Alexandre Voisin

Eric Levrat

Benoît Pierre Cocheteux

GENERIC PROGNOSIS MODEL FOR PROACTIVE MAINTENANCE DECISION SUPPORT: APPLICATION TO PRE-INDUSTRIAL E-MAINTENANCE TEST BED

Keywords: Maintenance, e-maintenance, prognosis, e-maintenance platform, MIMOSA

Generic prognosis model for proactive maintenance

decision support: application to pre-industrial e-maintenance test bed

Introduction

Even if maintenance is a necessity, maintenance has a negative image and suffers from a deficiency of understanding and respect. It is usually recognised as a cost, a necessary evil, not as a contributor. Most people think that the role of maintenance is "to fix things when they break" but when things break down maintenance has failed [START_REF] Dale | Reliability as a Strategic Initiative: To Improve Manufacturing Capacity[END_REF]. Moreover traditionally the scope of maintenance activities has been limited to the production vs. operation phase. But as the paradigm of manufacturing shift towards realizing a sustainable society, the role of maintenance has to change to take into account a lifecycle management oriented approach [START_REF] Takata | Maintenance: Changing Role in Life Cycle Management[END_REF] for enhancing the eco-efficiency of the product life [START_REF] Desimone | with The WBCSD[END_REF]. In that way, maintenance has to be considered not only in production vs. operation phase but also in product design, product disassembly, and product recycling … [START_REF] Van Houten | Product modelling for model-based maintenance[END_REF]. Thus the product can now play a major role in maintenance mainly when the product is "active" (i.e. Intelligent Product; Holon …) meaning able to support a part of its knowledge. The concept of "life cycle maintenance" [START_REF] Takata | Maintenance: Changing Role in Life Cycle Management[END_REF] emerged to stress this new role leading to develop a general maintenance value chain for which the engineering way is consistent with the system engineering one (INCOSE1 initiative). The value chain must be supported at each phase of the product life cycle, leading for each phase, to assign a (sub) objective to be fulfilled. Each objective and sub-objective is supported by Business Processes which transform input flows into output flows materialised by resources, information, energies… If all the sub-objectives (local objectives) are carried out, the global value chain is running well. This system view highlights that a not controlled deterioration of one of the product characteristics in a life cycle phase P can have important impact on the use of the product or on its expected service in the life cycle phase P+1 (after P). Consequently, the conventional strategies [START_REF] Wang | A survey of maintenance policies of deteriorating systems[END_REF] such as scheduled preventive maintenance strategies carried out some time too late in relation to the current status of the potential failure are not easily compatible with this maintenance vision. A failure can lead to a bad product or a bad service delivered by the product. Therefore degradation-based anticipation (the pro-activity in maintenance) becomes essential, at this stage, to avoid failing situation with negative impact on product condition (zero breakdown maintenance).

Pro-activity in maintenance makes emerging the E-maintenance philosophy [START_REF] Iung | Special issue on emaintenance[END_REF] to support the moving from "fail and fix" maintenance practices to "predict and prevent" strategies [START_REF] Lee | Intelligent prognostics tools and e-maintenance, Computers in Industry[END_REF]… while keeping Maintenance as an Enterprise process (holistic approach) -Integration concept (i.e. IEC/ISO 62264) for optimising performances. E-maintenance [START_REF] Muller | On the concept of e-maintenance: Review and current research[END_REF] is integrating the principles already implemented by Tele-maintenance which are added to the web-services and collaboration vs. synchronisation principles [START_REF] Iung | From remote maintenance to MAS-Based Emaintenance of an industrial process[END_REF] to support proactivity. Collaboration vs. synchronisation allows not only to share and exchange data and information but also knowledge and (e)-intelligence and this between all the actors (human, units, department…) all along the product life cycle. It leads to develop intelligent and cooperative maintenance architecture implementing all the Business Processes required within the new maintenance value chain.

The Business Processes the most important to develop anticipation action are monitoring, diagnosis, prognosis and decision-making processes. Among these modules, the prognosis process is often considered as the Achilles heel [START_REF] Wang | Fault Prognosis using dynamic wavelet neural networks[END_REF] while its goal is fundamental for implementing anticipation capabilities. « We simply do not know how to measure the performance degradation of components and machines; we lack the validated predictive models and tools that tell us what would happen when the process parameters decrease from values» [START_REF] Lee | Teleservice engineering in manufacturing: challenges and opportunities[END_REF].

Indeed, prognosis has to analyse the impact of degradation on the component itself and on the other items of the production system to predict future system failures and investigate (future maintenance) actions [START_REF] Jardine | A review on machinery diagnostics and prognostics implementing condition-based maintenance[END_REF]. Nevertheless most of the existing prognosis methods are component-oriented without really taking into account the system performances [START_REF] Provan | Prognosis and Condition-Based Monitoring : an open systems architecture[END_REF]. In addition, [START_REF] Lee | Intelligent prognostics tools and e-maintenance, Computers in Industry[END_REF] underlines that:

"In spite of the progress made, many fundamental Issues still remain:

• Most of the developed prognostics approaches are application or equipment specific. A generic and scalable prognostic methodology or toolbox does not exist. • Currently, methods are generally focused on solving the failure prediction problem. Tools for system performance assessment and degradation prediction have not been well addressed."

Indeed only few research work cover both these two issues. For example, (Muller, et al., 2008a) proposes a first step to move from prediction vision to prognosis one but without really generalising the process. Thus, the work presented in this paper concerns more a global formalization of the generic prognosis business process (prognosis of the failure) which can be used after, from an "instantiation procedure", to develop specific prognosis processes related to particular components or systems.

Then the specific prognosis processes should be implemented on intelligent vs. collaborative maintenance architecture to be integrated with the other maintenance business processes required to develop pro-activity.

In that way, the rest of the paper is organised, consistently with the e-maintenance framework (based on Zachman2 framework) proposed by [START_REF] Levrat | E-maintenance: Overview and Conceptual Framework[END_REF], as follows:

• section 2 and 3 gives a short overview on the "prognosis" business process definition and the related work already performed. • section 4 (business process section) presents the formalization of the generic "prognosis" business process. This formalization is based both on the process approach (processing view) and the objects vs. flows that are processed (data view). • section 5 (organisation section) describes the use of the generic prognosis process for developing the prognosis dedicated to the DYNAMITE3 E-maintenance platform. For example, a part of the prognosis in DYNAMITE has to be supported by Web Services.

• Section 6 (IT infrastructure section) gives a first description of the e-maintenance infrastructure for running well the specific prognosis process (activities and procedures) previously described (lab experimentation based on DYNAMITE context).

Finally, conclusions and prospective are developed in section 7.

Prognosis definition

The word 'prognosis' comes from the Greek 'progignôskein' which means "to know beforehand". This process associated with proactive maintenance has to predict the future state of a system or a component. Several definitions have been proposed to specify its aims.

For [START_REF] Byington | Prognostic Enhancements to Diagnostic Systems for Improved Condition-Based Maintenance[END_REF], "Prognosis is the ability to predict the future condition of a machine based on the current diagnostic state of the machinery and its available operating and failure history data." This definition specifies the needs for the prognosis to integrate its informational environment in order to obtain necessary data (current or past data). For [START_REF] Lebold | Open Standards for Condition-Based Maintenance and prognostic systems[END_REF] and [START_REF] Farrar | Damage prognosis: the future of structural health monitoring[END_REF], the prognosis has to project the current health of equipment into the future taking into account estimates of future usage. The future usage define a scenario, composed of the future evolution of data which influence the degradation/failure, maintenance actions… To anticipate 'at best' and to keep all benefits of the proactive maintenance, the prognosis of the future health of the system has to integrate the maximum of available information to be as exact as possible with respect to reality. That allows integrating future real events and not only statistical data. Thus one of the principal aspects of prognosis is the consideration at the prognosis time of all future possible scenarios about system load and interaction system/environment. For an industrial system, scenarios are principally composed of:

• the manufacturing schedule to obtain the future evolution of system loads (functioning mode),

• the environmental conditions and the information about the future evolution of the factors from the environment which influence the degradation/failure, • the maintenance schedule because maintenance actions influence the health of the system.

Since 2004, the standard ISO 13381-1 gives a "textual" framework of the prognosis. Prognosis is defined as: "estimation of time to failure and risk for one or more existing and future failure modes." This definition doesn't consider the needs to include a previous scenario, but it adds two important complementary aspects. First the standard specifies that the output of the prognosis process is composed of several remaining useful lives (RUL). Second, the ISO definition uses the notion of current failure mode and potential failure mode (its occurrence is in the future). These definitions are for the authors incomplete. Therefore the prognosis process will be considered in this article as the process which has to:

• predict the future health of the system,

• generate the different RULs (of the system, part of the system or component) for each detected (current) or potential degradation/failure mode, by taking into account the knowledge of the system (functional and dysfunctional), past information (background), current information (current state) and future information (scenario with manufacturing and maintenance data).

