Reinoud J Bootsma
email: reinoud.bootsma@univ-amu.fr

Simon Ledouit

Remy Casanova

Frank T J M Zaal

Fractional-order information in the visual control of lateral locomotor interception

Keywords: Perception-action, interception, movements, information, fractional order, model, dynamics, human

Previous work on locomotor interception of a target moving in the transverse plane has suggested that interception is achieved by maintaining the target's bearing angle (often inadvertently confused and/or confounded with the target-heading angle) at a constant value.

However, dynamics-based model simulations testing the veracity of the underlying control strategy of nulling optical velocity (i.e., the rate of change in bearing angle) have been restricted to limited conditions of target-motion and only few alternatives have been considered. Exploring a wide range of target motion characteristics with straight and curving ball trajectories in a virtual-reality setting, we examined how soccer goalkeepers moved along the goal line to intercept long-range shots on goal, a situation in which interception is naturally constrained to movement along a single dimension. Analyses of the movement patterns suggested reliance on combinations of optical position and velocity for straight trajectories and optical velocity and acceleration for curving trajectories. As an alternative to combining such standard integer-order derivatives, we demonstrate with a simple dynamical model that nulling a single informational variable of a self-tuned fractional (rather than integer) order efficiently captures the timing and patterning of the observed interception behaviors. This new perspective could fundamentally change the conception of what perceptual systems may actually provide, both in humans and other animals.

Introduction

Interceptive actions are paradigmatic examples of our behavioral interaction with dynamic elements of the environment. In the present contribution we address the functional organization of locomotor interception, operationally defined as making contact with a moving target by means of whole-body displacement. More specifically, we address the issue of the visual information used to organize such interceptive locomotor actions. While our study examines the example of soccer goalkeepers moving laterally to intercept balls shot at goal from long range, locomotor interception is of course encountered in many different situations and animal species, notably in the form of chasing prey, mates or invaders.

Studying freely moving agents chasing natural or artificial targets, a large body of locomotor interception work has focused on the correspondence of the behavioral patterns to one of two interception heuristics that we will refer to as classical pursuit and classical interception, respectively. In the case of unconstrained movement (see Fig. 1A), classical pursuit is characterized by moving, at every instant, in the current direction of the target. In the terms of [START_REF] Fajen | Behavioral dynamics of intercepting a moving target[END_REF], classical pursuit thus entails steering (i.e., changing heading) to align the locomotor velocity vector with the target's bearing direction such that the target-heading angle β (= heading angle ϕ -bearing angle θ) is equal to zero. For classical pursuit to allow the agent to make contact with the target, agent velocity needs to be larger than target velocity (i.e., v a > v t ). Classical pursuit has been documented in houseflies [START_REF] Land | Chasing behaviour of houseflies (Fannia canicularis)[END_REF], blowflies [START_REF] Boeddeker | Chasing a dummy target: smooth pursuit and velocity control in male blowflies[END_REF], honeybees [START_REF] Gries | Straight forward to the queen: pursuing honeybee drones (Apis mellifera L.) adjust their body axis to the direction of the queen[END_REF] and tiger beetles [START_REF] Gilbert | Visual control of cursorial prey pursuit by tiger beetles (Cicindelidae)[END_REF][START_REF] Haselsteiner | Tiger beetles pursue prey using a proportional control law with a delay of one half-stride[END_REF]. The alternative, classical interception, is characterized by moving in a direction ahead of the target "on a course that will keep the angular deviation of the target from the line of travel constant" (Lanchester & Mark, 1975, p. 628), that is, by maintaining target-heading angle 1 β at a constant non-zero value. For classical interception to allow the agent to make contact with the target, the agent's transverse velocity needs to match the target's transverse velocity (i.e., v t,a = v t,t ) while ensuring that the distance between target and agent decreases (i.e., v r,a > v r,t ) 2 . Such classical interception has been documented in teleost fish [START_REF] Lanchester | Pursuit and prediction in the tracking of moving food by a teleost fish (Acanthaluteres spilomelanurus)[END_REF], dragonflies [START_REF] Olberg | Prey pursuit and interception in dragonflies[END_REF], bats [START_REF] Ghose | Echolocating bats use a nearly time-optimal strategy to intercept prey[END_REF], falcons [START_REF] Kane | Falcons pursue prey using visual motion cues: new perspectives from animal-borne cameras[END_REF], hawks [START_REF] Kane | Falcons pursue prey using visual motion cues: new perspectives from animal-borne cameras[END_REF], dogs [START_REF] Shaffer | How dogs navigate to catch frisbees[END_REF]) and humans [START_REF] Fajen | Visual guidance of intercepting a moving target on foot[END_REF]. ***** Figure 1 about here ***** It has often been assumed that finding that over the course of interception the targetheading angle β remains close to zero or close to a particular non-zero value provides evidence for the operation of a control strategy based on, respectively, classical pursuit or classical interception heuristics. It is important to realize, however, that such evidence is but circumstantial. Direct evidence requires specification of how such a result is brought about, that is, specification of how the agent dynamically controls the locomotor velocity vector. If control is based on the state of target-heading angle β, as suggested by the classical pursuit and interception heuristics, then the steering dynamic driving target-heading angle β to zero (thereby instantiating a classical pursuit strategy) is provided by nulling β, and the steering dynamic driving target-heading angle β to a constant value (thereby instantiating a classical interception strategy) by nulling the rate of change of β. Simulations using such dynamic 1 Many studies have erroneously referred to the angle subtended at the point of observation by the target's angular position with respect to the agent's direction of locomotion as the target's bearing angle rather than the target-heading angle. However, by definition [START_REF] Fajen | Behavioral dynamics of steering, obstable avoidance, and route selection[END_REF]2007;[START_REF] Klatzky | Allocentric and egocentric spatial representations: Definitions, distinctions, and interconnections[END_REF], bearing is defined with respect to an exocentric (allocentric) frame of reference. 2 Following [START_REF] Fajen | Behavioral dynamics of intercepting a moving target[END_REF], we assume that v r is positive in the direction extending from the agent to the target, such that v r,a > 0 and v r,t < 0 in Fig. 1A.

steering models led [START_REF] Fajen | Behavioral dynamics of intercepting a moving target[END_REF] to conclude that a dβ/dt-based classical interception strategy was in fact underconstrained: it resulted in the agent leading the target under some conditions but lagging it under other conditions. The lag solution was not consistent with their empirical observations. [START_REF] Fajen | Behavioral dynamics of intercepting a moving target[END_REF] subsequently demonstrated that, consistent with their empirical observations, a steering dynamic based on nulling the rate of change in bearing angle θ reliably led the agent to steer onto a straight interception path leading the target. Of course, on a straight path (i.e., with a fixed heading angle ϕ) a constant value of bearing angle θ is accompanied by a constant value of targetheading angle β. Yet, [START_REF] Fajen | Behavioral dynamics of intercepting a moving target[END_REF] modeling work revealed that control (i.e., in their case how to get to the straight path and remain there) relies on nulling dθ/dt and not on nulling dβ/dt (also see [START_REF] Olberg | Prey pursuit and interception in dragonflies[END_REF].

