Velocity Propagation Between Robot Links 3/4

Transcription

1 eloty Propagaton etween obot nk / Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

2 Introduton eloty Propagaton Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

3 Jaoban Matrx - Introduton In the feld of robot the Jaoban matrx derbe the relatonhp between the jont angle rate ( ) and the tranlaton and rotaton elote of the end effetor ( ). Th relatonhp gen by: N x x J In addton to the eloty relatonhp, we are alo ntereted n deelopng a relatonhp between the robot jont torque ( fore and moment ( ) and the ) at the robot end effetor (Stat Condton). Th relatonhp gen by: F F T J F Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

4 Jaoban Matrx - Calulaton Method Dfferentaton the Forward Knemat Eq. Iterate Propagaton (elote or Fore / Torque) Jaoban Matrx Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

5 Summary Changng Frame of epreentaton near and otatonal eloty etor Form Matrx Form ngular eloty etor Form Matrx Form Q Q OG Q P Q Q OG Q P Q P T C C C C Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

6 Frame - eloty wth any etor, a eloty etor may be derbed n term of any frame, and th frame of referene noted wth a leadng uperrpt. eloty etor omputed n frame {} and repreented n frame {} would be wrtten epreented (eferene Frame) Q ( Q ) d dt P Q Computed (Meaured) OG Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

7 Poton Propagaton The homogeneou tranform matrx prode a omplete derpton of the lnear and angular poton relatonhp between adjaent lnk. Thee derpton may be ombned together to derbe the poton of a lnk relate to the robot bae frame {}. o T T T o T mlar derpton of the lnear and angular elote between adjaent lnk a well a the bae frame would alo be ueful. Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

8 Poton Propagaton 5 6 TT T T T T 5T 6T TT Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

9 Moton of the nk of a obot In onderng the moton of a robot lnk we wll alway ue lnk frame {} a the referene frame (Computed ND epreented). Howeer any frame an be ued a the referene (repreented) frame nludng the lnk own frame () Where: - the lnear eloty of the orgn of lnk frame () wth repet to frame {} (Computed ND epreented) - the angular eloty of the orgn of lnk frame () wth repet to frame {} (Computed ND epreented) Expreng the eloty of a frame {} (aoated wth lnk ) relate to the robot bae (frame {}) ung our preou notaton defned a follow: Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

10 elote - Frame & Notaton The elote dfferentate (omputed) relate to the bae frame {} are often repreented relate to other frame {k}. The followng notaton ued for th ondton k k k k k k k k Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

11 eloty Propagaton Gen: manpulator - han of rgd bode eah one apable of mong relate to t neghbor Problem: Calulate the lnear and angular elote of the lnk of a robot Soluton (Conept): Due to the robot truture (eral mehanm) we an ompute the elote of eah lnk n order tartng from the bae. The eloty of lnk + wll be that of lnk, plu whateer new eloty omponent were added by jont + Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

12 eloty of djaent nk - ngular eloty /5 Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

13 eloty of djaent nk - ngular eloty /5 From the relatonhp deeloped preouly C C we an re-agn lnk name to alulate the eloty of any lnk relate to the bae frame {} C y pre-multplyng both de of the equaton by,we an onert the frame of referene for the bae {} to frame {+} Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

14 eloty of djaent nk - ngular eloty /5 Ung the reently defned notaton, we hae - ngular eloty of frame {+} meaured relate to the robot bae, and expreed n frame {+} - eall the ar example - ngular eloty of frame {} meaured relate to the robot bae, and expreed n frame {+} - ngular eloty of frame {+} meaured relate to frame {} and expreed n frame {+} w Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

15 eloty of djaent nk - ngular eloty /5 ngular eloty of frame {} meaured relate to the robot bae, expreed n frame {+} Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

16 eloty of djaent nk - ngular eloty /5 ngular eloty of frame {+} meaured (dfferentate) n frame {} and repreented (expreed) n frame {+} umng that a jont ha only DOF. The jont onfguraton an be ether reolute jont (angular eloty) or prmat jont (near eloty). aed on the frame attahment onenton n whh we agn the Z ax pontng along the + jont ax uh that the two are onde (rotaton of a lnk preformed only along t Z- ax) we an rewrte th term a follow: + Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

