Three-Degree-of-Freedom Dynamic Model-Based Intelligent Nonsingular Terminal Sliding Mode Control...

IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 20, NO. 5, OCTOBER 2012 971

Three-Degree-of-Freedom Dynamic Model-BasedIntelligent Nonsingular Terminal Sliding Mode

Control for a Gantry Position StageFaa-Jeng Lin, Senior Member, IEEE, Po-Huan Chou, Chin-Sheng Chen, and Yu-Sheng Lin

Abstract—A three-degree-of-freedom (3-DOF) dynamic model-based intelligent nonsingular terminal sliding mode control(INTSMC) system is proposed in this study for the precision con-tours tracking of a gantry position stage. A Lagrangian equation-based 3-DOF dynamic model for the gantry position stage is de-rived first. Then, to minimize the synchronous error and track-ing error in the precision contours tracking, the 3-DOF dynamicmodel-based INTSMC system is proposed. In this approach, a non-singular terminal sliding mode control is designed for the gantryposition stage to achieve finite time tracking control. Moreover, toincrease the robustness and to improve the control performance,an interval type-2 recurrent fuzzy neural network, and asymmet-ric membership function, which combines the advantages of in-terval type-2 fuzzy logic system, recurrent neural network, andasymmetric membership function, is developed as an estimator toapproximate a lumped uncertainty. Finally, some experimental re-sults of the gantry position stage for optical inspection applicationare obtained to show the validity of the proposed control approach.

Index Terms—Asymmetric membership function (AMF), gantryposition stage, interval type-2 fuzzy logic system, Lyapunov sta-bility, neural network, three-degree-of-freedom (3-DOF) dynamicmodel.

I. INTRODUCTION

S LIDING mode control (SMC) is a well-known and power-ful control scheme that has been successfully and widely

applied for both linear and nonlinear systems [1]. In general, themost commonly used sliding surface is the linear sliding sur-face, which can guarantee the asymptotic stability and desiredperformance of the closed-loop control system by using linearsliding mode [1]. Although the parameters of the linear slid-ing surface can be adjusted appropriately to obtain the arbitraryconvergence rate, the system states cannot reach the equilibriumpoint in a finite time [2]. To overcome this drawback, terminalsliding mode control (TSMC) with nonlinear terminal sliding

Manuscript received March 24, 2011; revised August 3, 2011 and November29, 2011; accepted February 12, 2012. Date of publication March 20, 2012; dateof current version October 2, 2012. This work was supported by the NationalScience Council of Taiwan under Grant NSC 97-2221-E-008-098-MY3.

F.-J. Lin and Y.-S. Lin are with the Department of Electrical Engineering,National Central University, Chungli 320, Taiwan (e-mail: [email protected];[email protected]).

P.-H. Chou is with the Department of Mechatronics Control, Industrial Tech-nology Research Institute, Hsinchu 310, Taiwan (e-mail: [email protected]).

C.-S. Chen is with the Graduate Institute of Automation Technology,National Taipei University of Technology, Taipei 106, Taiwan (e-mail:[email protected]).

Digital Object Identifier 10.1109/TFUZZ.2012.2191412

surface has recently been proposed based on the concept ofa terminal attractor [2], [3]. Compared with the conventionalSMC with linear sliding surface, a TSMC offers some superiorproperties, such as faster tracking response, finite time conver-gence, and higher control precision [2]. However, there are twodisadvantages of TSMC: The singularity point problem and therequirement of the bound of the uncertainty. Fortunately, thefirst problem has been overcome by nonsingular terminal slid-ing mode control (NTSMC) [4], [5], and the second problemcan be solved by well-designed uncertainty estimator [6].

Fuzzy neural networks (FNNs) combine the capability offuzzy reasoning to handle uncertain information and the ca-pability of artificial neural networks to train from processes.Therefore, there were many researches using FNNs to repre-sent complex plants and construct advanced controllers [7], [8].However, because the membership functions (MFs) of type-1FNN (T1FNN) are obtained as the crisp value, the T1FNN onlyoffer limited scope for modeling uncertainty and cannot han-dle the high levels of uncertainty which are usually presentedin practical applications. On the other hand, an interval type-2FNN (IT2FNN), which consists of the interval type-2 fuzzy lin-guistic process as the antecedent part and a three-layer neuralnetwork as the consequent part, was developed in [9] and [10]to further improve the control performance of T1FNN. This isbecause the interval type-2 fuzzy logic controller (FLC) canbe regarded as a collection of many different embedded type-1FLCs, which allows for the detailed description of the uncer-tainty that cannot be achieved by the type-1 FLCs with the crispmembership. Moreover, the uncertainty represented in the inter-val type-2 fuzzy sets can cover the same range as type-1 fuzzysets with a smaller number of labels and fuzzy rules [11], [12].Hence, the problem of the overparameterization will not occurfor the interval type-2 FLC. However, a major drawback of theT1FNN and IT2FNN is that their applications are limited tostatic problems due to their feed-forward network structure. Forthis reason, the interval type-2 recurrent fuzzy neural network(IT2RFNN) had been proposed to solve this drawback [13].Since the neurons in the feedback layer of the IT2RFNN actas memory elements which endue the network with the abilityto cope with the temporal problems, the approximation accu-racy of the network is improved. Compared with the traditionalrecurrent neural network which adopted output feedback struc-ture, there were many research works using inner loop feedbackstructure to enhance training capability of the neural networksbecause it provides more dynamic information for the uncer-tainty modeling [14], [15].

1063-6706/$31.00 © 2012 IEEE

972 IEEE TRANSACTIONS ON FUZZY SYSTEMS, VOL. 20, NO. 5, OCTOBER 2012

In general, to simplify the design procedure of the FNN orIT2FNN, symmetric MFs (e.g., Gaussian or triangular func-tions) are commonly adopted in practical applications. How-ever, to achieve the specified approximation accuracy, a largenumber of rules should be applied [16]. On the other hand, sincethe dimensions of the standard Gaussian or triangular MF are di-rectly extended in asymmetric membership functions (AMFs),not only the training capability of the networks can be upgradedbut also the number of fuzzy rules can be optimized. Thus, AMFshave been adopted in several approaches to optimize the numberof fuzzy rules and improve the control precisions [17], [18].

The gantry position stages with high precision control per-formance have been widely adopted to facilitate the automatedprocesses in microelectronics, precision metrology, circuit as-sembly, printed circuit board manufacturing, and flat panel man-ufacturing and inspection [19]–[21]. In the configuration of agantry position stage, two motors are mounted on two parallellinear guides to drive a moving stage. In this configuration, thesynchronous error will affect the quality of the workpieces andeven lead to a stop of the working process due to overly cur-rent protection. Hence, the control of the synchronous error inthe gantry position stages has become a challenge with increas-ing demand for high-speed and high accuracy manufacture andinspection.

