CONTROL OF THE BALL AND BEAM USING KALMAN FILTER - A FLATNESS BASEDAPPROACH

JOSÉ ONIRAM DE AQUINO LIMAVERDE FILHO∗, EUGÊNIO LIBORIO FEITOSA FORTALEZA∗

∗GRACO - Grupo de Automação e ControleUniversidade de Brasília

Brasília, Distrito Federal, Brasil

Emails: joseoniram@ieee.org, efortaleza@unb.br

Abstract— The tracking control design problem for a nonlinear ball and beam system is addressed. In this paper, the flatnessproperty of the system is here exploited for design a state feedback control scheme aiming to stabilize the system’s trajectorytracking error with respect to off-line planned trajectories. Differentially flat nonlinear systems can be written in the Brunovskycanonical form, which allows to perform the state estimation using the Kalman filter. In order to validate the performance of theproposed tracking control, numerical simulations and experimental results are presented and discussed.

Keywords— Trajectory Tracking; Nonlinear Control; Differentially Flat Systems; Kalman Filter; Ball and Beam

Resumo— O problema de rastreamento de trajetória para um sistema não-linear barra-esfera é apresentado. Nesse trabalho,a planicidade diferencial do sistema é explorada tanto para desenvolver uma estrutura de controle por realimentação de estadosvisando estabilizar o erro de rastreamento de trajetória com respeito as trajetórias desejadas. Sistemas não-lineares diferencialmenteplanos podem ser escritos na sua forma canônica de Brunovsky, o que se permite realizar a estimação de estados através doFiltro de Kalman. Buscando avaliar a performance do controlador de rastreamento proposto, simulações numéricas e resultadosexperimentais são apresentados e discutidos.

Palavras-chave— Rastreamento de Trajetória; Controle Não-linear; Sistemas Diferencialmente Planos; Filtro de Kalman; Barra-Esfera

1 Introduction

Control of underactuated systems, which is de-fined as the one with fewer controls inputs than de-grees of freedom, is currently an active research fielddue to its theoretical challenges and their broad appli-cation in Robotics, Aerospace Vehicles, and MarineVehicles (Olfati-Saber, 2001). The motivation for thestudy of controllers for this systems is due the fact itallows us to reduce weight and cost as well as con-sider situations in which component failures can oc-cur (Reyhanoglu, 1996).

Belonging to the class of underactuated systems,the ball and beam system is one of the most popularand important benchmarks system for studying non-linear control design. This system is widely used be-cause of its simplicity to understand as a system andprovides the opportunity to analyze the control tech-niques (Amjad et al., 2010).

Many classical and modern control methods havebeen proposed to solve both stabilization and trackingproblems for the ball and beam system. In (Hauseret al., 1992), an approximate feedback linearization isproposed to resolve the fact the relative degree of theball and beam system is not well defined. A LinearQuadratic Regulator (LQR) was developed in (Rahmatand Wahab, 2000) as an optimal control strategy con-sidering the voltage of the motor as the input of thesystem. In (Chang et al., 2012), a tracking controlstrategy is designed using a pair of decoupled fuzzysliding-mode controllers.

Besides these techniques, tracking control ap-proaches based on differential flatness have grownsubstantially in recent years. Flatness property was

first proposed and developed in (Fliess et al., 1992).This property allows a complete parametrization ofall systems variables (states, inputs, outputs) in termsof a finite set of independent variables, called theflat outputs, and a finite number of their time deriva-tives (Sira-Ramírez and Agrawal, 2004).

Indeed, many physical systems are known to bedifferentially flat, as can be seen in (Murray et al.,1995). These systems can be directly linearize to acontrollable linear system in Brunovsky’s canonicalform by endogenous transformation. This feature al-lows to trivialize the trajectory planning tasks, withoutsolving differential equations, while simplifying thefeedback controller design problem to that of a set ofdecoupled linear time invariant systems (Sira-Ramírezand Agrawal, 2004).

The power of flatness is precisely that it does notconvert nonlinear systems into linear ones. When asystem is flat, it is an indication that the nonlinearstructure of the system is well characterized and onecan exploit that structure in designing control algo-rithms for motion planning, trajectory generation, andstabilization (Martin et al., 2012).

In (Rouchon et al., 1992), it was presented thata ball and beam system used in (Hauser et al., 1992)is non-differentially flat system, but a high frequencycontrol and averaging approach is proposed to approx-imate the system by means of a flat system. In (Sira-Ramírez, 2000), an approximate flat system is ob-tained by disregarding the centripetal force of the ball.The flatness property allowed to compute a suitableoff-line trajectory planning and designed an incremen-tal time-varying linear feedback controller.

