2014 IEEE/RSJ International Conference onIntelligent Robots and Systems (IROS 2014)
September 14-18, 2014, Chicago, IL, USA

Grasp Planning for Constricted Parts of Objects
Approximated with Quadric Surfaces
Tokuo Tsuji, Soichiro Uto, Kensuke Harada, Ryo Kurazume, Tsutomu Hasegawa and Ken’ichi Morooka
Abstract— This paper presents a grasp planner which allows
a robot to grasp the constricted parts of objects in our daily life.
Even though constricted parts can be grasped more firmly thanconvex parts, previous planners have not sufficiently focusedon grasping this part. We develop techniques for quadricsurface approximation, grasp posture generation, and stability
evaluation for grasping constricted parts. By modeling an
object into multiple quadric surfaces, the planner generates
a grasping posture by selecting one-sheet hyperbolic surfaces
or two adjacent ellipsoids as constricted parts. When a grasping
posture being generated, the grasp stability is evaluated basedon the distribution of the stress applied to an object by thefingers. We perform several simulations and experiments toverify the effectiveness of our proposed method.

I. I NTRODUCTION
Service robots are expected to pick up daily life objects
and deliver them to persons. For robots to execute this task,a stable grasping posture of an object has to be planned
to prevent the object from dropping off. To hold an object
firmly, a human often places his/her fingers on a constricted
part of the object. However, previous robot grasp planners
have not sufficiently focused on the surfaces of an object.
On the other hand, we propose a grasp planner which allows
a robot to grasp a constricted part of an object. By using
a parallel gripper whose fingers are covered with a soft
material, we will show that our method is very effective ingrasping an everyday object firmly.
Our proposed method first models an object using several
quadric surfaces [1][2]. As shown in Fig.1, we can find two
definitions of a constricted part of an object: one is definedas the constriction at the center of a one-sheet hyperbolic surface, and the other is defined as the constriction constitutedby two adjacent quadric surfaces. Since a constricted partcan be found by checking the quadric parameter of surfaces,
our proposed grasp planner can realize a grasping posture
where the fingers pinch the constricted part of an object.
The proposed grasp planner checks the grasp stability
based on the six dimensional wrench applied to an object
This work was supported by JSPS KAKENHI Grant Number 24700194.
T. Tsuji, R. Kurazume and K. Morooka are with Faculty of Information Science and Electrical Engineering, Kyushu University, 744 Motooka, Nishi-Ku, Fukuoka, 819-0395, Japan. tsuji, kurazume,morooka@irvs.kyushu-u.ac.jp
S. Uto is a student of Information Science and Electrical Engineering,
Kyushu University, 744 Motooka, Nishi-Ku, Fukuoka, 819-0395, Japan.

uto@irvs.ait.kyushu-u.ac.jp
K. Harada is a research group leader of National Institute of Advanced
Industrial Science and Technology, 1-1-1 Umezono, Tsukiba-Shi, Ibaragi,
305-8568, Japan. kensuke.harada@aist.go.jp
T. Hasegawa is a principal of Kumamoto National CollegeofTechnology,
Koshi-Shi,
Kumamoto,
861-1102,
Japan.

hasegawa@kumamoto-nct.ac.jp
978-1-4799-6933-3/14/$31.00 ©2014 IEEE

Fig. 1.

Grasping constrictions.

by considering the shape information of a quadric surface
around the contact point. For evaluating the grasp stabilityof two-fingered pinching, the frictional torque about thecontact normal plays an important factor. So far, the stress
distribution, which differs depending on the shapes of thecontact surfaces, has not been reflected in the frictional
torque when evaluating the grasp stability. On the other hand,
our method estimates the stress distribution on the basisof the quadric surface approximation. Hence, the frictional
torque changes depending on the shape of the surface aroundthe contact point. In our proposed method, the grasp stability
index is evaluated by using the frictional torque calculated
from the stress distributions. We show that a grasping postureis very stable if fingers pinch a constricted part of an object.
We verify the effectiveness of grasping a constricted partusing simulations and experiments with a robot. Several
everyday objects and toys are tested in the simulations. We
also show experimental results which show the difference ofgrasp between a convex part and a constricted part of an
object.
This paper is organized as follows: In the section 2,
related works are listed. The quadric surface approximation
method is explained briefly in the section 3. The method forgrasp posture generation is explained in the section 4. The
proposed grasp stability index is explained in the section 5.
This paper is concluded in the chapter 6.
II. R ELATED WORKS
Generally, grasp planning explores feasible grasp posturesin the configuration space of a robot hand. To efficiently finda good quality grasp, object shape approximation methods
using shape primitives have been proposed. Miller [3] proposed a method that uses spheres, cylinders, cones and boxes
as shape primitives. Yamanobe et al. [4] proposed a methodthat uses various primitive shapes such as spheres, cylinders,cones, boxes, tubular boxes, and tubular cylinders. Harada etal. [5] proposed a method where objects are approximated2447

using a plane. These primitives are difficult to represent a
constriction. Goldfeder et al. [6] proposed a method thatapproximates an object using multiple super quadric surfaces.
In general, a super quadric surface can express a wider variety of shapes than a quadric surface. However, this method
uses closed super ellipsoids and does not use opened super
quadric surfaces. For example, an unbounded hyperboloid
cannot be expressed as a closed surface. Therefore a bounded
approximated surface is suitable for representing various
shapes.
We divide an object into several bounded quadric surfaces[2]. These surfaces can express concave surfaces including
constrictions. In addition, a bounded quadric surface fits local
features more suitably than a closed quadric surface.
As a technique for evaluating grasps, several methods
are proposed based on force closure [7]-[9]. Force closure
is a concept that was originally introduced in kinematics[7] and was later introduced in the field of robot hands[9]. Ferrari and Canny [11] proposed an index for grasp
stability evaluation using a grasp wrench space. Nguyen[10] discussed a force closure constitution method for robot
hands.
Some researchers have already considered contact stability
problem depending on the curvature by assuming the contact
between a rigid body and rigid fingers. Montana [12][13]proposed a contact stability evaluation method that considers
the curvatures of both a hand and an object. Rimon andBurdick [14] proposed a mobility index that considered thecurvatures at contact points. Funahashi [15] also analyzed
grasp stability considering the curvatures at contact points.
These methods can only apply to the point contact between
a finger and an object. Also, these methods do not consider
the frictional torque of a surface contact model. Ciocarlie
et al. [16] proposed a force closure index that considers thefrictional torque for surface contact between objects and soft
fingers. Here, they just consider the case where the contact
area has an ellipsoidal shape. As far as we know, there has
been no research on grasp planning to grasp a constricted
part of an object. Also, there has been no research on grasp
stability which can evaluate the difference in the frictional
torque between a constricted shape and a convex shape.

III. Q UADRIC SURFACE APPROXIMATIONIn this section, techniques of quadric surface approximation of objects are described. Wu [20] proposed surface
recovery using quadric elements. Yan [21] proposed quadricsegmentation method considering surface shape and normal.
Our method [1][2] divides an object with bounded surfaces
which is approximated by quadric surfaces with consideration of the approximation error of the shapes. We assume
that the polygon model is given for each object. The quadricsurface equation is classified into types such as an ellipsoid, acylinder, and a hyperboloid. The detail of our approximation
is described in [2].

Fig. 2. Procedures of face clustering: (a)Total procedures for a cup, (b)Twomerging steps for a duck.

A. Least square fitting of quadric surface
We describe a method for delivering a quadric surface equation which approximates polygon meshes. Letf (x, y, z) = 0 be an implicit function of a quadric surface
defined as:
f (x, y, z)
where
a
p

= a·p

(1)

