Modeling Stochastic Data Using Copulas For Application in Validation of Autonomous Driving Katrin Lotto1Thomas Nagler2and Mladjan Radic1

2025-05-06 0 0 1.38MB 10 页 10玖币

侵权投诉

Modeling Stochastic Data Using Copulas For Application in Validation of

Autonomous Driving

Katrin Lotto,1Thomas Nagler,2and Mladjan Radic1

1ZF Friedrichshafen AG, Research and Development,

Graf-von-Soden-Platz 1, 88046 Friedrichshafen, Germany.

2Department of Statistics, LMU Munich, 80799 Munich, Germany

Veriﬁcation and validation of fully automated vehicles is linked to an almost interactable challenge

of reﬂecting the real world with all its interactions in a virtual environment. Inﬂuential stochastic

parameters need to be extracted from real-world measurements and real-time data, capturing all

interdependencies, for an accurate simulation of reality. A copula is a probability model that

represents a multivariate distribution, examining the dependence between the underlying variables.

This model is used on drone measurement data from a roundabout containing dependent stochastic

parameters. With the help of the copula model, samples are generated that reﬂect the real-time

data. Resulting applications and possible extensions are discussed and explored.

I. INTRODUCTION

Autonomous driving vehicles have a huge potential not only for reducing the number of road accidents but also from

an economic point of view, reducing emissions as well as increasing the eﬃciency of traﬃc usage by, e.g., robotaxis

or avoiding traﬃc jams. Thus autonomous driving already revolutionizes our view on mobility, traﬃc safety and how

time is spent during car journeys. The question arises, how to verify that an autonomous vehicle is safe enough.

There are many proposals of how many kilometers an autonomous vehicle has to drive accident-free before it may

be safely released on the road. The estimation reaches from several millions to even billions of kilometers. We want

to stress, that this is only for one system. Every update or change of the software, occuring errors, improvements

and transformations require a rerun. Every experiment of such dimension, if feasible at all, would be tremendously

cost-intensive. This motivates a simulation based testing of autonomous vehicles.

A simulation based framework aims to replicate the real-world traﬃc and to validate the automated driving func-

tionality accordingly. In this framework, the robustness, reliability and safety can be investigated in a much quicker

and cost-eﬃcient way. It is emphasized, that the validation results and investigations in the simulated environment

are only as trustworthy as the virtual environment reﬂects the real-world [1–4].

The simulation is fed with stochastic data, e.g. velocity/acceleration of the traﬃc participants, extracted from real

recordings of traﬃc events.

A huge challenge is that the stochastic parameters, such as velocities of every traﬃc participants are not independent.

There are a lot of factors, which have to be considered. The weather, visibility, amount of traﬃc participants are

correlated to the driving behaviour of the participants. Unfortunately, this correlation is unidentiﬁed and cannot

be assumed beforehand. In this work, dependencies in stochastic data are investigated and described by so called

copulas.

The latin word copulare may be translated in “to join” or “to connect”, which already reveals the meaning and

functionality of copulas. They are a great tool for modeling the (nonlinear) dependence of random variables and

to calculate the joint distribution. The ﬁrst appearance of the term copula may be found in [5]. Since then the

wide research and application of copulas can be seen in the ﬁelds of statistics [5–7], economics [8, 9], ﬁnance [10–13],

actuarial science [14, 15] or risk management [16–18], just to name a few.

The outline of this work is as follows. In Section 2, we will go into further depth of how to validate automated

vehicles and the theory of copulas. The construction and usage for resimulating data is described. Section 3 illustrates

how to extract data from observations and how to apply the theory on a concrete numerical example and application.

A conclusion and outlook for further research and development is given in Section 4.

II. PRELIMINARIES

A. Safety validation of automated vehicles

Unlike today’s cars, autonomous vehicles require additional regulations. A distinction is made between automation

levels, as in the SAE J3016 standard [19]: a Driver-Only-Vehicle without automation, assisting systems and semi-

automated systems. In all three levels the driver keeps the responsibility for the behavior of the vehicle. Furthermore,

arXiv:2210.13117v2 [stat.AP] 21 Nov 2022

we speak of fully automated vehicles when the driver no longer acts as a fallback [20]. In this case, the automated

systems should take over vehicle control permanently and should act safely in any environment.

The international organisation for standardisation (ISO) provides the ISO/PAS(Publicly Available Speciﬁcation)

21448 to ensure the safety of the functionalities [21] and the ISO 26262 to ensure, that no hazards are caused by

technical failure [22]. ISO/PAS 21448, safety of the intended funcionality (SOTIF), focuses on predictable misuse by

the driver, as well as accidents that are explicitly not caused by component failure, but situations that were not planned

for during development. Nevertheless, this standard does not provide a detailed strategy for identifying functional

deﬁciencies. In contrast, ISO 26262 focuses on safety in terms of intrinsic safety (protection of the environment from

the product), therefore functional safety. It is an ISO standard for safety-related electrical/electronic systems in

motor vehicles. It is not enough that the function has been executed correctly, but it must also be ensured that the

function has been executed in the correct context. For example, an airbag must not be triggered when driving too fast

over speed bumps. However, the minimum requirements for safeguarding an autonomous vehicle are not adequately

described by this standard. The goal is to minimize risks to a level that is “acceptable to society”.

