*Article* **Behavioral Dynamics of Pedestrians Crossing between Two Moving Vehicles**

#### **Soon Ho Kim 1, Jong Won Kim 2, Hyun-Chae Chung 3, Gyoo-Jae Choi <sup>4</sup> and MooYoung Choi 1,\***


Received: 25 December 2019; Accepted: 20 January 2020; Published: 26 January 2020

**Abstract:** This study examines the human behavioral dynamics of pedestrians crossing a street with vehicular traffic. To this end, an experiment was constructed in which human participants cross a road between two moving vehicles in a virtual reality setting. A mathematical model is developed in which the position is given by a simple function. The model is used to extract information on each crossing by performing root-mean-square deviation (RMSD) minimization of the function from the data. By isolating the parameter adjusted to gap features, we find that the subjects primarily changed the timing of the acceleration to adjust to changing gap conditions, rather than walking speed or duration of acceleration. Moreover, this parameter was also adjusted to the vehicle speed and vehicle type, even when the gap size and timing were not changed. The model is found to provide a description of gap affordance via a simple inequality of the fitting parameters. In addition, the model turns out to predict a constant bearing angle with the crossing point, which is also observed in the data. We thus conclude that our model provides a mathematical tool useful for modeling crossing behaviors and probing existing models. It may also provide insight into the source of traffic accidents.

**Keywords:** pedestrian behavior; human locomotion; human dynamics; traffic safety

#### **1. Introduction**

Pedestrians make up a large portion of traffic accident fatalities, particularly in areas of high population density [1,2]. Exploring the behavioral dynamics of road crossing may provide insight into the fundamental source of accidents [2–4]. The task of crossing a road involves a goal-directed movement, as the pedestrian desires to reach the other side of the street subject to the avoidance condition due to passing vehicles. This relates the problem to studies on human behavior during tasks of avoidance [3,5] as well as interception [6–8].

The constant bearing-angle model has gained attention as a possible strategy that humans employ in order to intercept moving objects [6–8]. This walking strategy is tied to how human locomotion is visually controlled [9–12]. In addition, studies involving movements through gaps have employed the concept of affordance, the possibilities for action constrained by the environment and physical conditions of the actor [13–15]. Statistical analyses of pedestrian inter-vehicle gap acceptance rates, which depend on the pedestrian's perception of affordance, have also been reported [16]; these studies, however, do not typically provide a dynamic model of action.

The purpose of this study is to develop a model of road crossing that can be used to analyze data and test hypotheses. Specifically, we examine how pedestrians cross a street between two moving vehicles. An experiment is constructed in which human participants cross a street within a virtual environment with a range of experimental conditions. A mathematical model is developed which accurately describes typical crossing patterns. The model is applied to experimental data via minimization of the root-mean-square deviation (RMSD) with respect to the model parameters. When the best fit parameters are found, the model equations give us the position of the pedestrian as a function of time.

Discussing the meaning of each parameter of the model, we examine how the average value of each parameter varies depending on crossing conditions. Specifically, we consider the effects of adjusting the gap size and the initial distance as well as the pedestrian's age, vehicle speed, and vehicle type. We further make use of our model to derive an inequality among the parameters that must be satisfied for successful crossing to occur, and, accordingly, to describe the affordance [13,15] of the crossing situation. The possibility of intercepting the gap between the cars has been mathematically modeled with the environmental factors (e.g., speed of cars, length of lanes) and the walker's capabilities such as walking speed and response time. It is also observed in the data that the bearing angle tends to be constant with respect to the crossing point, which is obtained analytically from the model equations.

In Section 2, we describe the experimental procedure and lay out our model. The concept of affordance and bearing angle are also formulated using the model. In Section 3, we draw conclusions from the fitting parameters of the model. The results are visualized and analyzed in the affordance and bearing-angle viewpoints. In Section 4, we discuss the implications of our results and we conclude in Section 5.

#### **2. Methods**

#### *2.1. Data Collection*

Sixteen children (of age 12.2 ± 0.8 yrs, i.e., mean age 12.2 years and standard deviation 0.8 years), sixteen young adults (of age 22.8 ± 2.6 yrs), and fourteen elderly people (of age 54.1 ± 4.9 yrs) with normal or corrected-to-normal vision were recruited for this experiment. Informed written consent was obtained from all individual participants. The experiment was conducted in accordance with the Declaration of Helsinki, and the protocol was approved by the Kunsan National University Research Board. Each subject was placed on a customized treadmill (of dimensions 0.67 m wide, 1.26 m long, and 1.10 m high) with four magnetic counters that track movements. A Velcro belt connected to the treadmill is worn to decrease vertical and lateral movements, and a handrail is placed for stability. The treadmill turns with minimal friction as the participants walk, and magnetic counters track rotations (left panel, Figure 1). Each participant wore an Oculus Rift (Menlo Park, CA, USA) DK1 virtual reality headset connected to a standard desktop PC. The headset portrayed a realistic view of a street with a crosswalk in 1280 × 800 resolution 3D stereoscopic visual images that respond realistically according to the participant's movements. Practice trials were performed prior to the experiments to familiarize each participant with the treadmill and virtual environment before the experiment. (see [4] for details).

A diagram of the virtual crossing situation is given in Figure 2. The participant, standing at rest before a car lane with a "ready" signal shown, was instructed to press a button when a "go" signal appears, and then to look at her/his left side, from which two vehicles, one in front of the other, were moving at equal constant speed *vc*. The participant was instructed to cross between the two vehicles if she/he believed she/he could cross successfully. If a collision with a vehicle occurred, the simulations halted. The experimenter recorded whether there was a successful crossing, a collision, or no crossing.

**Figure 1.** Images of experimental setup. Left panel: photograph of student participating in an experiment. Middle panel: cartoon crosswalk used for calibration. Right panel: Screenshot of the virtual crosswalk used in the experiment.

The experiment was designed to examine the participants' responses to a single gap, and so only two vehicles were present. We examine the responses of the participants to changing gap characteristics by varying experimental parameters. These include pedestrian starting position *y*0, gap time *tg* (or gap length *lg*), vehicle speed *vc*, and the vehicle type (sedan or bus). Details of age groups and a full list of experimental parameters are given in Table 1.

The gap center is defined to be the midpoint of the gap between the vehicles. While the width of the gap and speed of the vehicles are varied, the initial position *x*<sup>0</sup> of the gap center is always set in such a way that the gap center reaches the interception point at time *t* = 4 s. Accordingly, we have *x*<sup>0</sup> = −*vc* × 4 s.


