Stability Analysis of 3D Grasps by A Multifingered Hand

Stability Analysis of 3D Grasps by A Multifingered Hand

Takayoshi YAMADA*, Tarou KOISHIKURA*, Yuto MIZUNO*, Nobuharu MIMURA**, Yasuyuki FUNAHASHI*

*: Nagoya Institute of Technology, Gokiso-cho, Showa-Ku, Nagoya 466-8555, Japan

**: Niigata University, Ninomachi, Ikarashi, Niigata 950-2181, Japan yamat@eine.mech.nitech.ac.jp

Abstract

In this paper, we discuss stability of 3D grasps by us- ing a three-dimensional spring model (3D spring model).

Many works described stability of frictionless grasp.

These works used a one-dimensional spring model (1D spring model) for representation of frictionless contact as a matter of course. However, 1D spring model in- volves the following two problems. (i) Displacement of finger is restricted to one dimension along the initial normal at contact point. (ii)It is not clearly considered that a finger generates contact force to the object along the normal direction only. To overcome the problems and provide accurate grasp stability, we introduce 3D spring model. Then finger’s displacement is relaxed and finger’s contact force is precisely formulated. We ana- lyze grasp stability from the viewpoint of the potential energy method. From numerical examples, we show that there exists an optimum contact force for stable grasp.

Moreover, we also analyze frictional grasp by using con- tact kinematics of pure rolling. By comparing frictional grasp stability with frictionless one, we prove that fric- tion enhances stability of grasp.

1 Introduction

A multifingered robot hand has potential ability of fine and dexterous manipulation. Stability is the ten- dency of a system to return to an equilibrium state when displaced from this state. It is required for the hand to maintain the graspstable against external disturbances. This paper discusses stability of three- dimensional grasp.

Many works [1]-[7][9]-[12][14]-[21] explored stability of grasps. Grasp stability depends on whether fric- tion exists or not, because finger’s motion is different as shown in Fig. 1.

1.1 Frictional Grasps

The following papers discussed frictional grasp stability.

(a) External disturbance

(c) Frictional case (b) Frictionless case

Figure 1: Difference of finger’s displacement between frictionless contact and frictional contact for 2D grasp

Nguyen [14], Kaneko et al. [7] analyzed graspsta- bility by replacing a finger with a virtual spring model.

The stability is evaluated by positive definiteness of a hessian of potential energy stored in the grasp system.

The hessian is called a graspstiffness matrix.

Howard and Kumar [5], Funahashi et al. [4], Yamada et al. [20], and Choi et al. [2] included curvature effect of contact point. Sugar and Kumar [18] explored metrics of robustness of errors. Svinin [19] derived range of contact force for rotational stability.

Cutkosky and Kao [3] included the compliace of the finger joints and provided a systematic method for de- riving graspstability. Howard and Kumar [6] explored an enveloping grasp.

Montana [12], Shimoga [17], Xiong [21] discussed dyamamic stability of the grasp.

1.2 Frictionless Grasps

When the hand grasps a cake of soap or a cube of ice, a coefficient of contact friction is small. Hence, it is important to analyze frictionless grasp stability.

Refs. [4] [5] [6] [14] also explored frcitionless grasp stability. Rimon and Burdick [15], Lin et al. [9] [10]

presented systematic approach. These papers used 1D spring model for representation of frictionless contact as a matter of course (Fig. 2). However, these papers did not consider that contact force act along the normal

(a) 1D-spring model

(b) 2D-spring model

Figure 2: Difference of finger’s displacement betweem 1D spring model and 2D spring model for frictionless 2D grasp

direction, and tangential component of the contact force is equal to zero when the grasped object is displaced.

Ref. [2] did not considered the effect of initial grasping force.

To overcome the problems, Saha et al. [16] intro- duced 2D spring model as shown in Fig.2(b) and ex- plicitly formulated the frictionless contact force. Then, more accurate evaluation of graspstability was ob- tained.

