Enhancing precision performance of trajectory tracking controller for robot manipulators using RBFNN and adaptive bound

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Enhancing precision performance of trajectory trackingcontroller for robot manipulators using RBFNN and adaptivebound

Naveen Kumar a, Vikas Panwar b,⇑, Jin-Hwan Borm c, Jangbom Chai c

a Department of Mathematics, National Institute of Technology (NIT), Kurukshetra 136119, Haryana, Indiab Department of Applied Mathematics, Gautam Buddha University, Greater Noida 201308, Indiac Department of Mechanical Engineering, Ajou University, Suwon 443749, Republic of Korea

a r t i c l e i n f o

Keywords:Model based controllerRBF neural networkAdaptive boundReconstruction errorAsymptotically stable

a b s t r a c t

In this paper the design issues of trajectory tracking controller for robot manipulators areconsidered. The performance of classical model based controllers is reduced due to thepresence of inherently existing uncertainties in the dynamic model of the robot manipula-tor. An intermediate approach between model based controllers and neural network basedcontrollers is adopted to enhance the precision of trajectory tracking. The performance ofthe model based controller is enhanced by adding an RBF neural network and an adaptivebound part. The controller is able to learn the existing structured and unstructured uncer-tainties in the system in online manner. The RBF network learns the unknown part of therobot dynamics with no requirement of the offline training. The adaptive bound part isused to estimate the unknown bounds on unstructured uncertainties and neural networkreconstruction error. The overall system is proved to be asymptotically stable. Finally, thenumerical simulation results are produced with various controllers and the effectiveness ofthe proposed controller is shown in a comparative study for the case of a Microbot typerobot Manipulator.

� 2013 Elsevier Inc. All rights reserved.

1. Introduction

In the past three decades, the design of trajectory controllers for robot manipulators has evolved as an active area ofextensive research activities. Since the dynamics of robot manipulators are highly nonlinear in nature and involve uncertainelements such as friction, unknown payload mass and external disturbances, the researchers have investigated various con-trol schemes to achieve the precise tracking control of robot manipulators. Classically, many control schemes incorporatedPID type controllers due to their simple structure, ease of use and low cost [1,2]. Designs of enhanced version of nonlinearPID controller are also reported in the literature [3,4]. However, since the dynamics of robot manipulators are highly non-linear, the performance of linear controllers is very limited. To overcome the shortcomings of the linear controllers, modelbased nonlinear controllers like computed torque (CT) controller are also proposed [5]. Though for trajectory control of arobot manipulator, the model based control design requires an accurate dynamic model and precise parameters of themanipulator. In real world applications, every dynamic model is subject to various uncertainties of both kinds, structuredand unstructured. These uncertainties result in positioning and/or trajectory tracking errors and even cause instability ofthe system. The control of uncertain systems is usually accomplished using either an adaptive control scheme or a robust

0096-3003/$ - see front matter � 2013 Elsevier Inc. All rights reserved.http://dx.doi.org/10.1016/j.amc.2013.12.082

⇑ Corresponding author.E-mail addresses: [email protected] (N. Kumar), [email protected] (V. Panwar).

Applied Mathematics and Computation 231 (2014) 320–328

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control approach. In the robust approach, the controller has a fixed structure that yields an acceptable performance for plantsunder study [6–11]. The dynamics of the robot manipulator can be linearly parameterized. Exploiting this property manyadaptive learning control methods are proposed in literature [12,13]. However these methods require the knowledge of com-plicated regression matrices and face difficulties in case of unmodeled disturbances.

Recently, neural network technology has gained in popularity among control community due to their often cited paralleldistributed structure and learning capabilities. The neural network are extensively used to compensate for the full nonlineardynamics of robot manipulators with no knowledge of the dynamics of the system [14–16]. The ability of neural networks tocompensate for nonlinearities is due to the universal approximation property [17,18] that states that any sufficiently smoothfunction can be approximated by a suitable network for all inputs in a compact set and the resulting function reconstructionerror is bounded. The neural network weight estimates are generated using weight update law that incorporates the systemtracking error to modify the weights. The use of a prediction error containing the information for function reconstructionerror is also proposed for the purpose of weight update to achieve improved performance [19]. However, there has been lim-ited attempts to bridge the model based controllers and model learning controllers [20]. A hybrid approach is proposed fortrajectory tracking control of robot manipulators based on a model based controller and its fusion with RBF neural networkbased controller [21]. In the present study, we have modified the robust term of the controller [21] to avoid the singularitiesand we are able to enhance the precision performance of the trajectory tracking controller with the present approach.

