INTRODUCTION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

In recent years, it has become evident that the study of movement in two- and three-dimensional space introduces questions that are not apparent in the study of one-dimensional movement. With respect to arm movements, this has been demonstrated by various investigators who have recorded from motor and premotor cortical areas during movements to stationary targets and have shown movement direction to be prominently represented in the activity of these neurons (e.g., Fu et al. 1995; Georgopoulos et al. 1986; Kalaska et al. 1997; Schwartz et al. 1988). Much less is known about the control of arm movements for tasks in which the target itself is moving in space, such as tracking or intercepting a target moving in two dimensions (Johnson et al. 1999; Port et al. 1997; Viviani et al. 1987).

Sensory reception, neural computation, and motor output all require a finite amount of time. Therefore time delays are inherent in the task of manually tracking a moving target. However, during the normal tracking of a predictable target moving along one dimension, the tracking error can be very small (Poulton 1974), implying that predictive algorithms are employed by the nervous system. Studies of tracking in one dimension have shown that this predictive behavior is generated by a velocity error signal in combination with a positional error signal (Poulton 1974; Viviani et al. 1987; see also Lisberger et al. 1987). The question then can be posed: what is the form of the error signal for tracking in two dimensions?

The current study was undertaken to determine how speed and directional error signals are used in two-dimensional tracking. To this end, we asked subjects to track a target that moved initially in a straight line and then changed direction (and sometimes speed) abruptly. We identified a rather unexpected strategy. A conceptual model based on a constant time to intercept could predict the new direction of the finger motion as well as its maximum speed. This conceptual model provided the basis for a formal quantitative model in which direction and speed are the controlled variables.

	METHODS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Motor task

The manual tracking of targets moving in two dimensions was assessed in six different experimental conditions. Before describing each of the experimental conditions in detail, we will describe those aspects which were common to all experiments.

Subjects sat in front of a touch-sensitive computer video monitor. Seat height was adjusted such that each subject was comfortable and could easily reach all areas of the video screen. Room lighting was dimmed to increase the contrast of the display. No restrictions were imposed on head or eye movements. Each experiment typically consisted of 300-360 trials and lasted from 45 min to 1 h. The subjects gave their informed consent to the experimental procedures, which were approved by the Institutional Review Board of the University of Minnesota.

All subjects were right handed and were asked to track, with their right index finger, the motion of a target presented on the video monitor. In most experiments, a box 1.6 cm per side initially appeared 1.6 cm from the top edge of the screen to indicate the starting position for the subject's finger. When the subject placed his or her finger in the box, a round target 1.6 cm in diameter appeared at the same edge of the screen and began to move at a constant downward velocity toward the box (see Fig. 1). Subjects were to begin tracking the target as soon as it entered the box. In all cases, the target initially moved at a constant speed and in a direction that was constant from trial to trial. After the target had traveled a random distance of from 10.9 to 17.4 cm, it made a single abrupt change in direction. On average, the target motion changed direction when the target was in the middle of the screen ~1-2 s after the start of the trial.

View larger version (16K): [in this window] [in a new window]   Fig. 1. An example of tracking performance during a directional change in target motion. Top: results from a single trial. The thin line denotes the path of the target with the arrows indicating the direction of target motion. The open arrow (slightly below the start of the target's trajectory) marks the starting position of the finger. After the change in target direction, the position of the finger and position of the target are connected every 100 ms by a thin straight (isochronic) line. Inset: description of how angular changes, both of the target and of the finger, are defined with counterclockwise rotations from the vertical defined positive. Bottom: all 10 trials from this subject.

Experiment 1: tracking a target moving at a constant speed

In this experiment, the target appeared at the middle of the top edge of the screen, and moved straight downward at a speed of 10.8 cm/s. Then after the target had traveled a random distance, it made an abrupt change in direction to 1 of 24 equally spaced directions. These directions were varied randomly from trial to trial. The speed of the target remained constant throughout the experiment. Four subjects participated in this experiment.

Experiment 2: tracking an obliquely moving target

To determine whether the results from experiment 1 could be generalized to any initial direction, the experiment was repeated, with one modification: the initial direction of finger tracking was rotated counter-clockwise through 135°. Therefore the start box for the subject appeared in the lower left-hand corner of the video screen rather than at the top middle. The initial target motion was upward and rightward, rather than straight downward. As in experiment 1, target motion changed unpredictably to 1 of 24 equally spaced directions. Four subjects participated in this experiment.

