Design and Implementation of Integral Sliding-Mode Control on an Underactuated Two-Wheeled Mobile Robot

Abstract

This paper presents a novel implementation of an integral sliding-mode controller (ISMC) on a two-wheeled mobile robot (2 WMR). The 2 WMR consists of two wheels in parallel and an inverse pendulum, which is inherently unstable. It is the first time that the sliding-mode control method is employed for real-time control of a 2 WMR platform and several critical issues are addressed. First, the 2 WMR is underactuated, which uses only one actuator to achieve position control of the wheels while balancing the pendulum around the upright position. ISMC is suitable for control of the underactuated 2 WMR, because ISMC has an extra degree of freedom in control when sliding mode is achieved. In this paper, we utilize this extra degree of freedom to implement a linear nominal controller, which is found adequate in stabilizing the sliding manifold in a range around the equilibrium. Second, the 2 WMR system is in presence of both matched and unmatched uncertainties. The implemented ISMC, with an integral sliding surface and a switching term, is able to completely nullify the influence from the matched uncertainties. The implemented linear nominal controller stabilizes the sliding manifold that is subject to unmatched uncertainties. Third, references design are addressed when implementing ISMC on the 2 WMR. The effectiveness of ISMC is verified through intensive simulation and experiment results.

Full text

IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 7, JULY 2014 3671
Design and Implementation of Integral
Sliding-Mode Control on an Underactuated
Two-Wheeled Mobile Robot
Jian-Xin Xu, Fellow, IEEE, Zhao-Qin Guo, and Tong Heng Lee
Abstract—This paper presents a novel implementation of an
integral sliding-mode controller (ISMC) on a two-wheeled mobile
robot (2 WMR). The 2 WMR consists of two wheels in parallel
and an inverse pendulum, which is inherently unstable. It is the
first time that the sliding-mode control method is employed for
real-time control of a 2 WMR platform and several critical issues
are addressed. First, the 2 WMR is underactuated, which uses only
one actuator to achieve position control of the wheels while balanc-
ing the pendulum around the upright position. ISMC is suitable
for control of the underactuated 2 WMR, because ISMC has an
extra degree of freedom in control when sliding mode is achieved.
In this paper, we utilize this extra degree of freedom to implement
a linear nominal controller, which is found adequate in stabilizing
the sliding manifold in a range around the equilibrium. Second,
the 2 WMR system is in presence of both matched and unmatched
uncertainties. The implemented ISMC, with an integral sliding
surface and a switching term, is able to completely nullify the
influence from the matched uncertainties. The implemented linear
nominal controller stabilizes the sliding manifold that is subject to
unmatched uncertainties. Third, references design are addressed
when implementing ISMC on the 2 WMR. The effectiveness of
ISMC is verified through intensive simulation and experiment
results.
Index Terms—Integral sliding-mode controller (ISMC), linear
controller, steady-state error, trajectory planning, underactuated
system.
I. INTRODUCTION
THE development and control of 2 WMR or wheeled
inverted pendulum (WIP) is a popular research topic in
recent years [1]–[14]. However, most of the published works
are based on theoretical analysis and results are obtained by
simulations. Only few researchers have implemented their pro-
posed control algorithms on real-time platforms. Prototypes and
products of two-wheeled mobile vehicle or robot have been
designed in some universities and research institutes [1]–[9].
The 2 WMR usually consists of two wheels in parallel and
an inverse pendulum. The control objective of the 2 WMR is
to perform motion control of the wheels while stabilizing the
Manuscript received September 27, 2012; revised March 15, 2013 and
June 20, 2013; accepted August 13, 2013. Date of publication September 18,
2013; date of current version January 31, 2014.
The authors are with the Graduate School for Integrative Sciences and En-
gineering, National University of Singapore, Singapore 117456, and also with
the Department of Electrical and Computer Engineering, National University
of Singapore (NUS), Singapore 117583 (e-mail: guozhaoqin@gmail.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/TIE.2013.2282594
Fig. 1. Prototype of the two-wheeled mobile robot.
pendulum around the upright position that is an unstable equi-
librium point. This type of systems that have fewer numbers of
actuators than the degrees of freedom (DOF) to be controlled
are called underactuated systems.
Due to the difference in mechanical configuration, underac-
tuated 2 WMRs can be classified into the class without input
coupling and the class with input coupling [10]. Since the ex-
isting works mostly focus on studying control of underactuated
systems without input coupling, this work is devoted to the
development and control of an underactuated 2 WMR with
input coupling. A prototype of 2 WMR is built in our lab as
shown in Fig. 1. The motor shaft coupler is fixed at the center
of the wheel and the motor housing is rigidly connected to the
pendulum, thus the torque generated by the motor directly acts
on both the wheels and the pendulum with the same size but
opposite directions, which results in the input coupling of the 2
WMR system.
Stabilizing algorithms based on Lyapunov theory, passivity,
feedback linearization, etc., are developed for underactuated
systems in absence of uncertainties [15]–[18]. The controller
design and stability prove are based on the accurate mathe-
matical models without considering any uncertainties. How-
ever, uncertainties and model mismatch between the nominal
mathematical models and the real-life plants are inevitable. Fur-
thermore, some of the control algorithms are too complicated
to be implemented. Since uncertainties could affect system
performance or even devastate system stability, researchers are
motivated to explore robust control designs for underactuated
