ORIGINAL RESEARCH
published: 04 January 2019
doi: 10.3389/fpsyg.2018.02654

Hardness Perception Based on
Dynamic Stiffness in Tapping
Kosuke Higashi 1 , Shogo Okamoto 1*, Yoji Yamada 1 , Hikaru Nagano 2 and Masashi Konyo 2
1

Department of Mechanical Systems Engineering, Nagoya University, Nagoya, Japan, 2 Graduate School of Information
Sciences, Tohoku University, Sendai, Japan

Edited by:
Joachim Lange,
Heinrich Heine Universität Dsseldorf,
Germany
Reviewed by:
Pablo De Gracia,
Midwestern University, United StatesDavid Gueorguiev,
Max Planck Institute Stuttgart,
Germany*Correspondence:
Shogo Okamoto
okamoto-shogo@mech.nagoya-u.ac.jp
Specialty section:
This article was submitted toPerception Science,a section of the journal
Frontiers in Psychology
Received: 18 June 2018
Accepted: 10 December 2018
Published: 04 January 2019
Citation:
Higashi K, Okamoto S, Yamada Y,
Nagano H and Konyo M (2019)
Hardness Perception Based on
Dynamic Stiffness in Tapping.
Front. Psychol. 9:2654.
doi: 10.3389/fpsyg.2018.02654

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A human can judge the hardness of an object based on the damped natural vibration
caused by tapping the surface of the object using a fingertip. In this study, we investigated
the influence of the dynamic characteristics of vibrations on the hardness perceived bytapping. Subjectively reported hardness values were related to the dynamic stiffness ofseveral objects. The dynamic stiffness, which characterizes the impulsive response ofan object, was acquired across the 40–1,000 Hz frequency range for cuboids of 14
types of materials by administering a hammering test. We performed two psychophysicalexperiments—a ranking task and a magnitude-estimation tasks—wherein participants
rated the perceived hardness of each block by tapping it with a finger. We found thatthe perceptual effect of dynamic stiffness depends on the frequency. Its effect displayeda peak around 300 Hz and decreased or disappeared at higher frequencies, at whichhuman perceptual capabilities are limited. The acquired results help design hardness
experienced by products.
Keywords: hardness perception, dynamic stiffness, frequency, vibration, principal component analysis

1. INTRODUCTION
In daily life, a person may tap the surface of an object to judge its hardness when the object
is rigid and cannot be deformed by pinching or pushing. Thus far, the principle of hardness
perception by tapping has yet to be studied extensively. The general conclusion among previous
studies seems to be that the dominant frequency of the transient vibrations yielded by tapping
is a major cue for judging the hardness-related properties of objects (Okamura et al., 2001;
Kuchenbecker et al., 2006; Ikeda and Hasegawa, 2009; Higashi et al., 2015, 2018a). Greater
frequencies lead to greater perceived hardness. Other studies have revealed intriguing properties,
although repeated testing and confirmation of these properties in various conditions is necessary
before they can be widely accepted. For example, the presence of multiple frequency components
in a single tapping event was reported to produce realistic stimuli (Kuchenbecker et al., 2006).
The viscosity or damping properties of an object may also determine the hardness perceived
by tapping as well as the stiffness of the object (Higashi et al., 2016b). Despite these previous
investigations, few researchers have studied the vibrational frequency characteristics associatedwith the hardness perceived by tapping. In contrast, thus far, the contributions of force cue havebeen intensively studied. The maximum value and the rising steepness of the reaction force against
an impulsive input force influence the hardness perceived by striking an object (LaMotte, 2000;
Lawrence et al., 2000; Friedman et al., 2008; Han and Choi, 2010; Higashi et al., 2018b). Also,
when discriminating deformable objects by using a fingertip, material elasticity and static stiffness,
which is defined by the ratio of static pressing force to an object surface and its indentation