Related work

Prognosis is a process issued from aims and principle of proactivity firstly in the Condition Based Maintenance (CBM) and now in E-maintenance. In such visions, prognosis is in interaction-collaboration with some others business processes (condition monitoring, diagnosis, decision support…) by exchanging-sharing informationknowledge.

In OSA-CBM project, (Open Systems Architecture for Condition-Based Maintenance) [START_REF] Lebold | Open Standards for Condition-Based Maintenance and prognostic systems[END_REF], an architecture was defined (see Fig. 1). The structure is composed of seven layers which constitute a linear succession of sub-processes, ranging from the data acquisition to the Decision Support. The prognostics layer is located between the health assessment layer and the decision support layer. The health assessment layer includes the process which allows to define the current degradation/failure level (system or component), and the diagnosis process which gives the current degradation/failure mode(s). The decision support layer assists the decision making in order to choose a relevant maintenance action to restore the system in a previous state. The principal data exchanged between the different layers are identified and modeled within MIMOSA framework.

This OSA-CBM model identifies five informational flow classes like inputs of the prognosis layer (see Fig. 2):

• knowledge about the past functioning (operational usage/load, history and projection, historical failure rates, maintenance history, CM (Condition Monitor), HA (Health Assessment) and PR (Prognostics) history), • system information (application knowledge),

• future operational conditions (configuration set-up),

• current health (HA,CM, SP (Signal Processing) and SM (Sensor Module) input), • prognosis control (PL control vector). This model also defines the output classes of the prognosis layer:

• the prognosis result could take different forms: RUL, RUL distribution, future performance of the system (PLData), • the prognosis result tags allow to tag a prognosis result with its control data (PL control vector) and with the used algorithm and prognosis interval definition (PL Configuration), • explanation on the prognosis result (PL Explanation),

• update of the history (PR history).

In a more formalized way between the business processes [START_REF] Léger | Integration of maintenance in the enterprise: towards an enterprise modeling-based framework compliant with proactive maintenance strategy[END_REF] proposed an architecture for an integrated system of proactive maintenance (SIMP). This one is composed of three processes: the condition monitoring and diagnosis process, the prognosis process and the Decision Support process (DS). One of the particularities of a SIMP is to propose a loop between prognosis and DS. In this context the prognosis allows evaluating different maintenance or manufacturing options and allows DS to make the 'best' choice to prevent a failure. (Muller et al., 2008a) globally define the nature of exchanged data between the SIMP and the different processes of the environment (Fig. 3). In summary, the papers about the prognosis process modeling are very general and describe the links between input and output flows but don't propose an efficient formalization. Moreover the generic sub processes needed for the prognosis have not been fully identified. To face with this issue, our work aim at proposing a formalization of a generic prognosis process in the framework of an integrated system of proactive maintenance. This formalization based on a process approach allows making the exchanged data between prognosis and its environment explicit. It also allows to define generic sub processes and to define the data exchange between those sub processes. Thus the originality of our contribution is to lead to a generic model of prognosis (generic component) which can be instantiated to a lot of applications for developing specific and consistent prognosis (particular component based on generic one).

Formalization of the generic prognosis business process

The formalization of the prognosis business process is based on the concepts developed (a) in the process approach (process identification, process decomposition, identification of the flows produced/consumed by the process…) and (b) in the modeling of the objects/data linked to these flows. The process approach is supported by the MEGA Suite4 and the Object modeling is supported by UML classes and sequence diagrams.

The formalization is developed on five steps as described in this section:

• formalization of the prognosis process environment, which allows identifying the external partnerships of the prognosis and the external flows exchanged. The main output flow of the prognosis is the Remaining Useful Life (RUL), • formalization of the final purpose of the prognosis process which consists in defining the rules to calculate the RUL in a general way, • formalization of the functional decomposition of the prognosis process (sub processes identification), • formalization of the coordination of the sub processes needed to fulfill the prognosis mission (sequence view), • formalization of all the objects/data materializing the flows (internal and external to the process).

Formalization of prognosis process environment

The identification of the prognosis environment is based on an improvement of the results of (Muller et al., 2008a). Thus Fig. 4 is an extension of the SIMP concept Fig. 3, where is added several interactions with others enterprise processes. This extension leads to the modeling of the following business processes: "To acquire and to process signal", "To manage operations", "To manage maintenance", "To manage company". These business processes are representative of the operational, tactical and strategical levels. The interactions between the external business processes and the prognosis process could be summarized as follows:

Main output flow of the prognosis process:

• RUL + confidence level + future performance system.

Main input flows of the prognosis process:

• Fault localisation + Degradation/Failure mode (O1P2): (to be referred to the Fig. 6,7,8,9,10,11). This input informs about the current degradation/failure mode for the both level (component and functional). This information allows choosing the more pertinent degradation/failure model. Moreover, when a new degradation/failure mode occurs, a new prognosis is performed (the reference instant t=0 change).

•

Data about system and environment (O1P1-3). These data represent the current health (or degradation/failure level) of the system and of its components. This data are segmented and filtered by "To acquire and to process signal" in order to contain only the representative parts (e.g. data from a break are eliminated).

•

Company strategies (priority, costs, security, prognosis horizon, etc) (O1P7). This input specifies the prognosis horizon of the temporal window [t 0 ,T]. The horizon depends of the company's strategic choices. If a stock of spare parts exists or if a part of maintenance is contracting out, the horizon for the decision making and the prognosis is decreasing.

•

Historical data (O1DB). This input contains the historical knowledge of the system and allows obtaining models of degradation/failure mode.

•

Prognosis knowledge (I1P3). It is used during the engineering and development phase. This knowledge allows obtaining the degradation/failure mode from the historical data by identifying the parameters of the component degradation/failure mode and by modelling the impact of the component degradation/failure modes on the system degradation/failure mode, the impact of the maintenance actions, the impact of the operational conditions… Some of other input flows are used for developing the scenario of operation condition and maintenance which has to be known on the prognosis horizon [t 0 ,T]. These additional inputs are:

• Manufacturing schedule + environment on [t 0 ,T](O1P5).
This input is composed of the production loads and their durations. In addition the environmental conditions during [t 0 ,T] are able to influence the degradation/failure (e.g. the outside temperature, hygrometry, systems which are located near the system and which create vibrations or electromagnetic vibes, etc).

•

Maintenance schedule on [t 0 ,T](O1P6). This input is composed of the list of the maintenance actions which are initially planned (systematic maintenance). The interest of these data comes from the impact of these maintenance actions on the health.

•

Prognosis request (O1P4-3a) + option of scenario (O1P4-3b). This input performs the loop between the prognosis process and the DS process. It allows DS process to evaluate several options of maintenance actions or of manufacturing scheduling. So to prevent the failure, the scenario used by "To prognosticate" is composed from an original scenario with the initial manufacturing and maintenance schedules and from one option of scenario. Then the DS process proposes several options of scenario that the prognosis tests successively. The prognosis generates for each option a new list of RUL and then DS process can compare these options.

Formalization of the prognosis final purpose

As previously defined and according to the standard ISO 13381-1 the prognosis purpose is to generate the remaining useful life (RUL) of the system and of each component. The getting of RUL is carried out upon a temporal window [t 0 ,T] (t 0 is the reference instant of prognosis and T is the final time of the prognosis which corresponds to T=t c +∆T with t c the current time and ∆T the prognosis horizon). A RUL is defined thanks to an instant in the interval [t 0 ,T] for which the projected value reaches a threshold. Two classes of RUL have to be defined: a 'component-RUL' that corresponds to a failure of the component, and the 'functional-RUL' that corresponds to a loss of a function of the system (final purpose). Therefore there are two kinds of thresholds: the component threshold and the functional threshold. For a component, the values projected by the prognosis process are the degradation/failure mode levels and the performances. The component threshold corresponds to the shut-down of the component (ISO 13381-1). On the other hand, the functional threshold corresponds to the performance limit for which the final purpose of the function is not guaranteed (critical threshold). Usually, this threshold is defined by the system's requirements and functional specifications. Fig. 5

{ } { } { } MDm Cj MDl Ci MDk Ci RUL RUL RUL ∪ ∪ (1)
with:

• RUL CMD is the RUL of the component C with the degradation/failure mode MD,

• C i ∈Ă, Ă is the set of already degraded components which are detected by the diagnosis process,

• C j ∈Ŏ, Ŏ is the set of components, which are not degraded yet and therefore not detected by the diagnosis process, but for which prognostic predicts one or more apparition(s) of potential mode(s), • k∈{1,…,K i }, K i is the number of degradation/failure modes which are proved upon the component i, • l∈{1,…,L i }, L i is the number of degradation/failure modes which are susceptible to appear on the component i,

• m∈{1,…,M j }, M j
is the number of degradation/failure modes which are susceptible to appear on the component j.

