Frames of Reference and Control Parameters in Visuomanual Pointing

Transcription

1 Jonrnal of Experimental Psychology: Copyright 1998 by the American Psychological Association, Inc. Human Perception and Performance /98/$ , Vol. 24, No. 2, Frames of Reference and Control Parameters in Visuomanual Pointing Philippe Vindras University of Geneva Paolo Viviani University of Geneva and Vita-Salute University Three hypotheses concerning the control variables in visuomanual pointing were tested. Participants pointed to a visual target presented briefly in total darkness on the horizontal plane. The starting position of the hand alternated randomly among 4 points arranged as a diamond. Results show that during the experiment, movement drifted from hypometric to hypermetric. Final positions depended on the starting position. Theft average pattern reproduced the diamond of the starting points, either in same orientation (hypometric trials), or with a double inversion (hypermetric trials). The distribution of variable errors was elliptical, with the major axis aligned with the direction of the movement. Statistical analysis and Monte Carlo simulations showed that the results are incompatible with the final point control hypothesis (A. Polit & E. Bizzi, 1979). Better, but not fully satisfactory, agreement was found with the view that pointing involves comparing initial and desired postures (J. E Soechting & M. Flanders, 1989a). The hypothesis that accounted best for the results is that final hand position is coded as a vector represented in an extrinsic frame of reference centered on the hand. Reaching for an object, pressing a key, or pointing to a distant location are all familiar acts performed effortlessly under a variety of conditions and constraints. Yet, the underlying interplay between visual and motor mechanisms is still the subject of much debate (cf. Jeannerod, 1988). The key issue can be stated in relatively simple terms. On the one hand, despite eye, head, and body movements, vision affords a stable representation of the objects in the environment with respect to an extrinsic system of reference. On the other hand, kinesthesia affords information concerning the position of all body segments involved in manipulating, grasping, and pointing with respect to an intrinsic system of reference. Thus, the debate focuses on how this diverse information is set in register to establish a one-to-one correspondence between a posture and a spatial location. Most models of pointing derive from more general conceptions of movement control. One line of speculation (the equilibrium point hypothesis) holds that the motor plan involves the definition of a final stable posture. Along this line, the further strong suggestion has been made that this can be done disregarding the starting position of the limb (for a review, see Bizzi, Hogan, Mussa-Ivaldi, & Giszter, Philippe Vindras, Department of Psychobiology, Faculty of Psychology and Educational Sciences, University of Geneva, Carouge, Switzerland; Paolo Viviani, Department of Psychobiology, Faculty of Psychology and Educational Sciences, University of Geneva, Carouge, Switzerland, and Faculty of Psychology, Vim-Salute University, HSR, Milan, Italy. This work was partly supported by Fonds National Suisse pour la Recherche Scientifique Research Grant We wish to thank Anatol Feldman for the improvements suggested to the first draft of this article. Correspondence concerning this article should be addressed to Paolo Viviani, Department of Psychobiology, Faculty of Psychology and Educational Sciences, University of Geneva, 9 Route de Drize, 1227 Carouge, Switzerland. Electronic mail may be sent to viviani@uni2a.unige.ch. 1992). According to the gamma model (Merton, 1953), fusimotor efferences activate the stretch reflex driving the arm to the desired position. Alternatively, one may suppose that muscle tension (the final position [alphal control model; Polit & Bizzi, 1978, 1979) is specified in such a way that the skeletomuscular system has only one equilibrium point at which the limb matches the desired position. Neurophysiological evidence (Vallbo, 1970) proved damaging to Merton's gamma model. By contrast a more recent and elaborated version of the final position control model (the virtual equilibrium trajectory hypothesis; Bizzi, Accornero, Chapple, & Hogan, 1984; Flash, 1989; Hogan & Hash, 1987) still has considerable currency (cf. Bizzi et al., 1992; Jaric, Corcos, Gottlieb, Ilic, & Latash, 1994; see, however, Gomi & Kawato, 1996; Katayama & Kawato, 1993). Another version of the equilibrium point hypothesis is the so-called lambda model (Feldman, 1966a, 1966b, 1974, 1986; Feldman & Levin, 1993), which assumes that the control variable is the threshold length for motoneuron recruitment. By modifying this length, the motor control system sets the origin of a positional frame of reference for the sensorimotor system. Pointing movements would be generated by shifting the frame of reference from the initial to the desired final position and would involve a transition between stable equilibrium states (Feldman & Levin, 1995). This model predicts that changes in the initial position of the limb elicited by perturbations may not affect final precision (equifinality) provided that the participant does not change the pattern of control variables. In other models, exemplified by the work of Soechting, Flanders, and their collaborators (Flanders, Helms Tillery, & Soechting, 1992; Flanders & Soechting, 1990; Soechting & Flanders, 1989a, 1989b), both the initial and final desired posture of the hand are explicitly taken into account. According to Soechting and colleagues, retinal information, combined with eye and head position signals, yields a representation of the target position in a spherical, shoulder- 569

2 570 VINDRAS AND VIVIANI centered system of coordinates. This extrinsic representation is then converted into the set of arm and forearm orientations that would bring the hand to the target. Finally, the motor plan is set up by "taking the vectorial difference.., between the initial and final positions represented in terms of joint angles" (Soechting & Flanders, 1989b, p. 606). Rather than perceptual biases or inaccurate execution, the model ascribes pointing errors to the fact that the nervous system linearizes the mapping from the extrinsic to the intrinsic system of reference (Flanders et al., 1992). A third way of conceptualizing the operations involved in visuomanual pointing retains the idea that initial and final hand positions are both essential ingredients of the motor plan. However, unlike the postural model discussed earlier, the comparison is supposed to involve the Cartesian vector from hand to target (i.e., a geometrical entity represented in a frame of reference with the origin on the hand). Moreover, because the trajectories of most pointing movements are in fact well approximated by straight lines (Hollerbach & Flash, 1982; Morasso, 1981), the vectorial representation of the spatial error and the movement required to null the error are supposed to be closely connected, as in some models of saccadic capture of visual targets (Robinson, 1973; Schiller & Koerner, 1971; Schiller & Stryker, 1972). In fact, because the relation between articular angles and movement trajectory is nonlinear (Hollerbach & Atkeson, 1986), such a connection suggests that the path is the primary input to the motor plan and that the muscle synergies required to produce the appropriate covariation of the joint angles are specified at some later stage (Kalaska & Crammond, 1992). Support for the vectorial coding hypothesis comes from reaction time experiments (Bock & Arnold, 1992; Bonnet, Requin, & Stelmach, 1982; Rosenbaum, 1980) and from measurements of pointing accuracy (Bock & Arnold 1993; Bock, Dose, Ott, & Eckmiller, 1990; de Graaf, Denier van der Gon, & Sittig, 1996; Gordon, Ghilardi, Cooper, & Ghez, 1994; Gordon, Ghilardi, & Ghez, 1994; Rossetti, Desmurget, & Prablanc, 1995), all suggesting that the direction and extent of a movement are planned independently, as indeed one would expect if the vector from the starting hand position to the target were the basis for the motor plan. The work of Gordon, Ghilardi, and Ghez (1994) is particularly relevant to our research. Participants used the cursor of a digitizing tablet placed horizontally to reach targets displayed on a computer screen together with the cursor position. The hand was invisible throughout the experiment. Under these conditions, the spatial distribution of the final positions was elliptical, with a major axis oriented in the direction of the movement. The radial and tangential components of the variable errors were accounted for by assuming that the independence of direction and extent measured in a hand-centered system of reference is not only a feature of the early stages of planning but is in fact preserved throughout the ensuing stages. Except for those who endorse one or another version of the final position control hypothesis, most authors agree that reaching manually for a spatial location involves the assessment of a spatial mismatch between the initial and desired hand positions. Yet, disagreement remains on how the mismatch is sensed. Schematically, two major options seem available. Under the first option, exemplified by the work of Soechting and colleagues mentioned earlier, the mismatch involves postures and is coded intrinsically. The second option is exemplified by Bock and Eckmiller (1986), who showed that when participants pointed to a succession of targets in the same direction, errors added up. On this basis they concluded that the driving input to the motor system is a spatial mismatch (i.e., the distance between hand and target) estimated from visual cues and coded extrinsically (see also Bock et al., 1990). Note, however, that experiments with prisms (Rossetti et al., 1995) indicate that both visual and proprioceptive cues enter into the specification of the starting position. In short, several conceptual models--all supported by evidence--are being entertained to account for the properties of goal-directed movements. The purpose of our study was to assess the vector coding hypothesis for hand movements to visual targets. The hypothesis is common to various models with different degrees of physiological specificity and plausibility. Our aim was less to underwrite a specific proposal than to demonstrate that a vector coding stage is a necessary component of any realistic model of visuomanual pointing. The novel feature of the experiment with respect to other recent attempts to validate the hypothesis was the specific combination of conditions that were selected. On the one hand, displaying both hand and target position on a computer screen (Bock, 1992; Favilla, Hening, & Ghez, 1989; Ghilardi, Gordon, & Ghez, 1995; Gordon, Ghilardi, & Ghez, 1994) or display panels (Lepine, Glencross, & Requin, 1989; Rosenbaum, 1980) is likely to induce an allocentric, hand-centered perception of target position. If so, the support that some of these studies (e.g., Gordon, Ghilardi, & Ghez, 1994) provided to the vector coding hypothesis may not generalize to more naturalistic conditions. In our experiment we preserved the correspondence between visually estimated distances and movement extent that is normally present under everyday circumstances. On the other hand, because the competing hypotheses make contrasting predictions about errors, we emphasized the factors that are most conducive to obtaining a wellidentifiable pattern of constant and variable errors. The information on the target position was provided only briefly, before and during the early portion of the movement. Moreover, we eliminated all other visual cues from the environment, including those concerning the moving ann. Finally, no visual feedback on the final error was given at the end of the trial. Because position sense is known to be labile (Bedford, 1989; Paillard & Brouchon, 1968; Wann & Ibrahim, 1992), these three conditions should facilitate the occurrence of gain errors (i.e., proportional amplitude errors). The other potential source of systematic error was the initial position of the hand, which we manipulated systematically. According to the final position control hypothesis, the distribution of errors should be independent of the initial position. Thus, evidence of a correlation--possibly amplified by an incorrect gain calibration--would provide evidence directly against the hypothesis.