Whereas animal work has typically studied freely moving agents, human locomotor interception of targets moving in the transverse plane has often been studied in directionconstrained paradigms, with participants riding a tricycle along a ground-fixed rail [START_REF] Lenoir | Intercepting moving objects during self-motion: effects of environmental changes[END_REF][START_REF] Lenoir | Rate of change of angular bearing as the relevant property in a horizontal interception task during locomotion[END_REF], walking along a designated line [START_REF] Chohan | Postural adjustments and bearing angle use in interceptive actions[END_REF][START_REF] Chohan | Children's use of the bearing angle in interceptive actions[END_REF] or walking on a treadmill [START_REF] Bastin | Prospective strategies underlie the control of interceptive actions[END_REF][START_REF] Bastin | Testing the role of expansion in the prospective control of locomotion[END_REF][START_REF] Chardenon | The visual control of ball interception during human locomotion[END_REF][START_REF] Chardenon | The perceptual control of goal-directed locomotion: a common control architecture for interception and navigation[END_REF]2005;[START_REF] Morice | Environmental constraints modify the way an interceptive action is controlled[END_REF]. Compared to the situation where agents can move freely (Fig. 1A), constraining locomotor displacement to a fixed direction (Fig. 1B) has a number of consequences. First, as steering is no longer possible, the agent can only control locomotor speed v a . Second, the agent can no longer influence the location where contact with the target may be established; this location is now exclusively determined by the characteristics of the target's trajectory. Finally, for interception to be possible at all in a direction-constrained setting the target's trajectory must cross the agent's displacement axis. In other words, it must have an orthogonal velocity component v o,t reducing its distance to the agent's displacement axis. A direction-constrained setting thereby sets up a task-defined exocentric reference direction for locomotor interception that is orthogonal to the locomotor axis. (Fig. 1B).

In the light of the foregoing, it is perhaps surprising that studies of directionconstrained locomotor interception have systematically focused on the target-heading angle β rather than on the bearing angle θ. Moreover, all concluded that locomotor interception was (at least to a certain extent) controlled by nulling dβ/dt [START_REF] Bastin | Prospective strategies underlie the control of interceptive actions[END_REF]2008;[START_REF] Chardenon | The visual control of ball interception during human locomotion[END_REF]2004;2005;[START_REF] Chohan | Postural adjustments and bearing angle use in interceptive actions[END_REF]2008;[START_REF] Lenoir | Intercepting moving objects during self-motion: effects of environmental changes[END_REF][START_REF] Podlubny | Geometric and physical interpretation of fractional integration and fractional differentiation[END_REF][START_REF] Morice | Environmental constraints modify the way an interceptive action is controlled[END_REF] 3 . Not only does our analysis of the literature suggest that for a first-order interception strategy agents would rely on nulling dθ/dt rather than dβ/dt, but focusing on β also eradicates the potential use of a pursuit-like strategy. Indeed, in a direction-constrained paradigm nulling β is not a viable interception strategy: Unrelated to the agent's position (and thus to the establishment of contact), target bearing angle β reaches zero when the target crosses the agent's axis of displacement. Nulling θ, on the other hand, does allow instantiating a viable pursuit strategy, because it leads the agent to continuously track the target and thereby intercept it when the target crosses the agent's displacement axis. Finally, while for a stationary agent target-heading angle β is undefined, this is not the case for bearing angle θ.

In order to make the link with the literature on (direction-constrained) lateral manual interception studies [START_REF] Arzamarski | Lateral ball interception: hand movements during linear ball trajectories[END_REF][START_REF] Dessing | Bending it like Beckham: How to visually fool the goalkeeper[END_REF][START_REF] Dessing | A dynamical neural network for hitting an approaching object[END_REF][START_REF] Dessing | Prospective control of manual interceptive actions: comparative simulations of extant and new model constructs[END_REF][START_REF] Dessing | Visuomotor transformation for interception: catching while fixating[END_REF][START_REF] Dessing | How position, velocity, and temporal information combine in the prospective control of catching: Data and model[END_REF] 3 When moving over an earth-fixed ground surface target-heading angle β and bearing angle θ are of course complementary and nulling the one is therefore equivalent to nulling the other. [START_REF] Chardenon | The perceptual control of goal-directed locomotion: a common control architecture for interception and navigation[END_REF] however demonstrated that lateral translation of the visual ground surface during locomotor interception gave rise to adaptations in locomotor velocity that were consistent with the visually-induced change in θ but, under the assumption that heading direction was perceived from optic flow, not with the concomitant visually-induced change in β. Jacobs & Michaels, 2006;[START_REF] Ledouit | Prospective control in catching: the persistent angle-of-approach effect in lateral interception[END_REF][START_REF] Michaels | Lateral interception II: predicting hand movements[END_REF][START_REF] Montagne | Perception-action coupling in an interceptive task: First-order time-to-contact as an input variable[END_REF][START_REF] Montagne | Movement reversals in ball catching[END_REF][START_REF] Peper | Catching balls: how to get the hand to the right place at the right time[END_REF], the present study focused on lateral locomotor interception. As can be seen from Fig. 1C, the target's bearing angle θ coincides with its egocentric orientation in this particular situation. Using a large-scale virtual reality set-up participants were immerged in the setting of a soccer goalkeeper that could move laterally along the goal line (displacement axis) so as the intercept balls shot at goal from long range. In order to elucidate the information-based control strategy used in such lateral locomotor interception, we studied participant behavior when intercepting both straight and curving ball trajectories.

With the relevant information for lateral locomotor interception thus defined by the state of the optical angle θ, we can distinguish at least three potential control strategies based on the temporal-derivative order of θ that is being nulled: a zeroth-order control strategy based on nulling optical position θ (i.e., nulling d 0 θ/dt 0 ), a first-order control strategy based on nulling optical velocity dθ/dt (i.e., nulling d 1 θ/dt 1 ), and a second-order control strategy based on nulling optical acceleration (i.e., nulling d 2 θ/dt 2 ). Given that the latter strategy, proposed within the framework of catching fly balls4 , has been questioned on the basis of the human visual system's low sensitivity to acceleration [START_REF] Brouwer | Perception of acceleration with short presentation times: Can acceleration be used in interception?[END_REF][START_REF] Calderone | Visual acceleration detection: Effect of sign and motion orientation[END_REF][START_REF] Werkhoven | Visual processing of optic acceleration[END_REF][START_REF] Zaal | Base on balls for the Chapman strategy: reassessing Brouwer[END_REF], it does not seem reasonable to consider strategies based on even higher orders.

Based on the consensus in the literature [START_REF] Chardenon | The visual control of ball interception during human locomotion[END_REF]2004;2005;[START_REF] Chohan | Postural adjustments and bearing angle use in interceptive actions[END_REF]2008;[START_REF] Fajen | Behavioral dynamics of intercepting a moving target[END_REF][START_REF] Lenoir | Intercepting moving objects during self-motion: effects of environmental changes[END_REF][START_REF] Podlubny | Geometric and physical interpretation of fractional integration and fractional differentiation[END_REF], lateral locomotor interception of balls following straight trajectories would be expected to rely on a first-order dθ/dt-nulling strategy. However, studies on lateral manual interception have demonstrated that balls following straight trajectories starting from the same distance from the interception axis and arriving at the same lateral position after the same flight duration give rise to balltrajectory-dependent interception patterns [START_REF] Arzamarski | Lateral ball interception: hand movements during linear ball trajectories[END_REF][START_REF] Ledouit | Prospective control in catching: the persistent angle-of-approach effect in lateral interception[END_REF][START_REF] Montagne | Movement reversals in ball catching[END_REF]. If also present in locomotor interception, this angle-of-approach effect [START_REF] Ledouit | Prospective control in catching: the persistent angle-of-approach effect in lateral interception[END_REF] would militate against a dθ/dt-nulling strategy.

