# Data-Driven Robust Optimization for Energy-Aware Collision Avoidance of
Electric Vehicles

Simran Kumari, Ashish R. Hota, and Siddhartha Mukhopadhyay The authors are
with the Department of Electrical Engineering, IIT Kharagpur, West Bengal,
India. Email<EMAIL_ADDRESS><EMAIL_ADDRESS>Simran Kumari is supported by Ministry of Education (MoE), Govt. of India
under Prime Minister Research Fellowship (PMRF) scheme.

###### Abstract

In this paper, we simultaneously tackle the problem of energy optimal and safe
navigation of electric vehicles in a data-driven robust optimization
framework. We consider a dynamic model of the electric vehicle which includes
kinematic variables in both inertial and body coordinate systems in order to
capture both longitudinal and lateral motion as well as state-of-energy of the
battery. We leverage past data of obstacle motion to construct a future
occupancy set with probabilistic guarantees, and formulate robust collision
avoidance constraints with respect to such an occupancy set using convex
programming duality. Consequently, we present the finite horizon optimal
control problem subject to robust collision avoidance constraints while
penalizing resulting energy consumption. Finally, we show the effectiveness of
the proposed approach in reducing energy consumption and ensuring safe
navigation via extensive simulations involving curved roads and multiple
obstacles.

###### Index Terms:

Electric vehicle, Collision avoidance, Trajectory planning, Energy-aware
driving, Data-driven robust optimization.

##  I Introduction

Electric vehicles (EVs) have seen significant growth in the past few years
being a cleaner alternative to vehicles with internal combustion engines.
Advanced Driver Assist Systems (ADAS) for EVs need to fulfil two primary
objectives. First, it is essential for the ADAS to consider energy efficient
driving in order to be able to extend the driving range in the face of battery
storage limitations. The second objective is to ensure safe collision-free
operation of the vehicle in uncertain environment of roads with dynamic
obstacles represented by nearby vehicles. In this work, we propose a data-
driven robust receding horizon optimization framework for energy efficient and
collision-free navigation of EVs.

Several past works have explored the problem of energy-efficient driving in
EVs, and proposed eco-driving [1, 2, 3] and eco-routing [4] solutions in order
to improve driving range of EVs. However, these works consider only the effect
of longitudinal dynamics on energy consumption, and further, they do not
include safety related considerations. While a few recent works consider
energy efficiency as well as safety aspects under Ecological Driver Assist
System (EDAS) [5, 6], they mostly deal with the Adaptive Cruise Control (ACC)
mode, and consider only the effect of longitudinal dynamics on energy
consumption. In contrast, a few recent works have shown that lateral motion of
an EV has a significant impact on its energy consumption [7, 8, 9]. This
finding is highly relevant in the context of safe (collision-free) navigation
which results in lateral maneuver of the vehicle during overtaking and lane
changing operations, especially on curved roads.

Complementary to the above line of work, there exists a substantial body of
literature on path planning and collision avoidance for (autonomous) vehicles
[10, 11, 12]. Receding horizon or model predictive control (MPC) based
approaches are being used widely for ADAS functionalities such as lane change,
overtaking and collision avoidance, due to its real-time planning ability
while adhering to constraints imposed by vehicle dynamics and obstacles [12,
13, 14]. Specifically, [14] formulates collision avoidance constraints for
convex polytopic representation of obstacles. However, these works assume
obstacles to be static and ignore the dynamic evolution of their positions
over time.

Several recent papers have tackled the problem of safe navigation in presence
of dynamic obstacles using data-driven or stochastic optimization and MPC
techniques. The work [15] utilizes scenario MPC for trajectory planning during
lane change operation initiated by the driver. Similarly, [16] proposes
stochastic MPC formulation with collision avoidance constraints by
approximating the motion of surrounding vehicles by linear dynamics. The
authors in [17] combine physics-based and data-driven intention-based models
for future trajectory prediction of obstacles, and formulate a tube-based MPC
to track a given reference trajectory generated by considering a predicted
obstacle trajectory. Similarly, [18] presents a trajectory planning framework
which estimates intentions of dynamic obstacles. The work [19] predicts multi-
modal probabilistic occupancy of surrounding vehicles for a given maneuver
set, and generates target trajectory while maintaining safe distance from the
predicted occupancy set. Authors in [20] formulate distributionally robust MPC
for safe movement of the ego object when the obstacles have random
perturbations around a nominal position.

However, to the best of our knowledge, there has not been any major work that
incorporates the effect of energy consumption in the safe trajectory planning
problem. This is particularly important in the context of EVs since collision
avoidance often involves lateral motion (lane changing), which has a
significant impact on energy consumption, and therefore on the driving range
of EVs [7, 8].

In this work, we aim to address the above research gap by addressing the
problems of energy-aware and safe navigation simultaneously. We consider a
dynamic model of the EV that captures both longitudinal and lateral motion as
well as the dynamics of stored energy level in the battery. In a departure
from most past works that capture vehicle dynamics using a bicycle model and
assume vehicle motion along a straight road, we include the vehicle positions
in both the inertial and the body frames as state variables to capture motion
along curved roads. Instead of adopting a model-based approach to predict
future trajectory of the obstacles, we adopt a data-driven approach where past
data on the motion of surrounding obstacles is used to construct a polyhedral
occupancy sets for each obstacle over a finite prediction horizon. These sets
are constructed using Principal Component Analysis (PCA). The analysis in [21,
22] provides finite-sample guarantees on these occupancy (uncertainty) sets
containing the future positions of the obstacle with high probability. We then
derive robust collision avoidance constraints which guarantee that the EV will
not intersect with the future occupancy sets over the entire prediction
horizon. Finally, a finite horizon constrained optimization is presented which
minimizes control efforts, distance from a specified destination point, and
energy consumption from the battery, while satisfying constraints on robust
collision avoidance, and bounds on states and control inputs. This problem is
solved repeatedly in a receding horizon manner similar to MPC.

We evaluate the proposed approach via simulations on two navigation scenarios,
including motion on a curved road and multiple obstacles. Our results show
that under the proposed controller, the EV is able to reach the destination
while avoiding obstacles, and consumes significantly less amount of energy
compared to an energy-unaware scheme (where the penalty on energy consumption
is removed from the cost function) with a modest increase in travel time.

##  II Energy-Aware Safe Navigation Framework

We tackle the problem of energy-aware safe navigation as an optimal control
problem with robust collision avoidance constraints. The components of the
stated problem, including cost function, EV dynamics, and collision avoidance
constraints, formulated leveraging past data, are described in this section.