The novelty of the paper is the development of an asymptotically stable controller combining model based techniquesand neural network based learning approach for uncertainties. Generally, all neural network based controllers assume noprior knowledge about the system dynamics which is not the actual case. We are always equipped with at least partial infor-mation about the system dynamics. In the present paper we have utilized the available information about robot dynamics topropose a new controller. The resulting system is asymptotically stable compared to uniform ultimate boundedness formany NN based controllers. The numerical simulation results are produced with various controllers and the effectivenessof the proposed controller is shown in a comparative study for the case of a Microbot type robot Manipulator.

The paper is organized as follows. Section 2 presents the dynamic model of robot manipulator and its properties. The de-tails of control system design are given in Section 3. Section 4, presents the stability analysis of the system. In Section 5, thesimulation results are presented. Finally, conclusions are drawn in Section 6.

2. Dynamic model of robot manipulator

Based on Euler–Lagrangian formulation, the motion equation of an n-link rigid, nonredundant robot manipulator can beexpressed in joint space as

MðqÞ€qþ Vmðq; _qÞ _qþ GðqÞ þ Fð _qÞ þ sd ¼ s ð1Þ

where MðqÞ 2 Rn�n represents the inertia matrix, Vmðq; _qÞ 2 Rn�n represents the centripetal-coriolis matrix, GðqÞ 2 Rn repre-sents the gravity vector, Fð _qÞ 2 Rn represents the friction, sd 2 Rn represents a general nonlinear unmodeled disturbance,s 2 Rn represents the torque input control vector and qðtÞ; _qðtÞ; €qðtÞ 2 Rn represent link position, velocity and accelerationrespectively. The following fundamental properties of dynamic model (1) and assumptions will be exploited in the subse-quent controller development.

Property 1. The inertia matrix MðqÞ is symmetric positive definite matrix and bounded above and below i.e. there exist positiveconstants aM and bM such that aM 6 MðqÞ 6 bM.

Property 2. The matrix _MðqÞ � 2Vmðq; _qÞ is skew-symmetric i.e. for any y 2 Rn we have yTð _MðqÞ � 2Vmðq; _qÞÞy ¼ 0.

Assumption 1. kFð _qÞk ¼ aþ bk _qk for some unknown positive constants a and b.

Assumption 2. ksdk 6 c for some unknown positive constants c.

3. Controller design

In this section, the aim is to design a control torque input s to enhance the precision performance of tracking even in thepresence of structured and unstructured uncertainties in the system.

3.1. Error system development

Let qdðtÞ 2 Rn denote the desired trajectory in the joint space. Define the tracking error as follows

eðtÞ ¼ qdðtÞ � qðtÞ ð2Þ

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The filtered tracking error is defined as

sðtÞ ¼ _eðtÞ þ ^eðtÞ ð3Þ

where ^ ¼ ^T 2 Rn�n and is a positive definite matrix. Differentiating (3) and using (1), the robot manipulator dynamics maybe written in terms of the filtered tracking error as

M _sðtÞ ¼ �Vms� sþ hðxÞ þ Fð _qÞ þ sd ð4Þ

where hðxÞ ¼ MðqÞð€qd þ ^ _eÞ þ Vmðq; _qÞð _qd þ ^eÞ þ GðqÞ. The function hðxÞ can be written as combination of two parts repre-senting known dynamics hðxÞ and unknown dynamics ~hðxÞ respectively i.e.

