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ATC, ECE, Hong Kong University of Science and Technology

2nd International Summer School on Geometric Methods in Robotics, Mechanism Design and Manufacturing Research-Lecture 02 Rigid body motion

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Chapter 2 Rigid Body Motion 1 Chapter Lecture Notes for Rigid Body Motion A Geometrical Introduction to Rigid Body Transforma- Robotics and Manipulation tions Rotational motion in ℝ Richard Murray and Zexiang Li and Shankar S. Sastry CRC Press Rigid Motion in ℝ Velocity of a Rigid Body Zexiang Li and Yuanqing Wu Wrenches and Reciprocal Screws ECE, Hong Kong University of Science & Technology Reference May ,
Chapter 2 Rigid Body Motion 2 q Chapter Rigid Body Motion Chapter z zab Rigid Body Rigid Body Transformations Motion xab Rigid Body Transforma- Rotational motion in ℝ y tions x yab Rotational motion in ℝ Rigid Motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Velocity of a Rigid Body Wrenches and Reciprocal Screws Wrenches and Reciprocal Screws Reference Reference
Chapter 2 Rigid Body Motion 2.1 Rigid Body Transformations 3 § Notations: Chapter z p Rigid Body Motion px p Rigid Body p= py or p = p Transforma- tions y pz p Rotational x For p ∈ ℝn , n = , ( for planar, for spatial) motion in ℝ ⎡ p ⎤ ⎢ p ⎥ Rigid Motion Point: p = ⎢ ⎥, p = p + ⋯ + p in ℝ ⎢ ⎥ ⎢ pn ⎥ n ⎣ ⎦ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.1 Rigid Body Transformations 3 § Notations: Chapter z p Rigid Body Motion px p Rigid Body p= py or p = p Transforma- tions y pz p Rotational x For p ∈ ℝn , n = , ( for planar, for spatial) motion in ℝ ⎡ p ⎤ ⎢ p ⎥ Rigid Motion Point: p = ⎢ ⎥, p = p + ⋯ + p in ℝ ⎢ ⎥ ⎢ pn ⎥ n ⎣ ⎦ Velocity of a ⎡ p −q ⎤ Rigid Body ⎢ p −q ⎥ v Vector: v = p − q = ⎢ ⎥= v , v = v + ⋯ + vn Wrenches and ⎢ ⎥ Reciprocal ⎢ pn − qn ⎥ ⎣ ⎦ Screws Reference vn
Chapter 2 Rigid Body Motion 2.1 Rigid Body Transformations 3 § Notations: Chapter z p Rigid Body Motion px p Rigid Body p= py or p = p Transforma- tions y pz p Rotational x For p ∈ ℝn , n = , ( for planar, for spatial) motion in ℝ ⎡ p ⎤ ⎢ p ⎥ Rigid Motion Point: p = ⎢ ⎥, p = p + ⋯ + p in ℝ ⎢ ⎥ ⎢ pn ⎥ n ⎣ ⎦ Velocity of a ⎡ p −q ⎤ Rigid Body ⎢ p −q ⎥ v Vector: v = p − q = ⎢ ⎥= v , v = v + ⋯ + vn Wrenches and ⎢ ⎥ Reciprocal ⎢ pn − qn ⎥ ⎣ a ⎦ Screws vn Reference a ⋯ am Matrix: A ∈ ℝn×m , A = an an ⋯ anm
Chapter 2 Rigid Body Motion 2.1 Rigid Body Transformations 4 ◻ Description of point-mass motion: ⎡ x( ) ⎤ z ⎢ ⎥ Chapter p( ) = ⎢ y( ) ⎥: initial position Rigid Body ⎢ ⎥ Motion ⎢ z( ) ⎥ ⎣ ⎦ Rigid Body Transforma- tions Rotational motion in ℝ p( ) Rigid Motion in ℝ x y Velocity of a Rigid Body Figure 2.1 Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.1 Rigid Body Transformations 4 ◻ Description of point-mass motion: ⎡ x( ) ⎤ z ⎢ ⎥ Chapter p( ) = ⎢ y( ) ⎥: initial position Rigid Body ⎢ ⎥ Motion ⎢ z( ) ⎥ ⎣ ⎦ Rigid Body Transforma- p(t) ⎡ ⎤ tions ⎢ x(t) ⎥ p(t) = ⎢ y(t) ⎥ , t ∈ (−ε, ε) Rotational ⎢ ⎥ motion in ℝ ⎢ z(t) ⎥ p( ) ⎣ ⎦ Rigid Motion in ℝ x y Velocity of a Rigid Body Figure 2.1 Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.1 Rigid Body Transformations 4 ◻ Description of point-mass motion: ⎡ x( ) ⎤ z ⎢ ⎥ Chapter p( ) = ⎢ y( ) ⎥: initial position Rigid Body ⎢ ⎥ Motion ⎢ z( ) ⎥ ⎣ ⎦ Rigid Body Transforma- p(t) ⎡ ⎤ tions ⎢ x(t) ⎥ p(t) = ⎢ y(t) ⎥ , t ∈ (−ε, ε) Rotational ⎢ ⎥ motion in ℝ ⎢ z(t) ⎥ p( ) ⎣ ⎦ Rigid Motion in ℝ x y Velocity of a Rigid Body Figure 2.1 Wrenches and Reciprocal Definition: Trajectory ⎡ x(t) ⎤ ⎢ ⎥ Screws A trajectory is a curve p ∶ (−ε, ε) ↦ ℝ , p(t) = ⎢ y(t) ⎥ ⎢ ⎥ Reference ⎢ z(t) ⎥ ⎣ ⎦
Chapter 2 Rigid Body Motion 2.1 Rigid Body Transformations 5 ◻ Rigid Body Motion:q( ) z p( ) Chapter Rigid Body y Motion x Rigid Body Transforma- tions Rotational motion in ℝ Figure 2.2 Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.1 Rigid Body Transformations 5 ◻ Rigid Body Motion:q( ) z p( ) Chapter Rigid Body y Motion x q(t) Rigid Body Transforma- tions p(t) Rotational motion in ℝ Figure 2.2 p(t) − q(t) = p( ) − q( ) = constant Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.1 Rigid Body Transformations 5 ◻ Rigid Body Motion:q( ) z p( ) Chapter Rigid Body y Motion x q(t) Rigid Body Transforma- tions p(t) Rotational motion in ℝ Figure 2.2 p(t) − q(t) = p( ) − q( ) = constant Rigid Motion in ℝ Velocity of a Rigid Body Definition: Rigid body transformation Wrenches and Reciprocal g∶ℝ ↦ℝ Screws s.t. Length preserving: g(p) − g(q) = p − q Reference Orientation preserving: g∗ (v × ω) = g∗ (v) × g∗ (ω) † End of Section †
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 6 ◻ Rotational Motion: Chapter Rigid Body Motion Choose a reference frame A Rigid Body (spatial frame) Transforma- z tions Rotational motion in ℝ Rigid Motion o y in ℝ Velocity of a Rigid Body x A: o − xyz Wrenches and Reciprocal Screws Figure 2.3 Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 6 ◻ Rotational Motion: Chapter q Rigid Body Motion Choose a reference frame A Rigid Body (spatial frame) Transforma- tions Attach a frame B to the body z zab (spatial frame) Rotational motion in ℝ xab Rigid Motion o y in ℝ Velocity of a Rigid Body x yab A: o − xyz Wrenches and B: o − xab yab zab Reciprocal Screws Figure 2.3 xab ∈ ℝ : Reference coordinates of xb in frame A Rab = [xab yab zab] ∈ ℝ × : Rotation (or orientation) matrix of B w.r.t. A
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 7 ◻ Property of a Rotation Matrix: Let R = [r r r ] be a rotation matrix Chapter Rigid Body Motion Rigid Body Transforma- tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 7 ◻ Property of a Rotation Matrix: Let R = [r r r ] be a rotation matrix Chapter Rigid Body Motion Rigid Body ⇒ riT ⋅ rj = Transforma- i≠j tions Rotational i=j ⎡ rT ⎤ motion in ℝ ⎢ ⎥ or R ⋅ R = ⎢ rT ⎥ [r r r ] = I or R ⋅ RT = I Rigid Motion ⎢ T ⎥ T ⎢ r ⎥ in ℝ ⎣ ⎦ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 7 ◻ Property of a Rotation Matrix: Let R = [r r r ] be a rotation matrix Chapter Rigid Body Motion Rigid Body ⇒ riT ⋅ rj = Transforma- i≠j tions Rotational i=j ⎡ rT ⎤ motion in ℝ ⎢ ⎥ or R ⋅ R = ⎢ rT ⎥ [r r r ] = I or R ⋅ RT = I Rigid Motion ⎢ T ⎥ T ⎢ r ⎥ in ℝ ⎣ ⎦ Velocity of a Rigid Body Wrenches and Reciprocal det(RT R) = det RT ⋅ det R = (det R) = , det R = ± As det R = rT (r × r ) = ⇒ det R = Screws Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 8 Definition: Chapter SO( ) = R ∈ ℝ × RT R = I, det R = Rigid Body Motion and Rigid Body SO(n) = R ∈ ℝn×n RT R = I, det R = Transforma- tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 8 Definition: Chapter SO( ) = R ∈ ℝ × RT R = I, det R = Rigid Body Motion and Rigid Body SO(n) = R ∈ ℝn×n RT R = I, det R = Transforma- tions Rotational motion in ℝ Review: Group (G, ⋅) is a group if: Rigid Motion in ℝ g ,g ∈ G ⇒ g ⋅g ∈ G Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 8 Definition: Chapter SO( ) = R ∈ ℝ × RT R = I, det R = Rigid Body Motion and Rigid Body SO(n) = R ∈ ℝn×n RT R = I, det R = Transforma- tions Rotational motion in ℝ Review: Group (G, ⋅) is a group if: Rigid Motion in ℝ g ,g ∈ G ⇒ g ⋅g ∈ G Velocity of a Rigid Body ∃! e ∈ G, s.t. g ⋅ e = e ⋅ g = g, ∀g ∈ G Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 8 Definition: Chapter SO( ) = R ∈ ℝ × RT R = I, det R = Rigid Body Motion and Rigid Body SO(n) = R ∈ ℝn×n RT R = I, det R = Transforma- tions Rotational motion in ℝ Review: Group (G, ⋅) is a group if: Rigid Motion in ℝ g ,g ∈ G ⇒ g ⋅g ∈ G Velocity of a Rigid Body ∃! e ∈ G, s.t. g ⋅ e = e ⋅ g = g, ∀g ∈ G Wrenches and Reciprocal ∀g ∈ G, ∃! g − ∈ G, s.t. g ⋅ g − = g − ⋅ g = e Screws Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 8 Definition: Chapter SO( ) = R ∈ ℝ × RT R = I, det R = Rigid Body Motion and Rigid Body SO(n) = R ∈ ℝn×n RT R = I, det R = Transforma- tions Rotational motion in ℝ Review: Group (G, ⋅) is a group if: Rigid Motion in ℝ g ,g ∈ G ⇒ g ⋅g ∈ G Velocity of a Rigid Body ∃! e ∈ G, s.t. g ⋅ e = e ⋅ g = g, ∀g ∈ G Wrenches and Reciprocal ∀g ∈ G, ∃! g − ∈ G, s.t. g ⋅ g − = g − ⋅ g = e Screws g ⋅ (g ⋅ g ) = (g ⋅ g ) ⋅ g Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 9 Review: Examples of group Chapter (ℝ , +) Rigid Body Motion Rigid Body Transforma- tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 9 Review: Examples of group Chapter (ℝ , +) Rigid Body Motion ({ , }, + mod ) Rigid Body Transforma- tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 9 Review: Examples of group Chapter (ℝ , +) Rigid Body Motion ({ , }, + mod ) (ℝ, ×) Not a group (Why?) Rigid Body Transforma- tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 9 Review: Examples of group Chapter (ℝ , +) Rigid Body Motion ({ , }, + mod ) (ℝ, ×) Not a group (Why?) Rigid Body Transforma- (ℝ∗ ∶ ℝ − { }, ×) tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 9 Review: Examples of group Chapter (ℝ , +) Rigid Body Motion ({ , }, + mod ) (ℝ, ×) Not a group (Why?) Rigid Body Transforma- (ℝ∗ ∶ ℝ − { }, ×) tions S ≜ {z ∈ ℂ z = } Rotational motion in ℝ Rigid Motion in ℝ Property 1: SO( ) is a group under matrix multiplication. Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 9 Review: Examples of group Chapter (ℝ , +) Rigid Body Motion ({ , }, + mod ) (ℝ, ×) Not a group (Why?) Rigid Body Transforma- (ℝ∗ ∶ ℝ − { }, ×) tions S ≜ {z ∈ ℂ z = } Rotational motion in ℝ Rigid Motion in ℝ Property 1: SO( ) is a group under matrix multiplication. Velocity of a Rigid Body Proof : Wrenches and If R , R ∈ SO( ), then R ⋅ R ∈ SO( ), because (R R )T (R R ) = RT (RT R )R = RT R = I Reciprocal Screws Reference det(R ⋅ R ) = det(R ) ⋅ det(R ) =
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 9 Review: Examples of group Chapter (ℝ , +) Rigid Body Motion ({ , }, + mod ) (ℝ, ×) Not a group (Why?) Rigid Body Transforma- (ℝ∗ ∶ ℝ − { }, ×) tions S ≜ {z ∈ ℂ z = } Rotational motion in ℝ Rigid Motion in ℝ Property 1: SO( ) is a group under matrix multiplication. Velocity of a Rigid Body Proof : Wrenches and If R , R ∈ SO( ), then R ⋅ R ∈ SO( ), because (R R )T (R R ) = RT (RT R )R = RT R = I Reciprocal Screws Reference det(R ⋅ R ) = det(R ) ⋅ det(R ) = e=I ×
