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

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 July ,
  • 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) ⎡ x(t) ⎤ tions ⎢ ⎥ 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) ⎡ x(t) ⎤ tions ⎢ ⎥ 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 (body 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 y motion in R zb 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 y motion in R zb 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 motion in R Proof : a For a ∈ R , let a = −a Rigid Motion in R ˆ a −a −a 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 Rˆ R = (Rv) (prove it yourself) v 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 ∈ SO( ), R = r r 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 ∈ SO( ), R = r r 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 ∈ SO( ), R = r r 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 ∈ SO( ), R = r r 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 + ( Screws ˆ θ θ θ θ + ˆ − ˆ + ⋯)ω Reference ! ! ! ! = I + ω sin θ + ω ( − cos θ) ˆ ˆ
  • Chapter 2 Rigid Body Motion 2.2 Rotational Motion in R 15 Chapter Rodrigues’ formula for ω ≠ : eωθ = I + ( − cos ω θ) Rigid Body ˆ ˆ ω ˆ ω Motion sin ω θ + 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 Rodrigues’ formula for ω ≠ : eωθ = I + ( − cos ω θ) Rigid Body ˆ ˆ ω ˆ ω Motion sin ω θ + 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 Transforma- r r r 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 Rodrigues’ formula ⎡ ω vθ + cθ ⎤ Wrenches and ⎢ ⎥ Reciprocal ω ω vθ − ω sθ ω ω vθ + ω sθ = ⎢ ω ω vθ + ω sθ ⎢ ⎥ ⎥ Screws ωθ ˆ ⎢ ω ω v −ω s ⎥ e ω vθ + cθ ω ω vθ − ω sθ ⎣ ⎦ Reference θ θ ω ω vθ + ω sθ ω vθ + cθ (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 Rigid Motion in R Case : − < tr(R) < , θ = arccos ⇒ω= Velocity of a tr(R) − r −r Rigid Body r −r Wrenches and sθ r −r 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 Rigid Motion in R Case : − < tr(R) < , θ = arccos ⇒ω= Velocity of a tr(R) − r −r Rigid Body r −r sθ r − r 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 Motion Rigid Body Exp ∶ 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 Motion Rigid Body Exp ∶ 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 Motion Rigid Body Rx (φ) ∶= exφ = ˆ cos φ − sin φ Transforma- tions sin φ cos φ cos θ sin θ Rotational motion in R Ry (θ) ∶= eyθ = ˆ − sin θ 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 Screws cθ cψ −cθ sψ 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) Rigid Body cα cβ −sα cγ + cα sβ sγ sα sγ + cα sβ cγ 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 = π , sin β = , α + γ = const ⇒ singularity! motion in R Rigid Motion in R atan (y, x) 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 , q), P = (p , p) Rigid Motion QP = (q p − q ⋅ p, q p + p q + q × p) 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 , q), P = (p , p) Rigid Motion QP = (q p − q ⋅ p, q p + p q + q × p) in R Velocity of a 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: Chapter Given Q = (q , q), q ∈ R, q ∈ R , the vector part of QXQ∗ is Rigid Body Motion given by R(Q)x, recall that Rigid Body q = cos θ , q = ω sin θ Transforma- and the Rodrigues’ formula: eωθ = I + ω sin θ + ω ( − cos θ) tions ˆ ˆ ˆ Rotational motion in R then Rigid Motion R(Q) = I + q q + q in R ˆ ˆ ⎡ − (q + q ) − q q + qq q q + qq ⎤ ⎢ ⎥ Velocity of a =⎢ q q + qq ⎥ Rigid Body ⎢ − (q + q ) − qq + qq ⎥ ⎢ − q q + qq ⎥ Wrenches and ⎣ qq + qq − (q + q ) ⎦ Reciprocal Screws Reference where Q ≜ q + q + q + q = (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 , x sin )(cos , y sin )(cos , z sin ) ⇒ Motion φ φ θ θ ψ ψ Rigid Body