Thomas Leduc
email: cthomas.leduc@cerma.archi.fr

Fernando González Cortés
email: dfernando@fergonco.es

GearScape: an extensible free and open-source GIS adapted to the European Network of Territorial Intelligence main thematic

Keywords: GIS, geomatics, geo-processing, spatial SQL, GearScape, GGL SIG, géomatique, géo-traitement, SQL spatial, GearScape, GGL, geomática, geoprocesamiento, SQL espacial, GearScape, GGL

GearScape is a Free and Open-Source Software (FOSS). It is a Geographic Information System (GIS), that is a software dedicated to storage, retrieval, analysis and mapping of geographic data. More precisely, it is a platform focused on geo-processing that can be used in both interactive and batch modes. Its two main distinguishing features are: on the one hand it can be extended not only through the Application Programming Interface (by an advanced user with clear programming skill) but also through a simple spatial SQL based language called GGL (by a common end user), and one the other hand it relies on a generic layer that intends to uncouple the geo-processes with the specificities of each underlying spatial format. Extensibility and re-usability in a groupware context are the two main reasons that justify its clear appropriateness to the Territorial Intelligence main thematic.

Résumé:

Gearscape est un logiciel libre. C'est un Système d'Information Géographique (SIG), c'est-à-dire un outil logiciel adapté au stockage, à l'extraction, à l'analyse et à la représentation cartographique de données géographiques. Plus précisément, c'est une plate-forme orientée géo-traitements qui peut être utilisée en mode conversationnel comme en traitement par lot. Ses deux principales caractéristiques sont: l'extensibilité d'une part (non seulement par le biais de son interface programmative d'application pour les utilisateurs-développeurs d'application, mais aussi par le biais du langage GGL qui est simple et repose sur une version spatiale du SQL pour tous les autres), la généricité d'autre part (puisqu'il repose sur une couche applicative d'abstraction qui lui permet de s'affranchir du format des données sous-jacentes). Ces deux traits essentiels en font un outil de choix pour tous les traitements liés aux thématiques de l'Intelligence Territoriale.

Resumen:

GearScape es software de fuentes abiertas y libre. Es un Sistema de Información Geográfica (SIG) que permite el almacenamiento, recuperación análisis y visualización de datos geográficamente referenciados. Más concretamente, GearScape es una plataforma para la creación de geoprocesos que puede ser utilizada de manera interactive o por lotes. Sus dos principales características son: por una parte puede ser extendido a través de la Interfaz de Programación de Aplicaciones (por un usuario con conocimientos de programación) y también a través de un lenguaje de geoprocesamiento basado en SQL llamado GGL (por un usuario final); por la otra parte, GearScape está construido sobre una capa de acceso a datos que desacopla los geo-procesos de las características de los distintos formatos. Extensibilidad y reusabilidad en un contexto groupware son las dos mayores razones que justifican su clara adecuación la principal temática de la Inteligencia Territorial.

I. Introduction

One of the three current main research activities in progress in the European Network of Territorial Intelligence (ENTI) aims to develop the territorial information and spread the methods of territorial observation and spatial analysis. Indeed, as noticed by [START_REF] Franklin | An introduction to geographic information systems: linking maps to databases[END_REF], at the beginning of the 90s: about eighty percent of all data stored in corporate databases has a spatial component. Moreover, [START_REF] Clinton | Executive Order 12906 -Coordinating geographic data acquisition and access: The National Spatial Data Infrastructure[END_REF] wrote, in one of its key executive orders: Geographic information is critical to promote economic development, improve our stewardship of natural resources, and protect the environment. He established also that modern technology now permits improved acquisition, distribution, and utilization of geographic (or geospatial) data and mapping. Lastly, he emphasized the need to avoid wasteful duplication of effort and promote effective and economical management of resources. For all the reasons that will be briefly developed in this paper, we assume that the GearScape framework, as an extensible and re-usable GIS, is an essential component of the Territorial Intelligence. Indeed, this GIS software can be seen as a front-end able to access and address a distributed network of spatial databases, linked by common standards and protocols to ensure compatibility and interoperability of data and services.

II. Motivation

To sum up, the aim is to be able to integrate several geographic datasets from different data providers, and try to produce richer datasets, creating knowledge and raising analytical level. For easiness reason, this should be realized with an uniform framework so as to facilitate multi-disciplinary collaborations. Numerous systems, either open-source either commercial or proprietary, are nowadays available which cover all sectors of geospatial data handling. According to the GIS software typology presented in [START_REF] Steiniger | GIS Software -a description in 1000 words, Encyclopedia of Geography[END_REF], one could classified them all into different categories from WebGIS thin client to Spatial DataBase Management System (S-DBMS). In the huge set of existing GIS software solutions, the one we need has to be a: FOSS, portable and cross-platform, standalone, with geo-spatial batch scripting abilities. To rephrase all those conditions, one could say that the solution has to be developed in Java (so as to take benefits from the wide Java Virtual Machine dissemination and its well-known WORA slogan created by Sun Microsystems: "write once, run anywhere"), it must be a desktop GIS and has to provide a spatial SQL with parameterization ability.

To achieve such an objective, the GDMS layer [START_REF] Bocher | GDMS: an abstraction layer to enhance Spatial Data Infrastructures usability[END_REF][START_REF] Leduc | GDMSR: A mixed SQL to manage raster and vector data[END_REF] has been developed. It is both an API and a spatially enhanced SQL syntax, as close as possible to the SQL-92 standard and to the "Simple Features Specification for SQL" (Herring, 2006a;Herring, 2006b) to guarantee a smooth learning curve. Indeed, spatial languages are suitable tools to define, store and share geo-processing… and among all of them, the SQL (Structured Query Language) is probably the most common and the easiest to use. Indeed, the table paradigm -inherent to the SQL language -has been clearly identified as a very useful tool for Rapid Application Development (RAD). In the way from no interoperability to an ideal one, a syntactic level has therefore been reached. With the GearScape project and more precisely with the GearScape Geoprocessing Language (GGL), the GDMS layer has been enriched with parameterization capability. With GearScape and its GGL, expectation is to go one step further towards the semantic interoperability. Indeed, GGL is a geo-processing adapted development tool that lets users without programming skills create, execute and share their own geo-processes. It includes an editor to assist the geo-process authoring tasks, a wizard to execute the created geo-process and a model builder to link geo-processes so as to produce a bigger and composite one.

III. GearScape in a nutshell

GearScape is a Geographic Information System (GIS). It is software dedicated to storage, retrieval, analysis and mapping of geographic data, that is of spatial feature, with associated alphanumeric descriptive attributes and corresponding metadata such as the coordinate reference system. It is a platform that can be used either in interactive mode (as an example, it implements the WIMP1 paradigm in a classical but also renewed approach) either in batch mode (as an example, its command line interface has been deployed in a grid computing context).

GearScape is a multi-document application2 based on four major components that are the Geocatalog, the Geocognition, the layer view (called the "TOC") and the map view. The Geocatalog is a shared list of all datasources addressed by the GDMS layer that can be, as an example, dragged and dropped into the TOC or geo-processed through the spatial SQL in the GGL console. A datasource might be either a remote PostgreSQL/PostGIS table, either a local flat file such as a shapefile. The Geocognition component is a sort of project manager. It is the tool you need if you want to: retrieve an already existing geo-process or add a new one, open an already existing map or create a new one… The TOC is a textual representation of the current map view. It is a tree where each node corresponds to a layer name possibly displayed in the active map view.

One can notice also that, as shown in fig. 1,GearScape's user interface provides all the classical views and tools "required" in a desktop GIS and listed in [START_REF] Steiniger | GIS Software -a description in 1000 words, Encyclopedia of Geography[END_REF]. In the displayed map view, the selected path section in yellow is a component of a geographic layer called "path" (the red shape in the map and the selected entry in the TOC) that is tagged as editable. In this edition mode, a set of "editing tools" is enabled near the classical navigation tools.

Figure 1: GearScape provides a typical desktop GIS user interface with multi-views such as a map, a layer, an attribute or an information view. It also provides classical tools such as navigation or edition ones.

IV. An extensible geo-processing framework

As it is, GearScape is a quite comprehensive GIS. But, if you need some specific functionality that is not available, it can be extended with your own geo-processing script even if you are an end-user without clear programming skill (because of the easiness of use of the spatial SQL language). However, the analyses defined in spatial SQL are tightly coupled with the data they operate on and therefore it is not possible to apply the same analysis on several data sets without prior modification of the script. To solve this drawback, some extensions to the GearScape SQL implementation have been made that make possible the definition of the structure of the data to be analyzed by a script. Scripts defined this way do not reference concrete data sets but abstract. This abstract input definition can be useful for different tasks. When the script is executed, the system can analyze its abstract input requirements and open a wizard to let the user choose the data that matches the script parameters. Once the wizard is accepted, the script is executed against the data supplied by the user. This way, as no SQL edition is required to execute the script, users without SQL knowledge can benefit from scripts developed by other users.This solution makes possible the execution of a generic geoprocess chain on different data sets, making a generic problem from a concrete one. As a generic problem it may be shared by a greater number of people so the benefits of communication between users rise notably.

V. A simple use case

As a simple use case, it is possible to obtain an indicator of the building density liable to flooding because of an urban river. GearScape embeds, among some others, a geo-processing script called "density" which computes a grid containing the ratio of build areas per cell. Before using such an indicator, it is necessary to identify all the buildings that are near the river. It can be achieved using the spatial SQL join query presented in the GGL console of the fig. 2.

Conclusion

The GearScape framework presented in this paper is one more step towards one of the main ENTI's objective. As an efficient geo-processing system, it is a useful software tool to develop the territorial information and spread the methods of territorial observation. Indeed, the GGL language embedded in this platform, as a language that enhance spatial SQL with parameterization, is a convenient and relevant solution to document, share and develop geo-processes in a collaborative way.

Figure 2: Once the buildings liable to flooding are retrieved, it is possible to use a script shipped with GearScape that produces a grid with density values and apply some cartographic styles to paint in different colors the different densities.

In the human computer interaction context, the WIMP acronym stands for Window, Icon, Menu and Pointing device. It corresponds to a style of human interaction using all these elements.

Based on the InfoNode Docking Windows technology.


