INTRODUCTION

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It is generally recognized that the mechanical properties of a limb play a significant role in transforming the motor output of the CNS into a limb movement. The mechanical properties of a multi-jointed limb can be quite complex. For example, if the upper arm and forearm are free to move, the stiffness, viscosity, and inertia at the hand each vary systematically with the direction of imposed motion, i.e., they are anisotropic (Hogan 1985; Mussa-Ivaldi et al. 1985; Tsuji et al. 1995). Considerable attention has been given to the importance of the anisotropy of stiffnessusually represented by a stiffness ellipsein the context of the equilibrium-point hypothesis (e.g., Gomi and Kawato 1997; Shadmehr et al. 1993). Viscous and inertial anisotropies have been investigated to a lesser extent (Lacquaniti et al. 1993; Tsuji et al. 1995), although the importance of inertial anisotropy in the control of target-reaching movements has been shown by Gordon et al. (1994).

Whereas elastic and viscous anisotropies have their origin largely in the mechanical properties and arrangement of the muscles, inertial anisotropy reflects the relative involvement of the arm segments for hand movements in different directions. For example, when the elbow angle is near 90° and hand movement is in a direction perpendicular to the forearm, there is little movement of the upper arm segment, and therefore only the inertia of the forearm and hand matters, whereas a greater inertia comes into play when moving the hand in a direction for which there is considerable motion of the upper arm as well. That the inertia is of greater significance at higher speeds can be seen by comparing the requirements for producing the same movement at two different speeds: for example, for a system in which stiffness, viscosity, and inertia are constant, if the movement is to be performed at five times the speed, and thus in 1/5 of the time, the time course of each of the torque components would be speeded up by a factor of 5, and the magnitudes of the joint torques required to counter the stiffness, viscosity, and inertia would increase by factors of 1, 5, and 25, respectively. [As for the different inertial terms in the equations of motion related to the angular accelerations and to squares and products of angular velocities, they would each increase by a factor of 25 (Hollerbach and Flash 1982)]. Thus inertial anisotropy would play an increasingly important role compared with viscous and elastic anisotropies as the speed of movement increases.

Our general aim is to investigate the possible role of inertial anisotropy in causing errors or distortions in movement; this would provide insight into the extent to which motor planning does or does not take account of this mechanical property of the limb. Gordon et al. (1994) have demonstrated a lack of accounting for inertial anisotropy in the initiation of quick voluntary, target-reaching movements of the hand. The movements were performed in the absence of on-line visual feedback, and the targets were at the same distance but in different directions in the horizontal plane. The peak acceleration and velocity were found to vary systematically with direction, being largest for the direction of least inertia and smallest for the direction of greatest inertia. This suggests that the motor plan for launching the movement does not compensate for inertial anisotropy. Similar results have been obtained in monkeys who were allowed full vision: at least part of the directional variation in speed could be explained by inertial anisotropy (Turner et al. 1995).

Although the directional tuning of primary motor cortex cells is modified by the application of external forces (Kalaska et al.1989; Li et al. 2001), it is not known whether the motor plan at the cortical level takes account of the internal forces associated with inertial anisotropy. It has been shown that there is a nonuniform distribution of preferred directions of primary motor cortex cells that is correlated with joint power (Scott et al. 2001), a variable that reflects the effects of both the inertial anisotropy of the limb and the force-velocity relationship of contracting muscles. However, no study has shown a modulation of cell discharge that reflects inertial anisotropy.

If inertial anisotropy was ignored by the CNS in generating the motor output, one would expect that the movement direction would tend to veer toward the direction of least inertia. Although, as shown by Gordon et al. (1994), the resulting inaccuracy is quickly corrected in discrete, point-to-point movements, it is not known whether the same is true when a subject attempts to follow a curved path, for which the desired direction of movement varies continuously. In the case of a curved path, the direction of the acceleration must change continuously, thereby engaging a changing effective inertia. In particular, the attempt to draw a circle without accounting for inertial anisotropy would be expected to result in a figure elongated along the direction of least inertia. When subjects succeed in drawing circles that do not exhibit such elongation, they must either compensate for inertial anisotropy through prior planning and/or feedback corrections (Verschueren et al. 1999) or slow down sufficiently so that inertial effects become relatively unimportant. By asking subjects to draw circles at progressively faster speeds, one expects to see increasing distortion in the form of elongation along the direction of least inertia if inertial anisotropy is not fully compensated. In the present report, we focus on such movements, which have the additional advantage of being reasonably familiar to the subjects.

