Geometric Constraints


 Daniella Lewis
 3 years ago
 Views:
Transcription
1 Simulation in Computer Graphics Geometric Constraints Matthias Teschner Computer Science Department University of Freiburg
2 Outline introduction penalty method Lagrange multipliers local constraints University of Freiburg Computer Science Department Computer Graphics  2
3 Constraint Dynamics geometric constraints restrict the masspoint movement keep a mass point at a position keep two mass points at a distance keep a mass point on a curve, a surface or in a volume in constraint dynamics, mass points obey Newton's laws and geometric constraints constraint forces explicitly maintain a constraint or implicitly transform accelerations into consistent accelerations University of Freiburg Computer Science Department Computer Graphics  3
4 Applications constraint techniques are applied in robotics graphical manipulation keyframe animation collision handling, resting contact articulated rigid bodies University of Freiburg Computer Science Department Computer Graphics  4
5 Demos University of Freiburg Computer Science Department Computer Graphics  5
6 Global vs. Local Approaches global methods solve for all constraints simultaneously minimization techniques handle conflicting constraints useful for rigid bodies with a small number of degrees of freedom local methods constraints are handled independently efficient restricted to nonconflicting constraints useful for masspoint systems with a large number of degrees of freedom University of Freiburg Computer Science Department Computer Graphics  6
7 Classification penalty method global method introduces forces to meet a constraint Lagrange multiplier global method cancel force components that act against a constraint local constraints local method introduce forces to meet a constraint University of Freiburg Computer Science Department Computer Graphics  7
8 Outline introduction penalty method Lagrange multipliers local constraints University of Freiburg Computer Science Department Computer Graphics  8
9 Principle compute forces from geometric constraints (principle already known from generalized springs) define a constraint that depends on mass point positions iff the constraint is met University of Freiburg Computer Science Department Computer Graphics  9
10 Penalty Forces potential energy based on constraint iff the constraint is met iff the constraint is not met force at mass point based on the potential energy University of Freiburg Computer Science Department Computer Graphics  10
11 Example 2D pendulum mass point at distance to the origin penalty force corresponds to a spring force keeping at the given distance University of Freiburg Computer Science Department Computer Graphics  11
12 Discussion constraints are not accurately met penalty force is induced only if the constraint is not met effect of the penalty force depends on masses, velocities, external forces, and time step which are not considered in the force computation predictorcorrector or implicit schemes improve the accuracy compute a prediction step compute penalty forces based on predicted quantities apply penalty forces to the current time step (corrector step) University of Freiburg Computer Science Department Computer Graphics  12
13 Outline introduction penalty method Lagrange multipliers local constraints University of Freiburg Computer Science Department Computer Graphics  13
14 Principle instead of applying a penalty force too late, illegal force components are removed force components that act against a constraint are cancelled by the constraint force University of Freiburg Computer Science Department Computer Graphics  14
15 Principle if a constraint is met, then in order to meet the constraint in the future, the rate of change of a constraint should be in order to maintain the rate of change of a constraint, the second derivative should be legal position legal velocity legal acceleration compute a constraint force that ensures legal acceleration University of Freiburg Computer Science Department Computer Graphics  15
16 Example the usual pendulum (energy function slightly adapted to simplify further computations) if the constraint is met and its time derivative is zero, then a legal acceleration can be ensured by applying a constraint force University of Freiburg Computer Science Department Computer Graphics  16
17 Example for the constraint force, we get (one equation, two unknowns in 2D) for all solutions would cancel illegal force components for all solutions has the same direction, but with a varying magnitude additional boundary conditions are required University of Freiburg Computer Science Department Computer Graphics  17
18 Example the constraint force should not change the kinetic energy from and follows with being the Lagrange multiplier now, we have three equations for three unknowns University of Freiburg Computer Science Department Computer Graphics  18
19 Discussion the constraint force ensures that results in a legal acceleration intending to maintain a legal velocity and a legal position in contrast to the penalty approach, external forces, masses, and velocities are considered, which results in an improved accuracy accuracy can still be improved with implicit or predictorcorrector schemes University of Freiburg Computer Science Department Computer Graphics  19
