Control of a perturbed under-actuated mechanical system

Chadia Zayane-Aissa, Taous-Meriem Laleg-Kirati and Ahmed Chemori

Abstract— In this work, the trajectory tracking prob-lem for an under-actuated mechanical system in presenceof unknown input perturbation is addressed. The studiedinertia wheel inverted pendulum falls in the class of nonminimum phase systems. The proposed high order slidingmode structure composed of controller and differentiatorallows to track accurately the predefined trajectory andto stabilize the internal dynamics. The robustness of theapproach is illustrated through different perturbationand output noise configurations.

I. INTRODUCTION

Under-actuated systems are gaining an increas-ing interest especially in robotics application,aerospace and marine vehicles [1]. Their distinc-tive feature is the high number of degrees offreedom compared to the actuated ones, allowingfor a high degree of dexterity and a good con-figuration of the reachable space. Consequently,motion planning and control of robots can bedesigned more flexibly while guaranteeing certainoperational advantages such as minimizing weightand cost.Nevertheless, the generally nonlinear coupling be-tween the actuated and the non actuated degreesof freedom for under-actuated mechanical systems,may result in some difficulties related to the stabi-lization of internal dynamics. Consequently, whenit comes to robots control, the design of robustcontrol laws is of crucial importance.Additionally to the proposed variety of control ap-proaches ([2], [3], [1], . . . ), output feedback slidingmode controllers (SMC) ([4], [5], [6]) are knownfor their robustness and accuracy and can there-fore cope with such problems. Based on exactly

C. Zayane-Aissa and T.M. Laleg-Kirati are with Computer, Elec-trical and Mathematical Sciences and Engineering Division, KingAbdullah university of science and technology (KAUST), King-dom of Saudi Arabia. chadia.zayane@kaust.edu.sa,taousmeriem.laleg@kaust.edu.sa.

A. Chemori is with LIRMM, University of Montpel-lier 2 - CNRS, 161 rue Ada, 34392 Montpellier, France.ahmed.chemori@lirmm.fr

maintaining a well-chosen variable (generally thedifference between actual and reference outputs)at zero, SMC use high frequency switchings tosteer the system such that the desired constraint issatisfied. For the standard SMC, these switchingsare at the origin of undesired vibrations, alsocalled chattering effect ([7]), that may damage thesystem. This issue has been addressed in differentways, including dead zone and smooth approxima-tions of the sign function realizing the switchings.A more interesting perspective was to developSMC that allow not only to stabilize the slidingvariable at zero but also to steer its time derivativesto zero, which results in a smoother dynamics. Thisclass of modern controllers, namely High OrderSliding Mode (HOSM) controllers ([8], [9], [10],[11], . . . ), are more accurate than the first orderSMC in presence of sampling and measurementnoise and more robust against modeling uncer-tainties and disturbances. The implementation ofHOSM controllers requires good estimates of thederivatives, which can be achieved using HOSMdifferentiators ([9]).The main feature of these observers is to providehigh precision real time estimations of a givensignal derivatives without the need for direct differ-entiation of noisy measurements. More precisely,in the noise free case, a HOSM differentiatorprovides finite time exact estimates of the r firstderivatives of a given function provided that its(r+1) derivative is bounded by a known constant.In presence of an additive bounded noise, theestimates converge to the true derivatives valueswith optimal asymptotics.In this paper, we implement a HOSM controller to-gether with the underlying HOSM differentiator toachieve periodic trajectory tracking for an exampleof under-actuated mechanical system, in presenceof perturbations. The proposed approach is appliedto an inertia wheel inverted pendulum, where thependulum is subject to torque perturbations. The

accuracy of trajectory tracking and the robustnessof the method against perturbations and measure-ment noise are illustrated through examples ofdifferent torque profiles.This paper is organized as follows: Section 2introduces the inertia wheel inverted pendulum anddescribes its equations of motion. In Section 3, theHOSM controller and differentiator are presentedand their application to periodic trajectory trackingof the pendulum in presence of torque perturba-tions is discussed. Finally, Section 4 illustrates therobustness and accuracy of the developed approachthrough simulation examples of different configu-rations of the system and the perturbations.