Qiao Ma

Baptiste Fosty
email: baptiste.fosty@inria.fr

Carlos F Crispim-Junior
email: fernando.crispim_junior@inria.fr

François Brémond
email: francois.bremond@inria.fr

FUSION FRAMEWORK FOR VIDEO EVENT RECOGNITION

Keywords: object recognition and motion, architecture and implementation, Multisensor fusion, event recognition, Evidence theory

This paper presents a multisensor fusion framework for video activities recognition based on statistical reasoning and D-S evidence theory. Precisely, the framework consists in the combination of the events' uncertainty computation with the trained database and the fusion method based on the conflict management of evidences. Our framework aims to build Multisensor fusion architecture for event recognition by combining sensors, dealing with conflicting recognition, and improving their performance. According to a complex event's hierarchy, Primitive state is chosen as our target event in the framework. A RGB camera and a RGB-D camera are used to recognise a person's basic activities in the scene. The main convenience of the proposed framework is that it firstly allows adding easily more possible events into the system with a complete structure for handling uncertainty. And secondly, the inference of Dempster-Shafer theory resembles human perception and fits for uncertainty and conflict management with incomplete information. The cross-validation of real-world data (10 persons) is carried out using the proposed framework, and the evaluation shows promising results that the fusion approach has an average sensitivity of 93.31% and an average precision of 86.7%. These results are better than the ones when only one camera is used, encouraging further research focusing on the combination of more sensors with more events, as well as the optimization of the parameters in the framework for improvements.

Introduction

The number of older people living alone around the world is increasing, highlighting the importance of solutions for the treatment of this population health care conditions and their life quality improvement. [START_REF] Zouba | An activity monitoring system for real elderly at home: Validation study[END_REF] Human activity recognition is an important part of computer vision, and therefore an active research topic in video surveillance areas, like human activity monitoring in outdoor places or indoor (like metro stations, bank agencies, daily living places) environments [START_REF] Avanzi | Design and assessment of an intelligent activity monitoring platform[END_REF] .

The mono-sensor approaches are used in detection, tracking, and recognition of activities in the scene without the stereo match nor synchronization processes of sensors [3][4] . Methods based on mono-sensor rely on image analysis have been carried out using different approaches like probabilistic approach [ 5 -7 ] or constraint-based approach [START_REF] Ghallab | On chronicles: Representation, online recognition and learning[END_REF] [START_REF] Vu | Automatic video interpretation : A novel algorithm for temporal scenario recognition[END_REF] . The constraint-based approach proposed by Romdhane et al. [START_REF] Romdhane | A framework dealing with uncertainty for complex event recognition, Advanced Video and Signal Based Surveillance (AVSS)[END_REF] handles the uncertainty of the activity recognition in complex event using probabilistic reasoning. It provides a convenient mechanism of reasoning to handle event uncertainty.

But, there are cases where the scene cannot be covered only by a single sensor or the distance from the sensor to the target can compromise the level of detail necessary for desired task accomplishment.

In these cases, the combination (fusion) of multiple video-cameras can increase the level of details of a scene. The combination of several sensors can help improve the low accuracy caused by the long distance as well as some other problems in visual recognition situations. Especially, multisensor fusion shows the values of different target features of an event (e.g. distance, velocity, weight), the combination of which can help use obtain a more complete view of the event and therefore a more promising performance of the recognition task compared to job done by single sensor.

The data fusion can take place at three levels [14] : data level (fusion of pixels), feature level (fusion of feature vectors of each sensor), and decision level (fusion of event from each sensor). The decision level fusion involves the multisensor fusion after each sensor has made a preliminary detection of an event occurring in the scene. Since the sensors for fusion can be either homogeneous or heterogeneous, the fusion at decision level makes it possible that changing or adding different sensors without completely changing event models (see Section 3.1).

This paper is organized as follows: The state of the art is presented in Section 2. In Section 3 we present the experimental material and the proposed data fusion framework, whose evaluation procedure and respective results are presented in Section 4. Section 5 describes our contribution and conclusion.

Related work

The recognition of different human postures (herein mapped as primitive states, see Section 3.1) is one of the basic traits to be detected in the event recognition systems. Fusion of video camera and environmental sensors [START_REF] Chomat | Probabilistic recognition of activity using local appearance[END_REF] [START_REF] Zouba | Multisensor Fusion for Monitoring Elderly Activities at Home[END_REF] (like pressure, light, temperature, etc) are usually adopted in indoor places where specific sensors are already designed and installed. Data fusion between camera and inertial sensors has been also explored [12 -14] , as well as body sensor networks (BSNs) [ 15 -17 ] . These approaches have been successfully used on the combination of homogeneous or heterogeneous sensors for event recognition. However, the use of wearable devices could be considered intrusive, and not as convenient and generic as non-contact video cameras [START_REF] Moeslund | A Survey of Computer Vision-Based Human Motion Capture[END_REF] .

Decision-level fusion methods include (but not limited to) weighted decision methods [START_REF] Hall | Mathematical Techniques in Multisensor Data Fusion[END_REF] , Bayesian inference [START_REF] Iverson | Bayesian Statistical Inference[END_REF] , and Dempster-Shafer's method [ 21 ] [ 22 ] . Dempster-Shafer evidence theory extends the Bayesian inference theory using incomplete information to make knowledge fusion, which resembles human reasoning process, and therefore is widely used for uncertainty modeling in many applications [23] [24] . However, the rule of combination of evidences as claimed by Dempster could give unexpected fusion results when conflicts among evidences exist (see Section 3.3.3.2). Consequently, a variety of improving methods have been proposed upon the Dempster-Shafer evidence theory. Yager [START_REF] Yager | On the Dempster-Shafer framework and new combination rules[END_REF] and Dubois et al. [START_REF] Dubois | Representation and combination of uncertainty with belief functions and possibility measures[END_REF] proposed to assign the uncertainty of conflicting evidences to the frame of discernment, as a solution to the conflicting issue. Smets [START_REF] Smets | The combination of evidence in the transferable belief model[END_REF] proposes that the conflicting mass results from the non-exhaustively of the frame of discernment and Murphy [START_REF] Murphy | Combining belief functions when evidence conflicts[END_REF] improved the basic probability assignment distribution instead of modifying the combination rule of the evidence theory. New combination rules are also discussed by several authors [ 29 -31 ] in some specific applications.

Briefly, This paper proposes a decision-level multisensor fusion framework for event recognition by the combination of sensors on decision level, dealing with conflicting evidence by the adoption of Dempster-Shafer theory and statistical reasoning evidence method with the new combination rule proposed in [31]. Two spatially separated video cameras (a RGB camera and a RGB-D camera -Microsoft Kinect) are used, with each we have had managed to recognize the basic event (primitive state, see Section 3.1) occurring in the scene). Methods are used to deal with the basic event detection like "sitting" and "standing" human posture recognition when two sensors conflict. The framework proposed is tested with the realworld data by cross-validation and the evaluation shows promising results encouraging further research for improvements.

Material and Methods

Event modeling

This framework is based on a hierarchical approach for event modelling proposed by Vu et al. in [9] and extended by Zouba et al. in [11]. It categorizes events in respect to their complexity as follows:

The events are divided into four types: Primitive state (e.g. a person is standing) deals with the instantaneous values of a person. Composite state is a combination of primitive state. Primitive event corresponds to a change of primitive state (e.g. a person changes the posture), and Composite event is a combination of primitive states and/or primitive events. The model for event E includes all the physical objects in E, all the components involved in E, and a set of conditions (herein called "constraints") to be verified between physical objects and sub-events.

Briefly, a composite event stays at the top of the hierarchy in terms of complexity, and primitive state is the basic layer. A composite event recognition consists in the recognition of all the related primitive, composites states, and and/or primitive event described in the composite event model. Since the primitive state is the most basic layer of the hierarchy, we chose to fuse events at primitive state level to achieve an accurate recognition of higher level events.

Experimental set

An evaluation using videos of participants of is performed to verify the proposed framework. Experimental site is located in a test room of CHU Nice Hospital where ten videos of older person (5 females and 5 males) doing semi-directed activities like sitting and standing are taken.

In our case, two video cameras are used to record these experiments. The RGB camera is located from a wide view of the scenario but the long distance of detection leads to problems like noise, covering, etc. Besides, because of the Microsoft Kinect's 4-5 meters detection distance limitation, it has to be positioned nearer to the target person, and this leads to the problems of "missing" the objects and activities out of the view range. A data fusion is made between the RGB camera and the RGB-D camera which already have respectively the primitive state (sitting-standing posture in our case) recognition result. The original data reflect different sensors 'observations of the person's posture changes in a time interval (each is about 20 minutes). The scenarios performed in the experimental room are composed of a series of directed activities to access the person's physical profile (e.g. static and dynamic balance test, walking test) [14] . The goal of the proposed framework consists in managing conflicting evidences of spatially separated and time-synchronized sensors in the process of event recognition. It should be noticed, however, that the answer to the question that whether the fusion result of an event detection system can be improved or not compared to what it would have been if only one sensor is used will depend on the reliability and the performance of each sensor as well as the data fusion method to fuse the data from each sensor.

Proposed approach

As in our case, we focus on the fusion framework of the system, two preliminary works are assumed to be done before entering the fusion module: firstly, each sensor detects, tracks and recognizes the person in the scene and its target event by its own with the pre-defined a priori knowledge like event model and scene information; Secondly, the 2D camera and the Microsoft Kinect are time-synchronized with the event timestamp in [14]. The step before data fusion of the framework gives us a set of XML files indicating the preliminary recognition results of each sensor. These data are provided for the next step: the fusion module.

The proposed fusion framework architecture is presented in Figure 1. In our framework, we consider the event level fusion between two visual sensors mentioned in the previous section, although the framework can take into account more sensors by the use of an iterative approach.

For each time instant t, the framework takes the test video data, the groundtruth of event annotated by experts as inputs, as well as the data obtained by training. After the detection, tracking and event detection of each sensor, the conflicting events are evaluated on the fusion module (FM). To be precise, on every instant of each sensor, the data input to the FM is composed of a pair of vectors that indicate the current detected primitive states and the character features of the target person (e.g. recognized event "Standing" and person's instantaneous 3D height).

The FM computes the likelihood of every possible event and manages the conflicts. It starts with the computation of the instantaneous likelihood, with which the local temporal is obtained for probability reasoning in the Evidence Theory. Based on the standard Gaussian distribution of the person feature value, the instantaneous likelihood and the local reliability are computed for each instant, the D-S data fusion center makes a decision on which sensor's primitive state detection should be taken into account. For event data fusion, two aspects are considered related to the event likelihood computation: person bounding box height's "distance" to Gaussian parameters of person's height obtained on training at a time instant, and also the local temporal relationship in Figure 1. Proposed framework architecture FM respect to previous instants. Both aspects have an influence on the belief level for the detected event.