In general, the gantry position stages are controlled usingthe independent axis control without considering the effect ofinteraxis mechanical coupling, i.e., based on the independentsingle-input-single-output (SISO) model [22], [23]. However,considering the condition of the three-axes moving simulta-neously, although the robust synchronous or intelligent syn-chronous compensator was added to the middle between duallinear motors to improve the synchronous performance of duallinear motors [22], [23], the time-varying loading presented bythe translation of the slider along the gantry was not consid-ered which degenerates control performance of each axis andresults in synchronous error of dual linear motors. On the otherhand, a 3-DOF model was proposed to consider the interaxismechanical coupling force [24]. In this model, the proposed co-ordinate system not only corresponded to the linear positionsof the center of gravity of the gantry and slider but consideredboth rotation of the gantry and slider as well. However, it is verycomplicated to model the correlation between gantry bean andthe actuators using soft joint [24].

In this study, a 3-DOF dynamic model-based INTSMC sys-tem is proposed for the precision contours tracking of a gantryposition stage. First, to consider the effect of interaxis mechan-ical coupling, the Lagrangian equation-based 3-DOF dynamicmodel for the gantry position stage is derived. In this model,the new coordinate system considers not only the linear po-sitions of the center of gravity of the gantry and slider, butalso the rotation of gantry and slider. Moreover, the new coor-dinate system could achieve two control objectives simultane-ously by using simple geometric relations, i.e., position controlof the gantry and minimization of the synchronous error. Then, a3-DOF dynamic model-based NTSMC is designed for the gantryposition stage to achieve finite time tracking control. However,the bound of the lumped uncertainty is necessary in the design

Fig. 1. Structure of a 3-DOF dynamic model.

of NTSMC and is very difficult to obtain in advance in practicalapplications. Therefore, the 3-DOF dynamic model INTSMCis proposed to alleviate the aforementioned difficulties and toimprove the control performance of the gantry position stage. Inthis approach, an IT2RFNN-AMF estimator with superior ap-proximated ability is employed to estimate the unknown lumpeduncertainty online. Moreover, the adaptive laws for the onlinetraining of IT2RFNN-AMF are derived using the Lyapunov the-orem. Finally, the proposed 3-DOF dynamic model-based intel-ligent control approach is implemented in a control computerwhich is based on a 32-bit floating-point DSP, TMS320VC33.Some experimental results of optical inspection application arecarried out to investigate the effectiveness of the proposed con-trol scheme.

II. THREE-DEGREE-OF-FREEDOM DYNAMIC

MODEL FOR THE GANTRY POSITION STAGE

A. Brief Description of the Typical Gantry Stage

A typical gantry mechanism and the corresponding coordi-nate are shown in Fig. 1 where two servo motors in y-axis carrya gantry which a slider in x-axis holding an inspection tool ismounted. It can be considered as a 3-DOF servo mechanism.Moreover, two motors in the y-axis yield displacement dy1 anddy2 , respectively. These two displacements are equal when twoservo motors move synchronously, implying dy1 = dy2 . In prac-tice, the two displacements are different since the two servomotors are unavoidable due to the unbalanced forces, mecha-nism assembly variations, and various disturbances during theworking process. Furthermore, the central point C of the gantryis, thus, constrained to move along the central dashed line in 1DOF. Because of the deviation between dy1 and dy2 , the gantryalso rotates about the central point C [24], and the rotating angleis indicated as θ.

LIN et al.: THREE-DEGREE-OF-FREEDOM DYNAMIC MODEL-BASED INTELLIGENT NONSINGULAR TERMINAL SMC 973

B. Lagrangian-Based Three-Degree-of-FreedomDynamic Model

Let M1 and M2 denote the mass of the gantry and slider,respectively, l and 2w denote the length and width of the gantry,respectively, and IM 1 and IM 2 denote the moment of inertiaof the gantry and slider corresponding to the central point C,respectively. Then

IM 1 =M1

12

(l2 + (2w)2

)(1)

IM 2 = M2

(d2

x + (w + v)2)

. (2)

Moreover, the positions pM 1 and pM 2 corresponding to thecenter of gravity of M1 and M2 are represented as

pM 1 =[

0

dy

](3)

pM 2 =[

dy − (w + v) cos θ + dx sin θ

dx cos θ + (w + v) sin θ

](4)

where dy = dy1 + (dy2 − dy1)/2. The velocities νM 1 and νM 2

corresponding to the center of gravity of M1 and M2 are repre-sented as

νM 1 =[

0

dy

](5)

νM 2 =

[dy + (w + v)θ sin θ + dx sin θ + dxθ cos θ

dx cos θ − dxθ sin θ + (w + v)θ cos θ

]. (6)

Then, the translational and rotational kinetic energy of thegantry and slider are shown as follows:

KM 1 =12M1vT

M 1vM 1 +

12I1 θ

2 (7)

KM 2 =12M2vT

M 2vM 2 +

12I2 θ

2 . (8)

Thus, the total kinetic energy may be computed as

K = KM 1 + KM 2 =12M1 νT

M 1νM 1 +

12M2 νT

M 2νM 2

+12

(I1 + I2) θ2

=12

(M1 + M2) d2y +

12(I1 + I2 + M2d

2x

+M2(w + v)2) θ2

+12M2 d

2x + dy θM2 [(w + v) sin θ + dx cos θ]

+ dy dxM2 sin θ + θdxM2(w + v) (9)

which can be further written as

K =1

2X

T

DX (10)

where X = [dy θ dx ]T , and D is the inertia matrix given as in(11), shown at the bottom of the page.

Then, the government equation of the gantry mechanism canbe derived from the Lagrange equation

d

dt

[∂L

∂X

]− ∂L

∂X= F + U (12)

where L = K − V , K is the summation of kinetic energy, V ispotential energy, U is the driven force provided by motors, andF is the friction force. Next, the elements of the Coriolis andcentrifugal matrix C can be derived from

Cij =3∑

k=1

(cijk qk ) (13)

where q1 , q2 , and q3 represent the derivative of dy , θ, and dx ,respectively. The Christoffel symbols cijk are computed as

cijk =12

[∂dij (q)

∂qk+

∂dik (q)∂qj

+∂djk (q)

∂qi

](14)

where dij represents the element in the ith row and jth columnof the inertia matrix D. Substituting the inertia equation I1 andI2 into (11) and computing (14), the matrix C can be obtained,as in (15), shown at the bottom of the page.

Finally, the 3-DOF dynamic model of the gantry positionstage can be represented as DX + CX + BF = BU, where

B =

⎡⎢⎣

1 1 0

l cos θ −l cos θ 0

0 0 1

⎤⎥⎦ (16)

F =[Fy1 , Fy2 , Fx

]T(17)

U =[uy1 , uy2 , ux

]T(18)

D =

⎡⎢⎣

M1 + M2 M2(w + v) sin θ − M2dx cos θ M2 sin θ

M2(w + v) sin θ + M2dx cos θ I1 + I2 + M2d2x + M2(w + v)2 M2(w + v)

M2 sin θ M2(w + v) M2

⎤⎥⎦ . (11)

C = M2

⎡⎢⎢⎣

0 −dxθ sin θ + dθ cos θ + dx cos θ θ cos θ

−dxθ sin θ + dθ cos θ + dx cos θ (−dx sin θ + d cos θ) dy + 2dxdx 2dxθ + dy cos θ

θ cos θ 2dxθ + dy cos θ 0

⎤⎥⎥⎦ . (15)


Fig. 2. Configuration of a 3-DOF dynamic model-based NTSMC system.

where Fy1 , Fy2 , and Fx are the frictional forces, and uy1 , uy2 ,and ux are the generated mechanical forces along dy1 , dy2 , anddx , respectively. In this study, because the rotating angle θ ofthe x-axis is limited by the mechanical structure of the adoptedgantry position stage, the maximum rotating angle of the x-axis would be limited in ±5◦. Hence, according to the afore-mentioned mechanical limitation of the adopted gantry positionstage, the existence of inverse Bn can be guaranteed, i.e., thecase of cos(θ) = 0 (rotating angle θ = ±90◦) does not exist.

