Frequency-Domain System Identification and Simulation of a Quadrotor Controller
|
|
- Samantha Cook
- 6 years ago
- Views:
Transcription
1 AIAA SciTech January 2014, National Harbor, Maryland AIAA Modeling and Simulation Technologies Conference AIAA Frequency-Domain System Identification and Simulation of a Quadrotor Controller Wei Wei 1 University of Cincinnati, Cincinnati, Ohio Mark B. Tischler 2 Aviation Development Directorate AFDD US Army Research, Development and Engineering Command (AMRDEC) Moffett Field, California and Nicholas Schwartz 3, Kelly Cohen 4 University of Cincinnati, Cincinnati, Ohio Using frequency-domain system identification techniques, the closed-loop dynamic model of a quadrotor unmanned aerial vehicle is extracted using the computer program CIFER. The resulting dynamic model is verified by comparing its predicted response against previously collected flight-test data. The results show that an accurate model was successfully extracted, which could be utilized for simulations and control system development. Simulation of the extracted model and analysis of the transient response are also provided. = vertical velocity = lateral input = longitudinal input = directional input = vertical input Nomenclature I. Introduction n the aerospace industry today, there is a large focus on aircraft and rotorcraft simulation. Simulators not only Iprovide safety and efficiency when training pilots, but also greatly reduce costs when developing a control system for a particular vehicle. By having the ability to simulate an aircraft or rotorcraft response, flight controls engineers have the capability of designing and testing controller designs on the ground through computer simulations, greatly reducing the flying time and cost associated with controller development. In order to effectively utilize this capability, however, an accurate dynamic model of the system being analyzed is crucial. Recently, there has been a high interest in utilizing unmanned aerial vehicles (UAVs) for many different tasks. In particular, the demand for quadrotor UAVs has dramatically increased. Their ability to be easily scaled down and low cost make quadrotors an optimal choice for a variety of missions. As its configuration is naturally unstable, a feedback control system is required to stabilize the quadrotor so that it can be easily piloted. 1 Graduate Student, Department of Aerospace Engineering and Engineering Mechanics, Student Member AIAA. 2 Senior Scientist and Flight Control Group Lead, AFDD, Ames Research Center, T12B-2, Associate Fellow AIAA. 3 Undergraduate Student, Department of Aerospace Engineering and Engineering Mechanics, Student Member AIAA. 4 Associate Professor, Department of Aerospace Engineering and Engineering Mechanics, Associate Fellow AIAA. 1 Copyright 2014 by Wei Wei, Mark B. Tischler, Nicholas Schwartz, Kelly Cohen. Published by the, Inc., with permission.
2 Using traditional modeling methods, the aerodynamic, inertial, and structural characteristics of an aircraft or rotorcraft are analyzed to predict a dynamic model. After this preliminary model is acquired, it is then simulated and compared with flight-test data. Based on these comparisons, additional fine-tuning of the model is usually required to improve its accuracy. Though this traditional modeling method is practical for aircraft configurations, it is rather difficult for small-scale rotorcraft configurations their high vulnerability to turbulence and complex coupled aerodynamic forces greatly complicate the modeling process. In order to avoid these inaccuracies and minimize modeling time, a system identification approach is chosen to obtain a closed-loop dynamic model of a quadrotor configuration. II. System Identification Methodology The system identification process is one that is well suited and provides low cost when developing the dynamic model of a quadrotor. The method of system identification utilizes measured input and output time history data gathered during flight-testing to capture the dynamic characteristics of a particular system. 1 A generic mathematical architecture of the model is assumed prior to flight-testing. Flight-test data is then collected and processed through a system identification computer program, which obtains various dynamic coefficients through a process similar to statistical learning. A resulting dynamic mathematical model is then calculated and used to simulate the quadrotor dynamics. A frequency-domain system identification analysis was chosen for this project to obtain a linear representation of the quadrotor dynamics. Compared to time domain analyses, frequency-domain identification reduces the errors associated with bias effects and processing noise, resulting in a robust model. 1 Coherence functions also assist in determining the accuracy of the identified dynamic model throughout various frequencies of interest. For the scope of this project, it was desired to apply the method of system identification using the Comprehensive Identification from FrEquency Response (CIFER ) 1 program to develop a closed-loop dynamic model of a given quadrotor configuration. CIFER has been successfully implemented and applied in the system identification of commercial and military aircraft and rotorcraft configurations, including the XV-15, Bell-214ST, BO-105, AH-64, UH-60, V-22, AV-8 Harrier, and OH-58D. 1 In addition, CIFER has also been utilized to successfully extract dynamic models of scaled down model aircraft 2 and single-rotor model helicopter configurations. 3 When the frequency response is acquired in CIFER, it is imperative to check the validity of the data. In CIFER, the frequency response data validity is determined by evaluating its coherence, which is an indication of how well the output and input data are correlated. The definition of coherence is given as where,,, and represent the auto spectral densities of the input, output, and cross-spectral density of the input and output, respectively, and is the frequency point. A perfect correlation between input and output would result in a coherence value of unity, while poor coherence typically falls below a value of After the coherence of the data is validated, the data must be decoupled such that the inputs provided by off-axis commands are eliminated from the output on the axis of interest. During flight-testing it would be ideal to excite one axis while the other axes of interest remain trimmed. However, this very rarely occurs. In order to maintain total craft stability during a test maneuver, the pilot may be required to use slight inputs from the other axes. These slight inputs from the other axes must be removed from the data before system identification, so that they are not included in the extracted model. The multiple single output system estimation can be expressed in Eq. (2), where is the system estimation. (1) (2) Then the partial coherence of the th of inputs and the output can be written as (3) 2