Several studies of circle-drawing movements have reported distortions in the presence of full vision, but they have not attributed the distortions to inertial effects. Using three-dimensional forearm and upper arm motions, Soechting et al. (1986) observed systematic distortions in shape for large circles (ca. 30 cm diam) drawn freehand at relatively slow, self-determined frequencies (0.7-1.2 Hz). The distortions were attributed to inaccuracies in the mapping between the desired extrinsic, trajectory parameters and intrinsic, joint angle parameters (Soechting and Terzuolo 1986). If the distortions arise due to a linearization of the extrinsic-to-intrinsic mapping, as these authors argue, then the distortion should be considerably less for smaller circles. Moreover, distortions due to such a mapping inaccuracy, because they arise prior to any decisions concerning movement dynamics, should be independent of speed. In contrast, distortions due to inertial effects should be more pronounced at higher speeds. Indeed, Dounskaia et al. (2000) report increased elongation of circles into ovals as the drawing frequency was increased in steps over the range of approximately 1-5.3 Hz. Their subjects drew small (ca. 2.5 cm) circles in the horizontal plane with an inkless pen using wrist and finger movements while the forearm was immobilized. Because of the many kinematic degrees of freedom in the situation in which a pen is grasped in the fingers and the palm, it is not possible to determine whether the direction of elongation coincided with the direction of least inertia.

We studied circle-drawing movements at different speeds with full vision and with forearm and upper arm motions as the two available degrees of freedom in the horizontal plane. Because of the possibility that distortions in shape could arise from differences in the control of movements in different regions of the workspace, as has been shown for arm positioning and pointing tasks (e.g., Imanaka et al. 1995), circles drawn at different workspace locations were investigated. The purpose of the study was to test the idea that the distortions observed in circles drawn by neurologically normal subjects are consistent with those expected if the anisotropic nature of the inertia is not completely accounted for when generating the movement.

The following specific hypotheses were tested. Hypothesis 1: speed will affect the circularity of figures drawn by elbow and shoulder movements (as has previously been shown for wrist and finger movements) with faster movements associated with greater elongation into an oval shape; however, for comparable angles between the forearm and the upper arm, the position in the workspace will not affect the circularity of the figures. Hypothesis 2: the orientation of elongation of circles drawn at high speed will correspond to the direction of least inertia. The latter can be predicted on the basis of the inertial characteristics, i.e., mass and moment of inertia, of the arm segments. The expectation is that the orientation of the elliptical figures drawn at high speed will vary with workspace location, so that the long axis coincides with the direction of least inertia. In addition to testing these two hypotheses, which pertain to extrinsic space coordinates, we also investigated the amplitude and phase relations among the two joint angles (elbow and shoulder). Some of the findings have appeared previously in abstract form (Pfann et al. 2001).

	METHODS

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Six subjects were recruited for this study (4 male, 2 female; ages 22-41), who gave their informed consent in accordance with local Institutional Review Board guidelines. All subjects were free from known neurological or musculoskeletal abnormalities and were right handed. The basic task that subjects performed was to draw circles repetitively in the horizontal plane with the right hand near specified locations and at different speeds, using rotations at the shoulder and elbow joints.

Apparatus

A splint was fitted to the seated subject's right forearm and hand to immobilize the wrist with the index finger extended. The arm was supported in the horizontal plane by two plastic pylons, one under the forearm, and the other under the upper arm, which were free to slide on a table with a smooth surface. To minimize friction, the supporting pylons were floated just above the table surface by a continuous flow of compressed air (Karst and Hasan 1991). The forearm was maintained in a neutral position. A transparent pointer was attached to the base of the forearm support, slightly above the plane of the table, whose direction was aligned with the forearm and index finger. A black dot on the pointer was placed vertically below the tip of the index finger, whose movements therefore reflected the fingertip movements. This dot was used by the subject as the marker for circle drawing; it did not produce a trace of the movement path.

A Selspot active marker system with two cameras was used to record kinematic data. Pairs of infrared light-emitting diodes (LEDs) were placed along the long axes of the upper arm and forearm. Another pair, placed on the upper trunk, consisted of one LED on the right shoulder (acromion), and one along the axis between the right and left shoulders. The duration of the trial and sampling frequency was varied with the pace instruction so as to allow more than one cycle of circle-drawing movements to be recorded. The slow-pace trials were recorded for 6 s at a sampling frequency of 100 Hz, and the moderate- and fast-pace trials were recorded for 4 s at a sampling frequency of 200 Hz.

