IntellIgent AutomAtIon And Soft ComputIng, 2016http://dx.doi.org/10.1080/10798587.2015.1121618

Two adaptive control strategies for trajectory tracking of the inertia wheel pendulum: neural networks vis à vis model regressor

Javier Moreno-Valenzuelaa, Carlos Aguilar-Avelara, Sergio Puga-Guzmánb and Víctor Santibáñezc

aInstituto politécnico nacional-CItedI, Ave. Instituto politécnico nacional 1310, nueva tijuana, tijuana, B.C., méxico, 22435; btecnológico nacional de méxico, Instituto tecnológico de tijuana, Blvd. Industrial y Av. ItR tijuana S/n, Cd. Industrial de mesa de otay, tijuana, B.C., méxico, 22500; ctecnológico nacional de méxico, Instituto tecnológico de la laguna, Blvd. Revolución y Cuauhtémoc, torreón, Coah, méxico, 27000

ABSTRACTThe problem addressed in this paper is to achieve robust motion control of the inertia wheel pendulum (IWP). Specifically, trajectory tracking control of the pendulum of the IWP under the assumption of uncertain model is discussed. Two new robust algorithms are introduced whose design departs from a model-based input-output linearization controller. Then, the control problem is firstly solved by means of an adaptive neural network-based controller and secondly by an adaptive regressor-based controller. For both controllers, rigorous analysis of the respective closed-loop system is given, where Barbalat’s lemma is used to conclude asymptotic convergence of the pendulum tracking error. In addition, the wheel velocity and adaptive extension signals are shown to be bounded. An extensive real-time experimental study validates the introduced theory, where the performance of a classical linear PID controller and the two new adaptive schemes are compared.

© 2016 tSI® press

KEYWORDSInertia wheel pendulum; motion control; Adaptive control; neural networks; model regressor; Real-time experiments

CONTACT Javier moreno-Valenzuela moreno@citedi.mx*this work was supported by SIp-Ipn, tecnm, and ConACYt projects 176587 and 134534, mexico.

1. Introduction

Recently, the interest in the control of underactuated sys-tems has increased due to their complexity and applications. Various kinds of underactuated mechanical systems such as Furuta pendulum (Åström & Furuta, 2000), pendulum on a cart (Chung & Hauser, 1995), two wheeled pendulum (Kim, Kim, & Kwak, 2005), among others, have been used as reference models in control theory. A special underactuated mechanical system is the inertia wheel pendulum (IWP), which consists in a pendulum with a wheel attached to its tip (See Fig. 1). The wheel and the pendulum rotate in parallel planes, where the rotation axis of the pendulum is passive while a DC motor applies torque directly to the rotation axis of the wheel (Fantoni & Lozano, 2001).

The IWP is a classic testbed used to test new control algo-rithms and also to develop new control strategies and novel mechatronics systems that use inertial wheels as actuators, for example, a satellite (Chou & Liaw, 2014, 2011), robotic bike (Han, An, & Lee, 2013; Kim, An, Yoo, & Lee, 2013; Lee, Han, & Lee, 2013; Miková & Curilla, 2013), robotic sphere (Biswas & Seth, 2008), multibody pendulums (Sanyal & Goswami, 2014), three-dimensional-flywheel robots (Hsiao, Tsai, Tu, & Ting, 2015), the three dimensional inertia wheel pendulum (Mayr, Spanlang, & Gattringer, 2015), mobile robots (Larimi, Zarafshan, & Moosavian, 2014) to mention a few. Theses sys-tems use the change in the momentum of the inertial wheel to control the motion configuration.

The classic control problem addressed in the IWP is the stabilization problem, where the control objective is to reg-ulate the pendulum at the upward unstable position. The swing-up problem consists in designing a controller so that

the pendulum is moving from the downward position to the upward position. Finally, motion control problem consists in designing a torque input such that the pendulum exhibits time-varying motion. Next, the literature review on the control of the IWP is presented. Besides, the role of adaptive control schemes will be highlighted.

The problems of swinging-up and regulating the pendulum at the upright unstable position have been extensively studied in literature. A variety of control schemes have been proposed, e.g., using state-feedback and passivity based approaches (Spong, Corke, & Lozano, 2001), backstepping (Rudra et al., 2013), Lyapunov-based control (Srinivas & Behera, 2008), an equivalent-input-disturbance approach (She, Zhang, Lai, & Wu, 2012), variable structure control (Aguilar, Boiko, Fridman, & Freidovich, 2012; Hernández, 2003; Jepsen, Sborg, Pedersen, & Yang, 2009), optimal control (Kadam & Seth, 2011), among others. Other interesting problems in stabilization is the consideration of input saturation, which was addressed in (Aguilar-Ibáñez, Guitiérrez-Frias, & Suárez-Castañón, 2008, 2009; Santibáñez, Kelly, & Sandoval, 2005; Ye, Wang, & Wang, 2007).

Concerning the motion control problem of pendulum-like systems, motivation of introducing an oscillatory behavior has been well explained in (Canudas-de-Wit, Espiau, & Urrea, 2002; Shiriaev, Perram, & Canudas-de-Wit, 2005). In these documents, methodologies to induce oscillatory behavior in underactuated mechanical systems were introduced, where may also be applied to the IWP. A partial feedback lineari-zation based algorithm for the stable limit cycle generation on the IWP was reported in (Andary, Chemori, & Krut, 2009). Furthermore, the reference trajectory to be tracked is

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in pendulum-type systems are the following: In the work in (Zhang &Wang, 2001), recurrent neural networks were pro-posed for real-time synthesis and auto-tuning of feedback con-trollers for nonlinear output regulation systems. The scheme was demonstrated in an inverted pendulum on a cart system. In (Matsui, Nishimura, & Peper, 2005), an online training method for a class of neural networks was proposed in order to control a cart-pendulum system. Neural networks plus PID control were developed in (Cong & Liang, 2009) with good perfor-mance in single and double pendulums. A recent research was given (Yang, Li, Cui, & Xu, 2014), where adaptive neural net-works were proposed to control a wheeled inverted pendulum. Other interesting approaches are found in (Beigi, 1997; Chen & Ge, 2013; Jang, Chung, & Jeon, 2007; Nelson & Kraft, 1994; Sazonov, Klinkhachorn, & Klein, 2003; Wang, Sun, Xiang, & Miao, 2009; Xu, Shi, Yang, & Sun, 2014). An observation is that although there have been important advances in the development of neural network-based algorithms for different pendulum-like systems, not all of them were supported by rig-orous stability analysis. Although work has been done in the application of adaptive algorithms in many types of pendulums, motion control of the IWP via adaptive control seems to be unsolved until now.