systems with uncertainties [19]–[24].
Real-time control of 2 WMRs and similar underactuated
systems are presented in [1]–[9], [19], [20]. In [1]–[4], full-
state feedback linear controller is employed. However, the
robustness of the linear controller is limited. In [5], [6], a
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3672 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 7, JULY 2014
novel adaptive output recurrent cerebellar model articulation
controller is proposed, which is a model free design. Functions
are used to approximate the system model, thus, the designed
control algorithm is complex in mathematics and not evident
in physics idea, furthermore, there are plenty of controller pa-
rameters to be determined. In [7], a fuzzy traveling and position
control algorithm is proposed, however it is limited applicable
to the 2 WMR without input coupling. In [8], [9], two-wheeled
self-balancing vehicles are developed. The basic principle for
riding the two-wheeled vehicle is that the traveler’s body leads
forward to make the wheels accelerate and leads backward to
make the wheels slow down. Essentially, the mobility of the
scooter is not autonomous because the traveler is involved in
control.
Sliding-mode control (SMC) is a well-known robust control
approach for systems with model uncertainties and external
disturbances [19]–[31]. For control of underactuated systems,
SMC with a linear sliding surface has been proposed in [12],
[19], [20], [22]. However, the implementation of the SMC with
the linear sliding surface could be problematic. First, the sliding
surface parameters affect the system performance in a compli-
cated manner, thus, it is hard to predict the system responses
based on the information of the chosen parameters. Second,
the determination and tuning of the SMC parameters could be
challenging considering the non-affine structure of the sliding
manifold in the controller parameters. Other types of SMC
for controlling underactuated systems have been discussed in
[21], [23]–[25]. In [21], an SMC design based on the cascade
normal form is proposed, and the validity holds under certain
assumptions. However, the 2 WMR studied in our work does
not meet these assumptions. Second-order SMC designs for
underactuated systems are discussed in [23]–[25]. The design
of second-order SMC requires that the derivative of the de-
fined sliding variable is known. In [23]–[25], the SMC design
requires that the derivatives of all system states are known.
However, in this work, the derivatives of the velocity states are
not available because the 2 WMR system is in presence of both
parametric and external uncertainties.
Integral-type sliding-mode designs are proposed in [33] for
controlling systems with both matched and unmatched un-
certainties. The sliding mode exists from the very beginning,
therefore the system is more robust against perturbations than
the other SMC systems with reaching phase [33]. The ISMC
is constructed by a nominal control part and a switching term.
With the switching term, the matched uncertainties can be
perfectly rejected. With the freedom to design a nominal control
for the sliding manifold, ISMC can be easily incorporated with
other robust control methods, such as linear matrix inequality
(LMI), H∞, and linear quadratic regulator (LQR) to deal with
the unmatched uncertainties. Furthermore, ISMC provides one
more degree of freedom in choosing an appropriate projection
matrix to reduce the effect of the unmatched uncertainties.
In this paper, an ISMC is proposed for control of the 2 WMR.
First, an integral-type sliding surface is defined and the control
law is derived by using Lyapunov theory. The resulting sliding
manifold is still underactuated with a nominal controller to be
further designed. To make the control algorithm simple and
implementable, a linear controller is adopted as the nominal
Fig. 2. Model of the two-wheeled mobile robot.
controller. It is found that the linear controller is adequate to
stabilize the sliding manifold around the equilibrium.
In implementations, regulation and setpoint control of the
2 WMR are considered. In the existing works [2]–[9], the
control tasks are achieved only when the 2 WMRs are placed on
a flat surface. In this paper, the control tasks are achieved not
only when the 2 WMR is placed on a flat surface but also on
an inclined surface. The particular characteristics of the under-
actuated 2 WMR system are investigated, according to which,
references for both the wheel and the pendulum are designed.
The paper is organized as follows. In Section II, the 2 WMR
dynamic model is introduced. In Section III, the ISMC design is
detailed. In Section IV, intensive simulation investigations are
conducted to verify the effectiveness of the proposed controller.
In Section V, the implementation of ISMC on the real platform
is given. Conclusions are drawn in Section VI.
Throughout this paper, a function F(λ1,λ
2,...,λ
n)will be
written as F, where λ1,λ
2,...,λ
ncan be either parameters or
variables.
II. PROBLEM FORMULATION
A. System Model
Fig. 2 shows the model of the 2 WMR. The wheel motion
is defined along the surface. The wheels displacement and
velocity are denoted by xand ˙x, respectively, with rightward as
positive direction. θis the tilting angle of the pendulum with the
upright position as zero point and clockwise rotation as positive
direction. ˙
θis angular velocity of the pendulum. ϕis the slope
angle of the inclined road, for traveling on flat surface, ϕ=0.
fris the friction between the wheels and the ground. uis the
control input to the system and physically represents the torque
generated by the motor which acts on the wheels with clockwise
rotation as positive direction. Note that the motor driving the
wheel is directly mounted on the pendulum, there is a reaction
torque −uapplied to the pendulum. τfis the joint friction,
which also acts on both the wheel and the pendulum as τfand
−τf, respectively. Other system parameters are as: the mass of
the wheels mw=1.551 kg, the mass of the pendulum mp=
1.6kg, the rotation inertia of the wheels Iw=0.005 kg ·m2,
the rotation inertia of the pendulum Ip=0.027 kg ·m2,the