Moreover two kinds of components are distinguished: the components which can't be repaired, i.e. those components are replaced after failure, and the components which are repairable. In the former case, only one component-RUL is required and is given by the RULs of each degradation/failure mode of the component:

{ } { } ( ) { } ( )      ∈ ∈ ∪ = O C if RUL g A C if RUL RUL g RUL i l MD i C i m MD j C k MD i C i C ( ( (2)
where g is a function to aggregate the RULs. In the case of the form of RULs is a date or a number the function g is the function minimum because when the component has failed all degradation/failure modes stop their progression. In the latter case, the prognosis process has to give a list of RULs, where each RUL is a component-RUL for one degradation/failure mode (proved or potential) for one component. The list is composed of every RUL for each proved or potential degradation/failure mode. The knowledge of the RUL of each degradation/failure mode allows the DS process to propose maintenance actions to group together potential interventions in the global framework of opportunistic maintenance [START_REF] Thomas | Odds algorithm'-based opportunity-triggered preventive maintenance with production policy[END_REF]. For a given function of the system, the final purpose is characterised by demands on the product and its flows as well as demands on the manufacturing system. The demands on the product and its flows concern parameters of the production like the product quality, rating, etc. The demands on the manufacturing system concern the efficiency of means with respect to the result. For example, a system with a 'feedback loop' is susceptible to derive without visible effects upon its output. The final purpose of a function are affected either by an external cause (deviation of a input flow) or by an internal cause (degradation of a sub-function). Thus the degradation modes impact directly on the decrease of the performance of the functions and a RUL of the function Fi with the degradation/failure mode MD can be defined by: ) , , (

MDj Fik MDj Fi MDj Fi Perf Perf CV f RUL = (3)
with:

• RUL FMD is the RUL of the function F with the degradation/failure mode MD, • CV are the covariate which come from input flows, • Perf Fi MD j is the performance ok the function F i with the degradation/failure mode MD j , • Perf Fik MD j is the performance of the sub-function F ik with the degradation/failure mode MD j .

From these RUL a functional-RUL, affected by n degradation modes, can be obtained with a aggregation function:

] , 1 [ ) ( n j RUL f RUL MDj Fi Fi ∈ = . ( 4
)
where f is a function to aggregate the RULs. In the case of the form of RULs is a date or a number the function f is the function minimum.

Moreover for each RUL (component-or functional-RUL) a confidence level has to be associated (ISO 13381-1) and is defined as follow: "A confidence level is a figure of merit (percentage) that indicates the degree of certainty that the diagnosis/prognosis is correct. This value is essentially a figure representing the cumulative effect of error sources on the final certainty or confidence in the accuracy of the outcome. Such a figure can be determined algorithmically or via a weighted assessment system."

This definition looks restrictive because the confidence level is characterized by a confidence interval of level α or by a density function of the variable RUL.

In the specific output flow of "To prognosticate" which is named "RUL + Confidence level+Futures performances system (O1P3)", we need to complete the standard proposal by adding the system performances (object) to this result flow. The knowledge only composed of RULs do not allow the DS process to use a specific criterion upon the performance in addition to criteria related to costs, security... This criterion has to be assessed for all the options of the future scenario in order to classify them.

Formalization of the functional decomposition of the prognosis

For achieving the RUL objective in relation to its input flows the prognosis needs four sub processes (Fig. 6). The sub process "To pilot prognosis" coordinates the three others, ("To initialize state and performances", "To project" and "To compute RUL") which constitute the sequential stages to obtain a prognosis result.

"To pilot prognosis"

In addition to its coordination activity, the sub process provides the appropriated models in relation to the current situation, i.e. the proven degradation/failure modes. This process has two types of output flows:

• Request for action to be sent to the three others sub processes (initialisation (O2P1-2a), projection (O2P1-3a) and calculation requests (O2P1-4a)), • Appropriated models to be provided to the three others sub processes (functional and dysfunctional (O2P1-2b), projection (O2P1-3b), threshold models (O2P1-4b)).

… and several input flows :

• Fault localization + Degradation/Failure mode (O1P2), • Prognosis request (O1P4-3a). This input allows to this sub process to trigger only the two other sub processes ("To project" and "To compute RUL") when the DS process send a request (of prognosis) for a new option of scenario,

• Prognosis knowledge (I1P3).

"To initialize state and performances"

The purpose of this sub process is to define an initial situation for the projection. It updates from the last real data of the physical system, the image of the health of the system and the degradation/failure level. This image represents a synchronic view of both the system and its components. This process has one output flow composed of:

• the current health of components which is represented by the level of each proven degradation/failure mode, • the components or system performances given by indicators, which allow quantifying the activity of the system or the component in relation to it final purpose.

… and three input flows:

• Data about system and environment (O1P1-3),

• Initialisation request (O2P1-2a): it is generated by "To pilot prognosis",

• Functional and dysfunctional models of the system (O2P1-2b): these models are generated from a functional analysis based on a process approach and a dysfunctional analysis: FMECA associated with HAZOP (Muller et al., 2008a). The behaviour of a process is described by causal relations who joined inputs, physical support and outputs.

"To project"

This sub process determines the future evolution of the degradation process and the component or system performances (projection in the time, diachronic view of the system). The projection integrates the influence of the different data (operational condition, maintenance actions…) from the scenario. This process has one output flow composed of:

• Future level of degradation/failure • Uncertainty level (confidence degree) on the result … and the following inputs:

• Projection request(O2P1-3a): comes from "To pilot prognosis", • Current level of degradation/failure + Performances (O2P2) , • Projection models (O2P1-3b). The projection models have to respect three key points of the projection sub process. The first one concerns the modelling of the impact of the scenario. The second one is the projection of the future health and performances with the impact of the scenario from the first point. The last key point is the evaluation of the capacity of the potential degradation/failure mode to occur.

• Manufacturing schedule + environment on [t 0 ,T] (O1P5), • Maintenance schedule on [t 0 ,T] (O1P6),

• Option of scenario (O1P4-3b).

• Company strategies

"To compute RUL"

This last sub process computes the RULs which are the final purpose of the prognosis. For providing the RUL it requires three input flows:

• Calculation request (O2P1-4a),

• Future level of degradation/failure + uncertainty (O2P3): this input is the result of the projection sub process , • Threshold models (O2P1-4b). The thresholds are models because they integrate the uncertainty linked, for example, the variability between intercomponents. They can be represented under the form of distribution.

From the flows previously identified and mainly the event flow (requests) which allow to trigger each of the sub processes, the four processes have to be executed in a sequential and consistent way in order to achieve the prognosis objectives (RUL providing). This sequential way can be described in form of a sequence view and then modelled by a sequence diagram.

Sequence view of the Prognosis

The sequence view proposed here-after describes the three different stages needed for the prognosis but only in a textual way (a particularization of the formalized sequence diagram can be found Fig. 12). The prognosis is activated by means of the request corresponding to (a) an external demand or (b) an internal demand (SIMP) coming from the DS. (see Fig. 4).

According to the prognosis approach [START_REF] Byington | Prognostic Enhancements to Diagnostic Systems for Improved Condition-Based Maintenance[END_REF], t 0 is defined, e.g. in experience-based approach, t 0 is the last As Good As New maintenance action. It usually depends on maintenance action performed on the asset.

Hence the first step is to request the "To manage maintenance" process for knowing the past maintenance actions performed on the asset and determine t 0 . Then "To manage company" process is requested to assess the prognosis horizon ∆T which depends on company, operational and maintenance strategies. Knowing the degraded asset and the degradation mode, the "To manage maintenance" process is requested to provide the associated degradation model composed of equations, influence variables, degradation modes induced with their causality and occurrence probability levels. Nevertheless the degradation model is slightly different depending on the selected prognosis approach (data, reliability model…).

From the degradation model it is necessary now to define the prognosis scenario to be used for projection. The scenario is mandatory to obtain a realistic prognosis. In that way "To manage production" and "To manage maintenance" processes are requested for knowing the operational and environmental conditions but also the maintenance actions during the period [t 0 ,T]. All these data are then processed to build the prognosis scenario in a continuous time (for example, data associated to stopping times and maintenance times are deleted).

From the degradation model and the scenario, the projection step is launched. It consists in computing for each time from t 0 to T, a degradation (health) function, according to the impact of the operational, environmental and maintenance conditions on the degradation model. Uncertainty related both to modeling and forecasting are integrated in the degradation function (Fig. 5).

