Available online at www.sciencedirect.com

Journal of the Franklin Institute 351 (2014) 22612282

pendulum (WIP) system, which consists of a pendulum and two wheels in parallel. The control

www.elsevier.com/locate/jfranklin

nCorresponding author at: Graduate School for Integrative Sciences and Engineering, National University of

Singapore, Singapore 117456, Singapore.0016-0032/$32.00 & 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.jfranklin.2013.02.002

E-mail addresses: guozhaoqin@gmail.com (Z.-Q. Guo), elexujx@nus.edu.sg (J.-X. Xu),

eleleeth@nus.edu.sg (T.H. Lee).objective is to use only one actuator to perform setpoint control of the wheels while balance the

pendulum around the upright position, which is an unstable equilibrium. When designing the SMC

for the WIP system, various uncertainties are taken into consideration, including matched

uncertainties such as the joint friction, and unmatched uncertainties such as the ground friction,

payload variation, or road slope. The SMC proposed is capable of handling system uncertainties and

applicable to general underactuated systems with or without input coupling. For switching surface

design, the selection of the switching surface coefcients is in general a sophisticated design issue

because those coefcients are nonafne in the sliding manifold. In this work, the switching surface

design is transformed into a linear controller design, which is simple and systematic. By virtue of the

systematic design, various linear control techniques, such as linear quadratic regulator (LQR) or

linear matrix inequality (LMI), can be incorporated in the switching surface design to achieve

optimality or robustness for the sliding manifold. To further improve the WIP responses, the design

of reference signals is addressed. The reference position for the pendulum is adjusted according to the

actual equilibrium of the pendulum, which depends on the size of the friction and slope angle of the

traveling surface. A smooth reference trajectory for the setpoint of the wheel is applied to avoidDesign and implementation of a new sliding modecontroller on an underactuated wheeled

inverted pendulum

Zhao-Qin Guoa,b, Jian-Xin Xua,b,n, Tong Heng Leea,b

aGraduate School for Integrative Sciences and Engineering, National University of Singapore,

Singapore 117456, SingaporebDepartment of Electrical and Computer Engineering, National University of Singapore,

4 Engineering Drive 3, Singapore 117576, Singapore

Received 30 June 2012; received in revised form 21 December 2012; accepted 7 February 2013

Available online 15 February 2013

Abstract

In this paper, a sliding mode controller (SMC) is proposed for control of a wheeled inverted

directions, which results in the input coupling of the WIP system.

Z.-Q. Guo et al. / Journal of the Franklin Institute 351 (2014) 226122822262Considering that various uncertainties exist in the WIP system, for instance, the jointand the ground frictions, the slope angle of the traveling surface, robustness is addressed inthe control system design. SMC is a well known robust control approach for systems withmodel uncertainties and external disturbances and has been studied for control of wheeledinverted pendulum and similar underactuated mechanical systems [1018]. SMC utilizes aabrupt jumps in the system responses, meanwhile the reaching time of the switching surface can be

reduced. The effectiveness of the SMC is validated using intensive simulations and experiment

testings.

& 2013 The Franklin Institute. Published by Elsevier Ltd. All rights reserved.

1. Introduction

Systems that have fewer control inputs than the degrees of freedom (DOF) to becontrolled are dened as underactuated systems. Control of underactuated systems is apopular research topic due to its wide range of applications in robotics, underwatervehicles, aerospace vehicles, etc. [1,2]. From practical concerns such as cost reduction orweight reduction, many systems are designed to be underactuated. Some systems becomeunderactuated when actuator failure occurs. As benchmark examples of nonlinear andunderactuated systems, the cart-pendulum is often used to demonstrate and verify theeffectiveness of control algorithms.In recent years, the control of WIP systems has attracted attentions from both researchers

and engineers. The well known commercial product, two-wheeled SEGWAY, is a popularpersonal transporter. For research and education purposes, prototypes of autonomous WIPshave been designed in universities and research institutes [39]. The WIP usually consists oftwo actuated wheels in parallel and an unactuated inverse pendulum. The control objective ofthe WIP is to perform position or velocity control of the wheels while stabilize the pendulumaround the upright position that is an unstable equilibrium point. The WIP developed in [39]belongs to underactuated systems without input coupling [1]. The control input acts on thewheels only, while the balancing of the pendulum is achieved by the wheel motion. Themathematic model of the WIP system in [39] is essentially the same as that of the classic cart-pendulum system, therefore the control methods designed for cart-pendulum system can bedirectly applied to WIP system without input coupling.Due to the difference in mechanical conguration, underactuated WIPs can be classied

into the class without input coupling where the actuator is mounted on the wheel (class A),and the class with input coupling where the actuator is mounted on the pendulum orchassis (class B). Class A is more complex in mechanical construction but easier incontroller design owing to the absence of input coupling. In contrast, class B is easier inmechanical construction but more challenging in controller design due to the inputcoupling between the wheel and the pendulum. Since the existing works mostly focus onstudying control of underactuated systems without input coupling, this work is devoted tothe development and control of a WIP with input coupling. When building the prototypeof the WIP, the motor shaft coupler is xed at the center of the wheels and the motorhousing is rigidly connected to the inverse pendulum, thus the torque generated by themotor directly acts on both the wheels and the pendulum with the same size but oppositediscontinuous control law to drive system state trajectory into a designer specied

switching surface and to maintain the system state trajectory on this surface for all thesubsequent time. In standard SMC design for full actuated system, it is straightforward toconclude the convergence of the system states from the switching surface equation whensystem is in the sliding mode. However, for underactuated system, the standard SMCdesign and stability analysis are not applicable because the system has fewer inputs thanthe independent variables to be controlled. As a result, a nonlinear sliding manifold or ingeneral an internal dynamics must be stabilized by the proper selection of the switchingsurface coefcients.In [1013], coupled sliding mode control laws along with a linear coupled sliding surface

are proposed for controlling of the underactuated system, where the coupled linearswitching surface is designed by incorporating multiple independent state variables into ascalar switching surface. The single actuator can thus be used to manipulate the scalarswitching surface. Conclusions regarding the convergence of the states, however, cannot bedrawn directly from the convergence of the switching surface due to the presence of thesliding manifold or the internal dynamics, thus the stability of the sliding motion governedby the sliding manifold should be further investigated, namely, the switching surface designis required. In [1012], stable sliding manifolds are obtained with appropriately selectedswitching surface coefcients. In [13], a stabilizing control law is assumed existing. In [14],an SMC design based on the cascade normal form [1] is proposed, and the validity holdsunder certain assumptions. However, the WIP studied in our work does not meet theseassumptions. Second-order SMC designs for underactuated systems are discussed in[1518]. The design of second-order SMC requires that the derivative of the dened slidingvariable is known. In [1517], the SMC design requires that the derivatives of all systemstates are known. However, in this work, the derivatives of the velocity states are notavailable because the WIP system is in the presence of both parametric and externaluncertainties. In [19], a new nonlinear sliding surface is proposed to control a class ofnonminimum phase underactuated mechanical systems. The structure of the nonlinearsliding surface is more complex than that of a linear sliding surface.Design of optimal SMC algorithms has attracted particular interests recently [2128]. In

