Advanced Robotics

ISSN: 0169-1864 (Print) 1568-5535 (Online) Journal homepage: http://www.tandfonline.com/loi/tadr20

Friction model of fingertip sliding over wavy
surface for friction-variable tactile feedback panel
Yohei Fujii, Shogo Okamoto & Yoji Yamada
To cite this article: Yohei Fujii, Shogo Okamoto & Yoji Yamada (2016): Friction model offingertip sliding over wavy surface for friction-variable tactile feedback panel, Advanced
Robotics, DOI: 10.1080/01691864.2016.1208591
To link to this article: http://dx.doi.org/10.1080/01691864.2016.1208591

Published online: 22 Jul 2016.

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Date: 30 July 2016, At: 10:39

ADVANCED ROBOTICS, 2016
http://dx.doi.org/10.1080/01691864.2016.1208591

FULL PAPER

Friction model of fingertip sliding over wavy surface for friction-variable tactile
feedback panel
Yohei Fujii, Shogo Okamoto and Yoji Yamada

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Department of Mechanical Science and Engineering, Nagoya University, Nagoya, Japan

ABSTRACTARTICLE HISTORY

A friction-variable touch panel is capable of presenting virtual bumps and holes on its ﬂat surface
through the control of the surface friction when a ﬁngertip slides over it. To improve the presentation,
we developed a friction model of a ﬁngertip sliding over a sinusoidal surface with an amplitude of
0.5–2.5 mm and a spatial wavelength of 20–50 mm. When a metal ball rolls over a wavy surface
with a low friction and contact area, the ratio of the horizontal force to the normal force is equal
to the gradient of the surface (this is referred to as the ball bearing model) and is hardly aﬀectedby the normal load and rolling speed. In contrast, the proﬁle of the force ratio of a sliding ﬁnger issubstantially skewed and aﬀected by the sliding direction and normal force exerted by the ﬁnger.
To model this skewed force ratio, we formulated the asymmetric pressure distribution in the ﬁngersurface contact area and used the eﬀects of the adhesion friction to model the dependency of theforce ratio on the normal force and sliding direction. The developed model of a bare ﬁnger with thesefeatures was found to suﬃciently simulate the experimentally observed force ratios. The model canbe easily applied to friction-variable touch panels and enables the achievement of a wide variety ofhaptic contents with macroscopically concave or convex surfaces.

Received 3 August 2015
Revised 13 April 2016
Accepted 1 June 2016

1. Introduction
Tactile displays that manipulate the friction on their ﬂat
panels have attracted much attention in the digital industry mainly because of their good compatibility withtouch panels. Such tactile displays include the electrostatic type, which increases the surface friction by the
Coulomb forces produced by electric charges,[1,2] andthe squeeze-ﬁlm type, which decreases the surface friction
by ultrasound vibrations.[3,4] These friction-variable displays are eﬀective for presenting friction-dominant textures and physical interactions such as non-soft contact
by constraining a sliding ﬁnger [5] and virtual toggle
switching.[3]However, there are some aspects of tactile displays that
require further investigation. An example is the challengeof virtually expressing smooth bumps and holes as large
as or larger than a human ﬁngertip on the ﬂat panel. The
combination of spatially macroscopic surface proﬁles andﬁne textural information, which friction-variable tactile
displays are eﬀective for, enables the broadening of thehaptic content [6] and thus improves the commercial
value of the tactile feedback function. However, to thebest knowledge of the authors, a solution to this issue is
yet to be proposed, although a vibrotactile approach has
been developed.[7,8]
CONTACT Shogo Okamoto

okamoto-shogo@nagoya-u.jp© 2016 Taylor & Francis and The Robotics Society of JapanKEYWORDS

Tactile display; finger
friction; bump and holeAn idea for delivering virtual concave and convex
surfaces using friction-variable haptic displays waspresented by De-La-Torre and Hayward [9]. As shown in
Figure 1, they demonstrated that the perception of bumps
and holes can be attributed to normal and tangential
sliding forces rather than to the ﬁnger displacements
caused by the surface curvatures. This indicates that afriction model of the ﬁnger and wavy surface would enable the creation of a virtual surface proﬁle even on a
ﬂat touch panel. They used a customized force display
to demonstrate the perception mechanism. The utilized
friction model was simple because the main focus wason the perception phenomena. For the judgment of curvatures, the importance of friction or a resistive force
vector in sliding over a curvature was also suggested by
Christou and Wing [10]. Other researchers have noted
the signiﬁcant eﬀect of the pressure distribution in thecontact area between a ﬁngertip and a curved surface[11–13] or the center of pressure and the gradient of
the contact area [13] in the perception of macroscopic
surface proﬁles. One of the friction, gradient of the contact area, or pressure distribution may be eﬀective for thepresentation of virtual bumps. Nonetheless, because
friction-variable touch panels can only manipulate thesurface friction and not the contact area, we employed a

2

Y. FUJII ET AL.

Perceived curvature

X

Y
Force field on a plane producedby friction-variable tactile display

Figure 1. Virtual bump created by a friction field on a flat frictionvariable tactile display. Without the Y-direction displacement of
the finger or surface, the person would vividly feel a bump.