The subject's height, weight, age, gender, and limb segment lengths were recorded. Based on the anthropometric parameters, the limb segment masses, center of mass locations, and moments of inertia were estimated (Plagenhoef et al.1983). The positions of the LEDs on each limb segment were recorded with respect to the proximal and distal joints.

Protocol

Subjects drew circles repetitively with eyes open with three instructions concerning the pace and at four locations in the workspace. Prior to every trial, a circle of 5 cm diam was shown to the subject to indicate the size of the desired circles. We did not provide a template of a circle to the subject during the drawing task because tracing the template could have obscured the distortions we wished to study; moreover, Zelaznik and Lantero (1996) have shown that visual feedback does not serve as a source of form in circle drawing. For each trial, the subject was asked to start drawing circles continuously at the prescribed pace in a counterclockwise direction about the desired center point, which was marked by a dot on the table. Data acquisition was started after a few cycles, when the subject indicated his/her readiness verbally. Subjects were asked to continue drawing circles until the experimenter asked them to stop, 4 or 6 s later, at the end of the trial.

Three different instructions were given concerning the pace: as slow as necessary to draw perfect circles, at a moderate speed, and very fast. Subjects were free to interpret these instructions in terms of actual speed as knowledge of results was not provided. The frequency of cyclical circle-drawing movements ranged across subjects between 0.22 and 1.1 Hz (mean: 0.7 Hz) for the slow instruction, between 0.83 and 5.3 Hz (mean: 1.8 Hz) for the moderate speed instruction, and between 1.7 and 6.5 Hz (mean: 4.3 Hz) for the very fast instruction. Circles were drawn at all three paces in a fixed order (fast, moderate, slow) before changing locations. Twelve trials were recorded at each locationone with no movement, three fast paced, four moderate paced, and four slow paced.

Four different locations of the desired center of the circle were chosen to cover a significant range of the workspace in the horizontal plane. For three of the four locations, the elbow angle was in midrange (ca. 90°), with the center placed in the right (R), middle (M) or left (L) portion of the workspace. These desired locations, for which the fingertip was at approximately the same distance from the right shoulder, were defined with the shoulder horizontally abducted for R, adducted by 45° for M, and 90° for L, but the actual mean shoulder angles deviated from these values. These locations were chosen to span a range of the workspace while maintaining a similar extent of inertial anisotropy (as measured by the ratio of the effective inertias in the directions of greatest and least inertia), and therefore an expected similar effect of speed on circularity. The fourth location (E) was defined by a more extended elbow (bent 45° from full extension), with the shoulder adducted by 90°. The desired center for location E was therefore at a greater distance from the shoulder than for the other locations. This location was chosen not only to encompass a wider range of the workspace, but also because it was expected that the inertial anisotropy, as measured by the ratio of the effective inertias in the directions of greatest and least inertia, would be different with a more extended elbow compared with its value for the other three locations. The order in which the locations were presented was the same for all subjects (R, M, L, E).

Data analysis

The horizontal-plane orientations of the forearm, upper arm, and trunk were determined from the positions of the respective pairs of LED markers. From these orientations, the shoulder and elbow angles were determined as functions of time. The subjects, despite being strapped to the chair, often did not keep the trunk at the desired orientation in space, therefore the mean shoulder angle did not correspond to the location-specific desired value. This resulted in a smaller range of shoulder angles across the different workspace locations than was anticipated. To obtain a true indication of location in an egocentric frame of reference, the forearm and upper arm orientations (such as those shown in Fig. 1) were referenced to the trunk orientation, the latter being determined from the pair of LEDs placed on the upper trunk. We adopted the convention that the lateral (x axis) direction, defined by the trunk LEDs, corresponds to zero degree, and counterclockwise rotations (i.e., flexions) in the horizontal plane to positive values. Zero degree for the shoulder corresponds to the upper arm being oriented laterally, and for the elbow, to full extension.

View larger version (14K): [in this window] [in a new window]   Fig. 1. Representative trials of circles drawn by a subject (S4) at different locations in the workspace at a slow pace (A) and at a fast pace (B). For each of the 4 workspace locations [right (R), middle (M), left (L), and extended (E)], the circles drawn are shown along with the mean forearm and upper arm orientations, which are depicted by the stick figures. The x and y axes correspond respectively to the lateral and anterior directions referenced to the upper trunk, with (0, 0) corresponding to the acromion. In B, the direction of least inertia is also shown (thin lines) for each location, for comparison with the orientation of the elliptical figures actually drawn at the fast pace.

