MODELLING A SATELLITE CONTROL SYSTEM SIMULATOR

Size: px
Start display at page:

Download "MODELLING A SATELLITE CONTROL SYSTEM SIMULATOR"

Transcription

1 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 rd nternational Workshop and Advanced School - Spaceflight Dynamics and Control October 8-, University of Beira nterior - Covilhã, Portugal.

2 ntroduction Placing a satellite in orbit is a risky and expensive process. Space Projects must guarantee that satellite and/or its equipments work properly. Attitude Control System ACS should use new control techniques to improve reliability and performance. Experimental validation of new equipment and/or control techniques through simulators prototypes is one way to increase confidence and performance of the system.

3 Types of Simulators Basically, there are two types of simulators: The Planar one, with translational motion in one or two directions The spherical one, with rotation around one, two or three axes. The simulators consist of a platform supported on a plane or a spherical air bearing. The platform can accommodate various satellites components: like sensors, actuators, computers and its respective interface and electronic.

4 Example of Simulators Planar Simulador - Stanford University robotic arm 4

5 Example of Simulators Spherical Simulator - Georgia nstitute of Technology GT 5

6 DMC Lab ativities Brazilian Data Collection Satellite Prototype for experimental verification of its various sub systems 6

7 DMC Lab ativities 7

8 DMC Lab ativities Attitude Maneuvers Software for the China Brasil Earth Remote Sensing Satellite CBERS 8

9 DMC - Simulators DMC is responsible for constructing two simulators to test and implementing satellite ACS. A D simulator with rotation around the vertical axis with gyro as sensor and reaction wheel as actuator. 9

10 DMC - Simulators A D simulator with rotation around three axes, over which is possible to put satellite ACS components like sensors, actuators, computers, batteries and etc.

11 DMC Simulators

12 Objetives This talk presents the development of a D Satellite Attitude Control System Simulator Software Model. This simulator model allows to investigate fundamental aspects of the satellite dynamics and attitude control system.

13 Objetives From the Simulator Model One designs the simulator ACS based on a PD controller with gain obtained by the pole allocation method After that, using recursive least squares method the platform inertia parameters are estimated, considering data from the Simulator model.

14 Objetives Once the recursive least squares method has been checked. One uses it to estimate the D simulator inertia moment having experimental data from gyro and reaction wheel. 4

15 D Platform Equations of Motion The platform angular velocity is given by W pi qj rk w W The total angular moment is the sum of the base and reaction wheels angular moment r cg mg H B r W r dm R RW i R W ρ w i i ρ dm i w w Deriving the previously expression the equation of motion of the platform is given by r cg dh mg h dt r W h h W i r i i hi 5

16 D Platform Equations of Motion The reaction wheels equations of motion are T T T [ w p ] [ w q ] [ w r ] w W The kinematic equations considering Euler angles φ, θ, ψ in the sequence -- are w mg r cg w φ θ ψ p tan θ q cos φ r sin φ sec θ [ q sin φ r cos φ ] [ qsin φ r cos φ ] 6

17 7 [ ] [ ] cos sin cos sin cos cos sin tan sin cos sin sin cos cos cos sin cos cos T T T r q r q r q p mgr mgr p w q w p q pr qr pq mgr mgr p w r w r p qr pq pr mgr mgr q w r w q r pq pr qr w w w r q p y x xy yz xz yy xx z x xz xy yz xx zz z y yz xz xy zz xx zz yz xz yz yy xy xz xy xx φ φ θ φ φ φ φ θ θ θ φ θ θ φ θ φ θ φ ψ θ φ Putting together the previous equations of motion in matrix form yields D Platform Equations of Motion D Platform Equations of Motion

18 8 Control Law Design Control Law Design T T T r q p r q p zz yy xx ψ θ φ ψ θ φ CX Y Bu AX X u X The control gains are obtained applying the pole allocation method To design the control law, one needs the linear system. therefore, assuming small angles the equations of motion for designing purpose are

19 Simulation Results TABLE Typical Platform data used in the simulations Platform Platform Reaction wheel External torque xx. xy..5 Mgr x. yy. xz -..5 Mgr y.5 zz. yz..5 Mgr z.755 9

20 Simulation Results Using pole allocation method, one has defined three sets of poles p,,, in order to analyze the dynamic behavior of the system. p p p {.5 i.5 i. i. i. i. i} {. i. i.5. i.5. i.5. i.5. i} { } The first set of poles p is closer to the imaginary axis than the second set p and the third set p has only real part

21 Simulation Results 5 Simulacao nao linear p x t polos polos polos 5 velocidadedeg/s 5 5 Figures show the angular velocity p, q and r of the platform for the three set of poles p, p and p. velocidadedeg/s t s Simulacao nao linear q x t polos polos polos t s 8 7 Simulacao nao linear r x t polos polos polos 6 5 velocidadedeg/s t s

