Constrained motion control on a hemispherical …...Constrained motion control on a hemispherical...

Constrained motion control on a hemispherical surface: path planning

Sigal Berman,1 Dario G. Liebermann,2 and Joseph McIntyre3

1Department of Industrial Engineering and Management, Ben-Gurion University of the Negev, Beer-Sheva, Israel;2Department of Physical Therapy, Stantley Steyer School of Health Professions, Sackler Faculty of Medicine, Tel AvivUniversity, Tel Aviv, Israel; and 3Centre d’Etude de la Sensorimotricité, Institut des Neurosciences et de la Cognition, CentreNational de la Recherche Scientifique-Université Paris Descartes, Paris, France

Submitted 25 February 2013; accepted in final form 18 November 2013

Berman S, Liebermann DG, McIntyre J. Constrained motion controlon a hemispherical surface: path planning. J Neurophysiol 111: 954–968,2014. First published November 20, 2013; doi:10.1152/jn.00132.2013.—Surface-constrained motion, i.e., motion constraint by a rigid surface,is commonly found in daily activities. The current work investigatesthe choice of hand paths constrained to a concave hemisphericalsurface. To gain insight regarding paths and their relationship withtask dynamics, we simulated various control policies. The simulationsdemonstrated that following a geodesic path (the shortest path be-tween 2 points on a sphere) is advantageous not only in terms of pathlength but also in terms of motor planning and sensitivity to motorcommand errors. These stem from the fact that the applied forces liein a single plane (that of the geodesic path). To test whether humansubjects indeed follow the geodesic, and to see how such motioncompares to other paths, we recorded movements in a virtualhaptic-visual environment from 11 healthy subjects. The taskcomprised point-to-point motion between targets at two elevations(30° and 60°). Three typical choices of paths were observed froma frontal plane projection of the paths: circular arcs, straight lines,and arcs close to the geodesic path for each elevation. Based on themeasured hand paths, we applied k-means blind separation todivide the subjects into three groups and compared performanceindicators. The analysis confirmed that subjects who followedpaths closest to the geodesic produced faster and smoother move-ments compared with the others. The “better” performance reflectsthe dynamical advantages of following the geodesic path and mayalso reflect invariant features of control policies used to producesuch a surface-constrained motion.

constrained motion; geodesics; path planning

CONTROL OF PHYSICAL INTERACTION between the body and theenvironment is crucial in many tasks, e.g., writing, hammering,and cycling, where the environment constrains motion due tocontact with a rigid restraint. Everyday movements that areconstrained by a rigid surface, e.g., cleaning the floor, writingon a board, and carving a piece of wood, form a particular classof constrained motion in which the constraint is asymmetric.The hand is physically free to move up off of the surface, yetmaintaining contact is critical to success. The hand is alsophysically prevented from moving inside the surface, butexcessive forces against the constraint may lead to failure oreven destruction of the physical object. These factors make thecontrol of surface-constrained motion key for success in func-tional activities of daily living and of considerable interest forthe study of human motor behavior.

An oft-studied question in the field of human motor functionis that of trajectory planning. For the task of moving from onepoint to another, the task itself does not prescribe the path to befollowed by the hand, nor does it specify the time course of themovement along that path. In the case of free (unconstrained)movements of the hand, the problem of selecting a path for anygiven pair of endpoints is ill posed, with an infinite number ofpossible solutions. Several theories have been formulated re-garding how the central nervous system (CNS) resolves redun-dancy in the motor system, where there are many potentialsolutions to a given task. The minimum-jerk model, for in-stance, postulates that the derivative of acceleration is mini-mized over the course of the movement, and when the hand isotherwise unconstrained, it predicts bell-shaped velocity pro-files on straight-line paths (Flash and Hogan 1985). Thisoptimization happens to correspond to the shortest path be-tween the two points.

One can ask the same question about path planning ofmovements along a curved surface. In the case of a sphericalsurface, the shortest path between two points, i.e., the geodesic,is the shorter of the two arcs defined by the great circleconnecting them. By analogy with the problem of point-to-point movements in free space, one might therefore surmisethat humans will choose to move along the geodesic whenmoving from one point to another on a spherical surface, thusachieving an “optimal” (with respect to path length) endpointkinematic solution. In surface-constrained motion, however,planning endpoint trajectory kinematics may not be sufficient,and dynamics must also be taken into account. Combiningdesirable characteristics of the movement of the hand, e.g.,following a straight line, with constraints imposed by thesurface may cause conflicts because the control system be-comes overspecified. In such cases, attempting to follow apredefined path without taking surface characteristics intoaccount can be counterproductive. To successfully move alonga surface, the control system should instead “comply” withconstraints imposed by the physical constraint. This couldconceivably lead the subject to choose a different trajectorythan the shortest path between two points on the surface.Indeed, when moving the end of an inverted pendulum, i.e., afully constrained motion in a convex hemispherical surface,subjects seldom moved along geodesics, although they didmaintain minimum-jerk speed profiles along the chosen paths(Liebermann et al. 2008). However, when moving the handbetween points on a convex virtual hemispherical surface, i.e.,a surface-constrained motion in a convex hemispherical man-ifold, subjects tended to follow geodesics albeit with someresidual error (Sha et al. 2006). The question of how human

Address for reprint requests and other correspondence: S. Berman, Dept. ofIndustrial Engineering and Management, Ben-Gurion Univ. of the Negev, POBox 653, Beer-Sheva 84105, Israel (e-mail: [email protected]).

J Neurophysiol 111: 954–968, 2014.First published November 20, 2013; doi:10.1152/jn.00132.2013.

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subjects plan and execute movements along curved, rigidsurfaces therefore remains to be elucidated.

In this context, we asked how the physical interaction witha rigid constraint affects the choice of paths to be followed bythe hand. We addressed this question by simulating the dy-namics of various force profiles that could be used to move thehand from one point to another along different paths whilestaying in contact with a rigid, concave spherical surface. Wecompared these simulation results with measurements of actualmovements by human subjects performing the same task in avirtual haptic-visual environment. These experiments provideinsights into how the CNS takes the constraint into account andpotentially uses it to its advantage, e.g., to reduce the effortrequired to stay on the desired path and thus to perform fasterand smoother movements.

METHODS

For this study we defined a behavioral task in which human subjectswere asked to move an object with the hand along the inside, rigidsurface of a hemispherical bowl. We asked the question, “Whattrajectory would subjects follow to move from one point to another ifrequired to maintain contact with the interior surface of the bowl?” Togain insights regarding the possible solutions that subjects mightadopt, we first constructed a simplified mathematical model of thephysics of the task. We considered motion of a point mass subjectedto applied forces (e.g., forces exerted by the hand) and the forcesgenerated by the environment to keep the point mass on the surface.We used the results of these simulations to examine which trajectoriesmight be advantageous, in terms of kinematic properties or ease ofcontrol, and then used these simulation results to design and interpretthe experiments performed with human subjects.

Model of the Physical System

We simulated the dynamics of a point mass subjected to appliedforces F(t) and constrained to remain at a fixed distance from a definedcenter (Fig. 1). The equations of motion describing this system aregiven by

mp � F(t) � Fc(t), (1)

where m is mass, p � [x, y, z] is the position vector in a right-handcoordinate system, and the constraint force Fc(t) is given by

Fc(t) � A�[mb � AF(t)], (2)

where A � [x y z] and b � x2 � y2 � z2 are derived from the cons-traint equations describing movement confined to the surface of asphere (x2 � y2 � z2 � r2, where r is the sphere’s radius) and thesuperscript � indicates the Moore-Penrose pseudo-inverse. A descrip-tion of how these equations are derived from physical principles isprovided as an APPENDIX. The input variable of this dynamical modelwas a feedforward time series of force vectors, F(t), that would beapplied by the hand to the point mass in order to make a movement.In our simulations, the mass was arbitrarily set to 0.5 kg, and afourth-order Runge-Kutta method (implemented using MatLab v.11;The MathWorks, Natick, MA) was used to integrate Eqs. 1 and 2 tofind the trajectory (path and velocity) resulting from a given forceprofile.

