Thibault Charmet

Véronique Cherfaoui

Javier Ibanez-Guzman

Alexandre Armand

Overview of the Operational Design Domain Monitoring for Safe Intelligent Vehicle Navigation

The operational safety of intelligent vehicles (IV) is a central topic and a complex issue on which many research projects are focused. While there is no consensus yet on the most reliable way to validate driving automation systems, several methods exist such as scenario-based testing and real-time operational domain (OD) restriction based on the specification of the operational design domain (ODD). In our case, we focus on the role of the ODD concept as a safety guarantee for IVs and how, combined with real-time monitoring of their operational domain (OD) can act as a safeguard for complex mobile systems. We propose to analyze the related literature, standards, works and proposals made in the field of OD/ODD monitoring for operational safety and to assemble them into 3 categories : the first concerns the use of taxonomies and ontologies to represent the elements of the vehicle driving environment ; the second concerns the formal definition of the ODD through a dedicated language ; the third concerns techniques allowing the monitoring of the vehicles' OD to improve operational safety. By taking a step back from the identified methods, we highlight how they can be combined in a coherent manner, to progress towards a framework enabling ODD definition and OD/ODD monitoring.

I. INTRODUCTION

Passenger vehicles are becoming more complex through the introduction of advanced driving assistance systems (ADAS) and higher levels of autonomous driving functions (ADS), the need for assured safety becomes critical. Current vehicles could be divided into two types: Vehicles with ADAS functions, where drivers remain in the controlloop, represented by SAE levels 1 and 2, these functions only provide driver assistance. Vehicles with ADS, where drivers are temporarily or completely outside the controlloop, represented by SAE levels 3 to 5 [START_REF]Taxonomy and Definitions for Terms Related to Driving Automation Systems for On-Road Motor Vehicles[END_REF]. The design and deployment of autonomous vehicles by industry and academia have shown that ensuring their safe operation in all conditions is difficult, requiring millions of kilometers of validation drives [START_REF] Kalra | Driving to safety: How many miles of driving would it take to demonstrate autonomous vehicle reliability?[END_REF]. Their autonomous navigation environment remains limited to a given number of situations, depending on the capabilities of the vehicles [START_REF] Colwell | An Automated Vehicle Safety Concept Based on Runtime Restriction of the Operational Design Domain[END_REF]. These situations are represented by geography, roads infrastructure, traffic speed, environmental conditions, etc., may reflect technological limitations, vehicle design, risk management, business considerations, etc. They need to be defined formally for *This work has been carried out within SIVALab, a shared laboratory between Renault and Heudiasyc (UTC/CNRS), and financed by Renault and the ANRT (Association nationale de la recherche et de la technologie) as a CIFRE PhD. 1 Heudiasyc laboratory, Université Technologique de Compiègne (UTC) and Centre national de la recherche scientifique (CNRS), Compiègne, France name.surname@hds.utc.fr 2 Renault, Guyancourt, France name.surname@renault.com several reasons: to control the conditions under which the vehicle equipped with an ADS/ADAS would be able to navigate autonomously, to ensure the safety of passengers and other road users, etc. [START_REF]Taxonomy and Definitions for Terms Related to Driving Automation Systems for On-Road Motor Vehicles[END_REF].

In this context, within the last years the formalization of the working area of autonomous vehicles has emerged, the Operational Domain (OD), while introducing the concept of Operational Design Domain (ODD) [START_REF]Automated Driving Systems 2.0: A Vision for Safety[END_REF]. This corresponds to the set of situations and conditions for which an autonomous system has been designed, and therefore for which its operation and safety must be demonstrated and validated [START_REF]Taxonomy and Definitions for Terms Related to Driving Automation Systems for On-Road Motor Vehicles[END_REF]. For example, the ODD of an ADS may consist of autonomous navigation on a motorway, with full lane markings and during day conditions. If lane markings are poor, or it is at night or a bicycle is perceived, the ADS will be outside its ODD. The concepts are detailed in Section II.

Currently, there is much interest on the formalisation of the ODD through standards, verification processes, etc. as applied to ADS. The aim of OD monitoring, often refereed as ODD monitoring is to determine in real-time whether the current OD of the vehicle is compatible with the ODD for which it has been designed and validated. It means that, for OD monitoring, the environment in which the ADAS or ADS functions must operate has been defined (i.e. ODD determination). The monitoring is done in real-time to ensure the system remains within the defined ODD. That is, the driving conditions (OD) are being monitored against some specifications (ODD). In the paper, we refer to it, as OD monitoring.

The challenge is to determine in real-time whether or not a system is capable of operating in its current environment. For this purpose, different steps are needed: (i) Based on the system's capabilities, an ODD, fitting the situations where the system is working safely, must be established. (ii) This ODD must then be defined formally (i.e., a structured human and machine readable representation). (iii) It must be used in realtime to detect potentially unsafe situations (OD monitoring). However, if the ODD determination can be well established from test-based validation (Section V-B), nevertheless, the formalization of the ODD definitions and the OD monitoring are still under progress [START_REF] Zhang | Finding Critical Scenarios for Automated Driving Systems: A Systematic Mapping Study[END_REF]. The purpose of this paper is to seek methods that allow a verifiable formalization of the ODD and enable real-time OD monitoring. The paper shall provide a structured critical review leading to the formulation of the research problem and to outline the possible approaches to follow.

The paper is structured as follows: Section II, presents the context and definitions needed to understanding what OD monitoring is. We then review existing works centered around concepts relevant to OD monitoring: taxonomies and ontologies of ODD attributes, ODD definition languages and OD monitoring. Section III presents the taxonomies used to structure and define attributes describing the driving environment of intelligent vehicles. They are the basis of ODD definitions and constitute most of the related literature. Section IV looks into the languages used to define/describe the ODD. These formulate in a coherent manner the attributes defined in the ODD taxonomies. Section V looks into what constitutes OD monitoring and the methods used. Finally, Section VI concludes the paper, by summarizing findings and proposing the directions that future work will seek, to implement real-time OD monitoring, to determine whether the vehicle is operating within the environment for which it was designed.

II. DEFINITIONS

A. Operational Domain (OD)

The OD is the domain of operation of the vehicle i.e. the full range of real-life driving conditions. In practice, we observe and understand it through sensors and situation understanding algorithms, but it does not guarantee a complete representation of the real situation. Thus, it could be represented in many ways, the most common is as a finite set of attributes (like environmental, geographical or object information). This OD representation is a granular representation of the world. The attributes will be explained in section II-C.

B. Operational design domain (ODD)

a) Definition: The ODD, is defined by SAE J3016 [START_REF]Taxonomy and Definitions for Terms Related to Driving Automation Systems for On-Road Motor Vehicles[END_REF] as "Operating conditions under which a given driving automation system or feature thereof is specifically designed to function, including, but not limited to, environmental, geographical, and time-of-day restrictions, and/or the requisite presence or absence of certain traffic or roadway characteristics". For example, a traffic jam chauffeur should be able to drive inside slowed traffic (< 60km/h), on highways, with good illumination (no night, tunnels, etc), forgiving weather (no snow, storm, etc) and as long as it has access to network information including maps, alerts, etc. The ODD can be extended with other conditions include being able to drive under unfavorable weather (rain, etc) but at reduced max speed can be added. It is up to the system developers to define the ODD in terms of operational guarantees they want to provide. b) Motivation: There are three main reasons why it is necessary to understand and use the ODD.

First, for safe operation, i.e. all safety measures taken in real-time, during the operation of a system, to ensure its safe use. Because an ODD is a subset of all the driving conditions that can be encountered while driving (i.e. the OD). By restricting those conditions and controlling the driving conditions, it makes it easier to guarantee the safe operation of a system. It makes the task of creating highly autonomous driving automation systems much easier [START_REF]Taxonomy and Definitions for Terms Related to Driving Automation Systems for On-Road Motor Vehicles[END_REF] (fig. 1, based on constructors official information1 ). The OD monitoring is ensuring the safe operation of a system, by comparing the measurable OD representation of a given situation with pre-established specifications (i.e. the ODD) that must be respected. Fig. 1. SAE levels vs the number of driving conditions (ODD size). Level 4 autonomous shuttles, by limiting their ODD to predetermined paths, slow speeds, etc are able to reach high automation. Level 3 Mercedes Drive Pilot provides conditional automation by limiting the conditions to traffic jams under 60 km/h, daylight, no tunnels, etc. Level 2 Tesla Autopilot can be used in many situations but under the driver's liability. Because level 0 Automatic Emergency Braking (AEB) systems are simple, they can be used safely in almost any driving condition.

Second, for autonomous operation. The main difference between SAE level 2 and 3 is that the driving liability is transferred to the ADS when it is activated and within the ODD [START_REF]Taxonomy and Definitions for Terms Related to Driving Automation Systems for On-Road Motor Vehicles[END_REF]. It implies that the driving conditions (OD) must be monitored and compared to the ODD. Note that, even though, OD monitoring is necessary for level 3, it can be applied to any system in order to improve operation safety.

Third, for safe design, i.e. all safety measures taken during the conception of the system (validation, etc). An ODD can be seen as a set of high-level specifications for which the system has been designed to operate safely. It is common to see the constructors first use it as a set of operational domain specifications that the system should reach at the end of development. Then, as the project evolves, the ODD will be refined to match the actual tested and accepted driving conditions. Thus, at the end of the process, the ODD is the set of specifications for which the system is guaranteed to function.

