Adalberto Polenghi
email: adalberto.polenghi@polimi.it*

Irene Roda
email: irene.roda@polimi.it

Marco Macchi
email: marco.macchi@polimi.it

Alessandro Pozzetti
email: alessandro.pozzetti@polimi.it

Conceptual Framework for a Data Model to Support Asset Management Decision-Making Process

Keywords: data management, information management, asset management, decision framework, manufacturing

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

Introduction

The deployment of the disruptive technologies developed and used during the Industry 4.0 era is changing the way organisations are looking to their assets. The black box has been opened through the extensive installation of sensors to gather raw data. Therefore, databases are created for the collection of data from different sources to realise centralised data storages, even unstructured (also called data lakes). Moreover, software tools for data analytics aimed at catching information behind the Big Data are growing in number and capabilities [START_REF] Laney | 3D data management: Controlling data volume, velocity and variety[END_REF].

To acquire a leading role in this context, a successful digital transformation of the company is required and, to this end, capabilities are becoming central to manage information and data flows, recognise important data from waste (i.e. concept of Smart Data [START_REF] Siemens | From Big Data to Smart Data -Infographic Smart Data[END_REF]), and analyse data to extract information properly (from "simple" mathematical artefacts to machine learning techniques [START_REF] Cattaneo | Clarifying Data Analytics Concepts for Industrial Engineering[END_REF]).

During this digital transformation, the way the assets are managed receives new stimuli to improve their management policy, relying on increased knowledge and information through data collection. However, this technological jump has not been painless, and sometimes, results are not guaranteed.

In section 2, following an analysis of the most relevant works in this field performed, the main gaps encountered in the scientific literature are listed, with a special focus on physical assets within manufacturing systems. Then, in section 3, a summary of the AM fundamentals is provided, before introducing the proposed conceptual framework, aimed at guiding data modelling to support AM decision-making. Finally, some concluding remarks and future researches are stated in section 4.

2

Literature review on information and data management

The AM development, featuring an integrated approach along the asset lifecycle, is inherently geared towards sharing information and data between different databases, systems, and organisational functions, finally asking for an asset-centred orientation that relies on an effective asset data management [START_REF] Campos | A Big Data Analytical Architecture for the Asset Management[END_REF]. A lot of work has been done so far in this direction, not only in AM [START_REF] Kiritsis | Semantic technologies for engineering asset life cycle management[END_REF] but also in maintenance [START_REF] Matsokis | An Ontology-based model for providing Semantic Maintenance[END_REF], considered its natural precursor. However, two main extant gaps are recognised in the scientific literature when dealing with information and data for AM in manufacturing [START_REF] Polenghi | Asset Management in manufacturing: a systematic literature review[END_REF]:

• Information management and integration: consisting of the correct management and suitable integration of information in different asset control levels and between systems to support asset-related decisions [START_REF] Campos | A Big Data Analytical Architecture for the Asset Management[END_REF], [START_REF] Tyler | Asset management the track towards quality documentation[END_REF], [START_REF] Nuñez | An ontology-based model for prognostics and health management of machines[END_REF]; • Data to information transformation: consisting of the suitable exploitation of the data to derive information (and then knowledge) from the system [10]- [START_REF] Kortelainen | Asset Management Decisions-Based on System Thinking and Data Analysis[END_REF].

Overall, the information and data management results to be critical for a suitable AM system in manufacturing, especially when dealing with information integration. Different approaches are proposed to improve its body of knowledge while complying with AM fundamentals.

Among them, it is worth to notice the connection between AM and BIM (Building Information Modelling), which brought to the publication of the ISO 19650 (substituting the PAS 1192). Aligned with BIM, the asset information exchange is analysed in the ISO 19526, which took advantage from researches in the field of product data [START_REF] Kiritsis | Closed-loop PLM for intelligent products in the era of the Internet of things[END_REF]; despite the focus on process plants, it is adaptable to manufacturing [START_REF] Kiritsis | Semantic technologies for engineering asset life cycle management[END_REF]. Also, maintenance has taken the endeavour to face information and data within the wider scope of AM [START_REF]Maintenance -Maintenance within physical asset management[END_REF]: meaningful examples may derive from what developed in data management [START_REF] Tsang | Data management for CBM optimization[END_REF], or E-maintenance [START_REF] Muller | On the concept of e-maintenance: Review and current research[END_REF], [START_REF] Iung | Conceptual framework for e-Maintenance: Illustration by e-Maintenance technologies and platforms[END_REF]. In this very field, an interesting framework to guide information and data management has been developed in [START_REF] Guillén | A framework for effective management of condition based maintenance programs in the context of industrial development of E-Maintenance strategies[END_REF] but enclosed within the scope of maintenance without looking at the AM theory, which is the aim of this work.

As demonstrated by the literature, most of the data models adherent in terms of scope to AM derive from the maintenance field. Those data models help in structuring the maintenance decision-making process, dealing with alert generation for abnormal conditions of the assets, maintenance strategy definition for the assets, including CBM (condition-based maintenance) programs. Nevertheless, a first attempt to move towards AM decision-making process is recognised in [START_REF] Koukias | Semantic data model for operation and maintenance of the engineering asset[END_REF], which means enlarging the scope of decision-making along the asset lifecycle Moreover, data models based on object-oriented modelling and, as the next step, ontology, have been proven to be suitable to support problems related to information and data management, especially their integration along the lifecycle [START_REF] Negri | Requirements and languages for the semantic representation of manufacturing systems[END_REF], [START_REF] Colledani | Development of a Conceptual Reference Framework to Manage Manufacturing Knowledge Related to Products, Processes and Production Systems BT -Methods and Tools for Effective Knowledge Life-Cycle-Management[END_REF].

This paper aims at contributing to this promising direction. To this end, as a first methodological step (see [START_REF] Negri | Modelling internal logistics systems through ontologies[END_REF] for ontology development methodology), a conceptual framework to support data model development for AM decision-making is proposed in section 3, after a brief overview on AM fundamentals defined in the literature.

Proposed conceptual framework

The proposed conceptual framework starts from the work done in [START_REF] Guillén | A framework for effective management of condition based maintenance programs in the context of industrial development of E-Maintenance strategies[END_REF], and it paves the way to widen the scope towards AM. This goal is reached by firstly analysing which AM fundamentals must be considered to build decision-making coherent with AM theory. These fundamentals are summarised in a recent work published in 2018 [START_REF] Roda | A framework to embed Asset Management in production companies[END_REF]:

1. Asset lifecycle stages (BoL Beginning of Life, MoL Middle of Life, and EoL End of Life); 2. Asset control levels (strategic, tactical, operational); 3. AM principles (Lifecycle, System, Risk, Value orientation).

The underpinning goal of the conceptual framework is to enlarge the scope of the decision-making process (including decisions related to different areas such as capital investment, operations & maintenance, shutdown and outage, and others [START_REF]Asset Management -An Anatomy version 3[END_REF]): the decomposition into blocks of the proposed framework helps to reach this purpose.

The framework (in Fig. 1) has the aim of being a conceptual reference for the development of data models for the description of the AM decision-making processes.

On one side, the AM fundamentals are mapped considering the block in which they first appear (other blocks "receive" fundamentals due to cascade effect as well as information and data). In so doing, the conceptual framework aims at integrating these fundamental in the overall decision-making process, which starts from the asset (Physical description block) and ends up with the final decision (AM decision-making block).

On the other side, it promotes information integration, which happens at the block named Value-driven system analysis. This block is responsible for carrying out the analysis that supports the decision-making: information and data coming from different sources must be considered and integrated. Thus, the proposed conceptual framework wants to foster the need for a structured way to integrate information to better support the AM decision-making process.

Physical description

The Physical description block is devoted to the physical representation of the asset. The asset model may derive from the conceptualisation/idealisation activity in BoL (whose information and data are gathered in CAx systems), or from an installed asset in MoL / EoL (whose information and data are gathered in systems like CMMS, Computerised Maintenance Management Systems, besides CAx systems). Thus, in this block, the asset lifecycle stages are introduced since the asset is currently in BoL/MoL/EoL stage; this differs from the Lifecycle orientation, as explained after.

Logical description

The Logical description block aims at describing the function the asset must undertake; for example, in the maintenance context, this block describes how the asset works and how it fails. For this reason, this block enables modelling interdependencies and relationships between assets in the system (e.g. to logically represent the series or parallel configuration of two machines); so, it introduces the System orientation as AM principle. Moreover, the Lifecycle orientation is inserted here since "AM process should incorporate long-term objectives and performances to drive decision-making" [START_REF] Roda | A framework to embed Asset Management in production companies[END_REF], so the logical description enables the understanding of consequences of a decision. There is a difference between the asset lifecycle stages and the Lifecycle orientation: the former one only represents the stage in which the asset is (BoL, MoL, EoL), while the Lifecycle orientation represents how the decision is taken (driven by long-term objectives and asset, asset system, performance).

Information sources

The Information sources block represents the layer between the physical and logical description of the system, and the value-driven system analysis and AM decision-making. It is intended to represent the IT ecosystem or landscape (also called industrial software stack [START_REF] Koerber | Leveraging industrial software stack advancement for digital transformation[END_REF]), collecting the software tools adopted to support company decision-making processes, and AM among them.

The allocation of each software tool to a specific asset control level is a prerequisite to implementing an optimal AM system. Fig. 2. IT ecosystem for AM (adapted by [START_REF] Tucci | Methods and tools for the reliability engineering: a plant maintenance perspective[END_REF]).

Each software tool is so able to: i) provide/gather information and data to/from software tools belonging to other asset control levels or within the same one; ii) provide suitable information for the asset control level in which it is used to support the related decisions (e.g. asset health index for the development of strategies of asset replacement and maintenance, or Return On Investment for capital investment). This IT ecosystem has already been proven to be fundamental to implement a suitable strategy to manage the assets at best [START_REF] Campos | Development in the application of ICT in condition monitoring and maintenance[END_REF], especially to guide the analysis, as described in subsection 3.4.

