Robotic manipulation project
|
|
- Arleen Jordan
- 5 years ago
- Views:
Transcription
1 Robotic manipulation project Bin Nguyen December 5, 2006 Abstract Tis is te draft report for Robotic Manipulation s class project. Te cosen project aims to understand and implement Kevin Egan s non-convex constraint[5] as an extended Linear Complementarity problem[6]. Tis draft contains all te background teories along wit analysis needed to complete te project and te implementation progress. 1 Introduction Te main objective of tis project is to study Kevin Egan s metod[5] and apply it to properly model general 2D polyedral sapes using LCPs. Due to te limited time, I will just focus on simple test scene wic contains a moving spere and polyedral obstacles. Tat will simplify my collision detection. Also, in term of teories and setting LCPs, tis simple scene is similar to general polyedrons - polyedrons cases. Also, I will not consider friction because it will only complicate implementation. Te metod extends well to frictional cases. 1.1 Stewart-Trinkle LCP model Stewart-Trinkle [7] non-penetration constraints maintain a list of active pairs of vertex, edge and try to keep future position of te vertices from penetrating corresponding edges in contact normal direction. Tis is te same metod used in oter LCP models wic can be studied in [7],[8], [1],[2].Tese models linearized contact constraints suc tat eac contact constraint defines a alf-space in configuration space. Tus, tey can only model locally convex configuration space. Tis implicit assumption leads to artifacts in simulation. Note tat te artifacts only visible in in simulation wen two vertices are close enoug. So, te problems are bearable in some applications. A simple example tat exibits tis artifact is sowed in Figure 1 wen a point mass approaces a vertex in 2D. 1
2 Configuration space Point mass Obstacle Simulated position Correct position Obstacle Obstacle Figure 1: Stewart-Trinkle non-penetration constraints in non-convex configuration space. Te non-penetration constraints in tis case are simply te point mass cannot penetrate bot edges at te same time. Te point mass cannot penetrate edge 1: 0 W1nν T l+1 + ψl 1 Te point mass cannot penetrate edge 2: 1n 0 (1) In wic: 0 W T 2nν l+1 + ψl 2 2n 0 (2) ν l+1 generalized velocity at te end of time step [t l, t l+1 ]. time step size. W in in normal contact wrenc corresponds to edge it. unknown vector represents magnitude of impulse at contact it. ψ l i current distance between two contact features of contact it. If we define: W T inν l+1 + f l i ρl+1 (3) Ten ρ i is te projected distance between corresponding vertex and edge along current normal of contact it at time t l+1. 2
3 We can see tat Stewart-Trinkle metod limits te simulated possible position of te point mass in te area of ρ 1 0 ρ 2 0 wic is more restricted tan te pysically correct one. We can address tis problem by using Kevin Egan s non-convex non-penetration constraint. 1.2 Kevin Egan s model for a locally non-convex corner Kevin Egan s non-convex non-penetration constraints can model non-convex configuration space wit te price of more linear complementarity constraints. In tis 2D simple example, te possible position of te point mass is sowed in figure 2. Configuration space Point mass Correct position = Simulated position Obstacle Figure 2: Kevin Egan non-penetration constraints in non-convex configuration space. Kevin Egan s non-convex model as te ability to or te two conditions instead of and.so te possible future position of te point mass are ρ 1 0 ρ 2 0. I will spend te next section on ow to use linear complementarity problems to model or operator. 1.3 Extending Kevin Egan s model for group of vertices Figure 3 and 4 sow an arbitrary part of a polyedral sape. In order to model suc sape using Kevin Egan s metod, we will break te edges into groups follow te rules: 1. If a group contains edges {e i1, e i2,..., e ik } ten te configuration space corresponds to tat group will be modeled by te set of points tat satisfy ρ i1 0 ρ i ρ ik 0. In oter words, a group tat contains more tan one edge sould form a locally nonconvex configuration space or convex obstacle space. 3
4 2. Tere sould be no edge belong to two groups. Actually, tis rule is just for efficiency purpose. Te result sould still be te same if we enforce te same constraint more tan once. 3. Te final configuration space will be modeled by te set of points tat satisfy all groups s condition. So te configuration space will be modeled as te area tat satisfy: (ρ ρ 1k1 0) (ρ ρ 2k2 0)...(ρ n1 0...ρ nkp 0). 4. Usually te less number of groups, te better performance. We will see tis in later section wen we discuss about ow to model tese conditions. One ting wort mentioning is tat tese rules won t guarantee tat we will model an absolutely accurate configuration space. Te accuracy depends on ow we break te edges into groups. But tis metod provide us a muc more powerful tool to model locally nonconvex space. Also, tere are maybe oter sets of rules to partition te set of edges but tey are beyond te scope of tis project. Figure 3: A accurate way to group edges. Figure 4: An inaccurate way to group edges. 4
