Lawrence Makin

Gareth Barnaby
email: gareth.barnaby@gmail.com

Anne Roudaut
email: roudauta@gmail.com

L Watkin

Tactile and Kinesthetic Feedbacks Improve Distance Perception in Virtual Reality Les retours tactile et kinesthésique améliorent la perception de distance en réalité virtuelle

Keywords: CCS CONCEPTS, Hardware~Haptic devices Haptic, Force Feedback, Tactile Feedback, Depth Perception, Virtual Reality RÉSUMÉ

Research spanning psychology, neuroscience and HCI found that depth perception distortion is a common problem in virtual reality. This distortion results in depth compression, where users perceive objects closer than their intended distance. Studies suggested that cues, such as audio and haptic, help to solve this issue. We focus on haptic feedback and investigate how force feedback compares to tactile feedback within peripersonal space in reducing depth perception distortion. Our study (N=12) compares the use of haptic force feedback, vibration haptic feedback, a combination of both or no feedback. Our results show that both vibration and force feedback improve depth perception distortion over no feedback (8.3 times better distance estimation than with no haptic feedback vs. 1.4 to 1.5 times better with either vibration or force feedback on their own). Participants also subjectively preferred using force feedback, or a combination of force and vibration feedback, over no feedback.

INTRODUCTION

Inaccurate depth perception in virtual reality, where users incorrectly perceive the depth of virtual objects, is a welldocumented problem and ongoing concern [START_REF] Finnegan | An approach to reducing distance compression in audiovisual virtual environments[END_REF]. This distortion is intrinsically linked to the fact that human uses binocular disparities to extract depth information from twodimensional retinal images in stereopsis. And it tends to result in depth compression, i.e. users perceive objects closer than their intended distance and thus affecting virtual reality hardware and content creators alike, particularly for interactive applications such as gaming or training.

Research spanning over 20 years highlights the need for a multimodal approach in solving the problem of depth perception distortion. The vast majority of research appears to revolve around visual cues, such as increasing visual realism [START_REF] Ries | Presence, rather than prior exposure, is the more strongly indicated factor in the accurate perception of egocentric distances in real world co-located immersive virtual environments[END_REF], removing visual clutter [START_REF] Kruijff | Perceptual issues in augmented reality revisited[END_REF], simplifying virtual objects [START_REF] Rolland | Towards quantifying depth and size perception in virtual environments[END_REF] or using the presence of avatars [START_REF] Renner | The perception of egocentric distances in virtual environments-a review[END_REF]. Yet few studies tackle it through haptic feedback.

Only one series of studies (N=4) utilized force feedback (using a string-based device) [START_REF] Bouguila | Effect of coupling haptics and stereopsis on depth perception in virtual environment[END_REF]. This study revealed that it was significant in improving depth perception distortion.

However, as mentioned by the authors, the investigation lacks high level integration between haptics and stereopsis. It is probably due to the fact that the study was conducted 20 years ago with older 3D display technology. The stereoscopic image was displayed on a 120-inch large screen and participants viewed it by wearing liquid-crystalshuttered glasses. They also used simplistic visuals (random dot stereograms). Research of this type have not been replicated with more participants, and with more actual higher-resolution stereoscopic technologies such as virtual reality headsets. This motivates our paper.

To test the hypothesis that haptic feedback improves depth perception in virtual reality, we conducted a study (N=12) in which we compared force feedback, vibration feedback, and a combination of the two, against no haptic feedback. In particular we looked at depth distortion in the close peripersonal range (within 40cm) which corresponds to the workspace offered by most force feedback desktop devices.

RELATED WORK

We first discuss work explaining the depth perception distortion. We then go through several general solutions that have been proposed, in particular the ones relating to multisensory integration.

The problem of depth perception distortion

Prior research shows that a problem [START_REF] Kelly | Perceived space in the HTC vive[END_REF] exists where humans inaccurately perceive the depth of objects in virtual environments [START_REF] Finnegan | An approach to reducing distance compression in audiovisual virtual environments[END_REF]. The reasons are summarized by Lawson et al. [START_REF] Lawson | Multimodal Sensory Illusions for Improving Spatial Awareness in Virtual Environments[END_REF], which in turn are drawn from Renner et al.'s research [START_REF] Renner | The perception of egocentric distances in virtual environments-a review[END_REF]. In essence, the problem of depth perception distortion is highly complex, and has yet to be fully resolved. However, strides have been made through improved technology to rectify depth perception distortion. It is generally understood that distance is compressed in virtual environments [START_REF] Finnegan | An approach to reducing distance compression in audiovisual virtual environments[END_REF], [START_REF] Kelly | Perceived space in the HTC vive[END_REF], [START_REF] Siegel | Walking through a virtual environment improves perceived size within and beyond the walked space[END_REF], [START_REF] Ries | Presence, rather than prior exposure, is the more strongly indicated factor in the accurate perception of egocentric distances in real world co-located immersive virtual environments[END_REF], meaning users tend to perceive virtual objects in relation to themselves (egocentric space) as closer than their actual distance. Similarly, Rolland et al. [START_REF] Rolland | Towards quantifying depth and size perception in virtual environments[END_REF] found that using simple virtual objects such as a cube or a cylinder led to expanded perceptual distortion rather than compressed. It was also found that, in peripersonal space, object distances are more often over-estimated [START_REF] Armbrüster | Depth perception in virtual reality: distance estimations in peri-and extrapersonal space[END_REF].

Solutions to depth perception distortion

A visual solution to depth perception distortion is to use highly-detailed virtual environments with high resolution rendering to near-replicate the real-world environment. In a study by Ries et al. [START_REF] Ries | Presence, rather than prior exposure, is the more strongly indicated factor in the accurate perception of egocentric distances in real world co-located immersive virtual environments[END_REF] three separate virtual environments were implemented depicting the same scene, but of different sizes. The authors found that users make accurate judgments of egocentric distance when cognitively 'immersed' or in their terms "accept the virtual environment as being equivalent to the real world". Kruijff et al. [START_REF] Kruijff | Perceptual issues in augmented reality revisited[END_REF] discovered that visual clutter can worsen the effects of depth distortion, which is in stark contrast to Ries et al.

Another method to reduce depth perception distortion involves physically walking. Kelly et al. found perception of object size and distance was improved due to the constant visual feedback in relation to changing location in space [START_REF] Siegel | Walking through a virtual environment improves perceived size within and beyond the walked space[END_REF]. In large virtual environments, and provided users have real world space in which to 'walk' virtually, this approach could be useful. However, this cannot be assumed, and could pose a danger in terms of real world objects obscuring the scene travelled in the virtual environment.

