# SOFTWARE IN THE LOOP SIMULATION FOR MULTIROTOR POSITION TRACKING USING A LINEAR CONSTRAINED MODEL PREDICTIVE CONTROLLER

Size: px
Start display at page:

Download "SOFTWARE IN THE LOOP SIMULATION FOR MULTIROTOR POSITION TRACKING USING A LINEAR CONSTRAINED MODEL PREDICTIVE CONTROLLER"

## Transcription

2 the trajectory control by providing references of attitude comms d ref R 3, seeing that the aircraft needs to perform an inclination, in relation to the rotor s plane, to accelerate in a desired direction. Moreover, the outer loop computes the total thrust magnitude f R to control the vertical acceleration. On the other h, the attitude control block calculates the torque τ R 3 that makes the aircraft follow the attitude comms d ref. Simulations reduce the development time of Trajectory Control Attitude Control Multirotor Dynamics This section presents the tracking position problem its solution taking into account linear constraints set by using MPC. 2.1 Translational dynamic model Consider the multirotor vehicle the three Cartesian coordinate systems (CCS) illustrated in Figure 2. Assume that the vehicle has a rigid structure. The body CCS S B {X B, Y B, Z B } is fixed to the structure its origin coincides with the center of mass (CM) of the vehicle. The reference CCS S R {X R, Y R, Z R } is Earth-fixed its origin is at point O. Finally, the CCS S R {X R, Y R, Z R } is defined to be parallel to S R, but its origin is shifted to CM. Assume that S R is an inertial frame. Outer loop,, Inner loop Navigation System Sensors Figure 1: Block diagram structure. the project, as it allows the testing more quickly at a lower cost, thus it is possible to compare different control laws in the literature. For the development of control systems, a great choice is the software MATLAB/Simulink R, but is need the mathematical model of the vehicle to perform the simulation. These mathematical models are in some cases much as difficult to be generated or not available (Benini et al., 211). Furthermore, it is difficult to model mathematically complex robot behaviours in realistic environments. To solve this problem it is utilized a commercially available flight simulator that is able to simulate the vehicle dynamics as well as the flight environment. This paper refers to a scheme SIL simulation using MATLAB/Simulink R X-Plane R to face the position tracking problem under constraints on the total thrust vector. The corresponding constraint set is a piece of a cone with an inferior a superior spherical lids. This problem is solved using a linear state-space MPC, whose optimization is made hy by replacing the original conic constraint set by a circumscribed pyramidal space that renders a linear set of inequalities on the total thrust vector. The paper is organized as follows. Section II describes the multirotor translational dynamic defines the position tracking problem its solution. Section III describes Software-In-The-Loop simulation scheme. Section IV simulations tests Section V contains the conclusion suggestions for future work. 2 Tracking Position Problem Solution Figure 2: The Cartesian coordinate systems. Invoking the second Newton s law neglecting disturbance forces, the translational dynamics of the multirotor illustrated in Figure 2 can be immediately described in S R by the following second order differential equation: r = 1 m f + g, (1) where r [r x r y r z ] T R 3 is the position vector of CM, f [f x f y f z ] T R 3 is the total thrust vector, m is the mass of the vehicle, g is the gravitational acceleration. As illustrated in Figure 2, f is perpendicular to the rotor plane. Define the inclination angle φ R of the rotor plane to be the angle between Z B Z R. The angle φ can thus be expressed by φ cos 1 f z f, (2) where f f. Define the position tracking error r R 3 as r r r d, (3) where r d [r d,x r d,y r d,z ] T R 3 is a position comm. Problem 1. Let φ max R denote the maximum allowable value of φ f min R

3 f max R denote, respectively, the minimum maximum admissible values of f. The problem is to find a control law for f that minimizes r, subject to the inclination constraint φ φ max to the force magnitude constraint f min f f max. 2.2 State-Space Model Define the state vector x [r x ṙ x r y ṙ y r z ṙ z ] T R 6 the control input vector u R 3 intermediate perimeter u 1 m f. (4) g (a) (b) Let x(k) R 6, u(k) R 3 y(k) R 3 denote, respectively, the state vector, the control input vector the controlled output vector all described in discrete-time domain. Using the Euler discretization method with a sample time of T s = 3 ms, the discrete-time linear state-space model is given as x(k + 1) = A d x(k) + B d u(k) y(k) = C d x(k), (5) where A d = R 6 6, (6).13.5 B d =.13.5 R 6 3, (7) C d = 1 R 3 6 (8) Thrust Vector Constraints Using (4), the thrust magnitude constraint f min f f max can be rewritten in terms of u as f min mu + m g f max, (9) f min m u 2 x + u 2 y + (u z + g) 2 f max. (1) Assuming that φ max < π/2 rad, the inclination constraint φ φ max can be replaced by cos φ cos φ max. (11) Figure 3: (a) Real constraint space; (b) Linearized constraint space. Using (2), (11) can be rewritten as f z f cos φ max, (12) which in turn, using (4), can be rewritten in terms of u, yielding u z + g u 2x + u 2y + (u z + g) 2 cos φ max. (13) In short, (1) (13) give the nonlinear constraints expressed in terms of the components of the control vector u. From the original form of the constraints as defined in Problem 1, one can visualize them as a conic space with spherical inferior superior lids, as illustrated in Figure 3a. Note that, besides nonlinear, the original constraint space is non-convex Linear Constraints A linear suboptimal constraint space is now obtained as being the pyramidal space circumscribed in the original constraint space of Figure 3a. The new space is depicted in Figure 3b. From Figure 3b, the new constraint along the Z R axis can immediately expressed as f min f z f max cos φ max. (14) Consider an arbitrary f z let it represent the Z R axis coordinate of the dotted perimeter represented in Figure 3b. The corresponding projection on the X R Y R plane is given in Figure 4. The later figure defines the distances α β enables to expresses them immediately as α = 2β cos π 4, (15) β = f z tan φ max. (16)