1.3 Approach of this paper

This paper discusses stability of 3D grasps, based on Ref. [16]. 3D spring model is introduced into friction- less 3D grasps. The frictionless contact force in 3D is explicitly formulated. And grasp stability is determined by the potential energy method. From numerical exam- ples, it is shown that there exists a region of contact force for stable graspand an optimum contact force.

Moreover, frictional graspstability is also investigated by using contact kinematics of pure rolling. By com- paring frictional grasp stability with frictionless one, we prove that friction enhances stability of grasp.

2 Formulation

Suppose that an object is grasped by a multifingered hand as shown in Fig.3. We explore stability of the grasp.

2.1 Assumptions

In this paper, the following assumptions are considered.

(A1) The geometries of the fingertips and the ob- ject are known and rigid. Finger is shaped a sphere with curvature κf i. Local geometry of the object at the contact point is shaped a second-order model with primary curvatures κai and κbi whose principal axes

2EMHFW

)LQJHU +DQG

Figure 3: 3D graspsystem

k_xi k_yi k_zi

c_oi Σo

Finger Object

Σb

x_fi

x_o

c_fi

Figure 4: Coordinate frame

are denoted by r^zi and r^yi, respectively. (A2) Initial configuration is known and is in equilibrium. (A3) In- finitesimal displacement is occurred in the object due to external disturbances. The displacement is denoted by = [x, y, z, ξ, η, ζ]^T. (A4) Finger’s displacement is infinitesimal, and the relationshipbetween finger’s dis- placement and reaction force is replaced with 3D linear spring model which is fixed at center of the fingertip as shown in Fig.4. Stiffness of the spring is programmable.

One spring kxi(≥ 0) is set along the normal and the otherskyi(≥0) andkzi(≥0) are parallel to the tangent.

kyi and kzi are parallel to the primary curvatures κbi

andκai, respectively.

2.2 Notations

Reference coordinate and the i-th finger’s coordinate are denoted by Σb and Σf i, respectively. Σf iis aligned with the 3D spring model. xaxis of Σf i is aligned with the inward normal of the finger. Position and orientation of Σf i with respect to Σb are denoted by x^{f i} and Rf i :=

[r^xi,r^yi,r^zi]. Contact position on the fingertip is given

by

c^{f i}(u^{f i}) :=κ⁻¹f i



cosuf icosvf i

cosuf isinvf i

sinuf i



, (1)

whereu^{f i}= [uf i, vf i]^T is parameter of contact position on thei-th finger. The inward unit normal atc^{f i} is

n^{f i}(u^{f i}) :=−κf ic^{f i}(u^{f i}). (2) Object coordinate is denoted by Σo. Σb is aligned with the initial configuration of Σo. When the object is dis- placed by disturbances, position and orientation of Σo

is given by

x^o= [x, y, z]^T, Ro= Rot(ξ, η, ζ), (3) where Rot(•) is a 3D rotation matrix. Object local co- ordinate at contact point is represented by Σoi. Contact position of the object isc^oi. Position and Orientation of Σoi with respect to Σo is denoted bycⁱ andRci. Local shape around the contact point is approximated by the primary curvaturesκai andκbi.

coi(uci) :=ci+Rcidi(uci), (4) dⁱ(u^ci) := [uci, vci, (κaiu²ci+κbivci²)/2]^T, (5) Rci:= [r^zi,r^yi,−rxi], (6) where u^ci= [uci, vci]^T is parameter of contact position on the grasped object. The inward unit normal vector is denoted byn^oi.

n^oi(u^ci) = (d^iu⊗d^iv)/diu⊗d^iv

= 1

κ²aiu²ci+κ²_biv²ci+ 1



−κaiuci

−κbivci

1



, (7)

whered^iu :=∂di/∂uci,d^iv/=∂di/∂vci, and⊗is cross product.