The input to the model was a time series of the applied forces F(t),and the simulations performed were carried out as open loop withrespect to these applied forces. Because of the equality constraint thatis imposed by Eqs. 1 and 2, the simulated point mass was forced to lieon the spherical surface, regardless of the applied forces. Yet theactual constraint in our experiment was unbalanced; movement in onedirection was strictly impeded by the surface (one could not move intothe rigid surface), whereas movement in the other direction wascontrolled by the subject (one could break away from the surface,even if subjects were instructed not to). Equations 1 and 2 do notdistinguish between these two; they impose strict conformation to thesphere such that the constraint force pulled the point mass in case ofinsufficient applied force in the outward radial direction. Thus asimulation resulting in an outward radial constraint force wouldindicate a movement violating the constraint that in the physical worldwould cause the point mass to break away from the surface. Wetherefore examined the results post hoc for radially outward constraintforces indicating that the point mass would have lost contact with thesurface for that particular force profile (i.e., a failure to perform thetask).

To gain insight into the control problem faced by the CNS in orderto move along the surface of the sphere, we first simulated an ad hocstrategy in which the subject would attempt to move the hand along astraight line from start to end, initially without regard to the constraint.The modeled task of a point-to-point movement requires motiontermination at the target; thus the model had to simulate both accel-eration and deceleration. For instance, to achieve a minimum-jerkvelocity profile for unconstrained straight-line motion, the appliedforce profile followed Eq. 3:

F(t) �60m(pf � p0)

T2 · (� � 3�2 � 2�3), (3)

where p0 and pf are the initial and final positions, respectively; T is themovement duration, and � is the normalized time (t/T). We simulatedthis initial force profile and examined the forces that would begenerated by the constraint and the resulting hand path. We then usedthe results of this initial simulation to determine changes in the forceprofile that would be sufficient to maintain contact with the surfacewhile moving along this same path.

Force profiles that would generate movement along other paths onthe surface were computed as well, taking into account the inversedynamics of the mechanical system (in this case the simple pointmass) and the desired applied forces against the surface. One canfollow an infinite number of possible paths when moving from onepoint to another on the surface of a hemisphere. We considered asubset of such movements that can be described by the intersection ofa plane with the contour of the hemisphere, where both start and end

F(t)

Fc(t)

X Target

Fig. 1. A point mass moving on a spherical surface. F(t) is the force directingthe point mass toward the target (�), and Fc(t) is an additional force radial tothe surface assuring the point mass stays on the surface.

955CONSTRAINED MOTION CONTROL ON A HEMISPHERICAL SURFACE

J Neurophysiol • doi:10.1152/jn.00132.2013 • www.jn.org

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points lie within the plane (Fig. 2B). Even within this subset, there stillexist an infinite number of paths and each path is determined by therotation of the plane around the line connecting the start and endpoints. We considered three key examples of paths defined in thismanner. 1) The plane containing the center of the sphere and themovement start and end points generates the geodesic path. 2) A planerotated 90° around the start-end axis will generate a path that forms astraight line in visual space when viewed along the polar axis.3) When both start and end points lie at the same elevation angleabove the rim of the bowl, the plane perpendicular to the polar axiswill generate a circular arc on the surface of the hemisphere that runsparallel to the rim.

Note that the simulation of the open-loop behavior of thissystem was used not in an effort to reproduce the human behavior,but rather to highlight the control problems that result from theinteraction with a curved, rigid constraint and from the differentchoices that could be made about the path to be followed along thesurface. To illustrate the advantages and disadvantages of thechoice of hand path along the surface, we introduced small dis-crepancies from the ideal force profiles that would generate thesedifferent trajectories to demonstrate how sensitive the mechanicalsystem is to errors in the feedforward command.

Human Motion Experiment

Subjects. Eleven healthy subjects (age 23–56 yr, mean 34.1 yr, 6men) participated in the experiment. Subjects had normal or correct-ed-to-normal vision with no neurological, sensorimotor, or orthopedicimpairments. Ten subjects were right-hand dominant and one wasleft-hand dominant, but all subjects performed the experiment usingtheir right hand. All gave informed consent after receiving explana-tions about the study protocol. However, all were naive about thespecific hypotheses to be tested. The protocol was approved by theComité de Protection des Personnes (Institutional Review Board) “Ilede France II” (Project no. 2010-7).

Apparatus and procedures. The experimental task made use of aparallel three-dimensional (3D) haptic device (delta.3; Force Dimen-sion) to simulate force interactions between the hand and a rigidhemispherical surface (Fig. 3). The delta.3 device produced a maxi-mum force of 20 N, and the effective stiffness of the simulated surfacewas 2,000 N/m.

Subjects grasped the endpoint of the haptic device and were askedto make rapid movements between targets by sliding along the interiorof the hemisphere. The screen showing the hemispherical surface waspositioned 2 m away in front of the subject’s visual field in aneye-centered configuration. Depth illusion was created using shadingbased on a directional lighting coming from above the subject’s leftshoulder. Subjects performed hand movements on the concave hemi-spherical surface simulated by the haptic device that matched thevisual image. The orientation of the haptic device was aligned with theorientation of the hemispherical surface on the screen (parafrontal tothe subjects’ viewpoint while the handle was placed to the right of thesubject, aligned with the subjects’ right forearm). The position of thesubject’s hand within the virtual bowl created by the haptic devicewas indicated relative to the visual representation of the bowlon the screen by a gray spherical cursor. When visible, the cursortracked the 3D position of the hand, as measured by the haptic device.

Each individual movement was executed as follows. A yellowconical marker appeared on the interior surface of the hemisphere,signaling the movement starting point. The subject was instructed tomove the hand along the surface and to place the spherical cursor atthe position indicated by the cone. When the hand was correctlypositioned at the starting position (position error �1 cm), the conicalmarker turned green. If the hand was held at the start position for atleast 1 s, the green marker disappeared and a red conical markerappeared at one of the five target positions, marking the targetendpoint of the upcoming movement. The red target marker wasvisible for 1 s. To avoid paying excessive attention to final accuracy,the subject was instructed to start moving toward the target only after

Fig. 2. A: a simulated path drawn over pre-calculated curves for the 3 trajectories (geo-desic, straight-line projection, parallel arc).B: the intersection of a hemisphere at 30°with the 3 plane alternatives (straight-lineprojection, geodesic, parallel arc).

30° 60°A B

Fig. 3. Experimental setup. A: the subject isholding the handle of the 3-dimensional (3D)Force Dimension system while viewing thevirtual sphere on the screen. B: target pointsat 30° and 60°. The center was also includedas a target in both cases.

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the red target marker had disappeared. The subject was instructed tomove quickly and accurately to the remembered target position whilemaintaining contact with the hemispherical surface. Each movementset included several movement blocks. The movements in each blockwere executed consecutively, and every movement started at the targetpoint of the previous movement. Thus, 3 s after the target markerdisappeared, it reappeared, cueing the next starting point. If the handwas already within the 1-cm tolerance around the new startingposition, the marker would turn green and the next target would becued after 1 s. If the previous movement terminated more than 1 cmfrom the indicated target position, the marker appeared yellow and thesubject was instructed to move the hand to the indicated position andhold steady to trigger the subsequent trial.

Before the initiation of each set of trials subjects were allowed topractice for a few minutes until they felt comfortable with theexperimental protocol. During practice trials, the subjects receivedverbal feedback regarding the force they were applying on the hemi-spherical surface based on operator screening of force values (exces-sive, i.e., forces above 10 N; sufficient; etc.). The range of acceptableforces was above zero (required for maintaining contact) and belowthe haptic device limit of 25 N. Trials on each hemisphere (set) wereorganized in 5 blocks of 20 movements each. Subjects were allowedto rest at will between blocks. The position of the handle and the radialforce applied by the haptic device to the hand were recorded. Datawere sampled at a variable rate (due to the non-real-time nature of theWindows operating system), but all samples were spaced by no morethan 1 ms (i.e., sampling frequency �1 kHz).