4 α f y CM Y R α f x Π 4 β X R intermediate perimeter Figure 4: Constraints top view. Now, replace (16) into (15) to obtain α = 2f z tan φ max (17) note that the linear constraints along the horizontal axes are given by α/2 f x, f y α/2. To finally obtain in matrix form yields where Λ f λ, (18) tan φ max tan φ max Λ = tan φ max tan φ max 1 1 (19) λ =. (2) f min f max cos φ max Finally, using (4), (18) can be immediately reformulate in terms of u as where Λu λ, (21) Λ mλ (22) λ λ mλ g 2.4 Model Predictive Controller. (23) This paper use the incremental-input formulation. The MPC is responsible for solving a quadratic programming by minimizing a quadratic cost function subject to the aforementioned linear constraints set. Only the first element of the optimized control increments sequence is utilized. The control comm that is effectively applied on the plant is obtained by summing the last control input with the optimized control increments sequence The optimization process is repeated at the next sampling instant based on new sensors measurements. The mathematical formulation is found with more details in (Lopes et al., 211). 2.5 Computing Thrust Magnitude Attitude Comms After the optimization process of the control input, it is necessary to convert u to the total thrust f the attitude reference angles, which will be utilized as a set-point for some inner attitude control loop. Thus, rewriting (4) as f x f y f z = m u x u y u z + g, (24) Then, by taking the norm of (24), we can define f as f = m u 2 x + u 2 y + (u z + g) 2. (25) The attitude comms for the inner control loop need to be computed from the vector f. However, it is can be easily seen that there are infinite attitude of S B to represent the multirotor orientation in S R, taking in account when the Z B axis coincides with f. Therefore, it is necessary to specify an unique attitude to the aerial robot. It can be done utilizing the attitude represented by the principal Euler angle/axis (φ, e). Define the principal Euler axis e R 3, shown in Figure (5), as e = Z R f xy Z R f xy, (26) where f xy [f x f y ] T R 2 denotes the horizontal projection of f. Utilizing the (2) (26), it is possible to represent the attitude of S B in relation to S R utilizing, for example, Euler angles one obtains φ, θ, ψ. 3 SOFTWARE-IN-THE-LOOP SIMULATION SCHEME The developed scheme provides resources for the evaluation of a trajectory control using a linear constrained MPC applied to multirotors. The schematic simulation contains computers with computing power graphics to process calculations for the MPC.

6 X R [m] Y R [m] Z R [m] 4 2 r d,x r x r d,y r y r d,z r z Time [s] Figure 7: φ f. ure 1 shows that in the beginning of the simulation the total thrust magnitude was saturated on its upper bound to make the vehicle take off then stabilized in approximately 9.81N that represents the minimum thrust force to maintain the vehicle on a fixed altitude. Furthermore, the limits to φ were respected even with an oscillatory response. f [N] phi [deg] Total Thrust Magnitude Principal Euler Angle Time [s] Figure 8: φ f. 5 Conclusions This paper presents a scheme for Software-Inthe-Loop simulation applied to evaluate the performance of position tracking using MPC. The scheme is the integration of two software s. A software responsible for plant simulation the other to simulate the control system. A strategy was adopted in which the vehicle takes off follows a straight position. With the predictive control law that was tested in the environment SIL, it has become possible to evaluate the control law tracking position, a more realistic can be viewed movements of the aircraft in a flight simulator. As future work can exp the position tracking of a single vehicle for multiple vehicles, resulting in a cooperative control. Furthermore, one can carry out an expansion for a simulation Hardware-in-the-Loop (HIL). Acknowledgments The authors acknowledge the financial support of FAPEAM - Foundation for Research of the Amazon by means of Master Scholarships, as well as the ITA - Aeronautical Institute of Technology for the support needed to carry out this work. References Benini, A., Mancini, A., Minutolo, R., Longhi, S. Montanari, M. (211). A modular framework for fast prototyping of cooperative unmanned aerial vehicle, Journal of Intelligent Robotic Systems 65: Figueiredo, H. V. Saotome, O. (212a). Modelagem e simulação de veículo aéreo não tripulado (vant) do tipo quadricóptero uso o simulador x-plane e simulink, Anais do XIX Congresso Brasileiro de Automática, CBA 212, Campina Gre, Brazil. Figueiredo, H. V. Saotome, O. (212b). Simulation platform for quadricopter: Using matlab/simulink x-plane, XIV Simpósio de Aplicações Operacionais em Áreas de Defesa, SIGE 212, São José dos Campos, Brazil. Hoffman, G. M., Wasler, S. L. Tomlin, C. J. (28). Quadrotor helicopter trajectory tracking control, Proceedings of the AIAA Guidance, Navigation Control Conference Exhibit, Honolulu, Hawaii. Indriyanto, T. Jenie, Y. I. (21). Modeling simulation of a ducted fan unmanned aerial vehicle ( uav ) using x-plane simulation software, Regional Conference on Mechanical Aerospace Technology, Bali. Lopes, R. V., Santana, P. H. R. Q. A., Borges, G. A. Ishihara, J. Y. (211). Model predictive controle applied to tracking attitude stabilization of a vtol quadrotor aircraft, Proceedings of the 21st International Congress of Mechanical Engineering - COBEM, Natal, Brazil.