a typical SMC design, stability is the only concern in the switching surface design. Theoptimal SMC design aims at achieving both robust and optimal control, thus shows thesuperiority in practical applications. However, no results about optimal SMC design forunderactuated system have been shown in existing works. Thus, it would be meaningfuland interesting to explore optimal or suboptimal SMC designs for underactuated systems.In this paper, for the easiness in real-time implementation, an SMC along with a linear

switching surface is proposed for setpoint control of the WIP system. The linear switchingsurface is constructed by combining the two states of the wheel and two states of thependulum in a linear form [1013], which brings four coefcients associated with the fourstates. The SMC law is derived using Lyapunov theory, which guarantees the nitereaching time of the switching surface and leads to a sliding manifold with all the matcheduncertainties rejected. In the sliding mode, the sliding motion is determined by the fourcoefcients, however, in a complex and highly nonlinear form. Therefore, it is difcult todirectly choose or tune the coefcients to achieve the desired sliding motion. To simplifythe switching surface design, the sliding manifold is linearized around the desiredequilibrium point of the pendulum. Through a mathematical transformation, it is shownthat the linearized sliding manifold is equivalent to a normal linear system that is under a

Z.-Q. Guo et al. / Journal of the Franklin Institute 351 (2014) 22612282 2263full state linear feedback control with the freedom in choosing feedback gains. Now the

(iii)

TsystWIP platform. Conclusions are drawn in Section 7.

2. Problem formulation

Fig. 1 shows the prototype of the WIP. Since the lateral stability of the WIP isguaranteed, in this work we focus on longitudinal control of the WIP.

2.1. WIP model

Fig. 2 shows the model of the WIP. The wheel motion is dened along the road surface.The wheel displacement and velocity are denoted by x and _x, respectively, with rightwardas positive direction. y is the tilting angle of the pendulum with the upright position as zeroIn Sstudand2, . . . ,ln can be parameters or variables.he remainder of this paper is organized as follows. In Section 2, the underactuatedem model is given. In Section 3, SMC with linear switching surface design is detailed.ection 4, switching surface design is discussed. In Section 5, simulation based caseies are presented. Section 6 presents the implementation of the proposed SMC on theThl1,lSatisfactory responses are obtained for both regulation and setpoint control tasks.The particular characteristics of the underactuated WIP system with input couplingare investigated, according to which, references for both the wheel and the pendulumare designed to improve system performance.

roughout this paper, a function F l1,l2, . . . ,ln will be written as F, whereof freedom in control. By utilizing the extra degree of freedom, various linear controltechniques can be incorporated in the SMC design. Thanks to the easiness in tuning ofthe controller parameters, the SMC is successfully implemented on the WIP system.switching surface design becomes a nominal linear controller design, which is simple,systematic, and furthermore provides one extra degree of freedom in control. In this work,this degree of freedom is utilized to implement optimal or robust linear control techniques.Two alternative methods are adopted for the nominal linear controller design. One is basedon LQR method, which leads to a stable sliding manifold that also exhibits optimality interms of fast tracking convergence and control cost. The other is based on LMI methodand the resulting sliding manifold exhibits robustness with respecting to variousunmatched uncertainties. The resulting sliding manifold exhibits desirable propertiesbesides stability, such as optimality and robustness. The existing works on SMC design forunderactuated systems hitherto only focus on the stabilization of the sliding manifold.The main contributions of this paper are summarized as follows.

(i) An SMC is proposed to control an underactuated WIP system in the presence of bothmatched and unmatched uncertainties. The proposed control methods and theobtained results can be extended to general underactuated systems with or withoutinput coupling.

(ii) To make the control algorithm be simple and implementable, the SMC uses a linearswitching surface and the switching surface design is transformed into a nominal linearcontrol design, which is simple, systematic and furthermore provides one extra degree

Z.-Q. Guo et al. / Journal of the Franklin Institute 351 (2014) 226122822264clockwise rotation as positive direction. _y is angular velocity of the pendulum. j is the

Z.-Q. Guo et al. / Journal of the Franklin Institute 351 (2014) 22612282 2265slope angle of the inclined road, for traveling on at surface, j 0. fr is the groundfriction. t is the torque generated by the motor and acting on the wheel with clockwiserotation as positive direction, which is also the control input u to the system. Note that themotor driving the wheel is directly mounted on the pendulum, there is a reaction torque tapplied to the pendulum. tf is the joint friction, which also acts on both the wheel and thependulum as tf and tf , respectively. Other system parameters are as

mw 1:551 kg: the mass of the wheels;mp 1:6 kg: the mass of the pendulum;Iw 0:005 kgm2: the rotation inertia of the wheels;Ip 0:027 kgm2: the rotation inertia of the pendulum;r0.08 m: the radius of the wheel;l0.13 m: the distance between Center of Gravity (COG) of the pendulum and thecenter of the wheel;g9.81 m/s2: the acceleration of gravity.

Fig. 2. Model of the WIP system.

Fig. 1. Prototype of the WIP.

acbd b f ,The following assumptions are made in the formulation of the WIP dynamic model. (i)There is no rolling slip between the wheels and the ground. (ii) The wheels are always incontact with the ground.Lagrangian mechanics method is used to derive the mathematical model of the WIP

system [20], which leads to a second-order nonlinear model given by

a x b ympl siny j _y2 sin jmp mwg 1rt tfrf r, 1

b x c ymplg sin yttf , 2where a mw mp Iw=r2, b mpl cosx3 j and c Ip mpl2.

Remark 1. From the system viewpoint, for WIP with input coupling, the control input texists in both the wheel and the pendulum motion Eqs. (1) and (2). While for WIP withoutinput coupling, t only exists in motion equation of the wheel subsystem (1).

2.2. Control objective

The control objective is to achieve setpoint control of the wheel, while balance thependulum at an equilibrium (y ye, _y 0). Dene x x1, x2, x3, x4T x, _x, y, _yT andthe reference signal for x is chosen as r xr, vr,yr, 0T with _xr vr. We obtain the errorstates as e e1, e2, e3, e4T xr x1xr, x2vr, x3yr, x4T . Now the control objec-tive is to ensure the convergence of e. The error dynamic model of the WIP is obtained as

_e ge geu dme,t due,t, 3where g is the system nonlinear term, dm is the lumped matched uncertainties, du is thelumped unmatched uncertainties. We have

ge e2, Z1e, e4, Z2eT ,ge 0, g1e, 0, g2eT ,dme,t tf ,due,t 0, du1e,t, 0, du2e,tT ,

where

Z1 mpl

acb2 ce24 sine3 yr jbg sine3 yr

cmp mwg sin jacb2 _vr,

Z2 mpl

acb2 be24 sine3 yr j ag sine3 yr

bmp mwg sin jacb2 ,

g1 1

r

c

acb2 b

acb2 ,

g2 1

r

bacb2

aacb2 ,

du1 c

2fr,

u2acb2 r

Z.-Q. Guo et al. / Journal of the Franklin Institute 351 (2014) 226122822266and b mpl cose3 yr j.

