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Preprints of the 5th IFAC Symposium on Mechatronic Systems
Marriott Boston Cambridge
Cambridge, MA, USA, September 13-15, 2010

Synthesis of Stable Grasp by Four-Fingered
Robot Hand for Pick-and-Place ofAssembling Parts
Akira Nakashima ∗ Takashi Uno ∗ Yoshikazu Hayakawa ∗,∗∗
Toshiyuki Kondo ∗∗∗ Shinji Sawada ∗∗∗ Nobuhiro Nanba ∗∗∗∗Mechanical Science and Engineering, Graduate School ofEngineering, Nagoya University, Furo-cho, Chikusa-ku, Nagoya, Japan(e-mail: a nakashima@nuem.nagoya-u.ac.jp)∗∗
RIKEN-TRI Collaboration Center, RIKEN, 2271-103, Anagahora,
Shimoshidami, Moriyama-ku, Nagoya, Japan.
∗∗∗
Honda Engineering Co. Ltd., 6-1, Hagadai, Hagamachi, Hagagun,Tochigi, Japan

Abstract: This paper proposes a synthesis methodology of stable grasp by a four-fingered robot
hand in order to automate preparation for assembling work, which is composed of recognition,
grasp and carry of assembling parts. An optimization method of grasping forces, object
configuration and contact points is proposed. The criterion of the optimization is appropriate
normalized distances from the planes of a constraint polyhedron, which is derived by the friction
and joint torque limitations. An experimental result of grasp of a door mirror is shown to verify
the effectiveness of the proposed method.
Keywords: handling, robot control, stability criteria, industrial robots, optimization problems
1. INTRODUCTION

Stereo camera

There are press, welding, painting and assembling workin process of operation of product line for automobiles asshown in Fig. 1. The assembling work is carried out byhumans because of its complicacy while the first threeworks have been automated by robot manipulators in
the product lines. The automation of the assembling
work can reduce manufacturing cost and improve work
environment. The work to fix a door mirror to a door as
an example of the assembling work is illustrated in thebox of Fig. 1. The process of the operation is composedof the preparation for the fixing and the fixing of themirror. The preparation includes the recognition, grasp,
carry of the mirror. Humans use their own eyes, arms,
hands and machine tools in the assembling work. It iscomplex and difficult for robot arms to use the machine

Press work

Welding work

Painting work

Assembling work

Automobile

Automated work
Process of operation of Assembling work

Recognitionof Parts
Eyes

Grasp

Carry

Eyes
Arms, Hands

Eyes
Ar ms, Hands

Preparation for Fixing

Fig. 1. Product line of automobiles.

FixingEyes
Arms, Hands, Machine tools

Fixing of Mirror

Multi-fingered
robot hand

Dual arms

Assembling Parts
for multi-automobiles
within buckets. . ... . .

...

Assembling
Assemblingparts A
Assemblingparts B. . .. .. .
...
parts C

Assembling Parts
for one automobile

Fig. 2. Two-arms system for automating the preparation.
tools in the fixing. On the other hand, the machine tools
are not necessary and the eyes, arms and hands canbe alternated with vision-cameras, robot arms and robot
hands. Therefore, as the first step of the automation of
the assembling work, we aim to automate the preparation
for assembly by a two-arms system as shown in Fig. 2. It
is assumed that the assembling parts of multi-automobiles
are contained in the each blue bucket. The stereo cameraof the left arm detects the shapes and position/orientationof the parts in the buckets. The multi-fingered robot
hand of the right arm grasps the parts with the detected
information and collects the parts for one automobile up
in the red bucket. This system thus can devote humans tocarry out only the assembling work.
In this paper, we concentrate on grasp by a multi-fingered
robot hand. A multi-fingered robot hand allows a robot
arm to grasp a large class of objects with a single endeffector because the hand can grasp with multi contacts
and can control grasping forces via multi joint inputs. It is669

