Interactive Forces Caused by Scanning Wavy Surfaces
Yohei Fujii, Shogo Okamoto, and Yoji Yamada ∗Department of Mechanical Science and Engineering, Nagoya University, Japan.

A BSTRACT
Well-designed force fields can cause the perception of surface
shapes such as holes or bumps on a flat surface without involving
actual geometric displacements during exploratory hand motions.
To establish a better force field model, we observed the interactive
forces caused by scanning a sinusoidal wavy surface under point
and surface contact conditions. Under the point contact condition,
the acquired forces fit well into the model of earlier studies with the
exception of kinetic frictions. When the surface was scanned witha bare finger to obtain a surface contact area, the observed force
profiles significantly deformed compared with those of the earlier
static point-contact model. Such deformation of the force profile
can be explained by combining Hertz contact theory and the static
point-contact model. The results of this study will lead to the development of an interactive force model for surface exploration, that is
expected to improve force rendering algorithms.
1

I NTRODUCTION

A well-designed force field on a flat plane can produce the perception of displacements perpendicular to the plane [1, 2]. For example, Fig. 1 shows that delivering assistive or resistive forces to aperson’s finger causes the sense of shape along Y-axis as the finger
explores the X-Z plane. Such force fields can induce virtual surface
profiles that are perceptually equivalent to real ones [2, 3]. The expression of surface profiles using the force field on a plane is compatible with prevalent pointing interfaces represented by a mouseand touch panel. If these 2D or planar interfaces are equipped withforce feedback functions, they can allow users to feel the threedimensional characteristics of surfaces (e.g., [4, 5]), which can potentially become a major industrial or commercial application of
haptic interfaces.
The fact that a force field on a plane evokes the perception of
surface profiles is an important finding in terms of both practicalapplications of haptic interfaces and understanding human perceptual mechanisms. Improving this technique is commonly beneficial
for haptic researchers. There are still some gaps between earlier
force field models and the actual phenomenon; there are the resultof the complex interaction between a soft finger pad and geometric shapes. In this study, we focused on one of the most successful models used by Robles-De-La-Torre and Hayward [2]. As described in Sec. 2, this model assumes point and frictionless contact.
These assumptions are close to the conditions when a human slowly
scans a surface profile with a frictionless pointing probe such as aball bearing. However, they may somewhat differ from those when
exploration is carried out with a human finger pad.
The objective of this study was to observe interactive forces
caused by exploration of a wavy surface profile under two contact
conditions (point and surface contact) and two kinetic conditions(pseudo static and dynamic). We compared the observed force vectors with the earlier virtual shape model and specified the effects of∗ email: see http://www.mech.nagoya-u.ac.jp/asi/en/ This study was inpart supported by MEXT Shitsukan (#25135717).

IEEE Haptics Symposium 201423-26 February, Houston, Tx, USA
978-1-4799-3130-9/14/$31.00 ©2014 IEEE

Figure 1: Perception of virtual wavy surface by planar force field

the exploring conditions.
2 S TATIC POINT- CONTACT MODEL
As an example of the presentation of virtual shapes, Robles-DeLa-Torre and Hayward successfully demonstrated virtual bumps orholes using force fields on a plane [2]. In this method, they geometrically expanded the force vector received by a person from thesurface when s/he scans it with a probe. As shown in Fig. 2, thehorizontal force depends on the vertical force or gradient of the surface. In this figure, fs and fp are the force that a person applies tothe surface and its reaction force, respectively. When a probe orcontactor remains still on the surface at (x, y), the components offp satisfy
fpx = fpy tan(α(x))(1)

where α(x) is the angle of the surface at x. The relationship between the surface profile function y(x) and this angle istan(α(x)) =

dy
.
dx

(2)

Hence, the ratio of force components is equal to the gradient of thesurface profile, which is described by
fsx
dy
=
.
fsy
dx

(3)

Using the above force equilibrium, a force field can evoke theperception of surface displacements on a flat surface. The force
presented by a haptic display is denoted by fv and should satisfy fs + fv + fp = 0. The X-axis component of this equation
isfsx + fvx = 0 because fpx = 0 on the flat surface. The Y-axis
component is fsy + fpy = 0 because fvy = 0 in the case of a2-d.o.f. force display as shown in Fig. 1. Instead of fpx , whichis the reactive force from the angled surface, the force field fvx is
introduced. The horizontal force presented by the force displays isthen determined by
fvx = fpy tan(α(x)).

(4)

This force rendering algorithm successfully leads to the perceptionof wavy surfaces such as bumps or holes where the profiles are defined by α(x). Similar force rendering algorithms are also effective449

Figure 2: Force components of still pointing probe on slope

3.1 Equipment
Fig. 3 shows the experimental setup: the position of the finger orball bearing and the interactive forces between it and the sinusoidal
surface were recorded at 1 kHz. A triaxial force sensor (USL06-H5,
Tec Gihan Co. Ltd., nominal resolution: 19.6 × 10−3 N) locatedbeneath the surface measured the X- and Y-axis forces applied tothe surface. The position of the finger or bearing was measured
using two encoders (RE30E-500, Nidec Copal Electronics Corp.,
500 ppr): each was attached to a string that was fixed through apulley with a diameter of 13 mm. The encoders were placed at
the height of h = 250 mm with the interval w = 500 mm. The
resultant measurement resolution were 14.4 and 17.2 µm along theX- and Y-axes, respectively. The end of the string was fastened tothe tip of the ball bearing or to a ring worn on the index finger. The
diameter of the ball bearing was 0.5 mm, and it was well-lubricatedduring the experiments.
3.2 Sinusoidal surface
The sinusoidal wavy surface used in the experiment was made ofan ABS plastic board (170 mm × 50 mm). The interval between
two waves was set to 40 mm, and the height difference between thepeak and valley was 5 mm. The surface displacement along x was
y(x) = 2.5 sin (

2πx
)
40

(5)

in millimeters. The surface was polished with a fine sand cloth.
The participants coated their finger pads with talcum powder before
scanning the surface. This powder effectively prevented the finger
pads from sticking to the surface because of perspiration.

Figure 3: Experimental setup. Two encoders and force sensor for
position and force measurement. Two strings fixed to encoders bypulleys ends at ring on finger or tip of ball bearing. Plate with wavy
surface displacement was positioned on triaxial force sensor.

for presenting surface curvature in 3D environments involving actual displacements perpendicular to the surface [6].
There are some concerns when we compare the above force rendering algorithm and the phenomenon caused by interaction between a finger pad and wavy surface. First, the contact between
a finger pad and surface produces a contact area. Although the reactive force centers at a point of contact in the above algorithm, thepressure diffuses within the contact area in the case of a finger pad.
This effect is not ignorable because such information of contact areaor pressure distribution within the area contribute to the curvature
percepts [7, 8]. Second, the above model does not include frictionthat naturally exists between two objects. Such friction should be
fairly considered because the significant effect of friction on theperception of curvature is known [9]. These differences between
the model and finger pad’s contact potentially lead to a substantial
difference in the relationships of related force vectors.

3

E XPERIMENT

We observed the forces produced when a sinusoidal surface was
explored. In order to determine the effects of the contact conditions,
we compared the explorations by a finger pad and ball bearing withmoderately frictionless point contact. The exploratory speed wascontrolled at several different levels to investigate its influence on
the interactive forces.

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3.3 Procedures
One volunteer continuously explored the surface for 120 s while attempting to maintain contact between the finger pad or ball bearing
and the surface. Even if the contact was not maintained, irregularities were easily detected from the output of the force sensor. Whenthe volunteer used the ball bearing, he tried to maintain the postureof its grip handle perpendicular to the desktop. When exploringwith a bare finger, he again maintained the posture of his hand asmuch as possible. During the exploration, he was instructed to relax
so that extraordinarily large forces were not applied to the surface.
To observe the effect of scanning speeds i.e. quasi static or dynamics, three reference hand speed levels were used for exploration withthe ball bearing: 10, 30, and 60 mm/s. In contrast, only one reference speed was used for finger exploration: 30 mm/s. As describedlater, during analysis we excluded force samples observed at speeds
significantly far from these reference speeds.
4 R ESULTS
4.1 Data analysis
The positions of fingertip or ball bearing were computed from thedata acquired by the encoders. The position (x, y) should satisfy{( w2 + x)2 + (h − y)2 = l12(6)( w2 − x)2 + (h − y)2 = l22
where l1 and l2 are the distance between each encoder and ballbearing or fingertip. The velocities of the fingertip and ball bearing were computed from the position data. We classified the ballbearing velocities on X-Y plane into three levels: 0–20, 20–40, and
40–100 mm/s. Those above 100 mm/s were not used for the analysis. For the fingertip scan, we only used samples with velocities of
30–50 mm/s.
Because we were interested in the equilibrium of force ratios asdetermined by (3), we computed the force ratios fx (x)/fy (x) along
the X-axis after excluding outliers with weak fy – owing to irregularFigure 4: Average of observed force ratios fx (x)/fy (x). A sample ofthe finger scan.

