Abstract. This paper considers interaction of the humanarm with ``virtual'' objects simulated mechanically by aplanar robot. Haptic perception of spatial properties ofobjects is distorted. It is reasonable to expect that it maybe distorted in a geometrically consistent way. Threeexperiments were performed to quantify perceptualdistortion of length, angle and orientation. We foundthat spatial perception is geometrically inconsistentacross these perceptual tasks. Given that spatial percep-tion is distorted, it is plausible that motor behavior maybe distorted in a way consistent with perceptual distor-tion. In a fourth experiment, subjects were asked to drawcircles. The results were geometrically inconsistent withthose of the length perception experiment. Interestingly,although the results were inconsistent (statisticallydi�erent), this di�erence was not strong (the relativedistortion between the observed distributions wassmall). Some computational implications of this researchfor haptic perception and motor planning are discussed.

1 Introduction

Haptic spatial perception is distorted. When asked tomake judgments about spatial properties of objects suchas length, angle and orientation by feeling them, subjectsmake systematic errors. This paper reports an investi-gation of haptic interaction of the human arm with``virtual'' objects. Our goal was to understand moreabout the computational processes presumed to underlie

sensory-motor behavior. Objects were simulated using arobotic manipulandum (shown schematically in Fig. 11)which restricted movement to a horizontal plane. Inter-action forces were generated by the manipulandum'sactuators in response to commands computed at eachsampling instant based on measurements of position andvelocity and a model of the object to be simulated.

1.1 Geometric structure

We assumed that interaction of the human with themanipulandum over an interval of time is describedcompletely by a con®guration and force trajectory,�x�t�; f �t��. Such a trajectory shall be referred to as an(idealized) dynamic stimulus. The projection of acon®guration and force trajectory onto the con®gura-tion trajectory only, x�t�, shall be referred to as aspatiotemporal stimulus. The underlying path, x��,which is independent of the instantaneous velocities ofthe path (independent of the temporal parameteriza-tion), shall be referred to as the spatial stimulus.

During interaction, a�erent and e�erent informationis acquired. A�erent information comes from mec-hanoreceptors such as cutaneous and deep tissue sen-sors, muscle spindles, Golgi tendon organs and jointcapsule receptors. E�erent information is available fromso-called corollary discharge, information availablefrom motor areas of the central nervous system (CNS)that project onto sensory areas. Percepts of arm stateand of the simulated objects are formed based on af-ferent and e�erent information acquired during inter-action, and on prior knowledge (e.g., that the simulatedobject is a rectangle and not an arbitrary polygon). Forsimplicity, we assumed that a�erent information was afunction of dynamic stimulus only, and that e�erentinformation was a function of motor intent only.

Spatial perception can be viewed as an integrative,computational process in which spatial properties areinferred from instantaneously acquired e�erent and/ora�erent information, and prior knowledge. Spatialproperties of objects, such as length of segments, con-tinuity of paths, angles between surfaces, etc., can be

Biol. Cybern. 82, 69±83 (2000)

Haptic interaction with virtual objectsSpatial perception and motor control

Ernest D. Fasse1,*, Neville Hogan1,2, Bruce A. Kay2,**, Ferdinando A. Mussa-Ivaldi2,***1 Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA2 Department of Brain and Cognitive Sciences, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

Received: 12 February 1996 =Accepted in revised form: 6 May 1999

Correspondence to: E.D. Fasse

Present addresses:*Department of Aerospace and Mechanical Engineering, Tucson,AZ 85721-0119, USA**Department of Cognitive and Linguistic Sciences, Campus Box1978, Brown University, Providence, RI 02912, USA***Department of Physiology M211, Northwestern University,Medical School, 303 East Chicago Avenue, Chicago, IL 60611,USA

theoretically determined based on the spatial stimulusonly. It is therefore useful to think of the perceptualprocesses as implementing an underlying, abstract geo-metrical reasoning system. This led us to several inter-esting questions that we addressed experimentally: (1)what is the structure of the abstract geometrical rea-soning system implemented by the computational pro-cesses? (2) are the processes mutually consistent withrespect to the underlying system?

In our investigation, we considered the question of towhat degree there is a perceived metric of space, i.e., towhat degree humans are capable of perceiving the dis-tance traversed along paths. We assumed initially thatthe perceived distance depends only on the path (thespatial stimulus). This need not be the case, of course.The perceived distance might depend on the direction orspeed at which the path was traversed. It might dependon whether or not the path resulted from an unimpededmovement or from interaction with an object. If the pathresulted from interaction with an object, the perceiveddistance might depend on the texture of the object'ssurface, how hard the subject pressed against the sur-face, or prior knowledge about the object's shape. Al-though these factors and others no doubt signi®cantlyin¯uence perception of distance, we expected that thepath would be the most important factor and focusedour experiments on it.

We hypothesized that human behavior is metricallyconsistent. Consider a subject interacting with a simu-lated object: we hypothesized that the perceived extentof the object depends only on the spatial stimulus (thepath traversed by the hand). An idealized subject's dis-torted but metrically consistent perception of a spatialstimulus is a local, linear stretch of the stimulus. This isillustrated in Fig. 1.

This leads to testable predictions about human be-havior. One of the more interesting predictions is thatthe perception of length and angle is expected to be re-lated. For example, if a certain rectangle is perceived tobe square, then if the rectangle is cut in half along thediagonal, one would expect the acute angles of the re-sulting right triangle to be perceived as equal. In theexperiments described below we found that haptic spa-tial perception is not geometrically consistent; speci®-cally, we found that perception of an angle wasdistorted, but in a way that could not be predicted byknowing how perception of length was distorted. This issurprising given that, as Foley (1972) observed: ``If anobserver's words are any index of his thoughts, then

man (or at least the M.I.T. student!) is cognitively aEuclidean. In thinking about geometrical problems, hetends to make Euclidean assumptions.'' However, ourresults show that human haptic perception cannot bedescribed as Euclidean.

Four experiments were performed to test the hy-pothesis. Three experiments were performed to quantifyperceptual distortion of length, angle and orientation ata single arm con®guration. Subjects interacted withsimple, simulated objects such as rectangular and tri-angular holes, as shown schematically in Fig. 2. Subjectswere asked to make judgments about spatial propertiesof the objects: relative side lengths, relative angle mag-nitudes, and absolute object orientations.

A fourth experiment examined motor behavior.Given that spatial perception is distorted, it is plausiblethat motor behavior may be distorted in a way consis-tent with perceptual distortion. In the fourth experi-ment, subjects were asked to draw circles. The shapesdrawn were compared with those predicted given theresults of the length perception experiment.

