Introduction to Humanoid Robot Kinematics with the NAO H25 technical report
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1 Research Collection Report Introduction to Humanoid Robot Kinematics with the NAO H technical report Author(s): Wadehn, Federico; Diether, Salomon Publication Date: 01 Permanent Link: Rights / License: In Copyright - Non-Commercial Use Permitted This page was generated automatically upon download from the ETH Zurich Research Collection. For more information please consult the Terms of use. ETH Library
2 Technical Report Introduction to Humanoid Robot Kinematics with the NAO H Federico Wadehn Salomon Diether wadehn@ee.ethz.ch Institution: Swiss Federal Institute of Technology (ETH), Zurich, Switzerland. Date: 0/11/01
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7 ~q ~q =( 1,,..., n )
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9 ~r i ~r o m I 0! ~ (t) ~x (t) ~x (t) ~ (t) ~ ~ ~ (t) I 0 ~ = ~ ~ V ~
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11 ~f s : ~ J (t) 7! ~x T (t), f s,inv : ~x T (t) 7! ~ J (t) ~f v : ~ J (t) 7! ~x T (t), f v,inv : ~x T (t) 7! ~ J (t) ~f a : ~ J (t) 7! ~x T (t), f a,inv : ~x T (t) 7! ~ J (t) ~x T (t), ~x T (t), ~x T (t) R R ~ J (t), ~ J (t), ~ J (t) R n ~ =( 1,..., n ) n ~s R ~s = ~s ( ) ~ R ~ ~s x e = l 1 cos( 1 )+l cos( 1 + ) y e = l 1 sin( 1 )+l sin( 1 + ) xe = ~s ( 1, )=~s ( ) ~ y e
12 ~ ~q ~d =(d 1,...,d n ) O 0 X 0 Y 0 Z 0 O 1 X 1 Y 1 Z 1 O 1 X 1 Y 1 Z 1 O 0 X 0 Y 0 Z 0 (,, ) R ZXZ R ZXZ = R z, R x, R z, c s 0 s c c s 0 s c 1 0 c c s c s c s s c c s s s c + c c s s s + c c c c s s s s c c c s 0 s c A 1 A
13 R roll,pitch,yaw = R roll, R pitch, R yaw, 0 c 1 0 s 0 s c 0 c 0 s s 0 c c s 0 s c c c s c + c s s s s + c s c s c c c + s s s c s + s s c s c s c c 1 A 1 A L j c j L j
14 S j S j+1 ~r j,j ~r j,j+1 L j Sj Sj+1 ~e j ~e j+1 S j s j = {0, 1} s j =0 s j =1 S j S j d j ~r j,j ~r j 1,j ~e j j ) ~r j 1,j k ~e j L j 1 ~r j 1,j ~e j ~r j 1,j ~e j ~r j 1,j ) ~r j,j k ~e j L j 1 ~r j,j ~e j ~e j j L j ~d j ~ dj
15 ~ dj ~r j,j ~r j,j+1 ~e j ~e j+1 s j j S j j =1,...,n ~d j A j 1,j ~ dj = A j 1,j ~ dj A, A j = A 0,j = A 0,1 A 1,...A j 1,j ~ dj ~d j = A j ~ dj A j 1,j ~e j ~e j ~e j ~e j ~r j 1,j j =0 ~a j S j ~a j = ~e j (~r j 1,j ~e j ) ~e j (~r j 1,j ~e j ), ~a j = ~ e j (~r j 1,j ~e j ) ~e j (~r j 1,j ~e j ) ~ bj = ~e j ~a j ~ bj = ~e j ~a j, ~ bj = ~e j ~a j ~e j = A 0 j 1,j ~ e j, ~a j = A 0 j 1,j ~ a j, ~ bj = A 0 j 1,j~ bj, A 0 j 1,j j =0 A 0 j 1,j ~e j,~a j, ~ b j i h i h~e j ~a ~ j bj = A 0 j 1,j ~ e j ~a ~ j bj A 0 j 1,j A 0 j 1,j = A j 1,j (0) = h ih i ~e j ~a ~ j bj ~ e j ~a ~ 1 h. j bj = ~V j1 (0) V ~ j (0) V ~ i j (0) j ~ V jk ( j ),k=1,, ~V jk ( j )= ~ V jk (0) cos j +(1 cos j )(~e j ~ V jk (0)) ~e j + ~e j ~ V jk (0) sin j,k=1,, h A j 1,j ( j )= ~V j1 ( j ) V ~ j ( j ) V ~ j ( j )i S j j