3 where each quantity involves spectral matrix manipulations. 4 The partial coherence directly gives an evaluation of the on-axis input-output linear relation with the influence of other off-axis inputs removed. In the system identification process, the transfer functions of each axis will be acquired first, followed by state space representations, and complete system analysis. The Single Input-Single Output (SISO) transfer function identification cost function can be defined as [ ( ) ( ) ] (4) where represents the number of frequency points, and are the starting and ending frequencies of fit, and are the magnitude (db) and phase (degrees) at each frequency, and, and are the total, magnitude, and phase weights, respectively. and represent the estimated and actual value of each fit point, respectively. III. Experimental Setup For the experiment, an AeroQuad Cyclone quadrotor was selected for testing (Fig. 1). The frame is constructed with aluminum plates in the center that contain most of the electronic components. Four hollow aluminum square tubes construct the motor arms, having dimensions of 5/8 in. x 5/8 in. x 13 in. Four XXD A KV motors powered by 30 amp electric speed controllers (ESC) are attached at the ends of each motor arm. These motors directly drive four APC 12 x 3.8 propellers. Figure 1. AeroQuad UAV. A. Quadrotor Control For this project, the principal axes were positioned with the four rotor arms aligned with the X and Y axes, depicted in Fig. 2. For a quadrotor to achieve a stable hover, the rotational direction of the propellers must be altered such that the overall rotational torque is eliminated. To achieve this, the forward and aft motors rotate in a clockwise motion, while the port and starboard motors rotate in a counterclockwise motion. For a fixed propeller setup, the overall control of the quadrotor is achieved by directly altering the rotation speed of each individual motor. Figure 2. Quadrotor coordinate system and motor numbering. 3
4 B. Instrumentation and Data Collection The layout of the AeroQuad control system is outlined in Fig. 3. The sensors used onboard the AeroQuad include an ITG-3200 gyro, ADXL345 accelerometer, HMC5883L magnetometer, BMP085 barometer, and MaxSonars EZ0 ultrasonic sensor. These sensors provide adequate data-collection capability and record the necessary parameters required for closed-loop system identification. The onboard controller consists of an ATmega2560 Arduino -based platform. The radio controller used for this configuration is a Futaba T12MZ 2.4 GHz transmitter, coupled with a R6008HS receiver onboard the AeroQuad. For wireless flight data transmission, a pair of Digi XBee-PRO 900MHz modules was utilized. The data was transmitted through the XBee-PRO modules at a frequency of 40 Hz and was collected through MATLAB on a standalone computer. MOTOR1 MOTOR2 MOTOR3 MOTOR4 ESC1 ESC2 ESC3 ESC4 Quadrotor CPU RS-232 A/D IIC Figure 3. AeroQuad instrumentation and system layout. Due to its configuration, a great amount of vibration on the AeroQuad existed due to the rotating propellers. To suppress vibration noise so to not compromise onboard sensor performance, the motor mounts and electronic bay were isolated from the main frame using rubber vibration insulators. A traditional PID controller was applied to the longitudinal and lateral axes of the AeroQuad so that it could be easily piloted during flight-testing. Prior to flight-testing, the PID inputs were altered and tested such that good handling qualities of the AeroQuad resulted. After these gains were determined, they were not altered throughout the entire system identification process. Fig. 4 displays the resulting closed-loop system of the AeroQuad including this PID controller. With the instrumentation present on the AeroQuad, the pilot inputs, motor command, Euler angles (ϕ, θ, ), angular rates (p, q, r), linear accelerations (,, ), altitude, battery level, and elapsed time were able to be directly measured during flight-testing. For this paper, the angular rate responses of each axis were the only subjects of interest of the closed-loop dynamic model. As such, only the pilot commands, angular rates, and elapsed time were extracted from the flight-test data and used for the closed-loop system identification. XBEE IMU ULTRASONIC BAROMETER RECEIVER 900 MHz 2.4 GHz Figure 4. Quadrotor control system layout. C. Flight-test Procedure As CIFER was chosen to be the primary software used in the system identification process, a flight-test procedure was devised based on frequency-domain system identification guidelines provided by Tischler et. al. 1 For 4
5 frequency-domain system identification, the individual axes of the quadrotor were excited using frequency sweep maneuvers. In order to obtain a good data set, four frequency sweeps were carried out in the roll, pitch, and yaw axes during flight-tests. Based on a coherence analysis, the two best maneuvers for each axis of interest are selected for system identification. The selected input and output time history data is then concatenated and used in CIFER to generate the dynamic model. Fig. 5 below depicts an example of a roll sweep flight data with three axes inputs and outputs. Figure 5. Example roll sweep input. Figure 6. Roll rate power spectral density. Fig. 6 shows the power spectral density (PSD) of the roll rate signal given in Fig. 5. Integration under the PSD curve between 1-20 rad/s yields a root mean square (RMS) value of rad/s (i.e., about 20 deg/s), associated with the sum of the forced response content (i.e., signal) and noise. The PSD between 20 to 120 rad/s with an associated RMS of rad/s (i.e., about 2 deg/s), reflects the measurement noise content. Then, the associated coherence function can be expressed as a function of the noise to signal ratio 5 (5) where = / ( ) = is measurement noise to signal ratio. The resulting coherence is found to be 0.98, which is consistent with the high coherence of the roll rate frequency response shown in Fig. 7a, though there is additional loss in the frequency-response coherence due to process noise (e.g., atmospheric turbulence). This suggests a very low noise to signal ratio and is a very interesting result given the small scale of the vehicle and the inexpensive instrumentations onboard. In order to validate the accuracy of extracted models, it is desired to compare the predicted output provided by the dynamic model against the outputs of the actual system, using the same inputs and initial conditions. Therefore, it is required during flight-testing to perform doublet maneuvers in each axis of interest. Though not used in the system identification process, the doublet data collected will be used solely for model verification purposes. The accuracy of the model will be directly correlated to how well the response of the system is predicted. IV. Results A. System Identification Model Extraction After flight-testing, the dynamic models of the longitudinal, lateral, directional, and vertical axes were extracted from CIFER in the form of transfer functions. 5
