DESIGN AND TESTING OF TWO-WHEELED ROBOT WITH … · developed control algorithms, and construction of a physical model of a robot and verification of control algo-rithms. Both simulation

MODELOWANIE INŻYNIERSKIE 2016 nr 58 ISSN 1896-771X

24

DESIGN AND TESTING OF TWO-WHEELED

ROBOT WITH CHAOTIC PENDULUM

Kamil Fedio1, Maksym Kiełkowski1, Andrzej Katunin1a,

Wawrzyniec Panfil1b

1Institute of Fundamentals of Machinery Design, Silesian University of Technology [email protected] [email protected]

Summary This paper deals with a theoretical modeling of equations of motion, simulation motion of and control both in vir-

tual environments and in real-world conditions of a balancing two-wheeled robot with a double pendulum. By tak-

ing into consideration a motion of a double pendulum one needs to consider chaotic behavior of the whole system

resulted by this pendulum, which is a significant difficulty in development of control algorithms. The main goal of

the presented study is to reach dynamic balancing of a two-wheeled robot with a double pendulum under the cer-

tain scenarios of equilibrium disturbance. In order to apply appropriate control algorithms the following steps

were assumed during the development of a robot: theoretical modelling of a motion of the composite system of in-

verted and double pendulums, stability analysis, simulation of various scenarios in virtual environments using the

developed control algorithms, and construction of a physical model of a robot and verification of control algo-

rithms. Both simulation and experimental studies demonstrated the successful balancing performance.

Keywords: two-wheeled robot, inverted pendulum, double pendulum, chaotic motion, non-linear control,

balancing stability

PROJEKT I TESTOWANIE DWUKOŁOWEGO ROBOTA

Z WAHADŁEM CHAOTYCZNYM

Streszczenie Artykuł dotyczy teoretycznego modelowania równań ruchu, symulacji ruchu i sterowania balansującego robota

dwukołowego z podwójnym wahadłem zarówno w środowiskach wirtualnych, jak i w warunkach rzeczywistych.

Biorąc pod uwagę ruch podwójnego wahadła, należy uwzględnić chaotyczny sposób działania całego układu spo-

wodowany ruchem tego wahadła, co stanowi istotną trudność przy opracowywaniu algorytmów sterowania. Głów-

nym celem prezentowanej pracy jest uzyskanie stanu stabilności dynamicznej robota dwukołowego z podwójnym

wahadłem według poszczególnych scenariuszy zaburzenia jego równowagi. W celu zastosowania odpowiednich al-

gorytmów sterowania następujące etapy zostały założone podczas opracowania robota: teoretyczne modelowanie

ruchu układu złożonego z odwróconego i podwójnego wahadeł, analiza stabilności, symulacja różnych scenariuszy

w środowiskach wirtualnych z zastosowaniem opracowanych algorytmów sterowania oraz opracowanie modelu fi-

zycznego robota i weryfikacja algorytmów sterowania. Zarówno prace symulacyjne, jak i eksperymentalne wykaza-

ły zdolność do utrzymania równowagi.

Słowa kluczowe: robot dwukołowy, wahadło odwrócone, wahadło podwójne, ruch chaotyczny, sterowanie nie-

liniowe, stabilność dynamiczna

1. INTRODUCTION

Balancing systems are quite attractive for numerous

researchers since the static balancing control is one of

the crucial concepts used in applications of walking

control of humanoid robots [1,2]. The two-wheeled

KAMIL FEDIO, MAKSYM KIEŁKOWSKI, ANDRZEJ KATUNIN, WAWRZYNIEC PANFIL

25

system, which is an inverted pendulum in the physical

sense, since the center of mass is located above the

wheel axis, found wide popularity in numerous applica-

tions to-date, e.g. two-wheeled self-balanced vehicles

(Segway), Toyota’s personal transporter [3], and other

transportation systems [4,5]. An inverted pendulum

model is often used as a benchmark model for testing

control algorithms [6-8]. Therefore, the robustness of

such a system to external disturbances has a great

practical importance.

Several studies on stability and balancing control of

inverted pendulum-type systems were already performed

using various methods and control algorithms which

were experimentally verified in most cases. Yamakawa

described in [9] a high-speed fuzzy logic controller ap-

plied for stabilization of an inverted pendulum; the

authors of [4,10] developed a control approach based on

artificial neural networks; the authors of [11,12] used a

solution based on PID controller; while Chiu presented

an advanced controller – adaptive output recurrent

cerebellar model articulation controller used for the

problem of balancing control of inverted pendulum-type

systems [13]. The global stabilization studies with exper-

imental verification were performed by Srinivasan et al.

[14]. The more advanced cases of balancing control,

including parallel-type double inverted pendulum [15]

and multiple inverted pendulums [16-20], were also

studied and experimentally verified. These systems are

highly nonlinear, but still deterministic.