Value-driven system analysis

The Value-driven system analysis block aims at supporting the AM decision-making process by developing suitable PIs (Performance Indicators), which must be adequate and aligned for the use in the final decision to be taken. The development of such PIs happens through an analysis that must comply with AM fundamentals previously cited, and it must also consider the three main drivers guiding the decision-making process: cost, performance and risk (aligned with [START_REF]Asset management -Overview, principles and terminology[END_REF]). The definition of appropriate value elements is a core activity at this step of the decision-making process because it allows performing analyses aligned with corporate objectives [START_REF] Roda | A Framework for Implementing Value-Based Approach in Asset Management[END_REF].

The Value-driven system analysis block is the one that collects inputs from all the other blocks and in which the information integration happens. As an example, refer to [START_REF] Guillén | A framework for effective management of condition based maintenance programs in the context of industrial development of E-Maintenance strategies[END_REF], where this block corresponds to the symptom analysis. Here, the health status of the machine is used to finally understand and analyse the symptom of a failure mode that is in evolution. The symptom analysis performed required the integration of information coming from information sources (monitoring variables), but also from the logical description (information about the failure mode, which could have a symptom). The integration of this information supports the symptom analysis that allows making a final maintenance decision.

Summarising, this block is one responsible for information integration since all the analyses performed to sustain a final decision must rely not on one single information source, but different ones. Better structuring the connection between information sources and different analysis is valuable for an effective AM decision-making process.

AM decision-making

The enlargement of the scope towards the AM decision-making is represented by the last block that includes different asset-related decisions. The set of possible decisions to be considered within the scope of AM is large, but generally some classes of decisions could be recognised, from an asset user perspective [START_REF]Asset Management -An Anatomy version 3[END_REF], [START_REF] Brown | Asset management for transmission and distribution[END_REF]- [START_REF] Roda | TCO Evaluation in Physical Asset Management: Benefits and Limitations for Industrial Adoption[END_REF]: capital investment (evaluation of alternatives / suppliers, maintenance service contract selection, budget planning and cost control) in BoL; operations and maintenance (maintenance planning, operations planning, asset utilisation strategies), reconfiguration decision, and shutdown/turnaround/outage in MoL; reuse or rehabilitation strategy in EoL.

A suitable decision-making process enables these decisions, and they must rely on PIs developed in the previous block, i.e. Value-driven system analysis.

The correct integration of information in the previous analysis has a huge impact on the final asset-related decision.

Conclusions

The proposed conceptual framework claims to be a conceptual reference to develop data models, whose goals should be the integration of information to enable analysis on which decisions are taken. The framework serves as a guideline to enable a structured data modelling that could improve the AM decision-making process, including all the fundamentals. The framework is built looking at possible approaches to AM, in which maintenance plays a central role. Thus, starting from a framework developed for CBM programs, the proposed conceptual framework is built, whose decomposition into blocks helps in fulfilling and integrating AM fundamentals (asset control levels, asset lifecycle stages and AM principles), involving different decisions. Better structuring the relationship between different information sources and different analysis sustaining the decision is valuable to build a robust AM decision-making process. In so doing, thanks to the improvement of the information management and integration along with the AM decision-making, the expectation is to be capable of achieving more valueoriented decisions for the asset lifecycle. Future research will be focused on the development of data models for the different decisions in AM to support all the decisions set, rather than only maintenance, with the final aim of creating a comprehensive data model that may support an ontology study.

As a side effect, it will also enable to formalise AM decision-making process, currently not yet fully described by extant reference or standard models.

Fig. 1 .

 1 Fig. 1. Conceptual framework with a mapping of AM fundamentals.


Roman Senderek

Jan Kuntz

Volker Stich

Jana Frank

Drs Roman Senderek

Prof Volker Stich

Smart Service Engineering: Promising Approaches for a Digitalized Economy

Keywords: Smart services, service engineering, requirements. 1

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

Introduction

Over the past decades, service engineering models have proven to be valuable aids for the design and implementation of new services. They guide companies through the different steps that they need to take when developing new service offerings, starting from the initial generation of ideas and concluding with the service's market entry and subsequent adjustments or improvements. A great variety of service engineering models has been developed in the past, and many of them have displayed practical benefits in different contexts and industries. However, today's service industry differs significantly from the service industry of the late 20 th century when the first service engineering models were developed. One of the major factors inducing these changes has been the ongoing digitalization. While it has opened the door for numerous possibilities, it has also created multiple challenges. Among other things, the digitalization has led to an exponential rate of change in the way services are created and delivered [START_REF] Acatech | Smart Service Welt. Digitale Serviceplattformen -Praxiserfahrungen aus der Industrie. Best Practices[END_REF][START_REF] Höckmayr | Digitalisierung als Treiber für Faktenbasiertes Service-Systems-Engineering[END_REF], and it has lowered the barriers of market entry and thus paved the way for stronger competition and an increased supply [START_REF] Gotsch | Open Innovation Ansätze für den Dienstleistungsinnovationsprozess. Die Entwicklung eines Service Open Innovation Frameworks[END_REF]. Consequently, companies must now differentiate themselves from competitors by continuously delivering innovative solutions that speak to individual customer needs.

Existing service engineering models often lack pace and are too inflexible to succeed in the face of today's ever-changing market situation. Many older models also require excessive temporal and material resources, which makes them highly inefficient. Service engineering models that want to succeed in today's market situation must be highly flexible and efficient, economic in their use of resources and align the entire develop-ment process with the needs of the customers [START_REF] Meyer | Entwicklungspfad Service Engineering 2.0. Neue Perspektiven für die Dienstleistungsentwicklung[END_REF][START_REF] Leimeister | From touchpoints to journeys: Seeing the world as customers do[END_REF]. There are some promising approaches toward service engineering from recent years that may be able to succeed in the digitalized economy of the 21 st century. The present paper introduces three promising service engineering models and examines how well each of them suits today's market situation. For this purpose, the requirements that service engineering models need to fulfill today will be summarized in the following section. Afterwards, three selected models, each of which meets several of these requirements, will be described and compared to one another.

Present-day requirements to service engineering models

As hinted at in the introduction, today's digitalized economy poses specific challenges to companies who seek to develop new service offerings. Consequently, service engineering models must meet different requirements today than they had to meet in the past. Instead of prescribing a detailed, but inflexible approach to the development of services, they must now provide a highly flexible, adaptable and efficient course of action, allowing companies to quickly develop prototypes and to rework feedback or changing customer or market requirements at any step of the development process. A comparison of recommendations from recent literature and trends in service engineering reveals three best practices for the development of new services. A further specification of these best practices results in seven requirements for modern service engineering models. The first best practice, user centricity, pursues a co-creative approach in which the user is continuously integrated and their individuality considered in the design process. In a second step, a first prototype developed based on customer ideas can be introduced into a feedback loop until it meets user expectations. [6][START_REF]What is digital economy? Unicorns, transformation and the internet of things[END_REF][START_REF] Meyer | Understanding customer experience[END_REF] The second best practice is using service ecosystems to enable companies to collaborate and thus profit from shared resources. Today's digital infrastructure facilitates and enhances this exchange. [START_REF] Meyer | Entwicklungspfad Service Engineering 2.0. Neue Perspektiven für die Dienstleistungsentwicklung[END_REF][START_REF] Immonen | A service requirements engineering method for a digital service ecosystem[END_REF][START_REF] Lusch | Service innovation: A service-dominant logic perspective[END_REF] The third best practice follows a resource-efficient agile mindset [START_REF] Groll | The agile service management guide[END_REF][START_REF] Lamberth-Cocca | Towards a reference model for agile new service development using the example of e-mobility service systems[END_REF] combining adaptability, i.e. the design process being customized to meet any changing requirements while value is constantly being measured, with a lean mindset and crossfunctional development. Therefore, this approach allows for reduced waste and resources and a shorter time to market by developing minimum viable products. [START_REF] Blank | Why the lean start-up changes everything[END_REF][START_REF] Ries | The lean startup. How today's entrepreneurs use continuous innovation to create radically successful businesses[END_REF] For further information, please refer to the paper "Service Engineering Models: History and Present-Day Requirements" by Senderek and Kuntz.

Promising new service engineering models

While many service engineering models do not meet the requirements summarized above, some recent models display multiple characteristics of innovative, modern service engineering. The following section presents three recent models and points out to what extent they meet these requirements. Each model emphasizes at least one of the three categories of best practices and can therefore be considered a promising approach to developing services in today's economic environment.

Smart Service Engineering

The first model to be presented here is Smart Service Engineering (SSE), an approach for the development of data-driven services developed by FIR at RWTH Aachen University. The model relies on speed and an early positive market impact for its success, and therefore focuses on an agile customer centric approach. It consists of three consecutive loops. The loops are not designed for a linear process, but for a flexible movement between different tasks and multiple iterations of the same loop as needed. While there is a perceived starting and finishing point, they don't need to be the first or last task to be performed. Instead, the tasks can be completed as the circumstances of each project call for.

The main goal of the first SSE loop is to develop a strategy for the company to follow throughout the service engineering process. First, the company needs to analyze its ecosystem, which includes a differentiation of competitors, an identification of key customer segments and a consideration of both the company's current and desired market position. The second task is the development of user stories, which represent typical application scenarios and provide insight into the customer's goals and actions. Following, the company can begin to formulate value hypotheses for its new smart service, which is the third task belonging to the first phase.

The second phase of SSE is the creation of a prototype. This loop allows the company to quickly test its ideas following the principles of a minimum viable product (MVP). The first task here is to define the service's core functions that are required for user testing. The second task is to develop the core functions before they can be tested with the user, which is the third task of the loop. Following these tasks, the company can ascertain the strengths and weaknesses of the new service regarding user satisfaction and rework that feedback into a new prototyping iteration.