5 In figure 4, point P is incorrectly classified as valid because it satisfies condition ρ 3 0. Tis partition violates rule number 3. If tere is an algoritm to partition any polyedral sapes follow tese rules ten tis set of rules are complete. I leave tat algoritm to future work. Remark I don t ave a proof tat we will always be able to accurately model any 2D polyedral obstacle following te rules but I aven t found any counter example. 2 Modeling locally non-convex space using LCP Tis section discusses te metod to model te or operator using LCPs and extend it to model a group of non-convex constraints described in previous section. Kevin Egan reported in [5] two formulations: correct max and summation wit small error. But tese two formulations are superseded by a new diff formulation wic was developed later in a note by Kevin and Jong-Si Pang[3]. Te new formulation as two forms: one simpler but less accurate and one complicated and accurate. For eac form, I will first present metod to model a simple group wit only 2 sides and ten extend it to group wit n sides. 2.1 Less accurate form Te reason I call tis form is less accurate because altoug it properly models te configuration space described by te group it allows normal impulses exist at any edges in te group even if te distances corresponding to tose edges are not zero Simple case: group wit two sides If we name te group as {e 1, e 2 } ten te configuration space we want to model is te set of points tat satisfy condition: is: ρ 1 0 ρ 2 0 max{ρ 1, ρ 2 } 0. (4) So if we call normal impulse at edge 1 as p 1n ten tis form s non-penetration constraint 0 p 1n max{ρ 1, ρ 2 } 0 (5) 0 p 1n max{ρ 1 ρ 2 } + ρ 2 0 (6) We will use LCP to model te function max{a, 0} using following observation: Proposition 2.1 c = max{a, 0} if and only if 0 c a c 0. Proof We prove tis proposition by cecking all possible cases of a s value and make sure in all cases, te results of te two are te same. 5
6 Case 1: a < 0: max{a, 0} = 0 and c a > 0 c = 0. Case 2: a = 0: max{a, 0} = 0 and c a = c c = 0. Case 3: a > 0: max{a, 0} = a and because c a 0 c > 0 c = a So te penetration constraint can be written in LCPs form as: 0 c 1 ρ 1 + ρ 2 c c 1 + ρ 2 1n 0 Similarly, we can write constraint for edge number 2. Ten te final non-penetration constraints for te group are: 0 c 1 ρ 1 + ρ 2 c 1 0 (7) 0 c 1 + ρ 2 1n 0 (8) 0 c 1 + ρ 2 2n 0 (9) So te cost for tis formulation is one more LCP condition Extended case: group wit n sides In tis case, te group contains n edges so te configuration space is te set of points tat satisfy: ρ ρ n 0 max{ρ 1,..., ρ n } 0. (10) Ten te penetration constraint for edge 1 is: 0 p 1n max{ρ 1,, ρ n } 0 0 p 1n max{ρ 1, max{ρ 2,, max{ρ n 1, ρ n } }} 0 We can use n 1 max(a, b) to get max{ρ 1,, ρ n } so te penetration constraints for te group written in LCP form are: 6
7 0 c 2 ρ 2 + ρ 1 c c 3 ρ 3 + c 2 + ρ 1 c c k ρ k + c k c 2 + ρ 1 c k 0 0 c n ρ n + c n c 2 + ρ 1 c n 0 0 c 2 + c c n + ρ 1 1n 0 0 c 2 + c c n + ρ 1 2n 0 0 c 2 + c c n + ρ 1 nn 0 Te total number of linear complementarity conditions is 2n 1 so te price of modeling tis non-convex group is n 1 conditions. 2.2 Accurate metod Te previous formulation could properly model non-convex configuration space but it assumes tat normal impulses can exist in all edges in te group. But it doesn t mean tat we will always get normal impulse from every edges using previous equation, te LCP solver may properly coose te rigt edge. I present a more complicated formulation described in Kevin Egan and Jong-Si Pang s note[3] tat force normal impulse can only exist at te rigt edges wen te contacts maintained. Again, I will go troug simple 2 edges group first, ten extend it to many edges group Simple case: group wit two sides Te main difference is we now force p 1n > 0 only wen ρ i = 0. So te normal impulse at edge 1: p 1n > 0 (max{ρ 1, ρ 2 } = 0) (ρ 1 = 0) (11) We model tis equation using LCPs using following propositions: Proposition 2.2 (max{ρ 1, ρ 2 } = 0) (ρ 1 = 0) (max{ρ 1, ρ 2 } = 0) (max{ρ 1, ρ 2 } + min{ρ 1, 0} = 0). Proof We need to prove tat wen max{ρ 1, ρ 2 } = 0 ten min{ρ 1, 0} = 0 ρ 1 = 0. ρ 1 = 0 min{ρ 1, 0} = min{0, 0} = 0 7
8 min{ρ 1, 0} = 0 max{ρ 1, ρ 2 } = 0 ρ 1 = 0 We can also model min{0, 0} using LCP: Proposition 2.3 if min{a, 0} = b ten 0 b + a b 0. Proof Case a < 0: min{a, 0} = a and b + a 0 b > 0 a + b = 0 b = a. Case a = 0: min{a, 0} = 0 and 0 b b 0 b = 0 Case a > 0: min{a, 0} = 0 and b 0 a + b > 0 b = 0. So te final accurate formulation for non-penetration constraint at edge 1 is: 0 c 1 ρ 1 + ρ 2 c g 1 + ρ 1 g c 1 + ρ 2 + g 1 1n 0 0 c 1 + ρ 2 p 0 And for edge 2, we need to add two LCPs: 0 g 2 + ρ 2 g c 1 + ρ 2 + g 2 2n 0 So, te cost of accuracy in tis formulation is four linear complementarity conditions more tan te non-accurate one. In wic, two conditions are used for moodeling non-convexity, and two conditions for selecting te rigt edges were collisions appen Extended case: group wit n sides In tis case, we only need to add n LCPs to force impulse force only exist in constraints correspond to edges tat as distance zero. So te final LCPs formulation is: 8
9 0 c 2 ρ 2 + ρ 1 c c 3 ρ 3 + c 2 + ρ 1 c c k ρ k + c k c 2 + ρ 1 c k 0 0 c n ρ n + c n c 2 + ρ 1 c n 0 0 g 1 + ρ 1 g g 2 + ρ 2 g g n + ρ n g n 0 0 g 1 + c 2 + c c n + ρ 1 1n 0 0 g 2 + c 2 + c c n + ρ 1 2n 0 0 g n + c 2 + c c n + ρ 1 nn 0 0 c 2 + c c n + ρ 1 p 0 Te number of linear complementarity conditions to model a non-convex group of n edges using tis accurate formulation is 3n. In wic: n to model non-convex configuration space n to coose proper edges of contact n to model non-penetration constraints So in general, te price of accuracy in tis new formulation is 2n more linear complementarity conditions. 3 Implementation 3.1 Putting non-convex constraints in Stewart-Trinkle framework We know tat: ρ l+1 n = Ψl n + W n T ν l+1 + Ψl n So we can express ρ i n l+1 as function of ν l+1 to put tem in Stewart-Trinkle framework [8]. Below is te complete MLCP formulation I derived using Kevin Egan s accurate formulation 9
