2018 IEEE International Conference on Systems, Man, and Cybernetics

Stochastic relationships between the normal andshear interaction forces during tactile exploration of
textures
Hikaru Hasegawa, Shogo Okamoto, Hatem Elfekey, and Yoji Yamada
Department of Mechanical Systems Engineering
Nagoya University, Aichi, Japan
Abstract—Humans recognize textures through the interaction
induced by the normal and shear forces generated between theﬁngertip and the surface of objects when sliding their ﬁngers onmaterial surfaces. We investigated the stochastic relationships ofthese two axial forces during tactile exploration. The characteristic values of the elliptical distribution obtained by the variationsof the two axial forces were analyzed. It was found that thesizes, shapes, and gradients of the elliptical distribution of the
low and high frequency forces statistically varied among the 18
types of textures. These parameters may be the indicators to
distinguish the unevenness of the surface and the behavior offriction between the surface and the ﬁngertip. The fact that thestochastic relationships between the two axial forces depend on
the surface would help us understand human texture perception.
Index Terms—tactile exploration, normal and shear forces,
statistical values

I. I NTRODUCTION
Texture sensation depends on the interaction forces induced
between the surface of objects and the ﬁngertip or the deformation of the ﬁngertip during tactile exploration. Thusfar, researchers have attempted to investigate the relationships
between the ﬁnger-material interaction and the texture sensation. Many studies reported that the vibration components
of a force or deformation of the skin inﬂuence the haptic
sensation [1]–[6]. These studies discussed the relationships
between vibration frequency and haptic sensation for instance,by associating them with the frequency characteristics of themechanoreceptors. The vibration frequency conveyed to theﬁnger in the normal direction corresponds to the surface
proﬁle frequency of the objects and inﬂuences the roughness
sensation [1], [2]. Besides, the sensation of roughness isaffected by the shear force and the friction [3], [4] and thefriction models of the interaction between the surface andthe ﬁngertip have been investigated (e.g., [7], [8]). Although
the haptic sensation is affected by both the normal and shear
forces, the relationships between these forces and the haptic
sensation have not been investigated in a convergent way.
Previous studies found that stochastic information represented by the power spectrum of force acting on the ﬁngertip
or its skin deformation determine the tactile sensations experienced from materials (e.g., [9]–[11]). This idea has been
employed by the principles of tactile texture displays that
present virtual textures based on the strength of vibratory
2577-1655/18/$31.00 ©2018 IEEE
DOI 10.1109/SMC.2018.00515

cues applied to the ﬁngertip in the frequency domain [12]–[14]. These studies indicate inherent linkages between tactile
sensations of surfaces and the stochastic behaviors of contact
forces in the frequency domain.
The representative characteristic value regarding the two
axial forces is the friction coefﬁcient, which is deﬁned as
the ratio of normal force to shear force. Most of the previous studies that measured these two axial forces dealt withthis macroscopic value (e.g., [15]). However, other statisticalvalues calculated from the two axial forces have not been
investigated yet. For instance, a covariance matrix calculated
from the time series data of the two axial forces provides substantial information. Although Platkiewicz et al. [16] measured
the two axial forces, they did not focus on their stochastic
relationships, which have not been investigated thoroughly.
We measured the two axial forces induced by exploring
18 types of textures and calculated the statistical values inseveral frequency ranges. Then, the characteristic values ofthe material were extracted by applying a principal component
analysis. Few studies have investigated the stochastic relationships between the two axial forces during tactile exploration.
We obtained ﬁve components that indicate the power of two
axial forces, the correlation between the two axial forces, andfriction coefﬁcient. Especially, the power of two axial forces inhigh frequency range implies the power of the friction-induced
vibration. Also, the correlation of the two axial forces implies
the unevenness of the surface proﬁle. The results of this study
would help understand the texture sensation mechanism andcould be applied as a new method to detect the texture ofsurface by touch sensors.
II. F ORCE MEASUREMENT BETWEEN FINGERTIP AND
MATERIAL AND METHODS