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Perceived curvature

S
N
N
S
S
N
Magnetic force
Virtual hole
(Attractive)

Virtual bumps(Repulsive)

MagnetFigure 2. Demonstration of illusory bumps and holes using theforce of magnetic fields. The finger slides along a guide bar overa magnet placed on a non-magnetic plate beneath which othermagnets are attached. The generated attractive and repulsive
force fields, respectively, result in the perception of concave andconvex surfaces.

friction-based method. As shown in Figure 2, the virtual
bump based on resistive forces can be easily demonstrated by magnets. A plate with magnets attached to its
bottom surface is slid by a ﬁnger through another magnet.
The ﬁnger is moved along a linear guide and subjected torepulsive and attractive magnetic forces, resulting in theperception of a bump and hole, respectively, on the plate.
Although magnets produce vivid illusions, the development of a better model for computing the kinetic friction between a ﬁnger and a wavy surface would aﬀord
higher quality curvature presentation. In addition, thecomputational load should be made suﬃciently small for
implementation of the procedure in mobile computers asa background process. Many ﬁnger friction models havebeen studied, some of which are described in the following section. However, most of them were not intended
for application to kinetic friction between a ﬁnger andwavy proﬁles. Although some of them may be applied towavy proﬁles, attempts were not made to compare thesimulation results with the actual frictional forces. The
accuracies of the models thus remain uncertain.
In the present study, a model of the kinetic friction
between an elastic ﬁngertip and a wavy proﬁle over which
the ﬁnger slides was developed. By means of the model,
friction-variable tactile displays can present virtual
macroscopic bumps and holes. To approximate the actual
ﬁnger friction, the friction produced by an asymmetrically distributed pressure in the ﬁnger contact area wasexpressed in terms of two representative contact points
with load-dependent coeﬃcients of friction. In our experiments, the wavy surface was lubricated by talc to reduce the diﬀerence between the static and kinetic friction
to prevent stick-slip phenomena, which typically induce
uncomfortable touch sensations and should be avoided
in touch panels with tactile feedback functions. The developed model properly represents the sliding friction ofa ﬁngertip and is characterized by a low computationalcost, which makes it suitable for mobile terminals withfriction-variable tactile displays. The present study waspartly based on [14], its unique contributions being theextension of the adopted concept to variable ﬁnger loads
and comparison of the simulation and actual frictional
forces.

2. Related studies on finger friction
In the simulation of grasping using a robotic hand, surface contact between the ﬁngers and the object is notalways considered. However, without consideration of
the contact area, the frictional constraint would be
ineﬀective, resulting in unstable grasping in the virtualenvironment. To solve this problem, earlier researchers
formulated the conditions for stabilizing the grasp [15]and introduced frictional cones.[16,17] In addition, friction moments [18,19] were used to constrain rotation
about the normal to the contact area. However, the modeling of the friction around the contact center in the studies cited in the preceding sentences was diﬀerent from thefrictional behavior observed in the present study.
In some other previous studies, the friction of thesurface contact was modeled by multiple contact points.
In these other studies, the resultant interaction between
the hand and the object was deﬁned by the sum of theforces exerted on the individual contact points.[20,21]
For example, Talvas et al. formulated an ellipsoidal contact between a non-ﬂat surface and a virtual ﬁnger usingDownloaded by [Robotics Society of Japan] at 10:39 30 July 2016

ADVANCED ROBOTICS

Encoder
500 mm
170 mm
250 mmmultiple points that adjusted to the curved surface.[21]Although these studies did not discuss the kinetic friction
at the contact points, the idea of modeling a contact
surface using multiple contact points can also be used
to simulate the friction between a ﬁngertip and a curved
surface over which the ﬁnger slides. However, the results
obtained by these methods have not been compared withthe actual kinetic friction of a sliding ﬁnger. The methods
have thus not been used for direct solution of the problem
considered in the present study.
Many researchers have also formulated the friction
produced by the elastic deformation of a ﬁnger,[22–24]
although the formulations are not valid for a softﬁnger that touches an edged surface.[25] Moreover, anattempt has been made to estimate the friction of the
ﬁngertip based on the deformation of a circular contact
area.[26] These previous works discussed the constraint
caused by the ﬁnger deformation rather than the sliding
friction.
In reality, the actual friction of a sliding ﬁngertip ismore complex than those used in the simulation andmodeling of robotic grasping. A ﬁngertip is a ﬁnite
elastic body with epidermal ridges and sweat glands, andsome researchers have modeled its true frictional properties. However, the complicated friction phenomena constrained most of these studies to focus on the friction
between the ﬁngertip and a ﬂat body.[27] For example,
the coeﬃcient of friction under the adhesion and deformation of a ﬁngertip is not constant but a function of
the ﬁnger load, the sliding velocity and direction, and thesweat condition.[28–33] The dependency of the coeﬃcient of friction of a robotic ﬁngertip on the load has also
been investigated.[24,34] The sliding friction produced
by a textured surface is vibratory and is composed of
up to hundreds of Hertz.[35–37] Moreover, the stickslip phenomenon of a ﬁngertip is largely random.[38,39]
These complicated behaviors of the ﬁngertip are still being actively studied. Finite element models have likewise
been used to analyze ﬁnger friction,[40,41] although thecomputational cost may be too high for the utilizationof such models in the background processes of mobile
terminals.
As noted above, the friction models developed inprevious studies cannot be directly applied to thecomputation of the friction of a ﬁngertip sliding overa convex or concave surface, as in friction-variable tactile
displays used in mobile terminals. Although some of
the models are potentially applicable to this problem,
the goodness of their simulation of actual ﬁnger friction is yet to be established. To the best knowledge ofthe authors, the present study is the ﬁrst to model thesliding friction of a ﬁngertip on a macroscopically wavy
surface.

