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IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY 1

Nonlinear Disturbance Observer-Based Dynamic Surface Control

of Mobile Wheeled Inverted Pendulum

Jian Huang, Member, IEEE, Songhyok Ri, Lei Liu, Yongji Wang, Jiyong Kim, and Gyongchol Pak

Abstract— In this brief, a dynamic model of a mobile wheeled

inverted pendulum (MWIP) system is improved considering fric-

tion forces, and a nonlinear disturbance observer (NDO)-based

dynamic surface controller is investigated to control the MWIP

system. Using a coordinate transformation, this non-Class-I type

underactuated system is presented as a semistrict feedback

form, which is convenient for dynamic surface controller design.

A dynamic surface controller together with an NDO is designed

to stabilize the underactuated plant. The proposed approach

can compensate the external disturbances and the model

uncertainties to improve the system performance signiﬁcantly.

The stability of the closed-loop MWIP system is proved by

Lyapunov theorem. Experiment results are presented to illustrate

the feasibility and efﬁciency of the proposed method.

Index Terms—Dynamic surface control (DSC), mobile

wheeled inverted pendulum (MWIP), nonlinear disturbance

observer (NDO), robust control, underactuated mechanical

system.

I. INTRODUCTION

IN RECENT years, many approaches have been applied in

the control of mobile wheeled inverted pendulum (MWIP),

including the feedback linearization methods [1], fuzzy

control methods [2], neural network-based methods [3],

optimized adaptive control methods [4], and robust control

approaches [5], [6]. The backstepping control methods are

also applied for controlling the MWIP systems, in which

backstepping is often used in conjunction with other control

strategies [7]. An alternative control design method called

multiple sliding surface (MSS) control was developed.

However, designing an MSS controller may lead to an

explosion of terms problem.

Manuscript received November 14, 2014; revised February 3, 2015;

accepted February 7, 2015. Manuscript received in ﬁnal form

February 11, 2015. This work was supported in part by the International

Science and Technology Cooperation Program of China through the

Precision Manufacturing Technology and Equipment for Metal Parts under

Grant 2012DFG70640, in part by the Program for New Century Excellent

Talents in University under Grant NCET-12-0214, and in part by the National

Natural Science Foundation of China under Grant 61473130. Recommended

by Associate Editor N. K. Kazantzis.

J. Huang, L. Liu, and Y. Wang are with the Key Laboratory of

Ministry of Education for Image Processing and Intelligent Control,

School of Automation, Huazhong University of Science and Technology,

Wuhan 430074, China (e-mail: huang_jan@mail.hust.edu.cn; liulei@mail.

hust.edu.cn; wangyjch@mail.hust.edu.cn).

S. Ri is with the School of Automation, Huazhong University of Science

and Technology, Wuhan 430074, China, and also with the Department of

Control Science, University of Science, Pyongyang, D.P.R. of Korea (e-mail:

rishonghyok@163.com).

J. Kim and G. Pak are with the Department of Control Science,

University of Science, Pyongyang, D.P.R. of Korea (e-mail: kimjiyong@

163.com).

Color versions of one or more of the ﬁgures in this paper are available

online at http://ieeexplore.ieee.org.

Digital Object Identiﬁer 10.1109/TCST.2015.2404897

To avoid the drawback of the MSS controller mentioned

above, a robust nonlinear control technique called Dynamic

surface control (DSC) was developed in [8]. The DSC design

requires the strict or semistrict feedback form of the

model [8], [9]. Recently, some researchers applied the

DSC technique into the control of underactuated mechanical

systems, including the underactuated marine vessels [10] and

the inertia wheel pendulum [11]. Because the dynamics of a

Class-I underactuated mechanical system may be transformed

into a cascade nonlinear system in strict feedback form

(according to [12, Lemma 1]), most of aforementioned

studies discussed only the Class-I underactuated mechanical

system as deﬁned in [12, Deﬁnition 3.9.1]. Unfortunately, the

MWIP system does not belong to the Class-I underactuated

mechanical system.