Experimental design. The main experiment comprised 2 sets of 100movements performed on a hemispherical surface with a 6.33-cmradius. Each set included movements between five targets: center, left,right, high, and low. Within one set of trials the four peripheral targetswere located at 30°, whereas for the other set they were at 60°elevation with respect to the “equator” represented by the rim lying inthe frontoparallel plane, as shown in Fig. 2A. Target order wasrandomized within each set. Movements were performed between allpossible pairs of targets with at least three repetitions for each pair.The number of movements per pair was similar, but not identical,because each movement started at the end target of the previousmovement.

The four peripheral targets at 30° elevation were farther apart fromeach other in Cartesian space than the same four targets located at 60°(7.8 vs. 4.5 cm, respectively). To test whether any differences in pathsbetween the two sets of targets were due to this change in distance,rather than being due to the elevation on the sphere, subjects per-formed a third set of 100 movements to targets located at 30°elevation on a 3.66-cm-radius hemisphere, for a 3D inter-targetdistance of 4.8 cm.

Analysis. Our analyses focused on what paths were chosen by thesubjects between two points on the surface and on the effects that thechoice of path might have on other parameters such as the speed ofmovement or the smoothness of the trajectory. We analyzed the datain terms of the movement of the hand and the forces generated by thehaptic device to simulate the contact with a rigid bowl.

PREPROCESSING. Recorded position and force data were resampledto form time series at a constant rate of 1 kHz and then low-passfiltered (Wiener filter) with 6-Hz cutoff. Velocity was calculated bydifferentiating the position profiles. Movement onset (offset) wasdefined as the instant in time when the tangential velocity went above(below) 10% of the peak tangential velocity. Movements for whichmovement onset was found prior to the start cue were discarded(3.78% of the movements).

PLANE OF MOVEMENT. To characterize paths followed by the handalong the sphere, we computed for each trial the best-fit plane thatcontained the movement and further computed the angle between thenormal to the plane and the polar axis. To calculate such a plane, weapplied principal component analysis. We thus computed the 3 � 3

matrix of covariation of the movement around the mean and computedthe eigenvalues and eigenvectors of the covariance matrix. Theamount of variance accounted for by the first two principal compo-nents was taken as a measure of how well the movement could berepresented by the intersection of a plane and the sphere. Theeigenvector corresponding to the smallest eigenvalue (i.e., the thirdprincipal component) is the normal to the best-fit plane. We computedthe angle � between the normal vector and the XZ plane around theline connecting the start and end of the movement to describe theinclination of the movement plane.

We compared the orientations of the fitted planes to the orientationsof the three different planes (paths) considered in the simulation study.The plane angle was 0° for the arc in the frontoparallel (equatorial)plane and 90° for a straight line in the frontoparallel plane. The planegenerating the geodesic for oblique movements (i.e., movements thatdid not start from, end at, or pass through the center target) had aninclination of 39.2° between targets at 30° elevation (i.e., closer to therim) and 67.8° for movements between targets at 60° elevation (i.e.,closer to the pole). For movements between two targets lying on thecardinal axes in the frontoparallel plane (i.e., movements that started,ended, or passed through the center), the plane angle was 0° for theparallel arc and 90° for both the straight line and the geodesic.

To test whether subjects took into account the target elevation whenplanning the movement, we computed the movement plane modula-tion (MPM) as the difference between the movement plane angle �for oblique movements between targets at 30° and oblique movementsbetween targets at 60° on the large (6.33 cm) hemisphere (MPMelev).As a control, we also computed MPM between two different-sizedhemispheres for oblique movements between targets at the sameelevation (30°) (MPMradii). Statistical analysis of plane tilts and MPMwas conducted using the circular statistics toolbox for MatLab (Be-rens 2009).

ADDITIONAL MOVEMENT FEATURES. Subjects were instructed tomove rapidly and maintain contact with the surface of the spherethroughout the movement. Based on our simulation analysis, wepredicted that subjects who chose to follow the geodesic path tothe target position would have an easier time staying on path whilemaintaining contact, compared with any other choice of path. Becausesubjects could correct the movement on the basis of visual feedback,final reach accuracy was not used as a test of this hypothesis. Instead,we concentrated on parameters related to the speed and smoothness ofthe trajectory and on the interaction forces between the hand and theconstraint.

We computed the average speed (AS) and the zero force ratio(ZFR), i.e., the duration for which the radial force applied to thesphere was zero divided by the movement duration. A non-zero ZFRindicates difficulties in maintaining contact. Unless otherwise noted,we excluded from the analysis trials with a ZFR �0.5, which wastaken as an indicator of complete failure to perform the task. Onlycontact with the surface was required, rather than a specified pressure.We therefore computed the average force (AF) over the entire path,without characterizing variations in force within the trial. A loweraverage force can indicate a more efficient movement. We computedthe “straightness” of the path by computing the number of themovement plane crossing (MPC), i.e., the number of times the subjectchanged from moving on one side of the plane to the other and stayedat that side for at least 80 ms. Finally, we computed the number ofpeaks in the tangential velocity speed profile (SP). Both MPC and SPare indicators of movement smoothness; analogous to the straight-linepaths with bell-shaped velocity profiles that are typically observed forunconstrained point-to-point movements (Abend et al. 1982), a max-imally smooth trajectory on the sphere would have zero plane cross-ings (i.e., the entire movement would be in a single plane) and asingle-peaked velocity profile.

STATISTICAL ANALYSES. Based on the outcome of our numericalsimulations (see RESULTS), we concluded that movements destined to



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follow a path other than the geodesic would be more sensitive toerrors in the motor commands and thus should be more difficult tocontrol. We hypothesized that attempting to follow a path other thanthe geodesic would result in movements that are less smooth and moreprone to corrective movements and that subjects might therefore favormovement paths on or close to the geodesic.

Initial inspection of the raw hand paths indicated that for move-ments along the cardinal directions, all subjects followed hand pathscorresponding to movement planes relatively close to 90°. For obliquemovements, however, the average movement plane for differentsubjects varied considerably across the full range of possible tiltangles, although the within-subject variability was similar for bothtypes of movements (see RESULTS for details). We therefore hypothe-sized that individual subjects did not necessarily choose to follow theoptimal path for the oblique movements. Nevertheless, we set out totest whether those subjects that endeavored to move on or near thegeodesic would perform better, in terms of velocity and smoothness ofthe trajectory, than those who chose other paths.

Because any systematic variation could be nonmonotonic, a simpleregression analysis of performance measures vs. plane angles wouldnot be appropriate. We instead defined groups of subjects according totheir movement plane behavior and then looked to see if otherperformance measures correlated with that grouping. With that intentin mind, we used k-means clustering (Lloyd 1982) to categorizesubjects according to their respective average movement plane foroblique movements on the 6.33-cm-radius sphere, computed sepa-rately for the 30° and 60° targets. In other words, the variable vectorfor the k-means analysis was composed of the average movementplane orientation for each target elevation. As we will show (seeRESULTS), this analysis divided subjects into three different categories,which we termed the arc group, the line group, and the intermediategroup, based on the resemblance of each group’s behavior withrespect to the ideal.

To understand how the plane-tilt behavior differed between theidentified groups, we applied ANOVA to the movement plane mod-ulation between the 30° and 60° targets (MPMelev), with quadrant(upper left, upper right, lower left, lower right) and movementdirection (upward, downward) as within-subject factors and tilt group(arc, line, intermediate) as a between-subjects parameter. BecauseMPMelev is based on the same plane-tilt data used by the clusteranalysis, this test constitutes a complimentary post hoc assessment ofthe clustering. We compared the results to a similar ANOVA appliedto the movement plane modulation between hemispheres of differentradii for targets at 30° (MPMradii).

