JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015

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Phase difference between normal and shear forces during
tactile exploration represents textural features
Hikaru Hasegawa, Shogo Okamoto, and Yoji Yamada

Abstract—Contact forces and skin deformation induced during tactile
exploration have been investigated in the frequency domain to understandﬁnger–material interaction. Their power spectra are one of the representative feature quantities that have been associated with the surface
properties of materials. However, thus far, the phase information ofthese quantities has not been studied. Furthermore, most previous studies
focused on uni-dimensional signals such as forces in either the normal
or tangential directions. We investigated the phase differences between
normal and shear forces induced during tactile exploration. The results
showed that the phase differences between these two axial forces differ
among materials and that they exhibit features different from their power
spectra. These results indicate that the phase difference between two axial
forces should be taken into account to understand the ﬁnger–materialinteractions during tactile exploration.
Index Terms—Phase difference, Normal and shear forces, Power
spectrum, Tactile exploration.

I. I NTRODUCTION
UMANS perceive textures by sliding their ﬁngers on thematerial’s surface, and this process relies on ﬁnger–material
mechanical interaction. Different materials yield different interactions. Most previous studies analyzed the forces (e.g., [1], [2], [3], [4],[5], [6]) and skin deformation (e.g., [7], [8]) or vibration (e.g., [9],[10], [11], [12]) induced during tactile exploration, and found thatthey depended on the surface properties of the respective materials.
Furthermore, the relationships between the forces or skin deformationand the subjective evaluation of materials have been investigated [13],[14], [15]. Thus, as indicated above, many studies have analyzed themechanical phenomena associated with tactile exploration.
Mechanical signals such as contact forces or skin deformation
caused by tactile exploration have been investigated in the frequency
domain by many researchers. In particular, their power spectra are therepresentative feature quantities associated with the surface propertiesof the materials. For instance, it has been reported that 55 typesof materials have been classiﬁed by the power spectra of skinvibrations [11]. The differences in these spectra correspond to theperceptual differences among the materials [9], [14], [15], [16].
Moreover, the materials could be classiﬁed by the power spectra ofvibrations when they were scanned with a probe [17], [18], [19]or tactile sensors (e.g., [20], [21], [22], [23]). The differences inthe surface properties of materials can be represented by the power
spectra of mechanical signals observed when sliding the ﬁnger, probeor, tactile sensor on them.
Not only the power spectrum but also the phase spectrum can becomputed by frequency analysis. However, the latter aspect has notbeen investigated thoroughly. The phases of mechanical signals induced during tactile exploration have never been investigated whereassome related studies focused on discriminating phases of vibrotactile
components [24], [25], [26], [27]. It should be noted that the phasesdeﬁned in these studies are the shifts between two sinusoidal waves
with different frequencies that compose the stimuli. Some of theseH

All authors are with the Department of Mechanical Systems Engineering,
Nagoya University, Furo-cho, Chikusa-ku, Japan.
This work was partly supported by MEXT KAKENHI (17H04697 and
15H05923).

studies reported that the phases of high-frequency vibrotactile stimuli
are difﬁcult for humans to discriminate, and therefore, the phases offorces applied to the skin may not inﬂuence the perception of ﬁne
textures [24], [25]. The peak vibratory acceleration that is determinedby the phases may inﬂuence the discrimination of high-frequency
vibrotactile stimuli including several frequency components rather
than their amplitudes or frequencies [26]. Also, the phases of lowfrequency vibrotactile stimuli can be discriminated [24], [27]. Therefore, the effects of phases on texture perception may not be ignored.
Nonetheless, little knowledge exists about phase spectra involving
many frequency components; to date, no study has investigatedthe phases of forces or skin deformation induced during tactile
exploration.
Uni-dimensional signals were the analysis target in the studies
mentioned above. Although the forces or skin deformation are multidimensional or multidirectional, the relationships between them havenot been studied. Thus far, the power spectra of skin vibrations andforces in a single direction or the coefﬁcient of friction (deﬁned as theratio between the normal and shear forces) have been discussed. Our
previous study [28] showed that the stochastic relationships between
two axial forces differ among materials. This result suggests thatthe surface properties of materials may inﬂuence the relationships
between the mechanical signals in multidirections. However, again,
these relationships have not been investigated thoroughly. Consequently, the phase difference between multidirectional signals is a
new perspective in this area.
To ﬁll these gaps in the literature, we measured the normal andshear forces during tactile exploration in this study, and investigatedwhether their phase differences differ among materials. Then, we
compared the material classiﬁcation performance using phase differences between two axial forces and the power spectra of the forces
in a single direction (shear force was selected in this study). We
discussed the results from the viewpoint of phase differences, which
represent causal relationships in physics between two axial forces,and expanded on their statistical properties [28]. As stated previously,
despite its importance, phase difference, as an index, has not received
much academic attention; thus, this is the ﬁrst study to investigate therelationships between the phase difference between two axial forces
and materials.
II. E XPERIMENT
We measured the normal and shear forces as well as ﬁnger
positions during tactile exploration using one sample of each material.
The total number of samples amounted to 19. The apparatus andmaterials used in the measurements were the same as those in [28].
However, in this study, the measurements were conducted with alarger number of participants.
A. Measurement setup
The device shown in Fig. 1 was used for measuring the two axial
forces. Similar devices were used in previous studies [29], [30].
Two load cells (9313AA2, Kistler, Switzerland) were located on thebottom of the device for measuring the normal force. The shear

JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015

2

(b)

(a)
Material sheet(c)String is fixed to the fingertip

Encoder
Leaf spring

Normal force sensor(9313AA2 Kistler)

Shear force sensor(9218A Kistler)

Fig. 1. (a) Schematic view, and (b) and (c) photographs of the force measurement setup.

3

2

1

9

8

7

13

14

10

15

6

5

4

11

16

12

17

Fig. 2. Photographs of materials. The numbers correspond to those in Table I.
TABLE IM ATERIALS USED IN THE EXPERIMENT.
1. Wood
11. Fur
2. Coarse Japanese paper
12. Felt
3. Fine Japanese paper
13. Denim
4. Drawing paper
14. Polyester cloth
5. Coated paper
15. Polyethylene sheet
6. Crumpled paper
16. Rubber
7. Cork
17. Aluminum
8. Corrugated paper
18. Cork with talc
9. Woven resin wire mesh
19. Coarse Japanese paper with talc
10. Fake leather

force was measured by a high-precision load cell (9217A, Kistler,
Switzerland). The leaf spring ﬁxed to the side of the device decoupled
the two axial forces. The outputs of the normal and shear force
sensors were ampliﬁed by two charge ampliﬁers (5073A and 5015,
respectively; Kistler, Switzerland), and transferred to the computer at
the sampling frequency as 2 kHz. The material sheets were ﬁxed on
the top surface of the device.
The ﬁnger position was measured by two-wire encoders (MTL-12,
MTL Co., Japan). The wires were ﬁxed on the ﬁngertip, and thus, theencoders rotated along with the ﬁnger movements. The total tension
applied to the ﬁngertip by the two wires was canceled because theypulled in a crosswise direction.

B. Nineteen materials
We measured the forces using the materials that can be touchedin our daily lives to investigate the aspects associated with natural
tactile exploration. Fig. 2 shows the materials used in the experiment.
Among them, cork and coarse Japanese paper induced strong friction.
Thus, we prepared another pair of the same materials by lubricating
their surfaces with talc. The 19 materials that were used in this study
are shown in Table I.