10 for a non-convex group wit n sides. It s easy to extend to many groups by just duplicating complementarity conditions accordingly. 0 M W 1n W nn 0 ν l+1 ζ 2 (W1n T W2n) T 1 c 2 ζ 3 (W1n T W3n) T 1 1 c 3 ζ n (W1n T Wnn) T 1 1 c n γ 1 W1n T g 1 γ 2 = W2n T g 2 γ n W T ρ l+1 nn g n 1 W T ρ l+1 1n W2n T 1n n ρ l+1 n Wnn T W1n T nn 1 1 ρ l+1 + Ψ l 1n Ψ l 1n Ψ l 1n Mν l + p l ext + Ψl 1n + Ψl 1n + Ψl 1n Ψ l 1n Ψ l 2n Ψ l nn Ψ l 1n Ψ l 2n Ψ l nn Ψ l 1n Ψl 2n Ψl 3n Ψl nn + Ψl 1n + Ψl 2n + Ψl nn + Ψl 1n + Ψl 2n + Ψl nn + Ψl 1n + Ψl 2n + Ψl 3n + Ψl nn ; ζ 2 c 2 ζ 3 c 3 ζ n c n γ 1 g 1 γ 2 g 2 γ n g n ρ l+1 1 ρ l+1 1n 2 2n ρ l+1 n ρ l+1 nn (12) Te big matrix we usually called B for one n sides group is 3n + 2 3n Project s implementation I ave finised te coding part to test tese new formulations. Te demo program contains tree time-steppers: Original Stewart-Trinkle time-stepper 10
11 Kevin Egan s simple non-convex formulation Kevin Egan s accurate non-convex formulation Below is a screensot of te demo running wit simple non-convex formulation: Here are te list of some features built in tis demo: Read geometry from external file Allow simple user interaction Work wit moving particle and static polyedral sapes In sort, it can be used to test Kevin Egan s formulations. 4 Some interesting results In tis section, I will provide some interesting results from implementing Kevin Egan s formulation. 4.1 Simple formulation Te demo program sows tat te metod works as expected. We really get a non-convex configuration space as te particle could travel pass te virtual edge smootly. But once in contact wit te edges, te particle will automatically move along te edge as we allow normal impulses at all edges. Tis effect is sowed below: 11
12 Te particle can travel troug virtual edge to get in contact wit te edges. But once in contact, it will move along te edge. Te direction of te move depends on te edges s normal. It sows tat we can overcome te virtual edge s artifact caused by convex assumption of original Stewart-Trinkle time stepper but it leads to anoter problem. Actually, depend on applications, tis problem can be treated in some ways. One way of wic is cecking n for in contact event. If it s in contact ten switc to original Stewart-Trinkle time stepper. 4.2 Accurate formulation After a long time trying to implement tis more complicated formulation, I found out tat tere is one linear complementarity condition tat leads to singularity in our big matrix. And tis singularity make my LCP s solver, PATH [4], fails (crased) after some time steps. 12
13 Tat condition is: 0 c 2 + c c n + ρ 1 p 0 In wic, p is just a dummy variable and no oter conditions need p. Tat leads to a full column of 0 in te big matrix. I ave tried to introduce an small ɛ amount of p in te left and side but it makes te simulation unstable. Maybe tere is oter way of dealing wit it but in a sort time, I could not tink of any. Below is te screen-sot tat sows te big matrix in simulation: 4.3 Can Kevin Egan s metod model precisely all planar obstacle? Rigt now I don t ave an absolute answer for polyedron vs polyedral case but te answer for te case wit very simple implicit object: circle vs polyedron is NO. Below is te case were Kevin Egan s metod doesn t work wit circle: Tis is completely valid configuration but Kevin Egan s metod will report it as invalid because bot distances are negative. 5 Future works Tere are some areas we can explore in te future: 13
14 Using topology information of a polyedral sape in convex formulation. Advance into 3D. Work wit implicit surfaces by discretizing te surfaces on te fly and model it wit convex/non-convex obstacle. Find an algoritm to partition arbitrary polyedral sape into groups. Fix te accurate formulation References [1] M. Anitescu and F.A. Potra. A time-stepping metod for stiff multibody dynamics wit contact and friction. International Journal for Numerical Metods in Engineering, 55: , [2] D. Baraff. Issues in computing contact forces for non-penetrating rigid bodies. Algoritmica, pages , October [3] Kevin Egan. More tougts on nonconvex constraints [4] Micael C. Ferris and Todd S. Munson. Pat cp solver. URL ttp:// edu/cpnet/cpnetsoftware/ [5] S. Berard K.T. Egan and J.C. Trinkle. Modeling nonconvex constraints using linear complementarity. Tecnical Report 03-13, [6] Katta G. Murty. Linear Complementarity, Linear and Nonlinear Programming Internet Edition. Helderman-Verlag, URL ttp://ioe.engin.umic.edu/people/fac/ books/murty/linear_complementarity_webbook/. [7] D.E. Stewart and J.C. Trinkle. An implicit time-stepping sceme for rigid body dynamics wit inelastic collisions and coulomb friction. Int l Jrnl. of Numerical Metods in Engineering, pages , [8] D.E. Stewart and J.C. Trinkle. An implicit time-stepping sceme for rigid body dynamics wit inelastic collisions and coulomb friction. International Journal of Numerical Metods in Engineering, 39: ,
Physically Based Modeling: Principles and Practice Implicit Methods for Differential Equations
Pysically Based Modeling: Principles and Practice Implicit Metods for Differential Equations David Baraff Robotics Institute Carnegie Mellon University Please note: Tis document is 997 by David Baraff
More informationThe derivative function
Roberto s Notes on Differential Calculus Capter : Definition of derivative Section Te derivative function Wat you need to know already: f is at a point on its grap and ow to compute it. Wat te derivative
More informationConsider a function f we ll specify which assumptions we need to make about it in a minute. Let us reformulate the integral. 1 f(x) dx.