We then used the above categories (arc, line, intermediate) as anindependent factor in a mixed-model ANOVA to determine whetherperformance parameters other than the movement plane (i.e., AS, AF,MPC, SP) varied as a function of the movement plane behaviorexhibited by the different groups of subjects. For oblique movements,where the movement plane for the geodesic and for the visualstraight-line paths differed, movement parameters were subjected to

ANOVA with quadrant (upper left, upper right, lower left, lowerright), movement direction (upward or downward), and target eleva-tion (30° or 60°) as within-subject factors and tilt group (arc, line,intermediate) as a between-subjects factor. Finally, we applied amixed-model ANOVA with type (center-out or out-center), move-ment direction (upward, downward, leftward, rightward), and targetelevation (30° or 60°) as within-subject factors and tilt group (arc,line, intermediate) as a between-subjects factor to movements made inthe cardinal directions to and from the center on the 6.33-cm sphere.These movements were analyzed (and not movements passingthrough the center of the sphere) because their paths are of compara-ble lengths to the oblique movements. Each ANOVA was followed bypost hoc analysis adjusted using Bonferroni correction whenever maineffects or interactions reached significance.

In addition to the ANOVA tests described above, we used Pear-son’s correlation coefficient to test for correlations of each of themovement parameters between those measured for oblique move-ments and those measured for movements in the cardinal directions.We used these correlation analyses to ask whether any differencesbetween tilt groups are the result of criteria applied during movementplanning to all movements, or whether they could be attributed todiffering mechanical effects for the paths actually followed. In thecase of the cardinal directions, any difference between the interme-diate group and the line group should disappear, since the straightlines in the visual plane and the geodesic path were one and the samefor these pairs of targets.

RESULTS

Simulation Results

To understand how the dynamics of the interaction with aspherical surface might affect an endpoint trajectory, we sim-ulated a number of different force profiles that could be used todrive a point mass from the initial position to the target. First,we looked at the characteristics of the force profiles fordifferent possible paths along the surface. We then looked athow sensitive movements along different paths would be toerrors in the computed forces. We concentrated our efforts onoblique movements between two points on the surface locatedat the same elevation above the rim, because these movementsallowed for three possible “ideal” solutions: a straight line invisual space, a circular arc parallel to the rim of the bowl, andthe geodesic path between the two points.

The optimal path. Figure 4A shows a scaled version of theapplied forces that would drive the mass along a minimum-jerkvelocity profile in a straight line from start to end. Since theendpoints lie in the frontoparallel plane, the applied forcevaries only in Y and Z (because X was aligned with the sagittal

A

C

B

D

Fig. 4. Applied force (Fx, Fy, Fz; quadranttop) and constraint force (Fc; quadrant bot-tom) where shaded regions represent pulling(rather than pushing) forces. The appliedforces were scaled straight-line minimum-jerk force profile (A), scaled straight-lineminimum-jerk force profile with an addi-tional constant amplitude radial force (B),minimum norm force (C), and minimumnorm force with an additional constant am-plitude radial force (D).



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axis in our experiment). Note that the force profile that wouldgenerate a minimum-jerk profile for an unconstrained motionalong the straight line will stop short of the desired targetposition when constrained to lie on the sphere. However, if theamplitude of the minimum-jerk profile is scaled appropriately,as shown in Fig. 4A, this targeted movement can be achieved.

The combination of the constraint equations and a forceprofile designed to move directly toward the endpoint results ina movement on the sphere that follows the geodesic path (seeAPPENDIX). Simply applying the displacement force to the massis not, however, a valid solution to the real task of sliding alongthe curved surface, because the constraint force is actuallypulling the mass onto the surface (the constraint force ispositive). If we were to simulate an inequality constraint, thehand would leave the surface to follow the straight-line, point-to-point path in free space. To rectify this, one could add aconstant bias force in the radial direction without preciselytaking into account the force required to satisfy the constraint(Fig. 4B). If the constant radial bias force is always greater thanthe force required for pulling the hand to the surface, contactwith the surface will be maintained. Thus one strategy wouldbe to apply forces in the frontoparallel plane to drive the handin a straight line in visual space, and to add to that a constantradial force so as to assure contact with the surface in depth.However, with this strategy the constraint force will vary alongthe trajectory, with minima near the start and end of themovement and a maximum at the midpoint.

One can use the applied and constraint forces depicted inFig. 4A to compute the “minimum norm” modification to theapplied force that allows the mass to maintain contact with thesurface. Adding the constraint force (which is in the radialdirection) Fc(t) predicted by Eq. 2 to the original force profileF(t) generates a new force profile, Fbis(t) � F(t) � Fc(t), thatwould cause the mass to skim along the sphere following thegeodesic (Fig. 4C). Because this modification mimics the net

forces that would be achieved by a passive mechanical system,Fbis represents the force profile closest to the original profilethat will maintain contact with the surface. This is a validsolution, since the constraint force is never positive in thissituation, but it is not realistic in the sense that there would beno interaction forces at all with the surface. If, however, thehand applies an additional constant bias force in the radialdirection (outward with respect to the center of the sphere), thegreat circle will still be followed with a constant constraintforce (Fig. 4D).

Alternative paths. The geodesic connecting two points onthe sphere is not the only path that can be followed. Indeed,with proper knowledge of the surface geometry, and precisecontrol of the direction, amplitude, and timing of forces ap-plied, any path can be followed. The geodesic path is, however,more robust when the computed applied forces are erroneous,as we will show in the following simulations.

We considered three ideal trajectories on the surface of thesphere, all of which can be described as the intersection of aplane with the sphere 1) along a circular arc parallel to the rimof the sphere, 2) along a straight line in visual space, and3) along the great circle (geodesic) from start to end target (asdescribed above). For each, we computed a continuous ap-plied-force profile in three dimensions that would cause thepoint mass to move along the specified trajectory with aminimum-jerk velocity profile. When these profiles are numer-ically integrated in our simulation, with the constraint takeninto account, the nominal trajectory is achieved with zerointeraction force. We then added to each profile a constantforce precisely in the radial direction (normal to the surface) ateach point along the nominal surface (Fig. 5). Trivially, in allthree cases, the nominal trajectory is achieved with a constantmagnitude interaction force with the surface, because we usedcomplete and exact information to compute the applied forceprofile. It is interesting to note, however, that only for the great

GeodesicB CA Straight-line projectionParallel arc

Fig. 5. Applied force (top), constraint force (middle), and applied force direction with movement plane (bottom), where the 3D force profiles were parallel arc(A), geodesic (B) and straight-line projection (C). All profiles include a constant radial component.



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circle path do the force vectors in the radial direction and theforce vectors toward the target all lie in the plane of movement.For the parallel arc and the visual straight line, these forcevectors do not even lie in a single plane over the course of themovement, much less within the plane of motion itself (Fig. 5,bottom). This fundamental difference between the three differ-ent nominal paths is reflected in their sensitivity to errors in thecomputed force command, as we will demonstrate in thefollowing.

ERRORS IN FORCE MAGNITUDE. To produce the trajectories andforce profiles shown in Fig. 5, the applied force must beprecisely programmed in terms of amplitude and direction.Consider what happens if the applied radial force is applied ateach moment in the correct direction, but with the wrongamplitude. For the geodesic curve, the point mass will followthe intended path, but the path will undershoot (� � 1) orovershoot (� � 1) the target (Fig. 6A). However, for both thestraight-line projection and the parallel arc, amplitude errorswill take the point mass off of the desired path, for thestraight-line projection inward (� � 1) or outward (� � 1), andsimilarly but in opposite directions for the parallel arc.

ERRORS IN FORCE DIRECTION. Provided that the precomputedapplied force is normal to the surface along the nominal

trajectory (whatever that trajectory is), the interaction of theapplied radial force with the concave curved surface imparts astabilizing effect on the movement dynamics. Figure 6B de-picts effects of an applied radial force that is rotated around theX-axis �10° off of the true radial direction (normal to thespherical surface). Only when the radial force is appliedperpendicularly to the surface of the sphere does the hand stayon the desired path. If not, the applied radial force has theeffect of moving the hand upward and outward when the forceis incorrectly orientated outward in the radial direction inthe YZ plane and inward toward the pole of the sphere when theforce is incorrectly directed forward in depth along the polaraxis. Note that in this case, although the geodesic curve is morerobust to error than the parallel arc, it does not outperform thestraight-line projection.

