How does a baseball outfielder know where to run to catch a fly ball? The “outfielder problem” remains unresolved, and its solution would provide a window into the visual control of action. It may seem obvious that human action is based on an internal model of the physical world, such that the fielder predicts the landing point based on a mental model of the balls trajectory (TP). However, two alternative theories, Optical Acceleration Cancellation (OAC) and Linear Optical Trajectory (LOT), propose that fielders are led to the right place at the right time by coupling their movements to visual information in a continuous “online” manner. All three theories predict successful catches and similar running paths. We provide a critical test by using virtual reality to perturb the vertical motion of the ball in mid-flight. The results confirm the predictions of OAC but are at odds with LOT and TP.

*α,*or its projection on a vertical plane, see Figure 1). If the balls optical velocity (dtan

*α*/d

*t*) is increasing, the fielder should accelerate backward, and if it is decreasing, accelerate forward, thereby keeping the optical velocity approximately constant. Tangential movement perpendicular to the direction of the ball is controlled independently by matching the balls lateral position (Chapman, 1968), either by keeping the ball in a constant bearing direction (azimuth ψ) or by holding its horizontal angle to home plate (

*β*) constant (Fajen & Warren, 2007; McLeod et al., 2006; Tresilian, 1995).

*γ*relative to the horizontal. LOT has the virtue of not requiring that ball acceleration be detected; instead, LOT only requires that departures from a straight trajectory (i.e., constant

*γ*) be detected. However, LOT does not produce a unique solution or a well-defined mapping from optics to action but rather constrains the fielders movement to a family of paths; an additional constraint, such as that suggested by OAC to control radial movement, must be applied in order to generate a unique path. If such a constraint were applied, though, OAC and LOT would still differ in that LOT theory predicts that the fielders radial and tangential movements are coupled to maintain a constant

*γ,*implying that any factor affecting one would have an impact on the other. In OAC, radial and tangential movements are independent, which provides a means by which the two theories can be tested.

*α*/d

*t*shifts to a new constant value, whereas the tangential velocity will be unaffected so that the horizontal angle

*β*is unchanged. On the other hand, LOT predicts that the fielders radial and tangential velocities will be coupled to maintain

*γ,*preserving a constant ratio between tan

*α*and tan

*β*over the perturbation.

*α*increasing at a steadily decreasing rate (McLeod et al., 2006), and inconsistent with TP and LOT. However, the balls anterior–posterior and lateral landing positions were varied in separate experiments. In the present study, we vary the landing position in both directions simultaneously in a 12 m × 12 m virtual environment, eliciting large radial and tangential movements of the outfielder, to provide a stronger test of the three theories.

^{2}) and no air resistance. Perturbed fly balls followed a parabolic trajectory until the apogee (25.00 m), at which point the ball descended on a linear path with a constant vertical velocity (11.11 m/s); this altered its vertical optical acceleration but not its lateral optical motion.

*α, β, γ,*and

*ψ*( Figure 1) and their derivatives were calculated. Linear regressions were performed on tan

*α,*tan

*β, ψ,*and

*γ*vs. time and tan

*α*vs. tan

*β*for each trial. A mean

*R*

^{2}value across all trials was calculated for each experimental condition for each participant. Because no significant differences between fly balls to the right and left of the participant were observed, these trials were grouped together. To avoid transients at the beginning and end of each trial, only data between 1.00 s and 3.47 s were analyzed for the whole trial, while data between 1.00 s and 2.17 s were analyzed for the first half of the trial and between 2.30 s and 3.47 s for the second half. Comparison of

*R*

^{2}values between the first and second halves of trials, and between normal and perturbed fly balls, was made using a

*z*-test following a Fisher

*z*-transformation.

