Perception & Psychophysics1995,57 (6), 778-786

The effect of density and diameter on hapticperception of rod length

TIN-CHEUNG CHANThe Chinese University ofHong Kong, Shatin, New Territories, Hong Kong

Three experiments on the effect of density and diameter on haptic perception of rod length are re-ported. In Experiment 1,the subjects wielded visuallyoccluded rods of different densities. Perceivedlength was found to be affected by the density of the rod regardless of the actual length. In Experi-ment 2, three aluminum rods of different lengths with handles of four different diameters werewielded. Perceived length ofthe rod was found to be shorter as the diameter of the handle with whichit was wielded increased. A diameter-length illusion was thereby produced. In Experiment 3, visu-ally occluded rods of different diameters but of the same moment of inertia about the x-axis werewielded with the right hand, and tubes of different diameters were felt with the left hand. The sub-jects were instructed that their right hand was grasping a handle, and that the actual diameter of therod could be felt with the left hand. Rods were perceived to be shorter if a larger diameter was feltwith the left hand. The results showed that perceived length is not just a function of actual rod length,and that it is not accounted for by inertia only. The results are further discussed in terms of the na-ture of invariants and the effect of knowledge on perception.

It has been shown that the relative length of a visuallyoccluded rod can be judged quite accurately by wieldingit (Solomon & Turvey, 1988). Such a judgment is claimedto demonstrate the information-perception specification,whereby perception is specific to information and infor-mation is specific to the environment (Gibson, 1979;Kugler & Turvey, 1987). In vision, light reflected by theenvironment, called ambient light, carries with it the spa-tial and temporal patterning that can specify the layout ofthe environment. In the case of haptic perception of rodlength, the changing flux of stimulation in the musclesand tendons in wielding the rod provides the patterningthat specifies the length of the rod (Gibson, 1966). Thelength of the rod is not indicated by an image of length,but is specified by a force structure (not length) that pro-duces tissue deformation on muscle and tendon. More-over, it has been pointed out that because rod length is anenduring property of the environment, it has to be spec-ified by an invariant, an enduring pattern that remainsunchanged but that is revealed by changes in the stimu-lation. The invariant that specifies haptically perceivedlength has been found to be related to the moment of in-ertia-the resistance to rotational acceleration-experi-enced in wielding a rod.

Preparation of this article was supported by the Shaw College Stu-dent Campus Work Scheme (1993, fall) of the Chinese University ofHong Kong. The author would like to thank M. H. Bond for reading thefirst draft, and M. L. Braunstein, S. 1. Lederman, and another anony-mous reviewer for constructive criticism. Correspondence should beaddressed to T.-C. Chan, Department ofPsychology, The Chinese Uni-versity of Hong Kong, Shatin, New Territories, Hong Kong (e-mail:tcchan@cuhk.hk).

To wield a rod about the x-axis with angular velocityWI' an angular momentum L I (see Appendix) is pro-duced. If the motion is strictly along the yz plane (Fig-ure 1), it can be expressed by the equation L I = Ilw!,The quantity II is the moment of inertia of the rod aboutthe x-axis. Similarly,12 is the moment ofinertia about thez-axis, and h is the moment of inertia about the y-axis.In three-dimensional rotation, the equation ofmotion, inits simplest form, consists of the three equations to-gether. The three equations can be rewritten in matrixform as one equation, L = Iw, where I is called an iner-tia tensor expressed in a 3 X 3 matrix (Goldstein, 1980).The three moments of inertia are the diagonal compo-nents of the diagonalized inertia tensor (see Appendix).It is these moments of inertia that have been found tocomprise the invariant used in the judgment of rodlength.

Even so, moment of inertia is not the flux of stimula-tion (proximal stimulus or haptic array), but a propertyof the grasped rod (distal stimulus). It should be the an-gular momentum (or the rate of change of angular mo-mentum-the torque) and the velocity (or the rate ofchange of velocity-the acceleration) that produce theflux of stimulation in the muscles and joints.

