Hardness Perception through Tapping:Peak and Impulse of the Reaction Force Reflect
the Subjective Hardness
Kosuke Higashi1 , Shogo Okamoto1 , Yoji Yamada1 ,Hikaru Nagano2 , and Masashi Konyo2
1

2

Graduate School of Engineering, Nagoya University, Nagoya, Japan
Graduate School of Information Sciences, Tohoku University, Sendai, Japan

Abstract. Humans can judge the hardness of an object by tapping its
surface. To investigate physical indicators for estimating subjective hardness, we analyzed the short-time reaction force caused by tapping various
types of objects. We focused on five indicators in the time domain, including the peak force value, peak time, duration, maximum increase rate,and impulse of the reaction force. A strong correlation was observed between the peak force value, peak time, duration, and maximum increase
rate. We found that subjective hardness can be predicted by combining
the peak force value and impulse of the reaction force. Results suggestthat the hardness involving stiffness and damping factor of objects canbe estimated from the reaction force caused by tapping objects. Especially, the former and latter are, respectively, associated with the peakforce value and impulse of the reaction force.
Keywords: hardness perception, reaction force, impulse

1

Introduction

Humans can judge the hardness of an object by tapping or pushing its surface.
Hardness perception through tapping is characterized by a much shorter contact time with an object compared with pushing. Additionally, in the case oftapping, dynamic phenomena are dominant whereas pushing is based on quasistatic phenomena. In previous studies, hardness perception through tapping wasinvestigated by focusing on two types of dynamic phenomena: vibration of objects and reaction force caused by tapping. However, it has yet to be fully studied
how humans judge mechanical properties of objects.
Vibration induced by tapping an object was reported as the cue of hardness perception. Okamura et al. reported that vibration properties, including
amplitude, damping rate, and frequency of natural vibration caused by tapping,influenced material perception [13]. The effect of vibration frequency has been
especially well-tested and confirmed by multiple studies [1], [5], [9], [10], [13].
They agree that the greater frequency leads to the greater subjective hardness.
Additionally, hardness perception based on natural vibration caused by tappingobject surfaces has been studied in relation to the mechanical properties of objects such as stiffness and viscosity [6], [7], [8].
Regarding the reaction force, which we focused on in this research, somephysical indicators affecting hardness perception have been investigated in previous studies. For example, a greater magnitude and higher increase rate of thereaction force were observed when tapping harder real objects [11]. Hence, theincrease rate of the reaction force was proposed as the indicator of hardness perception presented on the haptic display [2], [12]. As investigated in the studies
on vibration cue, the reaction force should also be influenced by the mechanicalproperties of objects. Relationships between mechanical stiffness and indicators
based on the reaction force were discussed in some studies [3], [12]. However, theexact contribution of the damping factor is still unclear.
In this study, we investigated the relationship between the subjective hardness and reaction force produced when tapping various real objects. We focusedon five parameters, including the peak force value, peak time, duration, maximum increase rate, and impulse as the parameters characterizing the reaction
force. By focusing on the time characteristics of the reaction force, we attempted
to discuss the contribution of the mechanical properties of objects, such as thestiffness and damping factor, to the selected five indicators. Through a correlation analysis, we investigated the effective physical indicators for subjective
hardness. Furthermore, we computed a regression model for estimating subjective hardness by combining effects of multiple physical indicators. Describing
subjective hardness based on physical indicators enabled us to connect sensual
evaluation with quantitative design of materials and structures in the development of industrial products.

2

Hardness specimens

Eighteen types of specimens listed in Table 1 were used in the experiments. Eachspecimen was a block made of a material commonly available in daily life. Blocks
were rigid enough so as not to be deformed by pressing them with a fingertip.
The reaction force when tapping an object depends not only on its material
but also its structure. In order to investigate the effect of an object’s structure,
for three materials, we prepared full and hollow specimens. The hollow specimens
were blocks drilled from their bottom, leaving the top and side surfaces 5 mm
thick.

