Marcos Serrano

Thomas Pietrzak

Malika Auvray

Indira Thouvenin

J'ai eu la chance incroyable de faire ma thèse à l'ISIR encadré par Gilles Bailly et Sinan Haliyo et aussi à Segula Technologies encadré par Corentin Duboc. Je vous en remercie tous les trois ainsi que mon jury de thèse:

In this thesis, I investigate a visuo-haptic illusion, the hand redirection, consisting in injecting a position mismatch between the representation of the users' virtual hand and their real hand. This illusion may help to expand and improve haptic feedback in Virtual Reality (VR). I explore the design and implementation of the hand redirection as well as their use for learning with a focus on transfer of learning of motor control tasks.

I propose a new hand redirection technique that changes how the hand redirection is applied. In an empirical study, I evaluated the detectability of the illusion and compared it with a widely accepted technique. The interval of non-detection defines the maximum amplitude of redirection that can be applied to the users' hand without they noticing it. I also report on the difference in trajectories followed by the users with the new and previous technique. Results indicate the proposed new technique giving more flexibility to designers has a similar detection rate.

I also report on an empirical study measuring the amount of proactive transfer of learning from VR with the hand redirection to the unaltered real world. Proactive transfer is the influence a previously learned task has on users' performance a new task. Users had to pick and place cubes to imposed positions with a correct orientation while avoiding obstacles. Results show a strong positive transfer of learning and encourage the use of visuo-haptic illusions for skill learning in VR.

Based on the findings of this thesis, the design space of visuo-haptic illusions should receive more attention with two objectives: increase their interval of non-detection and offer more possibilities for designers.

Résumé

Dans cette thèse, j'étudie une illusion visio-haptique, la redirection de la main, qui consiste à introduire un décalage en position entre la main réelle d'un utilisateur et sa représentation virtuelle. Cette illusion permet d'étendre l'espace de design et d'améliorer le retour haptique. J'explore l'espace de design et l'implémentation de la redirection de la main, ainsi que leur utilisation pour la formation, en me concentrant sur le transfert d'apprentissage de tâches motrices.

Je propose une nouvelle technique de redirection de la main en changeant comment la redirection est appliquée à la main virtuelle de l'utilisateur. Dans une étude empirique, j'évalue sa détectabilité et la compare à une autre technique largement acceptée. L'interval de nondétection de l'illusion définit l'amplitude maximal de redirection que l'on peut appliquer à l'utilisateur sans qu'elle soit détectée. Je rapporte aussi des différences de trajectoires suivies par les utilisateurs entre la technique proposée et la précédente. Les résultats indiquent que cette nouvelle technique offrant plus de flexibilité aux designers a un intervalle de détection similaire à la technique comparée.

Je décris egalement une étude empirique mesurant la quantité de transfert d'apprentissage proactif de la réalité virtuelle lorsque l'utilisateur est exposé à la redirection de la main vers le monde réel non modifié. Le transfert proactif est l'influence qu'une tâche apprise précédement a sur la performance de l'utilisatuer sur une nouvelle tâche. Les utilisateur devait prendre et placer des cubes correctement dans une position et orientation imposées, tout en évitant des obstacles. Les résultats montrent un large transfert d'apprentissage positif et encourage l'utilisation d'illusions visuo-haptic pour l'apprentissage de compétences en réalite virtuelle. D'après les contributions de cette thèse, le design space devrait recevoir plus d'attention avec deux objectifs: augmenter l'interval de nondétection et offrir plus de libertés aux designeurs. T oday, a large collection of easily accessible tools, both as hard- ware (Virtual Reality Head Mounted Displays -HMD) and software (game engines, assets, etc), allow to create immersive environments quickly and cheaply. Virtual Reality (VR) is largely geared towards the entertainment and gaming market. It is also increasingly present in industrial/professional use cases such as Computer Aided Design (CAD), maintenance and training.

List of Figures

Immersion, multimodality and Visuo-haptic illusions

Probably, the primary aspect of a VR experience is immersion, the feeling of being in a different place than the physical one users are in [START_REF] Lessiter | A Cross-Media Presence Questionnaire: The ITC-Sense of Presence Inventory[END_REF]. To create better virtual environments, it is essential to understand why we feel immersed in VR and how to increase and exploit Figure 1.1: Haptic Retargeting, a visuo-haptic illusion used for repurposing a single physical cube (bottom images) to give haptic feedback for multiple virtual objects (top images) [START_REF] Azmandian | Haptic Retargeting: Dynamic Repurposing of Passive Haptics for Enhanced Virtual Reality Experiences[END_REF]. The users' virtual hand is redirected/translated away from their real hand position so that the real and virtual object appear in the same place.

the immersion. A factor that emerges is the multimodality of perception, giving users coherent feedback through multiple senses simultaneously. While it is straightforward to provide visual and auditory feedback in VR, it is more challenging to provide rich haptic feedback. Haptic feedback encompasses all tactile (texture, temperature...) and kinesthetic (weight...) sensations. As of today, haptic feedback in VR is sparse and is limited to vibrations in the controllers. Users cannot grasp and feel the texture of objects. Additionally, users often teleport instead of stroll due to the size difference between the physical space available and the virtual scene. Such interaction techniques tend to reduce the user's presence in a virtual scene. Sometimes, it even requires users to adopt these metaphors before they can comfortably interact in VR.

A proposed solution is passive haptic feedback [Insko, 2001] and visuo-haptic illusions [START_REF] Kohli | Combining passive haptics with redirected walking[END_REF], Azmandian et al., 2016]. Passive haptics consists in using physical props as proxies for similar or identical virtual objects, giving the user the haptic feedback necessary for immersion. It provides cheap and rich haptic feedback to users with little to no compromise on interactions and realism, giving both tactile feedback and kinesthetic feedback just like in real life. However, passive haptics on their own are limited: it would be cumbersome to have a prop for each of interactive object or touchable surface.

Visuo-haptic illusions allow, for instance, to re-purpose a single prop for multiple virtual objects as illustrated in Figure 1.1.

The range of amplitude the user cannot detect is relatively small and is the main limiting factor. Recent research focuses greatly on measuring the range of amplitude acceptable but the factors impacting it remain unclear. Furthermore, redefining the design space of visuo-haptic illusions may reveal new ways of improving visuo-haptic illusions.

Hence, my first research question is:

"How can we expand the design space of visuo-haptic illusions and what is the impact on immersion?"

Learning with Visuo-haptic Illusions

Another important criterion is learning. VR is used to train users on real world tasks such as maintenance or repair. The objective is for the trainees to transfer what they learned in VR to the real world. In general, in order to train or teach users a specific task, they need to occupy the productive space or there needs to be a physical reproduction of it. The main advantage of VR is the reusability of the technology: a single HMD and pair of controllers allow to reproduce a large variety of real-life scenarios without additional costs while keeping the productive space available. This is usually an effective way of training users on motor control tasks, however, it can be expensive. Haptic interfaces are costly especially when they are designed for large workspaces. Visuo-haptic illusions with passive haptics are much cheaper but are they a good tool to train users? Indeed, visuo-haptic illusions change the gestures and trajectories of trainees during learning. This might interfere with transfer of learning from VR to the real world. Although, results already show visuo-haptic illusions do not hinder learning and performance in VR [START_REF] Kohli | Redirected Touching: Training and adaptation in warped virtual spaces[END_REF], it remains unknown whether they influence transfer to the real world. Hence, my second specific research question is:

"Do visuo-haptic illusions favor transfer of learning in VR?"

General Problem

The objective of this thesis is to improve user interactions in Virtual Reality and promote the use of visuo-haptic illusions in a wider range of applications. One application is skill acquisition of motor control tasks. The global research question I address in this thesis is: "How to improve haptic feedback in virtual environments with visuohaptic illusions in order to promote immersion and transfer of learning on real world tasks?"

Research Approach

This thesis is at the crossroads of multiple domains: Human-Computer Interaction (HCI), Virtual Reality and Skill Acquisition. I adopt an HCI approach in this thesis. This domain focuses on designing, implementing and evaluating interaction systems. My work aims to expand the usability and flexibility of visuo-haptic illusions as an interaction techniques. I explore the design space of visuo-haptic illusions and propose a new implementation. I compare it with a previous well accepted implementation.

VR is an effective tool to immerse users in virtual environments. Research on VR focuses on finding what makes users feel present in a virtual environment and how to improve virtual experiences. Haptic feedback is a I use concepts from learning theory to better understand how visuohaptic illusions could be used to teach and train users in VR more effectively. I focus on the transfer of learning of motor skills from a virtual environment with a visuo-haptic illusions to the real world.

Context

This thesis is a CIFRE (industry funded) and is funded by SEGULA Technologies. Segula Technologies is an engineering group working in automotive, aerospace, defense, energy, naval and rail. Their need was to train operators on various motor control tasks. An example is in the pharmaceutical industry, operators in assembly lines need to perform maintenance (unjamming a machine, cleaning a broken sample, etc). They work with thick gloves in complex environments, if they pass over the product, the entire production line is wasted. They are under a strict time constraint since the production is paused while they operate.

These operators are trained in exact copies of the production line which is expensive and space consuming.

Their approach to solve this issue is to use VR to train operators. First to alleviate the need for real-sized models and second to have more control over the training: restarting, setting multiple different scenarios quickly. To train motor control tasks, they wanted to add haptic feedback to VR to have a more true to life simulation. They envisioned using haptic interfaces like the Omega from Haption [Haption, 2019] or more encountered-type haptic interfaces.

Contributions

The contributions of this thesis are the following:

-I expanded the design space of a visuo-haptic illusion by defining the important parameters and proposed to use different functions of redirection that controls the position of the users' virtual hand compared to their real hand. I proposed to use second order polynomial functions to redirect the users. They allow to apply the redirection more towards the start or towards the end of the users' gesture.

-I ran an empirical experiment testing the second order polynomial functions. I expanded on previous studies measuring the interval in which the users do not detect the illusion with a 2 Alternative Forced Choice (2AFC) design. This design allows to compare these new functions with the ubiquitously used linear function of redirection within the experiment and with previous ones.

-I performed the analysis of the results and computed the Interval of Non-Detection for each of the functions of redirection. I also analysed the trajectories followed by participants when exposed to the illusion and compared the results with previous studies. These results inform on the usability of the proposed function of redirection.

-I designed a experiment to measure the transfer of learning from VR with a visuo-haptic illusion. I chose a motor control tasks (pointing with obstacles) representative of typical real use-cases.

The design experiment was inspired from other widely accepted studies: between subjects, 24 hour rest period between a training phase and a test phase.

-I analysed the results by computing the amount of transfer as defined in the learning theory. I also compared participants that trained in the real world without an illusion with participants that trained in VR with the illusion. The results inform on how to use the visuo-haptic illusions to train users in VR for real world tasks.

Publications

I made three publications during this thesis. I explored the design of the hand redirection illusion and evaluated a new implementation in the proceedings of IHM'22 with Simon Besga, Flavien Lebrun and Gilles Bailly as co-authors [Geslain et al., 2022a].

I expanded the analysis of this paper in the proceedings of AVI'22 with Simon Besga, Flavien Lebrun and Gilles Bailly as co-authors [START_REF] Geslain | Generalizing the hand redirection function in virtual reality[END_REF].

I evaluated the hand redirection illusion for transfer of learning in the symposium of SUI'21 with Gilles Bailly and Sinan Haliyo as co-authors [START_REF] Geslain | Visuo-haptic Illusions for Motor Skill Acquisition in Virtual Reality[END_REF].

Overview of this thesis

• In chapter 2, I describe the related work on visuo-haptic illusions:

Body Redirection and World Redirection including and their various applications (prop repurposing, pseudo-haptic, infinite walking, etc). I have a particular focus on the different implementations of the Body Redirection techniques to highlight the potential design factors that can lead to the illusion being detected. I use what I discuss in this chapter to expand the design space of Body Redirection in chapter 4

• In chapter 3, I summarize the learning theory on skill transfer and discuss concepts, models and methods of transfer of learning. I also present previous work relating to VR and the specificity of transfer of learning in VR. From this chapter, I design an experiment measuring transfer of learning with visuo-haptic illusions in chapter 5.

• Using visuo-haptic illusions concepts and implementations described in chapter 2, I propose a novel approach on the Hand Redirection illusion using novel function of redirections in chap-ter 4. I describe an empirical study to evaluate this approach and compare it with preexisting implementations.

• From chapter 3 and transfer of learning methodology, in chapter 5, I explored the influence of the Hand Redirection illusion on skill transfer from VR to the real world with a user study.

T his chapter offers an overview of different visuo-haptic illusions. I first discuss the concepts and applications behind these different illusions organized into two categories: Body Manipulation and World Manipulation. I then describe the different implementations of these visuo-haptic illusions and the consequences on the interaction. Then, I introduce objective and subjective methods to evaluate these illusions. Finally, I describe a toolkit for visuo-haptic illusions I developed during my thesis.

Concepts & Design

Visuo-haptic illusions manipulate a user's perception by adding discrepancies between the visual and proprioceptive or haptic senses [START_REF] Kohli | Combining passive haptics with redirected walking[END_REF], Lécuyer, 2009]. This creates a conflict between senses that is generally solved by visual dominance [START_REF] Rock | Vision and Touch: An Experimentally Created Conflict between the Two Senses[END_REF], Egeth and Sager, 1977[START_REF] Colavita | Human sensory dominance[END_REF]. Visual dominance describes how humans put more trust in their vision over other senses [START_REF] Rock | Vision and Touch: An Experimentally Created Conflict between the Two Senses[END_REF] 1 . With visuo-haptic illusions, it is possible to provide richer haptic 1 Rock et al. had participants presented with a shape visually distorted with optics that they could also grasp and feel with their hands. When participants were asked to draw and describe the actual object, their results resembled more the distorted shape than the true shape of the object. However, visual dominance is not always the rule, in environments with low visibility (e.g. lack of light) participants shift their trust towards other senses [START_REF] Rock | Vision and Touch: An Experimentally Created Conflict between the Two Senses[END_REF].

c. b. a. shows that the virtual hand of the user is offset from their real one. The virtual and real hands are offset so that hands both reach the virtual and real targets [Zenner andKruger, 2019, Azmandian et al., 2016]. On figures b. and c., the virtual hand is progressively redirected to the virtual target so that the user's real hand touches the real target [START_REF] Abtahi | Beyond the force: Using quadcopters to appropriate objects and the environment for haptics in virtual reality[END_REF] (see section 4.1 for more details on implementation). feedback in VR by repurposing props or improving active haptic interfaces. They also allow to let the user interact in a larger virtual space than the real one. In general, visuo-haptic illusions increase the sense of immersion and presence in VR while the user remains unaware of it.

Many types of visuo-haptic illusions have been explored. I distinguish two main types: Body Redirection and World Redirection.

Body Redirection

Motivations

In general, providing haptic feedback when users act on the virtual environment improves their sense of presence. However, current VR technologies do not provide rich haptic feedback. Usually, the user has controllers for grasping an object by pressing a button or a trigger for example and haptic feedback is limited to vibrations which are uncommon in the real world (see Figure 2.2).

A simple approach to provide haptic feedback in VR is to use passive haptics, i.e. have physical objects that provide tactile and kinesthetic cues to the user (see Figure 2.3) [Insko, 2001]. Passive haptics works well with VR for several reasons. They offer true to life haptic feedback in terms of texture, weight, inertia, etc. They are cheap especially compared to more traditional haptic interfaces. Moreover, for effective use of passive haptics it is sufficient to use low fidelity physical objects that are similar enough in shape or weight to fit the visual cues [START_REF] Bouzbib | Survey of virtual reality interactions via haptic solutions: Revue de littérature des interactions en réalité virtuelle par le biais de solutions haptiques[END_REF]. For example, Hettiarachchi et al. used Figure 2.3: On both figures, on the right, a virtual environment with various furniture and objects, and on the right, the physical scene used to provide haptic feedback to the user with passive objects [Insko, 2001]. Every object the user can interact with in the virtual environment needs to have a physical counterpart. On figure a., the scene is simple, there are no tracked object, passive haptics is well suited here. On figure b. there are many moving parts (cubes) that need tracking. It requires specific equipment (an OptiTrack TM the picture) that can be expensive and/or cumbersome.

and Wigdor, 2016].

Although, passive haptics is a cheap and simple solution, they become quickly difficult and cumbersome to use in complex scenes. It can be tricky to track many objects at the same time and require manual intervention to reset (see Figure 2.3). They are often considered inflexible [Kohli, 2010]. To mitigate this issue and possibly improve presence and immersion in VR with passive haptics, they can be combined with the Body Redirection illusion [START_REF] Kohli | Combining passive haptics with redirected walking[END_REF].

Concept

Body Redirection illusions alter the representation of the user's body parts (typically their hand) and objects around them. A user's virtual representation of a body part is progressively offset from the position of the real body part. It can be any body part [START_REF] Kasahara | Malleable Embodiment: Changing Sense of Embodiment by Spatial-Temporal Deformation of Virtual Human Body[END_REF] but it is usually the user's hand, forearm and arm. In the case of Hand On the left, the real (in white) and virtual hands' (in black) trajectory while the user is reaching for the virtual target. On the right, a representation of the maximum distance between the real and virtual targets. While the virtual target is within the green area, the user does not detect the illusion. However, when they are too far away from each other, the illusion is broken.

Redirection, when users reach for an element in the virtual environment, their virtual hand is offset from their real hand position (see Figure 2.1). When this mismatch is small enough, users tend to believe their hands is where they see them (visual dominance) as is illustrated in Figure 2.4. Typically, when a user reaches for a virtual object offset from its real counterpart, their virtual hand is progressively shifted so that they touch the real object when they touch the virtual object. When the user touches the real object, it provides a haptic confirmation and gives the user a greater sense of control on the virtual environment. Different implementations have been proposed and are discussed in section 2.2.

Main application

The main application of the Hand Redirection illusion is to remap a virtual object onto another real object.

In early works on Hand Redirection, Kohli et al. matched surfaces oriented differently together (see Figure 2.5 [Kohli, 2009, Kohli, 2010]. The technique consisted in remapping the space around the physical surface to the virtual surface. Sparse haptics is another example where a The Redirected Touching technique where a 2D surface (top picture) can be mapped to differently shaped surfaces (bottom picture) [Kohli, 2009]. Each point of the real surface is made to correspond to a point on the virtual surface. The entire space between the user and the surface is remapped so the redirection is continous from the user's perspective.

physical facetted surface is used to represent a wide variety of virtual surfaces [START_REF] Cheng | Sparse Haptic Proxy: Touch Feedback in Virtual Environments Using a General Passive Prop[END_REF] (see . Similar approaches remap different shapes together like cubes to spheres [START_REF] Zhao | A Functional Optimization Based Approach for Continuous 3D Retargeted Touch of Arbitrary, Complex Boundaries in Haptic Virtual Reality[END_REF] or allow grasping differently sized virtual and real objects [START_REF] Bergström | Resized grasping in vr: Estimating thresholds for object discrimination[END_REF]. Azmandian et al. used a slightly different approach called Haptic Retargeting where participants could manipulate many virtual objects while manipulating a single real object [START_REF] Azmandian | Haptic Retargeting: Dynamic Repurposing of Passive Haptics for Enhanced Virtual Reality Experiences[END_REF] to essentially create a 1:n mapping between the real and virtual environment (see . This techniques combines well with low-fidelity props to use simpler, more reusable and fewer objects [START_REF] Han | Evaluating Remapped Physical Reach for Hand Interactions with Passive Haptics in Virtual Reality[END_REF].

Additionally to passive haptics, it is possible to combine Hand Redirection with active haptic interfaces [START_REF] Abtahi | Visuo-Haptic Illusions for Improving the Perceived Performance of Shape Displays[END_REF]. Gonzalez et al. used the Hand Redirection illusion to compensate for the latency of an encountered-type haptic device [START_REF] Gonzalez | Reach+: Extending the reachability of encountered-type haptics devices through dynamic redirection in vr[END_REF] (see Figure 2.6-d). When the system detects that the robot will not be able to reach the target position in front of the user, they redirect the hand towards the robot to reduce its travel time. Abtahi and Follmer improved a shape display [START_REF] Leithinger | Shape displays: Spatial interaction with dynamic physical form[END_REF] by scaling up or down and redirecting the direction of a user's gesture [Abtahi and Follmer, a. b.

c. d. shows the virtual environment with the virtual hand and target and superposed on the virtual environment the real hand and target [START_REF] Zenner | Estimating Detection Thresholds for Desktop-Scale Hand Redirection in Virtual Reality[END_REF]. b. shows the Sparse Haptic interface where facetted surfaces provide haptic feedback for various virtual surfaces in 2 different virtual environment [START_REF] Cheng | Sparse Haptic Proxy: Touch Feedback in Virtual Environments Using a General Passive Prop[END_REF]. c. is the Haptic Retargeting technique used to repurpose a single physical cube (bottom view) for three virtual cubes (top view) [START_REF] Azmandian | Haptic Retargeting: Dynamic Repurposing of Passive Haptics for Enhanced Virtual Reality Experiences[END_REF]. d. shows how Hand Redirection allows to improve the responsiveness of encountered-type haptic interface [START_REF] Gonzalez | Reach+: Extending the reachability of encountered-type haptics devices through dynamic redirection in vr[END_REF].

2018]. They managed to almost double the perceived size of interaction as well as drastically increase the speed perception in the vertical axis. It is the illusion I use and explore in this thesis.

Other applications

Another application of Hand Redirection is pseudo-haptic feedback. Pseudo-haptic feedback consists in providing or altering an existing haptic feedback [Lécuyer, 2009, Lécuyer et al., 2003].

