ME 115(b): Solution to Homework #1


 Ophelia Higgins
 3 years ago
 Views:
Transcription
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 bodytohybrid velocity transformation: JST h RST 0 JST b 0 R ST or recall that the columns of the hybrid Jacobian take the form: θ N JST h (1) ω 1 ω 2 ω N where the forward kinematics equations g ST ( θ) take the form: g ST ( θ) RST ( θ) p ST ( θ) and ω j is: ω j This solution will use the second approach. ( RST RST T θ j ). Manipulator (ii): While this manipulator has a rather odd geometry, it is relatively straightforward to tackle this problem by suitable choices of geometry in the Denavit Hartenburg approach. If β is the angle between the first and third joint axes when the manipulator lies in the configuration shown in the book, and if l 1 and l 2 are the two link lengths as shown in the book s figure, then the DH parameters for this manipulator are: a 0 0 α 0 0 d 1 0 θ 1 variable a 1 0 α 1 π 2 d 2 0 θ 2 variable a 2 l 1 cos β α 2 π 2 d 3 (l 1 + l 2 ) sin β θ 3 variable a 3 l 2 cos β α 3 0 d 4 0 θ 4 0 where we have assumed that the tool frame zaxis is parallel to joint axis 3. Recalling the the relationship between link frames in terms of the DH parameters: cos θ i+1 sin θ i+1 0 a i g i,i+1 sin θ i+1 cos α i cos θ i+1 cos α i sin α i d i+1 sin α i sin θ i+1 sin α i cos θ i+1 sin α i cos α i d i+1 cos α i
2 The forward kinematics of this mechanism can be found as: g ST g S,1 g 1,2 g 2,3 g 3,T c 1 s c 2 s c 3 s 3 0 a a 3 s 1 c d s 2 c s 3 c (2) (c 1 c 2 c 3 s 1 s 3 ) (c 1 c 2 s 3 + s 1 c 3 ) c 1 s 2 (a 3 c 1 c 2 c 3 + a 2 c 1 c 2 + d 3 c 1 s 2 a 3 s 1 s 3 ) (s 1 c 2 c 3 + c 1 s 3 ) ( s 1 c 2 c 3 + c 1 c 3 ) s 1 s 2 (a 3 s 1 c 2 c 3 + a 2 s 1 c 2 + d 3 s 1 s 2 + a 3 c 1 s 3 ) s 2 c 3 s 2 s 3 c 2 ( a 3 s 2 c 3 a 2 s 2 + d 3 c 2 ) Following Equation (1), the hybrid Jacobian is: J h ST ( v1 v 2 v 3 ω 1 ω 2 ω 3 θ 3 RST T ) (3) where: ( a 3 s 1 c 2 c 3 a 2 s 1 c 2 d 3 s 1 s 2 a 3 c 1 s 3 ) v 1 (a 3 c 1 c 2 c 3 + a 2 c 1 c 2 + d 3 c 1 s 2 a 3 s 1 s 3 ) 0 ( a 3 c 1 s 2 c 3 a 2 c 1 s 2 + d 3 c 1 c 2 ) v 2 ( a 3 s 1 s 2 c 3 a 2 s 1 s 2 + d 3 s 1 c 2 ) ( a 3 c 1 c 2 s 3 a 3 s 1 c 3 ) v 3 ( a 2 s 1 c 2 s 3 + a 3 c 1 c 3 ) ( a 3 c 2 c 3 a 2 c 2 d 3 s 2 ) (a 3 s 2 s 3 ) 0 s 1 c 1 s 2 ω1 ω 2 ω 3 0 c 1 s 1 s c 2 Manipulator (iv): This is the Stanford Manipulator. Recall from a previous homework solution that the DenavitHartenberg parameters and the forward kinematics are: a 0 0 α 0 0 d 1 0 θ 1 variable a 1 0 α 1 π 2 d 2 0 θ 2 variable a 2 0 α 2 π 2 d 3 variable θ 3 0 (constant) 2
3 g ST g S,1 g 1,2 g 2,3 g 3,T (4) c 1 s c 2 s s 1 c d s 2 c (5) c 1 c 2 s 1 c 1 s 2 d 3 c 1 s 2 s 1 c 2 c 1 s 1 s 2 d 3 s 1 s 2 s 2 0 c 2 d 3 c 2 RST (θ 1, θ 2, d 3 ) p ST (θ 1, θ 2, d 3 ) 0 T (6) Following Equation (1), the hybrid Jacobian is: J h ST ( RST T d 3 s 1 s 2 d 3 c 1 c 2 c 1 s 2 d 3 c 2 s 2 d 3 s 1 c 2 s 1 s 2 0 d 3 s 2 c 2 0 s c d 3 ) (7) (8) (9) Solution to Problem #2: To find the singularities of the regional part (just the first three joints) of the elbow manipulator, one can determine the conditions under which the Jacobian matrix of the manipulator loses rank. While one could use any Jacobian, for simplicity we will use the Hybrid Jacobian matrix. You could either recall from the class note (or derive) the forward kinematics of the Elbow manipulator: g ST ( θ) RST ( θ) p ST ( θ) c 1 c 23 s 1 c 1 s 23 c 1 (l 2 c 2 + l 3 c 23 ) s 1 c 23 c 1 s 1 s 23 s 1 (l 2 c 2 + l 3 c 23 ) s 23 0 c 23 (l 2 s 2 + l 3 s 23 ) (10) where c j cos(θ j ), s j sin(θ j ), c ij cos(θ i + θ j ), s ij sin(θ i + θ j ), etc. Recall that the hybrid Jacobian for the regional part of a manipulator is defined as: JST h (11) 3 pst θ