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 9 Review: Examples of group Chapter (ℝ , +) Rigid Body Motion ({ , }, + mod ) (ℝ, ×) Not a group (Why?) Rigid Body Transforma- (ℝ∗ ∶ ℝ − { }, ×) tions S ≜ {z ∈ ℂ z = } Rotational motion in ℝ Rigid Motion in ℝ Property 1: SO( ) is a group under matrix multiplication. Velocity of a Rigid Body Proof : Wrenches and If R , R ∈ SO( ), then R ⋅ R ∈ SO( ), because (R R )T (R R ) = RT (RT R )R = RT R = I Reciprocal Screws Reference det(R ⋅ R ) = det(R ) ⋅ det(R ) = e=I × R ⋅ R = I ⇒ R− = RT T
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 10 ◻ Configuration and rigid transformation: q Chapter Rab = [xab yab zab] ∈ SO( ) Rigid Body Motion Configuration Space z zab Rigid Body Transforma- tions Rotational xab motion in ℝ o y Rigid Motion in ℝ x yab Velocity of a A: o − xyz Rigid Body B: o − xab yab zab Wrenches and Figure 2.3 Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 10 ◻ Configuration and rigid transformation: q Chapter Rab = [xab yab zab] ∈ SO( ) Rigid Body Motion Configuration Space z zab Rigid Body Transforma- tions xb Rotational Let qb = yb ∈ ℝ : coordinates of q in B. xab o motion in ℝ zb y Rigid Motion qa = xab ⋅ xb + yab ⋅ yb + zab ⋅ zb in ℝ x yab xb Velocity of a = [xab yab zab ] yb = Rab ⋅ qb A: o − xyz B: o − xab yab zab Rigid Body zb Wrenches and Figure 2.3 Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 10 ◻ Configuration and rigid transformation: q Chapter Rab = [xab yab zab] ∈ SO( ) Rigid Body Motion Configuration Space z zab Rigid Body Transforma- tions xb Rotational Let qb = yb ∈ ℝ : coordinates of q in B. xab o motion in ℝ zb y Rigid Motion qa = xab ⋅ xb + yab ⋅ yb + zab ⋅ zb in ℝ x yab xb Velocity of a = [xab yab zab ] yb = Rab ⋅ qb A: o − xyz B: o − xab yab zab Rigid Body zb Wrenches and Figure 2.3 Reciprocal A configuration Rab ∈ SO( ) is also a transformation: Screws Rab ∶ ℝ → ℝ , Rab (qb ) = Rab ⋅ qb = qa Reference A config. ⇔ A transformation in SO( )
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 11 Property 2: Rab preserves distance between points and Chapter Rigid Body orientation. Rab ⋅ (pb − qb ) = pa − qa Motion R(v × ω) = (Rv) × Rω Rigid Body Transforma- tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 11 Property 2: Rab preserves distance between points and Chapter Rigid Body orientation. Rab ⋅ (pb − qb ) = pa − qa Motion R(v × ω) = (Rv) × Rω Rigid Body Transforma- tions Rotational −a motion in ℝ Proof : a For a ∈ ℝ , let a = a −a −a Rigid Motion in ℝ ˆ a Velocity of a Note that a ⋅ b = a × b ˆ follows from Rab (pb − pa ) = (Rab (pb − pa ))T Rab (pb − pa ) Rigid Body Wrenches and = (pb − pa )T RT Rab (pb − pa ) Reciprocal Screws ab = pb − pa Reference follows from RωR = (Rω) ˆ T ∧
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 12 ◻ Parametrization of SO( ) (the exponential coordinate): Review: S = {z ∈ ℂ z = } Chapter Rigid Body Motion Rigid Body Transforma- Im Euler’s Formula tions i eiφ = cos φ + i sin φ Rotational “One of the most remarkable, al- motion in ℝ φ sin φ most astounding, formulas in all Rigid Motion Re in ℝ cos φ of mathematics.” Velocity of a R. Feynman Rigid Body Wrenches and Figure 2.4 Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 12 ◻ Parametrization of SO( ) (the exponential coordinate): Review: S = {z ∈ ℂ z = } Chapter Rigid Body Motion Rigid Body Transforma- Im Euler’s Formula tions i eiφ = cos φ + i sin φ Rotational “One of the most remarkable, al- motion in ℝ φ sin φ most astounding, formulas in all Rigid Motion Re in ℝ cos φ of mathematics.” Velocity of a R. Feynman Rigid Body Wrenches and Figure 2.4 Reciprocal Screws Review: x(t) = ax(t) Reference ⇒ x(t) = eat x ˙ x( ) = x (Continues next slide)
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 13 r r r R ∈ SO( ), R = r r r r r r ⎧ ⎪ ⎪ , i≠j ri ⋅ rj = ⎨ Chapter ⎪ , i=j ← constraints ⎪ Rigid Body ⎩ Motion Rigid Body Transforma- ⇒ independent parameters! tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 13 r r r R ∈ SO( ), R = r ω r r q(t) r r r ⎧ ⎪ ⎪ , i≠j ri ⋅ rj = ⎨ Chapter ⎪ , i=j ← constraints ⎪ Rigid Body q(0) ⎩ Motion Rigid Body Transforma- ⇒ independent parameters! tions Consider motion of a point q on a rotating link q(t) = ω × q(t) = ωq(t) Rotational motion in ℝ ˙ ˆ Figure 2.5 q( ): Initial coordinates Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 13 r r r R ∈ SO( ), R = r ω r r q(t) r r r ⎧ ⎪ ⎪ , i≠j ri ⋅ rj = ⎨ Chapter ⎪ , i=j ← constraints ⎪ Rigid Body q(0) ⎩ Motion Rigid Body Transforma- ⇒ independent parameters! tions Consider motion of a point q on a rotating link q(t) = ω × q(t) = ωq(t) Rotational motion in ℝ ˙ ˆ Figure 2.5 q( ): Initial coordinates Rigid Motion in ℝ (ωt) (ωt) ⇒ q(t) = eωt q where eωt = I + ωt + + +⋯ Velocity of a ˆ ˆ ˆ ˆ Rigid Body ˆ Wrenches and ! ! Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 13 r r r R ∈ SO( ), R = r ω r r q(t) r r r ⎧ ⎪ ⎪ , i≠j ri ⋅ rj = ⎨ Chapter ⎪ , i=j ← constraints ⎪ Rigid Body q(0) ⎩ Motion Rigid Body Transforma- ⇒ independent parameters! tions Consider motion of a point q on a rotating link q(t) = ω × q(t) = ωq(t) Rotational motion in ℝ ˙ ˆ Figure 2.5 q( ): Initial coordinates Rigid Motion in ℝ (ωt) (ωt) ⇒ q(t) = eωt q where eωt = I + ωt + + +⋯ Velocity of a ˆ ˆ ˆ ˆ Rigid Body ˆ ! ! By the definition of rigid transformation, R(ω, θ) = eωθ . Let Wrenches and Reciprocal ˆ so( ) = {ω ω ∈ ℝ } or so(n) = {S ∈ ℝ S = −S} where ∧ ∶ Screws ˆ n×n T ℝ ↦ so( ) ∶ ω ↦ ω, we have: Reference ˆ Property 3: exp ∶ so( ) ↦ SO( ), ωθ ↦ eωθ ˆ ˆ
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 14 Rodrigues’ formula ( ω = ): Chapter eωθ = I + ω sin θ + ω ( − cos θ) ˆ ˆ ˆ Rigid Body Motion Rigid Body Transforma- tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 14 Rodrigues’ formula ( ω = ): Chapter eωθ = I + ω sin θ + ω ( − cos θ) ˆ ˆ ˆ Rigid Body Motion Proof : Rigid Body Let a ∈ ℝ , write a = ωθ, ω = (or ω = ), and θ = a Transforma- tions a a (ωθ) (ωθ) Rotational eωθ = I + ωθ + + +⋯ motion in ℝ ˆ ˆ ˆ Rigid Motion ˆ ! ! a = aa − a I, a = − a a in ℝ T Velocity of a As ˆ ˆ ˆ Rigid Body we have: Wrenches and Reciprocal eωθ = I + (θ − + − ⋯)ω + ( − + ⋯)ω Screws ˆ θ θ θ θ ˆ ˆ Reference ! ! ! ! = I + ω sin θ + ω ( − cos θ) ˆ ˆ
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 15 Chapter Rodrigue’s formula for ω ≠ : eωθ = I + sin ω θ + ( − cos ω θ) Rigid Body Motion ˆ ω ˆ ω ˆ Rigid Body ω ω Transforma- tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 15 Chapter Rodrigue’s formula for ω ≠ : eωθ = I + sin ω θ + ( − cos ω θ) Rigid Body Motion ˆ ω ˆ ω ˆ Rigid Body ω ω Transforma- tions Proof for Property 3: Rotational motion in ℝ Let R ≜ eωθ , then: ˆ (eωθ )− = e−ωθ = eω = (eωθ )T Rigid Motion T in ℝ ˆ ˆ ˆ θ ˆ ⇒ R− = RT ⇒ RT R = I ⇒ det R = ± Velocity of a Rigid Body From det exp( ) = , and the continuity of det function w.r.t. θ, Wrenches and Reciprocal we have det eωθ = , ∀θ ∈ ℝ Screws ˆ Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 16 Property 4: The exponential map is onto. Chapter Rigid Body Motion Rigid Body Transforma- tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 16 Property 4: The exponential map is onto. Chapter Proof : Given R ∈ SO( ), to show ∃ω ∈ ℝ , ω = and θ s.t. R = eωθ Rigid Body Motion ˆ Rigid Body Let r r r R= r Transforma- tions r r Rotational r r r motion in ℝ and vθ = − cos θ, cθ = cos θ, sθ = sin θ Rigid Motion in ℝ Velocity of a Rigid Body By Rodrigue’s formula ⎡ ω vθ + cθ ω ω vθ − ω sθ ω ω vθ + ω sθ ⎤ Wrenches and ⎢ ⎥ Reciprocal = ⎢ ω ω vθ + ω sθ ⎢ ω vθ + cθ ω ω vθ − ω sθ ⎥ ⎥ Screws ωθ ˆ e ⎢ ω ω v −ω s ω ω vθ + ω sθ ω vθ + cθ ⎥ ⎣ ⎦ Reference θ θ (continues next slide)
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 17 Taking the trace of both sides, Chapter tr(R) = r + r + r = + cos θ = Rigid Body Motion λi Rigid Body i= where λi is the eigenvalue of R, i = , , Transforma- tions Case : tr(R) = or R = I, θ = ⇒ ωθ = Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 17 Taking the trace of both sides, Chapter tr(R) = r + r + r = + cos θ = Rigid Body Motion λi Rigid Body i= where λi is the eigenvalue of R, i = , , Transforma- tions Case : tr(R) = or R = I, θ = ⇒ ωθ = Rotational motion in ℝ Case : − < tr(R) < , Rigid Motion in ℝ tr(R) − r −r θ = arccos ⇒ω= r −r Velocity of a r −r Rigid Body Wrenches and sθ Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 17 Taking the trace of both sides, Chapter tr(R) = r + r + r = + cos θ = Rigid Body Motion λi Rigid Body i= where λi is the eigenvalue of R, i = , , Transforma- tions Case : tr(R) = or R = I, θ = ⇒ ωθ = Rotational motion in ℝ Case : − < tr(R) < , Rigid Motion in ℝ tr(R) − r −r θ = arccos ⇒ω= r −r Velocity of a sθ r − r Rigid Body Case : tr(R) = − ⇒ cos θ = − ⇒ θ = ±π Wrenches and Reciprocal Screws Reference (continues next slide)
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 18 Following are possibilities: R= − ⇒ω= Chapter , − Rigid Body − Motion R= ⇒ω= Rigid Body , − Transforma- tions − R= − ⇒ω= Rotational motion in ℝ Rigid Motion in ℝ Note that if ωθ is a solution, then ω(θ ± nπ), n = , ± , ± , ... is Velocity of a Rigid Body Wrenches and Reciprocal also a solution. Screws Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 19 Definition: Exponential coordinate Chapter ωθ ∈ ℝ , with eωθ = R is called the exponential coordinates of R ˆ Rigid Body Exp ∶ Motion Rigid Body Transforma- tions Rotational so( ) ≅ ℝ motion in ℝ exp log Rigid Motion I SO( ) in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Figure 2.6 Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 19 Definition: Exponential coordinate Chapter ωθ ∈ ℝ , with eωθ = R is called the exponential coordinates of R ˆ Rigid Body Exp ∶ Motion Rigid Body Transforma- tions Rotational so( ) ≅ ℝ motion in ℝ exp log Rigid Motion I SO( ) in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Figure 2.6 Reference Property 5: exp is - when restricted to an open ball in ℝ of radius π.