A. Measurement setup
We used the two-axial-force sensing unit shown in Fig. 1 formeasuring the normal and shear interaction forces generated
between the ﬁngertip and a material sheet during tactile
exploration. A similar unit was used in [17]. We employed
two single-axial load cells (9313AA2, Kistler, Switzerland),
which were located on both sides of an aluminum base formeasuring the normal force. Furthermore, a high-precisionload cell (9217A, Kistler, Switzerland) was ﬁxed between
the two aluminum plates above the normal force sensors for3029

Fig. 1: Left: Schematic view of the force-sensing unit. Right: Measurement setup.
measuring the shear force. Leaf springs ﬁxed to both sides of
the upper aluminum plates decoupled the two axial forces. The
size of the top surface was 40 × 100 mm. The signals fromboth the normal and shear force sensors were ampliﬁed by
two charge ampliﬁers (5073A and 5015, Kistler, Switzerland)
and transmitted to a computer at the sampling frequency of
2 kHz, which is enough to detect the frequency response ofhuman touch.
The ﬁnger motion during tactile exploration was measured
by two optical encoders (RE30E, Nidec Copa Electronics
Co., Japan). A string that passed through a pulley which was
attached to the axis of each encoder was taped on the ﬁngertip.
The string was tensioned by small weights, thus the ﬁnger
position was measured by the encoders through the string. The
tension of this string was small enough to ignore its inﬂuenceon the ﬁnger movement.
B. Material

Fig. 2: To measure the force, 18 kinds of material were used.
Upper row (from left): wood, coarse Japanese paper, ﬁneJapanese paper, drawing paper, coated paper, and crumpled
paper. Middle row: cork, corrugated paper, woven resin wire
mesh, fake leather, fake fur, and felt. Bottom row: denim,
polyester cloth, polyethylene sheet, rubber, sticky rubber (rubber glove), and ﬂat aluminum

We employed 18 types of textures, as shown in Fig. 2,
for measuring the interaction forces. We selected textures that
we often touch in our daily life. Cork and coarse Japanese
paper induce intensive friction. Thus, we prepared another pairof these textures with talc on their surface for lubrication.
Consequently, we measured the interaction forces generated
between the ﬁngertip and the surface of 20 samples of textures.

C. Tasks and participantsParticipants stroked the material sheet ﬁxed on the uppersurface of the force-sensing unit, as shown in Fig. 3, with their
dominant index ﬁnger. They repeatedly slid their ﬁnger on thematerial surface in one direction for 20 s. They were instructed
to slide their ﬁnger as if they were investigating the qualityof the material. No instructions were provided regarding the
speed and force of the ﬁnger. Nonetheless, the force data while
the ﬁnger velocity was in the range between 30 mm/s to 100
mm/s were used for the latter analysis. Five voluntary students(all men, 21–23 years old, all right handed, unaware of theresearch objectives) participated in the task.

Fig. 3: Photograph of the force measurement setup.

III. DATA ANALYSIS
A. Extraction into frequency ranges
The vibration frequency conveyed to the ﬁnger during tactile
exploration inﬂuences the haptic sensation [1]–[3], [5], [6].