The Swamp illusion is a Pseudo-haptic illusion, one of the earlier example of a force feedback illusion: users move a virtual cube on the horizontal plane. The pseudo-haptic feedback give them the illusion that something is resisting against or helping them because the virtual shows an illustration of the Swamp illusion where participants controlled the blue cube and the yellow area slowed them [START_REF] Lecuyer | Pseudo-haptic feedback: can isometric input devices simulate force feedback?[END_REF]. c.

shows the physical setup to evaluate the stiffness of a spring and d. the visual representation of the spring they altered [START_REF] Lecuyer | Pseudo-haptic feedback: can isometric input devices simulate force feedback?[END_REF].

cube is sped up or down in a highlighted area (see Figure 2.7-b). As a consequence, users tend to apply more force on the device (a SpaceBall TM 2 ) giving them the illusion that something is resisting against or helping them.

Lécuyer et al. also managed to change the perceived stiffness of a spring by visually compressing more or less a virtual spring [START_REF] Lecuyer | Pseudo-haptic feedback: can isometric input devices simulate force feedback?[END_REF]. In a user study, participants press on a SpaceBall TM through a plastic extension (Figure 2.7-cd). They compare this virtual spring with three real springs at different stiffness. Authors find that participants can discriminate between two identical springs with one that has a modified apparent stiffness.

Many different instances of pseudo-haptic feedback emerged from these experiments. Similarly to changing the stiffness of a spring, the stiffness [START_REF] Punpongsanon | FleXeen: Visually Manipulating Perceived Fabric Bending Stiffness in Spatial Augmented Reality[END_REF], Punpongsanon et al., 2015] or elasticity [START_REF] Argelaguet | Elastic images: Perceiving local elasticity of images through a novel pseudo-haptic deformation effect[END_REF] of fabric and other materials can be modified.

Tactile images allow participants to feel simple surface deformations like bumps and holes [START_REF] Lécuyer | Feeling Bumps and Holes Without a Haptic Interface: The Perception of Pseudo-haptic Textures[END_REF]. When users explore a virtual surface with a passive device like a mouse, accelerating the cursor on an upward slope and decreasing it on a downward slope gives information about the surface deformation. Figure 2.8-b illustrates how the cursor behaves when crossing a bump. A simulator leveraged this technique to help medical students to find organs and avoid tissues and nerves for a loco-regional anaesthesia [START_REF] Bibin | SAILOR: A 3-D Medical Simulator of Locoregional Anaesthesia Based on Desktop Virtual Reality and Pseudohaptic Feedback[END_REF]. Organs feel harder when palping them under the skin so they rendered them as bumps and [START_REF] Lécuyer | Feeling Bumps and Holes Without a Haptic Interface: The Perception of Pseudo-haptic Textures[END_REF]. When the user's cursor enters the bump area, the cursor's speed is decreased to simulate the slope. The opposite is done when exiting the bump area by increasing the cursor's speed.

tissues as holes.

Weight perception can also be modified by altering the visual representation of the body or an object. Jauregui et al. used three techniques on a weight lifting exercise [Gomez Jauregui et al., 2014]. The user was represented by a visual avatar on a deported screen. The technique modifies the avatar inclination and the animation of the avatar during the lift. Authors show that users can correctly rank weights going from 2kg to 14kg with all techniques. Many recent studies focused on simulating weights in Virtual Reality [START_REF] Samad | Pseudo-Haptic Weight: Changing the Perceived Weight of Virtual Objects By Manipulating Control-Display Ratio[END_REF], Rietzler et al., 2018] as illustrated in Figure 2.8-a. These techniques seem to help users better feel and estimate the weight of the virtual object. Similarly to tactile images, there are medical applications to this illusion [START_REF] Li | Motion assistance and resistance using pseudo-haptic feedback for upper-limb rehabilitation[END_REF].

Haptic interfaces can be improved by pseudo-haptic feedback. For example, cutaneous fingertip interfaces render the stiffness of a material as the user touches it [START_REF] Chinello | A three revolute-revolute-spherical wearable fingertip cutaneous device for stiffness rendering[END_REF], Solazzi et al., 2010]. By changing the visual feedback, the pseudo-haptic illusion increases the range of stiffness the interface can procide by itself [START_REF] Jang | On utilizing pseudohaptics for cutaneous fingertip haptic device[END_REF]. Similarly, they improve feedback from vibro-tactile screens [START_REF] Hachisu | Pseudo-haptic feedback augmented with visual and tactile vibrations[END_REF].

These examples rely on existing haptic feedback and change users' perception. There are also examples of pseudo-haptic techniques that attempt to generate illusory haptic feedback using only visual cues. shows the world-in-miniature and the increased speed gains techniques [START_REF] Abtahi | Visuo-Haptic Illusions for Improving the Perceived Performance of Shape Displays[END_REF]. They either upscaled the user in the virtual environment or increased the movement speed of the user. b. shows a teleportation techniques using an HTC VIVE controller [START_REF] Xu | Object location memory error in virtual and real environments[END_REF]. The user can point to a location and be teleported there instantly (or after a fade to black). c. shows a stepper machine the user's hand in the flow's direction when it came in contact with it [START_REF] Pusch | HEMP-hand-displacement-based pseudo-haptics: A study of a force field application and a behavioural analysis[END_REF].

World Redirection

Motivations

Locomotion is an essential part of many tasks one experiences in VR.

A problem emerged concerning the lack of available physical space for a user to wander freely in. More often than not, the virtual environment is larger than the real one which prevents them from being able to actually walk unconstrained (real walking [START_REF] Steinicke | Taxonomy and Implementation of Redirection Techniques for Ubiquitous Passive Haptic Feedback[END_REF]).

To enable the user to explore the virtual environment, designers created new locomotion interactions. The most commonly used ones are:

• Controllers can be used to move in the virtual environment. For example, the user can translate or rotate using a joystick [Nabiyouni et al., 2015, Argelaguet andMaignant, 2016]. Also using a joystick, a pad or gestures, the user points a location to teleport to [START_REF] Cherni | Literature review of locomotion techniques in virtual reality[END_REF] illustrated in Figure 2.9-b. This is the most used technique as it is simple to implement. However, these a. c. shows the arm swinging technique [START_REF] Cherni | Literature review of locomotion techniques in virtual reality[END_REF] where swinging the arm forward moves the user forward. b. shows the walkingin-place [START_REF] Bozgeyikli | Locomotion in virtual reality for individuals with autism spectrum disorder[END_REF]. Walking-in-place consists in moving their feet up and down which moves the user forward in the virtual world. c. shows a passive treadmill in which the user's waist is fixed to the contraption and their feet slide along the surface [VR, 2021].

techniques do not promote immersion as they are far from natural walking. Moreover, they can require some training before being used effectively.

• Walking metaphors allow the user to mimic walking. Users can walk in place, i.e. do step-like movements while remaining stationary [START_REF] Bozgeyikli | Locomotion in virtual reality for individuals with autism spectrum disorder[END_REF]. Similarly, users can move forward by moving their arms as if they were walking or running [Boletsis, 2017, McCullough et al., 2015, Ferracani et al., 2016] illustrated in Figure 2.9-c. Users' limb movement are tracked or they step on input devices like the Stepper Machine illustrated in Figure 2.10-a and Figure 2.10-b. With treadmills, users walk on a surface that moves with their feet or on a smooth surface on which they glide (friction-free platforms) illustrated in Figure 2.10-c.

• Magical locomotion techniques allow the user to move in the virtual environment free of real world constraints. World-in-miniature or giant metaphors consist in changing the size of the virtual envi-ronment to increase the range of the user through walking [Abtahi et al., 2019a]. They are illustrated in Figure 2.9-a.

These methods allow users to move around but they are often not natural. They either use completely different control schemes than in the real world which can require a learning period before a user can use them effectively or they mimic the way, humans walk while removing some perception cues like acceleration. In general, they don't promote presence as much [Langbehn and Steinicke, 2018]. Indeed, real walking remains the best locomotion method when compared to them. More hardware-oriented solutions like using treadmills allow users to walk in any direction indefinitely [START_REF] Feasel | The integrated virtual environment rehabilitation treadmill system[END_REF], Powell et al., 2011, Powell, 2011]. However, those treadmills are expensive and cumbersome. An alternative to promote real walking is the World Redirection. Figure 2.12: On the top (a.), a visualization of a blinks and saccades based techniques [START_REF] Sun | Towards virtual reality infinite walking: dynamic saccadic redirection[END_REF]. The virtual scene is viewed from the point of view of the user on the left and from the point of view of the user after a blink. on the bottom (b.), an illustration of eye movements ("Saccades", red line) and points fixed by the user ("Fixations", green dots) [Langbehn and Steinicke, 2018]. The point of view of the user is redirected during saccades and blinks.

Concept

World manipulation is a visuo-haptic illusion that consists of altering a users' point of view (POV) in the scene in order to change their perception (see Figure 2.11). By subtly rotating their POV according to their movements, it is possible to increase the perceived size of the interactive space [START_REF] Razzaque | Redirected Walking[END_REF], Shen et al., 2020] or even to foster Infinite Walking by making the user walk in circles [START_REF] Nilsson | 15 years of research on redirected walking in immersive virtual environments[END_REF]. It can also enable physical prop repurposing so that a single physical prop may be used for multiple virtual objects [START_REF] Azmandian | Haptic Retargeting: Dynamic Repurposing of Passive Haptics for Enhanced Virtual Reality Experiences[END_REF] or hide unrealistic virtual elements [START_REF] Ciumedean | Mission impossible spaces: Using challenge-based distractors to reduce noticeability of selfoverlapping virtual architecture[END_REF].

Blinks, Saccades and Distractors Research also explored how to use user's temporary lack of attention to apply redirection. For example, Langbehn et al. rotated and shifted the user's field of view when they blinked [Langbehn et al., 2018]. This technique leverages the saccadic suppression and change blindness [START_REF] Kevin | [END_REF] that denotes the human inability to detect large changes in a scene after a short span of inattention like a blink or a saccade (rapid eye movement) [START_REF] Rensink | On the failure to detect changes in scenes across brief interruptions[END_REF], Rensink, 2002]. By placing an eye-tracker inside the VR HMD allows to detect eye blinks and saccades and to redirect the user undetected [Sun et al., 2018, Langbehn andSteinicke, 2018] (see Figure 2.12). Studies show that humans blink about 10 to 20 times a minute [START_REF] Kevin | [END_REF] and empirical results show that the VE can be rotated by 2°to 5°and translated by 4cm to 9cm during a blink without being detected [START_REF] Rensink | On the failure to detect changes in scenes across brief interruptions[END_REF]. Chen et al. actively distracted the user with virtual objects or characters appearing in the user's field of view [START_REF] Chen | Supporting free walking in a large virtual environment: Imperceptible redirected walking with an immersive distractor[END_REF], improving the control and better ensuring that the user will stay in the tracked area. Even sounds can be used to distract the user [START_REF] Rewkowski | Evaluating the effectiveness of redirected walking with auditory distractors for navigation in virtual environments[END_REF].

Main application

The main application of World Redirection is to promote real walking in VR. The techniques turns the user's virtual point of view compared to their actual orientation in the real world as illustrates Figure 2.13a. Early work by Razzaque et al. applied a factor to head rotations and slightly rotated the head when walking so that users think they walk a straight line while actually walking in a curved line [START_REF] Razzaque | Redirected Walking[END_REF] (see . They also showed that applying a small constant rotation of the head can go unnoticed in VR and curve even more their trajectory. Redirected walking is mostly used to allow users to walk naturally in a much smaller actual space than the virtual one. This ability is often called infinite walking and describes how a user can move without restrictions in an infinitely large virtual space.

Recent improvements have been made on this technique such as reducing even more the required physical area by allowing users to touch a wall while walking [START_REF] Matsumoto | Unlimited corridor: Redirected walking techniques using visuo haptic interaction[END_REF], Matsumoto et al., 2019] (see Figure 2.13-c). They concluded it provides a sort of haptic confirmation that they are walking a straight line which strengthen the illusion. This confirmation is similar to the one received with Hand Redirection after touching the target physical object. ), a representation of the virtual path followed in the virtual environment and the real path followed by the user. The user reaches a different virtual position in the virtual environment than in the physical world [START_REF] Rothacher | Visual capture of gait during redirected walking[END_REF]. On the top-right (b.), a top-down view of a virtual room (in blue) and the physical space (in red) with the path followed by a user while exposed to redirected walking. The real room is smaller than the virtual one and the path followed by the user is drastically altered in VR [START_REF] Razzaque | Redirected Walking[END_REF]. On the bottomright (c.), the unlimited corridor where users touch a curved wall in the real world to increase the amount of redirection that can be applied to the user without being noticed [START_REF] Matsumoto | Unlimited corridor: Redirected walking techniques using visuo haptic interaction[END_REF].

Other applications

Another application consists of creating the virtual scene given a certain amount of space available. This technique is called smooth mapping [START_REF] Sun | Mapping virtual and physical reality[END_REF]. The virtual environment is represented by polygons which are distorted and rearranged such that for every position inside the virtual scene corresponds a point in the physical space. Contrary to Redirected Walking, no discrepancy is created between the visual and proprioceptive senses. The result is shown Figure 2.14. Sun et al. also provided an additional parameter that reduces the amount of visual distortion and essentially using redirected walking to compensate the rest of the required redirection.

The virtual scene can be significantly distorted although participants Unmodified Camera-View After Smooth Mapping Figure 2.14: On the left, a side by side comparison of the virtual environment seen from the user point of view without any transformation added on the left, and with the smooth mapping algorithm applied on the right [START_REF] Sun | Mapping virtual and physical reality[END_REF]. On the top right, the path followed by the user in the virtual environment and on the bottom right, the path followed in the the physical space for a given virtual scene.

tend to rate highly the visual fidelity of the scene. Depending of the virtual and real environment, the results can vary: in Figure 2.14, the top scene corridor is very distorted and does not look like the original one, however, the bottom scene is less distorted.

Other works managed to overlap the virtual environment on itself to fit the physical space [START_REF] Suma | Impossible spaces: Maximizing natural walking in virtual environments with self-overlapping architecture[END_REF], Vasylevska et al., 2013]. This technique called Flexible Spaces, dynamically rearranges corridors so that the rooms of the virtual environment overlap saving space in the real world. This reduces the amount of redirection necessary and improves the maximum explorable virtual space.

World Redirection was also used to repurpose a physical prop for multiple virtual object similarly to Hand Redirection. Kohli et al. and Azmandian et al. did so by rotating the user's point of view to match a virtual object with a physical proxy [START_REF] Kohli | Combining passive haptics with redirected walking[END_REF], Azmandian et al., 2016]. This method requires that the virtual object and its physical proxy are equidistant to the user.

We retain there are two main types of visuo-haptic illusions. Hand Redirection allows to expand and improve haptic feedback in VR and combines particularly well with passive haptics. World Redirection can expand the space in which the user can interact in. Visuo-haptic illusions are a cheap solution to improve the user experience in VR.

Implementation

Implementing visuo-haptic illusions in real world scenarios requires (1) to know what the user intends to do, (2) applying the illusion in accordance and sometimes (3) implement some obstacle avoidance strategies.

Knowing the user's intention is essential to make the real and virtual environment coherent together. For example, in the case of repurposing real objects, the real and physical targets need to be known in advance before applying the redirection. Then, multiple implementations of the same visuo-haptic technique exist. In some cases, they can influence the user interaction and have direct implications on user presence. I revisit these implementations in chapter 4. Finally, since these illusions visually mismatch the virtual and real environment, the user can enter in contact with physical obstacles although they should not. In the case of World Redirection, the user should be redirected away from walls and not hit them.

In this section, I detail the various strategies and implementations used for visuo-haptic illusions.

Intention Detection Model

In the context of VR and visuo-haptic illusions, detecting the user intention consist in determining where the user is about to walk to or what object or surface the user is about to reach for. Unfortunately, most studies on visuo-haptic illusion use a deterministic scenario where they either know in advance what the user is going to do or tell them explicitly [START_REF] Azmandian | Haptic Retargeting: Dynamic Repurposing of Passive Haptics for Enhanced Virtual Reality Experiences[END_REF], Han et al., 2018]. In such scenarios, the real and virtual targets object are known. In contrast, in non-deterministic scenarios, the users have more freedom. These are more akin to real life. Determining which object the user will interact with or in which direction they want to move is essential for visuo-haptic illusions and requires the elaboration of an intention detection or prediction model. These models estimate the prior intention of a user before the movement starts [START_REF] Searle | Intentionality: An essay in the philosophy of mind[END_REF] and adjust this estimation during the users' movement [Cohen, 2015].

The output of such intention models is the probability that each potential target object of the scene to be reached. The output can have a finer level of granularity by indicating which area of the target object might be grasped (e.g. the top of the object, the side, etc.).

Several inputs have been considered for the elaboration of the intention models. A very practical one in VR is the direction of the users' head, assuming that the users are looking directly ahead [START_REF] Novak | Enhancing patient freedom in rehabilitation robotics using gaze-based intention detection[END_REF], Bouzbib et al., 2020, Sartori et al., 2011]; the distance from the target object [START_REF] Sartori | Does the intention to communicate affect action kinematics?[END_REF] or both [START_REF] Chen | Gaze-based interaction intention recognition in virtual reality[END_REF]. When available, the intention model can rely on the data of an eye-tracker to precisely capture the gaze direction [Cohen, 2015]. This approach is more and more convenient as some headsets now have a built-in eye tracker (HTC Vive Pro Eye) or have the possibility to install one. An alternative is to use the hand movement and the hand pose to detect intention [Cohen, 2015, Sartori et al., 2009].

Different strategies have been used to predict the most likely intended target given these inputs. For instance, Bouzbib et al. proposed a simple linear model weighting both head direction and distance from object to estimate the next object of interaction [START_REF] Bouzbib | Covr: A large-scale force-feedback robotic interface for nondeterministic scenarios in VR[END_REF]. [START_REF] Mugisha | Safe collaboration between human and robot in a context of intermittent haptique interface[END_REF] uses a k-d tree based model to predict user intention from hand movement and gaze direction.

Basic Implementation of Visuo-haptic Illusions

C/D Ratio

The Control to Display (C/D) ratio defines the ratio between the distance an input device moved by a user and the corresponding distance the controlled object (e.g. cursor) is moved on the display. In 2D pointing tasks with a computer mouse, the C/D ratio defines the amount of pixels the virtual cursor moves on screen according to the amount the mouse is moved. In the context of visuo-haptic illusions based on Hand Redirection, C/D ratio defines the ratio between the real and the virtual hand displacement. Zenner et al. proposed a C/D ratio implementation to redirect the user's hand in the depth axis [START_REF] Zenner | Estimating Detection Thresholds for Desktop-Scale Hand Redirection in Virtual Reality[END_REF]. When the user moves their real hand forward, their virtual hand displacement is scaled up or down depending on the objects position. This implementation is also largely used with Pseudo-Haptic: the swamp illusion modifies the C/D ration inside the highlighted area creating a sense of resistance or the opposite [START_REF] Lecuyer | Pseudo-haptic feedback: can isometric input devices simulate force feedback?[END_REF].

Since VR is 3D in nature, the C/D ratio as defined here is restricting the effect to the direction of movement. With visuo-haptic illusions, it is typical to redirect the user in a direction different than the user's movement. For example, if the user's hand is to be redirected to the right but the object to reach is forward, the C/D ratio is not enough. For this purpose, the C/D ratio was expanded into a more translation and/or rotation based dimensions as I describe now.

Dimensions: Translation/Rotation/Curvature Gain

Translation -based techniques of Hand Redirection consist in translating the virtual hand from the real hand position. The translation is defined by a vector ⃗ ∆ M [START_REF] Han | Evaluating Remapped Physical Reach for Hand Interactions with Passive Haptics in Virtual Reality[END_REF] as illustrated in Figure 2.15-left. The starting position is a fixed-point where no redirection is applied. It is a way of resetting the user's virtual and real hands positions. ⃗ λ defines the vector between the real and virtual targets, it is the maximum value of ⃗ ∆ M when the real hand reaches the real target. ⃗ ∆ M is computed using the distance from the target d, the total distance from the starting position to the target D and the distance between the targets λ. An example of such an implementation is shown in Equation 2.1.

⃗ ∆ M (d) = (1 - d D ) ⃗ λ (2.1)
This equation warps the user's hand as it gets close to the target and works well in 3D. This implementation is the one I used in my work and I expand on it further in chapter 4.

Similar examples tried to warp the user's hand as soon as it left the starting position. The redirection is then abrupt. Empirical results show that smoothing the redirection is more effective [START_REF] Han | Evaluating Remapped Physical Reach for Hand Interactions with Passive Haptics in Virtual Reality[END_REF]. On the left, the virtual hand position is computed using a position based technique as described in Equation 2.1. On the right, the virtual hand position is computed using a rotation based technique as described in Equation 2.2.

Translation-based techniques of

and the target. Zenner et al. used this implementation to redirect the user in the side (left/right) and vertical (up/down) axis [START_REF] Zenner | Estimating Detection Thresholds for Desktop-Scale Hand Redirection in Virtual Reality[END_REF]. A way to implement this is described in Equation 2.2:

⃗ P V (d) = R(θ, ϕ, 0) ⃗ P R (2.2)
⃗ P V and ⃗ P R are the positions of the Virtual and the Real hand centered on the starting position and R(θ, ϕ, 0) is a rotation matrix along the side axis and vertical axis of θ and ϕ angle. The resulting virtual hand trajectory is shown in Figure 2.15-right. The resulting function is equivalent to the position based-technique described in Equation 2.1 and illustrated in Figure 2.15-left.

Rotation based Hand Redirection implementation requires both the real and virtual targets to be on the circle centered on the starting position or they would not be matched with one another despite the redirection. To overcome this restriction, a C/D ratio based redirection needs to be applied in the radial direction on top of the rotation redirection. Zenner et al. used this combination to evaluate Hand Redirection [START_REF] Zenner | Estimating Detection Thresholds for Desktop-Scale Hand Redirection in Virtual Reality[END_REF].