Consequently validation of autonomous vehicles cannot be based on existing ISO standards and need extension.

Various approaches have been developed starting with Advanced Driver Assistant Systems (ADAS), where function-

based approaches and real-world testing operate well. In the function-based approach, requirements are deﬁned for the

operating system that are tested by simulation or on the test track. In real-world testing, a mileage-based evaluation

of functionality from ﬁeld tests with a driver is performed. For both methods, validating full autonomous systems is

economical infeasible in its current state. Furthermore there is the shadow mode, presented by Wang and Winner [23],

in which the automated driving function is excecuted passively in series production vehicles. The driving function

receives the (real) information from the sensors, but will not act. Its actions are evaluated afterwards by simulation.

Considering the real world as an open parameter space, with an inﬁnite number of traﬃc events, the scenario-based

approach tries to identify these traﬃc events and describe them in scenarios. It also attempts to exclude non-relevant

traﬃc events, where neither actions nor events are observed, and to cluster similar traﬃc events into a representative

scenario [24]. However, this leads to the question of how to ﬁnd the set of representative scenarios. A good overview

of the problem of identifying critical scenarios is provided by Neurohr et al, Riedmaier et al and Zhang et al. [25–27].

Menzel et al. [28] distinguish three categories of scenarios: functional, logical and concrete scenarios. In the case

of functional scenarios, the scenario space, in the same way as traﬃc events, is described on a sematic level by ”a

linguistic scenario annotation” [28]. For the logical scenarios, this is described at the state space level. Entities and

their relationships are described using parameter ranges in the state space and optionally speciﬁed using correlations

and numerical relationships. A traﬃc event is ﬁnally mapped explicitly to the state space given a concrete scenario.

Entities and their relationships are described using concrete values for each parameter. According to Zhang, the

identiﬁcation of critical scenarios can be done on all three levels of abstraction. This requires a clear deﬁnition of the

operational design domain (ODD), a deﬁnition of the operating conditions under which an AD system is attempted

to operate, and ”the formulation of a functional scenario to a logical scenario”.

The German research project Pegasus1followed the approach in [29] of modeling the environment in layers and

extended it to 6 layers describing the environment of a highway. In the follow-up project VVM2, the 6-layer model

was reﬁned, extended to the urban environment, provided with guidelines [30].

To describe the operating conditions of an AD system, the parameters of the six layers are used, such as weather,

number of road users, number of lanes or speed of a cyclist. The assessment of whether a scenario is critical or not

is based on the concrete values for these stochastic parameters. Some studies already consider realistic parameter

distributions [27] obtained from real-life driving databases.

In scenario-based testing, parameter distributions can be used to support the search for critical scenarios within

the scenario space. In doing so, they serve to model the mutuality of the scenarios. In [31], a risk index based

on a Gaussian Mixture Model is used to eﬃciently select critical traﬃc conditions. Thereafter, the scenarios are

replicated via simulation and used, for example, to evaluate the criticality of the AD system. For the simulation

of the scenarios, real trajectories, extracted from real data are used. In the same way, characteristic criteria of

scenarios are taken from the real data, e.g. parameter distributions or parameter dependencies, and used to reproduce

trajectories or parameters. Thus, parameter distributions are applied in estimating the failure rate of a scenario. For

example, Wagner et al [32] predict the behavior of road users based on conditional distributions obtained by analyzing

naturalistic driving study called euroFOT (large-scale European Field Operational Test on Active Safety Systems)

and determine the criticality for each predicted event.

1Pegasus - Project for the Establishment of Generally Accepted quality criteria, tools and methods as well as Scenarios and Situations

for the release of highly-automated driving functions, see https://www.pegasusprojekt.de/en/home

2VVM - Veriﬁcation and Validation Methods for Level 4 and 5 Automated Vehicles, see https://www.vvm-projekt.de/

文档加载中……请稍候！
如果长时间未打开，您也可以点击刷新试试。

下载文档到电脑，查找使用更方便

10 玖币 0人已下载

立即下载

摘要：

ModelingStochasticDataUsingCopulasForApplicationinValidationofAutonomousDrivingKatrinLotto,1ThomasNagler,2andMladjanRadic11ZFFriedrichshafenAG,ResearchandDevelopment,Graf-von-Soden-Platz1,88046Friedrichshafen,Germany.2DepartmentofStatistics,LMUMunich,80799Munich,GermanyVericationandvalidationoffull...

展开>> 收起<<

Modeling Stochastic Data Using Copulas For Application in Validation of Autonomous Driving Katrin Lotto1Thomas Nagler2and Mladjan Radic1.pdf

共10页,预览2页

还剩页未读，继续阅读

声明：本站为文档C2C交易模式，即用户上传的文档直接被用户下载，本站只是中间服务平台，本站所有文档下载所得的收益归上传人(含作者)所有。玖贝云文库仅提供信息存储空间，仅对用户上传内容的表现方式做保护处理，对上载内容本身不做任何修改或编辑。若文档所含内容侵犯了您的版权或隐私，请立即通知玖贝云文库，我们立即给予删除！

Modeling Stochastic Data Using Copulas For Application in Validation of Autonomous Driving Katrin Lotto1Thomas Nagler2and Mladjan Radic1

相关推荐

开通VIP享超值会员特权

作者详情

相关内容

热门标签

举报选择: