Lecture Topics. 6. Sensors and instrumentation 7. Actuators and power transmission devices. (System and Signal Processing) DR
|
|
|
- Alexandra Patrick
- 8 years ago
- Views:
Transcription
1 Lecture Tocs 1. Introducton 2. Basc knematcs 3. Pose measurement and Measurement of Robot Accuracy 4. Trajectory lannng and control 5. Forces, moments and Euler s laws 5. Fundamentals n electroncs and comutaton (System and Sgnal Processng) 6. Sensors and nstrumentaton 7. Actuators and ower transmsson devces DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
2 Knetc Robot dynamcs DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
3 Dervatves DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
4 Used Symbols Rotaton Matrx D R Identty Matrx E I DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
5 Dervatves I For the dervaton of a model to descrbe rgd-body dynamcs, t s of advantage to ntally analyse the quanttes veloctes and acceleratons n movng coordnate systems. Poston vectors descrbed n ther coordnate systems are subjected to changes n tme, whereas the relatons between them are exressed by tme varyng coordnate transformatons r D r t, wth O D R q q q q P q The tme-deendant rotaton matrx ossesses secal characterstcs, whch are gven secal attenton, snce they aear reeatedly when dervng the quanttes veloctes and acceleratons. DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
6 Dervatves II Pose q q r = D r + t P q O S r P P () t t() t () t Θ S q q t q O () t v() t t() t () t ω Θ() t Coordnate transformaton r D r t q q q P O r D r D r t q q q q P P O a() t t() t () t () t Θ() t () t t () t () t () t Θ() t DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
7 Dervatves III Orthonormalty crtera t -1 D D D D E t t D D -1 E t D D D D 0 t -1-1 t D D D D 0 t -1 1 D D D D -1 1 t t A B A B A B t t t t t t A B B A skew-symmetrc matrx DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
8 Angular velocty vector Current angular velocty tensor: z -1 ( ) z 0 Ω D D Ω ω ω 0 y x y x 0 ω t x y z D Ω D D DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
9 Snake-Oerator The ~-Oerator enables a comact wrtng-format for a cross roduct of matrx oeratons. The advantage of the matrx wth resect to the cross roduct notaton les wthn t s comatblty to matrx-calculus. Ω( ) r r r e e e x y z x y z r r r x y z r r 0 DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
10 Dervatves of hgher order I Ω D D D D Ω ω 0 y -1-1 ( ) z 0 x z y x 0 Commutablty DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
11 Dervatves of hgher order II D D D D D D D D D α α D D D D D D D D D D D D D D D D 2 D D D α β D 2 α D α D D DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
12 Dervatves of hgher order III t DD t α D D E t β D D 2 α α E DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
13 elocty DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
14 eloctes I r D r D r t q q q q P P O D ΩD q q q q q v, P Ω D rp D vp v O Rgd-body v 0 P v D r D v v q q q q q P P P O v r v v q q q q q P P P O DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
15 elocty II q q q q v v v r P P O P D := Geschwndgket elocty of ont des P relatve Punktes to P q-th relatv coordnate zum q-ten system Koordnatensystem := Geschwndgket elocty of ont des P relatve Punktes to P -th relatv coordnate zum -ten system Koordnatensystem := Geschwndgket elocty of orgn des of Ursrunges -th coordnate des system -ten Koordnate relatve to nsystems q-th coordnate relatv zum q-ten Koordnatensystem := Koordnaten Coordnates des of onts Punktes n m -th -ten coordnate Koordnatensystem := Rotatonsmatrx vom from -th -ten zum to q-th q-ten coordnate Koordnatensystem q Ω := Wnkelgeschwndgketstenso Angular velocty tensor from r -th vom to -ten q-th zum coordnate q-ten Koordnatensystem DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
16 Acceleraton DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
17 Acceleraton I v Ω D r D v v q q q q q P P P O q v Ω D r D v v t t q q q q q P P P O v Ω D r D v v t t t t q q q q P P P O a Ω D r Ω D r Ω D v q q q q q q q P P P P q q P q D v D a a O P DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
18 Acceleraton II Wth q q Ω Ω( ) and D Ω D q q q a Ω( ) D r Ω Ω D r Ω D a q q q q q q q q P P P P q q q P q Ω D v D a a O P Rgd-body vp 0, P a 0 DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
19 Acceleraton III classcal a D r D r D a q q q q q q q q P P P P D v D a q q q P q a O a r r a q q q q q q q q P P P P v q q q P q a O a P P DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
20 Acceleraton I q q q q a a a v r P P O P D Ω P := Beschleungung Acceleraton of ont des Punktes P relatve P relatv to q-th zum coordnate q-ten Koordnatensystem := Beschleungung Acceleraton of ont des Punktes P relatve P relatv to -th zum coordnate -ten Koordnatensystem := Beschleungung Acceleraton of orgn des Ursrunges of -th coordnate des -ten system Koordnatensy relatve to stems q-th relatv zum coordnate q-ten Koordnatensystem := Geschwndgket elocty of ont des P relatve Punktes to P -th relatv coordnate zum -ten system Koordnatensystem := Koordnaten Coordnates of des onts Punktes n -th m coordnate -ten Koordnatensystem := Rotatonsmatrx from vom -th en to zum q-th q-ten coordnate Koordnatensystem q := Angular Wnkelgeschwndgketstensor velocty tensor from -th vom to q-th -ten coordnate zum q-ten Koordnatensystem DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
21 Dynamcs Dfferental Newton-Euler-Equatons Knetc Energy Recursve Newton-Euler Equatons Jont Loads and Elastcty Generalzed Coordnates Knetc and Potental Energy Lagrange Equatons and Lagrange Equaton of Moton General Form of Equaton of Moton Soluton to lnearzed Equaton of Moton DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
22 Moment of nerta, angular momentum, ont mass and center of gravty DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
23 Robot Dynamcs What haens f the robot axes move? center of gravty and mass S 6D -th body-fxed coordnate system S Interta tensor 6D 6D The nature laws determne the comlexty! S 6D Pose Inertal- Frame 6D S 0 6D S -1 Inertal- Frame dm As smle as ossble, as comlcated as necessary (Easy to say and hard to do). DR Prof. Dr.-Ing. habl. Hermann Löddng PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. Wolfgang Hntze
24 olume Integral d = d z d y dx z z = h( x, y) ' 1 h( x, y) dz dx d y B g( x, y) dm d z = g( x, y) y B ' 1 h( x, y) g( x, y) dx d y ( xy, ) B 1 ' 1 h( x, y) m (x,y,z) dz dx d y B g( x, y) CAD DR PD Dr.-Ing. habl. Jörg Wollnack x m ( x, y, z ) z x y u v w u v w Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
25 Center of Gravty and Momentums Calculatng va dscreet volumes elements z = h( x, y) z = g( x, y) ( xy, ) B 1 DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
26 Moment of Inerta I Body d m = d F v P r r S S O r S S O DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
27 Moment of Inerta II Center of gravty n S 1 1 r r dm d, m m r S P P m d := Densty functon of -th Body := Boundary functon of body DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
28 Moment of nerta III Dfferental mass element dm d of rgd body moves wth the velocty v relatve to the ntal reference system S. P Dfferental momentum d v dm r dm P P Newton s 2 nd law d F d dt d DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
29 Moment of nerta I Forces can be categorzed nto nternal and external forces. Uon ntegratng all forces over the whole body, the nternal forces cancel out. The change n momentum s descrbed by the result of all external forces. d d d F F d r dm m r m v dt F m v m a S S Left-multlyng D, leads to F m v m a S S n the body fxed coordnate system. P P P DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
30 Angular momentum 1 The result s the angular momentum wth resect to ont : L r d r v q q q q O P P P dm q q q q The decomoston of the velocty vp vs r and q q q oston vector rp rs r of a general body movement leads to: ~ q q q LO rs r vp dm q q q q r S vp dm r vp dm O DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
31 Angular momentum 2 rather dm L r v r v r dm q q q q q q O P S q q q q q q q r v P r vs r r dm dm dm Part of angular momentum wth resect to dm r r v r v dm q q q q q q S S P S Part of angular momentum wth resect to I r r q q S dm O S by translaton by angular velocty q DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