II. THE INERTIA WHEEL INVERTED PENDULUM

The inertia wheel inverted pendulum is anexample of under-actuated systems, representinga classic system used to test different controlschemes (see [12] and [13] for example). Thereare different types of inverted pendulum systemshaving in common the fact that, unlike normalpendulums that are stable when hanging down,they are intrinsically unstable. Consequently, theunstable vertical equilibrium position needs to becontinuously balanced.The inertia wheel inverted pendulum consideredin this work consists of a rotating wheel that ismounted on an inverted pendulum as shown inFigure 1.

Fig. 1. Inertia wheel inverted pendulum

The angular position of the inertia wheel is ac-tuated through a torque generated by a DC motor.

The pendulum is steered as a consequence of thedynamic coupling of the inertia wheel and thependulum motions as shown in the next subsection.

A. Dynamics equations

Let θ1, θ1 and θ1 be respectively the angularposition, velocity and acceleration of the pendulumbody; θ2, θ2 and θ2 the angular position, velocityand acceleration of the inertia wheel as in theschematic description of Figure 2.

If C1 and C2 are respectively the perturbation

Fig. 2. Schematic representation of the inertia wheel invertedpendulum

torque applied to the pendulum and the torquedelivered by the DC motor directly to the inertiawheel, then the equations describing the dynamicbehavior of the system are as follows:{

(I + i2)θ1 + i2θ2−mlgsinθ1 =C1i2(θ1 + θ2) =C2

(1)

where i1 and i2 are the moment of inertia ofthe pendulum and the wheel towards the lowerextremity of the pendulum (taken as the origin anddenoted O), l is the length of the rod and g is thegravity constant. If l1 and l2 denote respectivelythe distances from the centre of gravity of thependulum and the inertia wheel to O, then themoment of inertia I of the whole system at theorigin is given by:

I = i1 +m1l21 +m2l2

2

where m1 and m2 are respectively the masses of thependulum and the wheel. Finally ml = m1l1+m2l2is the centre of mass of the system. In this study,

the main goal is to perform a periodic trajectorytracking based on the available measurement of thependulum angular position θ1. Choosing the statevector X = [x1,x2,x3]

T = [θ1, θ1, θ2]T , the system

(1) can be written in the following state spacerepresentation:

x1 = x2x2 =

1I mlgsin(x1)+u+ζ

x3 =−(1I mlgsin(x1)+(1+ I

i2)u+ζ )

y = x1

(2)

Where u =−C2I is the control input and ζ = C1

I isthe unknown input perturbation.

B. Control problem statement

It is clear from the second motion equation insystem (1) that the inertia wheel angular velocityθ2 is an internal dynamics of system (3) andthus the system is non minimum phase, which isa common feature of under-actuated mechanicalsystems.Due to physical limitations, stabilizing internaldynamics for this class of mechanical systems is apre-requisite for any control purpose.Choosing the state vector x = [x1,x2]

T = [θ1, θ1]T ,

the system (3) can be written in the following statespace representation:

x1 = x2x2 =

1I mlgsin(x1)+u+ζ

y = x1

(3)

Written in the form of system (3), the inertia wheelinverted pendulum dynamics is condensed in aminimum phase subsystem. This was possible be-cause the dynamics of the internal dynamics doesnot affect those of the remaining state components.Furthermore, it will be shown in next section thatthe designed control law allows to stabilize theinternal dynamics.

III. HIGH ORDER SLIDING MODE CONTROLLER

For the inertia wheel inverted pendulum (for-mulated in system (3)), the purpose is to design acontroller, that is robust to the perturbation ζ , thatenables the system to track a predefined trajectory(sine shape). The standard sliding mode approachcould be used to control the system efficientlyif the control input appears explicitly in the first

derivative of the output, that is the relative degreeof the system equals 1, which is not the case here.HOSM were designed to generalize the notion ofsliding mode to higher order derivatives of theoutput, thus allowing to deal with higher relativedegrees and to remove the restriction imposed bystandard sliding modes.Given the sliding surface σ ≡ 0, which is the dif-ference between the reference and reached outputs,the HOSM approach consists in deriving its firsttotal derivatives that are understood in the sense ofFilippov [7]. The number of these first derivativesdefines the class of sliding modes, satisfying thefollowing set of equations:

σ = σ = · · ·= σ(r−1) = 0

where r is the degree of smoothness in theneighborhood of the sliding surface, called theorder of the HOSM, or simply r-sliding. In thefollowing,we present a general class of HOSMcontrollers, namely the quasi-continuous ones andtheir implementation for inertia wheel invertedpendulum problem. Then we describe the cor-responding HOSM differentiator and provide anestimation of the unknown perturbation.

