Reliable Modeling for Safe Navigation of Intelligent Vehicles:

Analysis of First and Second Order Set-membership TTC

Nadhir Mansour Ben Lakhal

1,2

, Othman Nasri

2

, Lounis Adouane

3

and Jaleleddine Ben Hadj Slama

2

1

Institut Pascal, UCA/SIGMA - UMR CNRS 6602, Clermont Auvergne University, France

2

LATIS Lab, National Engineering School of Sousse (ENISo), University of Sousse, BP 264 Sousse Erriadh 1023, Tunisia

3

Heudiasyc UMR CNRS/UTC 7253, Université de Technologie de Compiègne, 60203 Compiègne, France

Keywords:

Intelligent Vehicles, Risk Management, Interval-based Modeling, Correlation Analysis, Interval Polynomial,

Second-order Time to Collision.

Abstract:

Developing high ﬁdelity models to compute the Time-To-Collision (TTC) between vehicles is addressed in this

work. A TTC interval value is over-approximated while considering several uncertainties via interval analysis.

Furthermore, to decrease modeling inaccuracy, a novel second-order set-membership TTC formalization is

introduced by solving a polynomial equation with interval coefﬁcients. This latter is derived from vehicles’

motion equations. Hence, an approach based on correlation analysis is exploited to improve the uncertainty

evaluation. The simulation results applied on an adaptive cruise control system of both high/low-order TTC

formalizations prove that the low-order model inaccuracy is compensated. Thanks to interval analysis and

correlation characterization, a great balance between modeling accuracy and simplicity is reached.

1 INTRODUCTION

Risk management should be inspected carefully to

employ autonomous vehicles in public roads (Nasri

et al., 2019). For the sake of safety, focus is currently

given to provide efﬁcient solutions for in-road risk

identiﬁcation. Thus, factors that stand behind the reli-

ability and accuracy of safety veriﬁcation techniques

should be analyzed.

Advanced perception/communication devices and

navigation scene analysis have been used to capture

in-road hazards (Abdi and Meddeb, 2018), (Kasmi

et al., 2019). Nonetheless, these tools are prone to se-

vere uncertainties. To overcome uncertainty impacts,

several methods have been proposed in the literature

(Lakhal et al., 2019a), (Lozenguez et al., 2011). The

uncertainty is propagated into the navigation process

via stochastic models such as the Kalman ﬁlter, etc.

A speciﬁc probability distribution, as the Gaussian

function, is assumed to describe the uncertainty evo-

lution. This assumption is controversial, and changes

in noise features may occur (Rigatos, 2012). Addi-

tionally, most uncertainty evolution models are sen-

sitive to non-linearity (Wang et al., 2018). On top

of that, an accurate knowledge of the initial states of

the studied system is required, which is not evident

(Nicola and Jaulin, 2018). Hence, it is important to

study alternative approaches that are less sensitive to

these errors.

Otherwise, risk management reliability depends

on the accuracy of models used to derive numer-

ous risk indicators. For instance, the Time To Col-

lision (TTC) has been widely used for risk identiﬁca-

tion (Iberraken et al., 2018), (Iberraken et al., 2019).

Tremendous attempts have been made to improve the

TTC precision. A comparative study between diverse

TTC formalizations could be found in (Hou et al.,

2014). A hidden Markov model has been used to pre-

dict the driving intention of nearby vehicles for more

accurate TTC estimation (Yang et al., 2020). Algo-

rithms computing distances between boxes bounding

vehicles were proposed to calculate TTC for com-

plex trafﬁc scenarios (Wang et al., 2018). A vehicle

motion-based concept, named looming, was exploited

to decrease the TTC false alarms (Ward et al., 2015).

Interval analysis is a reliable way to handle uncer-

tainties/modeling imperfections (Jaulin et al., 2001).

It turns standard data to intervals to bound uncer-

tainty impacting the studied system (Moore et al.,

2009). Correspondingly, interval analysis may con-

tribute strongly in characterizing the uncertainty evo-

lution into intelligent transportation systems. In pre-

vious work, an interval-based model to compute TTC

for a car-following scenario was proposed to handle

Ben Lakhal, N., Nasri, O., Adouane, L. and Slama, J.

Reliable Modeling for Safe Navigation of Intelligent Vehicles: Analysis of First and Second Order Set-membership TTC.