4 and thus substituting Equation (10) into Equation (11) yields: s 1 (l 2 c 2 + l 3 c 23 ) c 1 (l 2 s 2 + l 3 s 23 ) l 3 c 1 s 23 JST h c 1 (l 2 c 2 + l 3 c 23 ) s 1 (l 2 s 2 + l 3 s 23 ) l 3 s 1 s 23. (12) 0 (l 2 c 2 + l 3 c 23 ) l 3 c 23 Singularities will occur when the determinant of J h ST is zero: The singularities occur when: det(j h ST ) l 2 l 3 l 2 cos(θ 2 ) + l 3 cos(θ 2 + θ 3 ) sin(θ 3 ) θ 3 0. In this case, the arm is fully streteched out, and thus this singular configuration corresponds to the manipulator s outer workspace limit. θ 3 ±π. In this case, the arm is folded back on itself, and this singular configuration corresponds to the manipulator s inner workspace boundary. l 2 c 2 + l 3 c Note from the forward kinematics equations that in this case, x and y coordinates of the tool frame origin lie at x 0 and y 0. This occurs when the tool frame origin is placed anywhere along the first joint axis. Solution to Problem #3: Part (a): From Problem 1 above we know that regional forward kinematics that relates the joint variables (θ 1, θ 2, d 3 ) of the Stanford manipulator to the Cartesian location of the tool frame origin is: x d 3 c 1 s 2 p ST y d 3 s 1 s 2 (13) z d 3 c 2 T x where p ST y is the location of the tool frame origin. The inverse kinematic solution z can be derived by simple algebraic manipulation of these kinematic relationships. 1. Step #1: x 2 + y 2 + z 2 d 2 3 d 3 ± x 2 + y 2 + z 2. There are two solutions to this equation. In practice, one of these solutions will almost certainly be infeasible from a mechanical point of view. 2. Step #2: x 2 + y 2 d 2 3s 2 x 2 s 2 ± 2 +y 2. There are 4 possible solutions to x 2 +y 2 +z 2 this equation. For each different of the two possible d 3 solutions there are two different θ 2 solutions. 4
5 3. Step #3: From the forward kinematics, we have that: cos θ 1 x d 3 sin θ 2 ; sin θ 1 from which we can get the unique solution: θ 1 There is a single θ 1 solution for each (θ 2, d 3 ) combination. y d 3 sin θ 2 ; (14) y x Atan2,. (15) d 3 sin θ 2 d 3 sin θ 2 Part (b): This problem is most readily solved by realizing that manipulators with wrists have a unique forward kinematic (and therefore inverse kinematic) solution structure. Note that a wrist is defined as a serial kinematic chain of three revolute joints whose joint axes all intersect at a single point, which we will term the wrist center. The wrist center will also be the origin of the references frames (as defined by the DenavitHartenberg convention) of those links that make up the wrist. Let the form of the forward kinematic relationship be denoted by: g ST RST (θ 1,..., θ 6 ) p ST (θ 1,..., θ 6 ) R D ST p D ST where RST D and pd ST are the desired orientation and position of the tool frame. Let g 6,T denote the location of the tool frame relative to the DenavitHartenberg reference frame of link 6 (whose origin will lie at the wrist point). g 6,T is a constant transformation that accounts for any difference in the location of the tool relative to the link frame. The desired position and orientation of link frame 6 will thus be: g D S,6 R D ST p D ST R6,T p 6,T 1 R D S,6 p D S,6 (16). (17) Note that the forward kinematics formula which describes the origin of Link frame 6 (the wrist center point) is only a function of (θ 1, θ 2, d 3 ), and is in fact the formula used in part (a) of this problem. Thus, (θ 1, θ 2, d 3 ) can be found from part (a), using p D S,6 is the desired position of the wrist center point. Note that the forward kinematics relationship that describes the orientation of link frame 6 is: R D S,6 R D S,T R 1 6,T R S,1(θ 1 ) R 1,2 (θ 2 ) R 2,3 R 3,4 (θ 4 ) R 4,5 (θ 5 ) R 5,6 (θ 6 ). (18) But (θ 1, θ 2, d 3 ) have already been determined by the placement of the wrist point at its desired location. Hence, we can rearrange Equation (18) to isolate the unknown joint variables (θ 4, θ 5, θ 6 ): R 3,4 (θ 4 ) R 4,5 (θ 5 ) R 5,6 (θ 6 ) (R S,1 R 1,2 R 2,3 ) 1 RS,T D R 1 6,T. (19) All of the terms on the right hand side of Equation (19) are known from the inverse kinematic problem state, the knowledge of g 6,T, and the inverse kinematic solution of the regional structure. 5
