Robot Position from Wheel Odometry


 Clinton Welch
 1 years ago
 Views:
Transcription
1 Root Position from Wheel Odometry Christopher Marshall 26 Fe 2008 Astract This document develops equations of motion for root position as a function of the distance traveled y each wheel as a function of time (the odometry). The result is a set of equations to update the root position and heading (angle) with incremental changes in the wheel rotation. 1 The Prolem We wish to develop a method to determine the root position given the encoder tick distances for each wheel as a function of time. The inputs will e the linear distance traveled y each wheel or equivalently the encoder ticks traveled y each wheel. The output will e the position x= (x, y) and direction θ after each wheel position update. 2 Analysis 2.1 Coordinates and Parameters y L n x unit direction vector R Figure 1: Odometry Coordinates Figure 1 indicates the root position model eing used for this analysis. The eld coordinates are represented as a normal right handed axis system with positive x to the right and positive y to the left shown in gray. Some key parameters are 1
2 L,R linear distance of travel of the appropriate wheel (left or right) l, r the corresponding tick counts measured y the encoders aseline distance etween wheel centers n unit normal in the direction of the root heading = (cosθ, sin θ) Given the measured caliration parameters of distance per encoder tick count, δ L and δ R, we have and L = δ L l (1) R = δ R r (2) Since we expect the distance per tick to e equal for the left and right wheels, lets make a new representation of the relationship etween tick count and distance using parameters dened as δ = (δ R + δ L )/2 is the mean distance per tick γ = (δ R δ L )/2 is the asymmetry of the wheels Solving for δ L and δ R in terms of δ and γ we have δ R = δ + γ δ L = δ γ 2.2 Root Direction/Angle Update The general expression for the root direction (i.e. the heading angle with respect to the eld coordinate system) is θ = θ 0 + R L where θ is the current heading of the root, θ 0 is the initial heading of the root at the start of the match, L and R are the distance traveled y the left and right wheel, and is the aseline separation of the wheels on the root. This can e derived for general types of wheel motion ut can e most clearly seen y considering the specic case where L and R are the result of constant velocity wheel motion (linear functions of time). For these special cases, the motion is seen to e circular and the aove expression is immediately determined. Sustituting in the values for L and R from equations 1 and 2 we get: θ = θ 0 + δ Rr δ L l (δ + γ)r (δ γ)l = θ 0 + = θ 0 + δ (r l) + γ (r + l) (3) 2
3 We expect that γ δ since the wheel drive mechanisms and wheel diameters are the same. The natural units for θ are in terms of (δ/) so we dene a reduced variale y the relationship (δ/) θ = θ with which sustitution the update for heading ecomes ( δ ) θ = ( δ ) [ θ0 + (r l) + γ δ (r + l) ] θ = θ 0 + (r l) + γ (r + l) (4) δ θ is thus the direction angle in units of encoder ticks and, if we assume γ = 0, then the total heading change in these units is just the dierence etween the left and right tick counts. 2.3 Position Update Now consider the general formula for a position update for the root where the wheels have only moved a small amount. In this case, the change in angle is very small and the motion may e approximated y a straight line move of the root center y the mean change in wheel counts in the current direction of the root followed y an update of the heading from the aove expression. In the limit of innitesimal motion steps, this is exact. Let x= (x, y) e the root center position and θ (or equivalently, θ) the heading. The unit normal in the direction of the root heading is n = (cos θ, sin θ) = (cos δ θ, sin δ θ) (5) where x n, θ n are the root position and heading at the start of the small motion, and x n+1, θ n+1 are the nal position and heading. n n and n n+1 are the unit direction vectors corresponding to θ n and θ n+1 respectively. Dening r = r n+1 r n and l = l n+1 l n, we can write the position update as: ( ) x n+1 = x n δr r + δ L l + n n 2 = x n + 1 [δ( r + l) + γ( r l)] nn 2 The appearance of δ and γ as multipliers on r and l suggests a change to a reduced variale dened y δ x = x. The aove equation now ecomes x n+1 = x n + 1 [( r + l) + ( γ ] 2 δ )( r l) n n (6) which completes the equations required for position and heading update. From the equation 4 for θ we calculate the exact heading for all values of r, l. At the start of each step we know x n and θ n and can calculate the unit direction vector n n (cos θ n, sin θ n ). Now use the position update formula of equation 6 to get the new position, x n+1, and nish y updating the nal heading from the exact formula of equation 4. Iterating this process for each motion step will generate the position and heading of the root as a function of time. 3
4 2.4 Direction Vector Update We complete our analysis y determining an incremental update formula for the unit direction vector n n+1 as a function of r, l, and n n. The idea is to have an approximate solution that is accurate enough for our purposes ut does not required heavy oating point calculations or expensive transcendental function evaluations. Writing the exact vector expression for n n+1 we have where n n+1 = = = cos θn+1 n+1 sin θ cos{θ n + (δ/)( r l) + (γ/)( r + l)} sin{θ n + (δ/)( r l) + (γ/)( r + l)} cos sin n n sin cos = δ ( r l) + γ ( r + l) Now let δ = δ/ and γ = γ/ and we have = δ( r l) + γ( r + l) (7) where with the parameters of the 2008 root, we have δ < γ δ NOTE: for the measured values of δ and we expect to e 1 for small changes in the tick counts. For straight line motion or gentle turns, this condition will continue to hold through a larger range of tick count delta values. Therefore, we can use the small angle approximations for sin and cos : cos. = 1 2 /2 sin. = and sustituting into the exact expression for n n+1 we have the desired update formula: n n /2 = 1 2 n n /2 n n+1. (1 2 /2) n n = x n n y n n x + (1 2 /2) n n (8) y where n = (n x, n y ) and we may need to renormalize the direction vector so that (n x ) 2 + (n y ) 2 = 1 to correct for roundo or other numerical errors. For constant increments of the angle and direction vector, i.e. l = 0, ±1 and/or r = 0, ±1 we can precalculate the exact values for cos and sin and avoid the transcendental function call. The values of for these values are given in Figure 2. 4