= [a0 a1 a2 a3 a4 a5 a6 a7 a8 a9 ]T= [x2 y 2 z 2 xy yz zx x y z 1]TA least square fitting method acquires the parameter ato minimize the distance between the quadric surface
f (x, y, z) = 0 and original meshes. The constraint matrix
proposed by Taubin [17] is used for this fitting.
B. Face clustering using a quadric approximation
Garland et al. [19] proposed a method for decomposing asurface into connected triangles called “face clusters.” Thesame clustering method is used for dividing an object into
quadric surfaces. Examples of the process are shown in Fig.2.
In the initial state, each cluster has a triangle and all of
the triangles for the object are included in distinct clusters.
Then, an adjacent pair of clusters is regarded as a cluster
and approximated using a quadric surface. The pair withthe least approximation error in all pairs is merged andreplaced with a new cluster. By repeating this procedure
iteratively, clusters grow. If the approximation error exceeds
a threshold, the merging procedure stops. Then clusters with
an approximation error less than the threshold are generated.
The threshold is set so that the average distance between
clusters and quadric surfaces is less than 10 [mm]. The
generated clusters are used for grasp postures generation. In
addition, our planner continues merging process as shown in
Fig.2(b) and generated clusters with the larger approximation
error are also used for grasp postures generation. Then ourplanner generates various grasp postures for a object.
C. Classification of type of quadric surfaces
A quadric surface is defined as the general form of an
implicit equation:

2448

qx x 2 + qy y 2 + qz z 2 = 1(2)

TABLE I
C LASSIFICATION OF Q UADRIC SURFACES
type
Ellipsoid
Elliptic cylinder

One-Sheet Hyperboloid

condition
qx > 0, qy > 0, qz > 0
qx > 0, qy > 0, qz  0
qx > 0, qy  0, qz > 0
qx  0, qy > 0, qz > 0
qx > 0, qy > 0, qz < 0qx > 0, qy < 0, qz > 0
qx < 0, qy > 0, qz > 0

Fig. 4. Postures for grasping (a) ellipsoid, (b) elliptic cylinder and (c)one-sheet hyperboloid.

Fig. 3. Exclusion cases of grasp postures because of undesirable collisions.
The red lines indicate target surfaces.

Table.I lists the relation of the surface types and the conditions of qx , qy and qz . Ellipsoids, cylinders and one-sheet
hyperboloids are selected from the quadric surfaces.
IV. G ENERATION OF GRASP POSTURES
This section describes a process for generating a grasp
posture for a soft-finger parallel gripper. To show the effectiveness of our method, grasp postures for convex shapes
are also generated[2]. Several posture candidates for an
ellipsoid, an elliptic cylinder, a one-sheet hyperboloid, and aconstriction constituted by quadric surfaces are generated. In
this process, the postures which have an undesirable collision
between a finger and a quadric surface that is not a target
surface are excluded as shown in Fig.3.
A. Ellipsoid
Quadric surfaces whose parameters qx , qy and qz are
positive are selected from the quadric surfaces. For theselected ellipsoids, three principal axes r1 , r2 , and r3 are
calculated. The six directions r1 , r2 , r3 , −r1 , −r2 , and −r3
are used as the hand-approaching direction and the gripperclosing direction as shown in Fig.4(a). The approachingdirection can vary in six directions. The gripper closing
direction for each approaching direction can vary in four
directions, which do not include the approaching direction
and the negative one. Thus, the ellipsoid has 24 candidates
for grasp postures.
B. Elliptic cylinder
Elliptic cylinders are selected by checking quadric surfaces
parameters qx , qy , and qz . For the elliptic cylinders, two
principal axes r1 , and r2 of the ellipses of the end faces
are calculated. The approaching direction can vary in four
directions and the gripper closing direction can vary in two
directions. As shown in Fig.4(b), the upper, middle, and thelower points of the elliptic cylinder are set as grasping points.
Let l be the length between the end faces of the cylinder. The

Fig. 5.

Common cutting plane.

upper and lower points are set at positions that are moved+l/4 and −l/4 from the middle point, respectively. Thus,
the cylinder has 24 candidates for grasping postures.
C. One-sheet hyperboloid
A “constriction” is defined as an area that is thinner than
the neighboring areas in a certain direction. A one-sheet
hyperboloid is selected by checking parameters qx , qy , andqz . Vectors r1 and r2 are defined as the two principal axesof an ellipse with the smallest sectional area. The one-sheet
hyperboloid has eight candidates. Fig.4(c) shows an example
that a hand approaches along r2 and the gripper closes alongr1 .
D. Constriction between quadric surfaces
Previous grasp planners have generated grasp postures
for only one primitive. We generate grasp postures forconstrictions constituted by two adjacent quadric surfaces.
The boundary of two adjacent clusters is used for a grasp
posture parameter. The generation process has four steps asfollows:(1)
Derive the common cutting plane.
(2)
Check whether the boundary shape is an ellipse.
(3)
Check whether the boundary is a constriction.
(4)
Generate grasping postures.
(1) Common cutting plane: The plane that approximates
the boundary of two adjacent quadric surfaces is called the“Common Cutting Plane (CCP).” The derivation method for
the CCP is shown in Fig.5. The boundaries shared by twoclusters are approximated by a plane, which is called the
CCP (Fig.5(a),(b)). The coordinates of the i-th(= 1, · · · , n)boundary point is denoted as (xi , yi , zi ). A normal vector of
the plane is provided as the eigenvector ν of the smallest