DOI: 10.5220/0009890305450552

In Proceedings of the 17th International Conference on Informatics in Control, Automation and Robotics (ICINCO 2020), pages 545-552

ISBN: 978-989-758-442-8

Copyright

c

2020 by SCITEPRESS – Science and Technology Publications, Lda. All rights reserved

545

uncertainties and communication latencies (Lakhal

et al., 2019b) and (Lakhal et al., 2019c). Moreover,

the interval TTC over-approximation was optimized

via a data-driven characterization of correlation that

would relate the navigation system variables. In this

paper, we build on this previous work to analyze

much comprehensively the performances of the set-

membership modeling. The main contribution of this

work is to introduce a novel second-order interval-

based TTC over-approximation to consider more pa-

rameters intervening in the car-following scenario.

The high-order model consists in a quadratic poly-

nomial with interval coefﬁcients generated from ve-

hicles’ motion equations. Usually dedicated to bound

the rounding errors, interval polynomials have never

been used to build models for uncertainty evolution

into navigation systems. Afterwards, simulation is

elaborated on an Adaptive Cruise Control (ACC). The

performances of the interval high and low-order TTC

in conducting the risk worst-case analysis are com-

pared. The quality of the set-membership modeling

joined with the correlation analysis is evaluated in

terms of accuracy and simplicity.

The rest of this paper is arranged as follows:

Section 2 introduces the ﬁrst and second-order TTC

interval-based formalizations. Section 3 presents an

algorithm to ﬁnd roots for an interval polynomial to

approximate the TTC. Section 4 explains the correla-

tion analysis role in ameliorating the ﬁndings of the

TTC set-membership models. Section 5 presents the

simulation results. Section 6 concludes the results of

this work and discusses some future work.

2 SECOND ORDER

SET-MEMBERSHIP TTC

For a car-following scenario, the TTC is often ap-

proximated by the ratio between the distance separat-

ing two vehicles and their relative velocity. Instead,

the evolution of the spacing distance between the fol-

lower and the leader is used in this paper to perform

more accurate collision prediction. In this way, all the

interactions between vehicles are taken into account.

Let consider two vehicles i and j, which are respec-

tively the leader and the follower. V

i

, V

j

, p

i

and p

j

are

their respective velocities and vector positions. Ac-

cording to (Ward et al., 2015), the separation evolu-

tion between both vehicles is described at each instant

by:

• The separation distance:

d

i j

=

q

(p

i

− p

j

)

T

(p

i

− p

j

) (1)

• The change rate in the separation distance:

˙

d

i j

=

1

d

i j

(p

i

− p

j

)

T

(V

i

−V

j

) (2)

• The variation of the change rate in the separation

distance is governed by the following equation:

¨

d

i j

=

1

d

i j

(V

i

−V

j

)

T

(V

i

−V

j

) −

˙

d

2

i j

(3)

Equations (2) and (3) are obtained by the consecutive

differentiation of equation (1). In practice, d

i j

is mea-

sured in run-time thanks to diverse vehicular tools as

a LiDAR or laser scanner. Therefore, the authors in

(Ward et al., 2015) deﬁned T TC

1

as a ﬁrst order TTC:

T TC

1

= −

d

i j

˙

d

i j

(4)

However, equation (4) neglects parameter

¨

d

i j

. Model

simpliﬁcation is the main source of errors (Kheliﬁ

et al., 2018). In an effort to improve accuracy, the au-

thors in (Ward et al., 2015) upgraded the TTC approx-

imation to a second-order expression. When

¨

d

i j

6= 0,

a second-order TTC, denoted T TC

2

, is obtained by

solving the following polynomial:

d

i j

+

˙

d

i j

T TC

2

+

1

2

¨

d

i j

T TC

2

2

= 0 (5)

Note that equation (5) is derived from the vehicles’

motion equations. The polynomial roots underline at

which instants the two vehicles collide, and the sepa-

ration between them is zero. Accordingly, the authors

in (Ward et al., 2015) deﬁned the T TC

2

value depend-

ing on the roots of equation (5). When

¨

d

i j

= 0 or the

polynomial has no real roots, the low-order model is

used and T TC

2

= T TC

1

. In the case of two real posi-

tive roots, the lower value is attributed to T TC

2

since

it presents the ﬁrst collision time. If one of the roots

is positive and the other is negative, the positive one

is taken. Both roots can be also negative. In such a

situation, the root with the closest absolute value to

zero is selected because it consists of the most recent

interaction between the motions of both vehicles.