5 l r γ = γ δ γ 2 δ 0 δ 2 δ 0 δ γ 0 δ + γ δ 0 δ 1 2 δ δ + γ 2 γ 2 δ δ 0 Figure 2: values for single tick increments 2.5 Update Approximation Error Let us riey consider one type of error that may occur during the incremental position update. First, the approximation is exact in the limit of innitesimal position increments since the error terms are higher order in the step size than the update terms and so go to zero in the small step limit. What type of motion is generated y the single wheel update increment for constant velocity motion (i.e. oth wheels rotating at the same speed)? Assume, without loss of generality, that θ 0 = 0. In this case, ticks from the two sides will increment together in an alternating fashion L, R, L, R, L, R.... As a result, the position will alternately e moved straight ahead y δ, the angle will e changed y δ, then a move of δ at heading θ = δ, with the angle changed y + δ to the original heading at θ = 0. The result is the heading is correct after each pair of L, R increments ut the forward increment is less than the full amount y δ 2 /2 and there is a perpendicular shift to the left y δ. BUT, if the encoder ticks alternate starting on the right, the same shortening of the forward motion is otained ut now the perpendicular shift is to the right! What is the origin of this peculiar ehavior and what can e done to x it? The drift in the update position when driving straight comes from the nonphysical nature of the single encoder update steps. While in the limit as δ 0, the update approximation error does goes to zero, for nite steps, it amounts to a jerky, alternating wheel, step motion. In fact, the two cases of drifting to the left and drifting to the right represent the ounding, worst case errors for straight line motion. The x is to notice that although the ticks may increment discretely, the underlying wheel rotations, L(t) and R(t) are continuous and generally smooth functions. Rather than assuming the other wheel has a tick delta of zero, we should assume that it continues in motion with the current velocity. The result is that the forward motion will e larger due to the contriution from the alternate wheel and the direction error from one wheel will e reduced or canceled y the motion of the other wheel. As a nal point, the magnitude of the position error per step is less than 0.4% for each coordinate direction. This is small enough that if we could somehow remove the ias to a given side, that would resolve the lateral shift. The decrease in advance rate could e addressed y a scale factor. A simple approach would e to comine alternating encoder increments into a single, doule increment, straight ahead. A more sophisticated approach might e a predictorcorrector 5
6 algorithm where you do your est at each step, an on successive steps use the additional information to correct the error in the previous update. 3 Conclusions We have developed step method to update the current root position as the wheel tick counts change. By expressing position in units of tick length, δ, and heading angle in units of tick length over aseline length, δ = δ/, position an angle updates ecome integer additions and sutractions (exactly when wheels are the same size, γ = 0). The unit vector update is the only oating point required and it consists of constant rotations y small angles which should e amenale to xed point analysis. 3.1 Integer Parameter Sizes Given the use of appropriate tick ased units, what type of variales do we need to represent the odometer position and angle parameters? Position If we update the position after each encoder interrupt (in the handler itself) then the result that the propagation equations are all for tick deltas in { 1, 0, 1} so that the values for sin and cos may e precalculated. Since we are in the interrupt handler, only one of r or l is nonzero. For that tick delta, the expression for is an odd function of the delta count so we need only one value precalculated for each wheel and can get the rest y symmetry. For γ 0, we need to precalculate for two values, one for each wheel. Given the eld dimensions of 54 x 27 feet, and a measured δ = inches, the largest possile eld position is 57 12/ = 3811 which needs 12 inary digits to represent. If γ < δ/16 then another 4 inary digits are needed for 1 it of signicance. Therefore at least a 16 it integer is needed for eld position. To avoid roundo errors from the calculation, a 24 it representation should e used. 32 it integers should not e required ut an error analysis should e done. Note: this looks like a 24it add and a 24it for each component, since all tick deltas are ± Heading With the dimensionless units δ = δ/, the numer of counts in a 2π rotation is 2π/(δ/) or 6.28/(0.1795/22.5) = 787 which needs 10 inary digits for full representation. As aove, if γ < δ/16, than at least another 4 its resolution are required. A 16 it representation would give 6 its of precision in the update which might e sucient. We'll start with a 24 it representation which allows for update precision more than the angular integer position accuracy. A full error analysis needs to e performed. 6