hðxÞ ¼ hðxÞ þ ~hðxÞ ð5Þ

where hðxÞ ¼ MðqÞð€qd þ ^ _eÞ þ Vmðq; _qÞð _qd þ ^eÞ þ GðqÞ and ~hðxÞ ¼ ~MðqÞð€qd þ ^ _eÞ þ ~Vmðq; _qÞð _qd þ ^eÞ þ ~GðqÞ. The vector x is gi-ven by xðtÞ ¼ ½eT _eT qT

d_qT

d€qT

d �T

3.2. RBF Neural network

In this proposed work, a radial basis function (RBF) neural network is used to compensate for the unknown dynamics part.RBF neural network with fixed centers and widths provide a good choice for approximation of unknown dynamics part [18].Thus we may have the following approximation

~hðxÞ ¼WTfðxÞ þ �ðxÞ ð6Þ

where W 2 RN�n represents the matrix of neural network weights, fð�Þ : R5n ! RN is a vector of smooth basis functions(known), �ð�Þ : R5n ! Rn represents the neural network reconstruction error and N denotes the number of nodes of the net-work. The neural network reconstruction error can me made arbitrarily small if N is sufficiently large, i.e. k�ðxÞk < �N forsome �N > 0. The vector field fðxÞ is Gaussian type functions [22] defined element wise as:

fiðxÞ ¼ exp �kx� cik2

r2i

!1;2; . . . ;N ð7Þ

The centers ci 2 R5n and width ri 2 R are predetermined and local search techniques [23] can be appropriately used to chooseci and ri. Substituting for ~hðxÞ in (4), the robot dynamics can be rewritten as:

M _sðtÞ ¼ �Vms� sþ hðxÞ þWTfðxÞ þ Fð _qÞ þ sd þ �ðxÞ ð8Þ

3.3. Adaptive bound

Using assumptions (1) and (2) in Section 2 and considering the above mentioned bound �N on neural network reconstruc-tion error, we have

kFð _qÞ þ sd þ �ðxÞk 6 aþ bk _qk þ c þ �N ð9Þ

Define b ¼ aþ bk _qk þ c þ �N as adaptive bound and it is proposed that it may be written in the following form

b ¼ Q Tðk _qkÞ/ ð10Þ

where Q 2 Rk is a known vector function of joint velocities _q and / 2 Rk is a parameter vector, for some fixed positive numberk. considering the expressions in (8)–(10) simultaneously, the following control torque input is proposed to achieve the de-sired motion trajectory

s ¼ hþ Ksþ WTfðxÞ þ b2s

bksk þ dð11Þ

where _d ¼ �cd; dð0Þ ¼ design constant > 0; c > 0. b ¼ QT /, W and / are estimated values of neural network weights andparameter vector respectively provided by tuning algorithms. K ¼ KT 2 Rn�n is a positive gain matrix.

Substituting (11) in (8), the closed loop dynamics is becomes:

M _sðtÞ ¼ �ðK þ VmÞsþ ~WTfðxÞ þ Fð _qÞ þ sd þ �ðxÞ �b2s

bksk þ dð12Þ

where ~W ¼W � W .

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4. Stability analysis

Let the robot dynamics be represented by (1) and the control inputs to the system be designed as in (11) then with theadaptation laws given in (13) and (14) for neural network weights and adaptive bound respectively, the overall resultingsystem is asymptotically stable and the filtered tracking error sðtÞ and henceforth the tracking error eðtÞ goes to zero ast !1

_W ¼ CwfðxÞsT ð13Þ

_/ ¼ C/Qksk ð14Þ

where Cw ¼ CTw 2 RN�N and C/ ¼ CT

/ 2 Rk�k are positive definite matrices.

Proof. To prove the stability of the system, consider the following Lyapunov function candidate

L ¼ 12

sT Msþ 12

trð ~WTC�1w

~WÞ þ 12

trð~/TC�1/

~/Þ þ dc

ð15Þ

where ~W ¼W � W and ~/ ¼ /� /.The time derivative of the Lyapunov function gives

_L ¼ 12

sT _Msþ sT M _sþ trð ~WTC�1w

_~WÞ þ trð~/TC�1/

_~/Þ þ_dc

ð16Þ

Substituting (12) in (16) and using the fact that _~W ¼ � _W , _~/ ¼ � _/ and _d ¼ �cd, the following expression is obtained