**Table 1.** Table summarizing age groups and experimental parameters.

#### *2.2. Model*

The following model, called the simple crossing model, was used to analyze crossing data. The velocity is assumed to follow a logistic function of time in the form

$$v(t) = v\_{\text{max}} \frac{\exp[(t - t\_a)/\tau]}{1 + \exp[(t - t\_a)/\tau]},\tag{1}$$

which, upon integration, results in the position as a function of time:

$$y(t) = y\_0 + \upsilon\_{\text{max}} \tau \log \left\{ 1 + \exp[\left(t - t\_d\right)/\tau] \right\}.\tag{2}$$

Equation (2) is plotted in Figure 3a (red line). Constants *ta*, *τ*, *v*max are fitting parameters whose meanings can be understood as follows: The measurement begins at time *t* = 0. Assuming *ta* − 2*τ* > 0, we have the initial position and velocity of the pedestrian, *y*(*t*=0) ≈ *y*<sup>0</sup> (< 0) and *v*(*t*=0) ≈ 0, respectively. Then, the pedestrian accelerates smoothly until the maximum velocity *v*max is reached. The parameter *τ* then serves as a measure for the duration of this acceleration, the midpoint between which is given by *ta*. Note that, at time *t* = *ta* − 2*τ*, the velocity in Equation (1) becomes *v*(*t* = *ta*−2*τ*) = *<sup>v</sup>*max *<sup>e</sup>*<sup>−</sup>2/(<sup>1</sup> + *<sup>e</sup>*−2) ≈ 0.1 *<sup>v</sup>*max. While Equation (1) never gives *<sup>v</sup>* = 0 exactly, in practice, we may define *td* ≡ *ta* − 2*τ* to be the time at which the pedestrian begins to accelerate forward. If preferable, one may take alternatively *td* ≡ *ta* − 3*τ*, which corresponds to *v*(*t* = *td*) ≈ 0.01 *v*max.

A second model, called the two-step crossing model, is used to analyze crossings that have more than one acceleration event and thus do not fit the simple crossing model (Figure 3b). The two-step crossing model is discussed in the Appendix A.

Each piece of data classified as a simple crossing is fit to Equation (2) by minimizing RMSD with respect to the fitting parameters. We probe the effects of gap characteristics by examining how the distributions of parameters change with the variation of certain features of the gap, and discuss the results in Section 3.2. Those data displaying the two-step pattern are fitted separately to the extended model, and the results are discussed in the Appendix A.

#### *2.3. Affordance*

Affordance stands for the range of possible actions that the environment offers to the acting agent. In the crossing task, the affordance is determined by how long the gap overlaps with the participant's walking trajectory. Assuming the simple crossing model (i.e., Equations (1) and 2)), the affordance of the gap is described by the inequality

$$t\_f - \tau \log[e^{(-y\_0 - w/2)/v\_{\text{max}}\tau} - 1] < t\_d$$

$$t < t\_b - \tau \log[e^{(-y\_0 + w/2)/v\_{\text{max}}\tau} - 1]. \tag{3}$$

Here, *tf* ≡ |*x*<sup>0</sup> + *lg*/2|*v*−<sup>1</sup> *<sup>c</sup>* corresponds to the time at which the back bumper of the leading vehicle passes the intersection point and *tb* ≡ |*x*<sup>0</sup> − *lg*/2|*v*−<sup>1</sup> *<sup>c</sup>* corresponds to the time at which the front bumper of the trailing vehicle passes the point, while *w* denotes the width of the vehicles and equals 1.5 m in our experiment. *tf* is hence manifested in Figure 3a by the time coordinate of the right side of the box to the left (2.5 s), while *tb* is by that of the left side of the box to the right (5.5 s). Equation (3) thus describes the condition under which the pedestrian's trajectory passes between the two boxes in Figure 3a.

**Figure 3.** Pedestrian position-time plots illustrating typical crossing patterns. Black traces indicate example data; red traces indicate the corresponding model fits. Left and right boxes indicate the temporal and spatial area that the leading and the trailing vehicles occupy, respectively. Intersecting one of the box lines would indicate a collision. In both examples, the conditions are described by *y*<sup>0</sup> = −3.5 m, *tg* = 3.0 s, and *vc* = 30 km/h, with the vehicle type set to be a sedan. (**a**) an example of the simple cross with a single acceleration event followed by constant speed walking; (**b**) an example of the two-step cross with two acceleration events.

In general typical values of *τ* are small than the time scale of crossing, e.g., compared with (−*y*<sup>0</sup> ± *w*/2)/*v*max. (Note that *y*<sup>0</sup> < 0 in our coordinate system.) Accordingly, we may take the limit *τ* → 0, and reduce Equation (3) to

$$t\_f - \frac{1}{v\_{\text{max}}} \left( -y\_0 - \frac{w}{2} \right) < t\_a < t\_b - \frac{1}{v\_{\text{max}}} \left( -y\_0 + \frac{w}{2} \right). \tag{4}$$

This provides a simpler inequality involving two fitting parameters.

#### *2.4. Bearing Angle*

Dynamics of interceptive movement are often described in terms of the bearing angle [7,8], which refers to the angle between the velocity vector of the human subject and the line of sight between the subject and the object she/he hopes to intercept. In brief, this model asserts that people intercept a moving object by choosing such a trajectory that the bearing angle is kept constant.

Our case of crossing a road may be cast into an interception task: The pedestrian must "intercept" the empty gap between the vehicles [17]. We may hence apply the bearing angle approach to our crossing experiment and model. One difficulty with this approach is that the gap is not a point but a moving area. As an obvious choice, we may simply use the gap center *xg*(*t*), with respect to which the bearing angle is *θg*(*t*) = arctan *xg*(*t*)/*y*(*t*) . However, the pedestrian may not cross the gap center; it is thus more appropriate to examine the bearing angle with respect to the point within the moving gap that the pedestrian actually crosses. With *t* ∗ denoting the crossing time, we have *y*(*t* ∗) = 0 and let the position of the gap center at the crossing time be Δ *x*, i.e., *xg*(*t* ∗) = Δ *x*. We then define the crossing point,

$$\mathbf{x}\_{\mathcal{C}}(t) = \mathbf{x}\_{\mathcal{S}}(t) - \boldsymbol{\Delta} \,\mathbf{x} = \mathbf{x}\_0 - \boldsymbol{\Delta} \,\mathbf{x} + \boldsymbol{\upsilon}\_{\mathcal{C}} t = \boldsymbol{\upsilon}\_{\mathcal{C}}(t - t^\*), \tag{5}$$

and consider the angle with respect to *xc*:

$$\theta\_c(t) = \arctan\left[\frac{\mathbf{x}\_c(t)}{\mathbf{y}(t)}\right]. \tag{6}$$

Taking the time derivative of Equation (6) results in

$$\dot{\theta}\_{\varepsilon} = \frac{\mathbf{x}\_{\varepsilon}\mathbf{y}}{\mathbf{x}\_{\varepsilon}^{2} + \mathbf{y}^{2}} \left(\frac{\mathbf{x}\_{\varepsilon}^{\prime}}{\mathbf{x}\_{\varepsilon}} - \frac{\dot{\mathbf{y}}}{\mathbf{y}}\right). \tag{7}$$