7 3.1.3 Unit Direction Vector To match a 24 it representation of eld position (which implies a 12 it integer part and a 12 it fractional part) the coordinate values of n must have at least a 12 it representation. To avoid roundo error in the unit vector update step, we notice that 2 /2 is approximately which requires at least a 15 it fractional part to represent. A 24 it representation seems adequate to start. The update step then consists of 2 24it additions, 2 24it multiplies and a shift (maye) for each coordinate or a total of 4 additions and 4 multiplications per encoder interrupt. At the maximum encoder rate of 300 RPM with 128 counts/revolution and 2 encoders, we have a total of 1280 wheel updates per second maximum. From the PIC18 reference manual (and assuming that our model has the hardware multiply commandneed to check the assemly output), a rough estimate for a 24x24 signed integer multiply is aout 100 cycles. With a 10 MHz clock, that amounts to 7800 cycles/call Cycle Count From the maximum interrupt rate, we have aout 7800 CPU cycles to process each update. The position update takes aout 2 24it adds and 2 24it multiplies. The heading update consists of a single 24it add. The direction vector update takes 4 24it adds and 4 24it multiplies. If we call an add 1/5 of the multiply cycle time, we have a total of 7.4 multiplies or roughly 740 CPU cycles per interrupt. This works out to a duty cycle for position interrupt handling of aout 10% with a handler latency of < 0.08 msec/interrupt so it should e OK in terms of system response. This is the peak update load when driving at maximum speed (aout 10 ft/sec). For medium speed maneuvering, the load drops elow 5% or < 0.04 msec/interrupt. If the interrupt update rate ends up aecting the root responsiveness and other interrupt handling, it is possile to pack a it mask to with update encoder IDs and directions (the sign of the tick deltas). Then the it mask could e processed outside of the interrupt loops to update the position information. Additionally, since straight ahead motion is a common pattern (alternating left and right counts in the same direction), a single precalculated update could take care of multiple ticks at once. 3.2 Simple Odometry Update (γ = 0) From the aove analysis, we show an iterative position update process that may e used to track the root eld position in an incremental fashion. The steps are as follows: 1. Initialize motion parameters: x 0, θ 0, n 0 from the initial position and orientation 2. Each tick of the left or right wheel encoder, update the position, x,heading, θ,and unit direction vector, n,using equations 6, 4, and 8 7
8 By taking advantage of the appropriate units for position (δ) and angle ( δ) and computing the updates each tick change on each wheel separately, we get a very simple set of update equations for γ = 0: x n+1 = x n + n/2 θ n+1 = θ n + r l [ n n+1 = n n / /2 ] n n where r or l = ±1 and the other is zero. The left side of the tale of gure 2 shows the values for = ± δ and the appropriate sign depending on the wheel and direction. How could this e improved to handle the side shift error as discussed in section 2.5? 8
Section 8.5. z(t) = be ix(t). (8.5.1) Figure A pendulum. ż = ibẋe ix (8.5.2) (8.5.3) = ( bẋ 2 cos(x) bẍ sin(x)) + i( bẋ 2 sin(x) + bẍ cos(x)).
Difference Equations to Differential Equations Section 8.5 Applications: Pendulums MassSpring Systems In this section we will investigate two applications of our work in Section 8.4. First, we will consider
More informationFinQuiz Notes
Reading 9 A time series is any series of data that varies over time e.g. the quarterly sales for a company during the past five years or daily returns of a security. When assumptions of the regression
More informationExpansion formula using properties of dot product (analogous to FOIL in algebra): u v 2 u v u v u u 2u v v v u 2 2u v v 2
Least squares: Mathematical theory Below we provide the "vector space" formulation, and solution, of the least squares prolem. While not strictly necessary until we ring in the machinery of matrix algera,
More informationPolynomial Degree and Finite Differences
CONDENSED LESSON 7.1 Polynomial Degree and Finite Differences In this lesson, you Learn the terminology associated with polynomials Use the finite differences method to determine the degree of a polynomial
More informationSolving Systems of Linear Equations Symbolically
" Solving Systems of Linear Equations Symolically Every day of the year, thousands of airline flights crisscross the United States to connect large and small cities. Each flight follows a plan filed with
More information1 Caveats of Parallel Algorithms
CME 323: Distriuted Algorithms and Optimization, Spring 2015 http://stanford.edu/ reza/dao. Instructor: Reza Zadeh, Matroid and Stanford. Lecture 1, 9/26/2015. Scried y Suhas Suresha, Pin Pin, Andreas
More informationMATH 225: Foundations of Higher Matheamatics. Dr. Morton. 3.4: Proof by Cases
MATH 225: Foundations of Higher Matheamatics Dr. Morton 3.4: Proof y Cases Chapter 3 handout page 12 prolem 21: Prove that for all real values of y, the following inequality holds: 7 2y + 2 2y 5 7. You
More informationUNIT 5 QUADRATIC FUNCTIONS Lesson 2: Creating and Solving Quadratic Equations in One Variable Instruction
Lesson : Creating and Solving Quadratic Equations in One Variale Prerequisite Skills This lesson requires the use of the following skills: understanding real numers and complex numers understanding rational
More informationP = ρ{ g a } + µ 2 V II. FLUID STATICS
II. FLUID STATICS From a force analysis on a triangular fluid element at rest, the following three concepts are easily developed: For a continuous, hydrostatic, shear free fluid: 1. Pressure is constant
More informationModule 9: Further Numbers and Equations. Numbers and Indices. The aim of this lesson is to enable you to: work with rational and irrational numbers
Module 9: Further Numers and Equations Lesson Aims The aim of this lesson is to enale you to: wor with rational and irrational numers wor with surds to rationalise the denominator when calculating interest,
More informationROUNDOFF ERRORS; BACKWARD STABILITY
SECTION.5 ROUNDOFF ERRORS; BACKWARD STABILITY ROUNDOFF ERROR  error due to the finite representation (usually in floatingpoint form) of real (and complex) numers in digital computers. FLOATINGPOINT
More informationLuis Manuel Santana Gallego 100 Investigation and simulation of the clock skew in modern integrated circuits. Clock Skew Model
Luis Manuel Santana Gallego 100 Appendix 3 Clock Skew Model Xiaohong Jiang and Susumu Horiguchi [JIA01] 1. Introduction The evolution of VLSI chips toward larger die sizes and faster clock speeds makes
More information1Number ONLINE PAGE PROOFS. systems: real and complex. 1.1 Kick off with CAS
1Numer systems: real and complex 1.1 Kick off with CAS 1. Review of set notation 1.3 Properties of surds 1. The set of complex numers 1.5 Multiplication and division of complex numers 1.6 Representing
More informationSummary Chapter 2: Wave diffraction and the reciprocal lattice.