3

Hammering Test for Measuring the Impulsive ReactionForce

We investigated the reaction force produced by hammering the surface of eachspecimen. It should be noted that we measured the reaction force per tapping
speed following the indicator of hardness perception proposed in previous studies [1], [2], [12].

Table 1. Materials and dimensions of hardness specimens (partly used in [8])Material
Size [mm]ABS resin
60 × 60 × 30
ABS resin (hollow) 60 × 60 × 30
Acrylic resin
60 × 60 × 30
Acrylic resin (hollow) 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

3.1

Material
Size [mm]Wood
60 × 60 × 30
Wood (hollow) 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

Apparatus

The reaction force was measured by tapping specimens with an impulse hammer
because of the difficulty in directly measuring the force under a fingertip. In
order to mimic fingertip action, we selected a commercially available hammer
whose tip (half-sphere with the diameter of 6.4 mm) had physical characteristic
closely matching those of a fingertip. Fig. 1 shows the frequency characteristics ofreaction force when tapping the same specimen with the fingertip and hammer;
their responses were similar.
Fig. 2 shows the measurement apparatus used in the hammering test. The
apparatus was composed of an impulse hammer (GK-3100, Ono Sokki Co. Ltd.,
Japan, 140 g) and a rotator attached to the hammer. The reaction force wasmeasured by the load cell embedded in the hammer. The tapping speed v wascalculated as v = lω where l and ω are the hammer length and the angular velocity measured by a rotary encoder (MR Type L 225787, Maxon, Switzerland),
respectively. The force was sampled at 10 kHz using an oscilloscope.
3.2

Procedure

The hammer was lifted and released from a certain angle to collide with thespecimen. The collision position was the central part of the upper surface of thespecimen, and the collision angle was perpendicular to the surface. The contact
speed ranged 0.9–1.1 m/s such that the mean was approximately 1.0 m/s. The
specimen was fixed on a large metal plate (800 × 800 × 100 mm). Each specimen
was tested multiple times to acquire 10 valid data sets.
3.3

Results: Reaction Force per Tapping Speed

The reaction force per tapping speed, which is the speed of the hammer at
the moment of contact, was calculated from the force and velocity data. Whenthe relationship between the momentum of the input (hammer strike) and thereaction force as output is linear, this reaction force per tapping speed captures
the property of the striking system. Fig. 3 shows the reaction force per tappingForce (dB)0-10HammerFingerZŽƚĂƌǇĞŶĐŽĚĞƌ
/ŵƉƵůƐĞŚĂŵŵĞƌ

-20
-30
-40
2
10

^ƉĞĐŝŵĞŶ
3

Frequency (Hz) 10

Fig. 1. Frequency characteristics of the reaction force. Those observed when the fingertip and hammer tapped the same object
at similar speeds were similar.

Fig. 2. Measurement apparatus for thehammering test

speed computed for 18 types of specimens tested. For each specimen, the averageof 10 trials is shown. The peak height and width of the waveform were different
among the specimens. Notably, relatively soft specimens, made of cork or rubber,
exhibited low and wide waveforms.

4

Subjective Hardness of Specimens

In this study, we used the subjective hardness scores obtained from psychophysical experiments; results of an experiment conducted in a previous study wereused [8]. Herein, we introduce the outlines of the experimental protocol and thedata summary.
4.1

Outline of the Experiment

The participants compared and ranked the hardness of 18 types of specimens bytapping their surfaces. The test was performed twice by each participant with a
few-minutes break between tests. During the experiment, the visual and auditory
cues were blocked by using foggy glasses and headphones playing pink noise. The
participants were eight right-handed males in their 20s. All participants agreed
to participate in the study and provided informed consent.
4.2

Subjective Hardness Scores

The rank of each specimen was converted into a normalized rank [4]. Fig. 4 shows
the mean and standard error of the normalized ranking scores for each specimen.

Force/velocity (Ns/m)

140

ABS
ABS (hollow)

120

Acryl
Acryl (hollow)
Aluminum
Brick
Concrete
Cork
GraniteNylon
NitrilePolycarbonate
Stainless
Urethane (h)Urethane (s)Wax
Wood
Wood (hollow)

100
80
60
40
20
0
0

0.002

0.004
0.006
Time (s)0.008

Fig. 3. Reaction force per tapping speed for the hardness specimens

The highest scores were obtained for stone and metal specimens, followed byplastic, wood, and rubber specimens.

5

Analysis: Linear Regression between Subjective
Hardness and Reaction Force

5.1

Physical Indicators

In order to identify physical indicators affecting hardness perception, we focusedon five types of parameters that varied among specimens. They included thepeak force value fmax , peak time tmax , duration td , maximum increase rate
∆fmax (corresponding to the rate-hardness [2], [12]), and impulse fimp as shownin Fig. 5.
5.2

Correlations among Subjective Hardness and PhysicalIndicators

The middle column of Table 2 shows correlations between the subjective hardness
and each physical indicator. Positive correlation coefficients were observed forfmax (r = 0.69) and ∆fmax (r = 0.67). On the other hand, td (r = −0.68), tmax
(r = −0.67), and fimp (r = −0.50) exhibited negative correlation coefficients.
These correlation coefficients are statistically greater or smaller than 0 at p valueof 0.05 or smaller.
The right side of Table 2 shows correlations among the physical indicators.
Except for impulse, there was a strong correlation between all other parameters(|r| > 0.9). Thus, only the impulse would be a parameter independent from theothers.

Nylon
Acryl
Cork
Wood
Wax
Wood (Hollow)
Urethane (H)NitrileUrethane (S)0-2Concrete
Brick
Aluminum
Stainless steel
Polycarbonate
GraniteABS (Hollow)
Acryl (Hollow)
ABS

Normalized rank ofperceived hardness

2

Force pertapping speed(Ns/m)

Fig. 4. Subjective hardness scores of the specimens, including the means and standard
errors of the scores among the participants (modified from [8])

୫ୟ୶

୧୫୮

∆୫ୟ୶
୫ୟ୶

ୢ

Time (s)Fig. 5. Representative parameters characterizing the reaction force in the time domain

5.3

Multiple Regression Analysis to Estimate Subjective Hardness

We computed a regression model with multiple physical indicators and the subjective hardness score as explanatory variables and objective variable, respectively. To avoid unstable analytical results due to collinearity among explanatory variables, independent variables were selected from the five types of physicalindicators. As candidates for explanatory variables, we selected the peak force
value fmax , which displayed the strongest correlation to the subjective hardness,and the impulse fimp , which was independent from the other parameters. We
confirmed that the two explanatory variables significantly affect the hardness
score by using a stepwise method. The regression model was acquired as
Subj. hard. = 2.7 + 0.017fmax − 26fimp .

(1)

The p-values of the coefficients associated with fmax and fimp were p = 2.5 ×
10−19 and p = 2.5 × 10−5 , respectively. Fig. 6 shows the relationships between
the observed and estimated subjective hardness. The correlation coefficient was
r = 0.72.

Table 2. Correlation coefficients among subjective hardness and characteristic parameters of the reaction force

I. Peak force value
II. Peak time
III. Duration
IV. Maximum increase rate
V. Impulse

Subj.Hard. I II
III IVV
0.69
1.0 -0.95 -0.95 0.99 -0.43-0.67
1.0 0.99 -0.94 0.52
-0.68
1.0 -0.94 0.52
0.67
1.0 -0.43
-0.50
1.0

1.5

Estimated hardness

1

Nylon ABS
Wood

0.5

Wax

Urethane(H)

0

Nitrile

-0.5

Acryl

Wood
(hollow)

-1
-1.5
-2

-1.5

Brick
ABS Stainless(hollow)Acryl(hollow)

Cork

Urethane(S)-2GranitePolycarbonate
Aluminum
Concrete

-1

-0.5
0
0.5
Observed hardness

1

1.5

Fig. 6. Relationship between the observed and estimated hardness values. For eachspecimen, 10 hardness values were estimated from 10 times of reaction force measurements. The dotted line indicates equality of the observed and estimated values.