6 Note that the general form of the rotation matrix using the DenavitHartenberg convention is: cos θ i sin θ i 0 R i 1,i sin θ i cos α i 1 cos θ i cos α i 1 sin α i 1 (20) sin θ i sin α i 1 cos θ i sin α i 1 cos α i 1 For the Stanford manipulator, one choice 1 of the twist angles for the wrist substructure is: α 3 π/2, α 4 π/2, α 5 π/2. Hence: c 4 s 4 0 c 5 s 5 0 c 6 s 6 0 R 3, R 4, R 5, (21) s 4 c 4 0 s 5 c 5 0 s 6 c 6 0 and therefore: (c 4 c 5 c 6 s 4 s 6 ) (c 4 c 5 s 6 + s 4 c 6 ) c 4 s 5 R 3,4 R 4,5 R 5,6 s 5 c 6 s 5 s 6 c 5 (22) (s 4 c 5 c 6 + c 4 s 6 ) (s 4 c 5 s 6 c 4 c 6 ) s 4 s 5 Let the entries constant matrix R3,6 D (R S,1 R 1,2 R 2,3 ) 1 RS,T D R 1 6,T r 11 r 12 r 13 R3,6 D r 21 r 22 r 23. r 31 r 32 r 33 Equating terms, we can see that: cos θ 5 r 23 which has two solutions. Using these solutions, we see that: r 33 θ 4 Atan2, r 13 sin θ 5 sin θ 5 θ 6 Atan2 r 22 r 21, sin θ 5 sin θ 5 have the form: Hence, the wrist has two independent solutions, for each of the four different solutions to the regional structure inverse kinematics, yielding 8 different inverse kinematic solutions. angles 1 any other choice of twist angles that satisfy the DH convention will differ only in the signs of these 6
Simulation of Trajectories and Comparison of Joint Variables for Robotic Manipulator Using Multibody Dynamics (MBD)
Simulation of Trajectories and Comparison of Joint Variables for Robotic Manipulator Using Multibody Dynamics (MBD) Jatin Dave Assistant Professor Nirma University Mechanical Engineering Department, Institute
More informationC 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
More informationINTRODUCTION. Robotics is a relatively young field of modern technology that crosses traditional
1 INTRODUCTION Robotics is a relatively young field of modern technology that crosses traditional engineering boundaries. Understanding the complexity of robots and their applications requires knowledge
More informationRobot Modeling and Control
Robot Modeling and Control First Edition Mark W. Spong, Seth Hutchinson, and M. Vidyasagar JOHN WILEY & SONS, INC. New York / Chichester / Weinheim / Brisbane / Singapore / Toronto Preface TO APPEAR i
More informationConstraint satisfaction and global optimization in robotics
Constraint satisfaction and global optimization in robotics Arnold Neumaier Universität Wien and JeanPierre Merlet INRIA Sophia Antipolis 1 The design, validation, and use of robots poses a number of
More informationACTUATOR 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.Nagar388120, Gujarat, India 2 Parul Institute of Engineering & Technology, Limda391760,
More informationINSTRUCTOR WORKBOOK Quanser Robotics Package for Education for MATLAB /Simulink Users
INSTRUCTOR WORKBOOK for MATLAB /Simulink Users Developed by: Amir Haddadi, Ph.D., Quanser Peter Martin, M.A.SC., Quanser Quanser educational solutions are powered by: CAPTIVATE. MOTIVATE. GRADUATE. PREFACE
More informationMechanics 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 informationINTRODUCTION. Chapter 1. 1.1 Robotics
Chapter 1 INTRODUCTION 1.1 Robotics Robotics is a relatively young field of modern technology that crosses traditional engineering boundaries. Understanding the complexity of robots and their applications
More informationManipulator Kinematics. Prof. Matthew Spenko MMAE 540: Introduction to Robotics Illinois Institute of Technology
Manipulator Kinematics Prof. Matthew Spenko MMAE 540: Introduction to Robotics Illinois Institute of Technology Manipulator Kinematics Forward and Inverse Kinematics 2D Manipulator Forward Kinematics Forward
More informationIntelligent Submersible ManipulatorRobot, Design, Modeling, Simulation and Motion Optimization for Maritime Robotic Research
20th International Congress on Modelling and Simulation, Adelaide, Australia, 1 6 December 2013 www.mssanz.org.au/modsim2013 Intelligent Submersible ManipulatorRobot, Design, Modeling, Simulation and
More informationWe can use more sectors (i.e., decrease the sector s angle θ) to get a better approximation:
Section 1.4 Areas of Polar Curves In this section we will find a formula for determining the area of regions bounded by polar curves. To do this, wee again make use of the idea of approximating a region
More informationKinematics 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.........................................
More informationSection 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 informationIntroduction to the inverse kinematics of serial manipulators
Introduction to the inverse kinematics of serial manipulators Sandipan Bandyopadhyay Department of Engineering Design Indian Institute of Technology Madras, Chennai 600 036 Email: sandipan@iitm.ac.in Web:
More informationExam 1 Sample Question SOLUTIONS. y = 2x
Exam Sample Question SOLUTIONS. Eliminate the parameter to find a Cartesian equation for the curve: x e t, y e t. SOLUTION: You might look at the coordinates and notice that If you don t see it, we can
More informationLecture 14: Section 3.3
Lecture 14: Section 3.3 Shuanglin Shao October 23, 2013 Definition. Two nonzero vectors u and v in R n are said to be orthogonal (or perpendicular) if u v = 0. We will also agree that the zero vector in
More informationRobot Dynamics and Control
Robot Dynamics and Control Second Edition Mark W. Spong, Seth Hutchinson, and M. Vidyasagar January 28, 2004 2 Contents 1 INTRODUCTION 5 1.1 Robotics..................................... 5 1.2 History
More informationWe can display an object on a monitor screen in three different computermodel forms: Wireframe model Surface Model Solid model
CHAPTER 4 CURVES 4.1 Introduction In order to understand the significance of curves, we should look into the types of model representations that are used in geometric modeling. Curves play a very significant
More informationMetrics 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 informationPractical Work DELMIA V5 R20 Lecture 1. D. Chablat / S. Caro Damien.Chablat@irccyn.ecnantes.fr Stephane.Caro@irccyn.ecnantes.fr
Practical Work DELMIA V5 R20 Lecture 1 D. Chablat / S. Caro Damien.Chablat@irccyn.ecnantes.fr Stephane.Caro@irccyn.ecnantes.fr Native languages Definition of the language for the user interface English,
More informationdiscuss how to describe points, lines and planes in 3 space.
Chapter 2 3 Space: lines and planes In this chapter we discuss how to describe points, lines and planes in 3 space. introduce the language of vectors. discuss various matters concerning the relative position
More information1.7 Cylindrical and Spherical Coordinates
56 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE 1.7 Cylindrical and Spherical Coordinates 1.7.1 Review: Polar Coordinates The polar coordinate system is a twodimensional coordinate system in which the
More informationRoboAnalyzer: 3D Model Based Robotic Learning Software
International Conference on Multi Body Dynamics 2011 Vijayawada, India. pp. 3 13 RoboAnalyzer: 3D Model Based Robotic Learning Software C. G. Rajeevlochana 1 and S. K. Saha 2 1 Research Scholar, Dept.
More informationFURTHER 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: 97 Mathematics
More informationAn Application of Robotic Optimization: Design for a Tire Changing Robot
An Application of Robotic Optimization: Design for a Tire Changing Robot RAUL MIHALI, MHER GRIGORIAN and TAREK SOBH Department of Computer Science and Engineering, University of Bridgeport, Bridgeport,
More informationSection V.4: Cross Product
Section V.4: Cross Product Definition The cross product of vectors A and B is written as A B. The result of the cross product A B is a third vector which is perpendicular to both A and B. (Because the
More informationSynthesis of Constrained nr Planar Robots to Reach Five Task Positions
Synthesis of Constrained nr Planar Robots to Reach Five Task Positions Gim Song Soh Robotics and Automation Laboratory University of California Irvine, California 96973975 Email: gsoh@uci.edu J. Michael
More informationForce/position control of a robotic system for transcranial magnetic stimulation
Force/position control of a robotic system for transcranial magnetic stimulation W.N. Wan Zakaria School of Mechanical and System Engineering Newcastle University Abstract To develop a force control scheme
More information+ 4θ 4. We want to minimize this function, and we know that local minima occur when the derivative equals zero. Then consider
Math Xb Applications of Trig Derivatives 1. A woman at point A on the shore of a circular lake with radius 2 miles wants to arrive at the point C diametrically opposite A on the other side of the lake
More informationDesign of a six DegreeofFreedom Articulated Robotic Arm for Manufacturing Electrochromic Nanofilms
Abstract Design of a six DegreeofFreedom Articulated Robotic Arm for Manufacturing Electrochromic Nanofilms by Maxine Emerich Advisor: Dr. Scott Pierce The subject of this report is the development of
More informationParametric Curves. (Com S 477/577 Notes) YanBin Jia. Oct 8, 2015
Parametric Curves (Com S 477/577 Notes) YanBin Jia Oct 8, 2015 1 Introduction A curve in R 2 (or R 3 ) is a differentiable function α : [a,b] R 2 (or R 3 ). The initial point is α[a] and the final point
More informationMathematics Course 111: Algebra I Part IV: Vector Spaces
Mathematics Course 111: Algebra I Part IV: Vector Spaces D. R. Wilkins Academic Year 19967 9 Vector Spaces A vector space over some field K is an algebraic structure consisting of a set V on which are
More informationA PAIR OF MEASURES OF ROTATIONAL ERROR FOR AXISYMMETRIC ROBOT ENDEFFECTORS
A PAIR OF MEASURES OF ROTATIONAL ERROR FOR AXISYMMETRIC ROBOT ENDEFFECTORS Sébastien Briot, Ilian A. Bonev Department of Automated Manufacturing Engineering École de technologie supérieure (ÉTS), Montreal,
More informationSolving 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 informationInverse Kinematic Problem Solving: A Symbolic Approach Using MapleSim and Maple
Inverse Kinematic Problem Solving: A Symbolic Approach Using MapleSim and Maple 4 HighPerformance Physical Modeling and Simulation Inverse Kinematic Problem Solving: A Symbolic Approach Using MapleSim
More informationAn Application of Robotic Optimization: Design for a Tire Changing Robot
An Application of Robotic Optimization: Design for a Tire Changing Robot RAUL MIHALI, MHER GRIGORIAN and TAREK SOBH Department of Computer Science and Engineering, University of Bridgeport, Bridgeport,
More informationEuler Angle Formulas. Contents. David Eberly Geometric Tools, LLC Copyright c All Rights Reserved.
Euler Angle Formulas David Eberly Geometric Tools, LLC http://www.geometrictools.com/ Copyright c 19982016. All Rights Reserved. Created: December 1, 1999 Last Modified: March 10, 2014 Contents 1 Introduction
More informationLewis, F.L.; et. al. Robotics Mechanical Engineering Handbook Ed. Frank Kreith Boca Raton: CRC Press LLC, 1999
Lewis, F.L.; et. al. Robotics Mechanical Engineering Handbook Ed. Frank Kreith Boca Raton: CRC Press LLC, 1999 c 1999byCRCPressLLC Robotics Frank L. Lewis University of Texas at Arlington John M. Fitzgerald
More informationGRAPHING IN POLAR COORDINATES SYMMETRY
GRAPHING IN POLAR COORDINATES SYMMETRY Recall from Algebra and Calculus I that the concept of symmetry was discussed using Cartesian equations. Also remember that there are three types of symmetry  yaxis,
More informationCofactor 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
More information1 TRIGONOMETRY. 1.0 Introduction. 1.1 Sum and product formulae. Objectives
TRIGONOMETRY Chapter Trigonometry Objectives After studying this chapter you should be able to handle with confidence a wide range of trigonometric identities; be able to express linear combinations of
More informationLeastSquares Intersection of Lines
LeastSquares Intersection of Lines Johannes Traa  UIUC 2013 This writeup derives the leastsquares solution for the intersection of lines. In the general case, a set of lines will not intersect at a
More informationFACTORING ANGLE EQUATIONS:
FACTORING ANGLE EQUATIONS: For convenience, algebraic names are assigned to the angles comprising the Standard Hip kernel. The names are completely arbitrary, and can vary from kernel to kernel. On the
More informationLecture 7. Matthew T. Mason. Mechanics of Manipulation. Lecture 7. Representing Rotation. Kinematic representation: goals, overview
Matthew T. Mason Mechanics of Manipulation Today s outline Readings, etc. We are starting chapter 3 of the text Lots of stuff online on representing rotations Murray, Li, and Sastry for matrix exponential
More informationRecall the basic property of the transpose (for any A): v A t Aw = v w, v, w R n.
ORTHOGONAL MATRICES Informally, an orthogonal n n matrix is the ndimensional analogue of the rotation matrices R θ in R 2. When does a linear transformation of R 3 (or R n ) deserve to be called a rotation?
More informationCoefficient of Potential and Capacitance
Coefficient of Potential and Capacitance Lecture 12: Electromagnetic Theory Professor D. K. Ghosh, Physics Department, I.I.T., Bombay We know that inside a conductor there is no electric field and that
More informationDecember 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B. KITCHENS
December 4, 2013 MATH 171 BASIC LINEAR ALGEBRA B KITCHENS The equation 1 Lines in twodimensional space (1) 2x y = 3 describes a line in twodimensional space The coefficients of x and y in the equation
More information1 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
More informationMatrix Differentiation
1 Introduction Matrix Differentiation ( and some other stuff ) Randal J. Barnes Department of Civil Engineering, University of Minnesota Minneapolis, Minnesota, USA Throughout this presentation I have
More informationNumerical Analysis Lecture Notes
Numerical Analysis Lecture Notes Peter J. Olver 5. Inner Products and Norms The norm of a vector is a measure of its size. Besides the familiar Euclidean norm based on the dot product, there are a number
More informationx1 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(a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular.
Theorem.7.: (Properties of Triangular Matrices) (a) The transpose of a lower triangular matrix is upper triangular, and the transpose of an upper triangular matrix is lower triangular. (b) The product
More informationCALIBRATION OF A ROBUST 2 DOF PATH MONITORING TOOL FOR INDUSTRIAL ROBOTS AND MACHINE TOOLS BASED ON PARALLEL KINEMATICS
CALIBRATION OF A ROBUST 2 DOF PATH MONITORING TOOL FOR INDUSTRIAL ROBOTS AND MACHINE TOOLS BASED ON PARALLEL KINEMATICS E. Batzies 1, M. Kreutzer 1, D. Leucht 2, V. Welker 2, O. Zirn 1 1 Mechatronics Research
More information5.3 SOLVING TRIGONOMETRIC EQUATIONS. Copyright Cengage Learning. All rights reserved.
5.3 SOLVING TRIGONOMETRIC EQUATIONS Copyright Cengage Learning. All rights reserved. What You Should Learn Use standard algebraic techniques to solve trigonometric equations. Solve trigonometric equations
More informationGeometry 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 threedimensional space, we also examine the
More informationA vector is a directed line segment used to represent a vector quantity.
Chapters and 6 Introduction to Vectors A vector quantity has direction and magnitude. There are many examples of vector quantities in the natural world, such as force, velocity, and acceleration. A vector
More informationContinuous Groups, Lie Groups, and Lie Algebras
Chapter 7 Continuous Groups, Lie Groups, and Lie Algebras Zeno was concerned with three problems... These are the problem of the infinitesimal, the infinite, and continuity... Bertrand Russell The groups
More informationFigure 3.1.2 Cartesian coordinate robot
Introduction to Robotics, H. Harry Asada Chapter Robot Mechanisms A robot is a machine capable of physical motion for interacting with the environment. Physical interactions include manipulation, locomotion,
More informationSection 11.4: Equations of Lines and Planes
Section 11.4: Equations of Lines and Planes Definition: The line containing the point ( 0, 0, 0 ) and parallel to the vector v = A, B, C has parametric equations = 0 + At, = 0 + Bt, = 0 + Ct, where t R
More informationEquations Involving Lines and Planes Standard equations for lines in space
Equations Involving Lines and Planes In this section we will collect various important formulas regarding equations of lines and planes in three dimensional space Reminder regarding notation: any quantity
More informationA matrix over a field F is a rectangular array of elements from F. The symbol
Chapter MATRICES Matrix arithmetic A matrix over a field F is a rectangular array of elements from F The symbol M m n (F) denotes the collection of all m n matrices over F Matrices will usually be denoted
More information13 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 threespace, we write a vector in terms
More informationVector Algebra II: Scalar and Vector Products
Chapter 2 Vector Algebra II: Scalar and Vector Products We saw in the previous chapter how vector quantities may be added and subtracted. In this chapter we consider the products of vectors and define
More informationINVERSE KINEMATICS AND SINGULARITIES OF MANIPULATORS WITH OFFSET WRIST
INVERSE KINEMATICS AND SINGULARITIES OF MANIPULATORS WITH OFFSET WRIST Robert L. Williams II Department of Mechanical Engineering Ohio University Athens, OH 45701 IASTED International Journal of Robotics
More informationKinematics and Dynamics of Mechatronic Systems. Wojciech Lisowski. 1 An Introduction
Katedra Robotyki i Mechatroniki Akademia GórniczoHutnicza w Krakowie Kinematics and Dynamics of Mechatronic Systems Wojciech Lisowski 1 An Introduction KADOMS KRIM, WIMIR, AGH Kraków 1 The course contents:
More informationSection 9.5: Equations of Lines and Planes
Lines in 3D Space Section 9.5: Equations of Lines and Planes Practice HW from Stewart Textbook (not to hand in) p. 673 # 35 odd, 237 odd, 4, 47 Consider the line L through the point P = ( x, y, ) that
More information11.1. 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 informationOperational Space Control for A Scara Robot
Operational Space Control for A Scara Robot Francisco Franco Obando D., Pablo Eduardo Caicedo R., Oscar Andrés Vivas A. Universidad del Cauca, {fobando, pacaicedo, avivas }@unicauca.edu.co Abstract This
More informationDesignSimulationOptimization Package for a Generic 6DOF Manipulator with a Spherical Wrist
DesignSimulationOptimization Package for a Generic 6DOF Manipulator with a Spherical Wrist MHER GRIGORIAN, TAREK SOBH Department of Computer Science and Engineering, U. of Bridgeport, USA ABSTRACT Robot
More informationComputer 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,
More informationTHESIS REPORT. Ph.D. A Three Degree of Freedom Parallel Manipulator with Only Translational Degrees of Freedom. by R. E. Stamper Advisor: LW.
THESIS REPORT Ph.D. A Three Degree of Freedom Parallel Manipulator with Only Translational Degrees of Freedom by R. E. Stamper Advisor: LW. Tsai Ph.D. 974 ISR INSTITUTE FOR SYSTEMS RESEARCH Sponsored
More informationMath 241 Lines and Planes (Solutions) x = 3 3t. z = 1 t. x = 5 + t. z = 7 + 3t
Math 241 Lines and Planes (Solutions) The equations for planes P 1, P 2 and P are P 1 : x 2y + z = 7 P 2 : x 4y + 5z = 6 P : (x 5) 2(y 6) + (z 7) = 0 The equations for lines L 1, L 2, L, L 4 and L 5 are
More informationChapter 11 Equilibrium
11.1 The First Condition of Equilibrium The first condition of equilibrium deals with the forces that cause possible translations of a body. The simplest way to define the translational equilibrium of
More informationIntroduction to Computer Graphics MariePaule Cani & Estelle Duveau
Introduction to Computer Graphics MariePaule Cani & Estelle Duveau 04/02 Introduction & projective rendering 11/02 Prodedural modeling, Interactive modeling with parametric surfaces 25/02 Introduction
More informationDot product and vector projections (Sect. 12.3) There are two main ways to introduce the dot product
Dot product and vector projections (Sect. 12.3) Two definitions for the dot product. Geometric definition of dot product. Orthogonal vectors. Dot product and orthogonal projections. Properties of the dot
More informationVectors 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 informationMOVIES, GAMBLING, SECRET CODES, JUST MATRIX MAGIC
MOVIES, GAMBLING, SECRET CODES, JUST MATRIX MAGIC DR. LESZEK GAWARECKI 1. The Cartesian Coordinate System In the Cartesian system points are defined by giving their coordinates. Plot the following points:
More informationTypical Linear Equation Set and Corresponding Matrices
EWE: Engineering With Excel Larsen Page 1 4. Matrix Operations in Excel. Matrix Manipulations: Vectors, Matrices, and Arrays. How Excel Handles Matrix Math. Basic Matrix Operations. Solving Systems of
More information9 MATRICES AND TRANSFORMATIONS
9 MATRICES AND TRANSFORMATIONS Chapter 9 Matrices and Transformations Objectives After studying this chapter you should be able to handle matrix (and vector) algebra with confidence, and understand the
More informationSection 2.4: Equations of Lines and Planes
Section.4: Equations of Lines and Planes An equation of three variable F (x, y, z) 0 is called an equation of a surface S if For instance, (x 1, y 1, z 1 ) S if and only if F (x 1, y 1, z 1 ) 0. x + y
More information1 Scalars, Vectors and Tensors
DEPARTMENT OF PHYSICS INDIAN INSTITUTE OF TECHNOLOGY, MADRAS PH350 Classical Physics Handout 1 8.8.2009 1 Scalars, Vectors and Tensors In physics, we are interested in obtaining laws (in the form of mathematical
More informationGear Trains. Introduction:
Gear Trains Introduction: Sometimes, two or more gears are made to mesh with each other to transmit power from one shaft to another. Such a combination is called gear train or train of toothed wheels.