Shojaei and Shahri [13] proposed a dynamic surface

controller considering the actuator dynamics for trajectory

tracking of uncertain nonholonomic wheeled mobile robots.

In their study, however, the authors considered only two of

vehicle planar motions (yaw rotation and forward movement).

The balancing control problem of the wheeled mobile robot

was not discussed. Therefore, the dynamic model in their

study is not an underactuated mechanical system.

It should be pointed out that the balance of an MWIP system

is the prerequisite of its motion control tasks. Owing to this,

in this brief, we focus on the balancing control of the MWIP

system considering the degrees of freedom of yaw and tilt

motion. To facilitate the design of DSC for the MWIP system,

we transform the dynamics of an MWIP system into a cascade

nonlinear system in semistrict feedback form using a new

global change of coordinates. To the best our knowledge, it

might be the ﬁrst attempt of dealing with the DSC design for

the non-Class-I type underactuated systems.

It is found that using a disturbance observer can further

improve the robustness of DSC controller. Chen [14] proposed

a nonlinear disturbance observer (NDO) to cope with the

disturbance of nonlinear system. An NDO was proposed

in [15] considering both the constant and varying disturbances.

In this brief, we proposed a dynamic surface controller with

NDO (DSCNDO) for the balance control of an MWIP system.

The introduction of NDO enhances the robustness of closed-

loop MWIP system to model errors and external disturbances.

Moreover, the explosion term problem is also avoided in the

controller design.

The rest of this brief is organized as follows. In Section II,

an improved dynamic model of an MWIP system is proposed

considering friction forces and an NDO is obtained. The

detailed design procedure and stability analysis of DSCNDO

control strategy is given in Sections III and IV, respectively.

1063-6536 © 2015 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission.

See http://www.ieee.org/publications_standards/publications/rights/index.html for more information.

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.

2IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY

TAB L E I

NOTATIONS

Fig. 1. MWIP.

Section V veriﬁed the proposed methods by experiments.

Finally, we conclude our results in Section VI.

In the rest of this brief, (ˆ·)denotes a nominal value of (·).

II. SYSTEM FORMULATION

A. MWIP System Dynamic Model

Fig. 1 shows the structure of an MWIP system, where

ψrand ψlare the rotation angles of the right and left wheels,

respectively, and θis the inclination angle of the body. αis the

yaw angle of the MWIP system. To describe the parameters

of the MWIP system, some notations should be clariﬁed ﬁrst

(Fig. 1), which are listed in Table I.

Based on Euler–Lagrange equations, Pathak et al. [1]

derived a dynamic model of this system. However, they only

considered the kinetic energy and potential energy of the

whole system. In fact, the energy of the MWIP system itself

in the motion process due to factors such as friction will

dissipate. Thus, we can improve their model by considering

the dissipation energy of the whole system

D=1

2Dw˙

ψ2

r+1

2Dw˙

ψ2

l+1

2Db[(˙

θ−˙

ψr)2+(˙

θ−˙

ψl)2].

Therefore, the dynamic model of the MWIP system is

given by

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

m11 ¨

ψ+m12 cos(θ ) ¨

θ

=m12 sin(θ )( ˙

θ2+˙α2)−2Dw˙

ψ

+2Db(˙

θ−˙

ψ) +ur+ul+τext1

m12 cos(θ) ¨

ψ+m22 ¨

θ

=Ibl sin(θ ) cos(θ) ˙α2+Gbsin(θ )

−2Db(˙

θ−˙

ψ) −ur−ul+τext2

(Ibl sin2(θ ) +m33)¨α

=−2Ibl sin(θ ) cos(θ ) ˙α˙

θ−m12 sin(θ ) ˙α˙

ψ

−2b2

r2(Db+Dw)˙α+b

r(ur−ul)+τext3

(1)

where

ψ=1

2(ψr+ψl). (2)

Parameters m11,m12,m22 ,m33,Ibl ,andGbsatisfy

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎩

m11 =(mb+2mw)r2+2Iwa

m12 =mblr

m22 =mbl2+Iyyb

Gb=mbgl

Ibl =Izzb +mbl2

m33 =2Iwd+2b2

r2(Iwa+mwr2)

(3)

and τext =[τext1 τext2 τext3]Tare used to denote external

disturbances.