As shown in literature, most of the work in the control of the IWP is focused in the swing up and regulation problems. It should also be noted that the application of adaptive neu-ral networks and adaptive model regressor schemes to solve either regulation or motion control problems of the IWP has not been sufficiently studied. Hence the need to create motion controllers for the IWP with the advantage of the adaptive con-trol approach, either by neural networks or model regressor, motivates this research.

In this paper, two solutions for the trajectory tracking prob-lem in the IWP using new adaptive algorithms are provided: A neural network-based and a regressor-based. For the first time, this paper presents adaptive algorithms for the trajec-tory tracking of the IWP with a rigorous stability analysis. The introduced theory is validated by means of the experimental evaluation and the new controllers are compared with respect to a linear PID control. Experiments show that the pendu-lum tracking error converges to zero while the wheel velocity remains bounded for all time. The signals resulting from the adaptive extensions are also bounded.

The rest of this document is organized as follows: Section 2 is devoted to present the dynamic model of the IWP and to describe the control problem. In Section 3, the model-based controller is derived and the internal dynamics of the system is analyzed. In Section 4, the neural network-based control-ler is introduced and the corresponding convergence analysis is presented. In Section 5, the new adaptive regressor-based controller and the respective closed-loop analysis are given. Section 6 is devoted to present the experimental results. Finally, in Section 7 some concluding remarks are drawn.

2. IWP dynamic model and control goal

The dynamic model of the IWP in Euler-Langrange form is written as (Fantoni & Lozano, 2001),

where

(1)M�� + 𝐠(𝐪) + 𝐟v(��) = 𝐮,

� =

[q1q2

]and� =

[0

�

]

calculated at the beginning of each cycle in order to minimize the motor rotating speed. In (Andary, Chemori, Benoit, & Sallantin, 2012), a model-free controller for the stable limit cycle generation was presented. Again, the reference trajectory is optimized to minimize the motor rotating speed. A second order sliding mode tracking controller for IWP was presented in (Iriarte, Aguilar, & Fridman, 2013), where the desired tra-jectories are generated from a reference model. In the work reported in (Huber, Gruber, & Hofbaur, 2013), the motion con-trol problem was addressed by introducing a nonlinear model predictive scheme, which is experimentally tested to solve the swing-up and regulation problems. More recently, an intercon-nection and damping assignment passivity based controller for a class of port-controlled Hamiltonian systems, including the IWP has been proposed in (Ryalat, Laila, & Torbati, 2015). There, the motion control of the IWP is solved and numerical simulations are given.

On the other hand, adaptive control is a methodology to control a system without requiring the exact knowledge of the system. Adaptive control allows maintaining a desired level of control system performance when the parameters of the plant dynamic model are unknown. Adaptive control schemes have been applied to pendulum-type systems. Let us consider the following illustrative research examples, which clearly expose the advantge of the adaptive control in pen-dulum-type systems. An adaptive backstepping approach was proposed in (Ebrahim & Murphy, 2005) in order to robustly control an inverted pendulum. In the paper (Matsuda, Izutsu, Ishikawa, Furuta, & Åström, 2009), a study on adaptive con-trol of a cart-pendulum system was given, where the control aim is to maintain the pendulum at the upward position. The control development is based in energetic considerations while robustness tests are illustrated by simulations. In (Yang, Li, & Li, 2013), an adaptive control scheme was developed using a variable structure method to achieve motion control of a class of wheeled inverted pendulum vehicle models. More recently, an adaptive controller for a rotatory inverted pendulum was introduced in (Chen & Huang, 2014). The approach is based in a nonlinear transformation and tested by means of numer-ical simulations. It is worthwhile to say that adaptive neural networks has prompted the attention of many researchers and control practitioners. Neural networks have been applied to control nonlinear systems due to their approximation property (Haykin, 1999; Lewis, Jagannathan, & Yesildirek, 1999), which consists in the expansion of a smooth nonlinear function by means of weighted sigmoid functions plus an approximation error. Examples of application of adaptive neural networks

Figure 1. Inertia Wheel pendulum.

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InTellIGenT AuToMATIon AnD SoFT CoMPuTInG 3

are the vector of joint position and the torque input vector, respectively, being � ∈ ℝ the torque input of the wheel,

where M ∈ ℝ2×2 is the positive definite inertia matrix,

�(�) ∈ ℝ2 is the gravitational torque vector, �v(�) ∈ ℝ

2 is the vector containing the viscous friction terms of each joint, and φi with i = {1, ..., 5} are the parameters related with the phys-ical characteristics of the system. See Table 1 for the physical description of the parameters φi, where m1 and I1 represent the pendulum mass and inertia, respectively. Similarly, m2 and I2 represent the wheel mass and inertia, respectively. Moreover, l1 indicates the pendulum length and lc1 is the distance to the center of mass.

The physical meaning of the input vector � ∈ ℝ2 is that the

system is equipped with one actuator only, which delivers the torque input � ∈ ℝ.

The IWP dynamic model in (1) can be expressed as follows:

where �z(�, �) =[fz1 fz2

]T is the dynamics that is not

related with the control input, such that

and �z = [ gz1 gz2 ]T is the dynamics that is directly related with the control input, and is given by

The dynamic model (2) can be written explicitly in a space-state form as follows

M =

[�1 �2

�2 �2

],

�(�) =

[−�3g sin(q1)

0

],

𝐟v(��) =

[𝜑4q1𝜑5q2

],

(2)d

dt�� = 𝐟z(𝐪, ��) + 𝐠z� ,

(3)𝐟z(𝐪, ��) = M−1[−𝐠(𝐪) − 𝐟v(��)],

(4)�z =1

det(M)

[−�2

�1

].

(5)d

dtq1 = q1,

(6)d

dtq1 = fz1 + gz1𝜏 ,

(7)d

dtq2 = q2,

Table 1. physical description of the parameters φi.