Summary Chapter : Wave diffraction and the reciprocal lattice. In chapter we discussed crystal diffraction and introduced the reciprocal lattice. Since crystal have a translation symmetry as discussed
More informationragsdale (zdr82) HW7 ditmire (58335) 1 The magnetic force is
ragsdale (zdr8) HW7 ditmire (585) This printout should have 8 questions. Multiplechoice questions ma continue on the net column or page find all choices efore answering. 00 0.0 points A wire carring
More information2 discretized variales approach those of the original continuous variales. Such an assumption is valid when continuous variales are represented as oat
Chapter 1 CONSTRAINED GENETIC ALGORITHMS AND THEIR APPLICATIONS IN NONLINEAR CONSTRAINED OPTIMIZATION Benjamin W. Wah and YiXin Chen Department of Electrical and Computer Engineering and the Coordinated
More informationSample Solutions from the Student Solution Manual
1 Sample Solutions from the Student Solution Manual 1213 If all the entries are, then the matrix is certainly not invertile; if you multiply the matrix y anything, you get the matrix, not the identity
More informationTravel Grouping of Evaporating Polydisperse Droplets in Oscillating Flow Theoretical Analysis
Travel Grouping of Evaporating Polydisperse Droplets in Oscillating Flow Theoretical Analysis DAVID KATOSHEVSKI Department of Biotechnology and Environmental Engineering BenGurion niversity of the Negev
More informationSection 2.1: Reduce Rational Expressions
CHAPTER Section.: Reduce Rational Expressions Section.: Reduce Rational Expressions Ojective: Reduce rational expressions y dividing out common factors. A rational expression is a quotient of polynomials.
More informationPlanar Rigid Body Kinematics Homework
Chapter 2: Planar Rigid ody Kinematics Homework Chapter 2 Planar Rigid ody Kinematics Homework Freeform c 2018 21 Chapter 2: Planar Rigid ody Kinematics Homework 22 Freeform c 2018 Chapter 2: Planar
More informationNonLinear Regression Samuel L. Baker
NONLINEAR REGRESSION 1 NonLinear Regression 20062008 Samuel L. Baker The linear least squares method that you have een using fits a straight line or a flat plane to a unch of data points. Sometimes
More informationSolutions to Exam 2, Math 10560
Solutions to Exam, Math 6. Which of the following expressions gives the partial fraction decomposition of the function x + x + f(x = (x (x (x +? Solution: Notice that (x is not an irreducile factor. If
More informationQUADRATIC EQUATIONS EXPECTED BACKGROUND KNOWLEDGE
6 QUADRATIC EQUATIONS In this lesson, you will study aout quadratic equations. You will learn to identify quadratic equations from a collection of given equations and write them in standard form. You will
More informationFast inverse for big numbers: Picarte s iteration
Fast inverse for ig numers: Picarte s iteration Claudio Gutierrez and Mauricio Monsalve Computer Science Department, Universidad de Chile cgutierr,mnmonsal@dcc.uchile.cl Astract. This paper presents an
More informationShafts. Fig.(4.1) Dr. Salah Gasim Ahmed YIC 1
Shafts. Power transmission shafting Continuous mechanical power is usually transmitted along and etween rotating shafts. The transfer etween shafts is accomplished y gears, elts, chains or other similar
More informationPHY451, Spring /5
PHY451, Spring 2011 Notes on Optical Pumping Procedure & Theory Procedure 1. Turn on the electronics and wait for the cell to warm up: ~ ½ hour. The oven should already e set to 50 C don t change this
More informationGraphs and polynomials
1 1A The inomial theorem 1B Polnomials 1C Division of polnomials 1D Linear graphs 1E Quadratic graphs 1F Cuic graphs 1G Quartic graphs Graphs and polnomials AreAS of STud Graphs of polnomial functions
More informationTP A.18 Distance required for stun and natural roll to develop for different tip offsets
technical proof technical proof TP A.18 Distance required for stun and natural roll to develop for different tip offsets supporting: The Illustrated Principles of Pool and Billiards http://illiards.colostate.edu
More informationOptimal Routing in Chord
Optimal Routing in Chord Prasanna Ganesan Gurmeet Singh Manku Astract We propose optimal routing algorithms for Chord [1], a popular topology for routing in peertopeer networks. Chord is an undirected
More information3 Forces and pressure Answer all questions and show your working out for maximum credit Time allowed : 30 mins Total points available : 32
1 3 Forces and pressure Answer all questions and show your working out for maximum credit Time allowed : 30 mins Total points availale : 32 Core curriculum 1 A icycle pump has its outlet sealed with a
More informationChapter 7. 1 a The length is a function of time, so we are looking for the value of the function when t = 2:
Practice questions Solution Paper type a The length is a function of time, so we are looking for the value of the function when t = : L( ) = 0 + cos ( ) = 0 + cos ( ) = 0 + = cm We are looking for the
More informationERASMUS UNIVERSITY ROTTERDAM Information concerning the Entrance examination Mathematics level 2 for International Business Administration (IBA)
ERASMUS UNIVERSITY ROTTERDAM Information concerning the Entrance examination Mathematics level 2 for International Business Administration (IBA) General information Availale time: 2.5 hours (150 minutes).