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(a) Crumpled paper

Fig. 4: Finger position and two-axial force data during stroking
a woven resin wire mesh.
Therefore, we extracted the force data (e.g., Fig. 4) into ﬁve
frequency ranges and investigated their characteristics. The
spectra in the speciﬁc frequency ranges that were obtained byapplying the Fourier transform to the two-axial force data wereextracted and transformed into force data again by applying theinverse Fourier transform. This process plays a role of a ﬁlter
to extract the speciﬁc spectra. The data with the frequency
component in the 0–300 Hz range were extracted in ﬁve levels(0–1 Hz, 2–5 Hz, 5–30 Hz, 30–150 Hz, and 150–300 Hz),while the data with frequency over 300 Hz were ignored
because of the S/N ratio problem. Although these ranges were
determined in an arbitrary manner, in the latter analysis, wefound that they were ﬁnally divided into two ranges: 0-30 Hz
and 30-300 Hz.
B. Statistical values of two-axial force distribution
The two-axial force distribution shaped like an ellipsoid, as
shown in Fig. 5. Thus, we calculated the length of the major
and minor axes, gradient, and centroid of the estimated elliptical distribution shown in Fig. 6 to investigate the characteristic
values of the force data in each frequency range. These values
were computed by performing the eigenvalue decompositionof the covariance matrix Σ of the two-axial force data. This
calculation can be expressed by
λ1 0
Σ = (υ1 υ2 )(1)
(υ1 υ2 )T
0 λ2
where λ1 , λ2 (λ1 >λ2 ) are the eigenvalues of Σ and υ1 , υ2
∈ R2×1 are the corresponding eigenvectors (|υ1,2 |=1).
The major axis indicates the direction along which thetwo-axial force data vary the most and its length matches(b) Polyester clothFig. 5: Filtered raw data of the two-axial forces while stroking
a (a) crumpled paper and a (b) polyester cloth measured fromthe same participant. The component greater than 30 Hz are
not shown for visual clarity.

the volume of this variance. The major and minor axes are
orthogonal and their lengths√ can be calculated by the squareroot of each eigenvalue ( λ). The length of minor axis waslarger than the noise level caused during the measurement.
The gradient of the ellipsoid was deﬁned as the angle fromthe normal direction (∠υ1 ). The two-axial force distributionsof 18 types of material shaped inclined ellipsoids and noneof them shaped a perfect circle. Thus, the gradient of theellipsoid can be always determined and indicates the varianceratio of the normal and shear forces. The ratio between the
two axial lengths was also calculated as a variable that is
not inﬂuenced by how strong the touch is performed. Thisratio was calculated by dividing the length of the major axis
by the length of the minor axis (≥ 1). Thus, this indicatesthe proportion of force variation toward the directions of the
major and minor axes, which means the shape of the force
distribution. The shape becomes close to a circle when the ratio
is approximately 1 and becomes close to a narrow ellipsoid

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TABLE I: Euclidean distances among the vectors of statisticalvalues of each participant. Participants A, B, and C exhibiteda high similarity.
Participant

A

B

C

D

E

A
B
C
D

–
–
–
–

21.9
–
–
–

21.1
24.7
–
–

26.0
23.8
27.6
–

29.5
27.1
29.2
22.4

component analysis to the data of these three participants. We
also conﬁrmed that the results obtained by analyzing the dataof these three participants were similar to that by analyzing
the data of all participants.
Fig. 6: Statistical values calculated from the elliptical distribution.
when the ratio is large. Although the centroid of the elliptical
distribution was calculated as the mean value of the data, we
did not discuss about it because of its dependence on theindividual ﬁnger force.
Based on this, we applied in the latter analysis the two
axial lengths, their ratio, and gradient calculated from thetwo-axial force distribution in each frequency range as thestatistical values that indicate the characteristics of the data.
These values actually differed among the materials, and thus,
the shape of the two-axial force distribution differed between
arbitrary two materials, for instance, the crumpled paper andthe polyester cloth as shown in Fig. 5. For instance, orange
ellipsoids composed by the two-axial forces at 2-5 Hz weredistinct between these two types of materials. The ellipsoidmeasured for a crumpled paper was shorter and more circularand steeper than that of for polyester cloth. Furthermore,
the yellow ellipsoids at 5-30 Hz were substantially different
between a crumpled paper and polyester cloth.
C. Principal components of the statistical parameters
We investigated the covariance of the the statistical values
obtained from the two-axial force data to narrow them. The
principal component analysis is the representative method toreveal the covariance of the variables. Thus, we applied thismethod to four statistical values, which are the two axial
lengths, their ratio, and gradient in the ﬁve frequency ranges(in total 20 variables). The principal component coefﬁcients ofthe 20 variables, each of which was transformed into a z -score
were calculated. The calculated coefﬁcients were rotated in thepromax method for easier interpretation of the components.
The friction induced during tactile exploration differs
largely among individuals [18], and thus, we analyzed thedata of three participants that exhibited similar trends. We
judged the similarity among the data of the participants by
the Euclidean distance between the vectors, including 20
statistical values, as shown in Table I. The distances among
three participants (A, B, and C) were small, which meant thatthey exhibited high similarity. Thus, we applied the principal