The sampled positions of the two forearm LEDs, and the distances between them and the fingertip, were used to calculate fingertip position in Cartesian coordinates as a function of time as well as shoulder and elbow angle position in joint angle coordinates. Time-domain smoothing with a least-squares filterSavitzky-Golay, 2nd degree, with 10 neighboring points on each side (Press et al. 1992)was employed; this also yielded the fingertip velocity components. These time-series data were partitioned into individual cycles based on zero crossings of the velocity component in an arbitrarily chosen direction; incomplete cycles at the beginning or end of a trial were discarded. To avoid contamination by the effects of the drift in center position that often occurred during the cyclical movements, each cycle was analyzed separately. The parameters p₁-p₅ were estimated for the equation of the ellipse

(1)

by minimizing the sum of squares of the left hand side of Eq. 1 over all data points (x_i, y_i) in the cycle. The "fmins" algorithm in Matlab 5.2 yielded the best values of p₁-p₅. In addition, the average movement speed over the cycle was calculated. In three cycles of a total of 1,551, the path was qualitatively nonelliptical (e.g., a figure 8); these were discarded.

DIRECTION OF LEAST INERTIA. This was computed following the method of Lacquaniti et al. (1993) and Sabes and Jordan (1997). The procedure can be summarized as follows. The 2 × 2 inertia matrix, which relates the torques at the two joints to the angular accelerations of the two segments, was determined using the segment inertial parameters for the subject and the mean elbow angle for the given trial. The 2 × 2 Jacobian matrix, which relates increments in the Cartesian coordinates of the hand to increments in the segment angles, was also calculated, using the mean joint angles for the trial. The inertia matrix was then transformed, using the Jacobian, into a matrix that relates the force components to the acceleration components at the hand in Cartesian coordinates (Lacquaniti et al. 1993). The inverse of this matrix represents the "endpoint mobility tensor" (see APPENDIX IV in Hogan 1985). Its eigenvalues represent the reciprocals of the smallest and largest effective inertia at the hand. The corresponding eigenvectors represent the two mutually orthogonal directions of least and most inertia.

VARIABLES ANALYZED. Pertaining to extrinsic space coordinates. Location: a categorical variable with four values (R, M, L, and E). Speed: mean fingertip speed (cm/s) over a movement cycle. Circularity: the ratio of the length of the minor axis to the length of the major axis of the best-fitting ellipse for fingertip motion over a cycle. The possible range of this ratio is between 0.0 (a straight line) and 1.0 (a perfect circle). Orientation: the orientation of the major axis of the best-fitting ellipse for a cycle with respect to a fixed direction in the workspace. The fixed direction was the rightward lateral direction defined by the pair of LEDs placed on the trunk. The orientation measure was meaningful only when the circularity was not close to 1.0, and, in light of the circularity data, it was analyzed only for the fast pace trials. Relative orientation: this was defined for the fast-pace trials as the orientation of the major axis of the best-fitting ellipse for a cycle, relative to the direction of least inertia.

STATISTICAL ANALYSES. Mixed models (Brown and Prescott 1999) were used to investigate the following: the effects of location and instructed pace on speed, the effects of speed and location on circularity, elbow/shoulder ratio, and phase, and the effects of location on orientation and on relative orientation. For the relative orientation, we also tested the deviation of its values from zero. These models accounted for correlations among observations within the same subject and subject variability from the fixed effects in the models. All models controlled for trial and cycle regardless of significance.

	RESULTS

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Speed

The three instructions concerning the pace (slow, moderate, and fast) resulted in significantly different actual speeds of movement [13.0 ± 4.0, 35.1 ± 9.9, and 72.5 ± 15.0 (SD) cm/s, respectively; F(2,10) = 88.97, P < 0.0001]. There was no significant effect of workspace location on speed [F(3,15) = 0.30, P = 0.83].