A. Quasi-continuous HOSM controllers

Quasi-continuous controllers ([14]) represent aclass of the arbitrary order sliding modes. Thelatter controllers are defined in a recursive way asfollows:

u =−αΨr−1,r(σ , σ , · · · ,σ (r−1))

where r is the sliding order and α > 0 is the uniqueparameter to be adjusted for the controller.While a main characteristic of the sliding modeframework is the fact that developed control lawsare discontinuous, the quasi-continuous HOSMsprovide control laws for which discontinuities oc-cur only when the set of conditions σ = σ = · · ·=σ (r−1) = 0 are met. It is worth noting that anyHOSM controller is at least discontinuous whenthe previous set of equalities hold. Moreover, inpractice these conditions cannot be satisfied simul-taneously because of the imperfections and noises,thus the underlying quasi-continuous control law isindeed continuous when the sliding order is greaterthan 1.

The recursive procedure defining quasi-continuouscontrollers is the following:

Ψi,r =φi,r

Ni,r, i = 0, · · · ,r−1

where φi,r and Ni,r are obtained as follows:φ0,r = σ , N0,r = |σ |φi,r = σ (i)+ γiN

(r−1)/(r−i+1))i−1,r Ψi−1,r

Ni,r = |σ (i)|+ γiN(r−1)/(r−i+1))i−1,r

(4)

where the coefficients γi are predetermined inde-pendently from the studied system and have thestandard values: γ1 = 1, γ2 = 1, γ3 = 2, . . .For the chattering attenuation purpose, it has beenshown that increasing the relative degree by 1improves significantly the performance of the con-troller. Therefore, we consider a 3-sliding modecontroller instead of a second order one (practicalrelative degree equals 3).Denoting the sliding surface and its derivatives asfollows:

σ = θ1−θre f1

σ = ˆθ1− θ

re f1

σ = ˆθ1− θ

re f1

The proposed 3-sliding controller is obtained fromsystem (4) as follows:{

u = 0; 0≤ t ≤ t0u =−α

σ+2∗(|σ |+|σ |2/3)−1/2(σ+|σ |2/3sign(σ))

|σ |+2∗(|σ |+|σ |2/3)1/2 ; t > t0

The sliding surface derivatives are obtained via theHOSM presented below.

Remark 1: The initial offset applied to the con-trol law is used to allow the differentiator toconverge and thus to have accurate derivatives’estimations before applying the control.

B. HOSM differentiatorsHOSM differentiators were developed in order

to provide the derivatives required to implementHOSM controllers. They are known for their ro-bustness in presence of noise and their exactnessin its absence.Even if, in the case of the inertia wheel invertedpendulum, the variable of interest is provided

directly by the measurement, to implement theabove controller, we need to estimate the secondderivative of θ1. For this we use the second orderhigh order differentiator introduced by Levant [9],[?]: z0 = ν1,ν1 = z0−λ2L1/3|z0− y|2/3sign(z0−σ)

z1 = ν2,ν2 = z1−λ1L1/2|z1−ν0|1/2sign(z1−ν0)z2 =−λ0Lsign(z2−ν1)

(5)where y is the measurement (θ1), z0 its estimateand z1 and z2 represent the estimated first andsecond derivatives of the output to be used for thecontroller. A possible choice for the variables λiin system (5) is: λ0 = 1.1, λ1 = 1.5 and λ2 = 2.

C. Perturbation estimationThe designed controller allows to track the de-

sired trajectory and to reject the perturbation ζ . Soan estimate of the perturbation can be obtained asfollows:

ζ = z2−1I

mlgsin(x1)+u

In practice, for the perturbation to be estimated,the second derivative of the state variable θ needsto be considered as an additional state component.A common procedure to obtain z2 is to artificiallyincrease the differentiator order from a second to athird order, without modifying the control design.Let’s just recall that the perturbation estimationwas not emphasized in our study but could easilybe obtained.