From the degradation function, and taking into account the limit threshold provided by "Threshold models" (O2P1-4b Fig. 6), the elementary RULs associated to each degradation mode are computed. These RULs are then aggregated to form the global RUL and to compute the uncertainty.

The well execution of the stages previously described is based on a well exchange (of flows) between the different processes: interoperability [START_REF] Panetto | Interoperability of enterprise software and applications: Workshops of the INTEROP-ESA International Conference EI2N, WSI, ISIDI and IEHENA[END_REF] between processes i.e. maintenance ontology. It implies to formalize, not only on the syntaxic point of view but also on the semantic one, the objects/data materializing the flows.

Formalization of the Prognosis objects and Data

This formalization is based on MIMOSA-OSA/CBM Standards (Machinery Information Management Open System Alliance) [START_REF] Lebold | Open Standards for Condition-Based Maintenance and prognostic systems[END_REF]) for which we propose some extensions in order to be consistent with our generic prognosis model. The formalization uses UML classes diagram representation, and leads to a unique model which is presented below in five parts to be more readable.

The first part deals with the objects of the system architecture on which the prognosis has to be deployed. The occurrences of the object "segment" materialize the functions (and sub functions) fulfilled by the system. The occurrences of the object "asset" materialize the sub systems and components which support the functions. The objects "segment" and "asset" are associated to the "Fault localization +degradation/failure mode" flow (O1P2 in Fig. 4) for identifying the function or the component degraded.

The second part defines the diagnostic knowledge which encompasses information of degradation mode. It links the current degradation modes detected by the diagnostic process to its localization in a segment or an asset. Thus the object "DegradationMode" belongs also to the "Fault localization +degradation/failure mode" flow.

The third part concerns Maintenance and Production knowledge and Decision Support scenarios. It is related to the "Manufacturing scheduling+environmental" flow and "Maintenance scheduled" flow. The objects of these flows are the bases for building the scenario object (this object has been added to MIMOSA proposal) for which is also considered the objects coming from the decision making feedback.

A scenario is divided in several work orders. A work order corresponds to an atomic time sequence where a single functioning mode operates, i.e. production stop, maintenance stop, functioning … A work order pertains to assets and segments. According to the present time, past work order and future scheduled work order can be separated. When an alternative scenario has to be evaluated for the Decision Support, it contains the same past works order and alternative future works order. A work order is composed of several variables, related to segment or asset, and linked to their data according to their type. A special type is condition monitoring data if such prognosis approach is implemented.

The fourth part (prognosis knowledge) underlines two main objects for computing the RUL. The "Degradation level" object is associated to the "Current level of degradation/failure" flow (O2P2 Fig. 6) and "Future level of degradation/failure" flow (O2P3 Fig. 6). The "Model" object stores the models information needed for the projection, i.e. input, output, parameters and equation.

The ModelInput class is linked to the Variables class of the work order or to the "Degradationlevel" class.

The models are specific to a unique segment or asset and a unique degradation/failure mode. A 3-ary association links asset/segment, degradation mode and model since prognosis deals only with couple of degradation modeasset/segment for which a model is known.

The last part identifies the prognosis output model which is directly issued from the OSA-CBM model. It has been completed with association of the "Itemprognosis" class to the asset/segment, degradation mode and scenario objects. Such links are compulsory since it allows Decision Support to handle its several alternative scenarios and opportunistic maintenance [START_REF] Iung | Odds Algorithm'based Opportunistic Maintenance Task Execution for Preserving Product Conditions[END_REF].

This phase of Object formalization ends the proposal of the generic prognosis process modeling (conceptual view). The next step now is to show how this generic model can be used for a specific organization. In a theoretical way this use corresponds to an instantiation procedure of the generic model for developing specific model dedicated to a selected organization (organization view).

This phase is still under development implying that the next section is focusing more on the guidelines needed for starting instantiation.

Prognosis model for Dynamite E----maintenance organization

Defining a maintenance organization consists of assigning business processes (and their decompositions into activities) to the actors that will perform these activities on a specific site. An actor represents a person or a machine participating in business processes of the enterprise. A site could be internal or external with the company and can prejudge a subcontracting of competences. The sequence of operations (procedure) carried out by the actors provides the final purpose expected. During the development of this sequence if some activities of a same business process are distributed among different actors, located on different sites, it will require to create new operations in order to re-establish the right behavior of each business process. These new operations must carry out communication, storage, collaboration, negotiation… In that way, for developing a specific prognosis business process on an organization it is required:

• To identify the actors, • To identify sites on which the actors operate, • To identify activities (sub processes) performed by each actor, • To create if necessary new operations dedicated to business processes distribution, • To model the sequence of operations (organizational workflow between the actors), needed to fulfill the prognosis as a whole.

The organization we selected to show the deployment of our generic prognosis model is those of DYNAMITE E-maintenance organization.

The IP DYNAMITE n° 017498 project aims at promoting a major change in the focus of condition based maintenance, taking full advantage of recent advanced information technologies related to hardware, software and semantic information modeling. Special attention is given to the identification of cost-effectiveness related to the upgraded CBM strategies, as well as to the inclusion of innovative technologies within CBM. The DYNAWeb is the ICT architecture concerning software web services and communication architecture that intends to provide support to the new maintenance concept, related mainly to the operational layer of the maintenance system. The experimentation sites in DYNAMITE are dedicated in automotive (FIAT, VOLVO) and machine tools sectors (GORATU)

In DYNAMITE E-maintenance organization the actors are distributed on three different layers:

• a maintenance decisional layer considered as STRATEGIC level. • a maintenance management layer considered as TACTICAL level. • a maintenance operational (local) layer considered as OPERATIONAL level.

For each of the three maintenance layers the following maintenance actors could be identified: Among all these actors, those concerned by the deployment of the prognosis process are:

• Web actor, outside the company,

• The CMMS actor, located off site in the business area, • The CMOpS actor, located on site but not closed the machine,

•

The MES/SCADA actor, located on site for developing the right production actions (in synchronization with maintenance), • The PDA actor, located on site and very closed the machines,

The identification of the activities supported by each actor and consequently the sequence required to develop the prognosis are directly generated from the generic (functional and sequence) views. For example, the sequence diagram of an implementation of the prognosis in the case of DYNAMITE is illustrated in Fig. 12 and fully based on the textual description done in section "Sequence view of the Prognosis". In this diagram, the specific DYNAMITE actors exchange flows (containing objects), which are occurrences of the generic flows and which are composed of occurrences of the generic objects (Fig. 6,7,8,9,10,11). As these actors are not located on the same sites, it is necessary for running well the sequence to add communication operations for supporting interoperation.

E----maintenance infrastructure for supporting Prognosis

The prognosis generic model has now to be implemented on a particular infrastructure (intelligent vs. collaborative maintenance architecture). This infrastructure consists of a set of specific components (hardware, software, hybrid ...) supporting each part of the sequence and materialising the information technology (IT) means required for running applications and for enabling communication between these applications according to their distribution on sites.

The IT infrastructure is thus defined by all the resources (applications, services, protocols…) necessary to the execution of all the operations identified at the organizational level. In that way, it is composed of one or several networks with servers, workstations, applications, databases but also (smart) sensors, PDA… It is also characterized by its operating principles (wireless infrastructure, highly fault-tolerant, secured…) and the concrete implementation of a technological interoperability.

Several e-maintenance infrastructures already exist such as ICAS-AME, WSDF, PROTEUS [START_REF] Bangemann | PROTEUS -An Integration Platform for Distributed Maintenance Systems[END_REF], QUESTRA, ENIGMA… Based on the concepts of our TELMA platform (on which is implemented most of the e-maintenance principles), we developed in our lab a first new IT infrastructure for supporting DYNAMITE prognosis. Its specificity lies in wireless communication and web services technologies (Fig. 13) . At the field level, the communication is based on a Zigbee area for the interactions between sensors and PDA (HP iPAQ 4700) or CMOpS (TESSNET software of Tekniker5 ). At the business level, the communication is based on a Wifi/Ethernet area for the interactions between PDA or CMOpS and CMMS, MES, both supported by a MIMOSA6 database, and web services. A Gateway makes the link between the Wifi area and the Zigbee area. In accordance with sequence diagram (Fig. 12) each activity supported by an actor led to develop software module. Moreover, for communicating the required objects between the actors, the software module is linked to a communication software (a protocol) interacting with the wireless technology. For example, the HP iPAQ 4700 is equipped with a Wifi card but also a Zigbee CF card (i.e. link with sensors) and a Zigbee CF card (link with Smart Tag).

