IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED JANUARY, 2016

1

Bent Sheet Grasping Stability for Sheet
Manipulation
Yoshinori FUJIHIRA1 , Student, Member, IEEE, Toshihiro NISHIMURA2
and Tetsuyou WATANABE1 , Member, IEEE

Abstract—In this study, we focused on sheet manipulation withrobotic hands. This manipulation involves grasping the sides ofthe sheet and utilizing the convex area resulting from bending
the sheet. This sheet manipulation requires the development ofa model of a bent sheet grasped with fingertips. We investigated
the relationship between the grasping force and bending of the
sheet and developed a bent sheet model. We also performedexperiments on the sheet grasping stability with a focus on theresistible force, which is defined as the maximum external force atwhich a fingertip can maintain contact when applying an external
force. The main findings and contributions are as follows: 1)
After the sheet buckles, the grasping force only increases slightly
even if the fingertip pressure is increased. 2) The range ofthe applicable grasping forces depends on the stiffness of thefingertips. Stiffer fingertips cannot provide a small grasping force
but can resist large external forces. Softer fingertips can provide
a small grasping force but cannot resist large external forces. 3)
A grasping strategy for sheet manipulation is presented that isbased on controlling the stiffness of the fingertips.

Fig. 1. Example of sheet manipulation.

Index Terms—Grasping, Dexterous Manipulation, Manipulation Planning, Soft Material Robotics

I. INTRODUCTIONHEET manipulation is widely used in factories, medicine,and daily life. Examples include bonding a sheet-shaped
film to a plate in factories, bonding a cell sheet on heart/organs
or in a petri dish, and bonding a protective sheet on smartphones. Conventionally, specialized machines are used forsheet manipulation. If a multi-fingered robotic hand could be
used, the functions of robotic hands may be expanded.
Fig.1 shows a typical example of sheet manipulation, whichis considered as a manipulation that a multi-fingered robotic
hand can deal with. For the sake of convenience, the simplest
and the most fundamental mode of grasping, i.e., with two
opposing fingertips, was considered here. The manipulation isconducted by grasping the sides of sheet. It is assumed thatS

Manuscript received: August, 31, 2015; Revised December, 1, 2015;
Accepted January, 8, 2016.
This paper was recommended for publication by Editor Han Ding uponevaluation of the Associate Editor and Reviewers’ comments. This work
was supported by Tateisi Science and Technology Foundation Program (C)2157009.
1 Y. Fujihira is with the Graduated school of Natural science and Technology, Kanazawa University, Kakuma-machi, Kanazawa, 9201192 Japanfuji3134@stu.kanazawa-u.ac.jp
2 T. Nishimura is with the College of Science and Engineering, Kanazawa University, Kakuma-machi, Kanazawa, 9201192 Japan

to.nishimura@stu.kanazawa-u.ac.jp
3 T. Watanabe is with the Faculty of Mechanical Engineering, Institute of
Science and Engineering, Kanazawa University, Kakuma-machi, Kanazawa,
9201192 Japan te-watanabe@ieee.org
Digital Object Identifier (DOI): see top of this page.

one face of the sheet has a bonding agent. The manipulation
includes bending of the sheet not only for preventing bondingof the sheet to fingertips but also for precise placement of
the sheet. Without bending, the whole area of the sheet facewould need to be positioned at the same time. Bending allows
the (initial) contact area to be small by utilizing the convex
area resulted from bending, which makes precise positioning
and bonding easy. Planning the fingertip motions also becomeseasier. In the concrete case of bonding a protective sheet toa smartphone, additional entrained air between the sheet andsmartphone should be prevented. Our final goal in this study
was to conduct such a sheet manipulation. It should be notedthat some parts of this manipulation can correspond to steps
for pinching sheet objects such as envelops, papers, and cards.
The manipulation methods can also be applied to the taskof conveying large metal sheets with mobile robots. Thus,target manipulation is considered as one of the fundamental
tasks in order to realize widely varying sheet manipulations.
We focused on manipulation without external sensors suchas a camera. Manipulation with external sensors would be
expected to be more precise. The other assumptions are thatthe target objects are not bent by its own gravity and that the
manipulation is quasi-static. These assumptions indicate theapplicable types of sheets in this paper.
The merits of the sheet manipulation proposed here comparing to the conventional method utilizing suction systems are
as follows. First, the conventional method cannot always deal
with the shape and condition of the surface. Suction requires asmooth surface and cannot be used when there is a liquid such

2IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED JANUARY, 2016as a bonding agent on the surface. However, such a case canbe handled with the proposed sheet manipulation. Second, manipulation with suction requires a specific machine structure,and only specialized tasks can be carried out. The proposed
method can be conducted with many kinds of robotic hands
such as grippers and multi-fingered hands. In order to realize
the proposed sheet manipulation (without external sensors),
methods for estimating the sheet bending state from internalsensors such as force sensors embedded in fingertips are
required. This paper presents a sheet model for an estimation
based on the Euler-Bernoulli beam theory. The sheet model
can also give the characteristics of the sheet manipulation. The
design of appropriate fingertips and grasping stability analysis
are also important issues. Softer fingertips can envelop thesides of the sheet and should easily grasp the sheet even whenbending occurs. On the other hand, stiffer fingertips can apply
large grasping forces and resist large external forces. In sum,
there may be an appropriate fingertip stiffness depending oneach step of the proposed sheet manipulation. As a first step,
we investigated the relationship between the grasping force
and the bending of a sheet in order to construct a bent sheet
model. We also performed an experiment on the sheet graspingstability to consider the maximum resistible force, which isdefined as the maximum external force at which the fingertip
can maintain contact when applying an external force. Basedon the results, we developed a design of appropriate fingertips
and a grasping strategy for the proposed sheet manipulation.
The main contributions and findings are as follows:
1) After buckling, the grasping force increases only slightly
even as the fingertip pressure is increased. A bent sheet
model representing this phenomenon is presented.
2) The range of the applicable grasping force depends on
the stiffness of the fingertips. A stiffer fingertip cannot
provide a small grasping force but can resist large
external forces. A softer fingertip can provide a small
grasping force and grasping with small forces. However,large external forces cannot be resisted.
3) A grasping strategy for sheet manipulation is presented
that is based on controlling the stiffness of the fingertips.
A. Related works
To the best of our knowledge, there has been no researchthat dealt with sheet manipulation by explicitly utilizing bending. Belts are similar objects to sheet, and a model and manipulation strategy were presented by Hirai et al. [1], [2]. Ueda etal. [3] conducted the method of acquisition of a page turning
skill. M. Moll and L. E. Kavraki [4] introduced a methodof path planning for deformable linear object (e.g. flexible
wire). T. Bretl and Z. McCarthy [5] reported mechanics andmanipulation planning of chain with a fixed base in which eachjoint is an elastic element. L. Sun et al. [6] developed a system
which can flatten garments by accurate surface analysis using
RGB-D images. Odhner [7] proposed a model of robots having
planar elastic flexure joints by using the Euler-Bernoulli beam
theory.
In the material mechanics field, different research groups
have presented buckling models. Amirbayat and Hearle [8]presented the buckling theory of flexible sheet materials. Byklum [9] derived a computational model for buckling analysisof stiffened panels. However, the relation between the grasping
force and sheet deformation is unclear.
Watanabe et al. [10], [11], [12] investigated the effects offingertip stiffness on the grasping stability by performing experiments. These studies focused on the contact between rigid
objects and soft fingers. In the sheet manipulation considered
here, the elasticity of the bent sheet needs to be considered,and its phenomena are still unclear.
II. MECHANICS OF BENT SHEETThe grasping-style compression test of the sheet was used tounderstand the mechanics of the sheet. Three types of fingertips was used to investigate the effect of the stiffness aroundthe contact area: rigid material, elastic material, and fluid
fingertips. These were respectively made from ABS plastic,
silicon, and a rubber bag filled with machine oil. Based on
the experimental results, the sheet model is presented with the
Euler model for buckling and elastica theory. The theoreticaland experimental results were compared for validation.
A. Grasping-style compression test of sheet
Fig. 2 shows the experimental setup for the grasping-stylecompression test of the sheet. The fingertip on one side was
fixed, while the fingertip on the other side could be movedby the automatic positioning stage. The configuration was thesame as the gripper with two fingertips. Initially the sheet
located between the fingertips was on the stand. By closing
the fingertips, the sheet was compressed. The sheet was always
deflected upward because of the stand. The movable fingertip
was attached to a handmade load cell fixed on an automatic
positioning stage, and the grasping force (in the pushingdirection) was measured. A camera was used to observe thedeformation of the sheet. Fig. 3 shows photos of the sheet andfingertips. The sheet was made of PET and had dimensionsof 70 mm × 100 mm × 0.41 mm, which are equivalent tothe size of a screen for a smartphone. The rigid material,elastic material, and fluid fingertips were respectively madeof ABS plastic, silicon, and a rubber bag filled with machineoil (ISO VG 100). The surface of each fingertip was covered
with nitrile rubber to ensure the same friction conditions. The
fluid fingertip was the same as the one used for graspingfragile objects in our previous studies [13]. The stiffness ofFig. 2. Schematic illustration of the experimental setup for the grasping-stylecompression test of the sheet.