B. Nonlinear Disturbance Observer Design

This section illustrates the design procedure of an NDO in

the MWIP system.

To simplify the denotation, we rewrite (1) as vector form

M(q)¨q+N(q,˙q)+F(˙q)=u+τext (4)

where

q=[q1q2q3]T=[ψθα]T.

Consider that M(q)and N(q,˙q)are the corresponding

additive uncertainties presented in the model of the MWIP.

That is, we have

M(q)=ˆ

M(q)+M(q)

N(q,˙q)=ˆ

N(q,˙q)+N(q,˙q). (5)

It is assumed that model uncertainties and external

disturbances are all bounded. This makes that the lumped

disturbance vector is bounded and can be given by

τd≤

⎡

⎣

d1

d2

d3⎤

⎦

=τdmax.(6)

The effect of all dynamic uncertainties and external

disturbances is lumped into a single disturbance vector τd.

From (4), it can be seen that

ˆ

M(q)¨q+ˆ

N(q,˙q)=u+τd.(7)

To estimate the lumped disturbance τd, the NDO is designed as

˙

ˆτd=−L(q,˙q)ˆτd+L(q,˙q)( ˆ

M(q)¨q+ˆ

N(q,˙q)−u)(8)

This article has been accepted for inclusion in a future issue of this journal. Content is final as presented, with the exception of pagination.

HUANG et al.: NDO-BASED DSC OF MWIP 3

where L(q,˙q)is the observer gain matrix to be determined.

Deﬁning ˜τd=τd−ˆτdas the disturbance tracking error and

using (8), it is observed that we have

˙

ˆτd=L(q,˙q)˜τd(9)

or, equivalently

˙

˜τd=˙τd−L(q,˙q)˜τd.(10)

Let us deﬁne an auxiliary variable z=[z1z2z3]T=

ˆτd−p(q,˙q),where(d/dt)p(q,˙q)=L(q,˙q)ˆ

M(q)¨q.

Substituting it to (8), the observer can be represented by

˙z=L(q,˙q){ˆ

N(q,˙q)−u−p(q,˙q)−z}

ˆτd=z+p(q,˙q). (11)

The disturbance observer gain matrix L(q,˙q)and

vector p(q,˙q)are given by

L(q,˙q)=L(q)=X−1ˆ

M−1(q)

p(q,˙q)=p(˙q)=X−1˙q(12)

where X−1is a invertible matrix to be determined.

Lemma 1: Consider the dynamic model of the MWIP

system described by (7) in which the rate of change of

lumped disturbance is bounded. The disturbance observer is

given in (11) with the disturbance observer gain matrix L(q)

and the disturbance observer auxiliary vector p(˙q)deﬁned

in (12). The disturbance tracking error is globally uniformly

ultimately bounded if

X−1=1

2(ξ +2βσ2)I3(13)

where

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩

ξ=max ˆm12,Ibl |˙q2|

σ2=max ⎧

⎪

⎪

⎨

⎪

⎪

⎩

J1+J2+4ˆm2

12 J3

2,Ibl +ˆm33⎫

⎪

⎪

⎬

⎪

⎪

⎭

J1=ˆm2

11 +2ˆm2

12 +ˆm2

22

J2=ˆm2

11 −ˆm2

222

J3=ˆm11 +ˆm22 2.

(14)

βis the minimum convergence rate of the disturbance

tracking error, In∈Rn×nis the identity matrix.

Proof: The proof is similar to that of [15, Th. 2].

First, it is obvious from (13) that the matrix X−1is

invertible.

