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Stabilization of a Two-Wheeled Mobile Pendulum System using LQG and Fuzzy Control Techniques

Authors:
Ákos Odry at University of Szeged
Ákos Odry
János Fodor
János Fodor
János Fodor
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Peter Odry at University of Dunaújváros
Peter Odry
  • University of Dunaújváros
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Abstract and Figures

This paper studies the control performances of modern and soft-computing based control solutions. Namely, the stabilization of a naturally unstable mechatronic system will be elaborated using linear-quadratic-Gaussian and cascade-connected fuzzy control schemes. The mechatronic system is a special mobile robot (so called two-wheeled mobile pendulum system) that has only two contact points with the supporting surface and its center of mass is located under the wheel axis. Due to this mechanical structure, the inner body (which acts as a pendulum between the wheels) tends to oscillate during the translational motion of the robot, thus the application of feedback control is essential in order to stabilize the dynamical system. In the first part of the paper, the mechatronic system and the corresponding mathematical model are introduced, while in the second part the aforementioned control solutions are designed for the plant. The achieved control performances are analyzed both in simulation environment and on the real mechatronic system. At the end of the paper, a performance assessment of the elaborated control solutions is given based on transient response and error integral measurements.
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 1!
-:@;*!33
  1 !1 !!
   - 1 !  !  
!33
 !!     0-  
!1   -
   !  !!  
:A;   !     &
=!&B!!33=!
 &=!&B! 45CB< =!  
- 
  --1&
 :9;/!!
 -1
  -    -  D :E;
2!
&1=!!1!
#)+!:F;5CB&-
 -!
6&     1   
  -! -   
 0  - !   
2  !   5CB 
!      !!  
- :G; &- ! - :H; !
 !   :87; =!
IJ:88;1
!33 
 !     2-
        !
-=!1
K5K!:86;
!331 -
&
     !-!   L :8@; 
  !33    -!  
 - -  1 3 1
!  5  B33:8A; 2 
      
!33=!!&!
K !331!!
    1 - 1 L 
:89;-3!332
1-+3-4<&*!33-
-31&1 !!
- -! 
   :8F;K   -
1!33 1! 4! 
:86;:89;<1 !1
 1 4!   :8A;<     
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- -  !33    
!-!
-#  
        
-=! -!
-1- !
!!!
1 1 - !!  :8G; 4
-< 1       
3-!
4- <1-!1
!!1
     -     - 
-3  1 5CB =! -
 :8;1:8H;!33
-!3!:8;
:8H;- !
41-  !!!1-
  !<  -   1 -
 *      
4=!!!-<1- !
      3  1
+    !!   - 
!1+
 +J
  5CB  !33  =!   1
+ J  1  -   5CB 
1+J-!33
  !   !  
-     +J 
J   + M -   
-!!!
1+M!!!1

%*I.0IK%I0/+N+K
      - - 4
1&1-!!<
11-4<1
!!-I-
*!81-
-!-1
  1  ! ! !  
!!-- !!
-1    
   1  -  

+--
!- 121=!-!-
! /  -  !  -
=!-!1!
1 2  !    
--
--!
!- =!-!  1     
-3 -   1 2 I!  !-
=!-!   - !! 
--1
   !! *    
1-:67;
  !  -! ! 1 8E&-
!&1&1 2 ! K+A@7*6E8G
4K8K6<
  !3  -   !  @J
4'876A/77@+<!!-
*!- 1!1!
4DK<!2!0J9H6 
  !-3@@J 
!8!&45&<-I8EK%3
=!3!
+!1-!KB ?
K:68;  1 
1!3-
=!      
  I  1 
!1-!-+
K I:66;
KI%KII5K5
--
 --
K       
3  -  4!  1 ! 
   <  =! ! 1 
 !    !
 - +1      1
2!-!
D 1!
 1 1 1     
*!8--
I.5%I5+/+0+/%K.50/+
   #
I +K 5+@@85
B +K 5@BA677
! 2! /I8HG

 *!- I6&877
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!!4-<3
-  !3  - J   2 
11-!'1
- -

1-
5=!:6@;1!1
=!:8G;'
48<
1   @&- @    
  2   @&
!!
    4 <    5
!   2     48< 
-   2 *    3
1    
!!-
 !     3 2
48<=!!-
         
