Proceedings of the 45th IEEE Conference on Decision & Control FrIP5.9Manchester Grand Hyatt HotelSan Diego, CA, USA, December 13-15, 2006

Swing up and Balancing Control of Pendubot via Model OrbitStabilization: Algorithm Synthesis and Experimental Verification

Yury Orlov, Luis T. Aguilar, Leonardo Acho, and Adain Ortiz

Abstract- A model orbit stabilization approach to swing upcontrol of a two-link pendulum robot (Pendubot) is understudy. The quasihomogeneous control synthesis is utilized todesign a variable structure controller that drives the actuatedlink of the Pendubot to a periodic reference orbit in finitetime. A modified Van der Pol oscillator is involved into thesynthesis as an asymptotical generator of the periodic motion.Performance issues of the proposed synthesis are illustratedin an experimental study of the swing up/balancing controlproblem of moving the Pendubot from its stable downwardposition to the unstable inverted position and stabilizing it aboutthe vertical.

I. INTRODUCTION

A pendulum robot, typically abbreviated as Pendubot,is a simple underactuated mechanical manipulator, whosefirst link (shoulder) is actuated whereas the second one(elbow) is not actuated. In the present work it appears asa test bed for an experimental study of the capability ofthe recently developed model orbit-based synthesis [13] tomove the Pendubot from its stable downward position to theunstable upright position and stabilize it about the vertical.Throughout, the positions of both links of the Pendubotand their angular velocities are assumed to be available formeasurements.

Being verified experimentally, the proposed swingup/balancing control of Pendubot via the model orbit syn-thesis presents an interesting alternative to the energy-basedapproach from [2], [4].

Following the orbital transfer strategy from [13], a swing-ing controller is composed by an inner loop controller,partially linearizing the Pendubot, and an orbitally stabilizingouter loop controller. The quasihomogeneous variable struc-ture controller from [9], [10] is utilized to drive the Pendubotshoulder to a periodic reference orbit in finite time in spiteof the presence of external disturbances with an a prioriknown magnitude bound. The resulting controller exhibits aninfinite number of switches on a finite time interval, however,in contrast to standard sliding mode controllers, it does notrely on the generation of sliding motions on the switchingmanifolds but on their intersections.A modified Van der Pol oscillator from [11] is involved

into the quasihomogeneous synthesis as a reference model.The proposed modification still possesses a stable limitcycle, governed by a standard linear oscillator equation, and

Yury Orlov and Adan Ortiz are with CICESE Research Center, P.O. BOX434944, San Diego, CA, 92143-4944, yorlov{aortiz}@cicese.mx

Luis T. Aguilar and Leonardo Acho are with CITEDI-IPN, 2498 Roll Dr.#757, Otay Mesa, San Diego, CA, 92154, laguilar{leonardo}@citedi.mx

The work was supported by CONACYT under grant number 45900

therefore it constitutes an asymptotical harmonic generator.Another example of an asymptotical harmonic generator(nearly the only one available in the literature) is the vari-able structure Van der Pol oscillator from [14]. However,it is hardly possible to use that oscillator for generatinga reference signal because the system response would becontaminated by high frequency oscillations (a so-calledchattering effect) caused by fast switching the structure ofthe Van der Pol oscillator.

Switching from the swinging controller to a locally stabi-lizing one, when the Pendubot enters the attraction basinof the latter, yields a unified framework for the orbitaltransfer of the Pendubot from the downward position andits stabilization around the unstable equilibrium. The locallystabilizing controller is also obtained by applying the quasi-homogeneous robust synthesis.The paper is organized as follows. The model orbit-based

synthesis is developed in Section II. Experimental results onswing up/balancing control of Pendubot are given in SectionIII. Conclusions are collected in Section IV.