Experiment 3: the effect of target speed on tracking

In this experiment, the target again appeared at the top, middle section of the screen and initially moved straight downward. At a random point in time, the target then changed to one of six equally spaced directions. The speed of the target remained constant throughout any particular trial but was varied randomly across trials. Four speeds were used: 5.4, 10.8, 16.2, and 21.7 cm/s. Five subjects participated in this experiment.

Experiment 4: tracking during an abrupt change in speed

In experiments 1-3, during any particular trial, only the direction of the target's motion varied. Experiment 4 was conducted to determine whether the results from these three experiments generalize when target speed also changes unpredictably. In experiment 4, the target again appeared at the middle of the top edge of the screen, moved straight downward at a constant velocity of 10.8 cm/s, and then changed to 1 of 12 equally spaced directions. However, the target sometimes also changed speed (unpredictably) at the same time it changed direction. In one third of the trials, the target abruptly slowed from its initial speed of 10.8 cm/s to a speed of 5.4 cm/s. In another third, it abruptly increased speed to 16.2 cm/s. In the last third, it maintained its original speed of 10.8 cm/s. Six subjects participated in this experiment.

Experiment 5: constant vertical velocity/variable horizontal velocity

As in experiment 1, the target initially moved downward at a speed of 10.8 cm/s. The motion of the target then changed unpredictably to 1 of 12 directions. For 7 of these 12 directions, target speed also underwent a step change. In these instances, the vertical component of velocity was held constant throughout the entire trial and the horizontal velocity underwent a stepwise change from 0 to some new constant value. Seven values of horizontal velocity were used such that the resulting directional change of the target was 0, ±22.5, ±45, or ±67.5°. Consequently, the speed and direction of the target's motion changed simultaneously as in experiment 4. For the remaining five directions, the target's motion changed to one of five upward directions (180, ±150, and ±120° with respect to the downward direction); the target's speed being held constant throughout the trial (as in experiment 1). Four subjects participated in this experiment.

Experiment 6: the effect of target acceleration on tracking

The parameters for experiment 6 were nearly identical to those for experiment 4 except that acceleration was changed instead of speed. In experiment 6, the target either accelerated or decelerated at a rate of 5.4 cm/s² or maintained its original speed of 10.8 cm/s, at the time when its direction of motion changed. Five subjects participated in this experiment.

Recording system

The experiments were performed using a touch screen (Elo Touch Systems, TN) mounted over a standard 20-in computer monitor (Mitsubishi Diamond Scan 20 M). The touch screen has a spatial resolution of 0.08 mm. The target motion and recording of finger position were controlled by a laboratory computer using custom software. The position of the finger was recorded at a rate of 100 Hz. Target location was updated at a rate of 60 Hz, equal to the refresh rate of the video monitor. The output of the touch screen was scaled and aligned with the video image through the use of cubic polynomials and a rectangular reference grid of target positions (see Flanders and Soechting 1992).

Data analysis

Data were averaged by aligning the trials on the point at which the target changed direction. All subsequent analysis was performed on averaged data. Velocity was calculated by numerically differentiating the position data and digitally smoothed using a two-sided exponential filter with a cutoff frequency of 12 Hz. As will be shown in RESULTS, after the target changed direction, the finger maintained the original target direction for a reaction time period, changed direction, initially headed in a nearly straight line to intercept the target, and then finally curved to merge with the new target direction. This heading to intercept the target was defined by computing the inverse tangent of the ratio of the horizontal and vertical velocities at 350 ms after the change in target direction. This point in time was chosen because it is generally in the middle of the straight interception period, at a time when tracking speed was increasing (see Figs. 4B, 8, and 11).

To determine when two averages of either speed or direction began to differ from each other, we performed a t-test at each point in time. The averages were said to diverge once the 0.05% confidence level was reached and the two series remained separated by at least this level of confidence for the next 70 ms. To determine reaction time (defined as the interval between the time at which target motion changed direction and the first observable change in the finger's trajectory), a baseline period was defined by averaging both direction and speed data over the interval from 150 ms before the target changed direction to 100 ms after the target changed direction. The standard deviation of direction and speed was also computed for this same 250-ms interval. The subject's reaction time was then defined as the point in time beyond this baseline interval when either finger speed or direction exceeded the 2 standard-deviation limit and continued to exceed it for at least 30 ms.