20 Discussion still, the time step of the integration scheme is not considered resulting in socalled numerical drift note, that the constraint is not explicitly preserved if the constraint is not met, the approach does not care therefore, the Lagrange multiplier approach is commonly combined with penalty forces ensures a legal acceleration the constraint is not met at, but the approach assumes that the constraint is met University of Freiburg Computer Science Department Computer Graphics  20
21 General Approach and are vectors containing mass point positions and forces, is a diagonal mass matrix, is a vector representing constraints in the initial state and the acceleration is given as this acceleration is set to zero and we get the constraint force University of Freiburg Computer Science Department Computer Graphics  21
22 General Approach the constraint forces should not change the kinetic energy is one of our assumptions we get additional equations and unknowns (Lagrange multiplier) we get with being a square matrix University of Freiburg Computer Science Department Computer Graphics  22
23 Outline introduction penalty method Lagrange multipliers local constraints University of Freiburg Computer Science Department Computer Graphics  23
24 Motivation local constraint approaches are restricted to one constraint per mass point (nonconflicting constraints) since deformable masspoint systems consist of many mass points, these approaches are still quite flexible University of Freiburg Computer Science Department Computer Graphics  24
25 Motivation deformable objects larger number of degrees of freedom more than one nonconflicting constraint can be defined per object by distributing the constraints to different mass points allows for versatile constraints nonconflicting constraints can be solved locally efficient: timeconsuming iterative schemes are avoided accurate: information on the underlying integration scheme can be employed to improve accuracy University of Freiburg Computer Science Department Computer Graphics  25
26 Characteristics local approach for nonconflicting geometric constraints for deforming masspoint systems information on the numerical integration scheme is employed a generic scheme is presented to illustrate the incorporation of a variety of integration schemes very efficient in terms of memory and computing complexity very accurate, no numerical drift or other inaccuracies all constraints are accurately met at each simulation step University of Freiburg Computer Science Department Computer Graphics  26
27 Outline local constraints representation of integration schemes constraint forces pointtonail constraint pointstopoint constraint pointtoline, triangle, tetrahedron results University of Freiburg Computer Science Department Computer Graphics  27
28 Representation of Numerical Integration Schemes mass point at time with mass, position, velocity, force representation of numerical integration with system matrices state vector University of Freiburg Computer Science Department Computer Graphics  28
29 Example  Verlet position and velocity update rewrite velocity update University of Freiburg Computer Science Department Computer Graphics  29
30 Example  Verlet state vector system matrices University of Freiburg Computer Science Department Computer Graphics  30
31 Outline local constraints representation of integration schemes constraint forces pointtonail constraint pointstopoint constraint pointtoline, triangle, tetrahedron results University of Freiburg Computer Science Department Computer Graphics  31
32 Constraint Forces constraint forces are computed to meet a userdefined constraint on a position or velocity by substituting constraint forces are adapted according to the actually employed integration scheme University of Freiburg Computer Science Department Computer Graphics  32
33 Outline local constraints representation of integration schemes constraint forces pointtonail constraint pointstopoint constraint pointtoline, triangle, tetrahedron results University of Freiburg Computer Science Department Computer Graphics  33
34 PointtoNail Constraint constraint that enforces a given position constraint force University of Freiburg Computer Science Department Computer Graphics  34
35 Example  Verlet constraint force new position University of Freiburg Computer Science Department Computer Graphics  35
36 Outline local constraints representation of integration schemes constraint forces pointtonail constraint pointstopoint constraint pointtoline, triangle, tetrahedron results University of Freiburg Computer Science Department Computer Graphics  36
37 PointstoPoint Constraint hold points at a jointed position is not fixed and not userdefined equations are considered to compute unknown constraint forces University of Freiburg Computer Science Department Computer Graphics  37
38 Equations first equation: momentum of the masspoint system is preserved equations: all points have the same position in the next time step University of Freiburg Computer Science Department Computer Graphics  38
39 Resulting Constraint Forces in case of Verlet constraint forces move all points towards the center of mass of the next simulation step with respect to their position at the next simulation step University of Freiburg Computer Science Department Computer Graphics  39