With World Redirection, rotation-based techniques use head rotations to rotate the point of view of the user in the virtual environment [START_REF] Azmandian | Haptic Retargeting: Dynamic Repurposing of Passive Haptics for Enhanced Virtual Reality Experiences[END_REF], Razzaque et al., 2001]. There is a direct application of the C/D ratio on head rotation but rotations can also be added slowly over time without being noticed (Redirected Walking [START_REF] Razzaque | Redirected Walking[END_REF]). Additionally, rotations to the user's point of view can be applied using head translations. It is called Curvature Gain. It represents the bending of the virtual path walked compared with the real path walked. The curvature gain is defined as

g c = 1 r (2.3)
where r is the radius of the circular arc the user is meant to walk in the physical space. Once g c is computed, it is applied as

θ = g c d (2.4)
where θ is the orientation of the user's point of view along the vertical axis and d is the distance walked. If the curvature gain is null, the radius is infinite and the user will not be redirected. On the contrary, if r = 1, the curvature gain is g c = 1, hence if the user walks a distance of 2πm, they will be rotated by 2π and end up where they started.

Reversely, curvature gain can also describe translation offsets generated by head rotations [Steinicke et al., 2008a]. When the user turns their head, their point of view is translated in a chosen direction. Translations can be applied in all horizontal directions and ,depending on context, in the vertical direction.

In summary, there are 2 main techniques in Body Redirection: translation and rotation based techniques. Both are equivalent but translation based techniques are easier to implement in a real-world scenario.

And there are 3 main techniques in World Redirection:

• Translation gains where the user's movement is scaled in the virtual environment. This technique can use both the speed of displacement of the user or their relative displacement.

• Rotation gains where the user's rotation are amplified or reduced in the virtual environment.

• Curvature gains where participants point of view is rotated as they walk forward. This technique is particularly useful to control precisely the user's trajectory in the physical space.

Virtual POV Real POV On the left, the user thinks there's an object to the right of the target whereas the actual object is on the left of their real hand due to the illusion as shown on the right. This mismatch can create issues where the participants hits an object when they should not.

The limitations of gain-based Redirected Walking are that the largest unnoticeable gain is relatively small and the technique requires large physical environments in order to promote Infinite Walking.

Obstacle Management

Visuo-haptic illusions can be challenging to use when multiple physical target objects or obstacles are in the arena. Since, the mapping between the real and virtual environment is modified, some of the discrepancies can break the illusion.

Regarding Hand redirection, research mostly focuses on making the user reach the physical targets but not really on avoiding obstacles. When exposed to the illusion, the hand of the participant is not located where they think it is. If a physical obstacle is present on the trajectory of the real hand but not visible on the trajectory of the virtual hand, the user can make contact with it although they should not. This would "break" the illusion and decrease immersion in the virtual environment. Regarding World Redirection, to foster infinite walking, the user must be redirected away from the obstacles: the limits of physical space (walls), furniture, etc. This is challenging and four main strategies exist [Langbehn andSteinicke, 2018, Hodgson and[START_REF] Hodgson | [END_REF]:

Center Center Center

• Steer-To-Center • Steer-To-Orbit • Steer-To-Multiple-Targets • Steer-To-Multiple-Targets+Center
All strategies are illustrated in Figure 2.17, the user is always redirected towards a point or an orbit. The steer-to-center consists in redirecting the user towards the center of the physical space. The steer-toorbit consists in redirecting along an orbit/circle centered on the physical space to keep user parallel to the physical space limits. The Steer-To-Multiple-Targets aims to redirect the user to multiple targets. It's a particularly useful to repurpose props for multiple virtual objects. Finally, the Steer-To-Multiple-Targets+Center extends Steer-To-Multiple-Targets by including the center of the area. Of course, the shape of the physical room can influence those results and a strategy might be more appropriate for some use cases.

Evaluation Measures of Visuo-haptic illusions in Virtual Reality

Two main protocols are used to evaluate visuo-haptic illusion techniques: An objective one, based on a 2-Alternative Forced Choice (2AFC) experiment to measure the interval of non-detection of the illusion and a subjective one, based on questionnaires to measure presence.

Interval of Non-detection

A typical study measures the detection threshold of a given illusion based on a 2-Alternative Forced Choice (2AFC). Participants are exposed to the illusion technique at varying parameters: amplitude and direction of redirection (e.g. 2°, 4°, 6°and left or right). After each trial, participants are required to indicate in which direction the illusion occurred (e.g. the real hand is on the left side of the virtual hand) [START_REF] Zenner | Estimating Detection Thresholds for Desktop-Scale Hand Redirection in Virtual Reality[END_REF]. After enough repetitions, the probability of answering a choice (e.g. left) can be extracted for a given set of parameters. These probabilities follow a logistic function:

P (Left or Right | X) = 1 1 + e -X-a b (2.5)
Where P (Left or Right | X) is the probability of answering either choice (e.g. left) knowing the stimulus X in the experiment, a and b are the function's parameters. Plotting these probabilities gives a psychometric function like in Figure 2.18.

The Interval of Non-Detection (IND) is then defined as the range in the stimulus in which the illusion remains undetected. Usually, it is considered that when participants answer 75% of the time correctly that the illusion is detected [Lecuyer et al., 2000, Zenner and[START_REF] Zenner | [END_REF]. This method has been extensively tested and used to analyse psychometric results [Bonnet, 1986, Gescheider, 1985]. Hence the Left IND can be written as

P (Right | IN D L ) = 0.25 and the right IND as P (Right | IN D R ) = 0.75. Finally, the range of IN D is then written IN D = IN D L -IN D R .
In addition, the parameter a defines the Point of Subjective Equality (P ES), the point at which participants answer one choice as much as the other. It is also called the bias and represents the point where users have an equal probability of answering either answer. Parameter b represents the steepness of the function as it passes through the P ES. In practice, the IND are computed by fitting with Equation 2.5 the experimental points. Then, with the reciprocal of Equation 2.5: In the case of Hand Redirection, it is expected to be 0°(P (Right | 0 • ) = 50%) when no illusion is applied.

X = -b ln ( 1 P (Right | X) -1) + a (2.
I use a 2AFC protocol and the interval of non-detection in this thesis to evaluate a novel hand redirection technique in chapter 4.

Presence

Presence questionnaires have been extensively used to evaluate visuohaptic feedback. Presence defines the sense of "being physically present with visual, auditory or force displays generated by a computer" [Sheridan et al., 1992]. Witmer et al. defined four categories of factors contributing to presence [START_REF] Witmer | Measuring Presence in Virtual Environments: A Presence Questionnaire[END_REF]]:

• Control factors evaluate the immediacy of control: the environment must react when the user acts on it, predictability of the environment: being able to anticipate the outcome of an action and modify the environment.

• Sensory factors relate to the modalities used during the experience focusing primarily on vision and on a lesser extent on auditory, olfactory and haptic feedback. The sensory factors are the richness of the feedback: how much information is transmitted to a modality, and the coherence of multimodal feedback: the number of modalities stimulated and the coherence between the information received through each sensory channel.

• Distraction factors define the isolation of the user from the physical environment and the ability and willingness of the user to ignore unrelated stimuli from outside the virtual environment.

• Realism factors can relate to both life-like realism (texture, lightning of the field of view) or consistency of information (is the virtual environment consistent in its behaviour).

These questionnaires study whether the mismatch introduced by visuo-haptic illusions degrades the presence and interactivity in VR [START_REF] Kohli | Combining passive haptics with redirected walking[END_REF], Azmandian et al., 2016, Han et al., 2018, Matthews et al., 2019]. Empirical studies show it is often a viable way to reduce cost and complexity to set-up a virtual environment while maintaining a high sense of presence. However, these studies do not evaluate the influence of visuo-haptic illusions on motor skills. I will discuss in detail the learning of motor skills in VR environment involving visuo-haptic illusions in chapter 5.

Trajectory Analysis of Hand Redirection

Some insights into how users detect the illusion can be gained by studying the trajectories followed by the users from the starting position to the real target. Azmandian et al. did a qualitative analysis where they showed that participants tend to correct their trajectory towards the end [START_REF] Azmandian | Haptic Retargeting: Dynamic Repurposing of Passive Haptics for Enhanced Virtual Reality Experiences[END_REF]. A noticeable kink is visible a the end of the trajectory as is illustrated in Figure 2.19. It looks like there were two component to the trajectory. First, a ballistic phase where participants operate in an open loop in the robotics sense: they move towards the target without any adjustments according to their real hand position during the gesture. Finally, a correction phase where participants adjust their real hand position to correct for the error introduced by the Body Redirection illusion.

Lebrun et al. studied the trajectory shape according to the IN D [START_REF] Lebrun | A trajectory model for desktop-scale hand redirection in virtual reality[END_REF] (see Figure 2.20). They also found a similar trajectory than [START_REF] Azmandian | Haptic Retargeting: Dynamic Repurposing of Passive Haptics for Enhanced Virtual Reality Experiences[END_REF] with a ballistic and correction phase. They also found that the more the hand's trajectory deviates from a straight line between the starting position and the real target, the easier it is to detect the illusion.

Specific Measures of Infinite Walking

In addition to questionnaires, specific illusions such as infinite walking have been evaluated with specific objective measures: wall contacts and the required physical area.

Wall Contacts To compare the efficiency of the obstacle avoidance strategies, a simple measure is the number of wall contacts and the maximum distance from the center area [START_REF] Hodgson | Comparing four approaches to generalized redirected walking: Simulation and live user data[END_REF]. Hodgson et al. found that on largely unconstrained virtual environments the Steer-To-Center were the best strategy and outperformed other strategies in containing the user in the smallest area possible. However, in structured virtual environments with corridors, roads or straight lines in general the Steer-To-Orbit is better [START_REF] Hodgson | Performance of redirected walking algorithms in a constrained virtual world[END_REF].

Physical Area Required for Infinite Walking Techniques that promote infinite walking can be compared using the surface area required to promote infinite walking while the participants remain unaware of the

With illusion Without illusion

Starting Position

Figure 2.19: The trajectories of users when exposed or not to the illusion [START_REF] Azmandian | Haptic Retargeting: Dynamic Repurposing of Passive Haptics for Enhanced Virtual Reality Experiences[END_REF]. On the left, it is apparent that their is a sudden correction at the end of the trajectory. This correction is not visible when not exposed to the illusion (right figure).

Figure 2.20: The hand trajectories for 3 amplitude of redirection fitted with a cubic Bézier curve [START_REF] Lebrun | A trajectory model for desktop-scale hand redirection in virtual reality[END_REF] (for more details on Bézier curves, see chapter 4). The higher the amplitude of redirection, the curvier the trajectory gets as the third control points C 2 shows. The resulting trajectories are similar to [START_REF] Azmandian | Haptic Retargeting: Dynamic Repurposing of Passive Haptics for Enhanced Virtual Reality Experiences[END_REF]: the start of the trajectory is unaffected by the amplitude of the redirection (ballistic phase), then the users correct their trajectory (adjustment phase).

illusion. This measure relates to experiments showing that estimating a walked trajectory while blindfolded is difficult [START_REF] Bertin | Perception of two-dimensional, simulated ego-motion trajectories from optic flow[END_REF]

Toolkits

The ability to redirect a user in a virtual scene is very useful but it can be difficult and time consuming to implement. For this purpose, I implemented a toolkit of visuo-haptic illusion with SEGULA Technologies integrating both Hand Redirection and World Redirection illusions. I implemented all techniques discussed in section 2.2 in Unity3D. This toolkit integrates into the Unity3d interface by providing self-sufficient scripts that allow to select any illusion and apply the desired parameters (real and virtual objects location, starting point, etc) dynamically while the virtual environment is running. The combination of illusions such as the hybrid technique of Azmandian et al. is still a work in progress.

I used this toolkit to implement a demonstration with a real world use case. This demonstration was shown at multiple occasion like the science days of Sorbonne University. At the same time, Zenner et al. made a toolkit for Unity3D including the main visuo-haptic illusion techniques [START_REF] Zenner | Hart -the virtual reality hand redirection toolkit[END_REF]. They also added support for mouse and keyboard as well as many visualization tools such as a real world view in VR, trajectories, detection threshold warnings based on the literature, etc.

What is important in this chapter:

-Visuo-haptic illusions improve both haptic feedback and interaction in Virtual Reality. In particular, the hand redirection illusion combines well with passive haptics and reduce cost while providing life-like haptic feedback.

-There are multiple ways to implement visuo-haptic illusions.

Regarding Hand Redirection, the major implementations of the state of the art do not influence the resulting user experience. There are implementation parameters that have not been explored such as the distance between the user and the targets and the trajectory applied to the virtual hand.

-The interval of non-detection is the most important metric to evaluate a visuo-haptic illusion. It defines the stimulus interval in which users will largely not detect the illusion.

Chapter 3

Skill Transfer

Chapter Contents T his chapter offers an overview on skill transfer from the cognitive psychology literature. First we introduce general definitions and concepts related to learning and discuss how skill transfer is related to the learning phenomena. We then detail the definitions, theories, factors and methods to study skill transfer. Finally, we discuss a specific use case of skill transfer: the transfer of motor skill from VR environments to real world.

Most concepts discussed in this chapter are used to design an empirical study with the objective to measure the transfer of learning in VR with visuo-haptic illusions in chapter 5. Especially, I use extensively proactive transfer of learning, experimental protocols to measure the amount of transfer and how these concepts apply to transfer from VR to the real world.

Learning

Skill -Performance -Learning

To define learning, we first need to define skill and performance.

A skill is defined as human behaviours encompassing cognitive, perception and/or motor control elements [START_REF] Schmidt | Motor control and learning: A behavioral emphasis, 5th ed. Motor control and learning: A behavioral emphasis[END_REF]. Skills represent a wide range of activities such as typing, driving or playing sports. Skills have a goal, they allow to produce a document, move from a place to another and win a game. Most importantly, skills can be learned [Edwards, 2010]. With practice, users can improve at a task that requires skill. The degree of skill depends more on the degree of learning than on a innate capability.

Performance is the "qualitative or quantitative assessment of what can be observed during the execution of a skill" [Edwards, 2010]. It is the observation of behaviour: seeing someone riding a bike or typing on a keyboard. Performance is observed directly ranging from a single observation ("Mbappé ran 50m with the ball towards the goal") to a series of observation ("The PSG played really well this season"). It can be quantitative ("running 50m") or qualitative ("The PSG had a poor season").

Contrarily to performance, learning is "a relatively stable change in performance resulting from practice or experience" [Edwards, 2010].

It measures the capacity to perform a new skill and it is based on several observations of performance. For example, we say beginner players learn if they outperform themselves a few months after starting practice. Learning also means the performance is better some time after practice rather than before practice.

In summary, performance is the immediate observation of behaviour and can represent temporary change in behaviour. Learning is a relatively permanent change in behaviour and is measured from multiple measures of performance. Most importantly, it cannot be measured directly.

Acquisition -Retention -Transfer

There are three measurements of performance that determine if learning occured: (1) measurements of Acquisition, (2) Retention tests and (3) Transfer tests (also called ART measures) [Edwards, 2010, Schmidt andLee, 2011]. All three methods require repeated observations of performance and they are often used in combination, especially in motor control. They provide different insights on the nature and the amount of learning.

Acquisition

Acquisition refers to the direct measurement of performance observed over time or a series of trials [Edwards, 2010]. They are used to plot the performance curve. They inform on the rate of learning over the course of practice through a certain method and for a specific task (shooting free throws, parallel parking). However, they are not a direct measure of learning because they only measure a short term change in performance. Acquisition measurements are also referred as Intramodal Learning [Edwards, 2010].

Power law of practice When measuring performance during practice, we obtain a curve that expresses the diminishing returns of practice over time. This curve usually follows the Power Law of Practice (PLP). Although performance keeps increasing as long as practice continues, the rate of improvement gradually diminishes as illustrated in Figure 3.1. This law fits the performance curve very well in a wide variety of application [START_REF] Schmidt | Motor control and learning: A behavioral emphasis, 5th ed. Motor control and learning: A behavioral emphasis[END_REF]. It is very reliable and is the most widely accepted law in learning theory. It can be written in the following man- where P is the performance on the task (typically the completion time), x is the amount of practice time or the number of trials starting from 1, a is the amount to be learned, b is the learning rate, c is the asymptotic performance or ultimate performance.

The power law of practice can also be expresses in its exponential form:

P = ae -b(x-1) + c (3.2)
In this form, the parameters and variables have different values but they have the same interpretation.

In addition to express the diminishing returns of learning, this law shows an upper limit to learning that a learner progressively approaches but that is never reached. It is the asymptotic nature of learning and is also called the ultimate performance (parameter c). The ultimate performance only concerns the current learning technique and context and could potentially be exceeded.

The underlying measure of performance can increase to show an improvement (number of goals or points scored, number of targets reached, etc), and it can also decrease to show an improvement (task completion time, number of errors made, etc). In the case it increases (resp. decreases), the parameter b is negative (resp. positive).

It is generally accepted that when the performance curve follows the PLP, learning is occurring but it does not inform on the nature of learning by itself. This requires retention or transfer tests.

Retention

Retention tests refer to performance measurements conducted after acquisition measurements [START_REF] Schmidt | Motor control and learning: A behavioral emphasis, 5th ed. Motor control and learning: A behavioral emphasis[END_REF]. They measure the change of performance after a period without training [Edwards, 2010, Schmidt andLee, 2011]. For example, after a practice session, trainee drivers may be able to park parallel properly but may not be able to at the start of the next session.

Since performance is influenced by short term behaviour changes, additional observations of performance are made after a period of rest called the retention interval [Edwards, 2010, Schmidt andLee, 2011]. Experimentally, the retention interval can vary from a few minutes to a week or even months. The retention interval allows to retain only the relatively permanent changes in performance and measuring learning.

Predictive Performance Equation The PLP captures the increase of performance through practice. However, the performance declines when not practicing because users gradually forget. Although the precise function modelling this decline in not agreed on [Rubin andWenzel, 1996, Heathcote et al., 2000], it resembles a power law, a rapid loss after the last practice getting slower over time. The power law is similar to Equation 3.1 where a is the amount of performance loss, b is the decay rate and c is the final performance that tends to the initial performance after a long period of time.

The Predictive Performance Equation (PPE) is a computational model that predicts performance based on prior performance (retention) like the PLP and the distribution of practice that influences the amount of performance decay. The effects of practice and decay are multiplicative on activation M n :

M n = P LP T -d n (3.3)
where P LP is the PLP, T n is the elapsed time since the last practice session and d is the decay rate. n is the number of the current practice.

The predicted performance is expressed as a logistic function of activation M n :

P n = 1 1 + e ( τ -M n s ) (3.4)
τ and s control the slope and intercept of the logistic function. The logistic function is used to bound the result between 0 and 1. The parameters used to fit the model to experimental data are b, m, τ and s. Additional details are given about the PPE in Appendix A.

This model is able to predict the performance of users on retention after some practice sessions. Walsh et al. proposed it and found the model fitted well 14 different existing datasets [START_REF] Walsh | Evaluating the theoretic adequacy and applied potential of computational models of the spacing effect[END_REF].

Transfer

Transfer is last of the ART measures and the main focus of this chapter. Transfer of learning measures how effectively a person can transfer the learning of a previous skill to a new skill [Edwards, 2010]. Transfer is discussed in more detailed in the following sections.

In summary, acquisition tests measures the performance during learning, retention tests measure the somewhat permanent change in performance of a practiced skill, whereas transfer of learning measures the generalisability and adaptability of learning.

Transfer

Transfer of learning measures how effectively a person can transfer the learning of a previous skill to a new skill [Edwards, 2010]. Similarly to retention, the practice of the new skill is done after a period of retention. For instance, a transfer test could measure how learning to shoot a basketball from a given position transfers to shooting at a different angle from the same distance, or learning to type on a keyboard layout (e.g. QWERTY) can help someone learn a different layout (e.g. AZERTY). The main goal of transfer is to improve user's performance on a given task by applying an acquired skill to this new context. It is a measure of the strength of learning and on the adaptability and generalisability of what was learned.

In addition to transferring a skill to another, transfer can also occur form a context to another [START_REF] Perkins | Transfer of learning[END_REF]. For example, surgeon trainees practice a lot laparoscopy on robotic systems simulating lifelike interventions or on non human tissue. Additionally, in the industry, it is becoming more common to train operators on a maintenance task in VR before doing it in the real world. Similarly, airline pilots train many hours in flight simulators perfecting theirs skills, reflexes and procedures. Transfer of learning is not trivial, a trainee may have excellent performance in simulations but perform poorly on the real task. The learning process can only end once the skill transfers from VR or the simulation to the physical environment.

Transfer type

Transfer of learning can be split into two categories: Proactive transfer and retro-active transfer.

Proactive Transfer of Learning

The previous definition concerns specifically proactive transfer. Proactive transfer refers only to how a previous skill can influence learning a new skill. As a result of practicing the previous skill or in the previous context, performance on the new skill or in the new context changes as illustrated in Figure 3.3-a-b. A typical way to measure pro-active transfer is to have two groups of participants: one group learning a skill A and then learning skill B (A → B) and the other learning skill B without previous practice (Rest → B). More details are provided in section 3.6 on how to measure proactive transfer of learning.

Completion

Retro-active Transfer of Learning

In contrast, retroactive transfer of learning refers to how the acquisition of the new skill influences the performance of the previously learned skill [Bunch, 1946a, Raissi, 2021] (see Figure 3.4). Hence, it requires to measure the performance of a group that learns skill A, then skill B [START_REF] Raissi | Retroactive transfer phenomena in alternating user interfaces[END_REF]. According to their model, The design described above measures the first iteration of retroactive transfer, it is possible to measure n th alternation of retroactive transfer by repeating the alternation of task A and B [START_REF] Raissi | Retroactive transfer phenomena in alternating user interfaces[END_REF], Raissi, 2021]

(A → B → A → B → A → ...).
The second occurrence of task A is the first alternation of retroactive transfer, the third occurrence of task A is the second alternation of retroactive transfer, etc as illustrates Figure 3.5.

