Origins of the Unusual Space Shuttle Quaternion Definition

Size: px
Start display at page:

Download "Origins of the Unusual Space Shuttle Quaternion Definition"

Transcription

1 47th AIAA Aerospace Sciences Meeting Including The New Horizons Forum and Aerospace Exposition 5-8 January 2009, Orlando, Florida AIAA Origins of the Unusual Space Shuttle Quaternion Definition Douglas J. Yazell 1 Honeywell, Houston, Texas, This paper is offered to the history session of this conference. It relates to a mathematical convention which is fundamental to guidance, navigation and control, and is used by many other systems of a single spacecraft. The space shuttle program successfully pioneered the extensive use of quaternions for spacecraft. This program has a strong influence on conventions used by later space programs, but the American part of the International Space Station (ISS) uses a different convention, and most spacecraft now use a convention that agrees with ISS and not the space shuttle. This ISS convention has no negative sign in its quaternion definition, and the space shuttle quaternion has one negative sign, negating its vector part. The ISS convention does a better job of simplifying the mathematics. The benefit of re-use of space shuttle experience and products is offset by the fact that publicly available documentation of quaternion mathematics does not follow the space shuttle convention. Nomenclature δ = quaternion rotation angle u = quaternion rotation unit vector, with only two properties, magnitude and direction u1, u2, u3 = vector components of u q = quaternion S = scalar part of quaternion q V = vector part of quaternion q q -1 = quaternion inverse i = reference to the imaginary part of a complex number a + i b i, j, k = reference to the hyperimaginary part of a hypercomplex number a + (i b + j c + k d) A I. Introduction N Euler sequence of rotations, such as the familiar airplane yaw, pitch, then roll sequence, with all angles positive for a climbing, right-hand turn, is often replaced by a single rotation about a unit quaternion s rotation axis for spacecraft. Both space shuttle and the American part of the International Space Station (ISS) contain quaternion_inertial_to_body in their software and telemetry *. The names are spelled differently, but they are both inertial to body, not body to inertial. They are used in the same equation to solve for body coordinates when inertial coordinates are known. This same equation describes the orientation of the vehicle with respect to the inertial coordinate system. If body coordinates are known, the equation can easily be rearranged to solve for inertial coordinates, without changing any description of the coordinate system motion or the vehicle motion. Any quaternion can be divided into a scalar part and a 3-element vector part. In our work, the inverse is formed by negating its vector part. This quaternion transformation equation operates on a quaternion containing inertial coordinates, premultiplying by one quaternion and postmultiplying by a second quaternion. These quaternions, the pre- and postmultiplying quaternions, are the inverse of each other. The space shuttle quaternion negates its vector part and is used in the premultiply position. It then has two negative signs which cancel each other when it forms 1 Principal Systems Engineer, Defense & Space, 2525 Bay Area Blvd, Suite 200, AIAA Associate Fellow. * Refs. 1 and 2 for space shuttle solve for B coordinates from A coordinates, and a figure in each reference shows that B is the moving coordinate system and is moving away from coincidence with a fixed coordinate system. Ref. 3 for space station also solves for B coordinates from A coordinates. It does not show a sketch, but specifies that names such as inertial to body are helpful (as opposed to body to inertial, which it does not mention). 1 Copyright 2009 by Douglas J. Yazell. Published by the, Inc., with permission.

2 that quaternion inverse. The ISS quaternion has no negative signs in its definition and is used in the postmultiply position, so the same result for body coordinates is obtained by both spacecraft programs. One drawback to the space shuttle convention is the fact that, outside of that program, most documentation of quaternion mathematics does not have a negative sign in a quaternion definition until the quaternion inverse is formed. II. Quaternion Conventions A minimum number of equations are presented to familiarize an aerospace history audience with the vocabulary. It is helpful to start with an influential convention, the one used in robotics. Robotics has a fundamental coordinate transformation ( Moving to Fixed ) that obtains coordinates in a Fixed (Inertial) coordinate system from coordinates known in a Moving (Body) coordinate system: v Inertial Body = qv q 1 (1) where the quaternion has a scalar part (S) and a vector part (V): S = V and the quaternion inverse is formed by negating the vector part. q (2) S = V q 1 (3) This uses a pre-multiply convention, pre-multiplying by the quaternion and post-multiplying by the quaternion inverse. This quaternion is defined by a scalar part S and a vector part V which has three scalar parts. Both a quaternion and its inverse are quaternions. Our quaternion s inverse is formed by negating its vector part. The rules of quaternion multiplication 4 will not be presented here, but note, for example, that for a position vector whose coordinates are being transformed, a zero scalar part is assigned, and the position vector coordinates then become the vector part of that quaternion. This results in quaternions such as v Body and v Inertial in Eq. (1). To define the above terms S and V, = cos δ 2 S (4) V u δ = sin u 2 u (5) We use unit quaternions here, which have a norm of one, so that the square root of the sum of the squares of those four scalar elements is one. Three rotations such as yaw, pitch, and roll can be replaced by a single equivalent rotation about a single rotation unit vector u. The three coordinates of this vector are in Eq. (5), and they are the same in the two coordinate systems. All conventions (including space shuttle, ISS, and robotics) presented in this paper adhere to these four conventions for coordinate transformations and definition of the vehicle s orientation: (I) When the rotation angle is zero, the two coordinate systems have the same orientation. (II) The Body coordinate system rotates away from coincidence with the Inertial coordinate system, not into coincidence. (III) The rotation is right-handed. (IV) The coordinate systems are right-handed (and orthogonal) 2