Transforma- tions q = cos φ θ ψ φ θ ψ Rotational cos cos − sin sin sin ⎡ ⎤ motion in R ⎢ cos φ sin θ sin ψ + sin φ cos θ cos ψ ⎥ ⎢ ⎥ Rigid Motion ⎢ ⎥ in R ⎢ ⎥ q = ⎢ cos sin cos − sin cos sin ⎥ φ θ ψ φ θ ψ ⎢ ⎥ Velocity of a ⎢ ⎥ Rigid Body ⎢ ⎥ ⎢ cos φ cos θ sin ψ + sin φ sin θ cos ψ ⎥ Wrenches and ⎢ ⎥ Reciprocal ⎣ ⎦ 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 ⎢ ⎥ ⎣ ⎦ Rigid Body q 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 ⎢ ⎥ ⎣ ⎦ Rigid Body q Transforma- Vectors: ⎡ p ⎤ ⎡ q ⎤ tions p −q ⎢ p ⎥ ⎢ q ⎥ v v = p−q = ⎢ p ⎥−⎢ q ⎥= v v=p−q = p −q = ⎢ ⎥ ⎢ ⎥ v Rotational v p −q ⎢ ⎥ ⎢ ⎥ motion in R v ⎣ ⎦ ⎣ ⎦ 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 ⎢ ⎥ ⎣ ⎦ Rigid Body q Transforma- Vectors: ⎡ p ⎤ ⎡ q ⎤ tions p −q ⎢ p ⎥ ⎢ q ⎥ v v = p−q = ⎢ p ⎥−⎢ q ⎥= v v=p−q = p −q = ⎢ ⎥ ⎢ ⎥ v Rotational v p −q ⎢ ⎥ ⎢ ⎥ motion in R v ⎣ ⎦ ⎣ ⎦ 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 (g)− = RT −RT p Wrenches and Reciprocal 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 ˆ ω −ω × q p tions = p(t) p( ) or p= ξ ⋅ p ⇒ p(t) = e p( ) Rotational ˙ ˆ ˆ ξt motion in R ˆ (ξt) + ⋯ ˆ q ˆ where eξt =I + ξt + Rigid Motion 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 ˆ ω −ω × q p tions = p(t) p( ) or p= ξ ⋅ p ⇒ p(t) = e p( ) Rotational ˙ ˆ ˆ ξt motion in R ˆ (ξt) ˆ q ˆ where eξt =I + ξt + ! + ⋯ Rigid Motion 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 ˆ ω hω 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′ = , ξ′ = in R ˆ ˆ ω ˆ ˆ ω 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 = Rigid Body Transforma- ˆ p ˆ I 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 = Rigid Body Transforma- ˆ p ˆ I 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 = (I − eωθ )q + hθω Rigid Body p eωθ ˆ ˆ p Motion g⋅ ⇒ Rigid Body g= (I − eωθ )q + hθω Transforma- tions eωθ 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 39 Corresponding g ∈ SE( ): g ⋅ p = q + eωθ (p − q) + hθω ˆ Chapter = (I − eωθ )q + hθω Rigid Body p eωθ ˆ ˆ p Motion g⋅ ⇒ Rigid Body g= (I − eωθ )q + hθω Transforma- tions eωθ ˆ ˆ Rotational motion in R On the other hand... (I − eωθ )ω × v + ωωT vθ Rigid Motion eξθ = eωθ in R ˆ ˆ ˆ Velocity of a If we let v = −ω × q + hω, then Rigid Body Wrenches and (I − eωθ )(−ω q) = (I − eωθ )(−ωωT q + q) = (I − eωθ )q Reciprocal ˆ ˆ ˆ Screws ˆ Thus, eξθ = g Reference ˆ
  • Chapter 2 Rigid Body Motion 2.3 Rigid motion in R 39 Corresponding g ∈ SE( ): g ⋅ p = q + eωθ (p − q) + hθω ˆ Chapter = (I − eωθ )q + hθω Rigid Body p eωθ ˆ ˆ p Motion g⋅ ⇒ Rigid Body g= (I − eωθ )q + hθω Transforma- tions eωθ ˆ ˆ Rotational motion in R On the other hand... (I − eωθ )ω × v + ωωT vθ Rigid Motion eξθ = eωθ in R ˆ ˆ ˆ Velocity of a If we let v = −ω × q + hω, then Rigid Body Wrenches and (I − eωθ )(−ω q) = (I − eωθ )(−ωωT q + q) = (I − eωθ )q Reciprocal ˆ ˆ ˆ Screws ˆ Thus, eξθ = g Reference ˆ For pure rotation (h = ): ξ = (−ω × q, ω) For pure translation: g = I vθ , ⇒ ξ = (v, ), and eξθ = g ˆ
  • Chapter 2 Rigid Body Motion 2.3 Rigid motion in R 40 ◻ Screw associated with a twist: ξ = (v, ω) ∈ R ⎧ ωT v ⎪ Chapter ⎪ ⎪ Pitch: h = ⎨ ω , if ω ≠ Rigid Body Motion ⎪ ⎪ ∞, ⎪ if ω = Rigid Body ⎩ Transforma- ⎧ ω×v tions ⎪ ⎪ + λω, λ ∈ R, if ω ≠ Axis: l = ⎨ ω Rotational ⎪ + λv motion in R Rigid Motion ⎪ ⎩ λ ∈ R, if ω = ⎧ ⎪ in R ⎪ ω , if ω ≠ Magnitude: M = ⎨ Velocity of a ⎪ ⎪ v , if ω = Rigid Body Wrenches and Reciprocal ⎩ Screws Reference
  • Chapter 2 Rigid Body Motion 2.3 Rigid motion in R 40 ◻ Screw associated with a twist: ξ = (v, ω) ∈ R ⎧ ωT v ⎪ Chapter ⎪ ⎪ Pitch: h = ⎨ ω , if ω ≠ Rigid Body Motion ⎪ ⎪ ∞, ⎪ if ω = Rigid Body ⎩ Transforma- ⎧ ω×v tions ⎪ ⎪ + λω, λ ∈ R, if ω ≠ Axis: l = ⎨ ω Rotational ⎪ + λv motion in R Rigid Motion ⎪ ⎩ λ ∈ R, if ω = ⎧ ⎪ in R ⎪ ω , if ω ≠ Magnitude: M = ⎨ Velocity of a ⎪ ⎪ v , if ω = Rigid Body Wrenches and Reciprocal ⎩ Screws Special cases: h = ∞, Pure translation (prismatic joint) Reference h = , Pure rotation (revolute joint)
