Distributionally Robust Optimization of Adaptive Cruise Control Under

Uncertainty

Shangyuan Zhang

1,2 a

, Makhlouf Hadji

1 b

and Abdel Lisser

2 c

Institut de Recherche Technologique SystemX, 8 Avenue de la Vauve, 91120 Palaiseau, France

CentraleSupelec, L2S, Université Paris Saclay, 3 Rue Curie Joliot, 91190, Gif-sur-Yvette, France

Keywords:

Adaptive Cruise Control, Distributionally Robust Optimization, Stochastic Optimization, Autonomous

Vehicle.

Abstract:

Due to the recent advances in intelligent and connected vehicles, Adaptive Cruise Control (ACC) has become a

key functionality of advanced driver-assistant systems (ADAS) to enhance comfort and safety. The evaluation

of ACC’s efﬁciency and safety is also crucial for the industry to prove the reliability of its products. In our

paper, we propose a distributional robust optimization-based ACC reference generation model to produce

the optimal commands facing the uncertainty of sensors. By taking into account the uncertainty set with

knowledge of the ﬁrst and second moments, the original optimization problem with chance constraints can

be simpliﬁed and solved more efﬁciently. Numerical experiments in a driving simulator illustrate that the

robustness of the results is largely increased by minimizing the risks of violation of safety constraints.

1 INTRODUCTION

Over the past decade, there has been a marked

increase in the development of connected and

intelligent driving technologies, which has garnered

signiﬁcant interest from both industrial and academic

researchers. This results in various technological

breakthroughs and commercial solutions. The

development of intelligent applications in tandem

with new generations of cars (connected and semi-

autonomous) has changed the paradigm of driving

when ensuring and guaranteeing the safety of drivers

on different types of roads. Advanced Driver

Assistance System (ADAS) is one of the promising

applications that has been integrated into smart

vehicles in order to improve driving comfort and

safety. One of the most fundamental aspects of ADAS

is Adaptive Cruise Control (ACC), which has been

investigated extensively by researchers and engineers.

(Levine and Athans, 1966) is one of the oldest

articles invoking ACC and describing its objectives.

The authors of this reference claimed that the

objective of ACC is to maintain a safe distance

from a leading vehicle by adjusting the speed and

https://orcid.org/0000-0003-0230-8618

https://orcid.org/0000-0003-1048-753X

https://orcid.org/0000-0003-1318-6679

acceleration. In the situation in which there is no

leading vehicle, it should also maintain the desired

speed set by the driver or system. Nowadays, and

with the evolving of connected objects deployment,

ACC relies on two essential components represented

respectively by sensors to collect environmental

information and a controller to set inter-vehicle

spacing based on the collected information. Hence,

an ACC-enabled vehicle drives at a preset speed

(following a given trajectory) until detecting a leading

vehicle. Next to that, it switches the vehicle to

the distance regulation mode when activating its

ACC controller. This former will provide the

necessary safety distance and send other commands

to controllers.

In response to the need for improved driving

comfort and safety, researchers have put forth

numerous approaches to design adaptive cruise

control (ACC) systems that emulate the behavior

of expert human drivers. Therefore, ACC systems

are designed considering key factors such as energy

consumption, collision avoidance, etc. Then, it

becomes imperative to validate the proposed ACC

systems thoroughly and assess their performance

before deploying them in the market.

Nevertheless, the validation and evaluation process

comes with some drawbacks, such as real-world

road tests, which are expensive, time-consuming,

Zhang, S., Hadji, M. and Lisser, A.

Distributionally Robust Optimization of Adaptive Cruise Control Under Uncertainty.

DOI: 10.5220/0011670800003396

In Proceedings of the 12th International Conference on Operations Research and Enterprise Systems (ICORES 2023), pages 71-81

ISBN: 978-989-758-627-9; ISSN: 2184-4372

 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

and can only cover a limited number of driving

scenarios. Thus, we conduct the validation process in

a simulator that can generate scenarios automatically

based on our needs. This validation forms part of

the functional testing of advanced driver-assistance

systems (ADAS) in order to provide feedback on

the system’s performance. This feedback may

include information on avoided potential collisions,

the rationality of the system’s decision-making, and

the identiﬁcation of any potential ﬂaws. The goal

of this testing is to ensure that the ADAS functions

properly and can provide a useful aid to human

drivers.

In our work, we propose a reference generator

that is used to evaluate the performance of an

ACC system. This reference generator generates

commands that serve as a benchmark for the ACC

system’s output. We then compared the reference

commands generated using different frameworks.

Our results show that the trajectory generated using

the DRO approach was more robust compared to the

other frameworks. Figure 1 shows the ACC validation

process with the generated benchmark.

ADAS

Reference

Generator

Commands

Reference

Commands

Driving Scenario

Simulations

Sensors

Comparison

and Validation

Figure 1: Illustration of how the ADAS validation process

occurs.

There are plenty of approaches (with advantages

and drawbacks) to cope with the problem of

generating reference trajectories using, for example,

sampling-based methods, graph-based methods, and

optimization-based approaches. Our goal is to

improve the comfort and safety of autonomous

vehicle by using optimization-based approaches to

identify rational values for acceleration, speed, and

relative position. The optimization-based approaches

allow us to tailor the different criteria of the objective

function and facilitate the constraints description and

formulation according to different driving scenarios.

Hence, in this paper, we propose an optimization-

based model for the generation of ACC reference

commands.

Usually, an ACC system relies on various types

of sensors, including cameras, lidar, radars, etc. The

performance of each considered sensor is inﬂuenced

by certain factors like its maintenance and the

environment in which it is used (see (Rasshofer et al.,

2011) for more details). Hence, collected data from

these sensors is subject to inherent inaccuracies that

must be taken into account when computations are

performed. Under the assumption that the ﬁrst and

second moments of sensor errors are known, either

in full or in part, we propose a distributionally robust

optimization (DRO) model with chance constraints to

handle and integrate this uncertainty.

1.1 Our Contribution

According to the aforementioned details, we propose

in this paper two DRO approaches to formulate a

new ACC reference generator. Indeed, we formulate

both a deterministic optimization model (Zhang

et al., 2022) and twos DRO model which address

the challenge of sensor error uncertainty when the

ﬁrst and second moments of the error distribution

are partially known. Our optimizations are based

on quadratic programming (QP) to determine the

most appropriate command to optimize the distance

between two vehicles while jointly satisfying all the

problem constraints at the same time. We provide

a comprehensive comparison of the obtained results

using our generated driving data, which simulates

realistic driving scenarios. We also emphasize the

added value of the DRO model compared to the

deterministic formulation.

The remaining of this paper is organized as

follows. Section 2 addresses mainstream ACC

algorithms that can be found in the literature. We

describe and discuss a new ACC validation model in

Section 3. Section 4 is dedicated to assessing the

performance of the proposed new solution and its

comparison to other existing models. We conclude

the paper in Section 5.

2 RELATED WORK

Research on ACC has been conducted from a number

of perspectives to ensure its effectiveness and safety

in complex real-life driving scenarios, including its

design, implementation, and validation.

A large number of ACC systems use optimal

control methods to achieve the desired outcome

(Chehardoli, 2020; Jiang et al., 2020; Zhu et al.,

2018). In recent years, model predictive control

(MPC) has also gained popularity due to its receding

horizon approach (Takahama and Akasaka, 2018;

Li et al., 2010; Naus et al., 2010). Other

features have also been investigated, including

driver behavior modeling(Varotto et al., 2020),

ICORES 2023 - 12th International Conference on Operations Research and Enterprise Systems

string stability(Khound et al., 2021), and collision

avoidance(Magdici and Althoff, 2017).

As a crucial part of autonomous driving,

validation of the functionality of the ADAS is also

a very critical task. It applies not only to the ACC

but also to other modules that require evaluation.

A uniﬁed test architecture and validation process

are presented in (Lattarulo et al., 2017), which

presents a comprehensive framework for evaluating

path planning and control algorithms. Similar works

can also be found in (Lattarulo et al., 2018) and

(Alnaser et al., 2019).

In addition to evaluating the overall testing

framework, it is essential to carefully examine

the individual functionality such as adaptive cruise

control (ACC) system. An experimental platform for

evaluating and demonstrating an optimization-based

ACC controller is presented in (Mehra et al., 2015),

while (Djoudi et al., 2020) presents a simulation-

based tool chain for generating reference data and

analysing test results. A number of other insightful

studies concerning the testing and validation of ACC

can be found in (Schmied et al., 2015; Shakouri

et al., 2015). In this paper, a DRO model is proposed

for ACC reference generation that takes into account

uncertainties with partial information. The results

of the DRO model can then be incorporated into the

validation process. To the best of our knowledge, no

prior work has been conducted on the application of

DRO to ACC reference generation.

DRO was ﬁrst introduced by Scarf (Scarf, 1958)

in 1958 and widely applied in risk management. The

main requirement of this framework is to deﬁne a set

of probability measures, namely the uncertainty set,

that include the true stochastic model for the problem.

Furthermore, the problem is typically optimized using

a worst-case analysis over the choice of a distribution

from this set. This framework has attracted

considerable attention due to the advancement of

optimization techniques, and it has a wide range

of applications across a variety of ﬁelds nowadays

(Rahimian and Mehrotra, 2019). The uncertainty set

is normally constructed using historical data and is

known as a data-driven uncertainty set (Delage and

Ye, 2010; Mohajerin Esfahani and Kuhn, 2018; Miao

et al., 2021). By analyzing sensor errors in the ACC

system, we can apply the DRO framework to improve

the performance of our ACC reference generator.

3 PROBLEM FORMULATION

3.1 Overview

In the following, we rely on the usage of the provided

mathematical formulation of (Zhang et al., 2022) and

will address driving scenarios.

The objective of this section is to describe how we

model the ACC driving scenario and formulate the

related optimization problem. Note that the typical

ACC driving scenario involves two cars driving

simultaneously in one lane, namely the ego car and

the target car. It is the ego car that is equipped with

an ACC system, whereas the target car is the leading

vehicle positioned in front. Figure 2 illustrates the

driving scenario, as well as the states of two cars at

the moment t

. The purpose of the ACC reference

generation is to generate a sequence of acceleration

commands, that is, the decision variables in our

optimization problem. The ego car’s ACC system

is designed to remain at a distance from the target

car while taking into account a variety of factors,

such as vehicle dynamics, driving comfort, and trafﬁc

regulations.

Figure 2: ACC driving scenario at the moment t

Suppose that the total duration of a driving

scenario is T s composed of n sampling time dt,

i.e. T = ndt with a corresponding timestamp

,...t

,... t

] where t

i+1

= t

+ dt, ∀i ∈

{0,1,... n − 1}. At each moment t

, the ACC

of the ego car uses sensors to gather information

from the target car and generates the acceleration

commands. In the following, we list the parameters

and the decision variables used in our model.

The input parameters are given by the ego car

sensors, and the decision variables represent the ACC

optimal commands. The parameters of the ego car are

the initial position x

ego

, and the initial velocity v

ego

whilst the parameters of the target car are composed

of the position vector X

tgt

= (x

tgt

,... x

tgt

)

, the

velocity vector V

tgt

= (v

tgt

,... v

tgt

n−1

)

and the

acceleration vector A

tgt

= (a

tgt

,... a

tgt

n−1

)

in the

whole driving scenario. The decision variable is the

ACC ego car acceleration commands vector A

ego

,... a

ego

n−1

)

Given the decision variable and the initial state of

the ego car, we can derive the velocity and the position

Distributionally Robust Optimization of Adaptive Cruise Control Under Uncertainty

of the ego car by the equations of motion. The ego car

velocity v

ego

i+1

at time t

i+1

is given by the velocity at the

previous sample time v

ego

and the acceleration a

ego

i+1

= v

ego

+ a

ego

dt. (1)

The velocity for the whole driving scenario can be

written in matrix form as

ego







ego

n−1













ego

∑

k=i−1

k=0

ego

∑

k=n−2

k=0

ego







= dtK

ego

+ v

ego

(2)

where K

∈ R

n×n

and 1

∈ R

n×1







0 0 0 ... 0 0

1 0 0 ... 0 0

1 1 0 ... 0 0

1 1 1 ... 0 0

1 1 1 ... 1 0







(3)













. (4)

Similarly, the ego car position at time t

i+1

is given

ego

i+1

= x

ego

+ v

ego

dt +

ego

. (5)

The corresponding matrix format for all time steps

ego







ego













ego

+ v

ego

dt +

ego

∑

k=i−1

k=0

ego

dt +

∑

k=i−1

k=0

ego

∑

k=n−1

k=0

ego

dt +

∑

k=n−1

k=0

ego







= dtM

ego

+ x

ego

(6)

where M

∈ R

n×n







1 0 0 ... 0

1 1 0 ... 0

1 1 1 ... 0

1 1 1 ... 1







. (7)

We use Equation (2) to rewrite Equation (6) in

terms of the initial position, the initial velocity and

the acceleration vector, i.e.,

ego

= dtM

ego

+ x

ego

= dtM

(dtK

ego

+ v

ego

)

ego

+ x

ego

= dt

ego

+ v

ego

dtC

+ x

ego

(8)

where B

= M

·K

∈ R

n×n

and C

= M

·1

∈ R

n×1

These parameters are summarized in Table 1.

In the following, we use the position and the

velocity vector of the ego car to formulate our

optimization problem.

3.2 Mathematical Modeling

In the sequel, we outline how the generation of the

ACC reference can be viewed as an optimization

problem.

min

ego

||Q A

ego

+ P|| (9)

s.t. dt

ego

≤ X

tgt

− v

ego

dtC

− (x

ego

+ d

, (10)

− (v

max

+ v

ego

≤ dtK

ego

≤ (v

max

− v

ego

, (11)

− a

max

≤ A

ego

≤ a

max

, (12)

− j

max

dt1

≤ D

ego

≤ j

max

dt1

. (13)

The following part explains in detail how we

derive the objective function (9) and how constraints

(10, 11, 12, 13) are developed.

The objective of ACC is to maintain a safe

distance between the ego car and the target car. In

order to calculate the reference distance between the

ego car and the target car, we deﬁne two terms: the

inter-vehicle time tc (e.g., 3 seconds), which gives the

ego car enough time to brake and avoid a collision

with the target car, and the standstill distance δS to

ensure there is always enough room between the two

adjacent cars.

ICORES 2023 - 12th International Conference on Operations Research and Enterprise Systems

Table 1: Summary of used parameters and variables in our formulations.

Symbols Physical Meaning Relationship

Target Car

tgt

Acceleration proﬁle during simulation A

tgt

= (a

tgt

,... a

tgt

n−1

)

tgt

Speed proﬁle during simulation V

tgt

= (v

tgt

,... v

tgt

n−1

)

tgt

Position proﬁle during simulation X

tgt

= (x

tgt

,... x

tgt

)

Ego Car

ego

Acceleration proﬁle during simulation A

ego

= (a

ego

,... a

ego

n−1

)

ego

Speed proﬁle during simulation V

ego

= dtK

ego

+ v

ego

Position proﬁle during simulation dt

ego

+ v

ego

dtC

+ x

ego

Jerk proﬁle during simulation D

ego

At each moment t

, the reference distance of ACC

in platoons is deﬁned by

re f

= (v

ego

k−1

− v

tgt

k−1

)tc +

ego

k−1

− a

tgt

k−1

)tc

+ δS.

(14)

So the reference distance vector in the whole

driving scenario is :

re f

= tc(d tK

ego

+ v

ego

−V

tgt

)

ego

− A

tgt

) + δS1

= (dt ·tcK

I)A

ego

−tcV

tgt

−

tgt

+ (δS + v

ego

tc)1

(15)

Moreover, the current distance between the ego

car and target car is

vehicle

= X

tgt

− X

ego

= X

tgt

− [dt

ego

+ v

ego

dtC

+ x

ego

(16)

By combining (16) and (15), we obtain the

objective function (9):

min

ego

||D

vehicle

− D

re f

= min

ego

||X

tgt

− [dt

ego

+ v

ego

dtC

+ x

ego

] − [(dt ·tcK

I)A

ego

−tcV

tgt

−

tgt

+ (δS + v

ego

tc)1

]||

= min

ego

|| − (dt

+ dt ·tcK

I)A

ego

+ X

tgt

+tcV

tgt

− δS1

− v

ego

tc1

− x

ego

− v

ego

dtC

= min

ego

||Q A

ego

+ P||,

(17)

where Q = −(dt

+ dt · tcK

I),

P = X

tgt

+ tcV

tgt

− δS1

− v

ego

tc1

−

ego

− v

ego

dtC

and || ·|| is Euclidean norm.

In addition to the objective function (9), we

describe the following constraints

• Constraint (10) is the minimum distance

constraint that aims to prevent vehicles collisions.

It results from

vehicle

≥ d

. (18)

• Constraint (11) is the maximum velocity

constraint. Routes typically have a maximum

velocity limit which leads to the velocity

constraint. For a given speed limit v

max

, the

constraint is deduced from

||V

ego

∞

≤ v

max

. (19)

• Constraint (12) is the maximum acceleration

constraint. Car passengers’ comfort is impacted

by acceleration. Vehicle maneuverings like rapid

acceleration or braking should be avoided. Our

model proposes an acceleration limit of a

max

based on this motivation.

||A

ego

∞

≤ a

max

. (20)

• Constraint (13) is the maximum jerk constraint.

In jerk, we measure the acceleration variances,

which signiﬁcantly affect the comfort level of

passengers. A maximum limit j

max

is required for

this constraint.

||J

ego

∞

≤ j

max

(21)

Since j

= (a

ego

− a

ego

i−1

)/dt, the jerk constraint

can be simpliﬁed to (13) where D

∈ R

n×n







1 0 0 ... 0 0

−1 1 0 . . . 0 0

0 −1 1 ... 0 0

0 0 0 ... 1 0

0 0 0 ... −1 1







. (22)

Distributionally Robust Optimization of Adaptive Cruise Control Under Uncertainty

Given the form of the objective function and

the constraints, our model is a convex quadratic

optimization problem.

In the next section, we will discuss the uncertainty

involved in ACC and how to handle it by stochastic

modeling with chance constraints.

3.3 Distributionally Robust

Optimization Model

3.3.1 Uncertainty Set with First and Second

Moments

The model presented above is deterministic, i.e., all

input parameters are known in advance. Real-life

autonomous vehicle problems, however, may include

different sources of noise caused by external factors,

such as weather, which may affect these parameters.

Model uncertainties can be addressed in a variety of

ways. In our work, we apply the robust optimization

framework to manage distribution-free uncertainties

in models.

In the following, we model the ACC problem

as a chance-constrained optimization problem. Our

hypothesis is that the target car’s position information

tgt

contains some noise when obtained from the ego

car’s sensor, and we only know the ﬁrst two moments

of the distribution, the mean value µ

and the variance

, respectively. Mathematically, the target car’s

position x

tgt

is a random variable with a distribution

measure F

over its outcome space. Thus, the mean

and variance of the target car’s position is µ

and σ

where













(23)

and













(24)

The objective function for this stochastic

optimization problem is

min

ego

||E(D

vehicle

− D

re f

)||

= min

ego

||µ

+tcV

tgt

− δS1

− v

ego

tc1

− x

ego

− v

ego

dtC

− [dt

+ dt ·tcK

ego

= min

ego

||Q A

ego

+ P

||,

(25)

where P

= µ

+tcV

tgt

−δS1

−v

ego

tc1

−

ego

− v

ego

dtC

Let D

be the set of probability distributions with

mean µ

and variance σ

, and it’s deﬁned as

(

tgt

] = µ

[(x

tgt

− µ

)

] = σ

)

,i = 1,· · · , n.

(26)

The target car’s position x

tgt

∼ F

follows a

distributionF

with F

∈ D

Using Theorem 1 in (Ghaoui et al., 2003), the

minimum distance constraint (18) for each moment

can be expressed as a chance constraint (Prékopa,

2013) with a given a threshold α, i.e.,

inf

∈D

vehicle

≥ d

) ≥ α, ∀t

= inf

∈D

tgt

≥ x

ego

+ d

) ≥ α

= x

ego

+ d

≤ µ

− σ

1 − α

(27)

where the inf is taken with respect to all probability

distributions in D

For the whole driving scenario, the minimum

distance constraint in a matrix form is

ego

+ v

ego

dtC

ego

+ d

≤

tgt

(28)

where

tgt







− σ

1−α

− σ

1−α

− σ

1−α







. (29)

Since only the minimum distance constraint is

related to the position of the target car X

tgt

, all other

constraints remain unchanged.

ICORES 2023 - 12th International Conference on Operations Research and Enterprise Systems

3.3.2 Uncertainty Set with Unknown Moments

Here, we consider the case where the mean and

variance of the target car’s position x

tgt

are unknown

but limited in a speciﬁc range. This DRO model could

be applied to the scenario in which the sensor error is

unstable.

Supposing that the target car’s position x

tgt

is a

random variable, with an outcome space (Ω

) and

a distribution measure F

over the space. The mean of

tgt

lies in an interval of size 2

and centered at µ

and the upper bound of variance of x

tgt

is γ

. We

deﬁne the uncertainty set D

as follows:

(

tgt

] − µ

)

≤ γ

[(x

tgt

− µ

)

] ≤ γ

)

,i = 1,· · · , n.

(30)

The target car’s position x

tgt

∼ F

follows a

distributionF

with F

∈ D

As the mean position of the target car at time

is centered at µ

, we maintain the same objective

function as in Equation (25). By the result in (Peng

et al., 2021), the minimum distance constraint can be

transformed as follow

inf

∈D

vehicle

≥ d

) ≥ α, ∀t

= inf

∈D

tgt

≥ x

ego

+ d

) ≥ α

= x

ego

+ d

≤ µ

− σ

(

1 − α

(31)

For the whole driving scenario, the minimum

distance constraint in a matrix form is

ego

+ v

ego

dtC

ego

+ d

≤

tgt

(32)

where

tgt







− σ

(

1−α

)

− σ

(

1−α

)

− σ

(

1−α

)







. (33)

All other constraints remain unchanged since

uncertainty is not involved.

4 EXPERIMENTAL RESULTS

In numerical experiments, we randomly generate

various driving scenarios in a simulator and compare

the performance of the deterministic and the two

DRO models on those scenarios. To compare, we

evaluate the number of violated constraints in each

model on given driving scenarios. During the data

generation phase, different conﬁgurations of a driving

scenario are applied, including the ego car’s state and

the target car’s trajectory. The sensor error is also

included in the data. Once the driving scenarios are

prepared, we use a QP solver (Goldfarb and Idnani,

1983) to solve the formulated deterministic and DRO

models. To conclude, we compare the number of

violated constraints of the two models to demonstrate

the effectiveness of the DRO model.

For the generation of an ACC driving scenario,

two types of parameters are necessary: the parameters

related to the environment and to the vehicles.

The parameters related to the environment include

the simulation conﬁguration and vehicle regulations,

such as the total scenario duration, velocity limit,

collision avoidance limit, etc. Those parameters

reﬂect the real-life driving rules and simulation

setting, and therefore they are ﬁxed during numerical

experiments. The parameters related to the vehicles,

such as initial position, velocity and distance, vary in

each randomly generated instance due to the diversity

of driving scenarios. In order to simulate driving

situations realistically, the relationships between

randomly generated vehicle parameters should be

representative of real-world situations.

The sequel summarizes the parameters set up for

numerical simulations:

• Parameters related to the environment:

– Total duration of a scenario T : 2s.

– Sampling time step dt: 0.05s.

– Inter-vehicle time tc: 3s.

– Standstill distance σS: 3m.

– Minimum security distance d

: 10m.

– Maximum velocity v

max

: 30m/s.

– Maximum acceleration v

max

: 5m/s

– Maximum jerk j

max

: 5m/s

– DRO unknown moments parameter: γ

= 5 and

= 5.

– Conﬁdence level α: 0.9.

• Parameters related to the vehicles:

– Acceleration of the target car: independent

random variables following a normal

distribution with mean 0 and standard deviation

2, truncated from −5 to 5.

– Initial speed of target car and ego car:

independent random variables following a

normal distribution with mean 15 and standard

deviation 10, truncated from 5 to 25.

Distributionally Robust Optimization of Adaptive Cruise Control Under Uncertainty

Figure 3: Constraint function values of all instances for deterministic and DRO models.

– Standard deviation of target car position σ: 1.

– Initial position of target car: random variable

following a normal distribution with mean 200

and standard deviation 1.

– Speed and position of target car: random

variables following normal distributions with

mean calculated by an initial value and the

acceleration vector, and standard deviation 1.

– Initial position of ego car: the initial position

of the target car minus a random variable

following a normal distribution with mean 100

and standard deviation 20, truncated from 50 to

150.

In the previous section, the optimal solution in each

generated ACC driving scenario can be obtained by

solving a QP problem. Several techniques exist for

solving this QP problem, which can be divided into

two categories: active-set methods and interior point

methods. We use QP solver with Goldfarb–Idnani

algorithm (Goldfarb and Idnani, 1983), which is a

dual active set method, in order to obtain the optimal

solution for our QP problems.

Firstly, we generate 100 random driving scenarios

based on the conﬁguration above. Then our QP

solvers obtain the optimal reference of the scenarios

under the deterministic model and the two DRO

ICORES 2023 - 12th International Conference on Operations Research and Enterprise Systems

models, respectively. Considering the sensor error

in the input data, the obtained trajectory of our ego

car may violate the constraint (18) during the driving

scenario. Therefore, the performance of a model can

be measured by the number of times that constraint

(18) is violated throughout the scenario. A more

reliable model is one that produces fewer violations

of constraints statistically.

According to a numerical analysis of 100 random

instances solved with three models, only 50% of

the results are totally feasible, which means that the

constraint (18) is never violated during the scenario,

when solving with the deterministic model, whereas

98% of the results are totally feasible when solving

with the DRO model with known moments and 100%

when solving with the DRO model with unknown

moments.

Figure 3 shows a detailed analysis of

constraint violations across 100 driving scenarios.

Visualizations of the constraint violation value

− D

vehicle

, adapted from constraint (18), are

presented for the results of the three models. This

adapted constraint value must be less than or equal

to zero for the solution to be feasible (values greater

than zero cause a constraint violation). The constraint

value is shown in Figures 3(a), 3(b) and 3(c) for the

whole scenarios, whereas a partial zoom-in is shown

in Figures 3(d), 3(e) and 3(f) for easier reading. In

Figure 3, each curve in its own color displays the

constraint values of a driving scenario result, and

the x-axis represents the time index of constraints.

If the value at constraint index i exceeds 0, it means

that d

> D

vehicle

, i.e. the constraint (18) is violated

at this sampling time. It is evident from Figure 3

that the two DRO models produce signiﬁcantly fewer

violations than the deterministic model.

Figure 4: Maximal violated constraints under different

standard deviations.

In addition to the previous experiments, we

propose to evaluate the performance of the three

Figure 5: Average violated constraints under different

standard deviations.

proposed models under different sensor precision.

Keeping all other parameters unchanged, we vary the

standard deviation of the target car position, which

reﬂects the sensor’s precision, from 1 to 40. The

value of the standard deviation is gradually increased,

and we make 100 tests for each value. Figure 4 and

Figure 5 present the maximal and average number of

constraint violations for each model under different

standard deviations. In both cases, there are always

fewer constraint violations in the two DRO models

than in the deterministic model, which proves the

robustness of DRO approach for uncertainty.

5 CONCLUSION AND FUTURE

DIRECTIONS

The main contribution of this paper is a

distributionally robust optimization-based method

for ACC reference generation in driving scenarios

with uncertainties. In order to satisfy the safety

constraints, the reference generator takes into account

the sensor errors with partial information about its

distribution to produce optimal commands. The

results of the numerical simulations prove the

robustness of the DRO models by comparing their

performances with the deterministic model under

generated driving scenarios.

There are various further directions to explore for

the next step. For instance, we will consider other

assumptions for the sensor error and build an adapted

model for real-life scenarios. Additionally, we can

also apply this distributionally robust optimization

model in other modules of autonomous vehicles with

uncertainties in order to achieve better reliability for

some crucial functionalities.

Distributionally Robust Optimization of Adaptive Cruise Control Under Uncertainty

ACKNOWLEDGEMENT

This work was supported by the French government

under the "France 2030” program, as part of the

SystemX Technological Research Institute.

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