C. Tasks
Participants actively slid their dominant index ﬁngers over thematerials ﬁxed on the top surface of the device, as shown in Fig. 1 (b).
They were instructed to stroke the surface in one direction for 20 s foreach material. They wiped their ﬁngers before each trial to eliminate
surplus moisture. They could see the materials during trials, and thescanning speed and strength could be adjusted independently so as
to perceive the quality of each material’s surface and allow natural
tactile exploration. However, we only used data corresponding toﬁnger speeds ranging from 30–90 mm/s in the analysis that followed.
In this speed range, a satisfactory amount of force data were available
for computing the Fourier transforms for all participants.
D. Participants
Fourteen students (12 males and 2 females in their twenties) who
were unaware of the objective of the study participated voluntarily
in this experiment.
III. DATA ANALYSIS
We computed the phase differences between the normal and shear
forces. We also computed the power spectra of the shear forces
because they could be measured more precisely than those of thenormal forces. The data when the ﬁnger speed ranged from 30–
90 mm/s were analyzed. Frequency components higher than 300 Hz
were excluded from the analysis because of the problems resulting
from the signal-to-noise ratio.
A. Computation of phase difference between normal and shear forces
Let Fn (t) and Fs (t) be the time series data of the normal and shear
forces, respectively. Then, their phase difference ϕ can be computed
using
(
)
F {Fs (t)}ϕ(f ) = ∠,(1)
F {Fn (t)}where f and F denote the frequency and Fourier transform, respectively. The phase difference ϕ is deﬁned as the phase shift of the shear
force against the normal force, and its value varies in the range of(−π, π). When ϕ > 0, the phase of the shear force leads that of thenormal force. The sample numbers of Fn (t) and Fs (t) in each trial
were larger than 2000 points, and they were thus downsampled so thatthe frequency resolutions of their spectra became 1 Hz (∆f = 1 Hz).
To minimize the effect of noise, a moving average ﬁlter was applied
to the computed phase differences (ﬁlter size: 10). Fig. 3 (a) shows
an example of the phase difference between the two axial forces,
induced by stroking corrugated paper.

JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015

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Phase difference (rad)(a)

measure is the between-material scatter matrix Sb , which is deﬁned
as
c
∑
Sb =
(x̄j − x̄)(x̄j − x̄)T ,(3)

Raw
Filtered
2

j=1

0
−

2

−
0

50

200
150
100
Frequency (Hz)

250

300

0

50

200
150
100
Frequency (Hz)

250

300

(b)
Normalized power spectrum

where x̄ ∈ R
is the mean vector of all samples. Sb represents
the distances between the mean vectors of each material and all
samples; thus, the larger this value, the easier the classiﬁcation. LDA
determines the feature vectors that maximize the variance of Sbagainst Sw . This calculation involves the eigen value expansion of
−1
Sw
Sb .
We extracted the principal components of all samples using PCA
before conducting classiﬁcation with LDA to ﬁlter out the insigniﬁcant signals as a noise. We applied LDA to the sample set approximated by 19 principal components. They explained approximately
80% of the variance of the original data.
In this study, the feature vectors were computed from either orboth the phase differences between the two axial forces (m = 301
dimensions) and the power spectra of the shear forces (m = 300
dimensions) induced when the 14 participants stroked the 19 typesof materials (i.e., n = 266 samples). We evaluated the material
classiﬁcation performances using the ﬁrst 2–30 extracted feature
vectors (d = 2, · · · , 30) by applying LDA to the reduced data.
When both phase differences and power spectra were used, theirscales were uniﬁed before computing their feature vectors. The phasedifference ϕk ∈ R301×1 (k = 1, 2, · · · , 266) and power spectra pk ∈R300×1 (k = 1, 2, · · · , 266) in the k-th trial were transformed into
the z -score in each trial. Then, we applied the PCA and LDA to
m×1

1

0.5

0

Fig. 3. Examples of the phase and power spectra of forces induced during
tactile exploration. (a) Phase difference between two axial forces induced bystroking corrugated paper. The blue line denotes the raw spectrum. The orange
line refers to the spectrum after applying a tenth-order moving average ﬁlter.
(b) Normalized power spectrum of shear force induced by stroking cork.

B. Computation of power spectrum of shear force
We computed the power spectrum of shear force P (f ) with ∆f =1 Hz and the data in the range of 1–300 Hz. In order to compare the
proﬁles of the power spectra, P (f ) was normalized by the maximum
value in each trial. Fig. 3 (b) shows an example of a power spectrumof the shear force, which was induced by stroking cork.

