March 2015
Volume 15, Issue 3
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Article  |   March 2015
Optical angular constancy is maintained as a navigational control strategy when pursuing robots moving along complex pathways
Author Affiliations
  • Wei Wang
    Department of Psychology and Institute for Mind and Biology, The University of Chicago, Chicago, IL, USA
    wwang3@uchicago.edu
  • Michael K. McBeath
    Department of Psychology Arizona State University, Tempe, AZ, USA
    Michael.McBeath@asu.edu
  • Thomas G. Sugar
    The Polytechnic School, Ira A. Fulton Schools of Engineering, Arizona State University, Mesa, AZ, USA
    Thomas.Sugar@asu.edu
Journal of Vision March 2015, Vol.15, 16. doi:10.1167/15.3.16
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      Wei Wang, Michael K. McBeath, Thomas G. Sugar; Optical angular constancy is maintained as a navigational control strategy when pursuing robots moving along complex pathways. Journal of Vision 2015;15(3):16. doi: 10.1167/15.3.16.

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Abstract

The optical navigational control strategy used to intercept moving targets was explored using a real-world object that travels along complex, evasive pathways. Fielders ran across a gymnasium attempting to catch a moving robot that varied in speed and direction, while ongoing position was measured using an infrared motion-capture system. Fielder running paths were compared with the predictions of three lateral control models, each based on maintaining a particular optical angle relative to the robotic target: (a) constant alignment angle (CAA), (b) constant eccentricity angle (CEA), and (c) linear optical trajectory (LOT). Findings reveal that running pathways were most consistent with maintenance of LOT and least consistent with CEA. This supports that fielders use the same optical control strategy of maintaining angular constancy using a LOT when navigating toward targets moving along complex pathways as when intercepting simple ballistic trajectories. In those cases in which a target dramatically deviates from its optical path, fielders appear to simply reset LOT parameters using a new constant angle value. Maintenance of such optical angular constancy has now been shown to work well with ballistic, complex, and evasive moving targets, confirming the LOT strategy as a robust, general-purpose optical control mechanism for navigating to intercept catchable targets, both airborne and ground based.

Introduction
Past research exploring the navigational strategy used to intercept a projectile like a baseball or football has established that this task can be accomplished using simple optical navigational control heuristics (Babler & Dannemiller, 1993; Chapman, 1968; Fajen, 2013; Marken, 2014; McBeath, Nathan, Bahill, & Baldwin, 2008; McBeath, Shaffer, & Kaiser, 1995a; Michaels & Oudejans, 1992; Shaffer, McBeath, Kruchunas, & Sugar, 2008; Shaffer, Marken, Dolgov, & Maynor, 2013). This research examines whether the same navigational heuristics apply under vastly different physical conditions when attempting to intercept targets moving along complex, evasive, ground-based pathways in a real-world environment. 
Modeling vertical component of optical control
Chapman (1968) proposed an optical strategy, later framed as optical acceleration cancellation (OAC), for how baseball outfielders approach and intercept fly balls. He noted that in cases in which fielders are stationary or run directly forward or backward while heading toward the ball destination, the tangent of the vertical optical angle, tanα, increases at a constant rate. As shown in Figure 1a, the optical ball trajectory can be compared to that of an imaginary elevator rising from the home plate at a constant velocity and tilted by the amount that fielders run forward or backward (Babler & Dannemiller, 1993; McLeod & Dienes, 1993, 1996; McLeod, Reed, & Dienes, 2003, 2006; Michaels & Oudejans, 1992). Although some work examining acceleration discrimination has questioned OAC (Brouwer, Brenner, & Smeets, 2002; Zaal, Bongers, Pepping, & Bootsma, 2012), other research further supports that fielders maintain OAC even in extreme cases, such as pop-ups, in which balls can travel along dramatically nonparabolic trajectories (McBeath et al., 2008) and when catching fly balls in virtual environments (Fink, Foo, & Warren, 2009; Turvey & Fonseca, 2009; Zaal & Bootsma, 2011). In addition, further studies have verified that when fielders navigate to intercept ground balls, they also maintain a flipped version of OAC, in a downward-projected, horizontal direction, maintaining a constant-rate of decrease in cotα (Sugar, McBeath & Wang, 2006; see Figure 1b). 
Figure 1
 
Model of interception using optical acceleration cancellation (OAC). α indicates the vertical optical angle from the horizon to the ball. (a) Fly ball case. A fielder runs along a path that keeps the tangent of the vertical optical ball angle (tanα) increasing at a constant rate. Shown is a fielder (hat-eye) approaching an air-resistance shortened ball trajectory, in equal temporal intervals (t0–t7). (b) Ground ball case is a flipped, downward version of OAC. A fielder keeps eye level relatively constant while maintaining a relatively constant distance from the initial location of the ball and keeps the cotangent of the vertical optical angle (cotα) of the ball decreasing at a constant rate. Shown is a fielder (eye icon) and a ground ball approaching in equal temporal intervals (t0–t4).
Figure 1
 
Model of interception using optical acceleration cancellation (OAC). α indicates the vertical optical angle from the horizon to the ball. (a) Fly ball case. A fielder runs along a path that keeps the tangent of the vertical optical ball angle (tanα) increasing at a constant rate. Shown is a fielder (hat-eye) approaching an air-resistance shortened ball trajectory, in equal temporal intervals (t0–t7). (b) Ground ball case is a flipped, downward version of OAC. A fielder keeps eye level relatively constant while maintaining a relatively constant distance from the initial location of the ball and keeps the cotangent of the vertical optical angle (cotα) of the ball decreasing at a constant rate. Shown is a fielder (eye icon) and a ground ball approaching in equal temporal intervals (t0–t4).
Modeling lateral component of optical control
The OAC model is widely assumed to be the principal navigational mechanism specifying control of the vertical component of the optical angle to the target, and OAC appears to be relied upon in the two-dimensional (2D) case in which a target remains within a vertical plane bisecting the fielder (i.e., is heading straight toward the fielder). Yet there remains debate regarding the lateral navigational strategy used in the 3D cases in which fielders need to run not just forward and backward but laterally as well. 
In the present study, we compare three lateral angular control models that presumably work in conjunction with or generalize down to OAC in those cases in which the target remains in the sagittal plane headed toward the fielder. Model performance is tested by examining interception behavior with targets that change in speed and direction in a real-world navigational setting. The three models are maintenance of (a) constant alignment angle (CAA) with the target, (b) constant eccentricity angle (CEA) relative to the target, and (c) constant linear optical trajectory (LOT) relative to the background. 
Constant alignment angle
When introducing OAC, Chapman (1968) suggested that for balls headed to the side, fielders simply remain laterally aligned with the ball while performing OAC in the depth direction until interception. To achieve this, a fielder must simultaneously control both the lateral alignment and the rate of change of the vertical optical angle, α. Based on Chapman's suggestion, a lateral optical control strategy can be specified for intercepting a moving target headed off to the side by maintenance of a CAA. Mathematically, the CAA control model states that fielders select an interception path such that the top view of the vector between them and the moving target remains parallel to its initial world-based orientation (Figure 2). The lateral alignment angle, ϕ, is defined as the top view angle between the fielder and the target relative to the original alignment direction. 
Figure 2
 