Circularity

Figure 1A shows the circles drawn by a subject with the instruction to move slowly. Four trials are shown at the different locations in the workspace. Stick figures of the arm depict the mean orientations of the forearm and upper arm segments for each trial. The circles drawn were not perfect, but comparing them with the attempted circles drawn by the same subject under the fast instruction, shown in Fig. 1B, it is clear that the drawings made more rapidly departed much more from circularity. Moreover, at each location in the workspace, the distortion of the rapidly drawn circles took the form of a relative elongation in one direction and a relative shortening along the orthogonal direction. The orientation of the direction of elongation varied with the location in the workspace. For comparison with the direction of elongation, Fig. 1B also shows the computed direction of least inertia for each of the depicted trials. As workspace location varies, the orientation of the axis of elongation appears to vary similarly to the orientation of the direction of least inertia.

The impression given by Fig. 1, that the circularity decreases (distortion increases) with faster speeds, is confirmed by statistical analysis. A mixed model was used for the logit transformed circularity measure (see METHODS), in which the effects of mean cycle speed (continuous variable) and workspace location (categorical variable) were examined. The analysis showed that mean cycle speed had a statistically significant effect on circularity [F(1,5) = 127.61, P < 0.0001]. There was no statistically significant effect of workspace location [F(3,15) = 0.19, P = 0.90].

Figure 2 shows, for all the cycles in our data set, the circularity measure plotted against the mean speed. The trend of the relationship is brought out more clearly by the plot of the predicted circularity based on the statistical model, controlling for configuration, trial, and cycle number, shown by diamonds. Clearly, irrespective of workspace location, nearly circular forms are drawn at slow speeds (1 represents a perfect circle), and as speed increases the circle elongates (0 represents a straight line), as was hypothesized.

View larger version (26K): [in this window] [in a new window]   Fig. 2. The dependence of circularity on speed. Individual cycle data are shown (), for all the subjects, pace instructions, and locations in workspace. Instead of a regression line, which would be inappropriate for the curvilinear relationship exhibited by the data, the estimates of circularity based on the statistical model described in METHODS are shown (diamonds). The values correspond to the inverse logit transform of the predicted values stemming from significant components of the model.

Orientation of the major axis in fast trials

In Fig. 1B, the orientation of the long axis of the elongated circles appears to become more counterclockwise as the workspace is traversed from right to left. Figure 3A shows, for the different locations, the mean values (and model based standard errors) of the orientation of the major axis of the fitted ellipse with respect to the rightward lateral axis; positive values denote counterclockwise orientations with respect to this axis. The effect of workspace location was statistically significant [F(3,15) = 26.03, P < 0.0001]. Pairwise comparisons among the locations were all statistically significant except the difference between the orientations at locations M and E. 

View larger version (10K): [in this window] [in a new window]   Fig. 3. Dependence of the orientation of the major axis of the best-fitting ellipse on location in workspace. The mean parameter estimates and SE bars are shown based on 1-way ANOVAs for the fast movements. A: absolute orientation with respect to a fixed (rightward lateral) axis. B: relative orientation with respect to the direction of least inertia.

Relative orientation of the major axis in fast trials

Figure 3B shows the mean values and standard errors of the orientation of the long axis relative to the direction of least inertia. The estimated means range over less than 15° for the different workspace locations, suggesting that the long axis is fairly closely fixed relative to the direction of least inertia. The effect of workspace location on the orientation relative to the direction of least inertia, however, was statistically significant [F(3,15) = 25.55, P < 0.0001]. Pairwise comparisons revealed that the relative orientations at locations R, M, and L were each significantly different (P < 0.0001) only from the relative orientation at location E. (Subject S5 exhibited relative orientation values quite discrepant from the group means. Discarding the data for S5, however, did not alter the conclusion: The effect of location on the relative orientation remained significant, with location E being different from all others.)

In addition, the relative orientation was significantly different from zero at workspace locations R (P = 0.0033), M (P = 0.0108), and L (P = 0.0067) but not at location E (P = 0.9635). Figure 3B shows that the major axis of the best-fitting ellipse is oriented approximately 12° clockwise with respect to the direction of least inertia for three of the locations. The orientation of the major axis for location E, however, is not significantly different from the direction of least inertia.

