The derivation of motion equations of constrained spatial multibody system is an important problem of dynamics and control of parallel robots. The paper firstly presents an overview of the calculating the torque of the driving stages of the parallel robots using Kronecker product. The main content of this paper is to derive the inverse dynamics controllers based on the radial basis function (RBF) neural network control law for parallel robot manipulators. Finally, numerical simulation of the inverse dynamics controller for a 3-RRR delta robot manipulator is presented as an illustrative example.
SIAM-AG21-Topological Persistence Machine of Phase TransitionHa Phuong
Presentation at SIAM Conference on Applied Algebraic Geometry (AG21), Aug. 2021.
Abstract. The study of phase transitions using data-driven approaches is challenging, especially when little prior knowledge of the system is available. Topological data analysis is an emerging framework for characterizing the shape of data and has recently achieved success in detecting structural transitions in material science, such as the glass--liquid transition. However, data obtained from physical states may not have explicit shapes as structural materials. We thus propose a general framework, termed “topological persistence machine," to construct the shape of data from correlations in states so that we can subsequently decipher phase transitions via qualitative changes in the shape. Our framework enables an effective and unified approach in phase transition analysis without having prior knowledge about phases or requiring the investigation of the system with large size. We demonstrate the efficacy of the approach in terms of detecting the Berezinskii--Kosterlitz--Thouless phase transition in the classical XY model and quantum phase transitions in the transverse Ising and Bose--Hubbard models. Interestingly, while these phase transitions have proven to be notoriously difficult to analyze using traditional methods, they can be characterized through our framework without requiring prior knowledge of the phases. Our approach is thus expected to be widely applicable and will provide the prospective with practical interests in exploring the phases of experimental physical systems.
THE LEFT AND RIGHT BLOCK POLE PLACEMENT COMPARISON STUDY: APPLICATION TO FLIG...ieijjournal
It is known that if a linear-time-invariant MIMO system described by a state space equation has a number of states divisible by the number of inputs and it can be transformed to block controller form, we can design a state feedback controller using block pole placement technique by assigning a set of desired Block poles. These may be left or right block poles. The idea is to compare both in terms of system’s response.
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.
SIAM-AG21-Topological Persistence Machine of Phase TransitionHa Phuong
Presentation at SIAM Conference on Applied Algebraic Geometry (AG21), Aug. 2021.
Abstract. The study of phase transitions using data-driven approaches is challenging, especially when little prior knowledge of the system is available. Topological data analysis is an emerging framework for characterizing the shape of data and has recently achieved success in detecting structural transitions in material science, such as the glass--liquid transition. However, data obtained from physical states may not have explicit shapes as structural materials. We thus propose a general framework, termed “topological persistence machine," to construct the shape of data from correlations in states so that we can subsequently decipher phase transitions via qualitative changes in the shape. Our framework enables an effective and unified approach in phase transition analysis without having prior knowledge about phases or requiring the investigation of the system with large size. We demonstrate the efficacy of the approach in terms of detecting the Berezinskii--Kosterlitz--Thouless phase transition in the classical XY model and quantum phase transitions in the transverse Ising and Bose--Hubbard models. Interestingly, while these phase transitions have proven to be notoriously difficult to analyze using traditional methods, they can be characterized through our framework without requiring prior knowledge of the phases. Our approach is thus expected to be widely applicable and will provide the prospective with practical interests in exploring the phases of experimental physical systems.
THE LEFT AND RIGHT BLOCK POLE PLACEMENT COMPARISON STUDY: APPLICATION TO FLIG...ieijjournal
It is known that if a linear-time-invariant MIMO system described by a state space equation has a number of states divisible by the number of inputs and it can be transformed to block controller form, we can design a state feedback controller using block pole placement technique by assigning a set of desired Block poles. These may be left or right block poles. The idea is to compare both in terms of system’s response.
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.
Finite Element Analysis Lecture Notes Anna University 2013 Regulation NAVEEN UTHANDI
One of the most Simple and Interesting topics in Engineering is FEA. My work will guide average students to score good marks. I have given you full package which includes 2 Marks and Question Banks of previous year. All the Best
For Guidance : Comment Below Happy to Teach and Learn along with you guys
FEEDBACK LINEARIZATION AND BACKSTEPPING CONTROLLERS FOR COUPLED TANKSieijjournal
This paper investigates the usage of some sophisticated and advanced nonlinear control algorithms in order to control a nonlinear Coupled Tanks System. The first control procedure is called the Feedback linearisation control (FLC), this type of control has been found a successful in achieving a global exponential asymptotic stability, with very short time response, no significant overshooting is recorded and with a negligible norm of the error. The second control procedure is the approaches of Back stepping control (BC) which is a recursive procedure that interlaces the choice of a Lyapunov function with the design of feedback control, from simulation results it shown that this method preserves tracking, robust control and it can often solve stabilization problems with less restrictive conditions may been countered in other methods. Finally both of the proposed control schemes guarantee the asymptoticstability of the closed loop system meeting trajectory tracking objectives.
—This paper presents a new image based visual servoing (IBVS) control scheme for omnidirectional wheeled mobile robots with four swedish wheels. The contribution is the proposal of a scheme that consider the overall dynamic of the system; this means, we put together mechanical and electrical dynamics. The actuators are direct current (DC) motors, which imply that the system input signals are armature voltage applied to DC motors. In our control scheme the PD control law and eye-to-hand camera configuration are used to compute the armature voltages and to measure system states, respectively. Stability proof is performed via Lypunov direct method and LaSalle's invariance principle. Simulation and experimental results were performed in order to validate the theoretical proposal and to show the good performance of the posture errors. Keywords—IBVS, posture control, omnidirectional wheeled mobile robot, dynamic actuator, Lyapunov direct method.
Finite Element Analysis is a widely used computational method in most of the engineering domains. But still, its considered as a difficult topic by most students. This presentation is an effort to introduce the very basics of FEA so as to build an intuitive feel for the method. Enjoy !
Special Plenary Lecture at the International Conference on VIBRATION ENGINEERING AND TECHNOLOGY OF MACHINERY (VETOMAC), Lisbon, Portugal, September 10 - 13, 2018
http://www.conf.pt/index.php/v-speakers
Propagation of uncertainties in complex engineering dynamical systems is receiving increasing attention. When uncertainties are taken into account, the equations of motion of discretised dynamical systems can be expressed by coupled ordinary differential equations with stochastic coefficients. The computational cost for the solution of such a system mainly depends on the number of degrees of freedom and number of random variables. Among various numerical methods developed for such systems, the polynomial chaos based Galerkin projection approach shows significant promise because it is more accurate compared to the classical perturbation based methods and computationally more efficient compared to the Monte Carlo simulation based methods. However, the computational cost increases significantly with the number of random variables and the results tend to become less accurate for a longer length of time. In this talk novel approaches will be discussed to address these issues. Reduced-order Galerkin projection schemes in the frequency domain will be discussed to address the problem of a large number of random variables. Practical examples will be given to illustrate the application of the proposed Galerkin projection techniques.
The paper deals with the problem of control of continuous-time linear systems by the dynamic
output controllers of order equal to the plant model order. The design procedure is based on a
solution of the set of linear matrix inequalities and ensures the closed-loop stability using
Lyapunov approach. Numerical examples are given to illustrate the design procedure and
relevance of the methods as well as to validate the performances of the proposed approach
Adaptive Projective Lag Synchronization of T and Lu Chaotic Systems IJECEIAES
In this paper, the synchronization problem of T chaotic system and Lu chaotic system is studied. The parameter of the drive T chaotic system is considered unknown. An adaptive projective lag control method and also parameter estimation law are designed to achieve chaos synchronization problem between two chaotic systems. Then Lyapunov stability theorem is utilized to prove the validity of the proposed control method. After that, some numerical simulations are performed to assess the performance of the proposed method. The results show high accuracy of the proposed method in control and synchronization of chaotic systems.
THE LEFT AND RIGHT BLOCK POLE PLACEMENT COMPARISON STUDY: APPLICATION TO FLIG...ieijjournal1
It is known that if a linear-time-invariant MIMO system described by a state space equation has a number
of states divisible by the number of inputs and it can be transformed to block controller form, we can
design a state feedback controller using block pole placement technique by assigning a set of desired Block
poles. These may be left or right block poles. The idea is to compare both in terms of system’s response.
