International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
PID Controller Design for a Real Time Ball and Beam System – A Double Integra...idescitation
In this paper, the authors have discussed and shown
how to tune the PID controller in closed loop with time-delay
for the double integrator systems for a particular stability
margins. In math model it is assumed that time delay (ô) of
the plant is known. As a case study the authors have consid-
ered the mathematical model of the real-time beam and ball
system and analyzed the simulation and real time response.
Tuning of pid controller of inverted pendulum using genetic algorithmeSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Mechanism Design and Kinematics Analysis of Display Bracket Based on AdamsIJRESJOURNAL
ABSTRACT: In view of the problem of conventional display support equipment which has a poor intelligent, low technological level, easy to cause the body with sub-health, and can't meet the needs of high level office, designed a new type of display bracket mechanism. The principle and force analysis was carried out on the display bracket mechanism, and to simplify the original mechanism with three linkages for facilitating analysis and simulation. Forward kinematics analysis was carried out on the bracket mechanism based on DH method and obtained pose transformation matrix between each joint coordinate, and expressed the relationship between the end speed and the velocity of each joint by using Jacobian matrix. Establishing simplified virtual prototype and kinematics simulation was carried out on displacement, velocity, acceleration, angular velocity and acceleration of bracket mechanism based on the software of ADAMS which verified the correctness of theoretical analysis. The simulation results showed that the design of bracket mechanism complied with the requirements of free readjustment of display. Realizing the combination between high-tech and high emotion, improving the body with sub-health at the present situation of office, and meet the needs of advanced intelligent office by using the new display bracket mechanism
Analytical Model of Cage Induction Machine Dedicated to the Study of the Inne...IJECEIAES
This paper presents a new analytical model for inner bearing raceway defect. The model is based on the presentation of different machine inductances as Fourier series without any kind of reference frame transformation. The proposed approach shows that this model is able to give important features on the state of the motor. Simulation based on spectral analysis of stator current signal using Fast Fourier Transform (FFT) and experimental results are given to shed light on the usefulness of the proposed model.
PID Controller Design for a Real Time Ball and Beam System – A Double Integra...idescitation
In this paper, the authors have discussed and shown
how to tune the PID controller in closed loop with time-delay
for the double integrator systems for a particular stability
margins. In math model it is assumed that time delay (ô) of
the plant is known. As a case study the authors have consid-
ered the mathematical model of the real-time beam and ball
system and analyzed the simulation and real time response.
Tuning of pid controller of inverted pendulum using genetic algorithmeSAT Publishing House
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology.
Mechanism Design and Kinematics Analysis of Display Bracket Based on AdamsIJRESJOURNAL
ABSTRACT: In view of the problem of conventional display support equipment which has a poor intelligent, low technological level, easy to cause the body with sub-health, and can't meet the needs of high level office, designed a new type of display bracket mechanism. The principle and force analysis was carried out on the display bracket mechanism, and to simplify the original mechanism with three linkages for facilitating analysis and simulation. Forward kinematics analysis was carried out on the bracket mechanism based on DH method and obtained pose transformation matrix between each joint coordinate, and expressed the relationship between the end speed and the velocity of each joint by using Jacobian matrix. Establishing simplified virtual prototype and kinematics simulation was carried out on displacement, velocity, acceleration, angular velocity and acceleration of bracket mechanism based on the software of ADAMS which verified the correctness of theoretical analysis. The simulation results showed that the design of bracket mechanism complied with the requirements of free readjustment of display. Realizing the combination between high-tech and high emotion, improving the body with sub-health at the present situation of office, and meet the needs of advanced intelligent office by using the new display bracket mechanism
Analytical Model of Cage Induction Machine Dedicated to the Study of the Inne...IJECEIAES
This paper presents a new analytical model for inner bearing raceway defect. The model is based on the presentation of different machine inductances as Fourier series without any kind of reference frame transformation. The proposed approach shows that this model is able to give important features on the state of the motor. Simulation based on spectral analysis of stator current signal using Fast Fourier Transform (FFT) and experimental results are given to shed light on the usefulness of the proposed model.
This paper deals with the dynamic characterization of an
automotive shock absorber, a continuation of an earlier
work Vibration is undesirable, not only because of the
unpleasant motion, the noise and the dynamic stresses,
which may lead to fatigue and failure of the structure, but
also because of the energy losses and the reduction in
performance which accompany the vibrations [1].
Vibration analysis should be carried out as an inherent part
of the design because of the devastating effects, which
unwanted vibrations could have on machines and
structures.
Preliminary Research on Data Abnormality Diagnosis Methods of Spacecraft Prec...IJERA Editor
For precision measuring the satellite equipments, providing technical support for satellite assembly, combined
with satellite small size, complex structure, satellite equipment shapes vary, and other characteristics, presently,
indirect method that using electronic theodolite to measure cube mirror are commonly used to obtain the relative
attitude of the respective devices. But in the actual measurement process, there are measurement errors in the
measurement data. How to detect anomalies in the data is the focus of this study. This paper proposes two
methods to detect abnormal data, that is mathematical geometric method and outlier detection methods. This
paper analyzes their theoretical basis and verifies the feasibility of the two methods through part of the actual
measurement data to.
Numerical simulation of marangoni driven boundary layer flow over a flat plat...eSAT Journals
Abstract
A numerical algorithm is presented for studying Marangoni convection flow over a flat plate with an imposed temperature
distribution. Plate temperature varies with x in the following prescribed manner: where A and k are constants.
By means of similarity transformation, the original nonlinear partial differential equations of flow are transformed to a pair of
nonlinear ordinary differential equations. Subsequently they are reduced to a first order system and integrated using Newton
Raphson and adaptive Runge-Kutta methods. The computer codes are developed for this numerical analysis in Matlab
environment. Velocity profiles for various values of k, and temperature profiles for various Prandtl number and k are illustrated
graphically. The results of the present simulation are then compared with the previous works available in literature with good
agreement.
Keywords: Matlab, Marangoni Convection, Numerical Simulation, Surface Tension, Flat Plate.
Three-dimensional structure from motion recovery of a moving object with nois...IJECEIAES
In this paper, a Nonlinear Unknown Input Observer (NLUIO) based approach is proposed for three-dimensional (3-D) structure from motion identification. Unlike the previous studies that require prior knowledge of either the motion parameters or scene geometry, the proposed approach assumes that the object motion is imperfectly known and considered as an unknown input to the perspective dynamical system. The reconstruction of the 3-D structure of the moving objects can be achieved using just twodimensional (2-D) images of a monocular vision system. The proposed scheme is illustrated with a numerical example in the presence of measurement noise for both static and dynamic scenes. Those results are used to clearly demonstrate the advantages of the proposed NLUIO.
A Comparison Study between Inferred State-Space and Neural Network Based Syst...Ahmed Momtaz Hosny, PhD
In this paper, system identifications of an unmanned aerial vehicle (UAV) based on inferred state space and multiple neural networks were presented. In this work an optimization approach was used to conclude an inferred state space and the multiple neural networks system identifications based on the genetic algorithms separately. The UAV is a multi-input multi-output (MIMO) nonlinear system. Models for such MIMO system are expected to be adaptive to dynamic behavior and robust to environment.
CREATING A NEW CRITICAL DEPTH EQUATION FOR GRADUALLY VARIED FLOW IN CIRCULAR ...IAEME Publication
Drawing the water surfaces in open-channel for gradually varied flow is relatively
complicated and difficult. In order to identify which type of the water surfaces among
12 water-surface styles we have to base on critical depth (yc) and normal depth (yo). In
this case, to calculate the critical depth (yc) that particularly need to use the Semi
empirical equations.
This article generally the way to compute the critical depth; the way to compute
flow in circular sewers; analyze the application of existing formulas and then offering
a new equation to compute the critical depth. This new equation will help to have
more accurate result. Also it is more comfortable to non-uniform flow in the circular
section/ circular sewers.
OPTIMAL TRAJECTORY OF ROBOT MANIPULATOR FOR ENERGY MINIMIZATION WITH QUARTIC ...cscpconf
In this paper, a different way to find the trajectory of the robot manipulators for energyoptimization is presented. In our method, the joint angles of the manipulator are set as quadratic polynomial functions. Then, with them taken into the variational function of energy consumption, Finite Element Modelling is employed to optimize the unknown parameters of the fourth order joint angles
Computational model to design circular runner conduit for plastic injection m...eSAT Journals
Abstract Analytical solution quest for viscoelastic shear thinning fluid flow through circular conduit is a matter of great prominence as it directly evolvesmost efficient criteria to investigate various responses of independent parameters. Envisaging this facet present endeavour attempts to develop a computational model for designing runner conduit lateral dimension in a plastic injection mould through which thermoplastic melt gets injected. At outset injection phenomenon is represented by governing equations on the basis of mass, momentum and energy conservation principles [1]. Embracing Hagen Poiseuille flow problem analogous to runner conduit injection the manuscript uniquely imposes runner conduit inlet and outlet boundary conditions along with relative to appropriate assumptions; governing equations evolve a computation model as criteria for designing. To overwhelm Non-Newtonian’s abstruse Weissenberg-Rabinowitsch correction factor has been adopted byaccommodating thermoplastic melt behaviour towards the final stage of derivation. The resulting final computational model is believed to express runner conduit dimensions as a function of available type of injection moulding machine specifications, characteristics of thermoplastic melt and required features of component being moulded. Later the equation so obtained has been verified by using dimensional analysis method. Keywords: Computational modelling, Runner conduit design, Plastic injection mould, Hagen-Poiseuille flow
Applying Parametric Functional Approximations for Teaching Electromechanical ...IOSRJEEE
In this paper functional approximations for the parameters of a DC machine are proposed. These nonlinear algebraic approximations are based on the experimental data that are obtained by carrying out several steady state and transient tests, besides, this method links mathematics and practical analysis of electromechanical systems, allowing students to improve their academic performance and comprehension by comparing estimated values with measured data. An excellent dynamic model results of combining the well known state space description and these parametric functional approximations. This augmented model provides reliable results even during demanding large-excursion transient conditions
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Nonlinear Viscoelastic Analysis of Laminated Composite Plates – A Multi Scale...rtme
Laminated composite plates are widely used in modern structures. Resins of composites are almost made of
polymers which show time dependent and and in some cases stress dependent behaviour. In this paper, a
laminated composite plate is analysed using a multiscale method. At first, material properties of a lamina is
obtained using an analytical micromechanical approach called simplified unit cell method (SUCM) and
then in macromechanical level, Generalized Differential Quadrature Method (GDQM) is used to analyse
laminated composite plate. Schapery's integral is used to model nonlinear viscoelastic behaviour of the
matrix. Prony series is considered to define the compliance of matrix. Micromechanical process includes
obtaining overall properties of the composite by SUCM. Both geometrical and material nonlinearity are
taken into account in order to multiscale analysis of laminated composite plate.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
This paper deals with the dynamic characterization of an
automotive shock absorber, a continuation of an earlier
work Vibration is undesirable, not only because of the
unpleasant motion, the noise and the dynamic stresses,
which may lead to fatigue and failure of the structure, but
also because of the energy losses and the reduction in
performance which accompany the vibrations [1].
