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Perceived Hardness through Actual and VirtualDamped Natural Vibrations
Kosuke Higashi, Shogo Okamoto, and Yoji Yamada
Abstract—Humans perceive hardness of objects from damped natural vibrations by tapping their surfaces. We investigated the rolesof six parameters characterizing the vibration in virtual and real environments. The effects of some parameters differed between thetwo environments. This finding provides valuable information to the developers of haptic applications.
Index Terms—hardness, perception, tapping, vibration, virtual objects, real objects.

F

1

I NTRODUCTION

H

UMANS tap the surface of a rigid object to judge its
hardness, whereas they press or push a deformable
one for the same purpose. In the former situation, thevibrotactile signals caused by tapping are considered to be
major perceptual cues [1], [2]. Although the mechanism ofhardness perception by tapping has not been intensively
studied thus far, a number of current haptic rendering
algorithms and applications have utilized this perceptual
phenomenon, because simulated vibrotactile stimuli can beenhancements or alternatives to force displays to present
collisions with virtual objects [1], [2], [3], [4], [5], [6], [7].
In this study, we investigated vibration-based hardness perception by tapping while restricting or nullifying the effectof a static reaction force from objects, with the ultimate
goal of devising better rendering algorithms and elucidating
the perceptual mechanisms. In particular, we pursued theareas of consistency and inconsistency between the hardness
perception experienced by tapping actual objects versus
simulated vibrotactile stimuli. The results guide the effective usage of vibrotactile stimuli and raise problems withrendering methods which remained unsolved.
In general, the vibratory response of an object to an
impulsive force input, such as a tap, is characterized by its
amplitude, frequency, and attenuation. Among them, the vibratory frequency is known to affect the hardness perceived
through the vibration; this is the most reliable property,
and has been repeatedly proven in many studies. Higher
frequency leads to the perception of greater hardness [1],[6], [8]. Because the natural frequency of the vibration islargely influenced by the stiffness of the object, the stiffness
also influences not only the hardness perception by pressing but also by tapping. Furthermore, vibrations including
multiple frequency components are felt to be more realisticand harder than those constituted by a single frequency
component [2], [9].
In contrast, the perceptual effects of the amplitude andattenuation of the vibration have yet to be determined.
Using a model based on momentum conservation, described•

K. Higashi, S. Okamoto, and Y. Yamada are with the Department of
Mechanical Systems Engineering, Nagoya University, Nagoya, Japan.

below, the vibratory amplitude is determined by the term
related to the contact velocity of the tapping finger andthe term dependent on the object stiffness. As the stiffness
increases, the amplitude becomes smaller. However, in perceptual experiments using force displays, the harder objects
were either linked with the larger amplitudes [10], [11], orthe effect of the amplitude was minor [6].

Few studies have investigated the perceptual effects ofthe damping properties of the vibrotactile stimuli, except
for those on which the present study is based [8], [11].
The effects of the damping properties are still inconclusive.
In the presentation of simulated vibrotactile stimuli using
a commercially available force display, the effects of theviscosity of the virtual object, which largely determines thedamping property of the vibration, depended on individuals [11]. For some participants, the viscosity exhibited a positive correlation with the perceived hardness; however, forothers the viscosity exhibited either a negative correlationor an insignificant relationship with the perceived hardness.
In tapping actual objects, the viscosity coefficients were positively correlated with perceived hardness [8]. Evidently,theresults of these two studies on real and simulated tapping
events are inconsistent.

In the present study, we investigated the perceptual effects of three vibration properties, including the amplitude,
time constant, and frequency of the vibratory stimulus, andthree mechanical parameters including the mass, viscosity,and stiffness of the object to be tapped. Both the real object
samples and simulated vibratory stimuli were tested toinvestigate the consistency between the real and simulated
stimuli. These samples were tested such that only the vibrotactile cues affected judgment of hardness, and effectsof the static reaction force were nullified. The ranges of themechanical parameters of the virtual and real stimuli partly
overlapped, which enabled us to compare the results fromeach type of stimuli. All of the experimental procedures
were approved by the internal review board of the Schoolof Engineering in Nagoya University (#15-12).

