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2nd International Summer School on Geometric Methods in Robotics, Mechanism Design and Manufacturing Research-Lecture 06 Multifingered Hand

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[Download] rev chapter-6-june26th

  1. 1. Chapter 6 Multifingered Hand Modeling and Control 1 Chapter Mul- Lecture Notes for tifingered Hand Modeling and A Geometrical Introduction to Control Introduction Robotics and Manipulation Grasp Statics Kinematics of Richard Murray and Zexiang Li and Shankar S. Sastry Contact CRC Press Hand Kinematics Grasp Planning Zexiang Li and Yuanqing Wu Grasp Force Optimization ECE, Hong Kong University of Science & Technology Coordinated Control July ,
  2. 2. Chapter 6 Multifingered Hand Modeling and Control 2 Chapter 6 Multifingered Hand Modeling Chapter Mul- and Control tifingered Hand Modeling and Introduction Control Introduction Grasp Statics Grasp Statics Kinematics of Kinematics of Contact Contact Hand Hand Kinematics Kinematics Grasp Planning Grasp Planning Grasp Force Optimization Grasp Force Optimization Coordinated Control Coordinated Control
  3. 3. Chapter 6 Multifingered Hand Modeling and Control 6.1 Introduction 3 ◻ Hand function: Chapter Mul- tifingered Hand Modeling and Control Hand function: Introduction Interface with external Grasp Statics world Kinematics of Contact Hand operation: Hand Grasping Kinematics Dextrous manipulation Grasp Fine manipulation Planning Exploration Grasp Force Optimization Coordinated Control
  4. 4. Chapter 6 Multifingered Hand Modeling and Control 6.1 Introduction 4 Chapter Mul- ◻ History of hand design: tifingered Hand Modeling and Prosthetic devices (1509) Control Dextrous end-effectors Introduction Grasp Statics Multiple manipulator/agents coordination Kinematics of Human hand study Contact Hand Kinematics Grasp Planning Grasp Force Optimization Coordinated Control
  5. 5. Chapter 6 Multifingered Hand Modeling and Control 6.1 Introduction 5 ◻ Hand Design Issues: Chapter Mul- Mechanical systems tifingered Hand Sensor/actuators Modeling and Control Control hardware Introduction Grasp Statics ◻ A Sample List of Hand Prototypes: Kinematics of Contact Hand Kinematics Grasp Planning Grasp Force Optimization Coordinated Control The salisbury Hand (1982) The Utah-MIT Hand (1987)
  6. 6. Chapter 6 Multifingered Hand Modeling and Control 6.1 Introduction 6 Chapter Mul- tifingered Hand Modeling and Control Introduction Grasp Statics Kinematics of Contact Toshiba Hand (Japan) DLR hand (Germany, 1993) Hand Kinematics Grasp Planning Grasp Force Optimization Coordinated Control The HKUST Hand (1993) Micro/Nano Hand
  7. 7. Chapter 6 Multifingered Hand Modeling and Control 6.1 Introduction 7 Example: More Hands Chapter Mul- tifingered Hand Modeling and Control Introduction Grasp Statics Kinematics of Contact Hand Kinematics Grasp Planning Grasp Force Optimization Coordinated Control
  8. 8. Chapter 6 Multifingered Hand Modeling and Control 6.1 Introduction 8 ‘‘ e hand is indeed an instrument of creation par excellence.’’ Chapter Mul- tifingered Rodin (1840–1917) Hand Modeling and Control Introduction Grasp Statics Kinematics of Contact Hand Kinematics Grasp Planning Grasp Force Optimization Coordinated Control Rodin Hand (1898)
  9. 9. Chapter 6 Multifingered Hand Modeling and Control 6.1 Introduction 9 ◻ Lessons from Biological Systems: Chapter Mul- tifingered Hand Hip Modeling and % of motor cortex for Trunk Shoulder Elbow Wrist Control Kn nd e ee A Toe ng Ha ttl hand control: Ri dle x Li Introduction d e eck Mi Ind mb Now nk rs u Br le s ge Th ll Grasp Statics n yeba Fi nd e Human %∼ % ] lid a Face Kinematics of Eye V oc al iz at i o n Contact Lips Monkey %∼ % Jaw li vat i on] Hand Sw Tongu allo e n] Kinematics Dog < % win g io at c [ Sa sti [ Grasp [M a Planning Grasp Force Optimization Medial Lateral Coordinated Control Goal: A manipulation theory for robotic hand based on physical laws and rigorous mathematical models!
  10. 10. Chapter 6 Multifingered Hand Modeling and Control 6.2 Grasp Statics 10 ◻ Contact Models: Frictionless Point Contact (FPC) ⎡ ⎤ ⎢ ⎥ Chapter Mul- ⎢ ⎥ tifingered ⎢ ⎥ Hand F i = ⎢ ⎥ xi , xi ≥ Modeling and ⎢ ⎥ ⎢ ⎥ Control ⎢ ⎥ ⎣ ⎦ Introduction Grasp Statics Kinematics of Contact Hand Kinematics Grasp Planning Grasp Force Optimization Coordinated Control
