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IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS 1
TriPField: A 3D Potential Field Model and Its
Applications to Local Path Planning
of Autonomous Vehicles
Yuxiong Ji ,LantaoNi , Cong Zhao , Cailin Lei, Yuchuan Du ,Member, IEEE,
and Wenshuo Wang ,Senior Member, IEEE
Abstract— Potential fields have been integrated with local
path-planning algorithms for autonomous vehicles (AVs) to tackle
challenging scenarios with dense and dynamic obstacles. Most
existing potential fields are isotropic without considering the
traffic agent’s geometric shape and could cause failures due
to local minima. We propose a three-dimensional potential field
(TriPField) model to overcome this drawback by integrating an
ellipsoid potential field with a Gaussian velocity field (GVF).
Specifically, we model the surrounding vehicles as ellipsoids
in corresponding ellipsoidal coordinates, where the formulated
Laplace equation is solved with boundary conditions. Meanwhile,
we develop a nonparametric GVF to capture the multi-vehicle
interactions and then plan the AV’s velocity profiles, reducing
the path search space and improving computing efficiency.
Finally, a local path-planning framework with our TriPField is
developed by integrating model predictive control to consider
the constraints of vehicle kinematics. Our proposed approach is
verified in three typical scenarios, i.e., active lane change, on-
ramp merging, and car following. Experimental results show
that our TriPField-based planner obtains a shorter, smoother
local path with a slight jerk during control, especially in the
scenarios with dense traffic flow, compared with traditional
potential field-based planners. Our proposed TriPField-based
planner can perform emergent obstacle avoidance for AVs with
a high success rate even when the surrounding vehicles behave
abnormally.
Index Terms—Potential fields, Gaussian velocity fields, model
predictive control, local path planning, autonomous vehicles.
I. INTRODUCTION
PATH planning is one essential component for bridging
perception and motion control modules of autonomous
vehicles (AVs) [1], [2], which generally consists of global
and local stages. Global path planning is finding a viable path
to navigate an AV from origin to destination. With global
planning results, AVs must search for safe and efficient local
Manuscript received 26 June 2022; revised 13 November 2022; accepted
28 November 2022. This work was supported in part by the National Key
Research and Development Program of China under Grant 2021YFB1600400
and in part by the Innovation Program of Shanghai Municipal Education
Commission under Grant 2021-01-07-00-07-E00092. The Associate Editor
for this article was Y. Wang. (Corresponding author: Cong Zhao.)
Yuxiong Ji, Lantao Ni, Cong Zhao, Cailin Lei, and Yuchuan Du are with the
Key Laboratory of Road and Traffic Engineering of the Ministry of Education,
Tongji University, Shanghai 201804, China (e-mail: zhc@tongji.edu.cn).
Wenshuo Wang is with the Department of Civil Engineering, McGill
University, Montreal, QC H3A 0C3, Canada.
Digital Object Identifier 10.1109/TITS.2022.3231259
paths and motion control profiles to remain reactive to their
surroundings. However, local path planning still faces signif-
icant challenges in dynamic scenarios such as unstructured
environments with dense obstacles [3] regarding the success
rate and computational efficiency.
The earliest local path-planning for AVs relies on primarily
random search methods such as rapidly-exploring random
trees [4], [5] and Lattice [6]. Recent works mainly focused
on hybrid path planning by leveraging machine learning (e,g.,
reinforcement learning and deep learning) with random search
techniques [6], [7]. Hybrid path planning can improve robust-
ness when facing oscillations and data uncertainty. However,
they are not practical due to limited computational efficiency
and robustness. Moreover, many local path-planning methods
are limited to specific scenarios, such as automated valet
parking [8], obstacle avoidance [9], car-following [10] and
overtaking [11], and comfortable and energy-efficient driving
on rough pavements [12], [13].
Potential fields (PF) have benefited robot path planning [14],
[15] by following the gradient of a weighted sum of artifi-
cial potential functions composed of attraction to goals and
repulsion to obstacles. PF provides a computationally efficient
and probabilistically safe framework for path planning. Recent
works applied PF-based models for decision-making [16],
traffic flow modeling [17], multi-agent interactions [18], and
driving risk assessment [19]. For example, the PF-based car-
following model is in line with human drivers’ stimulus-
response mechanism [10]. Moreover, the dependence on the
number of vehicles can be alleviated by PF [18]. The above
works may fail to quantitatively and uniformly evaluate the
potentials when considering the factors of vehicles, roads,
environments, and other traffic participants. Reference [20]
and [19] proposed a unified driving safety field based on the
driver–vehicle–road interaction in a two-dimensional space.
However, most PF-based models cannot leverage the vehicle
kinematics constraints and generate a narrow velocity profile
during obstacle avoidance in challenging driving tasks. Thus,
the AV usually stops at unintended locations owing to the local
minima.
To address the above limitations, researchers designed
hybrid path-planning approaches to avoid obstacles and track
the planned trajectory dynamically by combining PF and con-
trol theory [21]. However, an isotropic-based PF model may
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2IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS
fail to represent the silhouettes and attitudes of surrounding
vehicles, generating low-feasibility paths and risking AVs in
dense traffic. In addition, most works also have limited variable
selection, for example, only optimizing the steering angle
of AVs under a constant velocity profile [22]. Some works
simultaneously generate acceleration and steering angle by
formulating it as a multi-objective optimization problem [23].
However, finding an optimal solution within the feasible
computing resources still needs to be solved. Therefore, the
challenges in efficiently using PF for local path planning
remain due to the intricate nature of traffic scenarios and the
model’s flaws. This paper aims to seek the solution to the
above challenges.
We solve the isotropic problem by introducing an ellipsoid-
based PF that can capture the differences in steering angles
and vehicle size, thus providing feasible reference local paths.
Moreover, the proposed model can alleviate the dependence
on the number of surrounding vehicles to capture agent inter-
actions [18]. We transformed the Cartesian coordinates into
ellipsoidal coordinates [24] to compute the potential energy
by solving the Laplace equation. We then integrate traditional
PF with a Gaussian velocity field to achieve velocity control
when planning local paths. We designed a new framework
combining the TriPField and model predictive control (MPC)
to improve the adaptability of local path planning across
various scenarios.
This paper aims to design a novel PF model suitable
for local path planning in complex traffic scenarios. The
contributions of this paper are three-fold:
•We developed an ellipsoid-based PF model capturing
the silhouettes and attitudes of vehicles. This model can
generate the gradient of potential functions and feasible
paths in an environment with dense obstacles.