Elbow/shoulder excursion ratio and phase difference

The mean values of the ratio of elbow and shoulder excursions for the locations R, M, L, and E were, respectively, 1.36, 1.38, 1.37, and 1.82. The corresponding phase differences (elbow relative to shoulder) were, respectively, 130.6, 128.9, 124.9, and 141.0°, the negative values indicating a phase lag. The effect of location was significant on the elbow/shoulder ratio [F(3,15) = 33.74, P < 0.0001] as well as on the phase [F(3,15) = 9.48, P = 0.0009]. Pairwise comparisons revealed that the elbow/shoulder ratio as well as the phase were different for location E compared with the other locations [P < (0.05/6) = 0.0083], but there were no significant differences when comparing the other pairs of locations. The effect of speed was not significant on the excursion ratio [F(1,5) = 3.42, P = 0.12]. Examining the data for individual subjects, however, we found that four of the six subjects exhibited increasing elbow/shoulder ratio with increasing speed, with significant positive correlations (r = 0.50 to 0.69, P < 0.0001). Of the remaining two subjects, subject S3 did not show a significant correlation of excursion ratio with speed and subject S5 showed a significant negative correlation (r = 0.37, P < 0.05). If the data for subject S5 were discarded, the effect of speed on the elbow/shoulder ratio reached significance at the criterion level [F(1,4) = 11.48, P = 0.03]. The effect of speed on the phase was significant [F(1,5) = 18.05, P = 0.0081], albeit with a modest slope of 0.204°/(cm/s), the positive sign indicating a decrease in the phase lag of the elbow angle with increasing speed. All six subjects exhibited significant correlations (r = 0.18 to 0.54, P < 0.05) of the same sign, and discarding the data for subject S5 did not the change the significant effect of speed on the phase [F(1,4) = 11.89, P = 0.03]. It appears, therefore that as the speed increases, individual subjects differ in how the shoulder and elbow motions are apportioned with the elbow/shoulder ratio increasing for the majority of subjects, but all subjects exhibit some decrease in the lag of the elbow with respect to the shoulder. With respect to the different locations in workspace, all subjects exhibit a greater elbow/shoulder excursion ratio and a greater phase lag for the more extended elbow position compared with the other three positions.

	DISCUSSION

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Tests of the specific hypotheses

Two hypotheses were advanced in the INTRODUCTION to test the idea that speed causes systematic distortions in drawing circles and that inertial anisotropy may be responsible. Hypothesis 1 was that speed will affect the circularity of figures drawn by elbow and shoulder movements (as has previously been shown for wrist and finger movements by Dounskaia et al. 2000), with faster movements associated with greater elongation into an oval shape, but the position in the workspace for comparable angle between the forearm and upper arm, will not affect the circularity of the figures. Our observations fully support this hypothesis. Position in workspace had no statistically significant effect on circularity not only for the three positions with similar inertial anisotropy (L, M, R) but also for the position with a different anisotropy (E).

Hypothesis 2 was that the orientation of the elliptical figures drawn at high speed will vary with workspace location such that the long axis coincides with the direction of least inertia. We found that indeed the orientation did not remain fixed in the workspace. The observations, however, revealed a small but consistent discrepancy of approximately 12° between the direction of elongation and the direction of least inertia for three of the four workspace locations tested (R, M, and L). At the fourth location (E), where the elbow was in a more extended position, the discrepancy was not significantly different from zero, which supports the hypothesis. This issue will be discussed in greater detail later on.

Before addressing our findings in the context of inertial anisotropy, it is important to consider other possible mechanisms, not based on the biomechanics of the arm, that can give rise to distortions in hand-drawn circles.

Possible nonmechanical explanations for distortion of circles

Location in workspace is known to influence the errors in arm positioning tasks; in particular, the left and right hemispaces differ in this regard when reaching with the same hand (Imanaka et al. 1995). This phenomenon, related presumably to cortical hemispheric effects, could also influence the errors in other tasks such as circle drawing. We found, on the contrary, that among the right, middle, and left positions located at comparable distances in the workspace, there was no significant change in the circularity, the relative orientation of the long axis of the distorted circle, the elbow/shoulder excursion ratio, or the phase. These observations weigh against an explanation of the distortion in terms of attentional or visual differences between locations in different directions. Moreover, Carey and Grace Otto-de Haart (2001) have argued that even the differences in arm positioning errors in different locations may be attributable to inertial anisotropy.

Errors due to inaccuracy in the transformation from the desired movement of the fingertip (in extrinsic coordinates) to the necessary motions of the segments (in intrinsic coordinates) have been reported, for pointing movements (Flanders et al. 1992) as well as for circle and ellipse drawing movements (Soechting et al. 1986). While distortion arising from this mechanism appears to be important for comparatively large circles, with diameter ca. 30 cm, it would have less effect when drawing smaller circles, ca. 5 cm, such as the ones we studied. More importantly, this mechanism cannot account for the observed effect of speed on the distortion. Therefore while inaccuracies in the transformation between extrinsic and intrinsic kinematic variables may well contribute to the distortion to some extent, the systematic speed-related increase in elongation cannot arise from this mechanism.

Some investigators consider the inter-segmental phase during periodic motion to be an expression of the phase preferred by the underlying dynamical system, and if there are discrete transitions in the preferred phase when the frequency of periodic motion is increased, these transitions are regarded as describing essential properties of the dynamical system (Kelso 1984; Kelso et al. 1991). From this point of view, the dynamical system could be so organized that it prefers in-phase or out-of-phase segmental motions, and a transition in the phase difference as the frequency is increased results in a circle changing into an ellipse. Neither Dounskaia et al. (2000) nor we, however, observed a sudden transition in circularity or in the phase as the speed was increased; moreover, the phase between the elbow and wrist angles (Dounskaia et al. 2000) or between the shoulder and elbow angles (our study) never came close to 0 or 180°. Therefore the dynamical systems approach in which a discrete transition occurs at a critical frequency does not provide an explanation for the observations.

Explanations based on mechanical properties

A circle can be distorted into an oval shape by a change in the relative excursions of the segments, in the inter-segmental phase, and in the precise waveforms of the segmental motions or by a combination of these three factors. We have investigated two of these factors. Our results indicate that all subjects decrease the phase lag of elbow motion with respect to shoulder motion when moving at higher speed, and a majority of them also increase the elbow/shoulder excursion ratio. Despite these individual differences, the change in circularity with speed is similar across subjects.

The fact that at lower speeds the circularity is high for all locations indicates that the anisotropy of stiffness is either of negligible importance or it is well accounted for by all subjects in generating the motor output. For faster speeds, none of the strategies adopted by different subjects succeeds in preventing the distortion of circles. This suggests that some speed-dependent mechanical property, related to inertia or viscosity, is not fully accounted for in generating the motor output, resulting in distortion at the higher speeds. Velocity-dependent forces, such as Coriolis forces, could lead to distortions. However, Lackner and colleagues have shown that, in reaching movements, subjects compensate automatically for self-generated Coriolis forces (e.g., reaching while turning the trunk) (Pigeon et al. 1999); moreover, subjects adapt over 8-15 trials, without visual or direct tactile feedback about reaching accuracy, to Coriolis forces that are experimentally imposed by rotating the room without the subject's knowledge (Lackner and DiZio 1994). Therefore we do not believe the distortions are due to Coriolis effects.

That the viscous properties of muscle affect the kinematics has been shown for single-joint movements (Jaric et al. 1998). The possibility of viscous anisotropy's contribution was not considered in the Introduction because it was assumed that at the highest speeds of movement the effect of inertial anisotropy will dominate the effects not only of stiffness anisotropy but also of viscous anisotropy. The observation that the direction of elongation of the ellipse is somewhat different from the direction of least inertia leads us to consider the possibility that anisotropy of viscosity may also be playing a role. This modification of the original postulate, we would argue, can help explain the consistent discrepancy we observed between the direction of elongation and the direction of least inertia for three of the four locations studied and can also explain the lack of discrepancy for the fourth location. The argument is presented in the following text.

Figure 4A depicts diagrammatically the stick figure of the arm in a middle location (M), along with the mean direction of elongation of the ellipse observed for this location. Also shown is the direction of least inertia (using the anthropometric parameters for a typical subject and the elbow angle depicted). The direction of elongation differs from the direction of least inertia by 11° clockwise, which is the mean value observed for location M. We would like to also compare the direction of elongation with the direction of least viscosity, but the latter is not known and therefore can only be approximated based on reports in the literature. Tsuji et al. (1995) found the direction of least viscosity to be close to the direction of least stiffness and Mussa-Ivaldi et al. (1985, their Fig. 7) show that for locations comparable to that of Fig. 4A, the maximum stiffness is approximately along the line joining the endpoint to the shoulder, and therefore the minimum stiffness direction is perpendicular to this line. This direction, which we assume is nearly unaltered between postural maintenance and voluntary movement (Gomi and Kawato 1997), is shown as a dotted line in Fig. 4A, which corresponds to a rough approximation of the direction of minimum stiffness and viscosity. It is noteworthy that the observed direction of elongation of the ellipse lies part way between the directions of least inertia and least viscosity. We interpret this to mean that the elongation arises not entirely because of lack of compensation for inertial properties but also partly from lack of compensation for viscous properties. The same interpretation is equally applicable to locations L and R (not depicted) because all the directions depicted in Fig. 4A will simply rotate with changes in arm orientation as long as the elbow angle remains the same. Indeed, we found the discrepancy between the direction of elongation and the direction of least inertia to be statistically indistinguishable among the three locations with comparable elbow angles.