Instantaneous likelihood function for primitive states

Based on the algorithms proposed in [10] dealing with the event recognition process, we propose an extension for human posture detection using a multi-sensor approach.

The likelihood function consists in assessing the reliability value (belief level) of each sensor's event (i.e. primitive state detection) based on a standard Gaussian distribution obtained by a training step.

A simplified model of primitive state such as sittingstanding posture could be simplified by thresholding of a person's detected height. Herein, besides to the use of a fixed height threshold to distinguish ''sitting'' and ''standing'', the likelihood function of ''sitting'' and ''standing'' is computed based on the ''distance'' between the detected 3D height of a person and the trained Gaussian parameters results for ''sitting'' and ''standing''. This computed likelihood can also be interpreted as ''belief level value''.

A primitive state involves one or several features: for example, ''Standing'' can be defined by the person's detected height, and ''in the zone of chair'' is recognized when the constraint of person's position and pre-defined zone in the scene has been satisfied.

The distribution of a person's height for sitting and standing is considered as a Gaussian distribution. For each sensor, the Gaussian parameters (the mean , and the standard variance ) for each posture state is learned by computing the average height of postures sitting or standing in the scene in respect to each sensor, with the training statistical data obtained a priori based on annotated data of person postures (Equation 11 and 2).

For all feature values including the example feature in our case (sitting-standing posture's height), the instantaneous likelihood of the test video is computed for each frame using Equation 3 [START_REF] Romdhane | A framework dealing with uncertainty for complex event recognition, Advanced Video and Signal Based Surveillance (AVSS)[END_REF] with the Gaussian parameters previously obtained.

Since the standard Gaussian distribution likelihood can be considered as a belief level value, this value is used as "how strongly we believe that the event result of the sensor is true at the evaluated time instant".

Local temporal reliability

Now another aspect of the likelihood computation is considered: whether sitting and standing is recognized or not should depend not only on likelihood based on person's detected feature value at this moment, but also the likelihood function values of previous instants in a pre-defined window size. The algorithm used can be the one computing the local temporal reliability values based on a fixed window size . (Herein a 5 seconds window is used.) As shown in Equation 4 and 5, the temporal reliability for current instant depends on the previous instants: the exponential of the time distance part is the cooling function of probability, which reinforces the near instants' effect and gives less importance to the far ones [START_REF] Romdhane | A framework dealing with uncertainty for complex event recognition, Advanced Video and Signal Based Surveillance (AVSS)[END_REF] . Generally, a primitive state is a continuous process which lasts seconds or minutes. The window size parameter depends on the domain application, and it should fit the minimum time interval of the person's primitive state.

The temporal reliability of each sensor for each target event is the input of the D-S evidence uncertainty reasoning. The reason why we choose this method in the fusion framework consists in its filter effect of instantaneous likelihood's uncertainty caused by segmentation and tracking errors during video processing. It should be noticed that is flexibly configurable parameter based on domain events.

Data fusion based on D-S Evidence Theory

The fusion architecture is illustrated in Figure 2. The event fusion engine serves as information process center for two sensors at each time, and the architecture can be used in an iterative way to fuse more than two sensors. The synchronization of each sensor uses their event timestamps and the "leave-accord-out" logic is adopted. Therefore, the engine only deals with conflicting events, as there is no need to make any fusion when the two sensors' events agree.

Proposed by Dempster [START_REF] Dempster | Generalization of Bayesian inference[END_REF] and improved by Shafer [START_REF] Shafer | A mathematical theory of evidence[END_REF] , the Evidence theory extends the Bayesian inference's application by allowing the uncertainty reasoning based on incomplete information. The support also comes from the possibility of distributing the imprecision to the combination of propositions such as "The person could be sitting and be in the zone chair". The evidence theory is used here to process uncertainty reasoning and to output the final event recognition result. In the next subsection, Dempster-Shafer evidence theory is briefly introduced, and then comes the fusion strategy in the proposed framework.

Dempster-Shafer sensor fusion algorithm

D-S theory is also called evidence theory which is statistical inference method used for modeling uncertainty. The evidence theory is based on a pre-defined set , called the frame of discernment, which contains the group of all the possible mutually exclusive evidences or hypothesis of interest:

The function related to a proposition satisfying: is defined as basic probability assignment (BPA). For any , m(A) is considered as the subjective confidence level on the event A. Accordingly, the whole body of evidence of one sensor is the set of all the BPAs greater than 0 under one frame of discernment.

The combination of multiple evidences defined on the same frame of discernment is the combination of the corresponding confidence level values based on BPAs (e.g., pre-defined by experts). Given two sensors (1 and 2), where each sensor has its body of evidence ( m1 and m2), which are the corresponding BPA functions of the frame of discernment. We can combine two bodies of evidences into a new one by applying the following combination rule:

where the sum in denominator is called the conflict factor of two bodies of evidence, and means the conflict between two propositions. This combination rule can be used iteratively because of its commutativity and associativity. Thus, the data fusion of more than two bodies of evidence is done by iterative pairwise process Dempster-Shafter evidence theory was chosen as our data fusion basic idea, because compared to Bayesian theory, its reasoning process resembles the human perception. Most importantly, based on our likelihood computation steps in the previous subsections, we can use those statistical-based results to assign the unknown parameters in D-S evidence theory, and thus deal with the data fusion problem by a more reliable uncertainty reasoning method.

Modified combination rule dealing with conflicts

The classical D-S combination rule can be implemented to fuse data from two sensors, but it can lead to illogical results when the conflict factor approaches 1. For example, Doctor A and doctor B's judgement of a patient's disease: a or b or c. Doctor A has 99% confidence on disease a, and 1% on b; doctor B has 99% on c and 1% on b. The classical combination rule gives the unexpected conclusion that the patient's disease is b. In strong contradictory situations like this example, the reason for the unrealistic result of D-S evidence theory is that it distributes the uncertainty of global conflict to the common evidence of the two bodies. [START_REF] Yager | On the Dempster-Shafer framework and new combination rules[END_REF] Varies alternatives or improvements of D-S combination rule have been put forward like: (Yager 1987), (Dubois et al. 1988), (Deng et al. 2004). These approaches consist in distributing the conflicting evidence probability to the whole set of propositions ( ) in the frame of discernment. But the limitations are mainly falls in the following two aspects:

 When there are a great number of propositions in the frame of discernment, the weighting factor in the improved rules has to be computed for every subset or combination of evidences, and this increases the arithmetic operations.  The associativity of the rule is not satisfied, which is vital for the local distributed algorithm structure in the reality applications. In our framework, the data fusion is mainly carried out and verified with the following strategies:

For posture recognition conflict management, a new combination rule proposed and verified by Ali et al. [START_REF] Ali | A new combination rule rule for conflict problem of Dempster-Shafer evidence theory[END_REF] is used for primitive state recognition's data fusion. As written in the Equation 7 and 8, this newly proposed rule has been demonstrated to be efficient for combining the evidences from two or more sensors, and can be extended to the application of more primitive states and complex events. (6) (7) (8) The event uncertainty for "being sitting" and "being standing" is computed for each instant and each sensor based on the described training process: mean and variance of person height during evaluated postures. Results of the fusion approach are compared to the result of individual cameras (fixed-threshold for standing/sitting discrimination) to show the improvements brought on event detection performance.

A leave-one out cross-validation is adopted to evaluate the proposed approach. Briefly, the evaluation takes 9 videos out of the database for training while the tenth is used for test, and this is repeated until every video in the database is tested as validation data.

A priori knowledge like event model files are shared by both sensors with some new created events: sittingfusion, standing-fusion, sitting-agreed, standing-agreed, which represents the different sensors situations (conflicting or consistent). For each instant of the validation video, the temporal reliability of each possible event result is computed based on a 5 seconds window size of historical reliabilities. Then, they are assigned to data fusion engine as BPAs to compute all the possible propositions' uncertainty after simple normalization for each sensor. At last, the output is the final decision of person's posture recognition result by comparing the fused uncertainty of every possible result.

The recognition performance of the proposed framework is measured in respect to the video annotation of experts, where TP (True positive) is assigned to the system evaluation when the system's recognition result is equal to the events annotated by experts; FP (False positive) is assigned when an event that doesn't occur in the annotated is detected; and FN (False negative) when the system misses an event that occurs in the annotated Groundtruth. ( Table 1 presents the comparison of the proposed framework with the individual recognition performance of each video camera. The precision and the sensitivity indicate the best of the ten test sets during the crossvalidation. Figure 3 (Detection view of RGB camera) shows a "standing" detection example of the RGB camera which is fixed at a corner of the experimental room, and Figure 4 shows the detection view of the RGB-D camera at the same time instant.

Conclusion and Future work

This paper proposes a framework for event fusion in activity recognition based on statistical learning and uncertainty reasoning using Dempster-Shafer theory. The decision-making process is implemented in an event recognition system to evaluate the applicability of the framework in real data. The proposed framework allows adding more sensors into the event recognition system by designing a proposition set and recomputing the weights. When it comes to a complex event, more primitive state computation can be implemented into the uncertainty reasoning process.

The experiments show that the fusion approach has a higher average sensitivity 93.31% than that of 2D camera and RGB-D camera. The averaged precision of 86.7% shows an improvement compared to the RGB camera, but a decline in respect of RGB-D camera. The reason may involve the fact that parameter factors like fixed windowsize value used in the framework still need to be optimized. The framework is not able to recover when segmentation and tracking errors happen, where sometimes the person not detected or something else is mis-tracked as a person.

Future work includes evaluating different window sizes for the temporal reliability computation, adding optimised weighting factors in the combination rule for each camera in respect to their recognition performance, and also the combination of more sensors in the recognition of more postures like lying and bending, and eventually complex events.

Figure 2 .

 2 Figure 2. Architecture of event fusion process

Figure 3 .

 3 Figure 3. RGB cam. view Figure 4. RGB-D cam. view

Table 1 .

 1 Comparison between mono-sensor using the threshold method and the fusion framework method: performance evaluation

	10)
	(11)

Compute for each e: target event, i: sensor, j: video data number, k: instant, and "av" means "average". The notation is used throughout this paper.


Keywords: partial differential equations, robust control, inclined cable, state space model, measurement feedback

Considering the partial differential equation model of the vibrations of an inclined cable, we are interested in applying robust control technics to stabilize the system with measurement feedback when it is submitted to external disturbances. This paper focuses indeed on the construction of a standard linear infinite dimensional state space system and an H ∞ feedback control of vibrations with partial observation of the state. The control and observation are performed using an active tendon.