### Onboard electronics of UAVs

AARMS Vol. 5, No. 2 (2006) 237 243 TECHNOLOGY Onboard electronics of UAVs ANTAL TURÓCZI, IMRE MAKKAY Department of Electronic Warfare, Miklós Zrínyi National Defence University, Budapest, Hungary Recent

### CONTRIBUTIONS TO THE AUTOMATIC CONTROL OF AERIAL VEHICLES

1 / 23 CONTRIBUTIONS TO THE AUTOMATIC CONTROL OF AERIAL VEHICLES MINH DUC HUA 1 1 INRIA Sophia Antipolis, AROBAS team I3S-CNRS Sophia Antipolis, CONDOR team Project ANR SCUAV Supervisors: Pascal MORIN,

### Lecture L5 - Other Coordinate Systems

S. Widnall, J. Peraire 16.07 Dynamics Fall 008 Version.0 Lecture L5 - Other Coordinate Systems In this lecture, we will look at some other common systems of coordinates. We will present polar coordinates

### Rotation: Moment of Inertia and Torque

Rotation: Moment of Inertia and Torque Every time we push a door open or tighten a bolt using a wrench, we apply a force that results in a rotational motion about a fixed axis. Through experience we learn

### Bead moving along a thin, rigid, wire.

Bead moving along a thin, rigid, wire. odolfo. osales, Department of Mathematics, Massachusetts Inst. of Technology, Cambridge, Massachusetts, MA 02139 October 17, 2004 Abstract An equation describing

### INSTRUCTOR WORKBOOK Quanser Robotics Package for Education for MATLAB /Simulink Users

INSTRUCTOR WORKBOOK for MATLAB /Simulink Users Developed by: Amir Haddadi, Ph.D., Quanser Peter Martin, M.A.SC., Quanser Quanser educational solutions are powered by: CAPTIVATE. MOTIVATE. GRADUATE. PREFACE

### Mechanics lecture 7 Moment of a force, torque, equilibrium of a body

G.1 EE1.el3 (EEE1023): Electronics III Mechanics lecture 7 Moment of a force, torque, equilibrium of a body Dr Philip Jackson http://www.ee.surrey.ac.uk/teaching/courses/ee1.el3/ G.2 Moments, torque and

### MODELLING A SATELLITE CONTROL SYSTEM SIMULATOR

National nstitute for Space Research NPE Space Mechanics and Control Division DMC São José dos Campos, SP, Brasil MODELLNG A SATELLTE CONTROL SYSTEM SMULATOR Luiz C Gadelha Souza gadelha@dem.inpe.br rd

### KINEMATICS OF PARTICLES RELATIVE MOTION WITH RESPECT TO TRANSLATING AXES

KINEMTICS OF PRTICLES RELTIVE MOTION WITH RESPECT TO TRNSLTING XES In the previous articles, we have described particle motion using coordinates with respect to fixed reference axes. The displacements,

### The Design and Implementation of a Quadrotor Flight Controller Using the QUEST Algorithm

The Design and Implementation of a Quadrotor Flight Controller Using the QUEST Algorithm Jacob Oursland Department of Mathematics and Computer Science South Dakota School of Mines and Technology Rapid

### Control and Navigation Framework for Quadrotor Helicopters

Control and Navigation Framework for Quadrotor Helicopters Amr Nagaty, Sajad Saeedi, Carl Thibault, Mae Seto and Howard Li Abstract This paper presents the development of a nonlinear quadrotor simulation