Capter 2 Integrals as sums and derivatives as differences We now switc to te simplest metods for integrating or differentiating a function from its function samples. A careful study of Taylor expansions
More informationPreface. Here are a couple of warnings to my students who may be here to get a copy of what happened on a day that you missed.
Preface Here are my online notes for my course tat I teac ere at Lamar University. Despite te fact tat tese are my class notes, tey sould be accessible to anyone wanting to learn or needing a refreser
More informationHow to Find the Derivative of a Function: Calculus 1
Introduction How to Find te Derivative of a Function: Calculus 1 Calculus is not an easy matematics course Te fact tat you ave enrolled in suc a difficult subject indicates tat you are interested in te
More informationCalculus I Homework: The Derivative as a Function Page 1
Calculus I Homework: Te Derivative as a Function Page 1 Example (2.9.16) Make a careful sketc of te grap of f(x) = sin x and below it sketc te grap of f (x). Try to guess te formula of f (x) from its grap.
More informationNotes on wavefunctions II: momentum wavefunctions
Notes on wavefunctions II: momentum wavefunctions and uncertainty Te state of a particle at any time is described by a wavefunction ψ(x). Tese wavefunction must cange wit time, since we know tat particles
More information2.8 The Derivative as a Function
.8 Te Derivative as a Function Typically, we can find te derivative of a function f at many points of its domain: Definition. Suppose tat f is a function wic is differentiable at every point of an open
More informationNUMERICAL DIFFERENTIATION. James T. Smith San Francisco State University. In calculus classes, you compute derivatives algebraically: for example,
NUMERICAL DIFFERENTIATION James T Smit San Francisco State University In calculus classes, you compute derivatives algebraically: for example, f( x) = x + x f ( x) = x x Tis tecnique requires your knowing
More informationExam 1 Review Solutions
Exam Review Solutions Please also review te old quizzes, and be sure tat you understand te omework problems. General notes: () Always give an algebraic reason for your answer (graps are not sufficient),
More informationContinuity and Differentiability Worksheet
Continuity and Differentiability Workseet (Be sure tat you can also do te grapical eercises from te tet- Tese were not included below! Typical problems are like problems -3, p. 6; -3, p. 7; 33-34, p. 7;
More informationTHE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Math 225
THE IDEA OF DIFFERENTIABILITY FOR FUNCTIONS OF SEVERAL VARIABLES Mat 225 As we ave seen, te definition of derivative for a Mat 111 function g : R R and for acurveγ : R E n are te same, except for interpretation:
More informationTime (hours) Morphine sulfate (mg)
Mat Xa Fall 2002 Review Notes Limits and Definition of Derivative Important Information: 1 According to te most recent information from te Registrar, te Xa final exam will be eld from 9:15 am to 12:15
More information3.4 Worksheet: Proof of the Chain Rule NAME
Mat 1170 3.4 Workseet: Proof of te Cain Rule NAME Te Cain Rule So far we are able to differentiate all types of functions. For example: polynomials, rational, root, and trigonometric functions. We are
More informationCombining functions: algebraic methods
Combining functions: algebraic metods Functions can be added, subtracted, multiplied, divided, and raised to a power, just like numbers or algebra expressions. If f(x) = x 2 and g(x) = x + 2, clearly f(x)
More informationLIMITATIONS OF EULER S METHOD FOR NUMERICAL INTEGRATION
LIMITATIONS OF EULER S METHOD FOR NUMERICAL INTEGRATION LAURA EVANS.. Introduction Not all differential equations can be explicitly solved for y. Tis can be problematic if we need to know te value of y
More information1 Limits and Continuity
1 Limits and Continuity 1.0 Tangent Lines, Velocities, Growt In tion 0.2, we estimated te slope of a line tangent to te grap of a function at a point. At te end of tion 0.3, we constructed a new function
More informationAverage Rate of Change
Te Derivative Tis can be tougt of as an attempt to draw a parallel (pysically and metaporically) between a line and a curve, applying te concept of slope to someting tat isn't actually straigt. Te slope
More informationREVIEW LAB ANSWER KEY
REVIEW LAB ANSWER KEY. Witout using SN, find te derivative of eac of te following (you do not need to simplify your answers): a. f x 3x 3 5x x 6 f x 3 3x 5 x 0 b. g x 4 x x x notice te trick ere! x x g
More informationDerivatives. By: OpenStaxCollege
By: OpenStaxCollege Te average teen in te United States opens a refrigerator door an estimated 25 times per day. Supposedly, tis average is up from 10 years ago wen te average teenager opened a refrigerator
More informationDynamics and Relativity
Dynamics and Relativity Stepen Siklos Lent term 2011 Hand-outs and examples seets, wic I will give out in lectures, are available from my web site www.damtp.cam.ac.uk/user/stcs/dynamics.tml Lecture notes,
More informationOrder of Accuracy. ũ h u Ch p, (1)