FUJIHIRA et al.: BENT SHEET GRASPING STABILITY FOR SHEET MANIPULATION

Fig. 3. Photo of the sheet and fingertips used in the experiments.

the fluid fingertip can be controlled by adjusting the fluidpressure inside the rubber bag. In this experiment, the initial
fluid pressures of the fluid fingertips were set to 3 and 6 kPa.
The moveable and fixed fingertips were the same type. The
movable fingertip was pushed against the sheet at a speed of
3.0 mm/s to minimize the influence of the deforming speed.
The test was conducted three times for each fingertip.
Fig. 4 shows the time series data of the grasping force forthe fluid fingertip with an initial fluid pressure of 3 kPa toclarify the behavior. Fig. 5 shows the corresponding photos.
The fingertip surface deformed until the force reached its peak(t = 5.67 s). The deformation mode changed due to buckling,
and the grasping force suddenly greatly decreased. A small
vibration was observed. After that, the grasping force slightly
increased thanks to the stiffness resulting from the bending
deformation of the sheet. Based on the experimental results,
we derived the relationship between the sheet deformation andgrasping force.
If totally considering, success rates for grasping sheet using
rigid material, elastic material, fluid (initial pressure: 6 kPa)
and fluid (initial pressure: 3 kPa) fingertips are 54.5%, 37.5%,
100% and 100% respectively. It should be noted that no
statistically significant difference is observed between the
success rates for grasping sheets, using rigid material andelastic material. It can be seen that softer (fluid) fingertips havethe advantage for realizing grasping. Note that if taking into
account rigid material fingertips got better results than elastic
material fingertips, there might be a threshold of stiffness offingertips to increase the success rate of grasping.
Fig. 6 shows the time series data for all cases when
the direction of initial small deformation was upward. After
buckling, the rate of increase in the grasping force was very
little. We present the model of this phenomenon in the next
subsection. The ascending order of the required time to reachbuckling was as follows: rigid material, elastic material, fluid(initial pressure: 6 kPa), and fluid (initial pressure: 3 kPa)
fingertips. This corresponded to the order of stiffness. A softer
fingertip decreased the rate of increase in the grasping force.
This indicates that, if a fingertip is stiff like the rigid material, the grasping force can easily reach the buckling force justafter contact. Therefore, it is difficult to control the grasping

3

Fig. 4. Time series data of the grasping force fgr for a fluid fingertip (initialpressure: 3 kPa). The peak (indicated by the arrow) is the critical bucklingload.

