6th

International Advanced Technologies Symposium (IATS11), 16-18 May 2011, Elaz, Turkey

6

AbstractThis paper presents a new pendulum experiment

setup for educational purposes. The pendulum is actuated by the

thrusts of two motorized propellers mounted back-to-back at its

free end. It can be positioned at any desired position including the

unstable upright. Setups hardware, control system and

mathematical model are described. Performance of the system in

single-thrust and differential-thrust modes is evaluated by

hardware-in-the-loop simulation experiments featuring feedback

linearization and discrete PD control.

Keywordspendulum, propeller, brushless motor, hardware-

in-the-loop simulation

I. INTRODUCTION

he inverted pendulum is a classic benchmark system for

control design and its various implementations have been

studied excessively in the literature [1-3]. The traditional

inverted pendulum systems are actuated by moving the axis of

rotation and the aim is to balance the system in its unstable,

upright balanced position. Due to its fairly simple dynamics,

the inverted pendulum is especially suited to the dynamics and

control systems laboratories of academic institutions.

This paper describes the development and testing of a new

pendulum experiment setup, designed to aid mechanical

engineering students in system dynamics and control related

courses. Unlike traditional systems, the axis of rotation is fixed

and the pendulum is actuated by the thrust of a motorized

propeller mounted at the free end. Similar propeller actuated

pendulums for educational purposes have been reported.

References [4-6] describe experiment kits utilizing a single

brushed dc-motor powered propeller. The pendulum is

balanced against gravity in a single direction and upright

balancing is not possible. In [7], a single motorized propeller

is rotated by a positioning servo around an axis parallel to the

pendulums axis of rotation. This way, the direction as well as

the magnitude of the thrust is varied to hold the pendulum at

any desired position. Our setup features two brushless dc-

motor powered propellers mounted back-to-back at the tip of

the pendulum. This provides two point forces of variable

magnitude and opposite direction that can be used to drive and

hold the pendulum at any position including the unstable

upright. The dual motor arrangement enables differential thrust

actuation which improves the responsiveness and stability of

the system.

Figure 1: Pendulum Setup

II. SETUP DESCRIPTION

The overall design of the setup is shown in Fig.1. Two

motorized propellers are attached back-to-back to the end of

an aluminum pendulum arm. Each of them is able to provide

thrust in a single direction, therefore they are referred to as

clockwise (CW) motor and counter-clockwise (CCW) motor

regarding the moment of their thrusts with respect to the

pendulums pivot point. The pendulum arm is suspended by a

free-to-rotate overhung axle which is coupled to a 2000

counts/revolution optical incremental encoder. The housing of

the axle bearings are clamped to a laboratory desk.

Figure 2: Circuit Board

Pendulum Positioning System Actuated by Dual

Motorized Propellers

Y. Gltekin1 and Y.Tacolu2

1TOBB University of Economics and Technology, Ankara, Turkey, ygultekin@etu.edu.tr

2TOBB University of Economics and Technology, Ankara, Turkey, ytascioglu@etu.edu.tr

T

Y. Gltekin, Y. Tacolu

7

Figure 3: Control Hardware Schematic

Figure 4: Thrust vs Control Input (a) CCW motor (b) CW motor

The control algorithm has been implemented on a PC by

using MATLAB/Simulink Real Time Windows Target

(RTWT) [8]. RTWT compiles Simulink models into C or C++

code and enables the execution of hardware-in-the-loop

simulations in real-time. A custom designed circuit board

(Fig.2) interfaces the control PC to the physical system via

standard RS-232 serial protocol. It includes a Microchip

PIC16F877A microcontroller for serial communications and

PWM (Pulse Width Modulation) signal generation for the

motors, a HCTL2022 integrated circuit for decoding and

counting incremental encoder pulses, and a serial level

converter circuit [9] for translating RS-232 signals from the

control PC to TTL signals suited to the boards

microcontroller. The data exchange between the control PC

and the microcontroller takes place at every 5 ms with 2-byte

packets. The PC receives 16-bit encoder position and sends

two 8-bit motor control signals (Fig.3).

The chosen brushless dc-motors are commonly used in

model aircrafts due to their small size and high speed (~20000

rpm) and they can deliver up to 5 N thrust with 8x4

propellers. The standard drive circuit for these types of motors

is called an ESC (Electronic Speed Controller). An ESC

adjusts the speed of the motor according to a special PWM

signal where the duty cycle period has to be between 1 ms to

2ms. As in this case; when the PWM frequency is 200 Hz, the

ESCs accept duty cycles from 20% to 40%. This range of duty

cycles is expressed with 8-bit resolution. That means the

motors stop at %20 duty cycle (0) and runs at full speed at

40% duty cycle (255). Even though both motors have identical

hardware, a small difference is expected between their

performances. Thrust measurements were performed for the

entire range of duty cycles and curves are fitted to the results

by using MATLABs Basic Fitting tool (Fig.4). The curve

equations are used for analytical approximation of control

inputs corresponding to the required thrusts.

III. MATHEMATICAL MODELING AND SIMULATION

The system is a simple pendulum with a point force at its

free end (see Fig.5). By using Newtons 2nd law and

D`Alembert principle, it can be modeled as follows:

TLcgLmmLmm BABA sin

2

1

3

1 2 (1)

where:

mA = mass of the arm (kg)

mB = total mass of the motor-propeller assembly attached at

point B (kg)

L = distance from the pivot O to point B (m)

mA = mass of the rod (kg)

= angle of the pendulum arm measured counter-

clockwise from rest (rad)

g = gravitational acceleration (m/s2)

c = total viscous damping coefficient (Nms/rad)

T = net thrust produced by the motorized propellers (N)

The first term of this equation is the moment of inertia of

the system. The second term is the moment due to the weight

of the pendulum assembly. The parameters mA, mB, and L are

0.21 kg, 0.16 kg and 0.6 m respectively. The third term is the

moment due to the viscous friction of the bearings and also the

aerodynamic drag caused by the propellers. An average value

for c is found to be 0.0074 Nms/rad by performing a simple

drop test while both motors are running at the same speed. The

last term of the equation is the moment about the pendulum

axle due to the net thrust of the motorized propellers.

ENCODER

DECODER

PC

ESC ESC

CCW

MOTOR

CW

MOTOR

ANGULAR POSITION

PIC16F877A

0 1 2 3 4 50

50

100

150

200

250

Thrust Force (N)

Contr

ol In

put

CCW Motor

Approximation

0 1 2 3 4 50

50

100

150

200

250

Thrust Force (N)

Contr

ol In

put

CW Motor

Approximation

(a)

(b)

Pendulum Positioning System Actuated by Dual Motorized Propellers

8

Figure 5: Pendulum Schematic

In order to cancel the non-linear term from the equation,

feedback linearization is applied in the form:

ugmmT BA sin2

1 (2)

The resulting system is linear and has a fairly simple

transfer function with two real poles:

smL

cs

mL

sU

s

2

2

1

(3)

where:

BA mmm3

1 (4)

Figure 6: Simulink Model

The Simulink model of the system is given in Fig. 6. Since

the system is already Type 1 (linearized system has 1 pole at

zero), a proportional-derivative (PD) controller is preferred. It

is known through thrust measurements (see Fig.4) that the

propellers are limited to apply just above 4.5 N of thrust.

Therefore a saturation block is also added to the model. The

system is simulated with a multiple step input from 0 to 150

in 30 increments. The proportional and derivative gains are

tuned with structured trial-and-error and selected to be 2 and

0.8 respectively. Fig.7 shows the results of the simulation.

0 10 20 30 40 50 600

20

40

60

80

100

120

140

160

time (s)

the

ta (

de

g)

reference

simulation

Figure 7: Simulation Result

IV. EXPERIMENTS AND RESULTS

The Simulink model shown in Fig.6 is also used for the

experiments but the mathematical model of the system is

replaced with the physical system thanks to RTWTs

hardware-in-the-loop capabilities. The blocks shown in Fig.8

are connected to the rest of the Simulink model instead of the

blocks inside the gray Plant box of Fig.6. Using RTWT, the

sampling time of 5 ms is also achieved on the PC side and RS-

232 communications are performed with the built in Packet-

Input, Packet-Output blocks. The Thrust to Control Input

Conversion block contains a MATLAB function which

calculates control inputs to the motors corresponding to the

required thrust based on the approximate curves of the Fig.4.

The Count to Radian Conversion block converts the

instantaneous encoder count to angle in radians.

Figure 8: RTWT Model

The first experiment is performed with the same multiple

step input and controller gains as in the simulation. Only the

CCW motor is used as a thrust source. Fig.9 shows the result

of this experiment in comparison to the simulation. The system

generally performs as expected; however, two shortcomings

can be identified. Firstly, a dead-band is present when the

motor starts from rest. This is the reason for the lag at the

initial response (t=13 s) to the first step reference. Secondly,

the lack of breaking mechanism causes system to overshoot

when the motor needs to slow down rapidly. This is apparent

in the transient at the last step (t=4855 s).

-K-

c

sin

Thrust

Saturation

pi

Target -K-

L

1

s

1

s

PID(z)

Discrete

PID Controller

-K-

1/

((mA/3+mB)*L^2)

-K-

(mA/2+mB)*g*L

-K-

(mA/2+mB)*g

Plant

ref

L

O

A

B

T

1

theta

T

theta

CW Motor

CCW Motor

Thrust to Control Input

Conversion

1

2

Output to

Serial Port

1

Input from

Serial Port

encoder theta

Count to Radian

Conversion

1

TPacket

Output Packet

Input

Y. Gltekin, Y. Tacolu

9

0 10 20 30 40 50 600

50

100

150

200

time (s)

the

ta (

de

g)

0 10 20 30 40 50 60-20

-10

0

10

20

time (s)

sim

ula

tio

n -

exp

erim

en

t(d

eg

)

simulation

experiment

Figure 9: Experiment Result - Single Thrust, Multiple Step

In the second set of experiments, the aim is to stabilize the

pendulum at the upright position ( = 180). Again a single

thrust source is active at any given time. That means, only the

CCW motor is active when the pendulum is at the right-hand-

side semi-circle (0 < < 180), and only the CW motor is

active when the pendulum is at the left-hand-side semi-circle

(180 < < 360). When the previously tuned gains are used,

the system oscillates around the reference with 18 amplitude.

The proportional gain is reduced until no further significant

reduction occurs in the oscillation amplitude. The best

performance is achieved with KP = 0.5, KD = 0.8, and it has

10 amplitude as shown in Fig.10.

0 5 10 15 20 25 300

20

40

60

80

100

120

140

160

180

200

time (s)

the

ta (

de

g)

Kp=2, Kd=0.8

Kp=0.5, Kd=0.8

Figure 10: Experiment Result - Single Thrust, Upright Balance

The last set of experiments also aims upright stabilization.

This time, both motors run at all times and actuation is

achieved by differential thrust. The idea is to shorten the time

required for the motors to achieve the desired speed in order to

minimize oscillations. Fig.11 shows the results of this

experiment. Using the same controller gains as in the previous

experiment, oscillation amplitude is reduced by 50% in

differential thrust mode. The proportional gain is further

reduced until the system becomes over damped. Finally, the

pendulum is stabilized at 179 with no oscillations.

0 5 10 15 20 25 300

20

40

60

80

100

120

140

160

180

200

time (s)

the

ta (

de

g)

Kp=0.5, Kd=0.8

Kp=0.2, Kd=0.8

Figure 11: Experiment Result - Differential Thrust, Upright Balance

V. CONCLUSION AND FURTHER WORK

Development and testing of a new pendulum experiment

setup for educational purposes is described. The setup features

two brushless dc-motor powered propellers mounted back-to-

back at the tip of the pendulum. Systems performance is

evaluated via hardware-in-the loop simulation experiments by

using Simulink RTWT with custom designed interface circuit.

The designed system is an ideal aid for teaching mathematical

modeling, parameter identification and control system design

at various levels. The further work will concentrate on

development of a compact, modular and portable setup based

on the prototype presented here. Such a system can be given to

students as a take-home hands-on project.

ACKNOWLEDGMENT

Authors thank to former undergraduate students smet Fatih

ekerolu and Elgin Oktay for their efforts in building the

system hardware.

REFERENCES

[1] J. Yi, N. Yubazaki, Stabilization fuzzy control of inverted pendulum

systems, Artificial Intelligence in Engineering, vol. 14, pp. 153-163,

2000.

[2] A. Siuka, M. Schberl, Applications of energy based control methods

for the inverted pendulum, Robotics and Autonomous Systems, vol. 57,

pp. 1012-1017, 2009.

[3] J. -J. Wang, Simulation studies of inverted pendulum based on PID

controllers, Simulation Modelling Practice and Theory, vol. 19, pp.

440-449, 2011.

[4] E. T. Enikov, V. Polyzoev, J. Gill, Low-cost take-home experiment on

classical control using Matlab/Simulink Real-Time Windows Target,

in Proc. of the ASEE/PSW-2010 Conf., Reno, NV, 2010.

[5] S.-F. Yang, J.-H. Chou, A mechatronic positioning system actuated

using a micro DC-motor-driven propellerthruster, Mechatronics, vol.

19, pp. 912-926, 2009.

[6] http://prism2.mem.drexel.edu/~paul/thrustTester/thrustTester.html (last

accessed 17.02.2011)

[7] K. Craig, M. Kleinigger, Mechatronics case study: Thrust vectoring

and control of an unstable system, Dept. Mech. Eng., Rensselaer

Polytechnic Institute, Troy, NY.

[8] http://www.mathworks.com/products/rtwt/ (last accessed 17.02.2011)

[9] http://picprojects.org.uk/projects/simpleSIO/ssio.htm (last accessed

17.02.2011)