DIEF, Department of Engineering Enzo Ferrari University of Modena e Reggio Emilia Italy Online Trajectory Planning for robotic systems


 Paulina Kennedy
 1 years ago
 Views:
Transcription
1 DIEF, Department of Engineering Enzo Ferrari University of Modena e Reggio Emilia Italy Online Trajectory Planning for robotic systems Luigi Biagiotti Luigi Biagiotti
2 Introduction to online planners Characteristics of online planners Optimal: minimum time trajectories Constraints must be considered Bounded velocities, accelerations, jerks, The user can assign initial i i an final interpolating i conditions i Positions, velocities, accelerations, Computationally efficient Classification Multidimensional planners Scalar planners p(u) Path u(t) Multidimensionality handled with the pathvelocity decomposition 2
3 Scalar problem goal: planning online timeoptimal trajectories compliant with prescribed constraints on the first n derivatives (nth order trajectory): Multisegment trajectories: es. Double S velocity trajectory asymmetric constraints The final position is specified by means of a rough reference input 3
4 Closedloop and openloop planners Closedloop structure Openloop structure 4
5 Closedloop generator Modular structure, based on the iteration of the following scheme ( ) and on variable structure controller L. Biagiotti and R. Zanasi, Timeoptimal regulation of a chain of integrators with saturated ated input and internal variables: ab An application to trajectory planning, in NOLCOS 2010, Bologna, Italia, 13 September
6 Closedloop generator: Third order filter Bou nd on x2 reac ched Bou nd on x2 not reach ed 6
7 Closedloop generator: Thirdorder filter The expression of the switching surface for the thirdorder filter with the switching surface satisfies the constraint (and obviously ) and forces the state variable to reach the value (and the des Too complex for higherorder trajectory generators 7
8 Openloop trajectory generator Additional condition symmetric ti constraints t where is the impulse response of the filter with the free parameters that depend on the temporal constraints 8
9 Openloop trajectory generator In order to guarantee that the trajectory is timeoptimal, the following constraints must hold true restriction to the feasible trajectories L. Biagiotti and C. Melchiorri, Fir filters for online trajectory planning pa with time and frequencydomain specifications, cat s, Control o Engineering Practice,
10 Openloop trajectory generator Evaluation of the trajectory by means of Finite Impulse Response (FIR) filters: discretization i i by backwarddifferences method Note that with Moving average filter 10
11 Openloop trajectory generator Computational complexity: The ith filter is characterized by the differences equation Note that therefore the evaluation of a trajectory of order n requires n multiplications 2n additions 11
12 Openloop trajectory generator: Applicative examples Multipoint optimal trajectories Consider a third order trajectory Where are the viapoints and Not constant 12
13 A comparison between c.l. and o.l. filters If the hypotheses mentioned in previous slides are met c.l. o.l. ol o.l. chattering 13
14 A comparison between c.l. and o.l. filters If the hypotheses mentioned in previous slides are not met: the input changes before the previous tracts has finished c.l. o.l. Bounds overcome 14
15 A comparison between c.l. and o.l. filters Closedloop generator: asymmetric constraints and variation of the bounds c.l. c.l. Asymmetric contraints Bound variation 15
16 A comparison between c.l. and o.l. filters Ramp input c.l. o.l. o.l. Tracking error null Tracking error constant 16
17 A comparison between c.l. and o.l. filters Generic input c.l. o.l. Velocity bounded Lowpass filter 17
18 Openloop trajectory generator Frequency characterization of the trajectory/filter: The Fourier transform of is where By properly choosing Ti it is possible to shape the spectrum of the output trajectory 18
19 Openloop trajectory generator: Applicative examples Time and frequency specifications Given a motion system with an elastic transmission, by properly choosing the parameters Ti it is possible to nullify the (acceleration) spectrum of the trajectory at the resonant frequency 19
20 From Filters for optimal trajectories to Bspline filters 20
21 Bsplines basics Bsplines of degree p are defined as where are the control points, and the basis functions of degree p: with knots 21
22 Bsplines basics Bsplines of degree p are defined as where are the control points, and the basis functions of degree p: with knots 22
23 Bsplines basics Bsplines of degree p are defined as where are the control points, and the basis functions of degree p. Geometrical interpretation 23
24 Bsplines basics Bsplines of degree p are defined as where are the control points, and the basis functions of degree p. Main (positive) features: Equivalence to splines in polynomial form Independance of the algorithms for control points computation from the spline degree p Possibility of generating local changes by modifying some control points Main drawback: Ealationof Evaluation of the function for a given value of t 24
25 Uniform Bsplines Particular case of Bsplines, defined for an equallyspaced distribution of the knots, i.e. For uniform Bsplines, the basis function of degree p can be defined as 25
26 From uniform basis functions to dynamic filters General case Laplace transform p filters 26
27 Openloop filter for Bsplines generation Particular case of Bsplines, defined for an equallyspaced distribution of the knots, i.e. They are equivalent to the output of a chain of filters fed by the sequence of control points p filters In an analytical aytca form (Laplace (apacedomain) 27
28 Selection of the control points Given a set of viapoints Goal: to design Interpolation a geometrich tihpath from a set of given points Approximation 28
29 Selection of control points: interpolation The interpolation of is achieved by imposing where are the time instants at which the spline crosses. If p is odd, we assume ; accordingly the values of are 29
30 Selection of control points For p=3 (cubic Bsplines) the equations for the control points computation are and lead to the following system The first and the last control points are equal to the related via points 30
31 General structure of trajectory planner The trajectory planner is composed by an algorithm that transforms the desired points in the set of control points ; a cascade of p moving average filters, where p is the desired degree of the Bspline (the resulting trajectory will be ). Between them, the Sequencer produces the piecewise constant function arranging in a line the control points, each one for a duration of T seconds. The continuity of the trajectory and its derivatives at the start/end point is guaranteed by the filters chain. A separated filters chain is needed for each trajectory component. 31
32 Numerical examples An industrial robot performing a welding operation 32
33 Numerical examples An industrial robot performing a welding operation (quintic B spline, i.e. p=5) Viapoints Bspline Control points Control polygon 33
34 Numerical examples An industrial robot performing a welding operation (quintic B spline, i.e. p=5) Viapoints Bspline Control points Control polygon 34
35 Numerical examples An industrial robot performing a welding operation (quintic B spline, i.e. p=5) By changing (and therefore ) it is possible to modify the derivatives of the trajectory (i.e. velocity, acceleration, jerk, etc.) according to 35
36 Numerical examples A mobile robot in an environment with obstacles (p=2) Viapoints Bspline Control points Control polygon 36
37 Numerical examples A mobile robot in an environment with obstacles (p=2) Viapoints Bspline Control points Control polygon Modified control points Local modification of the trajectory 37
38 Choice of the time T In the proposed trajectory planner, T is the unique free parameter (which determines the duration of the trajectory) In the literature the timedistance between knots (usually not constant) is determined with the purpose of: minimizing the trajectory duration and guarantying the compliance with temporal constraints on velocity/acceleration/jerk Proposed approach: to determine T, by taking into account frequency constraints (such as eigenfrequencies of the robotic system) L. Biagiotti and C. Melchiorri, Input shaping via bspline filters for 3d trajectory planning, in IEEE/RSJ International Conference on Intelligent Robots and Systems (IROS), San Francisco, California, September
39 Frequency analysis of Bspline trajectories From the analytical expression of the Bspline trajectory the spectrum (Fourier transform) immediately descends where In particular, the magnitude of the frequency response is 39
40 Frequency analysis of Bspline trajectories From the analytical expression of the Bspline trajectory the spectrum (Fourier transform) immediately descends where 40
41 Frequency analysis of Bspline trajectories Idea: to choose T in order to nullify the spectrum of the trajectory at the resonant frequency of vibratory systems This approach is very similar to input shaping technique, consisting in filtering i the reference input with a filter, that t depends on the system, in order to avoid residual vibrations 41
42 Filtering characteristics of Bspline generator When applied to an undamped second order system, with natural frequency, the Percent Residual Vibrations (PVR) is no residual vibration for low level l of vibration for 42
43 Comparison with standard Input Shapers PRV of Bspline cubic filter and standard Input Shapers, i.e. P = 1 P = 2 P = 3 P = 4 Strong asymmetry of cubic Bspline PRV characteristics Robustness w.r.t. errors (or changes) on estimation 43
44 Filtering characteristics of Bspline generator Robustness w.r.t. errors on estimation 44
45 Application to a robot with elastic joints Given the general model of a robot with elastic joints Hypotheses: Cartesian structure viscoelastic coupling An ideal control system is able to impose to the actuators the desired trajectory, that is robot dynamics motors dynamics 45
46 Application to a robot with elastic joints Given the general model of a robot with elastic joints Hypotheses: Cartesian structure An ideal control system is able to impose to the actuators the desired trajectory, that is The model of robot + control becomes with 46
47 Application to a robot with elastic joints The robot behaves like a secondorder system with The reference path is 47
48 Application to a robot with elastic joints Residual vibration with Bsplines along x axis 48
49 Application to a robot with elastic joints Residual vibration with a cubic Bspline along x and y directions limited vibrations if Residual vibrations as a function of the degree p 49
50 Application to a robot with elastic joints Bsplines vs. Input shapers: path tracking Cubic Bspline Quintic Bspline ZV IS ZVDD IS No interpolation of the viapoints 50
51 Conclusions Openloop and closed loop enjoy peculiar features that make each type of generator preferable for a specific application: openloop systems have a very simple structure and low computational costs but they suffers from some limitations (symmetric constraints on velocity, acceleration, etc.) closedloop trajectory generators allow to modify the limits in runtime and to start a new motion at any time. Openloop filters may be preferable in all those applications that are affected by vibrations and resonances, since the linear structure allows a precise frequency characterization of the output trajectory Filters for uniform Bsplines generation, that are characterized by a similar structure, can be designed with the purpose of properly shaping their frequency spectrum and suppressing residual vibrations, that may be present because elastic phenomena affecting the robotic system. 51
52 Spline trajectories ppform Multipoint trajectories obtained by joining n1 polynomial functions (of typical degree 3 or 5) that interpolate n given points and are continuous in the interior i (up to a given order). Splines are the interplating curves with minimum curvature, for a given levell of continuity. it 52
53 Spline trajectories ppform In order to guarantee the continuity of velocity and acceleration, polynomial of degree 3 are usually considered (cubic splines). Cubic splines are defined as It is necessary to compute 4 coefficients for each polynomial. Therefore there are 4(n1) parameters to be defined The constraints are : interpolating conditions continuity conditions on the velocity in the internal pointsintermedi continuity conditions on the accleration in the internal points There are degrees of freedom that can be used to set e.g. initial and final velocities Back to Bsplines 53
Computer Graphics. Geometric Modeling. Page 1. Copyright Gotsman, Elber, Barequet, Karni, Sheffer Computer Science  Technion. An Example.
An Example 2 3 4 Outline Objective: Develop methods and algorithms to mathematically model shape of real world objects Categories: WireFrame Representation Object is represented as as a set of points
More informationInput Shaping for Sway Control in Gantry Cranes
IOSR Journal of Mechanical and Civil Engineering (IOSRJMCE) ISSN : 22781684 Volume 1, Issue 2 (MayJune 2012), PP 3646 Input Shaping for Sway Control in Gantry Cranes Jeslin Thalapil 1 1 Department of
More information3. Interpolation. Closing the Gaps of Discretization... Beyond Polynomials
3. Interpolation Closing the Gaps of Discretization... Beyond Polynomials Closing the Gaps of Discretization... Beyond Polynomials, December 19, 2012 1 3.3. Polynomial Splines Idea of Polynomial Splines
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 informationPoint Lattices in Computer Graphics and Visualization how signal processing may help computer graphics
Point Lattices in Computer Graphics and Visualization how signal processing may help computer graphics Dimitri Van De Ville Ecole Polytechnique Fédérale de Lausanne Biomedical Imaging Group dimitri.vandeville@epfl.ch
More informationActive Vibration Isolation of an Unbalanced Machine Spindle
UCRLCONF206108 Active Vibration Isolation of an Unbalanced Machine Spindle D. J. Hopkins, P. Geraghty August 18, 2004 American Society of Precision Engineering Annual Conference Orlando, FL, United States
More informationCurve Fitting. Next: Numerical Differentiation and Integration Up: Numerical Analysis for Chemical Previous: Optimization.
Next: Numerical Differentiation and Integration Up: Numerical Analysis for Chemical Previous: Optimization Subsections LeastSquares Regression Linear Regression General Linear LeastSquares Nonlinear
More informationRobot coined by Karel Capek in a 1921 sciencefiction Czech play
Robotics Robot coined by Karel Capek in a 1921 sciencefiction Czech play Definition: A robot is a reprogrammable, multifunctional manipulator designed to move material, parts, tools, or specialized devices
More informationPRACTICAL GUIDE TO DATA SMOOTHING AND FILTERING
PRACTICAL GUIDE TO DATA SMOOTHING AND FILTERING Ton van den Bogert October 3, 996 Summary: This guide presents an overview of filtering methods and the software which is available in the HPL.. What is
More informationCHAPTER 1 Splines and Bsplines an Introduction
CHAPTER 1 Splines and Bsplines an Introduction In this first chapter, we consider the following fundamental problem: Given a set of points in the plane, determine a smooth curve that approximates the
More informationTexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA
2015 School of Information Technology and Electrical Engineering at the University of Queensland TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AAAAA Schedule Week Date
More informationEECS 556 Image Processing W 09. Interpolation. Interpolation techniques B splines
EECS 556 Image Processing W 09 Interpolation Interpolation techniques B splines What is image processing? Image processing is the application of 2D signal processing methods to images Image representation
More informationMOBILE ROBOT TRACKING OF PREPLANNED PATHS. Department of Computer Science, York University, Heslington, York, Y010 5DD, UK (email:nep@cs.york.ac.
MOBILE ROBOT TRACKING OF PREPLANNED PATHS N. E. Pears Department of Computer Science, York University, Heslington, York, Y010 5DD, UK (email:nep@cs.york.ac.uk) 1 Abstract A method of mobile robot steering
More informationIntroduction to time series analysis
Introduction to time series analysis Margherita Gerolimetto November 3, 2010 1 What is a time series? A time series is a collection of observations ordered following a parameter that for us is time. Examples
More informationIntroduction to Engineering System Dynamics
CHAPTER 0 Introduction to Engineering System Dynamics 0.1 INTRODUCTION The objective of an engineering analysis of a dynamic system is prediction of its behaviour or performance. Real dynamic systems are
More informationLecture Notes to Accompany. Scientific Computing An Introductory Survey. by Michael T. Heath. Chapter 10
Lecture Notes to Accompany Scientific Computing An Introductory Survey Second Edition by Michael T. Heath Chapter 10 Boundary Value Problems for Ordinary Differential Equations Copyright c 2001. Reproduction
More information3.3. Generating polygon Conclusion References
3.3. Generating polygon onclusion References [1]Y. Kanayama and B. Hartman, Smooth Local Path Planning for Autonomous Vehicles, Proc. Int. onf. on Robotics and Automation, pp 16164, Scottdale, Arizona,
More informationEstimating Dynamics for (DCmotor)+(1st Link) of the Furuta Pendulum
Estimating Dynamics for (DCmotor)+(1st Link) of the Furuta Pendulum 1 Anton and Pedro Abstract Here the steps done for identification of dynamics for (DCmotor)+(1st Link) of the Furuta Pendulum are described.
More informationThe Influence of Aerodynamics on the Design of HighPerformance Road Vehicles
The Influence of Aerodynamics on the Design of HighPerformance Road Vehicles Guido Buresti Department of Aerospace Engineering University of Pisa (Italy) 1 CONTENTS ELEMENTS OF AERODYNAMICS AERODYNAMICS
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 informationWe want to define smooth curves:  for defining paths of cameras or objects.  for defining 1D shapes of objects
lecture 10  cubic curves  cubic splines  bicubic surfaces We want to define smooth curves:  for defining paths of cameras or objects  for defining 1D shapes of objects We want to define smooth surfaces
More informationSeminar. Path planning using Voronoi diagrams and BSplines. Stefano Martina stefano.martina@stud.unifi.it
Seminar Path planning using Voronoi diagrams and BSplines Stefano Martina stefano.martina@stud.unifi.it 23 may 2016 This work is licensed under a Creative Commons AttributionShareAlike 4.0 International
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 informationFirst Order System. Transfer function: Response to a unit step input is: Partial Fraction Expansion leads to: Inverse Laplace transform leads to:
First Order System Transfer function: Response to a unit step input is: Partial Fraction Expansion leads to: Inverse Laplace transform leads to: First Order System At t = T, the output is: T represents
More informationa = x 1 < x 2 < < x n = b, and a low degree polynomial is used to approximate f(x) on each subinterval. Example: piecewise linear approximation S(x)
Spline Background SPLINE INTERPOLATION Problem: high degree interpolating polynomials often have extra oscillations. Example: Runge function f(x) = 1 1+4x 2, x [ 1, 1]. 1 1/(1+4x 2 ) and P 8 (x) and P
More informationScientific Computing: An Introductory Survey
Scientific Computing: An Introductory Survey Chapter 10 Boundary Value Problems for Ordinary Differential Equations Prof. Michael T. Heath Department of Computer Science University of Illinois at UrbanaChampaign
More informationPlanning Motion Trajectories for Mobile Robots Using Splines
Faculty of Applied Sciences Department of Computer Science Autonomous Intelligent Systems Lab Prof. Dr. Wolfram Burgard Student Project Planning Motion Trajectories for Mobile Robots Using Splines Christoph
More informationSolved with COMSOL Multiphysics 4.3
Vibrating String Introduction In the following example you compute the natural frequencies of a pretensioned string using the 2D Truss interface. This is an example of stress stiffening ; in fact the
More informationQNET Experiment #06: HVAC Proportional Integral (PI) Temperature Control Heating, Ventilation, and Air Conditioning Trainer (HVACT)
Quanser NIELVIS Trainer (QNET) Series: QNET Experiment #06: HVAC Proportional Integral (PI) Temperature Control Heating, Ventilation, and Air Conditioning Trainer (HVACT) Student Manual Table of Contents
More informationPHOTOGRAMMETRIC TRIANGULATION OF 3D CUBIC SPLINES INTRODUCTION
PHOTOGRAMMETRIC TRIANGULATION OF 3D CUBIC SPLINES Keith F. Blonquist, Research Engineer Robert T. Pack, Associate Professor Utah State University 4110 Old Main Hill Logan, UT 84322 ABSTRACT A common application
More informationSchool of Biotechnology
Physics reference slides Donatello Dolce Università di Camerino a.y. 2014/2015 mail: donatello.dolce@unicam.it School of Biotechnology Program and Aim Introduction to Physics Kinematics and Dynamics; Position
More informationPOTENTIAL OF STATEFEEDBACK CONTROL FOR MACHINE TOOLS DRIVES
POTENTIAL OF STATEFEEDBACK CONTROL FOR MACHINE TOOLS DRIVES L. Novotny 1, P. Strakos 1, J. Vesely 1, A. Dietmair 2 1 Research Center of Manufacturing Technology, CTU in Prague, Czech Republic 2 SW, Universität
More informationVéronique PERDEREAU ISIR UPMC 6 mars 2013
Véronique PERDEREAU ISIR UPMC mars 2013 Conventional methods applied to rehabilitation robotics Véronique Perdereau 2 Reference Robot force control by Bruno Siciliano & Luigi Villani Kluwer Academic Publishers
More informationparametric spline curves
parametric spline curves 1 curves used in many contexts fonts (2D) animation paths (3D) shape modeling (3D) different representation implicit curves parametric curves (mostly used) 2D and 3D curves are
More informationCover. SEB SIMOTION Easy Basics. Collection of standardized SIMOTION basic functions. FAQ April 2011. Service & Support. Answers for industry.
Cover SEB SIMOTION Easy Basics Collection of standardized SIMOTION basic functions FAQ April 2011 Service & Support Answers for industry. 1 Preface 1 Preface The SEB is a collection of simple, standardized
More informationI. Kaiser, B. Kurzeck Institut für Robotik und Mechatronik DLR Oberpfaffenhofen. SIMPACK User Meeting 2011 May 18th and 19th 2011, Salzburg
Enhancing the Modeling of TrainTrack Interaction by Including the Structural Dynamics of Wheelsets and Track Status of DLR SIMPACK Prototype Implementation I. Kaiser, B. Kurzeck Institut für Robotik und
More informationLabVIEW Based Embedded Design
LabVIEW Based Embedded Design Sadia Malik Ram Rajagopal Department of Electrical and Computer Engineering University of Texas at Austin Austin, TX 78712 malik@ece.utexas.edu ram.rajagopal@ni.com Abstract
More informationMean 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 expressions. Permutations and Combinations.
More informationGeometric Modelling & Curves
Geometric Modelling & Curves Geometric Modeling Creating symbolic models of the physical world has long been a goal of mathematicians, scientists, engineers, etc. Recently technology has advanced sufficiently
More informationHomework set 1: posted on website
Homework set 1: posted on website 1 Last time: Interpolation, fitting and smoothing Interpolated curve passes through all data points Fitted curve passes closest (by some criterion) to all data points
More informationFIR Filter Design. FIR Filters and the zdomain. The zdomain model of a general FIR filter is shown in Figure 1. Figure 1
FIR Filters and the Domain FIR Filter Design The domain model of a general FIR filter is shown in Figure. Figure Each  box indicates a further delay of one sampling period. For example, the input to
More informationReview of Key Concepts: 1.2 Characteristics of Polynomial Functions
Review of Key Concepts: 1.2 Characteristics of Polynomial Functions Polynomial functions of the same degree have similar characteristics The degree and leading coefficient of the equation of the polynomial
More informationModule 4. Contents. Digital Filters  Implementation and Design. Signal Flow Graphs. Digital Filter Structures. FIR and IIR Filter Design Techniques
Module 4 Digital Filters  Implementation and Design Digital Signal Processing. Slide 4.1 Contents Signal Flow Graphs Basic filtering operations Digital Filter Structures Direct form FIR and IIR filters
More informationOnline trajectory planning of robot manipulator s end effector in Cartesian Space using quaternions
Online trajectory planning of robot manipulator s end effector in Cartesian Space using quaternions Ignacio Herrera Aguilar and Daniel Sidobre (iherrera, daniel)@laas.fr LAASCNRS Université Paul Sabatier
More informationCATIA Kinematics TABLE OF CONTENTS
TABLE OF CONTENTS Introduction...1 Fitting Simulation...2 Pull Down Menus...3 Insert...3 Tools...4 Analyze...5 Window...5 Fitting Simulation Workbench...6 Manipulation...7 Recorder...8 Player...8 Bottom
More informationAdvantages of Autotuning for Servomotors
Advantages of for Servomotors Executive summary The same way that 2 years ago computer science introduced plug and play, where devices would selfadjust to existing system hardware, industrial motion control
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 informationBézier Curves and Splines
6.837 Computer Graphics Bézier Curves and Splines Wojciech Matusik MIT CSAIL vectorportal.com 6.837 Matusik 2 Before We Begin Anything on your mind concerning Assignment 0? Any questions about the course?
More informationAdvanced Signal Processing and Digital Noise Reduction
Advanced Signal Processing and Digital Noise Reduction Saeed V. Vaseghi Queen's University of Belfast UK WILEY HTEUBNER A Partnership between John Wiley & Sons and B. G. Teubner Publishers Chichester New
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 informationarxiv:cs/ v1 [cs.gr] 22 Mar 2005
arxiv:cs/0503054v1 [cs.gr] 22 Mar 2005 ANALYTIC DEFINITION OF CURVES AND SURFACES BY PARABOLIC BLENDING by A.W. Overhauser Mathematical and Theoretical Sciences Department Scientific Laboratory, Ford Motor
More informationCSE 167: Lecture 13: Bézier Curves. Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2011
CSE 167: Introduction to Computer Graphics Lecture 13: Bézier Curves Jürgen P. Schulze, Ph.D. University of California, San Diego Fall Quarter 2011 Announcements Homework project #6 due Friday, Nov 18
More informationthe points are called control points approximating curve
Chapter 4 Spline Curves A spline curve is a mathematical representation for which it is easy to build an interface that will allow a user to design and control the shape of complex curves and surfaces.
More informationModels and Filters for camerabased Multitarget Tracking. Dr.Ing. Mirko Meuter interactive Summer School 46 July, 2012
Models and Filters for camerabased Multitarget Tracking Dr.Ing. Mirko Meuter interactive Summer School 46 July, 2012 Outline: Contents of the Presentation From detection to tracking Overview over camera
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 informationUnderstanding CIC Compensation Filters
Understanding CIC Compensation Filters April 2007, ver. 1.0 Application Note 455 Introduction f The cascaded integratorcomb (CIC) filter is a class of hardwareefficient linear phase finite impulse response
More informationROUTH S STABILITY CRITERION
ECE 680 Modern Automatic Control Routh s Stability Criterion June 13, 2007 1 ROUTH S STABILITY CRITERION Consider a closedloop transfer function H(s) = b 0s m + b 1 s m 1 + + b m 1 s + b m a 0 s n + s
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 informationMoving Average Filters
CHAPTER 15 Moving Average Filters The moving average is the most common filter in DSP, mainly because it is the easiest digital filter to understand and use. In spite of its simplicity, the moving average
More informationDigital Filter Design (FIR) Using Frequency Sampling Method
Digital Filter Design (FIR) Using Frequency Sampling Method Amer Ali Ammar Dr. Mohamed. K. Julboub Dr.Ahmed. A. Elmghairbi Electric and Electronic Engineering Department, Faculty of Engineering Zawia University
More informationBICUBIC BSPLINE INTERPOLATION METHOD FOR TWODIMENSIONAL LAPLACE S EQUATIONS. Abstract
BICUBIC BSPLINE INTERPOLATION METHOD FOR TWODIMENSIONAL LAPLACE S EQUATIONS Nur Nadiah Abd Hamid 1, Ahmad Abd. Majid 2, Ahmad Izani Md. Ismail 3 13 School of Mathematical Sciences, Universiti Sains
More informationOptimization of PointtoPoint Positioning with a Nonlinear Mechanical Connection
Optimization of PointtoPoint Positioning with a Nonlinear Mechanical Connection Ken Brey. Technical Director, DMC. LEARN HOW DMC INCREASED THE SPEED OF A RATELIMITING PROCESS STEP FOR THE THYSSENKRUPP
More information3.5 Spline interpolation
3.5 Spline interpolation Given a tabulated function f k = f(x k ), k = 0,... N, a spline is a polynomial between each pair of tabulated points, but one whose coefficients are determined slightly nonlocally.
More informationThis is an authordeposited version published in: http://sam.ensam.eu Handle ID:.http://hdl.handle.net/10985/7459
Science Arts & Métiers (SAM) is an open access repository that collects the work of Arts et Métiers ParisTech researchers and makes it freely available over the web where possible. This is an authordeposited
More informationOpenFOAM Optimization Tools
OpenFOAM Optimization Tools Henrik Rusche and Aleks Jemcov h.rusche@wikkigmbh.de and a.jemcov@wikki.co.uk Wikki, Germany and United Kingdom OpenFOAM Optimization Tools p. 1 Agenda Objective Review optimisation
More informationEssential Mathematics for Computer Graphics fast
John Vince Essential Mathematics for Computer Graphics fast Springer Contents 1. MATHEMATICS 1 Is mathematics difficult? 3 Who should read this book? 4 Aims and objectives of this book 4 Assumptions made
More informationNumerical Analysis and Computing
Joe Mahaffy, mahaffy@math.sdsu.edu Spring 2010 #4: Solutions of Equations in One Variable (1/58) Numerical Analysis and Computing Lecture Notes #04 Solutions of Equations in One Variable, Interpolation
More informationSerendipity Basis Functions for Any Degree in Any Dimension
Serendipity Basis Functions for Any Degree in Any Dimension Andrew Gillette Department of Mathematics University of Arizona joint work with Michael Floater (University of Oslo) http://math.arizona.edu/
More informationGearbox Fault Diagnosis under Different Operating Conditions Based on Time Synchronous Average and Ensemble Empirical Mode Decomposition
Gearbox Fault Diagnosis under Different Operating Conditions Based on Time Synchronous Average and Ensemble Empirical Mode Decomposition Luyang Guan, Yimin Shao, Fengshou Gu, Bruno Fazenda, Andrew Ball
More informationINTEGRAL METHODS IN LOWFREQUENCY ELECTROMAGNETICS
INTEGRAL METHODS IN LOWFREQUENCY ELECTROMAGNETICS I. Dolezel Czech Technical University, Praha, Czech Republic P. Karban University of West Bohemia, Plzeft, Czech Republic P. Solin University of Nevada,
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 informationAn Instructional Aid System for Driving Schools Based on Visual Simulation
An Instructional Aid System for Driving Schools Based on Visual Simulation Salvador Bayarri, Rafael Garcia, Pedro Valero, Ignacio Pareja, Institute of Traffic and Road Safety (INTRAS), Marcos Fernandez
More informationFinite Element Formulation for Plates  Handout 3 
Finite Element Formulation for Plates  Handout 3  Dr Fehmi Cirak (fc286@) Completed Version Definitions A plate is a three dimensional solid body with one of the plate dimensions much smaller than the
More informationNovember 16, 2015. Interpolation, Extrapolation & Polynomial Approximation
Interpolation, Extrapolation & Polynomial Approximation November 16, 2015 Introduction In many cases we know the values of a function f (x) at a set of points x 1, x 2,..., x N, but we don t have the analytic
More informationCalculus 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
More informationThis is the 39th lecture and our topic for today is FIR Digital Filter Design by Windowing.
Digital Signal Processing Prof: S. C. Dutta Roy Department of Electrical Engineering Indian Institute of Technology, Delhi Lecture  39 FIR Digital Filter Design by Windowing This is the 39th lecture and
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 informationSMOOTH FREQUENCY RESPONSE DIGITAL AUDIO EQUALIZERS WITH SMALL RINGING IMPULSE RESPONSE
SMOOTH FREQUENCY RESPONSE DIGITAL AUDIO EQUALIZERS WITH SMALL RINGING IMPULSE RESPONSE LUCIAN STANCIU, CRISTIAN STANCIU, VALENTIN STANCIU Key words: Audio equalizers, Grouped Bspline functions, Smooth
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 informationLinear Models and Conjoint Analysis with Nonlinear Spline Transformations
Linear Models and Conjoint Analysis with Nonlinear Spline Transformations Warren F. Kuhfeld Mark Garratt Abstract Many common data analysis models are based on the general linear univariate model, including
More informationOPEN LOOP CONTROL OF FLEXIBLE BEAM PERIODIC MOTION VIA FREQUENCY RESPONSE ANALYSIS
ABCM Symposium Series in Mechatronics  Vol. 4  pp.778 Copyright Proceedings 2 of COBEM by ABCM 29 Copyright c 29 by ABCM 2th International Congress of Mechanical Engineering November 52, 29, Gramado,
More informationsince by using a computer we are limited to the use of elementary arithmetic operations
> 4. Interpolation and Approximation Most functions cannot be evaluated exactly: x, e x, ln x, trigonometric functions since by using a computer we are limited to the use of elementary arithmetic operations
More informationAlgorithms for RealTime Tool Path Generation
Algorithms for RealTime Tool Path Generation Gyula Hermann John von Neumann Faculty of Information Technology, Budapest Polytechnic H1034 Nagyszombat utca 19 Budapest Hungary, hermgyviif.hu Abstract:The
More informationDegree Reduction of Interval SB Curves
International Journal of Video&Image Processing and Network Security IJVIPNSIJENS Vol:13 No:04 1 Degree Reduction of Interval SB Curves O. Ismail, Senior Member, IEEE Abstract Ball basis was introduced
More informationFormulations of Model Predictive Control. Dipartimento di Elettronica e Informazione
Formulations of Model Predictive Control Riccardo Scattolini Riccardo Scattolini Dipartimento di Elettronica e Informazione Impulse and step response models 2 At the beginning of the 80, the early formulations
More informationT = 1 f. Phase. Measure of relative position in time within a single period of a signal For a periodic signal f(t), phase is fractional part t p
Data Transmission Concepts and terminology Transmission terminology Transmission from transmitter to receiver goes over some transmission medium using electromagnetic waves Guided media. Waves are guided
More informationNonlinear Iterative Partial Least Squares Method
Numerical Methods for Determining Principal Component Analysis Abstract Factors Béchu, S., RichardPlouet, M., Fernandez, V., Walton, J., and Fairley, N. (2016) Developments in numerical treatments for
More informationRoundoff Noise in IIR Digital Filters
Chapter 16 Roundoff Noise in IIR Digital Filters It will not be possible in this brief chapter to discuss all forms of IIR (infinite impulse response) digital filters and how quantization takes place in
More informationVALLIAMMAI ENGINEERING COLLEGE SRM Nagar, Kattankulathur603 203 DEPARTMENT OF ELECTRONICS AND COMMUNICATION ENGINEERING EC6502 PRINCIPAL OF DIGITAL SIGNAL PROCESSING YEAR / SEMESTER: III / V ACADEMIC
More informationUSE OF SYNCHRONOUS NOISE AS TEST SIGNAL FOR EVALUATION OF STRONG MOTION DATA PROCESSING SCHEMES
USE OF SYNCHRONOUS NOISE AS TEST SIGNAL FOR EVALUATION OF STRONG MOTION DATA PROCESSING SCHEMES Ashok KUMAR, Susanta BASU And Brijesh CHANDRA 3 SUMMARY A synchronous noise is a broad band time series having
More informationStability of time variant filters
Stable TV filters Stability of time variant filters Michael P. Lamoureux, Safa Ismail, and Gary F. Margrave ABSTRACT We report on a project to investigate the stability and boundedness properties of interpolated
More informationWhat is a Filter? Output Signal. Input Signal Amplitude. Frequency. Low Pass Filter
What is a Filter? Input Signal Amplitude Output Signal Frequency Time Sequence Low Pass Filter Time Sequence What is a Filter Input Signal Amplitude Output Signal Frequency Signal Noise Signal Noise Frequency
More information8. Linear leastsquares
8. Linear leastsquares EE13 (Fall 21112) definition examples and applications solution of a leastsquares problem, normal equations 81 Definition overdetermined linear equations if b range(a), cannot
More informationAPPLIED MATHEMATICS ADVANCED LEVEL
APPLIED MATHEMATICS ADVANCED LEVEL INTRODUCTION This syllabus serves to examine candidates knowledge and skills in introductory mathematical and statistical methods, and their applications. For applications
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 informationLecture 3: Quantization Effects
Lecture 3: Quantization Effects Reading: 6.76.8. We have so far discussed the design of discretetime filters, not digital filters. To understand the characteristics of digital filters, we need first
More informationLecture 2. Monday, April 4, 2005
Lecture Monday, April 4, 005 Supplementary Reading: Osher and Fedkiw, Sections 33 and 35; Leveque, Sections 67, 83, 10, 104 For a reference on Newton polynomial interpolation via divided difference tables,
More informationRobot Sensors. Outline. The Robot Structure. Robots and Sensors. Henrik I Christensen
Robot Sensors Henrik I Christensen Robotics & Intelligent Machines @ GT Georgia Institute of Technology, Atlanta, GA 303320760 hic@cc.gatech.edu Henrik I Christensen (RIM@GT) Sensors 1 / 38 Outline 1
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 informationA STRATEGIC PLANNER FOR ROBOT EXCAVATION' by Humberto RomeroLois, Research Assistant, Department of Civil Engineering
A STRATEGIC PLANNER FOR ROBOT EXCAVATION' by Humberto RomeroLois, Research Assistant, Department of Civil Engineering Chris Hendrickson, Professor, Department of Civil Engineering, and Irving Oppenheim,
More information