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depth, are known as major cues (Srinivasan and LaMotte, 1995;
Bicchi et al., 2000; Friedman et al., 2008; Bergmann Tiest andKappers, 2009; Scilingo et al., 2010; Kaim and Drewing, 2011).
Especially, for objects as soft as or softer than human skin, theskin deformation cue plays more important role than kinesthetic
one (Friedman et al., 2008; Bergmann Tiest and Kappers, 2009).
Another aspect that should be mentioned is the rapid growthof the usage of vibrotactile feedback functions in consumer
products. Compared with softness or compliance (Srinivasan
and LaMotte, 1995; Bicchi et al., 2000; Bergmann Tiest andKappers, 2009; Kaim and Drewing, 2011; Metzger and Drewing,
2017; Hauser and Gerling, 2018), hardness perception by tapping
has not been reported on extensively, potentially becauseof its limited commercial applications. However, vibrotactile
methods of expressing object collisions were proven to bevaluable (Okamura et al., 2001; Kuchenbecker et al., 2006; Caoet al., 2017; Culbertson and Kuchenbecker, 2017; Gongora et al.,2017; Hachisu and Kajimoto, 2017; Park and Choi, 2017) andare increasingly common in consumer products. Studying themechanisms of hardness perception by tapping will be helpful fordeveloping effective rendering techniques.
In previous studies in which force or vibrotactile displays havebeen used to present tapping events virtually based on transient
vibrations, vibratory stimuli have tended to be simplified todamped vibrations with single frequency components. However,in reality, the response of an object to an impulsive tapping
force includes a wide range of frequency components. Therefore,
in the present study, we investigated the perceptual effects
of frequency characteristics on hardness perception in a wide
range from 40 to 1,000 Hz. To this end, we used the dynamic
stiffness of a variety of objects. Dynamic stiffness is frequencydependent and exhibits different values depending on the speedof object deformation. Furthermore, it expresses the uniquefrequency characteristics associated with the hardness of theobject. We linked the dynamic stiffness and subjective hardnessof objects using multivariate analysis and determined the impactof the dynamic stiffness in each of several frequency bands. The
dynamic stiffness was investigated by performing a mechanicaltest using an impulse hammer, and the subjective hardness wasdetermined by conducting two types of psychophysical tests,
which were the ranking and magnitude estimation tasks, using
human participants.
This study is an extended version of an earlier study
conducted by the authors (Higashi et al., 2017), in which theconnection between dynamic stiffness and subjective hardness
was demonstrated. Nonetheless, they used a psychometrics thatwere converted based on a ranking task with strong assumptionsof the distribution of answers. Furthermore, they did not shutout
the information of weights and surface roughness of specimens
used in the experiment. The weight pertains to the density ofobject and apparently influences the subjective hardness. The
roughness perception is slightly correlated with the hardness
perception in several reports in a manner that the rougher objects
are felt the harder (Yoshida, 1968; Tamura et al., 2000; Picard
et al., 2003; Soufflet et al., 2004; Sakamoto and Watanabe, 2017).
In the present study, we conducted two types of psychophysicalexperiments (i.e., ranking task and magnitude estimation task) to

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TABLE 1 | Materials and sizes of hardness specimens.
Material

Size [mm]Acrylic resin

60 × 60 × 30

Polycarbonate resin

60 × 60 × 30

Nylon resin

60 × 60 × 30

Nitrile rubber

60 × 60 × 30

Urethane rubber (soft)

60 × 60 × 30

Urethane rubber (hard)

60 × 60 × 30

Wood

60 × 60 × 30

Wax

60 × 60 × 30

Stainless steel

60 × 60 × 30

Aluminum

60 × 60 × 30

Granite100 × 100 × 30

Brick

100 × 100 × 60

Concrete

200 × 100 × 60

Cork

200 × 100 × 60

calculate two types of psychometrics and investigated the effects
of dynamic stiffness. Furthermore, the surfaces of specimens were
finely finished, and the experimental protocol was designed suchthat participants would not know the weight of the specimen.

2. MATERIALS AND METHODS
2.1. Ethical Statements
This study was approved by the internal review board of the
School of Engineering, Nagoya University (#15-12).

2.2. Hardness Specimens
Fourteen types of specimens made of different materials wereused in both the mechanical and subjective tests. Each specimen
was a solid cuboid made of one of the 14 types of materials
listed in Table 1. The cuboids were composed of plastic, wood,
metal, rubber, and stone, which are popular materials in daily life.
Each of the materials was sufficiently rigid such that it would not
be deflected when its surface was pushed with a fingertip. Thiscondition blocked participants from judging hardness based on
the surface deformation. The surfaces of natural materials suchas cork were finely polished such that their surface textures would
not affect the material recognition.

2.3. Hammering Test for Determining
Dynamic Stiffness
2.3.1. Dynamic Stiffness
Static stiffness is defined as the ratio of the force acting on an
object to its surface displacement. In contrast, dynamic stiffness
is the force per unit vibratory displacement and is defined in thefrequency domain. The latter is frequency-dependent, whereas
the former is not. For example, provided that the dynamic
stiffness of an object is higher at 50 Hz than 100 Hz, the vibration
amplitude is smaller at 50 Hz than at 100 Hz, even if themagnitude of excitation force is the same. Thus, the transient
vibrations caused by tapping the object can be mathematically2

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decoupled into an impulsive force input by tapping and the
dynamic stiffness unique to that object.
The dynamic stiffness can be expressed in terms of theforce f (t) and surface displacement x(t), given that we used an
accelerometer to measure the acceleration ẍ(t) of the surface
vibrations, as follows
H(ω) =

F [f (t)]F [f (t)]=,F [x(t)]F [ẍ(t)]/(jω)2(1)

where F , ω, and j denote the Fourier transform, angularfrequency, and imaginary unit, respectively. Both ẍ(t) and f (t)were measured and transformed into values in the frequency
domain by performing Fourier transforms. Finally, we computed
the absolute value of H(ω) for later analysis.
In our experiment, the dynamic stiffness was calculated basedon the results of a hammering test. Considering the capabilitiesof the accelerometer that we used and human perception, we
measured the stiffness across the 40–1,000 Hz frequency range.

FIGURE 1 | Measurement apparatus for the hammering test. The hammer
was fixed to a rotator and released at a fixed angle.

2.3.2. Experimental Setup
Figure 1 shows the measurement apparatus used in thehammering test. An impulse hammer (GK-3100, Ono Sokki
Co. Ltd., Japan) and amplifier (480M96A, Ono Sokki Co. Ltd.)
intended for the load cell embedded in the hammer were
employed. One end of the hammer was fixed to a rotator andreleased from rest at a certain height. A high-precision piezo
accelerometer (2302B, Showa Sokki Co. Ltd., Japan, valid over
20 Hz) was fixed near the center of the surface that was to be
struck by the hammer, and its output was acquired through anamplifier (4035-50, Showa Sokki Co. Ltd., Japan). The force andacceleration data were sampled using an oscilloscope at 10 kHz.
Each specimen was fixed on a large metal plate (800 × 800 × 100mm) for the hammering test.
Ideally, the frequency response of the hammer tip should be
similar to that of a human fingertip. A few tips are commerciallyavailable for the impulse hammer we used. Among them, wechose the one whose frequency characteristics are the mostsimilar to those of a human fingertip. Figure 2 shows thefrequency spectra of the impulsive force when the two hammer
tips and a fingertip were used. Unfortunately, the spectra of thefingertip lay between those of the two hammer tips. We selected
the softer one, and it is not clear how the difference in spectra
between the hammer tip and fingertip affects the latter analysis.
It is intriguing to consider whether an actual fingertip can beused for the hammering test. However, in case using the actual
fingertip, the correct placement of the load cell is a difficult
challenge.

FIGURE 2 | Impulse force spectra obtained using two hammer tips andfingertip.

FIGURE 3 | Example of surface acceleration and force data acquired by
performing a hammering test (Hard urethane).

acceleration acquired during a strike. The contact periods of valid
trials were generally 2–3 ms.

2.3.3. Hammering Test Procedure
Each specimen was struck by the hammer at the center of its
largest surface. Furthermore, each of the specimens was tested
multiple times such that 10 valid data values were acquired.
Invalid trials most commonly occurred due to protracted single
strikes with double hammering or contact periods that were
too long. These invalid trials were easily detected by checking
the force records. Figure 3 shows an example of the force andFrontiers in Psychology | www.frontiersin.org

2.3.4. Dynamic Stiffness of Hardness Specimens
Figure 4 shows the dynamic stiffness computed for each of
the 14 specimens. The dynamic stiffness values agree with ourintuitive understanding of hardness. The specimens composedof materials that are typically perceived as hard, includingaluminum, stainless steel, concrete, and polycarbonate, display

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FIGURE 4 | Dynamic stiffness of the material specimens. Mean and standard deviations acquired from 10 hammer strikes.

the present study. It is not clear how heat transfer during a shortcontact period might affect material recognition. Furthermore,
to avoid judgment based on the density or heaviness of thespecimens, the experimenter lifted and moved the specimenson the desk following requests from each participant. During
the experiment, the participants wore headphones that playeda pink noise to shut out the sounds of tapping. Since theparticipants wore sunglasses with opaque film, judgment basedon the appearance of the specimens was impossible. Note thatthe specimen cuboids could barely be seen through the glasses. A
total of two sets of trials were performed by each participant witha break of a few minutes between sets. Most of the participants
were able to finish the ranking task within 10 min.

relatively large stiffness values across the wide frequency range.
In contrast, those composed of soft materials, including urethane
rubber, nitrile rubber, cork, and wood, exhibit smaller stiffness
values. The peak stiffness is different for each of the specimens,
especially for the softer materials. However, the hard materials
display similar dynamic stiffness values in the low frequency
range.

2.4. Psychophysical Experiments for
Subjective Ratings of Hardness Specimens
To investigate the effects of psychometrics, the subjective
hardness of each specimen was rated in two types ofpsychophysical experiments, i.e., ranking and magnitude
estimation tasks. The results of these two types of experiments
were separately analyzed and compared.

2.4.2.2. Data analysis
For better application of the multivariate analyses described
below, we converted the rank of each specimen assigned by theindividual participants into an interval scale using a normalized
ranking method (Harter, 1961). This method assumes that
random samples from a normal distribution with an average andstandard deviation of zero and one, respectively, are ranked in
the order of observed values. The method converts the rank ofa sample to its expected value on this assumption. The range
of normalized ranks depends on the number of samples, in thiscase ±1.70 based on 14 samples. The normalized ranks of eachspecimen were averaged across the participants and were used in
the subsequent analysis.

2.4.1. Participants
The participants were eight male university students who agreed
to participate in the study and provided informed consent. Allof the participants were in their 20s, were right-handed, anddeclared no haptic deficit. Two tasks were performed by all theparticipants. Half of the participants performed the ranking task
first, whereas the other half began with the magnitude estimation
task.

2.4.2. Experiment 1: Ranking Task
2.4.2.1. Procedures
Fourteen types of hardness specimens were randomly placed ona desk. The participants compared and ranked them by tapping
their surfaces. The number of taps was not specified, and aparticipant could assign the same ranks to multiple specimens(i.e., multiple specimens could be perceived to have the same
hardness).
Each participant was instructed to tap the center of the largest
area of each specimen using the index finger of his writing hand.
To avoid judgment based on the thermal properties, neither
placing a finger on the specimen nor pushing was allowed.
Nonetheless, the thermal cues could not be fully controlled in

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2.4.3. Experiment 2: Magnitude Estimation Task
2.4.3.1. Procedures
The participants evaluated the subjective hardness of thespecimens using the magnitude estimation method without amodulus. Specifically, each participant evaluated the subjective
hardness of each specimen as a numerical value by tapping its
surface. They could use positive real numbers, including fractions
and decimals. The participants followed the same instructions
regarding how to tap the specimens as in Experiment 1.
The visual and auditory cues were blocked by headphones

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playing pink noise and by sunglasses with opaque film.
The specimens were presented one-by-one in random order.
Fourteen specimens were evaluated in a single set, and a totalof three sets were performed by each participant. Most of theparticipants were able to finish each set within 10 min.

2.4.3.2. Data analysis
No significant differences in reported hardness scores were foundamong the three trials. The data from all trials were used forthe subsequent analysis. The scores were normalized acrossparticipants to bring all data onto a common scale and reducethe influence of the participants’ arbitrary choice of scale. The
hardness scores were then geometrically averaged among theparticipants for each specimen. These geometric averages wereused in the subsequent analysis.

FIGURE 5 | Correlation coefficients between the dynamic stiffness at eachfrequency and two types of subjective hardness. The effect of dynamic
stiffness on subjective hardness depends on the frequency.

2.5. Relation Between Dynamic Stiffness
and Perceived Hardness Based on
Multivariate Analysis

where sij , wj , and c are the stiffness at the jth representative
frequency for specimen i, its perceptual weight, and a constantthat represents the intercept of the hardness scores, respectively.
The first and lowest representative frequency was 40 Hz, while thelast and highest one was 1,015 Hz. The number of representative
frequencies was 26, and they were separated by equal intervals of
39 Hz: m = 26. The weight w that best relates hi to si suggests how
the dynamic stiffness influences the subjective hardness. Hence,
the objective of this analysis was to determine such w values.
For n dynamic stiffness samples, the simultaneous equations
can be expressed as

2.5.1. Correlation Between Single Component of
Dynamic Stiffness and Subjective Hardness
Although our main interest lies in the quantitative connections
between the magnitudes of dynamic stiffness and hardness
perception, herein, their simple correlation coefficients arechecked. Figure 5 shows correlation coefficients between thesubjective hardness and the dynamic stiffness values of specimens
for each frequency component. The correlation coefficients
calculated from the two types of subjective scores produced
similar profiles. The correlation coefficients varied in therange of about 0.5–0.7 depending on the frequency. Suchfrequency dependency suggested that the perceptual weightingsof frequencies are not equal across the whole frequency range. It
should be noted that these simple correlation coefficients do notdirectly indicate the relationships between the dynamic stiffnessand hardness perception when the dynamic stiffness values atdifferent frequencies are correlated with each other. Especially,
the correlations varied in a zigzag manner with peaks at 40 and
160 Hz with a valley at the center of these two peaks at 80 Hz.
There is no rationale for explaining this zigzag profile in terms ofthe characteristics of human’s vibrotactile perception.

h = Sw + c,

where S = [s1 , s2 , ..., sn ]T and h = [h1 , h2 , ..., hn ]T are thedynamic stiffness matrix and hardness score vector, respectively.
Matrix S ∈ Rn×m was composed of the stiffness values for m
representative frequencies for n types of specimens: n = 14.
Vector h ∈ Rn×1 comprised the subjective hardness scores of
the 14 specimens.
Because of the issue of the degree of freedom (i.e., n < m), (3)cannot be solved directly by using the generalized inverse matrixof S. Even if a number of samples are tested and n > m, thecollinearity property of the dynamic stiffness values is raised as aproblem. Collinearity makes the computation of the generalized
inverse matrix unstable because of the rank deficit caused by highcorrelation coefficients among explanatory variables, which are
the dynamic stiffness values. To avoid these issues, we employed
an approach described in the following sections.