C. Feature vector extraction from spectra
We regarded the phase differences and power spectra as multivariate data, and applied a combination of principal component analysis(PCA) and linear discriminant analysis (LDA) [31] (eig function ofMatlab 2019a, Mathworks Inc., MA) to compute the feature vectorsthat classify the materials. The PCA functions as a ﬁlter by removing
the noise from each sample, and the LDA classiﬁes them.
In LDA, the feature vectors that classify groups of samples arecomputed from two criteria measures. The ﬁrst measure concerns
the within-material scatter matrix Sw . Sw of a sample set composedof m dimensional vectors that store the spectra can be deﬁned as
Sw =

nc
c ∑
∑
(xji − x̄j )(xji − x̄j )T ,(2)

j=1 i=1

where xji ∈ Rm×1 is the vector of the i-th trial for material j,
and x̄j ∈ Rm×1 is the mean vector for material j. c is the numberof materials, and nc is the number of samples belonging to eachmaterial. Sw represent the variance of samples in each material; thus,
the smaller the variance, the easier the classiﬁcation. The second

Xϕp = [(ϕ1 ϕ2 · · · ϕ266 )T (p1 p2 · · · p266 )T ].

(4)

D. Material classiﬁcation
If the phase differences and power spectra represent the surface
properties of materials, then the materials can be classiﬁed using thisinformation. Therefore, we investigated whether the materials can beclassiﬁed in the feature vector spaces extracted by either or both thephase differences and the power spectra by using two popular but
distinctive approaches.
The ﬁrst approach was intended to statistically classify the materials. We applied multivariate analysis of variance (MANOVA) tothe pairs of materials in the feature vector space and judged whether
arbitrary pairs of two materials could be statistically distinguished.
When the results of MANOVA satisﬁed p < 0.05, we judged that thetwo materials could be distinguished. We calculated the proportionof the distinguished pairs among the whole pairs of the materials.
We performed this process for each d (= 2, · · · , 13). d = 13 is the
maximum dimension that is adaptable for MANOVA considering thedegree of freedom.
The second approach to classify the materials was based on the
k-means method, and in this case, we calculated the correct rate ofclassiﬁcation [11]. We randomly divided 14 participants into groups A
and B with replacement and calculated the centers of the distributionsof all 19 materials in each group. Each material in group A wasclassiﬁed as the material in group B that was distributed the closest(k-means method). This judgement was based on the Euclidean
distance between the centers of the distributions. This calculation
was repeated 100 times, and we computed the mean of the correct
rate among all the materials. This calculation was performed for eachd (= 2, · · · , 30).
IV. R ESULTS
Fig. 4 (a) exhibits the relationship between the number of feature
dimensions d and proportion of the pairs of materials classiﬁed by

JOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015

4

100
80
60
40
20
0
2

4 5 6 7 8 9 10 11 12 13
Number of feature dimensions (d)Power spectrum
Phase difference
Phase difference + power spectrum

3

100
80

60
40
20
0
2

26
22
18
14
10
Number of feature dimensions (d)Power spectrum
Phase difference
Phase difference + power spectrum
6

Feature 1
Feature 2

0.2

0

-0.2
0

50

200
150
100
Frequency (Hz)

30

250

300

(b)
0.4

(b) Material classification based on the k-means method
Correct rate ofmaterial classification (%)

Coefficients of feature vectors ofphase differences (arb. unit)(a)Coefficients of feature vectors of
power spectra (arb. unit)

Proportion ofclassified pairs of materials (%)(a) Material classification by MANOVA

Feature 1
Feature 2

0.2
0
-0.2
-0.4
-0.6
0

50

200
150
100
Frequency (Hz)

250

300

Fig. 4. Classiﬁcation performance in d-dimensional feature space. (a)Proportion of pairs of materials distinguished by applying MANOVA. (b)Correct rate of classiﬁcation based on the k-means method.

Fig. 5. Coefﬁcients of two feature vectors extracted from the spectra. The
feature vectors of (a) phase differences between the two axial forces and (b)power spectra of the shear forces.

MANOVA. At most, 57.9% (d = 4) and 80.1% (d = 5, 6) of material
combinations were distinguished by the phase differences and power
spectra, respectively. The corresponding result when both the phasedifferences and power spectra were used was 97.1% (d = 10). The
material classiﬁcation performance when using the power spectra
surpassed that of the classiﬁcation by the phase differences. It isnotable that the phase difference represents the differences among
the materials though it is inferior to the power spectra in terms ofclassiﬁcation performance. The classiﬁcation performance when boththe phase differences and power spectra were used was the highest
among the three conditions for all d.
Fig. 4 (b) exhibits the relationship between d and the correct
rate of material classiﬁcation based on the k-means method. The
correct rate converged to approximately 70% and 80% using the phasedifferences and power spectra, respectively, as d becomes larger. The
corresponding result was approximately 98% when both the phasedifferences and power spectra were used. Thus, the features of thematerials can be represented better using both the phase differences
and the power spectra.
Although similar analyses were performed for the phase differences
and power spectra at the frequency resolution ∆f = 2 Hz, the results
did not change considerably for both types of criteria. Thus, thefrequency resolution may not greatly inﬂuence material classiﬁcation
performance.
We introduce the ﬁrst two feature vectors as the representative
examples in the following paragraphs. The contribution ratios ofeach feature vector of the phase differences were 31.0% and 17.9%,
respectively. The corresponding results of the power spectra were

36.8% and 18.7%, respectively. Fig. 5 (a) shows the feature vectors
extracted from the phase differences between the two axial forces.
They were difﬁcult to interpret, and the coefﬁcients were either
positive or negative depending on the frequency range. Fig. 5 (b)shows the feature vectors extracted from the power spectra of theshear forces. They were apparently different up to 100 Hz.
Figs. 6 (a), (b), and (c) show the distribution of the studiedmaterials in the two-dimensional feature vector space computed using
phase differences, power spectra, and both, respectively. The sizes ofthe ellipsoids show the standard errors among all the participants. The
smaller the size, the smaller the individual differences. The colors
and numbers of the ellipsoids correspond to the types of materials.
Although the distribution of the materials differed between the feature
vector spaces of the phase differences and power spectra, similartypes of materials were found to be distributed together in both
spaces (e.g., clothes and papers were distributed in the third and ﬁrst
quadrants, respectively, in the phase difference space, and in the ﬁrst
and second quadrants, respectively, in the power spectrum space).
The distributions of the materials in Figs. 6 (b) and (c) were similar.
Thus, the effect of the power spectrum was dominant compared tothat of the phase difference in the material classiﬁcation.
Table II shows the breakdown of the number of material pairs
classiﬁed by MANOVA in the two-dimensional feature space extracted from either the phase differences or the power spectra. The
supplementary ﬁle shows the ﬁnal combinations of the classiﬁed
materials. As shown in Table II, among the 92 pairs of material
combinations that were classiﬁed by the phase differences, 74 pairs
were also classiﬁed by the power spectra. This number is equivalent toJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015

5(b) Power spectrum(a) Phase difference(c) Both phase difference and power spectrum
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0.5

19

18 17

14 1

10

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13 9

-0.5

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7

4 5

3

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-1

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36

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10 19
1
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-0.25

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1
0
-1
Score of feature 1

Wood
Coarse Japanese paper
Fine Japanese paper
Drawing paper
Coated paper

2

4
2

5

3 6 9

0
4

-2

7

-4

13
2
18

17 14
1
15
8

10 19

12
11

16

-6

-0.5
-2

1
2
3
4
5

5

0.25

Score of feature 2

0.5
0

13

2

Score of feature 2

Score of feature 2

1

× 10

0.4
0
-0.4
Score of feature 1-0.8

6 Crumpled paper
7 Cork8 Corrugated paper
9 Woven resin wire mesh
10 Fake leather

0.8

11 Fur
12 Felt
13 Denim
14 Polyester cloth
15 Polyethylene sheet

-6

-4

4
2
0
-2
Score of feature 1

6× 10

16 Rubber
17 Aluminum
18 Cork with talc
19 Coarse Japanese paper with talc

Fig. 6. Distribution of materials on the two-dimensional feature vector space computed using (a) phase differences of the two axial forces, (b) power spectraof the shear forces, and (c) both (a) and (b). The sizes of the ellipsoids show the standard errors among the participants. The material numbers correspond to
those in the list below.

TABLE IIN UMBER OF MATERIAL PAIRS CLASSIFIED IN TWO - DIMENSIONALFEATURE SPACE BY MANOVA USING EITHER PHASE DIFFERENCES OR
POWER SPECTRA .