Constant alignment angle (CAA) lateral strategy. A fielder chooses a lateral strategy that tries to keep the top view of the vector between him and the target object parallel or aligned to the initial direction. Shown are top views of a fielder (in purple) pursuing a target (in red) that moves from left to right. In the CAA strategy, the fielder tries to null out the optical angle, ϕ, created by the vector from the fielder to the target (in green) and a line parallel to the initial orientation of this vector. (a) Ideal alignment (ϕ always 0°). In the ideal case, the vertical alignment lines between the fielder and the target remain parallel at times t0, t1, and t2. The alignment offset angle is ϕ = 0° when ideal alignment is achieved. (b) Offset alignment (ϕ varies from 0°). The dotted vertical background lines show the locations that remain aligned with the fielder at times t0, t1, and t2. The alignment offset angle, ϕ, remains near 0° but varies somewhat in real-world cases.
Figure 2
 
Constant alignment angle (CAA) lateral strategy. A fielder chooses a lateral strategy that tries to keep the top view of the vector between him and the target object parallel or aligned to the initial direction. Shown are top views of a fielder (in purple) pursuing a target (in red) that moves from left to right. In the CAA strategy, the fielder tries to null out the optical angle, ϕ, created by the vector from the fielder to the target (in green) and a line parallel to the initial orientation of this vector. (a) Ideal alignment (ϕ always 0°). In the ideal case, the vertical alignment lines between the fielder and the target remain parallel at times t0, t1, and t2. The alignment offset angle is ϕ = 0° when ideal alignment is achieved. (b) Offset alignment (ϕ varies from 0°). The dotted vertical background lines show the locations that remain aligned with the fielder at times t0, t1, and t2. The alignment offset angle, ϕ, remains near 0° but varies somewhat in real-world cases.
The navigation strategy of alignment has been implied or suggested in a number of studies (Babler & Dannemiller, 1993; Chapman, 1968; McLeod, Reed, & Dienes, 2001; Michaels & Oudejans, 1992). Mathematically, this model is isomorphic to maintaining a constant bearing angle with respect to an exocentric cardinal direction (like north; e.g., Fajen, 2013), but it differs if the bearing angle is defined relative to a distant visible object or marker (i.e., the latter being the dominant definition of bearing angle in popular navigation usage, such as boating). Maintaining constant exocentric alignment relative to the original direction of the object need not require keeping track of another cardinal direction. Other researchers have used the phrase bearing angle to indicate the eccentricity angle described below (e.g., Chardenon, Montagne, Laurent, & Bootsma, 2004). To avoid potential confusion regarding different uses of the term bearing angle, we use the unambiguous term alignment angle
Constant eccentricity angle
For a target headed off to the side at a relatively constant velocity, another control strategy that can be used to achieve interception is to maintain a CEA, or in other words, to keep moving in a direction that bears ahead of the target by a constant angle (Bastin & Montagne, 2005; Chardenon, Montagne, Buekers, & Laurent, 2002; Pollack, 1995; Rushton & Allison, 2013; Shaffer & Gregory, 2009). Maintenance of this angular constancy guarantees that a fielder will converge upon any target that the fielder can keep up with in the lateral direction. Mathematically, the CEA control model states that fielders select an interception path that keeps constant the angle defined by the top view of the vector between the fielder and the target and the vector that specifies the fielder's ongoing direction of heading (Figure 3). The eccentricity angle, θ, can be thought of as the ongoing top view angle between a fielder's two arms when one points in the direction of travel and the other at the target. 
Figure 3
 
Constant eccentricity angle (CEA) lateral strategy. A fielder chooses a lateral strategy, attempting to keep constant the angle defined by the top view of the fielder's direction of travel and the vector between the fielder and the target (in green). Here, the optical eccentricity angle is specified as θ. Shown are top views of a fielder (in purple) pursuing a target (in red) that moves from left to right. (a) Ideal constant eccentricity angle (θ remains at θ0). In the ideal case, a fielder selects a path that keeps θ constant over time. (b) Nonconstant eccentricity angle (θ varies from θ0). Realistically, a fielder selects a path in which θ varies somewhat over time.
Figure 3
 
Constant eccentricity angle (CEA) lateral strategy. A fielder chooses a lateral strategy, attempting to keep constant the angle defined by the top view of the fielder's direction of travel and the vector between the fielder and the target (in green). Here, the optical eccentricity angle is specified as θ. Shown are top views of a fielder (in purple) pursuing a target (in red) that moves from left to right. (a) Ideal constant eccentricity angle (θ remains at θ0). In the ideal case, a fielder selects a path that keeps θ constant over time. (b) Nonconstant eccentricity angle (θ varies from θ0). Realistically, a fielder selects a path in which θ varies somewhat over time.
The navigational strategy of CEA is supported in some studies in which fielders intercept a moving ball (Bastin & Montagne, 2005; Chardenon et al., 2002; Chardenon, Montagne, Laurent, & Bootsma, 2004, 2005). Broader support for the use of the CEA strategy in interceptive tasks was found in the study tracking behavior of fish within one plane (Lanchester & Mark, 1975) and cruising behavior of flies (Collett & Land, 1975). 
Linear optical trajectory
McBeath, Shaffer, and Kaiser (1995a, 1995b) proposed a control strategy called linear optical trajectory (LOT), in which fielders direct themselves to the destination of a ball by maintaining a projected 2D optical trajectory of the ball that is linear. The observed 2D projection of the 3D trajectory can be divided into separate vertical and horizontal components within the fielder's picture plane, and fielders then control the observed angle of ball movement within this projection plane. In the fly ball case shown in Figure 4a, the optical trajectory of the ball continues to climb up in a straight line along the unchanging hypotenuse projection angles. Mathematically, the LOT control model states that fielders maintain a constant ratio between the rate of change in lateral optical angle to the target (/dt) and the rate of change in vertical optical angle (dα/dt), resulting in a constant optical projection angle slope of ψ = acot(/). Because the gaze of the fielder continuously tilts directly toward the moving target, the local projection plane near the target remains orthogonal to the direction of gaze (Postma, den Otter, & Zaal, 2014), and the ongoing changes in the vertical and lateral optical angles to the target (i.e., and ) are kept proportional within the local projection plane. Ideally, the lateral optical angle, β, will exhibit a perfect linear function of the vertical optical angle, α, which results in the target's moving along a straight line relative to local background scenery. Realistically, there will be variance between the ongoing value of β and the predicted proportion of α, resulting in offsets from keeping the target moving along a straight optical path. Because the fielder typically continues to foveate on the ball image, retinally the local background scenery moves linearly in the opposite direction (Postma et al., 2014; Wang, McBeath, & Sugar, 2015). 
Figure 4
 
Linear optical trajectory (LOT) strategy. When a fielder runs to the side, the observed position of the ball sweeps laterally from its former position. Over small periods of time, the magnitude of the lateral optical change in target position relative to the background can be designated as Δβ, and the lateral optical angle, β = Σ(Δβ). A LOT is achieved when the rate of change of β is matched to the rate of change in the vertical optical angle, α, or / = Constant, so ψ = acot(/) = Constant. Geometrically, the instantaneous triangular projections remain congruent. (a) Fly ball case. (b) Two-dimensional projection of the ball trajectory from the fielder's perspective, where the projection triangles are tiled to form a continuous straight optical trajectory. This illustrates how the fielder experiences the ball continuously moving along a straight optical path relative to the local background scenery. (c) Ground ball case. In aerial and ground ball cases, the vertical optical change is shown in red and the lateral change in blue. Similar to the optical acceleration cancellation (OAC) model for ground balls, the LOT model allows for fielder decreases in eye height over time (as depicted to the right), but such height changes are not necessary or specified by the model.
Figure 4
 