2.3 Spring Compression

From kinematics of contact, we have the following two equations. One is a condition of contact point.

x^o+Roc^oi=x^{f i}+Rf ic^{f i} (8) Another is a condition of the normal direction.

Ronoi=−Rf inf i (9) Due to object displacement, compression of the vir- tual spring is given by

δⁱ:=R^Tf i{xf i()−x^{f i}(0)}

=R^Tf i{xo+Roc^oi+κ⁻¹_{f i}Ron^oi−x^{f i}(0)}, (10) where note thatδⁱ=δⁱ(,u^ci) becausec^oi andn^oi are functions of u^ci. The relationshipbetween and u^ci depends on whether friction exists or not. This rela- tionshipis described in Section 3.1 and 3.2.

2.4 Potential Energy

The initial contact force with respect to Σf i is ex- pressed as fi := [fxi, fyi, fzi]^T. Initial compression of the spring for the contact force fi is denoted by δ^oi= [δxoi, δyoi, δzoi]^T.

fi=Kiδ^oi, Ki:= diag[kxi, kyi, kzi] (11) The potential energy stored in the grasp system is given by

U() := 1 2

n i=1

(δoi+δi)^TKi(δoi+δi). (12) Performing Taylor’s expansion, the function U() can be expressed as

U() =U(0) +^TG+1

2^TH+· · ·, (13) whereGandHare the gradient and the hessian, respec- tively. The graspsystem is stable at the initial configu- ration if and only if the potential energy reaches a local minimum. The function U() reaches a local minimum if the following two conditions are satisfied.

(i)G= 0,

(ii)H is positive difinite.

Condition (i) is satisfied by Assumption (A2). Hence, we can evaluate graspstability by eigenvalues of the hessian. The hessian is so-called a graspstiffness matrix.

Let us define ∇:=∂/∂, and we have the following forms: G=∇U|₀, H =∇∇^TU|₀. From Eq. (12), the hessianH is given by

H= n

i=1

Hi,

Hi:=kxi(∇δxi|0)(∇δxi|0)^T +fxi(∇∇^Tδxi|0) +kyi(∇δyi|0)(∇δyi|0)^T +fyi(∇∇^Tδyi|0)

+kzi(∇δzi|₀)(∇δzi|₀)^T +fzi(∇∇^Tδzi|₀), (14) where, from Eq. (10),

∇δxi|₀= r^xi

cⁱ⊗r^xi

,

∇δyi|₀=

r^yi cⁱ⊗r^yi+κ⁻¹f ir^zi

+κf i+κbi

κf i (∇vci|₀),

∇δzi|0=

r^zi cⁱ⊗r^zi−κ⁻¹_{f i}r^yi

+κf i+κai

κf i (∇uci|0).

∇∇^Tδxi|0, ∇∇^Tδyi|0, ∇∇^Tδzi|0 are omitted because they have long terms. In order to derive the hessian,

∇uci|0 and ∇vci|0 must be derived. These derivatives depend on whether friction exists or not.

3 Grasp Stability

3.1 Frictionless Grasp

In frictionless case, fyi=fzi = 0 because no tangential force occurs. Contact force generated by the displace- ment of the fingertipis given by fi = fi +Kiδi. In frictionless case, the direction of the contact force fi

is always along the normal direction at current contact point. So we have the following conditions about contact force.

Rf i(fi+Kiδi)⊥Rocoiu,Rf i(fi+Kiδi)⊥Rocoiv, where c^oiu = ∂coi/∂uci and c^oiv = ∂coi/∂vci are tan- gential vectors at the contact point. Hence, we have

hui(,u^ci) = (Roc^oiu)^T{Rf i(fi+Kiδⁱ)}= 0, hvi(,u^ci) = (Roc^oiv)^T{Rf i(fi+Kiδⁱ)}= 0. (15) From the derivation of Appendix A, the hessian H is given by

H^s= n i=1

(Hf i^s +Hci^s +Hkxi^s +Hkti^s ), (16)

where the superscript ”s” means frictionless contact (i.e.

slipor slide). Hf i^s expresses an effect of the initial con- tact forcefxi.

