Jacobians: Velocities and Static Force.

Transcription

1 Jaoban: Veote and Stat Fore mrkabr Unerty o ehnoogy Computer Engneerng Inormaton ehnoogy Department

2 Derentaton o poton etor Derate o a etor: V Q d dt Q m t Q( t t) t Q( t) We are auatng the derate o Q reate to rame.

3 Derentaton o poton etor eoty etor may be derbed n term o any rame: We may wrte t: ( ) V. Q Q d ( ) V Q Q dt V Speed etor a ree etor Spea ae: Veoty o the orgn o a rame reate to ome undertood unere reerene rame V C U V COG

4 Eampe 5. oth ehe are heedng n X dreton o U mph ed unera rame 3 mph U C C d dt ( ( U U V V P COG OG COG ) ) C C U V COG V C U COG C C U C U (Xˆ ) 3Xˆ. U V COG U C U C Xˆ. U 7Xˆ.

5 nguar eoty etor: Ω Lnear eoty attrbute o a pont nguar eoty attrbute o a body Sne we away attah a rame to a body we an onder anguar eoty a derbng ratona moton o a rame.

6 nguar eoty etor: Ω Ω derbe the rotaton o rame {} reate to {} Ω dreton o ndate ntantaneou a o rotaton Ω Magntude o ndate peed o rotaton In the ae whh there an undertood reerene rame: ω C U Ω C

7 Lnear eoty o a rgd body We wh to derbe moton o {} reate to rame {} I rotaton not hangng wth tme: Q V V OG VQ.

8 otatona eoty o a rgd body wo rame wth ondent orgn he orentaton o wth repet to hangng n tme. Ω {} {} Q Let onder that etor Q ontant a ewed rom. V Q

9 otatona eoty o a rgd body Q I perpenduar to Ω and Q Magntude o derenta hange : Q ( Q n )( Ω t) V Q Ω Q Vetor ro produt

10 otatona eoty o a rgd body In genera ae: V Q ( V ) Q Ω Q V Q V Q Ω Q.

11 We kp 5.4! Smutaneou near and rotatona eoty V Q V OG V Q Ω Q.

12 Moton o the Lnk o a obot Wrtten n rame t any ntant eah nk o a robot n moton ha ome near and anguar eoty.

13 Veoty o a Lnk emember that near eoty aoated wth a pont and anguar eoty aoated wth a body. hu: he eoty o a nk mean the near eoty o the orgn o the nk rame and the rotatona eoty o the nk

14 Veoty Propagaton From Lnk to Lnk We an ompute the eote o eah nk n order tartng rom the bae. he eoty o nk w be that o nk pu whateer new eoty omponent added by jont.

15 otatona Veoty otatona eote may be added when both w etor are wrtten wth repet to the ame rame. hereore the anguar eoty o nk the ame a that o nk pu a new omponent aued by rotatona eoty at jont.

16 Veoty Vetor o Neghborng Lnk ω Zˆ. ω

17 Veoty Propagaton From Lnk to Lnk y premutpyng both de o preou equaton to:. ˆ. ˆ Z Z ω ω ω ω Z Note that:

18 Lnear Veoty he near eoty o the orgn o rame {} the ame a that o the orgn o rame {} pu a new omponent aued by rotatona eoty o nk.

19 Lnear Veoty Q. V V V Q OG Q Ω Smutaneou near and rotatona eoty: ). ( ). ( P P ω ω y premutpyng both de o preou equaton to:. P ω

20 Prmat Jont Lnk For the ae that jont prmat:. ˆ ) ( Z d P ω ω ω

21 Veoty Propagaton From Lnk to Lnk ppyng thoe preou equaton ueuy rom nk to nk we an ompute the rotatona and near eote o the at nk.

22 Eampe 5.3 Cauate the eoty o the tp o the arm a a unton o jont rate? -nk manpuator wth rotatona jont

23 Eampe 5.3 Frame agnment or the two nk manpuator

24 Eampe 5.3 We ompute nk tranormaton:. 3

25 Eampe 5.3 Lnk to nk tranormaton. ) ( ω ω ω ω

26 Eampe 5.3 Veote wth repet to non mong bae. ) ( ) (

27 Derate o a Vetor Funton I we hae a etor unton r whh repreent a parte poton a a unton o tme t: r [ r r r ] y z dr dt dr dt dr dt y dr dt z