III. THREE-DEGREE-OF-FREEDOM DYNAMIC MODEL-BASED

INTELLIGENT NONSINGULAR TERMINAL SLIDING MODE

CONTROL SYSTEM

A. Three-Degree-of-Freedom Dynamic Model-BasedNonsingular Terminal Sliding Mode Control

In order to simplify the derived process of the proposed 3-DOF dynamic model-based NTSMC system shown in Fig. 2,the 3-DOF dynamic model of the gantry position stage shownin (15)–(18) can be rewritten as follows:

X = −D−1CX − D−1BF + D−1BU

= AnX + BnU − BnF (19)

where An = −D−1C, Bn = D−1B, and U is the control ef-fort. Then, considering the parameters variation and unknowndynamics of the gantry position stage, the 3-DOF dynamicmodel of the gantry position stage can be rewritten as follows:

X = (An + ΔA)X + (Bn + ΔB)U − BnF + FL

= AnX + BnU + H (20)

where ΔA and ΔB denote the uncertainties of An and Bn ,respectively, and the variations of these parameters are limitedby the mechanical structure of the adopted gantry position stage;FL is the external disturbance, and H is named the lumpeduncertainty and is defined by

H = ΔAX + ΔBU − BnF + FL . (21)

Here, the lumped uncertainty is assumed to be bounded

‖H‖ ≤ δ (22)

where δ is a given positive constant. Although the lumped un-certainty H can be assumed to be a relative bound upon thecontrol effort U as shown in (21), however, the control effort isalways limited by a limiter in practical applications. Therefore,the lumped uncertainty is assumed to be an absolute bound byδ as shown in (22).

In the gantry position stage, the control problem is to establisha control law so that the state X(t) can track the desired com-mand Xm (t) and guarantee the convergence of both the positiontracking error and synchronous error to zero simultaneously. Toachieve the control objective, the nonsingular terminal slidingsurface is defined as follows:

S = E +1λE

pq (23)

where E = Xm − X is the tracking error, λ is a designed posi-tive constant, and p and q are both positive odd integers whichshould satisfy the following condition:

q < p < 2q. (24)

Consider the nonsingular terminal sliding surface defined in(23); the condition shown in (24) is satisfied if the 3-DOF dy-namic model-based NTSMC control law UNTSMC is adoptedas the control effort U and designed as follows:

UNTSMC = B−1n

[Xm − AnX + λ

q

pE2− p

q + δsgn(S)]

.

(25)Hence, the tracking error E will reach the nonsingular termi-

nal sliding surface in a finite time Tr which satisfies [4], [5]

Tr ≤ ‖S(0)‖ζ

(26)

where ζ is a positive constant. Moreover, it is obvious that thethird term of the 3-DOF dynamic model-based NTSMC controllaw UNTSMC shown in (25) will not result in the negativepower as long as the condition q < p < 2q remains. Therefore,the singularity problem is solved, completely, in the NTSMC[4], [5]. Furthermore, one can observe that when S = 0, thesystem dynamic is equivalent to E = −λE

qp . Therefore, the

finite time Ts taken to reach the equilibrium point E = 0 of theNTSMC system can be obtained as follows:

Ts = −1λ

∫ 0

E(Tr )E− q

p dE =p

λ (p − q)‖E(Tr)‖

1− qp . (27)

Therefore, both the tracking error E and its derivative E willconverge to zero in a finite time, i.e., Tr + Ts by using theNTSMC system [4], [5].

Unfortunately, in the 3-DOF dynamic model-based NTSMCapproach, the required bound of the lumped uncertainty δ shownin (22) to keep the system state trajectory on the nonsingular ter-minal sliding surface is unknown in advance in practical appli-cations. To trade off the chattering and control precision, conser-vative δ is always selected by trial and error. Therefore, a 3-DOF


Fig. 3. Structure of an IT2RFNN with AMF.

dynamic model-based INTSMC system, in which an IT2RFNN-AMF estimator is designed to estimate the lumped uncertaintyonline, is proposed to alleviate the aforementioned drawbacks.In the proposed 3-DOF dynamic model-based INTSMC, sincethe bound of the lumped uncertainty is estimated online by us-ing the IT2RFNN-AMF estimator, not only can the trackingperformance be improved but the chattering phenomena can bereduced effectively as well.

B. Interval Type-2 Recurrent Fuzzy Neural Network WithAsymmetric Membership Function Estimator

In this study, the proposed IT2RFNN-AMF estimator is capa-ble of handling uncertain information with the powerful type-2FLS, improving the dynamic mapping ability with the internal-feedback loop in the membership layer and enhancing the train-ing capability of the network with AMF by extending the di-mensions of the Gaussian MF. The network structure of anIT2RFNN-AMF shown in Fig. 3 includes input layer (layer 1),membership layer (layer 2), rule layer (layer 3), type-reductionlayer (layer 4), and output layer (layer 5). Its dynamics areintroduced as follows.

Layer 1 (Input Layer): For every node i in this layer, the nodeinput and the node output are represented as

net1i (N) = x1i , y1

i = f 1i (net1i (N)) = net1i (N)

i = 1, . . . , 6. (28)

In this study, x1i is the tracking error E = Xm − X =

[dmy 0 dmx ]T − [dy θ dx ]T = [ey eθ ex ]T and its change E; Ndenotes the number of iterations.

Layer 2 (Membership Layer): In this layer, each node per-forms an interval type-2 asymmetric Gaussian MF shown inFig. 4. The interval type-2 asymmetric Gaussian MF is con-structed by a type-1 asymmetric Gaussian MF with an adjustableuncertain mean and an adjustable standard deviation [17], [18].Fig. 4 shows a 2-D type-2 asymmetric Gaussian MF with theadjustable uncertain mean and an adjustable standard deviation

Fig. 4. Interval type-2 fuzzy set with AMF.