22 Simulation Results 5 Simulacao nao linear phi x t polos polos polos 5 angulodeg Figures show the angles φ, θ, ψ of the platform for the three set of poles p, p and p. angulodeg t s Simulacao nao linear theta x t polos polos polos t s Simulacao nao linear psi x t polos polos polos - angulodeg t s

23 Simulation Results 4 Simulacao nao linear w x t polos polos polos velocidaderpm - Figures show the reaction wheel rotation ω, ω, ω for the three set of poles p, p and p. velocidaderpm t s Simulacao nao linear w x t polos polos polos t s Simulacao nao linear w x t polos polos polos - -4 velocidaderpm t s

24 Simulation Results Simulacao nao linear T x t polos polos polos - torquen.m Figures show the torques T,, applied by reaction wheel for the three set of poles p, p and p. torquen.m t s Simulacao nao linear T x t polos polos polos t s 5 Simulacao nao linear T x t polos polos polos torquen.m t s 4

25 Comments : Dynamics and Control The first set of poles red line have the undesirable low damping rate associated with great oscillation. The third set of poles blue line although it shows short time for damping the overshoots reach great values. The second set of poles green line, reduce the angular velocities and angles in short time, with small overshoot and the reaction wheels rotation are in acceptable levels. n the sequel the D platform model with the control law designed with the second set of poles are used to generated data to estimate the platform inertia parameters. 5

26 6 Parameters Estimation Parameters Estimation [ ]{ } { } Y X G [ ] G G G G { } Y Y Y Y n the estimation process the vector X has the inertia parameters and the location of the platform gravity center. The matrix G and vector Y contain angles, angular velocities, sensor measures and reaction wheels inertia which are known.

27 7 Parameters Estimation Parameters Estimation [ ] [ ][ ] { } [ ][ ] { } Y G P X G G P T T [ ] [ ][ ] [ ] [ ][ ][ ] [ ] [ ] [ ][ ] [ ] { } { } [ ] { } [ ]{ } T T X G Y L X X P G L P G P G G P L The recursive form of the least square method needs to satisfy the following equations :

28 8 Parameters Estimation Parameters Estimation [ ] T sin sin sin sin rq p p r p q r r p q pq r r q q p rq p rp q r pr rq pq q rq q p r p p G cos coscos cos cos cos θ θ φ θ θ φ θ φ θ φ { } wp w q w w p wr w wq wr w Y yz xz xy zz yy xx z y x mgr mgr mgr X,,,,,,,, The matrices G, Y and X are given by

29 Parameters Estimation The parameters are estimated with measures that have been done in time interval of 5s for simulation of s. The results are shown in the next Figures 9

30 Parameters Estimation 5 x 8 Mimimos quadrados recursivo 4 xx yy zz inercia kg.m t s Platform principal inertia moments estimation

31 Parameters Estimation 5 x 8 Mimimos quadrados recursivo 4 xy yz xz inercia kg.m t s Platform cross inertia moments estimation

32 Parameters Estimation 5 x 8 Mimimos quadrados recursivo 4 mgrx mgry mgrz Força x braço N.m t s External torque estimation

33 Comments : Parameters Estimation From the previous result, one observes that the recursive least square method is reliable. Therefore, it will be used to estimate the D simulator inertia parameter from experimental data.

34 nertia estimation - D Platform The previous recursive procedure is applied considering the simplification of the D equation of motion for rotation around the vertical axis which is given by r zz w J [ r ]{ } { Jw } zz [ G ]{ X } { Y} mg W r cg z x y w y z x Z X Y Where the experimental data come from gyros and reaction wheel. 4

35 nertia estimation - D Platform The equipments used to perform the experiments are : The air baring platform diameters : 65mm Sunspace reaction wheel Angular rotation : -/ 4 rpm Maximum torque : 5mNm Maximum angular moment :.65Nms nertia moment :.5E- gm.m Voltage : Vdc Sunspace Fiber Optics Gyroscope Field of measure : -/ 8º/s Freeware Radio-Modem ; MHz Rate : bps with RS- protocol Battery : Vdc National nstruments PC.6GHz nterface : RS-/RS-485 5

36 Experiment Procedure One stars with both angular velocities of the platform and reaction wheel equal to zero. Then one sends a commander to the reaction wheel so that it increases its angular velocity up to a certain value. That action makes the platform to move with opposite angular velocity. After that, one sends a commander to decrease the reaction wheel angular velocity up to zero. Again the platform will react with angular motion in opposite direction. During that process the platform is monitored by the gyroscope and the reaction wheel angular velocity is also measured. t is important to say that the platform friction has been neglected. 6