32 Angular momentum 3 The nerta tensor I r r q q S descrbes the roertes of nerta of a rgd body wth resect to t s mass center n the q-th nertal coordnate system Due to movement, the tensor s subjected to changes n tme. dm A descrton of the nerta tensor n the body-fxed coordnate system follows wth the tensor transformaton: I D I D q t S q S q DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
33 Angular momentum 4 I S r r d m, 0 z wth r z 0 y x y x 0 0 z y 0 z y r r z 0 x z 0 x y x 0 y x y z x y x z x y 2 2 x z y z x z y z x y 2 2 DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
34 Angular momentum 5 I S 2 2 y z x y x z 2 2 x y x z y z dm 2 2 x z y z x y 2 2 y z dm x y dm x z dm 2 2 x y dm x z dm y z dm 2 2 x z dm y z dm x y dm DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
35 Angular momentum 6 Inerta tensor I I I I I I I I xz I yz I z x xy xz S xy y yz Moment of nerta x 2 2 d 2 2 I y x z dm 2 2 z I y z m Moment of devaton I x y dm I xy x y dm I xz x z dm I yz y z dm DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
36 Angular momentum 7 Transformaton n body-fxed coordnate system I D I D ' ' q ' t S S ' S Egen-Decomoston Egen-ectors det I I E 0, Prncal axes (axes of symmetry uon homogeneous mass dstrbuton) S IS Ik E wk 0, k{1,2,3} k k k, k {1,2,3} Inerta tensor n coordnate system of rncal axes ' I dag I I I S e w w DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
37 Angular momentum 8 Alyng the angular momentum, the relatonsh between torques actng on a rgd body and the tme varyng changes n angular momentum n the q-th nertal coordnate system can be descrbed: d dt L r d F r d F q q q q O S S Provded the mass center of the -th body s chosen as the reference ont, nternal forces aear, but comensate each other. Hence, the change n torque s ultmately determned by external forces: d d M S LS IS ω dt dt DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
38 Angular momentum 9 d d M L I ω dt dt S S S A B A B A B M I ω I ω S S S d I D I D D I D D I D D I D dt t t t t S S S S S Tme nvarance I S 0 I D I D D I D t t S S S DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
39 Angular momentum 10 I D I D D I D D ω D ω D D ω t t S S S I ω D I D D I D ω t t S S S I ω I I ω S S S M ω I I ω ω I ω S S S S ω I ω I ω ω I ω S S S ω ω 0 DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
40 Angular momentum 11 Angular momentum M ω I ω I ω S S S The tme-varyng nerta tensor can be determned usng the tmeconstant nerta tensor: Euler s equaton of gyroscoc moton I D I D q ' t ' ' S S DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
41 Angular momentum 12 Descrton n body-fxed coordnate system M ω I ω I ω S S S M D ω I ω I ω S S S D ω I ω D I ω S S M D M S S D ω t ~ MS D D ω IS D ω D IS D ω t t t M I ω ω I ω S S S DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
42 Angular momentum 13 Descrton n body-fxed coordnate system of rncal axes M M M M t M I I I S x y z, x x x y z y z M I I I y y y z x z x M I I I z z z x y x y DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
43 Generalzed momentum and forces I Dfferental momentum d rp d d dt Dfferental angular momentum d rp d h rp d dt Generalzed dfferental momentum and generalzed forces (Newton-Euler-Equaton) d rp d F d d d dt = d d M dtd h dt d r P rp dt DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
44 Generalzed momentum and forces II The dfferental Newton-Euler-Equatons descrbe the fundamental equatons necessary for the analyss of moton. The tme-varyng change n momentum s therefore n drect relaton wth the generalzed forces. The transton to macroscoc bodes s erformed by ntegratng over the dfferental volume elements of the body. Hereby the densty functon, the body volume and the ntegraton boundares may be tme-varyng. Consderng a rgd body wth constant mass, the densty functon and the body volume are tme-nvarant. The descrton of the generalzed forces follows n the -th nertal coordnate system wth resect to t s orgn O. DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
45 Mass center, ont mass and momentum I d d r dt d r dt P P d d d( rs r) d rs d r d d d dt dt dt d S d dt r d r dt S d d dt r d Integraton r r r P S Rgd body Tme-nvarant Mass DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
46 Mass center, ont mass and momentum II d r dt S d Tme-nvarant coordnate of mass center zetnvarante Schwerunktkoordnate and body geometry und Körergeometre d r dt S 0 d r S d dt m v S? The result shows that the momentum of a rgd body wth tmenvarant mass can be descrbed by the concet of mass center vector and ont mass. DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
47 Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze PD Dr.-Ing. habl. Jörg Wollnack DR Mass center, ont mass, nerta tensor and angular momentum I P P d d d d t r h r Integraton P P d d d t r h r P S r r r ~ S S d d d t r r h r r S S ~ S S d d d d d d d d d d d d t t t t r r h r r r r r r Decomoston d r 0
48 Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze PD Dr.-Ing. habl. Jörg Wollnack DR Mass center, ont mass, nerta tensor and angular momentum II S S d d d d d d t t r h r r r S zetnvarante Schwerunktkoordnate und Körergeometre d 0 d t r S S d d d d d d t t r r h r r d m S S d d d m t r h r r r Tme-nvarant coordnate of mass center and body geometry
49 Mass center, ont mass, nerta tensor and angular momentum III d r h r r r d S m S dt? The result shows that the angular momentum of a rgd body wth tme-nvarant mass can be descrbed wth the concet of the mass center, ont mass and the nerta tensor. DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
50 Motvaton If we change the ose of the coordnate system, how change the momentum values? Does an coordnate system exst where the descrtons are not so comlex? Whch mact have symmetres? (mass and geometry) DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
51 Transformaton of nerta tensor DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
52 The nerta tensor Transformaton of nerta tensor I d r h r r r d S m S dt can ultmately be descrbed n an arbtrary coordnate system. Hereby I defnes the nerta vector wth resect to the mass center of the -th rgd body n the q-th coordnate system. For the two coordnate systems and j follows: h I h I j j j D j t j h D h j j I D I j j j DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
53 Transformaton of nerta tensor II I D I j j j D j t j I D I D j t j j Tensor transformaton I D I D j t j j DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
54 Parallel Axs Theorem (Stener) DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
55 Parallel axs theorem I F d m = d ρ v Translaton from O to O mantanng constant orentaton r P r S I r r O dm S O r r r S O r S S r r r S DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
56 Parallel axs theorem II I r r r r dm Multlcaton O S S I r r r r r r r r O S S S S S S S S dm r r dm r r dm r r dm r r dm I r r dm r r dm r dm r r r dm O S S S S I r r dm r r O S S dm r 1 S S m r dm 0 DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
57 Parallel axs theorem III I r r dm r r dm IS o S S dm r r I I r r dm O S S S DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
58 Robot Dynamcs r m S P m S 1 6D Pose r d Inertal- Frame dm center of gravty and mass S 6D S 0 6D 6D S -1 Inertal- Frame -th body-fxed coordnate system S Interta tensor dm 6D 6D I r r q q S I I r r dm O S S S j t I D j I D j M ω I ω I ω S S S dm DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
59 Rotaton Matrx and Tlde Oerator ~ r' D r t r' D r D r t t -1 D D D D E t -1 1 D D D D DD -1 0 z y ω z 0 x y x 0 D D r ' D r D r t I r r dm DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
60 Knetc Energy DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
61 Knetc Energy I Alyng the conservaton of energy rncle, mostly the dfferent system models can be couled va scalar quanttes, thus resultng n an overall descrton. Hence, the analyss of the knetc energy stored n a rgd body s useful. Hereby t becomes obvous that a descrton wth resect to the mass center of the rgd body makes sense. z = h( x, y) E 1 2 m v 2 d E v dm z ( xy, ) B 1 = g( x, y) d E 1 q t q vp v 2 P d DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
62 Knetc Energy II d E 1 q t q vp vp d Integraton 2 1 q t q E P P d 2 v v q q q q v v r P S 1 ( q q q t S ) ( q q q v r vs r ) d 2 E 1 1 v v d v r d q t q q t q q S S S q q t q 1 q q t q q S d r v r r d 2 2 DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
63 Knetc Energy III The thrd and fourth term dsaear, snce these comly wth the defnton of the mass center. 1 1 E v v r r d 2 2 q t q q q t q q S S d q t q q t q t q q S S A B B A t t t 1 1 v v d r r d a b b a v v d r r 2 2 q t q q t q t q q S S d 1 1 v v d r r d 2 2 q t q q t q t q q S S q q v S const const d d DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
