Chapter 2 Rigid Body Motion Rigid Body Transforma- tions Rotational motion in ℝ 3 Rigid Motion in ℝ 3 Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference Chapter 2 Rigid Body Motion 1 Lecture Notes for A Geometrical Introduction to Robotics and Manipulation Richard Murray and Zexiang Li and Shankar S. Sastry CRC Press Zexiang Li 1 and Yuanqing Wu 1 1 ECE, Hong Kong University of Science & Technology May 23, 2010
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Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
Chapter 2 Rigid Body Motion
1
Lecture Notes for
A Geometrical Introduction to
Robotics and Manipulation
Richard Murray and Zexiang Li and Shankar S. SastryCRC Press
Zexiang Li1 and Yuanqing Wu1
1ECE, Hong Kong University of Science & Technology
May 23, 2010
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
Chapter 2 Rigid Body Motion
2
Chapter 2 Rigid Body Motion
1 Rigid Body Transformations
2 Rotational motion in ℝ3
3 Rigid Motion in ℝ3
4 Velocity of a Rigid Body
5 Wrenches and Reciprocal Screws
6 Reference
z
yabx
xaby
zab
q
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.1 Rigid Body TransformationsChapter 2 Rigid Body Motion
3
§ Notations:
x
y
z p
p = [ pxpypz] or p = [ p1
p2p3]
For p ∈ ℝn, n = 2, 3(2 for planar, 3 for spatial)Point: p =
⎡⎢⎢⎢⎢⎣p1p2⋮pn
⎤⎥⎥⎥⎥⎦, ∥p∥ =√p21 +⋯+ p2n
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.1 Rigid Body TransformationsChapter 2 Rigid Body Motion
3
§ Notations:
x
y
z p
p = [ pxpypz] or p = [ p1
p2p3]
For p ∈ ℝn, n = 2, 3(2 for planar, 3 for spatial)Point: p =
⎡⎢⎢⎢⎢⎣p1p2⋮pn
⎤⎥⎥⎥⎥⎦, ∥p∥ =√p21 +⋯+ p2n
Vector: v = p − q = ⎡⎢⎢⎢⎢⎣p1 − q1p2 − q2⋮pn − qn
⎤⎥⎥⎥⎥⎦ = [v1v2⋮vn], ∥v∥ =√v21 +⋯+ v2n
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.1 Rigid Body TransformationsChapter 2 Rigid Body Motion
3
§ Notations:
x
y
z p
p = [ pxpypz] or p = [ p1
p2p3]
For p ∈ ℝn, n = 2, 3(2 for planar, 3 for spatial)Point: p =
0 0 ] ∈ ℝ4×4∣ v,ω ∈ ℝ3}is called the twist space. There exists a 1-1 correspondencebetween se(3) and ℝ
6, defined by ∧ ∶ ℝ6 ↦ se(3)ξ ∶= [ v
ω ]↦ ξ = [ ω v0 0 ]
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.3 Rigid motion in ℝ3Chapter 2 Rigid Body Motion
34
Definition:se(3) = {[ ω v
0 0 ] ∈ ℝ4×4∣ v,ω ∈ ℝ3}is called the twist space. There exists a 1-1 correspondencebetween se(3) and ℝ
6, defined by ∧ ∶ ℝ6 ↦ se(3)ξ ∶= [ v
ω ]↦ ξ = [ ω v0 0 ]
Property 6: exp ∶ se(3)↦ SE(3), ξθ ↦ eξθ
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.3 Rigid motion in ℝ3Chapter 2 Rigid Body Motion
34
Definition:se(3) = {[ ω v
0 0 ] ∈ ℝ4×4∣ v,ω ∈ ℝ3}is called the twist space. There exists a 1-1 correspondencebetween se(3) and ℝ
6, defined by ∧ ∶ ℝ6 ↦ se(3)ξ ∶= [ v
ω ]↦ ξ = [ ω v0 0 ]
Property 6: exp ∶ se(3)↦ SE(3), ξθ ↦ eξθ
Proof :Let ξ = [ ω v
0 0 ]If ω = 0, then ξ2 = ξ3 = ⋯ = 0, eξθ = [ I vθ
0 1 ] ∈ SE(3)(continues next slide)
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.3 Rigid motion in ℝ3Chapter 2 Rigid Body Motion
35
If ω is not 0, assume ∥ω∥ = 1.Define:
g0 = [ I ω × v0 1 ] , ξ′ = g−10 ⋅ ξ ⋅ g0 = [ ω hω
0 0 ]where h = ωT ⋅ v.
eξθ = eg−10 ⋅ξ′⋅g0 = g−10 ⋅ eξ′θ ⋅ g0and as
ξ′2 = [ ω2 00 0 ] , ξ′3 = [ ω3 0
0 0 ]we have
eξ′θ = [ eωθ hωθ
0 1]⇒ eξθ = [ eωθ (I − eωθ)ωv + ωωTvθ
0 1]
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.3 Rigid motion in ℝ3Chapter 2 Rigid Body Motion
36
p(θ) = eξθ ⋅ p(0)⇒ gab(θ) = eξθIf there is offset,
gab(θ) = eξθgab(0)(Why?)ω
θ
B
B′
Agab(0)
e ξθ
Figure 2.14
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.3 Rigid motion in ℝ3Chapter 2 Rigid Body Motion
37
Property 7: exp ∶ se(3) ↦ SE(3) is onto.
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.3 Rigid motion in ℝ3Chapter 2 Rigid Body Motion
37
Property 7: exp ∶ se(3) ↦ SE(3) is onto.Proof :
Let g = (p,R),R ∈ SO(3), p ∈ ℝ3
Case 1: (R = I) Letξ = [ 0
p∥p∥0 0
] , θ = ∥p∥⇒ eξθ = g = [ I p0 1 ]
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.3 Rigid motion in ℝ3Chapter 2 Rigid Body Motion
37
Property 7: exp ∶ se(3) ↦ SE(3) is onto.Proof :
Let g = (p,R),R ∈ SO(3), p ∈ ℝ3
Case 1: (R = I) Letξ = [ 0
p∥p∥0 0
] , θ = ∥p∥⇒ eξθ = g = [ I p0 1 ]
Case 2: (R ≠ I)eξθ = [ eωθ (I − eωθ)(ω × v) + ωωTvθ
0 1] = [ R p
0 1 ]⇒ { eωθ = R(I − eωθ)(ω × v) + ωωTvθ = p
Solve for ωθ from previous section. Let A = (I − eωθ)ω + wwTθ,Av = p. Claim:
A = (I − eωθ)ω +wwTθ ∶= A1 + A2
kerA1 ∩ kerA2 = ϕ⇒ v = A−1pξθ ∈ ℝ6: Exponential coordinates of g ∈ SE(3)
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.3 Rigid motion in ℝ3Chapter 2 Rigid Body Motion
38
◻ Screws, twists and screw motion:
Figure 2.15
x y
z
θ
ωq
p
d
Screw attributes Pitch: h = dθ (θ = 0, h =∞), d = h ⋅ θ
Axis: l = {q + λω∣λ ∈ ℝ}Magnitude: M = θ
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.3 Rigid motion in ℝ3Chapter 2 Rigid Body Motion
38
◻ Screws, twists and screw motion:
Figure 2.15
x y
z
θ
ωq
p
d
Screw attributes Pitch: h = dθ (θ = 0, h =∞), d = h ⋅ θ
Axis: l = {q + λω∣λ ∈ ℝ}Magnitude: M = θ
Definition:A screw S consists of an axis l, pitch h, and magnitude M. Ascrew motion is a rotation by θ =M about l, followed bytranslation by hθ, parallel to l. If h =∞, then, translationabout v by θ =M
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.3 Rigid motion in ℝ3Chapter 2 Rigid Body Motion
39
Corresponding g ∈ SE(3):g ⋅ p = q + eωθ(p − q) + hθωg ⋅ [ p
1 ] = [ eωθ (I − eωθ)q + hθω0 1
] [ p1 ]⇒
g = [ eωθ (I − eωθ)q + hθω0 1
]
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.3 Rigid motion in ℝ3Chapter 2 Rigid Body Motion
39
Corresponding g ∈ SE(3):g ⋅ p = q + eωθ(p − q) + hθωg ⋅ [ p
1 ] = [ eωθ (I − eωθ)q + hθω0 1
] [ p1 ]⇒
g = [ eωθ (I − eωθ)q + hθω0 1
]On the other hand...
eξθ = [ eωθ (I − eωθ)ω × v + ωωTvθ0 1
]If we let v = −ω × q + hω, then(I − eωθ)(−ω2q) = (I − eωθ)(−ωωTq + q) = (I − eωθ)qThus, eξθ = g
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.3 Rigid motion in ℝ3Chapter 2 Rigid Body Motion
39
Corresponding g ∈ SE(3):g ⋅ p = q + eωθ(p − q) + hθωg ⋅ [ p
1 ] = [ eωθ (I − eωθ)q + hθω0 1
] [ p1 ]⇒
g = [ eωθ (I − eωθ)q + hθω0 1
]On the other hand...
eξθ = [ eωθ (I − eωθ)ω × v + ωωTvθ0 1
]If we let v = −ω × q + hω, then(I − eωθ)(−ω2q) = (I − eωθ)(−ωωTq + q) = (I − eωθ)qThus, eξθ = g
For pure rotation (h = 0): ξ = (−ω × q,ω)For pure translation: g = [ I vθ
0 1 ], ⇒ ξ = (v, 0), and eξθ = g
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.3 Rigid motion in ℝ3Chapter 2 Rigid Body Motion
40
◻ Screw associated with a twist:ξ = (v,ω) ∈ ℝ6
1 Pitch: h =⎧⎪⎪⎪⎨⎪⎪⎪⎩
ωTv
∥ω∥2 , if ω ≠ 0∞, if ω = 0
2 Axis: l = ⎧⎪⎪⎨⎪⎪⎩ω × v∥ω∥2 + λω, λ ∈ ℝ, if ω ≠ 00 + λv λ ∈ ℝ, if ω = 0
3 Magnitude: M =⎧⎪⎪⎨⎪⎪⎩∥ω∥, if ω ≠ 0∥v∥, if ω = 0
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.3 Rigid motion in ℝ3Chapter 2 Rigid Body Motion
40
◻ Screw associated with a twist:ξ = (v,ω) ∈ ℝ6
1 Pitch: h =⎧⎪⎪⎪⎨⎪⎪⎪⎩
ωTv
∥ω∥2 , if ω ≠ 0∞, if ω = 0
2 Axis: l = ⎧⎪⎪⎨⎪⎪⎩ω × v∥ω∥2 + λω, λ ∈ ℝ, if ω ≠ 00 + λv λ ∈ ℝ, if ω = 0
3 Magnitude: M =⎧⎪⎪⎨⎪⎪⎩∥ω∥, if ω ≠ 0∥v∥, if ω = 0
Special cases:
1 h =∞, Pure translation (prismatic joint)
2 h = 0, Pure rotation (revolute joint)
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.3 Rigid motion in ℝ3Chapter 2 Rigid Body Motion
41
Screw Twist: ξθCase 1:Pitch: h =∞Axis: l = {q + λv∣∥v∥ = 1, λ ∈ ℝ}Magnitude:M
θ =M,
ξ = [ 0 v0 0 ]
Case 2:Pitch: h ≠∞Axis: l = {q + λω∣∥ω∥ = 1, λ ∈ ℝ}Magnitude:M
θ =M,
ξ = [ ω −ωq + hω0 0 ]
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.3 Rigid motion in ℝ3Chapter 2 Rigid Body Motion
41
Screw Twist: ξθCase 1:Pitch: h =∞Axis: l = {q + λv∣∥v∥ = 1, λ ∈ ℝ}Magnitude:M
θ =M,
ξ = [ 0 v0 0 ]
Case 2:Pitch: h ≠∞Axis: l = {q + λω∣∥ω∥ = 1, λ ∈ ℝ}Magnitude:M
θ =M,
ξ = [ ω −ωq + hω0 0 ]
Definition: Screw MotionRotation about an axis by θ =M, followed by translationabout the same axis by hθ
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.3 Rigid motion in ℝ3Chapter 2 Rigid Body Motion
42
1793–1880
Theorem 2 (Chasles):Every rigid body motion can be realized by a ro-tation about an axis combined with a translationparallel to that axis.
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.3 Rigid motion in ℝ3Chapter 2 Rigid Body Motion
42
1793–1880
Theorem 2 (Chasles):Every rigid body motion can be realized by a ro-tation about an axis combined with a translationparallel to that axis.