View larger version (10K): [in this window] [in a new window]   Fig. 4. Diagrammatic depiction of the arm (S: shoulder, E: elbow) for 2 different elbow angles (A for workspace location M, and B for location E). The average orientation of the elongation of circles drawn at fast speeds is shown by the thick short line at the endpoint. For comparison are shown the direction of least inertia (thin solid line), and the surmised direction of least viscosity and stiffness (dotted line).

For a more extended elbow, corresponding to location E, there is no significant discrepancy between the directions of elongation and of least inertia, as depicted in Fig. 4B. For this location compared with the other locations, the inertial anisotropy is more pronounced insofar as the ratio of the greatest and smallest inertias is larger. The larger inertial anisotropy can dominate any effects of viscous anisotropy, which would be consistent with the direction of elongation almost coinciding with the direction of least inertia.

There is evidence to indicate that subjects are aware of inertial anisotropy (Flanagan and Lolley 2001) and, indeed, seem to make use of the anisotropy in prescribing the hand-movement trajectory when avoiding an obstacle (Sabes and Jordan 1997; Sabes et al. 1998). It is surprising, then, that subjects do not account for it when launching a pointing movement (Gordon et al.1994) nor when drawing circles at high speeds. Comparing the response to inertial anisotropy with the response to externally added inertial loads, the initial motor output in the latter case too has been found to be independent of inertia (Gottlieb et al. 1989; Hong et al. 1994; Karst and Hasan 1991). When, for example, the inertial load is known to be greater, the CNS does not compensate by simply scaling up the entire motor output, although some compensation via motion-related feedback occurs during the movement (Gottlieb 1996; Sainburg et al. 1999). Rather, the movement time is prolonged, which is also the case without external loads when the hand moves in directions of greater inertia (Gordon et al. 1994). The limitation to compensation for inertia, both external and internal, may be a reflection of the delays in feedback correction that increase in significance at higher speeds. Whether the movement plan itself fails to account for mechanical anisotropies or whether there is not enough time for feedback corrections to compensate for the anisotropies, it is the mechanical properties that appear to cause distortions increasing consistently with speed of movement.

Conclusion

A good portion of the current literature on the control of multijoint movements focuses on inertial properties of the moving segments and the inter-segmental interactions that arise from them. If inertial properties were the only mechanical properties of significance, then the control of fast movements will be a simple matter of scaling the motor output that is appropriate for slow movements (Hollerbach and Flash 1982), resulting in identical movement paths at all speeds. Clearly this is not the case for the circle-drawing movements studied in this report or the ones studied by Dounskaia et al. (2000), nor is it the case for a variety of other cyclical movements including locomotion (Bianchi et al. 1998), mastication (Throckmorton et al. 2001), and arm swinging (Abe and Yamada 2001). Speed effects have also been described for noncyclical movements such as reaching (Fischer et al. 1997), drawing (Schillings et al. 1996), and handwriting (Wright 1993). Moreover, the CNS does not seem to take full account of the inertial properties even for such common movements as those involved in pointing to a fixed target. We tested whether the same is true for movements that necessarily entail a nonstraight path. Our results support this hypothesized lack of accounting for inertial properties to the extent that circles drawn at slow speeds turn into more elliptical shapes at higher speeds, with the long axis remaining close tobut not identical withthe direction of least inertia. The discrepancy, however, is consistent enough to demand explanation, and we have provided a possible explanation in terms of the concomitant influence of viscoelastic anisotropy.

An overall scheme that is consistent with our results is that the motor plan for repetitive movements is designedeither in advance or through sculpting by motion-dependent feedbackso as to be adequate for slow movements for which inertial effects are not dominant. When higher speed is required, the plan is altered, in a subject-dependent manner, but never to the extent needed to compensate fully for the now important inertial and viscous properties of the limb and therefore does not result in the same path as before. More rigorous tests of this scheme will involve measurements of the mechanical anisotropies along with the kinematics and kinetics of voluntary movements for the same subjects and limb configurations (cf. Lacquaniti et al. 1993) and the study of the influence of unusual loads that could possibly annul the inherent anisotropies.