Parameter Description unitsφ1 I1 + I2 + m1l

2c1 +m2l

21 kg ⋅m2

φ2 I2 kg ⋅m2

φ3 m1lc1 + m2l1 kg ⋅mφ4 fv1 N ⋅m ⋅ rad/sφ5 fv2 N ⋅m ⋅ rad/s

where

with

2.1. Control problem

The problem consists in designing a control signal τ(t) such that the pendulum position q1(t) tracks a desired reference trajec-tory qd(t), while the velocity of the wheel q2(t) remains bounded under the actuator constraints, this taking into account that the parameters of the dynamic model are unknown.

It is assumed that the desired trajectory qd(t) is a twice-dif-ferentiable signal and

Hence, defining the trajectory tracking error

the control signal τ(t) should guarantee

3. Model-based controller

The error dynamics is obtained by computing the first and sec-ond time-derivatives of the tracking error in ((10)), such that

Redefining the constant parameters

(8)d

dtq2 = fz2 + gz2𝜏 ,

fz1 =1

𝛽

(𝜑2𝜑5q2 − 𝜑2𝜑4q1 + g𝜑3𝜑2 sin(q1)

),

fz2 =1

𝛽

(−𝜑1𝜑5q2 + 𝜑2𝜑4q1 − g𝜑3𝜑2 sin(q1)

),

gz1 = −1

��2,

gz2 =1

��1,

� = �1�2 − �22.

(9)|qd(t)|, |qd(t)|,|||||∫

t

0

sin(qd(s))ds|||||< ∞.

(10)e1 = qd − q1,

(11)limt→∞

[e1(t)

e1(t)

]=

[0

0

].

(12)d

dte1 = e1,

(13)𝛽

𝜑2

d

dte1 =

𝛽

𝜑2

qd − 𝜑5q2 + 𝜑4q1 − g𝜑3 sin(q1) + 𝜏 .

(14)�

�2

= �1,

(15)�4 = �2,

(16)�5 = �3,

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such that the condition

is satisfied, where � =[�1 �2

]T, � =

[e1 e1 q2 q2

]T

and � =[0 g1 0 g2

]T. Then, by computing the time-

derivative of (27)-(28) and using the definitions (23)-(26), the internal dynamics is given by

The set of equations given by (23)-(24) and (31)-(32) results from the transformation of the system (23)-(26).

Given that the controller (22) guarantees that the output y(t) → 0 and y(t) → 0 when t → ∞, then the zero-dynamics is given by

In the rest of this discussion the state η2(t) is only considered, which represents the dynamics of the wheel velocity q2(t). Integrating (34) with respect to the time t > 0, yields

By choosing a desired trajectory qd(t) such that the condition (9) is satisfied for all t ≥ 0, then the solution of η2(t) in (35) satisfies |η2(t)| < ∞. Therefore, there exist enough conditions to ensure that the state η2(t) of the internal dynamics in (32) is bounded, at least for initial conditions in a compact set con-taining the origin.

4. Neural network-based controller

Given that the parameters of the dynamic model of the IWP given in (14)-(18) are unknown, in this section a neural net-work-based adaptive controller for the is proposed. This con-trol algorithm is derived taking into account the model-based controller in (22).

The output dynamics in (20)-(21) can be expressed as:

where the function

(30)y = e1,

��

���(�) = �,

(31)��1 =1

𝜃5(𝜂2 + 𝜃1y),

(32)��2 = 𝜃1(−qd −𝜃2𝜃5

𝛼(qd − y) + g

𝜃1𝜃5

𝛼sin(qd − y)).

(33)��1 =1

𝜃5𝜂2

(34)��2 = 𝜃1(−qd −𝜃2𝜃5

𝛼qd + g

𝜃1𝜃5

𝛼sin(qd)).

(35)

𝜂2(t) = 𝜂2(0) + 𝜃1(−qd(t) −𝜃2𝜃5

𝛼qd(t) + g

𝜃1𝜃5

𝛼 ∫t

0

sin(qd(s))ds).

(36)d

dty = y,

(37)𝜃1d

dty = f (�) + 𝜏 ,

(38)f (�) = 𝜃1qd − 𝜃3q2 + 𝜃2q1 − 𝜃4g sin(q1)

and defining the output function of the system as

the system (12)-(13) can be rewritten as

Taking into account the system (20)-(21), a model-based con-troller is proposed as

where kp, kdand� are positive constants. By substituting the controller (22) into (21), the output dynamics is given by

and the wheel dynamics is obtained from (7)-(8) as

The closed-loop system in terms of the output dynamics and the wheel dynamics is given by (23)-(24) and (25)-(26). It is possible to prove that y(t) and y(t) converge to zero as time t approaches to infinity, implying that the limit (11) is satisfied by the application of the model-based controller in (22).

3.1. Internal dynamics

In accordance with the Theorem 13.1 in (Khalil, 2002), the dynamic system given by the equations (23)-(24) and (25)-(26) can be represented in a normal form by using the coordinate transformation

(17)�3 = �4,

(18)�

�1

= �5,

(19)y(t) = e1(t),

(20)d

dty = y,

(21)𝜃1d

dty = 𝜃1qd + 𝜃2q1 − 𝜃3q2 − 𝜃4g sin(q1) + 𝜏 .

(22)

𝜏 = −kpy − kdy − 𝛿sign(y) − 𝜃1qd − 𝜃2q1 + 𝜃3q2 + 𝜃4g sin(q1),

(23)d

dty = y,

(24)𝜃1d

dty = −kpy − kdy − 𝛿sign(y),

(25)d

dtq2 = q2,

(26)𝜃5

d

dtq2 = −𝜃1qd − 𝜃2

𝜃1 − 𝜃5

𝜃1q1 + g𝜃4

𝜃1 − 𝜃5

𝜃1

sin(q1) − kpy − kdy − 𝛿sign(y).

(27)�1 = q2,

(28)𝜂2 =q2g2

−e1g1,

(29)y = e1,

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W is constant, then W = − W is satisfied. With W bounded in the sense ‖W‖ ≤ �, μ > 0, the output weights error dynamics is

The closed-loop system is given by the equations (31)-(32), (41)-(42) and (44).

In order to prove that the limit (11) is satisfied, the following positive definite function is proposed

where

guarantees that U is positive definite.Computing the time-derivative of U in (45), one obtains

By substituting (44) into (47) and simplifying,

(44)W = −(𝛼y + y

)N𝛔(VT ��) + 𝜅||𝛼y + y||NW .

