le

n

Zero dynamics 2010 Elsevier Ltd. All rights reserved.

1. Introduction

Wheeled inverted pendulums have attracted a lot of attentionrecently (Brooks et al., 2004; Gans & Hutchinson, 2006; Grasser,Arrigo, Colombi, & Rufer, 2002; Jung & Kim, 2008; Li & Luo,2009; Nasrallah, Michalska, & Angeles, 2007; Pathak, Franch, &Agrawal, 2005) as shown in Fig. 1. Similar systems like the cartand pendulums have been studied in the literature Ibanez, Frias,and Castanon (2005) and Zhang and Tarn (2002). The differencesfrom these systems are that the inverted pendulumsmotion in thepresent system is not planar and themotors driving thewheels aredirectly mounted on the pendulum body (Pathak et al., 2005).Motion of wheeled inverted pendulums is governed by under-

actuated configuration, i.e., the number of control inputs is lessthan the number of degrees of freedom to be stabilized (Isidori,Marconi, & Serrani, 2003), which makes it difficult to applythe conventional robotics approach to control EulerLagrangesystems. Due to these reasons, increasing effort has been madetowards control design that guarantees stability and robustness formobile wheeled inverted pendulums.

I This work is supported by Shanghai Pujiang Program under GrantNo. 08PJ1407000 and the Natural Science Foundation of China under GrantNos. 60804003, 60935001 and the New Faculty Foundation under GrantNo. 200802481003. The material in this paper was not presented at any con-ference. This paper was recommended for publication in revised form by AssociateEditor Shuzhi Sam Ge under the direction of Editor Miroslav Krstic. Corresponding author. Tel.: +86 21 34204616; fax: +86 21 34204616.E-mail addresses: zjli@ieee.org, lzjuec@hotmail.com (Z. Li).

Although wheeled inverted pendulums systems are intrinsi-cally nonlinear and their dynamics are described by nonlinear dif-ferential equations, if the system operates around an operatingpoint, and the signals involved are small, we can obtain a linearmodel approximating the nonlinear system in the region of op-eration. In Ha and Yuta (1996), motion control was proposed us-ing a linear state-space model. In Grasser et al. (2002), dynamicswas derived using a Newtonian approach and the control was de-signed based on the dynamic equations linearized around an op-erating point. In Salerno and Angeles (2003), dynamic equations ofthe inverted pendulum were studied involving pitch and rotationangles of the twowheels as the variables of interest, and in Salernoand Angeles (2004) a linear controller was designed for stabiliza-tion considering robustness as a condition. In Blankespoor andRoe-mer (2004), a linear stabilizing controller was derived by a planarmodelwithout considering yaw. In Kim, Kim, and Kwak (2005), theexact dynamics of a two-wheeled inverted pendulumwas investi-gated, and linear feedback control was developed on the dynamicmodel. In Pathak et al. (2005), a two-level velocity controller viapartial feedback linearization and a stabilizing position controllerwere derived.Based on the idea of linearization, a model-based approach

is generally utilized in dynamic control. If accurate knowledgeof the dynamic model is available, the model-based controlcan provide an effective solution to the problem. However,wheeled inverted pendulum control is characterized by unstablebalance and unmodelled dynamics, and subject to time varyingexternal disturbances, in the form of parametric and functionaluncertainties, which are generally difficult to model accurately.Therefore, traditional model-based control may not be the idealAutomatica 46 (2

Contents lists availa

Autom

journal homepage: www.els

Brief paper

Robust adaptive motion/force control forZhijun Li a,, Yunong Zhang ba Department of Automation, Shanghai Jiao Tong University, Shanghai, 200240, Chinab School of Information Science & Technology, Sun Yat-Sen University, Guangzhou, 510275

a r t i c l e i n f o

Article history:Received 14 June 2009Received in revised form9 February 2010Accepted 21 April 2010Available online 7 June 2010

Keywords:Wheeled inverted pendulumsNonholonomic constraintsUnmodelled dynamics

a b s t r a c t

Previous works for wheeledorder to make the control deas large as needed. Nevertheadaptive robust motion/forcand functional uncertaintieswheeled inverted pendulumdynamics. Based on Lyapunogiven bounded reference siguniform boundedness of allthrough extensive simulatio0005-1098/$ see front matter 2010 Elsevier Ltd. All rights reserved.doi:10.1016/j.automatica.2010.05.015010) 13461353

ble at ScienceDirect

atica

evier.com/locate/automatica

wheeled inverted pendulumsI

, China

inverted pendulums usually eliminate nonholonomic constraint force insign easier, under the assumption that the friction force from the ground isess, such an assumption is unfeasible in practical applications. In this paper,control for wheeled inverted pendulums is investigated with parametric. The proposed robust adaptive controls based on physical properties ofs make use of online adaptation mechanism to cancel the unmodelledv synthesis, the proposed controls ensure that the system outputs track thenals within a small neighborhood of zero, and guarantee the semi-globalclosed loop signals. The effectiveness of the proposed controls is verifieds.

cZ. Li, Y. Zhang / Automati

approach since it generally works best only when the dynamicmodel is known exactly. The presence of uncertainties anddisturbances would disrupt the function of the traditional model-based feedback control and lead to unstable balance.Moreover, the wheeled inverted pendulum is definitely differ-

ent from other nonholonomic systems subject to (i) only kinematicconstraints which geometrically restrict the direction of mobility,i.e., wheeledmobile robot (Ge,Wang, Lee, & Zhou, 2001; Ge,Wang,& Lee, 2003); (ii) only dynamic constraints due to dynamic balanceat passive degrees of freedom where no force or torque is applied,i.e., the manipulator with passive link (Arai & Tanie, 1998; Luca &Oriolo, 2002); (iii) both kinematic constraints and dynamic con-straints. It is obvious that the wheeled inverted pendulum is morecomplex than the former two cases, therefore, the controls suitablefor (i) and (ii) cannot be directly applied for (iii).A challenging problem is to control a mobile inverted pen-

dulum system whose cart is not constrained by guide rail likecartpendulum systems, but moves in its terrain while balancingthe pendulum. Therefore, the nonholonomic constraint force be-tween the wheels and the ground should be considered in order toavoid slipping or slippage. A similar application of interacting withenvironments can be found in Dong (2002), where motion/forcecontrol is considered for mobile manipulators under holonomicconstraints. Recent works, including Grasser et al. (2002), Jung andKim (2008), Li and Luo (2009), Nasrallah et al. (2007) and Pathaket al. (2005), where the wheeled inverted pendulummoves on theplanar plane or on the incline plane, do not consider nonholonomicconstraint force, i. e. friction force, while assuming beforehand thatthe ground can provide enough friction as needed, but in practi-cal applications, this assumption is difficult to satisfy. When theground friction cannot support motion, the control performanceby these controllers will be degraded.In this paper, by discovering and utilizing the unique physical

properties of wheeled inverted pendulums, we separate thezero-dynamics subsystem to simplify the model. Then, wepropose a robust adaptive motion/force control for wheeledinverted pendulums. Since the system except the zero-dynamicssubsystem is still a MIMO nonlinear system, we propose adaptiverobust controls to accommodate the presence of parametric andfunctional uncertainties in the dynamics of wheeled invertedpendulums.The main contributions of this paper are that: (i) adaptive

robust motion/force control is developed for wheeled invertedpendulums by using their physical properties with parametricand functional uncertainties; (ii) the nonholonomic constraintforce between the wheels and the ground is considered inorder to avoid slipping or slippage; (iii) based on Lyapunovsynthesis, motion/force stabilities are achieved and the input-to-state stability properties of the zero dynamics are used to derivebounds on the the tracking errors.

2. Preliminaries

In the following study, let denote the 2-norm, i.e. givenA = [aij] Rmn, A =

mi=1nj=1 |aij|2.

Lemma 2.1. Let e = H(s)r with H(s) representing a (n m)-dimensional strictly proper exponentially stable transfer function, rand e denoting its input and output, respectively. Then r Lm2

Lm

implies that e, e Ln2Ln, e is continuous, and e 0 as t .

If, in addition, r 0 as t , then e 0 (Ge, Lee, & Harris,1998).

Lemma 2.2. For x 0 and = 1+ 1 1with t > 0, we have

(1+t)2

ln(cosh(x))+ x.a 46 (2010) 13461353 1347

Fig. 1. Mobile wheeled inverted pendulum.