3XY

2a = 1, 3, 5 mm

= {20, 30, 40, 50} mm
XY

Figure 3. Experimental setup. Top: Schematic of theequipment. Middle: Sinusoidal profile. Bottom: Photograph of theexperimental scene.

3. Experiments to observe sliding friction of
fingertip on wavy surfaces
3.1. Equipment
We measured the force and position of the ﬁnger at
1 kHz using the equipment shown in Figure 3. The main
components of the equipment are the four strain-gaugetype load cells (Model 1004, Tedea Huntleigh, USA),
which were used to measure the bi-axial forces appliedon the wavy surface. The cells were installed such that
two were used to measure each axis of the forces. The
sinusoidal wavy surface was made from ABS plastic andwas ﬁnely polished. Its proﬁle was given by the following:

2πxy(x) = a cos
λ


(1)

where a and λ are the amplitude and spatial wavelength,
respectively. The parameters of the specimens used in theexperiment were a = {0.5, 1.5, 2.5} and λ = {20, 30, 40,50} mm. Two specimens were prepared for each parameter and alternately used. We targeted the surface curvatures that were as large as or larger than that of thehuman ﬁngertip because it is diﬃcult for friction-variabletactile displays to present ﬁner waves with more than one
curvature within the ﬁnger contact area.

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Y. FUJII ET AL.

Two encoders (RE30E-500, Nidec Copal Electronics
Corp., Japan) were placed on the sides of the wavy surface to measure the positions of the ﬁnger or the ball.
The encoders were wound through pulleys by a string
taped to the ﬁnger or the ball bearing, with resultant
position resolutions of 14.4 and 17.2 μm in the X- andY -direction, respectively.

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3.2. Participants and tasks
Three male volunteers with ages ranging between 22 and30 years participated in the experiment. The widths of
their index ﬁngertips were 15, 14, and 14 mm, respectively. As previously noted, talc was used to prevent theﬁnger from sticking to the test surface, thus enablingmeasurement and modeling of the kinetic friction of
the ﬁnger. The participants scanned the surface with
their index ﬁngers and an iron ball of diameter 6 mm.
They repeatedly slid the ﬁnger or ball from side to sideon the wavy surface for 30 s with arbitrary variation of
the scanning velocity and load. They were instructed tomaintain the ﬁnger posture as much as possible during
the sliding.

Figure 4. Friction ratio when a participant scanned the wavy
surface using his bare finger. The solid line represents the sample
average.
Physical surface y(x)
X

fbx
α

Y

fby fbFigure 5. Ball bearing model. The force applied by the ball on thewavy surface is equal to the reaction force of the surface. The ratio
between the X- and Y-direction forces is equal to the gradient of
the surface.

3.3. Analysis
For all the experimental trials, sample measurements withfy ≥ 0.6 N were regarded as valid. Although the test
surface was 170 mm long, the central ±30 mm samples
were used for the analysis. Although the participants
were allowed to scan the surface with any velocity andload, the analysis considered two ranges of the scanningvelocity and three ranges of the load to determine theeﬀects of the two parameters. The considered velocities
were 5–40 and 40–100 mm/s, and the loads were split
into 0.6–1.2, 1.2–1.8, and 1.8–3.0 N. As described below,
the scanning velocities were found to hardly aﬀect theresults under the present experimental conditions. The
samples for all the participants were analyzed together
because no signiﬁcant individual variations were observed. An example of such minor individualities is shown in
Section 6.4.
Because the ratios between the X and Y componentsof the forces are important,[9] we also examined theinteractive force with respect to the ratio. The ratio modelis practically beneﬁcial for the presentation of virtual
bumps and holes in friction-variable tactile displays because it can be used to control the tangential (X-direction)
surface friction with respect to the normal (Y -direction)
ﬁnger force. In the graph shown in Figure 4, the horizontal and vertical axes, respectively, represent the ﬁnger
or ball position along the X-axis and the force ratio fx /fy .
The plotted samples were averaged because of the naturalvariations.