(45)U =1

2

[y

y

]T[kp 𝛼𝜃1

𝛼𝜃1 𝜃1

][y

y

]+

1

2WTN−1W ,

(46)0 < 𝛼 <2

√kp

𝜃1

U = kpyy + 𝜃1yy + 𝛼𝜃1(yy + y2) + WTN−1 W

= −kdy2 − 𝛼kpy

2 − 𝛼kdyy + 𝛼𝜃1y2

−(𝛼y + y)𝛿sign(𝛼y + y) + (𝛼y + y)𝜀

(47)+(𝛼y + y)WT𝛔(VT ��) + WTN−1 W .

U = −

[y

y

]T[𝛼kp

1

2𝛼kd

1

2𝛼kd kd − 𝛼𝜃1

][y

y

]

−𝛿||𝛼y + y|| + (𝛼y + y)𝜀 + 𝜅||𝛼y + y||WT (W − W)

≤ −

[y

y

]T[𝛼kp

1

2𝛼kd

1

2𝛼kd kd − 𝛼𝜃1

][y

y

]

can be approximated by applying the universal approximation theorem (Haykin, 1999; Lewis et al., 1999),

where, W ∈ ℝL is the ideal output weights vector,

� = [ �1 �2 ... �L ] ∈ ℝL is the vector of activation

functions, V ∈ ℝ5×L is the constant input weights matrix,

� ∈ ℝ is the approximation error of the neural network, � = [ q1 q1 q2 qd ]T ∈ ℝ

4 is the input vector of the neu-ral network, � = [ 1 �T ]T ∈ ℝ

5 is the augmented input vec-tor of the neural network and L is the number of neurons in the neural network.

Taking into account the output dynamics (36)-(37), the fol-lowing neural network-based controller is proposed

where 𝛼, 𝛿 > 0, and

is the neural estimation of f (�) in (38), W ∈ ℝL is the esti-

mated output weights vector, wi and vij are the elements of the vector W and the matrix V, respectively. voi is the threshold value of each neuron in the hidden layer. The block diagram implementation of the neural network-based controller in (39) can be seen in Fig. 2. In the estimated function f (�) in (40), the elements of the matrix V ∈ ℝ

4×L are selected randomly.By substituting (39) into (37), the output dynamics in (36)-

(37), which is also the tracking error dynamics by the definition in (29), can be rewritten as

where W = W − W is the output weights error of the neural network.

The following adaptation law for the output weights of the neural network is proposed

where N ∈ ℝL×L is a diagonal positive definite matrix and κ is

a positive constant. Given that the ideal output weights vector

f (𝐱) = WT𝛔(VT ��) + �,

(39)𝜏 = −f (�) − kpy − kdy − 𝛿sign(𝛼y + y),

(40)f (𝐱) = WT𝛔(VT ��) =

L∑i=1

wi tanh(

4∑j=1

vijxj + voi)

(41)d

dty = y,

(42)𝜃1d

dty = WT𝛔(VT ��) − kpy − kdy − 𝛿sign(𝛼y + y) + 𝜀,

(43)W =(𝛼y + y

)N𝛔(VT ��) − 𝜅||𝛼y + y||NW ,

Figure 2. Block diagram Implementation of the neural network-based Controller.

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network-based controller, the solution η2(t) remains bounded for all t ≥ 0.

5. Regressor-based adaptive controller

In this section, a regressor-based adaptive controller for the trajectory tracking of the IWP is presented. Similarly, this con-trol algorithm is derived taking into account the model-based controller in (22).

Taking into account the output dynamics in (20)-(21), the following adaptive controller can be proposed.

where ��1(t), ��2(t), ��3(t) and ��4(t) are an estimated version of the IWP parameters θ1, θ2, θ3 and θ4 in (14)-(17). The block diagram implementation of the regressor-based controller in (51) can be seen in Fig. 3.

By defining the parameters adaptation errors as follows:

the adaptive controller in (51) can be rewritten as

Substituting this expression into the system (20)-(21), the out-put dynamics is given by

where

(51)𝜏 = −kpy − kdy − 𝛿sign(y + 𝛼y) − ��1qd − ��2q1

+ ��3q2 + ��4g sin(q1),

(52)𝜃1 = 𝜃1 − ��1,

(53)𝜃2 = 𝜃2 − ��2,

(54)𝜃3 = 𝜃3 − ��3,

(55)𝜃4 = 𝜃4 − ��4,

(56)𝜏 = −kpy − kdy − 𝛿sign(y + 𝛼y) − 𝜃1qd − 𝜃2q1 + 𝜃3q2

+ 𝜃4g sin(q1) + 𝜃1qd + 𝜃2q1 − 𝜃3q2 − 𝜃4g sin(q1).

(57)d

dty = y

(58)𝜃1d

dty = −kpy − kdy − 𝛿sign(y + 𝛼y) + Φ��,

(59)Φ =[qd q1 −q2 −g sin(q1)

],

It is possible to conclude that U ≤ 0 if the following conditions are satisfied

Thus, the function U is positive definite and U is negative sem-idefinite if (46), (48) and (49) are satisfied. Notice that (46) and (49) can be expressed as follows

Therefore, by satisfying these conditions the boundedness of the solutions y(t), y(t) ∈ ℝ and W(t) ∈ ℝ

L is ensured.By applying the Barbalat’s lemma (Khalil, 2002),

which implies that the limit (11) is satisfied.Considering the analysis of the internal dynamics for the

model-based controller presented in Section 3.1, it is also possible to conclude that, by the application of the neural

−𝜅||𝛼y + y||(‖‖W‖‖ −

𝜇

2

)2

−||(𝛼y + y)||(𝛿 − 𝜀 − 𝜅

𝜇2

4

)

≤ −

[y

y

]T[𝛼kp

1

2𝛼kd

1

2𝛼kd kd − 𝛼𝜃1

][y

y

]

−𝜅||𝛼y + y||(‖‖W‖‖ −

𝜇

2

)2

.

(48)𝛿 > 𝜀 + 𝜅𝜇2

4,

(49)0 < 𝛼 < min

⎧⎪⎨⎪⎩

kd𝜃1,

kpkd�1

4k2d + kp𝜃1

�⎫⎪⎬⎪⎭.