  11. 11. Chapter 6 Multifingered Hand Modeling and Control 6.2 Grasp Statics 10 ◻ Contact Models: Ci x z Frictionless Point Contact (FPC) ⎡ ⎤ y z y ⎢ ⎥ Chapter Mul- z O ⎢ ⎥ tifingered x ⎢ ⎥ Hand F i = ⎢ ⎥ xi , xi ≥ goci Modeling and ⎢ ⎥ y ⎢ ⎥ Control P ⎢ ⎥ ⎣ ⎦ Introduction gpo x Grasp Statics Figure 6.1 Point Contact with friction (PCWF) ⎡ ⎤ Kinematics of ⎢ ⎥ Contact ⎢ ⎥ ⎢ ⎥ Fi = ⎢ ⎥ xi , xi ∈ FCi Hand ⎢ ⎥ Kinematics ⎢ ⎥ ⎢ ⎥ Grasp ⎣ ⎦ Planning Grasp Force Optimization Coordinated Control FCi = {xi ∈ R ∶ xi, + xi, ≤ µi xi, , xi, ≥ } µi : Coulomb coefficient of friction
  12. 12. Chapter 6 Multifingered Hand Modeling and Control 6.2 Grasp Statics 11 So nger contact (SFC) ⎡ ⎤ ⎢ ⎥ Chapter Mul- friction ⎢ ⎥ cone ⎢ ⎥ tifingered Fi = ⎢ ⎥ xi , xi ∈ FCi Hand ⎢ ⎥ Modeling and ⎢ ⎥ Control ⎢ ⎥ ⎣ ⎦ Introduction object Grasp Statics surface Kinematics of Contact Hand Kinematics Figure 6.2 Grasp Elliptic Model: FCi = {xi ∈ R xi, ≥ , (xi, + xi, ) + xi, ≤ xi, } Planning Grasp Force Optimization µi µit Coordinated Linear Model: FCi = {xi ∈ R xi, ≥ , xi, + xi, + xi, ≤ xi, } Control µi µit µit : Torsional coefficient of friction
  13. 13. Chapter 6 Multifingered Hand Modeling and Control 6.2 Grasp Statics 12 Fi = Bi ⋅ xi , xi ∈ FCi , Bi ∈ R ×mi : wrench basis Property 1: Chapter Mul- tifingered Hand FCi is a closed subset of Rmi , with nonempty interior. ∀x , x ∈ FCi , αx + βx ∈ FCi , ∀α, β > Modeling and Control Introduction Grasp Statics Kinematics of Contact Hand Kinematics Grasp Planning Grasp Force Optimization Coordinated Control
  14. 14. Chapter 6 Multifingered Hand Modeling and Control 6.2 Grasp Statics 12 Fi = Bi ⋅ xi , xi ∈ FCi , Bi ∈ R ×mi : wrench basis Property 1: Chapter Mul- tifingered Hand FCi is a closed subset of Rmi , with nonempty interior. ∀x , x ∈ FCi , αx + βx ∈ FCi , ∀α, β > Modeling and Control Introduction Grasp Statics ◻ The Grasp Map: Kinematics of Single contact: Contact Roci Hand Fo = AdToc Fi = g− ˆ poci Roci Roci Bi xi , xi ∈ FCi , Gi = AdToc − Bi g i i Kinematics Grasp Gi Planning Multi ngered grasp: Grasp Force k x Gi xi = [AdToc − B , . . . , AdToc Bk ] ≜ G⋅x Optimization x Coordinated Fo = g g − Control i= k xk k x ∈ FC ≜ FC × ⋯FCk ⊂ Rm , m = mi i=
  15. 15. Chapter 6 Multifingered Hand Modeling and Control 6.2 Grasp Statics 13 Definition: Grasp map Chapter Mul- (G, FC) is called a grasp. tifingered Hand Modeling and Control Introduction Grasp Statics Kinematics of ⋯Ck Contact Hand Kinematics C o Grasp C Planning Grasp Force Optimization Coordinated Control Figure 6.3
  16. 16. Chapter 6 Multifingered Hand Modeling and Control 6.2 Grasp Statics 14 Example: Soft finger grasp of a box Chapter Mul- tifingered Hand Modeling and −r Control Introduction Rc = , pc = , x z y Grasp Statics − z y z Kinematics of Contact Rc = , pc = r , y x x Hand Kinematics ⎡ ⎤ Grasp ⎢ ⎥ ⎢ ⎥ Planning ⎢ ⎥ ⎢ ⎥ Grasp Force Roci ⎢ ⎥ Optimization Gi = ⎢ ⎥ ˆ poci Roci Roci ⎢ ⎥ Coordinated ⎣ ⎦ Control
  17. 17. Chapter 6 Multifingered Hand Modeling and Control 6.2 Grasp Statics 15 Example: Soft finger grasp of a box Chapter Mul- tifingered Hand Modeling and −r Control Introduction Rc = , pc = , x z y Grasp Statics − z y z Kinematics of Contact Rc = , pc = r , y x x Hand Kinematics ⎡ ⎤ Grasp ⎢ ⎥ ⎢ ⎥ Planning ⎢ ⎥ ⎢ ⎥ Grasp Force Roci ⎢ ⎥ Optimization Gi = ⎢ ⎥ ˆ poci Roci Roci ⎢ ⎥ Coordinated ⎣ ⎦ Control
  18. 18. Chapter 6 Multifingered Hand Modeling and Control 6.2 Grasp Statics 16 ⎡ ⎤ ⎢ ⎥ Chapter Mul- ⎢ − ⎥ tifingered ⎢ ⎥ Hand G=⎢ − ⎥ Modeling and ⎢ ⎥ ⎢ − ⎥ Control r ⎢ −r ⎥ ⎣ ⎦ Introduction Grasp Statics r Kinematics of x=[ x, ]∈R Contact x, x, x, x , x , x , x , FC = FC × FC Hand Kinematics Grasp (xi, + xi, ) + µit xi, ≤ xi, , xi, ≥ Planning Grasp Force FCi = xi ∈ R ,i = , Optimization µi Coordinated Control
  19. 19. Chapter 6 Multifingered Hand Modeling and Control 6.2 Grasp Statics 17 ◻ Friction Cone Representation: FPC Pi (xi ) = [xi ] Chapter Mul- tifingered Hand Modeling and µi xi, + xi, PCWF Pi (xi ) = Control µi xi, − xi, xi, Introduction xi, Grasp Statics xi, + xi, µi xi, µi − jxi, µit SFC Pi (xi ) = ,j = − Kinematics of xi, µi + jxi, µit xi, − xi, µi Contact Hand Kinematics Grasp P(x) ≜ Diag(P (xi ), . . . , Pk (xk )) ∈ Rm×m (m = k) ¯ ¯ ¯ x = (x , . . . , xm ) = (x , . . . , xk ) ∈ R Planning T T T m Grasp Force Optimization Coordinated Control
  20. 20. Chapter 6 Multifingered Hand Modeling and Control 6.2 Grasp Statics 18 Property 2: Chapter Mul- Pi = PiT , P = PT xi ∈ FCi (or x ∈ FC) ⇔ Pi (xi ) ≥ (or P(x) ≥ ) tifingered Hand Modeling and xi ∈ FCi ⇔ Pi (xi ) = Si + ∑mi Si xi,j Control j jT j Introduction j= ≥ , Si = Si , j = , . . . , mi = x ∈ FC ⇔ P(x) = S + ∑m Si xi ≥ , ST = Si , i = , . . . , m Grasp Statics Kinematics of i= i Contact = + ∑m Hand Kinematics Let Q(x) = S i= xi Si , ST = Si , then i Grasp AQ = {x ∈ Rm Q(x) ≥ } and BQ = {x ∈ Rm Q(x) > } are Planning = Grasp Force Optimization both convex. Coordinated Control