What is the relationship between actual and perceivedlengths, and what is the invariant force structure thatspecifies the perceived length? Since Solomon and Tur-vey's (1988) study, the first study ofperceived length ofa rod through rotation, there have been several claims.Solomon, Turvey, and Burton (1989), having shown thatthere must be a scaling function that maps the invariantonto a distance metric, claim, "It is noteworthy that theproposed scaling function yields an accurate estimate ofreaching distance when the hand-held object is of uni-

Copyright 1995 Psychonomic Society, Inc. 778

z

y '~mg

Figure 1. A hand-held rod is shown with respect to the coordinatesystem. The rod is grasped at one end, pivoted at the wrist. Theweight mg acts downward through the center of mass, and d is thevertical distance ofthe wrist from the center of the rod.

form mass distribution" (p. 66). Similarly, Burton andTurvey (1991) claim that "when the hand-held objecthas a uniform density, perceived extents can match ac-tu~l extent" (p. 129). Although Pagano and Turvey (1993)point out that actual length is not the same as perceivedlength, they assert that "perceived reachable distancewas tied to I in a manner closely similar to actual reach-able distance" (p. 147). It remains to be shown, however,whether this similarity between actual length and per-ceived length is an intrinsic relationship, or whether itjust results from the way in which the rods were used inthe previous experiments.

As to the identification of the invariant that specifiesperceived length, Solomon et al. (1989) ascribe the pick-ing up of the invariant to three successive operators: adifferential operator that describes the motion of thehand-and-rod system; an integral operator that producesfrom the description of the motion the moment of iner-tia tensor; and an orthogonal operator that picks out theinvariant from the inertia tensor. As to which elements inthe inertia tensor are picked out as the invariant andscaled to the length of the rod, Solomon and Turvey(1988) assert that for slender rods, only the moment ofinertia about the axis perpendicular to the rod has to beconsidered. It is claimed that "the dynamic behavior ofthe hand and rod system can be expressed in terms of asingle 1" (p. 419)-that is, II' the moment of inertiaabout the x-axis (see Figure 1). Pagano and Turvey(1992) also make a similar assertion. Yet Pagano andTurvey (1993), by using a special rod with perpendicu-lar branches, and by using multiple regression for theline of best fit, found the invariant to be made up of II(the moment of inertia about the x-axis) and 13 (the mo-ment of inertia about the y-axis).

Three questions remain to be answered definitively:(1) Is the perceived length similar to the actual length?(2) Is 13 actually involved in the perception of rodlength? And (3) are other parameters involved besidesthe principal components of moment of inertia thatwould affect the haptic perception of rod length? The

HAPTIC PERCEPTION OF ROD LENGTH 779

three experiments in the present study were aimed at an-swering these questions.

EXPERIMENT 1Is the Perceived Length, Lp, Similar to

the Actual Length?

That rod length is specified by the moment of inertiaII is incompatible with the claim that perceived lengthequals actual length. II is affected by variables other thanlength, such as diameter and density (see Appendix).Pagano and Turvey (1992) state clearly that perceivedlength is mainly specified by the moment of inertiaabout t~e axis perpendicular to the rod length. Simplemechamcs shows that moment of inertia increases withthe density and thickness ofa rod, variables that are un-related to the actual length of the rod. Thus, logically,equality between perceived and actual lengths seemsunlikely. Nor do empirical results support such a claim'indeed, in Solomon and Turvey's (1988) study, whe~solid aluminum rods were used, the subjects perceivedthe rods to be shorter than their actual lengths, but whendenser solid steel rods were used, the perceived lengthwas longer than the actual length. Although the differ-ence in judgment is small compared with the differencein density, a difference is clear. In experiments con-ducted by myself, light, hollow metal rods were usedand the subjects underestimated rod length tremen-dously (Chan, 1994; Chan & Turvey, 1991); therefore,perceived length could not be similar to the actual lengthof the rod.

To show that perceived length is not similar to the ac-tuallength, it is sufficient to show that perceived lengthis affected by density: For rods ofthe same length, then,denser rods will be perceived as longer. In the presentexperiment, four light, hollow steel rods and four solidaluminum rods were used. The rods were wielded be-hind an occluding board, and their lengths reported. Forthe same length ofrod, the set ofdenser solid aluminumrods was expected to be perceived as longer.

MethodSubjects. Ten undergraduates in an introductory psychology

~ourse ~t the Chinese University ofHong Kong participated in par-tial fulfillment of a course requirement.