Second, from [15, Th. 3] it can be seen that inequality

X+XT−XT˙

ˆ

M(q)X≥(15)

is equivalent to inequality Y+YT−ξI−YTY≥0,

where Y=X−1,is a positive deﬁnite and symmetric

matrix and ξis an upper bound of ˙

ˆ

M(q). Note that we

have

˙

ˆ

M(q)=λmax(˙

ˆ

MT(q)˙

ˆ

M(q))

=max{ˆm12|˙q2|,Ibl |˙q2|} ≤ max{ˆm12,Ibl }| ˙q2|.(16)

Thus, ξis chosen to be max{ˆm12,Ibl }| ˙q2|. To achieve a

tradeoff between the accuracy of the estimations and the noise

ampliﬁcation, an optimal Ycan be chosen to be Yoptimal =

(1/2)(ξ +2βσ2)I3[15, eq. (58)]. This leads to (13) which

ensures that inequality (15) holds.

Third, we have

ˆ

M(q)=λmax(ˆ

MT(q)ˆ

M(q))

=max ⎧

⎪

⎪

⎨

⎪

⎪

⎩

J1+J2+4ˆm2

12 J3

2,Ibl +ˆm33 ⎫

⎪

⎪

⎬

⎪

⎪

⎭

.(17)

Thus, from [15, eq. (17)] the relation σ2= ˆ

M(q)holds.

Then, from (14)–(17), the ﬁrst condition of [15, Th. 2] is then

satisﬁed.

According to [15, Th. 2], the disturbance tracking errors are

globally uniformly ultimately bounded by

⎧

⎨

⎩

|˜τd1|≤ζ1

|˜τd2|≤ζ2

|˜τd3|≤ζ3.

This completes the proof.

Remark 1: Unlike the theoretical analysis in [15], the

matrix Xgiven by (13) is not constant because ξis a function

of ˙q2. This loosens the bounded condition of the NDO by

removing the assumption that the velocity vector should lie in a

bounded set [15, eq. (3)]. At the same time, the velocity vector

of a real system is normally bounded since the kinematic

energy cannot be inﬁnite. For simplicity, in the practical

controller design we still use a constant matrix Xbasedonan

assumption that there is a maximum absolution value |˙q2|max.

III. CONTROLLER DESIGN

Let us introduce the following variables:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎩

x1=(ˆm11 +ˆm12 cos(x2))x4

+(ˆm22 +ˆm12 cos(x2))x3

x2=θ

x3=˙

θ=˙x2

x4=˙

ψ

x5=˙α.

(18)

For convenience of the mathematical derivation, we intro-

duce the following notations in advance:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

Mc12 =ˆm12 cos(x2)

Ms12 =ˆm12 sin(x2)

Mc22 =ˆm22 cos(x2)

Gs12 =ˆ

Gbsin(x2)

Ibl =ˆ

Izzb +ˆmbˆ

l2

A=ˆm11 ˆm22 −ˆm2

12cos2(x2)

¯

A=ˆm11 ˆm22 −ˆm2

12

B=ˆm33 +(ˆmbˆ

l2+ˆ

Izzb)sin2(x2).

(19)

Adding the ﬁrst equation of (1) to the second,

the MWIP system model can then be rewritten

4IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY

as

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

ˆm11 ˙x4+Mc12 ˙x3=Ms12x2

3+x2

5+ur+ul+τd1

(ˆm11 +Mc12)˙x4+(ˆm22 +Mc12)˙x3

=Ms12x2

3+x2

5+Ibl sin(x2)cos(x2)x2

5

+ˆ

Gbsin(x2)+τd1+τd2

(Ibl sin2(x2)+ˆm33)˙x5=−2Ibl sin(x2)cos(x2)x3x5

−Ms12x4x5+ˆ

b

ˆr(ur−ul)+τd3.

(20)

Lemma 2: For the MWIP system (20), the global change of

coordinates (18) transforms the dynamics of the system into

a cascade nonlinear system in semistrict feedback form

˙x1=−Ms12x3x4−x2

5+Ibl sin(x2)cos(x2)x2

5

+Gs12 +τd1+τd2(21a)