XU et al.: DESIGN AND IMPLEMENTATION OF INTEGRAL SLIDING-MODE CONTROL 3673
radius of the wheel r=0.08 m, the distance between Center
of Gravity (COG) of the pendulum and the center of the wheel
l=0.13 m, the acceleration of gravity g=9.81 m/s2.
Lagrangian mechanics method is used to derive the mathe-
matical model of the 2 WMR system, which leads to a second-
order nonlinear model given by
a¨x+b¨
θ−mplsin(θ+ϕ)˙
θ2+sinϕ(mp+mw)g
=1
r(u+τf−rfr)(1)
b¨x+c¨
θ−mplg sin θ
=−u−τf(2)
where a=mw+mp+(Iw/r2),b=mplcos(θ+ϕ)and c=
Ip+mpl2.
B. Control Objective
The control objective for the 2 WMR is to achieve setpoint
control of the wheel, while balance the pendulum at an equi-
librium (θ=θe,˙
θ=0). Define x=[x1,x
2,x
3,x
4]T=[x, ˙x,
θ, ˙
θ]Tand the reference signal for xis chosen as r=
[xr,v
r,θ
r,0]Twith ˙xr=vr. We obtain the error states as e=
[e1,e
2,e
3,e
4]T=x−r=[x1−xr,x
2−vr,x
3−θr,x
4]T.
Now the control objective is to ensure the convergence of e.
The error dynamic model of the 2 WMR is obtained as
˙
e=η(e)+g(e)[u+dm(e,t)] + du(e,t)(3)
where ηuis the system nonlinear term, dmis the lumped
matched uncertainties, duis the lumped unmatched uncertain-
ties. We have
η(e)=[e2η1(e)e4η2(e)]T
g(e)=[0 g1(e)0g2(e)]T
dm=τf
du(e,t)=[0 du1(e,t)0du2(e,t)]T
where
η1=mpl
ac −b2ce4
2sin(e3+θr+ϕ)−bg sin(e3+θr)
−c(mp+mw)gsin ϕ
ac −b2
η2=mpl
ac −b2−be4
2sin(e3+θr+ϕ)+ag sin(e3+θr)
+b(mp+mw)gsin ϕ
ac −b2
g1=1
r
c
ac −b2+b
ac −b2
g2=1
r
−b
ac −b2+−a
ac −b2
du1=−c
ac −b2fr,d
u2=b
ac −b2fr
and b=mplcos(e3+θr+ϕ).
C. Trajectory Planning
Without loss of generality, we consider a setpoint control task
for the 2 WMR, i.e., the 2 WMR is supposed to reach a desired
position xdand stop there. We simply use a linear segment
and two parabolic blends to construct a smooth trajectory for
the 2 WMR, which also yields a smooth reference signal for
the wheel velocity [10]. The reference inputs are computed
by the following equations:
vr(t)=⎧
⎪
⎨
⎪
⎩
vm
t1t, 0<t<t
1
vm,t
1≤t≤t2
vm−vm
t3−t2(t−t2),t
2≤t≤t3
0,t
3≤t≤ts
(4)
xr(t+Ts)=xr(t)+vrTs,if xr(t)<x
d
xd,if xr(t)≥xd
(5)
where xdis the desired setpoint, Tsis the sampling time.
For both simulations and experimental testings in the later
work, the parameters are specified as t1=1s, t2=15s, t3=
16 s, ts=20,vm=0.1m/s, xd=1.5m.
D. Analysis of the Pendulum Equilibrium Point
At the equilibrium point, the wheel acceleration is zero
(¨x=0), the pendulum angular velocity and acceleration are
zero ( ˙
θ=0,¨
θ=0), meanwhile the joint friction does not exist
(τf=0), the dynamic (1) and (2) become
sin ϕ(mp+mw)g=1
r(τ−rfr)
−mplg sin θ=−τ.
From the above equations, the pendulum equilibrium point is
obtained as
θe=arcsinrsin ϕ(mp+mw)g+rfr
mplg .(6)
Remark 1: The varying θeis an inherent characteristic of
the 2 WMR system with input coupling. The equilibrium θe
depends on the size of friction frand slope ϕ. When the 2
WMR travels under the same circumstance, θeis fixed and
irrelevant to controller parameters or control tasks.
Considering that our control objective is setpoint control,
the 2 WMR finally stops at the desired setpoint, thus we have
fr=0, and
θe=arcsinrsin ϕ(mp+mw)
mpl.(7)
It is reasonable to choose the reference position for the pen-
dulum as θr=θe. For 2 WMR traveling on a flat surface,
ϕ=0,wehaveθr=θe=0. For 2 WMR traveling on an
inclined surface, the value of θecan be calculated according
to (7) only when the system parameters involved are known.
In this paper, if part of the system parameters involved are
unknown, estimated values of the unknown parameters are used
to obtain the estimated equilibrium point, denoted as ˆ
θe. In such
a situation, the reference position for the pendulum is designed
as θr=ˆ
θe.

3674 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 7, JULY 2014
III. INTEGRAL SLIDING-MODE CONTROL DESIGN
Considering that the 2 WMR experiences modeling uncer-
tainties due to the unmodeled frictions and variation of system
parameters, robustness should be an important concern in the
controller design. The following nonlinear integral-type sliding
surface is proposed in [33] to handle systems with matched and
unmatched uncertainties:
σ(e,t)=se(t)−se(t0)−
t

t0
[sη(e)+sg(e)κ(e,t)] dτ =0
(8)
where κ(e,t)is a nominal control, sis a 1 ×4 projection vector
with freedom to design, and sg(e)=0. Here, we define s=
[s1,s
2,s
3,s
4], to satisfy sg(e)=0,wehavecs2−bs4=0.
A. ISMC for System With Unmodeled Frictions
First, we investigate the effect of frictions to the 2 WMR
system. From (3), we can see that the joint friction τfis a
matched uncertainty, while the ground friction fris an un-
matched uncertainty.
The control law is designed as
u(t)=κ(e,t)−ρ(e,t)sgn(sgσ)(9)
where the switching gain function is
ρ=ρm+ρu+ρ0(10)
where ρmis the upper bound of the matched uncertainty dm,ρu
is the upper bound of {sg}−1sdu, and ρ0is a positive constant.
Theorem 1: With the nonlinear integral-type sliding surface
(8) and the controller (9), the global attractiveness of the
sliding manifold is achieved. In the sliding mode, the matched
uncertainties will be completely nullified. Further, the influence
of unmatched uncertainties can be reduced with the freedom in
choosing the projection vector s.
Proof: Differentiating the sliding surface (8) with respect
to time tusing (3) one obtains
˙σ(t)=s˙
e(t)−sη(e)−sg(e)κ(e)
=sg dm+sdu
sg +u−κ.(11)
We choose a non-negative quadratic function V=σ2/2.Dif-
ferentiating Vwith respect to time tyields
˙
V=σ˙σ.
Substituting ˙σin (11) into the above we have
˙
V=σsg dm+sdu
sg +u−κ.(12)
Substituting the ISMC law (9) into the above we obtain
˙
V=σsg dm+sdu
sg −ρsgn(sgσ)
≤|σsg||dm|+