6 The resulting magnitude, phase and coherence of the above longitudinal, lateral, directional, and vertical transfer functions are shown in Fig. 7. An examination of Fig. 7 shows a good coherence ( ) for a frequency range of rad/s for the lateral and longitudinal axes, rad/s for the directional axis, and 2-9 rad/s for the vertical axis. This implies the dynamics of the system were well excited in these corresponding axis frequency ranges with excellent signal to noise ratio. Therefore, the system can be well represented by a linear model. a) b) c) d) Figure 7. Frequeny response of a) lateral, b) longtitudinal, c) directional, and d) vertical axes and flight-test data. The resulting transfer functions are displayed in Eqs. (6-9) below. Notice that the lateral and longitudinal transfer functions have similar layouts as expected considering the symmetrical configuration of the quadrotor. (6) (7) (8) (9) 6
7 The identified roll and pitch dynamics show a lightly damped behavior with very small time delay (about 60 ms). The identified directional and vertical models are simply first-order systems. Transfer function identification cost guidelines 1 are acceptable (cost<100), excellent (cost<50). The cost functions shown in Table 1 suggest that all parameters are identified with excellent confidence. The poles and zeros of each axis are depicted in Fig. 8. The corresponding natural frequencies and damping ratios for each pole are listed in Table 1. As the lateral and longitudinal dynamics were identified as third order transfer functions, each transfer function contains one real pole and a pair of complex poles. The identified directional dynamics is a simple first-order system, and therefore contains only one pole. The resulting pole locations indicate that all three axes are stable, which was expected from the closed-loop system. Figure 8. AeroQuad closed-loop poles and zeros. Table 1 Identified model characteristics. Pole Axis Undamped Natural Damping Frequency (rad/s) Ratio Cost 1 Lateral Lateral Longitudinal Longitudinal Directional Vertical B. Model Verification The extracted models were verified in the time-domain with doublet maneuvers collected during flight-testing. Again, these maneuvers were not used in the system identification process when extracting the dynamic models. The measured inputs and initial conditions were input into the models, which then predicted the outputs on each axis of interest. The simulation outputs are then compared with the outputs gathered during flight-testing. These verification results are presented below in Fig. 9. It is seen that the extracted dynamic models have excellent comparison to flight-test data in all axes. There exists a slight variation between the flight-test data and model prediction for the lateral and longitudinal axes at the higher frequencies. As these extracted models are only valid in a frequency range of rad/s, it is expected to see these small discrepancies at the higher frequencies. Due to the large responses of the small-scale quadrotor compared to full sized rotorcrafts, the verification cost is scaled by 1/5. The resulting costs are 0.4(directional), 0.72(lateral), 0.36(longitudinal), and 0.16(vertical), which is below the 1.0 to 2.0 guideline 1 for rotorcraft model verification. These results show that the closed-loop dynamic model of the AeroQuad was successfully extracted and is accurate to actual system response. 7
8 a) b) c) d) Figure 9. Input and response of a) lateral, b) longitudinal, c) directional, and d) vertical axes extracted transfer functions compared to flight-test data. C. Simulation A Simulink model was constructed using the extracted transfer function models from system identification. The simulation results are shown below in Fig. 10. Given the model is for trimmed hover flight conditions, initial states are set to be 0. A 10% lateral input is applied at time 1 s and the roll rate and roll angle responses are plotted. Because the quadrotor was set to be flying in the attitude mode, a step roll input will command the vehicle to corresponding roll angle. It can be seen from Fig. 10 that the roll angular rate response is influenced by both the lightly damped short period model and the roll subsidence mode. The roll subsidence mode has a time constant of 0.21 s, which makes the roll rate converge quickly. The roll angle settles in about 0.67 s after the command is applied. Figure 10. AeroQuad closed-loop poles. 8
9 V. Conclusions A closed-loop dynamic model of a quadrotor UAV was successfully extracted using the frequency-domain system identification program CIFER. This model was compared against flight-test data not used in the system identification process, and was shown to have good correlation to the actual quadrotor response. These results show that an accurate dynamic model of a closed-loop quadrotor system can be successfully extracted using system identification, thereby decreasing modeling time and complexity when compared to traditional modeling techniques based on first principles. Simulation of the identified model is presented with analysis of the transient response behavior. In future work, the process of system identification will be applied to extract a bare-airframe dynamic model of a quadrotor configuration, using the motor inputs and outputs. With an accurate bare-airframe model, a custom controller for the quadrotor wil be developed and the handling qualities will be analyzed. References 1 Tischler, M. B., and Remple, R. K., Aircraft and Rotorcraft System Identification: Engineering Methods with Flight-Test Examples, 2 nd ed., AIAA Education Series, AIAA, Reston, VA, Woodrow, P., Tischler M. B., Hagerott, S. G., Mendoza G. E., and Hunter, J. M., Low Cost Flight-Test Platform to Demonstrate Flight Dynamics Concepts using Frequency-Domain System Identification Methods, AIAA Atmospheric Flight Mechanics Conference, AIAA, Washington, DC, Bhandari, S, Flight Validated High-Order Models of UAV Helicopter Dynamics in Hover and Forward Flight using Analytical and Parameter Identification Techniques, Ph.D. Dissertation, Department of Aerospace Engineering, University of Kansas, ProQuest, Ann Arbor, MI, Otnes, R.K., and Enochson, L., Basic Techniques, Applied Time Series Analysis, Vol. 1, Wiley, New York, Bendat, J. S., and Piersol, A. G., Engineering Applications of Correlation and Spectral Analysis, 2 nd ed., Wiley, New York,
System Identification and Controller Optimization of a Quadrotor UAV
System Identification and Controller Optimization of a Quadrotor UAV Wei Wei Kelly Cohen Department of Aerospace Engineering & Engineering Mechanics, University of Cincinnati Cincinnati, OH, USA Mark B.
