Sliding Mode Control of Custom Built RotaryInverted Pendulum

M. Svec, S. Iles, J. MatuskoUniversity of Zagreb Faculty of Electrical Engineering and Computing, Zagreb, Croatia

Email: marko.svec, sandor.iles, jadranko.matusko@fer.hr

Abstract—Inverted pendulums have been used for a long timeas a typical laboratory setup for learning and testing of variouscontrol system techniques. In this paper, we present a new labora-tory rotary inverted pendulum setup. The setup is equipped witha microcontroller containing a code for communicating directlywith MATLAB/Simulink using Simulink Desktop Real Time. Inthis configuration students have the opportunity to implementa controller in MATLAB/Simulink or to implement it directlyon the microcontroller. Its capabilities are tested using slidingmode control, first using simulation and then verified on theexperimental setup.

Keywords—inverted pendulum, sliding mode control, Furutapendulum

I. INTRODUCTION

The rotary inverted pendulum is a commonly used bench-mark in control systems laboratories for learning and testing ofvarious control system techniques. It consists of an arm drivenby a motor which is able to rotate in the horizontal plane anda pendulum suspended from the arm which is able to rotatein the vertical plane. It is also called Furuta pendulum. It hasone input and two degrees of freedom and belongs to a classof nonlinear unstable underactuated mechanical systems [1],[2].

For Furuta pendulum a large number of papers are writ-ten presenting different linear and nonlinear controllers forswing-up and stabilizing the pendulum in the upright unstableposition.

In [3], [4] Linear Quadratic Regulator (LQR) is used for sta-bilizing the pendulum in the upright position, while additionalswing-up controller is used for swinging up the pendulum. Inpresence of nonlinearities, such as dead-zone caused by staticfriction, the performance of the controller degrades and a limitcycle can occur. For such case, the limit cycle can be avoidedby designing a linear controller in the frequency domain takinginto account nonlinearity described by describing function [5].A design based on differential flatness and root locus is usedfor selecting state feedback gains in [6].

Besides linear control techniques, various nonlinear tech-niques are also reported in the literature [7]. Model predictivecontrol is used for swinging up and/or stabilizing the pendulumin upright position [8]–[10]. Proportional retarted controller isreported successful in stabilizing the pendulum in case of timedelays [11].

Among nonlinear control techniques the sliding-mode con-trol is a popular nonlinear control technique due to robustness

against system uncertainties and external disturbances. How-ever, standard sliding-mode control is not directly applicablefor underactuated systems without selecting an appropriatesliding surface [12]. Sliding mode controllers are reported forboth swing-up and stabilization of Furuta pendulum in [12]–[14], proving semiglobal asymptotic stability. Furthermoresliding mode controllers are often used for comparison withother controllers such as active disturbance rejection controlcombined with differential flatness [15].

The simulation model can be written in terms of differentialequations representing dynamics of the system, which can beobtained by using Euler-Lagrange or Newton-Euler methods[16]–[18], while some authors resort to making a 3D modelin CAD software such as and exporting the simulation modeldirectly into SimMechanics [18]–[20].

Several papers presented building their own inverted pendu-lum systems (both rotary and pendulum on a cart) [19], [21],[22], concentrating on the mechanical design, design of thesimulation model and experiments. Papers use either a com-mercial rapid prototyping system, such as dSPACE DS1104,[19] or an inexpensive microcontroller, such as Arduino, toimplement the controller [22].

In this paper an experimental model of Furuta pendulumbuilt for experimental verification of various control algorithmstaught in undergraduate, graduate and postgraduate controlcourses, such as linear state feedback, model predictive controlor sliding-mode control is presented. The built experimentalsetup is equipped with a microcontroller with prepared codefor communicating directly with MATLAB/Simulink usingSimulink Desktop Real Time. The microcontroller controlstwo H-bridges with current sense for driving up to two DCmotors and it is able to provide either motor voltage or controlthe motor current (torque) directly. In the first phase, studentsuse the designed interface to run the control algorithms directlyfrom Simulink using Simulink Desktop Real Time, whilein a later phase they can exploit added JTAG connector onthe PCB to implement the control algorithm directly on themicrocontroller. Sliding mode control is selected to showthe possibility for testing a high bandwidth nonlinear controlalgorithm directly from MATLAB/Simulink.