3 VISUOMANUAL POINTING 571 Participants Method Twenty right-handed adults (14 women and 6 men; aged years) participated in the experiments and were paid 15 Swiss francs for their participation. Their height varied from 1.62 to 1.85 m. All participants had normal or corrected-to-normal vision and presented no evidence of a neurological disorder. The experimental protocol was approved by the ethical committee of the University of Geneva. Informed consent was obtained from all participants. Apparatus The experiments were conducted in a quiet, isolated booth kept in total darkness. Participants stood in front of a large ( m), translucid digitizing table (Model , Numonics Corporation, Montgomeryville, PA; nominal accuracy = mm; sampling frequency = 200 samples/s) mounted horizontally on a modified drawing board whose height could be adjusted individually at the level that the elbow takes in a comfortable writing posture (see Figure 1). Holding the recording pen (20 cm long, 1 cm in diameter, weight = 20 g) with the right hand, participants could point without effort to any location on the table within a distance of about 70 cm from the chest. The position of the pen's tip could be recorded continuously as long as it remained within 1 cm of the surface of the table. In addition, a pen-up/pendown signal was delivered when the pen was pressed gently on the table. Starting and target positions were identified by backprojecting a 4-mm-wide dim laser spot on the table. The spot position was controlled by two galvanometric mirrors (G300DT with CX660 amplifier, General Scanning Inc., Watertown, MA) driven by a 12-bit digital-to-analog converter. A computer controlled all phases of the experiment and provided the experimenter with real-time information about the data being acquired. Five positions were identified on the table (see Figure 1). Four of them (starting points) were placed at the vertices of a diamond. Points at Le Te De Re DT,, 180 mm LR,, 160 mm DP,, 120 mm the leftmost (L) and rightmost (R) vertex were 160 mm apart. Points at the proximal (P) and distal (D) vertices were 120 mm apart. The center of the diamond was at 250 mm from the table edge on the participant's midline. The fifth point was the target (T). It was placed sagittally at 490 mm from the edge. The distances from T to the starting points were as follows: D = 180 ram, R and L = 253 mm, and P = 300 ram. The lines LT and RT made an angle of 18.43" with respect to the sagittal line. Task and Experimental Procedure The task was introduced to the participant using written instructions detailing all phases of the experiment and the required behavior. After a phase of familiarization, during which the experimenter interacted with participants, the booth was sealed to eliminate all visual cues, and four additional warmup trials were administered before starting the experiment. Several times in the course of the experiment, we made sure that the participant was unable to perceive anything but the laser spot. Seven participants with unusually low luminance thresholds were eventually able to exploit the dim light coming from the spot to locate their ann. Data from these participants were eliminated. The experimental sequence was run to the end without allowing the participants to leave the booth. Each trial comprised the following steps. The laser spot indicated one initial position. A short beep signaled that the pen's tip was placed correctly (i.e., within a tolerance circle with a 2-ram radius). The participant initiated the trial by pressing the pen down. After a random delay distributed uniformly between 0.5 and 1.5 s, the laser spot moved to the target position in less than 5 ms, remained there for 200 ms, and then disappeared (the displacement was so fast that the path of the laser beam could not be seen). The participants had to move the pen to the target with a single straight movement. Although it was not necessary to keep the pen in touch with the table, participants were aware of the maximum distance compatible with continuous recording and were instructed not to raise the hand too much. The instructions placed equal emphasis on being accurate and fast. Because of the response latency, the movement began sometimes before and sometimes after the offset of the spot. However, because movement time considerably exceeded 200 ms, most of the displacement was covered in the absence of any visual cue. Trials were repeated whenever the movement anticipated the onset of the target or when the pen was accidentally lifted too much from the table. Participants remained in the final position for 5 s, until the laser beam indicated the starting position for the next trial. On average, a trial lasted 8 s. There were 160 trials, subdivided into 40 successive blocks of 4 trials. Within each block all four starting positions were presented in a different random order. The total duration of the experiment was 25 rain. To rest, which they were free to do, participants simply refrained from placing the pen on the starting position. Data Analysis Figure 1. Experimental setup. From four starting points (L = left; R = right; P = proximal; D = distal) participants pointed in complete darkness to a target (T) indicated by a brief (200-ms) laser spot. Reaction times and movements recorded by a digitizing table. The x- and y-coordinates of the movement were recorded for a period of 2 s starting at target onset. Before computing tangential velocities and accelerations, the samples were filtered (cutoff frequency = 8 Hz) with a 15-point digital convolution algorithm (Rabiner & Gold, 1975). Movement onset was defined as the first time the tangential velocity exceeded 3 cm/s and remained above this threshold for at least 50 ms (peak velocity ranged across participants from 60 to 180 cm/s). Likewise, the end of the movement was defined as the first time the tangential velocity remained 50 ms below a 3 cm/s threshold.

4 572 VtNDRAS AND VIVIANI Overview Results The salient results of the experiment are illustrated qualitatively in Figure 2 using typical examples from 2 participants. Each part of the figure shows the trajectories of the pointing movements for the indicated block of 4 successive trials. The first interesting finding was that, although movement amplitude varied from participant to participant (e.g., for Participant kd the value midway through the experiment exceeded the final value in Participant la), in almost all cases amplitude increased pad passu with the rank order of the blocks (compare the left and central panels showing the results of the first and penultimate block for Participant la). The second finding was that the final positions depended on the starting position in a way that was strongly reminiscent of a projective transformation through a fixed point placed in the proximity of the target. When the movements were hypometric (see the left panel of Figure 2), the arrangement of the final positions approximately reproduced that of the starting points. This also was true for strongly hypermetric movements (see the fight panel of Figure 2); in this case, however, the configuration was doubly inverted with respect to the configuration of starting points: Movements from the proximal starting point ended up farther away than those from the distal starting point, and movements from the fight starting point ended up to the left of those from the left starting point. Not every block of 4 trials exhibited such a neat correspondence. However, we show later that this projective pattern was conspicuous in most participants. The quantitative presentation of the results is organized as follows. First, we describe the kinematic characteristics of the movements as a function of the starting point and of trial rank order. In the following section we deal with the effect of the trial rank order on movement amplitude. Next, we demonstrate that final positions depended systematically on the starting position. In the fourth section, the results are compared with the predictions of four simple hypotheses concerning the variables represented and controlled by the motor plan. The last two sections concentrate on the distribution of the variable errors around the mean final positions and with the variability in the course of the movement, respectively. 300 la[1-4] la [ ] kd [85-88] p, R'~L' E~T L'!;R' ii" ~" 150 E "~ 100 ffl 50 / / /!.. \ D i o\ i R -50 P -100 I I I Frontal axis (rnm) I I I I I I I I I I I O0 Figure 2. Pointing movements. Typical examples from 2 participants (la and kd) illustrate the main qualitative findings. Each panel shows the pointing movements for four successive trials (one block), each starting from a different point. The average movement amplitude depended on the participant (compare middle and right) and tended to increase in the course of the experiment (compare left and middle). Movement end points (L', R', P', and D') reflect the geometrical arrangement of the starting points, either directly in hypometric blocks (left) or with a double mirror-image inversion in hypermetric blocks (fight). T = target.