  21. 21. Chapter 6 Multifingered Hand Modeling and Control 6.2 Grasp Statics 19 Proof of Property 2: For PCWF, σ(Pi ) = µi xi, ± xi, + xi, Chapter Mul- tifingered Hand Modeling and Control Thus, Introduction xi ∈ FCi ⇔ xi, ≥ , xi, + xi, ≤ µi xi, ⇔ σ(Pi ) ≥ Grasp Statics Kinematics of ⇔ Pi ≥ Contact Hand Kinematics Furthermore, Grasp Pi (xi ) = xi, + xi, + µi − Planning µi xi, Grasp Force Optimization Coordinated Exercise: Verify this for SFC. Control
  22. 22. Chapter 6 Multifingered Hand Modeling and Control 6.2 Grasp Statics 20 ◻ Force Closure: Definition: A grasp (G, FC) is force closure if ∀Fo ∈ Rp (p = or ), ∃x ∈ FC, s.t. Chapter Mul- tifingered Hand Modeling and Gx = Fo Control Introduction Grasp Statics Kinematics of Contact Hand Kinematics Grasp Planning Grasp Force Optimization Coordinated Control
  23. 23. Chapter 6 Multifingered Hand Modeling and Control 6.2 Grasp Statics 20 ◻ Force Closure: Definition: A grasp (G, FC) is force closure if ∀Fo ∈ Rp (p = or ), ∃x ∈ FC, s.t. Chapter Mul- tifingered Hand Modeling and Gx = Fo Control Introduction Problem 1: Force-closure Problem Determine if a grasp (G, FC) is force-closure or not. Grasp Statics Kinematics of Contact Hand Kinematics Grasp Planning Grasp Force Optimization Coordinated Control
  24. 24. Chapter 6 Multifingered Hand Modeling and Control 6.2 Grasp Statics 20 ◻ Force Closure: Definition: A grasp (G, FC) is force closure if ∀Fo ∈ Rp (p = or ), ∃x ∈ FC, s.t. Chapter Mul- tifingered Hand Modeling and Gx = Fo Control Introduction Problem 1: Force-closure Problem Determine if a grasp (G, FC) is force-closure or not. Grasp Statics Kinematics of Contact Hand Kinematics Problem 2: Force Feasibility Problem Grasp Given Fo ∈ Rp , p = or , determine if there exists x ∈ FC s.t. Gx = Fo Planning Grasp Force Optimization Coordinated Control
  25. 25. Chapter 6 Multifingered Hand Modeling and Control 6.2 Grasp Statics 20 ◻ Force Closure: Definition: A grasp (G, FC) is force closure if ∀Fo ∈ Rp (p = or ), ∃x ∈ FC, s.t. Chapter Mul- tifingered Hand Modeling and Gx = Fo Control Introduction Problem 1: Force-closure Problem Determine if a grasp (G, FC) is force-closure or not. Grasp Statics Kinematics of Contact Hand Kinematics Problem 2: Force Feasibility Problem Grasp Given Fo ∈ Rp , p = or , determine if there exists x ∈ FC s.t. Gx = Fo Planning Grasp Force Optimization Problem 3: Force Optimization Problem Coordinated Control Given Fo ∈ Rp , p = or , find x ∈ FC s.t. Gx = Fo and x minimizes Φ(x).
  26. 26. Chapter 6 Multifingered Hand Modeling and Control 6.2 Grasp Statics 21 Chapter Mul- Definition: internal force or xN ∈ (ker G ∩ FC). tifingered Hand Modeling and xN ∈ FC is an internal force if GxN = Control Introduction Grasp Statics Kinematics of Contact Hand Kinematics Grasp Planning Grasp Force Optimization Coordinated Control
  27. 27. Chapter 6 Multifingered Hand Modeling and Control 6.2 Grasp Statics 21 Chapter Mul- Definition: internal force or xN ∈ (ker G ∩ FC). tifingered Hand Modeling and xN ∈ FC is an internal force if GxN = Control Introduction Grasp Statics Kinematics of Property 3: (G, FC) is force closure iff G(FC) = Rp and ∃xN ∈ ker G s.t. xN ∈ int(FC). Contact Hand Kinematics Grasp Planning Grasp Force Optimization Coordinated Control
  28. 28. Chapter 6 Multifingered Hand Modeling and Control 6.2 Grasp Statics 22 Proof of Property 3: Chapter Mul- tifingered Sufficiency: ′ For Fo ∈ Rp , let x′ be s.t. Fo = Gx′ . Since limα→∞ x +αxN = xN ∈ Hand Modeling and int(FC), there ∃α ′ , sufficiently large, s.t. Control α Introduction Grasp Statics x ′ + α ′ xN Kinematics of ∈ int(FC) ⊂ FC Contact α′ ⇒ x = x′ + α ′xN ∈ int(FC) Hand Kinematics Grasp ⇒ Gx = Gx′ = Fo Planning Grasp Force Necessity: Optimization Choose x ∈ int(FC) s.t. Fo = Gx ≠ , and choose x ∈ FC s.t. Coordinated Control Gx = −Fo . Define xN = x + x , GxN = ⇒ xN ∈ int(FC)
  29. 29. Chapter 6 Multifingered Hand Modeling and Control 6.2 Grasp Statics 23 ◻ Solutions of the force-closure and the Chapter Mul- tifingered force-feasibility problems (P1&P2): Hand Modeling and Control Review: Introduction Proposition: Linear matrix inequality (LMI) property Grasp Statics Given Q(x) = S + ∑m xl Sl , where Sl = ST , l = , . . . , m, the sets AQ = {x ∈ Rm Q(x) ≥ } and BQ = {x ∈ Rm Q(x) > } are l= l Kinematics of Contact Hand convex. Kinematics Grasp Planning Grasp Force Optimization Coordinated Control