The advent of modern stereoscopic virtual reality headmounted displays (HMDs, such as the Oculus Rift and HTC Vive) has allowed for improved depth perception in virtual environments [START_REF] Kelly | Perceived space in the HTC vive[END_REF]. The improvement of depth perception distortion is potentially to the point that the HMDs provide "immersion and depth perception on a level that proved sufficiently realistic for healthy young adults to perform natural reaching movements" [START_REF] Gerig | Missing depth cues in virtual reality limit performance and quality of three dimensional reaching movements[END_REF]. Yet Lin et al. suggest that stereoscopic depth cues could be part of the problem of depth perception distortion [START_REF] Lin | Egocentric distance perception in real and HMD-based virtual environments: the effect of limited scanning method[END_REF]. These conflicting views could mean that depth perception distortion is harder to solve through HMDs, and thus visual stimuli, alone. Grossman et al [START_REF] Grossman | An evaluation of depth perception on volumetric displays[END_REF] also demonstrate that depth perception is improved when using a volumetric display compared to 3D displays. This result also potentially translate to levitated displays, e.g. [START_REF] Omirou | Floating charts: Data plotting using free-floating acoustically levitated representations[END_REF] as users can see real objects.

Solutions using multisensory feedback

Further research has found that the use of combined, or multimodal, sensory information (visual, aural, tactile) improves virtual environment presence and immersion [START_REF] Rosa | The Influence of visual cues on passive tactile sensations in a multimodal immersive virtual environment[END_REF], [START_REF] Jones | The effects of virtual reality, augmented reality, and motion parallax on egocentric depth perception[END_REF] and has a positive effect on depth perception [START_REF] Lawson | Multimodal Sensory Illusions for Improving Spatial Awareness in Virtual Environments[END_REF], [START_REF] Gerig | Missing depth cues in virtual reality limit performance and quality of three dimensional reaching movements[END_REF]. Hecht and Reiner suggest that audio and haptic stimuli are dominated by visual stimuli when only one or the other is coupled with the visual stimuli, but not both [START_REF] Hecht | Sensory dominance in combinations of audio, visual and haptic stimuli[END_REF] and are more likely to go unnoticed. They also note that when all three modalities are present together, any individual modality can dominate depending on its intensity compared to the other modalities. According to Lawson et al. there is also the possibility that multiple stimuli could conflict and thus reduce spatial awareness [START_REF] Lawson | Multimodal Sensory Illusions for Improving Spatial Awareness in Virtual Environments[END_REF]. This claim is further backed up by a study in the field of neuroscience, in which it was found that visual stimuli could be suppressed by accurate haptic stimuli [START_REF] Lunghi | Touch disambiguates rivalrous perception at early stages of visual analysis[END_REF].

These could be seen as a conflicting standpoint compared to that of Cooper et al. [START_REF] Cooper | The effects of substitute multisensory feedback on task performance and the sense of presence in a virtual reality environment[END_REF], who claims that auditory and tactile cues enhance the sense of immersion in virtual reality. However, it is worth pointing out that Hecht and Reiner's methodology implemented a non-virtual environment setup with at least decade-old technology, which could be behind the discrepancy. Conversely, Rosa et al. conclude that visual cues can have an effect on how users perceive vibrotactile feedback, albeit not necessarily accurately [START_REF] Rosa | The Influence of visual cues on passive tactile sensations in a multimodal immersive virtual environment[END_REF]. It is possible that audio cues can solve the problem of depth perception distortion through off-setting their spatial location in relation to the appropriate visual cue(s) [START_REF] Finnegan | An approach to reducing distance compression in audiovisual virtual environments[END_REF]. In other words, the audio cues are mapped differently to what would be expected in the real world. But with such conflicting views on the dominance of the various stimuli versus auditory and tactile stimuli improving virtual environment experience in tandem, it is difficult to put haptic stimuli in perspective.

Cooper et al. also state that auditory and tactile cues are useful when participants need to reach and grasp objects during the virtual reality interaction [START_REF] Jones | The effects of virtual reality, augmented reality, and motion parallax on egocentric depth perception[END_REF]. On the basis of this, they designed a virtual environment in which participants reach for and touch a virtual cube using a Phantom haptic device combined with a vibration motor.

Haptics, or tactile interaction, as a modality in virtual environments is under-researched in comparison to others [START_REF] Lawson | Multimodal Sensory Illusions for Improving Spatial Awareness in Virtual Environments[END_REF], [START_REF] Cooper | The effects of substitute multisensory feedback on task performance and the sense of presence in a virtual reality environment[END_REF], [START_REF] Azmandian | Haptic retargeting: Dynamic repurposing of passive haptics for enhanced virtual reality experiences[END_REF]. Azmandian et al. note that "studies have shown that using haptics can lead to significantly increased presence and spatial knowledge training transfer" [START_REF] Azmandian | Haptic retargeting: Dynamic repurposing of passive haptics for enhanced virtual reality experiences[END_REF]. Research since at least Bougila et al. [START_REF] Bouguila | Effect of coupling haptics and stereopsis on depth perception in virtual environment[END_REF] provides evidence for haptic feedback as a solution to depth distortion, but since then, very little research has been done in this area. Various forms of haptic feedback, such as passive, vibration, and force devices, have also been created since then.

EXPERIMENTAL DESIGN

The goal of this experiment is to investigate if haptic feedback enhances depth perception distortion, i.e. reduces the error made in estimating distance. Our secondary goal was to quantify the effects of different haptic conditions on depth perception distortion. We compared four conditions: tactile, force feedback, a combination of the two, and a nonhaptic baseline. We now describe the choice behind the experimental design and give details about our study implementation before discussing the results.

Apparatus

Figure 1 shows the haptic apparatus. We used a Phantom Premium haptic device to provide force feedback at a fingertip. We added a 3D printed finger rig to allow the participants to use their finger rather than the classical pen end-effector provided (Figure 2) in order to better represent direct manipulation tasks. The Phantom can also provide vibration feedback via its internal motors, although it does not focus on the fingertip of the user but instead transfers the vibration down the shaft of the haptic arm. Thus, we used a small disc vibration motor mounted at the end of the phantom end effector (Figure 3) to produce better vibrations. The vibrations were controlled by an Arduino Mega 2560 along with a Grove System vibration motor. We used an HTC Vive and a Leap Motion for tracking user's hands. We used Unity and a 3rd-party plugin from the Unity Asset Store allowing the Phantom to communicate its movements to Unity through its OpenHaptics Toolkit (Unity 5 Haptic Plugin for Geomagic OpenHaptics).

Virtual environment

Participants touched a virtual cube (Figure 4). We chose this shape following Rolland et al.'s assertion that simple shapes, such as cubes and cylinders, in virtual environments tend to be perceived as further away [START_REF] Rolland | Towards quantifying depth and size perception in virtual environments[END_REF] than the object's actual distance. We used a hand model from the Unity Asset Store to portray a sense of presence to the participants, rather than use the default virtual stylus provided with the OpenHaptics Toolkit Unity port (Figure 4). In Renner et al.'s review of egocentric distance perception in virtual reality [START_REF] Renner | The perception of egocentric distances in virtual environments-a review[END_REF], there is evidence to suggest that such a presence of an avatar aids in reducing depth distortion.