Robust Measurement Feedback Control of an Inclined Cable

Lucie Baudouin * Simon Neild * * Aude Rondepierre * * * David Wagg * *

INTRODUCTION

Inclined cables are critical components in many structures used in civil engineering -for instance in cable stayed bridges. Nonetheless, cable structures are not restricted to this range of applications and can also be found e.g. in telescopes or spacecraft, [START_REF] Smrz | Active damping of the camera support mast of a cherenkov gamma-ray telescope[END_REF]. One of the major issues related to such structures involving cables is the control of vibrations induced by any exterior perturbation.

Since cables are very flexible and lightly damped, cable structure systems usually have a range of dynamic problems. Their modelling is therefore very important in predicting and controlling the response to excitation. A good review about vibration suppression in civil structures and many references on this topic can be found in [START_REF] Song | Vibration control of civil structures using piezoceramic smart materials: A review[END_REF] and [START_REF] Preumont | Vibration control of active structures[END_REF]. This paper is devoted to the design of control laws for a vibrating system composed of an inclined cable connected at its bottom end to an active control input. We will present a linearized model using partial differential equations (PDE) and we aim at designing robust measurement feedback control laws for this kind of vibrating system.

In Section 2, we will first recall a linear PDE modeling of an inclined cable before detailing the standard state space model of infinite dimension describing the system, where the control and measurement of the vibrations of the inclined cable come from and active tendon connected to the cable, see Figure 1. Using a support motion in the cable axial direction is a natural choice of active control since the installation of the proper device can be done with small modification of the cable anchorage (see [START_REF] Fujino | An experimental study on active control of in-plane cable vibration by axial support motion[END_REF]). We hope to obtain good results when considering robust control with partial observation using an active tendon since the collocation of actuator and sensor has proved great effectiveness in active damping of cables, as in for example [START_REF] Bossens | Active tendon control of cable-stayed bridges: a large-scale demonstration[END_REF], [START_REF] Song | Vibration control of civil structures using piezoceramic smart materials: A review[END_REF]. In a second step, we will detail the state space model of infinite dimension we can derive from the PDE model and that we can use to perform the robust control of the inclined cable.

The H ∞ -control question for an infinite dimensional system with measurement feedback will be described in Section 3. In mechanics, the usual study of the vibrations of inclined cables is made through the consideration of a few structural modes. An important aspect of this work lies in the fact that we consider, as far as we can, the complete PDE model of the in-plane vibrations. Here, we will show that we can apply a robust control strategy based on modern control tools for distributed parameter systems, presented in [START_REF] Bensoussan | On the standard problem of H∞-optimal control for infinite-dimensional systems[END_REF][START_REF] Bensoussan | On the standard problem of H∞-optimal control for infinite-dimensional systems[END_REF][START_REF] Van Keulen | H∞-control with measurement-feedback for linear infinite-dimensional systems[END_REF]. We will indeed prove the H ∞ -control of the inclined cable, robust with respect to specific exterior perturbations, by the means of the possible resolution of Ricatti equations.

Among the numerous possibilities for modelling the motions of inclined cables with small sag, we adopt the PDE presented in [START_REF] Wagg | Nonlinear Vibration with Control: For Flexible and Adaptive Structures[END_REF] using the derivation from [START_REF] Warnitchai | A non-linear dynamic model for cables and its application to a cable structure system[END_REF].

The cable, which is of length ℓ, is supported at end points a and b and the direction of the chord line from a to b is defined as x, see Figure 1. The angle of inclination of the chord line relative to the horizontal is denoted θ and we set ρ the density of the cable, A the cross-sectional area, E Young's modulus and g gravity. We then define ̺ = ρg cos θ as the distributed weight perpendicular to the cable cord. The cable equilibrium sag position and the chord line both lie in the xz plane.

We assume that there is no significant dynamic response along the x-axis (the axial vibrations are usually excluded from models since the frequency of oscillations is much faster and of smaller amplitude than that in other directions) and that the sag is small in comparison to the length the cable.

Each dynamic variable is the sum of a quasi-static component ( q ) and a modal component ( m ):

• u = u q (x, t) + u m (x, t)
is the dynamic axial displacement of the cable ; • v is the dynamic out-of-plane transverse displacement; • w = w q (x, t) + w m (x, t) is the dynamic in-plane transverse displacement ; • T s is the static tension of the cable and is assumed to be constant (w.r.t. x and t) ; • T d = T q + T m is the dynamic tension of the cable and is assumed not to depend on x ;

• w s (x) = ̺A 2T s ℓx -x 2 is the static in-plane displaced shape of the cable.

As one can read (and find details about the modeling of the inclined cable movement) in [START_REF] Wagg | Nonlinear Vibration with Control: For Flexible and Adaptive Structures[END_REF], since we neglect the axial inertial force (∂ tt u = 0) we have the following nonlinear equation about the dynamic tension:

T d = AE ∂ x u + 1 2 (∂ x v) 2 + 1 2 (∂ x w) 2 + dw s dx ∂ x w .
In this current derivation, because of the intrinsic linearity of the setting we have to fit with, we will not consider the non-linearities that arise usually in modelling the vibrations of an inclined cable. For example, [START_REF] Warnitchai | A non-linear dynamic model for cables and its application to a cable structure system[END_REF] use a detailed non-linear PDE model which is decomposed into the first few vibration modes from which the precise non-linear coupling between in-plane and outof-plane vibrations can be seen. Neither do we consider a finite element modelling approach as in [START_REF] Preumont | Vibration control of active structures[END_REF], where one can find an introduction to active tendon control of cables.

We will actually work with a linearized equation of T d :

T d (t) = AE ∂ x u(x, t) + dw s dx (x)∂ x w(x, t) .
Considering that this tension due to the dynamics, is small compared to the static tension (i.e. T d << T s ), the equation of motion for the dynamic analysis of this inclined cable, are then given by, for all (x, t) in (0, ℓ) × (0, ∞),

ρA∂ tt v(x, t) = T s ∂ xx v(x, t), ρA∂ tt w(x, t) = T s ∂ xx w(x, t) + T d (t) d 2 w s dx 2 .

MODELING OF A CONTROLLED INCLINED CABLE

We are interested in the modeling of the robust feedback control of the vibrations of the cable subjected to perturbations that take the form of in-plane oscillations, using an active tendon as a control/measurement device. We use an infinite dimensional state space approach of the robust control, and it implies a linear model of the system. It is the reason why we linearized the dynamic tension T d . Nevertheless, it also means that we loose the (nonlinear) coupling between v and w. Therefore, the fact that the control and perturbations will only act in the sag plane does not allow us to consider the out-of-plane motion v anymore. It won't appear in the construction of our state space model since it satisfies a conservative wave equation that could only be influenced by coupling nonlinearities we don't deal with here.

Partial differential equation model of the inclined cable

One can read in [START_REF] Wagg | Nonlinear Vibration with Control: For Flexible and Adaptive Structures[END_REF] the details about how the decomposition of each displacement (u, v, w) into quasi-static and modal components helps to simplify the analysis of the inclined cable vibration. Anyway, imposing the boundary conditions corresponding to the support motion

u q (0, t) = 0, u q (ℓ, t) = u b (t), ∀t ∈ (0, ∞), w q (0, t) = 0, w q (ℓ, t) = w b (t), ∀t ∈ (0, ∞),
we can calculate the following quasi-static components corresponding to the motion of the cable without taking into account any dynamic response. We obtain

                     u q (x, t) = E q E u b (t) x ℓ - ̺Aℓ 2T s w b (t) x ℓ - x ℓ 2 + λ 2 E q 4E u b (t) x ℓ -2 x ℓ 2 + 4 3 x ℓ 3 w q (x, t) = w b (t) x ℓ - ̺E q ℓA 2 2T 2 s u b (t) x ℓ - x ℓ 2 T q (t) = AE q ℓ u b (t) (1) for (x, t) ∈ (0, ℓ) × (0, ∞), E q = E/(1 + λ 2 /12
) being the equivalent modulus of the cable and

λ 2 = E̺ 2 ℓ 2 A 3 /T 3 s , Irvine's parameter.
Then, assuming that the modal axial displacement is small (u m = 0), the modal dynamic tension satisfies

T m = ̺A 2 E 2T s (ℓ -2x) ∂ x w m
and the in-plane modal displacement is solution of the following PDE on (0, ℓ) × (0, ∞):

ρA∂ tt (w q + w m ) = T s ∂ xx w m -T m ̺A T s ,
submitted to homogeneous Dirichlet boundary conditions w m (0, t) = 0, w m (ℓ, t) = 0, ∀t ∈ (0, ∞) and initial conditions equal to zero. Since can calculate easily ∂ tt w q from (1), we obtain the self-contained equation:

∂ tt w m = T s ρA ∂ xx w m - ̺ 2 A 2 E 2ρT 2 s (ℓ -2x) ∂ x w m - x ℓ w ′′ b + ̺E q ℓA 2 2T 2 s x ℓ - x ℓ 2 u ′′ b .
(2)

Measurement and control of the cable

We are considering an inclined cable as in Figure 1 perturbed by in-plane oscilations (u b , w b ) and connected at its bottom end with an active tendon. As one can read in [START_REF] Preumont | Active tendon control of cable-stayed bridges[END_REF], an active tendon consists of a displacement actuator (e.g. piezoelectric) collocated with a force sensor. Here, a small discussion is necessary. A tendon is principally meant to have an axial movement, that corresponds to a control u c which is then an additive displacement term to the perturbation u b in equation ( 2). But we can also prove that if we only consider this inertial control, we only have access to the symmetric modes of vibration. And because of the linear framework, we loose the complementary parametric control (see [START_REF] Preumont | Vibration control of active structures[END_REF]). To overcome this, we will consider that the control also acts through the in-plane bottom displacement as a term w c which will be an added to the perturbation w b .

Besides, the force sensor allows to define the observation we will measure, in order to build our feedback, as the dynamic tension T d .