Remark 2. In the following work, control design and stability analysis are based on theexpression of dynamics model in Eq. (3), which can also be used to describe generalunderactuated systems. Thus, the SMC designed for the WIP system is extendable togeneral underactuated systems with or without input coupling.

2.3. Trajectory planning

The WIP is supposed to reach a desired position xd and stop there. However, using astep function as the desired trajectory for the WIP would cause the desired travelingvelocity to be discontinuous at the beginning of the motion. The discontinuousness wouldcause considerable jerk and pendulum swinging [6] in the response. High-orderpolynomials are used for computing smooth trajectory [6]. In this work, we simply use alinear segment and two parabolic blends to construct a smooth trajectory for the WIP,which also yields a smooth reference signal for the wheel velocity. The reference inputs are

Z.-Q. Guo et al. / Journal of the Franklin Institute 351 (2014) 22612282 2267computed by the following equations and shown in Fig. 3:

vrt

vm

t1t 0otot1,

vm t1rtrt2,vm

vm

t3t2tt2 t2rtrt3,

0 t3rtrts,

8>>>>>>>>>>>:

4

xrt Z t0

vr ds, 5

and xrts xd , where xd is the setpoint.

Remark 3. With the planned trajectory, the absolute value of initial e1 and e2 are zero, andthis is desirable in the SMC design. First, for system under feedback control, like SMC,large initial errors could result in a large control input at the beginning, which may lead tounstable responses. Second, later it can be found that the WIP system with smaller initial

0 5 10 15 200

0.5

1

1.5

time (s)

Ref

for W

heel

Pos

ition

(m) /

Vel

(m/s

)

xrvr

0 0.5 10

0.05

0.115 15.5 16 16.5

1.45

1.5

Fig. 3. Reference signals for wheel velocity and position as described in Eqs. (4) and (5) with t1 1 s, t2 15 s,

t3 16 s, ts20, vm 0:1 m=s, xd 1:5 m.

errors can reach the switching surface in a shorter time under the proposed SMC, whichmakes the control system more robust.

2.4. Equilibrium point of the WIP

At the equilibrium point, the wheel acceleration is zero ( x 0), the pendulum angularvelocity and acceleration are zero ( _y 0, y 0), meanwhile the joint friction does not exist(tf 0), the dynamic Eqs. (1) and (2) become

sin jmp mwg1

rtrf r, mplg sin yt:

From the above equations, the pendulum equilibrium point is

ye arcsinr sin jmp mwg rf r

mplg: 6

Remark 4. For the WIP with input coupling, the equilibrium of the pendulum varies withrespecting to the slope angle of the traveling surface and the ground friction. This isdifferent from WIP system without input coupling, for which the equilibrium of thependulum keeps at the upright position, i.e., ye 0.

We give the following explanation based on an intuitive observation. When the WIPwith input coupling is stabilized on a inclined surface (ja0) or traveling at a constantvelocity (fra0), a torque, denoted as ts, should be provided to overcome the effect ofgravity or the ground friction, meanwhile, the reaction torque ts acts on the pendulum.The balance of the pendulum can be reached only when the total torque acting on thependulum is zero. Thus, the pendulum tilts rightward or leftward from the upright positionsuch that the torque resulted from the gravity of the pendulum equals to the reactiontorque ts but with the opposite direction.Considering that our control objective is setpoint control, the WIP nally stops at the

desired setpoint, thus we have fr0, and ye arcsinr sin jmp mwg=mplg. To achieve azero steady-state error for the pendulum angle, we choose the reference position for thependulum as yr ye, that is

yr arcsinr sin jmp mw

mpl: 7

The above yr is applicable only if the system parameters involved are known.

3. Sliding mode controller design

3.1. SMC design for system with unmodeled frictions

The following linear switching surface is proposed

s ce 0, 8

Z.-Q. Guo et al. / Journal of the Franklin Institute 351 (2014) 226122822268where c is a constant row vector, and cg is uniformly invertible.

The SMC law is

u cZ r sgnscg

, 9

where sgn is a signum function andr rm ru r0, 10

with rmZjcgdmj, ruZjcduj, and r0 is a positive constant.Theorem 1. Under the SMC law (9), the WIP system can reach the defined switching surface(8) in a finite time and maintain on it afterwards. In the sliding mode, the matcheduncertainties will be completely nullified. Furthermore, we have the freedom to choose thevector c to stabilize the sliding manifold and meanwhile achieve other desirable properties.

Proof. The derivative of s is as

_s c_e cZ cgu dm cdu: 11A quadratic Lyapunov function candidate is chosen as

V1 1

2s2: 12

Differentiating V1 with respect to time t yields

_V1 s _s scZ cgu dm cdu: 13Substituting the control law (9) into Eq. (13), we have

_V1 sr sgnscgdmcdurr0jsjo0,which implies a nite reaching time to the switching surface, s 0, and the reaching timecan be calculated as treachrjs0j=r0. &Remark 5. Under the same r0, the reaching time treach reduces as js0j decreases. As westated in Section 2, the absolute value of the initial e1 and e2 are zero by applying theplanned trajectory (4) and (5), which yields a small js0j. Therefore, the switching surfacecan be reached in a shorter time.

After reaching the switching surface, the system is in sliding mode and s 0, _s 0.Accordingly the equivalent control is derived from _s 0, as

ueqt cZcgdm

cdu

cg:

Dene ed e1,d , e2,d , e3,d , e4,d T as the state vector in the sliding mode, and substitutethe above ueqt into Eq. (3), one obtains the sliding manifold as

_ed Zed ged cZcg

deq, 14

where

deq Igc

du: 15

Z.-Q. Guo et al. / Journal of the Franklin Institute 351 (2014) 22612282 2269cg

In the sliding manifold, the matched uncertainty dm is completely nullied. Furthermore,we have the freedom to choose the vector c to stabilize the sliding manifold and meanwhileachieve other desirable properties, such as robustness, optimality, etc.