More informationTurbulence Modeling of a Small Quadrotor UAS Using System Identification from Flight Data
Turbulence Modeling of a Small Quadrotor UAS Using System Identification from Flight Data Ondrej Juhasz Mark J.S. Lopez Research Associate Research Associate San Jose State University Ames Research Center
More informationQUADROTOR ROLL AND PITCH STABILIZATION USING SYSTEM IDENTIFICATION BASED REDESIGN OF EMPIRICAL CONTROLLERS
QUADROTOR ROLL AND PITCH STABILIZATION USING SYSTEM IDENTIFICATION BASED REDESIGN OF EMPIRICAL CONTROLLERS ANIL UFUK BATMAZ 1, a, OVUNC ELBIR 2,b and COSKU KASNAKOGLU 3,c 1,2,3 Department of Electrical
More informationFLCS V2.1. AHRS, Autopilot, Gyro Stabilized Gimbals Control, Ground Control Station
AHRS, Autopilot, Gyro Stabilized Gimbals Control, Ground Control Station The platform provides a high performance basis for electromechanical system control. Originally designed for autonomous aerial vehicle
More informationSTUDY OF FIXED WING AIRCRAFT DYNAMICS USING SYSTEM IDENTIFICATION APPROACH
STUDY OF FIXED WING AIRCRAFT DYNAMICS USING SYSTEM IDENTIFICATION APPROACH A.Kaviyarasu 1, Dr.A.Saravan Kumar 2 1,2 Department of Aerospace Engineering, Madras Institute of Technology, Anna University,
More informationControl System Design for Tricopter using Filters and PID controller
Control System Design for Tricopter using Filters and PID controller Abstract The purpose of this paper is to present the control system design of Tricopter. We have presented the implementation of control
More informationDesign of a Flight Stabilizer System and Automatic Control Using HIL Test Platform
Design of a Flight Stabilizer System and Automatic Control Using HIL Test Platform Şeyma Akyürek, Gizem Sezin Özden, Emre Atlas, and Coşku Kasnakoğlu Electrical & Electronics Engineering, TOBB University
More informationDevelopment of Hybrid Flight Simulator with Multi Degree-of-Freedom Robot
Development of Hybrid Flight Simulator with Multi Degree-of-Freedom Robot Kakizaki Kohei, Nakajima Ryota, Tsukabe Naoki Department of Aerospace Engineering Department of Mechanical System Design Engineering
More informationModule 2: Lecture 4 Flight Control System
26 Guidance of Missiles/NPTEL/2012/D.Ghose Module 2: Lecture 4 Flight Control System eywords. Roll, Pitch, Yaw, Lateral Autopilot, Roll Autopilot, Gain Scheduling 3.2 Flight Control System The flight control
More informationClassical Control Based Autopilot Design Using PC/104
Classical Control Based Autopilot Design Using PC/104 Mohammed A. Elsadig, Alneelain University, Dr. Mohammed A. Hussien, Alneelain University. Abstract Many recent papers have been written in unmanned
More informationDesign of Self-tuning PID Controller Parameters Using Fuzzy Logic Controller for Quad-rotor Helicopter
Design of Self-tuning PID Controller Parameters Using Fuzzy Logic Controller for Quad-rotor Helicopter Item type Authors Citation Journal Article Bousbaine, Amar; Bamgbose, Abraham; Poyi, Gwangtim Timothy;
More informationThe Next Generation Design of Autonomous MAV Flight Control System SmartAP
The Next Generation Design of Autonomous MAV Flight Control System SmartAP Kirill Shilov Department of Aeromechanics and Flight Engineering Moscow Institute of Physics and Technology 16 Gagarina st, Zhukovsky,
More informationSystem identification studies with the stiff wing minimutt Fenrir Flight 20
SYSTEMS TECHNOLOGY, INC 3766 S. HAWTHORNE BOULEVARD HAWTHORNE, CALIFORNIA 925-783 PHONE (3) 679-228 email: sti@systemstech.com FAX (3) 644-3887 Working Paper 439- System identification studies with the
More informationIntroducing the Quadrotor Flying Robot
Introducing the Quadrotor Flying Robot Roy Brewer Organizer Philadelphia Robotics Meetup Group August 13, 2009 What is a Quadrotor? A vehicle having 4 rotors (propellers) at each end of a square cross
More informationUAV: Design to Flight Report
UAV: Design to Flight Report Team Members Abhishek Verma, Bin Li, Monique Hladun, Topher Sikorra, and Julio Varesio. Introduction In the start of the course we were to design a situation for our UAV's
More informationOughtToPilot. Project Report of Submission PC128 to 2008 Propeller Design Contest. Jason Edelberg
OughtToPilot Project Report of Submission PC128 to 2008 Propeller Design Contest Jason Edelberg Table of Contents Project Number.. 3 Project Description.. 4 Schematic 5 Source Code. Attached Separately
More informationConstruction and signal filtering in Quadrotor
Construction and signal filtering in Quadrotor Arkadiusz KUBACKI, Piotr OWCZAREK, Adam OWCZARKOWSKI*, Arkadiusz JAKUBOWSKI Institute of Mechanical Technology, *Institute of Control and Information Engineering,
More informationTEAM AERO-I TEAM AERO-I JOURNAL PAPER DELHI TECHNOLOGICAL UNIVERSITY Journal paper for IARC 2014
TEAM AERO-I TEAM AERO-I JOURNAL PAPER DELHI TECHNOLOGICAL UNIVERSITY DELHI TECHNOLOGICAL UNIVERSITY Journal paper for IARC 2014 2014 IARC ABSTRACT The paper gives prominence to the technical details of
More informationFig m Telescope
Taming the 1.2 m Telescope Steven Griffin, Matt Edwards, Dave Greenwald, Daryn Kono, Dennis Liang and Kirk Lohnes The Boeing Company Virginia Wright and Earl Spillar Air Force Research Laboratory ABSTRACT
More informationTigreSAT 2010 &2011 June Monthly Report
2010-2011 TigreSAT Monthly Progress Report EQUIS ADS 2010 PAYLOAD No changes have been done to the payload since it had passed all the tests, requirements and integration that are necessary for LSU HASP
More informationImplementation of Nonlinear Reconfigurable Controllers for Autonomous Unmanned Vehicles
Implementation of Nonlinear Reconfigurable Controllers for Autonomous Unmanned Vehicles Dere Schmitz Vijayaumar Janardhan S. N. Balarishnan Department of Mechanical and Aerospace engineering and Engineering
More informationMulti-Axis Pilot Modeling
Multi-Axis Pilot Modeling Models and Methods for Wake Vortex Encounter Simulations Technical University of Berlin Berlin, Germany June 1-2, 2010 Ronald A. Hess Dept. of Mechanical and Aerospace Engineering
More informationSmall Unmanned Aerial Vehicle Simulation Research
International Conference on Education, Management and Computer Science (ICEMC 2016) Small Unmanned Aerial Vehicle Simulation Research Shaojia Ju1, a and Min Ji1, b 1 Xijing University, Shaanxi Xi'an, 710123,
More informationEstimation and Control of a Tilt-Quadrotor Attitude
Estimation and Control of a Tilt-Quadrotor Attitude Estanislao Cantos Mateos Mechanical Engineering Department, Instituto Superior Técnico, Lisboa, E-mail: est8ani@gmail.com Abstract - The aim of the present
More informationActive Vibration Isolation of an Unbalanced Machine Tool Spindle