Retroactive transfer has been measured for cognitive tasks [Briggs, 1954, Mareschal et al., 2002, Pallier et al., 2003], motor learning of discrete [START_REF] Koedijker | Interference effects in learning similar sequences of discrete movements[END_REF] or continuous movements [Brashers-Krug, 1995].

Visuo-motor Adaptation

Visuo-motor adaptation is a particular case of transfer. It consists in studying the effect of a large distortion of users' field of view which is easily perceived. Researchers then study how users adapt to this change in perception, for instance, how users adapt to a constant visual shift of their hand on a motor control task [Helmholtz and Powell Cocke, 1926]. The distortions are either a shift [START_REF] Krakauer | Learning of Visuomotor Transformations for Vectorial Planning of Reaching Trajectories[END_REF] or a rotation [Prager andContreras-Vidal, 2003, Shabbott and[START_REF] Shabbott | Learning a visuomotor rotation: simultaneous visual and proprioceptive information is crucial for visuomotor remapping. Experimental brain research[END_REF], Anglin The applied distortions degrade motor control performance because users have to adapt to this new perception-action scheme. A loss of performance is observed immediately when the distortion is introduced akin to proactive transfer of learning [START_REF] Krakauer | Adaptation to visuomotor transformations: consolidation, interference, and forgetting[END_REF]. Participants generally regain similar performance after some practice with the modified field of view. A similar loss of performance is observed when the distortion is removed. This is called an after-effect [Gibson, 1933, Prablanc et al., 2019[START_REF] Bunch | Retroactive inhibition or facilitation from interpolated learning as a function of time[END_REF] and is akin to a retroactive transfer [START_REF] Krakauer | Adaptation to visuomotor transformations: consolidation, interference, and forgetting[END_REF].

Factors Involved in Skill Transfer

This section discusses some factors that are responsible for skill transfer. These factors are important as they influence the short term performance introducing variability in the measures and can also change the long term performance. These factors also play a role regarding skill acquisition and retention

Practice and Contextual Interference

Practice is the most critical factor for skill learning. Practice is almost always beneficial, but the way it is organized is essential to improve future performance. For example, many practice schedule designs exist to plan a practice session of different skills (interskill practice 1 ). Blocked design consists in practicing a skill the required amount of trial before moving on to the subsequent skill. If A, B and C are three different skills (or conditions such as different context of learning), blocked practice is illustrated as follows: A, A, A, B, B, B, C, C, C. The Latin square design inspired from the mathematical Latin Square, consists in using combinations of the tested skills as illustrated in

Distribution of Practice

The distribution of practice concerns the spacing between practice sessions [Ebbinghaus, 2013]. In other words, the frequency at which the learner engages in practicing. Resting periods improve performance in general and longer rest periods promote better performance across multiple practice sessions. It also concerns the length of the practice sessions.

Researchers oppose two ways of practice distribution: massed practice and distributed practice as illustrated in Figure 3.7. Massed practice or blocked practice involves having a short period of rest time and longer practice session whereas distributed practice involves longer rest periods and shorter practice time. Distributed practice is often considered better for both performance and learning due to being a more restful 1 During practice, trainees rarely need to learn a single skill but rather multiple.

For example, a basketball player's interskill practice can be the practice of free throws, 3 points shooting and ball control All massed practice session take place the same day as opposed to the distributed session over 3 days with longer rest periods in between.

A B C C A B B C A
distribution of practice. Additionally, as time passes, users' memory decays and their performance decreases. This forgetting means there is a trade-off between this factor and the distribution of practice especially on the resting periods length.

Internal Factors

There are other factors such as attention and motivation which relates to how much the learner engages themselves in practice. Whether the learner receives feedback or not, how (written, spoken, etc) and when is the feedback provided (continuously, discretely, when the learner makes a mistake, etc).

Fatigue can also decrease learning. Overpracticing will induce fatigue but it can be mitigated by introducing breaks (promoting a more distributed practice), using multiple learning methodology and mediums.

Transfer theories

There are two dominant theories supporting the transfer effect: the Identical Elements Theory and the Transfer-Appropriate Processing Theory (TAPS)

Identical Elements Theory

The identical elements theory is intuitive and was theorized by [START_REF] Thorndike | The influence of improvement in one mental function upon the efficiency of other functions. ii. the estimation of magnitudes[END_REF]. It states that transfer occurs according to the common elements the tasks share. The more elements in common, the stronger the transfer. However, they did not explicitly define what a common element is. Still up to this day despite researchers proposing what identical elements could be, there is no clear consensus. [Royer, 1979] proposed that the elements concerned task stimuli received by the users. [Taatgen, 2013] proposed some languages are easier to learn thanks to common words like french and latin. Elements could also relate to motor control characteristics, trajectory constraints, etc [Edwards, 2010]. It is still not properly known which elements are the most relevant. [Osgood, 1949] introduced paired-associate list learning. It involves considering the stimulii and the corresponding responses as pairs. An example of a pair is typing the letter A requires to type the first letter of the second row on an AZERTY keyboard (A, [1,2]) but the first letter of the third row of a QWERTY keyboard (A, [1,3]) resulting in a different pair. The more similar the pairs, the more potential transfer of learning. No clear consensus has emerged and this theory and the subsequent theories it inspired helps understand transfer of learning but do not manage to explain it fully.

Transfer-Appropriate Processing Theory (TAPS)

TAPS proposes that it is the similarity in cognitive processes of the skills that allow transfer of learning [START_REF] Morris | Levels of processing versus transfer appropriate processing[END_REF], Bransford et al., 1982]. In this theory, the motor control aspect of the skill are not an important factor of transfer. Rather similarities in cognitive process like decision making, problem solving or attention determine the amount of transfer possible [Edwards, 2010]. According to TAPS, transfer is determined by the meaningfulness and depth of processing (learning to pronounce new words by focusing on the sounds opposed to how the mouth and lips move during pronunciation, the latter often being irrelevant) and how information is given and used by users (memorizing words by association with semantics or with rhymes).

It remains difficult to explain precisely what allows transfer to happen. Researchers introduced the terms "near transfer" to describe transfer between similar skills and far transfer to describe transfer between dissimilar tasks with few common elements. Only near transfers allow for strong transfer effects to occur [START_REF] Schmidt | Transfer of movement control in motor skill learning[END_REF].

Protocol and Measure to Study Transfer

In this section, I will focus on measuring proactive transfer of learning. The same measure can be used for retention and for each alternation of retroactive transfer of learning.

Protocol

As stated in section 3.3, a typical protocol to measure transfer requires two groups: a group practicing a skill, previous task A and then practicing another skill (or the same skill in a different context), new task B. A second group serving as control practicing only task B.

A period of rest, the retention interval is used to measure only the relatively permanent changes in performance. For most cases, it is enough to have a retention interval of 1 day, especially for motor control tasks.

Figure 3.8 illustrates a typical proactive transfer of learning protocol.

Amount of Transfer

The main measure of transfer is the amount of transfer [START_REF] Schmidt | Motor control and learning: A behavioral emphasis, 5th ed. Motor control and learning: A behavioral emphasis[END_REF]. It is the ratio between the performance increase during training and after a rest period (see also Figure 3.9):

T % = A -B C -B (3.5)
where The performance metric varies depending on the task studied. The most used metric is the completion time of a task but others exist like the number of mistakes made during a trial or the difference between a trajectory and an "ideal" reference.

Positive Transfer of Learning When learning a previous task enhances the performance on a new task, transfer is positive. Thus, the previous task has a beneficial effect on the performance of the latter task. The degree of positive transfer can be high or minimal, but as long as some improvements are observed in the new skill, it is considered as a positive transfer [Edwards, 2010]. For example, it is easier to learn a language with the same roots (like italian, spanish and french) as a previously learned language [Taatgen, 2013].

Negative Transfer of Learning On the contrary, when learning a previous task hinders the performance on a new task, transfer is negative. The previous task interferes with the new task and disturb learning in the initial phase of learning [START_REF] Perkins | Transfer of learning[END_REF]. The new task is also described as interfering with th previous task [START_REF] Bunch | Retroactive inhibition or facilitation from interpolated learning as a function of time[END_REF], Briggs, 1954]. Although the two skills seem similar and auspicious for transfer, they can interfere if they require a different action with the same stimulus [Osgood, 1949]. If these interference are large enough, practicing a task before practicing another task can lead to poorer performance. Keyboard layouts are a prime example as typing a letter may require to type a different key to type the same letter. To type the letter A on a QWERTY keyboard requires to type the second row of the first row of letters, but on an AZERTTY keyboard, it requires to type the first row of the first row of letters as illustrated in Figure 3.2. However, with practice and experience, it is possible to reduce and even eliminate the impact of interference.

Null Transfer of Learning When the performance of a new task is unchanged by a previously learned task, there is no transfer of learning or it is null [START_REF] Schmidt | Motor control and learning: A behavioral emphasis, 5th ed. Motor control and learning: A behavioral emphasis[END_REF]. This can happen when both tasks don't overlap enough and there's neither benefits nor interference.

Savings Time

Another measure is the savings time, it is the total amount of time an untrained group needs to train to reach the same performance than a group that previously trained on a slightly different task or context [START_REF] Schmidt | Motor control and learning: A behavioral emphasis, 5th ed. Motor control and learning: A behavioral emphasis[END_REF]. This measure is illustrated in Figure 3.9. It is highly relevant in VR because it represents the in-situ time saved allowing to keep workspaces productive and reduce the need for realsized model.

Skill Learning in Virtual Reality

[ [START_REF] Greeno | Transfer of situated learning[END_REF] hypothesized that transfer of learning occurs with task similarities in different environments such as virtual and real environments which they called situated transfer of learning in accordance with the Identical Elements theory and TAPS. Since then, fostering positive transfer has become a main objective of Virtual Reality (VR), i.e. to transfer skills learned in a virtual environment into the real world a.

b.

Figure 3.10: On the left, an example of visual guidance on a rowing task [START_REF] Sigrist | Terminal feedback outperforms concurrent visual, auditory, and haptic feedback in learning a complex rowing-type task[END_REF] similar to [START_REF] Rauter | Learning of Temporal and Spatial Movement Aspects: A Comparison of Four Types of Haptic Control and Concurrent Visual Feedback[END_REF]. The users are shown a reference trajectory and their last few trajectories for comparison. The blade rotations are also indicating by the ♦ symbol. On the right, a Falcon haptic interface (Novint) used to guide a user while drawing an arabic character on a screen [START_REF] Teranishi | Effects of full/partial haptic guidance on handwriting skills development[END_REF]. There were 2 guiding algorithms: full guidance which played the trajectory not requiring any user input and a partial guidance pushing the user back in the reference trajectory when needed.

without requiring real-sized models which are expensive, cumbersome and/or task-specific.

There are plenty of domains where VR has proven useful and effective for skill acquisition. In the medical field, medical staff and students train many difficult and precise procedure like a laparoscopy [Seymour, 2008, Gallagher et al., 2013], eye surgery [START_REF] Huang | Catar: A novel stereoscopic augmented reality cataract surgery training system with dexterous instruments tracking technology[END_REF], rehabilitation [START_REF] Lange | Designing informed game-based rehabilitation tasks leveraging advances in virtual reality[END_REF], etc. In the industry, operators can learn to assemble parts [START_REF] Vélaz | The Influence of Interaction Technology on the Learning of Assembly Tasks Using Virtual Reality[END_REF] or to realize a maintenance task [START_REF] Crison | The Use of Haptic and Pseudo-Haptic Feedback for the Technical Training of Milling[END_REF].

Factors of Transfer of Learning in VR

We have seen some general factors influencing transfer of learning like practice and its distribution, now I focus on factors specific to transfer of learning in VR. In the many studies of training in VR promoting transfer of learning, there many counter examples showing the transfer did not occur or the transfer was even negative [START_REF] Boud | Virtual reality and augmented reality as a training tool for assembly tasks[END_REF], Winther et al., 2020]. The most recurring factors I highlight here are the overuse of guidance and the lack of haptic feedback or the lack of multimodality.

Guidance

VR applications of haptic feedback focus on guiding healthy subject [Patton andMussa-Ivaldi, 2004, Lam et al., 2008] or with disabilities [START_REF] Adamovich | Sensorimotor training in virtual reality: a review[END_REF] during practice. There are 3 types of guidance: visual, haptic or a combination of the two. Visual guidance can remind the steps to follow [START_REF] Yang | Validating the Performance of Haptic Motor Skill Training[END_REF], show a reference trajectory to follow [Marchal Crespo and Reinkensmeyer, 2008] or indicate when an error was made [START_REF] Teranishi | Effects of full/partial haptic guidance on handwriting skills development[END_REF]. Haptic guidance often consists in correcting the users' trajectory [Marchal Crespo and Reinkensmeyer, 2008, Marchal-Crespo et al., 2015, Teranishi et al., 2017, Yang et al., 2008] (see Figure 3.10-a). Though, the literature has plenty of example of positive effect of haptic feedback, some studies show low or no positive effect [START_REF] Yang | Validating the Performance of Haptic Motor Skill Training[END_REF], Solis et al., 2003] (see Figure 3.10-b). This can be due to the lack of colocalisation 2 in some cases [START_REF] Solis | Validating a skill transfer system based on reactive robots technology[END_REF] but most importantly due to the guidance effect [START_REF] Yang | Validating the Performance of Haptic Motor Skill Training[END_REF], Solis et al., 2003]. Guidance can have the side effect of making users passive during the learning phase because the guidance is too present and makes the task too easy. Instead of guiding the user, VR allows to increase the task difficulty. For example, when learning a gesture or a trajectory, the users' mistakes are magnified to encourage learning and a stronger error compensation [START_REF] Patton | Evaluation of robotic training forces that either enhance or reduce error in chronic hemiparetic stroke survivors[END_REF].

Multimodal Feedback

Multimodal feedback in the VR literature often references the addition of haptic feedback to vision and hearing [Stone, 2000, Bouzbib, 2021], although, it can also reference olfactory [START_REF] Richard | Multi-modal virtual environments for education with haptic and olfactory feedback[END_REF] or taste [START_REF] Hoffman | Physically touching and tasting virtual objects enhances the realism of virtual experiences[END_REF]]. Haptic interfaces improve the feeling of presence in the virtual environment in a variety of task and applications [START_REF] Ramsamy | Using Haptics to Improve Immersion in Virtual Environments[END_REF] (see also chapter 2).

When haptic feedback is used to improve virtual environment in terms of realism and closes the gap between real and virtual interactions, it improves skill learning [START_REF] Adamovich | Sensorimotor training in virtual reality: a review[END_REF] and in the worst cases does not interfere with learning [START_REF] Yang | Validating the Performance of Haptic Motor Skill Training[END_REF].

Discussion

In this chapter, I described difference between performance and learning and the Acquisition -Retention -Transfer measures of learning theory.

2 Colocalisation is the superposition of the virtual environment and the physical one. It is often used to describe the haptic feedback given to the user. The colocalisation concept is part of the What You Can See is What You Can Feel paradigm [START_REF] Yokokohji | What you can see is what you can feel-development of a visual/haptic interface to virtual environment[END_REF].

I also enumerated some important factors that influence learning like practice and its distribution in time. Then, I focused on the two types of transfer of learning: proactive and retroactive and theories attempting to explain why transfer of learning occurs between two tasks. Finally, I discussed the special case of learning in VR. Now, I lay out previous work done on learning in VR with visuo-haptic illusions.

The impact of visuo-haptic illusions in virtual environments on learning has been studied [START_REF] Crison | The Use of Haptic and Pseudo-Haptic Feedback for the Technical Training of Milling[END_REF], Bibin et al., 2008, Kohli et al., 2013, Spillmann et al., 2013, Han et al., 2018]. However, all previous studies on this focused on the performance during practice [START_REF] Crison | The Use of Haptic and Pseudo-Haptic Feedback for the Technical Training of Milling[END_REF] or evaluated the retention in VR [START_REF] Han | Evaluating Remapped Physical Reach for Hand Interactions with Passive Haptics in Virtual Reality[END_REF]. Every previous study made the assumption that if learning occurs in VR when exposed to illusions then it should transfer to the real world. This is not trivial as there are many examples where transfer occurs poorly, not at all or is even negative. [START_REF] Crison | The Use of Haptic and Pseudo-Haptic Feedback for the Technical Training of Milling[END_REF] and [START_REF] Bibin | SAILOR: A 3-D Medical Simulator of Locoregional Anaesthesia Based on Desktop Virtual Reality and Pseudohaptic Feedback[END_REF] used deported screen and did not colocate multimodal feedback. [START_REF] Spillmann | Adaptive Space Warping to Enhance Passive Haptics in an Arthroscopy Surgical Simulator[END_REF] also did not have a colocated apparatus because there use-case was a arthroscopy procedure (a minimally invasive intervention similar to the laparoscopy). Both [START_REF] Kohli | Redirected Touching: Training and adaptation in warped virtual spaces[END_REF], Han et al., 2018] used VR headsets in a colocated setup, however all these studies focused on the acquisition or retention of skills in VR, they did not evaluate the performance of participants in a physical setup and did not measure transfer of learning. As such it is unknown whether Body Redirection or World Redirection illusions promote or hinder transfer of learning.

Studies measuring performance in VR with visuo-haptic illusions foa. b. cused on motor control tasks. This make senses since VR is most likely to influence motor control tasks since it modifies users' trajectories. Also, visuo-haptic illusions alter space perception and movements so cognitive tasks should be less impacted by them. However, motor control tasks can be affected by visuo-haptic illusions. In chapter 5, I quantify the amount of transfer of learning when users are exposed to visuo-haptic illusions in VR on a motor control task.

What is important in this chapter:

-Learning can be split into three concepts: Acquisition, Retention, Transfer (ART). In this thesis we mainly focus on skill transfer from VR to real world.

-Transferring skills from VR to the real world has many advantages, it is low cost, reusable and widely available.

-Haptic feedback favors acquisition and retention in VR. However, it is unclear whether it favors transfer of learning and specifically, I question if visuo-haptic illusions promote or interfere with transfer from VR to the real world.

Chapter 4

The I n chapter 2, I described the implementation of visuo-haptic illusions.

In particular, I highlighted that the implementation of the hand redirection techniques presented in literature relies on the same linear redirection function, which depends on the position of the virtual and physical objects. In this chapter, I explore other possible implementations of the hand redirection and study their effects on usability. The objective of exploring other redirection functions is to increase the range of redirection amplitude in which the users do not detect the redirection. A secondary objective is to give more control to the designer over the user's trajectory.

I propose to extend this redirection function by offering more flexibility on the shift between the virtual hand and the real hand. Instead of considering only linear redirection functions (first degree polynomials), we consider second degree polynomials. This gives a better control over the real hand trajectory when reaching the target.

The contents of this chapter was published in the proceedings of the IHM 2022 [Geslain et al., 2022a] and AVI 2022 [START_REF] Geslain | Generalizing the hand redirection function in virtual reality[END_REF].

Generalization of the Hand Redirection Function

Standard Linear Implementation

A common implementation of the hand redirection illusion is the Interpolated Reach [START_REF] Han | Evaluating Remapped Physical Reach for Hand Interactions with Passive Haptics in Virtual Reality[END_REF]. It defines the offset between the real hand and the virtual hand ⃗ ∆ M with a linear function as follows

⃗ ∆ M (d) = (1 - d D ) ⃗ λ (4.1)
where d is the distance between the real hand and the real target, D is the distance between the starting position and the real target and ⃗ λ is the vector between the real and virtual targets. These parameters are illustrated in Figure 4.1-left. On the left, the virtual hand position is computed as described in Equation 4.1. On the right, the virtual hand position is computed by applying a rotation by an angle θ the real hand around the starting point.

This equation is a product of a first order polynomial:

f 1 (d) = -d D +1 with ⃗
λ. Figure 4.1-left shows the trajectory followed by the virtual hand according to the real hand. An equivalent way to redirect the hand is to consider the angle created by the targets around the starting position and compute the virtual hand position as a rotation of the real hand around the starting position. As Figure 4.1 shows, both methods are equivalent as the trajectories are identical.

The linear function used to redirect the virtual hand might not be optimal for decreasing illusion detection and for design purposes. I propose to use higher polynomials as a redirection function: faster at the start or the end of the movement. The polynomial function of order n can be written as following:

⃗ ∆ M (d) = f n (d) ⃗ λ (4.
f n (d) = n ∑ i=0 (a i d i ) with n ≥ 1 (4.3)
with a i coefficients to be determined. When n = 1 (see Equation 4.1), there are no degrees of freedom, we have a 1 = -1 D and a 0 = 1 due to the continuity constraints at the target and the starting positions. With n > 1, the function has 1 or more degrees of freedom.

Second Order Polynomial Implementation

In this thesis, I focus on second order polynomials and their influence on the detection threshold of the hand redirection illusion. In this case, Equation 4.3 becomes

⃗ ∆ M (d) = f 2 (d) ⃗ λ = (a 2 d 2 + a 1 d + a 0 ) ⃗ λ (4.4)
This equation has 1 degree of freedom because there are only 2 constraints for 3 parameters. Figure 4.2 shows two examples satisfying the constraints. There are many ways to solve the equation. We chose to write the equation as a Bézier Curve. A Bézier curve is a polynomial parametric curve that was originally conceived to draw car body parts. Nowadays, it is used in many cases like drawing font characters, vector graphics and model hand trajectories [START_REF] Faraway | Modelling three-dimensional trajectories by using bézier curves with application to hand motion[END_REF].