The main goal of the third loop of the SSE model is to enter the market. The first task is to devise market entry strategies. Building upon the previously identified customer segments, the company can determine suitable sales channels and communication strategies. The second task to be performed here is to build up all resources required to deliver on a wide scale. The third task is to develop the business case for the new service, focusing on developing the cost structure and revenue streams to support the business model as well as consolidating the value proposition. [START_REF] Senderek | Smart service engineering. An agile approach to develop data-driven services[END_REF]

Multilevel Service Design

The second service engineering model to be discussed here is Multilevel Service Design (MSD), which stresses user centricity and the integration of multiple digital interfaces in the service design process. MSD is clearly designed for the digital age as it makes direct references to digital channels at certain design activities. MSD follows an iterative process consisting of four consecutive steps. This iterative character of MSD relates toward the individual tasks, as the model is designed to be executed in a linear fashion with a clear finish line in the form of the final task and outcome.

MSD begins the first step by a thorough examination of the customer experience. To this end, data is collected on three levels: the Value Constellation Experience (VCE), the Service Experience and the Service Encounter Experience (SEE). The ultimate goal of this step is for the company to achieve an in-depth understanding of the customer's desired experience on each level.

The second step is the design of the service concept. It begins by understanding the VCE, the collective co-created experience from all interactions between the customer and the service provider. Afterwards, the service concept is designed through the Customer Value Constellation (CVC), a construct that expands upon the VCE by putting the customer's given activity in the center and mapping all the services they rely on to fulfill said activity with their interrelations to one another. The CVC allows the company to analyze its surrounding service offerings and reflect on how it can reposition its service concept within this shared space to improve its position.

In the third step, the service system is designed. The first task here is to understand the customer's service experience, including his preferred interfaces and communication paths. Afterwards, the service system is designed with the help of the Service System Architecture (SSA) and the Service System Navigation (SSN). The company can thus easily view where it is active and add or remove elements of its service system to improve the service experience. In the design process, alternative service interfaces should be included without simply replicating service offerings on each interface. The SSN then adopts the SSA and describes how the customer may navigate through the service system, which allows the company to optimize the connections between the interfaces.

The final step is the design of the service encounter. Again, MSD begins with a comprehension task that aims at understanding the SEE. For each service encounter, the company analyzes how the customer interacts at each service interface to fulfill a specific service task. At this point, the critical experience factors, which can range from usability to interface aesthetics, need to be determined. Using this knowledge, the company designs the service encounter using the Service Experience Blueprint (SEB), a variation of the Service Blueprint by Shostack that captures each participant's actions in the service encounter, identifies service interface links and allows the company to enhance the SEE by exploring design alternatives. [START_REF] Patrício | Multilevel service design. From customer value constellation to service experience blueprinting[END_REF]

Design Thinking for Industrial Services

The third model to be introduced here is Design Thinking for Industrial Services (DETHIS). Apart from a service development model, DETHIS also includes a digital toolbox with practical Design Thinking methods that empowers SMEs to apply DETHIS without external help. DETHIS adopted the Design Thinking process by the HPI D-School [START_REF] Hpi D-School | What is design thinking? Design thinking process[END_REF] and added a phase at the beginning and at the end of the existing phase sequence. Hence, it consists of eight phases, each serving a singular purpose and including prescribed methods to complete it. Despite the seemingly linear outline, the model follows an iterative process which can be started or finished at any given phase.

Still, the process should be completed in the given order as this enables the Design Thinking mindset to be implemented successfully.

The first phase of DETHIS is the design challenge, during which the company defines what it wishes to accomplish. The design challenge forces them to obtain a more open and customer-centric perspective, which guarantees the company's chances of success as it initiates the subsequent Design Thinking process with the correct mindset. The second phase's goal is to understand the parameters of the problem at hand and to build empathy toward all stakeholders involved. This understanding is validated in the third phase, in which the company must observe and analyze the stakeholders closely.

In the fourth phase, the customer's point of view is defined. By incorporating all critical aspects of the customer's character into the persona, the company guarantees that it has a strong grasp of how the customer thinks and acts. In phase five, the company uses the previously established point of view to generate multiple ideas that address the customer's needs. The ideas are then clustered and weighed before the most promising one is selected for prototyping. In phase six, the prototype of the selected idea is created before it is tested with the customer in phase seven. All feedback is then reworked as need be. The last phase of DETHIS is an implementation phase, in which the broader implementation of the developed idea is planned out and integrated into the company. Given the iterative nature of the Design Thinking process, the steps described above may be repeated multiple times depending on customer feedback. [START_REF] Redlich | Design Thinking für das Service Engineering in kleinen und mittelständischen Unternehmen[END_REF]19]

Comparison

The service engineering models described above can be examined with regards to the seven requirements for innovative service design presented in chapter 2. This comparison lays open the individual strengths and weaknesses of each model. A qualitative evaluation for each requirement reveals the degree to which each model reflects a given requirement compared to the other three models. The results can be illustrated with the help of Harvey Balls (see Fig. 1). The evaluation discussed below is thus of relative and indicative nature and aims at contrasting the three reference models only.

The first model introduced above is Smart Service Engineering. In terms of user centricity, SSE utilizes user stories to form value hypotheses. While this promotes a user centric mindset, it is limited to the functional needs of the user and does not delve deeper into the bigger picture. On the other hand, the prototyping cycle at the heart of this model allows for the speedy testing of development hypotheses and ensures the offering is constantly aligned with the user. This prototyping cycle makes SSE the most agile of the three models, as it directly adopts MVP best practices. Prioritizing the development of the MVP is also in line with lean principles. Additionally, the iterative cycles allow the model to maintain a consistent pace of development and easily adapt to changing parameters. However, SSE falls short when it comes to utilizing service ecosystems. The full potential of partnerships or new forms of resource combinations are not explored. SSE stresses the importance of cross-functional project teams, promoting the dissemination of knowledge and agile development practices. Moreover, the model was derived from case studies and collaboration with industry partners, who tested the model and validated its agile approach. Thus, SSE can be said to display a high practical relevance. [START_REF] Senderek | Smart service engineering. An agile approach to develop data-driven services[END_REF] The second model introduced above is MSD, which particularly shines in its user centricity. The model simultaneously identifies other actors within the user's value constellation that can be partnered with to improve the offering. At the same time, this analysis provides the model with the insights into the types of interfaces that best suit the user, allowing it to adapt its forms of service delivery. MSD is cross-functional in its use of interdisciplinary methods such as the Service Experience Blueprint for the development process. However, the use of cross-functional teams is not explicitly mentioned. The main disadvantage of the MSD model lies in its limited adaptability. By basing the model on fixed interdependent methods of service design, it leaves little room for companies to improvise or adjust the design process. Its practical implementation in different industry contexts does, however, speak for its utility in practice. [START_REF] Patrício | Multilevel service design. From customer value constellation to service experience blueprinting[END_REF] DETHIS builds on a Design Thinking approach and therefore excels in its user centricity. The entire process is based on understanding the user's inherent needs and emotional drives, which puts DETHIS in a strong position to create resonant personal user experiences. Furthermore, it adopts an early and iterative prototyping and testing approach in order to validate its hypotheses and create measurable customer value. The Fig. 1. Comparison of the three selected models Design Thinking technique promotes creativity and collaboration among multiple disciplines, thus, benefiting from cross-functional knowledge and expertise. DETHIS, however, does not include any references or methods for utilizing service ecosystems. Indeed, it suggests conducting a stakeholder analysis in the early stages of understanding the user, yet this is not capitalized on later by other methods. [START_REF] Redlich | Design Thinking für das Service Engineering in kleinen und mittelständischen Unternehmen[END_REF] 4 Outlook

While many old service engineering models are no longer suitable for today's economic environment, there are some recent approaches to service engineering that might be suitable solutions for the challenges of the digital age. These newer models focus on user centricity, make use of service ecosystems and incorporate an agile mindset. All three models presented above meet several of the requirements for modern service engineering models, and some of them have been successfully tested in practice. They can therefore be considered promising approaches toward developing new service offerings in the complex and digitalized economy of the 21 st century.

Acknowledgment

This research and development project project Digivation constitutes the superordinate project of the funding line Service Innovation Based on Digitization and is funded by the German Federal Ministry of Education and Research within the research program Innovations for Tomorrow's Production, Services and Work under the registration number 02K14A221 and managed by the Project Management Agency Karlsruhe (PTKA). The author is responsible for the contents of this publication.


Stefan Wiesner

Jannicke Baalsrud Hauge

Paul Sonntag
email: paul.sonntag@uni-bremen.de

Klaus-Dieter Thoben

Applicability of agile Methods for dynamic Requirements in Smart PSS development

Keywords: Product-Service Systems, Smart Products, Smart Services, agile development, VUCA

Smart Product-Service Systems, i.e. solutions consisting of tangible and intangible components interacting with their environment through information and communication technology, are subject to various dynamic influences along the life cycle. Stakeholders to the solution may change, as well as their needs and technological capabilities. This makes the requirements for the solution volatile, uncertain, complex and ambiguous. The system scope and associated requirements are constantly changing. In this paper, it is discussed how agile methods can help to deal with these influences in the development phase. A literature review and an industrial case study are used for analysing the problem of dynamic requirements, and agile methods are identified that can be applied for Smart Product-Service Systems.

Introduction and Problem

For manufacturing companies, it becomes more and more relevant to provide additional services to their products and to enlarge the target market through innovative business models. This results in a transition from product-to service-oriented industries, in which the development process is increasingly based on participatory and co-creative design principles and thus makes an integrated products and services development process more relevant [START_REF] Qu | State-of-the-art of design, evaluation, and operation methodologies in product service systems[END_REF]. The implementation of Industry 4.0 can be seen as a catalyst and accelerator of this change. The gathering and evaluation of data support processes like design, operation or maintenance and can lead to optimized industrial value chains in the medium and long term [START_REF] Thoben | Industrie 4.0" and Smart Manufacturing -A Review of Research Issues and Application Examples[END_REF]. Manufacturing companies can create a unique selling point in an addressed market niche by offering an attractive bundle of Smart Products and Smart Services, using the potential of new technologies for sensors, actuators and data processing.