Order of Accuracy 1 Terminology We consider a numerical approximation of an exact value u. Te approximation depends on a small parameter, wic can be for instance te grid size or time step in a numerical
More information2.11 That s So Derivative
2.11 Tat s So Derivative Introduction to Differential Calculus Just as one defines instantaneous velocity in terms of average velocity, we now define te instantaneous rate of cange of a function at a point
More information5 Ordinary Differential Equations: Finite Difference Methods for Boundary Problems
5 Ordinary Differential Equations: Finite Difference Metods for Boundary Problems Read sections 10.1, 10.2, 10.4 Review questions 10.1 10.4, 10.8 10.9, 10.13 5.1 Introduction In te previous capters we
More informationFunction Composition and Chain Rules
Function Composition and s James K. Peterson Department of Biological Sciences and Department of Matematical Sciences Clemson University Marc 8, 2017 Outline 1 Function Composition and Continuity 2 Function
More informationA = h w (1) Error Analysis Physics 141
Introduction In all brances of pysical science and engineering one deals constantly wit numbers wic results more or less directly from experimental observations. Experimental observations always ave inaccuracies.
More informationPre-Calculus Review Preemptive Strike
Pre-Calculus Review Preemptive Strike Attaced are some notes and one assignment wit tree parts. Tese are due on te day tat we start te pre-calculus review. I strongly suggest reading troug te notes torougly
More informationExcerpt from "Calculus" 2013 AoPS Inc.
Excerpt from "Calculus" 03 AoPS Inc. Te term related rates refers to two quantities tat are dependent on eac oter and tat are canging over time. We can use te dependent relationsip between te quantities
More informationLines, Conics, Tangents, Limits and the Derivative
Lines, Conics, Tangents, Limits and te Derivative Te Straigt Line An two points on te (,) plane wen joined form a line segment. If te line segment is etended beond te two points ten it is called a straigt
More informationMath 34A Practice Final Solutions Fall 2007
Mat 34A Practice Final Solutions Fall 007 Problem Find te derivatives of te following functions:. f(x) = 3x + e 3x. f(x) = x + x 3. f(x) = (x + a) 4. Is te function 3t 4t t 3 increasing or decreasing wen
More information3.1 Extreme Values of a Function
.1 Etreme Values of a Function Section.1 Notes Page 1 One application of te derivative is finding minimum and maimum values off a grap. In precalculus we were only able to do tis wit quadratics by find
More information5.1 We will begin this section with the definition of a rational expression. We
Basic Properties and Reducing to Lowest Terms 5.1 We will begin tis section wit te definition of a rational epression. We will ten state te two basic properties associated wit rational epressions and go
More informationPolynomials 3: Powers of x 0 + h
near small binomial Capter 17 Polynomials 3: Powers of + Wile it is easy to compute wit powers of a counting-numerator, it is a lot more difficult to compute wit powers of a decimal-numerator. EXAMPLE
More informationHigher Derivatives. Differentiable Functions
Calculus 1 Lia Vas Higer Derivatives. Differentiable Functions Te second derivative. Te derivative itself can be considered as a function. Te instantaneous rate of cange of tis function is te second derivative.
More informationProblem Solving. Problem Solving Process
Problem Solving One of te primary tasks for engineers is often solving problems. It is wat tey are, or sould be, good at. Solving engineering problems requires more tan just learning new terms, ideas and
More informationFinite Difference Methods Assignments
Finite Difference Metods Assignments Anders Söberg and Aay Saxena, Micael Tuné, and Maria Westermarck Revised: Jarmo Rantakokko June 6, 1999 Teknisk databeandling Assignment 1: A one-dimensional eat equation
More informationExcursions in Computing Science: Week v Milli-micro-nano-..math Part II
Excursions in Computing Science: Week v Milli-micro-nano-..mat Part II T. H. Merrett McGill University, Montreal, Canada June, 5 I. Prefatory Notes. Cube root of 8. Almost every calculator as a square-root
More informationPolynomial Interpolation
Capter 4 Polynomial Interpolation In tis capter, we consider te important problem of approximatinga function fx, wose values at a set of distinct points x, x, x,, x n are known, by a polynomial P x suc
More information2.1 THE DEFINITION OF DERIVATIVE
2.1 Te Derivative Contemporary Calculus 2.1 THE DEFINITION OF DERIVATIVE 1 Te grapical idea of a slope of a tangent line is very useful, but for some uses we need a more algebraic definition of te derivative
More informationSolutions to the Multivariable Calculus and Linear Algebra problems on the Comprehensive Examination of January 31, 2014
Solutions to te Multivariable Calculus and Linear Algebra problems on te Compreensive Examination of January 3, 24 Tere are 9 problems ( points eac, totaling 9 points) on tis portion of te examination.