Apparatus. Four hollow homogeneous steel rods (0.94-cmdiam) of 50, 60, 70, and 80 em, with respective masses of 44.35,54.55, 61, and 70.5 g, and four solid aluminum rods (0.96-cmdiam) of 50, 60, 70, and 80 em, with respective masses of 102.77,122.50, 143.35, and 164.62 g, were used. The two sets of rods, witha .02-cm difference in diam, were regarded as equal in thickness.Twowooden boards (1.5 m high x 1.2 m wide), with a vertical slitof20 em in between, were used as screens to occlude the hand andthe grasped rods (Figure 2). A 0.8-m-high armrest (the back of achair) was placed against the back ofthe slit. A piece of black clothcovered the top part of the slit. For the subjects reporting length, alength measurer that was 1.3m in length was used. The length mea-surer had a sliding marker connected with a string mounted on aboard. By pulling the string up or down at the starting edge of theboard, the subjects could move the marker along a meter rule fixedat the top ofthe board. The perceived length of a rod could then be

780 CHAN

doth

(A) (8)Figure 2. (A) Arrangement of the apparatus. (B) Diagram of a subject wielding a rod,

with an ann inserted into the slot betweenthe twooccluding boards. The viewofthe lengthmeasurer as shown in A is blocked by the subject.

EXPERIMENT 2What Contributes to the Apparent Effect of/3?

Results and DiscussionThe means of the perceived forward length for differ-

ent densities and actual lengths are listed in Table 1. Ananalysis of variance (ANOVA) showed that the effect ofdensity was significant [F(1,12) = 78.84, P < .0001],indicating that density affects perceived length. The ef-fect of length was significant [F(3,36) = 77.93, p <.0001], indicating that length can be discerned. Togetherwith the effect of density, these results showed thatwithin each set of rod of equal density, longer rods areperceived as longer. There was an interaction betweendensity and length [F(3,36) = 9.58,p < .0002], indicat-ing a difference in the scaling functions of the actual-to-perceived length between the two sets ofrods (Figure 3).As shown in Figure 3, the regression coefficient b-weightfor the lighter hollow steel rods was 0.82, whereas thatof the solid aluminum rods was 1.23. Regression oflogLp on log II (Figure 4) yielded an R2 of .986, indicatingthat when the thickness of the rod was kept constant, per-ceived length was mainly dependent on II' as previouslyclaimed by Solomon and Turvey (1988) and Pagano andTurvey (1992).

Because 98.6% of the variance of perceived length isaccountable by II' the effect of13 should be insignificantfor length perception. Multiple regression of log per-ceived length on log II and log h yielded an R2 of .987,with partial Fs of78.1 (p = .0003) and 0.111 (p = .75)for log II and log h, respectively. h was shown to be ir-relevant to perceived length.

In sum, the results showed that because perceivedlength is mainly dependent on the first moment of inertia,perceived length is not similar to actual length ifdensityis allowed to vary. Denser rods are perceived as longer.

34.1639.5050.5757.91

PerceivedRod Length

(ern)

Hollow Steel Rod38.56 1.6167.42 1.98

101.83 2.21152.94 2.55

50607080

Table 1Perceived Lengths of the Two Sets of Rods in Experiment 1

Moment of Inertia(g . cm2/1 00)

translated to the distance of the marker from the starting edge. Thelength measurer was placed along the occluding board in front ofthe subject.

Procedure and Design. The subjects positioned their right armon the armrest, with their hand behind the screens. Rods wereplaced by the experimenter into the right hand of each subject, andthey were always gripped by the subjects so that the wrist wasabout 6 em above the rod axis at the rear end (Figure 1). The sub-jects were told to grip the rods firmly, with the thumb positionedadjacent to the index finger, to ensure that they could not move therod with the thumb independently of the wrist, which would haveenabled the wrist to act as the pivot point of the rod-hand system.The subjects swung the rod up and down and circled the rod to per-ceive its length. In each trial, the subjects moved the marker awayfrom themselves with their left hand to indicate the perceivedlength of the whole rod held in their right hand. They were en-couraged to take as much time as necessary to perceive the rodlength accurately. At the end of each trial, the rod was taken awayand the subjects moved the marker back to the starting edge. Be-fore the experimental trials, they were given a practice trial with arod that was not used in the experiment.

In the experiment, the eight rods from both sets were presented ran-domly in a block of trials. There were four blocks, with a total of 32trials. Perceived length of the whole rod was measured in each trial.

The experiment was a 2 X 4 (density of rod X rod length)within-subject design. Because perceived length of a rod is de-pendent on the moment of inertia of the rod about the x-axis, thedensity of a rod was expected to affect its perceived length.

Rod Length(em)

50607080

Solid Aluminum Rod89.34 3.71

151.41 4.42239.30 5.17357.12 5.94

45.7160.4073.8882.08

In Experiment 1, 13 was shown to be irrelevant to theperceived length in wielding homogeneous rods offixedthickness. In Pagano and Turvey's (1993) study, al-though perceived length was expressed as a function of

HAPTIC PERCEPTION OF ROD LENGTH 781

90

80

E 70~.c...