*α*(elevation angle) is constant on normal trials, implying a linear relationship between tan

*α*and time. We thus computed the linear regression of tan

*α*on time for each trial (see Figure 4 for a representative participant; data for all participants may be seen in the auxiliary files). In the normal conditions, mean

*R*

^{2}values exceeded 0.99 for both backward and forward fly balls ( Table 1). In the perturbed condition, however, the slope changed after the perturbation, yielding significantly lower

*R*

^{2}values in that condition (

*z*(72) = 8.61,

*p*< 0.0001, for forward and

*z*(72) = 6.63,

*p*< 0.0001, for backward fly balls). Because a shift in the slope of tan

*α*to a new constant value would be expected if the fielder cancelled the new optical acceleration caused by the perturbation, we performed separate regressions for the first and second halves of each trial. The mean

*R*

^{2}values were high on both halves of the trial (>0.99 in most conditions), confirming that slopes were also linear following the perturbation. The exception was the second half of the perturbed backward trials, in which the fit (

*R*

^{2}= 0.84) was significantly lower (

*z*(32) = 13.52,

*p*< 0.0001) due to the flat slope. The sum of squared error in this condition, indicative of departure from linearity, however, was not significantly different from the backward normal trials (

*t*(11) = 1.65,

*p*= 0.1272), which suggests that the linear increase in tan

*α*with time held, even in the perturbed condition. This confirms that the balls optical acceleration was approximately zero both before and after the perturbation, consistent with OAC.

Forward normal | Backward normal | Forward perturbed | Backward perturbed | |
---|---|---|---|---|

tan( α) vs. time | ||||

First | 0.9994 | 0.9999 | 0.9993 | 0.9998 |

Second | 0.9916 | 0.9933 | 0.9924 | 0.8408* ^{,}** |

Entire | 0.9931 | 0.9941 | 0.5914** | 0.8494** |

tan( β) vs. tan( α) | ||||

First | 0.9938 | 0.9961 | 0.9939 | 0.9954 |

Second | 0.8594* | 0.9553* | 0.8847* | 0.8347* ^{,}** |

Entire | 0.8511 | 0.9408 | 0.5217** | 0.6874** |

tan( β) vs. time | ||||

First | 0.9938 | 0.9961 | 0.9939 | 0.9954 |

Second | 0.9048* | 0.9558* | 0.9338* ^{,}** | 0.9593* |

Entire | 0.8618 | 0.9476 | 0.8766 | 0.9485 |

ψ vs. time | ||||

First | 0.8873 | 0.9780 | 0.9011 | 0.9634 |

Second | 0.9633* | 0.9519* | 0.9613* | 0.9297* |

Entire | 0.8657 | 0.8869 | 0.8746 | 0.9016 |

γ vs. time | ||||

First | 0.8892 | 0.9786 | 0.9001 | 0.9643 |

Second | 0.9639* | 0.9527* | 0.9626* | 0.9316* |

Entire | 0.8658 | 0.8871 | 0.8750 | 0.9020 |

*γ*) throughout a trial, which is equivalent to maintaining a linear relationship between tan

*α*and tan

*β*(between the balls elevation angle and horizontal angle to home plate). To test this, we performed a linear regression of tan

*β*on tan

*α*for each trial (Table 1, Figure 5). In the normal condition, the

*R*

^{2}values were high, with mean values of 0.8511 and 0.9408 for forward and backward fly balls, respectively. However, the fits were significantly lower in the perturbed condition, with mean

*R*

^{2}values of 0.5217 and 0.6874 for forward and backward fly balls (

*z*= 4.0887,

*p*< 0.0001 and

*z*= 5.417,

*p*< 0.0001, respectively). As before, this might be explained by a shift to a new linear slope following the perturbation, but this account appears unlikely for two reasons. First, separate regressions for the second half of the trial yielded significantly lower

*R*

^{2}values than for the first half in all conditions (

*z*≥ 4.92,

*p*< 0.0001). Second, an analysis of the horizontal angle tan

*β*over time (Table 1, Figure 6) revealed no differences in slope for the whole trial between normal and perturbed trials (

*z*= −0.36,

*p*= 0.7188, for forward and

*z*= 0.05,

*p*= 0.9601, for backward fly balls). Thus, contrary to LOT, the perturbation affected tan

*α*but not tan

*β,*indicating that the participants radial and tangential movements are not actually coupled.