The present study tested the adequacy of the first-order dθ/dt-nulling interception strategy to capture the behavioral patterns observed not only for straight but also for curving ball trajectories. The phenomena encountered when ball trajectories curve (e.g., under the influence of sidespin) indeed provide some qualitative evidence in favor of a first-order strategy. For instance, even top-level soccer players make errors in judging whether balls following laterally curving trajectories will end up in the goalmouth [START_REF] Craig | Judging where a ball will go: the case of curved free kicks in football[END_REF][START_REF] Craig | Optic variables used to judge future ball arrival position in expert and novice soccer players[END_REF]. Moreover, in direction-constrained paradigms, locomotor interception of curving ball trajectories has been demonstrated to give rise to ball-trajectory-specific changes in locomotor velocity [START_REF] Bastin | Prospective strategies underlie the control of interceptive actions[END_REF]2008;[START_REF] Morice | Environmental constraints modify the way an interceptive action is controlled[END_REF], sometimes even leading to reversal of movement direction [START_REF] Casanova | Perception of spin and the interception of curved football trajectories[END_REF][START_REF] Lenoir | Effects of contrasting colour patterns of the ball in the volleyball reception[END_REF]. Similar results have been reported for manual lateral interception [START_REF] Craig | How information guides movement: Intercepting curved free kicks in soccer[END_REF][START_REF] Dessing | Bending it like Beckham: How to visually fool the goalkeeper[END_REF]. However, the generally-drawn conclusion that participants relied on a first-order strategy for the interception of balls following curved trajectories is, for the moment, predominantly supported by the capability of such an interception strategy to capture certain qualitative aspects of the behavioral patterns observed. Modeling of a first-order strategy, on the other hand, yielded rather modest levels of fit, even when parameters were allowed to vary over experimental conditions [START_REF] Bastin | Prospective strategies underlie the control of interceptive actions[END_REF]2008;[START_REF] Morice | Environmental constraints modify the way an interceptive action is controlled[END_REF].

Going beyond the qualitative character of earlier studies focusing on the general form of the interception patterns observed, in the present contribution we also examined the potential of informational variables of different orders (i.e., θ, dθ/dt and d 2 θ/dt 2 ) to structure the movement characteristics at key moments of the interceptive action. As these analyses demonstrated that none of the three interception strategies considered (nulling d 0 θ/dt 0 ; d 1 θ/dt 1 or d 2 θ/dt 2 θ) could capture the behavioral patterns observed, we propose a new perspective for the control of interceptive actions based on nulling information of a fractional rather than integer temporal-derivative order [START_REF] Beek | Studying[END_REF][START_REF] Jacobs | The learning of visually guided action: an information-space analysis of pole balancing[END_REF][START_REF] Podlubny | Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications[END_REF][START_REF] Podlubny | Geometric and physical interpretation of fractional integration and fractional differentiation[END_REF].

Method

Participants

Twenty-one experienced male soccer players (age 25.8 ± 7.7 years) voluntarily participated.

All participants had at least four years of experience in competitive weekly matches.

Task and Procedure

Using a model incorporating aerodynamic Magnus-Robins lift and drag forces fully described earlier [START_REF] Craig | Judging where a ball will go: the case of curved free kicks in football[END_REF], we calculated a set of trajectories departing 30 m from the goal line at lateral distances of -4 m and +4 m with respect to the center of the goal and arriving in the goal at lateral distances of -1.8, -0.6, +0.6, and +1.8 m, at a height of around 1.2 m. For each combination of departure and arrival positions one straight (No Spin, NS) and two curving (10 rot•s -1 around the ball's vertical axis Clockwise Spin, CS, and Counter-Clockwise spin, CCS) trajectories were derived, for a total of 24 trajectories (see Fig. 2A). To allow sufficient player displacement, we slowed down the ball trajectories to flight durations of 2.66 ± 0.07 s, doubling the original ball flight durations using linear interpolation between successive positions. A block of trials included all 24 trajectories presented in random order.

Each participant performed five blocks of trials.

The experiment took place in a large virtual-reality facility (crvm.ism.univ-amu.fr) comprising four projection surfaces, each served by two projectors: a 3 x 3 m floor and three 4-m high x 3-m wide walls. The sidewalls were opened at 45° angles with respect to the front wall (theatre configuration) allowing participants to move laterally over a distance of 7.2 m.

Stereopsis was ensured with passive Infitec® filter technology. Participants' stereo glasses were equipped with a configuration of reflective markers. An eight-camera ART® optical system enabled real-time motion capture of head position. The visual scene was refreshed at 60 Hz, taking into account the position of the participant's eyes relative to the virtual environment.

Using in-house developed software (ICE) we simulated a soccer stadium with a grass pitch. Positioned within a regulation-sized goal (7.32 m wide and 2.44 m high), participants could see the pitch with the goal line and other regulation white chalk lines. At the start of each trial an arrow on the front screen guided the participant to the initial position on the goal line, halfway between the goalposts. After the participant had remained at the initial position for 1 s, the 0.22-m diameter plain white ball [START_REF] Casanova | Perception of spin and the interception of curved football trajectories[END_REF] started its flight trajectory. Participants were instructed to move along the goal line to try to intercept the ball with the torso (mid-line of the body), with the two arms jointly cupping the ball. While this did not prevent participants from using their hands every now and then when they realized that they were somewhat too late, a series of 20 randomly selected practice trials allowed them to correctly understand the whole-body movement task.

Data Analysis

The position of the participant's head (obtained by tracking the markers placed on the participant's stereo glasses and sampled at 100 Hz) was filtered using a dual-pass secondorder Butterworth filter with a cut-off frequency of 3 Hz. Movement initiation was determined by identifying the first 0.11-m (ball radius) change in position and then scanning backwards in time for the moment participant velocity exceeded a threshold of 0.05 m•s -1 . All data series were aligned with the moment of arrival of the ball in the goal. For each ball trajectory, ensemble averages of participant movement were obtained by averaging for each point in time all trials of the 21 participants (Fig. 2B). The target's bearing angle θ was defined as the azimuthal eccentricity [START_REF] Michaels | Lateral interception II: predicting hand movements[END_REF] of the ball with respect to the participant's straight-ahead (perpendicular to the goal line).

Results

Ball trajectory (Fig. 2A) systematically influenced players' interception movements (Fig. 2B

Straight ball trajectories

As can be seen from Fig. 2D, straight ball trajectories converging onto the same interception location (Fig. 2C) revealed influences of ball (departure) position on interception patterns, thereby confirming the presence of the angle-of-approach effect [START_REF] Ledouit | Prospective control in catching: the persistent angle-of-approach effect in lateral interception[END_REF] in locomotor interception. This observation was corroborated by an ANOVA on player position at 1.0 s before ball arrival that revealed not only the expected significant main effect of Ball Arrival Position (F(3, 60) = 256.1, p < .001, η 2 p = .93), but also a significant effect of Ball Departure Position (F(1, 20) = 132.9, p < .001, η 2 p = .87) as well as a significant interaction between the two (F(3, 60) = 3.2, p < .05, η 2 p = . Separating RM trials and NoRM trials5 for each ball trajectory revealed bimodal distributions of initiation times, indicating that RM trials were initiated considerably earlier than NoRM trials (see histogram insets in Figs. 3 and4). This pattern of results was not due to betweenparticipants differences: the presence of both RM and NoRM trials was observed within the individual participants. The number of trials with reversals varied over trajectories, for averages of 56.4% and 76.7% for balls arriving at the outer (±1.8 m) and inner (±0.6 m) arrival positions, respectively (see pie-chart insets in Figs. 3 and4).

Second, in NoRM trials the movement appeared to be driven by optical acceleration d 2 θ/dt 2 rather than by optical velocity dθ/dt: at the time of movement initiation the latter specified movement in the opposite direction (see indicator bars in Figs. 3 and4). This reliance on optical acceleration d 2 θ/dt 2 was observed in 95.4% of the NoRM trials. RM trials, on the other hand, appeared to be initially driven by optical velocity dθ/dt (consistently specifying movement away from the future arrival position at the time of movement initiation for the full set of RM trajectories). However, the subsequent reversing of movement direction again appeared to be driven by optical acceleration d 2 θ/dt 2 as even at the time of peak excursion optical velocity dθ/dt was still specifying movement in the same direction, away from the future ball arrival position (see indicator bars in Figs. 3 and4). This characteristic was observed in 79.2% of the RM trials.