_L ¼ 12

sT _Ms� sTðK þ VmÞsþ sT ~WTfðxÞ þ sTðFð _qÞ þ sd þ �ðxÞÞ �b2sT s

bksk þ d� trð ~WTC�1

w_WÞ � trð~/TC�1

/_/Þ � d ð17Þ

_L ¼ 12

sTð _M � 2VmÞs� sT Ksþ sT ~WTfðxÞ þ sTðFð _qÞ þ sd þ �ðxÞÞ �b2ksk2

bksk þ d� trð ~WTC�1

w_WÞ � trð~/TC�1

/_/Þ � d ð18Þ

If the tuning algorithms in (13) and (14) are chosen and the Property 2 in Section 2 is exploited, Eq. (18) can be rewritten as

_L ¼ �sT Ksþ sT ~WTfðxÞ þ sTðFð _qÞ þ sd þ �ðxÞÞ �b2ksk2

bksk þ d� trð ~WTfðxÞsTÞ � trð~/T QkskÞ � d ð19Þ

_L ¼ �sT Ksþ sTðFð _qÞ þ sd þ �ðxÞÞ �ðQ T /Þ2ksk2

ðQT /Þksk þ d� ~/T Qksk � d ð20Þ

Finally, using (9) and definition of adaptive bound b, one can find that

sTðFð _qÞ þ sd þ �ðxÞÞ 6 kskkðFð _qÞ þ sd þ �ðxÞÞk 6 bksk ¼ Q T/ksk ¼ Q Tð/þ ~/Þksk

Using the above inequality, (20) can be rewritten as

_L 6 �sT Ksþ ðQT /Þksk � ðQT /Þ2ksk2

ðQ T /Þksk þ d� d 6 �sT Ksþ dðQT /Þksk

ðQ T /Þksk þ d� d

_L 6 �kminksk26 0 ð21Þ

where kmin denote the minimum singular value of matrix K.Since, _LðsðtÞ; ~W; ~VÞ 6 0) LðsðtÞ; ~W; ~VÞ 6 Lðsð0Þ; ~W; ~VÞ one can find that sðtÞ; ~W; ~V are bounded. Define a function

DðtÞ ¼ kminksk26 � _L. Integrating the function DðtÞ with respect to time the following inequality can be deducedZ t

0DðtÞdt 6 Lð0Þ � LðtÞ ¼ Lðsð0Þ; ~W; ~VÞ � LðsðtÞ; ~W; ~VÞ ð22Þ

Now Lðsð0Þ; ~W; ~VÞ being a bounded function and LðsðtÞ; ~W; ~VÞ being decreasing and bounded function, one can obtain the fol-lowing result

limt!1

Z t

0DðtÞdt <1 ð23Þ

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Also, sðtÞ is bounded and therefore _sðtÞ and hence _DðtÞ is bounded. This establishes the uniform continuity of DðtÞ. Using Bar-balat’s Lemma [24], it can be shown that limt!1DðtÞ ¼ 0) sðtÞ ! 0 as t !1. This proves that the overall resulting system isasymptotically stable. Moreover, the tracking error eðtÞ goes to zero according to sðtÞ ! 0 with t !1.

5. Simulation results

In this Section, the simulation results are presented for a 3 degree of freedom Microbot robot manipulator. The details ofthe dynamic model of the manipulator can be found in [25]. Let q1ðtÞ; q2ðtÞ; q3ðtÞ denote the angles of the three joints;m1;m2;m3 and l1; l2; l3 denote the masses and lengths of the links 1, 2, 3 respectively. Cubic polynomial trajectories have beenplanned as desired trajectories for periodic motion between the following set of values for joint angles and joint velocities.The angle q1ðtÞ varies between 0 and 3p

4 ; the angle q2ðtÞ varies between 0 and p4 and the angle q3ðtÞ varies between � p

2 and p4.