Active Vibration Isolation of an Unbalanced Machine Tool Spindle David. J. Hopkins, Paul Geraghty Lawrence Livermore National Laboratory 7000 East Ave, MS/L-792, Livermore, CA. 94550 Abstract Proper configurations
More informationModeling And Pid Cascade Control For Uav Type Quadrotor
IOSR Journal of Dental and Medical Sciences (IOSR-JDMS) e-issn: 2279-0853, p-issn: 2279-0861.Volume 15, Issue 8 Ver. IX (August. 2016), PP 52-58 www.iosrjournals.org Modeling And Pid Cascade Control For
More informationEE 560 Electric Machines and Drives. Autumn 2014 Final Project. Contents
EE 560 Electric Machines and Drives. Autumn 2014 Final Project Page 1 of 53 Prof. N. Nagel December 8, 2014 Brian Howard Contents Introduction 2 Induction Motor Simulation 3 Current Regulated Induction
More informationIntegrated Navigation System
Integrated Navigation System Adhika Lie adhika@aem.umn.edu AEM 5333: Design, Build, Model, Simulate, Test and Fly Small Uninhabited Aerial Vehicles Feb 14, 2013 1 Navigation System Where am I? Position,
More informationEMBEDDED ONBOARD CONTROL OF A QUADROTOR AERIAL VEHICLE 5
EMBEDDED ONBOARD CONTROL OF A QUADROTOR AERIAL VEHICLE Cory J. Bryan, Mitchel R. Grenwalt, Adam W. Stienecker, Ohio Northern University Abstract The quadrotor aerial vehicle is a structure that has recently
More informationOPTIMAL AND PID CONTROLLER FOR CONTROLLING CAMERA S POSITION IN UNMANNED AERIAL VEHICLES
International Journal of Information Technology, Modeling and Computing (IJITMC) Vol.1,No.4,November 2013 OPTIMAL AND PID CONTROLLER FOR CONTROLLING CAMERA S POSITION IN UNMANNED AERIAL VEHICLES MOHAMMAD
More informationEEL 4665/5666 Intelligent Machines Design Laboratory. Messenger. Final Report. Date: 4/22/14 Name: Revant shah
EEL 4665/5666 Intelligent Machines Design Laboratory Messenger Final Report Date: 4/22/14 Name: Revant shah E-Mail:revantshah2000@ufl.edu Instructors: Dr. A. Antonio Arroyo Dr. Eric M. Schwartz TAs: Andy
More informationInternational Journal of Scientific & Engineering Research, Volume 8, Issue 1, January ISSN
International Journal of Scientific & Engineering Research, Volume 8, Issue 1, January-2017 500 DESIGN AND FABRICATION OF VOICE CONTROLLED UNMANNED AERIAL VEHICLE Author-Shubham Maindarkar, Co-author-
More informationManagement Process of a Frequency Response Flight Test for Rotorcraft Flying Qualities Evaluation
doi: 1.528/jatm.v8i3.644 Management Process of a Frequency Response Flight Test for Rotorcraft Flying Qualities Evaluation João Otávio Falcão Arantes Filho 1, Donizeti de Andrade 2 ABSTRACT: This paper
More informationDesign and Implementation of FPGA Based Quadcopter
Design and Implementation of FPGA Based Quadcopter G Premkumar 1 SCSVMV, Kanchipuram, Tamil Nadu, INDIA R Jayalakshmi 2 Assistant Professor, SCSVMV, Kanchipuram, Tamil Nadu, INDIA Md Akramuddin 3 Project
More informationDynamic Angle Estimation
Dynamic Angle Estimation with Inertial MEMS Analog Devices Bob Scannell Mark Looney Agenda Sensor to angle basics Accelerometer basics Accelerometer behaviors Gyroscope basics Gyroscope behaviors Key factors
More informationQUADROTOR STABILITY USING PID JULKIFLI BIN AWANG BESAR
QUADROTOR STABILITY USING PID JULKIFLI BIN AWANG BESAR A project report submitted in partial fulfillment of the requirement for the award of the Master of Electrical Engineering Faculty of Electrical &
More informationControl Servo Design for Inverted Pendulum
JGW-T1402132-v2 Jan. 14, 2014 Control Servo Design for Inverted Pendulum Takanori Sekiguchi 1. Introduction In order to acquire and keep the lock of the interferometer, RMS displacement or velocity of
More informationSELF STABILIZING PLATFORM
SELF STABILIZING PLATFORM Shalaka Turalkar 1, Omkar Padvekar 2, Nikhil Chavan 3, Pritam Sawant 4 and Project Guide: Mr Prathamesh Indulkar 5. 1,2,3,4,5 Department of Electronics and Telecommunication,
More information드론의제어원리. Professor H.J. Park, Dept. of Mechanical System Design, Seoul National University of Science and Technology.
드론의제어원리 Professor H.J. Park, Dept. of Mechanical System Design, Seoul National University of Science and Technology. An Unmanned aerial vehicle (UAV) is a Unmanned Aerial Vehicle. UAVs include both autonomous
More informationOS3D-FG MINIATURE ATTITUDE & HEADING REFERENCE SYSTEM MINIATURE 3D ORIENTATION SENSOR OS3D-P. Datasheet Rev OS3D-FG Datasheet rev. 2.
OS3D-FG OS3D-FG MINIATURE ATTITUDE & HEADING REFERENCE SYSTEM MINIATURE 3D ORIENTATION SENSOR OS3D-P Datasheet Rev. 2.0 1 The Inertial Labs OS3D-FG is a multi-purpose miniature 3D orientation sensor Attitude
More informationControl System Development and Flight Testing of the Tiger Moth UAV
Control System Development and Flight Testing of the Tiger Moth UAV Brian T. Fujizawa Mark B. Tischler Aeroflightdynamics Directorate (AMRDEC) U.S. Army Research, Development, and Engineering Command Moffett
More informationFlight Dynamics AE426
KING FAHD UNIVERSITY Department of Aerospace Engineering AE426: Flight Dynamics Instructor Dr. Ayman Hamdy Kassem What is flight dynamics? Is the study of aircraft motion and its characteristics. Is it
More informationLocation Holding System of Quad Rotor Unmanned Aerial Vehicle(UAV) using Laser Guide Beam
Location Holding System of Quad Rotor Unmanned Aerial Vehicle(UAV) using Laser Guide Beam Wonkyung Jang 1, Masafumi Miwa 2 and Joonhwan Shim 1* 1 Department of Electronics and Communication Engineering,
More informationFlight control Set and Kit
Flight control Set and Kit Quick Start Guide For MegaPirate NG Version 1.2 Thanks for choosing AirStudio flight control electronics. We have created it based on best-in-class software, hardware and our
More informationA New Perspective to Altitude Acquire-and- Hold for Fixed Wing UAVs
Student Research Paper Conference Vol-1, No-1, Aug 2014 A New Perspective to Altitude Acquire-and- Hold for Fixed Wing UAVs Mansoor Ahsan Avionics Department, CAE NUST Risalpur, Pakistan mahsan@cae.nust.edu.pk
More informationGeorgia Tech Aerial Robotics Team 2009 International Aerial Robotics Competition Entry
Georgia Tech Aerial Robotics Team 2009 International Aerial Robotics Competition Entry Girish Chowdhary, H. Claus Christmann, Dr. Eric N. Johnson, M. Scott Kimbrell, Dr. Erwan Salaün, D. Michael Sobers,
More informationOptimal Control System Design
Chapter 6 Optimal Control System Design 6.1 INTRODUCTION The active AFO consists of sensor unit, control system and an actuator. While designing the control system for an AFO, a trade-off between the transient
More informationEC6405 - CONTROL SYSTEM ENGINEERING Questions and Answers Unit - II Time Response Analysis Two marks 1. What is transient response? The transient response is the response of the system when the system
More informationVibration Control of Flexible Spacecraft Using Adaptive Controller.