contact. Naturally, these ratios fluctuated, as shown in Fig. 4. We
averaged them to determine the final values.
4.2 Exploration by ball bearing (pseudo point contact
and low friction)
Fig. 5 shows the force ratios observed when the ball bearing moved
on the surface from left to right. These figures show the data for
60 mm just above the force sensor. The top figure is the profileof the scanned surface, y(x). The middle and bottom ones are theforce ratios, fx /fy . All figures share the horizontal position axis.
We first examine the middle figure with two curves: curve determined by the model of (3), and the experimentally observed curveas the ball bearing explored the surface at the speed of less than
20 mm/s. The profiles of the two curves were fairly similar, whichmeans that (3) functions in the case of a ball bearing that was close
to the low-frictional point contact condition.
The exception between the model and observed curves was the
maximum peak values of the force ratios. The peak force ratioof the observed data was larger than that of the model curve, as
shown by the double-headed arrow in the figure. This gap was dueto the friction inside of the ball bearing. Although we considered
the exploration by ball bearing to be frictionless, tangential force
increased by friction.
The bottom of Fig. 5 shows the three observed force ratios at
different velocity levels. Considering the sampling deviation, we
concluded that there were no substantial differences among thesethree curves despite the different scanning speeds. Within the rangeof our experimental conditions, the ratios of the horizontal and normal forces did not depend on the probe speeds.
4.3 Exploration by index finger (surface contact withfriction)Fig. 6 shows the force ratios observed when the surface was explored by the index finger for which frictional surface contact was
assumed. The gray and black curves represent those observed when
the finger moved rightward and leftward, respectively. The dotted curve was determined by the static point-contact model of (3).
These curves significantly differed from each other unlike those observed using the ball bearing.
First, the base levels of the curves shifted. Compared with thestatic point-contact model (dotted curve), the gray curve significantly shifted upward. Similarly, the black curve shifted downward. These base-line shifts were due to the friction forces, which
increased the interactive forces along the moving direction. Whenthe kinetic friction was effective, the X-axis component of the interactive forces increased as described above.
Second, the profiles of the force ratios substantially deformed,
and they were no longer similar to cosine curves. For the static
point-contact model (dotted curve), the peak of the force ratio wasat the maximum gradient of the surface. However, the peak of theobserved curve (gray curve) shifted leftward compared with that of

Figure 5: Exploration by ball bearing: (Top) profile of sinusoidal surface; (Middle) comparison between static-point-contact model andobserved force ratio at low speed; shadow is width of average standard deviation; (Bottom) observed force ratios at three speed levels.

the dotted curve. Similarly, the bottom of the gray curve shiftedrightward compared with that of the dotted curve. Such deformation of the force ratio curves was due to the surface contact of thefinger pad. For the point contact model, the reaction force wassolely applied to the contact point; however, with surface contact,
the pressure distribution in the contact area varied depending on thegradient of the surface. We discuss the deformed profile of the force
ratio curves in the next section.