64 Knetc Energy I E 1 1 v m v I 2 2 q t q q t q q S S The total knetc energy results from the sueroston of the knetc energy of the ont mass n the mass center and the knetc rotaton energy of the nerta tensor wth resect to the mass center. DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
65 Recursve Newton-Euler Equatons DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
66 Recursve Newton-Euler Equatons I The relmnary consderatons have shown that, aart from the already exstng coordnate systems, further coordnate systems are needed for the analyss of the dynamcs of a rgd body system formed by jonts. Ths s the body fxed coordnate system formed by the mass centrod axs, where by a constant dagonal nerta tensor exsts. Furthermore, these are body fxed coordnate systems n whch hyscal quanttes are derved or measured. The analyss of the dynamcs s erformed assumng coordnate system of the bodes, as well as other body related rncal axs and sensor coordnate systems. Hereby S k s the descrbng nertal coordnate system, S H the rncal axs coordnate system, S K or S the body or jont coordnate systems resectvely and S B the measurement or observer coordnate system. DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
67 Recursve Newton-Euler Equatons II 6D 6D -ter -th Körer body T S 6D k ω S H k v S 6D S K -1 T H S -1 O -1 m S + I k r OS S K k k F M, -1, -1 k r O-1 k r S k g k r O S k k r O F M k k, +1, +1 DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
68 Recursve Newton-Euler Equatons III In ractce t s arorate to exress the further coordnate systems or the oses needed for the analyss of the dynamcs wth resect to the body coordnate systems. Thus, the followng oses are needed for the bodes: H 6 B 6 := Pose from rncal axs to body coordnate system := Pose from measurement to body coordnate system 1 6 K K K 1 := Pose from ( 1) to the -th body coordnate system Hence, the nerta tensor of the -th rgd body s reduced to dagonal form wth three arameters I 1, I 2 and I 3. The gravtatonal vector k g s gven n the nertal coordnate system S k. DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
69 Recursve Newton-Euler Equatons I Inertal generalzed forces F * S m v S M I ω ω I ω * S S S Equlbrum of forces and torques n the -th body coordnate system F F F m g 0 * S, 1, 1 ~ ~ * S, 1, 1 ( OS ), 1 OS 1,1 M M M r r F r F 0 The negatve sgn s a result of Newton s 2nd law. The equlbrum of forces leads the dynamc analyss to the rncle of equlbrum of the statc analyss, thus resultng n the negatve sgn for the dynamc terms. DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
70 Recursve Newton-Euler Equatons Generalzed forces of the (+1)-th rgd body F F F m g 0 * S, 1 1, ~ ~ * S, 1 1, ( OS ), 1 OS 1,1 M M M r r F r F 0 Accordng to the defnton of torque M rf, the vector sum of all lever arms must be couled wth the force vector n ooste drecton, hence resultng n a negatve sgn. From Newton s 3rd law follows: t t t t t F M F M, q q, t DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
71 Recursve Newton-Euler Equatons I S K -2 slt + k k F M -1-1 k k F S M S -1 S K -th Body T + +1= S k r T k r S k k F M S K k k F M slt S +1 S S w S k -th Lnk k g ( +1)-th Lnk k k F M +... S K +1 DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
72 Isolatng the (-1)-th generalzed force results n the recursvely descendng transformaton: * F,, 1 F 1, m g FS g D g k k ~ ~ * M, 1 M 1, ( r ros ) F, 1 ros F 1, MS Generalzed forces n the (-1)-th jont coordnate system F Recursve Newton-Euler Equatons II D F 1 1, 1, 1 M D M 1 1, 1, 1 The systems of equatons can be calculated from the endeffector recursvely. Provded the generalzed veloctes cancel out, these equatons are dentcal to the ones for the statc analyss. DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
73 Jont Loads and Elastcty DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
74 Jont loads and elastcty I The generalzed jont loads are the rojectons of the generalzed forces onto the corresondng jont axs. Hereby t s necessary to dfferentate between rotaton and lnear axes: M t t F e e wth { x, y, z} for rotaton axes for lnear axes Torson srng srng The rojectons of the generalzed forces to whch no actuator drven degree of freedom exst lead to jont loads. Generally, the generalzed force vector s decomosed nto jont and drve forces. The drve forces are generally known as actve forces. DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
75 Jont loads and elastcty II 6D 6D Rgd Body S 6D 6D Torson Srng S -1 DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
76 Jont loads and elastctes III At ths level the dfferental equatons of the drves can be ntegrated relatvely smle nto the system descrton. For ths urose, the equlbrum of forces A 0 can be used, whereby defnes the drve force. A The ntegraton of damng factors over M e d t D t F e d t for rotaton axes for lnear axes s also ossble, enablng a descrton of the mechancs as well as drves n the sense of a mechatronc system. DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
77 Jont loads and elastctes I These equatons can be used to descrbe the generalzed statc and dynamc forces of an oen knematc chan. In connecton wth the elastcty model, the dstortons of the seral robot model under the nfluence of statc and dynamc generalzed forces can be modeled. Ths allows the mrovement of the control of the statc and dynamc ose behavour. The advantage of ths s the ablty to dentfy the arameters of the elastcty model by statc measurement. Ths leads to the fact that dentfcaton of statc and dynamc arameters can be searated, whch favours the generally used mnmzaton rocedures. DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
78 Generalzed Coordnates DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
79 Generalzed coordnates I Sx locaton coordnates of a ose are necessary to descrbe the movement of a rgd body. Gven N rgd bodes, 6 N ose coordnates are needed for the locaton descrton. The overall system thus has 6 N degrees of freedom. In the resence of mechancal lnks between bodes or the envronment, the number of degrees of freedom reduces deendng on the tye of jont. Every rohbted degree of freedom results n generalzed reacton forces n the lnk. Hereby the constructve Desgn of the lnk determnes the number of geometrcal restrctons. These are usually tme-constant. DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
80 Generalzed coordnates II Aart from these geometrcal restrctons, other constrants may occur, whch defne coulngs between the generalzed veloctes (translatonal and angular veloctes). If the veloctes can be ntegrated, they can be dealt wth as geometrc restrctons or constrants. These constrants are called holonomc constrants. Generally, tme-varyng constrants may also be assumed, whch are then called rheonomc constrants. DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
81 Generalzed coordnates III Locaton vector of generalzed coordnate t... N q q, 1 qn N B f Locaton descrton w w w r (, ) O ro q t D w t D ( q, ) 3 33 B rgd bodes wth f degrees of freedom The bound generalzed coordnates dsaear uon dfferentatng, thus enablng the descrton of a system wth f degrees of freedom. w ndcates the world coordnate system (CS). The q as a suerscrt may lead to confuson wth the elements of the generalzed coordnate. DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
82 Generalzed coordnates elocty behavour I w w (, ) 3 ro ro q t total dfferental O N w w Ok r r O k k1 qk t w d r ( q, t) ( q, t) d q dt dervatve w d r ( q, t) r d r vo ( q, t) ( q, t) dt q dt t w N w w O O qk O k1 k N w w O r O ( q, t) qk r k1 qk t DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
83 Generalzed coordnates elocty behavour II w Matrx notaton r v q q q w w (, ) O O t r t O w w r O r w w O 3N JT q, JT, { x, y, z} t q k DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
84 Generalzed coordnates elocty behavour III w D w t D ( q, ) 33 Matrx notaton D w w w ω ( q, t) q q d t d w w w d w 3N JR q, JR, { x, y, z} t q k DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
85 Generalzed coordnates elocty behavour I w q D k w D w d q k w D t w D w d t Elements are column-vectors of the Jacobean matrx, whch exresses the angular veloctes assumng tme-varyng constrants. Gven a multbody system wth tme-constant (holonomc constrants), ths vector vanshes. DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
86 Generalzed coordnates Acceleraton behavour I w w w ro r ao( q, t) vo( q, t) q t t q t w w w (, ) r r r ao q t q q t q q t t t O O O O w r O w w q JT q v t q v a q v q J q q v q w w w O w O (, t) O(, t) T O O Sum and roduct rule w w ro r t q q t w w ro v t q q O O DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
87 Generalzed coordnates Acceleraton behavour II d t w w w ω( q, t) JR q w w w w w ( q, t) ω( q, t) JR q q q DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
88 Knetc Energy DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
89 Knetc Energy I E 1 1 v m v I 2 2 t t S S I D I D w w w t I w The nerta tensor s unlke tme-nvarant I 1 1 E v m v D I D 2 2 t t w w t S S DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
90 Knetc Energy II Generalzed velocty vector w w vs Jv Jω q E 1 2 t m t w w w w w t w JvS q JvS q Jω q D I D Jω q I 1 2 t w t w t w t w w t w q JvS m JvS q q Jω D I D Jω q I 1 2 t w t w t w t w w t w q JvS m JvS q q Jω D I D Jω q I DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
91 Knetc Energy III E 1 2 t w t w t w t w w t w q JvS m JvS q q Jω D I D Jω q I 1 2 t w E q M q M J m J q q J D I D J w w t w t w t w w t w vs vs ω ω I DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
92 Knetc Energy I The nertal mass matrx contans the Jacobean matrx of the oen unbranched knematc chan, hence t beng deendant of the confguraton. Ths matrx s symmetrc and ostve defnte. The quadratc form of the matrx results n the knetc energy always beng ostve. Elsewse a wrong system model s at hand. DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
93 Potental Energy DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
94 Potental Energy The otental energy of the oen unbranched knematc chan can be descrbed by the sum of all otental energes of the rgd bodes: E P w t w m g rs DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
95 Lagrange Equatons DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
96 Energy W =? F δr r δw F t δr r + δr DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
97 Lagrange Equatons I Accordng to the d Alembert s rncal, the dfferental work of all mass onts m, r over the vrtual moton r vansh. t δ δw m r F δr 0 Hereby only such movements satsfyng the constrants should be ermtted. In ths way, the decomoston nto reacton, drve and effectve force can be avoded. Usng generalzed coordnates the movement can smly be selected: r δr δq q DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
98 Lagrange Equatons II t δw m r F δr 0 t v t r δw m r F δq q q Wanted: r t v q v t v q r v v δr q q q δq Dervatve of velocty squared t 2 t v t v t v v v v v v 2 v q q q q q DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
99 Lagrange Equatons I t 2 t v t v t v v v v v v 2 v q q q q q v q r 2 t 2 v q r q v q 2 t r v 2 v t q t q DR PD Dr.-Ing. habl. Jörg Wollnack t 2 t v A B A B A B v r r r r v t q t q t q q t q r v t t 2 v v q q Term Isolatng 2nd term Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
100 Lagrange Equatons v 1 v v v v q 2 t q q t 2 t t δw m r F δr t t r δw m v v v F δq 2 t q q q Commutatve summaton and dfferentaton oerators t t δw m v m v v F r δq t q 2 q 2 q t m v m v F r δq t q 2 q 2 q DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
101 Lagrange Equatons I Knetc energy of mass ont 1 T m v 2 2 Total knetc energy of system T T r t δw T T F δq 0 t q q q Snce the vrtual dslacements are ndeendent of each other, the equaton can only be satsfed for vrtual dslacements, f the followng necessary condton s fulflled: T T Q Q F t k kk{1,..., f } k 0, k k t qk qk qk r DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
102 Lagrange Equaton of Moton DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
103 Lagrange Equaton of Moton I Lagrange equaton of moton L L Qk, L EK E t q q k k Dfference of knetc and otental energy E K und E P defned resectvely. P DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
104 Lagrange Equaton of Moton II In a multbody system wth rgd body comonents, nstead of mass dfferental mass elements and nstead of sum ntegrals over rgd body volumes, one obtans: T 1 2 m v I 2 t k k Sk k Sk k k The generalzed forces actng on a rgd body consttute of forces F k and torques M k. Snce the reacton forces and torques do not contrbute to the vrtual work done, the can be neglected, thus resultng n: t rk t sk Qk Fek Mek q q k k k DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
105 Lagrange Equaton of Moton III The actve forces can consst of conservatve and nonconservatve forces. Whle non-conservatve forces (e.g. frcton, flow forces) alter the total energy, conservatve forces do not result n nterchange between knetc and otental energy. DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
106 General Form of Equaton of Moton DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
107 General Form of Equaton of Moton I Knetc and otental energy Partal dervatve w.r.t. 1 L m q q m w t w j j S q q 2, j 1 2, j L g r m q q m g r m q j j w t w j j S q q j q t w t w q M q m g rs m q q m g r, j w t w j j S q w t w m g rs 0 DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
108 General Form of Equaton of Moton II Partal dervatve w.r.t. t L m j q j t q j t q m m q t t j j j j j j q m m q j j j j t j t m j k m q j k q k m L q q m q t q q j j k j j j k k j mj q k q j m j q j j, k qk j DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
109 General Form of Equaton of Moton III 1 L m q q m q q q g r w t w j j k Sk 2, j k 1 2 m q q g r, j j w t w j mk Sk q k q Snce the artal dervatve of the mass center vector s dentcal to the Jacobean matrx, t follows: q 1 m L q q m g J 2, j j w t w j k k q k DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
110 General Form of Equaton of Moton I L L Q t q q k k k mj L q q m q t q q j, k 1 m L q q m g J q 2, j k k j j j j j w t w j k k q k m 1 m g j j w t w qk q j m j q j q q j mk J k Q j, k qk j 2, j q k m 1 m j j w t w m j q j qk q j q q j mk J k Q j j, k qk 2, j q k g DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
111 General Form of Equaton of Moton m 1 m g j j w t w m j q j qk q j mk J k Q j j, k qk 2 q k The frst term of ths equaton defnes the nertal forces, the second term reresents the Corols and centrfugal forces, whereas the thrd term descrbes the effects to gravty. The balance has to result n the generalzed force of the corresondng generalzed coordnate n the sense of ths aquvalence statement. DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
112 General Form of Equaton of Moton I m 1 m g j j w t w m j q j qk q j mk J k Q j j, k qk 2 q k Mq G Q Sorted by forces and torque usng the Jacobean matrces M( q) q h( q, q) k g( q), drve M M J M dag{ E,..., E, I,..., I } m1 m 1,...,,,..., 1 1 J J J J J t t t t T T R R DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
113 Soluton to lnearzed Equaton of Moton DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
114 Soluton to Equaton of Moton I Equaton of moton wth f degrees of freedom M y D y K y f det M y M D y M K y M f Searatng y y M D y M K y M f Introducng of states y y 0 E y y M K M D y M f z A z f z DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
115 Soluton to Equaton of Moton II d z dt A z fz d z A z dtf z dt Total dfferental Integraton z() t t t d z A z dt f dt z t t z t t z( t) A z dt fz dt z t 0 0 t 0 DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
116 Soluton to Equaton of Moton III Lnear aroxmaton z( t ) z( t ) z( t ) ( t t ) R z A z f z z( t ) z( t ) A( t ) z( t ) f ( t ) ( t t ) R( t, t ) z Intal condton z( t ) z 0 0 s known Recurson z( t ) z( t 1) A( t 1) z( t 1) f z ( t 1) ( t t 1), {1,..., I} DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
117 Soluton to Equaton of Moton I The tme-dfferences t t 1 are to be chosen so the resdual terms R( t, t 1) become suffcently small to cancel out. An mrovement of the numercal ntegraton can be acheved usng the Runge-Kutta-method. Hereby the tme-nterval s beng subdvded to acheve a better redcton of the followng state. DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
118 END DR PD Dr.-Ing. habl. Jörg Wollnack Prof. Dr.-Ing. habl. Hermann Löddng Prof. Dr.-Ing. Wolfgang Hntze
NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING. Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582
NMT EE 589 & UNM ME 482/582 ROBOT ENGINEERING Dr. Stephen Bruder NMT EE 589 & UNM ME 482/582 7. Root Dynamcs 7.2 Intro to Root Dynamcs We now look at the forces requred to cause moton of the root.e. dynamcs!!