5 VISUOMANUAL Ponvr~c 573 Temporal and Kinematic Characteristics We begin by demonstrating that, as required by the task assignment, movements were indeed fast and straight. Across participants and conditions, reaction time averaged 291 ms. Individual averages and interquartile ranges are shown in the upper panel of Figure 3. Reaction times did not depend on the starting point (one-way analysis of variance with repeated measures on participants), F(3, 57) = 1.346, p =.269. They decreased instead with trial number, with a significant reduction across participants of 35 ms from the first to the last trial (linear regression performed on averages across participants, r 2 =.260), F (1, 158) = 55.5, p <.001. As for individual performances, 8 participants showed a significant decrease (p <.01) and none a significant increase. The trajectories of the movements were essentially rectilinear. The so-called "index of linearity" (Atkeson & Hollerbach, 1985) was used to estimate the amount of deviation from a straight path: For each trial, the maximum distance between the trajectory and the straight line joining starting and end points was divided by the amplitude of the movement. Across participants, the average index was (range = ), indicating a better approximation to a straight path than in other comparable studies using the same index (Atkeson & Hollerbach, 1985; Georgopoulos & Massey, 1988; Gordon, Ghilardi, & Ghez, 1994). The average movement time (MT) was 490 ms for a mean amplitude of 266 mm. There were large individual differences in MT (see the bottom of Figure 3) corresponding to the participants' varying ability to make fast movements without visual feedback (the standard deviation of individual means were 143 ms for MT and 32 mm for amplitude). MT also depended significantly on the starting point, F(3, 57) = 85.86, p <.001. In increasing order, the average MTs were as follows: D = 442 ms, L = 460 ms, R = 526 ms, and P = 531 ms. This order matched that of the corresponding average amplitudes: D = 195 mm, L = 265 ram, R = 279 ram, and P = 326 ram. In conjunction with the increase in amplitude already mentioned and displayed in Figure 2, the average MT increased with trial number 6OO 500 E I- n =,i i - 2 ai= bk he hu ib kd la Ic le mc nf np pb pm pz sk sl th It xv zr 1200 E O0 4OO = i = i bk he hu ib kd la Ic le mc nf np pb pm pz sk sl th tt xv zr Figure 3. Temporal characteristics of the movements: averages over all trials. Reaction times (RTs; top) and movement times (MTs; bottom) for all participants are arranged in alphabetic order. Dark bars indicate interquartile range (IQR). Whiskers extend to the extreme values or to 1.5 IQR, whichever is less. Horizontal lines indicate the population averages. In some cases, movements latency was shorter than target duration (200 ms).

6 574 VlNDRAS AND VIVIAN[ (r 2 =.069), F(1, 158) = 11.66, p <.001. In different participants the slope of the linear regression with trial number ranged from to 2.13 ms/trial, with an average of 0.24 ms/trial. For 9 participants the increase was significant, for 7 participants the tendency failed to reach significance, and for the remaining 4 participants MT tended to decrease in the course of the experiment. In most cases, the tangential velocity followed a slightly asymmetric bell-shaped curve. Across participants, the average velocity for all trials and all starting points ranged from 31.8 cm/s (Participant la) to 84.4 cm/s (Participant le), with the population mean being 54.4 cm/s. Along with the amplitude drift, the average velocity increased with trial number (r 2 =.106), F(1, 158) = 18.71, p <.001, with individual rates ranging from -.03 to 0.25 cm/s per trial. In spite of such large individual variations, there was a clear and systematic modulation of the velocity by the distance between the starting and final points. With no exception, movements from the starting point proximal to the participant were about 35% faster than those from the distal one. The strength of the amplitude-velocity relationship was estimated by the coefficient of isochrony (Viviani & McCollum, 1983). To take into account the hypermetric trend, we divided each complete sequence of movements into 10 groups of 16 consecutive trials (4 blocks). For each trial within a group, relative average velocity and relative average amplitude were then calculated by dividing each value by the group mean. Finally, we computed the isochrony coefficient as the slope of the linear regression between the logarithms of these two relative measures. Across participants the coefficient ranged from 0.46 to 0.95, with an average of 0.64, which is in excellent agreement with the values reported previously for other types of movements (Viviani & McCollum, 1983; Viviani & Schneider, 1991). However, a discrepancy emerged when comparing the movements from the left and right starting points (see Figure 4). Although the former were generally shorter than the latter (most participants pointed to the left of the target), movements from the left were faster than those from the right (paired t test) t(19) = 5.97, p <.001; the 99% confidence interval for the mean difference between left and right velocity was cm/s. The same asymmetry was reported by Gordon, Ghilardi, Cooper, and Gbez (1994). At the individual level, the left-right asymmetry was significant for 17 participants, with only one of them having faster movements from the right. In summary, despite total darkness, the temporal and kinematical characteristics were those typical of ballistic, uncorrected goaldirected movements. 100 Proximal,, Left o Right Distal._z, 0 _o (U la xv pb bk hu zr tt kd he nf th pm mc Ic sk ib sl np pz le Subjects Figure 4. Movement velocity. Average velocity was computed over all trials as a function of the starting point and participant. Bars encompass 2 SDs. Rank-order position of the participants (abscissa) was determined by the individual average over all starting points. Velocity was scaled with movement extent: Velocity from the proximal point (filled triangles) was always higher than the velocity from the distal point (filled circles). For all but 2 participants (he and sl), movements from the left (empty triangles) were larger and faster than movements from the right (empty circles).

7 VISUOMANUAL POINTING 575 Pointing Accuracy: Constant Errors For each movement, pointing accuracy was expressed in terms of gain and direction error (see Figure 5). The gain was defined as the ratio between the amplitude of the vector from the starting to the final position and the amplitude of the vector from the starting point to the target. The direction error (0) was defined as the angle between the two vectors. By convention, a positive direction error indicated that the actual movement was rotated counterclockwise with respect to the theoretical one. Individual gains for all trials (see Figure 6) indicated a general tendency for the amplitude of the movement to increase during the experiment. We measured the strength of this tendency for each participant by regressing gain values against trial number (see Table 1). The trend toward hypermetric movements was absent in Participant xv. The slope of the regression for the remaining 19 participants ranged from to For example, in Participants le and ib the gain increased by about.56 between the 1st and 160th trial, which amounted to an average amplitude A E E 15o W 5O 0 OL ,. R' ~ '~T _... "3. ".. Gain= RR' ',::',,, - RT OD OP "V,,"., I I I I Frontal axis (mm) Figure 5. Amplitude gain and direction errors. Amplitude gain is distance from starting to end point relative to target distance. Direction error is the angle 0 between the vector joining starting point and target, and the vector joining starting and end point. By convention, end points to the left of the target had positive direction errors. L = left; R = fight; T = target; D = distal; P = proximal; RR' = distance between points R and R'; RT = distance between points R and T. R increase of 138 mm. For half the participants, the percentage of variance explained by trial number exceeded 40%. As shown in Figure 6, in many cases the gain increase resulted into a transition from undershooting to overshooting. About 80% of the participants were hypometric on their first movements; a still larger proportion had become hypermetric on their last trials. Individual data for the direction error (see Figure 7) illustrate another general finding: In the vast majority of trials, participants pointed to the left of the target. A significant leftward bias was found for 19 participants; the corresponding errors ranged from 0.5 to 8.3. Across participants, the average direction error amounted to 3.24, and the 99% confidence interval was Unlike amplitude errors, direction errors did not drift consistently in the course of the experiment. A regression analysis of the errors averaged over participants revealed no significant trend as a function of trial number, F(1, 158) = 0.507, p =.477. The considerable gain drift present in almost all participants indicated that, as expected, the absence of visual feedback and the variability of the starting point set the stage for systematic errors to occur. In the next section, we examine how these errors depended on the starting position. Effect of the Starting Position on Constant Errors For each participant, Figure 8 shows the effect of the initial position on the constant error as a function of the trial rank order. For each group of 16 trials, we averaged the difference between the y-coordinates of the final positions from P and D starting points (triangles) and the difference between the x-coordinates of the final positions from L and R starting points (diamonds). Before averaging, the differences between the x- and y-coordinates were normalized to the difference of the corresponding coordinates of the starting points with the convention that positive differences indicate a mirror-image reversal of the final points with respect to the starting points. For instance, because R and L were 160 mm apart along the x-axis, a relative difference of 20% in the frontal direction indicates that the final point from R was 0.20 X 160 = 32 mm to the left of the final point from L. Because P and D were 120 mm apart along the y-axis, the same relative difference in the sagittal direction indicates that the final point from P was 24 mm beyond the final point from D. A regression analysis showed that in several participants, the final points tended to spread out increasingly with the trial rank order (cf. Table 2). A similar correlation existed with the amplitude gain error (i.e., gain - 1) averaged over groups of trials (the solid lines in Figure 8). Thus, the statistical significance of the effect of the starting position on the final point had to be tested separately for these participants and for those who showed no trend. When the individual trends were not significant, we performed separate t tests on the frontal and sagittal distances (the data from all blocks of trials were pooled). The test was not legitimate for the participants with a significant trend because their successive trials were correlated. However, the fact that

