NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING. Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582
|
|
- Kimberly Lee
- 5 years ago
- Views:
Transcription
1 NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582
2 4. Moton Knematcs 4.2 Angular Velocty Knematcs Summary From the last lecture we concluded that: If the jonts are all rotary then: ω () t zˆ zˆ zˆ J n 2 n 2 n However, f Jont "" s Prsmatc then the correspondng column n the angular velocty Jacoban (.e. J ) wll be ZERO!! For example f jont #2 s prsmatc we get: ω ( ) ˆ ˆ d d t z z J n n 2 2 n n Asde: What s the dfference between Tp B B A and Rp B B A? Pure translaton wll NOT gve rse to a change n orentaton 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 2 / 27
3 4. Moton Knematcs 4.2 Translatonal Velocty Knematcs Snce the jont varables could be θ (rotary jont) or d (prsmatc jont), let's represent the jont varable by the symbol q, whch could be ether rotary or prsmatc. If Jont "" s Prsmatc: (Assume that only one jont s movng) q, j j d n d d () t o Remember, only jont "" s movng, the others are not movng (q j =, j ). 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 3 / 27
4 4. Moton Knematcs 4.2 Translatonal Velocty Knematcs From the dagram, dn t d d q t dn ( ) ( ( )) ( ) d R d ( q ( t)) R d n (4.) CONSTANT Tme Varyng CONSTANT d n d n d d () t 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 4 / 27
5 4. Moton Knematcs 4.2 Translatonal Velocty Knematcs But, (neglectng the sub-sub-scrpt) d ( q ( t)) q ( t) z d Note that R d d as d T d = R refers to the orgn of {n} as seen from {}. d + d Substtutng equaton (4.2) nto (4.) and dfferentatng gves, q (4.2) d dt d dn t R d q t ( ) ( ( )) dt v ( t) R z q ( t) z q ( t) n (4.3) Note that R R v () t z = z as z means z, the z-axs of coordnate frame {} descrbed n terms of {}. The vector z represents the mappng of a "free vector". 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 5 / 27
6 4. Moton Knematcs 4.2 Translatonal Velocty Knematcs If Jont "" s Rotary: (Assume that only one jont s movng) d n d Agan only jont s movng, the others are not movng. 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 6 / 27
7 4. Moton Knematcs 4.2 Translatonal Velocty Knematcs From the dagram, d ( t) d ( d ) ( d ) n n d ( ( )) R d R t d n (4.4) Dfferentatng (4.4) gves, d n d dt RR d ( t) R R( ( t)) d n n R ( t) R( t) d n (4.5) a3 a 2 b a2b3 a3b2 R ( t) ( R( t) d ) a b n a3 a b 2 a3b ab 3 a b ( ar 2 a ( t)) ( R ( t b ) d n 3 a ) b2 a2b 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 7 / 27
8 4. Moton Knematcs 4.2 Translatonal Velocty Knematcs Now, recallng that ( t) ˆ k ( t) z ( t) allows us to smplfy the expresson R ( t) R z ( t) and z () t d n d n R( t) d d ( t) d n n d 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 8 / 27
9 4. Moton Knematcs 4.2 Translatonal Velocty Knematcs Substtutng back nto (4.5) yelds, (4.6) Note that R z = z as z means z, the z-axs of coordnate frame {} descrbed n terms of {}. The vector z represents the mappng of a "free vector". Equaton (4.6) s effectvely where and vn z t dn t d ( ) ( ( ) ) v r z r d d ( n ) 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 9 / 27
10 4. Moton Knematcs 4.2. Compoundng of Translatonal Veloctes (all jonts movng) As n the prevous case wth angular veloctes, f all the jonts are movng (prsmatc or rotary) the translatonal velocty at the tp becomes: v ( t) v ( t) v ( t) v ( t) n 2 n where agan the v 's are the ndvdual contrbutons compounded by equaton (4.3) as or v ( t) z q( t) f the th jont s prsmatc ( ) ( n ( ) ) ( ) v t z d t d q t f the th jont s rotary where the jont varable q (t) could be an angle (θ (t)) or a dsplacement (d (t)). 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde / 27
11 4. Moton Knematcs 4.3. The Drect Method To account for the fact that a jont varable could be θ(t) (rotary jont) or d(t) (prsmatc jont) we can use q(t) as a "generc" jont varable to represent both possbltes. In lght of our orgnal ntent, whch was to descrbe how the rate of change of jont varables effects the rate of change of tool pose (.e., n th coordnate frame moton), we can now wrte: vn () t n() t J q( t) q( t) J J v q() t q() t qt () J J2 J n q() t (4.7) 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde / 27
12 4. Moton Knematcs 4.3. The Drect Method If the th jont s rotary, then: If the th jont s prsmatc, then: ˆ z dn () t d org org J,,, n zˆ (4.8) J,,, n z ˆ (4.9) Thus, usng equatons (4.8) and (4.9) we can buld J. Ths s the Drect Method of Constructng the Jacoban 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 2 / 27