Hf i^s =fxi 0_3×3 [r^xi⊗]^T

[r^xi⊗] ¹₂([cⁱ⊗] [r^xi⊗]^T+ [r^xi⊗] [cⁱ⊗]^T)

(17) Hci^s means an effect of the local curvatures at the con- tact, κai, κbi, and κf i. Hc → 0 if κai → 0, κbi → 0, κf i→ ∞.

Hci^s =−fxi κbiκf i

κbi+κf i

×

r^yi cⁱ⊗r^yi+κ⁻¹_{f i}r^zi

r^yi cⁱ⊗r^yi+κ⁻¹_{f i}r^zi

T

−fxi κaiκf i

κai+κf i

×

r^zi cⁱ⊗r^zi−κ⁻¹_{f i}r^yi

r^zi cⁱ⊗r^zi−κ⁻¹_{f i}r^yi

T

+ fxi

κf i

0 0 0 r^yir^Tyi+r^zir^Tzi

(18) Hkxi^s is an effect of normal stiffnesskxi.

Hkxi^s =kxi

rxi

ci⊗rxi

rxi

ci⊗rxi

T

(19)

Hkti^s is an effect of tangential stiffness kyi and kzi. Hkti^s →0 ifkyi→ ∞and kzi → ∞.

Hkti^s =− 1 kyi−fxi

κ_biκ_fi κ_bi+κ_fi

fxi κbiκf i

κbi+κf i

₂

×

r^yi cⁱ⊗r^yi−κ⁻¹_bi r^zi

r^yi cⁱ⊗r^yi−κ⁻¹_bi r^zi

T

− 1

kzi−fxi κ_aiκ_fi κ_ai+κ_fi

fxi

κaiκf i

κai+κf i

₂

×

r^zi ci⊗rzi+κ⁻¹airyi

r^zi ci⊗rzi+κ⁻¹airyi

T

(20) From Appendix B, kyi, kzi, and fxi must be selected from the following regions for stable grasp.

kyi−fxi κbiκf i

κbi+κf i

>0 andkzi−fxi κaiκf i

κai+κf i

>0 (21) Eq. (21) is satisfied if convex shape (κ•<0) is in contact with concave shape (κ• >0), because ofkyi≥0, kzi ≥ 0, fxi≥0. If Eq. (21) is satisfied,Hkti^s is negative semi- difinite (Hkti^s ≤0) . Hence Hkti^s make unstable grasp.

However, stability can be made larger, if stiffness kyi

andkzi are stronger.

From Assumption (A4) about no rotation of finger- tip, offset of the spring model from contact point does not affect the hessianH.

We show the relationshipamong our result and pre- vious works. Nguyen [14] discussed the case of κf i →

∞, κai →0, κbi →0, kyi→ ∞, kzi → ∞. Hence, result of Ref. [14] is expressed as

HN guyen^s =Hf^s+Hkx^s .

Rimon and Burdick [15], Howard and Kumar [5], Funa- hashi et al. [4] treated the case of kyi → ∞, kzi → ∞. Consequently, their results are expressed as

HF unahashi^s , HHoward^s , HRimon^s =Hf^s+Hc^s+Hkx^s .

3.2 Frictional Grasp

If each finger is in contact with the object with friction, no slipoccurs at the contact point. Then we have the following condition of pure rolling.