Linear optical trajectory (LOT) strategy. When a fielder runs to the side, the observed position of the ball sweeps laterally from its former position. Over small periods of time, the magnitude of the lateral optical change in target position relative to the background can be designated as Δβ, and the lateral optical angle, β = Σ(Δβ). A LOT is achieved when the rate of change of β is matched to the rate of change in the vertical optical angle, α, or / = Constant, so ψ = acot(/) = Constant. Geometrically, the instantaneous triangular projections remain congruent. (a) Fly ball case. (b) Two-dimensional projection of the ball trajectory from the fielder's perspective, where the projection triangles are tiled to form a continuous straight optical trajectory. This illustrates how the fielder experiences the ball continuously moving along a straight optical path relative to the local background scenery. (c) Ground ball case. In aerial and ground ball cases, the vertical optical change is shown in red and the lateral change in blue. Similar to the optical acceleration cancellation (OAC) model for ground balls, the LOT model allows for fielder decreases in eye height over time (as depicted to the right), but such height changes are not necessary or specified by the model.
The LOT strategy specifies that fielders continuously maintain proportional monotonically ascending vertical and lateral optical angles for fly balls (Aboufadel, 1996; Marken, 2001, 2014; McBeath et al., 1995a; Shaffer et al., 2008; Shaffer, Kruchunas, Eddy, & McBeath, 2004; Shaffer & McBeath, 2002), whereas they continuously maintain proportional monotonically descending vertical and lateral optical angles for ground balls (Sugar, McBeath, Suluh, & Mundhra, 2006; Sugar, McBeath, & Wang, 2006; Figure 4). In the case of ground-based targets, if the optical ball trajectory curves inward, the fielder is headed too far to the side. If the optical trajectory curves outward, the fielder is not headed enough to the side. The fielder's task is essentially to discriminate and maintain a straight optical trajectory (i.e., optical angular constancy) versus a curved optical trajectory relative to local background scenery. As long as the fielder preserves a monotonically descending LOT, he or she will maintain control of ground balls and will travel to the correct destination. Maintaining a LOT is, in effect, a spatial method of achieving cancellation of optical acceleration, in that curvature can be defined in terms of orthogonal acceleration producing a directional change in velocity. Thus, for balls headed to the side, when a fielder maintains OAC along the ball's vertical optical axis of movement and also maintains a LOT to prevent orthogonal lateral optical curvature, then acceleration is cancelled both vertically and laterally. For balls headed within the sagittal plane, directly toward the fielder, the lateral optical movement is eliminated and the optical trajectory converges and simplifies down to just vertical optical constancy specified by OAC. The LOT control strategy has also been confirmed with mobile robots intercepting balls (Sugar & McBeath, 2001; Sugar, McBeath, Suluh et al., 2006), with moving backgrounds (Wang, McBeath, & Sugar, 2015), and a segmented version of the LOT has been found with dogs catching Frisbees that change in direction (Shaffer, Kruchunas, Eddy & McBeath, 2004), and with human fielders chasing randomly moving toy helicopters (Shaffer, Marken et al., 2013). 
Purpose of current research
In most previous studies, speed of the target is kept relatively constant and the pathway is straight or ballistic. When this is true, the fielder typically runs along a near-optimal straight path to interception and CAA, CEA, and LOT are all maintained, so it is difficult to establish which control strategy is being used (Figure 5). Only when the target moves in a complex manner does fielder behavior diverge for the three strategies. In other words, these models converge when targets move along simple, relatively constant-velocity pathways but should differ when the path is complex. 
Figure 5
 
Interception control models when pursuing a simple constant velocity target. Shown is a top view of a fielder approaching a target. The approach follows a straight path and maintains much perceptual regularity: (i) Lateral alignment with the target (green vertical lines parallel), (ii) constant eccentricity angle to the target (blue angle, θ, constant), (iii) linear optical trajectory (triangular optical projection remains congruent, see Figure 4b). Tests can tease apart which of these control strategies dominate only by using targets with complex, changing paths.
Figure 5
 
Interception control models when pursuing a simple constant velocity target. Shown is a top view of a fielder approaching a target. The approach follows a straight path and maintains much perceptual regularity: (i) Lateral alignment with the target (green vertical lines parallel), (ii) constant eccentricity angle to the target (blue angle, θ, constant), (iii) linear optical trajectory (triangular optical projection remains congruent, see Figure 4b). Tests can tease apart which of these control strategies dominate only by using targets with complex, changing paths.
These control strategies have a major advantage over the more classic, predictive strategies (Adair, 1995; Saxberg, 1987a, 1987b) in which the fielder predicts the final destination point and time and then moves toward that point at the appropriate speed. Here, the fielder does not need to predict in advance when or where the target will arrive, which research has shown they are very poor at doing (Shaffer & McBeath, 2005; Todd, 1981). These kinds of active control strategies appear generally promising in a broad range of behavioral domains in which individuals interact with and guide themselves through environments (Gigerenzer & Goldstein, 2011; Marken, 2014). 
The current study tests the suitability of the three lateral control strategies, CAA, CEA, and LOT, in the cases of intercepting targets that move along complex, evasive pathways. Each model is based on minimization of the variance of a particular optical angle. In CAA, the fielder minimizes variance of the alignment angle, ϕ. In CEA, the fielder minimizes variance of the eccentricity angle, θ. In LOT, the fielder minimizes variance of the lateral optical angle, β, around the linear function specified by the change in the vertical angle, α, which effectively minimizes variance in the optical projection angle, ψ. This allows us to compare the three control models by simply calculating the variances in each functional lateral optical angle as specified by the model. Whichever control model exhibits significantly less variance in the lateral optical angle will be supported as the superior general-purpose interception strategy. Meanwhile, we can also confirm that OAC is simultaneously in operation in the vertical direction, which corresponds to the depth component of interception, when pursuing targets moving along complex, evasive, ground-based pathways. 
Methods
Participants
Three male volunteers from Arizona State University participated in the experiment, allowing us to record a total of N = 60 motion-capture interception trials. All had normal or corrected-to-normal vision. None were trained athletes, but all were moderately skilled in ball-catching sports and were consistently able to intercept the moving robot object under testing conditions. All were naïve to the purpose of the experiments and completed consent forms in accordance with the policy of Institutional Review Board. 
Apparatus
An eight-camera high-speed motion-capture system was used to record the running paths of both participant and moving target (Figure 6). The motion-capture system coded marker positions each 60th of a second in a fixed, global 3D coordinate system. 
Figure 6
 
Experimental setup. (a) Shown is a top-view diagram of the experimental setup: the four fielder starting positions, the target robot starting position, and the positions of the eight motion-capture cameras arranged in a U-shaped arc. The initial position of the robot is in front of the central four cameras, whereas the four initial fielder positions are designated far right, far left, near right, and near left (relative to the robot). Starting distances were 12 m for the far positions and 6 m for the close ones. (b) Shown is a photograph in which a fielder is about to begin a trial in front of the target robot and the motion-capture cameras. The robot has a ball suspended above it that the fielder will pursue and grab once the robot starts moving. Both that ball and the fielder's helmet are marked with infrared reflective tape so that their ongoing positions are recorded by the motion-capture system.
Figure 6
 
