There were a total of 50 trials. On 6 trials, either the helicopter was not intercepted by the pursuer (2), or the data were not codable due to interference with the signal from the cameras (4). On 3 trials, the pursuer did not move for at least one half of the trial due to the helicopter making only small movements or hovering essentially in the same place. This was typically due to signal interference between the remote control and the toy helicopter. Thus, we analyzed data from 41 trials. We derived the optical coordinates of the helicopter from the angle between the x-, y-, and z-coordinates of the pursuer and helicopter. The variation in these optical coordinates during a trial is the optical trajectory traced out by the helicopter; the variation in the ground coordinates of the pursuer during a trial is the ground path traced out by the pursuer. We analyzed the optical trajectory and ground path data until the last 0.5 s of each trial, where it has been shown that other cues are used for interception (Regan, 1997; Regan & Gray, 2001).
Comparing observed and predicted optical trajectories for SLOT, OAC, and GOAC
We compared the OAC, GOAC, and SLOT models in terms of their ability to account for the observed optical trajectories. LOT was not included in this comparison because the observer optical trajectories were clearly nonlinear. The comparison was done using the same methods that had been employed in earlier comparisons of these models (e.g., Fink et al., 2009; McLeod et al., 2008; Shaffer et al., 2008). Our aim was to see whether the use of objects with highly unpredictable trajectories (the helicopter) would allow a clearer distinction between these models.
In order to perform this analysis, we calculated optical trajectories in terms of the vertical and lateral optical angles based on the observed trajectories of the helicopter and movements of the pursuer on each trial. To evaluate OAC, we plotted tanα versus time. The results of this analysis showed that, averaging over 41 trials, OAC accounted for 64.58 % of the variance in observed optical trajectories (SD = 32 %, range = 96.7 %). Representative trials are shown in Fig. 2. This figure shows that OAC typically fails to account for a significant proportion of the variance in tanα. OAC failed for one of two reasons: Either (1) tanα decreased, as it did in the top two panels, or (2) α came close to and, in some cases (as in the bottom panel), actually reached 90°, at which point the pursuer is looking virtually directly over their head at the helicopter and the tangent function runs to positive infinity.
In order to analyze the vertical component of GOAC, we first applied regression lines to plots of the velocity of the vertical optical angle, dα/dt, against time. We then calculated the correlation coefficient, identified whether it was positive or negative, and tested to see whether it was statistically significant. Since a plot can look relatively constant but can still be significantly increasing or decreasing statistically, and to fairly assess “constancy,” we also adopted a criterion where the magnitude of the correlation coefficient had to be at least r = .30 to be considered “significantly” increasing or decreasing. This corresponds to R
2 = .09, which is considered a moderate effect size (Cohen, 1988). If dα/dt decreased according to our criterion, it would be evidence in favor of GOAC.
Analysis of dα/dt against time revealed that dα/dt was maintained as constant (24 trials) or significantly increasing (1 trial) as often as it was maintained as significantly decreasing (16 trials), χ
2(1) = 1.2, p > .1. Representative plots of dα/dt versus time are shown in the left panels of Fig. 3. The top left panel shows da/dt significantly decreasing, while the bottom two panels show dα/dt staying constant, as it did for the majority of trials. The bottom two panels serve as examples of how GOAC failed to explain the vertical component of the optical trajectory.
We used the same strategy for analyzing the lateral component of GOAC, dβ/dt, as we did for analyzing the vertical component, except that, here, we graphed the lateral optical angle versus time. If dβ/dt was maintained as constant according to our criterion, it would be evidence in favor of GOAC. Analysis of dβ/dt against time revealed that dβ/dt was maintained as constant (37 trials) significantly more than it was maintained as significantly increasing (3 trials) or decreasing (1 trial), χ
2(1) = 26.56, p < .005. Representative plots of dβ/dt versus time are shown in the right panels of Fig. 3.
To evaluate SLOT, we used regression analysis to determine how much of the variance in the observed optical trajectories could be accounted for by a set of optimally oriented linear segments.Footnote 2 The SLOT accounted for a mean of 95.1 % of the variance (SD = 2.4 %, R = 9.1 %). Representative plots of SLOT—the lateral optical angle plotted against the vertical optical angle—are shown in Fig. 4. The graphed trials are the same as those in Fig. 2.
A paired-samples t-test was performed on the variance accounted for by OAC and SLOT strategies (no such comparisons could be made for GOAC, since R
2s were not part of the analysis). SLOT accounted for significantly more variance than did OAC, t(40) = 6.14, p < .001. This was a large effect, R
2 = .49. We also tested the variability (coefficient of variation, or CV) of both OAC and LOT as a measure of reliability and found that the OAC was almost 20 times as variable in explaining interceptive behavior as was a SLOT (CVOAC = 49.55; CVSLOT = 2.52).
Testing working models of object interception—COV and GOAC
We compared both the observed optical trajectories and pursuer’s ground paths with those produced by a working model of object interception. It was a “working” model in the sense that it was implemented as a computer program that produced object interception behavior (two-dimensional ground tracks) on the basis of the actual helicopter paths (temporal variations in the x-, y-, z-coordinates of the helicopter) observed in the experiment. The coordinates of the running space were rotated and translated so that, on each trial, the model pursuer was at the origin of the x- and y-axes that defined this space and the helicopter launch point was mapped to the 0 position of the x-axis.
We tested two versions of the model that differed in terms of the optical information that was the basis for the object interception behavior. One version implemented the COV model. It used the vertical optical velocity, dα/dt, and lateral optical angle, β, of the target as the basis of object interception behavior. The other version of the model, which is equivalent to the GOAC model specificallyˆgy using OAC in the vertical direction, rather than more generally using decreasing vertical optical velocity, used vertical optical acceleration, d
α/dt, and lateral optical velocity, dβ/dt, as the basis of interception. So the only difference between the two working models was in the informational basis of the object interception behavior produced by the models.