  • Chapter 2 Rigid Body Motion 2.3 Rigid motion in R 41 Screw ˆ Twist: ξθ Case : Pitch: h = ∞ Chapter θ = M, Rigid Body Axis: l = {q + λv v = , λ ∈ R} Motion Rigid Body Magnitude: M ξ= ˆ v Transforma- Case : Pitch: h ≠ ∞ tions θ = M, Axis: l = {q + λω ω = , λ ∈ R} Rotational ξ = ω −ωq + hω ˆ ˆ motion in R ˆ Rigid Motion Magnitude: M in R Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • Chapter 2 Rigid Body Motion 2.3 Rigid motion in R 41 Screw ˆ Twist: ξθ Case : Pitch: h = ∞ Chapter θ = M, Rigid Body Axis: l = {q + λv v = , λ ∈ R} Motion Rigid Body Magnitude: M ξ= ˆ v Transforma- Case : Pitch: h ≠ ∞ tions θ = M, Axis: l = {q + λω ω = , λ ∈ R} Rotational ξ = ω −ωq + hω ˆ ˆ motion in R ˆ Rigid Motion Magnitude: M in R Velocity of a Rigid Body Definition: Screw Motion Rotation about an axis by θ = M, followed by translation Wrenches and Reciprocal Screws about the same axis by hθ Reference
  • Chapter 2 Rigid Body Motion 2.3 Rigid motion in R 42 Theorem 2 (Chasles): Chapter Every rigid body motion can be realized by a ro- Rigid Body Motion tation about an axis combined with a translation Rigid Body parallel to that axis. 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 42 Theorem 2 (Chasles): Chapter Every rigid body motion can be realized by a ro- Rigid Body Motion tation about an axis combined with a translation Rigid Body parallel to that axis. Transforma- tions – Rotational motion in R Proof : For ξ ∈ se( ): ˆ Rigid Motion in R Velocity of a Rigid Body ξ=ξ +ξ = ˆ ˆ ˆ ˆ ω −ω × q + hω [ ξ , ξ ] = ⇒ eξθ = eξ θ eξ θ Wrenches and ˆ ˆ ˆ ˆ ˆ Reciprocal Screws Reference † End of Section †
  • Chapter 2 Rigid Body Motion 2.4 Velocity of a Rigid Body 43 Review: Point-mass velocity Chapter q(t) ∈ R , t ∈ (−ε, ε), v = q(t) ∈ R , a = q(t) = v(t) ∈ R Rigid Body d d d Motion Rigid Body dt dt dt 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 2.4 Velocity of a Rigid Body 43 Review: Point-mass velocity Chapter q(t) ∈ R , t ∈ (−ε, ε), v = q(t) ∈ R , a = q(t) = v(t) ∈ R Rigid Body d d d Motion Rigid Body dt dt dt Transforma- tions Rotational ◻ Velocity of Rotational Motion: Rab (t) ∈ SO( ), t ∈ (−ε, ε), qa (t) = Rab (t)qb motion in R Rigid Motion V a = qa (t) = Rab (t)qb = Rab (t)RT (t)Rab (t)qb = Rab RT qa in R d ˙ ˙ ˙ ab Velocity of a ab dt Rab(t)RT (t) = I ⇒ Rab RT + Rab RT = , Rab RT = −(Rab RT )T Rigid Body Wrenches and ab ˙ ab ˙ ab ˙ ab ˙ ab Reciprocal Screws z q Reference z zab p(t) xab y o A:o − xyz x p( ) y Figure 2.1 x yab B:o − x ab y ab z ab Figure 2.3
  • Chapter 2 Rigid Body Motion 2.4 Velocity of a Rigid Body 44 Denote spatial angular velocity by: Chapter Rigid Body Motion ˆ ab ˙ ab ωs = Rab RT , ωab ∈ R Rigid Body Transforma- tions Then Rotational V a = ωs ⋅ qa = ωs × qa ˆ ab ab motion in R Rigid Motion Body angular velocity: in R Velocity of a Rigid Body ωb = RT ⋅ Rab , vb ≜ RT ⋅ va = ωb × qb ˆ ab ab ˙ ab ab Wrenches and Reciprocal Screws Relation between body and spatial angular velocity: ωb = RT ⋅ ωs or ωb = RT ωs Rab Reference ab ab ab ˆ ab ab ˆ ab
  • Chapter 2 Rigid Body Motion 2.4 Velocity of a Rigid Body 45 ◻ Generalized Velocity: Rab (t) pab (t) , q (t) = g (t)q Chapter Rigid Body Motion gab = a ab b Rigid Body qa (t) = gab(t)qb = gab ⋅ gab ⋅ gab ⋅ qb = Vab ⋅ qa − Transforma- d ˆs tions ˙ ˙ Rotational dt motion in R Rigid Motion in R Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • Chapter 2 Rigid Body Motion 2.4 Velocity of a Rigid Body 45 ◻ Generalized Velocity: Rab (t) pab (t) , q (t) = g (t)q Chapter Rigid Body Motion gab = a ab b Rigid Body qa (t) = gab(t)qb = gab ⋅ gab ⋅ gab ⋅ qb = Vab ⋅ qa − Transforma- d ˆs tions ˙ ˙ Rotational dt motion in R Rigid Motion in R ˆs ˙ − Vab = gab ⋅ gab = ˙ ˙ Rab pab RT −RT pab ab ab Velocity of a Rigid Body Wrenches and Reciprocal = RabRT −Rab RT pab + pab ˙ ab ˙ ab ˙ Screws Reference = ωs −ωs × pab + pab ˆ ab ab ˙ ≜ ω s vs ˆ ab ab
  • Chapter 2 Rigid Body Motion 2.4 Velocity of a Rigid Body 46 ◻ (Generalized) Spatial Velocity: Chapter −ωs × pab + pab Vab = = vs ˙ (Rab RT )∨ Rigid Body s ab ab Motion ωs ab ˙ ab vq a = Rigid Body Transforma- tions ωs ab × qa + vs ab − − vqb = gab ⋅ vqa = gab ⋅ gab ⋅ qb = Vab ⋅ qb Rotational motion in R Note: ˙ ˆb Rigid Motion in R Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • Chapter 2 Rigid Body Motion 2.4 Velocity of a Rigid Body 46 ◻ (Generalized) Spatial Velocity: Chapter −ωs × pab + pab Vab = = vs ˙ (Rab RT )∨ Rigid Body s ab ab Motion ωs ab ˙ ab vq a = Rigid Body Transforma- tions ωs ab × qa + vs ab − − vqb = gab ⋅ vqa = gab ⋅ gab ⋅ qb = Vab ⋅ qb Rotational motion in R Note: ˙ ˆb Rigid Motion in R Velocity of a ◻ (Generalized) Body Velocity: Rigid Body − Vab = gab gab = Wrenches and Reciprocal ˆb ˙ RT Rab RT pab ab ˙ ab ˙ ≜ ω b vb ˆ ab ab Screws Reference Vab = = b vb RT pab ˙ (Rab Rab )∨ ab ab T ˙ ωb ab
  • Chapter 2 Rigid Body Motion 2.4 Velocity of a Rigid Body 47 ◻ Relation between body and spatial velocity: Chapter − − − ˆb − Vab = gab ⋅ gab = gab ⋅ gab ⋅ gab ⋅ gab = gab ⋅ Vab ⋅ gab Rigid Body Motion ˆs ˙ ˙ Rigid Body = Rab pab ω b vb ˆ ab ab RT −RT pab Transforma- tions ab ab Rotational = Rab pab ωb RT −ωb RT pab + vb motion in R ˆ ab ab ˆ ab ab ab Rigid Motion in R Velocity of a Rigid Body = Rab ωb RT −Rab ωb RT pab + Rab vb ˆ ab ab ˆ ab ab ab Wrenches and Reciprocal Screws Reference
  • Chapter 2 Rigid Body Motion 2.4 Velocity of a Rigid Body 47 ◻ Relation between body and spatial velocity: Chapter − − − ˆb − Vab = gab ⋅ gab = gab ⋅ gab ⋅ gab ⋅ gab = gab ⋅ Vab ⋅ gab Rigid Body Motion ˆs ˙ ˙ Rigid Body = Rab pab ω b vb ˆ ab ab RT −RT pab Transforma- tions ab ab Rotational = Rab pab ωb RT −ωb RT pab + vb motion in R ˆ ab ab ˆ ab ab ab Rigid Motion in R Velocity of a Rigid Body = Rab ωb RT −Rab ωb RT pab + Rab vb ˆ ab ab ˆ ab ab ab Vab = = Wrenches and s vs ab ˆ Rab pab Rab b Reciprocal Screws ωs Rab Vab ab Reference Adg × Adg = ˆ R pR R ∈R , for g = (p, R)
  • Chapter 2 Rigid Body Motion 2.4 Velocity of a Rigid Body 48 ◻ Properties of Adjoint mapping: Chapter g− = RT −RT p ⇒ Rigid Body (−RT p)∧ RT Motion Rigid Body Adg − = RT Transforma- RT tions = R −RT p = (Adg )− T Tˆ Rotational motion in R R Rigid Motion in R and Adg ⋅g = Adg ⋅ Adg Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • Chapter 2 Rigid Body Motion 2.4 Velocity of a Rigid Body 48 ◻ Properties of Adjoint mapping: Chapter g− = RT −RT p ⇒ Rigid Body (−RT p)∧ RT Motion Rigid Body Adg − = RT Transforma- RT tions = R −RT p = (Adg )− T Tˆ Rotational motion in R R Rigid Motion in R and Adg ⋅g = Adg ⋅ Adg Velocity of a The map Ad ∶ SE( ) ↦ GL(R ), Ad(g) = Adg is a group Rigid Body Wrenches and Reciprocal Screws homomorphism Matrix Rep Vector Rep Reference ˆ ∈ se( ) ξ ξ∈R g ⋅ ξ ⋅ g − ∈ se( ) Adg ξ ∈ R ˆ
  • Chapter 2 Rigid Body Motion 2.4 Velocity of a Rigid Body 49 Chapter Example: Velocity of Screw Motion Rigid Body Motion gab(θ) = eξθ(t) gab ( ), = ξ θ(t)eξθ(t) = θ(t)eξθ(t) ξ Rigid Body ˆ d ξθ(t) ˆ ˙ ˆ ˆ ˙ ˆ ˆ Transforma- e tions dt − Vab = gab ⋅ gab = ( ξ θeξθ(t) gab ( )) ⋅ (g − ( )e− ξθ(t) ) ˆ˙ ˆ ˆ Rotational motion in R ˆs ˙ ab Rigid Motion in R = ξ θ ⇒ Vab = ξ θ ˆ˙ s ˙ Vab = gab ⋅ gab = gab ( )e− ξθ ⋅ eξθ ξ θgab( ) − − ˆ ˆ ˆ Velocity of a ˆb ˙ ˙ Rigid Body Wrenches and Reciprocal = gab ( ) ξ θgab ( ) = (Adg − ( ) ξ)∧ θ ⇒ Vab = Adg − ( ) ξ θ − ˆ˙ ˙ b ˙ ab ab Screws Reference
  • Chapter 2 Rigid Body Motion 2.4 Velocity of a Rigid Body 50 ◻ Metric Property of se( ): Chapter Let gi (t) ∈ SE( ), i = , , be representations of the same motion, Rigid Body Motion obtained using coordinate frame A and B. Then, Rigid Body g (t) = g ⋅ g (t) ⋅ g − ⇒ V s = Adg ⋅ V s Transforma- tions Rotational motion in R Rigid Motion g g (t) in R A B Velocity of a Rigid Body Wrenches and g (t) g Reciprocal Screws Reference Figure 2.2 (Continues next slide)
  • Chapter 2 Rigid Body Motion 2.4 Velocity of a Rigid Body 51 Chapter Vs = (Adg ⋅ V s )T (Adg ⋅ V s ) = (V s )T AdT ⋅ Adg ⋅ V s g Rigid Body Motion RT AdT ⋅ Adg = R pR ˆ −RT p RT Rigid Body Transforma- g ˆ R tions Tˆ I R pR −RT p R I − RT p R Rotational motion in R = ˆ ˆ Rigid Motion in R In general, V s ≠ V s , or there exists no bi-invariant Velocity of a Rigid Body metric on se( ). Wrenches and Reciprocal Screws Reference
  • Chapter 2 Rigid Body Motion 2.4 Velocity of a Rigid Body 52 ◻ Coordinate Transformation: gac(t) = gab (t) ⋅ gbc (t) Chapter Rigid Body gab gbc Motion B Rigid Body Transforma- tions A gac C − Vac = gac ⋅ gac Rotational motion in R ˆs ˙ Figure 2.12 − − = (˙ab ⋅ gbc + gab ⋅ gbc )(gbc ⋅ gab ) Rigid Motion in R g ˙ − − − ˆs − Velocity of a Rigid Body = gab ⋅ gab + gab ⋅ gbc ⋅ gbc ⋅ gab = Vab ˙ ˙ ˆs + gabVbc gab ⇒ Vac = Vab + Adgab Vbc Wrenches and Reciprocal s s s Screws Reference
  • Chapter 2 Rigid Body Motion 2.4 Velocity of a Rigid Body 52 ◻ Coordinate Transformation: gac(t) = gab (t) ⋅ gbc (t) Chapter Rigid Body gab gbc Motion B Rigid Body Transforma- tions A gac C − Vac = gac ⋅ gac Rotational motion in R ˆs ˙ Figure 2.12 − − = (˙ab ⋅ gbc + gab ⋅ gbc )(gbc ⋅ gab ) Rigid Motion in R g ˙ − − − ˆs − Velocity of a Rigid Body = gab ⋅ gab + gab ⋅ gbc ⋅ gbc ⋅ gab = Vab ˙ ˙ ˆs + gabVbc gab ⇒ Vac = Vab + Adgab Vbc Wrenches and Reciprocal s s s Screws Vac = Adg − Vab + Vbc Reference b b b Similarly: Vbc = ⇒ Vac = Vab , Vab = ⇒ Vac = Vbc bc s s s b b b Note:
  • Chapter 2 Rigid Body Motion 2.4 Velocity of a Rigid Body 53 Example: θ2 Chapter θ1 ⎡ cθ ⎤ Rigid Body C ⎢ ⎥ Motion l −sθ ⎢ ⎥ B gab(θ ) = ⎢ sθ cθ ⎥ Rigid Body ⎢ l ⎥ l0 ⎢ ⎥ Transforma- l1 ⎣ ⎦ tions ⎡ ⎤ Figure 2.16 l ⎤ ⎡ A Rotational ⎡ cθ ⎤ ⎢ ⎥ ⎢ ⎥ motion in R ⎢ −sθ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ l ⎥, V s = ⎢ ⎥˙ ⎢ ⎥˙ gbc (θ ) = ⎢ sθ ⎥ ab ⎢ ⎥ θ , Vbc = ⎢ ⎥ θ Rigid Motion cθ ⎢ l ⎥ ⎢ ⎥ ⎢ ⎥ in R s ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎢ ⎥ ⎢ ⎥ Velocity of a ⎣ ⎦ ⎣ ⎦ Rigid Body ⎡ ⎤ ⎡ l cθ ⎤ ⎢ ⎥ ⎢ ⎥ Wrenches and ⎢ ⎥ ⎢ ls ⎥ Reciprocal ⎢ ⎥˙ ⎢ θ ⎥ Screws Vac = Vab + Adgab ⋅ Vbc = ⎢ ⎥θ +⎢ ⎢ ⎥θ ⎥ ⎢ ⎥ s s s ˙ ⎢ ⎥ Reference ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ ⎦ ⎣ ⎦ † End of Section †
  • Chapter 2 Rigid Body Motion 2.5 Wrenches & Reciprocal Screws 54 Review: Dual space and dual vectors Chapter V: n-dimensional vector space V ∗ : dual space of V, also n-dimensional 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 2.5 Wrenches & Reciprocal Screws 54 Review: Dual space and dual vectors Chapter V: n-dimensional vector space V ∗ : dual space of V, also n-dimensional Rigid Body V ∗ ∋ ϕ ∶ V ↦ R: Linear functional Motion Rigid Body Transforma- (ϕ + ψ)(x) = ϕ(x) + ψ(x) tions (αϕ)(x) = αϕ(x) Rotational motion in R Rigid Motion in R Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • Chapter 2 Rigid Body Motion 2.5 Wrenches & Reciprocal Screws 54 Review: Dual space and dual vectors Chapter V: n-dimensional vector space V ∗ : dual space of V, also n-dimensional Rigid Body V ∗ ∋ ϕ ∶ V ↦ R: Linear functional Motion Rigid Body Transforma- (ϕ + ψ)(x) = ϕ(x) + ψ(x) tions (αϕ)(x) = αϕ(x) Rotational motion in R Rigid Motion (e , . . . , en ): Basis of V in R (e , . . . , en ): Dual basis of V ∗ , with Velocity of a Rigid Body ⎧ ⎪ Wrenches and ⎪ i=j Reciprocal ej ∶ V ↦ R, ej (ei ) = δij = ⎨ Screws ⎪ ⎪ ⎩ else Reference
  • Chapter 2 Rigid Body Motion 2.5 Wrenches & Reciprocal Screws 55 Chapter Example: Rigid Body V = R ≅ (so( )) Motion Rigid Body Transforma- V ∗ ∋ τ ∶ R ↦ R ∶ ω ↦ ⟨τ, ω⟩, tions Rotational motion in R Rigid Motion in R Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • Chapter 2 Rigid Body Motion 2.5 Wrenches & Reciprocal Screws 55 Chapter Example: Rigid Body V = R ≅ (so( )) Motion Rigid Body Transforma- V ∗ ∋ τ ∶ R ↦ R ∶ ω ↦ ⟨τ, ω⟩, tions Rotational motion in R Rigid Motion V = R (≅ se( )) in R Velocity of a Rigid Body Wrenches and V∗ ∋ F ∶ R ↦ R ∶ v ω ↦ ⟨f , v⟩ + ⟨τ, ω⟩ Reciprocal Screws Reference
  • Chapter 2 Rigid Body Motion 2.5 Wrenches & Reciprocal Screws 56 ◻ Duality between twist and wrench: Twist: Chapter gac = gab ⋅ gbc Rigid Body Motion g bc C = Adg − Rigid Body b b B Transforma- ⇒ Vac ⋅ Vab bc = Adgbc ⋅ Vac tions b b Rotational or Vab A motion in R = vb ˆ Rbc pbc Rbc vb Rigid Motion ab ac Figure 2.17 in R ωb ab Rbc ωb ac Velocity of a Rigid Body Wrenches and Reciprocal Screws Reference
  • Chapter 2 Rigid Body Motion 2.5 Wrenches & Reciprocal Screws 56 ◻ Duality between twist and wrench: Twist: Chapter gac = gab ⋅ gbc Rigid Body Motion g bc C = Adg − Rigid Body b b B Transforma- ⇒ Vac ⋅ Vab bc = Adgbc ⋅ Vac tions b b Rotational or Vab A motion in R = vb ˆ Rbc pbc Rbc vb Rigid Motion ab ac Figure 2.17 in R ωb ab Rbc ωb ac Velocity of a Special case: Rbc = I Rigid Body vb ac ωb ac Wrenches and A Reciprocal C = Screws vb ab vb + pab × ωb ac ac ωb ab Reference ωb ab ωbac B p bc v b = v b + p bc × ω b ab ac ac Accounts for body rotation (Continues next slide) Figure 2.18
  • Chapter 2 Rigid Body Motion 2.5 Wrenches & Reciprocal Screws 57 Wrench: Fb = τ b , Fc = fb fc τb Chapter τc τc fc = RT fb Rigid Body Motion fb bc g bc C τc = + (−RT pbc ) × (RT fb ) Rigid Body fc B Transforma- RT τb bc bc bc = tions Rotational RT τb bc Tˆ − Rbc pbc fb A motion in R ⇒ Fc = fc = Rigid Motion RT bc fb Figure 2.19 in R τc −RT pbc RT ˆ τb bc bc = Velocity of a Rigid Body AdTbc Fb g Wrenches and Reciprocal Screws Reference
  • Chapter 2 Rigid Body Motion 2.5 Wrenches & Reciprocal Screws 57 Wrench: Fb = τ b , Fc = fb fc τb Chapter τc τc fc = RT fb Rigid Body Motion fb bc g bc C τc = + (−RT pbc ) × (RT fb ) Rigid Body fc B Transforma- RT τb bc bc bc = tions Rotational RT τb bc Tˆ − Rbc pbc fb A motion in R ⇒ Fc = fc = Rigid Motion RT bc fb Figure 2.19 in R τc −RT pbc RT ˆ τb bc bc = Velocity of a Rigid Body AdTbc Fb g fc Wrenches and Special case: Rbc = I A Reciprocal Screws p bc C fb Reference = fc fb B τb τc −pbc × fb + τb τ c = τ b − p bc × f b Accounts for body force Figure 2.20
  • Chapter 2 Rigid Body Motion 2.5 Wrenches & Reciprocal Screws 58 Chapter ◻ Alternative Derivation: Principle of Rigid Body Motion Virtual Power: δW = Fc Vac = Fb Vab Rigid Body Transforma- T b T b tions Rotational motion in R = Fb Adgbc Vac T b = AdTbc Fb T b b Rigid Motion in R g Vac , ∀Vac Velocity of a ⇒ Fc = AdTbc Fb Rigid Body Wrenches and g Reciprocal Screws Reference
  • Chapter 2 Rigid Body Motion 2.5 Wrenches & Reciprocal Screws 59 Review: Screw coordinates for a twist Chapter Twist ξ = ⇔ screw S(l, h, M) Rigid Body Motion p v ω Rigid Body ω Transforma- q ⎧ ω × v + λω, λ ∈ R ⎪ if ω ≠ ⎪ ω z ⎨ tions ⎪ + λv, λ ∈ R θ l: ⎪ if ω = d ⎩ Rotational ⎧ ωT v motion in R ⎪ ⎪ if ω ≠ y ⎪ x Rigid Motion ⎨ ω in R ⎪ Figure 2.15 h: ⎪∞ ⎪ if ω = ⎩ Velocity of a ⎧ Rigid Body ⎪ ⎪ ω if ω ≠ ⎨ Wrenches and ⎪ v if ω = M: ⎪ Reciprocal ⎩ Screws Reference Given a screw S, how to generate a wrench associated with S?
  • Chapter 2 Rigid Body Motion 2.5 Wrenches & Reciprocal Screws 60 ◻ Screw coordinates for a wrench: Generate a wrench associated with S: (h ≠ ∞): force of mag. M along l, and torque of mag. hM Chapter Rigid Body Motion about l. (h = ∞): pure torque of mag. M about l Rigid Body Transforma- tions Rotational ⎧ ⎡ ⎤ motion in R ⎪ ⎢ ⎪ ⎢ ⎥ ⎪M ⎪ ⎢ ω ⎥ h≠∞ Rigid Motion ⎪ ⎪ ⎢ −ω × q + hω ⎥ ⎪ in R ⎪ ⎥ ω q F=⎨ ⎣ ⎦ τ ⎪ ⎡ ⎤ Velocity of a ⎪ ⎢ ⎥ Rigid Body ⎪M ⎢ ⎪ ⎢ ⎥ h=∞ ⎪ ⎪ ⎢ ω ⎥ ⎪ Wrenches and ⎪ ⎥ f ⎩ ⎣ Reciprocal A ⎦ Screws Reference Figure 2.21 F: wrench along the screw S. (Continues next slide)
  • Chapter 2 Rigid Body Motion 2.5 Wrenches & Reciprocal Screws 61 ◻ Screw coordinates for a wrench (Continued): Screw coordinates of F: Chapter Case : f = = Rigid Body Motion M ω τ Rigid Body ⇒M = τ ,ω = ,h = ∞ Transforma- τ tions τ Case : f ≠ Rotational motion in R = Rigid Motion ω f in R M −ω × q + hω τ M = f ,ω = Velocity of a Rigid Body f Wrenches and M f Tτ h= ,q = Reciprocal Screws f Reference f f F= f τ ⇔ Apply torque τ of mag. hM along l,l and a a force f of mag. M about (Continues next slide)
  • Chapter 2 Rigid Body Motion 2.5 Wrenches & Reciprocal Screws 62 ◻ Screw coordinates for a wrench (Continued): ⎧ f Tτ Chapter Pitch: ⎪ ⎪ Rigid Body ⎪ ⎪ if f ≠ Motion h=⎨ f ⎪ Rigid Body ⎪∞ ⎪ ⎪ if f = Transforma- ⎩ tions Rotational Axis: ⎧f × τ motion in R ⎪ ⎪ + λf , λ ∈ R if f ≠ ⎪ Rigid Motion l=⎨ f in R ⎪ ⎪ + λτ, λ ∈ R ⎪ if f = Velocity of a ⎩ Rigid Body Wrenches and Magnitude: ⎧ Reciprocal ⎪ f ⎪ if f ≠ Screws M=⎨ ⎪ τ Reference ⎪ ⎩ if f =
  • Chapter 2 Rigid Body Motion 2.5 Wrenches & Reciprocal Screws 63 Theorem 3 (Poinsot): Every collection of wrenches applied to a rigid body Chapter Rigid Body is equivalent to a force applied along a fixed axis Motion plus a torque about the axis. 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 2.5 Wrenches & Reciprocal Screws 63 Theorem 3 (Poinsot): Every collection of wrenches applied to a rigid body Chapter Rigid Body is equivalent to a force applied along a fixed axis Motion plus a torque about the axis. Rigid Body Transforma- tions - Rotational motion in R ◻ Multi-fingered grasp: Rigid Motion in R Velocity of a Rigid Body k Cj Fo = Ci O Wrenches and Reciprocal AdToc ⋅ Fci g− i Screws i= Reference Si P Sj Figure 2.22
  • Chapter 2 Rigid Body Motion 2.5 Wrenches & Reciprocal Screws 64 ◻ Reciprocal screws: V= , F= Chapter Rigid Body v f Motion ω τ Rigid Body α F⋅V =f ⋅v+τ ⋅ω T T Transforma- ω1 tions ω2 q1 Rotational motion in R ↓ ↓ S1 d Rigid Motion S S q2 in R S2 Velocity of a α = atan ((ω × ω ) ⋅ n, ω ⋅ ω ) Rigid Body Figure 2.23 Wrenches and S = M M ((h + h ) cos α − d sin α) Reciprocal Screws S = Reference if reciprocal (continues next slide)
  • Chapter 2 Rigid Body Motion 2.5 Wrenches & Reciprocal Screws 65 Given V = M q × ωω+ h ω ω q ×ω +h ω Chapter Rigid Body , F=M , Let q = q + dn, then Motion Rigid Body Transforma- tions V ⋅ F = M M (ω ⋅ (q × ω + h ω ) + ω ⋅ (q × ω + h ω )) = M M (ω ⋅ (q × ω ) + h ω ⋅ ω Rotational motion in R + ω ⋅ ((q + dn) × ω ) + h ω ⋅ ω ) Rigid Motion in R Velocity of a Rigid Body = M M ((h + h ) cos α − d sin α) Wrenches and Reciprocal Screws Reference
  • Chapter 2 Rigid Body Motion 2.5 Wrenches & Reciprocal Screws 66 Example: Basic joints Chapter Revolute joint: ξ = −ωω q × Rigid Body Motion ξ⊥ = span ωi ωi ∈ S , i = , , Rigid Body q × ω i , vj vj ⋅ ω = , j = , : -system Transforma- tions Rotational motion in R Rigid Motion in R Velocity of a Rigid Body ω(ω ) Revolute Wrenches and Reciprocal Screws q Reference ω (v ) ω (v ) Figure 2.24
  • Chapter 2 Rigid Body Motion 2.5 Wrenches & Reciprocal Screws 66 Example: Basic joints Chapter Revolute joint: ξ = −ωω q × Rigid Body Motion ξ⊥ = span ωi ωi ∈ S , i = , , Rigid Body q × ω i , vj vj ⋅ ω = , j = , : -system Transforma- tions Rotational v motion in R Prismatic joint: ξ = ωi ωi ⋅ v = , i = , ξ⊥ = span Rigid Motion in R q × ω i , vj vj ∈ S , j = , , : -system Velocity of a Rigid Body ω(ω ) v(v ) Revolute Prismatic Wrenches and Reciprocal Screws q q Reference ω (v ) ω (v ) ω (v ) ω (v ) Figure 2.24 Figure 2.25
  • Chapter 2 Rigid Body Motion 2.5 Wrenches & Reciprocal Screws 67 Example: Basic joints (continued) Chapter Spherical joint: ξ = span −ωi × q ω ∈ S , i = , , Rigid Body ωi i Motion ⊥ = span ωi Rigid Body ξ q × ωi ωi ∈ S , i = , , : -system Transforma- tions Rotational motion in R Rigid Motion in R Velocity of a Rigid Body Spherical ω Wrenches and Reciprocal Screws q Reference ω ω Figure 2.26
  • Chapter 2 Rigid Body Motion 2.5 Wrenches & Reciprocal Screws 67 Example: Basic joints (continued) Chapter Spherical joint: ξ = span −ωi × q ω ∈ S , i = , , Rigid Body ωi i Motion ⊥ = span ωi Rigid Body ξ q × ωi ωi ∈ S , i = , , : -system Transforma- tions Universal joint: ξ = span q × x , q × y Rotational motion in R x y Rigid Motion ξ⊥ = span ωi in R q × ωi , z ωi ∈ S , i = , , : -system Velocity of a Rigid Body Spherical ω z(ω ) Universal Wrenches and Reciprocal Screws q q Reference ω ω x(ω ) y(ω ) Figure 2.26 Figure 2.27
  • Chapter 2 Rigid Body Motion 2.5 Wrenches & Reciprocal Screws 68 Example: Kinematic chains Universal-Spherical Dyad: ξ = span q ×x , q ×y q × ωi ωi ∈ S , i = , , Chapter Rigid Body Motion x y ωi v q −q Rigid Body Transforma- ξ⊥ = span q ×v v= q −q tions Rotational motion in R Rigid Motion in R q Velocity of a Rigid Body v(ω ) Wrenches and Reciprocal Screws q Reference x(ω ) y(ω ) Figure 2.28
  • Chapter 2 Rigid Body Motion 2.5 Wrenches & Reciprocal Screws 68 Example: Kinematic chains Universal-Spherical Dyad: ξ = span q ×x , q ×y q × ωi ωi ∈ S , i = , , Chapter Rigid Body Motion x y ωi v q −q Rigid Body Transforma- ξ⊥ = span q ×v v= q −q tions Revolute-Spherical Dyad: Rotational motion in R zero pitch screws passing through the center of the sphere, lie Rigid Motion on a plane containing the axis of the revolute joint: -system in R q Velocity of a Rigid Body v(ω ) Wrenches and Reciprocal Screws q Reference x(ω ) y(ω ) Figure 2.28 Figure 2.29 † End of Section †
  • Chapter 2 Rigid Body Motion 2.6 References 69 ◻ Reference: [1] Murray, R.M. and Li, Z.X. and Sastry, S.S., A mathematical introduc- Chapter tion to robotic manipulation. CRC Press, 1994. Rigid Body [2] Ball, R.S., A treatise on the theory of screws. University Press, 1900. Motion [3] Bottema, O. and Roth, B. , Theoretical kinematics. Dover Publica- Rigid Body tions, 1990. Transforma- tions [4] Craig, J.J., Introduction to robotics : mechanics and control, 3rd ed. Prentice Hall, 2004. Rotational [5] Fu, K.S. and Gonzalez, R.C. and Lee, C.S.G., Robotics : control, motion in R sensing, vision, and intelligence. CAD/CAM, robotics, and computer Rigid Motion vision. McGraw-Hill, 1987. in R [6] Hunt, K.H., Kinematic geometry of mechanisms. 1978, Oxford, New Velocity of a York: Clarendon Press, 1978. Rigid Body [7] Paul, R.P., Robot manipulators : mathematics, programming, and Wrenches and control. The MIT Press series in artificial intelligence. MIT Press, 1981. Reciprocal [8] Park, F. C., A first course in robot mechanics. Available online, 2006. Screws [9] Tsai, L.-W., Robot analysis : the mechanics of serial and parallel Reference manipulators. Wiley, 1999. [10] Spong, M.W. and Hutchinson, S. and Vidyasagar, M. , Robot model- ing and control. John Wiley & Sons, 2006. [11] Selig, J., Geometric Fundamentals of Robotics. Springer, 2008.