Experimental setup. (a) Shown is a top-view diagram of the experimental setup: the four fielder starting positions, the target robot starting position, and the positions of the eight motion-capture cameras arranged in a U-shaped arc. The initial position of the robot is in front of the central four cameras, whereas the four initial fielder positions are designated far right, far left, near right, and near left (relative to the robot). Starting distances were 12 m for the far positions and 6 m for the close ones. (b) Shown is a photograph in which a fielder is about to begin a trial in front of the target robot and the motion-capture cameras. The robot has a ball suspended above it that the fielder will pursue and grab once the robot starts moving. Both that ball and the fielder's helmet are marked with infrared reflective tape so that their ongoing positions are recorded by the motion-capture system.
A custom-made mobile robot controlled by a PC was used as the moving target. It is capable of traveling up to speeds of 10 m/s. A marker-based helmet worn by participants was used to indicate ongoing fielder position. A marked ball on a pole was mounted on the robot and designated as the target to be grasped. 
Procedure
The study took place in a gymnasium at Arizona State University. The four positions were starting from either the left or the right side relative to the robot and starting from either far away (12 m) or close (6 m), as shown in Figure 6. The robot typically moved at speeds of about 5 m/s, along a random, evasive pathway that varied in speed and direction. A random variety of robot movement paths were selected and were remotely preprogrammed before the start of each trial, so target paths were independent of ongoing fielder tracking behavior. Approximately one third of the selected paths were intentionally simple, and the remaining ones were intentionally more complex to maintain task diversity. 
For each trial, fielders started from one of four designated initial positions relative to the location of the robot. They were instructed to “go all out” and grasp the ball mounted on the robot as rapidly as possible, and they were informed that the robot might randomly change its speed and direction. In each position, each fielder performed five trials, navigating to grasp the ball mounted on the robot. These resulted in 20 trials for each fielder, for a total of 60 trials. Each trial typically lasted about 2 to 3 s. Fielders were successful in touching or grasping the target ball atop the robot on every trial. 
Results
Among all 60 trials, there were 18 more complex ones in which the robot dramatically changed speed or direction and fielders appeared to discontinuously readjust their approach behavior immediately following the dramatic deviation. We defined these discontinuity points to be located at the beginning point of a turn when a fielder rapidly changed running path direction more than 45°. In these trials, trajectories were split into sections at the initial point of discontinuity, and the model parameters were reset (Figure 7). In the remaining 42 trials, only one set of model parameters was used. 
Figure 7
 
Top view of an extreme parameter-resetting case. When dramatic discontinuities in robot (red) and fielder (blue) behavior occurred, the control model parameters were allowed to reset at the discontinuity. Shown is an extreme example in which a trial was split into three parts. (a) Top view of the whole trial (with hat icon designating the fielder start point and the ball icon designating the robot start point). (b) Top view of Part 1. (c) Top view of Part 2. (d) Top view of Part 3. Note how the fielder dramatically changes direction corresponding to earlier dramatic changes in robot direction.
Figure 7
 
Top view of an extreme parameter-resetting case. When dramatic discontinuities in robot (red) and fielder (blue) behavior occurred, the control model parameters were allowed to reset at the discontinuity. Shown is an extreme example in which a trial was split into three parts. (a) Top view of the whole trial (with hat icon designating the fielder start point and the ball icon designating the robot start point). (b) Top view of Part 1. (c) Top view of Part 2. (d) Top view of Part 3. Note how the fielder dramatically changes direction corresponding to earlier dramatic changes in robot direction.
Figures 8 and 9 illustrate graphs of the data from the typical trials. For each trial, there are five graphs: one showing a top view of the fielder approaching the robot, a second showing the fit for the vertical OAC model, and the last three illustrating the fit for each of the three lateral control models. 
Figure 8
 
Typical trial without parameter resetting. (a) Top view of the ongoing position of the fielder (blue) and the target robot (red), with the hat icon designating the fielder start point and the ball icon designating the robot start point. The remaining diagrams shown in (b), (c), (d), and (e) are all graphs depicting the observed optical angles to the target from the fielder's perspective (blue) and the best-fit angular values corresponding to each optical control model (red). (b) The cotangent of the vertical optical angle (blue) and model prediction of optical acceleration cancellation (OAC) control mechanism (red). (c) The alignment angle (blue) and prediction of the constant alignment angle (CAA) control mechanism (red). (d) The eccentricity angle (blue) and the prediction of the constant eccentricity angle (CEA) control mechanism (red). (e) The optical projection angle or optical trajectory of the target (blue) and prediction of the linear optical trajectory (LOT) control mechanism (red). The LOT model specifies values of the lateral optical angle, β, as a function of the vertical optical angle, α.
Figure 8
 
Typical trial without parameter resetting. (a) Top view of the ongoing position of the fielder (blue) and the target robot (red), with the hat icon designating the fielder start point and the ball icon designating the robot start point. The remaining diagrams shown in (b), (c), (d), and (e) are all graphs depicting the observed optical angles to the target from the fielder's perspective (blue) and the best-fit angular values corresponding to each optical control model (red). (b) The cotangent of the vertical optical angle (blue) and model prediction of optical acceleration cancellation (OAC) control mechanism (red). (c) The alignment angle (blue) and prediction of the constant alignment angle (CAA) control mechanism (red). (d) The eccentricity angle (blue) and the prediction of the constant eccentricity angle (CEA) control mechanism (red). (e) The optical projection angle or optical trajectory of the target (blue) and prediction of the linear optical trajectory (LOT) control mechanism (red). The LOT model specifies values of the lateral optical angle, β, as a function of the vertical optical angle, α.
Figure 9
 
Typical trial with parameter resetting. (a) Top view of the ongoing position of fielder (blue) and target robot (red), with the hat icon designating the fielder start point and the ball icon designating the robot start point. Below this graph is a blow up of the bold inset box elucidating the point at which the fielder turned around, which occurred very close to the 2-s point in time. (b–e) The remaining four diagrams are all graphs depicting the observed and predicted optical angles of the models as explained in Figure 8. The red lines showing the three lateral control models CAA, CEA, and LOT are each allowed to discontinuously reset parameters at the 2-s point, corresponding with the clear behavioral change of the fielder at that time.
Figure 9
 