The rest of the paper is organized as follows. In Section IIthe mathematical model of the pendulum is presented, SectionIII presents the experimental setup, Section IV presents thedesigned control algorithm and Section V presents simulationand experimental results. Section VI concludes the paper.

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II. PENDULUM MODEL

The Furuta pendulum, shown in Fig 1, consists of an armwhich is able to rotate in the horizontal plane and a pendulumsuspended from the arm which is able to rotate in the verticalplane. The pendulum angle is denoted by α and arm angle isdenoted by θ, respectively, while their meaning is shown inFig 1. The arm is driven directly by an electric motor.

Fig. 1. Furuta pendulum model

Dynamic equations of the system can be obtained usingEuler-Lagrange or Newton-Euler methods. The full nonlinearsystem dynamics for the Furuta pendulum with a completeinertia tensor is reported in [16]. This model can be simplifiedfor simulation purposes by considering only the total momentof inertia of the pendulum arm and the pendulum about theirpivot points: Jr = Jr + mpl

2r and Jp = Jp + 0.25mpl

2p,

respectively. In those expressions, Jr represents the momentof inertia of the arm around z axis, Jp moment of inertia ofthe pendulum around its center of mass, mp pednulum mass,lr length of the arm and lp length of the pendulum.

The equations are [16]:(Jr + Jp sin2 α

)θ − 1

2mplplr cosαα+

Jp sin(2α)θα+1

2mplplr sinαα2 = τ −Br θ,

(1)

−1

2mplplr cosαθ + Jpα−

1

2Jp sin(2α)θ2 − 1

2mplpg sinα = −Bpα.

(2)

Br is DC motor and Bp pendulum friction coefficient. Thetorque τ is generated by the DC motor and can be expressedas

τ =ncm(u− nceθ)

Ra. (3)

Parameter Value Unitmr 5.3 · 10−3 kglr 0.09 mJr 4.6 · 10−3 kg ·m2

mp 0.107 kglp 0.3395 mJp 3.5 · 10−6 kg ·m2

Br 0.056 kg ·m2/sBp 2.5 · 10−4 kg ·m2

g 9.81 m/s2

cm 0.0134 Nm/Ace 0.0134 V s/radRa 3.33 Ωn 54.7945

TABLE IPARAMETERS OF THE ROTARY PENDULUM SIMULATION MODEL

In the upper equation, n is the gearbox ratio, Ra armature re-sistance, ce and cm are back-EMF and torque motor constants.

III. EXPERIMENTAL SETUP

A custom experimental setup shown in Fig 2 has beendesigned. It consist of a DC motor with gearbox equippedwith a magnetic encoder for driving the arm and measuringits position and 1024 ppr incremental encoder for measuringthe position of the pendulum.

A custom made PCB is designed, which consist of amicrocontroller, dual H-bridge with current sense, USB port,connectors for connecting 3 incremental encoders, and 2 mo-tors. A custom code is written to communicate with MATLABS-funcition and a Simulink block diagram has been made.However, the PCB is equipped with a JTAG connector toenable students downloading their own custom code. Further-more, H-bridge with current sense enables for two differentmodes of operation. The control input can be either the DCmotor voltage or armature current (motor torque). The latter,enables students to use equations of motion directly obtainedfrom Euler-Lagrange equations.