3 With a complete description of coordinate system rotation, the description of the vehicle s orientation is obvious. Equations of the type in Eq. (1) can also be interpreted as vector rotation in a single coordinate system, but that interpretation is avoided in this paper. This paper will not present the familiar matrix equations for sequences of Euler rotations, but a few notes about them will be made in order to compare the order of stacking for matrices and quaternions when the equations describe successive rotations. The classic airplane Euler sequence is a familiar example, the yaw, pitch, and then roll angles, rotating from, for example, a North-East-Down coordinate system, so that the first angle is the heading, the second angle measures how far up or down the nose moves, and the third angle measures a roll angle about a line down the long axis of the airplane, with all angles positive for a climbing, right-hand turn. The robotics matrix equation for this motion of the end effector or link orientation stacks from left to right: the 3x3 yaw rotation matrix, then the pitch rotation matrix, then the roll rotation matrix, premultiplying a 3 x 1 column matrix of Body (Moving) coordinates to obtain the Inertial (Fixed) coordinates. The corresponding quaternions in the premultiply position in a modified Eq. (1) stack left to right: the yaw quaternion, the pitch quaternion, and then the roll quaternion. The corresponding inverses would then appear in the postmultiply position from right to left, but stacking refers to the multiplication order of these quaternions for successive rotations, not the order of these quaternion inverses. Both sides of Eq. (1) can be pre-multiplied by the quaternion inverse and post-multiplied by the quaternion q to obtain Eq. (6). This is the equation used by the American ISS software 3, and the ISS quaternion Inertial to Body is in the post-multiply position. v Inertial = q v q (6) Body 1 These mathematical operations used to obtain Eq. (6) do not change the description of the motion of the coordinate system or the vehicle orientation. For both Eq. (1), used in robotics, and Eq. (6), used in aerospace, conventions I, II, III, and IV apply. Also, for both Eqs. (1) and (6), Eqs. (2) through (5) apply. The matrix version of Eq. (6) for the classic airplane Euler sequence, yaw, pitch, and then roll, is familiar to aerospace engineers. The matrix order is right to left: the yaw matrix, then the pitch matrix, and then the roll matrix, and they premultiply the 3 x 1 column matrix of Inertial (Fixed) coordinates to obtain the Body (Moving) coordinates. This describes the vehicle orientation with respect to the Inertial coordinate system, but the robotics version of that equation, which is described in an earlier paragraph, describes the same orientation with the reverse order of matrices. Just as Eq. (6) is obtained easily from Eq. (1), the robotics matrix equation is obtained easily from the aerospace matrix equation, noting that the inverse of these rotation matrices are the transposes. This results in the robotics rotation matrix being the transpose of the aerospace rotation matrix for all rotations: in this case, yaw, pitch, and roll. Successive rotations in Eq. (6) stack from left to right in the postmultiply position. Multiplication order ( stacking ) for successive rotations Matrices which can premultiply 3x1 column matrices Quaternions (q) ( = q -1 ) Quaternion premultiply or postmultiply Coordinate transformation equation Robotics Left to Right Left to Right Premultiply Moving to Fixed (Body to Inertial) Shuttle (STS) Right to Left Right to Left Premultiply Inertial to Body ISS and most of aerospace Right to Left Left to Right Postmultiply Inertial to Body Table 1 A summary of operations for successive rotations The robotics paradigm solves for Fixed (Inertial) coordinates and the aerospace paradigm solves for Moving (Body) coordinates. Space shuttle orbital flights began in 1981 and ISS on-orbit operations began in Ref. 2 (1975), which is cited in Ref. 1 (1982), states very early in the paper that the (Fixed to Moving, which we can call Inertial to Body ) space shuttle quaternion coordinate transformation equation is written this way: v (7) Body Inertial 1 = qsts v qsts 3

4 For the space shuttle, that places quaternion Inertial to Body in the pre-multiply position. Ref. 2 (1975), then proceeds at length to show that given Eq. (7) and the definitions in Eqs. (4) and (5), we conclude This can be seen by comparing Eq. (7) to Eq. (6). 1 = q (8) As Ref. 4 (2003) points out, Ref. 2 (1975) can use Eqs. (7) and (8), but could also have written the following equation in place of Eq. (7): v Body 1 Inertial = qsts v qsts (9) That would have led Ref. 2 (1975) to conclude, in place of Eq. (8): = q (10) This can be seen by comparing Eq. (9) to Eq. (6). Ref. 4 (2003) points out that the post-multiply convention used Eqs. (6) and (9) is more common in aerospace than the pre-multiply convention of Eq. (7). The author of Ref. 2 (1975) does not demonstrate any awareness of the option presented in Eqs. (9) and (10). Ref. 2 (1975) thus ends with these two space shuttle definitions in place of Eqs. (4) and (5), using Eq. (8) in Eqs. (2) and (3): S = V (11) 1 = S ( V ) (12) Although the two negative signs cancel out in Eq. (12), it is an awkward way to define the inverse of the fundamental quaternion for the given application. A general rule seems to have been adopted by later aerospace programs, including the American part of ISS: Define the application s fundamental quaternion with no negative signs. Since we always simultaneously premultiply by a quaternion rotational operator and postmultiply by its inverse, it is always easy to implement this rule. III. Program History Despite the repeated space shuttle interaction with ISS, they carry these different definitions for this fundamental subject. Astronauts are trained to read quaternions in a readout of four numbers using ISS onboard computers. With practice, it s as easy as reading a display of three Euler angles. It is rather complex to then explain that the space shuttle program defines its quaternion in a different way. Quite a few space shuttle engineers were familiar only with that quaternion convention in recent years. When they moved on to other aerospace programs, it was sometimes a painful surprise to find that most other aerospace programs use a quaternion convention that is the same as the American ISS quaternion convention. IV. Mathematics History While writing this paper, a two-dimensional (2D) example was considered, since cos(δ) +/- i sin(δ) is a rotational operator in the complex plane. For example, any complex number a + ib can be viewed as a vector in a plane with a horizontal real axis and a vertical imaginary axis. For our example, we are using vector coordinate transformation, not vector rotation, so let s examine a1 + i a2, corresponding to Inertial vector coordinates transposed as [a1 a2] in 2D and [a1 a2 0] in 3D. When solving for Body (Moving) coordinates from Inertial (Fixed) 4