More informationDifferential Calculus (The basics) Prepared by Mr. C. Hull
Differential Calculus Te basics) A : Limits In tis work on limits, we will deal only wit functions i.e. tose relationsips in wic an input variable ) defines a unique output variable y). Wen we work wit
More informationlecture 26: Richardson extrapolation
43 lecture 26: Ricardson extrapolation 35 Ricardson extrapolation, Romberg integration Trougout numerical analysis, one encounters procedures tat apply some simple approximation (eg, linear interpolation)
More informationLab 6 Derivatives and Mutant Bacteria
Lab 6 Derivatives and Mutant Bacteria Date: September 27, 20 Assignment Due Date: October 4, 20 Goal: In tis lab you will furter explore te concept of a derivative using R. You will use your knowledge
More information2.3 Algebraic approach to limits
CHAPTER 2. LIMITS 32 2.3 Algebraic approac to its Now we start to learn ow to find its algebraically. Tis starts wit te simplest possible its, and ten builds tese up to more complicated examples. Fact.
More informationChapter 2 Ising Model for Ferromagnetism
Capter Ising Model for Ferromagnetism Abstract Tis capter presents te Ising model for ferromagnetism, wic is a standard simple model of a pase transition. Using te approximation of mean-field teory, te
More informationMAT244 - Ordinary Di erential Equations - Summer 2016 Assignment 2 Due: July 20, 2016
MAT244 - Ordinary Di erential Equations - Summer 206 Assignment 2 Due: July 20, 206 Full Name: Student #: Last First Indicate wic Tutorial Section you attend by filling in te appropriate circle: Tut 0
More informationSin, Cos and All That
Sin, Cos and All Tat James K. Peterson Department of Biological Sciences and Department of Matematical Sciences Clemson University Marc 9, 2017 Outline Sin, Cos and all tat! A New Power Rule Derivatives
More informationMaterial for Difference Quotient
Material for Difference Quotient Prepared by Stepanie Quintal, graduate student and Marvin Stick, professor Dept. of Matematical Sciences, UMass Lowell Summer 05 Preface Te following difference quotient
More informationNumerical Differentiation
Numerical Differentiation Finite Difference Formulas for te first derivative (Using Taylor Expansion tecnique) (section 8.3.) Suppose tat f() = g() is a function of te variable, and tat as 0 te function
More informationNew Streamfunction Approach for Magnetohydrodynamics
New Streamfunction Approac for Magnetoydrodynamics Kab Seo Kang Brooaven National Laboratory, Computational Science Center, Building 63, Room, Upton NY 973, USA. sang@bnl.gov Summary. We apply te finite
More informationIntroduction to Derivatives
Introduction to Derivatives 5-Minute Review: Instantaneous Rates and Tangent Slope Recall te analogy tat we developed earlier First we saw tat te secant slope of te line troug te two points (a, f (a))
More information1 Solutions to the in class part
NAME: Solutions to te in class part. Te grap of a function f is given. Calculus wit Analytic Geometry I Exam, Friday, August 30, 0 SOLUTIONS (a) State te value of f(). (b) Estimate te value of f( ). (c)
More informationMath 31A Discussion Notes Week 4 October 20 and October 22, 2015
Mat 3A Discussion Notes Week 4 October 20 and October 22, 205 To prepare for te first midterm, we ll spend tis week working eamples resembling te various problems you ve seen so far tis term. In tese notes
More informationPractice Problem Solutions: Exam 1
Practice Problem Solutions: Exam 1 1. (a) Algebraic Solution: Te largest term in te numerator is 3x 2, wile te largest term in te denominator is 5x 2 3x 2 + 5. Tus lim x 5x 2 2x 3x 2 x 5x 2 = 3 5 Numerical
More informationContinuity and Differentiability of the Trigonometric Functions
[Te basis for te following work will be te definition of te trigonometric functions as ratios of te sides of a triangle inscribed in a circle; in particular, te sine of an angle will be defined to be te
More informationChapter 5 FINITE DIFFERENCE METHOD (FDM)
MEE7 Computer Modeling Tecniques in Engineering Capter 5 FINITE DIFFERENCE METHOD (FDM) 5. Introduction to FDM Te finite difference tecniques are based upon approximations wic permit replacing differential
More informationHOMEWORK HELP 2 FOR MATH 151
HOMEWORK HELP 2 FOR MATH 151 Here we go; te second round of omework elp. If tere are oters you would like to see, let me know! 2.4, 43 and 44 At wat points are te functions f(x) and g(x) = xf(x)continuous,
More information1. Consider the trigonometric function f(t) whose graph is shown below. Write down a possible formula for f(t).
. Consider te trigonometric function f(t) wose grap is sown below. Write down a possible formula for f(t). Tis function appears to be an odd, periodic function tat as been sifted upwards, so we will use
More informationVolume 29, Issue 3. Existence of competitive equilibrium in economies with multi-member households
Volume 29, Issue 3 Existence of competitive equilibrium in economies wit multi-member ouseolds Noriisa Sato Graduate Scool of Economics, Waseda University Abstract Tis paper focuses on te existence of
More informationSolution. Solution. f (x) = (cos x)2 cos(2x) 2 sin(2x) 2 cos x ( sin x) (cos x) 4. f (π/4) = ( 2/2) ( 2/2) ( 2/2) ( 2/2) 4.