sdu
sg 



−ρ
≤−ρ0|σsg|<0.
Since σ(e(t0),t
0)=0, we can conclude that the controller (9)
using the gain function (10) guarantees that the sliding mode
σ=0can be maintained ∀t∈[t0,∞).
In the sliding mode, σ(t)=0,˙σ(t)=0, and define edas
the state vector in the sliding mode. The equivalent control is
derived from ˙σ=0, which is
ueq(t)=κ−dm−sdu
sg .
Substituting the above ueq(t)into (3), one obtains the sliding
manifold as
˙
ed(t)=η(ed)+g(ed)κ(ed)+δ(13)
where δis the resulting unmatched uncertainty and
δ=⎡
⎢
⎣
0
δ1
0
δ2
⎤
⎥
⎦=I−gs
sg du=g2du1−g1du2
s2g1+s4g2
⎡
⎢
⎣
0
s4
0
−s2
⎤
⎥
⎦.
(14)
From (3) and (13), it can be seen that the matched uncertainty
dmis completely nullified.
We can choose s2and s4to minimize the effect of the
unmatched uncertainty δin the sliding manifold. Referring to
(14), when s2=0and s4=0, the unmatched uncertainties in
the sliding manifold only exist in the wheel subsystem, when
s4=0and s2=0, the unmatched uncertainties only exist in
the pendulum subsystem. Since the pendulum subsystem is
much more sensitive to uncertainties than the wheel subsystem,
it is preferred to choose s=[0,0,0,s
4]and s4=0.
B. ISMC for System With Parameter Uncertainties
From the practical point of view, the load of the pendulum
mp,COGland slope angle of the traveling surface ϕare most
likely to vary. Define p=[mp,l,ϕ]and ˆ
pas the estimation of
p=[ˆmp,ˆ
l, ˆϕ], for the 2 WMR system with parameter uncer-
tainties, the nonlinear system term η(e)and the input vector
g(e)in (3) are expressed as η(e,p)and g(e,p). The dynamic
model of the 2 WMR system with parameter uncertainties is
expressed as
˙
e=η(e,p)+g(e,p)(u+dm)+du.(15)
Define ˆ
η(e,ˆ
p)and ˆ
g(e,ˆ
p), the estimation of η(e,p)and
g(e,p), respectively, we have the estimation errors caused by
the parameter uncertainties as Δη(e,ˆ
p,p)=η(e,p)−η(e,ˆ
p)
and Δg(e,ˆ
p,p)=g(e,p)−g(e,ˆ
p), and the system dynamic
model (15) becomes
˙
e=ˆ
η(e,ˆ
p)+Δη(e,ˆ
p,p)
+[
ˆ
g(e,ˆ
p)+Δg(e,ˆ
p,p)] (u+dm)+du.(16)

XU et al.: DESIGN AND IMPLEMENTATION OF INTEGRAL SLIDING-MODE CONTROL 3675
The integral sliding surface is designed as
σ(e,t)=se(t)−se(t0)−
t

t0
[sˆ
η(e,ˆ
p)+sˆ
g(e,ˆ
p)κ(e,t)] dτ
=0.(17)
The control law is
u(t)=κ(e,t)−ρ(e,t)sgn(sgσ)(18)
where the switching gain function is as (10) with
ρm≥|dm|(19)
ρu≥



sdu
sg 



+



sΔη
sg 



+



sΔgκ
sg 



.(20)
In (18), although the vector g(e,p)is unknown, we can
choose appropriate projection vector sto make the sign of
sg(e,p)be available and fixed. For instance, by choosing s=
[0,0,0,s
4],wehave
sgn [sg(e,p)] = sgn [s4·g2(e,p)] .
Refer to the expression of g2in Section II-B, we have
g2=−a+b
r1
aIp+mpl2(mw+Iw/r2)<0
for (θ+ϕ)∈(−π/2,π/2), thus
sgn(sg)=−sgn(s4)
which is known and irrelevant to the system parameters.
Theorem 2: For system with parameter uncertainties, the
sliding surface (17) and the ISMC (18) with the new switching
gain guarantee the existence of the sliding mode. In the sliding
mode, the desirable properties stated in Theorem 1 also hold.
Proof: Differentiating the sliding surface (17) with re-
spect to time using (16) one obtains
˙σ(t)=s˙
e(t)−sˆη(e,ˆ
p)−sˆ
g(e,ˆ
p)κ(e,t)
=sg (u+dm)+s(du+Δη)−sˆ
gκ
s(ˆ
g+Δg).(21)
We choose a non-negative quadratic function V=σ2/2.Dif-
ferentiating Vwith respect to time tyields
˙
V=σ˙σ=sgσ(u+dm)+ s(du+Δη)−sˆ
gκ
s(ˆ
g+Δg).
Substituting the control law (18) into the above we obtain
˙
V=sgσ[κ−ρsgn(sgσ)+dm]+ s(du+Δη)−sˆ
gκ
s(ˆ
g+Δg)
=sgσdm−ρsgn(sgσ)+ s(du+Δη)
sg +sΔgκ
sg 
≤|sgσ||dm|+



sdu
sg 



+



sΔη
sg 



+



sΔgκ
sg 



−ρ
≤−ρ0|sgσ|<0.
Since σ(x(t0),t
0)=0, we can conclude that the sliding mode
σ=0can be maintained ∀t∈[t0,∞).
In the sliding mode, the equivalent control is derived from
˙σ=0, which is
ueq(t)= sˆ
g
sg κ−dm−s(du+Δη)
sg .
Substituting the above ueq(t)into (16), one obtains the sliding
manifold as
˙
ed(t)=ˆη(ed)+ˆ
g(ed)κ(ed)+δ(22)
with
δ=I−gs
sg [Δη+du−ˆ
gκ].(23)