The formulation of the function of redirection as a Bézier curve has multiple advantages. First, it is simpler than a more standard polynomial representation. It only requires to select a few control points to define the curve. Second, this formalism provides a simple and convenient geometrical representation of the trajectory.

For a polynomial of the n th order, the corresponding Bézier curve has n + 1 points and is defined by its parametric representation:

⃗ P (t) = n ∑ i=0 B n i (t) ⃗ P Ci (4.5)
where B n i (t) are the Bernstein polynomials and ⃗ P Ci are the control points. The Bernstein polynomials are the following [Farouki, 2012]:

B n (t) = n ∑ i=0 ( n i )(1 -t) n-i t i (4.6)
where ( n i ) are the binomial coefficients. In 2D, ⃗ P (t) is a 2D vector that defines each point of the curve according to t, the Bézier parameter varying from 0 to 1.

There are an infinite kind of Bézier curve (see Figure 4.3):

• The linear Bézier curve (order 1) has 2 control points, it is a straight line

[P 0 , P 1 ]. Its parametric function is ⃗ P (t) = (1 -t) ⃗ P C0 + t ⃗ P C1 .
• The quadratic Bézier curve (order 2) has 3 control points, it is a second order polynomial starting from point ⃗ P C0 ending in point ⃗ P C2 . ⃗ P C1 defines the tangents of the curve at the starting and ending points as

[ ⃗ P C0 , ⃗ P C1 ] and [ ⃗ P C2 , ⃗ P C1 ]. Its parametric function is (1 -t) 2 ⃗ P C0 + 2t(1 -t) ⃗ P C1 + t 2 ⃗ P C2 .
• The cubic Bézier curve (order 3) has ([ ⃗ P C3 , ⃗ P C2 ]) of the curve. This parametric function is the most widely used as it allows to ensure continuity between cubic curve. For this, it's enough to set the last point of a curve as the first point of the next curve. Additionally, the tangents can bet set to ensure a continuous and smooth curve.

• Bézier curves with order greater than 3 are rarely used. Usually, multiple cubic functions can be stitched together creating a more complex curve.

I used a quadratic Bézier, there are 3 control points: P C0 the starting point, P C1 the free roaming point, P C2 the virtual target relative to the real target. The parametric representation is then

P (t) = ( D -d || ⃗ ∆ M || ) = (1 -t) 2 ⃗ P C0 + 2t(1 -t) ⃗ P C1 + t 2 ⃗ P C2 (4.7)
The first value x of P (t) is the distance from the starting position to the real hand Dd and the second value y is the offset between the real and virtual hands || ⃗ ∆ M ||. n is the order of the polynomial and t the Bézier variable varying between 0 and 1. The first and last control points correspond respectively to the starting position and the virtual target position relative to the real target. The other control points allow to visually control the speed at which the redirection is applied as the user approaches the target. To actually compute the value of ∆ M , we solve the 2 equation system that Figure 4.4: The offset between the real and its avatar ∆ M according to the distance between the real hand and the physical target d. The red curve is the standard linear redirection function L = 0; the six additional curves are the second degree polynomial functions. Above the red curve L < 0, the redirection is applied faster at the beginning of the movement. Below the red curve L > 0, the redirection is applied faster at the end of the movement. In this case, the redirection is applied for an offset ∆ M of 8.75cm or 5°as shown in the left axis.

P O L = -3 L = -2 L = -1 L = 1 L = 0 L = 2 L = 3 0 P V C P R C =3 =2 =1 =0 =-1 =-2 =-3 L L L L L L L C = 2 C =
1 is Equation 4.7. First, solving the first equation to determine t and then computing the offset ∆ M . The function behaviour for six P 1 points is illustrated in Figure 4.4. Note that for L < 0, the virtual hand is redirected away from the real hand faster than with the linear function and the opposite for L > 0. Finally, for all functions, the virtual and real hands are at the same position when the real hand is at the starting point and offset by ⃗ λ when the real hand touching the targets.

In this section, I proposed a new set of function of redirection: second 1 Although, the L=1 and L=-1 look like they are on one of the function of redirection, they are actually the 2 nd control points of the first functions above and below the linear function (in red). These control points are unrelated to the redirection function they are on.

order polynomials. These functions offer the flexibility to redirect users more when close or far to the target. I used Bézier curves to characterize and describe them. In the next section, I will describe an experimental study I ran in order to evaluate these novel functions.

User Study

As discussed in chapter 2, in order to evaluate the detectability of different redirection functions, we ran a user study. The main objective of this user study is to evaluate the influence of non-linear redirection functions on the interval of non-detection (IND) of the hand redirection illusion. We chose an unobstructed pointing task where the user reaches for a virtual target offset from the real target. This task is widely used in HCI [START_REF] Akamatsu | Movement characteristics using a mouse with tactile and force feedback[END_REF], Murata, 1998, Fischer, 2003] and in particular with Hand Redirection illusions [START_REF] Kohli | Redirected touching: The effect of warping space on task performance[END_REF], Kohli et al., 2013, Azmandian et al., 2016]. We focus on second order polynomials illustrated in Figure 4.4 as described in the previous section. The IND is defined as described in subsection 2.3.1, it is the interval in which the distance between the real and virtual targets remains undetected by the user. The experimental protocol is similar to [START_REF] Lebrun | A trajectory model for desktop-scale hand redirection in virtual reality[END_REF] where participants realize a pointing task in VR. After each repetition, participants select a 2 Alternative Forced Choice (2AFC) about the direction of redirection (Left or Right) as well as if they detected the illusion (Yes or No).

Participants and Apparatus

The study was conducted with 20 participants (10 male, 10 female), aged from 20 to 29 years old (23.8 years old average). Among them 2 were left-handed, 1 ambidextrous and 14 right-handed. All participants had a normal or corrected to normal eyesight (5 participants wore glasses or contact lenses). 3 of them had a regular VR experience, 11 had previous VR experience and 3 had no experience at all. No participants had musculoskeletal or proprioception disorders.

This user experiment was approved by the ethics comity of Sorbonne University n°2020-020.

Participants were seated in front of a table wearing a VR HMD (HTC VIVE Pro [START_REF] Htc | Vive pro full kit[END_REF]) and a hand harness with a VIVE Tracker attached on it (see Figure 4.5). The hand harness restrains the user's hand in a pointing position and the VIVE Tracker tracks the user's hand po-

Decoys

Virtual hand

Real hand sition and orientation with a precision 1mm and 0.3°. This experiment could have been done with a similar HMD like the Oculus Rift [START_REF] Oculus | Oculus rift[END_REF], the HTC VIVE Pro was chosen because it was the most easily available.

A blue haptic marker is placed on the table as a starting position. 4 targets are arranged in an arc of a circle 56cm away from the starting point and offset by 15°from each other. We choose this distance between the targets and the starting position so that all participants could reach the targets while remaining seated and comfortable. The angle between the real and virtual targets varies. The largest angle between a real and a virtual target (14°) is set to be easily detected by participants. Similarly, the smallest angle is set to 0°. Using intermediate angle values ensures that the IND is included inside the tested range. Targets on the far right and far left are dummies and participants never interact with them. They are here to conceal the illusion from participants seeing the real setup before putting the HMD on. Targets have a push-button on top for the users to press and validate a trial before they must answer the 2AFC.

The virtual scene illustrated in Figure 4.5 in the middle is a copy of the real environment. It was created in the Unity3D editor. The hand is represented by a static hand with a fixed index pointing forward. During a trial, only a single virtual target is shown to the user, the one they should reach. The users selected the direction of redirection by pressing one of two buttons. They were virtual buttons with no real counterpart. As soon as the user virtual hand enter in contact with the button, the choice was selected. Participants had the choice to correct their answer verbally if they desired.

Experimental Design

Task and Stimulus

The experiment is a pointing task with a 2AFC as described in subsection 2.3.1. The task starts with the user's finger on the starting position, a virtual target then appears. The participant reaches the target, presses the physical button and goes back to the starting position. We ask that participant move naturally and avoid sudden movements and do the entire movement in 4 seconds. Once done, participants have to choose if they had detected the illusion or not and if their real hand was on the left or on the right of their virtual hand. This first question is an extension of the more typical protocol.

When participants make a mistake with the largest amplitude of redirection (14°et -14°), the ground becomes red to notify them. From preliminary results and the state of the art [START_REF] Zenner | Estimating Detection Thresholds for Desktop-Scale Hand Redirection in Virtual Reality[END_REF], participants should always be able to correctly guess the redirection direction in this case. We consider that participants had a drop in attention when this happened. No further information is given to participants.

Conditions

There are 3 independant variables (see Figure 4.6). The first variable is the Function of redirection L with 7 values: {-3, -2, -1, 0, 1, 2, 3} (see Figure 4.4). Note that L = 0 is the linear reference function commonly used in the literature [Zenner andKruger, 2019, Han et al., 2018].

The second variable is the Redirection Amplitude θ. It is the angle formed by the real target, the starting position and the virtual target (see Figure 4.1-right). We consider 9 values: {-14

• , -10

• , -6 • , -2 • , 0 • , 2 • , 6 • , 10 • , 14 • }.
The extreme values should be easily detectable (14°and to some extent, 10°) whereas the closest value to 0°, 2

• should be very hard to detect. There was a compromise between (1) having values covering a large enough spectrum to have easily detectable values, (2) enough points to fit a psychometric curve on our data and (3) to restrict the length of the experiment per participant (∼ 1h). The last variable is the Physical Target: {Left or Right} target, introducing variability in the user's gesture. The leftmost and rightmost physical targets are not used in the experiment.

Each participants did 2 repetitions of every combination of these 3 independant variables and trials were randomized for each for each participant.

Hypothesis

I propose an hypothesis supporting the novel functions of redirection. The hypothesis is that redirecting at the begining of the trajectory will increase the interval of non detection. This is supporting by the Two-Component Model. The result trajectory of user reaching for a cube while being exposed to the body redirection technique on the left and without illusion on the left [START_REF] Azmandian | Haptic Retargeting: Dynamic Repurposing of Passive Haptics for Enhanced Virtual Reality Experiences[END_REF]. The kink at the end of the trajectory circled in red is a correction the users make when they reach the redirected targets.

participants reach for the target in an open loop, and a second phase of adjustment where the user corrects their trajectory to compensate for the mismatch created by the redirection [START_REF] Azmandian | Haptic Retargeting: Dynamic Repurposing of Passive Haptics for Enhanced Virtual Reality Experiences[END_REF] (see Figure 4.7). This observation fits well with the two-component model [Woodworth, 1899] that describes those two phases when realizing a pointing or reaching task. A way of improving the Hand Redirection illusion is to soften the adjustment phase by redirecting the user more at the start of the trajectory. That way, the adjustment phase can start earlier and may be less noticeable.

Procedure

Participants are first informed about the objective of the study and the task they need to perform. Careful attention was brought on explaining the concept of hand redirection and how to correctly answer the 2AFC: "Your virtual hand is represented in the virtual scene. During the task, an offset is added gradually between this virtual hand and your real hand. This offset shifts the virtual hand either on the left or on the right of your real hand." Participants then put on the hand harness and the HMD. They had time to accommodate with the illusion and the virtual scene by doing 4 training trials where their real hand was visible in the virtual hand as well as their shifted virtual hand. The offset values for the training were -10°, -2°, 2°et 10°. After each trial, participants were informed of the direction of redirection. They then realize the full experiment without seeing their real hand position. At the end, they filled a demographic questionnaire.

Design

This study follows a within-subject design. Each participant did 2 repetitions of each combination of the 9 Redirection Amplitudes, 7 Redirection Functions and 2 Physical Targets, i.e. 2 blocks of 126 trials. The different conditions are pseudo-randomized in each bloc. In summary, the experimental design is 17 Participants × 2 blocks × 9 Redirection Amplitudes × 7 Redirection Functions × 2 Physical Targets = 4284 repetitions.

Dependant Variable

For each trial, we log the participant's answer for the direction of the redirection (Left/Right) according to the 2AFC design.

We also log the position of the user's hand to inform on the trajectory followed by the participant during every trial. The position of the user's hand is logged every frame displayed by the HMD at ∼90Hz.

Analysis

A typical analysis of this kind of experiment (2AFC) consists in plotting the psychometric curve of the population to estimate the detection threshold of the illusion. The psychometric curve is explained in more details in chapter 2.

Psychometric Curve

A psychometric curve (see Figure 4.8) consists in plotting the probability of answering one choice of the 2AFC over a controlled parameter. The resulting curve can be fitted with a sigmoid:

P (X) = 1 1 + e -X-a b (4.8)
where a and b are two parameters used to adjust the fit. X is the controlled parameter and P (X) is the probability to answer one of the two force choices. In our case, it's the probability that the participant chose Left over the Amplitude of Redirection P (X = θ). Intuitively, P (θ) closes in on 100% when the virtual hand is indeed on the left of the real hand in higher amplitudes and inversely, closes in on 0% when the virtual hand is on the right of the real hand.

Point of Subjective Equality

The point of Subjective Equality P SE is where participants perceive the real gesture as equal to the virtual one. Graphically, it is where the curve crosses the 50% probability. Essentially, participants answer randomly to the 2AFC. It is expected that this value is close to a Amplitude of Redirection of 0°.

Detection Threshold and Interval of Non-Detection

The psychometric curve allows to determine detection thresholds by looking for θ so that [START_REF] Zenner | [END_REF], Lecuyer et al., 2000]. IN D G is the detection threshold when the virtual hand is redirected on the left of the real hand and vice-versa for IN D D . We use the reciprocal function of equation 4.8:

P (θ = IN D G ) = 25% and P (θ = IN D D ) = 75% [Zenner
X = -b ln ( 1 P (IN D) -1) + a (4.9) The Interval of Non-Detection (IND) is then IN D = IN D D -IN D G .
It represents the span of Amplitude of Redirection where participants don't detect the hand redirection illusion.

The main aim of this analysis is to estimate if and how the IND is influenced by the Function of Redirection.

Results

Outliers

We removed participants from the analysis when any of the following conditions were met:

• The left or right non-detection interval IND for the linear redirection function (L = 0) is less than -14°or greater than 14°, similar to [START_REF] Benda | Determining detection thresholds for fixed positional offsets for virtual hand remapping in virtual reality[END_REF].

• The correct response percentage is less than 80% for the 14°and -14°angles and for the linear redirection function (L = 0).

Therefore, 5 participants were excluded for a detection interval of 15.0°, 16.2°and 28.5°and 2 participants for a correct response rate of 60%. The results presented here are therefore those of the 13 remaining participants. The experimental design is therefore : 13 Participants × 2 blocks × 9 Redirection Amplitude × 7 Redirection Functions × 2 Physical Targets = 3276 repetitions. ). These values are comparable to the results of Zenner et al. in [START_REF] Zenner | Estimating Detection Thresholds for Desktop-Scale Hand Redirection in Virtual Reality[END_REF] (see Table 4.1). The distance between the source and the target is within the same magnitude and their results are very close with a IN D = 8.19

Interval of Non-Detection

• (IN D G = -4.38 • , IN D D = 3.81 • )
. The results reported by [START_REF] Ogawa | Effect of avatar appearance on detection thresholds for remapped hand movements[END_REF] and [START_REF] Benda | Determining detection thresholds for fixed positional offsets for virtual hand remapping in virtual reality[END_REF] show higher IND (respectively IN D = 26.7

• and IN D = 44.6

• ). However, the distance between the origin and the real target is much smaller: around 20cm for the first one, and variable value for the second one (around 24cm ± 7, 5cm), against 56cm in our study. Bootstrapping is a resampling method that estimates the properties of an estimator 2 such as the standard deviation and the confidence interval in this case. The curve tends to be constant, which would indicate that the detection interval appears to be invariant to the redirection function.

L = -3 L = -2 L = -1 L = 0 L = 1 L = 2 L = 3
A One-Way ANOVA of the Function of Redirection over the IND reveals no significant difference: F 8,13 = 0.702, p > 0.05. An equivalence test carried out on all pairs of L Function of Redirection, the Two One-sided T-test (TOST), confirms the equivalence with the lower limit -4°and the upper limit +4°(p < 0.05). Nevertheless, the TOST does not allow us to conclude on the equivalence of the IND with -2°and +2°b ounds. We therefore did not find any significant effect of the Function Results in [START_REF] Zenner | Estimating Detection Thresholds for Desktop-Scale Hand Redirection in Virtual Reality[END_REF] are given for comparison purpose.

Point of Subjective Equality

Figure 4.10 shows the P SE for the different redirection functions. For the linear case L = 0, the P SE is -0.877°. As for IND, our value of P SE is similar to the results obtained by Zenner et al. [START_REF] Zenner | Estimating Detection Thresholds for Desktop-Scale Hand Redirection in Virtual Reality[END_REF], with a P SE = -0.28

• . A Two-Way ANOVA (Physical Targets × Redirection Function) on the P SE indicates no significant effect of Physical Targets (F 1,13 = 2.859, p > 0.05) nor interaction effect (F 6,13 = 1.751, p > 0.05).

Physical Target and Redirection Function Effect on Detection

We calculated the non-detection interval (IND) for each of the two physical targets, for each redirection function and for each amplitude of redi-rection. A Two-Way ANOVA (Physical Targets × Redirection Functions) on the IN D indicate no significant effect of the Physical Targets (F 1,13 = 0.003, p > 0.05), of the Redirection Functions (F 6,13 = 0.087, p > 0.05), nor interaction effects (F 6,13 = 0.984, p > 0.05).

Participant Analysis

Figure 4.11 shows the non-detection interval for each user (one color per participant) depending on the redirection function. We do not observe any particular pattern such as groups of behaviours.

Conclusion

The main tool used to analyze the data is the psychometric curve and more precisely the IND. The results showed that only the amplitude of Redirection has a significant effect on the IND among the independant variables. Both the physical target and the function of redirection showed no effect on the IND. Our results confirmed previous studies on illusion detection with similar results on IND and P SE as [START_REF] Zenner | Estimating Detection Thresholds for Desktop-Scale Hand Redirection in Virtual Reality[END_REF] for example.

Hand Trajectory analysis

We logged the positions of the hand over time to analyse the real hand trajectory when reaching for the target as a function of the redirection function. We first removed trajectories were the hand tracking was not optimal (<5%) and then grouped trajectories for all participants by amplitude of redirection and redirection function.

Figure 4.12 illustrates all the trajectories as a function of the function of redirection for the redirection amplitude 14°. We observe that the trajectory shape is influenced by the redirection function L: a strong redirection at the beginning of the movement (L < 0) generates a smooth reversed C shaped trajectory (Figure 4.12-Left). In contrast, a strong redirection at the end of the movement (L > 0) generates a trajectory with a stronger correction at the end of the movement. (Figure 4.12-Right).

Figure 4.12: A visualization of participants trajectories for each redirection function evaluated. These trajectories come from all trials with a redirection amplitude of 14°on both physical targets. The top row represents redirection functions where the redirection is applied more at the start of the gesture and respectively at the end the trajectory for the bottom row.

Discussion

Findings

This chapter explores the design space of redirection functions for the hand redirection technique: we generalized the current implementation by considering polynomials of degree 2. The main objective of the user study was then to investigate the influence of the redirection functions on the interval of non-detection of the illusion. We considered 6 redirection functions with different different dynamics (a shift at the beginning or at the end of the movement). Our results show no significant effect of the redirection function on the interval of non detection. Moreover, a Two-One Sided T-test (TOST) suggests the equivalence of the non-detection intervals for the different redirection functions.

By generalising the redirection functions, we can choose whether the offset between the real and the virtual hand mainly increases during the ballistic phase or the correction phase. Indeed, with L < 0, the offset is likely to increase during the ballistic phase (beginning of the movement) while while L > 0, the offset is likely to increase during the correction phase (end of the movement). A difference regarding the detection of the illusion between L < 0 and L > 0 would suggest that users are more sensitive to the offset increase in one of the two phases. In particular, we thought that the amplitude of the offset at the end of the ballistic phase would influence the detection of the illusion.

Surprisingly, our results do not show differences regardless of the value of L. Our hypothesis based on the Two-Component Model is not verified here. One explanation could be that users are not sensitive to when the offset increases, but only to the maximum offset amplitude at the end of the trial, i.e. when the users touch the physical target. Another explanation might be that the ballistic phase is much shorter than expected. Participants' movements could have been too slow due to our instructions favoring correction movements. Participants could also have focused too much on their virtual hand, rather than on the target to accurately answer the 2AFC choice. As future work, we plan to study possible interaction effects between movement speed and redirection function. Indeed, faster movements and potentially more contrasted redirection functions could make some phenomena more salient. Future work should also control the effect of the distance from the real hand to the body as the visual and proprioceptive accuracy might vary in the ballistic (currently close to the body) and correction phase (far from the body). For instance, we will consider a reverse movement where the starting position of the hand is away from the body and the target is close to the body. Another possible explanation is that an effect is present but not visible due to the lack of data points. Future work should consider a larger number of participants and/or a larger number of samples per participant. The instructions might have been more difficult to interpret than expected as some participants might have noticed the visuo-haptic mismatch but were not able to correctly indicate the direction of the mismatch.

Implications for Design

Our results have implications for design interactions. They suggest the designer can better control the real hand trajectory of the users during the hand redirection. For example, in a complex scene with multiple physical objects that can be accidentally hit, the designer can use an appropriate function to avoid any obstacles along the real hand's path as illustrated in Figure 4.13. Also, many interaction techniques revolve Virtual POV Real POV On the left, the user thinks there's an object to the right of the target whereas the actual object is on the left of their real hand due to the illusion as shown on the right. This mismatch can create issues where the participants hits an object when they should not.

around using actuated robots to provide haptic feedback. It is conceivable to alter the hand's trajectory to avoid unwanted contact with the robot.