13 4. Moton Knematcs An Alternatve Method As noted earler the tool poston s d n () t and therefore the rate of change of tool poston s d dn () v ( t) dn ( q( t)) q( t) dt q n 2 Therefore, the translatonal part of the Jacoban can be obtaned as Jv q( t) q( t) Jv Jv Jv n q( t) J v dn () q (4.2) 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 3 / 27
14 4. Moton Knematcs An Alternatve Method For the orentatonal part of the Jacoban we could obtan the angular velocty by recallng equaton (4.4), whch allows us to wrte.e., ( t) R( t) R T ( t) n n n (4.2) z y ωn z x n n T ( t) R( t) R ( t) y x (4.22) Now equatng the correspondng terms gves R( t) R ( t) T n n 3,2 x T y ωn ( t) n R( t) n R ( t),3 z T nr( t) nr ( t) 2, 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 4 / 27
15 4. Moton Knematcs An Alternatve Method Then factorng out terms n q,..., q n from the rhs of the above gves rse to ( t ωn ) q ( t ) o Choosng between the approaches descrbed n Sectons 4.3. and s somewhat dependent on the problem at hand! 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 5 / 27
16 4. Moton Knematcs A Jacoban Example Example: The ADEPT robot s a so called SCARA (Selectvely Complant Assembly Robot Arm) type robot. VRML Smulaton 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 6 / 27
17 4. Moton Knematcs A Jacoban Example The Denavt-Hartenberg table: D-H params. - a - d d a 2 a d t 4 3 () d 4 () t () t 2 () t 4 Usng MATHEMATICA to generate the T-matrces. 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 7 / 27
18 4. Moton Knematcs A Jacoban Example C S S C T(,,, ( t)) C2 S2 a S2 C2 2T(, a,, 2( t)) a2 T(8, a, d ( t),) -d3( t) C4 S4 3 S4 C4 4T(,, d4, 4( t)) d , Dr. Stephen Bruder Thur 27th Sept 22 Slde 8 / 27
19 4. Moton Knematcs A Jacoban Example Note that: q( t) ( t), ( t), d ( t), ( t) T (4.23) C2 S2 Ca S C S a R d xˆ yˆ zˆ d T T 2T C2 S2 C2a2 Ca S C S a S a R d T 2T 3T d3( t) C2C4 S2S4 S2C4 C2S4 C2a2 Ca S C C S C S S S a S a R d T 3T 4T d4 d3( t) 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 9 / 27
20 4. Moton Knematcs A Jacoban Example Usng the Drect Method of Secton 4.3. Snce jont s rotary then: J C2a2 Ca S2a2 Sa S S a a C a Ca zˆ ˆ ( ) z d4 d d4 d3 t Snce jont 2 s rotary then: J 2 C2a2 S2a2 S C a a zˆ ˆ ( ) z2 d4 d2 d4 d3 t 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 2 / 27
21 4. Moton Knematcs A Jacoban Example Snce jont 3 s prsmatc then: J 3 zˆ 3 Snce jont 4 s rotary then: J 4 ˆ z4 d4 d4 zˆ 4 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 2 / 27
22 4. Moton Knematcs A Jacoban Example Puttng ths all together yelds the Jacoban as Hence, J v J J q() t J J J J () t v4() t 2() t J q( t) J q( t) q( t) 4() t d3() t 4() t S2a2 Sa S2a2 C a C a C a (4.24) 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 22 / 27
23 4. Moton Knematcs A Jacoban Example Asde: Recall that ω4(t) = J ω q and v4(t) = J v q, hence 4( t) ( t) 2( t) 4( t) ( t) 2( t) 4( t) Usng the Alternatve Method of Secton Let us frst look at the translatonal part of the Jacoban. Now, d v ( t) d ( q( t)) dt 4 4 d 4 ( ) d4 ( ) d4 ( ) d4 ( ) q ( t) q ( t) q ( t) q ( t) q q q q , Dr. Stephen Bruder Thur 27th Sept 22 Slde 23 / 27
24 4. Moton Knematcs A Jacoban Example d Evaluatng the ndvdual partals gves: C2a2 Ca S S a a d4 d3() t S2a2 Sa d4 () C2 2 C a a d d d d 3 4 () () () S a C 2a2 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 24 / 27
25 4. Moton Knematcs A Jacoban Example Therefore, J v S2a2 Sa S2a2 C2a2 Ca C2a2 (4.25) Let us next look at the rotatonal part of the Jacoban. Now, T ( t) ( t) R( t) R ( t) (4.26) C C S S S C C S S C C S C C S S R C24 S24 S24 C24 where C 2 4 Cos(θ + θ 2 θ 4 ) and S 2 4 Sn(θ + θ 2 θ 4 ). 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 25 / 27
26 4. Moton Knematcs A Jacoban Example Usng the substtuton θ 2 4 = (θ + θ 2 θ 4 ) for the sake of clarty: S24 C24 d 4 R C2 4 S dt S24 C24 C24 S24 T ωn ( t) n R( t) n R ( t) C S S C , Dr. Stephen Bruder Thur 27th Sept 22 Slde 26 / 27
27 4. Moton Knematcs A Jacoban Example ( ) ( ) 2 Jq( t) d 3 4 ω t n (4.27) Puttng together the results of equaton (4.25) for J v and equaton (4.27) for J ω we get, J q() t S2a2 Sa S2a2 C2a2 Ca C2a2 Jv q() t J q() t Both methods agree!!! 2, Dr. Stephen Bruder Thur 27th Sept 22 Slde 27 / 27
NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING. Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582
NMT EE 589 & UNM ME 48/58 ROBOT ENGINEERING Dr. Stephen Bruder NMT EE 589 & UNM ME 48/58 7. Robot Dynamcs 7.5 The Equatons of Moton Gven that we wsh to fnd the path q(t (n jont space) whch mnmzes the energy