σ. For the jth node

net2j (N )

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

−12

[x2

i + rj y2j (N − 1) − m

( l)ij

]2(σ

( l)ij

)2 , for x2i ≤ m

( l)ij

1, for m( l)ij ≤ x2

i ≤ m(r )ij

−12

[x2

i + rj y2j (N − 1) − m

(r )ij

]2(σ

(r )ij

)2 , for m(r )ij ≤ x2

i

net2j (N )

=

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

−12

[x2

i + rj y2j (N − 1) − m

( l)ij

]2(σ

( l)ij

)2 , for x2i ≤ m

( l)ij

1, for m( l)ij ≤ x2

i ≤ m(r )ij

−12

[x2

i + rj y2j (N − 1) − m

(r )ij

]2(σ

(r )ij

)2 , for m(r )ij ≤ x2

i

y2j (N ) = M j

i (x2i ) =

{f 2

j

(net

2j (N ))

f 2j

(net2

j (N ))

=

{exp(net

2j (N ))

exp(net2

j (N )) =

{y2

j (N )

y2j(N )

, j = 1, . . . , s (29)

{m

( l)ij ≤ m

( l)ij ≤ m

(r )ij ≤ m

(r )ij

σ( l)ij ≤ σ

( l)ij ≤ σ

(r )ij ≤ σ

(r )ij

(30)

where y2j (N) is the output of layer 2; m

(l)ij , m

(l)ij , m

(r)ij , m

(r)ij ,

σ(l)ij , σ

(l)ij , σ

(r)ij , and σ

(r)ij are, respectively, the mean and the

standard deviation of the asymmetric Gaussian function in thejth term of the ith input linguistic variable x2

i to the node oflayer 2; rj is defined as the recurrent weight of the membershiplayer; and s is the number of the linguistic values with respect


to each input node. As shown in Fig. 4, type-2 AMFs can berepresented as an interval bound by upper AMF λA (x) andlower AMF λA (x). Therefore, the output of layer 2 y2

j (N) isalso represented as [y2

j(N), y2

j (N)].Layer 3 (Rule Layer): Each node k in this layer is denoted by∏and for the kth rule node

net3k (N ) =

∏j

w3j k y2

j (N )

y3k (N ) = f 3

k

(net3

k (N ))

= net3k (N )

⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩

y3k (N ) =

n∏j=1

(w3j k y2

j )

y3k(N ) =

n∏j=1

(w3j k y2

j)

k = 1, . . . , n (31)

where y3k (N) is the output of layer 3, w3

jk are weights betweenthe membership layer and the rule layer and are set to be unityto simplify the implementation for real-time control, and n isthe number of rules. Similar to layer 2, the output of layer 3 isrepresented as [y3

k(N), y3

k (N)].Layer 4 (Type-Reduction Layer): This layer is used to im-

plement the type reduction. Using Zadeh’s extension princi-ple [9], [10], type-1 defuzzification can derive a crisp outputfrom type-1 fuzzy set; similarly, for a higher type set as type-2,this operation derives the type-2 sets to a type-1 set. Moreover,the center-of-set type-reduction algorithm [9], [10] is adoptedin this paper. The process of this layer is described as follows:

net4l (N) =∑n

k=1 w4lk y3

k (N)∑nk=1 y3

k (N)

y4l (N) = f 4

l

(net4l (N)

)= net4l (N)

=

⎧⎪⎪⎨⎪⎪⎩

y4Rl =

∑nk=1 w4

Rlk y3Rk (N)∑n

k=1 y3Rk (N)

= WTRYR

y4Ll =

∑nk=1 w4

Llk y3Lk (N)∑n

k=1 y3Lk (N)

= WTL YL

, l = 1, 2, . . . , p

(32)

where y4l (N) is the output of layer 4; w4

k ∈ [w4Rlkw4

Llk ] is thecentroid of the type-2 interval consequent set

WR =

⎡⎢⎢⎢⎢⎣

w4R11 w4

R12 · · · w4R1n

w4R21 w4

R22 · · · w4R2n

......

. . ....

w4Rp1 w4

Rp2 · · · w4Rpn

⎤⎥⎥⎥⎥⎦

T

WL =

⎡⎢⎢⎢⎢⎢⎣

w4L11 w4

L12 · · · w4L1n

w4L21 w4

L22 · · · w4L2n

......

. . ....

w4Lp1 w4

Lp2 · · · w4Lpn

⎤⎥⎥⎥⎥⎥⎦

T

YR =[

y3R1(N)∑n

k=1 y3Rk (N)

y3R2(N)∑n

k=1 y3Rk (N)

· · · y3Rn (N)∑n

k=1 y3Rk (N)

]T

and

YL =[

y3L1(N)∑n

k=1 y3Lk (N)

y3L2(N)∑n

k=1 y3Lk (N)

· · · y3Ln (N)∑n

k=1 y3Lk (N)

]T

and p is the number of nodes in layer 4. The weighting intervalset [w4

Rlk , w4Llk ], (k = 1, . . . , n) should be set first before the

computation of y4l (N). The center-of-set type-reduction algo-

rithm [9, 10] decides the value of k for y3Rk and y3

Lk to separatetwo sides by the number R and L, respectively. Therefore, theoutput of this layer shown in (32) can be restated as

y4l (N ) =

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

y4R l =

∑n

k=1 w4R lk y3

Rk (N )∑n

k=1 y3Rk (N )

=

∑R

k=1 w4R lk y3

k(N ) +

∑n

k= R+1 w4R lk y3

k (N )∑R

k=1 y3k(N ) +

∑n

k= R +1 y3k (N )

y4L l =

∑n

k=1 w4L lk y3

Lk (N )∑n

k=1 y3Lk (N )

=

∑L

k=1 w4L lk y3

k (N ) +∑n

k= L+1 w4L lk y3

k(N )

∑L

k=1 y3k (N ) +

∑n

k= L+1 y3k(N )

l = 1, 2, . . . , p. (33)

Layer 5 (Output Layer): This layer performs the linear com-bination of y4

Rl and y4Ll . Thus, the output of layer 5 y5

l (N) canbe represented as follows:

y5l =

y4Rl + y4

Ll

2. (34)

For ease of discussion, the output of the IT2RFNN-AMFestimator is rewritten as follows:

Y5 =12(WT

RYR + WTL YL ) =

12WT Y(x1 ,m,σ, r) (35)

where Y5 = UIT2RFNN-AMF is the output ofIT2RFNN-AMF, Y = [YRYL ]T , m = [m(r)

1 1· · · m(r)

6S m(l)1 1

· · · m(l)6S m(r)

1 1· · ·m(r)

6S m(l)1 1

· · ·m(l)6S ] , σ = [σ(r)

1 1· · · σ(r)

6S σ(l)1 1

· · ·σ

(l)6S σ

(r)11 · · ·σ(r)

6S σ(l)11 · · ·σ(l)

6S ],r = [r1 · · · rs ], x1 = [ey eθ ex

ey eθ ex ], and W = [WTR WT

L ]T .