(50)0 < 𝛼 < min

⎧⎪⎨⎪⎩

�kp

𝜃1,kd𝜃1,

kpkd

kp𝜃1 +1

4k2d

⎫⎪⎬⎪⎭.

limt→∞

[y(t)

y(t)

]=

[0

0

],

Figure 3. Block diagram Implementation of the Regressor-based Adaptive Controller.

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is satisfied, then the function U is positive definite and U is negative semidefinite. By applying the Barbalat’s lemma,

which implies that the limit (11) is satisfied.Considering the analysis of the internal dynamics for the

model-based controller presented in Section 3.1, it is also pos-sible to conclude that, by the application of the regressor-based adaptive controller, the solution η2(t) remains bounded for all t ≥ 0.

6. Experimental results

In this section, the results of the experimental implementation of the neural network-based controller in (39) and the regres-sor-based controller in (51) are presented.

6.1. Experimental platform

The experimental tests have been conducted in an experimen-tal platform built at Instituto Politécnico Nacional-CITEDI research center.

The hardware components of the IWP experimental plat-form were as follows: PC with dual-core processor. PCI data acquisition (DAQ) board Sensoray 626. Optical encoder US Digital model S1-1,000-236-IE-B-D for the pendulum posi-tion measurement. 24 V DC motor with embedded encoder PITTMAN model 8222S003 as actuator on the rotating wheel. Servo amplifier AMC model 12A8 to power the DC motor.

The PC runs Windows XP operative system, Matlab 2007a and Simulink, which interacts with the DAQ through the Real-Time Windows Target libraries at 1 × 10−3 [s] sampling rate. Please, see Fig. 4 for an actual picture of the IWP experimental platform.

It is assumed that the control input of the system is equal to the input voltage of the servo-amplifier υ(t), as can be seen in Fig. 4. Without loss of generality, the electrical dynamics of the actuator can be neglected, and it is possible to consider that υ(t) = τ(t).

6.2. Description of the experiment

The following linear PID control has been experimentally tested for comparing the performance of the proposed adap-tive controllers:

limt→∞

[y(t)

y(t)

]=

[0

0

],

being 𝛷( ⁄qd , q1, q2, q1) the regression matrix and q(t) the adap-tation error vector.

The following positive definite function is proposed for system (57)-(58)

where � ∈ ℝ4×4 is a symmetric and positive definite matrix.

The condition 𝛼 <

√kp

𝜃1 ensures that the function U is positive

definite and radially unbounded.By computing the time-derivative of U in (61), one obtains

which suggest the adaptation law 𝛉 defined as

Applying the adaptation law (63) into the U in (62), the time-derivative of U can be upper-bounded by

where the matrix Q is symmetric and positive definite if the following condition is satisfied

Hence, if

(60)�� =[𝜃1 𝜃2 𝜃3 𝜃4

]T,

(61)U =1

2

[y

y

]T[kp 𝛼𝜃1

𝛼𝜃1 𝜃1

][y

y

]+

1

2��TΓ−1��,

U = −

[y

y

]T[𝛼kp

1

2𝛼kd

1

2𝛼kd kd − 𝛼𝜃1

][y

y

]

(62)−𝛿|y + 𝛼y| + ��T [ΦT [y + 𝛼y] + Γ−1 𝛉],

(63)𝛉 = −ΓΦT [y + 𝛼y].

(64)

U ≤ −

[y

y

]T[𝛼kp

1

2𝛼kd

1

2𝛼kd kd − 𝛼𝜃1

][y

y

]= −

[y

y

]T

Q

[y

y

].

(65)𝛼 <kpkd

kp𝜃1 + 1∕4k2d.

(66)0 < 𝛼 < min

⎧⎪⎨⎪⎩

�kp

𝜃1,kd𝜃1,

kpkd

kp𝜃1 +1

4k2d

⎫⎪⎬⎪⎭

Figure 4. description of the experimental platform. Without loss of generality, it is Assumed that the Servo-amplifier Input Voltage υ(t) is equal to the Control Input τ(t).

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Figure 6. The applied control voltage υ(t) is shown in Figure 7. Finally, in Figures 8 and 9 the estimated output weights W(t) of the neural network-based controller and the estimated param-eters q(t) of the regressor-based adaptive controller are shown, respectively.

In order to observe and compare the performance of each algorithm, the same control gains were used. In Figures 5–6 the convergence of the tracking error e1(t) = y(t) can be seen, where the performance of the proposed adaptive controllers is clearly superior than the performance of the PID algorithm. Some remaining error can be observed after some time, which is attributed to the discrete implementation of the control algorithms and the estimations of the velocity signals from the quantified position measurements. Also it is possible to observe that the neural network-based controller presents a slower adaptation with respect to the regressor-based control-ler, which may possibly be corrected by tunning the adaptation gains N for the output weights estimation. The boundedness of the wheel velocity q2(t) can be verified from the input voltage signals υ(t) in Figure 7, which confirms the introduced theoret-ical results. It is worth mentioning that despite the saturation of the applied voltage υ(t) at the beginning of the experiments, both adaptive algorithms achieve the control objective with

where

and

The desired trajectory is proposed as

with amplitude a = 0.25 [rad], angular frequency ω = 10 [rad/s] and offset b = π [rad]. With this values, the desired trajectory (69) represents a periodic oscillation around the downward pendulum position. The signal qd(t) satisfies the assumption in (9). The initial conditions for the system were q1(0) = π [rad], q2(0) = 0 [rad], q1(0) = 0 [rad/s] and q2(0) = 0 [rad/s].

The new controllers (39) and (51) were implemented with the PD gains in (68) and

For controller (39), the number of neurons in the hidden layer of the neural network was L = 10. The adaptation gains N and the constant κ for the neural network-based controller in (39) were

Finally, the adaptation gains for the regressor-based controller in ((51)) were selected as

6.3. Results

Experimental results of the implementation of the neural network-based controller (NNC) in (39), the regressor-based adaptive controller (RAC) in (51) and the PID algorithm in (67) are depicted in Figures 5–10. Five runs of each control algorithm have been carried out, where Figures 5–10 show the results of the experiment with best performance achieved by each controller.