Its definition must be understandable by people outside the field (ADS users, regulatory authorities, decision-makers, etc) because its purpose is to be communicated with the public, autonomous car users, authorities and other manufacturers. It must also be precise enough to describe the majority of the attributes of the environment.

In other fields, equivalent concepts are used, such as Operational Envelope (OE) for maritime and Operational Context (OC) for railroad [START_REF] Tonk | Operational Design Domain or Operational Envelope: Seeking a suitable concept for autonomous railway systems[END_REF].

C. ODD attributes

As the ODD definition must be understandable by people outside the field and machine readable, ODD attributes (or OD representation attributes) are commonly used as the base elements of the OD representation and the ODD definition. They must be agreed upon before defining an ODD, so every reader can understand exactly what they mean in real life. In this document the usage of the terms OD representation attributes and ODD attributes are interchangeable as the attributes of the representation of the OD and the ones used to define an ODD are generally the same.

Let's take an example. Like the T-junction presented in the ASAM OpenODD concept paper (fig. 2) [START_REF] Asam | OpenODD: Concept Paper[END_REF]. A vehicle moving in this area would have to deal with a 2 way lane, a T-junction, a traffic light, a zebra crossing, a speed bump, signs for 50km/h speed limit, a pedestrian crossing, streetlamps, and some vegetation, environmental conditions like the weather, the illumination, and road users' information like the presence or absence of car/pedestrian traffic. This is the OD of this particular area, at a specific time, expressed in natural language. Fig. 2. Illustration of a T-junction from ASAM OpenODD concept paper [START_REF] Asam | OpenODD: Concept Paper[END_REF]. With a 2 way lane, a traffic light, a zebra crossing, a speed bump, signs for 50km/h speed limit, a pedestrian crossing, streetlamps, and some vegetation.

In order to represent this OD in a numerical way, we can create a set of attributes from the perceived environment. This set of attributes is the OD representation of this particular area at a specific time, expressed in a formal way.

III. TAXONOMIES AND ONTOLOGIES FOR INTELLIGENT VEHICLES

In order to describe and structure the driving environment of vehicles, it is common to use attributes organized into taxonomies. A taxonomy is a type of hierarchical classification, often organized in tree-like structure, categorizing things into groups and subgroups (like for biological classification), such that there is a logical semantic relation between the attributes and their parents. Taxonomies are a special type of ontology. They both share the same goal of representing information, knowledge and concepts in a formal, explicit, structured format [START_REF] Studer | Knowledge engineering: Principles and methods[END_REF]. The main difference is that ontologies are used to represent various relations between the concepts (i.e. possible interactions between multiple attributes). Ontologies are used in many domains to represent expert's knowledge, complex interactions, etc. [START_REF] Armand | Ontology-based context awareness for driving assistance systems[END_REF] describes an ontology-based reasoning for situation awareness that can be used to model spatio-temporal relations between the road users and the infrastructure.

Most of the works around ODD definition and labelling will tend to use taxonomies for their simple and intuitive tree-like structure. While most of the works around scenariobased validation will use ontologies, as their more complex graph-like structure can represent interactions between the elements of the environment, thus making scenario generation possible.

A. Taxonomies

In the autonomous driving domain, multiple attribute taxonomies have been proposed (summarized in table I). In their 2017 report [START_REF]Automated Driving Systems 2.0: A Vision for Safety[END_REF], the NHTSA encourages car manufacturers to define and document the ODD for their ADS. They only provide some top-level elements as example minimal information. The following year, they propose a more extensive ODD taxonomy [START_REF] Thorn | A Framework for Automated Driving System Testable Cases and Scenarios[END_REF], multiple levels deep. Their paper also provides an extensive literature review of ADS features, Object and Event Detection and Response capabilities (OEDR), Fail Operational (FO) and Fail Safe (FS) strategies based on around 50 sources. The same year [START_REF] Czarnecki | Operational Design Domain for Automated Driving Systems -Taxonomy of Basic Terms[END_REF], [START_REF] Czarnecki | Operational world model ontology for automated driving systems-part 1: Road structure[END_REF], [START_REF]Operational world model ontology for automated driving systems-part 2: Road users, animals, other obstacles, and environmental conditions[END_REF] proposed an Operational World Model Ontology with elements like road structure, road users, animals and obstacles, road user behavior, environmental conditions, etc. It has a tree-like taxonomic structure without formal definition of the relations between attributes. Thought, is mostly aimed at driving scenarios' specifications rather than ODD specification. It can be seen as an extended ODD attributes taxonomy including behavioral information divided into two categories. Behavioral factors are information like traffic rules, informal rules, user skills, capabilities, personality, emotions, etc. behavioral models are information related to the type of model used to infer road users' behavior. This last category couldn't be used for an ODD definition, as it is not dependent on the driving environment, but from the functional architecture of the ego driving system. In 2020, two taxonomies are proposed. One thought a report [START_REF]AVSC Best Practice for Describing an Operational Design Domain: Conceptual Framework and Lexicon[END_REF] aiming to establish best practices for ODD description and proposing some terminology that can be used to describe the ODD. The other, PAS 1883 [START_REF] Bsi | Operational Design Domain (ODD) Taxonomy for an Automated Driving System (ADS) -Specification[END_REF], often mentioned, proposed a minimal taxonomy for ODD attributes. Technical standards such as ISO 34503 [START_REF]ISO/DIS 34503 : Road Vehicles -Test scenarios for automated driving systems -Taxonomy for operational design domain[END_REF] and J3259 [START_REF]J3259 Taxonomy & Definitions for Operational Design Domain (ODD) for Driving Automation Systems[END_REF] are still under development. ISO 34503 should specify requirements for the ODD attributes taxonomy, along with a natural language definition format for use by regulators and non-coders.

All of these taxonomies attempt to list fairly exhaustively the attributes describing the driving conditions. They are often the result of extensive research in the field of transportation safety and public transport engineering, and many elements are present in all of them. The main difference between the cited taxonomies is their organization into groups and subgroups, which is more subjective. This is because one attribute can be part of multiple groups, but has to be organized under one branch. Therefore, based on our analysis of the taxonomies, we can illustrate them with a representative taxonomy containing the attributes commonly found, organized in the following distinct toplevel categories:

• Road structure (road type, surface, geometry, edge, lanes, specificities, junctions, signage, etc) • Objects and road users (road users, animals, non-static objects, etc)

B. Ontologies

Ontologies are used as knowledge representation structures for others tasks like scenario generation, situation awareness, etc. They could serve, like taxonomies, as a knowledge base for ODD attributes. In 2018, [START_REF] Bagschik | Ontology based Scene Creation for the Development of Automated Vehicles[END_REF] used ontologies as knowledge-base for scene generation, as a basis for a scenario creation, mentioning OpenDRIVE and OpenSCENARIO as future simulation data format. In 2022, [START_REF] Jannis Erz | Towards an Ontology That Reconciles the Operational Design Domain, Scenario-based Testing, and Automated Vehicle Architectures[END_REF] proposed an ontology based on elements from ODD taxonomies, scenario-based testing, and ADS architecture, in order to help the creation of scenarios for scenario based testing. ASAM is developing OpenXOntology2 , an ontology combining all the attributes/objects common to their OpenX standards (mainly OpenSCENARIO 2.0, OpenLABEL and OpenODD). The goal is to have a compatible set of objects that can be referenced by any OpenX definition.

C. Taxonomy and ontology implementation

Taxonomies and ontologies need to be implemented in order to be used by downstream tasks. Choosing a language to represent and query the taxonomy is a necessary step toward it. Through the efforts to create an ODD definition language (see section IV), multiple ways to represent the ODD attribute taxonomies or ontologies were used. JSONlike languages (YAML, etc) are best for simple taxonomies representations. They can represent simple tree-like data structures with values and hierarchy. [START_REF] Whiteside | ASAM OpenXOntology ProposalWorkshop 5 Ontologies and ODDS at Five[END_REF] uses a DSL which is a simplification of a Ontology Web Language (OWL). ASAM's OpenXOntology needs to represent complex relations between objects that occur in scenarios, and thus is being built on an OWL base. OpenODD will probably be based on OpenXOntology. Other projects like [START_REF] Bagschik | Ontology based Scene Creation for the Development of Automated Vehicles[END_REF] are OWLbased and use the Java-based OWL-API. The ontology built by [START_REF] Jannis Erz | Towards an Ontology That Reconciles the Operational Design Domain, Scenario-based Testing, and Automated Vehicle Architectures[END_REF] uses a UML based representation.

IV. OPERATIONAL DESIGN DOMAIN DEFINITION LANGUAGE

The ODD definition is often based on the considered use cases and written in a tabular format. Using instead an ODD definition language, may be a better way to represent the bounded driving conditions in a way:

• that is both shareable with others (because it can be written in an independent file), that can be understood both by non-developers, and by the machine. • that is automatable, because it is understandable by the machine and can be used for OD monitoring. • that is exhaustive because it can be based on a selected and shared set of attributes, more or less detailed as needed. As pointed out by multiple papers, today, there is no method to completely define an ODD, whether in a formalized way or not. This means that actual ODD definitions available in papers are insufficient for other downstream tasks like scenario generation [START_REF] Zhang | Finding Critical Scenarios for Automated Driving Systems: A Systematic Mapping Study[END_REF], [START_REF] Irvine | A Two-Level Abstraction ODD Definition Language: Part I[END_REF], [START_REF] Rohne | Implementing ODD as single point of knowledge to support the development of automated driving[END_REF].