Vibration analysis should be carried out as an inherent part
of the design because of the devastating effects, which
unwanted vibrations could have on machines and
structures.
Preliminary Research on Data Abnormality Diagnosis Methods of Spacecraft Prec...IJERA Editor
For precision measuring the satellite equipments, providing technical support for satellite assembly, combined
with satellite small size, complex structure, satellite equipment shapes vary, and other characteristics, presently,
indirect method that using electronic theodolite to measure cube mirror are commonly used to obtain the relative
attitude of the respective devices. But in the actual measurement process, there are measurement errors in the
measurement data. How to detect anomalies in the data is the focus of this study. This paper proposes two
methods to detect abnormal data, that is mathematical geometric method and outlier detection methods. This
paper analyzes their theoretical basis and verifies the feasibility of the two methods through part of the actual
measurement data to.
Numerical simulation of marangoni driven boundary layer flow over a flat plat...eSAT Journals
Abstract
A numerical algorithm is presented for studying Marangoni convection flow over a flat plate with an imposed temperature
distribution. Plate temperature varies with x in the following prescribed manner: where A and k are constants.
By means of similarity transformation, the original nonlinear partial differential equations of flow are transformed to a pair of
nonlinear ordinary differential equations. Subsequently they are reduced to a first order system and integrated using Newton
Raphson and adaptive Runge-Kutta methods. The computer codes are developed for this numerical analysis in Matlab
environment. Velocity profiles for various values of k, and temperature profiles for various Prandtl number and k are illustrated
graphically. The results of the present simulation are then compared with the previous works available in literature with good
agreement.
Keywords: Matlab, Marangoni Convection, Numerical Simulation, Surface Tension, Flat Plate.
Three-dimensional structure from motion recovery of a moving object with nois...IJECEIAES
In this paper, a Nonlinear Unknown Input Observer (NLUIO) based approach is proposed for three-dimensional (3-D) structure from motion identification. Unlike the previous studies that require prior knowledge of either the motion parameters or scene geometry, the proposed approach assumes that the object motion is imperfectly known and considered as an unknown input to the perspective dynamical system. The reconstruction of the 3-D structure of the moving objects can be achieved using just twodimensional (2-D) images of a monocular vision system. The proposed scheme is illustrated with a numerical example in the presence of measurement noise for both static and dynamic scenes. Those results are used to clearly demonstrate the advantages of the proposed NLUIO.
A Comparison Study between Inferred State-Space and Neural Network Based Syst...Ahmed Momtaz Hosny, PhD
In this paper, system identifications of an unmanned aerial vehicle (UAV) based on inferred state space and multiple neural networks were presented. In this work an optimization approach was used to conclude an inferred state space and the multiple neural networks system identifications based on the genetic algorithms separately. The UAV is a multi-input multi-output (MIMO) nonlinear system. Models for such MIMO system are expected to be adaptive to dynamic behavior and robust to environment.
CREATING A NEW CRITICAL DEPTH EQUATION FOR GRADUALLY VARIED FLOW IN CIRCULAR ...IAEME Publication
Drawing the water surfaces in open-channel for gradually varied flow is relatively
complicated and difficult. In order to identify which type of the water surfaces among
12 water-surface styles we have to base on critical depth (yc) and normal depth (yo). In
this case, to calculate the critical depth (yc) that particularly need to use the Semi
empirical equations.
This article generally the way to compute the critical depth; the way to compute
flow in circular sewers; analyze the application of existing formulas and then offering
a new equation to compute the critical depth. This new equation will help to have
more accurate result. Also it is more comfortable to non-uniform flow in the circular
section/ circular sewers.
OPTIMAL TRAJECTORY OF ROBOT MANIPULATOR FOR ENERGY MINIMIZATION WITH QUARTIC ...cscpconf
In this paper, a different way to find the trajectory of the robot manipulators for energyoptimization is presented. In our method, the joint angles of the manipulator are set as quadratic polynomial functions. Then, with them taken into the variational function of energy consumption, Finite Element Modelling is employed to optimize the unknown parameters of the fourth order joint angles
Computational model to design circular runner conduit for plastic injection m...eSAT Journals
Abstract Analytical solution quest for viscoelastic shear thinning fluid flow through circular conduit is a matter of great prominence as it directly evolvesmost efficient criteria to investigate various responses of independent parameters. Envisaging this facet present endeavour attempts to develop a computational model for designing runner conduit lateral dimension in a plastic injection mould through which thermoplastic melt gets injected. At outset injection phenomenon is represented by governing equations on the basis of mass, momentum and energy conservation principles [1]. Embracing Hagen Poiseuille flow problem analogous to runner conduit injection the manuscript uniquely imposes runner conduit inlet and outlet boundary conditions along with relative to appropriate assumptions; governing equations evolve a computation model as criteria for designing. To overwhelm Non-Newtonian’s abstruse Weissenberg-Rabinowitsch correction factor has been adopted byaccommodating thermoplastic melt behaviour towards the final stage of derivation. The resulting final computational model is believed to express runner conduit dimensions as a function of available type of injection moulding machine specifications, characteristics of thermoplastic melt and required features of component being moulded. Later the equation so obtained has been verified by using dimensional analysis method. Keywords: Computational modelling, Runner conduit design, Plastic injection mould, Hagen-Poiseuille flow
Applying Parametric Functional Approximations for Teaching Electromechanical ...IOSRJEEE
In this paper functional approximations for the parameters of a DC machine are proposed. These nonlinear algebraic approximations are based on the experimental data that are obtained by carrying out several steady state and transient tests, besides, this method links mathematics and practical analysis of electromechanical systems, allowing students to improve their academic performance and comprehension by comparing estimated values with measured data. An excellent dynamic model results of combining the well known state space description and these parametric functional approximations. This augmented model provides reliable results even during demanding large-excursion transient conditions