II. ORBITAL PENDUBOT STAB!! IZATION

A. Problem Statement

The state equation of the Pendubot, depicted in Fig. 1, isgiven by [18, p. 55]:

M(q)q + N(q, q)= T + W (1)

where

M(q) = inll i2) N(q, 0

T( WiW2

(N1ATN2J

) (2)

and

Tnil

M12

01

03 cos(q - q2)M22 = 02

N1

N2

(3)

03 sin(q1- q2)2 -g04 sin(ql),-03 sin(q -q2) - g05 sin(q2), (4)

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By substituting the orbit equation (9) into (8) one con-cludes that the limit cycle of the modified Van der Polequation (8) is remarkably generated by a standard linearharmonic oscillator

2+j 0,

m2122 + J2,m1ll + m2L1, (5)

Here, ml is the mass of link 1, m2 the mass of link 2, L,and L2 are respectively the lengths of link 1 and link 2; 11and 12 are the distances to the center of mass of link 1 andlink 2; J1 and J2 are the moments of inertia of link 1 andlink 2 about their centroids; Jm is the motor inertia, Ti isthe control torque, w is the external disturbance, and g is thegravity acceleration.We assume throughout that the external disturbance is of

class Loo (0, ox) with an a priori known norm bound K > 0,i.e.,

ess sup llw(t)ll < Ktc [0,x)

where stands for the standard Euclidean norm.Our objective is to design a controller that causes

actuated link of the Pendubot to track a reference traject

lim [qi(t) + x(t)] 0,

while attenuating the effect of an admissible external disbance (6).

In the above relation the reference trajectory x (t)generated by the modified Van der Pol oscillator

* 2+E[(x2 + 2) _ p]±+2=

proposed in [11]. This oscillator has been shown in [11possess a stable limit cycle, governed by

* 22 X 2

x + 2 = p

where the parameter p stands for the amplitude of the licycle whereas p is for its frequency.

initialized on (9). Thus, we arrive at a nonlinear asymptoticalharmonic generator (8) which naturally exhibits an idealsinusoidal signal (10) in its limit cycle (9). The amplitudeand frequency of this sinusoidal signal can be varied at willby tuning the parameters p and ,u of the harmonic generator(8).

In the sequel the Van der Pol modification (8) is used as areference model in the orbital stabilization of the Pendubot.In this regard, it is worth of noting that using the asymptoticalharmonic generator (8) instead of its linear counterpart (10)allows one to modify not only the frequency of the referencesignal but also its amplitude on-line by simply changing theparameters of the reference model.

B. Control Strategy

In order to present a control strategy that allows us toachieve the above objective let us partially linearize the Pen-dubot dynamics in accordance with [16]. For this purpose,let us rewrite equation (1) in the form

Tin q1 +ml2 q2 +N1 = Ti + Wl

Then using (12), the following equation is derived:

(6)Now substituting equation (13) into (11) yields

(7) Finally, setting lM i1=inl- T 2Tm- 1M2 and

where u is the new control input, the desired linearization isobtained:

q1 = u + AM-l1[w -ml2m221w2] (16)(8) q2 -m2-21{m12 [U+ Al (w1 m12m22Tw2)]

+N2 -W2}

In the above relations the positive definiteness of the inertiamatrix M(q) has been used to ensure that IMI 74 0. Since

(9) system (16), (17) describes the linearized actuated jointmodel it is referred to as collocated linearization [15].

imit The control strategy is now formalized as follows. Thecontrol input (15) is composed by an inner loop controller,

6139

A

M2

{g

Fig. 1. Pendubot

(10)

010305

mTln2 + m2L2 + Ji +Jm, 02m2L,12, 04m2l 2.

M12 q1 +m22 q2 +N2 = W2-

(1 1)

(12)

q2= -n22 [iM12 q1 +N2 -W2]- (13)

(mll- ml2m22 M12) q1

-ml2m221 (N2 -W2) + N1 = Ti + W1. (14)

Ti= M u -ml2m2T-lN2 + N1 (15)

(17)

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partially linearizing the Pendubot, and an outer loop con-troller u to be constructed. Given the system output

y(t) = qC(t) + x(t), (18)

that combines the actuated state ql(t) of the system and thereference variable x(t) governed by the modified Van derPol equation (8), the outer loop controller u is to drive thesystem output (18) to the surface y = 0 in finite time andmaintain it there in spite of a uniformly bounded externaldisturbance w, affecting the system.