###  II-A Plant Model

We consider a bicycle model [23] of a rear-wheel-driven EV with states given
by coordinate of the center of gravity (COG) along longitudinal axis of EV
($s_{x}$) in the body frame, lateral deviation ($e_{y}$) and relative yaw
angle ($e_{\psi}$) with respect to center-line of road, position of COG of the
EV ($p_{x},p_{y}$) in the inertial frame, velocity along longitudinal and
lateral axes of the EV ($v_{x},v_{y}$) in the body frame, rotation of the
vehicle frame about the $z$-axis ($\psi$) in the inertial frame, yaw rate
($r$) in the body frame, and state-of-energy of the battery ($\gamma$). There
are three inputs being applied to the EV: acceleration along longitudinal axis
($a$), braking deceleration along longitudinal axis ($d$), and steering angle
($\delta$). The continuous-time dynamics of EV with state vector
$x:=[s_{x},e_{y},e_{\psi},v_{x},v_{y},r,\gamma,p_{x},p_{y},\psi]^{\intercal}\in\mathbb{R}^{10}$
and input vector $u:=[a,\delta,d]^{\intercal}\in\mathbb{R}^{3}$ are given by

$\displaystyle\dot{s}_{x}$ $\displaystyle=v_{x},$ (1a)
$\displaystyle\dot{e}_{y}$ $\displaystyle=v_{x}+e_{\psi}v_{y},$ (1b)
$\displaystyle\dot{e}_{\psi}$ $\displaystyle=v_{y}-v_{x}\rho(s_{x}),$ (1c)
$\displaystyle\dot{v_{x}}$
$\displaystyle=\frac{F_{x}}{m}-\left(\frac{F_{F,y}\sin\delta-
mv_{y}r}{m}\right)$ (1d) $\displaystyle=a+d-\left(\frac{F_{F,y}\sin\delta-
mv_{y}r}{m}\right),$ $\displaystyle\dot{v_{y}}$
$\displaystyle=\frac{(F_{F,y}\cos\delta+F_{R,y}-mv_{x}r)}{m},$ (1e)
$\displaystyle\dot{r}$ $\displaystyle=\frac{(F_{F,y}l_{F}\cos\delta-
F_{R,y}l_{R})}{I_{z}},$ (1f) $\displaystyle\dot{\gamma}$
$\displaystyle=\frac{-\eta_{b}P_{b}}{E_{b}},$ (1g) $\displaystyle\dot{p}_{x}$
$\displaystyle=v_{x}\cos\psi-v_{y}\sin\psi,$ (1h) $\displaystyle\dot{p}_{y}$
$\displaystyle=v_{x}\sin\psi+v_{y}\cos\psi,$ (1i) $\displaystyle\dot{\psi}$
$\displaystyle=r.$ (1j)

Figure 1: Schematic of different kinematic and dynamic variables associated with the bicycle model. Table I: Nomenclature Parameters |  Symbols
---|---
Mass |  $m$
Yaw polar inertia |  $I_{z}$
Longitudinal distance from COG to front axle |  $l_{F}$
Longitudinal distance from COG to rear axle |  $l_{R}$
Front tire slip angle |  $\alpha_{F}$
Rear tire slip angle |  $\alpha_{R}$
Front tire corner stiffness |  $C_{F}$
Rear tire corner stiffness |  $C_{R}$
Front wheel lateral force |  $F_{F,y}$
Rear wheel lateral force |  $F_{R,y}$
Longitudinal force acting at COG of vehicle |  $F_{x}$
Curvature of road |  $\rho$
Slip angle of EV |  $\beta_{dyn}$
Differential gear ratio |  $N_{d}$
Wheel radius |  $r_{w}$
Drivetrain efficiency |  $\eta_{i}$
Battery charging/discharging efficiency |  $\eta_{b}$
Energy storage capacity of the battery |  $E_{b}$
Battery power |  $P_{b}$
Motor Torque |  $\tau_{m}$
Motor speed |  $\omega_{m}$

The notation used in the above equations for different EV parameters are
summarized in Table I, and a schematic of the model is presented in Figure 1.
In particular, the force terms $F_{F,y}$ and $F_{R,y}$ are given by

$\displaystyle F_{F,y}$ $\displaystyle=-2C_{F}\alpha_{F},\quad
F_{R,y}=-2C_{R}\alpha_{R},$ (2a) $\displaystyle\alpha_{F}$
$\displaystyle=\arctan\left(\frac{v_{y}+l_{F}r}{v_{x}}\right)-\delta,$ (2b)
$\displaystyle\alpha_{R}$
$\displaystyle=\arctan\left(\frac{v_{y}-l_{R}r}{v_{x}}\right).$ (2c)

For a highly efficient inverter, the battery power $P_{b}$ (assumed to be
positive during discharging and negative during charging/regeneration) can be
approximated in terms of motor input power and motor power loss, both of which
depend on the motor torque $\tau_{m}$ and motor speed $\omega_{m}$.
Specifically,

$\displaystyle\tau_{m}$
$\displaystyle=\frac{mar_{w}}{\eta_{i}N_{d}},\quad\omega_{m}=N_{d}\frac{v_{x}}{r_{w}},$
(3a) $\displaystyle P_{b}$
$\displaystyle=\tau_{m}\omega_{m}+P_{\mathtt{Loss}},$ (3b) $\displaystyle
P_{\mathtt{Loss}}$
$\displaystyle=k_{c}\tau_{m}^{2}+k_{i}\omega_{m}+k_{w}\omega_{m}^{3}+k_{f}\omega_{m}^{2}+k_{0},$
(3c)

where $P_{\mathtt{Loss}}$ represents power loss in the motor assumed to be a
function of $\tau_{m}$ and $\omega_{m}$ with coeffcient of copper loss
$k_{c}$, iron loss coefficient $k_{i}$, the windage loss coefficient $k_{w}$,
the friction loss coefficient $k_{f}$, and $k_{0}$ being a constant component
[24]. Consequently, we approximate battery power as a cubic polynomial of the
motor speed and motor torque given by

$\displaystyle P_{b}(\omega_{m},\tau_{m})$
$\displaystyle=c_{1}+c_{2}\omega_{m}+c_{3}\tau_{m}+c_{4}\omega_{m}^{2}+c_{5}\omega_{m}\tau_{m}$
$\displaystyle\quad+c_{6}\tau_{m}+c_{7}\omega_{m}^{3}.$ (4)

The coefficients are obtained using least squares regression. For simplicity,
we assume that efficiency during charging and discharging operation of the
battery are identical.

Note that in order to tackle navigation along curved roads, we have included
position and orientation in both inertial and body coordinate systems in our
state variables. This is in contrast with most prior works which consider
vehicle motion along straight roads only. Previewed road curvature $\rho$ is
assumed to be a function of $s_{x}$ given by

$\rho(s_{x}):=k_{1}s_{x}^{2}+k_{2}s_{x}+k_{3},$ (5)

with the coefficients obtained using least square regression.

The continuous-time model (1) can be represented in compact form as
$\dot{x}=f(x,u)$. The discrete-time EV model is obtained after Euler
discretization of the continuous-time model with sampling time $T_{s}$, and is
given by

$\displaystyle x_{k+1}$ $\displaystyle=x_{k}+T_{s}f(x_{k},u_{k}).$ (6)

###  II-B Nominal Constraints

The EV dynamic variables need to satisfy the following constraints on the
inputs and states:

$\displaystyle a_{\min}$ $\displaystyle\leq a\leq a_{\max},$ (7a)
$\displaystyle d_{\min}$ $\displaystyle\leq d\leq d_{\max},$ (7b)
$\displaystyle\delta_{\min}$ $\displaystyle\leq\delta\leq\delta_{\max},$ (7c)
$\displaystyle v_{x,\min}$ $\displaystyle\leq v_{x}\leq v_{x,\max},$ (7d)
$\displaystyle e_{y,\min}$ $\displaystyle\leq e_{y}\leq e_{y,\max},$ (7e)
$\displaystyle\gamma_{\min}$ $\displaystyle\leq\gamma\leq\gamma_{\max}.$ (7f)

These include actuation constraints (7a), (7b), (7c), longitudinally forward
movement of EV (7d), movement within road boundary (7e), and safe operating
region of battery (7f). Note that constraints on rate of the change of the
above variables can also be included without increasing the complexity of the
optimization problem.