5 coordinates, using conventions (I), (II), (III), and (IV), cos(δ) i sin(δ) obtains the correct answer for Body coordinates, which is a1 cos(δ) + a2 sin(δ) for its first coordinate and a1 sin(δ) + a2 cos(δ) for its second coordinate. And cos(δ) + i sin(δ) obtains the wrong answer. This is a surprising result, since it provides a bit of support for the space shuttle convention: it is natural to want to extend this to define the space shuttle quaternion convention to be cos(δ/2) sin(δ/2)(i u1 + j u2 + k u3) instead of cos(δ/2) + sin(δ/2)(i u1 + j u2 + k u3). As noted earlier, the 2D vector is a1 + i a2, and the 3D quaternion is 0 + sin(δ/2)(i a1 + j a2 + k 0). In 1843, William Rowan Hamilton invented quaternions using i, j, and k in place of i, extending complex numbers to hypercomplex numbers. He defined the simple rules for multiplying with i, j, and k, and not just i. He needed 15 or more years to extend that 2D example with complex numbers to 3D with his invention of quaternions. But in the 2D case, multiplication commutes, and only one of the two operators gives the right answer. In the 3D case, we simultaneously premultiply and postmultiply by the quaternion operator and its inverse, so, unlike Ref. 2 (1975) for space shuttle work, we have two choices that give the same correct answer for body coordinates. In selecting our definition of quaternion_inertial_to_body, we can premultiply with the space shuttle quaternion or postmultiply with the ISS quaternion. My formal training with quaternions, ending in 1992, was in a graduate study program including teachers and texts in the field of robotics. I was not familiar with the space shuttle quaternion convention. Despite my work with quaternions, from 1988 to 1992, in many applications on Space Station Freedom guidance, navigation, and control, I was surprised to see the 1997 American ISS quaternion convention: this was something completely new in my experience. That s an example of the sometimes dramatic difference in vocabularies and points of view for different fields such as engineering, physics, mathematics, and robotics. Equally strong differences sometimes exist in fields such as aerospace navigation and flight control. And some aerospace engineers and documents view these equations first and foremost as coordinate transformations, while others view the same equations first and foremost as a definition of vehicle orientation. In my limited experience, these latter two viewpoints do not conflict, but communication among professionals can be difficult because of these different points of view. It s natural to ask if one of these two quaternion conventions (space shuttle and ISS) is better than the other, though they achieve the same result. Quaternion mathematics used in these programs is more complex than shown here, so the more common and simpler ISS convention is preferred in that sense. On the other hand, re-use of the space shuttle convention will sometimes be helpful. V. Conclusions This history paper begins a public conversation about the mysterious origin of the space shuttle quaternion convention. My conclusion, which I conclude was later supported when I recently read Ref. 2 for the first time, is that responsible space shuttle engineers were familiar with the robotics premultiply convention of Eq. (1), and they took that to be the form of the equation (premultiply, not postmultiply) for coordinate transformation and attitude definition: they rearranged the robotics equation as required for aerospace tradition to solve for Body coordinates and not Inertial coordinates, but they kept the premultiply form, as shown in Eq. (7). With conventions (I) through (IV), as explained above, only one choice remained for them in They defined the quaternion inverse of Eq. (6) to be their quaternion in Eq. (7). As the aerospace world moved away from the pioneering space shuttle quaternion convention, documents began describing the space shuttle quaternion as a left-handed quaternion, since it has a negative sign in the vector part of its definition. But the most common space shuttle application 1,2 is a right-handed coordinate system rotation used for coordinate transformation and attitude definition. Refs. 1 and 2 never describe the space shuttle quaternion as lefthanded. The space shuttle quaternion convention appears to be result of choosing the pre-multiply convention for coordinate transformation when the post-multiply convention was not considered 2. This could easily have been the influence of the use of quaternions in robotics in 1975 and before. Since the space shuttle was pioneering the extensive use of quaternions in spacecraft, a strong influence from a field other than aerospace would not be a surprise. Re-use of the space shuttle quaternion convention for later programs has obvious benefits, but there are bad effects, too. Good quaternion documentation is still hard to find, and extra work is required to compare it to the space shuttle example. That complexity is not present with the ISS quaternion example 3, which Ref. 4 states is the most common example in aerospace. Imagine the ISS engineers selecting their quaternion convention despite the influence of the space shuttle quaternion convention. There must have been great resistance, but somehow the ISS engineers chose and 5

6 implemented a convention prior to the 1998 ISS launch which Ref. 4 described in 2003 as being the most common choice in aerospace. A general rule seems to have been adopted in aerospace: Define the application s fundamental quaternion with no negative signs. This rule corresponds well with quaternion use in the American part of ISS, which matches most aerospace programs, and it corresponds well with quaternion use in robotics. Modern spacecraft often combine robotics and aerospace vocabularies and communities. Acknowledgments Rodolfo Gonzalez at NASA Johnson Space Center was generous with his time, resources, and expertise as we had a few short conversations just before the final draft of this paper was completed, though he did not see this paper in any form. All quaternion conventions mentioned in this paper use right-hand Eigen information, which refers to the quaternion unit vector and the quaternion rotation angle. Mr. Gonzalez also mentioned the order in which quaternions and matrices stack when successive rotations are described. This subject is now developed a bit more in this paper. I apologize in advance for any undetected wrong content in this paper. I thank other authors for their publications on this slippery subject. References 1 Schletz, B., Use of Quaternions in Shuttle Guidance, Navigation and Control, AIAA , Carroll, J. V., The Notation and Use of Quaternions for Shuttle Ascent Steering, Memo 10C-75-47, CSDL, Appendix A (a two-page appendix): QUATERNION CONVENTION DEFINITION, part of the Technical Description Document for the PG-1 Guidance, Navigation & Control System, Product Group 1 (PG-1), McDonnell Douglas Aerospace, Space & Defense Systems, Space Station Division, 17 January 1997, Document Number MDC 95H0223, REVISION D. 4 Stevens, B. L., and Lewis, F. L., Aircraft Control and Simulation, 2 nd ed., John Wiley & Sons, 2003, Chap. 1, which was a free download from the publisher s web site (http://www.wiley.com/wileycda/wileytitle/productcd html), pg. 18 6