The parametric uncertainties, resulting from the variation of
mp,land ϕ, would not destroy the stability of the closed-
loop system owing to the robustness of the ISMC. However,
those parametric uncertainties would affect the steady-state
responses, due to their unmatched nature. The detailed analysis
of the steady-state responses is provided in the later work.
C. Linear Controller Design for the Sliding Manifold
To stabilize the obtained sliding manifold (13) or (22), which
is still nonlinear and underactuated, the nominal controller
κshould be further designed. In this paper, considering the
feasibility and simpleness in real implementation, a linear full
state feedback controller is adopted as
κ=−kTe(24)
where k=[k1,k
2,k
3,k
4]T.
The linear controller design is based on a linearized dynamic
model at the desired equilibrium point by assuming sin e3≈e3,
e2
4≈0and cos e3≈1.
Remark 2: The linearization assumptions stand when e3
and e4are small enough. For 2 WMR traveling on an in-
clined surface with unknown slope angle, when the pendulum
stays around the equilibrium, e3≈θe−θr=θe−ˆ
θe, where
ˆ
θehighly depends on the estimation accuracy of ϕ. Thus, to
make the linearization assumptions stand, it should be assumed
that the difference between the estimation and the actual value
of the slope angle is small enough.
The linearized dynamic model is as
˙
e=A0e+g0(κ+ηm)(25)
where
A0=⎡
⎢
⎢
⎣
01 0 0
00−b0mplg cos θr
ac−b020
00 0 1
00 amplg cos θr
ac−b0
20
⎤
⎥
⎥
⎦
,g0=⎡
⎢
⎣
0
g1,0
0
g2,0
⎤
⎥
⎦
ηm=−r(mp+mw)gsin ϕ

3676 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 7, JULY 2014
g1,0=c
r(ac −b2
0)+b0
ac −b0
2
g2,0=−b0
rac −b0
2+−a
ac −b2
0
with b0=m2lcos(θr+ϕ). For system with parameter uncer-
tainties, ηm=−r(ˆmp+mw)gsin ˆϕand b0=ˆmplcos(θr+ˆϕ).
The matched term ηmreflects the effect of the gravity when
the 2 WMR travels on an inclined surface. A constant torque
should be generated to overcome the effect of the gravity. Since
ηmis matched to the control input, it can be compensated
directly by introducing −ηmin the control input. The nominal
controller becomes
κ=−kTe+ηm.(26)
Various methods could be applied to obtain the control
feedback gains such that the closed-loop matrix (A0−g0kT)is
Hurwitz, which ensures the stability of the desired equilibrium
point.
D. Steady-State Analysis
We define the-steady state error vector as es=[e1,s ,e
2,s,
e3,s,e
4,s]T, in steady state, e2,s =0,e4,s =0. From (13), we
can obtain
η2+g2κs+δ2=0 (27)
where
η2=amplg sin θe
ac −b2+b(mp+mw)gsin ϕ
ac −b2.(28)
Substituting θein (7) into the above equation, we have
η2=−g2r(mp+mw)gsin ϕ. (29)
Refer to (14), for s=[0,0,0,s
4],wehaveδ2=0in (30), thus
equation (27) becomes as
g2[κs−r(mp+mw)gsin ϕ]=0.(30)
In steady state, the nominal controller (26) becomes
κs=−k1e1,s −k3e3,s +ηm=r(mp+mw)gsin ϕ. (31)
Thus,
e1,s =−k3
k1
e3,s
and e3,s =θe−θr.
For systems without parameter uncertainties, θecan be com-
puted according to equation (7). Let θr=θe,wehavee3,s =0
and e1,s =0. For system with parameter uncertainties, from
(22), we have
ˆη2+ˆg2κ+δ2=0.
Similarly, by choosing s=[0,0,0,s
4],wehaveδ2=0.
At steady state, the above sliding motion equation becomes
ˆη2+ˆg2(−k1e1,s −k3e3,s +ηm)=0
where e3,s =θe−ˆ
θeand ηm=−r(ˆmp+mw)gsin ˆϕ.From
the above equation, it can be concluded that
e1,s =ˆη2
ˆg2k1
−k3e3,s −ηm
k1
=0.(32)
To make e1,s =0, a compensation term is added to the nominal
controller, the new nominal controller is designed as
κ=−kTe+ηm+γc.(33)
The sliding motion equation in steady state becomes
ˆη2+ˆg2(−k1e1,s −k3e3,s +ηm+γc)=0.(34)
A databased approach is proposed to determine the value of γc.
First, the nominal controller in (33) is applied with γc=0.The
value of e1,s is obtained from simulation or experiment results,
denoted as e1,s|γc=0 .From(32),wehave
−k1e1,s|γc=0 =−ˆη2
ˆg2
+k3e3,s −ηm.
Next, let
γc=−k1e1,s|γc=0 =−ˆη2
ˆg2
+k3e3,s −ηm(35)
and substitute the above equation into (34), we can conclude
e1,s =0.
IV. NUMERICAL VALIDATIONS
For simulation, fris modeled as a combination of viscous
friction and Coulomb friction, that is, fr=fv˙x+fcsgn( ˙x),
where fvis a viscous-friction constant, fcis a Coulomb-friction
constant, and sgn(·)is a signum function. Similarly, τfis
modeled as τf=τv˙
θ+τcsgn( ˙
θ), where τvis a viscous-friction
constant, τcis a Coulomb-friction.
A. Linear Controller for Nominal System
The ISMC proposed in this work consists of a nominal
linear controller and a switching term. As we can see, the
linear controller is designed to stabilize the sliding manifold,
which is without any matched uncertainty. First we conduct
the simulation by applying a linear controller to the 2 WMR in
absence of uncertainties, i.e., fr=0,τf=0and all the system
parameters are known.
LQR method is used to design the feedback gains. The
objective is to minimize the performance index
J=1
2
∞