Proof. If x 0, we have x0 2e2+1d < x0 2e2 d = 1 e2x n , only mvariables of z(t) can be controlled, the control objective can bespecified as: design a controller that ensures the tracking errorsof zi(1 i m) from their respective desired trajectories zid(t) tobe within a small neighborhood of zero, i.e.,

|zi(t) zid(t)| i, i = 2, 3 (18)where i > 0. Ideally, i should be the threshold of measurablenoise, while the constraint force error ( d) is bounded in acertain region. At the same time, n variables of z(t) are to be keptbounded. The variables zi, (1 i m), can be thought of as theoutput equation of the system.

Assumption 4.1. The desired reference trajectory zid(t), (1 i m) is assumed to be bounded and uniformly continuous, and hasbounded and uniformly continuous derivatives up to the secondorder. The desired d(t) is bounded and uniformly continuous.

Remark 4.1. Since we can plan and design the desired trajectoryfor zid(t) before implementing control, it is reasonable and feasiblethat we give the trajectory satisfying the Assumption 4.1.

5. Robust adaptive motion/force control

5.1. z2 and z3-subsystems motion control

Let us define the following notations as e = d, e = d,r = d1e , s = e+1e , where d = [z3d, z2d]T , r = [z3r , z2r ]Tis the reference signal described in internal state space, and 1 ispositive diagonal. Apparently, we have

= r + s. (19)From the dynamic equation (17) together with (19), we have

Ms = VsMr V r D +B1U. (20)Let M0, V0, D0 and B10 be nominal parameter vectors whichgive the corresponding nominal functionsM0r + V0r +D0 and(B10)

1, respectively. There exist some finite positive constantsci > 0 (1 i 6), such that q, q Rn, M M0 c1,VV0 c2+c3q, DD0 c4+c5q, B1B10 c6.The proposed control for the system is given as

U = u1 + u2 (21)where u1 is the nominal control

u1 = B110 Kps+B110 (M0r + V0r +D0) (22)where Kp is a diagonal positive constant, and u2 is designedto compensate for the parameter errors and the functionapproximation errors arising from approximating the unknownfunction as

u2 = u21 + u22 + u23 + u24 + u25 + u26 (23)u21 = 1b sgn(s)(ln(cosh(1))+ ), (24)

u22 = 1b sgn(s)(ln(cosh(2))+ ), (25)u23 = 1b sgn(s)(ln(cosh(3))+ ), (26)a 46 (2010) 13461353 1349

u24 = 1b sgn(s)(ln(cosh(4))+ ), (27)

u25 = 1b sgn(s)(ln(cosh(5))+ ), (28)

u26 = 1b sgn(s)(ln(cosh(6))+ ), (29)with sgn(s) = ss , which are adaptively tuned according toc1 = 1c1 + 1sr, c1(0) > 0 (30)c2 = 2c2 + 2sr, c2(0) > 0 (31)c3 = 3c3 + 3srq, c3(0) > 0 (32)c4 = 4c4 + 4s, c4(0) > 0 (33)c5 = 5c5 + 5sq, c5(0) > 0 (34)c6 = 6c6 + 6su1, c6(0) > 0 (35)with 1 = c1r, 2 = c2r, 3 = c3qr, 4 = c4,5 = c5q, 6 = c6u1, and i > 0 being design parametersand satisfying

limti = 0 (36) 0i(t)dt = % 0 as design parameter.

Remark 5.1. As we discuss in Section 1, a wheeled inverted pen-dulum subject to kinematic constraints and dynamic constraints isapparently different from generalized nonholonomic systems, thecontrol (23) is proposed based on the obtained reducedmodel (17),for the generalized nonholonomic system, if we can obtain the cor-responding reduced model similar to (17), the control (23) is alsoapplicable.