Discussion

Overall, the above analyses demonstrated that the control of lateral interception seemed to require concurrent reliance on several informational variables.

For straight ball trajectories the behavioral effects observed (Fig. 2C and2D) implicated both optical position θ and optical velocity dθ/dt. Interestingly, similar results were reported by Fajen andWarren (2004, 2007; also see Land & Collet, 1974). Fajen andWarren (2004, 2007) noted that in their experimental setting targets following straight trajectories that led them to cross the participants' initial movement direction gave rise to S-shaped bends in the interception paths. In order to capture such (target-position driven) effects within their steering model of locomotor interception, they incorporated a 0.5-s sigmoidal latency function on dθ/dt suggested to correspond to "a visual delay to detect that the target is moving and a locomotor delay to overcome the inertia of the body" (Fajen & Warren, 2007, p. 311). When balls moved along straight trajectories in the present study, participants initiated their movements after 0.97 s of ball flight on average, for a range of 0.82 to 1.21 s. Thus, without considerably lengthening of its rise time (well beyond 1 s), the latency function proposed by [START_REF] Fajen | Behavioral dynamics of intercepting a moving target[END_REF] cannot explain the angle-of-approach effects observed here. The question why a combination of optical position θ and optical velocity dθ/dt information would be used over an extended period of time, rather than the presumably available dθ/dt information per se, therefore remains unanswered.

For curving ball trajectories the behavioral effects observed (Figs. 3 and4) implicated both optical velocity dθ/dt and optical acceleration d 2 θ/dt 2 . While these observations could be captured by a model of control based on (perhaps time-evolving) combinations of dθ/dt and d 2 θ/dt 2 information, such a combinatorial approach appears to lack a principled foundation. It also begs the question why interception of curving trajectories would not simply rely on d 2 θ/dt 2 information: if indeed accessible, this would allow avoiding the advent of reversal movements altogether. Given the consensus in the literature that optical acceleration cannot be detected accurately [START_REF] Brouwer | Perception of acceleration with short presentation times: Can acceleration be used in interception?[END_REF][START_REF] Calderone | Visual acceleration detection: Effect of sign and motion orientation[END_REF][START_REF] Werkhoven | Visual processing of optic acceleration[END_REF][START_REF] Zaal | Base on balls for the Chapman strategy: reassessing Brouwer[END_REF], it is not clear how d 2 θ/dt 2 could be used at all, either alone or in combination with other informational variables.

We suggest an alternative account. Rather than limiting potential informational variables to integer-valued time derivatives of θ, we propose that information may cover the full continuum of states offered by fractional-order time derivatives [START_REF] Beek | Studying[END_REF][START_REF] Jacobs | The learning of visually guided action: an information-space analysis of pole balancing[END_REF][START_REF] Podlubny | Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications[END_REF][START_REF] Podlubny | Geometric and physical interpretation of fractional integration and fractional differentiation[END_REF]. In this perspective, interception is controlled by nulling an informational variable d α θ/dt α of a trajectory-specific fractional order α, where α ∈R + .

Model

We developed the following model as a proof of concept for movement control based on trajectory-specific fractional-order information. We started from the attractor dynamics:

d 2 X dt 2 = G ⋅t c α ⋅ d α θ Δt dt α (1)
in which d 2 X/dt 2 is player acceleration along the interception axis, G is a gain coefficient, and t c is a characteristic time constant raised to the power α; d α θ Δt /dt α is the α th -order time derivative of θ at a visuomotor delay ∆t earlier. The gain coefficient G has the same dimension as d 2 X/dt 2 . Because α determines the dimension of d α θ/dt α , including t c α on the right-hand side of the equation balances dimensions.

For each trajectory the fractional order α is determined at each point in time by having α start at 0 and increase linearly (slope s α ) over time until the magnitude of d α θ/dt α becomes smaller than a criterion value:

d α θ dt α < ϕ t c α (2)
in which φ is the optical size of the ball, increasing during approach. Note that prior to movement initiation the instantaneous value of d α θ/dt α is determined by the way the ball trajectory unfolds with respect to the stationary player. After movement initiation the characteristics of ball and player motion co-determine the instantaneous value of d α θ/dt α .

In parallel to the order progression that leads to the fractional order that the system settles on for the specific ball trajectory, after a dead period t m , the moment of movement initiation is determined when the magnitude of required player acceleration exceeds a threshold:

d 2 X dt 2 > k ini t -t m (3)
in which k ini is a gain coefficient, t is time and t m is the duration of an initial dead period during which movement cannot be initiated.

In order to implement the model, fractional derivatives of θ needed to be calculated.

For reasons of simplicity we used an algorithm based on Euler's generalization [START_REF] Dalir | Applications of fractional calculus[END_REF]:

d n x m dx n = Γ(m + 1) Γ(m -n + 1) x m-n (4)
in which Γ is the gamma function. Fractional derivatives of time series of θ can thus directly be computed using polynomials 6 . To this end, for each point in time, a fourth-order polynomial was fitted through the values of θ at the preceding time steps. The coefficients c m of the obtained polynomial were used to compute the fractional derivative according to:

d α dt α θ(t) = c m Γ(m + 1) Γ(m -α + 1) t m-α ⎛ ⎝ ⎜ ⎞ ⎠ ⎟ m=0 4 ∑ (5)
in which α is the fractional order of the time derivative of θ, t is time, and m is the order of each term in the polynomial. Until sufficient values were available after onset of ball motion, the polynomial was fit through the first 34 time steps (during which the player never moved) for the computation of the fractional derivative at each time step. ***** Figure 5 about here *****

The model (i.e., Equations 1 to 3) was fitted to the ensemble averages of the interception patterns for all 24 ball trajectories, including only the RM trials for trajectories with RM and NoRM behaviors. Two parameters were set at fixed values (Δt = 0.1 s and t m = 0.23 s) while the remaining four parameters were optimized (s α = 5.4 s -1 , G = 160 m•s -2 , t c = 0.23 s and k ini = 0.255 m•s -1 ) using sequential quadratic programming. Subsequent model simulations using this unique set of parameter values provided adequate overall results (Fig. 5). Simulations satisfactorily reproduced the interception patterns observed for all trajectories (R 2 = 0.976 ± 0.038 and RMSE = 0.054 ± 0.021 m). Additionally, timing was accurately captured. For each ball trajectory, predicted initiation time closely matched observed initiation time, R 2 = 0.950 and mean error = -0.014 ± 0.046 s. For the trajectories giving rise to movement reversals, the moment of peak excursion was accurately predicted, R 2 = 0.973 and mean error = 0.031 ± 0.029 s. When the model was initiated at the mean movement initiation times of the NoRM trials, the simulated trajectories showed no reversal movements. ***** Table 1 about here ***** In the model, order α generally increased monotonically before settling onto a trajectory-specific value that was then maintained throughout the interceptive action (see Table 1 for final values of α). However, due to the spatiotemporal characteristics of certain ball trajectories, the temporal evolution of α revealed a short-duration, transitory plateau for the four CS ball trajectories departing from -4 m and for their mirror counterparts, the four CCS trajectories departing from +4 m. This plateauing of α at values between 0.7 and 0.8 observed for these particular (curving) ball trajectories resulted from small absolute d α θ/dt α values occurring in the early stage of ball flight, when the ball was on one side of the player while its velocity was driving it to the other side. As over these trajectories the ball quickly approached and subsequently crossed the (still stationary) player's initial position, the magnitude of d α θ/dt α rapidly began to grow again. Driven by Equation ( 2), α therefore started to increase again and continued to do so until it settled onto a trajectory-specific value that was then maintained throughout the interceptive action.