This part presents previous and current works on the domain. Table II summarizes the ODD definition language projects and it highlights some of the most important aspects: the ODD attributes used (taxonomies, etc), the type of representation used for those attributes (YAML, OWL, etc), the type of the definition language (DSL, etc), the possibility to define quantitative values (in addition to qualitative values), possible conditional definitions of the attributes, the presence of consistent rules toward hierarchy and the definition order, permissive and restrictive modes, if maneuvers, behaviors and probabilities are included, availability of the full language specifications, the availability of the language implementation if any.

There have been few propositions to create an ODD language. In a Five AI's presentation [START_REF] Whiteside | ASAM OpenXOntology ProposalWorkshop 5 Ontologies and ODDS at Five[END_REF], 3 Domain Specific Language (DSL) are presented, including one for ODD

✗ ✓ ✓ ✓ ✓ ✗ ✗ ✗ ✗ [23] ✓ ✓ ✓ ✓ ✗ ✗ ✓ ✓ ✗ [22] ✓ ✓ ? ? ✗ ✗ ✓ ✗ ✗ [7] ✓ ✓ ✓ ✓ ✗ ✗ later ✓ ✗
definition and one for "high-level scene annotation" (i.e. OD representation for scenarios). Quantitative values can be used to define qualitative attributes in the ontology, then only qualitative attributes are used in the ODD definition. Maneuvers are part of the ODD attributes. Conditional definitions are implemented using a global/local fashion. And it is partially compatible with OpenX.

In this pair of papers [START_REF] Irvine | A Two-Level Abstraction ODD Definition Language: Part I[END_REF], [START_REF] Schwalb | A Two-Level Abstraction ODD Definition Language: Part II[END_REF], the authors introduce a specific language that allows to define the ODD, at 2 levels of representation: semantic, and formal. For the semantic language, they propose a SQL-like query language composed of statements like: CLASS, ACCEPT, etc. Conditional definitions of attributes can be made with "WHEN" and "EXCEPT" keywords. Operators can be used for value comparison as long as units are specified. There is consistency between the statements when they are imported. Once an attribute has been accepted or rejected in some conditions, it cannot be changed by subsequent statements by mistake. Attribute accepted uncertainty can be represented using probabilities and the "QUALIFIER" and "@" keywords. e.g. "pedestrian@P8,E6" would mean a pedestrian occurrence less than once per 10 8 hours, or a pedestrian detection error less than 2 -6 (> 98.5% accuracy). This initiative, although interesting, remains to this day only theoretical. Indeed, no publication presents whether this language has been implemented and used in real conditions.

Recent works such as [START_REF] Rohne | Implementing ODD as single point of knowledge to support the development of automated driving[END_REF] are starting to build ODD frameworks. Their ODD definition language is based on [START_REF] Schwalb | A Two-Level Abstraction ODD Definition Language: Part II[END_REF] and the paper generally inspired by it. They made their own ODD attributes taxonomy composed of around 760 terminal nodes (i.e. attributes) inspired by [START_REF]ISO/DIS 34503 : Road Vehicles -Test scenarios for automated driving systems -Taxonomy for operational design domain[END_REF], thought the definition format used is unknown. They propose a method for validating defined ODD against situations datasets. They use situations datasets to estimate the OD in different areas, which can be then compared to an ODD. The ODD, YAMLbased, is monitored against a small subset of ODD attributes. Conditional statements can be made and combined logically with "WHEN" and "EXCEPT" keywords. There is little in-formation about the consistency of the definition when doing multiple statements and imports and restrictive/permissive modes are not mentioned. Even if the paper is proposing a prototype implementation, there is little reproducibility as neither the code nor the taxonomy are given.

The most recent work is the OpenODD [START_REF] Asam | OpenODD: Concept Paper[END_REF]. If the V1.0.0 is still under progress, they already proposed two syntax for an ODD language. There is a quite exhaustive list of 15 requirements for the language including human/machine readability, composability, conditional statement, extensibility of ontology, binary boundary, units, and a general compatibility with OpenX standards. It should support quantitative values and even variables storing values. Uncertainties and probabilities should be added in later versions. They clearly define that behavior information are reserved for scenarios and not ODD. The project is backed by members from many companies and universities such as the University of Warwick, Five AI, etc. There is a quite extensive research done about what functionalities must be included in the ODD definition language (user stories, requirements, etc). The release date is due for January 2024, however, no official implementation of OpenODD is expected by ASAM as they are a standardization group.

The ISO 34503 standard [START_REF]ISO/DIS 34503 : Road Vehicles -Test scenarios for automated driving systems -Taxonomy for operational design domain[END_REF] can also be mentioned here. This standard, still under development, will provide an ODD natural language definition format for use by regulators and non-coders. As no computer readable format is planned, this wasn't added to the list with others.

V. OPERATIONAL DOMAIN MONITORING

OD monitoring is not a new concept, as any autonomous system would have to limit their functionalities to some conditions. Commercial level 3 ADS like Honda's traffic jam pilot for their Legend Hybrid EX and Mercedes's traffic jam pilot, restricted to 60km/h speed limits, in traffic jams, good lightning conditions, are good examples. If the best practice publications mention the ODD of their system, it is very rare to find the methods that have been used for the OD monitoring. In this part, we present some of the methods used.

A. Geofencing

The simplest way to define the ODD and to then monitor the OD in real time is to use geofencing as an approximation of the driving conditions. Geofences are virtual perimeters describing real geographic areas. They can be used by ADS to bound the areas inside which the system can be used, or inside maps, to describe areas or interest (e.g. school zones, low emission zones, etc) [START_REF] Maiouak | Dynamic Maps for Automated Driving and UAV Geofencing[END_REF]. As simple restrictions on vehicle location are easier to implement and understand than high-dimensional ODD descriptions, this is the most common approach. In their guidelines [START_REF]Implementation of Autonomous Systems Sub-Committee[END_REF], the Global Mining Guidelines Group (GMG) restricts conditional automation to "designated autonomous area". Ohmio level 4 shuttles, and robotaxis in general, evolve in very specific geofenced areas, at restricted speed limits.

B. Measuring the ODD

Some works focus on identifying and defining the ODD through data collection or depending on system capabilities (maneuvers or object and event detection and response). [START_REF] Colwell | An Automated Vehicle Safety Concept Based on Runtime Restriction of the Operational Design Domain[END_REF] define a Restricted OD (ROD) when some system capabilities are lost (Degraded Operation Mode, DOM). They consider using OWL as a structured language to represent the ODD attributes ontology in future works. They mention the ontology of [START_REF] Czarnecki | Operational world model ontology for automated driving systems-part 1: Road structure[END_REF] as a base to build and ODD. [START_REF] Lee | Identifying the Operational Design Domain for an Automated Driving System through Assessed Risk[END_REF] are identifying a geographical and environmental ODD by evaluating the conditions in which the ADS is underperforming. For each combination of ODD attributes, the estimated risk is computed based on expected severity, contextual exposure to the attribute, and system-measured performance in those conditions. Attribute combinations associated with high risk are removed from the initial permissive ODD. In a NIST publication [START_REF] Griffor | Automated Driving System Safety Measurement Part I: Operating Envelope Specification[END_REF], the idea of measurable OD representation is developed through what they call an Operating Envelope Specification (OES). It focuses on the operational safety side of the ODD, with OD reasoning against and ODD (OD monitoring), but do not detail about how is should be done and formalized. [START_REF] Sun | Operational Design Domain Monitoring and Augmentation for Autonomous Driving[END_REF] proposes a way to determine the ODD of a system based on collected data, and proposes a realtime assessment for perception algorithms. They also do robust training for perception using data augmentation. [START_REF] Torfah | Learning Monitorable Operational Design Domains for Assured Autonomy[END_REF] have a data-driven, out-of-ODD research via counter example trained machine learning (ML). They focus on few temporal attributes that can be predicted over time for the environment, and use specific rules corresponding to faulty behaviors to find out-of-ODD situations.

C. ODD for safety validation

Most of the works around ODD and using taxonomies and ontologies are for design safety (i.e. ADS safety validation). The main focuses being scenario generation [START_REF] Bagschik | Ontology based Scene Creation for the Development of Automated Vehicles[END_REF] and scenario database advanced labelling. Deepen AI and the Warwick Manufacturing Group (WMG) worked in collaboration on SafetyPool™, a public scenario database repository with multiple features 34 . They allow the creation and filtering of scenarios based on multiple label including ODD attributes from BSI PAS 1883. They have their own scenario definition language that can be converted to an OpenSCENARIO format. They generally try to be compatible with OpenX standards. They can also generate CARLA 5 simulations from those scenarios. They plan to add testing and other new safety related features in order to provide a platform with a full safety validation workflow. In their case the ODD attributes are used as labels for scenarios. This is one use case that allow them, for example, to display on a map all the roads corresponding to a given set of ODD attributes. Those attributes can be compared to the information available on the map to determine which one corresponds and which one don't. This is the same method that would be used by level 4 path planning systems to avoid road outside a given ODD.