37 Experiment Procedure The reaction wheel angular velocity 7

38 Experiment Procedure The platform angular velocity 8

39 nertia moment estimation zz.495 gm.m 9

40 Summary This talk presents a mathematical model of a platform that simulates a satellite ACS in -D with three reaction wheels as actuators and three gyros as sensors. A control law based on a PD controller using poles allocation method is designed and its performance is evaluated. That model is used to generated data to estimate the inertia parameters of the platform, using the least square recursive method. The simulations has shown that the recursive method is reliable for the simulator objectives. The D platform inertia moment is estimated using real data using the recursive method. 4

41 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 Thank you! rd nternational Workshop and Advanced School - Spaceflight Dynamics and Control October 8-, University of Beira nterior - Covilhã, Portugal. 4

PHY121 #8 Midterm I 3.06.2013

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

More information

CHAPTER 2 ORBITAL DYNAMICS

CHAPTER 2 ORBITAL DYNAMICS 14 CHAPTER 2 ORBITAL DYNAMICS 2.1 INTRODUCTION This chapter presents definitions of coordinate systems that are used in the satellite, brief description about satellite equations of motion and relative

More information

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

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

More information

Manufacturing Equipment Modeling

Manufacturing Equipment Modeling QUESTION 1 For a linear axis actuated by an electric motor complete the following: a. Derive a differential equation for the linear axis velocity assuming viscous friction acts on the DC motor shaft, leadscrew,

More information

11. Rotation Translational Motion: Rotational Motion:

11. Rotation Translational Motion: Rotational Motion: 11. Rotation Translational Motion: Motion of the center of mass of an object from one position to another. All the motion discussed so far belongs to this category, except uniform circular motion. Rotational

More information

Lecture L30-3D Rigid Body Dynamics: Tops and Gyroscopes

Lecture L30-3D Rigid Body Dynamics: Tops and Gyroscopes J. Peraire, S. Widnall 16.07 Dynamics Fall 2008 Version 2.0 Lecture L30-3D Rigid Body Dynamics: Tops and Gyroscopes 3D Rigid Body Dynamics: Euler Equations in Euler Angles In lecture 29, we introduced

More information

Rotation: Moment of Inertia and Torque

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

More information

Midterm Solutions. mvr = ω f (I wheel + I bullet ) = ω f 2 MR2 + mr 2 ) ω f = v R. 1 + M 2m

Midterm Solutions. mvr = ω f (I wheel + I bullet ) = ω f 2 MR2 + mr 2 ) ω f = v R. 1 + M 2m Midterm Solutions I) A bullet of mass m moving at horizontal velocity v strikes and sticks to the rim of a wheel a solid disc) of mass M, radius R, anchored at its center but free to rotate i) Which of

More information

PHYSICS 111 HOMEWORK SOLUTION #10. April 10, 2013

PHYSICS 111 HOMEWORK SOLUTION #10. April 10, 2013 PHYSICS 111 HOMEWORK SOLUTION #10 April 10, 013 0.1 Given M = 4 i + j 3 k and N = i j 5 k, calculate the vector product M N. By simply following the rules of the cross product: i i = j j = k k = 0 i j

More information

RAJALAKSHMI ENGINEERING COLLEGE MA 2161 UNIT I - ORDINARY DIFFERENTIAL EQUATIONS PART A

RAJALAKSHMI ENGINEERING COLLEGE MA 2161 UNIT I - ORDINARY DIFFERENTIAL EQUATIONS PART A RAJALAKSHMI ENGINEERING COLLEGE MA 26 UNIT I - ORDINARY DIFFERENTIAL EQUATIONS. Solve (D 2 + D 2)y = 0. 2. Solve (D 2 + 6D + 9)y = 0. PART A 3. Solve (D 4 + 4)x = 0 where D = d dt 4. Find Particular Integral:

More information

Quadcopter Dynamics, Simulation, and Control Introduction

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

More information

Chapter 11. h = 5m. = mgh + 1 2 mv 2 + 1 2 Iω 2. E f. = E i. v = 4 3 g(h h) = 4 3 9.8m / s2 (8m 5m) = 6.26m / s. ω = v r = 6.

Chapter 11. h = 5m. = mgh + 1 2 mv 2 + 1 2 Iω 2. E f. = E i. v = 4 3 g(h h) = 4 3 9.8m / s2 (8m 5m) = 6.26m / s. ω = v r = 6. Chapter 11 11.7 A solid cylinder of radius 10cm and mass 1kg starts from rest and rolls without slipping a distance of 6m down a house roof that is inclined at 30 degrees (a) What is the angular speed

More information

Physics 1A Lecture 10C

Physics 1A Lecture 10C Physics 1A Lecture 10C "If you neglect to recharge a battery, it dies. And if you run full speed ahead without stopping for water, you lose momentum to finish the race. --Oprah Winfrey Static Equilibrium

More information

Simple Harmonic Motion

Simple Harmonic Motion Simple Harmonic Motion 1 Object To determine the period of motion of objects that are executing simple harmonic motion and to check the theoretical prediction of such periods. 2 Apparatus Assorted weights