The joint velocities are taken as zero values at the periodic end points of the planned trajectories. The periodic time intervalis assumed to be 10 s e.g. for t 2 ½0;10� the following desired trajectory is considered

q1dðtÞ ¼ 9p400 t2 � 3p

2000 t3; q1dð0Þ ¼ 0; q1dð10Þ ¼ 3p4

_q1dð0Þ ¼ 0; _q1dð10Þ ¼ 0

q2dðtÞ ¼ 3p400 t2 � p

2000 t3; q2dð0Þ ¼ 0; q2dð10Þ ¼ p4

_q2dð0Þ ¼ 0; _q2dð10Þ ¼ 0

q3dðtÞ ¼ � p2 þ 9p

400 t2 � 3p2000 t3; q3dð0Þ ¼ � p

2 ; q3dð10Þ ¼ p4

_q3dð0Þ ¼ 0; _q3dð10Þ ¼ 0

The remaining part of planned trajectories for t 2 ð10;45� can be found similarly. It is assumed that the manipulator picks upan unknown payload at time instants t ¼ 10 s and t ¼ 30 s. It releases the payload at instants t ¼ 20 s and t ¼ 40 s. For sim-ulation purpose the payload mass is taken as 1 unit. The whole system is actually simulated for 45 s.

The friction and unknown disturbance terms are taken asFðqÞ ¼ 5 _qþ sgnð _qÞ; sd ¼ ½4cosð2tÞ; sinðtÞ þ cosð2tÞ; 2sinðtÞ�T .The following parameter values are considered for the modelm1 ¼ 10 kg; m2 ¼ 5 kg; m3 ¼ 4 kg; l1 ¼ 5 m; l2 ¼ 4 m; l3 ¼ 2:5 m; g ¼ 9:8m=sec2

0 5 10 15 20 25 30 35 40 45−0.2

−0.1

0

0.1

0.2

time (sec)

trac

king

err

or (

radi

an)

Joint angle 1

0 5 10 15 20 25 30 35 40 45−0.2

−0.1

0

0.1

0.2

time (sec)

trac

king

err

or (

radi

an)

Joint angle 2

0 5 10 15 20 25 30 35 40 45−0.2

−0.1

0

0.1

0.2

time (sec)

trac

king

err

or (

radi

an)

Joint angle 3

Fig. 1. Tracking error with CT controller.

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0 5 10 15 20 25 30 35 40 45−0.2

−0.1

0

0.1

0.2

time (sec)

trac

king

err

or (

radi

an)

Joint angle 1

0 5 10 15 20 25 30 35 40 45−0.2

−0.1

0

0.1

0.2

time (sec)

trac

king

err

or (

radi

an)

Joint angle 2

0 5 10 15 20 25 30 35 40 45−0.2

−0.1

0

0.1

0.2

time (sec)

trac

king

err

or (

radi

an)

Joint angle 3

Fig. 2. Tracking error with FFNN controller.

0 5 10 15 20 25 30 35 40 45−0.2

−0.1

0

0.1

0.2

time (sec)

trac

king

err

or (

radi

an)

Joint angle 1

0 5 10 15 20 25 30 35 40 45−0.2

−0.1

0

0.1

0.2

time (sec)

trac

king

err

or (

radi

an)

Joint angle 2

0 5 10 15 20 25 30 35 40 45−0.2

−0.1

0

0.1

0.2

time (sec)

trac

king

err

or (

radi

an)

Joint angle 3

Fig. 3. Tracking error with proposed RBFNN controller.

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0 5 10 15 20 25 30 35 40 45−0.4

−0.2

0

0.2

0.4

time (sec)

join

t vel

ociti

es (

radi

an/s

ec)

Joint 1

0 5 10 15 20 25 30 35 40 45−0.4

−0.2

0

0.2

0.4

time (sec)

join

t vel

ociti

es (

radi

an/s

ec)

Joint 2

0 5 10 15 20 25 30 35 40 45−0.4

−0.2

0

0.2

0.4

time (sec)

join

t vel

ociti

es (

radi

an/s

ec)

Joint 3

Fig. 4. Joint velocities with CT controller.