Proof :For ξ ∈ se(3):
ξ = ξ1 + ξ2 = [ ω −ω × q0 0 ] + [ 0 hω
0 0 ][ξ1 , ξ2] = 0⇒ eξθ = eξ1θeξ2θ
† End of Section †
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.4 Velocity of a Rigid BodyChapter 2 Rigid Body Motion
43
♢ Review: Point-mass velocity
q(t) ∈ ℝ3, t ∈ (−ε, ε), v = d
dtq(t) ∈ ℝ3, a = d2
dt2q(t) = d
dtv(t) ∈ ℝ3
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.4 Velocity of a Rigid BodyChapter 2 Rigid Body Motion
43
♢ Review: Point-mass velocity
q(t) ∈ ℝ3, t ∈ (−ε, ε), v = d
dtq(t) ∈ ℝ3, a = d2
dt2q(t) = d
dtv(t) ∈ ℝ3
◻ Velocity of Rotational Motion:Rab(t) ∈ SO(3), t ∈ (−ε, ε), qa(t) = Rab(t)qbVa = d
dtqa(t) = Rab(t)qb = Rab(t)RT
ab(t)Rab(t)qb = RabRTabqa
Rab(t)RTab(t) = I ⇒ RabR
Tab + RabR
Tab = 0, RabR
Tab = −(RabR
Tab)T
x y
z
p(t)
p(0) Figure 2.1
z
yabx
xaby
zab
q
o A:o − xyzB:o − xabyabzab Figure 2.3
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.4 Velocity of a Rigid BodyChapter 2 Rigid Body Motion
44
Denote spatial angular velocity by:
ωsab = RabR
Tab,ωab ∈ ℝ
3
ThenVa= ωs
ab ⋅ qa = ωsab × qa
Body angular velocity:
ωbab = R
Tab ⋅ Rab, v
b≜ RT
ab ⋅ va = ωbab × qb
Relation between body and spatial angular velocity:
ωbab = R
Tab ⋅ ωs
ab or ωbab = R
Tabω
sabRab
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.4 Velocity of a Rigid BodyChapter 2 Rigid Body Motion
45
◻ Generalized Velocity:
gab = [ Rab(t) pab(t)0 1 ] , qa(t) = gab(t)qb
d
dtqa(t) = gab(t)qb = gab ⋅ g−1ab ⋅ gab ⋅ qb = V s
ab ⋅ qa
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.4 Velocity of a Rigid BodyChapter 2 Rigid Body Motion
45
◻ Generalized Velocity:
gab = [ Rab(t) pab(t)0 1 ] , qa(t) = gab(t)qb
d
dtqa(t) = gab(t)qb = gab ⋅ g−1ab ⋅ gab ⋅ qb = V s
ab ⋅ qa
V sab = gab ⋅ g−1ab = [ Rab pab
0 0] [ RT
ab −RTabpab
0 1]
= [ RabRTab −RabRT
abpab + pab0 0
]= [ ωs
ab −ωsab × pab + pab
0 0] ≜ [ ωs
ab vsab0 0 ]
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.4 Velocity of a Rigid BodyChapter 2 Rigid Body Motion
2.5 Wrenches & Reciprocal ScrewsChapter 2 Rigid Body Motion
55
AB
g0
Vb1 F1
Vb2 F2
Adg0
Ad∗g−10
Figure 2.19
V s2 = Adg−10 ⋅V s
1
(Vb2 = Adg−10 ⋅Vb
1 )⇒ Vb
1 = Adg0 ⋅Vb2
F2 = Ad∗g0F1
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.5 Wrenches & Reciprocal ScrewsChapter 2 Rigid Body Motion
56
◻ Screw coordinates for a wrench:
S: l: {q + λω∣λ ∈ ℝ}h: pitchM: Magnitude∥f ∥ =M, ∥τ∥ = hMIf h =∞, Pure τ, ∥τ∥ =M
A
qω
f
τ
Figure 2.20Associate with S a wrench:
F =⎧⎪⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎪⎩
M
⎡⎢⎢⎢⎢⎣ω−ω × q + hω
⎤⎥⎥⎥⎥⎦ h ≠∞M
⎡⎢⎢⎢⎢⎣
0
ω
⎤⎥⎥⎥⎥⎦
h =∞
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.5 Wrenches & Reciprocal ScrewsChapter 2 Rigid Body Motion
57
Also, given F = [ fτ ] (What a force/torque sensor at A
measures), construct S as follows:
⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
Case 1: (f = 0, pure torque)M = ∥τ∥,ω = τ
M , h =∞Case 2: (f ≠ 0)
M = ∥f ∥,ω = fM
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.5 Wrenches & Reciprocal ScrewsChapter 2 Rigid Body Motion
57
Also, given F = [ fτ ] (What a force/torque sensor at A
measures), construct S as follows:
⎧⎪⎪⎪⎪⎪⎪⎪⎨⎪⎪⎪⎪⎪⎪⎪⎩
Case 1: (f = 0, pure torque)M = ∥τ∥,ω = τ
M , h =∞Case 2: (f ≠ 0)
M = ∥f ∥,ω = fM
Solve for q, h from M(q × ω + hω) = τ⇒ h = f Tτ
∥f ∥2 , q =f × τ∥f ∥2
F = [ fτ ]⇔ Apply a force f of mag. M along l,
and a torque τ of mag. hM about l
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.5 Wrenches & Reciprocal ScrewsChapter 2 Rigid Body Motion
58
1777-1859
Theorem 3 (Poinsot):Every collection of wrenches applied to a rigid bodyis equivalent to a force applied along a fixed axisplus a torque about the axis.