More informationIterative General Dynamic Model for Serial-Link Manipulators
EEL6667: Knematcs, Dynamcs and Control of Robot Manpulators 1. Introducton Iteratve General Dynamc Model for Seral-Lnk Manpulators In ths set of notes, we are gong to develop a method for computng a general
More informationχ x B E (c) Figure 2.1.1: (a) a material particle in a body, (b) a place in space, (c) a configuration of the body
Secton.. Moton.. The Materal Body and Moton hyscal materals n the real world are modeled usng an abstract mathematcal entty called a body. Ths body conssts of an nfnte number of materal partcles. Shown
More informationWeek3, Chapter 4. Position and Displacement. Motion in Two Dimensions. Instantaneous Velocity. Average Velocity
Week3, Chapter 4 Moton n Two Dmensons Lecture Quz A partcle confned to moton along the x axs moves wth constant acceleraton from x =.0 m to x = 8.0 m durng a 1-s tme nterval. The velocty of the partcle
More informationMEV442 Introduction to Robotics Module 2. Dr. Santhakumar Mohan Assistant Professor Mechanical Engineering National Institute of Technology Calicut
MEV442 Introducton to Robotcs Module 2 Dr. Santhakumar Mohan Assstant Professor Mechancal Engneerng Natonal Insttute of Technology Calcut Jacobans: Veloctes and statc forces Introducton Notaton for tme-varyng
More informationPhysics 181. Particle Systems
Physcs 181 Partcle Systems Overvew In these notes we dscuss the varables approprate to the descrpton of systems of partcles, ther defntons, ther relatons, and ther conservatons laws. We consder a system
More informationChapter 3. r r. Position, Velocity, and Acceleration Revisited
Chapter 3 Poston, Velocty, and Acceleraton Revsted The poston vector of a partcle s a vector drawn from the orgn to the locaton of the partcle. In two dmensons: r = x ˆ+ yj ˆ (1) The dsplacement vector
More information(c) (cos θ + i sin θ) 5 = cos 5 θ + 5 cos 4 θ (i sin θ) + 10 cos 3 θ(i sin θ) cos 2 θ(i sin θ) 3 + 5cos θ (i sin θ) 4 + (i sin θ) 5 (A1)
. (a) (cos θ + sn θ) = cos θ + cos θ( sn θ) + cos θ(sn θ) + (sn θ) = cos θ cos θ sn θ + ( cos θ sn θ sn θ) (b) from De Movre s theorem (cos θ + sn θ) = cos θ + sn θ cos θ + sn θ = (cos θ cos θ sn θ) +
More informationNEWTON S LAWS. These laws only apply when viewed from an inertial coordinate system (unaccelerated system).
EWTO S LAWS Consder two partcles. 1 1. If 1 0 then 0 wth p 1 m1v. 1 1 2. 1.. 3. 11 These laws only apply when vewed from an nertal coordnate system (unaccelerated system). consder a collecton of partcles
More informationPHYS 705: Classical Mechanics. Newtonian Mechanics
1 PHYS 705: Classcal Mechancs Newtonan Mechancs Quck Revew of Newtonan Mechancs Basc Descrpton: -An dealzed pont partcle or a system of pont partcles n an nertal reference frame [Rgd bodes (ch. 5 later)]
More informationT f. Geometry. R f. R i. Homogeneous transformation. y x. P f. f 000. Homogeneous transformation matrix. R (A): Orientation P : Position
Homogeneous transformaton Geometr T f R f R T f Homogeneous transformaton matr Unverst of Genova T f Phlppe Martnet = R f 000 P f 1 R (A): Orentaton P : Poston 123 Modelng and Control of Manpulator robots
More informationSpring Force and Power
Lecture 13 Chapter 9 Sprng Force and Power Yeah, energy s better than orces. What s net? Course webste: http://aculty.uml.edu/andry_danylov/teachng/physcsi IN THIS CHAPTER, you wll learn how to solve problems
More informationRigid body simulation
Rgd bod smulaton Rgd bod smulaton Once we consder an object wth spacal etent, partcle sstem smulaton s no longer suffcent Problems Problems Unconstraned sstem rotatonal moton torques and angular momentum
More informationELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM
ELASTIC WAVE PROPAGATION IN A CONTINUOUS MEDIUM An elastc wave s a deformaton of the body that travels throughout the body n all drectons. We can examne the deformaton over a perod of tme by fxng our look
More informationLagrange Multipliers. A Somewhat Silly Example. Monday, 25 September 2013
Lagrange Multplers Monday, 5 September 013 Sometmes t s convenent to use redundant coordnates, and to effect the varaton of the acton consstent wth the constrants va the method of Lagrange undetermned
More informationThe classical spin-rotation coupling
LOUAI H. ELZEIN 2018 All Rghts Reserved The classcal spn-rotaton couplng Loua Hassan Elzen Basher Khartoum, Sudan. Postal code:11123 louaelzen@gmal.com Abstract Ths paper s prepared to show that a rgd
More informationTranslational Equations of Motion for A Body Translational equations of motion (centroidal) for a body are m r = f.