2449

Fig. 7.

Fig. 6.

Check of constriction.

A. Force closure index on surface contact

eigenvalue in the covariance matrix R∈R3×3 of boundary
points. Let u be the center of the boundary points. The
equation of the common cutting pane is expressed as follows:(x, y, z)ν = uT ν(3)(2) Ellipse check: The equation of the intersection line
between the CCP and the quadric surface is calculated using(2)-(3) as follows:qx x2 + qy y 2 = 1.

Difference of the shapes of contact surfaces.

(4)

The planner checks whether the intersection line between
CCP and the quadric surface is an ellipse. If qx and qy arepositive, the line is an ellipse. Two intersection lines for thetwo quadric surfaces are checked.
(3) Constriction check: Let s1 and s2 be the areas ofthe ellipses generated in the previous step. CCP’ is generated
by moving the CCP half a finger width to the quadricsurface center in the direction ν as shown in Fig.6. The
area inside the intersection line between the CCP’ and thequadric surfaces is defined as s1 and s2 , If the conditions
s1 < s1 and s2 < s2 are satisfied, there is a constriction.
(4) Generation of grasping postures: Let r1 andr2 be the axes of the ellipse. One axis is used as theapproaching direction and the other is used as the gripper
closing direction. Since there are two quadric surfaces andeight candidates per quadric surface, 16 grasping posture
candidates are generated for a constriction.

The relation between the tangential force and the frictional
torque is expressed in the following form:
τn 2≤μ2 p2en 2

ft 2 +(5)

where ft is the magnitude of the tangential contact force, pis the magnitude of a total load, μ is the frictional coefficient,and τn is the magnitude of a frictional moment. en is referred
to as the eccentricity parameter and is calculated using thefollowing equation:en =max(τn )max(ft )(6)

en is calculated at each contact point. The stability index
is calculated using the values of en .
B. Calculation of the maximum static frictional torque
We show the calculation method for the maximum static
frictional torque max(τn ) in (6). max(τn ) is derived byintegrating the static frictional moment over the contact
surface. max(τn ) is shown in the following equations:  max(τn ) =(7)
x2 + y 2 μs(x, y)dxdyD max(ft ) =μs(x, y)dxdy(8)D

where s(x, y) is the stress distribution, which depends on
the shape of the grasped object. The integral range D isdetermined by a condition s(x, y) > 0 and the width of thefinger. As shown in (7)(8), when a high stress is generated
far from the center of the contact surface, en becomes high.
C. Quadratic approximation of stress distribution

V. E VALUATION OF GRASP STABILITY
We use a gripper whose surface is made of a soft material.
The gripper affects an object with a frictional torque, which
is a moment in the normal direction at the contact surface.
Frictional torque is an important factor for grasp stability.
When frictional torque is small, a grasped object may rotateand slip off. Ciocarlie et al. [16] proposed a force closureindex using a frictional torque for a surface contact model.
A Hertzian contact and Winkler elastic foundation are usedas the stress distribution model. The stress distribution isexpressed as shown in Fig.7(a). We use different models ofstress distribution depending on a shape of contact regions,as shown in Fig.7(b).