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.5 Wrenches & Reciprocal ScrewsChapter 2 Rigid Body Motion
58
1777-1859
Theorem 3 (Poinsot):Every collection of wrenches applied to a rigid bodyis equivalent to a force applied along a fixed axisplus a torque about the axis.
◻ Multi-fingered grasp:
Fo = k∑i=1
AdTg−1oci⋅ Fci Ci CjO
Si SjP
Figure 2.21
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.5 Wrenches & Reciprocal ScrewsChapter 2 Rigid Body Motion
59
◻ Reciprocal screws:
V = [ vω ] , F = [ f
τ ]F ⋅V = f T ⋅ v + τT ⋅ ω↓ ↓
S2 S1
ω2
S1
q1
d
q2
S2
ω1
α
Figure 2.22α = atan2((ω1 × ω2) ⋅ n,ω1 ⋅ ω2)
S1 ⊙ S2 =M1M2((h1 + h2) cos α − d sin α)= 0 if reciprocal
(continues next slide)
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.5 Wrenches & Reciprocal ScrewsChapter 2 Rigid Body Motion
2.5 Wrenches & Reciprocal ScrewsChapter 2 Rigid Body Motion
63◇ Example: Kinematic chains
x(ω1) y(ω2)
v(ω3)
q1
q2
Figure 2.27
● Universal-Spherical Dyad:ξ = span {[ q1 × x
x ] , [ q1 × yy ] [ q2 × ωi
ωi]∣ωi ∈ S2 , i = 1, 2, 3}
ξ⊥ = span{[ vq1 × v ]∣ v = q2−q1∥q2−q1∥}
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.5 Wrenches & Reciprocal ScrewsChapter 2 Rigid Body Motion
63◇ Example: Kinematic chains
x(ω1) y(ω2)
v(ω3)
q1
q2
Figure 2.27
● Universal-Spherical Dyad:ξ = span {[ q1 × x
x ] , [ q1 × yy ] [ q2 × ωi
ωi]∣ωi ∈ S2 , i = 1, 2, 3}
ξ⊥ = span{[ vq1 × v ]∣ v = q2−q1∥q2−q1∥}● Revolute-Spherical Dyad:
zero pitch screws passing through the center of the sphere, lieon a plane containing the axis of the revolute joint: 2-system
Figure 2.28† End of Section †
Chapter2 Rigid BodyMotion
Rigid BodyTransforma-tions
Rotationalmotion in ℝ
3
Rigid Motion
in ℝ3
Velocity of aRigid Body
Wrenches andReciprocalScrews
Reference
2.6 ReferencesChapter 2 Rigid Body Motion
64
◻ Reference:[1] Murray, R.M. and Li, Z.X. and Sastry, S.S., A mathematical introduction to robotic manipu-lation. CRC Press, 1994.[2] Ball, R.S., A treatise on the theory of screws. University Press, 1900.[3] Bottema, O. and Roth, B. , Theoretical kinematics. Dover Publications, 1990.[4] Craig, J.J., Introduction to robotics : mechanics and control, 3rd ed. Prentice Hall, 2004.[5] Fu, K.S. and Gonzalez, R.C. and Lee, C.S.G., Robotics : control, sensing, vision, and intelli-gence. CAD/CAM, robotics, and computer vision. McGraw-Hill, 1987.[6] Hunt, K.H., Kinematic geometry of mechanisms. 1978, Oxford, New York: Clarendon Press,1978.[7] Paul, R.P., Robot manipulators : mathematics, programming, and control. The MIT Pressseries in artificial intelligence. MIT Press, 1981.[8] Park, F. C., A first course in robot mechanics. Available online:http://robotics.snu.ac.kr/files/_pdf_files_publications/a_first_coruse_in_robot_mechanics.pdf ,2006.[9] Tsai, L.-W., Robot analysis : the mechanics of serial and parallel manipulators. Wiley, 1999.[10] Spong, M.W. and Hutchinson, S. and Vidyasagar, M. , Robot modeling and control. JohnWiley & Sons, 2006.