Lesson 12: Equatons o Moton Newton s Laws Frst Law: A artcle remans at rest or contnues to move n a straght lne wth constant seed there s no orce actng on t Second Law: The acceleraton o a artcle s roortonal
More information12. The Hamilton-Jacobi Equation Michael Fowler
1. The Hamlton-Jacob Equaton Mchael Fowler Back to Confguraton Space We ve establshed that the acton, regarded as a functon of ts coordnate endponts and tme, satsfes ( ) ( ) S q, t / t+ H qpt,, = 0, and
More informationLesson 5: Kinematics and Dynamics of Particles
Lesson 5: Knematcs and Dynamcs of Partcles hs set of notes descrbes the basc methodology for formulatng the knematc and knetc equatons for multbody dynamcs. In order to concentrate on the methodology and
More informationAn Algorithm to Solve the Inverse Kinematics Problem of a Robotic Manipulator Based on Rotation Vectors
An Algorthm to Solve the Inverse Knematcs Problem of a Robotc Manpulator Based on Rotaton Vectors Mohamad Z. Al-az*, Mazn Z. Othman**, and Baker B. Al-Bahr* *AL-Nahran Unversty, Computer Eng. Dep., Baghdad,
More information1 Matrix representations of canonical matrices
1 Matrx representatons of canoncal matrces 2-d rotaton around the orgn: ( ) cos θ sn θ R 0 = sn θ cos θ 3-d rotaton around the x-axs: R x = 1 0 0 0 cos θ sn θ 0 sn θ cos θ 3-d rotaton around the y-axs:
More informationConservation of Angular Momentum = "Spin"
Page 1 of 6 Conservaton of Angular Momentum = "Spn" We can assgn a drecton to the angular velocty: drecton of = drecton of axs + rght hand rule (wth rght hand, curl fngers n drecton of rotaton, thumb ponts
More informationWeek 11: Chapter 11. The Vector Product. The Vector Product Defined. The Vector Product and Torque. More About the Vector Product
The Vector Product Week 11: Chapter 11 Angular Momentum There are nstances where the product of two vectors s another vector Earler we saw where the product of two vectors was a scalar Ths was called the
More informationMathematical Preparations
1 Introducton Mathematcal Preparatons The theory of relatvty was developed to explan experments whch studed the propagaton of electromagnetc radaton n movng coordnate systems. Wthn expermental error the
More informationcoordinates. Then, the position vectors are described by
Revewng, what we have dscussed so far: Generalzed coordnates Any number of varables (say, n) suffcent to specfy the confguraton of the system at each nstant to tme (need not be the mnmum number). In general,
More informationEE 570: Location and Navigation: Theory & Practice
EE 570: Locaton and Navgaton: Theory & Practce Navgaton Sensors and INS Mechanzaton Tuesday 26 Fe 2013 NMT EE 570: Locaton and Navgaton: Theory & Practce Slde 1 of 14 Navgaton Sensors and INS Mechanzaton
More informationLinear Momentum. Center of Mass.
Lecture 6 Chapter 9 Physcs I 03.3.04 Lnear omentum. Center of ass. Course webste: http://faculty.uml.edu/ndry_danylov/teachng/physcsi Lecture Capture: http://echo360.uml.edu/danylov03/physcssprng.html
More informationKinematics in 2-Dimensions. Projectile Motion
Knematcs n -Dmensons Projectle Moton A medeval trebuchet b Kolderer, c1507 http://members.net.net.au/~rmne/ht/ht0.html#5 Readng Assgnment: Chapter 4, Sectons -6 Introducton: In medeval das, people had
More informationNMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING. Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582
NM EE 589 & UNM ME 48/58 ROBO ENGINEERING Dr. Stephen Bruder NM EE 589 & UNM ME 48/58 5. Robot Dynamics 5. he Microbot Arm Dynamic Model A Second Dynamic Model Example: he Microbot Robot Arm he Denavit-Hartenberg
More informationWeek 9 Chapter 10 Section 1-5
Week 9 Chapter 10 Secton 1-5 Rotaton Rgd Object A rgd object s one that s nondeformable The relatve locatons of all partcles makng up the object reman constant All real objects are deformable to some extent,
More informationIntegrals and Invariants of Euler-Lagrange Equations
Lecture 16 Integrals and Invarants of Euler-Lagrange Equatons ME 256 at the Indan Insttute of Scence, Bengaluru Varatonal Methods and Structural Optmzaton G. K. Ananthasuresh Professor, Mechancal Engneerng,
More informationC/CS/Phy191 Problem Set 3 Solutions Out: Oct 1, 2008., where ( 00. ), so the overall state of the system is ) ( ( ( ( 00 ± 11 ), Φ ± = 1
C/CS/Phy9 Problem Set 3 Solutons Out: Oct, 8 Suppose you have two qubts n some arbtrary entangled state ψ You apply the teleportaton protocol to each of the qubts separately What s the resultng state obtaned
More informationENGN 40 Dynamics and Vibrations Homework # 7 Due: Friday, April 15
NGN 40 ynamcs and Vbratons Homework # 7 ue: Frday, Aprl 15 1. Consder a concal hostng drum used n the mnng ndustry to host a mass up/down. A cable of dameter d has the mass connected at one end and s wound/unwound
More information8.592J: Solutions for Assignment 7 Spring 2005
8.59J: Solutons for Assgnment 7 Sprng 5 Problem 1 (a) A flament of length l can be created by addton of a monomer to one of length l 1 (at rate a) or removal of a monomer from a flament of length l + 1
More informationKinematics of Fluids. Lecture 16. (Refer the text book CONTINUUM MECHANICS by GEORGE E. MASE, Schaum s Outlines) 17/02/2017
17/0/017 Lecture 16 (Refer the text boo CONTINUUM MECHANICS by GEORGE E. MASE, Schaum s Outlnes) Knematcs of Fluds Last class, we started dscussng about the nematcs of fluds. Recall the Lagrangan and Euleran
More informationMoments of Inertia. and reminds us of the analogous equation for linear momentum p= mv, which is of the form. The kinetic energy of the body is.