0
(eTQe+Ru2)dt
with Q≥0and R>0. The optimal solution is
k=R−1Pg0

XU et al.: DESIGN AND IMPLEMENTATION OF INTEGRAL SLIDING-MODE CONTROL 3677
Fig. 3. Time responses of x,θand uunder linear control. In simulations, the
2 WMR system is considered in absence of frictions and system uncertainties.
Fig. 4. Time responses of x,θand uunder linear control. In simulations, the
2 WMR system is considered in presence of frictions, τf=0.2˙
θ+0.3sgn( ˙
θ)
and fr=0.5˙x+sgn(˙x).
where Pis the solution of the Riccati equation
AT
0P+PA
0+Q−Pg0R−1gT
0P=0.
Choosing {q1,q
2,q
3,q
4}={50,0.1,500,1},R=1, we ob-
tain the feedback gains as k=[−7.0711,−9.6708,−27.0228,
−2.8418]T. The initial states of the 2 WMR are as x=
[0,0,0.1,0]T. The simulation results are shown in Fig. 3.
The wheel reaches the desired setpoint smoothly with a small
overshoot, the pendulum angular stays around zero.
Next, the linear controller is applied to the 2 WMR system
in presence of the frictions, τf=0.2˙
θ+0.3sgn( ˙
θ)and fr=
0.5˙x+sgn(˙x). The simulation results are shown in Fig. 4. It is
found that the pendulum and the wheel keep vibrating around
the desired positions, which are not satisfactory responses and
indicates the limited robustness of the linear controller.
B. ISMC for System With Matched Uncertainties
We consider the joint friction exists in the system and τf=
0.2˙
θ+0.3sgn( ˙
θ), which is a matched uncertainty. ISMC is
applied with s=[0,0,0,1],ρ=0.1+0.2|x4|+0.3, and the
nominal linear controller κuses the same feedback gains as
in the pervious subsection. We set θr=0 and γc=0 since
fr=0and ϕ=0. The simulation results are shown in Fig. 5.
The 2 WMR reaches the desired setpoint smoothly and the
pendulum is balanced at θe=0. The 2 WMR responses are
almost the same as in Fig. 4 despite the system is in presence
of the joint friction τf, which demonstrates the effectiveness of
ISMC in rejecting matched uncertainties. It is noted that control
signal shows switching behavior, which can be explained as
the following. In the ideal sliding mode, we have σ=0.To
make the system states stay on the switching surface, an infinite
Fig. 5. Time responses of x,θ,uand σunder the ISMC. In simulations, the
2 WMR system is considered with the joint friction τf=0.2˙
θ+0.3sgn( ˙
θ),
which is a matched uncertainty.
switching frequency is needed, which is impossible to achieve
in any digital implementation. Due to the finite sampling
frequency in implementations, the “chattering” phenomenon
occurs.
Remark 3: In this paper, the DC motor is controlled by
a discontinuous pulse width modulation (PWM) signal. The
characteristic of the PWM control is its switching (on–off) op-
eration mode, which is achieved by electronic power switchers.
Therefore, the implementation of the switching type control
signal is not a problem. Furthermore, it may even be more
advantageous to employ the ISMC than other continuous con-
trollers because the ISMC naturally generates a discontinues
control signal while other continuous controllers are designed
to generate continuous signals which however are forced to
become discontinuous in real implementation [31], [32].
C. ISMC for System With Both Matched Uncertainties and
Unmatched Uncertainties
Two type of unmatched uncertainties exist in the system, one
is due to the external disturbance and the other is due to the
uncertain system parameters.
First, we consider the 2 WMR system with the ground
friction fr=0.5˙x+sgn(˙x)and the joint friction τf=0.2˙
θ+
0.3sgn( ˙
θ). ISMC is applied with ρ=0.1+0.2|x4|+0.3+
br/(b+ar)(0.5|x2|+1) and the nominal controller in (33).
Other control parameters are chosen the same as in the pre-
ceding simulation. The simulation results are shown in Fig. 6.
The 2 WMR reaches the desired setpoint at t=20 s and the
pendulum is finally balanced at the upright position, i.e., θ=
0, which indicates that the proposed ISMC is also robust to
unmatched unknown friction.
At the time interval 3 ∼15 s, the 2 WMR reaches a steady
state that the 2 WMR travels with the constant speed 0.1 m/s,
the pendulum is balanced at θ=0.041 rad and the tracking
error of the wheel position exists. The results are consistent
with the analysis in Section II-D. When fr=0, the equilibrium
of the pendulum is not the upright position, but related with
the size of the ground friction. Since the ground friction is

3678 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 7, JULY 2014
Fig. 6. Time responses of x,θand uunder the ISMC with and without the
compensation term γc. In simulations, the 2 WMR system is considered in
presence of the joint friction τf=0.2˙
θ+0.3sgn( ˙
θ), and the ground friction
fr=0.5˙x+sgn(˙x)which is an unmatched uncertainty.
unknown to the designer, θr=0is used in the controller design,
which yields e3=θ−θr=0and e1=0. Although the ground
friction brings a tracking error of the wheel position during the
traveling, the system performance is still satisfactory. When the
2 WMR stops at the desired setpoint at t=20 s, the ground
friction disappears, so does the tracking error of the wheel
position.
Next, the 2 WMR system with parameter uncertainties is
considered. The actual values of the uncertain parameters are as
[mp,l,ϕ]=[2.0kg,0.18 m,π/15 rad], which are assumed to
be unknown to the designer. Estimated values of the uncertain
parameters, [ˆmp,ˆ
l, ˆϕ]=[1.6kg,0.13 m,0rad], are used in
sliding surface and controller designs. The frictions are also
considered to exist in the system. ISMC is applied with θr=0
and the nominal controller is designed as in (33). First, γc=0is
applied. ISMC shows the robustness to the parameter uncertain-
ties. The pendulum balances at a new equilibrium position θ=
0.26 rad. However, the tracking performance of the 2 WMR
is not satisfactory. The tracking error of the wheel position in
steady state is e1,s|γc=0 =−0.9259 m. Next, γc=−6.5471 is
computed according to (35) and used in (33). The simulation
results for the two cases, with and without the compensation
term γc, are shown in Fig. 7. By adding the compensation term
to the control input, the 2 WMR tracks the planned trajectory
better and reaches the desired setpoint without steady-state
error. The simulation results are consistent with the theoretical
analysis in Sections II-D and III-D.
V. I MPLEMENTATION AND EXPERIMENT RESULTS
In simulations, an ideal model of the 2 WMR is used. To
stabilize the 2 WMR system in absence of uncertainties, the
feedback gains for the nominal linear controller can be chosen
in a wide range as long as A0−g0kTis Hurwitz. However,
considering the existence of mismatch between the real-time
system model and the mathematical model (1) and (2), the
feedback gains obtained from simulations may not function
well on the real-time platform, thus need to be adjusted through
experimental testings on the 2 WMR prototype.
Fig. 7. Time responses of x,θand uunder ISMC with and without the
compensation term γc. In simulations, the 2 WMR system is considered with
uncertain parameters mp,land ϕ.
Fig. 8. Experimental testing results for regulation task: time responses of x,
θand uunder ISMC and linear controller. The mobile robot is placed on flat
surface.
A. Regulation Task
For implementation, first, we consider a simple regulation
task that is to balance the robot at the original position on
a flat surface, i.e., xr=0,vr=0, and ϕ=0. Since there
exists backlash in the driving mechanism of the 2 WMR
[10], strong vibrations are observed by applying the linear
controller with the feedback gains obtained from simulations.
To reduce the vibrations, the feedback gains are adjusted to
k=[−10,−0.5,−35,−3]T.
ISMC is applied with the projection vector as s=[0,0,0,
0.05] and the feedback gains for the nominal linear controller
as k=[−10,−0.5,−35,−3]T. For comparison, the linear con-
troller alone is also applied to the 2 WMR. Fig. 8 shows
the experimental results for the 2 WMR under the ISMC and

XU et al.: DESIGN AND IMPLEMENTATION OF INTEGRAL SLIDING-MODE CONTROL 3679
Fig. 9. Experimental testing results for regulation task: time responses of x
and θunder the ISMC proposed in this work. The 2 WMR is placed on a flat
surface. A disturbance is added to the system at t=18s.
the linear controller. When the linear controller is applied,
the 2 WMR is stabilized at the first few seconds, however,
becomes unstable in 10 seconds. By applying the ISMC, the
2 WMR is consistently stabilized. The wheels stay around the
original place and the pendulum is balanced around θ=0,
which verifies the effectiveness of the ISMC in handling system
uncertainties.
A testing is conducted to check the robustness of the ISMC
with respect to an exceptional disturbance. The experiment
results are shown in Fig. 9. At t=18s, we push the 2 WMR
to the right about 0.15 m. The 2 WMR is finally stabilized
around the original position and the transit responses show
small oscillations.
For comparison, several other existing methods, including
the fuzzy traveling and position controller (FTPC) proposed in
[7], and the sliding-mode controller proposed in [19], [20], are
used to control the 2 WMRs. The experimental results for the
2 WMR system under the FTPC [7] and SMC [19], [20] can
be found in [10] (Figs. 11 and 12). By comparing the results,
it is evident that the ISMC proposed in this work provides a
better performance than the existing methods [7], [19], [20]
when controlling the 2 WMR.
Next, the robot is placed on an inclined surface and the
slope angle ϕis unknown. ISMC is applied with θr=0and
the nominal linear controller is designed as in (33). For the
first trial, we set γc=0. The pendulum is balanced around
θe=0.1rad, however, steady-state error of the wheel position
exists, and e1,s|γc=0 =−0.35 m. For the second trial, we use
γc=−3.5, which is computed according to (35). Experiment
results for the two cases, with and without the compensation
term, are shown in Fig. 10. The steady-state error for the wheel
position is eliminated under ISMC with the compensation term,
which is consistent with the theoretical analysis and simulation
results.
B. Reaching a Setpoint
First, we consider the mobile robot travels on a flat surface,
i.e., ϕ=0. The planned trajectory for the wheeled mobile robot
Fig. 10. Experimental testing results for regulation task: time responses of x,
θand uunder ISMC with and without the compensation term γc.The2WMR
is placed on an inclined surface.
Fig. 11. Experimental testing results for setpoint task: time responses of x,θ
and uunder ISMC. The mobile robot travels on a flat surface. The pre-planned
reference trajectory (5) is applied.
is the same as we used for simulation. ISMC is applied with
θr=0,γc=0. All other controller parameters are chosen the
same as for the regulation task. Experiment results are shown
in Fig. 11. The 2 WMR reached the desired setpoint and stays
there afterward. ISMC shows the effectiveness for setpoint
control of the 2 WMR system. However, it is observed that the
trajectory of the wheels x1is not smooth enough. When the real
position of the 2 WMR x1surpasses the given reference xr,
the 2 WMR would stop for a while or travel backwards, which
are not the desired motions. Considering our objective for the
2 WMR is traveling forward to arrive the desired position, a