The combination of Smart Products and Smart Services to an integrated solution can be seen as a "Smart Product-Service Systems" (Smart PSS). During the life cycle of such a system, stakeholders, scope and configuration will change. This means that also the dependencies and interactions between the product and the service elements may vary, and consequently this affects the overall solution design [START_REF] Wiesner | Interactions between Service and Product Lifecycle Management[END_REF]. Therefore, the engineering of Smart PSS is evolving from a temporal development process for individual solutions towards permanent orchestration of distributed product, service and information technology elements adapted to a dynamic environment [START_REF] Wiesner | Cyber-Physical Product-Service Systems[END_REF]. This requires that these systems are aligned to the environment of stakeholders, technology and constraints throughout the life cycle, which implies that also the requirements towards the Smart PSS are dynamic.

It is therefore relevant to question the methods used for designing and developing such systems. Conventional approaches aim to generate static requirements documents, which limit the reactivity to unforeseen changes in needs. Tools and methods are required that support the gathering and analysis of dynamic requirements throughout the development phase and (prototypical) operation. In the software domain, similar challenges have led to the introduction of agile methods, which rely on continuous feedback loops to catch instantly new requirements [START_REF] Beck | others: Manifesto for agile software development[END_REF]. Thus, our main research question is: Are methods of agile software development transferable to Smart PSS in order to support agile system development there? The aim of the paper is to design a procedure model for agile system development of Smart PSS based on an analysis of the applicability of agile methods from software development. More specifically, this paper looks into the suitability of SCRUM and Design Thinking.

The paper is structured as follows: The next section explains the research approach and methodology, followed by an overview of the state-of-the art in the fields of dynamic development environments and agile development methods. Section 4 presents a procedure model for the agile development of Smart PSS, which is applied in a case study for Smart PSS development in automotive plant engineering in Section 5. The following section 6 illustrates the application results for the identified tools and methods in the use case. Finally, in section 7 we discuss next steps and future work.

Methodology

The main objective of our work is two-fold, thus we have used a mixed method approach for our research approach: a literature review in combination with action based research, in this instance a case study. A literature review in the field of agile design methods and design of smart PSS was planned, conducted and reported. This analysis showed that there is a research gap in the need for specific application areas of agile methods for Smart PSS. In order to dig deeper into the problems and to shed some light on under which circumstances agile methods can be used in an industrial environment and how they can form the basis for agile development processes, empirical research was carried out in an automotive plant-engineering case study according to Sein, Henfridsson et al. [START_REF] Sein | Action Design Research[END_REF].

State-of-the-Art

This section summarizes the results of the literature review and describes the current challenges in Smart PSS development originating from an environment characterized by volatility, uncertainty, complexity and ambiguity. Furthermore, agile development methods are introduced as an approach to deal with these influences.

Smart PSS Development in a dynamic Environment

Smart PSS are complex systems that have to be aligned to an environment of stakeholders, technology and constraints. This environment is dynamic and changes have an impact on the requirements for the solution. Its characteristics influencing Smart PSS development can be described with the elements of volatility, uncertainty, complexity and ambiguity (VUCA). Volatility denotes strong fluctuations of a state over a relatively short period, making it hardly predictable. Uncertainty means that causal relationships of the system under consideration are known, but not their probability of occurrence to forecast future developments. Complexity describes the unpredictability of system behavior due to the abundance of elements and connections. Ambiguity refers to the obscurity of causal relationships, when an event cannot be clearly assigned to a potential effect, leading to false assumptions. [START_REF] Bennett | What a difference a word makes: Understanding threats to performance in a VUCA world[END_REF][START_REF] Scheller | Auf dem Weg zur agilen Organisation. Wie Sie Ihr Unternehmen dynamischer, flexibler und leistungsfähiger gestalten[END_REF] Developing Smart PSS poses particular difficulties under these conditions. Influences from the volatile system environment have a direct impact through technological interfaces, e.g. the real time processing of Big Data. As future operation scenarios are often vague and can only be described by probabilities, there is uncertainty about the requirements for the Smart PSS. Furthermore, they are complex systems with a large number of different elements and connections, making it impossible to predict precisely the behavior of the system. As Smart PSS are by nature one-of-a-kind solutions, there is no pattern to derive requirements, leading to ambiguous specifications.

In order to cope with a VUCA environment in Smart PSS development, an approach has to be able to handle these conditions. It has to be agile enough to react to volatile changes and unlock additional information sources to reduce uncertainty. Furthermore, processes should be restructured to match system complexity and involve experimentation with prototypes to reduce ambiguity. Such methods and tools have been introduced in computer science as agile software development. [START_REF] Bennett | What a difference a word makes: Understanding threats to performance in a VUCA world[END_REF]

Agile Development Methods

Agile development methods address the challenges of VUCA-influenced development environments with an iterative and incremental way of working [START_REF] Schmidt | REAL EFFECTS OF AGILE DEVELOPMENT OF PHYSICAL PRODUCTS: APPORTIONING THE HYPE[END_REF]. They start as quickly as possible with value-creating development tasks in order to achieve a socalled Minimal Viable Product (MVP), which is repeatedly subjected to customer feedback, leading to a participatory design approach. Compared to plan-driven product developments there is no need to draw up specifications, but new requirements are continuously added. An agile approach is consequently suitable for Smart PSS develop-ments, in which the complete specification of requirements at the beginning of the project is impossible. Based on reviews of agile development approaches in industry [START_REF] Komus | Study Report: Status Quo Agile 2016/2017[END_REF][START_REF] Preußig | Agiles Projektmanagement. Scrum, User Stories, Task Boards & Co[END_REF][START_REF] Klein | Agiles Engineering im Maschinen-und Anlagenbau[END_REF][START_REF] Abrahamsson | Agile Software Development Methods: Review and Analysis[END_REF], two methods with a high relevance for physical product development are examined more closely -Scrum and Design Thinking.

Scrum is the most widespread agile approach [START_REF] Komus | Study Report: Status Quo Agile 2016/2017[END_REF], with a focus on small, highly efficient teams. As shown in Fig. 1, product development is orchestrated in the form of events that are held at fixed intervals and with fixed durations. An iteration in the development is called "Sprint" and is timed between 1 and 4 weeks, depending on the project. The Development Team plans the development activities and scope of the Sprint with the Product Owner and discusses the progress in a Daily Scrum meeting up to a maximum of 15 minutes. Each iteration ends with the Sprint Review, where the implemented functions are presented by the Development Team and accepted by the Product Owner. A reflection on the methodical work is carried out in the team within the Sprint Retrospective. [START_REF] Sutherland | The Scrum Guide™[END_REF] Fig. 1. Scrum Framework, following [START_REF] Sutherland | The Scrum Guide™[END_REF] Design Thinking is an agile method to promote innovation focused on the intuitive thinking processes of interdisciplinary developers by approaching the problem from several perspectives [START_REF] Grots | Design Thinking -Kreativität als Methode[END_REF]. The Design Thinking process combines empathy for the problem context with creative problem solving and rational analysis of proposed solutions, as illustrated below.

Fig. 2. Design Thinking Process, based on [15]

In the first phase, the development team builds understanding for the problem. In the second phase, the developers acquire the user perspective by recording their needs in direct interactive exchange as well as indirectly through observations. The synthesis defines the team-wide view of the problem from the findings. Based on this common view, ideas are then generated and finally implemented in prototypes. From then on, the prototypes offer the possibility to test the functions and thus to recognize false assumptions at an early stage. Based on the test results, the development team generates a new, deeper understanding of the problem and starts the design thinking cycle again. By repeating this process, the problem and ultimately the solution space are narrowed down. [START_REF] Grots | Design Thinking -Kreativität als Methode[END_REF] 4

Procedure Model for agile Smart PSS Development

According to [START_REF] Preußig | Agiles Projektmanagement. Scrum, User Stories, Task Boards & Co[END_REF], the application of Scrum is suitable in the early steps of product development in automotive plant engineering due to the availability of detailed descriptions how to perform the method. Scrum is also the method most widely used in an industrial context, so that benchmarking would be possible [START_REF] Komus | Study Report: Status Quo Agile 2016/2017[END_REF]. A high user acceptance is also probable. A disadvantage of the exclusive application of Scrum is the absence of a methodical development of an initial product vision [START_REF] Sutherland | The Scrum Guide™[END_REF].

In contrast, the use of Design Thinking methods in the early steps of product development make it possible to develop an initial product vision [START_REF] Grots | Design Thinking -Kreativität als Methode[END_REF]. In plant-engineering, consistent usage of Design Thinking in the development phase could have an effect on the modernization of the service portfolio and organizational structures. The integration of the Scrum method into the Design Thinking process is thus proposed as a concept for the realization of the procedure model for agile development of Smart PSS and is illustrated in Fig. 4 below.

Case Study

The thyssenkrupp System Engineering subsidiary of thyssenkrupp AG manufactures systems for automotive body and engine assembly, as well as the associated test stations. The change to electric drive and shorter product cycles make the automotive plant-engineering environment extremely volatile, while the preference for different power and energy storage technologies remains uncertain. While the complexity of automotive production lines is increasing, their bespoke design causes ambiguous requirements. Therefore, a training system was developed to examine whether future employees could be familiarized with their work processes already in the design phase. The standard training method so far has been to introduce employees to the basic welding process in the welding school in order to train aluminum welding, and then receive practical training in production on the real work piece. This procedure repeatedly led to high training effort and consumption of the training components during the learning phases. In order to get better learning results at lower costs with fewer resources, a virtual training environment has been implemented as a part of the Scrum phase (prototyping). This integrated virtual training environment as an integrated part of the prototyping phase can be seen as a Smart PSS. The virtual training project was set up to check whether the use of Virtual Reality (VR) technology and the CAD models of the assemblies would make it possible to familiarize the trainees with complex work processes. A concept was developed to simulate the station and work processes using VR. The objective of the project was to review firstly the concept for internal purposes. After this, the system can be offered to external customers.

Fig. 4. Virtual Reality Application and virtual Workstation

During the introduction of agile development, the team and the project manager were introduced to the basics of the Scrum framework. The differences between the roles of a classical project manager and the Product Owner were explicitly addressed. The Scrum methodology was applied in the development phase. Two sprints of three weeks each were run through. Additional techniques such as testing and the virtual task board were integrated in the course of development.

The sprint length of 3 weeks at thyssenkrupp System Engineering is commonly used across all industries. In contrast to the cross-industry practice of daily scrums, in the case study scrum meetings were held every two days. This deviation was due to the sector-specific organizational structure of a weak matrix organization and the resulting design of the Scrum process. In the context of the final retrospective, the work with the Scrum methodology was evaluated. At thyssenkrupp System Engineering, 15 of the Scrum users answered an online questionnaire, which is partly based on the questions of the study Status Quo Agile 2017 [START_REF] Komus | Study Report: Status Quo Agile 2016/2017[END_REF].

Results

At thyssenkrupp System Engineering, 15 of the employees with Scrum experience answered an anonymous online questionnaire, derived from the questions of the Status Quo Agile 2017 study [START_REF] Komus | Study Report: Status Quo Agile 2016/2017[END_REF]. The resulting feedback was compared with the statements of the cross-industry study in order to enable conclusions about the applicability especially for automotive Smart PSS. The overall performance of Scrum was assessed positively, with 80% of the users rating the performance as good or very good. This is close to the cross-industry ratings with 86% positive feedback. The thyssenkrupp System Engineering Scrum users rated the development method in each criterion (transparency, innovation potential and speed) better than the participants in the comparative study did. It could be proven that cross-industry studies and the application in the case study show similar tendencies. The survey shows that the Scrum development method is accepted by thyssenkrupp System Engineering developers. Through the exemplary application and the feedback of the Scrum users, the applicability of Scrum in the case study development project could be demonstrated. The applicability of Scrum in the early product development of an internal Smart PSS R&D project could be shown.

Conclusions and Future Work

The pre-selection and evaluation of the agile methods was carried out based on industry-independent literature. The theoretical applicability of Scrum and Design Thinking was derived, which was used to build a procedure model for agile product development in automotive Smart PSS. The investigation of the applicability of agile methods for the development of Smart PSS in automotive plant engineering was conducted as a pioneering activity. The resulting procedure model supports agile system development by applying two agile methods, Scrum and Design Thinking. The Scrum part of the model could be implemented exemplarily in a case study. The evaluation of the Scrum method exceeded those of a cross-industry comparative study. In the test phase of the procedure model, a high result quality could be confirmed and the need as well as the readiness for user integration into the development process was validated. The procedure model developed represents an approach for the industry-specific application of agile methods for Smart PSS. In future research, the process model will be validated including the ideation phase with the application of Design Thinking activities. Based on the results from the case study and literature review, better results can be expected for the early development phase. In addition, it is necessary to examine the applicability in the medium and long-term, as well as the transferability to other Smart PSS developments with a high ratio of physical components.

Fig. 3 .

 3 Fig. 3. Phases of the Procedure Model for agile Smart PSS Development


Essomanda Konzou
email: essomanda.konzou@univ-lorraine.fr

Efoévi Koudou
email: efeovi.koudou@univ-lorraine.fr

Kossi E Gneyou
email: kossigneyou@yahoo.fr

Bounds of derivatives of the solution of the Stein equation for the generalized inverse Gaussian and Kummer distributions

Keywords: Generalized Inverse Gaussian distribution, Kummer distribution, Stein characterization, Stein equation

come

Bounds of derivatives of the solution of the Stein equation for the generalized inverse Gaussian and Kummer distributions

Essomanda Konzou, Efoevi Angelo Koudou, Kossi

Introduction

The generalized inverse Gaussian (hereafter GIG) distribution with parameters p ∈ R, a > 0, b > 0 has density x a-1 (1 + x) -a-b e -cx , x > 0 where ψ is the confluent hypergeometric function of the second kind.

For details on GIG and Kummer distributions see for example [START_REF] Jøgensen | Statistical properties of the generalized inverse gaussian distribution[END_REF][START_REF] Koudou | Characterizations of gig laws: a survey[END_REF][START_REF] Koudou | Independence properties of the matsumoto-yor type[END_REF][START_REF] Piliszek | Change of measure technique in characterizations of the gamma and kummer distributions[END_REF].

In this paper, these two distributions are considered in the context of Stein's method introduced by Stein in [START_REF] Stein | A bound for the error in the normal approximation to the distribution of a sum of dependent random variables[END_REF]. A first instance of the method was found in [START_REF] Stein | A bound for the error in the normal approximation to the distribution of a sum of dependent random variables[END_REF], where Stein showed that a random variable X has a standard normal distribution if and only if, for all real-valued absolutely continuous function f such that E |f (Z)| < ∞, with Z a random variable following the standard normal distribution,

E [f (X) -Xf (X)] = 0. The corresponding Stein equation is f (x) -xf (x) = h(x) -Eh(Z) (1)
where h is a bounded function. The corresponding Stein operator is f → (T f ) (x) = f (x) -xf (x). If f h solves equation [START_REF] Chatterjee | Exponential approximation by stein's method and spectral graph theory[END_REF], then for any random variable X, we have

|E [f h (X) -Xf h (X)]| = |Eh(X) -Eh(Z)| .
Thus, we can bound |Eh(X) -Eh(Z)| given h, by finding a solution f h of the Stein equation ( 1) and bounding the left-hand side of the previous equation. For more details on Stein's method, see [START_REF] Chen | Normal approximation by stein's method, Probability and its Applications[END_REF][START_REF] Ross | Fundamentals of stein's method[END_REF]. Solving the Stein equation for other distributions than the standard normal distribution, and bounding the solution and its derivatives has become a challenge often addressed in the literature (see for example [START_REF] Chen | Poisson approximation for dependent trials[END_REF] for the Poisson distribution, [START_REF] Luk | Stein's method for the gamma distribution and related statistical applications[END_REF] for the gamma distribution, [START_REF] Chatterjee | Exponential approximation by stein's method and spectral graph theory[END_REF] for the exponential distribution). A bound of the solution of Stein equation and its first and second derivatives for GIG and Kummer distributions is derived in [START_REF] Konzou | About the stein equation for the generalized inverse gaussian and kummer distributions[END_REF] by using the density Stein approach [START_REF] Chen | Normal approximation by stein's method, Probability and its Applications[END_REF][START_REF] Ley | Stein's density approach and information inequalities[END_REF][START_REF] Schoutens | Orthogonal polynomials in stein's method[END_REF].

The purpose of this paper is to complete the results of [START_REF] Konzou | About the stein equation for the generalized inverse gaussian and kummer distributions[END_REF] by providing upper bound of the third derivative of these two solutions. We also express a bound of the k-th derivative for k ≥ 4 of these solutions by doing a stepwise calculation. Sections 2 and 3 present the main results obtained for GIG and Kummer distributions respectively. Section 4 contains the proofs.

2 About the bounds of the derivatives of the solution of the Stein equation for the generalized inverse Gaussian distribution

Let s(x) = x 2 and τ p,a,b (x) = b 2 + (p + 1)x - a 2 x 2 . (2)
Then, as observed in [START_REF] Gaunt | A stein characterisation of the generalized hyperbolic distribution[END_REF], the GIG density g p,a,b satisfies

(s(x)g p,a,b (x)) = τ p,a,b (x)g p,a,b (x). (3)
This enables us to apply Theorem 2.1 in [START_REF] Konzou | About the stein equation for the generalized inverse gaussian and kummer distributions[END_REF] to retrieve the following Stein characterization of the GIG distribution (Proposition 3.1. in [START_REF] Konzou | About the stein equation for the generalized inverse gaussian and kummer distributions[END_REF]) given in [START_REF] Koudou | Characterizations of gig laws: a survey[END_REF][START_REF] Gaunt | A stein characterisation of the generalized hyperbolic distribution[END_REF]: g p,a,b (x)f (x) = 0, and such that the following expectation exists, we have:

Proposition 2.
E X 2 f (X) + b 2 + (p + 1)X - a 2 X 2 f (X) = 0.
The corresponding Stein equation is

x 2 f (x) + b 2 + (p + 1)x - a 2 x 2 f (x) = h(x) -Eh(W ) ( 4
)
where h is a bounded function and W a random variable following the GIG distribution with parameters p, a, b. Theorem 3.2 in [START_REF] Konzou | About the stein equation for the generalized inverse gaussian and kummer distributions[END_REF] shows that the bounded solution of Stein's equation ( 4) is given by

f h (x) = 1 s(x)g p,a,b (x) x 0 g p,a,b (t) [h(t) -Eh(W )] dt = -1 s(x)g p,a,b (x) +∞ x g p,a,b (t) [h(t) -Eh(W )] dt. (5)
Previous work in [START_REF] Konzou | About the stein equation for the generalized inverse gaussian and kummer distributions[END_REF] provided explicit bounds of f h , f h and f h . We complete this result by providing an upper bound of the third derivative f

(3) h of f h and propose a bound of k-th derivative for k ≥ 4 by doing a stepwise calculation. The following proposition established in [START_REF] Konzou | About the stein equation for the generalized inverse gaussian and kummer distributions[END_REF] will be useful to prove the results obtained.

Proposition 2.2 Consider a density function g > 0 on (0, ∞) satisfying the equation (s(x)g(x)) = τ (x)g(x), where s and τ are polynomial functions such that s > 0 on (0, ∞). Assume that lim x→0 s(x)g(x) = lim x→∞ s(x)g(x) = 0.

Assume also that τ (0) > 0 and, for some β > 0, τ is increasing on (0, β], decreasing on [β, ∞) and has a unique zero α on [β, ∞). Let h be a bounded continuous function. Then

||f h || ≤ N 0 ||h(.) -Eh(W )|| , (6)
where

N 0 = max 1 τ (0) ; 1 s(β)g(β) and f = sup x>0 |f (x)| .
In the remainder of this section, we define

α k = p + 2k + 1 + (p + 2k + 1) 2 + ab a , τ k (x) = b 2 + (p + 2k + 1)x - a 2 x 2 , ∀k ≥ 1; h k (x) = h (k) (x) -k(p + k -ax)f (k-1) (x) + 3 + (k -3)(k + 2) 2 af (k-2) (x) ∀k ≥ 3
and we denote by g k := g p+2k,a,b the density of the GIG(p + 2k, a, b) distribution. We have

(s(x)g k (x)) = τ k (x)g k (x) and lim x→0 s(x)g k (x) = lim x→∞ s(x)g k (x) = 0 ∀k ≥ 1. ( 7
)
We have the following results.

Theorem 2.1 Let h : (0, ∞) → R be a bounded function, three times differentiable such that h , h and h (3) are bounded. Then the third derivative f

(3) h of the solution f h of Stein equation (4) is such that f (3) h ≤ M 3 =    max(G 3 1 , G 3 2 ) if p ≤ -7 max(G 3 3 , G 3 4 , G 3 2 ) if p > -7 (8)
where

G 3 1 = h (3) + 3 max |p + 3|, (4 + (p + 7) 2 + ab) M + 3aM 1 α 2 3 g 3 (α 3 ) α3 0 g 3 (t)dt, G 3 2 = 5a + a (p + 7) 2 + ab α 2 3 p + 7 + (p + 7) 2 + ab ||h || + 6a + a (p + 7) 2 + ab 6p + 42 + 6 (p + 7) 2 + ab h(.) -Eh(W ) + 2 α 2 3 ( h + aM h(.) -Eh(W ) ) , G 3 3 = 2 b h (3) + 3 max (4, |p + 3|) M + 3aM , G 3 4 = h (3) + 3 max |p + 3|, (4 + (p + 7) 2 + ab) M + 3aM a 2 (p + 7) 2 g 3 p + 7 a ,
W is a random variable following GIG distribution with parameters p, a, b, the constants M , M and M are given in [START_REF] Konzou | About the stein equation for the generalized inverse gaussian and kummer distributions[END_REF] and are such that

f h ≤ M , f h ≤ M and f h ≤ M .
The proof of Theorem 2.1 will be given in Section 4.

Theorem 2.2 Let h : (0, ∞) → R be a bounded function, k times differentiable such that h (k) is bounded for all k ≥ 1. The k-th derivative f (k) h of the solution f h of Stein equation (4) is such that f (k) h ≤ M k =    max(G k 1 , G k 2 ) if p ≤ -2k -1 max(G k 3 , G k 4 , G k 2 ) if p > -2k -1 (9)
where

G k 1 = H × 1 α 2 k g k (α k ) α k 0 g k (t)dt, G k 2 = 2 α 2 k h k-1 , h k-1 ≤ h (k-1) + a(k -1) k + 2 + (p + 2k + 1) 2 + ab 4p + 8k + 4 + 4 (p + 2k + 1) 2 + ab h k-2 + a 3 + (k -4)(k + 1) 2 M k-3 , G k 3 = 2 b h (k) + k max (k + 1, |p + k|) M k-1 + a 3 + (k -3)(k + 2) 2 M k-2 , G k 4 = H × a 2 (p + 2k + 1) 2 g k p + 2k + 1 a , H = h (k) + k max |p + k|, k + 1 + (p + 2k + 1) 2 + ab M k-1 + a 3 + (k -3)(k + 2) 2 M k-2 ,
and

M i is such that, f (i) k ≤ M i for all i ≥ 1.
The proof of Theorem 2.2 will be given in Section 4.

3 About the bounds of the derivatives of the solution of the Stein equation for the Kummer distribution

Let s(x) = x(1 + x) and τ (x) = (1 -b)x -cx(1 + x) + a. (10)
Then the density d a,b,c of Kummer distribution satisfies

(s(x)d a,b,c (x)) = τ (x)d a,b,c (x).
Theorem 4.1 in [START_REF] Konzou | About the stein equation for the generalized inverse gaussian and kummer distributions[END_REF] give the following Stein characterization of the Kummer distribution.

Theorem 3.1 A random variable X follows the Kummer distribution with density d a,b,c if and only if, for all differentiable function f such that the expectation exists,

E [X(1 + X)f (X) + [(1 -b)X -cX(1 + X) + a] f (X)] = 0.
The corresponding Stein equation is

x(x + 1)f (x) + [(1 -b)x -cx(1 + x) + a] f (x) = h(x) -Eh(W ) ( 11
)
where h is a bounded function and W has K(a,b,c) distribution.

The bounded solution of equation ( 11) is

f h (x) = 1 x(1 + x)d a,b,c (x) x 0 d a,b,c (t) [h(t) -Eh(W )] dt = -1 x(1 + x)d a,b,c (x) +∞ x d a,b,c (t) [h(t) -Eh(W )] dt. (12)
In [START_REF] Konzou | About the stein equation for the generalized inverse gaussian and kummer distributions[END_REF], the bounds of f h , f h and f h have been proposed. We complete this result by providing a bound of the third derivative f

(3) h of f h and we propose a bound of k-th derivative for k ≥ 4 as in the case of GIG distribution. In the remainder of this section, we define

α k = 2k + 1 -b -c + (2k + 1 -b -c) 2 + 4c(a + k) 2c , ∀k ≥ 1; τ k (x) = (2k + 1 -b)x -cx(1 + x) + a + k, ∀k ≥ 1; h k (x) = h (k) (x) -k(k -b -c -2cx)f (k-1) (x) + k(k -1)cf (k-2) (x) ∀k ≥ 3
and we denote by

d k := d a+k,b-2k,c the density of the K(a + k, b -2k, c) distribution. We have (s(x)d k (x)) = τ k (x)d k (x) and lim x→0 s(x)d k (x) = lim x→∞ s(x)d k (x) = 0 ∀k ≥ 1. ( 13
)
Using similar arguments as in the case of GIG distribution, we obtain the following results

Theorem 3.2 Let h : (0, ∞) → R be a bounded, three times differentiable function such that h , h and h (3) are bounded. Then the third derivative f

(3) h of the solution f h given by equation ( 12) is such that

f (3) h ≤ N 3 =    max(K 3 1 , K 3 2 ) if 7 -b -c ≤ 0 max(K 3 3 , K 3 4 , K 3 2 ) if 7 -b -c > 0 (14)
where

K 3 1 = h (3) + 3 max |3 -b -c|, 4 + (7 -b -c) 2 + 4c(a + 3) K + 6cK × 1 α 2 3 d 3 (α 3 ) α3 0 d 3 (t)dt, K 3 2 = 2 α 3 (1 + α 3 ) ( h + 2cK h(.) -Eh(W ) ) + 10c + 2c (7 -b -c) 2 + 4c(a + 3) α 3 (1 + α 3 ) 7 -b + (7 -b -c) 2 + 4c(a + 3) × h + 6c + c (7 -b -c) 2 + 4(a + 3) 21 -3b + 3 (7 -b -c) 2 + 4(a + 3) h(.) -Eh(W ) , K 3 3 = 1 a + 3 h (3) + 3 max (4, |3 -b -c|) K + 6cK , K 3 4 = h (3) + 3 max |3 -b -c|, 4 + (7 -b -c) 2 + 4c(a + 3) K + 6cK × 4c 2 (7 -b -c)(7 -b + c)d 3 7 -b -c 2c , W ∼ K(a, b, c
), K, K and K are given in [START_REF] Konzou | About the stein equation for the generalized inverse gaussian and kummer distributions[END_REF] and are such that

f h ≤ K, f h ≤ K and f h ≤ K . Noticing that f (k) h
satisfies the differential equation

x(1 + x)f (k+1) (x) + [(2k + 1 -b)x -cx(1 + x) + a + k] f (k) (x) = h k (x), (15)
we have

Theorem 3.3 Let h : (0, ∞) → R be a bounded, k times differentiable function such that h (k) is bounded for all k ≥ 1. The k-th derivative f (k) h of the solution f h of Stein equation (11) is such that f (k) h ≤ N k =    max(K k 1 , K k 2 ) if 2k + 1 -b -c ≤ 0 max(K k 2 , K k 3 , K k 4 ) if 2k + 1 -b -c > 0 (16)
where

K k 1 = T × 1 α k (1 + α k )d k (α k ) α k 0 d k (t)dt, K k 2 = 2 α k (α k + 1) h k-1 , h k-1 ≤ h (k-1) + c(k -1) k + 2 + (2k + 1 -b -c) 2 + 4c(a + k) 4k + 2 -2b + 2 (2k + 1 -b -c) 2 + 4c(a + k) h k-2 + (k -1)(k -2)cN k-3 , K k 3 = 1 a + k h (k) + k max (k + 1, |k -b -c|) N k-1 + k(k -1)cN k-2 , K k 4 = T × 4c 2 (2k + 1 -b -c)(2k + 1 -b + c)d k 2k + 1 -b -c 2c , T = h (k) + k max |k -b -c|, k + 1 + (2k + 1 -b -c) 2 + 4c(a + k) N k-1 + k(k -1)cN k-2 ,
and N j is such that, f

≤ N j for all j ≥ 1.

Proofs

Proof of Theorem 2.1

First, assume that p ≤ -7. f

h satisfies the differential equation

x 2 f (4) (x) + b 2 + (p + 7)x - a 2 x 2 f (3) (x) = h (3) (x) -3(p + 3 -ax)f (x) + 3af (x). ( 17
) Let h 3 (x) = h (3) (x) -3(p + 3 -ax)f (x) + 3af (x). Then, by equation (17), f (3)
h solves the differential equation For p ≤ -7, τ 3 is decreasing on (0, ∞) and its only zero is

x 2 f (4) (x) + b 2 + (p + 7)x - a 2 x 2 f (3) (x) = h 3 (x). ( 18
) Since Eh 3 (Y ) = E Y 2 f (4) (Y ) + b 2 + (p + 7)Y -a 2 Y 2 f (3) (Y ) = 0 if Y ∼GIG(p + 6,
f (3) h (x) = 1 s(x)g 3 (x) x 0 g 3 (t)h 3 (t)dt = -1 s(x)g 3 (x)
α 3 = p+7+ √ (p+7) 2 +ab a . If x ≤ α 3 and if h 3 is bounded, then f (3) h (x) ≤ h 3 1 s(x)g 3 (x) x 0 g 3 (t)dt. Let (x) = 1 s(x)g 3 (x) x 0 g 3 (t)dt. is differentiable on (0, ∞) and (x) = -(s(x)g 3 (x)) (s(x)g 3 (x)) 2 x 0 g 3 (t)dt + 1 s(x) ≥ - (s(x)g 3 (x)) (s(x)g 3 (x)) 2 1 τ 3 (x) x 0 τ 3 (t)g 3 (t)dt + 1 s(x) 1 ≤ τ 3 (t) τ 3 (x) ∀t ≤ x = - 1 s(x) + 1 s(x) = 0 (s(x)g 3 (x)) = τ 3 (x)g 3 (x) and lim t→0 s(t)g 3 (t) = 0 .
Then for x ≤ α 3 , (x) ≤ (α 3 ).

|h 3 (x)| = h (3) (x) -3(p + 3 -ax)f h (x) + 3af h (x) ≤ h (3) + 3|p + 3 -ax| f h + 3a f h ≤ h (3) + 3 max |p + 3|, 4 + (p + 7) 2 + ab f h + 3a f h (0 < x < α 3 ) ≤ h (3) + 3 max |p + 3|, 4 + (p + 7) 2 + ab M + 3aM
Hence for p ≤ -7 and x ≤ α 3 ,

f (3) h ≤ h (3) + 3 max |p + 3|, 4 + (p + 7) 2 + ab M + 3aM 1 α 2 3 g 3 (α 3 ) α3 0 g 3 (t)dt.
For p ≤ -7 and x > α 3 , consider the following Stein equation solving by f h :

s(x)f (3) (x) + τ 2 (x)f (x) = h 2 (x) = h (x) -2(p + 2 -ax)f (x) + af (x). ( 20
)
By equation (20), one obtains

f (3) h (x) = h 2 (x) s(x) - τ 2 (x)f h (x) s(x) f (3) h (x) ≤ h 2 s(x) + |τ 2 (x)f h (x)| s(x) .
For p ≤ -7 and x > α 3 , τ 2 is decreasing and negative. We have

f h (x) = -1 s(x)g 2 (x) +∞ x g 2 (t)h 2 (t)dt |τ 2 (x)f h (x)| = -τ 2 (x) s(x)g 2 (x) +∞ x g 2 (t)h 2 (t)dt ≤ h 2 -τ 2 (x) s(x)g 2 (x) +∞ x g 2 (t)dt (τ 2 is negative) ≤ h 2 -τ 2 (x) s(x)g 2 (x) +∞ x τ 2 (t) τ 2 (x) g 2 (t)dt τ 2 (t) τ 2 (x) ≥ 1 ∀x ≤ t = h 2 -1 s(x)g 2 (x) +∞ x (s(t)g 2 (t)) dt = h 2 lim t→0 s(t)g 2 (t) = 0 . |h 2 (x)| = |h (x) -2(p + 2 -ax)f h (x) + af h (x)| ≤ h + 2 |(p + 2 -ax)f h (x)| + a f h ; f h (x) = -1 s(x)g 1 (x) +∞ x g 1 (t)h 1 (t)dt |(p + 2 -ax)f h (x)| ≤ h 1 -(p + 2 -ax) s(x)g 1 (x) +∞ x g 1 (t)dt ≤ h 1 p + 2 -ax τ 1 (x) . Let κ(x) = p + 2 -ax b/2 + (p + 3)x -ax 2 /2 = p + 2 -ax τ 1 (x) . κ is differentiable on (α 3 , ∞) and κ (x) = - ab + 2(p + 2)(p + 3) -2a(p + 2)x + a 2 x 2 2τ 2 1 (x) = - ab + (p + 2) 2 + 2(p + 2) + (p + 2 -ax) 2 2τ 2 1 (x)
.

Thus for p ≤ -3 or p ≥ -2, κ (x) ≤ 0. For -3 < p < -2, if 1 -ab -(p + 3) 2 > 0, then the only possible positive

zero of x → ab + 2(p + 2)(p + 3) -2a(p + 2)x + a 2 x 2 is p + 2 + 1 -ab -(p + 3) 2 a
, which is smaller than α 3 . Hence for all p ∈ R, κ is decreasing on (α 3 , ∞) and κ(x) ≤ κ(α 3 ) = 5a + a (p + 7) 2 + ab 4p + 28 + 4 (p + 7) 2 + ab .

The function

h 1 (x) = h (x) -(p + 1 -ax)f h (x) is such that |h 1 (x)| ≤ h + -(p + 1 -ax) s(x)g p,a,b (x) +∞ x g p,a,b (t)(h(t) -Eh(W ))dt ≤ h + h(.) -Eh(W ) p + 1 -ax τ p,a,b (x) . v : x → p + 1 -ax τ p,a,b (x) is decreasing on (α 3 , ∞), as a consequence, for all x > α 3 v(x) ≤ v(α 3 ) = 6a + a (p + 7) 2 + ab 6p + 42 + 6 (p + 7) 2 + ab .
We obtain

h 2 ≤ h + 5a + a (p +
( h + aM h(.) -Eh(W ) ) . Now assume that p > -7 -On 0, p+7 a , τ 3 is such that τ3(t) τ3(0) ≥ 1, ∀0 ≤ t ≤ p+7 a . Since |h 3 (x)| ≤ h (3) + 3|p + 3 -ax| f h + 3a f h ≤ h (3) + 3 max (4, |p + 3|) f h + 3a f h ≤ h (3) + 3 max (4, |p + 3|) M + 3aM , we have |f (3) h (x)| ≤ h 3 1 s(x)g 3 (x) x 0 g 3 (t)dt ≤ h 3 1 τ 3 (0)s(x)g 3 (x) x 0 τ 3 (t)g 3 (t)dt = 1 τ 3 (0) h 3 = 2 b h 3 and f (3) h ≤ 2 b h (3) + 3 max (4, |p + 3|) M 2 + 3aM 1 .
-On p+7 a , ∞ , the similar arguments as in the case p ≤ -7 and Proposition 2.2 lead to the result.

Proof of Theorem 2.2

Assume p ≤ -2k -1. The function f

(k) h
satisfies the following differential equation

x 2 f (k+1) (x) + b 2 + (p + 1 + 2k)x - a 2 x 2 f (k) (x) = h k (x) (21)
where

h k (x) = h (k) (x) -k(p + k -ax)f (k-1) (x) + 3 + (k -3)(k + 2) 2 af (k-2) (x). ( 22
)
Equation ( 21) shows that f

f (k) h (x) = 1 s(x)g k (x) x 0 g k (t)h k (t)dt = -1 s(x)g k (x) +∞ x g k (t)h k (t)dt. ( 23
)
For p ≤ -2k -1, τ k is decreasing on (0, ∞) and has unique zero α k . If x ≤ α k and if h k is bounded, we have

f (k) h (x) ≤ h k 1 α 2 k g k (α k ) α k 0 g k (t)dt.
h k is such that

|h k (x)| = h (k) (x) -k(p + k -ax)f (k-1) h (x) + 3 + (k -3)(k + 2) 2 af (k-2) h (x) ≤ h (k) + k|p + k -ax| f (k-1) h + a 3 + (k -3)(k + 2) 2 f (k-2) h ≤ h (k) + k max |p + k|, k + 1 + (p + 2k + 1) 2 + ab f (k-1) h + a 3 + (k -3)(k + 2) 2 f (k-2) h .
As consequence, for p ≤ -2k -1 and x ≤ α k ,

f (k) h ≤ H × 1 α 2 k g k (α k ) α k 0 g k (t)dt
where

H = h (k) + k max |p + k|, k + 1 + (p + 2k + 1) 2 + ab M k-1 + a 3 + (k -3)(k + 2) 2 M k-2 .
For p ≤ -2k -1 and x > α k , τ k-1 is decreasing and negative, hence

τ k-1 (t) τ k-1 (x) ≥ 1 ∀x ≤ t.
Replacing k by k -1 in equation ( 21), we have 3) .