40 Outline local constraints representation of integration schemes constraint forces pointtonail constraint pointstopoint constraint pointtoline, triangle, tetrahedron results University of Freiburg Computer Science Department Computer Graphics  40
41 PointtoLine, Face, Tetrahedron mass point and a corresponding position on a line with on a triangle with in a tetrahedron with in general, is given as with being Barycentric coordinates University of Freiburg Computer Science Department Computer Graphics  41
42 Constraint Force constraint force guarantees constraint force constraint force is computed for and the negative of the constraint force is distributed to the University of Freiburg Computer Science Department Computer Graphics  42
43 Outline local constraints representation of integration schemes constraint forces pointtonail constraint pointstopoint constraint pointtoline, triangle, tetrahedron results University of Freiburg Computer Science Department Computer Graphics  43
44 Performance University of Freiburg Computer Science Department Computer Graphics  44
45 Results University of Freiburg Computer Science Department Computer Graphics  45
46 Conclusion local constraints for deformable objects numerical integration scheme is employed to efficiently and accurately solve the constraints a generic scheme can incorporate various schemes no iterative solvers, no stabilization techniques no numerical drift, no other inaccuracies restricted to nonconflicting constraints University of Freiburg Computer Science Department Computer Graphics  46
47 Summary penalty method introduce forces to meet a constraint handles conflicting constraints efficient and inaccurate or unstable Lagrange multipliers cancel force components that act against a constraint handles conflicting constraints less efficient (linear system) and drift problem local constraints introduce forces to meet a constraint restricted to one constraint per degree of freedom efficient and accurate University of Freiburg Computer Science Department Computer Graphics  47
48 References Andrew Witkin, Kurt Fleischer, Alan Barr, "Energy Constraints on Parameterized Models", Proc. ACM SIGGRAPH '87, pp , David Baraff, "LinearTime Dynamics using Lagrange Multipliers", Proc. ACM SIGGRAPH '96, pp , Andrew Witkin, David Baraff, Michael Kass, "Physicallybased modeling", ACM SIGGRAPH '01, Course Notes, Marc Gissler, Markus Becker, Matthias Teschner, "Local Constraint Methods for Deformable Objects", Eurographics / ACM SIGGRAPH SCA '06, Posters and Demos, Marc Gissler, Markus Becker, Matthias Teschner, "Local Constraint Methods for Deformable Objects", Proc. VriPhys '06, pp , University of Freiburg Computer Science Department Computer Graphics  48
2.5 Physicallybased Animation
2.5 Physicallybased Animation 320491: Advanced Graphics  Chapter 2 74 Physicallybased animation Morphing allowed us to animate between two known states. Typically, only one state of an object is known.
More informationAn Introduction to Physically Based Modeling: Constrained Dynamics
An Introduction to Physically Based Modeling: Constrained Dynamics Andrew Witkin Robotics Institute Carnegie Mellon University Please note: This document is 1997 by Andrew Witkin. This chapter may be freely
More informationSOFA an Open Source Framework for Medical Simulation
SOFA an Open Source Framework for Medical Simulation J. ALLARD a P.J. BENSOUSSAN b S. COTIN a H. DELINGETTE b C. DURIEZ b F. FAURE b L. GRISONI b and F. POYER b a CIMIT Sim Group  Harvard Medical School
More informationPosition Based Dynamics
3 rd Workshop in Virtual Reality Interactions and Physical Simulation "VRIPHYS" (2006) C. Mendoza, I. Navazo (Editors) Position Based Dynamics Matthias Müller Bruno Heidelberger Marcus Hennix John Ratcliff
More informationSingle and Double plane pendulum
Single and Double plane pendulum Gabriela González 1 Introduction We will write down equations of motion for a single and a double plane pendulum, following Newton s equations, and using Lagrange s equations.
More informationSOFA. Flexible Plugin. Benjamin Gilles, François Faure, Maxime Tournier, Matthieu Nesme
SOFA Flexible Plugin Benjamin Gilles, François Faure, Maxime Tournier, Matthieu Nesme Training Days, Montpellier 22 OCTOBRE 2013 Objectives Deformable solid simulation Modularity Unification of meshbased
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 informationApplied Finite Element Analysis. M. E. Barkey. Aerospace Engineering and Mechanics. The University of Alabama
Applied Finite Element Analysis M. E. Barkey Aerospace Engineering and Mechanics The University of Alabama M. E. Barkey Applied Finite Element Analysis 1 Course Objectives To introduce the graduate students
More information2D 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 Matrixvector multiplication Matrixmatrix multiplication
More informationPhysics 9e/Cutnell. correlated to the. College Board AP Physics 1 Course Objectives
Physics 9e/Cutnell correlated to the College Board AP Physics 1 Course Objectives Big Idea 1: Objects and systems have properties such as mass and charge. Systems may have internal structure. Enduring
More 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 informationSTATICS. Introduction VECTOR MECHANICS FOR ENGINEERS: Eighth Edition CHAPTER. Ferdinand P. Beer E. Russell Johnston, Jr.