Despite its accuracy, the high-order TTC is still

sensitive to uncertainty and communication latencies.

To overcome this issue, interval analysis is adopted

in this paper. Data representation is extended to

intervals. Mathematical operations (+, −,∗,/) and

functions (sin, cos, etc.) are extended to handle inter-

vals (Jaulin et al., 2001). Subsequently, the obtained

interval-based models provide over-approximations

of results that deﬁnitely enclose the exact outputs.

Henceforth, [x] = [x,x] is a real interval, where x and

x are its lower and upper bounds. The width of [x]

ICINCO 2020 - 17th International Conference on Informatics in Control, Automation and Robotics

546

underlines the uncertainty extent. Accordingly, equa-

tions (4) and (5) are represented as:

[T TC

1

] = −

[d

i j

]

[

˙

d

i j

]

(6)

[d

i j

] + [

˙

d

i j

][T TC

2

] +

1

2

[

¨

d

i j

][T TC

2

]

2

= 0 (7)

Since it describes the real behavior, the second-order

set-membership TTC is expected to be more accurate

than the ﬁrst-order one. Equation (7) is a quadratic

polynomial with perturbed coefﬁcients. Its roots are

intervals enclosing the collision exact time. Solving

this polynomial is not feasible by standard analyti-

cal approaches. A speciﬁc interval polynomial solver

must be used. Before doing so, a methodological

manner to quantify uncertainties attributed to each in-

terval measurement is introduced. The environmen-

tal circumstances, where more uncertainties are ex-

pected, are examined. At ﬁrst, the following assump-

tions, which are based on the conﬁdence intervals of

sensors and communication devices, are admitted:

• The localization inaccuracy is assessed via a sig-

nal strength indicator that considers the signal at-

tenuation in the navigation zone.

• The accumulated error impacting the separation

distance measurement is considered by an uncer-

tainty range of ±1% from the measured d

i j

.

• The follower speed V

j

is assumed to be exact, and

no uncertainty is attributed to this parameter.

• The leader speed V

i

is assumed to be erroneous

with a range of ±0.5% due to measurement im-

precision.

Afterwards, several latencies can slow down the auto-

motive system operation and prohibit the quick man-

agement of risks. For that reason, it is advisable to

consider such latencies by [T TC]. In this work, the

follower car is expected to receive the V

i

value via

a Vehicle-to-Vehicle (V2V) communication. Hence-

forth, latencies impacting the V2V communication

are characterized through interval [T

V 2V

]. The un-

certainty attributed to [T

V 2V

] is appraised through

the empirical research work depicted in (Dey et al.,

2016). Min/max values of latencies that may happen

were provided in (Dey et al., 2016). These bounds

were presented as a function of the vehicle speed and

the number of connected vehicles in close proximity

(communication conﬂicts increase delays). Besides,

[T

L

] is a constant interval that takes into account laten-

cies due to update time of sensors and the data prop-

agation into the embedded system. Consequently, the

TTC set-membership formalization must consider ex-

plicitly the aforementioned uncertainty sources:

[T TC

1

] = −

[d

i j

]

[

˙

d

i j

]

− [T

V 2V

] − [T

L

] (8)

[T TC

2

] = [ℜ] − [T

V 2V

] − [T

L

] (9)

where [ℜ] is the polynomial root of equation (7). Sim-

ilar to the deterministic case detailed above (cf. equa-

tion (5)), [ℜ] is the root corresponding to the ﬁrst col-

lision time. Figure 1 illustrates the main instructions

of the proposed uncertainty quantiﬁcation strategy to

over-approximate the ﬁrst/second order TTC.

Figure 1: Interval-based risk management.

3 SOLVING QUADRATIC

INTERVAL POLYNOMIAL

Finding roots for interval polynomials has been

widely discussed in the literature. Several numerical

branch and bound algorithms were introduced for this

aim (Fan et al., 2008). Despite their accuracy, the cal-

culation time of these approaches was unpredictable.