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 20 Theorem 1 (Euler): Any orientation is equivalent to a rotation about a fixed axis ω ∈ ℝ through an angle θ ∈ [−π, π]. Chapter Rigid Body Motion Rigid Body Transforma- tions – ω Rotational motion in ℝ B Rigid Motion in ℝ A Velocity of a Rigid Body Wrenches and Reciprocal Screws Figure 2.7 Reference SO( ) can be visualized as a solid ball of radius π.
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 21 ◻ Other Parametrizations of SO( ): Chapter XYZ fixed angles (or Roll-Pitch-Yaw angle) Rigid Body Motion Rigid Body θ-Pitch Transforma- y tions Rotational motion in ℝ Rigid Motion in ℝ Roll-φ Velocity of a Rigid Body x Wrenches and ψ-Yaw Reciprocal Screws z Reference Figure 2.8 (continues next slide)
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 22 XYZ fixed angles (or Roll-Pitch-Yaw angle) Continued Chapter Rigid Body cos φ − sin φ Motion Rigid Body Rx (φ) ∶= exφ = ˆ Transforma- tions sin φ cos φ cos θ sin θ Ry (θ) ∶= eyθ = ˆ − sin θ Rotational motion in ℝ cos θ cos ψ − sin ψ Rz (ψ) ∶= ezψ = Rigid Motion in ℝ ˆ sin ψ cos ψ Velocity of a Rigid Body Rab = Rx (φ)Ry (θ)Rz (ψ) Wrenches and Reciprocal −cθ sψ Screws cθ cψ sθ = sφ sθ cψ + cφ sψ −sφ sθ sψ + cφ cψ −sφ cθ Reference −cφ sθ cψ + sφ sψ cφ sθ sψ + sφ cψ cφ cθ
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 23 ZYX Euler angle z A′ A′′ Chapter Rigid Body A B Motion Rigid Body Transforma- tions Rotational motion in ℝ x y Rigid Motion in ℝ Velocity of a Rigid Body Figure 2.9 Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 23 ZYX Euler angle z(z ) ′ z′ A′ A′′ Chapter Rigid Body A B Motion Rigid Body Transforma- tions Rotational α y′ y′ motion in ℝ x y x′ x′ Rigid Motion in ℝ Velocity of a Rigid Body Figure 2.9 Raa′ = Rz (α) Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 23 ZYX Euler angle z(z ) ′ z′ A′ A′′ Chapter Rigid Body A B Motion z ′′ z ′′ Rigid Body Transforma- tions Rotational α y′ y′ (y′′ ) y′′ motion in ℝ x y x′ x′ β Rigid Motion in ℝ x′′ x′′ Velocity of a Rigid Body Figure 2.9 Raa′ = Rz (α) Ra′ a′′ = Ry (β) Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 23 ZYX Euler angle z(z ) ′ z′ A′ A′′ Chapter Rigid Body A B Motion z ′′ z ′′′ z ′′ Rigid Body γ Transforma- y′′′ tions Rotational α y′ y′ (y′′ ) y′′ motion in ℝ x y x′ x′ β Rigid Motion in ℝ x′′ (x′′′ ) x′′ Velocity of a Rigid Body Figure 2.9 Raa′ = Rz (α) Ra′ a′′ = Ry (β) Ra′′ b = Rx (γ) Wrenches and Reciprocal Screws Reference Rab = Rz (α)Ry (β)Rx (γ) (continues next slide)
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 24 Chapter ZYX Euler angle (continued) cα cβ −sα cγ + cα sβ sγ sα sγ + cα sβ cγ Rigid Body Rab (α, β, γ) = sα cβ cα cγ + sα sβ sγ −cα sγ + sα sβ cγ Motion Rigid Body Transforma- −sβ c β sγ c β cγ tions Rotational Note: When β = , sin β = , α + γ = const ⇒ singularity! motion in ℝ Rigid Motion atan (y, x) in ℝ β = atan (−r , r +r ) Velocity of a Rigid Body α = atan (r cβ , r cβ ) Wrenches and y Reciprocal γ = atan (r cβ ) Screws Reference cβ , r x Figure 2.10
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 25 § Quaternions: Chapter Q =q +q i+q j+q k where i = j = k = − , i ⋅ j = k, j ⋅ k = i, k ⋅ i = j Rigid Body Motion Rigid Body Transforma- tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 25 § Quaternions: Chapter Q =q +q i+q j+q k where i = j = k = − , i ⋅ j = k, j ⋅ k = i, k ⋅ i = j Rigid Body Motion Rigid Body Property 1: Define Q∗ = (q , q)∗ = (q , −q), q ∈ ℝ, q ∈ ℝ Transforma- tions Rotational motion in ℝ Q = QQ∗ = q + q + q + q Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 25 § Quaternions: Chapter Q =q +q i+q j+q k where i = j = k = − , i ⋅ j = k, j ⋅ k = i, k ⋅ i = j Rigid Body Motion Rigid Body Property 1: Define Q∗ = (q , q)∗ = (q , −q), q ∈ ℝ, q ∈ ℝ Transforma- tions Rotational motion in ℝ Q = QQ∗ = q + q + q + q Property 2: Q = (q , ⃗), P = (p , ⃗) Rigid Motion q p QP = (q p − ⃗ ⋅ ⃗, q ⃗ + p ⃗ + ⃗ × ⃗) in ℝ Velocity of a q p p q q p Rigid Body Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 25 § Quaternions: Chapter Q =q +q i+q j+q k where i = j = k = − , i ⋅ j = k, j ⋅ k = i, k ⋅ i = j Rigid Body Motion Rigid Body Property 1: Define Q∗ = (q , q)∗ = (q , −q), q ∈ ℝ, q ∈ ℝ Transforma- tions Rotational motion in ℝ Q = QQ∗ = q + q + q + q Property 2: Q = (q , ⃗), P = (p , ⃗) Rigid Motion q p QP = (q p − ⃗ ⋅ ⃗, q ⃗ + p ⃗ + ⃗ × ⃗) in ℝ Velocity of a q p p q q p Rigid Body Wrenches and Reciprocal Property 3: (a) The set of unit quaternions forms a group (b) If R = eωθ , then Q = (cos , ω sin ) Screws ˆ θ θ Reference (c) Q acts on x ∈ ℝ by QXQ∗ , where X = ( , x)
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 26 ◻ Unit Quaternions: Given Q = (q , q), q ∈ ℝ, q ∈ ℝ , the vector part of QXQ∗ is Chapter Rigid Body given by R(Q)x, Motion Rigid Body R(Q) = I + q q + q ˆ ˆ ⎡ − (q + q ) ⎤ Transforma- ⎢ − q q + qq q q + qq ⎥ tions =⎢ q q + qq ⎢ − (q + q ) − qq + qq ⎥ ⎥ Rotational ⎢ − q q + qq − (q + q ) ⎥ motion in ℝ Rigid Motion ⎣ qq + qq ⎦ in ℝ Velocity of a Rigid Body where Q ≜ q + q + q + q = Wrenches and Reciprocal Screws Reference (continues next slide)
Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 27 ◻ Quaternions (continued): Chapter Conversion from Roll-Pitch-Yaw angle to unit quaternions: Rigid Body Q = (cos Motion φ φ θ θ ψ ψ Rigid Body , x sin )(cos , y sin )(cos , z sin ) ⇒ Transforma- tions q = cos − sin φ θ ψ φ θ ψ Rotational cos cos sin sin motion in ℝ ⎡ ⎤ Rigid Motion ⎢ cos φ sin θ sin ψ + sin φ cos θ cos ψ ⎥ in ℝ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ q = ⎢ cos sin cos − sin cos sin Velocity of a Rigid Body φ θ ψ φ θ ψ ⎥ ⎢ ⎥ Wrenches and ⎢ ⎥ ⎢ ⎥ Reciprocal ⎢ cos φ cos θ sin ψ + sin φ sin θ cos ψ ⎥ Screws ⎢ ⎥ Reference ⎣ ⎦ Conversion from unit quaternions to roll-pitch-yaw angles (?) † End of Section †
Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 28 z Chapter Rigid Body pab B Motion A Rigid Body Transforma- tions qa y qb x Rotational motion in ℝ q Rigid Motion in ℝ Figure 2.11 Velocity of a pab ∈ ℝ ∶ Coordinates of the origin of B Rab ∈ SO( ) ∶ Orientation of B relative to A Rigid Body Wrenches and SE( ) ∶ {(p, R) p ∈ ℝ , R ∈ SO( )} ∶ Configuration Space Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 28 z Chapter Rigid Body pab B Motion A Rigid Body Transforma- tions qa y qb x Rotational motion in ℝ q Rigid Motion in ℝ Figure 2.11 Velocity of a pab ∈ ℝ ∶ Coordinates of the origin of B Rab ∈ SO( ) ∶ Orientation of B relative to A Rigid Body Wrenches and SE( ) ∶ {(p, R) p ∈ ℝ , R ∈ SO( )} ∶ Configuration Space Reciprocal Screws Reference Or...as a transformation: gab = (pab , Rab ) ∶ ℝ ↦ ℝ qb ↦ qa = pab + Rab ⋅ qb
Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 29 ◻ Homogeneous Representation: Chapter Points: ⎡ q ⎤ ⎢ q ⎥ Rigid Body q=⎢ q ⎥∈ℝ q q= ∈ℝ ⎢ ⎥ Motion q q ⎢ ⎥ Rigid Body ⎣ ⎦ Transforma- tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 29 ◻ Homogeneous Representation: Chapter Points: ⎡ q ⎤ ⎢ q ⎥ Rigid Body q=⎢ q ⎥∈ℝ q q= ∈ℝ ⎢ ⎥ Motion q q ⎢ ⎥ Rigid Body ⎣ ⎦ Transforma- Vectors: ⎡ p ⎤ ⎡ q ⎤ tions p −q ⎢ p ⎥ ⎢ q ⎥ v v = p−q = ⎢ p ⎥−⎢ q ⎥= v v=p−q = p −q = ⎢ ⎥ ⎢ ⎥ v Rotational v ⎢ ⎥ ⎢ ⎥ v motion in ℝ p −q v Rigid Motion ⎣ ⎦ ⎣ ⎦ in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 29 ◻ Homogeneous Representation: Chapter Points: ⎡ q ⎤ ⎢ q ⎥ Rigid Body q=⎢ q ⎥∈ℝ q q= ∈ℝ ⎢ ⎥ Motion q q ⎢ ⎥ Rigid Body ⎣ ⎦ Transforma- Vectors: ⎡ p ⎤ ⎡ q ⎤ tions p −q ⎢ p ⎥ ⎢ q ⎥ v v = p−q = ⎢ p ⎥−⎢ q ⎥= v v=p−q = p −q = ⎢ ⎥ ⎢ ⎥ v Rotational v ⎢ ⎥ ⎢ ⎥ v motion in ℝ p −q v Rigid Motion ⎣ ⎦ ⎣ ⎦ in ℝ Point-Point = Vector Velocity of a Rigid Body Vector+Point = Point Wrenches and Reciprocal Vector+Vector = Vector Screws Reference Point+Point: Meaningless (continues next slide)
Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 30 qa = pab + Rab ⋅ qb gab = (pab , Rab ) = Rab pab qa qab Chapter Rigid Body Motion g ab qa = g ab ⋅ qb g ab = Rigid Body Transforma- Rab pab tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 30 qa = pab + Rab ⋅ qb gab = (pab , Rab ) = Rab pab qa qab Chapter Rigid Body Motion g ab qa = g ab ⋅ qb g ab = Rigid Body Transforma- Rab pab tions Rotational motion in ℝ Rigid Motion ◻ Composition Rule: in ℝ Velocity of a Rigid Body gab gbc Wrenches and qb = g bc ⋅ qc B qa = g ab ⋅ qb Reciprocal Screws A C Reference Figure 2.12
Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 30 qa = pab + Rab ⋅ qb gab = (pab , Rab ) = Rab pab qa qab Chapter Rigid Body Motion g ab qa = g ab ⋅ qb g ab = Rigid Body Transforma- Rab pab tions Rotational motion in ℝ Rigid Motion ◻ Composition Rule: in ℝ Velocity of a Rigid Body gab gbc Wrenches and qb = g bc ⋅ qc B qa = g ab ⋅ qb = g ab ⋅ g bc ⋅qc Reciprocal Screws A C gac Reference g ac Figure 2.12 g ac = g ab ⋅ g bc = Rab Rbc Rab pbc + pab
Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 31 ◻ Special Euclidean Group: SE( ) = R p ∈ℝ × p ∈ ℝ , R ∈ SO( ) Chapter Rigid Body Motion Rigid Body Transforma- tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 31 ◻ Special Euclidean Group: SE( ) = R p ∈ℝ × p ∈ ℝ , R ∈ SO( ) Chapter Rigid Body Motion Rigid Body Transforma- Property 4: SE( ) forms a group. tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 31 ◻ Special Euclidean Group: SE( ) = R p ∈ℝ × p ∈ ℝ , R ∈ SO( ) Chapter Rigid Body Motion Rigid Body Transforma- Property 4: SE( ) forms a group. tions Rotational motion in ℝ Proof : g ⋅ g ∈ SE( ) Rigid Motion in ℝ e=I Velocity of a Rigid Body Wrenches and Reciprocal (g)− = RT −RT p Screws Reference Associativity: Follows from property of matrix multiplication
Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 32 § Induced transformation on vectors: Chapter Rigid Body v v v =s−r = , g ∗ v = gs − gr = = Motion v R p v Rv Rigid Body v v Transforma- tions The bar will be dropped to simplify notations Rotational motion in ℝ Property 5: An element of SE( ) is a rigid transformation. Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 33 Exponential coordinates of SE( ): ω p(t) Chapter For rotational motion: ˙ p(t)=ω × (p(t) − q) Rigid Body Motion ˙ Rigid Body Transforma- p ˙ ω −ω × q ˆ p tions = p(t) p( ) or p= ξ ⋅ p ⇒ p(t) = e p( ) Rotational ˙ ˆ ˆ ξt motion in ℝ ˆ (ξt) + ⋯ q where eξt =I + ξt + Rigid Motion ˆ ˆ in ℝ ! A Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference Figure 2.13
Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 33 Exponential coordinates of SE( ): ω p(t) Chapter For rotational motion: ˙ p(t)=ω × (p(t) − q) Rigid Body Motion ˙ Rigid Body Transforma- p ˙ ω −ω × q ˆ p tions = p(t) p( ) or p= ξ ⋅ p ⇒ p(t) = e p( ) Rotational ˙ ˆ ˆ ξt motion in ℝ ˆ (ξt) q where eξt =I + ξt + ! + ⋯ Rigid Motion ˆ ˆ in ℝ A For translational motion: p(t) p(t)=v Velocity of a ˙ Rigid Body ˙ Wrenches and p(t) ˙ v p v Reciprocal Screws = p(t) p( ) p(t)= ξ ⋅ p(t) ⇒ p(t) = eξt p( ) Reference ˙ ˆ ˆ ˆ v ξ= A Figure 2.13
Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 34 Definition: Chapter se( ) = ω v ∈ ℝ × v, ω ∈ ℝ ˆ Rigid Body Motion is called the twist space. There exists a - correspondence Rigid Body between se( ) and ℝ , defined by ∧ ∶ ℝ ↦ se( ) ξ ∶= ω ↦ ξ = ω v Transforma- tions v ˆ ˆ Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 34 Definition: Chapter se( ) = ω v ∈ ℝ × v, ω ∈ ℝ ˆ Rigid Body Motion is called the twist space. There exists a - correspondence Rigid Body between se( ) and ℝ , defined by ∧ ∶ ℝ ↦ se( ) ξ ∶= ω ↦ ξ = ω v Transforma- tions v ˆ ˆ Rotational motion in ℝ Rigid Motion Property 6: exp ∶ se( ) ↦ SE( ), ξθ ↦ eξθ in ℝ ˆ ˆ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 34 Definition: Chapter se( ) = ω v ∈ ℝ × v, ω ∈ ℝ ˆ Rigid Body Motion is called the twist space. There exists a - correspondence Rigid Body between se( ) and ℝ , defined by ∧ ∶ ℝ ↦ se( ) ξ ∶= ω ↦ ξ = ω v Transforma- tions v ˆ ˆ Rotational motion in ℝ Rigid Motion Property 6: exp ∶ se( ) ↦ SE( ), ξθ ↦ eξθ in ℝ ˆ ˆ Velocity of a Rigid Body Proof : Let ξ = ω v ˆ ˆ Wrenches and Reciprocal Screws If ω = , then ξ = ξ = ⋯ = , eξθ = I vθ ∈ SE( ) Reference ˆ ˆ ˆ (continues next slide)
Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 35 Chapter If ω is not , assume ω = . Rigid Body Motion Define: Rigid Body g = I ω×v , ξ′ = g − ⋅ ξ ⋅ g = ˆ ˆ ω hω ˆ where h = ωT ⋅ v. Transforma- tions ⋅ ξ ′ ⋅g eξθ = eg = g − ⋅ eξ θ ⋅ g Rotational ˆ − ˆ ˆ′ motion in ℝ Rigid Motion and as ξ′ = ω , ξ′ = ω in ℝ ˆ ˆ ˆ ˆ Velocity of a Rigid Body we have Wrenches and Reciprocal eξ θ = ˆ′ eωθ ˆ hωθ ⇒ eξθ = ˆ eωθ ˆ (I − eωθ )ωv + ωωT vθ ˆ ˆ Screws Reference
Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 36 p(θ) = eξθ ⋅ p( ) ⇒ gab (θ) = eξθ ˆ ˆ Chapter Rigid Body Motion If there is offset, Rigid Body gab(θ) = eξθ gab ( )( Why?) Transforma- tions ˆ Rotational motion in ℝ ω Rigid Motion in ℝ Velocity of a Rigid Body θ B′ Wrenches and Reciprocal ˆ Screws B e ξθ Reference gab ( ) A Figure 2.14
Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 37 Property 7: exp ∶ se( ) ↦ SE( ) is onto. Chapter Rigid Body Motion Rigid Body Transforma- tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 37 Property 7: exp ∶ se( ) ↦ SE( ) is onto. Proof : Let g = (p, R), R ∈ SO( ), p ∈ ℝ Chapter (R = I) Let Rigid Body Motion Case 1: p ξ= , θ = p ⇒ eξθ = g = I p Rigid Body ˆ Transforma- ˆ p tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 37 Property 7: exp ∶ se( ) ↦ SE( ) is onto. Proof : Let g = (p, R), R ∈ SO( ), p ∈ ℝ Chapter (R = I) Let Rigid Body Motion Case 1: p ξ= , θ = p ⇒ eξθ = g = I p Rigid Body ˆ Transforma- ˆ p tions Rotational Case 2: (R ≠ I) = (I − eωθ )(ω × v) + ωωT vθ = motion in ℝ ωθ ˆ ˆ e ˆ ξθ e R p Rigid Motion eωθ = R in ℝ ˆ (I − eωθ )(ω × v) + ωωT vθ = p Velocity of a ⇒ ˆ Rigid Body Wrenches and Solve for ωθ from previous section. Let A = (I − eωθ )ω + wwT θ, ˆ ˆ Av = p. Claim: Reciprocal Screws Reference A = (I − eωθ )ω + wwT θ ∶= A + A ˆ ˆ ker A ∩ ker A = ϕ ⇒ v = A− p ξθ ∈ ℝ : Exponential coordinates of g ∈ SE( )
Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 38 ◻ Screws, twists and screw motion: p Chapter Rigid Body ω Motion z q Rigid Body Figure 2.15 Transforma- θ tions d Rotational motion in ℝ x y h = d (θ = , h = ∞), d = h ⋅ θ Rigid Motion in ℝ Screw attributes Pitch: l = {q + λω λ ∈ ℝ} θ Velocity of a Axis: M=θ Rigid Body Magnitude: Wrenches and Reciprocal Screws Reference
Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 38 ◻ Screws, twists and screw motion: p Chapter Rigid Body ω Motion z q Rigid Body Figure 2.15 Transforma- θ tions d Rotational motion in ℝ x y h = d (θ = , h = ∞), d = h ⋅ θ Rigid Motion in ℝ Screw attributes Pitch: l = {q + λω λ ∈ ℝ} θ Velocity of a Axis: M=θ Rigid Body Magnitude: Wrenches and Reciprocal Screws Definition: A screw S consists of an axis l, pitch h, and magnitude M. A screw motion is a rotation by θ = M about l, followed by Reference translation by hθ, parallel to l. If h = ∞, then, translation about v by θ = M
Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 39 Corresponding g ∈ SE( ): g ⋅ p = q + eωθ (p − q) + hθω ˆ Chapter g⋅ p = eωθ (I − eωθ )q + hθω p Rigid Body ˆ ˆ Motion ⇒ Rigid Body g= eωθ (I − eωθ )q + hθω Transforma- ˆ ˆ tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
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[Download] Rev-Chapter-2

  • 1. Chapter 2 Rigid Body Motion 1 Chapter Lecture Notes for Rigid Body Motion A Geometrical Introduction to Rigid Body Transforma- Robotics and Manipulation tions Rotational motion in ℝ Richard Murray and Zexiang Li and Shankar S. Sastry CRC Press Rigid Motion in ℝ Velocity of a Rigid Body Zexiang Li and Yuanqing Wu Wrenches and Reciprocal Screws ECE, Hong Kong University of Science & Technology Reference May ,
  • 2. Chapter 2 Rigid Body Motion 2 q Chapter Rigid Body Motion Chapter z zab Rigid Body Rigid Body Transformations Motion xab Rigid Body Transforma- Rotational motion in ℝ y tions x yab Rotational motion in ℝ Rigid Motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Velocity of a Rigid Body Wrenches and Reciprocal Screws Wrenches and Reciprocal Screws Reference Reference
  • 3. Chapter 2 Rigid Body Motion 2.1 Rigid Body Transformations 3 § Notations: Chapter z p Rigid Body Motion px p Rigid Body p= py or p = p Transforma- tions y pz p Rotational x For p ∈ ℝn , n = , ( for planar, for spatial) motion in ℝ ⎡ p ⎤ ⎢ p ⎥ Rigid Motion Point: p = ⎢ ⎥, p = p + ⋯ + p in ℝ ⎢ ⎥ ⎢ pn ⎥ n ⎣ ⎦ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • 4. Chapter 2 Rigid Body Motion 2.1 Rigid Body Transformations 3 § Notations: Chapter z p Rigid Body Motion px p Rigid Body p= py or p = p Transforma- tions y pz p Rotational x For p ∈ ℝn , n = , ( for planar, for spatial) motion in ℝ ⎡ p ⎤ ⎢ p ⎥ Rigid Motion Point: p = ⎢ ⎥, p = p + ⋯ + p in ℝ ⎢ ⎥ ⎢ pn ⎥ n ⎣ ⎦ Velocity of a ⎡ p −q ⎤ Rigid Body ⎢ p −q ⎥ v Vector: v = p − q = ⎢ ⎥= v , v = v + ⋯ + vn Wrenches and ⎢ ⎥ Reciprocal ⎢ pn − qn ⎥ ⎣ ⎦ Screws Reference vn
  • 5. Chapter 2 Rigid Body Motion 2.1 Rigid Body Transformations 3 § Notations: Chapter z p Rigid Body Motion px p Rigid Body p= py or p = p Transforma- tions y pz p Rotational x For p ∈ ℝn , n = , ( for planar, for spatial) motion in ℝ ⎡ p ⎤ ⎢ p ⎥ Rigid Motion Point: p = ⎢ ⎥, p = p + ⋯ + p in ℝ ⎢ ⎥ ⎢ pn ⎥ n ⎣ ⎦ Velocity of a ⎡ p −q ⎤ Rigid Body ⎢ p −q ⎥ v Vector: v = p − q = ⎢ ⎥= v , v = v + ⋯ + vn Wrenches and ⎢ ⎥ Reciprocal ⎢ pn − qn ⎥ ⎣ a ⎦ Screws vn Reference a ⋯ am Matrix: A ∈ ℝn×m , A = an an ⋯ anm
  • 6. Chapter 2 Rigid Body Motion 2.1 Rigid Body Transformations 4 ◻ Description of point-mass motion: ⎡ x( ) ⎤ z ⎢ ⎥ Chapter p( ) = ⎢ y( ) ⎥: initial position Rigid Body ⎢ ⎥ Motion ⎢ z( ) ⎥ ⎣ ⎦ Rigid Body Transforma- tions Rotational motion in ℝ p( ) Rigid Motion in ℝ x y Velocity of a Rigid Body Figure 2.1 Wrenches and Reciprocal Screws Reference
  • 7. Chapter 2 Rigid Body Motion 2.1 Rigid Body Transformations 4 ◻ Description of point-mass motion: ⎡ x( ) ⎤ z ⎢ ⎥ Chapter p( ) = ⎢ y( ) ⎥: initial position Rigid Body ⎢ ⎥ Motion ⎢ z( ) ⎥ ⎣ ⎦ Rigid Body Transforma- p(t) ⎡ ⎤ tions ⎢ x(t) ⎥ p(t) = ⎢ y(t) ⎥ , t ∈ (−ε, ε) Rotational ⎢ ⎥ motion in ℝ ⎢ z(t) ⎥ p( ) ⎣ ⎦ Rigid Motion in ℝ x y Velocity of a Rigid Body Figure 2.1 Wrenches and Reciprocal Screws Reference