Fig. 5. Photos of side views when the sheet was grasped with the fluid
fingertips (initial pressure: 3 kPa): t = 0.00 s the fingertips contact the sheet; t= 5.67 s the fingertips deform; t = 5.70 s the sheet buckles; t = 5.77 s the sheet
vibrations settle; t > 5.77 s the sheet is bent.The circles show the (derived)
endpoints of the sheet.

force just after contact. Additional small perturbations in thecontact position can make the grasping force exit the friction
cone because the sheet has a small contact area. Thus, slip
can easily occur and make it hard to keep grasping the sheet.
If the fingertips are softer like the fluid fingertip, the rate ofincrease in the grasping force is small, and the sheet aroundthe contact area is enveloped by the fingertips. Deviations in
the contact position can be absorbed. Therefore, the grasping
force can be located inside friction cone, and the grasping canbe easily maintained.
Fig. 7 shows the displacement of the endpoint of the sheet
δ in each case when the vibration by buckling settled. The
endpoint was defined as the endpoint of the sheet whenviewing from the side (see Fig. 5). The displacement of the4IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED JANUARY, 2016

B. Mechanics of bent sheet
We considered a two-dimensional model with symmetricproperties. First, we considered the critical buckling load fcr
at which sheet deformation mode changes (see Fig. 4). As thegrasping force increases, the deformation mode changes when
the load reaches a certain value (critical buckling load fcr ),and buckling occurs (see Fig. 5). By utilizing Euler’s column
formula [14], the critical buckling load fcr can be represented
by
fcr =

Fig. 6. Time series data for the grasping force fgr of each fingertip. The
ascending order for the required time to reach buckling was as follows: rigidmaterial, elastic material, fluid (initial pressure: 6 kPa), and fluid (initialpressure: 3 kPa) fingertips.

π2
EI
l02

(1)

where l0 is the length of the sheet, I is the second momentof the area of the sheet, and E is Young’s modulus of the
sheet.
Next, we focused on the bending deformation of the sheet
after buckling in order to model bending. The contact between
the fingertip and the sheet can be regarded as rotationally free,because rotations at the contact point can be observed, as seenin the photo of the experiment (see Fig. 5). We then assumed
the bent beam model shown in Fig. 9 for the sheet. Bending
moment M (x) at x can be represented by
M (x) = fgr y(x)(2)

where y(x) is the deflection of the sheet at x. The bending
moment M (x) causes the bending deformation of the sheet.
The relationship between the bending moment M (x) and thedeflection of the sheet y(x) is represented by

Fig. 7. Displacement of the endpoint of the sheet δ when the vibration frombuckling settled. The ascending order of the displacement was as follows:
rigid material, elastic material, fluid (initial pressure: 6 kPa) and fluid (initialpressure: 3 kPa) fingertips.

d2 y(x)= −M (x)(3)
dx2This equation is the beam bending equation based on the
Euler-Bernoulli beam theory [14]. From (2) and (3), we obtainEI

d2 y(x)+ fgr y(x) = 0(4)dx2
Let A be the deflection at the center of sheet. Let l be thedistance between the endpoints of the sheet. Note that, if the
sheet bends, l does not always equal l0 . From (4) and theboundary conditions at both endpoints of the sheet (y(0) =y(l) = 0), we obtain
EI

Fig. 8. Relationship between the displacement of the endpoint of the sheetδ related with the grasping force fgr and the displacement of the fingertip
δdsp .

endpoint of the sheet δ was acquired from the photos of sideviews when the sheet was grasped. It should be noted thatδ = δdsp −δdf m where δdsp is the displacement of the fingertip
and δdf m is the deformation of the fingertip at the endpoints.
Fig. 8 shows the relationship between the displacement of theendpoint of the sheet δ and the displacement of the fingertip
δdsp . Softer fingertips produced larger sheet deformations. The
required time for buckling increased for softer fingertips. This
indicates that the distance between fingertips was small forbuckling and that the distance between the extreme points ofthe sheet are small. If a small deformation of the sheet isrequired after buckling, stiffer fingertips are preferable.