1.2 Previous work: the tangential-radial distortion

The perceptual distortion most relevant to this work isthe tangential-radial distortion (also referred to as thetangential-radial e�ect or illusion). Experimentally, it isobserved that the perceived length of a line segmentdepends on its position and orientation with respect tothe subject. In particular, line segments oriented radiallyfrom the shoulder are perceived as being longer than linesegments oriented tangentially to circles centered at theshoulder (KuÈ nnapas 1955; Davidon and Cheng 1964;Day and Avery 1970; Deregowski and Ellis 1972; Dayand Wong 1973; Wong 1977, 1979; von Collani 1979;Marchetti and Lederman 1983; Kay et al. 1989a,b;Fasse et al. 1990; Hogan et al. 1990; Fasse 1992). Wong(1977, 1979) hypothesized that the tangential-radialdistortion is caused by a di�erence of inertia in thetwo directions; distance is inferred from temporal cues,that is, subjects try to move at a constant velocity and

Fig. 1. An idealized observer's distorted but metrically consistentperception can be thought of as a linear stretch of the stimulus. Theobserver's (distorted) perception of the shape on the left is identical toan undistorted perception of the shape on the right

Fig. 2. a In the ®rst experiment, subjects were asked to judge therelative lengths of simulated rectangular containers. b Angle percep-tion is expected to be related to length perception. In the secondexperiment, subjects were asked to judge the relative magnitude ofangles of simulated triangular containers

70

infer distance from time. Marchetti and Lederman(1983) claimed to disprove this hypothesis. Deregowskiand Ellis (1972) showed that the perception of shapedepends on the orientation of the shape. Experimentallythey observed distortions in the order of 12%.

Kay et al. (1989a,b) (see also Hogan et al. 1990)conducted a series of experiments to study haptic per-ception using the same apparatus used in this investi-gation. They looked at perception of length, force andsti�ness at three di�erent locations in the workspace.Simulated, compliant objects were centered at 25%,50% and 75% of the maximum reach from the subjects'shoulders. They observed that the amount of distortionis con®guration dependent; distortion becomes morepronounced as the center of the object moves away fromthe shoulder.

1.3 Mathematical preliminaries

The following overview is presented to help clarify thedesign and interpretation of the experiments.

1.3.1 Geometrical analysisThe prior work on perceptual distortion shows thatEuclidean geometry is not applicable. Some alternativemathematical tools are necessary to describe subjects'non-Euclidean percepts of simulated objects and basicRiemannian geometry is useful to this end. Riemanniangeometry is a mathematically simple extension of Eu-clidean geometry based on a measure of alignment oftangent vectors known as an inner product. The innerproduct of vectors v and w is denoted hv;wi � vtGw,where G is a symmetric, positive-de®nite matrix. Thisinner product can, in theory, vary from location tolocation and, in general, perceptual distortion is knownto be location dependent. However, in these experiments,we were concerned only with perceptual distortion in asmall region and, in that case, we may assume the innerproduct is e�ectively constant. Inner products inducenorms of vectors, measures of length, and measures ofangle. For example, the norm of a vector v is the squareroot of the inner product of that vector with itself,kvk � hv; vi1=2. An inner product can be used to generategeometrical shapes similar to Euclidean shapes. Forexample, a Riemannian circle of radius r can be identi®edwith the set of displacement vectors from its center oflength r, fv j vtGv � r2g. This is the equation of an ellipse.

An idealized observer's perception may be charac-terized by the set of ellipses that are perceived as circu-lar. Observers will be considered equivalent if they agreewhich ellipses are circular. (They generally will not agreeabout the radius of any particular ellipse.) Equivalentobservers are said to belong to the same metric class. Ametric class can be represented by any of the charac-teristic ellipses.

1.3.2 Statistical analysisThis graphic representation cannot be used to showstatistical distribution; a set of coordinates is needed.Two possible coordinate systems are shown in Fig. 3.

An obvious choice of coordinates (shown on the left inFig. 3) is ellipse eccentricity, �, (the ratio of major andminor axis lengths) and major axis angle, h. Unfortu-nately, this set of coordinates has poor statisticalproperties. The major axis angle is not de®ned for aneccentricity of one and, for eccentricities near one, themajor axis angle is expected to have a nearly ¯atdistribution. Also, the two distributions are not expectedto be statistically independent; since there is moreangular uncertainty for eccentricities near one, theinterpretation of angular measurements is dependenton the corresponding eccentricity measurements.

A better choice of coordinates is ln(ratio) coordinates,de®ned on the right in Fig. 3. The ratio of lengths in anytwo, ®xed directions is expected to have an approxi-mately log-normal statistical distribution. Thus, thelogarithm of the ratio of lengths in any two, ®xed di-rections is expected to be approximately normally dis-tributed. Furthermore, these distributions are expectedto be statistically independent. The ln(ratio) coordinatesused are denoted by ln�r0� and ln�r45�.

The statistical distribution of experimental data re-ported here will be given in ln(ratio) plots such as thatshown in Fig. 4. Each point corresponds to a metricclass and can be identi®ed with a shape such as an el-lipse. The origin corresponds to the Euclidean metricclass. The set of metric classes has an interesting struc-ture, including a natural metric. This structure is dis-cussed further by Fasse and Hogan (1993) (see alsoFasse 1992).

Fig. 3. Two possible sets of coordinates: on the left, ellipseeccentricity and major axis angle; on the right, ln(ratio) coordinates

Fig. 4. ln(ratio) plots are used to show the statistical distribution.Shown is the response distribution of a single subject for the ®rstexperiment

71

The icons next to the ln�r0� and ln�r45� axes showwhich length ratios are to be compared. Variables ln�r0�and ln�r45� are assumed to be normally distributed andindependent. The error bars indicate plus or minus onestandard deviation from the mean.

Figure 4 also illustrates the measures of perceptualdistortion and perceptual uncertainty as used in thiswork. Consider perception of a spatial property such aslength. When asked to make judgments about spatialproperties, humans will give a distribution of responses.Perceptual distortion is de®ned to be the systematic biasof the response distribution from the objective descrip-tion. Perceptual uncertainty is de®ned to be a measure ofthe size or shape of the distribution that is independentof the systematic component. Perceptual uncertaintymay be attributed to di�erent sources. It may be theresult of unquanti®ed external in¯uences or a temporalvariation of physical properties of the perceptual appa-ratus. It can also be the result of inherent ambiguity ofthe perceptual process Bennett et al. (1989). Figure 4depicts the response distribution of an observer exhib-iting both perceptual distortion and uncertainty. Thisobserver's perception is distorted because the means arenot zero; the observer's perception is uncertain becausethe standard deviations are not zero.

2 Length judgment experiment

The goal of this experiment was to quantify perceptualdistortion of length. This was done by simulating smallrectangular containers with a robotic manipulandumand asking subjects to decide which pair of sides of thecontainer was longer. A schematic picture of a subjectinteracting with a simulated container is shown in ofFig. 2a.

2.1 Method

Results are presented for eight, 20±35-year-old, right-handed subjects; seven were male, one was female.Subjects in all experiments were unpaid volunteers withbackgrounds in science and engineering and with noknown neurological or physical impairment.

All experiments were performed using a man-ipulandum designed by Faye (1986). The manipulandumis a planar, two-link, serial linkage actuated by a pair ofPMI JR16M4CH motors. The motors are driven byPMI 00-88007-003 pulse-width-modulated ampli®ers.The ®rst link is driven directly by its motor, whereas thesecond link is driven semi-directly (i.e., with an angulartransmission ratio of 1.0) by a parallelogram linkage.

Position sensing was originally provided by Litton70SSB 12000-1-2-1A optical encoders. These encodersare incremental encoders with phase detection circuitrycapable of 8000 counts (13 bits) per revolution. Theencoders were later replaced with Teledyne Gurley 25/04S-NB17-IA-PPA-QAR1S encoders, which are 17-bitabsolute optical encoders. Their accuracy is comparableto their resolution.