Particularly, the time-evolution of the pendulum position q1(t) and the desired trajectory qd(t) are shown in Figure 5. The time-evolution of the output function y(t) = e1(t) is shown in

(67)𝜏 = −kpy − ki ∫t

0

y(𝜌)d𝜌 − kdy,

(68)kp = 10, kd = 3,

ki = 10.

(69)qd = a sin(�t) + b,

� = 2, � = 0.1.

N = diag10{1}, � = 0.1.

Γ = diag{0.01, 0.00001, 0.00001, 1.0}.

Figure 5. time-evolution of the Angular position q1(t) and the desired trajectory qd(t) for the neural network-based Controller in (39), the Regressor-based Adaptive Controller in (51) and the pId Algorithm in (67).

Figure 6.  time-evolution of the output function y(t) for the neural network-based Controller in (39), the Regressor-based Adaptive Controller in (51) and the pId Algorithm in (67).

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Furthermore, Table 2 shows the RMS value of the tracking error e1(t) and the RMS value of the applied control voltage υ(t) for the best run of each control algorithm. Only the data of the last five seconds of experiment have been used, i.e., the RMS values for 5 ≤ t ≤ 10 are considered. It is worth pointing out that, for the neural network-based controller, the best performance was ∥e1(t)∥RMS = 0.0105, for the regressor-based controller was ∥e1(t)∥RMS = 0.0116, and for the PID was ∥e1(t)∥RMS = 0.0809. This means that the RMS error for the NNC is 9.48\ smaller with respect to the RMS error for the RAC, the RMS error for the RAC is 85.66\ smaller with respect to the RMS error for the PID, and the RMS error for the NNC is 87.02\ smaller with respect to the RMS error for the PID. Finally, Figure 10 shows the RMS value of the tracking error e1(t) for five runs of the introduced controllers, showing that the results presented above are reliable.

7. Concluding remarks

In this work, a neural network-based controller and a regres-sor-based controller for the trajectory tracking of the IWP have been proposed. The structure of the controllers is inspired by a model-based controller derived from the input-output lineari-zation technique. The rigorous stability analysis for the output dynamics, which is equal to the tracking error dynamics, was presented. Furthermore, conditions to ensure that the inter-nal dynamics is bounded were given, which implies that the velocity of the wheel remains under the actuator constraints. The convergence of the tracking error trajectories e1(t) was ver-ified by means of experiments. The boundedness of the output weights W(t) of the neural network, the estimated parameters q(t) of the regressor-based controller, and the velocity of the wheel q2(t) were also verified.

Real-time experiments were conducted in an experimental set-up built at our laboratory. The new adaptive controllers

good performance. Also the boundedness of the estimated out-put weights wi(t) and the estimated parameters ��i(t) is verified in Figures 8–9.

Figure 7. Control Voltage υ(t) Applied to the Wheel for the neural network-based Controller in (39), the Regressor-based Adaptive Controller in (51) and the pId Algorithm in (67).

Table 2.  Comparison of the RmS Values of the tracking error and the Applied Control Voltage during the time-window 5 ≤ t ≤ 10.

PID rAC nnC‖e1(t)‖RMS 0.0809 0.0116 0.0105‖�(t)‖

RMS2.6321 3.7409 4.2025

Figure 8.  time-evolution of the estimated output Weights W(t) of the neural network Controller.

Figure 9.  time-evolution of the estimated parameters q(t) of the Regressor-based Adaptive Controller.

Figure 10. RmS Value of the error e1(t) for five Runs of each Control Algorithm.

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nested saturation functions. 2008 American Control Conference, 3434–3439. doi:10.1109/ACC.2008.4587024

Aguilar-Ibáñez, C., Guitiérrez-Frias, O., & Suárez-Castañón, M. (2009). Controlling the strongly damped inertia wheel pendulum via nested saturation functions. Computación y Sistemas, 12, 436–449.

Andary, S., Chemori, A., Benoit, M., & Sallantin, J. (2012). A dual model-free control of underactuated mechanical systems, application to the inertia wheel inverted pendulum. 2012 American Control Conference, 1029–1034. doi:10.1109/ACC.2012.6315492

Andary, S., Chemori, A., & Krut, S. (2009). Control of the underactuated inertia wheel inverted pendulum for stable limit cycle generation. Advanced Robotics, 23, 1999–2014. doi:10.1163/016918609X12529279062438.

Åström, K. J., & Furuta, K. (2000). Swinging up a pendulum by energy control. Automatica, 36, 287–295. doi:10.1016/S0005-1098(99)00140-5.

Beigi, H. S. (1997). New adaptive and learning-adaptive control techniques based on an extension of the generalized secant method. Intelligent Automation & Soft Computing, 3, 171–184. doi:10.1080/10798587.1997.10750700.

Biswas, J., & Seth, B. (2008). Dynamic stabilization of a reaction-wheel actuated wheel-robot. International Journal of Factory Automation, Robotics and Soft Computing, 4, 96–101.

Canudas-de-Wit, C., Espiau, B., & Urrea, C. (2002, July). Orbital stabilization of underactuated mechanical systems. Proc. of IFAC World Congress, Bareclona, Spain.

Chen, M., & Ge, S. S. (2013). Direct adaptive neural control for a class of uncertain nonaffine nonlinear systems based on disturbance observer. IEEE Transactions on Cybernetics, 43, 1213–1225. doi:10.1109/TSMCB.2012.2226577.

Chen, Y. F., & Huang, A. C. (2014). Adaptive control of rotary inverted pendulum system with time-varying uncertainties. Nonlinear Dynamics, 76, 95–102. doi:10.1007/s11071-013-1112-4.

Chou, M. C., & Liaw, C. M. (2011). Dynamic control and diagnostic friction estimation for an SPMSM-driven satellite reaction wheel. IEEE Transactions on Industrial Electronics, 58, 4693–4707. doi:10.1109/TIE.2011.2107712.

Chou, M. C., & Liaw, C. M. (2014). PMSM-driven satellite reaction wheel system with adjustable DC-link voltage. IEEE Transactions on Aerospace and Electronic Systems, 50, 1359–1373. doi:10.1109/TAES.2014.110694.

Chung, C. C., & Hauser, J. (1995). Nonlinear control of a swinging pendulum. Automatica, 31, 851–862. doi:10.1016/0005-1098(94)00148-C.