More information

AP Physics C. Oscillations/SHM Review Packet

AP Physics C. Oscillations/SHM Review Packet AP Physics C Oscillations/SHM Review Packet 1. A 0.5 kg mass on a spring has a displacement as a function of time given by the equation x(t) = 0.8Cos(πt). Find the following: a. The time for one complete

More information

SOLID MECHANICS TUTORIAL MECHANISMS KINEMATICS - VELOCITY AND ACCELERATION DIAGRAMS

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

More information

Columbia University Department of Physics QUALIFYING EXAMINATION

Columbia University Department of Physics QUALIFYING EXAMINATION Columbia University Department of Physics QUALIFYING EXAMINATION Monday, January 13, 2014 1:00PM to 3:00PM Classical Physics Section 1. Classical Mechanics Two hours are permitted for the completion of

More information

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 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

More information

Adding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors

Adding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors 1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number

More information

Rotation. Moment of inertia of a rotating body: w I = r 2 dm

Rotation. Moment of inertia of a rotating body: w I = r 2 dm Rotation Moment of inertia of a rotating body: w I = r 2 dm Usually reasonably easy to calculate when Body has symmetries Rotation axis goes through Center of mass Exams: All moment of inertia will be

More information

Torque Analyses of a Sliding Ladder

Torque Analyses of a Sliding Ladder Torque Analyses of a Sliding Ladder 1 Problem Kirk T. McDonald Joseph Henry Laboratories, Princeton University, Princeton, NJ 08544 (May 6, 2007) The problem of a ladder that slides without friction while

More information

EE 402 RECITATION #13 REPORT

EE 402 RECITATION #13 REPORT MIDDLE EAST TECHNICAL UNIVERSITY EE 402 RECITATION #13 REPORT LEAD-LAG COMPENSATOR DESIGN F. Kağan İPEK Utku KIRAN Ç. Berkan Şahin 5/16/2013 Contents INTRODUCTION... 3 MODELLING... 3 OBTAINING PTF of OPEN

More information

KINEMATICS OF PARTICLES RELATIVE MOTION WITH RESPECT TO TRANSLATING AXES

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,

More information

The dynamic equation for the angular motion of the wheel is R w F t R w F w ]/ J w

The dynamic equation for the angular motion of the wheel is R w F t R w F w ]/ J w Chapter 4 Vehicle Dynamics 4.. Introduction In order to design a controller, a good representative model of the system is needed. A vehicle mathematical model, which is appropriate for both acceleration

More information

Chapter 9 Rigid Body Motion in 3D

Chapter 9 Rigid Body Motion in 3D Chapter 9 Rigid Body Motion in 3D Rigid body rotation in 3D is a complicated problem requiring the introduction of tensors. Upon completion of this chapter we will be able to describe such things as the

More information

Power Electronics. Prof. K. Gopakumar. Centre for Electronics Design and Technology. Indian Institute of Science, Bangalore.

Power Electronics. Prof. K. Gopakumar. Centre for Electronics Design and Technology. Indian Institute of Science, Bangalore. Power Electronics Prof. K. Gopakumar Centre for Electronics Design and Technology Indian Institute of Science, Bangalore Lecture - 1 Electric Drive Today, we will start with the topic on industrial drive

More information

Onboard electronics of UAVs

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

More information

Solutions to old Exam 1 problems

Solutions to old Exam 1 problems Solutions to old Exam 1 problems Hi students! I am putting this old version of my review for the first midterm review, place and time to be announced. Check for updates on the web site as to which sections

More information

Lecture L29-3D Rigid Body Dynamics

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

More information

Angular acceleration α

Angular acceleration α Angular Acceleration Angular acceleration α measures how rapidly the angular velocity is changing: Slide 7-0 Linear and Circular Motion Compared Slide 7- Linear and Circular Kinematics Compared Slide 7-

More information

Chapter 11 Equilibrium

Chapter 11 Equilibrium 11.1 The First Condition of Equilibrium The first condition of equilibrium deals with the forces that cause possible translations of a body. The simplest way to define the translational equilibrium of

More information

Lab 9 Magnetic Interactions

Lab 9 Magnetic Interactions Lab 9 Magnetic nteractions Physics 6 Lab What You Need To Know: The Physics Electricity and magnetism are intrinsically linked and not separate phenomena. Most of the electrical devices you will encounter

More information

Dynamics of Rotational Motion

Dynamics of Rotational Motion Chapter 10 Dynamics of Rotational Motion PowerPoint Lectures for University Physics, Twelfth Edition Hugh D. Young and Roger A. Freedman Lectures by James Pazun Modified by P. Lam 5_31_2012 Goals for Chapter