One more category of approaches used polynomial

factorization and cumbersome mathematical calcula-

tion as an inverting interval matrix (Zhang and Deng,

2013). Other fast methods were developed (Ferreira

et al., 2001). Nevertheless, these approaches provided

just a prior estimate for the space containing the real

roots. In this work, real roots with sharp bounds of

interval polynomials are obtained by studying the in-

terval polynomial boundary functions.

Let consider a quadratic polynomial with the fol-

lowing shape:

P([x]) = [a]x

2

+ [b]x + [c] (10)

Intuitively, P([x]) can be expressed within its bound-

ary functions, where: P([x]) = [P([x]),P([x])]. For

such a polynomial, P([x]) and P([x]) may be derived

through all possible combinations between the coef-

ﬁcient bounds. Indeed, eight real single-valued poly-

nomials are given from these combinations:

f

1

= ax

2

+ bx + c; f

2

= ax

2

+ bx + c

f

3

= ax

2

+ bx + c; f

4

= ax

2

+ bx + c

f

5

= ax

2

+ bx + c; f

6

= ax

2

+ bx + c

f

7

= ax

2

+ bx + c; f

8

= ax

2

+ bx + c

(11)

Reliable Modeling for Safe Navigation of Intelligent Vehicles: Analysis of First and Second Order Set-membership TTC

547

By interpreting the dominant term of P([x]), it is ev-

ident that P([x]) and P([x]) are respectively enclosed

between ( f

1

, f

2

, f

3

, f

4

) and ( f

5

, f

6

, f

7

, f

8

). It is clear

also that:

f

1

≤ f

2

; f

3

≤ f

4

f

6

≥ f

5

; f

7

≥ f

8

(12)

Subsequently, we can ﬁgure out that:

P(x) =

(

P

1

= ax

2

+ bx + c, if x ≥ 0

P

2

= ax

2

+ bx + c, if x ≤ 0

(13)

and

P(x) =

(

P

3

= ax

2

+

bx + c, if x ≥ 0

P

4

= ax

2

+ bx + c, if x ≤ 0

(14)

Note that P

i=1..4

= (P

1

, P

2

, P

3

, P

4

) represents non-

interval real boundary functions associated to P([x]).

To illustrate such a notion, Figure 2 presents two ex-

amples of quadratic polynomials with perturbed coef-

ﬁcients.

Figure 2: Examples of interval polynomials.

An efﬁcient way to ﬁnd polynomial roots is to de-

termine sets where: P([x]) ≤ 0 ≤ P([x]). Eventually,

estimating sharp bounds of this intersection should be

accomplished by solving P

i=1..4

. Once the roots of the

boundary functions P

i=1..4

are calculated, it remains

to clarify how to join these roots to formulate a pre-

cise enclosure of P([x]) solutions. Contrary to non-

interval polynomials, P([x]) may have at maximum

three distinct interval roots, including semi-inﬁnite in-

tervals (see Figure 2).

In this work, a simple algorithm is presented to ex-

tract P([x]) interval roots using P

i=1..4

. It is based on

the theoretical results obtained in (Hansen and Wal-

ster, 2002) and (Hansen and Walster, 2003), where

the P([x]) coefﬁcient bounds are analyzed to ﬁgure

out the shape and orientation of P(x) and P(x). Con-

sequently, the right number and values of the interval

roots are appropriately determined.

First, the number of subcases that must be checked

to resolve P([x]) is decreased by admitting a > 0. In

the opposite case, the sign of P([x]) must be simply

reversed. Therefore, each P

i

must be solved by inter-

val arithmetic. The readers must distinguish between

solving interval polynomials and isolating real roots

of standard polynomials. Several set-membership al-

gorithms resolve the non-interval polynomials in or-

der to bound rounding errors.

In this work, the real roots of P

i=1..4

are computed

numerically via a speciﬁc interval computation pack-

age. The isolated real roots associated to each P

i

, in-

cluding multiple roots, are added to list L. Since func-

tions P

1

and P

2

bound P([x]) only for x ≥ 0 (see equa-

tion (13)), any negative root or part of a root must

indeﬁnitely be discarded from L. Seemingly, posi-

tive roots or parts of roots associated to P

3

and P

4

are

dropped.

Otherwise, there are some particular cases that

must be considered while arranging L. Indeed, a dou-

ble root is obtained at x = 0 for (P

1

,P

2

) when c = 0

and respectively for (P

3

,P

4

) if c = 0. For both cases,

this root must be entered just one time into the list.