IJRET : International Journal of Research in Engineering and Technology is an international peer reviewed, online journal published by eSAT Publishing House for the enhancement of research in various disciplines of Engineering and Technology. The aim and scope of the journal is to provide an academic medium and an important reference for the advancement and dissemination of research results that support high-level learning, teaching and research in the fields of Engineering and Technology. We bring together Scientists, Academician, Field Engineers, Scholars and Students of related fields of Engineering and Technology
Nonlinear Viscoelastic Analysis of Laminated Composite Plates – A Multi Scale...rtme
Laminated composite plates are widely used in modern structures. Resins of composites are almost made of
polymers which show time dependent and and in some cases stress dependent behaviour. In this paper, a
laminated composite plate is analysed using a multiscale method. At first, material properties of a lamina is
obtained using an analytical micromechanical approach called simplified unit cell method (SUCM) and
then in macromechanical level, Generalized Differential Quadrature Method (GDQM) is used to analyse
laminated composite plate. Schapery's integral is used to model nonlinear viscoelastic behaviour of the
matrix. Prony series is considered to define the compliance of matrix. Micromechanical process includes
obtaining overall properties of the composite by SUCM. Both geometrical and material nonlinearity are
taken into account in order to multiscale analysis of laminated composite plate.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
IJERA (International journal of Engineering Research and Applications) is International online, ... peer reviewed journal. For more detail or submit your article, please visit www.ijera.com
STABILIZATION AT UPRIGHT EQUILIBRIUM POSITION OF A DOUBLE INVERTED PENDULUM W...ijcsa
A double inverted pendulum plant has been in the domain of control researchers as an established model for studies on stability. The stability of such as a system taking the linearized plant dynamics has yielded satisfactory results by many researchers using classical control techniques. The established model that is analyzed as part of this work was tested under the influence of time delay, where the controller was fine tuned using a BAT algorithm taking into considering the fitness function of square of error. This proposed
method gave results which were better when compared without time delay wherein the calculated values
indicated the issues when incorporating time delay
Attitude Control of Quadrotor Using PD Plus Feedforward Controller on SO(3)IJECEIAES
This paper proposes a simple scheme of Proportional-Derivative (PD) plus Feedforward controller on SO(3) to control the attitude of a quadrotor. This controller only needs the measurement of angular velocity to calculate the exponential coordinates of the rotation matrix. With rotation matrix as an error variable of the controller, the simulation shows that the controller is able to drive the attitude of the quadrotor from hovering condition to desired attitude and from an attitude condition goes to the hovering condition, despite the system is disturbed. When the system is convergent, the rotation error matrix will be a 3 3 identity matrix.
Using real interpolation method for adaptive identification of nonlinear inve...IJECEIAES
In this paper, we investigate the inverted pendulum system by using real interpolation method (RIM) algorithm. In the first stage, the mathematical model of the inverted pendulum system and the RIM algorithm are presented. After that, the identification of the inverted pendulum system by using the RIM algorithm is proposed. Finally, the comparison of the linear analytical model, RIM model, and nonlinear model is carried out. From the results, it is found that the inverted pendulum system by using RIM algorithm has simplicity, low computer source requirement, high accuracy and adaptiveness in the advantages.
Magnetic bearing is well-known for its advantage of reducing friction in rotary machines and is placing conventional bearings where high-speed operations and cleanliness are essential. It can be shown that the AM is a nonlinear system due to the relation between the magnetic force and current/rotor displacement. In this paper, a mathematical model of a 4-DOF AMB supported by four dual electric magnets is presented. The control objective is placed in a view of control input saturation and output limitation that are meaningful aspect in practical applications. Backstepping algorithm based control strategy is then adopted in order to achieve the high dynamic performance of the bearing. The control is designed in such a way that it takes input and output constraints into account by flexibly using hyperbolic tangent and barrier Lyapunov functions. Informative simulation studies are carried out to understand the operations of the machine and evaluate the controller quality.
Comparison of a triple inverted pendulum stabilization using optimal control ...Mustefa Jibril
In this paper, modelling design and analysis of a triple inverted pendulum have been done using
Matlab/Script toolbox. Since a triple inverted pendulum is highly nonlinear, strongly unstable
without using feedback control system. In this paper an optimal control method means a linear
quadratic regulator and pole placement controllers are used to stabilize the triple inverted
pendulum upside. The impulse response simulation of the open loop system shows us that the
pendulum is unstable. The comparison of the closed loop impulse response simulation of the
pendulum with LQR and pole placement controllers results that both controllers have stabilized
the system but the pendulum with LQR controllers have a high overshoot with long settling time
than the pendulum with pole placement controller. Finally the comparison results prove that the
pendulum with pole placement controller improve the stability of the system.
Body travel performance improvement of space vehicle electromagnetic suspensi...Mustefa Jibril
Electromagnetic suspension system (EMS) is mostly used in the field of high-speed vehicle. In this paper, a space exploring vehicle quarter electromagnetic suspension system is modelled, designed and simulated using linear quadratic optimal control problem. Linear quadratic Gaussian and linear quadratic integral controllers are designed to improve the body travel of the vehicle using bump road profile. Comparison between the proposed controllers is done and a promising simulation result have been analyzed.