According to DYNAMITE architecture, the prognosis service must be implemented as a web service that is located outside the enterprise. Such architecture is of interest since it allows to share prognosis knowledge for several equipments from several sites and to use the most up to date available services. But, there are 2 main drawbacks to such view. The first is that the data about the processes are confidential and must not be made available outside the enterprise. So the enterprise information system is protected using Firewall meaning that the web service cannot access directly intra-enterprise data and that specific contextualized data must not be sent outside the enterprise. And secondly, the huge amount of data to be send to the web service for a complete prognosis of a system is not well fitted to web service architecture. Under these hypothesises, the functional view of figure 6 has to be mapped to the DYNAMITE architecture. The prognosis service is split into 2 services: a distant web service and a local agent. The local agent will take in charge part of the functions to pilot prognostic (2P1) and to initialise state and performances (2P2) and the web service the remaining functions. In this sense, the web service serves as a basic prognosis function dedicated to prognosticate a simple asset/function RUL for a single failure/degradation mode. In this implementation, the management for prognosticate sub-function, function or asset from basic asset/component has to be performed by the local agent. Moreover, it will not be possible to consider interaction between concurrent failure/degradation modes.

The agent has to manage all the local "point of view" which defines the context such as asset/segment informations, models, data… maintenance actions, production and environmental conditions. The agent must extract this information from the MIMOSA database and structure it to call the web service. It must also keep a data structure to manage the prognosis of the whole system/sub system under consideration. This is performed in the MIMOSA database since in the DYNAMITE project, a unique scenario, i.e. the production/maintenance plan, is investigated and the prognosis output information can be stored in the MIMOSA database. The information send to the web service do not embed any link to specific process/asset and can be safely send to the web service. Its structure is derived from the generic UML class diagram and is presented figure 14.

The Web Service is invocated by the agent and compulsory asset/function data (Fig. 14) is provided by mean of a XML file since it represent high level information and not only input parameters of the web service. In the same way, the web service returns the result back to the client using a XML file. The web service is then composed of three stages: to parse out XML input data into usable format, to prognosticate and to parse in XML output data.

Conclusion

In consistence with the e-maintenance framework proposed by [START_REF] Levrat | E-maintenance: Overview and Conceptual Framework[END_REF], this paper develops the formalization of the generic "prognosis" business process. This formalization is based both on the process approach (processing view) and on the objects vs. flows that are processed (data view). In particular, initial OSA-CBM models are completed with specific prognosis classes presented in Fig. 7, 8, 9, 10, 11 and they are used to define the basis of a domain model or ontology of prognosis in a CBM system.

Then the generic process is used (by a first step of instantiation) for developing a specific prognosis dedicated to the DYNAMITE E-maintenance organisation. It gave a first description of the e-maintenance infrastructure for running well the specific prognosis process (lab experimentation based on DYNAMITE context). By using the required IT components, a first set of experimentation is currently in progress in order to assess the performances of the prognosis implemented trough the infrastructure, but also the flexibility of such IT architecture to support different levels of proactivity.

The main prospective is to end (a) the scientific work on instantiation phase but also (b) the experimentations to show the real added value (and the constraints) of this prognosis approach.

FIGURE'S CAPTION

Fig. 1 The architecture OSA-CBM [START_REF] Lebold | Open Standards for Condition-Based Maintenance and prognostic systems[END_REF]. Fig. 3 Integrated system of proactive maintenance [START_REF] Muller | Formalization of a new prognosis model for supporting proactive maintenance implementation on industrial system[END_REF] based on [START_REF] Léger | Integration of maintenance in the enterprise: towards an enterprise modeling-based framework compliant with proactive maintenance strategy[END_REF].

Prognosis process

Aided-decision making process

Selected maintenance plan

Operating knowledge

Fig. 3 Integrated system of proactive maintenance [START_REF] Muller | Formalization of a new prognosis model for supporting proactive maintenance implementation on industrial system[END_REF] based on [START_REF] Léger | Integration of maintenance in the enterprise: towards an enterprise modeling-based framework compliant with proactive maintenance strategy[END_REF].

Fig. 2

 2 Fig. 2 Input / output of the "Prognosis Layer" from OSA-CBM model.

Fig. 4

 4 Fig.4Environment of the process "To prognosticate".

Fig. 5

 5 Fig. 5 Component and functional thresholds.

Fig. 6

 6 Fig. 6 Sub processes of "To prognosticate".

Fig. 7

 7 Fig. 7 Process knowledge data.

Fig. 8

 8 Fig. 8 Diagnostic knowledge.

Fig. 9

 9 Fig. 9 Maintenance and production knowledge.

Fig. 10

 10 Fig. 10 Prognosis knowledge.

Fig. 11

 11 Fig. 11 Item prognosis.

Fig. 12

 12 Fig. 12 Sequence Diagram of the prognosis service.

Fig. 13 A

 13 Fig.13A first hardware and software lab infrastructure for supporting prognosis.

Fig. 14

 14 Fig. 14 Information sent to the prognosis Web Service.

Fig. 1

 1 Fig.1The architecture OSA-CBM[START_REF] Lebold | Open Standards for Condition-Based Maintenance and prognostic systems[END_REF]

Fig. 4 Fig. 5

 45 Fig. 4 Environment of the process "To prognosticate"

FigFig. 9

 9 Fig. 8 Diagnostic knowledge

Fig. 12

 12 Fig. 12 Sequence Diagram of the prognosis service

http://www.incose.org

http://www.zifa.com

European Integrated Project DYNAMITE 017498 (Dynamic Decisions in Maintenance)

http://www.mega.com

http://www.tekniker.es

http://www.mimosa.org

Acknowledgments

The research reported here is partially supported by the European Integrated Project DYNAMITE (Dynamite Decision in Maintenance). The authors are grateful to all the partners who contributed by constructive discussions to the emergence of this work.


Valerian Guelpa

Patrick Sandoz

Cédric Clevy

Nadine Piat

Miguel Asmad Vergara

Guillaume J Laurent

Valérian Guelpa

Cédric Clévy

Nadine Le Fort-Piat

2D Visual Micro-Position Measurement based on Intertwined Twin-Scale Patterns

Keywords: displacement sensing, subpixelic, direct phase computation, two-axes pattern, nanometric precision

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

Introduction

Position measurement raises specific issues in the microworld. Often the research of accuracy, range or overall dimensions needs compromises at this scale. Numerous sensors have the drawback to be in touch with the moving object (lot of capacitive or piezoelectric sensors for example) and are limited to one-direction measurement. Many sensors without contact, like interferometers or confocal sensors, are also limited to one direction. In this context, a measurement method by vision could be the solution of choice for 2D measurements.

A wide variety of visual methods exist and are able to detect sub-pixelic motions (typically with a resolution between 0.1 and 0.01 pixel). Some can be qualified as areabased methods: they work on the entire image without extracting particular features. It is also possible to use image features, for example with gradient-based motion estimation [START_REF] Lu | Micro-Vision-Based Displacement Measurement with High Accuracy[END_REF] or phase-correlation method like Huang et al. [START_REF] Huang | Displacement Measurement with Nanoscale Resolution using a Coded Micro-Mark and Digital Image Correlation[END_REF] who apply correlation to a coded but non-periodic structure and measure displacements with a resolution of 60 nm on a several millimeters range. Other methods use specific features in the image and measure their displacements, like Kim et al. who evaluate displacement of MEMS with the tracking of markers [START_REF] Kim | Design of MEMS Vision Tracking System Based on a Micro Fiducial Marker[END_REF], or Kokorian et al. who detect picometric displacements with a curve-fitting based method [START_REF] Kokorian | In-Plane Displacement Detection with Picometer Accuracy on a Conventional Microscope[END_REF].

Nevertheless these methods are limited in range by the size of the field of observation, that is why they only allow incremental measurements and hence a relative knowledge of the position. But another category of visual methods solves this problem: pattern-based methods have a range dependent on the used pattern, not on the field of observation.

With simple periodic patterns, it is possible to measure position accurately by feature-detection, as Clarck et al. [START_REF] Clark | A Vision-Based Measurement Algorithm for Micro/Nano Manipulation[END_REF], or by Fourier-like processing, as Yamahata et al. [START_REF] Yamahata | Subpixel Translation of MEMS Measured by Discrete Fourier Transform Analysis of CCD Images[END_REF], but in these cases the absolute range is limited to one period. The most common solution to improve the range is to combine a large encoding to the periodicity. Various codes are usable, as the Manchester code used by Masa et al. [START_REF] Masa | Nanometric Resolution Absolute Position Encoder[END_REF] or a pseudo-periodic binary pattern for 2D measurement of Galeano-Zea et al. [START_REF] Galeano Zea | Position-Referenced Microscopy for Live Cell Culture Monitoring[END_REF]. It is also possible to use moiré fringes to improve the range, as Ri et al. [START_REF] Ri | Accurate Full-Field Optical Displacement Measurement Technique using a Digital Camera and Repeated Patterns[END_REF] or Sugiura et al. [START_REF] Sugiura | On-Chip Method to Measure Mechanical Characteristics of a Single Cell by Using Moiré Fringe[END_REF].