The observed angle-of-approach effect for balls following straight trajectories to the same arrival position while coming from different departure positions (Fig. 5C andD) resulted from the presence of trajectories with α < 1. The apparent initial dependence on optical velocity dθ/dt and subsequent switch to optical acceleration d 2 θ/dt 2 underlying the observed RM behavior for curving trajectories in fact resulted from continuous reliance on information with 1 < α < 2.

General Discussion

The purpose of the model developed in the present contribution was to provide proof of concept for our hypothesis of control of interception based on fractional-order information nulling. Equation ( 1) captures the information-based attractor dynamics, defining how player acceleration (d 2 X/dt 2 ) depends on information (d α θ/dt α ), in its simplest form: beyond a general gain G it only includes a time constant t c raised to the power α that is required to maintain dimensional homogeneity. Equation ( 1) is the part of the model that can be compared to existing behavioral dynamics models of interception, in both their directionconstrained (Chardenon et al., 2005, Eq. 1, p. 53;Bastin et al., 2006, Eq. 1, p. 720;Bastin et al., 2008, p. 302;Morice et al., 2010, Eq. 1, p. 398) and speed-constrained versions (Fajen & Warren, 2007, Eq. 5, p. 310 combined with the latency function described in Eq. 6, p. 311).

While most models [START_REF] Bastin | Prospective strategies underlie the control of interceptive actions[END_REF]2008;[START_REF] Fajen | Behavioral dynamics of intercepting a moving target[END_REF][START_REF] Morice | Environmental constraints modify the way an interceptive action is controlled[END_REF] incorporate a damping term in the attractor dynamics, ours (currently) does not, for a simple reason: Fitting systematically yielded a damping coefficient of negligible size. What makes our model conceptually different from existing models is not only that we open up the potential information space from isolated (category-like) integer time derivatives 7 Finally, [START_REF] Fajen | Behavioral dynamics of intercepting a moving target[END_REF] model also incorporates a distance term (weighting the influence of a distant target more heavily) to avoid sluggish turns when the target is far away. Note that they demonstrate that inclusion of the distance term is not required for the range of target distances and speeds explored in their interception experiments (see [START_REF] Fajen | Visual guidance of intercepting a moving target on foot[END_REF].

to the continuum of fractional-order time derivatives [START_REF] Jacobs | The learning of visually guided action: an information-space analysis of pole balancing[END_REF]. Taking the idea of a continuum seriously 8 , we also allow different situations to give rise to reliance on information of a different fractional order. Thus, in our model we suggest that on each occasion the system tunes in to (i.e., self-selects) the appropriate order of the information to be nulled. This self-tuning to an appropriate fractional order of the information is captured in our model by Equation ( 2), stating that the order α of information increases when d α θ/dt α exceeds a threshold, defined by the ratio of current optical target size φ and t c α . Note that substitution of Equation (2) in Equation ( 1) reveals that the threshold corresponds to a (required) player acceleration of G times φ. Thus, when the acceleration required to intercept the target is too large, the order α of the information is increased. Because very short stimulus exposure times only allow detection of position, in building our model we assumed that α starts from zero. The choice to increase α following a linear function, characterized by slope s α , is at present also but a simplicity assumption. For the set of conditions explored in the present study, after movement onset the value of α nulled by model did not change over the course of movement. However, a change in the characteristics of ball motion (such as a sudden change in ball speed and/or direction) may be expected to lead to a change in α. The present formulation of the α-tuning mechanism (Equation ( 2)) allows for an increase but not for a decrease in α during the interceptive action. Future work will have to determine whether α may indeed change when ball trajectory characteristics change and, if so, whether α then may decrease as well as increase. Experimental evidence in favor of a decrease in α over the course of an interceptive action would, of course, require a reformulation of the α-tuning mechanism.

8 When limited to isolated integer-order derivatives, it may seem parsimonious to choose, for a given set of conditions, between categorically different interception strategies such as classical pursuit (nulling zeroth-order information) and classical interception (nulling first-order information). Allowing for fractional-order derivatives eliminates these categorical differences and, thereby, the need of having to choose between them.

With the information-based attractor dynamics being defined by Equations ( 1) and (2) at each moment in time for each ball trajectory, we were able to determine an effective criterion for movement initiation captured by Equation (3), the last element of our model: the player begins to move when the magnitude of required acceleration exceeds a threshold value.

Incorporating a dead period during which movement simply cannot be initiated, the threshold value decreases over time so that movement is eventually initiated even for a ball trajectory requiring only a small amount of movement for interception. Note that such an actionrelevant criterion [START_REF] Bootsma | On the information-based regulation of movement: What Wann (1996) may want to consider[END_REF] allows the whole action sequence (from not moving, via starting to move at a particular moment, to moving in a particular way) to be understood within the same logic. It is indeed this logic that allows our model to reproduce the observed behavioral patterns (Fig. 5). Surprisingly, most existing studies (and a forteriori models) of locomotor interception remain silent on the timing of movement initiation (see [START_REF] Collett | How hoverflies compute interception courses[END_REF], for an exception); at best they incorporate a sigmoidal time-dependent activation function [START_REF] Bastin | Prospective strategies underlie the control of interceptive actions[END_REF]2008;[START_REF] Morice | Environmental constraints modify the way an interceptive action is controlled[END_REF]. We suggest that this lack of interest with respect to the timing characteristics of locomotor interception is related to the specifics of the experimental paradigms used in which analysis begins when agents are already moving (e.g., [START_REF] Bastin | Prospective strategies underlie the control of interceptive actions[END_REF]2008;[START_REF] Boeddeker | Chasing a dummy target: smooth pursuit and velocity control in male blowflies[END_REF][START_REF] Chardenon | The visual control of ball interception during human locomotion[END_REF]2004;2005;[START_REF] Fajen | Visual guidance of intercepting a moving target on foot[END_REF][START_REF] Haselsteiner | Tiger beetles pursue prey using a proportional control law with a delay of one half-stride[END_REF][START_REF] Kane | Falcons pursue prey using visual motion cues: new perspectives from animal-borne cameras[END_REF][START_REF] Kane | When hawks attack: animal-borne video studies of goshawk pursuit and prey-evasion strategies[END_REF][START_REF] Lenoir | Intercepting moving objects during self-motion: effects of environmental changes[END_REF][START_REF] Podlubny | Geometric and physical interpretation of fractional integration and fractional differentiation[END_REF][START_REF] Morice | Environmental constraints modify the way an interceptive action is controlled[END_REF][START_REF] Olberg | Prey pursuit and interception in dragonflies[END_REF][START_REF] Tucker | Curved flight paths and sideways vision in peregrine falcons (Falco peregrinus)[END_REF]. Moreover, for walking participants the data are generally binned over considerable time intervals in order to remove the cyclical variations of the step cycle; Fajen andWarren (2004, 2007) in fact timenormalized their data. Thus, by including a criterion for movement initiation such as the one provided by Equation (3), we hope that our model may incite future work on locomotor interception to consider not only the patterning but also the timing of behavior.

We conclude that control based on fractional-order information nulling provides a unifying framework, explaining the ball-trajectory effects observed here and in the literature for straight [START_REF] Chardenon | The perceptual control of goal-directed locomotion: a common control architecture for interception and navigation[END_REF]2005;[START_REF] Fajen | Visual guidance of intercepting a moving target on foot[END_REF]) and curving trajectories [START_REF] Bastin | Prospective strategies underlie the control of interceptive actions[END_REF]2008;[START_REF] Casanova | Perception of spin and the interception of curved football trajectories[END_REF][START_REF] Lenoir | Effects of contrasting colour patterns of the ball in the volleyball reception[END_REF][START_REF] Morice | Environmental constraints modify the way an interceptive action is controlled[END_REF].