IV. R ESULTS
By applying the principal component analysis to the dataof three participants who exhibited high similarity, ﬁve components were extracted. These components explained 80%of the data variation. Table II summarizes the results obtained by rotating the coefﬁcients in the promax method. The
oblique rotation angle between each component after rotationin promax method was sufﬁciently small. This afﬁrms thatthe orthogonality between the components was still kept evenafter the rotation. Thus, the ﬁve components extracted by the
analysis are independent to each other.
From the component coefﬁcient, the ﬁrst component implied
the length of the two axes of the elliptical distribution in
the high frequency range (30–300 Hz). These coefﬁcientswere of the same sign; thus, the variances of the two axial
lengths were linked, e.g., increasing and decreasing together.
The second component was the ratio of the two axial lengthsof the elliptical distribution in the low frequency range (0–30 Hz). The third and ﬁfth components were the gradientsof the elliptical distribution in the high and low frequencies,
respectively. The fourth component was the length of the twoaxes of elliptical distribution in the low frequency range (0–30Hz). Similar results were obtained by analyzing the data of allparticipants.
V. D ISCUSSION : E XPLANATION OF COMPONENTS
The axial length of the elliptical distribution indicates theforce variation in the axis direction. This corresponds tothe amplitude when the behavior of the two axial forces is
considered as a vibration in the time domain. In addition, thiscorresponds to the power density in the frequency domain,which was extensively investigated in previous studies. The
low-frequency component (comp. 4) and the high-frequency
component (comp. 1) were extracted independently in thisstudy. Thus, we suggest that the interaction forces during
tactile exploration can be discussed separately in the low andhigh frequencies. This result is supported by the fact thatthe mechanical receptor unit FAI, which respond to the lowfrequency component, and FAII, which respond to the highfrequency component, work selectively by the frequency range[19]. Furthermore, the two-axial force distribution in high3032

TABLE II: Component coefﬁcients after rotation in the promax method. Relatively small values are gray.
Variables

Frequency [Hz]Component 1
Power ofhigh freq.

Component 2
Regularity of
two axial forces

Component 3(Friction coeff.
at high freq.)

Component 4
Power oflow freq.

Component 5
Friction coeff.
at low freq.

Major axis (Length)

0–1
2–5
5–30
30–150150–300

-0.10
-0.09
0.19
0.46
0.52

0.46
0.47
0.18
0.00
-0.06

0.20
0.05
-0.11
0.04
0.10

0.17
0.37
0.30
-0.01
-0.03

-0.08
0.11
0.00
0.02
-0.06

Minor axis (Length)

0–1
2–5
5–30
30–150150–300

-0.02
-0.09
0.09
0.42
0.48

0.03
-0.05
-0.06
-0.08
-0.05

-0.03
-0.01
0.03
0.04
0.09

0.54
0.46
0.40
0.04
-0.05

-0.16
0.08
0.00
-0.01
0.03

Ratio of two axes

0–1
2–5
5–30
30–150150–300

-0.03
-0.02
0.07
0.19
-0.07

0.37
0.45
0.38
0.21
0.07

0.09
0.06
-0.24
-0.20
0.11

-0.19
-0.15
0.00
-0.10
-0.02

0.00
0.02
0.03
-0.04
-0.31

Gradient (Angle)