5

S KEWED PROFILE OF FORCE RATIO CAUSED BY ASYM METRIC PRESSURE DISTRIBUTION WITHIN CONTACT SUR FACE

5.1 Asymmetry of pressure distribution in contact area
According to the Hertz contact theory [10], a circular contact surface with a diameter of 2a is obtained from the contact of two spherical objects. As shown in Fig. 7, the pressure is distributed on thearea in a parabolic manner with the maximum pressure pmax at thecenter of the area. R is the radius of the object. The pressure at r

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Figure 8: Reaction forces concentrated on two points within the contact surface in an asymmetric manner

Using (9), the ratio of asymmetric pressures at two points, the force
ratio received by two points at x is described by
fx
1
√
=√
×
fy
−r−R sin(α) 2
r−R sin(α) 2
1−(
)
+
1
−
(
)
a
a
√
−r − R sin(α) 2) × tan(α(x − r))+
{ 1−(
a
√
r − R sin(α) 2
1−() × tan(α(x + r))}.
a

Figure 6: Force component ratios when surface was scanned by index finger at 30–50 mm/s

Figure 7: Pressure distribution within the contact surface

from the circular center is expressed as√rp(r) = pmax 1 − ( )2 .
a

(7)

By definition, the pressure distribution is symmetric to the center:p(r) = p(−r).
When the surface is inclined at an angle of α, the surface centermoves as much as R sin(α). Accordingly the pressure changes to√p(r) = pmax

1−(r − R sin(α) 2
) .
a

(9)

With a positive angle of the surface, p(r) > p(−r). In contrast,with a negative angle, p(r) < p(−r). On the half of the surface
closer to the top of the bump, the pressures are larger than those on
the other side of the surface that is closer to the valley of the wavy
surface.
5.2 Simulation of skewed profile of force ratios
Here, numerical simulation is conducted to see the effects of asymmetric pressure distribution within the contact area. As shown in
Fig. 8, two representative points that are apart 2r from each otherapproximate the asymmetric normal force. The point at the higher
place receives larger force than the point at the lower place on both
the rising and descending slopes. We assume that the force equilibrium of each point follows the static point-contact model of (3).

452

Taking account of the dimensions of a human finger, we determined
the pertinent parameters such that the profile of this force ratio resembles the observed ones. The parameters were set to R = 7,
r = 3.4, and a = 6 mm. Fig. 9 shows the one from this equation and those from the exploration using the index finger that are
the same as those in Fig. 6. The above equation fairly expresses
the phase shifts of their peaks and the profile that resembles theobserved force ratio curves.
Fig. 10 shows the effect of the parameter. Larger r values deformthe profile of force ratios more because r determines the degree ofasymmetry of pressures. From this simulation, we suggest that the
skew of the force profile is mainly due to the asymmetric pressure
distribution within the finger pad. Still, the base levels of thesesimulated and observed curves are largely different. As discussed
before, such base line shifts are potentially explained by kinetic
friction. Hence, the differences in the profiles of force ratios between the static point-contact model and the exploration of finger
pad mainly originate from the surface contact and friction.

(8)

Hence, the pressures at p(r) and p(−r) are no longer the same.
Their ratio becomesp(−r) : p(r) =√√−r − R sin(α) 2r − R sin(α) 21−() : 1−(
) .
a
a

(10)

6

C ONCLUSION

A well-designed planar force field evokes the perception of shapeswith displacement perpendicular to the plane. This is a beneficial
perceptual mechanism that allows 3D shapes to be presented byhaptic displays with limited degrees of freedom. Establishing better algorithms to produce force fields based on the observation of
actual haptic exploration is an important challenge. In this study,
we observed interactive forces when a sinusoidally displaced surface was scanned at different contact conditions and speeds. Whenthe surface was explored with a ball bearing, the profiles of the ratios of horizontal and vertical forces were similar to those of thestatic point-contact model with the exception of larger force ratios
originating from friction at the rising slope. We did not observe any
differences in the force ratios among the different speed levels of
exploration. When the surface was explored with a bare finger, theprofile of the force ratios substantially deformed. Such deformation
can be explained by the surface contact between the finger pad andsurface. An asymmetrical pressure distribution on the contact area
was produced by the inclined surface, which skewed the force ratio
profile. Understanding the force interaction of finger exploration
will lead to better force rendering techniques in the future.

[10] T. Kato and M. Masuko, Basis of Tribology, 2nd ed. Baihukan, 2006.

Figure 9: Simulated force ratio curve and curves observed by finger
exploration

Figure 10: Effect of r on the skew of force ratio profile

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