3.2. SMC design for system with parameter variations

From the practical point of view, the load of the pendulum mp and slope angle of thetraveling surface j are most likely to vary. The system dynamic model with parameteruncertainties is expressed as

_e ge DZe,p ge Dge,pdm u due,t, 16where Z and g are known nominal parts, p represents the uncertain parameters, DZ and Dgare uncertain parts. Dene constants mp,0, j0 the estimation values of mp and j,respectively. The known parts are Z Ze,mp,0,j0 and g ge,mp,0,j0, respectively. Theunknown parts are DZ Ze,mp,jZe,mp,0,j0 and Dg ge,mp,jge,mp,0,j0,respectively.The SMC in Eq. (9) is applied with the following switching gain:

r 1ebrm ru r0, 17

where

rmZjcg Dgdmj, ruZjcduj jcDZj cDgcZcg

and r040:

Theorem 2. For system with parameter uncertainties, under the SMC law (9) with theswitching gain (17), the WIP system can reach the switching surface (8) in a finite time andmaintain on it afterwards, under the condition jcDgcg1jo1eb eb40. In the sliding mode,the desirable properties stated in Theorem 1 also hold.

Proof. Differentiating the switching surface (8) with respect to time using Eq. (16) oneobtains

_st c_e cZ DZ cg Dgu dm cdu: 18Substituting the SMC law (9) into the above we have

_st cZ DZ cg Dg cZ r sgnscg

dm

cdu

r sgns cDgcg

r sgns cg Dgdm cdu DZcDgcg

cZ:

Differentiating the nonnegative quadratic function in Eq. (12) with respect to time t yields

_V 1 s _s s r sgns cDg

cgr sgns cg Dgdm cdu DZ

cDgcg

cZ

rrjsj rjsj cDg

jscg Dgdmj jscduj jscDZj cDg cZs

Z.-Q. Guo et al. / Journal of the Franklin Institute 351 (2014) 226122822270cg cg

Substituting the switching gain in Eq. (17) to the above inequality, we have

_

established constraints. However, since the vector c affects the system performance in a

complicated manner, it is hard to predict the system responses from the information of c,which is the main drawback of the existing switching surface designs. Another majordrawback or difculty in the switching surface design is the nonafne structure of thesliding manifold in the coefcients c, as shown in Eq. (14). To avoid the drawbacks, in thiswork, the switching surface coefcients are determined indirectly, by transforming thecoefcients determination problem into a nominal linear controller design. Hence theswitching surface design becomes simple, systematic, and furthermore provides one extradegree of freedom in control. Feedback gains for the nominal controller can be determinedthrough various systematic linear control design methods, which makes the systemresponses predictable. In the transformation, relations are established between theswitching surface coefcients and the feedback gains. With the obtained feedback gains,the switching surface coefcients can be determined.

4.1. Problem transformation

We linearize the sliding manifold (14) around the desired equilibrium point by assuminge3,d 0, sin e3,d e3,d and e24,d 0. The obtained linearized sliding manifold isV 1rr0jsjo0:Similarly, we conclude that the switching surface can be reached in a nite time astreachrjs0j=r0. In the sliding mode, the equivalent control is derived from _s 0, which is

ueqt cZ DZcg Dg dm

cdu

cg Dg :

Substituting the above ueq(t) into Eq. (16), one obtains the sliding manifold as Eq. (14) with

deq Ig Dgccg Dg

du DZu g

cZcg

: 19

In the sliding manifold, the matched uncertainty dm is completely nullied. Sliding motionis determined by the vector c. &

4. Switching surface design

Under the proposed SMC in Eq. (9), the WIP system can reach the switching surface in anite time. When the system is in sliding mode, from Eq. (14), we can see that the slidingmotion is directly determined by the vector c. In [1012], the stabilization of the slidingmanifold is achieved by choosing the switching surface coefcients according to severaljsj r cDgcg

rjcg DgdmjjcdujjcDZj cDgcg cZ

rjsj ebrjcg DgdmjjcdujjcDZjcDgcg

cZ

:

Z.-Q. Guo et al. / Journal of the Franklin Institute 351 (2014) 22612282 2271_ed Aed g0u0 deq,0, 20

where

A

0 1 0 0

0 0 a23 0

0 0 0 1

0 0 a43 0

26664

37775, g0

0

g1,0

0

g2,0

266664

377775, deq,0 I

g0c

cg0

du,

with

a23 b0mplg cos yr

acb20, a43

amplg cos yracb20

,

g1,0 c

racb20 b0

acb20, g2,0

b0racb20

aacb20

,

b0 mpl cosj yr,and the system control input is as

u0 c1

c2g1,0 c4g2,0e2,d

c2a23 c4a43c2g1,0 c4g2,0

e3,dc3

c2g1,0 c4g2,0e4,d : 21

When the WIP system is in the sliding mode, we have s c1e1,d c2e2,d c3e3,dc4e4,d 0, thus,

e1,d c2

c1e2,d

c3

c1e3,d

c4

c1e4,d : 22

Let

c1

c2g1,0 c4g2,0 k2

k1

c1c2, 23

c2a23 c4a43c2g1,0 c4g2,0

k3k1

c1c3, 24

c3

c2g1,0 c4g2,0 k4

k1

c1c4, 25

the system control input (21) can be rewritten as

u0 k2k1

c1c2

e2,d k3

k1

c1c3

e3,d k4

k1

c1c4

e4,d

k1c2

c1e2,d

c3

c1e3,d

c4

c1e4,d

k2e2,dk3e3,dk4e4,d

k1e1,dk2e2,dk3e3,dk4e4,d : 26

To stabilize the nominal linear system, i.e., the linearized sliding manifold (20), variouslinear controller design methods could be applied to obtain the feedback gainsk k1, k2, k3, k4, and c12c4 can be solved from relations (23)(25).

Remark 6. The four coefcients c12c4 are constrained by three Eqs. (23)(25), therefore

Z.-Q. Guo et al. / Journal of the Franklin Institute 351 (2014) 226122822272there are innumerable solutions of c for a given k. However, from Eqs. (23)(25), it can be

q1 and q2, respectively.

The relative value of the weighting matrix Q and R expresses the relative importance of

keeping e and u near zero. If we place more importance on the convergence of e, then wecan select Q to be relatively large to R, and so forth. Although we are interested inminimizing JLQR in Eq. (27), the actual value of JLQR is usually not of interest, which alsomeans that we can set either Q or R to be xed for the convenience of parameter tuningbecause it is their relative weight that is important. In this work, since Q is matrix and R isa scalar, it is better to x Q and tune R to achieve desired performance. A smaller R resultsin a larger feedback gain and faster convergence of e, however, a larger magnitude of u.Thus, the selection of R should achieve a compromise between these effects.The control system based on LQR design not only shows optimality, but also

robustness. Applying the LQR based linear controller, the phase margin of the closed-loopconcluded that the ratios between c12c4 are xed. In this work, we choose c1 be the freeparameter, then c22c4 can be decided from Eqs. (23)(25) once c1 is set.

4.2. Nominal linear controller design

Two alternative linear control design methods are introduced in this work. For WIPsystem to travel in a safe environment, a LQR based optimal linear controller is adopted toachieve an stable and optimal sliding manifold. For the WIP system to travel in a severeenvironment with various uncertainties, robustness of the system is the main concern, thusa LMI based robust design is employed to address unmatched uncertainties.

