MODELOWANIE INŻYNIERSKIE 2016 nr 58 ISSN 1896-771X
DESIGN AND TESTING OF TWO-WHEELED ROBOT WITH CHAOTIC PENDULUM
Kamil Fedio
1, Maksym Kiełkowski
1, Andrzej Katunin
1a, Wawrzyniec Panfil
1b1Institute of Fundamentals of Machinery Design, Silesian University of Technology
aandrzej.katunin@polsl.pl
bwawrzyniec.panfil@polsl.pl
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
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
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 m1=m2=m and lengths
l l
l1= 2= 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:
V E
L= − , (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)
2 1 2 3 3 2
2 0
1 Mv mv mv mv
E= + + + , (2)
where
2 2
0
∂
= ∂ t
v x , (3)
2 3 3 3 2
3 3 2 3
3 sin
2 cos 1
2
1
∂
− ∂
+
∂ + ∂
∂
= ∂ θ θ
θ θ
l t l
t t
v x ,(4)
2 1 1 3 3 3 2
1 cos
2
cos 1
∂ + ∂
∂ +∂
∂
= ∂ θ θ
θ θ
l t l
t t v x
2 1 1 3 3
3 sin
2
sin 1
∂
− ∂
∂
−∂
+ θ θ
θ θ
l t l
t , (5)
2 2 2 1 1 3 3 3 2
2 cos
2 cos 1
cos
∂ + ∂
∂ +∂
∂ +∂
∂
= ∂ θ θ
θ θ θ θ
l t l
t l
t t v x
2 2 2 1 1 3 3
3 sin
2 sin 1
sin
∂
− ∂
∂
−∂
∂
−∂
+ θ θ
θ θ θ θ
l t l
t l
t ,(6)
while the potential energy is given by:
−
+
= 33 3 3 3 cos 1
2 cos 1 cos
2
1ml
θ
m lθ θ
V
− −
+ 3 3 1 cos 2
2 cos 1
cos
θ
lθ
lθ
l
m . (7)
Using (1) one can obtain the Lagrange equation. Using the Lagrange equation of a second kind:
∂ =
− ∂
∂
∂
∂
∂
∂ =
− ∂
∂
∂
∂
∂
∂ =
− ∂
∂
∂
∂
∂
∂ =
−∂
∂
∂
∂
∂
0 0
0
2 2 1 1
3 3
θ θ
θ θ θ θ
L L t
L L t
L L t
F x 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:
∂ + ∂
∂
− ∂
∂
+ ∂ 21
2 2 1
2 2 2 2
2 2
2 sin 3cos
cos 2 1
t t
t
lm θ
θ θ θ θ
θ
F t =
∂
− ∂ 21
2
sin 1
3 θ
θ , (9a)
( )
l( (
m m)
t m m t
x
l cos 4 sin 8
2 32 2 3 3
2 3 2 2 3
2 3
3 +
∂ +∂ +
∂
∂ θ θ
θ
( ) )
∂ + ∂
+
+ 22
2 3 2 3
3 3
2 8 2 sin sin
cos
t m
ll m
m θ
θ θ θ
( )
+
∂
− ∂
∂ + ∂
+
∂ +
∂ lm lm
t t
m m M t
x
3 3 2 3
3 2 2 3
3 2 3 2 3
2
2 2 sin 1
cos
2 θ
θ θ θ
KAMIL FEDIO, MAKSYM KIEŁKOWSKI, ANDRZEJ KATUNIN, WAWRZYNIEC PANFIL
2 1 2 3 2 1
1 2 3 2 1
2 2 3
2sin 3sin sin 3cos sin
cos
t t
t ∂
+ ∂
∂ + ∂
∂
+ ∂ θ
θ θ θ
θ θ θ
θ θ
2 1 2 3 2 1
2 2 3 2 2
2 2 3
2cos sin cos 3cos cos
cos
t t
t ∂
+ ∂
∂
− ∂
∂
+ ∂ θ
θ θ θ
θ θ θ
θ θ
( )
0sin 2 cos
cos
3 3 3 3
2 1 2 3
1 − + =
∂
− ∂ l m m
tθ θ
θ
θ ,(9b)
(
1 3 1 32 3 3 2 2 2
1 sin sin cos cos
cos
6 θ θ θ θ θ
θ +
∂ +∂
∂
∂ l
t t lm x
)
1)
3 1 3
1cos cos sin sin
sinθ θ − θ θ + θ
+
(
1 2 1 2 1 2 1 2)
2 2 2
2 sin sin cos cos sin cos cos sin
2 θ θ θ θ θ θ θ θ θ
− +
+
∂ + ∂
t m l
(
sin cos)
05 2 1 2 1
2 1 2
2 + =
∂
+ ∂ θ θ θ
t m
l , (9c)
(
2 3 2 32 3 3 2 2 2
2 sin sin cos cos
cos
2 θ θ θ θ θ
θ +
∂ +∂
∂
∂ l
t t lm x
)
2)
3 2 3
2cos cos sin sin
sinθ θ + θ θ + θ
+
(
1 2 1 2 1 2 1 2)
2 1 2
2 sin sin cos cos sin cos cos sin
2 θ θ θ θ θ θ θ θ θ
− +
∂ + + ∂
t m l
(
sin cos 2)
02 2 2 2
2 2
2 + =
∂
+ ∂ θ θ θ
t m
l . (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
3 0
3=θ =
θ & (see the scheme of assumed coordinate system in Fig.2). For the double chaotic pendulum there are four critical points (0,0), (0, ), ( ,0), and ( , ), 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 , , , , ,
, 2 3 1 2 3
1
A A A A x
π π π θ π θ θ θ θ
θ & & & (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: θ1=θ2 =θ3=5, x=0, and
3 0
2
1= & = & =x&=
& θ θ
θ . The control variable during each simulation had a constant value and variable sense, depending on the angle θ3. Exemplary results of simu- lation are presented in Fig.3 in the form of Poincaré sections of θ3−θ&3.
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
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
KAMIL FEDIO, MAKSYM KIEŁKOWSKI, ANDRZEJ KATUNIN, WAWRZYNIEC PANFIL
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 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
KAMIL FEDIO, MAKSYM KIEŁKOWSKI, ANDRZEJ KATUNIN, WAWRZYNIEC PANFIL
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-
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
KAMIL FEDIO, MAKSYM KIEŁKOWSKI, ANDRZEJ KATUNIN, WAWRZYNIEC PANFIL
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, = 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.
DESIGN AND TESTING OF TWO-WHEELED ROBOT WITH CHAOTIC PENDULUM
Fig. 5. Poincaré sections for the first, second and third limbs, respectively, using the first considered control algorithm
Fig. 6. Poincaré sections for the first, second and third limbs, respectively, using the second considered control algorithm
Fig. 7. Poincaré sections for the first, second and third limbs, 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.
KAMIL FEDIO, MAKSYM KIEŁKOWSKI, ANDRZEJ KATUNIN, WAWRZYNIEC PANFIL
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.
KAMIL FEDIO, MAKSYM KIEŁKOWSKI, ANDRZEJ KATUNIN, WAWRZYNIEC PANFIL
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-
Fig. 10. Poincaré sections for the first, second and third limbs, 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.
KAMIL FEDIO, MAKSYM KIEŁKOWSKI, ANDRZEJ KATUNIN, WAWRZYNIEC PANFIL
Fig. 10. Poincaré sections for the first, second and third limbs, 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.
DESIGN AND TESTING OF TWO-WHEELED ROBOT WITH CHAOTIC PENDULUM
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
KAMIL FEDIO, MAKSYM KIEŁKOWSKI, ANDRZEJ KATUNIN, WAWRZYNIEC PANFIL
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
KAMIL FEDIO, MAKSYM KIEŁKOWSKI, ANDRZEJ KATUNIN, WAWRZYNIEC PANFIL
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.
KAMIL FEDIO, MAKSYM KIEŁKOWSKI, ANDRZEJ KATUNIN, WAWRZYNIEC PANFIL
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
DESIGN AND TESTING OF TWO
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
OF TWO-WHEELED ROBOT WITH CHAOTIC PENDULUM
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.
WHEELED ROBOT WITH CHAOTIC PENDULUM
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.
KAMIL FEDIO, MAKSYM KIEŁKOWSKI, ANDRZEJ KATUNIN, WAWRZYNIEC PANFIL
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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