Rotation Kinematics, Moment of Inertia, and Torque
Rotaton Knematcs, Moment of Inerta, and Torque Mathematcally, rotaton of a rgd body about a fxed axs s analogous to a lnear moton n one dmenson. Although the physcal quanttes nvolved n rotaton are qute
where the coordinates are related to those in the old frame as follows.
Chapter 2 - Cartesan Vectors and Tensors: Ther Algebra Defnton of a vector Examples of vectors Scalar multplcaton Addton of vectors coplanar vectors Unt vectors A bass of non-coplanar vectors Scalar product
Goals Rotational quantities as vectors. Math: Cross Product. Angular momentum
Physcs 106 Week 5 Torque and Angular Momentum as Vectors SJ 7thEd.: Chap 11.2 to 3 Rotatonal quanttes as vectors Cross product Torque expressed as a vector Angular momentum defned Angular momentum as a
Politecnico di Torino. Porto Institutional Repository
Poltecnco d orno Porto Insttutonal Repostory [Proceedng] rbt dynamcs and knematcs wth full quaternons rgnal Ctaton: Andres D; Canuto E. (5). rbt dynamcs and knematcs wth full quaternons. In: 16th IFAC
Lagrangian Dynamics: Virtual Work and Generalized Forces
Admssble Varatons/Vrtual Dsplacements 1 2.003J/1.053J Dynamcs and Control I, Sprng 2007 Paula Echeverr, Professor Thomas Peacock 4/4/2007 Lecture 14 Lagrangan Dynamcs: Vrtual Work and Generalzed Forces
1 What is a conservation law?
MATHEMATICS 7302 (Analytcal Dynamcs) YEAR 2015 2016, TERM 2 HANDOUT #6: MOMENTUM, ANGULAR MOMENTUM, AND ENERGY; CONSERVATION LAWS In ths handout we wll develop the concepts of momentum, angular momentum,
Rotation and Conservation of Angular Momentum
Chapter 4. Rotaton and Conservaton of Angular Momentum Notes: Most of the materal n ths chapter s taken from Young and Freedman, Chaps. 9 and 0. 4. Angular Velocty and Acceleraton We have already brefly
Efficient Computation of Optimal, Physically Valid Motion
Vol. xx No. xx,.1 5, 200x 1 Effcent Comutaton of Otmal, Physcally Vald Moton Anthony C. Fang 1 and Nancy S. Pollard 2 1 Deartment of Comuter Scence, Natonal Unversty of Sngaore 2 Robotcs Insttute, Carnege
Inertial Field Energy
Adv. Studes Theor. Phys., Vol. 3, 009, no. 3, 131-140 Inertal Feld Energy C. Johan Masrelez 309 W Lk Sammamsh Pkwy NE Redmond, WA 9805, USA jmasrelez@estfound.org Abstract The phenomenon of Inerta may
Homework: 49, 56, 67, 60, 64, 74 (p. 234-237)
Hoework: 49, 56, 67, 60, 64, 74 (p. 34-37) 49. bullet o ass 0g strkes a ballstc pendulu o ass kg. The center o ass o the pendulu rses a ertcal dstance o c. ssung that the bullet reans ebedded n the pendulu,
Recurrence. 1 Definitions and main statements
Recurrence 1 Defntons and man statements Let X n, n = 0, 1, 2,... be a MC wth the state space S = (1, 2,...), transton probabltes p j = P {X n+1 = j X n = }, and the transton matrx P = (p j ),j S def.
v a 1 b 1 i, a 2 b 2 i,..., a n b n i.
SECTION 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS 455 8.4 COMPLEX VECTOR SPACES AND INNER PRODUCTS All the vector spaces we have studed thus far n the text are real vector spaces snce the scalars are
Detailed Analysis of SCARA-Type Serial Manipulator on a Moving Base with LabView
Internatonal Journal of Advanced Robotc Systems ARTICLE Detaled Analyss of SCARA-Type Seral Manpulator on a Movng Base wth LabVew Regular Paper Alrıa Kalel 1,*, Ahmet Dumlu 1, M. Fath Çorapsı 1 and Köksal
Chapter 11 Torque and Angular Momentum
Chapter 11 Torque and Angular Momentum I. Torque II. Angular momentum - Defnton III. Newton s second law n angular form IV. Angular momentum - System of partcles - Rgd body - Conservaton I. Torque - Vector
Support Vector Machines
Support Vector Machnes Max Wellng Department of Computer Scence Unversty of Toronto 10 Kng s College Road Toronto, M5S 3G5 Canada wellng@cs.toronto.edu Abstract Ths s a note to explan support vector machnes.
The difference between voltage and potential difference
Slavko Vjevć 1, Tonć Modrć 1 and Dno Lovrć 1 1 Unversty of Splt, Faclty of electrcal engneerng, mechancal engneerng and naval archtectre Splt, Croata The dfference between voltage and potental dfference
An Integrated Semantically Correct 2.5D Object Oriented TIN. Andreas Koch
An Integrated Semantcally Correct 2.5D Object Orented TIN Andreas Koch Unverstät Hannover Insttut für Photogrammetre und GeoInformaton Contents Introducton Integraton of a DTM and 2D GIS data Semantcs
Damage detection in composite laminates using coin-tap method
Damage detecton n composte lamnates usng con-tap method S.J. Km Korea Aerospace Research Insttute, 45 Eoeun-Dong, Youseong-Gu, 35-333 Daejeon, Republc of Korea yaeln@kar.re.kr 45 The con-tap test has the
Dynamics of Earth Orbiting Formations
AIAA Gudance, Navgaton, and Control Conference and Exhbt 16-19 August 004, Provdence, Rhode Island AIAA 004-5134 Dynamcs of Earth Orbtng Formatons Scott R. Ploen, Danel.P. Scharf, Fred Y. Hadaegh, A. Behcet
I. INTRODUCTION. 1 IRCCyN: UMR CNRS 6596, Ecole Centrale de Nantes, Université de Nantes, Ecole des Mines de Nantes
he Knematc Analyss of a Symmetrcal hree-degree-of-freedom lanar arallel Manpulator Damen Chablat and hlppe Wenger Insttut de Recherche en Communcatons et Cybernétque de Nantes, rue de la Noë, 442 Nantes,
2.5 1.5 0.5. I(λ ) 0.5 1.5
NONCOLOCATION EFFECTS ON THE RIGID BODY ROTORDYNAMICS OF ROTORS ON AMB Gancarlo Genta Department of Mechancs, Poltecnco d Torno, Torno, Italy, genta@polto.t Stefano Carabell Department of Automatc Control,
Chapter 9. Linear Momentum and Collisions
Chapter 9 Lnear Momentum and Collsons CHAPTER OUTLINE 9.1 Lnear Momentum and Its Conservaton 9.2 Impulse and Momentum 9.3 Collsons n One Dmenson 9.4 Two-Dmensonal Collsons 9.5 The Center of Mass 9.6 Moton
BERNSTEIN POLYNOMIALS
On-Lne Geometrc Modelng Notes BERNSTEIN POLYNOMIALS Kenneth I. Joy Vsualzaton and Graphcs Research Group Department of Computer Scence Unversty of Calforna, Davs Overvew Polynomals are ncredbly useful
8.5 UNITARY AND HERMITIAN MATRICES. The conjugate transpose of a complex matrix A, denoted by A*, is given by
6 CHAPTER 8 COMPLEX VECTOR SPACES 5. Fnd the kernel of the lnear transformaton gven n Exercse 5. In Exercses 55 and 56, fnd the mage of v, for the ndcated composton, where and are gven by the followng
Applied Research Laboratory. Decision Theory and Receiver Design
Decson Theor and Recever Desgn Sgnal Detecton and Performance Estmaton Sgnal Processor Decde Sgnal s resent or Sgnal s not resent Nose Nose Sgnal? Problem: How should receved sgnals be rocessed n order
Point cloud to point cloud rigid transformations. Minimizing Rigid Registration Errors
Pont cloud to pont cloud rgd transformatons Russell Taylor 600.445 1 600.445 Fall 000-014 Copyrght R. H. Taylor Mnmzng Rgd Regstraton Errors Typcally, gven a set of ponts {a } n one coordnate system and
SELF BALANCING SYSTEM FOR ROTATING MECHANISMS
Rev. Fac. Ing. - Self Unv. balancng Tarapacá, system vol. for Nº rotatng, 005, mechansms pp. 59-64 SEF BAANCING SSTEM FOR ROTATING MECHANISMS Marco Antono Meraz Andrés ánez Carlos Jménez Raúl Pchardo Recbdo
www.engineerspress.com Neural Network Solutions for Forward Kinematics Problem of Hybrid Serial-Parallel Manipulator
www.engneersress.com World of Scences Journal ISSN: 307-307 Year: 03 Volume: Issue: 8 Pages: 48-58 Aahmad Ghanbar,, Arash ahman Deartment of Mechancal Engneerng, Unversty of Tabrz, Tabrz, Iran School of
Chapter 7 Symmetry and Spectroscopy Molecular Vibrations p. 1 -
Chapter 7 Symmetry and Spectroscopy Molecular Vbratons p - 7 Symmetry and Spectroscopy Molecular Vbratons 7 Bases for molecular vbratons We nvestgate a molecule consstng of N atoms, whch has 3N degrees
Linear Circuits Analysis. Superposition, Thevenin /Norton Equivalent circuits
Lnear Crcuts Analyss. Superposton, Theenn /Norton Equalent crcuts So far we hae explored tmendependent (resste) elements that are also lnear. A tmendependent elements s one for whch we can plot an / cure.
