# The highs and lows of short-term anticipation in models for pedestrian dynamics: From groping to probing the imminent future

Alexis Raulin-Foissac, Alexandre Nicolas

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Alexis Raulin-Foissac, Alexandre Nicolas. The highs and lows of short-term anticipation in models for pedestrian dynamics: From groping to probing the imminent future. Traffic and Granular Flow 2024 (TGF'24), Dec 2024, Lyon, France. pp.04023, 10.1051/epjconf/202533404023. hal-05364799

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# The highs and lows of short-term anticipation in models for pedestrian dynamics: From groping to probing the imminent future

Alexis RAULIN-FOISSAC¹,\* and Alexandre NICOLAS¹,\*\*

¹Université Claude Bernard Lyon 1, CNRS, Institut Lumière Matière, UMR 5306, F-69100, Villeurbanne, France

**Abstract.** Anticipation manifests itself at various levels in pedestrian dynamics. This paper aims to clarify its impact on the modeling of local navigation, by shedding light on two facets of anticipation over the near future, one related to the others' motion and the other one related to the agent's own prospective motion. We start by observing that in simple ('reactive') models anticipation is completely missing, whereas in velocity-based models, it generally rests on linear extrapolation, with the use of e.g. time-to-collision metrics. Focusing on a crossing scenario, in which a pedestrian has to cross a static crowd, we pinpoint some of the ensuing limitations. We conclude by presenting preliminary data on a modeling framework that attempts to bridge the gap between the above mainstream models and full blown game theory.

## 1 Introduction

Every step a pedestrian takes is a step into the future. Envisioning and framing the misty future that lies just a few steps ahead is so ingrained in our habits that we can do so without racking our brains. In contrast, the simulated agents in state-of-the-practice pedestrian dynamics models often conspicuously struggle to envision the situation they will face in the next couple of seconds, which may result in unrealistic behavior [1, 2].

Anticipation thus seems to drive a wedge between physical particles or reactive systems, where the dynamic depends solely on the current state, and actual pedestrians. Strangely enough, though, even the dynamics of classical physical particles can be deduced from the minimization of an *action* that extends into the future, i.e., a path-integral over time. In fact, this extremal approach is equivalent to writing Newton's equation of motion and (complemented with stochastic fluctuations) it was shown to quantitatively replicate pedestrian dynamics in simple scenarios with dilute crowds [3, 4].

However, unlike physical particles, pedestrians are (for all practical purposes) *autonomous* decisional entities; there is no reason why a *global* potential should govern their individual dynamics. Accordingly, in order to navigate smoothly and (more or less) in harmony with their surroundings, even in crowded settings, they need to adapt their behavior by anticipating the near future and, in particular, the motion of their neighbors. Over the past

<footnote>\*e-mail: alexis.raulin-foissac@univ-lyon1.fr</footnote>

<footnote>\*\*e-mail: alexandre.nicolas@cnrs.fr</footnote>
years, several studies have evinced that failing to do so in models leads to unrealistic predictions, in dense bidirectional flows in a corridor [5] or in scenarios involving the crossing of a dense crowd [6, 7].

In this contribution, we shed light on a distinction between two facets of anticipation over the next few seconds, namely, the forecast of the others' moves and the simulation of one's own move (in one's mind) and we pinpoint the limitations that are encountered when these facets are not properly accounted for in models, notably in the context of crossing a static crowd. These limitations are interpreted geometrically, from a space-time perspective. Finally, we lay the first bricks of a modeling framework that bridges the gap between full blown game theory, where the anticipatory dimension is fully unrolled, and state-of-the-practice approaches [10–13], where it is reduced to its bare minimum.

## 2 The two facets of short-term anticipation and their reflection in mainstream models

In pedestrian dynamics, anticipation has at least two facets. On the one hand, each agent should plan their own future movements and envision their trajectory in the next seconds. On the other hand, each agent should forecast the future state of their environment depending on the other agents' behaviors, so that they can adapt to changes and effectively interact with their surroundings. (Of course, the two facets are connected, in that the agent's actions influence those of their neighbors).