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

˙

X1=d

dt x2

α

=f11x1,XT

1

f12x1,XT

1+g1

11x1,XT

1g1

12x1,XT

1

g1

21x1,XT

1g1

22x1,XT

1X2

˙

X2=d

dt x4

x5

=f21x1,XT

1,XT

2

f22x1,XT

1,XT

2+1τd,XT

1,XT

2

2τd,XT

1,XT

2

+g2

11x1,XT

1,XT

2g2

12x1,XT

1,XT

2

g2

21x1,XT

1,XT

2g2

22x1,XT

1,XT

2ur

ul

(21b)

where

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

f11x1,XT

1=x1

ˆm22 +Mc12

g1

11x1,XT

1=−ˆm11 +Mc12

ˆm22 +Mc12

g1

12x1,XT

1=0

f12x1,XT

1=0

g1

21x1,XT

1=0

g1

22x1,XT

1=1

f21x1,XT

1,XT

2

=1

Aˆm22Ms12 x1−(ˆm11+Mc12 )x4

ˆm22 +Mc12 2+x2

5

−Ms12 Ibl cos2(x2)x2

5−Mc12Gs12

g2

11x1,XT

1,XT

2=ˆm22 +Mc12

A

g2

12x1,XT

1,XT

2=ˆm22 +Mc12

A

1τd,XT

1,XT

2=1

Aˆm22τd1−Mc12τd2

f22x1,XT

1,XT

2=1

B −2Ibl cos(x2)

ˆm22 +Mc12

·[x1−(ˆm11 +Mc12)x4]− ˆm12x4sin(x2)x5

g2

21x1,XT

1,XT

2=ˆ

b

ˆrB

g2

22x1,XT

1,XT

2=− ˆ

b

ˆrB

2τd,XT

1,XT

2=τd3

B.

Similar to [9], after coordinate transformation the MWIP

system model is represented in a semistrict feedback

form as cascade of a outer (21b) and a core (21a)

subsystem.

Our purpose is to design a control urand ulforcing x2and α

to be stabilized around zero. Together with the proposed

disturbance observer, for MWIP system (21) we design a new

DSCNDO as follows:

¯u=ur

ul=uDSC +ud=urDSC

ulDSC +urd

uld (22)

where

urd

uld =⎡

⎢

⎢

⎣

−ˆm22 ˆτd1+Mc12 ˆτd2

2(ˆm22 +Mc12)−ˆr

2ˆ

bˆτd3

−ˆm22 ˆτd1+Mc12 ˆτd2

2(ˆm22 +Mc12)+ˆr

2ˆ

bˆτd3

⎤

⎥

⎥

⎦

.(23)

The pure DSC component of DSCNDO can be obtained

through the following procedure.

Step 1: Design the virtual control law ¯x4and ¯x5.

1) Deﬁne the ﬁrst dynamic surface

S1=S11

S12 =x2

α−0

0=x2

α.(24)

Then, from the ﬁrst equation of (21b) the derivative of

S1can be expressed as

˙

S1=˙

S11

˙

S12 =˙x2

˙α=x3

x5

=⎡

⎣

x1

ˆm22 +Mc12 −ˆm11 +Mc12

ˆm22 +Mc12 x4

x5⎤

⎦.(25)

2) Select the virtual control law ¯x4and ¯x5as

¯x4

¯x5=⎡

⎣

ˆm22 +Mc12

ˆm11 +Mc12 k11S11 +x1

ˆm22 +Mc12

−k12S12 ⎤

⎦(26)

where k11 >0, k12 >0.

3) Input ¯x4and ¯x5to a ﬁrst-order ﬁlter, respectively, then

we have

#T11 ˙x4d+x4d=¯x4,x4d(0)=¯x4(0),T11 >0

T12 ˙x5d+x5d=¯x5,x5d(0)=¯x5(0),T12 >0(27)

where T11 >0andT12 >0 are the ﬁlter time constants.

The ﬁlter errors are deﬁned as follows:

e=e1

e2=x4d−¯x4

x5d−¯x5.(28)

Step 2: Design the actual control law.

1) Deﬁne the second dynamic surface

S2=S21

S22 =x4−x4d

x5−x5d.(29)

HUANG et al.: NDO-BASED DSC OF MWIP 5

Fig. 2. Block diagram of the MWIP system with DSCNDO.