Fig. 2. The experimental setup

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IV. CONTROLLER DESIGN

Fig. 3. Simulink model

For controller synthesis, the following simplified model isused:

θ = −amθ + ceu,

α = − CJpα+

mpglp

2Jpsinα+

K

Jpθ,

(4)

where Jp = Jp + 0.25mpl2p. The upper equation represents

simplified model of the permanent magnet DC motor whichis driving the rotating base, where the constant am includesback-EMF and friction effects and ce effect of the input voltageon the angular acceleration of the base. The bottom equationdescribes the dynamics of the pendulum. C is the pendulumfriction, mp and Jp pendulum mass and moment of inertia andlp pendulum length. K is a proportionality constant which isK > 0 for inverted and K < 0 for non-inverted position. Theonly control input is the DC motor voltage u [23].

The system (4) can be rewritten as

θ = −amθ + ceu,

α = − CJpα+

mpglp

2Jpsinα+

K

Jpamθ +

K

Jpceu.

(5)

The next step is to transform the system (5) into the newform using [23]

y = θ − JpKα,

x = y − C

Kα,

(6)

which results in the following system formulation:

x = −mpglp2K

sinα,

y = x+C

Kα.

(7)

Parameter Value Unitam 20 1/sce 20.142 rad/s2/Vmp 0.107 kglp 0.3395 mJp 3.5 · 10−6 kg ·m2

C 2 · 10−3 kg ·m2/sK 1.9 · 10−3 sign(cosα) kg ·m2

g 9.81 m/s2

TABLE IIPARAMETERS OF THE ROTARY PENDULUM CONTROL MODEL

A. Sliding Mode Controller

Lets take the state component θ1 in the upper equation ofthe system (7) as control input. If the last term of this equationsatisfies

sinα = −λ1(x+ y) (8)

and

λ1 = − 2K

mpglp,

mpglp2C

>π

2, K > 0, (9)

then the system (7) is asymptotically stable. Consequently,the system (5) is asymptotically stable. For more informationregarding system stability, refer to [23].

Equation (8) holds if the function

s1 = sinα+ λ1(x+ y) = 0. (10)

The condition s1 = 0 will be satisfied, if the function s1satisfies the linear first order differential equation

s1 = −λs1, λ > 0. (11)

The expression (11) will be satisfied if the sliding mode isenforced in the switching surface

s = s1 + λs1 = cosαα+ λ1(x+ y) + λs1 = 0. (12)

The time derivative of s equals

s =Kce

Jpcosαu+ Ψ(x, y, α, α), (13)

where Ψ is a nonlinear function of the system states. For thecontroller to achieve a stable behaviour, the functions s and sneed to have opposite signs. This is achieved if

s = −η sign(s), η > 0, (14)

in which case the control input equals

u =−Jp

Kce cosα(Ψ + η sign(s)). (15)

However, such formulation can cause unwanted chattering. Inorder to reduce this effect, (14) was replaced by

s = −η tanh(s/2), η > 0. (16)

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In the following chapters, control law realized with (14) willbe referred to as ”Controller 1” and the one realized with (16)as ”Controller 2”.

B. Swing-Up Controller

To bring the pendulum in the upright position, we useenergy control strategy described in [24]. The energy of thependulum is

E =1

2Jpα

2 +1

2mpglp(1 + cosα), (17)

while the reference energy equals maximum potential energy

E0 = mpglp. (18)

There are various control laws which can be used to achievereference energy (18). The one we use in this work keeps theamplitude of control signal constant, while changing only itssign, and is given by

u = −µ sign((E − E0)α cosα). (19)

Swing-up controller is active until the condition |α| < 30

is satisfied, when it switches to sliding mode control law (15).

V. SIMULATION AND EXPERIMENT RESULTS

Verification of described algorithms has been done usingboth simulation and experimental tests. Controller parametersare λ = 10, η = 40 and swing-up gain µ = 12. At the be-ginning of the experiment, pendulum is in its stable stationaryposition with all the system states [θ θ α α]T = [0 0 0 0]T .