	RESULTS

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

Response to a change in target direction (experiment 1)

To study manual tracking in two dimensions, we began with the simplest experiment, that of a target moving downward at a constant speed and making a single change in direction. Figure 1, top, shows the result for one trial of tracking. The path of the target is shown by the thin line. The direction of target motion is indicated by arrows. In this example, the target changed direction to  = 135°. The path of the finger is shown by the thick line. (During the vertical segment of the target's motion, the path of the finger obscures the path of the target.) To demonstrate how the motion of the finger with respect to the target evolved over time, the position of the finger and the position of the target were joined by a thin (isochronic) line, every 100 ms, starting at the point where the target changed direction. After this point in time, the finger maintained its original downward trajectory for a period of time, slowed in speed to change direction, then accelerated to reacquire the target (see Fig. 4B). For this example, the reaction time was found to be 230 ms using directional data and 210 ms using speed. For this subject, for all directions, the average reaction time was 239 ± 25 (SD) ms for direction and 235 ± 26 ms for speed. Averaging across all directions for all subjects, the reaction time was 229 ± 24 ms for direction and 227 ± 28 ms for speed. There was a statistically significant effect of target direction on reaction time (ANOVA, P < 0.05). However, a post hoc comparison (Tukey HSD) showed that the reaction times did not depend significantly on the amount by which the target changed direction for changes exceeding 30°. For smaller changes in target direction, the estimated reaction times were about 30 ms longer, but these estimates are not as reliable because the signal to noise ratio is much smaller in these cases (see Fig. 4). These reaction times are in general agreement with previous findings (Hanneton et al. 1997; Poulton 1974).

Figure 1 illustrates a general aspect of our results: after the finger's motion changed direction, the hand initially headed in a nearly straight line before curving to merge with the path of the target. It is clear that this new heading is not directed toward the current location of the target. (The target's location can be noted by considering the isochronic line that most closely connects finger and target location at the time of the directional change.) Rather, the finger heads in a direction anticipating the future location of the target.

In Fig. 1, bottom, all trials from this subject for this direction are shown with the trials aligned on the point at which the target changed direction. The consistency in the handpaths illustrated was typical of the results we obtained in this and the other subjects. We computed the deviation of finger position (the square root of the sum of the variances in X and in Y). For this example, the average deviation of the finger position after the change in target direction was 0.56 cm. This deviation was not significantly related to target direction (slope not significantly different from 0, P > 0.36). Across all subjects and directions the average deviation was 0.63 ± 0.1 cm. Since this value was fairly small, we restricted our analysis to averaged data.

Figure 1 shows the results from one out of 24 target directions tested in this experiment. Figure 2 summarizes the results that were obtained from one subject for 8 of the 24 directions, the dotted lines denoting the paths of the target. The finger trajectories begin to diverge from a common point, and during the reacquisition of the target, the path of the finger appears to be reasonably straight for a considerable period of time. Finally, once most of the positional error has been eliminated, the path of the finger curves to merge with the path of the target. Therefore while the movement is continuous, for the purpose of discussion, we may consider it as occurring in four steps: a "reaction phase," in which the subject continued on the original target heading, a change in direction, a "reacquisition phase," in which the subject moved in a relatively straight, anticipatory path to reduce the error between the target and the finger, and a gradual "merging" of finger velocity with target velocity.

View larger version (21K): [in this window] [in a new window]   Fig. 2. Tracking paths for 8 of 24 target directions. Dotted lines represent the path of the target and the thick lines represent the average trajectories of the finger. Inset: approximation to the behavior: the finger trajectory is represented by 2 straight lines. With this approximation, the finger appears to intercept the target at a constant time, represented by the circle centered on the point where the target changes direction. Thin lines emanate from the finger position at the end of a constant reaction time (250 ms) and intersect the target paths on the perimeter of the circle. Note that these lines generally parallel the straight line portion of the finger path following the reaction time. (The fit between the model and experimental data are better for target changes to the right, as the finger position on the touch screen deviates slightly for right-handed subjects moving toward the left hand of the screen.)

Simple model for tracking directional changes in target motion

In an attempt to gain some understanding as to how the reacquisition of the target is controlled by the nervous system, we developed a simple geometrical model of the subjects' behavior. In particular, we approximated the finger path by two straight lines, one representing the movement of the finger during the reaction phase, the other representing the movement of the finger during the reacquisition phase (see Fig. 2, inset). We also assumed that the reaction time did not depend on the amount by which target motion changed direction (see preceding text). These are clearly oversimplifications because the finger does not change direction instantaneously, because there is some curvature in the path of the finger, and because there may be some slight variability in the reaction time. However, this conceptual model allowed us to test a hypothesis that was suggested by the data: the path of the finger merges with the path of the target at a constant time after the target changes in direction, independent of the amount of the directional change. (In this experiment, the target moved a constant distance in a constant time. Accordingly, the predicted point of interception can be represented by a circle centered on the time of the target's change in direction, as in Fig. 2.)

Because the path of the hand merged gradually with the path of the target, we could not measure this time with any confidence. However, using the simple model we could test a corollary of the hypothesis: the motion of the finger during the "reacquisition phase" is in a direction such as to intercept the target at a constant time. For the reacquisition phase of the finger, we predicted the direction of travel () such that the points of interception were a constant distance (d_t) away from the point in space at which the target changed direction, assuming a constant distance of travel (d_r) during the reaction time. Therefore by trigonometry, for each target direction the distance the finger must travel to intercept the target is simply

(1)

Furthermore, the direction to intercept the target is given by

(2)

The straight lines in Fig. 2 represent the idealized finger paths of the model, from the point representing the reaction distance (d_r) to the point where the path of the target intercepts the circle. The model fits the data reasonably well. In this example, the fit appears better for counterclockwise changes in target direction. The larger error for clockwise rotations may be due to a slight rotation of finger position on the touch screen. This typically occurred during leftward movements of the right-handed subjects.

To evaluate the extent to which this model fit the experimental data, we used an error minimization algorithm (Nelder and Mead 1964) to compute and compare the predicted heading of the finger (Eq. 2) with the heading measured experimentally. Figure 3A shows how well the model was able to predict the heading of the finger. The plot illustrates the angular difference between the initial heading of the finger and the heading of the target during the reacquisition phase plotted as a function of target direction. The crosses represent the average angular differences recorded for the 24 target directions for one subject. The smooth curve represents the angular differences calculated by the model. For this subject, a ratio of target reacquisition distance (d_t) to reaction distance (d_r) of 3.63 provided the best fit. [Assuming a reaction time of 250 ms, this would imply a time to reacquire the target (reaction time + interception time) of 900 ms.] The model accounted for 94% of the variance in angular differences across directions for this subject. For the four subjects, the ratio of d_t/d_r ranged from 2.51 to 3.63 (see Table 1), and the variance accounted for by the model was above 0.85 in each case.

View larger version (19K): [in this window] [in a new window]   Fig. 3. A comparison between the model and experimental performance. Experimental data (+) for 1 subject and the model's prediction are shown for the direction (A) and maximum velocity (B) of the finger's trajectory. In A, angular difference is defined as the difference between the angular change of the finger and the angular change of the target. Finger heading was determined by taking the inverse tangent of the ratio of horizontal and vertical velocities at 350 ms after the target changed direction. In B, the straight line represents the speed of the target.

The simple conceptual model also makes clear predictions about the maximum speed of the finger during the reacquisition phase. If one is to intercept a target moving at a constant speed at a constant time, the average speed of the finger must scale proportionally to the distance traveled (d_i). If the speed profile is similar for each direction (e.g., is bell-shaped), then the maximum speed should be proportional to d_i. Therefore the variation in maximum finger speed with direction should be predicted by Eq. 1. Figure 3B displays the average peak speed reached by the finger for another subject as a function of target direction (crosses). The smooth curve represents the prediction of the model. Finally, the horizontal line indicates the target speed (10.8 cm/s). For this subject, the scaling factor relating interception distance to peak velocity was 2.73, while the variance in peak speed accounted for by the scaled distance was 0.93. For the four subjects, the scaling factor relating interception distance to peak velocity ranged from 2.70 to 4.54 (Table 1), and the variance in peak speed during the reacquisition period accounted for by the distance to the reacquisition circle ranged from 0.89 to 0.94.