We use a quadratic approximation for the stress distributions, similar to Winkler’s elastic foundation. We categorize the stress distributions into the following models: aparaboloid of revolution, a parabolic cylinder and a hyperbolic paraboloid. The stress distribution models are shownin Fig.8 The stress distributions are classified depending on
the target of quadric surfaces, as listed in Table.II.
(1) Paraboloid of revolution: An ellipsoid grasp belongs
this category. The stress model of Winkler is shown in
Fig.8(a), and the stress distribution s(x, y) is shown as afollowing equation: 2  2 
y
x
−
(9)
s(x, y) = pmax 1 −
ar
br

2450

TABLE IIC LASSIFICATION OF STRESS DISTRIBUTION .
Stress distribution
Paraboloid of revolution
Parabolic cylinder
Hyperbolic paraboloid

Grasped quadric surface
Ellipsoid
Elliptic cylinder
One-sheet hyperboloid
Constriction constituted by quadric surfaces

Fig. 9.

Considering for en .

in a following equation:


1
en =
ac bc (22a2c − 3b2c ) a2c + b2c
380ac bc
bc + a2c + b2c2+ 8ac logacac + a2c + b2c
322+ bc (20ac + 3bc ) log
bcFig. 8.

Stress distribution models.

where each of ar and br is two radiuses of elliptic contact
surface and pmax is the maximum stress. en derived by (9)is shown as approximate expression [22]:8 4(ar − br )2 + π 2 ar br
en =
15π

(10)

(2) Parabolic cylinder: An elliptic cylinder grasp belongs
this category. The stress model is shown in Fig.8(b), and thestress distribution s(x, y) is shown as a following equation:s(x, y) = pmax


1−

x
ac

2 
(11)

where ac is the half region length in parallel to the generatrixof the cylinder, and bc is the half length in the other direction
on the contact surface. en is derived using (11) and is shown(12)(3) Hyperbolic paraboloid: A one-sheet hyperbolic and aconstriction constituted by two quadric surfaces belongs thiscategory. The stress model is shown in Fig.8(c)(d). When thenumber of contact regions is one as shown in Fig.8(c), thecondition is called ”Contact Condition 1 (CC1)”, and when
the number of contact surface is two as shown in Fig.8(d),the condition is called ”Contact Condition 2 (CC2)”. The
stress distribution of the hyperbolic paraboloid is shown in
the following equation:
 2
 2
y
x
+ (pmax − pmin )s(x, y) = pmin − pmax
ah
bh
(13)
where pmin is the maximum stress at the constriction. If
pmin > 0, the condition is CC1. If pmin < 0, the condition
is CC2. ah is parallel to the generatrix of the hyperboloid,and bh is perpendicular to ah on the contact surface. The
resulting en using (13) is too lengthy to be shown in thispaper.
D. Discussion
In the following, we discuss en to confirm the validityof the evaluation equation. For the en of a paraboloidof revolution, Ciocarlie et al. [16] also derived en using
Winkler’s elastic foundation. They derived the following
equation:8
a r bren =15
ar br is proportional to the area of the ellipse, and en is
proportional to the square root of the ar br . If the shape of thecontact surface is a circle, this method and our method output
the same value for en . In our method, en can be calculated
accurately in a case where the shape of the contact surface
is an ellipse. The difference is shown in Fig.9(b) where ar br
is fixed at 1 and the ratio√ between√ ar and br√is changed√as (ar , br ) = {(1, 1), ( 2, 1/ 2), (2, 1/2), ( 8, 1/ 8)}.
When the contact region is narrow as a line contact, the

2451

Bowl

Salt shaker

PET bottle 1

Duck

Tea pod

Ellipsoid

Ellipsoid5.35 / 0.0266

5.80 / 0.0338

Ellipsoid

Ellipsoids

3.94 / 0.0316

2.97 / 0.0433

Elliptic cylinder

Elliptic cylinder

2.54 / 0.0321

2.57 / 0.0323

Squeeze bobottleTokkuri

PET bottle 2

Ellipsoid

Ellipsoids

4.05 / 0.0321

0.565, 0.307 / 0.0577

Ellipsoid

Hyperboloid

4 70 / 00.0342
034
42
4.70

Cat

90 0
265 / 00.04670467
0.290,
0.265
Fig. 10.