Let us now incorporate these facts into the PDE model. Since the perturbations of the system u b and w b will be proportional to cos(ωt), we can assume that u ′′ b (t) = -ω 2 u u b (t) and w ′′ b (t) = -ω 2 w w b (t) and obtain the following state equation

∂ tt w m = T s ρA ∂ xx w m - ̺ 2 A 2 E 2ρT 2 s (ℓ -2x) ∂ x w m + ̺E q ℓA 2 2T 2 s x ℓ - x ℓ 2 u ′′ c - x ℓ w ′′ c -ω 2 u ̺E q ℓA 2 2T 2 s x ℓ - x ℓ 2 u b + ω 2 w x ℓ w b , (3)
with the measurement output

T d = T q + T m = AE q ℓ u b + ̺A 2 E 2T s (ℓ -2x) ∂ x w m . (4)

State space model of an inclined cable's vibrations

A linear infinite-dimensional state space model derived from the PDE model presented above will be used in the sequel. It will take the usual shape [START_REF] Doyle | Robust and Optimal Control Robust and Optimal Control Robust and Optimal Control[END_REF])

     X ′ (t) = AX(t) + B 1 W (t) + B 2 U (t), ∀t ≥ 0 X(0) = 0, Z(t) = C 1 X(t) + D 12 U (t), Y (t) = C 2 X(t) + D 21 W (t).
(5)

In order to fit in this formalism, the following notations are introduced:

• The state vector is X = (w m , ∂ t w m ) ⊤ ; • The exogenous disturbance is W = (W mod , u b , w b ) ⊤
where W mod gathers uncertainty on the model (e.g. the neglected nonlinearities) ; • The control vector U = (u ′′ c , w ′′ c ) ⊤ is the acceleration of the active tendon actuator ;

• The measurement vector Y = T d is the dynamic tension of the cable; • The controlled vector Z = (w m , U ) ⊤ gathers the inplane vibrations and the control input vector;

Moreover, A should be the infinitesimal generator of a C 0semigroup T (t) = e At on a real separable Hilbert space X (see [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF]). Let be the following linear bounded operators We consider the system (5) where the state X(t) belongs to X , and where the control input U (t) ∈ U, the disturbance input W (t) ∈ W, the measured output Y (t) ∈ Y and the "to be controlled output" Z(t) ∈ Z are formally linked in a closed-loop system sketched by the standard description in Figure 2. Using (3) and ( 4), the operator matrices involved in the system (5) are therefore given by:

B 1 ∈ L(W, X ), B 2 ∈ L(U, X ), C 1 ∈ L(X , Z), C 2 ∈ L(X , Y), D 12 ∈ L(U,
A =   0 I T s ρA ∂ xx - ̺ 2 A 2 E 2ρT 2 s (ℓ -2x) ∂ x 0   , B 1 =   0 0 0 d 1 -ω 2 u ̺E q ℓA 2 2T 2 s x ℓ - x ℓ 2 ω 2 w x ℓ   , B 2 =   0 0 ̺E q ℓA 2 2T 2 s x ℓ - x ℓ 2 - x ℓ   , C 1 = I 0 0 0 0 0 C 2 = ̺A 2 E 2T s (ℓ -2x) ∂ x 0 , D 12 = 0 0 1 0 0 1 , D 21 = d 2 AE q ℓ 0 ,
with the tuning parameters d 1 , d 2 ∈ L 2 (0, ℓ). The appropriate functional Hilbert spaces associated with the infinite-dimensional model are the state space X = H 1 0 (0, ℓ) × L 2 (0, ℓ) and the input or output spaces

U = R 2 , W = R 3 , Y = L 2 (0, ℓ), Z = L 2 (0, ℓ) × R 2 .
In order to prove that A is the infinitesimal generator of a C 0 -semigroup on X , one can rely on the classical theory of semi-groups as in [START_REF] Pazy | Semigroups of linear operators and applications to partial differential equations[END_REF] or refer to the book [START_REF] Tucsnak | Observation and Control for Operator Semigroups[END_REF] or the article [START_REF] Tadmor | Worst-case design in the time domain: the maximum principle and the standard H∞ problem[END_REF]. Since equation ( 2) is essentially a damped wave equation with a source term -∂ tt w q , we do not give details of the proof here. This equation can also be seen as a non-homogeneous wave equation perturbed by a bounded operator (see [START_REF] Tucsnak | Observation and Control for Operator Semigroups[END_REF]). Either way, one can prove existence, uniqueness and regularity of the solution when the initial conditions are w(t = 0) = w 0 ∈ H 1 0 (0, ℓ) and ∂ t w(t = 0) = w 1 ∈ L 2 (0, ℓ) and ∂ tt w q ∈ L ∞ (0, ∞; H 2 (0, ℓ)) (obvious with u b , w b ∈ W 2,∞ (0, ∞)): then there exists a unique solution w of (2), that belongs to C([0, ∞), H 1 0 (0, ℓ)) ∩ C 1 ([0, ∞), L 2 (0, ℓ)). This regularity also explains why the operator C 2 ∈ L(H 1 0 (0, ℓ) × L 2 (0, ℓ), L 2 (0, ℓ)) is a bounded one.

H ∞ CONTROL WITH MEASUREMENT-FEEDBACK

This section is devoted to recalling the H ∞ -robust control theorem proved in [START_REF] Bensoussan | On the standard problem of H∞-optimal control for infinite-dimensional systems[END_REF] and/or van [START_REF] Van Keulen | H∞-control with measurement-feedback for linear infinite-dimensional systems[END_REF] that we want to apply on the PDE model we derived. This result gives an equivalence between the H ∞ -robust control with measurementfeedback of an infinite-dimensional system and the solvability of two Ricatti equations. We will explain later how this theorem can be applied to the model we consider, following the assumptions recalled below. For a survey of the H ∞ -control theory with state-feedback in the infinitedimensional case, one can also read van Keulen et al. (1993). The main results in all these articles are a generalization of the finite-dimensional regular H ∞ -control problem (see for instance [START_REF] Doyle | Robust and Optimal Control Robust and Optimal Control Robust and Optimal Control[END_REF] and [START_REF] Tadmor | Worst-case design in the time domain: the maximum principle and the standard H∞ problem[END_REF]) presented in a standard state-space approach.

Let us now make precise what is meant by H ∞ -optimal control (or robust control) with measurement feedback. The description (5) of the system we consider implies that, as in Figure 2, the plant P is to be controlled under the cost function (related to the output Z)

J 0 (U, W ) = ∞ 0 |C 1 X| 2 + |D 12 U | 2 dt and the partial observation Y = C 2 X + D 21 W . The point is to construct a feedback controller K = (L, M, N, R) of the form Φ ′ (t) = (A + M )Φ(t) + N Y (t), Φ(0) = 0, U (t) = LΦ(t) + RY (t). ( 6
)
where Φ, the adjoint state, depends on the measurement Y and leads to the control U . The coupled system is then as follows:

X ′ = (A + B 2 RC 2 )X + B 2 LΦ + B 1 W Φ ′ = (A + M )Φ + N C 2 X + N D 21 W
and introduces an operator

Λ = A + B 2 RC 2 B 2 L N C 2 A + M .
The goal is to find a dynamic measurement-feedback controller K that exponentially stabilizes this system (meaning that Λ is exponentially stable, and also yield to a finite cost J 0 (LΦ + RY, W )) and ensures that the influence of the disturbances on the "to be controlled output", i.e. the ratio

ρ(K) = sup W ∈W J 0 (LΦ + RY, W ) W 2
W is smaller than some specific bound.

The main result we apply to the dynamic measurement feedback control of an inclined cable is the following. Theorem 1. [Proof to be read in [START_REF] Bensoussan | On the standard problem of H∞-optimal control for infinite-dimensional systems[END_REF] or van [START_REF] Van Keulen | H∞-control with measurement-feedback for linear infinite-dimensional systems[END_REF]] Let γ > 0 and assume that the pair A, B 1 is stabilizable and that A, C 1 is detectable.

The following assertions are equivalent:

(i) The γ 2 -robustness property with partial observation hold for the system (5) ; (ii) There exists an exponentially stabilizing dynamic output-feedback controller K of the form ( 6) such that Λ is exponentially stable and ρ(K) < γ 2 ; (iii) There exist two nonnegative definite symmetric operators P, Σ ∈ L(X ) solutions of the Riccati and compatibility equations:

• ∀X ∈ D(A), P X ∈ D(A * ), P A + A * P + P (B 2 B * 2 -γ -2 B 1 B * 1 )P + C * 1 C 1 X = 0 (7) and A -(B 2 B * 2 -γ -2 B 1 B * 1 )
P generates an exponentially stable semigroup ;

• ∀X ∈ D(A * ), P X ∈ D(A), ΣA * + AΣ + Σ(C * 2 C 2 -γ -2 C * 1 C 1 )Σ + B 1 B * 1 X = 0 (8) and A * -(C * 2 C 2 -γ -2 C * 1 C 1 )
Σ generates an exponentially stable semigroup ;

• I -γ -2 P Σ is invertible and Π = Σ I -γ -2 P Σ -1 ≥ 0. ( 9
)
Moreover, if the conditions ( 7)-( 9) hold, then the feedback controller K given by

M = -(B 2 B * 2 -γ -2 B 1 B * 1 )P -ΠC * 2 C 2 , N = ΠC * 2 , L = -B * 2 P, R = 0 (10)
give an exponentially stable operator Λ and guarantees that ρ(K) < γ 2 . Finally, if the solutions to the Riccati equations exists, then they are unique.

One can notice that the feedback controller K given in ( 10) is actually sub-optimal and known as the central controller.

Since B 1 , B 2 , C 1 , C 2 , D 12 and D 21 are bounded operators well defined in the appropriate spaces we will be allowed to apply Theorem 1 if we can confirm that (A, B 1 ) is stabilizable and (A, C 1 ) is detectable.

The proof that (A, B 1 ) is controllable (implying stabilizability) relies on the study of the controllability through u b and w b of the wave equation

∂ tt w m - T s ρA ∂ xx w m + ̺ 2 A 2 E 2ρT 2 s (ℓ -2x) ∂ x w m = -ω 2 u ̺E q ℓA 2 2T 2 s x ℓ - x ℓ 2 u b + ω 2 w x ℓ w b . (11)
The main difficulty comes from the fact that the control input has a prescribed shape in space. The control u b (t) alone is not sufficient because of the symmetry property of the function in x in front of it (see the α i in section 4). We do not wish to give the proof here, but the additional control w c (t) is used to overcome this restriction. One can consult [START_REF] Tucsnak | Observation and Control for Operator Semigroups[END_REF] for details about observability results for the wave equation (even with a first order term), and to see the proof that observability is also true with a control acting only on the boundary of the domain. Therefore, we have (A, B 1 ) stabilizable. Finally, since C 1 X = w m on the whole domain, one proves easily an observability estimate and we obtain that (A, C 1 ) is detectable. This closes the verification of the assumptions of Theorem 1.

Considering that we have now a well-posed robust control problem in infinite dimension, we would like to perform some numerical experiments to illustrate the results we can obtain.