8 576 Vn,~DRAS AND VIVIANI bk he hu ib..." kd 1.50" t*...,, :~',,.._~"".,-.~. *.,. "~'o " la IC f! l e :" "-:. "J~ mc. Y nl. 1.75" np pb pm pz sk 1.50" 1.25".".~.'t J' "0 1.00" 0.75". 1.75" th It zr r " 1.25" 1.00-,,o ~, ". ~ -, 0.75" Trial number Figure 6. Gain changes from trial to trial (individual results). Regression analysis as a function of trial number (continuous lines through the data points) demonstrated a significant increasing trend for 19 participants. In most cases, the gain at the end of the experiment exceeded the reference value (dashed lines). Table 1 Linear Regression of Amplitude Gain Against Trial Number Participant Slope R 2 F(1,158) bk ** he "* hu "* ib ** kd * la "* lc ** le ** me "* nf * np pb ** 200.2** pm * pz "* sk ** sl ** th "* tt ** xv zr "* Note. R 2 is the amount of variance accounted for by linear regression. For 19 participants the slope was significantly different from zero at the.01 (*) or.001 (**) level. there was a significant trend showed that even in this case, final points from different origins did not overlap Data from only Participants pb and xv had no trend and failed to reach significance at the.01 level for either direction. In conclusion, the results of Table 2 indicate a significant relation between the starting and final points. A synthetic description of this relation was obtained using the following averaging procedure. First, for each participant we computed the average amplitude gain over groups of four successive blocks of trials (i.e., over four repetitions for each starting point). On the basis of the distribution of the resulting 20 participants 10 groups = 200 average gain values, we defined six contiguous intervals: , , , , , and oo. The boundaries were chosen so that each interval contained at least five average gains from at least 2 participants (the actual distribution was 7, 52, 90, 38, 6, and 7). Finally, using this partition criterion, for each starting point we computed the average x (frontal) and y (sagittal) final coordinate pooling all trials within a group. The resulting six sets of 4 average final points are shown in Figure 9. For instance, the upper cluster is the set of averages computed from 112 trials from 2 participants. The averages in Figure 9 were based on values from different participants. Because we could not assume the homogeneity of the parent distributions, we could not rigorously test the significance of the differences between points. However, the

9 VZSUOM~a~ Poncrn'qG ls bk --," ":...= ~-.: Ja he Ic hu ~,. -,'%~'~':_. le ib o -~-o11%=.-~ - o ~ ~_....-'o~.~..=.- mc kd nf. ". ~':..-.",,,,.,~ ~ "~ ":',~*.%&".~o dr.,,..o. i... "'" I %o ""...,% " ~., ~ :. ~ ". ~" "t %.. o %. " -I S np *'o' = " ". s" pb sl. 2 th tt.... t "d '? ~o "=t ~'% ~ ~ "'~t. ~, :..,... :.,;,.."-~,..., pm pz sk *" o i,, o.... -_._.~-.,_~ xv ~. - '-"-",".',-:::":,i.'..',;* ~ " "....~..~',, _-;. -~. =. = ~-~-. =,...-~.. ~ zr.~ -5 r", -10 ' GO () lc () I00 1,50 Trial number Figure 7. Direction error for each trial (individual results). For most participants, regression analysis (continuous lines) showed no significant trend as a function of trial number. For all but 1 participant (le) movement direction was consistently biased toward the left. size of these differences was estimated for each cluster using the 95% confidence intervals for the frontal distance between left and right final points and for the sagittal distance between proximal and distal final points. Nine of these 12 confidence intervals did not include zero. The most robust result was based on 90 groups of 16 trials from 18 participants (the third cluster from the bottom); the 95% confidence limits were mm for the frontal difference and mm for the sagittal difference. The data in Figure 9 show that all hypermetric clusters presented a double inversion of the final positions with respect to starting positions. When the gain exceeded one, movements from L landed to the right of movements from R and vice versa. Similarly, movements from P landed farther than movements from D. The converse was true when gain was less than one. Nevertheless, the results were not consistent with the strict projective rule: If movement direction were accurate and errors were attributable only to an inaccurate gain, the points of the three upper clusters should have fallen on the dotted lines connecting the starting points to the target. In the next section we discuss the significance of the results and the possible reason for.this discrepancy within the context of a simple geometrical model. Modeling End-Point Positions The fact that final points were closer to each other than predicted by the projective rule means that trajectories in successive trials did not always converge toward the target. In fact, for some hypermetric blocks of trials, the approximate crossing points of the trajectories were as far as 100 mm beyond the target. This suggests that constant errors resulted from the conjunction of two factors. On the one side participants represented the target as being either farther away or closer than it really was. On the other side, with respect to this mislocated target, they either overestimated or underestimated the distance to be traveled by as much as 15%-20%. The distinction between a factor that we may suppose to be perceptual and a factor related to motor execution was formalized and tested in the form of a simple geometric "cross-point model." Of course, the details of the model depend on the specific experimental design, but the basic idea may have a more general significance. The model assumes that for each block of 4 consecutive trials, participants pointed to a fixed point C (hereafter referred to as a "cross-point") not necessarily coincident with the target (see Figure 10). The gain G is supposed to be computed with respect to C and to be the same for the 4 trials: By denoting with P', L', D', and R' the end points corresponding to the four starting points, we assume that PP'/PC = LL'/LC = DD'/DC = RR'/RC = G. Together, the cross-point and the gain determine uniquely the 4 final points. Conversely, gain and cross-point are overdetermined by the final points. Using a standard simplex algorithm (Gay, 1984), we calculated the cross-point coordinates and the gain that minimized the average square deviations between

10 578 VINORAS AND vrw~d~ri 60.bk he hu ib kd ," t-..-..r:..- -,~ L 60 mc nf ~ A O O A A A A ~-i O O, ~, ~ -20 '.....,,,,-' 60.rip pb pm pz sk ,...,, ~ t -,:---" ~. ~.. ~.. -"" :._~ -20 t~ t~ 60 sl th XV zr 8 4o 1-2o.~ o "~ -20 I" " A.A_ A... "- " -**,,"...,, ~'"'--... _ ~,_,... ~ ;-,--'A,~. _ ~ ~ o- _ - o-. ~_- ~ ~. i i i i i i i i t Group number Figure 8. Individual data showing end-point separation as a function of the starting point. The averages are over groups of four successive blocks of trials. The abscissas show the group number. The ordinates show the difference between the x-coordinates for left and right starting points (diamonds) and the y-coordinates for proximal and distal starting points (triangles). Differences are normalized to the corresponding difference for the starting points and are expressed in percentages. Also shown is the evolution of the amplitude error (amplitude gain - 1; solid lines) and of the cross-point error (cross-point gain - 1; dotted lines) defined in text (of. Figure 10), again expressed in percentages. For most participants, amplitude and cross-point errors had similar trends. actual and predicted final points (one estimate for each block). The residual errors with respect to the model were not independent of the experimental variables. For each participant we performed a multivariate regression of the residuals using as independent variables the block number and three dummy variables standing for the starting points. Significant correlations (<.01) with starting point were found for 18 participants (see Table 3). Moreover, for 9 participants there was a significant interaction between block number and starting point. A post hoc analysis revealed two significant departures from the model's predictions. For 10 participants there was a tendency for movements from the distal starting point to end up to the fight of the model's prediction. For 8 participants the actual final points for movements from the left starting point were to the left of the predictions. Fitting the model to the individual data demonstrated a significant mismatch between the target and cross-point positions. The mismatch was estimated using the normalized difference (i.e., by the ratio [OC - OT]/OT; see Figure 10) between the distance of cross-point and that of the target distance from the center of gravity O of the starting points. The dashed lines in Figure 8 are the average over four successive blocks of this estimate. They help to show the distinction between the two factors that, according to the model, determined the final points. In particular, they explain how movements that, on average, were not hypermetric could nevertheless present end-point inversions (e.g., for Participant sl). In fact, in this case the cross-point was closer to the participant than the target (the dashed line was below the zero reference), and the performance was conceptually similar to that of a participant who perceived the target

