# Metrics on SO(3) and Inverse Kinematics

Save this PDF as:

Size: px
Start display at page:

## Transcription

1 Mathematical Foundations of Computer Graphics and Vision Metrics on SO(3) and Inverse Kinematics Luca Ballan Institute of Visual Computing

2 Optimization on Manifolds Descent approach d is a ascent direction

3 Optimization on Manifolds Given the current point Compute the directional derivative for each direction, i.e. for each curve Determine the is maximum for which Move along this

4 Optimization on Manifolds Given the current point Compute the directional derivative for each direction, i.e. for each curve Determine the is maximum for which I Move along this

5 Last Lecture Rigid transformations (maps which preserve distances and space orientation) isomorphic Angle/axis representation Rigid rotations I = Tangent space at the identity Lie algebra of the nxn skewsymmetric matrices isomorphic = k manifold of the rotation matrices immerse in Lie group smooth map coincides with the matrix exponential GL(n) O(n) = k+n Manifold defined as Lie group SE(n) O(n)/SO(n) SO(n)

6 Exponential Map The exponential map is a function proper of a Lie Group For matrix groups I For SO(3), Rodrigues rotation formula: Smooth Surjective not Injective not Linear (not an isomorphism)

7 Logarithm Map Since is surjective it exists at least an inverse I The inverse of is For SO(3), Rodrigues rotation formula:

8 Properties Identity Inverse in general not Linear (different from the standard log in ) Derivative *

9 Last Lecture There exists a famous local chart for any rotation matrix in SO(3) can be describe as a non-unique combination of 3 rotations (e.g. one along the x-axis, one on the y-axis, and one on the z-axis) Although it is widely used, this representation has some problems Euler-Angle representation (cube) Topology is not conserved (, ) Metric is distorted Derivative is complex (although people use it) Intuitive (easy to visualize) Easy to set constraints Gimbal Lock

10 Content Interpolation in SO(3) Metric in SO(3) Kinematic chains

11 Interpolation in SO(3) Given two rotation matrices, one would like to find a smooth path in SO(3) connecting these two matrices. SO(3) smooth

12 Interpolation in SO(3) Approach 1: Linearly interpolate R1 and R2 in one of their representation Euler angles: R1, R2 too far -> not intuitive motion Topology is not conserved

13 Interpolation in SO(3) Approach 1: Linearly interpolate R1 and R2 in one of their representation Angle-Axis: Interpolate on a plane and then project on a sphere The movement is not linear with a constant speed. It gets faster the more away it is from the Identity

14 Interpolation in SO(3) Approach 2: Linearly interpolate R1 and R2 as matrices it needs to be projected back on the sphere Not an element of SO(3) because it is a multiplicative group not an additive one if R1, R2 are far away from each other, the speed is not linear at all

15 Interpolation in SO(3) Approach 3: use the geodesics of SO(3) Lie Groups: a line passing through 0 in the Lie algebra maps to a geodesic of the Lie group through the identity I so(3) consequently the curve is a geodesic of SO(3) passing through I This holds only for any line passing through 0 and consequently for any geodesic passing through the identity

16 Interpolation in SO(3) To find the geodesic passing through and we need to rotate the ball SO(3) by geodesic between I and SLERP (spherical linear interpolation) geodesic between and The resulting motion is very intuitive and it is performed at uniform angular speed in

17 Interpolation in SO(3) On a vector space with Euclidean metric, the geodesic connecting would have corresponded to the straight line and

18 Questions? Given two rotations R1 and R2, interpolate along the geodesic starting from R1 passing n times through R2 and R1 and ending in R2. from [jacobson 2011] something like this but not limited to a single axis

19 A word about quaternions isomorphic isometry = the hypersphere in quotient the antipodal points. (the hemisphere) quaternion multiplication 3-manifold 3-manifold exp Group of the unit quaternions 3-manifold Lie group Geodesics I = (1,0) Tangent space at the identity Lie Algebra Imaginary numbers Exponential map quaternion exponential they are the same as the geodesics in PRO: easy to compute SLERP CON: difficult to perform derivatives in this space