4. Force ratio of ball rolling on wavy surface
Here, we discuss the force ratios of a ball rolling on thesinusoidal surface. The contact area between the ball andthe surface is small and can be considered as a point contact. The rolling friction is also very small. Under these
conditions, the ratio between the X- and Y -direction
forces is mainly determined by the gradient of the surface,
and is described by the simple model used in [9]. Thismodel is referred to as the ball bearing model.
4.1. Ball bearing model
Figure 5 shows the contact between a ball and a sinusoidal
surface. The force exerted by the ball ( f b ) is equivalent
to the reaction force of the surface. Under this condition,
the ratio of the X-direction force fbx to the Y -direction
force fby is equal to the surface gradient, and the following
holds:
fbx (x)dy| f | sin (α(x))== b= tan (α(x)).
fby (x)dx| f b | cos (α(x))(2)

Here, α(x) is the gradient of the surface at the contact
point x.
4.2. Friction ratio for ball bearing
Figure 6 shows the force ratio when the ball bearingrolls on the surface with fy = 1.2–1.8 N. The left andADVANCED ROBOTICS

Leftward

fx / fy

0.2

Force ratio

Rightward0

a = 0.5 mm

v = 5—40 mm/s

Ball bearing model5

v = 40—100 mm/sa=
1.5 mma=
1.5 mm

-0.2
-20

0

20

-20

0
Position x [mm]

-20

20

0

20

5. Force ratio of bare finger sliding over wavy
surface
Here, we discuss the force ratio of a ﬁngertip sliding overthe surface and show some sample results.
Figure 7 shows the force ratios for the sinusoidal proﬁlewith a = 1.5 mm. Firstly, it should be noted that the
proﬁle of the force ratio is not actually sinusoidal but
skewed. Secondly, the force ratio is dependent on the sliding direction. The force ratio for the positive X-direction
sliding is larger than that determined by the ball bearing
model, while that for the negative X-direction sliding issmaller. Furthermore, the force ratio is aﬀected by thenormal load of the ﬁnger, decreasing with increasing fy .
Below, we discuss the reasons for these observations andestablish a friction model for a bare ﬁnger sliding over asinusoidal surface.

fy = 1.8—3.0 N
fy = 0.6—1.2 N

fx / fy

Force ratio

middle ﬁgures correspond to surface amplitude a valuesof 0.5 and 1.5 mm, respectively. The dotted and solid
curves represent the gradient of the surface (dy/dx) andthe observed force ratio, respectively. The red and blue
solid curves represent the force ratios when the surface
was scanned in the positive and negative X-directions,
respectively. The right ﬁgure shows the force ratio foreach of the two considered velocity ranges when a =1.5 mm. From the ﬁgures and other data not shown in theﬁgures, it was observed that the force ratio matched theprediction of the dotted ball bearing model irrespectiveof the amplitude of the sinusoidal proﬁle and the rolling
direction and velocity. When the contact is nearly a point
contact and the friction is small, the ball bearing model
fairly represents the actual surface exploration.

Ball-bearing model

0.5
0
-0.5

Surface profile
Y [mm]

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Figure 6. Model-predicted (dotted) and experimentally determined (solid) force ratios of a ball bearing rolling over a sinusoidal surface.
Left: For amplitude a = 0.5 mm. Middle: For amplitude a = 1.5 mm. Right: For 1.5-mm surface and two velocity ranges. The blue andred curves correspond to rightward and leftward rolling of the ball, respectively.

1.5
-1.5
-20

0
Position x [mm]20

Figure 7. Measured and model-predicted friction ratios of a barefinger sliding over a wavy surface with a = 1.5 and λ = 40 mm.
The profiles are not sinusoidal. The absolute measured ratios
are mostly larger than those predicted by the ball bearing
model and are dependent on fy . The solid curves correspondto rightward sliding over the surface, while the dotted coloredcurves correspond to leftward sliding. The bottom panel shows
the profile of the wavy surface.

5.1. Skewed force ratio: eﬀect of surface contact
The observed force ratio of the sliding ﬁnger is skewed
compared to a sinusoidal curve with the rising phaserapid and shorter than the descending phase. The force
ratio sharply increases when the ﬁnger moves up a slope,and slowly decreases as the top of the curvature is approached and the ﬁnger moves down the slope. It is speculated that the deviation of the proﬁle for the ﬁnger fromthat predicted by the ball bearing model is due to thediﬀering contact areas, which are characterized by surface
and point contact, respectively. As shown in Figure 8,

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Y. FUJII ET AL.

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on an angled slope, the force distribution in the contact
area is asymmetric about the contact center, and this is
the cause of the skewed force ratio. A reasonably simple
model that expresses this asymmetry is the two-point
model that approximates the surface contact between aﬁngertip and the gradient surface by two representative
balls. The model does not express the change in thecontact area, but the change in the associated pressure
distribution. Nonetheless, because friction-variable tactile displays do not manipulate the contact area or thepressure distribution within it, the purpose of the modelis the computation of the resultant frictional forces, toenable the tactile display to render virtual curvatures.
The total force f f that the ﬁnger exerts on the surface
at position x is composed of the f b forces of the two balls.
Hence,f f (x) = f b (x + r) + f b (x − r).

-r x rf b (x − r )Two ball bearings
Figure 8. Two balls used to represent the surface contact of a barefinger. The higher ball exerts a larger load.