Vol. 2 (2012) No. 1 ISSN: 2088-5334 Vibration Control of Flexible Spacecraft Using Adaptive Controller. V.I.George #, B.Ganesh Kamath #, I.Thirunavukkarasu #, Ciji Pearl Kurian * # ICE Department, Manipal
More informationThrust estimation by fuzzy modeling of coaxial propulsion unit for multirotor UAVs
2016 IEEE International Conference on Multisensor Fusion and Integration for Intelligent Systems (MFI 2016) Kongresshaus Baden-Baden, Germany, Sep. 19-21, 2016 Thrust estimation by fuzzy modeling of coaxial
More informationHopper Spacecraft Simulator. Billy Hau and Brian Wisniewski
Hopper Spacecraft Simulator Billy Hau and Brian Wisniewski Agenda Introduction Flight Dynamics Hardware Design Avionics Control System Future Works Introduction Mission Overview Collaboration with Penn
More informationTesting Autonomous Hover Algorithms Using a Quad rotor Helicopter Test Bed
Testing Autonomous Hover Algorithms Using a Quad rotor Helicopter Test Bed In conjunction with University of Washington Distributed Space Systems Lab Justin Palm Andy Bradford Andrew Nelson Milestone One
More informationCATEGORY 7 - NAVIGATION AND AVIONICS A. SYSTEMS, EQUIPMENT AND COMPONENTS
Commerce Control List Supplement No. 1 to Part 774 Category 7 page 1 CATEGORY 7 - NAVIGATION AND AVIONICS A. SYSTEMS, EQUIPMENT AND COMPONENTS N.B.1: For automatic pilots for underwater vehicles, see Category
More informationVarious levels of Simulation for Slybird MAV using Model Based Design
Various levels of Simulation for Slybird MAV using Model Based Design Kamali C Shikha Jain Vijeesh T Sujeendra MR Sharath R Motivation In order to design robust and reliable flight guidance and control
More informationFlapping Wing Micro Air Vehicle (FW-MAV) State Estimation and Control with Heading and Altitude Hold
Flapping Wing Micro Air Vehicle (FW-MAV) State Estimation and Control with Heading and Altitude Hold S. Aurecianus 1, H.V. Phan 2, S. L. Nam 1, T. Kang 1 *, and H.C. Park 2 1 Department of Aerospace Information
More informationAIRCRAFT CONTROL AND SIMULATION
AIRCRAFT CONTROL AND SIMULATION AIRCRAFT CONTROL AND SIMULATION Third Edition Dynamics, Controls Design, and Autonomous Systems BRIAN L. STEVENS FRANK L. LEWIS ERIC N. JOHNSON Cover image: Space Shuttle
More informationHardware-in-the-Loop Simulation for a Small Unmanned Aerial Vehicle A. Shawky *, A. Bayoumy Aly, A. Nashar, and M. Elsayed
16 th International Conference on AEROSPACE SCIENCES & AVIATION TECHNOLOGY, ASAT - 16 May 26-28, 2015, E-Mail: asat@mtc.edu.eg Military Technical College, Kobry Elkobbah, Cairo, Egypt Tel : +(202) 24025292
More informationTeleoperation of a Tail-Sitter VTOL UAV
The 2 IEEE/RSJ International Conference on Intelligent Robots and Systems October 8-22, 2, Taipei, Taiwan Teleoperation of a Tail-Sitter VTOL UAV Ren Suzuki, Takaaki Matsumoto, Atsushi Konno, Yuta Hoshino,
More informationAn Overview of MIMO-FRF Excitation/Averaging Techniques
An Overview of MIMO-FRF Excitation/Averaging Techniques Allyn W. Phillips, PhD, Research Assistant Professor Randall J. Allemang, PhD, Professor Andrew T. Zucker, Research Assistant University of Cincinnati
More informationARKBIRD-Tiny Product Features:
ARKBIRD-Tiny Product Features: ARKBIRD System is a high-accuracy autopilot designed for fixed-wing, which has capability of auto-balancing to ease the manipulation while flying. 1. Function all in one
More informationA Comparison of MIMO-FRF Excitation/Averaging Techniques on Heavily and Lightly Damped Structures
A Comparison of MIMO-FRF Excitation/Averaging Techniques on Heavily and Lightly Damped Structures Allyn W. Phillips, PhD Andrew T. Zucker Randall J. Allemang, PhD Research Assistant Professor Research
More informationBW-IMU200 Serials. Low-cost Inertial Measurement Unit. Technical Manual
Serials Low-cost Inertial Measurement Unit Technical Manual Introduction As a low-cost inertial measurement sensor, the BW-IMU200 measures the attitude parameters of the motion carrier (roll angle, pitch
More informationARDUINO BASED CALIBRATION OF AN INERTIAL SENSOR IN VIEW OF A GNSS/IMU INTEGRATION