Loop Parallelization
- - Loop Parallelzaton C-52 Complaton steps: nested loops operatng on arrays, sequentell executon of teraton space DECLARE B[..,..+] FOR I :=.. FOR J :=.. I B[I,J] := B[I-,J]+B[I-,J-] ED FOR ED FOR analyze
The Virtual Movable Human Upper Body for Palpatory Diagnostic Training
Paper Number 26-1-232 Sesson DHM5 The Vrtual Movable Human Upper Body for Palpatory Dagnostc Tranng Meng-Yun Chen, Robert L. Wllams II, Robert R. Conatser Jr. and John N. Howell Interdscplnary Insttute
benefit is 2, paid if the policyholder dies within the year, and probability of death within the year is ).
REVIEW OF RISK MANAGEMENT CONCEPTS LOSS DISTRIBUTIONS AND INSURANCE Loss and nsurance: When someone s subject to the rsk of ncurrng a fnancal loss, the loss s generally modeled usng a random varable or
Portfolio Loss Distribution
Portfolo Loss Dstrbuton Rsky assets n loan ortfolo hghly llqud assets hold-to-maturty n the bank s balance sheet Outstandngs The orton of the bank asset that has already been extended to borrowers. Commtment
Conversion between the vector and raster data structures using Fuzzy Geographical Entities
Converson between the vector and raster data structures usng Fuzzy Geographcal Enttes Cdála Fonte Department of Mathematcs Faculty of Scences and Technology Unversty of Combra, Apartado 38, 3 454 Combra,
Laws of Electromagnetism
There are four laws of electromagnetsm: Laws of Electromagnetsm The law of Bot-Savart Ampere's law Force law Faraday's law magnetc feld generated by currents n wres the effect of a current on a loop of
Risk-based Fatigue Estimate of Deep Water Risers -- Course Project for EM388F: Fracture Mechanics, Spring 2008
Rsk-based Fatgue Estmate of Deep Water Rsers -- Course Project for EM388F: Fracture Mechancs, Sprng 2008 Chen Sh Department of Cvl, Archtectural, and Envronmental Engneerng The Unversty of Texas at Austn
Inner core mantle gravitational locking and the super-rotation of the inner core
Geophys. J. Int. (2010) 181, 806 817 do: 10.1111/j.1365-246X.2010.04563.x Inner core mantle gravtatonal lockng and the super-rotaton of the nner core Matheu Dumberry 1 and Jon Mound 2 1 Department of Physcs,
+ + + - - This circuit than can be reduced to a planar circuit
MeshCurrent Method The meshcurrent s analog of the nodeoltage method. We sole for a new set of arables, mesh currents, that automatcally satsfy KCLs. As such, meshcurrent method reduces crcut soluton to
21 Vectors: The Cross Product & Torque
21 Vectors: The Cross Product & Torque Do not use our left hand when applng ether the rght-hand rule for the cross product of two vectors dscussed n ths chapter or the rght-hand rule for somethng curl
Faraday's Law of Induction
Introducton Faraday's Law o Inducton In ths lab, you wll study Faraday's Law o nducton usng a wand wth col whch swngs through a magnetc eld. You wll also examne converson o mechanc energy nto electrc energy
RELIABILITY, RISK AND AVAILABILITY ANLYSIS OF A CONTAINER GANTRY CRANE ABSTRACT
Kolowrock Krzysztof Joanna oszynska MODELLING ENVIRONMENT AND INFRATRUCTURE INFLUENCE ON RELIABILITY AND OPERATION RT&A # () (Vol.) March RELIABILITY RIK AND AVAILABILITY ANLYI OF A CONTAINER GANTRY CRANE
Comparison of Control Strategies for Shunt Active Power Filter under Different Load Conditions
Comparson of Control Strateges for Shunt Actve Power Flter under Dfferent Load Condtons Sanjay C. Patel 1, Tushar A. Patel 2 Lecturer, Electrcal Department, Government Polytechnc, alsad, Gujarat, Inda
The Noether Theorems: from Noether to Ševera
14th Internatonal Summer School n Global Analyss and Mathematcal Physcs Satellte Meetng of the XVI Internatonal Congress on Mathematcal Physcs *** Lectures of Yvette Kosmann-Schwarzbach Centre de Mathématques
Inter-Ing 2007. INTERDISCIPLINARITY IN ENGINEERING SCIENTIFIC INTERNATIONAL CONFERENCE, TG. MUREŞ ROMÂNIA, 15-16 November 2007.
Inter-Ing 2007 INTERDISCIPLINARITY IN ENGINEERING SCIENTIFIC INTERNATIONAL CONFERENCE, TG. MUREŞ ROMÂNIA, 15-16 November 2007. UNCERTAINTY REGION SIMULATION FOR A SERIAL ROBOT STRUCTURE MARIUS SEBASTIAN
CHAPTER 8 Potential Energy and Conservation of Energy
CHAPTER 8 Potental Energy and Conservaton o Energy One orm o energy can be converted nto another orm o energy. Conservatve and non-conservatve orces Physcs 1 Knetc energy: Potental energy: Energy assocated
Generalizing the degree sequence problem
Mddlebury College March 2009 Arzona State Unversty Dscrete Mathematcs Semnar The degree sequence problem Problem: Gven an nteger sequence d = (d 1,...,d n ) determne f there exsts a graph G wth d as ts
Form-finding of grid shells with continuous elastic rods
Page of 0 Form-fndng of grd shells wth contnuous elastc rods Jan-Mn L PhD student Insttute of Buldng Structures and Structural Desgn (tke), Unversty Stuttgart Stuttgar, Germany quantumamn@gmal.com Jan
Chapter 3: Dual-bandwidth Data Path and BOCP Design
Chater 3: Dual-bandwdth Data Path and BOCP Desgn 3. Introducton The focus of ths thess s on the 4G wreless moble Internet networks to rovde data servces wthn the overlang areas of CDA2000-WLA networks.