In addition, in order to operate the motor in a high efficiency region of the
motor torque-speed curve [25], we impose the following constraints on the
motor torque ($\tau_{m}=\frac{mr_{w}}{\eta_{i}N_{d}}a$):

$\displaystyle-\tau_{m,\max}(v_{x})\leq\frac{mr_{w}}{\eta_{i}N_{d}}a\leq\tau_{m,\max}(v_{x}),\quad\text{where}$
(8a)
$\displaystyle\tau_{m,\max}(v_{x})=\xi_{1}v_{x}^{3}+\xi_{2}v_{x}^{2}+\xi_{3}v_{x}+\xi_{4}.$
(8b)

The above bound on the motor torque is motivated by typical motor torque-speed
efficiency curve which can be approximated by a cubic polynomial of motor
speed $\omega_{m}$ (and thus, longitudinal speed $v_{x}$ which is the state
variable). The coefficients are determined using least squares regression.

###  II-C Data-Driven Robust Collision Avoidance

We now formulate the collision avoidance constraints. We assume that the EV is
a point object represented by its position
$p_{k}:=[p_{x,k},p_{y,k}]^{\intercal}\in\mathbb{R}^{2}$ at a discrete time
instant $k$. Let there be $M$ number of obstacles, and
$\mathbb{O}^{(m)}_{k}\subset\mathbb{R}^{2}$ be the space occupied by $m$-th
obstacle at time instant $k$. Then, to guarantee collision free navigation
over a finite horizon of length $N$, we require the following condition to
hold:

$\displaystyle\\{p_{k}\\}\cap\mathbb{O}^{(m)}_{k}=\emptyset,\quad\forall
m\in[M],\forall k\in\\{0\\}\cup[N].$ (9)

where $[S]:=\\{1,2,\ldots S\\}$ for brevity of notation. The safety constraint
can be formulated in terms of a distance function:

$\displaystyle\mathtt{dist}(p_{k},\mathbb{O}^{(m)}_{k})\\!=\\!\min_{t}\\{||t||_{2}:p_{k}+t\in\\!\mathbb{O}^{(m)}_{k}\\}\\!>\\!d_{\mathtt{safe}},$
(10)

where $d_{\mathtt{safe}}>0$ represents a safety margin on the distance between
the EV and the obstacle. We omit the superscript $(m)$ for better readability
since the discussion is applicable for any of the $M$ obstacles. For
simplicity, we assume a constant value of $d_{\mathtt{safe}}$ for each
obstacle.

At a given time, denoted $k=0$ for ease of notation, the area enclosing the
obstacle is assumed to be a polyhedron given by

$\displaystyle\mathbb{O}_{0}$ $\displaystyle=\\{y\in\mathbb{R}^{2}:Ay\leq
b\\}.$ (11)

Since we need to impose collision avoidance constraints over a prediction
horizon of finite length $N$, we need to make suitable assumptions on how the
occupancy set of such an obstacle evolves over time. In particular, we assume
that orientation of this obstacle in the inertial frame remains unchanged, but
this set $\mathbb{O}_{0}$ may translate by a random vector in
$\mathbb{R}^{2}$. Specifically, the occupied space at discrete time instant
$k+1$, denoted $\mathbb{O}_{k+1}$, is represented as

$\displaystyle\mathbb{O}_{k+1}=\mathbb{O}_{0}\oplus\Omega_{k},$ (12)

where, $\oplus$ denotes Minkowski sum of sets, and $\Omega_{k}$ is the support
of random variable that denotes $k$ step translation of the obstacle. We now
present a data-driven approach to construct the set $\Omega_{k}$ from past
data on obstacle position. We then derive robust collision avoidance
constraints for $\Omega_{k}$ constructed from past-data.

####  II-C1 Data-Driven Uncertainty Set Generation

At each time, we observe the current position of the obstacle, and obtain the
change in its position from its position $k$ time steps earlier for $k\in[N]$.
Formally, let $\omega^{k}_{i}\in\mathbb{R}^{2}$ denote the difference between
the current position of the obstacle and the position of the obstacle $k$
steps prior to the current time. For each $k\in[N]$, we collect $N_{s}$ number
of such samples or scenarios denoted by
$W^{k}=\\{\omega^{k}_{1},\ldots,\omega^{k}_{N_{s}}\\}\subseteq\mathbb{R}^{2}$
from past $N_{s}$ time points. We now construct a polyhedral uncertainty set
in $\mathbb{R}^{2}$, denoted by $\Omega(W^{k})$, using Principal Component
Analysis (PCA) following [21, 22]. The superscript $k$ is suppressed for
better readability.

We first compute the sample mean of $W$ as
$\bar{\omega}:=\frac{1}{N_{s}}\sum_{j=1}^{N_{s}}\omega_{j}$. We then subtract
this mean from each scenario to obtain the “centered” data
$\omega_{j0}=\omega_{j}-\bar{\omega}$ for every sample $j\in[N_{s}]$. We
construct the data matrix $\textbf{Z}_{0}\in\mathbb{R}^{N_{s}\times 2}$, where
$j$-th row represents $\omega^{\intercal}_{j0}\in\mathbb{R}^{1\times 2}$, and
compute the sample covariance matrix

C $\displaystyle=\frac{1}{N_{s}-1}\textbf{Z}^{\intercal}_{0}\textbf{Z}_{0}.$

With $\textbf{Z}^{\intercal}_{0}=U\Sigma V^{\intercal}$, we perform eigen-
decomposition on C as

C
$\displaystyle=U\left(\frac{\Sigma\Sigma^{\intercal}}{N_{s}-1}\right)U^{\intercal}=U\Lambda
U^{\intercal},$

where $U=[d_{1},d_{2}]\in\mathbb{R}^{2\times 2}$, $V\in\mathbb{R}^{N_{s}\times
N_{s}}$ and $\Sigma\in\mathbb{R}^{2\times N_{s}}$. The columns of $U$, $d_{1}$
and $d_{2}$, are principal directions of the centered data. Following [21,
22], we obtain a polyhedral uncertainty set approximation given by

$\displaystyle\Omega(W)=\biggl{\\{}y:y=\bar{\omega}+\sum_{i=1}^{2}\left(\alpha_{i}\frac{\bar{\omega}_{i}}{||d_{i}||}d_{i}+(1-\alpha_{i})\frac{\underline{\omega}_{i}}{||d_{i}||}d_{i}\right),$
$\displaystyle\quad\qquad\qquad 0\leq\alpha_{i}\leq 1,\forall
i\in\\{1,2\\}\biggl{\\}},\quad\text{where}$
$\displaystyle\bar{\omega}_{i}=\max_{1\leq j\leq
N_{s}}\left\\{\frac{\omega_{j0}^{\intercal}d_{i}}{||d_{i}||}\right\\}\in\mathbb{R},\quad\underline{\omega}_{i}=\min_{1\leq
j\leq
N_{s}}\left\\{\frac{\omega_{j0}^{\intercal}d_{i}}{||d_{i}||}\right\\}\in\mathbb{R}.$

We now present the following result from [21] which states probabilistic
guarantees on $\Omega(W)$ defined above.