4.2.1. LQR based optimal design

LQR is a widely used method owing to its good properties in terms of stability andoptimality. An LQR problem is to minimize an index

JLQR 1

2

Z 10

eT Qe uT Ru dt, 27

with QZ0 and R40.The solution for the optimal control gain is as

k R1gT0 P1, 28where P1 is the solution of the following Riccati equation

P1A AT P1P1g0R1gT0 P1 Q 0:In this work, we choose the weighting matrix Q in diagonal form, i.e., Q diagfq1,

q2,q3,q4g, and R a scalar, where qi and R are the weighting factors for ei (i1, 2, 3, 4) andu, respectively.The relative values of qi represent the relative weighting among ei. If q1 is bigger than q2,

there is higher penalty on error e1 than e2 and control tries to make smaller e21 than e

22, and

vice versa. For control of the WIP system, the main objective is the convergence of e1 ande3, thus it is nature to select q1 and q3 to be relatively larger than q2 and q4. Furthermore,considering that the balancing of the pendulum is more important than the trackingperformance of the wheel, it is reasonable to select q3 and q4 to be relatively larger than

Z.-Q. Guo et al. / Journal of the Franklin Institute 351 (2014) 22612282 2273system is at least 60 degrees and the gain margin is from 1/2 to innity.

4.2.2. LMI based robust design

Let k wP, we have u0 wPe. The nominal linear system (20) becomes_ed Aedg0wPed deq,0: 29

Dene a Lyapunov function V2 eTd Ped , differentiating V2 with respect to t yields_V 2 _eTd Ped eTd P_ed

Aedg0wPed deq,0T Ped eTd PfAedg0wPed deq,0g 30_V 2 eTd Ag0wPT P PAg0wPed dTeq,0Ped eTd Pdeq,0: 31

For deq,0 0, to make _Vo0, we have the following sufcient condition:Ag0wPT P PAg0wP mIo0, 32

with mZ0.Assume the unmatched uncertainties deq,0 is bounded by b1JedJ b2 with b1,b240, we

have

dTeq,0Ped eTd Pdeq,0r2lmaxPb1JedJ22 b2JedJ2, 33thus

_V 2omJedJ22 2lmaxPb1JedJ22 b2JedJ2:For system with only vanishing unmatched uncertainties, i.e., b1a0 and b2 0, we have

_Vo0 if b1om=2lmaxP. It can be concluded that the desired equilibrium ed 0 is locallyasymptotically stable. For system with both vanishing and nonvanishing unmatcheduncertainties, i.e., b1a0 and b2a0, we have _V 2 is negative outside the set fJedJr2b2lmaxP=m2b1lmaxPg, under the condition that b1om=2lmaxP. We can concludethat JedJ is ultimately bounded by

JedJr2b2lmaxP

m2b1lmaxP b2m

2lmaxPb1

:

Based on the above analysis, we seek for solutions of w and P which can maximizem=2lmaxP such that the system could be robust against unmatched uncertainties,meanwhile the ultimate bound of JedJ can be minimized.Pre and post multiplying Eq. (32) by P1, and letting P P1, we have

PAT APwTgT0g0w mP2o0, 34

which can be casted in the LMI form as

maxm

2lmaxPPAT APwTgT0 g0w

m

pP

mp

P I

" #o0

Po0

8>>>>>>>>>>>:

: 35

Z.-Q. Guo et al. / Journal of the Franklin Institute 351 (2014) 226122822274The above LMIs can be solved numerically.

5. Numerical validations

For simulation, fr is modeled as a combination of viscous friction and Coulomb friction,that is, fr fv _x fc sgn _x, where fv0.2 and fc0.3. Similarly, tf is modeled astf tv _y tc sgn _y, where tv 0:2 and tc 0:3. Both frictions are vanishing terms andassumed to be unknown. From the dynamic Eq. (3), it is evident that tf is the matcheduncertainty to the control input while fr is the unmatched uncertainty. Initial states for theWIP system are x 0,0,0:1,0T .

Case 1. In this case, a linear controller based on LQR method [29], is applied to the WIPsystem with and without the joint friction tf . The WIP travels on a at surface. Choose fq1, q2,q3, q4g f50, 0:1, 500, 1g, R0.8. We obtain the feedback gains as k 7:9057,10:7948, 29:9739,3:1183T . The results are shown in Fig. 4. For WIP system withoutthe joint friction, the LQR based linear controller shows effectiveness that the wheel reaches thedesiHowthewhi

C

is aparpen

Z.-Q. Guo et al. / Journal of the Franklin Institute 351 (2014) 22612282 2275(a) To compare the WIP responses under the SMC and the LQR based linear controller,the weighting matrices Q and R are chosen the same as in Case 1. The feedback gainsare obtained as k 7:9057, 10:7948, 29:9739, 3:1183. Set c1 1, solving Eqs.(23)(25) yields c2, c3, c4 1:330530, 3:454665, 0:2738175. The results are shown inFig. 5. The WIP reaches the desired setpoint smoothly and stays still afterwards, thependulum is balanced at ye 0 and the control signal shows switching behavior. Theswitching surface is reached at t0.87 s, which is a fairly short time.Comparing the results in Case 2(a) with the results in Case 1 when tf 0, theperformances are almost the same in terms of tracking error proles and control

0 10 20 300

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

time(s)

Whe

el P

ositi

onx

(m)

ff 0

reference

0 10 20 300.04

0.02

0

0.02

0.04

0.06

0.08

0.1

time(s)

Pen

dulu

m T

iltin

g A

ngul

ar

(rad

)

f=0

f 0

0 10 20 300.5

0

0.5

1

1.5

2

2.5

3

time(s)

Con

trol S

igna

l u (N

m)

f=0

f 0

Fig. 4. Case 1: time responses of x, y and u under linear controller based on LQR. WIP travels on a at surface.gainSystred setpoint smoothly with a small overshoot, and the pendulum angle stays around zero.ever, the LQR based linear controller cannot function well when the joint friction exists inWIP system. The pendulum and the wheel keep vibrating around the desired positions,ch does not meet the desired performance.ase 2. In this case, we consider only the joint friction tf exists in the WIP system, whichmatched uncertainty. The WIP travels on a at surface. SMC is applied with

ameters designed as the following. Refer to Eq. (7), the reference position for thedulum is yr 0 since j 0, the switching gain is r 0:1 jcgj0:3 0:2jx4j. Feedbacks for the nominal controller u0 are obtained based on LQR method.em with and without the joint friction tf are considered.