Recently, the great interest is paid to highly nonlinear

systems which reveal chaotic behavior globally or under

certain conditions. Ones of the simplest systems that

reveal chaotic behavior are double, triple and multiple

mathematical pendulums. A series of original studies on

the dynamics and stability of such chaotic systems were

performed by Awrejcewicz and his team (see e.g. [21-23])

and other researchers [24-26], including previous studies

of the authors’ team [27]. In order to control the motion

of such chaotic systems a different class of control

algorithms was developed. Generally, two types of

approaches of control chaotic systems can be distin-

guished: feedback control and non-feedback control,

which focus on periodization of chaotic motion of multi-

ple pendulum-type systems [28-30]. Another problem of

controlling chaotic systems is tending to stabilization of

a system. For solving a class of problems of stabilization

of chaotic oscillations several approaches were proposed:

de Korte et al. [31] used semi-continuous control meth-

od, Guan et al. [32] proposed an impulsive control

method, while Awrejcewicz et al. [33] used a feedback

control approach.

The system investigated in the present study is a two-

wheeled robot with double chaotic pendulum which can

be considered, from the point of view of its kinematics,

as a composite of inverted pendulum and double chaotic

pendulum. In the best of the authors’ knowledge, such

system was not previously investigated elsewhere.

2. MOTIVATION

AND ASSUMPTIONS

The main goal of the designed robot was to develop

effective and simple control algorithms which allow

reaching dynamic balancing of a robot without any

external loading and under the certain scenarios of

equilibrium disturbance. This goal was reached by

performing consequent steps in the performed study,

namely: theoretical modeling of a system with further

analysis of its stability and simulation tests using vari-

ous control algorithms, design of a robot and performing

simulation tests in a virtual environment, and finally,

physical implementation of a robot and verification tests

of implemented algorithms.

At the beginning of development of both mathemati-

cal model and mechatronic physical system four most

important groups of assumptions were applied.

The assumptions of theoretical model were as fol-

lows: simplification of a kinematic model due to sym-

metry (from 3D to 2D); discretization of a model of

composite system of inverted and double chaotic pendu-

lums to the form of three limbs with point masses

located in their geometric centers; the limbs are perfect-

ly rigid, and the last two of them are of the same

lengths and masses; inertial forces of each limb are high

enough to influence on each other limb.

The functional assumptions covered a condition of

holding the vertical position of a whole robot considering

scenarios when the stability of the robot is disturbed

(including reaction on motion of chaotic pendulum), and

minimization of sliding during balancing.

The third group covered hardware assumptions of

the developed robot, namely: wheels of a robot should

be rigid enough to neutralize the effect of gravitational

deflection and their diameter should be at least the same

as the width of the robot frame, and the motor should

be a high-speed in order to react on disturbances timely

with a possibility of mounting encoders.

The last group of assumptions was concerned to the

control hardware system and covered the following ones:

the applied platform should allow rapid control proto-

typing; quick and low-cost components for control

system (regulators, sensors, cable connections); the only

sensors are the accelerometer and the gyroscope mount-

ed in the axis of rotation of wheels should allow satisfy-

ing the measurement data stream enough for effective

control.

Following the presented groups of assumptions the

theoretical model as well as hardware implementation of

the designed robot were performed.

DESIGN AND TESTING OF TWO-WHEELED ROBOT WITH CHAOTIC PENDULUM

26

3. THEORETICAL MODEL

3.1 EQUATIONS OF MOTION

Considering the assumptions to the mathematical model

presented above, the following notion should be intro-

duced: the same masses mmm == 21 and lengths

lll == 21 of a chaotic pendulum limbs. Since the system

has four degrees of freedom, then four generalized coor-

dinates should be introduced: translation coordinate x,

and three rotation coordinates for each limb: �1, �2, �3.

The kinematic scheme of the considered system is pre-

sented in Fig.1.

In order to obtain equations of motion of the analyzed

system one need to minimize the action functional

which, in consequence, is a Lagrangian:

VEL −= , (1)

where E and V are the kinetic and potential energies of

a system, respectively.

Fig. 1. Kinematic scheme of the analyzed system

Considering the equations of inverted pendulum (see e.g.

