Search results for: motion controller

Authors: Salman Hameed

Abstract:

In this paper, a self-tuning fuzzy PI controller (STFPIC) is proposed for thyristor controlled series capacitor (TCSC) to improve power system dynamic performance. In a STFPIC controller, the output scaling factor is adjusted on-line by an updating factor (α). The value of α is determined from a fuzzy rule-base defined on error (e) and change of error (Δe) of the controlled variable. The proposed self-tuning controller is designed using a very simple control rule-base and the most natural and unbiased membership functions (MFs) (symmetric triangles with equal base and 50% overlap with neighboring MFs). The comparative performances of the proposed STFPIC and the standard fuzzy PI controller (FPIC) have been investigated on a multi-machine power system (namely, 4 machine two area system) through detailed non-linear simulation studies using MATLAB/SIMULINK. From the simulation studies it has been found out that for damping oscillations, the performance of the proposed STFPIC is better than that obtained by the standard FPIC. Moreover, the proposed STFPIC as well as the FPIC have been found to be quite effective in damping oscillations over a wide range of operating conditions and are quite effective in enhancing the power carrying capability of the power system significantly.

Keywords: genetic algorithm, power system stability, self-tuning fuzzy controller, thyristor controlled series capacitor

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Authors: P. Sawatnatee, S. Udomsuk, K-N. Areerak, K-L. Areerak, A. Srikaew

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The paper presents how to design the indirect vector control of three-phase induction motor drive systems using the artificial intelligence technique called the adaptive tabu search. The results from the simulation and the experiment show that the drive system with the controller designed from the proposed method can provide the best output speed response compared with those of the conventional method. The controller design using the proposed technique can be used to create the software package for engineers to achieve the optimal controller design of the induction motor speed control based on the indirect vector concept.

Keywords: indirect vector control, induction motor, adaptive tabu search, control design, artificial intelligence

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Authors: Jingjing Lan, Jun Yu, Muthukumaraswamy Annamalai Arasu

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This paper presents a digital non-linear pulse-width modulation (PWM) controller in a high-voltage (HV) buck-boost DC-DC converter for the piezoelectric transducer of the down-hole acoustic telemetry system. The proposed design controls the generation of output signal with voltage higher than the supply voltage and is targeted to work under high temperature. To minimize the power consumption and silicon area, a simple and efficient design scheme is employed to develop the PWM controller. The proposed PWM controller consists of serial to parallel (S2P) converter, data assign block, a mode and duty cycle controller (MDC), linearly PWM (LPWM) and noise shaper, pulse generator and clock generator. To improve the reliability of circuit operation at higher temperature, this design is fabricated with the 1.0-μm silicon-on-insulator (SOI) CMOS process. The implementation results validated that the proposed design has the advantages of smaller size, lower power consumption and robust thermal stability.

Keywords: DC-DC power conversion, digital control, high temperatures, pulse-width modulation

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Authors: A. Elnady

Abstract:

This paper presents an improved Direct Power Control (DPC) scheme applied to the multilevel inverter that forms a Distributed Generation Unit (DGU). This paper demonstrates the performance of active and reactive power injected by the DGU to the smart grid. The DPC is traditionally operated by the hysteresis controller with the Space Vector Modulation (SVM) which is applied on the 2-level inverters or 3-level inverters. In this paper, the DPC is operated by the PI controller with the Phase-Disposition Pulse Width Modulation (PD-PWM) applied to the 5-level diode clamped inverter. The new combination of the DPC, PI controller, PD-PWM and multilevel inverter proves that its performance is much better than the conventional hysteresis-SVM based DPC. Simulations results have been presented to validate the performance of the suggested control scheme in the grid-connected mode.

Keywords: direct power control, PI controller, PD-PWM, and power control

Procedia PDF Downloads 240

Authors: Martins Olatunbosun Babatunde, Muhammed Bashir Muazu, Emmanuel Adewale Adedokun

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This paper presents the development of a double-loop trajectory-tracking controller for the ball and plate system (BPS) using the Normalized Right Coprime Factorization (NRCF) scheme.The Linear Algebraic (LA) method is used to design the inner loop required to stabilize the ball, while H-infinity NRCF method, that involved the lead-lag compensator design approach, is used to develop the outer loop that controls the plate. Simulation results show that the plate was stabilized at 0.2989 seconds and the ball was able to settle after 0.9646 seconds, with a trajectory tracking error of 0.0036. This shows that the controller has good adaptability and robustness.

