Forward Kinematics 1

Transcription

1 Forward Knemat

2 lnk 2 Lnk and Jont jont 3 jont 4 jont n- jont n jont lnk 3... lnk n- lnk lnk n jont 2 lnk n jont, n + lnk lnk fxed (the bae) q d revolute prmat jont onnet lnk to lnk lnk move when jont atuated 2

3 Forward Knemat gven the jont varable and dmenon of the lnk what the poton and orentaton of the end effetor? p? a 2 y 2 a x 3

4 Forward Knemat beaue the bae frame and frame have the ame orentaton, we an um the oordnate to fnd the poton of the end effetor n the bae frame (a o + a 2 o ( + 2 ), a n + a 2 n ( + 2 ) ) y a 2 ( a 2 o ( + 2 ), y 2 x a 2 n ( + 2 ) ) a x ( a o, a n ) 4

5 Forward Knemat from earler n the oure p = (a o + a 2 o ( + 2 ), a n + a 2 n ( + 2 ) ) x 2 = (o ( + 2 ), n ( + 2 ) ) y 2 x 2 y 2 = (-n ( + 2 ), o ( + 2 ) ) y a 2 2 a x 5

6 Frame y 2 x 2 y a 2 y x 2 a x 6

7 Forward Knemat ung tranformaton matre R z D x, a, R z D x 2, a 2,

8 lnk 2 Lnk and Jont jont 3 jont 4 jont n- jont n jont lnk 3... lnk n- lnk lnk n jont 2 lnk n jont, n + lnk lnk fxed (the bae) q d revolute prmat jont onnet lnk to lnk lnk move when jont atuated 8

9 Forward Knemat attah a frame {} to lnk all pont on lnk are ontant when expreed n {} f jont atuated then frame {} move relatve to frame { - } moton derbed by the rgd tranformaton the tate of jont a funton of t jont varable q (.e., a funton of q ) th make t eay to fnd the lat frame wth repet to the bae frame ( q ) 2 n n 2 3 n 9

10 Forward Knemat more generally j j2 I f j f j j j j f j the forward knemat problem ha been redued to matrx multplaton

11 Forward Knemat Denavt J and Hartenberg RS, A knemat notaton for lowerpar mehanm baed on matre. ran ASME J. Appl. Meh, 23:25 22, 955 derbed a onventon for tandardzng the attahment of frame on lnk of a eral lnkage ommon onventon for attahng referene frame on lnk of a eral manpulator and omputng the tranformaton between frame

12 Denavt-Hartenberg 2 x a x d z z R R,,,, d a a jont angle lnk offet lnk twt lnk length d a

13 Denavt-Hartenberg 3

14 Denavt-Hartenberg 4 note the form of the rotaton omponent th doe not look lke t an repreent arbtrary rotaton an the DH onventon atually derbe every phyally poble lnk onfguraton?

15 Denavt-Hartenberg ye, but we mut hooe the orentaton and poton of the frame n a ertan way (DH) (DH2) xˆ zˆ xˆ nteret zˆ lam: f DH and DH2 are true then there ext unque number a, d,, uh that R z, Dz, d Dx, a R x, 5

16 Denavt-Hartenberg proof: on blakboard n la 6

17 7 Denavt-Hartenberg

18 DH Parameter a : lnk length dtane between z - and z meaured along x : lnk twt angle between z - and z meaured about x d : lnk offet dtane between o - to the ntereton of x and z - meaured along z - : jont angle angle between x - and x meaured about z - 8

19 Example wth Frame Already Plaed 9

20 Step 5: Fnd the DH parameter Lnk a d d * 2-9 d 2 * 3 d 3 * * jont varable 2

21 Denavt-Hartenberg Forward Knemat RPP ylndral manpulator 2

22 Denavt-Hartenberg Forward Knemat How do we plae the frame? 22

23 Step : Chooe the z-ax for eah frame 23 reall the DH tranformaton matrx x a x d z z R R,,,, d a a ˆ x ˆ y ˆ z

24 Step : Chooe the z-ax for eah frame ẑ ax of atuaton for jont + lnk lnk + lnk lnk + ẑ jont + jont + ẑ 24

25 Step : Chooe the z-ax for eah frame Warnng: the pture deevng. We do not yet know the orgn of the frame; all we know at th pont that eah z pont along a jont ax 25

26 Step 2: Etablh frame {} plae the orgn o anywhere on z often the hoe of loaton obvou hooe x and y o that {} rght-handed often the hoe of dreton obvou 26

27 Step 2: Etablh frame {} 27

28 Step 3: Iteratvely ontrut {}, {2},... {n-} ung frame {-} ontrut frame {} DH: x perpendular to z - DH2: x nteret z - 3 ae to onder dependng on the relatonhp between z - and z 28

29 Step 3: Iteratvely ontrut {}, {2},... {n-} Cae z - and z are not oplanar (kew) ˆ z hortet lne between zˆ and ẑ a ẑ (out of page) xˆ o pont of ntereton angle from z - to z meaured about x 29

30 Step 3: Iteratvely ontrut {}, {2},... {n-} Cae 2 z - and z are parallel ( = ) ˆ z ẑ o a xˆ o pont of ntereton note that th hoe reult n d = 3

31 Step 3: Iteratvely ontrut {}, {2},... {n-} Cae 3 z - and z nteret ( a = ) ˆ z o pont of ntereton xˆ (out of page) ẑ 3

32 Step 3: Iteratvely ontrut {}, {2},... {n-} 32

33 Step 3: Iteratvely ontrut {}, {2},... {n-} 33

34 Step 4: Plae the end effetor frame ldng normal approah 34

35 Step 4: Plae the end effetor frame 35

36 Step 5: Fnd the DH parameter a : dtane between z - and z meaured along x : angle between z - and z meaured about x d : dtane between o - to the ntereton of x and z - meaured along z - : angle between x - and x meaured about z - 36

37 Step 5: Fnd the DH parameter Lnk a d d * 2-9 d 2 * 3 d 3 * * jont varable 37

38 38 More Denavt-Hartenberg Example

39 Step 5: Fnd the DH parameter Lnk a d d * 2-9 d 2 * 3 d 3 * * jont varable 39

40 Step 6: Compute the tranformaton 4 one the DH parameter are known, t eay to ontrut the overall tranformaton Lnk a d d * 2-9 d 2 * 3 d 3 * * jont varable,,,, d R R x a x d z z

41 Step 6: Compute the tranformaton 4 Lnk a d d * 2-9 d 2 * 3 d 3 * * jont varable 2,,,, d R R x a x d z z

42 Step 6: Compute the tranformaton 42 Lnk a d d * 2-9 d 2 * 3 d 3 * * jont varable 3,,,, d R R x a x d z z

43 Step 6: Compute the tranformaton d d d d

44 Spheral Wrt 44

45 Spheral Wrt 45

46 Spheral Wrt: Step 46

47 Spheral Wrt: Step 2 47

48 Spheral Wrt: Step 2 48

49 Spheral Wrt: Step 4 49

50 Step 5: DH Parameter Lnk a d * * 6 d 6 6 * * jont varable 5

51 Step 6: Compute the tranformaton d d d

52 RPP + Spheral Wrt 52

53 RPP + Spheral Wrt z y x d r r r d r r r d r r r d d d d r z

54 Stanford Manpulator + Spheral Wrt Lnk a d -9 * 2 9 d 2 2 * 3 d 3 * * * 6 d 6 6 * * jont varable 54

55 SCARA + DOF Wrt 55 Lnk a d a d * 2 a * 3 d 3 * 4 d 4 4 * * jont varable