December 09, 20 Calculus PracticeTest s Name: (4 points) Find te absolute extrema of f(x) = x 3 0 on te interval [0, 4] Te derivative of f(x) is f (x) = 3x 2, wic is zero only at x = 0 Tus we only need
More information3.4 Algebraic Limits. Ex 1) lim. Ex 2)
Calculus Maimus.4 Algebraic Limits At tis point, you sould be very comfortable finding its bot grapically and numerically wit te elp of your graping calculator. Now it s time to practice finding its witout
More informationDefinition of the Derivative
Te Limit Definition of te Derivative Tis Handout will: Define te limit grapically and algebraically Discuss, in detail, specific features of te definition of te derivative Provide a general strategy of
More information. If lim. x 2 x 1. f(x+h) f(x)
Review of Differential Calculus Wen te value of one variable y is uniquely determined by te value of anoter variable x, ten te relationsip between x and y is described by a function f tat assigns a value
More informationQuantum Numbers and Rules
OpenStax-CNX module: m42614 1 Quantum Numbers and Rules OpenStax College Tis work is produced by OpenStax-CNX and licensed under te Creative Commons Attribution License 3.0 Abstract Dene quantum number.
More information2.3 More Differentiation Patterns
144 te derivative 2.3 More Differentiation Patterns Polynomials are very useful, but tey are not te only functions we need. Tis section uses te ideas of te two previous sections to develop tecniques for
More informationKrazy Katt, the mechanical cat
Krazy Katt, te mecanical cat Te cat rigting relex is a cat's innate ability to orient itsel as it alls in order to land on its eet. Te rigting relex begins to appear at 3 4 weeks o age, and is perected
More information1 The concept of limits (p.217 p.229, p.242 p.249, p.255 p.256) 1.1 Limits Consider the function determined by the formula 3. x since at this point
MA00 Capter 6 Calculus and Basic Linear Algebra I Limits, Continuity and Differentiability Te concept of its (p.7 p.9, p.4 p.49, p.55 p.56). Limits Consider te function determined by te formula f Note
More informationLesson 6: The Derivative
Lesson 6: Te Derivative Def. A difference quotient for a function as te form f(x + ) f(x) (x + ) x f(x + x) f(x) (x + x) x f(a + ) f(a) (a + ) a Notice tat a difference quotient always as te form of cange
More informationThe Laplace equation, cylindrically or spherically symmetric case
Numerisce Metoden II, 7 4, und Übungen, 7 5 Course Notes, Summer Term 7 Some material and exercises Te Laplace equation, cylindrically or sperically symmetric case Electric and gravitational potential,
More information4. The slope of the line 2x 7y = 8 is (a) 2/7 (b) 7/2 (c) 2 (d) 2/7 (e) None of these.
Mat 11. Test Form N Fall 016 Name. Instructions. Te first eleven problems are wort points eac. Te last six problems are wort 5 points eac. For te last six problems, you must use relevant metods of algebra
More informationCopyright c 2008 Kevin Long
Lecture 4 Numerical solution of initial value problems Te metods you ve learned so far ave obtained closed-form solutions to initial value problems. A closedform solution is an explicit algebriac formula
More informationEfficient algorithms for for clone items detection
Efficient algoritms for for clone items detection Raoul Medina, Caroline Noyer, and Olivier Raynaud Raoul Medina, Caroline Noyer and Olivier Raynaud LIMOS - Université Blaise Pascal, Campus universitaire
More informationPolynomial Interpolation
Capter 4 Polynomial Interpolation In tis capter, we consider te important problem of approximating a function f(x, wose values at a set of distinct points x, x, x 2,,x n are known, by a polynomial P (x
More informationMATH1131/1141 Calculus Test S1 v8a
MATH/ Calculus Test 8 S v8a October, 7 Tese solutions were written by Joann Blanco, typed by Brendan Trin and edited by Mattew Yan and Henderson Ko Please be etical wit tis resource It is for te use of
More informationSection 15.6 Directional Derivatives and the Gradient Vector
Section 15.6 Directional Derivatives and te Gradient Vector Finding rates of cange in different directions Recall tat wen we first started considering derivatives of functions of more tan one variable,
More information1watt=1W=1kg m 2 /s 3
Appendix A Matematics Appendix A.1 Units To measure a pysical quantity, you need a standard. Eac pysical quantity as certain units. A unit is just a standard we use to compare, e.g. a ruler. In tis laboratory
More informationSECTION 1.10: DIFFERENCE QUOTIENTS LEARNING OBJECTIVES
(Section.0: Difference Quotients).0. SECTION.0: DIFFERENCE QUOTIENTS LEARNING OBJECTIVES Define average rate of cange (and average velocity) algebraically and grapically. Be able to identify, construct,
More information1 1. Rationalize the denominator and fully simplify the radical expression 3 3. Solution: = 1 = 3 3 = 2
MTH - Spring 04 Exam Review (Solutions) Exam : February 5t 6:00-7:0 Tis exam review contains questions similar to tose you sould expect to see on Exam. Te questions included in tis review, owever, are
More informationQuantum Theory of the Atomic Nucleus
G. Gamow, ZP, 51, 204 1928 Quantum Teory of te tomic Nucleus G. Gamow (Received 1928) It as often been suggested tat non Coulomb attractive forces play a very important role inside atomic nuclei. We can
More informationSome Review Problems for First Midterm Mathematics 1300, Calculus 1
Some Review Problems for First Midterm Matematics 00, Calculus. Consider te trigonometric function f(t) wose grap is sown below. Write down a possible formula for f(t). Tis function appears to be an odd,