Keywords: ball and plate system, normalized right coprime factorization, linear algebraic method, compensator, controller, tracking.

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Authors: Raunak Munjal, Mohammad Akif Ali, Prithiv Raj

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This study investigates the relative effectiveness of various flight controllers for drone obstacle avoidance. To assess ultrasonic sensors' performance in real-time obstacle detection, they are integrated with ESP32 and Arduino Nano controllers. The study determines which controller is most effective for this particular application by analyzing important parameters such as accuracy (mean absolute error), standard deviation, and mean distance range. Furthermore, the study explores the possibility of incorporating state-driven algorithms into the Arduino Nano configuration to potentially improve obstacle detection performance. The results offer significant perspectives for enhancing sensor integration, choosing the best flight controller for obstacle avoidance, and maybe enhancing drones' general environmental navigation ability.

Keywords: ultrasonic distance measurement, accuracy and consistency, flight controller comparisons, ESP32 vs arduino nano

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Authors: T. Vamsee Kiran, A. Gopi

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The undesired torque and flux ripple may occur in conventional direct torque control (DTC) induction motor drive. DTC can improve the system performance at low speeds by continuously tuning the regulator by adjusting the Kp, Ki values. In this differential evolution (DE) is proposed to adjust the parameters (Kp, Ki) of the speed controller in order to minimize torque ripple, flux ripple, and stator current distortion.The DE based PI controller has resulted is maintaining a constant speed of the motor irrespective of the load torque fluctuations.

Keywords: differential evolution, direct torque control, PI controller

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Authors: Fabiola M. Villalobos-Castaldi, Nicolás C. Kemper, Esther Rojas-Krugger, Laura G. Ramírez-Sánchez

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This paper presents the classification results about exploring the combination of a motion based approach with a local appearance method to describe the facial motion caused by the muscle contractions and expansions that are presented in facial expressions. The proposed feature extraction method take advantage of the knowledge related to which parts of the face reflects the highest deformations, so we selected 4 specific facial regions at which the appearance descriptor were applied. The most common used approaches for feature extraction are the holistic and the local strategies. In this work we present the results of using a local appearance approach estimating the correlation coefficient to the 4 corresponding landmark-localized facial templates of the expression face related to the neutral face. The results let us to probe how the proposed motion estimation scheme based on the local appearance correlation computation can simply and intuitively measure the motion parameters for some of the most relevant facial regions and how these parameters can be used to recognize facial expressions automatically.

Keywords: facial expression recognition system, feature extraction, local-appearance method, motion-based approach

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Authors: Bharti Gupta, Alaukik Sharma

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The first aim is to determine the effect of the Earth's magnetic field on the motion of a conductor to evaluate the variations of the orbital elements of the conductor due to these effects. The effects of Earth's magnetic field on the motion of conductors have been studied at different heights, longitudes and latitudes. When the conductor cut the geomagnetic line of force, then an electro-motive force (EMF) is induced across to the conductor. Due to this induced EMF, an induced current will flow through the conductor. Resulting, a Lorentz force will be applied on the conductor who opposes the motion of the conductor. So our second aim is to determine the accurate value of Induced EMF and induced Lorentz Force at different heights, longitudes and latitudes.

Keywords: induced EMF, Lorentz force, geomagnetic lines of force, moving conductor

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Authors: Meng Wu

Abstract:

Motion planning is a common task required to be fulfilled by robots. A strategy combining Ant Colony Optimization (ACO) and gravity gradient inversion algorithm is proposed for motion planning of mobile robots. In this paper, in order to realize optimal motion planning strategy, the cost function in ACO is designed based on gravity gradient inversion algorithm. The obstacles around mobile robot can cause gravity gradient anomalies; the gradiometer is installed on the mobile robot to detect the gravity gradient anomalies. After obtaining the anomalies, gravity gradient inversion algorithm is employed to calculate relative distance and orientation between mobile robot and obstacles. The relative distance and orientation deduced from gravity gradient inversion algorithm is employed as cost function in ACO algorithm to realize motion planning. The proposed strategy is validated by the simulation and experiment results.

Keywords: motion planning, gravity gradient inversion algorithm, ant colony optimization

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Authors: Suzanne Ahmed

Abstract:

Nanomotors, also commonly referred to as nanorobotics or nanomachines, have garnered considerable research attention due to their numerous potential applications in biomedicine, including drug delivery and microsurgery. Nanomotors typically consist of inorganic or polymeric particles that are powered to undergo motion. These artificial, man-made nanoscale motors operate in the low Reynolds number regime and typically have no moving parts. Several methods have been developed to actuate the motion of nanomotors including magnetic fields, electrical fields, electromagnetic waves, and chemical fuel. Since their introduction in 2012, ultrasonically powered nanomotors have been explored in biocompatible fluids and even within living cells. Due to the common use of ultrasound within the biomedical community for both imaging and therapeutics, the introduction of ultrasonically propelled nanomotors holds significant potential for biomedical applications. In this work, metallic nanomotors are electrochemically plated within porous anodic alumina templates to have a diameter of 300 nm and a length that is 2-4 µm. Nanomotors are placed within an acoustic chamber capable of producing bulk acoustic waves in the ultrasonic range. The motion of nanomotors within biomimetic confines is explored. The control over nanomotor motion is exerted by virtue of the properties of the acoustic signal within these biomimetic confines to control speed, modes of motion and directionality of motion. To expand the range of control over nanorod motion within biomimetic confines, external forces from biocompatible magnetic fields, are exerted onto the acoustically propelled nanomotors.

Keywords: nanomotors, nanomachines, nanorobots, ultrasound

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Authors: Anton V. Doroshin

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This article contains a description of main ideas for the attitude reorientation of spacecraft (small dual-spin spacecraft, nanosatellites) using properties of its chaotic attitude motion under the action of internal perturbations. The considering method based on intentional initiations of chaotic modes of attitude motion with big amplitudes of the nutation oscillations, and also on the redistributions of the angular momentum between coaxial bodies of the dual-spin spacecraft (DSSC), which perform in the purpose of system’s phase space changing.

Keywords: spacecraft, attitude dynamics, control, chaos

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Authors: Samah Riache, Madjid Kidouche

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In this paper, we propose a nonlinear controller for Aerial Manipulator (AM) consists of a Quadrotor equipped with two degrees of freedom robotic arm. The kinematic and dynamic models were developed by considering the aerial manipulator as a coupled system. The proposed controller was designed using Nonsingular Terminal Sliding Mode Control. The objective of our approach is to improve performances and attenuate the chattering drawback using an adaptive algorithm in the discontinuous control part. Simulation results prove the effectiveness of the proposed control strategy compared with Sliding Mode Controller.

Keywords: adaptive algorithm, quadrotor, robotic arm, sliding mode control

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Authors: R. M. Kalayappan, N. Kathiravan

Abstract:

In automobile industries, Controller Area Network (CAN) is widely used to reduce the system complexity and inter-task communication. Therefore, this paper proposes the hardware implementation of data frame communication between one controller to other. The CAN data frames and protocols will be explained deeply, here. The data frames are transferred without any collision or corruption. The simulation is made in the KEIL vision software to display the data transfer between transmitter and receiver in CAN. ARM7 micro-controller is used to transfer data’s between the controllers in real time. Data transfer is verified using the CRO.

Keywords: control area network (CAN), automotive electronic control unit, CAN 2.0, industry

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Authors: Sabir Messalti, Bilal Boudjellal, Azouz Said

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In this paper, power system Voltage improvement using wind turbine is presented. Two controllers are used: a PI controller and Artificial Neural Networks (ANN) controllers are studied to control of the power flow exchanged between the wind turbine and the power system in order to improve the bus voltage. The wind turbine is based on a doubly-fed induction generator (DFIG) controlled by field-oriented control. Indirect control is used to control of the reactive power flow exchanged between the DFIG and the power system. The proposed controllers are tested on power system for large voltage disturbances.

Keywords: artificial neural networks controller, DFIG, field-oriented control, PI controller, power system voltage improvement

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Authors: V. Sankaranarayanan, V. Amrita Sundari, Sunit P. Gopal

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This paper presents the design and implementation of a nonlinear controller for the point to point control of a mobile inverted pendulum (MIP). The controller is designed based on the kinematic model of the MIP to stabilize all the four coordinates. The stability of the closed-loop system is proved using Lyapunov stability theory. The proposed controller is validated through numerical simulations and also implemented in a laboratory prototype. The results are presented to evaluate the performance of the proposed closed loop system.

Keywords: mobile inverted pendulum, switched control, nonlinear systems, lyapunov stability

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Authors: Doo W. Lee, Soo J. Lee, Bye R. Yoon, Jae Y. Jho, Kyehan Rhee

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Hand exoskeletons have been developed in order to assist daily activities for disabled and elder people. A figure exoskeleton was developed using ionic polymer metal composite (IPMC) actuators, and the performance of it was evaluated in this study. In order to study dynamic performance of a finger dummy performing pinching motion, force generating characteristics of an IPMC actuator and pinching motion of a thumb and index finger dummy actuated by IMPC actuators were analyzed. The blocking force of 1.54 N was achieved under 4 V of DC. A thumb and index finger dummy, which has one degree of freedom at the proximal joint of each figure, was manufactured by a three dimensional rapid prototyping. Each figure was actuated by an IPMC actuator, and the maximum fingertip force was 1.18 N. Pinching motion of a dummy was analyzed by two video cameras in vertical top and horizontal left end view planes. A figure dummy powered by IPMC actuators could perform flexion and extension motion of an index figure and a thumb.

Keywords: finger exoskeleton, ionic polymer metal composite, flexion and extension, motion analysis

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Authors: T. D. Tran, H. Schlegel, R. Neugebauer

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Ball screws are an important component in machine tools. In mechatronic systems and machine tools, a ball screw has to work usually at a high speed. Otherwise the axial compliance of the ball screw, in combination with the inertia of the slide, the motor, the coupling and the screw, will cause an oscillation resonance, which limits the systems bandwidth and consequently influences performance of the motion controller. In this paper, the modal analysis method by measuring and analysing the vibrating parameters of the ball screw system to determine the dynamic characteristic of existing structures is used. On the one hand, the results of this study were obtained by the theoretical analysis and the modal testing of a ball screw system test station with the help of an impact hammer, respectively using excitation by motor. The experimental study showed oscillating forms of the ball screw for each frequency and obtained eigenfrequencies of the ball screw system. On the other hand, in this research a simulation with the help of the numerical modal analysis in order to analyse the oscillation and to find the eigenfrequencies of the ball screw system is used. Furthermore, the model order reduction by modal reduction and also according to Guyan is carried out. On the basis of these results a secure and also rapid commissioning of the control loops with regard to operating in their optimal function is targeted.

Keywords: modal analysis, ball screw, controller system, machine tools

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Authors: R. A. Petre, S. E. Nichifor, A. Craifaleanu, I. Stroe

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The relative motion of a robotic arm formed by homogeneous bars of different lengths and masses, hinged to each other is investigated. The first bar of the mechanism is articulated on a platform, considered initially fixed on the surface of the Earth, while for the second case the platform is considered to be in rotation with respect to the Earth. For both analyzed cases the motion equations are determined using the Lagrangian formalism, applied in its traditional form, valid with respect to an inertial reference system, conventionally considered as fixed. However, in the second case, a generalized form of the formalism valid with respect to a non-inertial reference frame will also be applied. The numerical calculations were performed using a MATLAB program.

Keywords: Lagrange equations, relative motion, inertial reference frame, non-inertial reference frame

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Authors: Jinsiang Shaw, Pik-Hoe Chen

Abstract:

This paper aims to integrate human recognition, motion recognition, and object tracking technologies without requiring a pre-training database model for motion recognition or the unknown object itself. Furthermore, it can simultaneously track multiple users and multiple objects. Unlike other existing human motion recognition methods, our approach employs a rule-based condition method to determine if a user hand is approaching or departing an object. It uses a background subtraction method to separate the human and object from the background, and employs behavior features to effectively interpret human object-grabbing actions. With an object’s histogram characteristics, we are able to isolate and track it using back projection. Hence, a moving object trajectory can be recorded and the object itself can be located. This particular technique can be used in a camera surveillance system in a shopping area to perform real-time intelligent surveillance, thus preventing theft. Experimental results verify the validity of the developed surveillance algorithm with an accuracy of 83% for shoplifting detection.