2.5.2. Prediction Model and Analysis Strategy
Since the frequency of the damped vibrations produced bytapping influences the perceived hardness (Okamura et al., 2001;
Kuchenbecker et al., 2006; Ikeda and Hasegawa, 2009; Higashi
et al., 2016a) and the perception of a vibrotactile stimulus
significantly depends on its frequency (Bolanowski et al., 1988),
the effect of dynamic stiffness on hardness perception should also
depend on the frequency of the stimulus. We related the dynamic
stiffness values si ∈ Rm×1 of m representative frequencies for
specimen i to the subjective hardness hi using
hi = sTi w + cmX
sij wj + c,=

2.5.3. Multiple Regression Analysis Using Principal
Components of Dynamic Stiffness
To avoid the abovementioned computational problem, we used
the principal components of the observed dynamic stiffness.
With a varimax rotation applied, S was decomposed into the′component scores A ∈ Rn×m of all trials and the principal′
component vectors B ∈ Rm ×m by(2)
S ∼ AB,j

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(3)

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(4)

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Based on (5), (7) is identical toh ∼ S[b1 b3 ]+ [z1 z3 ]T + c,

(8)where + indicates the generalized inverse matrix. Here, w ∼[b1 b3 ]+ [z1 z3 ]T is regarded as the perceptual weights against the
stiffness for m representative frequencies.

3. RESULTS
3.1. Subjective Hardness Scores
Figure 7 (left) shows the mean and standard error of thenormalized ranking scores for each specimen. The standard error
for each specimen was small compared to the differences intheir means, which indicates that the individual differences inthe scores were small. The stone and metal cuboids were ranked
the highest, followed by plastic, wood, and rubber. The hardness
ranks were largely consistent with our intuitive understandingof hardness. That is, ostensibly hard specimens were ranked assuch, and vice versa. In the subsequent analysis, the mean scores
were used as representative subjective hardness values. Figure 7(right) shows the geometric means and standard errors of thereported hardness scores for all of the specimens. The stone andmetal cuboids were scored the highest, followed by the plastic,
wood, and rubber ones. The hardness ranks of the specimens
were largely consistent between the two types of psychometrics.

FIGURE 6 | Loading vectors of three principal components for the dynamic
stiffness of 14 specimens. The loadings are scaled by their eigenvalues. The
first component includes stiffness below 800 Hz with the peak around 300 Hz.
The second component includes the stiffness at 800–900 Hz, while the third
component weights the stiffness above 900 Hz.

where m′ < m. The number of principal components wasdetermined to be three such that the cumulative contribution
ratio reached 95%: m′ = 3.
Hence, by using vectors, (4) can be rewritten as
S ∼ [a1 a2 a3 ][b1 b2 b3 ]T ,(5)

where ak ∈ Rn×1 is the score vector of principal componentbk ∈ Rm×1 . The three components, b1 , b2 , and b3 , are shownin Figure 6. The first component, b1 , was largely loaded by thestiffness in the middle frequency range of approximately 200–500 Hz. In contrast, the second component, b2 , was loaded athigh frequencies of nearly 900–1,000 Hz. The third component,
b3 , lay between b1 and b2 , and was loaded by the stiffness at
700–900 Hz.
We then performed multiple regression analysis with theprincipal component scores and subjective hardness scores
being explanatory and objective variables, respectively. For thisanalysis, the subjective hardness h was modeled as a product of a′vectors and regression coefficients z = [z1 z2 z3 ]T ∈ Rm ×1 andwas determined based onh ∼ [a1 a2 a3 ][z1 z2 z3 ]T + c.