17 eloty of djaent nk - ngular eloty 5/5 The reult a reure equaton that how the angular eloty of one lnk n term of the angular eloty of the preou lnk plu the relate moton of the two lnk. Sne the term depend on all preou lnk through th reuron, the angular eloty ad to propagate from the bae to ubequent lnk. Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

18 eloty of djaent nk - near eloty /6 Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

19 eloty of djaent nk - near eloty /6 Smultaneou near and otatonal eloty The derate of a etor n a mong frame (lnear and rotaton elote) a een from a tatonary frame etor Form Matrx Form Q Q OG Q P Q Q OG Q P Q P Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

20 eloty of djaent nk - near eloty /6 From the relatonhp deeloped preouly (matrx form) we re-agn lnk frame for adjaent lnk ( and +) wth the eloty omputed relate to the robot bae frame {} y pre-multplyng both de of the equaton by,we an onert the frame of referene for the left de to frame {+} Q Q OG Q P C P Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

21 eloty of djaent nk - near eloty /6 Whh mplfe to Fatorng out from the left de of the frt two term P P P Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

22 eloty of djaent nk - near eloty /6 - near eloty of frame {+} meaured relate to frame {} and expreed n frame {+} umng that a jont ha only DOF. The jont onfguraton an be ether reolute jont (angular eloty) or prmat jont (near eloty). aed on the frame attahment onenton n whh we agn the Z ax pontng along the + jont ax uh that the two are onde (tranlaton of a lnk preformed only along t Z- ax) we an rewrte th term a follow: P d Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

23 eloty of djaent nk - near eloty 5/6 d P T Multply by Matrx Defnton Defnton Defnton Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

24 eloty of djaent nk - near eloty 6/6 The reult a reure equaton that how the lnear eloty of one lnk n term of the preou lnk plu the relate moton of the two lnk. Sne the term depend on all preou lnk through th reuron, the angular eloty ad to propagate from the bae to ubequent lnk. d P Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

25 eloty of djaent nk - Summary ngular eloty near eloty d P - Prmat Jont - eolute Jont Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

26 ngular and near elote - obot - Example For the manpulator hown n the fgure, ompute the angular and lnear eloty of the tool frame relate to the bae frame expreed n the tool frame (that, alulate and ). Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

27 ngular and near elote - obot - Example Frame attahment Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

28 ngular and near elote - obot - Example DH Parameter a d 9 Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

29 ngular and near elote - obot - Example From the DH parameter table, we an pefy the homogeneou tranform matrx for eah adjaent lnk par: d d a T T T T T Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

30 ngular and near elote - obot - Example Compute the angular eloty of the end effetor frame relate to the bae frame expreed at the end effetor frame. For = Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

31 ngular and near elote - obot - Example For = For = For = Note Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

32 ngular and near elote - obot - Example Compute the lnear eloty of the end effetor frame relate to the bae frame expreed at the end effetor frame. Note that the term nolng the prmat jont ha been dropped from the equaton (t equal to zero). d P Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

33 ngular and near elote - obot - Example For = For = P P Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

34 ngular and near elote - obot - Example For = ) ( P Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

35 ngular and near elote - obot - Example For = ) ( ) ( ) ( P Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

36 ngular and near elote - obot - Example Note that the lnear and angular elote ( ) of the end effetor where dfferentate (meaured) n frame {} howeer repreented (expreed) n frame {} In the ar example: Oberer ttng n the Car Oberer ttng n the World Sole for and by multply both de of the queton from the left by k k k k k k k k, C W C C W W Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

37 ngular and near elote - obot - Example Multply both de of the equaton by the nere tranformaton matrx, we fnally get the lnear and angular elote expreed and meaured n the tatonary frame {} T T T T T T T Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

38 ngular and near elote - obot - Example Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

39 ngular and near elote - obot - Example Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

40 ngular and near elote - obot - Example Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

41 ngular and near elote - obot - Example Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

42 ngular and near elote - obot - Example Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

43 ngular and near elote - obot - Example Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

44 ngular and near elote - obot - Example ) ( Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC

45 ngular and near elote - obot - Example ) ( ) ( Intrutor: Jaob oen daned obot - ME 6D - Department of Mehanal & eropae Engneerng - UC