Cong, S., & Liang, Y. (2009). PID-like neural network nonlinear adaptive control for uncertain multivariable motion control systems. IEEE Transactions on Industrial Electronics, 56, 3872–3879. doi:10.1109/TIE.2009.2018433.

Ebrahim, A., & Murphy, G. V. (2005). Adaptive backstepping controller design of an inverted pendulum. Thirty-Seventh Southeastern Symposium on System Theory, SSST’05, 172–174. doi:10.1109/SSST.2005.1460900

Fantoni, I., & Lozano, R. (2001). Non-linear control for underactuated mechanical systems. Berlin: Springer-Verlag.

Han, I. W., An, J. W., & Lee, J. M. (2013). Balancing control of unicycle robot. Advances in Intelligent Systems and Computing, 193, 663–670. doi:10.1007/978-3-642-33926-4_63.

Haykin, S. (1999). Neural Networks: A Comprehensive Foundation. Upper Saddle River: Prentice Hall.

Hernández, V. M. (2003). A combined sliding mode generalized PI control scheme for swinging up and balancing the inertia wheel pendulum. Asian Journal of Control, 5, 620–625. doi:10.1111/j.1934-6093.2003.tb00178.x.

Hsiao, C. C., Tsai, C. E., Tu, J. Y., & Ting, Y. K. (2015). Development of a three-dimensional-flywheel robotic system. International Journal of Mechanical, Aerospace, Industrial, Mechatronic and Manufacturing Engineering, 9, 544–549. Retrieved from http://waset.org/publications/10001039

Huber, J., Gruber, C., & Hofbaur, M. (2013). Online trajectory optimization for nonlinear systems by the concept of a model control loop—Applied to the reaction wheel pendulum. IEEE International Conference on Control Applications, 2013 CCA, 935–940. doi:10.1109/CCA.2013.6662871

Iriarte, R., Aguilar, L. T., & Fridman, L. (2013). Second order sliding mode tracking controller for inertia wheel pendulum. Journal of the Franklin Institute, 350, 92–106. doi:10.1016/j.jfranklin.2012.10.013.

Jang, J. O., Chung, H. T., & Jeon, G. J. (2007). Neuro-fuzzy controller for a XY positioning table. Intelligent Automation & Soft Computing, 13, 153–169. doi:10.1080/10798587.2007.10642957.

were compared with respect to a linear PID controller. Better tracking accuracy was obtained with new adaptive control-lers, in spite of the uncertainties of the real-time experimental platform.

Disclosure statementNo potential conflict of interest was reported by the authors.

Notes on contributors

Javier Moreno-Valenzuela received a B.Sc. degree in Electronics Engineering from the Instituto Tecnológico de Culiacán, Mexico, in 1997, and a Ph.D. degree in Automatic Control from CICESE Research Center, Ensenada, Mexico, in 2002. He was a Postdoctoral Fellow at the Université de Liège, Belgium, from 2004 to 2005. He is with the Instituto Politécnico Nacional-CITEDI, Tijuana, Mexico and the author of more than 40 journal papers. He has

served as reviewer of a number of prestigious scientific journals. He was invited editor of International Journal of Advanced Robotic Systems and Mathematical Problems in Engineering. His research interests include nonlinear systems, complex systems, mechatronics and intelligent systems.

Carlos Aguilar-Avelar received a B.Sc. degree in electronics engineering with specialization in digital systems from the Instituto Tecnológico de Los Mochis, Los Mochis, Mexico, in 2011 and an M.Sc. degree in digital systems with specialization in control systems from the Instituto Politécnico Nacional-CITEDI, Tijuana, Mexico, in 2013, where he is currently pursuing a Ph.D. degree in digital systems at the Laboratory of Systems and Control. His research

interests include the analysis and control of underactuated mechatronic systems, nonlinear control, adaptive control and neural network-based control.

Sergio Puga-Guzmán received a B.Sc. degree in elec-tronics and communications engineering from the Instituto Tecnológico y de Estudios Superiores de Monterey, Monterey, Mexico, in 1988, and a Ph.D. degree digital systems with specialization in control systems from the Instituto Politécnico Nacional-CITEDI, Tijuana, Mexico, in 2014. He is currently with the Tecnológico Nacional de México, Tecnológico de Tijuana, Tijuana, Mexico. His research interests

include nonlinear systems, analysis and control of underactuated sys-tems, adaptive control, and neural network-based control.

Víctor Santibáñez received B.E. and M.Sc. degrees in electronic engineering from the Laguna Institute of Technology, Torreon, Coahuila, Mexico, and a Ph.D. degree from CICESE, Mexico, in 1977, 1984, and 1997, respectively. He has coauthored two books on robot control: Control of Robot Manipulators (in Spanish, Prentice Hall, 2003) and Control of Robot Manipulators in Joint Space (Springer, 2005). He is currently Professor with Instituto Tecnológico de la

Laguna. His research interests are robot control, nonlinear systems con-trol, fuzzy control and adaptive control.

ReferencesAguilar, L. T., Boiko, I. M., Fridman, L. M., & Freidovich, L. B. (2012).

Generating oscillations in inertia wheel pendulum via two-relay controller. International Journal of Robust and Nonlinear Control, 22, 318–330. doi:10.1002/rnc.1696.

Aguilar-Ibáñez, C., Guitiérrez-Frias, O., & Suárez-Castañón, M. (2008). Stabilization of the strongly damping inertia wheel pendulum by a

Dow

nloa

ded

by [

Javi

er M

oren

o-V

alen

zuel

a] a

t 15:

14 1

7 M

arch

201

6

InTellIGenT AuToMATIon AnD SoFT CoMPuTInG 11

Ryalat, M., Laila, D. S., & Torbati, M. M. (2015). Integral IDA-PBC and PID-like control for port-controlled Hamiltonian systems. 2015 American Control Conference, 5365–5370. doi:10.1109/ACC.2015.7172178

Santibañez, V., Kelly, R., & Sandoval, J. (2005). Control of the inertia wheel pendulum by bounded torques. 44th IEEE Conference on Decision and Control and 2005 European Control Conference, CDC-ECC’05, 8266–8270. doi:10.1109/CDC.2005.1583500

Sanyal, A. K., & Goswami, A. (2014). Dynamics and balance control of the reaction mass pendulum: A three-dimensional multibody pendulum with variable body inertia. Journal of Dynamic Systems, Measurement, and Control, 136, 1–10. doi:10.1115/1.4025607.