UNMODELED FRICTION. Figure 7A shows the consequences ofunexpected resistance due to friction, modeled either as dy-namic Coulomb friction (i.e., a constant magnitude force in thedirection opposite the instantaneous velocity) or as viscousfriction (i.e., a force proportional to, but in the oppositedirection, of the instantaneous velocity). The effect of suchfriction is to slow the movement with respect to the nominalplan. If the applied forces are preprogrammed, the mass will

A

B

Geodesic Straight-line projectionParallel arc

Fig. 6. Testing path sensitivity for 3D trajec-tories: parallel arc (left), geodesic (middle),and straight-line projection (right). Trajecto-ries are viewed from an oblique angle on theside so that errors in depth can be discerned.A: amplitude sensitivity: top, � � 1.2; bot-tom, � � 0.8. B: direction sensitivity: radialforce amplitude � 0.5; rotation about theX-axis: top, �10°; bottom, �10°.

A

B

Geodesic Straight-line projectionParallel arc

Fig. 7. Testing path sensitivity for 3D trajectories:parallel arc (left), geodesic (middle), and straight-lineprojection (right) viewed from an oblique angle.A: friction sensitivity: top, Coulomb; bottom, viscous.B: submovements.



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not be at the expected location when the force at any time t isapplied. Because all the programmed forces for the nominalgeodesic trajectory lie in the plane of movement, any devia-tions from the nominal trajectory resulting from the timingerrors will also lie in the plane. Again, the great circle is lesssensitive to errors for unmodeled forces of this type. The massundershoots the target but remains on the desired path. Not sofor the parallel-arc and visual straight-line paths. For theparallel arc and visual straight line, without real-time recalcu-lation of the force direction to take into account the deviationoff the nominal path, the actual and desired trajectoriesdiverge.

PLANAR SEGMENTATION OF THE FORCE COMMAND. The advan-tages of the geodesic path noted above stem from the fact thatthe applied forces required to stay on this path all lie in a singleplane, that of the geodesic path itself. In essence, one canexploit the fact that extraneous forces, directed perpendicularto the surface, have no effect on the trajectory of the mass. Thischaracteristic can be exploited in other ways to simplify theplanning of the geodesic movement. For instance, one canimplement the nominal force command of a geodesic by afinite number of planar segments. Indeed, one can achieve thegeodesic movement with a single such segment consisting of astraight-line driving force plus a radial bias force within themovement plane of constant amplitude, as shown in Fig. 4.Despite the approximation to the ideal force profile, the massremains on the geodesic path, albeit with large variations in theinteraction force with the surface. The parallel-arc and visualstraight-line paths are not so easily approximated. If only twoplanar segments are used to approximate the ideal force com-mand, the mass will deviate significantly from the nominalpath (Fig. 7B). The planarity of the force command for thegeodesic lends itself to other ad hoc strategies such as bang-bang control, where two force impulses are sufficient to startand stop the movement along the desired path. The continuallyvarying direction of the applied force required to stay on theparallel-arc or visual straight-line paths does not allow for such“ballistic” control of the movement on the sphere.

Summary of simulations. The analysis of oblique movementsdemonstrated the inherent benefits of following the geodesicbetween two points on the surface. Other pairs of targets werealso considered, including starting and ending at differentelevations, starting and ending points on the rim of the bowl,and starting and ending paths that lie on a line passing throughthe center of the sphere. The results of these simulations (notshown) were the same: the geodesic was more robust to errorsin the precomputed motor command.

Human Motion

Equipped with the insights provided by the simulation ofvarious paths between two points on a hemisphere and theforce profiles required to produce them, we then analyzed theempirical data from the experiments performed by humansubjects. The main analysis was focused on the oblique move-ments between targets at either 30° or 60° elevation on the6.33-cm hemisphere. For these movements, the parallel arc, thevisual straight line, and the geodesic differed from one anotherand so allowed for the greatest level of discrimination in termsof the path chosen by the subject. As a control we alsocompared oblique movements between targets at 30° elevation

on the larger (6.33 cm) and smaller (3.66 cm) hemispheres, andin a separate analysis, we quantified trajectory characteristicsfor movements along the cardinal directions (up, down, left,right) that started, ended, or passed through the center target.

Qualitative observations. When given the option to followthe path of their own choosing, subjects produced a variety ofmovement strategies between oblique targets in terms of handtrajectories. Figure 8 shows raw data collected from threetypical subjects. Plots of the hand trajectory are shown as seenfrom the viewer’s vantage point (frontoparallel plane view) andfrom above (horizontal plane view). Red and blue lines showhand paths for movements in the upward and downwarddirections, respectively. Alongside the position traces, velocityand force traces are depicted for movements in an obliquedirection for one set of targets.

These three particular subjects were chosen to illustrate thefull range of path choices that we observed for oblique move-ments in our experiments. Subject 1 (Fig. 8, top) producedtrajectories that, when projected into the visual plane, formedcircular arcs. One can see from the top view, however, that thehand did, in general, dip inside the bowl (i.e., out of thefrontoparallel plane) in the middle of the movement, only tomove back out again to rejoin the frontoparallel plane contain-ing the starting and ending target positions. These paths werenot, therefore, perfect arcs in the visual plane. Subject 3 (Fig.8, bottom) manifested a different overall strategy, performingmovements that, when projected into the visual plane, tendedto follow a straight line connecting the starting and endingpoints. To stay on the surface while attempting to follow astraight-line visual projection, however, the hand must neces-sarily make a significant movement in depth. Subject 2 (Fig. 8,middle) manifested a combination of the two strategies ofsubjects 1 and 3, producing paths at 30° on either hemispherethat formed arcs in the frontoparallel plane while producingprojected paths for targets at 60° that were somewhatstraighter. From the velocity and force traces it seems thatsubject 3 used more force and moved slower than the other twosubjects, whereas subject 2 seemed to move fastest.

Quantitative measures of task success. The proportion oftrials where contact with the surface was never lost (ZFR � 0)and the proportion of the trials where contact was maintainedfor at least half the movement time (ZFR � 0.5) were calcu-lated. One subject (subject 5) had difficulties maintainingsurface contact throughout the movement, exhibiting manymovements with ZFR � 0 (78.2%) and a high rate of trialswith contactless durations (average ZFR � 42.3%). We there-fore excluded subject 5 from further analysis. All other sub-jects succeeded in maintaining contact with the surfacethroughout the movement (ZFR � 0) in most of their move-ments (median success rate 93.6%). For these subjects, whenloss of contact did occur it was usually short (average ZFR �3.1%). Success at maintaining contact for at least half of themovement (ZFR � 0.5) was very common (median successrate 99.2%). We excluded individual movements for whichZFR exceeded 0.5.

Hand paths. We characterized the choice of path adopted byeach subject by computing the movement plane that best fit thedata from each trial. The average total variance explained bythe first two principal components extracted during the plane-fitting analysis (see METHODS) was 99.84 � 0.15% (mean �



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SD), which attests to the validity of fitting a plane to themovement trajectories.

Consider first the movements along the cardinal axes, i.e.,movements that started, ended, or passed through the centraltarget. For these target combinations, the geodesic path and thepath that forms a straight line in the visual plane are one andthe same. Here we observed fairly consistent behavior acrosssubjects, with most movement plane orientations close to 90°

(92 � 27°). Paired t-tests demonstrated, however, a small butstatistically significant difference between upward and down-ward movements (P � 0.05). The downward movements didnot deviate significantly from 90° (mean 89.7°, 95% confi-dence interval [85.8°, 93.5°]), but the upward movements wereslightly biased to the right (mean 83.9°, 95% confidenceinterval [81.5°, 86.4°]). Plane orientations of horizontal move-ments dipped below the horizontal plane (mean 96.4°, 95%

Force [N]

Velocity [cm/s]

Control30° 60°

Fig. 8. Recorded data from 3 subjects from the arc (top), intermediate (middle), and line groups (bottom), showing front (frontoparallel plane) and top views(horizontal plane) of the hand trajectory for the 2 elevations (30° and 60°) on the large sphere (6.33 cm) and force and velocity traces (far right) for upwardmovements on the upper left quadrant of the large sphere. Small insets in the column marked “control” show data for movement to targets at 30° on the smallsphere (3.66 cm). Red lines indicate upward movements, and blue downward. All movements are shown for the large sphere, elevation 30°. For clarity, onlyoblique movements are shown for the control and 60° target configurations.