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an important issue to find the set of the contact points forthe grasping forces to be balanced with the gravity force.
Investigation methods of the contact points have been
proposed by Omata (1990, 1993); Trinkle and Paul (1990);
Li et al. (2002). It is also another important issue to determine the grasping forces with criteria since the grasping
forces have the redundancy, i.e., the internal force (Salisbury and Roth (1983)). Kerr and Roth (1986) proposed an
optimization method for the grasping forces with respect
to the distances from the planes of a constraint polyhedron
derived from the friction and joint torque limitations.
Nakamura et al. (1989) considered a safety factor whichmeans that the small grasping force is better with being
included in the frictional domain. Yoshikawa and Nagai(1991) proposed a decomposition of the grasping forces tomanipulating and grasping forces with compatible physical meaning. Nguyen (1988) considered an investigation
method of the contact points to achieve the force-closure.
The force-closure is the ability for the grasping forces toresist any external disturbances to the grasped object andis a major criterion for the grasp stability. Li and Sastry(1988) proposed the criterion where the disturbance force
and moment is evaluated by the volume of the ellipsoid
derived from the singular value decomposition of the grasp
map. Markenscoff and Papadimitriou (1989) investigated
the set of the contact points minimizing the grasping forces
with any normalized disturbance forces. Magialardi et al.
(1996) derived the set of the contact points minimizing thegrasping forces by considering the disturbance force andmoment in a sequential order. Watanabe and Yoshikawa(2003b) optimized the set of the contact points minimizing the grasping forces with a required external force set
which is an extended criterion of any disturbance forces.
The force limitation and the object region with respect
to the fingers have been not considered in these studies.
Watanabe and Yoshikawa (2003a) extended their previous
problem to the one with required acceleration of the object
and considered the force limitation and the object region.
The methods and algorithms in the mentioned studies
are not feasible in two points of view of the optimization
criterion and the finger configuration. The vector norm of
the grasping forces is minimized in the studies of the forceclosure while Watanabe and Yoshikawa (2003b) minimized
the vector norm of one grasping force which is the maximum in all the grasping forces. On the other hand, Kerrand Roth (1986) maximized the minimum of the distancesof the grasping forces from the planes of the constraint
polyhedron. The planes represent the bounds derived fromthe friction and torque limitations. The criterion of the
distances can be better than the vector norm of the grasping forces because the former criterion denotes the safety
factor not to break the contacts. However, in the study of
Kerr and Roth (1986), since the distances from the planes
are not normalized to appropriate scales, the optimized
grasping forces can be close to some of the bounds. We consider the distances normalized by the norms of the normal
vectors of the planes. Watanabe and Yoshikawa (2003a)
considered the region of the position and orientation of theobject with respect to the fingers. However, they did notshow a method to obtain the object region, which depends
on the finger configuration and angle limitation. We show
a method to derive the object region with respect to afour-fingered robot hand.

Joint 1Motor
for J1

Joint 2

Joint 3Motorsfor J3Motor
for J4

Motor
for J2

Joint 4
Motor
for J5Finger 2

UrethanerubberFinger 1

Finger 3

Joint 5Finger 0(a) An overview of the four-fingered robot hand(b) The configuration of the joints and motors

Fig. 3. The four-fingered robot hand.
Separated
spacesBucket(a) Grasp of a large object(b) Grasp of a small object(c) Adjustment for narrow spaces

Fig. 4. The adjustment of the hand.
2. SYSTEM AND BASIC ANALYSES
2.1 Four-Fingered Robot Hand
The four-fingered robot hand for grasping assembling parts
is shown in Fig. 3 where the fingers are numbered by
i = 0, · · · , 3 as the figure (a). The finger 0 is shown in the
figure (b) where the 1st and 5th joints are prismatic andthe 2nd, 3rd, 4th joints are rotational. The other fingers
are the same as the finger 0 expect for the 1st joints andthe fingers 2 and 3 do not has the 1st prismatic joint. The
1st and 2nd joints are driven by a ball screw (the pitchof 1 [mm]) and a rack and pinion gear (the gear ratio of
1:4). The gear ratios of the 2nd–4th joints are 1:2, 1:4 and1:4 respectively and note that the joint 3 is driven by thetwo motors. The motors of the 1st–4th joints and the 5th
joint are the AC servo actuators of RSF-5A and RSF-3A(Harmonic Drive Systems, Inc.). The maximum torquesof the joints are 1.4 and 0.3 [Nm] respectively. Urethane
rubbers are bonded on the surfaces of the 5th links for thegrasp. The hand can adjust the configuration to both small
and large objects by changing the 1st joints of the fingers
0 and 1 as in Fig. 4 (a) and (b). The hand can also adjust
the lengths of the 5th links to separated narrow spaces by
the 5th joints. The 2nd, 3rd and 4th joints are used forcontrolling the grasping forces.
The frame configuration is illustrated in Fig. 5. Σh is thereference frame and Σi0 is the base frame of the finger i.
X

Finger 2Z

Z

Finger 1

X
Z
X
X
Z
Finger 3

L2 minX

Z

Finger 0

L2 maxFig. 5. The frame configuration of the hand.

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The frame configurations of all the fingers are the same
except for the 5th frames. The X5 -axes are outside andparallel to the long sides of the 5th frames. The Z5 axes are opposite each other in the cases of i = 0, 2
and i = 1, 3 in order for the Z5 -axes to be outside andnormal to the urethane rubbers. L1 = 176 [mm] is the
distance between the fingers 0 and 1 and L2 min = 48
and L2 max = 158 [mm] are the minimum and maximum
distances between the fingers 0 and 3. Figure 6 shows theframe configuration of the fingers. The link parameters are
Z12 = 62, X34 = 61.5, Y34 = 24 (i = 0, 2), −24 (i = 1, 3),
X45 = 220, Y45 = −24 (i = 0, 2), 24 (i = 1, 3) and
Z45 = −13.9 [mm]. The links depicted by the dottedlines represent the cases of the finger 1 and 3. The joints
are represented by qi1 , qi5 [mm] and qi2 –qi4 [rad]. The
positions of the centers of mass of the 3rd–5th links are(X3c , Y3c ) = (22, 10.5) (i = 0, 2), (22, −10.5) (i = 1, 3),(X4c , Y4c ) = (29.7, 5.9) (i = 0, 2), (29.7, −5.9) (i = 1, 3)
and X5c = −124 [mm] expressed in the 3rd–5th frames.
The mass of the 3rd–5th links are m3 = 0.408, m4 =0.073 and m5 = 0.081 [kg]. The joint limitations are
0 ≤ qi5 ≤ 110 [mm] (i = 0, 1), −26 ≤ qi2 ≤ 90 (i = 0, 2),−90 ≤ qi2 ≤ 26 (i = 1, 3), −94 ≤ qi4 ≤ 139 (i = 0, 2),−139 ≤ qi4 ≤ 94 (i = 1, 3) [deg] and −54 ≤ qi5 ≤ 0[mm]. Furthermore, the sum of qi3 and qi4 is bounded as|qi3 + qi4 | ≤ π2 .
2.2 Kinematic and Force Analyses
In the latter, the script i is dropped for notation simplicity
if it is not necessary. The forward kinematics is firstly
shown. The position of the 5th frame with respect to thebase frame Σ0 is given by
px =q̄5 c2 s34 +Y45 c2 c34 +Z45 s2 + c2 (X34 s3 +Y34 c3 )py =q̄5 c34 − Y45 s34 + X34 c3 − Y34 s3 + Z12(1)
pz =q̄5 s2 s34 + Y45 s2 c34 − Z45 c2 + s2 (X34 s3 +Y34 c3 ) + q1 ,where q̄5 := X45 + q5 , s2 := sin q2 , c2 := cos q2 , s34 :=sin(q3 + q4 ) and c34 := cos(q3 + q4 ).
The inverse kinematics is secondly analyzed. The contact
points and the corresponding normal vectors on the object
are assumed to be given. Since the contact surfaces of thefingers are soft and flat, it necessary to satisfy the following
constraints:
X5T n = 0, Z5T n ≤ cos θ̄
(2)
X0
Y0

Z0

X1

X0

Z1

Z0
Y0

Y1

Z1
X1

Y1

q1
Z3

Y3

q2
Z2

X3
Z4

X2

q3

Y2

q4

Y4

X2
X 34

X 45

− q5

Z2
Y2

Z4
Y4

X4
Z 45

Y3

Z 12

Y 45

Y4

X4

Y5

Z5 Z5

X5

Y34

Fig. 6. The frame configuration of the finger.

Z3
X3

Z4
X4

Y5
X5

θ

n

θ

Z5

Y5

X5

Z5
X5

Fig. 7. The finger joint constraints.
where

"

#
"
#
"
#
c2 s34c2 c34c2 c34
X5 := c34 , Z5 := − −s34 (i = 0, 2), −s34 (i = 1, 3).
s2 s34s2 c34s2 c34
X5 , Z5 ∈ R3 are the vectors in the X5 , Z5 -axes and n ∈ R3
is the normal vector at the contact point. The constraints(2) is illustrated in Fig. 7. The first of (2) denotes that thelong side of the urethane rubber is normal to the surface.
The second of (2) denotes that the angle between the
narrow side and the surface is allowed within θ̄ because
the urethane can be deformed and adapted on the surface
despite the difference θ̄. The joint ranges to satisfy (2) isgiven by the following theorem:
Theorem 1. Suppose |q3 + q4 | ≤ π2 , |θ̄| < π2 and |ny | ≤cos θ̄. Then, the constraints of (2) are solved as½(π − φ34 )sgn(φ34 ) (i = 0, 2)
q3 + q4 = Q34 (q2 ), Q34 :=−φ34(i = 1, 3)
(

(3)
−1

−π + sin
−1

sin
where

−1

l2 − φ2 ≤ q2 ≤ − sin−1l2 − φ2 ≤ q2 ≤ π − sinφ2 := atan2(nx , nnz ), l2 :=l2 − φ2 (i = 0, 2)

l2 − φ2(i = 1, 3)(4)

q(cos2 θ̄ − n2y )/(n2x + n2z )φ34 (q2 ) := atan2(ny , nx cos q2 + nz sin q2 ).
Proof . The first of (2) is calculated asq
X5T n = (nx cos q2 +nz sin q2 )2 +n2y sin(q3 +q4 +φ34 ) = 0.
(5)
Eq. (5) leads to
q3 + q4 = mπ − φ34 (m = 0, ±1, ±2, · · · )(6)The second of (2) is calculated by (6) and cos θ̄ > 0 asq
Z5T n = (nx cos q2 + nz sin q2 )2 + n2y ≥ cos θ̄ (7)½
cos(q3 + q4 + φ34 ) = −1 (i = 0, 2).
(8)
cos(q3 + q4 + φ34 ) = 1 (i = 1, 3)
Considering (8) and |q3 + q4 | ≤ π2 yields to m = 1 (φ34 >0), m = −1 (φ34 < 0) (i = 0, 2) and m = 0 (i = 1, 3). This
result and (6) lead to (3). (7) is rewritten using |ny | ≤ cos θ̄as¡¢2[nx nz ][c2 s2 ]T ≥ cos2 θ̄ − n2y ≥ 0(9)Note that [c2 s2 ]T represents the direction of X2 -axisof the 2nd frame in (X, Y )-plane. Since the X2 -axis is
opposite to (i = 0, 2) or same as (i = 1, 3) the X5 -axisof the 5th frame in (X, Y )-plane from the definition of
the frame configuration (See Fig. 6), the inner productbetween [nx nz ]T and [c2 s2 ]T is negative (i = 0, 2) or
positive (i = 1, 3). Therefore, (9) is considered as

671

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Mechatronics'10
Cambridge, MA, USA, September 13-15, 2010Csin(q2 + φ2 ) ≤ −l2 (i = 0, 2), sin(q2 + φ2 ) ≥ l2 (i = 1, 3).
(10)(4) can be easily obtained by solving (10).
The joint angles for the origin the 5th frame to contact ona desired point on the object is given by the theorem:
Theorem 2. Suppose that q1 is a constant of q1∗ , a desired
contact point (px , py , pz ) is given and Theorem 1. Then,the solutions q2∗ –q5∗ of (1) are given by
!
Ã
Z
45
+ ψ2∗
(11)
q2∗ = cos−1 p
p2x + (pz − q1∗ )2
Ã
!
l3∗
∗
−1
p
q3 = cos+ ψ3∗(X34 + Y34 t∗34 )2 + (Y34 − X34 t∗34 )2
(12)
∗
∗
∗
q4 = Q34 (q2 ) − q3(13)q5∗ = (py + Y45 s∗34 − X34 c∗3 + Y34 s∗3 − Z12 )/c∗34 −X45 (14)where t34 := tan(q3 + q4 ) andψ2∗ := atan2(px , q1∗ − pz )l3∗ := px c∗2 + (pz − q1∗ )s∗2 − Y45 c34 − t∗34 (py + Y45 s∗34 − Z12 )ψ3∗ := atan2(X34 + Y34 t∗34 , Y34 − X34 t∗34 ).
Proof . Eqs. (1) is rewritten in the following matrix form:
·
¸ ·
¸·
¸
px
c2 s2 Y45 c34 + X45 s34 + X34 s3 + Y34 c3=
pz − q1∗
s2 −c2
Z45
·
¸ ·
¸·
¸
Y45 c34 + X45 s34 + X34 s3 + Y34 c3
c2 s2
px
⇔
=
Z45
s2 −c2 pz − q1∗(15)
The second equation of (15) is transformed toZ45
cos(q2 − ψ2 ) = p.
(16)2px + (pz − q1∗ )2(16) leads to (11). Substituting q2∗ into the first equation of(15) and (1) and combining the resultant equations withrespect to (X45 + q5 ) yield(X34 + Y34 t∗34 )s3 + (Y34 − X34 t∗34 )c3 = l3∗ .
(17)
∗
∗
∗
(17) leads to (12). (13) is given by q2 , q3 , Q34 (q2 ) of (3).
(14) can be easily obtained by q2∗ , q3∗ , Q34 (q2∗ ) and (1).
Example 1. Figure 8 shows an example of the grasp of
a door mirror. The figure (a) is the discrete data on the
mirror plotted by the blue dots. The green circles and thered arrows represent the contact points and the normal
vectors. The mirror frame Σm is attached to the center ofmass. The contact points and the normal vectors in figure(a) are expressed in Σm as
#
"
#
"
#
"
#
"
59
−69
−67
69
pA = −21 , pB = −21 , pC = −21 , pD = −21
−45
−33
26
27
#
#
"
#
"
#
"
"
−0.83
0.92
0.95
−0.94
nA = 0.19 , nB = 0.28 , nC = 0.16 , nD = 0.04 .
0.55
0.37
0.17
0.29
The figure (b) shows the configuration of the hand and themirror. The axes of (Xh , Yh , Zh ) coincides with the axes of(Zm , Ym , Xm ). Figure 9 shows the reachable areas of thefingers in the case of q11 = q21 = 0. The black areas are thereachable areas in the (Xm , Zm )-plane ((Xh , Zh )-plane).
The red circle is the origin of Σh , the green circles is thecontact positions and the position of the center of mass is(5, −35) [mm]. It is confirmed that all the contact points