Assuming that *y* follows the simple crossing model (i.e., Equation (2)), |*y*˙| is small when *t* < *ta*. Considering the signs of variables (especially, *xc* < 0 and *x*˙*<sup>c</sup>* > 0), we thus have that ˙*θ<sup>c</sup>* < 0, indicating a decreasing bearing angle. When *t* > *ta* + 2*τ*, the speed approaches the maximum: *y*˙ ≈ *v*max, so that we have *<sup>x</sup>*˙*<sup>c</sup>*

$$\frac{\dot{x}\_c}{\infty} - \frac{\dot{y}}{y} \approx \frac{v\_c}{v\_c(t - t^\*)} - \frac{v\_{\text{max}}}{v\_{\text{max}}(t - t^\*)} = 0,\tag{8}$$

which, upon substituting into Equation (7), yields ˙*θ<sup>c</sup>* = 0 or a constant bearing angle. The model thus predicts that the bearing angle should decrease at the first stage of crossing and remain constant thereafter. The constant value *θ*∗ *<sup>c</sup>* that the bearing angle approaches can be estimated by

$$\begin{split} \lim\_{\Delta t \to 0} \theta\_{\varepsilon}(t^\* - \Delta t) &= \lim\_{\Delta t \to 0} \arctan \left( \frac{v\_{\varepsilon} \Delta t}{v\_{\max} \Delta} \right) \\ &= \arctan \left( \frac{v\_{\varepsilon}}{v\_{\max}} \right) . \end{split} \tag{9}$$

#### **3. Results**

#### *3.1. Data Analysis*

Tables 2 and 3 show the percentage of successful crossings in this group and the proportions of two-step crossings to the total successful crossings. The success rate drops significantly when the gap length is made small at 20.8 m but still stays above 80%. The highest proportion of two-step crossings occurs when the gap is the shortest and the walking distance is the furthest.


**Table 2.** Proportion of successful crossings to all crossing attempts for several values of *y*<sup>0</sup> and *lg*, when *vc* = 30 km/h and vehicle type is sedan.

**Table 3.** Proportion of two-step crossings to all successful crossings for several values of *y*<sup>0</sup> and *lg*, when *vc* = 30 km/h and vehicle type is sedan.


Equation (2) was fit to simple crossings with an average RMSD of 0.068 m. The low RMSD values indicate that the model accurately describes the majority of crossings. Two-step crossings were also found to be accurate, and are discussed in the Appendix A. Examples of the model equations fit to simple and two-step crossing time series are given in Figure 3a,b, respectively.

#### *3.2. Behavioral Response to Gap Features*

Restricting the analysis to simple crossings, we consider the variations of the parameters to changing crossing conditions. Experimental parameters *y*<sup>0</sup> and *lg* affect directly the affordance of the gap by changing the temporal window of the gap or the distance the pedestrian needs to traverse to reach the gap. Effects of the experimental parameters on the three fitting parameters *v*max, *ta*, and *τ* have been examined; only *ta* has turned out to respond significantly. Figure 4 shows the distribution of *ta* obtained for several values of *y*<sup>0</sup> and *lg*. It is observed that *ta* generally increases as *y*<sup>0</sup> approaches zero. This can be understood intuitively as follows: Recall that *y*<sup>0</sup> denotes the distance the pedestrian must traverse to reach the gap. The larger the distance, the earlier they must begin walking. However, when the initial position is farther, namely, when *y*<sup>0</sup> is made larger, this trend disappears and *ta* tends to stay at slightly over one second (*ta* - 1 *s*). This is likely to result from the minimum response time. Namely, the pedestrian may not cross earlier than the earliest timing at which they can reasonably begin to walk. On the other hand, an increase in the gap size appears to lower *ta*. This indicates that, when the gap is accessible earlier, the pedestrian tends to cross earlier. The distributions of the other two parameters *v*max and *τ* have also been examined. While the average value of *v*max tends generally to increase with *y*0, the trend is not statistically significant. No significant trends have been observed for *τ*.

Contrary to *y*<sup>0</sup> and *lg*, the vehicle speed *vc* and the vehicle type are manipulated without changing the gap affordance. These experimental parameters affect the visual perception of the gap without changing its temporal window of availability. Figure 5 displays the effects of the vehicle speed and type on *ta* when the gap time *tg* is set to be 3 s and *y*<sup>0</sup> to be −3.5 m. Doubling the vehicle speed results in a significant increase in *ta*. Moreover, in several cases, buses resulted in a greater value of *ta* than sedans did.

**Figure 4.** Distributions of parameter *ta* for varying *y*<sup>0</sup> and *lg*. Here, *vc* = 30 km/h and vehicle type is sedan. Columns indicate the average values of *ta* in the data for given experimental conditions while error bars represent standard deviations. Pairs of samples, marked with asterisks, are presumed to belong to different distributions (*p* < 0.05) according to the Mann-Whitney *U* test. (Note here that not all such pairs are marked.)

**Figure 5.** Distributions of parameters *ta* (left) and *td* (right) for varying vehicle speeds *vc* and for two vehicle types. Other parameters are set to *tg* = 3 s and *y*<sup>0</sup> = −3.5 m. Columns indicate the average values in the data for given experimental conditions while error bars represent standard deviations. Pairs of samples, marked with asterisks, are presumed to belong to different distributions (*p* < 0.05) according to the Mann–Whitney *U* test.

On the other hand, when the same comparison is made for data with *y*<sup>0</sup> < −3.5 m, there arises no significant shift in *ta* or *td* upon changing the vehicle type. For *y*<sup>0</sup> < −4.5 m, no significant shift is observed upon changing the vehicle speed as well. This suggests that, when the initial distance is sufficiently far, pedestrian's judgement of the gap is hardly affected by the vehicle type or speed.

Finally, we examine differences among age groups. According to the Mann–Whitney *U* test, the difference in the distribution of *v*max is found to be significant (*p* < 0.05) when either the young adult group or the elderly group is compared with the child group. Both the young adult and elderly groups consistently have higher average values of *v*max across all crossing conditions, by about 0.3 m/s. While children have generally slightly lower values of *ta*, perhaps a sign of earlier start up times to compensate for their lower speeds, the differences are not found to be statistically significant. The young adult and elderly groups do not show significant differences.

#### *3.3. Parameters Fall within Affordance*

Due to the accuracy of the simple crossing model, we expect Equations (3) and (4) to hold for the fitting parameters derived from the data. Figure 6a presents the affordance boundaries in the 3D parameter space. The curved surfaces depict the boundaries specified by Equation (3), and each data point plots the parameters corresponding to a single crossing. For comparison, data for two-step crossings (crosses) as well as simple crossings (circles) are displayed. It is observed that the circles, corresponding to simple crossings indeed lie within the volume between the boundaries. In contrast, most of the crosses are located outside, above the higher surface. This indicates that the subject is on route to collide with the leading vehicle and therefore deceleration is necessary.

(a)

(b)

*t*

**Figure 6.** Data plotted with boundaries representing the affordance of the gap. (**a**) data plotted with surfaces in the three-dimensional parameter space (*τ*, *v*max, *ta*) defined by Equation (3). Data points for *lg* = 25 m and *y*<sup>0</sup> = −3.5 m are plotted for all age groups; dots represent simple crossings and crosses represent two-step crossings; (**b**) data plotted with boundaries on the two-dimensional plane (*v*max, *ta*) defined by Equation (4). To illustrate the effects of a shift in affordance, data and boundaries for *lg* = 25 m and *y*<sup>0</sup> = −3.5 m (black) are plotted with those for *lg* changed to 33.3 m and for *y*<sup>0</sup> changed to −6.5 m are shown in red and in blue, respectively. Triangles indicate average parameter values.