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confidence interval [90.6°, 102.2°]) and did not differ signifi-cantly between leftward and rightward movements. Yet,whereas leftward movements did not significantly differ from90° (mean 94.2°, 95% confidence interval [85.9°, 102.6°]),rightward movements were biased toward a dip below thehorizontal plane (mean 98.5°, 95% confidence interval [90.4°,106.5°]).

The paths followed by the hand for the oblique movementsvaried much more between subjects. Examination of the aver-age movement plane inclination for each subject on eachtarget/sphere configuration revealed that a large range of av-erage plane inclinations could be observed, from 15° to 93°,with an overall average of 50° and a standard deviation of�40°. Note, however, that each subject tended to produceoblique movements within a more limited range of movementplane angles (average standard deviation of movement planesby subject 28 � 9°), similar to the within-subject dispersion forthe cardinal directions (27 � 7°). In other words, subjectsmoved consistently near to their own nominal trajectory, evenif different subjects chose vastly different paths, on average,between the oblique targets. Note that some subjects showedintermediate average plane inclinations that could be inter-preted as representative of motor plans meant to follow thegeodesic path.

To test whether the choice of nominal movement planeaffected the control of the trajectory, as predicted by thesimulations, we divided subjects into groups by conducting ablind separation using the average movement planes for 30°and 60° targets as the variable vector in a k-means clusteringalgorithm (see METHODS). When the subjects were clustered intothree groups, the percentage of variance explained was 92%,whereas when the subjects were clustered into two groups, thepercentage of variance explained was only 79%. We thereforeset the number of clusters to three (Fig. 9). The algorithmclustered the subjects into consistent groups, which we termedthe arc group, the line group, and the intermediate group.

Note that the clustering analysis was based on the planeangle data for two different target elevations, but the arc lengthbetween the targets at the two elevations also differed. As such,the identified groups could reflect differences in average planeangles due to arc length, or the distinction between groupsmight, in part, have been based on plane angle differencesbetween target elevations. We therefore asked whether indeedthe different groups of subjects took into account the elevationangle of the targets on the sphere when performing oblique

movements. We applied ANOVA post hoc to the clusteranalysis to ask whether movement plane modulation betweentargets at 30° vs. 60° for the 6.33-cm hemisphere variedbetween groups (MPMelev, Fig. 10). As a control condition, weconducted a separate ANOVA on the change in plane angle for30° targets between the 3.66-cm and 6.33-cm hemispheres(MPMradii). These targets differ in arc length (the arc length forthe 30° targets on the 3.66 sphere is similar to the arc length forthe 60° targets on the 6.33 sphere) but not in elevation.

Fig. 9. Movement plane tilt. A: mean tilt for each subject for 30° (blue) and 60°(red) targets on the large sphere (6.33 cm; main experiment) and the groupingresulting from k-means analysis. B: mean tilt for each group for targets at 30° (blue) and 60° (red) on the large sphere (6.33 cm) and for the control conditionwith targets at 30° on the small sphere (3.66 cm; green).

Fig. 10. Average movement plane modulation (MPM) for each subject group.Left: MPM between targets on the large sphere (6.33 cm) at 2 different targetelevations, 30° and 60° (MPMelev); right: MPM between targets at the sameelevation (30°) on spheres of 2 different radii, 6.33 and 3.66 cm (MPMradii).Top: oblique movements; bottom: cardinal movements (to and from thecenter).



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• MPMelev between target elevations was positive (tilt atelevation 60° higher than tilt at elevation 30°) for all threegroups such that the 95% confidence interval did notinclude 0. MPMelev between different target elevationsdiffered, however, between the tilt groups (F � 8.32, P �0.001); MPMelev of the intermediate group was higherthan that of both the arc and the line groups (Parc � 0.001,Pline � 0.015). MPMelev of the arc group did not differfrom that of the line group.

• MPMradii between different sphere radii for targets withthe same elevation was not significantly different fromzero (the 95% confidence interval included 0) and did notdiffer between the tilt groups.

We then looked to see if the identified groups performeddifferently in terms of performance parameters other thanmovement plane tilt. Because the division of subjects intogroups reflects a choice of path planning strategy, we testedperformance indicators (AS, AF, MPC, and SP) of movementsto get insights regarding the difficulty of performing the task

under each strategy, where MPC and SP were taken as mea-sures of movement smoothness. Mean values for the obliquemovements are presented in Fig. 11, top. A mixed-modelANOVA was performed with subject as a random effect, targetelevation, quadrant, and movement direction as the within-subject factors, and tilt group as the between-subjects factor.Again, the analysis was followed by a pairwise Bonferroni-corrected post hoc analysis of tilt group.

• The overall speed of movement differed between thegroups (F � 24.32, P � 0.001), where AS for theintermediate group was higher than AS for both the arcand the line groups (Parc � 0.001, Pline � 0.001). AS ofthe arc group did not differ from that of the line group.

• The force applied against the constraint was relativelyconsistent across subjects: AF did not differ between thetilt groups.

• The planarity of the movements varied as a function of tiltgroup. MPC differed between the groups (F � 11.07, P �0.001), where the MPC of the line group was higher than

Fig. 11. Average speed (AS), average con-straint force (AF), number of movement planecrossings (MPC), and number of peaks in thetangential velocity speed profile (SP) as a func-tion of target elevation and tilt group. Top:oblique movements; bottom: cardinal move-ments. *P � 0.05; **P � 0.001.



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the MPC of both the arc and the intermediate groups (Parc �0.001, Pintermediate � 0.02). The MPC of the arc group didnot differ from that of the intermediate group.

• Movements varied in terms of smoothness betweengroups, as measured by the number of peaks in thetangential velocity speed profile. SP differed between thegroups (F � 15.91, P � 0.001), where the SP of allgroups differed one from the other (Parc-intermediate � 0.05,Parc-line � 0.05, Pintermediate-line � 0.001). The SP of theline group was the highest, whereas the SP of the inter-mediate group was the lowest.

Figure 11, bottom, also shows the performance measures formovements in the cardinal directions. Recall that this analysiswas restricted to movements inward to or outward from thecenter target, excluding movements that passed through thecenter target, because the former were of similar length tothe oblique movements. First, an ANOVA applied to the tiltangle for the cardinal movements showed no difference be-tween the tilt subgroups that were identified for the obliquemovements. This further confirms the observation that for thecardinal movements, all subjects followed the same nominalhand path corresponding to both the visual straight line and thegeodesic. An ANOVA then applied to the cardinal movementsshowed no significant difference between the three tilt groupsfor any of the measurements analyzed above (AS, AF, MPC,SP). It is interesting to note, however, that the average value ofAS was highest and the average values of MPC and SP werelowest for the intermediate group, compared with the other twogroups, for both the oblique and cardinal movements. Pear-son’s correlation coefficient confirmed that there was a statis-tically significant correlation between cardinal movements andoblique movements for AS and SP, but not for MPC. Thussubjects that moved slower and generated more submovements(more velocity peaks) on the oblique movements also movedslower and generated more submovements on movements inthe cardinal directions, even though all such movements wereclose to the geodesic.

DISCUSSION

In the current study we investigated how the physical inter-action with a rigid constraint affects the choice of paths to befollowed by the hand. To do so we started with a mathematicalanalysis and numerical simulation of moving along a curvedsurface. The model we chose was highly simplified, consistingof a point mass being pushed along a rigid sphere by apredefined force profile. We thus ignored the complex dynam-ics of the limb [gravity, centrifugal, and Coriolis forces; inter-action torques (Hollerbach and Flash 1982)] and considered thehand as a pure force generator. In the data from the movementsalong the cardinal axes, however, where there is little doubtthat all subjects attempted to follow the straight-line pathbetween targets and thus the geodesic, we saw evidence fordynamical effects. Biases of the movement plane to one side orthe other of the vertical, depending on movement direction(upward or downward), were most likely due to interactionsbetween the limb segments and gravity during the verticalmovements, whereas the downward dip of horizontal move-ments most likely reflected an effect of gravity on the chosenpath. We also ignored the viscoelastic properties of the musclesand the possibility for online corrections to the motor com-

mand, opting to simulate the effects of a preplanned forceapplied in an open-loop manner. Yet subjects who did notmove along the geodesic for the oblique movements dideventually arrive at the target, rather than drifting off to someother location, as purely open-loop simulations would predict.Thus, online correction most certainly played a role in redi-recting the hand to the target. Nevertheless, neither limbdynamics nor online corrections can explain the significantdifferences in movement characteristics between those subjectswho followed the geodesic path for movements in obliquedirections and those that did not. Indeed, the simulations andexperiments that we performed were chosen to highlight theeffects of the interaction between the hand and the surfaceduring a constrained, curved motion.