Investigation of auto-oscilational regimes of the system by dynamic nonlinear...IJECEIAES
The paper proposes a method for the analysis and synthesis of self-oscillations in the form of a finite, predetermined number of terms of the Fourier series in systems reduced to single-loop, with one element having a nonlinear static characteristic of an arbitrary shape and a dynamic part, which is the sum of the products of coordinates and their derivatives. In this case, the nonlinearity is divided into two parts: static and dynamic nonlinearity. The solution to the problem under consideration consists of two parts. First, the parameters of self-oscillations are determined, and then the parameters of the nonlinear dynamic part of the system are synthesized. When implementing this procedure, the calculation time depends on the number of harmonics considered in the first approximation, so it is recommended to choose the minimum number of them in calculations. An algorithm for determining the self-oscillating mode of a control system with elements that have dynamic nonlinearity is proposed. The developed method for calculating self-oscillations is suitable for solving various synthesis problems. The generated system of equations can be used to synthesize the parameters of both linear and nonlinear parts. The advantage is its versatility.
Projective and hybrid projective synchronization of 4-D hyperchaotic system v...TELKOMNIKA JOURNAL
Nonlinear control strategy was established to realize the Projective Synchronization (PS) and Hybrid Projective Synchronization (HPS) for 4-D hyperchaotic system at different scaling matrices. This strategy, which is able to achieve projective and hybrid projective synchronization by more precise and adaptable method to provide a novel control scheme. On First stage, three scaling matrices were given in order to achieving various projective synchronization phenomena. While the HPS was implemented at specific scaling matrix in the second stage. Ultimately, the precision of controllers were compared and analyzed theoretically and numerically. The long-range precision of the proposed controllers are confirmed by third stage.
Finite Element Analysis Lecture Notes Anna University 2013 Regulation NAVEEN UTHANDI
One of the most Simple and Interesting topics in Engineering is FEA. My work will guide average students to score good marks. I have given you full package which includes 2 Marks and Question Banks of previous year. All the Best
For Guidance : Comment Below Happy to Teach and Learn along with you guys
FEEDBACK LINEARIZATION AND BACKSTEPPING CONTROLLERS FOR COUPLED TANKSieijjournal
This paper investigates the usage of some sophisticated and advanced nonlinear control algorithms in order to control a nonlinear Coupled Tanks System. The first control procedure is called the Feedback linearisation control (FLC), this type of control has been found a successful in achieving a global exponential asymptotic stability, with very short time response, no significant overshooting is recorded and with a negligible norm of the error. The second control procedure is the approaches of Back stepping control (BC) which is a recursive procedure that interlaces the choice of a Lyapunov function with the design of feedback control, from simulation results it shown that this method preserves tracking, robust control and it can often solve stabilization problems with less restrictive conditions may been countered in other methods. Finally both of the proposed control schemes guarantee the asymptoticstability of the closed loop system meeting trajectory tracking objectives.
—This paper presents a new image based visual servoing (IBVS) control scheme for omnidirectional wheeled mobile robots with four swedish wheels. The contribution is the proposal of a scheme that consider the overall dynamic of the system; this means, we put together mechanical and electrical dynamics. The actuators are direct current (DC) motors, which imply that the system input signals are armature voltage applied to DC motors. In our control scheme the PD control law and eye-to-hand camera configuration are used to compute the armature voltages and to measure system states, respectively. Stability proof is performed via Lypunov direct method and LaSalle's invariance principle. Simulation and experimental results were performed in order to validate the theoretical proposal and to show the good performance of the posture errors. Keywords—IBVS, posture control, omnidirectional wheeled mobile robot, dynamic actuator, Lyapunov direct method.
Finite Element Analysis is a widely used computational method in most of the engineering domains. But still, its considered as a difficult topic by most students. This presentation is an effort to introduce the very basics of FEA so as to build an intuitive feel for the method. Enjoy !
Special Plenary Lecture at the International Conference on VIBRATION ENGINEERING AND TECHNOLOGY OF MACHINERY (VETOMAC), Lisbon, Portugal, September 10 - 13, 2018
http://www.conf.pt/index.php/v-speakers
Propagation of uncertainties in complex engineering dynamical systems is receiving increasing attention. When uncertainties are taken into account, the equations of motion of discretised dynamical systems can be expressed by coupled ordinary differential equations with stochastic coefficients. The computational cost for the solution of such a system mainly depends on the number of degrees of freedom and number of random variables. Among various numerical methods developed for such systems, the polynomial chaos based Galerkin projection approach shows significant promise because it is more accurate compared to the classical perturbation based methods and computationally more efficient compared to the Monte Carlo simulation based methods. However, the computational cost increases significantly with the number of random variables and the results tend to become less accurate for a longer length of time. In this talk novel approaches will be discussed to address these issues. Reduced-order Galerkin projection schemes in the frequency domain will be discussed to address the problem of a large number of random variables. Practical examples will be given to illustrate the application of the proposed Galerkin projection techniques.
The paper deals with the problem of control of continuous-time linear systems by the dynamic
output controllers of order equal to the plant model order. The design procedure is based on a
solution of the set of linear matrix inequalities and ensures the closed-loop stability using
Lyapunov approach. Numerical examples are given to illustrate the design procedure and
relevance of the methods as well as to validate the performances of the proposed approach
Adaptive Projective Lag Synchronization of T and Lu Chaotic Systems IJECEIAES
In this paper, the synchronization problem of T chaotic system and Lu chaotic system is studied. The parameter of the drive T chaotic system is considered unknown. An adaptive projective lag control method and also parameter estimation law are designed to achieve chaos synchronization problem between two chaotic systems. Then Lyapunov stability theorem is utilized to prove the validity of the proposed control method. After that, some numerical simulations are performed to assess the performance of the proposed method. The results show high accuracy of the proposed method in control and synchronization of chaotic systems.
THE LEFT AND RIGHT BLOCK POLE PLACEMENT COMPARISON STUDY: APPLICATION TO FLIG...ieijjournal1
It is known that if a linear-time-invariant MIMO system described by a state space equation has a number
of states divisible by the number of inputs and it can be transformed to block controller form, we can
design a state feedback controller using block pole placement technique by assigning a set of desired Block
poles. These may be left or right block poles. The idea is to compare both in terms of system’s response.
Investigation of auto-oscilational regimes of the system by dynamic nonlinear...IJECEIAES
The paper proposes a method for the analysis and synthesis of self-oscillations in the form of a finite, predetermined number of terms of the Fourier series in systems reduced to single-loop, with one element having a nonlinear static characteristic of an arbitrary shape and a dynamic part, which is the sum of the products of coordinates and their derivatives. In this case, the nonlinearity is divided into two parts: static and dynamic nonlinearity. The solution to the problem under consideration consists of two parts. First, the parameters of self-oscillations are determined, and then the parameters of the nonlinear dynamic part of the system are synthesized. When implementing this procedure, the calculation time depends on the number of harmonics considered in the first approximation, so it is recommended to choose the minimum number of them in calculations. An algorithm for determining the self-oscillating mode of a control system with elements that have dynamic nonlinearity is proposed. The developed method for calculating self-oscillations is suitable for solving various synthesis problems. The generated system of equations can be used to synthesize the parameters of both linear and nonlinear parts. The advantage is its versatility.
Projective and hybrid projective synchronization of 4-D hyperchaotic system v...TELKOMNIKA JOURNAL
Nonlinear control strategy was established to realize the Projective Synchronization (PS) and Hybrid Projective Synchronization (HPS) for 4-D hyperchaotic system at different scaling matrices. This strategy, which is able to achieve projective and hybrid projective synchronization by more precise and adaptable method to provide a novel control scheme. On First stage, three scaling matrices were given in order to achieving various projective synchronization phenomena. While the HPS was implemented at specific scaling matrix in the second stage. Ultimately, the precision of controllers were compared and analyzed theoretically and numerically. The long-range precision of the proposed controllers are confirmed by third stage.
Back-Stepping Control of Free-Floating Space Robot based on Adaptive Neural N...TELKOMNIKA JOURNAL
Trajectory tracking control problems of the free-floating space robot are considered by the paper,
back-stepping control method based on adaptive neural network is put forward. The complex system is
decomposed into several simple sub-systems. The control laws are designed by derived, so that closedloop
stability can be obtained by each subsystem; Because of the influence of interference and the
measurement level limitation, accurate mathematical model is difficult to be obtained. Neural network
controller of good nonlinear approximation ability is designed to compensate the uncertainty of system
model. Adaptive learning laws are designed to ensure that weights can be adjusted online real-time. The
system uniformly ultimately bounded (UUB) is proved based on the Lyapunov theory. Simulation
experiments show that the control method can fast track the desired trajectory, and has a good application
value for space robotic manipulators with uncertainty.
ANALYSIS AND SLIDING CONTROLLER DESIGN FOR HYBRID SYNCHRONIZATION OF HYPERCHA...IJCSEA Journal
Hybrid synchronization of chaotic systems is a research problem with a goal to synchronize the states of master and slave chaotic systems in a hybrid manner, namely, their even states are completely synchronized (CS) and odd states are anti-synchronized. This paper deals with the research problem of hybrid synchronization of chaotic systems. First, a detailed analysis is made on the qualitative properties of hyperchaotic Yujun system (2010). Then sliding controller has been derived for the hybrid synchronization of identical hyperchaotic Yujun systems, which is based on a general hybrid result derived in this paper.MATLAB simulations have been shown in detail to illustrate the new results derived for the hybrid synchronization of hyperchaotic Yujun systems. The results are proved using Lyapunov stability theory.
On tracking control problem for polysolenoid motor model predictive approach IJECEIAES
The Polysolenoid Linear Motor (PLM) have been playing a crucial role in many industrial aspects due to its functions, in which a straight motion is provided directly without mediate mechanical actuators. Recently, with several commons on mathematic model, some control methods for PLM based on Rotational Motor have been applied, but position, velocity and current constraints which are important in real systems have been ignored. In this paper, position tracking control problem for PLM was considered under state-independent disturbances via min-max model predictive control. The proposed controller forces tracking position errors converge to small region of origin and satisfies state including position, velocity and currents constraints. Further, a numerical simulation was implemented to validate the performance of the proposed controller.
Evaluation of Vibrational Behavior for A System With TwoDegree-of-Freedom Und...IJERA Editor
Analysis of the vibrational behavior of a system is extremely important, both for the evaluation of operating conditions, as performance and safety reason. The studies on vibration concentrate their efforts on understanding the natural phenomena and the development of mathematical theories to describe the vibration of physical systems. The purpose of this study is to evaluate an undamped system with two-degrees-of-freedom and demonstrate by comparing the results obtained in the experimental, numerical and analytical modeling the characteristics that describe a structure in terms of its natural characteristics. The experiment was conducted in PUC-MG where the data were acquired to determine the natural frequency of the system. We also developed an experimental test bed for vibrations studies for graduate and undergraduate students. In analytical modeling were represented all the important aspects of the system. In order, to obtain the mathematical equations is used MATLAB to solve the equations that describe the characteristics of system behavior. For the simulation and numerical solution of the system, we use a computational tool ABAQUS. The comparison between the results obtained in the experiment and the numerical was considered satisfactory using the exact solutions. This study demonstrates that calculation of the adopted conditions on a system with two-degrees-of-freedom can be applied to complex systems with many degrees of freedom and proved to be an excellent learning tool for determining the modal analysis of a system. One of the goals is to use the developed platform to be used as a didactical experiment system for vibration and modal analysis classes at PUC Minas. The idea is to give the students an opportunity to test, play, calculate and confirm the results in vibration and modal analysis in a low-cost platform
Solution of Inverse Kinematics for SCARA Manipulator Using Adaptive Neuro-Fuz...ijsc
Solution of inverse kinematic equations is complex problem, the complexity comes from the nonlinearity of joint space and Cartesian space mapping and having multiple solution. In this work, four adaptive neurofuzzy networks ANFIS are implemented to solve the inverse kinematics of 4-DOF SCARA manipulator. The implementation of ANFIS is easy, and the simulation of it shows that it is very fast and give acceptable
error.
Integral Backstepping Approach for Mobile Robot ControlTELKOMNIKA JOURNAL
This paper presents the trajectory tracking problem of a unicycle-type mobile robots. A robust output tracking controller for nonlinear systems in the presence of disturbances is proposed, the approach is based on the combination of integral action and Backstepping technique to compensate for the dynamic disturbances. For desired trajectory, the values of the linear and angular velocities of the robot are assured by the kinematic controller. The control law guarantees stability of the robot by using the lyapunov theorem. The simulation and experimental results are presented to verify the designed trajectory tracking control.
SLIDING MODE CONTROLLER DESIGN FOR GLOBAL CHAOS SYNCHRONIZATION OF COULLET SY...ijistjournal
This paper derives new results for the design of sliding mode controller for the global chaos synchronization of identical Coullet systems (1981). The synchronizer results derived in this paper for the complete chaos synchronization of identical hyperchaotic systems are established using sliding control theory and Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve global chaos synchronization of the identical Coullet systems. Numerical simulations are shown to illustrate and validate the synchronization schemes derived in this paper for the identical Coullet systems.
SLIDING MODE CONTROLLER DESIGN FOR GLOBAL CHAOS SYNCHRONIZATION OF COULLET SY...ijistjournal
This paper derives new results for the design of sliding mode controller for the global chaos synchronization of identical Coullet systems (1981). The synchronizer results derived in this paper for the complete chaos synchronization of identical hyperchaotic systems are established using sliding control theory and Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve global chaos synchronization of the identical Coullet systems. Numerical simulations are shown to illustrate and validate the synchronization schemes derived in this paper for the identical Coullet systems.
SLIDING MODE CONTROLLER DESIGN FOR SYNCHRONIZATION OF SHIMIZU-MORIOKA CHAOTIC...ijistjournal
This paper investigates the global chaos synchronization of identical Shimizhu-Morioka chaotic systems (Shimizu and Morioka, 1980) by sliding mode control. The stability results derived in this paper for the complete synchronization of identical Shimizu-Morioka chaotic systems are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve global chaos synchronization of the identical Shimizu-Morioka chaotic systems. Numerical simulations are shown to illustrate and validate the synchronization schemes derived in this paper for the identical Shimizu-Morioka systems.
SLIDING MODE CONTROLLER DESIGN FOR SYNCHRONIZATION OF SHIMIZU-MORIOKA CHAOTIC...ijistjournal
This paper investigates the global chaos synchronization of identical Shimizhu-Morioka chaotic systems (Shimizu and Morioka, 1980) by sliding mode control. The stability results derived in this paper for the complete synchronization of identical Shimizu-Morioka chaotic systems are established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the sliding mode control method is very effective and convenient to achieve global chaos synchronization of the identical Shimizu-Morioka chaotic systems. Numerical simulations are shown to illustrate and validate the synchronization schemes derived in this paper for the identical Shimizu-Morioka systems.
ACTIVE CONTROLLER DESIGN FOR THE GENERALIZED PROJECTIVE SYNCHRONIZATION OF TH...ijait
This paper discusses the design of active controllers for generalized projective synchronization (GPS) of identical Wang 3-scroll chaotic systems (Wang, 2009), identical Dadras 3-scroll chaotic systems (Dadras and Momeni, 2009) and non-identical Wang 3-scroll system and Dadras 3-scroll system. The synchronization results (GPS) derived in this paper for the 3 scroll chaotic systems have been derived using active control method and established using Lyapunov stability theory. Since the Lyapunov exponents are not required for these calculations, the active control method is very effective and convenient for
achieving the generalized projective synchronization (GPS) of the 3-scroll chaotic systems addressed in this paper. Numerical simulations are provided to illustrate the effectiveness of the GPS synchronization results derived in this paper.
Universal Approximation Property via Quantum Feature Maps
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The quantum Hilbert space can be used as a quantum-enhanced feature space in machine learning (ML) via the quantum feature map to encode classical data into quantum states. We prove the ability to approximate any continuous function with optimal approximation rate via quantum ML models in typical quantum feature maps.
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Contributed talk at Quantum Techniques in Machine Learning 2021, Tokyo, November 8-12 2021.
By Quoc Hoan Tran, Takahiro Goto and Kohei Nakajima
Chaos synchronization in a 6-D hyperchaotic system with self-excited attractorTELKOMNIKA JOURNAL
This paper presented stability application for chaos synchronization using a 6-D hyperchaotic system of different controllers and two tools: Lyapunov stability theory and Linearization methods. Synchronization methods based on nonlinear control strategy is used. The selecting controller's methods have been modified by applying complete synchronization. The Linearization methods can achieve convergence according to the of complete synchronization. Numerical simulations are carried out by using MATLAB to validate the effectiveness of the analytical technique.
Similar to Radial basis function neural network control for parallel spatial robot (20)
Amazon products reviews classification based on machine learning, deep learni...TELKOMNIKA JOURNAL
In recent times, the trend of online shopping through e-commerce stores and websites has grown to a huge extent. Whenever a product is purchased on an e-commerce platform, people leave their reviews about the product. These reviews are very helpful for the store owners and the product’s manufacturers for the betterment of their work process as well as product quality. An automated system is proposed in this work that operates on two datasets D1 and D2 obtained from Amazon. After certain preprocessing steps, N-gram and word embedding-based features are extracted using term frequency-inverse document frequency (TF-IDF), bag of words (BoW) and global vectors (GloVe), and Word2vec, respectively. Four machine learning (ML) models support vector machines (SVM), logistic regression (RF), logistic regression (LR), multinomial Naïve Bayes (MNB), two deep learning (DL) models convolutional neural network (CNN), long-short term memory (LSTM), and standalone bidirectional encoder representations (BERT) are used to classify reviews as either positive or negative. The results obtained by the standard ML, DL models and BERT are evaluated using certain performance evaluation measures. BERT turns out to be the best-performing model in the case of D1 with an accuracy of 90% on features derived by word embedding models while the CNN provides the best accuracy of 97% upon word embedding features in the case of D2. The proposed model shows better overall performance on D2 as compared to D1.
Design, simulation, and analysis of microstrip patch antenna for wireless app...TELKOMNIKA JOURNAL
In this study, a microstrip patch antenna that works at 3.6 GHz was built and tested to see how well it works. In this work, Rogers RT/Duroid 5880 has been used as the substrate material, with a dielectric permittivity of 2.2 and a thickness of 0.3451 mm; it serves as the base for the examined antenna. The computer simulation technology (CST) studio suite is utilized to show the recommended antenna design. The goal of this study was to get a more extensive transmission capacity, a lower voltage standing wave ratio (VSWR), and a lower return loss, but the main goal was to get a higher gain, directivity, and efficiency. After simulation, the return loss, gain, directivity, bandwidth, and efficiency of the supplied antenna are found to be -17.626 dB, 9.671 dBi, 9.924 dBi, 0.2 GHz, and 97.45%, respectively. Besides, the recreation uncovered that the transfer speed side-lobe level at phi was much better than those of the earlier works, at -28.8 dB, respectively. Thus, it makes a solid contender for remote innovation and more robust communication.
Design and simulation an optimal enhanced PI controller for congestion avoida...TELKOMNIKA JOURNAL
In this paper, snake optimization algorithm (SOA) is used to find the optimal gains of an enhanced controller for controlling congestion problem in computer networks. M-file and Simulink platform is adopted to evaluate the response of the active queue management (AQM) system, a comparison with two classical controllers is done, all tuned gains of controllers are obtained using SOA method and the fitness function chose to monitor the system performance is the integral time absolute error (ITAE). Transient analysis and robust analysis is used to show the proposed controller performance, two robustness tests are applied to the AQM system, one is done by varying the size of queue value in different period and the other test is done by changing the number of transmission control protocol (TCP) sessions with a value of ± 20% from its original value. The simulation results reflect a stable and robust behavior and best performance is appeared clearly to achieve the desired queue size without any noise or any transmission problems.
Improving the detection of intrusion in vehicular ad-hoc networks with modifi...TELKOMNIKA JOURNAL
Vehicular ad-hoc networks (VANETs) are wireless-equipped vehicles that form networks along the road. The security of this network has been a major challenge. The identity-based cryptosystem (IBC) previously used to secure the networks suffers from membership authentication security features. This paper focuses on improving the detection of intruders in VANETs with a modified identity-based cryptosystem (MIBC). The MIBC is developed using a non-singular elliptic curve with Lagrange interpolation. The public key of vehicles and roadside units on the network are derived from number plates and location identification numbers, respectively. Pseudo-identities are used to mask the real identity of users to preserve their privacy. The membership authentication mechanism ensures that only valid and authenticated members of the network are allowed to join the network. The performance of the MIBC is evaluated using intrusion detection ratio (IDR) and computation time (CT) and then validated with the existing IBC. The result obtained shows that the MIBC recorded an IDR of 99.3% against 94.3% obtained for the existing identity-based cryptosystem (EIBC) for 140 unregistered vehicles attempting to intrude on the network. The MIBC shows lower CT values of 1.17 ms against 1.70 ms for EIBC. The MIBC can be used to improve the security of VANETs.
Conceptual model of internet banking adoption with perceived risk and trust f...TELKOMNIKA JOURNAL
Understanding the primary factors of internet banking (IB) acceptance is critical for both banks and users; nevertheless, our knowledge of the role of users’ perceived risk and trust in IB adoption is limited. As a result, we develop a conceptual model by incorporating perceived risk and trust into the technology acceptance model (TAM) theory toward the IB. The proper research emphasized that the most essential component in explaining IB adoption behavior is behavioral intention to use IB adoption. TAM is helpful for figuring out how elements that affect IB adoption are connected to one another. According to previous literature on IB and the use of such technology in Iraq, one has to choose a theoretical foundation that may justify the acceptance of IB from the customer’s perspective. The conceptual model was therefore constructed using the TAM as a foundation. Furthermore, perceived risk and trust were added to the TAM dimensions as external factors. The key objective of this work was to extend the TAM to construct a conceptual model for IB adoption and to get sufficient theoretical support from the existing literature for the essential elements and their relationships in order to unearth new insights about factors responsible for IB adoption.
Efficient combined fuzzy logic and LMS algorithm for smart antennaTELKOMNIKA JOURNAL
The smart antennas are broadly used in wireless communication. The least mean square (LMS) algorithm is a procedure that is concerned in controlling the smart antenna pattern to accommodate specified requirements such as steering the beam toward the desired signal, in addition to placing the deep nulls in the direction of unwanted signals. The conventional LMS (C-LMS) has some drawbacks like slow convergence speed besides high steady state fluctuation error. To overcome these shortcomings, the present paper adopts an adaptive fuzzy control step size least mean square (FC-LMS) algorithm to adjust its step size. Computer simulation outcomes illustrate that the given model has fast convergence rate as well as low mean square error steady state.
Design and implementation of a LoRa-based system for warning of forest fireTELKOMNIKA JOURNAL
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Impact of CuS counter electrode calcination temperature on quantum dot sensit...TELKOMNIKA JOURNAL
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A progressive learning for structural tolerance online sequential extreme lea...TELKOMNIKA JOURNAL
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Radial basis function neural network control for parallel spatial robot
1. TELKOMNIKA Telecommunication, Computing, Electronics and Control
Vol. 18, No. 6, December 2020, pp. 3191~3201
ISSN: 1693-6930, accredited First Grade by Kemenristekdikti, Decree No: 21/E/KPT/2018
DOI: 10.12928/TELKOMNIKA.v18i6.14913 3191
Journal homepage: http://journal.uad.ac.id/index.php/TELKOMNIKA
Radial basis function neural network control for parallel spatial
robot
Nguyen Hong Quang1
, Nguyen Van Quyen2
, Nguyen Nhu Hien3
1,3
Thai Nguyen University of Technology, Vietnam
2
Hanoi University of Science and Technology, Vietnam
Article Info ABSTRACT
Article history:
Received Dec 9, 2019
Revised May 30, 2020
Accepted Jun 25, 2020
The derivation of motion equations of constrained spatial multibody system is
an important problem of dynamics and control of parallel robots. The paper
firstly presents an overview of the calculating the torque of the driving stages
of the parallel robots using Kronecker product. The main content of this paper
is to derive the inverse dynamics controllers based on the radial basis function
(RBF) neural network control law for parallel robot manipulators. Finally,
numerical simulation of the inverse dynamics controller for a 3-RRR delta
robot manipulator is presented as an illustrative example.Keywords:
Inverse dynamics controller
Kronecker product
Numerical simulation
Parallel robot manipulator
RBF neural network control This is an open access article under the CC BY-SA license.
Corresponding Author:
Nguyen Hong Quang,
Thai Nguyen University of Technology,
No. 666, 3/2 Street, Tich Luong Ward, Thai Nguyen City, Vietnam.
Email: quang.nguyenhong@tnut.edu.vn
1. INTRODUCTION
In the past three decades, the theory on dynamics of constrained multibody systems has been
developed to a high degree of maturity [1-4]. The parallel robot manipulators are constrained multibody
structures [5-7]. The equations of motion for a multibody system are obtained as the end result of a sequence
of mathematical operators. In general, the known methods to derive the equations of motion of multibody
systems are Lagrange’s equations, Newton–Euler equations, Kane’s equations. Among these methods,
the approach using Lagrange’s equations with multipliers has become an attractive method to derive
the equations of motion of constrained multibody systems. This approach provides a well analytical and orderly
structure that is very useful for control purposes.
The control of treelike multibody systems is of interest to a number of research communities in a very
of applications areas. Many advanced methods for control of robot manipulators based on the Lagrange’s
equations have been developed [8-19]. The application of modern control methods such as sliding mode control
method, the neural network control method for controller design of the treelike robot manipulators is presented
in the works [20-30]. In contrast to the rapid progress in control theory of treelike robot manipulators,
the development of the control theory for parallel robots is still limited. Modern control methods have also
been used in the control problem of plane parallel manipulators [31-34]. One has used the control methods
such as the proportional derivative (PD) control and proportional integral derivative (PID) control for designing
some controllers of spatial parallel robot manipulators [35-38]. However, the application of modern control
methods such as sliding mode control method, the radial basis function (RBF) neural network control method
for controller design of the spatial parallel robot manipulators is a new problem that has not been investigated.
2. ISSN: 1693-6930
TELKOMNIKA Telecommun Comput El Control, Vol. 18, No. 6, December 2020: 3191 - 3201
3192
Recently, N. H. Quang [39-41] proposed a control method using model predictive approach.
AL-Azzawi [42] address the control problem for a class of nonlinear dynamical systems based on linear
feedback control strategies. S. Riache [43] proposed adaptive nonsingular terminal super-twisting controller
consists of the hybridization of a nonsingular terminal sliding mode control and an adaptive super twisting.
Simulations with nonsingular terminal super-twisting control to prove the superiority and the effectiveness of
the proposed approach. In [44], a new compound hierarchical sliding mode control and fuzzy logic control
scheme has been proposed for a class of underactuated systems with mismatched uncertainties.
In the present study, we present a control method using neural network for controller design of spatial
parallel robot manipulators. In the section 2, the application of the new matrix form of Lagrangian equations
with multipliers for constrained multibody systems to establish a new expression for calculation of the driving
torques of parallel robots will be discussed. The inverse dynamics controller for the parallel robot manipulator
is considered in the section 3. In the section 4, numerical simulation of the inverse dynamics controller for a
3-RRR delta parallel spatial robot manipulator is presented as an illustrative example.
2. INVERSE DYNAMICS OF CONSTRAINED MULTIBODY SYSTEMS
Let us consider a scleronomic multibody system of a
f n degree of freedom containing 𝑝
rigid-bodies with r holonomic constraints. Let 𝑠 = [𝑠1, 𝑠2, . . . , 𝑠 𝑛] 𝑇
be the vector of generalized coordinates,
the motion equations of constrained holonomic multibody systems can be written as:
𝑀(𝑠)𝑠̈ + 𝐶(𝑠, 𝑠̇) 𝑠̇ + 𝑔(𝑠) + 𝛷𝑠
𝑇
(𝑠)𝜆 + 𝑑(𝑠, 𝑠̇) = 𝜏 (1)
𝑓(𝑠) = 0 (2)
where 𝑀(𝑠) is the n n mass matrix, 𝐶(𝑠, 𝑠̇) is the n n coriolis/centripetal matrix, 𝑓 is 1r vector of
constraint equations, 𝛷𝑠(𝑠) is the r n Jacobian matrix of the vector 𝑓 ,d is the 1n vector of friction force
and disturbance, is the 1n vector of driving forces/torques, is the 1r vector of Lagrangian
multipliers. The Coriolis/Centripetal matrix 𝐶(𝑠, 𝑠̇) is determined from the mass matrix according the following
formula [45, 46].
𝐶(𝑠, 𝑠̇) =
𝜕𝑀(𝑠)
𝜕𝑠
(𝐸 𝑛 ⊗ 𝑠̇) −
1
2
[
𝜕𝑀(𝑠)
𝜕𝑠
(𝑠̇ ⊗ 𝐸 𝑛)]
𝑇
(3)
The Jacobian matrix 𝛷𝑠(𝑠) of the constrained equations is determined by the following formula;
𝛷𝑠 =
𝜕𝑓
𝜕𝑠
= [
𝜕𝑓1
𝜕𝑠1
. . .
𝜕𝑓1
𝜕𝑠 𝑛
. . . . . . . . .
𝜕𝑓𝑟
𝜕𝑠1
. . .
𝜕𝑓𝑟
𝜕𝑠 𝑛
] (4)
Firstly, the generalized coordinates in vectors are divided into two subgroups: independent coordinates 𝑞 𝑎, and
redundant coordinates 𝑧. Then we have;
𝑠 = [𝑞 𝑎
𝑇
𝑧 𝑇] 𝑇
, 11
, ......
T
r
T
a f
z zq qq z , 𝑛 = 𝑓 + 𝑟 (5)
By differentiating in (2) with respect to vectors 𝑠, 𝑞 𝑎, 𝑧, respectively, we obtain the following Jacobian matrices,
, ,r n r r r f
s z a
a
f f f
s z q
, 𝛷𝑠 = [𝛷𝑎 𝛷𝑧] (6)
By introducing the projection matrix [47]:
𝑅(𝑠) = [
𝐸
−𝛷𝑧
−1
𝛷𝑎
] ∈ 𝑅 𝑛×𝑓
(7)
one has:
𝑅 𝑇(𝑠)𝛷𝑠
𝑇
(𝑠) = 0, (8)
3. TELKOMNIKA Telecommun Comput El Control
Radial basis function neural network control for parallel spatial robot (Nguyen Hong Quang)
3193
where E is the 𝑓 × 𝑓 identity matrix.