SIMULATION OF THERMAL AND CHEMICAL RELAXATION IN A POST-DISCHARGE AIR CORONA REACTOR
XVIII Internatonal Conference on Gas Dscharges and Ther Applcatons (GD 2010) Grefswald - Germany SIMULATION OF THERMAL AND CHEMICAL RELAXATION IN A POST-DISCHARGE AIR CORONA REACTOR M. Mezane, J.P. Sarrette,
1. Fundamentals of probability theory 2. Emergence of communication traffic 3. Stochastic & Markovian Processes (SP & MP)
6.3 / -- Communcaton Networks II (Görg) SS20 -- www.comnets.un-bremen.de Communcaton Networks II Contents. Fundamentals of probablty theory 2. Emergence of communcaton traffc 3. Stochastc & Markovan Processes
ON THE ACCURACY, REPEATABILITY, AND DEGREE OF INFLUENCE OF KINEMATICS PARAMETERS FOR INDUSTRIAL ROBOTS
Internatonal Journal of Modellng and Smulaton, Vol. 22, No. 3, 2002 ON THE ACCURACY, REPEATABILITY, AND DEGREE OF INFLUENCE OF KINEMATICS PARAMETERS FOR INDUSTRIAL ROBOTS P.S. Shakolas, K.L. Conrad, and
Energy-based Design of Steel Structures According to the Predefined Interstory Drift Ratio 1
Dgest 01, December 01, 1573-1593 Energy-based Desgn of Steel Structures Accordng to the Predefned Interstory Drft Rato 1 Onur ERTER* Özgür BOZDAĞ** ustafa DÜZGÜ*** ABSTRACT The methods whch take lace n
Causal, Explanatory Forecasting. Analysis. Regression Analysis. Simple Linear Regression. Which is Independent? Forecasting
Causal, Explanatory Forecastng Assumes cause-and-effect relatonshp between system nputs and ts output Forecastng wth Regresson Analyss Rchard S. Barr Inputs System Cause + Effect Relatonshp The job of
Code_Aster ( ) D8.01.03 Charter for the realization of the mathematical formulas in the documentation of the Code_Aster ( )
Ttre : Charte graphque pour la réalsaton des formules [...] Date : //008 Page : /5 Organzaton (S): Manual EDF/IMA/MMN of data-processng Descrpton D8.0 Booklet: Presentaton of documentaton Document graphc
O(n) mass matrix inversion for serial manipulators and polypeptide chains using Lie derivatives Kiju Lee, Yunfeng Wang and Gregory S.
Robotca 7) volume 5, pp 739 75 7 Cambrdge Unversty Press do:7/s6357477385 Prnted n the Unted Kngdom On) mass matrx nverson for seral manpulators and polypeptde chans usng Le dervatves Ku Lee, Yunfeng Wang
An Alternative Way to Measure Private Equity Performance
An Alternatve Way to Measure Prvate Equty Performance Peter Todd Parlux Investment Technology LLC Summary Internal Rate of Return (IRR) s probably the most common way to measure the performance of prvate
Multiple stage amplifiers
Multple stage amplfers Ams: Examne a few common 2-transstor amplfers: -- Dfferental amplfers -- Cascode amplfers -- Darlngton pars -- current mrrors Introduce formal methods for exactly analysng multple
Research Article Competition and Integration in Closed-Loop Supply Chain Network with Variational Inequality
Hndaw Publshng Cororaton Mathematcal Problems n Engneerng Volume 2012, Artcle ID 524809, 21 ages do:10.1155/2012/524809 Research Artcle Cometton and Integraton n Closed-Loo Suly Chan Network wth Varatonal
A New Technique for Vehicle Tracking on the Assumption of Stratospheric Platforms. Department of Civil Engineering, University of Tokyo **
Fuse, Taash A New Technque for Vehcle Tracng on the Assumton of Stratosherc Platforms Taash FUSE * and Ehan SHIMIZU ** * Deartment of Cvl Engneerng, Unversty of Toyo ** Professor, Deartment of Cvl Engneerng,
Elastic Systems for Static Balancing of Robot Arms
. th World ongress n Mechans and Machne Scence, Guanajuato, Méco, 9- June, 0 _ lastc Sstes for Statc alancng of Robot rs I.Sonescu L. uptu Lucana Ionta I.Ion M. ne Poltehnca Unverst Poltehnca Unverst Poltehnca
Chapter 2. Lagrange s and Hamilton s Equations
Chapter 2 Lagrange s and Hamlton s Equatons In ths chapter, we consder two reformulatons of Newtonan mechancs, the Lagrangan and the Hamltonan formalsm. The frst s naturally assocated wth confguraton space,
Rate Monotonic (RM) Disadvantages of cyclic. TDDB47 Real Time Systems. Lecture 2: RM & EDF. Priority-based scheduling. States of a process
Dsadvantages of cyclc TDDB47 Real Tme Systems Manual scheduler constructon Cannot deal wth any runtme changes What happens f we add a task to the set? Real-Tme Systems Laboratory Department of Computer
Jet Engine. Figure 1 Jet engine
Jet Engne Prof. Dr. Mustafa Cavcar Anadolu Unversty, School of Cvl Avaton Esksehr, urkey GROSS HRUS INAKE MOMENUM DRAG NE HRUS Fgure 1 Jet engne he thrust for a turboet engne can be derved from Newton
05 Enthalpy of hydration of sodium acetate
05 Enthaly of hydraton of sodum acetate Theoretcal background Imortant concets The law of energy conservaton, extensve and ntensve quanttes, thermodynamc state functons, heat, work, nternal energy, enthaly,
Brigid Mullany, Ph.D University of North Carolina, Charlotte
Evaluaton And Comparson Of The Dfferent Standards Used To Defne The Postonal Accuracy And Repeatablty Of Numercally Controlled Machnng Center Axes Brgd Mullany, Ph.D Unversty of North Carolna, Charlotte
What is Candidate Sampling
What s Canddate Samplng Say we have a multclass or mult label problem where each tranng example ( x, T ) conssts of a context x a small (mult)set of target classes T out of a large unverse L of possble
Dynamic Load Balancing of Parallel Computational Iterative Routines on Platforms with Memory Heterogeneity
Dynamc Load Balancng of Parallel Comutatonal Iteratve Routnes on Platforms wth Memory Heterogenety Davd Clare, Alexey Lastovetsy, Vladmr Rychov School of Comuter Scence and Informatcs, Unversty College
Basic Equations of Fluid Dynamics
Basc Equatons of Flud Dynamcs Sergey Pankratov A lecture to the Practcal Course Scentfc Computng and Vsualzaton (June 17, 2004) Sergey Pankratov, TU München 1 Focus areas Euler and Lagrange descrptons
TENSOR GAUGE FIELDS OF DEGREE THREE
TENSOR GAUGE FIELDS OF DEGREE THREE E.M. CIOROIANU Department of Physcs, Unversty of Craova, A. I. Cuza 13, 2585, Craova, Romana, EU E-mal: manache@central.ucv.ro Receved February 2, 213 Startng from a
Analysis and Modeling of Buck Converter in Discontinuous-Output-Inductor-Current Mode Operation *
Energy and Power Engneerng, 3, 5, 85-856 do:.436/ee.3.54b63 Publshed Onlne July 3 (htt://www.scr.org/journal/ee) Analyss and Modelng of Buck Converter n Dscontnuous-Outut-Inductor-Current Mode Oeraton
MOLECULAR PARTITION FUNCTIONS
MOLECULR PRTITIO FUCTIOS Introducton In the last chapter, we have been ntroduced to the three man ensembles used n statstcal mechancs and some examples of calculatons of partton functons were also gven.
arxiv:quant-ph/0609012v3 4 Jan 2007
Two-way and three-way negatvtes of three qubt entangled states S. Shelly Sharma Deto. de Físca, Unversdade Estadual de Londrna, Londrna 8605-990, PR Brazl N.. Sharma Deto. de Matemátca, Unversdade Estadual
Module 2 LOSSLESS IMAGE COMPRESSION SYSTEMS. Version 2 ECE IIT, Kharagpur
Module LOSSLESS IMAGE COMPRESSION SYSTEMS Lesson 3 Lossless Compresson: Huffman Codng Instructonal Objectves At the end of ths lesson, the students should be able to:. Defne and measure source entropy..
