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Article Not peer-reviewed version
Roundabout Trajectory Planning:
Integrating Human Driving Models
for Autonomous Vehicles
Salvatore Leonardi and Natalia Distefano *
Posted Date: 24 October 2023
doi: 10.20944/preprints202310.1441.v1
Keywords: Autonomous vehicles; Human driving models; Roundabouts; Speed profiles; Traffic-free planning
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Article
Roundabout Trajectory Planning: Integrating Human
Driving Models for Autonomous Vehicles
Salvatore Leonardi and Natalia Distefano *
Department of Civil Engineering and Architecture - University of Catania, Viale Andrea Doria, 6 - 95125 Ca-
tania (Italy)
* Correspondence: natalia.distefano@unict.it; Tel.: +390957382226
Abstract: This research investigates the utilization of human driving models in autonomous vehicles, particu-
larly in scenarios with minimal or no interactions with other vehicles. Human driving models provide valuable
insights into driver behavior and play a crucial role in shaping the behavior of autonomous vehicles, enhancing
their performance and user experience. The primary focus of the study is the creation of a planning model for
autonomous vehicles when navigating roundabouts in the absence of traffic. This model seeks to emulate hu-
man driving behavior, ensuring predictability, safety, optimization of traffic flow, and adaptation to various
roundabout geometries. To achieve this, the research introduces a trajectory model that takes into account ge-
ometric attributes and speed variations within roundabouts. The model is calibrated using empirical data and
generalizes parameters through statistical regression methodologies. This model is referred to as “MRoundabout”
and is evaluated for its consistency in generating plans that closely mimic human driving behavior within
roundabouts. While the study presents a promising approach, it acknowledges limitations related to the mod-
el's reliance on geometric attributes and its inability to account for external factors like weather conditions. This
research underscores the importance of bridging the gap between theoretical research and practical applica-
tion, with the aim of enhancing safety and the overall user experience in real-world driving scenarios.
Keywords: autonomous vehicles; human driving models; roundabouts; speed profiles; traffic-free
planning
1. Introduction
Autonomous vehicles can benefit from human driving models in various ways, whether they
need to interact with other vehicles in complex situations or if interactions with other vehicles are
rare or limited [1,2]. Here's how these models can be useful in both cases:
1) Interaction with other vehicles: In contexts where autonomous vehicles have to interact with other
vehicles on the roads, human driving models can provide valuable insights into driver behavior
and driving dynamics. By studying and learning from human driver data, autonomous vehicles
can learn how to behave in complex situations such as standard intersections, roundabouts, or
curves. For example, human driving models can provide information on trajectory choices, ap-
propriate speeds in certain situations, and common driving habits. This information can be used
by autonomous vehicles to make safer and more predictable decisions during interactions with
other vehicles on the road.
2) Limited interaction scenarios: Even when interactions with other vehicles are rare or limited, hu-
man driving models can be useful for autonomous vehicles. For example, in autonomous driving
situations in rural areas or areas with low traffic density, vehicles encounter fewer or no vehicles.
Nevertheless, human driving models can provide information on how to handle certain road
elements with conditional geometry (curves, roundabouts, highway ramps, etc.) or traffic signs
or adverse weather conditions. Additionally, human driving models can be used to provide a
more comfortable and familiar driving experience for passengers. For instance, if a human driver
prefers gradual acceleration or gentle braking in certain situations, the autonomous vehicle can
learn such habits and replicate them to provide a more human-like driving experience.
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2
In this context, speed profiles obtained through naturalistic observations can be extremely useful
for the human driving models employed in the learning phases of autonomous vehicles [3,4]. Here's
why:
a) Real-world data: Naturalistic speed profiles are based on real data collected from vehicles in real
driving conditions. These data represent the actual behavior of human drivers in real-world sit-
uations, allowing the human driving models to learn from authentic experiences. This helps
make the models more accurate and adaptable to various road situations.
b) Contextual variation: Naturalistic speed profiles capture the variation of speed in different driv-
ing situations and contexts. This includes information about average speeds, maximum speeds,
and typical decelerations/accelerations in certain areas or types of roads. Learning from these
variations allows the driving models to guide autonomous vehicles to behave more realistically
and consistently with human drivers in different scenarios, improving safety and efficiency in
the autonomous driving system.
c) Consideration of individual preferences: Naturalistic speed profiles can also reflect individual
driver preferences regarding speed and driving style. These preferences can be learned and
taken into account by the human driving models during the learning process of autonomous
vehicles. This enables autonomous vehicles to adapt to the preferences of human drivers or pas-
sengers, providing a more familiar and personalized driving experience.
d) Performance enhancement: Using naturalistic speed profiles can contribute to overall perfor-
mance improvements in autonomous vehicles. For example, they can be used to fine-tune con-
trol algorithms, improve trajectory planning, or optimize acceleration and deceleration strate-
gies. Integrating real driver data into the human driving models helps autonomous vehicles
learn from realistic driving examples and develop more effective driving strategies.
Specifically with respect to the approach of autonomous vehicles to roundabouts, the collection
of speed profiles obtained through naturalistic surveys may be particularly useful for the following
reasons [5–8]:
• These profiles help autonomous vehicles understand appropriate speeds, acceleration, and de-
celeration required for safe driving.