To analyze closed loop stability for the z2 and z3 subsystems,consider the following Lyapunov function candidate

V1 = 12 sTMs+ 1

2

6i=1ci1i ci (38)

where ci = ci ci, i = 1, . . . , 6, therefore, c i = c i. Its timederivative is given by

V1 = sT(Ms+ 1

2Ms)+

6i=1ci1i c i. (39)

Considering Property 3.2, and substituting (20) into (39), we have

V1 = sT[B1U

(Mr + V r +D

)]+ 6i=1ci1i c i. (40)

Integrating (22) and (23) into (40), we have

V1 = sT[(B1 B10)u1 +B1u2 +B10u1 Mr V r D

]+

6i=1ci1i c i

= sT [(B1 B10) u1 +B16i=1u2i Kps (M M0)r

6

(V V0)r (D D0)] +

i=1ci1i c i

c1350 Z. Li, Y. Zhang / Automati

= sTKps+ sT[B1u21 (M M0)r

]+ sT [B1(u22 + u23) (V V0)r]+ sT [B1(u24 + u25) (D D0)]

+ sT [B1u26 + (B1 B10)u1]+6i=1

1ici c i. (41)

Considering (24) and (30), and Lemma 2.2, the second right-hand term of (41) is bounded by

sT[B1u21 (M M0)r

]+ 11c1 c1

c1sr s(ln(cosh(1))+ )+ 11c1 c1

c1sr c1sr + c1[1

1c1 sr

]= 11c1c1 = 1

1

(c1 12 c1

)2+ 141c21 (42)

Considering (25), (26), (31), (32), and Lemma 2.2, the thirdright-hand term of (41) is bounded by

sT[B1(u22 + u23) (V V0)r

]+ 3i=2

1ici c i

c2sr + c3sqr 3i=2s(ln(cosh(i))+ )

+3i=2

i

ici c i

c2sr + c3sqr c2sr c3sqr+ c2

[2

2c2 sr

]+ c3

[3

3c3 sqr

]

=3i=2

i

icici =

3i=2

i

i

(ci 12 ci

)2+ i4ic2i (43)

Similarly, considering (27), (28), (33), (34) and Lemma 2.2, thefourth right-hand term of (41) is bounded by

sT [B1(u24 + u25) (D D0)]+5i=4

1ici c i

c4s + c5sq 5i=4s(ln(cosh(i))+ )

+5i=4

1ici c i

c4s + c5sq c4s c5sq+ c4

[4

4c4 s

]+ c5

[5

5c5 sq

]

=5i=4

i

icici =

5i=4

i

i

(ci 12 ci

)2+ i4ic2i . (44)

Similarly, considering (29) and (35) and Lemma 2.2, the fifth right-hand term of (41) is bounded by

sT [B1u26 + (B1 B10)u1]+ 16c6 c6[ ] c6su1 c6su1 + c6 66c6 su1a 46 (2010) 13461353

= 66c6c6 = 6

6

(c6 12 c6

)2+ 646c26 . (45)

Combining (42)(45), we could obtain

V1 min(Kp)s2 6i=1

i

i

(ci 12 ci

)2+

6i=1

i

4ic2i .

Therefore

V1 min(Kp)s2 +6i=1

i

4ic2i . (46)

Considering (36) and (37),6i=1

i4ic2i is bounded, there exists

T > t1,6i=1

i4ic2i %1 with the finite constant %1, when

s

%1min(Kp)

, then V1 0. From all the above, s convergesto a small set : s

%1

min(Kp)containing the origin

as T . Integrating both sides of (46) gives V1(t) V1(0)

T0 min(Kp)s2dt +

T0

6i=1

i4ic2i dt . Since ci and i

are constants, moreover,0 idt = %i , we can rewrite (46) as

V1(t)V1(0) T0 min(Kp)s2dt+

6i=1

%i4ic2i

cZ. Li, Y. Zhang / Automati

Proof. From (14)(16), we choose the following function as

V2 = V1 + ln(cosh(z1)). (50)Differentiating (50) along (14) gives

V2 = V1 + tanh(z1)z1= V1 + tanh(z1)m111 (1 v1 g1 d1 m12z2 m13z3). (51)

From (16), we have

1 = m11m131 (lz2 + jz3 + f + k). (52)Integrating (52) into (51), we have

V2 = V1 + tanh(z1)(m131 (lz2 + jz3 + f + k)+m111 (v1 g1 d1 m12z2 m13z3))

= V1 [tanh(z1)(m131 l+m111 m12)tanh(z1)(m131 j+m111 m13)

]T [z2z3

] tanh(z1)(f + k) tanh(z1)m111 (v1 + g1 + d1).