Besides, the inﬁnite interval endpoints ±∞ must be

placed if necessary in L. Referring to (Hansen and

Walster, 2003), once the following cases are satisﬁed,

a lower endpoint −∞ is added to L :

a < 0 ∨ (a = 0 ∧ b > 0) ∨ (a = 0 ∧ b = 0 ∧ c ≤ 0)

(15)

Likewise, +∞ is added to L only if:

a < 0 ∨ (a = 0 ∧ b > 0) ∨ (a = 0 ∧ b = 0 ∧ c ≤ 0)

(16)

At this stage, L contains intervals that certainly

present a lower or upper end-point of the ﬁnal inter-

val roots of P([x]). Thus, it is necessary to recog-

nize which are the lower and upper ones. Let denote

[S

i

] = [S

i

,S

i

] the set of intervals held in L. All intervals

[S

i

] are sorted such that S

i

≤ S

i+1

. It is worth mention-

ing that the adopted algorithm requires to consider

±∞ as degenerate intervals. Hence, n denotes the

number of intervals included in L (no more than six

roots 0 ≤ n ≤ 6). The ﬁnal step from the root ﬁnding

strategy consists in arranging the solution according

to the obtained n. Table 1 summarizes all probable

shapes of the interval roots associated to P([x]). Fi-

nally, all necessary steps to solve the interval polyno-

mial are recapitulated in Algorithm 1.

ICINCO 2020 - 17th International Conference on Informatics in Control, Automation and Robotics

548

Table 1: Interval roots according to n.

Interval roots

n = 0 ∅

n = 2 [S

1

,S

2

]

n = 4 [S

1

,S

2

],[S

3

,S

4

]

n = 6 [−∞,S

2

],[S

3

,S

4

],[S

5

,+∞]

Algorithm 1: Solving Interval Polynomial.

Require: [a], [b] and [c]

Ensure : Solve P([x]) = [a]x

2

+ [b]x + [c]

1 -Deﬁne P

i=1..4

(cf. equations (12) and (13)).

2 -Find interval roots of P

i=1..4

.

3 -Put results in L.

4 -Add inﬁnite entries ±∞ to L, if needed (cf.

equations (15) and (16)).

5 -Sort the interval elements in L (S

i

≤ S

i+1

).

6 -Check the length of L to deﬁne roots of

P([x]).

4 CORRELATION-BASED

OPTIMIZATION STEP

In this work, the interval-based TTC formalizations

are dedicated to ensure safety for a modern ACC sys-

tem. At every sample time, an interval enclosure for

the position of target assigned to the ACC-equipped

vehicle (follower) is deﬁned proportionally to the cur-

rent [T TC], which is calculated via the ﬁrst or second-

order formalization. Then, a reference distance, de-

noted d

re f

, is maintained from the in-front vehicle ac-

cording to the worst-case risk indicated by the target

enclosure (cf. Figure 3).

Figure 3: Proposed ACC risk management principle.

Nevertheless, the interval over-approximations ob-

tained through assumptions deﬁned in section 2 are

too conservative. The occurrence of the worst cases of

uncertainties for all parameters considered in the TTC

computation is quite unrealistic. The main task of

ACC systems is to optimize the distance between ve-

hicles to prevent congestion and trafﬁc disturbances.

In this sense, the model developed for risk identiﬁca-

tion must make a trade off between safety and accu-

racy. As an enhancement for the interval-based mod-

els, a data-driven optimization step was introduced in

the previous work of (Lakhal et al., 2019b), (Lakhal

et al., 2019c) and (Ben Lakhel et al., 2016). Accord-

ingly, this approach is joined to the TTC second-order

model to obtain more compact bounds of ﬁndings.

In this section, the proposed data-driven optimization

approach is brieﬂy recalled.

Correlation is a relevant statistical parameter to

describe the operation of systems. More focus is

given currently to correlation analysis to study the

performances of autonomous vehicles (Wu et al.,

2020). The main idea behind the proposed data-

driven-optimization step is to examine the correlation

progression over time. During the navigation run-

time, substantial and brutal changes in vehicle dy-

namics are unrealistic in few sampling periods. Based

on this understanding, the evolution of the correlation

states should be smooth. Only uncertainties and erro-

neous measurements may invoke an irregular progres-

sion of correlation. Various data-driven approaches

have relied on this assumption to capture faults or to

regress outliers (Xia et al., 2017), (Chen et al., 2019).