Roc˙^oi=Rf ic˙^{f i} (22) Frictional graspstability is formulated by considering Eqs. (9) and (22). From Appendix C, the hessianH^ris given by

H^r= n i=1

(Hf i^r +Hci^r +Hkxi^r +Hkti^r ), (23) where

Hf i^r =fxi

03×3 03×3

03×3 1

2(cⁱr^Txi+r^xic^Ti)−(c^Tir^xi)I3

+fyi

03×3 03×3

03×3 1

2(cir^Tyi+ryic^Ti)−(c^Tiryi)I3

+fzi

0_3×3 0_3×3

0_3×3 ¹₂(cⁱr^Tzi+r^zic^Ti)−(c^Tir^zi)I₃

,

Hci^r =fxi

03×3 03×3

03×3 1

κ_ai+κ_firyir^Tyi+_κ ¹

bi+κ_firzir^Tzi

+fyi

03×3 03×3

03×3 −₂₍κ_ai¹+κ_fi)(rxir^Tyi+ryir^Txi)

+fzi

0_3×3 0_3×3

0_3×3 −_2(κbi¹_+κfi)(r^xir^Tzi+r^zir^Txi)

,

Hkxi^r =kxi

r^xi cⁱ⊗r^xi

r^xi cⁱ⊗r^xi

T

,

Hkti^r =kyi

r^yi cⁱ⊗r^yi

r^yi cⁱ⊗r^yi

T

+kzi

r^zi cⁱ⊗r^zi

r^zi cⁱ⊗r^zi

T

.

3.3 Effect of Friction

We make an effect of friction clear by comparing friction- less graspand frictional one under the same condition.

For this reason, let us set asfyi=fzi = 0. The differ- ence of stability between frictionless graspand frictional one is given as

H^d:=H^r−H^s, (24) where

Hi^d= k²yi

kyi−fxi κ_biκ_fi κ_bi+κ_fi

ryi

cⁱ⊗r^yi−_kyi(^κ_κ^fi_bi^f₊^xi_κfi)r^zi

× r^yi

cⁱ⊗r^yi−_kyi(^κ_κ^fi_bi^f₊^xi_κfi)r^zi T

+ kzi²

kzi−fxi κ_aiκ_fi κ_ai+κ_fi

r^zi cⁱ⊗r^zi+_k ^κfifxi

zi(κ_ai+κ_fi)r^yi

× r^zi

cⁱ⊗r^zi+_k ^κfifxi

zi(κ_ai+κ_fi)r^yi T

. (25)

Hi^d is positive semi-difinite if Eq. (21) is satisfied. And λ(H^s) = −∞ if Eq. (21) is not satisfied, where λ(•) represents eingenvalues of •. Therefore, friction en- hances graspstability because frictional graspstability is greater than frictionless one.

4 Numerical Examples

Numerical examples show effects of spring stiffness and contact force on the graspstability. For a simple exam- ple, we consider a symmetric 3-finger grasp such as

kxi=kx, kyi=ky, kzi=kz, fxi=fx, κai=κa, κbi =κb, κf i=κf, i= 1,· · · , n c1= [lc,0,0]^T,

c2= [lccos(2π/3), lcsin(2π/3),0]^T, c2= [lccos(4π/3), lcsin(4π/3),0]^T, Rf1= [rx1,ry1,rz1] =Rz(0), Rf2=Rz(2π/3), Rf3=Rz(4π/3), Rz(θ) =



cosθ −sinθ 0 sinθ cosθ 0

0 0 1



.

In case of frictionless grasp, the hessian Hs is given as

H^s= diag[H₁₁^s, H₂₂^s, H₃₃^s, H₄₄^s , H₅₅^s, H₆₆^s], (26) H₁₁^s =H₂₂^s =−

3 2

kyfx κ_bκ_f κ_b+κ_f

ky−fx κ_bκ_f κ_b+κ_f

+3 2kx, H₃₃^s =−3 kzfx

κ_aκ_f κ_a+κ_f

kz−fx κ_aκ_f κ_a+κ_f

, H₄₄^s =H₅₅^s

= 3

2

(lc−κ⁻¹a ){kz(lc+κ⁻¹_f )−fx}fx κ_aκ_f κ_a+κ_f

kz−fx κ_aκ_f κ_a+κ_f

,

H₆₆^s = 3(lc−κ⁻¹_b ){ky(lc+κ⁻¹_f )−fx}fx κ_bκ_f κ_b+κ_f

ky−fx κ_bκ_f κ_b+κ_f

. In case of frictional grasp, the hessianH^ris obtained as

H^r= diag[H₁₁^r , H₂₂^r, H₃₃^r, H₄₄^r , H₅₅^r , H₆₆^r], (27) H₁₁^r =H₂₂^r = 3(kx+ky)/2,