6

Discussion

6.1

Maximum Reaction Force Linked to Object Stiffness

The maximum reaction force per tapping speed showed a positive correlation
with subjective hardness. We speculated that humans judge the stiffness of objects from the maximum reaction force caused by tapping. Here, in order tofocus on the relationship between the maximum reaction force and an object’s
stiffness, we considered the simplest object model with stiffness k, as shown in
Fig. 7. The fingertip of equivalent mass m collides with the object at an initial
velocity v at t = 0. The displacement of the object’s surface x(t) is described asrx(t) = v

r
m
k
sin(
t)
k
m

(2)

supposing that the mass vibrates at the tip of the stiffness element. The maximum reaction force per tapping speed fmax ismax(x(t))k
√ v
= mk.

fmax =(3)(4)

fmax is proportional to the square root of the object’s stiffness. Thus, humans canestimate the stiffness of objects from fmax . It is known that hardness perception
by tapping increases with increasing stiffness of the object [6], [7].

v

m

k

x(t)

m

k

Fig. 7. Physical model of tapping an object

6.2

Impulse Reflects the Damping Factor

Since impulse exhibited a negative correlation to subjective hardness, we speculated that it is affected by the energy dissipation due to the damping factor ofan object. Without the damping factor effect, based on the law of conservationof momentum, the impulse received from the object is equal to the momentumof the tapping finger before contact. On the other hand, considering the effect ofthe damping factor, the impulse decreases as the energy dissipation due to thedamping factor becomes larger. Thus, humans can estimate the damping factorof objects from the impulse. It has been reported that the damping factor of objects affects hardness perception through tapping [7] the way greater damping
factors lead to greater hardness perception.
6.3

To Improve the Prediction

By combining the effects of the maximum reaction force and impulse, the estimation accuracy for subjective hardness was improved. However, some specimens
exhibited a large gap between the estimate and experimental result as shown in
Fig. 6. For example, the estimates for the top-four specimens in Fig. 6 (concrete,
brick, aluminum, and stainless-steel specimens) were almost the same and didnot match the observations. In other words, subjective hardness was different
even though the maximum reaction force and impulse were the same among
these specimens. This suggests that factors other than maximum reaction force
and impulse contribute to the discrimination of hardness through tapping. Sinceother physical indicators listed in Table 2 have a strong correlation to the maximum reaction force, it is necessary to investigate other independent parametersin order to improve the estimation model. Considering the frequency-dependent
aspect of hardness perception [8], the analysis in the frequency domain will help
us find other physical indicators for subjective hardness.

6.4

Consistency and Inconsistency in the Effects of Structural
Properties

Comparing the full-block and hollow specimens, the experimental results varied
depending on the structure of the specimen. As results of the hammering test
show, the physical indicators consistently changed depending on the structure of
specimens. The peak time, duration, and impulse when tapping hollow specimens
were greater than those of full-block specimens. On the other hand, the peakforce value and maximum increase rate of the hollow specimens were smaller thanthose for the full blocks. Considering the effects of physical indicators analyzed in
Section 5.2, hollow specimens should be felt as softer than full-block specimens.
However, the effect on subjective hardness was inconsistent among specimens.
Wooden specimens with a hollow structure were felt as softer than full blocks;
however, ABS and acrylic resins exhibited the opposite trend. Results suggest
the existence of perceptual cues other than those investigated.