IV. NUMERICAL RESULTS

The HOSM differentiator and the 3-slidingmode controller described in the previous sectionwere applied to the inverted pendulum system witha time step Te = 5ms for a total simulation durationof 7s. The reference trajectory has a sine shapewith a period of 3 seconds and the system wastaken initially to be in a non equilibrium positioncorresponding to θ1(t = 0) = 2 ˚ .The parameters of the HOSM differentiator weretaken as follows: λ0 = 1.1, λ1 = 1.5 and L = 100.The sign function was approximated by a sigmoidt 7→ t

|t|+εwhere ε = 1e−4.

The parameter of the controller is α =−2, the mi-nus sign is added since the control is taken with aminus sign in the system dynamics equations (this

choice is not straightforward, but it corresponds toa physical torque).The results corresponding to different perturbationsignals (zero, constant and sine) are shown below.Figures 3, 4 and 5 illustrate respectively the statevector components, the phase diagram (to show thelimit cycle) and the corresponding control inputin the ideal case, i.e absence of perturbation andmeasurement noise.

[sec]

0 1 2 3 4 5 6 7

[rad]

-0.1

-0.05

0

0.05

0.1θ1

[sec]

0 1 2 3 4 5 6 7

[rad/s

ec]

-0.2

-0.1

0

0.1

0.2θ1

[sec]

0 1 2 3 4 5 6 7

[rad/s

ec]

-100

0

100

200

300θ2

Fig. 3. (1) & (2) State vector: reference (red) and achieved (blue)trajectories. (3) Internal dynamics.

θ1

-0.06 -0.04 -0.02 0 0.02 0.04 0.06

θ1

-0.15

-0.1

-0.05

0

0.05

0.1

0.15

0.2Phase diagram

Fig. 4. Phase diagram

Then a white measurement noise, with a stan-dard deviation value corresponding to 3.8% of thesine amplitude, has been added to the measuredpendulum angular position. Figure 6 shows the

[sec]

0 1 2 3 4 5 6 7

[N/m

]

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6u

Fig. 5. Control law

results corresponding to a constant perturbation,while Figure 7 illustrates the sine perturbationcase.

[sec]

0 1 2 3 4 5 6 7

[ra

d]

-0.1

-0.05

0

0.05

0.1

θ1

θref1

[sec]

0 1 2 3 4 5 6 7-0.1

-0.05

0

0.05

0.1

ζ

y

[sec]

0 1 2 3 4 5 6 7

[ra

d/s

ec]

0

100

200

300

400

ˆθ2

Fig. 6. (1): Perturbation and noisy measurement. (2)Reference(red) and achieved (blue) trajectories. (3) Internal dynamics.

In both figures, the noisy measurements togetherwith the injected perturbation are shown in thefirst subfigure, the convergence of the pendulumangular position to the reference one in the secondsubfigure and the corresponding wheel velocity inthe last one.It is important to note that the proposed controllaw stabilizes the internal dynamics of the system(the inertia wheel angular position) and allowsan accurate and fast convergence of controlled

[sec]

0 1 2 3 4 5 6 7

[ra

d]

-0.1

-0.05

0

0.05

0.1

θ1

θref1

[sec]

0 1 2 3 4 5 6 7-0.1

-0.05

0

0.05

0.1ζ

y

[sec]

0 1 2 3 4 5 6 7

[ra

d/s

ec]

0

100

200

300

400

ˆθ2

Fig. 7. (1): Perturbation and noisy measurement. (2)Reference(red) and achieved (blue) trajectories. (3) Internal dynamics.

variable to the reference trajectory.

V. CONCLUSIONSIn this paper, a quasi-continuous high-order

sliding mode controller has been proposed toperform trajectory tracking for an under-actuatedmechanical system with unknown perturbations.The HOSM controller coupled to the correspond-ing HOSM differentiator, providing the requiredoutput derivatives, allow to perform a robust andaccurate periodic trajectory tracking for an inertiawheel inverted pendulum. Furthermore, the systemwas subject to different profiles of torque pertur-bations, considered as unknown inputs, for whichthe implemented controller/differentator providedestimations.The good performance of the studied method wasillustrated through numerical simulations of dif-ferent perturbation profiles in presence of mea-surement noise. The implementation of the quasi-continuous HOSM control law for an inertia wheelinverted pendulum prototype is under investiga-tion.

ACKNOWLEDGMENTResearch reported in this publication was sup-

ported by the King Abdullah University of Scienceand Technology (KAUST).

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