Robot Manipulators. Position, Orientation and Coordinate Transformations. Fig. 1: Programmable Universal Manipulator Arm (PUMA)

Robot Manipulators. Position, Orientation and Coordinate Transformations. Fig. 1: Programmable Universal Manipulator Arm (PUMA) Robot Manipulators Position, Orientation and Coordinate Transformations Fig. 1: Programmable Universal Manipulator Arm (PUMA) A robot manipulator is an electronically controlled mechanism, consisting of

More information

3D Tranformations. CS 4620 Lecture 6. Cornell CS4620 Fall 2013 Lecture 6. 2013 Steve Marschner (with previous instructors James/Bala)

3D Tranformations. CS 4620 Lecture 6. Cornell CS4620 Fall 2013 Lecture 6. 2013 Steve Marschner (with previous instructors James/Bala) 3D Tranformations CS 4620 Lecture 6 1 Translation 2 Translation 2 Translation 2 Translation 2 Scaling 3 Scaling 3 Scaling 3 Scaling 3 Rotation about z axis 4 Rotation about z axis 4 Rotation about x axis

More information

We have just introduced a first kind of specifying change of orientation. Let s call it Axis-Angle.

We have just introduced a first kind of specifying change of orientation. Let s call it Axis-Angle. 2.1.5 Rotations in 3-D around the origin; Axis of rotation In three-dimensional space, it will not be sufficient just to indicate a center of rotation, as we did for plane kinematics. Any change of orientation

More information

Quaternion Math. Application Note. Abstract

Quaternion Math. Application Note. Abstract Quaternion Math Application Note Abstract This application note provides an overview of the quaternion attitude representation used by VectorNav products and how to convert it into other common attitude

More information

Lecture 6 : Aircraft orientation in 3 dimensions

Lecture 6 : Aircraft orientation in 3 dimensions Lecture 6 : Aircraft orientation in 3 dimensions Or describing simultaneous roll, pitch and yaw 1.0 Flight Dynamics Model For flight dynamics & control, the reference frame is aligned with the aircraft

More information

Standard Terminology for Vehicle Dynamics Simulations

Standard Terminology for Vehicle Dynamics Simulations Standard Terminology for Vehicle Dynamics Simulations Y v Z v X v Michael Sayers The University of Michigan Transportation Research Institute (UMTRI) February 22, 1996 Table of Contents 1. Introduction...1

More information

Quaternions and Rotations

Quaternions and Rotations CSCI 520 Computer Animation and Simulation Quaternions and Rotations Jernej Barbic University of Southern California 1 Rotations Very important in computer animation and robotics Joint angles, rigid body

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

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

Rotation in the Space

Rotation in the Space Rotation in the Space (Com S 477/577 Notes) Yan-Bin Jia Sep 6, 2016 The position of a point after some rotation about the origin can simply be obtained by multiplying its coordinates with a matrix One

More information

Physics 235 Chapter 1. Chapter 1 Matrices, Vectors, and Vector Calculus

Physics 235 Chapter 1. Chapter 1 Matrices, Vectors, and Vector Calculus Chapter 1 Matrices, Vectors, and Vector Calculus In this chapter, we will focus on the mathematical tools required for the course. The main concepts that will be covered are: Coordinate transformations

More information

Abstract. Introduction

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

More information

Orthogonal Projections

Orthogonal Projections Orthogonal Projections and Reflections (with exercises) by D. Klain Version.. Corrections and comments are welcome! Orthogonal Projections Let X,..., X k be a family of linearly independent (column) vectors

More information

Essential Mathematics for Computer Graphics fast

Essential Mathematics for Computer Graphics fast John Vince Essential Mathematics for Computer Graphics fast Springer Contents 1. MATHEMATICS 1 Is mathematics difficult? 3 Who should read this book? 4 Aims and objectives of this book 4 Assumptions made

More information

13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions.

13 MATH FACTS 101. 2 a = 1. 7. The elements of a vector have a graphical interpretation, which is particularly easy to see in two or three dimensions. 3 MATH FACTS 0 3 MATH FACTS 3. Vectors 3.. Definition We use the overhead arrow to denote a column vector, i.e., a linear segment with a direction. For example, in three-space, we write a vector in terms

More information

Helpsheet. Giblin Eunson Library MATRIX ALGEBRA. library.unimelb.edu.au/libraries/bee. Use this sheet to help you:

Helpsheet. Giblin Eunson Library MATRIX ALGEBRA. library.unimelb.edu.au/libraries/bee. Use this sheet to help you: Helpsheet Giblin Eunson Library ATRIX ALGEBRA Use this sheet to help you: Understand the basic concepts and definitions of matrix algebra Express a set of linear equations in matrix notation Evaluate determinants

More information

WEIGHTLESS WONDER Reduced Gravity Flight

WEIGHTLESS WONDER Reduced Gravity Flight WEIGHTLESS WONDER Reduced Gravity Flight Instructional Objectives Students will use trigonometric ratios to find vertical and horizontal components of a velocity vector; derive a formula describing height

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

Solving Simultaneous Equations and Matrices

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

More information

Magnetometer Realignment: Theory and Implementation

Magnetometer Realignment: Theory and Implementation Magnetometer Realignment: heory and Implementation William Premerlani, Octoer 16, 011 Prolem A magnetometer that is separately mounted from its IMU partner needs to e carefully aligned with the IMU in

More information

A Brief Primer on Matrix Algebra

A Brief Primer on Matrix Algebra A Brief Primer on Matrix Algebra A matrix is a rectangular array of numbers whose individual entries are called elements. Each horizontal array of elements is called a row, while each vertical array is

More information

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

11.1. 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

More information

Aircraft Flight Dynamics!

Aircraft Flight Dynamics! Aircraft Flight Dynamics Robert Stengel, Princeton University, 2014 Course Overview Introduction to Flight Dynamics Math Preliminaries Copyright 2014 by Robert Stengel. All rights reserved. For educational

More information

3 Orthogonal Vectors and Matrices

3 Orthogonal Vectors and Matrices 3 Orthogonal Vectors and Matrices The linear algebra portion of this course focuses on three matrix factorizations: QR factorization, singular valued decomposition (SVD), and LU factorization The first

More information

Motion Sensors Introduction

Motion Sensors Introduction InvenSense Inc. 1197 Borregas Ave., Sunnyvale, CA 94089 U.S.A. Tel: +1 (408) 988-7339 Fax: +1 (408) 988-8104 Website: www.invensense.com Document Number: Revision: Motion Sensors Introduction A printed

More information

Lecture L6 - Intrinsic Coordinates

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

More information

Module 8 Lesson 4: Applications of Vectors

Module 8 Lesson 4: Applications of Vectors Module 8 Lesson 4: Applications of Vectors So now that you have learned the basic skills necessary to understand and operate with vectors, in this lesson, we will look at how to solve real world problems

More information

Vectors 2. The METRIC Project, Imperial College. Imperial College of Science Technology and Medicine, 1996.

Vectors 2. The METRIC Project, Imperial College. Imperial College of Science Technology and Medicine, 1996. Vectors 2 The METRIC Project, Imperial College. Imperial College of Science Technology and Medicine, 1996. Launch Mathematica. Type

More information

1111: Linear Algebra I

1111: Linear Algebra I 1111: Linear Algebra I Dr. Vladimir Dotsenko (Vlad) Lecture 3 Dr. Vladimir Dotsenko (Vlad) 1111: Linear Algebra I Lecture 3 1 / 12 Vector product and volumes Theorem. For three 3D vectors u, v, and w,

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

Quick Reference Guide to Linear Algebra in Quantum Mechanics

Quick Reference Guide to Linear Algebra in Quantum Mechanics Quick Reference Guide to Linear Algebra in Quantum Mechanics Scott N. Walck September 2, 2014 Contents 1 Complex Numbers 2 1.1 Introduction............................ 2 1.2 Real Numbers...........................

More information

Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture 7 Transformations in 2-D

Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture 7 Transformations in 2-D Computer Graphics Prof. Sukhendu Das Dept. of Computer Science and Engineering Indian Institute of Technology, Madras Lecture 7 Transformations in 2-D Welcome everybody. We continue the discussion on 2D

More information

28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. v x. u y v z u z v y u y u z. v y v z

28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE. v x. u y v z u z v y u y u z. v y v z 28 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.4 Cross Product 1.4.1 Definitions The cross product is the second multiplication operation between vectors we will study. The goal behind the definition

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

How to add sine functions of different amplitude and phase

How to add sine functions of different amplitude and phase Physics 5B Winter 2009 How to add sine functions of different amplitude and phase In these notes I will show you how to add two sinusoidal waves each of different amplitude and phase to get a third sinusoidal

More information

Lecture L17 - Orbit Transfers and Interplanetary Trajectories

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

More information

Vector algebra Christian Miller CS Fall 2011

Vector algebra Christian Miller CS Fall 2011 Vector algebra Christian Miller CS 354 - Fall 2011 Vector algebra A system commonly used to describe space Vectors, linear operators, tensors, etc. Used to build classical physics and the vast majority

More information

MAC 1114. Learning Objectives. Module 10. Polar Form of Complex Numbers. There are two major topics in this module:

MAC 1114. Learning Objectives. Module 10. Polar Form of Complex Numbers. There are two major topics in this module: MAC 1114 Module 10 Polar Form of Complex Numbers Learning Objectives Upon completing this module, you should be able to: 1. Identify and simplify imaginary and complex numbers. 2. Add and subtract complex

More information

x1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0.

x1 x 2 x 3 y 1 y 2 y 3 x 1 y 2 x 2 y 1 0. Cross product 1 Chapter 7 Cross product We are getting ready to study integration in several variables. Until now we have been doing only differential calculus. One outcome of this study will be our ability

More information

The basic unit in matrix algebra is a matrix, generally expressed as: a 11 a 12. a 13 A = a 21 a 22 a 23

The basic unit in matrix algebra is a matrix, generally expressed as: a 11 a 12. a 13 A = a 21 a 22 a 23 (copyright by Scott M Lynch, February 2003) Brief Matrix Algebra Review (Soc 504) Matrix algebra is a form of mathematics that allows compact notation for, and mathematical manipulation of, high-dimensional

More information

Introduction to Matrix Algebra

Introduction to Matrix Algebra Psychology 7291: Multivariate Statistics (Carey) 8/27/98 Matrix Algebra - 1 Introduction to Matrix Algebra Definitions: A matrix is a collection of numbers ordered by rows and columns. It is customary

More information

1 KINEMATICS OF MOVING FRAMES

1 KINEMATICS OF MOVING FRAMES 1 1 KINEMATICS OF MOVING FRAMES 1.1 Rotation of Reference Frames We denote through a subscript the specific reference system of a vector. Let a vector expressed in the inertial frame be denoted as γx,

More information

2D Geometric Transformations. COMP 770 Fall 2011

2D Geometric Transformations. COMP 770 Fall 2011 2D Geometric Transformations COMP 770 Fall 2011 1 A little quick math background Notation for sets, functions, mappings Linear transformations Matrices Matrix-vector multiplication Matrix-matrix multiplication

More information

Portable Assisted Study Sequence ALGEBRA IIA

Portable Assisted Study Sequence ALGEBRA IIA SCOPE This course is divided into two semesters of study (A & B) comprised of five units each. Each unit teaches concepts and strategies recommended for intermediate algebra students. The first half of

More information

Section 1.1. Introduction to R n

Section 1.1. Introduction to R n The Calculus of Functions of Several Variables Section. Introduction to R n Calculus is the study of functional relationships and how related quantities change with each other. In your first exposure to

More information

This week. CENG 732 Computer Animation. The Display Pipeline. Ray Casting Display Pipeline. Animation. Applying Transformations to Points

This week. CENG 732 Computer Animation. The Display Pipeline. Ray Casting Display Pipeline. Animation. Applying Transformations to Points This week CENG 732 Computer Animation Spring 2006-2007 Week 2 Technical Preliminaries and Introduction to Keframing Recap from CEng 477 The Displa Pipeline Basic Transformations / Composite Transformations

More information

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS

December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in two-dimensional space (1) 2x y = 3 describes a line in two-dimensional space The coefficients of x and y in the equation

More information

The Solution of Linear Simultaneous Equations

The Solution of Linear Simultaneous Equations Appendix A The Solution of Linear Simultaneous Equations Circuit analysis frequently involves the solution of linear simultaneous equations. Our purpose here is to review the use of determinants to solve

More information

Introduction to Matrices for Engineers

Introduction to Matrices for Engineers Introduction to Matrices for Engineers C.T.J. Dodson, School of Mathematics, Manchester Universit 1 What is a Matrix? A matrix is a rectangular arra of elements, usuall numbers, e.g. 1 0-8 4 0-1 1 0 11

More information

Metrics on SO(3) and Inverse Kinematics

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

More information

2.5 Complex Eigenvalues

2.5 Complex Eigenvalues 1 25 Complex Eigenvalues Real Canonical Form A semisimple matrix with complex conjugate eigenvalues can be diagonalized using the procedure previously described However, the eigenvectors corresponding

More information

CHAPTER 12 MOLECULAR SYMMETRY

CHAPTER 12 MOLECULAR SYMMETRY CHAPTER 12 MOLECULAR SYMMETRY In many cases, the symmetry of a molecule provides a great deal of information about its quantum states, even without a detailed solution of the Schrödinger equation. A geometrical

More information

Vector Math Computer Graphics Scott D. Anderson

Vector Math Computer Graphics Scott D. Anderson Vector Math Computer Graphics Scott D. Anderson 1 Dot Product The notation v w means the dot product or scalar product or inner product of two vectors, v and w. In abstract mathematics, we can talk about

More information

Vector Representation of Rotations

Vector Representation of Rotations Vector Representation of Rotations Carlo Tomasi The vector representation of rotation introduced below is based on Euler s theorem, and has three parameters. The conversion from a rotation vector to a

More information

Syntax Description Remarks and examples Also see

Syntax Description Remarks and examples Also see Title stata.com permutation An aside on permutation matrices and vectors Syntax Description Remarks and examples Also see Syntax Permutation matrix Permutation vector Action notation notation permute rows

More information

Description Syntax Remarks and examples Also see

Description Syntax Remarks and examples Also see Title stata.com permutation An aside on permutation matrices and vectors Description Syntax Remarks and examples Also see Description Permutation matrices are a special kind of orthogonal matrix that,

More information

Geometry of Vectors. 1 Cartesian Coordinates. Carlo Tomasi

Geometry of Vectors. 1 Cartesian Coordinates. Carlo Tomasi Geometry of Vectors Carlo Tomasi This note explores the geometric meaning of norm, inner product, orthogonality, and projection for vectors. For vectors in three-dimensional space, we also examine the

More information

Chapter 2 Coordinate Systems and Transformations

Chapter 2 Coordinate Systems and Transformations Chapter 2 Coordinate Systems and Transformations 2.1 Introduction In navigation, guidance, and control of an aircraft or rotorcraft, there are several coordinate systems (or frames intensively used in

More information

geometric transforms

geometric transforms geometric transforms 1 linear algebra review 2 matrices matrix and vector notation use column for vectors m 11 =[ ] M = [ m ij ] m 21 m 12 m 22 =[ ] v v 1 v = [ ] T v 1 v 2 2 3 matrix operations addition

More information

Images and Kernels in Linear Algebra By Kristi Hoshibata Mathematics 232

Images and Kernels in Linear Algebra By Kristi Hoshibata Mathematics 232 Images and Kernels in Linear Algebra By Kristi Hoshibata Mathematics 232 In mathematics, there are many different fields of study, including calculus, geometry, algebra and others. Mathematics has been

More information

The Inverse of a Matrix

The Inverse of a Matrix The Inverse of a Matrix 7.4 Introduction In number arithmetic every number a ( 0) has a reciprocal b written as a or such that a ba = ab =. Some, but not all, square matrices have inverses. If a square

More information

Chapter 2. Mission Analysis. 2.1 Mission Geometry

Chapter 2. Mission Analysis. 2.1 Mission Geometry Chapter 2 Mission Analysis As noted in Chapter 1, orbital and attitude dynamics must be considered as coupled. That is to say, the orbital motion of a spacecraft affects the attitude motion, and the attitude