B

Ym

C

D

Σm

Σm

C
B

D
A

Ym

Zm

Zm

Σm

Zm

A

B

Xm

D

Xm

A

Finger 2

Xm

Finger 1

Ym
Yh

Σh

Zh

Xh

Finger 3(a) The contact points and normal of the mirror

Finger 0(b) Configuration of the finger and mirror

Fig. 8. An example of the grasp of a door mirror.
Finger 2: The normal of the finger
coincides with the normal vector

3rd joint

Finger 1

Σh

Yh
Xh

Zh

Finger 3

The center of massof the mirror
Finger 0

Orbit of4th joint

Reachable area of the 5th link

Fig. 9. The reachable area of the fingers.
are included in the reachable area. The aqua lines meanthat the normals of the finger coincide with the contact
normal. It is expected that the hand can easier grasp the
mirror by adjusting the relative configuration between thehand and the mirror in order for the contact points to beclose to the aqua lines. The red area in the left figure is
the reachable area of the 5th link. The distance of thecontact points from the hand in the Yh -axis should be
middle within the red area.
The static force is finally analyzed. The joint torque isdefined as τ := [τ2 τ3 τ4 τ5 ]T ∈ R4 . The grasping forcef ∈ R3 is expressed as f = f nf , where f is the magnitudeand nf ∈ R3 is the normalized direction. The static
relation between τ and f is given byτ = J (q)T f ,(18)
where J ∈ R3×5 is the jacobian matrix defined as J :=(∂p/∂q)T with p := [px py pz ]T and (1). Let us consider
the maximum of the grasping force fmax when nf is given.
fmax is given by the following theorem:
Theorem 3. Suppose that the direction of the force nf
and the maximum values of the torque τi max are given.
Then, fmax is solved as
fmax = min |τimax /ui | ,(19)
iwhere ui is the component of
u := J T nf
(20)
Proof . Combining f = f nf , (18) and (20) leads toτi = ui f.
(21)
Substituting (21) into |τi | ≤ τi max results in|f | ≤ |τimax /ui | .
(22)
Therefore, fmax is given by the minimum of |τi max /ui |.

672

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Hand famepm, xzObject position

Vectorsv
in the directions kof the edges

Prismatic joints

q s = [q11 q12 ]T

Lengths ofthe edges

f max, kγDistancesfrom boundplanesfi

Contact
normalObjectconfig.

pm , Rm(a) Friction cone

Graspingfforce(c) Constraint polyhedron

Fig. 11. A friction cone and its approximation.

Objectframe
Contact m
pc
points(b) Polygonal cone (Octagon)

where

h
i
Ae := G(p0 ) · · · G(p3 ) , be := −G(pm )mghiT
G(x) := I3 S(x)T , S(x)y := x × y.

CMContact mncnormalsFig. 10. The configuration of the grasp optimization.
Theorem 3 is necessary for the calculation of the distancesof the grasping forces from the constraint bounds of thetorque limitation.
3. GRASP OPTIMIZATION
3.1 Problem Setting
Figure 10 illustrates the variables and parameters in thegrasp optimization. pm ∈ R3 and Rm ∈ R3×3 are theposition vector and the rotation matrix of the object frameΣm . The contact point expressed in Σh is related to theone in Σm as
pci = pm + Rm m pci ,(23)
3mwhere pci ∈ R is the ith contact point expressed in Σm .
In the latter, position vectors without any left superscripts
are expressed inΣh . Suppose that the contact normalsmnc ∈ R12 , the y-position and orientation of the object,i.e., pm,y and Rm are given. are given. The grasp is
optimized with respect to the grasping forces f ∈ R12 ,the object position in the (Xh , Zh )-plane pm,xz ∈ R2 , theprismatic joints qs := [q11 q21 ]T and the contact pointsmpc ∈ R12 . The force and position vectors are expressedin the hand frame Σh . The optimization is formulated asmax
V (f , pm,xz , qs , m pc ), (24)m
f ∈F ,pm,xz ∈Pm,xz ,qs ∈Qs , pc ∈Pcwhere V is an optimization criterion and F, Pm,xz , Qs and
Pc are constraint sets. In the latter, these are shown. Notethat the optimization problem (24) is considered in the3D situation although the (x, z) components of the object
position are only optimized. This is because the contact
points on the object is not restricted in 2D space.

Note that S(x) ∈ R3×3 is the skew-symmetric matrix tobe equivalent to the cross product of x.
Friction and Torque Limitations Figure 11 illustrates afriction cone. The friction cone is defined as
T
T
T
|tT
(27)
xi fi | ≤ µni fi , |tyi fi | ≤ µni fi ,where txi , tyi ∈ R3 are the tangent vectors, tT
xi tyi = 0 andµ is the friction coefficient between the finger and object.
Note that the angle γ in the figure (a) is defined as γ :=tan−1 µ. The cone is approximated to a polygonal cone totouch to the cone internally as in the figure (b), which is an
example of an octagonal cone. vk is the unit vector in the
direction of the kth edge of the cone. The force limitation
can be considered by obtaining the maximum force fmax,kin the direction of vk using Theorem 3. Therefore, the conein the figure (b) is modified to the constraint polyhedron.
Note that the lengths of the edges of vk of the polyhedron.
The constraint polyhedron of ith force is formulated as
Afi fi ≤ bfi ,(28)
where Afi ∈ R(K+NK )×3 , bfi ∈ R(K+NK ) , NK := K C3
and K is the number of the apexes the polygon of the approximated cone. The first K numbers of the inequalities
mean that the force is inside the approximated cone andthe second NK numbers of the inequalities mean that theforce satisfies the force limitation.
Example 2. Figure 12 shows an example of an approximated friction cone. The all joints are 0 and the contact
normal n = [0 0 − 1]T . The friction coefficient is µ = 2.86
Zh

Xh

Xh
Yh

n

Zh

3.2 Constraint Sets
Equilibrium of Forces The grasping forces are balancedwith the gravity force:
3
3
XX
f i + mg = 0,
pi × fi + pm × mg = 0,
(25)
i=0

Yh

Yh

i=0

where m is the mass of the object and g ∈ R3 is the gravity
vector. (25) is transformed to the linear matrix form:
Ae f = be ,
(26)

Zh

Xh

Fig. 12. An example of an approximated friction cone.

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Finger 2

Finger 1

where afl ∈ R1×2 is the lth row vector of Afi and dil isthe lth component of
dbi := bf − Af fi(35)

Ar ,i ,1 p f i ≤ br,i ,1i

Ar ,i ,2 p f i ≤ br ,i ,2ikafl k−1 is the scaling operator and dil is the distance ofthe grasping force fi from the boundary plane of the finger
i. The optimization criterion V is defined as
V := min dil .
(36)
Finger 3i,lFinger 0(b) An object area to be grasped(a) Approximation to the linear convex areas