Velocity sensing was provided by an electromechan-ical tachometer integral with the motor. Force sensingwas provided by a Lord FT series force transducer. Themanipulandum was originally controlled by a DECPDP-11/73 computer. This was later replaced by a DellSystem 325 PC.

Position and velocity were sampled continually by thecontrolling computer. Desired interaction forces werecomputed at each sampling instant based on this infor-mation and a model of the object to be simulated. Theseforces were generated by the actuators of the man-ipulandum; actual interaction forces were approximatelyequal to the desired interaction forces.

2.2 Procedure

Subjects were seated and grasped a vertically orientedhandle on the manipulandum. Since a palmar grasp wasused, subjects interacted with the manipulandum usingprimarily arm motion, rather than wrist or ®nger action.The manipulandum restricted hand motion to a hori-zontal plane. The height was adjusted so that the heightof the subject's shoulder was approximately that of themanipulandum. Each subject's arm was supported by asling that restricted elbow motion to a horizontal plane.A plastic panel was mounted horizontally above themanipulandum. A marker was placed on this panel toindicate the desired initial position of the hand.

The objects simulated were rectangular containers(rectangular holes in a sti� wall). In earlier experiments(Kay et al. 1989a,b; Hogan et al. 1990), the objectssimulated were rectangular blocks. We found thatmoving along the inside of a rectangular container iseasier than moving along the outside because the handlecannot slide o� the corner. The rectangles were simu-lated as four intersecting walls. Each wall had a sti�nessof 1 N/mm and a viscous damping coe�cient of 10N � s �mmÿ1 perpendicular to the wall surface. Each wallhad a sti�ness of zero and a damping of zero parallel tothe wall surface. The sti�ness and damping of themanipulandum were varied using an impedance con-troller (Hogan 1985). Force feedback was not used in theimpedance controller so that the e�ective inertia of themanipulandum was the actual inertia.

The orientation of a rectangle is only de®ned modulo90�; a rectangle with orientation h is also a rectanglewith orientation �h� 90��. For this reason, only orien-tations of 0±90� need be considered. Rectangles werepresented to the subject in two orientations. Half werepresented parallel to the plane of the subject's torso (byde®nition 0�). The other half were presented at 45� tothis plane.

Directions x and y were de®ned as shown in Fig. 2a.Let lx be the length of the x side and ly be the length ofthe y side. The area of the rectangles was kept at aconstant lxly � 5000mm2. Fifteen di�erent aspect ratioswere presented with ln�lx=ly� ranging from ÿ0:40 to 0.70in increments of D ln�lx=ly� � 0:07857.

Each container was presented for 10 s, during whichtime the subject traced around the bounding walls. The

72

number of times subjects interacted with each object isgiven in the trials column of the histograms of Fig. 5(e.g., four 50� 53 mm rectangles at an angle of 0�, four50� 53 mm rectangles at 45�, ten 45� 57 mm rectan-gles at an angle of 0�, etc.). The order in which the ob-jects were presented was random, although all subjectswere presented with the same sequence of objects. In all,188 objects were presented.

Objects in all experiments were simulated in the sameregion of the manipulandum's workspace. The seat po-sition was adjusted so that simulated objects were cen-tered at a point located in the sagittal plane on anoutward radius originating at the shoulder, 75% of thedistance of maximum reach from the shoulder.

Each subject was tested during a single session thatlasted 1±2 h. There was a brief break halfway throughthe session. Subjects were instructed to keep their eyesclosed during the experiment, except for brief periodsbetween trials when they could open them to return tothe starting position marked on the plastic panel abovethe manipulandum.

Subjects were told that they would be presented witha sequence of rectangles, that rectangles would be pre-sented in two orientations, and that each rectanglewould be available for ten seconds. Two directionsnamed ``x'' and ``y'' were de®ned for each orientation.After each trial, subjects reported which side was longer,x or y.

Figure 5 shows the histograms obtained for one ofthe subjects. The ``ln(r)'' listed in the histograms is the

natural logarithm of the ratio of x length to y length ofthe rectangle.

A cumulative Gaussian distribution function was ®tto the experimental data as follows. Let i index each rowof the histogram. Let pi be the value of ln�r� in row i. Letfi be the frequency of response observed for row i. Let nibe the number of trials corresponding to row i. Letcdf��p; r; p� be a cumulative Gaussian distribution func-tion with mean �p and standard deviation r evaluated atp. The cost function used for optimal ®tting was

C��p; r� �X

i

ni fi ÿ cdf��p; r; pi�� �2 : �1�

The cumulative distribution function that minimizes thiscost function is the function plotted in Fig. 5 along withthe histogram; the mean and standard deviation asso-ciated with the ®tted distribution are given above thehistogram. The solution was found numerically1.

The statistical information determined from the his-tograms of Fig. 5 is plotted in Fig. 4. The subject's dataare represented by indicating the observed distortions[mean ln�r0�, mean ln�r45�] and deviations. Error barsindicate the observed standard deviations in the twodirections. In this case, the subject exhibited a distortion

Fig. 5. Histograms of a singlesubject

1 A more common procedure is to transform the data by mappingthe interval (0,1) to the real line and to then ®nd the minimum of aquadratic cost function of the reals. This e�ectively weights datapoints with frequencies near 0 and 1 much heavier than frequenciesnear 0.5. The cost function above is more appropriate.

73

of 0.1856 at 0�, a distortion of 0.0491 at 45�, a standarddeviation of 0.0714 at 0� and a standard deviation of0.1482 at 45�.

2.3 Results

The distributions observed for all eight subjects areshown in Fig. 6. An estimate of average distortion wasobtained for the eight subjects by combining all theirdata. The resultant distribution is highlighted in Fig. 6.The subjective circle2 consistent with the averagedistortion has an eccentricity of 1.29 and a major axisangle of 17� (See Fig. 10a). The observed distortion wassigni®cant in that the mean ln�r0� was 1.3 standarddeviations away from 0.0, and the mean ln�r45� was 0.9standard deviations away from 0.0.

The results of this experiment show the following.

1. Perceptual distortion of length is signi®cant, in theorder of 30% at the location studied.

2. This perceptual distortion is similar among individu-als in that all observed means of ln�r0� and ln�r45�were greater than zero.

3 Angle judgment experiment

The goal of this experiment was to test the hypothesisthat a single, ideal metric relates perceived angles toperceived lengths. Subjects were presented with anumber of triangles and asked to judge the relativemagnitudes of two angles of the triangle. Correspondinglength judgments were inferred from the subjects'responses and compared with the results of the ®rstexperiment.

To understand qualitatively how length and angleperception are expected to be related, consider the tri-angle of Fig. 2b, which shows schematically a subjectinteracting with a simulated triangle. Assume that thesubject perceived the sides parallel to the x-axis andy-axis to be of equal length. This subject would beexpected to perceive that angles 1 and 2 have equalmagnitudes.

A plausible strategy for measuring the relativemagnitude of angles of a triangle is to measure thelength of the opposite sides. Allowing this direct mea-surement of length would have defeated the purpose ofthe experiment. This strategy was made impossible bypreventing the subject from ever reaching the top cor-ner. The object was simulated in such a way that thesubject felt like he was tracing around a triangular holewith the curious property that the top corner could notbe reached.

3.1 Method

Results are presented for seven, male, 20±35-year-old,right-handed subjects. All seven subjects participated inboth the length and angle judgment experiments.