A TRUNCATED MODEL FOR NUMERICAL DESIGN

Model of finite dimension

The goal of this section is to define an appropriate finitedimensional model of the PDE system (5) whose state representation can be written as:

     X ′ N (t) = A N X N (t) + B 2,N U (t) + B 1,N W (t), X N (0) = 0 Y N (t) = C 2,N X N (t) + D 21,N W (t) Z N (t) = C 1,N X N (t) + D 12,N U (t), (12)
where the operators of system (5) are replaced by realvalued matrices computed on a truncated basis of the N first eigenfunctions precisely defined below. X N ∈ R 2N is the state vector, Y N ∈ R N is the measurement output vector, Z N ∈ R N +2 is the to-be-controlled output vector, W ∈ R 3 and U ∈ R 2 are still the exogenous perturbation and the control. The truncation of the PDE system can be seen as a way of coming back to the structural vibrations of the system.

In order to compute these objects, we choose to use everywhere the Hermitian base of L 2 (0, ℓ) given by the eigenfunctions of the (compact self-adjoint) operator Ts ρA ∂ xx . The orthonormal base (φ i ) i∈N * is indeed defined by:

φ i (x) = 2 ℓ sin iπ x ℓ , ω i = iπ ℓ T s ρA
and satisfies for all x ∈ (0, ℓ)

and i ∈ N * , T s ρA ∂ xx φ i (x) = -ω 2 i φ i (x).
This approach meets the Galerkin method used in (Wagg and Neild, 2010, chap 7) and the point is that every y ∈ L 2 (0, ℓ) can be written

y(x) = i≥1 y i φ i (x),
(y i ) i∈N * being a sequence of real numbers satisfying

y i = y, φ i := ℓ 0 y(x)φ i (x) dx and i≥1 y 2 i < ∞.
Given N ∈ N, we compute A N , B 1,N , B 2,N , C 1,N , C 2,N , D 12,N and D 21,N using the truncated basis {φ 1 , . . . , φ N }.

We make the assumption that the tuning parameters d = (d i ) are vectors of real numbers and recall that it is a weighting function of the disturbance signal W mod that corresponds for instance to the "forgotten" non-linearities. We obtain:

A N = block i,j 0 δ ij -ω 2 i δ ij - ̺ ρT s a ij 0 ,
where δ ij is the Kronecker symbol and

a ij = ̺A 2 E 2T s (ℓ -2x) ∂ x φ j , φ i ; B 1,N = vect i 0 0 0 d 1 i -ω 2 u α i ω 2 w β i , B 2,N = vect i 0 0 α i -β i ,
where

α i = ̺E q ℓA 2 2T 2 s x ℓ - x ℓ 2 , φ i , β i = x ℓ , φ i ; C 2,N = block i,j ([ a ij 0 ]) , D 21,N = vect i d 2 i AE q ℓ 1, φ i L 2 0 .
When computing all the terms a ij , one will observe that we get terms different from zero in the non diagonal blocks of the matrices A N and C 2,N . This is due to the choice of the eigenfunctions of the operator Ts ρA ∂ xx which is different from the global operator of the damped wave equation in w m . Besides, one can notice that the calculation of the term α i describe the effective influence of the control u ′′ c on the i-th vibration mode: since α 2j = 0 only the symmetric modes are controllable (odd index i). Therefore, the terms β i = 0 (thus the control w ′′ c ) have a crucial importance to stabilize the other modes.

The numerical aim now is to find a dynamic measurementfeedback controller that exponentially stabilizes the system (12) and makes the influence of the perturbations W on the output vector Z N small in a sense to define. In this short paper we choose to focus on the attenuation of the N first modes of vibrations with controls of small amplitude:

C 1,N = diag i 1 0 0 0 ⊤ , D 12,N = 0 0 1 0 0 1

Numerical simulations

In the numerical simulations we use the parameter values given in Table 1 as chosen in [START_REF] Gonzalez-Buelga | Modal stability of inclined cables subjected to vertical support excitation[END_REF] to approximately match a typical full-scale bridge cable inclined at θ = 20 • to the horizontal, of length 400m, mass per unit length 130kg.m -1 and tension 8000kN: This gives the following parameters: E q = 174×10 9 N.m -2 , λ 2 = 1.74 and ̺ = ρg cos θ = 12.35 × 10 6 kg.s -2 .m -2 . These values correspond to an inclined steel cable experiment that could be used in a next step to implement our controllers. We also choose the perturbation frequencies ω u = ω w = 55.4rad.s -1 , i.e. close to the second linear inplane natural frequency mode where the studies usually focus when studying the effect of the nonlinear modal coupling (not considered here).

We can illustrate the results of the closed loop control based on this truncated state model of the PDE modelling in several ways. Through the singular values of the frequency response, Figure 3 presents the attenuation of the first modes of vibrations for the case N = 5, with respect to the uncontrolled open loop. The H ∞ optimal controller is computed using the hinfsyn Matlab function. The goal of this article was to study the robust measurement feedback control of an infinite-dimensional state space model of an inclined cable. After dealing with several modelling issues, we were able to ensure the H ∞ -control of the system under the condition of solvability of two Riccati equations. This allowed us to perform some numerical computations on a truncated model in order to illustrate the action of the robust controller. We examined the possibility of connecting the inclined cable to an active tendon in order to bring active damping into the cable structure and as far as we know, there exists no such study of the robust control of an inclined cable when the partial differential equation model is used.

We can mention several new directions that we intend to pursue as future extensions of this study. Some of them concern the modeling of the controlled inclined cable. First we would propose a more sophisticated modeling of the output Z N (tuning the attenuation of the modes for instance) and of the H ∞ channel W → Z N but also to compare the control results we manage to obtain applied to the non-linear model of 2 or more modes that one can read in [START_REF] Wagg | Nonlinear Vibration with Control: For Flexible and Adaptive Structures[END_REF]. Then the next challenging step of this preliminary study would be to propose a multi-objective design problem, where the feedback controller has to respond favorably to several performance specifications: typically we could consider the active tendon control as a mixed H 2 /H ∞ synthesis problem, where the H ∞ channel is used to enhance the robustness and the H 2 channel guarantees good performance despite e.g. measurement noise or uncertainty on the cable dynamic. Last but not least, improvements in the modeling of the inclined cable itself could be studied. Indeed when linearizing the infinite dimensional model, we lost the parametric control and the nonlinear coupling between w m and v m and therefore needed another control w c and neglected the out-of-plane motion v m . Another objective would indeed be to find a way, maybe using a modified version of the dynamic tension that carries the coupling, to include some nonlinearities in the control loop in order to define a more complete system to stabilize. This will be considered in future studies.

  Fig. 1. Inclined Cable

Fig. 2 .

 2 Fig.2. Closed-loop system. Plant P, controller K.

Fig. 3 .

 3 Fig. 3. Singular values of the frequency respond of the truncated model in open and closed loop 5. DISCUSSION AND CONCLUSION

Table 1 .

 1 Cable characteristics

	Cable length	ℓ	1.98 m
	Density	ρ	1.34 × 10 6 kg.m -3
	Cross sectional area	A	0.5 × 10 -6 m 3
	Static tension	Ts	205 N
	Steel Young's modulus	E	200 × 10 9 N.m -2


Jean-Marc Azaïs
email: jean-marc.azais@math.univ-toulouse.fr

Li-Vang Lozada-Chang

A toolbox on the distribution of the maximum of Gaussian processes

Keywords: AMS Subject Classification: Primary 60G15, Secondary 60G60 Distribution of the maximum, Ornstein-Uhlenbeck process, Brownian motion, Sine-Cosine process, Bessel process, matlab toolbox

In this paper we are interested in the distribution of the maximum, or the maximum of the absolute value, of certain Gaussian processes for which the result is exactly known.

Introduction

This paper is devoted to a unified presentation of some classical results on the distribution of the maximum of stochastic processes. These results have a rather simple form and can be put in a toolbox. Apart from some general methods that give bound or asymptotic expansions for some classes of stochastic process like, for example, the Rice method [START_REF] Azaïs | Level sets and extrema of random processes and fields[END_REF]. There exist a short list of cases where the distribution is known exactly. The simplest and the most famous case is the Brownian motion (see below). A full list is given in [START_REF] Azaïs | Level sets and extrema of random processes and fields[END_REF] p. 3-4. Among it, some results are very complicated and are not included in the present presentation, some are very simple and some other need clarification see, for example, Section 2.4. After recalling these classical results, adding some new ones, we give in Section 4 a description of the toolbox. Our intention is to put a set of dispersed results at the disposal of the community. Throughout the paper, X((t) (or U (t) ,W (t)) is a random process with real values and 1 parameters (except for the Brownian motion in dimension n, see Section 2.4). Set

M t = sup 0<s<t X(s), M * t = sup 0<s<t |X(s)|.
The aim of the paper is to compute P{M t ≤ u} and P{M * t ≤ u} or similar expression.

2 Some results on Gaussian processes

The Brownian motion

Most of the result of this section and the next one are an application of the reflection principle and a good reference among many others is [START_REF] Dudley | Real Analysis and Probability[END_REF] for all the results cited, see also [START_REF] Borodin | Handbook of Brownian Motion: Facts and Formulae[END_REF].

Let W (t) be a standard Brownian motion, we mean the 1-dimensional Brownian motion starting from zero (also called Wiener process).

Unilateral case: It is well-known that for u > 0,

P {M t ≤ u} = 2Φ u √ t -1,
where Φ stands for the standard normal cumulative distribution function [START_REF] Dudley | Real Analysis and Probability[END_REF]p.459].

We turn now to the bilateral case. The distribution of maximum absolute value of the Brownian motion is given by

P {M * t ≤ u} = ∞ k=-∞ (-1) k Φ u(2k + 1) √ t -Φ u(2k -1) √ t = 4 π ∞ k=0 (-1) k 2k + 1 exp - (2k + 1) 2 π 2 t 8u 2 .
The second expression is less known, we refer the reader to [START_REF] Feller | An Introduction to probability theory and its Applications[END_REF]p. 342]. A detailed proof can be found in [START_REF] Chung | A Course in Probability Theory[END_REF]p. 223]. This infinite sum has two advantages respect to the first one. Firstly, it involves only the exponential function. It avoids the computation of the cumulative distribution function of the normal distribution. Secondly, it converges very quickly for large values of t.

Extensions : The most simple is the case of a Brownian motion with a drift µ. In such a case

P{ sup 0<s<t W (s) + µs ≥ u} = Φ u -µt √ t + exp(2µu)Φ u -µt √ t
where Φ = 1 -Φ. A proof can be found in [START_REF] Borodin | Handbook of Brownian Motion: Facts and Formulae[END_REF]. A generalization of this result to the case of two different drifts is given in [START_REF] Boukai | An explicit expression for the distribution of the supremum of Brownian motion with a change point[END_REF].