Eighth E CHAPTER VECTOR MECHANICS FOR ENGINEERS: STATICS Ferdinand P. Beer E. Russell Johnston, Jr. Introduction Lecture Notes: J. Walt Oler Texas Tech University Contents What is Mechanics? Fundamental
More informationVector 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
More informationRigid Body Dynamics (I)
Rigid Body Dynamics (I) COMP768: October 4, 2007 Nico Galoppo From Particles to Rigid Bodies Particles No rotations Linear velocity v only 3N DoFs Rigid bodies 6 DoFs (translation + rotation)
More informationLecture L20  Energy Methods: Lagrange s Equations
S. Widnall 6.07 Dynamics Fall 009 Version 3.0 Lecture L0  Energy Methods: Lagrange s Equations The motion of particles and rigid bodies is governed by ewton s law. In this section, we will derive an alternate
More information(a) We have x = 3 + 2t, y = 2 t, z = 6 so solving for t we get the symmetric equations. x 3 2. = 2 y, z = 6. t 2 2t + 1 = 0,
Name: Solutions to Practice Final. Consider the line r(t) = 3 + t, t, 6. (a) Find symmetric equations for this line. (b) Find the point where the first line r(t) intersects the surface z = x + y. (a) We
More informationThe equivalence of logistic regression and maximum entropy models
The equivalence of logistic regression and maximum entropy models John Mount September 23, 20 Abstract As our colleague so aptly demonstrated ( http://www.winvector.com/blog/20/09/thesimplerderivationoflogisticregression/
More informationDynamic Simulation of Nonpenetrating Flexible Bodies
Dynamic Simulation of Nonpenetrating Flexible Bodies David Baraff Andrew Witkin Program of Computer Graphics School of Computer Science Cornell University Carnegie Mellon University Ithaca, NY 14853 Pittsburgh,
More informationDynamics. Basilio Bona. DAUINPolitecnico di Torino. Basilio Bona (DAUINPolitecnico di Torino) Dynamics 2009 1 / 30
Dynamics Basilio Bona DAUINPolitecnico di Torino 2009 Basilio Bona (DAUINPolitecnico di Torino) Dynamics 2009 1 / 30 Dynamics  Introduction In order to determine the dynamics of a manipulator, it is
More informationVector 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 informationVolumetric Meshes for Real Time Medical Simulations
Volumetric Meshes for Real Time Medical Simulations Matthias Mueller and Matthias Teschner Computer Graphics Laboratory ETH Zurich, Switzerland muellerm@inf.ethz.ch, http://graphics.ethz.ch/ Abstract.
More informationLecture L19  Vibration, Normal Modes, Natural Frequencies, Instability
S. Widnall 16.07 Dynamics Fall 2009 Version 1.0 Lecture L19  Vibration, Normal Modes, Natural Frequencies, Instability Vibration, Instability An important class of problems in dynamics concerns the free
More informationGraphics. Computer Animation 고려대학교 컴퓨터 그래픽스 연구실. kucg.korea.ac.kr 1
Graphics Computer Animation 고려대학교 컴퓨터 그래픽스 연구실 kucg.korea.ac.kr 1 Computer Animation What is Animation? Make objects change over time according to scripted actions What is Simulation? Predict how objects
More informationMAT Solving Linear Systems Using Matrices and Row Operations
MAT 171 8.5 Solving Linear Systems Using Matrices and Row Operations A. Introduction to Matrices Identifying the Size and Entries of a Matrix B. The Augmented Matrix of a System of Equations Forming Augmented
More informationThe Basics of FEA Procedure
CHAPTER 2 The Basics of FEA Procedure 2.1 Introduction This chapter discusses the spring element, especially for the purpose of introducing various concepts involved in use of the FEA technique. A spring
More informationNumerical Analysis An Introduction
Walter Gautschi Numerical Analysis An Introduction 1997 Birkhauser Boston Basel Berlin CONTENTS PREFACE xi CHAPTER 0. PROLOGUE 1 0.1. Overview 1 0.2. Numerical analysis software 3 0.3. Textbooks and monographs
More informationC O M P U C O M P T U T E R G R A E R G R P A H I C P S Computer Animation Guoying Zhao 1 / 66 /
Computer Animation Guoying Zhao 1 / 66 Basic Elements of Computer Graphics Modeling construct the 3D model of the scene Rendering Render the 3D model, compute the color of each pixel. The color is related
More informationAppendix 12.A: OneDimensional Collision Between Two Objects General Case An extension of Example in the text.