Typical trial with parameter resetting. (a) Top view of the ongoing position of fielder (blue) and target robot (red), with the hat icon designating the fielder start point and the ball icon designating the robot start point. Below this graph is a blow up of the bold inset box elucidating the point at which the fielder turned around, which occurred very close to the 2-s point in time. (b–e) The remaining four diagrams are all graphs depicting the observed and predicted optical angles of the models as explained in Figure 8. The red lines showing the three lateral control models CAA, CEA, and LOT are each allowed to discontinuously reset parameters at the 2-s point, corresponding with the clear behavioral change of the fielder at that time.
Each of the three models was analyzed by first computing the residual difference between the actual and predicted control optical angle on each recorded frame and then determining the standard deviation (σ) of these residuals over the entire trial. Thus, the analysis of all trials resulted in 60 σ's for the dependent optical angle of each control model, in which a value of zero indicates a perfect model fit. For the CAA, this resulted in standard deviation (in degrees) about a constant alignment angle (σCAA). For the CEA, this resulted in a standard deviation (in degrees) about a constant eccentricity angle (σCEA). For the LOT, this resulted in a standard deviation (in degrees) of the lateral optical angle, β, around its linear optical fit with the vertical optical angle, α (σLOT). The mean standard deviations for the three model optical angles are: 
  1.  
    CAA: mean standard deviation, σCAA = 2.74°
  2.  
    CEA: mean standard deviation, σCEA = 19.12°
  3.  
    LOT: mean standard deviation, σLOT = 1.56°.
The mean standard deviations of the predictive optical angles of each of the three models were compared to each other using matched-pairs t tests, so that the predictor angle deviations for each trial were matched in the analyses of model comparisons. 
CAA versus CEA
Residual error from CEA exhibited significantly larger standard deviations than that from CAA (mean σCEAσCAA = 16.38°, t59 = 15.44, p < 0.001, Cohen's d = 4.02). Each of the fielders also individually produced the same trend, with all three reaching statistical significance (Fielder 1: mean σCEAσCAA = 18.04°, t19 = 8.30, p < 0.001, Cohen's d = 3.81; Fielder 2: mean σCEAσCAA = 18.27°, t19 = 9.64, p < 0.001, Cohen's d = 4.43; Fielder 3: mean σCEAσCAA = 12.83°, t19 = 11.81, p < 0.001, Cohen's d = 5.42). Splitting the data into simple and complex trials also produced the same trend, with the difference between the CAA and CEA models reaching significance in both conditions. Results for both the simple trials (mean σCEAσCAA = 16.18°, t41 = 12.37, p < 0.001, Cohen's d = 3.86) and the complex trials (mean σCEAσCAA = 16.86°, t17 = 9.17, p < 0.001, Cohen's d = 4.45) specified strong effects differentiating the models. 
LOT versus CAA
Residual error from CAA exhibited significantly larger standard deviations than that from LOT (mean σCAAσLOT = 1.18°, t59 = 5.27, p < 0.001, Cohen's d = 1.37). Each of the fielders also individually produced the same trend, with all three reaching statistical significance (Fielder 1: mean σCAAσLOT = 0.98°, t19 = 1.76, p < 0.05, Cohen's d = 0.81; Fielder 2: mean σCAAσLOT = 1.51°, t19 = 7.85, p < 0.001, Cohen's d = 3.60; Fielder 3: mean σCAAσLOT = 1.05°, t19 = 3.15, p < 0.01, Cohen's d = 1.45). Splitting the data into simple and complex trials also produced the same trend, with the difference between the LOT and CAA models reaching significance in both conditions. Results for both the simple trials (mean σCAAσLOT = 1.16°, t41 = 4.45, p < 0.001, Cohen's d = 1.39) and the complex trials (mean σCAAσLOT = 1.21°, t17 = 2.76, p < 0.01, Cohen's d = 1.34) specified strong effects differentiating the models. 
LOT versus CEA
Residual error from CEA exhibited significantly larger standard deviations than that from LOT (mean σCEAσLOT = 17.56°, t59 = 16.48, p < 0.001, Cohen's d = 4.29). Each of the fielders also individually produced the same trend, with all three reaching statistical significance (Fielder 1: mean σCEAσLOT = 19.02°, t19 = 9.12, p < 0.001, Cohen's d = 4.18; Fielder 2: mean σCEAσLOT = 19.78°, t19 = 10.11, p < 0.001, Cohen's d = 4.64; Fielder 3: mean σCEAσLOT = 13.88°, t19 = 12.14, p < 0.001, Cohen's d = 5.57). Splitting the data into simple and complex trials also produced the same trend, with the difference between the LOT and CEA models reaching significance in both conditions. Results for both the simple trials (mean σCEAσLOT = 17.34°, t41 = 13.42, p < 0.001, Cohen's d = 4.19) and for the complex trials (mean σCEAσLOT = 18.08°, t17 = 9.39, p < 0.001, Cohen's d = 4.55) specified strong effects differentiating the models. 
Statistically, the LOT strategy exhibits the best fit to fielder's performance of the three lateral optical angle control algorithms, whereas CEA exhibits the worst fit. On average, the LOT model accounted for 95.2% of the variance in lateral optical target movement. 
In addition, the mean standard deviation for the OAC model is 1.05°. On average, the OAC model accounted for 99.3% of the variance in vertical optical target movement. The data fit the OAC model quite well and confirm the assumption that OAC occurs in tandem with one of the three lateral control strategies. 
Discussion
The goal of this work was to determine which optical control model best accounts for the lateral navigational behavior of fielders chasing and intercepting targets moving along complex, evasive ground-based pathways in a real-world environment. Results support that the strategy of maintaining a LOT relative to the background best matches the fielder behavior. This is consistent with past findings of fielders intercepting ballistic airborne baseballs (Aboufadel, 1996; Marken, 2001; McBeath et al., 1995a; Shaffer & McBeath, 2002; Wang et al., 2015), more complex moving airborne targets such as Frisbees and toy helicopters (Shaffer et al., 2004; Shaffer et al., 2008; Shaffer, Marken et al., 2013), as well as ballistic ground balls (Sugar, McBeath, & Wang, 2006), but in the case of ground-based trajectories, fielders maintain a flipped version of the same optical angular control heuristics. Thus, despite the vast differences in forces applied on airborne versus ground-based targets and resultant differences in target behavior, in both cases fielders selected running paths that maintained constancy of optical speed and angle of the target. The present study is novel in that it addresses what happens when fielders pursue targets that move erratically along complex, evasive pathways. In our analysis, fielders are allowed to reset angular control parameters during cases when they dramatically change running direction. The findings support that the LOT interception strategy used in simple ballistic cases also generalizes well to these ecologically valid, real-world types of cases in which targets move along complex, erratic pathways. 
The LOT model has the additional benefit that it is a strategy that integrates vertical and lateral control components in a manner that seamlessly combines with the OAC model. For an optimal OAC strategy, the fielder maintains zero optical acceleration in the vertical direction, whereas for the optimal LOT model, the fielder maintains zero optical acceleration in both the vertical and horizontal directions. Thus, common acceleration detecting mechanisms could potentially be used, but in the case of LOT, the 2D acceleration manifests as spatial curvature. Because discrimination of spatial curvature has been found to be easier for observers than discrimination of temporal acceleration, this can account for the curious finding that fielders consider balls headed off to the side to be easier to track down than ones headed directly toward them, despite the added burden of an additional lateral control mechanism (McBeath et al., 1995a; Shaffer & McBeath, 2005). This type of coupled control strategy is the hallmark of several recent general-purpose dynamic perception-action models (Gigerenzer & Goldstein, 2011; Lee, 2014; Marken, 2014). 
McLeod et al. (2003, 2006) introduced a version of an alignment angle model that allows it to change linearly over time. They argued that the rate of change in the alignment angle and the rate of change in the vertical optical angle should be viewed as independent. Yet our data support that the two rates of change are not independent but remain proportional to each other to produce a LOT. When the OAC is maintained, the fielder keeps the vertical image of the target changing at a constant rate. If the fielder also keeps the lateral alignment angle changing at a constant rate, then the two remain proportional, and that control model becomes functionally equivalent to the LOT. In other words, if an image moves vertically at a constant optical rate and horizontally at a constant optical rate, then it will also travel along a straight optical projection line. As long as changes in both the vertical and lateral control angles are defined as local changes within a projection plane perpendicular to the fielder's angle of gaze, then the OAC and LOT models converge to the same solution. With our use of high-resolution 3D motion-capture measurements, the current study confirms that the optical path of the target also remains linear within the continuously tilting gaze projection plane, and that this is true for ground-based targets moving along complex pathways. Describing the control model as a LOT has the advantage that it both converges to OAC as the target angle becomes more vertical and accounts for why fielders often find it more difficult to catch balls headed directly toward them, despite the seeming simplicity of eliminating the need for a lateral control mechanism. 
In recent years, there has been some disagreement regarding the generalizability of the LOT model, primarily in studies using either virtual environments or with thought experiments that vary in the manner in which optical linearity is defined (Adair, 1995; Fink et al., 2009; Lee, 2014; Turvey & Fonseca, 2009; Zaal & Bootsma, 2011). In the current study, we clarify that we define optical linearity as maintaining the same instantaneous ratio of changes in the lateral versus vertical optical position of the target. We additionally note that although tests of catching in virtual environments have the advantage of being able to fully specify stimulus parameters, they also have the disadvantages of limitations in display resolution and reaction speed, and they have been shown to fundamentally distort the perceived environment (Knapp & Loomis, 2004; Renner, Velichkovsky, & Helmert, 2013; Tompson et al., 2004). A big issue with the depth cue and the temporal reaction time limitations is that they are likely to weaken the weighting of rapid interactive perception-action control mechanisms and promote the weighting of slower, more cognitive-level mechanisms. So a fundamental advantage of the real-world test used in the current study is that it allows participants a natural, full-range, ecologically valid environment. Other research findings examining navigational strategy in real-world environments with somewhat complex moving targets confirmed findings consistent with usage of the LOT control mechanism (Shaffer, Dolgov, Maynor, & Reed, 2013; Shaffer, Marken et al., 2013). 