Since tanh(z1) 1, m131 , m12 and m13 are all bounded. Let[m131 l+m111 m12m131 j+m111 m13

]T 1, and m111 2, where 1 and 2 arebounded constants. Considering Assumption 5.1 and Remark 5.3,we have

V2 12min(Kp)s2 +

6i=1

i

4ic2i

+ 1(d + 2)+ L2 (d + 2)+ L2f+ 2(L1 (d + 1)+ L1f ). (53)

Let =6i=1 i4i 2i +1(d+2)+ L2 (d+2)+ L2f +2(L1 (d+1)+ L1f ) and it is apparently bounded positive, wehave V2 0, when s

2

min(Kp), we can choose the proper

Kp such that s can be arbitrarily small. Therefore, we can obtainthat the internal dynamics is stable with respect to the outputz1. Therefore, the z1-subsystem (14) is globally asymptoticallystable.

Theorem 5.1. Consider the system (15)(16) with Assumption 4.1,under the action of control laws (21). For compact set , where(z2(0), z3(0), z2(0), z3(0)) , the tracking errors converges to thecompact set defined in (18), and all the signals in the closed loopsystem are bounded.

Proof. From the results (18), it is clear that the tracking error sconverges to the compact set defined by (18). From Lemma 2.1,we can know e , e are also bounded. From the boundedness ofz2d, z3d in Assumption 4.1, we know that z2, z3 are bounded. Sincez2d, z3d are also bounded, it follows that z2, z3 are bounded. FromLemma5.1,we know that the z1-subsystem (15) is stable, and z1, z1are bounded. This completes the proof.

5.3. Force control

The force control input b is designed as

b = c1ZR+T zd + d Kf e (54)where e = d, Kf is a constant matrix of proportional controlfeedback gains. Substituting the control (21) and (54) into thereduced order dynamics (12) yields

(I + Kf )e = Z[(M(q)R(q)+ C(q, q)R(q)z)+ G+ F a] + b

= ZR+TM1z + c1ZR+TM1zd. (55)a 46 (2010) 13461353 1351

Since z2 z2d, z3 z3d, z1 is bounded from Theorem 5.1,ZR+TM1z + c1ZR+TM1zd is also bounded, therefore, the size ofe can be adjusted by choosing the proper gain matrix Kf . Since s,z, z, r , r , and e are all bounded, it is easy to conclude that isbounded from (21) and (54).

6. Simulation

Let us consider a mobile wheeled inverted pendulum as shownin Fig. 1. The following variables have been chosen to describe thevehicle (see also Fig. 1): l, r : the torques of the left and rightwheels; : the tilt angle of the pendulum; : the direction angleof the mobile platform; r: the radius of the wheels; d: the distancebetween the two wheels; 2l: the length of the pendulum; m: themass of the mobile pendulum; Mw: the mass of each wheel; Im:the moment of inertia of the mobile pendulum; Iw: the inertiamoment of each wheel; g: gravity acceleration; : the motionfriction coefficient of the ground.The wheeled inverted pendulum is subject to the following

constraints x sin y cos = 0. Using the Lagrangian approach,we can obtain the reduced dynamics for qv = [x, y, ]T , q = ,J = [sin , cos , 0, 0], and z = [, , ]T in (13) as

M1 =

m11 0

12ml cos

0 m22 012ml cos 0 ml2 + Im

, G1 =[ 0

0mgl sin

]

V1 =

0 0 ml sin0

12ml2 sin 2

12ml2 sin 2

0 12ml2 sin 2 0

,F1 =

[1 sin t 2 cos t 3 sin t

]T (56)where m11 = 1r2 (2Mwr2 + 2Iw + mr2) and m22 = d

2

4r2(2Mwr2 +

2Iw + 4r2d2 Im + 4mr2 l2

d2sin2 ).

In the simulation, we assume the parameters Iw = 0.5 kg m2,Mw = 0.2 kg, Im = 1.0 kg m2, m = 10.0 kg, l = 1.0 m,d = 1.0 m, r = 0.5 m, 1 = 2 = 3 = 1.0,z(0) = [0,0.2, pi/180]T , z(0) = [0.1, 0.0, 0.0]T . The desiredtrajectories are chosen as d = 0.05t rad, d = 0.0 rad,initial velocity is 0.1 m/s. The external disturbances are set as1.0 sin(t) and 1.0 cos(t). The desired nonholonomic constraintforce is set as 10 N. The system state is observed through thenoisy linear measurement channel, zero-mean Gaussian noisesare added to the state information. All noises are assumed to bemutually independent. The noises have variances correspondingto a 5% noise to signal radio. The design parameters of the abovecontrollers are: Kp = diag[10.0, 800.0], 1 = diag[1.0, 5.0], i =0.5, i = = 1/(t + 1)2, B10 = [{1.0, 0}; {1.0, 1.0}], b =0.9, [c1(0), . . . , c6(0)]T = [1.0, 1.0, 10.0, 20.0, 10.0, 10.0]T , Kf =10.0.The trajectory tracking by the proposed control approach is

shown in Fig. 8, and direction angle, the tilt angles for thedynamic balance and the stable velocities are shown in Figs. 25, respectively. The input torques are shown respectively inFig. 6. The nonholonomic constraint force error with the desiredforce is shown in Fig. 7. Therefore, the slipping and slippagecannot happen since the friction force converges within theregion of 10 N. From these figures, even if without the priorknowledge of the system, we can obtain good performance bythe proposed adaptive robust control. Robust adaptive approach is

tolerant of modeling errors because accurate modeling of wheeledinverted pendulums dynamics is difficult, and time-consuming

c1352 Z. Li, Y. Zhang / Automati

Fig. 2. Tracking the direction angle.

Fig. 3. The direction angle error.

Fig. 4. The tilt angle tracking error.

and uncertain. The presence of parametric errors is a commonproblem since the identification of dynamic parameters is error-prone. The robust adaptive control presented in this paper is notsusceptible to this problem, since the unknown parameters arelearned during thewheeled inverted pendulumoperation in actualconditions. Although the parametric uncertainties and the externaldisturbances are both introduced into the simulation model, themotion/force control performance of system, under the proposedcontrol, is not degraded. The simulation results demonstrate theeffectiveness of the proposed adaptive control in the presence ofunknown nonlinear dynamic systems and environments.

7. Conclusions

In this paper, robust adaptive motion/force control design iscarried out for dynamic balance and stable tracking of desired tra-

jectories of a mobile wheeled inverted pendulum, in the presenceof unmodelled dynamics, or parametric/functional uncertaintiesa 46 (2010) 13461353

Fig. 5. The stable velocity.

Fig. 6. Input torques.

Fig. 7. The nonholonomic constraint force error with d = 10 N.Fig. 8. The produced trajectory.

cZ. Li, Y. Zhang / Automati

and nonholonomic constraint force. The control is mathematicallyshown to guarantee semi-global uniformly bounded stability, andthe steady state compact sets to which the closed loop error sig-nals converge are derived. The size of compact sets for motion andforce can be made small through appropriate choice of control de-sign parameters. Simulation results demonstrate that the systemis able to track reference signals satisfactorily, with all closed loopsignals uniformly bounded.

References

Arai, H., & Tanie, K. (1998). Nonholonomic control of a three-dof planar underactu-atedmanipulator. IEEE Transactions on Robotics and Automation, 14(5), 681694.

Blankespoor, A., & Roemer, R. (2004). Experimental verification of the dynamicmodel for a quarter size self-balancing wheelchair. In American control confer-ence, Boston, MA (pp. 488492).

Brooks, R., Aryanada, L., Edsinger, A., Fitzpatrick, P., Kemp, C. C., & OReilly, U.(2004). Sensing andmanipulating built-for-human environments. InternationalJournal of Humanoid Robotics, 1(1), 128.

Dong, W. (2002). On trajectory and force tracking control of constrained mobilemanipulators with parameter uncertainty. Automatica, 38(8), 14751484.

Gans, N.R., & Hutchinson, S.A. (2006). Visual servo velocity and pose control of awheeled inverted pendulum through partial-feedback linearization. In Proc.IEEE/RSJ int. conf. intelligent robots and systems (pp. 38233828).

Ge, S. S., Lee, T. H., & Harris, C. J. (1998). Adaptive neural network control of robotmanipulators. London: World Scientific.

Ge, S. S., Wang, J., Lee, T. H., & Zhou, G. Y. (2001). Adaptive robust stabilizationof dynamic nonholonomic chained systems. Journal of Robotic System, 18(3),119133.

Ge, S. S., Wang, Z., & Lee, T. H. (2003). Adaptive stabilization of uncertain nonholo-nomic systems by state and output feedback. Automatica, 39(8), 14511460.

Grasser, F., Arrigo, A., Colombi, S., & Rufer, A. C. (2002). Joe: a mobile, invertedpendulum. IEEE Transactions on Industrial Electronics, 49(1), 107114.

Ha, Y. S., & Yuta, S. (1996). Trajectory tracking control for navigation of the inversependulum type self-contained mobile robot. Robotics and Autonomous System,17, 6580.

Ibanez, C. A., Frias, O. G., & Castanon, M. S. (2005). Lyapunov-based controller forthe inverted pendulum cart system. Nonlinear Dynamics, 40(4), 367374.

Isidori, A., Marconi, L., & Serrani, A. (2003). Robust autonomous guidance: an internalmodel approach. New York: Springer.

Jung, S., & Kim, S. S. (2008). Control experiment of a wheel-driven mobile in-verted pendulum using neural network. IEEE Transactions on Control SystemsTechnology, 16(2), 297303.

Kim, Y., Kim, S. H., & Kwak, Y. K. (2005). Dynamic analysis of a nonholonomic two-wheeled inverted pendulum robot. Journal of Intelligent and Robotic Systems,44, 2546.a 46 (2010) 13461353 1353

Li, Z., & Luo, J. (2009). Adaptive robust dynamic balance and motion controls ofmobile wheeled inverted pendulums. IEEE Transactions on Control SystemsTechnology, 17(1), 233241.

Luca, A. D., & Oriolo, G. (2002). Trajectory planning and control for planar robotswith passive last joint. The International Journal of Robotics Research, 21(56),575590.

Nasrallah, D. S., Michalska, H., & Angeles, J. (2007). Controllability and posturecontrol of a wheeled pendulum moving on an inclined plane. IEEE Transactionson Robotics, 23(3), 564577.

Pathak, K., Franch, J., & Agrawal, S. K. (2005). Velocity and position control of awheeled inverted pendulum by partial feedback linearization. IEEE Transactionson Robotics, 21(3), 505513.

Salerno, A., & Angeles, J. (2003). On the nonlinear controllability of a quasiholonomicmobile robot. In Proc. IEEE int. conf. robotics and automation (pp. 33793384).

Salerno, A., & Angeles, J. (2004). The control of semi-autonomous two-wheeledrobots undergoing large payload-variations. In Proc. IEEE int. conf. robotics andautomation (pp. 17401745).

Zhang, M., & Tarn, T. (2002). Hybrid control of the pendubot. IEEE/ASME Trans.Mechatronics, 7(1), 7986.

Zhijun Li received Dr. Eng. Degree in Mechatronics,from Shanghai Jiao Tong University, PR China, in 2002.From 2003 to 2005, he was a postdoctoral fellow inDepartment of Mechanical Engineering and Intelligentsystems at the University of Electro-Communications,Tokyo, Japan. From 2005 to 2006, he was a research fellowin the Department of Electrical and Computer Engineeringat the National University of Singapore, and NanyangTechnological University, Singapore. Currently, he is anassociate professor in the Department of Automation,Shanghai Jiao Tong University, PR China. Dr. Li is IEEE

Senior Member and his current research interests are adaptive/robust control,mobile manipulators, nonholonomic systems, etc.

Yunong Zhang received B.S., M.S. and Ph.D. degreesrespectively from Huazhong University of Science andTechnology (HUST), South China University of Technology(SCUT) and the Chinese University of Hong Kong (CUHK),respectively, in 1996, 1999 and 2003. He is currentlya professor at the School of Information Science andTechnology, Sun Yat-Sen University (SYSU), Guangzhou,China. Before joining SYSU in 2006, he had been withthe National University of Ireland (NUI), University ofStrathclyde, and National University of Singapore (NUS)since 2003. His main research interests include neural

networks, robotics and Gaussian processes. His web-page is now available athttp://www.ee.sysu.edu.cn/teacher/detail.asp?sn=129.

Robust adaptive motion/force control for wheeled inverted pendulumsIntroductionPreliminariesSystem descriptionDynamics of mobile wheeled inverted pendulumsReduced dynamics and state transformationPhysical properties

Problem statementControl objectives

Robust adaptive motion/force control z2 and z3 -subsystems motion control z1 -subsystem stabilityForce control

SimulationConclusionsReferences