### 2. Dynamics, Control and Trajectory Following

2. Dynamics, Control and Trajectory Following This module Flying vehicles: how do they work? Quick refresher on aircraft dynamics with reference to the magical flying space potato How I learned to stop

### Design-Simulation-Optimization Package for a Generic 6-DOF Manipulator with a Spherical Wrist

Design-Simulation-Optimization Package for a Generic 6-DOF Manipulator with a Spherical Wrist MHER GRIGORIAN, TAREK SOBH Department of Computer Science and Engineering, U. of Bridgeport, USA ABSTRACT Robot

### Lecture L6 - Intrinsic Coordinates

S. Widnall, J. Peraire 16.07 Dynamics Fall 2009 Version 2.0 Lecture L6 - Intrinsic Coordinates In lecture L4, we introduced the position, velocity and acceleration vectors and referred them to a fixed

### Force/position control of a robotic system for transcranial magnetic stimulation

Force/position control of a robotic system for transcranial magnetic stimulation W.N. Wan Zakaria School of Mechanical and System Engineering Newcastle University Abstract To develop a force control scheme

### Chapter 24 Physical Pendulum

Chapter 4 Physical Pendulum 4.1 Introduction... 1 4.1.1 Simple Pendulum: Torque Approach... 1 4. Physical Pendulum... 4.3 Worked Examples... 4 Example 4.1 Oscillating Rod... 4 Example 4.3 Torsional Oscillator...

### Quadcopters. Presented by: Andrew Depriest

Quadcopters Presented by: Andrew Depriest What is a quadcopter? Helicopter - uses rotors for lift and propulsion Quadcopter (aka quadrotor) - uses 4 rotors Parrot AR.Drone 2.0 History 1907 - Breguet-Richet

### Homework 4. problems: 5.61, 5.67, 6.63, 13.21

Homework 4 problems: 5.6, 5.67, 6.6,. Problem 5.6 An object of mass M is held in place by an applied force F. and a pulley system as shown in the figure. he pulleys are massless and frictionless. Find

### Figure 1.1 Vector A and Vector F

CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have

### PID, LQR and LQR-PID on a Quadcopter Platform

PID, LQR and LQR-PID on a Quadcopter Platform Lucas M. Argentim unielargentim@fei.edu.br Willian C. Rezende uniewrezende@fei.edu.br Paulo E. Santos psantos@fei.edu.br Renato A. Aguiar preaguiar@fei.edu.br

### HYDRAULIC ARM MODELING VIA MATLAB SIMHYDRAULICS

Engineering MECHANICS, Vol. 16, 2009, No. 4, p. 287 296 287 HYDRAULIC ARM MODELING VIA MATLAB SIMHYDRAULICS Stanislav Věchet, Jiří Krejsa* System modeling is a vital tool for cost reduction and design

### Research Methodology Part III: Thesis Proposal. Dr. Tarek A. Tutunji Mechatronics Engineering Department Philadelphia University - Jordan

Research Methodology Part III: Thesis Proposal Dr. Tarek A. Tutunji Mechatronics Engineering Department Philadelphia University - Jordan Outline Thesis Phases Thesis Proposal Sections Thesis Flow Chart

### Animations in Creo 3.0

Animations in Creo 3.0 ME170 Part I. Introduction & Outline Animations provide useful demonstrations and analyses of a mechanism's motion. This document will present two ways to create a motion animation

### 5.2 Rotational Kinematics, Moment of Inertia

5 ANGULAR MOTION 5.2 Rotational Kinematics, Moment of Inertia Name: 5.2 Rotational Kinematics, Moment of Inertia 5.2.1 Rotational Kinematics In (translational) kinematics, we started out with the position

### CHAPTER 11. The total energy of the body in its orbit is a constant and is given by the sum of the kinetic and potential energies

CHAPTER 11 SATELLITE ORBITS 11.1 Orbital Mechanics Newton's laws of motion provide the basis for the orbital mechanics. Newton's three laws are briefly (a) the law of inertia which states that a body at

### 1.10 Using Figure 1.6, verify that equation (1.10) satisfies the initial velocity condition. t + ") # x (t) = A! n. t + ") # v(0) = A!

1.1 Using Figure 1.6, verify that equation (1.1) satisfies the initial velocity condition. Solution: Following the lead given in Example 1.1., write down the general expression of the velocity by differentiating

### Response to Harmonic Excitation Part 2: Damped Systems

Response to Harmonic Excitation Part 2: Damped Systems Part 1 covered the response of a single degree of freedom system to harmonic excitation without considering the effects of damping. However, almost

### 11.1. Objectives. Component Form of a Vector. Component Form of a Vector. Component Form of a Vector. Vectors and the Geometry of Space

11 Vectors and the Geometry of Space 11.1 Vectors in the Plane Copyright Cengage Learning. All rights reserved. Copyright Cengage Learning. All rights reserved. 2 Objectives! Write the component form of

