Inverse Kinematics 1 1/21/2018

Transcription

1 Invere Kinemati 1

2 Invere Kinemati 2 given the poe of the end effetor, find the joint variable that produe the end effetor poe for a -joint robot, given find 1 o R T 3 2 1,,,,, q q q q q q

3 RPP + Spherial Writ 3

4 RPP + Spherial Writ olving for the joint variable diretly i hard z y x d r r r d r r r d r r r T T T d d d d r z

5 Kinemati Deoupling for -joint robot where the lat 3 joint intereting at a point (e.g., lat 3 joint are pherial writ) there i a impler way to olve the invere kinemati problem 1. ue the interetion point (writ enter) to olve for the firt 3 joint variable invere poition kinemati 2. ue the end-effetor poe to olve for the lat 3 joint variable invere orientation kinemati

6 RPP Cylindrial Manipulator d 3 d 2 o d 1 z y x Given o = x y z find θ 1, d 2, d 3

7 RPP Cylindrial Manipulator d 3 d 2 o d 1 z y x 7

8 RPP Cylindrial Manipulator d 3 d 2 o d 1 z y x 8

9 RPP Cylindrial Manipulator d 3 d 2 o d 1 z y x 9

10 RPP Cylindrial Manipulator d 3 d 2 o d 1 z y x 1

11 RRP Spherial Manipulator Given o = x y z find θ 1, θ 2, d 3 11

12 RRP Spherial Manipulator 12

13 RRP Spherial Manipulator 13

14 RRP Spherial Manipulator 1

15 RRP Spherial Manipulator 1

16 Spherial Writ Link a i a i d i q i -9 q * 9 q * d q * * joint variable 1

17 Spherial Writ d d d T T T T

18 Spherial Writ o o d o o d R 1 18

19 Invere Kinemati Reap 1. Solve for the firt 3 joint variable q 1, q 2, q 3 uh that the writ enter o ha oordinate o 2. Uing the reult from Step 1, ompute R 3 3. Solve for the writ joint variable q, q, q orreponding to the rotation matrix R o d R T R 3 3 R 1 19

20 Spherial Writ 2 for the pherial writ d d d T T T T neg po, 1 atan2, 1 atan2 if r r r r q q

21 Spherial Writ d d d T T T T po, atan2, atan2, for r r r r q q q

22 Spherial Writ d d d T T T T neg, atan2, atan2, for r r r r q q q

23 Spherial Writ 23 if θ = d d d T T T T 1 1 d

24 Spherial Writ 2 ontinued from previou lide 1 1 d 1 1 d only the um θ +θ an be determined

25 Uing Invere Kinemati in Path Generation 2

26 Path Generation a path i defined a a equene of onfiguration a robot make to go from one plae to another a trajetory i a path where the veloity and aeleration along the path alo matter 2

27 Joint-Spae Path a joint-pae path i omputed onidering the joint variable link 2 link 1 end effetor path 27

28 Joint-Spae Path Joint Angle linear joint-pae path link 1 link 2 28

29 Joint-Spae Path given the urrent end-effetor poe and the deired final end-effetor poe find a equene of joint angle that generate the path between the two poe idea T f T olve for the invere kinemati for the urrent and final poe to get the joint angle for the urrent and final poe interpolate the joint angle 29

30 Joint-Spae Path 3 T T f q n q q Q 2 1 n f f q q q Q 2 1 invere kinemati invere kinemati

31 Joint-Spae Path find Q from T find f Q from f T t = 1 / m Q = f Q Q for j = 1 to m t j = j t j Q = Q + t j Q et joint to j Q end 31

32 Joint-Spae Path linearly interpolating the joint variable produe a linear joint-pae path a non-linear Carteian path depending on the kinemati truture the Carteian path an be very ompliated ome appliation might benefit from a imple, or well defined, Carteian path 32

33 Carteian-Spae Path a Carteian-pae path onider the poition of end-effetor link 2 link 1 end effetor path 33

34 Carteian-Spae Path Joint Variable 1 non-linear joint-pae path 3

35 Carteian-Spae Path Joint Variable 2 non-linear joint-pae path 3

36 Iue with Carteian-Spae Path 3

37 Joint Veloity Iue onider the RR robot hown below aume that the eond joint an rotate by ±18 degree 37

38 Joint Veloity Iue what happen when it i ommanded to follow the traight line path hown in red? 38

39 Joint Veloity Iue 39

40 Joint Veloity Iue jump diontinuity in firt derivative = infinite rotational aeleration teep lope = high rotational veloity

41 Workpae the reahable workpae of a robot i the volume wept by the end effetor for all poible ombination of joint variable i.e., it i the et of all point that the end effetor an be moved to 1

42 Workpae onider the RR robot hown below aume both joint an rotate by 3 degree

43 Workpae rotating the eond joint through 3 degree weep out the et of point on the dahed irle

44 Workpae rotating the firt and eond joint through 3 degree weep out the et of all point inide the outer dahed irle

45 Workpae workpae onit of all of the point inide the gray irle

46 Workpae workpae onit of all of the point inide the gray irle

47 Workpae onider the RR robot hown below where the eond link i horter than the firt aume both joint an rotate by 3 degree

48 Workpae rotating the eond joint through 3 degree weep out the et of point on the dahed irle

49 Workpae workpae onit of all of the point inide the gray area

50 Workpae onider the following traight line path hown in red tart point, end point, and all point in between are reahable

51 Workpae onider the following traight line path hown in red tart point and end point are reahable, but ome point in between are not reahable

52 Path atifying end point ontraint 2

53 Joint-Spae Path a joint-pae path i omputed onidering the joint variable link 1 link 2 end effetor path 3

54 Joint-Spae Path Joint Angle linear joint-pae path link 1 link 2

55 Contraint in the previou example we had two ontraint for joint 1: θ 1 = f θ 1 = 27 the implet path atifying thee ontraint i the traight line path if we add more ontraint then a traight line path may not be able to atify all of the ontraint

56 Veloity ontraint a ommon ontraint i that the robot tart from a tationary poition and top at a tationary poition in other word, the joint veloitie are zero at the tart and end of the movement 3. dθ1 dt = θ 1 =. f dθ1 dt = f θ 1 = more generally, we might require non-zero veloitie 3.. dθ1 dt f dθ1 dt = θ 1 = v = f θ 1 = f v

57 Aeleration ontraint for mooth motion, we might require that the aeleration at the tart and end of the motion be zero. d 2 θ 1 dt 2 = θ 1 =. f d 2 θ 1 dt 2 = f θ 1 = more generally, we might require non-zero aeleration. d 2 θ 1 dt 2 = θ 1 = α. f d 2 θ 1 dt 2 = f θ 1 = f α 7

58 Satifying the ontraint given ome et of ontraint on a joint variable q our goal i to find q t that atifie the ontraint there are an infinite number of hoie for q t it i ommon to hooe imple funtion to repreent q t 8

59 Satifying the ontraint with polynomial uppoe that we hooe q t to be a polynomial if we have n ontraint then we require a polynomial with n oeffiient that an be hoen to atify the ontraint in other word, we require a polynomial of degree n 1 9

60 Satifying the ontraint with polynomial uppoe that we have joint value and joint veloity ontraint 1. q t = q 2. q t f = q f 3.. q t = v q t f = v f we require a polynomial of degree 3 to repreent q t q t = a + bt + t 2 + dt 3 the derivative of q t i eay to ompute q t = b + 2t + 3dt 2

61 Satifying the ontraint with polynomial equating q t and q t to eah of the ontraint yield: 1. q t = q = a + bt + t 2 + dt 3 2. q t f = q f = a + bt f + t f 2 + dt f q t = v = b + 2t + 3dt 2 q t f = v f = b + 2t f + 3dt f 2 whih i a linear ytem of equation with unknown (a, b,, d) 1

62 Example onider the following ontraint where the robot i tationary at the tart and end of the movement 1. q t = θ = 1 2. q t f = θ 3 = q t = θ = q t f = θ 3 = 2

63 Example: Joint angle ubi θ 3 = 8 θ = 1 3

64 Example: Joint veloity quadrati θ = θ 3 =

65 Example: Joint aeleration linear

66 Satifying the ontraint with polynomial uppoe that we have joint value, joint veloity, and joint aeleration ontraint 1. q t = q 2. q t f = q f q t = v q t f = v f q t = α q t f = α f

67 Satifying the ontraint with polynomial we require a polynomial of degree to repreent q t q t = a + bt + t 2 + dt 3 + et + ft the derivative of q t are eay to ompute q t = b + 2t + 3dt 2 + et 3 + ft q t = 2 + dt + 12et 2 + 2ft 3 7

68 Satifying the ontraint with polynomial equating q t, q t, and q t to eah of the ontraint yield: 1. q t = q = a + bt + t 2 + dt 3 2. q t f = q f = a + bt f + t f 2 + dt f q t 2 = v = b + 2t + 3dt q t f 2 = v f = b + 2t f + 3dt f q t = α = 2 + dt + 12et ft q t f = α f = 2 + dt f + 12et 2 3 f + 2ft f whih i a linear ytem of equation with unknown (a, b,, d, e, f) 8

69 Example onider the following ontraint where the robot i tationary at the tart and end of the movement, and the joint aeleration are zero at the tart and end of the movement 1. q t = θ = 1 2. q t f = θ 3 = q t = θ = q t f = θ 3 = q t = θ = q t f = θ 3 = 9

70 Example: Joint angle quinti θ 3 = 8 θ = 1 7

71 Example: Joint veloity quarti θ = θ 3 = 71

72 Example: Joint aeleration ubi θ = θ 3 = 72