  • 8. Chapter 2 Rigid Body Motion 2.1 Rigid Body Transformations 4 ◻ Description of point-mass motion: ⎡ x( ) ⎤ z ⎢ ⎥ Chapter p( ) = ⎢ y( ) ⎥: initial position Rigid Body ⎢ ⎥ Motion ⎢ z( ) ⎥ ⎣ ⎦ Rigid Body Transforma- p(t) ⎡ ⎤ tions ⎢ x(t) ⎥ p(t) = ⎢ y(t) ⎥ , t ∈ (−ε, ε) Rotational ⎢ ⎥ motion in ℝ ⎢ z(t) ⎥ p( ) ⎣ ⎦ Rigid Motion in ℝ x y Velocity of a Rigid Body Figure 2.1 Wrenches and Reciprocal Definition: Trajectory ⎡ x(t) ⎤ ⎢ ⎥ Screws A trajectory is a curve p ∶ (−ε, ε) ↦ ℝ , p(t) = ⎢ y(t) ⎥ ⎢ ⎥ Reference ⎢ z(t) ⎥ ⎣ ⎦
  • 9. Chapter 2 Rigid Body Motion 2.1 Rigid Body Transformations 5 ◻ Rigid Body Motion:q( ) z p( ) Chapter Rigid Body y Motion x Rigid Body Transforma- tions Rotational motion in ℝ Figure 2.2 Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • 10. Chapter 2 Rigid Body Motion 2.1 Rigid Body Transformations 5 ◻ Rigid Body Motion:q( ) z p( ) Chapter Rigid Body y Motion x q(t) Rigid Body Transforma- tions p(t) Rotational motion in ℝ Figure 2.2 p(t) − q(t) = p( ) − q( ) = constant Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • 11. Chapter 2 Rigid Body Motion 2.1 Rigid Body Transformations 5 ◻ Rigid Body Motion:q( ) z p( ) Chapter Rigid Body y Motion x q(t) Rigid Body Transforma- tions p(t) Rotational motion in ℝ Figure 2.2 p(t) − q(t) = p( ) − q( ) = constant Rigid Motion in ℝ Velocity of a Rigid Body Definition: Rigid body transformation Wrenches and Reciprocal g∶ℝ ↦ℝ Screws s.t. Length preserving: g(p) − g(q) = p − q Reference Orientation preserving: g∗ (v × ω) = g∗ (v) × g∗ (ω) † End of Section †
  • 12. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 6 ◻ Rotational Motion: Chapter Rigid Body Motion Choose a reference frame A Rigid Body (spatial frame) Transforma- z tions Rotational motion in ℝ Rigid Motion o y in ℝ Velocity of a Rigid Body x A: o − xyz Wrenches and Reciprocal Screws Figure 2.3 Reference
  • 13. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 6 ◻ Rotational Motion: Chapter q Rigid Body Motion Choose a reference frame A Rigid Body (spatial frame) Transforma- tions Attach a frame B to the body z zab (spatial frame) Rotational motion in ℝ xab Rigid Motion o y in ℝ Velocity of a Rigid Body x yab A: o − xyz Wrenches and B: o − xab yab zab Reciprocal Screws Figure 2.3 xab ∈ ℝ : Reference coordinates of xb in frame A Rab = [xab yab zab] ∈ ℝ × : Rotation (or orientation) matrix of B w.r.t. A
  • 14. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 7 ◻ Property of a Rotation Matrix: Let R = [r r r ] be a rotation matrix Chapter Rigid Body Motion Rigid Body Transforma- tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • 15. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 7 ◻ Property of a Rotation Matrix: Let R = [r r r ] be a rotation matrix Chapter Rigid Body Motion Rigid Body ⇒ riT ⋅ rj = Transforma- i≠j tions Rotational i=j ⎡ rT ⎤ motion in ℝ ⎢ ⎥ or R ⋅ R = ⎢ rT ⎥ [r r r ] = I or R ⋅ RT = I Rigid Motion ⎢ T ⎥ T ⎢ r ⎥ in ℝ ⎣ ⎦ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • 16. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 7 ◻ Property of a Rotation Matrix: Let R = [r r r ] be a rotation matrix Chapter Rigid Body Motion Rigid Body ⇒ riT ⋅ rj = Transforma- i≠j tions Rotational i=j ⎡ rT ⎤ motion in ℝ ⎢ ⎥ or R ⋅ R = ⎢ rT ⎥ [r r r ] = I or R ⋅ RT = I Rigid Motion ⎢ T ⎥ T ⎢ r ⎥ in ℝ ⎣ ⎦ Velocity of a Rigid Body Wrenches and Reciprocal det(RT R) = det RT ⋅ det R = (det R) = , det R = ± As det R = rT (r × r ) = ⇒ det R = Screws Reference
  • 17. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 8 Definition: Chapter SO( ) = R ∈ ℝ × RT R = I, det R = Rigid Body Motion and Rigid Body SO(n) = R ∈ ℝn×n RT R = I, det R = Transforma- tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • 18. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 8 Definition: Chapter SO( ) = R ∈ ℝ × RT R = I, det R = Rigid Body Motion and Rigid Body SO(n) = R ∈ ℝn×n RT R = I, det R = Transforma- tions Rotational motion in ℝ Review: Group (G, ⋅) is a group if: Rigid Motion in ℝ g ,g ∈ G ⇒ g ⋅g ∈ G Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • 19. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 8 Definition: Chapter SO( ) = R ∈ ℝ × RT R = I, det R = Rigid Body Motion and Rigid Body SO(n) = R ∈ ℝn×n RT R = I, det R = Transforma- tions Rotational motion in ℝ Review: Group (G, ⋅) is a group if: Rigid Motion in ℝ g ,g ∈ G ⇒ g ⋅g ∈ G Velocity of a Rigid Body ∃! e ∈ G, s.t. g ⋅ e = e ⋅ g = g, ∀g ∈ G Wrenches and Reciprocal Screws Reference
  • 20. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 8 Definition: Chapter SO( ) = R ∈ ℝ × RT R = I, det R = Rigid Body Motion and Rigid Body SO(n) = R ∈ ℝn×n RT R = I, det R = Transforma- tions Rotational motion in ℝ Review: Group (G, ⋅) is a group if: Rigid Motion in ℝ g ,g ∈ G ⇒ g ⋅g ∈ G Velocity of a Rigid Body ∃! e ∈ G, s.t. g ⋅ e = e ⋅ g = g, ∀g ∈ G Wrenches and Reciprocal ∀g ∈ G, ∃! g − ∈ G, s.t. g ⋅ g − = g − ⋅ g = e Screws Reference
  • 21. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 8 Definition: Chapter SO( ) = R ∈ ℝ × RT R = I, det R = Rigid Body Motion and Rigid Body SO(n) = R ∈ ℝn×n RT R = I, det R = Transforma- tions Rotational motion in ℝ Review: Group (G, ⋅) is a group if: Rigid Motion in ℝ g ,g ∈ G ⇒ g ⋅g ∈ G Velocity of a Rigid Body ∃! e ∈ G, s.t. g ⋅ e = e ⋅ g = g, ∀g ∈ G Wrenches and Reciprocal ∀g ∈ G, ∃! g − ∈ G, s.t. g ⋅ g − = g − ⋅ g = e Screws g ⋅ (g ⋅ g ) = (g ⋅ g ) ⋅ g Reference
  • 22. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 9 Review: Examples of group Chapter (ℝ , +) Rigid Body Motion Rigid Body Transforma- tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • 23. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 9 Review: Examples of group Chapter (ℝ , +) Rigid Body Motion ({ , }, + mod ) Rigid Body Transforma- tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • 24. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 9 Review: Examples of group Chapter (ℝ , +) Rigid Body Motion ({ , }, + mod ) (ℝ, ×) Not a group (Why?) Rigid Body Transforma- tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • 25. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 9 Review: Examples of group Chapter (ℝ , +) Rigid Body Motion ({ , }, + mod ) (ℝ, ×) Not a group (Why?) Rigid Body Transforma- (ℝ∗ ∶ ℝ − { }, ×) tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • 26. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 9 Review: Examples of group Chapter (ℝ , +) Rigid Body Motion ({ , }, + mod ) (ℝ, ×) Not a group (Why?) Rigid Body Transforma- (ℝ∗ ∶ ℝ − { }, ×) tions S ≜ {z ∈ ℂ z = } Rotational motion in ℝ Rigid Motion in ℝ Property 1: SO( ) is a group under matrix multiplication. Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • 27. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 9 Review: Examples of group Chapter (ℝ , +) Rigid Body Motion ({ , }, + mod ) (ℝ, ×) Not a group (Why?) Rigid Body Transforma- (ℝ∗ ∶ ℝ − { }, ×) tions S ≜ {z ∈ ℂ z = } Rotational motion in ℝ Rigid Motion in ℝ Property 1: SO( ) is a group under matrix multiplication. Velocity of a Rigid Body Proof : Wrenches and If R , R ∈ SO( ), then R ⋅ R ∈ SO( ), because (R R )T (R R ) = RT (RT R )R = RT R = I Reciprocal Screws Reference det(R ⋅ R ) = det(R ) ⋅ det(R ) =
  • 28. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 9 Review: Examples of group Chapter (ℝ , +) Rigid Body Motion ({ , }, + mod ) (ℝ, ×) Not a group (Why?) Rigid Body Transforma- (ℝ∗ ∶ ℝ − { }, ×) tions S ≜ {z ∈ ℂ z = } Rotational motion in ℝ Rigid Motion in ℝ Property 1: SO( ) is a group under matrix multiplication. Velocity of a Rigid Body Proof : Wrenches and If R , R ∈ SO( ), then R ⋅ R ∈ SO( ), because (R R )T (R R ) = RT (RT R )R = RT R = I Reciprocal Screws Reference det(R ⋅ R ) = det(R ) ⋅ det(R ) = e=I ×
  • 29. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 9 Review: Examples of group Chapter (ℝ , +) Rigid Body Motion ({ , }, + mod ) (ℝ, ×) Not a group (Why?) Rigid Body Transforma- (ℝ∗ ∶ ℝ − { }, ×) tions S ≜ {z ∈ ℂ z = } Rotational motion in ℝ Rigid Motion in ℝ Property 1: SO( ) is a group under matrix multiplication. Velocity of a Rigid Body Proof : Wrenches and If R , R ∈ SO( ), then R ⋅ R ∈ SO( ), because (R R )T (R R ) = RT (RT R )R = RT R = I Reciprocal Screws Reference det(R ⋅ R ) = det(R ) ⋅ det(R ) = e=I × R ⋅ R = I ⇒ R− = RT T
  • 30. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 10 ◻ Configuration and rigid transformation: q Chapter Rab = [xab yab zab] ∈ SO( ) Rigid Body Motion Configuration Space z zab Rigid Body Transforma- tions Rotational xab motion in ℝ o y Rigid Motion in ℝ x yab Velocity of a A: o − xyz Rigid Body B: o − xab yab zab Wrenches and Figure 2.3 Reciprocal Screws Reference
  • 31. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 10 ◻ Configuration and rigid transformation: q Chapter Rab = [xab yab zab] ∈ SO( ) Rigid Body Motion Configuration Space z zab Rigid Body Transforma- tions xb Rotational Let qb = yb ∈ ℝ : coordinates of q in B. xab o motion in ℝ zb y Rigid Motion qa = xab ⋅ xb + yab ⋅ yb + zab ⋅ zb in ℝ x yab xb Velocity of a = [xab yab zab ] yb = Rab ⋅ qb A: o − xyz B: o − xab yab zab Rigid Body zb Wrenches and Figure 2.3 Reciprocal Screws Reference