The COV model
The COV version of the model is shown in Fig. 5. The model consists of two control systems, one controlling the vertical optical velocity (dα/dt) of the pursued object (the helicopter) by producing outputs, o
, that move the pursuer forward or backward in the x-dimension and the other controlling the lateral optical angle, β, of the helicopter by producing outputs, o
, that move the pursuer laterally in the y-dimension. This is a closed-loop control model of object interception behavior. The horizontal line running through the middle of the figure separates the two control systems that pursue the helicopter from the environment that contains the moving helicopter.
The variables controlled by each control system, dα/dt and β, are perceptions computed as a function of temporal variations in the observed position—changes in the x-, y-, z-coordinates over time—of the helicopter. The neural networks that compute these perceptions are represented by the boxes labeled p
v = dα/dt and p
l = β. The perceptions, p
v and p
l, are compared, via subtraction, to reference signals (r
v and r
l, respectively) that specify the goal values for these perceptions. The difference between each perception and its reference signal is an error signal, e, that drives the model outputs o
The movement of the pursuer relative to the object pursued has immediate feedback effects on the vertical and lateral optical angles, q
, of the pursued object relative to the pursuer. These feedback effects are indicated in Fig. 5 by the arrows connecting the pursuer’s outputs, o
, to the optical angles q
. The optical angles are the inputs to the perceptual functions that compute the perceptual variables, p
v and p
l, that are under control.
Note that each output in Fig. 5 has feedback effects on both optical angles. This does not create a conflict between the systems, because the effect of an output, such as o
, on the optical variable that it is not controlling, q
in this case, is simply treated as a disturbance that is opposed by the output, o
, of the system that is controlling q
. Thus, there is no conflict between the two control systems.Footnote 3
The GOAC model
The GOAC version of the model differs from COV only in the assumption that one basis of object interception is vertical optical acceleration, d
α/dt, rather than vertical optical velocity, dα/dt, and that the other is lateral optical velocity, dβ/dt, rather than lateral optical angle, β.
Analysis of ground tracks for COV and GOAC
We compared the working version of GOAC and COV in terms of their ability to account for the observed variance in the two-dimensional ground tracks of pursuers that were observed on each trial of the experiment. The SLOT model was not included in this analysis because there is currently no working version of the model that can act—by moving appropriately on the ground—so as to change the slope of the linear segments of optical trajectory appropriately.
We used regression analysis to determine how much of the variance in the observed ground tracks was accounted for by the ground tracks produced by the best-fitting GOAC and COV models. The results of this analysis showed that, averaging over 41 trials, the GOAC model accounted for an average of 75 % of the variance in observed ground tracks (SD = 26 %, R = 99 %), while the COV model accounted for a mean of 93 % of the variance (SD = 6 %, R = 27 %). The fit of the COV model to the ground tracks on four experimental trials (“Takes”) is shown in Fig. 6.
A paired-samples t-test revealed that COV accounts for significantly more variance in the ground tracks than does GOAC, t(40) = 4.77, p < .001. This was a large effect, R
2 = .36. We also tested the variability of both GOAC and COV as a measure of reliability and found that the GOAC was more than 5 times as variable in explaining interceptive behavior as was COV (CVOAC = 34.67, CVCOV = 6.45).
The relative success of the two models can also be measured in terms of RMS deviation of model behavior from observed ground tracks. Again averaging over 41 trials, the RMS deviation of the GOAC model from observed ground tracks was 30.18 cm (SD = 19.02 cm), and the RMS deviation of the COV model from observed ground tracks was 16.51 cm (SD = 10.61 cm). A paired-samples t-test revealed that COV had far less error in predicting ground tracks than did GOAC, t(40) = −5.39, p < .001. This was also a large effect, R
2 = .42. So, the COV model was almost twice as accurate as the GOAC model at predicting the ground tracks that controlled the respective types of optical information during pursuit.
It should be noted that the fit of both the COV and OAC/GOAC versions of the model was obtained by estimating only four free parameters (the gain and slowing factors for the output functions of the two control systems). Thus, the superiority of the COV version of the model in terms of accounting for the observed ground tracks can be attributed only to the fact that the informational basis of the behavior of this model differed from that of the OAC/GOAC version.
Analysis of optical trajectories for COV and GOAC
It was also possible to compare the optical trajectories of α and β produced by the working versions of the COV and GOAC models with those observed in the experiment. The fit of model to actual trajectories was measured in terms of R
2, which is the proportion of variance in the observed values of α and β accounted for by the corresponding values of α and β produced by the model. The average R
2 was calculated for 41 experimental trials. The COV model accounted for 84 % (SD = 13 %), while the GOAC model accounted for 75 % (SD = 15 %) of the variance in the optical trajectories. A paired-samples t-test revealed that COV accounts for significantly more variance in the optical trajectories than does GOAC, t(40) = 4.44, p < .001. Nevertheless, the COV model accounted for considerable less variance in the optical trajectories (84 %) than did the SLOT model (95.1 %).
The comparison of the ability of the SLOT and COV models to account for the optical trajectory data is complicated by the fact that the models differed in terms of the number of free parameters used to make these predictions. The COV model used four free parameters to estimate the optical trajectories on all trials. The SLOT model uses two free parameters (slope, intercept) for each linear segment fit to the observed optical trajectory; thus, the number of free parameters used to account for the data depends on the number of segments found in each trajectory.