0–1
2–5
5–30
30–150150–300

-0.04
0.00
0.00
0.07
0.14

0.05
0.03
-0.04
0.11
0.04

0.06
-0.04
0.11
0.66
0.62

-0.12
0.00
0.03
-0.08
0.06

0.57
0.52
0.49
0.13
-0.08

frequency range of slippery materials (e.g., drawing paper andcork with powder) were relatively small ellipsoid as shown in
Fig. 7 (top). On the other hand, those of the relatively sticky
materials (e.g., rubber, coated paper and polyethylene sheet)were large ellipsoids. Thus, the size of the two-axial force
distribution in high frequency (comp. 1) implies the frictioninduced vibration. They are typically greater than 50 Hz [20].
The ratio between the two axial lengths which was extracted as the second component indicates the shape of the
distribution. Especially, whether the force distribution is sharp
or round depends on whether the relationships between the
two axial forces are regular or not. When the shape issharp, the correlation between the two axial forces is large,
and the relationships between the two axial forces are moreregular than when the distribution is round, which means thatthe relationships between the two axial forces are random.
Therefore, this component indicates the correlation between
the variation of normal and shear forces. We suggest that thiscorrelation explains the unevenness of the surface because
the phase difference between the normal and shear force
become large when the surface is undulated [8], [21]. In
fact, the two-axial force distribution of uneven material (e.g.,
woven resin wire mesh and corrugated paper) were roundand the distribution of relatively even material (e.g., coatedpaper, cork, and aluminum) were sharp ellipsoid (see Fig.
7 (middle)). Thus, this component may be related to themacroscopic roughness of the material.
The gradients of the elliptical distribution, which wereextracted in the third and ﬁfth components, indicated the ratios
of variation of the two axial forces, and they correspond to thefriction coefﬁcient. Especially, the variation ratio in the two
axial forces in the low frequency range corresponds to thekinetic friction coefﬁcient. A larger gradient corresponds to alarger friction coefﬁcient. In fact, the material that induce large
friction (e.g., sticky rubber and aluminum) and the material
that induce small friction (e.g., coarse Japanese paper, wood,and drawing paper) were distinguished by the score of thiscomponent (see Fig. 7 (bottom)). Moreover, the variation ratio
in the high frequency range was not discussed in the previous
studies. Although the meaning of this component is unclear,
it was deﬁnitely extracted as an independent factor from othercomponents.
As stated above, we determined not only the variation of theeach of the two axial forces but also the shape and gradient of
the two-axial force distribution, which have not been discussed
in previous studies. We acquired the results by investigating
the stochastic relationships between the two axial forces. The
relationships between the ﬁve components obtained in thisstudy and the haptic sensation of materials should be further
investigated.
VI. C ONCLUSION
In the present study, we investigated the stochastic relationships between the normal and shear forces induced between
the ﬁngertip and 18 types of material sheets by measuring
these forces. We calculated four statistical values in ﬁve
frequency ranges (0–1 Hz, 2–5 Hz, 5–30 Hz, 30–150 Hz,
and 150–300 Hz) obtained by the variation of the force
data. These were the lengths of the major and minor axes,
their ratio, and gradient of the elliptical distribution fromthe two-axial force data. A principal component analysis wasapplied to the data of participants to extract either covariant or
independent information among these statistical values. As aresult, it was revealed that the sizes and gradients of the twoaxial force distributions in both the low- and high-frequency
ranges and the shape of the distribution in the low frequency

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Fig. 7: The distribution of 20 samples of material used in theforce measurement on the component axis. Mean componentscores among the three participants. The two components are
not shown because of their unclarity and page limitation.

range differed among the materials. Especially, the size of the
two-axial force distribution in high frequency range implies
the power of the friction-induced vibration and the shapeof this distribution implies the unevenness of the surface.
Also, the gradient of the two-axial force distribution in low
frequency range indicates the friction coefﬁcient. Moreover,the gradients of the two axial forces at high frequency and theirdistribution shapes are new parameters that can be revealed byinvestigating the stochastic relationship between the two axial
forces. The results of this study indicate the importance offocusing on the stochastic relationships between the normal
and shear forces induced during tactile exploration to reveal
the mechanism of texture sensations.
ACKNOWLEDGEMENTS
This work was partly supported by MEXT KAKENHI(17H04697).

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