11 VISUOMANUAL POINTING 579 position correctly and made hypermetric movements (e.g., the last groups of trials for Participant bk). In the Discussion section, the cross-point model will provide the basis for arguing in favor of the vector coding hypothesis. Thus, we have to show that this particular scheme captures the salient features of the results significantly better than simpler alternative schemes. The model was contrasted with three other hypotheses. The first hypothesis is that for each block of 4 trials, the final points P', L', D', and R' would be randomly scattered around their center of gravity. In this case the coordinates of the center of gravity are the only two parameters to be estimated. The second hypothesis is that, over 4 consecutive trials, participants will point with a constant gain to a cross-point that deviates from the real target only in direction (relative distance error = 0). Also, in this case there are only two free parameters, namely the direction error of the cross-point and the gain. The third possibility is that direction errors will be a linear function of the correct direction and that amplitudes will be scaled by a constant gain. Like the cross-point model, this last hypothesis involves three free parameters. If the residual errors after fitting any of these models to the data of all trials are independent samples from a Gaussian distribution, their sum of squares should follow a chi-square distribution. The distributions for the cross-point model and for the third alternative have 200 dfs; those for the other two hypotheses 240 dfs. Thus, for each participant, the ratio between the mean square of the residuals for the cross-point Table 2 Linear Regression Between the Differences of Coordinates of the End Points From Opposite Starting Points and Trial Number Sagittal difference Frontal difference Participant Slope M Slope M bk he 0.667** ** "* 0.513" 9.76** hu " ib kd " 0.837* "* 13.68"* la lc 0.678* "* 11.56" "* 6.03* -8.89* le "* 0.695* 17.13"* mc " 0.589* 12.02"* nf ** "* np "* * pb pm ** 0.519" pz ** 0.510"* "* sk "* " 8.28** sl ** 1.557"* 20.28** th "* tt "* xv zr * " Note. For most participants, the mean positions of the end point from opposing starting points were significantly different. Distances between end points increased significantly with trial number. *p <.01. **p <.001. A E g i o} O ~D T- O O4 *1-,- 8,p. g o O O4 O O =- o el el& o,~," e. ~ / '.,. ~ / o ",!..' "~' Target,... -.,....; Frontal axis (mm) / / / / Prox. & Left o Right Distal Figure 9. End-point evolution results for all participants. Data points are the average end-point positions corresponding to the indicated starting points. The 200 available groups of four successive blocks were partitioned into six unequal sets using amplitude gain as a criterion. Average positions were computed over all trials within sets. If end points were to follow a strict projective rule, they would fall on the dotted lines connecting the target to the starting points (not shown). Note the dominant leftward bias. Prox. = proximal. model and the mean square for the other models should be distributed as F(200, 240) (Hypotheses 1 and 2) or as F(200, 200) (Hypothesis 3). To test the superiority of the crosspoint model (see Table 4), we counted the number of participants for whom the ratio was significantly greater than either F(200, 240) = 1.249, p <.05, or F(200, 200) = 1.263, p <.05 (thus, for instance, according to the Bernoulli law, finding at least 5 such participants is equivalent to a global test at the.0026 level). In fact, the number of significant ratios was much higher against all alternatives (see Table 4). Having demonstrated that the cross-point scheme accounted for the data significantly better than the three alternatives considered earlier, we used this scheme to address a question that will turn out to be relevant in the Discussion section, namely the relation between the drift of the amplitude gain and the drift of the cross-point (both documented in Figure 8). To this end, we defined the cross-point gain for a block of trials as the ratio of the amplitude of the vectors CO and TO (see Figure 10). Then, for every participant, we regressed independently the cross- 40

12 580 VINDPa~S AND VIVIANI point and amplitude gains (averaged over 4 trials) versus the block number. Finally, the slopes of the two linear regressions for all participants were regressed against each other (normal regression). The results (see Figure 11) showed a strong linear relation between the two drifts, as demonstrated by the fact that the principal axis (solid line) explained 86.3% of the variance. This suggests that a common mechanism affected both the perceptual localization of the target (cross-point drift) and the planning and execution of the movement (amplitude drift). In the next section we consider the scatter of the errors around their mean. Variable Errors The analysis of gain and directional errors (see Figures 6 and 7, respectively) and the satisfactory fitting of the cross-point model can both be taken to suggest that amplitude and direction are control parameters of the planning and execution processes. If so, for geometrical reasons, this should be reflected in the way the compounded variability of kd [85-88] p, R'"...!D:,, L' ',A~: T "_.,._ attributable to the gain drift was eliminated by interpolating / ii '-./ o "'"-.. Table 3 Cross-Point Model: Wilks's Lambda Test of Residual Error Dependency on Block Number and Starting Point Start Block Number Participant point Start Point bk.780**.856** he.394**.897 hu.678**.874* ib.708**.874* kd.747**.957 la.696**.833** lc.672**.944 le.847**.910 mc.678**.955 nf.638**.870* np.855**.944 pb.858**.741"* pm * pz.270**.940 sk.653**.970 sl.517"*.839** th.517"*.904 u.913".914 xv.868**.893* zr Note. For most participants, the accuracy of the significantly on the starting point. *p <.01. **p <.001. fitting depended these parameters contributes to the scatter of the final points around their center of gravity. Specifically, if amplitude and direction are controlled independently, the distribution of the final points should be elongated either in the radial or in the tangential direction. Conversely, if the principal axes of the error distribution are aligned with the radial and tangential directions, the polar coordinates of the final points are independent. To check this prediction, one must remove the effects of the trial rank order on the variable error. The trend the x- and y-coordinates of the final points with 6-df smooth splines for each participant and each starting point independently. Then, using principal-components analysis, we calculated the.95 ellipses of tolerance and the principal axes for the error distribution. The results for the left starting point and for each participant are shown in Figure 12. The shape of the distributions and the orientation of the principal axes demonstrate unequivocally that the variability was much greater in the direction of the movement than in orthogonai direction (across participants, the average normal coefficient of correlation was.50; i.e., about 75% of the variance was explained by the major axis)) Similar results were found for Figure 10. Schematic representation of the cross-point model. End points are supposed to follow a projective rule with respect to a cross-point that does not coincide with the target. Cross-point gain is defined as the ratio CO/TO. P = proximal; R = right; L = left; T = target; D = distal; CO = distance of the crosspoint (C) from the center of the starting points (O); TO = distance of the target (T) from the center of the starting points (O). 1 The so-called "normal coefficient of correlation" is defined as rn = (R 2-1)/(R 2 + 1), where R 2 is the ratio of the larger to the smaller eigenvalue of the variance--covariance matrix. By definition, the amount of variance explained by the major axis of the tolerance ellipse is (ru + 1)/2; rn coincides with the Pearson coefficient of linear correlation when the major axis of the tolerance ellipse has a slope of 1.

13 VISUOMANUAL POINTING 581 Table 4 Comparison of Four Models for Accounting for Constant Pointing Errors: Residual Variances and Fisher Tests of Their Ratio Total Participant variance M1 M2 M3 M4 F12 F13 F14 bk 179,633 15,270 25,125 21,868 19, " 1.432" he 153,871 51,275 56,244 76,095 70, " 1.372" hu 85,832 26,621 33,575 32,961 29, " ib 348,860 41,824 51,265 85,296 67, * 1.618" kd 96,546 26,248 40,650 37,153 31, " 1.415" la 121,267 21,706 29,926 28,527 25, " 1.314" lc 170,444 25,495 43,934 74,648 33, " 2.928* 1.301" le 388,103 41,617 65, ,039 72, " 3.221" 1.732" mc 97,318 19,213 31,472 29,488 17, " 1.535" nf 98,711 31,539 47,132 48,982 43, " 1.553" 1.389" np 119,113 45,992 60,822 63,859 53, " 1.388" pb 67,893 11,409 16,563 19,016 15, " 1.667" 1.339" pm 43,353 22,798 29,600 32,595 28, " 1.430" pz 65,399 24,360 27,352 40,671 34, " 1.430" sk 57,969 18,213 29,194 40,814 32, " 2.241" 1.784" sl 123,843 37,150 71,103 83,633 71, " 2.251" 1.916" th 95,298 22,669 28,573 62,227 30, " 2.745* 1.357" tt 66,704 26,081 31,923 33,679 29, " xv 50,832 16,695 20,843 25,954 22, " 1.343" zr 69,593 22,258 28,614 29,969 26, " 1.346" Note. MI to M4 = residual variance for the four models. F12, Fl3, and F14 = variance ratios for comparing the cross-point model (1) with the alternatives. In most cases the cross-point model was superior. *p <.01. the right starting point (for 17 participants, the major axis was tilted to the left). A strong elongation also is present in the distributions for the proximal and distal starting points, which were well aligned with the sagittal direction. Because of the high consistency across individuals, we pooled data from all participants (see Figure 13). The average normal coefficient of correlation for all starting points was found to be.455, meaning that 73% of the variance could be explained by the major axis. Note that in all four cases, the major axes (solid lines) were aligned almost perfectly with the movement direction required for reaching the target (dashed lines). Because these findings were not likely to be merely the result of biomechanical factors (Gordon, Ghilardi, & Ghez, 1994), they strongly support the hypothesis that movement planning and execution includes at least one stage in which amplitude and direction are represented and controlled independently. The gain and direction error data (see Figures 6 and 7) had provided no indication that practice affected the variability of the performance. The absence of practice effects was confirmed by the analysis of the variable errors. For each block of 16 trials, we computed the sum of the squares of the distances of the final points with respect to the mean for trials having the same starting point. By regressing this index of variability against block number, we found a significant slope (p <.05) only for 2 participants. Variability increased in Participant pm and decreased in Participant pb. Time Course of the Variability That direction and amplitude are controlled independently does not imply that they also are monitored simultaneously. Showing that the variability amplitude and direction follow different time courses during the movement would strengthen the contention that separate processes are involved in controlling these parameters. For each trajectory we considered 10 intermediate points corresponding to equal fractions of the total MT. The drifts documented in Figures 6 and 7 also affected these intermediate points. Thus, to obtain the desired time course of the variability, we had to neutralize these drifts. For each intermediate point we computed its distance and direction with respect to the starting point, and we interpolated separately the 40 pairs of measures available for each participant and each starting point (one pair for each trial block) with a cubic spline (thus reducing by 6 the number of degrees of freedom). We then computed the residuals with respect to the corresponding interpolations and converted the direction residuals into linear units (in millimeters) by multiplying them by the fitted amplitude. Finally, we measured the standard deviation of these residuals over the 40 blocks of trials. Because the individual data were as highly consistent as those for the final points (see Figure 12), we pooled them (see Figure 14). The results were similar for all starting points. They confirmed that amplitude variability (filled data points) was always higher than direction variability (open data points).

14 582 VINDRAS AND VIVIANI le / ib o " t- d _8.Q Ic bk he U3 IO 8 _= d Q. E v 19 L _o 09 sl np 18 ehu th *tt zr pb sk O d exv 0.0 i i Slope (cross-point gain vs block number) Figure 11. Relation between amplitude gain and cross-point gain. Gains are computed for 10 groups of four successive blocks of four trials. One data point is shown for each participant. The abscissa is the slope of the linear regression between the cross-point gain and group number. The ordinates are the slope of the linear regression between amplitude gain and group number. The two gains are highly correlated (normal coefficient of correlation rn =.726). More important, the time course of the two components was different. Direction variability increased monotonically, whereas amplitude variability rose steeply at the onset of the movement, peaked midway through, and, for most participants, dropped by as much as 25% when approaching the end. The difference between the two components was most evident for the proximal starting point (i.e., for the longest and fastest movements). This suggests a systematic effect of target distance on the components of the variability. To partial out this distance effect, we divided the coordinates of the intermediate points by the corresponding splineinterpolated values of the amplitude. Figure 15 (data for all participants) shows that the time course of these normalized components of the variability became almost identical for all starting points. Relative direction variability dropped soon after movement onset, reaching a stable level before 50% of MT had elapsed. By contrast, relative amplitude variability continued to decrease at a slower rate until at least 80% of MT. Paired t tests confirmed that this contrast between components was significant. At the.05 confidence level, differences between two successive estimates of the direction variability were no longer significant beyond 50% of MT, t(19) = 1.31, p =.100, for the 50%-60% difference. Differences for the amplitude component ceased to be significant beyond 80% of MT, t(19) = 0.588, p =.280, for the 80%-90% difference. Discussion In the introduction we argued that, to contrast alternative hypotheses about the control parameters in pointing movements, the experimental design must strain the capacities of the visuomanual coordination system. Suppressing all visual cues during the movement and varying the starting position of the hand was expected to impair the ability that people normally have to keep pointing accuracy largely independent of the initial posture. The design proved effective. We demonstrated that the initial posture of the hand was reflected in a principled way in both constant and variable errors. In addition, we exposed two other forms of inaccuracy that turned out to be useful for identifying the control parameters: (a) Pointing movements became progressively larger in the course of the sequence of trials and (b) the direction of the movement was biased toward the left with respect to the sagittal axis. We begin by considering these two findings.

15 VISUOMANUAL POINTING 583 e~ ~'0 la 0.~ Ic O. i Is nf O. go i o. t,o, xv o Frontal variable error (mm~ Figure 12. Individual results of the distribution of the variable errors for movements from the left starting point. Data points in each square indicate end-point positions relative to the average position computed on 40 trials. Error distributions are surrounded by the 95% tolerance ellipses. Also indicated in the inset is the normal coefficient of correlation for the cluster. Note the general rightward trend of the axes. Amplitude and Directional Biases The data in Figure 6 indicate that the process in which target position was mapped into the desired arm posture could not be maintained in stable operating conditions. Even considering that during the entire experiment participants never saw their arm or the environment, overshooting errors were surprisingly large compared with the distance between starting points and target (on average, 246 mm): Over the last group of 16 movements the average sagittal error reached 153 mm for Participant le, 93 mm for Participant ib, and exceeded 40 mm for 7 others. These values are much larger than those commonly found in the literature, with the only exception being the 100-mm undershooting errors reported by Flanders et al. (1992). To explain how such a conspicuous drift is compatible with the cross-point model, we can offer a plausible argument. The argument is suggested by the well-known fact that visuomotor calibration can be altered by position errors induced with prisms (Jeannerod, 1994) and even by manipulating motor images (Finke, 1979). Suppose that in one trial there is a mismatch between the visual estimation of target distance and the motor commands issued for pointing, so that the distance traveled is either larger or smaller than the actual target distance. In the absence of visual feedback, if the spatial error is small enough, finger position at the end of the movement, as provided by the position sense, will nevertheless be associated with target position. We assume, then, that although it has eluded consciousness, and precisely for that reason, the original mismatch recalibrates the relationship between target distance perceived visually and motor commands. Provided that the cause of the mismatch remains active, this sequence of events will repeat itself in the next trial. Although each adjustment is small, the cumulative effect over a series of trials may indeed result in the observed large drift of both the cross-point and the final positions. Our assumption that a common factor was responsible for both drifts was motivated by the positive correlation across participants between cross-point and amplitude gain (see Figure 11). At the onset of the experiment some participants undershot the target, some overshot it, and some were accurate. However, for 19 of 20 participants, the loss of stability manifested itself as a progressive increase of the distance traveled by the hand. Thus, the validity of our reasoning hinges on the further assumption that the original mismatch was biased in the hypermetric direction independently of the initial sensorimotor calibration.

16 584 VINDRAS AND VIVIANI ~. Proximal "~ ~o~ Distal , ii. :.. ~.~ ~'~' oo~ u,~..' Right "t 0.43 ~ Left 0.45 ")i i..'/ " Frontal variable error (mm) Figure 13. Distribution of relative errors Shown are the results for all participants and all starting points (cf. Figure 12). There are 800 data points in each distribution. The major axes of the 95% tolerance ellipses (continuous lines) coincide almost perfectly with the lines connecting the starting points to the target (dashed lines). Normal coefficients of correlations are indicated in the inset All participants but one pointed to a region of the workplane to the left of the target, a tendency that remained almost unchanged throughout the sequence of trials (see Figure 7). Such a directional bias has been documented previously (Berkinblit, Fookson, Adamovich, & Poizner, 1992; de Graaf et al., 1996; de Graaf, Sittig, & Denier van der Gon, 1991; Soechting & Flanders, 1989a) and is apparently related to the hand used for pointing. In fact, Berkinblit et al. (1992) reported that right-handed participants pointing with their left hand exhibited the opposite bias Moreover, testing 3 additional left-handed participants who used the dominant hand, we also found a significant shift to the right with respect to the main experiment (the average directional error was -0.36* vs. 3.22, respectively), t(21) = -2.82, p <.01, one-tailed. This modest deviation from the sagittal plane is likely the consequence of biomechanical factors that are normally compensated for by visual cues Alternatively, these biases could reflect different motor habits in left- and right-handers. Ghilardi et al. (1995) found that participants trained to point to targets in the left or right portion of the workspace made fight- and leftward directional errors, respectively, when requested to point to the center of the workspace. Thus, our directional errors, which are in quantitative agreement with those of Ghilardi et al. (1995, Figure 1C), could well have resulted from an entrenched preference to use the hemispace ipsilateral to the dominant hand Distal Right 15 E lo ~ 5 /I "6 0 P r o x i m ~ Left - ffl lo Percentage of movement time Figure 14. Variability of the trajectories. The standard deviations of the residuals for amplitude (filled data points) and direction (empty data points) were computed at 10 intermediate points along the trajectory. Amplitude variability was higher than direction variability and had a different time course.

17 VISUOMANUAL POINTING 5 85 Distal Right 0.20 A 0.15 "12_ ~ 0.'10._> to "o E to 0.20 "6 Proximal Left a) 0.15 to ~" 0.10 G) Percentage of movement time Figure 15. Variability of the trajectory. The data from Figure 14 were normalized to the amplitude of the ongoing movement. Most of the corrections in direction occur early on in the movement. Amplitude corrections remain relatively large throughout the movement. Evidence Against the Final Position Control Hypothesis At least two results conflict directly with the final position control hypothesis. The hypothesis (Polit & Bizzi, 1979) and its later qualifications (Bizzi et al., 1984) lead to a clear-cut prediction concerning the initial posture. If target position is mapped into fixed settings of muscle stiffness, and movement results from the spontaneous tendency of the muscle synergies to reach the new equilibrium configuration compatible with these settings, final positions ought to be independent of the initial posture. Indeed, one of the motivations for assuming that "posture and movement are subserved by the same processes" (Bizzi et al., 1992, p. 613) was precisely to account for such a purported independence. A similar independence also is predicted by the more general virtual trajectory hypothesis because the final point of this trajectory should coincide with the (invariable) target position regardless of the starting position. The large and significant influence of the starting point on the final position is inconsistent with this key prediction of the theory. Both the multiple regression analyses and the analysis of variance performed on the last half of the trial sequence extended to planar movements the conclusion reached previously in the one-dimensional case (Bock & Eckmiller, 1986; Cruse & Dean, 1992) that the program of goal-directed movements encodes movement amplitude rather than target position. Note that the errors documented here cannot be explained by the properties of the oculomotor system (Honda, 1984; Jeannerod, 1988, pp ) because our experiment, unlike that of Bock and Eckmiller, did not involve multiple fixations. The elongation of the distribution of variable errors in the direction of hand movement (see Figures 12 and 13) also speaks against the final position control hypothesis, which, at least in its basic version, predicts a roughly symmetrical distribution. As for the additional stipulations necessary to eliminate the discrepancy, note that the anisotropy of arm compliance within its working space (Flash & Mussa-Ivaldi, 1990; Lacquaniti, Carrozzo, & Borghese, 1993; Mussa-Ivaldi, Hogan, & Bizzi, 1985) cannot be invoked in this context because hand compliance depends only on the final posture of the arm and is actually minimum along all radial directions centered on the shoulder. Thus, if anything, compliance anisotropy would predict a larger spread of the errors along the tangential direction. Imbalance in muscle contractions, internal viscous forces, and frictional forces caused by the rubbing of the hand on the working surface are other potential causes for the increased variability in the direction of movement. However, as noted by Gordon, Ghilardi, and Ghez (1994) in discussing a similar elongated distribution of errors, unbalanced contractions, coupled with the inertial anisotropy of the hand (Hogan, 1985), would cause the hand to veer away from the intended course. In fact, hand trajectories in our experiment did not deviate significantly from a straight line.

18 586 VlNORAS AND VIVIANI Coding Direction and Amplitude: Evidence From Constant and Variable Errors There are many formally equivalent ways of representing the location of a point in space. Although the biological embodiments of these systems of representation entail different operations, no measure of performance can expose the difference as long as the systems work accurately. Each of them, however, may have its own characteristic way of going wrong. In particular, in a vectorial mode of representation, the radial and tangential components of the variability will not be balanced unless direction and amplitude errors are functionally related in a highly specific way. The main motivation of the experimental design was to exploit the discriminating power of the error pattern. Suppressing visual feedback enhanced both constant and variable errors with respect to normal conditions. Even if amplitude gain had not drifted, the fact that average final positions (constant error) reproduced the spatial disposition of the starting points (with or without double reflection) would already speak for vector coding. The case for this mode of coding, however, is made much stronger by the correlation between final position and amplitude gain, particularly if one considers that the transition from hypometric to hypermetric behavior is accompanied by a double reflection. On the basis of these findings, the simplest functional description of the motor plan for pointing is one in which the relative position of the hand with respect to the target is represented by a vector originating from the hand, and the movement is designed to null this vector. We emphasize that the different time course of amplitude and direction variability (see Figures 14 and 15) would be hard to explain if they were not also distinct parameters during the execution of the movement. Such a distinction is consistent with the finding that the activity of many neurons in the motor cortex, combined through the population vector algorithm (Georgopoulos, Caminiti, Kalaska, & Massey, 1983), yields a time-varying vector that points in the direction of the physical movement and has a length proportional to the movement speed. In fact, recent neurophysiological results in alert behaving monkeys (Schwartz, 1994) provide support for the strong hypothesis that the vector corresponds to a central representation of the intended trajectory, including both its spatial and temporal characteristics (Kalaska & Crammond, 1992). If target position were estimated accurately, pointing errors in the radial direction (gain errors) should be credited entirely to the computation of movement amplitude. The fact that, although significantly different, the final positions from the four starting positions remained closer to each other than predicted by a strict projective rule was taken to suggest that gain errors were compounded with an erroneous estimation of the target position with respect to the hand. The crosspoint model captured this distinction by admitting that the point that movements aim to may not coincide with the true target. Modeling Variable Errors The elongation of the distributions of pointing errors was found to be inconsistent with the hypothesis that target coordinates are translated directly into the appropriate set of joint angles (Soechting & Flanders, 1989b). To argue that the analysis of variable errors instead provides support to the vector coding hypothesis, we begin by spelling out the hypothesis in further detail. The complete process leading from target perception to hand positioning includes at least three steps: (a) coding and storing in short-term memory the extrinsic coordinates of the target (the nature of the reference system, Cartesian, polar, etc., is irrelevant here); (b) identifying the intrinsic coordinates (joint angles or limb orientations) that code the same target location in postural space; and (c) delivering the motor commands for reaching the desired final posture (see Figure 16C). The gist of the vector coding hypothesis is to postulate an additional processing stage in between the first and second steps (see Figure 16B) that transforms the extrinsic coordinates to represent the relative position of the target with respect to the starting position as a vector (i.e., by an amplitude and a direction). Crucially, this implies that at some stage the motor plan takes into account both the starting and end point of the movement. The vector and the joint angles corresponding to the initial posture are then used to solve the problem of defining the angles required for correct pointing (inverse kinematics). However one chooses to describe the complete process, it is safe to assume that a source of error is associated to each intervening stage, so that the final position is better seen as a two-dimensional random variable. The distribution of this variable depends on the statistical properties of the noise sources as well as on the specific sequence in which they come into play. Clearly, the number of degrees of freedom involved makes it impossible to infer the noise properties and the sequence of processing stages from the distribution of the variable errors. However, by making the simplifying assumption that all the noise sources have a Gaussian distribution with unspecified means and variances, it is possible to show that the two schemes in B Figure 16 (opposite). Modeling end-point variability. A: Schematic representation of the arm and forearm. There is a one-to-one correspondence between end-point position and joint angles. B: Block diagram representation of the end-point control hypothesis. The coordinates of the target are mapped directly into the desired joint angles. C: Block diagram representation of the vector coding hypothesis. The target position is coded as a Cartesian vector originating from the initial hand position. D: Distributions of the variable errors simulated according to the vector coding Scheme C. E: Distribution of the variable errors simulated according to the end-point control Scheme B. Compare these simulations with the corresponding data in Figure 13.