The CEA model has been promoted in a broad range of settings varying from ball playing (Chardenon et al., 2005), to vehicular navigation of boats and planes (Beall & Loomis, 1997; Pollack, 1995), to usage by biological predators such as fish and insects (Collett & Land, 1975; Lanchester & Mark, 1975). Although it is a mathematically proven strategy, it was the least favored of the ones that we compared in our test setting with a complex, moving target. One possibility is that CEA may generally operate at a higher cognitive level, as an additional mechanism to periodically augment or adjust more rapid, basic, lower-level perceptual control mechanisms such as LOT and CAA. In any case, our findings support that CEA is the least well-fitting angular control mechanism for intercepting rapid, complexity-moving, ground-based targets. 
Because the control strategies that we compare are based on maintenance of optical angles, there is not a unique optimal running path for each strategy, but rather each specifies a family of running paths, determined by the angular parameters the fielder selects. This may seem to be a disadvantage of such models, but this level of diversity more accurately characterizes actual variance found between running paths in cases in which fielders run to catch fly balls (McBeath et al., 1995a, 1995b; McLeod et al., 2006; Shaffer et al., 2008; Wang et al., 2015). This expected variance is particularly notable in circumstances in which either an airborne target is at a very high vertical angle such as infield pop flies or a ground-based target is very distant and near the horizon (McBeath et al., 2008; Sugar, McBeath, Suluh, et al., 2006). Under such circumstances, movements of the fielders have little effect on the optical position of the target relative to the background, so optical control strategies predict increased variance in running path choice. Studies examining running paths of identically lofted pop fly balls confirm this fielder variance, verifying that some fielders were more aggressive, sped ahead earlier, and were at a different angle compared with others who lagged behind and had to make up distance at the end (McLeod et al., 2001; Shaffer et al., 2008). Because all three of the optical angular control strategies being compared here specify behavior only relative to the specific angular variables in that model, each allows for a range of deviation in running paths. A feature of the design of this study is that all three strategies being compared are entirely based on a single optical angle that fielders use as their principal control variable, so the statistical comparison is well matched. 
Other research has explored angular control mechanisms within complex scenes using considerably more elaborate behavioral models. Warren (2006) developed and promoted a behavioral dynamics model of locomotor control designed to account for running path nonlinearities due to inertial factors. Warren modeled locomotor behavior as a nonlinear dynamical system and specified a set of equations. The model contains four independent variables, including one for the distance between the individual and the target, which often may not be perceptually known or easy to directly discern. Warren's model appears useful for determining the point at which a fielder may elect to abandon heading in one direction to precede in another and has the nice feature of accounting for inertial factors of movement. The models that we compared are considerably simpler than Warren's yet can still typically account for more than 95% of the variance in ongoing lateral optical angle. The current research demonstrates that a good match of real-world fielder behavior can be achieved with a locomotive model that uses fewer variables and is based only on maintaining simple geometric angular principles during interception. These simple geometric principles can be used to intercept targets that follow not only ballistic trajectories but also active ones that move along complex, evasive pathways. We suggest that the extra parameters included by Warren's inertial model may be needed only when the complexity of the environment promotes fielder usage of higher-level navigational mechanisms, particularly in virtual environments, which may be more likely to require inertial correction factors. 
Findings from several studies of animals and insect prey interception are consistent with usage of the LOT control mechanism coupled with OAC (Fux & Eilam, 2009; Tucker, Tucker, Akers, & Enderson, 2000). Shaffer et al. (2004) confirmed that the LOT model reliably describes the behavior of dogs chasing Frisbees and noted that in cases when the Frisbee dramatically changed direction near the end of the trajectory, dogs appeared to establish a second LOT with new parameter values or a segmented LOT. Shaffer, Marken et al. (2013) later found the same segmented LOT (or SLOT) pattern with fielders chasing toy helicopters. The current work replicates this pattern of fielders resetting control parameters when targets dramatically change direction, but in this case with ground-based targets. Ghose, Horiuchi, Krishnaprasad, and Moss (2006) examined the strategy of bats catching prey moving along complex, evasive flight paths and modeled the pursuit trajectories by using simple delay differential equations. They concluded that, in order to minimize the time to intercept an unpredictably moving target, bats use a constant absolute target direction strategy during pursuit and not a constant eccentricity angle strategy. Hence, they also appear to favor a strategy that is similar to maintenance of both CAA and LOT. That work verifies that the same geometric navigational principles that are visually controlled in our task can be achieved using ultrasonic auditory control mechanisms by bats. 
This research is directly applicable to the development of superior navigation control algorithms in the design and operation of truly autonomous mobile robots, specialized for either interception or collision avoidance. The simple geometric control strategies that we tested are easily programmed as optical control parameters that guide mobile robotics to navigate naturally toward or away from moving target objects (Sugar & McBeath, 2001; Sugar, McBeath, Suluh et al., 2006). The interception geometry is identical to that shown in Figure 4. When the target optical trajectory begins to accelerate vertically, the robot adjusts forward and backward to cancel that, and when it begins to accelerate laterally, the robot adjusts leftward and rightward to cancel that. This work holds promise for automated navigation and interception tasks. For example, in transportation applications, the use of an optical angular control mechanism can contribute to automated collision-avoidance systems for vehicles or handicapped pedestrians, such as the blind. 
The current study also advances the methodology of determining dynamic optical angles. Most past research has measured the optical angle either by modeling the trajectory and indirectly calculating angles from the fielder and the inferred ball positions (McLeod & Dienes, 1996; McLeod et al., 2006) or by measuring the angular changes relative to the local background scenery with the use of head cams (McBeath et al., 1995a; Shaffer et al., 2004). The present usage of a rapid-refresh, multicamera, 3D motion-capture system allows us make indisputable, real-time, high-resolution angle and position measurements. 
More generally, the types of control models that our findings support can serve as archetypical examples of control theory in the area of perception-action as well as the broader arena of psychology. Over the years, the OAC model has been framed as an example in which a controlling, pursuing, perceiver-actor (i.e., fielder) maintains relative control over a target by not allowing its optical image to accelerate vertically, either upward or downward. As long as the controlling fielder is able to approach the target and is fast enough to react and maintain cancellation of any vertical acceleration, a collision will occur, independent of the actual vertical movement of the target. Similarly, when targets travel both vertically and laterally, the LOT strategy specifies the same control principal of eliminating optical acceleration but this time in both the vertical and lateral directions, again independent of the actual 3D trajectory of the target. Thus, in examples such as the one shown in Figure 4, despite the actual curvy 3D path of the target, the observed optical trajectory remains a straight line, exhibiting neither vertical nor lateral acceleration from the perspective of the fielder. In practice, we have found some variance in the actual optical angle chosen by fielders, depending on factors such as running speed, but typically fielders try to maintain a constant projection angle somewhat close to the one that is initially presented to them. In cases like in the present study in which targets change direction enough so that fielders are unable to maintain the initial LOT angle, they appear to simply recalibrate to the new optical angle to which the target has presented itself. Not all linear optical trajectory angles are viable, but maintenance of ones near the initial LOT angle of the target tends to lead to easier, lower-energy pursuit paths. This type of acceleration cancelation control mechanism can in theory apply to virtually any domain in which the relationship parameters between controller and target can be operationalized (Gigerenzer & Goldstein, 2011; Marken, 2014). 
This study compared three simple lateral control algorithms to see which generalized best to interceptive behavior in a complex navigational pursuit task. Our findings support that fielders use the same simple optical control mechanism to navigate toward targets with complex, evasive movements as they do when intercepting targets with simple ballistic movements but with the ability to reset parameters following dramatic changes in target behavior. Maintenance of a linear optical trajectory appears to be a superior, general-purpose strategy for maintaining angular constancy and navigating to intercept either airborne or ground-based targets that move laterally, either ballistically or along complex, evasive pathways. 
Acknowledgments
This research was supported in part by NSF grants BCS-0318313 and CISE-0403428. 
Commercial relationships: none. 
Corresponding author: Michael K. McBeath. 
E-mail: Michael.McBeath@asu.edu. 
Address: Department of Psychology, Arizona State University, Tempe, AZ, USA. 
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Figure 1
 