Breaking away from the traditional combinatorial approach (e.g., [START_REF] Beek | Modelling the control of interceptive actions[END_REF]; also see the sequence of model developments in [START_REF] Dessing | Prospective control of manual interceptive actions: comparative simulations of extant and new model constructs[END_REF]2004;2005;2009), fractional derivatives fill the space between classical integer-order derivatives, thereby providing a powerful tool to identifying intermediate informational states [START_REF] Jacobs | The learning of visually guided action: an information-space analysis of pole balancing[END_REF]. The neurophysiological plausibility of the fractional-order dynamics was first demonstrated by [START_REF] Anastasio | The fractional-order dynamics of brainstem vestibulo-oculomotor neurons[END_REF]1998;[START_REF] Anastasio | Velocity leakage" in the pigeon vestibuloocular reflex[END_REF] in his work on the vestibulo-ocular reflex. More recently, the firing rate of neocortical pyramidal neurons has been reported to encode slowly varying stimulus statistics through fractional-order differentiation [START_REF] Lundstrom | Fractional differentiation by neocortical pyramidal neurons[END_REF]. We therefore conclude that reliance on fractional-order information as proposed in the present contribution is, at least, neurophysiologically plausible.

Fractional-order information may also be used in the locomotor interception of balls following curving trajectories along the sagittal plane, for which a heuristic of keeping constant the rate of change of the tangent of optical elevation (often referred to as optical velocity) has been proposed [START_REF] Chapman | Catching a baseball[END_REF][START_REF] Fink | Catching fly balls in virtual reality: A critical test of the outfielder problem[END_REF][START_REF] Mcleod | The generalized optic acceleration cancellation theory of catching[END_REF][START_REF] Michaels | The optics and actions of catching fly balls: Zeroing out optical acceleration[END_REF]. However, the plausibility of a control strategy implementing this interception heuristic in the form of nulling vertical optical acceleration [START_REF] Mcleod | The generalized optic acceleration cancellation theory of catching[END_REF][START_REF] Michaels | The optics and actions of catching fly balls: Zeroing out optical acceleration[END_REF] has been questioned on the basis of the human visual system's low sensitivity to optical acceleration [START_REF] Brouwer | Perception of acceleration with short presentation times: Can acceleration be used in interception?[END_REF][START_REF] Calderone | Visual acceleration detection: Effect of sign and motion orientation[END_REF][START_REF] Werkhoven | Visual processing of optic acceleration[END_REF][START_REF] Zaal | Base on balls for the Chapman strategy: reassessing Brouwer[END_REF]. Human observers' ability to detect changes in optical velocity has led to the formulation of alternative perceptual candidates, generally based on changes in velocity scaled to average velocity [START_REF] Babler | Role of image acceleration in judging landing location of free-falling projectiles[END_REF][START_REF] Brouwer | Perception of acceleration with short presentation times: Can acceleration be used in interception?[END_REF][START_REF] Calderone | Visual acceleration detection: Effect of sign and motion orientation[END_REF][START_REF] Schmerler | The visual perception of accelerated motion[END_REF]. To date, however, the variable that the visual system relies on to detect changes in optical velocity has not been identified [START_REF] Zaal | Base on balls for the Chapman strategy: reassessing Brouwer[END_REF]. The current framework intimates that reliance on information of a fractional order of about 1.8, as identified for the laterally curving trajectories of the present study, might characterize the informational variable underlying detection of change in velocity.

The proposed account suggests that combinations of ball trajectory characteristics and timing constraints may lead to fractional orders of the informational variable filling a continuum of possibilities. Generalizing, this implies that conceptual models of motion perception [START_REF] Van Doorn | Temporal properties of the visual detectability of moving spatial white noise[END_REF] could benefit from considering sensitivity to fractional (rather than exclusively integer) order derivatives. The account may even provide a simple solution to the 40-year old enigma that constant velocity is not perceived as such [START_REF] Runeson | Constant velocity-not perceived as such[END_REF].

Figure Captions

  ), for both straight and curving ball trajectories. ***** Figure 2 about here *****

  14). Post-hoc Newman-Keuls analysis of this interaction demonstrated a significant influence of Ball Departure Position on player position at each of the four Ball Arrival Positions (p's < .001), even though the effect was somewhat smaller for balls arriving at the inner (±0.6 m) than at the outer (±1.8 m) arrival positions. The angle-of-approach effect observed on player position was accompanied by variations in the moment of movement initiation over straight ball trajectories arriving at the same lateral distance (Fig. 2C): an ANOVA on movement initiation time demonstrated a significant main effect of Ball Arrival Position (F(3, 60) = 24.8, p < .001, η 2 p = .55) as well as a significant interaction between the factors of Ball Arrival Position and Ball Departure Position (F(3, 60) = 9.4, p < .001; η 2 p = .32). Post-hoc Newman-Keuls analysis revealed that movement was initiated later (p's < .05) for balls arriving at the inner (±0.6 m) than at the outer (±1.8 m) arrival positions. While for each ball arrival position balls departing from the opposite side of the player's initial position tended to give rise to longer movement initiation times, this effect reached significance only for the ball arrival position +0.6 m (p < .001). Because at least up to movement initiation the time course of optical velocity dθ/dt was nearly identical for balls converging along straight trajectories onto the same arrival position, the observation of systematic trajectory-specific differences in movement characteristics at each ball arrival position demonstrated that participants did not fully rely on optical velocity dθ/dt; that is, optical position θ also appeared to be involved. Curving ball trajectories Inspection of Fig. 2B revealed that 12 of the 16 curving trajectories led to reversals of movement direction during the players' interceptive actions. For four of these 12 trajectories this change in direction occurred toward the end of the movement, following a slight overshoot. The eight remaining trajectories gave rise to players initially moving away from the future ball arrival position before reversing movement direction. Focusing on the characteristics of these latter reversal movements brought out the following points (Fig. 3 and 4). ***** Figures 3 and 4 about here ***** First, while present in the ensemble averages of the full data set presented in Fig. 2B, reversal movements (RM) were not always observed at the level of the individual trials.

  Future work will have to demonstrate whether the inconsequential role of damping observed in the present context is related to task constraints or to other factors. As mentioned earlier, in order to capture the observed target-position-dependent effects on interception patterns, Fajen and Warren (2007) included a latency function on dθ/dt in their steering model. Our model based on fractional-order information does not require such an extension 7 .

Fig. 1 :

 1 Fig. 1: Plan view of an agent moving through an environment containing a target

Fig. 2 :

 2 Fig. 2: Ball trajectories and corresponding interception movements. (A) Bird's-eye view

Fig. 3 :Fig. 4 :

 34 Fig. 3: Details of interception movements for the four curving trajectories with ball

Fig. 5 :

 5 Fig. 5: Ball trajectories and corresponding observed and simulated interception

Table 1 :

 1 Values of the fractional order α being nulled for the 24 ball trajectories resulting from the model simulations. Ball Departure Position. BAP: Ball Arrival Position. NS: No spin. CS: Clockwise spin. CCS: Counter-clockwise spin. Note the mirror symmetry in the α values corresponding to the mirror symmetry in the trajectories departing from -4 m and +4 m.

			BDP -4 m			BDP +4 m
			BAP (m)			BAP (m)
		-1.8	-0.6	+0.6	+1.8	-1.8	-0.6	+0.6	+1.8
	NS	1.03	0.97	0.97	0.92	0.92	0.97	0.97	1.03
	CS	1.78	1.78	1.73	1.67	1.84	1.78	1.78	1.73
	CCS	1.73	1.78	1.78	1.84	1.67	1.73	1.78	1.78
	BDP:

Rotating the reference frame of Fig.1Bby 90° around the agent's displacement axis allows examining the situation of target motion in the sagittal plane[START_REF] Chapman | Catching a baseball[END_REF][START_REF] Mcleod | Do fielders know where to go to catch the ball or only how to get there?[END_REF][START_REF] Mcleod | How fielders arrive in time to catch the ball[END_REF][START_REF] Michaels | The optics and actions of catching fly balls: Zeroing out optical acceleration[END_REF][START_REF] Todd | Visual information about moving objects[END_REF][START_REF] Zaal | The information for catching fly balls: Judging and intercepting virtual balls in a CAVE[END_REF]. Because gravity will eventually lead inanimate targets initially moving upward to fall back onto the agent's locomotor surface, the task still defines an appropriate exocentric reference direction (the gravity-defined vertical).