20 Content Interpolation in SO(3) Metric in SO(3) Kinematic chains

21 Metric in SO(3) We talk about geodesics, but what was the used metric? a metric tells how close two rotations are it is necessary to evaluate an estimator w.r.t. a ground truth

22 Metric in SO(3) We talk about geodesics, but what was the used metric? Riemannian/Geodesic/Angle metric (= to the length of the geodesic connecting R1 and R2) * I I =

23 Metric in SO(3) We talk about geodesics, but what was the used metric? Riemannian/Geodesic/Angle metric (= to the length of the geodesic connecting R1 and R2) Hyperbolic metric similar to the Riemannian if R1=I I I Hyperbolic metric Riemannian metric

24 Metric in SO(3) We talk about geodesics, but what was the used metric? Riemannian/Geodesic/Angle metric (= to the length of the geodesic connecting R1 and R2) Hyperbolic metric I I Frobenius/Chordal metric not similar to Hyperbolic similar to the Riemannian if R1 and R2 are close to each other Hyperbolic metric Frobenius metric

25 Metric in SO(3) We talk about geodesics, but what was the used metric? Riemannian/Geodesic/Angle metric (= to the length of the geodesic connecting R1 and R2) Hyperbolic metric Frobenius/Chordal metric Quaternion metric (related to the space of quaternions, not specifically to the sphere of unit quaternions) Similar to the Hyperbolic one

26 Filtering in SO(3) Given n different estimation for the rotation of an object how can I get a better estimate of? Object at unknown rotation

27 Filtering in SO(3) Given n different estimation for the rotation of an object how can I get a better estimate of? Object at unknown rotation Solution: which of these is the best? Average the rotation matrices? (not rotation) Average the Euler angles of each? Average the angle-axes of each? Average the quaternions related to each? (rotation matrices) Why average?

28 Filtering in SO(3) Why average? By saying average, I m implicitly assuming that the error in the measurements is Gaussian with zero mean Average Median This can be generalized using metrics instead of norms

29 Filtering in SO(3) Why average? By saying average, I m implicitly assuming that the error in the measurements is Gaussian with zero mean Average Median This formulas can be applicable only to neither to in case of SO(3), which metric do we use here?

30 Filtering in SO(3) Geometric mean Average of the angle-axes of each * Similar to the projection of

31 Filtering in SO(3) Matrix mean Similar to the projection of Average of the each matrix element

32 Filtering in SO(3) Fréchet/Karcker mean No close form solution Solve a minimization problem when the solution R is close to I = Geometric mean when the close together are all = Matrix mean

33 Filtering in SO(3) Fréchet/Karcker mean Why is so different? we need to find the rotation R such that the squared sum of the lengths of all the geodesics connecting R to each is minimized The geodesics should start from R and not from the identity (like in the geometric mean) * we need to find the tangent space such that the squared sum of the lengths of all the geodesics of each is minimized

34 Fréchet mean Gradient descent on the manifold J. H. Manton, A globally convergent numerical algorithm for computing the centrer of mass on compact Lie groups, ICARCV 2004 Set Matrix or Geometric mean Compute the average on the tangent space of Move towards

35 Content Interpolation in SO(3) Metric in SO(3) Kinematic chains

36 Special Euclidean group SE(3) Special Euclidean group of order 3 for simplicity of notation, from now on, we will use homogenous coordinates A way of parameterize SE(3) is the following Translation Angle/axis representation of the rotation is called twist, and usually indicated with the symbol This is not the real exponential map in SE(3) (but it is more intuitive)

37 Composition of Rigid Motions Transform p Transform the transformation of p p p p is expressed in local coordinates relative to the framework induced by The second transformation is actually performed on the twist

38 Kinematic Chain A kinematic chain is an ordered set of rigid transformations 0 1 A B 2 3 C Each is called bone (A,B,C) Each is called joint (0,1,2,3) joint 0 is called base/root (and assumed to be fixed) joint 3 is called end effector

39 Kinematic Chain A kinematic chain is an ordered set of rigid transformations 0 1 A B 2 3 C Each bone has its own coordinate system orientation of its local axes determining its position in the space and the the bones A, B, C are oriented accordingly to the x-axis of the reference system The base of each bone corresponds to a joint Each reference system is an element of SE(3) determined by a twists (,,, ) the twists,,, and all together determine completely the configuration of the kinematic chain

40 Kinematic Chain The base twist has the form represents the coordinates of the joint A B 2 3 C determine the orientation of the reference system of bone A All the internal twists ( and ) are defined as the translation is applied only along the x-axis with amount and denote the length of the bone A and B, respectively

41 Kinematic Chain The end effector twist has the form 0 1 A B 2 3 C denote the length of the bone C The orientation of the end effector is the same as the bone C

42 Kinematic Chain: Summary 0 1 A B 2 C 3 determines the position of joint 0 and the orientation of bone A determine the position of joint 1, the length of bone A, and the orientation of bone B w.r.t. the reference system of joint 0 determine the position of joint 2, the length of bone B, and the orientation of bone C w.r.t. the reference system of joint 1 determine the position of joint 3 and the length of bone C

43 Kinematic Chain: DOF 0 1 A B 2 C 3 Given the constraints the actual DOFs of this particular kinematic chain are 3x3 DOF (ball joints) +3 DOF if the base can move +3x1 DOF if the bone is extendible (prismatic joints)

44 Kinematic Chain Problems Given a kinematic chain 0 1 A B 2 C 3 A Forward Kinematics Problem consists in finding the coordinates of the end effector given a specific kinematic chain configuration Forward Kinematics of the end effector

45 Kinematic Chain Problems Given a kinematic chain 0 1 A B 2 C 3 Target q An Inverse Kinematics Problem consists in finding the configuration of the kinematic chain for which the distance between the end effector and a predefined target point q is minimized fixed/or not fixed/or not

46 Inverse Kinematics Problem Given a kinematic chain 0 1 A B 2 C 3 Target q Minimize the distance between where the end effector is and where it should be Generative approach to IK Generative model for p I IK O = Forward Kinematics Forward Kinematics

47 Inverse Kinematics Problem fixed/or not fixed/or not it is equivalent to a non-linear least square optimization problem (it is equivalent to the squared norm and this is ) (note: here it does not matter if the norm is squared or not, later it will) * The problem is under-constrained, 3 equations and (at least) 9 unknowns If q is reachable by the kinematic chain, there are infinite solutions to the problem If q is not reachable, the solution is unique up to rotations along the bones axes

48 A Possible Solution Newton s method let denote with our unknowns let be the current estimate for the solution compute the Taylor expansion of around can be computed using SVD, or approximated as if speed is critical

49 The Jacobian of the Forward Kinematics Given the forward kinematic assuming fixed fixed and the Jacobian of the forward kinematic is 1x3 column vector only one term depends on

50 The Jacobian of the Forward Kinematics

51 The Jacobian of the Forward Kinematics and so on (all the other derivatives are computed in a similar way) The Jacobian of forward kinematic is very easy to compute if the angle/axis representation is used. On the contrary, if quaternions are used instead, the Jacobian is not as trivial

### Kinematical Animation. lionel.reveret@inria.fr 2013-14

Kinematical Animation 2013-14 3D animation in CG Goal : capture visual attention Motion of characters Believable Expressive Realism? Controllability Limits of purely physical simulation : - little interactivity

### Computer Animation. Lecture 2. Basics of Character Animation

Computer Animation Lecture 2. Basics of Character Animation Taku Komura Overview Character Animation Posture representation Hierarchical structure of the body Joint types Translational, hinge, universal,

### 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

### 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

### 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

### 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

### 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

### 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

### 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

### Introduction to Computer Graphics Marie-Paule Cani & Estelle Duveau

Introduction to Computer Graphics Marie-Paule Cani & Estelle Duveau 04/02 Introduction & projective rendering 11/02 Prodedural modeling, Interactive modeling with parametric surfaces 25/02 Introduction

### SO(3) Camillo J. Taylor and David J. Kriegman. Yale University. Technical Report No. 9405

Minimization on the Lie Group SO(3) and Related Manifolds amillo J. Taylor and David J. Kriegman Yale University Technical Report No. 945 April, 994 Minimization on the Lie Group SO(3) and Related Manifolds

### 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

### 521493S Computer Graphics. Exercise 2 & course schedule change

521493S Computer Graphics Exercise 2 & course schedule change Course Schedule Change Lecture from Wednesday 31th of March is moved to Tuesday 30th of March at 16-18 in TS128 Question 2.1 Given two nonparallel,

### Interactive Computer Graphics

Interactive Computer Graphics Lecture 18 Kinematics and Animation Interactive Graphics Lecture 18: Slide 1 Animation of 3D models In the early days physical models were altered frame by frame to create

### Mean value theorem, Taylors Theorem, Maxima and Minima.

MA 001 Preparatory Mathematics I. Complex numbers as ordered pairs. Argand s diagram. Triangle inequality. De Moivre s Theorem. Algebra: Quadratic equations and express-ions. Permutations and Combinations.

### Advanced Techniques for Mobile Robotics Compact Course on Linear Algebra. Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz

Advanced Techniques for Mobile Robotics Compact Course on Linear Algebra Wolfram Burgard, Cyrill Stachniss, Kai Arras, Maren Bennewitz Vectors Arrays of numbers Vectors represent a point in a n dimensional

### 4. Factor polynomials over complex numbers, describe geometrically, and apply to real-world situations. 5. Determine and apply relationships among syn

I The Real and Complex Number Systems 1. Identify subsets of complex numbers, and compare their structural characteristics. 2. Compare and contrast the properties of real numbers with the properties of

### Inverse Kinematics. Ming Yao

Mathematics for Inverse Kinematics 15 464: Technical Animation Ming Yao Overview Kinematics Forward Kinematics and Inverse Kinematics Jabobian Pseudoinverse of the Jacobian Assignment 2 Vocabulary of Kinematics

### animation animation shape specification as a function of time

animation animation shape specification as a function of time animation representation many ways to represent changes with time intent artistic motion physically-plausible motion efficiency control typically

### Precalculus REVERSE CORRELATION. Content Expectations for. Precalculus. Michigan CONTENT EXPECTATIONS FOR PRECALCULUS CHAPTER/LESSON TITLES

Content Expectations for Precalculus Michigan Precalculus 2011 REVERSE CORRELATION CHAPTER/LESSON TITLES Chapter 0 Preparing for Precalculus 0-1 Sets There are no state-mandated Precalculus 0-2 Operations

### Senior Secondary Australian Curriculum

Senior Secondary Australian Curriculum Mathematical Methods Glossary Unit 1 Functions and graphs Asymptote A line is an asymptote to a curve if the distance between the line and the curve approaches zero

### Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE. M.Sc. in Advanced Computer Science. Friday 18 th January 2008.

COMP60321 Two hours UNIVERSITY OF MANCHESTER SCHOOL OF COMPUTER SCIENCE M.Sc. in Advanced Computer Science Computer Animation Friday 18 th January 2008 Time: 09:45 11:45 Please answer any THREE Questions

### Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum

Calculus C/Multivariate Calculus Advanced Placement G/T Essential Curriculum UNIT I: The Hyperbolic Functions basic calculus concepts, including techniques for curve sketching, exponential and logarithmic

### Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks

Thnkwell s Homeschool Precalculus Course Lesson Plan: 36 weeks Welcome to Thinkwell s Homeschool Precalculus! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

### Prerequsites: Math 1A-1B, 53 (lower division calculus courses)

Math 151 Prerequsites: Math 1A-1B, 53 (lower division calculus courses) Development of the rational number system. Use the number line (real line), starting with the concept of parts of a whole : fractions,

### WASSCE / WAEC ELECTIVE / FURTHER MATHEMATICS SYLLABUS

Visit this link to read the introductory text for this syllabus. 1. Circular Measure Lengths of Arcs of circles and Radians Perimeters of Sectors and Segments measure in radians 2. Trigonometry (i) Sine,

### 3D geometry basics (for robotics) lecture notes

lecture notes Marc Toussaint Machine Learning & Robotics lab, FU Berlin Arnimallee 7, 495 Berlin, Germany October 30, 20 This document introduces to some basic geometry, focussing on 3D transformations,

### Identify examples of field properties: commutative, associative, identity, inverse, and distributive.

Topic: Expressions and Operations ALGEBRA II - STANDARD AII.1 The student will identify field properties, axioms of equality and inequality, and properties of order that are valid for the set of real numbers

### Geometric Modelling Summer 2016

Geometric Modelling Summer 2016 Prof. Dr. Hans Hagen http://gfx.uni-kl.de/~gm Prof. Dr. Hans Hagen Geometric Modelling Summer 2016 1 Ane Spaces, Elliptic Prof. Dr. Hans Hagen Geometric Modelling Summer

### DISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION. Keenan Crane CMU (J) Spring 2016

DISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION Keenan Crane CMU 15-869(J) Spring 2016 PART III: GEOMETRIC STRUCTURE DISCRETE DIFFERENTIAL GEOMETRY: AN APPLIED INTRODUCTION Keenan Crane CMU 15-869(J)

### Math 497C Sep 9, Curves and Surfaces Fall 2004, PSU

Math 497C Sep 9, 2004 1 Curves and Surfaces Fall 2004, PSU Lecture Notes 2 15 sometries of the Euclidean Space Let M 1 and M 2 be a pair of metric space and d 1 and d 2 be their respective metrics We say

### CIS 536/636 Introduction to Computer Graphics. Kansas State University. CIS 536/636 Introduction to Computer Graphics

2 Lecture Outline Animation 2 of 3: Rotations, Quaternions Dynamics & Kinematics William H. Hsu Department of Computing and Information Sciences, KSU KSOL course pages: http://bit.ly/hgvxlh / http://bit.ly/evizre

### 14.11. Geodesic Lines, Local Gauss-Bonnet Theorem

14.11. Geodesic Lines, Local Gauss-Bonnet Theorem Geodesics play a very important role in surface theory and in dynamics. One of the main reasons why geodesics are so important is that they generalize

### ME 115(b): Solution to Homework #1

ME 115(b): Solution to Homework #1 Solution to Problem #1: To construct the hybrid Jacobian for a manipulator, you could either construct the body Jacobian, JST b, and then use the body-to-hybrid velocity

### animation shape specification as a function of time

animation 1 animation shape specification as a function of time 2 animation representation many ways to represent changes with time intent artistic motion physically-plausible motion efficiency typically

### The Three Reflections Theorem

The Three Reflections Theorem Christopher Tuffley Institute of Fundamental Sciences Massey University, Palmerston North 24th June 2009 Christopher Tuffley (Massey University) The Three Reflections Theorem

### 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

### α = u v. In other words, Orthogonal Projection

Orthogonal Projection Given any nonzero vector v, it is possible to decompose an arbitrary vector u into a component that points in the direction of v and one that points in a direction orthogonal to v

### 6. Representing Rotation

6. Representing Rotation Mechanics of Manipulation Matt Mason matt.mason@cs.cmu.edu http://www.cs.cmu.edu/~mason Carnegie Mellon Lecture 6. Mechanics of Manipulation p.1 Lecture 6. Representing Rotation.

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

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

### CS 4620 Practicum Programming Assignment 6 Animation

CS 4620 Practicum Programming Assignment 6 Animation out: Friday 14th November 2014 due: : Monday 24th November 2014 1 Introduction In this assignment, we will explore a common topic in animation: key

### 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 =,

### MATHEMATICS (CLASSES XI XII)

MATHEMATICS (CLASSES XI XII) General Guidelines (i) All concepts/identities must be illustrated by situational examples. (ii) The language of word problems must be clear, simple and unambiguous. (iii)

### Section 12.1 Translations and Rotations

Section 12.1 Translations and Rotations Any rigid motion that preserves length or distance is an isometry (meaning equal measure ). In this section, we will investigate two types of isometries: translations

### Dimension Theory for Ordinary Differential Equations

Vladimir A. Boichenko, Gennadij A. Leonov, Volker Reitmann Dimension Theory for Ordinary Differential Equations Teubner Contents Singular values, exterior calculus and Lozinskii-norms 15 1 Singular values

### MCE/EEC 647/747: Robot Dynamics and Control

MCE/EEC 647/747: Robot Dynamics and Control Lecture 4: Velocity Kinematics Jacobian and Singularities Torque/Force Relationship Inverse Velocity Problem Reading: SHV Chapter 4 Mechanical Engineering Hanz

### Content. Chapter 4 Functions 61 4.1 Basic concepts on real functions 62. Credits 11

Content Credits 11 Chapter 1 Arithmetic Refresher 13 1.1 Algebra 14 Real Numbers 14 Real Polynomials 19 1.2 Equations in one variable 21 Linear Equations 21 Quadratic Equations 22 1.3 Exercises 28 Chapter

### PURE MATHEMATICS AM 27

AM Syllabus (015): Pure Mathematics AM SYLLABUS (015) PURE MATHEMATICS AM 7 SYLLABUS 1 AM Syllabus (015): Pure Mathematics Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs)

### PURE MATHEMATICS AM 27

AM SYLLABUS (013) PURE MATHEMATICS AM 7 SYLLABUS 1 Pure Mathematics AM 7 Syllabus (Available in September) Paper I(3hrs)+Paper II(3hrs) 1. AIMS To prepare students for further studies in Mathematics and

### Module 1 : A Crash Course in Vectors Lecture 2 : Coordinate Systems

Module 1 : A Crash Course in Vectors Lecture 2 : Coordinate Systems Objectives In this lecture you will learn the following Define different coordinate systems like spherical polar and cylindrical coordinates

### Geometric Transformations

Geometric Transformations Definitions Def: f is a mapping (function) of a set A into a set B if for every element a of A there exists a unique element b of B that is paired with a; this pairing is denoted

### NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS

NEW YORK STATE TEACHER CERTIFICATION EXAMINATIONS TEST DESIGN AND FRAMEWORK September 2014 Authorized for Distribution by the New York State Education Department This test design and framework document

### Diploma Plus in Certificate in Advanced Engineering

Diploma Plus in Certificate in Advanced Engineering Mathematics New Syllabus from April 2011 Ngee Ann Polytechnic / School of Interdisciplinary Studies 1 I. SYNOPSIS APPENDIX A This course of advanced

### Vector Spaces; the Space R n

Vector Spaces; the Space R n Vector Spaces A vector space (over the real numbers) is a set V of mathematical entities, called vectors, U, V, W, etc, in which an addition operation + is defined and in which

### Official Math 112 Catalog Description

Official Math 112 Catalog Description Topics include properties of functions and graphs, linear and quadratic equations, polynomial functions, exponential and logarithmic functions with applications. A

### arxiv: v1 [math.mg] 6 May 2014

DEFINING RELATIONS FOR REFLECTIONS. I OLEG VIRO arxiv:1405.1460v1 [math.mg] 6 May 2014 Stony Brook University, NY, USA; PDMI, St. Petersburg, Russia Abstract. An idea to present a classical Lie group of

### AP Calculus BC. All students enrolling in AP Calculus BC should have successfully completed AP Calculus AB.

AP Calculus BC Course Description: Advanced Placement Calculus BC is primarily concerned with developing the students understanding of the concepts of calculus and providing experiences with its methods

### Kinematics of Robots. Alba Perez Gracia

Kinematics of Robots Alba Perez Gracia c Draft date August 31, 2007 Contents Contents i 1 Motion: An Introduction 3 1.1 Overview.......................................... 3 1.2 Introduction.........................................

### Common Core Unit Summary Grades 6 to 8

Common Core Unit Summary Grades 6 to 8 Grade 8: Unit 1: Congruence and Similarity- 8G1-8G5 rotations reflections and translations,( RRT=congruence) understand congruence of 2 d figures after RRT Dilations

### MATH Mathematics-Nursing. MATH Remedial Mathematics I-Business & Economics. MATH Remedial Mathematics II-Business and Economics

MATH 090 - Mathematics-Nursing MATH 091 - Remedial Mathematics I-Business & Economics MATH 094 - Remedial Mathematics II-Business and Economics MATH 095 - Remedial Mathematics I-Science (3 CH) MATH 096

### 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

### Fourier Analysis. A cosine wave is just a sine wave shifted in phase by 90 o (φ=90 o ). degrees

Fourier Analysis Fourier analysis follows from Fourier s theorem, which states that every function can be completely expressed as a sum of sines and cosines of various amplitudes and frequencies. This

### Least-Squares Intersection of Lines

Least-Squares Intersection of Lines Johannes Traa - UIUC 2013 This write-up derives the least-squares solution for the intersection of lines. In the general case, a set of lines will not intersect at a

### The shortest path between two points, Geodesics and Mechanics

Raynaud, X., 2010: The shortest path between two points, Geodesics and Mechanics, In: W. Østreng (ed): Transference. Interdisciplinary Communications 2008/2009, CAS, Oslo. (Internet publication, http://www.cas.uio.no/publications_/transference.php,

### Rotations and Quaternions

Uniersity of British Columbia CPSC 414 Computer Graphics Rotations and Quaternions Week 9, Wed 29 Oct 2003 Tamara Munzner 1 Uniersity of British Columbia CPSC 414 Computer Graphics Clipping recap Tamara

### Liouville Quantum gravity and KPZ. Scott Sheffield

Liouville Quantum gravity and KPZ Scott Sheffield Scaling limits of random planar maps Central mathematical puzzle: Show that the scaling limit of some kind of discrete quantum gravity (perhaps decorated

### We shall turn our attention to solving linear systems of equations. Ax = b

59 Linear Algebra We shall turn our attention to solving linear systems of equations Ax = b where A R m n, x R n, and b R m. We already saw examples of methods that required the solution of a linear system

### Numerical Methods for Civil Engineers. Linear Equations

Numerical Methods for Civil Engineers Lecture Notes CE 311K Daene C. McKinney Introduction to Computer Methods Department of Civil, Architectural and Environmental Engineering The University of Texas at

### Students will understand 1. use numerical bases and the laws of exponents

Grade 8 Expressions and Equations Essential Questions: 1. How do you use patterns to understand mathematics and model situations? 2. What is algebra? 3. How are the horizontal and vertical axes related?

### B4 Computational Geometry

3CG 2006 / B4 Computational Geometry David Murray david.murray@eng.o.ac.uk www.robots.o.ac.uk/ dwm/courses/3cg Michaelmas 2006 3CG 2006 2 / Overview Computational geometry is concerned with the derivation

### Mathematics Course 111: Algebra I Part IV: Vector Spaces

Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 1996-7 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are

### Lecture 19 Camera Matrices and Calibration

Lecture 19 Camera Matrices and Calibration Project Suggestions Texture Synthesis for In-Painting Section 10.5.1 in Szeliski Text Project Suggestions Image Stitching (Chapter 9) Face Recognition Chapter

### 1 Introduction. 2 The G 4 Algebra. 2.1 Definition of G 4 Algebra

1 Introduction Traditionally translations and rotations have been analysed/computed with either matrices, quaternions, bi-quaternions, dual quaternions or double quaternions depending of ones preference

### Thinkwell s Homeschool Algebra 2 Course Lesson Plan: 34 weeks

Thinkwell s Homeschool Algebra 2 Course Lesson Plan: 34 weeks Welcome to Thinkwell s Homeschool Algebra 2! We re thrilled that you ve decided to make us part of your homeschool curriculum. This lesson

### 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

### ROBOTICS: ADVANCED CONCEPTS & ANALYSIS

ROBOTICS: ADVANCED CONCEPTS & ANALYSIS MODULE 5 - VELOCITY AND STATIC ANALYSIS OF MANIPULATORS Ashitava Ghosal 1 1 Department of Mechanical Engineering & Centre for Product Design and Manufacture Indian

### C is a point of concurrency is at distance from End Effector frame & at distance from ref frame.

Module 6 : Robot manipulators kinematics Lecture 21 : Forward & inverse kinematics examples of 2R, 3R & 3P manipulators Objectives In this course you will learn the following Inverse position and orientation

### PSS 27.2 The Electric Field of a Continuous Distribution of Charge

Chapter 27 Solutions PSS 27.2 The Electric Field of a Continuous Distribution of Charge Description: Knight Problem-Solving Strategy 27.2 The Electric Field of a Continuous Distribution of Charge is illustrated.