### Rotational inertia (moment of inertia)

Rotational inertia (moment of inertia) Define rotational inertia (moment of inertia) to be I = Σ m i r i 2 or r i : the perpendicular distance between m i and the given rotation axis m 1 m 2 x 1 x 2 Moment

### On Motion of Robot End-Effector using the Curvature Theory of Timelike Ruled Surfaces with Timelike Directrix

Malaysian Journal of Mathematical Sciences 8(2): 89-204 (204) MALAYSIAN JOURNAL OF MATHEMATICAL SCIENCES Journal homepage: http://einspem.upm.edu.my/journal On Motion of Robot End-Effector using the Curvature

### Fluid Mechanics Prof. S. K. Som Department of Mechanical Engineering Indian Institute of Technology, Kharagpur

Fluid Mechanics Prof. S. K. Som Department of Mechanical Engineering Indian Institute of Technology, Kharagpur Lecture - 20 Conservation Equations in Fluid Flow Part VIII Good morning. I welcome you all

### An inertial haptic interface for robotic applications

An inertial haptic interface for robotic applications Students: Andrea Cirillo Pasquale Cirillo Advisor: Ing. Salvatore Pirozzi Altera Innovate Italy Design Contest 2012 Objective Build a Low Cost Interface

### ACTUATOR DESIGN FOR ARC WELDING ROBOT

ACTUATOR DESIGN FOR ARC WELDING ROBOT 1 Anurag Verma, 2 M. M. Gor* 1 G.H Patel College of Engineering & Technology, V.V.Nagar-388120, Gujarat, India 2 Parul Institute of Engineering & Technology, Limda-391760,

### Chapter. 4 Mechanism Design and Analysis

Chapter. 4 Mechanism Design and Analysis 1 All mechanical devices containing moving parts are composed of some type of mechanism. A mechanism is a group of links interacting with each other through joints

### Stability Of Structures: Basic Concepts

23 Stability Of Structures: Basic Concepts ASEN 3112 Lecture 23 Slide 1 Objective This Lecture (1) presents basic concepts & terminology on structural stability (2) describes conceptual procedures for

### A Simple Model for Ski Jump Flight Mechanics Used as a Tool for Teaching Aircraft Gliding Flight

e-περιοδικό Επιστήμης & Τεχνολογίας 33 A Simple Model for Ski Jump Flight Mechanics Used as a Tool for Teaching Aircraft Gliding Flight Vassilios McInnes Spathopoulos Department of Aircraft Technology

### Orbital Mechanics. Angular Momentum

Orbital Mechanics The objects that orbit earth have only a few forces acting on them, the largest being the gravitational pull from the earth. The trajectories that satellites or rockets follow are largely

### Model Predictive Control Lecture 5

Model Predictive Control Lecture 5 Klaus Trangbæk ktr@es.aau.dk Automation & Control Aalborg University Denmark. http://www.es.aau.dk/staff/ktr/mpckursus/mpckursus.html mpc5 p. 1 Exercise from last time

### ELEMENTS OF VECTOR ALGEBRA

ELEMENTS OF VECTOR ALGEBRA A.1. VECTORS AND SCALAR QUANTITIES We have now proposed sets of basic dimensions and secondary dimensions to describe certain aspects of nature, but more than just dimensions

### Chapter 13, example problems: x (cm) 10.0

Chapter 13, example problems: (13.04) Reading Fig. 13-30 (reproduced on the right): (a) Frequency f = 1/ T = 1/ (16s) = 0.0625 Hz. (since the figure shows that T/2 is 8 s.) (b) The amplitude is 10 cm.

### Units and Vectors: Tools for Physics

Chapter 1 Units and Vectors: Tools for Physics 1.1 The Important Stuff 1.1.1 The SI System Physics is based on measurement. Measurements are made by comparisons to well defined standards which define the

### Pierre Bigot, Valdemir Carrara, Philipe Massad National Institute for Space Research- INPE, Av. dos Astronautas, 1758 São José dos Campos, SP, Brasil

NUMERICAL CALCULATION ALGORITHM OF ATTITUDE AND MOVEMENT OF A BOOMERANG Pierre Bigot, Valdemir Carrara, Philipe Massad National Institute for Space Research- INPE, Av. dos Astronautas, 1758 São José dos

### IMU Components An IMU is typically composed of the following components:

APN-064 IMU Errors and Their Effects Rev A Introduction An Inertial Navigation System (INS) uses the output from an Inertial Measurement Unit (IMU), and combines the information on acceleration and rotation

### Spacecraft Dynamics and Control. An Introduction

Brochure More information from http://www.researchandmarkets.com/reports/2328050/ Spacecraft Dynamics and Control. An Introduction Description: Provides the basics of spacecraft orbital dynamics plus attitude

### Lecture L2 - Degrees of Freedom and Constraints, Rectilinear Motion

S. Widnall 6.07 Dynamics Fall 009 Version.0 Lecture L - Degrees of Freedom and Constraints, Rectilinear Motion Degrees of Freedom Degrees of freedom refers to the number of independent spatial coordinates