More informationLinear algebra and the geometry of quadratic equations. Similarity transformations and orthogonal matrices
MATH 30 Differential Equations Spring 006 Linear algebra and the geometry of quadratic equations Similarity transformations and orthogonal matrices First, some things to recall from linear algebra Two
More information1.3. DOT PRODUCT 19. 6. If θ is the angle (between 0 and π) between two nonzero vectors u and v,
1.3. DOT PRODUCT 19 1.3 Dot Product 1.3.1 Definitions and Properties The dot product is the first way to multiply two vectors. The definition we will give below may appear arbitrary. But it is not. It
More informationThe 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, highdimensional
More informationKinematics and force control for a GantryTau robot
ISSN 02805316 ISRN LUTFD2/TFRT5850SE Kinematics and force control for a GantryTau robot Johan Friman Department of Automatic Control Lund University January 2010 Lund University Department of Automatic
More informationFigure 1.1 Vector A and Vector F
CHAPTER I VECTOR QUANTITIES Quantities are anything which can be measured, and stated with number. Quantities in physics are divided into two types; scalar and vector quantities. Scalar quantities have
More informationInverse 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
More information1.5 Equations of Lines and Planes in 3D
40 CHAPTER 1. VECTORS AND THE GEOMETRY OF SPACE Figure 1.16: Line through P 0 parallel to v 1.5 Equations of Lines and Planes in 3D Recall that given a point P = (a, b, c), one can draw a vector from
More informationAdding vectors We can do arithmetic with vectors. We ll start with vector addition and related operations. Suppose you have two vectors
1 Chapter 13. VECTORS IN THREE DIMENSIONAL SPACE Let s begin with some names and notation for things: R is the set (collection) of real numbers. We write x R to mean that x is a real number. A real number
More informationMotion Control of 3 DegreeofFreedom DirectDrive Robot. Rutchanee Gullayanon
Motion Control of 3 DegreeofFreedom DirectDrive Robot A Thesis Presented to The Academic Faculty by Rutchanee Gullayanon In Partial Fulfillment of the Requirements for the Degree Master of Engineering
More informationThe 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 informationDesign Project 2. Sizing of a Bicycle Chain Ring Bolt Set. Statics and Mechanics of Materials I. ENGR 0135 Section 1040.
Design Project 2 Sizing of a Bicycle Chain Ring Bolt Set Statics and Mechanics of Materials I ENGR 0135 Section 1040 November 9, 2014 Derek Nichols Michael Scandrol Mark Vavithes Nichols, Scandrol, Vavithes
More informationHumanlike Arm Motion Generation for Humanoid Robots Using Motion Capture Database
Humanlike Arm Motion Generation for Humanoid Robots Using Motion Capture Database Seungsu Kim, ChangHwan Kim and Jong Hyeon Park School of Mechanical Engineering Hanyang University, Seoul, 133791, Korea.
More information1 Review of complex numbers
1 Review of complex numbers 1.1 Complex numbers: algebra The set C of complex numbers is formed by adding a square root i of 1 to the set of real numbers: i = 1. Every complex number can be written uniquely
More informationLengths in Polar Coordinates
Lengths in Polar Coordinates Given a polar curve r = f (θ), we can use the relationship between Cartesian coordinates and Polar coordinates to write parametric equations which describe the curve using
More information5 Systems of Equations
Systems of Equations Concepts: Solutions to Systems of EquationsGraphically and Algebraically Solving Systems  Substitution Method Solving Systems  Elimination Method Using Dimensional Graphs to Approximate
More informationLINEAR ALGEBRA W W L CHEN
LINEAR ALGEBRA W W L CHEN c W W L Chen, 1997, 2008 This chapter is available free to all individuals, on understanding that it is not to be used for financial gain, and may be downloaded and/or photocopied,
More information