  30. 30. Chapter 6 Multifingered Hand Modeling and Control 6.2 Grasp Statics 23 ◻ Solutions of the force-closure and the Chapter Mul- tifingered force-feasibility problems (P1&P2): Hand Modeling and Control Review: Introduction Proposition: Linear matrix inequality (LMI) property Grasp Statics Given Q(x) = S + ∑m xl Sl , where Sl = ST , l = , . . . , m, the sets AQ = {x ∈ Rm Q(x) ≥ } and BQ = {x ∈ Rm Q(x) > } are l= l Kinematics of Contact Hand convex. Kinematics Grasp Review: Planning LMI Feasibility Problem: Grasp Force Optimization Determine if the set AQ or BQ is empty or not. Coordinated Control
  31. 31. Chapter 6 Multifingered Hand Modeling and Control 6.2 Grasp Statics 24 Review: Convex constraints on x ∈ Rm (e.g., linear, (convex) quadratic, or Chapter Mul- tifingered matrix norm) can be transformed into LMI’s. Hand x ≥ ⇔ diag(x) ≥ Modeling and Control x= xm Introduction For A ∈ Rn×m , b ∈ Rn , d ∈ R, (CT x + d)I Ax + b Grasp Statics Ax + b ≤ CT x + d ⇔ (Ax + b)T CT x + d Kinematics of ≥ Contact Hand Kinematics Grasp Planning Grasp Force Optimization Coordinated Control
  32. 32. Chapter 6 Multifingered Hand Modeling and Control 6.2 Grasp Statics 24 Review: Convex constraints on x ∈ Rm (e.g., linear, (convex) quadratic, or Chapter Mul- tifingered matrix norm) can be transformed into LMI’s. Hand x ≥ ⇔ diag(x) ≥ Modeling and Control x= xm Introduction For A ∈ Rn×m , b ∈ Rn , d ∈ R, (CT x + d)I Ax + b Grasp Statics Ax + b ≤ CT x + d ⇔ (Ax + b)T CT x + d Kinematics of ≥ Contact Hand Kinematics Grasp Review: Condition on force-closure Planning Rank(G) = Grasp Force Optimization xN ∈ Rm , P(xN ) > and GxN = Coordinated Control Let V = [v , ⋯, vk ] whose columns are the basis of ker G xN = Vw, w ∈ Rk P(w) = P(Vw) = ∑k wl Sl , Sl = ST , l = , . . . , k ˜ l= ˜ ˜ l
  33. 33. Chapter 6 Multifingered Hand Modeling and Control 6.2 Grasp Statics 25 Review: Proposition: Given Q(x) = S + ∑m xl Sl , where Sl = ST , l = , . . . , m. Let x = Az + b where A ∈ Rm×n , b ∈ Rm , then i= l Chapter Mul- Q(z) = Q(Az + b) = S + ∑n zl Sl , where Sl = ST , l = , . . . , n tifingered Hand ˜ ˜ l= ˜ ˜ ˜ Modeling and l Control Introduction Grasp Statics Kinematics of Contact Hand Kinematics Grasp Planning Grasp Force Optimization Coordinated Control
  34. 34. Chapter 6 Multifingered Hand Modeling and Control 6.2 Grasp Statics 25 Review: Proposition: Given Q(x) = S + ∑m xl Sl , where Sl = ST , l = , . . . , m. Let x = Az + b where A ∈ Rm×n , b ∈ Rm , then i= l Chapter Mul- Q(z) = Q(Az + b) = S + ∑n zl Sl , where Sl = ST , l = , . . . , n tifingered Hand ˜ ˜ l= ˜ ˜ ˜ Modeling and l Control Introduction Theorem 1 (force-closure): k (G, FC) is force-closure iff BP = {w ∈ Rk wl Sl > } is non-empty. Grasp Statics ˜ ˜ Kinematics of Contact l= Hand Kinematics Force feasibility Problem Grasp Gx = Fo ⇒ xo = G† Fo (may not lie in FC) Planning generalized inverse x = G† Fo + Vw, span(V) = ker G, Grasp Force Optimization P(w) = P(G† Fo + Vw) Coordinated Control ¯ k =S + ¯ Sl wl , where Sl = ST , l = , . . . , k ¯ ¯ l l=
  35. 35. Chapter 6 Multifingered Hand Modeling and Control 6.2 Grasp Statics 26 Theorem 2 (force-feasibility): The force-feasibility problem for a given Fo ∈ R is solvable iff AP (w) = {w ∈ Rk S + ∑k Sl wl > } is non-empty. Chapter Mul- tifingered ¯ ¯ ¯ Hand l= Modeling and Control Introduction Grasp Statics Kinematics of Contact Hand Kinematics Grasp Planning Grasp Force Optimization Coordinated Control
  36. 36. Chapter 6 Multifingered Hand Modeling and Control 6.2 Grasp Statics 26 Theorem 2 (force-feasibility): The force-feasibility problem for a given Fo ∈ R is solvable iff AP (w) = {w ∈ Rk S + ∑k Sl wl > } is non-empty. Chapter Mul- tifingered ¯ ¯ ¯ Hand l= Modeling and Control Review: LMI Feasibility Problem as Introduction Optimization Problem Grasp Statics Kinematics of Q ≥ ⇔ ∃t ≤ s.t. Q + tI ≥ ⇔ t ≥ −λmin (Q) Contact Hand ⇔ λmin (Q) ≥ Kinematics Grasp ′ Planning Problem 2 : min t subject toQ(x) + tI ≥ Grasp Force Optimization Coordinated If t ∗ ≤ , then the LMI is feasible. Control
  37. 37. Chapter 6 Multifingered Hand Modeling and Control 6.2 Grasp Statics 27 ◻ Constructive force-closure for PCWF: pc × nc . . . pck × nck , FC = {x ∈ R xi ≥ } nc ... nck k G= Chapter Mul- tifingered Hand G(FC) = R ⇔ Positive linear combination of column of G = R Modeling and Control Introduction Grasp Statics Kinematics of Contact Hand Kinematics Grasp Planning Grasp Force Optimization Coordinated Control
  38. 38. Chapter 6 Multifingered Hand Modeling and Control 6.2 Grasp Statics 27 ◻ Constructive force-closure for PCWF: pc × nc . . . pck × nck , FC = {x ∈ R xi ≥ } nc ... nck k G= Chapter Mul- tifingered Hand G(FC) = R ⇔ Positive linear combination of column of G = R Modeling and Control Definition: v , . . . , vk , vi ∈ Rp is positively dependent if ∃αi > such that Introduction ∑ α i vi = Grasp Statics Kinematics of v , . . . , vk , vi ∈ Rn positively span Rn if ∀x ∈ Rn , ∃αi > , such Contact that ∑ αi vi = x Hand Kinematics Grasp Planning Grasp Force Optimization Coordinated Control