Model of interception using optical acceleration cancellation (OAC). α indicates the vertical optical angle from the horizon to the ball. (a) Fly ball case. A fielder runs along a path that keeps the tangent of the vertical optical ball angle (tanα) increasing at a constant rate. Shown is a fielder (hat-eye) approaching an air-resistance shortened ball trajectory, in equal temporal intervals (t0–t7). (b) Ground ball case is a flipped, downward version of OAC. A fielder keeps eye level relatively constant while maintaining a relatively constant distance from the initial location of the ball and keeps the cotangent of the vertical optical angle (cotα) of the ball decreasing at a constant rate. Shown is a fielder (eye icon) and a ground ball approaching in equal temporal intervals (t0–t4).
Figure 1
 
Model of interception using optical acceleration cancellation (OAC). α indicates the vertical optical angle from the horizon to the ball. (a) Fly ball case. A fielder runs along a path that keeps the tangent of the vertical optical ball angle (tanα) increasing at a constant rate. Shown is a fielder (hat-eye) approaching an air-resistance shortened ball trajectory, in equal temporal intervals (t0–t7). (b) Ground ball case is a flipped, downward version of OAC. A fielder keeps eye level relatively constant while maintaining a relatively constant distance from the initial location of the ball and keeps the cotangent of the vertical optical angle (cotα) of the ball decreasing at a constant rate. Shown is a fielder (eye icon) and a ground ball approaching in equal temporal intervals (t0–t4).
Figure 2
 
Constant alignment angle (CAA) lateral strategy. A fielder chooses a lateral strategy that tries to keep the top view of the vector between him and the target object parallel or aligned to the initial direction. Shown are top views of a fielder (in purple) pursuing a target (in red) that moves from left to right. In the CAA strategy, the fielder tries to null out the optical angle, ϕ, created by the vector from the fielder to the target (in green) and a line parallel to the initial orientation of this vector. (a) Ideal alignment (ϕ always 0°). In the ideal case, the vertical alignment lines between the fielder and the target remain parallel at times t0, t1, and t2. The alignment offset angle is ϕ = 0° when ideal alignment is achieved. (b) Offset alignment (ϕ varies from 0°). The dotted vertical background lines show the locations that remain aligned with the fielder at times t0, t1, and t2. The alignment offset angle, ϕ, remains near 0° but varies somewhat in real-world cases.
Figure 2
 
Constant alignment angle (CAA) lateral strategy. A fielder chooses a lateral strategy that tries to keep the top view of the vector between him and the target object parallel or aligned to the initial direction. Shown are top views of a fielder (in purple) pursuing a target (in red) that moves from left to right. In the CAA strategy, the fielder tries to null out the optical angle, ϕ, created by the vector from the fielder to the target (in green) and a line parallel to the initial orientation of this vector. (a) Ideal alignment (ϕ always 0°). In the ideal case, the vertical alignment lines between the fielder and the target remain parallel at times t0, t1, and t2. The alignment offset angle is ϕ = 0° when ideal alignment is achieved. (b) Offset alignment (ϕ varies from 0°). The dotted vertical background lines show the locations that remain aligned with the fielder at times t0, t1, and t2. The alignment offset angle, ϕ, remains near 0° but varies somewhat in real-world cases.
Figure 3
 
Constant eccentricity angle (CEA) lateral strategy. A fielder chooses a lateral strategy, attempting to keep constant the angle defined by the top view of the fielder's direction of travel and the vector between the fielder and the target (in green). Here, the optical eccentricity angle is specified as θ. Shown are top views of a fielder (in purple) pursuing a target (in red) that moves from left to right. (a) Ideal constant eccentricity angle (θ remains at θ0). In the ideal case, a fielder selects a path that keeps θ constant over time. (b) Nonconstant eccentricity angle (θ varies from θ0). Realistically, a fielder selects a path in which θ varies somewhat over time.
Figure 3
 
Constant eccentricity angle (CEA) lateral strategy. A fielder chooses a lateral strategy, attempting to keep constant the angle defined by the top view of the fielder's direction of travel and the vector between the fielder and the target (in green). Here, the optical eccentricity angle is specified as θ. Shown are top views of a fielder (in purple) pursuing a target (in red) that moves from left to right. (a) Ideal constant eccentricity angle (θ remains at θ0). In the ideal case, a fielder selects a path that keeps θ constant over time. (b) Nonconstant eccentricity angle (θ varies from θ0). Realistically, a fielder selects a path in which θ varies somewhat over time.
Figure 4
 
Linear optical trajectory (LOT) strategy. When a fielder runs to the side, the observed position of the ball sweeps laterally from its former position. Over small periods of time, the magnitude of the lateral optical change in target position relative to the background can be designated as Δβ, and the lateral optical angle, β = Σ(Δβ). A LOT is achieved when the rate of change of β is matched to the rate of change in the vertical optical angle, α, or / = Constant, so ψ = acot(/) = Constant. Geometrically, the instantaneous triangular projections remain congruent. (a) Fly ball case. (b) Two-dimensional projection of the ball trajectory from the fielder's perspective, where the projection triangles are tiled to form a continuous straight optical trajectory. This illustrates how the fielder experiences the ball continuously moving along a straight optical path relative to the local background scenery. (c) Ground ball case. In aerial and ground ball cases, the vertical optical change is shown in red and the lateral change in blue. Similar to the optical acceleration cancellation (OAC) model for ground balls, the LOT model allows for fielder decreases in eye height over time (as depicted to the right), but such height changes are not necessary or specified by the model.
Figure 4
 