### Chapter 18 Static Equilibrium

Chapter 8 Static Equilibrium 8. Introduction Static Equilibrium... 8. Lever Law... Example 8. Lever Law... 4 8.3 Generalized Lever Law... 5 8.4 Worked Examples... 7 Example 8. Suspended Rod... 7 Example

### Operational Space Control for A Scara Robot

Operational Space Control for A Scara Robot Francisco Franco Obando D., Pablo Eduardo Caicedo R., Oscar Andrés Vivas A. Universidad del Cauca, {fobando, pacaicedo, avivas }@unicauca.edu.co Abstract This

### Unified Lecture # 4 Vectors

Fall 2005 Unified Lecture # 4 Vectors These notes were written by J. Peraire as a review of vectors for Dynamics 16.07. They have been adapted for Unified Engineering by R. Radovitzky. References [1] Feynmann,

### Differential Relations for Fluid Flow. Acceleration field of a fluid. The differential equation of mass conservation

Differential Relations for Fluid Flow In this approach, we apply our four basic conservation laws to an infinitesimally small control volume. The differential approach provides point by point details of

### Vector Algebra II: Scalar and Vector Products

Chapter 2 Vector Algebra II: Scalar and Vector Products We saw in the previous chapter how vector quantities may be added and subtracted. In this chapter we consider the products of vectors and define

### Abstract. Introduction

SPACECRAFT APPLICATIONS USING THE MICROSOFT KINECT Matthew Undergraduate Student Advisor: Dr. Troy Henderson Aerospace and Ocean Engineering Department Virginia Tech Abstract This experimental study involves

### CHAPTER 1 INTRODUCTION

CHAPTER 1 INTRODUCTION 1.1 Background of the Research Agile and precise maneuverability of helicopters makes them useful for many critical tasks ranging from rescue and law enforcement task to inspection

### IMPORTANT NOTE ABOUT WEBASSIGN:

Week 8 homework IMPORTANT NOTE ABOUT WEBASSIGN: In the WebAssign versions of these problems, various details have been changed, so that the answers will come out differently. The method to find the solution

### An Introduction to Applied Mathematics: An Iterative Process

An Introduction to Applied Mathematics: An Iterative Process Applied mathematics seeks to make predictions about some topic such as weather prediction, future value of an investment, the speed of a falling

### Lecture L3 - Vectors, Matrices and Coordinate Transformations

S. Widnall 16.07 Dynamics Fall 2009 Lecture notes based on J. Peraire Version 2.0 Lecture L3 - Vectors, Matrices and Coordinate Transformations By using vectors and defining appropriate operations between

### Modeling and Autonomous Flight Simulation of a Small Unmanned Aerial Vehicle

13 th International Conference on AEROSPACE SCIENCES & AVIATION TECHNOLOGY, ASAT- 13, May 26 28, 2009, E-Mail: asat@mtc.edu.eg Military Technical College, Kobry Elkobbah, Cairo, Egypt Tel : +(202) 24025292

### Chapter 2 Lead Screws

Chapter 2 Lead Screws 2.1 Screw Threads The screw is the last machine to joint the ranks of the six fundamental simple machines. It has a history that stretches back to the ancient times. A very interesting

### MATHEMATICS CLASS - XII BLUE PRINT - II. (1 Mark) (4 Marks) (6 Marks)

BLUE PRINT - II MATHEMATICS CLASS - XII S.No. Topic VSA SA LA TOTAL ( Mark) (4 Marks) (6 Marks). (a) Relations and Functions 4 () 6 () 0 () (b) Inverse trigonometric Functions. (a) Matrices Determinants

### Hardware In The Loop Simulator in UAV Rapid Development Life Cycle

Hardware In The Loop Simulator in UAV Rapid Development Life Cycle Widyawardana Adiprawita*, Adang Suwandi Ahmad = and Jaka Semibiring + *School of Electric Engineering and Informatics Institut Teknologi

### 1.7 Cylindrical and Spherical Coordinates

56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a two-dimensional coordinate system in which the

### CONTROL SYSTEMS, ROBOTICS AND AUTOMATION Vol. XVI - Fault Accomodation Using Model Predictive Methods - Jovan D. Bošković and Raman K.

