Haptic Exploration of Spheres: Techniques and Initial Experiments

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Haptic Exploration of Spheres: Techniques and Initial Experiments Blake Hannaford, Jesse Dosher, and Sugandhan Venkatachalam Department of Electrical Engineering The University of Washington

I. A BSTRACT Haptic perception of properties of objects in the hand is not fully understood. One aspect is the precision with which subjects can detect geometrical features. In this experiment, subjects discriminated between pairs of steel spheres in which one was a precisely manufactured ball bearing and the other was distorted by various degrees. Precise distortions of spheres provide a very sensitive but practical way to create controlled experiments. In one experiment, we reduced the diameter along a single axis and smoothed and polished the resulting shape. In another, the reduced diameter was held constant, and local curvature was varied. Using these spheres, subjects were able to detect subtle changes in sphere diameter along one axis, but were not as sensitive to changes in local curvature. For diameter, a just-noticable-difference (JND) between 2% and 5% was measured. II. I NTRODUCTION Haptic exploration is a complex sensory/motor activity involving a high number of degrees of freedom, multiple sensing modalities, and integration of sensory information over space and time. Lederman and Klatzky created a taxonomy of “Exploratory Procedures” which are stereotyped hand and finger movements which people preferentially use to make specific sensory discriminations and object identifications. Their experiments were mostly aimed at either 1) identification of objects through haptic interaction and 2) assessment of a property such as temperature or roughness. The work reported here studies haptic perception of sphericity. If an object is placed in the fingers, how well can humans determine whether or not it is a sphere? Starting from nearly perfect spheres, such as for example, precision industrial balls, a variety of increasingly subtle distortions can be made. Psychophysical literature relevant to the present study has been reviewed in detail in [1] and [2]. Weber fraction (the minimum value of ∆x/x which can be reliably perceived) has been measured for the radius of spheres indenting the immobilized finger (10%, r ≈ 6.3mm)[3] and for surface textures actively swept with the finger (5%, λ ≈ 1mm)[4]. The authors would like to acknowledge National Science Foundation grant IIS-0303750, and valuable suggestions from annonymous referees.

If a subject is to detect deviations from perfect sphericity, they may use a variety of cues, depending on how the reference spheres are distorted. These cues may include • Eccentricity (non-constant diameter) • Texture (local variations in radius) • Changes in curvature (the curvature is constant at all points on a perfect sphere) To define these experiments more precisely, we wish to distort the shape but not the texture of some test spheres in precisely defined ways, ideally to determine a sensory threshold for a specific attribute. However, these cues are not variable independently. Since a sphere is defined as a surface of constant local curvature, any amount of eccentricity would also change local curvature. Texture also changes local curvature. We will define texture as radius and curvature changes on a very small scale compared to the sphere’s nominal radius which average out to zero within each small region. III. M ETHODS A set of seven distorted spheres (test objects) were produced starting from highly polished steel ball bearings 0.4996 inches (12.69mm) in diameter. The factory-new balls appear quite perfect with a highly polished mirror finish. Seven additional new balls were set aside as controls. The seven test spheres were ground down on two opposing sides (the “North” and “South” poles) to achieve a range of diameters along a single axis of the sphere. Initially, a flat face was ground on each side. Then the ball was ground by hand to approximately minimize the curvature over the area between the center of the flat and a circle 45◦ below the center of the flat. A description of the grinding procedure is given in [5]. After grinding, the balls were polished using a rotary tumbler with a series of finer grits and polishing compounds to achieve uniform texture around the entire surface of the balls[6]. The factory new control balls were included in the tumbler processing so that they had identical texture to the modified balls. Each ball was etched with a 3 digit randomized serial number on its equator using a diamond stylus. A. Mathematical Description of Sphere Distortion An equation describing the desired ball contour is derived as follows. We will parameterize the smoothing

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Fig. 1. Plot of the profile of a smoothly distorted sphere and an ideal sphere to illustrate the type of desired distortion for r = 0.25, d = 0.01, γ = π/4. For this distortion function is only used √ model, the √ between the points −r 2/2 < x < r 2/2 or −0.177 < x < 0.177.