H₃₃^r = 3kz, H₄₄^r =H₅₅^r = 3

2

kzl²c−fxlc+ fx

κa+κf

, H₆₆^r = 3

kyl²c−fxlc+ fx

κb+κf

.

Note that, in this example, ky is included in H11, H22,H66and kz is included inH33, H44,H55.

0 50 100 150 200 -75

-50 -25 0 25 50 75 100

3D spring model without friction

Spring stiffness k_y H11,H22

with friction

1D spring model

0 50 100 150 200

-2 -1 0 1 2 3 4

3D spring model without friction

Spring stiffness k_y H66

with friction

1D spring model

0 50 100 150 200

-75 -50 -25 0 25 50 75 100

3D spring model without friction

Spring stiffness ky

H11,H22

with friction 1D spring model

0 50 100 150 200

-2 -1 0 1 2 3 4

3D spring model without friction

Spring stiffness ky

H66

with friction 1D spring model

0 50 100 150 200

-75 -50 -25 0 25 50 75 100

3D spring model without friction

Spring stiffness k_y H11,H22

with friction 1D spring model

0 50 100 150 200

-2 -1 0 1 2 3 4

3D spring model without friction

Spring stiffness k_y H66

with friction

1D spring model

(a) Grasp (I)

(b) Grasp (II)

(c) Grasp (III)

Figure 5: Effect of spring stiffnessky

4.1 Effect of Spring Stiffness k

_y

and k

_z

Fig.5 shows an effect of spring stiffness ky, kz on H11, H22,H66whenkx= 1.0,fx= 1.0,lc= 0.1 and

Grasp(I) κa =κb= 1/0.02, κf = 1/0.02, Grasp(II)κa=κb=−1/0.04, κf = 1/0.02, Grasp(III)κa=κb= 1/0.02, κf =−1/0.04.

SinceH33,H44,H55are similar to Fig.5, we omit them.

In case of Grasp(I), the range of ky ≤25 or kz ≤ 25 does not satisfy Eq. (21). Grasp(I) is unstable for any stiffness ofkyandkz, because both finger and object are convex. Grasp(II) and (III) become unstable if spring stiffness is small. The range of spring stiffness, which makes the hessian positive definiteness and makes the graspstable, depend on whether friction exists or not.

In case of frictionless contact,kyandkzmust be selected from

k^sy> fx/(lc+κ⁻¹f ), k^sz> fx/(lc+κ⁻¹f ).

In case of frictional contact,ky andkzmust be selected from

k^ry> fxlc(κb+κf)−1

lc²(κb+κf) , k^rz> fxlc(κa+κf)−1 lc²(κa+κf) .

4.2 Effect of Contact Force f

x

Fig.6 shows an effect of contact force fx on H11, H22, andH66 of Grasp(II). There exists contact force which

0 1 2 3 4

-10 0 10 20 30 40 50 60

ky =20 ky =30

ky =10 without friction

Contact force fx

H11,H22

with friction 1D spring model

0 1 2 3 4

-0.2 0.0 0.2 0.4 0.6 0.8 1.0

ky =20 ky =30

ky =10 without friction

Contact force fx

H66

with friction 1D spring model

Figure 6: Effect of contact forcefxfor Grasp(II)

makes unstable graspabout rotation. For this reason, we must obtain the range which makes stable grasp. In case of frictionless contact,fxmust be selected from