Task

Our task was based on Bouguila et al. [START_REF] Bouguila | Effect of coupling haptics and stereopsis on depth perception in virtual environment[END_REF] reaching experiment. Bouguila et al. designed a reaching experiment utilizing the big SPIDAR [START_REF] Sato | Development of string-based force display: SPIDAR[END_REF] haptic device, where a seated participant would align their hand with where they perceived a virtual square rendered in front of them in a 3D display. Participants would state when they thought they had reached the square, and the distance travelled by their hand would be recorded at this point. The square was moved between tasks, and participants were asked to close their eyes briefly between tasks to prevent training.

We developed a series of eight reaching zones (Figure 5). Each dot represents a reaching task. The black dots represent the central point location of the bottom edge of the visible virtual cube's front face. The red dots represent the same, except for the cube being invisible in this case, with the corresponding visible cube still in place behind the red dot. The dots to either side of the central ones are offset by 45 degrees in the forward direction, to prevent training.

The distance between the red and black dots in the forward position is 10 cm. This is the distance we arbitrarily chose to test for depth perception against the visible cube at any given position. We choose 10 cm because we wanted to have a large enough distance but not too large to fit within the dimension of our apparatus. For each reaching task, we apply one of four feedback types: no haptic feedback (as a control), vibration feedback, force feedback, and a combination of vibration and force feedback. If depth perception distortion exists at 10 cm, participants should at least occasionally feel as though they are touching the visible cube, even though they are 10 cm in front of it when a haptic force is applied.

Procedure

The participants were sitting at a table. They were asked to move the hand they had placed in the rig as close to themselves as the Phantom would allow. We manually guided their arm to ensure the Phantom arm was as straight as possible for the task. They practiced with the system. When a trial started, the participants first had to close their eyes. The virtual cube would then appear in a predefined pseudo-random position, at which time we asked the participant to open their eyes and reach towards the center of the cube's front face. We specifically asked this as if a participant reached around the edges of the cube, they would gain more perspective of whether or not they had touched its surface. If a force was applied, either on contact with the cube, or 10 cm before it (unbeknownst to the participant), we asked the participants if they thought they had touched the cube. We then asked them to hold their hand steady so that we could submit their distance data and answer via another button in the Unity scene. We then asked the participant to close their eyes, at which point we repeated the process, until all 32 tasks had been completed. The haptic device arm was moved back to its original position after each trial.

Counterbalancing

We randomized the order of positions for each haptic feedback, so that the order between the sets was not the same, to prevent training. We used a Latin square to counterbalance the presentation of the feedback (no haptic

Dependent variables

Each participant was asked to stop moving the Phantom arm at the point they felt they were touching the virtual cube (if no haptic force was applied) or they felt a haptic force being applied. They were then asked if they thought they had reached the virtual cube or if they were experiencing the haptic force too early. The experimenter clicked on a button within the Unity scene to submit the participant's answer, and at that moment the distance travelled by the Phantom arm, the distance of the virtual cube from the participant, and the current haptic feedback type, were logged. The raw data per participant included the distance they reached per task, the distance of the actual front face of the virtual cube, and whether or not the participant thought they'd success-fully reached and touched the front face of the cube.

We also designed a post-study questionnaire with answers on 7-point Likert scales. Each question asked the participant their opinion of a specific aspect of the interactive part of the study with answers ranging from "Strongly Disagree" to "Strongly Agree". The purpose was to determine whether there is any subjective user preference of the four methods of haptic feedback provided during the reaching tasks.

Calibration

To ensure the virtual environment was calibrated to be as accurate as possible in terms of scale and rotation, we performed calibration of the Phantom arm and the Vive headset for each participant. Calibration of the Phantom is relatively straightforward as it utilizes its own software. For the calibration of the Leap Motion we asked each participant to grasp the edge of the real desk in front of them and see if the Leap Motion hand tracking models lined up with the virtual desk. It could take a few attempts to make sure the participants were seated close enough and central to the midpoint of the real desk to reduce inaccuracy of their position in relation to the virtual desk.

Note that the HMD, Leap Motion, and Phantom have their own coordinate frames. We manually aligned these and hardcoded the offsets in software. Note that we found that the leap motion tracking was inconsistent between users and that an offset was detected. We had to realign the Leap Motion coordinate frame with the HMD and the Phantom for each user. We also observed that the phantom can impact the Leap Motion tracking in some capacity. However, it only affects the finger attached to the Phantom and not the others finger and the hand which can be used to ensure a good tracking. We also did some informal test and found that the Phantom interferes with tracking when curling the finger. In this case the Leap Motion detects it as being outstretched or partially curled. But this did not cause issues in our study as the finger was kept outstretched during the entire experiment.

Participants 12 participants (2 female) (1 left-handed, 1 ambidextrous) (from 20 to 35 years old) were recruited in our institution and received no intensive for their participation. They had never used an actuated arm nor a force feedback device before but were familiar with virtual reality technologies.

RESULTS AND DISCUSSION

Of the 384 total tasks (32 tasks per participant), we filtered out the mistakes and excluded them from the analysis. The mistakes occurred when participants failed to keep their reaching hand steady at the point of contact. In total, the mistakes rate was 10, or approximately 2.6% of all tasks. in meters between the virtual cube and where the participants thought the cube was while trying to touch it.

Quantitative results

Figure 6 shows the overall data gathered for each type of feedback. We used non-parametric tests since a Shapiro-Wilk test did not validate the assumption of normality. We ran a Friedman test to determine if there were differences in distances recorded when participants felt they had touched a virtual cube when a haptic force was applied (vibration, force feedback, and a combination of the two) versus no force applied. Pairwise comparisons were performed with a Bonferroni correction for multiple comparisons. There was a significant difference in distances measured between vibration feedback and all other feedback types, χ2(3) = 34.496, p < .0005. Post hoc analysis revealed statistically significant differences in distances recorded with vibration feedback applied (Mdn = -0.012) and no haptic feedback (Mdn = -0.0141) (p < .0005), force feedback (Mdn = 0.0) (p < .0005), and combined force and vibration feedback (Mdn = 0.0) (p < .0005) respectively. No significance was found between any comparison of no haptic feedback, force feedback, and combined feedback. We ran a second Friedman test on the cases where participants felt they hadn't touched the virtual cube, to see if there is any difference in distances over the different feedback types. Pairwise comparisons were performed with a Bonferroni correction for multiple comparisons. There was a significant difference in distances measured between the fake point of contact of the cube when vibration is applied too soon, and the fake point of contact of both force feed-back and the combined feedback, χ2(2) = 41.738, p < .0005. Post hoc analysis revealed statistically significant differences in distances recorded with vibration feedback applied too early (Mdn = -0.1231) and force feedback (Mdn = -.1) (p < .0005), and combined feedback (Mdn = -.1) (p < .0005) respectively. No significant difference was found between force feedback and combined feedback distances when the forces were applied too soon.