*β*) or of the bearing direction of the ball (azimuth

*ψ*). We tested these strategies by regressing tan

*β*and

*ψ*on time during the first and second halves of each trial (Figures 6 and 7, Table 1). Mean

*R*

^{2}values were high for both variables (>0.866), with significant differences between the first and second halves of the trials in both normal and perturbed conditions (∣

*z*∣ ≥ 2.0297, all

*p*≤ 0.05);

*R*

^{2}increased in the backward conditions for

*ψ*and decreased in the other conditions. The strong fits both before and after the perturbation suggest that either variable could be used to control the fielders tangential movement.

*α*(elevation) over time, with a different slope. This linear relationship was quite strong, with

*R*

^{2}values exceeding 0.99 in most conditions, both before and after the perturbation. Moreover, participants maintained this relationship while moving both radially and tangentially. The recovery of linearity after the perturbation is surprising because it requires a large correction in running speed to restore a constant optical ball velocity and a varying running speed to maintain it. This finding provides strong evidence that participants cancel the balls optical acceleration to control their radial movements.

*α*.

*α*(elevation angle) was changed by the perturbation, tan

*β*(horizontal angle to home plate) was not affected, and a linear relationship in time for tan

*β*was observed throughout the entire trial, regardless of the perturbation. The change in tan

*α*but not tan

*β*resulted in a change of

*γ*(trajectory angle) that cannot be accounted for by LOT. Instead, it appears that radial and tangential movements of the fielders are controlled independently, consistent with OAC.

*α*and tan

*β*are independent, how are the fielders tangential movements controlled? We tested two versions of Chapmans (1968) hypothesis that the fielder matches the lateral position of the ball, either by nulling the rate of change in the horizontal angle between the ball and home plate (tan

*β*) or in the bearing direction of the ball (

*ψ*). The high correlations indicate that either variable could be used to control the fielders tangential movement. This is not surprising because they are closely related, but the bearing direction

*ψ*is more general because “home plate” need not be in view to determine the angle.

*α*increasing at a steadily decreasing rate, whereas in Chapmans (1968) original version, this is accomplished by maintaining a constant rate of increase in tan

*α*. The results in perturbed fly ball conditions seem somewhat at odds with the generalized version of OAC: after the perturbation,

*α*and tan

*α*actually decrease in our experiment (top of Figure 4), or increase at an increasing rate (bottom two panels of Figure 5 in McLeod et al., 2008). In addition, McLeod et al. (2008) needed to introduce a memory component to explain the tendency for d

*α*/d

*t*to return toward its pre-perturbation trajectory, whereas if tan

*α*is used to guide movement, this tendency is simply a consequence of keeping dtan

*α*/d

*t*constant at its post-perturbation value, without having to invoke a memory component. We therefore believe that keeping dtan

*α*/d

*t*constant provides a better control principle for catching fly balls.

*α*), while simultaneously and independently controlling tangential velocity by nulling the velocity of tan(

*ψ*) similar to the constant bearing model of target interception (Fajen & Warren, 2007). The model treats the fielder as analogous to a damped spring with a fixed point at d

^{2}tan

*α*/d

*t*

^{2}= 0 and dtan

*ψ*/d

*t*= 0. The equations of motion are given as follows:

*b*

_{r}and

*b*

_{t}are damping terms,

*k*

_{r}and

*k*

_{t}are stiffness terms, and

*c*

_{r}and

*c*

_{t}are thresholds that approximate a sigmoidal response to the optical variables.

*x*- and

*z*-positions of the model and the data, summed over all time points in all catching trials. The mean

*R*

^{2}value was 0.5994 for the

*x*-position and 0.4174 for the

*z*-position (the best possible values were 0.8068 and 0.6836, respectively, determined by calculating

*R*

^{2}values between the mean trajectory of each participant and all trials). Fits of the model are shown in Figures 4, 5, 6, and 7 with thick magenta and cyan lines for the normal and perturbed conditions respectively, using the parameters derived for the representative participant (

*b*

_{r}= 19.38 1/s,

*b*

_{t}= 0.19 1/s,

*k*

_{r}= 116.48 m/s,

*k*

_{t}= 223.52 m/s

^{2},

*c*

_{r}= 4 rad/s

^{2},

*c*

_{t}= 4 rad/s). The model thus approximates the fielders movements when catching a fly ball by directly coupling optical variables to action.