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TOH.2018.2841820, IEEE
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2

v
M

m+M

m
x(t)
c

k

k

c

Fig. 1. Tapping of the object modeled as a one degree-of-freedom
spring-mass-damper system.
Fig. 2. One degree-of-freedom force display with direct DC motor drive.

2

E XPERIMENT 1: S UBJECTIVE HARDNESS OF

VIRTUAL OBJECTS

2.1

Modeling of tapping an object

We modeled the damped natural vibration as the impulse
response of the one degree-of-freedom spring-mass-damper
system shown in Fig. 1. Note that we did not reproduce
the quasi-static force based on Hooke’s law in this model.
The hand with equivalent mass M collides with the object
characterized by mass m, viscosity c, and stiffness k at avelocity v , and the impulse response is caused by the momentum of the contact object M v . The mass and hand thencollectively vibrate as one object with mass m′ (= m + M ).
The displacement of the vibration x(t) at time t, with t = 0
being the moment of impact, is represented by−t(1)
sin ωtτ
2M v
A = A′ v = √(2)k(4m′ k − c2 )
2m′
τ=
(3)
c
√
4m′ k − c2
ω = 2πf =(4)2m′where A, A′ , τ , ω , and f are the amplitude, amplitude
per contact speed, time constant, angular frequency, andfrequency, respectively. The amplitude of the resultant vibration A is proportional to the contact speed of the hand,and can be decomposed into A′ and v , where A′ is fully
determined by the mechanical parameters of the object
and the hand mass. Hence, the vibration amplitude A wasdetermined by both A′ and v in the experiment.
Although the models for vibration feedback proposed
in previous research slightly differ from each other, ourvibration model reflects their physical soundness. In ourmodel, the vibration amplitude is proportional to the contact
speed, which is consistent with the previous models [1], [2],[7], [10]. The natural frequency increases with the increasein the stiffness of the object, which is true in our and former
models [2], [10]. We also introduced the inverse proportional
relationship between the time constant and the viscosity ofthe object [10] into our vibration model.
x(t) = A exp

2.2

Force display

We assembled a haptic display, shown in Fig. 2. The main
components of this apparatus are a dial, DC motor (RE40,

Maxon, Switzerland), rotary encoder (Encoder MR Type L,
Maxon, Switzerland), current controller (ADS50/10, Maxon,
Switzerland), and microcomputer (mbed LPC1768, ARM,
United Kingdom). The participant was able to experiencea vibration stimulus by moving their fingertip forward torotate the dial to a certain angle, which was measured by arotary encoder installed on the DC motor. The output shaftof the motor was directly connected to the dial with no gearsuch that the rotation resistance remained very low whenfreely rotating. The control system employed a feedforward
control method operated at 5 kHz. In order to reduce therisk of malfunction when the movable part was rotated athigh speed, we adopted a dial, rather than an arm-shaped
manipulator. The vibration was presented in the tangential
direction of the dial. The tangential displacement of thedial can be described as x(t) = rθ(t) where r and θ are
the radius of the dial and its rotational angle, respectively.
We presented the various stimuli by changing the massm, viscosity c, and stiffness k of the virtual object, andthe amplitude per tapping speed A′ , time constant τ , andfrequency f of the damped vibration.
In order to design the open loop control system, we
identified the characteristics of the display system, including the tapping fingertip. We regarded the user’s finger as
the load mass of the motor actuation and its mechanicalmodel as the one degree-of-freedom spring-mass-damper
system. The output torque T necessary for actuating the dialat acceleration ẍ(t) is given by

T = rF (t)F (t) = M ẍ(t) + C ẋ(t) + Kx(t)