• Contextualized speed profiles help autonomous vehicles make more informed decisions based
on specific road contexts. For example, they enable speed adjustment based on the presence of
other vehicles in roundabouts or traffic conditions.
• Naturalistic speed profiles allow autonomous vehicles to adapt to different roundabout geome-
tries, and to understand and respect the laws of physics governing the dynamic behavior of
vehicles. This includes managing centripetal force, optimizing tire friction and grip, and main-
taining stability during curves.
• Naturalistic speed profiles can be used for validation and testing of autonomous driving sys-
tems. They allow comparing the behavior of autonomous vehicles with known human speed
profiles to evaluate the effectiveness of the autonomous system and identify any necessary im-
provements.
This study proposes a model for planning the crossing of a single-lane roundabout, based sim-
ultaneously on the specification of the geometric curve and on the generation of the speed plane. The
aim is to achieve a “human-like” planning, i.e. a planning based on the imitation of the human driv-
ing behavior within the limits of a safe driving mode.
It should be noted that the crossing trajectory considered is the fastest trajectory in a roundabout,
i.e., the crossing trajectory without significant interference from other user categories (vehicles, pe-
destrians, bicyclists, etc.). Therefore, traffic-free design results in a crossing trajectory that considers
only the geometric elements that make up a roundabout (diameter, entry radius, exit radius, deflec-
tion angle, lane width) and ignores other road users and other obstacles. Therefore, the crossing tra-
jectory planning model must be able to generate a dynamically feasible trajectory based on aspects of
human driving behavior that does not involve traffic impacts.
It is believed that this type of modeling is useful for at least the following four aspects:
Predictable behavior: The fastest crossing trajectory with no interaction with other road users
represents a predictable and safe behavior model that has been solidified by human driver
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experience. Validation and testing of autonomous vehicles on this route allows them to learn
and adopt behaviors that humans recognize as effective.
Safety: crossing roundabouts quickly and efficiently can contribute to road safety. Modeling au-
tonomous vehicles on the fastest crossing route can verify that the autonomous system can main-
tain an appropriate speed and perform the proper maneuvers to safely traverse the roundabout,
avoiding slowdowns and potential hazards.
Optimizing traffic flow: the correct behavior of autonomous vehicles when passing through sin-
gle-lane roundabouts can help optimize traffic flow. If autonomous vehicles follow the fastest
crossing trajectory without interacting with other road users, they can help reduce roundabout
crossing times and improve traffic flow.
Adaptation to road conditions: Naturalistic speed profiles based on the fastest crossing trajectory
can vary depending on the geometric characteristics of roundabouts. The model to be proposed
allows autonomous vehicles to adapt to different roundabout configurations, such as diameter,
entry radius, exit radius, deflection angle, and lane width. In this way, autonomous vehicles can
learn the appropriate behavior and be able to negotiate single-lane roundabouts safely and effi-
ciently, regardless of the specific geometric specifications.
In order to attain the aforementioned objective, this work has been structured into the subse-
quent sections: Section 2 outlines the review of relevant literature. Section 3 introduces suitable path
and speed models aimed at capturing human driving behavior. Section 4 elaborates on the calibration
process of the speed model. Section 5 details the experimental investigations. Sections 6 and 7 are
dedicated to presenting the outcomes of the learning process, summarizing the study, and offering
recommendations for future research.
The framework of the proposed analysis method is shown in Figure 1.
Figure 1. Framework of the proposed analysis.
2. Literature review
In recent decades, autonomous vehicle trajectory planning efforts have generally focused on first
finding an optimal and safe spatial maneuver and then using rule-based speed assignment to form a
target trajectory that is evaluated for collision avoidance [9,10].
Some studies have investigated how vehicle movement in different road scenarios is also deter-
mined by road geometry, i.e., curvy roads force the driver to slow down the vehicle to reduce the
driving discomfort caused by increasing lateral accelerations. In one study, a geometry-based speed
planning approach was proposed by generating a reference path for the autonomous vehicle by com-
bining the smooth and peak-reduced curve and a parameterized speed model fitted from human
driving data [11].
In addition, a combined behavior planning and trajectory planning method was proposed in
which the sampled trajectories were grouped according to topological properties in the spatial-tem-
poral domain to generate different high-level maneuver patterns [12].
For the case of road intersections, a temporal speed planning approach was developed using the
behavior patterns learned from human drivers in a simulator experiment. The speed profiles from an
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intersection scenario were extracted to develop temporal behavior plans using the k-means clustering
technique [13].
Other studies have attempted to understand human driving behavior and strategies to support
autonomous driving decision making in complex traffic scenarios. Since experienced human drivers
have shown that they can adapt their longitudinal speed behaviors and strategies to effectively man-
age complex traffic scenarios, human-inspired longitudinal speed control is seen as a promising av-
enue for autonomous vehicle application. This approach has two advantages: First, it makes autono-
mous driving more natural [14], which helps it fit seamlessly into environments with other semi-
autonomous and human-controlled vehicles, and second, it will improve the driving experience, es-
pecially in scenarios that tend to involve stop-start movements or roads with severe curvature [15].
In one such study, a risk-aware decision-making approach was used to select human-like longitudi-
nal behavior profiles for navigation in a roundabout scenario. First, the speed profiles were generated
using patterns learned from human driving behavior, and then they were adapted to the dynamic
characteristics of the scenario. There are two new contributions in this work, firstly, the naturalistic
profile generation for human-like navigation and secondly, the risk-aware multi-criteria decision-
making approach that considers driving comfort and performance in addition to driving safety. The
performance of the proposed solution was compared with human driving data from experimental
studies, which showed encouraging benefits [16].
Developing an advanced driver assistance system also means learning from human behavior to
increase driving safety [17–19]. Entering roundabouts smoothly is a challenge even for human drivers
[20–25]. Several studies have proposed different approaches for defining a new decision models
based on imitation learning to provide recommendations for entering a roundabout.
A study shows that an Adaptive Tactical Behavior Planner (ATBP) for an autonomous vehicle is
able to plan human-like movement behavior for navigating a roundabout by combining naturalistic
behavior planning and tactical decision algorithms [26].
Another study presents a multi-grid image processing approach based on multiple cameras at
roundabouts that include different grid sizes to increase accuracy and protect the autonomous vehi-
cle when entering a roundabout. By using multiple cameras, the system can mimic the vision and
perception of the real driver when approaching the roundabout so that a human-like decision can be
made [27].
In another work, a strategy is presented to generate different speed profiles for a set of path
candidates to achieve a merging maneuver on roundabouts according to the current traffic situation.
The autonomous driving system proposed in this work was tested under real-world conditions, and
the results showed that the automated vehicle was able to enter roundabouts with narrow gaps while
maintaining both comfort and safety [28].
Based on travel data, a study proposes numerical optimization to minimize travel time and com-
fort through motion planning and speed profiling. The driving risks in roundabouts were also ana-
lyzed to apply the driving behavior for autonomous vehicles, which can improve the driving comfort
and road safety [29]. In addition, a machine learning model was used and trained to determine the
safe movement and possible exits of vehicles, and an optimal control method was developed to min-
imize travel time and increase energy efficiency, taking into account the constraints of collision avoid-
ance when crossing roundabouts [30].
To coordinate autonomous vehicles in roundabouts, several researchers have developed control
strategies that incorporate artificial intelligence (AI) approaches and models to ensure safe traffic.
Support vector machine, linear regression, and deep learning algorithms have been compared in pre-
dicting vehicle speed and steering angle in roundabouts with different geometry for drivers, and
action rules for autonomous vehicles to perform maneuvers in roundabouts have been developed
[31]. Vehicle motion prediction algorithms combining dynamic Bayesian networks and sequential
neural network models are also used in another research [32]. In addition, the adversarial multiagent
reinforcement learning method is applied to coordinate the passage of autonomous vehicles through
roundabouts by considering behaviors such as those of human drivers [33]. This method improves
the travel time and average speed of vehicles. A fuzzy behavior-based roundabout coordination
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algorithm has also been developed to calculate speed profiles for different vehicles to achieve more
comfortable driving profiles and reduce congestion [34].
There are several research works that use Model Predictive Control (MPC) strategies based on
analytical calculation of travel time and speed profile design, which has been shown to be efficient
for safe driving of autonomous vehicles in roundabouts. Various constraints such as speed regula-
tions, acceleration limits, and maximum curve speeds are incorporated into the control design to en-
sure safe operation [35]. In one study, a method to solve the roundabout merging problem was pro-
posed by considering a target trajectory generated by Bezier curves in combination with the MPC
method [36]. In another study, a controller for roundabout trajectory tracking was presented. Given
the choice of exits, the MPC tracking controller is used to test the effects of weight parameters and
target speed on tracking controller performance [37].
3. Model design
In this section, models are proposed that adapt to human maneuver data and efficiently generate
both the geometric curve for crossing a single-lane roundabouts and the speed plan. In order to effi-
ciently achieve these two objectives, the path and speed models are developed and explained inde-
pendently.
3.1. Path model
The autonomous vehicle must always be able to detect the presence of the roundabout on the
road using sensors such as cameras, LiDAR, or radar. This allows the system to detect the position
and shape of the roundabout to adapt to the predefined path. The following steps must be followed
to model the path:
1) A-priori definition of the path: It is necessary to define in advance the path that the autonomous
vehicle must follow to cross the roundabout. The path can be mapped based on appropriate
assumptions, e.g., assuming that the vehicle must follow a curved path within the lane, main-
taining a constant distance from the inner edges of the circulatory roadway. To achieve this goal,
this study uses the fastest path for crossing a single-lane roundabout, as described in accordance
with guidance in NCHRP Report 672 [38].
2) Waypoints generation: The predefined path can be represented by a series of control points
(waypoints) that indicate the ideal position of the vehicle along the path. These waypoints can
be generated manually or by algorithms that take into account the geometry of the roundabout
and the formulated assumptions. The waypoints should be arranged to ensure smooth and safe
navigation through the roundabout.
3) Least squares optimization: Using the Levenberg-Marquardt algorithm, it is possible to opti-
mize the through path based on the least squares formula. In this case, the objective is to mini-
mize the difference between the desired path (represented by the waypoints) and the actual path
of the autonomous vehicle. The Levenberg-Marquardt algorithm iteratively updates the model
parameters to approach the optimal solution.
Operationally, the three steps described above can be translated into the following procedural
process:
Path definition: The path for crossing the roundabout is determined following the NCHRP
model of path [38]. This model is based on the so-called “fastest through path” allowed by the
geometry, which determines the negotiation speed for the respective movement into, through
and out of the roundabout. It is the smoothest, flattest path possible for a single vehicle, in the
absence of other traffic and ignoring all lane markings. The fastest path is drawn for a vehicle
traversing through the entry, around the central island, and out the relevant exit. Figure 2 illus-
trates the construction of the fastest vehicle path at a single-lane roundabout. The fastest path
for the through movement is a series of reverse curves (i.e., a curve to the right followed by a
curve to the left followed by a curve to the right). When drawing the path, a tangent should be
drawn between consecutive curves to account for the time it takes for a driver to turn the steering
wheel. In particular, there are 3 radii that fully define the fastest through path that are defined
as “critical radii”. R1, the entry path radius, is the minimum radius on the fastest through path
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prior to the entrance line. R2, the circulating path radius, is the minimum radius on the fastest
through path around the central island. R3, the exit path radius, is the minimum radius on the
fastest through path into the exit. In the case of a vehicle, it is considered to have a width of 2
meters and to maintain a minimum clearance of 0.5 meters from the centerline of the roadway
or a concrete curb, while also aligning itself with a painted edge line. Consequently, the center-
line of the vehicle's path is marked with specific distances from various geometric features as
follows:
• 1.0 m from a painted edge line,
• 1.5 m from a concrete curb,
• 1.5 m from a roadway centerline.
Figure 2. Construction of the fastest through path according to the NCHRP model. In this example: a
= 1.0 m, b = 1.5 m, c = 1.5 m.
Parametric representation of the path: To describe the path through the roundabout, a para-
metric representation is used (Figure 3). In polar coordinates, where r is the radial distance from
the center of the roundabout and θ is the angle relative to the horizontal axis, the path is ex-
pressed as:
r(θ)= r0+a⋅θ2
Here, r0 represents the initial distance from the roundabout, a governs the curvature of the
path, and θ varies from the entry angle (θentry) to the exit angle (θexit).
Figure 3. Parametric representation of the fastest through path.
Least squares formulation: The objective is to minimize the error between the calculated path
(pcalculated) and the desired path (pdesired). The objective function is the sum of squared differences:
Objective = p(
θ
)−p(
θ
)
Where N is the number of sampled points along the through path.
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Levenberg-Marquardt Algorithm: The Levenberg-Marquardt algorithm is employed to mini-
mize the objective. During iteration k, the model parameters (r0 and a) are updated using the
Jacobian matrix of partial gradients. The updates are given by:
Δr
(
)=J
()∙J
()+
λ
() ∙diagJ
()∙J
() ∙J
()∙Δr
Δa() =
J
()∙J
()+
λ
() ∙diagJ
(
)∙J
()
∙J
()∙Δa
Here, J(k)calculated is the Jacobian matrix in iteration k, λ(k) is the regularization parameter, and Δr
and Δa are the parameter changes in iteration k.
Iteration and convergence: The Levenberg-Marquardt algorithm continues iteratively until ac-
ceptable convergence is achieved or a maximum number of iterations is reached. In each itera-
tion, parameters are updated according to the algorithm's formulas, and the objective is gradu-
ally reduced.
Path Model output: After completing the optimization with the Levenberg-Marquardt algo-
rithm, optimal values of parameters r0 and a which define the fastest through path of the round-
about are obtained. These parameters constitute the optimized path for the autonomous vehicle.
The final output of the path model consists of these optimal values, enabling the vehicle to safely
and efficiently cross the roundabout while adhering to the NCHRP model of path [38].
3.2. Speed model
The uniform reference path generated for crossing a single-lane roundabout represents the main
input for generating the specific speed model for that path.
A two-stage speed model is designed to reflect human driving behavior. In the first stage, infor-
mation about the geometry of the reference path is used to create basic speed profile typical of the
through maneuver in the case of a single-lane roundabout.
In particular, the most appropriate speed profile in response to the geometry of the path must
take into account the sequence of 3 critical radii that characterizes the path itself (Figure 2). In these
cases, in order to model the driving behavior, it is necessary to take into account that the decelerations
at entry, due to the effect of the critical radius R1, occur from a distance from the entry that varies
according to the caution adopted by the human drivers, and that the accelerations required to ap-
proach the circulation radius (R2) and the exit radius (R3) are also very different according to the
behavior of the drivers when approaching these maneuvers.
To model this guidance pattern, the various path features are first marked to position the speed
profile by referring to the NCHRP template described in section 3.1. In particular, the proposed
MRoundabout speed model is based on the speed profile shown in the lower part of Figure 4, whose
construction requires the following steps:
1) the entire path is divided into 3 main “Turning Regions”: TR1 starts at the point before entering
the roundabout, from which the vehicle decelerates, and ends in the middle of the section passed
under acceleration, between the first and second circular arcs of the path; TR2 starts at the end
of the previous region and ends in the middle of the section that is passed under acceleration,
between the second and third circular arcs of the path; TR3 starts at the end of the previous region
and ends at the point where the user varies his speed (accelerates) after passing the last section
with constant curvature of the crossing path;
2) for each of the “Turning Regions” the longitudinal distance Lsi is defined (i = 1,2,3) between the
starting point of the region and the point where the speed value si is reached (with i =1, 2,3),
which on average remains constant along the maneuver radius within the region itself;
3) for each of the “Turning Regions”, the longitudinal distance ∆Si (i = 1,2,3) is defined between the
central point of the region and the point where the travel starts at constant speed si (i = 1, 2,3).
Thus: ∆Si = TRi/2 - LSi;
4) in addition:
• s0 = characteristic speed of the road before and after the roundabout. It could also be indi-
cated by vertical signs, and for a particular design there may be other relationships. For
example, a curve before the entrance (with radius R0) can determine the speed that can be
reached at the entrance. An entry coming from a parking lot may have a much lower speed
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than an entry coming from a high-speed rural road, even with the same entry geometry
[38]. Therefore, the speed s0 may be a constraint that the autonomous vehicle must learn;
• d01 = deceleration from speed s0 to speed s1 typical of the circumference of radius R1 within
region TR1;
• a12 = acceleration from speed s1 to speed s2 typical of the circumference of radius R2 within
region TR2;
• a23 = acceleration from speed s2 to speed s3 typical of the circumference of radius R3 within
region TR3.
Finally, the formulation of the velocity profile {si} proposed to describe the crossing trajectory of
a single-lane roundabout in parametric form:
s=M
(r(
θ
),𝐐)
Where:
• {r(θ)} describes the entire through path, and
• Q =[s0, s1, s2, s3, d01, a12, a23, ∆S1, ∆S2, ∆S3]T defines the shape of the speed profile.
Figure 4. Speed model (MRoundabout) to describe the speed profile during the crossing maneuver of a
single-lane roundabout.
Note that the proposed speed model (MRoundabout) potentially yields speed profiles with non-
smooth (high value of longitudinal jerk) transition points connecting linear segments. To improve
speed smoothing, the second step iteratively constrains the numerically estimated jerk until the max-
imum jerk value falls below a specific threshold (jerklon).
|s
|≤jerk
4. Model calibration
The presented model necessitates the calibration of numerous parameters using data obtained
from human driving. Specifically, the path model is tasked with optimizing two criteria associated
with the geometric attributes of the path through a roundabout: smoothness and the seamless con-
nection between the three successive turning regions within the fastest crossing path. It is assumed
that a typical human driving pattern inherently seeks to optimize these two aspects. Consequently,
the adjustment of the path model (followed by subsequent learning) becomes unnecessary in such
circumstances. As a result, this section is primarily dedicated to ascertaining the parameters for the
MRoundabout speed model.
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The parameters for the MRoundabout model, denoted as Q, can be determined through an optimiza-
tion process aimed at minimizing the least square error between the MRoundabout model and the human
driving data: 𝐐^ = argmin
𝐐
s
−M
(r(
θ
),𝐐)
In this equation, {sihuman} represents the dataset comprising human driving speeds along the fast-
est through path. The parameter si (where i = 1, 2, 3) in Q can be readily derived by examining the
human driving data for fitting purposes. The previous equation is then employed to determine the
remaining six parameters in Q.
The “argmin” function used in the equation signifies the argument that minimizes the subse-
quent expression. In this context, it identifies the set of parameters Q (denoted as Q^) that results in
the smallest least square error when comparing the MRoundabout model's predictions with the observed
human driving data. This optimization process effectively tunes the parameters of the MRoundabout
model to align it as closely as possible with real-world human driving behavior along the specified
path.
In Figure 5, an illustrative through maneuver in a roundabout, utilizing the aforementioned op-
timization routine, is presented. The human driving data concerning the roundabout crossing ma-
neuver illustrated in the figure were obtained from the dataset collected as a result of the experi-
mental study described in Section 5 of this manuscript. It is evident that the profiles generated by the
model align quite closely with the actual speed data collected. Notably, the incorporation of jerk
smoothing further enhances the precision and faithfulness of this alignment, as evidenced by the
distinctive blue curve. This underscores the effective adaptation of the proposed speed model (MRound-
about) to human driving behavior, even though it cannot fully replicate it.
Figure 5. Example of fitting a speed profile by the proposed model (MRoundabout).
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5. Experimental investigation
To substantiate the claims made in this paper, an experimental investigation was conducted. The
purpose of this investigation is twofold: 1) to explain the results of parameter learning for the pro-
posed speed model (MRoundabout); 2) to evaluate the MRoundabout by comparing the results of the model
with actual human driving data.
5.1. Selection of roundabouts
The roundabouts of the experimental investigation are located in Italy in the district of San Gio-
vanni Galermo (northwest of Catania) and in the municipality of Mascalucia (which is part of the
metropolitan city of Catania and about 12 km from it). In the area of S. Giovanni Galermo are located
3 of the 5 roundabouts (respectively named “Roundabout n.1”, “Roundabout n.2” and “Roundabout
n.3”). These roundabouts are arranged along the same route in a rather homogeneous territorial con-
text. The other two roundabouts (“Roundabout n.4” and “Roundabout n.5”), located in the munici-
pality of Mascalucia, are also arranged one after the other and are located along the Provincial Road
10 (SP10), the so-called “Via Alcide de Gasperi”, which is used by many inhabitants of the neighbor-
ing municipalities.
The geometrical characteristics of the roundabouts are shown in Table 1. In detail, the following
parameters are given: Number of legs, diameter, width of the circulatory roadway. In addition, the
widths of the entry leg and exit leg in Table 1 refer to the legs of interest for the crossing paths ana-
lyzed in this study.
Figure 6 shows the aerial photographs of the five roundabouts subject of the experimental in-
vestigation.
Table 1. Geometrical characteristics of the roundabouts subject of the experimental investigation.
Roundabout Through Path Number
of legs
Diameter
(m)
Circulatory
roadway
width (m)
Entry
width
(m)
Exit
width
(m)
1 Leg A – Leg B 3 40 6.70 3.70 4.10
2 Leg C – Leg A 3 30 5.50 3.40 3.70
3 Leg B – Leg D 4 40 7.50
3.70 3.90
Leg D – Leg B 3.60 4.50
4 Leg B – Leg D 4 35/33 7.00
4.30 4.70
Leg D – Leg B 4.30 4.70
5 Leg B – Leg D 4 35 8.00
4.20 5.10
Leg D – Leg B 4.40 6.60
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Figure 6. Aerial photographs of the 5 roundabouts subject of the experimental investigation.
5.2. Data collection
The experimental study on crossing maneuvers in roundabouts was conducted with a sample
of 15 drivers between the ages of 23 and 62 (7 men and 8 women).
The drivers were recruited from the University of Catania. In particular, an advertisement was
published on the website of the Department of Civil Engineering and Architecture, which included
information about the study and a questionnaire to recruit the participants. Participants were selected
from all those who responded to the advertisement. Drivers were required to be between the ages of
twenty-one and sixty-five and to have held a driver's license for at least three years.
Participants gave their informed consent to participate in the experiment. They were informed
that all data collected would be kept confidential and used for research purposes only. Participants
were also informed that their driving skills would not be assessed and that the only purpose of the
study was to collect data on paths traveled (trajectories and speed). The study was conducted in ac-
cordance with the Declaration of Helsinki and the protocol was approved by the DISS - Center for
Road Safety of the University of Parma (decision of the Steering Committee—prot. 211117/2021 of
February 24, 2021).
The 15 test drivers performed the planned routes in summer and at low-traffic times, i.e. in the
time windows between 10:00 and 11:00 a.m. and between 3:00 and 4:00 p.m., so that the conditions
for performing the maneuvers were always little affected by interactions with other vehicles. Survey
data (position, direction and curvature of the path, longitudinal speed and accelerations) were col-
lected using a tracking system based on the differential GPS placed in the center of the rear axle of
the vehicle and used from time to time by the driver involved in the test. It should be noted that each
driver performed the test using their own vehicle so that their behavior was as natural as possible.
The surveys, conducted in the time windows indicated above, spanned several days until the com-
plete database of all 15 drivers was available. Each driver performed all the crossing maneuvers
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indicated in Table 2. In cases where the maneuvers were affected by other vehicles or external events,
drivers were asked to repeat the maneuver. A total of 9 trajectories per driver were validly recorded,
for a total of 135. In detail:
each test driver performed all 8 planned crossing maneuvers. The resulting 120 trajectories were
considered for the parameter learning phase of the model (see Section 5.3);
each test driver was asked to repeat one maneuver from those already performed. In this way,
the parameters for an additional 15 trajectories were acquired to be used for the evaluation phase
of the model (see Section 5.4). These additional maneuvers are adequately specified in Table 2.
Table 2. Main data from the experimental investigation.
Roundabout Through path Number of trajectories acquired
For learning For evaluation
1 Leg A – Leg B 15 1
2 Leg C – Leg A 15 2
3 Leg B – Leg D 15 2
Leg D – Leg B 15 1
4 Leg B – Leg D 15 3
Leg D – Leg B 15 1
5 Leg B – Leg D 15 2
Leg D – Leg B 15 3
Total 120 15
6. Results and discussions
6.1. Learning results and discussion
The proposed speed model incorporates parameters such as s1, s2, s3, d01, a12, a23, ∆S1, ∆S2, ∆S3, all
contingent upon the reference path that underpins the model, as elucidated in the paper's initial sec-
tion. The objective of this study is to discern and unveil the fundamental attributes intrinsic to the
human driving model. This facilitates the adjustment of the theoretical speed model, referred to as
MRoundabout, to accommodate the fluctuations in human behavior observed during navigation through
the three turning regions delineated within the path model.
Consequently, it becomes imperative to identify the nine parameters within the MRoundabout speed
model that can be tailored to align with the driving conditions that result from the approaches taken
by human drivers when maneuvering through various crossing paths. To achieve this, a statistical
regression methodology was employed, wherein the nine parameters for model adaptation, denoted
by an asterisk, were statistically derived using data from the experimental study outlined in prior
sections.
Symbolically, this can be represented as:
Q* = [s*1, s*2, s*3, d*01, a*12, a*23, ∆*S1, ∆*S2, ∆*S3]T
From an operational point of view, the learning took place as follows:
1) for each of the 120 crossing paths experimentally performed by the test drivers, the three char-
acteristic radii of the path (R1, R2 e R3) that characterize the curvature of the three turning regions
(TR1, TR2 e TR3) were evaluated, again as a function of the geometric characteristics of the round-
abouts;
2) with respect to each of the trajectories obtained through the experimental study, the average
speed values (s1, s2, s3) corresponding to the radii that make up the turning regions were also
obtained;
3) for each trajectory, the average values of the deceleration (d01) characteristic of the first turning
region and of the accelerations (a12 and a13) for the two following turning regions were calculated;
4) the distances ∆S1, ∆S2 and ∆S3 were also determined starting from the data of the trajectories ob-
tained experimentally, after determining for each trajectory the extent of each turning region
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and the transition points between the sections covered with deceleration/acceleration and those
covered with constant speed;
5) the nine parameters of the speed model evaluated experimentally for each of the three curve
regions were plotted in different scatter plots as a function of the values of the characteristic radii;
6) linear regressions were used to learn the correlations between the parameters of interest and,
consequently, to explain the formulations describing the variability of the statistically learned
parameters.
Figure 7 shows, for each of the 3 turning regions, the scatter plots on the 9 parameters assessed
after the experimental tests and the corresponding regression lines with the associated R2 determina-
tion coefficients.
Figure 7. Scattering plot of the speed model MRoundabout parameters. From subfigure (a) to (i), blue sym-
bols are the scattered parameter values after model fitting. Black lines are the results after linear re-
gression.
Below are the various representative formulations of the linear regressions obtained for the sta-
tistically learned parameters: s
∗(R)= 2.619 ∙R− 40.65
s
∗(R)= 1.202 ∙R− 7.488
s
∗(R)= 0.824 ∙R+ 3.586
d
∗(R)= −0.0928 ∙R+ 3.227
a
∗(R)= 0.0386 ∙R− 0.571
a
∗(R)= 0.0299 ∙R− 0.596
∆
∗(R)= −0.1193 ∙R+ 10.284
Δ
∗(R)= −0.1059 ∙R+15.882
Δ
∗(R)= −0.0968 ∙R+20.371
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The analyzed parameters showed correlations ranging from very strong to very weak.
The closer their coefficients of determination (R2) are to 1, the stronger linearity can be con-
cluded. Upon examination of scatter plots for each parameter (Figure 7), certain discernible human
driving patterns emerge, which can be statistically elucidated:
Strong Linearity in s1, s2, and s3: The analysis shows a strong linear correlation between the
speeds s1, s2, and s3 and the radii of curvature characterizing the three turning regions. In other
words, as logically expected, as the radius of curvature decreases, the speed of execution of the
maneuver also decreases [23–25]. Specifically, it is observed that the highest degree of linearity
is evident at speed s1 when the radius values R1 fall within the range of 25 meters to 28 meters
(with a variation of ΔR1 = 3 meters). This corresponds to a speed change ranging from 23 km/h
to 40 km/h (with a variation of ΔS1 = 17 km/h). With respect to speed s2, the strongest linearity
occurs for values of R2 between 34 m and 44 m (∆R2 = 10 m), where the speed varies between 27
km/h and 46 km/h (∆S2 = 19 km/h). When examining speed s3, it becomes evident that no pro-
nounced linearity exists within the spectrum of radius values R3 (with a total variation of ΔR3
equivalent to 21 meters). As a result, the speed variance in the third turning region spans the
complete range between 25 km/h and 51 km/h (with a total variation of ΔS3 equal to 26 km/h).
Thus, it is noteworthy that the ΔS and ΔR intervals, which are linked to the growing significance
of the correlations between radii and speed, exhibit substantial expansion as the final turning
region is approached. This means that road users feel more influenced on the first approach and
therefore adopt a more cautious behavior. Conversely, drivers approaching the last part of the
trajectory (the exit part) feel less constrained and therefore tend to adopt variable speeds within
a very wide range.
Low Linearity in d01, a12, and a23: For the MRoundabout model, there is a low level of linearity ob-
served in the parameters d01, a12, and a23. Moreover, the values of the coefficients of determina-
tion R2 for all three correlations are almost identical in the scattering diagrams (d, e, and f) shown
in Figure 7. A closer analysis of the three diagrams also shows that in none of the diagrams is
there such a density of data that highlights a stronger linearity in one part of the diagram com-
pared to other parts. This result suggests that accelerations/decelerations can vary significantly
and may be influenced by factors that are hard to quantify, such as the driver's mood. This find-
ing aligns with the argument presented in [39].
Weak Linear Correlation in ∆s1, ∆s2, and ∆s3: The analysis shows a weak linear correlation between
∆s1, ∆s2, ∆s3, and their respective reference radii. This indicates that as the radius of curvature of
the crossing trajectory decreases drivers tend to behave more cautiously by slowing down ear-
lier. Essentially, when faced with smaller radii, human drivers tend to exhibit more conservative
driving behavior. This also means that drivers are less predictable when navigating through
turning regions with small radii. These considerations are stronger in the case of turning ma-
neuvers that take place in the first region. Indeed, diagram 7-g shows a lower dispersion of the
data and a higher R2 coefficient compared to the behavioral situations described in diagrams 7-
h and 7-i. This confirms that all the correlations found describe more accurately the human be-
havior in the first approach phase to roundabouts, i.e. in the entry phase.
6.2. Speed model evaluation and discussion
The evaluation of the learned model, known as MRoundabout, involves the generation of speed plans
and their comparison with the behavior observed in 15 human driving processes learned during the
experimental study phase (as described in section 5.2, the test drivers were used for further trajectory
measurements, in addition to the 120 whose results were used for learning the parameters). Parame-
ter Q, representing this model, is specifically tailored to capture the characteristics of navigation
through roundabouts. The objective of this evaluation is to gauge the model's ability to replicate hu-
man driving behavior.
In Figure 8, the outcomes of the comparison between the speed plans generated by the MRoundabout
model and the driving data of test drivers No. 3, No. 10, No. 13, and No. 14 are presented. The re-
markable alignment between the speed profiles generated by the model and the actual profiles is
readily apparent. Notably, for drivers 3 and 10, the congruence between the speed profiles derived
from the model and the real counterparts is nearly flawless. Conversely, for drivers No. 13 and No.
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14, there are more substantial deviations between the two curves, even if the degree of overlap is still
very considerable.
This study highlights an important aspect: despite the difficulties of achieving a perfect one-to-
one match with individual human data points, traffic-free reference plans are successfully generated.
These reference plans are designed to capture the essential characteristics of human behavior
during roundabout crossing maneuvers, albeit in a statistical sense. In other words, while the gener-
ated plans may not precisely replicate any specific human instance, they encompass the core features
commonly observed in such situations.
Furthermore, the use of the learned model MRoundabout offers significant advantages in the context
of autonomous vehicle planning. By incorporating statistical representations of human-like behavior
into the planning algorithms, the resulting plans are characterized by fluidity and predictability. This
means that autonomous vehicles can navigate roundabout crossings in a manner that not only aligns
with common human patterns but also ensures a smooth and predictable driving experience.
In summary, while the generated plans may not achieve perfect matches with random human
data, the study demonstrates the ability to capture essential human driving characteristics statisti-
cally. This contributes to the development of autonomous vehicle planning systems that prioritize
smooth and predictable driving maneuvers during roundabout crossings, ultimately enhancing both
safety and the overall driving experience for passengers and other road users.
Figure 8. Learned speed model evaluation results. In the subfigures from (a) to (d), some emblematic
examples are shown for the comparison between four generated speed plans and as many human
driving processes.
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7. Conclusions
This research is an exploration into enhancing the user experience within the domain of auton-
omous driving. The primary objective is the development of a planning model that operates without
the influence of traffic, aiming to mimic human driving behavior, especially when navigating round-
abouts.
The central focus of this research primarily revolves around traffic-free planning, which entails
a deep understanding of how individuals navigate roundabouts in isolation from other vehicles. To
accomplish this, a trajectory model has been introduced as a solution to accurately represent the path
that a vehicle takes within a roundabout, taking into account variations in speed. The process in-
volves the identification of the various parameters governing this trajectory model. This undertaking
is structured as a least square optimization problem, aiming to derive parameter values that align
optimally with empirical data observations. Subsequently, statistical regression methodologies are
employed to generalize these parameters. This stage serves to define the speed model (referred to as
MRoundabout) and assesses the sensitivity of each parameter to the inherent unpredictability that char-
acterizes human driving behavior. The model's effectiveness is then assessed by subjecting it to a new
dataset, with the goal of gauging its consistency in generating plans that closely emulate human driv-
ing behavior within roundabouts.
However, it is of paramount importance to acknowledge a substantial limitation inherent in this
study. The model's reference speed is solely determined by the geometric attributes of the rounda-
bout and does not account for external factors, such as weather conditions or the state of the road
surface, which can significantly impact human driving behavior. As a result, the model's adaptability
to real-world driving scenarios may be subject to constraints.
In terms of future research directions, the authors intend to outline several procedural and op-
erational initiatives. Foremost among these is a clear intention to transition from a theoretical research
concept to practical implementation. This entails the integration of the model into the planning sys-
tem of an operational autonomous vehicle to contribute to tangible advancements in autonomous
driving technology. Additionally, there is a recognized need to explore the development of planning
models that take into account the presence of traffic and are designed to seamlessly adapt to dynamic
environments where other vehicles are concurrently in operation. This adaptability is particularly
crucial in scenarios where effective vehicular interactions and decision-making are fundamental as-
pects of a comprehensive and adaptable autonomous driving system.
These proposed intentions underscore the significance of bridging the divide between research
endeavors and practical implementation. They emphasize the urgency of addressing the intricate
complexities that characterize real-world driving scenarios, characterized by varying traffic dynam-
ics and dynamic environments. The pursuit of these intentions aims to advance the ultimate objective
of creating a resilient and versatile autonomous driving system that enhances safety and enriches the
overall user experience.
Author Contributions: Conceptualization, N.D. and S.L.; methodology, N.D. and S.L.; software, N.D. and S.L.;
validation, S.L.; formal analysis, N.D.; investigation, N.D. and S.L.; resources, N.D. and S.L.; data curation, S.L.;
writing—original draft preparation, N.D. and S.L.; writing—review and editing, N.D. and S.L.; visualization,
N.D.; supervision, S.L. All authors have read and agreed to the published version of the manuscript.
Funding: This research received no external funding.
Institutional Review Board Statement: The study was conducted according to the guidelines of the Declaration
of Helsinki, and the protocol was approved by the DISS-Center for Road Safety of the University of Parma (De-
liberation of the Steering Committee—prot. 211117/2021 of February 24, 2021).
Informed Consent Statement: Informed consent was obtained from all subjects involved in the study. Written
informed consent has been obtained from the patient(s) to publish this paper.
Data Availability Statement: The data presented in this study are available on request from the corresponding
author. The data are not publicly available due to confidentiality issues.
Conflicts of Interest: The authors declare no conflict of interest.
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17
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