19 VISUOMANUAL POINTING 587 I' U ~,_~ ~ ~ LU " 0. ~~, ' ~,..,,,~ ~,,;..,, ~,~ ~.. ",~,~ m ~ L o u e ~ a~,l~,,~'-~-.,..~..~-,, ~ ~..~,~ ~, :, ~.,,... ~.~. : ~_~.. ~.~ :: -~ ~._.~" -. t"~ " T o x v ~0 0 n o ~ -I o= 0,~ u ~ =~ o- x,- ~ a3 C_~

20 588 VINDRAS AND VIVIANI and C of Figure 16 lead to contrasting qualitative predictions about the error distribution. For the purpose of demonstrating this point, we schematize the moving limb as a 2-dfbiomechanical chain (see Figure 16A) whose parameters are the indicated joint angles a and b. Considering that the elbow angle cannot exceed 180, the inverse kinematic problem of calculating the intrinsic angular coordinates corresponding to any given hand position has a unique solution. We performed a Monte Carlo simulation of the movements from all starting points by computationally implementing the two functional schemes of Figure 16. For Scheme C the simulation included the following steps: (a) The coordinates of the perceived target (i.e., the crosspoint) were computed by adding an uncorrelated two-dimensional Gaussian noise N(mr, st) to the true target coordinates. (b) After calculating amplitude and direction of the oriented vector from the starting point to the cross-point, and multiplying the amplitude by a fixed gain, we added to both amplitude and direction independent Gaussian noises with zero means and standard deviations SA and so, respectively. (c) From the vector and from the initial joint angles we calculated the final joint angles and, once again, added a zero-mean Gaussian noise with equal variance sc to each intrinsic coordinate. (d) Finally, the final position was calculated back using the forward equations shown in Figure 16. For Scheme B, which does not include a vector coding stage, we simply skipped the second step and solved the inverse kinematic problem directly from the cross-point coordinates. For each starting point, we simulated the same number of trials (800) that were actually performed by all participants. For the simulation of the vector scheme, the biomechanical parameters were estimated from the anthropometric data of a sample of participants: L1 = 35 cm, L2 = 50 cm, x0 = 30 cm, Y0 = - 12 cm. The amplitude gain was set to 1.2, a value typical of late trials. The noise parameters were then selected by a trial-and-error procedure to mimic the results of Figure 13. The mean and variances were set as follows: mr = 0, sr = 0.5, SA = 0.05, SO = 1.2, and sc = As shown in Figure 16D, this choice of parameters reproduced accurately both amplitude and direction of the principal axes of the ellipses of tolerance for all starting points. For the simulation of the second scheme, we scaled up the variances of the two remaining noise sources to equate the surface of the ellipses of tolerance to that of the data. The results (see Figure 16E) were grossly at variance with the experimental data. Moreover, different partitions of the noise yielded even worse approximations. Apparently, the nonlinearity of the equations relating extrinsic and intrinsic coordinates was insufficient to account for the elongation of the error distributions in the direction of the movement. The inadequacy of our streamlined computational version of the hypothesis that planning occurs in joint-coordinate space does not, of course, rule out the possibility that a better fit can be obtained with the help of additional stipulations on the correlation among noise sources. This, however, should be motivated by direct evidence that is not yet available. Couching the cross-point model in simple geometrical terms was sufficient to support the hypothesis of a vector coding stage within the complete process of motor planning. The main conclusions, however, are likely to remain valid if one were to consider more physiologically minded qualifications of the hypothesis. Specifically, the recent development of the lambda model (Feldman & Levin, 1995) postulates that, in pointing to a stationary target, the control variable lambda produces movement by shifting an extrapersonal frame of reference from the starting to the final location. In extrapersonal space the control variable can be depicted as a velocity vector associated with the more distal segment of the arm (Flanagan, Ostry, & Feldman, 1993). Thus, by a judicious choice of its parameters, the lambda model may well predict a distribution of the variable errors similar to those shown in Figure 13. It is not clear, however, whether the lambda model would also predict the dissociation between amplitude and direction documented in Figures 14 and 15. We offer a final comment concerning the lambda model. The notion of equilibrium state plays a central role within the dynamical framework of this model, which in fact conceives of pointing movements as transitions between two such states. Because of its limited scope, our simple geometrical analysis did not take into account the processes by which a stable posture is maintained before and after the movement. However, to the extent that--as argued earlier-- the shift of the frame of reference postulated by the lambda model is consistent with our analysis, the main conclusions reached in this study should not depend on how stable postures are maintained. An Alternative View: Postural Matching Postulating an intermediate processing stage in which the relative position of the target with respect to the hand is coded vectorially permitted us to account naturally for certain features of the constant and variable errors. Yet, we found it worthwhile to explore another possibility to account for the pattern of errors that was different from both the final position control and the vector coding schemes. As summarized in the introduction, Flanders et al. (1992) proposed that the target position, initially coded in an extrinsic system of reference centered on the shoulder, is subsequently recoded into a four-dimensional postural reference by calculating, separately for the forearm and the arm, the yaw and elevation angles that would bring the hand to the target. The motor plan for pointing is then supposed to involve (a) comparing this postural representation of the target with the initial posture of the limb, measured kinestheticauy and represented in the same reference, and (b) computing the motor commands that transform the second posture into the first. We tested a somewhat similar hypothesis again based on the observation that in two-dimensional movements, postural angles are uniquely defined by the hand position. We assumed that the perceived target position would be transformed into the shoulder and elbow angles that would make hand and target overlap and that the corresponding angles for the initial posture also would be available. Then,

21 - (b'~ VISUOMANUAL POINTING 589 at least in principle, the motor plan can set up directly in terms of the commands that would null the difference between the initial and final angles. The variable errors predicted by this scheme depend critically on the assumptions concerning the sources of noise. Because assessing the initial and final postures, and computing the commands are conceptually separate processes, it is natural to suppose that noise affects each process independently. Specifically, with reference to the scheme shown in Figure 16A, we supposed that the actual final angles are expressed as as = (a~ + Nal) + (1 + Nac)[a~ + NaF) -- (a~ + N~I)] be = (b* + Nbl) + (1-4- Nbc)[b} + NbF) - + Nbi)] where (a% b*i) and (a'f, b'f) are the theoretical joint angles corresponding to the initial and final posture, respectively. N,a, Nbt, NaF, and NbF are uncorrelated Gaussian random variates with unspecified means and unspecified but equal variances. Nac and Nbc are correlated Gaussian random variates with an unspecified mean and variance. The additive noise terms Na~, NbI, NaF, and NbF affecting the theoretical angles compound the variability associated with the perceptual specification of the target and the variability introduced by the translation of positional information from extrinsic into intrinsic coordinates. The multiplicative noises Nac and Nbc are instead supposed to capture the variability of the motor command, which is proportional to the angle differences (a% + Nat:) - (a*l + NaI) and (b* F + NbF ) -- (b*l.4- Nbi). Using the same biomechanical parameters of the previous analyses, we performed a similar Monte Carlo simulation of this scheme to determine the noise parameters that resulted in the best approximation to the experimental error distributions shown in Figure 13. The salient results were as follows: Concerning the constant errors, the existence of a cross-point, as well as the observation that final positions could either undershoot or overshoot the crosspoint, could be simulated qualitatively by appropriately choosing the means of the additive and multiplicative noises, respectively. Concerning the variable errors, when the multiplicative noises Nac and Nbc were either independent or weakly correlated, the predicted distributions for all starting points shared a roughly common direction that was orthogonal to the direction of the movement. Thus, they were at variance with the data. Only when the correlation approached one did the orientation of the distributions tend to be aligned with the direction of the movement. Even in this case, however, the fit to the data was less accurate than the one obtained with the vectorial scheme. Moreover, the assumption that the fluctuations in the motor commands for flexing shoulder and elbow would be highly correlated is somewhat dubious because these commands may originate from distinct cortical areas (Brinkman & Kuypers, 1972, 1973). Conclusion We contrasted three general hypotheses concerning the planning and execution of pointing movements performed in the absence of visual feedback. A control strategy based exclusively on the desired final posture could not explain the systematic relation between starting points and constant errors. Moreover, it failed to account for the distribution of variable errors, both at the end and during the movement. The hypothesis that the motor plan is based on the comparison between the initial and final joint angles at the shoulder and elbow fared better because it was compatible with the observed pattern of constant errors. 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