  • 32. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 10 ◻ Configuration and rigid transformation: q Chapter Rab = [xab yab zab] ∈ SO( ) Rigid Body Motion Configuration Space z zab Rigid Body Transforma- tions xb Rotational Let qb = yb ∈ ℝ : coordinates of q in B. xab o motion in ℝ zb y Rigid Motion qa = xab ⋅ xb + yab ⋅ yb + zab ⋅ zb in ℝ x yab xb Velocity of a = [xab yab zab ] yb = Rab ⋅ qb A: o − xyz B: o − xab yab zab Rigid Body zb Wrenches and Figure 2.3 Reciprocal A configuration Rab ∈ SO( ) is also a transformation: Screws Rab ∶ ℝ → ℝ , Rab (qb ) = Rab ⋅ qb = qa Reference A config. ⇔ A transformation in SO( )
  • 33. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 11 Property 2: Rab preserves distance between points and Chapter Rigid Body orientation. Rab ⋅ (pb − qb ) = pa − qa Motion R(v × ω) = (Rv) × Rω Rigid Body Transforma- tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • 34. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 11 Property 2: Rab preserves distance between points and Chapter Rigid Body orientation. Rab ⋅ (pb − qb ) = pa − qa Motion R(v × ω) = (Rv) × Rω Rigid Body Transforma- tions Rotational −a motion in ℝ Proof : a For a ∈ ℝ , let a = a −a −a Rigid Motion in ℝ ˆ a Velocity of a Note that a ⋅ b = a × b ˆ follows from Rab (pb − pa ) = (Rab (pb − pa ))T Rab (pb − pa ) Rigid Body Wrenches and = (pb − pa )T RT Rab (pb − pa ) Reciprocal Screws ab = pb − pa Reference follows from RωR = (Rω) ˆ T ∧
  • 35. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 12 ◻ Parametrization of SO( ) (the exponential coordinate): Review: S = {z ∈ ℂ z = } Chapter Rigid Body Motion Rigid Body Transforma- Im Euler’s Formula tions i eiφ = cos φ + i sin φ Rotational “One of the most remarkable, al- motion in ℝ φ sin φ most astounding, formulas in all Rigid Motion Re in ℝ cos φ of mathematics.” Velocity of a R. Feynman Rigid Body Wrenches and Figure 2.4 Reciprocal Screws Reference
  • 36. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 12 ◻ Parametrization of SO( ) (the exponential coordinate): Review: S = {z ∈ ℂ z = } Chapter Rigid Body Motion Rigid Body Transforma- Im Euler’s Formula tions i eiφ = cos φ + i sin φ Rotational “One of the most remarkable, al- motion in ℝ φ sin φ most astounding, formulas in all Rigid Motion Re in ℝ cos φ of mathematics.” Velocity of a R. Feynman Rigid Body Wrenches and Figure 2.4 Reciprocal Screws Review: x(t) = ax(t) Reference ⇒ x(t) = eat x ˙ x( ) = x (Continues next slide)
  • 37. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 13 r r r R ∈ SO( ), R = r r r r r r ⎧ ⎪ ⎪ , i≠j ri ⋅ rj = ⎨ Chapter ⎪ , i=j ← constraints ⎪ Rigid Body ⎩ Motion Rigid Body Transforma- ⇒ independent parameters! tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • 38. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 13 r r r R ∈ SO( ), R = r ω r r q(t) r r r ⎧ ⎪ ⎪ , i≠j ri ⋅ rj = ⎨ Chapter ⎪ , i=j ← constraints ⎪ Rigid Body q(0) ⎩ Motion Rigid Body Transforma- ⇒ independent parameters! tions Consider motion of a point q on a rotating link q(t) = ω × q(t) = ωq(t) Rotational motion in ℝ ˙ ˆ Figure 2.5 q( ): Initial coordinates Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • 39. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 13 r r r R ∈ SO( ), R = r ω r r q(t) r r r ⎧ ⎪ ⎪ , i≠j ri ⋅ rj = ⎨ Chapter ⎪ , i=j ← constraints ⎪ Rigid Body q(0) ⎩ Motion Rigid Body Transforma- ⇒ independent parameters! tions Consider motion of a point q on a rotating link q(t) = ω × q(t) = ωq(t) Rotational motion in ℝ ˙ ˆ Figure 2.5 q( ): Initial coordinates Rigid Motion in ℝ (ωt) (ωt) ⇒ q(t) = eωt q where eωt = I + ωt + + +⋯ Velocity of a ˆ ˆ ˆ ˆ Rigid Body ˆ Wrenches and ! ! Reciprocal Screws Reference
  • 40. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 13 r r r R ∈ SO( ), R = r ω r r q(t) r r r ⎧ ⎪ ⎪ , i≠j ri ⋅ rj = ⎨ Chapter ⎪ , i=j ← constraints ⎪ Rigid Body q(0) ⎩ Motion Rigid Body Transforma- ⇒ independent parameters! tions Consider motion of a point q on a rotating link q(t) = ω × q(t) = ωq(t) Rotational motion in ℝ ˙ ˆ Figure 2.5 q( ): Initial coordinates Rigid Motion in ℝ (ωt) (ωt) ⇒ q(t) = eωt q where eωt = I + ωt + + +⋯ Velocity of a ˆ ˆ ˆ ˆ Rigid Body ˆ ! ! By the definition of rigid transformation, R(ω, θ) = eωθ . Let Wrenches and Reciprocal ˆ so( ) = {ω ω ∈ ℝ } or so(n) = {S ∈ ℝ S = −S} where ∧ ∶ Screws ˆ n×n T ℝ ↦ so( ) ∶ ω ↦ ω, we have: Reference ˆ Property 3: exp ∶ so( ) ↦ SO( ), ωθ ↦ eωθ ˆ ˆ
  • 41. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 14 Rodrigues’ formula ( ω = ): Chapter eωθ = I + ω sin θ + ω ( − cos θ) ˆ ˆ ˆ Rigid Body Motion Rigid Body Transforma- tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • 42. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 14 Rodrigues’ formula ( ω = ): Chapter eωθ = I + ω sin θ + ω ( − cos θ) ˆ ˆ ˆ Rigid Body Motion Proof : Rigid Body Let a ∈ ℝ , write a = ωθ, ω = (or ω = ), and θ = a Transforma- tions a a (ωθ) (ωθ) Rotational eωθ = I + ωθ + + +⋯ motion in ℝ ˆ ˆ ˆ Rigid Motion ˆ ! ! a = aa − a I, a = − a a in ℝ T Velocity of a As ˆ ˆ ˆ Rigid Body we have: Wrenches and Reciprocal eωθ = I + (θ − + − ⋯)ω + ( − + ⋯)ω Screws ˆ θ θ θ θ ˆ ˆ Reference ! ! ! ! = I + ω sin θ + ω ( − cos θ) ˆ ˆ
  • 43. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 15 Chapter Rodrigue’s formula for ω ≠ : eωθ = I + sin ω θ + ( − cos ω θ) Rigid Body Motion ˆ ω ˆ ω ˆ Rigid Body ω ω Transforma- tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • 44. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 15 Chapter Rodrigue’s formula for ω ≠ : eωθ = I + sin ω θ + ( − cos ω θ) Rigid Body Motion ˆ ω ˆ ω ˆ Rigid Body ω ω Transforma- tions Proof for Property 3: Rotational motion in ℝ Let R ≜ eωθ , then: ˆ (eωθ )− = e−ωθ = eω = (eωθ )T Rigid Motion T in ℝ ˆ ˆ ˆ θ ˆ ⇒ R− = RT ⇒ RT R = I ⇒ det R = ± Velocity of a Rigid Body From det exp( ) = , and the continuity of det function w.r.t. θ, Wrenches and Reciprocal we have det eωθ = , ∀θ ∈ ℝ Screws ˆ Reference
  • 45. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 16 Property 4: The exponential map is onto. Chapter Rigid Body Motion Rigid Body Transforma- tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • 46. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 16 Property 4: The exponential map is onto. Chapter Proof : Given R ∈ SO( ), to show ∃ω ∈ ℝ , ω = and θ s.t. R = eωθ Rigid Body Motion ˆ Rigid Body Let r r r R= r Transforma- tions r r Rotational r r r motion in ℝ and vθ = − cos θ, cθ = cos θ, sθ = sin θ Rigid Motion in ℝ Velocity of a Rigid Body By Rodrigue’s formula ⎡ ω vθ + cθ ω ω vθ − ω sθ ω ω vθ + ω sθ ⎤ Wrenches and ⎢ ⎥ Reciprocal = ⎢ ω ω vθ + ω sθ ⎢ ω vθ + cθ ω ω vθ − ω sθ ⎥ ⎥ Screws ωθ ˆ e ⎢ ω ω v −ω s ω ω vθ + ω sθ ω vθ + cθ ⎥ ⎣ ⎦ Reference θ θ (continues next slide)
  • 47. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 17 Taking the trace of both sides, Chapter tr(R) = r + r + r = + cos θ = Rigid Body Motion λi Rigid Body i= where λi is the eigenvalue of R, i = , , Transforma- tions Case : tr(R) = or R = I, θ = ⇒ ωθ = Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • 48. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 17 Taking the trace of both sides, Chapter tr(R) = r + r + r = + cos θ = Rigid Body Motion λi Rigid Body i= where λi is the eigenvalue of R, i = , , Transforma- tions Case : tr(R) = or R = I, θ = ⇒ ωθ = Rotational motion in ℝ Case : − < tr(R) < , Rigid Motion in ℝ tr(R) − r −r θ = arccos ⇒ω= r −r Velocity of a r −r Rigid Body Wrenches and sθ Reciprocal Screws Reference
  • 49. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 17 Taking the trace of both sides, Chapter tr(R) = r + r + r = + cos θ = Rigid Body Motion λi Rigid Body i= where λi is the eigenvalue of R, i = , , Transforma- tions Case : tr(R) = or R = I, θ = ⇒ ωθ = Rotational motion in ℝ Case : − < tr(R) < , Rigid Motion in ℝ tr(R) − r −r θ = arccos ⇒ω= r −r Velocity of a sθ r − r Rigid Body Case : tr(R) = − ⇒ cos θ = − ⇒ θ = ±π Wrenches and Reciprocal Screws Reference (continues next slide)
  • 50. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 18 Following are possibilities: R= − ⇒ω= Chapter , − Rigid Body − Motion R= ⇒ω= Rigid Body , − Transforma- tions − R= − ⇒ω= Rotational motion in ℝ Rigid Motion in ℝ Note that if ωθ is a solution, then ω(θ ± nπ), n = , ± , ± , ... is Velocity of a Rigid Body Wrenches and Reciprocal also a solution. Screws Reference
  • 51. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 19 Definition: Exponential coordinate Chapter ωθ ∈ ℝ , with eωθ = R is called the exponential coordinates of R ˆ Rigid Body Exp ∶ Motion Rigid Body Transforma- tions Rotational so( ) ≅ ℝ motion in ℝ exp log Rigid Motion I SO( ) in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Figure 2.6 Reference
  • 52. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 19 Definition: Exponential coordinate Chapter ωθ ∈ ℝ , with eωθ = R is called the exponential coordinates of R ˆ Rigid Body Exp ∶ Motion Rigid Body Transforma- tions Rotational so( ) ≅ ℝ motion in ℝ exp log Rigid Motion I SO( ) in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Figure 2.6 Reference Property 5: exp is - when restricted to an open ball in ℝ of radius π.