-r r

R
pmax
p(r)

(3)

The distance between the contact center and the ball isdenoted by r. On an angled slope, the force distribution
in the contact area is asymmetric about the contact centerand this is the cause of the skewed force ratio. The normalloads applied by the two balls are not equal; the ball at
the higher position on the wavy surface applies a larger
normal load:fby (x + r) > fby (x − r) if y(x + r) > y(x − r). (4)The force distribution in the contact area of a ﬁngerlike elastic object is largely bell-shaped.[41] Hence, weused the idea of Hertzian contact to represent the force
distribution, although a sliding human ﬁnger does notexactly exhibit Hertzian contact. As shown in Figure 9,
the pressure distribution in the contact area between aspherical and ﬂat body is given by

 2
r
(5)
p(r) = pmax 1 −swhere pmax and s are the maximum pressure at the centerof contact and the radius of the contact area, respectively.
When a sphere contacts a ﬂat surface, two points awayfrom the contact center x by ±r experience equal pressures; that is, p(x + r) = p(x − r). When the surface
is inclined at α, the point of maximum pressure shifts
by R sin (α) as shown in the right panel in Figure 9. R
is the radius of the sphere. Based on the ratio between
the pressures at the two points, the ratio between the
Y -direction forces at the two points is
fby (x − r) : fby (x + r)






−r − R sin (α(x)) 2
r − R sin (α(x)) 2
: 1−
.
= 1−
s
s

(6)

f b (x+ r )s r
Figure 9. Left: Parabolic pressure distribution on a flat surface.
Right: Shift of the center of pressure on an inclined surface.

Hence, the total X-direction force acting on the ﬁnger is
ffx (x) = fbx (x + r) + fbx (x − r)
= fby (x + r) tan (α(x + r))+ fby (x − r) tan (α(x − r)).

(7)

In terms of the Y -direction forces, ffy (x) = fby (x − r) +fby (x + r). Hence,
fby (x ± r) = ffy (x)±r − R sin (α(x)) 2
s
×
2 
 .


−r − R sin (α(x))
r − R sin (α(x)) 2
1−
+ 1−
s
s


1−

(8)
The ratio between the X- and Y -direction forces isthus given by
fby (x + r)
fx (x)=tan (α(x + r))fy (x)
ffy (x)fby (x − r)+tan (α(x − r)).
ffy (x)(9)

Figure 10 shows the force ratios for the combinedforces of the two balls. With the assumption that thewidth of the index ﬁngers of the participants is approximately 14 mm, the radius R of the ﬁngertip was set toADVANCED ROBOTICS

fx / fy

0.5

a = 1.5

r = 4.4
r = 3.6

r = 2.5
r=0With frictionfx
f= tanα = bx
fy
fby

fx
f
> tanα = bx
fy
fby

0
x

1

fr

y

-0.5

fx / fy

Without friction

0

0

fby

α

fb

fby

fb + fr
fb

Figure 11. Left: Statically stable ball bearing model without
considering friction. Right: Ball bearing model considering
friction. f r is the sliding friction between the ball and the surface
when the ball slides rightward. The resultant force exerted on thesurface is the sum of f b and f r .

-1
-20

fbx

fbx

α

r=0
r = 2.5
r = 3.4

a = 2.5

7

20

Figure 10. Skewed force ratio. The representation of the fingertip
contact by two balls skews the force ratio. 2r is the distance
between the two balls. The r = 0 curve corresponds to the ballbearing model. R = 7 mm and s = 6 mm.

1

fx / fy

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Position x [mm]07 mm. Considering the minor axis radius of the ellipticalcontact area between the ﬁngertip and the ﬂat surface,s was set to 6 mm. Because of the increasing value ofr, which is the distance between the representative balland the center of contact, the proﬁle of the force ratio
becomes more skewed and similar to that for the bareﬁnger.
5.2. Eﬀect of sliding friction
Here, we introduce sliding friction, which is neglectedin the ball bearing model. The sliding friction, which isshown in Figure 7, increases the force ratio compared tothat obtained by the ball bearing model when the ballslides rightward, whereas it decreases the ratio when theball slides leftward. Figure 11 shows the forces exerted on
the surface with and without considering sliding friction.
In the latter case, shown in the left ﬁgure, the force ratio
is determined by the gradient of the tangent to the curve.
In the other case, the interaction force is composed of f b ,which is the equivalent force exerted by the ball bearing,and the sliding friction f r . If the coeﬃcient of friction
is denoted by μ, the sliding friction can be expressedasf r = μ| f b |[cos α, sin α]T(10)Therefore, from (2) and (10), the force ratio is obtainedasfx (x) | f b | sin α + μ| f b | cos α=fy (x) | f b | cos α − μ| f b | sin α
tan α + μ.
=
1 − μ tan α

(11)

-20

0

20

= 0.5
= 0.4
= 0.3
= 0.2
= 0.1
=0

Position x [mm]Figure 12. Force ratio of a ball bearing sliding on a wavy surface
with varying friction coefficients. When μ = 0, the ratio is equal
to that obtained by the ball bearing model.