Plotting Equation (4) on the 2D parameter plane (*v*max, *ta*) corresponds to the projection of the 3D plot in Figure 6a onto the plane defined by *τ* = 0. This results in Figure 6b, where two more cases have been included in addition to the case of the gap length *lg* = 25 m and the initial position *y*<sup>0</sup> = −3.5 m presented in Figure 6a. Namely, to probe how the distribution of parameter values shifts with *lg* and *y*0, we consider the data for a larger gap *lg* = 33.3 m and for a farther initial position *y*<sup>0</sup> = −6.5 m. Accordingly, Figure 6b presents data for three sets of the gap length and initial position together with the corresponding boundaries (lines instead of surfaces in Figure 6a) determined by Equation (4). Specifically, the cases of (*lg* = 33.3 m, *y*<sup>0</sup> = −3.5 m) and (*lg* = 25 m, *y*<sup>0</sup> = −6.5 m) are plotted in red and in blue, respectively, as well as the case of (*lg* = 25 m, *y*<sup>0</sup> = −3.5 m) plotted in black. It is observed that, as the gap is widened, the average behavior (designated by red triangles) shifts toward smaller *ta* and larger *v*max. This reflects the tendency of the pedestrian to cross early before the gap center when possible. On the other hand, in the case that the initial position becomes farther from the intersection point, the pedestrian must compensate by either beginning to walk earlier or walking faster. Data in blue indeed exhibit on average a decrease in *ta* and a slight increase in *v*max (which is, however, not statistically significant). We remark that Figure 6b corresponds directly to Figures 3C and 3D of [13] while Figure 6a is a generalization.

#### *3.4. Bearing Angle Analysis*

Finally, we examine the bearing angle of the data and compare it with the model predictions. Figure 7 shows the bearing angle as a function of time for two sets of data (colored lines). The bearing angle tends to decrease in the first few seconds of crossing and to remain constant thereafter, as predicted by Equation (9). In addition, the analytical results given by Equation (6) are plotted with the average parameter values (black line). Both the theory (analytical result from the model) and the experiment (result computed from data) show that a constant bearing angle is held once the pedestrian starts moving at a nearly constant speed. It is shown in Figure 7 that the time interval during which the constant angle is observed is significantly shorter for *y*<sup>0</sup> = −3.5 m (red) than for *y*<sup>0</sup> = −6.5 m (cyan). This reflects the smaller value of *ta* in the latter case, when the initial position is farther from the interception point.

**Figure 7.** Time evolution of the bearing angle *θc*, defined by Equation (6). The gap length is *lg* = 25 m and the initial position is *y*<sup>0</sup> = −3.5 m and −6.5 m for red and cyan lines, respectively. Colored lines are computed from data, while black lines show the analytical results using average parameter values. The time axis is given in terms of time before crossing *t* − *t* ∗, so that the zero point is equal to the intersection time of the run (i.e., the time at which *y*(*t*) = 0).

We note that the fluctuations of the data in Figure 7 are due to measurement error, which is magnified immediately before the pedestrian meets the crossing point. This can be seen in Equation (5) by considering that *y*(*t*) → 0 as *t* → *t* ∗, causing the error in the argument of the arctan function to become magnified.

#### **4. Discussion**

In this study, we have proposed a model for pedestrian crossing and utilized it to extract information from experimental data. The model fit the data with high accuracy, allowing for applications of different methods. In particular, the model allows us to visualize the affordance of each gap and see whether the data lies within it. The model also predicts a constant bearing angle, which has been observed in the data.

The fitting parameters of the model, *ta*, *v*max and *τ*, provide a physically intuitive interpretation of the data. Analysis has revealed that pedestrians respond to shifting gap affordances primarily by timing their accelerations, rather than changing their walking speeds, as shown by the distributions of *ta*. Moreover, shifts in *ta* have been observed in response to the speed of the gap and the size of the surrounding vehicles, even if the gap affordance remains the same, indicating that these environmental factors can change the visual perception of the gap. However, this trend disappeared when the initial distance was greater, suggesting that a greater distance from the road tends to offer a more accurate visual perception of the gap. It has also been observed that children's lower walking velocities indirectly shrank the affordances of the gaps and in certain situations failed to compensate.

Due to the simplicity of the equations, this methodology offers a versatile method to analyze pedestrian behavior. While the velocity equation does not necessarily need to follow a logistic function in particular, the high accuracy with which the equation fits the data and its ease of manipulation makes it an ideal tool for such an analysis.

It should be noted that the accuracy of affordance judgments in virtual environments has been questioned in previous studies [18]. In addition, it is not straightforward to measure walking speeds on treadmills, and a different method was proposed [19]. These factors should be considered when interpreting the results, but we expect them to affect neither qualitative results nor the efficacy of the model. Moreover, it would be desirable to include more general walking scenarios where the pedestrian is not constrained to walk in a straight line. This is left for future study.

#### **5. Conclusions**

We have utilized a newly developed model for crossing behavior to extract the control parameter used for adjusting to changing gaps, visualizing crossings within the gap affordance, and testing the bearing angle hypothesis. The model thus serves as a useful tool with which pedestrian behavior can be understood. It can also be used in the context of previous modeling frameworks and may further be developed to extract elements of crossing behavior which leads to collision.

**Author Contributions:** Conceptualization, M.C., J.W.K. and H.-C.C.; methodology, M.C. and J.W.K.; software, S.H.K.; visualization, S.H.K.; investigation, H.-C.C. and G.-J.C.; writing–original draft preparation, S.H.K.; writing–review and editing, J.W.K. and M.C.; All authors have read and agreed to the published version of the manuscript.

**Funding:** This research was funded by National Research Foundation of Korea through the Basic Science Research Program (Grant No. 2019R1F1A1046285) and by Korea Institute for Advancement of Technology and Ministry of Trade, Industry, and Energy (Grant No. 10044775).

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **Appendix A. Analysis of Two-Step Crossings**

In order to model two-step crossings in which there are two acceleration events (Figure 3b), we extend the model in the following way: We first take the acceleration equation

$$
\dot{y} = \frac{\dot{y}}{\pi} \left( 1 - \frac{\dot{y}}{v\_{\text{max}}} \right),
\tag{A1}
$$

which is equivalent to Equations 1 and 2 with appropriate initial conditions. In the two-step crossing, after acceleration (i.e., at time *t* > *ta* + 2*τ*), the pedestrian will decelerate at a point *ys* and stop until she/he accelerates again at time *ts*. This behavior may be described by adding two terms in Equation (A1), which leads to

$$\begin{split} \dot{y} &= \frac{\dot{y}}{\tau} \left( 1 - \frac{\dot{y}}{\upsilon\_{\text{max}}} \right) - r\_s \dot{y} \exp \left[ -\frac{(y - y\_s)^2}{\sigma\_s^2} \right] \theta(t\_s - t) \\ &+ \upsilon\_s \delta(t - t\_s), \end{split} \tag{A2}$$

where the second and the third terms of the right-hand side represent repulsion and impulse force, respectively. The repulsion is centered at position *ys* with range *σs*; *ys* may be interpreted as the point beyond which the pedestrian perceives to be unsafe, due to the incoming traffic. Accordingly, *ys* is the position of the flat region of the curve in Figure 3b, i.e., *ys* = −2.3 m in this example. The Heaviside step function *θ*(*ts* − *t*) effectively "turns off" the repulsion force at time *ts*, thus removing the potential for collision after the vehicle has passed. The parameter *rs* adjusts the overall strength of the repulsion. The impulse term is necessary for the model to undergo sharp acceleration from rest, so that the pedestrian starts to walk again at time *ts*. In the example of Figure 3b, the time at which the subject begins the second acceleration is given by *ts* = 2.0 s. The magnitude *vs* of the impulse is a fraction of *v*max, and determines how quickly the model regains the maximum velocity.