[11]) and a double chaotic pendulum (see e.g. [27]), the

kinetic energy for the analyzed system takes a form:

( )22

21

233

20

2

1mvmvvmMvE +++= , (2)

where

2

20

∂

∂=

t

xv , (3)

2

333

2

3332

3 sin2

1cos

2

1

∂

∂−+

∂

∂+

∂

∂= θ

θθ

θl

tl

tt

xv ,(4)

2

11

3332

1 cos2

1cos

∂

∂+

∂

∂+

∂

∂= θ

θθ

θl

tl

tt

xv

2

11

333 sin

2

1sin

∂

∂−

∂

∂−+ θ

θθ

θl

tl

t, (5)

2

22

11

3332

2 cos2

1coscos

∂

∂+

∂

∂+

∂

∂+

∂

∂= θ

θθ

θθ

θl

tl

tl

tt

xv

2

22

11

333 sin

2

1sinsin

∂

∂−

∂

∂−

∂

∂−+ θ

θθ

θθ

θl

tl

tl

t,(6)

while the potential energy is given by:

−+= 133333 cos

2

1coscos

2

1θθθ lmlmV

−−+ 2133 cos

2

1coscos θθθ lllm . (7)

Using (1) one can obtain the Lagrange equation. Using

the Lagrange equation of a second kind:

=∂

∂−

∂

∂

∂

∂

=∂

∂−

∂

∂

∂

∂

=∂

∂−

∂

∂

∂

∂

=∂

∂−

∂

∂

∂

∂

0

0

0

22

11

33

θθ

θθ

θθ

LL

t

LL

t

LL

t

Fx

L

x

L

t

&

&

&

&

(8)

where F is an excitation force of a mass M in the direc-

tion of a vector x. The dots in (8) mean derivatives of

particular variables over a time. Using (2)-(7) in (8) one

obtains the system of equations as follows:

∂

∂+

∂

∂−

∂

∂+

2

12

12

22

22

22

2 cos3sincos2

1

ttt

lmθ

θθ

θθ

θ

Ft

=

∂

∂−

2

12

1sin3θ

θ , (9a)

( ) ( )( mmlt

mmt

xl 8sin4cos2 33

2232

32

32

2

33 +∂

∂++

∂

∂θ

θθ

( ))

∂

∂+++

2

22

323332

sinsin28cost

mllmmθ

θθθ

( )

+

∂

∂−

∂

∂+++

∂

∂mlml

tt

mmM

t

x

3332

32

32

32

332

2

22

1sincos2

θθ

θθ


27

2

12

312

12

312

22

32 sincos3sinsin3sincosttt ∂

∂+

∂

∂+

∂

∂+

θθθ

θθθ

θθθ

2

12

312

22

322

22

32 coscos3cossincoscosttt ∂

∂+

∂

∂−

∂

∂+

θθθ

θθθ

θθθ

( ) 0sin2coscos3 3332

12

31 =+−

∂

∂− mml

tθ

θθθ ,(9b)

( 313132

32

2

2

1 coscossinsincos6 θθθθθ

θ +∂

∂+

∂

∂l

tt

xlm

) )13131 sinsincoscossin θθθθθ +−+

( )212121212

22

2sincoscossincoscossinsin2 θθθθθθθθ

θ−++

∂

∂+

tml

( ) 0cossin5 12

12

2

12

2 =+∂

∂+ θθ

θ

tml , (9c)

( 323232

32

2

2

2 coscossinsincos2 θθθθθ

θ +∂

∂+

∂

∂l

tt

xlm

) )23232 sinsincoscossin θθθθθ +++

( )212121212

12

2sincoscossincoscossinsin2 θθθθθθθθ

θ−++

∂

∂+

tml

( ) 0cossin 22

22

2

22

2 =+∂

∂+ θθ

θ

tml . (9d)

Solving the system of equations (9) one gets the equa-

tions of motion of the considered robot.

3.2 STABILITY ANALYSIS

Analyzing the composite system of inverted and double

chaotic pendulums one can consider the stability points

of these pendulums separately. In the case of inverted

pendulum there is only one critical point which repre-

sents dynamical equilibrium occurred when for M

033 == θθ & (see the scheme of assumed coordinate

system in Fig.2). For the double chaotic pendulum there

are four critical points (0,0), (0,P), (P,0), and (P,P), where

only first one guarantees a stable equilibrium.

Fig. 2. A scheme of assumed coordinate system

Following this, the stability conditions for the whole

investigated system can be described as follows:

( )

=

0,0,0,0,,0,,

0,0,0,0,,0,0,

0,0,0,0,,0,,0

0,0,0,0,,0,0,0

,,,,,, 321321

A

A

A

A

x

ππ

π

πθθθθθθ &&& (10)

where A stands for arbitrary parameter.

All of the coordinates are time-dependent. In practice,

the most stable critical point is the first one from (10).

Several simulations were performed in order to examine

the modeled system. In order to excite different types of

oscillation modes the initial conditions of the system

were assumed as follows: 5321 === θθθ , 0=x , and

0321 ==== x&&&& θθθ . The control variable during each

simulation had a constant value and variable sense,

depending on the angle 3θ . Exemplary results of simu-

lation are presented in Fig.3 in the form of Poincaré

sections of 33 θθ &− .

DESIGN AND TESTING OF TWO

Fig. 3. Exemplary Poincaré sections for m3 for

b) F = 12 N, c) F = 14.5 N, d) F = 15 N

OF TWO-WHEELED ROBOT WITH CHAOTIC PENDULUM

28

Fig. 3. Exemplary Poincaré sections for m3 for a) F = 8 N,

Presented Poincaré sections show a typical chaotic

behavior of a system, i.e. even small change of a force

value cause significant changes of dynamics of the

system. If the control force is too small (see Fig.3a,b),

then for a small period of time the attractors appear,

which reflect periodic oscillation of a point of location of

m3; however, finally, the observed point became oscillate

chaotically. Similarly, if F is too large (see Fig.3d), few

attractors appear. In the third case (Fig.3c) the

Poincaré section reveals an occurrence of period

doubling bifurcation which, in fact, denotes that oscill

tions of m3 are quasi-periodic. This, however, does not

mean that the behavior of the whole system is the same.

Obviously, reaching the stable equilibrium poi

system is not possible in practice, however, using appr

priate control methods the balancing stability around

the first stable point from (10) is possible.

4. BALANCING CONTROL

SIMULATION

4.1 CONCEPT OF AUTOMATIC

CONTROL SYSTEM

In order to ensure a possibility of balancing control

of the robot the automatic feedback control system

is proposed following the scheme presented in

Fig.4.

Fig. 4. A scheme of the control system:

2) motors, 3) a composite of inverted and double

pendulums, 4) accelerometer and gyroscope; signals: w(t)

input, v(t) – feedback, y(t) – output, e(t)

tion, u(t) – control signal, u*(t) –

ances.

The regulation system presented in Fig.4. works as

follows: the input signal w(t) is a value of

zero) is compared with a value of a feedback signal

i.e. the angle measured by sensors. The resulting devi

tion between these signals e(t) become an input to the

microcontroller, where, on its basis, a control signal

is generated which is responsible for the motion of

motors. The motors generate a torque which is an

excitation u*(t) that acts on the controlled system.

Various disturbances z(t), like air resistance, may infl

ence on the controlled object. An excitation causes the

change of a slope of the robot which is registered by

sensors, and which, in turn, begins the next regulation

loop.

WHEELED ROBOT WITH CHAOTIC PENDULUM

sections show a typical chaotic

behavior of a system, i.e. even small change of a force

value cause significant changes of dynamics of the

system. If the control force is too small (see Fig.3a,b),

then for a small period of time the attractors appear,

ch reflect periodic oscillation of a point of location of

; however, finally, the observed point became oscillate

is too large (see Fig.3d), few

attractors appear. In the third case (Fig.3c) the

section reveals an occurrence of period-

doubling bifurcation which, in fact, denotes that oscilla-

periodic. This, however, does not

mean that the behavior of the whole system is the same.

Obviously, reaching the stable equilibrium point for this

system is not possible in practice, however, using appro-

priate control methods the balancing stability around

the first stable point from (10) is possible.

BALANCING CONTROL

CONCEPT OF AUTOMATIC

re a possibility of balancing control

of the robot the automatic feedback control system

is proposed following the scheme presented in

Fig. 4. A scheme of the control system: 1) microcontroller,

motors, 3) a composite of inverted and double chaotic

4) accelerometer and gyroscope; signals: w(t) –

output, e(t) – regulation devia-

excitation, z(t) – disturb-

The regulation system presented in Fig.4. works as

) is a value of �3 (equaled

zero) is compared with a value of a feedback signal v(t),

i.e. the angle measured by sensors. The resulting devia-

) become an input to the

microcontroller, where, on its basis, a control signal u(t)

is generated which is responsible for the motion of

motors. The motors generate a torque which is an

) that acts on the controlled system.

), like air resistance, may influ-

ence on the controlled object. An excitation causes the

change of a slope of the robot which is registered by

sensors, and which, in turn, begins the next regulation


4.2 SIMULATOR AND CONTROL

ALGORITHMS

The simulator was developed and implemented in the

Matlab®/Simulink® environment. The simulator consists

of four main blocks: controlled object, sensors

erometer and gyroscope, and a control system with three

control algorithms. The block of a contr

tionally contains a complementary filter which is used

for preconditioning of measurement signals from sensors

before using them in control algorithms. For the testing

purposes three simple control algorithms were impl

mented.

The first control algorithm was based on comparison of

the angle �3, determined by the robot subsystems, to

zero. If the measured angle is greater than zero (robot

inclined to the right), then the control signal is equal to

the constant value F, otherwise, if the measured angle is

lower than zero (robot inclined to the left), then the

control signal is equal to the constant value

case when the angle of inclination equals zero, the

control signal is zero-valued.

The second algorithm is based on a proporti

This means, that the value of a control signal, in spite of

the first algorithm, is proportional to the angle of incl

nation, i.e. if �3 P 0, then the value of the control signal is

�3·F.

The third algorithm is a slight modification of the

second one which was caused by a limited precision of

determination of an inclination angle and a fact, that

the system cannot reach static equilibrium state (see

section 3.2). Following this, the inclination angle

this algorithm is compared not to zer

small angle �. Therefore, if the measured value of

or �3 < –�, then then the value of the control signal is

�3·F, otherwise, if �3 is in <–�, �>, then the control

signal is 0.

4.3 RESULTS OF SIMULATIONS

Since the simulator is an idealized model of a considered

system, its application has several advantages and

difficulties. The main difficulty is that the conditions of

the experiment cannot be the same as assumed during

simulation studies, e.g. omitting the influence of friction

and sliding in theoretical model. However, the initial

and boundary conditions (e.g. initial values of inclin

tion angles and velocity of rotation of all limbs as well

as a value of the control signal) can be precisely set,

similarly as in the theoretical model. In order to prepare

the simulator to testing particular scenarios the sensors

need to be calibrated. The calibration is necessary in

order to reflect their realistic operation, including a

simulation of measurement errors. For this purpose a

signal prefiltering is necessary which requires calibration


29

SIMULATOR AND CONTROL

The simulator was developed and implemented in the

environment. The simulator consists

of four main blocks: controlled object, sensors – accel-

erometer and gyroscope, and a control system with three

control algorithms. The block of a control system addi-

tionally contains a complementary filter which is used

for preconditioning of measurement signals from sensors

before using them in control algorithms. For the testing

purposes three simple control algorithms were imple-

ol algorithm was based on comparison of

, determined by the robot subsystems, to

zero. If the measured angle is greater than zero (robot

inclined to the right), then the control signal is equal to

measured angle is

lower than zero (robot inclined to the left), then the

control signal is equal to the constant value –F. In the

case when the angle of inclination equals zero, the

The second algorithm is based on a proportional control.

This means, that the value of a control signal, in spite of

the first algorithm, is proportional to the angle of incli-

P 0, then the value of the control signal is

The third algorithm is a slight modification of the

cond one which was caused by a limited precision of

determination of an inclination angle and a fact, that

the system cannot reach static equilibrium state (see

section 3.2). Following this, the inclination angle �3 in

this algorithm is compared not to zero, but to some

. Therefore, if the measured value of �3 > �

, then then the value of the control signal is

>, then the control

RESULTS OF SIMULATIONS

idealized model of a considered

system, its application has several advantages and

difficulties. The main difficulty is that the conditions of

the experiment cannot be the same as assumed during

simulation studies, e.g. omitting the influence of friction

nd sliding in theoretical model. However, the initial

and boundary conditions (e.g. initial values of inclina-

tion angles and velocity of rotation of all limbs as well

as a value of the control signal) can be precisely set,

odel. In order to prepare

the simulator to testing particular scenarios the sensors

need to be calibrated. The calibration is necessary in

order to reflect their realistic operation, including a

simulation of measurement errors. For this purpose a

refiltering is necessary which requires calibration

of filters. The performed calibration studies show a very

good convergence of simulated and determined signals

for both considered sensors.

In order to test the control algorithms using developed

simulator the following initial conditions were assumed:

velocity and acceleration of rotation of all limbs of

pendulum and of wheels of a robot were assumed to be

zero, while the angles of inclination were assumed as

�1 = �2 = 0°, and �3 = 10°. Before testing the

of considered algorithms were determined empirically:

• first algorithm: F = 8,

• second algorithm : F = 13,

• third algorithm: F = 13,

The results of simulation tests are presented in the form

of Poincaré sections for all limbs following the scheme

presented in Fig.1. It should be noticed that in the case

of a physical robot the Poincaré section can be obtained

only for the third limb (according to the scheme

Fig.1), since the inclination angle is observ

sensors for this limb only. For the easier interpretation

and comparison the values on axes in all presented cases

are the same. The horizontal axis represents an inclin

tion angle from the assumed zero

radians, and vertical axis represents an angular velocity

of the given limb. The results for all considered control

algorithms are presented in Figs. 5


of filters. The performed calibration studies show a very

good convergence of simulated and determined signals

In order to test the control algorithms using developed

r the following initial conditions were assumed:

velocity and acceleration of rotation of all limbs of

pendulum and of wheels of a robot were assumed to be

zero, while the angles of inclination were assumed as

= 10°. Before testing the parameters

of considered algorithms were determined empirically:

= 13,

= 13, P = 1°.

The results of simulation tests are presented in the form

sections for all limbs following the scheme

presented in Fig.1. It should be noticed that in the case

of a physical robot the Poincaré section can be obtained

only for the third limb (according to the scheme –

Fig.1), since the inclination angle is observed by a set of

sensors for this limb only. For the easier interpretation

and comparison the values on axes in all presented cases

are the same. The horizontal axis represents an inclina-

tion angle from the assumed zero-positions (see Fig.2) in

vertical axis represents an angular velocity

of the given limb. The results for all considered control

algorithms are presented in Figs. 5-7, respectively.


30

Fig. 5. Poincaré sections for the first, second and third limbs,

respectively, using the first considered control algorithm


respectively, using the second considered control algorithm


respectively, using the third considered control algorithm

From the preliminary observations of the Poincaré

sections for tested control algorithms one can conclude

that the first algorithm does not allow for balancing

control of the considered system: the trajectories of

every limb diverge after certain amount of time and the

system loses its stability.

In order to compare the second and third control algo-

rithm the time realizations for a duration of 60 s from

the initial disturbance of a stability are presented in

Fig.8 and Fig.9, respectively. One can observe, both on

Poincaré sections and time realizations, that the third

algorithm increase the magnitudes of oscillations around

the stability point, however, the damping of oscillations

is better for this algorithm.


Fig. 8. Variability of �3 during application of the second control

algorithm

Fig. 9. Variability of �3 during application of the

algorithm

Due to the occurrence of a chaotic motion of the consi

ered system it is also necessary to perform a simulation

when initial conditions generate chaotic motion of a

system at the beginning. For this purpose the following

initial conditions were assumed: �1 = �3

20°. The time of simulation was extended to 210

the considered case is more complex than the previous

ones. The results of simulation for all limbs are presen

ed in Fig.10.


31

during application of the second control

during application of the third control

Due to the occurrence of a chaotic motion of the consid-

ered system it is also necessary to perform a simulation

when initial conditions generate chaotic motion of a

r this purpose the following

= 0°, and �2 =

20°. The time of simulation was extended to 210 s, since

the considered case is more complex than the previous

ones. The results of simulation for all limbs are present-


respectively, using the third considered control algorithm with

modified initial conditions

From the Poincaré sections presented in Fig.10 one can

observe that the system reveal dynamic stability at the

defined critical point. The stabilization of the system

and tending to the equilibrium can be observed on the

time realization for this case which is presented in

Fig.11. One can see that after 140 s the system is stab

lized, i.e. the magnitudes of �3 become lower; after 180 s

the system reveal low-magnitude periodic oscillations

which proves the stabilization of the system.

Fig. 11. Variability of �3 during application of the third control

algorithm with modified initial conditions

5. HARDWARE

IMPLEMENTATION

AND TESTING

5.1 CAD DESIGN AND VIRTUAL

TESTING

Irrespective of simulations in Matlab

ronment a 3D virtual model of the robot was

and simulations in V-REP virtual environment were

carried out.

In order to prepare the 3D model of the robot in V

(Fig.12) it was necessary to make models of the robot

parts using CAD software. Then, the models in

format were imported to V-REP, and revolute joints

between robot base, wheels, pendulum limbs were made.



respectively, using the third considered control algorithm with

From the Poincaré sections presented in Fig.10 one can

veal dynamic stability at the

defined critical point. The stabilization of the system

and tending to the equilibrium can be observed on the

time realization for this case which is presented in

Fig.11. One can see that after 140 s the system is stabi-

become lower; after 180 s

magnitude periodic oscillations

which proves the stabilization of the system.

during application of the third control

algorithm with modified initial conditions

IMPLEMENTATION

CAD DESIGN AND VIRTUAL

Irrespective of simulations in Matlab®/Simulink® envi-

ronment a 3D virtual model of the robot was prepared

REP virtual environment were

In order to prepare the 3D model of the robot in V-REP

(Fig.12) it was necessary to make models of the robot

parts using CAD software. Then, the models in stl

REP, and revolute joints

between robot base, wheels, pendulum limbs were made.


32

Fig. 12. Virtual 3D model of the robot in V-REP simulator

(visible red bars representing revolute joints)

Controlling of the robot movement and obtaining infor-

mation from virtual sensors were realized using a remote

API for Matlab®. As the output from simulator served

an angle of the robot base with respect to the ground

(�3), the input was the angular velocities of both wheels.

These velocities were determined using several algo-

rithms.

5.2 EXPERIMENTAL SETUP

Mechanical part of the real robot consists of: robot base

frame, chaotic pendulum, motors with attached wheels,

and other parts (bolts, fastenings, bearings, etc.). Final

version of the robot visible is on Fig.13. Some parts

(mainly limbs of the pendulum) of the robot have been

made using 3D printing technology.

A control system of the robot was composed using

market-available rapid control prototyping parts, i.e.:

• prototyping platform DfRobot Mega2560 (Ar-

duino Mega2560),

• motor driver Roboclaw 2×15 A,

• Inertial Measurement Unit MPU-9150,

• TTL voltage converter.

The robot was supplied using 11.1 V Li-Po batteries. To

drive the robot there were applied gearmotors

(285 RPM (4.75 s-1), 0.42 Nm) with encoders. The IMU

consisted of a 3-axis gyroscope (range up to

±2000°/sec), a 3-axis accelerometer (range up to ±16 g)

and a magnetometer. It is important to notice that this

IMU uses Digital Motion Processor™ (DMP™) in order

to process MotionFusion algorithms. DMP allows to

obtain information about orientation of the robot (an-

gles Yaw, Pitch, Roll) releasing a main control unit from

this task. It also takes an advantage of resetting meas-

urements and calibration.

Fig. 13. Experimental robot

5.3 VERIFICATION CASES

Verification tests of the control algorithms included

three cases (Fig.14). The first one was the simplest. If

the tested algorithm manages with the first case, it will

be tested for other two cases.

Case 1: The robot is placed in a stable position, then it

is released.

Case 2: The same situation as in the Case 1, but

additionally the robot is disturbed from equilibrium by

an external force.

Case 3: The same situation as in the Case 1, but

additionally the pendulum is disturbed from equilibri-

um.

Fig. 14. Three test cases

In the V-REP simulation environment and on the real

robot all the three algorithms described in section 4.2.

were tested. Additionally, fourth algorithm based on

PID control scheme was tested.

5.4 SIMULATION TESTS IN V-REP

Verification of the control algorithms included test

Cases 1 and 3, not 2, because it was impossible to apply


in simulation an additional force acting on the robot.

The tests shown that only fourth algorithm is able to

control the robot. Fig.15 presents results of testi

second and third algorithms in the Case 1

Fig. 15. Results of verification of the first, second and third

algorithms for Case 1 in V-REP (blue line

motors velocity (max.1000 representing 4,75s-

It can be noted that none of these algorithms can cope

with robot oscillations. Due to this fact all these three

algorithms were not tested in the test Case 3

Analyzing results of testing the first algorithm (first plot

in Fig.15), it can be seen that direction of wheels rot

tion changes according to the sign (+/-

inclination �3. It is also important to notice, that max


33

in simulation an additional force acting on the robot.

The tests shown that only fourth algorithm is able to

control the robot. Fig.15 presents results of testing first,

Case 1.

Results of verification of the first, second and third

REP (blue line - �3[°], red line – -1))

It can be noted that none of these algorithms can cope

with robot oscillations. Due to this fact all these three

Case 3.

Analyzing results of testing the first algorithm (first plot

direction of wheels rota-

-) of the angle of

. It is also important to notice, that maxi-

mum velocity of motors had to be reduced, because the

velocities changed rapidly and higher velocities caused

turnover of the robot.

Looking at Fig.15 (green rectangles on second plot), one

can see an exemplary situations when oscillations of the

pendulum remarkably influence on the robot oscillations.

Second and third plots in Fig.15 show results obtained

using proportional algorithms. An expectable situation,

when the velocity of the motors changes proportionally

to the angle of inclination �3 is observed

senting motors velocities and robot inclination almost

overlap.

Only fourth PID algorithm was able to ma

robot oscillations in Case 1, so then it was tested in

Case 3 (Fig.16).

Fig. 16. Results of verification of the fourth (PID) algorithm for

Case 1 and Case 3 in V-REP (blue line

velocity (max.1000 representing 4,75s

During the simulation the variability of

±25° and the velocity of the wheels was up to 50% of

maximum velocity. Looking at results shown on Fig.16

it is quite easy to notice that sometimes oscillations of

the pendulum intensify oscillations

sometimes suppress, which is resulted from the chaotic

nature of these oscillations.


mum velocity of motors had to be reduced, because the

velocities changed rapidly and higher velocities caused

Looking at Fig.15 (green rectangles on second plot), one

can see an exemplary situations when oscillations of the

pendulum remarkably influence on the robot oscillations.

Second and third plots in Fig.15 show results obtained

nal algorithms. An expectable situation,

when the velocity of the motors changes proportionally

is observed – lines repre-

senting motors velocities and robot inclination almost

Only fourth PID algorithm was able to manage with

, so then it was tested in

Results of verification of the fourth (PID) algorithm for

REP (blue line - �3[°], red line – motors

velocity (max.1000 representing 4,75s-1))

During the simulation the variability of �3 was about

±25° and the velocity of the wheels was up to 50% of

maximum velocity. Looking at results shown on Fig.16

it is quite easy to notice that sometimes oscillations of

the pendulum intensify oscillations of the robot, but

sometimes suppress, which is resulted from the chaotic


5.5 TESTS ON THE REAL ROBOT

Tests of the first, second and third algorithms on the

real robot were very similar to those conducted in

virtual simulation in V-REP. These algorithms were not

able to control efficiently the robot even in

consequently they were not tested for other test cases.

Fig. 17. Results of verification of the PID algorithm for cases 1,

2 and 3 (blue line - �3[°], red line – motors velocity (PWM,

max.1024 representing 4,75s-1))

The reason of such high inconsistencies between the

theoretical model presented in Sections 3 and 4, and the

real robot are backlashes of the gearboxes used in the

constructed robot, i.e. in the cases of the control alg

rithms presented in Section 4 these backlashes create so


34

TESTS ON THE REAL ROBOT

Tests of the first, second and third algorithms on the

real robot were very similar to those conducted in

REP. These algorithms were not

able to control efficiently the robot even in Case 1, so

other test cases.

Results of verification of the PID algorithm for cases 1,

motors velocity (PWM,

The reason of such high inconsistencies between the

presented in Sections 3 and 4, and the

real robot are backlashes of the gearboxes used in the

constructed robot, i.e. in the cases of the control algo-

rithms presented in Section 4 these backlashes create so

big range of velocity values that the control val

often placed inside this range.

Results obtained for fourth PID algorithm during tests

carried out on the real robot are presented in the Fig.17.

First plot in Fig.17 shows that robot oscillates slightly

(±6°) near the equilibrium state, so the al

to sufficiently control the robot.

The second plot in Fig.17 presents behavior of the robot

controlled by PID algorithm in Case 2

the oscillation of the robot is about ±10°. A green

rectangle on the second plot indicates a

robot is pushed out from equilibrium by an external

force (to be precise – by a hand of a testing person).

One can see that the algorithm is able to easily reduce a

huge (almost 40°) deviation of regulation, and then the

robot oscillates around the equilibrium point.


big range of velocity values that the control values were

Results obtained for fourth PID algorithm during tests

carried out on the real robot are presented in the Fig.17.

First plot in Fig.17 shows that robot oscillates slightly

(±6°) near the equilibrium state, so the algorithm is able

The second plot in Fig.17 presents behavior of the robot

Case 2. One can see that

the oscillation of the robot is about ±10°. A green

rectangle on the second plot indicates a situation when

robot is pushed out from equilibrium by an external

by a hand of a testing person).

One can see that the algorithm is able to easily reduce a

huge (almost 40°) deviation of regulation, and then the

ound the equilibrium point.


35

Fig. 18. Snapshots of tests of constructed robot: a) balancing on

equilibrium, b) disturbance by external force, c) oscillations of

the pendulum, d) equilibration, e) disturbance of the pendulum,

f) equilibration

The snapshots of the tests of the constructed robot

which correspond with the testing cases presented in

Fig.14 were stored in the Fig.18. The frames presented

in Fig.18 can be compared to the signals shown in

Fig.17 for all tested cases.

6. CONCLUSIONS

In the presented paper the results of theoretical model-

ling and physical implementation of the two-wheeled

balancing robot with a double chaotic pendulum were

analyzed. The control routines for the robot with vari-

ous scenarios were tested theoretically, in virtual simula-

tion environment, and on the physical model of the

robot. The mathematical model of the robot was devel-

oped by merging equations of motion of inverted pendu-

lum and double chaotic pendulum. By solving this

system of equations analytically and defining initial and

boundary conditions one achieves Poincaré sections

based on which the dynamic stability of convergence to

the equilibrium of the investigated system of pendulums

was analyzed under various scenarios. Afterwards, the

analysis in V-REP simulation software was performed.

The results of analyzes eliminates simple control algo-

rithms applied on mathematical model, since the loss of

stability was observed for the analyzed system. This can

be explained by high degree of simplification of the

mathematical model with respect to the real robot (one

axis of motion, assumption of concentrated masses,

weightless limbs, etc.). At the final stage the control

algorithms were tested on the real model of a robot. The

comparative studies show that the physical model has

even worth controllability than its virtual simulation.

Besides the mentioned problem of simplification of the

theoretical model, the backlashes were observed on the

gearboxes, which eliminates all previously applied con-

trol algorithms. By applying PID algorithm it was

possible to achieve control on motion of the robot, even

for the considered test cases, two of which assumes

throwing of balance of the robot.

The developed platform, since it reveals complex behav-

ior and very weak stability, is an outstanding physical

benchmark to test new control algorithms in future

studies. The new control algorithms for control of such

system, as well as further development of a complexity

of this system is planned in the future studies.

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DESIGN AND TESTING OF TWO-WHEELED ROBOT WITH … · developed control algorithms, and construction of a physical model of a robot and verification of control algo-rithms. Both simulation