3680 IEEE TRANSACTIONS ON INDUSTRIAL ELECTRONICS, VOL. 61, NO. 7, JULY 2014
Fig. 12. Experimental testing results for setpoint task: time responses of x,θ
and uunder ISMC. The mobile robot travels on a flat surface. The modified
reference trajectory (36) is applied.
modified reference trajectory xr,n for the wheel position is
applied [10], as the following:
xr,n (t+Ts)
=⎧
⎨
⎩
xr,n (t)+vrTs,if x1(t)≤xr,n (t)<x
d
x1(t)+vrTs,if xr,n (t)<x
1(t)<x
d
xd,if xr,n (t)≥xdor x1(t)≥xd.
(36)
Another test is conducted with applying the modified refer-
ence trajectory, and the design of ISMC is the same as in the
preceding test. Experiment results are shown in Fig. 12. We can
see that the response is much smoother and the 2 WMR arrives
the desired position in a shorter time.
Next, we consider the mobile robot travels on an inclined
surface and the slope angle ϕis unknown. ISMC is applied
with θr=0 and the nominal linear controller is designed
as in (33). For the first trial, we set γc=0, the experiment
results are shown in Fig. 13. The pendulum is balanced around
0.05 rad. However, steady-state error exists for the wheel posi-
tion, and e1,s|γc=0 =−0.17 m. For the second trial, γc=−1.7
is computed according to (35) and applied in (33). To have
a smooth response, similarly, the modified trajectory in (36)
is applied. The experiment results are shown in Fig. 14. We
can see the robot reaches the desired position smoothly without
steady-state error.
VI. CONCLUSION
In this paper, an ISMC is proposed for regulation and setpoint
control of an underactuated 2 WMR system with both matched
and unmatched uncertainties. The ISMC is constructed by a
nominal control part and a switching term. With the switching
term, the matched uncertainties are perfectly rejected. With the
freedom to design a nominal control for the sliding manifold,
Fig. 13. Experimental testing results for setpoint task: time responses of x,θ
and uunder ISMC with γc=0. The mobile robot travels on an inclined surface
with ϕ=2.5◦. The pre-planned reference trajectory (5) is applied.
Fig. 14. Experimental testing results for setpoint task: time responses of x,θ
and uunder ISMC with γc=−1.7. The mobile robot travels on an inclined
surface. The modified reference trajectory (36) is applied.
ISMC is incorporated with a linear controller. The main contri-
bution of this paper is that for the first time ISMC is applied
to a real time platform of 2 WMR. Regulation and setpoint
control tasks are achieved not only when the 2 WMR is placed
on a flat surface but also on an inclined surface. Strategies have
been proposed to handle many practical problems regarding
the implementation such as trajectory planning, eliminating
the steady-state error. Simulation and experiment results are
provided to validate the effectiveness and robustness of the
ISMC.

XU et al.: DESIGN AND IMPLEMENTATION OF INTEGRAL SLIDING-MODE CONTROL 3681
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Jian-Xin Xu (M’92–SM’98–F’12) received the
Ph.D. degree from The University of Tokyo, Tokyo,
Japan, in 1989.
In 1991, he joined the National University of
Singapore, Singapore, where he is currently a Pro-
fessor in the Department of Electrical and Computer
Engineering. His research interests lie in the fields of
learning theory, intelligent control, nonlinear and ro-
bust control, robotics, and precision motion control.
He has published over 160 journal papers and five
books in the area of systems and control.
Zhao-Qin Guo received the B.S. degree from
Huazhong University of Science and Technology,
Wuhan, China, in 2008 and the Ph.D. degree from the
National University of Singapore, Singapore, in 2013.
She is currently a Research Fellow in the De-
partment of Electrical and Computer Engineering,
National University of Singapore. Her research inter-
ests include control theory and applications, partic-
ularly fuzzy logic control and sliding-mode control
with application to underactuated systems and power
systems.
Dr. Guo was a recipient of the NUS Graduate School for Integrative Sciences
and Engineering Research Scholarship in 2008–2012.
Tong Heng Lee received the B.A. degree (with first
class honors) in engineering tripos from the Univer-
sity of Cambridge, Cambridge, U.K., in 1980 and the
Ph.D. degree from Yale University, New Haven, CT,
USA, in 1987.
He is the Past Vice President for Research of the
National University of Singapore, Singapore, where
he is currently a Professor with the Department of
Electrical and Computer Engineering and also a Pro-
fessor with the Graduate School for Integrative Sci-
ences and Engineering. He is currently an Associate
Editor of Control Engineering Practice. He is the Deputy Editor-in-Chief of the
IFAC journal Mechatronics. His research interests are in the areas of adaptive
systems, knowledge-based control, intelligent mechatronics, and computational
intelligence.
Dr. Lee is currently an Associate Editor of the IEE E T RANSACTIONS ON
SYSTEMS,MAN,AND CYBERNETICS and the IEEE TRANSACTIONS ON
INDUSTRIAL ELECTRONICS.