π
x
(5)
l
Next, we derived the deflection at the center of the sheet A.
The deflection of the sheet is considered to mainly be causedby bending (compressive deformation in the direction along
the curve of the bent sheet is negligible). The length of thecurve for the bent sheet S is then the same as the length of
the sheet l0 , and we gety(x) = A sin

l0 = S

√

)2
dy(x)
1+
dx
=
dx
0
√
(
)2
∫ l
Aπ
π
1+
=
cos2 xdx
l
l
0
∫ l

(

(6)

FUJIHIRA et al.: BENT SHEET GRASPING STABILITY FOR SHEET MANIPULATION

5

Fig. 9. Schematic illustration of the bent sheet model.

Here the deviation of S is based on differential geometry[15]. By utilizing this equation, A can be numerically obtained.
Finally, we derived the relation between the grasping force
fgr and the displacement of the endpoint of the sheet δ
based on the potential energy. Assuming that the sheet canbe represented by simply-supported Euler’s column (see Fig.
9), from elastica theory [16], the potential energy of the sheet
U is represented by∫ ( )2
EI l0 dθU=ds − fgr δ
(7)
2 0
ds
where s denotes the coordinate along the curve of thebent sheet, and θ is the rotation of the column cross section (the deflection angle) of the bent sheet at s. The firstterm corresponds to the strain energy and the second term
corresponds to the work done by fgr . In Fig.6, the energy
dissipation was observed at buckling. Letting fdis be thecorresponding dissipative force, the potential energy of the
sheet after buckling Uaf ter buckling can be represented by

Uaf ter buckling

)2
dθ
=
ds
ds
0
(
)
∫ l0
cos θds−fgr l0 −EI2∫ l0 (

0

(
+fdis

l0 −

∫ l0

Fig. 11. Displacement of the endpoint of the sheet δ versus the deflection of
the center of the sheet A. The dashed line is the theoretical value from (6).

integral terms in (6) was integrated numerically, and (6) was
solved by being formulated as an optimization problem withconstraints with respect to A. fgr was calculated utilizing (9),A derived from (6), and fcr −fdis obtained from experimental
results. fcr − fdis was set to be the mean grasping force whenA = 0 after buckling in the experiments shown in Fig. 6. The
calculation was performed by using MATLAB⃝ (Mathworks).
Fig. 10 shows the critical buckling loads obtained experimentally and derived from (1). The theoretically derived
value was close to the experimental results. No statisticallysignificant difference is observed between the buckling loads
for the type or stiffness of the fingertips.
Fig. 11 shows the relation between the displacement of theendpoint of the sheet in the x direction δ and the deformation
at the center of the sheet A. Note that the data is the one
after buckling and in the range of bending deformation (seeFig. 4). The theoretical value derived from (6) was very close
to the experimental results. As we assumed, the length of the
sheet did not change, and the sheet deformation was mainly
dominated by bending. The deflection of sheet A is determinedby the displacement of the endpoint of the sheet δ, irrespectiveof the stiffness or deformation of fingertips. The stiffness
or deformation of the fingertips is indirectly related with A
through δ.
Fig. 12 shows the relation between the displacement of theR)cos θds

Fig. 10. Comparison of the experimental critical buckling loads fcr and thetheoretical value from (1).