Left multiplication of the motion in (1) with the matrix T
R s yields,
𝑅 𝑇[𝑀(𝑠)𝑠̈ + 𝐶(𝑠, 𝑠̇) 𝑠̇ + 𝑔(𝑠) + 𝛷𝑠
𝑇(𝑠)𝜆 + 𝑑(𝑠, 𝑠̇)] = 𝑅 𝑇
[
𝜏 𝑎
𝜏 𝑧
]
= [𝐸 [𝛷𝑧
−1
𝛷𝑎(𝑠)] 𝑇] [
𝜏 𝑎
𝜏 𝑧
]
= 𝜏 𝑎 − [𝛷𝑧
−1
𝛷𝑎(𝑠)] 𝑇
𝜏 𝑧
(9)
where a
is the vector of the driving forces/torques in active joints and 𝜏 𝑧 is the vector of the forces/torques in
passive joints. Making use of in (8) and assuming that 𝜏 𝑧 = 0, the driving torques can be deduced from (9) as,
𝜏 𝑎 = 𝑅 𝑇
[𝑀(𝑠)𝑠̈ + 𝐶(𝑠, 𝑠̇) 𝑠̇ + 𝑔(𝑠) + 𝑑(𝑠, 𝑠̇)] (10)
3. ADAPTIVE RBF NEURAL NETWORK CONTROL BASED ON INVERSE DYNAMICS FOR
PARALLEL ROBOTS
3.1. Transformation of motion equations
To study the stability of the control algorithms, the motion equations of parallel robots are transformed into
a suitable form. Let us consider a scleronomic constrained multibody system. From the constrained in (2) we get;
( ) ( , ) ,a a a z
f s f q z 0 f q z 0 (11)
Assuming that the Jacobian matrix 𝛷𝑧 is nonsingular, det (𝛷𝑧) 0 . From (11) one may obtain,
𝑧̇ = −𝛷𝑧
−1
𝛷𝑎 𝑞̇ 𝑎 (12)
It is noted that,
𝑞 𝑎 = 𝐸𝑞 𝑎 (13)
Combining (12) with (13) yields the following differential equation:
𝑠̇ = 𝑅(𝑠)𝑞̇ 𝑎 (14)
Differentiating in (14) with respect to time gives the acceleration relation as;
𝑠̈ = 𝑅(𝑠)𝑞̈ 𝑎 + 𝑅̇(𝑠, 𝑠̇) 𝑞̇ 𝑎 = 𝑅(𝑠)𝑞̈ 𝑎 +
𝜕𝑅(𝑠)
𝜕𝑠
(𝐸 𝑝 ⊗ 𝑠̇) 𝑞̇ 𝑎 (15)
Substituting in (14) and (15) into to (9) yields;
( )
( ) ( ( ) ( ) ) ( ) + ( , ),a p a a a
T s
s s q E s q s q d s s
s
R
R M s R C s s R g s (16)
To simplify the description, we define;
𝑀̄ (𝑠) ≔ 𝑅 𝑇(𝑠)𝑀(𝑠)𝑅(𝑠)𝐶̄(𝑠, 𝑠̇)
≔ 𝑅 𝑇(𝑠) [𝑀(𝑠)
𝜕𝑅(𝑠)
𝜕𝑠
(𝐸 𝑝 ⊗ 𝑠̇) + 𝐶(𝑠, 𝑠̇) 𝑅(𝑠)] 𝑔̄( 𝑠):
= 𝑅 𝑇(𝑠)𝑔(𝑠) 𝑑̄(𝑠, 𝑠̇): = 𝑅 𝑇
(𝑠)𝑑(𝑠, 𝑠̇)
(17)
In (9) and (11) now can be rewritten as follows;
𝑀̄ (𝑠)𝑞̈ 𝑎 + 𝐶̄(𝑠, 𝑠̇) 𝑞̇ 𝑎 + 𝑔̄( 𝑠) + 𝑑̄(𝑠, 𝑠̇) = 𝜏 𝑎 (18)
𝑓(𝑠) = 0 (19)
The motion equations of parallel robots (18) and (19) are called the motion equations in mixture form.
Where s is the vector of redundant generalized coordinates and 𝑞 𝑎 is the vector of independent coordinates.
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We will use this equation as the basis for designing the controller for parallel robots. For this purpose, we prove
the following properties [33]:
− 𝑀̄ is a symmetric positive definite matrix: 𝑀̄ 𝑇
= 𝑀̄ ,
− 𝑀̄̇ − 2𝐶̄ is a skew-symmetric matrix: (𝑀̄̇ − 2𝐶̄) 𝑇
= −(𝑀̄̇ − 2𝐶̄).
Due to the symmetry of the matrix M is symmetric, one has;
𝑀̄ 𝑇(𝑠) = [𝑅 𝑇
(𝑠)𝑀(𝑠)𝑅(𝑠)] 𝑇
= 𝑅 𝑇
(𝑠)𝑀(𝑠)𝑅(𝑠) = 𝑀̄ (𝑠)
Since 𝑀(𝑠) is positive definite, 𝑀̄ (𝑠) is also a positive definite matrix. Using the (17), one obtains;
𝑀̄̇ (𝑠) − 2𝐶̄(𝑠, 𝑠̇) = 𝑅̇ 𝑇
𝑀𝑅 + 𝑅 𝑇
𝑀̇ 𝑅 + 𝑅 𝑇
𝑀𝑅̇ − 2𝑅 𝑇
(𝑀𝑅̇ + 𝐶𝑅)
= 𝑅̇ 𝑇
𝑀𝑅 + 𝑅 𝑇
𝑀̇ 𝑅 − 𝑅 𝑇
𝑀𝑅̇ − 2𝑅 𝑇
𝐶𝑅
= 𝑅 𝑇
(𝑀̇ − 2𝐶)𝑅 − 𝑅 𝑇
𝑀𝑅̇ + 𝑅̇ 𝑇
𝑀𝑅 (20)
Since 𝑀̇ − 2𝐶 is skew-symmetric [8], from in (20) one has;
[𝑀̄̇ (𝑠) − 2𝐶̄(𝑠, 𝑠̇)]
𝑇
= [𝑅 𝑇
(𝑀̇ − 2𝐶)𝑅]
𝑇
− (𝑅 𝑇
𝑀𝑅̇) 𝑇
+ (𝑅̇ 𝑇
𝑀𝑅) 𝑇
= −𝑅 𝑇
(𝑀̇ − 2𝐶)𝑅 − 𝑅̇ 𝑇
𝑀𝑅 + 𝑅 𝑇
𝑀𝑅̇ = −[𝑀̄̇ (𝑠) − 2𝐶̄(𝑠, 𝑠̇)] (21)
Thus, 𝑀̄̇ (𝑠) − 2𝐶̄(𝑠, 𝑠̇) is a skew symmetric matrix.
3.2. RBF neural network control law and stability analysis
In practice, the perfect robot model could be difficult to obtain, and external disturbances are always
present in practice. The uncertain motion equations of parallel robots with a
n f active joints (18) can be
described in the following form;
𝑀̂(𝑠)𝑞̈ 𝑎 + 𝐶̂(𝑠, 𝑠̇) 𝑞̇ 𝑎 + 𝑔̂(𝑠) + 𝑑̂(𝑠, 𝑠̇) = 𝜏 𝑎 (22)
where 𝑀̂(𝑠) is an f f inertia matix, 𝐶̂(𝑠, 𝑠̇) is an f f matrix containing the centrifugal and Coriolis terms,
ˆ( )g s is an 1f vector containing gravitational forces and torques, s is the vector of generalized coordinates,
a
q is active joint coordinates, and ˆd denotes disturbances. It is supposed that,
𝑀̂(𝑠) = 𝑀̄ (𝑠) + 𝛥𝑀̄ (𝑠)
𝐶̂(𝑠, 𝑠̇) = 𝐶̄(𝑠, 𝑠̇) + 𝛥𝐶̄(𝑠, 𝑠̇) (23)
𝑔̂(𝑠) = 𝑔̄( 𝑠) + 𝛥𝑔̄( 𝑠)
𝑑̂(𝑠, 𝑠̇) = 𝑑̄(𝑠, 𝑠̇) + 𝛥𝑑̄(𝑠, 𝑠̇)
where , , ,M C g d are the prior-known components and , , , M C g d are modeling errors of
ˆ ˆ ˆ, , M C g and ˆd respectively. Assume that the modeling errors are bounded by some finite constants as;
0 0 0 0
, , d,m c gM C g d (24)
where 0 0 0 0
, , ,m g dc are known constants. Substituting in (23) into to (22) yields.
(𝑀̄ + 𝛥𝑀̄ )𝑞̈ 𝑎 + (𝐶̄ + 𝛥𝐶̄)𝑞̇ 𝑎 + 𝑔̄ + 𝛥𝑔̄ + 𝑑̄ + 𝛥𝑑̄ = 𝜏 𝑎 (25)
From (25) one has;
𝑀̄ (𝑠)𝑞̈ 𝑎 + 𝐶̄(𝑠, 𝑠̇) 𝑞̇ 𝑎 + 𝑔̄( 𝑠) + 𝑑̄(𝑠, 𝑠̇) + ℎ̄(𝑠, 𝑠̇) = 𝜏 𝑎 (26)
where ℎ̄(𝑠, 𝑠̇) is the sum of unknown terms of the dynamic system.
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ℎ̄(𝑠, 𝑠̇) = 𝛥𝑀̄ 𝑞̈ 𝑎 + 𝛥𝐶̄ 𝑞̇ 𝑎 + 𝛥𝑔̄ + 𝛥𝑑̄ (27)
Assume that‖ℎ̄(𝑠, 𝑠̇)‖ ≤ ℎ0. The sliding mode function is selected as;
𝑛(𝑡) = 𝑒̇ 𝑎(𝑡) + 𝛬𝑒 𝑎(𝑡) (28)
where is the positive diagonal matrix.