Triangles were simulated by simulating a number ofintersecting walls. Each wall had a sti�ness of 1 N/mmand a viscous damping coe�cient of 10 N � s �mmÿ1perpendicular to the wall and had a sti�ness of zeroand viscous damping coe�cient of zero parallel to thewall.3

Subjects were prevented from measuring length di-rectly by making it impossible to reach the third corner.This was accomplished by presenting only two sides at atime. Referring to Fig. 2b, the base side (unlabeled) wasactive at all times. After leaving the base, either side x orside y was active, whichever was contacted ®rst. Re-turning to the base reset the state, making it possible tointeract with either adjacent side. The object can bethought of as having three non-parallel sides and twocorners (a biangle). There is no such Euclidean object.Since the object cannot exist physically, it must be sim-ulated.

The length of the bases of the triangles was a constant50 mm. Triangles were right triangles with two wallsparallel to the x-axis and y-axis. Similar to the lengthperception experiment, triangles were presented with onewall (x) parallel to the plane going through the subjectstorso or at a 45� angle to this plane. Two angles (cor-ners), named 1 and 2, were de®ned for each orientation.

Fig. 6. Response distribution of eight subjects, together with theaverage distribution estimated by combining individual data

2 This is the ellipse that would be perceived to be circular by anidealized observer, one for whom length perceptual distortion forany pair of axes was consistent with the distortions measured forthe pairs of axes tested in this experiment. Of course, this need notbe the case for the real observer.

3 Simulating a corner with two intersecting walls is satisfactory aslong as the corner angle is right or obtuse. For acute angles, theresulting sti�ness from the superimposed sti�ness ®elds is low in thedirection of the bisector of the angle, allowing signi®cant dis-placements into the walls. To remedy this problem, corners weresimulated by presenting three intersecting walls instead of two. Thethird wall was perpendicular to the bisector of the angle to besimulated. This sti�ened the corner, making the corner feel moredistinct. Though imperfect, the e�ect is much better than thatachieved using only two intersecting walls.

74

Corner 1 was the corner opposite the side oriented in thex direction; corner 2 was the corner opposite the sideoriented in the y direction.

In a pilot experiment, the triangles used were thosegenerated by bisecting the rectangles used in the lengthperception experiment along the diagonal. This provedto be too di�cult for subjects, so a larger range of ratioswas used. Let lx be the length of the x side and ly be thelength of the y side. Nineteen di�erent aspect ratios werepresented with ln�lx=ly� ranging from ÿ1:61 to 0.92 inincrements of D ln�lx=ly� � 0:01403. Each object waspresented for 15 s. The order in which the objects werepresented was random, although all subjects were pre-sented with the same sequence of objects. In all, 264objects were presented.

3.2 Procedure

Subjects were tested in two sessions, with a period of 2±10 days between sessions in most cases. Each sessionlasted between 1±2 h. There was a brief break halfwaythrough each session. Subjects were instructed to keep

their eyes closed during the experiment, except for briefperiods between trials when they could open them toreturn to the starting position marked on the plasticpanel above the manipulandum.

Subjects were told that they would be presented witha sequence of biangles (triangles of which only two sideswould be simultaneously present). They were told thatthe objects would be presented in two orientations, andthat each object would be available for 15 s. Angles(corners) 1 and 2 were de®ned for each orientation.After each trial, subjects reported which angle waslarger, 1 or 2.

Two histograms of responses were determined foreach subject, similar to the histograms of Fig. 5. A cu-mulative distribution function of Gaussian type was ®-ted to each histogram. Figure 7 shows the histogramsobtained for one of the subjects. The ``ln(r)'' listed in thehistograms is the natural logarithm of the ratio of xlength to y length of the sides of the triangle. The ``%responses x'' listed in the histogram is actually the ``%responses 1'', but is so labeled to emphasize the expectedcorrespondence of histograms obtained in the length andangle perception experiments.

Fig. 7. Histograms of a singlesubject

75

3.3 Results

As mentioned, each experiment was split into two sets oftrials. The intervals between trial sets for the sevensubjects were 69, 8, 2, 10, 5, 3 and 5 days, respectively.Although there was considerable variation in theresponse distributions observed among subjects, theresponse distribution of individuals was stable over time.Subject 1's unusually long hiatus was due to anoversight; the results were still consistent. Distributionsmeasured for two individuals on di�erent days areshown in Fig. 8. The response distributions of all sevensubjects are shown in Fig. 9. The distributions shownare those obtained by combining data from the two trialsets for each subject.

An estimate of average distortion was obtained forthe seven subjects by combining all their data4. Theaverage distortion was 30%. The resultant responsedistribution is highlighted in Fig. 9. The mean ln�r0�was 0.4 standard deviations away from 0.0, and themean ln�r45� was 0.5 standard deviations away from 0.0.Also shown in the ®gure is the estimate of the averagedistribution obtained in the length perception experi-ment. The distributions are inconsistent. Both the ob-served distortions and uncertainties were signi®cantlydi�erent in the two experiments [signi®cances P0,P45 < 1% as determined by Student's t-test for normallydistributed data sets with unequal variances on the dis-tributions of individual mean ln�r0� and mean ln�r45�[24]]. The subjective circles consistent with the observeddistortions are shown in Fig. 10a.

Figure 10b illustrates how an ellipse is related to anangle measurement. The non-Euclidean protractor onthe left would be perceived as the Euclidean protractoron the right. The marks shown along the ellipse are atincrements that would be perceived to be 10� apart. Sincethis experiment makes no predictions about the perceivedorientation of a single line segment, only about the per-ceived angle between line segments, the marks should notbe interpreted as measures of orientation.

Summarizing, the results of this experiment show thefollowing.

1. Perceptual distortion of an angle is of the order 30%at the location studied.

2. The observed angle perceptual distortion was signi®-cantly di�erent to that expected given the observedlength perceptual distortion.

3. There is more angle perceptual uncertainty amongindividuals than would be predicted from the lengthperceptual scatter; the standard deviations were con-sistently higher than those of the length experiment.

4. Response distributions observed for individuals wereconsistent over time.

Fig. 8. a Two sessions 69 days apart. b Two sessions 8days apart

Fig. 9. Response distributions of seven subjects, together with theaverage distribution estimated by combining individual data. Shownalso is the average distribution determined from the length experiment

Fig. 10. a Average subjective circles as determined by (1) the angleexperiment, with � � 1:28, h � ÿ62� and (2) the length experiment,with � � 1:29, h � 17�. b The results of the angle judgmentexperiment are consistent with an observer using the protractorshown on the left. This protractor would be perceived as the normal,Euclidean protractor shown on the right

4 A similar estimate of the means may be obtained by averagingthe means across subjects, but yields no estimate of the standarddeviation.

76

4 Orientation judgment experiment

Section 3 explained why it is reasonable to expect lengthand angle perceptions to be related. In experiment 3, weagain looked at the consistency of length and angleperception, albeit indirectly. The goal of the experimentwas to determine locally which directions are perceivedas being oriented ``straight ahead'' and ``straight side-ways''. This was done by simulating short slots (seg-ments inside a solid boundary) and asking subjects todecide whether or not the end of the slot farthest awayfrom the subject pointed forward and to the left, orforward and to the right. Figure 11a shows schemati-cally a subject interacting with a simulated slot. Thehypothesis that spatial perception is metrically consis-tent does not predict which directions will be perceivedas ``straight ahead'' and ``straight sideways'', but it doespredict that those two orientations will be perceived asbeing at right angles.