This distinction has bearing on the equations of motion that are implemented in pedestrian dynamics models. To illustrate the first aspect of anticipation, we examine a crossing scenario in which an agent navigates through a crowd of stationary pedestrians. Although the second aspect, involving mutual anticipation among moving agents, could also be explored with a dynamic crowd, here we focus on the static case. Nevertheless, we briefly address the dynamic case in Sec. 2.2 to lay the groundwork for the interpretation presented in Sec. 2.3.

Model predictions will be compared to the controlled experiments of Wang et al. [8], who tracked pedestrian participants as they moved through static groups of varying densities. These groups are made of pedestrians located in a square enclosure of 5 meters by 5 meters, and asked to stand still. The crossing pedestrians walk one by one, from the right to the left, before returning to the waiting area. Some of their trajectories are presented in Figure 1 (a, d). Numerically, this scenario will be replicated using pedestrian models already implemented in the UMANS (Unified Microscopic Agent Navigation Simulator) software developed at INRIA [9]. The mass of static agents is set to an almost infinite value, to prevent them from moving. In all models that are discussed below, agents are handled as disks. We have found that, with mainstream models, setting the radius to $r = 0.2$ m (to reflect the shoulder breadth) leads to many cases of entrapment or deadlocks. Accordingly, to avoid penalizing existing models excessively, we have set the radius of static pedestrians to $r = 0.1$ m (as a reflection of the chest depth).

### 2.1 Anticipation based on one's own move

Coming back to our discussion on anticipation, we make the following (seemingly reasonable) assumptions: Each agent has a goal and strives to achieve it by following a trajectory that meets their individual criteria, such as maintaining a reasonable speed, avoiding collisions, and keeping a safe distance from others. This involves planning one's movements over a specific time frame. The degree of accuracy of the plan depends not only on the planning horizon, but also on the current level of attention — from a focus on the immediate future to the anticipation of multiple future events along a complex trajectory.
<figure><img src="image_2.png" /><figcaption>**Figure 1.** Crossing of a crowd composed of 35 (a-c) or 45 (d-f) static pedestrians following the experimental setup of Wang et al [8]. Blue circles are at scale and represent static pedestrians. Black lines correspond to the experimentally delimited enclosure. Orange lines represent each of the 50 trajectories observed experimentally (a, d) or simulated with different agent-based models (b, c, e, f) using the UMANS (Unified Microscopic Agent Navigation Simulator) software [9].</figcaption></figure>

When anticipation is reduced to its minimum, the model becomes reactive and agents literally walk step by step. As we will see, this locality in the decisions may result in jagged trajectories and a higher likelihood of being stranded at a certain position in a cluttered environment. This is directly mirrored by simulations of the crossing scenario with the vanilla Social Force model [SFM] (i.e., the circular specification of [10]), in Fig. 1b, even in the lower-density situation, with 35 static pedestrians. While the observed trajectories (Fig. 1a) exhibit limited diversity and are mostly straight (insofar as pedestrians see a simple path and understand that it will remain open because they know that the crowd is static), the model yields a significantly different pattern. Admittedly, approximately half of the simulated trajectories align with the primary path observed experimentally, but the remaining trajectories are irregular and are characterized by frequent directional changes; some of them even fail to reach the target. This is a consequence of the reactive nature of the model, in which the agent does not plan a trajectory in advance. Relying solely on local collision avoidance leads to suboptimal path planning and, in some cases, increases the likelihood of subsequent collisions. Furthermore, it is clear that the trajectories reaching the target are significantly longer than those observed in the experimental data.

A popular alternative model inspired from robotics, ORCA (Optimal Reciprocal Collision Avoidance) [11], produces results much closer to the experimental ones, as shown in Fig. 1c. Since the crowd's staticity is taken into account, the crossing agent only needs to adopt velocities that do not make it collide with an obstacle; grazing trajectories (making it pass very close to an obstacle) are acceptable. As the density is fairly low, the model suc-
ceeds in finding a path almost straight across the crowd. ORCA is reflective of a broader
class of models, which go beyond strictly reactiveness and in which agents seek the optimal
velocity among all possible directions and speeds. For each test velocity, agents extrapolate
their future positions under the assumption of a uniform rectilinear trajectory; on this basis,
they can assess quantities such as the occurrence or not of a collision within a given time
window, the anticipated time to collision [12] or the time required to reach the target. Beyond
ORCA [11], these general ideas that penalize test velocities leading to a collision are used
in RVO (Reciprocal Velocity Obstacles) [13], in Karamouzas's power-law model [12] and in
ANDA (ANTicipatory Dynamics Algorithm) [2]. Compared to the step-by-step approach, the
extrapolation of the agent's future trajectory is pushed one order higher in terms of Taylor
expansion: instead of just considering the current position (zeroth order), the current velocity
(first order) is also taken into account.

However, there is no doubt that the uniform rectilinear assumption also has some limitations: Under this assumption, agents are unable to consider trajectories that require rapid turning because they cannot predict collision-free pathways with sharp directional changes. Indeed, in the crossing scenario, the numerical results change dramatically when the static crowd becomes denser and includes 45 pedestrians, instead of 35; anticipating and selecting one of the few feasible paths then becomes critical. Using the ORCA model, only a few crossing agents successfully reach the target, to the left of the static crowd (Fig. 1f). For sure, this is more convincing than the results obtained with the vanilla SFM (Fig. 1e), in which case all trajectories converge to the same endpoint and get trapped there, due to the strong repulsive forces encountered at this position. Nevertheless, the experimental results are quite different: pedestrians adopt less linear trajectories and make more detours, but all successfully cross the crowd (Fig. 1d). Accordingly, even first-order extrapolations may not suffice to consistently find successful paths in crowded settings. Importantly, the consequences of this failure at the operational level are not just inaccurate trajectory predictions, but a flawed tactical outcome, as the agents may fail to cross the crowd. These findings underscore the necessity for pedestrian models to incorporate longer-term anticipation to connect the operational and tactical levels, especially when navigating through congested environments filled with other pedestrians or various obstacles.

## 2.2 Anticipation based on the others' moves

Anticipating one's own movements is essential for effective navigation in a static and known
environment such as in Figure 1; however, when dynamic (i.e., not static) neighbors are
introduced, it becomes critically important to also predict and account for their trajectories.
As for the foregoing representation of one's own move, predicting the others' movements can
be carried out at different levels.

At the most rudimentary level, other agents are treated as static entities: only their current
positions are considered. Once again, this approach is exemplified by the vanilla SFM (circular specification), where interaction forces are calculated solely based on the relative distances between agents. Since velocities are left out of the picture, the avoidance of head-on
encounters is only undertaken at the very last moment. Conversely, it may lead to excessive
repulsion in scenarios where agents are not getting closer to each other, resulting in unnecessarily conservative navigation. This conservative behavior can be observed in Figure 1b
where the simulated agents make excessive detours to avoid the static individuals who would
present low risk of collision because of their (zero) speed.

A natural way to improve on this description is to take into account other agents' ve-
locities (which, incidentally, was already considered in the original paper introducing social
forces [14]). Most velocity-based models predict the timing of potential collisions with neigh-
boring agents by assuming that each agent maintains a constant velocity in the immediate fu-
ture. Based on this forecast, the agent’s velocity is adjusted to postpone or avoid impending
collisions. (To avoid the duplication of the avoidance maneuver in the agents risking colli-
sion in decentralized algorithms, smart methods to share the effort have been proposed [13]).
This constant-velocity extrapolation performs well in open environments with minimal veloc-
ity heterogeneity, but displays conspicuous limitations in some other (still relatively simple)
scenarios. For instance, consider a pedestrian who approaches a crosswalk; they may see a
decelerating car, infer its intention to stop, and proceed accordingly. In contrast, these models
would strive to avoid the car despite its manifest intention to stop, leading to suboptimal or
unrealistic behavior.

## 2.3 Interpreting anticipation with tubes in space-time

Both the self- and the neighbor-anticipation paradigms can be better understood from a space-
time perspective, using tubes to represent trajectories. Figure 2a depicts this perspective: each
horizontal plane is a snapshot at a given time, with agents represented by their geometric
shapes (typically a circle); these planes are stacked along the vertical axis, which represents
time. Connecting the instantaneous agents’ shapes gives rise to tubes, whose spatial projec-
tions, depicted in Figure 2b, are the spatial trajectories described earlier. The angle between
the tube and the temporal direction is related to the speed: a vertical tube corresponds to
a static agent (its position does not change as time passes) while a tube close to horizontal
corresponds to a very fast agent. Therefore, the curvature of the tube is directly related to ac-
celeration: moving at constant velocity tilts the agents’ tube, whereas accelerating or turning
bends it.

This perspective sheds a different light on the previous two facets of anticipation:

* **Reactive** models, such as the vanilla SFM, do not evolve any trajectory extrapolation be-
  beyond the current positions. This results in a significant loss of information; as illustrated in
  Fig. 2a, which may lead to a lack of realism.

* **Velocity-obstacle** models [11, 13] (and their likes [12]) aim to determine velocities that prevent future collisions with neighboring agents by taking their motion into account. This can be interpreted as finding a combination of directions for the rigid tubes such that no two tubes intersect in the future, or at least ensuring that intersections occur at a distant future time, as depicted Figure 2. More precisely, in Karamouzas’s model, collisions are penalized by a potential $\propto \tau^{-2} \exp(-\tau/\tau_0)$ where $\tau$ is the time-to-collision, while in ORCA, collisions are avoided at all costs over a short but finite time window, which corresponds to considering a cylinder with a finite extension along the time axis.

As the pedestrian density increases, the number of tubes also rises, making it significantly
more challenging to find a configuration in which none of the cylinders intersect. This high-
lights the necessity of anticipating one’s own future motion with a deformable tube rather
than a rigid cylinder: not only the immediate velocity, but the full prospective trajectory
should then be predicted, which naturally raises the question of cognitive (or computational)
abilities. In Fig. 2, this marks the transition from the second column to the third one.

## 3 Going beyond the linear extrapolation of trajectories

To go beyond the foregoing zeroth-order (reactive modeling) or first-order (constant-velocity
assumption) treatments of anticipation, we introduce a modeling framework that relaxes the
constraints on anticipated trajectories in a stepwise fashion.
<figure><img src="image_3.png" /><figcaption>**Figure 2.** Schematic representation of different pedestrian modeling paradigms based on their extrap-
olation of the future (trajectories). Pedestrian trajectories are depicted as circular tubes in space-time,
as pedestrians are modeled as disks in the plane. The red tube corresponds to the agent whose trajec-
tory is optimized. Reactive models include the vanilla SFM; Velocity-based models include approaches
such as RVO, ORCA, Karamouzas’ model, and ANDA; and Game-theoretic approaches align with the
framework described in Section 3.1. (a) Space-time representation of the trajectories. (b) Spatial trajec-
tories included past and anticipated movements. Figure inspired from [15].</figcaption></figure>

3.1 Game theory formalism

This broader view on anticipation suggests the recourse to game theory. Game theory has
already been applied to pedestrian dynamics several times [16–18], in many forms. In this
framework, each pedestrian is modeled as a rational agent seeking to optimize their strategy
to achieve individual goals, while accounting for the decision-making of other pedestrians
taking place simultaneously, in parallel. The resulting dynamics can be interpreted as a game
in which players, the pedestrians, aim to minimize a cost function that depends on both their
own future actions and the behaviors they anticipate from their neighbors. From a space-
time perspective, this corresponds to the situation in the last column of Fig. 2 (also see [19]),
where agents optimize their trajectories, represented by non-linear tubes, while anticipating
what others will do to have a better grasp of the future. In general, the trajectory that an agent
perceives as optimal obeys:

$$
\mathbf{r}_{1,\text{opt}} = \underset{\mathbf{r}:[0,T] \to \mathbb{R}^2}{\operatorname{argmin}} C[\mathbf{r}] \quad \text{with} \quad C[\mathbf{r}_I] = \int_0^T c(t, \mathbf{r}_I, \{\mathbf{r}_J\}_{J \neq I}) dt + C_{f,I}(\mathbf{r}_I(T)). \quad (1)
$$

where **r**<sub>*I*</sub> represents the trajectory of the *I*-th agent, *c* denotes a local cost function that incorporates factors such as energy expenditure, proximity to other agents, and collision avoidance, while *C*<sub>*f*,I</sub> encapsulates the terminal cost associated with achieving the desired target. This problem is inherently challenging to solve due to the coupled nature of the optimizations. However, in practice, pedestrians are rational only up to a certain bound and they simplify this process through experiential learning, which affords efficient anticipation in most cases.
## 3.2 Stepwise relaxation of the rigid-rod constraint

The representation of tubes in space-time of Fig. 2 provides inspiration to push anticipation beyond the aforementioned limitations of mainstream models, while circumventing the complexity of full-blown game theory. In the velocity-obstacle and velocity-based models of Section 2.3, combinations of non-intersecting rigid cylinders (rods) were searched for. Fully relaxing the constraint of rigidity would lead to a complex optimization of tubes of arbitrary curvature. Instead, here, we relax the cylinder constraint in a stepwise fashion, by considering tubes shaped as piecewise linear trajectories (broken rods). The length of each rod segment controls the complexity of the trajectory under consideration, while introducing more segments extends the temporal horizon of anticipation. It should be noted that these lengths are not necessarily related to the physical step length of the agent. Clearly, this method has the potential to bridge the gap in trajectory anticipation between the velocity-obstacle approach (equivalent to having tubes with just one segment) and full-blown game theory (in the limit of an infinite number of vanishingly short segments).

## 3.3 Implementation

We have implemented these concepts in a model that searches one local minimum of the cost function defined in Equation 1, using a sequence of piecewise linear segments instead of an arbitrarily shaped tube. The segmentation leads to the following expression for the cost:

$$C[\mathbf{r}_I] = \sum_{i=1}^{N} c^{(i)}(\mathbf{r}_I, \{\mathbf{r}_J^{(i)}\}_{J \neq I}) + \sum_{\text{contacts}} c_{\text{contact}} + C_{f,I}(\mathbf{r}_I). \quad (2)$$

where the indices $i = 1...N$ refer to successive segments of the trajectory. The instantaneous cost $c^{(i)}$ is the sum of two contributions, $c_{\text{energy}}^{(i)}$ and $c_{\text{repulsion}}^{(i)}$, where $c_{\text{energy}}^{(i)} \propto v^2$ represents the energy expenditure [20] and $c_{\text{repulsion}}^{(i)}$ encapsulates the agent's desire to keep clear of their neighbors. For this term, we adopt an expression similar to the vanilla SFM, namely, a decreasing exponential with a visual prefactor:

$$c_{\text{repulsion},I}^{(i)} \propto \sum_{J \neq I} f(\theta_{IJ}^{(i)}) \exp \left( - \frac{r_{IJ}^{(i)} - R_I - R_J}{\sigma} \right). \quad (3)$$

where $\sigma > 0$ is the typical interaction length, $R_I$ and $R_J$ are the radii of the respective pedestrians, $r_{IJ}^{(i)}$ the distance between them at timestep $i$ and $f$ weights the cost based on the vision angle $\theta_{IJ}^{(i)}$.

The second sum in $C[\mathbf{r}_I]$ prevents interpenetration by penalizing contacts (between agents and with obstacles) along the trajectory, with a Hertzian potential $c_{\text{contact}} \propto d_{\text{contact}}^2$, where $d_{\text{contact}}$ is the interpenetration distance.

Finally, the terminal cost $C_{f,I}(\mathbf{r}_I)$ combines the attraction into a Gaussian well and the penalization of the distance of the last rod to the target:

$$C_{f,I}(\mathbf{r}_I) \propto - \sum_{i=1}^{N} \exp \left( -\nu (\mathbf{r}_I^{(i)} - \mathbf{r}_I^*)^2 \right) + \alpha |\mathbf{r}_I^{(N)} - \mathbf{r}_I^*| \quad (4)$$

where $\nu > 0$ characterizes the width of the well, $\mathbf{r}_I^*$ is the target of pedestrian $I$ and $\alpha$ is a constant. This term ensures that pedestrians reach their goal if possible, even though the anticipation time window is shorter than the travel time.
We differ the remaining technical details to a future, more comprehensive publication. Here, we present some first results obtained with this method, focusing on the crowd-crossing scenario.

## 3.4 Numerical results

<figure><img src="image_4.png" /><figcaption>**Figure 3.** Crossing of a static crowd simulated with our proposed method, whereby future trajectories are divided into multiple consecutive segments of durations *dt* = 2, 1 and 0.5 second. The settings are identical to those described in Figure 1 (d-f). The trajectories have been obtained with a cost optimization acting directly on the trajectory.</figcaption></figure>

The model was put to the probe in a static crowd scenario involving 45 pedestrians, where both the SFM and ORCA failed to replicate experimentally observed behaviors. The results are presented in Figure 3, where three different segment lengths were tested. For each segment length, a sample of simulated trajectories is displayed to illustrate the diversity of possible paths. This diversity stems from two sources: (1) differences in initial conditions, as pedestrians began from slightly distinct positions, and (2) stochastic noise incorporated in the optimization process to facilitate exploration of alternative trajectories - reflecting the fact that no single path is uniquely optimal.

In the first case, with long (2-second) segments, the simulated trajectories closely resemble the ORCA results shown in Fig. 1c for 35 pedestrians. Most trajectories have minimal, or no, turns; the bottom trajectory coincides with the (only) ORCA trajectory that successfully reached the opposite side in Fig. 1f. This suggests that when trajectories are discretized into long broken-line segments, the model effectively reproduces the global behavior of ORCA.

As the broken-line segments get shorter, the simulated trajectories exhibit greater flexibility, allowing agents to navigate around static pedestrians more effectively. This reduction in segment length also leads to an increased diversity in chosen paths, in broad agreement with the experimental spectrum of possibilities.

## 4 Conclusion

In summary, we have explored the multifaceted role of anticipation in pedestrian dynamics and provided a critical assessment of the capabilities and limitations of current modeling approaches. In doing so, two facets of anticipation are distinguished: (1) the ability of agents
to anticipate their own future movements, and (2) their capacity to predict and adapt to the anticipated behaviors of neighboring individuals. The emphasis was put on the first part and through this analysis, we highlighted significant shortcomings in existing models, particularly in scenarios involving the crossing of a static crowd. For instance, models such as the vanilla SFM exhibit erratic and, at times, non-convergent trajectories; both SFM and ORCA fail to successfully cross the crowd as its density increases. These limitations highlight the need for more robust and adaptive self-anticipation mechanisms, while also revealing the nuanced interplay between the operational and tactical levels of decision-making in pedestrian dynamics.

To address these challenges, we proposed a geometric interpretation of anticipation by conceptualizing trajectories as tubes in space-time. Building on this visualization, we proposed a new framework inspired by game-theoretic principles, which seeks to optimize trajectories by breaking them into smaller segments and minimizing a defined cost function with constraints. Preliminary results demonstrate its potential to extend anticipation beyond the capabilities of mainstream agent-based models in the context of crowd-crossing. These results will be elaborated in an upcoming publication, along with methods inspired from condensed-matter physics to expedite the computation of the optimal segmented tubes. This work thus lays the foundation for a new branch of pedestrian dynamics models, which integrates sophisticated anticipation strategies and achieves more realism in complex environments.

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