•We proposed a TriPField to characterize vehicle interac-
tions in highly dynamic scenarios and designed a velocity
discriminator to reduce the search space and improve
computing efficiency.
•We proposed a new framework for local path planning
by combining TriPField and MPC. Their effectiveness
and robustness are verified with experimental analysis in
different driving scenarios.
The remainder of this paper is organized as follows.
Section II describes related work. Section III details our
proposed TriPField. Section IV develops a novel local path
planning framework. Section V presents testing scenarios and
experiments, followed by conclusions in Section VI.
II. RELATED WORK
A. Potential Fields (PF)
Khati [25] proposed the potential field, which has attracted
extensive attention in traffic flow, driving behavior, and driving
risk assessment. Ni [17] applied the physical field to traffic
flow modeling and demonstrated the objectivity and universal-
ity of the field but did not discuss the meaning of numerical
value in the field. Wang et al. [19], [20] defined the potentials
and fields in road traffic scenarios as a unified driving safety
field considering the risk factors of drivers, vehicles, roads, and
environments. Tian et al. [26] compressed the safety parame-
ters in different directions using an ellipse correction formula,
making PF more realistic to consider the micro-characteristics
of driving behavior. Li et al. [10] proposed a driving risk
potential field model under the connected and automated
vehicle environment that considers the vehicle’s acceleration
and steering angle. However, all the above PF-based models
are primarily rough and static. In traffic flow applications, cars
are always treated as particles with isotropy, which could cause
large prediction errors in dense traffic scenarios. In addition,
the total PF is viewed as the superposition of multiple PF
sources but ignores the vehicle interactions. This paper pro-
poses a new TriPField model composed of an ellipsoid-based
2D-PF and a GVF, which is anisotropic and dynamic and thus
can effectively overcome these limitations.
B. PF-Based Local Path Planning
Many works on local path planning with PF have been
done, including traditional path planning and hybrid path
planning [27]. Researchers usually build a PF-based path
planner by designing an attraction function for destination
and a repulsive function for obstacles [28]. The planned local
path is vulnerable to passing through a scenario with dense
obstacles without collisions [29]. Some researchers improved
PF by leveraging the relative distance between AVs and the
target vehicles to present a new repulsive potential function
to improve the model stability [30], [31]. However, they still
might fall into the local minimum and cause collisions.
On the other hand, some works developed hybrid
path-planning methods using nature-inspired computation
algorithms such as ant colony [32], particle swarm opti-
mization [33], and genetic algorithms [34]. These hybrid
path-planning methods can obtain a globally optimal path
planning but fail to leverage the vehicle kinematics constraints.
Recently, control theory with dynamic constraints of road and
vehicle states can ensure that the planned path is feasible
and optimal. Some works formulated the local path-planning
problem as a multi-objective optimization issue regarding the
vehicles’ expected position, velocity, and acceleration [35].
Meanwhile, some studies also developed cooperative and
heuristic solvers by selecting a larger region and reducing
the optimal local value to improve computing efficiency [36].
However, the combination of MPC with PF has limited
velocity control. When planning velocity and steering angle
simultaneously, a large search space will make the models
and algorithms difficult to obtain the optimal solution in the
feasible calculation time.
In this paper, we proposed to combine the TriPField with
MPC to achieve local path planning in dense and dynamic
traffic scenarios while improving computing efficiency. The
optimal reference local path is generated via a 2D-PF model,
and GVF is used to monitor the velocity changes of surround-
ing vehicles. Our method has the advantages of closed-loop
control and adapting to multiple scenarios.
III. 3D POTENTIAL FIELDS
This section presents the TriPField model to calcu-
late potential energy from surrounding vehicles. First, an
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JI et al.: TriPField: A 3D PF MODEL AND ITS APPLICATIONS TO LOCAL PATH PLANNING OF AVs 3
Fig. 1. Vehicle modeling based on the ellipsoid. (a) The vehicle model and
(b) the ellipsoid model.
ellipsoids-modified 2D-PF model is presented by considering
the vehicle’s silhouette and attitude. Then, GVF is integrated
into the ellipsoid-based PF to capture vehicle interactions.
A. Ellipsoid-Based PF
The silhouette and attitude of vehicles are critical for a
small relative distance between vehicles. As shown in Fig. 1,
combined with the real vehicle model, the ellipsoid describes
the three-dimensional contour information of driving risks
around the vehicle. We estimate the ellipsoid’s potential energy
to characterize the vehicle, including its virtual mass, coor-
dinate system conversion, solving the Laplace equation, and
modeling vehicle orientation.
1) Virtual Mass: The virtual mass Mimeasures the poten-
tial energy caused by the vehicle i’s attributes (mass, type,
and velocity). Potential energy is the potential loss caused by
the collision between the AV and its surroundings. A high
velocity reflects a great virtual mass and a high potential loss
for vehicles with the same weight. Meanwhile, at the same
speed, the potential loss caused by collision depends on the
vehicle’s actual weight. The empirical formula of virtual mass
is
Mi=αmi|vi|βTi+γ(1)
where miis the actual weight of the vehicle i,viis the velocity,
and Tiis the vehicle type correction factor. α,β,andγare the
fitting parameters that can be calibrated from actual accident
data [19].
2) Coordinates Conversion: Treating the surrounding vehi-
cle as an ellipsoid cannot estimate the associated potential
energy in the Cartesian coordinates [37]. To solve this prob-
lem, we convert the Cartesian coordinates into ellipsoidal
coordinates to solve the Laplace equation with boundary
conditions. The coordinates transformation is realized by using
the quadratic confocal plane, i.e., (x,y,z)→ξe
j,η
e
j,ζe
j.
For more details, see Appendix A.
Under the ellipsoidal coordinates, the potential energy EPEe
j
generated by the surrounding vehicle jfor the AV satisfies the
Laplace equation [37]
ηe
j−ζe
jRξ
∂
∂ξe
jRξ
∂EPEe
j
∂ξe
j
+ζe
j−ξe
jRη
∂
∂ηe
jRη
∂EPEe
j
∂ηe
j
+ξe
j−ηe
jRζ
∂
∂ζe
jRζ
∂EPEe
j
∂ζe
j=0(2)
with
Rλ=λ+lj2λ+wj2λ+hj2
where Rξ,Rηand Rζare the numerical solutions of Rλ
when λis taken as ξe
j,ηe
jand ζe
j.
3) Solving Laplace Equation: By simplifying the Laplace
equation, we obtained the integral expression of EPE (see
Appendix B) in ellipsoidal coordinates as
EPEe
jξe
j=2ρ
lj2−hj2
F(P,) (3)
with
P=lj2−wj2
lj2−hj2
=arcsin
lj2−hj2
(ξe
j)k1+lj2
where ρis the integral constant determined by boundary
conditions. F(P,) is the first kind of elliptic integral queried
in the mathematics manual. Pis the influence factor of vehicle
size. is the potential energy distance coefficient. k1is the
potential energy distance amplification factor. Converting the
potential energy solution into elliptic integral can improve
computing efficiency and meet the requirements of real-time
implementations.
In what follows, we will estimate ρusing the boundary con-
ditions. When the surrounding vehicle jhas a large distance
from the AV, we have
ξe
j→x−xj2+y−yj2+z−zj2=rj2.(4)
When rj→∞, the potential energy generated by the
ellipsoid-based PF is similar to the traditional PF [19]. When
vj=0, the value of ρis
ρ=KMjMe
(k1−1),v
j=0(5)
where Meis the virtual mass of the AV, and Kis the potential
energy proportional coefficient.
We have discussed the ellipsoid-based PF with a stationary
surrounding vehicle. However, a moving surrounding vehicle
would cause an uneven distribution of potential energy at the
moving direction. A high relative velocity would generate
a small distance to the surrounding vehicle, thus having
significant potential energy. Inspired by [20], we define a
velocity vector operator
V=k2−|ve
j|cos θe
j(6)
where k2is the velocity deviation coefficient. A large value
of k2indicates that the potential energy tends to the driving
direction of the surrounding vehicle j.ve
jis the relative
velocity vector of the surrounding vehicle jto the AV. θe
jis the
angle between the distance vector of the surrounding vehicle j
and ve
j. Then, the potential energy EPEe
jin the ellipsoid-based
PF formed by the surrounding vehicle jis
EPEe
j=ρ
V
2
lj2−hj2
F(P,) (7)
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4IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS
Fig. 2. Vehicle orientation adjustment.
Fig. 3. Equipotential projection of (a) the traditional PF and (b)-(d)the
ellipsoid-based PF with different vehicle sizes and poses.
where the calculation result EPEe
jof (7) is positive. We dis-
cretized the road into a two-dimensional grid, thus resulting
in a two-dimensional ellipsoid-based PF. Note that we should
discuss the vehicle center when z=zjfor EPEe
j.Itis
convenient for comparative analysis of the traditional and
ellipsoid-based PF.
4) Vehicle Orientation: Intheabove,wehavediscussed
the calculation of EPEe
jwhen the heading angle ϕjof the
surrounding vehicle jis zero. A non-zero heading angle will
get equation (7) deflected. For example, during lane-changing
tasks, the vehicle is usually not parallel to the road profile,
thus the ellipsoid-based PF should be deflected to capture
such behavior. Fig. 2 illustrates a surrounding vehicle jat
the position xj,yjwith a heading angle ϕj.Werotate
the ellipsoid-based PF counter-clockwise around the straight
line x=xj,y=yjto transfer the original coordinates
x,y,EPEe
jto x,y,EPEe
j
by
x=x−xjcos ϕj−y−yjsin ϕj+xj(8a)
y=x−xjsin ϕj−y−yjcos ϕj+yj(8b)
EPEe
j
=EPEe
j(8c)
Fig. 3 shows the equipotential projection of the traditional
PF and the ellipsoid-based PF with different vehicle sizes
and motion states, and white arrows represent the vehicle
velocity. Black contour represents a horizontal curve formed
by projecting points with the same potential energy onto a
plane. The heat map indicates the intensity of the potential
field. The potential fields across various vehicle sizes and
motion states indicate that the potential energy of the potential
field reaches the maximum at the location of the surrounding
vehicle. The AV at this location will cause a high risk and even
collision. Meanwhile, enlarging the distance between vehicles
will increase the potential energy of the region.
In addition to these standard features, the ellipsoid-based
PF has other different elements from the traditional PF.
Figs. 3(a) and (b) illustrate the simulation results with the
surrounding vehicle in the same size and motion state.
The ellipsoid-based PF can dynamically change the potential
energy adapting to the vehicle size. Specifically, the potential
energy distribution in the region adjacent to the surrounding
vehicle is ellipsoidal, and the potential energy along the X-axis
is greater than that along the Y-axis. The potential energy
distribution is circular in the region far from the surrounding
vehicle, similar to the traditional PF. This implies that the
ellipsoid-based PF is sensitive to the vehicle size when the
relative distance is small. On the contrary, the traditional PF
is isotropic in different directions, which deviates from the
actual driving scenario. The ellipsoid-based PF can capture
the driving risk changes of the AV during lane changing (e.g.,
active lane-changing, on-ramp merging), thereby planning a
better path. Fig. 3(c) indicates that the change of the surround-
ing vehicle heading angle also leads to the change of potential
energy at the same angle.
There is no doubt that the ellipsoid-based PF will improve
the driving safety compared with the traditional PF. One more
critical finding in Fig. 3(d) is that a surrounding truck would
make the potential energy in its driving direction be a forward
incline trend. This implies that the AV will face greater driving
risks in the driving direction of trucks, especially at a close
distance.
B. 3D Potential Fields
PF-based local path planning aims to output the minimum
value of its potential function to ensure that AVs can avoid
collisions dynamically. The characteristics of PF indicate that
the driving risk in the descending direction of the gradient is
low. However, the gradient descent direction of the PF usually
cannot leverage vehicle kinematics constraints. A potential
solution to this problem is combining PF with MPC, using
PF to generate a reference path tracked by an MPC module.
This framework needs to control the velocity and steering
angle at the same time. However, the simple steering angle
change needs to meet the requirements of driving conditions
in complex tasks. It ignores the velocity control, causing the
failure of local path planning. For example, when the front
vehicle slows down in a dynamic scenario with dense vehicles,
the AV should adjust the velocity in time rather than choose
obstacle avoidance, falling into local optimization, or even
the vehicle collision risk. Aiming at improving the velocity
control of these PFs in local path-planning, we propose
a TriPField model using GVF to describe the interactions
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JI et al.: TriPField: A 3D PF MODEL AND ITS APPLICATIONS TO LOCAL PATH PLANNING OF AVs 5
between vehicles. By estimating the velocity distribution in
the plane, GVF is not sensitive to vehicle position error and
noise and can represent different interaction modes between
vehicles. In general, the dimension of this TriPField is similar
to the traditional PF, but the number of channels is different,
i.e., w×h×3, where 3 represents the components of the
potential energy of xposition and yposition and velocities.
1) Gaussian Process (GP): For a set of random variables,
G=[g1,...,gn]and corresponding functional outputs
f(G)=[f(g1),..., f(gn)], if any partition of functional
outputs with associated collections of random variables still
obey a multivariate Gaussian distribution, then fis a Gaussian
process, expressed as
f(G)∼N(μ(G), κ (G,G)) (9)
where μ(G):Rn→Rnis the mean function that returns
the mean value of each dimension. κ(G):Rn×Rn→
Rn×nreturns the covariance matrix between the dimensions
of two variables for the covariance function, known as kernel
functions. A GP is uniquely defined as a mean function and
covariance function, and the finite-dimensional subsets of a
GP obey a multivariate Gaussian distribution. According to the
definition of GP, any finite collection of the random variables
f(G∗)conditional on the training set G,f(G)(with the input
set G=[g1,...,gn],nis the number of observations) can be
treated as a conditional Gaussian distribution, and their joint
distribution is also a Gaussian distribution as
f(G)
f(G∗)∼N0
0,κ(G,G)κ(G,G∗)
κ(G∗,G)κ(G∗,G∗) (10)
with
κ(G,G)=⎡
⎢
⎣
k(g1,g1)··· k(g1,gn)
.
.
.....
.
.
k(gn,g1)··· k(gn,gn)
⎤
⎥
⎦
Other kernel matrices can be calculated. Therefore, the
conditional distribution of f(G∗)on f(G)is
f(G∗)|G,f(G), G∗∼N(μG∗,
G∗)(11)
with
μG∗=κ(G∗,G)κ(G∗,G)−1f(G)
G∗=κ(G∗,G∗)−κ(G∗,G)κ(G,G)−1κ(G,G∗)
The kernel function is the core of GP, which influences the
properties of a GP. The most commonly used kernel function
is the Gaussian kernel function
K(gi,gj)=σ2exp −||gi−gj||2
2χ2(12)
where σand χare the hyperparameters of the Gaussian kernel.
GP can maximize the probability under these two hyperpara-
meters for the kernel function parameters and find the opti-
mal parameters by maximizing the marginal log-likelihood.
Fig. 4-B1(a) and (b) compare the results without and with
hyper-parametric optimization. Model hyperparameters have
significant influences on the GP’s performance.
Fig. 4. Comparison of GP regression in two-dimensional space (a) with
and (b) without hyper-parametric optimization. Red dots are the training data
points {G,f(G)}.
2) Gaussian Velocity Field: Inspired by GP, the velocity
of each position in the field can be treated as a random
variable and specified by its mean and covariance. GVF is a
random field of velocity, which thus can estimate the velocity
distribution of the entire area according to the observed
velocity of all vehicles. We define f(·)(g)v(·)tby assuming
that the velocity estimates in the xand ydirections are
independent, where (·)can be xor y. By implementing (11),
vt(x,y)=[vx,t,vy,t]can be estimated anywhere within the
area.
Fig. 5(a)-(b) illustrates the simulated results of the potential
energy in multi-vehicle scenarios using the traditional PF and
the proposed ellipsoid-based PF. The total PF of the AV is
composed of the superposition of sub-PF formed by multiple
surrounding vehicles. Fig. 5(a) shows that the traditional PF
does not consider the surrounding vehicles’ size and pose
information. The surrounding vehicles are treated as particles,
thus the potential energy is isotropic and only provides limited
information. Fig. 5(b) indicates that the AV can intuitively
judge the driving risk at each position through the contour
distribution of potential energy based on the ellipsoid-based
PF. The size and driving direction of surrounding vehicles
can influence the potential energy to build a more accurate
driving area for the AV. Therefore, the AV can dynamically
judge the range of the driving area at every moment based on
the proposed ellipsoid-based PF, overcoming the shortcomings
of the traditional PF and solving the problem of where to go.
Fig. 5(c) shows the velocity field based on a GP with
the RBF kernel. The black arrows represent the observed
relative velocity vector of the AV and the surrounding vehi-
cles. Their velocity directions are independent of each other.
The setting of the velocity field provides a reference for
the AV speed control, i.e., acceleration or deceleration. The
simulation results show a backward incline in the relative
speed of surrounding vehicles and the AV in this scenario.
At this time, the AV should decelerate adequately to meet
the consistency of the overall traffic flow and avoid dangerous
driving situations, such as being relatively close to surrounding
vehicles. Therefore, we can use the velocity field to solve the
problem of how to go.
TriPField is generated through numerical superposition,
as shown in Fig. 5(d), enabling AVs dynamically to generate
a stable and safe reference local path and reference velocity.
The ellipsoid-based 2D-PF aims to solve the problem of where
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6IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS
Fig. 5. Illustration of (a) traditional PF, (b) ellipsoid-based PF, (c) GVFs, and
(d) our TriPField. The red rectangle is the AV with six vehicles surrounding
it. The bright color indicates a great potential energy value.
to go for the AV. Adding a dimension of the velocity field of
surrounding vehicles can solve the problem of how to go.
IV. TRIPFIELD-BASED LOCAL PATH PLANNING
FRAMEWORK
This section introduces a new local path-planning frame-
work for AVs with a combination of TriPField and MPC.
First, a vehicle kinematics model is presented. Then, the
local path planning is formulated as an MPC problem based
on potential energy generated from the TriPField model and
vehicle kinematics.
A. Vehicle Module
Fig. 6 illustrates the kinematics of a bicycle model we used.
The control inputs are the steering angle of the front wheel
δand acceleration a. The AV’s states seand inputs ueare
defined as
se=[xe,ye,v
e,ϕ
e](13a)
ue=[ae,δ
e](13b)
Fig. 6. Vehicle kinematics model.
where (xe,ye)are the position, ϕeis the yaw angle, veis
the velocity, aeis acceleration, and δeis the steering angle.
The vehicle center is the midpoint of the rear axle. Thus, the
vehicle kinematic model can be formulated as [38]
˙xe=vecos(ϕe)(14a)
˙ye=vesin(ϕe)(14b)
˙ve=ae(14c)
˙ϕe=vetan(δe)
Le
(14d)
where Leis the vehicle wheelbase. After derivation (see
Appendix C), the state prediction equation can be calculated
as
se(t+1)=Ase(t)+Bue(t)+C(15)
with
A=⎡
⎢
⎢
⎣
10cos(ϕe)t−vsin(ϕe)t
01sin(ϕe)tvcos(ϕe)t
00 1 0
00 tan(δe)
Let1
⎤
⎥
⎥
⎦
B=⎡
⎢
⎢
⎣
00
00
t0
0ve
Lecos2(δe)t
⎤
⎥
⎥
⎦
C=⎡
⎢
⎢
⎣
vesin(ϕe)ϕet
−vecos(ϕe)ϕet
0
veδe
Lecos2(δe)t
⎤
⎥
⎥
⎦
where tis the step size of discrete sampling.
B. MPC Implementation
Fig. 7 shows the implementation flow of MPC. First,
the vehicle’s start position, velocity, steering angle, and end
position are initialized using the actual trajectory dataset.
Second, the reference local path and speed are obtained from
TriPField and fed into the MPC controller. The points with
the smallest potential function are connected to form the local
reference path of AV in the future, as shown in Fig. 8. For
the reference speed, we set a step size to dynamically adjust
the reference speed of the AV according to the relative speed
generated from GVF. Then, the MPC is run to meet the vehicle
kinematics constraints to search for the optimal steering angle
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JI et al.: TriPField: A 3D PF MODEL AND ITS APPLICATIONS TO LOCAL PATH PLANNING OF AVs 7
Algorithm 1 Reference State Generate Based on TriPField
Input: AV current state(x,y,v,ϕ), surrounding vehicles
states(xj,yj,vj,ϕj)
Output: AV reference state(xref ,yref ,vref ,ϕref )
1: Initialize AV state (x0,y0,v
0,ϕ
0)
2: for i=0→2πdo
3: motions ←[cos i,sin i]
4: end for
5: min_motion
6: ←POTENTIALENERGY(xj,yj,vj,ϕj,motions)
7: reference_local_path(xref,yref ,ϕref )
8: ←GETSPLINEPATH (min_motion)
9: relative_velocity ←GVF(xj,yj,vj)
10: if relative_velocity then ≥vmax
11: vref -= ω
12: end if
13: if relative_velocity ≤−vmax then
14: vref += ω
15: end if
16: return xref ,yref ,vref ,ϕref
and velocity output, which means path tracking. Finally, select
the first control step as the control output. It represents the
path tracking and generates the actual local path, as shown in
Fig. 8. The above steps are iterated in a cycle to ensure the
AV reaches its destination smoothly and safely.
In order to simulate local path planning, we set the destina-
tion before starting. The repulsive potential energy of a single
surrounding vehicle is calculated by (7). The total repulsive
potential energy Erep of the AV is the superposition of each
vehicle
Erep =
Snum
j=1
EPEe
j(16)
where EPEe
jis the potential energy generated by the jsur-
rounding vehicle, and Snum is the total number of surrounding
vehicles. The attractive potential energy Eatt is calculated by
the following formula
Eatt =Kp(x−xdes)2+(y−ydes)2(17)
where (xdes,ydes)is the local path’s destination. Kpis the
coefficient of amplification of the attractive potential energy.
Therefore, the total potential energy Esum is
Esum =Erep +Eatt (18)
The properties of PF [23] indicate that the summed poten-
tials the AV faces should be minimized to ensure driving
safety. Therefore, the vehicle’s local reference path is defined
as the minimum value (i.e., the valley) of Esum .
After initialization, we should provide the MPC controller
with a reference state to complete the local path planning
and tracking task. To solve this problem, we propose a multi-
stage algorithm Reference State Generate based on TriPField
summarized in Algorithm 1. This algorithm divides the ref-
erence states into local reference paths and speed. As shown
in Fig. 7, the PF generates the local reference path and the
GVF generates the reference velocity. First, we initialize and
discretize the vehicle motion space to obtain all motions.
Second, the discrete point set min_motion of the minimum
potential energy in the valley is calculated by (18). Then,
we use the spline curve [39] to connect the discrete points into
a curve and get reference_local_path (xref ,yref ,ϕref ).
Then, we use relative_velocity to adjust vref of AV
by step ω. Finally, the set of reference state [xref ,yref ,vref ,
ϕref ] is returned. After obtaining the reference state, we aim
to minimize the error between the vehicle states and the
references:
J1=xerr(ti)=
Np
i=0
(x(ti)−xref (ti))2(19)
J2=yerr(ti)=
Np
i=0
(y(ti)−yref (ti))2(20)
J3=verr(ti)=
Np
i=0
(v(ti)−vref(ti))2(21)
J4=ϕerr(ti)=
Np
i=0
(ϕ(ti)−ϕref(ti))2(22)
where Npis the prediction horizon, xerr (ti),yerr(ti),verr (ti)
and ϕerr(ti)are the errors related to the AV’s longitudinal and
lateral position, velocity, and yaw angle in the tiprediction
step, respectively. x(ti),y(ti),v(ti),andϕ(ti)denote the lateral
position, longitudinal position, velocity, and heading angle of
the vehicle in the tiprediction step. vref (ti)and ϕref (ti)are the
vehicle’s reference velocity and yaw in the tiprediction step.
The vehicle is controlled to travel along the valley of TriPField
through J1and J2. We shall reduce the AV’s sudden turning,
acceleration, and deceleration to reach stability and comfort
performance. To this end, we consider the variation of steering
angle and acceleration by
J5=
Nc
j=0a(tj+1)−a(tj)2(23)
J6=
Nc
j=0δ(tj+1)−δ(tj)2(24)
where Ncis the maximum control steps, a(tj)and δ(tj)are
the acceleration and steering angle at the tjcontrol step.
In addition, we leverage the constraints into the optimization
problem to avoid exceeding the limit of vehicle kinematics,
min
u,ε
6
k=1
τkJk+εt2(25a)
sub.to s(t+1)=As (t)+Bu(t)+C(25b)
0≤δ(tj)≤δmax (25c)
−adec ≤a(tj)≤aacc (25d)
−ϕmax ≤ϕ(ti)≤ϕmax (25e)
0≤v(ti)≤vmax (25f)
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8IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS
Fig. 7. The local path planning framework based on PF and MPC.
Fig. 8. The valley of PF and discrimination of reference velocity by GVF.
where τkwith k=1,...,6 are the weight parameters, εtare
the vectorized slack variables at tsteps ahead of the current
time. δmax is the maximum steering angle, aacc and adec are
the maximum acceleration and deceleration, respectively. ϕmax
is the maximum yaw, and vmax is the maximum velocity.
At each sampling step, the optimal control quantity is
obtained by solving the optimization problem of (25a). The
state quantity of the vehicle is solved by substituting the
control quantity into the prediction (15). Repeating the above
calculation obtains a dynamic planning vehicle path.
V. E XPERIMENTS AND RESULTS
This section defines three typical test scenarios (i.e., active
lane changing, ramp merging, and car following) to evaluate
our developed local path planning framework based on TriP-
Field. The multi-vehicle simulation analysis is based on the
natural trajectory in the HighD dataset.
A. Data Pre-Processing
The HighD trajectory dataset was collected by a drone in
Germany [40], and the sampling frequency is 25 Hz. To meet
TAB L E I
SETUP OF SIMULATION PARAMETERS
the requirements of our MPC controller, we re-sampled data
as 10 Hz. Before starting the experiment, we processed the
dataset for different scenarios. We take lane change as an
example. Firstly, we selected the lane-changing vehicle accord-
ing to Lane_id in the dataset. Secondly, the number of vehicles
around the same frame is used to evaluate the traffic flow. The
lane-changing vehicle is viewed as the AV. The information
of the AV and its surroundings in the same frame is extracted,
as shown in Fig. 9. We also process the merging scenario at
highway on-ramps and the front vehicle deceleration scenarios
with the same procedure. Note that these two scenarios require
Lane_id. In the on-ramp merging scenario, we must ensure that
the AV’s starting lane is on the ramp. Thus, Lane_id denotes
the maximum or minimum value in the interval rather than the
intermediate value. In the front vehicle deceleration scenario,
we must ensure that the AV’s Lane_id remains unchanged to
complete the car-following task.
After data preprocessing, we obtained multiple groups of
data from three driving scenarios. The AV path lengths vary
in these scenarios: 240 m in the lane-changing scenario and
180 m in the ramp-merging scenario. Their duration time
is about 8 s. The deceleration time of the front vehicle
deceleration scenario is about 6 s.
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JI et al.: TriPField: A 3D PF MODEL AND ITS APPLICATIONS TO LOCAL PATH PLANNING OF AVs 9
Fig. 9. Data splicing process to obtain driving scenarios data.
B. Simulation Setup
The proposed TriPField-based planner (TriPField-MPC) is
simulated via real vehicle trajectory data of HighD. We input
the AV’s starting and end positions and output the AV’s
position and speed at each time to compare and evaluate
the performance with baselines, i.e., the planners based on
traditional PF and MPC (PF-MPC) and ellipsoid-based PF
and MPC (EPF-MPC). Simulation parameters are shown in
Table I. The simulated time step is 0.1 s.
The planners are conducted in three typical scenar-
ios to verify their obstacle avoidance and maneuverability
performances, including active lane change (Scenario 1),
merge on-ramp (Scenario 2), and abnormal driving behaviors
(e.g., sudden deceleration of the front vehicle, Scenario 3).
We design three indicators according to [5] to quantify these
models’ performance in lane-changing scenarios.
1) Curve Length: During the local path planning for obsta-
cle avoidance, the curve length Lpis the critical metric to
evaluate the planner performance, which is defined as
Lp=
n
t=1(xe(t+1)−xe(t))2+(ye(t+1)−ye(t))2(26)
where (xe(t),ye(t))is the position of the AV at time t.
2) Curve Smoothness: We introduce roughness Rpto eval-
uate the smoothness. A high Rpindicates a rough planned
path challenging to track.
Rp=1
Lp
n
t=1
|δe(t+1)−δe(t)|(27)
where δe(t)is the AV steering angle at time t.
3) Driving Comfort: Driving comfort is also a critical eval-
uation metric. A high acceleration change rate indicates that
the vehicle accelerates and decelerates rapidly and the comfort
performance is poor. Thus, we use the average acceleration
change rate ARpto describe the driving stability.
ARp=1
n
n
t=1
|ae(t+1)−ae(t)|(28)
where ae(t)is the acceleration at time t.
Fig. 10. (a) An active lane-change driving scenario (Scenario 1). (b) A ramp
merging driving scenario (Scenario 2).
TAB L E I I
SIMULATION RESULTS OF SCENARIO 1AND SCENARIO 2
C. Scenario 1 – Lane Change
Fig. 10(a) depicts the active lane-changing and overtaking
task, and the AV must change the lane from Lane 1 to Lane 2
by overtaking the surrounding green vehicle. The entire lane-
changing process is divided into three steps. First, the AV
accelerates into Lane 1 and passes the surrounding green
vehicle. Second, the AV deviates from the centerline of Lane 1
to the right lane boundary, crosses the boundary, and changes
the lane to Lane 2. Finally, the AV adjusts its lateral position
and velocity to maintain Lane 2 and complete the overtaking
task.
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10 IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS
Fig. 11. Testing results with different planners in Scenario 1: (a) PF-MPC,
(b) EPF-MPC, and (c) TriPField-MPC.
Fig. 12. The AV’s steer angle using TriPField-MPC, EPF-MPC, and PF-MPC
in Scenario 1.
Fig. 11 shows all vehicles’ trajectories with different plan-
ners. The same marker indicates the same time step for
vehicles. Expressly, squares, diamonds, lower triangles, and
circles represent the position of vehicles at 0 s, 2.5 s, 5 s,
and 7.5 s, respectively. Colors distinguish vehicle trajectories;
for example, red represents the AV, and others represent the
surrounding vehicles. These three planners can succeed in
changing lanes without collisions. The PF-MPC controller
has the longest path and the largest curvature, resulting in
an excessive safety margin when avoiding obstacles, which
does not benefit vehicle stability control. While EPF-MPC
Fig. 13. Testing results of the AV and surrounding vehicles in Scenario 2
using (a) PF-MPC, (b) EPF-MPC, and (c) TriPField-MPC.
and TriPField-MPC can keep the path smoother and shorter
while successfully avoiding obstacles because the ellipsoid
can finely calculate the potential function of surrounding
vehicles, as discussed in Section III (Fig. 5). Table II lists the
quantitative results. When the vehicle actively changes lanes,
PF-MPC generates the longest and the roughest local path
with average values of 264.26 m and 0.217 rad/m, respectively.
Both EPF-MPC and TriPField-MPC show advantages in the
active lane change. Table II shows that the path length is
reduced by about 10% and the roughness is reduced by about
11%. There is no significant difference in path length and
roughness for TriPField-MPC and EPF-MPC. However, due
to poor speed control, EPF-MPC produces steering angle
fluctuation in the second and third parts of lane changing and
has poor stability (Fig. 12). In addition, the average change
rate of acceleration (i.e., jerk) by EPF-MPC is 0.34 m/s3,
which is higher than that of TriPField-MPC. For TriPField-
MPC, GVF obtains the interaction mode between vehicles,
which can control the AV more stably and comfortably.
D. Scenario 2 – On-Ramp Merge
Fig. 10(b) depicts Scenario 2 of the ramp merging task,
consisting of three steps. First, the AV deviates from the
centerline of Lane 3 to the left lane boundary. Second, the
AV crosses the left lane boundary by choosing an appropriate
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JI et al.: TriPField: A 3D PF MODEL AND ITS APPLICATIONS TO LOCAL PATH PLANNING OF AVs 11
Fig. 14. The AV’s steer angle using TriPField-MPC, EPF-MPC, and PF-MPC
in Scenario 2.
Fig. 15. MPC lateral error of different planners in Scenarios 1 and 2.
gap and changing the lane to Lane 2. Finally, the AV adjusts its
lateral position and velocity to maintain Lane 2 and complete
the ramp merging task. Fig. 13 shows the planning results with
different planners in scenario 2. The decision-making space
of the AV is limited due to the restriction of road conditions.
There is no significant difference in the paths and steer angles
controlled by the three planners (Fig. 14). Table II shows that
the controlled paths of TriPField-MPC and EPF-MPC planners
are still smoother, with a value of 0.110 m/rad on average.
With no collisions, TriPField-MPC and EPF-MPC slightly
outperform PF-MPC. In addition, the acceleration change rate
of TriPField-MPC is 0.16 m/s3on average, which is lower
than that of the other two planners, indicating that the GVF
can improve the planner’s stability.
Fig. 15 compares the MPC control errors for the three
planners in Scenario 1 and Scenario 2. TriPField-MPC obtains
the slightest MPC error, reduced by about 20% compared
with PF-MPC. In Scenario 2, the limitation of road conditions
makes the MPC control error of TriPField-MPC slightly
smaller than that of the other two planners. In ramp merging,
these three planners have little difference in the driving paths
of vehicles since they need to meet the constraints of road
boundaries and the length of the merging area. Therefore, the
reference paths of different planners generate similar paths.
E. Scenario 3 – Car-Following
We demonstrated the effectiveness of adding an ellipsoid
to PF, mainly in curve length and smoothness in Scenarios
1 and 2. Adding a velocity field improves the vehicle’s speed
Fig. 16. A car-following driving scenario and the front vehicle drives
abnormally (Scenario 3).
Fig. 17. Testing results in Scenario 3: (a) Local path using EPF-MPC of the
AV and surrounding vehicles. (b) Local path by TriPField-MPC of the AV
and surrounding vehicles.
control in terms of driving comfort. The above discussion is
limited to general driving scenarios to prove the applicability
to abnormal driving scenarios. To this end, we would compare
EPF-MPC and TriPField-MPC in Scenario 3, which mimics
the sudden deceleration of the front vehicle during the car-
following process, as shown in Fig. 16. The AV moves forward
in Lane 2 by following a leading truck. The truck will suddenly
decelerate to a low speed.
Fig. 17 shows the controlled paths for the AV form
EPF-MPC and TriPField-MPC planners. Blue and red lines
represent the paths of the front truck and the AV, respectively.
The TriPField-MPC planner decelerates successfully, and the
steering angle and acceleration are within the constraints. The
front truck suddenly decelerates (Fig. 18), making the posi-
tive velocity deviation among the agents and then gradually
increasing, which causes a sharp increase in 1/TTC (time-to-
collision, TTC) and a collision risk. Then, the AV with our
TriPField-MPC planner decelerates in time so that the velocity
deviation reaches the inflection point at 1.2 s, resulting in the
gradual reduction of 1/TTC with a similar trend. To sum up,
our TriPField-MPC planner can help AV to slow down to
maintain a safe distance on the lane even when the leading
truck makes a sudden deceleration.
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12 IEEE TRANSACTIONS ON INTELLIGENT TRANSPORTATION SYSTEMS
Fig. 18. 1 / TTC and velocity difference of the AV and in Scenario 3.
Velocity control mainly causes the failures of the EPF-MPC
planner. When surrounded by multiple vehicles, the AV should
slow down and keep the lane as the front vehicle slows
down. However, the EPF-MPC planner tries to avoid obstacles
greedily when the front vehicle decelerates in Fig. 17(a), which
causes local optimization and collides with the surrounding
vehicles in the adjacent lane.
In summary, the simulation results indicate that the proposed
TriPField-based planner can help AVs to safely, efficiently,
and comfortably succeed in all three challenging scenarios
(e.g., lane-changing and car-following). In Scenarios 1 and 2,
the proposed EPF-based planner outperforms the traditional
PF-based planner in guiding vehicles to avoid obstacles
dynamically in a dense traffic flow. In scenario 3, the GVF can
guide the AV to make acceleration or deceleration decisions
quickly under highly dynamic traffic conditions, especially
with some risky or abnormal driving behaviors.
VI. CONCLUSION
This paper presents a 3D potential field model called
TriPField that can simultaneously characterize position and
velocity information with a composition of an ellipsoid-based
2D potential field and a Gaussian velocity field. The TriPField
allows us to plan a smooth and safe local reference trajectory
of longitudinal and lateral positions and steering angles in a
computationally efficient way for downstream modules such
as the MPC controller. This benefits from integrating the
Gaussian velocity field and the traditional potential field.
Moreover, a systematic planning and control framework with
an account for vehicle kinematics is proposed. The effective-
ness of our proposed TriPField is validated and verified in
three typical but challenging driving tasks, including actively
changing lanes, merging on-ramps and following a leading car.
Experimental results demonstrate that our proposed TriPField
can reduce the path’s length, roughness, acceleration change
rate, and MPC tracking errors by 10%, 11%, 24%, and 20%,
respectively.
Our proposed TriPField achieves an expected performance
in controlled simulations. In the future, we will focus on
real vehicle experiments on roads. We will conduct field
tests to validate our proposed framework integrating with the
TriPField model. We also intend to fuse roadside sensory
information [41] using graph attention networks [42] to predict
the surrounding vehicle trajectory more accurately to benefit
the planning and decision-making of autonomous vehicles.
APPENDIX
A. Coordinates Conversion
To convert the Cartesian coordinate system to an ellipsoidal
coordinate system, a quadratic confocal surface makes the
conversion between two coordinate systems with the ellipsoid.
The quadratic confocal surface with the ellipsoid of the
surrounding vehicle jin the Cartesian coordinates is
x−xj2
lj2+λe
j
+y−yj2
wj2+λe
j
+z−zj2
hj2+λe
j
=1
where (x,y,z)is any point in the three-dimensional space
of the Cartesian coordinates. For the surrounding vehicle j,
we denote (xj,yj,zj)as the position and lj,wjand hjas
its length, width, and height. Note that the quadratic confocal
surface with the ellipsoid is a cubic equation of parameter
λe
j. Given a set of values of (x,y,z),λe
jhas three different
real roots: λe
1,j=ξe
j,λ
e
2,j=ηe
j,λ
e
3,j=ζe
j.Setle
j>w
e
j>
he
j, the three roots are arranged as ξe
j>−hj2>η
e
j>
−wj2>ζ
e
j>−lj2. Three kinds of quadratic confocal
surfaces corresponding to ξe
j,ηe
jand ζe
jare as:
x−xj2
lj2+ξe
j
+y−yj2
wj2+ξe
j
+z−zj2
hj2+ξe
j
=1
x−xj2
lj2+ηe
j
+y−yj2
wj2+ηe
j
+z−zj2
hj2+ηe
j
=1
x−xj2
lj2+ζe
j
+y−yj2
wj2+ζe
j
+z−zj2
hj2+ζe
j
=1
For each point (x,y,z), three kinds of surfaces are orthog-
onal to each other, forming an orthogonal surface coordinate,
called ellipsoidal coordinates. ξe
jis the scale parameter of the
confocal ellipsoid of the surrounding vehicle j.Foragiven
value of ξe
j,ηe
jand ζe
jare the variables that determine the
spatial position of the surrounding vehicle jin the ellipsoidal
coordinates ξe
j,η
e
j,ζe
j.
B. Solving Laplace Equation
From the ellipsoid view of the surrounding vehicle j,EPE
e
j
is a constant independent of ηe
jand ζe
j.IfEPE
e
jis a function
of ξe
jbut independent of ηe
jand ζe
j, the boundary condition
is satisfied. Thus, the Laplace equation is rewritten as
d
dξe
jRξ
dEPEe
j
dξe
j=0
and the solution is
EPEe
jξe
j=ρ∞
ξe
j
dξe
j
ξe
j+lj2ξe
j+wj2ξe
j+hj2
where ρis the integral constant determined by the boundary
condition. Through formula derivation, it can be transformed
into:
EPEe
jξe
j=ρ2
lj2−hj2
F(P,)
where F(P,) is the first kind of elliptic integral.
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JI et al.: TriPField: A 3D PF MODEL AND ITS APPLICATIONS TO LOCAL PATH PLANNING OF AVs 13
C. State Prediction
Based on the vehicle bicycle model, the change rate ˙seof
seis
˙se=A1se+B1ue
where A1is the Jacobian of the state and B1is the Jacobian
of the control input
A1=⎡
⎢
⎢
⎣
00cos(ϕe)−vsin(ϕe)
00sin(ϕe)vcos(ϕe)
00 0 0
00 tan(δe)
Le0
⎤
⎥
⎥
⎦
B1=⎡
⎢
⎢
⎣
00
00
10
0ve
Lecos2
⎤
⎥
⎥
⎦
We use the sampling time tto discretize the system and
transform the model into the discrete-time analysis. The state
of the next time step can be calculated as
se(t+1)=se(t)+˙set
Using Taylor series expansion to the first order, it can be
simplified as
se(t+1)=Ase(t)+Bue(t)+C
with
A=⎡
⎢
⎢
⎣
10cos(ϕe)t−vsin(ϕe)t
01sin(ϕe)tvcos(ϕe)t
00 1 0
00 tan(δe)
Let1
⎤
⎥
⎥
⎦
B=⎡
⎢
⎢
⎣
00
00
t0
0ve
Lecos2(δe)t
⎤
⎥
⎥
⎦
C=⎡
⎢
⎢
⎣
vesin(ϕe)ϕet
−vecos(ϕe)ϕet
0
veδe
Lecos2(δe)t
⎤
⎥
⎥
⎦
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Yuxiong Ji received the B.S. and M.S. degrees
in transport engineering from Tongji University in
2001 and 2004, respectively, and the Ph.D. degree
in civil engineering from The Ohio State University,
Columbus, OH, USA, in 2011.
He is currently a Professor with the College
of Transportation Engineering, Tongji University.
His research interests include cooperative vehicle-
infrastructure systems, data mining, smart transit,
and traffic operation and control.
Lantao Ni received the bachelor’s degree in trans-
portation from Fuzhou University, Fujian, China,
in 2020. He is currently pursuing the master’s degree
with the College of Transportation Engineering,
Tongji University. His research interests include
autonomous driving perception and decision-making
and driving risk quantification.
Cong Zhao received the B.S., M.S., and Ph.D.
degrees in transportation engineering from Tongji
University, Shanghai, China, in 2014, 2017,
and 2020, respectively. He was a Visiting Stu-
dent Researcher with California PATH, University
of California at Berkeley, Berkeley, CA, USA,
from 2018 to 2019. He is currently an Associate
Research Professor with the College of Transporta-
tion Engineering, Tongji University. His research
interests include intelligent transportation systems,
connected and automated vehicles, infrastructure
enabled automation, and machine learning.
Cailin Lei received the master’s degree in trans-
portation engineering from Chongqing Jiaotong Uni-
versity, Chongqing, China, in 2019. He is currently
pursuing the Ph.D. degree with the College of
Transportation Engineering, Tongji University. His
research interests include the driving behavior analy-
sis and evolutionary mechanism of driving risk.
Yuchuan Du (Member, IEEE) received the B.S.
and M.S. degrees in road engineering and the Ph.D.
degree in traffic engineering from Tongji Univer-
sity, Shanghai, China, in 1998, 2001, and 2004,
respectively.
From 2003 to 2006, he was an Assistant Professor
with the College of Transportation Engineering,
Tongji University, where he was an Associate
Professor, from 2006 to 2010. He is currently a Pro-
fessor with the College of Transportation Engineer-
ing, Tongji University. His current research interests
include innovative technologies for smart transportation infrastructure and
intelligent transportation systems.
Wenshuo Wang (Senior Member, IEEE) received
the Ph.D. degree in mechanical engineering from
the Beijing Institute of Technology, Beijing, China,
in 2018. Currently, he is a Post-Doctoral Researcher
at McGill University, Canada. Before joining McGill
University, he was a Post-Doctoral Research Asso-
ciate with Carnegie Mellon University and UC
Berkeley from 2018 to 2020. He was also a Research
Assistant at UC Berkeley and the University of
Michigan from 2015 to 2018. His research inter-
ests include Bayesian learning, reinforcement learn-
ing, and optimization and their applications to human driving behavior,
human–vehicle interaction, ADAS, and autonomous vehicles.
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