C. Switched Control Synthesis

Due to (8), (16), (18), the output dynamics is given by

y = u+ IM - [w- ml2m22 w2]* 2

[(2 + X:-( 12) p2]C _u2X.

The following control law

.22 X 2] 2u [(x + 2) P ]x+P x

-asign(y) -3sign(rj) -hy p' (20)

with the parameters such that

h,p>O, />(M 1 + mo2m2-23)K (21)

is proposed.The closed-loop system (19), (20) is then feedback trans-

formed to the one

y = 1[wl-m2m22 w2]

-asign(y) -3sign(j) - hy (22)

with piece-wise continuous right-hand side. Throughout,solutions of such a system are defined in the sense of Filippov[5] as that of a certain differential inclusion with a multi-valued right-hand side.

Relating the quasihomogeneous synthesis from [10], theabove controller has been composed of the nonlinear com-pensator

2

uc = [(x2 + -) -p2]C + i2x (23)2)

the homogeneous switching part (the so-called twisting con-troller from [6], [7])

Uh =-asign(y) -3sign(j),and the linear remainder

ui =-hy- pY

that vanishes in the origin y = y = 0. By Theorem 4.2 from[9] the quasihomogeneous system (22) with the parametersubordination (21) is finite time stable regardless of which

external uniformly bounded disturbance subject to (6) affectsthe system. The control objective is thus achieved.

In the remainder, capabilities of the proposed synthesisprocedure are tested in an experimental study of the swingup/balansing control problem.

III. SWING UP CONTROL AND STABILIZATION

In this section, an orbitally stabilizing controller is de-signed to swing up the Pendubot from its downward positionto the upright position and it is then switched to a quasihomo-geneous controller from [12], locally stabilizing the Pendubotabout the vertical. The hybrid control strategy, to be testedin an experimental study, is to select the amplitude p and thefrequency ,u of the model limit cycle (9) reasonably smalland the parameter E, controlling the speed of the limit cycletransient in the modified Van der Pol equation (8), reasonablylarge to ensure that the Pendubot enters the attraction basin ofthe quasihomogeneous locally stabilizing controller. Properswitching from the swinging quasihomogeneous controllerto the stabilizing one yields the generation of a swing upmotion, asymptotically stable about the vertical.

A. Pendubot prototypePerformance issues of the quasihomogeneous synthesis

are tested on the laboratory Pendubot, manufactured byMechatronics Systems Inc. and installed in the CICESEResearch Center. The values of the Pendubot parameters (5),supplied by the manufacturer [1], are listed in Table 1.

TABLE IPARAMETERS OF THE PENDUBOT.

Notation Value Units01 0.0308 kg m202 0.0106 kg m203 0.0095 kg m204 0.2087 kg m05 0.0630 kg m

B. Swinging controller designIn order to apply the orbitally stabilizing synthesis (8),

(15), (20) to swinging the Pendubot up from the down-ward position to the upright position the cumulative energy,pumped into the Pendubot, should be of an appropriate level.According to the energy-based approach [4], such a level isto ensure that the total energy

E(q, 2)=2TM(q)Q+ 04gcosqQ+ 05g cos q2 (24)

of the closed-loop system near the upright position ap-proaches the nominal energy value

Eo = (04 + 05)g (25)

that corresponds to the inverted equilibrium of the Pendubot.Being crucial to a successful swing up, this is achieved bytuning both the controller parameters a, Q, h, p and thereference parameters E, p, p of the Van der Pol modification

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(8). Appropriate values of the parameters to be tuned arecarried out in successive experiments where the total energy(24) of the Pendubot, driven by the developed controller (20),is required to enter an interval BR (Eo) of a sufficiently smallradios R, centered around the nominal energy value (25).

In our experimental study the controller gains were setto a = 140, Q = 40, h = 0, p 0 whereas thereference parameters were tuned to E 8.7, p = 0.013,,u = 10. With these parameters the total energy (24) of thePendubot, driven by the developed controller (20), entersthe ball BR(EO) of the radios R = 0.127, centered atEo = 2.665. Detailed experimental results, supporting theorbitally stabilizing synthesis, are presented in SubsectionIII.E.