Another solution, inherited from interferometry, consists in using two fringe sets with different periods [START_REF] Creath | Step Height Measurement using Two-Wavelength Phase-Shifting Interferometry[END_REF], [START_REF] Born | Principles of Optics: Electromagnetic Theory of Propagation, Interference and Diffraction of Light, 7th[END_REF]. The combination of those two fringe sets forms a larger pattern while being an accurate tool. This method was already applied to displacement measurement along one direction in a previous work [START_REF] Guelpa | Subpixelic Measurement of Large 1D Displacements: Principle, Processing Algorithms, Performances and Software[END_REF]. We also applied it to the measurement of a force through the use of a compliant structure [START_REF] Guelpa | Vision-Based Microforce Measurement With a Large Range-to-Resolution Ratio Using a Twin-Scale Pattern[END_REF]. Experiments showed a 5 nm error for a range of 168 µm and real-time processing.

This paper presents a novel visual method to measure displacements at nano-scale along two directions through the observation of a single pattern. This pattern results from the intertwining of periodic grids that requires a specific algorithm to be used. The method combines accuracy and range along two-axes with monocular vision using a regular microscope.

The next sections present the principle of the method, firstly for 1-D measurement and then along two axes in section 3. Section 4 describes the experimental validation of the method and the obtained performances. Section 5 concludes the paper.

Twin-scale measurement principle along one direction

We previously presented a displacement measurement method based on pattern [START_REF] Guelpa | Subpixelic Measurement of Large 1D Displacements: Principle, Processing Algorithms, Performances and Software[END_REF], [START_REF] Guelpa | Vision-Based Microforce Measurement With a Large Range-to-Resolution Ratio Using a Twin-Scale Pattern[END_REF]. Despite a high performance level, this method is limited to one-direction measurement. The aim of this paper is to move beyond by doing 2D measurements with a similar principle. This section introduces the 1D measurement principle prior to develop the 2D method.

Single-grid position measurement

The use of a periodic pattern to evaluate the position of an object is not a new concept. A method consists in using phase measurement to obtain an estimation of the position, modulo the period of the pattern.

Fundamentally, the method is based on the phase-todisplacement relationship:

F(f (x -δ)) = e -2πiδξ • F(f (x)) (1)
where F corresponds to the Fourier transformation, f (x) is the distribution of pixel intensity (or more generally the space function), x the initial spatial coordinate, δ the spatial displacement and ξ the reciprocal variable of x. In this way, the spatial displacement δ induces a phase shift Φ in the frequency domain equal to 2πδξ. This relationship applied to an image I of a periodic pattern allows to measure its displacement with high precision. Moreover, a short computation time could be obtained through the use of a single-frequency spectral computation applied to the spatial frequency of the pattern used, instead of a complete discrete Fourier transform. In Details are in [START_REF] Guelpa | Subpixelic Measurement of Large 1D Displacements: Principle, Processing Algorithms, Performances and Software[END_REF].

our case, it corresponds to calculate the phase Φ 1 as the argument of the dot product between the pixel intensity vector V (obtained after averaging of the image along the stripe direction) and a complex analysis vector Z notably composed of a Gaussian window:

Φ 1 = arg(V • Z) (2)
with :

Z(k) = e - k-Ñ /2 Ñ /4.5 2 • e - 2iπ(k-Ñ /2) L (3)
with k the pixel index, Ñ the image width in pixels and L the stripe period in pixels1 obtained with a Fourier transform method.

The target displacement is determined with:

δ = Φ 1 • L 2π + n L (4)
where n is an unknown integer and L the period in meters. Indeed the pattern periodicity involves a modulo L on the knowledge of its position by vision. Since a displacement equal to an entire number of periods does not change the image obtained.

Twin-scale principle

In order to improve the range of the method, we use a twin-scale pattern [START_REF] Guelpa | Subpixelic Measurement of Large 1D Displacements: Principle, Processing Algorithms, Performances and Software[END_REF]. A second grid, with a slightly different period, is joined to the first grid. The phase measurement Φ 2 of this grid provides another phase data that allows to remove phase ambiguities on the global pattern (see Fig. 1): The synthetic period of the pattern is given by:

Φ Λ = Φ 1 -Φ 2 (5)
L Λ = L 1 • L 2 | L 1 -L 2 | (6)
where L 1 and L 2 are the periods of the two strip sets. The accuracy of the method is also improved by a calculus trick: the position is not directly evaluated with the phase Φ Λ but with Φ 1 and the entire number m of periods L 1 in the period L Λ :

m ≈ L Λ Φ Λ -L 1 Φ 1 2πL 1 (7)
Finally the displacement is given by:

δ = Φ Λ L Λ 2π + p L Λ = L 1 2π Φ 1 + m L 1 + p L Λ ( 8
)
where p is an unknown integer (can be set to 0 if the range of displacement is below L Λ ). The new ambiguity is equal to L Λ rather than L 1 . The algorithm is summarized Fig. 2.

When the periods L 1 and L 2 of the patterns are known, the method provides a direct conversion from pixels to meters. This self-calibration does not depend on experimental parameters such as magnification or field of view. High-frequency noises are also filtered by the spectral filtering involved in the phase measurement. The method presents low sensitivity to sharpness or brightness variation, at least as long as the contrast is sufficient to extract spacial frequencies of interest. The determinant parameter to obtain accuracy is finally the resolution of the camera (see [START_REF] Guelpa | Subpixelic Measurement of Large 1D Displacements: Principle, Processing Algorithms, Performances and Software[END_REF]).

Table 1 gathers the performances of the method using a micro-machined pattern with twin-scales of periods 8 µm and 8.4 µm. The range-to-resolution ratio of the method is larger than 10 6 with a 8-bits camera. Its working principle and the evaluation of these performances are detailed in [START_REF] Guelpa | Subpixelic Measurement of Large 1D Displacements: Principle, Processing Algorithms, Performances and Software[END_REF] as well as discussions about the influence of the parameters.

Extension to two-direction measurement

In view of performances obtained for one axis displacement measurement, we worked on the evolution of the method to be able to measure displacements along two axes with the objective of similar performances in terms of range and accuracy.

Basically, a 2D measurement could be done through the use of two 1D sensors correctly placed. In the case of the method previously exposed, two patterns could be placed perpendicularly on the object of interest and observed by two cameras during a planar displacement, what would result in 2D measurement. However in numerous cases, and more particularly in the micro-world, the bulk of the sensing setup is a critical point. That is why a single 2D sensor is preferable to a pair of 1D sensors.

The use of the new pattern presented below allows to carry out direct 2D measurement with a single camera. This section introduces this pattern and describes the measurement process. This process is divided in two points: the measurement algorithm itself and the necessary preprocessing steps.

Pattern design

A new pattern was designed and micro-machined in clean room to measure displacements along two directions simultaneously. Its shape results from the fusion of two twin-scale patterns intertwined perpendicularly by inclusive disjunction (see Fig. 3). The result is a chessboardlike pattern that stores the entire data of the two previous patterns and suits for measurement displacements with a large unambiguous range along the two axes.

In the experimental case L x,1 = L y,1 = L 1 = 8 µm and L x,2 = L y,2 = L 2 = 8.3 µm, resulting in an unambiguous range of 221.33 µm. The heights W x and W y of the pattern are larger than the periods L (32 µm) with the aim to not affect measurements. It is noteworthy that these periods can be adapted to the needs.

In practice, the pattern is realized by photolithography on a thin aluminum layer deposited on a glass substrate (see Fig. 3.c). The elementary, non-ambiguous range of 221.33 × 221.33 µm 2 is duplicated several times in both directions and the actual full pattern size is 10 × 10 mm 2 .

Algorithm

The new pattern requires a new algorithm to be used. Both horizontal and vertical displacements have to be measured by phase evaluation applied to specific areas of the image, similarly to the method presented in section 2.1. The major problem is now the choice of these areas. Only the case of horizontal measurement will be detailed here since the vertical one is similar. The global measurement algorithm, detailed below, is presented in Fig. 5.

The first step consists in the vertical filtering of the image:

I m = I * C (9)
with C a column vector of size fy = round( Ly,1 + Ly,2 )/2 with each component equal to 1/ fy . The aim of this step is to cancel high frequency noises and recover a 1D-pattern shape in the 2D-pattern image (see Fig. 4).

x-axis pattern y-axis pattern The second step consists in the extraction of four horizontal and equidistant lines, named V A ,V B ,V C et V D (see Fig. 4). The aim of these multiple lines is to identify a couple of lines corresponding to periods L 1 and L 2 respectively. Indeed at least one of the couples AC or BD is suitable, i.e. without stripe set overlapping, because of the choice of a W/4 distance between these lines.

L 1,x L 2,x W x L 1,y L 2,y
Then eight products are then computed:

P Ki = x V K (x) • Z x,i (x, Lx,i ) (10)
with K ∈ [A, B, C, D], i ∈ [1, 2] and Z x,i the analysis function (similar to that of equation 3) such as:

Z x,i (x, Lx,i ) = e - x-Ñ /2 Ñ /4.5 2 • e - 2iπ(x-Ñ /2) Lx,i (11)
To determine which is the best couple and who is who in this couple (who corresponds to period L x,1 and who corresponds to period L x,2 ), four cross-modulus are calculated:

M A1C2 = |P A1 | • |P C2 | M A2C1 = |P A2 | • |P C1 | M B1D2 = |P B1 | • |P D2 | M B2D1 = |P B2 | • |P D1 |
The largest cross-modulus corresponds to the best couple, noted M best (since stripe overlapping or period mismatch would reduced the modulus drastically).

The penultimate step consists in calculating the phases of the two grids of the chosen couple and to correct them to take into account the slight residual misalignment between the pattern and the camera pixel frame (see Fig. 5). It requires the rate θ rad/p,x of the rotation of the pattern with respect to the pixel frame (in radian per pixel). For example if M best = M B1D2 , we know that V B has a period L 1 and V D has a period L 2 , so we obtain: φ B1 = arg(P B1 ) + (ỹ A -ỹB ).θ rad/p,x φ D2 = arg(P D2 ) + (ỹ A -ỹD ).θ rad/p,x

Finally these two phases are used to determinate the position with the same method as for one degree of free-dom measurement (see equation 8). The same algorithm is applied to the y-axis.

It is important to mention that some parameters have to be identified before the measurement: the periods of the pattern (in pixel) Lx,1 , Lx,2 , Ly,1 , Ly,2 ; its heights Wy and Wx ; the analysis functions Z x,1 , Z x,2 , Z y,1 and Z y,2 ; the rates θ rad/p,x and θ rad/p,y . These parameters are determined once by a pre-processing step, as described with full details in next section.

Parameters identification

The pre-processing step allows to evaluate ten elements useful for the measurement step: the four grid periods, the two grid widths, the four analysis functions and the two angular corrections. The global calibration algorithm is shown in Fig. 6 for the half of the elements (the second half is obtained similarly by inverting x-axis and y-axis.

In a first instance, the pre-processing is made on filtered images (as the measurement step), allowing to reduce high frequency noises (see equation 9). The four periods in pixels ( Lx,1 , Lx,2 , Ly,1 , Ly,2 ) are firstly determined thanks to a Fourier transform applied to the selected vectors. Next an iterative algorithm allows to find these periods with subpixelic resolution (inferior to 0.01 pixel). The analysis functions Z x,1 , Z x,2 , Z y,1 and Z y,2 are calculated thanks to these periods (see equation 11).

The widths of the grids ( Wx , Wy ) are obtained with the same algorithm but applied to vectors made by specific filtering of the image to bring to light the widths.

Finally the necessary angular correction is calculated by considering the deviation of the phase variation, horizontally or vertically. For example if the rows of an image are scanned to measure the horizontal position of the pattern according to the ordinate, the phase curve is a line tilted by the angle of the pattern. Thus the phase error θ rad/p,x (in radian by pixels) due to the orientation of the pattern can be calculated. However this evaluation of the orientation is limited to small angles, due to residual static misalignment.

Globally most of the pre-processing time corresponds to the selection of four rectangular areas by the operator to obtain the coordinates x1 , x2 , ỹ1 and ỹ2 . The size of the filter applied to the image ( fx and fy ) can also be set. This time is of course user-dependent and it was evaluated to be close to 30 s by averaging from different users (extremes 20 s -45 s). In comparison, the actual preprocessing computation time of 1.3 s is almost negligible.

Experimental validations

An experimental setup was built to test the method (see Fig. 7). It primarily consists of a pattern fixed on a nano-stage and observed through a microscope. Evaluated performances are resolution, range, linearity, repeatability and trueness.

I m,y VA • Zx,1 VA • Zx,2 V A Z x,1 Z x,2 P A1 P A2 VB • Zx,1 VB • Zx,2 V B Z x,1 Z x,2 P B1 P B2 VC • Zx,1 VC • Zx,2 V C Z x,1 Z x,2 P C1 P C2 VD • Zx,1 VD • Zx,2 V D Z x,1 Z x,2 P D1 P D2
Step

M A1C2 M C1A2 M B1D2 M D1B2
Step 3: Computation of the actual displacement Argument Argument

P α1 P β2 Φ1 = Φα1 -(ỹα -ỹA)θrad/p,x Φ2 = Φβ2 -(ỹβ -ỹA)θrad/p,x Φ α1 Φ β2 θ rad/p,x θ rad/p,x ỹA ỹA ỹα ỹβ Global phase ΦΛ = Φ1 -Φ2 Φ 1 Φ 2
Refined position computation (Eq. 8)

Φ Λ Φ 1 δ L 1 L 2

Experimental setup

The aim of the experimental setup is to observe a moving pattern to evaluate the performances of the method, thus it is important to control precisely the displacement. We use for this purpose a piezo nanopositioner P-753.1CD from Physik Instrumente (specifications in table 3).

This piezoactuator is mounted on a 6-axis manual positioning device (three translations and three rotations) allowing micro-displacements. This device is used to place the pattern in front of the microscope without out-of-plane tilt and to realize displacements larger than the piezoactuator range.

The visual setup is composed of a Fireware camera (Allied Vision Technology Stingray F-125, 8 bits, 1292 × 964 pixels) and a 10× microscope lens. The computer used is a common Intel Core2 Quad CPU Q9550 2.83 GHz, running under Windows 7.

The performances of the visual measurement method are determined by observing the pattern fixed on the actuator, so these performances are highly dependent on noises that move the pattern with respect to the microscope. To reduce mechanical noises, the setup is fixed on an antivibration table placed in a metrologic room with an isolated slap. Thermal noises are reduced by the temperature control of the room (variations below 0.5 degrees).

Resolution evaluation

In this work, we refer to resolution definition provided by the "Joint Committee for Guides on Metrology" [START_REF]Joint Committee for Guides on Metrology, International vocabulary of metrology -Basic and general concepts and associated terms (VIM)[END_REF] as the "smallest change in a quantity being measured that causes a perceptible change in the corresponding indication". So the important is to detect a displacement as small as possible, even if it is lost within a noisy background of much larger magnitude. In this way, the experiment sends a picometric control input to the piezoactuator and detects it with the vision (at 20 frames per second) to evaluate the resolution.

The chosen input signal is a pulse sequence of period 1 s with an amplitude of 0.5 nm. This amplitude is near to the minimal displacement we can observe with our visual method, as demonstrated in [START_REF] Guelpa | Subpixelic Measurement of Large 1D Displacements: Principle, Processing Algorithms, Performances and Software[END_REF]. However a so little displacement will be lost in the measurement noise (here approximately 5 nm, as observed in the following experiments). To detect the displacement despite noises, we use the Fourier transform applied to the temporal data of position measurement, to detect the periodicity of this sequence of displacement. Indeed if the peak of the resulting curve is found at the expected frequency, the displacement is detected so the visual method has a resolution at least equal to its amplitude. Fig. 8.a shows the control input (with an amplitude of 0.5 nm) and the corresponding capacitive sensor measurement. We can clearly see the repeatability of the capacitive is around 1 nm. We can also check the effective motion by averaging the measures of low and high levels. We found a difference of 0.4937 nm. Since the resolution of the capacitive sensor is ten times lower, we are confident that Figure 8: Results of the resolution experiment. The fact that the magnitude of FFT of both visual method and capacitive sensor have a peak which corresponds to the period of the input command proves that the resolution of the method is at least 500 pm.

a motion of 0.5 nm is produced. The question is then to detect the motion with the visual method. Fig. 8.b and c illustrate respectively the magnitude of the FFT of the temporal visual measurement along the x-axis and of the capacitive sensor. We can see distinctly the peaks for a frequency of 1 Hz, so the resolution of the visual method is (at least) equal to 0.5 nm. This value is to be considered only with this pattern and this visual setup, for an analysis using 2000 images (theoretical best values for other setups are discussed in [START_REF] Guelpa | Subpixelic Measurement of Large 1D Displacements: Principle, Processing Algorithms, Performances and Software[END_REF]). This result is widely below the others performances evaluated experimentally further (linearity, repeatability, trueness), and should thus be seen as an absolute measurement limitation in our experimental conditions.