Moments of Inerta Suppose a body s movng on a crcular path wth constant speed Let s consder two quanttes: the body s angular momentum L about the center of the crcle, and ts knetc energy T How are these
More informationLecture 20: Noether s Theorem
Lecture 20: Noether s Theorem In our revew of Newtonan Mechancs, we were remnded that some quanttes (energy, lnear momentum, and angular momentum) are conserved That s, they are constant f no external
More informationEN40: Dynamics and Vibrations. Homework 7: Rigid Body Kinematics
N40: ynamcs and Vbratons Homewor 7: Rgd Body Knematcs School of ngneerng Brown Unversty 1. In the fgure below, bar AB rotates counterclocwse at 4 rad/s. What are the angular veloctes of bars BC and C?.
More informationClassical Mechanics ( Particles and Biparticles )
Classcal Mechancs ( Partcles and Bpartcles ) Alejandro A. Torassa Creatve Commons Attrbuton 3.0 Lcense (0) Buenos Ares, Argentna atorassa@gmal.com Abstract Ths paper consders the exstence of bpartcles
More informationNovember 5, 2002 SE 180: Earthquake Engineering SE 180. Final Project
SE 8 Fnal Project Story Shear Frame u m Gven: u m L L m L L EI ω ω Solve for m Story Bendng Beam u u m L m L Gven: m L L EI ω ω Solve for m 3 3 Story Shear Frame u 3 m 3 Gven: L 3 m m L L L 3 EI ω ω ω
More informationLie Group Formulation of Articulated Rigid Body Dynamics
Le Group Formulaton of Artculated Rgd Body Dynamcs Junggon Km 11/9/2012, Ver 1.01 Abstract It has been usual n most old-style text books for dynamcs to treat the formulas descrbng lnearor translatonal
More informationSolution of Linear System of Equations and Matrix Inversion Gauss Seidel Iteration Method
Soluton of Lnear System of Equatons and Matr Inverson Gauss Sedel Iteraton Method It s another well-known teratve method for solvng a system of lnear equatons of the form a + a22 + + ann = b a2 + a222
More informationTHE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructions
THE VIBRATIONS OF MOLECULES II THE CARBON DIOXIDE MOLECULE Student Instructons by George Hardgrove Chemstry Department St. Olaf College Northfeld, MN 55057 hardgrov@lars.acc.stolaf.edu Copyrght George
More informationPhysics 5153 Classical Mechanics. D Alembert s Principle and The Lagrangian-1
P. Guterrez Physcs 5153 Classcal Mechancs D Alembert s Prncple and The Lagrangan 1 Introducton The prncple of vrtual work provdes a method of solvng problems of statc equlbrum wthout havng to consder the
More informationCanonical transformations
Canoncal transformatons November 23, 2014 Recall that we have defned a symplectc transformaton to be any lnear transformaton M A B leavng the symplectc form nvarant, Ω AB M A CM B DΩ CD Coordnate transformatons,
More information11. Dynamics in Rotating Frames of Reference
Unversty of Rhode Island DgtalCommons@URI Classcal Dynamcs Physcs Course Materals 2015 11. Dynamcs n Rotatng Frames of Reference Gerhard Müller Unversty of Rhode Island, gmuller@ur.edu Creatve Commons
More informationENGI9496 Lecture Notes Multiport Models in Mechanics
ENGI9496 Moellng an Smulaton of Dynamc Systems Mechancs an Mechansms ENGI9496 Lecture Notes Multport Moels n Mechancs (New text Secton 4..3; Secton 9.1 generalzes to 3D moton) Defntons Generalze coornates
More information10. Canonical Transformations Michael Fowler
10. Canoncal Transformatons Mchael Fowler Pont Transformatons It s clear that Lagrange s equatons are correct for any reasonable choce of parameters labelng the system confguraton. Let s call our frst
More informationModeling of Dynamic Systems
Modelng of Dynamc Systems Ref: Control System Engneerng Norman Nse : Chapters & 3 Chapter objectves : Revew the Laplace transform Learn how to fnd a mathematcal model, called a transfer functon Learn how
More informationDARWIN-OP HUMANOID ROBOT KINEMATICS. Robert L. Williams II, Ph.D., Mechanical Engineering, Ohio University, Athens, Ohio, USA
Proceedngs of the ASME 1 Internatonal Desgn Engneerng Techncal onferences & omputers and Informaton n Engneerng onference IDET/IE 1 August 1-15, 1, hcago, IL, USA DET1-765 DARWIN-OP HUMANOID ROBOT KINEMATIS
More informationCHAPTER 6. LAGRANGE S EQUATIONS (Analytical Mechanics)
CHAPTER 6 LAGRANGE S EQUATIONS (Analytcal Mechancs) 1 Ex. 1: Consder a partcle movng on a fxed horzontal surface. r P Let, be the poston and F be the total force on the partcle. The FBD s: -mgk F 1 x O
More informationSection 8.3 Polar Form of Complex Numbers
80 Chapter 8 Secton 8 Polar Form of Complex Numbers From prevous classes, you may have encountered magnary numbers the square roots of negatve numbers and, more generally, complex numbers whch are the
More informationTransfer Functions. Convenient representation of a linear, dynamic model. A transfer function (TF) relates one input and one output: ( ) system
Transfer Functons Convenent representaton of a lnear, dynamc model. A transfer functon (TF) relates one nput and one output: x t X s y t system Y s The followng termnology s used: x y nput output forcng
More informationMEEM 3700 Mechanical Vibrations
MEEM 700 Mechancal Vbratons Mohan D. Rao Chuck Van Karsen Mechancal Engneerng-Engneerng Mechancs Mchgan echnologcal Unversty Copyrght 00 Lecture & MEEM 700 Multple Degree of Freedom Systems (ext: S.S.
