Multi-joint kinematics and dynamics

Transcription

1 ut-ont nematcs and dnamcs Emo odorov pped athematcs omputer Scence and Engneerng Unverst o Washngton Knematcs n generazed vs. artesan coordnates generazed artesan dm() euas the number o degrees o reedom (DOFs) dm()>dm().e. artesan coordnates are over-compete orward nematcs: awas we-dened mappng rom to = h() nverse nematcs: usua not we-dened but one can resove redundanc va optmzaton: o a satg = h() or gven pc the one est to a preerred * arg mn ˆ : h ˆ ˆ *

2 Knematcs n D ttach a spata rame to each bod - usua at the center o mass or at the center o the ont connectng t to ts parent bod. Rotatons can be epressed as R e e e R R detr ransormaton between rames: o R rame e e o e rame ddton and mutpcaton can be combned ug homogeneous coordnates: R o ˆ : ˆ ˆ he spata reatons b between bod rames depend d on the h ont parameters. Let (n) be the parameters specng the ont between bod n and ts parent. hen the correspondng transormaton s parameterzed as parentn n n omputng the orward nematcs nvoves a orward recurson: word n n word nr ˆ parent nr ˆ parent n Numerca orward nematcs s more accurate ug uaternons nstead o matrces. Rgd-bod dnamcs (Newton-Euer) ~ r v r v see Featherstone s sdes on Spata Vector gebra

3 Newtonan mechancs wth mpct constrants Newton s second aw or a scaar pont mass s m mi For a set o n pont masses n D we have whch n vector notaton s D mni n n Now consder a set o m postona euat constrants dened mpct as he coud spec that some masses beong to the same rgd bod or that some rgd bodes are constraned b onts etc. he constrants emnate m DOFs and create a n-m dmensona conguraton manod parameterzed b. nu space he constrant orces can on act wthn the nu space whch s spanned b the rows o the acoban matr. hus or some m-dmensona vector λ tot ound db tang nto account the derentated d constrants: where he constraned dnamcs D tot are he constraned dnamcs are the souton to the near n euaton D D D D 5 onstraned nerta and the Gauss prncpe 6 When the sstem s statonar the constraned dnamcs smp to where s the nverse o the constraned nerta matr: D D here s no acceeraton n the nu space: D D s guar wth ran() = dm(). D D Ug the matr nverson emma we can represent as hus the constraned nerta s D D D D m D whch oows rom " D " and s nnte n the nu space. he same resuts can be obtaned rom the more genera Gauss prncpe: the constraned acceeraton s the souton to the mnmzaton probem a D a s.t. a b arg mn a s the unconstraned acceeraton; b can encode genera constrants.

4 he mpct-constraned dnamcs D D D are epressed n over-compete artesan coordnates () whch s oten undesrabe. Instead t s better to epress the dnamcs n generazed () coordnates. hs s done through epct constrants gven b the orward nematcs uncton = h() Derentatng the constrants twce eds Epct constrants he dnamcs are D c where c are the constrant orces. Snce the coumns o span the tangent space to the manod c 7 ssembng these euatons we obtan a sstem whch s near n D I I c c he constraned dnamcs are c where D c D oordnate transormatons 8 onsder an set o coordnates reated to as = h() Veoctes n the two coordnate sstems reate as Let and τ denote the same orce epressed n and coordnates respectve. Power s coordnate-ndependent: Snce ths hods or an veoct orces n the two coordnate sstems reate as Let D and denote the same nerta epressed n and coordnates respectve. e Knetc energ s coordnate-ndependent: Snce ths hods or an veoct nertas n the two coordnate sstems reate as D D D

5 Euat constrants are handed as n the case o pont-mass dnamcs: we sove the near n euaton Euat constrants n generazed coordnates 9 c Here the constrants are and the acoban s Euat constrants are oten used to create nematc oops (e.g. hodng hands) In smuatons the constrants can be voated numerca due to ntegraton errors. hus t s necessar to ntroduce constrant stabzaton (resembng PD contro). Eampe: -n arm m m m m D () ( ) m m Impct constrants: ( ) Epct constrants: h h