7

Conclusions

We investigated the relationship between the subjective hardness and the reaction force when tapping a variety of real objects. The peak force value andmaximum increase rate of the reaction force exhibited positive correlation tosubjective hardness; whereas, the peak time, duration, and impulse exhibited
negative correlation. Results suggest that humans can judge the stiffness anddamping factor of objects based on the reaction force caused by tapping. By
combining the effects of the maximum reaction force and impulse, we computed
an estimation model of hardness perception with a higher accuracy than a model
based on any single indicator. Nonetheless, the peak time, duration, and maximum rate of the reaction force can be a substitute of the peak force, because
they exhibited high correlation coefficients and are highly associated with eachother. This research may contribute to the quantitative estimation of subjective hardness for designing industrial products and stimuli rendered on haptic
displays.

Acknowledgment
This study was in part supported by ImPACT (Tough Robotics) and MEXT
KAKENHI (15H05923).

References
1. Culbertson, H., Kuchenbecker, K.: Importance of matching physical friction, hardness, and texture in creating realistic haptic virtual surfaces. IEEE Trans. Haptics
10(1), 63–74 (2017)
2. Han, G., Choi, S.: Extended rate-hardness: a measure for perceived hardness. In:
Kappers, A.M.L., Van Erp, J.B., Tiest, W.M.B., Van Der Helm, F.C. (eds.) Haptics: generating and perceiving tangible sensations, vol. 6191, pp. 117–124 (2010)
3. Harper, R., Stevens, S.: Subjective hardness of compliant materials. Quarterly
Journal of Experimental Psychology 16(3), 204–215 (1964)
4. Harter, H.L.: Expected values of normal order statistics. Biometrika 48(1), 151–165(1961)
5. Higashi, K., Okamoto, S., Nagano, H., Konyo, M., Yamada, Y.: Perceived hardness
by tapping: The role of a secondary mode of vibration. In: Hasegawa, S., Konyo,
M., Kyung, K.U., Nojima, T., Kajimoto, H. (eds.) Haptic Interaction, Lecture
Notes in Electrical Engineering, vol. 432, pp. 27–32. Springer (2016)
6. Higashi, K., Okamoto, S., Nagano, H., Yamada, Y.: Effects of mechanical parameters on hardness experienced by damped natural vibration stimulation. In: Proc.
IEEE International Conference on Systems, Man, and Cybernetics. pp. 1539–1544(2015)
7. Higashi, K., Okamoto, S., Yamada, Y.: What is the hardness perceived by tapping?
In: Bello, F., Kajimoto, H., Visell, Y. (eds.) Haptics: Perception, Devices, Control,and Applications, Lecture Notes in Computer Science, vol. 9774, pp. 3–12. Springer(2016)
8. Higashi, K., Okamoto, S., Yamada, Y., Nagano, H., Konyo, M.: Hardness perception by tapping: Effect of dynamic stiffness of objects. In: Proc. IEEE World
Haptics Conference. pp. 37–41 (2017)
9. Ikeda, Y., Hasegawa, S.: Characteristics of perception of stiffness by varied tapping
velocity and penetration in using event-based haptic. In: Proc. 15th Joint Virtual
Reality Eurographics Conference on Virtual Environments. pp. 113–116 (2009)
10. Kuchenbecker, K.J., Fiene, J., Niemeyer, G.: Improving contact realism throughevent-based haptic feedback. IEEE Trans. Visualization and Computer Graphics
12(2), 219–230 (2006)
11. LaMotte, R.H.: Softness discrimination with a tool. Journal of Neurophysiology
83(4), 1777–1786 (2000)
12. Lawrence, D.A., Pao, L.Y., Dougherty, A.M., Salada, M.A., Pavlou, Y.: Ratehardness: A new performance metric for haptic interfaces. IEEE Trans. Robotics
and Automation 16(4), 357–371 (2000)
13. Okamura, A.M., Hage, M.W., Cutkosky, M.R., Dennerlein, J.T.: Improving realitybased models for vibration feedback. In: Proc. ASME Dynamic Systems & Control
Conference. vol. 69, pp. 1117–1124 (2000)