:8G;=!--
=! 46< 1  !   !  
1
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1!/ !1
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.  48<  &    1&
1 !!-D
1&=!:8G;'
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=!4A<11 !
5CBE& -
 11!!1
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    !     -
- 1 !!3 
4!-=!-! < -
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1BB !
 1    -     1
!-    =!  
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 :9;  -   
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!  - !- 
!--2
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International Journal
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5I>!!33:@;-2
  1& !  k& l1  1
!!  m& n *!33  -  
!331!
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%/ ! ! %      
     !  
2!
      1 
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&*!33'K -!
!33 m& n!33 -
 -   !33 ! - 
- !   !33 T  1
op1Jm& n11!
&'!33*&%/!
  -1   
=!     
& !!!33
! !
 !!
1!!
!33  -!334 
!33!!!<
&!33' K -  !! !33  
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 1 
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201
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J+
K5I/
0
+5+
 !      1
 KI5I.+!   !33 
 1  -     *!33 5
-2KI5I.!!   
- *!9
*-
    1       !

   - 1     
      1 
1'
&
B
k&0‡&&0&lm>žn
&
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k&0”& 0&lm„…ˆžn
I.5

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!
. /
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%   >  % 0 
0  > / !  > /
 > / /   
I.5


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23
 ,#  . * 
K/ KIM K/ KIM
D
 
*!AK-!*5
*!@.!33
228
International Journal
o
n Advances in Intell
igent Systems, vol
9
no
1
&
2
1
6
, http://www.iariajournals.org/intelligent_systems/
201
6
, © Copyright by authors, Published under agreement with IARIA
-
www.iaria.org
! !1 - 5CB 
   & !33  
-3      -   
-  !! !  
4 
1!-!<!
-4!-<
12.~ &&•¬d1=!
=!
JK5K/I/0+5+
   1    ! K6
1   1    ! !
4K<          4
   ! + < 
    •• Ÿ K8
!! {~ &•-1••
4-!
! -  <   !
!!4!-<1
ŸL
K8 2!     -  
 !     
••Ÿ K6  2  1 
!     --
!
  !  !   
 !  !    DK 
*!!!.!
! 1  =!.~ &&¬d I B 1 
KI5I.!
0  5CB   ! 
-488<486< 
-!! 
-1!!
*51-!33!1
!!  !  ! -
&!--
K --
--!!
*!E*
-
1-
11 -
 -    !!
!-!
    -  1   2 
   1   -  &0‡•>ž
™•„…ˆž   4  *! E< 2
1-=!-
! 
1 -   -   !  !
!
MKI0+/*%/05+0IB+
*--
     
-311-
  -     *  =!
! ! 
-  !      -
 !  !  +! -!
4+I< +! =! 4++< +!  
&1 -!  4+I<   +! 
&1=!-48G<48H<
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 1
   !  4 ¡ «<  
*!9- !-
4+!!<


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, © Copyright by authors, Published under agreement with IARIA
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  1     !  
- ©¨³4
   3  + J<  1  
 !  =!&! 2 -! 
! !1!
!3!332
 
.   !   ! 
=!  -  1
!3! -J1{´µ¶·
{ƒ¸¹ºž0
!  --   I  
!!  5CB  
  -  1   
  * - J    -   
- !33     =!
1!- 477@EO7AO
<  !   &     
   -     
0!--
P 4
2!<1-169&@7
!!1!
      
! !      5CB 
 !     
  0  !33    -
 =!!!4!!*!
E<   1   3  1
   !33   . 3
!!!-
1=!42! 1!
99<
I   !  - !  
   1   -  5CB
=!
 - !33   !   
!3
!! 
--F
(1  4   !  
<1 !33 
---!
! K    -      
  4A<   -   
 4 !< !    4
2<4!
  !   <  3 
-1! !!
       -  
- J  1    -1 
!!12
    3  1 
!!!
5CB!
 =! 41   2 1<
!-
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 ! =!!!48G<
 48H<  !3  - J  !  
 !!! 
 !    -   
*!E5CB!33
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International Journal
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n Advances in Intell
igent Systems, vol
9
no
1
&
2
1
6
, http://www.iariajournals.org/intelligent_systems/
201
6
, © Copyright by authors, Published under agreement with IARIA
-
www.iaria.org
 -!33
!  - J      +I
+I++=!!!3
!33! !
1-J!
1! !
  / « ©¨  
 !  -  - !  !-&
 !2+I'
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 5CB     -  
 1   -   
1 -&!33
!   1 !   2
!1 
 !-
!-!  -!
=! ! ! 48G<  48H<  
3! 3!-
-!!
-3- 
M/5+/I/*0D0L
   5CB  !33   1
-1&1-!!
-   !! !  
   -  -3   -
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International Journal
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no
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6
, http://www.iariajournals.org/intelligent_systems/
201
6
, © Copyright by authors, Published under agreement with IARIA
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!   AG 6778  F9F S FE9 '
87887HOA8H@FA7F
:@;5 I >R*!33Q   G
8HE9@@GS@9@'87878EO+778H&HH9G4E9<H76A8&M
:A;% . J-!   K .!# R*!33  
 'D4-<-!
*!33+Q*!33+ H78HHF
898S8E7'87878EO+78E9&788A4HF<777G8&M
:9;B**)1I&/*-
   +   % 678A
+./'HFG&7&8@@AH&E9H&G
:E;I -)DT#&
 T     +
  9 8HHF  6EH & 6FG '
87887HOGF9F6869
:F;)&L )  +&L +! TL   5C - 
 -!6&
  T   !
A667769EA&9F8'87887HOA8AF9AHE
:G;+)TD !!
-' -T
O0+)0-
+ 40+ 677F< 677F  8H@6 & 8H@F '
87887HO0+677FA@HGHE8
:H;+ >!  5  T.    !
T 6H    4< 6787 
@697&@69A+./'HFG&8&A6AA&E6E@&E
:87;5N&-5D&3!+CT 5CB
!- !
 T    
!4<67886GH&6H6
'87887HO6788E7EFG87
:88;I/I!T*!=!IJ
5C  %UT LI    
4/05< 678A  8@@ & 8@G '
87887HO/05678AEH8986G
:86; K5  % K! R% &'
 5C0 !33&  !& Q
 I  I   88
8HHGA88SA8G'87878EO+7H96&8HFE4HG<77779&7
:8@;    / L R    
  !33 &-   1 -
-Q  + 
8A677E978&987'87887HO+677EGF69@E
:8A; + 5  0 J B33 R*!33 ! 
!&!Q
)!K+ 66677G
8AF9&8AG6'87877FO8667E&77G&7A6A&F
:89; L    R3    *!33&
  +! K  +3-4<& D
0-Q )!  I  0-+
678A'879FF6O9H876
:8E;K + J 5V3  * K R !33
   =!Q   
+L14+L6787<
67878A8&8AE'87887HO+L67879EG7G86
:8F;&% %! D&)! D  &%! !
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232
International Journal
o
n Advances in Intell
igent Systems, vol
9
no
1
&
2
1
6
, http://www.iariajournals.org/intelligent_systems/
201
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, © Copyright by authors, Published under agreement with IARIA
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www.iaria.org
... The gear ratio (k) defines the connection between the rotor (motor) and gear (load) sides, that is, the angular position of the rotor is θ kθ = r g , where θ g denotes the shaft position on the gear side. The fundamental electrical and mechanical equations on the load side are summarized as [46,49] ...
... The derivation of system dynamics starts with the determination of the total kinetic (translational and rotational) and potential energies of both the wheels and the IB. Based on the kinetic (K ) and potential (P) energies, the Lagrangian of the system is formed as q K P ( ) = −  and given in the appendix [46,49]. Let θ 1,2 and θ 3 indicate the angular displacement of the wheels and the angular position of the IB (i.e., pendulum angle), respectively. ...
... simulation with initial and estimated parameters: motor speeds and applied excitation 3-dimensional vector term including the Coriolis, centrifugal, and potential force terms. The parameters of these matrices are described in [46,49]. ...
Fuzzy control of self-balancing robots: A control laboratory project
Article
  • Mar 2020
This paper presents a novel control laboratory project that provides hands‐on experience in feedback control concepts (embedded control systems) through dedicated assignments, with a particular focus on the design and implementation of fuzzy control. The project is structured around an inexpensive, portable self‐balancing robot (SBR), whose embedded system is realized using commercially available breakout boards as the first assignment. For the stabilization of the plant, students are guided to execute the essential stages of control system design, from system modeling and parameter optimization, over basic or advanced control strategy design in the MATLAB/Simulink environment, to both implementation and validation of the closed loop on the real robot. To demonstrate and foster the application of fuzzy logic, the second part of the paper introduces a simple control strategy based on fuzzy logic controllers. Then, a lookup table‐based implementation technique is described for the demonstration of manual interfacing and embedded coding of fuzzy control strategies. The proposed methods are clear and straightforward; they highly foster the understanding of feedback control techniques and allow students to gain vast knowledge in the practical implementations of control systems.
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... In references [13]- [16] the proposed fuzzy control approaches provided superior performance over using linear controllers, however the opposite outcome is sometimes claimed as well [17]. Similarly, the study in [18] compared fuzzy control and linearquadratic-regulator (LQR) approaches for the stabilization of a mobile robot. The measurement results have shown that the former solution provided better overall control performance, and the latter approach showed faster system dynamics for transient events. ...