Limits and Future Work

The main limitation of this work is the choice of functions of redirection.

Han et al. showed that using a square function of redirection severely reduces immersion [START_REF] Han | Evaluating Remapped Physical Reach for Hand Interactions with Passive Haptics in Virtual Reality[END_REF]. Users noticed the illusion, although they did not measure the IND, it is safe to say it was also reduced. It would be interesting to measure the IND with functions spanning from the squared function to one of our second order polynomial or to the linear function. This way, we could discover at which point the IND starts increasing. To better explore this design space, cubic Bézier curve (third order polynomials) could be used. They allow for more flexibility in choosing the tangent at the starting and ending point. A different class of functions such as exponential functions may also yield interesting results.

Another limitation is the noise inside the collected data, Figure 4.11 shows some participants have very large IND that are probably incorrect. These large values may be due to each user's data being fitted to the logistic function curve without enough points. Having all participants doing more trials may reveal trends or effects we could not see. The issue would be the time necessary for each participants to do the experiment. Also adding more participants may help smooth out some noise in the results.

What is important in this chapter:

-We evaluate new functions of redirection for the Hand Redirection illusion. The new functions are second order polynomials, they allow to redirect the user more when the user is closer to or farther from the real target.

-The novel second order function of redirection did not change the interval of non-detection. Our results showed the IND remains the same than with the linear function.

-These new functions allow more freedom for the designer to include Hand Redirection illusions in the virtual environment.

I n this chapter, I investigate the potential of using visuo-haptic illusions to transfer motor control skills from VR to the real world. Based on chapter 3, we saw that learning happens in VR when users experience visuo-haptic illusions: empirical studies show a strong acquisition and retention effect. However, there is no empirical data on the transfer of learning to the real world.

I first describe a user experiment aimed at measuring the transfer of learning from VR when exposed to visuo-haptic illusions to the real world. I also aim to compare training in VR with visuo-haptic illusion with the best context of learning: practicing in the real world. For this, I compare the retention performance of users that practiced in the real world with the users that practiced in VR. Results show that the amount of transfer is large and positive and also that the performance after training in VR with illusion and training in the real world is very similar.

The contents of this chapter was published at ACM SUI 2021 [START_REF] Geslain | Visuo-haptic Illusions for Motor Skill Acquisition in Virtual Reality[END_REF].

Motivations

Product assembly, repairing or maintaining a mechanical machine are complex real-world tasks. These tasks encapsulate several actions such as placing objects precisely, avoiding obstacles and following trajectories with time constraints. This class of tasks is classified as procedural involving both motor and cognitive skills [START_REF] Rodríguez | Training of procedural tasks through the use of virtual reality and direct aids[END_REF]. Such tasks are targeted by training assistance systems [Pathomaree andCharoenseang, 2005, Henderson andFeiner, 2011], however they remain difficult to implement in a virtual environment due to the limitations of haptic feedback in VR.

chapter 2 informed that visuo-haptic illusions have been shown to improve presence and interactivity in VR [START_REF] Azmandian | Haptic Retargeting: Dynamic Repurposing of Passive Haptics for Enhanced Virtual Reality Experiences[END_REF], Han et al., 2018, Matthews et al., 2019]. They appear as a viable way to reduce cost and complexity to set-up a virtual training environment while providing life-like haptic feedback. However, when motor skills are involved, they may induce a negative effect as suggested by studies in visuo-motor adaptation [Helmholtz and Powell Cocke, 1926]. Indeed, in visuo-motor adaptation studies, the users' field of view is partially or entirely altered (translated or rotated) similarily to visuo-haptic illusions except the alterations are not hidden and easily detected by users. As seen in chapter 3, visuo-motor adaptation studies showed that users have a loss of performance when adding (proactive transfer) or removing (after-effect or retro-active transfer) a change in perception. Although there are some clear differences 1 between these changes in perception and visuo-haptic illusions, there may be an adaptation phase when exposed to visuo-haptic illusions. If an adaptation occurs, it could be detrimental to the transfer of learning.

As seen in chapter 3, previous studies explored motor learning in VR with visuo-haptic illusions, specifically hand redirection illusions [START_REF] Kohli | Redirected Touching: Training and adaptation in warped virtual spaces[END_REF], Bibin et al., 2008, Spillmann et al., 2013]. These studies focused on acquisition and retention tests (two of the three ART measures) and there is no data on the transfer of learning.

The following sections describe an emprirical study that aim to explore transfer of learning from VR with visuo-haptic illusions to the real world.

User Study

To operationalize complex tasks in a laboratory setting, it is common to use a a multi-object pick-and-place task [START_REF] Roldán | A training system for industry 4.0 operators in complex assemblies based on virtual reality and process mining[END_REF], Gutiérrez et al., 2010] where users pick a labeled cube and place it on a designated target. This experiment thus relies on this modus operandi. Moreover, this task is particularly appropriate for visuo-haptic illusions as it is possible to implement a n:1 virtual to physical mappings [START_REF] Azmandian | Haptic Retargeting: Dynamic Repurposing of Passive Haptics for Enhanced Virtual Reality Experiences[END_REF], i.e. one physical prop for several virtual objects.

To study proactive transfer of learning, we use a well-established methodology where the experiment is segmented in a training phase and a testing phase with a 24 hours rest time between [Edwards, 2010, Schmidt andLee, 2011] as described in chapter 3. We compare the performance of three groups (Figure 5.3):

• The first group (baseline) only performs the test in the real environment with no training.

• The second group also performs the test in the real environment, but only after practicing in the virtual environment exploiting hand redirection. The first two groups are the same as the ones described in subsection 3.6.1. They allow to compute the amount of transfer 1 Visuo-motor adaptation distortions are constant during the task repetitions unlike visuo-haptic illusions mismatch (especially for Hand Redirection illusions). Indeed, the shift or rotation of visuo-haptic illusions are progressive and can vary inside a task [START_REF] Azmandian | Haptic Retargeting: Dynamic Repurposing of Passive Haptics for Enhanced Virtual Reality Experiences[END_REF], Lebrun et al., 2021] (see also chapter 4). Hence, it is not trivial whether the adaptation effect applies to visuo-haptic illusions.

from VR with illusion to the real world as described in chapter 3 and in section 5.3.

• The third group also performs the test in the real environment, but only after practicing in the real environment. This last group is a ceiling reference, they train in the ideal conditions and inform on the retention on the task chosen. The performance of the second group can at best equal the performance of this group. By computing the amount of transfer for both groups, we get information about the loss of performance due to transfer compared to retention.

This experiment has been approved by the IRB of Sorbonne University with request n°2020-20.

Participants and Apparatus

20 participants (11 male, 9 female, aged 28.4, 2 left handed) took part in the present study. 15 played video games regularly and all had previous experience with VR. Participants were compensated with candies after each session.

Participants are seated on a chair in front of a table. The experiment consists in picking in sequence 6 cubes laying on predefined positions on a complex rectangular shape, and placing them on designated square targets. All cubes and their targets are marked with the same number, which also gives the order of execution (Figure 5.1-a). The cubes are 8 cm in size and a cube is considered correctly placed when its center is within 3 cm of the target center, with an alignment error of less than 10°. This precision is similar to [START_REF] Azmandian | Haptic Retargeting: Dynamic Repurposing of Passive Haptics for Enhanced Virtual Reality Experiences[END_REF]. The location of the real cubes were chosen in order to maximize the movement distances given the available space during the training phase. The virtual cubes are placed so that the amount of maximum redirection (warping distance) is large and spans both horizontal (X and Z) directions (cf table 1). A screen displaying the instructions stands in front of the user, behind the table. A motion capture system (Optitrack ) tracks the hand trajectory and the 6 cubes. Participants wear 3 reflective markers on their dominant hand. An exact replica of this scene is built in VR using Unity (Figure 5.1-c).

Participants training in VR experience the scene through an Oculus Rift S Head-Mounted Display (HMD). Those are not made aware that they will be subject to a visuo-haptic illusion. They put on the HMD only after taking the seat, hence observe the real set-up beforehand. At this point, the experimenter, unseen to the participant, removes discreetly

Experimental Design

Task and stimulus

The task is a multi-object pick-and-place task [START_REF] Roldán | A training system for industry 4.0 operators in complex assemblies based on virtual reality and process mining[END_REF], Gutiérrez et al., 2010], involving obstacle avoidance. The participants are informed that the experiment will evaluate their performance and they are asked to be as accurate and fast as possible. They are instructed to pick the cube whose number is shown on the display and place it on the designated target, with the correct orientation. They press a green button located on the table to initiate the trial. The screen displays instructions and the participants pick the corresponding cube with their dominant hand and place it on the target. They then re-press the button to mark completion. If the cube is not placed accurately enough at the correct position or orientation (Figure 5.2), the feedback on the screen requests them to rectify it. At the end of the sequence of 6 cubes, the execution time is displayed.

Conditions

We have three conditions: • Test only. This condition measures the performance of participants performing the test phase in a real environment, without any training, on the full set-up (Figure 5.1-a).

• Real Training + Test. Participants first practice in the real environment in the same setup as the test phase (Figure 5.1-a). They then perform the test in the real environment 24 hours later, identically to the previous condition.

• Virtual Training + Test. Participants first practice in a virtual environment exploiting a visuo-haptic illusion. They then perform the test in the real environment 24 hours later. The virtual scene used in Virtual Training + Test is a replica of the setting used in test as shown in Figure 5.1-c. Virtual obstacles, cubes and screen have exactly the same geometry, and are reasonably well colocated with the real set-up. The real setting is however modified without their knowledge, after they put on the HMD (Figure 5.1b). All cubes except the number 1 are removed, as well as most real obstacles. This reflects the possibility that the training set-up may be significantly simpler than the real task. A mismatch between the virtual scene and the real setup hence occurs. Indeed, participants are made to believe they manipulate 6 real cubes while there is only one (1:6 mapping). This illusion is implemented through a hand redirection technique: participants virtual hand's position is shifted from their real hand's. In consequence, participants do not perform the same trajectory in the training phase and the test phase.

The training consists in 15 blocks of 6 trials. The test consists in 5 blocks of 6 trials, 24 hours after the training when applicable (see Figure 5.3)

Hand Redirection Implementation

Our implementation of Hand Redirection relies on "Interpolated Reach" detailed in [START_REF] Han | Evaluating Remapped Physical Reach for Hand Interactions with Passive Haptics in Virtual Reality[END_REF], also see chapter 2 for more details on implementation. The shift between the real hand and the virtual hand is applied inside a sphere of radius R centered on the real target. It is a continuous function, maximum at the target location and null at the surface of the sphere and beyond. It is tuned so that users place down the real cube not on the real target but where the participant will pick up the next cube meanwhile the virtual hand places the cube on the virtual target. This way, in the next step, participants are tricked into picking up the same cube they've just placed down. Hence, the illusion initiates with a null shift once the cube is picked up, increases on the way to the target, and decreases when the hand goes back to the button. The same real cube is thus used in subsequent manipulations. The position of the shifted objects (i.e. the virtual hand and the virtual cube) depends on the distance d to the target and is computed as follows:

⃗ P V = ⎧ ⎪ ⎪ ⎪ ⎪ ⎨ ⎪ ⎪ ⎪ ⎪ ⎩ ⃗ P R + (1 -d D ) × ⃗ λ if d ⩽ D, ⃗ P R otherwise. ⎫ ⎪ ⎪ ⎪ ⎪ ⎬ ⎪ ⎪ ⎪ ⎪ ⎭ (5.1) with ⃗ λ = ⃗ T V -⃗ T R
⃗ P V and ⃗ P R are the virtual and real hand positions, d represents the distance between the real and virtual hand and D (mean = 57.5cm, std = 13.7cm) defines the illusion sphere centered on the target square where the cube should be placed. ⃗ T V and ⃗ T R are the virtual and real positions of the target, T 0 is the starting position (the button). The maximum shift (mean = 10.7cm, std = 5.0cm), i.e. the distance between the virtual cube and the real cube at the end of the manipulation varies between trials. These values, summarized on Table 5.1, are generally higher than the literature [START_REF] Zenner | Estimating Detection Thresholds for Desktop-Scale Hand Redirection in Virtual Reality[END_REF], as participants were not informed about the illusion. A pilot study had confirmed that the illusion remains undetected when participants are not notified. Our shifts are thus large enough to affect users' trajectory but small enough to not be detected and maintaining a high level of immersion [START_REF] Azmandian | Haptic Retargeting: Dynamic Repurposing of Passive Haptics for Enhanced Virtual Reality Experiences[END_REF].

Procedure

Each block contains 6 manipulations/trials of different amplitudes and directions to vary the difficulty and to introduce variability. The sequence of manipulations (i.e. the order of the trials) remains the same from one block to another so that each block is identical to each other. Table 5.1: The direction and distance for hand trajectories and warping for all trials. R is the same value as in Equation 5.1. X represents the sideways horizontal direction (positive to the left), Y represents the vertical direction (positive upwards) and Z is the depth horizontal direction (positive forward).

This reflects a sequence of operations in the real world to assemble or repair a product. Because all the blocks contain the same sequence of manipulations, we can study learning phenomena by comparing performance between blocks. Participants first received written instructions explaining the experiment procedure and the task. They then performed a pre-test consisting of picking a cube and placing it on a target in the real environment to ensure that the different groups of participants achieve a similar level of initial performance and understood the task properly. We measured the Time as described in subsubsection 5.2.2.6 to realize the task and found no significant difference with ANVOA: F 1,19 = 0.25, p = 0.59. This manipulation was not present in the main experiment. Participants then performed the main experiment. They have a 20 seconds break between each block. When realizing the task in the real or virtual environment, the breaks give time to the experimenter to reset cubes' position. After they complete the entire training phase, i.e. at the end of the first day, participants filled a questionnaire and performed a semi-structured interview. In the questionnaire, participants are asked to rate the perceived difficulty of each of the six manipulations (Likert scale from 1 to 7). They also answer two additional questions related to the illusion:

• How many real cubes did you think you handled?

• My hand moved as I intended. (1-7: "Fully agree/disagree")

The semi-structured interview comprised of the questions:

• How difficult did you find the Test compared to the Training?

• Would you like to train in a similar VR simulator?

The other questions such as "Did a particular manipulation seem more difficult during the test?" depended on the answers to the two first questions. Figure 5.3 summarizes the procedure.

Design

This study follows a between-subject design. All groups perform a pretest and a test. Both groups assigned to the conditions Virtual Training + Test and Real Training + Test performed 15 blocks of 6 trials respectively in VR and in the real environment during the training phase. During the test phase, the second day, all three groups performed 5 blocks of 6 trials in the real environment. The order of the trials within blocks is fixed.

In summary, the design is 2 groups × 10 participants × 15 blocks × 6 trials for training + 2 groups × 10 participants × 5 blocks × 6 trials + 1 group × 10 participants × 5 blocks × 6 trials for test = 2700 manipulations.

Dependent variables

For each trial, we log Time, the time to perform a trial (time measured between 2 button presses). We can then derive our main dependent variable, the amount of proactive transfer (T % ) based on the Equation 5.2 [START_REF] Schmidt | Motor control and learning: A behavioral emphasis, 5th ed. Motor control and learning: A behavioral emphasis[END_REF].

We also log the position of the cubes at 60Hz to estimate the evolution of the trajectory Variability i.e. how much different are the trajectories between each other. To achieve this, we used the Dynamic Time Warping (DTW) distance [START_REF] Berndt | Using dynamic time warping to find patterns in time series[END_REF]. More on DTW distance in subsection 5.3.2.

Analysis

Transfer of Learning

As seen in chapter 3, we define the amount of transfer as: where A is the performance on the new task after training of the previous task, B is the performance on the new task without training on the previous task and C is the ultimate performance reached after a long training on the new task (see Figure 5.4).

T % = A -B C -B (
We use the competition Time as our performance measure to compute the amount of transfer. To compare the performance of the Real Training + Test and the Virtual Training + Test groups, we computed the amount of transfer for both groups. For the Real Training + Test group, it relates more to an amount of retention. Both are still comparable with each other because the retention interval and the task is the same for both groups.

Dynamic Time Warping Distance

It is not trivial to compare trajectories (defined as a sequence of positions) followed by participants because they have not the same length and are not synced together. Participants realized the task at different speeds resulting in uncomparable sequences using the euclidean distance. To extract information on the trajectories of participants, we used the Dynamic Time Warping (DTW) distance.

DTW produces a discrete matching between existing elements of two series before estimating the distance between them. It is capable of realigning a sequence with another which makes it more useful than the euclidean distance for temporal series [Müller, 2007, Petitjean, 2011]. The computed DTW distance is independent of time and the more similar the series, the lower the distance will be (see Figure 5.5). DTW computations are made using the python dtaidistance library.

We computed the DTW distance in two ways. On the test phase (1) to measure acquisition, we computed the DTW distance on every trial compared with the last training trial for each participant. We consider the last trial as the ultimate performance and use it as a reference. Hence, we obtain 14 points for each participants that we averaged for each group. The resulting curve should inform on the convergence or not of the users' trajectory over time. On the training phase (2) to measure transfer, we computed the DTW distance for every combination of trials (pair-wise). We then averaged the values resulting in a measure of the variability of the trajectory during the test.

Results

Pro-active Transfer and Test Performance

Figure 5.6-left shows the performance for Time for all three conditions during the test phase. We used the Equation 5.2 to calculate the percentage of proactive transfer, which is 78.5% (std=18. 2.33, p = 0.13). These results suggest the hand redirection illusion had little to no negative impact on the performance during the test.

Training Performance

While our main objective is the initial performance during the test (proactive transfer), we also analyzed data collected during the training. Figure 5.7 shows the intramodal learning improvement, time and variability, during the training phase. We use the Power Law of Practice (PLP) model [START_REF] Schmidt | Motor control and learning: A behavioral emphasis, 5th ed. Motor control and learning: A behavioral emphasis[END_REF] The reduction of variability of the trajectory followed by the users according to the PLP is important as it confirms motor learning, i.e. a gradual reduction of variability necessary to optimize the accuracy and efficiency [START_REF] Schmidt | Motor control and learning: A behavioral emphasis, 5th ed. Motor control and learning: A behavioral emphasis[END_REF].

Our results thus show that the proactive transfer for the Virtual Training + Test condition is both positive and large (78.5%) and similar to the Real Training + Test. Our setup thus favors learning and immersion while using a high amount of redirection (12°).

Questionnaire and interviews

The median result to the affirmation "My hand moved as I intended" for the Virtual Training + Test condition is 2 ("Agree") for both conditions. This result was confirmed in the interview as well, suggesting participants (8/10) did not notice the illusion and did not notice that there was only a single cube during training. When it was revealed to participants that they trained with only one cube, they reacted very enthusiastically: "It's amazing, I would never have noticed there was only one [real] cube! " (Participant 2) or "This works very well, you fooled me properly" (Participant 14). Two participants noticed there was only one cube because they went to grab cubes during the 30 seconds break despite the experimenter telling them otherwise or because they had to correct the position of a cube and their hand passed through one of the virtual cubes. The findings do not change when excluding these two participants.

During the semi-structured interview when asked about how participants of the Virtual Training + Test condition felt when placing a cube, participants sometimes mentioned that something was a bit off. They all agreed to say this sensation didn't bother them and they would gladly train again in a similar VR simulator.

Finally, A Kruskal-Wallis Test on difficulty rating indicates that the manipulations 2 and 4 are significantly more difficult to perform than the other manipulations (time comprised between 3.2 seconds and 5.5 seconds, except for cube 2 and 6: statistic= 3.03, p = 0.08). The feedback from participants during the interview are in line with the results of the questionnaire. be due to participants moving the cube faster at the costs of less efficient trajectories. Moreover, several trajectories in Virtual Training have a kink toward the end of the movement (red cross). This result is similar to what has been observed in previous retargeting studies [START_REF] Azmandian | Haptic Retargeting: Dynamic Repurposing of Passive Haptics for Enhanced Virtual Reality Experiences[END_REF] and indicates that participants had to adjust their trajectory towards the end to compensate the illusion.

Qualitative Analysis of the Trajectory

This kink is less noticeable at the end of the training phase (block 15), indicating there might be an adaptation to the illusion. On the contrary, during the first block of the test (after virtual training), we do not observe an after-effect in the sense of visuo-motor adaptation (i.e. a kink in the inverse direction). Additionally, it seems that the adaptation had little to no effect on the transfer of learning.

Discussion

Findings

The main objective of this study was to investigate the transfer of learning when training with the hand redirection illusion. Our results show a large positive transfer of learning on Time (T %V irtual = 78.5%, std = 18.5%). Moreover, the transfer is close to the upper bound, i.e. the optimal setup (Real Training + Test condition): T %Real = 82.4% (std = 14.2%). VR training with hand redirection also reduces movement variability by 45% similarly to what was observed after training in the real environment (35%). We were not able to measure an effect between the Virtual Training + Test and Real Training + Test conditions so if there is an effect of the illusion on transfer of learning, it is likely to be small. These results thus motivate the use of visuo-haptic illusions during virtual training. Finally, participants expressed enthusiasm during the virtual training. Indeed, only 2 out of 10 participants noticed the illusion despite using a large amount of redirection above the detection thresholds recommended in the literature [START_REF] Zenner | Estimating Detection Thresholds for Desktop-Scale Hand Redirection in Virtual Reality[END_REF].

