Rotations. Space and Body  Space coordinates are an inertial system. Fixed in spaceFixed in space  Body coordinates are a non- inertial system. Move.

Slides:

Advertisements

Similar presentations

Projective Geometry- 3D

Advertisements

Chris Hall Aerospace and Ocean Engineering

Tensors. Transformation Rule  A Cartesian vector can be defined by its transformation rule.  Another transformation matrix T transforms similarly. x1x1.

Rotation of Coordinate Systems A x z y x* z* y* Rotation Matrix.

Euler Rotation. Angular Momentum  The angular momentum J is defined in terms of the inertia tensor and angular velocity. All rotations included  The.

Ch. 2: Rigid Body Motions and Homogeneous Transforms

Ch. 4: Velocity Kinematics

Chapter 3 Determinants and Matrices

Angular Motion. Measuring a Circle  We use degrees to measure position around the circle.  There are 2  radians in the circle. This matches 360°This.

Mechanics of Rigid Bodies

Euler Equations. Rotating Vector  A fixed point on a rotating body is associated with a fixed vector. Vector z is a displacement Fixed in the body system.

Body System. Space and Body  Space coordinates are an inertial system. Fixed in spaceFixed in space  Body coordinates are a non- inertial system. Move.

Euler Angles. Three Angles  A rotation matrix can be described with three free parameters. Select three separate rotations about body axesSelect three.

Review (2 nd order tensors): Tensor – Linear mapping of a vector onto another vector Tensor components in a Cartesian basis (3x3 matrix): Basis change.

Rotations and Translations

8.1 Vector spaces A set of vector is said to form a linear vector space V Chapter 8 Matrices and vector spaces.

Theory of Machines Lecture 4 Position Analysis.

Rotations and Translations

Transformations Jehee Lee Seoul National University.

Sect 5.4: Eigenvalues of I & Principal Axis Transformation

Linear algebra: matrix Eigen-value Problems

1 Fundamentals of Robotics Linking perception to action 2. Motion of Rigid Bodies 南台科技大學電機工程系謝銘原.

Rotational Motion 2 Coming around again to a theater near you.

Chapter 8: Rotational Kinematics Essential Concepts and Summary.

§ Linear Operators Christopher Crawford PHY

EEE. Dept of HONG KONG University of Science and Technology Introduction to Robotics Page 1 Lecture 2. Rigid Body Motion Main Concepts: Configuration Space.

KINEMATIC CHAINS & ROBOTS (I).

Rotation matrices 1 Constructing rotation matricesEigenvectors and eigenvalues 0 x y.

1 Dynamics Differential equation relating input torques and forces to the positions (angles) and their derivatives. Like force = mass times acceleration.

Linear Algebra Diyako Ghaderyan 1 Contents:  Linear Equations in Linear Algebra  Matrix Algebra  Determinants  Vector Spaces  Eigenvalues.

Physics 3210 Week 10 clicker questions. Consider a Foucault pendulum in the northern hemisphere. We derived the motion of the pendulum in the absence.

The Spinning Top Chloe Elliott. Rigid Bodies Six degrees of freedom:  3 cartesian coordinates specifying position of centre of mass  3 angles specifying.

Geometric Algebra Dr Chris Doran ARM Research 3. Applications to 3D dynamics.

Linear Algebra Diyako Ghaderyan 1 Contents:  Linear Equations in Linear Algebra  Matrix Algebra  Determinants  Vector Spaces  Eigenvalues.

INTRODUCTION TO DYNAMICS ANALYSIS OF ROBOTS (Part 4)

Physics 3210 Week 11 clicker questions. Multiplication of a vector by a matrix A is a linear transformation What happens to an eigenvector of A under.

10/22/2014PHY 711 Fall Lecture 241 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 24: Rotational.

1 VARIOUS ISSUES IN THE LARGE STRAIN THEORY OF TRUSSES STAMM 2006 Symposium on Trends in Applications of Mathematics to Mechanics Vienna, Austria, 10–14.

Sect. 4.9: Rate of Change of a Vector Use the concept of an infinitesimal rotation to describe the time dependence of rigid body motion. An arbitrary.

Computer Graphics I, Fall 2010 Transformations.

INTRODUCTION TO DYNAMICS ANALYSIS OF ROBOTS (Part 1)

10/24/2014PHY 711 Fall Lecture 251 PHY 711 Classical Mechanics and Mathematical Methods 10-10:50 AM MWF Olin 103 Plan for Lecture 25: Rotational.

Sect. 4.5: Cayley-Klein Parameters 3 independent quantities are needed to specify a rigid body orientation. Most often, we choose them to be the Euler.

1 Part B Tensors Or, inverting The graph above represents a transformation of coordinates when the system is rotated at an angle  CCW.

PHY 711 Classical Mechanics and Mathematical Methods

Review of Linear Algebra

Rotation Matrices and Quaternion

Ch. 2: Rigid Body Motions and Homogeneous Transforms

Lecture 3 Jitendra Malik

Problem 1.5: For this problem, we need to figure out the length of the blue segment shown in the figure. This can be solved easily using similar triangles.

Matrices and vector spaces

Lecture Rigid Body Dynamics.

Lecture 16 Newton Mechanics Inertial properties,Generalized Coordinates Ruzena Bajcsy EE

PHY 711 Classical Mechanics and Mathematical Methods

MiniSkybot: Kinematics

5. Direct Products The basis  of a system may be the direct product of other basis { j } if The system consists of more than one particle. More than.

Rotation / Angle of Rotation / Center of Rotation

Manipulator Dynamics 2 Instructor: Jacob Rosen

Christopher Crawford PHY

2D Geometric Transformations

Physics 321 Hour 31 Euler’s Angles.

Outline: 5.1 INTRODUCTION

Concepts of stress and strain

Outline: 5.1 INTRODUCTION

Derivation of the 2D Rotation Matrix

PHY 711 Classical Mechanics and Mathematical Methods

Rigid Body Transformations

Physics 319 Classical Mechanics

Linear Algebra: Matrix Eigenvalue Problems – Part 2

Translation in Homogeneous Coordinates

Presentation transcript:

Rotations

Space and Body  Space coordinates are an inertial system. Fixed in spaceFixed in space  Body coordinates are a non- inertial system. Move with rigid body x1x1 x2x2 x3x3 x1x1 x2x2 x3x3

Matrix Form  A linear transformation connects the two coordinate systems.  The rotation can be expressed as a matrix. Use matrix operationsUse matrix operations  Distance must be preserved. Matrix is orthogonalMatrix is orthogonal Product is symmetricProduct is symmetric Must have three free parametersMust have three free parameters x1x1 x2x2 x3x3

Axis of Rotation  An orthogonal 3 x 3 matrix will have one real eigenvalue. Real parameters Cubic equation in s  The eigenvalue is unity. Matrix leaves length unchanged  The eigenvector is the axis of rotation. x1x1 x2x2 x3x3 +1 for right handedness

Single Rotation  The eigenvector equation gives the axis of rotation. Eigenvalue = 1Eigenvalue = 1  The trace of the rotation matrix is related to the angle. Angle of rotation Angle of rotation  Trace independent of coordinate systemTrace independent of coordinate system

Rotating Vector  A fixed point on a rotating body is associated with a fixed vector. Vector z is a displacement Fixed in the body system  Differentiate to find the rotated vector. x1x1 x2x2 x3x3

Angular Velocity Matrix  The velocity vector can be found from the rotation.  The matrix  is related to the time derivative of the rotation. Antisymmetric matrixAntisymmetric matrix Equivalent to angular velocity vectorEquivalent to angular velocity vector

Matching Terms  The terms in the  matrix correspond to the components of the angular velocity vector.  The angular velocity is related to the S matrix.

Body Rotation  The angular velocity can also be expressed in the body frame. Body version of matrixBody version of matrix x1x1 x2x2 x3x3