Article

Apparent Randomness of the Normal-Force Dependence of the
Coefficient of Friction between a Bare Finger and Artificial Skin
under Active Tactile Exploration
Koki Inoue 1,† , Shogo Okamoto 2, *,† , Yasuhiro Akiyama 1 and Yoji Yamada 1
1

2

*
†

Citation: Inoue, K.; Okamoto, S.;
Akiyama, Y.; Yamada, Y. Apparent
Randomness of the Normal-Force
Dependence of the Coefficient ofFriction between a Bare Finger andArtificial Skin under Active Tactile
Exploration. Appl. Mech. 2022, 3,
649–662. https://doi.org/10.3390/
applmech3020038
Received: 11 March 2022

Department of Mechanical Systems, Nagoya University, Nagoya 486-0814, Japan;
inoue.koki@j.mbox.nagoya-u.ac.jp (K.I.); yasuhiro.akiyama@mae.nagoya-u.ac.jp (Y.A.);
yoji.yamada@mae.nagoya-u.ac.jp (Y.Y.)
Department of Computer Science, Tokyo Metropolitan University, Hino 191-0065, Japan
Correspondence: okamotos@tmu.ac.jp
These authors contributed equally to this work.

Abstract: When a finger actively slides over a surface, contact conditions including the contact area,
sliding speed, and finger moisture naturally fluctuate. These random fluctuations lead to an apparentchange of frictional properties and influence tactile pleasantness. Nonetheless, this probabilistic
behavior has not been explicitly analyzed in previous studies on human fingertips. This study
investigates the dependence of the coefficients of kinetic friction on the normal force produced bysliding a bare finger over different artificial skins with seven levels of hardness. The coefficient offriction was modeled as a power function of the normal force. An experimental study that involved
sliding a finger over artificial skin surfaces was carried out under two conditions: the fingertip being
wiped by a dry cloth or a cloth soaked in ethanol. Although the exponential term was assumed
to be nearly constant for identical tribological conditions, we observed that the exponent varied
randomly and could be negative, zero, or positive. This can be attributed to the variation of grossfinger deformation that was not controlled during the observation. The probability density functionof the exponent depended on the moisture content of the finger and object hardness. The variabilityof the exponent was higher for a soft material than it was for a harder material. In other words,
for the softer materials, the exponent appears more random. Furthermore, the exponent tended
to be positive and the coefficient of friction increased with the normal force when the finger was
wiped with ethanol. These findings play an important role in understanding the frictional forces
produced during skin–skin contact in terms of determining the root cause of random variations in
the dependence of the coefficient of friction on the normal force.
Keywords: fingertip; artificial skin; friction

Accepted: 17 May 2022
Published: 21 May 2022
Publisher’s Note: MDPI stays neutral
with regard to jurisdictional claims in

1. Introduction

published maps and institutional affil-

Humans use their fingers to touch their skin in day-to-day activities. To develop
skincare products, artificial skins, or human-friendly surfaces, it is important to study
the friction generated while rubbing the skin with fingers because friction is a major
determinant of pleasantness in touch [1–3]. During active tactile exploration in our daily
lives, contact conditions naturally fluctuate. Understanding the effects of this random
fluctuation on frictional properties would be helpful to control the pleasantness of touch.
This study focuses on the relationship between the coefficient of friction (COF) and thenormal force produced when a human finger slides on artificial skin surfaces.
The frictional force acting on elastic bodies such as the skin deviates from the Amontons–
Coulomb friction laws because the coefficient of kinetic friction depends on the normal force
and sliding speed [4–6]. The frictional force on the skin surface is considered a combinationof two mechanisms, namely adhesion and deformation frictions [7–9]. In adhesion friction,

iations.

Copyright: © 2022 by the authors.
Licensee MDPI, Basel, Switzerland.
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Appl. Mech. 2022, 3, 649–662. https://doi.org/10.3390/applmech3020038

https://www.mdpi.com/journal/applmech

Appl. Mech. 2022, 3

650

the adhesive force acting on the real contact area Ar between two elastic bodies is overcomeby the interfacial shear strength τ [7]. If a sphere, imitating a finger pad, applies a load
f n on an elastic plane and the apparent contact area A, derived from the Hertzian contact
theory, and real contact area Ar are identical, the coefficient of adhesion friction µ ad is
proportional to the function of the negative exponent of the normal force, as shown below.
µ ad

=

τAr
fn

∝

fn − 3 .

1(1)

In contrast, deformation friction is produced due to the restoring force of the deformedmaterial. The deformation friction force f de f is proportional to the exponent of the normal
force [7,10], as shown below.
f de f∝4

fn 3 .

(2)

Thus, the coefficient of deformation friction, µde f , is proportional to the positive
exponent of the normal force.
µde f

∝

f de ffn∝

fn 3 .

1(3)

Hence, the coefficient of friction, µ, is generally expressed as an exponential functionof the normal force, as shown below.
µ = α fn β.

(4)According to Equations (1) and (3), β is negative if the adhesion friction is dominant
and positive if the deformation friction is dominant, respectively, as in Figure 1. Various
studies investigating the frictional forces acting on human skin focus on their dependence
on the normal force, i.e., the exponent β in (4) [4,11–17]. Because adhesion friction is adominant force acting on dry human skin [7,18–20], the real contact area responsible forthe adhesion friction and its dependence on the normal force are two important factors
as well [21]. The β values reported in these studies do not necessarily follow the range
shown in Figure 1 partly because the Hertzian contact theory is not accurately applicableto human finger contact.
Adhesion friction
is dominant

1
3

Amonton’s Deformation friction
law
is dominant

0

1
3

ߚ

Figure 1. Relationship between the exponent β and the frictional mechanisms.

Several studies have analyzed the COF between the skin and a rigid probe and the
COF between a finger and a rigid plane. Most studies have reported that the COF decreased
with increasing normal force. The exponent of the normal force, β, in (4) was approximately
equal to −1/3 [4,13,16,18]. Mahdi et al. investigated the friction between a skin model anda rigid probe and observed that the ratio of the adhesion friction to deformation friction
was approximately 8:2 [20]. Adhesion friction was dominant under dry conditions, i.e., βfor the friction between the skin and rigid bodies was negative under dry conditions.
Several studies have calculated the frictional force for different surface roughness values, lubrication conditions of the contact surfaces, and contact states. Derler et al. measured
the friction between the human skin and glass plates for different surface roughness values

Appl. Mech. 2022, 3651

and lubrication conditions. They observed that β ranged from approximately −0.5 to −0.2
when adhesion friction was dominant, and from −0.1 to 0.2 when deformation friction wasprominent [11]. Ules et al. studied the variation of the frictional force when a finger slid
over a glass surface with varying roughness values. The value of β was approximately 0 fora rough surface and approximately −0.5 for a smooth surface [17]. Han et al. measured the
maximum coefficient of static friction between an acrylic plate and a finger while varying
the angle of contact of the finger. They observed that β ranged from −0.8 to −0.6 and wasdependent on the finger angle and the participants [14]. These values were less than thevalue of β calculated from the Hertzian contact theory, which was equal to −1/3. According
to these studies, the dependence of the COF on the normal force differs for different surfaces
and conditions, such as roughness and lubrication. Nevertheless, in part, because of therandom variation of the gross finger deformation, β actually varies under the identicalsurface conditions and its probabilistic variation has not been thoroughly discussed.
Although previous studies have analyzed the friction between the skin and rigid
bodies, very few studies have focused on the frictional force when a finger or fingerlike probes slide over the skin [22–24]. For example, Nonomura et al. investigated thefrictional phenomena between the skin and soft materials under non-stationary sliding
conditions [22,24]. Experiments were performed in the present study on artificial skinsamples that replicated the hardness and texture of the human skin and studied thedependence of the COF on the normal force. We studied the dependence of the COF on
the normal force by applying (4), which has been used in previous studies to calculate thefriction between the skin and rigid bodies. We observed in our previous study [25] that βrandomly assumed positive and negative values while calculating the friction between afinger and artificial skin, even across identical tribological conditions. For example, β can
attain a positive value during a single active sliding motion of the finger over artificial skin;however, if a similar motion is replicated, the value of β could be positive, zero, or negative.
This variation was completely random. The random variation of β while calculating humanskin friction has not been discussed in previous studies. The objective of this study is toinvestigate the apparent randomness of β in the coefficient of kinetic friction that changes
for each active finger slide.
This study builds upon our previous one [25], where random fluctuation of β was
observed by investigating the dependence of the randomness of β on the hardness of theartificial skin. The friction acting on the surface of the skin changes with varying moisture
contents at the contact interface [7,11,12,26,27]. The random values of β can be attributed
to the transient nature of skin sweat. Hence, we experimented on two differently treatedskin surfaces and compared the results obtained under both conditions. In the first case,the fingertip was wiped with a dry cloth before each trial. The second case involved wiping
the fingertip with a cloth soaked in ethanol and drying it before each trial to moderately
control the moisture content on the finger surface. Notably, some of the data presented inthis study were acquired in our previous study [28].
2. Methods
2.1. Measurement of the Two-Axial Forces and Sliding Speed
The normal and shear forces generated by rubbing artificial skin with a finger were
measured in self-assembled axial force sensing units (Figure 2) [29]. The normal force wasmeasured by using two uniaxial force sensors (9313AA2, Kistler, Winterthur, Switzerland) installed in the lower part of the device, and the shear force was measured using auniaxial high-sensitivity force sensor (9217A, Kistler, Switzerland) centrally attached to
two metal pieces of the device. The signals from both normal and shear force sensors were
amplified by two charge amplifiers (5073A2 and 5015, Kistler, Switzerland; nominal drifts
were 0.005 N/s and 0.0003 N/s, respectively). Finger movement was measured using two
wired encoders (MTL-12, MTL. Co., Yokohama, Japan ) by winding a wire, attached to theencoders, around the finger. The resolution of this device was nominally 0.01 mm. The
sampling frequency was set to 2 kHz.

Appl. Mech. 2022, 3

652

Linear encoder(MLS-12, MTL)

Wire

Artificial skin
(Bioskin, Beaulax)

Leaf spring
to cancel
the moment

60 mm

100 mmShear force sensor(9217A, Kistler)

Normal force sensor(9313AA2, Kistler)

Figure 2. Measurement setup. The normal force was measured using two uniaxial force sensors
installed in the lower part of the device, and the shear force was measured using a uniaxial force
sensor fixed in the upper part of the device. The finger movement was measured through two wire
encoders by winding a wire attached to the encoders around a finger.

2.2. Artificial Skins Used in the Experiment
Commercially available artificial skins (Bioskin, Beaulax Ltd., Saitama, Japan ), with sevenlevels of hardness, were used in the experiment. The softest level of hardness was represented by 1, whereas the hardest was represented by 7. Their thicknesses were 5 mm,
and their surfaces were covered with a thin film of thermoplastic polyurethane. The surface
asperity of this film is similar to that of human skin. The average roughness value, R a ,of the artificial skin was 7.6 ± 1.9 µm [30]. Table 1 summarizes the shore AO hardness
and Young’s moduli of artificial skins and finger pads. For the measurements, we used adurometer (GS-721N, Teclock, Japan) following the ISO 7619–1 guideline and an indentation tester (Yawasa, Tech Gihan Co. Ltd., Kyoto, Japan). The finger pad values account for
the average and standard error among all the participants (11 participants). As listed inTable 1, we tested artificial skins softer than, close to, and harder than the finger pads.
Table 1. Young’s moduli (E) and Shore AO hardness of artificial skin models and human fingers.
For the finger, the mean and standard error are shown.
Hardness level
Shore AO Hardness
E (kPa)

1
3.2
69.4

2
7.8
74.2

3
9.6
79.6

4
11.0
85.2

Hardness level
Shore AO Hardness

5
16.9

6
18.0

7
19.1

Finger
7.9 ± 2.1

E (kPa)

122.4

131.7

141.8

74.4 ± 18.6

2.3. Task
The participants rubbed the surface of the artificial skin attached to the top of themeasurement instrument using the index finger of their right hand. They were asked to
rub the surface in a manner identical to that of an individual examining the quality of thematerial of the surface. The participants were instructed to lay down their fingers and rub
the surface of the artificial skin with their finger pad while maintaining the same posture
as much as possible (see discussion about the potential effects of contact angle). There were
no instructions regarding the finger speed and force to be applied. The experiments wereconducted under two conditions: (1) where participants wiped the finger and artificial
skin using a dry cloth after each trial and (2) where participants wiped the finger using
a cloth soaked in ethanol and then dried to lower the moisture content. The results forcase 1 were sourced from our previous study [28]. The measurement periods for cases 1

Appl. Mech. 2022, 3

653

and 2 were 10 s and 5 s, respectively. The measurement time for case 2 was less than thatof case 1, to minimize the effect of sweat on the fingertips during measurement. Threetrials were performed for case 1 and 10 trials were performed for case 2 using each skinmodel. This increased the amount of data collected for case 2, thereby improving theexperimental accuracy. The experiments for the two cases were conducted on the same day
for each participant in an air-conditioned room at 20 ◦ C. We did not control and record thehumidity. The participants rubbed the artificial skin surface on multiple occasions during
the measurement process. Seven types of skin models were tested randomly.
2.4. Participants
Eleven university students participated in the experiment. All of them participatedin case 1. However, one of them did not participate in case 2. The participants were all
males and in their early 20s. The participants provided written informed consent and were
unaware of the objectives of the study before the experiment was performed.
2.5. Analysis
Before data analysis, we removed the linear drift component that was defined by thezero force levels before and after a slide. The measured normal and shear forces were
subjected to zero-phase filtering using a low-pass FIR filter with a passband frequency of
20 Hz. Only the data extracted from the trials that involved clear sliding of the finger on theartificial skins were used to analyze the kinetic friction. This ensured that the instantaneous
normal force f n and shear force f s were at least 0.05 N, and the sliding velocity v was atleast 20 mm/s. Evident stick-slip phenomena that could be visually detected by checking
the shear force profiles were not observed in our setup. The value of the COF µ was
calculated as a ratio of f s to f n for each moment. We plotted the instant values of f n andµ for each finger slide as in Figure 3 and approximated their relationship according to (4)using the Curve Fitting Toolbox of MATLAB 2019a (Mathworks Inc., Natic, Massachusetts,
USA). Only the valid slides, i.e., slides in which f n ≥ 0.05 N, f s ≥ 0.05 N, and v ≥ 20 mm/s
during a single sliding motion, were used for the subsequent analysis. We analyzed 1130
and 725 slides for cases 1 and 2, respectively. The minimum and maximum f n values
in each valid slide were 0.74 ± 0.49 N and 1.84 ± 0.61 N (mean and standard deviation),
respectively. This range covers that reported for rubbing fine abrasive papers [31]. The
number of slides obtained within the stipulated time for each case varied from individual
to individual. In general, the sliding velocity affects the COF [26,32], but its effect is small
for the artificial skins used in this study. For example, for the softest and the hardest
artificial skins (hardness levels 1 and 7) in case 1, the COF only changed by 0.04 and 0.01,
respectively, when the sliding velocity changed by 10 mm/s [28], and we did not consider
its effects.

Coefficient of friction: ߤ

(a)

(b)

1.2

0.6
0.8
0.4
0.4
0

0.8

0.2
1.2

1.6

Normal force: ݂௡ (N)

2

0

0.5

1

1.5

2

2.5

Normal force: ݂௡ (N)

Figure 3. Data plots and fitting results of the relationship between µ and f n . Red curves represent thefitting curves, and the blue dots correspond to the observed coefficients of friction. (a) β is positive(β = 0.24). (b) β is negative (β = −0.12).

The exponent β was classified into three categories, namely, significantly positive,
significantly negative, and not significantly different from zero. The proportions of theseAppl. Mech. 2022, 3654

categories in the results were then calculated. The β values were classified according toa significance level of 5% and the effect size, i.e., Cohen’s d [33]. The β values that did
not meet the conditions for either the significance level or effect size were assumed to beequal to 0. The two criteria for the effect size were set as d = 0.8 and d = 0.5 to ensurethat the results were not dependent on the criterion of the chosen effect size. In addition,a chi-square test was conducted to compare the proportions of the β values between thetwo cases for each of the seven artificial skins. If the chi-square test indicated a significant
difference, a residual analysis was performed to analyze the β category which differedsignificantly between the two cases. The significance level was set to 5%, and the Bonferroni
correction was applied in these tests.
The average R2 values of curve fitting for an effect size of d = 0.8 are listed below. For
the slides with β > 0 in case 1, the maximum average R2 value was 0.67 for a hardness levelof 1. The minimum value of R2 in case 1 was 0.55 for a hardness level of 4. In the case ofslides with β < 0 in case 1, the maximum average value of R2 was 0.69 for a hardness levelof 3 whereas the minimum value was 0.65 for a hardness level of 5. The slides with β > 0in case 2 reported a maximum R2 value of 0.70 for a hardness level of 6 and a minimum
value of 0.58 for a hardness level of 3. The slides with β < 0 in case 2 reported maximum
and minimum R2 values of 0.69 for a hardness level of 2 and 0.60 for a hardness levelof 7, respectively. The R2 value is the square value of correlation coefficient for whichempirically 0.7–0.9 are considered a strong correlation (e.g., [34]). We consider that R2
values greater than 0.49 are acceptable.
The probability density function of β was calculated via kernel density estimation
because the sample distribution of β did not appear to be popular among the distributionfunctions. The kernel function was a normal distribution, and the bandwidth was definedby Silverman’s method. For this purpose, we used all the participants’ trials to compute thedensity function for each hardness level because each density function needs to be drawn
based on a satisfactory number of samples.
3. Results
3.1. Probability Density Functions of β
The probability density functions of β for each hardness value in cases 1 and 2 areshown in Figures 4 and 5, respectively. The horizontal axis indicates the value of β, and thevertical axis indicates the estimated probability density of β. The solid lines represent theprobability density functions calculated from the slides by artificial skin hardness. The
red, blue, and black dotted lines represent the probability density functions of the positive,
negative, and insignificant β categories, respectively. The effect size d = 0.8 was used tocategorize the β values. It is evident from Figures 4 and 5 that despite sliding a finger over
the artificial skin under identical friction conditions, regardless of the material hardness
and conditions, the value of β was not constant and varied probabilistically.
3.2. Second Moments and Interquartile Range of β
The second moments σ2 and the interquartile ranges of the probability distributions of
β for all artificial skins are shown in Table 2 and Figure 6 based on the AO hardness scoresof the artificial skins. Negative correlations were observed between the second moments
and the AO hardness scores at r = −0.54 and r = −0.86 for cases 1 and 2, respectively. The
correlation was significant in case 2 (p < 0.05). Negative correlations were also observed
between the interquartile ranges and AO hardness scores at r = −0.82 and r = −0.32 forcases 1 and 2, respectively. The correlation was significant in case 1 (p < 0.05). We observedthat the magnitude of the random variation of β tended to decrease with the increasing
hardness of the artificial skin. However, this finding is inconclusive because the degree ofnegative correlation changes depending on the type of the variability index.

Appl. Mech. 2022, 3

Probability density

Probability density

655

1

1
0.5

0
2

0.5

1

0

1

2

ー1

0

1

ߚ

Hardness lv. 3(3rd softest) (n=145)

Hardness lv. 4(Middle) (n=168)

2

2
1.5
1

0.5

0.5

2

1

0

1

2

0
2

0

1

ߚ

Hardness lv. 5

Hardness lv. 6(2nd hardest) (n=177)(3rd hardest) (n=157)

1.5

1

1

0.5

0.5
1

0

1

2

2

1.5

0
2

1

ߚ

2



Probability density

0
2

ߚ

1

0
2

Probability density

Hardness lv. 2(2nd softest) (n=176)

Hardness lv. 1(Softest) (n=145)

0
2

1

0

1

2



Hardness lv. 7(Hardest) (n=162)
2

All slides
Slides with   0
Slides with ~0Slides with   0

1.5
1
0.5
0
2

1

0

1

2


Figure 4. Probability density functions of β under case 1, in which the finger pad was wiped by a dry
cloth. Solid lines include all the slides, and red, blue, and black dotted lines are the distributions ofpositive, negative, and insignificant β (β ∼ 0), respectively. d = 0.8 for categorizing the sign of β.

Appl. Mech. 2022, 3

Probability density

Probability density

656

2

1

1.5
1

0.5

0.5
0
2

1

0

1

2

1

Hardness lv. 3(3rd softest) (n=107)

Hardness lv. 4(Middle) (n=93)

2

1.5

1.5

2

1

1

0.5

0.5
1

0

1

2

0
2

1

0

1

ߚ

ߚ

Hardness lv. 5

Hardness lv. 6(2nd hardest) (n=107)(3rd hardest) (n=112)

2

1.5

1

1

0.5

0.5

0
2

0

ߚ

2

1.5

1

ߚ

1

0

1

2



Probability density

0
2

2.5

0
2

Probability density

Hardness lv. 2(2nd softest) (n=97)

Hardness lv. 1(Softest) (n=103)

0
2

1

0

1

2



Hardness lv. 7(Hardest) (n=106)
3

All slides
Slides with   0
Slides with ~0Slides with   0

2
1
0
2

1

0

1

2


Figure 5. Probability density functions of β under case 2 (ethanol condition). Solid lines include all theslides, and red, blue, and black dotted lines are the distributions of positive, negative, and insignificant
β (β ∼ 0), respectively. d = 0.8 for categorizing the sign of β.

Appl. Mech. 2022, 3

657

Table 2. Second moment (and interquartile range) of β under cases 1 (dry cloth) and 2 (ethanol cloth).
A higher AO hardness value corresponds to a harder material, while higher values of the second
moment and the interquartile range indicate greater variability of β values.
Lv.

AO Hardness

Case 1Case 2

1
2
3
4
5
6
7

3.2
7.8
9.6
11.0
16.9
18.0
19.1

0.15 (0.54)
0.24 (0.47)
0.19 (0.50)
0.11 (0.29)
0.15 (0.35)
0.12 (0.33)
0.10 (0.31)

0.32 (0.30)
0.23 (0.50)
0.19 (0.23)
0.15 (0.31)
0.18 (0.36)
0.15 (0.32)
0.053 (0.18)

GatheredDistributed
GatheredInterquartile range

2nd moment of ߚ: ߪଶDistributed

Case 1 (Dry wiping)
0.6

Case 2 (Ethanol wiping)  0.54

0.6

0.4

0.4

0.2

0.2

0

0

5

10

15

20

0

Soft AO hardness Hard

5
10
15
20
Soft AO hardness Hard

Case 1 (Dry wiping)

Case 2 (Ethanol wiping)0.6

  0.82

0.6

0.4

0.4

0.2

0.2

0

0

5

10

15

20

Soft AO hardness Hard0

  0.86

00  0.32

5
10
15
20
Soft AO hardness Hard

Figure 6. Scatter plots between the second moments (dots) and the interquartile ranges (circles),and AO hardness values of artificial skins. The left and right section represent the results of case 1(dry-wipe condition) and case 2 (ethanol-wipe condition), respectively. The vertical axis indicates thesecond moments and the interquartile ranges, with greater values indicating greater variability in β.

3.3. Proportions of Positive, Negative, and Insignificant β Values
The proportions of the positive, negative, and insignificant β values for effect sizesof d = 0.8 and d = 0.5 are shown in Figures 7 and 8, respectively. The bars represent the
proportions of the positive, negative, and insignificant β values from left to right. The upperand lower figures indicate the classification results for cases 1 and 2 (ethanol condition),
respectively. The colors of the bars correspond to different participants. β randomly
assumed positive, negative, and insignificant values for all artificial skins, regardless of theexperimental conditions and the effect size criteria, as shown in Figures 7 and 8.

Participant 2

Participant 3

Participant 4

Participant 5

Participant 7

Participant 8

Participant 9

Participant 10

Participant 11

Hardness lv. 3(3rd softest)

1

0.8

0.8

0.8

0.8

0.6

0.6

0.6

0.6

0.4

0.4

0.4

0.4

0.2

0.2

0.2

0.2

0

Positive Negative00Positive Negative00Positive Negative

0

0

1

1

1

1

0.8

0.8

0.8

0.8

0.6

0.6

0.6

0.6

0.4

0.4

0.4

0.4

0.2

0.2

0.2

0.2

1

0

0

Positive Negative

Hardness lv. 5(3rd hardest)

100

Positive Negative

Hardness lv. 6(2nd hardest)

1

0.8

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0

Positive Negative00Positive Negative

0

0

1

1

1

0.8

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0

Positive Negative00Positive Negative

0

0

0
0

Positive Negative0

Positive Negative0

Hardness lv. 7(Hardest)

Case 1(wiped by dry paper)

Positive Negative

Probability of occurrence0

Hardness lv. 4(Middle)1Case 1(wiped by dry paper)

Hardness lv. 2(2nd softest)

1

Case 2(wiped by ethanol cloth)

Hardness lv. 1(Softest)

Participant 6

Positive Negative0

Positive Negative

0

Case 2(wiped by ethanol cloth)

1

658

Participant 1

Probability of occurrence

Probability of occurrence

Probability of occurrenceAppl. Mech. 2022, 3

Figure 7. Probability of occurrence of positive, negative, and insignificant β values. The upper andlower sections represent the results under case 1 (wiped by a dry cloth) and case 2 (wiped by ethanol),respectively. Cohen’s d = 0.8. Asterisks * and ** indicate statistical differences in the proportions
between cases at p < 0.05 and p < 0.01, respectively. The sum of the proportions for all individuals is
1. Note that the number of slides differs from participant to participant.

The proportion of negative β values in case 2 was less than that of case 1 for several
hardness levels. This trend was consistent for the two arbitrarily selected criteria of effectsize. In particular, there was a significant reduction in the proportion of the negative βvalues for hardness levels of 1, 3, 4, and 5 when the criterion for the effect size was d = 0.8.

Participant 2

Participant 3

Participant 7

Participant 8

Participant 9Hardness lv. 1(Softest)

Hardness lv. 2(2nd softest)

1

Participant 4

Participant 5

Participant 10

Participant 11

Hardness lv. 3(3rd softest)

1

0.8

0.8

0.8

0.6

0.6

0.6

0.6

0.4

0.4

0.4

0.4

0.2

0.2

0.2

0.2

Positive Negative00Positive Negative00Positive Negative

0

0

1

1

1

1

0.8

0.8

0.8

0.8

0.6

0.6

0.6

0.6

0.4

0.4

0.4

0.4

0.2

0.2

0.2

0.2

Positive Negative

1

0
0Positive Negative

Hardness lv. 5(3rd hardest)

1

0

0

Positive Negative

Hardness lv. 6(2nd hardest)

1

0.8

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0

Positive Negative00Positive Negative

0

0

1

1

1

0.8

0.8

0.8

0.6

0.6

0.6

0.4

0.4

0.4

0.2

0.2

0.2

0

Positive Negative00Positive Negative

0

0

0
0

Positive Negative0

Positive Negative0

Hardness lv. 7(Hardest)

Case 1(wiped by dry paper)

0

Hardness lv. 4(Middle)

1

0.8

0

Participant 6

Case 1(wiped by dry paper)

Participant 1

Case 2(wiped by ethanol cloth)

1659

Positive Negative0

Positive Negative

0

Case 2(wiped by ethanol cloth)

Probability of occurrence

Probability of occurrenceAppl. Mech. 2022, 3

Figure 8. Probability of occurrence of positive, negative, and insignificant β values. The upper andlower sections represent the results under case 1 (wiped by a dry cloth) and case 2 (wiped by ethanol),respectively. Cohen’s d = 0.5. Asterisks * and ** indicate statistical differences in the proportions
between cases at p < 0.05 and p < 0.01, respectively. The sum of the proportions for all individuals is
1. Note that the number of slides differs from participant to participant.

4. Discussion
It was observed that the exponential component of the normal force β, which was
obtained from (4), was not constant and varied randomly, despite the same finger rubbing
the same artificial skin surface under identical lubrication conditions. One of the reasons forthis random variation can be a change in the real contact area due to transient changes in
the amount of moisture on the fingertips. Because the fingertips were wiped with ethanoland dried in case 2, we expected a reduction in the value of the variability of the probability
density function of β, thereby indicating that the variability of β was smaller in case 2 thanAppl. Mech. 2022, 3660

in case 1. However, a comparison between the variability of β in both the cases, as shownin Table 2 and Figure 6, proved that there was no reduction in the variability of β in case
2. This implies that the transient variation of the moisture content on the fingertips was
not responsible for the random variation of β. Nonetheless, it is noted that wiping with an
ethanol-soaked cloth only moderately dried the fingertip.
However, the fingertip moisture influences the dependence of the COF on the normal
force. The application of alcohol reduced the average proportion of the negative β values
for all artificial skins from 27.5% in case 1 (wiped by a dry cloth) to 16.8% in case 2 (wipedby ethanol). The fingertips were considered to be naturally moisturized in case 1. Under
such conditions, the COF is proportional to the negative exponent of the normal force [8,27],
as shown in (1). In case 2, the contribution of adhesion friction to the total frictional
force was decreased, whereas the contribution of deformation friction was increased. The
value of the deformation friction was proportional to the positive exponent of the normal
force, as shown in (3). Thus, the proportion of negative β values is reduced, whereas
the proportions of β = 0 and β > 0 are increased when the moisture on the fingertips ismoderately controlled in case 2.
One reason for the randomness of β could be the differences in the manner of rubbing.
Although participants were instructed not to change their rubbing mannerisms during
the experiment, minor variations were unavoidable. Han et al. demonstrated that βvaried within a small range of approximately −0.8–−0.6 for a variation of 30 degrees in
the contact angle between a finger and an object [14]; however, they did not report that β
assumed positive and negative values. In addition, the real contact area Ar , which is a major
determinant of friction [35], can vary randomly even for the same normal force [21,36].
Such random variations of Ar might have affected the random variation of β. Further,
the present experiment did not control the rate of normal force and finger velocity. In otherwords, the frictional phenomena were not steady-state, and the variation of β might havebeen caused by the viscoelastic properties of finger pads. There is no evidence that thefingers were constantly sliding on the artificial skins and they might have rolled or twisted
partially. Tribological phenomena during rolling or twisting are not represented by (4),based on which our analysis was. These aforementioned factors could have collectively
caused the random variation β.
There were negative correlations between the variability, i.e., second moment andinterquartile range, and AO hardness for both cases, as shown in Table 2 and Figure 6.
When rubbing soft objects, a slight variation in the normal force led to a change in thecontact status i.e., the contact state was mutable, and β was likely to assume a different
value for each finger slide. Rubbing a hard, artificial skin resulted in minimal deformationof the artificial skin with a less mutable contact status, irrespective of any variation in thenormal force. As a result, the likelihood of β assuming random values is reduced. The
variation in contact stability due to the different hardness values of the artificial skin isreflected in the relationship between the variability of the normal force dependence of the
COF and the hardness. Note that we did not find any singular phenomenon when thefinger slid over the artificial skins of levels 2 or 3, of which hardness were close to those of
the participants’ finger pads.
We tested several types of artificial skins manufactured by the same company. Hence,
we are not sure whether similar findings can be found for other artificial skins or objects.
However, there is no special reason to deny potentially similar observations for otherobjects as long as the major cause of the random variation of β is the variation of the contact
status between the fingertip and object surfaces.
5. Conclusions
The current study investigated how the dependence of the COF on the normal force
fluctuates while rubbing an artificial skin sample with a finger. Participants were made toactively slide their fingers on different kinds of artificial skin under two different conditions,
namely, wiping the finger with a dry cloth and wiping the finger with a cloth soaked inAppl. Mech. 2022, 3

661

ethanol to control the moisture content on the finger. The dependence of the COF on
the normal force varied randomly with each finger slide, despite maintaining identicaltribological conditions. The randomness was more prominent for the softer artificial skins.
The moisture content of the fingertips influenced the dependence of the COF on the normal
force. The COF either increased or remained constant with increasing normal force when
the finger pad was wiped using ethanol. Although the dependence of the COF on thenormal force may be influenced by the variation of the contact state between the artificial
skin and a finger and transient changes in the moisture content of the fingertip, there are
additional factors that are responsible for the random variation of the dependence of the
COF on the normal force. These findings help in designing surfaces, such as artificial skins
and cosmetic products with stably low friction, which leads to pleasant and preferred
textures [1–3].
Author Contributions: Conceptualization, K.I. and S.O.; methodology, K.I. and S.O.; software, K.I.;
validation, K.I. and S.O.; formal analysis, K.I. and S.O.; investigation, K.I. and S.O.; resources, K.I.; data
curation, K.I.; writing—original draft preparation, K.I. and S.O.; writing—review and editing, K.I.,
S.O., Y.A. and Y.Y.; supervision, S.O., Y.A. and Y.Y.; project administration, S.O.; funding acquisition,
S.O. All authors have read and agreed to the published version of the manuscript.
Funding: This research was funded by MEXT Kakenhi (20H04263 and 21H05819).
Institutional Review Board Statement: The study was conducted in accordance with the Declarationof Helsinki, and approved by the Institutional Review Board of the School of Engineering, Nagoya
University (#15-12).
Informed Consent Statement: Written informed consent was obtained from all subjects involved in
the study.
Data Availability Statement: Data are accessible by contacting the corresponding author.
Conflicts of Interest: The authors declare no conflict of interest.

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