More information

APPLIED MATHEMATICS ADVANCED LEVEL

APPLIED MATHEMATICS ADVANCED LEVEL APPLIED MATHEMATICS ADVANCED LEVEL INTRODUCTION This syllabus serves to examine candidates knowledge and skills in introductory mathematical and statistical methods, and their applications. For applications

More information

Electric Motors and Drives

Electric Motors and Drives EML 2322L MAE Design and Manufacturing Laboratory Electric Motors and Drives To calculate the peak power and torque produced by an electric motor, you will need to know the following: Motor supply voltage,

More information

3600 s 1 h. 24 h 1 day. 1 day

3600 s 1 h. 24 h 1 day. 1 day Week 7 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

More information

ACTUATOR DESIGN FOR ARC WELDING ROBOT

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,

More information

Physics 201 Homework 8

Physics 201 Homework 8 Physics 201 Homework 8 Feb 27, 2013 1. A ceiling fan is turned on and a net torque of 1.8 N-m is applied to the blades. 8.2 rad/s 2 The blades have a total moment of inertia of 0.22 kg-m 2. What is the

More information

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

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

More information

Physics Notes Class 11 CHAPTER 5 LAWS OF MOTION

Physics Notes Class 11 CHAPTER 5 LAWS OF MOTION 1 P a g e Inertia Physics Notes Class 11 CHAPTER 5 LAWS OF MOTION The property of an object by virtue of which it cannot change its state of rest or of uniform motion along a straight line its own, is

More information

dspace DSP DS-1104 based State Observer Design for Position Control of DC Servo Motor

dspace DSP DS-1104 based State Observer Design for Position Control of DC Servo Motor dspace DSP DS-1104 based State Observer Design for Position Control of DC Servo Motor Jaswandi Sawant, Divyesh Ginoya Department of Instrumentation and control, College of Engineering, Pune. ABSTRACT This

More information

Center of Gravity. We touched on this briefly in chapter 7! x 2

Center of Gravity. We touched on this briefly in chapter 7! x 2 Center of Gravity We touched on this briefly in chapter 7! x 1 x 2 cm m 1 m 2 This was for what is known as discrete objects. Discrete refers to the fact that the two objects separated and individual.

More information

Rotation Matrices and Homogeneous Transformations

Rotation Matrices and Homogeneous Transformations Rotation Matrices and Homogeneous Transformations A coordinate frame in an n-dimensional space is defined by n mutually orthogonal unit vectors. In particular, for a two-dimensional (2D) space, i.e., n

More information

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

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

More information

PHYSICS 111 HOMEWORK SOLUTION #10. April 8, 2013

PHYSICS 111 HOMEWORK SOLUTION #10. April 8, 2013 PHYSICS HOMEWORK SOLUTION #0 April 8, 203 0. Find the net torque on the wheel in the figure below about the axle through O, taking a = 6.0 cm and b = 30.0 cm. A torque that s produced by a force can be

More information

Lab 8: Ballistic Pendulum

Lab 8: Ballistic Pendulum Lab 8: Ballistic Pendulum Equipment: Ballistic pendulum apparatus, 2 meter ruler, 30 cm ruler, blank paper, carbon paper, masking tape, scale. Caution In this experiment a steel ball is projected horizontally

More information

CH-205: Fluid Dynamics

CH-205: Fluid Dynamics CH-05: Fluid Dynamics nd Year, B.Tech. & Integrated Dual Degree (Chemical Engineering) Solutions of Mid Semester Examination Data Given: Density of water, ρ = 1000 kg/m 3, gravitational acceleration, g

More information

USING MS EXCEL FOR DATA ANALYSIS AND SIMULATION

USING MS EXCEL FOR DATA ANALYSIS AND SIMULATION USING MS EXCEL FOR DATA ANALYSIS AND SIMULATION Ian Cooper School of Physics The University of Sydney i.cooper@physics.usyd.edu.au Introduction The numerical calculations performed by scientists and engineers

More information

Lecture L3 - Vectors, Matrices and Coordinate Transformations

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

More information

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

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

More information

Chapter 18 Static Equilibrium

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

More information

Practice Exam Three Solutions

Practice Exam Three Solutions MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department of Physics Physics 8.01T Fall Term 2004 Practice Exam Three Solutions Problem 1a) (5 points) Collisions and Center of Mass Reference Frame In the lab frame,

More information

L 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has

L 2 : x = s + 1, y = s, z = 4s + 4. 3. Suppose that C has coordinates (x, y, z). Then from the vector equality AC = BD, one has The line L through the points A and B is parallel to the vector AB = 3, 2, and has parametric equations x = 3t + 2, y = 2t +, z = t Therefore, the intersection point of the line with the plane should satisfy:

More information

Acceleration due to Gravity

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

More information

DYNAMICS OF A TETRAHEDRAL CONSTELLATION OF SATELLITES-GYROSTATS

DYNAMICS OF A TETRAHEDRAL CONSTELLATION OF SATELLITES-GYROSTATS 7 th EUROMECH Solid Mechanics Conference J. Ambrosio et.al. (eds.) Lisbon, Portugal, 7 11 September 2009 DYNAMICS OF A TETRAHEDRAL CONSTELLATION OF SATELLITES-GYROSTATS Alexander A. Burov 1, Anna D. Guerman

More information

Motion Control of 3 Degree-of-Freedom Direct-Drive Robot. Rutchanee Gullayanon

Motion Control of 3 Degree-of-Freedom Direct-Drive Robot. Rutchanee Gullayanon Motion Control of 3 Degree-of-Freedom Direct-Drive Robot A Thesis Presented to The Academic Faculty by Rutchanee Gullayanon In Partial Fulfillment of the Requirements for the Degree Master of Engineering

More information

EXPERIMENT: MOMENT OF INERTIA

EXPERIMENT: MOMENT OF INERTIA OBJECTIVES EXPERIMENT: MOMENT OF INERTIA to familiarize yourself with the concept of moment of inertia, I, which plays the same role in the description of the rotation of a rigid body as mass plays in

More information

Simulation of Trajectories and Comparison of Joint Variables for Robotic Manipulator Using Multibody Dynamics (MBD)

Simulation of Trajectories and Comparison of Joint Variables for Robotic Manipulator Using Multibody Dynamics (MBD) Simulation of Trajectories and Comparison of Joint Variables for Robotic Manipulator Using Multibody Dynamics (MBD) Jatin Dave Assistant Professor Nirma University Mechanical Engineering Department, Institute

More information

Dynamics. Basilio Bona. DAUIN-Politecnico di Torino. Basilio Bona (DAUIN-Politecnico di Torino) Dynamics 2009 1 / 30

Dynamics. Basilio Bona. DAUIN-Politecnico di Torino. Basilio Bona (DAUIN-Politecnico di Torino) Dynamics 2009 1 / 30 Dynamics Basilio Bona DAUIN-Politecnico di Torino 2009 Basilio Bona (DAUIN-Politecnico di Torino) Dynamics 2009 1 / 30 Dynamics - Introduction In order to determine the dynamics of a manipulator, it is

More information

ANALYTICAL METHODS FOR ENGINEERS

ANALYTICAL METHODS FOR ENGINEERS UNIT 1: Unit code: QCF Level: 4 Credit value: 15 ANALYTICAL METHODS FOR ENGINEERS A/601/1401 OUTCOME - TRIGONOMETRIC METHODS TUTORIAL 1 SINUSOIDAL FUNCTION Be able to analyse and model engineering situations

More information

Midterm Exam 1 October 2, 2012

Midterm Exam 1 October 2, 2012 Midterm Exam 1 October 2, 2012 Name: Instructions 1. This examination is closed book and closed notes. All your belongings except a pen or pencil and a calculator should be put away and your bookbag should

More information

3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field

3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field 3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field 77 3.8 Finding Antiderivatives; Divergence and Curl of a Vector Field Overview: The antiderivative in one variable calculus is an important

More information

Utfordringer i Arktis: Offshore installasjoner og infrastruktur i kystnære strøk

Utfordringer i Arktis: Offshore installasjoner og infrastruktur i kystnære strøk Nordområdekonferansen 2012, 27-29 november, Longyearbyen Utfordringer i Arktis: Offshore installasjoner og infrastruktur i kystnære strøk Prof. Sveinung Løset, NTNU www.ntnu.edu/samcot The Petroleum Industry

More information

Useful Motor/Torque Equations for EML2322L

Useful Motor/Torque Equations for EML2322L Useful Motor/Torque Equations for EML2322L Force (Newtons) F = m x a m = mass (kg) a = acceleration (m/s 2 ) Motor Torque (Newton-meters) T = F x d F = force (Newtons) d = moment arm (meters) Power (Watts)

More information

Orbital Mechanics. Angular Momentum

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

More information

Review Sheet for Test 1

Review Sheet for Test 1 Review Sheet for Test 1 Math 261-00 2 6 2004 These problems are provided to help you study. The presence of a problem on this handout does not imply that there will be a similar problem on the test. And

More information

Fundamentals of Computer Animation

Fundamentals of Computer Animation Fundamentals of Computer Animation Quaternions as Orientations () page 1 Visualizing a Unit Quaternion Rotation in 4D Space ( ) = w + x + y z q = Norm q + q = q q [ w, v], v = ( x, y, z) w scalar q =,

More information

Design of a Universal Robot End-effector for Straight-line Pick-up Motion

Design of a Universal Robot End-effector for Straight-line Pick-up Motion Session Design of a Universal Robot End-effector for Straight-line Pick-up Motion Gene Y. Liao Gregory J. Koshurba Wayne State University Abstract This paper describes a capstone design project in developing