Uncertainties assigned to interval measurements

can be over-estimated. This fact may entail a brutal

variation in the correlation progression between two

successive instants: t

k−1

and t

k

. Hence, the proposed

approach narrows recursively intervals until obtaining

an acceptable progression in the correlation between

variables. Narrowing is interpreted once the corre-

lation relating the new tightened intervals matches

reference values characterized off-line. Let denote

by C

([X],[Y ])|k

the correlation relating interval variables

[X] and [Y ] at instant t

k

. The overall process to esti-

mate the correlation values for interval variables is de-

tailed in (Lakhal et al., 2019b). Thereafter, the gap in

the correlation between instants t

k

and t

k−1

, denoted

γ

k|k−1

, is estimated through equation (17):

γ

k|k−1

= C

([X],[Y ])|k

−C

([X],[Y ])|k−1

(17)

Interval widths must be narrowed in a recursive way

to adapt the value of γ

k|k−1

in run-time and elimi-

nate over-estimated uncertainties. For each couple of

interval-valued variables intervening in the TTC com-

putation, the interval with the largest width is con-

cerned with iterative narrowing. After that, narrowing

is aborted at two conditions:

• Condition 1. When γ

k|k−1

decreases from one

iteration to another and suddenly starts to raise;

i.e., the interval is narrowed as much as possible.

Reliable Modeling for Safe Navigation of Intelligent Vehicles: Analysis of First and Second Order Set-membership TTC

549

Extra-narrowing may cause an undesirable modi-

ﬁcation in the correlation structure.

• Condition 2. Once γ

k|k−1

exceeds the minimum

variation of correlation, which is recorded during

the off-line simulation of a normal system opera-

tion.

Algorithm 2: [TTC] Optimized Estimation.

Input : p

i

, p

j

, V

i

, V

j

, d

i, j

,[T

V 2V

] and [T

L

].

Output: [T TC

1

] and [T TC

2

].

1 while Navigation process is running do

2 -Deﬁne [d

i, j

], [

˙

d

i j

], [

¨

d

i j

], [V

i

], [p

i

] and [p

j

].

3 for each couple of variables between

instants t

k

and t

k−1

do

4 repeat

5 -Calculate C

([X],[Y ])|k

.

6 -Estimate γ

k|k−1

(equation (17)).

7 -Narrow the interval, if needed.

8 until Condition 1 or 2 is satisﬁed

9 end

10 -Evaluate [T TC

1

] and [T TC

2

] (see

equations (10) and (11))

11 end

5 SIMULATION RESULTS

In this section, the reliability of the proposed interval-

based models to compute the TTC is demonstrated.

The quantitative results obtained from the conducted

simulation are analyzed to provide a qualitative com-

parison between the performances of the ﬁrst-order

and second-order TTC formalizations.

5.1 Test Scenario and Simulation Setups

The overall set-membership TTC-based risk manage-

ment is tested under a MATLAB freeway naviga-

tion simulator. Vehicle motions implicated in the test

phase are simulated through the well-known tricycle

kinematic model. The elaborated test scenario con-

sists of a car-following scenario in a highway road.

In addition, a model of a highway-road segment is

selected as the test-scene. Otherwise, a white Gaus-

sian noise is injected in the exact measurements of the

navigation dynamics during simulation.

As already said, the follower vehicle is equipped

with an ACC system. This latter exploits bounds of

the interval TTC (according to the ﬁrst or second-

order model) to take precaution of the risk worst cases

and adapt the reference distance from the vehicle

ahead. Full details about the ACC operation princi-

ple are available in (Lakhal et al., 2019c).

Technically speaking, the interval computation is

ensured by the reliable computation package INT-

LAB (Rump, 1999). All the simulation work depicted

in this section is carried out under MATLAB on an

Intel i5 Processor with 3.5 GHz and 16 GB memory.

More conﬁgurations involved in the established sim-

ulation are recapitulated in Table 2.

Table 2: Simulation setups.

Parameter Value

Sampling step 0.1 (s)

Sensors update time 0.01 (s)

Follower embedded system delay 0.025 (s)

Leader maximum velocity 22 (m/s)

Follower maximum velocity 23 (m/s)

5.2 Results and Discussion

At ﬁrst, the role of the correlation analysis in provid-

ing more sharp bounds of TTC values is inspected.