60CQc.!!~ 50..

0 aluminium=-'il 40 y = ·14 + 1.23x RA2 = .985....... • hoUowsteelQ, 30 Y= • 8+ 0.82x RA2 = .98420 -1-----...---...----..----r-----1

40 ~ 60 m ~ 00actual length (em)

Figure 3. The scaling function of actuai-to-perceived length is dif-ferent for rods of different densities (Experiment 1).

I} and 13(perceived length = 389Ij42 h- .O}), the effect ofh was not significant. However, in Fitzpatrick, Carello,and Turvey's (1994) study, h was significant. Even so,three different equations were found in three experi-ments when wielding homogeneous rods with or withouthandles. With rods of different diameters and lengths,but of the same material, perceived length was found tobe 4.74If.36 r;0.25. With rods made of materials of dif-ferent densities inserted in handles of constant thick-ness, perceived length was found to be 2.15IP.37 h-O. 17.With lengths of rods of the same material, but attachedto handles of different diameters, perceived length wasfound to be 1.03If.54 1;°·34. In fact, with the data listed,in the first and second experiments, I} was highly corre-lated with 13, a condition of collinearity in which multi-ple regression should not be conducted. In addition, hwas not significant in the second experiment.

The effect of13 on perceived length was shown to beinsignificant in the present Experiment I. It is uncertainas to what the apparent effect ofh is really attributed by

2.0.....-------------------,

Fitzpatrick et al. (1994). In Pagano and Turvey's (1993)study, with the diameter of the rod held constant and 13systematically varied with I} such that II and 13 were notcorrelated, the effect ofh was still insignificant. A re-view of Fitzpatrick et al.'s (1994) two experiments, inwhich h was significant, shows that, in both cases, the di-ameter of the rod was varied. It is suspected that the ap-parent effect of h was due to its correlation with the di-ameter, d, which really affected the perceived length. Infact, a multiple regression ofaverage log Lp on log handlog d in the third experiment yielded a significant R2 of.98, with partial Fs of 133.52 (p = .000I) for log I} and127.46 (p = .0001) for log d. There are reasons why 13could not be an appropriate predictor. In the previous ex-periments, the calculated power coefficient for 13 fluc-tuated greatly. The effect of13 was significant only for afew specific conditions. Moreover, an increase ofactuallength in homogeneous rods should be concomitant withan increase of13, and yet the power coefficient of13 inperceived length was negative.

To show that perceived rod length is affected by thediameter of the rod rather than by h, handles of differ-ent diameters but of similar 13 can be used. An unre-ported pilot experiment (Chan, 1990) showed that whenidentical rods were wrapped with different thicknessesof cardboard paper as handles, rods with handles ofthicker diameters were perceived as shorter. Because thepaper wrapping was light, both of the changes in I] and13 produced by the paper wrapping were too small to af-fect the perceived length. Moreover, the distance of thewrist from the axis of the rod (d in Figure I) with in-creasing thickness of the wrapping did not change. Theshortening of the perceived length could only be a resultof the change in the diameter of the paper handle.

In this experiment, four aluminum handles of differ-ent diameters were used with three lengths of aluminumrod. The four handles were hollowed so that the differ-ence in mass, and thus the difference in the moment ofinertia, was kept to a minimum. With handles of largerdiameters, the perceived length was expected to beshorter.

MethodSubjects. Ten undergraduates in an introductory psychology

course at the Chinese University ofHong Kong participated in par-tial fulfillment of a course requirement.

Apparatus. Three solid aluminum rods (0.96-cm diam), 60, 70,and 80 em in length and with respective masses of 122.5, 143.4,and 164.7 g, and four hollowed handles (12 em long), 2, 3,4, and5 em in diam and with respective masses of73.3, 115.7, 115.5, and115.5 g, were used. In each trial, one of the handles was fixed tothe end ofeach rod (Figure 5). The occluding board and the lengthmeasurer were the same as those used in Experiment 1.

Procedure and Design. The subjects wielded the occluded rodto judge the length of the rod as in Experiment 1, except that ineach trial, handles with the rod attached were placed into theirright hand. The handles were grasped such that the wrist was about6 em above the axis at the rear end. The subjects wielded the rodwith their right hand and reproduced the perceived whole length ofthe rod on the length measurer with their left hand. They were told

2.62.41.61.5+--...L:.-----,,....----,.---r---,------l

1.4 1.8 2.0 2.2

log 11 (g.cm2)

Figure 4. For rods with similar thickness, perceived length ismainly accounted for by the moment of inertia about the axis per-pendicular to the rod, I, (Experiment 1).

782 CHAN

Figure 5. Cross-section to show the fitting of a rod into a handle(Experiment 2).

that what they grasped was a handle fixed to the end of each rod,and that all handles were ofthe same length and were only slightlylonger than the width of their palm.

The 12 rod -handle combinations were presented randomly in ablock of trials. There were four blocks, with a total of 48 trials.Perceived length of the whole rod was measured in each trial.

This experiment was a 4 X 3 (handle X rod length) within-subjectdesign. It was expected that the diameter of the handle would af-fect the perceived rod length.

Results and DiscussionThe means of the perceived rod length for different

handle thicknesses and moments of inertia are listed inTable2. An ANOVA on mean perceived rod length showedthat the effect of handle thickness was significant [F(3,27)= 42.1, p < .0001]. Together with the mean perceivedlength, the results showed that rods with thicker handleswere perceived as shorter. The effect of length was sig-nificant [F(2,27) = 48.77,p < .0001], indicating that rodlength can be discerned. There was a marginal inter-action between handle thickness and length [F(6,54) =2.28, p < .05].

To find out whether 13 and II can be used together inthe multiple regression equation log I = alogil +blogI3 + log c (a log transformation ofthg equation I =cIf If), correlations between log II' log h, and log d {di-ameter of the rod) were found. The correlation betweenlog II and log 13 was .579, between log II and log d it was0, and between log h and log d it was .78. Because ofcollinearity, it is inappropriate to regress log Lp on log II

and log L3; yet log II and log d can be used in the sameequation .

Regression of log Lp on log II alone yielded an R2 of

only .64 (Lp = 0.042IP.5I). The variance in L intro-duced by changing d is orthogonal to the variance intro-duced by changing h (Figure 6). It is this variance in-troduced by changing d that reduces the R2 from .986 inExperiment I to .64. Multiple regression oflog L on logI. and log d yielded an R2 of .98 [F(2,9) = 244:04, p <.0002], with partial Fs of 327.463 (p < .0002) and170.88 (p < .0002) for log II and log 13, respectively(~p = .0?8IP.52 d - O.l 8) . Because log L p correlated nega-tively With log d (r = -0.57), a negative power coeffi-cient is appropriate.

Why does diameter give a shrinking illusory effect onperceived rod length? This question can be addressed byreferring to the study of Armstrong and Chaffin (1978),who show that a specific degree of joint rotation of thefingers corresponds to a specific amount of tendon dis-placement at the wrist. When the fingers are stretchedout, as when holding a larger object, there is greater ten-don displacement, resulting in greater muscle elongationand, thus, greater wrist stiffness. This is compatible withthe findings of Davis and Brickett (1977) that indicatethat in preparing to lift a larger can, greater muscle ten-sion is produced, resulting in the size-weight illusion.These findings suggest that with increased wrist stiff-ness and muscle tension in preparing to lift a rod oflarger diameter, the experienced moment of inertia inwielding the rod would be reduced, resulting in shorterperceived length.

Ellis and Lederman (1993) report similar findings ofa size-weight illusion in holding objects, and claim thata haptic volume cue is sufficient to produce such an il-lusion. Lederman, Ganeshan, and Ellis (in press), usinghorizontal rods of different diameters held stationary,show that subjects perceived rods of larger diameters aslighter and shorter simultaneously. It is thus suggestedthat the diameter-length illusion may be mediated by aweight percept. In other words, the subjects used per-ceived weight to judge rod length. Since rods of largerdiameters were perceived as lighter, they were also per-ceived as shorter. This weight is defined as a function ofthe gravitational pull (force) on the rod and the torqueproduced by the rod on the grasping hand. It is the re-ferred weight rather than the actual weight of the rod,and it can vary even though the actual weight remainsconstant. Though such an explanation is possible, it isnot yet certain, in holding rods stationary, whether per-ceived length is mediated through the referred weight,which is a function of the torque felt in holding the rod,or whether it is affected directly by the felt torque. Fur-ther experimentation may have to be conducted for clari-fication .