Fig. 13. Approximation of the reachable area of the finger.
of the one between the urethane and the mirror. The
figures are plotted in the hand frame Σh . It is confirmedthat the friction cone is modified to the polyhedron.
Constrained Area of Object Position The reachable areasof the fingers are depicted by Theorem 1 with given thecontact points and normal vectors as in Fig. 9 of theexample 1. The coordinates on the bounds of the reachable
areas in the (Xh , Zh )-plane are given by (3) and (4).
Therefore, the reachable area in the (Xh , Zh )-plane of
the finger is approximated to some linear convex areas asshown in Fig. 13 (a):Ar,i,αi pci ,xz ≤ br,i,αi , αi = 1, 2, · · · ,(29)where pci ,xz ∈ R2 is the contact position in the (Xh , Zh )plane and αi is the number of the independent convex
areas. In the cases of the fingers 0 and 1, (29) is modified
asAr,i,αi (pci ,xz − [0 qi1 ]T ) ≤ br,i,αi , αi = 1, 2, · · · . (30)
The object area for the finger i to reach at the contact
point pci ,xz is easily calculated by (23) as
Ar,i,αi (pm,xz + pmci ,xz ) ≤ br,i,αi⇔Ar,i,αi pm,xz ≤ br,i,αi − Ar,i,αi pmci ,xz ,(31)
where pmci ,xz ∈ R2 is the x and z components of Rm m pci .
(30) is also transformed to the similar form. Note that (29)and (30) are moved in parallel to areas around the centerof mass as in the figure (b). Considering the product sets
between (29) and (30) with respect to the number i as
illustrated in the figure (b), we obtain the object areas for
all the fingers to reach at the contact points:
Am,β x ≤ bm,β , β = 1, 2, · · · ,
(32)
where
£
¤
T T
x := pT∈ R4m,xz qs
and β is the number of the product sets of (31).
Searching Areas of Contact Points Suppose the discrete
contact points on the object are given. The searching areaof the ith contact point m pci is the circle with the originm
pci :
Pci = {p | p ∈ C, kp − pci k ≤ rci },(33)
where C is the set of the points on the object.
3.3 Optimization Criterion
The distances of the grasping force fi from the constraint
polyhedron are considered as the optimization criterion asshown in Fig. 11 (c). The distances are calculated asdil := kaf k−1 dbil ,(34)lSuppose that the optimized value of V is dmin . Let us
consider the allowable disturbance force from dmin . From(26), the force equilibrium with the disturbance fd is givenbyAe (f + δf ) = be − G(pm )fd ,(37)
where δf ∈ R12 denotes the grasping forces to balance to
fd . Substituting (26) into (37) leads to
Ae δf = −G(pm )fd .
(38)
Solving (38) with respect to fd results in
fd = −G(pm )+ Ae δf ,(39)+
where G is the pseudo matrix of G. With the singularvalue decomposition of G(pm )+ Ae (39) is rewritten as£¤
fd = −U Σ 03×9 V T δf ,(40)
where U ∈ R3×3 , V ∈ R12×12 and
"
#
σ1
0
σ2
Σ=
.
0
σ3
Since δf represents the allowable forces, δf can be expressed as
δf = dmin e, kek = 1,
(41)
where e ∈ R12 is an arbitrary vector. Substituting (41)
into (40), we get"#dmin σ10dmin σ2
fd = −U
ev1 ,
(42)
0
dmin σ3
3
9
T T
Twhere [eTv1 ev2 ] := V e, ev1 ∈ R and ev2 ∈ R . Therefore, fd is represented as the ellipsoid with the main axes of
the colums in U and the sizes of (dmin σ1 , dmin σ2 , dmin σ3 )in the major axes. The allowable magnitude of fd is givenbyξ := min(dmin σ1 , dmin σ2 , dmin σ3 ).
(43)

3.4 Optimization Problem
The problem (24) is considered by the following two
separated problems:max
max V (f , pm,xz , qs ; m pc )(44)
pm,xz ∈Pm ,x∈X f ∈Fm

max max V (f , pc ; pm,xz , qs ).

m p ∈P f ∈F
c

(45)

These problems are solved by turns and iteratively.
Example 3. The initial contact points and normal vectors are
#
#
"
#
"
#
"
"
53
−52
−72
76
pA = −31 , pB = −37 , pC = −28 , pD = −24
−49
−49
27
27
#
"
#
"
#
"
#
"
−0.75
0.44
0.99
−0.99
nA = 0.02 , nB = 0.06 , nC = 0.11 , nD = 0.01 .
0.66
0.89
0.01
0.12
The configuration of the contact points are similar to
Fig. 8. The initial 1st joints of the fingers 0, 1 are

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Mechatronics'10
Cambridge, MA, USA, September 13-15, 2010

Grasping forces
Contact normals

Initial Grasp Point

Finger 2

Object Area

Xh

Robot arm

Finger 1

Hand frame

Finger 1

Finger 2

StereocameraXh
Finger 3

Finger 0

Zh

Hand

Finger 3

Finger 0

MirrorMirror frame

Reference frameZhFig. 14. The optimization result of the grasp of the mirror.
T

qs = [0 0] [mm]. The initial object position is pm,xz =[10 63]T [mm]. The grasping forces are optimized by the
initial values as
"
#
"
#
"
#
"
#
−5.6
10
1
−5.6
f0 = −3.7 , f1 = −11 , f2 = 0.80 , f3 = −8.9
−2.1
−8.9
6.7
−4.3
and the allowable disturbance is ξ = 7.1 [N]. The optimalvalues of the 1st joints and the object position are qs =[36 13] and pm,xz = [−2.8 76] [mm]. The values of thegrasping forces are
"
#
"
#
"
#
"
#
−20
15
13
−7.4
f0 = −8.1 , f1 = −6.6 , f2 = −5.4 , f3 = −2.6 .
−13
−8.5
11
−10.4
It is confirmed that the internal forces become stronger
because the magnitudes of the z components of the grasping forces. The optimization result is shown in Fig. 14.
The left figure shows the initial and optimal contact points
and object position and the object area. As for the object
position, the black cross of the initial point moves tothe purple cross of the inner optimal point in the blue
object area. As for the contact points, the black circlesof the initial points of the finger 0 and 1 moved to thegreen circles of the optimal points. It is confirmed thatthe symmetry of the configuration of the optimal contact
points is better than that of the initial contact points.
The right figure illustrates the grasping forces and friction
cones. The red lines of the grasping forces are close to theblue lines of the contact normals. The allowable magnitudeof the disturbance is optimized from ξ = 7.1 to ξ = 22 [N].
Furthermore, dmin σ1 , dmin σ2 and dmin σ3 are almost same.
This means that the ellipsoid of fd is approximated to thesphere with the radius ξ by the optimization.
4. CONTROL METHOD
The grasping forces are realized by a position-based controller (e.g., See Murakami et al. (1993)). In view of fault
tolerance in product lines, the forces should be controlled
without any force sensors. Then, a control method is shownbriefly, which consists of a force estimation and a force
controller.
The static relation of (18) with the effect of the joint
friction is given byτ ± τf = J T f ,(46)

Fig. 15. Overview of the experimental environment.
where τf ∈ R4 is the joint friction. Solving (46) yields
f = (J T )+ (τ + τf ),
T +

(47)

3×4

is the pseudo inverse matirx of J T .
where (J ) ∈ R
τfi is defined asτfi ∈ Υi , Υi := {τ | − τfmi ≤ τ ≤ τfpi },(48)
where τfmi and τfpi are the maximum friction in thepositive and negative directions of the joints. Note that
it is impossible to determine the frictions generated at
the joints because the static frictions of the joints are
hyperstatic. Therefore, we use the mean value of (47) as
the estimated grasping force.
The force controller consists of the following two parts:Zτ = −Kp (q − qd )−Kv (q̇− q̇d )−Ki (q − qd )dt (49)qd := g(pd ), pd := p0 + ∆pd
Z∆pd := −Kfp (f − fd ) − Kfi (f − fd )dt,

(50)

where qd ∈ R4 and fd ∈ R3 are the references of thejoint angles and grasping forces and p0 ∈ R3 is the fingertip position when the fingers contact on the object. Notethat the subscript i is dropped here for simple notation.
K∗ is the positive definite matrix of the control gain.
q : R3 → R4 represents the inverse kinematics with theconstraint q3 + q4 = Q34 (q20 ), where q20 is the 2nd joint
angle in initial contact situation. ∆pd ∈ R3 is the reference
variation of pd due to the force error.
5. EXPERIMENT
Figure 15 shows the overview of the experimental environment. The hand is attached to the robot arm (VS-6556G,
DENSO Co. Ltd.). The reference frame Σw is set to thebase of the arm. The position and orientation of the handwith respect to Σw , (w ph , Rwh ), are related to (h pm , Rhm )with respect to Σh byw
pm = w ph + Rwh h pm , Rwm = Rmh Rhm ,(51)whwhere ( p, Rwm ) are preliminarly set and ( pm , Rhm )are the optimized values. The arm is used for adjusting(w ph , Rwh ) to the values calculated by (51). The mirror is
supported by a seat in order to be similar orientation as
in Fig. 8. The orientation of the hand is also moved overthe mirror in order to be similar orientation as in Fig. 8.
The position and orientation is calculated by the feature

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Mechatronics'10
Cambridge, MA, USA, September 13-15, 2010(a) Initial state(b) Approach(c) Grasp(d) Pick up(e) Rotated(f) Banked

Fig. 16. Snapshots of the experiment of the grasp of the mirror.
points on the mirror detected by the stereo camera. The
optimal values are same as in Example 3.
Figure 16 shows snapshots of the experiment of the graspof the mirror. The hand is at the initial position in the
figure (a) approaches the mirror by the arm in the figure(b). The fingers contact on and grasp the mirror in the
figure (c). After 10 [s] for the converge of the controlled
grasping forces, the mirror is picked up by the handand arm in the figure (d). The grasped mirror is rotatedand banked as in the figure (e) and (f), which show theeffectiveness of the proposed grasp optimization.
6. CONCLUSIONSThis paper proposed the synthesis methodology of stablegrasp by the four-fingered robot hand in order to automate
the preparation for assembling work. The optimization
method of the grasping forces, the object configurationand the contact points was proposed. The criterion of
the optimization was the appropriate normalized distancesfrom the planes of the constraint polyhedron, which isderived by the friction and joint torque limitations. The
experimental result of the grasp of the door mirror was
shown to verify the effectiveness the proposed method.
In our optimization method, the y coordinate of the object
position and the object orientation have not been considered. Integrating these degrees of freedom can improve theoptimization method. Furthermore, it is necessary to verify
the grasp of another variety shaped object.
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