  39. 39. Chapter 6 Multifingered Hand Modeling and Control 6.2 Grasp Statics 27 ◻ Constructive force-closure for PCWF: pc × nc . . . pck × nck , FC = {x ∈ R xi ≥ } nc ... nck k G= Chapter Mul- tifingered Hand G(FC) = R ⇔ Positive linear combination of column of G = R Modeling and Control Definition: v , . . . , vk , vi ∈ Rp is positively dependent if ∃αi > such that Introduction ∑ α i vi = Grasp Statics Kinematics of v , . . . , vk , vi ∈ Rn positively span Rn if ∀x ∈ Rn , ∃αi > , such Contact that ∑ αi vi = x Hand Kinematics Grasp Planning Grasp Force Definition: A set K is convex if ∀x, y ∈ K, λx + ( − λ)y ∈ K, λ ∈ [ , ] Optimization Coordinated Control Given S=v , . . . , vk , vi ∈ Rp , the convex hull of S: co(S) = {v = ∑ αi vi , ∑ αi = , αi ≥ }
  40. 40. Chapter 6 Multifingered Hand Modeling and Control 6.2 Grasp Statics 28 Property 4: Let G = {G , . . . , Gk }, the following are equivalent: (G, FC) is force-closure. Chapter Mul- tifingered Hand Modeling and Control e columns of G positively span Rp , p = , Introduction e convex hull of Gi contains a neighborhood of the origin. Grasp Statics ere does not exist a vector v ∈ Rp , v ≠ s.t. ∀i = , . . . , k, v ⋅ Gi ≥ Kinematics of Contact Hand Kinematics v2 Grasp v1 Planning v3 co(S) Grasp Force v5 Optimization v6 v4 Coordinated Control separating c hyperplane (a) (b)
  41. 41. Chapter 6 Multifingered Hand Modeling and Control 6.3 Kinematics of Contact 29 ◻ Surface Model: c ∶ U ⊂ R → R , c(U) ⊂ S Chapter Mul- tifingered Hand Modeling and Control Introduction Grasp Statics Kinematics of ∂c ∂u Contact v Hand ∂c Kinematics cu = ∈R ∂u Grasp ∂c c u Planning cv = ∈R Grasp Force ∂v O Optimization R2 Coordinated Control cT cu cT cv u u First Fundamental form: Ip = cT cu cT cv v v
  42. 42. Chapter 6 Multifingered Hand Modeling and Control 6.3 Kinematics of Contact 30 Chapter Mul- tifingered Orthogonal Coordinates Chart: cT cv = (assumption) u = Mp ⋅ Mp Hand Modeling and cu Control Ip = cv Introduction Metric tensor: Grasp Statics cu Mp = c Kinematics of v cu × cv Contact Gauss map: N ∶ S → s ∶ N(u, v) = ∶= n Hand cu × cv Kinematics Grasp Planning 2nd Fundamental form: cT nu cT nv u u ∂n ∂n Grasp Force IIp = , nu = , nv = Optimization cT nu cT nv v v ∂u ∂v Coordinated Control
  43. 43. Chapter 6 Multifingered Hand Modeling and Control 6.3 Kinematics of Contact 31 Chapter Mul- tifingered ⎡ ⎤ Hand Curvature tensor: ⎢ ⎥ Modeling and cT nu cT nv =⎢ ⎥ u u ⎢ ⎥ Control −T − cu cu cv ⎢ ⎥ Kp = Mp IIp Mp T T ⎣ ⎦ Introduction cv nu cv nv Grasp Statics cu cv cv Gauss frame: [x, y, z] = xT Kinematics of cu cv nu nv Contact n , Kp = Hand cu cv yT cu cv Kinematics Torsion form: x x Grasp Tp = y T u cu v cv (Mp , Kp , Tp ):Geometric parameter of the surface. Planning Grasp Force Optimization Coordinated Control
  44. 44. Chapter 6 Multifingered Hand Modeling and Control 6.3 Kinematics of Contact 32 Example: Geometric parameters of a sphere in R Chapter Mul- tifingered z Hand Modeling and ρ cos u cos v Control c(u, v) = ρ cos u sin v Introduction ρ sin u U = {(u, v) − π < u < π , −π < v < π} −ρ sin u cos v Grasp Statics cu = −ρ sin u sin v Kinematics of u Contact v ρ cos v −ρ cos u sin v Hand Kinematics Grasp cv = ρ cos u cos v x y Planning Grasp Force Optimization cT cv = ⎡ ⎤ u ⎢ ⎥ K=⎢ ⎥, M = Coordinated ρ tan v ⎢ ⎥ Control ρ ρ cos u , T = ⎣ ⎦ ρ ρ
  45. 45. Chapter 6 Multifingered Hand Modeling and Control 6.3 Kinematics of Contact 33 ◻ Gauss Frame: Chapter Mul- tifingered Hand Modeling and Control no Introduction F Grasp Statics O Kinematics of ∂co ψ goc (t) ∈ SE( ) ∂uo Contact ∂cf Hand ∂uf b Kinematics Voc =? poc (t) = p(t) = c(α(t)), Roc (t) = [x(t), y(t), z(t)] = cu cv c u ×c u Grasp cu cv c u ×c v ⎡ xT ⎤ ⎡ xT ⎤ Planning ⎢ ⎥ ∂c ⎢ ⎥ voc = RT poc (t) = ⎢ yT ⎥ ∂α α = ⎢ yT ⎥ [ cu cv ] α = cu ⎢ ⎥ ˙ ⎢ ⎥ Grasp Force ˙ Mα oc ˙ ⎢ zT ⎥ ⎢ zT ⎥ ˙ cv = ⎣ ⎦ ⎣ ⎦ Optimization ⎡ xT ⎤ ⎡ xT y xT z ⎤ Coordinated ⎢ T ⎥ ⎢ ˙ ⎥ ωoc = RT Roc = ⎢ y ⎥ [ x y z ] = ⎢ yT x ˙ Control ˙ ⎢ ⎥ ˙ ˙ ˙ ⎢ yT z ⎥ ˙ ⎥ ⎢ zT ⎥ ⎢ zT x zT y ⎥ ˆ pc ˙ ⎣ ⎦ ⎣ ˙ ˙ ⎦
  46. 46. Chapter 6 Multifingered Hand Modeling and Control 6.3 Kinematics of Contact 34 yT x = yT [xu xv ]α = TM α Chapter Mul- tifingered Hand ˙ ˙ ˙ Modeling and Control xT z ˙ xT xT [ nu nv ] α = KM α Introduction = z= ˙ ˙ ˙ Grasp Statics yT z ˙ yT yT ⎡ −TM α ⎤ Kinematics of ⎢ ˙ KM α ⎥ ωoc = ⎢ TM α ˙ ⎥ Contact ⎢ ˙ ⎥ ⎢ ⎥ Hand ˆ ⎣ −(KM α )T ⎦ Kinematics ˙ Grasp Planning , Grasp Force Optimization voc = ˙ Mα Coordinated Control