Then, from the second equation of (21b), (22),

(23), and (27), the derivative of S2can be expressed as

˙

S21 =1

Aˆm22Ms12 $x1−(ˆm11 +Mc12)x4

ˆm22 +Mc12 2

+x2

5%

−Ms12 Ibl cos2(x2)x2

5−Mc12Gs12

+ˆm22 +Mc12

A(urDSC +ulDSC)

+1

A(ˆm22 ˜τd1−Mc12 ˜τd2)−¯x4−x4d

T11 (30)

˙

S22 =1

B−2Ibl cos(x2)

ˆm22 +Mc12 (x1−(ˆm11 +Mc12)x4)−ˆm12 x4

×sin(x2)x5+ˆ

b

ˆrB(urDSC −ulDSC)

+˜τd3

B−¯x5−x5d

T12 .(31)

2) Select the control law uDSC as follows:

#urDSC =1

2(uA+uB)

ulDSC =1

2(uA−uB)(32)

where

uA=−ˆm22 Ms12 (x1−(ˆm11 +Mc12)x4)2

(ˆm22 +Mc12)3

+Ms12(−ˆm22 +Ibl cos2(x2))x2

5+Mc12Gs12

ˆm22 +Mc12

+A(¯x4−x4d)

(ˆm22 +Mc12)T11 −Ak21S21

ˆm22 +Mc12 (33)

uB=ˆr

ˆ

b2Ibl cos(x2)

ˆm22 +Mc12 (x1−(ˆm11 +Mc12)x4)+ˆm12 x4

×sin(x2)x5+ˆrB(¯x5−x5d)

ˆ

bT12

−ˆrBk

22S22

ˆ

b(34)

and k21 >0, k22 >0.

The whole control system block diagram is shown in Fig. 2.

IV. STABILITY ANALYSIS

The stability analysis of the whole system is concluded in

the following theorem.

Theorem 1: Considering (21) with modeling errors,

external disturbance, unknown payloads, and frictions, there

exists a set of the surface gains k11,k12,k21,andk22,the

ﬁlter time constant T11 and T12 satisfying

γ=min(a1,a2,a3,a4,a5,a6)>0,∃γ(35)

where

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

a1=k11 −ˆm11 +ˆm12

ˆm22

a2=k12 −1

a3=k21 −k11

2−ˆm11 +ˆm12

2ˆm22

a4=k22 −k12

2−1

2

a5=1

T11 −3k11

2−ˆm11 +ˆm12

2ˆm22

a6=1

T12 +k12

2−1

2

(36)

such that the NDO-based dynamic surface controller

guarantees: Based on the control law (22), (23), and (32),

all signals in the closed-loop system are uniformly and

ultimately bounded and exponentially converge to a small

ball containing the origin.

Proof: Choose the following Lyapunov function

candidate:

V=1

2ST

1S1+1

2ST

2S2+1

2e2

1+1

2e2

2.(37)

From (24)–(26), (28), and (29), the derivative of S11 can be

written as

˙

S11 =−ˆm11 +Mc12

ˆm22 +Mc12 (S21 +e1)−k11S11 (38)

and

˙

S12 =x5=S22 +e2+¯x5=S22 +e2−k12S12.(39)

Then, from (30)–(32), the derivative of S21 and S22 are

given by

˙

S21 =−k21 S21 +1

A[(ˆm22 +Mc12)˜τd1−Mc12 ˜τd2](40)

and

˙

S22 =−k22 S22 +˜τd3

B.(41)

From (21a) and (25)–(27), the derivative of e1is given by

˙e1=−e1

T11 +k11(S21 +e1)−Gs12

ˆm11 +Mc12

−τd1+τd2

ˆm11 +Mc12 +(ˆm22 +Mc12)k2

11S11

ˆm11 +Mc12

−(ˆm12 +Ibl cos(x2)) sin(x2)(S22 +e2−k12 S12)2

ˆm11 +Mc12

−Ms12(S21 +e1+k11S11)(S21 +e1)

ˆm22 +Mc12

−Ms12k11 S11(S21 +e1+k11 S11)

ˆm11 +Mc12 .(42)

In addition, from (26)–(28) and (39), the derivative of e2is

given by

˙e2=˙x5d−˙

¯x5=− e2

T12 −k12 ˙

S12

=−e2

T12 −k12S22 −k12e2+k2

12S12.(43)

6IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY

From (38)–(43) and according to the Young,s inequality,

it follows that:

˙

V≤−

k11 −ˆm11 +ˆm12

ˆm22 S2

11 −(k12 −1)S2

12

−k21 −k11

2−ˆm11 +ˆm12

2ˆm22 S2

21

−k22 −k12

2−1

2S2

22

−1

T11 −3k11

2−ˆm11 +ˆm12

2ˆm22 e2

1

−1

T12 +k12

2−1

2e2

2+ˆm22 +ˆm12

2¯

A˜τ2

d1

+ˆm12

2¯

A˜τ2

d2+˜τ2

d3

2ˆm33 +τ2

d1+τ2

d2

2ˆm11 +ϕ1(·)(44)

where ϕ1(·)is a nonnegative continuous function satisfying

0≤&&&&

k2

12S12e2−Gs12e1

ˆm11 +Mc12

+(ˆm22 +ˆm12)S2

21

2¯

A+S2

22

2ˆm33 +e2

1

ˆm11

−(ˆm12 +Ibl cos(x2)) sin(x2)e1(S22 +e2−k12S12)2

ˆm11 +Mc12

+(ˆm22 +Mc12)k2

11S11e1

ˆm11 +Mc12

−Ms12(S21 +e1+k11 S11)(S21 +e1)e1

ˆm22 +Mc12

−Ms12k11 S11(S21 +e1+k11 S11)e1

ˆm11 +Mc12 &&&&

≤ϕ1(k11,k12,S11,S12,S21,S22,e1,e2). (45)

Given any p>0, let us introduce a set

={(S11,S12,S21,S22,e1,e2):V(t)≤p}. Apparently, set

is compact in R6. Therefore, the continuous function ϕ1(·)

has a maximum, say Mon . It follows that:

˙

V≤−

k11 −ˆm11 +ˆm12

ˆm22 S2

11 −(k12 −1)S2

12

−k21 −k11

2−ˆm11 +ˆm12

2ˆm22 S2

21

−k22 −k12

2−1

2S2

22

−1

T11 −3k11

2−ˆm11 +ˆm12

2ˆm22 e2

1

−1

T12 +k12

2−1

2e2

2+ˆm22 +ˆm12

2¯

Aζ2

1

+ˆm12

2¯

Aζ2

2+ζ2

3

2ˆm33 +d2

1+d2

2

2ˆm11 +M.

If the following inequalities are satisﬁed:

ai>0,i=1,2,3,4,5,6 (46)

then, we have

˙

V≤−2γV+M1(47)

Fig. 3. Photograph of the MWIP system.

TAB L E I I

EXPERIMENTAL PARAMETERS OF MWIP SYSTEM

where

M1=M+ˆm22 +ˆm12

2¯

Aζ2

1+ˆm12

2¯

Aζ2

2

+ζ2

3

2ˆm33 +d2

1+d2

2

2ˆm11 (48)

γ=min(a1,a2,a3,a4,a5,a6)>0.(49)

After solving the differential inequality (47), we have

V(t)≤V(0)e−2γt+M1/(2γ )(1−e−2γt)∀t∈[0,∞).

By the selections of k11,k12,k21 ,andk22 and T11 and T12,

we can make γ>M1/2p. This results in ˙

V≤0onV=p.

Thus, V≤pis an invariant set, i.e., if V(0)≤pthen

V(t)≤pfor all t≥0. Therefore, V(t)is bounded. This

implies that (S11,S12,S21,S22,e1,e2)are uniformly and

ultimately bounded and exponentially converge to a small

ball containing the origin.

V. EXPERIMENT STUDY

A. Hardware Implementation

Fig. 3 shows the overall robot system used in the

experiments.

A container is ﬁxed on the base of the robot body and

a weight of 0.35 kg is placed in the box as test load.