Identification of RM and NoRM trials was based on the criterion for movement initiation described in the methods section: trials were qualified as RM when an initial movement of at least 0.11 m was made in a direction away from the future ball arrival position.

In order to ascertain that our results did not depend on the specific algorithm used for calculating the fractionalorder derivatives of θ, we also ran the model with another, computationally more intensive matlab routine (fgl_deriv.m; MathWorks reference 45982-fractional-derivative) based on a vectorized Grunwald-Letnikov definition[START_REF] Podlubny | Fractional differential equations: an introduction to fractional derivatives, fractional differential equations, to methods of their solution and some of their applications[END_REF]. The algorithm used did not qualitatively affect the results obtained.

perceptual-motor actions from mutually constraining perspectives. In A. A. Post, J. R.

Pijpers, P Bosch, & M. S. J. Boschker (Eds.), Models in Human Movement Science (pp.

Acknowledgments

Exchanges between the authors were facilitated by a French-Dutch collaborative travel grant (PHC Van Gogh 29353PF) awarded to Reinoud Bootsma and Frank Zaal. We thank Joanne Smith and Laure Fernandez for their help in data collection and in initial analyses. Cedric Goulon provided essential support in implementing the virtual reality environment and experimental conditions. Viktor Jirsa's constructive comments are gratefully acknowledged.

Reinoud Bootsma is a senior member of the Institut Universitaire de France.


Denis Dennehy

Anastasia Griva

Nancy Pouloudi

Yogesh K Dwivedi

Ilias Pappas

Matti Mäntymäki

Conference Chairs

Preface

This book presents the proceedings of the 20th International Federation of Information Processing (IFIP) Conference on e-Business, e-Services, and e-Society (I3E), which was hosted in Galway, Ireland, 1-3 September 2021. The annual I3E conference is a core part of Working Group 6.11, which aims to organize and promote exchange of information and co-operation related to all aspects of e-business, e-services, and e-society (the three Es). The I3E conference series is truly interdisciplinary and welcomes contributions from both academics and practitioners alike.

The central theme of the 2021 conference was Responsible AI and Analytics for an Ethical and Inclusive Digitized Society although, in line with the inclusive nature of the I3E series, all papers related to e-Business, e-Services, and e-Society were welcome. Digital technologies (e.g., AI, Blockchain, Big Data Analytics), and ICT in general, create opportunities and unintended or negative consequences for individuals and society [START_REF] Gupta | Questioning racial and gender bias in AI recommendations: do individual-level cultural values matter?[END_REF][START_REF] Majchrzak | Designing for digital transformation: lessons for information systems research from the study of ICT and societal challenges[END_REF]. These opportunities and consequences have not been evenly distributed. Therefore, the aim of the conference was to bring together a community for the advancement of knowledge regarding the adoption, use, impact, and value of digital technologies across e-business, e-services, and e-society.

Despite the many personal, economic, and societal benefits offered by AI and analytics [START_REF] Dennehy | Ireland after the pandemic: utilising AI to kick-start a sustainable economic recovery[END_REF][START_REF] Pappas | Big data and business analytics ecosystems: paving the way towards digital transformation and sustainable societies[END_REF], their use raises a variety of ethical concerns that need to be addressed in order to create a "good AI society" (Fossa [START_REF] Wamba | Are we preparing for a good AI society? A bibliometric review and research agenda[END_REF]. Ethics permeates the entire analytics process, from what data to use, to how to represent the extracted knowledge and exploit the insights to create economic and social value. Ethical concerns (e.g., illegitimate surveillance, invasion of privacy, unemployment, malicious use, etc.) are frequently used to portray AI and other digital technologies as a danger to humanity. For example, digital exclusion is part of the overall challenge of exclusion, a growing phenomenon which carries with it a series of deteriorations in life paths (e.g., poor lifelong earnings and an increased risk of marginalization). There are many who are currently excluded for reasons of low income and education, location, culture, trust and confidence levels or various disabilities. These concerns warrant the attention of the academic community to ensure the information society is built on a foundation in which integrity and rigor for good science will promote quality systems, and good ethics will promote good professional practice (Calzarossa et al., 2009). Hence, in order to be able to practice in an ethical manner, professionals must see vistas beyond technology [START_REF] Stoodley | Expanding ethical vistas of IT professionals[END_REF].

At the same time, we acknowledge that AI and other digital technologies can offer transformational power across sectors, namely, public (Alshahrani et al., 2021), private [START_REF] Mikalef | Artificial intelligence capability: conceptualization, measurement calibration, and empirical study on its impact on organizational creativity and firm performance[END_REF], and not-for-profit [START_REF] Dennehy | Supply chain resilience in mindful humanitarian aid organizations: the role of big data analytics[END_REF], ranging from enhanced business operations and supply chains [START_REF] Cadden | Understanding the influential and mediating role of cultural enablers of AI integration to supply chain[END_REF] to reinventing business models (Duan et al., 2019) to decision-making [START_REF] Paschen | Collaborative intelligence: how human and artificial intelligence create value along the B2B sales funnel[END_REF] to changing the nature of work (Schwartz et al., 2019) to enhanced human capabilities (e.g., AI-enabled recruiting) [START_REF] Dwivedi | Artificial intelligence (AI): multidisciplinary perspectives on emerging challenges, opportunities, and agenda for research, practice and policy[END_REF].

The Call for Papers solicited submissions in two main categories: full research papers and short research-in-progress papers. Each submission was reviewed by at least two knowledgeable academics in the field, in a double-blind process. The 2021 conference received submissions from more than 33 countries across the world, including China, Mexico, India, Pakistan, Japan, Kenya, Ghana, Morocco, South Africa, Finland, Sweden, Norway, Greece, the Netherlands, the UK, Papua New Guinea, Canada, and the USA to name a few. The best papers were selected for inclusion in a special issue of Information Systems Frontiers or their authors were given the opportunity to enhance the manuscript for fast-track review and publication in International Journal of Information Management and Journal of Decision Systems. The final set of 57 full papers and 8 short papers submitted to I3E 2021 and appearing in these proceedings were clustered into twelve groups, each of which are outlined below.

Part I encapsulates a core theme of the conference, with nine manuscripts that address the adoption and diffusion of AI for digital transformation and public good.

Part II contains five manuscripts relating to AI and analytics for decision making.

Part III continues the core theme of the conference, drawing together seven manuscripts related AI philosophy, ethics, and governance.

Part IV complements the previous clusters, with five manuscripts related to privacy and transparency in a digitized society.

Part V captures five manuscripts focused on digitally enabled sustainable organizations and societies.

Part VI dovetails with the theme of the previous cluster, with five manuscripts that address digital technologies and organizational capabilities.

Part VII consists of four manuscripts that investigate the role of AI and analytics in digitizing supply chains.

Part VIII contains five manuscripts that address customer behavior and e-business.

Part IX is made up of four manuscripts that examine the opportunities afforded by blockchain technology.

Part X consists of three manuscripts that explore the growing use of AI and analytics in the context of information systems development.

Part XI draws together eight manuscripts that explore social media analytics in a variety of contexts.

Part XII is the final cluster of these proceedings, with five manuscripts focused on AI and analytics in the context of teaching and learning.

In addition to the above papers, we were delighted to welcome Professor Katina Michael, Professor H. Raghav Rao, and Professor Dinesh Kumar as our keynote speakers.