  47. 47. Chapter 6 Multifingered Hand Modeling and Control 6.3 Kinematics of Contact 35 ◻ Contact Kinematics: pt ∈ S ↦ pf (t) ∈ Sf Chapter Mul- tifingered Hand Modeling and Control Local coordinate: Introduction Grasp Statics c ∶U ⊂R →S no cf ∶ Uf ⊂ R → Sf Kinematics of F Contact O Hand ∂co α (t) = c− (p (t)) Kinematics ∂uo ψ ∂cf ∂uf αf (t) = c− (pf (t)) Grasp Planning f Grasp Force Optimization Angle of contact: ϕ Coordinated Control Contact coordinates: η = (αf , α , ϕ)
  48. 48. Chapter 6 Multifingered Hand Modeling and Control 6.3 Kinematics of Contact 36 Rotation about the z-axis of Co by −ϕ aligns the x axis of Cf Chapter Mul- tifingered Hand Modeling and with that of Co Control cos ϕ − sin ϕ ⇒ Rco cf = − sin ϕ − cos ϕ Introduction , pco cf = ∈ R − Grasp Statics Kinematics of Angle of contact. Contact xf xf Hand xo y f xo y f Kinematics ϕ Grasp zf zo Planning zf zo y Grasp Force o Optimization yo Coordinated Control
  49. 49. Chapter 6 Multifingered Hand Modeling and Control 6.3 Kinematics of Contact 37 Define L (τ): Chapter Mul- At τ = t, L (τ) coincide with the Gauss frame at p (t). Lf (τ) ∶ coincide with Cf (t) at τ = t tifingered Hand vl lf = (vx , vy , vz ), Modeling and Control Lo (τ1 ) ω l lf = (ω x , ω y , ω z ), Introduction Grasp Statics ω y ∶ Rolling velocities Lo (t) = Co ωx O Kinematics of Contact vy ∶ Sliding velocities vx Hand vz ∶ Linear velocity in the normal direction Kinematics Lo (τ2 ) Grasp = Adgfl Vof ∶ Velocity of the nger relative to the object Planning Vl lf Grasp Force f Optimization Coordinated Control
  50. 50. Chapter 6 Multifingered Hand Modeling and Control 6.3 Kinematics of Contact 38 Define:K = Rϕ K Rϕ ∶Curvature of O relative to Cf ˜ Chapter Mul- Kf + K ˜ ∶ Relative Curvature. tifingered Hand Modeling and Control Theorem 3 (Montana Equations of contact): ⎧ ⎪ αf = Mf− (Kf + Ko )− −ωy ⎪˙ ωx − Ko vy vx ⎪ Introduction ⎪ ˜ ˜ ⎪ ⎪ ⎪ −ωy Grasp Statics ⎪ α = M − R(K + K )− ⎪ ˙o ⎪ ωx + Kf vy Kinematics of ˜o ˜ vx ⎨ o f ⎪ Contact ⎪ ˙ ψ Hand ⎪ ψ = ω z + Tf M f α f + To M o α o ⎪ ⎪ ⎪ ⎪ ˙ ˙ ⎪ Kinematics ⎪ vz = ⎪ ⎩ Grasp Planning Grasp Force Optimization Coordinated Control
  51. 51. Chapter 6 Multifingered Hand Modeling and Control 6.3 Kinematics of Contact 39 Proof of Theorem 3: Chapter Mul- Vf lf = tifingered Vf cf = Adg − Vf lf + Vlf cf = Vlf cf Hand Modeling and Control lf cf Introduction Voco = Adg − Volo + Vlo co = Vlo co lo co Grasp Statics Kinematics of z Contact f f f o o o lo lf Hand Kinematics Grasp Planning (2) Grasp Force co cf Optimization Coordinated Control f O (1)
  52. 52. Chapter 6 Multifingered Hand Modeling and Control 6.3 Kinematics of Contact 40 (1). At time t, Plf cf = , Rlf cf = I ⇒ Vlo cf = Vlo lf + Vlf cf RTo cf Vlo co + Vco cf Chapter Mul- c tifingered (2) pco cf = ⇒ Vlo cf = Hand RTo cf c Modeling and Control RTo cf ⇒ Vlo lf + Vf cf = Voco + Vco cf c Introduction RTo cf c Grasp Statics Rψ − Kinematics of Contact pco cf = ⇒ Vco cf = Rco cf = ⇒ ω co cf = ˙ ψ ⎡ ⎤ ⎢ ⎥ Hand vx ⎢ vy ⎥ Kinematics =⎢ ⎢ ⎥,Vf = Mf αf ⎥ cf Grasp vz ˙ ⎢ ⎥ Vlo lf ωx ⎢ ⎥ Planning ωy Grasp Force ⎣ ωz ⎦ ⎡ −Tf Mf αf ⎤ ⎢ Kf Mf αf ⎥ Optimization ˙ =⎢ ˙ ⎥ ⎢ ⎥ Coordinated ˙ Tf M f α f ⎢ ⎥ Control ˆ ω f cf ⎣ −(Kf Mf αf )T ˙ ⎦
  53. 53. Chapter 6 Multifingered Hand Modeling and Control 6.3 Kinematics of Contact 41 ⎡ −To Mo αo K M α ⎤ ⎢ ⎥ , ωoco = ⎢ To Mo αo ⎥ ˙ o o ˙o ⎢ ⎥ ˙ Mo α o ˙ ⎢ ⎥ voco = ˆ Chapter Mul- tifingered ⎣ −(Ko Mo αo ) ˙ T ⎦ vx + vy = Rψ Mo αo Hand ˙ Mf α f ˙ Modeling and Linear component: Control vz ωy + −ωx = − Introduction ˙ Kf Mf αf ˙ Rψ Ko Mo αo Grasp Statics ˙ Tf M f α f ωz ˙ ψ ˙ To M o α o Kinematics of Contact ⇒ Theorem result Hand Kinematics Grasp Planning Grasp Force Optimization Coordinated Control
  54. 54. Chapter 6 Multifingered Hand Modeling and Control 6.3 Kinematics of Contact 41 ⎡ −To Mo αo K M α ⎤ ⎢ ⎥ , ωoco = ⎢ To Mo αo ⎥ ˙ o o ˙o ⎢ ⎥ ˙ Mo α o ˙ ⎢ ⎥ voco = ˆ Chapter Mul- tifingered ⎣ −(Ko Mo αo ) ˙ T ⎦ vx + vy = Rψ Mo αo Hand ˙ Mf α f ˙ Modeling and Linear component: Control vz ωy + −ωx = − Introduction ˙ Kf Mf αf ˙ Rψ Ko Mo αo Grasp Statics ˙ Tf M f α f ωz ˙ ψ ˙ To M o α o Kinematics of Contact ⇒ Theorem result Hand Kinematics Grasp Planning Corollary: Rolling contact motion. −ωy ˙ Grasp Force αf = Mf− (Kf + Ko )− Optimization ˙ ˜ − ωx αo = Mo Rψ −ωy ˙ Coordinated (Kf + Ko )− ω x ψ = Tf M f α f + To M o α o Control ˜ ˙ ˙
  55. 55. Chapter 6 Multifingered Hand Modeling and Control 6.3 Kinematics of Contact 42 Example: A sphere rolling on a plane Chapter Mul- tifingered Hand ρ cos uf cos vf cf (u, v) = Modeling and Control ρ cos uf sin vf F Introduction ρ sin uf uo Grasp Statics co (u, v) = vo Kinematics of O ⎡ ⎤ Contact ⎢ ⎥ , Kf = ⎢ ρ ⎥ Hand ⎢ ⎥ Kinematics Ko = Grasp ⎣ ρ ⎦ Planning ρ Grasp Force Mo = , Mf = ρ , To = [ ] , Tf = − ρ tan uf Optimization Coordinated Control
  56. 56. Chapter 6 Multifingered Hand Modeling and Control 6.3 Kinematics of Contact 43 ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ Chapter Mul- ⎢ uf ˙ ⎥ ⎢ ⎥ ⎢ − ⎥ tifingered ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ Hand sec uf ⎥ ωx + ⎢ −ρ cos ψ ⎥ ωy vf ˙ ⎢ ⎥=⎢ −ρ sin ψ Modeling and ⎢ ⎥ ⎢ −ρ cos ψ ⎥ ⎢ ρ sin ψ ⎥ ⎥ ⎢ ⎥ Control uo ˙ ⎢ ⎥ ⎢ − tan uf ⎥ ⎢ ⎥ ⎢ ⎥ ⎣ vo ˙ ⎦ ⎣ ⎦ Introduction Grasp Statics ⎣ ψ˙ ⎦ η = g (η) ωx +g (η) ωy (∗) Kinematics of Contact ˙ u (t) Hand Kinematics u (t) Grasp Planning Q:Given η , ηf , how to find a path u:o[0, T]→ R Grasp Force Optimization so that solution of (*) links η to ηf ? Coordinated A question of nonholonomic motion planning! Control
  57. 57. Chapter 6 Multifingered Hand Modeling and Control 6.4 Hand Kinematics 44 gpo = gpfi (θ i )gfi lfi glfi loi gloi Vpo = Adg − Vpfi + Adg − Vlfi loi Chapter Mul- tifingered Hand fi o loi o Modeling and Adglo o Vpo = Adg − Vpfi + Vlfi loi Control Introduction i fi loi PCWF: Vz = ⇒ [ ]Vlfi loi = BT Vlfi loi = Grasp Statics Kinematics of i Contact BT Adglo o Vpo = Adgfi lo Vpfi Hand Kinematics i i i Ji (θ i )θ i Grasp Planning ˙ Grasp Force Optimization Coordinated Control
  58. 58. Chapter 6 Multifingered Hand Modeling and Control 6.4 Hand Kinematics 45 ⎡ BT Ad ⎤ ⎡ Adg − J (θ ) ⎤⎡ ⎢ ⎢ ⎥ ⎥ ⎢ ⎢ ⎥⎢ ⎥⎢ θ ⎤ ˙ ⎥ ⎥ glo o ⎢ ⎥ Vpo = ⎢ ⋱ ⎥⎢ ⎥ i fi lo ⎢ BT Ad ⎥ ⎢ Adgf− l Jk (θ k ) ⎥ ⎢ ⎢ k glo o ⎣ k ⎥ ⎦ ⎢ ⎣ k ok ⎥⎣ ⎦ θk ⎥ ˙ ⎦ Chapter Mul- GT (η)Vpo = Jh (θ, x , η)θ tifingered Hand ˙ Modeling and Control k Introduction θ = (θ , . . . , θ k ) ∈ Rn , n = ni , Jh ∈ Rm×n : Hand Jacobian Grasp Statics i= Kinematics of Contact Hand Kinematics Grasp Planning Grasp Force Optimization Coordinated Control
  59. 59. Chapter 6 Multifingered Hand Modeling and Control 6.4 Hand Kinematics 45 ⎡ BT Ad ⎤ ⎡ Adg − J (θ ) ⎤⎡ ⎢ ⎢ ⎥ ⎥ ⎢ ⎢ ⎥⎢ ⎥⎢ θ ⎤ ˙ ⎥ ⎥ glo o ⎢ ⎥ Vpo = ⎢ ⋱ ⎥⎢ ⎥ i fi lo ⎢ BT Ad ⎥ ⎢ Adgf− l Jk (θ k ) ⎥ ⎢ ⎢ k glo o ⎣ k ⎥ ⎦ ⎢ ⎣ k ok ⎥⎣ ⎦ θk ⎥ ˙ ⎦ Chapter Mul- GT (η)Vpo = Jh (θ, x , η)θ tifingered Hand ˙ Modeling and Control k Introduction θ = (θ , . . . , θ k ) ∈ Rn , n = ni , Jh ∈ Rm×n : Hand Jacobian Grasp Statics i= Definition: Ω = (G, FC, Jh ) is called a multifingered grasp. Kinematics of Contact Hand Kinematics finger contact object Jh GT Grasp velocity Planning ˙ θ xc ˙ b Vpo domain Grasp Force Optimization Coordinated Control force τ fc Fo domain T Jh G
  60. 60. Chapter 6 Multifingered Hand Modeling and Control 6.4 Hand Kinematics 46 Definition: Chapter Mul- A multifingered grasp Ω = (G, FC, Jh ) is manipulable at a configuration (θ, x ) if for any object motion, Vpo ∈ R , tifingered Hand ∃θ ∈ Rn s.t. GT Vpo = Jh θ. Modeling and Control ˙ ˙ Introduction Grasp Statics Kinematics of Contact Hand Kinematics Grasp Planning Grasp Force Optimization Coordinated Control
  61. 61. Chapter 6 Multifingered Hand Modeling and Control 6.4 Hand Kinematics 46 Definition: Chapter Mul- A multifingered grasp Ω = (G, FC, Jh ) is manipulable at a configuration (θ, x ) if for any object motion, Vpo ∈ R , tifingered Hand ∃θ ∈ Rn s.t. GT Vpo = Jh θ. Modeling and Control ˙ ˙ Introduction Grasp Statics Property 6: Γ is manipulable at (θ, x○ ) iff Im(GT ) ⊂ Kinematics of Contact Hand Im(Jh (θ, x )) Kinematics Grasp Planning Grasp Force Optimization Coordinated Control force closure not force-closure not force-closure not manipulable manipulable not manipulable
  62. 62. Chapter 6 Multifingered Hand Modeling and Control 6.4 Hand Kinematics 47 Table 5.4: Grasp properties. Chapter Mul- Property Definition Description tifingered Hand Force-closure Can resist any applied G(F C) = Rp Modeling and Control wrench Manipulable Can accommodate any R(GT ) ⊂ R(Jh ) Introduction object motion Grasp Statics Internal forces Contact forces fN which fN ∈ N (G) ∩ int(F C) Kinematics of Contact cause no net object wrench Hand Kinematics Internal motions Finger motions θN˙ ˙ θN ∈ N (Jh ) Grasp which cause no object Planning motion T Grasp Force Structural forces Object wrench FI which G+ FI ∈ N (Jh ) Optimization causes no net joint Coordinated torques Control
  63. 63. Chapter 6 Multifingered Hand Modeling and Control 6.4 Hand Kinematics 48 Example: Two SCARA fingers grasping a box Chapter Mul- tifingered Soft finger ⎡ ⎤ Hand ⎢ − ⎥ Modeling and ⎢ ⎥ Control ⎢ ⎥ G = ⎢ −r +r ⎥ Introduction ⎢ ⎥ ⎢ − ⎥ Grasp Statics ⎢ +r −r ⎥ Kinematics of Contact ⎣ ⎦ ⎡ ⎤ 2r Hand ⎢ ⎥ ⎢ ⎥ Kinematics ⎢ ⎥ l1 l2 Bc i = ⎢ ⎥ x y Grasp ⎢ ⎥ C1 z Planning z O ⎢ ⎥ C2 y x ⎢ ⎥ l3 Grasp Force z z ⎣ ⎦ a Optimization y y Coordinated S1 x P S2 x Control Rpo = I, ppo = b b a