28 Vetor Derate We e een how to take a derate o a etor. aar What about the derate o a etor. etor?

29 Jaoban Jaoban a etor derate wth repet to another etor I we hae () the Jaoban a matr o parta derate- one parta derate or eah ombnaton o omponent o the etor he Jaoban uuay wrtten a j() but you an reay jut thnk o t a d/d

30 Jaoban ( ) N M M N J

31 Parta Derate he ue o the ymbo ntead o d or parta derate reay jut mpe that t a nge omponent n a etor derate

32 Jaoban. ). ( ) ( ) ( ) ( X X F Y y y y X F Y y y y δ δ δ δ δ δ δ δ δ δ δ δ δ δ L M L L M J(X) Chan rue

33 Jaoban In the ed o robot we generay peak o Jaoban whh reate jont eote to Cartean eote o the tp o the arm.. ) (. ) ( Θ Θ J X X J Y ω V

34 Jaoban For a 6 jont robot the Jaoban 66. a 6 and 6. he number o row n Jaoban equa to number o degree o reedom n Cartean pae and the number o oumn equa to the number o jont. V J ( Θ) Θ

35 Jaoban In eampe 5.3 we had:. ) ( ) ( Θ ) ( J hu: Θ 3 ) ( J nd ao:

36 Jaoban Jaoban mght be ound by drety derentatng the knemat equaton o the mehanm or near eoty howeer there no 3 orentaton etor whoe derate rotatona eoty. hu we get Jaoban ung uee appaton o: ˆ Z ω ω ( ) P ω

37 Snguarte Gen a tranormaton reatng jont eoty to Cartean eoty then I th matr nertbe? ( I t non nguar) V J ( Θ) Θ J ( ) det[ J] :nguarty det[ J] : nonnguarty

38 Snguarte Snguarte are ategorzed nto two a: Workpae boundary nguarte: Our when the manpuator uy tarhed or oded bak on te. Workpae nteror nguarte: re away rom workpae boundary and are aued by two or more jont ae nng up. manpuator hae nguarty at boundare o ther workpae. In a nguar onguraton one or more degree o reedom ot. ( moement mpobe )

39 Eampe 5.4 In eampe 5.3 we had: Θ 3 ) ( J Θ Θ o 8 o. ) ( )] ( [ J J DE Workpae boundary nguarte

40 Eampe ) ( J Θ the arm trethe out toward both jont rate go to nnty

41 Stat Fore n Manpuator Fore and moment propagaton o oe or jont torque n tat equbrum ore eerted on nk by nk - n torque eerted on nk by nk -

42 Stat Fore n Manpuator Soe or the jont torque whh mut be atng to keep the ytem n tat equbrum. Summng the ore and ettng them equa to zero n Summng the torque about the orgn o rame n P

43 Stat Fore n Manpuator.. P n n P n n Workng don rom at nk to the bae we ormuate the ore moment epreon Stat ore propagaton rom nk to nk:. ˆ. ˆ Z Z n τ τ Important queton: What torque are needed at the jont to baane reaton ore and moment atng on the nk?

44 Eampe 5.7

45 Work-energy Prnpe he hange n the knet energy o an objet equa to the net work done on the objet.

46 Prnpe o Vrtua Work Eterna rtua work equa the nterna rtua tran energy.

47 Jaoban n the Fore Doman Work the dot produt o a etor ore or torque and a etor dpaement F δx τ δ It an be wrtten a: Θ he denton o jaoban δx jδθ So we hae F τ J τ F δx τ δθ JδΘ τ δθ F J F. F. J τ.

48 Cartean ranormaton o Veote and Stat Fore Genera eoty o a body V ω 3 near eoty Genera ore o a body F F N 3 anguar eoty 3 ore etor 3 moment etor 6 6 tranormaton map thee quantte rom one rame to another.

49 Cartean ranormaton o Veote and Stat Fore Sne two rame are rgdy onneted ˆ. Z ω ω (5.45) Where the ro produt the matr operator OG w P w y z y z p p p p p p P

50 Cartean ranormaton o Veote and Stat Fore We ue the term eoty tranormaton OG P ω ω Derpton o eoty n term o when gen the quantte n OG P ω ω

51 Cartean ranormaton o Veote and Stat Fore ore-moment tranormaton OG N F P N F F F Wth marty to Jaoban

52 Eampe 5.8 Frame o nteret wth a ore enor F S S F S S P S SOG S S.

53 Net Coure: Manpuator Dynam mrkabr Unerty o ehnoogy Computer Engneerng Inormaton ehnoogy Department