C. Locally stabilizing controller designThe quasihomogeneous synthesis, used above for the

orbital stabilization, is now applied to the Pendubot block-canonical form, well-known from [18], to derive a robustcontroller, locally stabilizing the Pendubot around its uprightposition.As reported in [18], the disturbance-free Pendubot dynam-

ics (1) is locally minimum phase if its output is given by

z = sin q2 + klw + k2tb (26)

F = cos q2f2 -sin(q2 )j2 + k16+k2 [r1 sin(v) + rj cos(v)v>] (wb + _yji7)2

+ 2k2r sin(v)(Lb + y>7)(6 + yr->)-2yk2r sin(v)i2-A k2r,2 cos(v)iA

+ AkA sin(q2) + Ak2T cos(q2)>22-yk2r 2sinv [b+ 7(-1)] (f- f2),

(36)

) =2yk2rT2 sin v [wb +l(yr 1)]- ml2 cos q2, (37)

f, (q ) =m12(C21j1 + G2) -m22(Cl2A2 + G1)

f2 (q ) m12(Cl2j2 + G1) -miT (C2ij1 + G2)A/ '

andA\ = ml1m22-m 22 > 0

(38)

(39)

(40)

because the inertia matrix M is positive definite.Taking into account (31), (32), one concludes that

where the output parameters k, and k2 are positive, and D(q1, q2.qlqh2) (0,0,0,0) 7 0

w(qi, q2) = q2 -o(q -q2),cp(v) -v

+2 -

tan-l (V tan ("2)IFor+lateruse, let us denote

For later use, let us denote

(27) which is why the locally minimum phase system (1), (26)has relative degree 2 at the origin. Thus, the quasihomoge-neous synthesis from [10] becomes applicable to the local

(28) stabilization of the Pendubot around its upright position.The following quasihomogeneous control law is then

proposed:

A m2g12m2L112'

1q -q2 , T =

a/ + cos vi

u(qj C)-_y72] +Ar sin(q2),

-F(q, q) -a sigtn z- /31 sigtn-hlz -plz

the controller gains are such that

V> = ql q-2 = T2 sin(V)(Ql-q2)p1,hi>O,>a, > 1 >0,

(31)

and by virtue of (27), (28)

w q2 -/COS(V)(ql -q2). (31

Then differentiating (26) along the solutions of (1) yields

z = cos(q2)q2 + klLb + k26, (3

z = F(qi,q2, 1, 2) +u (3'

where

u = 1(ql, q2, 1l, 42)T,

F(q, q) is governed by (36), and z, z are viewed as func-tions of (q,q), which are defined by relations (26)-(33).

2) In analogy to the orbitally stabilizing controller (15), the

above controller is composed by a partially stabilizing innerloop controller and an outer loop controller that nullifies thesystem output (26) in finite time.

3) By applying [10, Theorem 2]), the disturbance-free Pen-4) dubot dynamics (1), enforced by the quasihomogeneous state

feedback (42)-(44), is shown to be locally asymptoticallystable about the upright position, uniformly in matcheddisturbances wl(t), whose magnitude is less than a1 -1.

5) Apart from this, the closed-loop system proves to have a

6141

(41)

(29)

T(q, C) = D (q, C)u(q, C)

where

where

(42)

(30)

(43)

(44)

(tb + _6 = TIsinv .T,7.) 2

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45th IEEE CDC, San Diego, USA, Dec. 13-15, 2006

Fig. 2. Block diagram of the closed-loop system.

certain degree of robustness against vanishing mismatcheddisturbances w2 (t).

Robustness features of the proposed locally stabilizingcontroller were tested in an experimental study. Quite im-pressive experimental results were obtained for the local sta-bilization of the Pendubot about the upright position with thecontroller parameters a1 = 40, 1 = 20, hi = 0, P1 = 0and the output parameters k, = 0.4, k2 = 0.2. Those resultscan be observed in Subsection III.E where the proposedcontroller is involved into a hybrid synthesis of swinging thePendubot up and balancing it about the vertical. The size ofthe attraction domain of the locally stabilizing controller wasadditionally evaluated by adding quick disturbances using ametal stick.