This force ratio depends on the gradient of the wavy
surface and the friction, while the ratio obtained by theball bearing model is determined by the gradient only, asin (2). It should be noted that the ratio does not dependon the magnitude of the force exerted by the ball, namely,| f b |.
Figure 12 shows the force ratios when the sliding friction is applied to the ball bearing model. With increasing
coeﬃcient of friction when the ball slides in the positive
X direction, the force ratio is strained and shifts upward.
The ratio shifts downward when the ball slides in thenegative direction.
5.3. Dependency of friction coeﬃcient on normalforce
As previously described, Figure 7 shows that the force
ratio depends on the Y direction force. An increase in thisforce when the ﬁnger slides rightward decreases the force
ratio. This is because of the dependency of the coeﬃcientof friction on the force. In the case of a ﬁnger that slides
over a smooth rigid body, as in the present experiment,
the friction is mainly composed of the deformation andadhesion friction. The behaviors of the coeﬃcients of friction for these two types of frictions diﬀer. The coeﬃcientof friction for the adhesion friction is a negative powerfunction of the force applied by the ﬁnger, whereas that

8

Y. FUJII ET AL.

for the deformation friction is a positive power function of the ﬁnger force.[28,29] Regarding the contributions of these two types of frictions for a sliding ﬁnger,
the adhesion friction is dominant under dry conditions,[30,42] whereas the deformation friction plays a slightly
more dominant role under lubricated conditions.[30]Hence, the general form of the coeﬃcient of friction canbe written as
μ ∝ | f b |n(12)where n can be either positive, negative, or zero depending on the tribological system.

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5.4. Bare ﬁnger model
By incorporating the above characteristics, the force ratio
when a bare ﬁnger slides over a sinusoidal surface can be
described as follows based on (9), (11), and (12):fx (x) fby (x + r) tan (α(x + r)) + μ| f b |n=fy (x)
ffy (x) 1 − μ| f b |n tan (α(x + r))+

fby (x − r) tan (α(x − r)) + μ| f b |n
. (13)
ffy (x) 1 − μ| f b |n tan (α(x − r))

6. Comparison of simulation and observed
friction ratios
Here, we compare the experimentally observed force
ratio with that determined by simulation using the bareﬁnger model.
6.1. Parameters of simulation force ratio
The parameters of the model were set as follows: R =7 mm, r = 3.4 mm, s = 6 mm, μ = 0.4, and n = −0.3.
With the exception of the R and s values, the parameters
were searched for such that the simulation and actualforce ratios would be comparable for λ = 40 mm.
As previously noted, a larger r further skews the force
ratio. We evaluated the degree of skewness based on the
proportion of the downward curve of the force ratio to the
upward curve. The value of r that made the proportion forthe simulation very similar to that for the experiment was
searched for within 0–6 mm using increments of 0.1 mm.
Furthermore, the coeﬃcient of friction μ, whichmainly determines the degree of the base shift of theforce ratio, and the power value n, which indicates thedependency of the coeﬃcient of friction on the ﬁnger
load, were varied within 0–0.5 and −0.5–0.5, respectively,
to make the center levels of the simulation and actualforce ratios as close as possible. The ﬁnal values were μ =0.4, which is within the range of the coeﬃcients of friction

between the human skin and common materials,[27] andn = −0.3.
6.2. Comparison of general features with varying
wave heights and ﬁnger loads
Figure 13 shows the force ratio for each amplitude of thesinusoidal proﬁle for λ = 40 mm.
As previously mentioned, the proﬁles of the force
ratios for a bare ﬁnger and a ball bearing diﬀer significantly. The force ratio for a ﬁnger rises rapidly andfalls slowly, whereas that for a ball bearing has equallylong rising and falling phases. Although our bare ﬁnger
model fairly exhibits the skewed trend, its force ratio
does not completely match the actual value, especiallyfor a = 2.5 mm, in which case the actual force ratio islarger than that of the model.
Friction shifts the base line of the force ratio. Whenthe ﬁnger slides in the positive X direction, the force
ratio shifts upward and vice versa. This trend was clearlyobserved in the experiment, and was well represented by
the bare ﬁnger model.
In addition, the actual force ratio depends on the ﬁnger
load, with a smaller load increasing the force ratio. Under
our experimental conditions, the coeﬃcient of friction is anegative power function of the load because the adhesion
friction is dominant. The adhesion friction between asphere and an elastic plate is proportional to the two-third
power of the load,[28] while the coeﬃcient of friction is
proportional to the negative one-third power of the load.
The exponent n used above was −0.3, which is close to−1/3. Consequently, the bare ﬁnger model produced anegative correlation between the normal load and themagnitude of the force ratio.
In the experiment, the force ratio was hardly aﬀectedby the sliding velocity. This was not unexpected because
the adhesion friction between elastic and rigid bodies
rarely depends on the sliding velocity.[30]6.3. Comparison for various surface wavelengths
In this section, we compare the simulation and experimental results for λ = 20, 30, and 50 mm. The abovementioned results were for λ = 40 mm.
As can be observed from Figure 14, the simulation
proﬁles also agree well with the experimental results for
λ = 30 and 50 mm (middle and right). The same simulation parameters used for λ = 40 mm were adopted forthese cases. As can be seen from the plots, when the widthof the bumps or holes is larger than that of the ﬁngertip,or the surface gradient is small, our model successfully
predicts the actual interaction forces. This is because themodel presumes suﬃciently small surface gradients such

ADVANCED ROBOTICS

fy = 0.6—1.2 N

fy = 1.2—1.8 N

fy = 1.8—3.0 N

-20

20

9

Ball-bearing model

Force ratio
fx / fy

1.0
0.5
0

-1.0
1.5
-1.5
-20

0
Position x [mm]200Position x [mm]

-20

0
20
Position x [mm]Figure 13. Force ratio of sliding finger. The fine curves represent the simulation results, while the solid and dotted bold curves,
respectively, correspond to rightward and leftward sliding of the finger. The black dotted curves represent the force ratio of the ballbearing model. Left: a = 0.5 mm. Center: a = 1.5 mm. Right: a = 2.5 mm. The other simulation parameters were as follows: R = 7 mm,
r = 3.4 mm, s = 6.0 mm, μ = 0.4, and n = −0.3. The employed value of fy for the simulation was the median of each range.

Force ratio
fx / fy

1.0
0.5
0
-0.5
Surface profile
Y [mm]

Downloaded by [Robotics Society of Japan] at 10:39 30 July 2016

Surface profile
Y [mm]

-0.5

-1.0
1.5
-1.5
-20

0
Position x [mm]

20

-20

0
Position x [mm]

20

-20

0
Position x [mm]20

Figure 14. Force ratio of sliding finger. The fine curves represent the simulation results, with the solid and dotted bold curves,
respectively, corresponding to rightward and leftward sliding of the finger. The black dotted curves represent the force ratio of theball bearing model. Left: λ = 20 mm (r = 2.2 mm, s = 5.0 mm). Center: λ = 30 mm (r = 3.4 mm, s = 6.0 mm). Right: λ = 50 mm(r = 3.4 mm, s = 6.0 mm). The height a of the wavy surfaces was 1.5 mm. The other simulation parameters were as follows: R = 7 mm,
μ = 0.4, and n = −0.3. The fy value for the simulation was 1.0 N, which is the median of the finger loads used for the computation.

that the contact area of the ﬁnger does not signiﬁcantly
vary while scanning the bumpy surfaces.
However, for a smaller bump width, as in the case of
λ = 20 mm, diﬀerent simulation parameters were needed
to be selected. Using the procedure described in Section
6.1, the simulation parameters for this case were set tor = 2.2 and s = 5.0 mm. These values are smaller thanthose used for the other cases with larger λ values. Because
the surface gradient for λ = 20 mm was steeper thanthose for the other cases, the parameters related to thecontact area were needed to be adjusted accordingly.

proﬁle for λ = 40 mm and a = 0.5 mm with fy = 1.2–
1.8 N. One additional participant (Participant D) with aﬁngertip width of 14 mm was invited for the purpose,
i.e. in addition to the three participants of the previous experiments. The diﬀerences among the individual
force ratiocurves are apparently minor although there
are subtle biases that may be attributed to the varying
forces applied by the participants. Such relatively minuteindividual diﬀerences might have been due to the control of the experimental conditions. Particularly, talc waseﬀective for maintaining dry conditions.

6.4. Individual variance of force ratios

7. Discussion

Finally, in Figure 15, we present the individuality of theforce ratio when the participants scanned the sinusoidal

We modeled the ratio between the X- and Y -direction
forces applied by a ﬁngertip sliding over a sinusoidal

10

Y. FUJII ET AL.

Force ratio: fx / fy

1.0
0.5

Part APart B
Part CPart D
Ball-bear.

0-0.5b) Involvement of finger
rotation on a high bump

-1.0
-20

Downloaded by [Robotics Society of Japan] at 10:39 30 July 2016a) Hole as large as a finger pad

0
Position x [mm]20

Figure 15. Force ratio curves for the four participants. A sinusoidal
surface with λ = 40 mm and a = 0.5 mm was explored with afinger load fy of 1.2–1.8 N.

c) Small hole that causes a gap
between finger pad and surface

surface. Whereas the model can still be improved, it isuseful for the presentation of macroscopic surface proﬁles on friction-variable tactile panels.

Figure 16. Cases with limited applicability of the proposedmodel. (a) The fingertip fits into a hole, resulting in irregularincrease in the contact area. (b) The fingertip rotates on a large
bump, resulting in violation of the model assumption that thefingertip remains horizontal. (c) The interval between neighboringbumps is smaller than the fingertip, resulting in the finger pad
simultaneously contacting more than one bump.

7.1. Expression of asymmetric pressure distribution
We considered that the skewed force ratio of a sliding ﬁnger indicates that the pressure in the contact area
is not symmetric about the center of contact. We useda pressure distribution model based on Hertzian theory to describe this asymmetry although the physicalplausibility or correctness of this modeling is uncertain
and other approaches may be available. For example, thedeformation of the contact surface of an elastic bodyunder a tangential force has been formulated,[43] andsuch a formula is used to deﬁne the asymmetric pressure
distribution as a function of the tangential force, whereas
it was determined from the gradient of the surface in thepresent study.
7.2. Limitations of friction model
As stated in Section 6.3, the prediction of the actual interaction forces using our model requires adjustment of thesimulation parameters with respect to the surface proﬁle.
The detailed relationship between these parameters andthe surface proﬁle requires further study. Additionally,
our proposed method is not applicable to some surfaces
because of the assumptions that the model is based on.
First, the skewness of the actual force ratio for a surface
amplitude of 2.5 mm was unexpectedly large, comparedto the predictions of our model. This discrepancy can be
explained by the fact that, when the amplitude of thewavy surface is as large as 2.5 mm, the ﬁngertip may
ﬁt into the hole as shown in Figure 16(a), thereby increasing the contact area, and hence the coeﬃcient offriction. Consequently, the values of s and μ may varywith the macroscopic surface proﬁle, whereas constantvalues were employed in our simulation.