Calculation of Sampling Weights
Perre Foy Statstcs Canada 4 Calculaton of Samplng Weghts 4.1 OVERVIEW The basc sample desgn used n TIMSS Populatons 1 and 2 was a two-stage stratfed cluster desgn. 1 The frst stage conssted of a sample
Review C: Work and Kinetic Energy
MASSACHUSETTS INSTITUTE OF TECHNOLOGY Department o Physcs 8.2 Revew C: Work and Knetc Energy C. Energy... 2 C.. The Concept o Energy... 2 C..2 Knetc Energy... 3 C.2 Work and Power... 4 C.2. Work Done by
Product-Form Stationary Distributions for Deficiency Zero Chemical Reaction Networks
Bulletn of Mathematcal Bology (21 DOI 1.17/s11538-1-9517-4 ORIGINAL ARTICLE Product-Form Statonary Dstrbutons for Defcency Zero Chemcal Reacton Networks Davd F. Anderson, Gheorghe Cracun, Thomas G. Kurtz
A Multi-Camera System on PC-Cluster for Real-time 3-D Tracking
The 23 rd Conference of the Mechancal Engneerng Network of Thaland November 4 7, 2009, Chang Ma A Mult-Camera System on PC-Cluster for Real-tme 3-D Trackng Vboon Sangveraphunsr*, Krtsana Uttamang, and
PERRON FROBENIUS THEOREM
PERRON FROBENIUS THEOREM R. CLARK ROBINSON Defnton. A n n matrx M wth real entres m, s called a stochastc matrx provded () all the entres m satsfy 0 m, () each of the columns sum to one, m = for all, ()
Reduced magnetohydrodynamic equations with coupled Alfvén and sound wave dynamics
PHYSICS OF PLASMAS 14, 10906 007 Reduced magnetohydrodynamc equatons wth coupled Alfvén and sound wave dynamcs R. E. Denton and B. Rogers Department of Physcs and Astronomy, Dartmouth College, Hanover,
Computational Fluid Dynamics II
Computatonal Flud Dynamcs II Eercse 2 1. Gven s the PDE: u tt a 2 ou Formulate the CFL-condton for two possble eplct schemes. 2. The Euler equatons for 1-dmensonal, unsteady flows s dscretsed n the followng
Multi-Robot Tracking of a Moving Object Using Directional Sensors
Mult-Robot Trackng of a Movng Object Usng Drectonal Sensors Xaomng Hu, Karl H. Johansson, Manuel Mazo Jr., Alberto Speranzon Dept. of Sgnals, Sensors & Systems Royal Insttute of Technology, SE- 44 Stockholm,
Automated information technology for ionosphere monitoring of low-orbit navigation satellite signals
Automated nformaton technology for onosphere montorng of low-orbt navgaton satellte sgnals Alexander Romanov, Sergey Trusov and Alexey Romanov Federal State Untary Enterprse Russan Insttute of Space Devce
Extending Probabilistic Dynamic Epistemic Logic
Extendng Probablstc Dynamc Epstemc Logc Joshua Sack May 29, 2008 Probablty Space Defnton A probablty space s a tuple (S, A, µ), where 1 S s a set called the sample space. 2 A P(S) s a σ-algebra: a set
The Distribution of Eigenvalues of Covariance Matrices of Residuals in Analysis of Variance
JOURNAL OF RESEARCH of the Natonal Bureau of Standards - B. Mathem atca l Scence s Vol. 74B, No.3, July-September 1970 The Dstrbuton of Egenvalues of Covarance Matrces of Resduals n Analyss of Varance
Energies of Network Nastsemble
Supplementary materal: Assessng the relevance of node features for network structure Gnestra Bancon, 1 Paolo Pn,, 3 and Matteo Marsl 1 1 The Abdus Salam Internatonal Center for Theoretcal Physcs, Strada
Actuator forces in CFD: RANS and LES modeling in OpenFOAM
Home Search Collectons Journals About Contact us My IOPscence Actuator forces n CFD: RANS and LES modelng n OpenFOAM Ths content has been downloaded from IOPscence. Please scroll down to see the full text.
The Development of Web Log Mining Based on Improve-K-Means Clustering Analysis
The Development of Web Log Mnng Based on Improve-K-Means Clusterng Analyss TngZhong Wang * College of Informaton Technology, Luoyang Normal Unversty, Luoyang, 471022, Chna wangtngzhong2@sna.cn Abstract.
Traffic State Estimation in the Traffic Management Center of Berlin
Traffc State Estmaton n the Traffc Management Center of Berln Authors: Peter Vortsch, PTV AG, Stumpfstrasse, D-763 Karlsruhe, Germany phone ++49/72/965/35, emal peter.vortsch@ptv.de Peter Möhl, PTV AG,
Ring structure of splines on triangulations
www.oeaw.ac.at Rng structure of splnes on trangulatons N. Vllamzar RICAM-Report 2014-48 www.rcam.oeaw.ac.at RING STRUCTURE OF SPLINES ON TRIANGULATIONS NELLY VILLAMIZAR Introducton For a trangulated regon
Network Security Situation Evaluation Method for Distributed Denial of Service
Network Securty Stuaton Evaluaton Method for Dstrbuted Denal of Servce Jn Q,2, Cu YMn,2, Huang MnHuan,2, Kuang XaoHu,2, TangHong,2 ) Scence and Technology on Informaton System Securty Laboratory, Bejng,
HÜCKEL MOLECULAR ORBITAL THEORY
1 HÜCKEL MOLECULAR ORBITAL THEORY In general, the vast maorty polyatomc molecules can be thought of as consstng of a collecton of two electron bonds between pars of atoms. So the qualtatve pcture of σ
4 Cosmological Perturbation Theory
4 Cosmologcal Perturbaton Theory So far, we have treated the unverse as perfectly homogeneous. To understand the formaton and evoluton of large-scale structures, we have to ntroduce nhomogenetes. As long
Optimization-based Motion Retargeting Integrating Spatial and Dynamic Constraints for Humanoid
Optmzaton-based Moton Retargetng Integratng Spatal and Dynamc Constrants for Humanod Thomas Moulard, Ech Yoshda and Shn chro Nakaoka CNRS-AIST JRL (Jont Robotcs Laboratory), UMI38/CRT ({thomas.moulard,
Implementation of Deutsch's Algorithm Using Mathcad
Implementaton of Deutsch's Algorthm Usng Mathcad Frank Roux The followng s a Mathcad mplementaton of Davd Deutsch's quantum computer prototype as presented on pages - n "Machnes, Logc and Quantum Physcs"
DECOMPOSITION OF MEASURED GROUND VIBRATIONS INTO BASIC SOIL WAVES
DECOMPOSITION OF MEASURED GROUND VIBRATIONS INTO BASIC SOIL WAVES D. Macjausas Department of Scence & Technology, Unversty of Luembourg, Luembourg S. Van Baars Department of Scence & Technology, Unversty
An interactive system for structure-based ASCII art creation
An nteractve system for structure-based ASCII art creaton Katsunor Myake Henry Johan Tomoyuk Nshta The Unversty of Tokyo Nanyang Technologcal Unversty Abstract Non-Photorealstc Renderng (NPR), whose am
Fixed income risk attribution
5 Fxed ncome rsk attrbuton Chthra Krshnamurth RskMetrcs Group chthra.krshnamurth@rskmetrcs.com We compare the rsk of the actve portfolo wth that of the benchmark and segment the dfference between the two
Stochastic Six-Degree-of-Freedom Flight Simulator for Passively Controlled High-Power Rockets
Stochastc Sx-Degree-of-Freedom Flght for Passvely Controlled Hgh-Power s Smon Box 1 ; Chrstopher M. Bshop 2 ; and Hugh Hunt 3 Downloaded from ascelbrary.org by TECHNISCHE UNIVERSITEIT DELFT on 2/7/13.