(5)
(6)

where r, F , M , C , and K are the radius of the dial,tangential force, equivalent mass, viscosity, and stiffness ofthe display system plus fingertip, respectively. We derived
the transfer function between F (t) and x(t) by a vibrating
test within the frequency range used in the experiment, inwhich the participants were three right-handed males intheir twenties. Each placed an index finger on the edge ofthe dial with a relaxed posture, placing their elbow on anarmrest. M , C , and K values were then estimated from thetransfer function. The resulting mean values of M , C , andK were 0.02 kg, 4.5 Ns/m, and 1, 000 N/m, respectively.
The values of the viscosity and stiffness might have beenmostly from the finger skin because they were close to those

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3

TABLE 1
Stimuli used in Experiment 1α  200 Hz

0

Reference Stimulus
0.05

-100

0.1
CommandAcceleration [m/s2]

Observed
100

Test Stimulus  100 Hz
0
0.05
-100

0.1
Time [s]Reference Stimulusof the shear direction of fingertip skin reported in a past
study [12].
Fig. 3 shows the examples of vibratory stimuli producedby the force display with a fingertip on the dial at 100 and
200 Hz, which was the maximum frequency used in theexperiment.

Test Stimulus

Stimuli

We prepared three stimulus groups α, β , and γ . In group α,
the vibration parameters (A′ , τ , and f ) were independently
varied, whereas the mechanical parameters (m, c, k ) wereindependently varied in group β and γ . Such stimulus sets
are ideal to conduct multiple regression analysis where explanatory variables are preferably independent. The stimuli
in groups α and β included vibration only, whereas those ingroup γ included vibration and a virtual wall that causeda reaction force depending on the indentation depth. The
stiffness of the wall was fixed to 5 N/m for all stimuli ingroup γ . This group allowed us to investigate the effectsof vibration cue when the vibration and reaction force cues
coexist.
The A′ , τ , and f values were independently varied ingroup α as listed in Table 1. The values of the reference
stimulus, which was the center point for the variation of thestimuli, were set to A′ = 0.13 × 10−3 s, τ = 25 × 10−3 s,and f = 100 Hz. We prepared thirteen test stimuli, including twelve stimuli with one of their vibration properties
changed from the reference values, and the reference stimulus itself. The amplitude per contact speed A′ and√√ frequency
f of the test stimuli varied by factors of 1/2, 1/ 2, 2, and2 times those of the reference stimulus. The time constantτ of the test stimuli was quartered, halved, doubled, andquadrupled from that of the reference stimulus.
Group β consisted of stimuli with different mass m,
viscosity c, and stiffness k as listed in Table 2. The physicalproperties of the reference stimulus were set to m = 0.1 kg,
c = 10 Ns/m, and k = 50, 000 N/m. We prepared thirteen
test stimuli, including twelve stimuli with one of their physical properties changed from the reference values, and theτ [ms]
25
25
25
25
25
25
100
50
12.5
6.25
25
25
25
25

f [Hz]
100
100
100
100
100
100
100
100
100
100
50
71
141
200

TABLE 2
Stimuli used in Experiment 1β and 1γ

Fig. 3. Examples of vibratory stimuli produced by the force display witha fingertip on the dial at 200 Hz (top) and 100 Hz (bottom). The time
constant was τ = 0.02 s. The peak acceleration value was set to
120 m/s2 .

2.3

A′ [ms]
0.13
0.13
0.065
0.092
0.18
0.26
0.13
0.13
0.13
0.13
0.13
0.13
0.13
0.13

m [kg]
0.1
0.1
0.025
0.05
0.2
0.4
0.1
0.1
0.1
0.1
0.1
0.1
0.1
0.1

c [Ns/m]
10
10
10
10
10
10
2.5
5
20
40
10
10
10
10

k [N/m]
50000
50000
50000
50000
50000
50000
50000
50000
50000
50000
12500
25000
100000200000reference stimulus itself. The physical properties of the test
stimuli were quartered, halved, doubled, and quadrupledfrom the reference values. These three parameters wereindependently varied. More than two vibration properties
were changed when a change was made in either m, c, or k ,as expressed by (2), (3), and (4).
Group γ included the same stimuli as group β ; however,those in group γ were presented with a virtual wall with aspring coefficient of 5 N/m.
2.4

Tasks

Participants evaluated the subjective hardness of the test
stimulus based on the magnitude estimation method, withthe reference stimulus being a modulus. The hardness valueof the modulus was set to 10. Participants operated thehaptic display and compared the hardness of a test andreference stimulus. They were able to switch the presented
stimuli using a keyboard and experience each stimulus as
many times as desired to enable us to acquire answers withlittle fluctuation from each participant. The participants
wore headphones playing pink noise to block out auditory
cues. Thirteen test stimuli were presented in random orderin a single set of each stimulus group. A total of three sets
were performed for each participant with a several minutebreak between sets. The experiments using stimulus groups
α–γ were performed in random order. In total, 117 stimuli(13 stimuli × 3 sets × 3 stimulus groups) were tested byeach participant. We sacrificed the number of trials in order

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4

8

8

8

4

4

4

0

0

0

-4

-4

-4

ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ
P1
P2
P3
P4
P5
P6
P7
P8
P9

(i) Gradient along amplitude (A′ )ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ
P1

P2

P3

P4

P5

P6

P7

P8

P1

P9

(ii) Gradient along time constant (τ )

8

8

8

4

4

4

0

0

0

-4

-4

-4

ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ
P1
P2
P3
P4
P5
P6
P7
P8
P9

(iv) Gradient along mass (m)ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ
P1
P2
P3
P4
P5
P6
P7
P8
P9

(v) Gradient along viscosity (c)

P2

P3

P4

P5

P6

P7

P8

P9

(iii) Gradient along frequency (f )ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ ĮȕȖ
P1
P2
P3
P4
P5
P6
P7
P8
P9

(vi) Gradient along stiffness (k)Fig. 5. Gradients of H along m, c, k, A′ , τ , and f in Experiment 1 for individual participants. α, β , and γ indicate the experimental conditions or
stimuli sets. Digits indicate participants 1–9. Error bars indicate 90% confidence intervals. Colored bars indicate a statistically significant gradient(p < 0.10).

to remain participants motivated considering the required
time for experiment which was 2.5–3 hours. The small
number of trials limited the interpretation of the results withthe standard significance level of 0.05 or smaller when the
results were analyzed for individual participants.

3
2
1
0
-1

2.5

-2

ParticipantsParticipants were eight right-handed males and one righthanded female in their twenties. All signed a written consent form and were unaware of the objectives of the experiments beforehand.
2.6

Analysis

The answered values for perceived hardness for each participant were fitted to the approximate function of perceived
hardness H(m, c, k) and H(A′ , τ, f ) as follows:

H(m, c, k) = p0 + p1 m + p2 c + p3 k
H(A′ , τ, f ) = p4 + p5 A′ + p6 τ + p7 f

(7)
(8)

where H and pi were the values of the perceived hardness
and regression coefficients, respectively. These linear equations were intended to express the subjective hardness ina small parameter space. We did not simultaneously use
the six types of variables as explanatory variables because
the mechanical parameters (m, c, and k ) and vibration
properties (A′ , τ , and f ) are not independent as indicatedby (2), (3), and (4).
For these regression analyses, the values of the perceived
hardness were geometrically averaged for the individuals,
and the values for each physical parameter were normalized to z-scores. Because the parameters could be mutuallyconverted between the physical characteristics of the object
and the vibration properties using (2), (3), and (4), the results
could be analyzed in both dimensions.

Amplitude Time Frequency Mass
(f)
(A’)
(m)
constant(߬)

Viscosity Stiffness(k)(c)

Fig. 4. Mean gradients of H along each parameter in Experiment
1 (virtual stimuli). Error bars indicate the standard errors among theparticipants. Orange and blue bars are significantly greater and smaller
than zero, respectively (p < 0.001).

We discussed the contribution of each parameter tohardness perception from the gradients of H along eachparameter in the vicinity of the parameters of a reference
stimulus. These gradients along the parameters correspondto the regression coefficients of the parameters.
2.7

Results

To check how well the H functions represented the observed
data, we calculated the correlation coefficients between thesubjective hardness values answered by participants andthose estimated by H functions. When H(m, c, k) wasused, for each group of α–γ , the mean and standard deviation of correlation coefficients among participants werer = 0.78 ± 0.10, 0.69 ± 0.08, and 0.44 ± 0.16. Moreover,when H(A′ , τ, f ) was used, the r values were 0.83 ± 0.09,
0.78 ± 0.11, and 0.64 ± 0.18. Comparing the r values ofgroups α and β (vibration only) and γ (vibration with avirtual wall as the quasi-static reaction force), the existenceof the reaction force led to the decrease in the effects of

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TOH.2018.2841820, IEEE
Transactions on Haptics
IEEE TRANSACTIONS ON HAPTICS, VOL. ***, NO. ***, *** ***the parameters. This is reasonable because participants usedboth the reaction force from the wall and vibratory cues
to judge the hardness for the stimuli in group γ . The
contributions of vibratory cues were smaller for group γthan for other groups.
As mentioned in Section 1, we expected that the effectsof virtual stimuli would depend on individual participants.
Nonetheless, we first show the mean effects of vibration andphysical parameters across all participants. Fig. 4 shows themeans and standard errors of the gradient of H along eachparameter. Except for the viscosity and time constant, thegradients of parameters were significantly greater or smaller
than zero (p < 0.001). The amplitude and frequency of thevibration and stiffness of virtual objects exhibited positiveeffects on hardness perception, whereas the mass negatively
influenced the same. In contrast, the time constant andviscosity seem to have no effects on hardness perception.
Fig. 5 shows the gradients of H along each vibration
property for each individual participant. The error bars
indicate 90% confidence intervals. Orange and blue bars
indicate statistically significant positive and negative effects,
respectively, on hardness perception (p < 0.10). Gray bars
are insignificant gradients judged by statistical tests of theregression coefficient (p > 0.10). Although some of thegradients are significant and others are not, the gradientsalong the vibration properties and mechanical parameters
tended to have the same sign among experiments for groups
α–γ . The six types of parameters can be classified into two
types: those that exhibited individual differences in the signof the coefficient, and those that did not. The mass, stiffness,
amplitude, and frequency matched the sign of the gradientamong all participants. The stiffness, amplitude, and frequency exhibited positive gradients, while the mass had anegative gradient. On the other hand, the viscosity and time
constant exhibited individual differences in the sign of thegradient. For the participants in one group, their perceived
hardness was positively affected by the viscosity, whereas
the participants of the other group found the perceived
hardness to be negatively affected by the viscosity. The time
constant had the opposite effect on the perceived hardness.

5ůƵŵŝŶƵŵWůĂƚĞ
^ƉƌŝŶŐ
ĂŵƉĞƌ

Fig. 6. Spring-damper system consisting of mechanical components.
The subject tapped the upper surface of the aluminum plate that was
supported by a spring and damper.
TABLE 3
Measured parameters of stimuli used in Experiment 2
Sample
I
II
III
IV
V

E XPERIMENT 2: S UBJECTIVE HARDNESS OF

REAL OBJECTS

3.1 Spring-damper specimens
We assembled spring-damper specimens [8] to present thedamped vibration of a single-frequency component. As
shown in Fig. 6, the spring-damper specimen was composed of two aluminum plates, a linear spring (SWY-31-40,
SWU-31-40, and SWR-31-40, MISUMI, Japan), and a shock
absorber (EMACN1212A, B, and C, MISUMI, Japan). Eachaluminum plate was 80 × 80 mm in size, and the upperand lower plates were 3 mm and 5 mm thick, respectively.
Three types of springs and shock absorbers with different
strengths could be used. Thus, there were five types of
specimens, as listed in Table 3. The springs were sufficiently
rigid to not be deflected when pushed with a fingertip.
Furthermore, the specimens were placed upon a buffer material to muffle their static reaction forces. This made it evenmore difficult for participants to judge the hardness basedon the static deflection of the specimen. From observation

Time Constant
[ms]
120
200
69
130
140

Stiffness
[N/m]
15700
15700
15700
7830
24700

Viscosity[Ns/m]
0.047
0.026
0.078
0.024
0.047

using tapping tests and an accelerometer on the aluminum
surface, the vibration caused by tapping included a single
major frequency that varied with the combination of aspring and damper.
3.2

Tasks

Participants tapped the specimens and experienced theirdamped vibrations. As in Experiment 1, they evaluated thesubjective hardness of a test stimulus as a positive real
number based on the magnitude estimation method. The
hardness value of the reference stimulus, which was sample I, was defined as 10. Participants were able to experienceeach stimulus as many times as desired. The stimuli were
presented in random order. Five stimuli were tested in asingle set and a total of three sets were performed for eachparticipant.
3.3

3Frequency
[Hz]
381
384
381
354
440

Participants

The same participants were used as in Experiment 1. They
wore headphones playing pink noise to block out audio
cues. The order in which Experiments 1 and 2 were performed was balanced among the participants.
3.4

Analysis

The answered values for perceived hardness were fitted tothe approximate function for perceived hardness H(c, k)
and H(τ, f ) as follows:H(c, k) =H(τ, f ) =q0 + q1 c + q2 k
q3 + q4 τ + q5 f

(9)
(10)

where qi was the regression coefficient. The values of theperceived hardness were geometrically averaged for individuals, and the physical parameters of the vibration stimuli were normalized to z-scores. Similar to Experiment 1,
we discussed the contribution of each parameter from thegradients of H along each parameter in the vicinity of theparameters of a reference stimulus.

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3
2
1
0
-1
-2
Viscosity(c)

Time
Stiffness Frequency
constant
(k)
(f)
(߬)

Fig. 7. Gradients of H along c, k, τ , and f in Experiment 2 (actual
stimuli). Error bars indicate the standard errors among the participants.

3.5

Results

Similar to Experiment 1, to check how well H functions
fit with the observations, we calculated the correlation coefficients between the answered and estimated subjective
hardness values for individual participants. For H(c, k) andH(τ, f ), the means and standard deviations of the correlation coefficients among participants were r = 0.72 ± 0.21
and 0.70 ± 0.29. Both H functions equally estimated theanswered hardness values.
Fig. 7 shows the gradients of H along each parameter.
Because no individual differences were observed, the results
for all participants are averaged and shown in a single
figure. Error bars indicate the standard errors among theparticipants. The stiffness, frequency, and viscosity exhibited positive gradients and the time constant exhibited anegative gradient. The effects of the stiffness and frequency
matched those in the virtual environment. There were no
individual differences in the sign of the coefficient on theviscosity and time constant, unlike the case of the virtual
object.

4

D ISCUSSION

Here, we discuss the effect of each parameter on the perceived hardness. The consistency or inconsistency of theeffect of each parameter among participants in virtual andreal environments is shown in Table 4. When the signsof all significant gradients along a specific parameter in
Fig. 5 are the same, we judged the effect of that parameteras consistent. Note that individual differences in the effectof some parameters were observed only in the experimentusing virtual stimuli.
4.1

Consistent effect of frequency

Vibration stimuli with higher frequency induced greater
perceived hardness. Regardless of whether the object wasreal or virtual, the effect of the frequency was consistentamong participants. This result for the frequency agrees
with previous studies [1], [2]. The vibration frequency is aneffective parameter for controlling the perceived hardnessof real and virtual objects.
4.2