Summary of the quantitative results

Our results confirm that haptic force feedback does significantly improve depth perception distortion when compared to no haptic feedback. We also found that a combination of vibration feedback and force feedback does significantly improve depth perception distortion compared to no haptic feedback. Overall we found that with force feedback or vibration only, participants estimated distances around 1.4 to 1.5 times better than with no haptic. But with both feedbacks combined they were able to estimate distance 8.3 time better than with no haptic.

Note that in both sets of reaching tasks involving force feedback, the median distance travelled tended towards the exact point of contact of the surface of the virtual cube (Mdn = .0) or the 'fake' cube (Mdn = .1), depending on where the force was applied. In contrast, the medians for both vibration feedback (Mdn = -.0120) and no haptic feedback (Mdn = -.0141) tended towards -1.5 cm, in other words beyond the front surface of the virtual cube.

In the data where participants felt they had not reached the virtual cube when a force was applied, we found a significant difference in accuracy of vibration feedback compared to both force feedback and the combination feedback. This is not unexpected, as in both sets of reaching tasks involving force feedback, participants could not move the Phantom arm beyond the 'fake' cube surface. Conversely, nothing was in place to stop participants from moving through the same 'fake' cube with only the vibration feedback active, allowing for a discrepancy in distance travelled.

Subjective results

Figure 7 shows the results of the post-questionnaire. The three last questions (displayed on the right) were general questions: "Did you feel that force feedback felt more realistic compared to vibration feedback? Did you enjoy your virtual reality and haptic feedback experience? Do you think virtual reality could benefit from haptic feedback?". We asked these questions in order to ensure that none of our participants had issues with the overall set-up and to have an overall opinion of the feeling about force feedback and vibration feedback. Our results confirm that our participants had a good experience and had a positive feeling about the haptic feedback used in the study. We then looked at the four first questions which are comparable: "Did you feel that (no haptic, vibration, force feedback, both) feedback made it easy to touch the cube (4 questions)?". Our questionnaire produces non-parametric ordinal data, so we looked at them using a non-parametric Wilcoxon signed-rank test. However, testing the difference of values between question variables yielded an asymmetrical distribution. We thus began by using a Spearman's Rank Order Correlation, but scatter graph plots showed that the variable relationships were not monotonic. Thus, we used a Sign Test. We compared the first four questions in pairs as seen in the following.

Vibration feedback vs. no haptic feedback

Of the 12 participants, 9 felt vibration feedback made touching the cube easier, 2 felt no haptic feedback was easier, and 1 felt there was no difference between the two. Overall, participants felt that vibration feedback made touching the cube easier (Mdn = 5.5) compared to no haptic feedback (Mdn = 2.0), a statistically insignificant increase in the median differences of 3.5, p = .065.

Force feedback vs. no haptic feedback

9 participants felt force feedback made touching the cube easier, 1 felt no haptic feedback was easier, and 2 felt no difference between the two. Participants felt that force feedback made touching the cube easier (Mdn = 7.0) 37 compared to no haptic feedback (Mdn = 2.0), a statistically significant increase in the median differences of 5, p = .021.

Force feedback vs. vibration feedback

6 participants felt force feedback made touching the cube easier, 1 felt vibration feedback was easier, and 5 felt there was no difference between the two. Overall, participants felt that force feedback made touching the cube easier (Mdn = 7.0) compared to vibration feedback (Mdn = 5.5), a statistically insignificant increase in the median differences of 1.5, p = 0.125.

Combined feedback vs. no haptic feedback

9 participants felt the combination of force and vibration feedback made touching the cube easier, 2 felt no feedback was easier, and 1 felt there was no difference between the two. Overall, participants felt that the combined feedback made touching the cube easier (Mdn = 6.5) compared to none (Mdn = 2.0), a statistically significant increase in the median differences of 4.5, p = .021.

Combined feedback vs. vibration feedback

8 participants felt the combination of force and vibration feedback made touching the cube easier, 0 felt vibration feedback was easier, and 4 felt there was no difference between the two. Overall, participants felt that the combined feedback made touching the cube easier (Mdn = 6.5) compared to vibration (Mdn = 5.5), a statistically significant increase in the median differences of 1, p = .008.

Combined feedback vs. force feedback

4 participants felt the combination of force and vibration feedback made touching the cube easier, 4 that force feedback was easier, and 4 felt no difference between the two. Overall, participants felt that the combined feedback did not make touching the cube easier (Mdn = 6.5) compared to force (Mdn = 7), a statistically insignificant decrease in the median differences of 0.5, p = 1.000.

Summary of qualitative results

Vibration feedback was not seen to be significantly better than no haptic feedback in the ease of touching a virtual cube, while force feedback did show significance (p = .021) over no haptic feedback, as did the combined haptic feedback (p = .021). However, there was no statistical significance in force vs. vibration, while there was significance in the combined haptic feedback vs. vibration. Yet there was no significance between the combined haptic feedback and force feedback whatsoever (p = 1.000).

This outcome seems to show that there is a subtle difference in preference of haptic feedback types, with a general trend towards either force feedback or a combination of vibration of force feedback. We can draw the conclusion that vibration is not preferred over no haptic feedback, and that force feedback is preferred over no feedback. There is no significant difference in preference between force feedback and the combined force and haptic feedback. Participants enjoyed the haptic experience, generally felt it was beneficial to virtual reality, and felt force feedback is more realistic compared to vibration feedback. Thus, in our opinion force feedback is the common factor in haptic preference to touch virtual objects, and from a purely interactive enjoyment aspect should be the method to use going forward.

DISCUSSION

Haptic feedback does improve depth perception in virtual reality, but we did not find that force feedback does significantly improve it compared to vibration feedback. The fact that there was still depth distortion expansion (albeit only around 1 to 1.5 cm) for both vibration feedback and no haptic feedback tasks, suggests that both Naceri et. al [START_REF] Naceri | Depth perception within peripersonal space using head-mounted display[END_REF] and Rolland et al. [START_REF] Rolland | Towards quantifying depth and size perception in virtual environments[END_REF] are correct in their assessment of depth distortion expansion in short distances, and simple objects such as cubes, respectively. Furthermore, it could be argued that using the hand model avatar could have reduced depth perception distortion in the experiment, as suggested by Renner et al. [START_REF] Renner | The perception of egocentric distances in virtual environments-a review[END_REF].

There was a significant difference in force feedback compared to vibration and the combination of feedback when the force was applied at 10 cm before the surface of the cube. This may indicate that vibration introduces a certain level of inaccuracy in depth perception, as in comparison the distances travelled under force feedback conditions was very accurate. This is unsurprising considering the arm stops almost completely when touching a virtual object with force feedback. Our qualitative analysis indicated a preference for force feedback or combined force over no feedback in terms of how easy the haptic feedback made touching a virtual cube.

There was also a strong opinion in favor of using haptic in virtual reality experiences.