(b)

RemCassysttheswicoegaininstsign

C

systonacsholarger than the one in Case 2(a), which is due to the higher feedback gains used for thenominal linear controller in Case 2(b). The results are consistent with the discussions inSection 4.

ark 7. It is noticeable that the vector c used in Case 2(b) is quite close to the one incaused by the matched uncertainty, and then the switching surface is designed by LQRfor the sliding manifold. Thus, our new SMC approach achieves both robust andoptimal properties.To illustrate our discussion in Section 4 that the selection of the weighting matricesdirectly affects the system performance, we select the weighting matrix Q to be thesame as in Case 2(a), while the weighting factor for the control input to be R0.08,which is smaller than the one used in Case 2(a). The obtained feedback gains arek 25:0000, 33:7972, 90:3320, 8:6977, which are larger than in Case 2 (a). Letc1 1, solving Eqs. (23)(25) yields c2, c3, c4 1:307674, 3:338392, 0:200306. Theresults are shown in Fig. 6. The WIP system is stabilized and reached the desiredsetpoint. We can see the switching amplitude of the control signal in Case 2(b) is muchproles. However, in Case 1 the control system is directly designed by LQR, whereas inCase 2(a), the control system is rst designed by SMC in order to eliminate the effect

0 10 20 300

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

time (s)

Whe

elPo

sitio

nx

(m)

x1xr

0 10 20 300.04

0.02

0

0.02

0.04

0.06

0.08

0.1

time (s)

Pend

ulum

Tilti

ngA

ngul

ar

(rad)

0 10 20 300.6

0.4

0.2

0

0.2

0.4

0.6

0.8

time (s)

Con

trolS

igna

lu(N

m)

0 10 20 300.1

0

0.1

0.2

0.3

0.4

time (s)

Slid

ing

Man

ifold

Fig. 5. Case 2 (a): time responses of x, y, u and s under SMC. WIP is considered traveling on a at surface. Onlymatched uncertainty, the joint friction tf 0:2 _y 0:3 sgn _y, exists in the system. R0.8 is used in SMC design.

Z.-Q. Guo et al. / Journal of the Franklin Institute 351 (2014) 226122822276e 2(a). In other words, a minor change of the vector cmight lead to large changes in theem responses and the changes are unpredictable, which indicates the difculty in tuningvector c directly to achieve desired responses and shows the drawback of the existingtching surface designs. With our proposed design method, the switching surfacefcients are determined indirectly. The sliding manifold is determined by the feedbacks of the nominal linear controller, which can be tuned in a systematic way. Forance, we can choose more of control penalty in R so as to prevent overlarge controlals. The advantage of our proposed method is immediately obvious.

ase 3. In this case, we consider both frictions, tf and fr, exist in the WIP system, i.e., theem is in the presence of both matched and unmatched uncertainties. The WIP travelsa at surface. SMC is applied with r 0:1 jcgj0:3 0:2jx4j jcc2 bc4=b2j 0:3 0:2jx2j, and all other parameters the same as in Case 2. The results are

wn in Fig. 7. The WIP reaches the desired setpoint and the pendulum is nally balanced

1.2

1.4

1.6

(m)

0.08

0.1

lar

(rad

)

0.4

0.6

0.8(N

m) 0.3

0.4

d

0 10 20 300

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

time(s)

Whe

el P

ositi

on x

(m)

x1xr

0 10 20 300.04

0.02

0

0.02

0.04

0.06

0.08

0.1

time(s)

Pen

dulu

m T

iltin

g A

ngul

ar

(rad

)

0 10 20 301.5

1

0.5

0

0.5

1

1.5

time(s)

Con

trol S

igna

l u (N

m)

0 10 20 300.1

0

0.1

0.2

0.3

0.4

time(s)

Slid

ing

Man

ifold

Fig. 6. Case 2 (b): time responses of x, y, u and s under SMC. WIP is considered traveling on a at surface. Onlymatched uncertainty, the joint friction tf 0:2 _y 0:3 sgn _y, exists in the system. R0.08 is used in SMC design.

Z.-Q. Guo et al. / Journal of the Franklin Institute 351 (2014) 22612282 2277at the upright position, i.e., ye 0, which indicates that the proposed SMC is also robust tounmatched uncertainties.

Case 4. In this case, the WIP traveling on an inclined surface is considered. The slopeangle is known as j p=15 rad. Both frictions tf and fr exist in the system. SMC is appliedwith parameters designed as the following. Refer to Eq. (7), the reference position for thependulum is yr 0:2547 rad. Feedback gains k 7:9057,10:7535,30:0154,3:1275for the nominal controller u0 are obtained based on LQR method, where Q and R arechosen the same as in Case 1. Next, let c1 1, solving Eqs. (23)(25) yieldsc2,c3,c4 1:327784, 3:470293, 0:283022. The switching gain is the same as in Case 3.The results are shown in Fig. 8, the WIP reaches the desired setpoint, while the pendulumis balanced at the new equilibrium point ye yr 0:2547 rad. The simulation results areconsistent with the theoretical analysis in Section 2.4.

Case5. In this case, the WIP travels on the same surface as in Case 4. However, the slopeangle is assumed to be unknown to the designer, thus j0 0 and yr 0 are used in thecontroller design. Both frictions tf and fr exist in the system. SMC is applied. Feedbackgains k 7:4802, 11:2445, 26:9865, 5:5473 for the nominal controller u0 areobtained based on LMI method. Next, let c1 1, solving Eqs. (23)(25) yieldsc2, c3, c4 1:241766, 2:113164, 0:1890710. The results are shown in Fig. 9. The

0 10 20 300

0.2

0.4

0.6

0.8

1

time (s)

Whe

el P

ositi

on x

x1

xr

0 10 20 300.02

0

0.02

0.04

0.06

time (s)

Pend

ulum

Tilti

ngA

ngu

0 10 20 300.6

0.4

0.2

0

0.2

time (s)

Con

trolS

igna

l u

0 10 20 300.1

0

0.1

0.2

time (s)

Slid

ing

Man

ifol

Fig. 7. Case 3: time responses of x, y, u and s under SMC. WIP is considered traveling on a at surface. Bothmatched uncertainty, the joint friction tf 0:2 _y 0:3 sgn _y, and unmatched uncertainty, the ground frictionfr 0:2 _x 0:3 sgn _x, exist in the system.

0 10 20 300.5

0

0.5

1

1.5

time(s)

Whe

el P

ositi

on x

(m)

x1xr

0 10 20 300.1

0.15

0.2

0.25

0.3

0.35

time(s)

Pen

dulu

m T

iltin

g A

ngul

ar

(rad

)

0 10 20 300.5

0

0.5

1

1.5

time(s)

Con

trol S

igna

l u (N

m)

0 10 20 300.6

0.5

0.4

0.3

0.2

0.1

0

0.1

time(s)

Slid

ing

Man

i fol

d

Fig. 8. Case 4: time responses of x, y, u and s under SMC. WIP is considered traveling on an inclined surface withknown slope angle j p=15. Both matched uncertainty, the joint friction tf 0:2 _y 0:3 sgn _y, and unmatcheduncertainty, the ground friction fr 0:2 _x 0:3 sgn _x, exist in the system.