More informationMAT 145. Type of Calculator Used TI-89 Titanium 100 points Score 100 possible points
MAT 15 Test #2 Name Solution Guide Type of Calculator Used TI-89 Titanium 100 points Score 100 possible points Use te grap of a function sown ere as you respond to questions 1 to 8. 1. lim f (x) 0 2. lim
More informationLogarithmic functions
Roberto s Notes on Differential Calculus Capter 5: Derivatives of transcendental functions Section Derivatives of Logaritmic functions Wat ou need to know alread: Definition of derivative and all basic
More informationPhase space in classical physics
Pase space in classical pysics Quantum mecanically, we can actually COU te number of microstates consistent wit a given macrostate, specified (for example) by te total energy. In general, eac microstate
More informationMA455 Manifolds Solutions 1 May 2008
MA455 Manifolds Solutions 1 May 2008 1. (i) Given real numbers a < b, find a diffeomorpism (a, b) R. Solution: For example first map (a, b) to (0, π/2) and ten map (0, π/2) diffeomorpically to R using
More informationSolution for the Homework 4
Solution for te Homework 4 Problem 6.5: In tis section we computed te single-particle translational partition function, tr, by summing over all definite-energy wavefunctions. An alternative approac, owever,
More information1. Questions (a) through (e) refer to the graph of the function f given below. (A) 0 (B) 1 (C) 2 (D) 4 (E) does not exist
Mat 1120 Calculus Test 2. October 18, 2001 Your name Te multiple coice problems count 4 points eac. In te multiple coice section, circle te correct coice (or coices). You must sow your work on te oter
More informationDifferentiation in higher dimensions
Capter 2 Differentiation in iger dimensions 2.1 Te Total Derivative Recall tat if f : R R is a 1-variable function, and a R, we say tat f is differentiable at x = a if and only if te ratio f(a+) f(a) tends
More information1 + t5 dt with respect to x. du = 2. dg du = f(u). du dx. dg dx = dg. du du. dg du. dx = 4x3. - page 1 -
Eercise. Find te derivative of g( 3 + t5 dt wit respect to. Solution: Te integrand is f(t + t 5. By FTC, f( + 5. Eercise. Find te derivative of e t2 dt wit respect to. Solution: Te integrand is f(t e t2.
More informationQuantum Mechanics Chapter 1.5: An illustration using measurements of particle spin.
I Introduction. Quantum Mecanics Capter.5: An illustration using measurements of particle spin. Quantum mecanics is a teory of pysics tat as been very successful in explaining and predicting many pysical
More informationMath 212-Lecture 9. For a single-variable function z = f(x), the derivative is f (x) = lim h 0
3.4: Partial Derivatives Definition Mat 22-Lecture 9 For a single-variable function z = f(x), te derivative is f (x) = lim 0 f(x+) f(x). For a function z = f(x, y) of two variables, to define te derivatives,
More informationReflection Symmetries of q-bernoulli Polynomials
Journal of Nonlinear Matematical Pysics Volume 1, Supplement 1 005, 41 4 Birtday Issue Reflection Symmetries of q-bernoulli Polynomials Boris A KUPERSHMIDT Te University of Tennessee Space Institute Tullaoma,
More informationf a h f a h h lim lim
Te Derivative Te derivative of a function f at a (denoted f a) is f a if tis it exists. An alternative way of defining f a is f a x a fa fa fx fa x a Note tat te tangent line to te grap of f at te point
More informationMTH-112 Quiz 1 Name: # :
MTH- Quiz Name: # : Please write our name in te provided space. Simplif our answers. Sow our work.. Determine weter te given relation is a function. Give te domain and range of te relation.. Does te equation
More informationDifferential equations. Differential equations
Differential equations A differential equation (DE) describes ow a quantity canges (as a function of time, position, ) d - A ball dropped from a building: t gt () dt d S qx - Uniformly loaded beam: wx
More informationDerivatives of Exponentials
mat 0 more on derivatives: day 0 Derivatives of Eponentials Recall tat DEFINITION... An eponential function as te form f () =a, were te base is a real number a > 0. Te domain of an eponential function
More informationChapter 2 Limits and Continuity
4 Section. Capter Limits and Continuity Section. Rates of Cange and Limits (pp. 6) Quick Review.. f () ( ) () 4 0. f () 4( ) 4. f () sin sin 0 4. f (). 4 4 4 6. c c c 7. 8. c d d c d d c d c 9. 8 ( )(
More informationAMS 147 Computational Methods and Applications Lecture 09 Copyright by Hongyun Wang, UCSC. Exact value. Effect of round-off error.
Lecture 09 Copyrigt by Hongyun Wang, UCSC Recap: Te total error in numerical differentiation fl( f ( x + fl( f ( x E T ( = f ( x Numerical result from a computer Exact value = e + f x+ Discretization error
More informationThe Verlet Algorithm for Molecular Dynamics Simulations
Cemistry 380.37 Fall 2015 Dr. Jean M. Standard November 9, 2015 Te Verlet Algoritm for Molecular Dynamics Simulations Equations of motion For a many-body system consisting of N particles, Newton's classical
More informationCubic Functions: Local Analysis
Cubic function cubing coefficient Capter 13 Cubic Functions: Local Analysis Input-Output Pairs, 378 Normalized Input-Output Rule, 380 Local I-O Rule Near, 382 Local Grap Near, 384 Types of Local Graps
More informationMath Spring 2013 Solutions to Assignment # 3 Completion Date: Wednesday May 15, (1/z) 2 (1/z 1) 2 = lim
Mat 311 - Spring 013 Solutions to Assignment # 3 Completion Date: Wednesday May 15, 013 Question 1. [p 56, #10 (a)] 4z Use te teorem of Sec. 17 to sow tat z (z 1) = 4. We ave z 4z (z 1) = z 0 4 (1/z) (1/z
More information