Simulation, bifurcation, and stability analysis of a SEPIC converter control...IJECEIAES
This article presents some results of SEPIC converter dynamics when controlled by a center pulse width modulator controller (CPWM). The duty cycle is calculated using the ZAD (Zero Average Dynamics) technique. Results obtained using this technique show a great variety of non-linear phenomena such as bifurcations and chaos, as parameters associated with the switching surface. These phenomena have been studied in the present paper in numerical form. Simulations were done in MATLAB.
International Journal of Engineering Research and Applications (IJERA) is an open access online peer reviewed international journal that publishes research and review articles in the fields of Computer Science, Neural Networks, Electrical Engineering, Software Engineering, Information Technology, Mechanical Engineering, Chemical Engineering, Plastic Engineering, Food Technology, Textile Engineering, Nano Technology & science, Power Electronics, Electronics & Communication Engineering, Computational mathematics, Image processing, Civil Engineering, Structural Engineering, Environmental Engineering, VLSI Testing & Low Power VLSI Design etc.
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Hh3312761282
1. Nguyen Thanh Phuong, Ho Dac Loc, Tran Quang Thuan / International Journal of Engineering
Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 3, Issue 3, May-Jun 2013, pp.1276-1282
1276 | P a g e
Control of Two Wheeled Inverted Pendulum Using Silding Mode
Technique
Nguyen Thanh Phuong, Ho Dac Loc, Tran Quang Thuan
Ho Chi Minh City University of Technology (HUTECH) Vietnam
Abstract
In this paper, a controller via sliding
mode control is applied to a two-wheeled inverted
pendulum, which is an inverted pendulum on a
mobile cart carrying two coaxial wheels. The
controller is developed based upon a class of
nonlinear systems whose nonlinear part of the
modeling can be linearly parameterized. The
tracking errors are defined, then the sliding
surface is chosen in an explicit form using
Ackermann’s formula to guarantee that the
tracking error converge to zero asymptotically.
The control law is extracted from the reachability
condition of the sliding surface. In addition, the
overall control system is developed. The
simulation and experimental results on a two-
wheel mobile inverted pendulum are provided to
show the effectiveness of the proposed controller.
Key-Words: - Sliding mode controller, two
wheeled inverted pendulum.
1 Introduction
The two-wheeled inverted pendulum is a
novel type of inverted pendulum. In 2000, Felix
Grasser et al. [1] built successfully a mobile inverted
pendulum JOE. SEGWAY HT was invented by
Dean Kamen in 2001 and made commercially
available in 2003. The basic concept was actually
introduced by Prof. Kazuo Yamafuzi in 1986.
In practice, various inverted pendulum systems
have been developed. Inverted pendulum systems
always exhibit many problems in industrial
applications, for example, nonlinear behaviors under
different operation conditions, external disturbances
and physical constraints on some variables.
Therefore, the task of real time control of an unstable
inverted pendulum is a challenge for the modern
control field.
Chen et al. [2] proposed robust adaptive
control architecture for operation of an inverted
pendulum. Though the stability of the control
strategies can be guaranteed, some prior knowledge
and constraints were required to ensure the stability
of the overall system. Huang et al. [3] proposed a
grey prediction model combined with a PD controller
to balance an inverted pendulum. The control
objective is to swing up the pendulum from a stable
position to an unstable position and to bring its slider
back to the origin of the moving base. However, the
stability of this control scheme cannot be assured.
On the other hand, there are many
literatures on sliding mode control theory, which is
one of the effective nonlinear robust control
approaches. Juergen et al. [4] designed a sliding
mode controller based on Ackermann’s Formula.
Their simulation results prove that the mobile
inverted pendulum can be balanced by this controller.
There is no tracking controller. So tracking controller
of mobile inverted pendulum is deeply needed.
In this paper, a practical controller via
sliding mode control is applied to control two-
wheeled mobile inverted pendulum. The sliding
surface is chosen in an explicit form using
Ackermann’s formula, and the control law is
extracted from the reachability condition of the
sliding surface [5]. Finally, the simulation and
experimental results on computer are presented to
show the effectiveness of the proposed controller.
The two-wheeled mobile inverted pendulum
prototype is shown in Fig. 1. It is composed of a cart
carrying a DC motor coupled to a planetary gearbox
for each wheel, the microcontroller used to
implement the controller, the incremental encoder
and tilt sensor to measure the states, as well as a
vertical pendulum carrying a weight.
Fig. 1 Two-wheeled mobile inverted pendulum
2 System Modelling
In this paper, it is assumed that the wheels
always stay in contact with the ground and that there
will be no slip at the wheel’s contact patches.
The motor dynamics have been considered
[5]. Fig. 2 shows the conversion of the electrical
energy from the DC power supply into the
mechanical energy supplied to the load.
2. Nguyen Thanh Phuong, Ho Dac Loc, Tran Quang Thuan / International Journal of Engineering
Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 3, Issue 3, May-Jun 2013, pp.1276-1282
1277 | P a g e
Fig. 2 The diagram of DC motor
The motor model is given as follows [6],
m m e
m
K K K
T V
R R
(1)
Parameters are given in Appendix A.
Straight Motion Modeling
In straight motion modeling of mobile inverted
pendulum, the left and right wheels are driven under
identical velocity, i.e., rRl xxx .
The modeling can be linearized by
assuming , where represents a small angle
from the vertical direction,
2
1 0
d
cos ,sin ,
dt
(2)
The dynamics equation in the straight motion is
as follows [6],
x Ax bu (3)
22 23 2
42 43 4
0 1 0 0 0
0 0
0 0 0 1 0
0 0
r
r
x
a a bx
A ,x ,b ,
a a b
where 22 23 42 43 2 4a ,a ,a ,a ,b ,b and d are defined as a
function of the system’s parameters, which are given
in Appendix B.
Tracking error is defined as
1
2
3
4
r d
r d
d
d
e x x ,
e x x ,
e ,
e
(4)
where dx is desired value, d is desired inverted
pendulum angle.
Fig. 3 Free body of motion modeling
From Eq. (4), the followings are obtained.
1
2 1
2
3
4 3
4
r d
r d d
r d
d
d d
d
x x e
x x e x e
x x e
e
e e
e
(5)
From Eq. (5), the following can be obtained
2 1
4 3
e e
e e
(6)
Substituting Eq. (5) into Eq. (3), the following can be
obtained.
1 1
2 2
3 3
4 4
d d
d d
d d
d d
x e x e
x e x e
A bu
e e
e e
(7)
Rearranging Eq. (7), the following can be obtained
1 1
2 2
3 3
4 4
d d
d d
d d
d d
x xe e
x xe e
A bu A
e e
e e
(8)
where
rx
3. Nguyen Thanh Phuong, Ho Dac Loc, Tran Quang Thuan / International Journal of Engineering
Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 3, Issue 3, May-Jun 2013, pp.1276-1282
1278 | P a g e
22 23
42 43
22 23 2
42 43 4
0 0
0 0
d d d d
d d d d d
d d d d
d d d d d
d d d
d d d
x x x x
x x a x a x
A
a x a
a x a x d
a x a d
From Eq. (8), the following error dynamic equation
of mobile inverted pendulum is obtained:
e Ae bu d (9)
where
1
2 2 2
3
4 4 4
0 0
0 0
e
e b d
e ,b ,d
e
e b d
.
Rearranging Eq. (9), the following can be obtained
1e Ae b u d Ae bu (10)
where 1u u d .
Proof of Eq. (10)
bd d (11)
Because b is not square matrix, the following is
obtained by pseudo inverse of b
1
T T
d b b b d
for 0bbT
(12)
where 2 2 2
2 4 2 4
4
0
0 0
0
T b
b b b b b b
b
.
If 2 0b and 4 0 0b b , Eq. (10) is given as
e Ae (13)
If
2 2
2 4 0b b ,
1
2 2 4 4
2 2
2 4
T T b d b d
d b b b d
b b
(14)
In this paper,
2 2
2 4 0b b , dx and d are consider as
constant, so
2 23 4 43
2 2 4 4
d
b a b a
d
b b b b
(15)
where d is constant, so d is also constant.
3 Controller Design
The control law u is defined as
1 2au u d u u (16)
where static controller au contributes the design of
sliding surface, and dynamic controller 2u directly
makes sliding surface attractive to the system state.
The static system is nominal system as follows:
ae Ae bu (17)
The static controller is given by Ackermann’s
formula as follows:
T
au k e (18)
12 3
0 0 0 1T T T
k h P( A),h , , , b,Ab,A b,A b
,
1 2 3 4P( ) ( )( )( )( )
where 1 2 3 4, , , are the desired eigenvalues and
P( ) is characteristic polynomial of Eq. (17).
Substituting Eq. (18) into Eq. (17), the equation of
closed loop system is obtained as
T
e A bk e (19)
The sliding surface is chosen as
T T T
1S C e,C h P ( A) (20)
where 1 1 2 3P( ) ( )( )( ) ,
2 3
1 2 3p p p .
Proof of Eq. (20)
The left eigenvector T
C of T
A bk associated with
4 satisfies
4
T T T
C ( A bk ) C (21)
Rearranging Eq. (21) can be obtained as
4
T T T
C ( A I ) C bk (22)
1 2 3
T T
C b h ( A I )( A I )( A I )b (23)
12 3
2 3
1 2 3
0 0 0 1
1
1
T
, , , b,Ab,A b,A b
b Ab A b A b p p p
Substituting Eqs. (18), (20) and (23) into Eq. (22)
yields
4 1 4
T T T
k C ( A I ) h P( A)( A I )
T T
k h P( A)
Substituting Eqs. (16) and (18) into Eq. (10), the
dynamic controller as perturbed system is obtained
as follows:
2
T
e ( A bk )e bu (24)
A new variable z is defined as
1
0
T
e I
z e Te
S C
(25)
where 1
1 2 3
T
e e ,e ,e is the first three error
variables of e and
T
S C e becomes the last state
variable of z .
4. Nguyen Thanh Phuong, Ho Dac Loc, Tran Quang Thuan / International Journal of Engineering
Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 3, Issue 3, May-Jun 2013, pp.1276-1282
1279 | P a g e
The new coordinate system can be obtained as
1
2 2
T
z T( A bk )T z Tbu Az Bu
(26)
The transformed system is
1 1 1
1 1 2e A e a S b u (27)
4 2S S u (28)
where 1
1 2 3
T
b b ,b ,b ,
1
1 11
40 1
T A a b
A T( A bk )T ,B Tb
.
A is shown in details in Appendix C
Proof of (28)
Differentiating Eq. (20) and substituting Eqs. (10),
(21) and (23), the following can be reduced into
2
T T T
S C e C A bk e bu
2
T T T
C A bk e C bu
4 2 4 2
T
C e u S u (29)
The dynamic controller is chosen as
2u M(e,t )sign( S ) (30)
where 4
T
M(e,t ) C e .
Proof of (30)
According to reachability condition
0
0
S
lim S S
,
the followings must be satisfied,
i) S 0 and S 0
4 2 0T
S C e u
2 4
T
u C e
4
T
M(e,t )sign( S ) C e
4
T
M( e,t ) C e
ii) S 0 and S 0
4 2 0T
S C e u
2 4
T
u C e
4
T
M(e,t )sign( S ) C e
4
T
M(e,t ) C e
From Eqs. (16) and (18), the control law is given
as
2
T
u k e u d (31)
4 Hardware Design
The control system is specially designed for
two-wheeled mobile inverted pendulum. The control
system is based on the integration of three
PIC16F877s: two for servo controllers and one for
main controller. The main controller which is
functionalized as master links to the two servo
controllers, as slave, via I2C communication. The
total configuration of the control system is shown in
Fig. 4.
Fig. 4 The configuration of control system
5 Simulation and Experimental Results
The parameters for two-wheeled mobile inverted
pendulum used for simulation are given in Table 1.
Table 1 System numerical values
Parameters Values Units
r 0.05 m
pM 1.13 Kg
pI 0.004 2
Kg m
eK 0.007 /Vs rad
mK 0.006 /Nm A
R 3
wM 0.03 Kg
wI 0.001 2
Kg m
L 0.07 m
g 9.8 2
m / s
Substituting system’s parameters into Eqs. (24), the
state space equation can be rewritten as follows:
0 1 0 0
0 4228 0 8809 2 1459 0 8822
0 0 0 1
4 3309 9 1060 34 5772 9 1191
. . . .
e e
. . . .
2
0
0 1989
0
2 0565
.
u
.
(32)
where 1 2 3 41 2 3 4, , , .
It is readily shown that the rank of the controllability
matrices is full.
Simulation has been done for the proposed controller
to be used in the practical field. In addition, the
controller was applied to the two-wheeled inverted
pendulum for the experiment. It is shown that the
proposed controller can be used in the practical field.
In the case of straight motion modeling, the
parameters of the sliding surface are
5. Nguyen Thanh Phuong, Ho Dac Loc, Tran Quang Thuan / International Journal of Engineering
Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 3, Issue 3, May-Jun 2013, pp.1276-1282
1280 | P a g e
1 1 , 2 32 3, ; the parameters of the
control law is 0 40M ; the initial values rx , are
zero, and initial errors are 1 1 0e . m and
3 0 1e . rad .
Fig. 5 shows that the cart position error has the
convergent time of 5 seconds. Fig. 6 shows that the
inverted pendulum angle error is bounded around
zero after 4 seconds. Fig. 7 shows the sliding surface
to converge zero very fast in simulation.
0 5 10 15
-1
-0.8
-0.6
-0.4
-0.2
0
0.2
0.4
Time(s)
CartPositionError
Cart Position Error(m)
Fig. 5 Cart position error 1 1 0e . m
0 5 10 15 20 25 30
-0.15
-0.1
-0.05
0
0.05
Time(s)
InvertedPendulumAngleError
Inverted Pendulum Angle Error(rad)
Fig. 6 Inverted pendulum angle error
3 0 1e . rad
0 2 4 6 8 10
-0.25
-0.2
-0.15
-0.1
-0.05
0
0.05
0.1
Time(s)
SlidingSurface
Fig.7 Sliding surface
Fig. 8 shows controller 1u versus time. Fig. 9 shows
controller 2u versus time. Fig. 10 presents control
law u of mobile inverted pendulum to track
reference.
0 2 4 6 8 10
-40
-30
-20
-10
0
10
20
30
40
50
Time(s)
Input Fig. 8 Dynamic controller 1u
0 2 4 6 8 10
-40
-30
-20
-10
0
10
20
30
40
Time(s)
Input
Fig. 9 Dynamic controller 2u
The experimental results in stable condition are
given after errors converge through Figs.(11) ~ (13).
The encoder voltage is shown in Fig. 11. Fig. 12
presents the output voltage of tilt sensor. Fig. 13
presents motor PWM output under the proposed
controller u in experiment.
0 1 2 3 4 5 6 7 8 9 10
-40
-30
-20
-10
0
10
20
30
40
50
60
Time(s)
ControlInputu
Fig. 10 Controller input u
6. Nguyen Thanh Phuong, Ho Dac Loc, Tran Quang Thuan / International Journal of Engineering
Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 3, Issue 3, May-Jun 2013, pp.1276-1282
1281 | P a g e
Fig. 11 Output voltage of motor encoder
Fig. 12 Output voltage of tilt sensor
Fig. 13 PWM output of DC motor under control
law
6 Conclusion
This paper presents a sliding mode tracking
controller for the mobile inverted pendulum. The
controller is developed based upon a class of
nonlinear systems whose nonlinear part of the
modeling can be linearly parameterized. The tracking
errors are defined, and then the sliding surface is
chosen in an explicit form using Ackermann’s
formula to guarantee that the tracking error converge
to zero asymptotically.
The simulation and experimental results show
that the proposed controller is applicable and
implemented in the practical field.
REFERENCES:
[1] F. Grasser, A. D’Arrigo, S. Colombi and A.
C. Rufer, JOE: a Mobile, Inverted
Pendulum, IEEE Trans. Industrial
Electronics, Vol. 49, No. 1(2002), pp.
107~114.
[2] C.S. Chen and W.L. Chen: Robust Adaptive
Sliding-mode Control Using Fuzzy
Modeling for an Inverted-pendulum
System, IEEE Trans. Industrial Electronics,
Vol. 45, No. 2(1998), pp. 297~306.
[3] S.J. Huang and C.L. Huang: Control of an
Inverted Pendulum Using Grey Prediction
Model, IEEE Trans. Industrial Applications,
Vol. 36, No. 2(2000), pp. 452~458.
[4] J. Ackermann and V. Utkin: Sliding Mode
Control Design Based on Ackermann’s
Formula, IEEE Trans. Automatic Control,
Vol. 43, No. 2(1998), pp. 234~237.