FAULT ACCOMMODATION USING MODEL PREDICTIVE METHODS Scientific Systems Company, Inc., Woburn, Massachusetts, USA. Keywords: Fault accommodation, Model Predictive Control (MPC), Failure Detection, Identification

### CORE STANDARDS, OBJECTIVES, AND INDICATORS

Aerospace Engineering - PLtW Levels: 11-12 Units of Credit: 1.0 CIP Code: 14.0201 Core Code: 38-01-00-00-150 Prerequisite: Principles of Engineering, Introduction to Engineering Design Test: #967 Course

### Interactive Motion Simulators

motionsimulator motionsimulator About the Company Since its founding in 2003, the company Buck Engineering & Consulting GmbH (BEC), with registered offices in Reutlingen (Germany) has undergone a continuous

### Whole-body dynamic motion planning with centroidal dynamics and full kinematics

Sep. 18. 2014 Introduction Linear Inverted Pendulum compute ZMP with point-mass model linear system, analytical solution co-planar contact solve kinematics separately Our approach dynamic constraint for

### OUTCOME 2 KINEMATICS AND DYNAMICS TUTORIAL 2 PLANE MECHANISMS. You should judge your progress by completing the self assessment exercises.

Unit 60: Dynamics of Machines Unit code: H/601/1411 QCF Level:4 Credit value:15 OUTCOME 2 KINEMATICS AND DYNAMICS TUTORIAL 2 PLANE MECHANISMS 2 Be able to determine the kinetic and dynamic parameters of

### Physics 111: Lecture 4: Chapter 4 - Forces and Newton s Laws of Motion. Physics is about forces and how the world around us reacts to these forces.

Physics 111: Lecture 4: Chapter 4 - Forces and Newton s Laws of Motion Physics is about forces and how the world around us reacts to these forces. Whats a force? Contact and non-contact forces. Whats a

### MATH 304 Linear Algebra Lecture 24: Scalar product.

MATH 304 Linear Algebra Lecture 24: Scalar product. Vectors: geometric approach B A B A A vector is represented by a directed segment. Directed segment is drawn as an arrow. Different arrows represent

### Universal Law of Gravitation

Universal Law of Gravitation Law: Every body exerts a force of attraction on every other body. This force called, gravity, is relatively weak and decreases rapidly with the distance separating the bodies

### THE UNIVERSITY OF TEXAS AT AUSTIN Department of Aerospace Engineering and Engineering Mechanics. EM 311M - DYNAMICS Spring 2012 SYLLABUS

THE UNIVERSITY OF TEXAS AT AUSTIN Department of Aerospace Engineering and Engineering Mechanics EM 311M - DYNAMICS Spring 2012 SYLLABUS UNIQUE NUMBERS: 13815, 13820, 13825, 13830 INSTRUCTOR: TIME: Dr.

### SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS

SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS This work covers elements of the syllabus for the Engineering Council exams C105 Mechanical and Structural Engineering

### An Introduction to Using Simulink. Exercises

An Introduction to Using Simulink Exercises Eric Peasley, Department of Engineering Science, University of Oxford version 4.1, 2013 PART 1 Exercise 1 (Cannon Ball) This exercise is designed to introduce

### Lab 7: Rotational Motion

Lab 7: Rotational Motion Equipment: DataStudio, rotary motion sensor mounted on 80 cm rod and heavy duty bench clamp (PASCO ME-9472), string with loop at one end and small white bead at the other end (125

### Metrics on SO(3) and Inverse Kinematics

Mathematical Foundations of Computer Graphics and Vision Metrics on SO(3) and Inverse Kinematics Luca Ballan Institute of Visual Computing Optimization on Manifolds Descent approach d is a ascent direction

### Linear Motion vs. Rotational Motion

Linear Motion vs. Rotational Motion Linear motion involves an object moving from one point to another in a straight line. Rotational motion involves an object rotating about an axis. Examples include a

### Acceleration due to Gravity

Acceleration due to Gravity 1 Object To determine the acceleration due to gravity by different methods. 2 Apparatus Balance, ball bearing, clamps, electric timers, meter stick, paper strips, precision

### Solution: (a) For a positively charged particle, the direction of the force is that predicted by the right hand rule. These are:

Problem 1. (a) Find the direction of the force on a proton (a positively charged particle) moving through the magnetic fields as shown in the figure. (b) Repeat part (a), assuming the moving particle is

### Lecture L29-3D Rigid Body Dynamics

J. Peraire, S. Widnall 16.07 Dynamics Fall 2009 Version 2.0 Lecture L29-3D Rigid Body Dynamics 3D Rigid Body Dynamics: Euler Angles The difficulty of describing the positions of the body-fixed axis of

### EDUMECH Mechatronic Instructional Systems. Ball on Beam System

EDUMECH Mechatronic Instructional Systems Ball on Beam System Product of Shandor Motion Systems Written by Robert Hirsch Ph.D. 998-9 All Rights Reserved. 999 Shandor Motion Systems, Ball on Beam Instructional

### Lecture L19 - Vibration, Normal Modes, Natural Frequencies, Instability

S. Widnall 16.07 Dynamics Fall 2009 Version 1.0 Lecture L19 - Vibration, Normal Modes, Natural Frequencies, Instability Vibration, Instability An important class of problems in dynamics concerns the free

### Solving Simultaneous Equations and Matrices

Solving Simultaneous Equations and Matrices The following represents a systematic investigation for the steps used to solve two simultaneous linear equations in two unknowns. The motivation for considering

### The Stern-Gerlach Experiment

Chapter The Stern-Gerlach Experiment Let us now talk about a particular property of an atom, called its magnetic dipole moment. It is simplest to first recall what an electric dipole moment is. Consider

### Isaac Newton s (1642-1727) Laws of Motion

Big Picture 1 2.003J/1.053J Dynamics and Control I, Spring 2007 Professor Thomas Peacock 2/7/2007 Lecture 1 Newton s Laws, Cartesian and Polar Coordinates, Dynamics of a Single Particle Big Picture First

### Physics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives

Physics 9e/Cutnell correlated to the College Board AP Physics 1 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring

### Let s first see how precession works in quantitative detail. The system is illustrated below: ...

lecture 20 Topics: Precession of tops Nutation Vectors in the body frame The free symmetric top in the body frame Euler s equations The free symmetric top ala Euler s The tennis racket theorem As you know,

### Matlab and Simulink. Matlab and Simulink for Control

Matlab and Simulink for Control Automatica I (Laboratorio) 1/78 Matlab and Simulink CACSD 2/78 Matlab and Simulink for Control Matlab introduction Simulink introduction Control Issues Recall Matlab design

### Lecture 07: Work and Kinetic Energy. Physics 2210 Fall Semester 2014

Lecture 07: Work and Kinetic Energy Physics 2210 Fall Semester 2014 Announcements Schedule next few weeks: 9/08 Unit 3 9/10 Unit 4 9/15 Unit 5 (guest lecturer) 9/17 Unit 6 (guest lecturer) 9/22 Unit 7,

### Lecture L17 - Orbit Transfers and Interplanetary Trajectories

S. Widnall, J. Peraire 16.07 Dynamics Fall 008 Version.0 Lecture L17 - Orbit Transfers and Interplanetary Trajectories In this lecture, we will consider how to transfer from one orbit, to another or to

### AP1 Gravity. at an altitude equal to twice the radius (R) of the planet. What is the satellite s speed assuming a perfectly circular orbit?

1. A satellite of mass m S orbits a planet of mass m P at an altitude equal to twice the radius (R) of the planet. What is the satellite s speed assuming a perfectly circular orbit? (A) v = Gm P R (C)

### PHY121 #8 Midterm I 3.06.2013

PHY11 #8 Midterm I 3.06.013 AP Physics- Newton s Laws AP Exam Multiple Choice Questions #1 #4 1. When the frictionless system shown above is accelerated by an applied force of magnitude F, the tension

### Lecture L22-2D Rigid Body Dynamics: Work and Energy

J. Peraire, S. Widnall 6.07 Dynamics Fall 008 Version.0 Lecture L - D Rigid Body Dynamics: Work and Energy In this lecture, we will revisit the principle of work and energy introduced in lecture L-3 for

### Penn State University Physics 211 ORBITAL MECHANICS 1

ORBITAL MECHANICS 1 PURPOSE The purpose of this laboratory project is to calculate, verify and then simulate various satellite orbit scenarios for an artificial satellite orbiting the earth. First, there

### When the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid.

Fluid Statics When the fluid velocity is zero, called the hydrostatic condition, the pressure variation is due only to the weight of the fluid. Consider a small wedge of fluid at rest of size Δx, Δz, Δs

### Physics 403: Relativity Takehome Final Examination Due 07 May 2007

Physics 403: Relativity Takehome Final Examination Due 07 May 2007 1. Suppose that a beam of protons (rest mass m = 938 MeV/c 2 of total energy E = 300 GeV strikes a proton target at rest. Determine the

### Flight dynamics helicopter model validation based on flight test data

757 Flight dynamics helicopter model validation based on flight test data A Hiliuta and R M Botez Department of Automated Production Engineering, Ecole de Technologie Supérieure, Montréal, Québec, Canada

### AOE 3134 Complete Aircraft Equations

AOE 3134 Complete Aircraft Equations The requirements for balance and stability that we found for the flying wing carry over directly to a complete aircraft. In particular we require the zero-lift pitch

### Chapter 10 Rotational Motion. Copyright 2009 Pearson Education, Inc.

Chapter 10 Rotational Motion Angular Quantities Units of Chapter 10 Vector Nature of Angular Quantities Constant Angular Acceleration Torque Rotational Dynamics; Torque and Rotational Inertia Solving Problems

### Quadcopter Dynamics, Simulation, and Control Introduction

Quadcopter Dynamics, Simulation, and Control Introduction A helicopter is a flying vehicle which uses rapidly spinning rotors to push air downwards, thus creating a thrust force keeping the helicopter

### Stabilizing a Gimbal Platform using Self-Tuning Fuzzy PID Controller

Stabilizing a Gimbal Platform using Self-Tuning Fuzzy PID Controller Nourallah Ghaeminezhad Collage Of Automation Engineering Nuaa Nanjing China Wang Daobo Collage Of Automation Engineering Nuaa Nanjing