(8)

0

∫l
where δ = l0 − 0 0 cos θds was used. The third term
corresponds to the work done by fdis , namely the dissipatedenergy. A perturbation method [16] was applied to obtain therelation between fgr and δ. By perturbing the potential energy
U around the buckling point (fgr = fcr and θ = 0), we get()2
1 πA(δ)fgr=1+(9)
fcr − fdis
8l0
C. Comparison between theory and experimental results
We numerically calculated the relations between the grasping force fgr , the displacement of the endpoint of the sheet in
the x direction δ, and the deflection at the center of the sheet A.
We set the Young s modulus of the sheet to E = 3 GPa. The

6IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED JANUARY, 2016

The speed of the automatic positioning stage for applying
the external force was set to 1.0 mm/s. We repeated theexperiments three times for each setting.
B. Results and discussion

Fig. 12. Displacement of the endpoints of the sheet δ versus the grasping
force fgr . The dashed line is the theoretical value from (9).

endpoint of the sheet in the x direction δ and the grasping
force fgr . Note that the data is the one after buckling andin the range of bending deformation (see Fig. 4). Even as δ
increased, fgr did not increase very much. This indicates thatthe normal way of grasping by increasing the pressure on theobject does not work well for grasping a sheet. Conventional
grasping methods (for example, see [17], [18]) may not work
for sheet grasping. In the comparison of the theoretical andexperimental results, the theoretical results were close to theexperimental results.
III. GRASPING STABILITY ANALYSISThe balancing of a large external force is one metric forevaluating the grasping stability (e.g., see [17], [18]). In oursituation of sheet grasping, the external force in the directionof the gravitational force is critical, as the effect of gravity is
maximum in this direction. The direction of the gravitational
force is orthogonal to the grasping force, and during theprocesses of picking and placing, the external force can beapplied mainly in this direction. Thus, the resistible force canbe considered to be minimum in the critical direction. We
investigated how a large external force in that direction canbe applied to the sheet while maintaining a grasp with three
types of fingertips: rigid material, elastic material, and fluid.

Fig. 14 shows the time series data for the grasping forceof the elastic fingertip and δdsp = 5 mm. We denote the force
with no external force as the initial grasping force and theforce with the maximum external force (corresponding to the
maximum balanced external force) as the critical grasping
force. The grasping force decreased when the external force
was increased because the deformation/deflection of the sheet
was increased by the external force, and the endpoints of the
sheet were displaced toward the center of the sheet.
Fig. 15 shows the initial grasping force versus the resistible
force by the grasp, which is defined as the maximum external
force at which the fingertip can maintain contact with theapplication of an increasing external force. For the fluid
fingertip, the resistible force was small even when the initial
grasping force was increased. For the rigid material fingertip,
the resistible force was greatly increased as the initial grasping
force was increased. As shown in Fig. 6, buckling occurred
when the pushing amount was 15 mm with a fluid fingertip(3 kPa). Therefore, the results for the fluid fingertips include
the cases of before and after buckling. Including the previous
experimental results (see Fig. 6), the rigid material fingertip
required large grasping forces but could resist large external
forces. The fluid fingertips required small grasping forces but
could not resist large external forces. In sum, various levels ofsoftness for the fingertips have both merits and demerits, andthe type of fingertip should be selected according to the task.
The conventional metric evaluating grasping (for example, see[17], [18]) evaluates based on convex theory. Therefore, if
the magnitude of grasping forces was set to be twice, themagnitude of resistible force also got to be twice, irrespectiveof stiffness of fingertips. However, this way is not applicable
to sheet manipulation, because the range applicable grasping
force varies with stiffness of fingertips as it can be seen from
Fig. 15. These results correspond to our previous results, which

A. Set up of experimentFig. 13 shows a schematic view of the experimental setup.
It is based on the experimental setup shown in Fig. 1, but adevice for providing an external force was added. The device
was constructed by attaching a force gauge to the automatic
positioning stage. By moving the stage in the vertical direction,
an external force in the direction of the gravitational force
could be applied, and its value could be measured by theforce gauge. The fingertips and sheet were the same as in theprevious experiment. The initial pressure of the fluid inside
the fluid fingertip was set to 3 kPa. Supposing the position
for the movable fingertip is zero when the grasping force fgr
is zero, we moved the movable fingertip by δdsp and grasped
the sheet. After that, the external force in the direction of thegravitational force was applied to the center of the sheet. We
considered the cases of δdsp = 5, 10, 15, 20, and 25 mm.