Implications for design

Our results demonstrate the potential of using visuo-haptic illusions for motor-skill acquisition. Indeed, designers can use an excessive warping, higher than what's recommended in previous studies [START_REF] Zenner | Estimating Detection Thresholds for Desktop-Scale Hand Redirection in Virtual Reality[END_REF] on visuo-haptic illusions, and still favor skill transfer on task requiring the users to interact with multiple similar parts (pieces of an assembly, buttons) or tools (screwdrivers). These results are rather unexpected. Indeed, users do not perform the same trajectories in the virtual and real environment and visuo-motor adaptation literature suggests a loss of performance when such a distortion is introduced. A main difference is that in our setup most users do not notice the illusion despite the intensive warping, and a high level of immersion is maintained. It is especially interesting given the practicality of visuo-haptic illusions such as providing rich haptic feedback while reducing the total number of props in comparison with the number of virtual objects (6 virtual objects for 1 real object in our study). Our findings thus invite designers and researchers to consider visuo-haptic illusions to help operators to learn motor skills in an immersive, cost-effective, and less-maintenance environment.

Limits and Future Work

The main limitation of this work is the lack of a condition training in VR without the hand redirection and then testing in the real world. This condition could provide more information on the impact the visuo-haptic illusion has during training. It would split the confounding factor of VR + Visuo-haptic illusion of the current user experiment. The results of this new condition may be closer to the Real Training + Test which would indicate the illusion is causing the slight drop in performance we observed or closer to the Virtual Training + Test condition indicating that training such a task in VR is slightly less effective than in the real world whether there is an illusion or not.

This user study evaluates a single Body Redirection illusion, the hand redirection for repurposing props, it does not inform on the influence pseudo-haptic or the World Redirection illusion or even hybrid techniques [START_REF] Azmandian | Haptic Retargeting: Dynamic Repurposing of Passive Haptics for Enhanced Virtual Reality Experiences[END_REF] would have on transfer of learning. Additional studies are required to have a broader understanding of the visuo-haptic illusion and their usability for training.

Finally, involving more participants to this study may yield more significant results and reduce noise in our data. Although 20 users participated in this study, only 10 participants per group where kept which is not many for the statistical tools I used.

What is important in this chapter:

-I conducted a study evaluating the amount of transfer on a motor control task from a VR environment with the hand redirection illusion to the real world. I also compared the amount of transfer with the retention measure of a group training in the real world.

-I found that (1) the amount of transfer is strongly positive and (2) the amount of transfer is almost as good as the amount of retention (not significantly different). The hand redirection illusion seems like a viable addition to training in VR expanding the possible applications of visuo-haptic illusions.

Chapter 6

Conclusions

Chapter Contents V isuo-haptics has been shown to improve the perception feedback and to promote natural interactions in several VR contexts. In this thesis, I proposed a novel implementation of the hand redirection illusion in chapter 4. I conducted an experiment investigating the detectability of illusion by comparing the proposed implementation with existing ones. I also evaluated the usability of the hand redirection for learning in VR in chapter 5. For this purpose I conducted an empirical study measuring the transfer of learning from a virtual environment with the illusion to the real world.

In this chapter, I summarize my findings and discuss possible future works from short term to long term.

Progress on the Research Problems

The main objective of this thesis was to improve haptic feedback in virtual environments using visuo-haptic illusions. I focused mainly on the hand redirection illusion and I used two different approaches: expanding the design space of hand redirection and quantifying the transfer of learning when exposed to the hand redirection.

Hand Redirection Design

The main challenge of the hand redirection illusion (and visuo-haptic illusions in general) is to not be detected by the users while maximizing the amplitude of illusion. The main measure for this is the interval of Non-Detection (IND): the higher the IND, the more difficult it is to notice the illusion. The IND can be obtained only empirically.For linear hand redirection, [START_REF] Zenner | Estimating Detection Thresholds for Desktop-Scale Hand Redirection in Virtual Reality[END_REF] found that the IND was 8.19

• horizontally and 8.88

• vertically when no distractions (visual or auditory). In chapter 4, I found a similar horizontal IND (9.89

• ) in a comparable setup. I have then proposed and evaluated 6 new redirection functions, using second order polynomials. The results, summarised in Table 6.1 show slightly higher INDs for all cases in comparison with [START_REF] Zenner | Estimating Detection Thresholds for Desktop-Scale Hand Redirection in Virtual Reality[END_REF] are however still inside the margin of error. No significant differences emerged in these results. Further analysis on the trajectory shows users followed very different paths depending on redirection function.

These results show that redirection function does not impact the de-IND PSE L=-3 [START_REF] Geslain | Generalizing the hand redirection function in virtual reality[END_REF] 11.41 [START_REF] Zenner | Estimating Detection Thresholds for Desktop-Scale Hand Redirection in Virtual Reality[END_REF] are given as comparison point.

tection of the hand redirection illusion in the studied range. It implies that designers have the liberty to use these new functions in virtual environments without impacting the user experience and breaking immersion. These give more control on users' hand trajectory to make them avoid obstacles for example which may arise form unused props, no longer colocalised due to the redirection.

Training with Hand Redirection

Visuo-haptic illusions are a great way to expand the capabilities of VR.

One application of VR is skill learning. As seen in chapter 3, research explored on acquisition and retention with visuo-haptic illusions. While it is important that users are able to learn in VR with illusions, it is also essential to know whether what is learned is useful in a real world context. In chapter 5, I measured the amount of transfer of learning when users train in VR and are exposed to the hand redirection illusion and tested in the real world. I found the transfer of learning was positive and strong: 78.5% as well as very similar to the amount of retention: 82.4% (not significantly different). The strong transfer of learning observed encourages the use of hand redirection and visuo-haptic illusions for training in VR on motor control tasks.

6.2 Short Term Future Work In both studies made in chapter 4 and chapter 5, the empirical data obtained exhibit some noise. Differences between tested conditions could be significant with more data points. For example in chapter 5, we observed a difference in retention and transfer of learning between the groups that trained but it is not significant. More participants would help alleviate this uncertainty as was noted by reviewers of the related publication. Similarly, in chapter 4, the 95% confidence interval on the IND per functions of redirection is large and some participants had a high IND compared to the largest amplitude of redirection. Hence, it would make sense to have more participants and more repetitions per participant. In this thesis, I focused on the hand redirection illusion used for prop repurposing. Other illusions like pseudo-haptic or world redirection may have a different impact on the transfer of learning than hand redirection. Evaluating more illusions may give a broader understanding of visuohaptic illusions and how to better use them in virtual environments.

Considering Additional Conditions

Exploring a Larger Design Space

Other Second-order Redirection Functions

As illustrated in Figure 6.1, the linear function in green dashes is the most used and tested for performance, immersion and detectability [START_REF] Han | Evaluating Remapped Physical Reach for Hand Interactions with Passive Haptics in Virtual Reality[END_REF], Azmandian et al., 2016, Zenner and Kruger, 2019]. In chapter 4, I studied a set of new functions of redirection: second order polynomials. The chosen functions expanded the design space of visuo-haptic illusions but they did not span the entire design space. I can thus consider additional functions corresponding to the gray area of Figure 6.1. This area corresponds to the redirection functions with L > 3 and L < -3.

Step Redirection Function

Other classes of redirection functions can be considered. For instance, Han et al. studied the step function that consists in redirecting the user hand instantly as the users' hand leave the starting position [START_REF] Han | Evaluating Remapped Physical Reach for Hand Interactions with Passive Haptics in Virtual Reality[END_REF]. They found it decreased immersion and made the illusion easier to detect. Our design space includes continuous functions. It could be extended to no-continuous (or step) functions.

Relative -Absolute Redirection

To further expand the design space of the hand redirection illusion, it is possible to classify every visuo-haptic illusion according to how the redirection is applied. In most examples discussed in this thesis, the hand redirection uses an absolute redirection: the virtual hand position is computed according to the distance between the real hand position and the real target (called d throughout this thesis). On the contrary, world redirection illusions are based on a relative redirection as the point of view is computed according to instantaneous translations and rotations of the head. Similarly, pseudo-haptics often rely on a relative redirection. For example, to increase the perception of inertia, the weight illusion reduces the speed of movement when lifting an object. A relative redirection depends on translations independently on their direction, speed or rotations of the user whereas an absolute redirection depends on the distance between the user and the target. I propose to use the speed or the instantaneous translation of the real hand to redirect the user towards the targets. A benefit from this technique would be to have additional control over the users' trajectory and the rate of redirection. The rate of redirection would define how much the virtual hand is shifted away or towards the real hand in respect to the displacement of the real hand. This technique would also allow changing targets more dynamically as there is no need to have a starting position or a radius of activation.

Long Term Future Work -Modelling Transfer of Learning

I plan to build a computational model of transfer of learning. The Predictive Performance Equation (PPE) explains retention and decay of performance. This model cannot be applied to transfer as it does not account for similarities between the previous and the new tasks. According to Identical Elements Theory and Transfer-Appropriate Processing Theory, similarities in cognitive processes and/or in motor control improve the degree of transfer. This concept could be integrated in the PPE by increasing the decay rate when there are few similarities.

Final word

Illusions are a powerful tool in VR. They affect how the users interpret virtual objects' physical properties that they interact with. Possible uses are enlarging the apparent workspace, with redirected walking, or in conjunction with passive haptics, increasing the number of haptic and tactile cues in the scene, as well as distorting apparent inertia or viscosity. These illusions work from body-scale to hand-scale, and can increase the quality of immersion and enjoyment in a wide range of VR use-cases. This thesis has focused on the hand-scale illusions, and explored the hand redirection with passive haptics. Implications of its use on learning transfer and acquisition were explored, which would help a more widespread adoption of VR systems for motor learning. The use of different redirection functions in order to alter users' hand trajectory is a also novel idea emerging from this work. This can help designers to more easily exploit physical props in their scenario and provide a better immersion to their users. Note that most of the approaches developed here at hand-scale can be adapted to body-scale. This opens quite interesting perspectives on mapping illusions in VR and is a great motivation for future work.

Here, b offsets the decay by a parameter intercept, m is the decay slope, lag j represents the time since the last practice session j and e = 2.782 is Euler's number

1 . The decay is described here as a linear function of the history of practice (the n -1 last practice sessions). The terms inside the summation tends to 0 when the lag is long reducing the decay to the asymptotic value b and the summation tends to 1 when the lag is short increasing the decay to b + m. Finally, the predicted performance is expressed as a logistic function of activation M n :

P n = 1 1 + e ( τ -M n s ) (A.5)
τ and s control the slope and intercept of the logistic function. The logistic function is used to bound the result between 0 and 1 which is practical to scale to experimental data. The parameters used to fit the model to experimental data are b, m, τ and s.

This model is able to predict the performance of users on retention after some practice sessions. Walsh et al. proposed it and found the model fitted well 14 different existing datasets [START_REF] Walsh | Evaluating the theoretic adequacy and applied potential of computational models of the spacing effect[END_REF]. They compared the PPE with two other models: an extension of ACT-R [START_REF] Pavlik | Practice and forgetting effects on vocabulary memory: An activation-based model of the spacing effect[END_REF] and Search of Associative Memory (SAM) model [Raaijmakers, 2003] that both take spacing of practice and retention into account. Among other comparisons, they evaluated their model on a dataset of a memorisation task of letter triplets [Bregman, 1967] and obtained results in They found that their model best fitted the results (lower RMSE and hight R²) and better predicted the retention performance in various tasks. [Bregman, 1967]. Participants had to memorize consonant-vowel-consonant triplets under 3 conditions of practice: massed, intermediate and spaced (more distributed). On this dataset, the PPE seems to better predict the lower performance due to the massed distribution of practice.

4. 1

 1 Illustration of the hand redirection technique trajectories with a linear redirection function . . . . . . . . . . . . . . 4.2 The modified hand redirection technique with a 2 nd order polynomial function . . . . . . . . . . . . . . . . . . . . . 4.3 Examples of Bézier curves with different numbers of control points. . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 The offset between the real and its avatar ∆ M according to the distance between the real hand and the physical target d. The linear case is represented as well as 4 novel redirection functions . . . . . . . . . . . . . . . . . . . . . 4.5 Experimental setup . . . . . . . . . . . . . . . . . . . . . . 4.6 The three independant variables of the user study . . . . . 4.7 trajectory of user reaching for a cube while being exposed to the body redirection technique or not . . . . . . . . . . 4.8 An illustration of a psychometric function showing the probability to answer "Right" according to the stimulus provided. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.9 The psychometric function for each redirection function according to the angle of redirection. Vertical lines show the interval of non-detection IND for a redirection on the right (25%) and on the left (75%). . . . . . . . . . . . . . 4.10 Interval of non-detection and point of subjective equality . 4.11 The interval of non-detection IND for each function of redirection and for each participant . . . . . . . . . . . . . 4.12 A visualization of participants trajectories for each redirection function evaluated . . . . . . . . . . . . . . . . . . xii 4.13 Illustration of obstacle issue with visuo-haptic illusions . . 5.1 The task set-up for the experiment . . . . . . . . . . . . . 5.2 Target position and orientation . . . . . . . . . . . . . . . 5.3 User Study Procedure . . . . . . . . . . . . . . . . . . . . 5.4 A transfer performance curve showing relevant performance points to compute the amount of transfer. . . . . . 5.5 DTW algorithm . . . . . . . . . . . . . . . . . . . . . . . . 5.6 Performance during the test phase. Left: Time performance; Right: Variability. Areas / bars shows the 95% confidence intervals. . . . . . . . . . . . . . . . . . . . . . 5.7 Intramodal learning improvement during the virtual training and real training phase over Time (left) and DTW distance (right) performance . . . . . . . . . . . . . . . . . 5.8 Cubes trajectories for the first and last blocks of training and test for both training conditions . . . . . . . . . . . . 6.1 The new design space of the function of redirection for the hand redirection illusion . . . . . . . . . . . . . . . . . . . A.1 The results of an experimental study comparing the Predictive Performance Equation with an extension of ACT-R and Search of Associative Memory . . . . . . . . . . . . xiii List of Tables 3.1 An example of a Latin square. It is constructed by shifting the first row to the right and wrapping the last element to the left. . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.1 Summary of our results: Interval of Non-Detection IND and Point of Subjective Equality PSE as a function of the function of redirection L. Results in [Zenner and Kruger, 2019] are given for comparison purpose. . . . . . . . . . . , multimodality and Visuo-haptic illusions . . . 1 1.2 Learning with Visuo-haptic Illusions . . . . . . . . . . . . 3 1.3 General Problem . . . . . . . . . . . . . . . . . . . . . . . 4 1.4 Research Approach . . . . . . . . . . . . . . . . . . . . . . 4 1.5 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.6 Contributions . . . . . . . . . . . . . . . . . . . . . . . . . 5 1.7 Publications . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.8 Overview of this thesis . . . . . . . . . . . . . . . . . . . . 6

Figure 2 . 1 :

 21 Figure 2.1: Examples of a visuo-haptic illusion: The Hand Redirection. a. and b.shows that the virtual hand of the user is offset from their real one. The virtual and real hands are offset so that hands both reach the virtual and real targets[Zenner and Kruger, 2019, Azmandian et al., 2016]. On figures b. and c., the virtual hand is progressively redirected to the virtual target so that the user's real hand touches the real target[START_REF] Abtahi | Beyond the force: Using quadcopters to appropriate objects and the environment for haptics in virtual reality[END_REF] (see section 4.1 for more details on implementation).

Figure 2 . 2 :

 22 Figure 2.2: Two examples of off-the-shelf VR controllers, the Oculus controller on the left and the Valve Index on the right. The user can interact with the virtual environment through them but they provide very little feedback. They can vibrate, which can be used as a metaphor for contact with an object or a haptic confirmation. They are limited regarded touching things, grabbing objects, etc.

  simple objects like a tin can to provide feedback for similar objects like a flashlight [Hettiarachchi a. b.

  Figure 2.4: On the left, the real (in white) and virtual hands' (in black) trajectory while the user is reaching for the virtual target. On the right, a representation of the maximum distance between the real and virtual targets. While the virtual target is within the green area, the user does not detect the illusion. However, when they are too far away from each other, the illusion is broken.

Figure 2

 2 Figure 2.5:The Redirected Touching technique where a 2D surface (top picture) can be mapped to differently shaped surfaces (bottom picture)[Kohli, 2009]. Each point of the real surface is made to correspond to a point on the virtual surface. The entire space between the user and the surface is remapped so the redirection is continous from the user's perspective.

Figure 2 . 6 :

 26 Figure 2.6: Examples of Hand Redirection applications. a.shows the virtual environment with the virtual hand and target and superposed on the virtual environment the real hand and target[START_REF] Zenner | Estimating Detection Thresholds for Desktop-Scale Hand Redirection in Virtual Reality[END_REF]. b. shows the Sparse Haptic interface where facetted surfaces provide haptic feedback for various virtual surfaces in 2 different virtual environment[START_REF] Cheng | Sparse Haptic Proxy: Touch Feedback in Virtual Environments Using a General Passive Prop[END_REF]. c. is the Haptic Retargeting technique used to repurpose a single physical cube (bottom view) for three virtual cubes (top view)[START_REF] Azmandian | Haptic Retargeting: Dynamic Repurposing of Passive Haptics for Enhanced Virtual Reality Experiences[END_REF]. d. shows how Hand Redirection allows to improve the responsiveness of encountered-type haptic interface[START_REF] Gonzalez | Reach+: Extending the reachability of encountered-type haptics devices through dynamic redirection in vr[END_REF].

Figure 2 . 7 :

 27 Figure 2.7: Figure a. shows the SpaceBall TM input device. figure b.

Figure 2 . 8 :

 28 Figure 2.8: Figure a. shows a weight illusion, participants had to lift an object. Its weight was virtually increased (resp. decreased) by showing its virtual representation lower (resp. higher) than it really is [Rietzler et al., 2018]. Figure b shows how Lécuyer et al. created bumps and holes on flat 2D surfaces[START_REF] Lécuyer | Feeling Bumps and Holes Without a Haptic Interface: The Perception of Pseudo-haptic Textures[END_REF]. When the user's cursor enters the bump area, the cursor's speed is decreased to simulate the slope. The opposite is done when exiting the bump area by increasing the cursor's speed.

  For example, Push et al. simulated a liquid flow through a pipe by displacing

Figure 2 . 9 :

 29 Figure 2.9: Examples of locomotion techniques in large VR environments. a. shows the world-in-miniature and the increased speed gains techniques[START_REF] Abtahi | Visuo-Haptic Illusions for Improving the Perceived Performance of Shape Displays[END_REF]. They either upscaled the user in the virtual environment or increased the movement speed of the user. b. shows a teleportation techniques using an HTC VIVE controller[START_REF] Xu | Object location memory error in virtual and real environments[END_REF]. The user can point to a location and be teleported there instantly (or after a fade to black). c. shows a stepper machine

Figure 2 .

 2 Figure 2.10: Examples of locomotion techniques in large VR environments. a. shows the arm swinging technique[START_REF] Cherni | Literature review of locomotion techniques in virtual reality[END_REF] where swinging the arm forward moves the user forward. b. shows the walkingin-place[START_REF] Bozgeyikli | Locomotion in virtual reality for individuals with autism spectrum disorder[END_REF]. Walking-in-place consists in moving their feet up and down which moves the user forward in the virtual world. c. shows a passive treadmill in which the user's waist is fixed to the contraption and their feet slide along the surface[VR, 2021].

Figure 2 .

 2 Figure 2.11: Examples of how World Redirection changes the Point Of View (POV) of the user[Nilsson et al., 2018a]. On the left, a head rotation is scaled down. In the middle, a user's translation is scaled up. On the right, the path of the user is bent. These examples allow to trick users into misinterpreting their actual position in space and allows to have different real and virtual space without the user noticing for example.

Figure 2 .

 2 Figure2.13: On the left (a.), a representation of the virtual path followed in the virtual environment and the real path followed by the user. The user reaches a different virtual position in the virtual environment than in the physical world[START_REF] Rothacher | Visual capture of gait during redirected walking[END_REF]. On the top-right (b.), a top-down view of a virtual room (in blue) and the physical space (in red) with the path followed by a user while exposed to redirected walking. The real room is smaller than the virtual one and the path followed by the user is drastically altered in VR[START_REF] Razzaque | Redirected Walking[END_REF]. On the bottomright (c.), the unlimited corridor where users touch a curved wall in the real world to increase the amount of redirection that can be applied to the user without being noticed[START_REF] Matsumoto | Unlimited corridor: Redirected walking techniques using visuo haptic interaction[END_REF].

Figure 2 .

 2 Figure2.15: An illustration of the hand redirection technique trajectories with a linear redirection function. On the left, the virtual hand position is computed using a position based technique as described in Equation2.1. On the right, the virtual hand position is computed using a rotation based technique as described in Equation2.2.

Figure 2 .

 2 Figure 2.16: Illustration of obstacle issue with visuo-haptic illusions.On the left, the user thinks there's an object to the right of the target whereas the actual object is on the left of their real hand due to the illusion as shown on the right. This mismatch can create issues where the participants hits an object when they should not.

Figure 2 .

 2 Figure 2.17: Four Redirected Walking methods: (a) Steer-To-Center, (b) Steer-To-Orbit, (c) Steer-To-Multiple-Targets, (d) Steer-To-Multiple-Targets+Center.

Figure 2 .

 2 Figure 2.16 illustrates this problem.Regarding World Redirection, to foster infinite walking, the user must be redirected away from the obstacles: the limits of physical space (walls), furniture, etc. This is challenging and four main strategies exist[Langbehn and Steinicke, 2018, Hodgson and[START_REF] Hodgson | [END_REF]]:

Figure 2 . 18 :

 218 Figure 2.18: An illustration of a psychometric function showing the probability to answer "Right" according to the stimulus provided.

  6) where P (Right | IN D L ) = 25% and P (Right | IN D R ) = 75%, we get both IN D.

  Figure 3.1: The Power Law of Practice curve. The completion time represents the time needed to realize the task. The performance of users increases with practice. It reaches an asymptote called the ultimate performance. Such a performance curve shows the acquisition measurements.

Figure 3 . 2 :

 32 Figure 3.2: The differences in QWERTY and AZERTY keyboard layouts.

  Figure 3.3: Conceptual curves representing acquisition measurements and proactive transfer of learning. The thicker curve show the alternation between the previous skill or context (thin and blue) and the new skill or context (bold and orange).

Figure 3 . 5 :

 35 Figure 3.5: The alternation model between a previous task P (method) in blue and a new task N in orange and the ART model: a 2 the acquisition measures, b the proactive transfer of learning, c the first alternation of retroactive transfer of learning and d the second alternation of retroactive transfer of learning[Raissi, 2021].

Figure 3

 3 Figure3.6: a. prisms used to shift users' field of view[START_REF] Prablanc | Adapting terminology Clarifying prism adaptation vocabulary, concepts, and methods[END_REF]. b. prisms mounted on HMDs or glasses[START_REF] Prablanc | Adapting terminology Clarifying prism adaptation vocabulary, concepts, and methods[END_REF]. c. Optical devices rarely used recently in favor of HMDs and glasses[START_REF] Shabbott | Learning a visuomotor rotation: simultaneous visual and proprioceptive information is crucial for visuomotor remapping. Experimental brain research[END_REF] Sainburg, 2010, Prablanc et al., 2019]

Figure 3

 3 Figure 3.8: Top: proactive transfer experimental design. where participants practice a task A before practicing task B whereas another group practices task B without previous training. Bottom: a retroactive transfer experimental design with a single alternation.

  Figure 3.9: Transfer of learning curves from task A to task B. The point A represents the performance on task B after practicing task A. The point B represents the performance on task B without practicing task A. The point C represents the ultimate performance on task B. The amount of transfer is then given by Equation3.5. The savings time is the time necessary for a group not practicing task A to reach the same performance on task B than the group that practiced task A prior.

Figure 3 . 11 :

 311 Figure 3.11: Some examples of haptic interfaces: a. the Phantom interface, exerting forces in all 3 axis with no couple feedback when the user holds the handle [Massie and Salisbury, 1994], b. the H-Wall producing feedback for surfaces like walls, doors, whiteboards, etc[START_REF] Kim | Encountered-type haptic display for large vr environment using perplane reachability maps[END_REF], c. Sparse Haptics, a primitive surface used in combination with visuohaptic illusions[START_REF] Cheng | Sparse Haptic Proxy: Touch Feedback in Virtual Environments Using a General Passive Prop[END_REF].

Figure 4 . 1 :

 41 Figure 4.1: An illustration of the hand redirection technique trajectories with a linear redirection function. On the left, the virtual hand position is computed as described in Equation4.1. On the right, the virtual hand position is computed by applying a rotation by an angle θ the real hand around the starting point.

2 )

 2 with f n (d) a n th order polynomial partially defined by continuity constraints: f n (D) = 0 et f n (0) = 1. These functions are flexible and allow to create a large variety of redirection behaviour like redirecting

Figure 4 . 2 :

 42 Figure 4.2: The modified hand redirection technique with a 2 nd order polynomial function. On the left, the function (L < 0) redirects the user faster at the start of the movement rather than at the end. Inversely, on the right, the function (L > 0) redirects the user faster at the end of the movement rather than at the start. L is an arbitrary parameter that defines the direction and eccentricity of the curve from the linear function. The function of redirection is further away from the linear function the higher L.

Figure 4 . 3 :

 43 Figure 4.3: Examples of Bézier curves. On the left, a linear curve defined by 2 control points. In the middle, a quadratic curve with 3 control points and a cubic curve with 4 control points. On the right, a close approximation of half a circle created using 2 Cubic Bézier curve. This last example shows how cubic Bézier curves can create complex curve when stitched together.

  hand and the physical target -d (%)

Figure 4 . 5 :

 45 Figure 4.5: On the left, the table with 4 targets (including 2 decoys), the haptic marker and a participant's hand with the VIVE tracker. In the middle, what the user sees during the experiment: a single target, the hand avatar and the haptic marker. On the right, what the user sees during the 4 training trials with their real hand represented in white.

Figure 4 . 6 :

 46 Figure 4.6: The three independant variables of the user study. a. represents the Function of Redirection which defines the trajectory of the virtual hand according to the real hand. In addition to the linear function, 6 new second degree polynomials are evaluated. b. represents the Amplitude of Redirection that defines the angle offset between the real and the virtual target. Figure c. shows the 2 physical targets used to provide haptic confirmation to the user.

  Two-Component Model Azmandian et al. found the participant's trajectory exposed to the illusion were two-fold, a ballistic phase where

Figure 4 .

 4 Figure 4.7:The result trajectory of user reaching for a cube while being exposed to the body redirection technique on the left and without illusion on the left[START_REF] Azmandian | Haptic Retargeting: Dynamic Repurposing of Passive Haptics for Enhanced Virtual Reality Experiences[END_REF]. The kink at the end of the trajectory circled in red is a correction the users make when they reach the redirected targets.

Figure 4 . 8 :

 48 Figure 4.8: An illustration of a psychometric function showing the probability to answer "Right" according to the stimulus provided.

Figure 4 .

 4 Figure 4.9 shows psychometric curve for each Function of Redirection. The vertical bars indicate the Left and Right Intervals of Non-Detection. They vary on the left from -7.75°to -3.45°and on the right from 3.99°to 6.79°. Non-detection interval in the linear case L = 0 is IN D = 9.42 • (IN D G = -4.61 • and IN D D = 4.81 •). These values are comparable to the results of Zenner et al. in[START_REF] Zenner | Estimating Detection Thresholds for Desktop-Scale Hand Redirection in Virtual Reality[END_REF] (see Table4.1). The distance between the source and the target is within the same magnitude and their results are very close with a IN D = 8.19• (IN D G = -4.38 • , IN D D = 3.81 • ). The results reported by[START_REF] Ogawa | Effect of avatar appearance on detection thresholds for remapped hand movements[END_REF] and[START_REF] Benda | Determining detection thresholds for fixed positional offsets for virtual hand remapping in virtual reality[END_REF] show higher IND (respectively IN D = 26.7• and IN D = 44.6• ). However, the distance between the origin and the real target is much smaller: around 20cm for the first one, and variable value for the second one (around 24cm ± 7, 5cm), against 56cm in our study.

Figure 4 . 9 :

 49 Figure 4.9: The psychometric function for each redirection function according to the angle of redirection. Vertical lines show the interval of non-detection IND for a redirection on the right (25%) and on the left (75%).

Figure 4 .

 4 Figure 4.10 represents Intervals of Non-Detection according to the redirection functions L. The area around the curve represents the 95% confidence interval calculated with the 1000-sample Bootstrap technique.Bootstrapping is a resampling method that estimates the properties of an estimator 2 such as the standard deviation and the confidence interval

Figure 4 .

 4 Figure 4.10: On the left, the interval of non-detection IND according to the function of redirection. Each point is computed by fitting a sigmoid curve with points acquired with the corresponding Redirection Function, and computing the IND. This curve doesn't show an effect of Redirection Function as it is mostly flat. On the right, the point of subjective equality P SE according to the function of redirection. The blue-colored area shows the 95% confidence intervals computed with the bootstrap method.

  -detection -IND (°)

Figure 4 .

 4 Figure 4.11: The interval of non-detection IND for each function of redirection and for each participant. For each point, the IND is computed using the 2 repetitions of each Physical Target and Redirection Amplitude amounting to 36 points fitted on a sigmoid logical function. No particular trend emerge although the variability among the population is high.

Figure 4 .

 4 Figure 4.13: Illustration of obstacle issue with visuo-haptic illusions.On the left, the user thinks there's an object to the right of the target whereas the actual object is on the left of their real hand due to the illusion as shown on the right. This mismatch can create issues where the participants hits an object when they should not.

Figure 5 . 1 :

 51 Figure 5.1: The task set-up for the experiment. (a) The setting with 6 cubes used in Test Only condition, (b) the real set-up with a single cube for the Virtual Training + Test condition and (c) the virtual replica of the setting used in the Test Only condition. (d) Participant's right hand wearing reflective markers.

Figure 5 . 2 :

 52 Figure 5.2: Target placement of a cube. In (a), the cube is in the right position but wrong orientation. In (b), the cube is in the wrong position but in the right orientation. In (c), the cube is both in the right position and orientation. The sequence moves ahead only if the cube is correctly placed like in (c).

Figure 5 . 3 :

 53 Figure 5.3: The procedure of the user study in two days: one day of test for the Test only and one day of training and one day of test for the Real Training + Test conditions and Virtual Training + Test.,

Figure 5 . 4 :

 54 Figure 5.4: A transfer performance curve showing relevant performance points to compute the amount of transfer.

Figure 5 . 5 :

 55 Figure 5.5: Two sinus signals with different phase realigned with the DTW algorithm[Petitjean, 2011]. The sequences are aligned using an optimisation function minimizing the distance between them. The top curve (blue) is offset forward compared to the bottom one (red). The algorithm matched the first few points of the top curve (resp. the last few points of the bottom curve) with the first point of the bottom curve (resp. the last point of the top curve) so that all the other points are connected with the corresponding points on the other curve, essentially putting back in phase the two signals.

  as described in chapter 3 to analyze it:where T is the performance (e.g. time in seconds), P is the trial number starting from 1. a, b and c are three parameters representing respectively the amount to be learned, the rate of learning, the asymptotic performance (i.e. ultimate performance). See chapter 3 for more details.We first fitted the learning curve on the evolution of Time and obtained a = 6.64, b = 0.51, c = 4.17 for Virtual Training + Test and a = 4.77, b = 0.49, c = 3.84 for Real Training + Test. The Mean Square Error (virtual: M SE = 0.22, real: M SE = 0.01) and R-Squared (virtual: R 2 = 0.94, real: R 2 = 0.92) confirm that the PLP reflects well users' behavior during the training phase.

Figure 5 . 6 :

 56 Figure 5.6: Performance during the test phase. Left: Time performance; Right: Variability. Areas / bars shows the 95% confidence intervals.

Figure 5 . 7 :

 57 Figure 5.7: Intramodal learning improvement during the virtual training and real training phase over Time (left) and DTW distance (right) performance. The areas show the 95% confidence intervals. All curves are fitted with the PLP so we can determine the speed of learning b and the ultimate performance c.

Figure 5 .

 5 Figure 5.8 illustrates the trajectories in the horizontal plane of all participants for the first/last block of the training/test phase for the two training conditions for the fifth manipulation. The variability reduces with practice during the training, but not during the test, which might

Figure 5 . 8 :

 58 Figure 5.8: Cubes trajectories for the first and last blocks of training and test for both training conditions. The black dot indicates the start position, the red/blue cross indicates the target position in respectively the Real / VR Training. The call-out, bottom left highlights the end of the movement and the kink observed in VR Training.

6. 2 . 1

 21 Expanding and Improving on User Studies 6.2.1.1 Getting More Data and Replicating Studies

Also in chapter 5 ,

 5 exploring a new training condition -training in VR without the hand redirection -and then testing in the real world could provide more information on the impact the visuo-haptic illusion has during training. It would resolve the confounding factor of VR + Visuohaptic illusion of the current user experiment. The results of this new condition may be closer to the Real Training + Test which would indicate the illusion is causing the slight drop in performance we observed or closer to the Virtual Training + Test condition indicating that training such a task in VR is slightly less effective than in the real world whether there is an illusion or not.

Figure 6 . 1 :

 61 Figure 6.1: The new design space of the function of redirection for the hand redirection illusion. The linear function (green tight dashes) and the second order polynomial function (green plain dashes) have a similar IND and are equally usable in virtual environments. The step function (red dashes) decreases the IND making easier to detect and less appropriate to use. The grey area remains unexplored and more specifically, the classes of functions where the detectability starts increasing is unknown.

  Figure A.1.

Figure A. 1 :

 1 Figure A.1: The results of an experimental study comparing the Predictive Performance Equation with an extension of ACT-R and Search of Associative Memory[Bregman, 1967]. Participants had to memorize consonant-vowel-consonant triplets under 3 conditions of practice: massed, intermediate and spaced (more distributed). On this dataset, the PPE seems to better predict the lower performance due to the massed distribution of practice.

  1.1 Haptic Retargeting, a visuo-haptic illusion . . . . . . . . . 2.1 Examples of a visuo-haptic illusion . . . . . . . . . . . . . 2.2 Two examples of off-the-shelf VR controllers . . . . . . . . 2.3 A virtual environment and its physical scene to provide haptic feedback . . . . . . . . . . . . . . . . . . . . . . . . 2.4 (non) detectection of the illusion . . . . . . . . . . . . . . 2.5 Redirect Touching technique . . . . . . . . . . . . . . . . . 2.6 Examples of Hand Redirection applications . . . . . . . . 2.7 Pseudo-haptic feedback devices . . . . . . . . . . . . . . . 2.8 Pseudo haptic feedback . . . . . . . . . . . . . . . . . . . 2.9 Examples of locomotion techniques . . . . . . . . . . . . . 2.10 Other examples of locomotion techniques . . . . . . . . . . 2.11 World redirection . . . . . . . . . . . . . . . . . . . . . . . 2.12 World redirection implementations . . . . . . . . . . . . . 2.13 World redirection paths . . . . . . . . . . . . . . . . . . . 2.14 World redirection applications . . . . . . . . . . . . . . . . 2.15 Illustration of the hand redirection technique trajectories with a linear redirection function . . . . . . . . . . . . . . 2.16 Virtual vs real POV . . . . . . . . . . . . . . . . . . . . . 2.17 Redirected walking methods . . . . . . . . . . . . . . . . . 2.18 An illustration of a psychometric function showing the probability to answer "Right" according to the stimulus provided. . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.19 Trajectories of users when exposed or not to the illusion . 2.20 Hand trajectories for 3 amplitude of redirection fitted with a cubic Bézier curv . . . . . . . . . . . . . . . . . . . . . . 3.1 Power Law of practice . . . . . . . . . . . . . . . . . . . . 3.2 The differences in QWERTY and AZERTY keyboard layouts. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Proactive transfer of learning . . . . . . . . . . . . . . . . 3.4 Retroactive transfer of learning . . . . . . . . . . . . . . . 3.5 The alternation model between a previous task P (method) in blue and a new task N in orange and the ART model . 3.6 Visuo-motor adaptation setups . . . . . . . . . . . . . . . 3.7 A comparison of a massed and distributed practice schedule. All massed practice session take place the same day as opposed to the distributed session over 3 days with longer rest periods in between. . . . . . . . . . . . . . . . . . . . 3.8 proactive transfer vs retroactive transfer experimental design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.9 transfer of learning measures . . . . . . . . . . . . . . . . 3.10 Visual guidance . . . . . . . . . . . . . . . . . . . . . . . . 3.11 Examples of haptic inerfaces . . . . . . . . . . . . . . . . .

  Transfer . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Transfer type . . . . . . . . . . . . . . . . . . . . . . . . . Transfer theories . . . . . . . . . . . . . . . . . . . . . . . 3.5.1 Identical Elements Theory . . . . . . . . . . . . . . 3.5.2 Transfer-Appropriate Processing Theory (TAPS) . 3.6 Protocol and Measure to Study Transfer . . . . . . . . . . 3.6.1 Protocol . . . . . . . . . . . . . . . . . . . . . . . . 3.6.2 Amount of Transfer . . . . . . . . . . . . . . . . . 3.6.3 Savings Time . . . . . . . . . . . . . . . . . . . . . 3.7 Skill Learning in Virtual Reality . . . . . . . . . . . . . . Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 54

	Visuo-haptic Illusions in VR
	3.8
	3.7.1 Factors of Transfer of Learning in VR . . . . . . .
	3.7.1.1 Guidance . . . . . . . . . . . . . . . . . .
	3.7.1.2 Multimodal Feedback . . . . . . . . . . .

3.1 Learning . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1.1 Skill -Performance -Learning . . . . . . . . . . . 3.1.2 Acquisition -Retention -Transfer . . . . . . . . . 3.1.2.1 Acquisition . . . . . . . . . . . . . . . . . 3.1.2.2 Retention . . . . . . . . . . . . . . . . . . 3.1.2.3 Transfer . . . . . . . . . . . . . . . . . . . 3.2 3.3.1 Proactive Transfer of Learning . . . . . . . . . . . 3.3.2 Retro-active Transfer of Learning . . . . . . . . . . 3.3.3 Visuo-motor Adaptation . . . . . . . . . . . . . . . 3.4 Factors Involved in Skill Transfer . . . . . . . . . . . . . . 3.4.1 Practice and Contextual Interference . . . . . . . . 3.4.2 Distribution of Practice . . . . . . . . . . . . . . . 3.4.3 Internal Factors . . . . . . . . . . . . . . . . . . . . 3.5

  Table 3.1. The Latin square design would look like: A, B, C, C, A, B, B, C, A. Finally, random design where each skill is presented randomly illustrated as follows: B, A,A, C, B, C, C, A, B. The latter design is more challenging to the learner because of contextual interference. Contextual interference is how the arrangement of trials can influence the performance during practice). In experimental protocols evaluating a task with another, the Latin square or random design are preferred to avoid possible interactions between conditions. In these examples, designs resembling random practice often results in better learning despite a lower performance during practice[START_REF] Schmidt | Motor control and learning: A behavioral emphasis, 5th ed. Motor control and learning: A behavioral emphasis[END_REF].

Table 3 .

 3 1: An example of a Latin square. It is constructed by shifting the first row to the right and wrapping the last element to the left.Figure 3.7: A comparison of a massed and distributed practice schedule.

		Day 1	Day 2	Day 3
	Distributed Practice	Session 1	Session 2	Session 3
	Massed Practice	Session 1 Session 2 Session 3

  Hand Trajectory analysis . . . . . . . . . . . . . . . . . . 75 4.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . 77 4.6.1 Findings . . . . . . . . . . . . . . . . . . . . . . . . 77 4.6.2 Implications for Design . . . . . . . . . . . . . . . . 78 4.7 Limits and Future Work . . . . . . . . . . . . . . . . . . . 79

	4.5
	Detection of Hand
	Redirection Illusion
	Chapter Contents

  . Twenty participants are split into two groups of ten assigned to the Real Training + Test condition (5 male, 5 female, aged 29) and Virtual Training + Test condition (6 male, 4 female, aged 27.7). The beginning of the training phase of the Real Training + Test condition is identical to the Test only condition. Hence, we did not assign participants to this group and used the data from the first five blocks of the Real Training + Test condition for the Test only condition. This allowed us to have less participants overall while using the same experiment design. We expect participants in the Real Training + Test condition to perform best. The condition Real Training + Test provides an upper bound in terms of performance as users perform the training and the test in the same setting. So, our objective is to study how far is the condition Virtual Training + Test from this upper bound.

  5) for Virtual Training + Test and 82.4% (std=14.2) for Real Training + Test. ANOVA does not reveal an effect of Training (Virtual vs. Real) on the amount of transfer (F 1,19 = 0.25, p = 0.62). As expected, the amount of transfer is higher for the Real Training + Test but it is nonetheless very high for the Virtual Training + Test condition. Figure 5.6-right shows the performance for Variability for all three conditions during the test phase. The variability during the test phase for Virtual Training + Test is 1.38 and for Real Training + Test is 1.49. ANOVA reveals a significant difference between Virtual Training + Test and Test only (F 1,19 = 4.92, p = 0.027) and no significant difference between the Real Training + Test and the other conditions (Virtual Training + Test: F 1,19 = 0.11, p = 0.74, Test Only: F 1,19 =

  6.1 Progress on the Research Problems . . . . . . . . . . . . . 101 6.1.1 Hand Redirection Design . . . . . . . . . . . . . . 101 6.1.2 Training with Hand Redirection . . . . . . . . . . . 102 6.2 Short Term Future Work . . . . . . . . . . . . . . . . . . . 102 6.2.1 Expanding and Improving on User Studies . . . . . 102 6.2.1.1 Getting More Data and Replicating Studies102 6.2.1.2 Considering Additional Conditions . . . . 103 6.2.2 Exploring a Larger Design Space . . . . . . . . . . 103 6.2.2.1 Other Second-order Redirection Functions 103 6.2.2.2 Step Redirection Function . . . . . . . . . 103 6.2.2.3 Relative -Absolute Redirection . . . . . 104 6.3 Long Term Future Work -Modelling Transfer of Learning 105 6.4 Final word . . . . . . . . . . . . . . . . . . . . . . . . . . . 105

  Summary of the 2AFC study results on the Interval of Non-Detection IND and Point of Subjective Equality PSE for redirection function L. Results of

		•	-0.89 •
	L=-2 [Geslain et al., 2022b]	10.87 •	-0.18 •
	L=-1 [Geslain et al., 2022b]	9.97 •	1.08 •
	L= 0 [Geslain et al., 2022b]	9.89 •	-0.23 •
	L= 0 [Zenner and Kruger, 2019]	8.19 •	-0.28 •
	L= 1 [Geslain et al., 2022b]	10.26 •	1.38 •
	L= 2 [Geslain et al., 2022b]	9.37 •	-0.13 •
	L= 3 [Geslain et al., 2022b]	10.51 •	-0.41

•

Table 6.1:

This parameter is added to the lag to ensure the natural logarithm of their sum remains positive.

Remerciements

Chapter 5

Transfer of Learning and Visuo-haptic Illusions Chapter Contents

Appendix A

Predictive Performance Equation

The Predictive Performance Equation (PPE) is a computational model that predicts performance based on prior performance (retention) like the PLP and the distribution of practice that influences the amount of performance decay. The effects of practice and decay are multiplicative on activation M n :

where P LP is the PLP, T n is the elapsed time since the last practice session, d is the decay rate. n is the number of the current practice.

The temporal decay T n is measured as the weighted sum of the elapsed time since each practice session.

This means all previous practice sessions are considered but they are weighted down as the elapsed time grows:

The decay rate d expresses the the idea that distributed practice produces better, more lasting learning: .4)