C. Three-Degree-of-Freedom Dynamic Model-BasedIntelligent Nonsingular Terminal Sliding Mode Control System

The configuration of the proposed 3-DOF dynamic model-based INTSMC system is shown in Fig. 5. According tothe universal approximation property, there exists an opti-mal IT2RFNN-AMF estimator U∗

IT2RFNN-AMF to learn thelumped uncertainty H such that

U∗IT2RFNN-AMF = H + ε =

12W∗T Y∗(x1 ,m∗,σ∗, r∗) + ε

(36)


Fig. 5. Configuration of a 3-DOF dynamic model-based INTSMC system.

where ε is the minimum reconstructed error; and W∗, Y∗,m∗, σ∗, and r∗ are, respectively, the optimal parameters of W,Y, m, σ, and r in the IT2RFNN-AMF. Moreover, the actualIT2RFNN-AMF estimator can be obtained as

UIT2RFNN-AMF =12WT Y(x1 , m, σ, r) + UR (37)

where UR is a robust controller, which is designed tocompensate the difference between U∗

IT2RFNN-AMF andUIT2RFNN-AMF ; and W, Y, m, σ, and r are, respectively,the estimates of the optimal parameters W∗, Y∗, m∗, σ∗, andr∗. Then, subtracting (36) from (37), the approximation errorUIT2RFNN-AMF is denoted as

UIT2RFNN-AMF = UIT2RFNN-AMF − UIT2RFNN-AMF

=12W∗T Y∗ − 1

2WT Y + ε − UR

=12W∗T

(Y + Y

)− 1

2

(W∗T − WT

)Y + ε − UR

=12W∗T Y +

12WT Y + ε − UR (38)

where W = W∗ − W and Y = Y∗ − Y. In this study, a con-trol methodology is proposed to guarantee the asymptotical sta-bility of the closed-loop control system and to achieve the per-fect tracking performance with the parameters of the IT2RFNN-AMF tuning online. To achieve this goal, a linearization tech-nique is applied to transform the nonlinear IT2RFNN-AMF intopartially linear form to obtain the expansion of Y in a Taylor

series

Y =

⎡⎢⎢⎢⎢⎣

Y1

Y2

...

Yk

⎤⎥⎥⎥⎥⎦

=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

∂Y1

∂m∂Y2

∂m...

∂Yk

∂m

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

|m=m (m∗ − m)

+

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

∂Y1

∂σ∂Y2

∂σ...

∂Yk

∂σ

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

|σ=σ(σ∗ − σ)

+

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

∂Y1

∂r∂Y2

∂r...

∂Yk

∂r

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

|r= r(r∗ − r) + N

≡ YTmm + YT

σ σ + YTr r + N (39)

where

Ym =[∂Y1

∂m∂Y2

∂m· · · ∂Yk

∂m

]|m=m ∈ R4j×k

Yσ =[∂Y1

∂σ

∂Y2

∂σ· · · ∂Yk

∂σ

]|σ=σ ∈ R4j×k,σ = σ∗−σ

Yr =[∂Y1

∂r∂Y2

∂r· · · ∂Yk

∂r

]|r= r ∈ Rj×k

m = m∗ − m, σ = σ∗ − σ, r = r∗ − r

and N is a vector of higher order terms. Rewriting (39), it canbe obtained that

UIT2RFNN-AMF = UIT2RFNN-AMF − UIT2RFNN-AMF

=12WT(Y − YT

m m − YTσ σ − YT

r r)

+12WT(YT

m m + YTσ σ + YT

r r)

+ F − UR (40)

where the unknown nonlinear function F is designed as

F =12WT N +

12WT(YT

mm∗ + YTσ σ∗ + YT

r r∗ + N)

+ ε.

(41)Theorem 1: Considering the 3-DOF dynamic model of the

gantry position stage represented by (20), if the proposed 3-DOF dynamic model-based INTSMC control law UINTSMC

in (42), shown below, which is composed of the IT2RFNN-AMF estimator UIT2RFNN-AMF with adaptive laws designed


as (43)–(48), shown below, and the robust controller Ur de-signed as (49), shown below, with adaptive law (50), shownbelow, is adopted as the control effort U of the gantry posi-tion stage, then the asymptotic stability of the proposed 3-DOFdynamic model-based INTSMC system can be guaranteed:

UINTSMC = B−1n

[Xm − AnX + λ

q

pE2− p

q

−UIT2RFNN-AMF

](42)

˙W1 = −η112λ

p

q

(ey +

1λ

epqy

)e

pq −1y

× (Y − YTmm − YT

σ σ − YTr r) (43)

˙W2 = −η112λ

p

q

(eθ +

1λ

epq

θ

)e

pq −1θ


σ σ − YTr r) (44)

˙W3 = −η112λ

p

q

(ex +

1λ

epqx

)e

pq −1x


σ σ − YTr r) (45)

˙mT

= −η212λ

p

qST diag

(E

pq −1)WT YT

m (46)

˙σT

= −η312λ

p

qST diag

(E

pq −1)WT YT

σ (47)

˙rT

= −η412λ

p

qST diag

(E

pq −1)WT YT

r (48)

Ur = −F + S (49)

˙FT

= −η512λ

p

qST diag

(E

pq −1)

(50)

where η1 , η2 , η3 , η4 , and η5 are positive constants, F is anonline estimated value of the unknown nonlinear function F,and W = [W1 W2 W3 ].

Remark: To simplify the description of the proposedIT2RFNN-AMF shown in Section III, the connected weightswere presented as general matrix form W. On the other hand,to derive the three adaptation laws of the connected weights ofthe proposed IT2RFNN-AMF, the general matrix form shouldbe represented as the individual matrix form

W =[W1 W2 W3

]=

⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣

w4R11 w4

R12 w4R13

w4R21 w4

R22 w4R23

......

...

w4Rp1 w4

Rp2 w4Rp3

w4L11 w4

L12 w4L13

w4L21 w4

L22 w4L23

......

...

w4Lp1 w4

Lp2 w4Lp3

⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦

by using linear algebra to describe the three connected weights,respectively.

Proof: To minimize the error function and to derive the adap-tive laws of W, m, σ, r, and F, the following Lyapunov functioncandidate is selected:

V =12ST S +

12η1

(WT

1 W1 + WT2 W2 + WT

3 W3

)

+1

2η2

(mT m

)+

12η3

(σT σ)

+1

2η4

(rT r)

+1

2η5FT F

(51)

where F = F − F. Then, taking the time derivative of the Lya-punov function and using (42)–(50), one can obtain

V = ST S − 1η1

(WT

1˙W1 + WT

2˙W2 + WT

3˙W3

)

− 1η2

(˙m

Tm)

− 1η3

(˙σ

Tσ)− 1

η4

(˙rTr)− 1

η5

˙FT

F

=[−1

λ

p

qST diag

(E

pq −1)(1

2WT(Y − YT

mm − YTσ σ

−YTr r)+

12WT(YT

mm − YTσ σ − YT

r r)

+ F − Ur

)]

− 1η1

(WT

1˙W1 + WT

2˙W2 + WT

3˙W3

)− 1

η2

(˙m

Tm)

− 1η3

(˙σ

Tσ)− 1

η4

(˙rTr)− 1

η5

˙FT

F

= −1λ

p

q

[12

(sy e

pq −1y WT

1 + sθ epq −1θ WT

2 + sx epq −1x WT

3

)

×(Y − YT


r r +12ST diag

(E

pq −1)

× WT

(YT


r r)

+ ST diag(E

pq −1)F − ST diag

(E

pq −1)Ur

]

− 1η1

(WT

1˙W1 + WT

2˙W2 + WT

3˙W3

)

− 1η2

(mT ˙m

)− 1

η3

(σT ˙σ)− 1

η4

(rT ˙r)− 1

η5

˙FT

F

= −1λ

p

qST diag

(E

pq −1)S ≤ 0. (52)

Since V (S(t),W, m, σ, r, F(t)) ≤ 0,V (S(t),W, m, σ, r,F(t)) is negative semidefinite [i.e., V (S(t),W,m, σ, r, F(t)) ≤ V (S(0),W, m, σ, r, F(0))], which im-plies S(t), W, m, σ, r, and F(t) is bounded. Now, define


the following term:

P (t)≡ 12λ

p

qST diag

(E

pq −1)S ≤ −V (S(t),W, m, σ, r, F(t)).