Nevertheless, results in this area of research showvery clearly that in wielding rods of identical moment ofinertia II' rods with greater diameters would be per-ceived as shorter. Let us call this factor the muscularconstraint. In addition, when a rod is held, the diameter

21.7326.8732.7517.8724.6628.3716.9122.3324.8215.3719.6223.74

PerceivedLength(em)

15.80 0.7324.60 0.8136.39 0.8816.22 0.9025.02 0.9736.81 1.0516.25 0.9325.05 1.0036.84 1.0816.27 0.9625.06 1.0436.85 1.I1

Total (g . em2) /104

I J 13607080607080607080607080

RodLength(em)

Table 2Perceived Lengths of the Three Rods With

Different Handle Thicknesses

3

2

5

4

rod

HandleDiameter

(em)

HAPTIC PERCEPTION OF ROD LENGTH 783

1.6-r--------------------,

y = • 1.4+ 0.5x RA2 = 0.64 o

perceived lengths reported. The subjects were informedthat what they were holding was just the handle, whichwas about the length of their palm, while the actual di-ameter of the rod could be felt with the left hand. Plas-tic tubes of three different diameters were felt by sub-jects using their left hand. It was expected that wriststiffness (muscular constraint) and knowledge of the di-ameter (geometric constraint) would affect perceivedrod length independently. Greater diameters felt with theleft hand would make the perceived length of the rodwielded in the right hand shorter.

MethodSubjects. Eight undergraduates in an introductory psychology

course at the Chinese University of Hong Kong participated in par-tial fulfillment of a course requirement.

Apparatus. Three hollow homogeneous metal rods ofdifferentthickness and length, but of equal principle moment of inertiaabout the x-axis, II (gripped at 5 em from one end), were wieldedwith the right hand. Detailed specifications of the three rods areshown in Table 3. Three plastic tubes of 1.3,2.2, and 3.77 cm diamwere felt by the left hand. The occluding board and the length mea-surer were the same as those used in Experiment 1.

Procedure and Design. The subjects wielded the occludedrod to judge the length of the rod as in Experiment 1, except thatin each trial, while they wielded the rod grasped with the righthand, one of the three plastic tubes ofdifferent diameters was pre-sented in random order to be felt by the left hand. The subjectswere instructed that what they were grasping with their right handwas the handle of a rod, and that the diameter of the rod could befelt with their left hand. They were also instructed not to use log-ical deduction, but to perceive the length directly and intuitively.The experimenter held on to the plastic tube so that the subjectscould not sense the weight of the tube. After perceiving the lengthofthe metal rod, while still wielding the metal rod with their righthand, the subjects reported perceived length by adjusting thelength measurer with the left hand. There were nine combinationsof the three metal rods with the three plastic tubes, presented ran-domly in a block of trials. There were four blocks of trials, with atotal of32 trials. Before the experimental trials, the subjects weregiven a practice trial with a rod that was not used in the experi-ment.

5.65.11.1+---"'T""--""T'"--"""T"--.....,.----,,-----i

5.0 ~2 ~3 5Alog II (g.cm2)

Figure 6. With variation of rod thickness, an additional varianceof perceived length orthogonal to the variance produced by 11 is pro-duced (Experiment 2).

EXPERIMENT 3The Effect ofPerceived Diameter on

Perceived Rod Length

of the rod is known. The known diameter is purely geo-metric in nature, and may have an effect on perceivedlength. To differentiate the effect of the grasped diame-ter from the effect of the known diameter, let us call thelatter effect the geometric constraint. To investigatewhether knowing the diameter of a rod actually has aneffect on the perceived length, the following experimentwas conducted.

To separate the muscular constraint from the geomet-ric constraint, in this experiment, information about thediameter of the rod was felt by the left hand indepen-dently of the right hand, which wielded the rod. Threerods of equal II but of different diameters were wieldedwith the right hand behind an occluding board and their

Table 3Perceived Lengths of the Three Rods With Identical Moment of Inertia 11,

but With Rods of Different Diameters Felt With the Left Hand

Length (em)

Dimensions of Rods Wielded

Mass (gm) Diameter (em)

Diameter ofRods Felt with

Left Hand(em)

Perceived Lengthof Rod(em)

40.5

68.5

80.0

Thick313.0 2.50

Medium99.0 1.25

Thin69.5 1.00

12.71

12.70

12.70

3.77 34.452.21 40.771.31 43.58

3.77 41.802.21 47.781.31 53.99

3.77 48.762.21 52.531.31 57.47

Note-The rods were all grasped with the wrist at 5 em from the rear end. The moments of inertia of thethree rods are roughly the same.