As shown in Figure 4, the TTC enclosures are efﬁ-

ciently narrowed for both [T TC

1

] and [T TC

2

]. For

the ﬁrst-order set-membership formalization, initial

amounts of uncertainties are minimized with an aver-

age range of 60.3%. Similarly, the average reduction

in the width of [T TC

2

] due to the correlation-based

optimization step is about 65.79%.

Figure 4: T TC

1

and T TC

2

enclosures with/without opti-

mization step.

ICINCO 2020 - 17th International Conference on Informatics in Control, Automation and Robotics

550

More importantly, the results of the interval high-

order TTC computation model are more conservative

than the low-order one. In average, the widths of

[T TC

1

] and [T TC

2

] are respectively about 1.25s and

1.579s. This fact can be explained by the “depen-

dency effect" characterizing the interval arithmetic

(Moore et al., 2009). Indeed, variables occurring sev-

eral times in one expression are assumed as indepen-

dently varying over their enclosures, which may lead

to an additional pessimism in the results. Hence, more

pessimism is entailed by upgrading the ﬁrst-order-

model to a second-order formalization since the num-

ber of the involved variables is increased.

In a second place, Figure 5 illustrates the evolution

of the exact T TC

1

and T TC

2

. These exact values of

T TC

1

and T TC

2

are obtained in a deterministic way

(respectively via equations (4) and (5)) without any

noise injection during the simulation. All along the

simulation run-time, the results of the two developed

interval-based formalizations of the TTC enclose per-

fectly the reference values provided by the exact evo-

lution of T TC

1

and T TC

2

. Correspondingly, the con-

sistency of the set-membership modeling joined with

the correlation analysis is proven. Even more, the

ﬁrst-order interval-based TTC is more accurate than

the second-one since it provides sharp bounds and si-

multaneously encompasses the exact and real values

of the TTC. This fact optimizes implicitly the naviga-

tion trafﬁc ﬂow because it decreases the reference dis-

tance maintained between vehicles. Eventually, the

interval-based uncertainty quantiﬁcation method con-

tributes to compensate the inaccuracy presented by

the ﬁrst-order TTC resulting from the modeling sim-

pliﬁcation.

Figure 5: T TC

1

and T TC

2

enclosures compared with exact

results.

Another advantage of the proposed approach is the

reduction in the computational cost of the risk man-

agement of intelligent vehicles. Using simple mod-

els, which handle efﬁciently all possible uncertainties,

helps to respect the real-time constraints. In our case

of study, solving the interval quadratic polynomial to

compute an over-approximation to the TTC requires

0.09s as an average execution time. Therefore, the

T TC

1

is more efﬁcient as a risk indicator than the

T TC

2

especially in terms of computational demands.

Additionally, the accuracy-level ensured by the T TC

1

is sufﬁcient to guarantee the navigation safety since it

handles properly uncertainties and modeling errors.

6 CONCLUSIONS

The reliability of models employed for the risk man-

agement of intelligent navigation systems is of ut-

most importance. In this work, ﬁrst and second-order

interval-based models to compute the TTC between

two vehicles are introduced. Set-membership mod-

eling allows considering various uncertainties and la-

tencies that can emphasize collision risks. Moreover,

the evolution of the correlation that relates variables

is characterized to improve the uncertainty evalua-

tion. Then, the performances of both proposed mod-

els are compared. The results of the ﬁrst-order TTC

are more compact and permit handling efﬁciently all

uncertainties. Fast risk analysis with the same accu-

racy level of the second-order TTC is ensured via the

simple low-order model. A trade off between accu-

racy and simplicity is ensured by joining the interval-

based computation with correlation analysis. Accord-

ingly, the need for sophisticated models for intelligent

vehicles’ motions to make the risk management suc-

cessful is discarded, while mastering all uncertainty-

induced risks.

Otherwise, the proposed method should be inte-

grated in the future on a real vehicle and applied for

more critical maneuvers such as lane changes.

ACKNOWLEDGEMENTS

The present work is supported by the WOW (Wide

Open to the World) program of the CAP 20-25

project. It receives also the support of IMobS3 Labo-

ratory of Excellence (ANR-10-LABX-16-01).

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