... The fuzzy control structure has been introduced in [18], [25] and its optimization was elaborated in [26]. The control structure is composed of three cascade-connected FLCs, i.e., a linear speed controller (FLC1), an anti-sway controller (FLC2), and a yaw rate controller (FLC3). ...
Comparison of Optimized PID and Fuzzy Control Strategies on a Mobile Pendulum Robot
Conference Paper
  • Apr 2018
This paper investigates the optimized control performances of fuzzy and proportional-integral-derivative (PID) control schemes developed for the stabilization of an under-actuated mobile robot. The fuzzy control strategy had been designed in an earlier paper, its equivalent PID controller-based scheme is established first. Then a complex cost function is defined that evaluates the reference tracking performance, the efficiency of system oscillations suppression and the average current consumption in the motor drive system. The particle swarm optimization (PSO) is applied to tune both control schemes under the same circumstances by minimizing the formulated cost function. Results demonstrate that the optimized fuzzy control strategy provides the same reference tracking quality with significantly better suppression of system oscillations and current peaks compared to the optimized PID control.
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... Accelerometers measure linear acceleration along one or more axes, while gyroscopes measure angular velocity around one or more axes. These sensors can be applied for control [12][13][14], position and orientation estimation [15][16], etc. ...
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... Magnetometers are sensors that measure the Earth`s magnetic field and can be used in different applications, like compasses, position and orientation estimation in mobile robotics [1][2][3][4], vehicle detection [5][6], etc. They can be also used in various pattern recognition systems, where classification is done based on different features, which are extracted from the signal waveforms. ...
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... Brushless Direct Current Motor has the advantages of high speed dynamic response, simple structure, high stability, high efficiency, low noise and high torque, so it is widely used in industrial production. However, in the specific design of the Brushless Direct Current Motor speed control system, the design process will be very complex, and there will be a lot of problems in this process, because of the difficulties in the preparation of the underlying hardware driver, the design of the motor phase change program and the compilation of the control system program and so on [1][2][3][4][5][6]. Model based design method of developing control system don't need to be proficient in various programming languages, it will work tedious code generation to the computer to complete, and there are all kinds of function module and driving module for designers to use, only need to design personnel to understand the principle of work and focus on the algorithm itself, thus to ensure the efficiency and reliability of the code, and greatly reduces the difficulty of embedded system development, significantly shorten the product development cycle, greatly improving the work efficiency. ...
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... The fuzzy controller can provide a more comprehensive solution compared to the PID controller [46]. This is confirmed by our previous studies: ...
Multi-Scenario Multi-Objective Optimization of a Fuzzy Motor Controller for the Szabad(ka)-II Hexapod Robot
Article
  • Jan 2019
The aim of this research was to develop a robust motor controller for the Szabad(ka)-II hexapod robot. A Fuzzy-PI controller that utilized a lookup table was chosen because of its reported promising performance and its ability to be embedded in the microcontrollers of the robot. The variables of the controller were defined by a particle swarm optimization method to minimize the five quality objectives related to the walking of the robot. The preferences of the five objectives were successfully expressed by a biased weighted geometric mean utility function. The resulting optimal solutions were significantly altered by changing the bias and exponential weights of preferences. Therefore, we checked the robustness of the solution against the controller’s variables as a secondary objective.
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... The strength of different brain wave types can be calculated by FFT (Fast Fourier Transform) algorithm, which is a mathematical method used in EEG analysis to investigate the composition of an EEG signal. In this way, the frequency distributions (as a spectrum of brain waves) can be observed by using FFT algorithm after filtering, for example Kalman filters [47]. The spectrum of brain waves is very sensitive to the mental and emotional states of the brain, so this attribute is used to monitor mental activity [19]. ...
The evaluation of BCI and PEBL-based attention tests
Article
Full-text available
  • Jun 2018
New technological advances of the 20 th and 21 st Centuries provide several new opportunities for engineers in the future. Modern engineering methods feasible by these new technical solutions, however, cannot meet expectations in all fields and situations without adequate knowledge and competence. Spreading new and modern engineering methods as well as acquiring new applications is a crucial issue in education. In brain research, which is one of the most significant research areas of the past decades, many new results and meters have appeared that could be used in engineering methods, too. Based on brain activity observation, new meters open up new horizons in engineering applications. Electroencephalogram-based brain activity observation processes are very promising and have been used in several engineering research primarily for implementation of control tasks. In this paper, an EEG-based engineering research work is demonstrated, which supports the acquirement of practical knowledge and can measure cognitive ability with a device capable of brain activity observation. In the engineering research task, a brain-computer interface (BCI) had to be developed for the measurement of the average level of attention. The results of the BCI have been compared and contrasted to the results of two tests applied in cognitive psychology, the PEBL Continuous Performance Test (pCPT) and the PEBL Test of Attentional Vigilance (pTOAV). It can be stated that the results of the procession developed in this research and the results of the pCPT and the pTOAV tests are in correlation.
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... These sensors are generally applied to solve the relative localization problem, where the positions and orientation information of moving objects are determined using a MEMS-based inertial measurement unit (IMU) composed of the aforementioned MEMS sensors. The MEMS-based localization approach has been widely investigated and is widely utilized in robotics [1][2][3][4][5][6][7], human position tracking [8,9], medical fields [10], sports equipment [11,12], and automated driving [13][14][15][16][17] both in industry and in scientific research. Independent of the specific application, it is always advantageous to https://doi.org/10.1016/j.ymssp.2018.03.053 0888-3270/Ó 2018 Elsevier Ltd. ...
Kalman filter for mobile-robot attitude estimation: Novel optimized and adaptive solutions
Article
  • Mar 2018
This paper proposes two novel approaches to estimate accurately mobile robot attitudes based on the fusion of low-cost accelerometers and gyroscopes. The first part of the paper demonstrates the use of a special test bench that both enables simulations of various dynamic behaviors of wheeled robots and measures their real attitude angles along with the raw sensor data. These measurements are applied in a simulation environment and we outline an offline optimization of Kalman filter parameters. The second part of the paper introduces a novel adaptive Kalman filter structure that modifies the noise covariance values according to the system dynamics. The instantaneous dynamics are characterized regarding the magnitudes of both the instantaneous vibration and the external acceleration. The proposed adaptive solution measures these magnitudes and utilizes fuzzy-logic to modify the filter parameters in real time. The results show that the adaptive filter improves the overall filter convergence by a remarkable 10.9% over using the optimized Kalman filter, thereby demonstrating its efficacy as an accurate and robust attitude filter. The proposed filter performances are also benchmarked against other common methods indicating that the flexibility of the developed adaptive filter allowed it to compete and even outperform the benchmark filters.
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... Using and control the motor current beside the motor speed by a fuzzy logic controller is a key element to reach more robust controlling [9], [11], [12], [13]. Our fuzzy logic controller also take the motor current as the second input beside the first speed error. ...
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Development of a Fuzzy-LQR and Fuzzy-LQG Stability Control for a Double Link Rotary Inverted Pendulum
Article
Full-text available
  • Sep 2020
  • J FRANKLIN I
In this paper, a Fuzzy based Linear Quadratic Regulator (FLQR) and Linear Quadratic Gaussian (FLQG) controllers are developed for stability control of a Double Link Rotary Inverted Pendulum (DLRIP) system. The aim of this work is to study dynamic performance analysis of both FLQR and FLQG controllers and to compare them with the classical LQR and LQG controllers, respectively. A dynamic mechanical simulation model of the DLRIP was obtained using both the numerically SimMechanics toolbox in MATLAB and the analytically dynamic nonlinear equations. To determine the control performance of the controllers, Settling Time (Ts), Peak Overshoot (PO), Steady-State Error (Ess), and the total Root Mean Squared Errors (RMSEs) of the joint positions are computed. Furthermore, the dynamic responses of the controllers were compared based on robustness analysis under internal and external disturbances. To show the control performance of the controllers, several simulations were conducted. Based on the comparative results, the dynamic responses of both FLQR and FLQG controllers produce much better results than the dynamic responses of the classical LQR and LQG controllers, respectively. Moreover, the robustness results indicate that the FLQR and FLQG controllers under the internal and external disturbances were effective.
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