Micrometric measurements

The second experiment is thought to allow the evaluation of the range, linearity, repeatability and trueness of the method. The input command is a ramp function (see Fig. 9). The pattern is always placed tilted beside the actuator to allows measurements simultaneously along x-axis and y-axis (see Fig. 7).

One series of measurements is presented in details below, but five others were carried out with the same proce- dure. Each series includes 2000 points allocated between the 17 positions of a ramp function. The results are presented at the end of this section: Table 2 gives a general vision of these results; Table 3 summarizes these results and collates them to the performances of the piezo-actuator and its capacitive sensor. Also notable is that the calculation method is able to process more than 100 image per second under Matlab, meaning video-rate processing with common cameras.

Accuracy estimation

We choose to characterize the accuracy of our method with three elements: its linearity, its repeatability and its trueness.

The linearity is defined as the maximum deviation of the measured quantity values from the least square line of the measured values during a linear displacement. Fig. 10 shows the corresponding error values and the resulting linearity of 19 nm, which corresponds to 0.16 % relatively to the explored range during this experiment.

The repeatability (or measurement precision under repeatability conditions) is defined as the "closeness of agreement between indications or measured quantity values obtained by replicate measurements on the same or similar objects under specified conditions" [START_REF]Joint Committee for Guides on Metrology, International vocabulary of metrology -Basic and general concepts and associated terms (VIM)[END_REF]. Fig. 11 expresses this definition by the evaluation of three times the standard deviation of the measurement for each input command. The result is a repeatability below 10.5 nm.

The trueness is defined as the "closeness of agreement between the average of an infinite number of replicate measured quantity values and a reference quantity value" [START_REF]Joint Committee for Guides on Metrology, International vocabulary of metrology -Basic and general concepts and associated terms (VIM)[END_REF]. In our case the reference is the capacitive sensor included Error between measured position and its linear fitting Gap of linearity Figure 10: Error between the visual measurement and its least square line (serie 1, see Fig. 14). The linearity is the maximum deviation of this error, here equal to 19 nm.

in the piezo-actuator and each trueness value is the average of more than hundred errors. Fig. 12 presents these trueness according to the input command, never bigger than 18.5 nm. The accuracy of the actuator is better than that of visual sensor; in this way the wavy shape is due to edge effect on the pattern that induce a numerical error on the phase measurement.

Range evaluation

The range of the method is determined by the periods of the pattern according to Eq. 6. Thus with L 1 = 8 µm and L 2 = 8.3 µm the complete range is equal to L Λ = 221.33 µm. To illustrate this range, two 1D-axis with a larger range (M-111.1DG from Physik Instrumente) are used to move the pattern along the two axes. Their control inputs are sine curves with slightly different periods, what generates the trajectory of Fig. 13.

To illustrate the fact that performances are homogeneous along the whole range, the initial setup (with the more accurate actuator) is used. The manual positioning table was used to change the x, y and θ positions, then five measurement series (similar to that one used to evaluate accuracy) are done. Fig. 8 shows the 6 trajectories of the pattern and the maximal range. Results are summarized in Table 2 and Table 3.

To conclude on the range issue, it is important to mention that displacements larger than L Λ can be measured by performing phase compensation; i.e. adjusting parameter p in equation 8. In this case the global range of the method is limited by the size of the pattern. It is a common method used to improve the range of application of a sensor. Figure 11: Repeatability of the visual measurement (serie 1, see Fig. 14). The repeatability is equal to 3σ (with σ the standard deviation), calculated with all the measures obtained for a specific input command. It corresponds coarsely to the dispersion of each group of point in Fig. 10. Here the repeatability is always below 10.5 nm and on average equal to 5.5 nm. Figure 12: Trueness of the visual measurement (serie 1, see Fig. 14). Each trueness value is the error between the average of visual measurements and the average of capacitive measurements for a specified input command. Here the default of trueness is mainly due to the linearity default (similar to Fig. 10).

Conclusion

The visual measurement method presented in this paper provides a satisfactory solution to the initial issue, namely the 2D position measurement with a large rangeto-resolution ratio. The new micro-pattern intertwines properly the twin-scale grids along the two axes. The new algorithm is adapted to its exploitation, selecting correctly which areas of the image are suited to realize measurements.

We experimentally characterized the performances of the method with a specific setup. Large absolute range (221.33 µm), sub-nanometric resolution (0.5 nm) and good linearity (0.16 %), repeatability (13.5 nm) and trueness (18.5 nm) are observed (with a capacitive sensor as reference). Moreover the computation time enables video-rate use. The method constitutes thus a new tool for the calibration of micro-actuators as well as a supplementary visual sensor in the field of automation at micro-scale. The method is also easily and quickly calibrated. Finally it is important to mention that the visual method could be adapted to others scales: general enlargement or shortening of the pattern; modifications of L x,1 , L y,1 , L x,2 or L y,2 to modify its resolution or range; using of a more powerful camera and adjustment of the lens magnification accordingly; etc. Future works will be focused on the use of a similar pattern to realize spatial position measurement, always aiming large range-to-resolution ratio.

Figure 1 :

 1 Figure 1: Working principle of the twin-scale visual measurement [14].

Figure 2 :

 2 Figure 2: Block diagram of the twin-scale measurement method. Details are in [13].

  Design principle of the 2D pattern. (b) Mask used for the photolithography of the pattern in clean room. Here L x,1 = L y,1 = 8 µm, L x,2 = L y,2 = 8.3 µm and Wx = Wy = 32 µm.(c) Typical 1280x960 image of the pattern captured with the visual setup.

Figure 3 :

 3 Figure 3: The 2D pattern, from principle to actual device.

Figure 4 :

 4 Figure 4: An example of selection of four horizontal lines used to find the best working area. The image is averaged vertically beforehand. Each line is chosen Wx/4 away from the previous line.

Figure 5 :Figure 6 :

 56 Figure 5: Block diagram of 2D measurement. Here only the measurement along the x-axis is presented, but the y-axis case is similar.

Figure 7 :

 7 Figure 7: The experimental setup, placed in a metrological room to reduce mechanical and thermal noises. The 1 DoF piezo-actuator is placed obliquely to generate a pattern displacement along both x and y axes.

  The input command is a pulse of period 1 s.

  the FFT (arbitrary unit) (b-c) Magnitude of the FFT of visual measurement along x-axis (left) and from the capacitive sensor measurement (right). The peak at frequency of 1Hz is perceived more easily with the capacitive sensor because the signal is less noisy.

Figure 9 :

 9 Figure 9: Typical position measurement during the micrometric experiment (here for serie 1, see Fig. 14). The input command is a ramp function of amplitude 12 µm. The global series is ten times longer than what is illustrated by the figure, with 2000 points.

Figure 13 :Figure 14 :

 1314 Figure 13: Illustration of the range of the visual sensor during a large-range measurement. Two actuators are used with sinusoidal control inputs.

Table 1 :

 1 Performances of twin-scale visual measurement with a 8 µm periodic pattern and 640x480 8-bits camera (from[START_REF] Guelpa | Subpixelic Measurement of Large 1D Displacements: Principle, Processing Algorithms, Performances and Software[END_REF]).

	Property	Value
	Travel range	168 µm
	Theoretical resolution	55 pm
	Experimental repeatability (3-σ)	5 nm
	Bandwidth	>1500 Hz

Table 2 :

 2 Overview of the experimental results for the six series of measurements by vision.

	Performances	Serie 1 Serie 2 Serie 3 Serie 4 Serie 5 Serie 6
	Linearity (%)	0.16	0.13	0.16	0.12	0.12	0.16
	Repeatability (nm)	10.5	10	13.5	6.5	12	11
	Trueness (nm)	18.5	11	11	10	8	15

Table 3 :

 3 Summary of results and comparison with the piezo-actuator performances.

	Performance	Visual method	Capacitive sensor
		Worst case Average value
	Resolution (nm)	0.5		0.05
	Linearity (%)	0.16	0.14	0.04
	Repeatability (nm)	13.5	7.22	1
	Trueness (nm)	18.5	6.9	reference
	Range (µm)	221.33		15

In this article, spacial dimensions are metric by default. The dimensions in pixels are wrote with a tilde, like L.

Acknowledgments

This work was supported by Labex ACTION project (ANR-11-LABX-01-01) and by Région de Bourgogne-Franche-Comté. Authors acknowledge the French RENATECH network through its FEMTO-ST technological facilities MIMENTO, ROBOTEX and ENSMM for its metrological room. M. Asmad Vergara was supported by Student Grant Huiracocha Program-PUCP (Peru).