1 2 na i
diag , , , , 0 ; 1,2, , a
i n (29)
In (28) we define,
d
a a a
t t te q q (30)
where 𝑞 𝑎
𝑑(𝑡) is the vector of desired trajectory and 𝑞 𝑎(𝑡) is the vector of real trajectory. The function ℎ̄(𝑠, 𝑠̇)
can be rewritten as:
ℎ̄(𝑛): = ℎ̄(𝑠, 𝑠̇) (31)
The function ℎ̄(𝑛) is the main reason for the degradation of the control quality. If this effect is compensated,
the control accuracy can then be improved. According to Stone-Weierstrass theorem [23-24] one can choose
an appropriate artifical neural network (ANN) with a limited number of neurals that can approximate an
unknown nonlinear function with a given accuracy. For approximating function ℎ̄(𝑛)we choose the following
simple structure ANN:
ℎ̄(𝑛) = 𝑊𝑠 + 𝑒 = ℎ̂(𝑛) + 𝑒 (32)
where 𝑊 is the 𝑛 𝑎 × 𝑛 𝑎 matrix, ℎ̂(𝑛) = [ℎ̂1, ℎ̂2, . . , ℎ̂ 𝑛𝑎]
𝑇
= 𝑊𝑠 is the approximation of ℎ̄(𝑛), 𝑒 is
the approximation error. If ‖ℎ̄(𝑛)‖ ≤ ℎ0, we have ‖𝑒‖ ≤ 𝜀0. Assuming that the matrix 𝑊 has 𝑛 𝑎 column
vectors 𝑤𝑖, we have;
ℎ̂ = [ℎ̂1, ℎ̂2, . . , ℎ̂ 𝑛𝑎]
𝑇
= 𝑊𝜎 = ∑ 𝑛 𝑎
𝑖=1 𝜎𝑖 𝑤𝑖 (33)
In this paper, the radial basis function (RBF) neural network was used as shown in Figure 1. This structure has
been proved to satisfy the Stone-Weierstrass theorem [23]. If we choose the Gaussian activation function 𝜎𝑖
according to the formula,
𝜎𝑖 = 𝑒𝑥𝑝 [−
‖𝒗−𝒄 𝑖‖2
𝜒𝑖
2 ] (34)
where the vector 𝑐𝑖 represents the coordinate value of the center point of the Gaussian function of neural net 𝑖,
and 𝜒𝑖 is derivation parameter which is freely choosen, the function approximation ˆh has the following form;
1
ˆ , 1..,
a
j
n
i j a
j
i
h i nw (35)
where 𝑤𝑗𝑖 are the weights to be updated of the approximating neural network.
Figure 1. RBF neural network structure
1
ˆh
2
ˆh
ˆ
na
h
1
2
na
11
w
12
w
2na
w
1na
w
1
2
na
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The control problem is now to find the control torque u and learning algorithm of 𝑤𝑗𝑖 of the neural
network so that 𝑛 → 0 and position errors 𝑒 → 0, granting 𝑞 𝑎(𝑡) → 𝑞 𝑎
𝑑(𝑡).
Theorem: The trajectory 𝑞 𝑎(𝑡) of dynamic system defined by (3.34) with RFB neural network
according to (3.60), (3.62), and the sliding surface (3.40) will track the desired trajectory 𝑞 𝑎
𝑑(𝑡) with error
𝑒(𝑡) = 𝑞 𝑎(𝑡) − 𝑞 𝑎
𝑑(𝑡) → 0 if the control law u the learning algorithm i
w are chosen as follows;
𝑢 = 𝑀̄ (𝑠)𝑞̈ 𝑎
𝑑
+ 𝐶̄(𝑠, 𝑠̇) 𝑞̇ 𝑎
𝑑
+ 𝑔̄ + 𝑑̄ − 𝑀̄ (𝑠)𝛬𝑒̇ 𝑎 − 𝐶̄(𝑠, 𝑠̇) 𝛬𝑒 𝑎 − 𝐾𝑛 − 𝛾
𝑛
‖𝑛‖
+ (1 + 𝜂)𝑊𝑠 (36)
𝑤̇ 𝑖 = −𝜂𝜎𝑖 𝑛 (37)
where 𝐾 is a 𝑛 𝑎 × 𝑛 𝑎 symmetric positive matrix, and 𝜂 > 0, 𝛾 > 0. Noting that the states 𝑞 𝑎, 𝑞̇ 𝑎 in the control
law (36) are measured.
Proof: This theorem can be proved using the Lyapunov direct method. We choose the Lyapunov
function as;
𝑉(𝑡) =
1
2
[𝑛 𝑇
𝑀̄ 𝑛 + ∑ 𝑛 𝑎
𝑖=1 𝑤𝑖
𝑇
𝑤𝑖] (38)
Since 𝑀̄ (𝑠) is symmetric and positive definite 𝑉(𝑡) > 0 for 𝑛 ≠ 0, 𝑤𝑖 ≠ 0 and 𝑉(𝑡) = 0 if and only if
𝑛 = 0, 𝑤𝑖 = 0.
The derivative of the function 𝑉(𝑡) is,
𝑉̇ (𝑡) = 𝑛 𝑇
𝑀̄ 𝑛̇ +
1
2
𝑛 𝑇
𝑀̄̇ 𝑛 + ∑ 𝑛 𝑎
𝑖=1 𝑤𝑖
𝑇
𝑤̇ 𝑖 (39)
Using the skew-symmetry of the matrix 𝑀̄ (𝑠) − 2𝐶̄(𝑠, 𝑠̇), we have
𝑛 𝑇
(𝑀̄̇ − 2𝐶̄) 𝑛 = 0 → 𝑛 𝑇
𝑀̄̇ 𝑛 = 2𝑛 𝑇
𝐶̄ 𝑛 (40)
Substituting (40) into (39) gives:
𝑉̇ (𝑡) = 𝑛 𝑇(𝑀̄ 𝑛̇ + 𝐶̄ 𝑛) + ∑ 𝑛 𝑎
𝑖=1 𝑤𝑖
𝑇
𝑤̇ 𝑖 (41)
If we choose a
u , from (38) and (26) one has,
𝑀̄ 𝑛̇ + 𝐶̄ 𝑛 = − [𝐾𝑛 + 𝛾
𝑛
‖𝑛‖
− (1 + 𝜂)𝑊𝑠 + ℎ̄(𝑛)] (42)
Substituting in (42) into to (41) yields,
𝑉̇ (𝑡) = 𝑛 𝑇
[−𝐾𝑛 − 𝛾
𝑛
||𝑛||
+ 𝜂𝑊𝜎 − 𝑒] + ∑ 𝑛 𝑎
𝑖=1 𝑤𝑖
𝑇
𝑤̇ 𝑖 (43)
Using the learning algorithm (37), the last term in (43) has the following form,
∑ 𝑛 𝑎
𝑖=1 𝑤𝑖
𝑇
𝑤̇ 𝑖 = −𝜂 ∑ 𝑛 𝑎
𝑖=1 𝑤𝑖
𝑇
𝑛𝜎𝑖 = −𝜂𝑛 𝑇
𝑤𝑠 (44)
Substituting (44) into to (43) yields,
𝑉̇ (𝑡) = −𝑛 𝑇
𝐾𝑛 − 𝛾
𝑛 𝑇 𝑛
‖𝑛‖
− 𝑛 𝑇
𝑒 (45)
If we select 0
, and 0 , one obtains: 𝑉̇ (𝑡) = −𝑛 𝑇
𝐾𝑛 − 𝛿‖𝑛‖ − (𝜀0‖𝑛‖ + 𝑛 𝑇
𝑒) (46)
Since ‖𝑒‖ < 𝜀0, 𝑉̇ (𝑡) < 0 for all 𝑛 ≠ 0, and 𝑉̇ (𝑡) = 0 if and only if 𝑛 = 0. It follows from Lyapunov’s theory
that the system is asymptotically stable, or 𝑛 → 0 as 𝑡 → ∞, therefore,
t d
a a a
t te q q 0 (47)
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4. SIMULATION EXAMPLE
From the RBF neural network control law presented in the above section, the resulting block scheme
is illustrated in the Figure 2. For the use of the control lawu based on (36) and (37), actual signals 𝑞 𝑎, 𝑞̇ 𝑎, 𝑞̈ 𝑎
are assumed to be known. Therefore, one can obtain actual values of generalized coordinates, velocities and
accelerations 𝑠, 𝑠̇, 𝑠̈. The numerical simulation may be a possible way to suggest an alternative choice for actual
values of generalized coordinates, velocities and accelerations. Considering the dynamic equations of parallel
robot manipulator as;
𝑀(𝑠)𝑠̈ + 𝐶(𝑠, 𝑠̇) 𝑠̇ + 𝑔(𝑠) + 𝛷𝑠
𝑇
(𝑠)𝜆 + 𝑑(𝑠, 𝑠̇) = 𝜏 (48)
𝑓(𝑠) = 0 (49)
where,
𝜏 = 𝜏(𝑡, 𝑠, 𝑠̇, 𝑠 𝑑
, 𝑠̇ 𝑑
) = [𝑢 𝑇
, 0 𝑇
] 𝑇
(50)
Differentiating in (49) with respect to time gives,
𝑓̇(𝑠) =
𝜕𝑓
𝜕𝑠
𝑠̇ = 𝛷𝑠 𝑠̇ = 0 (51)
𝑓̈(𝑠, 𝑠̇) = 𝛷𝑠 𝑠̈ + 𝛷̇ 𝑠 𝑠̇ = 0 (52)
where [45, 46]:
𝛷̇ 𝑠(𝑠) =
𝜕𝛷
𝜕𝑠
(𝐸 𝑛 ⊗ 𝑠̇) (53)
Define,
𝑝1(𝑠, 𝑠̇, 𝑠 𝑑
, 𝑠̇ 𝑑
, 𝑡) = 𝜏 − 𝐶(𝑠, 𝑠̇) 𝑠̇ − 𝑔(𝑠) − 𝑑(𝑠, 𝑠̇) (54)
𝑝2(𝑠, 𝑠̇, 𝑡) = −𝛷̇ 𝑠(𝑠)𝑠̇ = − [
𝜕𝛷 𝑠
𝜕𝑠