D. Interactions with the ODD

Other works may focus on the interactions between the ODD definitions, users and the infrastructure. [START_REF] Vreeswijk | Cooperative Automated Driving for managing Transition Areas and the Operational Design Domain (ODD)[END_REF] mention, for future works, the use of ODD for managing Transition Areas in a Cooperative Automated Driving context to mitigate handovers and minimal risk maneuvers. There is the Traffic Management for Connected and Automated Driving (TM4CAD) project 6 , expected to finish in March 2023. The project is targeted toward regulatory bodies, in particular the National road authority (NRA), responsible for the national road network in Ireland. They mainly provide insights and recommendations about the ODD management (via perceived and exchanged ODD-relevant information by the infrastructure and CAVs) from the point of view of the infrastructure providers. Basically, they are trying to answer the question "how traffic management systems and CAD vehicles can best interact to improve traffic operations" ? They give insights about the detectability or availability of the ODD attributes (e.g. today's CAV may not be able to measure the road surface friction and will never be able to detect roadside obstructions in advance without the information from the infrastructure), the change frequency of ODD attribute information (e.g. weather changes faster than the road geometry).

E. Monitoring strategies

There are very few publications proposing actual frameworks for OD monitoring. In their paper, [START_REF] Gyllenhammar | Towards an Operational Design Domain That Supports the Safety Argumentation of an Automated Driving System[END_REF] divide OD monitoring into 4 strategies corresponding to 4 groups of ODD attributes. Direct internal conditions (like speed) on which the vehicle should have total control on ; conditions that can be guaranteed beforehand (like a priori information about the path, available in advance) ; conditions that can be guaranteed beforehand depending on geographical or temporal values (similar by more geospacial-based like weather forecast in some areas) ; and dynamic conditions (like road friction, real time rain). This approach helps to monitor each attribute with different adapted monitoring strategies. They also propose to combine those strategies for a same attribute. For example, coupling a weather forecast with a sensor in order to detect strong rain. A first constraint is placed on rain probability. The weather forecast gives a predicted probability distribution of rain, and the ODD decides if the probability is low enough. A second constraint is placed on rain strength. A rain sensor measure in real time the intensity and the ODD decides if it is accepted. Though this approach is interesting, this might be equally simple to have 2 attributes "rain forecast" and "rain intensity" that are both restricted by the ODD.

VI. DISCUSSION

We can state, that multiple elements are needed to ensure ODD representation and OD monitoring. They can be referred as the ODD framework and can be described as follows:

• ODD attributes. Preferably organized into a taxonomy or an ontology. • ODD attributes representation. Machine readable attributes implemented with a structured language (e.g., a tree like structure language; JSON, YAML, etc.) or with more complete ontology representations (e.g., based on a web ontology languge -OWL, OpenXOntology, etc.). • ODD definition language, machine and human readable.

• Intermediate OD representation.

• Algorithm for OD monitoring. As found in [START_REF] Gyllenhammar | Towards an Operational Design Domain That Supports the Safety Argumentation of an Automated Driving System[END_REF], it might be useful to take the divide and conquer approach and apply multiple monitoring algorithms for different types of information sources (e.g., dynamic, a priori, etc). The core of the algorithm will be the ODD and OD comparison, however, the difficult task resides on having a valid OD representation in the first place.

Finally, most studies and publications are best practices for human-defined ODD, conceptual taxonomies. Few address techniques to implement an ODD framework either for development or the system validation phase, or on-board use in real-time ; to determine if the driving conditions in which the vehicle evolves (the OD) is compatible with the ODD. There are also few propositions to represent the knowledge uncertainty inside the ODD framework.

Whilst the ODD is referred as part of several projects, there is no method for general and automatic OD monitoring. To date, the OpenODD project can be considered as the most advanced ODD definition language. For human, naturallanguage ODD definition,the ISO 34503 should become the norm. With regards to ODD attributes, any taxonomy or combination of taxonomies can be relevant. The simplest and one of the earliest would be PAS 1883 and the most complete SAE J3259 (in preparation). The simplest taxonomy representation would be YAML and the more advanced an OWL-based representation ideally compatible with OpenXOntology.

VII. CONCLUSION

In this paper, we identified three methods needed to effect real-time monitoring of the operational domain: taxonomies and ontologies to represent the driving environment, formal operational design domain definition languages, and methods for operational domain monitoring. So far, only specifications, and high-level ODD taxonomies have been published, little information is available on the manner how they should be formalised so they can be integrated into ADS. Formats from other domains, like ontologies used for scenario generation, could be applied. Currently there is work to provide a structure capable of representing machinereadable ODD definitions as part of the OpenODD, ASAM standard. Different approaches are being proposed towards OD monitoring, these define the ODD automatically including strategies to monitor ODD attributes. By using insights from all of these works, it would be possible to formulate a full ODD definition and develop an OD monitoring framework beyond conventional geofencing based boundaries, and towards attribute-rich operational domain monitoring. While taking into account deployment constrainsts such as perception uncertainty and the non-observability of some attributes. This operational domain monitoring, will be a needed piece to ensure safe navigation of intelligent vehicles in the future.

TABLE I ODD

 I TAXONOMIES PROPOSED IN THE LITERATURE.

	Name	Authors	Date	# attributes
	ADS 2.0 [4]	NHTSA	2017	approx. 10
	DOT HS 812 623 [10]	NHTSA	2018	approx. 170
	Operational world model ontology for ADS [11]	Czarnecki	2018	approx. 100
	AVSC00002202004 [12]	AVSC	2020	approx. 100
	PAS 1883 [13]	BSI	2020	approx. 150
	ISO 34503 [14]	ISO	WIP	-
	SAE J3259 [15]	SAE	WIP	-

https://easymile.com/technology/ezfleet https://www.mercedes-benz.de/passengercars/ technology/drive-pilot.html https://www.tesla.com/support/autopilot

ASAM OpenXOntology, https://www.asam.net/standards/ asam-openxontology/

https://www.safetypool.ai/

Demo of SafetyPool Usecase for ODD Scenario Tagging, Siddartha Khastgir (ASAM Webinar), https://youtu.be/lSycwx44Glg

https://carla.org/

https://tm4cad.project.cedr.eu/


domain: spi phys

text:
Caroline Nore
email: caroline.nore@limsi.fr

Wietze Herreman
email: wietze.herreman@limsi.fr

Sabrina Benard
email: sabrina.benard@ens-paris-saclay.fr

Paolo Personnettaz
email: p.personnettaz@hzdr.de

Loic Cappanera
email: loic.cappanera@gmail.com

Jean-Luc Guermond
email: guermond@math.tamu.edu

Sabrina Bénard

Loïc Cappanera

MIXING IN LIQUID METAL BATTERIES

Keywords: Liquid metal batteries, Magnetohydrodynamics, Numerical simulations

de niveau recherche, publiés ou non, émanant des établissements d'enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

Liquid Metal Batteries (LMB) are composed of three layers of fluids (liquid metal electrode-electrolyteliquid metal electrode) of different densities lying over each other and stabilized by gravity. The composition of the positive/bottom electrode of an LMB may become very inhomogeneous over charge/discharge cycles, which limits the battery efficiency. Yet rotational flows in electrically conducting fluids can be generated by the Electro-Vortex-Flow (EVF) phenomenon, i.e. the interaction of a non-uniform current with the magnetic field it generates [START_REF] Herreman | Numerical simulation of electrovortex flows in cylindrical fluid layers and liquid metal batteries[END_REF]. Using axisymmetric and fully three-dimensional numerical simulations with the multiphase magnetohydrodynamic code SFEMaNS [START_REF] Cappanera | Momentum-based approximation of incompressible multiphase fluid flows[END_REF], we investigate whether the mixing induced by the EVF mechanism can help homogenize the positive electrode against buoyancy during charge/discharge cycles.

Figure 1 :

 1 Figure 1: Molar fraction of Li atoms in the positive/bottom electrode during the charge of a Li Pb liquid metal battery after a discharge.


David Robbe
email: david.robbe@inserm.fr

Mostafa Safaie

Neural Dynamics Hot times for the dorsal striatum

Using temperature manipulation, Monteiro and colleagues found a way to bidirectionally alter the speed of neuronal dynamics in the dorsal striatum of anesthetized rats. The same temperature manipulation selectively slowed down or sped up time perception, providing novel insights on the mechanisms of time-based decisions.

Main Text

Time is at the heart of most of our daily decisions and actions. As we stand at a pedestrian crossing, we often must make quick estimates as to whether we have enough time to make it across before an oncoming vehicle approaches. Well-timed behaviors are obviously not exclusive to humans; many other animals, as they forage, appear to wait or exploit food resources for precise durations or at regular time intervals. From such examples, it seems reasonable to assume that animals have an inner sense of time to make decisions based on duration. Still, the quest for the neuronal underpinning of time estimation, especially in the range of a few seconds, has been a wild one! At first, researchers assumed that animals tracked duration using internal clocks [START_REF] Church | Properties of the Internal Clock[END_REF] , potentially centered around the dorsal striatum and its dopaminergic modulation [START_REF] Buhusi | What makes us tick? Functional and neural mechanisms of interval timing[END_REF] . More recently, the realization that time is inherently encoded in the dynamics of neuronal activity led to the proposal that duration could be intrinsically represented by neuronal ensembles distributed over multiple brain areas activated in a given timing task [START_REF] Paton | The Neural Basis of Timing: Distributed Mechanisms for Diverse Functions[END_REF] . A difficulty with such a framework is to establish a causal relation between the evolving activity of a neuronal population and decisions based on duration. It is that precise challenge that Monteiro et al. have tackled in their recent study [START_REF] Monteiro | Using temperature to analyse the neural basis of a time-based decision[END_REF] , building on two important lines of work. On the one hand, population activity in the striatum has been previously shown to predict time-based binary decisions in rats [START_REF] Gouvêa | Striatal dynamics explain duration judgments[END_REF] . On the other hand, manipulating the temperature of brain tissues has been previously shown to alter not only the speed of specific behavioral patterns [START_REF] Long | Using temperature to analyse temporal dynamics in the songbird motor pathway[END_REF] but also the dynamics of neural circuit activity while preserving its overall structure [START_REF] Banerjee | Using focal cooling to link neural dynamics and behavior[END_REF] . Thus, to investigate the neural bases underlying time judgment, the authors asked two audacious questions. First, could striatal population dynamics be finely tuned by temperature? And second, would temperature manipulation of striatal tissue in behaving animals affect duration-based decisions? To address these questions, Monteiro and colleagues developed a custom-built probe, based on the Peltier effect, to bidirectionally and quickly manipulate the temperature of the brain. They first used this probe to warm up (+1.5°) and cool down (two levels: -2.7° and -5.1°) the tissue and examined the impact of such manipulation on striatal neural activity evoked by stimulation of a sensory region of the thalamus. These experiments were performed under anesthesia to minimize neural modulations due to ongoing behavior. Importantly, the evoked response of striatal neurons lasted hundreds of milliseconds, most likely reflecting population dynamics across loops formed by the basal ganglia with thalamus and cortex. Using dimensionality reduction methods to represent striatal population dynamics as a trajectory in a 3D state space, the authors showed that cooling the striatum slowed down the evoked "trajectory" whereas warming accelerated it. Critically, this monotonic modulation of the population dynamics was observed despite an overall reduction in striatal firing rates during both warming and cooling regimes. Next, the authors examined whether striatal population dynamics could provide information about time-based decisions. They used a categorization task (Figure 1A) in which rats heard two brief tones separated by an interval between 0.6 and 2.4 s. Upon presentation of the second tone and after estimating whether the interval was shorter or longer than an arbitrary reference set at 1.5 s, animals expressed their judgment by poking toward the left (long) or right (short) port. Rats were very good at discriminating between very short and long intervals, but their accuracy decreased for the two intervals nearest to the reference duration. The authors focused on those trials and replicated their previous observation that, during short trials misjudged as long, the striatal population dynamics had evolved more quickly [START_REF] Gouvêa | Striatal dynamics explain duration judgments[END_REF] . They then move to the more exciting part of their study: testing whether temperature manipulation could bias the interval duration judgment of near-boundary intervals. As predicted from the above electrophysiological experiments in behaving and anesthetized animals, warming the striatum led to a bias towards longer temporal judgments, while cooling the striatum had the opposite effect (Figure 1B-D). Although only one level of warming was performed due to technical limitations, larger manipulations during cooling trials resulted in larger decision biases. Critically, the monotonic influence of temperature was specific to the dorsal striatum, as manipulating the temperature of the overlying motor cortex failed to produce a similar effect. Considering the vast body of evidence indicating the involvement of the dorsal striatum in controlling movement and decision speed [START_REF] Robbe | The Basal Ganglia invigorate actions and decisions[END_REF] , the authors finally examined whether temperature manipulation affected the kinematics of the movements toward the reward ports, and reported that both cooling and warming the striatum led to slower movements and prolonged reaction times. Because temperature manipulation had non-monotonic effect on the firing rates of striatal neurons under anesthesia, Monteiro et al. concluded that the speed of population dynamics in the dorsal striatum regulates the judgment of time, while the average activity of its single neurons may provide a low-dimensional gain-like signal controlling the vigor of actions. Altogether, the authors proposed the view that the speed of population dynamics in the dorsal striatum guides the when and what of action selection but does not control movement kinematics on a moment-to-moment basis. Such a conclusion is both exciting and provocative. It is stimulating because it may provide a way to bridge systems neuroscience in rodents with clinical observation in humans. Indeed, it is well known that patients with Parkinson's disease, which results in a malfunctional dorsal striatum, are capable, in specific contexts, of producing perfectly skilled actions, a phenomenon referred to as kinesia paradoxa [START_REF] Banou | Kinesia paradoxa: a challenging Parkinson's phenomenon for simulation[END_REF] . This would make sense if the striatum controlled latent decisional processes rather than movements themselves. It is also provocative as there has been recent experimental evidence supporting the view that the sensorimotor region of the striatum continuously controls the kinematics of skilled movements [START_REF] Dhawale | The basal ganglia control the detailed kinematics of learned motor skills[END_REF] . In addition, electrophysiological recordings across the cortex and basal ganglia in non-human primates suggested that the basal ganglia influence the speed of deliberation but not what action to select (i.e., the when but not the what of action selection [START_REF] Thura | The Basal Ganglia Do Not Select Reach Targets but Control the Urgency of Commitment[END_REF] . At this stage, we can only speculate about how to reconcile the distinct views on the striatal contribution to motor control and decision-making derived from the present and aforementioned studies. The devil, as often, might be in the details. Even though the authors provided evidence that gross movements did not directly influence timebased decisions, estimating time independently of movements remains largely an assumption as it is well known that animals, when forced to wait for specific durations, tend to produce stereotyped rituals that can serve as clocks [START_REF] Robbe | Lost in time: relocating the perception of duration outside the brain[END_REF] . Such rituals can include orofacial movements and could easily go unnoticed while animals were maintaining fixations. Critically, this type of brief ritualistic movements has been previously shown to be drastically impaired by lesions of the sensorimotor region of striatum [START_REF] Dhawale | The basal ganglia control the detailed kinematics of learned motor skills[END_REF] . The presence of habit-like movements during interval presentation, controlled by distinct processes than full-body reward-oriented movements, could provide a unifying explanation to apparently contradictory interpretations on the striatal function in decision making and motor control. Indeed, striatal population dynamics may control a latent process that can bidirectionally control the speed of decision and execution of automatized actions, while average activity in the brain region would control the vigor of reward-oriented actions. Of course, the above possibility might sound quite speculative but one should not underestimate the ability of animals to solve tasks in different ways than assumed by their human experimenters [START_REF] Despret | Thinking Like a Rat[END_REF] . Thus, as any important study, the one of Monteiro et al. engenders new questions. It certainly demonstrates the importance and power of using subtle alterations of network dynamics to tackle functional questions in neuroscience. Reconciling the seemingly contradictory views on striatal function over the past years will require new experiments with different approaches, debates, and passionate disagreements; in other words, the normal and necessary business of science in the making [START_REF] Latour | Science in action: How to follow scientists and engineers through society[END_REF] .

Figure 1 .

 1 Figure 1. Temperature manipulation biases time-based decisions. A) Schematic representation of the binary time interval categorization task. B) Warming the dorsal striatum increased the distance traveled in neuronal space leading to categorizing near-boundary time intervals as long ones. C) Cooling the striatum led to opposite effects. D) In a spacetime representation, population dynamics associated with striatal warming (cooling) appear contracted (dilated).


Yirmeyahu Kaminski

Pierre Lochak

On the Equilibrium Locus of a Parameterized Dynamical System with Independent First Integrals

For a family of dynamical systems with k > 0 independent first integrals evolving in a compact region of an Euclidean space, we study the equilibrium locus. We show that under mild and generic conditions, it is a smooth manifold that can be viewed as the total space of a certain fiber bundle and that this bundle comes equipped with a natural connection. We then proceed to show parallel transport for this connection does exist and explore some of its properties. In particular, we elucidate how one can to some extent measure the variation of the system eigenvalues restricted to a given fiber.

Introduction

We study the equilibrium locus of a family of dynamical systems with k > 0 independent first integrals; in other words we explore the zero set of a vector field in Euclidean space, with at least one first integral, giving rise to a foliation of the ambient space. More precisely the system is assumed to evolve inside a compact set V ⊂ R n of maximal dimension, whose boundary ∂V has the structure of a stratified manifold.

We show that if the vector field is analytic and satisfies some genericity conditions, the equilibrium locus E is a smooth manifold of dimension m + k, where m is the number of parameters and k the number of independent first integrals. Then we focus on the projection from E onto the space of parameters Λ. We prove that this projection has the structure of a fiber bundle, that we call the equilibrium bundle , and that an Erhesmann connection can be naturally associated to it. We explicitly prove that parallel transport exists for this connection and discuss its holonomy group. Then we prove that it can be made equivalent to a Riemannian submersion provided one chooses an appropriate metric on E. Finally we show how one can extract some information about the variation of the (nonzero) eigenvalues of the Jacobian matrix of the vector field along a loop embedded in a fiber of the above projection.

The viewpoint developed in this article does not seem to appear as such in the literature. Yet, it is clearly connected with works in dynamical systems in general, more specifically differential equations. It is worth in this respect quoting papers by the Russian school. In particular, several articles by Valery Romanovski and coworkers provide a useful entry point ( [Zhou-all(2020)]). Equilibria of systems admitting at least one first integral have been studied in particular in [Dudoladov(1996)] and related papers (see the reference list of [Dudoladov(1996)]). A more topological viewpoint is developed in [Dancer-all(1982)] and other papers by these authors (see again their reference list). It is also worth noting that the stability problem for equilibria is pervasive in the literature, at least since ... Poincaré. In the present paper it is addressed in section 7. Finally we remark that a natural generalization of the considerations of the present paper would consist in addressing the same issues for more general trajectories, starting with the periodic ones.

The paper is organized as follows. In sections 2 and 3, we lay down all the definitions and assumptions, and we deduce the first consequences. The other sections are the core of our work. Sections 4 and 5 deal with the definition of the equilibrium bundle, the natural connection on it and the construction of the parallel transport. In section 6, the connection is analyzed from the perspective a Riemannian submersion. Section 7, we discuss the variation of the non zero eigenvalues of the vector field along a loop embedded in a fiber of the equilibrium bundle.

In section 8, we are finally in a position to discuss examples and possible applications. The first example was investigated in [Kaminski(2018)]. It is a dynamical system that describes the behavior of circular net of cells. Our work can therefore be seen as a vast generalization of this particular case. We also discuss other possible examples and applications.

Definitions and first properties

Let us consider an n-dimensional (n > 0) smooth dynamical model ẋ = f (λ, x), depending on m parameters λ (m > 0). We make the following assumptions:

1. λ ∈ Λ, where Λ is a connected open set of R m (m > 0), 2. x ∈ V , where V ⊂ R n , is a compact submanifold of R n of codimension zero, whose
boundary ∂V is a stratified manifold. The system is assumed to stay inside V , that is on ∂V , the vector field either points inward or is tangent to ∂V . 3. There exist k smooth independent first integrals h

1 , • • • , h k (k > 0), independent
of the parameter(s) λ, so that for every i, h i : R n → R is a smooth function. Here the independence of the functions h 1 , • • • , h k means that their gradients are linearly independent at every points.

Writing M = Λ × V , for (λ, x) ∈ M , f (λ, x) ∈ T x V ≃ R n
, the tangent space of V at x, so that we may and do consider f as a smooth map M → R n giving the components of the "velocity" at the point x ∈ V with parameter λ ∈ Λ. We denote by E ⊂ M the equilibrium locus defined by the vanishing condition f (λ, x) = 0 and assume throughout that E is not empty.

Proposition 1. In the setting described by assumption 3 above, the Jacobian matrix ∂f ∂x has rank at most n -k on E.

Proof. Consider 1 ≤ l ≤ k and λ ∈ Λ. Then a solution x(t) of the system satisfies h l (x(t)) = Cst and taking the derivative with respect to t, we get: f (λ, x(t)) • ∇h l (x(t)) = 0 where the dot denotes the usual scalar product on R n . Since through every point x ∈ V and any λ ∈ Λ, there is an integral curve of the vector field f (λ, .) passing through x, this is valid for every λ ∈ Λ and x ∈ V :

∀λ ∈ Λ, ∀x ∈ V, < f (λ, x), ∇h l (x) >= 0.
Taking a further derivative, we get:

n i=0 ∂f i ∂x j ∂h l ∂x i + f i ∂ 2 h l ∂x j ∂x i = 0,
for every j (1 ≤ j ≤ n) where we write f componentwise:

f = (f 1 , • • • , f n ).
This is equivalent to writing:

∂f ∂x T ∇h l + D 2 h l f = 0, (1)
where D 2 h l denotes the Hessian matrix of h l and the superscript T denotes transposition (both ∇h l and f are regarded as n-vectors). Now at an equilibrium point (f = 0), the second term vanishes and thus the kernel of ∂f ∂x T contains the k vectors ∇h l (x). Since the first integrals h l (l = 1, . . . , k) are assumed to be independent, that kernel has at least dimension k and rank ∂f ∂x ≤ n -k.

Observe now that the functions h 1 , • • • , h k being independent of λ, the vanishing identities ∂f ∂λ T ∇h l = 0 are also obtained for l = 1, . . . , k. We may therefore require the further: Non-degeneracy Conditions: i) The rank of the matrix ∂f ∂λ is constant equal to min(m, n-k); ii) The rank of the matrix ∂f ∂x is constant equal to n-k for (λ, x) ∈ E and n-p(λ, x) at points (λ, x) ̸ ∈ E, where p(λ, x) ≤ k; iii) For every (λ, x) ∈ E, ker ∂f ∂x (λ, x) ⊕ im ∂f ∂x (λ, x) = R n .

Remarks: Conditions i) and ii) simply express the fact that there are no further degeneracies, beyond the one exhibited in Proposition 2.1. The third condition iii) requires that the kernel and image of the partial Jacobian matrix ∂f ∂x be transverse subspaces of R n , which is true for a dense open set of the matrices with a fixed a given rank.

Examples:

1. Consider the following vector field in R 2 : f = (v(λ, x, y), 0), such that for all λ ∈ Λ = R, v(λ, x, y)x ≤ 0 when x 2 + y 2 = 1, so that V can be taken to be the Euclidean unit ball of R 2 . Obviously, h 1 = y is a first integral and

∂f ∂x = ∂v ∂x ∂v ∂y 0 0 . Of course, ∂f ∂x T ∇h 1 = 0. A similar result is obtained if one consider 1 2 y 2 instead of y. The set of equilibrium points E = {(λ, x, y)|v(λ, x, y) = 0} has dimension 2.
2. Consider now a vector field f is R 3 , such that h 1 = x 2 +y 2 +z 2 and h 2 = 4x 2 +4y 2 +z 2 /4 are first integrals. These functions are obviously differentially independent, outside the union of the plane z = 0 and the line x = y = 0. Take V to be the union of the points {(x, y, z)} such that

h 1 = R 1 and h 2 = R 2 , with R 1 , R 2 ∈ [1, 3] × [5, 15] and R 2 ≥ 5R 1 . Then let us consider f = (-λy(z -x), λx(z -x), 0)) with Λ = R * + ; it is straightforward to compute ∂f ∂x =   λy -λ(z -x) -λy λz -2λx 0 λx 0 0 0   and E = {(λ, x, y, z) ∈ Λ × V |x = z}.
Then for (λ, x, y, z) ∈ E, as expected:

∂f ∂x T ∇h 1 = ∂f ∂x T ∇h 2 = 0. Also, obviously, dim(E) = 3.
Proposition 2. Consider the map f : M = Λ × V → R n and assume the nondegeneracy conditions above hold true. If f is an analytic vector field, the equilibrium locus

E = f -1 (0) is a smooth real analytic variety of dimension m + k.
Proof. Since f is analytic, E is analytic by definition. Moreover the Jacobian matrix of f has constant rank along E. Indeed let J be this Jacobian matrix: J = ∂f ∂λ , ∂f ∂x . Then by the non-degeneracy condition, the kernel of the transpose J T is precisely the span of

∇h l for 1 ≤ l ≤ k. Therefore rank(J) = n -k along E which is a smooth analytic variety of dimension m + n -(n + k) = m + k.
This proposition fails to be true if the field f is smooth but not analytic. Indeed for such a field there may exist an open set U ⊂ M included in f -1 (0). In that case, the equilibrium locus will have dimension m + n > m + k.

We assume from now on that f is analytic and write π : E ⊂ Λ × V → Λ. This is an analytic and proper map whose image we denote by Λ 0 ⊂ Λ.

On the nature of Λ 0

We start with the following observation.

Lemma 1. The set Λ 0 is closed in Λ.
Proof. This is a direct consequence of the fact that π is proper. Alternatively in a more down-to-earth and detailed way, consider λ ∈ Λ \ Λ 0 , i.e. for every x in V , f (λ, x) ̸ = 0. By continuity, there are open sets U x and V x respectively in Λ and V , such that for all

(µ, y) ∈ U x × V x , f (µ, y) ̸ = 0. The collection {V x } x∈V is an open cover of V . Since V compact we can extract a finite subcover, V x 1 , • • • , V xp . Writing U = U x 1 ∩ • • • ∩ U xp , for all (µ, x) ∈ U × V , f (µ, x) ̸ = 0 which shows that Λ \ Λ 0 is open in Λ. Lemma 2. For λ ∈ Λ 0 , the fiber E λ of π over λ is a smooth real analytic k-dimensional manifold. Proof. Given λ ∈ Λ 0 let f λ = f (λ, •) : V → R n . By the non-degeneracy condition the rank of f λ is constant equal to n -k over E λ = f -1 λ (0)
which is therefore as in the statement of the lemma. Proposition 3. The closed set Λ 0 has non empty interior in Λ.

Proof. Assume Λ 0 had empty interior and consider a point (λ, x) where π has a maximal rank. Then this rank is constant over a neighborhood U of this point. If the rank is m, then π restricted to U is a submersion, so that π(U ) is open in Λ, which is impossible if Λ 0 has empty interior. Therefore the rank of π is always strictly smaller than m and consequently the dimension of the fibers has to be at least k + 1, contradicting lemma 2 above.

From now on, we will focus on the case where λ lies in the interior of Λ 0 . Restricting attention to a connected component of Λ if need be, we may and will assume in the sequel that Λ 0 is open and connected. Finally and for the sake of notational simplicity, we rename Λ 0 just Λ. In other words we henceforth assume that Λ is open and connected in R m and that the projection π : E → Λ is surjective.

Fibers over the parameter space

Let us refine lemma 2 slightly Proposition 4. For every λ ∈ Λ the fiber E λ = π -1 (λ) is a compact smooth real analytic variety of dimension k whose boundary (if any) is contained in ∂V .

Proof. With the notations used in the proof of lemma 2, we know that

E λ = f -1 λ (0) is indeed a smooth real analytic variety of dimension k. Being a closed set in V , it is compact.
Its boundary is nonempty if and only if f λ vanishes on the boundary ∂V of V . Indeed if x 0 is a point in E λ not in ∂V , then by the implicit function theorem, E λ is locally the graph of a function of k variables, so x 0 cannot lie on the boundary of E λ . Note that this result is consistent with lemma 2 and proposition 4.

Proposition 6. The triplet (E, π, Λ) defines a fiber bundle.

Proof. By Erhesmann's theorem it is enough to show that π is a proper surjective submersion to conclude that it is a fiber bundle. Now it is surjective by definition, and we already know it is proper (because V is compact). Finally it is a submersion by proposition 5.

Since Λ is assumed to be connected, following section 3, we get the following corollary.

Corollary 2. The fibers of E λ of E are all diffeomorphic.

In the sequel the bundle (E, π, Λ) will be called the equilibrium bundle.

5 Parallel transport of the fibers

Connection

In this section, we aim at defining a connection on the equilibrium bundle and showing that parallel transport exists. In a very general setting, a connection on a bundle π : E → B is defined by an horizontal distribution, that is a subbundle of the tangent bundle T E → E, say H, such that for every u ∈ E, we have:

T u E = V u ⊕ H u ,
where V u is the vertical summand of the tangent space, that is the set of tangent vectors X at u, such that π * (X) = 0. In other words V u ⊂ T u E is spanned by the vectors which are tangent to the fiber over π(u). We will denote by V the vertical subbundle of T E.

Transversality between fibers and integral manifolds

Given u = (λ 0 , x 0 ) ∈ E, a basis of ker ∂f ∂x (u)

T is given by (∇h l (x 0 )) 1≤l≤k . Therefore the family (∇h l (x 0 )) 1≤l≤k is also a basis of the subspace orthogonal to im ∂f ∂x (u) , so that:

im ∂f ∂x (u) = {v ∈ R n | v T ∇h l (x 0 ) = 0, l = 1, . . . , k} (2)
Let a = h(x 0 ) ∈ R k where h = (h 1 , • • • , h k )
and let N a be the manifold defined by

N a = V ∩ ∩ k l=1 Z(h l -a l ) i.e. h l = a l on N a , with a = (a 1 , • • • , a k ). Equation (2) simply means that: im ∂f ∂x (u) = T x 0 N a (3)
From this equation, one can state:

Proposition 7. Assuming the vector field f (λ, x) satisfies the genericity condition iii) in §2, the manifold N h(x) and the fiber E λ are transverse at x in V for every (λ, x) ∈ Λ × E λ .

Lifting curves

We now show how a curve in Λ can be naturally lifted to E. So let λ(t) : [0, 1] → Λ define a curve from λ(0) = λ 1 to λ(1) = λ 2 in Λ (both ends are included). Given x ∈ E λ 1 we want to build a curve γ :

[0, 1] → E ∩ N h(x) starting from x = γ(0). So γ(t) must satisfy: ∀t ∈ [0, 1], π(γ(t)) = λ(t) ∀i, h i (γ(t)) = h i (x)
This yields the following differential system:

∂f ∂λ (λ(t), γ(t)) λ(t) + ∂f ∂x (λ(t), γ(t)) γ(t) = 0 ∀i, ∇h i (γ(t)) T γ(t) = 0
which can be written more compactly as:

A(t) γ(t) = b(t), (4)
with

A(t) = ∂f ∂x (λ(t), γ(t)) ∂h ∂x (γ(t)) , b(t) = -∂f ∂λ (λ(t), γ(t)) λ(t) 0 , where h = (h 1 , • • • , h k ).
Proposition 8. The differential system defined by equation ( 4), has a unique solution defined on the closed interval [0, 1].

Proof. By proposition 7, the matrix A(t) has full rank n for every t ∈ [0, 1]. Therefore equation ( 4) can be written:

γ(t) = A(t) T A(t) -1 A(t) T b(t)
where the right-hand side actually depends on γ(t). The functions entering on the righthand side being smooth, a fortiori Lipschitz, elementary results about differential equations (the Cauchy-Kovaleska theorem) garantee the local existence (in time) and uniqueness of a solution with given initial condition, thus also over a compact interval.

Theorem 1. For any λ ∈ Λ and any x ∈ E λ , the intersection E λ ∩ N h(x) consists in a finite set of points. Moreover the number of points in

E λ ∩ N h(x) is independent of λ.
Proof. By proposition 7, the intersection E λ ∩ N h(x) is discrete, hence finite since E λ is compact. Let λ 1 , λ 2 ∈ Λ and consider a curve λ(t), defined on [0, 1] joining these two points.

For every x ∈ E λ 1 , there is a unique lift of λ(t) drawn on N h(x) and starting at x, which ends at some y ∈ E λ 2 . Lifts starting from a point in E λ 2 ∩ N h(x) are also uniquely defined.

This sets up a one-to-one correspondence between the points of E λ 1 ∩ N h(x) and those of

E λ 2 ∩ N h(x) .

The natural connection

Let π ′ : E → R n be the projection to the second summand. Consider again the function

h : R n → R k , (h = (h 1 , • • • , h k )) and note that h • π ′ is a submersion. Now for a ∈ R k , if (h • π ′ ) -1 ̸ = ∅ it is a submanifold of E of dimension m. Let u = (λ, x) ∈ (h • π ′ ) -1 (a).
Then the tangent space of (h

• π ′ ) -1 (a) at u is ker [0 k,m , ∂h ∂x (x)] .
Lemma 3. The tangent space of (h

• π ′ ) -1 (a) at u = (λ, x) is the complement of the tangent space of E λ at x in T u E.
Proof. This is a direct consequence of equation ( 3) and the non-degeneracy condition iii).

This results shows that we can define a connection on E choosing as the horizontal space:

H u = T u ((h • π ′ ) -1 (h(x))).
We call it the natural connection of the equilibrium bundle.

Theorem 2. The natural connection of the equilibrium bundle is flat (i.e. its curvature vanishes).

Proof. The curvature of a connection measures the failure of the horizontal distribution to be integrable. Here however, for u ∈ E the subspace H u is tangent to an embedded submanifold of E. Therefore the distribution (H u ) u is indeed integrable and the curvature vanishes.

Theorem 3. Parallel transport exists for the natural connection on the equilibrium bundle.

Proof. It is well-known that when the fibers are compact, parallel transport exists (see [Michor(2008), 204] for details). By proposition 4, the fibers in our case are indeed compact, so this result applies. Here in fact proposition 8 makes parallel transport explicit. Consider two points λ 1 , λ 2 ∈ Λ and a curve λ(t) joining them. Consider a point x in the fiber E λ 1 and the manifold (h • π ′ ) -1 (h(x)). By proposition 8, there exists a curve γ(t) on E ∩ (h • π ′ ) -1 (h(x)) starting at x, such that ∀t, π(γ(t)) = λ(t) and γ(t) ∈ H (λ(t),γ(t)) , which explicitly shows that parallel transport exists.

The connection being flat, assuming the base manifold Λ is simply connected, the holonomy groups vanish, which we record in the following theorem.

Theorem 4. Assuming that the base manifold Λ is simply connected, the holonomy groups of the equilibrium bundle are trivial and the fiber bundle is trivial.

Proof. The horizontal lift of a smooth path in Λ entirely lies in (h • π ′ ) -1 (a) for some a. Since every fiber intersects h -1 (a) in the same finite number of points, there are finitely many horizontal lifts of a smooth loop. Therefore the holonomy groups are finite. Therefore they are Lie transformation groups of the fibers. Now since the curvature of the connection identically vanishes, the restricted holonomy groups are in fact trivial, applying a result of [START_REF] Reckziegel | How the curvature generates the holonomy for a connection in an arbitrary fibre bundle[END_REF]] which generalizes the Ambrose-Singer theorem to general fiber bundles.

Since the base space Λ is assumed to simply connected, the holonomy group at a point is identical to the restricted holonomy group at this point. This leads to the conclusion.

The natural connection via a Riemannian submersion

Let us Φ be the natural connection form, Φ ∈ Ω 1 (E, T E) (that is for each p ∈ E , we have Φ p ∈ End(T p E)), which is the projection over the vertical bundle, so that for each p ∈ E, we have Φ p • Φ p = Φ p and im(Φ) = V, i.e. the vertical bundle.

Let us define a Riemann metric on E as follows:

g(X, Y ) =< Φ(X), Φ(Y ) > 1 + < π * (I -Φ(X)) , π * (I -Φ(Y )) > 2 ,
where < ., . > 1 and < ., . > 2 respectively denote the standard inner product in R m × R n and in R m . When E is endowed with this metric, as we shall assume from now, the projection π becomes a Riemannian submersion. Indeed H u is the orthogonal complement of V u in T u E and the restriction π * : H u → T π(u) Λ is an isometry.

Therefore the natural connection coincides with the connection defined by π as a Riemannian submersion.

There is apparently no reason for which the parallel transport would define an isometry between two different fibers with the given metric.

However, since by theorem 4, the holonomy groups are trivial if we further assume that Λ is simply connected, then in that case the parallel transport doesn't depend on the chosen curve on Λ. Therefore, we have a well defined diffeomorphism γ λ 2 λ 1 : E λ 1 ∼ → E λ 2 between any two fibers, such that:

γ λλ = Id E λ γ λ 3 λ 1 = γ λ 3 λ 2 • γ λ 2 λ 1 (5)
This observation allows transporting the metric from one fiber to another such that the parallel transport becomes an isometry between these two fibers. Choosing one fiber and transporting its metric to other fibers makes the parallel transport between any two fibers an isometry, thanks to the cocycle condition (5).

semi-direct product π orb 1 (X/G) = S p ⋉ (π 1 (Y )) p where again the action of S p on π 1 (Y ) p is by permutations of the factors. In our case π 1 (Y ) = Z and so returning to the orginal situation we get a natural map π orb 1 (e) : π 1 (E λ ) → S p ⋉ Z p which records both the monodromy of the individual (nonzero) eigenvalues and their permutations.

What does this tell us about stability in this very general setting? Let us restrict attention to the topological fundamental group π 1 (µ) = Z p (p = n -k). Then for any nonzero element (m 1 , . . . , m p ) in the image of π 1 (e), with a nonzero component -say -m i for some index i, we can assert that the eigenvalue µ i has crossed the imaginary axis (Re(z) = 0) at least 2m i times, so that the sign of its real part has changed at least that many times. Although at this level of generality it seems difficult to extract more information from the map e, this may be feasible in specific, concrete cases.

Examples and Applications

Our first and motivating example concerns mathematical biology and consists in a circular network of cells i.e. a finite set of cells that are connected along a ring such that the last cell is connected to the first. An instance of this kind of model is the ribosome flow model on a ring, which has been introduced in [START_REF] Raveh | Ribosome Flow Model on a Ring[END_REF]].

Such a model is a parametric dynamical system defined as follows:

             ẋ1 = λ n x n (1 -x 1 ) -λ 1 x 1 (1 -x 2 ) ẋ2 = λ 1 x 1 (1 -x 2 ) -λ 2 x 2 (1 -x 3 ) . . . ẋn-1 = λ n-2 x n-2 (1 -x n-1 ) -λ n-1 x n-1 (1 -x n ) ẋn = λ n-1 x n-1 (1 -x n ) -λ n x n (1 -x 1 )
The parameters λ 1 , • • • , λ n are real strictly positive numbers. They define the degree of diffusion between the cells. As shown in [START_REF] Raveh | Ribosome Flow Model on a Ring[END_REF]], if the initial point lies in the hypercube [0, 1] n , then the system always stays within these limits. Then this system models the occupancy levels of a circular chain of n sites (for example a circular DNA), while λ 1 , • • • , λ n are transition rates. In [Kaminski(2018)], the author analysed the structure of the equilibrium locus of this system.

Coming to dynamical systems in general one is looking for systems such that a) the trajectories are confined to a compact region of phase space b) they possess more or less natural first integrals (conserved quantities) and c) they display varieties of equlibria of positive dimensions. Requirements a) and b) are quite common and easy to satisfy : think of parametrized Hamiltonian systems and the conservation of energy, global momentum, possibly angular momentum, or conserved quantities coming from some continuous group of geometric symmetries, via Noether's theorem.

Requirement c) is less commonly fulfilled. The system which first comes to mind, stemming from celestial mechanics, the oldest and in some sense motivating domain in dynamical systems, is the so-called N -body problem, i.e. the system of differential equations which governs the motion (in ordinary three dimensional Euclidean space) of N > 1 massive bodies interacting according to Newton's law of gravitation. This horrendously difficult problem (for N > 2) was first studied in "modern" times by Lagrange and Euler, who isolated by now famous discrete sets of relative equilibria. Because the problem is so old, so natural and so difficult, literally dozens of variants of all kinds have been produced. All of them satisfy a) and b) and so does the original system. Some of them, especially those which involve blowups of certain singularities, also satisfy c) (see [Devaney(1982)]). A precise investigation of such systems would lead us too far afield. We hope to return to such and similar problems at some later time.

Finally we remark that it would also be interesting to explore the field of mathematical economy, where similar systems may exist in some abundance. Again such an exploration is postponed to later studies, possibly by other, more specialized authors.

In addition to these examples, one would consider the application of our results to control theory. Having first integrals, the system is not controllable. However lying on the manifold defined by fixed values a = (a 1 , • • • , a k ) ∈ R k of the first integrals, there is a non denumerable set E a = E ∩ ∩ k i=1 π ′-1 • h -1 i (a i ) of possible equilibrium points according to the value of the parameter vector. The notations are defined in section 5.4. According to the discussion there, this set is actually a manifold of dimension m.

By proposition 8, every curve in Λ can be lifted into a curve that lies in E a . Therefore if the parameters are considered as commands, one can change the set of available equilibrium point(s) continuously. Then the stabilization around an equilibrium is a classical control problem that we do not recall here.

Corollary 1 .

 1 If k = 1, the fibers over Λ have connected components which are diffeomorphic either to circles or to line segments. In the latter case, the ends of the segment lie on ∂V . Proposition 5. The rank of the projection π is constant equal to m. It is a surjective submersion. Proof. The tangent space of E at u = (λ, x) is ker(J) = ker ∂f ∂λ , ∂f ∂x . Moreover (a, b) ∈ T u (E) ∩ ker(π * ) if and only if ∂f ∂λ a + ∂f ∂x b = 0 and π * (a, b) = a = 0. Hence, ker(π * ,u ) = {(0, b)| ∂f ∂x b = 0}. Therefore dim(ker(π * ,u )) = k and rank(π * ,u ) = m + k -k = m.

Corollary 3. If parallel transport is an isometry between fibers, the fibers E λ for all λ ∈ Λ are totally geodesic in E for the Levi-Civita connection induced by the metric on E.

Proof. Since parallel transport defines an isometry, between fibers, these latter are totally geodesic. See [Ziller(2009), proposition 1.1].

The eigenvalues along the fibers

The stability of an equilibrium point is controlled by the sign of the real parts of the eigenvalues of the Jacobian matrix. In this section we set up, in the general framework of our discussion, a map at the level of the fundamental groups which can provide some information on these signs, or rather their changes. First, note that the stability of an equilibrium is determined by the non-zero eigenvalues, since the system is constrained to evolve along the submanifold defined by the k first integrals. We shall now consider the variation of the nonzero eigenvalues along a homotopy class of loops for a given value of the parameter.

Consider a given value of λ ∈ Λ. It is natural to look at the map e : (λ, x) ∈ E λ → µ where µ = [µ 1 , . . . , µ n-k ] denotes the set of the nonzero eigenvalues. In other words µ ∈ (C ⋆ ) n-k /S n-k where for p > 0, S p denotes the permutation group on p objects and the action here is by permutations of the factors.

Slightly more generally consider a manifold Y and its p-fold product X = Y p for some p > 0. Let S = S p act by permutation of the factors and let Z = X/S denote the quotient. (Here and below, for the sake of simplicity, the topological objects and morphisms are assumed to be smooth.) One may consider Z either as a topological space, or as an orbifold. In this section we will make use of both categories. The case at hand is relatively elementary because we are dealing with a global quotient under the action of a finite group. The theory of orbifolds is more general and adapted to the proper discontinuous action of a topological groups. Given that in our example the application is fairly intuitive, we refrain from recalling the details, referring instead to [Thurston(1978)] for much more.

Given a (smooth) map f : E → Z where E is a manifold, we may consider the topological fundamental group (or functor rather) denoted π top 1 or just π 1 , giving rise to a map

where Z is considered as a topological space. We may also consider the orbifold fundamental group π orb 1 and the attending map

) because E is just a manifold (a "trivial" orbifold). So we get two maps π 1 (f ) and π orb 1 (f ) with the same source, namely π 1 (E), and respective targets π 1 (Z) and π orb 1 (Z) according to whether Z is considered in the category of topological spaces or that of orbifolds.

There remains to compute the groups π 1 (Z) and π orb 1 (Z), which can be done easily (in our case), using [Armstrong(1968), [START_REF][END_REF]] to which we refer the reader. Indeed, assume that G is a topological group acting properly discontinuously on a connected smooth manifold X. The quotient Z = X/G exists as a topological space and π 1 (X/G) is then an extension of G/I by π 1 (X), where I (the letter stands for "inertia") is the normal subgroup of G generated by the elements whose action has fixed points in X. If X = Y p and G = S p acts by permutations, it is easy to see that I = G and so π 1 (X/G) = π 1 (X) = (π 1 (Y )) p = Z p if Y = C ⋆ .

On the other hand, π orb 1 (X/G) is (essentially by definition) an extension of G itself by π 1 (X). Now if again X = Y p and G = S p the extension splits and π orb 1 (X/G) is given as a