3.2. Weightings of the Dynamic Stiffness
Figure 8 shows the perceptual weights w of the dynamic stiffness
values from 40 to 1,015 Hz. This figure includes the results
obtained using the two types of psychometrics. A constantvalue c was −0.75 and 28.8 for the ranking and magnitude
estimation tasks, respectively. The shaded regions represent the
maximum and minimum values of variation of w found in thevalidation in which one of the 14 types of hardness specimens
was removed from the analysis to determine whether the results
were sensitive to or robust against the sample data. The typeof psychometrics hardly influenced the results, and peaks wereobservable around 300 Hz for both types of psychometrics. The
w values calculated from the two types of subjective scoresproduced similar profiles.
We conducted a leave-one-out cross validation method
where each model estimated the hardness score of a specimenthat was not involved to establish the model. Figures 9A,B
compare the hardness scores reported by the participants
and those estimated based on the dynamic stiffness of thespecimens. The correlation coefficients between the observed
and estimated values were determined to be 0.69, when
the normalized ranked scores were used and 0.72, whenthe values from the magnitude estimation method wereused.
When performing regression analysis, the regression
coefficients z1 and z3 in (8) were 6.0E-9 and 2.6E-9, respectively,
for the normalized rank scores, and they were 1.0E-7 and2.4E-8, respectively, for the scores of the magnitude estimation
task. Hence, the first principal component influenced thesubjective score more than the third component did.

(6)

The use of principal components allowed us to avoid overfitting
because the number of explanatory variables was merely m′ =3. Furthermore, this method does not include the problem
of collinearity because the explanatory variables are mutuallyindependent.
Using the stepwise method, we selected ai vectors thatsignificantly affected h. As a result, a1 [t0 (138) = 11.6, p < 0.001]
and a3 [t0 (138) = 2.43, p = 0.016], which were the scores for the
first and third principal components b1 and b3 , respectively, were
found to influence the hardness scores statistically. Hence, using
the scores for these two components, the regression equation canbe rewritten ash ∼ [a1 a3 ][z1 z3 ]T + c.

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FIGURE 7 | Perceived hardness values of the specimens, including the means and standard errors of the hardness scores among the participants. (Left) Values
based on the ranking task. (Right) Values based on the magnitude estimation task.

Nonetheless, both components were good predictors. The
correlation coefficients between the subjective hardness
scores and each of the two components were high. The
first component exhibited a correlation coefficient of 0.69
for the normalized ranks and 0.72 for the scores of themagnitude estimation task. The third component exhibiteda correlation coefficient of 0.67 for both types of scores.
Both principal components were equally correlated with thehardness scores. Although these results apparently suggest thecontribution of the third component, which mainly includes
the dynamic stiffness at 800–900 Hz, as discussed later insection 4, the resultant perceptual weights shown in Figure 8
does not support the effects of the dynamic stiffness above
800 Hz.
The hardness values of certain specimens were not accurately
predicted. These specimens were the urethane, concrete, andbrick cuboids, which were reported to be the softest or hardest
among all the specimens in the psychophysical experiments.
This behavior is typical for a linear prediction model thatlinearly approximates limited parts of nonlinear phenomena.
Our linear model, (3), is applicable to a limited set ofmoderately hard specimens. It would be intriguing to search for anonlinear estimation model that suits a wide range of specimens;
however, the present experiments were not designed for such a
purpose.

FIGURE 8 | Frequency characteristics of the weight function for the subjective
hardness and dynamic stiffness. Weights for the two types of subjective
hardness are shown with shades being the maximum and minimum range.
The perceptual weights of dynamic stiffness values at approximately 300 Hz
are the greatest whereas those above 800 Hz are nearly zero.