Implications

Our results have implication for the design of haptic system. Currently most haptic systems are designed with tactile (vibration) or force feedback. New force feedback systems are emerging every year (e.g. the Mantis force feedback system [START_REF] Barnaby | Mantis: A Scalable, Lightweight and Accessible Architecture to Build Multiform Force Feedback Systems[END_REF]) and our study suggests that there is a benefit in combining force feedback with some form of tactile feedback.

In addition, haptic force feedback devices are particularly used in applications requiring high-precision haptic rendering such as teleoperation, medical training and of course gaming in immersive environments, in order to offer users, the ability to touch and sense in the digital world.

Being able to enhance the perception of distance and thus increase the accuracy of such task is thus, we believe, a substantial insight for designers and researchers working within those areas.

Limitations

One limitation of our study is the short range of the Phantom (40 cm with the 3D-printed finger rig). This may be one of the reasons why depth distortion when reaching with no haptic feedback was small (around 2 cm away from the surface on average). This short range compressed the usable depth into about 20 cm, as there needed to be free space both before and behind the virtual cube and the 'fake' virtual cube at any reaching task position. Without the free space, participants would not be able to move the Phantom arm forward during the closer reaching tasks. We think a haptic device such as the LHIfAM [5] would be better, as its range of forward depth only starts around 30 cm but continues on to 1.5 m, which we believe would extrapolate the depth distortion effects found in the study.

One of the participants mentioned that the vibration feedback could be more realistic if it was briefer rather than continuous. We choose to produce the vibration in the same manner as the Phantom's force feedback, i.e. continuous as long as a collision is detected. The potential issue with this is that the user will feel a vibration at any point within the collider of the cube, which is the cube's entire volume. This could explain why the travelled distances during the vibration reaching tasks were the least accurate. Participants continued moving the virtual hand forward as no force was stopping them, yet vibration continued. It is possible that this made them believe they were still touching the surface of the cube rather than moving inside it. Further studies thus need to investigate different types of vibration patterns.

CONCLUSION

Our work investigates the effect of haptic feedback on the improvement of depth perception distortion in virtual reality. Our study using a virtual reality headset and four different types of haptic feedback demonstrates that haptic modality improves depth perception. With force feedback or vibration only, participants estimated distances around 1.4 to 1.5 times better than with no haptic; but with both feedbacks combined they were able to estimate distance 8.3 time better than with no haptic. We also found that participants subjectively preferred using force feedback, or a combination of force and vibration feedback, over no feedback. We think our work can benefit to virtual reality hardware and content creators when creating interactive applications that require precision in depth perception (e.g. gaming, surgery training).

Figure 1 .

 1 Figure 1. Overview of the apparatus.

Figure 2 .

 2 Figure 2. The 3D-printed finger rig with gimbal and finger clamp, used with the Phantom Premium haptic device.

Figure 3 .

 3 Figure 3. The vibration motor used at the fingertip.

Figure 4 .

 4 Figure 4. The virtual environment used in the user study: (left) a 50cm cube on top of a 140cm(W) X 72cm(H) X 80cm(D) desk created in Blender; (right) a virtual hand model posed in the general shape expected from participants' hands once in the Phantom's finger rig.

Figure 5 .

 5 Figure 5. The black dots represent the mid-point of the front face of the cube for each reaching task. The red dots are 'fake', invisible cube where forces are applied 10 cm in front of the visible cube.

Figure 6 .

 6 Figure 6. Descriptive statistics for the quantitative user study data, measuring the difference in distances recorded in meters between the virtual cube and where the participants thought the cube was while trying to touch it.

Figure 7 .

 7 Figure 7. Stacked bar graph representation of the questionnaire data.

IHM' 19 ,

 19 r a t i o n + f o r c e f o r c e v s . v i b r a t i o n h a p t i c e n j o y m e n t h a p t i c b e n e f i t s December 10-13, 2019, Grenoble, France L. Watkin, G. Barnaby, and A. Roudaut 2:6

  feedback, force feedback, vibration and force feedback, vibration feedback). As there are 4 feedback types and 8 dot positions, that means a total of 8 sets of 4 tasks, or 32 tasks per participant. The study lasted for ~30 minutes including breaks that could be taken between conditions.

	IHM'19, December 10-13, 2019, Grenoble, France	L. Watkin, G. Barnaby, and A. Roudaut
	2:4

ACKNOWLEDGMENTS

This work was supported by the Engineering and Physical Sciences Research Council (grant number EPSRC EP/P00s4342/1) and the Leverhulme Trust.


W V Bienvenut

P-A Charbit

J-P Scarpelli

T Meinnel

C Giglione

Keywords: 95% vs, 91, % Combined predictions: 88% vs, 75%

Collecting experimental data : the eNergiomeDB.

Data validation: based on data redundancy and manual validation

Datamining for transit peptide cleavage site prediction.

Hits % Hits % Hits % Hits %

Mitochondria transit peptide cleavage site

Chloroplast transit peptide cleavage site prediction

Transit Peptide length Prediction Results

Project overview

Protein N-terminal maturation is essential for protein activity, sub-cellular location and half-life… Huge interest to know the exact status and position of proteins' N-termini but…

Main issues:

Experimental data are not available for all proteins… Prediction tools are available but not always reliable…

Sample preparation and data processing [1-2]

	A. thaliana (cells and seedling)	Chloroplast enrichment	H. sapiens (cell lines)	Mitochondria enrichment
				EnCOUNTer
				xml export file
							N-terPred
						Train	Test		Train	Test
						dataset	dataset	dataset	dataset
				Total hits	371	-	160	-	371	-	160	-
				Top	True 315 85% 104 65% 204 55% 66 41%
				Pred *	False 56 15% 56 35% 167 45% 94 59%
				Extended	True 336 91% 115 72% 235 63% 86 54%
				Pred **	False 58 16%	58 36%	58 16%	58 36%

ChloroP

novel tools to improve the prediction of plastidic and mitochondrial mature N-termini,

Data reprocessing : Data collected from repositories

				e.g. PXD002069 & PXD002690 [3-5]
		Literature data
		(manual annotation)
			900 stromal mature N-term
	Plastidic localisation prediction	4500 at protein N-terminus (Pos 1-2) of which 176/117 Hs/At mitochondrial proteins	70% Train	30% Test
	Randomized dataset using 1400 proteins per training round	and 100 plastidic proteins
	(700 plastid/700 cytoplasmic candidates) 70% 30% Train Test	4000 downstream mature N-termini including: 280/400 Mitochondrial N-term (Hs /At) 2300 plastidic (At) N-term
		1700 N-termini used for prediction tool training :
		125/225 mitochondrial (H sapiens/A. thaliana)
		900 plastidic (A. thaliana)
		2300 N-term quantified for Acetylation yield :
		1230 at protein N-term (Pos 1&2)
		1114 downstream of the protein N-term (Pos >2)

eNergiomeDB protein screen: collected data (experimental data, prediction…) and validated mature N-termini.

eNergiomeDB main screen: Filtered protein list selection eNergiomeDB overview More than 10000 proteins/entries: 3000 H. sapiens, 6000 A. thaliana, 700 S. lycopersicum, 500 E. coli

More than 8500 distinct N-termini :

Data mining using k-nearest neighbors Network

combining few predictors results (TargetP, WolfPSort, Predotar)

Datamining using Position Weight Matrix Localization Prediction Results

N-terPred prediction screen:


William Fauriat
email: william.fauriat@centralesupelec.fr

Enrico Zio
email: enrico.zio@polimi.it

Estimation of the value of prognostic information for condition-based and predictive maintenance

Keywords: Condition-Based Maintenance, Predictive Maintenance, Value of Information, Prognostics, Maintenance optimization, Remaining Useful Life

For components subject to degradation, cost-efficient maintenance is necessary. Periodic or continuous collection of information, reducing uncertainty on the component's state of health, generally leads to a better-informed and, thus, more efficient maintenance. Processing condition monitoring data to estimate the current and future health states of the component, can prove valuable. In this paper, it is proposed to quantify the Value of Information (VoI) that may be obtained from state estimation and prediction procedures, with known precision, applied for conditionbased and predictive maintenance. VoI is computed numerically using gamma process paths and on the basis of the optimization of the parameters of different maintenance strategies.

Introduction

Proper maintenance is necessary to keep a component undergoing degradation in a functional state and, thus, limit the risk and costs associated to its failure. Maintenance operations, such as repairs, replacements or inspections, are carried out in view of optimizing life-cycle performance and minimizing costs in an uncertain environment. Collection of additional information on the component's State of Health (SoH), thereby reducing the associated uncertainty, allows performing better maintenance.

In this practical context of condition-informed decision-making for maintenance, it is worth considering seminal works on optimal decisionmaking under uncertainty proposed in the sixties by [START_REF] Raiffa | Applied statistical decision theory[END_REF], [START_REF] Degroot | Uncertainty, information, and sequential experiments[END_REF], [START_REF] Howard | Information value theory[END_REF]. The latter offer a framework wherein the value of a particular piece of information depends on its ability to 'guide our decision'. Formally, the metric of Value of Information (VoI) is defined as the difference in expected utility when a decision is made with and without the possession of additional information. For maintenance planning and optimization, such a metric may be used to 'evaluate the benefit of collecting additional information to reduce or eliminate uncertainty in a specific decision-making context', according to [START_REF] Pozzi | Assessing the value of information for long-term structural health monitoring[END_REF].

Maintenance optimization approaches developed over the years along with probabilistic models and methods, see e.g. early works of [START_REF] Barlow | Mathematical theory of reliability[END_REF], [START_REF] Abdel-Hameed | A gamma wear process[END_REF]. Exact resolution of the maintenance optimization problem is sometimes possible using renewal theory. Good reviews of the extensive literature are given by [START_REF] Dekker | Applications of maintenance optimization models: a review and analysis[END_REF], [START_REF] Wang | A survey of maintenance policies of deteriorating systems[END_REF], [START_REF] Frangopol | Probabilistic models for lifecycle performance of deteriorating structures: review and future directions[END_REF], [START_REF] Van Noortwijk | A survey of the application of gamma processes in maintenance[END_REF]. With the progress of computer power, maintenance optimization may also be carried out numerically, see e.g. [START_REF] Marseguerra | Condition-based maintenance optimization by means of genetic algorithms and monte carlo simulation[END_REF]. This is the approach used in this paper.

With the reduction in the cost of monitoring devices, an increasing amount of conditionmonitoring (CM) data is available. The latter may not directly represent information on the component's state and may have to be processed using adapted methods for state estimation or prediction. In the field of Prognostics and Health Management (PHM), extracting information on a component's state for the purpose of improving maintenance decisions is a growing concern, see good reviews of [START_REF] Jardine | A review on machinery diagnostics and prognostics implementing condition-based mainte-nance[END_REF], [START_REF] Heng | Rotating machinery prognostics: state of the art, challenges and opportunities[END_REF], [START_REF] Si | Remaining useful life estimationa review on the statistical data driven approaches[END_REF], or approaches of Bayesian filtering in [START_REF] Myotyri | Application of stochastic filtering for lifetime prediction[END_REF], [START_REF] Zio | Particle filtering prognostic estimation of the remaining useful life of nonlinear components[END_REF].

In this paper it is proposed to investigate, with numerical simulation and using concepts from the VoI framework, the effect of state estimation and prognostics information obtained from condition monitoring data, on condition-based and predictive maintenance policies. Similar considerations 1 have been studied by [START_REF] Huynh | Maintenance decision-making for systems operating under indirect condition monitoring: value of online information and impact of measurement uncertainty[END_REF], and the present paper will adopt the same approach, albeit using only numerical tools and a particular model for specifying the precision of state estimation and prognostics.

The paper is organized as follows. In section 2, a brief history of maintenance policies involving information collection are given. The different maintenance strategies that are considered and compared in this paper are, then, described in section 3. The proposed numerical experimentation and resolution of the maintenance planning problem with different strategies, for gamma degradation processes, is described in section 4. The effect of information in different decision settings is discussed in 5. Conclusions and future perspectives are detailed in section 6.

Review on maintenance strategies involving information collection

Early cases of inspection policies were considered by [START_REF] Thoft-Christensen | Optimal strategy for inspection and repair of structural systems[END_REF], [START_REF] Park | Optimal continuous-wear limit replacement under periodic inspections[END_REF], [START_REF] Mori | Maintaining reliability of concrete structures. ii: optimum inspection/repair[END_REF] and [START_REF] Van Noortwijk | Optimal inspection decisions for the block mats of the eastern-scheldt barrier[END_REF]. In these works, the objective is mostly to optimize the time interval between costly inspections, generally using renewal theory.

The formulation of the maintenance planning issue as a sequential decision problem, not involving resolution schemes based on renewal theory and using Markov Decision Process (MDP), was proposed by [START_REF] Madanat | Optimal infrastructure management decisions under uncertainty[END_REF] and latter extended by [START_REF] Ellis | Inspection, maintenance, and repair with partial observability[END_REF]. In these works, inspections are not necessarily perfect and may be affected by measurement uncertainty.

Interest in so-called Condition-Based Maintenance (CBM) policies, and associated formalization and resolution approaches, started growing, as they generally lead to improved maintenance outcomes, see e.g. [START_REF] Christer | A simple condition monitoring model for a direct monitoring process[END_REF], [START_REF] Wang | Towards a general condition based maintenance model for a stochastic dynamic system[END_REF], [START_REF] Grall | A condition-based maintenance policy for stochastically deteriorating systems[END_REF] or [START_REF] Dieulle | Sequential condition-based maintenance scheduling for a deteriorating system[END_REF]. In many cases, it is assumed that the current condition of the component is known perfectly through inspections or continuous monitoring. In particular, the choice of control-limit strategies, popular in CBM, is used in the present paper.

Considering limited resources and imperfect inspection outcomes, risk-based inspection planning was studied by [START_REF] Faber | Risk-based inspection: the framework[END_REF], [START_REF] Straub | Risk based inspection planning for structural systems[END_REF] and [START_REF] Kallen | Optimal maintenance decisions under imperfect inspection[END_REF].

The notion of Value of Information (VoI) has recently received increasing attention for inspection prioritization, sensor placement or complex policies for maintaining systems, especially in Structural Health Monitoring (SHM) where sensors or inspections can be expensive, see works of [START_REF] Pozzi | Assessing the value of information for long-term structural health monitoring[END_REF], [START_REF] Straub | Value of information analysis with structural reliability methods[END_REF], [START_REF] Zonta | Value of information: impact of monitoring on decision-making[END_REF], [START_REF] Konakli | Numerical investigations into the value of information in lifecycle analysis of structural systems[END_REF] and [START_REF] Memarzadeh | Value of information in sequential decision making: component inspection, permanent monitoring and system-level scheduling[END_REF].

On the other hand, developments of numerical schemes for the resolution of sequential decision problems using MDP and Partially Observable Markov Decision Process (POMDP), support the progress of more advanced maintenance policies, e.g. with decision structures that may be more dynamic, see e.g. [START_REF] Papakonstantinou | Planning structural inspection and maintenance policies via dynamic programming and markov processes. part i: theory[END_REF], [START_REF] Srinivasan | Value of condition monitoring in infrastructure maintenance[END_REF] and [START_REF] Memarzadeh | Value of information in sequential decision making: component inspection, permanent monitoring and system-level scheduling[END_REF]. For such policies, different information collection strategies can be compared on the basis of a VoI metric, notably for resource prioritization and sub-optimal and heuristic resolution schemes.

In this paper, it is proposed to consider VoI as a metric for evaluating the interest of processing CM data with state estimation and prognostics approaches, with different levels of precision. Said precision must be specified in advance.

Maintenance strategies and information processing

Maintenance policies

For a component undergoing continuous degradation, the elaboration of maintenance strategies generally consists in specifying a schedule or condition-based decision rules for repair, replacement or inspection actions. Preventive maintenance (PM) strategies only rely on timebased actions whereas condition-based maintenance (CBM) involves decisions rules dependent on the component's state, either known exactly or with uncertainty and periodically or non-periodically, see reviews e.g. in [START_REF] Dekker | Applications of maintenance optimization models: a review and analysis[END_REF], [START_REF] Wang | A survey of maintenance policies of deteriorating systems[END_REF] and van Noortwijk (2009).

Here, three strategies are considered, namely:

• Block Replacement (BR)

• Periodic Inspection and Replacement (PIR)

• Predicted Quantile Replacement (PQR)

Block replacement policy (BR) is purely timebased. The component is preventively replaced at cost c p at time T BR or correctively replaced, at cost c c > c p , when failed, whichever occurs first. Failure of the component is observed without need of inspection and component is immediately replaced upon failure. These hypotheses may not represent all conceivable situations in practice, but the purpose here is not to compare all possible situations.

Periodic inspection and replacement policy (PIR) involves inspections with a given cost c i . At each inspection date, with time-interval ∆T , the component is preventively replaced if its condition is above a specified threshold or control-limit M . The component is correctively replaced if it fails before the next inspection date.

A more detailed description of these policies can be found for example in [START_REF] Huynh | Maintenance decision-making for systems operating under indirect condition monitoring: value of online information and impact of measurement uncertainty[END_REF]. In this latter reference, more complex policies with dynamic decision rules, e.g. inspection dates that are not fixed in advance, are proposed. Here it is proposed to focus on studying the effect of additional information from state estimation and prognostics.

The efficiency of the policy, in the sense of lifecycle cost, is quantified by the average cost per unit time C ∞ . The latter is estimated through the use of renewal theory, which states that it may be calculated as the ratio of the average renewal cycle cost other the average renewal cycle length.

C ∞ = lim t→∞ C(t) t = E[C(S)] E[S] (1)
where S is the length of a given renewal cycle, which brings the system back to an 'as good as new' condition and C(S) is the cost associated to the cycle.

State-estimation-based and prognostics-based policy

Processing the data from inspections or continuous monitoring into an estimate of the current or future state of health (SoH) of the component may provide information allowing policy decisions that are better adapted to the component's degradation path.

It is proposed to consider a policy where the decision to proceed to a preventive replacement is piloted via a control-limit level M and via a quantile value α ∈ [0, 1[. The latter is associated to the uncertain distribution of the current or future SoH of the component, known from the processing of condition data collected and/or processed periodically at dates separated by a time interval ∆T . The policy is denoted as Predicted Quantile Replacement (PQR).

A prediction is made for the SoH of the component on the current interval [t, t + ∆T ]. If for a given quantile α ∈ [0, 1[ of the predicted distribution of the SoH and for any date t α in the time interval, the value is above the threshold M , a random value X tα is drawn from the former distribution and considered as a possible realization of the component's SoH at time t α . If X tα correspond to a failed state, the component is correctively replaced at cost c c , otherwise it is preventively replaced at time t α and cost c p . Thus the parameter α of the decision rule, somehow controls the risk of having a prediction that underestimates the true state.

Model for precision evolution in time

It is assumed that the precision of the state estimation or prognostics approach that is used to process condition data has been previously evaluated, with techniques outside of the scope of the present paper. Estimation of such precision is often known through the difference between the predictions Xi and the true values X i , for a set of labeled examples and generally involving crossvalidation. The Root Mean Square Error (RMSE) metric can be used as an estimate of the deviation between the predicted and true state:

RMSE(h) = i ( Xt+h,i -X t+h,i ) 2 (2)
where Xt+h is the prediction of the state of the component at time t + h that is made using the available information on [0, t[, h ∈ [0, ∆T ] is the prediction horizon and X t+h is the true state of the component at time t + h.

In this paper, a model for the evolution of the precision of the prediction approach with time is specified. While, the representativeness of this model may be debatable or hard to assess in practice, it is nonetheless interesting to study the influence of additional prediction information on the outcome of the maintenance policy based on such prediction information.

It is, then, assumed that the true state of the component on [t, t + ∆T ] is distributed normally X h ∼ N ( Xh , δ(h) Xh ), where h ∈ [0, ∆T ], Xh is the prediction obtained from the processing of CM data and δ(h) is the coefficient of variation which controls the prediction precision and may be related to a RMSE evaluated outside the framework of this paper. Here, the following model is proposed and two parameters are used to control the estimation and prediction uncertainty, namely e u and p u . The coefficient of variation δ(h) is assumed to evolve linearly according to:

δ(h) = e u + p u h ∆T (3)
It is worth noting that when h = 0, this model, even if not strictly, can approximate a PIR policy with measurement uncertainty. Indeed, there is the possibility that high measurement uncertainty (here through e u ), with a significant chance (say α) of crossing the threshold M , will trigger preventive replacement.