(53)Then

∫ t

0P (τ)dτ ≤ V (S(0),W, m, σ, r, F(0))

− V (S(t),W, m, σ, r, F(t)). (54)

Since V (S(0),W, m, σ, r, F(0)) is bounded andV (S(t),W, m, σ, r, F(t)) is nonincreasing and bounded,the following results can be concluded:

limt→∞

∫ t

0P (τ)dτ < ∞. (55)

Moreover, P (t) is uniformly continuous. Using Barbalat’slemma [1], the following results can be obtained:

limt→∞

P (t) = 0. (56)

Thus, it can imply that S(t), W, m, σ, r, and F(t) will con-verge to zero as t → ∞. Therefore, the 3-DOF dynamic model-based INTSMC system can guarantee the asymptotic stabilityof both the position tracking error and the synchronous errorsimultaneously, even if parametric uncertainty, external forcedisturbance, and friction force exist. On the other hand, sincethe input excitations of the proposed IT2RFNN-AMF do notsatisfy the persistency of excitation condition [1], the guaran-teed convergence of tracking error to zero does not imply theconvergence of the estimated value of the lumped uncertaintyto its real value. The physical interpretation of the persistentexcitation condition is that the input space could span the fulldimensions to guarantee the convergence of the estimated valueto its real value. However, it will result in the undesired compu-tational load due to the increase of the input space.

IV. EXPERIMENTAL RESULTS

The block diagram of the DSP-based computer-controlledgantry position stage is shown in Fig. 6. A TMS320VC33floating-point DSP is the core of the control computer. More-over, the control computer includes multichannels of analog-to-digital converters, digital-to-analog converters (DACs), par-allel input/output, and encoder interface. The resolution of thelinear scales for the permanent magnet linear synchronous mo-tor (PMLSM) is 1μm. Furthermore, the proposed control algo-rithms are realized in the DSP using the “C” and “Assembly”languages. All the programs are developed in the PC under Win-dows environment and then downloaded to the Flash ROM ofthe DSP.

The methodology proposed for the implementation of thereal-time 3-DOF dynamic model-based INTSMC system con-sists of a main program and one interrupt service routine (ISR).In the main program, parameters and input/output initializationare set first. Next, the interrupt interval for the ISR is set. Afterenabling the interrupt, the ISR with 1-ms sampling rate is used

Fig. 6. Block diagram of a DSP-based controlled gantry position stage.

Fig. 7. Experimental setup.

for the encoder interface and DACs. The ISR first reads the posi-tions of the three PMLSMs of the gantry position stage from theencoders and the velocities of the three PMLSMs of the gantryposition stage are derived from the positions. Then, the ISR cal-culates the tracking errors and generates the control efforts i∗qsx ,i∗qsy1 , and i∗qsy2 , i.e., the ui , according to the proposed controlalgorithms. Finally, the calculated commands are sent to thex-axis, y1-axis and y2-axis motor servo drives via three DACs.Photos of the experimental setup including the gantry positionstage, the PMLSM drives, and the DSP-based control computerare presented in Fig. 7. Moreover, the mechanical specificationsof the gantry position stage are shown in Table I.

A. Scanning Contour Planning

For the automatically optical inspection application, the scan-ning, typically, is carried out line by line in a zigzag contour,as shown in Fig. 8, to reduce illumination bias. The capturearea of inspection tool using CCD is 30.24 mm2 in hardware


TABLE IMECHANICAL SPECIFICATIONS OF A GANTRY POSITION STAGE

Fig. 8. Zigzag scanning contour.

TABLE IIEXPERIMENTAL CASES

setup. To investigate the effectiveness of the proposed 3-DOFdynamic model-based control system with different testing scan-ning frequency and load condition, two cases, as shown inTable II, are considered. In the low scanning frequency case(0.25 Hz), the Case 1, the trajectories of x- and y-axis areshown in Figs. 9(b), 9(c), 11(b), and 11(c), and the velocities ofx- and y- axis are 5 and 8 mm/s, respectively. The whole scan-ning area 453.6 mm2 can be obtained in this case. In the highscanning frequency case (0.5 Hz), the Case 2, the trajectories ofx- and y-axis are shown in Figs. 10(b), 10(c), 12(b), and 12(c),and the velocities of x- and y-axis are 10 and 16 mm/s, respec-tively. In this case, the whole scanning area 907.2 mm2 can beobtained.

B. Performance Measures

To measure the control performance of the proposed controlsystem, the maximum tracking error TM , the average trackingerror m, and the standard deviation of the tracking error TS forthe trajectory tracking are defined as follows [12]:

TM = maxN

|Ti(N)| , where Ti(N) = dm (N) − di(N)

i = x, y1 , y2 (57)

m =k∑

N =1

|Ti(N)| /k (58)

Fig. 9. Experimental results of 3-DOF dynamic model-based INTSMC withsymmetric MF at Case 1. (a) Tracking response of a gantry position stage. (b)Tracking response of x-axis. (c) Tracking responses of y1 -axis and y2 -axis. (d)Control effort of x-axis. (e) Control efforts of y1 -axis and y2 -axis. (f) Trackingerror of x-axis. (g) Tracking errors of y1 -axis and y2 -axis. (h) Synchronous errorof dual linear motors. (i) Output of IT2RFNN with symmetric MF of x-axis. (j)Outputs of IT2RFNN with symmetric MF of y1 -axis and y2 -axis.


Fig. 10. Experimental results of 3-DOF dynamic model-based INTSMC atCase 1. (a) Tracking response of a gantry position stage. (b) Tracking responseof x-axis. (c) Tracking responses of y1 -axis and y2 -axis. (d) Control effort ofx-axis. (e) Control efforts of y1 -axis and y2 -axis. (f) Tracking error of x-axis.(g) Tracking errors of y1 -axis and y2 -axis. (h) Synchronous error of dual linearmotors. (i) Output of IT2RFNN-AMF of x-axis. (j) Outputs of IT2RFNN-AMFof y1 -axis and y2 -axis.

TS =

√√√√ k∑N =1

(|Ti(N)| − m)2/k (59)

where k is the total number of the iterations. Moreover, to mea-sure the synchronous control performance of the proposed con-trol systems, the maximum synchronous error TMS , the averagesynchronous error mS , and the standard deviation of the syn-chronous error TSS for the trajectory tracking are defined asfollows:

TMS = maxN

|TS (N)| , where TS (N) = dy1 (N) − dy2 (N)

(60)

mS =k∑

N =1

|TS (N)| /k (61)

TSS =

√√√√ k∑N =1

(|TS (N)| − mS )2/k. (62)

The comparison of the control performance can be easilyillustrated using the aforementioned performance measures.