  • 53. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 20 Theorem 1 (Euler): Any orientation is equivalent to a rotation about a fixed axis ω ∈ ℝ through an angle θ ∈ [−π, π]. Chapter Rigid Body Motion Rigid Body Transforma- tions – ω Rotational motion in ℝ B Rigid Motion in ℝ A Velocity of a Rigid Body Wrenches and Reciprocal Screws Figure 2.7 Reference SO( ) can be visualized as a solid ball of radius π.
  • 54. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 21 ◻ Other Parametrizations of SO( ): Chapter XYZ fixed angles (or Roll-Pitch-Yaw angle) Rigid Body Motion Rigid Body θ-Pitch Transforma- y tions Rotational motion in ℝ Rigid Motion in ℝ Roll-φ Velocity of a Rigid Body x Wrenches and ψ-Yaw Reciprocal Screws z Reference Figure 2.8 (continues next slide)
  • 55. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 22 XYZ fixed angles (or Roll-Pitch-Yaw angle) Continued Chapter Rigid Body cos φ − sin φ Motion Rigid Body Rx (φ) ∶= exφ = ˆ Transforma- tions sin φ cos φ cos θ sin θ Ry (θ) ∶= eyθ = ˆ − sin θ Rotational motion in ℝ cos θ cos ψ − sin ψ Rz (ψ) ∶= ezψ = Rigid Motion in ℝ ˆ sin ψ cos ψ Velocity of a Rigid Body Rab = Rx (φ)Ry (θ)Rz (ψ) Wrenches and Reciprocal −cθ sψ Screws cθ cψ sθ = sφ sθ cψ + cφ sψ −sφ sθ sψ + cφ cψ −sφ cθ Reference −cφ sθ cψ + sφ sψ cφ sθ sψ + sφ cψ cφ cθ
  • 56. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 23 ZYX Euler angle z A′ A′′ Chapter Rigid Body A B Motion Rigid Body Transforma- tions Rotational motion in ℝ x y Rigid Motion in ℝ Velocity of a Rigid Body Figure 2.9 Wrenches and Reciprocal Screws Reference
  • 57. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 23 ZYX Euler angle z(z ) ′ z′ A′ A′′ Chapter Rigid Body A B Motion Rigid Body Transforma- tions Rotational α y′ y′ motion in ℝ x y x′ x′ Rigid Motion in ℝ Velocity of a Rigid Body Figure 2.9 Raa′ = Rz (α) Wrenches and Reciprocal Screws Reference
  • 58. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 23 ZYX Euler angle z(z ) ′ z′ A′ A′′ Chapter Rigid Body A B Motion z ′′ z ′′ Rigid Body Transforma- tions Rotational α y′ y′ (y′′ ) y′′ motion in ℝ x y x′ x′ β Rigid Motion in ℝ x′′ x′′ Velocity of a Rigid Body Figure 2.9 Raa′ = Rz (α) Ra′ a′′ = Ry (β) Wrenches and Reciprocal Screws Reference
  • 59. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 23 ZYX Euler angle z(z ) ′ z′ A′ A′′ Chapter Rigid Body A B Motion z ′′ z ′′′ z ′′ Rigid Body γ Transforma- y′′′ tions Rotational α y′ y′ (y′′ ) y′′ motion in ℝ x y x′ x′ β Rigid Motion in ℝ x′′ (x′′′ ) x′′ Velocity of a Rigid Body Figure 2.9 Raa′ = Rz (α) Ra′ a′′ = Ry (β) Ra′′ b = Rx (γ) Wrenches and Reciprocal Screws Reference Rab = Rz (α)Ry (β)Rx (γ) (continues next slide)
  • 60. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 24 Chapter ZYX Euler angle (continued) cα cβ −sα cγ + cα sβ sγ sα sγ + cα sβ cγ Rigid Body Rab (α, β, γ) = sα cβ cα cγ + sα sβ sγ −cα sγ + sα sβ cγ Motion Rigid Body Transforma- −sβ c β sγ c β cγ tions Rotational Note: When β = , sin β = , α + γ = const ⇒ singularity! motion in ℝ Rigid Motion atan (y, x) in ℝ β = atan (−r , r +r ) Velocity of a Rigid Body α = atan (r cβ , r cβ ) Wrenches and y Reciprocal γ = atan (r cβ ) Screws Reference cβ , r x Figure 2.10
  • 61. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 25 § Quaternions: Chapter Q =q +q i+q j+q k where i = j = k = − , i ⋅ j = k, j ⋅ k = i, k ⋅ i = j Rigid Body Motion Rigid Body Transforma- tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • 62. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 25 § Quaternions: Chapter Q =q +q i+q j+q k where i = j = k = − , i ⋅ j = k, j ⋅ k = i, k ⋅ i = j Rigid Body Motion Rigid Body Property 1: Define Q∗ = (q , q)∗ = (q , −q), q ∈ ℝ, q ∈ ℝ Transforma- tions Rotational motion in ℝ Q = QQ∗ = q + q + q + q Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • 63. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 25 § Quaternions: Chapter Q =q +q i+q j+q k where i = j = k = − , i ⋅ j = k, j ⋅ k = i, k ⋅ i = j Rigid Body Motion Rigid Body Property 1: Define Q∗ = (q , q)∗ = (q , −q), q ∈ ℝ, q ∈ ℝ Transforma- tions Rotational motion in ℝ Q = QQ∗ = q + q + q + q Property 2: Q = (q , ⃗), P = (p , ⃗) Rigid Motion q p QP = (q p − ⃗ ⋅ ⃗, q ⃗ + p ⃗ + ⃗ × ⃗) in ℝ Velocity of a q p p q q p Rigid Body Wrenches and Reciprocal Screws Reference
  • 64. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 25 § Quaternions: Chapter Q =q +q i+q j+q k where i = j = k = − , i ⋅ j = k, j ⋅ k = i, k ⋅ i = j Rigid Body Motion Rigid Body Property 1: Define Q∗ = (q , q)∗ = (q , −q), q ∈ ℝ, q ∈ ℝ Transforma- tions Rotational motion in ℝ Q = QQ∗ = q + q + q + q Property 2: Q = (q , ⃗), P = (p , ⃗) Rigid Motion q p QP = (q p − ⃗ ⋅ ⃗, q ⃗ + p ⃗ + ⃗ × ⃗) in ℝ Velocity of a q p p q q p Rigid Body Wrenches and Reciprocal Property 3: (a) The set of unit quaternions forms a group (b) If R = eωθ , then Q = (cos , ω sin ) Screws ˆ θ θ Reference (c) Q acts on x ∈ ℝ by QXQ∗ , where X = ( , x)
  • 65. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 26 ◻ Unit Quaternions: Given Q = (q , q), q ∈ ℝ, q ∈ ℝ , the vector part of QXQ∗ is Chapter Rigid Body given by R(Q)x, Motion Rigid Body R(Q) = I + q q + q ˆ ˆ ⎡ − (q + q ) ⎤ Transforma- ⎢ − q q + qq q q + qq ⎥ tions =⎢ q q + qq ⎢ − (q + q ) − qq + qq ⎥ ⎥ Rotational ⎢ − q q + qq − (q + q ) ⎥ motion in ℝ Rigid Motion ⎣ qq + qq ⎦ in ℝ Velocity of a Rigid Body where Q ≜ q + q + q + q = Wrenches and Reciprocal Screws Reference (continues next slide)
  • 66. Chapter 2 Rigid Body Motion 2.2 Rotational Motion in ℝ 27 ◻ Quaternions (continued): Chapter Conversion from Roll-Pitch-Yaw angle to unit quaternions: Rigid Body Q = (cos Motion φ φ θ θ ψ ψ Rigid Body , x sin )(cos , y sin )(cos , z sin ) ⇒ Transforma- tions q = cos − sin φ θ ψ φ θ ψ Rotational cos cos sin sin motion in ℝ ⎡ ⎤ Rigid Motion ⎢ cos φ sin θ sin ψ + sin φ cos θ cos ψ ⎥ in ℝ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ q = ⎢ cos sin cos − sin cos sin Velocity of a Rigid Body φ θ ψ φ θ ψ ⎥ ⎢ ⎥ Wrenches and ⎢ ⎥ ⎢ ⎥ Reciprocal ⎢ cos φ cos θ sin ψ + sin φ sin θ cos ψ ⎥ Screws ⎢ ⎥ Reference ⎣ ⎦ Conversion from unit quaternions to roll-pitch-yaw angles (?) † End of Section †
  • 67. Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 28 z Chapter Rigid Body pab B Motion A Rigid Body Transforma- tions qa y qb x Rotational motion in ℝ q Rigid Motion in ℝ Figure 2.11 Velocity of a pab ∈ ℝ ∶ Coordinates of the origin of B Rab ∈ SO( ) ∶ Orientation of B relative to A Rigid Body Wrenches and SE( ) ∶ {(p, R) p ∈ ℝ , R ∈ SO( )} ∶ Configuration Space Reciprocal Screws Reference
  • 68. Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 28 z Chapter Rigid Body pab B Motion A Rigid Body Transforma- tions qa y qb x Rotational motion in ℝ q Rigid Motion in ℝ Figure 2.11 Velocity of a pab ∈ ℝ ∶ Coordinates of the origin of B Rab ∈ SO( ) ∶ Orientation of B relative to A Rigid Body Wrenches and SE( ) ∶ {(p, R) p ∈ ℝ , R ∈ SO( )} ∶ Configuration Space Reciprocal Screws Reference Or...as a transformation: gab = (pab , Rab ) ∶ ℝ ↦ ℝ qb ↦ qa = pab + Rab ⋅ qb
  • 69. Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 29 ◻ Homogeneous Representation: Chapter Points: ⎡ q ⎤ ⎢ q ⎥ Rigid Body q=⎢ q ⎥∈ℝ q q= ∈ℝ ⎢ ⎥ Motion q q ⎢ ⎥ Rigid Body ⎣ ⎦ Transforma- tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • 70. Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 29 ◻ Homogeneous Representation: Chapter Points: ⎡ q ⎤ ⎢ q ⎥ Rigid Body q=⎢ q ⎥∈ℝ q q= ∈ℝ ⎢ ⎥ Motion q q ⎢ ⎥ Rigid Body ⎣ ⎦ Transforma- Vectors: ⎡ p ⎤ ⎡ q ⎤ tions p −q ⎢ p ⎥ ⎢ q ⎥ v v = p−q = ⎢ p ⎥−⎢ q ⎥= v v=p−q = p −q = ⎢ ⎥ ⎢ ⎥ v Rotational v ⎢ ⎥ ⎢ ⎥ v motion in ℝ p −q v Rigid Motion ⎣ ⎦ ⎣ ⎦ in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • 71. Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 29 ◻ Homogeneous Representation: Chapter Points: ⎡ q ⎤ ⎢ q ⎥ Rigid Body q=⎢ q ⎥∈ℝ q q= ∈ℝ ⎢ ⎥ Motion q q ⎢ ⎥ Rigid Body ⎣ ⎦ Transforma- Vectors: ⎡ p ⎤ ⎡ q ⎤ tions p −q ⎢ p ⎥ ⎢ q ⎥ v v = p−q = ⎢ p ⎥−⎢ q ⎥= v v=p−q = p −q = ⎢ ⎥ ⎢ ⎥ v Rotational v ⎢ ⎥ ⎢ ⎥ v motion in ℝ p −q v Rigid Motion ⎣ ⎦ ⎣ ⎦ in ℝ Point-Point = Vector Velocity of a Rigid Body Vector+Point = Point Wrenches and Reciprocal Vector+Vector = Vector Screws Reference Point+Point: Meaningless (continues next slide)