To better demonstrate the modulation of the speed profile as a function of interception distance, finger speed was plotted in two different formats in Fig. 4. Figure 4A shows how speed varied with time for one subject for 12 of the 24 target directions. The traces all begin at the time the target changed direction; the baseline indicates the speed of the target. In each case, initially the speed of the finger was slightly slower than the target's speed, indicating that this subject was slowly starting to fall behind the target (perhaps anticipating the change in target direction). Then after a nearly uniform period of time (the subject's reaction time), the finger's speed changed, first slowing to allow for the change in direction of the finger, then accelerating to reacquire the target. (For 0°, there was no change in target direction and the finger maintained its original velocity throughout.) As the amount by which target motion changed direction became greater, the amount of deceleration as well as acceleration also increased such that the time to reacquire the target remained constant.

View larger version (25K): [in this window] [in a new window]   Fig. 4. The temporal profile of finger speed during target reacquisition. A: the finger speed for 12 of the 24 target directions for 1 subject. In each case, the baseline indicates the speed of the target. The horizontal and vertical scales are identical for each trace. Each trace begins at the point when the target changed direction. The variations in speed have been superimposed for 13 of the 24 total target directions (half of the circle) in B. Line thickness increases with increasing angular changes of the target.

Figure 4B, where the traces for the 13 directions ranging from 0 to 180° (the right hemisphere) have been superimposed, provides further support for our hypothesis. One can observe that there are only minor variations in the reaction time. Furthermore the duration of the reacquisition phase does not depend on target direction; after ~750 ms the finger has re-assumed the velocity of the target. In summary, since the distance to intercept the target is not constant, it appears that both the direction and speed of the finger's motion are coordinated in such a manner that the time to intercept, or possibly the distance the target travels before interception, is held constant.

Obliquely moving targets (experiment 2)

The initial path of the finger in the first experiment was always straight downward. It is possible that the tracking of a downward moving target is somehow unique or that gravity might have an effect on tracking performance. To determine whether the results from experiment 1 can be generalized to other tracking directions, the experiment was repeated with the target initially moving obliquely at an angle of 135°. Figure 5 shows the paths of the target and of the finger to 8 of the 24 directions (left). As was the case in experiment 1, the hand generally headed in a straight line to intercept the target. Also as was true for initial downward target motion, the simple model was able to account for the angular differences between finger heading and target heading as well as modulation in the finger's peak speed (right).

View larger version (22K): [in this window] [in a new window]   Fig. 5. Tracking obliquely moving targets. Left: path of the target, which always started in the lower-left corner of the screen and the average path of the finger during tracking to 8 of the 24 directions tested. Right: comparison between the model and tracking performance of the obliquely moving targets. The conventions are the same as those used in Fig. 3.

For the four subjects, the ratio of target interception distance (d_t) to reaction distance (d_r) varied from 2.84 to 3.77. In this experiment, the variance accounted for by the model ranged from 0.92 to 0.97. There was one subject who participated in this experiment as well as in experiment 1. The results for this subject were almost identical, with ratios of 3.71 and 3.63 respectively. Again, in this second experiment, the peak speed for all subjects was well correlated to the target interception distance. For the four subjects in this experiment, the factor relating distance to peak speed ranged from 3.17 to 4.44 with the variance in peak speed accounted for by the distance to reacquisition ranging from 0.86 to 0.95.

Effect of target speed on tracking performance (experiment 3)

In the first two experiments, the speed of the target was the same for all trials. We observed that the initial heading of the hand () was such as to reacquire the target at a constant time. Since the target moved at a constant speed, a constant distance to intercept (d_t, Eqs. 1 and 2) is equally consistent with the data. By examining manual tracking at different target speeds, it was possible to differentiate these two possibilities. To accomplish this, we repeated experiment 1 varying the speed of the target randomly from trial to trial over a fourfold range. As will be demonstrated in the following text, if the goal is to keep time to intercept constant, the initial heading of the finger () should not depend on target speed. However, if the goal is to keep distance (d_t) constant,  should depend in a predictable manner on target speed, becoming larger for faster speeds (to make up for the extra distance traveled during the reaction time).

Figure 6, A and B, shows the results from one subject for two different target directions. The thin line represents the path of the target, which was the same independent of target speed. The bold lines represent the average finger paths for each of the different target speeds. There was a small but statistically significant (ANOVA, averaged data for all subjects and all directions, P < 0.01) effect of speed on reaction time. A post hoc pairwise comparison showed that the reaction time for the slowest speed was longer (13%) than for the other three speeds. This difference in time was small (33 ms) compared with the mean reaction time of ~250 ms. Since the reaction time is approximately constant, the distance traveled during the reaction time is approximately a linear function of the target speed.