Chapter 1 Appendices Appendix 1A: OneDimensional Collision Between Two Objects General Case An extension of Example 11 in the text Appendix 1B: TwoDimensional Elastic Collisions Between Two Objects with
More informationSimple Harmonic Motion Concepts
Simple Harmonic Motion Concepts INTRODUCTION Have you ever wondered why a grandfather clock keeps accurate time? The motion of the pendulum is a particular kind of repetitive or periodic motion called
More informationSPECIAL PERTURBATIONS UNCORRELATED TRACK PROCESSING
AAS 07228 SPECIAL PERTURBATIONS UNCORRELATED TRACK PROCESSING INTRODUCTION James G. Miller * Two historical uncorrelated track (UCT) processing approaches have been employed using general perturbations
More informationCG T17 Animation L:CC, MI:ERSI. Miguel Tavares Coimbra (course designed by Verónica Orvalho, slides adapted from Steve Marschner)
CG T17 Animation L:CC, MI:ERSI Miguel Tavares Coimbra (course designed by Verónica Orvalho, slides adapted from Steve Marschner) Suggested reading Shirley et al., Fundamentals of Computer Graphics, 3rd
More informationSouth Carolina College and CareerReady (SCCCR) PreCalculus
South Carolina College and CareerReady (SCCCR) PreCalculus Key Concepts Arithmetic with Polynomials and Rational Expressions PC.AAPR.2 PC.AAPR.3 PC.AAPR.4 PC.AAPR.5 PC.AAPR.6 PC.AAPR.7 Standards Know
More informationUnit 4: Science and Materials in Construction and the Built Environment. Chapter 14. Understand how Forces act on Structures
Chapter 14 Understand how Forces act on Structures 14.1 Introduction The analysis of structures considered here will be based on a number of fundamental concepts which follow from simple Newtonian mechanics;
More informationIntersection of Objects with Linear and Angular Velocities using Oriented Bounding Boxes
Intersection of Objects with Linear and Angular Velocities using Oriented Bounding Boxes David Eberly Geometric Tools, LLC http://www.geometrictools.com/ Copyright c 19982016. All Rights Reserved. Created:
More informationGeomechanical restoration is a quantitative method of modelling strain during geological deformation.
Geomechanical Modelling Geomechanical restoration is a quantitative method of modelling strain during geological deformation. Geomechanical methods incorporate the elastic properties of the rock and therefore
More informationProof of the conservation of momentum and kinetic energy
Experiment 04 Proof of the conservation of momentum and kinetic energy By Christian Redeker 27.10.2007 Contents 1.) Hypothesis...3 2.) Diagram...7 3.) Method...7 3.1) Apparatus...7 3.2) Procedure...7 4.)
More informationCHAPTER 3. INTRODUCTION TO MATRIX METHODS FOR STRUCTURAL ANALYSIS
1 CHAPTER 3. INTRODUCTION TO MATRIX METHODS FOR STRUCTURAL ANALYSIS Written by: Sophia Hassiotis, January, 2003 Last revision: February, 2015 Modern methods of structural analysis overcome some of the
More informationThe Kinetic Theory of Gases Sections Covered in the Text: Chapter 18
The Kinetic Theory of Gases Sections Covered in the Text: Chapter 18 In Note 15 we reviewed macroscopic properties of matter, in particular, temperature and pressure. Here we see how the temperature and
More informationSCHOOL DISTRICT OF THE CHATHAMS CURRICULUM PROFILE
CONTENT AREA(S): Mathematics COURSE/GRADE LEVEL(S): Honors Algebra 2 (10/11) I. Course Overview In Honors Algebra 2, the concept of mathematical function is developed and refined through the study of real
More informationOffline Model Simplification for Interactive Rigid Body Dynamics Simulations Satyandra K. Gupta University of Maryland, College Park
NSF GRANT # 0727380 NSF PROGRAM NAME: Engineering Design Offline Model Simplification for Interactive Rigid Body Dynamics Simulations Satyandra K. Gupta University of Maryland, College Park Atul Thakur
More informationElasticity Theory Basics
G22.3033002: Topics in Computer Graphics: Lecture #7 Geometric Modeling New York University Elasticity Theory Basics Lecture #7: 20 October 2003 Lecturer: Denis Zorin Scribe: Adrian Secord, Yotam Gingold
More informationPerformance Driven Facial Animation Course Notes Example: Motion Retargeting
Performance Driven Facial Animation Course Notes Example: Motion Retargeting J.P. Lewis Stanford University Frédéric Pighin Industrial Light + Magic Introduction When done correctly, a digitally recorded
More informationPiecewise Cubic Splines
280 CHAP. 5 CURVE FITTING Piecewise Cubic Splines The fitting of a polynomial curve to a set of data points has applications in CAD (computerassisted design), CAM (computerassisted manufacturing), and
More informationDINAMIC AND STATIC CENTRE OF PRESSURE MEASUREMENT ON THE FORCEPLATE. F. R. Soha, I. A. Szabó, M. Budai. Abstract
ACTA PHYSICA DEBRECINA XLVI, 143 (2012) DINAMIC AND STATIC CENTRE OF PRESSURE MEASUREMENT ON THE FORCEPLATE F. R. Soha, I. A. Szabó, M. Budai University of Debrecen, Department of Solid State Physics Abstract
More informationKinematics & Dynamics
Overview Kinematics & Dynamics Adam Finkelstein Princeton University COS 46, Spring 005 Kinematics Considers only motion Determined by positions, velocities, accelerations Dynamics Considers underlying
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 informationNonlinear analysis and formfinding in GSA Training Course
Nonlinear analysis and formfinding in GSA Training Course Nonlinear analysis and formfinding in GSA 1 of 47 Oasys Ltd Nonlinear analysis and formfinding in GSA 2 of 47 Using the GSA GsRelax Solver
More informationHW 7 Q 14,20,20,23 P 3,4,8,6,8. Chapter 7. Rotational Motion of the Object. Dr. Armen Kocharian
HW 7 Q 14,20,20,23 P 3,4,8,6,8 Chapter 7 Rotational Motion of the Object Dr. Armen Kocharian Axis of Rotation The radian is a unit of angular measure The radian can be defined as the arc length s along
More information521493S 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 1618 in TS128 Question 2.1 Given two nonparallel,
More informationTHEORETICAL MECHANICS
PROF. DR. ING. VASILE SZOLGA THEORETICAL MECHANICS LECTURE NOTES AND SAMPLE PROBLEMS PART ONE STATICS OF THE PARTICLE, OF THE RIGID BODY AND OF THE SYSTEMS OF BODIES KINEMATICS OF THE PARTICLE 2010 0 Contents
More informationCHAPTER 9 MULTIDEGREEOFFREEDOM SYSTEMS Equations of Motion, Problem Statement, and Solution Methods
CHAPTER 9 MULTIDEGREEOFFREEDOM SYSTEMS Equations of Motion, Problem Statement, and Solution Methods Twostory shear building A shear building is the building whose floor systems are rigid in flexure
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 informationAP Physics Energy and Springs
AP Physics Energy and Springs Another major potential energy area that AP Physics is enamored of is the spring (the wire coil deals, not the ones that produce water for thirsty humanoids). Now you ve seen
More informationAbstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of matrix multiplication).
MAT 2 (Badger, Spring 202) LU Factorization Selected Notes September 2, 202 Abstract: We describe the beautiful LU factorization of a square matrix (or how to write Gaussian elimination in terms of matrix
More informationLecture PowerPoints. Chapter 7 Physics: Principles with Applications, 6 th edition Giancoli
Lecture PowerPoints Chapter 7 Physics: Principles with Applications, 6 th edition Giancoli 2005 Pearson Prentice Hall This work is protected by United States copyright laws and is provided solely for the
More informationEXAMPLE 8: An Electrical System (MechanicalElectrical Analogy)
EXAMPLE 8: An Electrical System (MechanicalElectrical Analogy) A completely analogous procedure can be used to find the state equations of electrical systems (and, ultimately, electromechanical systems
More informationAnimation. 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
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 information3.6 Solving Problems Involving Projectile Motion
INTRODUCTION 12 Physics and its relation to other fields introduction of physics, its importance and scope 15 Units, standards, and the SI System description of the SI System description of base and
More informationComputational Optical Imaging  Optique Numerique.  Deconvolution 
Computational Optical Imaging  Optique Numerique  Deconvolution  Winter 2014 Ivo Ihrke Deconvolution Ivo Ihrke Outline Deconvolution Theory example 1D deconvolution Fourier method Algebraic method
More informationVectors and the Inclined Plane
Vectors and the Inclined Plane Introduction: This experiment is designed to familiarize you with the concept of force as a vector quantity. The inclined plane will be used to demonstrate how one force
More informationLinear Threshold Units
Linear Threshold Units w x hx (... w n x n w We assume that each feature x j and each weight w j is a real number (we will relax this later) We will study three different algorithms for learning linear
More information(Refer Slide Time: 1:42)
Introduction to Computer Graphics Dr. Prem Kalra Department of Computer Science and Engineering Indian Institute of Technology, Delhi Lecture  10 Curves So today we are going to have a new topic. So far
