Two-Dimensional Inverse Dynamics

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Robyn Spencer
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1 Notes_8_11 1 of 9 wo-dmensonal Inverse Dynamcs Knematcally drven Must use centrodal coordnate frames! { { { Q { r φ { r φ on { F on on [ M m m J G Sngle body [ M { ( { Q on { Q on { Qon { Qon CONSRAIN ALIED

2 Notes_8_11 of 9 System of multple bodes { { [ M { { 3 { 4 { { 3 { 4 [ M [ 3x3 [ 3x3 [ 3x3 [ M 3 [ 3x3 [ [ [ 3x3 3x3 M 4 [ M [ 3x3 [ 3x3 [ 3x3 [ M 3 [ 3x3 [ [ [ M { Q 3x3 3x3 [ M { { Q 4 ( { Q on ( { Qon 3 ( { Q on 4 { { 3 { 4 ( { Q on ({ Qon 3 ({ Qon 4 { Q { Q ALIED + { Q CONSRAIN { CONSRAIN { KINEMAIC { DRIVER { Q CONSRAIN { Q KINEMAIC { Q DRIVER

3 Notes_8_11 3 of 9 Vrtual work [ M{ { Q { { ([ M{ { Q { Q { Q ALIED + { Q CONSRAIN { { Q for knematc consstency [ { { CONSRAIN { ([ M{ { Q subect to [ { { ALIED Vrtual work for one revolute { { Q? { CONSRAIN φ { { F { on F on { Q φ { Fon ([ B { s ' { F { Q { F on ([ { { B s ' F on on on on { { Q + { { Q? on on { F on φ ([ B { s ' { F on + { Fon φ ([ B { s ' { F on? { F + ( φ [ B { s ' { F + { F + ( φ [ B { s ' { F? on on on on + φ [ B { s ' + φ [ B { s ' ({ r { F + ( { F? OK on on

4 Notes_8_11 4 of 9 Lagrange multpler theorem general problem { { x { b { x { λ [ A{ x b subect to [ A { x { + usng Lagrange multplers { λ for any arbtrary { x vrtual work { ([ M{ { Q subect to [ { { ALIED { x { { b [ M{ { Q ALIED [ A [ ([ M{ { Q { + { λ [ { ALIED for arbtrary sze but knematcally consstent{ ([ M{ { Q + [ { λ { ALIED [ M { + [ { λ { Q ALIED [ { λ each row n { λ s multpled tmes correspondng column n [ each row n { λ corresponds to matchng row n [ and { Lagrange multplers [ M { + [ { λ { Q ALIED [ M{ { [ { λ Q ALIED [ M { { Q { Q ALIED + { Q CONSRAINS { Q [ { λ CONSRAINS { Q ( { Qon ( { Qon 3 ( { Qon 4 { Q CONSRAINS ( { Q on CONSRAINS ( { Qon 3 CONSRAINS ({ Qon 4 CONSRAINS

5 Notes_8_11 5 of 9 Euatons of moton (EOM [ M { + [ { λ { Q ALIED [ { { γ { { [ M { Q { { γ { λ [ n x 1 n x 1 n x n ALIED n x 1 n number of generalzed coordnates nk number of knematc constrants nd number of drver constrants nc total number of constrants (nc nk + nd nc x n Inverse dynamcs knematcally drven [ 1 solve knematcs { [ { γ must have full rank nc n 1 compute constrant forces { λ ([ { [ M{ ( Q ALIED { { { KINEMAIC DRIVER { λ { λ KINEMAIC { λ DRIVER Inverse dynamcs smultaneous EOM matrx [ M{ + [ { λ { Q and [ { { γ ALIED [ M [ n x n [ [ nc x n n x nc nc x nc { n x 1 { λ { Q n ALIED x 1 { γ [ EOM [ M [ [ [ ( nc+ n x ( nc+ n { { λ [ M [ [ [ 1 { Q ALIED { γ

6 Notes_8_11 6 of 9 Statcs { { [ M { + [ { λ { Q ALIED 1 { λ ([ { Q ALIED Lagrange multplers for specfc constrants { Fon [ r { λ CONSRAIN CONSRAIN CONSRAIN ( r { λ CONSRAIN ( ([ B { s ' [ [ on CONSRAIN CONSRAIN φ { Fon [ r { λ CONSRAIN CONSRAIN CONSRAIN ( ([ B { s ' [ [ CONSRAIN ( r { λ CONSRAIN on CONSRAIN CONSRAIN φ CONSRAIN Revolute { { r { r { Note: Haug uses { r { r x1 { F { λ at { r on ( on { F { λ at { r on ( on check body [ [ [ [ { I φ B s ' OK r check body [ [ [ [ { r I φ B s ' OK

7 Notes_8_11 7 of 9 Double revolute _ for and and and { d { d L { d { r { r { d { d { r { r { d { r { r L cons tan t length { F { d λ at { r on _ ( on CONSRAIN _ { F { d λ at { r on _ ( on CONSRAIN _ arallel vectors (planar parallel-1 { a parallel to { a AREL for { a [ R { a Q Q { a { r { r and { a { r { r { F { on AREL x1 ( on ( norm{ a ( norm{ a λarel AREL { F { on AREL x1 ( on ( norm{ a ( norm{ a λarel AREL n-n-slot (planar parallel-

8 Notes_8_11 8 of 9 { a parallel to { d IN _ SLO for and { a [ R { d Q { d { r { r and { a { r { r { d { d { d from above { F [ R{ a λ at { r on IN _SLO IN _SLO ( on ( norm{ a ( norm{ d λin _ SLO CONSRAIN { F [ R{ a λ at { r on IN _ SLO ( on IN _ SLO Relatve angle drver IN _ SLO φ φ C f (t C cons tan t { F { on x1 ( λ on { F { on x1 ( on λ Gear par (chan/sprockets, belt/pulleys GEAR φ Kφ external gears { F { on GEAR x1 C K ρ ( on K λ GEAR CONSRAIN { F { on GEAR x1 / ρ, K cons tan t, C nt ernal gears K +ρ cons tan t / ρ

9 Notes_8_11 9 of 9 ( on λgear CONSRAIN Gear par on rotatng lnk k ( φ φ K( φ φ C K cons tan t, C cons tan t from above GEAR _ ON _ K k k { F { on GEAR x1 ( on K λ GEAR CONSRAIN { F { on GEAR x1 ( on λgear CONSRAIN { F { on k GEAR x1 ( ( 1 K λ GEAR on k CONSRAIN

Lesson 5: Kinematics and Dynamics of Particles

Lesson 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