Elliptic cylinder

Hyperboloid

4.27 / 0.0415

0.134,0.134 / 0.0444

Ellipsoid

Ellipsoids

4.70 / 0.0342

0.948 / 0.0504

Elliptic cylinder

Hyperboloid

2.75
75 / 00.0325032

3.05 / 0.0448

Ellipsoid

Ellipsoids

4 43 / 00.03390339
4.43

0.0429, 0.613 / 0.0571

Surface type

Clustering
ResultGrasp
PostureName

Area [cm2] / Stability index

Result of grasp planning.

previous method returns a small value, whereas our method
returns the frictional torque that is proportional to the lengthof line.
We next consider the relationship between the stress
distribution and en . For this consideration, we use the valueof en calculated using s(x, y) for several stress distributions.
A graph comparing the value of en where a is fixed at
0.01 m and b is changed is shown in Fig.9(a). The en ofa hyperbolic paraboloid has the highest value because thestress is high at the points that are far from the center of thecontact surface. In other words, the static frictional torqueof grasping constriction shapes is higher than that whengrasping convex shapes. Therefore, a constriction grasp ismore stable than a convex grasp.
VI. S IMULATIONS AND EXPERIMENTS
We confirm the effectiveness of our grasp planning forconstriction parts in simulation. We verify the feasibility of
our method using experiments with a robot.

A. Simulation of the grasp planning
As a robot hand, we use a model with a two-finger parallel
gripper on the tip of PA10 (7 DOF arm). The model of actual
everyday objects and toys are used in simulation. The friction
coefficient μ is set to 0.5. Results of nine objects are shownin Fig.10. In these objects, a Tokkuri (sake decanter) and a
Tea pod are Japanese style. The gripper closes its fingers to
2 mm depth from the surface of the target object.
The models with color coding applied to each of thequadric surfaces are shown at left side in Fig.10. Eachquadric surface has a different color. We only show aclustering result with an approximation error for each object. Grasping postures are also generated for clusters withthe different approximation error in the process of cluster
merging.
Two grasping postures for an object are shown in at theright side in Fig.10. The contact area and the stability index
are shown under each image of a grasping posture. If agrasping posture has contact condition 2, two contact areas

2452

R EFERENCES(a) Grasping Tokkuri

Fig. 11.

(b) Grasping PET bottle

Results of grasping a tokkuri and a pet bottle.

are shown. The type of quadric surfaces is shown on the topof each image of grasping posture. The type of “Ellipsoids”
means a constriction constituted by two ellipsoids. Hyperboloid and Ellipsoids for grasping constrictions are indicatedby red characters.
When the stability index is 0.03, the maximum frictional
torque is approximately 3[cm · N ] with 1[N] contact force.
The stability index is higher, the gripper needs less force forfixing a object.
One of the PET bottles has a side approximated by anelliptic cylinder, and the other has a side approximated bya one-sheet hyperboloid. We can confirm that the grasp ofa PET bottle constriction is more stable. The grasp stability
index of grasping a constriction is confirmed to be generally
higher in spite of the contact area is small.
Calculation time is measured using a PC (CPU: Intel(R) Core (TM) i7-2600 3.4GHz, Memory: 4.00GB). Surface
approximation costs approximately 4[s] with 10000 triangular faces. Grasp stability evaluation costs less than 0.2[ms].
Hand position optimization for fitting the fingers to the object
surface costs approximately 5[s].
B. Experiment using an actual robot
We compare grasping a constricted shape and a convex
shape. The object position is assumed to be known in advance. The planner selects a grasping posture by consideringcollisions and joint limitations.
The result of grasping a tokkuri (268 [g]) is shown in
Fig.11(a). The gripper grasps the constriction of the tokkuri
firmly where the stability index is 0.050. The gripper can notgrasp a convex part of a tokkuri because of joint limitationof the gripper. When the gripper grasps a cap of the PET
bottle (350 [g]), the bottle slips where the stability index is0.032. When an object moves in a hand, it is difficult to beplaced securely. The conditions such as friction coefficient,and relative positions of gravity center are different in theseexperiments. The tokkuri can be held firmly, regardless of asmaller contact area, a smaller friction coefficient comparedwith the ones of the PET bottle.
VII. C ONCLUSION
In this paper, we proposed a grasp planning method
for a constricted shape. The planner approximates a target
object using quadric surfaces. One-sheet hyperboloids, orconstrictions constituted by two quadric surfaces are selected,
and several grasp postures are generated. We also proposedan evaluation method for grasp stability, that reflects thestress distribution.

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