More informationa b a b ab b b b Math 154B Elementary Algebra Spring 2012
Math 154B Elementar Algera Spring 01 Stud Guide for Eam 4 Eam 4 is scheduled for Thursda, Ma rd. You ma use a " 5" note card (oth sides) and a scientific calculator. You are epected to know (or have written
More informationVectors in Physics. Topics to review:
Vectors in Physics Topics to review: Scalars Versus Vectors The Components of a Vector Adding and Subtracting Vectors Unit Vectors Position, Displacement, Velocity, and Acceleration Vectors Relative Motion
More informationDeterminants of generalized binary band matrices
Determinants of generalized inary and matrices Dmitry Efimov arxiv:17005655v1 [mathra] 18 Fe 017 Department of Mathematics, Komi Science Centre UrD RAS, Syktyvkar, Russia Astract Under inary matrices we
More informationab is shifted horizontally by h units. ab is shifted vertically by k units.
Algera II Notes Unit Eight: Eponential and Logarithmic Functions Sllaus Ojective: 8. The student will graph logarithmic and eponential functions including ase e. Eponential Function: a, 0, Graph of an
More informationProblem 3 Solution Page 1. 1A. Assuming as outlined in the text that the orbit is circular, and relating the radial acceleration
Prolem 3 Solution Page Solution A. Assug as outlined in the text that the orit is circular, and relating the radial acceleration V GM S to the graitational field (where MS is the solar mass we otain Jupiter's
More information#A50 INTEGERS 14 (2014) ON RATS SEQUENCES IN GENERAL BASES
#A50 INTEGERS 14 (014) ON RATS SEQUENCES IN GENERAL BASES Johann Thiel Dept. of Mathematics, New York City College of Technology, Brooklyn, New York jthiel@citytech.cuny.edu Received: 6/11/13, Revised:
More informationDynamic Optimization of Geneva Mechanisms
Dynamic Optimization of Geneva Mechanisms Vivek A. Suan and Marco A. Meggiolaro Department of Mechanical Engineering Massachusetts Institute of Technology Camridge, MA 09 ABSTRACT The Geneva wheel is the
More informationPARAMETER IDENTIFICATION, MODELING, AND SIMULATION OF A CART AND PENDULUM
PARAMETER IDENTIFICATION, MODELING, AND SIMULATION OF A CART AND PENDULUM Erin Bender Mechanical Engineering Erin.N.Bender@RoseHulman.edu ABSTRACT In this paper a freely rotating pendulum suspended from
More informationComparison of Numerical Method for Forward. and Backward Time Centered Space for. Long  Term Simulation of Shoreline Evolution
Applied Mathematical Sciences, Vol. 7, 13, no. 14, 16173 HIKARI Ltd, www.mhikari.com http://dx.doi.org/1.1988/ams.13.3736 Comparison of Numerical Method for Forward and Backward Time Centered Space for
More informationA converse Gaussian Poincaretype inequality for convex functions
Statistics & Proaility Letters 44 999 28 290 www.elsevier.nl/locate/stapro A converse Gaussian Poincaretype inequality for convex functions S.G. Bokov a;, C. Houdre ; ;2 a Department of Mathematics, Syktyvkar
More informationUpper Bounds for Stern s Diatomic Sequence and Related Sequences
Upper Bounds for Stern s Diatomic Sequence and Related Sequences Colin Defant Department of Mathematics University of Florida, U.S.A. cdefant@ufl.edu Sumitted: Jun 18, 01; Accepted: Oct, 016; Pulished:
More informationEstimating a Finite Population Mean under Random NonResponse in Two Stage Cluster Sampling with Replacement
Open Journal of Statistics, 07, 7, 834848 http://www.scirp.org/journal/ojs ISS Online: 6798 ISS Print: 678X Estimating a Finite Population ean under Random onresponse in Two Stage Cluster Sampling
More informationExploring Lucas s Theorem. Abstract: Lucas s Theorem is used to express the remainder of the binomial coefficient of any two
Delia Ierugan Exploring Lucas s Theorem Astract: Lucas s Theorem is used to express the remainder of the inomial coefficient of any two integers m and n when divided y any prime integer p. The remainder
More informationInteger Sequences and Circle Chains Inside a Circular Segment
Forum Geometricorum Volume 18 018) 47 55. FORUM GEOM ISSN 15341178 Integer Sequences and Circle Chains Inside a Circular Segment Giovanni Lucca Astract. We derive the conditions for inscriing, inside
More informationIN this paper, we consider the estimation of the frequency