Consistent effect of stiffness

Greater stiffness enhanced the perceived hardness for boththe virtual and real objects. In our experiment, the stiffness of the objects contributed only to the characteristics

6of damped natural vibration, and not to the static reaction force. Even in the experiment using real objects, theparticipants were unlikely to use the static reaction forces
to judge hardness because they were muffled by the soft
buffer material. Therefore, humans can discriminate between differences in hardness without the static reaction
force, by using the vibration stimulus caused by tapping.
This result supports the method of presenting the stiffnesschange of virtual objects based on vibratory feedbacks with
ungrounded devices that cannot present a static reaction
force.
Nonetheless, according to Fig. 5(vi), for virtual stimuli,
only a few participants felt that the stiffness k exhibited
positive effects on the perceived hardness. For other participants, the effects of k were insignificant. This phenomenon
can be explained as follows. As in (4), the increase in k
increases the vibratory frequency f , which leads to a greater
perceived hardness. Simultaneously, as in (2), the increasein k decreases the vibration amplitude A′ . According to
Fig. 5(i), a decrease in A′ leads to a smaller perceived
hardness. Hence, the effects of f and A′ mitigate the effectsof each other on perceived hardness, and as a result, theeffect of k was moderate.
4.3 Inconsistent effects of vibration amplitude between
virtual and real objects
In this experiment, the vibration amplitude was the productof the contact speed v and the amplitude per contact speed
A′ . Here, we discuss the effect of A′ .
In the virtual object, the perceived hardness increased
as the amplitude of the vibration stimulus increased. The
results show that the amplitude is an effective parameter forcontrolling the hardness of the virtual object. In the case of
the real object, the higher stiffness decreases the amplitudeof the damped natural vibration of the object surface. Here,
we find that the amplitude has an opposite effect on the realand virtual cases.
The above mentioned inconsistency was potentiallycaused by the gap between the vibration model of (1) andthe fingertip’s vibration caused by tapping real objects. In
the vibration model, the fingertip and target object collectively vibrate as one object; however, the displacements of
their surfaces may not match in real situations [13], unlike
the model used in the present study. For example, whentapping harder objects with a tool, the vibration amplitude
on the tool surface increases [6] in a manner opposite
to (1). The selection of an appropriate model to cause aperceptually sound effect is still an open question.
4.4 Inconsistent effects of time constant between virtual and real objects
The effect of the time constant exhibited individual differences in the virtual environment. When the time constant increased, five of the nine participants reported anincrease in the perceived hardness, and the others reported
its decrease. Therefore, the time constant is not an effective
parameter for stable control of the perceived hardness ofvirtual objects. However, since the effect of the time constantis significant for each individual, the time constant can
express the difference in the material of virtual objects.

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TOH.2018.2841820, IEEE
Transactions on Haptics
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TABLE 4
Perceptual effects of six types of parameters on subjective hardness. “Positive” or “negative” indicate that the parameter is positively or negatively
correlated with the subjective hardness. “Inconsistent” indicates that the effect of the parameter depends on individuals.
Frequency (f )Virtual stimulus

Positive

Real specimen

Positive

Time constant (τ )Inconsistentamong participants
Negative

Stiffness (k)The reason for the individual differences is unclear. Our
experimental design was not to investigate the reason for theindividual differences in the virtual environment. We had
considered that such individual differences were attributed
to a virtual wall, however, we did not find any substantial
differences between conditions with and without the virtual
wall. The same perceptual phenomena were observed in ourpast study [11] where a commercial haptic interface and arendering algorithm with the effect of a virtual wall wereused. Hence, the observation of these differences was not
unexpected in the present study.
On the other hand, in the case of the real damped natural
vibration, the effect of the time constant did not exhibit
individual differences. A longer time constant decreased theperceived hardness for all participants.
4.5

