Chapter 6: Modeling Multifingered Hands

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

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


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


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


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)


6.1 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


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


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)


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!


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


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


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


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


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=


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


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


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


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


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


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


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


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


6.2 Grasp Statics
20
◻ Force Closure:
Deﬁnition:
Chapter Mul-
tifingered
Hand
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


6.2 Grasp Statics
20
◻ Force Closure:
Deﬁnition:
Chapter Mul-
tifingered
Hand
Control

Introduction
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


6.2 Grasp Statics
20
◻ Force Closure:
Deﬁnition:
Chapter Mul-
tifingered
Hand
Control

Introduction
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).


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


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


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)


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
l= l
Kinematics of
Contact

Hand
convex.
Kinematics

Grasp
Planning

Grasp Force
Optimization

Coordinated
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
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


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


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


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


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=


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


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


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


6.2 Grasp Statics
27
nc ... nck k
G=
Chapter Mul-
tifingered
Hand
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


6.2 Grasp Statics
27
nc ... nck k
G=
Chapter Mul-
tifingered
Hand
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 ≥ }


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)


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


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


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


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


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 ˙
⎣ ⎦ ⎣ ˙ ˙ ⎦


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


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)


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


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


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


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)


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
˙ ⎦


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


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
˜ ˙ ˙


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


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


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


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


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


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


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


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


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


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

⎢
⎣ ⎦ ⎢
⎢ −
⎥
⎥
⎢
⎢
⎥
⎥
⎢
⎢
⎥
⎥
⎣ ⎦ ⎣ ⎦ ⎣ ⎦


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


6.5 Grasp Planning
50
Chapter Mul-
tifingered
Hand

(Σ) = pci × nci
Modeling and nci
Control ci ∈ Σ
Introduction
O
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


6.5 Grasp Planning
50
Chapter Mul-
tifingered
Hand

(Σ) = pci × nci
Modeling and nci
Control ci ∈ Σ
Introduction
O
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 +


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


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.


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.


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

Chapter 6: Modeling Multifingered Hands

Chapter 6: Modeling Multifingered Hands

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Chapter 6: Modeling Multifingered Hands