f (k) h (x) = h k-1 (x) s(x) - τ k-1 (x)f k-1 h (x) s(x) f (k) h (x) ≤ h k-1 s(x) + τ k-1 (x)f k-1 h (x) s(x) . τ k-1 (x)f k-1 h (x) = -τ k-1 (x) s(x)g k-1 (x) +∞ x g k-1 (t)h k-1 (t)dt ≤ h k-1 -τ k-1 (x) s(x)g k-1 (x) +∞ x g k-1 (t)dt ≤ h k-1 -τ k-1 (x) s(x)g k-1 (x) +∞ x τ k-1 (t) τ k-1 (x) g k-1 (t)dt = h k-1 -1 s(x)g k-1 (x) +∞ x (s(t)g k-1 (t)) dt = h k-1 . |h k-1 (x)| ≤ h (k-1) + (k -1) (p + k -1 -ax)f (k-2) (x) + 3 + (k -4)(k + 1) 2 a f (k-
(p + k -1 -ax)f (k-2) (x) = -(p + k -1 -ax) s(x)g k-2 ∞ x g k-2 (t)h k-2 (t)dt ≤ h k-2 -(p + k -1 -ax) s(x)g k-2 ∞ x g k-2 (t)(t)dt ≤ h k-2 -(p + k -1 -ax) s(x)g k-2 τ k-2(x) ∞ x τ k-2 (t)g k-2 (t)(t)dt = h k-2 -(p + k -1 -ax) s(x)g k-2 τ k-2(x) ∞ x (s(t)g k-2 (t)(t)) dt = h k-2 p + k -1 -ax τ k-2(x) . Let ϕ(x) = p + k -1 -ax τ k-2(x) = p + k -1 -ax b 2 + (p + 2k -3)x -a 2 x 2 . ϕ is differentiable on (α k , ∞) and ϕ (x) = - ab + 2(p + k -1)(p + 2k -3) -2a(p + k -1)x + a 2 x 2 2τ 2 k-2 (x) = - ab + (p + k -1) 2 + 2(p + k -1)(k -2) + (p + k -1 -ax) 2 2τ 2 k-2 (x)
. -On 0, p+2k+1 a , τ k is such that τ k (t) τ k (0) ≥ 1 , ∀0 ≤ t ≤ p+2k+1 a , since

For p ≤ -2k + 3 or p ≥ -k + 1, ϕ (x) ≤ 0 and ϕ(x) ≤ ϕ(α k ) for all x > α k . For -2k + 3 < p < -k + 1, if (k -2) 2 -(p + 2k -3) 2 -ab > 0, then the only possible positive zero of x → ab + 2(p + k -1)(p + 2k -3) -2a(p + k -1)x + a 2 x 2 is p + k -1 + (k -2) 2 -(p + 2k -3) 2 -ab a < α k .
|h k (x)| ≤ h (k) + k|p + k -ax| f k-1 h + a 3 + (k -3)(k + 2) 2 f k-2 h ≤ h (k) + k max (|p + k|, k + 1) f k-1 h + a 3 + (k -3)(k + 2) 2 f k-2 h ≤ h (k) + k max (|p + k|, k + 1) M k-1 + a 3 + (k -3)(k + 2) 2 M k-2 ,
we have

|f (k) h (x)| ≤ h k 1 s(x)g k (x) x 0 g k (t)dt ≤ h k 1 τ k (0)s(x)g k (x) x 0 τ k (t)g k (t)dt = 1 τ k (0) h k = 2 b h k .
As a consequence,

f (k) h ≤ 2 b h (k) + k max (|p + k|, k + 1) M k-1 + a 3 + (k -3)(k + 2) 2 M k-2 .
-On p+2k+1 a , ∞ , similar arguments as in the previous cases lead to the result.

Remark : These results can be used to estimate the rate of convergence via Stein's approach of:

• a continuous fraction with entries gammas to the generalized inverse Gaussian or Kummer distributions;

• the total resistance of an infinite tree, whose edges are equipped with independent resistances that are inverse Gaussian or reciprocal inverse Gaussian distributions to the reciprocal inverse Gaussian distribution (the inverse Gaussian and reciprocal inverse Gaussian distributions are respectively the GIG distribution with p = -1/2 and p = 1/2);

• the generalized inverse Gaussian distribution to the inverse gamma distribution.

-1 2 (

 2 ax+b/x) , x > 0 where K p is the modified Bessel function of the third kind. For a > 0, b ∈ R, c > 0, the Kummer distribution K(a,b,c) with parameters a, b, c has density d a,b,c (x) = 1 Γ(a)ψ(a, a -b + 1; c)

  a, b) by the Stein characterization of GIG(p + 6, a, b) distribution, equation (18) is the Stein equation for GIG(p + 6, a, b) distribution. We have

+∞ x g 3

 3 (t)h 3 (t)dt.

solves

  Stein equation for GIG(p + 2k, a, b) distribution with second member h k given by (22). Let g k be the density of the GIG(p + 2k, a, b) distribution. Then

2 M k- 3 .

 23 Hence for all x > α k and for all p ∈ R,ϕ(x) ≤ ϕ(α k ) = (k + 2)a + a (p + 2k + 1) 2 + ab 4p + 8k + 4 + 4 (p + 2k + 1) 2 + ab .As a consequence for x > α k ,h k-1 ≤ h (k-1) + a(k -1) k + 2 + (p + 2k + 1) 2 + ab 4p + 8k + 4 + 4 (p + 2k + 1) 2 + ab h k-2 + a 3 + (k -4)(k + 1)Now, assume p > -2k -1.

Acknowledgement: The first author acknowledges financial support from the French Embassy in Togo and additional support by the Institut Elie Cartan de Lorraine in the framework of his PhD programme.


Tim P Pusch

Lennart Karstensen

Tobias Behr

Towards using SOFA to train a controller for neurovascular catheter interventions

Introduction

Neurovascular catheters are thin flexible tubes used for the treatment of strokes and other vascular diseases. During the procedure, the catheter instrument is navigated through the blood vessels of the patient to a target site in the brain where the surgeon performs the intervention (e.g., a mechanical thrombectomy) by feeding instruments or drugs through the catheter. Navigating catheters through tortuous blood vessels to the brain is challenging due to their small size and flexibility, which often lead to unexpected and unintuitive catheter motions. Thus, a number of researches have proposed robotic assistance to aid the surgeon in the control of such catheters [START_REF] Rafii-Tari | Current and emerging robot-assisted endovascular catheterization technologies: A review[END_REF].

In previous work, we proposed a closed-loop control system for the navigation of catheter instruments based on Deep Reinforcement Learning (DRL) [START_REF] Behr | Deep Reinforcement Learning for the Navigation of Neurovascular Catheters[END_REF]. The system uses the SOFA framework -specifically, the BeamAdapter (BA) plugin [START_REF] Duriez | New approaches to catheter navigation for interventional radiology simulation[END_REF] -to generate training data for a neural network-based controller. By means of simulations, we showed that this approach is feasible for a simplified vessel phantom.

Here, we present our ongoing work on the development of said control system. We show that a DRL-based controller trained only in a simulated environment (SOFA + BA plugin) can successfully be transferred to a physical test bench.

Methods and Materials

Our setup consists of the controller (DRL-Agent), a simulated environment created in SOFA and a physical test bench (Fig. 1). The simulation in SOFA is used to generate training data for our DRL-Agent. Once training has been finished, we transfer the controller to the test bench for evaluation.

For training the controller, we use the Deep Reinforcement Learning framework 'Nervana Systems Coach'. The Deep Deterministic Policy Gradient (DDPG) algorithm along with Hindsight Experience Replay (HER) are used to train the artificial neural networks of our controller. Catheter instrumentshere, only the guidewire -are simulated in SOFA with the help of the BeamAdapter plugin. Our SOFA scene consists of a rigid phantom with evacuated lumen and a single beam instrument with a curved tip (i.e., the guidewire) that is moved through the lumen. The geometric and material parameters of the simulated instrument were set to loosely resemble its physical counter part. The test bench consists of a custom-made robotic manipulator (two degrees of freedom), a 3D-printed vessel phantom and a camera for tracking the position of the guidewire in the vessel phantom. The robotic manipulator is used to translate and rotate the guidewire according to the controller commands it receives. For real-time tracking of the guidewire in the phantom, we use MeVisLab (MeVis Medical Solutions AG, Bremen). The communication framework OpenIGTLink [START_REF] Tokuda | OpenIGTLink: an open network protocol for image-guided therapy environment[END_REF] is used to exchange information between the DRL framework and the simulation and test bench, respectively.

Our training routine consists of the DRL-Agent sending motion commands ('action', i.e., velocity, direction, travel distance) to the SOFA simulation, which computes the resulting pose of the instrument. The coordinates of the instrument tip ('state') are then sent back to the DRL-Agent. The goal of the training is for the guidewire tip to reach a specified target in the vessel phantom. During the training, this target is moved gradually to areas in the phantom that are intuitively harder to reach, for example, further away from the starting position. Thereby, the difficulty of the task is increased during the training.

In our first experiments, we consider the controller to be sufficiently trained once no significant improvement in its performance can be observed. The trained controller is then transferred to the test bench where it is tasked to steer the guidewire to randomly chosen points in the phantom. No additional training on the test bench is performed.

Results and Discussion

Our first qualitative results show that our DRL-based controller yields a success rate of about 90-95 % in both the simulated and the physical environment. Although so far only qualitative results have been obtained, transfer from simulated to physical environment did not require further training of the controller. However, when changing the 3D-printed phantom we noticed a drop in the success rate. This is likely due to differences in the surface roughness between the two vessel phantoms caused by manufacturing tolerances. We were able to reproduce our initial results by adjusting material parameters in the SOFA simulation and then repeating parts of the training procedure.

Training the controller in the simulated environment took approximately one day on a computer with an AMD Ryzen 7 2700X, GPU GeForce RTX 2070 and 48 GB RAM.

Simulating the guidewire using SOFA together with the BeamAdapter plugin proved sufficiently accurate for our use case. Apart from a few minor adjustments to the BeamAdapter code, the software could be used 'out of the box' for our purposes.

Conclusion

In this ongoing work, we present first promising results of our DRL-based catheter controller trained using SOFA. We are able to demonstrate that a controller trained only in SOFA with the BeamAdapter plugin can reliably navigate a guidewire through a simple 3D-printed vessel phantom.

In the future, we plan to expand on this work by successively increasing the complexity of our simulation and test bench with the goal of approximating the real world application better.

Fig. 1 .

 1 Fig. 1. System overview showing the interface between the DRL framework, the SOFA simulation and the physical test bench. Training is performed only in the simulated environment.