Effect of viscosity via effect on the time constant

The increase in the viscosity of real objects had a consistenteffect of increasing the perceived hardness for all participants. On the other hand, the effect of the viscosity exhibited
individual differences in the experiment using virtual objects. The inconsistency of the effect of the viscosity change
in virtual objects was observed in our previous studies using
a commercial force display [8], [11]. As shown in Fig. 5(ii)and (v), all participants reported that the viscosity and time
constant influenced their perceived hardness in the opposite
manner. These results were consistent with the inversely
proportional relationship between the viscosity and time
constant in (3). Thus, it is assumed that the effect of theviscosity is caused by the resultant time constant change.
4.6

Effect of mass of the virtual object

The virtual object with the smaller mass was felt as theharder object. As shown in (2), (3), and (4), the decreasein the mass increases the amplitude and frequency of thevibration stimuli, and decreases its time constant. The increase in the frequency and amplitude enhances the perceived hardness, as mentioned in previous sections. On theother hand, the decrease in the time constant can result ineither the increase or decrease in the perceived hardness
depending on the participant. In summary, an increase in themass induces greater perceived hardness with the composite
effect of three vibration parameters.

Positive
Positive

Viscosity (c)Inconsistentamong participants
Positive

Amplitude (A′ )Mass (m)Positive

Negative--

effects of the vibration properties. This is because the combination of the vibration and force results in the decrease ofeach contribution ratio against the whole of the perceived
hardness.

5

C ONCLUSION

We researched the effects of parameters related to thedamped natural vibration by investigating the perceived
hardness based on tapping real and virtual objects.
For virtual objects, vibration stimuli were produced
through a momentum-conservation model where the virtual
fingertip and object collectively vibrated after the collision.
The decrease in the virtual mass and increase in the virtual stiffness, vibration amplitude, and frequency induced
greater perceived hardness perception. The viscosity andtime constant of the virtual stimuli exhibited inconsistenteffects among participants and influenced perceived hardness in the opposite manner for each participant.
We validated the effects of the viscosity, stiffness, time
constant, and frequency by using the real objects. The results
showed that the effects of the stiffness and frequency wereconsistent between the real and virtual objects. On the other
hand, the effects of the viscosity and time constant weredifferent from those of the virtual objects. When real objects
were used, an increase in the viscosity and decrease in thetime constant led to greater hardness perception.
The stiffness and frequency are effective indexes for describing the perceived hardness of real/virtual environment
and individuals. The inconsistency of the results between
the virtual and real objects suggests that there remains a
gap in the dynamic phenomena between the vibration of
the real object case and the reproduced stimulus on thehaptic interface. Developers of haptic applications should
remember that the general physical model that induces
perceptually consistent effects with real situations remains
unsolved.

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

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This article has been accepted for publication in a future issue of this journal, but has not been fully edited. Content may change prior to final publication. Citation information: DOI 10.1109/TOH.2018.2841820, IEEE
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Kosuke Higashi received bachelor’s and master’s degrees in engineering from Nagoya University in 2016 and 2018, respectively. He
is currently working for Brother Industries, Ltd. His research interests
comprise haptic interactions through vibration stimuli.

Shogo Okamoto received a Ph.D. in information sciences in 2010 fromthe Graduate School of Information Sciences, Tohoku University. Since
2010, he has been with the Graduate School of Engineering, Nagoya
University. Currently, he is an associate professor at the Departmentof Mechanical Systems Engineering, Nagoya University. His researchinterests include haptics and human-assistive technology.

Yoji Yamada received a Ph.D. from Tokyo Institute of Technology in1990. He had been with Toyota Technological Institute since 1983.
In 2004, he joined Intelligent Systems Research Institute of AIST asa group leader. In 2009, he moved to Nagoya Unversity as a professor. He is currently studying safety and intelligence technology inhuman/machine systems and their robotic sensing and control.

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