  64. 64. Chapter 6 Multifingered Hand Modeling and Control 6.4 Hand Kinematics 49 Jh = J J ⎡ ⎤ ⎢ ⎥ Chapter Mul- J = ⎢ −b + r −b +l rs + l c −b +l rs++ lc s+ l c ⎥ tifingered ⎢ ⎥ l ⎢ ⎥ Hand ⎣ ⎦ Modeling and Control ⎡ b−r b−r+l c b−r+l c +l c ⎤ ⎢ ⎥ Introduction J =⎢ ⎢ −l s −l s − l s ⎥ ⎥ Grasp Statics ⎢ ⎥ ⎣ ⎦ Kinematics of Contact Hand Kinematics The grasp is not manipulable, as ⎡ ⎤ ⎡ ⎤ ⎡ ⎤ Grasp ⎡ ⎤ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Planning ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Grasp Force T⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎥=⎢ ⎥ ∈ Im(Jh ), θ N = ⎢ ⎥ , θN = ⎢ ⎥ Optimization G ⎢ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ˙ ˙ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Coordinated ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ ⎢ ⎥ Control ⎢ ⎣ ⎦ ⎢ ⎢ − ⎥ ⎥ ⎢ ⎢ ⎥ ⎥ ⎢ ⎢ ⎥ ⎥ ⎣ ⎦ ⎣ ⎦ ⎣ ⎦
  65. 65. Chapter 6 Multifingered Hand Modeling and Control 6.5 Grasp Planning 50 ◻ Bounds on number of required contacts: Chapter Mul- tifingered Consider PCWF, let Σ = ∂O Hand (Σ) = pci × nci Modeling and nci Control ci ∈ Σ Introduction O Grasp Statics be the set of all wrenches, where nci is Kinematics of Contact inward normal. Hand Kinematics Grasp Planning Grasp Force Optimization Coordinated Control
  66. 66. Chapter 6 Multifingered Hand Modeling and Control 6.5 Grasp Planning 50 ◻ Bounds on number of required contacts: Chapter Mul- tifingered Consider PCWF, let Σ = ∂O Hand (Σ) = pci × nci Modeling and nci Control ci ∈ Σ Introduction O Grasp Statics be the set of all wrenches, where nci is Kinematics of Contact inward normal. Hand Definition: Exceptional surface The convex hull of ⋀(Σ) does not contain a neighborhood of Kinematics Grasp Planning o in Rp . Grasp Force Optimization E.g. a Sphere or a circle. Coordinated Control
  67. 67. Chapter 6 Multifingered Hand Modeling and Control 6.5 Grasp Planning 50 ◻ Bounds on number of required contacts: Chapter Mul- tifingered Consider PCWF, let Σ = ∂O Hand (Σ) = pci × nci Modeling and nci Control ci ∈ Σ Introduction O Grasp Statics be the set of all wrenches, where nci is Kinematics of Contact inward normal. Hand Definition: Exceptional surface The convex hull of ⋀(Σ) does not contain a neighborhood of Kinematics Grasp Planning o in Rp . Grasp Force Optimization E.g. a Sphere or a circle. Coordinated Control Theorem 4 (Caratheodory): If a set X = (v , . . . , vk ) positively spans Rp , then k ≥ p +
  68. 68. Chapter 6 Multifingered Hand Modeling and Control 6.5 Grasp Planning 51 ⇒ Lower bound on the number of fingers. Chapter Mul- tifingered v1 v1 v1 Hand v2 v2 v2 Modeling and Control Introduction Grasp Statics v5 v5 v4 v3 v3 v4 v3 Kinematics of Contact (a) (b) (c) Hand Figure 5.11: Sets of vectors which positively span R2 . Kinematics Grasp Planning Grasp Force Optimization Coordinated Control
  69. 69. Chapter 6 Multifingered Hand Modeling and Control 6.5 Grasp Planning 51 ⇒ Lower bound on the number of fingers. Chapter Mul- tifingered v1 v1 v1 Hand v2 v2 v2 Modeling and Control Introduction Grasp Statics v5 v5 v4 v3 v3 v4 v3 Kinematics of Contact (a) (b) (c) Hand Figure 5.11: Sets of vectors which positively span R2 . Kinematics Grasp Planning Theorem 5 (Steinitz): If S ⊂ Rp and q ∈ int(co(S)), then there exists X = (v , . . . , vk ) ⊂ S such that q ∈ int(co(X)) and k ≤ p. Grasp Force Optimization Coordinated Control ⇒ upper bound on the number of minimal fingers.
  70. 70. Chapter 6 Multifingered Hand Modeling and Control 6.5 Grasp Planning 52 Table 5.3: Lower bounds on the number of fingers required to grasp an object. Chapter Mul- tifingered Space Object type Lower Upper FPC PCWF SF Hand Planar Exceptional 4 6 n/a 3 3 Modeling and Control (p = 3) Non-exceptional 4 3 3 Introduction Spatial Exceptional 7 12 n/a 4 4 Grasp Statics (p = 6) Non-exceptional 12 4 4 Kinematics of Polyhedral 7 4 4 Contact Hand Kinematics ◻ Constructing force-closure grasps: Grasp Planning Theorem 1 (Planar antipodal grasp): Grasp Force Optimization A planar grasp with two point contacts with friction is Coordinated force-closure iff the line connecting the contact point lies Control inside both friction cones.
  71. 71. Chapter 6 Multifingered Hand Modeling and Control 6.5 Grasp Planning 53 Theorem 2 (Spatial antipodal grasps): A spatial grasp with two soft-finger contacts is force-closure Chapter Mul- tifingered iff the line connecting the contact points lies inside both Hand Modeling and friction cones. Control Introduction Grasp Statics Kinematics of Contact Hand Kinematics Grasp Planning Grasp Force Optimization Coordinated Control

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