Iterative Frequency Estimation y Interpolation on Fourier Coefficients Elias Aoutanios, MIEEE, Bernard Mulgrew, MIEEE Astract The estimation of the frequency of a complex exponential is a prolem that is
More informationarxiv:math/ v1 [math.sp] 27 Mar 2006
SHARP BOUNDS FOR EIGENVALUES OF TRIANGLES arxiv:math/0603630v1 [math.sp] 27 Mar 2006 BART LOMIEJ SIUDEJA Astract. We prove that the first eigenvalue of the Dirichlet Laplacian for a triangle in the plane
More informationFinding Complex Solutions of Quadratic Equations
y  y    x x Locker LESSON.3 Finding Complex Solutions of Quadratic Equations Texas Math Standards The student is expected to: A..F Solve quadratic and square root equations. Mathematical Processes
More informationSchool of Business. Blank Page
Equations 5 The aim of this unit is to equip the learners with the concept of equations. The principal foci of this unit are degree of an equation, inequalities, quadratic equations, simultaneous linear
More informationMATH 131P: PRACTICE FINAL SOLUTIONS DECEMBER 12, 2012
MATH 3P: PRACTICE FINAL SOLUTIONS DECEMBER, This is a closed ook, closed notes, no calculators/computers exam. There are 6 prolems. Write your solutions to Prolems 3 in lue ook #, and your solutions to
More informationZeroing the baseball indicator and the chirality of triples
1 2 3 47 6 23 11 Journal of Integer Sequences, Vol. 7 (2004), Article 04.1.7 Zeroing the aseall indicator and the chirality of triples Christopher S. Simons and Marcus Wright Department of Mathematics
More informationTopological structures and phases. in U(1) gauge theory. Abstract. We show that topological properties of minimal Dirac sheets as well as of
BUHEP9435 Decemer 1994 Topological structures and phases in U(1) gauge theory Werner Kerler a, Claudio Rei and Andreas Weer a a Fachereich Physik, Universitat Marurg, D35032 Marurg, Germany Department
More informationExploring the relationship between a fluid container s geometry and when it will balance on edge
Exploring the relationship eteen a fluid container s geometry and hen it ill alance on edge Ryan J. Moriarty California Polytechnic State University Contents 1 Rectangular container 1 1.1 The first geometric
More informationExact Free Vibration of Webs Moving Axially at High Speed
Eact Free Viration of Wes Moving Aially at High Speed S. HATAMI *, M. AZHARI, MM. SAADATPOUR, P. MEMARZADEH *Department of Engineering, Yasouj University, Yasouj Department of Civil Engineering, Isfahan
More informationTOPICAL PROBLEMS OF FLUID MECHANICS 17 ONEDIMENSIONAL TEMPERATURE DISTRIBUTION OF CONDENSING ANNULAR FINS OF DIFFERENT PROFILES
TOPICAL PROBLEMS OF FLUID MECHANICS 17 ONEDIMENSIONAL TEMPERATURE DISTRIBUTION OF CONDENSING ANNULAR FINS OF DIFFERENT PROFILES A. Bouraaa 1, 2, M. Saighi 2, K. Salhi 1, A. Hamidat 1 and M. M. Moundi
More informationGraphs and polynomials
5_6_56_MQVMM  _t Page Frida, Novemer 8, 5 :5 AM MQ Maths Methods / Final Pages / 8//5 Graphs and polnomials VCEcoverage Areas of stud Units & Functions and graphs Algera In this chapter A The inomial
More informationJWKB QUANTIZATION CONDITION. exp ± i x. wigglyvariable k(x) Logical Structure of pages 611 to 614 (not covered in lecture):
71 JWKB QUANTIZATION CONDITION Last time: ( ) i 3 1. V ( ) = α φ( p) = Nep Ep p 6m hα ψ( ) = Ai( z) zeroes of Ai, Ai tales of Ai (and Bi) asymptotic forms far from turning points 2. SemiClassical Approimation
More informationA framework for the timing analysis of dynamic branch predictors
A framework for the timing analysis of dynamic ranch predictors Claire Maïza INP Grenole, Verimag Grenole, France claire.maiza@imag.fr Christine Rochange IRIT  CNRS Université de Toulouse, France rochange@irit.fr
More information7.8 Improper Integrals
CHAPTER 7. TECHNIQUES OF INTEGRATION 67 7.8 Improper Integrals Eample. Find Solution. Z e d. Z e d = = e e!! e = (e ) = Z Eample. Find d. Solution. We do this prolem twice: once the WRONG way, and once
More informationSEG/New Orleans 2006 Annual Meeting. Nonorthogonal Riemannian wavefield extrapolation Jeff Shragge, Stanford University
Nonorthogonal Riemannian wavefield extrapolation Jeff Shragge, Stanford University SUMMARY Wavefield extrapolation is implemented in nonorthogonal Riemannian spaces. The key component is the development
More informationPROBLEM SET 1 SOLUTIONS 1287 = , 403 = , 78 = 13 6.
Math 7 Spring 06 PROBLEM SET SOLUTIONS. (a) ( pts) Use the Euclidean algorithm to find gcd(87, 0). Solution. The Euclidean algorithm is performed as follows: 87 = 0 + 78, 0 = 78 +, 78 = 6. Hence we have
More informationHash Table Analysis. e(k) = bp(k). kp(k) = n b.
August 17, 2018 Hash Tale Analysis Assume that we have a hash tale structured as a vector of lists, resolving collisions y sequential search of uckets, as in the generic hash tale template HashTale
More informationMath 216 Second Midterm 28 March, 2013
Math 26 Second Midterm 28 March, 23 This sample exam is provided to serve as one component of your studying for this exam in this course. Please note that it is not guaranteed to cover the material that
More information1.17 Triangle Numbers
.7 riangle Numers he n triangle numer is nn ( ). he first few are, 3,, 0, 5,, he difference etween each numer and e next goes up y each time. he formula nn ( ) gives e ( n ) triangle numer for n. he n
More informationDivideandConquer. Reading: CLRS Sections 2.3, 4.1, 4.2, 4.3, 28.2, CSE 6331 Algorithms Steve Lai
DivideandConquer Reading: CLRS Sections 2.3, 4.1, 4.2, 4.3, 28.2, 33.4. CSE 6331 Algorithms Steve Lai Divide and Conquer Given an instance x of a prolem, the method works as follows: divideandconquer
More informationLocalization. Howie Choset Adapted from slides by Humphrey Hu, Trevor Decker, and Brad Neuman
Localization Howie Choset Adapted from slides by Humphrey Hu, Trevor Decker, and Brad Neuman Localization General robotic task Where am I? Techniques generalize to many estimation tasks System parameter
More informationChaos and Dynamical Systems
Chaos and Dynamical Systems y Megan Richards Astract: In this paper, we will discuss the notion of chaos. We will start y introducing certain mathematical concepts needed in the understanding of chaos,
More informationAs shown in the text, we can write an arbitrary azimuthallysymmetric solution to Laplace s equation in spherical coordinates as:
Remarks: In dealing with spherical coordinates in general and with Legendre polynomials in particular it is convenient to make the sustitution c = cosθ. For example, this allows use of the following simplification
More informationParallel Recombinative Simulated Annealing: A Genetic Algorithm. IlliGAL Report No July 1993
Parallel Recominative Simulated Annealing: A Genetic Algorithm IlliGAL Report No. 93006 July 1993 (Revised Version of IlliGAL Report No. 92002 April 1992) Samir W. Mahfoud Department of Computer Science
More informationFree Water Surface Oscillations in a Closed Rectangular Basin with Internal Barriers
Scientia Iranica, Vol. 15, No. 3, pp 315{3 c Sharif University of Technology, June 008 Research Note Free Water Surface Oscillations in a Closed Rectangular Basin with Internal Barriers A.R. KairiSamani
More informationA stochastic method for solving Smoluchowski's. Institute of Computational Mathematics and Mathematical Geophysics
A stochastic method for solving Smoluchowski's coagulation equation A. Kolodko, K. Saelfeld ; and W. Wagner Institute of Computational Mathematics and Mathematical Geophysics Russian Academy of Sciences,
More informationC) 2 D) 4 E) 6. ? A) 0 B) 1 C) 1 D) The limit does not exist.
. The asymptotes of the graph of the parametric equations = t, y = t t + are A) =, y = B) = only C) =, y = D) = only E) =, y =. What are the coordinates of the inflection point on the graph of y = ( +
More informationInverse Functions Attachment
Inverse Functions Attachment It is essential to understand the dierence etween an inverse operation and an inverse unction. In the simplest terms, inverse operations in mathematics are two operations that
More informationMathematics Background
UNIT OVERVIEW GOALS AND STANDARDS MATHEMATICS BACKGROUND UNIT INTRODUCTION Patterns of Change and Relationships The introduction to this Unit points out to students that throughout their study of Connected
More informationSolution: (a) (b) (N) F X =0: A X =0 (N) F Y =0: A Y + B Y (54)(9.81) 36(9.81)=0
Prolem 5.6 The masses of the person and the diving oard are 54 kg and 36 kg, respectivel. ssume that the are in equilirium. (a) Draw the freeod diagram of the diving oard. () Determine the reactions at
More informationEssential Maths 1. Macquarie University MAFC_Essential_Maths Page 1 of These notes were prepared by Anne Cooper and Catriona March.
Essential Maths 1 The information in this document is the minimum assumed knowledge for students undertaking the Macquarie University Masters of Applied Finance, Graduate Diploma of Applied Finance, and
More informationHomework 7 Solutions to Selected Problems
Homework 7 Solutions to Selected Prolems May 9, 01 1 Chapter 16, Prolem 17 Let D e an integral domain and f(x) = a n x n +... + a 0 and g(x) = m x m +... + 0 e polynomials with coecients in D, where a
More informationSimple Examples. Let s look at a few simple examples of OI analysis.
Simple Examples Let s look at a few simple examples of OI analysis. Example 1: Consider a scalar prolem. We have one oservation y which is located at the analysis point. We also have a ackground estimate
More informationLecture 12: Grover s Algorithm
CPSC 519/619: Quantum Computation John Watrous, University of Calgary Lecture 12: Grover s Algorithm March 7, 2006 We have completed our study of Shor s factoring algorithm. The asic technique ehind Shor
More informationThe first property listed above is an incredibly useful tool in divisibility problems. We ll prove that it holds below.
1 Divisiility Definition 1 We say an integer is divisile y a nonzero integer a denoted a  read as a divides if there is an integer n such that = an If no such n exists, we say is not divisile y a denoted
More informationThe Mean Version One way to write the One True Regression Line is: Equation 1  The One True Line
Chapter 27: Inferences for Regression And so, there is one more thing which might vary one more thing aout which we might want to make some inference: the slope of the least squares regression line. The
More informationFigure 3: A cartoon plot of how a database of annotated proteins is used to identify a novel sequence
() Exact seuence comparison..1 Introduction Seuence alignment (to e defined elow) is a tool to test for a potential similarity etween a seuence of an unknown target protein and a single (or a family of)
More informationMTH 65 WS 3 ( ) Radical Expressions
MTH 65 WS 3 (9.19.4) Radical Expressions Name: The next thing we need to develop is some new ways of talking aout the expression 3 2 = 9 or, more generally, 2 = a. We understand that 9 is 3 squared and
More information1530 Chapter 15: Homework Problems
1530 hapter 15: Homework Prolems 8.1 Let s start with a simple prolem. Determining the forces on the front and rear tires of car that weighs 2400 ls. You may assume that a = 2.15ft. and = 2.65ft.. What
More informationDynamical Systems Solutions to Exercises
Dynamical Systems Part 56 Dr G Bowtell Dynamical Systems Solutions to Exercises. Figure : Phase diagrams for i, ii and iii respectively. Only fixed point is at the origin since the equations are linear
More informationDepth versus Breadth in Convolutional Polar Codes
Depth versus Breadth in Convolutional Polar Codes Maxime Tremlay, Benjamin Bourassa and David Poulin,2 Département de physique & Institut quantique, Université de Sherrooke, Sherrooke, Quéec, Canada JK
More informationDesign Variable Concepts 19 Mar 09 Lab 7 Lecture Notes
Design Variale Concepts 19 Mar 09 La 7 Lecture Notes Nomenclature W total weight (= W wing + W fuse + W pay ) reference area (wing area) wing aspect ratio c r root wing chord c t tip wing chord λ taper
More information" $ CALCULUS 2 WORKSHEET #21. t, y = t + 1. are A) x = 0, y = 0 B) x = 0 only C) x = 1, y = 0 D) x = 1 only E) x= 0, y = 1
CALCULUS 2 WORKSHEET #2. The asymptotes of the graph of the parametric equations x = t t, y = t + are A) x = 0, y = 0 B) x = 0 only C) x =, y = 0 D) x = only E) x= 0, y = 2. What are the coordinates of
More informationOmm AlQura University Dr. Abdulsalam Ai LECTURE OUTLINE CHAPTER 3. Vectors in Physics
LECTURE OUTLINE CHAPTER 3 Vectors in Physics 31 Scalars Versus Vectors Scalar a numerical value (number with units). May be positive or negative. Examples: temperature, speed, height, and mass. Vector
More informationAir and Heat Flow through Large Vertical Openings
Air and Heat Flow through Large Vertical Openings J L M Hensen *, J van der Maas #, A Roos * * Eindhoven University of Technology # Ecole Polytechnique Federale de Lausanne After a short description of
More information8.1. Measurement of the electromotive force of an electrochemical cell
8.. Measurement of the electromotive force of an electrochemical cell Ojectives: Measurement of electromotive forces ( the internal resistances, investigation of the dependence of ) and terminal voltages
More informationERASMUS UNIVERSITY ROTTERDAM
Information concerning Colloquium doctum Mathematics level 2 for International Business Administration (IBA) and International Bachelor Economics & Business Economics (IBEB) General information ERASMUS
More informationSU(N) representations
Appendix C SU(N) representations The group SU(N) has N 2 1 generators t a which form the asis of a Lie algera defined y the commutator relations in (A.2). The group has rank N 1 so there are N 1 Casimir
More informationTP B.1 Squirt angle, pivot length, and tip shape
technical proof TP B.1 Squirt angle, pivot length, and tip shape supporting: The Illustrated Principles of Pool and Billiards http://illiards.colostate.edu y David G. Alciatore, PhD, PE ("Dr. Dave") technical
More informationImplementation of Galois Field Arithmetic. Nonbinary BCH Codes and ReedSolomon Codes
BCH Codes Wireless Information Transmission System La. Institute of Communications Engineering g National Sun Yatsen University Outline Binary Primitive BCH Codes Decoding of the BCH Codes Implementation
More informationUnit IV: Introduction to Vector Analysis
Unit IV: Introduction to Vector nalysis s you learned in the last unit, there is a difference between speed and velocity. Speed is an example of a scalar: a quantity that has only magnitude. Velocity is
More informationMATHEMATICS (Three hours and a quarter)
MTHEMTIS (Three hours and a quarter) (The first 5 minutes of the eamination are for reading the paper onl. andidates must NOT start writing during this time). Total marks: 00 
More information