Journal of Young Scientist, Volume IV, 2016 ISSN 2344-1283; ISSN CD-ROM 2344-1291; ISSN Online 2344-1305; ISSN-L 2344 1283 ARDUINO BASED CALIBRATION OF AN INERTIAL SENSOR IN VIEW OF A GNSS/IMU INTEGRATION
More informationEmbedded Robust Control of Self-balancing Two-wheeled Robot
Embedded Robust Control of Self-balancing Two-wheeled Robot L. Mollov, P. Petkov Key Words: Robust control; embedded systems; two-wheeled robots; -synthesis; MATLAB. Abstract. This paper presents the design
More information302 VIBROENGINEERING. JOURNAL OF VIBROENGINEERING. MARCH VOLUME 15, ISSUE 1. ISSN
949. A distributed and low-order GPS/SINS algorithm of flight parameters estimation for unmanned vehicle Jiandong Guo, Pinqi Xia, Yanguo Song Jiandong Guo 1, Pinqi Xia 2, Yanguo Song 3 College of Aerospace
More informationIPRO 312: Unmanned Aerial Systems
IPRO 312: Unmanned Aerial Systems Kay, Vlad, Akshay, Chris, Andrew, Sebastian, Anurag, Ani, Ivo, Roger Dr. Vural Diverse IPRO Group ECE MMAE BME ARCH CS Outline Background Approach Team Research Integration
More informationHardware in the Loop Simulation for Unmanned Aerial Vehicles
NATIONAL 1 AEROSPACE LABORATORIES BANGALORE-560 017 INDIA CSIR-NAL Hardware in the Loop Simulation for Unmanned Aerial Vehicles Shikha Jain Kamali C Scientist, Flight Mechanics and Control Division National
More informationJurnal Teknologi IMPROVEMENT OF QUADROTOR PERFORMANCE WITH FLIGHT CONTROL SYSTEM USING PARTICLE SWARM PROPORTIONAL-INTEGRAL-DERIVATIVE (PS-PID)
Jurnal Teknologi IMPROVEMENT OF QUADROTOR PERFORMANCE WITH FLIGHT CONTROL SYSTEM USING PARTICLE SWARM PROPORTIONAL-INTEGRAL-DERIVATIVE (PS-PID) Andi Adriansyah a*, Shamsudin H. M. Amin b, Anwar Minarso
More informationDESIGN AND TEST OF FLIGHT CONTROL LAWS FOR THE KAMAN BURRO UNMANNED AERIAL VEHICLE
DESIGN AND TEST OF FLIGHT CONTROL LAWS FOR THE KAMAN BURRO UNMANNED AERIAL VEHICLE AIAA--45 Chad R. Frost * NASA Mark B. Tischler U.S. Army Aeroflightdynamics Directorate (AMRDEC) Mike Bielefield Troy
More informationZJU Team Entry for the 2013 AUVSI. International Aerial Robotics Competition
ZJU Team Entry for the 2013 AUVSI International Aerial Robotics Competition Lin ZHANG, Tianheng KONG, Chen LI, Xiaohuan YU, Zihao SONG Zhejiang University, Hangzhou 310027, China ABSTRACT This paper introduces
More informationTrajectory Tracking and Payload Dropping of an Unmanned Quadrotor Helicopter Based on GS-PID and Backstepping Control
Trajectory Tracking and Payload Dropping of an Unmanned Quadrotor Helicopter Based on GS-PID and Backstepping Control Jing Qiao A Thesis in The Department of Mechanical, Industrial and Aerospace Engineering
More information3DM -CV5-10 LORD DATASHEET. Inertial Measurement Unit (IMU) Product Highlights. Features and Benefits. Applications. Best in Class Performance
LORD DATASHEET 3DM -CV5-10 Inertial Measurement Unit (IMU) Product Highlights Triaxial accelerometer, gyroscope, and sensors achieve the optimal combination of measurement qualities Smallest, lightest,
More informationCDS 101/110a: Lecture 8-1 Frequency Domain Design
CDS 11/11a: Lecture 8-1 Frequency Domain Design Richard M. Murray 17 November 28 Goals: Describe canonical control design problem and standard performance measures Show how to use loop shaping to achieve
More informationDesign of Missile Two-Loop Auto-Pilot Pitch Using Root Locus
International Journal Of Advances in Engineering and Management (IJAEM) Page 141 Volume 1, Issue 5, November - 214. Design of Missile Two-Loop Auto-Pilot Pitch Using Root Locus 1 Rami Ali Abdalla, 2 Muawia
More informationDEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING BANGLADESH UNIVERSITY OF ENGINEERING & TECHNOLOGY EEE 402 : CONTROL SYSTEMS SESSIONAL
DEPARTMENT OF ELECTRICAL AND ELECTRONIC ENGINEERING BANGLADESH UNIVERSITY OF ENGINEERING & TECHNOLOGY EEE 402 : CONTROL SYSTEMS SESSIONAL Experiment No. 1(a) : Modeling of physical systems and study of
More informationF-16 Quadratic LCO Identification
Chapter 4 F-16 Quadratic LCO Identification The store configuration of an F-16 influences the flight conditions at which limit cycle oscillations develop. Reduced-order modeling of the wing/store system
More informationME 5281 Fall Homework 8 Due: Wed. Nov. 4th; start of class.
ME 5281 Fall 215 Homework 8 Due: Wed. Nov. 4th; start of class. Reading: Chapter 1 Part A: Warm Up Problems w/ Solutions (graded 4%): A.1 Non-Minimum Phase Consider the following variations of a system:
More informationVECTOR CONTROL SCHEME FOR INDUCTION MOTOR WITH DIFFERENT CONTROLLERS FOR NEGLECTING THE END EFFECTS IN HEV APPLICATIONS
VECTOR CONTROL SCHEME FOR INDUCTION MOTOR WITH DIFFERENT CONTROLLERS FOR NEGLECTING THE END EFFECTS IN HEV APPLICATIONS M.LAKSHMISWARUPA 1, G.TULASIRAMDAS 2 & P.V.RAJGOPAL 3 1 Malla Reddy Engineering College,
More informationArtificial Neural Networks based Attitude Controlling of Longitudinal Autopilot for General Aviation Aircraft Nagababu V *1, Imran A 2
ISSN (Print) : 2320-3765 ISSN (Online): 2278-8875 International Journal of Advanced Research in Electrical, Electronics and Instrumentation Engineering Vol. 7, Issue 1, January 2018 Artificial Neural Networks
More informationJUNE 2014 Solved Question Paper
JUNE 2014 Solved Question Paper 1 a: Explain with examples open loop and closed loop control systems. List merits and demerits of both. Jun. 2014, 10 Marks Open & Closed Loop System - Advantages & Disadvantages
More informationAnalysis of Handling Qualities Design Criteria for Active Inceptor Force-Feel Characteristics
Analysis of Handling Qualities Design Criteria for Active Inceptor Force-Feel Characteristics Carlos A. Malpica NASA Ames Research Center Moffett Field, CA Jeff A. Lusardi Aeroflightdynamics Directorate
More informationDevelopment of a Low Cost Autonomous Indoor Aerial Robotics System V1.0 1 June 2009
Development of a Low Cost Autonomous Indoor Aerial Robotics System V1.0 1 June 2009 Zack Jarrett Pima Community College Christopher Miller Pima Community College Tete Barrigah University of Arizona Huihong
More informationDESIGN & FABRICATION OF UAV FOR DATA TRANSMISSION. Department of ME, CUET, Bangladesh
Proceedings of the International Conference on Mechanical Engineering and Renewable Energy 2017 (ICMERE2017) 18 20 December, 2017, Chittagong, Bangladesh ICMERE2017-PI-177 DESIGN & FABRICATION OF UAV FOR
More informationFrequency Response Analysis and Design Tutorial
1 of 13 1/11/2011 5:43 PM Frequency Response Analysis and Design Tutorial I. Bode plots [ Gain and phase margin Bandwidth frequency Closed loop response ] II. The Nyquist diagram [ Closed loop stability
More informationA Mini UAV for security environmental monitoring and surveillance: telemetry data analysis
A Mini UAV for security environmental monitoring and surveillance: telemetry data analysis G. Belloni 2,3, M. Feroli 3, A. Ficola 1, S. Pagnottelli 1,3, P. Valigi 2 1 Department of Electronic and Information
More information3DM-GX4-45 LORD DATASHEET. GPS-Aided Inertial Navigation System (GPS/INS) Product Highlights. Features and Benefits. Applications
LORD DATASHEET 3DM-GX4-45 GPS-Aided Inertial Navigation System (GPS/INS) Product Highlights High performance integd GPS receiver and MEMS sensor technology provide direct and computed PVA outputs in a
More informationHeterogeneous Control of Small Size Unmanned Aerial Vehicles
Magyar Kutatók 10. Nemzetközi Szimpóziuma 10 th International Symposium of Hungarian Researchers on Computational Intelligence and Informatics Heterogeneous Control of Small Size Unmanned Aerial Vehicles
More informationUniversity of Bristol - Explore Bristol Research. Peer reviewed version. Link to published version (if available): /6.
Araujo-Estrada, S., Gong, Z., Lowenberg, M., Neild, S., & Goman, M. (216). Wind tunnel manoeuvre rig: a multi-dof test platform for model aircraft. In 54th AIAA Aerospace Sciences Meeting [AIAA 216-2119]
More informationA3 Pro INSTRUCTION MANUAL. Oct 25, 2017 Revision IMPORTANT NOTES
A3 Pro INSTRUCTION MANUAL Oct 25, 2017 Revision IMPORTANT NOTES 1. Radio controlled (R/C) models are not toys! The propellers rotate at high speed and pose potential risk. They may cause severe injury
More informationThe Pennsylvania State University. The Graduate School. College of Engineering
The Pennsylvania State University The Graduate School College of Engineering INTEGRATED FLIGHT CONTROL DESIGN AND HANDLING QUALITIES ANALYSIS FOR A TILTROTOR AIRCRAFT A Thesis in Aerospace Engineering
More informationNautical Autonomous System with Task Integration (Code name)
Nautical Autonomous System with Task Integration (Code name) NASTI 10/6/11 Team NASTI: Senior Students: Terry Max Christy, Jeremy Borgman Advisors: Nick Schmidt, Dr. Gary Dempsey Introduction The Nautical
More informationCHAPTER 6. CALCULATION OF TUNING PARAMETERS FOR VIBRATION CONTROL USING LabVIEW
130 CHAPTER 6 CALCULATION OF TUNING PARAMETERS FOR VIBRATION CONTROL USING LabVIEW 6.1 INTRODUCTION Vibration control of rotating machinery is tougher and a challenging challengerical technical problem.
More informationExperiment design for MIMO model identification. Marco Lovera Dipartimento di Scienze e Tecnologie Aerospaziali Politecnico di Milano
Experiment design for MIMO model identification Marco Lovera Dipartimento di Scienze e Tecnologie Aerospaziali Politecnico di Milano Introduction Experiment design: fundamental role in the practice of
More informationConventional geophone topologies and their intrinsic physical limitations, determined
Magnetic innovation in velocity sensing Low -frequency with passive Conventional geophone topologies and their intrinsic physical limitations, determined by the mechanical construction, limit their velocity
More informationWIND VELOCITY ESTIMATION WITHOUT AN AIR SPEED SENSOR USING KALMAN FILTER UNDER THE COLORED MEASUREMENT NOISE
WIND VELOCIY ESIMAION WIHOU AN AIR SPEED SENSOR USING KALMAN FILER UNDER HE COLORED MEASUREMEN NOISE Yong-gonjong Par*, Chan Goo Par** Department of Mechanical and Aerospace Eng/Automation and Systems
More informationPosition Control of AC Servomotor Using Internal Model Control Strategy
Position Control of AC Servomotor Using Internal Model Control Strategy Ahmed S. Abd El-hamid and Ahmed H. Eissa Corresponding Author email: Ahmednrc64@gmail.com Abstract: This paper focuses on the design
More informationRobot Joint Angle Control Based on Self Resonance Cancellation Using Double Encoders
Robot Joint Angle Control Based on Self Resonance Cancellation Using Double Encoders Akiyuki Hasegawa, Hiroshi Fujimoto and Taro Takahashi 2 Abstract Research on the control using a load-side encoder for
More informationSRV02-Series Rotary Experiment # 3. Ball & Beam. Student Handout
SRV02-Series Rotary Experiment # 3 Ball & Beam Student Handout SRV02-Series Rotary Experiment # 3 Ball & Beam Student Handout 1. Objectives The objective in this experiment is to design a controller for
More informationDesign of a Miniature Aircraft Deployment System
Project Customer Prof. Eric Frew Project Advisors Prof. Bill Emery Prof. Kurt Maute Design of a Miniature Aircraft Deployment System http://www.colorado.edu/aerospace/mads Leah Crumbaker Jason Farmer Michael
More informationPenn State Erie, The Behrend College School of Engineering
Penn State Erie, The Behrend College School of Engineering EE BD 327 Signals and Control Lab Spring 2008 Lab 9 Ball and Beam Balancing Problem April 10, 17, 24, 2008 Due: May 1, 2008 Number of Lab Periods:
More information