Advantages of the Geodesic

Intuitively, one can conclude that the geodesic has an ad-vantage, being the shortest distance between two points on asphere. However, the geodesic is shown here to be additionallyadvantageous in terms of motor planning and sensitivity toerrors in the motor commands. A motor command designed tofollow a geodesic path, rather than some other path, is morerobust in terms of errors in the magnitude of applied forces. Italso requires less precise timing of the motor command vari-ations, making it easier to stay on the desired path even in theface of unanticipated frictional forces. The increased robust-ness of the geodesic derives from the fact that the entire forceprofile required to produce such a trajectory lies in a plane.Other paths between the same two points on the surface of thesphere would require more complex modulation of force vec-tors in 3D.

The fact that radial forces in the movement plane have noeffect on the path followed by the hand for the geodesic meansthat there is a reduction of degrees of freedom to be controlled.It is much easier to approximate the motor command for ageodesic path because only two linear movement segmentsmay be sufficient. Also, the application of a finite number offorce impulses can be used to perform different forms ofballistic control of the movement. Attempting to follow anyother (nongeodesic) path on the sphere requires much moremeticulous control of the direction of the applied force as afunction of where one actually is on the sphere at any givenmoment.

Given the clear advantages described above, one mightexpect human subjects to optimize their behavior on ourexperimental task by choosing to follow the geodesic path.This was not the case in our experiment. Instead, we observeda wide range of behaviors for the oblique movements, rangingfrom subjects who performed arclike movement in the fronto-parallel plane, parallel to the rim of the spherical bowl, tosubjects whose movement planes were perpendicular to theline of sight, leading to trajectories that formed straight lineswhen projected into the frontoparallel plane but that dippedmaximally in and out in depth. We did, however, identify agroup of subjects that appeared to have profited from theadvantageous characteristics of the geodesic.

Between-subjects variation in the plane orientation wasexpected (e.g., shoulder configuration could have a largerinfluence on the hand for some subjects but not for others;similarly, individual experience could predetermine prefer-



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ences for one or another plane tilt), but an unbiased clusteranalysis divided subjects into three groups that could be easilybe differentiated by the average movement-plane inclination.These groups could be linked to the three nominal choices forhand path that we identified in our mathematical analysis:1) the arc group, who tended to move parallel to the rim (lowinclinations); 2) the line group, who followed straight-linepoint-to-point paths in visual space; and 3) the intermediategroup, whose movement planes were closer to the planedefining the geodesic. Note that one cannot, with so fewsubjects, conclude from this analysis that there is indeedclustering of behaviors within the population, but that was notour intent. Rather, we used the k-means clustering algorithm asan objective means to define groups of subjects within our dataset that exhibited similar behaviors in terms of the movementplane and then asked whether other movement parameterscorrelated with that grouping. Within the overall population ofhuman subjects, however, behavior may very well follow acontinuous distribution, rather than exhibiting significant clus-tering behavior.

Of the three identified groups within our subject pool, onlythe intermediate group, i.e., those who moved closest to thegeodesic on the oblique movements, seemed to be aware of theinherent advantage of adjusting the chosen path according towhere the targets were positioned on the sphere. Whereasneither the arc group nor the line group showed a significantmodulation of the movement plane between targets at 30° and60° elevation (Fig. 10), the intermediate group did show asignificant change in this parameter, as would be expected ifone were attempting to follow the geodesic specific to eachtarget configuration. These “enlightened” subjects were appar-ently able to take advantage of the dynamical properties of thegeodesic path, moving much faster on average than subjects inboth the arc group and the line group. The intermediate groupalso produced smoother trajectories compared with those whoattempted to produce visually straight lines, as measured by thenumber of plane crossings, and the trajectories produced by theintermediate group were composed of fewer submovements,compared with both the arc and line groups, as indicated by thenumber of peaks in the tangential velocity speed profile.

Factors Affecting the Choice of Path

Compared with subjects in other studies, our subjects as awhole appeared less inclined to follow the geodesic whenmoving from point to point on a spherical surface. This may bedue to the fact that they worked against the inside surface of thespherical bowl. As we have shown, pressing outward againstthis concave surface tends to stabilize the movement, whateverthe selected path, making the movement less sensitive to errorsin the direction of the applied force. Pressing against theoutside, convex surface of a sphere (Mussa-Ivaldi et al. 2003;Sha et al. 2006) or pressing downward against an invertedpendulum (Liebermann et al. 2008) is naturally unstable. Sub-jects in these other experiments may therefore have had moreincentive to search for optimized paths and to control repeatedmovements more tightly.

Another factor may have been the affordances of the task.With vision in depth inside the bowl being less precise,subjects in our task may have been more inclined to follow themore easily identified circular contour of the rim or to follow

a straight-line path in visual space, as humans are known to do(Abend et al. 1982). Following a visually guided path in thefrontoparallel plane while applying constant pressure in theforward direction might seem to be a reasonable strategy thatmight have been further encouraged by our virtual realityexperimental setup. Although the rendering on the screenincluded many cues as to the 3D nature of the visual scene(directional lighting, shading, etc.), there were no stereoscopiccues as to the movement in depth. Naive physics (McCloskeyand Kohl 1983) might also have come into play, where somesubjects may have intuitively felt that moving in a circular arcwhile pressing radially outward toward the rim would be astable solution, even though this is not a valid strategy for thetargets not on the equator. Policies such as these would haveproduced the suboptimal behaviors observed for the arc andline groups.

Finally, as our computational analysis shows, integration ofposition and force information on a curved surface requiresprecise coordination that can only be achieved through practiceand learning. This interpretation is supported by the findings ofTorres (2010), who argued that spatiotemporal alignment be-tween (internal and external) constraints are learned to preservethe map between intended and actual hand action dynamics.Learning to move on a curved surface as in our experimentmay imply the forming of a correlation between the sensory(visual) space (underlying the shortest hand path) and the handforce (haptic) space where a force application should corre-spond in time with endpoint location. Internal feedback abouttime lags is needed for such learning (i.e., perception of timedifferences between the formation of geodesics in the visualand haptic spaces). Indeed, one may speculate that such acorrelation process could be mediated via cerebellar loops,where intended and actual motion seem to be matched (Marr1970) and temporal aspects of movement (e.g., perceivedmovement gaps) may be learned and stored (Raymond et al.1996). It remains to be seen whether additional practice oradditional incentive to perform precise, repeatable movementswould push our naive subjects to adopt the more robust motorplan of following the geodesic.

Implications for Motor Planning

It is interesting to note that movements along the cardinaldirections showed similar patterns between the tilt groups foraverage speed, peaks in the velocity profile, and even in theaverage force applied to the constraining surface. This wasconfirmed by correlation analysis. It appears that subjects whomoved more quickly and more smoothly on the obliques alsomoved more quickly and more smoothly for the cardinaldirections. Since all three groups followed the geodesic for thecardinal movements, one cannot explain these patterns basedsimply on the mechanical effects that occur during the produc-tion of a geodesic vs. nongeodesic hand path. The differencesin performance between groups, in terms of greater speed andfewer submovements, would instead appear to reflect charac-teristics of motor planning.