784 CHAN

The Nature of Invariants in HapticPerception ofRod Length

An invariant is the consistent pattern of stimulationthat specifies a perceived property such as length (Gib-son, 1979; Turvey, Solomon, & Burton, 1989). The aimof the present research, in concert with the previouswork by other researchers, is to find out which is the in-variant that actually specifies perceived length in thehaptic sense. The invariant is expected to be a covaria-tion ofseveral physical variables giving rise to perceivedlengths close to actual lengths. Perceived length hasbeen shown to be proportional to the variable II (Sol-omon & Turvey, 1988), which increases with densityand thickness in a way that is unrelated to the actuallength of the rod. In order that perceived length is simi-lar to actual length, there must be a covariable to reducethe effect ofII due to density and thickness. 13 is the bestcandidate, and it has indeed been picked by previous re-searchers as the covariant of II (Fitzpatrick et aI., 1994;Pagano & Turvey, 1992). Nevertheless, there is no con-vincing empirical evidence that 13 is a covariant ofII forperception of rod length.

Experiments 2 and 3 showed, however, that d, the di-ameter of the handle, is a covariant of II in the hapticperception of rod length. Such covariation would con-strain perceived length to its actual magnitude withthicker rods of the same material. It is interesting thatthere are two independent factors to constrain the per-ceived length due to variation in thickness, as revealed inExperiments 2 and 3-namely, wrist stiffness andknowledge of rod thickness. Such covariation cannot,however, eliminate the variation due to the density ofthematerial of the rod. Thus, with denser rods, perceivedlength is longer, as shown in Experiment 1.

Thus, unlike the perception of looming (Lee, 1976),where a specific invariant specifies the time-to-contact,in haptic perception of rod length, there is no invariantthat specifies perceived length that is identical to actuallength; evolution does not provide humans with an in-variant to specify length haptic ally with accuracy. Itseems that it is primarily the functions ofknowledge andpractice that compensate for such inadequacy.

perceived length. The latter component was separatedout in Experiment 3. When the rod wielded in the righthand was implied to be greater in diameter, the lengthwas perceived to be shorter. These results have impor-tant implications as far as two issues are concerned-thenature of invariants in haptic perception of rod length,and the effect of knowledge on perception.

The Effect ofKnowledge on PerceptionIn Experiment 3, the diameter felt with the left hand

altered the perceived length of the rod wielded in theright hand. The results indicated that variables experi-enced with different parts of the body can be combinedto specify one single rod length. Such integration hasbeen demonstrated in a study by Carello, Fitzpatrick,Domaniewicz, Chan, and Turvey (1992), in which sub-

thin rod

mediumrod

thick: rod

30..L..------,..------r----T"'"""---......I

GENERAL DISCUSSION

In summary, the results of Experiment 1 showed thatin wielding cylindrical rods of similar thickness madefrom different materials, actual length is not similar toperceived length. Denser rods are perceived as longer,depending on the magnitude ofII. 13 is not involved inthe perception of the whole rod length. The results ofExperiment 2 showed that the thickness ofhandle has aneffect on perceived length: Perceived length is reducedwhen grasping rods ofgreater diameters. This can be ex-plained by an increase in wrist stiffness when grasping ahandle ofgreater diameter, producing a diameter-lengthillusion similar to the size-weight illusion. Yet there maybe another interrelated component-the knowledge ofthe diameter of the handle or the rod which also affects

The experiment was a 3 x 3 (felt diameter X wielded rod)within-subject design. Both variables were of fixed effect. It waspredicted that with the three metal rods of equal II' rods ofgreaterdiameter would be perceived as shorter. It was also predicted thatwhen plastic tubes ofgreater diameter were felt with the left hand,the rods wielded in the right hand would be perceived as shorter.

1.3 2.2 3.77felt diameter (em)

Figure 7. Perceived length is affected by the grasped diameter andthe knowledge ofthe diameter ofthe rod (Experiment 3).

60 ....----------------.....,

Results and DiscussionThe means ofperceived length are plotted in Figure 7.

An ANOVA showed that the effect of diameter of thewielded rod was significant [F(2,14) = 9.637,p < .0025].This can be explained by the wrist-stiffening effect withhandles oflarger diameter. Further, the effect ofthe diam-eter ofthe plastic tubes felt by the left hand ofthe subjectswas also significant [F(2,14) = 14.654,p < .0005]. Thismust be explained in terms ofknowledge of the diameterof the rod held. No interaction was shown [F(2,28) < 1],indicating that the two effects are orthogonal.

These results clearly indicate that haptic perceptionof rod length depends not on inertia tensor alone, but ontwo additional orthogonal variables that were revealed byusing rods ofdifferent diameters-namely, muscular andgeometric constraints. Different constraints can be ap-plied to different parts of the body for a single perception.

jects perceived the length of rods held at one end with adownward thrust on the leg and an upward thrust on thefinger. The feeling of the diameter ofthe rod in the pres-ent Experiment 3, however, has a unique characteristic:It is independent of the wielding of the rod by the sub-ject with his or her right hand. Indeed, such a presenta-tion can be replaced by a statement or a picture. Buck(1990) called such statements descriptive knowledge.The results showed that such knowledge modifies theperceived length of a rod.

Knowledge has been shown to affect perception invarious modalities: Lindauer (1989) has shown that ex-pectation alters ambiguous figure-ground perception;Van der Velde, Van der Heijden, and Schreuder (1989)have shown that knowledge affects visual word per-ception in context-dependent migration; Elman andMcClelland (1988) have shown that knowledge in En-glish affects speech perception with lexically restoredphonemes; and Appelle and Countryman (1986) haveshown that knowledge of the standard would reduce thehaptic oblique effect. It appears that any theory in per-ception must be able to account for the effect of know1-edge on perception.

Traditionally, the notion that knowledge is requiredfor perception would mean that with sensory inputs im-pinging on the sensory receptors, specific memorieshave to be retrieved from storage to fuse with the sen-sory input. According to this view, perceptual knowl-edge is not primary. We can only understand what we seenow in terms of the past.

Gibson (1979) rejects such a dichotomy between thepast and the present (see also Turvey & Shaw, 1979;Turvey,Shaw,Reed, & Mace, 1981), and instead regardsinformation as coming in a stream. What is available vi-sually with one saccade at any putative moment is onlya fraction of a surface that happens to be in the field ofvision. This snapshot is not taken as the basis ofpercep-tion. Gibson (1979) suggests that the perceptual systemhas an input-output loop such that invariants can be de-tected over time, and the invariant, once detected, will beavailable for succeeding processes. Invariants detectedlater can be integrated with previous invariants. Withoutthe dichotomy between past and present, the extractionof invariants has no boundaries. According to this view,knowledge is the extraction of invariants independentlyof the stimulus flux. Conception is an extension of per-ception (Turvey et al., 1989). To perceive the environ-ment and to conceive it are different in degree but not inkind; one is continuous with the other.

The results of the present research show that in hapticperception of rod length, there is no physical invariantthat specifies the actual length. Perception of the lengthof a rod when wielded is affected by moment of inertiaas a physical invariant, by grasped diameter as a muscu-lar constraint, and by known diameter as a geometricconstraint. According to the view presented above, themuscular and geometric constraints can be consideredas supplementary constraints, and perception due to de-tection of a physical invariant is modified by various

HAPTIC PERCEPTION OF ROD LENGTH 785

constraints. Such an explanation, I believe, is consistentwith the ecological view of perception.

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LEDERMAN, S. J., GANESHAN, S. R., & ELLIS, R. E. (in press). Hapticperception of the extent of statically held rods. Journal of Experi-mental Psychology: Human Perception & Performance.

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APPENDIXDynamic Variables Used in This Article

Variables with Definitions and Formulae

Angular velocity (w): The velocity of rotation.

w = vir, where r is the radius of rotation and v is the linear velocity.

Angular momentum (L): A conserved quantity of angular motion.

L = r X p, where r is the radius of the trajectory, p is the linear momentum, and Xis the cross product for vector multiplication.

Inertia tensor (I): A property of the rod that characterizes the resistance of rotation ofthe rod in three dimensions; it is represented by a 3 X 3 matrix.

In its simplest form, it is:

The equation L = Iw is equivalent to:

Moment of inertia (IJ Resistance to rotational acceleration of an object.With respect to Figure I, for a cylindrical rod of mass m, length L, and radius agrasped at one end at a distance r from the center of mass with the wrist at a distanced from the center of the rod, the moment of inertia about the axis through the wristand paral1el to the x-axis is:

I) = ~ (3a 2+ L2) + m(r 2+ d 2) .

If the rod is thin,

For a cylindrical rod, the moment of inertia (12) about the axis through the wrist andparallel to the z-axis is the same as 11,The moment of inertia about the axis through the wrist and parallel to the rod is:

13 = ma2 + md? for hollow rods,

or

= ma2/2 + md 2 for solid rods.

Since m equals p7fa2L , where p is the density of the rod, 11 and 12 are proportional tothe density of the rod, the diameter squared, and the length to the third power. Sim-larly,I3 is proportional to the density of the rod and the diameter to the fourth power.

(Manuscript received May 23, 1994;revision accepted for publication January 3, 1995.)

Units

rad/sec

g : em?

g : em?