6–6 Stewart platform using algebraic elimination. Mech Mach Theory 2001;36:1073–85. [17] Zhao JS, Yun Y, Wang LP, Wan JS, Dong JX. Investigation of the forward kinematics manipulator with natural coordinates. Int J Adv Manuf Technol 2006;30:700–16. [18] Zanganeh KE, Angeles J. Real-time direct kinematics of general six-degree-offreedom parallel manipulators with minimum-sensor data. J Robot Syst 1995;12(12):833–44. [19] Dieudonne J.E., Parrish R.V., Bardusch R.E. An actuator extension transformation for a motion simulator and an inverse transformation applying Newton– Raphson’s method. NASA Langley Research Center, Hampton, VA, Tech Rep NASA TND-7067, 1972. [20] Merlet JP. Direct kinematics of parallel manipulator. IEEE Trans Robot Autom 1993;9(6):842–6. [21] Ascher M. Cycling in the Newton–Raphson algorithm. Int J Math Educ Sci Technol 1974;5(2):229–35. [22] Wang YF. A direct numerical solution to forward kinematics of general Stewart–Gough platforms. Robotica 2007;25(1):121–8. [23] Geng Z, Haynes LS. Neural network solution for the forward kinematics problem of a Stewart platform. Robotics Comput Integr Manuf 1992;9(6):485–95. [24] Yee CS, Lim KB. Forward kinematics solution of Stewart platform using neural networks. Neurocomputing 1997;16(4):333–49. [25] Li, J. Design of 3-dof parallel manipulators for micro-motion applications, University of Ontario Institute of Technology, Master Thesis, 2009. [26] Vapnik V. Statistical learning theory. New York: Wiley; 1998. [27] Zha S., Mei T., Luo M., Wang D. Direct kinematics problem of parallel robot based on LS-SVM, In: IEEE Proceedings of the World Congress on Intelligent Control and Automation (WCICA), 2006, p. 8958–61. [28] Zhang D. Parallel robotic machine tools. New York, NY: Springer; 2009. [29] Elish KO, Elish MO. Predicting defect-prone software modules using support vector machines. J Syst Software 2008;81(5):649–60. [30] Haykin S. Neural networks—a comprehensive foundation. Prentice Hall; 1999. [31] Collobert R, Benegio S. SVM torch: support vector machines for large-scale regression problems. J Mach Learn Res 2001;1:143–60. [32] Smola AJ, Scholkopf B. A tutorial on support vector regression. Neuro COLT technical report NC-TR-98-030. UK: Royal Holloway College, University of London; 1998. [33] Howard D, Beale M, Hagan M. Matlab neural network toolbox 6 user guide. The MathWorks, Inc.; 2009. [34] Cherkassky V., Ma Y. Selection of meta-parameters for support vector regression. In: Proceedings of the International Conference on Artificial Neural Networks (ICANN), 2002, Madrid, Spain. p. 687–93.