0 5 10 15 20 25 30 35 40 45−0.4

−0.2

0

0.2

0.4

time (sec)

join

t vel

ociti

es (

radi

an/s

ec)

Joint 1

0 5 10 15 20 25 30 35 40 45−0.4

−0.2

0

0.2

0.4

time (sec)

join

t vel

ociti

es (

radi

an/s

ec)

Joint 2

0 5 10 15 20 25 30 35 40 45−0.4

−0.2

0

0.2

0.4

time (sec)

join

t vel

ociti

es (

radi

an/s

ec)

Joint 3

Fig. 5. Joint velocities with FFNN controller.

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The control gain design matrices are taken to be K ¼ 10I3; ^ ¼ 10I3. The RBF neural network is composed of 10 nodes. Thecentral positions of the Gaussian functions ci are selected from ½�2;2�15 and spread factors ri are taken to be 0.2. The positivedefinite matrices are taken as Cw ¼ 10I10; C/ ¼ 5I4. The controller is simulated in MATLAB environment using ODE45 solver.The performance of the proposed controller is shown with the Figs. 1–6. The performance of Computed Torque (CT) control-ler is shown in Fig. (1). The figure shows the inability of CT controller to deal with existing uncertainties and varying payloadin the robot manipulator system. The performance of feedforward neural network (FFNN) based model free controller andthe proposed controller are shown in Figs. (2) and (3) respectively. The improvement of the controller performance is evidentin both cases. However the enhanced precision performance in the case of proposed controller is obvious by the comparisonof Figs. (2) and (3). The joint velocities are shown in Figs. 4–6 for CT controller, FFNN based model free controller and theproposed controller respectively.

A scaler valued L2 norm has been used as an performance index for an entire error curve. The L2 norm measures the root-mean-square (RMS) average of the tracking errors, which is given by

L2½e� ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1

tf � t0

Z tf

t0

keðtÞk2dt

sð24Þ

where t0; tf 2 Rþ are the initial and final times, respectively. A smaller L2½e� represents lesser tracking error and it indicatesthe better performance of the controller.

From Table 1 it is clear that the proposed controller has achieved better accuracy than FFNN based controller and the CTcontroller. The steady state error for FFNN based model free controller and the proposed controller is shown in Table 2. FromTable 2 it is clear that for the case of the proposed controller the steady state error is improved by 100 to 1000 times whencompare to FFNN based controller.

0 5 10 15 20 25 30 35 40 45−0.4

−0.2

0

0.2

0.4

time (sec)

join

t vel

ociti

es (

radi

an/s

ec)

Joint 1

0 5 10 15 20 25 30 35 40 45−0.4

−0.2

0

0.2

0.4

time (sec)

join

t vel

ociti

es (

radi

an/s

ec)

Joint 2

0 5 10 15 20 25 30 35 40 45−0.4

−0.2

0

0.2

0.4

time (sec)

join

t vel

ociti

es (

radi

an/s

ec)

Joint 3

Fig. 6. Joint velocities with proposed RBFNN controller.

Table 1Root Mean Square (RMS) average of the tracking error.

L2½e1� L2½e2� L2½e3�

CT Controller 0.0317 0.0560 0.0744FFNN Controller 0.0147 0.0157 0.0128RBFNN Controller 0.0121 0.0153 0.0103

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6. Conclusion

In this paper, a hybrid approach, consisting of a model based controller and its fusion with RBF neural network based con-troller, is proposed to enhance precision performance of trajectory tracking control of robot manipulators. The controller isable to learn the existing structured and unstructured uncertainties in the system in online manner. The RBF network learnsthe unknown part of the robot dynamics with no requirement of the offline training. The adaptive bound part is used to esti-mate the unknown bounds on unstructured uncertainties and neural network reconstruction error. The overall system isproved to be asymptotically stable using Lyapunov function generated by weighting matrices. Finally, the numerical simu-lation results are produced with various controllers and the effectiveness of the proposed controller is shown in a compar-ative study for the case of a Microbot type robot Manipulator. Through simulation study it is concluded that the proposedcontroller is able to enhance precision performance as compared to model free neural network based controller.

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Table 2Steady state error.

e1 e2 e3

FFNN Controller Oð10�5Þ Oð10�3Þ Oð10�4ÞRBFNN Controller Oð10�7Þ Oð10�6Þ Oð10�6Þ

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