More informationwhere v means the change in velocity, and t is the
1 PHYS:100 LECTURE 4 MECHANICS (3) Ths lecture covers the eneral case of moton wth constant acceleraton and free fall (whch s one of the more mportant examples of moton wth constant acceleraton) n a more
More informationSpin-rotation coupling of the angularly accelerated rigid body
Spn-rotaton couplng of the angularly accelerated rgd body Loua Hassan Elzen Basher Khartoum, Sudan. Postal code:11123 E-mal: louaelzen@gmal.com November 1, 2017 All Rghts Reserved. Abstract Ths paper s
More informationPHYS 705: Classical Mechanics. Calculus of Variations II
1 PHYS 705: Classcal Mechancs Calculus of Varatons II 2 Calculus of Varatons: Generalzaton (no constrant yet) Suppose now that F depends on several dependent varables : We need to fnd such that has a statonary
More information10/23/2003 PHY Lecture 14R 1
Announcements. Remember -- Tuesday, Oct. 8 th, 9:30 AM Second exam (coverng Chapters 9-4 of HRW) Brng the followng: a) equaton sheet b) Calculator c) Pencl d) Clear head e) Note: If you have kept up wth
More informationHow Differential Equations Arise. Newton s Second Law of Motion
page 1 CHAPTER 1 Frst-Order Dfferental Equatons Among all of the mathematcal dscplnes the theory of dfferental equatons s the most mportant. It furnshes the explanaton of all those elementary manfestatons
More information2D Motion of Rigid Bodies: Falling Stick Example, Work-Energy Principle
Example: Fallng Stck 1.003J/1.053J Dynamcs and Control I, Sprng 007 Professor Thomas Peacock 3/1/007 ecture 10 D Moton of Rgd Bodes: Fallng Stck Example, Work-Energy Prncple Example: Fallng Stck Fgure
More informationAP Physics 1 & 2 Summer Assignment
AP Physcs 1 & 2 Summer Assgnment AP Physcs 1 requres an exceptonal profcency n algebra, trgonometry, and geometry. It was desgned by a select group of college professors and hgh school scence teachers
More informationPhysics 40 HW #4 Chapter 4 Key NEATNESS COUNTS! Solve but do not turn in the following problems from Chapter 4 Knight
Physcs 40 HW #4 Chapter 4 Key NEATNESS COUNTS! Solve but do not turn n the ollowng problems rom Chapter 4 Knght Conceptual Questons: 8, 0, ; 4.8. Anta s approachng ball and movng away rom where ball was
More informationτ rf = Iα I point = mr 2 L35 F 11/14/14 a*er lecture 1
A mass s attached to a long, massless rod. The mass s close to one end of the rod. Is t easer to balance the rod on end wth the mass near the top or near the bottom? Hnt: Small α means sluggsh behavor
More informationON MECHANICS WITH VARIABLE NONCOMMUTATIVITY
ON MECHANICS WITH VARIABLE NONCOMMUTATIVITY CIPRIAN ACATRINEI Natonal Insttute of Nuclear Physcs and Engneerng P.O. Box MG-6, 07725-Bucharest, Romana E-mal: acatrne@theory.npne.ro. Receved March 6, 2008
More informationProblem Points Score Total 100
Physcs 450 Solutons of Sample Exam I Problem Ponts Score 1 8 15 3 17 4 0 5 0 Total 100 All wor must be shown n order to receve full credt. Wor must be legble and comprehensble wth answers clearly ndcated.
More informationSo far: simple (planar) geometries
Physcs 06 ecture 5 Torque and Angular Momentum as Vectors SJ 7thEd.: Chap. to 3 Rotatonal quanttes as vectors Cross product Torque epressed as a vector Angular momentum defned Angular momentum as a vector
More informationPhysics 2A Chapter 3 HW Solutions
Phscs A Chapter 3 HW Solutons Chapter 3 Conceptual Queston: 4, 6, 8, Problems: 5,, 8, 7, 3, 44, 46, 69, 70, 73 Q3.4. Reason: (a) C = A+ B onl A and B are n the same drecton. Sze does not matter. (b) C
More informationROTATIONAL MOTION. dv d F m m V v dt dt. i i i cm i
ROTATIONAL MOTION Consder a collecton of partcles, m, located at R relatve to an nertal coordnate system. As before wrte: where R cm locates the center of mass. R Rcm r Wrte Newton s second law for the
More information= 1.23 m/s 2 [W] Required: t. Solution:!t = = 17 m/s [W]! m/s [W] (two extra digits carried) = 2.1 m/s [W]
Secton 1.3: Acceleraton Tutoral 1 Practce, page 24 1. Gven: 0 m/s; 15.0 m/s [S]; t 12.5 s Requred: Analyss: a av v t v f v t a v av f v t 15.0 m/s [S] 0 m/s 12.5 s 15.0 m/s [S] 12.5 s 1.20 m/s 2 [S] Statement:
More informationLecture 23: Newton-Euler Formulation. Vaibhav Srivastava
Lecture 23: Newton-Euler Formulaton Based on Chapter 7, Spong, Hutchnson, and Vdyasagar Vabhav Srvastava Department of Electrcal & Computer Engneerng Mchgan State Unversty Aprl 10, 2017 ECE 818: Robotcs
More informationDifference Equations
Dfference Equatons c Jan Vrbk 1 Bascs Suppose a sequence of numbers, say a 0,a 1,a,a 3,... s defned by a certan general relatonshp between, say, three consecutve values of the sequence, e.g. a + +3a +1
More informationCOMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD
COMPOSITE BEAM WITH WEAK SHEAR CONNECTION SUBJECTED TO THERMAL LOAD Ákos Jósef Lengyel, István Ecsed Assstant Lecturer, Professor of Mechancs, Insttute of Appled Mechancs, Unversty of Mskolc, Mskolc-Egyetemváros,
More informationPhysics 5153 Classical Mechanics. Principle of Virtual Work-1
P. Guterrez 1 Introducton Physcs 5153 Classcal Mechancs Prncple of Vrtual Work The frst varatonal prncple we encounter n mechancs s the prncple of vrtual work. It establshes the equlbrum condton of a mechancal
More informationMathematics Intersection of Lines
a place of mnd F A C U L T Y O F E D U C A T I O N Department of Currculum and Pedagog Mathematcs Intersecton of Lnes Scence and Mathematcs Educaton Research Group Supported b UBC Teachng and Learnng Enhancement
More informationChapter 11 Angular Momentum
Chapter 11 Angular Momentum Analyss Model: Nonsolated System (Angular Momentum) Angular Momentum of a Rotatng Rgd Object Analyss Model: Isolated System (Angular Momentum) Angular Momentum of a Partcle
More information8.6 The Complex Number System
8.6 The Complex Number System Earler n the chapter, we mentoned that we cannot have a negatve under a square root, snce the square of any postve or negatve number s always postve. In ths secton we want
More informationPhysics 111: Mechanics Lecture 11
Physcs 111: Mechancs Lecture 11 Bn Chen NJIT Physcs Department Textbook Chapter 10: Dynamcs of Rotatonal Moton q 10.1 Torque q 10. Torque and Angular Acceleraton for a Rgd Body q 10.3 Rgd-Body Rotaton
More informationHow Strong Are Weak Patents? Joseph Farrell and Carl Shapiro. Supplementary Material Licensing Probabilistic Patents to Cournot Oligopolists *
How Strong Are Weak Patents? Joseph Farrell and Carl Shapro Supplementary Materal Lcensng Probablstc Patents to Cournot Olgopolsts * September 007 We study here the specal case n whch downstream competton
More informationRobot Modeling and Kinematics Errata R. Manseur. List of errors and typos reported as of 4/1/2007:
Robot Modelng and Knematcs Errata R. Manseur. Lst of errors and typos reported as of 4//7:. Fle CF4.wrl mentoned on page 45 s not on the CD. Fle ZYZ_Sm.wrl mentoned on page 67 s not on the CD. 3. Pg. 43.
More informationStudy Guide For Exam Two
Study Gude For Exam Two Physcs 2210 Albretsen Updated: 08/02/2018 All Other Prevous Study Gudes Modules 01-06 Module 07 Work Work done by a constant force F over a dstance s : Work done by varyng force
More informationLecture 12: Discrete Laplacian
Lecture 12: Dscrete Laplacan Scrbe: Tanye Lu Our goal s to come up wth a dscrete verson of Laplacan operator for trangulated surfaces, so that we can use t n practce to solve related problems We are mostly
More informationCHAPTER 10 ROTATIONAL MOTION
CHAPTER 0 ROTATONAL MOTON 0. ANGULAR VELOCTY Consder argd body rotates about a fxed axs through pont O n x-y plane as shown. Any partcle at pont P n ths rgd body rotates n a crcle of radus r about O. The
More informationPHYS 705: Classical Mechanics. Canonical Transformation II
1 PHYS 705: Classcal Mechancs Canoncal Transformaton II Example: Harmonc Oscllator f ( x) x m 0 x U( x) x mx x LT U m Defne or L p p mx x x m mx x H px L px p m p x m m H p 1 x m p m 1 m H x p m x m m
More informationPoisson brackets and canonical transformations
rof O B Wrght Mechancs Notes osson brackets and canoncal transformatons osson Brackets Consder an arbtrary functon f f ( qp t) df f f f q p q p t But q p p where ( qp ) pq q df f f f p q q p t In order
More informationIntroduction to Antennas & Arrays
Introducton to Antennas & Arrays Antenna transton regon (structure) between guded eaves (.e. coaxal cable) and free space waves. On transmsson, antenna accepts energy from TL and radates t nto space. J.D.
More informationNotes on Analytical Dynamics
Notes on Analytcal Dynamcs Jan Peters & Mchael Mstry October 7, 004 Newtonan Mechancs Basc Asssumptons and Newtons Laws Lonely pontmasses wth postve mass Newtons st: Constant velocty v n an nertal frame
More informationWork is the change in energy of a system (neglecting heat transfer). To examine what could
Work Work s the change n energy o a system (neglectng heat transer). To eamne what could cause work, let s look at the dmensons o energy: L ML E M L F L so T T dmensonally energy s equal to a orce tmes
More informationOpen Systems: Chemical Potential and Partial Molar Quantities Chemical Potential
Open Systems: Chemcal Potental and Partal Molar Quanttes Chemcal Potental For closed systems, we have derved the followng relatonshps: du = TdS pdv dh = TdS + Vdp da = SdT pdv dg = VdP SdT For open systems,
More informationStatistical mechanics handout 4
Statstcal mechancs handout 4 Explan dfference between phase space and an. Ensembles As dscussed n handout three atoms n any physcal system can adopt any one of a large number of mcorstates. For a quantum
More informationPHYS 1441 Section 002 Lecture #16
PHYS 1441 Secton 00 Lecture #16 Monday, Mar. 4, 008 Potental Energy Conservatve and Non-conservatve Forces Conservaton o Mechancal Energy Power Today s homework s homework #8, due 9pm, Monday, Mar. 31!!
More informationModule 9. Lecture 6. Duality in Assignment Problems
Module 9 1 Lecture 6 Dualty n Assgnment Problems In ths lecture we attempt to answer few other mportant questons posed n earler lecture for (AP) and see how some of them can be explaned through the concept
More informationCHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE
CHAPTER 5 NUMERICAL EVALUATION OF DYNAMIC RESPONSE Analytcal soluton s usually not possble when exctaton vares arbtrarly wth tme or f the system s nonlnear. Such problems can be solved by numercal tmesteppng
More informationIn this section is given an overview of the common elasticity models.
Secton 4.1 4.1 Elastc Solds In ths secton s gven an overvew of the common elastcty models. 4.1.1 The Lnear Elastc Sold The classcal Lnear Elastc model, or Hooean model, has the followng lnear relatonshp
More information5.04, Principles of Inorganic Chemistry II MIT Department of Chemistry Lecture 32: Vibrational Spectroscopy and the IR
5.0, Prncples of Inorganc Chemstry II MIT Department of Chemstry Lecture 3: Vbratonal Spectroscopy and the IR Vbratonal spectroscopy s confned to the 00-5000 cm - spectral regon. The absorpton of a photon
More informationScrew and Lie group theory in multibody kinematics
Multbody Syst Dyn DOI 10.1007/s11044-017-9582-7 Screw and Le group theory n multbody knematcs Moton representaton and recursve knematcs of tree-topology systems Andreas Müller 1 Receved: 10 Aprl 2016 /
More informationPhysics 141. Lecture 14. Frank L. H. Wolfs Department of Physics and Astronomy, University of Rochester, Lecture 14, Page 1
Physcs 141. Lecture 14. Frank L. H. Wolfs Department of Physcs and Astronomy, Unversty of Rochester, Lecture 14, Page 1 Physcs 141. Lecture 14. Course Informaton: Lab report # 3. Exam # 2. Mult-Partcle
More informationσ τ τ τ σ τ τ τ σ Review Chapter Four States of Stress Part Three Review Review
Chapter Four States of Stress Part Three When makng your choce n lfe, do not neglect to lve. Samuel Johnson Revew When we use matrx notaton to show the stresses on an element The rows represent the axs
More informationDesigning Information Devices and Systems II Spring 2018 J. Roychowdhury and M. Maharbiz Discussion 3A
EECS 16B Desgnng Informaton Devces and Systems II Sprng 018 J. Roychowdhury and M. Maharbz Dscusson 3A 1 Phasors We consder snusodal voltages and currents of a specfc form: where, Voltage vt) = V 0 cosωt
More informationModel of cyclotorsion in a Tendon Driven Eyeball: theoretical model and qualitative evaluation on a robotic platform
03 IEEE/RSJ Internatonal Conference on Intellgent Robots and Systems (IROS) November 3-7, 03. Tokyo, Japan Model of cyclotorson n a Tendon Drven Eyeball: theoretcal model and qualtatve evaluaton on a robotc
More informationPhysics 4B. Question and 3 tie (clockwise), then 2 and 5 tie (zero), then 4 and 6 tie (counterclockwise) B i. ( T / s) = 1.74 V.
Physcs 4 Solutons to Chapter 3 HW Chapter 3: Questons:, 4, 1 Problems:, 15, 19, 7, 33, 41, 45, 54, 65 Queston 3-1 and 3 te (clockwse), then and 5 te (zero), then 4 and 6 te (counterclockwse) Queston 3-4
More information