More information

A Simulation Study on Joint Velocities and End Effector Deflection of a Flexible Two Degree Freedom Composite Robotic Arm

A Simulation Study on Joint Velocities and End Effector Deflection of a Flexible Two Degree Freedom Composite Robotic Arm International Journal of Advanced Mechatronics and Robotics (IJAMR) Vol. 3, No. 1, January-June 011; pp. 9-0; International Science Press, ISSN: 0975-6108 A Simulation Study on Joint Velocities and End

More information

Unit - 6 Vibrations of Two Degree of Freedom Systems

Unit - 6 Vibrations of Two Degree of Freedom Systems Unit - 6 Vibrations of Two Degree of Freedom Systems Dr. T. Jagadish. Professor for Post Graduation, Department of Mechanical Engineering, Bangalore Institute of Technology, Bangalore Introduction A two

More information

Rigid body dynamics using Euler s equations, Runge-Kutta and quaternions.

Rigid body dynamics using Euler s equations, Runge-Kutta and quaternions. Rigid body dynamics using Euler s equations, Runge-Kutta and quaternions. Indrek Mandre http://www.mare.ee/indrek/ February 26, 2008 1 Motivation I became interested in the angular dynamics

More information

Exemplar Problems Physics

Exemplar Problems Physics Chapter Eight GRAVITATION MCQ I 8.1 The earth is an approximate sphere. If the interior contained matter which is not of the same density everywhere, then on the surface of the earth, the acceleration

More information

Human-like Arm Motion Generation for Humanoid Robots Using Motion Capture Database

Human-like Arm Motion Generation for Humanoid Robots Using Motion Capture Database Human-like Arm Motion Generation for Humanoid Robots Using Motion Capture Database Seungsu Kim, ChangHwan Kim and Jong Hyeon Park School of Mechanical Engineering Hanyang University, Seoul, 133-791, Korea.

More information

CONTRIBUTIONS TO THE AUTOMATIC CONTROL OF AERIAL VEHICLES

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,

More information

Precise Modelling of a Gantry Crane System Including Friction, 3D Angular Swing and Hoisting Cable Flexibility

Precise Modelling of a Gantry Crane System Including Friction, 3D Angular Swing and Hoisting Cable Flexibility Precise Modelling of a Gantry Crane System Including Friction, 3D Angular Swing and Hoisting Cable Flexibility Renuka V. S. & Abraham T Mathew Electrical Engineering Department, NIT Calicut E-mail : renuka_mee@nitc.ac.in,

More information

Force on Moving Charges in a Magnetic Field

Force on Moving Charges in a Magnetic Field [ Assignment View ] [ Eðlisfræði 2, vor 2007 27. Magnetic Field and Magnetic Forces Assignment is due at 2:00am on Wednesday, February 28, 2007 Credit for problems submitted late will decrease to 0% after

More information

Lecture 17. Last time we saw that the rotational analog of Newton s 2nd Law is

Lecture 17. Last time we saw that the rotational analog of Newton s 2nd Law is Lecture 17 Rotational Dynamics Rotational Kinetic Energy Stress and Strain and Springs Cutnell+Johnson: 9.4-9.6, 10.1-10.2 Rotational Dynamics (some more) Last time we saw that the rotational analog of

More information

Response to Harmonic Excitation Part 2: Damped Systems

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

More information

Computing Euler angles from a rotation matrix

Computing Euler angles from a rotation matrix Computing Euler angles from a rotation matrix Gregory G. Slabaugh Abstract This document discusses a simple technique to find all possible Euler angles from a rotation matrix. Determination of Euler angles

More information

1 Differential Drive Kinematics

1 Differential Drive Kinematics CS W4733 NOTES - Differential Drive Robots Note: these notes were compiled from Dudek and Jenkin, Computational Principles of Mobile Robotics. 1 Differential Drive Kinematics Many mobile robots use a drive

More information

CS100B Fall 1999. Professor David I. Schwartz. Programming Assignment 5. Due: Thursday, November 18 1999

CS100B Fall 1999. Professor David I. Schwartz. Programming Assignment 5. Due: Thursday, November 18 1999 CS100B Fall 1999 Professor David I. Schwartz Programming Assignment 5 Due: Thursday, November 18 1999 1. Goals This assignment will help you develop skills in software development. You will: develop software

More information

Lecture L5 - Other Coordinate Systems

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

More information

Problem 6.40 and 6.41 Kleppner and Kolenkow Notes by: Rishikesh Vaidya, Physics Group, BITS-Pilani

Problem 6.40 and 6.41 Kleppner and Kolenkow Notes by: Rishikesh Vaidya, Physics Group, BITS-Pilani Problem 6.40 and 6.4 Kleppner and Kolenkow Notes by: Rishikesh Vaidya, Physics Group, BITS-Pilani 6.40 A wheel with fine teeth is attached to the end of a spring with constant k and unstretched length