Second, in our model, the posture of the ﬁnger remains
horizontal during the scanning. However, this assumption is unlikely to hold when the surface gradient is large,
as shown in Figure 16(b), in which case the ﬁnger may
rotate on the surface. Hence, it is speculated that thefriction model will not accurately present the force ratio
for a large bump.
Finally, as previously mentioned, our model targets
macroscopic surface roughness or proﬁles, for which thewidth of a bump is larger than that of the ﬁngertip. For
this purpose, it is assumed that the ﬁnger pad contacts
the surface with no gap. As the surface wave becomesﬁner, the ﬁnger simultaneously contacts multiple bumps
and a non-contacted gap is developed, as shown in Figure
16(c). In this case, our model is inapplicable.
7.3. Presentation of surface bumps and holes
through friction-variable touch panel
Our model was aimed at presenting virtual bumps andholes by means of a friction-variable touch panel. For
this purpose, the friction force is computed by multiplying the measured normal load by the force ratio
determined by the developed model. Although most currently available commercial touch panels do not haveforce detection functions, such functions are expected
to become part of the general speciﬁcations when tactile feedback displays become prevalent. One advantage
of curvature presentation using our model is that thepanel does not need to be able to increase and decreaseits surface friction from its neutral value. In the ballbearing model, the force ratio range includes zero andDownloaded by [Robotics Society of Japan] at 10:39 30 July 2016

ADVANCED ROBOTICS

the touch panel therefore needs to output both resistive and assistive friction. Unfortunately, suchbidirectional friction variation can only be realized by
combining more than two diﬀerent mechanisms.
In contrast, the actual force ratio of a ﬁngertip does
not necessarily cross zero and can be presented by simply increasing the friction. From this perspective, anelectrostatic-type friction-variable display is suitable forthe presentation of virtual waves. Indeed, there has beena previous study [44] in which virtual bumps were rendered using the ball bearing model and an electrostatictype display with limited capabilities. Furthermore, theamplitudes of virtualbumps or holes perceived by friction
may be smaller than the values employed in a model
because a touch panel does not induce normal displacements on its surface. However, it is obvious that the
combined use of a frictional cue by a tactile display and avisual cue would facilitate eﬀective presentation of virtual
bumps and holes.

11

Disclosure statement
No potential conﬂict of interest was reported by the authors.

Funding
This study was in part supported by MIC SCOPE [27-J-J6238];
KAKENHI Shitsukan [15H05923].

Notes on contributors
Yohei Fujii received a MS degree inengineering from Nagoya University in2016. His research interests compriseenergy engineering and haptics.

8. Conclusion
Based on the illusory perception of macroscopic concaveand convex surfaces by the tangential force of a sliding ﬁnger without movement in the normal direction,
wedeveloped a friction model for the presentation ofvirtual bumps and holes on the ﬂat surface of a frictionvariable tactile feedback panel. No previous model could
be directly used for this purpose. Although a numberof studies on ﬁnger friction have been conducted,mostof them did not address the kinetic friction of a ﬁngertip sliding over a macroscopically wavy surface. In
addition, while a few of the developed models may be
extended to serve our purpose, the predictions of themodels were not compared with the actual sliding friction of the ﬁnger. In the present study, we observed theratios between the X- and Y -direction forces appliedby an iron ball and a ﬁngertip sliding over a surface
with a sinusoidal proﬁle. The force ratio for the ball,which had a very small contact area and rolling friction
matched the gradientof the surface irrespective of themagnitudes of the forces and the rolling velocity. This
force ratio was referred to as the ball bearing model.
The force ratio of the ﬁngertip did not match the ballbearing model but had a skewed proﬁleand a shiftedbase level that depended on the sliding direction andﬁnger force. We consequently developed a bare ﬁnger
model that described the skewed force ratio based on
the asymmetric pressure distribution in the contact area.
The model further incorporated the adhesion friction toconsider the dependency of the ratio on the normal force.
By manual adjustment of its parameters, the bare ﬁnger
model fairly represents the actual force ratio of the humanﬁnger.

Shogo Okamoto received MS and PhD
degrees in information sciences in 2007
and 2010, respectively, from the Graduate
School of Information Sciences, Tohoku
University. Since 2010, he has been with
the Department of Mechanical Science andEngineering, Nagoya University, Japan. His
research interests comprise haptics, humanassistive technology, and wearable robots.

Yoji Yamada received a PhD degree fromthe Tokyo Institute of Technology in1990. He has been an associate professor
at the Toyota Technological Institute,Japan since 1983. In 2004, he joined the
National Institute of Advanced Industrial
and Science Technology (AIST), as a groupleader of the Safety Intelligence ResearchGroup at the Intelligent Systems ResearchInstitute. In 2008, he moved to the Graduate
School of Engineering, Nagoya University,
as a professor.

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