6 Fast recursve computaton o and c omputng D and c D drect s necent. Instead one can use aster agorthms epotng the structure o nematc trees. Let s be the 6D moton vector o the (-do) ont connectng bod to ts parent. omposte Rgd od agorthm or computng the nerta matr () () bacward recurson: () set: comp s D s descendants comp comp D D D chdren( ) comp s D s descendants otherwse Recursve Newton-Euer agorthm or computng the nverse dnamcs () orward recurson: () bacward recurson: () set: s parent s s parent D D runnng ths agorthm wth eds c chdren( ) s Once and c are computed we can compute c and ntegrate. Dnamcs n generazed coordnates c g where c c g nerta matr oros and centruga orces gravtatona orces apped/contro orces hrstoe smbos hs can be derved rom the Euer-Lagrange euaton: d dt L L where the Lagrangan s the netc energ mnus the potenta energ: L K P K P 9.8m h g n n n P I does not depend on then c= and we have Newton s second aw: g

7 Hamtonan ormuaton he same dnamcs can be obtaned rom the euvaent Hamtonan ormuaton based on the Hamtonan H = K + P nstead o the Lagrangan L = K P. Now the state s represented n terms o and the generazed momentum p H and L are reated b the Legendre transormaton Knetc energ n the new coordnates s Hamton s euatons are: K H p p H p p H p L p p p K he rate o change o the Hamtonan (.e. the tota energ) euas power: d dt H H p H H p H H p H p p p H In the absence o eterna orces the Hamtonan s conserved. anods and metrcs Q s a derentabe manod and Q the tangent space at pont. Q * Q denotes the co-tangent (or dua) space. u metrc denes a dot-product on the tangent space: u v u v u v u v (Ensten) Q he manod s Remannan () s s.p.d. or a. he dot-product on the co-tangent space s dened b the nverse o : u* v * u * v* u v where u u u* u he metrc provdes the mappng between the two spaces: u* u u u*; n coordnates u u u u angent and co-tangent vectors are mutped drect: u v* u v u v ppcaton to mut-ont dnamcs: he conguraton space o a mut-ont sstem s a Remannan manod wth metrc gven b the ont-space nerta matr (). he tangent vectors are veoctes. he co-tangent vectors are orces and momenta p. p s netc energ; s power.

8 ovarant dervatves and geodescs he tangent bass vectors are assocated wth parta dervatves: e he co-tangent bass vectors are assocated wth derenta orms: d I () s scaar and v v e s a tangent vector then v s the drectona dervatve: v v e v v grad 5 connecton speces how nearb coordnate rames connect.e. how the bass vectors change over the manod. he usua vector drectona dervatve s repaced wth the covarant dervatve dened n coordnates b the hrstoe smbos e e e u For genera vectors u u e v v e the covarant dervatve s vu v u v connecton s at when. In that case we recover the reguar dervatve. For a Remannan manod wth metrc () there ests a unue metrc-preservng torson-ree connecton (the Lev-vta connecton) wth hrstoe smbos: s s s s s s geodesc s a curve γ(t) such that.e. or a. hs s caed the geodesc euaton. e Unorced motons as geodescs 6 he unorced motons o a mut-ont sstem sats whch we can rewrte (ug the act that s s.p.d.) as c c Recang the epresson or c ths can be wrtten n component orm as c s s where s s s s and s hus we have recovered the geodesc euaton he Lev-vta connecton or the Remannan metrc dened b the nerta matr s caed the mechanca connecton. Its geodescs are the unorced motons. Wth eterna orces and gravt the dnamcs become g hs s euvaent to Newton s s second aw wth the covarant dervatve n pace o the reguar dervatve: d dt