Fig. 13. Schematic illustration of the experimental setup for the graspingstability test focusing on the resistible force, which is defined as the maximum
external force at which the fingertip can maintain contact when applying an
increasing external force.

FUJIHIRA et al.: BENT SHEET GRASPING STABILITY FOR SHEET MANIPULATION

7

the sheet in the pushing direction (δ) with respect to thepressing amount is small, as shown in Fig. 6. Then, softer
fingertips can precisely control the bending and positioning
while providing grasping ability with a small grasping force.
Therefore, stiffness-variable fingertips [19], [20] are preferable
for sheet manipulation. In particular, fingertips with a twolayer structure where a rigid component is located inside thefluid (as presented in [20]) may be good idea.

Fig. 14. Time series data for grasping force fgr for elastic fingertip and thedisplacement of the movable fingertip δdsp = 5 mm.

Fig. 16. Grasping strategy for sheet manipulation.

V. CONCLUSION
Fig. 15. Initial grasping force fg0 versus resistible force frst .

showed that stiffer fingertips have a higher resistible force(frictional force) [11], [12].
IV. I DEAL GRASPING STRATEGY FOR SHEETMANIPULATIONThe results of the two experiments showed that stiff andsoft fingertips provide different advantages. Ideally, we should
control the stiffness of the fingertip according to the task.
Then, for a case where the stiffness is controllable, an ideal
grasping strategy for sheet manipulation can be constructed asshown in Fig. 16. When making contact with a sheet, a softer
fingertip like the fluid fingertip is desirable. Softer fingertips
can easily grasp the sheet with a small force and are also
robust against perturbations of the contact position. Whenpicking up or transferring a sheet, external forces such as
inertia are applied. In that case, stiffer fingertips are preferable
because they can resist large external forces. When placing asheet, stiffer fingertips are preferable until contact in orderto resist the impulsive force at contact. After contact, thebending of the sheet should be released while the sheet isplaced precisely. Softer fingertips are preferable in this case.
The rate of increase in the displacement of the endpoint ofIn order to realize sheet manipulation (see Fig. 1), we
considered grasping by bending the sheet and evaluated thegrasping stability in terms of resistance to a large external
force. The main results are as follows:
1) We theoretically and experimentally showed that the rateof increase in the grasping/pushing force with respect
to the displacement of the endpoint of the sheet in
the pushing direction (δ) is small after buckling. The
conventional method of displacing the fingertip toward
the object to increase the grasping force is not available
for bent sheet grasping.
2) The available range of the grasping force varies with thefingertip stiffness. Stiffer fingertips can only provide alarge grasping force while resisting a large maximum resistible force. Softer fingertips provide a small grasping
force with a small maximum resistible force.
3) An ideal grasping strategy for sheet manipulation where
the stiffness of the fingertips changes according to thetask is presented based on the above analysis. A suitablefingertip may be a two-layer structure with a rigid
component located inside the fluid, as presented in [20].
The other findings and contributions are as follows:• The buckling load was the same regardless of thetype/stiffness of fingertip.

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IEEE ROBOTICS AND AUTOMATION LETTERS. PREPRINT VERSION. ACCEPTED JANUARY, 2016

Softer fingertip can grasp the endpoints of the sheet
easily, while stiffer fingertips cannot.
• Sheet models (1), (6), (9), which represent the above
phenomenon and provide the estimation of sheet bending,were presented.
• In stiffer fingertips, the maximum resistible force canbe greatly increased by increasing the (initial) grasping
force. For softer fingertips, this approach is not available.
•

In the future, we plan to develop suitable fingertips for sheet
manipulation.
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