D. Hybrid Controller DesignIn order to accompany swinging up the Pendubot by

the subsequent stabilization around the upright position themodel orbit-based swinging controller, presented in Sub-section III.B, is switched to the locally stabilizing con-troller from Subsection III.C whenever the Pendubot entersthe basin of attraction, experimentally found for the lattercontroller. The problem of choosing a proper switchingtime moment is thus logically resolved. While being not

studied in details, the capability of the closed-loop systemof entering the attraction basin of the locally stabilizingcontroller is supported by experiments. The block-diagramof the Pendubot, driven by the proposed hybrid controller, isdepicted in Fig. 2.

E. Experimental verification

The initial conditions of the Pendubot position and thoseof the modified Van der Pol oscillator, selected for allexperiments, were ql(O) = 3.14 rad, q2(0) = 3.14 rad, andx(0) =-3.14 rad, whereas all the velocity initial conditionswere set to zero.To begin with, we separately implemented the orbitally

stabilizing controller from Subsection 4.2. In order to test therobustness of this controller an external disturbance, similarto that of [17], was randomly added by lightly hitting thelinks of the Pendubot. For demonstrating the capability of thecontroller to move the Pendubot from one orbit to another bymodifying the orbit parameters we then introduced a randomtime instant to (it was to - lOs in the experiment), whenthe amplitude p of the model limit cycle was changed fromits initial value p= 0.013 to the new one p= 0.5.

Finally, the hybrid controller from Subsection III.D wasimplemented to swing the Pendubot up and stabilize it aboutthe vertical. To better demonstrate robustness features of theproposed hybrid synthesis some external disturbances, againsimilar to those of [17], were randomly added by lightlyhitting the links of the Pendubot. For robustness comparison,the same hybrid controller was additionally applied to thePendubot while a mass of 0.0542kg was detached from itsshoulder. Since the manufacturer's user manual [1] identifiedthe Pendubot parameters (5) only, while the physical parame-ters of the Pendubot such as 11,12, Jl, J2 remained unknown,parameters (5), corresponding to the modification of thePendubot, appeared unavailable for tuning the controllergains.

Experimental results for the resulting Pendubot motion,enforced by the orbitally stabilizing controller and the hybridcontroller, are depicted in Fig. 3. This figure demonstratesthat the hybrid controller swings the Pendubot up andstabilizes it about the upright position while also attenuatingthe parameter variations and external disturbances. Switchingfrom the model orbit-based swinging controller to the locallystabilizing one occurred at the time instant t,5 2.1s. Asopposed to the hybrid controller from [17], whose applicationto a modified model of the Pendubot required the knowledgeof the modified parameters to tune the controller parameters,our controller is successfully applied not only to the nominalPendubot model but also to its modification.

IV. CONCLUSIONS

Model orbit synthesis of a Pendubot, presenting a simpleunderactuated (two degrees-of-freedom, one actuator) manip-ulator, is under study. The quasihomogeneity based controlsynthesis is utilized to design a switched controller thatdrives the Pendubot to a desired zero dynamics manifold

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2 4 6Time [s]

'Lo Disturbances

0o0.

-1

8 10 0 2 4 6 8 10Time [s]

- d/dt q1d/dt q2

2 4 6Time [s]

-1508 10 0

15

10

155zo

-5

a -10

0

--15

-20

-251

2 4 6 8 10Time [s]

2 4 6 8Time [s]

2 4 6 8 2 4 6 8Time [s] Time [s]

Fig. 3. Swing up and stabilization of the Pendubot: (a) left column forthe nominal model and (b) right column for the modified model, operatingunder randomly added quick disturbances.

in finite time and maintains it there in sliding mode in spiteof the presence of external disturbances.The modified Van der Pol oscillator is involved into the

synthesis as an asymptotical generator of the periodic mo-

tion. The resulting closed-loop system is capable of movingfrom one orbit to another by simply changing the parametersof the modified Van der Pol oscillator.

