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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.
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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)
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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
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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)
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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)
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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.
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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,
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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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