More information

1.4 Velocity and Acceleration in Two Dimensions

1.4 Velocity and Acceleration in Two Dimensions Figure 1 An object s velocity changes whenever there is a change in the velocity s magnitude (speed) or direction, such as when these cars turn with the track. 1.4 Velocity and Acceleration in Two Dimensions

More information

FURTHER VECTORS (MEI)

FURTHER VECTORS (MEI) Mathematics Revision Guides Further Vectors (MEI) (column notation) Page of MK HOME TUITION Mathematics Revision Guides Level: AS / A Level - MEI OCR MEI: C FURTHER VECTORS (MEI) Version : Date: -9-7 Mathematics

More information

Chapter 2 Rotation and Orientation

Chapter 2 Rotation and Orientation Chapter Rotation and Orientation Abstract Rotation about an arbitrary axis is described by the use of Rodrigues s formula. Orientation of a coordinate frame with respect to another frame is expressed with

More information

Attitude Control and Dynamics of Solar Sails

Attitude Control and Dynamics of Solar Sails Attitude Control and Dynamics of Solar Sails Benjamin L. Diedrich A thesis submitted in partial fulfillment of the requirements for the degree of Master of Science in Aeronautics & Astronautics University

More information

Unified Lecture # 4 Vectors

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,

More information

1 Spherical Kinematics

1 Spherical Kinematics ME 115(a): Notes on Rotations 1 Spherical Kinematics Motions of a 3-dimensional rigid body where one point of the body remains fixed are termed spherical motions. A spherical displacement is a rigid body

More information

9 Multiplication of Vectors: The Scalar or Dot Product

9 Multiplication of Vectors: The Scalar or Dot Product Arkansas Tech University MATH 934: Calculus III Dr. Marcel B Finan 9 Multiplication of Vectors: The Scalar or Dot Product Up to this point we have defined what vectors are and discussed basic notation

More information

Basic Principles of Inertial Navigation. Seminar on inertial navigation systems Tampere University of Technology

Basic Principles of Inertial Navigation. Seminar on inertial navigation systems Tampere University of Technology Basic Principles of Inertial Navigation Seminar on inertial navigation systems Tampere University of Technology 1 The five basic forms of navigation Pilotage, which essentially relies on recognizing landmarks

More information

A Introduction to Matrix Algebra and Principal Components Analysis

A Introduction to Matrix Algebra and Principal Components Analysis A Introduction to Matrix Algebra and Principal Components Analysis Multivariate Methods in Education ERSH 8350 Lecture #2 August 24, 2011 ERSH 8350: Lecture 2 Today s Class An introduction to matrix algebra

More information

Algebra and Linear Algebra

Algebra and Linear Algebra Vectors Coordinate frames 2D implicit curves 2D parametric curves 3D surfaces Algebra and Linear Algebra Algebra: numbers and operations on numbers 2 + 3 = 5 3 7 = 21 Linear algebra: tuples, triples,...

More information

The General Linear Model: Theory

The General Linear Model: Theory Gregory Carey, 1998 General Linear Model - 1 The General Linear Model: Theory 1.0 Introduction In the discussion of multiple regression, we used the following equation to express the linear model for a

More information

Overview. Essential Questions. Precalculus, Quarter 3, Unit 3.4 Arithmetic Operations With Matrices

Overview. Essential Questions. Precalculus, Quarter 3, Unit 3.4 Arithmetic Operations With Matrices Arithmetic Operations With Matrices Overview Number of instruction days: 6 8 (1 day = 53 minutes) Content to Be Learned Use matrices to represent and manipulate data. Perform arithmetic operations with

More information

15.062 Data Mining: Algorithms and Applications Matrix Math Review

15.062 Data Mining: Algorithms and Applications Matrix Math Review .6 Data Mining: Algorithms and Applications Matrix Math Review The purpose of this document is to give a brief review of selected linear algebra concepts that will be useful for the course and to develop

More information

Vector has a magnitude and a direction. Scalar has a magnitude

Vector has a magnitude and a direction. Scalar has a magnitude Vector has a magnitude and a direction Scalar has a magnitude Vector has a magnitude and a direction Scalar has a magnitude a brick on a table Vector has a magnitude and a direction Scalar has a magnitude

More information

Quaternions & IMU Sensor Fusion with Complementary Filtering!

Quaternions & IMU Sensor Fusion with Complementary Filtering! !! Quaternions & IMU Sensor Fusion with Complementary Filtering! Gordon Wetzstein! Stanford University! EE 267 Virtual Reality! Lecture 10! stanford.edu/class/ee267/! April 27, 2016! Updates! project proposals

More information

Basic numerical skills: EQUATIONS AND HOW TO SOLVE THEM. x + 5 = 7 2 + 5-2 = 7-2 5 + (2-2) = 7-2 5 = 5. x + 5-5 = 7-5. x + 0 = 20.

Basic numerical skills: EQUATIONS AND HOW TO SOLVE THEM. x + 5 = 7 2 + 5-2 = 7-2 5 + (2-2) = 7-2 5 = 5. x + 5-5 = 7-5. x + 0 = 20. Basic numerical skills: EQUATIONS AND HOW TO SOLVE THEM 1. Introduction (really easy) An equation represents the equivalence between two quantities. The two sides of the equation are in balance, and solving

More information

We know a formula for and some properties of the determinant. Now we see how the determinant can be used.