Sazonov, E. S., Klinkhachorn, P., & Klein, R. L. (2003). Hybrid LQG-Neural controller for inverted pendulum system. 35th Southeastern Symposium on Theory, 206–210. doi:10.1109/SSST.2003.1194559

She, J., Zhang, A., Lai, X., & Wu, M. (2012). Global stabilization of 2-DOF underactuated mechanical systems–An equivalent-input-disturbance approach. Nonlinear Dynamics, 69, 495–509. doi:10.1007/s11071-011-0280-3.

Shiriaev, A., Perram, J. W., & Canudas-de-Wit, C. (2005). Constructive tool for orbital stabilization of underactuated nonlinear systems: Virtual constraints approach. IEEE Transactions on Automatic Control, 50, 1164–1176. doi:10.1109/TAC.2005.852568.

Spong, M. W., Corke, P., & Lozano, R. (2001). Nonlinear control of the reaction wheel pendulum. Automatica, 37, 1845–1851. doi:10.1016/S0005-1098(01)00145-5.

Srinivas, K. N., & Behera, L. (2008). Swing-up control strategies for a reaction wheel pendulum. International Journal of Systems Science, 39, 1165–1177. doi:10.1080/00207720802095137.

Wang, Y., Sun, W., Xiang, Y., & Miao, S. (2009). Neural network-based robust tracking control for robots. Intelligent Automation & Soft Computing, 15, 211–222. doi:10.1080/10798587.2009.10643026.

Xu, B., Shi, Z., Yang, C., & Sun, F. (2014). Composite neural dynamic surface control of a class of uncertain nonlinear systems in strict-feedback form. IEEE Transactions on Cybernetics, 44, 2626–2634. doi:10.1109/TCYB.2014.2311824.

Yang, C., Li, Z., Cui, R., & Xu, B. (2014). Neural network-based motion control of an underactuated wheeled inverted pendulum model. IEEE Transactions on Neural Networks and Learning Systems, 25, 2004–2016. doi:10.1109/TNNLS.2014.2302475.

Yang, C., Li, Z., & Li, J. (2013). Trajectory planning and optimized adaptive control for a class of wheeled inverted pendulum vehicle models. IEEE Transactions on Cybernetics, 43, 24–36. doi:10.1109/TSMCB.2012.2198813.

Ye, H., Wang, H., & Wang, H. (2007). Stabilization of a PVTOL aircraft and an inertia wheel pendulum using saturation technique. IEEE Transactions on Control Systems Technology, 15, 1143–1150. doi:10.1109/TCST.2007.899718.

Zhang, Y., & Wang, J. (2001). Recurrent neural networks for nonlinear output regulation. Automatica, 37, 1161–1173. doi:10.1016/S0005-1098(01)00092-9.

Jepsen, F., Sborg, A., Pedersen, A. R., & Yang, Z. (2009). Development and control of an inverted pendulum driven by a reaction wheel. 2009 International Conference on Mechatronics and Automation, ICMA 2009, 2829–2834. doi:10.1109/ICMA.2009.5246460

Kadam, S. N., & Seth, B. (2011). LQR controller of one wheel robot stabilized by reaction wheel principle. 2nd International Conference on Instrumentation Control and Automation (ICA), 299–303. doi:10.1109/ICA.2011.6130176

Khalil, H. K. (2002). Nonlinear Systems. Upper Saddler River: Prentice Hall.

Kim, H. W., An, J. W., Yoo, H. D., & Lee, J. M. (2013). Balancing control of bicycle robot using PID control. 13th International Conference on Control, Automation and Systems (ICCAS 2013), 145–147. doi:10.1109/ICCAS.2013.6703879

Kim, Y., Kim, S. H., & Kwak, Y. K. (2005). Dynamic analysis of a nonholonomic two-wheeled inverted pendulum robot. Journal of Intelligent and Robotic Systems, 44, 25–46. doi:10.1007/s10846-005-9022-4.

Larimi, S. R., Zarafshan, P., & Moosavian, S. A. A. (2014). A new stabilization algorithm for a two-wheeled mobile robot aided by reaction wheel. Journal of Dynamic Systems, Measurement, and Control, 137, 1–8. doi:10.1115/1.4027852.

Lee, J., Han, S., & Lee, J. (2013). Decoupled dynamic control for pitch and roll axes of the unicycle robot. IEEE Transactions on Industrial Electronics, 60, 3814–3822. doi:10.1109/TIE.2012.2208431.

Lewis, F. L., Jagannathan, S., & Yesildirek, A. (1999). Neural network control of robot manipulators and nonlinear systems. Taylor and Francis. Oxford UK.

Matsuda, N., Izutsu, M., Ishikawa, J., Furuta, K., & Åström, K. J. (2009). Swinging-up and stabilization control based on natural frequency for pendulum systems. 2009 American Control Conference, 5291–5296. doi:10.1109/ACC.2009.5160431

Matsui, N., Nishimura, H., & Peper, F. (2005). An examination of qubit neural network in controlling an inverted pendulum. Neural Processing Letters, 22, 277–290. doi:10.1007/s11063-005-8337-2.

Mayr, J., Spanlang, F., & Gattringer, H. (2015). Mechatronic design of a self-balancing three-dimensional inertia wheel pendulum. Mechatronics, 30, September 2015, Pages 1–10 doi:10.1016/j.mechatronics.2015.04.019.

Miková, L., & Curilla, M. (2013). Possibility of the kinematics arrangement of a mobile mechatronic system. American Journal of Mechanical Engineering, 1, 390–393. doi:10.12691/ajme-1-7-47.

Nelson, J., & Kraft, L. G. (1994). Real-time control of an inverted pendulum system using complementary neural network and optimal techniques. 1994 American Control Conference, 3, 2553–2554. doi:10.1109/ACC.1994.735019

Rudra, S., Barai, R. K., Maitra, M., Mandal, D., Ghosh, S., Dam, S., Bhattacharya, P., & Dutta, A. (2013, January). Global stabilization of a flat underactuated inertia wheel: A block backstepping approach. 2013 International Conference on Computer Communication and Informatics (ICCCI), 1–4. doi:10.1109/ICCCI.2013.6466318

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