Knowing only one pair of orientations that are per-ceived to be at right angles is not enough to determine aconsistent metric. For this reason, the results cannot becompared directly with those of the other experiments,and strong conclusions about metric consistency cannot

be drawn. Nonetheless, indirect comparison is possibleand the results are interesting.

4.1 Method

Results are presented for seven, male, 20±35-year-old,subjects. Six subjects were right-handed, one subject wasleft-handed. Six of the seven subjects participated in thelength, angle and orientation judgment experiments.

Line segments were presented as short slots in a sti�wall. The same container simulator used in the lengthperception experiment was used to simulate slots. A slotwas simulated as a container with zero width and ®nitelength. Line segments were presented to subjects in 5�increments from ÿ90� to 85�. Positive angles are mea-sured clockwise from a sagittal orientation. Segmentswere all 5 cm and centered at the standard starting po-sition. Five segments of each of the 36 orientations werepresented to each subject, for a total of 180 segments.Segments were presented in a random order, but allsubjects were presented with the same sequence. Eachsegment was available for 10 s, during which time thesubject traced back and forth inside the slot.

Fig. 11. a Subjects interacted withsimulated slots and were asked tojudge which quadrant the slot wasin. b Response distribution of asingle subject. c Subjective orienta-tions determined from the responsedistribution of b

77

4.2 Procedure

Subjects were tested in a single session that lasted about45 min. Subjects stopped for a brief break halfwaythrough the experiment. Subjects were instructed to keeptheir eyes closed during the experiment, except for briefperiods between trials when they could open them toreturn to the starting position marked on the plasticpanel above the manipulandum.

Subjects were told that they would be presented witha sequence of slots (short line segments), each of whichwould be available for 10 s. They were asked to deter-mine ``whether each line segment is pointing forwardand to the left, or forward and to the right of straightahead''. Subjects gave an abbreviated response of ``left''or ``right''.

A histogram of responses was obtained for eachsubject. One such histogram is shown in Fig. 11b. It isprobably obvious that it is di�cult to judge if the slotorientation is near 12 o'clock but it may be less obviousthat it is di�cult to judge if the slot orientation is near 3o'clock. There are two stimulus regions in the histogramfor which the response frequency is approximately 50%;one corresponds to the subjective 12 o'clock orientation,the other to the subjective 3 o'clock orientation. Theseorientations were obtained using the following proce-dure. In each of the two crossover regions, a linearfunction, f �h�, was ®t to the histogram using a standardlinear regression. The orientation, hc, for whichf �hc� � 50%, was the orientation of subjective 12o'clock or 3 o'clock, as appropriate. Figure 11c showsthe resultant orientations of subjective 12 o'clock and 3o'clock.

4.3 Results

Figure 12a shows the subjective orientations from allseven subjects, with the data from the left-handed subjectreversed. One subject exhibited a qualitatively di�er-ent distribution of responses from the other subjects.

We are interested in metric consistency, and not ori-entation perception per se. For this reason, it is onlyrelevant that for each subject there is a known pair ofsegments that are (presumably) perceived to be orthog-onal. Consider any two pairs of segments. For ease ofvisualization, assume that one pair corresponds to a``normal'' subject, such as the pair shown in Fig. 11c.Assume that the other pair corresponds to the qualita-tively di�erent ``outlier'' of Fig. 12a. Are these responsesmetrically inconsistent? Not necessarily; the two subjectswould disagree about the absolute orientations of eachother's segments, but they might agree that both pairs ofsegments were orthogonal. For example, the outliermight perceive that the normal subject's segments wereoriented at 2 o'colck and 5 o'clock, and thus orthogonal.It is thus not reasonable to simply average the subjects'data. Instead, to make a comparison with the other ex-periments, we used the results of (1) the length experi-ment and (2) the angle experiment to derive twopredictions of the perceived angle between ``straightahead'' and ``straight sideways''.

Predicted, perceived angles were determined as fol-lows. A mean distortion with eccentricity of 1.29 and amajor axis angle of 17� was determined in the lengthjudgment experiment. A corresponding inner productmatrix G is

G � R�17�� 1 0

0 1:292

� �R�ÿ17�� �2�

� 1:0568 ÿ0:1857ÿ0:1857 1:6073

� �; �3�

where

R�h� � cos�h� ÿ sin�h�sin�h� cos�h�� �

: �4�

For each subject, vectors corresponding to subjective 3o'clock and 12 o'clock were determined. For example,the vectors corresponding to the orientations ÿ6� andÿ88� are

v � ÿ0:10450:9945

� �; w � ÿ0:9994

0:0349

� �: �5�

The predicted, perceived angle between these vectors isthen

��v;w� � arccoshv;wikvk kwk� �

� 75� : �6�

Using the inner product of (2), predicted, perceivedangles were computed for each subject. These angleswere averaged using circular statistical methods [1],resulting in an average of 75� (�10�, 99% con®dence).This is signi®cantly di�erent from the expected 90�.

The results of the orientation judgment experimentare more consistent with the results of the angle per-ception experiment. For each subject, predicted, per-ceived angles were computed using an inner productmatrix derived from the results of the angle perceptionexperiment. The average of these angles was 91� (�10�,

Fig. 12. Estimates of subjective 12 o'clock and 3 o'clock directions ofseven subjects

78

99% con®dence). This angle is obtuse; the acute angle is89�. Both are statistically similar to the expected 90�.5

To understand these results, consider Fig. 13 whichmay provide qualitative insight on the implications ofthe statistical analysis. Figure 13a shows an idealizedstimulus-percept pair. The stimulus consists of an ellipseand two line segments corresponding to the vectors of(3). An ideal observer whose perceptions were consistentwith the results of the length perception experimentwould perceive the ellipse to be circular. The observerwould perceive the relative, acute angle between the twoline segments as being 75�. Figure 13b shows a di�erentstimulus-percept pair. The stimulus is a di�erent ellipsewith tick marks around its circumference, and again thetwo line segments. An ideal observer whose perceptionswere consistent with the results of the angle perceptionexperiment would perceive the tick marks to be at equal,10� intervals. The observer would perceive the relative,obtuse angle between the two line segments as being 93�.

The conclusions drawn from this experiment are asfollows.

1. The observed orientational distortion was not con-sistent with the observed length perceptual distor-tion.

2. The observed orientational distortion was consistentwith the observed angular perceptual distortion.

5 Circle drawing experiment

Humans make errors when performing motor tasks suchas drawing. For example, we cannot draw perfect circleswithout special tools. This error is often thought of asresulting from failure of the motor control system toachieve the intended behavior but alternatively, some ofthis error may also be ascribed to a distorted intent.

Assume, for example, that a subject was asked todraw a circle. It is reasonable to assume that it is thesubject's intent to draw a shape that is perceived to becircular. If the subject's perception is distorted so that

Euclidean circles are not perceived as being circular, it isunlikely that the subject will draw Euclidean circles,even if no error is introduced by the motor controlsystem. Motor behavior may thus be expected to berelated to perceptual behavior. More speci®cally, motorintent may be expected to be geometrically consistentwith perceptual distortion. The goal of the fourth ex-periment was to determine if the shapes that people drewcould be predicted, given the results of the length per-ception experiment.

5.1 Method

Results are presented for 11, male, 20±35-year-old, right-handed subjects. Two of the 11 subjects participated inboth the length judgment and the circle drawingexperiments.

The manipulandum was used solely as a data acqui-sition device in this experiment. No controller was ac-tive, so the primary in¯uence of the manipulandum wasits inertia and friction, both of which were su�cientlysmall to have little or no e�ect on movement.

5.2 Procedure

Subjects were asked to draw circles continually. Thecircles were to be centered at the standard startingposition (75% of maximum reach), approximately 10 cmin diameter. The circles could be drawn in a clockwise orcounterclockwise direction, but once a direction hadbeen chosen, all circles were to be drawn using thatdirection. Nine of the 11 subjects chose to draw circlescounterclockwise.

Subjects performed the experiment with their eyesclosed, although they could open their eyes periodicallyto make sure the circles were centered properly. Datawas sampled at 50 Hz and stored in a ring bu�er thatheld 4 s of data. Movement speeds were such that atleast one complete circle was drawn in 4 s. Subjects weretold to slow down if their rates approached one circleper 2 s. Data collection was halted at the subject's reportthat a ``good'' circle had been drawn. Subjects drew 20circles in all. The experiment lasted about 15 min.

Ellipses were ®t to the data by ®nding a metric forwhich the data looked most circular. The procedure usedis described in the appendix. Ellipses ®tted using thisprocedure are shown in Fig. 14.

Fig. 13. a The percept of the stimulus is assumed to be a linear stretchof the stimulus consistent with the length perception experiment. bOrientation perception results are more consistent with angularperception results

Fig. 14. Two examples of ellipses ®tted to data

5 Note that the estimated �10�, 99% con®dence intervals of thepredicted, perceived angles take into account only statistical vari-ation of the orientation judgment data and not of the length orangle judgment data, which were assumed to be described perfectlyby their means.

79

This procedure is sensitive to drift in the center of thedata because it assumes that the center of the circles isstationary. Attempts to ®t a single ellipse to data that isbest described as multiple circles with di�erent centerswill lead to erroneous results. The center of the ellipsestended to drift toward the subjects' midlines. For thisreason, the data recording duration was chosen so thatsubjects had time to draw only one or two circles.

Measures of statistical distribution analogous tothose used in the perceptual experiments were obtainedby assuming that the distributions of ln�r0� and ln�r45�of the ellipses were independent and distributed nor-mally. The mean values of ln�r0� and ln�r45� thus ob-tained determined an average ellipse.

5.3 Results

Distributions obtained for each of the 11 subjects areshown in Fig. 15a; most of the means are in the ®rstquadrant. Figure 15b shows the resultant distributionfrom combining the data from all 11 subjects. Discretepoints in the plot correspond to ellipses drawn bysubjects. The distribution measured in the lengthperception experiment is presented for comparison.Although the distributions were statistically distinguish-able [signi®cances P0 < 1%, P45 � 45% as determinedby Student's t-test for normally distributed data setswith unequal variances on the distributions of individualmean ln�r0� and mean ln�r45�], the di�erence of meansis small.

The results of this experiment show the following.

1. Motor distortion of length is signi®cant, in the orderof 13% on average.

2. This motor distortion is similar among individuals.3. The observed motor distortion was inconsistent with

the observed length perceptual distortion.

It is interesting to note that the relative distortionbetween the distributions obtained in the length percep-tion and circle drawing experiments was much smallerthan the relative distortion between the distributionsobtained in the length perception and angle perceptionexperiments (compare Fig. 9 with Fig. 15b). Thus,despite their inconsistency (statistical distinctness), itappears that motor distortion and length perceptualdistortion may be similar.

6 Discussion

6.1 Validity of the results

The experiments reported here were performed on``virtual'' objects simulated mechanically by a robot tomimic certain properties of real objects, but the mimicryis inevitably imperfect. Could our observations havebeen due to some di�erence between real and virtualobjects? Further work would be required to conclusivelyrule out this possibility but the similarity of our lengthperception results to earlier investigations of the tan-gential-radial e�ect makes it unlikely.

In the length perception experiment, subjects werelimited to 10 s of interaction before judgment, whichmight have a�ected the results. In the length perceptionexperiment reported by Kay et al. (1989a,b) subjects weregiven unlimited time of interaction before judgment. Ourresults were similar to those of Kay et al. so the limitedduration of interaction seems not to have been important.

Could our observations be peculiar to the experi-mental task we used? Task context is undoubtedly im-portant. Haptic perception may depend upon theparticular motor and sensory apparatus involved. Sub-jectively, this task is analogous to feeling inside a shapemade of sti� foam6 with a stick held in a palmar grasp.As humans usually explore objects with the ®ngersrather than with the palm, we might ®nd di�erent resultsif we con®ned our subjects to a ®ngertip pincer grasp.Once again, the similarity of our results to prior work onthe tangential-radial e�ect (which was not restricted topalmar interaction) suggests that our observations arenot peculiar to the grasp our subjects used but furtherwork is required to investigate this possibility. Subjectswere allowed to open their eyes periodically, so that theywould remain alert; these intermittent visual cues mayhave a�ected the results. Furthermore, given that visionplays a dominant role in spatial perception, di�erentresults might be found if the experiments were per-formed with the eyes open. Without further research wecannot rule out these possibilities, but we note that thereare realistic everyday situations in which haptic per-ception occurs in the absence of visual information. Theobserved haptic perceptual distortions were similar for

Fig. 15. a Ellipse distributions of 11 subjects. b Distri-butions obtained from both the circle drawing experi-ment and the length perception experiment. Pointscorrespond to ellipses drawn by all 11 subjects

6 Due to limitations of the apparatus, objects with perfectly rigidsides could not be simulated.

80

di�erent subjects and consistent over time, indicatingthat they re¯ect a robust perceptual phenomenon, in-teresting in its own right.

Could our observations be an artifact of our experi-mental design? While the order of stimulus presentationmight have been varied to eliminate possible order ef-fects, in each experiment we chose to use the same trialsequence for all subjects. This was done to enhanceconsistency between subjects by ensuring that all sub-jects experienced the same set of trials but alternativedesigns may be appropriate for further research.

In the orientation perception experiment, subjectswere allowed only two possible responses: forward andto the left, forward and to the right. As a result, we couldnot determine a metric directly from this experiment andhad to use the indirect method described above to testwhether orientation perception was consistent withlength perception (it was not) and angle perception (itwas). However, if subjects were given, say, four possibleresponses, four subjective orientations might be deter-mined. That would be more than enough information todetermine a metric, so that it would then be possible tocompare the experimental results directly. This wouldalso make it possible to take into account the statisticalvariation of the length and angle judgment data, whichwas not possible with the indirect method we used.

Each of our experiments used a relatively smallnumber of subjects, and not all the same subjects. Forexample, only two subjects participated in both thelength perception and the circle drawing experiments.Finally, in the length and angle perception experiments,only two object orientations were used to determine a``perceptual metric''. In e�ect, it was assumed, in ac-cordance with basic Riemannian geometry, that lengthperception for all other object orientations could bederived from these two measurements. This, of course,need not be the case and further experimentation wouldbe required to validate this assumption.

Nevertheless, even with these caveats, our experi-ments yielded statistically reliable results that admit ameaningful interpretation.

6.2 Relation to prior work

These experiments con®rm and extend prior observa-tions of a tangential-radial distortion of haptic percep-tion. The e�ect is quite pronounced. For example, lengthperception is distorted in the order 25±30% in theworkspace location studied. It becomes even morepronounced as the center of the object moves awayfrom the shoulder (Kay et al. 1989a,b; Hogan et al.1990). However, it appears that the e�ect is not bestrepresented as a combination of radial and tangentiale�ects. Our results indicate that the greatest lengthperception distortion (indicated by the minor axis of theellipse of Fig. 10) is not radial (i.e., along a radiuscentered on the shoulder) but in a direction orientedmore towards midline. Contrary to the hypothesisproposed by Deregowski and Ellis (1972) we observedsigni®cantly distorted perception of objects at 45�.

Our experiments also extend prior work to quantifyhaptic perception of other spatial properties of objectssuch as angles of corners. We found that haptic angleperception was also signi®cantly distorted, in the orderof 30% in the workspace location studied.

By quantitatively comparing the perceptual distortionof di�erent object properties such as angles and lengths,we were able to investigate the geometric structure ofhaptic perception. We believe this fundamental ap-proach may ultimately reveal important features of thecomputations underlying perception.

6.3 Geometric inconsistency

The ®rst result of our study is experimental evidence thathaptic spatial perception was not geometrically consis-tent or, more speci®cally, not consistent with a Riem-annian mathematical model. However, this does notimply that haptic perception is unstructured. The secondresult of our study is our experimental evidence thatperception of orientation was largely consistent withperception of angle. This suggests that there may bemore commonality between the computational resources(or modules, or regions of the CNS) used in perceivingorientation and relative angle than between those used inperceiving orientation and length. There appears to bean internal orientation perceptual apparatus unrelatedto the apparatus used to perceive length and distance.Put another way, there appears to be an internal,metrically inconsistent compass.

What does it mean for a compass to be metricallyinconsistent? A compass is a device for measuring theabsolute orientation of a single line segment. A pro-tractor is a metrically consistent device for measuring therelative angle between pairs of line segments. The com-passes used in everyday navigation are metrically con-sistent in that there exist metrically consistent angularrelations between the various bearings on the compass.For example, the orientation East is at 90� to the orien-tation North, which is in turn at 45� to the orientationNorthwest. Metrical consistency is not a logical neces-sity, it is a logical convention. It is possible to navigateusing compasses that are not metrically consistent.7

To our knowledge, there is no obvious reason whyhumans would have developed an apparatus for mea-suring angles between line segments (an internal pro-tractor). There is, however, ample reason to expect thathumans would have developed an orientation perceivingapparatus (an internal compass) as it would be useful inmaking directed movements.

6.4 Perception of length and production of movement

Since haptic perception requires an interplay betweensensation and action, we studied the relation between

7 The sidereal compass used for navigation in certain Micronesiancultures is an example of a metrically inconsistent compass(Hutchins 1983).

81

haptic perceptual distortions and movement production.The third result of our study is experimental evidencethat distortion of motor production is similar to thedistortion of length perception (though a small, statis-tically signi®cant di�erence was observed). This isfurther evidence of geometric structure in haptic per-ception. It also implies that commonly observed errorsin production of motor behavior may, at least in part, beattributed to central processes underlying the produc-tion of motor commands rather than to an imperfectexecution of motor commands.

One interpretation consistent with all of our experi-mental results is that there may exist both an ``internalcompass'' and an ``internal ruler''. The internal ruler isused in length perception; the internal compass is used inorientation perception. The compass and the ruler arefunctionally independent, or the results of the length andorientation perception experiments would have beenconsistent. In estimating relative angles between seg-ments, humans refer to their internal compass, and nottheir internal ruler. This is supported by the geometricconsistency between the angle and orientation percep-tion experiments. In the generation of paths, humansrefer to their internal ruler. This is supported by thesimilar geometric distortion observed in the length per-ception and circle drawing experiments.

This hypothesis is in accord with results in motorneuroscience that show that direction of movement isencoded independent of velocity, notably the work ofGeorgopoulos and colleagues (Kalaska et al. 1983; Ge-orgopoulos et al. 1986, 1988; Taira et al. 1990; Ge-orgopoulos 1991). It is also consistent with thephenomenon of dysmetria, where individuals move thehand in an appropriate direction towards a target, butovershoot or undershoot the distance (Meador et al.1986). Of course, the results of the experiments pre-sented here are not strong enough to verify these spec-ulations but further investigation is clearly warranted toexplore the rami®cations of these ideas and test the hy-pothesis that independent CNS processes are used fororientation and length perception.

7 Concluding remarks

In the experiments reported here, we found that humansmisperceive geometric properties of felt objects. Webelieve it is important to understand and quantify thisphenomenon in order to better design and developdevices to interact with humans. This is especiallyrelevant for devices to simulate the touch and feel ofreal objects, an important part of virtual environmenttechnology. Furthermore, the analytical and experimen-tal methods described here may prove useful forobjective assessment of virtual environment technology,e.g., the ``real-ness'' of a particular display (see Fasseand Hogan 1993, for further discussion).

Conversely, our experiments illustrate the value ofvirtual environment technology for psychophysical in-vestigations. Even the relatively crude device we useda�orded a useful enhancement of psychophysical

methods previously used to investigate haptic percep-tion. By providing enhanced control of experimentalstimuli, virtual environment technology provides signi-®cant new opportunities for studying the transformationfrom sensation to action.

Finally, the psychophysical techniques used in theseexperiments are potentially useful for the evaluation ofpatients with neurological disorders, for example, re-covering stroke patients. A ``haptic examination'',analogous to a vision or hearing examination, could beadministered using simple robot technology (see, e.g.,Krebs et al. 1998) and may provide useful insight intothe nature and extent of the perceptual abilities anddisabilities of patients. Ultimately, a haptic examinationmight be useful for clinical diagnosis or evaluation ofrecovery.

Acknowledgements. The research reported in this paper was sup-ported in part by National Institutes of Health Grant AR40029,O�ce of Naval Research Grant N00014/88/K/0372, and by theFairchild Foundation. E.D. Fasse is now with the Dept. of Aero-space and Mechanical Engineering at the University of Arizona.B.A. Kay is now with the Dept. of Cognitive and Linguistic Sci-ences at Brown University. F.A. Mussa-Ivaldi is now with theDept. of Physiology at the Northwestern University MedicalSchool.

