# Predicting pedestrian trajectories at different densities: A multi-criteria
empirical analysis

Raphael Korbmacher  Huu-Tu Dang  and Antoine Tordeux

###### Abstract

Predicting human trajectories is a challenging task due to the complexity of
pedestrian behavior, which is influenced by external factors such as the
scene’s topology and interactions with other pedestrians. A special challenge
arises from the dependence of the behaviour on the density of the scene. In
the literature, deep learning algorithms show the best performance in
predicting pedestrian trajectories, but so far just for situations with low
densities. In this study, we aim to investigate the suitability of these
algorithms for high-density scenarios by evaluating them on different error
metrics and comparing their accuracy to that of knowledge-based models that
have been used since long time in the literature. The findings indicate that
deep learning algorithms provide improved trajectory prediction accuracy in
the distance metrics for all tested densities. Nevertheless, we observe a
significant number of collisions in the predictions, especially in high-
density scenarios. This issue arises partly due to the absence of a collision
avoidance mechanism within the algorithms and partly because the distance-
based collision metric is inadequate for dense situations. To address these
limitations, we propose the introduction of a novel continuous collision
metric based on pedestrians’ time-to-collision. Subsequently, we outline how
this metric can be utilized to enhance the training of the algorithms.

###### keywords:

Pedestrian trajectory prediction, deep learning, high-density, collision,
time-to-collision

††journal: Physica A: Statistical Mechanics and its Applications

[label1]organization=University of Wuppertal, Department for traffic safety
and reliability,city=Wuppertal, postcode=42119, country=Germany
[label2]organization=UMR 5505 IRIT, University Toulouse 1
Capitole,city=Toulouse, country=France

Empirical comparison of traditional physics-based models and state-of-the-art
deep learning algorithms for predicting pedestrian trajectories.

The accuracy is tested across a range of different pedestrian densities.

Comparing by using three accuracy criteria: a Euclidean distance-based metric,
a distance-based collision metric, and our novel time-to-collision based
metric.

Improve the predictions by incorparating the time-to-collision based metric
into the loss function of the deep learning algorithms.

##  1 Introduction

The task of predicting pedestrian trajectories has emerged as a critical
component in a variety of real-world applications, ranging from autonomous
vehicles [1] and human-robot interactions [2] to the design of events,
infrastructure, and buildings [3], especially in the case of evacuation [4].
This topic has been addressed within academia from the two distinct
disciplinary perspectives of pedestrian dynamics and data science [5].

On one hand, the discipline of pedestrian dynamics applies a _knowledge-based_
(KB) approach, developing mathematical models that encapsulate the inherent
rules governing pedestrian behavior [6]. These models are utilized to conduct
simulations in which pedestrian trajectories are computed. The difficulty lies
in identifying fundamental mechanisms and parameters that induce realistic
pedestrian behavior. On the other hand, computer scientists employ a _data-
based_ (DB) approach, collecting extensive data and training sophisticated
algorithms intended to predict pedestrian trajectories. Their focus
predominantly lies in devising efficient algorithm architectures and
meticulously fine-tuning the hyperparameters of DB algorithms.

While KB models cater to a broad array of applications and include
macroscopic, mesoscopic, and microscopic models [7], DB algorithms are
primarily deployed for microscopic trajectory predictions in low-density
scenes [5]. Low-density scenes denote situations with a medium pedestrian
presence, where individuals possess a high degree of freedom and exhibit long-
range interactions. This paper seeks to explore the efficacy of DB algorithms
in high-density situations and compare the results with those derived from
traditional KB models. One significant challenge inherent in such a comparison
lies in devising a fair and comprehensive evaluation. While prior studies have
demonstrated the superior performance of DB algorithms in terms of prediction
accuracy, these evaluations have exclusively pertained to low-density data
[8], focusing solely on distance metrics such as Average Displacement Error
(ADE) [9] and Final Displacement Error (FDE) [10]. We propose to extend this
evaluation by incorporating two additional metrics: a binary distance-based
collision metric as proposed by Kothari et al. [11], and an original
continuous time-to-collision-based metric.

Findings indicate that the DB algorithms surpass the KB models across all
tested densities in terms of distance metrics. However, the DB algorithm
predictions generate a significantly higher number of collisions when compared
to the real trajectories and the KB models, which are typically designed with
collision avoidance mechanisms.

##  2 Related work

In the following chapter, an extensive review of the KB models Section 2.1 and
DB algorithms Section 2.2 is conducted to provide a robust foundation for the
current study.

###  2.1 Knowledge-based models

KB models have a rich history in pedestrian dynamics that dates back to the
middle of the 20th century. These models apply principles from physics, such
as force fields and particles, to understand and predict the behavior of
pedestrians. Currently, KB pedestrian models range from macroscopic,
mesoscopic, and microscopic models among others modeling scale
characteristics. Macroscopic and mesoscopic approaches are borrowed from
continuous fluid dynamics or gas-kinetic models describing the dynamics at an
aggregated level, while microscopic approaches model individual pedestrian
motions [12]. For pedestrian trajectory predictions macroscopic and mesoscopic
models are less relevant, which is why in the following we focus on the
microscopic models.