Using differential flatness theory, this paper pro-

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poses a different control approach for trajectory track-ing of the flat nonlinear ball and beam system pre-sented in (Sira-Ramírez, 2000). Transforming thenonlinear system to the Brunovsky’s canonical form,one can design a feedback control law based onthe flat output ensuring convergence of the track-ing error to zero. In addition, following the resultsin (Rigatos, 2012), state estimation of the flat outputand its time derivatives is performed by applying thestandard Kalman Filter recursion to the equivalent lin-ear system.

The paper is organized as follows. The mathe-matical model describing the dynamics of a nonlinearball and beam model is introduced in Section 2. InSection 3, a brief mathematical description of differ-entially flat systems is given. The flatness property ofthe system is used to determine the nominal trajecto-ries for all system variables. Then, the steps to derivethe tracking controller is presented. Section 4 containsthe state estimation procedure for the associated lin-earized system by applying Kalman Filter. The perfor-mance of the proposed controller is evaluated throughnumerical simulations and experimental tests in Sec-tion 5. The conclusions and suggestions for furtherresearch are presented in the last section.

2 Ball and Beam Model

Figure 1: Schematic of the ball and beam system.

The nonlinear model for the ball and beam systemconsidered in this paper was presented in (Quanser,2008). The mathematical model of the system, illus-trated in Figure 1, is given by:

r =mrarmgR

2

Lbeam(mR2 + Jb)sin(θ)

θ = −1

τθ +

K1

Vm

(1)

where θ and r are the beam angle and the ball position,respectively. Also, τ is the time-constant, K1 is thesteady-state gain, Lbeam is the length of the beam, mand Jb are the mass and moment of inertia of the ball,

respectively. Furthermore,R is the radius of the ball, gis the acceleration due to gravity, rarm is the distancebetween screw and motor gear. The system input isgiven by the voltage of the motor Vm.

Let (x1, x2, x3, x4)T = (r, r, θ, θ)T . The sys-tem (1) can be represented by the following state spaceequations:

x1 = x2

x2 = Kbb sin(x3)

x3 = x4

x4 = β1x4 + β2Vm

(2)

where

β1 = −1

τ

β2 =K1

τ

Kbb =mrarmgR

2

Lbeam(mR2 + Jb).

(3)

3 Differentially Flat Systems

Differentially flat system is a system whose inte-gral curves (curves that satisfy the system equations)can be mapped in a one-to-one way to ordinary curves(which need not satisfy any differential equation) in asuitable space, whose dimension is equal to the num-ber of input vector (Lévine, 2010).

More precisely, the flatness condition reduce tofind a set of specific variables such that all states andinputs can be parametrize from these outputs and theirderivatives without integration. If a nonlinear systemx = f(x, u) has states x ∈ Rn, and inputs u ∈ Rm,then the system is said to be differentially flat, if thereexists a flat outputs z ∈ Rm such that:

z = H(x, u, u, · · · , u(n)) (4)

x = α(x, x, · · · , x(n)) (5)

u = λ(x, x, · · · , x(n), x(n+1)) (6)

where the components are differentially independentand two functions α and λ are smooth functions.

3.1 Off-line Trajectory Planning

Equation (4) implies that, if we want to constructa trajectory whose x(0) and u(0) are known, onecan find the initial and final values of the flat out-puts and their time derivatives by the surjectivity of(α, λ). Thus, it suffices then to construct a trajectoryt 7−→ z(t) at least n + 1 times differentiable that sat-isfies these initial and final conditions since x(t) andu(t) are obtained by (5) and (6) (Lévine, 2010).

Knowing the system (2) is found to be differen-tially flat with flat output represented by the ball po-sition x1, the differential parametrization associated

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with the flat output and its time derivatives is givenby:

x2 = x1 (7)

x3 = arcsin

(x1Kbb

)(8)

x4 =x(3)1

Kbb

[1− x21

K2bb

]1/2 (9)

Vm =

[K2bb − x21]x

(4)1 + x1(x

(3)1 )2

K3bb

[1− x21

K2bb

]3/2 − β1x4

β2(10)

Thus, it means that we can directly obtain thenominal trajectories x∗2(t), x

∗3(t), x

∗4(t) and V ∗

m(t) ifx∗1(t) and its time derivatives are known. It shouldbe emphasized that (8) is only valid when x1/Kbb be-longs to the domain of the function arcsin, and that (9)and (10) present singularities when x21 = K2

bb.

3.2 A Trajectory Tracking Feedback Controller

As immediate consequence of the differentialparametrization, the system (2) can be transformed inthe Brunovsky’s canonical form (Rigatos, 2012):

x(4)1 = υ = f(xT ) + g(xT )Vm (11)

with

xT =[x1 x1 x1 x

(3)1

](12)

f(xT ) =

β1x4K3bb

[1− x21

K2bb

]3/2− x1[x(3)1 ]2

[K2bb − x21]

(13)

g(xT ) =

β2K3bb

[1− x21

K2bb

]3/2[K2

bb − x21](14)

It means that system (2) is equivalent to a fourthorder chain of integrators with the state vector xT con-taining the flat output and its time derivatives.

Thus, the linear system (11) can be written instate-space representation:

xT = AxT +Bυ

y = CxT(15)

with

A =

0 1 0 0

0 0 1 0

0 0 0 1

0 0 0 0

, B =

0

0

0

1

, CT =

1

0

0

0

(16)

Then, initially, assume that the state vector xT canbe measured. If the nominal trajectories for the x1and its time derivatives are known, we can denote bye = x1− x∗1(t) the tracking error. An endogenous dy-namic feedback can be computed for the system (15)as follows (Lévine, 2010):

υ = υ∗ +3∑

i=0

k(i+1)e(i) (17)

where υ∗ = x∗(4)1 and the gains ki being chosen suchthat the polynomial p(s) = s4+k4s

3+k3s2+k2s+k1

are Hurwitz polynomials. The tracking error e con-verges to 0, which implies convergence of the flat out-put and all its time derivatives to their respective nom-inal trajectories. Thus, from (5) and (6), we concludethat the x1, x2, x3, x4 and Vm locally exponentiallyconverge to their reference.

From (11), the final expression therefore forVm(t) is given by replacing (17) in (10).

4 Kalman Filter for Nonlinear Flat Systems

All state variables usually cannot be measured inmany control applications, therefore, state estimatorscan be a practical or economical alternative solution.For linear systems subject to Gaussian measurementor process noise, the Kalman Filter (KF) provides anefficient computational (recursive) solution for stateestimation using a predictive-corrective structure. Fig-ure 2 illustrates a classic discrete-time Kalman Filter.

Figure 2: Schematic diagram of the discrete-timeKalman Filter (Bishop and Welch, 2001).

where the matrices Q and R correspond to the pro-cess and measurement noise covariance matrices, re-spectively. For further details, refer to (Bishop andWelch, 2001).

For nonlinear systems such as ball and beam sys-tem, the Extended Kalman Filter (EKF) is commonlyused for obtaining estimates of the state vector throughthe fusion of observed measurements. Although EKFis efficient in several sensorless control, it is charac-terized by cumulative errors due to the local lineariza-tion assumption, and this may affect the accuracy ofthe state estimation or even risk the stability of theobserver-based control loop (Rigatos, 2012).

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In (Rigatos, 2012), it is proposed a derivative-freeKF based on the differential flatness theory. For non-linear flat systems, state estimation of the flat outputand its time derivatives is performed by applying thestandard Kalman Filter recursion to the equivalent lin-ear model in Brunovsky’s canonical form without theneed for calculation of Jacobian matrices. In contrastto EKF, this method improves the accuracy of stateestimation for nonlinear system avoiding linearizationapproximations.

Based on that, for the ball and beam system, itallowed us to estimate the state vector xT of the lin-ear system (15), which is used to compute the trackingcontroller described by the schematic shown in (17).The control design using KF therefore can be summa-rized in Figure 3.

Figure 3: Tracking control design for the ball andbeam system.

5 Simulation and Experimental Results

The experimental apparatus is a Quanser ball andbeam as can be seen in Figure 4. The beam consistsof a metal beam with a built in potentiometer sensorthat detects the ball role along the beam and its posi-tion, whereas an encoder is used to measure the beamangle. By applying a torque at the end of the beam,an DC servomotor drives the beam to a desired angle.Furthermore, input saturation was added to the system.

Figure 4: Ball and beam system setup

Numerical simulations were initially carried outto validate the performance of the proposed trackingcontroller for two different set-points:

I) Constant Trajectory

x∗1(t) = 0 (18)

II) Polynomial Trajectory

x∗1(t) = (1.2× 10−10)t5 − (3× 10−8)t4

+ (2× 10−6)t3 − 0.1 (19)

where the second trajectory starts at −0.1m and endsat 0.1m after 50 seconds. For each trajectory, the cor-responding nominal trajectories for x∗2(t), x

∗3(t), x

∗4(t)

and V ∗m(t) were directly computed by (7-10).

From (Quanser, 2008), the ball and beam param-eters were Lbeam = 0.4255 m, rarm = 0.0254 m,R = 0.0127 m, g = 9.81 m/s2, m = 0.064 kg,Jb = 4.129 × 10−6 kg · m2, K1 = 1.5286 rad/sVand τ = 0.0248 s.

The initial conditions for the system were set as:

x1(0) = −0.2125m x2(0) = 0m/s

x3(0) = −0.9774 rad x4(0) = 0 rad/s(20)

The control parameters were chosen to be:

k4 = 20 k3 = 150 k2 = 500 k1 = 625 (21)

where all closed-loop poles are equal to -5.The ball velocity and beam angular velocity are

estimated by the KF described in Section 4. Using asampling period Ts = 0.001s, the discrete-model ofthe system (15) can be represented as follows:

xT (k+1) = AkxT (k) +Bkυk (22)

with

Ak = exp(ATs) (23)

Bk = BTs (24)

Then, the matrices Q and R were defined as:

Q =

0.1 0 0 00 0.1 0 00 0 0.1 00 0 0 0.1

, R =

[0.01 00 0.01

](25)

Figures (5-6) and (8-9) depicts the ball positionand beam angle for constant and polynomial trajec-tory, respectively. The time evolution of the controlinputs is shown in Figure 7 and Figure 10.

Both simulations show the ball and beam sys-tem perfectly following the trajectories specified pre-viously, thereby ensuring that the tracking errors con-verging to zero as expected. Then, it allows us to val-idate the tracking controller in the Quanser ball andbeam.

0 5 10 15 20 25 30 35 40 45 50−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15Ball Position

t [s]

R [m

]

ReferenceBall Position

Figure 5: Ball position for constant trajectory.

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0 5 10 15 20 25 30 35 40 45 50−1

−0.5

0

0.5

1Beam Angle

t [s]

θ [r

ad]

ReferenceBeam Angle

Figure 6: Beam angle for constant trajectory.

0 5 10 15 20 25 30 35 40 45 50−8

−6

−4

−2

0

2

4

6

8

10Input Signal

t [s]

Vm

[V]

ReferenceInput Signal

Figure 7: Input signal for constant trajectory.

0 5 10 15 20 25 30 35 40 45 50−0.25

−0.2

−0.15

−0.1

−0.05

0

0.05

0.1

0.15Ball Position

t [s]

R [m

]

ReferenceBall Position

Figure 8: Ball position for polynomial trajectory.

0 5 10 15 20 25 30 35 40 45 50−1

−0.5

0

0.5

1Beam Angle

t [s]

θ [r

ad]

ReferenceBeam Angle

Figure 9: Beam angle for polynomial trajectory.

0 5 10 15 20 25 30 35 40 45 50−4

−2

0

2

4

6

8

10Input Signal

t [s]

Vm

[V]

ReferenceInput Signal

Figure 10: Input signal for polynomial trajectory.

By using the same control parameters previ-ously defined, experimental tests were carried out forx∗1(t) = −0.05m, x∗1(t) = 0m and x∗1(t) = 0.05m.Figures (11-13) present the ball position for each de-sired trajectory.

0 50 100 150 200 250 300 350 400 450 500

−0.2

−0.1

0

0.1

0.2

Ball Position

t [s]

R [m

]

ReferenceBall Position

Figure 11: Performance of the tracking control forx∗1(t) = −0.05m.

0 50 100 150 200 250 300 350 400 450 500

−0.2

−0.1

0

0.1

0.2

Ball Position

t [s]

R [m

]

ReferenceBall Position

Figure 12: Performance of the tracking control forx∗1(t) = 0m.

0 50 100 150 200 250 300 350 400 450 500

−0.2

−0.1

0

0.1

0.2

Ball Position

t [s]

R [m

]

ReferenceBall Position

Figure 13: Performance of the tracking control forx∗1(t) = 0.05m.

As it can be seen, the proposed controller has agood performance for all three cases. The main con-trol design issue is choosing appropriate values for thegains of the closed-loop controller, since it was real-ized that the state estimation is directly influenced bytheir values.

6 Conclusions

In this work, a nonlinear ball and beam model ispresented as differentially flat. This property immedi-ately allows to establish the equivalence of the model,by means of dynamic state feedback, a controllable

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linear system in Brunovsky’s canonical form. Precom-puting the necessary off-line trajectory, the approachis based on showing a systematic feedback controllerdesign to converge trajectory tracking error to zero ex-ponentially.

Unlike other estimation methods, it was shownthat Kalman Filter can be apply in nonlinear flatsystems avoiding linearization approximations, whichimproves the accuracy of estimation of the systemstate variables. Numerical simulations and experimen-tal tests illustrates the validity of the designed motionplanning and tracking control scheme for different tra-jectories.

In further studies, we expect to propose a flatness-based controller design for the ball and beam sys-tem assuming the influence of centrifugal force term,which implies that the system is no longer flat. How-ever, a new approach based on the concept of flat in-puts could be used to redesign the input vector field ofthe system such that the given output becomes the flatoutput, and allows to use the method discussed here.

Acknowledgment

The authors would like to acknowledge theBrazilian institutions: ANP, FINEP, MCT and Petro-bras for supporting the present study and the programPRH-PB 223.

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