View larger version (14K): [in this window] [in a new window]   Fig. 6. The effect of target speed on tracking performance. Thin lines indicate the path of the target traveling at 4 speeds (5.4, 10.8, 16.2, and 21.6 cm/s), changing by 135° (left) and 60° (right). Bold lines demonstrate the average path of the finger for each of the 4 speeds. After the reaction time, finger paths are parallel to each other, intercepting the target at a constant time.

After the finger changed direction, the four traces of the finger position during the reacquisition phase were parallel with each other. An ANOVA showed that for four of the five subjects, there was no effect of speed on reacquisition direction (P > 0.05). For the one remaining subject, linear regression showed that for only one of six target directions was the relation between speed and reacquisition direction significantly different from zero (with a slope of 0.44°/cm/s, corresponding to a 7° difference in heading between the slowest and fastest target speeds). In conclusion, the distance traveled by the finger during the reaction time scaled with target speed and the directions of the finger paths during the reacquisition phase were parallel to each other at the four speeds. Therefore from the law of similar triangles, the finger intercepted the path of the target at a distance that also scaled with target speed. Accordingly, since the target traveled at a constant speed throughout each trial, the time not the distance to reacquisition remained constant, irrespective of the speed of the target.

Response to an abrupt change in target speed (experiment 4)

In all of the experiments described so far the speed of the target was constant and thus predictable throughout any given trial. The question can then be posed, do the results generalize when speed as well as direction changes unpredictably during the trial? We explored this question by introducing, in some trials, a step change in the speed of the target at the same time its direction changed. Since the speed during the initial downward tracking segment was constant, the reaction distance remained relatively constant as well. This is evident in Fig. 7A where the traces for all three averages overshoot the change in target direction by nearly the same amount.

View larger version (19K): [in this window] [in a new window]   Fig. 7. The effect of changing target direction and target speed simultaneously. The thin line in A represents the path of the target. Target speed was constant (10.8 cm/s) during the downward phase. When the target changed direction, target speed either decreased (5.4 cm/s), increased (16.2 cm/s), or remained constant. Bold lines show the average path of the finger during tracking. The scale along the target path demonstrates the position of the target at a constant time for the 3 speed levels. B: a comparison of the direction of finger motion with that predicted by the constant time to intercept model. The angular difference between target and finger heading is shown for each target speed. Smooth curves represent the angular difference predicted by the model assuming the same time to intercept for all 3 speeds.

The direction of the finger motion during the reacquisition phase depended on the new speed of the target. Figure 7B shows the results of fitting Eq. 2 to these data. The three curves of the model were calculated such that the time to intercept was the same for all three speeds (i.e., d_t was proportional to target speed). For this subject, the model explained 89% of the variance for those trials where the target abruptly slowed, 95% of the variance for those trials where speed did not change, and 99% of the variance for those trials where the target abruptly increased speed.

Therefore it appears that the subject was able to detect the change in speed of the target and scale the speed of the interception trajectory such that time to contact remained constant. This can also be appreciated in Fig. 7A. The three tick marks are equally spaced and represent the position of the target at the modeled interception time for the three speeds. The location of these ticks correspond well to where the extrapolated linear motion of the finger intercepts the path of the target. The results in Fig. 7 are representative of the results for the five subjects who participated in this experiment. For all subjects, the constant time to intercept model was able to fit the data at all three speeds with a considerable amount of fidelity (Table 2).

Figure 8 demonstrates that the reaction time for a change in target speed is similar to the reaction time to a change in target direction. The plots provide a comparison of the two instances in which the target either increased or decreased in speed at the time of the directional change. The dashed line indicates the time at which the target changed both direction and speed. The top panel describes the speed of the target, while the bottom panels show the speed and direction of the finger for both conditions. We performed a statistical comparison between the finger data for the two target speeds to determine the time at which the two curves first diverged (P < 0.05, see METHODS). In Fig. 8, this time is denoted (). For this subject, the speed traces were found to diverge 250 ms after the target changed speed. Note that finger speed decreases at about the same time (~175 ms) for both target speeds, but that the finger begins to accelerate earlier when the target speed is increased. Across all subjects and directions the average time at which the speed of the finger first differed significantly (P < 0.05) for slow and fast target speeds was 270 ± 40 ms after the change in target speed, slightly larger than the reaction time found for finger speed to change in response to a change in target direction.

View larger version (17K): [in this window] [in a new window]   Fig. 8. Reaction time to a change in target speed. - - -, point in time at which target speed and direction were varied. The target either increased to 16.2 cm/s or decreased to 5.4 cm/s (top). Bottom traces: time profiles of finger speed and direction. , time at which the 2 movements diverged from each other.

Direction of finger motion also depended on target speed, with a reaction time that was again comparable to that found for a change in target direction. In this instance (Fig. 8), the direction of the finger diverged 180 ms after the target changed speed. Across all subjects and directions, finger direction for the two target speeds diverged on average 280 ± 80 ms after the target changed speed. This value is comparable with the reaction time for the finger direction to change in response to a change in target direction.

Tracking a constant downward velocity (experiment 5)

There has been considerable work, both in terms of experimentation and modeling, attempting to understand manual tracking performance in one dimension (cf. Poulton 1974; Viviani et al. 1987). One might conjecture that tracking in two dimension (2-D) is equivalent to two simultaneous cases of one dimensional tracking occurring in the X and Y directions. To ascertain whether this viewpoint was viable, we conducted the following experiment. As in previous experiments, target direction changed randomly through angles encompassing all 360°. However, for those cases in which the change in direction was <90°, (maintaining a downward directional component), the speed of the target was modified such that the vertical (Y) velocity remained constant, and the horizontal (X) velocity underwent a step change. If manual tracking can be decomposed into two cases of independent tracking along orthogonal axes, then one would expect that the addition of an X component should not perturb the tracking in the vertical dimension.

Figure 9, left, shows the path of the target as well as the average path of the finger for one of these downward trajectories. Figure 9, right, displays the X and Y components of the target and finger velocity. - - - indicates the time at which the target changed direction. For the X component of finger motion, after ~200 ms the finger accelerated, exceeded, and eventually matched the velocity of the target. At a comparable latency (~200 ms), the Y velocity of the finger first decreased, then increased to reacquire the target, even though the Y velocity of the target did not change. The same general pattern was seen in all four subjects for all angles. It is therefore clear that tracking behavior in two dimensions cannot adequately be expressed as two independent cases of one-dimensional tracking occurring simultaneously along orthogonal axes.

View larger version (21K): [in this window] [in a new window]   Fig. 9. Tracking a target with a constant downward velocity. Left: the average target and finger paths for 1 direction for trials in which the vertical velocity was constant. Right: horizontal and vertical motion of the finger and the target vs. time. - - -, time at which the target changed direction.

Response to target acceleration (experiment 6)

In the preceding (experiment 4), we showed that target speed influenced the direction of finger motion. Can target acceleration also influence the direction of the reacquisition movement? To address this question, in a final experiment, the target either maintained its original speed or accelerated or decelerated, coincident with the change in target direction. For the two examples shown in Fig. 10, A and B, there is no discernable effect of target acceleration on the direction of finger motion. Figure 10, C-E, shows how the movement described in the left-hand panels evolved over time for the cases of accelerating and decelerating targets. The speed of the finger for the two movements did not begin to diverge until 590 ms after the target changed direction (arrow in Fig. 10D). For all subjects and all directions, target acceleration did not begin to have an effect on finger speed until 490 ± 80 ms after the target changed direction. The direction of finger motion was nearly identical for both the accelerating and decelerating targets (Fig. 10E). For this subject, target acceleration had no significant effect on the direction of finger motion. In only 11 of 50 cases was there a statistically significant divergence in finger direction at any point in the movement, at an average time of 450 ± 120 ms.

View larger version (22K): [in this window] [in a new window]   Fig. 10. The effect of target acceleration on tracking performance. A and B: 2 examples of tracking an accelerating target. Thin lines indicate the path of the target for 3 acceleration profiles. During the downward phase of target motion, the target maintained a constant speed of 10.8 cm/s. At the point at which the target changed direction, the target speed either accelerated or decelerated at a rate of 5.4 cm/s2 or remained constant. The thick lines represent the finger paths for each of these acceleration profiles. C-E: increasing and decreasing target accelerations. The dotted line indicates the time at which the target changed direction and accelerated. The arrow indicates the time at which the finger speed profiles for accelerating and decelerating targets diverged. For this subject, finger direction did not differ for the 2 target motions.