C. Experiment

To investigate the improved control performance of the pro-posed 3-DOF dynamic model-based INTSMC, the 3-DOF dy-namic model-based INTSMC with symmetric MF, in which theproposed IT2RFNN estimator with AMF shown in Fig. 5 isreplaced by the IT2RFNN estimator with symmetric MF, is im-plemented in the experiment for comparison of control perfor-mance. All the parameters of the 3-DOF dynamic model-basedINTSMC with symmetric MF are the same as the ones of theproposed 3-DOF dynamic model-based INTSMC. Moreover,the training-rate parameters of the weighting interval factor,mean, and standard deviation of the proposed 3-DOF dynamicmodel-based INTSMC are given as follows:

η1 = 0.065, η2 = 0.042, η3 = 0.015, η4 = 0.025

η5 = 0.025

λ = 4.2, p = 11, q = 7. (63)

Since the IT2RFNN-AMF has a complicated network struc-ture with a high computational iterative type-reduction pro-cess, in order to simplify the network structure, the developedIT2RFNN-AMF is designed only with three rules. Each rulehas two antecedent parts and one consequent part. Therefore,there are six neurons in the input layer, 18 neurons in the mem-bership layer, three neurons in the rule layer, six neurons in thetype-reduction layer, and three neurons in the output layer in theproposed IT2RFNN-AMF shown in Fig. 3. All the parametersin the proposed control system are chosen to achieve the besttransient control performance in the experiment considering therequirement of stability. The values of all the connecting weightsare initialized to zero.

Figs. 9 and 10 depict the experimental results of commandtracking due to periodical scanning reference trajectories of the


3-DOF dynamic model-based INTSMC with symmetric MF.The tracking responses of the gantry position stage at Cases 1and 2 are shown in Figs. 9(a) and 10(a); the tracking responsesof the x-axis, y1-axis, and y2-axis at Cases 1 and 2 are shown inFigs. 9(b), 9(c), 10(b), and 10(c); the control efforts of the x-axis,y1-axis, and y2-axis at Cases 1 and 2 are shown in Figs. 9(d),9(e), 10(d), and 10(e); the position tracking errors of the x-axis,y1-axis, and y2-axis at Cases 1 and 2 are shown in Figs. 9(f),9(g), 10(f), and 10(g); the synchronous errors of the dual linearmotors at Cases 1 and 2 are shown in Figs. 9(h) and 10(h);the outputs of IT2RFNN with symmetric MF of the x-axis,y1-axis, and y2-axis at Cases 1 and 2 are shown in Figs. 9(i),9(j), 10(i), and 10(j). From the experimental results, althoughfavorable tracking responses and small synchronous errors ofthe dual linear motors can be obtained at Case 1 for the 3-DOFdynamic model-based INTSMC with symmetric MF shown inFig. 9(f)–(h), the tracking error and synchronous error of the3-DOF dynamic model-based INTSMC with symmetric MF atCase 2 are degraded, as shown in Fig. 10(f)–(h).

Figs. 11 and 12 depict the experimental results of commandtracking due to periodical scanning reference trajectories of theproposed 3-DOF dynamic model-based INTSMC. The trackingresponses of the gantry position stage at Cases 1 and 2 are shownin Figs. 11(a) and 12(a); the tracking responses of the x-axis,y1-axis, and y2-axis at Cases 1 and 2 are shown in Figs. 11(b),11(c), 12(b), and 12(c); the control efforts of the x-axis, y1-axis,and y2-axis at Cases 1 and 2 are shown in Figs. 11(d), 11(e),12(d), and 12(e); the position tracking errors of the x-axis, y1-axis, and y2-axis at Cases 1 and 2 are shown in Figs. 11(f),11(g), 12(f), and 12(g); the synchronous errors of the dual linearmotors at Cases 1 and 2 are shown in Figs. 11(h) and 12(h);the outputs of the IT2RFNN with AMF of the x-axis, y1-axis,and y2-axis at Cases 1 and 2 are shown in Figs. 11(i), 11(j),12(i), and 12(j). From the experimental results, good trackingresponses and rather small synchronous errors of the dual linearmotors can be obtained at the two testing cases as shown inFigs. 11(f)–(h), 12(f)–(h). Comparing the experimental resultsshown in Figs. 9–12, the position tracking errors of the proposed3-DOF dynamic model-based INTSMC are smaller than the3-DOF dynamic model-based INTSMC with symmetric MF.Furthermore, the synchronous errors of the dual linear motorsare also much reduced for the proposed 3-DOF dynamic model-based INTSMC.

To investigate the improved control performance of the pro-posed 3-DOF dynamic model-based INTSMC, the 3-DOF dy-namic model-based NTSMC shown in Fig. 2 and the 3-DOFdynamic model-based INTSMC with T1FNN, in which theIT2RFNN-AMF shown in Fig. 5 is replaced by the T1FNN,are also implemented for the comparison of the control per-formance. All the parameters of the 3-DOF dynamic model-based NTSMC and the 3-DOF dynamic model-based INTSMCwith T1FNN are the same as the ones of the proposed 3-DOFdynamic model-based INTSMC. The performance measuresof the 3-DOF dynamic model-based NTSMC, the 3-DOF dy-namic model-based INTSMC with T1FNN, the 3-DOF dynamicmodel-based INTSMC with symmetric MF, and the proposed3-DOF dynamic model-based INTSMC for the tracking control

Fig. 11. Experimental results of 3-DOF dynamic model-based INTSMC withsymmetric MF at Case 2. (a) Tracking response of a gantry position stage.(b) Tracking response of x-axis. (c) Tracking responses of y1 -axis andy2 -axis. (d) Control effort of x-axis. (e) Control efforts of y1 -axis andy2 -axis. (f) Tracking error of x-axis. (g) Tracking errors of y1 -axis andy2 -axis. (h) Synchronous error of dual linear motors. (i) Output of IT2RFNNwith symmetric MF of x-axis. (j) Outputs of IT2RFNN with symmetric MF ofy1 -axis and y2 -axis.


Fig. 12. Experimental results of proposed 3-DOF dynamic model-basedINTSMC at Case 2. (a) Tracking response of a gantry position stage. (b) Trackingresponse of x-axis. (c) Tracking responses of y1 -axis and y2 -axis. (d) Controleffort of x-axis. (e) Control efforts of y1 -axis and y2 -axis. (f) Tracking errorof x-axis. (g) Tracking errors of y1 -axis and y2 -axis. (h) Synchronous errorof dual linear motors. (i) Output of IT2RFNN-AMF of x-axis. (j) Outputs ofIT2RFNN-AMF of y1 -axis and y2 -axis.