Human vibrotactile sensitivity is the highest around 250 Hz
and deteriorates above and below this frequency (Bolanowski
et al., 1988). At higher frequencies near 1 kHz, the perceptionof vibrotactile stimuli is difficult. Such sensitivity largely agrees
with the weights derived from the principal component analysis.
The weights were the greatest around 300 Hz, the range
in which vibrotactile perception is the most sensitive. Aboveapproximately 800 Hz, the range in which vibrotactile perception
is unlikely to operate, the weights were nearly zero, suggestingthat stiffness does not affect hardness perception in this frequency
range.
As described in the introduction, higher frequency leads to
the greater hardness perception, according to previous studies
where the frequency used for vibrotactile stimuli was at most300 Hz (Okamura et al., 2001; Ikeda and Hasegawa, 2009; Higashi
et al., 2015, 2018a; Culbertson and Kuchenbecker, 2017). As
shown in Figure 7, the weight monotonically increases up to
approximately 300 Hz, which means that (at least beneath thisfrequency) a higher frequency is more influential on hardness
perception. Hence, the derived weights are consistent with thefindings in previous studies.
It should be noted that the generality of the findings remains
to be studied. In this study, we used cuboids made of several typesof materials. The dimensions of some were different from the

4. DISCUSSION
Figures 9A,B demonstrate that the dynamic stiffness of an object
is an effective predictor of the hardness perceived by tapping.
In other words, hardness perception is likely to be based on therelationship between the force exerted on the object surface andits vibrations. The perceptual weight was positive or zero for themost frequency range, which indicates that greater stiffness led
to greater perceived hardness. For the tapping of the object, theperceived hardness is positively correlated with the static stiffness
defined by the quasi-static relationship between the force anddeflection (Higashi et al., 2016b). The results of our study indicatethat this correlation also holds for dynamic stiffness.
The weight values varied across the frequency range,
suggesting that the perceptual effect of dynamic stiffness depends
on the frequency. This dependence may be due to humanvibrotactile characteristics, which are also frequency-dependent.

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FIGURE 9 | Correspondence between the observed (reported) and estimated hardness values. (A) Ranking task estimates. (B) Magnitude estimation method
estimates. The dotted lines indicate the equality of the observed and estimated values.

cuboids. We found that the dynamic stiffness across the 40–
800 Hz frequency range can be used effectively to estimate theperceived hardness with the peak weightings around 300 Hz.
The frequency-dependent contributions to perception can bereasonably interpreted based on the vibrotactile characteristics
of humans. These findings help understand the mechanism ofhardness perception in tapping objects.

others. Both the material and structure of an object influence thesoftness perceived by pinching or pushing it (Bergmann Tiest andKappers, 2009). Similarly, dynamic stiffness of an object depends
on both the material of which it is composed and its dimensions,because an object’s dimensions influence its vibration modes.
Hence, we did not exclusively investigate the differences among
the materials. To investigate the effects of dimensions, however,
we need a set of specimens made of the same materials but ofdifferent sizes. Such specimens would help us test the generalityof this study.

AUTHOR CONTRIBUTIONSKH and SO equally contributed to the work including
the design, preparation, and execution of the experiments,
analysis, and manuscript preparation. YY, HN, and MK equallycontributed to advising the study and proof-reading of themanuscript.

5. CONCLUSION
Humans judge the hardness of objects using vibratory cues
generated by tapping. However, little is known about howhumans leverage such transient vibrotactile cues. To search
for the relationships between the mechanical characteristics ofobjects and the hardness perceived by tapping in a wide frequency
range, we investigated the relationship between the dynamic
stiffness values and subjective hardness scores of a variety of

FUNDING
This study was partly supported by JSPS Kakenhi (15H05923)
and ImPACT (Tough Robotics Challenge).

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Conflict of Interest Statement: The authors declare that the research was
conducted in the absence of any commercial or financial relationships that could
be construed as a potential conflict of interest.
Copyright © 2019 Higashi, Okamoto, Yamada, Nagano and Konyo. This is an openaccess article distributed under the terms of the Creative Commons Attribution
License (CC BY). The use, distribution or reproduction in other forums is permitted,
provided the original author(s) and the copyright owner(s) are credited and that theoriginal publication in this journal is cited, in accordance with accepted academic
practice. No use, distribution or reproduction is permitted which does not comply
with these terms.

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