16 A j ( 1,,..., n )=A j ( ~ ) A j ( ~ ) ~x = ~ f( 1,,..., n )= ~ f( ~ ), ~x R /R, ~ R n Z j S j X j L j L j 1 L j A j,j 1 =[Z j X j ] A n A n =[Z 1 X 1 ] [Z X ]... [Z n X n ] S j z j 1 z j x j L j z j y j x j z j Z j X j L j O j 1 x j 1 y j 1 z j 1 O j x j y j z j d j x j 1 x j j z x j 1 x j r j z j 1 z j j z j 1 z j
17 X j L j r r j X j = 6 0 cos j sin j sin j cos j Z j Z j = 6 4 S j cos j sin j 0 0 sin j cos j d j j d j z j 1 S j A j,j 1 O j 1 x j 1 y j 1 z j 1 O j x j y j z j A j,j 1 = 6 4 cos j sin j cos j sin j sin j r j cos j sin j cos j cos j cos j sin j r j sin j 0 sin j cos j d j = 6 4 R T A 1 j,j 1 = 6 4 R T R T T A j,j 1 R T R T ij = R ji i, j L i,l j A 1 j,j 1 7
18 z j 1 z j d j
19 ~s = ~s ( ~ )= ~ f( ~ ) ~ = ~ f 1 (~s ) ~f 1 st 1 st 1 st
20 ~f( ~ ) ~s = J( ~ ) ~ ~s, ~,~t ~t ~ ~, ~ + ~ ~s t J ~ ~s ( ~ )= ~ f( ~ )= ~ f( ~ 0 )+J f( ~ 0) (~ ~ 0 )+o(( ~ ~ 0 ) )=) ~s ( ~ ) ~s ( ~ 0 ) t J f( ~ 0) (~ ~ 0 ) ~f( ~ ) ~s t J f( ~ 0) ~ ~ ~s t ~e. ~e ~ v e ~. J( ~ ) ~v e = J( ~ ) ~ = J ~ = J J n n ~ s J det(j) =0. J J J det(j)
21 A~x = ~ b A~x ~ b A T A~x = A T ~ b ~ b A~x A J R k n J = UDV T k k D k n V n n i r i 6= 0 J
22 A~x = ~ b ~x = A ~ b A AA A = A A AA = A (AA ) = AA AA (A A) = A A A A A = lim!0 (A A + E) 1 A = lim!0 (A (AA + E) 1 ) A = VD U ~s = ~e = J f( ~ 0) ~ =) ~ = J f( ~ 0) ~e J k = {, } J J R n J ~ = J f( ~ 0) ~e ~e J f( ~ 0) ~ = J f( ~ 0) ~e ~ = ~e ~e J ~ ~e
23 (I J J) J. ~ = J f( ~ 0) ~e +(I J J)~v ~v J(I J J)~v =0 ~v ~ J T ~ = J T ~e J T J J T ~s ~e ~s = JJ T ~e < ~s, ~e > > 0 <~e, ~s > = ~e (~e J J T )=~e (J T ~e ) T J T = (J T ~e ) T (J T ~e )=<J T ~e, J T ~e > = J T ~e > 0. ~s t ~e. = <~e,jjt ~e > <JJ T ~e, J J T ~e > ~s = JJ T ~e t ~e ~s = <~e,jjt ~e > <JJ T ~e, J J T ~e > JJT ~e = <JJT ~e, J J T ~e > <JJ T ~e, J J T ~e > ~e = ~e =0. min J ~ ~e + ~ ~ ~ J ~ = ~e
24 min ~ J I ~ ~e 0 ~ J I T J I J ~ = I T ~e 0 () (J T J + I) ~ = J T ~e (J T J + I) 1 J T = J T (JJ T + I) 1 k k ~ = J T (JJ T + I) 1 ~e () J T 1 ~ = J T (JJ T + I) 1 ~e JJ T + I J U J = USV T =) J T J + I =(VSU T )(USV T )+ I = U(SS T + I)U T SS T + I i +.! f = J T 1 ~ (JJ T + I) ~ f = ~e J T ~ f = ~. =
25 j ˆx j j ~x cos( j )= ~p e ~p j p e p j ~p t ~p j p t p j ~p j ~p t ~p e ~r = ~p e ~p j ~p e ~p j ~p t ~p j ~p t ~p j 1.