Capabilities of the quasihomogeneous synthesis and itsrobustness against parameter variations and external distur-bances are illustrated in an experimental study of the swingup control problem of moving the Pendubot from its stabledownward position to the unstable inverted position andstabilizing it about the vertical.

Eliminating undesirable chattering oscillations, caused byfast switching in the implemented hybrid controller, is amongother problems calling for further investigation. While beingnon-trivial, this problem is however well-understood from theexisting literature (see, e.g., [3], [8] and references therein)and hopefully general methods of chattering reduction applyhere as well.

REFERENCES

[1] The Pendubot user's manual. Mechatronics Systems Inc., Champaign,IL, 1998.

[2] Astrom K. J. and Furuta K. (2000). Swinging up a pendulum by energycontrol. Automatica, 36, 287-295.

[3] Bartolini G., Ferrara A. and Usai E. (2000). Chattering avoidance bysecond-order sliding mode control. IEEE Trans. Autom. Contr, 43,241-246.

[4] Fantoni I., Lozano R. and Spong M. (2000). Energy Based Control ofthe Pendubot. IEEE Trans. Aut. Contr, 45, 4, 725-729.

[5] Filippov A. F. (1988). Differential equations with discontinuous right-hand sides. Dordrecht: Kluwer Academic Publisher.

[6] Fridman L. and Levant A. (1996). Higher order sliding modes as anatural phenomenon in control theory, in Robust Control via variablestructure and Lyapunov techniques, Garafalo and Glielmo (eds.),Lecture notes in control and information science, Berlin, Springer,217, 107-133.

[7] Fridman L. and Levant A. (2002). Higher order sliding modes, inSliding mode control in engineering, W. Perruquetti and J.-P. Barbout(eds.), New York: Marcel Dekker, 53-102.

[8] Hirschorn R. (2006) Generalized sliding mode control for multi-inputnonlinear systems. IEEE Trans. Autom. Contr (to be published).

[9] Orlov Y. (2005). Finite-time stability and robust control synthesisof uncertain switched systems. SIAM Journal on Optimization andControl, 43, 1253-1271.

[10] Orlov Y. (2005). Finite time stability and quasihomogeneous controlsynthesis of uncertain switched systems with application to underac-tuated manipulators. Proc. of the 44th Conference on Decision andControl, Seville, Spain, 4566-4571.

[11] Orlov Y., Acho L., and Aguilar L. (2004). Quasihomogeneity approachto the pendubot stabilization around periodic orbits. Proc. 2nd IFACSymposium on Systems, Structure and Control, Oaxaca, Mexico, 448-453.

[12] Orlov Y., Aguilar L., and Acho L. (2005). Zeno mode control ofunderactuated mechanical systems with application to the Pendubotstabilization around the upright position Proc. 16th IFAC WorldCongress, CD-ROM, Prague, Czech.

[13] Orlov Y., Aguilar L., and Acho L. (2004). Model orbit robust stabi-lization (MORS) of Pendubot with application to swing up control.Proc. of the 44th Conference on Decision and Control, Seville, Spain,6164-6169.

[14] Sira-Ramirez H. (1987). Harmonic response of variable-structure-controlled Van der Pol oscillators. IEEE Trans. Circuits and Systems,34, 103-106.

[15] Spong M.W. (1995). The Swing Up Contol Problem for the Acrobot.IEEE Control Systems Magazine, 49-55.

[16] Spong M.W. and Praly L. (1997). Control of Underactuated Mechani-cal Systems Using Switching and Saturation. Lecture Notes in Controland Information Sciences 222, Springer Verlag, London, 163-172.

[17] Zhang M. and Tarn T.J. (2002). Hybrid control of the Pendubot. IEEETrans. Mechatronics, 7, 79-86.

[18] Utkin V.I., Guldner J. and Shi J. (1999). Sliding modes in Electrome-chanical Systems. London: Taylor and Francis.

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