More information

Problem Set #13 Solutions

Problem Set #13 Solutions MASSACHUSETTS INSTITUTE OF TECHNOLOGY Physics Department 8.0L: Physics I January 3, 06 Prof. Alan Guth Problem Set #3 Solutions Due by :00 am on Friday, January in the bins at the intersection of Buildings

More information

Sittiporn Channumsin Co-authors

Sittiporn Channumsin Co-authors 28 Oct 2014 Space Glasgow Research Conference Sittiporn Channumsin Sittiporn Channumsin Co-authors S. Channumsin Outline Background Objective The model Simulation Results Conclusion and Future work 2 Space

More information

Kyu-Jung Kim Mechanical Engineering Department, California State Polytechnic University, Pomona, U.S.A.

Kyu-Jung Kim Mechanical Engineering Department, California State Polytechnic University, Pomona, U.S.A. MECHANICS: STATICS AND DYNAMICS Kyu-Jung Kim Mechanical Engineering Department, California State Polytechnic University, Pomona, U.S.A. Keywords: mechanics, statics, dynamics, equilibrium, kinematics,

More information

8.012 Physics I: Classical Mechanics Fall 2008

8.012 Physics I: Classical Mechanics Fall 2008 MIT OpenCourseWare http://ocw.mit.edu 8.012 Physics I: Classical Mechanics Fall 2008 For information about citing these materials or our Terms of Use, visit: http://ocw.mit.edu/terms. MASSACHUSETTS INSTITUTE

More information

Pre-lab Quiz/PHYS 224 Magnetic Force and Current Balance. Your name Lab section

Pre-lab Quiz/PHYS 224 Magnetic Force and Current Balance. Your name Lab section Pre-lab Quiz/PHYS 224 Magnetic Force and Current Balance Your name Lab section 1. What do you investigate in this lab? 2. Two straight wires are in parallel and carry electric currents in opposite directions

More information

Tips For Selecting DC Motors For Your Mobile Robot

Tips For Selecting DC Motors For Your Mobile Robot Tips For Selecting DC Motors For Your Mobile Robot By AJ Neal When building a mobile robot, selecting the drive motors is one of the most important decisions you will make. It is a perfect example of an

More information

Isaac Newton s (1642-1727) Laws of Motion

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

More information

discuss how to describe points, lines and planes in 3 space.

discuss how to describe points, lines and planes in 3 space. Chapter 2 3 Space: lines and planes In this chapter we discuss how to describe points, lines and planes in 3 space. introduce the language of vectors. discuss various matters concerning the relative position

More information

Solutions for Review Problems

Solutions for Review Problems olutions for Review Problems 1. Let be the triangle with vertices A (,, ), B (4,, 1) and C (,, 1). (a) Find the cosine of the angle BAC at vertex A. (b) Find the area of the triangle ABC. (c) Find a vector

More information

MEMs Inertial Measurement Unit Calibration

MEMs Inertial Measurement Unit Calibration MEMs Inertial Measurement Unit Calibration 1. Introduction Inertial Measurement Units (IMUs) are everywhere these days; most particularly in smart phones and other mobile or handheld devices. These IMUs

More information

Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices

Linear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two

More information

13.4 THE CROSS PRODUCT

13.4 THE CROSS PRODUCT 710 Chapter Thirteen A FUNDAMENTAL TOOL: VECTORS 62. Use the following steps and the results of Problems 59 60 to show (without trigonometry) that the geometric and algebraic definitions of the dot product

More information

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

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

More information

Torsion Pendulum. Life swings like a pendulum backward and forward between pain and boredom. Arthur Schopenhauer

Torsion Pendulum. Life swings like a pendulum backward and forward between pain and boredom. Arthur Schopenhauer Torsion Pendulum Life swings like a pendulum backward and forward between pain and boredom. Arthur Schopenhauer 1 Introduction Oscillations show up throughout physics. From simple spring systems in mechanics

More information

Physics 9 Fall 2009 Homework 2 - Solutions

Physics 9 Fall 2009 Homework 2 - Solutions Physics 9 Fall 009 Homework - s 1. Chapter 7 - Exercise 5. An electric dipole is formed from ±1.0 nc charges spread.0 mm apart. The dipole is at the origin, oriented along the y axis. What is the electric

More information

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

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,

More information

SOLID MECHANICS BALANCING TUTORIAL BALANCING OF ROTATING BODIES

SOLID MECHANICS BALANCING TUTORIAL BALANCING OF ROTATING BODIES SOLID MECHANICS BALANCING TUTORIAL BALANCING OF ROTATING BODIES This work covers elements of the syllabus for the Edexcel module 21722P HNC/D Mechanical Principles OUTCOME 4. On completion of this tutorial

More information