0< fx^s<min{ky(lc+κ⁻¹f ), kz(lc+κ⁻¹f )}. (28) In case of frictional contact,fx must be selected from

0< fx^r<min{ kzl²c(κa+κf)

lc(κa+κf)−1, kyl²c(κb+κf)

lc(κb+κf)−1}. (29) There exists contact force which maximizes rotational graspstability. From∂H₄₄^s/∂fx= 0 and∂H₆₆^s/∂fx= 0, it is given at

fx^s=kz

κ⁻¹a +κ⁻¹_f +κ⁻¹a

κ⁻¹_f (κa+κf)(1−lcκa)

, fx^s=ky

κ⁻¹_b +κ⁻¹_f +κ⁻¹_b

κ⁻¹_f (κb+κf)(1−lcκb)

.

5 Conclusions

This paper discussed stability of 3D grasps from the viewpoint of potential energy method.

We introduced 3D spring model into frictionless 3D grasps in order to obtain accurate grasp stability. The relation between object’s displacement and finger’s one is derived. It was precisely formulated that a finger gen- erates contact force along the normal direction at cur- rent contact point. The hessian H^s is obtained, and the frictionless graspstability is evaluated by the eigen- values of H^s. The conditions ofkyi,kzi, fxi for stable grasp is also provided. From numerical examples, we showed that there exist the range of contact force for stable graspand an optimum contact force.

Moreover, in frictional case, the hessianH^ris estab- lished by using contact kinematcs of pure rolling. It is proved that friction enhances grasp stability by compar- ing frictional graspstability with frictionless one.

If a coefficient of contact friction is unknown, the graspis assumed to be frictionless one. The graspis made stable by using the eigenvalues of H^s. Then, it

is guaranteed that the graspis stable, whether friction exists or not.

While Kerr and Roth [8] and Nakamura et al. [13]

reduced the problem of determining optimum contact force to a linear or nonlinear programming problem, our analysis about frictionless graspsuggested that op- timum contact force can be determined from the view- point of the grasp stability.

References

[1] H. Bruyninckx, S. Demey, and V. Kumar, Generalized Stability of Compliant Grasps,Proc. of IEEE Int. Conf.

on Robotics and Automation, pp. 2396-2402, (1998).

[2] K. K. Choi, S. L. Jiang, and Z. Li, Grasping with Elastic Finger Tips,Proc. of IEEE Int. Conf. on Robotics and Automation, pp. 920-925, (1999).

[3] M. R. Cutkosky and I. Kao, Computing and Controlling the Compliance of a Robotic Hand, IEEE Trans. on Robotics and Automation, Vol. 5, No. 2, pp. 151-165, (1989).

[4] Y. Funahashi, T. Yamada, M. Tate, and Y. Suzuki, Grasps stability analysis considering the curvatures at contact points, Proc. of IEEE Int. Conf. on Robotics and Automation, pp. 3040-3046, (1996).

[5] W. S. Howard and V. Kumar, On the Stability of Grasped Objects, IEEE Trans. on Robotics and Au- tomation, Vol. 12, No. 6, pp. 904-917, (1996).

[6] W. S. Howard and V. Kumar, Modeling and Analysis of the Compliance and Stability of Enveloping Grasps, Proc. of IEEE Int. Conf. on Robotics and Automation, pp. 1367-1372, (1995).

[7] M. Kaneko, N. Imamura, K. Yokoi, and K. Tanie, A re- alization of stable graspbased on virtual stiffness model by robot fingers,Proc. of IEEE Int. Workshop on Ad- vanced Motion Control, pp. 156-163C(1990).

[8] J. Kerr and B. Roth, Analysis of multifingered hands, Int. Journal of Robotics Research, Vol. 4, No. 4, pp.

3-17, (1986).