[5] D. Necsulescu: Mechatronics, Prentice-Hall,
Inc. (2002), pp. 57~60.
[6] M.T. Kang: M.S. Thesis, Control for
Mobile Inverted Pendulum Using Sliding
Mode Technique, Pukyong National
University, (2007).
Appendix A
Nomenclature
Variable Description Units
V Motor voltage V
i Current armature A
Rotor angular velocity rad/s
eV Back electromotive force
voltage
V
eK Back electromotive
voltage coefficient
Vs/rad
RI Inertia moment of the
rotor
2
Kg m
mT Magnetic torque of the
rotor
Nm
mK Magnetic torque
coefficient
Nm/A
T Load torque of the motor Nm
fK Viscous frictional
coefficient of rotor shaft
Nms/ rad
R Nominal rotor resistance
H Nominal rotor inductance H
cx Cart position
perpendicular to the
wheel axis
m
Inverted pendulum angle rad
d Desired inverted
pendulum angle
rad
L Distance between the
wheel axis and the
pendulum’s center
m
D Lateral distance between
two coaxial wheels
m
7. Nguyen Thanh Phuong, Ho Dac Loc, Tran Quang Thuan / International Journal of Engineering
Research and Applications (IJERA) ISSN: 2248-9622 www.ijera.com
Vol. 3, Issue 3, May-Jun 2013, pp.1276-1282
1282 | P a g e
L RT ,T Torques acting on the left
and the right wheel
Nm
L LH ,P Reaction force between
left wheel and the
inverted pendulum
N
R RH ,P Reaction force between
right wheel and the
inverted pendulum
N
Lw Rw, Rotation angles of the left
and right wheel
rad
L Rx ,x left and right wheel
position
m
fL fRH ,H Friction forces between
the left and right wheel
and the ground
N
r Wheel radius m
pM Pendulum mass Kg
pI Inertia moment of the
inverted pendulum around
the wheel axis
2
Kg m
wM Wheel mass with DC
motor
Kg
wI Inertia moment of wheel
with rotor around the
wheel axis
2
Kg m
g Gravitation acceleration 2
m / s
Appendix B
2
22 2
2 m e p p pK K ( M Lr I M L )
a
Rr
,
2 2
23
pM gL
a
,
42 2
2 m e pK K ( r M L )
a
Rr
,
43
pM gL
a
,
2
2
2 m p p pK ( I M L M Lr )
b
Rr
,
4
2 m pK ( M L r )
b
Rr
,
where 2
2
2 w
w p
I
( M M )
r
,
2
2
2 w
p p w
I
I M L ( M )
r
.
Appendix C
11 12 13 14 1 1 1 2 1 3 1 4
21 22 23 24 2 1 2 2 2 3 2 4
31 32 33 34 3 1 3 2 3 3 3 4
41 42 43 44 4 1 4 2 4 3 4 4
T
a a a a b k b k b k b k
a a a a b k b k b k b k
A bk
a a a a b k b k b k b k
a a a a b k b k b k b k
(C1)
11 1 1 12 1 2 13 1 3 14 1 4 11 12 13 14
21 2 1 22 2 2 23 2 3 24 2 4 21 22 23 24
31 3 1 32 3 2 33 3 3 34 3 4 31 32 33 34
41 4 1 42 4 2 43 4 3 44 4 4 41 42 43 44
a b k a b k a b k a b k A A A A
a b k a b k a b k a b k A A A A
a b k a b k a b k a b k A A A A
a b k a b k a b k a b k A A A A
11 12 13 14
21 22 23 241
31 32 33 34
31 2
1 2 3 4 41 42 43 44
4 4 4 4
1 0 0 0
1 0 0 0
0 1 0 0
0 1 0 0
0 0 1 0
0 0 1 0
1
T
A A A A
A A A A
A T A bk T
A A A A
cc c
c c c c A A A A
c c c c
31 2 14
11 14 12 14 13 14
4 4 4 4
31 2 24
21 24 22 24 23 24
4 4 4 4
3 341 2
31 34 32 34 33 34
4 4 4 4
31 2 4
1 4 2 4 3 4
4 4 4 4
cc c A
A A A A A A
c c c c
cc c A
A A A A A A
c c c c
c Ac c
A A A A A A
c c c c
cc c Q
Q Q Q Q Q Q
c c c c
(C2)
where
1 1 11 2 21 3 31 4 41Q c A c A c A c A (C3)
2 1 12 2 22 3 32 4 42Q c A c A c A c A
3 1 13 2 23 3 33 4 43Q c A c A c A c A
4 1 14 2 24 3 34 4 44Q c A c A c A c A .
From (21), (C1) and (C3), the followings can be
obtained
1 2 3 4
T T
Q Q Q Q Q C A bk
4 1 4 2 4 3 4 4 4
T
C c c c c (C4)
1 1
1 4 1 4 4 4
4 4
0
c c
Q Q c c
c c
2 2
2 4 2 4 4 4
4 4
0
c c
Q Q c c
c c
3 3
3 4 3 4 4 4
4 4
0
c c
Q Q c c
c c
4 4 4
4
4 4
Q c
c c
(C5)
Substituting (C5) into (C2) can be rewritten as
follows:
31 2 14
11 14 12 14 13 14
4 4 4 4
31 2 24
21 24 22 24 23 24 1 1
4 4 4 4
4
3 341 2
31 34 32 34 33 34
4 4 4 4
4
0
0 0 0
cc c A
A A A A A A
c c c c
cc c A
A A A A A A A a
c c c cA
c Ac c
A A A A A A
c c c c