zone in terms of an angle from the pole, γ. The boundary of the smoothing zone will then be all the points “North” of 90◦ − γ or the intersection of the sphere with a cone of apex angle 2γ. There will be a similar zone at the “South” pole. Considering a section through the sphere, a function which can be used is a polynomial which intersects the sphere at a point ±γ from the pole, and which is tangent to the sphere at these points, and further passes through the point x = 0, y = r − d where d is the amount by which the sphere is ground down such that diameter, D = 2y. This must be an even function and a suitable candidate is y = ax4 + bx2 + c (1) y˙ = 4ax3 + 2bx From this and the boundary conditions we can get: c=r−d z1 y1 − r + d b= +2 2x1 x21 a=

z1 + 2bx1 −4x21

where x1 = r sin(γ), y1 = r cos(γ), and z1 = tan(γ). The parameters a, b, c were derived as functions of γ, r, and d. However, our hand fabrication methods did not guarantee that this actual profile could be achieved for a given diameter. For the example of r = 0.250, γ = π/4, d = 0.020, the sphere and this function are plotted in Figure 1. B. Ball Measurement Machine We designed a machine for measurement of the profile of processed balls. A Mitutoyo digital indexing gauge with 0.0001 inch resolution was used. The machine (Figure 2) consists of a bearing spindle with a cylindrical ball holder. The bearing is provided by an old hard drive, whose chassis is mounted below the top plate of the machine. The ball holder has a cylindrical hole in it which supports the ball with a slightly beveled inside edge 7.5mm in diameter. The spherical tip of the indexing gauge contacts the ball at its equator. A 360◦ protractor

Fig. 2. Measurement machine for ball profilometry. Protractor disk is rotated by hand to each measurement angle.

six inches in diameter is affixed to the ball holder. The protractor is rotated by hand and the value of the dial gauge is read. A flap folds down onto the ball and makes point contact with the top of the ball via a polished metal surface. A 280 gram weight is placed on the flap to make sure the ball stays seated firmly in the ball holder and resists side forces from the indexing gauge. The side force applied by the gauge was about 100 grams. C. Calibration The machine was calibrated with an unmodified factory sphere of diameter 0.4996 inches. The indexing gauge was zeroed with the protractor at zero degrees. Relative measurements were taken at 10◦ intervals. Three measurements were repeated and the results are plotted in Figure 3. The average of the six measurements including those of Figure 3 is subtracted from subsequent measurements of test balls. A single test ball measurement, corrected in this manner showed peak error magnitude of 0.0003 inches. D. Curvature Ball measurements were processed with the Scilab numerical analysis package [7]. We want to find a measure of the curvature of the ball at each point around its circumference. Curvature is defined as the inverse of the radius of a circle which describes the curve at

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The interpolation and the width of the convolution window undoubtedly affect the curvature computation accuracy. Further analysis is ongoing to verify that a reasonable compromise has been achieved between noise sensitivity and curvature accuracy. Our curvature measurements are constrained by the fact that the test objects are closed contours. Thus, curvature must integrate out to an average of close to 0.25. Each distortion therefore caused both a local increase and a local decrease in curvature at different points on the sphere.

Fig. 3. Three ball profile measurements taken with a reference ball. Maximum error is 0.0031 in. Average of the three measurements is shown as green triangles.

each point1 . In polar coordinates, the curvature can be computed as K(θ) =

r2 + 2r˙ 2 − r¨ ˙r (r2 + r˙ 2 )3/2

where r˙ =

dr(θ) dθ

d2 r(θ) dθ2 An important aspect of this computation is that when the curve r(θ) is not available in closed form, the derivatives r, ˙ r¨ must be evaluated numerically. Numerical evaluation of derivatives is complicated by the tendency of derivative estimates to have infinite bandwidth and amplify measurement noise and quantization noise. 1) Interpolation: It is time consuming to acquire 360 data points around the sphere with our current manual machine. Instead measurements were acquired only every 10 degrees. We used the Scilab interpolation function to interpolate values between the 10 degree samples to 1 degree resolution. 2) Estimation of Derivatives: The Scilab script used the convolution operator to estimate derivatives and simultaneously filter the data to reduce the effect of noise[8]. The convolution operator h =[-1, -1, 0, 0] was applied to a data set, rm which is sampled every 1 degree. Thus to generate estimates of the two derivative functions we use ˆr˙ = h ∗ rm ˆr¨ = h ∗ ˆr˙ r¨ =