Fig. 4 shows simulation results for both controller versions.In the upper graph, it can be seen that Controller 1 yieldsfaster θ angle dynamics, but is unable to hold the base of thependulum in the zero position, whereas Controller 2 causesoscilations and results in zero steady state tracking error.Pendulum angle α dynamics is shown in the lower graph andno difference in dynamics between the two controllers can benoticed. Both stabilize pendulum very fast and with no steadystate error.

Input signals from the simulation can be seen in the Fig.5. Swing-up phase lasts for cca. 1 second, after which thesliding mode is turned on. There, one can notice alreadymentioned chattering effect, which causes input signal to haveextremely fast switching behaviour and causes high jerk whenapplied to an electromechanical system. Controller 2 doesn’tcause chattering, since around s = 0 it behaves similarly to aproportional controller and no switching is enforced.

Fig. 6 shows measured angles θ and α. It can be seen that, inreality, neither of controllers is able to hold zero steady state θerror. Controller 1 makes the system oscillate slightly aroundθ ≈ 100, while Controller 2 results in a better behaviourand shifts those oscillations around θ = 0. This behaviour isconsistent with observations made based on simulation, butwith higher steady state error because of modelling errorsand mechanical disturbances (e.g. friction, gearbox backlashetc.). On the other hand, both controllers show almost identical

0 2 4 6 8 10 12 14 16 18 20

−100

0

100

200

θ( )

s = −η sign(s)s = −η tanh(s/2)

0 2 4 6 8 10 12 14 16 18 20

0

100

200

300

t(s)

α()

s = −η sign(s)s = −η tanh(s/2)

Fig. 4. Simulation results for both controller versions (angles).

0 2 4 6 8 10 12 14 16 18 20

−10

0

10

t(s)

u(V

)

s = −η sign(s)s = −η tanh(s/2)

Fig. 5. Simulation results for both controller versions (input).

behaviour when it comes to α angle control. Behaviour is verysimilar to the one obtained from the simulation.

Input signals in Fig. 7 show that swing-up time in ex-periment lasted similarly long as in the simulation. Afterthe sliding mode control has been activated, Controller 1behaves very similar to the simulated one, causing a lotof chattering. However, in reality Controller 2 also causeschattering, although smaller than Controller 1.

VI. CONCLUSION

In this paper a new rotary inverted pendulum experimentalsetup has been proposed. The proposed experimental setuphas a custom built PCB board equipped with a microcon-troller with a code for communicating directly with MAT-LAB/Simulink using Simulink Desktop Real Time. In this con-figuration students have opportunity to implement controller inMATLAB/Simulink or implement the controller directly on themicrocontroller. To show the capability of implementing highbandwith control algorithms, a sliding mode controller hasbeen synthesised and implemented in MATLAB/Simulink. Thecontroller has been tested in simulations and experimentallyfor two different versions of the controller. The first versionuses sign function in the computation of the control signalwhile the second version uses tanh for reducing the chattering

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0 2 4 6 8 10 12 14 16 18 20

−100

0

100

θ( )

s = −η sign(s)s = −η tanh(s/2)

0 2 4 6 8 10 12 14 16 18 20

0

100

200

300

t(s)

α()

s = −η sign(s)s = −η tanh(s/2)

Fig. 6. Experimental results for both controller versions (angles).

0 2 4 6 8 10 12 14 16 18 20

−10

0

10

t(s)

u(V

)

s = −η sign(s)s = −η tanh(s/2)

Fig. 7. Experimental results for both controller versions (input).

problem. In both simulation and experimental results thechattering has been reduced by using tanh function. Theresults on the proposed experimental setup closely resemblesimulation results, which shows that the experimental setupcan be used for testing various advanced nonlinear controlalgorithms.

ACKNOWLEDGMENT

This research has been supported by the European RegionalDevelopment Fund under the grant KK.01.1.1.01.0009 (DAT-ACROSS).

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