Studies of free movements of the hand (Pellizzer et al. 1992;Soechting and Terzuolo 1987a, 1987b; Viviani and Cenzato1985) and force profiles in isometric tasks (Gordon and Ghez1987; Pellizzer et al. 1992) reveal “invariants” of motor plansthat are reflected in neural discharge patterns in motor cortex



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(Georgopoulos et al. 1992; Schwartz and Moran 1999). Indeed,it is likely that neuronal processes are tuned to conform todynamical interactions with the environment. Analyses ofunconstrained hand movements in 3D indicate that the CNS infact breaks down complex movements into a finite number ofsegments (submovements), with each segment constrained tolie in a plane (Soechting and Terzuolo 1987a). If one general-izes this principle to the construction of both the hand trajec-tory and the applied forces in a constrained motion task (i.e., toencompass kinematic and kinetic features of the motor com-mand), this means that the geodesic solution should be pre-ferred because displacements of the hand and its driving forcesall lie in a single plane. To the extent that geodesic paths on asphere may represent optimized solutions [e.g., minimal en-ergy in a joint space corresponding with geodesics in a mani-fold endowed with a kinetic energy metric (Biess et al. 2007,2011), or in a broader sense, as an emergent property of thesystem regardless of the choice of metric], a generalization ofmovement decomposition, both constrained and unconstrained,into piecewise geodesics on a sphere suggests a new hypothesisabout how and why the nervous system programs hand move-ments in a particular fashion.

Conclusions

In a study of sliding movements of the hand inside a concavehemispherical surface, we have characterized the trajectoriesproduced by human subjects in terms of the paths chosen alongthe surface. Although the chosen paths varied widely fromsubject to subject, certain paths were produced at a higherspeed and more smoothly than others. Through modeling of themovement dynamics, we have shown why these particulartrajectories, close to the geodesic, are advantageous in terms ofinteractions between the hand and the environment. We pos-tulate that these physical constraints may underlie the forma-tion of movement primitives in the nervous system encompass-ing both hand kinematics and applied forces for the class ofmovements known as “constrained motion” that require controlof both of these movement parameters.

APPENDIX

For a surface-constrained task, the physical constraint is repre-sented by an inequality function, since although one cannot move intothe surface, it is possible to move away from it. Inequality constraintsare nonholonomic constraints that do not lend themselves easily tocommon modeling. A tractable problem that includes an equalityholonomic constraint is that of simulating the motion of a point massstrictly constrained to lie at a fixed distance from the center (i.e., ona sphere) and then considering post hoc instances where the constraintpulls the point mass down onto the surface as situations where thepoint mass has lifted off of the surface.

Movement Dynamics on a Sphere

Lagrangean mechanics divide forces into applied forces (e.g., theforces applied by the hand to slide a block of wood on a tabletop) andconstraint forces (the forces that keep the object from sinking into thesurface). The problem to be solved is that of computing the constraintforces that arise for a given mechanical system and the forces andtorques applied to it. D’Alembert’s principle asserts that the totality ofthe constraint forces (i.e., internal forces due to the constraint andopposing applied forces) does not contribute to acceleration of a pointmass. This principle is the counterpart of the third law in Newtonian

mechanics. Using Bernoulli’s concept of virtual displacement (looselydefined, virtual displacements are imagined infinitesimal displace-ments, not violating the constraints, which take place in zero time),D’Alembert’s principle states that the totality of constraint forces doesno virtual work. These principles are used in Lagrangean mechanicsto develop the equations of motion using Lagrangean multipliers(Rosenberg 1977).

Gauss’s principle is equivalent to D’Alembert’s and Bernoulli’sprinciples and can be directly derived from them (Udwadia andKalaba 1996). It asserts that among all the accelerations that a systemmay have at any given time that are compatible with the constraints,the one that materializes is the one that minimizes the Gaussian, G:

G(p) � (p�a)TM(p�a), (A1)

where p � [x, y, z] is the position vector in a right-hand coordinatesystem, a is the acceleration related to the applied force (the acceler-ation of the system if the constraint was absent), and M is the massmatrix. This in fact means that the motion evolves at each instant intime such that the deviation of the acceleration of the constrainedsystem from the acceleration it would have had, had there been noconstraint, is directly proportional to the extent to which the acceler-ation of the unconstraint motion at that instant violates the constraints.

The set of h holonomic constraints can be described by

f(p, t) � 0, i � 1, 2, . . . h . (A2)

The Pfaffian representation can be obtained by differentiation(using the chain rule):

�j�1

3

dij(p, t)dpj � gj(p, t)dt � 0, i � 1, 2, . . . h , (A3)

where p � [p1, p2, p3]T (p1 � x, p2 � y, p3 � z), dij � fi(p, t)/pi,and gi(p, t) � fi(p, t)/t.

Equations with the same form as Eq. A3 can also representnonholonomic constraints, yet in such a case they are nonintegrable.Provided that dij(p, t) and gi(p, t) are sufficiently smooth, the Pfaffianform may be differentiated as follows:

�j�1

3

dij(p, t)pj � �j�1

3

�k�1

3 dij(p, t)

pkpkpj � �

j�1

3 dij(p, t)

tpj

��j�1

3 gi(p, t)

pjpj �

gi(p, t)

t� 0, i � 1, 2, . . . m ,

(A4)

where m is the number of holonomic and Pfaffian nonholonomicconstraints. These equations can be expressed in matrix forms as

A(p, p, t)p � b(p, p, t), (A5)

where A is an m � 3 matrix (m is the number of constraints) and b isan m-vector. In such cases, applying the minimum norm solution tominimize the Gaussian, one obtains

p � a � M�1⁄2(AM�1⁄2)�(b � Aa), (A6)

where the superscript � indicates the Moore-Penrose generalizedinverse (Udwadia and Kalaba 2003). Simplifying for the case of apoint mass and rearranging, one can see that the equation of motion isgiven by

mp � F(t) � mA�(b � Aa)

�F(t) � A�[mb � AF(t)]

�F(t) � Fc(t),

(A7)

where F(t) is the applied force and Fc(t) � A�[mb � AF(t)] is theconstraint force.

For motion confined to a spherical surface, the constraint is de-scribed by



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x2 � y2 � z2 � r2, (A8)

where r is the sphere’s radius. Differentiating twice, the constraintequation can be written as

�x y z � �x

y

z� � � x2 � y2 � z2, (A9)

which has a form equivalent to Eq. A5. One can therefore compute themotion of the point mass on the sphere due to an applied force F(t) byintegrating Eq. A7 with A � [x y z] and b � x2 � y2 � z2.

To test the simulation, we asked what path would result fromsimply driving the mass with an applied force that would generate astraight-line, minimum-jerk profile if executed without the sphericalconstraint. In the dynamic simulations we observed that in such acase, i.e., application of the holonomic constraint such that the massstays on the sphere, the trajectory follows the geodesic from initial tofinal position (Fig. 4). This complies with the fundamental principlesof motion due to Gauss’s principle. A straight line between two pointson the sphere clearly violates the spherical constraint, and the viola-tion direction is orthogonal to the spherical surface. The demand thatthe acceleration of the constraint system be directly proportional to theextent to which the unconstrained acceleration violates the constraintmeans that the path of the constrained system is a projection in theradial direction (orthogonal to the surface at every point) of thestraight line on the spherical surface. Since by definition great circlesare formed by an intersection of the sphere and a plane containing thecenter of the sphere, this projection is in fact a projection onto a greatcircle. Therefore, if one imposes the equality spherical constraint ontop of a displacement force that drives the hand along a straight linetoward the target in visual space, the system will follow a geodesic onthe spherical surface from the starting point to the target position.

GRANTS

The current research was partially supported by the Israel-France ResearchNetworks Program in Neuroscience and Robotics, the Paul Ivanier Center forRobotics Research and Production Management, and the European UnionProject “STIFF” (FP7 Grant Agreement 231576).

DISCLOSURES

No conflicts of interest, financial or otherwise, are declared by the authors.

AUTHOR CONTRIBUTIONS

S.B., D.G.L., and J.M. conception and design of research; S.B., D.G.L., andJ.M. performed experiments; S.B. and J.M. analyzed data; S.B., D.G.L., andJ.M. interpreted results of experiments; S.B. and J.M. prepared figures; S.B.drafted manuscript; S.B., D.G.L., and J.M. edited and revised manuscript; S.B.,D.G.L., and J.M. approved final version of manuscript.

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