In these models the individual pedestrian behavior is described according to
certain rules and mechanisms ground on physical social, or psychological
factors [13]. These rules and mechanisms are formulated in hand-crafted
dynamic equations based on Newton’s laws of motion. Given the input
information about the initial status of the pedestrians like position,
velocity, and acceleration a forward simulation of KB models can be used to
predict the future trajectories. Depending on the modelling order of the
model, they can be classified into _decision-based_ (zeroth order), _velocity-
based_ (first order), and _acceleration-based_ models (second order) [5]. In
acceleration-based models, typically force-based models, the movement of
pedestrians is defined by a superposition of exterior forces. Most
acceleration-based models are force-based consisting of a relaxation term to
the desired direction and an interaction term [5]. This last term is generally
the sum of repulsion with the neighbours and obstacles. This is also the case
in the most famous force-based model, the _Social Force model_ (SF) from
Helbing and Molnar [14], where the interaction force is an exponential
gradient of a distance-based potential.

Velocity-based models are speed functions, depending on the position
differences with neighbors and obstacles. In opposite to the acceleration-
based models that are based on second-order differential equations, the
velocity-based models rely on first-order equations. Many of these models are
based on collision avoidance techniques and are formulated as optimisation
problems on some ensemble of feasible trajectories devoid of collisions. The
most famous models are the Reciprocal Velocity Obstacle model [15, 16] and the
_Optimal Reciprocal Collision Avoidance_ (ORCA) [17].

In the last class of models, the decision-based or rule-based models, the
pedestrian behavior is not modeled based on differential equations, but on
rules or decisions determining the new agent positions, velocities, etc [13].
The time is considered to be discrete for this class of decision-based models,
which are typically Cellular automata [18, 19, 20, 21].

KB models in pedestrian dynamics encompass various approaches, including
microscopic models, which can be used for trajectory predictions. Most of
these models focus on the interactions between pedestrians and the environment
and, as a result, are fundamentally based on collision avoidance mechanisms.

###  2.2 Data-based algorithms

The previously outlined approach is fundamentally grounded in a theoretical
modeling framework. Essential mechanisms are pre-identified and expressed in
the form of equations, equipped with a handful of significant parameters that
require calibration and validation. The operational functionality of these
models extends to the simulation of pedestrian scenes, enabling the prediction
of future trajectories as a consequential byproduct. In the DB approach, the
prediction of trajectories is not a secondary outcome, but the main objective.
The parameters (or coefficients) of the algorithm have no physical meaning and
can not be interpreted. These parameters are determined through a process of
training the algorithm with data, with the goal of minimizing a predefined
cost function. The common cost function for trajectory predictions is the
displacement error metrics ADE or FDE. The trained algorithms are then tested
on new data, i.e. data which was previously not used to train the algorithm
(cross-validation).

Over the past decade, a multitude of studies employing various data-based
methodologies have been published with the aim of predicting pedestrian
trajectories. For a comprehensive overview, please refer to the following
reviews [5, 11, 22, 23]. The majority of these studies use supervised deep
learning techniques with either _Long Short-Term Memory_ (LSTM) or _Generative
Adversarial Networks_ (GAN) architectures. Among the most influential works in
this field are _Social-LSTM_ by Alahi et al. [8] and _Social-GAN_ by Gupta et
al. [24]. These groundbreaking papers have served as the inspiration for a
number of subsequent studies, which have extended these initial algorithms to
incorporate elements such as scene information [25], attention mechanisms
[26], graph neural networks [27, 28], and heterogeneity among pedestrians
[29]. In addition, it’s noteworthy to mention the utilization of convolutional
neural networks, which can be trained on video data rather than trajectory
data [30, 31, 32]. There are also studies using reinforcement learning
algorithms, which do not require explicit data, but rather operate based on a
defined reward function [33, 34].

##  3 Method

In this section, we provide a detailed description of the datasets used in
this study (Section 3.1). We then define the models and algorithms employed in
the following analysis (Section 3.2) and introduce distance and collision-
based evaluation metrics that will be used to assess the performance of the
different models and algorithms (Section 3.4).

###  3.1 Pedestrian trajectory data

In recent years, a large number of datasets have been collected and made
publicly available from an extensive range of studies, which mainly include
real-world trajectories of scenarios with low pedestrian densities ranging
from 0.1 to 0.4 ped/m2. For a comprehensive overview, one may refer to [35].
Interestingly, datasets corresponding to higher-density situations are
noticeably absent, possibly due to the challenges associated with the data
collection (i.e. trajectory extraction).

####  3.1.1 Low-density dataset

Initially, we assess the performance of the models and algorithms using low-
density datasets, which typically feature long-range interactions and
scenarios involving less than 0.5 ped/m2 [5]. Pedestrians in these scenes have
high degrees of freedom, and their behaviour is primarily influenced by a few
neighbouring people.

Given their emergence as benchmark datasets in pedestrian studies over recent
years, we have selected the ETH [9] and UCY [10] datasets for the analysis.
The ETH dataset comprises a total of 750 trajectories, divided into two
subsets: ETH and Hotel. Fig. 1 (a) shows an example of a segment from the
hotel dataset. The UCY dataset, on the other hand, has been subdivided into
three subsets: ZARA01, ZARA02, and UNIV, collectively containing 786
trajectories. An example of ZARA02 is shown in Fig. 1 (b). Both datasets,
collected in outdoor environments, encapsulate a variety of pedestrian traffic
patterns, including unidirectional, bidirectional, and multidirectional. These
datasets have been recorded at a framerate of 2.5 frames per second.

(a) Example from ETH.

(b) Example from ZARA02.

Figure 1: Illustrative examples of trajectory samples from low-density
datasets (ETH and UCY sets).

####  3.1.2 High-density datasets

High-density data refers to pedestrian situations characterized by more than 2
ped/m2, commonly known as crowds. The generation of accurate pedestrian
trajectory data in these situations is a challenging task due to the
difficulties in the automatic pedestrian identification and tracking. The
problems that usually occur are that the pedestrians often gather together,
occlude each other, and result in overlapping in pedestrian shapes [36]. These
factors contribute to the scarcity of real-world high-density pedestrian
trajectory datasets. Nevertheless, there are some rare examples available,
which can be found in [37, 38, 39]. In addition to these real-world datasets,
Forschungszentrum Jülich has conducted various laboratory experiments,
including HERMES, BaSiGo, CroMa, and CrowdDNA, which provide high-density
trajectory data [40, 41, 42]. Furthermore, other experimental datasets can be
found in [43, 44].

In this study, we primarily focus on the corridor experiments with
bidirectional flow from the Forschungszentrum Jülich, as they encompass a
diverse array of interactions. The experiment setup incorporates two starting
points or entrances, from where pedestrian start their walk. The size of the
recording area is a = 10 m, and $b_{corr}$ = 4 m. We utilize data gleaned from
six distinct bidirectional corridor settings, as showcased in Table 1. For
illustrative purposes, we depict the trajectories from the experiment
exhibiting the third-highest density (bidi3), alongside those from the fifth-
highest density (bidi5) in Fig. 2. In total, the dataset comprises 3096
trajectories, recorded at a framerate of 16 frames per second.

(a) Example from bidi3.

(b) Example from bidi5.

Figure 2: Illustrative examples of trajectory samples from high-density datasets (Juelich experiments). Table 1: Overview of pedestrian trajectory datasets. Dataset | Setting | Number of | Average | Maximum
---|---|---|---|---
|  | Pedestrians | Density | Density
ETH |  |  |  |
ETH | Outdoor | 365 | 0.14 | 0.35
HOTEL | Outdoor | 420 | 0.13 | 0.32
UCY |  |  |  |
ZARA01 | Outdoor | 148 | 0.21 | 0.51
ZARA02 | Outdoor | 204 | 0.27 | 0.48
UNIV | Outdoor | 434 | 0.38 | 0.52
JUELICH |  |  |  |
bidi1 | Lab | 141 | 0.38 | 0.55
bidi2 | Lab | 259 | 0.58 | 0.75
bidi3 | Lab | 480 | 1.00 | 1.15
bidi4 | Lab | 743 | 2.32 | 3.03
bidi5 | Lab | 643 | 2.64 | 3.275
bidi6 | Lab | 830 | 3.0 | 3.775

###  3.2 Models and Algorithms

In the subsequent trajectory predictions, we focus on two crucial elements:
predictions across varying pedestrian densities and the utilization of diverse
models/algorithms. We systematically use the notation $x_{i}\in\mathbb{R}^{2}$
and $v_{i}\in\mathbb{R}^{2}$ to represent the position and velocity of the
$i$-th pedestrian. The Euclidean distance is denoted as $|\cdot|$, while $x$
and $v$ refer to the vectors of pedestrian positions and velocities. These
vectors have a dimension of $2N$ for $N$ pedestrians. All variables, including
$x(t)$ and $x_{i}(t)$, depend on the time $t$.

We first select two contemporary knowledge-based models to facilitate this.
The first is the Social Force model (SF) by Helbing and Molnar [14], and the
second is the Optimal Reciprocal Collision Avoidance (ORCA) approach by van
den Berg et al. [45]. The SF model is a widely adopted method that simulates
pedestrian movement, treating individuals as particles influenced by various
forces. The acceleration within this model is calculated from the summation of
three forces as demonstrated in Equation 1

$m_{i}\frac{dv_{i}}{dt}=m_{i}\frac{v^{0}_{i}-v_{i}}{\tau}+\sum_{j\neq i}\nabla
U(x_{j}-x_{i})+\sum_{W}\nabla V(x_{W}-x_{i})$ (1)

where $m_{i}$, $v_{i}$ and $v^{0}_{i}$are the mass, current velocity and
preferred velocity of the $i$-th pedestrian, respectively, while $U$ and $V$
are distance-based interaction potential, e.g.,

$U(d)=Ae^{-\|d\|/B},\qquad A,B>0.$ (2)

The first term of Equation 1 denotes the driving force that the $i$-th
pedestrian experiences to achieve their desired speed and direction within the
reaction time $\tau>0$. The second term represents the summation of the social
forces derived from the repulsive effects of pedestrians maintaining distance
from each other. The third force denotes the cumulative interaction forces
between pedestrian i and obstacles.

In contrast, ORCA is centered on efficiently determining collision-free
velocities for multiple agents within a shared environment [45]. It is based
on a geometric approach to model agent interaction, identifying the range of
velocities that guarantees collision avoidance within a specified time horizon
by extrapolating linearly (i.e., assuming constant the velocity) the
trajectories. These computations result in collision cones between the
pedestrians, that can be empty. Further mechanisms are taken into account to
avoid unrealistic oscillation effects and makes the agents acting
independently without communicating with each other. Ultimately, each agent
subsequently selects the optimal velocity $v_{i}$ that is closest to its ideal
(preferred) velocity $v^{0}_{i}$ within the feasible velocity region [46]
excluding collisions, as shown in Equation 3

$\displaystyle v_{i}(t+dt)=\text{arg}\min_{v\,\in\,\cap_{j\neq
i}\text{ORCA}_{ij}(t)}\|v-v^{0}_{i}\|,$ (3)

with $\text{ORCA}_{ij}$ the set of feasible (collision-free) velocities of the
$i$-th pedestrian with the $j$-th neighboring pedestrian, and $dt$ a (small)
time step (typically equal to $dt=0.01$ s).

For the DL approach, we adopt the LSTM network, which is widely used in
pedestrian trajectory prediction. In this architecture, we leverage two
algorithms. The first, referred to as the Vanilla LSTM, considers the
historical trajectory over $[t-T_{o},t]$, with $T_{o}>0$ the observation time,
to predict the trajectory over $[t,t+T_{p}]$ with $T_{p}$ the prediction time

$x_{i}(t+t_{p})=\text{LSTM}\big{(}t+t_{p},\big{(}x_{i}(t-t_{o}),~{}t_{o}\in\mathbb{[}0,T_{o}]\big{)}\big{)},\qquad\forall
t_{p}\in\mathbb{[}0,T_{p}].$ (4)

This algorithms is grid-based which means that the input is discretised in a
local grid constructed around the pedestrian. The second algorithm, known as
the Social-LSTM [8], incorporates a social pooling mechanism to aggregate
information about neighboring entities within the grid, as illustrated in
Equation (5)

$x_{i}(t+t_{p})=\text{SLSTM}\big{(}i,t+t_{p},\big{(}x(t-t_{o}),~{}t_{o}\in\mathbb{[}0,T_{o}]\big{)}\big{)},\qquad\forall
t_{p}\in\mathbb{[}0,T_{p}].$ (5)

With this mechanism, the model can use the historical trajectories of
surrounding pedestrians $x$ over $[t-T_{o},t]$, enabling the consideration of
interactions in the predictions. Table 2 lists the various types of
information required by each model or algorithm to make predictions. The first
approach in the Table 2 is the constant velocity model. It is the most simple
approach making prediction assuming the pedestrian velocities remain constant

$x_{i}(t+t_{p})=x_{i}(t)+t_{p}v_{i}(t),\qquad\forall
t_{p}\in\mathbb{[}0,T_{p}].$ (6)

###  3.3 Implementation details

The DL algorithms are trained with a learning rate of 0.0015, and a RMS-prop
is used as the ADAM optimizer. The batch size is 8, and we train for 12
epochs. As a loss function, the mean squared error is used. For the validation
and testing, we use a hold-out validation strategy. 15 % of the data is used
for validation, 15 % for testing and the rest for training. The computations
are performed using the PyTorch library111http://pytorch.org. Two different
observation and prediction times are employed in the study. For 1.2-second
predictions, a 1-second observation period is utilized. For 4.8-second
predictions, the observation length extends to 3.6 seconds. We employ the
Stochastic Gradient Descent (SGD) algorithm to fit the parameters of the KB
models to the training data by minimizing the ADE metric. For the SF, we
optimize the preferred velocity, the interaction potential, and the reaction
time, according to [47]. For the ORCA we optimize the distance to pedestrians
that are taken into account and the corresponding reaction time. It is worth
noting that the reaction time plays a significant role in the model’s
behavior. A shorter reaction time prompts a quicker response to the presence
of other agents but reduces the pedestrian’s freedom in choosing their
velocities, as mentioned in [17]. On the other hand, the CV model stands apart
as it does not require any calibration or training due to its parameter-free
nature. An overview of the most strinking differences between the approaches
is presented in Table 2. The displayed information includes the input of
primary pedestrians and their neighbors, as well as the optimization method
and the parameters being optimized.

Table 2: Overview of important features. Approach |  Primary Input |  Neighbor Input |  Optim. Parameters |  Optim. Method
---|---|---|---|---
CV |  Current State |  None |  None |  None
SF |  Current State |  Yes |  Preferred Velocity, Interaction Potential, Reaction Time |  SGD
ORCA |  Current State |  Yes |  Neighbor Distance, Reaction Time |  SGD
Vanilla-LSTM |  Past Trajectory |  None |  Large number of coefficients |  ADAM
Social-LSTM |  Past Trajectory |  Yes |  Large number of coefficients |  ADAM

###  3.4 Evaluation

####  3.4.1 Distance-based metrics

A crucial question that emerges when employing pedestrian trajectory
prediction algorithms in high-density scenarios is the appropriate method of
evaluation. This query is applicable more broadly to the field of pedestrian
dynamics. Without an objective metric for evaluation, it is impossible to
definitively determine the best-fitting model or algorithm. In low-density
trajectory predictions, two metrics based on Euclidean distance are commonly
utilized. The first is the Average Displacement Error (ADE) [9], which
measures the distance between the predicted trajectory and the ground truth
trajectory at a set number of points

$\textbf{ADE}=\frac{1}{NT}\sum_{i=1}^{N}\sum_{t=1}^{T}\|\hat{x}_{i}(t)-x_{i}(t)\|,$
(7)

$x_{i}(t)$ being the actual position of the $i$-th pedestrian at time $t$
while $\hat{x}_{i}(t)$ is the predicted position.

####  3.4.2 Discrete distance-based collision metric

These two distance metrics are effective in guiding the algorithm to make
predictions that match actual trajectories. However, a significant drawback is
their focus on distance, leading to an underestimation of the repulsive forces
that arise between pedestrians. Consequently, these metrics do not account for
potential overlaps or collisions between pedestrians.

To tackle this problem, the following distance-based collision metric has been
presented by Kothari et al. [11]

$\textbf{Col}=\frac{1}{|S|}\displaystyle\sum_{\hat{Y}\in S}Col(\hat{Y}),$ (8)

with

$Col(\hat{Y})=\min\bigg{(}1,\sum_{t=1}^{T}\sum_{i=1}^{N}\sum_{j>i}^{N}\big{[}||\hat{x}_{i}(t)-\hat{x}_{j}(t)||\leq
2R\big{]}\bigg{)},$ (9)

where $S$ includes all scenes in the test set, $\hat{Y}$ represents a scene
prediction containing $N$ agents, and $\hat{y}_{i}$ is the prediction of agent
$i$ over the prediction time of $T$, while $[\cdot]$ is the Iverson bracket

$\big{[}P\big{]}=\left\\{\begin{array}[]{ll}1&\text{if $P$ is true,}\\\
0&\text{otherwise.}\end{array}\right.$ (10)

This last metric counts a prediction as a collision when a predicted
pedestrian trajectory intersects with neighboring trajectories, thus
indicating the proportion of predictions where collisions occur. A vital
factor in this calculation is the chosen pedestrian size (radius), represented
by the variable $R$ in equation (9). An increase in $R$ will likewise increase
the number of collisions.

####  3.4.3 New TTC-based error metric

The collision metric Col discussed so far has been designed to mitigate
overlapping and collisions between pedestrians. This is based on the principle
that pedestrians inherently strive to avoid physical contact with others, an
aspect not sufficiently captured by ADE. Nevertheless, this collision metric
isn’t without its shortcomings, which we aim to address. One drawback is that
the metric is based on binary collision identifications. As such, it doesn’t
distinguish between minor instances of contact, such as a shoulder brush
between two pedestrians walking side-by-side, and significant collisions such
as a head-on crash. Another limitation is its inability to account for
scenarios where a prediction results in multiple collisions. Additionally, the
metric models pedestrians as circles, represented by radius $R$, although an
elliptical representation would be more accurate. The proposed solution is to
introduce a collision metric based on the concept of Time-to-Collision (TTC)
between two pedestrians. In this system, a low TTC implies an impending
collision.

The TTC is estimated as the time remaining before two moving pedestrians,
donated as $i$ and $j$, collide based on their current velocities. Suppose
that $R_{i}$ and $R_{j}$ are the radius of $i$-th and $j$-th pedestrians,
respectively. Given the relative distance and relative velocity between the
two pedestrians

$x_{ij}=x_{i}-x_{j}\qquad\text{and}\qquad v_{ij}=v_{i}-v_{j},$ (11)

a collision between $i$-th and $j$-th pedestrians occurs if there exists a
time $\tau>0$ such that $x_{ij}+v_{ij}\tau$ lies within a circle centered at
$(0,0)$ with radius $2R$. Mathematically, this condition can be expressed as
$\|x_{ij}+v_{ij}t\|<2R$ where $\|\cdot\|$ denotes Euclidean norm. It turns out
to solve the quadratic inequality in $t$, and $\tau$ is the smallest positive
root:

$\tau_{ij}=\frac{\displaystyle-x_{ij}\cdot v_{ij}-\sqrt{(x_{ij}\cdot
v_{ij})^{2}-||v_{ij}||^{2}(||x_{ij}||^{2}-4R^{2})}}{||v_{ij}||^{2}}.$ (12)

In this scheme, if no collision is imminent, then $\tau_{ij}$ is not real-
value or is negative. We set in this case $\tau_{ij}=\infty$. Conversely, we
assume by convention that $\tau_{ij}=0$ if the pedestrians are already in
collision. To be able to compare the performances of different prediction
approaches, the inverse of average TTC (ITTC) is calculation according to

$\displaystyle
ITTC=\frac{NT}{\displaystyle\sum_{i=1}^{N}\sum_{t=1}^{T}\min_{j\neq
i}\big{\\{}\tau_{ij}(t),\tau_{\text{max}}\big{\\}}},$ (13)

with $\tau_{\text{max}}=12$ seconds a maximal TTC threshold value.

##  4 Results

In the upcoming chapter, the objective is to showcase the effectiveness of
different approaches in making predictions across diverse time intervals and
densities. To accomplish this, we will conduct comprehensive evaluations of
the predictions using ADE (Section 4.1), the distance-based collision metric
Col (Section 4.2.1), and the ITTC (Section 4.2.2). Additionally, in Section
4.3, we will demonstrate how the TTC metric can be leveraged to enhance
trajectory predictions.

###  4.1 Distance metric

At first, we will focus on analyzing the distance-error metrics for the low-
density datasets. The outcomes of the predictions are presented in Fig. 3. As
previously mentioned, we utilize five datasets for the low-density
predictions, with densities ranging between 0.13 and 0.38 ped/m2. In Fig. 3,
the x-axis displays the average densities of each dataset, while the y-axis
represents the ADE metric. On the left side a prediction time $T_{p}$ of 1.2
seconds is chosen and on the right side a prediction time of 4.6 seconds.

(a) Low-density data with $T_{p}=1.2$ sec.

(b) Low-density data with $T_{p}=4.8$ sec.

Figure 3: Distance error-metric (ADE) for low-density datasets.

The Fig. 3 initially demonstrates that the algorithms consistently outperform
the models across nearly all low-density datasets, showcasing their superior
predictive capabilities. The Social-LSTM exhibits slightly better performance
compared to the Vanilla-LSTM.
Furthermore, notable differences can be observed between the two prediction
horizons. While the CV approach performs reasonably well at shorter prediction
times, the error increases significantly, nearly doubling compared to the
other approaches, as the prediction time extends. The substantial error of the
CV approach on the ETH dataset highlights its challenging nature, with the
highest deviation from keeping speed and direction constant. Interestingly,
the most complex approach, namely SLSTM, demonstrates the best performance on
this dataset, indicating its effectiveness in handling complexity. Moreover,
it is worth noting that an increase in density does not necessarily result in
higher prediction errors for the low-density dataset. For instance, Zara02,
despite having a higher density than ETH, exhibits a lower average ADE.

In the next step, the same analysis of the different approaches with different
prediction horizons is done for the high-density data. The results are
presented in Fig. 4.

(a) High-density data with $T_{p}=1.2$ sec.

(b) High-density data with $T_{p}=4.8$ sec.

Figure 4: Distance error-metric (ADE) for high-density datasets.

As in Fig. 3, the algorithms consistently outperform the models, irrespective
of the dataset density. However, it is noteworthy that, surprisingly, the
VLSTM exhibits superior performance compared to the more complex SLSTM.
Additionally, it is notable that the error for SLSTM tends to increase with
rising density, whereas for the VLSTM, it is lower within the higher density
range of 2-3 ped/m2. This unexpected outcome suggests that higher complexity
does not necessarily lead to improved results when dealing with high
densities. This assumption gains further support from the observed progression
of the ADE for the CV approach, where the error appears to decrease with
higher densities.

Similar to the results presented in Fig. 3, we observed that the SF model
performs better for longer prediction horizons compared to other approaches.
Conversely, the ORCA model struggles to make accurate predictions at densities
exceeding 2 ped/m2. This limitation is attributed to the ”freezing problem”
highlighted in Luo et al. [48], which becomes prominent at higher densities.
For the longer prediction time, the performance of ORCA declines with higher
density, while the performance of the SF improves with increasing density.

###  4.2 Collision metrics

In this section, we will address an important challenge regarding collisions.
In certain instances, pedestrians fail to avoid one another, contrary to what
one would expect. This leads to overlapping or collision events, contravening
one of the most critical physical criteria that a realistic prediction should
satisfy. To assess the magnitude of this phenomenon, we will employ the two
collision metrics Col and ITTC described in Section 3.

####  4.2.1 Distance-based collision metric

An essential aspect of the collision metrics is accurately defining when
unrealistic behaviors, such as overlapping, occur. Hence, the shape of the
pedestrians plays a crucial role. In the following, we present predictions
made for two distinct radii: 0.1 meters and 0.2 meters. The y-axis represents
the percentage of predictions demonstrating collisions, as determined by the
distance-based collision metric. The black line has not been presented in the
figures before and it displays the percentage of collisions in the real
datasets.

(a) Low-density data, $R=0.1$.

(b) Low-density data, $R=0.2$.

Figure 5: Distance-based collision metric for different radius sizes.

In Fig. 5, it is evident that, as anticipated, the percentage of collisions is
significantly higher for the larger radius. This observation holds true for
all approaches and the real data, with the exception of the ORCA model, which
demonstrates no collisions for either radius. Regarding the real data,
collisions at a radius of 0.1 meters only occur within the Zara2 dataset,
whereas collisions occur in the HOTEL, ETH, Zara2, and UCY datasets for a
radius of 0.2 meters. This finding is surprising because collisions should not
occur in the real trajectories. Collision between pedestrian occur very rarely
in the real world. However, upon animating the real trajectories and plotting
the colliding trajectories, it becomes apparent that the distance-based
collision metric has a drawback: it sometimes considers grouping behavior as
collision behavior. When pedestrians walk in groups, they occasionally come so
close together that even at a radius of 0.1 meters, they sometimes overlap. In
the next Fig. 6 the distance-based collision metric for the high-density data
is presented.

(a) Bidirectional data, $R=0.1$.

(b) Bidirectional data, $R=0.2$.

Figure 6: Distance-based collision metric for different radius sizes.

Substantial disparities are evident between the two diagrams representing
collision occurrences at varying radii. The predicted collision frequency is
nearly double at a radius of 0.2 meters as compared to a smaller radius. The
black line, indicative of actual collision data, suggests a negligible number
of collisions at high-density datasets for a radius of 0.1 meters. However,
when the radius is increased to 0.2 meters, the predictions indicate almost 50
% more collisions in higher densities. These observations suggest that a
radius of 0.2 meters is excessively large for such scenarios. Pedestrians at
these densities are in such close proximity that they often overlap when
represented as circular objects. The ORCA model predicts no collisions for
either radius, but this results in extraordinarily high ADE values for large
prediction horizons, as depicted in Fig.4, right panel. The CV model
underperforms and displays the highest collision rate in its predictions. The
SLSTM shows comparable performance at lower densities as demonstrated in Fig.
6, but its effectiveness diminishes at higher densities. The VLSTM model
exhibits superior performance to the SLSTM in terms of collision metrics. The
percentage of predicted collision of the SF model most closely aligns with
actual trajectories.

####  4.2.2 TTC-based collision metric

In this section, we will discuss the results of the predictions centered
around the ITTC. As observed in the preceding figures, the x-axis represents
the density of the data. On the other hand, the y-axis denotes ITTC of the
predictions (see equation 13). It is important to note that the maximum TTC
value was arbitrarily set to 12 seconds. A higher TTC can be interpreted as an
indication of a prediction that is not at risk of collision. Consequently, a
lower ITTC value is preferable for collision avoidance compared to a higher
one.

(a) Low-density data, $R=0.1$.

(b) Low-density data, $R=0.2$.

Figure 7: TTC collision metric for different radius sizes.

Upon initial observation of Fig. 7, it is evident that the differences between
Fig. 7 (a) and (b) are considerably smaller compared to those seen in the
distance-based collision metric. This observation suggests that the TTC-based
collision metric is less sensitive to changes in the radius.

Despite this difference, the overall trend of the lines bears a resemblance to
that seen in Fig. 5. However, the information provided by the continuous
collision metric is richer and appears to be more accurate. The KB models
exhibit lower inverse ATTC values than the algorithms, with the CV model
demonstrating the least optimal performance. Notably, the inverse ATTC of
ORCA’s prediction most closely aligns with the actual trajectories. Proceeding
further, we will present the ITTC for high-density data in Fig. 8.

(a) High-density data, $R=0.1$.

(b) High-density data, $R=0.2$.

Figure 8: TTC collision metric different radius sizes.

Again, it is noteworthy, that the variations between Fig. 8 (a) and (b) are
not as substantial as those seen in Fig. 6. In the latter virtually no
collisions are observable in the actual trajectories (black line) for a radius
of 0.1 meters, yet the collision rates for a radius of 0.2 meters are markedly
high. In contrast, in Fig. 8, the difference between (a) and (b) for real
trajectories is much less pronounced. Despite this, the trends in Fig. 6 and
Fig. 8 do not exhibit significant discrepancies, with the KB models
consistently surpassing the algorithms and the SLSTM model underperforming in
comparison to the VLSTM. In some instances, SLSTM even demonstrates worse
performance than the CV model.

###  4.3 TTC Metric for improving performance of the algorithm

The preceding section demonstrates the promising attributes of the ITTC
collision metric. It provides reasonable results that have the advantage of
continuity and exhibit less dependence on the pedestrians’ shape compared to
the distance-based collision metric Col. As such, we have incorporated the TTC
into the cost function to see if the prediction can be improved. The cost
function quantifies the disparity between the prediction and the real
observation. It provides a single indicator $L_{i}$ that will be minimized
during the training. The idea is to penalize predictions with exceptionally
low TTC values. Keeping all configurations the same ($T_{p}=4.8$ seconds and
R=0.2), we have integrated the TTC into the cost function of the SLSTM, as
illustrated in equation 14

$\textbf{L}_{i}=\sum_{t=1}^{T}\|x_{i}(t)-\hat{x}_{i}(t)\|^{2}+\lambda\sum_{t=1}^{T}f(\min_{j\neq
i}\\{\tau_{ij}\\}).$ (14)

The first part of the equation 14 shows the distance-based metric ADE that is
used for training. It compares the actual position of the pedestrian $x_{i}$
to the predicted one $\hat{x_{i}}$. The subsequent segment utilizes a sigmoid
penalty function $f$ (see Equation 15), which results in a high penalty for
low TTC values

$f(\tau)=\frac{1}{1+e^{s(\tau-\delta)}},$ (15)

where $s$ and $\delta$ are slope and threshold parameters, respectively.
The parameter $\lambda\geq 0$ determines the weight to be given to the second
part of the cost function. In Fig. 9 the results of the predictions in terms
of ADE and Col are shown for different settings of $\lambda$.

(a) Low-density data.

(b) High-density data.

Figure 9: Illustrating the improvements achieved by incorporating TTC into the
training function for both low-and high-density data.

The green line represents the Col metric, while the blue line shows ADE. In
the first observation, the value of $\lambda$ is zero, which means that this
model is identical to the SLSTM without a TTC term in the cost function. It
can be clearly shown, that the TTC term in the cost function helps to improve
avoidance behavior. There is a strong relationship between the value of
$\lambda$ and the number of collisions in the predictions. In the case of low-
density data, it is possible to halve the number of collisions without an
accompanying increase in ADE. For the high-density data, a reduction in the
Col metric by 20 % is achievable, which also leads to a decrease in ADE. The
enhancements afforded by the incorporation of TTC into our algorithm are
discernible both quantitatively and qualitatively. When visualizing the
predicted trajectories, those produced by the TTC-incorporated model appear
more realistic, exhibiting superior collision avoidance characteristics. Fig.
10 presents two scenes from the low density data. On the left side the
predictions made by the algorithm trained with TTC are shown and on the right
side the predictions with the SLSTM.

(c) Prediction with TTC in cost function.

(d) Prediction with SLSTM.

Figure 10: Example of trajectory predictions based on the algorithm that was
trained with TTC in the cost function and the SLSTM.

In both scenes, the SLSTM’s predictions result in collisions. In the upper
image, pedestrians do not navigate around the stationary individuals in the
middle. Meanwhile, in the lower image, the pedestrians form a group walking so
close to each other that it registers as a collision. In the predictions
generated using TTC, pedestrians within a group maintain a greater distance
from each other. Additionally, in the upper image, pedestrians navigate
successfully around those stationary in the center, further demonstrating the
benefits of incorporating TTC into trajectory predictions.

##  5 Conclusion

In this paper, we have conducted a detailed empirical analysis comparing
various pedestrian trajectory prediction approaches. The investigation
underscores that the task of predicting pedestrian trajectories is
intrinsically complex, with different densities posing additional challenges.
While the SLSTM demonstrates excellent performance in low-density scenarios,
it struggles to maintain similar accuracy in high-density situations. A
particular limitation of the DB algorithms, namely, a significant incidence of
collisions in high-density predictions, is addressed by introducing an
innovative continuous collision metric that calculates the time-to-collision
between pedestrians. This new metric presents a valuable instrument to assess
the performance of the approaches, enhancing the overall trajectory prediction
accuracy realism feature in terms of hardcore body exclusion. Future work will
continue to optimize this metric and further explore its potential in
improving the safety and efficiency of pedestrian movement predictions. This
investigation thus serves as a foundation for refining and expanding the
capabilities of data-based algorithms for predicting and modelling pedestrian
behavior at high densities.

## Acknowledgment

The authors acknowledge the Franco-German research project MADRAS funded in
France by the Agence Nationale de la Recherche (ANR, French National Research
Agency), grant number ANR-20-CE92-0033, and in Germany by the Deutsche
Forschungsgemeinschaft (DFG, German Research Foundation), grant number
446168800.

## Author Contributions

R.K and A.T formulated the conceptualization and methodology. R.K wrote the
manuscript, performed the formal analysis, and visualized the results. R.K and
T.D developed the software and performed the data curation. A.T assumed the
supervision and funding acquisition. All authors reviewed the article.

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