Quantitative modeling of manual tracking in two dimensions

A simple conceptual model in which the direction of finger motion changes abruptly in response to a change in the direction of target motion so as to intercept the target at a constant time was able to account for a large body of experimental data. This model is compatible with the idea of intermittent control during tracking (Miall et al. 1986; Young and Stark 1963). Specifically, our results might be interpreted to imply that there is one major correction in the finger's trajectory, secondary corrections perhaps occurring as the finger's trajectory eventually merges with that of the target. However, tracking behavior is more commonly modeled to be under continuous control (Krauzlis and Lisberger 1994; Lisberger et al. 1987; Viviani et al. 1987). The question then arises: can the observed behavior be a consequence of the workings of continuous feedback control? We will show that this is indeed the case.

A CARTESIAN MODEL. We begin with analytical models of tracking similar to models for tracking in one dimension that have been used previously to account for two-dimensional manual tracking (Viviani et al. 1987). In this form of model, the acceleration of the finger (_F) is related to a positional (e_p) and a velocity (e_v) error signal

(3)

where

The subscripts _T and _F denote the target and the finger, respectively and p is a vectorial representation of the position {p_x, p_y}. The coefficients a₁ and a₂ are constants, as are the two time delays _p and _v. This model leads to two uncoupled equations, one in x and one in y

(4)

where e_px and e_py are the x and y components of the position error vector and _Fx and _Fy are the x and y components of the finger acceleration.

View larger version (27K): [in this window] [in a new window]   Fig. 11. Comparison of the behavior of 2 analytical models in capturing tracking behavior in trials in which the target changed direction (experiment 1). In model 1, the finger acceleration vector is related to vectorial error signals of position and velocity. In model 2, acceleration is defined by its normal and tangential component. The tangential component is related to a vectorial error signal in position and velocity (as in model 1), but the normal component is related to a directional error signal. Results of the modeling for 3 directions are shown for subject 2. For each direction, the paths of the finger and the target are shown to the left and the temporal variation in speed and direction is shown to the right. The dotted traces represent the performance of the models. Note that model 2 provides a better fit to the time course of the change in direction of finger motion as well as the time course of the variations in speed.

EXTENSIONS TO THE CARTESIAN MODEL. The errors in matching finger speed suggest that a nonlinear model might give an improved fit, i.e., that finger acceleration would depend on the square and/or the cube of the error terms (e_p and e_v) in Eq. 3. We can rule out a quadratic nonlinearity since it would violate the mirror symmetry that was obtained for targets moving to the right or to the left (Fig. 2). We did try a model including a cubic nonlinearity (which does not violate mirror symmetry) for the data for one subject. This nonlinear model gave a negligible improvement in the fit (<1% reduction in error).

(5)

MODELING TRACKING BEHAVIOR IN CURVILINEAR COORDINATES. As an alternative to the Cartesian model, we investigated models in which the error signals are described in a curvilinear coordinate system fixed to the hand (Flanders et al. 1992). Specifically, we defined feedback error signals in directions tangential and perpendicular to the finger's trajectory at each point in time. In such a description, acceleration is defined by the rate of change of speed and direction

(6)

where  is the speed,  is the direction of motion, and and are unit vectors in the tangential and normal directions, respectively (see Fig. 12A). (The normal component of the acceleration can also be written as ²/R, where R is the radius of curvature.) We chose this frame of reference because it makes speed and direction explicit parameters in the model and because the experimental data suggested these two parameters were important controlled variables.

View larger version (9K): [in this window] [in a new window]   Fig. 12. Schematic defining the error signals in model 2. A: the path of the finger, where and denote the tangential and normal directions and  denotes the direction of finger movement. B: how the directional target signal  is defined. pT and pF denote the position of the target and of the finger, and T indicates the target velocity. The angle  is defined by a vector from the finger to target positions (pT  pF) plus the target velocity vector.

(7)

(8)

(9)

(10)

(11)

View larger version (16K): [in this window] [in a new window]   Fig. 13. Performance of model 2 in matching tracking behavior in trials in which target speed underwent a step change. The data are the same as in Fig. 8 and are from subject 7.

View larger version (20K): [in this window] [in a new window]   Fig. 14. Performance of model 2 in matching tracking behavior in trials in which target speed accelerated or decelerated. The data are the same as in Fig. 10 and are from subject 10.

	DISCUSSION

TOP ABSTRACT INTRODUCTION METHODS RESULTS DISCUSSION REFERENCES

When targets made an abrupt change