Extensions are possible considering the case of the maximum of a Brownian motion W (s) on an interval [0, t], subject to the condition W (T ) = η where t and T are distinct and η is non-zero. We consider here the case t < T and refer to [START_REF] Beghin | On the maximum of the generalized Brownian bridge[END_REF] for the other case. Regression formulas imply that the conditional process can be represented as

X(s) = W (s) - s T (W (T ) -η).
This formula implies, in particular, that a linear drift µt can be added to the Brownian motion without modifying the expression of X(s). The main results of [START_REF] Beghin | On the maximum of the generalized Brownian bridge[END_REF] are

P sup 0≤s≤t X(s) ≤ u = exp -2 u(u -η) T Φ u(2t -T ) -ηt T t(T -t) + Φ uT -ηt T t(T -t) ,
and for the bilateral case

P sup 0≤s≤t |X(s)| ≤ u = +∞ k=-∞ (-1) k exp -2 ku(ku -η) T × Φ uT -2ku(T -t) -ηt T t(T -t) -Φ -uT -2ku(T -t) -ηt T t(T -t) .

The Brownian Bridge

The simplest way of defining of the Brownian bridge is considering the distribution of W (s), 0 ≤ s ≤ 1 conditional to W (1) = 0. It can be written as X(s) = W (s)-sW (1), 0 ≤ s ≤ 1.

For the unilateral case it holds:

P {M 1 ≤ u} = 1 -exp(-2u 2 ),
which corresponds to the Rayleigh distribution. By a direct scaling argument we can extend the result to an interval [0, t],

P sup 0≤s≤t W (s) - s t W (t) ≤ u = 1 -exp -2 u 2 t .
Bilateral case: the distribution of the maximum of the absolute value is given by the infinite sum

P sup 0≤s≤t W (s) - s t W (t) ≤ u = ∞ k=-∞ (-1) k exp -2 k 2 u 2 t .

The Sine-Cosine process

Let X(t) be the process defined by

X(t) = ξ 1 cos(ωt) + ξ 2 sin(ωt), (2.1)
where ω is a positive parameter and ξ 1 , ξ 2 are two standard normal variables. This process can be written equivalently as

X(t) = Z cos(ωt + θ),
where Z and θ are independent random variables with Rayleigh and uniform (on [0, 2π]) distribution respectively. First it is clear that for t > 2π/ω, M t = M * t = Z. For u > 0 and 0 < t ≤ π/ω, the distribution of M t was given by Berman [START_REF] Berman | Excursions above high levels for stationary Gaussian processes[END_REF]. Delmas [START_REF] Delmas | Projections on spherical cones, maximum of Gaussian fields and Rice's method[END_REF] gives it for π/ω < t < 2π/ω. Putting these results together, we obtain the following general statement. For u > 0,

P{M t ≤ u} =          Φ(u) -tω 2π e -u 2 /2 , 0 < t ≤ π/ω, Φ(u) - tω 2π e -u 2 2 + 1 2π tω π exp - u 2 1 -cos(s) sin 2 (s) ds, π/ω < t < 2π/ω, 1 -e -u 2 2 , t ≥ 2π/ω.
Observe that in fact the second case is the most general and it can be extended, with some precautions, to the case u < 0 which is anyway not very relevant.

For M * t we can establish a similar result using the method of Delmas. As far as we know, this result is new. See detailed proof in Section 3.

Proposition 2.1. For u > 0

P{M * t ≤ u} =      2Φ(u) -1 - tω π e -u 2 2 + 1 π tω 0 exp - u 2 1 + cos(s) sin 2 (s) ds, 0 ≤ t ≤ π/ω 1 -e -u 2 2 , t > π/ω.

The normalized Bessel process

Let W n (t) be a n-dimensional Brownian motion. The normalized Bessel U n (t) process of order n is defined as

U n (t) = W n (t) √ t ,
where • stands for the corresponding Euclidean norm. Set

G n t (u) = P sup 1<s<t U n (s) ≤ u .
DeLong [START_REF] David | Crossing probabilities for a square root boundary by a Bessel process[END_REF] gives a formula for G n t (u). The explicit expression, where we have corrected a little sign error in (2.4) as pointed out also by Estrella [START_REF] Estrella | Critical values and p values of Bessel process distributions: computation and application to structural break tests[END_REF], is

G n t (u) = ∞ i=1 α i (u)t -β i (u) (2.2)
where β i (u) is the i th root as a function of x of

M x + n 2 , n 2 , - u 2 2 = 0, (2.3)
M represents the confluent hypergeometric function (see [START_REF] Stegun | Handbook of mathematical functions with formulas, graphs, and mathematical tables[END_REF]§13]) and

α i (u) = - e -u 2 /2 (u 2 /2) n/2 M (β i + n/2, n/2 + 1, -u 2 /2) Γ(n/2 + 1)β i (u) ∂ ∂E M (E + n/2, n/2, -u 2 /2) E=β i (u)
.

(2.4)

For n = 1 it is possible to consider the one-sided problem. More precisely, now we focus on calculating

P max 1<t<t W (s) s 1/2 ≤ u (2.5)
where W (t) is the 1-dimensional standard Brownian motion. Tables of values computed from (2.5) were presented by Delong [START_REF] David | Crossing probabilities for a square root boundary by a Bessel process[END_REF] but without giving the explicit formula he used. Following the DeLong's procedure, Estrella and Rodrigues [ER05, formula (19)] obtain

P max 1<s<t W (s) s 1/2 ≤ u = ∞ k=1 -t -v k /2 φ(u)D v k -1 (-u) v k ∂ ∂v D v (-u) v=v k (2.6)
where D is the parabolic cylinder function (see [AS72, §19]), v i is i-th root of D v (-u) = 0 as a function of v and φ stands for the standard normal density.

The Ornstein-Uhlenbeck process

In this paper the Ornstein-Uhlenbeck(OU) process X(t) will be defined as the centered Gaussian stationary process with covariance r(t) = exp(-|t|). This process and the Bessel process are related by the following. When n = 1, the following representation holds

|X(t)| D = U 1 (e 2t ).
Moreover if X 1 (t), ..., X n (t) are n independent copies of the OU process,

X 2 1 (t) + • • • + X 2 n (t) D = U 2 n (e 2t ).
Let define

F t (u) = P sup s∈[0,t] X(t) ≤ u , F * t (u) = P sup s∈[0,t] |X(t)| ≤ u .
We have in particular

F * t (u) = G 1 exp(2t) (u) = ∞ i=1 α i (u) exp -β i (u)t (2.7)
where G is defined in (2.2). The first results in the literature concern the unilateral case and dates back to Slepian [START_REF] Slepian | The one-sided barrier problem for gaussian noise[END_REF]. A direct application of the reflection principle for the level u = 0 shows that

F t (0) = 1 2 -2P X(0) < 0, X(t) > 0 = 2 π arcsin e -t .
Another interesting and rather unknown result is due to Newell and Rosenblatt [START_REF] Newell | Zero crossing probabilities for Gaussian stationary processes[END_REF].

They give an expression of F t (u) under the form of a series with non very explicit terms.

In fact Formula (2.6) implies that

F t (u) = - ∞ k=1 φ(u)D v k -1 (-u) v k ∂ ∂v D v (-u) v=v k exp(-tv k ).
3 Proof of proposition 2.1

Without loss of generality we can assume that ω = 1. We have, since we are considering at most a half period

P{M * t > u} = P{|X(0)| > u} + P{|X(0)| < u, u u > 0} + P{|X(0)| < u, d -u > 0} = P{|X(0)| > u} + 2P{|X(0)| < u, u u > 0}
where d -u and u u are the numbers of up-crossings of u. The variable u u 1I {|X(0)|≤u} takes the value 0 or 1. Thus by the Rice formula

P{|X(0)| < u, u u > 0} = E u u 1I |X(0)|≤u = t 0 ds u -u E (X (s)) + X(0) = x, X(s) = u p X(0),X(s) (x, u)dx (3.1)
Using Formula (2.1), it is easy to check that under the condition

X(0) = x, X(s) = u, X (s) = 1 sin(s) [u cos(s) -x].
On the other hand

p X(0),X(s) (x, u) = 1 2π sin(s) exp - x 2 + u 2 -2 cos(s)xu 2 sin 2 (s) .
The right hand side of (3.1) is equal to

t 0 ds u cos(s) -u 1 2π e -u 2 /2 [u cos(s) -x] 2 sin 2 (s) exp - 1 2 u cos(s) -x sin 2 (s) dx = 1 2π e -u 2 /2 t 0 1 -exp - u 2 (1 + cos(s)) 2 2 sin 2 (s) ds = t 2π e -u 2 /2 - 1 2π t 0 exp - u 2 (1 + cos(s)) sin 2 (s) ds.

Writing a toolbox

Since most of the special functions are available in Matlab, the second author made a toolbox MAXGPBOX that is available at http://www.math.univ-toulouse.fr/˜azais/softwares.php and that permits to make all these computations. The more complex formulae we work with in this paper are those concerning the Bessel and OU process. Therefore, before describe the toolbox, it is worth discussing some few aspects about the numerical computations involved.

4.1 Computing (2.2) and (2.6)

In the computation of (2.2) there is involved the finding of the roots of equation (2.3). This part is the more complex and computational time demanding. Following Estrella (see [START_REF] Estrella | Critical values and p values of Bessel process distributions: computation and application to structural break tests[END_REF] for details) we compute the roots using the Matlab standard function. The initial zero approximation is giving by steps of length 1. Propositions of [START_REF] Estrella | Critical values and p values of Bessel process distributions: computation and application to structural break tests[END_REF] prove that this choice is optimal to capture all the roots. On the other hand, the partial sums in the RHS of (2.2) converge very quickly. Thus only a few roots are necessary to obtain a accurate approximation of the infinite sum, and consequently of the computation of G n t . Typical with 5 roots an approximation of order 10 -5 is obtained. To compute (2.6), once again, the challenging part is the calculation of the roots. Although, there is not similar Propositions to those of [START_REF] Estrella | Critical values and p values of Bessel process distributions: computation and application to structural break tests[END_REF] to localize the roots, in [START_REF] Estrella | One-sided test for an unknown breakpoint: theory, computation, and application to monetary theory[END_REF] the authors use a similar grid strategy to find the zeros with good results. In our experience, we need to use a grid of large 0.5 to getting all the roots. As pointed by Estrella and Rodrigues we notice that the computation of (2.6) is more complex than (2.2). See [START_REF] Estrella | One-sided test for an unknown breakpoint: theory, computation, and application to monetary theory[END_REF] for the reasons of this. To evaluate our implementation of algorithms we compute the values of G n t to compare with the ones in [START_REF] Donald | Tests for parameter instability and structural change with unknown change point[END_REF], [START_REF] David | Crossing probabilities for a square root boundary by a Bessel process[END_REF], [START_REF] Estrella | Critical values and p values of Bessel process distributions: computation and application to structural break tests[END_REF] and [START_REF] Estrella | One-sided test for an unknown breakpoint: theory, computation, and application to monetary theory[END_REF]. We also ran simulations to calculated the values. In general, for the two-sided problem our results matched Estrella's.

In the case of DeLong's the differences correspond to those found and explained by Estrella, i.e. n = 1, u = 1. In the case of the one-sided problem, our computations permit to recover the critical values in Table 1 pxx: Gives the value of the distribution function at u of the maximum the process.

qxx: Gives the inverse of the distribution function at p of the maximum the process.

pnxx: Gives the value of the distribution function at u of the maximum norm of the process.

qnxx: Gives the inverse of the distribution function at u of the maximum norm of the process.

For details, about the routine call, and parameters see the comments inside each program.

  of [ER05] and the values in DeLong's Table 1 [DeL81, p. 2210]. Throughout the section the suffix xx in the name of routines represents any of the suffixes in the previous table.

	4.2 Toolbox description
	suffix stands for
	bb	Brownian Bridge
	bm	Brownian Motion
	bp	normalized Bessel Process
	ou	Ornstein-Uhlenbeck process
	sc	Sine-Cosine process
	Routines:


Thi Thuong

Huyen Nguyen

Thierry Duval
email: thierry.duval@irisa.fr

3-Point++: a new Technique for 3D Manipulation of Virtual Objects

Keywords:

Manipulation in immersive Virtual Environments (VEs) is often difficult and inaccurate because humans have difficulty in performing precise positioning tasks or in keeping the hand motionless in a particular position without any help of external devices or haptic feedback. To address this problem, we propose a set of four manipulation points attached to objects (called a 3-Point++ tool, including three handle points and their barycenter), by which users can control and adjust the position of objects precisely. By determining the relative position between the 3-Point++ tool and the objects, and by defining different states of each manipulation point (called locked/unlocked or inactive/active), these points can be freely configured to be adaptable and flexible to enable users to manipulate objects of varying sizes in many kinds of positioning scenarios.

INTRODUCTION

Object position and orientation manipulation are among the most fundamental and important interactions between humans and environments in VR applications. Many approaches have been developed to maximize the performance and the usability of 3D manipulation. However, each manipulation metaphor has its own limitations and cannot be generally used in a broad variety of VEs. We are especially interested in precise manipulation of large objects in VEs. This task is usually difficult because of user's view obstruction caused by the size of objects during their positioning stages [START_REF] Bowman | An evaluation of techniques for grabbing and manipulating remote objects in immersive virtual environments[END_REF]. It is more difficult when the user has to place large objects with required precision. Some approaches [START_REF] Pierce | Voodoo dolls: Seamless interaction at multiple scales in virtual environments[END_REF][START_REF] Poupyrev | The go-go interaction technique: non-linear mapping for direct manipulation in vr[END_REF][START_REF] Stoakley | Virtual reality on a wim: Interactive worlds in miniature[END_REF] have been proposed to manipulate objects at a distance by creating their miniature models or by expanding the user's virtual arm. These propositions have an advantage for large objects manipulation: when the user has an overall view of objects or of environment, it is easier for him to know how to move these objects to the target position and orientation without worrying about obstruction problems. However, their main issue is that small movements of the miniature models or of the user's virtual hand from a distance are often magnified in the environment, making precise positioning difficult. A reasonable distance, at which the size of objects is not too disturbing and the user can still determine where to place them, is hardly found sometimes.

Some approaches have been proposed to manipulate objects more precisely and efficiently. PRISM [START_REF] Frees | Precise and rapid interaction through scaled manipulation in immersive virtual environments[END_REF] proposes a dynamical adjustment method which determines the relationship between physical hand movements and the controlled object motion, making it less sensitive to the user's hand movements. However, the user may feel a sense of incompatibility because of the difference between the visual feedback and motor control. Osawa [START_REF] Osawa | Two-handed and one-handed techniques for precise and efficient manipulation in immersive virtual environments[END_REF] proposes a technique adding a viewpoint adjustment stage to enlarge the scene when the hand grasping the virtual object is moving slowly. Nevertheless, this adjustment may influence the user's immersion and may cause fatigue when the user manipulates large objects. In addi- tion, there is no orientation manipulation in this technique, it means that it is incomplete for object manipulation in general. Precise manipulation is also often required in assembly modeling systems through constraint-based manipulation. These systems usually demand knowledge-based descriptions to be integrated in advance into a constrained behavior manager [START_REF] Liu | Constrained behavior manipulation for interactive assembly in a virtual environment[END_REF]. This manager realizes assembly relationship recognition, constraint solving and constrained motion. It can improve the precision of manipulation but it is quite complicated to apply them in the general case of manipulation.

The 3-hand manipulation technique [START_REF] Aguerreche | 3-Hand Manipulation of Virtual Objects[END_REF] may facilitate the 3D manipulation by determining the position of objects through the position of three non-aligned manipulation points on a plane. However, this technique is mainly devoted to multi-user collaborative manipulation and it is quite difficult for one user to manipulate objects. In this paper, we want to enhance this technique for a general purpose of manipulating objects of varying sizes, especially large objects, in many kinds of positioning scenarios.

3-POINT++ TECHNIQUE FOR DIRECT MANIPULATION

We present an interaction technique which improves significantly the 3-Hand manipulation technique [START_REF] Aguerreche | 3-Hand Manipulation of Virtual Objects[END_REF] for manipulating objects precisely and efficiently to a target position and orientation. This 3-Point++ tool consists in a set of four points (three handle points and their barycenter), that is attached to an object. This technique is based on the idea that for any kind of object, its precise position and orientation are always explicitly described by the position of the three handle points of this tool. On the other hand, by determining the position of this 3-Point++ tool, the position of the object can be calculated. By moving these points, the position and the orientation of the object are controlled. The barycenter can be used for approximate positioning to control the object directly without constraint, while the three handle points are used for precise positioning. This is the reason why the barycenter has 6 Degrees Of Freedom (6-DOF) in manipulation, while the three handle points has only 3-DOF. When the position of one of three handle points changes, the position and the orientation of the barycenter are recalculated and so are the position and the orientation of the attached object. If one handle point is manipulated, the object is rotated around an axis created by the two other handle points (Figure 1). If two handle points are manipulated at the same time, the object is rotated around the third handle point.

Configuring the offset of manipulation points relatively to the manipulated object

In order to adapt the 3-Point++ tool to objects of varying sizes, we propose an extension for configuring the shape formed by the three handle points as well as the offset distance between their barycenter and the object. This stage can be executed in the first place because depending on the shape of object, the relative position of the tool and the object need to be refined earlier for a better manipulation. The adjustment of these manipulation points can neither move nor rotate the attached object. It only allows a user to place the handle points relatively to the object to manipulate. During this adjustment, the offset distance between the barycenter of the manipulated object and the barycenter of the three handle points is re-computed. This offset will be used for the further real manipulation.

Locking and unlocking manipulation points

In the normal mode of manipulation, all the manipulation points are unlocked. When these points move, the object moves respectively. But the object movement depends on how many points are controlled and how they are controlled. If there is no locked manipulation point, there is a different effect on the movement of the attached object depending whether the movement is caused by the handle points or by their barycenter. The barycenter of the handle points changes its position with 6-DOF and the position of the object changes the same way. It means the position and the orientation of the manipulated object are determined by the position and the orientation of the barycenter itself. When one or two of the handle points are controlled, the other handle points stay at the same position. Because the position and the orientation of object are always calculated by the position and the orientation of the barycenter and the triangle shape formed by the three handle points can change, the translation of their barycenter (and also the translation of the object) is different from the translation of the controlled handle points. It causes the slide of the object on the surface of the tool and it makes the object move imprecisely. This is why we add another manipulation mode where the user can lock one or two handle points and control the other points to get the target position (Figure 2). By combining the configuration of relative position as mentioned above and this mode of locking/unlocking points, the user can decide which part of the object he does not want to move and which other part to be controlled to reach the wanted position. In addition, using such locked points allows the user to focus on the active points and makes the user more comfortable to adjust the orientation and the position of the manipulated object.

PRELIMINARY STUDY

We conducted an experiment to evaluate the usefulness and efficiency of our proposition for manipulating virtual objects. The task of 18 subjects was to place three marks on each large 3D alphabet letter onto three corresponding targets on a support frame respectively at three different levels of difficulty. These levels were represented by the distance threshold d th between one mark on each letter and its corresponding target. The more this threshold decreases, the more the manipulation needs to be precise. We tested the two manipulation techniques with three distance thresholds: 10, 6 and 3 centimeters which are considered easy, medium, and difficult level respectively. We measured the mean time to accomplish the task by the 3-Point++ technique and by a well-known technique using a 3D cursor to control an object directly with 6-DOF. P values of average manipulation time of the two techniques at different levels of difficulty were calculated using repeated measures ANOVA and Tukey-Kramer post hoc analysis. The result revealed that there was no significant difference between the two techniques at the three levels of difficulty as well as in the general case. These statistical results could be explained by the fact that 3-Point++ technique usually took more time than using the 6-DOF technique because it was more complicated for users to change the mode of control, to reconfigure the offset distance between the manipulation points and object. This fact was also represented in the subjective estimation. So to improve our technique, its ease of use and intuitiveness need to be upgraded.

Figure 1 :

 1 Figure 1: The handle point P2 is manipulated, the object is rotated around an axis created by the two handle points P1 and P3

Figure 2 :

 2 Figure 2: Two examples of locking and manipulating handle points

CONCLUSION AND FUTURE WORK

We have presented the 3-Point++ manipulation technique, a new metaphor for the 6-DOF manipulation of virtual objects by controlling three manipulation points and their barycenter. In the future, the ease of use of our technique need to be improved to make it easier to use. We will have to enhance our approach for release adjustment to enable a more precise release in manipulation. This adjustment could improve the problem of quick release by estimating the position that seems to be the intended release position.