Keywords: Automatic Tracking, Back Projection, Motion Recognition, Shoplifting

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Authors: Mohammad Javad Maghsoudi, Z. Mohamed, A. R. Husain

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Input shaping has been utilized for vibration reduction of many oscillatory systems. This paper presents an optimal closed-loop input shaping scheme for control of a three dimensional (3D) gantry crane system including. This includes a PID controller and Zero Vibration shaper which consider two control objectives concurrently. The control objectives are minimum sway of a payload and fast and accurate positioning of a trolley. A complete mathematical model of a lab-scaled 3D gantry crane is simulated in Simulink. Moreover, by utilizing PSO algorithm and a proposed scheme the controller is designed to cater both control objectives concurrently. Simulation studies on a 3D gantry crane show that the proposed optimal controller has an acceptable performance. The controller provides good position response with satisfactory payload sway in both rail and trolley responses.

Keywords: 3D gantry crane, input shaping, closed-loop control, optimal scheme, PID

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Authors: Bandreddy Anand Babu, Mandadi Srinivasa Rao, Chintala Pradeep Reddy

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The main theme of this paper is to design fuzzy logic Proportional Integral Derivative controller for controlling of Pulse Width Modulator (PWM) based DCDC buck converter in continuous conduction mode of operation and comparing the results of FPID and ANFIS. Simulation is done to fuzzy the given input variables and membership functions of input values, creating the interference rules linking the input and output variables and after then defuzzfies the output variables. Fuzzy logic is simple for nonlinear models like buck converter. Fuzzy logic based PID controller technique is to control, nonlinear plants like buck converters in switching variables of power electronics. The characteristics of FPID are in terms of rise time, settling time, rise time, steady state errors for different inputs and load disturbances.

Keywords: fuzzy logic, PID controller, DC-DC buck converter, pulse width modulation

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Authors: Abderraouf Gaaloul, Faouzi Msahli

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The use of standard sliding mode controller, usually, leads to the appearing of an undesirable chattering phenomenon affecting the control signal. Such problem can be overcome using a higher-order sliding mode controller (HOSMC) which preserves the main properties of the standard sliding mode and deliberately increases the control smoothness. In this paper, we propose a new HOSMC for a class of uncertain multi-input multi-output nonlinear systems. Based on high gain and integral sliding mode paradigms, the established control scheme removes theoretically the chattering phenomenon and provides the stability of the control system. Numerical simulations are developed to show the effectiveness of the proposed controller when applied to solve a control problem of two water levels into a quadruple-tank process.

Keywords: nonlinear systems, sliding mode control, high gain, higher order

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Authors: Haodong Huang, Zida Zhao, Shilong Sun, Chiyao Li, Wenfu Xu

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The control methods for legged robots based on deep reinforcement learning have seen widespread application; however, the inherent black-box nature of neural networks presents challenges in understanding the decision-making motives of the robots. To address this issue, we propose a fully interpretable deep reinforcement learning training method to elucidate the underlying principles of legged robot motion. We incorporate the dynamics of legged robots into the policy, where observations serve as inputs and actions as outputs of the dynamics model. By embedding the dynamics equations within the multi-layer perceptron (MLP) computation process and making the parameters trainable, we enhance interpretability. Additionally, Bayesian optimization is introduced to train these parameters. We validate the proposed fully interpretable motion control algorithm on a legged robot, opening new research avenues for motion control and learning algorithms for legged robots within the deep learning framework.

Keywords: deep reinforcement learning, interpretation, motion control, legged robots

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Authors: Yao Jie, Yeo Khoon Seng

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In recent decades, flapping wing aerodynamics has attracted great interest. Understanding the physics of biological flyers such as birds and insects can help improve the performance of micro air vehicles. The present research focuses on the aerodynamics of insect-like flapping wing flight with the approach of numerical computation. Insect model of hawkmoth is adopted in the numerical study with rigid wing assumption currently. The numerical model integrates the computational fluid dynamics of the flow and active control of wing kinematics to achieve stable flight. The computation grid is a hybrid consisting of background Cartesian nodes and clouds of mesh-free grids around immersed boundaries. The generalized finite difference method is used in conjunction with single value decomposition (SVD-GFD) in computational fluid dynamics solver to study the dynamics of a free hovering hummingbird hawkmoth. The longitudinal dynamics of the hovering flight is governed by three control parameters, i.e., wing plane angle, mean positional angle and wing beating frequency. In present work, a PID controller works out the appropriate control parameters with the insect motion as input. The controller is adjusted to acquire desired maneuvering of the insect flight. The numerical scheme in present study is proven to be accurate and stable to simulate the flight of the hummingbird hawkmoth, which has relatively high Reynolds number. The PID controller is responsive to provide feedback to the wing kinematics during the hovering flight. The simulated hovering flight agrees well with the real insect flight. The present numerical study offers a promising route to investigate the free flight aerodynamics of insects, which could overcome some of the limitations of experiments.

Keywords: aerodynamics, flight control, computational fluid dynamics (CFD), flapping-wing flight

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Authors: Mario Gianni, Manuel A. Ruiz Garcia, Fiora Pirri

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In this work, we present a Bayesian non-parametric approach to model the motion control of ATVs. The motion control model is based on a Dirichlet Process-Gaussian Process (DP-GP) mixture model. The DP-GP mixture model provides a flexible representation of patterns of control manoeuvres along trajectories of different lengths and discretizations. The model also estimates the number of patterns, sufficient for modeling the dynamics of the ATV.

Keywords: Dirichlet processes, gaussian mixture models, learning motion patterns, tracked robots for urban search and rescue

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Authors: Jianrong Wu, Xuan Fu, Akihiro Higashi, Zhiming Tan

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CMOS image sensors with a rolling shutter are used broadly in the digital cameras embedded in mobile devices. The rolling shutter suffers the flicker artifacts from the fluorescent lamp, and it could be observed easily. In this paper, the characteristics of illumination flicker in motion case were analyzed, and two efficient detection methods based on matching fragment selection were proposed. According to the experimental results, our methods could achieve as high as 100% accuracy in static scene, and at least 97% in motion scene.

Keywords: illumination flicker, embedded camera, rolling shutter, detection

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Authors: Mandar Ghodekar, Sharad Jadhav, Sangram Jadhav

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Speed control of rotor during startup and under varying load conditions is one of the most difficult tasks of gas turbine operation. In this paper, power plant gas turbine (GE9001E) is considered for this purpose and fuzzy and fuzzy-PI rotor speed controllers are designed. The goal of the presented controllers is to keep the turbine rotor speed within predefined limits during startup condition as well as during operating condition. The fuzzy controller and fuzzy-PI controller are designed using Takagi-Sugeno method and Mamdani method, respectively. In applying the fuzzy-PI control to a gas-turbine plant, the tuning parameters (Kp and Ki) are modified online by fuzzy logic approach. Error and rate of change of error are inputs and change in fuel flow is output for both the controllers. Hence, rotor speed of gas turbine is controlled by modifying the fuel ƒflow. The identified linear ARX model of gas turbine is considered while designing the controllers. For simulations, demand power is taken as disturbance input. It is assumed that inlet guide vane (IGV) position is fixed. In addition, the constraint on the fuel flow is taken into account. The performance of the presented controllers is compared with each other as well as with H∞ robust and MPC controllers for the same operating conditions in simulations.

Keywords: gas turbine, fuzzy controller, fuzzy PI controller, power plant

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Authors: N. Bachok, N. L. Aleng, N. M. Arifin, A. Ishak, N. Senu

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The problem of laminar fluid flow which results from the shrinking of a permeable surface in a nanofluid has been investigated numerically. The model used for the nanofluid incorporates the effects of Brownian motion and thermophoresis. A similarity solution is presented which depends on the mass suction parameter S, Prandtl number Pr, Lewis number Le, Brownian motion number Nb and thermophoresis number Nt. It was found that the reduced Nusselt number is decreasing function of each dimensionless number.

Keywords: Boundary layer, nanofluid, shrinking sheet, Brownian motion, thermophoresis, similarity solution

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Authors: Divij Gupta

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Pendular systems have a range of both mathematical and engineering applications, ranging from modelling the behaviour of a continuous mass-density rope to utilisation as Tuned Mass Dampers (TMD). Thus, it is of interest to study the differential equations governing the motion of such systems. Here we attempt to generalise these equations of motion for the plane compound pendulum with a finite number of N point masses. A Lagrangian approach is taken, and we attempt to find the generalised form for the Euler-Lagrange equations of motion for the i-th bob of the N -bob pendulum. The co-ordinates are parameterized as angular quantities to reduce the number of degrees of freedom from 2N to N to simplify the form of the equations. We analyse the form of these equations up to N = 4 to determine the general form of the equation. We also develop a MATLAB program to compute a solution to the system for a given input value of N and a given set of initial conditions.

Keywords: classical mechanics, differential equation, lagrangian analysis, pendulum

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