We fit Equation (A2) to the data for the case *y*<sup>0</sup> = −3.5 m and *lg* = 25 m, yielding *ys* = −2.29 ± 0.22 m. This implies that participants who performed two-step crossings walked forward about 1.2 m before stopping, which amounts to about 1.5 m from the path of the vehicles. We also have the impulse magnitude *vs*/*v*max = 0.67 ± 0.22 and time *ts* = 2.41 ± 0.26 s, which corresponds to the time for the pedestrian to start walking again. This range includes the time (about 2.5 s) at which the leading vehicle passes the interception point. Other parameters, repulsion strength *rs* and range *σs*, which suffer from large fluctuations due to the very limited sample number, are obtained as *rs* = <sup>520</sup> ± 480 s−<sup>1</sup> and *<sup>σ</sup><sup>s</sup>* = 0.26 ± 0.24 m. However, the sample size of two-step crossings was insufficient to derive statistically meaningful results. A possible behavioral interpretation of this type of crossing is as exchanging the additional energy required for deceleration and acceleration in favor of more safety or security.

#### **References**


c 2020 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## *Article* **Functional Roles of Saccades for a Hand Movement**

#### **Yuki Sakazume, Sho Furubayashi and Eizo Miyashita \***

School of Life Science and Technology, Tokyo Institute of Technology, Tokyo 152-8550, Japan; sakazume.y.aa@m.titech.ac.jp (Y.S.); furubayashi.s.aa@m.titech.ac.jp (S.F.)

**\*** Correspondence: miyashita.e.aa@m.titech.ac.jp; Tel.: +81-45-924-5573

Received: 6 March 2020; Accepted: 27 April 2020; Published: 28 April 2020

**Abstract:** An eye saccade provides appropriate visual information for motor control. The present study was aimed to reveal the role of saccades in hand movements. Two types of movements, i.e., hitting and circle-drawing movements, were adopted, and saccades during the movements were classified as either a leading saccade (LS) or catching saccade (CS) depending on the relative gaze position of the saccade to the hand position. The ratio of types of the saccades during the movements was heavily dependent on the skillfulness of the subjects. In the late phase of the movements in a less skillful subject, CS tended to occur in less precise movements, and precision of the movement tended to be improved in the subsequent movement in the hitting. While LS directing gaze to a target point was observed in both types of the movements regardless of skillfulness of the subjects, LS in between a start point and a target point, which led gaze to a local minimum variance point on a hand movement trajectory, was exclusively found in the drawing in a less skillful subject. These results suggest that LS and some types of CS may provide positional information of via-points in addition to a target point and visual information to improve precision of a feedforward controller in the brain, respectively.

**Keywords:** eye movements; reaching arm movements; eye-hand coordination; visual information; movement precision; movement segment; forward model; control model of the brain

#### **1. Introduction**

Complex movement is considered to have consecutive segments of movement [1,2]. A connecting point of each segment can be called a via-point [3]. Thus, previous studies have focused on hitting or reaching movements with the hand as an elemental movement or a movement segment to reveal behavioral characteristics [4,5] and to investigate neuronal activity patterns [6,7]. Such studies often include a drawing movement [8]. Investigation of generation mechanism of the via-point may lead to understanding a complex movement.

In daily life, movements depend heavily on visual information, which is used for feedforward and feedback controls. For example, goal-directed movements of the hand are usually accompanied by an eye saccade that precisely directs an individual's gaze to an observed target location in case of stational target [9–13] or a predicted target location in case of moving target [14,15] before the hand starts to move. This type of saccades has been thought to provide visual information about the observed or predicted target location to guide a hand movement. Feedback control of the reaching movements also relies substantially on visual information [16–19]. Additionally, another type of saccade was identified during the performance of a simple line-drawing movement in which a sequence of small eye saccades closely followed the trajectory of a pencil [20]. The author of this study suggested that this type of saccade contributed to feedback control. Therefore, at least two types of saccades may be associated with hand movement: a saccade that directs an individual's gaze to a target position, which has been observed during a reaching movement, and a saccade that directs an individual's gaze to a hand position, which has been observed during the drawing of a simple line.

The goal of the present study was to reveal the relationship between the two types of saccades and hand movements, especially its feedforward control, and to get an insight into the role of visual information acquired by the saccades on movement control of the hand. We adopted hitting and circle-drawing movements as discrete and continuous movements, respectively. These two types of movements were selected instead of reaching and circle-tracking movements to increase the relative contribution of feedforward control to feedback control. Saccades during the movements were quantitatively classified as either a leading saccade (LS) or catching saccade (CS) depending on the relative gaze position of the saccade to a cursor position that represented the hand position: LS and CS directed the gaze to the cursor position in the direction of the cursor movement and to around the current cursor position, respectively. Precision of the hand movements was analyzed in relation to the saccades.

#### **2. Materials and Methods**

Two Japanese monkeys (*Macaca fuscata*; male: Monkey H, 6.7 kg, female: Monkey U, 6.6 kg) were used in the present study. Using a robotic arm manipulandum, the monkeys were trained to execute hitting and drawing tasks. During the tasks, hand and gaze positions of the monkey were recorded. All experimental procedures were performed in accordance with the Guidelines for Proper Conduct of Animal Experiments of Science Council of Japan and approved by the Committee for Animal Experiment at Tokyo Institute of Technology.

#### *2.1. Subjects and Apparatus*

To fixate the head position during the tasks, a head holder was installed on each monkey. Inducted by ketamine (10 mg/kg, intramuscular injection (i.m.)) and xylazine (2 mg/kg, i.m.), the monkey was deeply anesthetized with pentobarbital (25 mg/kg for an initial dose and 12.5 mg/kg for a supplementary dose when required by intravenous injection), a surgical operation was performed under aseptic conditions while monitoring heart rate and blood saturated oxygen concentration. The head holder was installed on the monkey skull and fixed with dental cement after cortical screws were implanted into the skull.

After an appropriate recovery period from the surgical operation, the monkeys were well trained to perform behavioral tasks (Figure 1). The monkey sat comfortably in a primate chair with its head fixated. RANARM, a robotic arm manipulandum for normal and altered reaching movements [21], was used to control the tasks and record the hand position of the monkeys, and a 19-inch computer monitor was set 61 cm away from the monkeys' eyes. A cursor on the monitor provided visual feedback regarding the current spatial location of a gripper of RANARM that the monkey held in its hand. Gaze position was estimated by measuring the position and shape of the pupil at a sampling rate of 1000 Hz with EyeLink (SR Research, Ottawa, ON, Canada).

**Figure 1.** Experimental setup and time consequence of the tasks are illustrated. (**A**) A monkey directs a cursor on a monitor by manipulating a robotic arm with its right hand on a horizontal plane. The hand position and its eye position are recorded during the tasks. (**B**) Hitting task: monkeys are required to move the cursor from the bottom red home point to the top blue target point within a designated amount of time.