[9] Q. Lin, J. Burdick, and E. Rimon, A Quality Mea- sure for Compliant Grasps, Proc. of IEEE Int. Conf.

on Robotics and Automation, Vol. 1, pp. 86-92, (1997).

[10] Q. Lin, J. Burdick, and E. Rimon, Computation and Analysis of Compliance in Grasping and Fixturing, Proc. of IEEE Int. Conf. on Robotics and Automation, Vol. 1, pp. 93-99, (1997).

[11] H. Maekawa, K. Tanie, and K. Komoriya, Kinematics, Statics and Stiffness Effect of 3D Graspby multifin- gered hand with rolling contact at the fingertip,Proc.

of IEEE Int. Conf. on Robotics and Automation, p p . 78-85, (1997).

[12] D. J. Montana, Contact Stability for Two-Fingered Grasps,IEEE Trans. on Robotics and Automation, Vol.

8, No. 4, pp. 421-430, (1992).

[13] Y. Nakamura, K. Nagai, and T. Yoshikawa, Dynam- ics and Stability in Coordination of Multiple Robotic Mechanisms,Int. Journal of Robotics Research, Vol. 8, No. 2, pp. 44-61, (1989).

[14] V. D. Nguyen, Constructing stable grasps,Int. Journal of Robotics Research, Vol. 7, No. 3, pp. 3-16, (1988).

[15] E. Rimon and J. W. Burdick, Mobility of bodies in con- tact - Part II: How Forces are Generated by Curvature Effects, IEEE Trans. of Robotics and Automation, p p . 709-717, (1998).

[16] S. K. Saha, T. Yamada, N. Mimura, and Y. Funahashi, Stability analysis of frictionless planar grasps with 2D spring model,Int. Journal of JSME, Series C, Vol. 42, No. 4, pp. 974-981, (1999).

[17] K. B. Shimoga, Robot graspsynthesis algorithms: A survey.Int. Journal of Robotics Research, Vol. 15, No.

3, pp. 230-266, (1996).

[18] T. G. Sugar and V. Kumar, Metrics for Analysis and Optimization of Grasps and Fixtures, Proc. of IEEE Int. Conf. on Robotics and Automation, pp. 3561-3566, (2000).

[19] M. M. Svinin, K. Ueda, and M. Kaneko Analytical Conditions for the Rotational Stability of an Object in Multi-Finger Grasping,Proc. of IEEE Int. Conf. on Robotics and Automation, Vol. 1, pp. 257-262, (1999).

[20] T. Yamada, S. K. Saha, N. Mimura, and Y. Funahashi, Stability analysis of grasping an arbitrary shaped planar object,Trans. of JSME, Series C, Vol. 64, No. 621, pp.

1736-1742, (1998), (in Japanese).

[21] C. H. Xiong, Y. F. Li, H. Ding, and Y. L. Xiong, On the Dynamic Stability of Grasping,Int. Journal of Robotics Research, Vol. 18, No. 9, pp. 951-958, (1999).

Appendix A: Frictionless Grasp

From Eq. (15), we have

∇hui+∂hui

∂uci

∇uci+∂hui

∂vci

∇vci= 0,

∇hvi+∂hvi

∂uci

∇uci+∂hvi

∂vci

∇vci= 0.

Then∇uciand ∇vci are formulated as [∇uci ∇vci] =−[∇hui ∇hvi]

∂h_ui

∂u_ci

∂h_vi

∂u_ci

∂h_ui

∂v_ci

∂h_vi

∂v_ci

₋₁ . Considering initial conditions yields

∇uci|₀= 1

κaiκf ifxi−(κai+κf i)kzi

×

κf ikzi

r^zi cⁱ⊗r^zi

+ (κf ifxi−kzi) 0

r^yi ,

∇vci|0= 1

κbiκf ifxi−(κbi+κf i)kyi

×

κf ikyi

ryi

ci⊗ryi

−(κf ifxi−kyi) 0

rzi .