  • 72. Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 30 qa = pab + Rab ⋅ qb gab = (pab , Rab ) = Rab pab qa qab Chapter Rigid Body Motion g ab qa = g ab ⋅ qb g ab = Rigid Body Transforma- Rab pab tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • 73. Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 30 qa = pab + Rab ⋅ qb gab = (pab , Rab ) = Rab pab qa qab Chapter Rigid Body Motion g ab qa = g ab ⋅ qb g ab = Rigid Body Transforma- Rab pab tions Rotational motion in ℝ Rigid Motion ◻ Composition Rule: in ℝ Velocity of a Rigid Body gab gbc Wrenches and qb = g bc ⋅ qc B qa = g ab ⋅ qb Reciprocal Screws A C Reference Figure 2.12
  • 74. Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 30 qa = pab + Rab ⋅ qb gab = (pab , Rab ) = Rab pab qa qab Chapter Rigid Body Motion g ab qa = g ab ⋅ qb g ab = Rigid Body Transforma- Rab pab tions Rotational motion in ℝ Rigid Motion ◻ Composition Rule: in ℝ Velocity of a Rigid Body gab gbc Wrenches and qb = g bc ⋅ qc B qa = g ab ⋅ qb = g ab ⋅ g bc ⋅qc Reciprocal Screws A C gac Reference g ac Figure 2.12 g ac = g ab ⋅ g bc = Rab Rbc Rab pbc + pab
  • 75. Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 31 ◻ Special Euclidean Group: SE( ) = R p ∈ℝ × p ∈ ℝ , R ∈ SO( ) Chapter Rigid Body Motion Rigid Body Transforma- tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • 76. Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 31 ◻ Special Euclidean Group: SE( ) = R p ∈ℝ × p ∈ ℝ , R ∈ SO( ) Chapter Rigid Body Motion Rigid Body Transforma- Property 4: SE( ) forms a group. tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • 77. Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 31 ◻ Special Euclidean Group: SE( ) = R p ∈ℝ × p ∈ ℝ , R ∈ SO( ) Chapter Rigid Body Motion Rigid Body Transforma- Property 4: SE( ) forms a group. tions Rotational motion in ℝ Proof : g ⋅ g ∈ SE( ) Rigid Motion in ℝ e=I Velocity of a Rigid Body Wrenches and Reciprocal (g)− = RT −RT p Screws Reference Associativity: Follows from property of matrix multiplication
  • 78. Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 32 § Induced transformation on vectors: Chapter Rigid Body v v v =s−r = , g ∗ v = gs − gr = = Motion v R p v Rv Rigid Body v v Transforma- tions The bar will be dropped to simplify notations Rotational motion in ℝ Property 5: An element of SE( ) is a rigid transformation. Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • 79. Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 33 Exponential coordinates of SE( ): ω p(t) Chapter For rotational motion: ˙ p(t)=ω × (p(t) − q) Rigid Body Motion ˙ Rigid Body Transforma- p ˙ ω −ω × q ˆ p tions = p(t) p( ) or p= ξ ⋅ p ⇒ p(t) = e p( ) Rotational ˙ ˆ ˆ ξt motion in ℝ ˆ (ξt) + ⋯ q where eξt =I + ξt + Rigid Motion ˆ ˆ in ℝ ! A Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference Figure 2.13
  • 80. Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 33 Exponential coordinates of SE( ): ω p(t) Chapter For rotational motion: ˙ p(t)=ω × (p(t) − q) Rigid Body Motion ˙ Rigid Body Transforma- p ˙ ω −ω × q ˆ p tions = p(t) p( ) or p= ξ ⋅ p ⇒ p(t) = e p( ) Rotational ˙ ˆ ˆ ξt motion in ℝ ˆ (ξt) q where eξt =I + ξt + ! + ⋯ Rigid Motion ˆ ˆ in ℝ A For translational motion: p(t) p(t)=v Velocity of a ˙ Rigid Body ˙ Wrenches and p(t) ˙ v p v Reciprocal Screws = p(t) p( ) p(t)= ξ ⋅ p(t) ⇒ p(t) = eξt p( ) Reference ˙ ˆ ˆ ˆ v ξ= A Figure 2.13
  • 81. Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 34 Definition: Chapter se( ) = ω v ∈ ℝ × v, ω ∈ ℝ ˆ Rigid Body Motion is called the twist space. There exists a - correspondence Rigid Body between se( ) and ℝ , defined by ∧ ∶ ℝ ↦ se( ) ξ ∶= ω ↦ ξ = ω v Transforma- tions v ˆ ˆ Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • 82. Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 34 Definition: Chapter se( ) = ω v ∈ ℝ × v, ω ∈ ℝ ˆ Rigid Body Motion is called the twist space. There exists a - correspondence Rigid Body between se( ) and ℝ , defined by ∧ ∶ ℝ ↦ se( ) ξ ∶= ω ↦ ξ = ω v Transforma- tions v ˆ ˆ Rotational motion in ℝ Rigid Motion Property 6: exp ∶ se( ) ↦ SE( ), ξθ ↦ eξθ in ℝ ˆ ˆ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • 83. Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 34 Definition: Chapter se( ) = ω v ∈ ℝ × v, ω ∈ ℝ ˆ Rigid Body Motion is called the twist space. There exists a - correspondence Rigid Body between se( ) and ℝ , defined by ∧ ∶ ℝ ↦ se( ) ξ ∶= ω ↦ ξ = ω v Transforma- tions v ˆ ˆ Rotational motion in ℝ Rigid Motion Property 6: exp ∶ se( ) ↦ SE( ), ξθ ↦ eξθ in ℝ ˆ ˆ Velocity of a Rigid Body Proof : Let ξ = ω v ˆ ˆ Wrenches and Reciprocal Screws If ω = , then ξ = ξ = ⋯ = , eξθ = I vθ ∈ SE( ) Reference ˆ ˆ ˆ (continues next slide)
  • 84. Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 35 Chapter If ω is not , assume ω = . Rigid Body Motion Define: Rigid Body g = I ω×v , ξ′ = g − ⋅ ξ ⋅ g = ˆ ˆ ω hω ˆ where h = ωT ⋅ v. Transforma- tions ⋅ ξ ′ ⋅g eξθ = eg = g − ⋅ eξ θ ⋅ g Rotational ˆ − ˆ ˆ′ motion in ℝ Rigid Motion and as ξ′ = ω , ξ′ = ω in ℝ ˆ ˆ ˆ ˆ Velocity of a Rigid Body we have Wrenches and Reciprocal eξ θ = ˆ′ eωθ ˆ hωθ ⇒ eξθ = ˆ eωθ ˆ (I − eωθ )ωv + ωωT vθ ˆ ˆ Screws Reference
  • 85. Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 36 p(θ) = eξθ ⋅ p( ) ⇒ gab (θ) = eξθ ˆ ˆ Chapter Rigid Body Motion If there is offset, Rigid Body gab(θ) = eξθ gab ( )( Why?) Transforma- tions ˆ Rotational motion in ℝ ω Rigid Motion in ℝ Velocity of a Rigid Body θ B′ Wrenches and Reciprocal ˆ Screws B e ξθ Reference gab ( ) A Figure 2.14
  • 86. Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 37 Property 7: exp ∶ se( ) ↦ SE( ) is onto. Chapter Rigid Body Motion Rigid Body Transforma- tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • 87. Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 37 Property 7: exp ∶ se( ) ↦ SE( ) is onto. Proof : Let g = (p, R), R ∈ SO( ), p ∈ ℝ Chapter (R = I) Let Rigid Body Motion Case 1: p ξ= , θ = p ⇒ eξθ = g = I p Rigid Body ˆ Transforma- ˆ p tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • 88. Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 37 Property 7: exp ∶ se( ) ↦ SE( ) is onto. Proof : Let g = (p, R), R ∈ SO( ), p ∈ ℝ Chapter (R = I) Let Rigid Body Motion Case 1: p ξ= , θ = p ⇒ eξθ = g = I p Rigid Body ˆ Transforma- ˆ p tions Rotational Case 2: (R ≠ I) = (I − eωθ )(ω × v) + ωωT vθ = motion in ℝ ωθ ˆ ˆ e ˆ ξθ e R p Rigid Motion eωθ = R in ℝ ˆ (I − eωθ )(ω × v) + ωωT vθ = p Velocity of a ⇒ ˆ Rigid Body Wrenches and Solve for ωθ from previous section. Let A = (I − eωθ )ω + wwT θ, ˆ ˆ Av = p. Claim: Reciprocal Screws Reference A = (I − eωθ )ω + wwT θ ∶= A + A ˆ ˆ ker A ∩ ker A = ϕ ⇒ v = A− p ξθ ∈ ℝ : Exponential coordinates of g ∈ SE( )
  • 89. Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 38 ◻ Screws, twists and screw motion: p Chapter Rigid Body ω Motion z q Rigid Body Figure 2.15 Transforma- θ tions d Rotational motion in ℝ x y h = d (θ = , h = ∞), d = h ⋅ θ Rigid Motion in ℝ Screw attributes Pitch: l = {q + λω λ ∈ ℝ} θ Velocity of a Axis: M=θ Rigid Body Magnitude: Wrenches and Reciprocal Screws Reference
  • 90. Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 38 ◻ Screws, twists and screw motion: p Chapter Rigid Body ω Motion z q Rigid Body Figure 2.15 Transforma- θ tions d Rotational motion in ℝ x y h = d (θ = , h = ∞), d = h ⋅ θ Rigid Motion in ℝ Screw attributes Pitch: l = {q + λω λ ∈ ℝ} θ Velocity of a Axis: M=θ Rigid Body Magnitude: Wrenches and Reciprocal Screws Definition: A screw S consists of an axis l, pitch h, and magnitude M. A screw motion is a rotation by θ = M about l, followed by Reference translation by hθ, parallel to l. If h = ∞, then, translation about v by θ = M
  • 91. Chapter 2 Rigid Body Motion 2.3 Rigid motion in ℝ 39 Corresponding g ∈ SE( ): g ⋅ p = q + eωθ (p − q) + hθω ˆ Chapter g⋅ p = eωθ (I − eωθ )q + hθω p Rigid Body ˆ ˆ Motion ⇒ Rigid Body g= eωθ (I − eωθ )q + hθω Transforma- ˆ ˆ tions Rotational motion in ℝ Rigid Motion in ℝ Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference