Morgan Le Ch Én Échal
email: morgan.lechenechal@b-com.com

Irt B<>com

Bruno Arnaldi
email: bruno.arnaldi@irisa.fr

Thierry Duval
email: thierry.duval@irisa.fr

From 3D Bimanual Toward Distant Collaborative Interaction Techniques: An Awareness Issue

Keywords: A.1 [Introductory and Survey], H.5.3 [Information Interfaces and Presentations]: Group and Organization Interfaces-Computer-supported cooperative work, I.3.6 [Computer Graphics]: Methodology and Techniques -Interaction techniques, I.3.7 [Computer Graphics]: Three-Dimensional Graphics and Realism-Virtual reality

These two figures state the question: How close are 3D bimanual interactions to collaborative interactions?

INTRODUCTION

Human-Computer Interfaces (HCI) have often taken advantage of the natural bimanual skills of humans. In the late of eighty's, pioneers as Zimmerman imagined advanced interfaces such as the Data Glove TM [START_REF] Zimmerman | A hand gesture interface device[END_REF] enabling natural gestures of the hands as interface. This kind of interface paved the way to first work about natural bimanual interactions for VR. It took place fifteen years ago and proposed a set of 3D manipulation techniques [START_REF] Cutler | Two-handed direct manipulation on the responsive workbench[END_REF] as intuitive and immersive as possible. HCI using both hands rather than a single one provide interaction techniques closer to the real ones and can increase the accuracy of the interactions. These research work on bimanual interactions are strongly based on natural skills of humans including the proprioceptive sense as an essential part. Thus, some work proposed asymmetric interactions [START_REF] Hinckley | Two-handed virtual manipulation[END_REF] (e.g considering a different role for the dominant hand and the non-dominant one) inspired by physiological results and proved that it is an efficient and natural way to interact using both hands. In this paper, we restrict bimanual interactions to this kind of systems, using free motion and gestures of the hands to interact.

Another way to improve performances is to provide multi-user systems in order to cross the skills of each collaborators. Computersupported cooperative work (CSCW) is a powerful tool to help people to collaborate in a distant and efficient way. Distant users can share some virtual contents to collaborate, analyze and act on and with them in real-time. CSCW systems aim to virtually reduce the real distance between collaborators by sharing a common virtual space. For instance, users can experiment teleconferencing experiences as if they were in the same room [START_REF] Beck | Immersive groupto-group telepresence[END_REF] or share 3D models in a common virtual environment (VE) [START_REF] Rekimoto | Transvision: A hand-held augmented reality system for collaborative design[END_REF]. Users are not always collaborating in a remote way: they can also collaborate locally, which obviously simplify the understanding of the other's actions (e.g face-to-face setup [START_REF] Mulder | A modular system for collaborative desktop vr/ar with a shared workspace[END_REF] or co-located setup [START_REF] Simon | Multi-viewpoint images for multi-user interaction[END_REF]). But when distant users are collaborating, specific feedbacks must be provided to let users be aware of the other participants, their actions, their focus and their interaction capabilities. These types of feedback, as well as every available feedbacks about user's actions on the system, are encapsulated in the term awareness.

Virtual reality (VR) systems are widely used in conception, training or simulation and provide useful feedback in immersive setups. In this context, these feedbacks are essential because the user is D R A F T immersed in a virtual world and isolated from the real world in order to have a sensory-motor and cognitive activity in an interactive virtual world [START_REF] Fuchs | Virtual reality: concepts and technologies[END_REF]. Research on CSCW and VR converge in the field of collaborative virtual environment (CVE). Indeed, CVE have focused in both interaction techniques and awareness provided in CSCW and VR applications. Thus, CVE propose shared virtual environments to perform immersive virtual tasks such as scientific data analysis [START_REF] Fleury | Evaluation of remote collaborative manipulation for scientific data analysis[END_REF] or training tasks [START_REF] Luna | Collaborative virtual en-vironments for training: A unified interaction model for real humans and virtual humans[END_REF] in a natural way. CVE aim to improve performances in many ways, such as usability, task completion time or environment perceptions. But CVE require adequate interaction techniques to support collaborative interactions and specific metaphors for concurrent interactions handling as well as awareness features.

In this survey, we try to analyze both fields, 3D bimanual and collaborative interaction techniques, and we aim to draw a parallel between some of these techniques as extensions from bimanual toward collaborative interactions. We will address the main issue to extend bimanual toward distant collaborative interactions that is the lack of awareness. Section 2 presents an overview of both bimanual and CVE applications motivating our analysis. Then, section 3 classifies the interactions according to the main interactive features. Section 4 presents the principles used to enhance interaction performances proposed by the literature in bimanual setups. Section 5 explains the need for CVE to overcome the limitations of bimanual interactions developing collaborative interaction techniques and the involving awareness issues that will be addressed in section 6. Finally, section 7 concludes this paper.

OVERVIEW & MOTIVATIONS

Let us consider two users in collaborative interactions, if each user interacts with only one hand, we state that it can be compared to bimanual interactions that can be considered as a collaboration between both hands 1 . Indeed, bimanual interactions handle two different inputs of the same user that could be, in some cases, one input per user for two different users. Figure 1 illustrates this concept. This parallel is even more obvious if considering tabletop/surfacebased interactions (e.g tactile interactions) handling several inputs regardless of the number of users (locally [START_REF] Bérard | The magic table: Computer-vision based augmentation of a whiteboard for creative meetings[END_REF] or remotely [START_REF] Coldefy | DigiTable: an interactive multiuser table for collocated and remote collaboration enabling remote gesture visualization[END_REF]). Thus, bimanual interactions can be a great entry point to understand key issues about collaboration. Several work have been done in bimanual VR applications with advanced features about 3D interaction techniques and awareness. Some of these features can certainly be useful in CVE and extended to handle more than two inputs with more than one user. This will lead to propose enhanced awareness features for collaboration.

First, subsection 2.1 focuses on bimanual requirements, then subsection 2.2 lists all CVE requirements. Last, subsection 2.3 highlights similarities between collaborative and bimanual interactions, and subsection 2.4 sums up the section.

Bimanual requirements

Basically, systems providing bimanual interactions must consider both hands of the user to control a single or a couple of hardware interfaces. In our study, we only consider bimanual interactions managed through two 6DOF tracking of the hands. We are not interested about static or 2D interfaces because they do not allow the use of advanced 3D interaction techniques based on proprioception.

In real bimanual interactions, people act with little cognitive effort [START_REF] Cutler | Two-handed direct manipulation on the responsive workbench[END_REF], almost seamlessly with a millimeter precision [START_REF] Hinckley | Two-handed virtual manipulation[END_REF]. Ideally, this accuracy and simple cognitive task aspect must also be obtained in bimanual interactive applications. To achieve it, physiological studies were performed in bimanual interactions and revealed the significant part of the kinaesthetic feedback and the 1 With some limitations such as the use of two dominant hands in the collaborative case.

proprioceptive sense when interacting collaboratively with both hands [START_REF] Balakrishnan | The role of kinesthetic reference frames in two-handed input performance[END_REF]. The cognitive load can also be influenced by the switching between modes of interaction (uni-manual vs. bimanual [START_REF] Cutler | Two-handed direct manipulation on the responsive workbench[END_REF]).

Moreover, as any interactive system, bimanual systems must handle interactive features such as responsiveness and awareness. However, dealing with two independent inputs rather than a single one involves more difficulties. The user must be aware of the individual action of both hands in the system, as well as the common action applied by the merging of the two inputs. Thus, enhanced interactions (enabling infeasible tasks otherwise) compared with a uni-manual setup can be proposed, and the system must consider them and let the user to be aware of these additional interaction capabilities.

CVE requirements

To begin, we can paraphrase Joslin et al. [START_REF] Joslin | Collaborative virtual environments: from birth to standardization[END_REF] saying that "CVE is VR plus lots of features". Indeed, it aims to propose VR applications for several users acting thanks to collaborative interactions. Users can be distant, thus a network layer has to handle consistency between the different sites. Moreover, it has to provide multi-modal communication and has to handle the virtual and real spaces of the heterogeneous users.

The deep analysis of networking issues is not the purpose of this paper, a good survey can be found in [START_REF] Fleury | Architectures and mechanisms to efficiently maintain consistency in collaborative virtual environments[END_REF]. The network layer is directly developed from networking research results and we can consider that it provides us a reliable framework. It deals with shared objects over the network that can be dynamically modified by different users. Consequently it has to handle concurrency access, as well as latency issues due to the real-time constraint. Object distribution paradigms, such as the mirror/reference distribution paradigm, have been proposed. They adapt dynamically the distribution of objects on the network to enhance local interaction and distant collaboration. Indeed, CVE have to handle access rights policy, and choose on which site the behavior of an object is executed. This choice is influenced by network performances and current interactions. The network latency involves inconsistencies between distant sites in co-dependent interaction phase, and the distributed system must minimize the impact of inconsistency to not disturb remote co-dependent interactions.

The other features are related to interaction techniques. The multi-user aspect involves the need for specific interaction techniques, such as a 3-hand manipulation technique [START_REF] Aguerreche | Short paper: 3-hand manipulation of virtual objects[END_REF] to be used by two or three users. Moreover, if the system is aware of the real users space (and of their hardware setup), it can be represented in the VE in order to make the other users perceive it [START_REF] Duval | Improving awareness for 3D virtual collaboration by embedding the features of users' physical environments and by augmenting interaction tools with cognitive feedback cues[END_REF]. Thus, VE handle virtual spaces [START_REF] Snowdon | Collaborative virtual environments: Digital spaces and places for CSCW: an introduction[END_REF] to help the users to have a better mental representation of the whole 3D environment including the remote users.

Last, the remote aspect involves the need for communication between the different users. The multi-modality of the communication is important to enhance users' performances [START_REF] Nguyen | Guiding techniques for collaborative exploration in multi-scale shared virtual environments[END_REF]. It helps synchronization between distant interactions and provides some kind of awareness. The audio channel is a primordial one, but other techniques can be employed such as the use of symbols that users can put in the VE [START_REF] Nguyen | Guiding techniques for collaborative exploration in multi-scale shared virtual environments[END_REF] or even realistic [START_REF] Beck | Immersive groupto-group telepresence[END_REF] and emotive [START_REF] Freeman | Can virtual reality be used to investigate persecutory ideation?[END_REF] avatars for non-verbal communication.

Shared involvements

First, bimanual as well as collaboration enhances interaction performances in many tasks too complex to be performed using a single input [START_REF] Murayama | SPIDAR G&G: a two-handed haptic interface for bimanual VR interaction[END_REF]. For example, Latulipe et al. [START_REF] Latulipe | symSpline: symmetric two-handed spline manipulation[END_REF] proposed a system to draw complex spline-curves increasing performances using users' both hands in a symmetric way. In CVE, interaction performances can be increased in many ways, such as providing multiple viewpoints available by the different users in a 3D selection task.

D R A F T

Another similarity is the availability of multiple input-based interactions to achieve infeasible task in uni-manual setting. For instance the rotation of a cubic prop in [START_REF] Frohlich | Cubicmouse-based interaction in virtual environments[END_REF] can be done only using both hands to hold it in a different orientation. In CVE, collaborators can act in a complementary way for example in finding objects in a complex VE [START_REF] Nguyen | Guiding techniques for collaborative exploration in multi-scale shared virtual environments[END_REF].

With regard to these multiple input-based interaction techniques, several bimanual metaphors can be extended for collaboration, especially for object 3D manipulation. Indeed, some metaphors enabling a user to control an object manipulated with both hands are based on an asymmetric pattern where each hand controls a set of degrees of freedom (DOF) of that object. This principle of splitting DOF can consistently be reused for collaboration letting each user control only a subset of the DOF.

Last, Cutler et al. [START_REF] Cutler | Two-handed direct manipulation on the responsive workbench[END_REF] raise the issue about the transitions between uni-manual and bimanual interactions. These transitions must not disturb the user and can be defined as explicit or implicit ones according to the context. They have to be seamless and natural. The same issue is encountered in CVE where transitions between mono and multi-user interactions have to be as natural as possible [START_REF] Pinho | Cooperative object manipulation in collaborative virtual environments[END_REF].

Summary

In this section we presented an overview of both fields we are interested in: bimanual interactions and CVE. We developed all requirements to build a CVE and explain why it is a complex task. Last, we highlighted some similarities encountered in both fields. These similarities induce same kinds of issues when developing bimanual or collaborative interaction techniques. Thus, both fields can be studied in parallel in order to capitalize on common solutions proposed in each fields.

INTERACTION CLASSIFICATION

The literature proposes classifications for bimanual interactions as well as for collaborative ones. These classifications are presented respectively in subsection 3.1 and subsection 3.2, then a comparative discussion is provided in subsection 3.3.

Bimanual classification

Bimanual interactions can be split into two main classes [29]:

• symmetric interactions,

• asymmetric interactions.

First, symmetric interactions do not make distinction between the role of both hands. For example, "bring a ball" [START_REF] Gaucher | A novel 3D carousel based on pseudo-haptic feedback and gestural interaction for virtual showcasing[END_REF] or the two-hand "SkeweR" technique [START_REF] Duval | Skewer: a 3d interaction technique for 2-user collaborative manipulation of objects in virtual environments[END_REF] are symmetric interactions.

Second, asymmetric interactions differentiate the role of each hand. In this case, the dominant hand is the skillful hand (the right hand for right-handed people) that can perform fine grain interactions, while the non-dominant hand is used for coarser interactions. Asymmetric bimanual interaction has been deeply studied by Guiard [START_REF] Guiard | Asymmetry in bimanual skills. Manual asymmetries in motor performance[END_REF] and its principles are detailed in subsection 4.1.

Collaboration classification

In the literature several authors classified collaborative interactions. We present our classification based on a mix between the ones presented in [START_REF] Margery | A general framework for cooperative manipulation in virtual environments[END_REF][START_REF] Luna | Collaborative virtual en-vironments for training: A unified interaction model for real humans and virtual humans[END_REF]. It is a hierarchical classification from the basic features to the more advanced ones.

• Level 1: Awareness of team members' presence (avatars and communication channels).

• Level 2: Individual interactions on the scene.

• Level 3: Collaborative/codependent interactions (at the same time).

-Level 3.1: Modification of different attributes.

-Level 3.2: Modification of the same attribute.

Level 2 is not restrained to collaboration because it is equivalent to a single user interaction in a VE. However, it is useful to explicit it because some collaborative applications only provide this level of interaction (and the Level 1) and do not handle codependent interaction (Level 3). Level 3 is the more advanced feature offered by CVE, and it involves complex issues linked firstly to the handling of multiple inputs and secondly to the network latency and/or failures.

Comparative discussion

In some bimanual manipulation tasks as the "bring a ball" presented in [START_REF] Gaucher | A novel 3D carousel based on pseudo-haptic feedback and gestural interaction for virtual showcasing[END_REF], the interaction is based on a symmetric collaboration between both hands. Level 3.2 is the collaborative equivalence of this bimanual interaction paradigm. The same way, Level 3.1 is based on a partial DOF control by each user that is used in most of the bimanual manipulation tasks based on asymmetric interaction. Thus, these classifications are a starting point of a comparison between bimanual and collaborative interactions. We develop this idea of comparable interaction paradigm further in the next section, especially in subsection 5.2.

BIMANUAL INTERACTIONS

Improved performances in bimanual interactions are not only due to the use of additional inputs into the system, but also to the user proprioceptive sense. A human always knows where his both hands are relatively to each other and to his body. This natural proprioceptive information provides essential cues to be able to act seamlessly without looking at his hands all the time. Exploiting proprioception and the different skill capabilities of each hand leads to propose enhanced interaction techniques. First, subsection 4.1 presents bimanual interactions based on an asymmetric use of each hand, then subsection 4.2 provides some counter-examples based on symmetric interactions.

Asymmetric paradigm: bimanual interactions based on a frame of reference

Bimanual interaction has been well-studied by physiological researchers. HCI community used this research to design better their interaction techniques according to important findings about the asymmetry of human bimanual gestures. In [START_REF] Cutler | Two-handed direct manipulation on the responsive workbench[END_REF], Cutler et al. use a right-handed writer as an example to summarize the following main principles:

• Right-to-left reference: the writing right hand moves according to the left hand motion that holds the paper.

• Asymmetrical scales of motion: the right hand motion is much more accurate than the left hand motion (writing task vs. move the paper).

• Left-hand precedence: the left hand begins positioning paper, then the right hand starts to write.

These principles have been used for various interactions such as selection task [START_REF] Ulinski | Two handed selection techniques for volumetric data[END_REF] or 3D manipulation [START_REF] Hinckley | Two-handed virtual manipulation[END_REF]. This last work by Hinckley et al. perfectly illustrates interactions based on a dynamic frame of reference. The user brings two props in each hands, the non-dominant one representing the world-frame of reference in their context, and the dominant one controlling an object moving in the VE. In their surgical application, the moving object must be precisely placed into the 3D VE. Better performances are obtained with the participation of the left hand as a frame of reference. Accuracy is enhanced thanks to proprioception compared with the same uni-manual task.

D R A F T

Counter-examples: bimanual symmetric interactions

According to bimanual interaction principles proposed by Guiard [START_REF] Guiard | Asymmetry in bimanual skills. Manual asymmetries in motor performance[END_REF], bimanual interactions are intrinsically asymmetric. However, some tasks proved to be more usable with symmetric bimanual interactions [START_REF] Murayama | SPIDAR G&G: a two-handed haptic interface for bimanual VR interaction[END_REF][START_REF] Latulipe | symSpline: symmetric two-handed spline manipulation[END_REF]. In particular, Conan et al. [START_REF] Clay | Bi-manual 3D painting: an interaction paradigm for augmented reality live performance[END_REF] proposed an immersive setup to perform artistic 3D painting while dancing in which the symmetry is an essential part of the interaction to let the user dance in a artistic way without constraints from the system. In selection tasks, Ulinski et al. [START_REF] Ulinski | Two handed selection techniques for volumetric data[END_REF] performed evaluations that demonstrate a better accuracy with a symmetric interaction compared to two asymmetric ones. Authors explain this result due to the decreasing cognitive demand compared with an asymmetric interaction. Results show that the symmetric technique is the best if the time of the task is relatively short, else asymmetric techniques become better since they decrease arm strain issues. Last, Balakrishnan et al. [START_REF] Balakrishnan | Symmetric bimanual interaction[END_REF] showed that a poor design of a symmetric interaction technique can lead to a bimanual task performed in a sequential way. In this case, the interaction can be considered as asymmetric, and the performances of the bimanual task decrease due to the loss of parallel collaboration between both hands. Thus, symmetric interactions can be classified as a borderline case of asymmetric interactions.

To conclude, it appears that the choice between a symmetric or an asymmetric interaction technique is task-dependent. Indeed, according to the task, better interaction performances will be obtained in one setting or another. This choice must be guided by user studies that reveal the most appropriate technique depending on the required task.

COLLABORATIVE INTERACTIONS IN CVE

Extending bimanual interactions toward collaborative interactions involves difficulties linked to the increasing complexity of the interactions and to the coordination of potentially remote collaborators. First, subsection 5.1 focuses on interests to provide collaborative interaction techniques, then subsection 5.2 explains how some metaphors can easily be used in one setup or another without important modifications. Last, subsection 5.3 addresses specific difficulties implied by CVE and subsection 5.4 sums up this section

Interests of collaborative interaction techniques

Compared to uni-manual interactions, bimanual interactions have had a great success because they allow easier and more natural interactions in a lot of applications, for instance 3D manipulation [START_REF] Hinckley | Two-handed virtual manipulation[END_REF] or virtual design [START_REF] Le Chénéchal | Poster: Bimanual design of deformable objects thanks to the multi-tool visual metaphor[END_REF] (cf. figure 2). In everyday life, there are still lots of interactions that are not wrapped into this class of interaction (e.g bimanual interactions). In particular, some tasks need multiple users acting at the same time on the same object to be achieved successfully. For example, some industrial manipulations [START_REF] Aguerreche | Reconfigurable tangible devices for 3D virtual object manipulation by single or multiple users[END_REF][START_REF] Salzmann | Collaborative interaction in co-located two-user scenarios[END_REF] cannot be done alone but need at least two collaborators to manipulate voluminous objects. In the same way, teaching scenarios involving a teacher and at least a learner [START_REF] Luna | Collaborative virtual en-vironments for training: A unified interaction model for real humans and virtual humans[END_REF] intrinsically require CVE. Moreover, other kinds of scenarios are performed better in a collaborative way. For instance, virtual exploration of large environment [START_REF] Nguyen | Guiding techniques for collaborative exploration in multi-scale shared virtual environments[END_REF] can be enhanced by the help of a collaborator with a different point of view on the VE to guide the explorer.

Collaborative interaction techniques as extensions from bimanual interaction techniques

Most of the time, collaborative interactions use the split of DOF paradigm [START_REF] Pinho | Cooperative object manipulation in collaborative virtual environments[END_REF] exactly as bimanual asymmetric interactions. For example, several 3D manipulation techniques are asymmetric and can be used in both contexts such as the trackball or the grab-and-scale techniques. In the same way, symmetric interaction techniques such as the slide-and-turn or the grab-and-twirl are implemented in both bimanual and collaborative setups. Figure 3 illustrates these manipulation techniques. These kinds of interaction techniques based on multiple inputs share the same need for a controller. Controllers are the interfaces between several inputs and an interactive object. They generate merged outputs based on a specific policy. In this context, the controller's goal is to mediate the parallel and concurrent inputs in order to co-manipulate an object through multiple points of manipulation [START_REF] Aguerreche | Short paper: 3-hand manipulation of virtual objects[END_REF][START_REF] Duval | Skewer: a 3d interaction technique for 2-user collaborative manipulation of objects in virtual environments[END_REF]. This mediation depends on the implemented manipulation policy that can be based on a symmetric interaction paradigm (e.g average method) or on an asymmetric one (e.g split of DOF). Controllers of these metaphors can always be reused or extended in a consistent way, and thus we note a strong parallel between bimanual and collaborative interactions: interaction metaphors are based on similar concepts of control. At this step, we could even reverse our approach noting that in fact a generic framework for (remote or local) collaboration could be used in bimanual scenarios seamlessly, e.g that bimanual is a specialization of collaboration for two local hands.

Involved difficulties

Comparisons between bimanual and collaborative interactions in asymmetric-based paradigm suffer from a strong limitation. Indeed, the role of proprioception in this bimanual interaction [START_REF] Balakrishnan | The role of kinesthetic reference frames in two-handed input performance[END_REF] is predominant and essential to successfully accomplish these kinds of complex tasks. However, in collaborative setups, collaborators cannot be aware of this internal sensory data. In order to overcome this limitation, additional feedbacks can be proposed to let the users perceive an external representation of this personal feedback. To achieve this, multiple sensory channels can be leveraged. Aguerreche et al. proposed a reconfigurable tangible device to be used in local co-manipulation tasks [START_REF] Aguerreche | Reconfigurable tangible devices for 3D virtual object manipulation by single or multiple users[END_REF]. This device helps the users to feel the actions of others users by providing a direct and tangible link between the collaborators acting as a passive haptic interface. Sallnãs et al. used active haptic feedback that extends the feeling of other actions to distant collaborators [START_REF] Sallnäs | Supporting presence in collaborative environments by haptic force feedback[END_REF]. In a similar way, vibrotactile [START_REF] Naud | Effect of tactile feedback and viewpoint on task performance in a collaborative virtual environmment[END_REF] and pseudo-haptic [START_REF] Sreng | Using visual cues of contact to improve interactive manipulation of virtual objects in industrial Assembly/Maintenance simulations[END_REF] feedbacks can enhance interac-D R A F T tions by providing direct data about others actions. All these sensory feedbacks (e.g haptic, visual and auditory) allow collaborators to have an external and potentially distant representation of other users actions who use their own internal proprioceptive sensory to act into the VE.

These direct feedbacks provide a representation of the distant users' activities. Thus, it helps collaborators coordination, especially in co-dependent interactions that request a strong synchronization of each user actions in a common task. It provides a collaborative aspect to the system interactivity, and is essential to let the shared VE be usable in a collaborative way.

Summary

We argued that bimanual and collaborative interactions follow the same goal that is to enhance interactions between a virtual world and one or several final users. Thus, we explained the similarities that can be found between the proposed interaction techniques in both fields. However, bimanual interactions are only a first step toward multiple input-based interactions. Collaboration is the next one and involves more complex and complete interaction techniques.

Bimanual interactions take into account only one user and are strongly based on the proprioceptive sensory of the user to enhance performances. In collaborative scenarios, the issue is to let collaborators perceive the other actions. This can be achieved through different feedback techniques that are wrapped in the term awareness. These awareness features are the difficult part to extend bimanual interactions toward collaborative interactions as explained in the next section.

THE AWARENESS ISSUE

Awareness is essential to let the shared VE be usable in potentially distant collaborative tasks. It allows to perceive other actions and to understand the collaborators activities. Subsection 6.1 presents a classification of awareness features, then subsection 6.2 describes awareness solutions used to enhance the perception of the VE and its content. Subsection 6.3 develops some awareness techniques used in bimanual interaction techniques that could be reused in CVE. Last, subsection 6.4 describes some awareness solutions especially designed to enhance collaboration in CVE, and subsection 6.5 provides a final discussion.

Awareness classification

Classically in VR, systems' interactivity is defined as a 3-step loop that is: perception, decision, action. Another way to express it follows an interactive loop defined in three steps: awareness, action, feedback [START_REF] Goebbels | Design and evaluation of team work in distributed collaborative virtual environments[END_REF]. The first step2 indicates to the user what can be done in the VE, e.g what interactions are possible. The second step represents the actual interaction of the user with the VE, then the third step provides a feedback to let the user be aware of the consequences of his actions on the VE. Then, the loop can go on. Without it, there would be no mean for the user to directly understand his actions and the application would not be interactive enough. The lack of direct feedback drastically decreases the usability of the system. Goebbels et al. [START_REF] Goebbels | Design and evaluation of team work in distributed collaborative virtual environments[END_REF] proposed an extension of this interactive loop to be used in CVE. In particular, they extend the awareness part and divided it as follows:

1. Proprioception, e.g perception of his own actions.

Perception of his own physical and virtual input devices and

of the virtual data set.

3. Perception of co-presence, e.g of other collaborators.

4. Perception of co-physical and virtual input devices and of covirtual data set (e.g shared data).

5. Perception of co-knowledge and co-status.

These five steps are the ordered steps perceived by users at each step of the awareness part of the interactive loop defined previously. We note that steps 1 and 2 are the same in a single-user or a multi-user setup, and steps 3 to 5 specifically address awareness features in collaborative settings. The term presence, in single-user settings, defines "the feeling of existence within a given environment" [START_REF] Zahorik | Presence as being-in-the-world[END_REF]. Sanchez et al. [START_REF] Sanchez-Vives | From presence towards consciousness[END_REF] claim that presence is a level of consciousness above awareness. Co-presence [START_REF] Goebbels | Design and evaluation of team work in distributed collaborative virtual environments[END_REF] extends this consciousness feeling for collaborators in shared worlds. The previous last 3 steps enable coordination between users and improve the use of collaborative interactions trying to compensate the lack of direct collaborators' proprioception feedback. Awareness is not only studied by computer science researchers, it is inspired by work from the social science community, and adapted to the context of CSCW and CVE. Social science community provides an exhaustive classification of awareness, but CSCW and CVE do not require every features of this classification. A survey about similarities between awareness in CSCW and social sciences can be found in [START_REF] Gross | User-centered awareness in computer-supported cooperative work-systems: Structured embedding of findings from social sciences[END_REF]. We rather present a specific classification of awareness for CVE taken from [START_REF] García | Enhancing collaborative manipulation through the use of feedback and awareness in CVEs[END_REF] and [START_REF] Pinho | Cooperative object manipulation in immersive virtual environments: framework and techniques[END_REF]. Thereby, awareness in CVE can be split into 6 classes:

1. Object: is it selectable? selected? who is its owner? can we modify some of its attributes? This need of informations must be well-studied for each application and must not disturb the use of the system. For instance we must avoid to provide too many data that could hide some important parts of the VE. Rather, awareness must improve interaction performances by providing necessary visual, auditive or haptic cues together with a direct verbal communication channel at least. This must lead to enhance the whole perception of the VE including remote users, their current actions and their interaction capabilities.

Awareness basics: VE and content

First, VE must provide some ways to users to have an accurate perception of the whole environment. In immersive settings (e. Echoes managing network troubles [START_REF] Duval | Managing network troubles while interacting within collaborative virtual environments[END_REF], (c) Rays from the owners to the manipulated object [START_REF] Aguerreche | A description of a dialog to enable interaction between interaction tools and 3D objects in collaborative virtual environments[END_REF].

thus better perceive their environment and the collaborators' location and action. Second, VE can let users be aware of the objects' interaction capabilities to improve their usability. For instance, interactive objects can display their current state by changing some of its appearance attributes when they are selected or manipulated. Another way to represent it is to display a semi-transparent bounding-box (cf. figure 4a). Note that this method can also be extended using echoes in order to inform users about network troubles concerning shared virtual objects [START_REF] Duval | Managing network troubles while interacting within collaborative virtual environments[END_REF] (cf. figure 4b). Moreover, when an object is selected or manipulated by a user named an owner, it is interesting for other users to know who is this owner. Typical technique consists in displaying the owner's name next to the interactive object, but more visual awareness techniques propose to display a ray (or an extended arm) from the owner location to the object (cf. figure 4c). Concerning interaction capabilities, objects can provide visual cues such as manipulation arrows in order to inform the users which interactions are possible.

Last, a famous approach considered two distinct spaces for each user named the focus and the nimbus [START_REF] Benford | Awareness, focus, and aura: A spatial model of interaction in virtual worlds[END_REF]. In [START_REF] Benford | Managing mutual awareness in collaborative virtual environments[END_REF], focus and nimbus are "subspaces within which an object chooses to direct either its presence or its attention". More specifically, "the more an object is within your focus, the more aware you are of it", and "the more an object is within your nimbus, the more aware it is of you". It means that, in a generic way, interactive objects can react to user position and focus by changing appearances for example, and also inform others about user's attention and interaction capabilities.

Bimanual awareness solutions

In single-user setup, awareness is a key feature too. Particularly in bimanual interactions, the system can provide awareness through three different channels. First, visual feedback of the interactive tool allows the user to better anticipate the consequences of his actions. It is essential in immersive setups as in [START_REF] Le Chénéchal | Poster: Bimanual design of deformable objects thanks to the multi-tool visual metaphor[END_REF] to improve the immersive feeling of the user in the VE. Second, auditory feedback can enhance user perception of the VE. For instance, physical behaviors can be simulated thanks to specific sounds, such as contacts and friction in [START_REF] Le Chénéchal | Poster: Bimanual design of deformable objects thanks to the multi-tool visual metaphor[END_REF]. Third, haptic feedback improves kinaesthetic feedback and helps the user to coordinate his movements. In [START_REF] Murayama | SPIDAR G&G: a two-handed haptic interface for bimanual VR interaction[END_REF], Murayama et al. used two SPIDAR devices, one for each hand. They provided separate haptic feedback for each hand and increased the bimanual interaction abilities thanks to the feeling of forces applied by hands in the VE. Other pseudo-haptic feedbacks such as proposed in [START_REF] Sreng | Using visual cues of contact to improve interactive manipulation of virtual objects in industrial Assembly/Maintenance simulations[END_REF] can provide visual or auditory cues to overcome the lack of haptic device.

Awareness techniques for CVE

In the literature, many solutions have been implemented to enhance collaboration in CVE thanks to awareness. We do not present an exhaustive list of these solutions, rather we discuss some of them that we think they are widely used and generic enough.

Awareness of collaborators

First, users have to be able to perceive their collaborators into the shared VE. To achieve it, the typical solution is to use an avatar animated according to the user motion. In the simplest case, only the body of the avatar is animated, but more advanced research worked on realistic avatar [START_REF] Beck | Immersive groupto-group telepresence[END_REF] textured as the real user, and even emotive avateering [START_REF] Freeman | Can virtual reality be used to investigate persecutory ideation?[END_REF] in order to enhance non-verbal communication between remote users. In this field, an interesting modality is gaze tracking that enables eye-contact and awareness of others' gaze [START_REF] Steptoe | Eye-tracking for avatar eye-gaze and interactional analysis in immersive collaborative virtual environments[END_REF]. Avatars with gaze direction allow fine and informative non-verbal communication and can reproduce natural social interactions.

Second, users should perceive others' interactions capabilities to be able to collaborate effectively. These capabilities can depend on hardware setups because users can interact with heterogeneous setups, such as an immersive visualization room with full body tracking, a semi-immersive 3D display with a 6DOF tracker input or a head-mounted display with tracked hands. Figure 5 gives an illustration of two heterogeneous setups. This heterogeneity leads to different interaction capabilities for each user. Collaboration becomes more effective if everyone is aware of these limitations. Duval et al. [START_REF] Fleury | A generic model for embedding users' physical workspaces into multiscale collaborative virtual environments[END_REF] proposed the Immersive and Interactive Virtual Cabin (IIVC) as a generic concept to handle this awareness feature. It allows to define the real 3D volumes that represent different workspaces linked to a specific modality (e.g motion, visual, auditory,...) and to let other users perceive these spaces. These virtual spaces can be linked to some tracking limitation or screen size, and to be aware of that avoid misunderstanding between collaborators, for example representing the view frustum [START_REF] Fraser | Supporting awareness and interaction through collaborative virtual interfaces[END_REF]. Figure 6 explicates the generic concept of the IIVC.

Collaborative awareness through communication

In this paragraph, we present general awareness techniques providing ways to users to communicate. First, CVE must allow direct and D R A F T Figure 6: Explicative scheme of the different workspaces composing the IIVC and their relations [START_REF] Fleury | A generic model for embedding users' physical workspaces into multiscale collaborative virtual environments[END_REF]. natural communication. Several channels can be employed such as an audio channel. The audio communication has been proved to be the most important and essential for effective collaboration. Video [START_REF] Pauchet | Mutual awareness in collocated and distant collaborative tasks using shared interfaces[END_REF] or haptic [START_REF] Chellali | Haptic communication to enhance collaboration in virtual environments[END_REF] channels can also be used. Second, symbolic communication is another way to allow communication between remote users. For instance, users can deposit arrows, compass or lighting into the VE to inform collaborators about interesting spots [START_REF] Nguyen | Guiding techniques for collaborative exploration in multi-scale shared virtual environments[END_REF]. They can also create notes and post-it to add informations they collect in the VE and to share it with collaborators.

Discussion

Awareness techniques used in systems based on bimanual interactions providing visual feedback of the interactive tools, as auditory and haptic feedback, can be used in collaborative setups too. In this context, they allow remote users to perceive interactive tools and actions of other users [START_REF] Pauchet | Mutual awareness in collocated and distant collaborative tasks using shared interfaces[END_REF]. In the same way, it enhances coordination, especially in co-manipulation tasks through visual [START_REF] García | Enhancing collaborative manipulation through the use of feedback and awareness in CVEs[END_REF], auditory and haptic/tactile [START_REF] Naud | Effect of tactile feedback and viewpoint on task performance in a collaborative virtual environmment[END_REF] cues. Experiments proved that these awareness techniques improve task performances in collaborative settings too.

Table 7 summarizes main aspects of similarities that can be found between the two fields we are interested in, and highlights some of the CVE challenges. These challenges are mainly due to the remote aspect and the loss of proprioceptive feedback compared with bimanual interactions. Figure 7: Synthesis table illustrating the similarities between bimanual and collaborative interaction techniques. Some features have an equivalent in both fields, others are shared, and last, some are specific to CVE.

CONCLUSION AND PERSPECTIVES

This preliminary study shows that multiple input-based metaphors used for bimanual interactions can partially inspire the design of collaborative interaction techniques. However, even if bimanual interactions bring large improvements compared to uni-manual interactions, collaborative interactions are yet another step beyond. Thus, they imply more complexity and have to be specifically designed for this purpose. However, some guidelines can be picked from the existing bimanual interaction techniques such as the choice between symmetric or asymmetric collaboration according to the task. As well, some awareness issues have been handled by bimanual interaction techniques such as the tools representation or the action explicitness by different means (visual, haptic, auditive,...) and can be re-employed in CVE. However, this awareness aspect is more important in CVE and then must propose new techniques to let the users have an accurate perception of the shared VE including remote users. An important feature improving coordination is the availability to perceive distant activities to be able to anticipate collaborators' actions. Thus, co-dependent interactions can take place in a comprehensive way. This collaborative awareness aspect will drive our future work and we will try to propose innovative solutions to let remote users have a perception of the collaborators' activity through specific feedbacks based on immersion, awareness and presence.

Figure 2 :

 2 Figure2: Illustration of a virtual design immersive system based on natural bimanual interactions proposed in[START_REF] Le Chénéchal | Poster: Bimanual design of deformable objects thanks to the multi-tool visual metaphor[END_REF]

Figure 3 :

 3 Figure 3: Illustrations of 3 manipulation techniques based on 2 6DOF inputs. Asymmetric: (1) trackball, (2) grab-and-scale. Symmetric: (3) slide-and-turn (≃ grab-and-twirl)

Figure 4 :

 4 Figure 4: Illustrations of visual cues used to enhance awareness in CVE: (a) Bounding boxes indicating a specific object property [16], (b)Echoes managing network troubles[START_REF] Duval | Managing network troubles while interacting within collaborative virtual environments[END_REF], (c) Rays from the owners to the manipulated object[START_REF] Aguerreche | A description of a dialog to enable interaction between interaction tools and 3D objects in collaborative virtual environments[END_REF].

Figure 5 :

 5 Figure 5: An illustration of a collaborative session with two users.The one on the left works in an immersive setup, while the one on the right uses a desktop setup[START_REF] Duval | Improving awareness for 3D virtual collaboration by embedding the features of users' physical environments and by augmenting interaction tools with cognitive feedback cues[END_REF].

The first step of the loop is named awareness in the literature. However, it only refers to the interaction capabilities awareness part of the whole awareness that the system can provide (that includes general feedbacks).


F R Petruzielo

Julien Toulouse
email: julien.toulouse@upmc.fr

C J Umrigar
email: cyrusumrigar@cornell.edu

Approaching Chemical Accuracy with Quantum Monte Carlo

A quantum Monte Carlo study of the atomization energies for the G2 set of molecules is presented. Basis size dependence of diffusion Monte Carlo atomization energies is studied with a single determinant Slater-Jastrow trial wavefunction formed from Hartree-Fock orbitals. With the largest basis set, the mean absolute deviation from experimental atomization energies for the G2 set is 3.0 kcal/mol. Optimizing the orbitals within variational Monte Carlo improves the agreement between diffusion Monte Carlo and experiment, reducing the mean absolute deviation to 2.1 kcal/mol. Moving beyond a single determinant Slater-Jastrow trial wavefunction, diffusion Monte Carlo with a small complete active space Slater-Jastrow trial wavefunction results in near chemical accuracy. In this case, the mean absolute deviation from experimental atomization energies is 1.2 kcal/mol. It is shown from calculations on systems containing phosphorus that the accuracy can be further improved by employing a larger active space.

I. INTRODUCTION

Quantum Monte Carlo (QMC) [1] is touted by devotees as a "very accurate" method. However, previous QMC studies of the atomization energies of the molecules in the G2 set [2] have not obtained chemical accuracy [3,4], defined as 1 kcal/mol. These studies, which are limited to a single determinant Slater-Jastrow (SJ) trial wavefunction and a fixed set of orbitals obtained via a quantum chemistry calculation, produce a mean absolute deviation (MAD) from experimental atomization energies of about 3 kcal/mol. This work aims to improve upon both of those shortcomings. As a starting point, a single determinant SJ trial wavefunction composed of Hartree-Fock (HF) orbitals is used to compute the diffusion Monte Carlo (DMC) [5] atomization energies for the G2 set. These calculations, which are performed for double-zeta (2z), triple-zeta (3z), and quintuple-zeta (5z) bases, demonstrate the convergence of the DMC atomization energies with respect to basis size. The MAD from experiment for the 5z basis is 3.0 kcal/mol, in agreement with previous QMC studies [3,4].

Next, the restriction to a fixed set of quantum chemistry orbitals is relaxed. The orbitals for each system and basis are optimized in variational Monte Carlo (VMC) via the linear method [6][7][8]. Employing the single determinant SJ trial wavefunction with optimized orbitals, DMC yields a MAD from experiment of 2.1 kcal/mol for the 5z basis.

Finally, the restriction of a single determinant SJ trial wavefunction is relaxed. With a complete active space (CAS) SJ trial wavefunction formed from just an s and p valence orbital active space, DMC produces atomization energies of near chemical accuracy. The MAD from experimental atomization energies is 1.2 kcal/mol. This lends some backing to the claim that QMC is "very accurate". It is found that the MAD can be further reduced by including valence d orbitals in the active space for the heavier systems that are underbound in DMC.

This paper is organized as follows. In Section II, the computational setup is described. In Section III, results of the computations are described. Concluding remarks are in Section IV.

II. COMPUTATIONAL SETUP

All QMC calculations performed for this work use the Burkatzki-Filippi-Dolg (BFD) pseudopotentials [9,10] in the QMC package CHAMP [START_REF] Champ | a quantum Monte Carlo program written[END_REF]. The 2z and 3z basis sets are the recently developed atomic natural orbital Gauss-Slater (ANO-GS) bases [START_REF] Petruzielo | [END_REF]13]. For the 5z basis, the Gaussian BFD basis set [9,10] is used, omitting the g and h functions. In the course of this study, it was determined that the hydrogen pseudopotential produced unreliable atomization energies. A significantly improved pseudopotential for hydrogen was developed by Filippi and Dolg, and is used in this work. Also, 2z and 3z ANO-GS basis sets, and a 5z Gaussian basis set appropriate for this pseudopotential have been constructed for this work. The improved pseudopotential and corresponding basis sets are available in the supplementary material [14].

A combination of experimental and theoretical molecular geometries are used in this study [15][16][START_REF]NIST Computational Chemistry Comparison and Benchmark Database, NIST Standard Reference Database Number 101[END_REF][START_REF] Kalemos | [END_REF]. The zero point energies and experimental atomization energies are from Feller et al. [15,19]. The geometries, zero point energies, and experimental atomization energies for each molecule are available in the supplementary material [14].

For single determinant SJ trial wavefunctions, the initial orbitals are generated in GAMESS [20] via spinrestricted Hartree-Fock calculations. The Jastrow parameters, and when applicable, the orbital parameters, are then optimized in VMC via the linear method [6][7][8].

For CAS SJ trial wavefunctions, the initial orbitals and initial configuration state function (CSF) coefficients are generated in GAMESS via multi-configurational selfconsistent field theory (MCSCF) calculations. The Jastrow, orbital, and CSF parameters are then optimized in VMC via the linear method. The active space consists of the 1s orbital for hydrogen, the 2s and 2p orbitals for the first row atoms, and the 3s and 3p orbitals for the second row atoms, and the corresponding orbitals for the molecules.

Additionally, not all of the CSFs generated by the MC-SCF calculations are included in the QMC calculations. Instead, a dual criterion for selecting CSFs is employed. If the magnitude of a CSF coefficient is at least 0.005 or a CSF is a double excitation from the HF CSF, then it is included in the trial wavefunction. This dual criterion is employed in contrast to the usual single criterion based only on the magnitude of CSF coefficients because the optimal CSF coefficients in QMC can differ greatly from the coefficients generated via MCSCF. Although the magnitude of most CSF coefficients decrease upon optimization in VMC due to the Jastrow factor's effectiveness in describing electronic correlations, there are systems for which the magnitude of the coefficients for a few double excitations increase considerably. This dual selection criterion results in a relatively modest number of CSFs. The largest number employed is for C 2 H 6 and Si 2 H 6 . These trial wavefunctions consist of 650 CSFs comprising 1700 unique determinants, whereas the MC-SCF calculation generates 1.4 million CSFs.

Finally, all DMC calculations are performed with a 0.01 H -1 time step. The walker populations are large enough for a negligible population control bias and furthermore the small population control bias is eliminated using the method described in Refs. [5,21]. For all systems except LiH, BeH, CH2 ( 3 B 1 ), LiF, C 2 H 2 , CN, HCN, HCO, NaCl the locality approximation [22] is employed for the nonlocal pseudopotential. The aforementioned systems suffer from instabilities with the locality approximation, so those computations are performed with the size-consistent version of the T-moves approximation [23]. Note that for these systems the atomic energies are also calculated with T-moves so that atomization energies are always calculated in a consistent manner. All DMC calculations are performed with a sufficient number of Monte Carlo steps such that the statistical error bar on the atomization energy of each system is about 0.1 kcal/mol.

III. RESULTS

The raw data for all calculations presented here are available in the supplementary material [14].

The deviation of the DMC atomization energies from experiment for a single determinant SJ trial wavefunction composed of HF orbitals is shown in Figure 1. The results for 2z, 3z, and 5z basis sets demonstrate the convergence of the atomization energies with respect to basis size. The MAD from experiment for the 2z, 3z, and 5z bases are 4.5 kcal/mol, 3.2 kcal/mol, and 3.0 kcal/mol, respectively. The 5z result agrees with previous QMC studies [3,4] which had a MAD from experiment of about 3 kcal/mol. Note that Nemec et. al performed all-electron DMC calculations with HF orbitals [4] whereas Grossman employed the Stevens-Basch-Krauss pseudopotentials [24] with MCSCF natural orbitals [3].

Although orbitals from a quantum chemistry calculation are a reasonable starting point for a QMC calculation, they are certainly not optimal due to the presence of a Jastrow factor in the QMC wavefunction. Consequently, more accurate results are obtained by optimizing the orbitals in VMC. The deviation of the DMC atomization energies from experiment for a single determinant SJ trial wavefunction composed of VMC optimized orbitals is shown in Figure 2. Again, the results for 2z, 3z, and 5z basis sets demonstrate the convergence of the atomization energies with respect to basis size. The MAD from experiment for the 2z, 3z, and 5z bases are 3.1 kcal/mol, 2.3 kcal/mol, and 2.1 kcal/mol, respectively.

As seen in Figure 3, the orbital optimized results are noticeably better than previous QMC studies [3,4] which produce a MAD from experiment of about 3.0 kcal/mol. The gains in MAD from orbital optimization are 1.4 kcal/mol, 0.9 kcal/mol, and 0.9 kcal/mol for the three bases, respectively. Although, the largest gain is for the 2z basis, it is evident that the benefits of orbital optimization remain for even the largest basis set. It is worth pointing out that using optimized orbitals and a 2z basis produces results of similar quality to HF orbitals with a 5z basis.

Although orbital optimization provides significant improvements to the atomization energy, the results are still a long way off from chemical accuracy. To approach chemical accuracy, it is necessary to move beyond a single determinant SJ trial wavefunction because orbital optimization alone does not provide sufficient flexibility in the nodal surface of the trial wavefunction. Since the MAD of atomization energies from experiment for the 3z basis is only 0.2 kcal/mol higher than that of the 5z basis, and the cost of performing orbital optimization scales quadratically with the number of basis functions, the 3z basis used here represents a compromise between accuracy and computational efficiency. The deviation of the DMC atomization energies from experiment for the s and p valence CAS SJ trial wavefunctions is shown in Figure 4. The 5z single determinant results are included to demonstrate the benefit of using a CAS SJ trial wave- function. This modest basis and CSF expansion results in a MAD from experiment of 1.2 kcal/mol, a significant step forward for QMC.

As seen with the single determinant SJ results, both increasing the basis size and optimizing the orbitals have the effect of increasing the atomization energies for every system, since the energy gain is larger for the molecule than its constituent atoms. Since the small basis, single determinant SJ DMC results in most systems are underbound, this on average reduces the MAD of the atomization energies. On the other hand, going from single determinant to CAS trial wavefunctions increases the at- [3,4] were obtained with HF orbitals and MCSCF natural orbitals, respectively. The MAD from experiment for Nemec et al. and Grossman are 3.1 and 2.9 kcal/mol, respectively. omization energies for some systems and decreases it for others, but on average in the correct direction to reduce the MAD. For example, the atomization energies of CH and CH 2 ( 1 A 1 ) are increased and that of CH 2 ( 3 B 1 ) reduced, but all of these changes result in better agreement with experiment. However, using the CAS trial wavefunctions certainly does not always improve agreement with experiment, e.g. LiF and CO 2 .

QMC can do yet better. Using a larger active space will certainly help, as the largest impediment for QMC is the fixed-node error. The choice of the modest s and p valence CAS allows for the possibility of scaling up to larger systems. However, for some systems an s and p valence CAS may not be sufficient to properly describe the nodal structure. To explore this, further study is performed on the phosphorous containing systems of the G2 set: PH 2 , PH 3 , P 2 . Each of these systems is underbound for the s and p valence CAS. As shown in Figure 5, using s, p, and d valence CAS improves agreement between DMC atomization energies and experiment. The MAD from experiment for these three systems is 3.7, 2.3, and 1.6, for single determinant, s and p valence CAS, and s, p, and d valence CAS, respectively. Although using a larger active space for the phosphorus systems is beneficial, a large active space becomes impractical as system size increases. Even though the number of CSFs included in a QMC calculation via the dual criterion described in Section II is much smaller than the total number of CSFs for a given active space, it is impractical to even perform the initial MCSCF calculation for large systems. Some options for alleviating this problem are obtaining the initial trial wavefunction from less expensive configuration interaction (CI) rather than MC-SCF calculations, or, from restricted active space rather than complete active space calculations.

It is likely that some of the deviations of our results from experiment are due to using pseudopotentials. These deviations could be evaluated by performing a similar study with the all-electron couloumbic potential. However, there are some advantages to using pseudopotentials too. First, all-electron calculations for molecules containing second and higher row atoms are expensive. Second, it is possible that the fixed-node error for a given active space is larger for all-electron calculations. Finally, the use of pseudopotentials provides a simple way of including the scalar relativistic corrections.

Additionally, some of the deviations of our results from experiment are likely due to errors in the experimental atomization energies or zero point energies. In particular, as seen in Figure 4, systems containing both Si and H systematically overbind. Additionally, very accurate allelectron frozen-core coupled cluster calculations which produce sub-1 kcal/mol MAD from experiment for the G2 set [19] also systematically overbind these systems. In particular, Feller et al. overbinds SiH 2 ( 1 A 1 ), SiH 2 ( 3 B 1 ), SiH 3 , SiH 4 , Si 2 H 6 by 1.3 kcal/mol, 1.1 kcal/mol, 0.2 kcal/mol, 1.6 kcal/mol, 3.5 kcal/mol, respectively.

IV. CONCLUSION

A QMC study of the atomization energies for the G2 set of molecules was presented. Basis size dependence of DMC atomization energies was studied with a single determinant SJ trial wavefunction formed from HF orbitals. With the largest basis set, the mean absolute deviation from experimental atomization energies for the G2 set was found to be 3.0 kcal/mol, in agreement with previous QMC studies.

It was determined that optimizing the orbitals within VMC improved the agreement between DMC and experiment, reducing the mean absolute deviation to 2.1 kcal/mol. In fact, using optimized orbitals and a 2z basis produced results of similar quality to HF orbitals with a 5z basis.

Finally, DMC results for a CAS SJ trial wavefunction were near chemical accuracy with MAD from experimental atomization energies of 1.2 kcal/mol. Although a MAD of 1.2 kcal/mol is a significant step forward for QMC, this result is still lackluster compared to all-electron frozen-core coupled cluster calculations which produce sub-1 kcal/mol results for the G2 set [15,19]. Several directions for improving upon the current results are larger active spaces, backflow transformations [25], yet more accurate pseudopotentials, or all-electron calculations.

FIG. 5 .

 5 FIG.5. Deviation of the DMC atomization energies from experiment for a single determinant SJ trial wavefunction composed of VMC optimized orbitals, a CAS SJ trial wavefunction with an s and p active space, and a CAS SJ trial wavefunction with an s, p, and d active space. The MAD from experiment for the phosphorous containing systems of the G2 set with these trial wavefunctions is 3.7, 2.3, and 1.6, respectively.

  Comparison of the deviation of the DMC atomization energies from experiment for a single determinant SJ trial wavefunction. The results from this work are for a 5z basis and VMC optimized orbitals. The MAD from experiment for this work is 2.1 kcal/mol. The results of Nemec et al. and Grossman

		10																																																Grossman
																																																			Nemec
																																																		This Work
	Deviation of Atomization Energy From Exp. (kcal / mol)	-10 -5 0 5
		-15
		LiH	BeH	CH	CH 2 ( 3 B 1 ) 1 CH 2 ( A 1 )	CH 3	CH 4	NH	NH 2	NH 3	OH	H 2 O	HF	SiH 2 ( 1 A 1 ) 3 SiH 2 ( 1 ) B	SiH 3	SiH 4	PH 2	PH 3	H 2 S	HCl	Li 2	LiF	C 2 H 2	C 2 H 4	C 2 H 6	CN	HCN	CO	HCO	H 2 CO	H 3 COH	N 2	N 2 H 4	NO	O 2	H 2 O 2	F 2	CO 2	Na 2	Si 2	P 2	S 2	Cl 2	NaCl	SiO	CS	SO	ClO	ClF	Si 2 H 6	CH 3 Cl	H 3 CSH	HOCl	SO 2
	FIG. 3.

  Deviation of the DMC atomization energies from experiment for a single determinant SJ trial wavefunction composed of VMC optimized orbitals and a CAS SJ trial wavefunction. The MAD from experiment for the single determinant SJ trial wavefunction is 2.1 kcal/mol. The MAD from experiment for the CAS SJ trial wavefunction is 1.2 kcal/mol.

																																	5z 1CSF
																																	3z CAS
	Deviation of Atomization Energy From Exp. (kcal / mol)
	H 2 S	HCl	Li 2	LiF	C 2 H 2	C 2 H 4	C 2 H 6	CN	HCN	CO	HCO	H 2 CO	H 3 COH	N 2	N 2 H 4	NO	O 2	H 2 O 2	F 2	CO 2	Na 2	Si 2	P 2	S 2	Cl 2	NaCl	SiO	CS	SO	ClO	ClF	Si 2 H 6	CH 3 Cl	H 3 CSH	HOCl	SO 2
	FIG. 4.

V. ACKNOWLEDGMENTS

We thank Claudia Filippi for very valuable discussions pertaining to the T-moves approach for treating nonlocal pseudopotentials, and, for sending us the revised hydrogen pseudopotential. This work was supported in part by NSF grant DMR-0908653. Computations were performed in part at the Computational Center for Nanotechnology Innovation at Rensselaer Polytechnic Institute.


Denis V Efimov

Vladimir O Nikiforov

Hossein Javaherian

Supervisory Control of Air-Fuel Ratio in Spark Ignition Engines

The problem of air-fuel ratio stabilization in spark ignition engines is addressed in this paper. The proposed strategy consists of proper switching among two control laws to improve quality of the closed-loop system. The first control law is based on an a priori off-line identified engine model and ensures robust and reliable stabilization of the system at large, while the second control law is adaptive, it provides on-line adaptive adjustment to the current fluctuations and improves accuracy of the closed-loop system. The supervisor realizes a switching rule between these control laws providing better performance of regulation. Results of implementation on two vehicles are reported and discussed.

I. INTRODUCTION

The requirement on vehicle tailpipe emissions is one of the main restrictions for engine development and certification.

Three-way catalytic converters (TWC) installation in exhaust manifold aims at oxidizing HC and CO and reducing NOx concentration. Usually TWC peak efficiency is guaranteed if fuel-air ratio (FAR) is close to the stoichiometric value and the conversion efficiency of TWC is significantly reduced away from the stoichiometric value. Therefore, the primary objective of the FAR control system is to maintain the fuel injection in stoichiometric proportion to the ingested air flow (exception to this occurs in heavy load situations where a rich mixture is required to avoid premature detonation or for more power).

Variations in the air flow affected by the driver serve as an exogenous disturbance to the system. Due to its importance, the problem of FAR regulation has attracted significant attention during the last few decades [START_REF] Cook | Control, computing and communications: Technologies for the Twenty-First century Model T[END_REF].

Adaptive control theory [START_REF] Ault | Adaptive air-fuel ratio control of a spark ignition engine[END_REF], [START_REF] Franceschi | An Adaptive Delay-Compensated PID Air Fuel Ratio Controller[END_REF], [START_REF] Powell | Observer-based air fuel ratio control[END_REF], [START_REF] Turin | Model-reference adaptive A/F-ratio control in an SI engine based on Kalman-filtering techniques[END_REF], robust control approaches [START_REF] Brandstetter | Robust Air-Fuel Ratio Control For Combustion Engines[END_REF], fuzzy control systems theory [START_REF] Ghaffari | Adaptive Fuzzy Control for Air-Fuel Ratio of Automobile Spark Ignition Engine[END_REF], neural network techniques [START_REF] Huang | Neural sliding mode control of engine torque[END_REF], [START_REF] Zhai | RBF based feedforward feedback control for air-fuel ratio of SI engines[END_REF] and learning approach [START_REF] Andrianov | Spark ignition engine control strategies for minimising cold start fuel consumption under cumulative tailpipe emissions constraints[END_REF] are successfully tested in this particular application. However, the complexity of the problem and growing demands on FAR regulation quality require new solutions. These solutions have to combine reliability and performance of robust control approaches and the accuracy and insensitivity to changes of dynamics of adaptation methods. In addition, for implementation purposes, they should have a small number of tuning parameters and clear design guidelines. Switching control theory gives a solution to this problem.

There exist many good reasons and practical motivations to use a set of controllers to regulate a single plant as opposed to one controller [START_REF] Hespanha | Certainty equivalence implies detectability[END_REF], [START_REF] Huang | Neural sliding mode control of engine torque[END_REF], [START_REF] Morse | Control using logic-based switching[END_REF]. In such a case the problem of trade off the advantages and disadvantages of each subsystem for modeling and control is appearing. The theory of switched systems addresses this issue proposing the proper switching laws between controllers. Application of a supervisory (switched) control algorithm may seriously improve performance of the system regulation [START_REF] Efimov | On Input-to-Output Stability of Switched Nonlinear Systems[END_REF]. In addition, in order to solve a complex control problem, it can be decomposed on several simpler ones with design of control laws for each of subproblems, then proper supervisor ensures switching among the controls and solution of the initial problem.

In this work, the problem of FAR regulation problem is solved considering switching between two control laws. The first one is based on robust model-based control algorithm, which ensures stability for all ranges of the system parameters and inputs, but may have accuracy shortcomings. The second control law is adaptive, it is directed at improving the quality of transient response on a dynamic fluctuation around the reference model (used in the first control). Supervisor performs activation of the adaptive control when unsatisfactory quality of the reference model is detected and, hence, improvement of the robust control is needed. Theoretical stability conditions of the developed supervisory control are established, and the results of implementation are reported confirming efficiency of the proposed solution.

The outline of this work is as follows. In section 2 the detailed problem statement and some preliminaries are presented. Section 3 contains descriptions of the control algorithms. Supervisor equations are introduced in section 4. Results of implementation are reported in section 5. Concluding remarks are given in Section 6.

II. PROBLEM STATEMENT

It is a well-known fact that an automotive engine is a highly nonlinear multi-variable system and derivation of its precise model is a complex process. This is a reason why the simplified models of engines are very popular in practice. These models can take into account the main features of engine processes, like the presence of time delays and nonlinearities, which are important for controller design or fault detection applications. In this work nonlinear autoregressive (NARX) model is chosen for FAR dynamics description (in this context FAR refers to the non-dimensional engine-out fuel-air ratio sometimes known as ):

1 0 0 ( ) ( ) [ ( )] ( ) ( ) ( ), p p k T T i j j i j j y m a y m i m j u m j m j v m = = = = -+ - -+ -+ å å å b f r d (1)
where y R Î is FAR (the regulated output), respectively. Then, applying standard approaches [START_REF] Ljung | System Identification: Theory for the User[END_REF] the vectors of coefficients a , B and R can be obtained as off-line approximations of a , B and R . Substituting a , B and R in (1) we represent the dynamics of FAR loop (1) with a suf- ficient accuracy. The residual error can be assumed bounded and modeled as a part of the exogenous disturbance v . The coefficients a , B can be derived ensuring stability of the model (1) as well as stability of its inverse with respect to the control input (that corresponds to the physical nature of the engine).

[ ... ] T k k a a R = Î a , ( 1)
A s s u m p t i o n 1 . Polynomials defined by the vectors of coefficients a and c , where

, 1 q j j k k c = = å b for 0 j p £ £ ,
have all zeros with norms smaller than one.

Requirement on stability of the polynomial a corresponds to a physical restriction that an engine has a stable dynamics.

Under this assumption and with substitution ( ) 1 m j -= f , stabilizing controls for the system (1) can be designed applying simple inversion of its equation (inverse system is stable and, thus, the control algorithm will be realizable). The choice

( ) 1 m j -= f
is the basic one, but some other normalized inputs may also be included in f .

The problem is to design control

III. CONTROL ALGORITHMS

In this section description of robust model-based and adaptive controls are presented.

A. Model-based control algorithm

The following is the condition of the control applicability.

A s s u m p t i o n 2 . For all Î f F it holds 0 0 T ¹ b f .

Since the vector f is composed by measured engine variables or their nonlinear functions and products, which all have some sets of admissible values, then assumption 2 can be easily checked for Î f F and the vector of coefficients 0 b . For instance, ( ) i f , 0 i ³ and elements of 0 b can be all positive (that may be guaranteed by proper approximation of (1)).

Under assumptions 1 and 2, the control law is calculated as a simple inversion of the model ( 1) with respect to u :

1 0 1 0 1 ( ) ( ) ( 1) ( ) ( ) [ ( ) ] ( ) , ( ) p p T T k d PID i j j i j j T U m y m U m a y m i m j m j u m j m = = = ù é = - -- -- -- - - ë û å å å r d b f b f (2)
where due to the presence of the disturbance v (which reflects possible unmodeled dynamics, measurement noise and approximated model errors) it is required to use an internal feedback in the form of a nonlinear PID:

3 1 2 3 4 5 0 ( ) ( ) ( ) [ ( ) ( 1) ] ( ( ) ) ( ) ,
( ) ( ) ( 1) ( ) d PID y m y m U m v m = - -+ .
Without PID U the control (2) forms the so-called feedforward part of the regulator, that does not contain any feedback errors (it depends on the current and past values of the inputs and outputs of the engine dynamics and the approximated coefficients of the model).

The control (2) cannot be realized in practice since there exist constraints on admissible control amplitudes, i.e. it should be within the following bounds:

min max u u u £ £ . The implementation of a simple saturation [ ] ( ) ( ) u m sat U m = , min min max max if ; ( ) if ; otherwise, u x u sat x u x u x < ì ï = > í ï î (4)
for stable plants provides a solution to the problem. Define the control actuator error as follows

0 ( ) ( )[ ( ) ( ) ] T m m u m U m d = - b f ,
then the closed loop dynamics of ( 1), ( 4) takes the form:

( ) ( ) ( 1) ( )
d PID y m y m U m v m = - -+ % , (5)
( ) ( ) ( ) v m v m m = + d % . P r o p o s i t i o n 1 .
Under assumptions 1 and 2 there exist constants j k , 1, 5 j = such that for any solutions of the system ( 1), ( 2), ( 3), [START_REF] Cook | Control, computing and communications: Technologies for the Twenty-First century Model T[END_REF]

with Î d D , Î f F for all 0 i ³ : 1 1 [ 0, ] | ( ) | ( | ( 0 ) |, ) ( || || ) i e i e i v £ b + g % , [ 0, ] 0 || || sup | ( ) | i j i v v j £ £ = % % , 1 b ÎKL , 1 g ÎK . P r o o f .
From assumption 1 the system (1) produces bounded solutions for any bounded control min max u u u £ £ and

Î d D , Î f F .
Moreover, the equations ( 5) can be rewritten as follows

1 3 1 2 3 4 5 0 ( ) ( 1) ( ) [ ( 1) ( 2)] ( ( 1)) ( 1) ( ).
1 1 [ 0, ] | ( ) | | ( 0 ) | exp[ ln( ) ] (1 ) || || i e i e k i k v - £ + - % , 0 i ³ ,
which is an example of the desired one.

Additionally, adjusting values of the coefficients j k , 2, 5 j = one can improve the performance of the control (4). For example, coefficient 2 k provides insensitivity to static errors (integral part of PID), coefficient 4 k cancels disturbances with amplitudes less than 4 k , and coefficient 5 k may help on large deviations of the error.

Proposition 1 establishes a qualitative result on the error convergence, the estimate obtained in proposition 1 is close to input-to-state stability property introduced in the paper [START_REF] Jiang | Input-to-state stability for discrete-time nonlinear systems[END_REF].

B. Adaptive control algorithm

The control (4) has feedbacks aimed at attenuation of disturbances, approximation errors and measurement noise. In spite of this, in some cases an additional adaptation of the control is further needed. An important issue is that the model (1) has been approximated on a large a priori collected dataset, and the coefficients a , B and R suit well for all Î d D , Î f F .

However for some operating conditions, which were not well presented in the dataset, there exists another set of coefficients a % , B % and R % which represents dynamics of FAR more accurately. In fact, for almost all modes of engine operation there exist such coefficients locally working better than the global ones a , B and R .

Thus the problem of the coefficients a % , B % and R % identification with posterior update of the control can be posed. To solve the problem it is proposed to use the conventional identification algorithm [START_REF] Fradkov | Nonlinear and Adaptive Control of Complex Systems[END_REF] denoting

1 0 0 ( ) ( ) [ ( ) ] ( ) ( ) ( ) p p k T T i j j i j j y m a y m i m j u m j m j m = = = = -+ - -+ -= w q å å å b f r d % % % , (6)
where

0 0 [ ... ... ] T T T T T T p p q = a b b r r % % % % % and ( ) [ ( 1)... ( ) ( ) ( )... ( ) ( ) ( ) ... ( ) ] T T T T m y m y m k m u m m p u m p m m p w = - - - - - f f d d
is the row of regressors. Then we obtain the following parameterization for the identification error

( ) ( ) ( ) ( )[ ( ) ] m y m y m m m e = - = w q -q ) )
,

where 0 0 [ ... ... ] T T T T T T p p q = a b b r r ) ) ) )
) )

is the adjustable vector of estimates for q and ( ) y m ) is the output of the adaptive observer:

1 0 0 ( ) ( ) ( ) ( ) ( ) [ ( ) ( )] ( ). p p k T T i j j i j j
y m a m y m i m m j m m j u m j

= = = = -+ -+ - - å å å r d b f ) ) ) ) (7)
Note that the model ( 6) has a form similar to (1). However, for the coefficients a % , B % and R % it is assumed that ( ) 0 v i = , 0 i ³ (the coefficients locally approximate the system dynamics exactly). The observer (7) also has the form (1) under q substitution instead of q ) , thus ( ) y m ) is an estimate of the output ( ) y m .

To design the adaptation algorithm for q ) let us choose the conventional quadratic error functional 2 ( ) 0.5 ( )

Q m m e = e ,
whose minimization is ensured by the following gradient adaptation algorithm [START_REF] Fradkov | Nonlinear and Adaptive Control of Complex Systems[END_REF]:

( ) ( 1) ( ) ( 1) ( 1) T m m m m m q = q -+ g w - e - ) ) , (8)
where ( ) 0 m g > is a design parameter. To specify conditions of the algorithm (8) applicability we impose the following restrictions on the engine dynamics.

A s s u m p t i o n 3 . For all Î d D , Î f F it holds:

there exist series k a % , k B % and k R % and

0 k D ³ , 0 k ³ such that the model (6) is valid with k a % , k B % and k R % for all 1 k k j i j + £ £ for all 0 k ³ with 1 k k k j j + = + D , 0 0 j = ;
for any

0 i ³ there exist 0 K £ ( k K < D for all 0 k ³ ), 0 V > and 0 1 < r < such that 1 i K j j i P + - = £ r Õ I , 2 1 ( | ( ) | ) ( ) ( ) T j P j j j - = -V+ w w w I ,
where I is the identity matrix of the corresponding dimension.

This assumption has two parts. First, it is assumed that the time range of the system operation can be decomposed on sub-

intervals 1 k k j i j + £ £ , 0 k ³ ,
where the model ( 6) is valid for some k a % , k B % and k R % . Secondly, a variant of persistency of excitation condition is introduced, that is required for the convergence of the adjusted parameters to 0, ,

... ] T T T T T T k k k p k k p k q = a b b r r % % % % % 0, , [ ...
. This part can be verified numerically on-line (during experiments/implementation) or offline on the given a priori dataset.

P r o p o s i t i o n 2 . In the observer [START_REF] Franceschi | An Adaptive Delay-Compensated PID Air Fuel Ratio Controller[END_REF] with the adaptation algorithm ( 8) for

2 1 ( ) ( | ( ) | ) m j - g = V+ w
, the parameter identification error ( ) ( )

k k i i q = q -q ) %
has the following properties:

if the first part of Assumption 3 holds, then

| ( 1) | | ( ) | k k i i q + £ q % % , 1 k k j i j + £ £ , 0 k ³ ;
if Assumption 3 holds, then the estimate is satisfied:

| ( ) | | ( ) |exp[ ln( ) ( mod ) ] k k k k i j i j K q £ q r - % % , 1 k k j i j + £ £ , 0 k ³ . P r o o f . From (8) we have ( ) ( 1) ( ) ( 1) ( 1)
T k k m m m m m q -q = q --q + g w - e - )
) ,

and the identification error has form (

]

k m y m y m m m e -= -- -= w -q -q - ) )
. Then

2 1 1 ( ) ( 1) ( | ( ) | ) ( 1) ( 1) ( 1) ( 1)
.

T k k k m k m m j m m m P m - - q = q --V+ w w - w -q -= q - % % % % (9)
The matrices

m k k P m m -q -= q - % % is equivalent to ( 1) ( 1)
0 m e -= ). This fact means that the error k q % is not an increasing function of time. From (9) we have:

1 1 ( ) ( 1)
K k m j k j m K P m + - = q + = q - Õ % % and | ( ) | | ( 1) | k k m K m q + £ r q - % %
from assumption 3, which implies the estimate.

Under conditions of proposition 2, the parametric error ( ) k m q % asymptotically converges to zero, then taking control

1 0 0 1 1 ( ) ( ( ) ), ( ) ( ) ( 1) ( ) ( ) ( ) [ ( ) ] ( ) k a a d PID i i T p p T T j j j j u m sat U m U m y m U m a y m i m m j m j u m j = = = é = = - -- -- ë ù - -- - -û å å å b f r d b f ) ) ) ) (10)
it is possible to ensure the model ( 6) stabilization, where

PID U mis defined by (3).

P r o p o s i t i o n 3 .

Under assumption 3 there exist constants j k , 1, 5 j = such that for any solution of the system (1), ( 7), ( 8), [START_REF] Hespanha | Supervision of integral-input-to-state stabilizing controllers[END_REF]

with Î d D , Î f F for all 1 k k j i j + £ £ , 0 k ³ ( 2 b ÎKL , 2 g ÎK ): 2 2 | ( ) | ( | ( ) |, ) ( | ( ) | ) k k k k e i e j i j j £ b - + g q % . P r o o f . Since the control ( ) ( ) ( ) u m W m w m = +
, where

1 0 0 1 ( ) ( 1) ( ) 1 ( ) ( ) ( ), ( ) , ( ) ( ) [ ( ) ] ( ) k d PID i i p p T T T j j j j y m U m a y m i w m u m W m W m m m j m j u m j = = = é ù - -- -- ê ú ê ú = - = ê ú - -- - - ê ú ë û å å å b f r d b f % % % %
the system (1), [START_REF] Hespanha | Supervision of integral-input-to-state stabilizing controllers[END_REF] has the following closed loop dynamics:

1 3 1 2 3 4 5 0 ( ) ( 1) ( ) [ ( 1) ( 2 ) ] ( ( 1) ) ( 1) ( )
| ( ) | (| ( ) |, ) (|| || ) k k k j i e i e j i j w £ b - + g % for all 1 k k j i j + £ £ , 0 k ³ . Since for all Î d D , Î f F there exists 0 g > such that | ( ) | | ( ) | k w m g m £ q % % , then [ , ] || || | ( ) | k j i k k w g j = q % %
, that implies the result for 2 ( ) ( )

s g s g = g , 0 s ³ .
The advantage of the control [START_REF] Hespanha | Supervision of integral-input-to-state stabilizing controllers[END_REF]

is that ( ) 0 k i q ® %
with i ® ¥ according to proposition 2, therefore, if the adaptive algorithm is active for sufficiently long time (constants 0 k D ³ from assumption 3 are large enough), then the adaptive control ( 7), ( 8) and ( 10) ensures an exact regulation of FAR dynamics at the stoichiometric value.

In this section two control algorithms are proposed, which ensure input-to-state stabilization of FAR dynamics. Both controls are based on the measurement information (the first one designed for the approximated off-line model, the second one for the on-line model). The issue of the improvement of the closed-loop system quality with the use of special switching between these control laws is discussed in the following section.

IV. THE SUPERVISOR

All proposed control algorithms from section 3 possess their own advantages. The control algorithm designed off-line is rather reliable (it ensures stability for all operating modes of the engine) and robust (it is not sensitive to disturbances and unmodeled dynamics), but it may fail to ensure good accuracy over the entire range of operating conditions. The adaptive control has some transients after which it is tuned to compensate all errors at a particular engine operating condition. The switching algorithm executed in the supervisor has to combine the advantages of these controls neglecting their shortages providing the closed loop control with an improved performance. For this purpose note that the main difference between these controls consists in the models on which they are based. The following performance functionals evaluate accuracy of models on the last 0 L ³ steps:

1 2 1 1 ( ) ( ) m q m L J m L e q - = - = å , 1 2 2 2 ( ) ( ) m q m L J m L e q - = - = å , ( 11
) 1 1 0 0 ( ) ( ) ( ) [ ( ) ] ( ) ( ), p p k T T i j j i j j
e q y q a y q i q j u q j q j

= = = = - -- - -- - å å å b f r d 2 1 0 0 ( ) ( ) ( ) ( ) [ ( ) ( )] ( ) ( ) ( ). p p k T T i j j i j j
e q y q a m y q i m q j u q j m q j

= = = = - -- - -- - å å å b f r d ) ) ) y u u 1 u 2 y d , , a b R , , a b R
) ) )

Plant

(1) Robust control [START_REF] Ault | Adaptive air-fuel ratio control of a spark ignition engine[END_REF] Adaptive control [START_REF] Hespanha | Supervision of integral-input-to-state stabilizing controllers[END_REF] Supervisor [START_REF] Huang | Neural sliding mode control of engine torque[END_REF] Adaptation alg. ( 8)

Fig. 1. The structure scheme of the supervisory control system of the engine

In this case activation of the control with the most accurate model looks reasonable, that is the idea of supervision algorithm in this work. The width of the time window L used for evaluation of accuracy of both models is a design parameter (it is also related with the rate of convergence of the adaptive observer ( 7), ( 8)). However, the switching among nonlinear systems is not so trivial. Even if the systems are asymptotically stable or input-to-state stable as in our case, an inappropriate switching strategy may lead to instability [START_REF] Liberzon | Switching in Systems and Control. Systems and Control: Foundations and Applications[END_REF].

The problem of switching among input-to-state systems has been addressed in the previous works [START_REF] Efimov | On Input-to-Output Stability of Switched Nonlinear Systems[END_REF], [START_REF] Hespanha | Supervision of integral-input-to-state stabilizing controllers[END_REF], [START_REF] Hespanha | Certainty equivalence implies detectability[END_REF]. The main idea there in the dwell-time mechanism application. Dwell-time constant restricts the rate of switching between the controls and for a sufficiently slow rate the stability of the closed loop system is guaranteed. For the rest of the section let 1 ( ) u m be defined by (2) and 2 ( ) u m be given as in [START_REF] Hespanha | Supervision of integral-input-to-state stabilizing controllers[END_REF]. The structure scheme of the control system is presented in Fig. 1.

T h e o r e m 1 . Let assumptions 1 3 hold and for some 0 1 < l < there exist dwell-time constant where w s is the instant of w th switch, then in the system (1) with the control

0 D t > such that ( , ) i D r r b t £ l for all 0 r ³ ; D k t < D £ D for all 0 k ³ for some 0 D > ; ln( ) / 2 ( ) D r e r l D t b £ l for all 0 k ³ and 0 r ³ , where 1 2 ( ) max{ ( , 0 ), ( , 0 )} s s s b = b b . If 1 w w D s s + -³ t , 0 w ³ ,
( ) ( ) ( ) w z s u m u m = , ( ) {1,2} w z s Î
for any solutions the following estimate is satisfied for some b ÎKL and g ÎK :

[ 0, ] | ( ) | (| ( 0) |, ) (|| || ) i e i e i l £ b + g , | ( ) | , ( ) 1; ( ) | ( ) | , ( ) 2. w w k w w w v i if i s z s l i s if i s z s ³ = ì = í q ³ = î % % P r o o f . Consider 1 k k j i j + £ £
. Since assumptions 1 and 2 hold, while ( ) 1

w z s = , 0 w ³ the estimate 1 1 [ , ] | ( ) | (| ( ) |, ) ( || || ) w w w s i e i e s i s v £ b - + g % , [ , ] || || sup | ( ) | w w s i s j i v v j £ £ = % % , 1 b ÎKL , 1 g ÎK
from proposition 1 is true. Under assumptions 1 and 3 the estimate from proposition 3 is valid for all ( ) 2 The stated restriction D t < D implies that switching between the controls in the supervisor must be faster than the change in the operating regime. The dwell-time switching algorithm ensures boundedness of the system trajectories and the theorem presents the worst-case estimate on the closed loop error behavior. Dwell-time switching algorithm still leaves room to further supervisor rule design focusing on the improvement of transient behavior. Keeping in mind the performance functionals [START_REF] Hespanha | Certainty equivalence implies detectability[END_REF] and dwell-time rule from theorem 1, the following supervision algorithm is proposed: An additional logic can be added to [START_REF] Huang | Neural sliding mode control of engine torque[END_REF] for an emergent switching back to control 1 u , when the regulation error with the control 2 u in the loop becomes suddenly high. In this case a kind of average dwell-time condition would be ensured, under which the result of theorem 1 can be confirmed after a mild modification of arguments. Such a modification is omitted for brevity of presentation.

w z s = , 0 w ³ : 2 2 | ( ) | ( | ( ) |, ) (| ( ) | ) w w k w e i e s i s s £ b - + g q % , where ( ) ( ) k k m m q = q -q ) % with 1 k w k j s j + £ £ for all 0 k ³ , 2 b ÎKL
( ) ( ) ( ) w z s u m u m = , ( ) {1, 2} w z s Î , 0 w ³ , ( 12
) 2 1 1 1 2 arg inf { ( ) ( )} if 1; arg inf { ( ) ( )} if 2 , w D w D m s w w m s w J m J m s s J m J m s ³ +t + ³ +t < = ì ï = í < = ï î 1 ( ) 3 (

V. APPLICATION RESULTS

The proposed switching control has been tested in two vehicles with V8 engines: Chevrolet Tahoe with 5.7 engine and GMC Yukon with a 5.3 engine (these vehicles and such a type of engines are rather popular on US market, in addition, the emission improvement for vehicles with a big volume of engines is of great practical importance). The schedule of testing was as follows. At the second step the controls (2) and (10) are calculated. Assumption 2 is verified for the control (2) on the a priori collected dataset. Coefficients j k , 1, 5 j = are chosen zero initially, and after some experimentations they are tuned to some values providing acceptable performance. For the control [START_REF] Hespanha | Supervision of integral-input-to-state stabilizing controllers[END_REF] the same values of coefficients j k , 1, 5 j = are chosen. The assumption 3 is valid (after a verification on the dataset). After that, the system is ready for experimental testing.

During the experiments, the control algorithms and the supporting schemes have been loaded in the board computer using dSpace, where these codes have replaced the production controllers. Next, tests in different operating conditions (e.g. and u respectively (in these experiments 1 was selected as the desired value for the fuel-air ratio, which corresponds to the stoichiometric value), the part (c) plots the values of the functionals 1 J and 2 J ). In these results, the supervisor switches between two control algorithms based on quality functionals, the instants of switches almost everywhere correspond to verification of the conditions (12) (dwell-time constant has been assigned sufficiently small). The adaptive regulator provides better quality of control of FAR dynamics. A change in modes of engine operation results in backward activation of the model (1) based controls (in Fig. 3 the adjusted model has almost everywhere a better accuracy, in Fig. 4 in some cases activation of the control 1 u happens, while the adaptive observer needs a time for adjustment after an operating mode change). For both vehicles, experiments confirm applicability of the proposed approach. A comparison of the obtained results with a reference controller implemented in both vehicles shows a good potential of the proposed supervisory controller. If the adaptively adjusted model has better accuracy of FAR dynamics representation during some number of previous events, then the adaptive control is switched on. Such a scheme allows us to combine the reliability of robust control (which was intensively tested and it ensures admissible quality of FAR regulation for all operating regimes) and the flexibility of the adaptive control (which can improve the performance due to the higher local accuracy of the FAR dynamics approximation). Stability conditions are established. Practical implementation and intensive tests demonstrate applicability of the approach. Augmentation of an anti-windup scheme for the developed control can be a subject of future researches.

  are the model (1) constant parameters. The advantage of NARX model consists in availability of various methods for its approximation and simplicity of control design. It is assumed that the variables d and f are independent in the control variable u and available for design, therefore, the model (1) is affine in control.It is assumed that a dataset is given, that a priori has collected measured information on y , u (and other variables involved in the vectors d , f ) for various regimes of engine operation obtained for a preliminary control. Based on the given dataset, the compact sets which define admissible values for the vectors d and f

  the regulation error, j k , 1, 5 j = are control parameters, which have to be determined based on real or computer experiments. The terms proportional to 1 k and 5 k are responsible for proportional feedback ( 1 k for local regulation, and 5 k for big deviations of ( ) e m , appearance of two gains helps to improve quality of feedback). The terms proportional to 2 k and 3 k correspond to integral and differential actions respectively. The term with 4 k allows small matched disturbances to be compensated. The control (2) under substitution ( ) ( ) u m U m = ensures the model inversion and the following closed loop dynamics:

  = this system becomes linear and it has the estimate

1

 1

  the time instant of the last switch; 0 D t > is the dwell-time that prevents chattering (high frequency switching between the control algorithms) and ensures the closed-loop stability. For 0 1 s = the system starts with the model-based control (2), if accuracy of the adaptation model (7) is better ( the adaptive control[START_REF] Hespanha | Supervision of integral-input-to-state stabilizing controllers[END_REF] has to be activated. If after dwell-time the accuracy of the model (1) again becomes better ( would be switched on giving time for adaptation algorithm to adjust the observer.

Fig. 2 .

 2 Fig. 2. FAR model accuracy verification

Federal

  Test Program) have been performed. The sampling time is time-varying and it is proportional to the speed of engine rotation. The results of the tests are shown in figures 3 and 4 for the vehicles with the 5.7 and 5.3 engines, respectively (the parts (a) and (b) represent the variables 1 FAR e FAR =

Fig. 3 .

 3 Fig. 3. Trajectories for the vehicle with a 5.7 engine

Fig. 4 .

 4 Fig. 4. Trajectories for the vehicle with a 5.3 engine

  , if it is from class K for the first argument for any fixed second one, and it is strictly decreasing to zero by the second argument for any fixed first one.

							( ) [ u i Î	u	min max , u	]	,	0 i ³ ensuring practical output regulation to a given reference
	( ) y i , d	0 i ³ , i.e., the property | ( ) y i	-	( ) | y i d	£ D should be satisfied for all	0 i ³ and Î d D , Î f F for some prescribed
	0 D > providing that | ( 0 ) y	-	y	d	( 0 ) |	£ D .
	To this end, recall that a continuous function : R + s	®	R +	belongs to class K if it is strictly increasing and ( ) 0 0 s = ;
	additionally it belongs to class ¥ K if it is also radially unbounded; and continuous function : R R + + b ´®	R +	is from class

KL

  , 2 g ÎK . According to Lemma 2 of[START_REF] Efimov | On Input-to-Output Stability of Switched Nonlinear Systems[END_REF] for dwelltime switched input-to-state systems, the upper estimate follows for all

																1 j i j + k k £ £	and all	0 k ³ :
	| ( ) | e i	1 £ l b -	(| ( ) |) k e j	e	ln( )( l -i j	k	)/	t	D	2 + -l g (1 ) -	[ 0, ] (|| || i l	)	,	1 ( ) max{ ( ), ( )} 2 s s s g = g g	.
	Now we may use the same technique considering	0 D ³ D > as a dwell-time between switches among different operating k
	modes of the system, that under theorem conditions gives:
	| ( ) |	2	(| (0) |)	ln( ) /		(1 )	4	[ 0, ] ( || ||	)

The research was supported by the General Motors Corporation. D.V. Efimov and V.O. Nikiforov are with Department of Information and Control Systems, Saint Petersburg State Institute of Information Technology, Mechanics and Optics, Russia (nikiforov@mail.ifmo.ru). D.V. Efimov is also with Non-A project, Inria, Parc Scientifique de la Haute Borne 40, avenue Halley, 59650 Villeneuve d'Ascq, France (Denis.Efimov@inria.fr). H. Javaherian. is with Powertrain Systems Research Laboratory, GM Research and Development Center, Warren, Michigan, USA (hossein.javaherian@gm.com).

ACKNOWLEDGMENT

We would like to thank Dr. Man-Feng Chang of GM R&D for his continued support of the project and many useful discussions during the course of this study.