Linear optical trajectory (LOT) strategy. When a fielder runs to the side, the observed position of the ball sweeps laterally from its former position. Over small periods of time, the magnitude of the lateral optical change in target position relative to the background can be designated as Δβ, and the lateral optical angle, β = Σ(Δβ). A LOT is achieved when the rate of change of β is matched to the rate of change in the vertical optical angle, α, or / = Constant, so ψ = acot(/) = Constant. Geometrically, the instantaneous triangular projections remain congruent. (a) Fly ball case. (b) Two-dimensional projection of the ball trajectory from the fielder's perspective, where the projection triangles are tiled to form a continuous straight optical trajectory. This illustrates how the fielder experiences the ball continuously moving along a straight optical path relative to the local background scenery. (c) Ground ball case. In aerial and ground ball cases, the vertical optical change is shown in red and the lateral change in blue. Similar to the optical acceleration cancellation (OAC) model for ground balls, the LOT model allows for fielder decreases in eye height over time (as depicted to the right), but such height changes are not necessary or specified by the model.
Figure 5
 
Interception control models when pursuing a simple constant velocity target. Shown is a top view of a fielder approaching a target. The approach follows a straight path and maintains much perceptual regularity: (i) Lateral alignment with the target (green vertical lines parallel), (ii) constant eccentricity angle to the target (blue angle, θ, constant), (iii) linear optical trajectory (triangular optical projection remains congruent, see Figure 4b). Tests can tease apart which of these control strategies dominate only by using targets with complex, changing paths.
Figure 5
 
Interception control models when pursuing a simple constant velocity target. Shown is a top view of a fielder approaching a target. The approach follows a straight path and maintains much perceptual regularity: (i) Lateral alignment with the target (green vertical lines parallel), (ii) constant eccentricity angle to the target (blue angle, θ, constant), (iii) linear optical trajectory (triangular optical projection remains congruent, see Figure 4b). Tests can tease apart which of these control strategies dominate only by using targets with complex, changing paths.
Figure 6
 
Experimental setup. (a) Shown is a top-view diagram of the experimental setup: the four fielder starting positions, the target robot starting position, and the positions of the eight motion-capture cameras arranged in a U-shaped arc. The initial position of the robot is in front of the central four cameras, whereas the four initial fielder positions are designated far right, far left, near right, and near left (relative to the robot). Starting distances were 12 m for the far positions and 6 m for the close ones. (b) Shown is a photograph in which a fielder is about to begin a trial in front of the target robot and the motion-capture cameras. The robot has a ball suspended above it that the fielder will pursue and grab once the robot starts moving. Both that ball and the fielder's helmet are marked with infrared reflective tape so that their ongoing positions are recorded by the motion-capture system.
Figure 6
 
Experimental setup. (a) Shown is a top-view diagram of the experimental setup: the four fielder starting positions, the target robot starting position, and the positions of the eight motion-capture cameras arranged in a U-shaped arc. The initial position of the robot is in front of the central four cameras, whereas the four initial fielder positions are designated far right, far left, near right, and near left (relative to the robot). Starting distances were 12 m for the far positions and 6 m for the close ones. (b) Shown is a photograph in which a fielder is about to begin a trial in front of the target robot and the motion-capture cameras. The robot has a ball suspended above it that the fielder will pursue and grab once the robot starts moving. Both that ball and the fielder's helmet are marked with infrared reflective tape so that their ongoing positions are recorded by the motion-capture system.
Figure 7
 
Top view of an extreme parameter-resetting case. When dramatic discontinuities in robot (red) and fielder (blue) behavior occurred, the control model parameters were allowed to reset at the discontinuity. Shown is an extreme example in which a trial was split into three parts. (a) Top view of the whole trial (with hat icon designating the fielder start point and the ball icon designating the robot start point). (b) Top view of Part 1. (c) Top view of Part 2. (d) Top view of Part 3. Note how the fielder dramatically changes direction corresponding to earlier dramatic changes in robot direction.
Figure 7
 
Top view of an extreme parameter-resetting case. When dramatic discontinuities in robot (red) and fielder (blue) behavior occurred, the control model parameters were allowed to reset at the discontinuity. Shown is an extreme example in which a trial was split into three parts. (a) Top view of the whole trial (with hat icon designating the fielder start point and the ball icon designating the robot start point). (b) Top view of Part 1. (c) Top view of Part 2. (d) Top view of Part 3. Note how the fielder dramatically changes direction corresponding to earlier dramatic changes in robot direction.
Figure 8
 
Typical trial without parameter resetting. (a) Top view of the ongoing position of the fielder (blue) and the target robot (red), with the hat icon designating the fielder start point and the ball icon designating the robot start point. The remaining diagrams shown in (b), (c), (d), and (e) are all graphs depicting the observed optical angles to the target from the fielder's perspective (blue) and the best-fit angular values corresponding to each optical control model (red). (b) The cotangent of the vertical optical angle (blue) and model prediction of optical acceleration cancellation (OAC) control mechanism (red). (c) The alignment angle (blue) and prediction of the constant alignment angle (CAA) control mechanism (red). (d) The eccentricity angle (blue) and the prediction of the constant eccentricity angle (CEA) control mechanism (red). (e) The optical projection angle or optical trajectory of the target (blue) and prediction of the linear optical trajectory (LOT) control mechanism (red). The LOT model specifies values of the lateral optical angle, β, as a function of the vertical optical angle, α.
Figure 8
 
Typical trial without parameter resetting. (a) Top view of the ongoing position of the fielder (blue) and the target robot (red), with the hat icon designating the fielder start point and the ball icon designating the robot start point. The remaining diagrams shown in (b), (c), (d), and (e) are all graphs depicting the observed optical angles to the target from the fielder's perspective (blue) and the best-fit angular values corresponding to each optical control model (red). (b) The cotangent of the vertical optical angle (blue) and model prediction of optical acceleration cancellation (OAC) control mechanism (red). (c) The alignment angle (blue) and prediction of the constant alignment angle (CAA) control mechanism (red). (d) The eccentricity angle (blue) and the prediction of the constant eccentricity angle (CEA) control mechanism (red). (e) The optical projection angle or optical trajectory of the target (blue) and prediction of the linear optical trajectory (LOT) control mechanism (red). The LOT model specifies values of the lateral optical angle, β, as a function of the vertical optical angle, α.
Figure 9
 
Typical trial with parameter resetting. (a) Top view of the ongoing position of fielder (blue) and target robot (red), with the hat icon designating the fielder start point and the ball icon designating the robot start point. Below this graph is a blow up of the bold inset box elucidating the point at which the fielder turned around, which occurred very close to the 2-s point in time. (b–e) The remaining four diagrams are all graphs depicting the observed and predicted optical angles of the models as explained in Figure 8. The red lines showing the three lateral control models CAA, CEA, and LOT are each allowed to discontinuously reset parameters at the 2-s point, corresponding with the clear behavioral change of the fielder at that time.
Figure 9
 
Typical trial with parameter resetting. (a) Top view of the ongoing position of fielder (blue) and target robot (red), with the hat icon designating the fielder start point and the ball icon designating the robot start point. Below this graph is a blow up of the bold inset box elucidating the point at which the fielder turned around, which occurred very close to the 2-s point in time. (b–e) The remaining four diagrams are all graphs depicting the observed and predicted optical angles of the models as explained in Figure 8. The red lines showing the three lateral control models CAA, CEA, and LOT are each allowed to discontinuously reset parameters at the 2-s point, corresponding with the clear behavioral change of the fielder at that time.
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