######  Theorem 1 ([21],Theorem 2).

If
$N_{s}\geq\lceil\frac{1}{\epsilon}\frac{e}{e-1}(2m-1+\ln\frac{1}{\beta})\rceil$,
then we have $1-\beta$ confidence that any realisation $\omega$ belongs to the
uncertainty set $\Omega(W)$ with the probability of at least $1-\epsilon$,
i.e.

$\displaystyle\mathbb{P}_{W}\\{\mathbb{P}_{\omega}\\{\omega\in\Omega(W)\\}\geq
1-\epsilon\\}\geq 1-\beta,$

where $\epsilon\in(0,1),\beta\in(0,1)$.

Thus, for each $k\in[N]$, we construct a polyhedral uncertainty set
$\Omega(W^{k})$ that contains $k$-step change in position of the obstacle with
high probability from past data. The set $\Omega(W^{k})$ is oriented along the
principal directions of the centered data and encloses all points in the
dataset $W^{k}$. It can be equivalently expressed in the form

$\displaystyle\Omega(W^{k})=\\{\omega\in\mathbb{R}^{2}:G_{k}\omega\leq
h_{k}\\}.$ (13)

####  II-C2 Formulation of Robust Collision Avoidance Constraints

In this subsection, we formulate the robust collision avoidance constraints
for the EV. We treat the data-driven uncertainty set $\Omega(W^{k})$ defined
in (13) to be the support of the random variable that represents $k$-step
translation of the obstacle in (12). We now reformulate the safety constraint
(10).

######  Proposition 1.

Assume that at time $k+1$, the obstacle occupancy set
$\mathbb{O}_{k+1}:=\mathbb{O}_{0}\oplus\Omega(W^{k})$, the EV position is
given by $p_{k+1}$, and let $d_{\mathtt{safe}}>0$ be a desired safety margin
between the controlled object and the obstacle. Then, we have

$\displaystyle\mathtt{dist}(p_{k+1},\mathbb{O}_{k+1})>d_{\mathtt{safe}}$
$\displaystyle\Leftrightarrow$
$\displaystyle\mathtt{dist}(p_{k+1},\mathbb{O}_{0}+\omega_{k})>d_{\mathtt{safe}},\qquad\forall\omega_{k-1}\in\Omega(W^{k})$
$\displaystyle\Leftrightarrow$ $\displaystyle\exists\lambda_{k}\geq
0,\mu_{k}\geq
0:d_{\mathtt{safe}}-(Ap_{k+1}-b)^{\intercal}\lambda_{k}+\mu_{k}^{\intercal}h_{k}<0,$
$\displaystyle\qquad\qquad\qquad\quad||A^{\intercal}\lambda_{k}||_{2}\leq
1,A^{\intercal}\lambda_{k}=G_{k}^{\intercal}\mu_{k},$

where $\Leftrightarrow$ indicates “if and only if” condition.

###### Proof.

For a given translation $\omega^{k}\in\Omega(W^{k})$ of the obstacle, we
define the distance between the EV and the obstacle as

$\displaystyle\mathtt{dist}(p_{k+1},\mathbb{O}_{0}\\!\oplus\\!\\{\omega^{k}\\})\\!=\min_{t}\\{||t||:A(p_{k+1}+t-\omega^{k})\leq
b\\}$
$\displaystyle=\max_{\lambda_{k}}\\{(Ap_{k+1}-A\omega^{k}-b)^{\intercal}\lambda_{k}:\lambda_{k}\geq
0,||A^{\intercal}\lambda_{k}||_{2}\leq 1\\},$

which follows from [14, Proposition 1]. The condition for collision avoidance
is given by

$\displaystyle
d_{\mathtt{safe}}-\mathtt{dist}(p_{k+1},\mathbb{O}_{k+1})<0\quad\forall\omega^{k}\in\Omega(W^{k})$
$\displaystyle\Leftrightarrow$
$\displaystyle\max_{\begin{subarray}{c}\omega^{k}\in\Omega(W^{k})\end{subarray}}[d_{\mathtt{safe}}-\max_{\lambda_{k}}\\{(Ap_{k+1}-A\omega^{k}-b)^{\intercal}\lambda_{k}:$
$\displaystyle\qquad\lambda_{k}\geq 0,||A^{\intercal}\lambda_{k}||_{2}\leq
1\\}]<0.$ (14)

The L.H.S. of the above equation can be stated as

$\displaystyle\max_{\begin{subarray}{c}\omega^{k}\in\Omega(W^{k})\end{subarray}}\min_{\begin{subarray}{c}\lambda_{k}\geq
0,\\\ ||A^{\intercal}\lambda_{k}||_{2}\leq
1\end{subarray}}\left[d_{\mathtt{safe}}-(Ap_{k+1}-A\omega^{k}-b)^{\intercal}\lambda_{k}\right]$
$\displaystyle\leq\min_{\begin{subarray}{c}\lambda_{k}\geq 0,\\\
||A^{\intercal}\lambda_{k}||_{2}\leq
1\end{subarray}}\max_{\begin{subarray}{c}\omega^{k}\in\Omega(W^{k})\end{subarray}}\left[d_{\mathtt{safe}}-(Ap_{k+1}-A\omega^{k}-b)^{\intercal}\lambda_{k}\right]$
$\displaystyle=\min_{\begin{subarray}{c}\lambda_{k}\geq 0,\\\
||A^{\intercal}\lambda_{k}||_{2}\leq
1\end{subarray}}d_{\mathtt{safe}}-(Ap_{k+1}-b)^{\intercal}\lambda_{k}+\max_{\omega^{k}\in\Omega(W^{k})}(\lambda_{k}^{\intercal}A)\omega^{k},$

where the first inequality is the standard min-max inequality
$\left(\max\;\min\;[\;\;]\right)\leq\left(\min\;\max\;[\;\;]\right)$. Since,
$\omega^{k}\in\Omega(W^{k})=\\{\omega^{k}\in\mathbb{R}^{2}:G_{k}\omega^{k}\leq
h_{k}\\}$, strong duality in linear programming yields

$\displaystyle\max_{\omega^{k}\in\Omega(W^{k})}(\lambda^{\intercal}_{k}A)\omega^{k}\Longleftrightarrow\min_{\begin{subarray}{c}\mu_{k}\geq
0,\\\
A^{\intercal}\lambda_{k}=G_{k}^{\intercal}\mu_{k}\end{subarray}}\mu_{k}^{\intercal}h_{k}.$

Therefore, the L.H.S. of (14) is upper bounded by the optimal value of the
following optimization problem:

$\displaystyle\min_{\lambda_{k},\mu_{k}}\quad$ $\displaystyle
d_{\mathtt{safe}}-(Ap_{k+1}-b)^{\intercal}\lambda_{k}+\mu_{k}^{\intercal}h_{k}$
subject to $\displaystyle\lambda_{k}\geq
0,\quad||A^{\intercal}\lambda_{k}||_{2}\leq 1,$ $\displaystyle\mu_{k}\geq
0,\quad A^{\intercal}\lambda_{k}=G_{k}^{\intercal}\mu_{k}.$

Therefore, the following is a sufficient condition for the collision avoidance
constraint to hold in a robust manner for all perturbations that belong to the
support set $\Omega(W^{k})$:

$\displaystyle\exists\lambda_{k}\geq 0,\mu_{k}\geq 0:$ $\displaystyle
d_{\mathtt{safe}}-(Ap_{k+1}-b)^{\intercal}\lambda_{k}+\mu_{k}^{\intercal}h_{k}<0,$
$\displaystyle||A^{\intercal}\lambda_{k}||_{2}\leq
1,A^{\intercal}\lambda_{k}=G_{k}^{\intercal}\mu_{k}.$

This concludes the proof. ∎

The constraints derived in above proposition guarantee that at time $k+1$, the
EV is at least $d_{\mathtt{safe}}$ distance away from the occupancy region of
the obstacle for all possible change in its position from the data-driven
uncertainty set $\Omega(W^{k})$. Since the true displacement of the obstacle
is expected to lie within the uncertainty set with high probability, it
follows from Theorem 1 that the distance between the ego vehicle and the
obstacle is guaranteed to be larger than $d_{\mathtt{safe}}$ with high
probability.

A static obstacle remains at its initial position for all time. In that case,
uncertainty set $\Omega(W^{k}):=\\{0\\}$. Consequently, constraints in
Proposition 1 reduce to

$\displaystyle\mathtt{dist}(p_{k+1},\mathbb{O}_{0})>d_{\mathtt{safe}}$
$\displaystyle\Leftrightarrow$ $\displaystyle\exists\lambda_{k}\geq
0:(Ap_{k+1}-b)^{\intercal}\lambda_{k}>d_{\mathtt{safe}},||A^{\intercal}\lambda_{k}||_{2}\leq
1,$

which coincides with the constraints developed in [14] for static obstacles.

###  II-D Constrained Optimal Control Problem Formulation

We now formulate the constrained optimal control problem for energy-aware and
safe navigation of the EV. The goal of the EV is to reach a desired
destination whose coordinates ($s_{x,\mathtt{dest}},e_{y,\mathtt{dest}})$ are
assumed to provided by route level planner module or the driver while avoiding
collision. The $e_{y,\mathtt{dest}}$ component captures geometry of the road.

Let the current time be denoted $k=0$ for ease of notation. Let the current
state $x_{0}$ be known. Let $u_{0:(N-1)}:=\\{u_{0},u_{1},\ldots,u_{N-1}\\}$
denote a sequence of control inputs over a finite horizon of length $N$, and
let $x_{1:N}:=\\{x_{1},x_{2},\ldots,x_{N}\\}$ be the resultant state
trajectory. Consequently, we define the cost incurred by this trajectory as

$\displaystyle J$
$\displaystyle=\sum_{k=0}^{N-1}\Big{[}Q_{1}\Bigl{(}\frac{e_{y,k}\\!-\\!e_{y,\mathtt{dest}}}{e_{y,\max}}\Bigl{)}^{2}+Q_{2}\Bigl{(}\frac{e_{\psi,k}}{e_{\psi,\max}}\Bigl{)}^{2}$
$\displaystyle\quad+Q_{3}(\gamma_{k}\\!-\\!\gamma_{\max})^{2}\\!+\\!R_{1}\Bigl{(}\frac{a_{k}}{a_{\max}}\Bigl{)}^{2}\\!+\\!R_{2}\Bigl{(}\frac{\delta_{k}}{\delta_{\max}}\Bigl{)}^{2}$
$\displaystyle+\\!R_{3}\Bigl{(}\frac{d_{k}}{d_{\min}}\Bigl{)}^{2}\Big{]}\\!+P_{1}\Bigl{(}\frac{s_{x,N}\\!-\\!s_{x,\mathtt{dest}}}{s_{x,\mathtt{dest}}}\Bigl{)}^{2}\\!\\!+\\!P_{2}(\gamma_{N}\\!-\\!\gamma_{\max})^{2},$
(15)

where $Q_{i},R_{i}$ and $P_{i}$ denote the coefficients of the cost function
pertaining to stage cost, cost of control inputs and terminal cost,
respectively. The first two terms minimize distance from the destination and
road boundary, the third term increases the cost when state of energy of the
battery $\gamma$ decreases from the maximum allowed value $\gamma_{\max}$, the
next three terms penalize large control efforts, and the last two terms
penalize terminal distance from the destination and deviation from
$\gamma_{\max}$.

The resulting finite horizon constrained optimal control problem can be
compactly stated as:

$\displaystyle\min\quad$
$\displaystyle\sum_{k=0}^{N-1}\Big{[}||x_{k}-x_{\mathtt{dest}}||_{Q}+||u_{k}||_{R}\Big{]}+||x_{N}-x_{\mathtt{dest}}||_{P},$
(16a) w.r.t. $\displaystyle
x_{1:N},u_{0:(N-1)},\lambda^{(1)}_{0:(N-1)}\ldots\lambda^{(M)}_{0:(N-1)},$
(16b) $\displaystyle\mu^{(1)}_{0:(N-1)}\ldots\mu^{(M)}_{0:(N-1)}$ s.t.
$\displaystyle x_{k+1}=x_{k}+T_{s}f(x_{k},u_{k}),$ (16c) $\displaystyle
u_{\min}\leq u_{k}\leq u_{\max},$ (16d) $\displaystyle x_{\min}\leq x_{k}\leq
x_{\max},$ (16e)
$\displaystyle-\tau_{m,\max}(x_{k})\leq\tau_{m,k}\leq\tau_{m,\max}(x_{k}),$
(16f)
$\displaystyle(A^{(m)}p_{k+1}-b^{(m)})^{\intercal}\lambda^{(m)}_{k}-\mu_{k}^{(m)^{\intercal}}h_{k}^{(m)}>d_{\mathtt{safe}},$
(16g) $\displaystyle\mu_{k}^{(m)}\geq
0,\quad\mu_{k}^{(m)^{\intercal}}G_{k}^{(m)}=\lambda^{(m)^{\intercal}}_{k}A^{(m)},$
(16h) $\displaystyle\lambda^{(m)}_{k}\geq
0,\quad||A^{(m)^{\intercal}}\lambda^{(m)}_{k}||_{2}\leq 1,$ (16i)
$\displaystyle x_{0}\quad\text{given},$ (16j)

where the notation $||v||_{W}$ in (16a) denotes quadratic function
$v^{\intercal}Wv$, the equality constraint representing EV dynamics in (16c)
holds for $k\in\\{0,1,\ldots,N-1\\}$, the constraints on input (16d) encode
(7a)-(7c), the constraint (16f) encodes (8a) (and is a function of
longitudinal acceleration input $a$), and hold for $k\in\\{0,1,\ldots,N-1\\}$.
Similarly, the bounds on state variables (16e) encode (7d)-(7f) and these
constraints hold for $k\in[N]$. Finally, the constraints (16g)-(16i) encode
robust collision-avoidance constraints derived in Proposition 1 for $M$
dynamic obstacles and hold for all $m\in[M]$ and $k\in\\{0,1,\ldots,N-1\\}$.

At every sampling instant, the position and orientation of each obstacle is
obtained, and the set $\mathbb{O}_{0}$ is constructed. Then, a set of new
samples are collected after computing the deviation of the obstacles from
their past positions. Then, the occupancy sets (13) are recomputed using the
PCA approach. Then, the above optimization problem, encoding robust collision
avoidance constraints and penalizing energy consumption, is solved. The first
component of the control input is applied to the EV, the state and obstacles
are allowed to evolve, and the process repeats in a receding horizon manner.

The proposed MPC controller is in contrast with approaches where a path
planning algorithm is used to compute a collision-free path which is then
tracked by a low level tracking controller. Essentially, we combine both
trajectory planning and path tracking functionalities in the proposed
approach. We clarify that while we impose robust collision avoidance
constraints on the MPC optimization problem, the proposed approach is distinct
from robust MPC where robustness is with respect to uncertainty in the plant
dynamics.

(a) Navigation without energy consideration

(b) Energy aware navigation

(c) EV state trajectory

Figure 2: (a) Subplots (downwards) show EV trajectory evolution under the
energy-unaware scheme at time instants $9.1$, $9.8$, $10.5$ and $11.2$
seconds. (b) Subplots (downwards) show EV trajectory evolution under the
energy-aware scheme at time instants $9.3$, $10.2$, $11.1$ and $12$ seconds.
For (a) and (b), Magenta: Trajectory of EV, Red: Obstacle, Yellow: Future
occupancy set. (c) EV state evolution with time. Blue: energy consumption is
not penalized and Red: energy consumption is penalized. In last subplot, Grey:
$d_{\mathtt{safe}}$, Blue/Red: $d_{\mathtt{EO}}$.

##  III Simulation Results

We now illustrate the effectiveness of the presented energy-aware collision-
free navigation framework through simulations. The EV model is developed in
Matlab 2021b based on reference applications [26], [27]. The values of various
parameters related to vehicle dynamics and cost function in (16) used in the
simulation are given in Table II.

Parameters | Values | Parameters | Values
---|---|---|---
$m$ |  $1500$ kg | $a_{\max},a_{\min}$ |  $7.4,-4$ $\text{m}/\text{s}^{2}$
$I_{z}$ |  $3000$ kg$\text{m}^{2}$ | $\delta_{\max},\delta_{\min}$ |  $0.5,-0.5$ rad
$l_{F}$ |  $1.188$ m | $v_{x,\max},v_{x,\min}$ |  $30,0$ m
$l_{R}$ |  $1.512$ m | $p_{y,\max},p_{y,\min}$ |  $4,-4$m
$C_{\alpha F}$ | $6.3\times 10^{4}\text{N}/\text{rad}$ | $d_{\max},d_{\min}$ |  $0,-5.75$$\text{m}/\text{s}^{2}$
$N_{d}$ | $7.94$ | $\gamma_{\min},\gamma_{\max}$ | $0.1,0.9$
$C_{\alpha R}$ | $6.3\times 10^{4}\text{N}/\text{rad}$ | $N,N_{s}$ | $20,85$
$r_{w}$ |  $0.33$ m | $T_{s}$ |  $0.1$s
$E_{b}$ |  $54.28$ kWh | $\eta_{b}$ | $1$
$R$ | $\mathtt{diag}(0.0025,0.0125,0.05)$
$P$ | $\mathtt{diag}(0.4,0.0048,5\times 10^{-4},0,0,0,10,0,0,0)$
$Q$ | $\mathtt{diag}(0,0.0048,5\times 10^{-4},0,0,0,10,0,0,0)$
Table II: Simulation Parameters

The coefficients for $P_{b}$ as a function of motor torque ($\tau_{m}$) and
motor speed ($\omega_{m}$) as stated in (4), and the coefficients of the
bounds on motor torque $\tau_{m,\max}$ as a function of $v_{x}$ are obtained
through least squares curve fitting. These coefficients and accuracy of fit
are given in Tables III and IV, respectively. The coefficients $c_{6}$ and
$c_{7}$ of $P_{b}$ are approximately zero and not included in the table.

Coefficients | $c_{1}$ | $c_{2}$ | $c_{3}$ | $c_{4}$ | $c_{5}$
---|---|---|---|---|---
Values ($\times 10^{-14}$) | $2238$ | $-2.63$ | $8.64$ | $2.06\times 10^{-5}$ | $10^{14}$
R-squared value | 0.99
Table III: Estimated coefficients of the polynomial approximation of battery output power $P_{b}$ in (4). Coefficients | $\xi_{1}$ | $\xi_{2}$ | $\xi_{3}$ | $\xi_{4}$
---|---|---|---|---
Values | $0.0036$ | $-0.3661$ | $3.663$ | $454.2$
R-squared value | 0.974
Table IV: Estimated coefficients of the polynomial approximation of motor
maximum torque limit $\tau_{m,\max}$ in (8a).

The Automated Driving Toolbox of Matlab is used to generate scenarios for
evaluating the performance of the proposed framework. In order to ensure that
stated MPC formulation remains feasible at all time steps and the algorithm
terminates without exceeding maximum allowed iterations, slack variables are
included in the safety constraints (16g) for each obstacle. These slack
variables are penalized with very high cost in the cost function. At every
time instant, The uncertainty set parameters $G$ and $h$ are obtained using
Multi Parametric Toolbox [28]. Nonlinear solver IPOPT [29] is used to solve
the optimization problem in Python using Casadi modelling language [30]. All
computations were carried out on a Workstation Desktop with $128$ GB RAM and
Intel Xeon(R) Gold 6146 CPU @ $3.20\text{GHz}\times 48$ Processor.

The value of $N_{s}$ is chosen to be $85$ to ensure $90\%$ confidence
($\beta=0.1$) for actual change in obstacle position to lie in $\Omega(W)$
with the probability of $0.9$ ($\epsilon=0.1$) in accordance with Theorem 1.
The EV plans and follows a trajectory which avoids static as well as dynamic
obstacles while trying to keep the level of stored energy $\gamma$ close to
$\gamma_{\max}=0.9$. We have chosen sampling time $T_{s}=0.1$ second and
prediction horizon $N=20$ for the MPC formulation (16).

We now discuss the efficacy of proposed approach for two different driving
scenarios.

####  III-1 Overtaking

In this scenario, the obstacle is moving ahead of the EV in the same lane and
decelerates to stop after some time. This could potentially represent a moving
obstacle that slows down to cause a road block on a particular lane. The
target or destination state, $x_{\mathtt{dest}}$, for the EV is set to
$[170,1.3,0,0,0,0,0.9,0,0,0]^{\intercal}$. We compare the energy consumption
and time taken to reach the destination under the proposed energy-aware scheme
with another MPC-based controller that does not penalize energy in its cost
function (with $(\gamma_{k}-\gamma_{\max})^{2}$ terms removed), but maintains
collision avoidance constraints to ensure a fair comparison. The latter scheme
is termed as the energy-unaware scheme.

Figure 3: Lateral power dissipation profile for EV overtake maneuver.

Figure 4: Power flow profile for (top) energy unaware and (bottom) energy
aware EV overtake maneuver.

(a)

(b)

Figure 5: (a) Subplots (left $\rightarrow$ right) show EV trajectory evolution
when energy consumption is not considered at time instants $8.4s$, $10.2s$,
$12s$, $13.9s$, $15.7s$ and $17.5s$. (b) Subplots (left $\rightarrow$ right)
show EV trajectory evolution when energy consumption is considered at time
instants $8.4s$, $10.6s$, $12.9s$, $15.1s$, $17.4s$ and $19.6s$. For (a) and
(b), Magenta: Trajectory of EV, Red, Cyan, Green: Obstacle, Yellow: Future
occupancy sets.

The trajectories of EV under both energy-unaware and energy-aware schemes are
shown in Figures 2(a) and 2(b), respectively. It is evident that the EV
overtakes the obstacle (which is slowing down) while avoiding collision.
Figure 2(c) shows the evolution of EV states for both cases. We found that the
energy consumed under the energy-aware scheme is $27.4$ Wh over a distance of
$150$ meter, while energy consumed is $65.8$ Wh when the penalty term
involving state-of-energy is removed. The vehicle covered this distance in
$11.9$ seconds in the energy-unaware scheme as it performs aggressive
maneuver, while it took $12.8$ seconds in the energy-aware case where the EV
performs smooth longitudinal and lateral maneuvers to reduce energy
consumption. Thus, the proposed scheme was able to achieve a significant
reduction in energy consumption with a minimal increase in travel time. This
observation is consistent with those reported in the literature [31, 32, 33].
The variation of the distance between the EV and the obstacle
($d_{\mathtt{EO}}$) is given in the last subplot of Figure 2(c) which shows
that $d_{\mathtt{EO}}$ always remains greater than $d_{\mathtt{safe}}$ (chosen
to be $2$m).

Figure 3 shows that the amount of lateral dissipation during overtaking is
significant in the energy-unaware case, while it is considerably smaller in
the energy-aware case, which plays a major role in reducing overall energy
consumption. Figure 4 shows the power flow profiles. Under the energy-aware
scheme, a relatively large proportion of maneuver energy is recuperated during
deceleration in comparison to energy-unaware case as shown in the figure.

####  III-2 Multiple Obstacles

In this scenario, the EV is expected to maneuver and reach target states in
presence of multiple obstacles (both static and as dynamic) on a curved road.
Figure 5 shows EV maneuver for energy-unaware (left) and energy-aware (right)
schemes, respectively. The green obstacle is a static obstacle, the cyan
obstacle is far and moves ahead of EV near center of the road, and the red
obstacle is moving ahead of EV in same lane of the EV. Initial target state,
$x_{\mathtt{dest}}$, for EV is set to
$[250,1.3,0,0,0,0,0.9,0,0,0]^{\intercal}$. Despite this challenging
environment, the EV avoids collision with every obstacle and complete the
maneuver in $17.9$s and $20$s for energy-unaware and energy-aware cases,
respectively. Lateral power dissipation and state-of-energy corresponding to
both schemes are shown in Figure 6. There is power dissipation for energy-
unaware case at approximately $10$s and $13$s as the EV overtakes green and
red obstacle respectively. In contrast, a smaller amount of power is
dissipated for the energy-aware case as the EV overtakes the green obstacle.
In this case, the EV follows the red obstacle and does not overtake it due to
energy consideration. Energy consumed under the energy-aware scheme is $47.4$
Wh, while energy consumed is $76.7$ Wh for the energy-unaware scheme for this
example.

Figure 6: Lateral power dissipation and battery state-of-energy profile for EV
maneuver shown in Figure 5. Figure 7: Computation time of the energy aware
safe trajectory planning approach. On the $x$-axis, the number $1$ and $3$
indicate number of obstacles while $a$ indicates horizon $N=10$ and $b$
indicates $N=20$.

The mean computation time for solving the optimization problem (16) together
with confidence intervals for different number of obstacles and prediction
horizons are shown in Figure 7. Although the size of the optimization problem
scales with increase in number of obstacles, the resulting increase in the
computation time is not significant. We emphasize that the computation time
reported here is to illustrate the relative impact of prediction horizon and
number of obstacles. Our optimization program is written in Python which is a
high level programming language. A native C++ implementation together with
multi-core programming to parallelize the task of constructing uncertainty
sets is likely to improve computation time by an order of magnitude.

##  IV Discussion and Conclusion

In this paper, we presented a data-driven robust optimization framework for
energy-aware collision avoidance of EVs in scenarios which consist of static
as well as dynamic obstacles. Simulations performed on a variety of realistic
scenarios illustrate that the proposed formulation achieves a significant
reduction in energy consumption with a minimal increase in travel time
compared to an energy-unaware scheme.

The primary assumption underlying the proposed formulation is that the
orientation of the obstacles are assumed to remain unchanged throughout the
prediction horizon. It is further assumed that battery power consumption is a
function of motor torque and angular velocity, and power loss at the inverter
is negligible. In addition to relaxing the above assumptions, there are
several possible directions in which the present work can be strengthened.
These include (i) handling model uncertainty or mismatch between the nominal
and high fidelity EV models, (ii) formulating the collision avoidance
constraints in terms of chance constraints or conditional-value-at-risk, and
(iii) implementing the proposed approach in real-time hardware-in-loop
simulation environments.

## References

  * [1] H. Lee, N. Kim, and S. W. Cha, “Model-based reinforcement learning for eco-driving control of electric vehicles,” IEEE Access, vol. 8, pp. 202886–202896, 2020.
  * [2] D. Shen, D. Karbowski, and A. Rousseau, “A minimum principle-based algorithm for energy-efficient eco-driving of electric vehicles in various traffic and road conditions,” IEEE Transactions on Intelligent Vehicles, vol. 5, no. 4, pp. 725–737, 2020.
  * [3] W. Dib, A. Chasse, P. Moulin, A. Sciarretta, and G. Corde, “Optimal energy management for an electric vehicle in eco-driving applications,” Control Engineering Practice, vol. 29, pp. 299–307, 2014.
  * [4] Z. Yi and P. H. Bauer, “Optimal stochastic eco-routing solutions for electric vehicles,” IEEE Transactions on Intelligent Transportation Systems, vol. 19, no. 12, pp. 3807–3817, 2018.
  * [5] S. A. Sajadi-Alamdari, H. Voos, and M. Darouach, “Nonlinear model predictive control for ecological driver assistance systems in electric vehicles,” Robotics and Autonomous Systems, vol. 112, pp. 291–303, 2019.
  * [6] M. Kamal, M. Mukai, J. Murata, and T. Kawabe, “Ecological driver assistance system using model-based anticipation of vehicle–road–traffic information,” IET Intelligent Transport Systems, vol. 4, no. 4, pp. 244–251, 2010.
  * [7] S. Kumari, S. Ghosh, A. Hota, and S. Mukhopadhyay, “Energy consumption of electric vehicles: Effect of lateral dynamics,” 97th IEEE Vehicular Technology Conference, 2023 (to appear).
  * [8] F. Ding and H. Jin, “On the optimal speed profile for eco-driving on curved roads,” IEEE Transactions on Intelligent Transportation Systems, vol. 19, no. 12, pp. 4000–4010, 2018.
  * [9] G. N. Bifulco, A. Coppola, S. G. Loizou, A. Petrillo, and S. Santini, “Combined energy-oriented path following and collision avoidance approach for autonomous electric vehicles via nonlinear model predictive control,” in 2021 IEEE International Conference on Environment and Electrical Engineering and 2021 IEEE Industrial and Commercial Power Systems Europe (EEEIC/I&CPS Europe), pp. 1–6, IEEE, 2021.
  * [10] S. Lefèvre, D. Vasquez, and C. Laugier, “A survey on motion prediction and risk assessment for intelligent vehicles,” ROBOMECH journal, vol. 1, no. 1, pp. 1–14, 2014.
  * [11] D. González, J. Pérez, V. Milanés, and F. Nashashibi, “A review of motion planning techniques for automated vehicles,” IEEE Transactions on Intelligent Transportation Systems, vol. 17, no. 4, pp. 1135–1145, 2015.
  * [12] S. Dixit, U. Montanaro, M. Dianati, D. Oxtoby, T. Mizutani, A. Mouzakitis, and S. Fallah, “Trajectory planning for autonomous high-speed overtaking in structured environments using robust MPC,” IEEE Transactions on Intelligent Transportation Systems, vol. 21, no. 6, pp. 2310–2323, 2019.
  * [13] C. Zhang, D. Chu, S. Liu, Z. Deng, C. Wu, and X. Su, “Trajectory planning and tracking for autonomous vehicle based on state lattice and model predictive control,” IEEE Intelligent Transportation Systems Magazine, vol. 11, no. 2, pp. 29–40, 2019.
  * [14] X. Zhang, A. Liniger, and F. Borrelli, “Optimization-based collision avoidance,” IEEE Transactions on Control Systems Technology, vol. 29, no. 3, pp. 972–983, 2020.
  * [15] G. Schildbach and F. Borrelli, “Scenario model predictive control for lane change assistance on highways,” in 2015 IEEE Intelligent Vehicles Symposium (IV), pp. 611–616, IEEE, 2015.
  * [16] T. Brüdigam, M. Olbrich, D. Wollherr, and M. Leibold, “Stochastic model predictive control with a safety guarantee for automated driving,” IEEE Transactions on Intelligent Vehicles, vol. 8, no. 1, pp. 22–36, 2021.
  * [17] T. Qie, W. Wang, C. Yang, Y. Li, Y. Zhang, W. Liu, and C. Xiang, “An improved model predictive control-based trajectory planning method for automated driving vehicles under uncertainty environments,” IEEE Transactions on Intelligent Transportation Systems, vol. 24, no. 4, pp. 3999 – 4015, 2023.
  * [18] T. Benciolini, D. Wollherr, and M. Leibold, “Non-conservative trajectory planning for automated vehicles by estimating intentions of dynamic obstacles,” IEEE Transactions on Intelligent Vehicles, 2023.
  * [19] J. Zhou, B. Olofsson, and E. Frisk, “Interaction-aware motion planning for autonomous vehicles with multi-modal obstacle uncertainties using model predictive control,” arXiv preprint arXiv:2212.11819, 2022.
  * [20] A. Hakobyan and I. Yang, “Wasserstein distributionally robust motion control for collision avoidance using conditional value-at-risk,” IEEE Transactions on Robotics, vol. 38, no. 2, pp. 939–957, 2021.
  * [21] M. Cheramin, R. L.-Y. Chen, J. Cheng, and A. Pinar, “Data-driven robust optimization using scenario-induced uncertainty sets,” arXiv preprint arXiv:2107.04977, 2021.
  * [22] C. Shang and F. You, “A data-driven robust optimization approach to scenario-based stochastic model predictive control,” Journal of Process Control, vol. 75, pp. 24–39, 2019.
  * [23] J. Kong, M. Pfeiffer, G. Schildbach, and F. Borrelli, “Kinematic and dynamic vehicle models for autonomous driving control design,” in 2015 IEEE Intelligent Vehicles Symposium (IV), pp. 1094–1099, IEEE, 2015.
  * [24] J. Larminie and J. Lowry, Electric vehicle technology explained.  John Wiley & Sons, 2012.
  * [25] Matlab, “Powertrain blockset-mapped motor,” 2017.  https://in.mathworks.com/help/autoblks/ref/mappedmotor.html.
  * [26] “EV reference application.” https://in.mathworks.com/help/autoblks/ug/electric-vehicle-reference-application.html.  Accessed: 2023-06-18.
  * [27] “Double lane change reference application.” https://in.mathworks.com/help/vdynblks/ug/double-lane-change-reference-application.html.  Accessed: 2023-06-18.
  * [28] M. Herceg, M. Kvasnica, C. Jones, and M. Morari, “Multi-Parametric Toolbox 3.0,” in Proc. of the European Control Conference, (Zürich, Switzerland), pp. 502–510, July 17–19 2013.  http://control.ee.ethz.ch/~mpt.
  * [29] A. Wächter and L. T. Biegler, “On the implementation of an interior-point filter line-search algorithm for large-scale nonlinear programming,” Mathematical programming, vol. 106, pp. 25–57, 2006.
  * [30] J. A. E. Andersson, J. Gillis, G. Horn, J. B. Rawlings, and M. Diehl, “CasADi – A software framework for nonlinear optimization and optimal control,” Mathematical Programming Computation, vol. 11, no. 1, pp. 1–36, 2019.
  * [31] M. Kamal, M. Mukai, J. Murata, and T. Kawabe, “Model predictive control of vehicles on urban roads for improved fuel economy,” IEEE Transactions on Control Systems Technology, vol. 21, no. 3, pp. 831–841, 2012.
  * [32] Y. He, Q. Zhou, M. Makridis, K. Mattas, J. Li, H. Williams, and H. Xu, “Multiobjective co-optimization of cooperative adaptive cruise control and energy management strategy for phevs,” IEEE Transactions on Transportation Electrification, vol. 6, no. 1, pp. 346–355, 2020.
  * [33] A. Hadjigeorgiou and S. Timotheou, “Optimizing the trade-off between fuel consumption and travel time in an unsignalized autonomous intersection crossing,” in 2019 IEEE Intelligent Transportation Systems Conference (ITSC), pp. 2443–2448, IEEE, 2019.