### Linear Systems and Gaussian Elimination

Eivind Eriksen Linear Systems and Gaussian Elimination September 2, 2011 BI Norwegian Business School Contents 1 Linear Systems................................................ 1 1.1 Linear Equations...........................................

### State of Stress at Point

State of Stress at Point Einstein Notation The basic idea of Einstein notation is that a covector and a vector can form a scalar: This is typically written as an explicit sum: According to this convention,

### The calibration problem was discussed in details during lecture 3.

1 2 The calibration problem was discussed in details during lecture 3. 3 Once the camera is calibrated (intrinsics are known) and the transformation from the world reference system to the camera reference

### Special Orthogonal Groups and Rotations

Special Orthogonal Groups and Rotations Christopher Triola Submitted in partial fulfillment of the requirements for Honors in Mathematics at the University of Mary Washington Fredericksburg, Virginia April

### Animation. Animation. What is animation? Approaches to animation

Animation Animation CS 4620 Lecture 20 Industry production process leading up to animation What animation is How animation works (very generally) Artistic process of animation Further topics in how it

### Appendix 3 IB Diploma Programme Course Outlines

Appendix 3 IB Diploma Programme Course Outlines The following points should be addressed when preparing course outlines for each IB Diploma Programme subject to be taught. Please be sure to use IBO nomenclature

### Modern Geometry Homework.

Modern Geometry Homework. 1. Rigid motions of the line. Let R be the real numbers. We define the distance between x, y R by where is the usual absolute value. distance between x and y = x y z = { z, z

### 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,

### Basic methods in Computer Animation

jo a r Ju r ik e nd sk. g scc @ k eri d on Lesson 02 Basic methods in Computer Animation Lesson 02 Outline Key-framing and parameter interpolation Skeleton Animation Forward and inverse Kinematics Procedural

### 1 VECTOR SPACES AND SUBSPACES

1 VECTOR SPACES AND SUBSPACES What is a vector? Many are familiar with the concept of a vector as: Something which has magnitude and direction. an ordered pair or triple. a description for quantities such

### Items related to expected use of graphing technology appear in bold italics.

- 1 - Items related to expected use of graphing technology appear in bold italics. Investigating the Graphs of Polynomial Functions determine, through investigation, using graphing calculators or graphing

### 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

### Cofactor Expansion: Cramer s Rule

Cofactor Expansion: Cramer s Rule MATH 322, Linear Algebra I J. Robert Buchanan Department of Mathematics Spring 2015 Introduction Today we will focus on developing: an efficient method for calculating

### Chapter 1 Quadratic Equations in One Unknown (I)

Tin Ka Ping Secondary School 015-016 F. Mathematics Compulsory Part Teaching Syllabus Chapter 1 Quadratic in One Unknown (I) 1 1.1 Real Number System A Integers B nal Numbers C Irrational Numbers D Real

### MyMathLab ecourse for Developmental Mathematics

MyMathLab ecourse for Developmental Mathematics, North Shore Community College, University of New Orleans, Orange Coast College, Normandale Community College Table of Contents Module 1: Whole Numbers and

### 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

### 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

### Chapter 18 Electric Forces and Electric Fields. Key Concepts:

Chapter 18 Lectures Monday, January 25, 2010 7:33 AM Chapter 18 Electric Forces and Electric Fields Key Concepts: electric charge principle of conservation of charge charge polarization, both permanent

### Algebra II Pacing Guide First Nine Weeks

First Nine Weeks SOL Topic Blocks.4 Place the following sets of numbers in a hierarchy of subsets: complex, pure imaginary, real, rational, irrational, integers, whole and natural. 7. Recognize that the