26 J = VD U T = rx i=1 1 i v i u T i J T (JJ T + I) 1 = VD T (DD T + I) 1 U T = V diag( i i + ) U J T (JJ T + I) 1 = rx i=1 i i + v i u T i D =0 1 i i! 0. i!!!!!!!!
27 !!!!!!!! 1
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29 ~e
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32 f clamp ( ~e ) = ( E max, for ~e >E max ~e, else
33 ~s = ~ f( ) =) ~v = ~s ( ~ )= d dt ~ f( ~ )=J ~f( ~ ) ~ ~v ~ = J 1 ~f( ~ ) ~v ~a (t) = d dt ~v (t) =(d dt J ~ f( ~ ) ) + J ~f( ~ ) ~s = J ~f( ~ ) ~ + A( ~, ~ ) ~ ~F = m~a = m ~x = I 0! = I 0
34 T 0,crit T 0! T 0 v max a max v max
35 Ĩ ~q ~ = d dt (Ĩ~!) ~! ~F = m ~q ˆ I = V ~ (~r ) ~ r? dv ~ (~r ) ~r? Ĩ = 4 I 11 I 1 I 1 I 1 I I I 1 I I I 11 = P N k=1 m k(y k + z k ) I = P N k=1 m k(x k + z k ) I = P N k=1 m k(x k + y k ) I 1 = I 1 = I = P N P N k=1 m k(x k y k ) k=1 m k(x k z k ) P N k=1 m k(y k z k )
36 0 I COM ~a a 1 a a 1 A r I parallel = I COM + mr Ĩ 0 Ĩ parallel = ĨCOM + a + a a 1 a a 1 a a 1 a a 1 + a a a a 1 a a a a 1 + a m 1 A P L L = K P K d q j, q j q j j, q j, q j j q j j =1,...,n ~f ~ m 0 = 0 Ĩ c ~q ~! + ~0 ~! Ĩc ~! Ĩ c ~ f ~ ~! ~q. i i +1. ~a c,i ~! i i 0
37 ~ i i 0 ~g i ~f i+1 i i +1 ~ i+1 i i +1 Ri+1 i i +1 i m i i Ĩ i ~r i,ci ~r i+1,ci i +1 i ~f i R i i+1 ~ f i+1 + m i ~g i = m i ~a c,i ~ i R i i+1~ i+1 + ~ f i ~r i,ci (R i i+1 ~ f i+1 ) ~r i+1,ci = ~ i + ~! i (Ĩi~! i ) ~r ~ F ~f i+1 ~r ~ F R i i+1 ~s ~s ~s ~ 1 A B ~ n C A 7! 0 V 1 V n 1 C A.
38 ~f n+1 =0 ~ n+1 =0 i +1. i, i, i a c,i,~! i,~ i ~! 0 =0,~ 0 =0
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43 L j L j 1 L j 1 L j L j 1 L j 1 L j 1 L j x z y y
44 8.1 Kinematic chain model - Aldebaran NAO H z y x A (0,1,-1) Joint 9 (0,-1,-1) Joint Remark: All joints with (1,0,0) are displayed like this without further comment. 4
45 z Kinematic chain model - Aldebaran NAO H x y e 14 e 18 r 1,14 r 1,18 id = 1 torso m = 179. g r o1 = (0, 18, 140) mm r o = (0, -18, 140) mm r o = (0, 7, -100) mm r o4 = (0, -7, -100) mm r 1, A r 1,8 COM here because actual masses of Head, 1, 14 and 18 combined! id = right pelvis m = 06. g r i = (0, -4, -) mm r o = (0, 8, 10) mm e r, e r, (1,0,0) Joint e r, r,4 (0,-1,-1) Joint (0,1,-1) Joint COM here because actual mass of and combined! id = right hip (zero length) m = 0 g r i = (0, 0, 0) mm r o = (0, 0, 0) mm e e 8 COM here because actual mass of 8 and 9 combined! id = 9 left hip (zero length) m = 0 g r i = (0, 0, 0) mm r o = (0, 0, 0) mm e 8 (1,0,0) Joint e 9 r 9,9 r 8,8 id = 8 left pelvis m = 06. g r i = (0, 4, -) mm r o = (0, -8, 10) mm r 8,9 e 9 r 9,10 e 10 e 4 e 4 e 10 r 4,4 id = 4 right thigh id = 10 left thigh r 10,10 m = 94. g r i = (0, -, -0) mm r o = (0, -, 0) mm m = 94. g r i = (0,, -0) mm r o = (0,, 0) mm r 4, e e 11 r 10,11 e e 11 id = right tibia m = 40. g r i = (0, 0, -64) mm r o = (0, 0, 9) mm r, r,6 COM here because actual mass of and 6 combined! COM here because actual mass of 11 and 1 combined! r 11,11 r 11,1 id = 11 left tibia m = 40. g r i = (0, 0, -64) mm r o = (0, 0, 9) mm e 6 44 e 1
46 z Kinematic chain model - Aldebaran NAO H x y id = 6 id = 1 right ankle left ankle (zero length) e 6 (zero length) e 1 m = 0 g r i = (0, 0, 0) mm r o = (0, 0, 0) mm r 6,6 m = 0 g r i = (0, 0, 0) mm r o = (0, 0, 0) mm r 1,1 (1,0,0) Joint r 6,7 (1,0,0) Joint r 1,1 e 7 e 1 (1,0,0) Joint (1,0,0) Joint e 7 e 1 id = 7 right foot r 7,7 id = 1 left foot r 1,1 m = g r i = (, -, -) mm m = g r i = (,, -) mm e 14 id = 14 right shoulder (zero length) id = 18 left shoulder (zero length) e 18 r 14,14 m = 0 g r i = (0, 0, 0) mm r o = (0, 0, 0) mm m = 0 g r i = (0, 0, 0) mm r o = (0, 0, 0) mm r 18,18 r 14,1 r 18,19 (1,0,0) Joint e 1 (1,0,0) Joint e 19 (1,0,0) Joint e 1 r 1,1 e 16 r 1,16 e 16 id = 1 right biceps m = 18.7 g r i = (0, -6, -40) mm r o = (0, 0, 6) mm COM here because actual mass of 1 and 16 combined! id = 16 right elbow (zero length) m = 18.7 g r i = (0, -6, -40) mm r o = (0, 0, 6) mm id = 0 left elbow (zero length) id = 19 left biceps (1,0,0) Joint e 19 e 0 e 0 r 19,19 r 19,0 COM here because actual mass of 19 and 0 combined! r 16,16 r 16,17 m = 0 g r i = (0, 0, 0) mm r o = (0, 0, 0) mm m = 0 g r i = (0, 0, 0) mm r o = (0, 0, 0) mm r 0,0 r 0,1 e 17 e 1 e 17 e 1 id = 17 right hand (unmotorized) r 17,17 r 1,1 id = 1 left hand (unmotorized) m = 18 g r i = (0, 0, -64) mm m = 18 g r i = (0, 0, -64) mm 4
47 [g mm ] I 1 = 4 I = I = I 4 = 4 I = I 6 = I 7 = I 8 = I 9 = I 10 =
48 I 11 = I 1 = I 1 = I 14 = I 1 = I 16 = I 17 = I 18 = I 19 = I 0 = I 1 =
49
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