1.5 0.35

ad) 6 0.3

Z.-Q. Guo et al. / Journal of the Franklin Institute 351 (2014) 226122822278pendulum is balanced at the new balanced position ye, which is around y 0:2547 rad,thus steady state error for e3 exist as e3,s yeyra0. From s ce 0, steady-state errorfor e1 also exist as e1,s c3e3,s=c1 0:5382 m, which meets the simulation results.Comparing the results in Case 4 and Case 5, we can see the necessity of adjusting thereference position of pendulum when the WIP travels on an inclined surface. yr cannot becomputed according to Eq. (7) if system parameters involved are unknown. When taskrepeats, we obtain yr ye, which can be incorporated in the controller design to eliminatesteady-state errors.

6. Implementation and experiment results

In simulations, an ideal model of the WIP is used. To stabilize the WIP system inabsence of uncertainties, the feedback gains for the nominal linear controller can be chosenin a wide range as long as Ag0k is Hurwitz. However, considering the existence ofmismatch between the real-time system model and the mathematical model (1) and (2), thefeedback gains obtained from simulations may not function well on the real-time platform,thus need to be adjusted through experimental testings on the WIP prototype.

0 10 20 300.5

0

0.5

1

time(s)

Whe

el P

ositi

on x

(m)

x1xr

0 10 20 300

0.05

0.1

0.15

0.2

0.25

0.3

time(s)

Pen

dulu

m T

iltin

g A

ngul

ar

(r

0 10 20 304

2

0

2

4

time(s)

Con

trol S

igna

l u (N

m)

0 10 20 30

0

0.05

0.05

0.1

0.15

0.2

0.25

time(s)

Slid

ing

Man

ifold

Fig. 9. Case 5: time responses of x, y, u and s under SMC. WIP is considered traveling on an inclined surface withunknown slope angle. Both matched uncertainty, the joint friction tf 0:2 _y 0:3 sgn _y, and unmatcheduncertainty, the ground friction fr 0:2 _x 0:3 sgn _x exist in the system.

6.1. Regulation task

For implementation, rst, we consider a simple regulation task that is to balance theWIP at the original position on a at surface, i.e., xr0, vr0, and j 0. In order to ndappropriate feedback gains, rst a linear controller is applied with the feedback gainsobtained from simulations. However, strong vibrations are observed, which can beexplained as the following. For systems having backlash in the driving mechanism, largefeedback gains could easily incur vibrations [9]. In our WIP system, the backlash isproduced by the gearbox. To reduce or avoid vibrations, feedback gains for the velocityterms, k2 and k4, should be minimized [9]. From experimental testing, we observe that thesystem vibration reduces signicantly as k2 and k4 decrease. It is also found that largefeedback gains are necessarily needed for the position terms, hence, k1 and k3 should bekept at the original designed level. If k3 is too small, the pendulum could easily fall downbecause the torque generated by the motor is not large enough to overcome the effect ofthe gravity. If k1 is too small, the position control of the wheels fails that a steady stateerror exists in the response of the wheel position. After several trials, the feedback gains areadjusted to k 7:9057, 0:5, 29:9739, 1:8.For implementing the SMC, the obtained feedback gains are used for the nominal linear

controller u0. Let c1 1, solving Eqs. (23)(25) yields c2, c3, c4 0:026051, 3:627349,0:092766. Fig. 10 shows the experimental results for the WIP under the SMC and the

Z.-Q. Guo et al. / Journal of the Franklin Institute 351 (2014) 22612282 2279linear controller. When the linear controller is applied, the WIP becomes unstable in 10 s.By applying the SMC, the WIP is stabilized. The wheels stay around the original place andthe pendulum is balanced around y 0, which veries the effectiveness of the SMC inhandling system uncertainties. The value of the dened sliding variable s switchesaround zero.

0 5 10 15 20 25 30time (s)

x(m

)

SMCLC

0 10 20 30

(rad)

time (s)

SMCLC

0 5 10 15 20 25 30

1

0.5

0

0.5

20

10

0

10

time (s)

u (N

m)

SMCLC

0 10 20 30

0.5

0

0.5

0.1

0.05

0

0.05

0.1

time (s)

Fig. 10. Experimental testing results for regulation task: time responses of x, y, u and s under SMC and linear

controller. The WIP is placed on at surface.

0 5 10 15 20 25time (s)

x(m

) x1xr

0 5 10 15 20 25

(ra

d)

time (s)

0 5 10 15 20 25

1

0

1

2

5

0

5

time (s)

u(N

m)

0 5 10 15 20 25

0.2

0.1

0

0.1

0.2

0.2

0.1

0

0.1

0.2

time (s)

Fig. 11. Experimental testing results for setpoint task: time responses of x, y, u and s under SMC. The WIP isplaced on a at surface. The reference trajectory (5) is applied.

Z.-Q. Guo et al. / Journal of the Franklin Institute 351 (2014) 2261228222806.2. Reaching a setpoint

We consider the WIP travels on a at surface, i.e., j 0. The reference trajectory for theWIP is the same as in simulations. SMC is applied and the sliding surface coefcients arechosen the same as for the regulation task. Experiment results are shown in Fig. 11.The WIP reached the desired setpoint x1.5 m and stays there afterwards. The pendulumis stabilized around the upright position. SMC shows effectiveness for setpoint control ofthe WIP system.

7. Conclusion

In this work, control of an underactuated WIP system with input coupling is studied. Anovel design of SMC is presented. First, a linear switching surface and the SMC isintroduced. The system reaches the switching surface in a nite time under the proposedSMC. Next, after the system reaches the switching surface, the switching surface design isdiscussed, which mainly focuses on choosing the switching surface coefcients to stabilizethe sliding manifold. To avoid the complexity on tuning the coefcients directly, wetransform the problem into a simple nominal linear controller design problem, which notonly simplies the tuning process, but also provides one extra degree of freedom in control.By utilizing the extra degree of freedom, optimal and robust linear control techniques areincorporated in the SMC design. Intensive simulations and experiment testings areconducted to verify the effectiveness of the proposed SMC and satisfactory results areachieved. Thanks to the easiness in tuning the feedback gains for the nominal linear

[21] V. Azhmyakov, On the set-valued approach to optimal control of sliding mode processes, Journal of theFranklin Institute-Engineering and Applied Mathematics 349 (4) (2012) 13231336.

[22] M. Basin, P. Rodriguez-Ramirez, Sliding mode controller design for linear systems with unmeasured states,

Journal of the Franklin Institute-Engineering and Applied Mathematics 349 (4) (2012) 13371349.

[23] M. Basin, P. Rodriguez-Ramirez, A. Ferrara, D. Calderon-Alvarez, Sliding mode optimal control for linearcontroller, the proposed SMC shows realizability and superiority regarding theimplementation on real-time platforms of underactuated mechanical systems.