Where ∗ is the discrete convolution operator. The estimate of curvature is thus 2

r2 + 2ˆr˙ − ˆr˙ ˆr¨ ˆ K(θ) = m 2 2 +ˆ (rm r˙ )3/2 1 http://wikipedia.org/Curvature

E. Weight Brand new 0.4996” balls weighed an average of 8.17g. After processing in the tumbler, seven control balls weighed an average of 8.10 grams. Weighed individually, all of the test balls weighed either 8.0 or 8.1 grams (0.1g scale resolution) (see Table I). F. Final Calibration With the machine calibrated and the curvature estimate validated with simulated data, we could procede to final calibration of the curvature estimates. The machine measurement error curve (Figure 3) was measured on different days and found to be repeatable. The Scilab script was modified to substract the measured systematic error from the data set[5], [6], [8]. First, we made three measurements of a factorynew sphere and analyzed them as described above. The results are given in Figure 4. The deviation from the factory specified radius of 0.2498 inches is plotted on an expanded scale in Figure 5. The measurements are all within 0.2497 < r < 0.2505 Note that we are making a relative measurement compared to the radius at θ = 0◦ . Thus the absolute diameter of the balls was measured separately with calipers. The minimum and maximum curvature was 3.70 and 4.57 inches−1 respectively. The noise in Figures 5 and 4 (Bottom) indicates either noise in the measurement or imperfections in the factory balls. Next, three sets of measurements taken every 10 degrees around a tumbler-polished control ball were acquired and processed. The results are given in Figure 6. The minimum and maximum curvature was 3.2 and 5.0 inches−1 respectively, slightly greater curvature than the factory new spheres. A similar test on one of the modified balls is given in Figure 7. G. Experimental Procedures The protocol below was approved by the University of Washington Human Subject Committee. 13 subjects between the ages of 22 and 53 years were recruited with 65% male. Each subject read a printed set of instructions. A plastic box with 7 numbered bins contained 7 pairs of balls, one test and one control. Test ball diameters were assigned randomly to the bins (Table I) and bin 2 contained two control balls. These

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Fig. 5. Magnified Ball profile (radius vs. angle) from three measurements of a factory-new ball. Error is bounded by 0.5×10−4 inches.

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D (inches) 0.4770 ctl 0.4695 0.4925 0.4835 0.4980 0.4880

Weight (g) 8.0 8.1 8.0 8.1 8.0 8.1 8.0

Serial 733 268 814 498 784 155 532

Kmin 1.54 2.88 -0.3 2.89 1.97 2.86 2.16

Kmax 6.13 4.94 7.01 4.85 5.89 4.84 5.22

TABLE I C HARACTERISTICS OF MODIFIED BALLS . C ONTROL BALLS WEIGHED 8.10 GRAMS . R ESOLUTION LIMIT OF BALANCE WAS 0.1 G .

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selection, subjects read the engraved number and placed a checkmark on a data form by that serial number in the current trial column. Subjects did not receive knowledge of the correctness of their results, however it may have been possible for them to see the distortion on some balls. Learning effects were reduced by ensuring that the subjects completed all ball pairs before moving to the next trial.

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Fig. 4. Analysis of data from three measurements of a factory-new control ball. (Top to bottom): Sphere profile (measurements shown with open circles. Interpolation plotted as solid line); first two derivatives plotted on the circle; calculated curvature.

two identical undistorted balls were designed to detect hidden sources of bias in the experiment as they should be reported at chance level of 50%. The subjects were instructed to take the pair from bin 1 first, and work through the bins in order. One test of the entire set was termed a trial. They completed 10 trials for a total of 70 discriminations. The subjects were instructed to randomize each pair of balls by shaking in their cupped hands. They were allowed to touch and manipulate them with one or both hands without looking at the balls and then asked to select the distorted ball. After making the

IV. R ESULTS 1) Diameter Experiment: A boxplot of the experiment data (Figure 8) showed a clear trend towards lower recognition rates as the distorted ball diameter approached 0.499 inches (the control size). For diameters between 0.4925 and 0.4990 inches, the subjects reported at chance levels. Another possible cue is curvature. The same data as in Figure 8 can be plotted against Minimum or Maximum curvature (Figure 9). Since each ball has a unique diameter and Kmin , Kmax , it might seem redundant to plot all three. However it is interesting that the median recognition rates (solid horizontal bars inside boxes) are related in a monotone manner to the ordinate only for the Maximum curvature feature (Figure 9). Another way to visualize the data is by plotting a point for each ball and coloring it according to recognition rate (Figure 10). Each ball is plotted as a point in the plane defined by Kmin and Kmax . Each ball is colored according to its average recognition rate. There is a clear trend of increasing recognition rates from the lower right

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Fig. 6. Analysis of data from three measurements of a tumblerpolished control ball. Plot characteristics same as Figure 4.

(an ideal perfect sphere) to the upper left, our most distorted sphere with Kmax = 7 and Kmin = −0.2. These spheres lie in a line because of the curvature constraint described above. However there is a significant confound in this experiment design between curvature and ball diameter which will be addressed below. The normality of the data was evaluated with the R statistical software package using the qnorm() command and found to be normal except for a saturation at 100% recognition. One-way Analysis of Variance was computed and the effect of ball diameter was highly significant compared to the 0.05 level.

Fig. 7. Measurements of test ball 814. Plot characteristics same as Figure 4.

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Fig. 8. Distributions of correct discrimination rate vs. test ball diameter. Ball on right (D=0.499) was identical to the control ball.

Kmax 7.33 4.00 6.54 5.26 5.27 5.76

Kmin 0.93 4.00 1.60 1.94 2.37 1.73

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TABLE II P ROPERTIES OF TEST BALLS USED IN THE CURVATURE EXPERIMENT.

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Serial 341 ctl 993 041 193 878

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2) Curvature Experiment: Seven new balls were fabricated and polished, all with the same diameter. The selected diameter was 0.4835 which corresponded to an average recognition rate of 0.7 — half way between chance and perfect recognition. After the initial flats were ground and the diameter verified, these balls were individually modified to have varying degrees of maximum curvature. After measurement of the new balls, two were found to be very close in curvature to others and were discarded. Five test balls were thus paired with controls and a sixth ball was a double control. Properties of these balls are given in Table II. Ten subjects handled the test balls of Table II, each matched with a control. The percentage correct for each ball is plotted in Figure 11. Control ball (Kmax = 4.0) was reported at the 50% chance level. Although mean recognition rates are slightly higher for the higher curvature balls, this trend was not statistically significant at the 5% level, even when the control ball was included in the analysis. Because Kmax and Kmin are highly correlated (see Figure 10), the plot of recognition rate vs. Kmin was similar to Figure 11, and is not shown.

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Fig. 9. Distributions of correct discrimination rate vs. test ball minimum and maximum curvature. The ball with Kmax = 4.94 and Kmin = 2.88 was identical to the control ball.

V. D ISCUSSION This experiment showed that when attempting to judge sphericity of eccentric spheres of 0.4996 inches nominal diameter, a threshold probably exists for 0.4770 ≤ D ≤ .4880. Diameters above that range were reported at chance levels by most of the subjects and below that range were almost always detected by all subjects. In terms of Weber fraction (WF), this is equivalent to 2.3% ≤ WF ≤ 4.7% These figures are rather low compared to values of 6-9% obtained in force discrimination experiments by others[9], [10]. Goodwin’s experiment[3] was most similar to ours (including using spheres of the same nominal radius) except that the spheres were indented into the immobile finger by experimenter action.

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Fig. 10. Recognition rate (dot color) plotted along axes of Kmin and Kmax. Note that approximately Kmin + Kmax = a constant, and that recognition rate increases with increasing deviation of curvature. Ideal perfect sphere has constant curvature everwhere (lower right).

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0.4835 ≤ D ≤ .4880 To control for a possible confound due to changing curvature with diameter, we made a second experiment in which only the min and max curvature was varied among the test balls. In this experiment (Figure 11), there was no significant max curvature effect. At least two major cues are present in this experiment: kinesthetic effects of diameter, and local skin deformation changes due to curvature changes. Our results suggest that the kinesthetic cues are dominant. This was a somewhat surprising result to us since the diameter changes we induce are less than 5% yet the curvature changes are in a range of 2:1. Since only one diameter was tested in the curvature experiment, a possible interaction effect between diameter and curvature cannot be ruled out. A potential limitation is a potential confound between ball weight and diameter. Computing volume from the equation of Section III-A, the predicted weight change is 1.5% or less for all the spheres (consistent with the measurements of Table I). This is unlikely to be detectable. Another potential limitation of the experiment is the limited degree of geometric control of the test spheres which could be achieved by our hand fabrication methods. We are currently investigating machining techniques which can better control the ball contours. Future work includes better control and characterization of the test objects and video taping and analysis of the types of finger movements used by the subjects to make the discrimination.

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Fig. 11. Recognition rates for five balls with D=0.4835 (see Table II) and max curvature ranging from 4.0 to 7.33. Kmax = 4.0 was control ball.

Our results of less than one half their 10% WF suggest a key role for active haptic exploration. The definition of sensory threshold is somewhat arbitrary. It must be greater than 50% to be above chance, and it must be less than 100% to account for human fallibility. A commonly accepted method, the adaptive 1-up 2-down method converges to a threshold estimate at which the subject responds correctly about 71% of the time[11]. Using this value, our measurement suggests a threshold for perception of non-sphericity between

[1] Steven C. Venema. Experiments in Surface Perception using a Fingertip Haptic Display. PhD thesis, University of Washington, Department of Electrical Engineering, 1999. [2] KB Shimoga. A survey of perceptual feedback issues in dexterous telemanipulation. i. finger force feedback. Proc. Virtual Reality Annual International Symposium, pages 263–270, Sept 1993. [3] AW Goodwin, KT John, and AH Marceglia. Tactile discrimination of curvature by humans using only cutaneous information from the fingerpads. Experimental brain research, 86(3):663–672, 1991. [4] JW Morley, AW Goodwin, and I. Darian-Smith. Tactile discrimination of gratings. Experimental brain research, 49(2):291–299, 1983. [5] Blake Hannaford. Making distorted spheres for psychophysical experimentation. Technical Report 2009-0001, University of Washington Electrical Engineering Department, Rev April- 2009. [6] Blake Hannaford and Jesse Dosher. Polishing and measuring distorted spheres for psychophysical experimentation. Technical Report 2009-0004, University of Washington Electrical Engineering Department, April 24 2009. [7] www.scilab.org. [8] B. Hannaford and S. Venkatachalam. Characterization of test balls for psychophysical experiments. Technical Report UWEETR2009-0006, University of Washington, Electrical Engineering, June 5 2009. [9] L.A. Jones. Perception and control of finger forces. Proceedings Haptics Symposium, ASME Dynamic Systems and Control Division, DSC-64:133–137, 1998. [10] S. Allin, Y. Matsuoka, and R. Klatzky. Measuring just noticeable differences for haptic force feedback: Implications for rehabilitation. pages 299–302, 2002. [11] J. C. Stevens, E. Foulke, and M. Q. Patterson. Tactile acuity, aging, and braille readings in long-term blindness. Journal of Experimental Psychology: Applied, 2(2):91–106, 1996.