(𝐸 𝑛 ⊗ 𝑠̇)] 𝑠̇ (55)
In (48) and (52) now can be written in the following form,
𝑀(𝑠)𝑠̈ + 𝛷𝑠
𝑇(𝑠)𝜆 = 𝑝1(𝑠, 𝑠̇, 𝑠 𝑑
, 𝑠̇ 𝑑
, 𝑡) (56)
𝛷𝑠(𝑠)𝑠̈ = 𝑝2(𝑠, 𝑠̇, 𝑡) (57)
Left multiplication of (56) with the matrix 𝑅 𝑇
yields,
𝑅 𝑇(𝑠)𝑀(𝑠)𝑠̈ + 𝑅 𝑇(𝑠)𝛷𝑠
𝑇(𝑠)𝜆 = 𝑅 𝑇(𝑠)𝑝1(𝑠, 𝑠̇, 𝑡) (58)
According to (8), (58) becomes:
𝑅 𝑇(𝑠)𝑀(𝑠)𝑠̈ = 𝑅 𝑇(𝑠)𝑝1(𝑠, 𝑠̇, 𝑠 𝑑
, 𝑠̇ 𝑑
, 𝑡) (59)
In (59) is a system of f second-order differential equations. Combining in (59) with (57) yields,
[
𝑅 𝑇
(𝑠)𝑀(𝑠)
𝛷𝑠(𝑠)
] 𝑠̈ = [
𝑅 𝑇
(𝑠)𝑝1
𝑝2
] (60)
If the matrix,
𝐴(𝑠) = [
𝑅 𝑇(𝑠)𝑀(𝑠)
𝛷𝑠(𝑠)
] (61)
is nonsigular, from (60) one obtains the following diferential equation system,
𝑠̈ = 𝑠̈( 𝑠, 𝑠̇, 𝑠 𝑑
, 𝑠̇ 𝑑
, 𝑡) (62)
Then, solving the (62) we find 𝑠, 𝑠̇ [48]. Therefore we can calculate control law according to (36).
8. ISSN: 1693-6930
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3198
Figure 2. Block scheme of joint space control
A 3-RRR spatial parallel robot manipulator shown in Figure 3 is utilized in this study to verify
the effectiveness of the proposed control scheme. The mechanical model for the 3-RRR delta robot manipulator
is a system of rigid bodies connected by joints as Figure 4. The parallelogram mechanisms that connect the
driving links to the mobile platform are modeled as homogeneous rods with universal and spherical joints at
two ends. From Figures 4 and 5 it is followed that the configuration of the 3-RRR delta spatial parallel robot
manipulator is represented by a vector of generalized coordinates as:
1 2 3 1 2 3
,
T
P P P
x y zs
The differential-algebraic equations of the system are given in the Appendix. The kinematic and dynamic
parameters of the robot manipulator are given in the Table 1. In the simulation, the center of the moving
platform will be controlled to track the given trajectory defined by,
0.3cos 2 ; 0.3sin 2 ; 0.7 ( )P P P
x t y t z m
The parameters of the neural network control law are chosen as follows,
1 2 3 1 2 3
80,80,80 ; (80,80,80); 1.1; 200;
1; 2; 3; 0.01; 0.02; 0.03;
diag diag
c c c
K L
Figure 3. Delta robot with three parallelogram mechanisms
Figure 4. Model of 3-RRR Delta robot Figure 5. Position of the i i
B D rod in the space
Inverse
kinematics
Control law Parallel
robot
( ), (t)d d
ts s ( ), (t)ts s( )tu( )d
tx
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Table 1. The parameters of the Delta robot in Figure 3
𝐿1 𝐿2 𝑅 𝑟 r 𝛼1 1
𝛼2 𝛼3 𝑚1 1
m 𝑚2 2
m 𝑚 𝑃
0.3 (𝑚) 0.8 (𝑚) 0.266 (𝑚) 0.04 (𝑚) 0 (𝑟𝑎𝑑)
2𝜋
3
(𝑟𝑎𝑑)
4𝜋
3
(𝑟𝑎𝑑) 0.42 (𝑘𝑔) 2 ⋅ 0.2 (𝑘𝑔) 0.75 (𝑘𝑔)
In this paper the modeling errors are chosen to be 20% of the prior-known values of the nominal
model as,
𝛥𝑀(𝑠) = 20%𝑀(𝑠); 𝛥𝐶(𝑠, 𝑠̇) = 20%𝐶(𝑠, 𝑠̇); 𝛥𝑔(𝑠) = 20%𝑔(𝑠)
The disturbance vector is chosen as 1 1 6 6
sin20 cos20 ... sin20 cos20
T
t t t td . Some simulation
results are given in the Figures from 6 to 9. The position errors of the moving platform are shown in
Figures 6 and 7. The control torques are shown in Figures 8 and 9. The stationary errors in position of
the platform are kept about 10-4
mm.
Figure 6. Position errors of the moving platform without the modeling errors and disturbance
Figure 7. Position errors of the moving platform with the modeling errors and disturbance
Figure 8. Control torques without the modeling errors and disturbance
10. ISSN: 1693-6930
TELKOMNIKA Telecommun Comput El Control, Vol. 18, No. 6, December 2020: 3191 - 3201
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Figure 9. Control torques with the modeling errors and disturbance
5. CONCLUSIONS
Many modern methods for control of robot manipulators based on the Lagrangian multipliers have been
developed. In contrast to the rapid progress in control theory of treelike robot manipulators, the development of
the modern control theory for parallel robot manipulators is still limited. This paper presented the application of
the RBF neural network control law to compensate uncertainties in the parallel robot manipulators. The new matrix
form of Lagrangian equations with multipliers for constrained multibody systems was used to derive dynamic
equations of spatial parallel robot manipulators based on computer software packages. Using adaptive RBF neural
network control method, the controller for spatial robot manipulators based on inverse dynamics was developed.
The stability of the control law using adaptive RBF neural network method for the control problem of spatial robot
manipulators based on inverse dynamics was proven. Using Simulink program, the numerical simulation of
the adaptive RBF neural network controller for a 3-PRR spatial parallel robot manipulator is studied. The application
of modern methods for motion control of the constrained spatial multibody systems and spatial parallel robot
manipulators will be presented in other works.
ACKNOWLEDGEMENTS
This research was supported by Research Foundation funded by Thai Nguyen University of Technology.
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