References

[1] O.-S. Reza, Nonlinear Control of Underactuated Mechanical Systems with Application to Robotics and

Aerospace Vehicles, MIT PhD Thesis, 2001.

[2] I. Fantoni, R. Lozano, Nonlinear Control for Underactuated Mechanical Systems, 2002.

[3] F. Grasser, A. DArrigo, S. Colombi, A.C. Rufer, JOE: a mobile, inverted pendulum, IEEE Transaction on

Industrial Electronics 49 (1) (2002) 107114.

[4] J. Solis, A. Takanishi, Development of a wheeled inverted pendulum robot and a pilot experiment with

master students, in: Proceedings of the 7th International Symposium on Mechatronics and its Applications

(ISMA10), Sharjah, UAE, 2010, pp. 16.

[5] J. Solis, R. Nakadate, T. Yamamoto, A. Takanishi, Introduction of mechatronics to undergraduate students

based on robotic platforms for education purposes, in: Proceedings of the 18th IEEE International

Symposium on Robot and Human Interactive Communication Toyama, Japan, 2009, pp. 693698.

[6] P. Oryschuk, A. Salerno, A.M. Al-Husseini, J. Angeles, Experimental validation of an underactuated two-

wheeled mobile robot, IEEE/ASME Transactions on Mechatronics 14 (2) (2009) 252257.

[7] C.-H. Chiu, Y.-W. Lin, C.-H. Lin, Real-time control of a wheeled inverted pendulum based on an intelligent

model free controller, Mechatronics 21 (2011) 523533.

[8] C.-H. Huang, W.-J. Wang, C.-H. Chiu, Design and implementation of fuzzy control on a two-wheel inverted

pendulum, IEEE Transactions on Industrial Electronics 58 (7) (2011) 29883001.

[9] T. Takei, R. Imamura, S. Yuta, Baggage transportation and navigation by a wheeled inverted pendulum

mobile robot, IEEE Transactions on Industrial Electronics 56 (10) (2009) 39853994.

[10] M.-S. Park, D. Chwa, Swing-up and stabilization control of inverted-pendulum systems via coupled sliding-

mode control Method, IEEE Transactions on Industrial Electronics 56 (9) (2009) 35413555.

[11] J. Huang, Z.-H. Guan, T. Matsuno, T. Fukuda, K. Sekiyama, Sliding-mode velocity control of mobile-

wheeled inverted-pendulum systems, IEEE Transactions on Robotics 26 (4) (2010) 750758.

[12] M.-S. Park, D. Chwa, Orbital stabilization of inverted-pendulum systems via coupled sliding-mode control,

IEEE Transactions on Industrial Electronics 56 (9) (2009) 35563570.

[13] H. Ashrauon, R.S. Erwin, Sliding mode control of underactuated multibody systems and its application to

shape change control, International Journal of Control 81 (12) (2008) 18491858.

[14] R. Xu, U. Ozguner, Sliding mode control of a class of underactuated systems, Automatica 44 (2008) 233241.

[15] S. Kurode, P. Trivedi, B. Bandyopadhyay, P.S. Gandhi, Second order sliding mode control for a class of

underactuated systems, in: Proceedings of the 12th IEEE International Workshop on Variable Structure

Systems (VSS12), 2012, pp. 458462.

[16] S. Riachy, Y. Orlov, T. Floquet, R. Santiesteban, J. Richard, Second order sliding mode control of

underactuated mechanical systems I: local stabilization with application to an inverted pendulum,

International Journal of Robust Nonlinear Control 18 (4/5) (2007) 529543.

[17] R. Santiesteban, T. Floquet, Y. Orlov, S. Riachy, J. Richard, Second-order sliding mode control of

underactuated mechanical systems II: orbital stabilization of an inverted pendulum with application to swing

up/balancing control, International Journal of Robust Nonlinear Control 18 (4/5) (2007) 544556.

[18] R. Iriarte, L. T. Aguilar, L. Fridman, Second order sliding mode tracking controller for inertia wheel

pendulum, Journal of the Franklin Institute, 2012, http://dx.doi.org/10.1016/j.jfranklin.2012.10.013.

[19] M. Lopez-Martinez, J.A. Acosta, J.M. Cano, Non-linear sliding mode surfaces for a class of underactuated

mechanical systems, IET Control Theory and Applications 4 (10) (2010) 21952204.

[20] J.-X. Xu, Z.-Q. Guo, T.H. Lee, Synthesized design of a fuzzy logic controller for an underactuated unicycle,

Fuzzy Sets and Systems 207 (2012) 7793.

Z.-Q. Guo et al. / Journal of the Franklin Institute 351 (2014) 22612282 2281systems, Journal of the Franklin Institute-Engineering and Applied Mathematics 349 (4) (2012) 13501363.

[24] B. Mirkin, P.-O. Gutman, Y. Shtessel, Coordinated decentralized sliding mode MRAC with control cost

optimization for a class of nonlinear systems, Journal of the Franklin Institute-Engineering and Applied

Mathematics 349 (4) (2012) 13641379.

[25] A. Polyakov, Minimization of disturbances effects in time delay predictor-based sliding mode control

systems, Journal of the Franklin Institute-Engineering and Applied Mathematics 349 (4) (2012) 13801396.

[26] T.R. Oliveira, A.J. Peixoto, L. Hsu, Global real-time optimization by output-feedback extremum-seeking

control with sliding modes, Journal of the Franklin Institute-Engineering and Applied Mathematics 349 (4)

(2012) 13971415.

[27] A.F. de Loza, M. Jimenez-Lizarraga, L. Fridman, Robust output nash strategies based on sliding mode

observation in a two-player differential game, Journal of the Franklin Institute-Engineering and Applied

Mathematics 349 (4) (2012) 14161429.

[28] J.-X. Xu, A quasi-optimal sliding mode control scheme based on control Lyapunov function, Journal of the

Franklin Institute-Engineering and Applied Mathematics 349 (4) (2012) 14451458.

[29] S.Y. Seo, S.H. Kim, S.-H. Lee, S.H. Han, H.S. Kim, Simulation of attitude control of a wheeled inverted

pendulum, in: Proceedings of the International Conference on Control, Automation and Systems, Korea,

2007, pp. 22642269.

Z.-Q. Guo et al. / Journal of the Franklin Institute 351 (2014) 226122822282

Design and implementation of a new sliding mode controller on an underactuated wheeled inverted pendulumIntroductionProblem formulationWIP modelControl objectiveTrajectory planningEquilibrium point of the WIP

Sliding mode controller designSMC design for system with unmodeled frictionsSMC design for system with parameter variations

Switching surface designProblem transformationNominal linear controller designLQR based optimal designLMI based robust design

Numerical validationsImplementation and experiment resultsRegulation taskReaching a setpoint

ConclusionReferences