The parameters of the MWIP mechanical platform are given

in Table II.

Fig. 4 shows the corresponding control hardware for the

MWIP, which consists of a main control circuit board, a three-

axis gyro and an accelerometer. The main control circuit board

is designed based on a 32-b ARM Cortex-M3 microcontroller

(LM3S2965, Texas Instruments), the working frequency of

which is 25 MHz. The system sampling rate was designed

as T=5ms.

HUANG et al.: NDO-BASED DSC OF MWIP 7

Fig. 4. Control hardware for the MWIP system.

TABLE III

EXPERIMENT PARAMETERS OF CASES 1TO 2

Fig. 5. Balance control results of the MWIP system by employing LQR, DSC,

and DSCNDO control strategies with nominal parameters. (a) Tilt angles.

(b) Yaw angles.

B. Experimental Results

The experiments were implemented in the practical

MWIP-based autonomous robot system. Three cases

(Case 0, Case 1, and Case 2) were studied in the experiments.

In the ﬁrst two cases, the robot was controlled to keep its

balance on the ﬂat ground with different group of parameters.

While the balance control of robot was studied on a slope

in Case 2. The physical parameters of all cases are given

in Table III.

The parameters of Case 0 were used to represent a nominal

case (Table II), from which the DSCNDO is derived. Based

on the physical parameters of MWIP-based autonomous robot

system, the parameters of proposed DSCNDO are chosen as

k11 =k12 =k21 =k22 =35

T11 =T12 =0.015,X−1=0.0345I3.

To investigate whether the proposed DSCNDO achieves

better performance in comparison with alternative control

Fig. 6. Balance control results of the MWIP system by employing LQR,

DSC, and DSCNDO control strategies with relative large mass and height of

the container. (a) Tilt angles. (b) Yaw angles.

Fig. 7. Balance control results of the MWIP system by employing

LQR, DSC, and DSCNDO control strategies with nominal parameters in

sloped plane. (a) Tilt angles. (b) Yaw angles.

approaches, an Linear Quadratic Regulator (LQR) controller

was also applied for the balance control of the robot. The

LQR control coefﬁcients were chosen as

K11 =10.1196,K12 =0.3844,K13 =1.0584,K14 =−0.3177

K21 =10.1196,K22 =0.3844,K23 =1.0584,K24 =0.3177

and Q=5I4,R=25I2. Therefore, the LQR control is given

by

uLQRr=K11θb+K12 ˙

ψ+K13 ˙

θb+K14 ˙α

uLQRl=K21θb+K22 ˙

ψ+K23 ˙

θb+K24 ˙α.

The balance control results of the robot system by

employing LQR, DSC, and DSCNDO are shown in Figs. 5–7.

The rms errors of the tilt angles and yaw angles are shown

in Table IV. From the experiment results of Cases 0 to 2,

8IEEE TRANSACTIONS ON CONTROL SYSTEMS TECHNOLOGY

TAB L E I V

RMS ERRORS OF THE TILT ANGLES AND YAW ANGLES

it turns out that the control performance and disturbance

suppression effect of the proposed DSCNDO is superior to

those of conventional LQR and DSC methods. Note that there

still exists steady-state error when using DSCNDO, which may

be caused by the time-varying model uncertainties or external

disturbances in the real robot system.

VI. CONCLUSION

Compared with pure DSC and conventional LQR

controller, the new controller presents better performance

which is veriﬁed by experiments. In summary, the major

contributions of this brief can be listed as follows.

1) Based on the improved dynamic model of the

MWIP system, novel global coordinate transformation

was proposed to achieve the DSC control of an

MWIP system. In general, it is difﬁcult to design a

DSC controller for a non-Class-I type underatcuated

mechanical system because DSC requires strict or

semistrict feedback form of the dynamic model.

2) To compensate for parametric uncertainties in a

real MWIP-based robot system as well as external

disturbances, we combined the proposed DSC controller

and an NDO.

3) Experimental studies were undertaken by employing

LQR, DSC, and DSCNDO for balance control of the

MWIP system.

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