Katina His interests are in the areas of management information systems, decision support systems, e-business, emergency response management systems, and information assurance. He has chaired sessions at international conferences and presented numerous papers. He also has co-edited four books, including Information Assurance Security and Privacy Services and Information Assurance in Financial Services. He has authored or co-authored more than 200 technical papers, of which more than 125 are published in archival journals. H. R. Rao was the inaugural recipient of The Bright Internet Award for his contributions to the information systems discipline by KMIS, the Korea Society of Management Information Systems. In 2018, H. R. Rao was awarded the International Federation for Information Processing (IFIP) Outstanding Service Award for significant service contributions to the field of information systems and information systems security. In November 2016, H. R. Rao received the prestigious Information Systems Society Distinguished Fellow Award (Class of 2016) for outstanding intellectual contributions to the information systems discipline. Rao's work has received best paper and best paper runner up awards at ISR, AMCIS, and ICIS. He has received funding for his research from the National Science Foundation, the Department of Defense, and the Canadian Embassy. He also received the Fulbright fellowship in 2004. Rao is a past chair of IFIP WG 8.11/11.13, the working group for Information Systems Security Research. He is co-editor-in-chief of Information Systems Frontiers, advisory editor of Decision Support Systems, associate editor of ACM TMIS, and senior editor at MIS Quarterly.

U Dinesh Kumar is a professor in decision sciences area and also the chairperson of DCAL at the Indian Institute of Management, Bangalore (IIMB). Dinesh Kumar holds a Ph.D. in Mathematics from IIT Bombay and has over two decades of teaching and consulting experience. He has been recognized as one of the top 10 most prominent analytics academicians in India for his extensive research in big data analytics. He has spearheaded the analytics education industry in India. IIMB was one of the first education institutes in the country to offer a regular long-duration certification program on Business Analytics & Intelligence (BAI) in the year 2010. U. Dinesh Kumar is also the Programme Director of the Big Data Analytics certification program. He has published several research articles in reputed academic journals such as the European Journal of Operational Research, Annals of Operations Research, the International Journal of Production Economics, The Journal of Operational Research Society, Computers and Operations Research, IEEE Transactions on Reliability, and the International Journal of Reliability, Quality and Safety Engineering. He has also published more than 30 case studies on business analytics and machine learning algorithms based on Indian and multinational organizations at Harvard Business Publishing. He has authored 3 books and his most recent book titled "Business Analytics -The Science of Data Driven Decision Making" has been recommended by the All India Council for Technical Education (AICTE). He is the Founder-President of the Analytics Society of India (ASI). U. Dinesh Kumar regularly conducts corporate training programs in analytics and has trained many professionals in the field of analytics in the last 11 years. He has provided analytics consulting services to organizations such as Boston Consulting Group, GE Healthcare, General Motors, Hindustan Aeronautics Limited, Indian Army, TVS Motors, Wipro, and so on. He has conducted in-house training programs on analytics for several organizations including Accenture, Aditya Birla Group, Allianz Benelux, Ashok Leyland, Bank of America, CISCO, Fidelity, Honeywell, and ITC Infotech.

The conference schedule included the IFIP 6.11 Committee Meeting (day 1), live traditional Irish Sean-nós (old style) dancing (day 1), virtual coffee house meet ups (days 1-3), Dr. John Oredo chaired a panel discussion with renowned academics and practitioners on the topic of ethical AI (day 2), Best Paper award (day 2), and the conference concluded with a closing ceremony that included a presentation about I3E 2022 (day 3). Supplementary to the conference was a one-day doctoral symposium that involved presentations from 10 Ph.D. candidates and discussions with the symposium committee.

The success of the 20th IFIP I3E conference was a result of the enormous effort of numerous people and organizations. Firstly, this conference was only made possible by the continued support of WG 6.11 for this conference series and for selecting Galway to host I3E 2021, and for this we are extremely grateful. We are privileged to have received so many good quality submissions from authors across the globe and the biggest thank you must go to them for choosing I3E 2021 as the outlet for their current research. We are indebted to the Program Committee who generously gave up their time to provide constructive reviews and facilitate enhancement of the manuscripts submitted. Finally, we extend our sincere gratitude to everyone involved in organizing the conference, to our esteemed keynote speakers, and to Springer LNCS as the publisher of these proceedings, which we hope will be of use for continued development of research related to the three Es and social media in particular.

  Michael has recently moved to Arizona State University, holding a joint appointment in the School for the Future of Innovation in Society and the School of Computing, Informatics, and Decision Systems Engineering. She is also the director of the Centre for Engineering, Policy, and Society. Katina Michael is also affiliated with the School of Computing and Information Technology at the University of Wollongong. Until recently she was the Associate Dean -International, in the Faculty of Engineering and Information Sciences. Katina was formerly the long-standing IEEE vi Preface Technology and Society Magazine editor-in-chief (2012-2017), and is presently an IEEE Consumer Electronics Magazine senior editor. Since 2008 she has been a board member of the Australian Privacy Foundation, and was formerly the Vice-Chair. Her research focuses on the socio-ethical implications of emerging technologies. She has written and edited six books, guest edited numerous special issue journals on themes related to radio-frequency identification (RFID) tags, supply chain management, location-based services, innovation, and surveillance/uberveillance. In 2017, Katina was awarded the prestigious Brian M. O'Connell Award for Distinguished Service to the IEEE Society on the Social Implications of Technology (IEEESSIT). H. Raghav Rao was named the AT&T Distinguished Chair in Infrastructure Assurance and Security at The University of Texas at San Antonio College of Business in January 2016. He also holds a courtesy appointment as full professor in the UTSA Department of Computer Science. Prior to working at UTSA, H. R. Rao was the SUNY Distinguished Service Professor at the University at Buffalo. He graduated from Krannert Graduate School of Management at Purdue University.

Preface vii

Organization xiii

Pat Doody Munster Technological University, Ireland Patrick Mikalef Norwegian University of Science and Technology, Norway Peter Saba EMLV Business School Paris, France Pieter Toussaint Norwegian University of Science and Technology, Norway Polyxeni Vassilakopoulou University of Agder, Norway Rakhi Tripathi FORE School of Management, India Randsome Bawack Toulouse Business School, France Rennie Naidoo University of Pretoria, South Africa Rob Gleasure Copenhagen Business School, Denmark Rogier Van de Wetering Open University, The Netherlands Rónán Kennedy National University of Ireland Galway, Ireland Saad Alshahrani University of Tabuk, South Arabia Sachin Modgil International Management Institute, Kolkata Samrat Gupta Indian Institute of Management Ahmedabad, India Samuli Laato University of Turku, Finland Sarah Sabbaghan London South Bank University, UK Savvas Papagiannidis Newcastle University, UK Shang Gao Örebro University, Sweden Sobah Abbas Petersen Norwegian University of Science and Technology, Norway Sofia Papavlasopoulou Norwegian University of Science and Technology, Norway Stavros Lounis Athens University of Economics and Business, Greece Stephen McCarthy Cork University Business School, Ireland Stratos Baloutsos Athens University of Economics and Business, Greece Sunet Eybers University of South Africa, South Africa Sven Laumer Otto-Friedrich-University Bamberg, Germany Tania Prinsloo University of Pretoria, South Africa Tendani Mawela University of Pretoria, South Africa Trevor Cadden University of Ulster, UK Umair Ul Hassan Maynooth University, Ireland Vasiliki Chronaki Athens University of Economics and Business, Greece Vasiliki Koniakou University of Turku, Finland Vasilis Stavrou Athens University of Economics and Business, Greece Vigneswara Ilavarasan Indian Institute of Technology Delhi, India Vincent Dutot Ipag Business School, France Yashoda Karki University of South-Eastern Norway Ying Tueanrat Newcastle University, UK