More informationMechanics 1: Conservation of Energy and Momentum
Mechanics : Conservation of Energy and Momentum If a certain quantity associated with a system does not change in time. We say that it is conserved, and the system possesses a conservation law. Conservation
More informationLecture 07: Work and Kinetic Energy. Physics 2210 Fall Semester 2014
Lecture 07: Work and Kinetic Energy Physics 2210 Fall Semester 2014 Announcements Schedule next few weeks: 9/08 Unit 3 9/10 Unit 4 9/15 Unit 5 (guest lecturer) 9/17 Unit 6 (guest lecturer) 9/22 Unit 7,
More informationBasic numerical skills: EQUATIONS AND HOW TO SOLVE THEM. x + 5 = 7 2 + 52 = 72 5 + (22) = 72 5 = 5. x + 55 = 75. 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 informationThnkwell 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
More informationanimation 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 physicallyplausible motion efficiency control typically
More informationProblem definition: optical flow
Motion Estimation http://www.sandlotscience.com/distortions/breathing_objects.htm http://www.sandlotscience.com/ambiguous/barberpole.htm Why estimate motion? Lots of uses Track object behavior Correct
More informationInterpolating and Approximating Implicit Surfaces from Polygon Soup
Interpolating and Approximating Implicit Surfaces from Polygon Soup 1. Briefly summarize the paper s contributions. Does it address a new problem? Does it present a new approach? Does it show new types
More informationMathematical Modeling and Engineering Problem Solving
Mathematical Modeling and Engineering Problem Solving Berlin Chen Department of Computer Science & Information Engineering National Taiwan Normal University Reference: 1. Applied Numerical Methods with
More informationCurves and Surfaces. Goals. How do we draw surfaces? How do we specify a surface? How do we approximate a surface?
Curves and Surfaces Parametric Representations Cubic Polynomial Forms Hermite Curves Bezier Curves and Surfaces [Angel 10.110.6] Goals How do we draw surfaces? Approximate with polygons Draw polygons
More informationIntroduction Epipolar Geometry Calibration Methods Further Readings. Stereo Camera Calibration
Stereo Camera Calibration Stereo Camera Calibration Stereo Camera Calibration Stereo Camera Calibration 12.10.2004 Overview Introduction Summary / Motivation Depth Perception Ambiguity of Correspondence
More informationDipartimento di Tecnologie dell Informazione Università di MIlano. Physics Engines LOGO
Dipartimento di Tecnologie dell Informazione Università di MIlano Physics Engines LOGO Computer game physics Computer animation physics or game physics involves the introduction of the laws of physics
More informationThe dynamic equation for the angular motion of the wheel is R w F t R w F w ]/ J w
Chapter 4 Vehicle Dynamics 4.. Introduction In order to design a controller, a good representative model of the system is needed. A vehicle mathematical model, which is appropriate for both acceleration
More informationToday. Keyframing. Procedural Animation. PhysicallyBased Animation. Articulated Models. Computer Animation & Particle Systems
Today Computer Animation & Particle Systems Some slides courtesy of Jovan Popovic & Ronen Barzel How do we specify or generate motion? Keyframing Procedural Animation PhysicallyBased Animation Forward
More informationOverview of Math Standards
Algebra 2 Welcome to math curriculum design maps for Manhattan Ogden USD 383, striving to produce learners who are: Effective Communicators who clearly express ideas and effectively communicate with diverse
More informationParametric Curves (Part 1)
Parametric Curves (Part 1) Jason Lawrence CS445: Graphics Acknowledgment: slides by Misha Kazhdan, Allison Klein, Tom Funkhouser, Adam Finkelstein and David Dobkin Parametric Curves and Surfaces Part 1:
More informationAN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS
AN INTRODUCTION TO NUMERICAL METHODS AND ANALYSIS Revised Edition James Epperson Mathematical Reviews BICENTENNIAL 0, 1 8 0 7 z ewiley wu 2007 r71 BICENTENNIAL WILEYINTERSCIENCE A John Wiley & Sons, Inc.,
More informationAN INTRODUCTION TO THE FINITE ELEMENT METHOD FOR YOUNG ENGINEERS
AN INTRODUCTION TO THE FINITE ELEMENT METHOD FOR YOUNG ENGINEERS By: Eduardo DeSantiago, PhD, PE, SE Table of Contents SECTION I INTRODUCTION... 2 SECTION II 1D EXAMPLE... 2 SECTION III DISCUSSION...