Classiﬁed bypower spectra

Yes
No
Total

Classiﬁed by phase differences
Yes
No
74
54
18
25
92
79

Total
128
43

approximately 80% of the material combinations that were classiﬁedby the phase differences. However, the remainder of the 18 pairs(∼20%) were classiﬁed by the phase differences but not by the power
spectra. The phase difference represents differences among materials
in a manner unlike the power spectrum, even though its material
classiﬁcation performance was inferior to that of the latter.
V. D ISCUSSION
More types of materials were classiﬁed using both phase differences and power spectra than using only the latter. The cloth materials
were especially well classiﬁed when using the power spectra, whereas
the other materials were classiﬁed well when using both the power
spectra and phase differences (Figs. 6 (b) and (c)). This suggests thatthe features of the materials that cannot be represented by the power
spectra can be supplementally represented by the phase differences.
Although it is unclear what kinds of properties are represented by thephase difference, the surface property information of materials can beconsidered to be contained in both the power spectrum and the phasedifference, and the latter deserves more focus. Furthermore, when
we computed the phase spectra of either the normal or the shear
forces during the tactile exploration, they appeared to be random
and non-locked, unlike the results seen in Fig. 3 (a). The differencesamong the materials appear with regard to the phase informationof multidimensional signals, such as the phase difference between
the normal and shear forces, but not with regard to that of unidimensional signals. It is thus important to investigate not only unidimensional signals but also the relationships between these multidimensional signals.

In this study, the materials were classiﬁed by the phase differences
and power spectra even though the scanning speeds of the participants’ ﬁngers showed some variability (30–90 mm/s) in the analysis.
In [11], 55 types of materials were classiﬁed using the power spectraof skin vibration induced by scanning the materials at a constant
speed under a passive touch condition. Their classiﬁcation accuracies
were 80–93%. In [30], six types of materials were classiﬁed using thepower spectra of the shear forces while the ﬁnger speed was in the
narrow ranges under an active touch condition. Their classiﬁcation
accuracies exceeded 90%. The corresponding result of ours was 80%,
which may be smaller than those in [11], [30]. The number andtypes of materials were different among these three studies. Thus, wecannot compare these results directly but in our study the variability
in the scanning speed might have had a negative impact on theclassiﬁcation performance.
The classiﬁcation performances by MANOVA using either thephase differences or the power spectra were maximized at d = 4 − 6and decreased as d exceeded these values (Fig. 4 (a)). This result
may be attributed to signal noise extraction as high-dimensionalfeature vectors. In general, it is difﬁcult to conduct classiﬁcation
using MANOVA when the data include large amounts of noise. This
problem may be similar to overﬁtting, which may not have occurred
when both the phase differences and power spectra were used because
the number of variables was larger in this case than when only oneof them was used. Thus, the classiﬁcation performance might havebeen maintained for a large d. On the other hand, the correct rate ofmaterial classiﬁcation based on the k-means method was high for alarge d in all three conditions (Fig. 4 (b)). The noise may not likely
inﬂuence the k-means method because it uses the distances between
the centers of the material distributions.
As shown in Fig. 3 (a), ϕ took both positive and negative values.
This result means that the phase of the shear force leads or lags thatof the normal force. The former can be explained geometrically. The
shear force induced by stroking an uneven surface can be expressedas the derivative of surface displacement [32], [33]. Simultaneously,
the normal force can be expressed as the function of surface displacement. This means that the phase of the shear force leads that of
the normal force. The delay in the phase of the shear force may be
attributed to the behavior of the coupled system in the normal andJOURNAL OF LATEX CLASS FILES, VOL. 14, NO. 8, AUGUST 2015

tangential directions with viscoelastic resistance.
VI. C ONCLUSION
Most previous studies demonstrated that the power spectra ofcontact forces or skin deformation induced during tactile exploration
represent differences among materials. However, their phase spectra
were disregarded because they are not locked and behave randomly.
It is reasonable to suppose that the phase differences of two axial
forces can be physically determined. Our experiments to this effectshowed that the phase differences between them are not random anddiffer statistically among materials. Among the 19 types of materials,
80% and 58% of the pairs were statistically distinguished by thepower spectra and phase differences, respectively. The corresponding
results of classiﬁcation accuracy were 80% and 70%, respectively.
Some materials were only classiﬁed by the phase difference, andthus, its importance should be reconsidered for understanding tactile
material perception and developing tactile texture displays. The
correspondence between the phase difference of two axial forces andperception should also be directly investigated in the future.
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