TABLE IIIPERFORMANCE MEASURES AT CASE 1

TABLE IVPERFORMANCE MEASURES AT CASE 2

due to the scanning reference trajectories at Cases 1 and 2 areshown in Tables III and IV, respectively. From the performancemeasures, comparing with the 3-DOF dynamic model-basedNTSMC and the 3-DOF dynamic model-based INTSMC withT1FNN, the tracking errors and the synchronous errors of the3-DOF dynamic model-based INTSMC with symmetric MFand the proposed 3-DOF dynamic model-based INTSMC areboth much reduced due to their capability to handle uncertaininformation using the type-2 FLS and their capability for cop-ping the temporal problems using the internal-feedback loopin the membership layer. Moreover, the control performance ofthe proposed 3-DOF dynamic model-based INTSMC is superiorthan the 3-DOF dynamic model-based INTSMC with symmetricMF because it directly extended the dimensions of the GaussianMF to enhance the training capability of the networks by usingAMF. Therefore, the proposed 3-DOF dynamic model-basedINTSMC system possesses both robust and accurate controlperformance for the gantry position stage.

To further investigate the effectiveness and improvement ofthe proposed 3-DOF dynamic model-based INTSMC controlsystem, the adopted intelligent estimators, the number of fuzzyrules and adjustable parameters, and the required computationaltime of various controllers implemented in this study are listedin Table V. From the controller factors shown in Table V,


TABLE VCONTROLLER FACTORS

compared with the 3-DOF dynamic model-based INTSMC withT1FNN and the 3-DOF dynamic model-based INTSMC withsymmetric MF, although the number of adjustable parameters ofthe proposed 3-DOF dynamic model-based INTSMC increases52% (121 to 184) and 84% (100 to 184), respectively, the differ-ences of the computational time are only 18% (0.625–0.738 ms)and 2.1% (0.723–0.738 ms), respectively, due to the ability tominimize the number of fuzzy rules using the AMF and the inter-val type-2 FLC. On the other hand, from the performance mea-sures at Case 2 shown in Table IV, the average of synchronouserror and the tracking errors of the x-axis, y1-axis, and y2-axis ofthe proposed 3-DOF dynamic model-based INTSMC are signif-icantly reduced by 96.6%, 72%, 67.3%, and 73%, respectively,compared with the 3-DOF dynamic model-based INTSMC withT1FNN and 81%, 29%, 57%, and 49%, respectively, comparedwith the 3-DOF dynamic model-based INTSMC with symmet-ric MF. Therefore, the proposed IT2RFNN-AMF can providesignificant improvement of the tracking performance with slightincrease of the computational effort.

V. CONCLUSION

This study, successfully, demonstrated the application of a3-DOF dynamic model-based INTSMC system to control agantry position stage composed of three PMLSMs for the preci-sion contours tracking. The proposed 3-DOF dynamic model fora gantry position stage based on Lagrangian equation consider-ing the interaxis mechanical coupling was derived first. Then,the theoretical bases and the stability analyses of the proposed3-DOF dynamic model-based INTSMC system were describedin detail. Moreover, in the proposed 3-DOF dynamic model-based INTSMC approach, the IT2RFNN-AMF estimator wasproposed to estimate the unknown lumped uncertainty of thegantry position stage online. Furthermore, the adaptive trainingalgorithms, which can train the parameters of the IT2RFNN-AMF online, were derived using the Lyapunov stability theorem.In addition, since the 3-DOF dynamic model was incorporatedinto the proposed intelligent control scheme for the gantry po-sition stage, both the position tracking error and synchronouserror will converge to zero simultaneously. Finally, some exper-imental results of optical inspection application were carried outusing different reference contours to test the effectiveness of theproposed control scheme. From the comparison of performance

measures, the proposed 3-DOF dynamic model-based INTSMCsystem possessed the most favorable control performance.

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Faa-Jeng Lin (M’93–SM’99) received the B.S. andM.S. degrees in electrical engineering from NationalCheng Kung University, Tainan, Taiwan, in 1983 and1985, respectively, and the Ph.D. degree in electri-cal engineering from National Tsing Hua University,Hsinchu, Taiwan, in 1993.

From 1993 to 2001, he was an Associate Professorand then a Professor with the Department of Electri-cal Engineering, Chung Yuan Christian University,Chung Li, Taiwan. From 2001 to 2003, he was theChairperson and a Professor with the Department of

Electrical Engineering, National Dong Hwa University, Hualien, Taiwan, wherehe was the Dean of Research and Development from 2003 to 2005 and the Deanof Academic Affairs from 2006 to 2007. He is currently the Chair Professor withthe Department of Electrical Engineering, National Central University, ChungLi. His research interests include fuzzy systems (FSs), neural networks and ge-netic algorithms’ control theories, nonlinear control theories, ac and ultrasonicmotor drives, digital signal processing-based computer control systems, powerelectronics, microgrid, and smart grids.

Dr. Lin received the Crompton Premium Best Paper Award from the Insti-tution of Electrical Engineers, U.K., in 2002; the Outstanding Research Awardfrom the National Science Council, Taiwan, in 2004 and 2010, respectively; theOutstanding Research Professor Award from the National Dong Hwa Universityin 2004; and the Outstanding Professor of Electrical Engineering Award fromthe Chinese Electrical Engineering Association, Taiwan, in 2005. He was theChair of the Power Engineering Division, National Science Council, Taiwan,and the Chair of the IEEE Industrial Electronics and Power Electronics SocietyTaipei Chapter from 2007 to 2009. He is the Chair of Task Force on FS onRenewable Energy, the FSs Technical Committee of the IEEE ComputationalIntelligence Society (CIS), and the Chair of the IEEE CIS Taipei Chapter. Healso received the Distinguished Professor Award and Chair Professor Awardfrom National Central University in 2008 and 2010, respectively. He is a Fellowof the Institution of Engineering and Technology.

Po-Huan Chou was born in Taipei, Taiwan, in 1982.He received the B.S. degree from Feng Chia Uni-versity, Taichung, Taiwan, in 2005 and the M.S. andPh.D. degrees from National Dong Hwa University,Hualien, Taiwan, in 2007 and 2011, respectively, allin electrical engineering.

He is currently with the Department of Mechatron-ics Control, Industrial Technology Research Institute,Hsinchu, Taiwan. His research interests include per-manent magnet linear synchronous motor servo drivesystems, intelligent control, digital signal processing-

based control system, and motion control.

Chin-Sheng Chen received the Ph.D. degree in me-chanical engineering from National Chiao Tung Uni-versity, Hsinchu, Taiwan, in 1999.

He was a Researcher with Sintec TechnologyCompany Ltd., from 1999 to 2000, and an R&D man-ager of TECO Electric & Machinery Company Ltd.,from 2000 to 2002. In 2002, he joined the GraduateInstitute of Automation Technology, National TaipeiUniversity of Technology, Taipei, Taiwan, as an As-sistant Professor, where he is currently an AssociateProfessor. His research interests include motion con-

trol, mechatronics, and machine version systems.

Yu-Sheng Lin was born in I-Lan, Taiwan, in 1986.He received the B.S. degree in electrical engineeringfrom Dong Hwa University, Hualien, Taiwan, in 2009and the M.S. degree in electrical engineering from theCentral University, Taoyuan, Taiwan, in 2011.

He is currently with the Department of ElectricalEngineering, National Central University, Chungli,Taiwan. His current research interests include per-manent magnet linear synchronous motor servo drivesystems, intelligent control, digital signal processing-based control systems, and motion control.

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