Appendix: Ellipse ®tting

This appendix describes the procedure that was used to ®t ellipsesto experimental data. Ellipses were ®t to data by ®nding a metricfor which the data was most circular. Let G be a matrix repre-senting an arbitrary metric, so that

hv;wi � vtGw � vt gxx gxy

gxy gyy

� �w : �7�

A circle with respect to this metric, centered at c with radius r, is theset of points fp j �p ÿ c�tG�p ÿ c� � r2g. Real data will not be cir-cular with respect to any metric. The deviation from being circularfor a data point pi can be expressed by

�i � �kpi ÿ ck2 ÿ r2�2 �8�� �gxxp2xi ÿ 2gxxcxpxi � gxxc2x � 2gxypxipyi

ÿ 2gxycxpyi ÿ 2gxypxicy � 2gxycxcy

� gyyp2yi ÿ 2gyycypyi � gyyy2c ÿ r2�2 : �9�

The cost function used for optimal ®tting was V �Pi �i. De®nea0 � gxx, a1 � 2gxy , a2 � gyy , a3 � ÿ2gxxcx ÿ 2gxycy , and

a4 � ÿ2gxycx ÿ 2gyycy . De®ne a5 � gxxc2x � 2gxycxcy � gyyc2y ÿ r2 �kck2 ÿ r2. Substitution into (9) yields

V �X

i

�a0p2xi � a1pxipyi � a2p2yi � a3pxi � a4pyi � a5�2 : �10�

As stated, the problem is overspeci®ed. There are six parameters, a0

through a5, which are, in turn, functions of gxx, gxy , gyy , cx, cy andr2. An ellipse requires only ®ve parameters for description. Theradius, r, can be chosen arbitrarily without a�ecting the solution.Unfortunately, the resultant optimization problem does not havean obvious closed-form solution and has to be solved iteratively.This approach was tried initially using a gradient descent algo-rithm, but was found to give unsatisfactory results on elliptical testpatterns. The convergence properties of the algorithm were poor,

82

and at times unstable. The algorithm could not be tuned to givesatisfactory estimates of the elliptical test patterns.

If instead a5 � kck2 ÿ r2 is chosen arbitrarily, the problem has aclosed-form solution, as given below. Empirically, this proceduregave better estimates in general, and found the exact solutions ofelliptical test patterns in particular. This is the procedure that wasultimately used for ®tting ellipses to experimental data. Althoughthe resultant ®ts were empirically excellent, this procedure shouldbe used with caution. It is not obvious how ®xing kck2 ÿ r2 a�ectsthe solution. Assume, then, that a5 is constant. Let

a � a0 a1 a2 a3 a4� �t ; �11�

b � ÿX

i

� p2xi pxipyi p2yi pxi pyi �t �12�

and

M �X

i

p4xi p3xipyi p2xip2yi p3xi p2xipyi

p3xipyi p2xip

2yi pxip3yi p2

xipyi pxip2yi

p2xip

2yi pxip3

yi p4yi pxip2yi p3yi

p3xi p2xipyi pxip2yi p2xi pxipyi

p2xipyi pxip2

yi p3yi pxipyi p2yi

2666664

3777775 : �13�

The optimal a is a � Mÿ1b. Knowing a it is easy to solve for G andc, which fully describe the ellipse. Ellipses ®tted using this proce-dure are shown in Fig. 14.

References

Batschelet E (1981) Circular statistics in biology. Academic Press,San Diego, California

Bennett BM, Ho�man DD, Prakash C (1989) Observer mechanics:a formal theory of perception. Academic Press, San Diego,California

Davidon RS, Cheng MFH (1964) Apparent distance in a hori-zontal plane with tactile-kinesthetic stimuli. J Exp Psychol16:277±281

Day RH, Avery GC (1970) Absence of the horizontal-vertical il-lusion in haptic space. J Exp Psychol 83:172±173

Day RH, Wong TS (1973) Radial and tangential movement di-rections as determinants of the haptic illusion in an 1 ®gure.J Exp Psychol 87:19±22

Deregowski J, Ellis HD (1972) E�ect of stimulus orientation uponhaptic perception of the horizontal-vertical illusion. J ExpPsychol 95:14±19

Fasse ED (1992) On the use and representation of sensory infor-mation of the arm by robots and humans. PhD Thesis, MIT,Cambridge, Massachusetts

Fasse ED, Hogan N (1993) Quantitative assessment of humanperception of simulated objects. In: Kazerooni H, ColgateJE, Adelstein BD (eds) Advances in robotics, mechatronicsand haptic interfaces, vol 49, ASME, The American Societyof Mechanical Engineers, New York, pp 89±97

Fasse ED, Kay BA, Hogan N (1990) Human haptic illusions invirtual object manipulation. In: Prod 12th Annu Conf IEEE,Eng Med Biol Soc, vol 12 IEEE pp 1917±1918

Faye (1986) An impedance controlled manipulandum for humanmovement studies. MSc Thesis, MIT, Cambridge, Massachu-setts

Foley FM (1972) The size-distance relation and intrinsic geometryof visual space: implications for processing. Vis Res 12:323±332

Georgopoulos AP (1986) On reaching. Annu Rev Neurosci 9:147±170

Georgopoulos AP (1991) Higher order motor control. Annu RevNeurosci 14:361±377

Georgopoulos AP, Kottner RE, Schwartz AB (1988) Primatemotor cortex and free arm movements in three-dimensionalspace II: coding of the direction of movement by a neuronalpopulation. J Neurosci 8:2928±2937

Hogan N (1985) Impedance control: An approach to manipulation.ASME J Dynam Syst Measurement and Control 107:1±24

Hogan N, Kay BA, Fasse ED, Mussa-Ivaldi FA (1990) Hapticillusions: experiments on human manipulation and perceptionof ``virtual objects''. Cold Spring Harbor Symp Quant Biol55:925±931

Hutchins E (1983) Understanding micronesian navigation. Law-rence Erlbaum Association, Mahwah, New Jersey, pp 191±225

Kalaska JF, Caminiti R, Georgopoulos AP (1983) Cortical mech-anisms related to the direction of two-dimensional arm move-ments: relations in parietal area 5 and comparison with motorcortex. Exp Brain Res 51:247±260

Kay BA, Hogan N, Mussa-Ivaldi FA, Fasse ED (1989a) perceivedproperties of objects using kinesthetic sense depend on work-space location. Soc Neurosci Abstr 15:173

Kay BA, Hogan N, Mussa-Ivaldi FA, Fasse ED (1989b) Perceivingthe properties of objects using arm movements: workspace-depend e�ects. In: Proc 11th Annu Conf IEEE, Eng Med BiolSoc vol 11, New York p 1522

Krebs HI, Hogan N, Aisen ML, Volpe BT (1998) Robot-aidedneuro-rehabilitation. IEEE Trans Rehab Eng 6:75±87

KuÈ nnapas TM (1955) An analysis of the ``vertical- horizontal il-lusion'' J Exp Psyschol 45:134±140

Marchetti FM, Lederman SJ (1983) The haptic radial-tangentiale�ect: two tests of Wong's ``moments-of-inertial'' hypothesis.Bull Psychon Soc 21:43±46

Meador KJ, Watson RT, Bowers D, Heilman KM (1986) Hypo-metria with hemispatial and limb motor neglect. Brain109:293±305

Sachs L (1984) Appl Stat Springer, Berlin New York HeidelbergTaira M, Miner S, Georgopoulos AP, Murata A, Sakata S (1990)

Parietal cortex neurons of the monkey related to the visualguidance of hand movements. Exp Brain Res 83:29±36

von Collani G (1979) An analysis of illusion components with 1and ?-®gures in active touch. J Exp Psychol 31:241±248

Wong TS (1977) Dynamic properties of radial and tangentialmovements as determinants of the haptic horizontal-verticalillusion with an 1 ®gure. J Exp Psychol: Hum Percep Perform3:151±164

Wong TS (1979) Developmental study of a haptic illusion in relationto piaget's centration theory. Exp Child Psychol 27:489±500

83