(**C**) Circle drawing task: a prompt shows the required direction (blue: clockwise, green: counter-clockwise (not shown)) of drawing while the monkey holds the cursor within the start point. Eight evenly spaced start points are used. The monkey can begin drawing at any time and without any time restrictions on movement time.

#### *2.2. Behavioral Tasks*

Two tasks were executed by each monkey with the right hand: a hitting task and a circledrawing task.

In the hitting task, a red home point appeared 0.100 m below the center of the monitor at the start of each trial. The monkey was required to move the cursor to the home point and then hold it there for a random period of time that ranged from 0.50 to 1.00 s. Next, a blue target point appeared 0.075 m above the center of the monitor, and the monkey was required to move the cursor to the target point within a designated amount of time. If the monkey hit the target point in the allotted time, the target point disappeared, and the monkey received a drop of water as a reward. If the monkey did not hit the target in time, the trial was considered a failure. To ensure quick and precise hand movements, the size of the home and target points, and the required task time (reaction time plus movement time) were regulated so that the ratio of successful trials to failure trials was equal to approximately 1.

In the circle-drawing task, a red start point (radius: 0.030 m) appeared at one of eight predetermined positions located at 45◦ intervals around a circle (radius: 0.100 m) in the center of the monitor. The monkey was required to move the cursor (radius: 0.010 m) into the start point and to hold the cursor there for 1 sec. Then, a prompt consisting of a circle was quickly drawn beginning from the start point and using different colors to identify the direction of movement (blue: clockwise, green: counterclockwise). During this period, the monkey was required to hold the cursor within the start point and was then able to begin its drawing movement at any time after the prompt drawing was completed. There were no time restrictions on the movement, and the trajectory of the cursor during hand movement was recorded. Three indices were used to determine a successful trial: 1) the drawing direction of the monkey was the same as that of the prompt drawing; 2) the center of cursor passed within 0.070 m of the center of the prompt trajectory; and 3) the cursor returned to the start point. The monkey got a drop of water as a reward after a successful trial.

#### *2.3. Data Recording and Analysis*

Gaze and hand positional data were outputted to a home-made data recording system using LabView (National Instruments, Austin, TX, USA) through a 16-bit analog to digital converter and saved as a binary file. Data were analyzed using MATLAB (The MathWorks, Natick, MA, USA).

#### 2.3.1. Saccade

The eye saccades were detected from the gaze positional data that were filtered with a fourth-order Butterworth low-pass filter at 50 Hz. The onset and offset of the saccades were determined using the following criteria: the gaze speed exceeded a threshold criterion of 61◦/s or it was lower than a threshold criterion of 3◦/s after the saccade onset. We classified the saccades into two types depending on the relative gaze position at offset of the saccade to the hand position: a leading saccade (LS) and a catching saccade (CS). LS and CS directed the gaze to the cursor position in the direction of the cursor movement and to around the current cursor position, respectively. We analyzed CS in the hitting task and both LS and CS in the circle-drawing task from a viewpoint of relationships between saccades and precision of the hand movement.

#### 2.3.2. Hand Movement

The cursor positional data that corresponded to the hand position were obtained using RANARM and were filtered with a fourth-order Butterworth low-pass filter at 10 Hz. Precision of the hand movement was evaluated.

In the hitting task, it was evaluated at three different points during the movement where the hand acceleration was maximum, its velocity was maximum, and its acceleration was minimum, each of which represents the initial, middle, and terminal phases of the movement. Movement precision was defined based on the positional variance of the cursor's *x*-coordinate. At the acceleration maximum point, another index, which was the directional variance of tangential movement direction (Dm) of the cursor, was adopted. We expected that Dm was more robust to variation of the start position of the cursor and was a more appropriate index for a very early phase of the movement.

In the circle-drawing task, the position of the cursor was expressed in polar coordinates, and precision of the hand movement was defined as the positional variance of the cursor in the radial direction at a given phase. As another index of the precision, we also adopted curvature of the hand movement, which was calculated as follows: . *x* .. *<sup>y</sup>* <sup>−</sup> .. *x* . *y* ( . *x* <sup>2</sup> + . *y* 2 ) −3/2 , where *x* and *y* are the cursor's coordinates of its position in Cartesian coordinates with the horizontal direction and the center of the monitor as its x-axis and origin, respectively.

#### **3. Results**

#### *3.1. Hitting Task*

Monkeys had to control its hand quickly and precisely to execute the task. For *Monkey H*, the radii of the home and target points were set at 0.012 and 0.014 m, respectively, and the task time was designated at 0.75 s. This subject completed a total of 780 trials, of which 401 (52%) were successful. For *Monkey U*, the radii of the home and target points were both set at 0.006 m, and the task time was designated at 0.54 s. This subject completed a total of 560 trials, of which 403 (58%) were successful.

Although success ratio of the task was more or less the same in the two monkeys, movement characteristics of the hand in the hitting task varied depending on the monkey. *Monkey U* showed better performance than *Monkey H* with smaller target diameter, shorter movement time (0.47 ± 0.04 s vs 0.33 ± 0.03 s) (mean ± standard deviation [SD]), higher peak hand velocity (0.54 ± 0.05 m/s vs 0.90 ± 0.07 m/s) (mean ± SD), all of which indicate more precise feedforward control of the hand.

In the following analysis, trials that did not reach 0.125 m in y-coordinate within the designated task time were excluded; 762 trials in Monkey H and 556 trials in Monkey U were analyzed as a result. In almost all of the hitting task trials, both monkeys made a saccade directly to the target point upon its appearance and prior to the onset of hand movement. Subsequently, two different types of gaze control were observed. In one type, the monkey kept its gaze on the target point until the cursor hit it (Figure 2A), and in the other type, the monkey returned its gaze from the target to the cursor and made a couple of saccades around the cursor while following its movement (Figure 2B). LS and CS that execute these two types of gaze control were analyzed. We defined LS and CS as follows: LS was a saccade that directed the gaze in the range from 0.135 to 0.200 m in y-coordinate; CS was one that was not LS and difference of gaze position directed by that and the cursor position was within ± 0.050 m in y-coordinate.

The LS was observed in both monkeys in more than 90% of the trials. Inversely correlated with the performance of the task, interestingly, the number of the CS in Monkey H was prominently higher than that in Monkey U (Table 1). Therefore, the CS in Monkey H was analyzed in the following analysis. In the trial with the CS in Monkey H, the subject executed the CS along the hand trajectory (Figure 3A). The gaze points following the CS in Monkey H seemed bimodally distributed in the direction of cursor movement with a border point at 0.11 m from the center of the start point (Figure 3B), which roughly corresponded to the cursor velocity maximum point. We tentatively classified the CS into two types; the CS in hand acceleration phase (CSa) and the CS in hand deceleration phase (CSd).

**Figure 2.** Two types of saccades during the hitting task are depicted. (**A**) leading saccade (LS) occurs almost simultaneously at the onset of acceleration of the cursor. (**B**) catching saccade (CS) follows LS in this trial.

**Table 1.** Occurrence probability of each type of saccades during the hitting task.


<sup>1</sup> CSa: CS in cursor acceleration phase. <sup>2</sup> CSd: CS in cursor deceleration phase. <sup>3</sup> Both CSa and CSd were observed in 24 trials.

**Figure 3.** Cursor trajectories of trials with CS and distribution of gaze points directed by CS. (**A**) Cursor trajectories of the hitting trials with CS and gaze points directed by CS in *Monkey H*. (**B**) Distribution of gaze points directed by CS in y-coordinate. Broken line indicates the border point (0.11 m) of CS in hand acceleration phase (CSa) and CS in hand deceleration phase (CSd).

To analyze the relationship between the CS and precision of the hand movement, the hitting task trials were grouped into three; those without the CS, those with CSa, and those with CSd, and precision of the hand movement among the groups was compared. The trials with CSd showed a tendency towards larger variance of the movement than the other two groups from the middle phase of the movement and showed significantly larger variance than those without the CS both at the middle (F-test, p = 0.0144) and deceleration (F-test, p = 0.0158) phases of the movement (Figure 4A). Although there was no significant difference in the variance of Dm among the three groups, we found that only the subsequent trials to ones with CSd showed a tendency towards smaller variance than the current trials (Figure 4B). This was indicative of influence of CSd on improvement of precision of

the subsequent movement in its initial phase, i.e. feedforward control component. Like in the task performance indices, e.g. size of the target and peak hand velocity, precision of the movement by *Monkey U* was superior to that by *Monkey H* in all the movement phases.

**Figure 4.** Effects of the CS on accuracy and precision of the hand movement in the hitting task are shown. Trials are grouped into three depending on the types of saccades during the trial in Monkey H; those without (w/o) CS (520 trials), with CSa (199 trials), and with CSd (67 trials). As a comparison, the result of all trials in Monkey U (556 trials) is also shown. (**A**) Top and bottom panels: average and variance of the cursor position in x-coordinate are shown, respectively (\* *p* < 0.05). They are evaluated at cursor acceleration maximum (Amax), velocity maximum (Vmax), and acceleration minimum (Amin) point during the hand movement. (**B**) Top and bottom panels: average and variance of movement direction (Dm) are shown, respectively. The current and subsequent trials are shown as filled boxes and white oblique-lined boxes, respectively. An error bar represents a standard deviation in all panels.

#### *3.2. Circle-Drawing Task*

Following a training period, both monkeys were able to smoothly draw a circle from all eight start points in both directions of rotation at success rates up to around 90%. In total, 2,000 trials (two rotation directions × eight start points × 125 trials) were recorded from each monkey. All of the following analyses were applied to the successful trials in which cursor trajectory was within ± 2 SD range from the average and the saccades that directed the gaze within ± 3 SD range from the average trajectory of the cursor.

Monkey H performed the task with a movement time of 0.90 ± 0.14 s and 0.97 ± 0.11 s (mean ± SD) in the counterclockwise and clockwise directions, respectively. *Monkey U* did it with a movement time of 0.77 ± 0.08 s and 0.71 ± 0.07 s (mean ± SD) in the counterclockwise and clockwise directions, respectively.

Similar to the hitting task, both monkeys made several saccades throughout the drawing task (Figure 5A). We focused on the saccades that led the gaze on the way to the target. Therefore, we excluded the saccade within the final 45◦ range of the phase and plotted the distribution of phase difference of the saccade against the cursor (Figure 5B). In Monkey H, the distribution seemed to be bimodal with peaks around at 0◦ and –100◦ in both rotation directions. In contrast, in *Monkey U*, it seemed to be unimodal with a peak around at 0◦ in both rotation directions. We defined LS and CS in the drawing task as a saccade that had the phase difference smaller than −35◦ and within ±35◦, respectively. More than 40% of the saccades were the LS in Monkey H (46% in the counterclockwise

rotation direction, 44% in the clockwise rotation direction). In contrast, the CS dominated in Monkey U (45% in the counterclockwise rotation direction, 76% in the clockwise rotation direction) (Figure 5C) (Table 2).

**Figure 5.** Two types of saccades during the circle-drawing task are depicted. (**A**) Example trials of the circle-drawing task from the 315◦ start point toward the clockwise direction in *Monkey H*. (**B**) Distributions of gaze points directed by saccades relative to the hand position are shown. Broken lines indicate the borders (–35◦ and 35◦) of LS, CS, and other saccades. (**C**) Distributions of gaze points directed by LS, CS, and the other saccade are plotted in relation to a start point (a positive sign is assigned to the phase when the gaze point is in the cursor movement direction). Bin width is 5◦ in all the panels.


**Table 2.** Occurrence probability of each type of saccades during the circle-drawing task.

<sup>1</sup> CCW: counterclockwise, CW: clockwise. <sup>2</sup> Midway: saccades except the final 45◦ range of the phase. <sup>3</sup> Total = LS + CS + Other saccades.

We then analyzed the relationship of these saccades and precision of the hand movement. In Monkey H, variance of the movement tended to be small at a specific phase around 30◦–60◦ irrespective of the eight different start points and two different rotation directions, toward which point the LS tended to be directed (Figure 6A). When the LS was aligned at the local minimum phase of the variance in the range of 0◦–100◦, it clearly showed that the LS directed the gaze 0◦ (counterclockwise rotation direction) and –30◦ (clockwise rotation direction) in advance to the local minimum point of the variance (Figure 6B). In Monkey U, the number of the LS was relatively small (29% in the counterclockwise rotation direction, 7% in the clockwise rotation direction), and no obvious relationship of the LS and precision of the hand movement was found.

**Figure 6.** Relationships of the cursor positional variance and gaze points directed by saccades in between (within 45◦–315◦ phase range relative to the start point) the start and target points in the circle-drawing task are shown. (**A**) The relationships are plotted in absolute phase (phase in polar coordinates with its origin at the center of the monitor). Cursor variance and number of each saccade are indicated by the solid lines and bars, respectively. Note that the cursor variance gets local minimum at the phase of around 30◦–60◦ for almost all starting point regardless of the rotational directions in *Monkey H*. (**B**) Gaze points directed by saccades are plotted against the local minimum phase of the cursor variance in *Monkey H* (a negative sign is assigned to the phase when the gaze leads the cursor movement). Bin width is 5◦ in all the panels.

Finally, the CS was analyzed in terms of precision of the drawing movement. We applied the same method as in the hitting task; precision of the movement was compared between trials with CSd within the phase range of 60◦ to the final target (CSt) and ones without it. As an index of the movement precision, curvature of the cursor was evaluated at the local minimum of the acceleration of the cursor. In *Monkey H*, the trials with CSt showed significantly higher variance of the curvature than those without CSt in the counterclockwise (F-test, p <sup>=</sup> 1.6 <sup>×</sup> <sup>10</sup><sup>−</sup>7) and clockwise (F-test, p <sup>=</sup> 3.1 <sup>×</sup> <sup>10</sup><sup>−</sup>3) rotation directions, respectively (Figure 7). The analysis was not applied to Monkey U because there were a few CSt found in both rotation directions.

**Figure 7.** Effects of CS close to the target (CSt) on the cursor movement accuracy and precision during the circle-drawing task Monkey H are shown. They are evaluated at the cursor acceleration minimum point close to the target (**A**) Average of curvature of the cursor trajectory. An error bar represents a standard deviation. (**B**) Variance of the curvature. \*\* *p* < 0.01. \*\*\* *p* < 0.001.

#### **4. Discussion**

The relationship between saccades and hand movement has been studied using both pointing and drawing movements. For example, goal-directed pointing tasks were used to assess the type of saccades that preceded the onset of hand movement [9–15]. This type of saccades has been thought to provide visual information about the observed or predicted target location to guide a hand movement. Additionally, a simple line-drawing task was used to identify another type of saccade in which the gaze closely followed a pencil trajectory with a sequence of small saccades [20]. This type of saccade may be related to the feedback control of hand movement.

In the present study, saccades during the movements were quantitatively classified as either a leading saccade (LS) or catching saccade (CS) depending on the relative gaze position of the saccade to a cursor position that represented the hand position, and relationships between the saccades and feedforward control component of hand movements were analyzed.

#### *4.1. Functional Roles of the Saccades*

We found CS during the hitting task in addition to the circle-drawing task. Although CS during the circle-drawing task was predictable considering that a similar type of saccade was observed in a previous study using a line-drawing task [20], CS during the hitting task was a novel finding.

CS during the hitting task was exclusively observed in the monkey that showed lower precision of the movement than the other. While CS was found both in the hand acceleration phase and in the hand deceleration phase (CSd), precision of the middle and late phases of the movement in the trials with CSd was significantly lower than that in the trials without CSd. Since the subject had to heavily rely on feedforward control to accomplish the task because of its designated short movement time, the observed precision is considered to largely reflect that of the feedforward control component. Taking the following two results into account, therefore, we assume that CSd during the hitting task may arise from the imprecision of the feedforward control component to improve its precision. Firstly, the trials with CSd were significantly less precise than those without CSd at the middle and deceleration phases of the movement. Secondly, the initial movement direction, which represents the feedforward control component, in the subsequent trial tended to be more precise than that in the current trial with CSd, while no significant difference of the precision was found between trials without CSd. This assumption may support the previous study that showed a faster adaptation rate in a novel task in a trial with continuous visual feedback of the cursor position than in a trial with post-trial knowledge of task performance [22]. Although we could not detect any evidence suggesting contribution of CS to online error correction but find those suggesting contribution of CSd on offline error correction ("error" is used in terms of not systematic error but accidental error), it should be further carefully

investigated which saccade, i.e. LS or CS, provides visual information for the online feedback error correction [17,23,24].

In the drawing task, two types of CS were also observed like in the hitting task; one occurred in early phase of the movement (CSe) and the other in late phase of the movement, especially within a phase close to the target (CSt). While the monkey that showed higher precision in the hitting task had not CSt but exclusively CSe, the one that showed lower precision had both CSt and CSe. Like CSd in the hitting task, CSt in the circle-drawing task was observed exclusively in the monkey that showed lower precision in the hitting task. Furthermore, precision at the deceleration local minimum phase of the movement close to the target (curvature was used as an index) in the trials with CSt was significantly lower than that in the trials without CSt, which coincides with the finding in the hitting task. Taking this analogy between CSd and CSt into account, we assume that CSt during the circle-drawing task may also arise from the imprecision of the feedforward controller to improve its precision. As for CSe, we presume that CSe may have a role on precise control of the hand movement because CSe was predominantly observed in the monkey that executed more precise control of the hitting task although we could not find any valuable index to evaluate the function of CSe.

The present study also found LS during the circle-drawing task that directed the gaze to a point in between the start and target points although there was no explicit presentation of a via-point. This type of gaze control has been reported in walking in natural terrain [25], in which a saccade directed the gaze to a future point of foot placement about 1.5 ms in advance. In the present study, the LS (LSm) was always followed by a fixated gaze until the cursor passed the area and the gaze points of LSm were concentrated at a point with 40◦ or 0◦ phase lead depending on the rotation direction where the positional variance of the cursor reached a local minimum. These findings suggest that the gaze point by LSm is directed to close to an internally set via-point. These points were set in working coordinates, i.e. external coordinates because the local minimum points of the variance of the cursor were more or less consistent irrespective to the start point of the drawing movement. LSm was observed in the monkey that executed less precise control of the hitting task, suggesting that the skillful monkey regarded the circle-drawing movement as single segment of movement and the less skillful monkey divided the movement into multiple segments setting via-points.

#### *4.2. The Saccades and the Control Model of an Arm Movement*

An optimal feedback control model was proposed as one for pointing or reaching arm movements [26] and has successfully explained various characteristics of those behaviors [27–30]. According to the model, an optimal controller produces a control input to muscles based on the estimated state of an arm, i.e., a predicted online state that may be corrected by observed information. Furthermore, a forward model generates the predicted state using the control input. Thus, the optimal controller is able to serve as a controller for a segmental movement, and to control an object even without feedback information if the forward model is accurate and precise enough.

Based on the minimum intervention principle of the model, it predicts that variance of trajectories of movement becomes minimum at an aimed point, i.e. a via-point in multiple segmental movement. Therefore, the local minimum points of the variance of the cursor in the circle-drawing task can be reasonably regarded as a via-point. In the case of movements that are more complicated than the circle-drawing, it is possible that the brain internally sets via-points [3] toward which LS directs an individual's gaze and, in turn, sequentially feeds information to some type of neurological optimal controller.

The model does not inherently implement a mechanism that adaptively changes the forward model depending on a change of a control object. Since there has been accumulated evidence suggesting that the brain can adaptively change its feedforward control signal depending on a change of an environment [31–34], the forward model in the brain must adaptively changes. For the adaptive change of the forward model, some types of CS, e.g., CSd in the hitting task and CSt in the circle-drawing task may provide visual information.

Finally, a higher-level mechanism that governs the way of adaptation or learning has been proposed as meta-learning, which is mainly investigated in the framework of reinforcement learning [35,36]. An apparent different strategy to the drawing task found in two subjects in the present study might be a resultant outcome of a kind of meta-learning because the subjects had acquired the skill to perform the task through a reward-based training process. If this is the case, we suggest that precision or reliability of the forward model should be considered as an important factor to determine a strategy in the meta-learning. In other words, a more reliable forward model may lead to a strategy to perform a complex movement with a smaller number of moment segments.

**Author Contributions:** Conceptualization, E.M.; Formal analysis, Y.S. and S.F.; Investigation, Y.S.; Methodology, E.M.; Supervision, E.M.; Visualization, Y.S. and S.F.; Writing—original draft, E.M.; Writing—review & editing, Y.S., S.F. and E.M. All authors have read and agree to the published version of the manuscript.

**Funding:** This work was supported in part by Grant-in-Aid for Scientific Research on Innovative Areas, "The study on the neural dynamics for understanding communication in terms of complex hetero systems (No. 4103)" (211200112).

**Acknowledgments:** The authors would like to thank Tomohiro Nabe (MEng), Shinichiro Murakami (MEng), and Zhiwei He (MEng), who were master course students at Tokyo Institute of Technology, for their help of the animal experiment.

**Conflicts of Interest:** The authors declare no conflict of interest.

#### **References**


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