We know a formula for and some properties of the determinant. Now we see how the determinant can be used. Cramer s rule, inverse matrix, and volume We know a formula for and some properties of the determinant. Now we see how the determinant can be used. Formula for A We know: a b d b =. c d ad bc c a Can we

More information

Classroom Tips and Techniques: The Student Precalculus Package - Commands and Tutors. Content of the Precalculus Subpackage

Classroom Tips and Techniques: The Student Precalculus Package - Commands and Tutors. Content of the Precalculus Subpackage Classroom Tips and Techniques: The Student Precalculus Package - Commands and Tutors Robert J. Lopez Emeritus Professor of Mathematics and Maple Fellow Maplesoft This article provides a systematic exposition

More information

An inertial haptic interface for robotic applications

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

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

Fundamentals of Rocket Stability

Fundamentals of Rocket Stability Fundamentals of Rocket Stability This pamphlet was developed using information for the Glenn Learning Technologies Project. For more information, visit their web site at: http://www.grc.nasa.gov/www/k-12/aboutltp/educationaltechnologyapplications.html

More information

Lecture L1 - Introduction

Lecture L1 - Introduction S. Widnall 16.07 Dynamics Fall 2009 Version 2.0 Lecture L1 - Introduction Introduction In this course we will study Classical Mechanics and its application to aerospace systems. Particle motion in Classical

More information

Near Space Balloon Performance Predictions

Near Space Balloon Performance Predictions 48th AIAA Aerospace Sciences Meeting Including the New Horizons Forum and Aerospace Exposition 4-7 January 2010, Orlando, Florida AIAA 2010-37 Near Space Balloon Performance Predictions Joseph P. Conner,

More information

Eigenvalues and Eigenvectors

Eigenvalues and Eigenvectors Chapter 6 Eigenvalues and Eigenvectors 6. Introduction to Eigenvalues Linear equations Ax D b come from steady state problems. Eigenvalues have their greatest importance in dynamic problems. The solution

More information

Satellite Breakup Risk Mitigation

Satellite Breakup Risk Mitigation Satellite Breakup Risk Mitigation Darrin P. Leleux 1 and Jason T. Smith 2 Orbit Dynamics Branch, Johnson Space Center, Houston TX 77058 Many satellite breakups occur as a result of an explosion of stored

More information

Solution: 2. Sketch the graph of 2 given the vectors and shown below.

Solution: 2. Sketch the graph of 2 given the vectors and shown below. 7.4 Vectors, Operations, and the Dot Product Quantities such as area, volume, length, temperature, and speed have magnitude only and can be completely characterized by a single real number with a unit

More information

SIGNAL PROCESSING & SIMULATION NEWSLETTER

SIGNAL PROCESSING & SIMULATION NEWSLETTER 1 of 10 1/25/2008 3:38 AM SIGNAL PROCESSING & SIMULATION NEWSLETTER Note: This is not a particularly interesting topic for anyone other than those who ar e involved in simulation. So if you have difficulty

More information

Plotting and Adjusting Your Course: Using Vectors and Trigonometry in Navigation

Plotting and Adjusting Your Course: Using Vectors and Trigonometry in Navigation Plotting and Adjusting Your Course: Using Vectors and Trigonometry in Navigation ED 5661 Mathematics & Navigation Teacher Institute August 2011 By Serena Gay Target: Precalculus (grades 11 or 12) Lesson

More information

Principles of inertial sensing technology and its applications in IHCI

Principles of inertial sensing technology and its applications in IHCI Principles of inertial sensing technology and its applications in IHCI Intelligent Human Computer Interaction SS 2011 Gabriele Bleser Gabriele.Bleser@dfki.de Motivation I bet you all got in touch with

More information

[1] Diagonal factorization

[1] Diagonal factorization 8.03 LA.6: Diagonalization and Orthogonal Matrices [ Diagonal factorization [2 Solving systems of first order differential equations [3 Symmetric and Orthonormal Matrices [ Diagonal factorization Recall:

More information

DRAFT. Further mathematics. GCE AS and A level subject content

DRAFT. Further mathematics. GCE AS and A level subject content Further mathematics GCE AS and A level subject content July 2014 s Introduction Purpose Aims and objectives Subject content Structure Background knowledge Overarching themes Use of technology Detailed

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

v 1 v 3 u v = (( 1)4 (3)2, [1(4) ( 2)2], 1(3) ( 2)( 1)) = ( 10, 8, 1) (d) u (v w) = (u w)v (u v)w (Relationship between dot and cross product)

v 1 v 3 u v = (( 1)4 (3)2, [1(4) ( 2)2], 1(3) ( 2)( 1)) = ( 10, 8, 1) (d) u (v w) = (u w)v (u v)w (Relationship between dot and cross product) 0.1 Cross Product The dot product of two vectors is a scalar, a number in R. Next we will define the cross product of two vectors in 3-space. This time the outcome will be a vector in 3-space. Definition

More information

Mathematics Notes for Class 12 chapter 3. Matrices

Mathematics Notes for Class 12 chapter 3. Matrices 1 P a g e Mathematics Notes for Class 12 chapter 3. Matrices A matrix is a rectangular arrangement of numbers (real or complex) which may be represented as matrix is enclosed by [ ] or ( ) or Compact form

More information

Gauss s Law for Gravity

Gauss s Law for Gravity Gauss s Law for Gravity D.G. impson, Ph.D. Department of Physical ciences and Engineering Prince George s Community College December 6, 2006 Newton s Law of Gravity Newton s law of gravity gives the force

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

2. Dynamics, Control and Trajectory Following

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

More information

Lecture 4. Vectors. Motion and acceleration in two dimensions. Cutnell+Johnson: chapter ,

Lecture 4. Vectors. Motion and acceleration in two dimensions. Cutnell+Johnson: chapter , Lecture 4 Vectors Motion and acceleration in two dimensions Cutnell+Johnson: chapter 1.5-1.8, 3.1-3.3 We ve done motion in one dimension. Since the world usually has three dimensions, we re going to do

More information