More informationChapter 4 One Dimensional Kinematics
Chapter 4 One Dimensional Kinematics 41 Introduction 1 4 Position, Time Interval, Displacement 41 Position 4 Time Interval 43 Displacement 43 Velocity 3 431 Average Velocity 3 433 Instantaneous Velocity
More informationNumerically integrating equations of motion
Numerically integrating equations of motion 1 Introduction to numerical ODE integration algorithms Many models of physical processes involve differential equations: the rate at which some thing varies
More information5 Day 5: Newton s Laws and Kinematics in 1D
5 Day 5: Newton s Laws and Kinematics in 1D date Friday June 28, 2013 Readings Knight Ch 2.47, Ch 4.6, 4.8 Notes on Newton s Laws For next time: Knight 5.38 lecture demo car on a track, freefall in
More informationABSTRACT 1. INTRODUCTION
Generation of Flexible 3D Objects Adela C. González Universidad Nacional de San Luis Lab. de Investigación y Desarrollo en Int. Artificial 1 Ejército de los Andes 950 San Luis, 5700, Argentina email:
More informationBehavioral Animation Modeling in the Windows Environment
Behavioral Animation Modeling in the Windows Environment MARCELO COHEN 1 CARLA M. D. S. FREITAS 1 FLAVIO R. WAGNER 1 1 UFRGS  Universidade Federal do Rio Grande do Sul CPGCC  Curso de Pós Graduação em
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 informationGravitational Potential Energy
Gravitational Potential Energy Consider a ball falling from a height of y 0 =h to the floor at height y=0. A net force of gravity has been acting on the ball as it drops. So the total work done on the
More informationConcept Review. Physics 1
Concept Review Physics 1 Speed and Velocity Speed is a measure of how much distance is covered divided by the time it takes. Sometimes it is referred to as the rate of motion. Common units for speed or
More informationAlgebra Unpacked Content For the new Common Core standards that will be effective in all North Carolina schools in the 201213 school year.
This document is designed to help North Carolina educators teach the Common Core (Standard Course of Study). NCDPI staff are continually updating and improving these tools to better serve teachers. Algebra
More informationTypes of Elements
chapter : Modeling and Simulation 439 142 20 600 Then from the first equation, P 1 = 140(0.0714) = 9.996 kn. 280 = MPa =, psi The structure pushes on the wall with a force of 9.996 kn. (Note: we could
More informationKinematical Animation. lionel.reveret@inria.fr 201314
Kinematical Animation 201314 3D animation in CG Goal : capture visual attention Motion of characters Believable Expressive Realism? Controllability Limits of purely physical simulation :  little interactivity
More informationNEW 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
More informationSample Questions for the AP Physics 1 Exam
Sample Questions for the AP Physics 1 Exam Sample Questions for the AP Physics 1 Exam Multiplechoice Questions Note: To simplify calculations, you may use g 5 10 m/s 2 in all problems. Directions: Each
More information1 of 7 9/5/2009 6:12 PM
1 of 7 9/5/2009 6:12 PM Chapter 2 Homework Due: 9:00am on Tuesday, September 8, 2009 Note: To understand how points are awarded, read your instructor's Grading Policy. [Return to Standard Assignment View]
More informationEMOT Evolutionary Approach to 3D Computer Animation
EMOT Evolutionary Approach to 3D Computer Animation Halina Kwasnicka, Piotr Wozniak 1 Institute of Applied Informatics, Wroclaw University of Technology Abstract. Keyframing and Inverse Kinematics are
More informationFACTORING QUADRATICS 8.1.1 and 8.1.2
FACTORING QUADRATICS 8.1.1 and 8.1.2 Chapter 8 introduces students to quadratic equations. These equations can be written in the form of y = ax 2 + bx + c and, when graphed, produce a curve called a parabola.
More informationKyuJung Kim Mechanical Engineering Department, California State Polytechnic University, Pomona, U.S.A.
MECHANICS: STATICS AND DYNAMICS KyuJung Kim Mechanical Engineering Department, California State Polytechnic University, Pomona, U.S.A. Keywords: mechanics, statics, dynamics, equilibrium, kinematics,
More informationEMOT Evolutionary Approach to 3D Computer Animation
Proceedings of the International Multiconference on Computer Science and Information Technology pp. 111 121 ISSN 18967094 c 2006 PIPS EMOT Evolutionary Approach to 3D Computer Animation Halina Kwasnicka,
More informationLinear Algebraic and Equations
Next: Optimization Up: Numerical Analysis for Chemical Previous: Roots of Equations Subsections Gauss Elimination Solving Small Numbers of Equations Naive Gauss Elimination Pitfalls of Elimination Methods
More informationVisualization of General Defined Space Data
International Journal of Computer Graphics & Animation (IJCGA) Vol.3, No.4, October 013 Visualization of General Defined Space Data John R Rankin La Trobe University, Australia Abstract A new algorithm
More information