Let us also note that a normal distribution of the prediction may create degradation paths that are not strictly non-decreasing. This is not a serious concern if high quantiles α of the distribution are considered and compared to preventive replacement and failure thresholds.

Comparison of maintenance polices

Simulation-based optimization and evaluation

As described in section 2, different approaches can be considered for the optimization of main-tenance strategies. Here it is proposed to evaluate C ∞ numerically, using (1). The average length and costs of renewal cycles E[S] and E[C(S)] will be dependent on the design parameters of the selected maintenance strategies, namely:

• For BR: the time of replacement T BR • For PIR: the interval between inspections ∆T and the preventive replacement threshold M • For PQR: the interval between collections/predictions ∆T , the preventive replacement threshold M and the quantile of the prediction uncertainty α Multiple random realizations of renewal cycles are obtained through the generation of degradation paths, based on a gamma process model described hereinafter, and the application of maintenance strategies described in section 3. The estimated average cost per unit time C ∞ is the quantity to be minimized. The minimization problem is solved using the bounded and non-constrained stochastic optimization technique of simulated annealing, see [START_REF] Kirkpatrick | Optimization by simulated annealing[END_REF].

Gamma degradation process

As continuous-time representations of degradation, Gamma processes are very often considered, see e.g. an extensive review in van Noortwijk (2009). They are quite amenable to mathematical treatment and the fact that they are nondecreasing makes them an appropriate representation for many monotonous degradation mechanisms. A gamma process X(t) is a stochastic process with independent, non-negative, random increments, having the following distribution:

X(m) -X(n) ∼ Ga(v(m) -v(n), u) (4)
where Ga : x → Ga(x, v, u) is the gamma probability density function, v is the shape parameter and u is the scale parameter. At any time, the expectation and variance of the degradation can be derived according to:

E[X(t)] = v(t)/u (5) Var[X(t)] = v(t)/u 2 (6)
The specification of the mean function may allow the description of different degradation trends. If v(t) = vt, ∀t ∈ [0, ∞[ the gamma process is said to be stationary. Power laws are also used v(t) = vt b , where the exponent b may be quite different for different degradation mechanisms, e.g. mechanical, chemical, etc., see details in [START_REF] Van Noortwijk | A survey of the application of gamma processes in maintenance[END_REF].

Results and discussion

Comparison of BR and PIR policies

First, let us insist on the fact that, the metric that will be denoted as VoI here, does not correspond to the formal definition of VoI but is in fact the 'net gain of information'. Indeed, the cost of acquiring information is already included in the optimization of the policy. Thus, as opposed to the formal definition of VoI, see [START_REF] Raiffa | Applied statistical decision theory[END_REF], the calculated metric can be negative here. A negative VoI indicates that it is not worth collecting information at cost per inspection c i .

The comparison of the outcome of the optimization of the parameters of both the BR and PIR policies is given in Figure 1. The parameters used to describe what will be referred to as the decision context, namely, the degradation process and the cost and outcome of actions (here perfect) is given in Table 1. In this particular configuration of the decision context, the (net) VoI obtained from inspection is 0.07.

Comparison with PQR policy

A visualization of the implementation of the PQR policy is displayed in Figure 2. On [t, t + ∆T ], maintenance actions are taken based on a conditional decision rule which is linked to a quantile of the predicted distribution of the component's SoH. For the specific example of Figure 2, both the PQR and PIR decision rules would provide quite similar outcomes. On the segment [200,250], the predicted quantile first crosses the preventive replacement level M, thus triggering a random generation of a value from the normal distribution X tα ∼ N ( Xtα , δ(h) Xtα ), which is in that case, below the failure level and the component is subsequently preventively replaced at time t α .

Here, the prediction mean X is directly taken to be the degradation path, which is supposedly unknown. This is obviously a debatable choice, as the latter is not necessarily representative of the mean value of the degradation phenomenon but only of a given degradation path. Yet, the purpose here is not to develop a prediction tool and such a choice has been made for lack of a better, and nonetheless simple, alternative.

The result in terms of average outcome of policies, whose parameters are optimized numerically through simulated annealing, is given in Figure 3. It is seen that the use of prediction is valuable from the perspective of expected outcome. As could be expected, the advantage of a predictive policy does no longer hold when the prediction uncertainty p u increases. Indeed, a high prediction uncertainty may have the effect of making the predicted quantile value (for fixed α) cross the threshold level M too early. Thus, the decision rule, in a case of poor confidence in the prediction model, can be overly conservative.

Influence of the decision context on the value of prognostic information

The interest of using prognostic information extracted from the collected CM data in order to improve maintenance outcomes is further studied by repeating the estimation of VoI with different values of the parameters of the decision context. The comparison between PQR policy and BR and PIR policies is displayed in Figures 4 and5 for different decision contexts and varying prognostic uncertainty p u . As can be expected, when the variability of the degradation process is more significant, it is more valuable to gather information on the condition of the component. With low variability, it may be enough to plan maintenance actions with a time-based schedule and, thus, inspections and predictions become less valuable.

The interest a prognostics-based policy is also evaluated for different values of inspection/monitoring cost in Figure 6. It is seen that with an increase in inspection cost, overall it becomes more advantageous to make the most out of the available information. When the cost of inspection/monitoring decreases, it may be more valuable to inspect more and to rely less on predictions that may be imperfect.

Let us point out again that, here the evaluated metric denoted as VoI, is in fact the net gain between two policies with specific structures, thus not necessarily optimal (in a very large action space), with one policy involving additional information in the form of processed condition data, i.e. prognostic information.

Conclusions

In this paper, different maintenance strategies have been compared in terms of overall maintenance cost. They can be based on the combination of the available stochastic information on deterioration, the outcome of periodic inspections and on the information that can be obtained by processing condition-monitoring data through adapted state estimation and prognostics procedures, with known precision. The parameters of maintenance policies have been optimized to achieve minimal cost. All estimations have been carried out numerically, using a sample of simulated paths from a gamma degradation process. It is seen that the Value of Information (VoI) concept can provide a relevant metric to quantitatively asses if the processing of condition monitoring data into predictions of the current and future states of the component has a positive effect on the maintenance policy. The reasoning behind the present study is that such assessment may provide insights into the need and interest of further investing in the development of a more precise prediction tool, in a particular maintenance decision context.

For illustration purposes, rather simple settings, in terms of maintenance cost and consequences, have been considered in this paper but the same overall approach could be used for more complex settings. In the end, the interest of using additional information is always dependent on the considered decision context, namely the degradation process and the cost and outcomes of maintenance actions.

Perspectives of the present work include studies of predictive policies involving specific prediction tools from the Prognostics and Health Management domain, especially in a setting where the degradation process can evolve in time, thus requiring dynamic updating of models and decisions. Another interesting perspective is to look at a more strict application of the VoI concept, which involves a comparison between policies that are optimal, given a certain state of knowledge. This may not be the case with predefined specific policy structures, thus demanding that the complete sequential decision problem is formulated, e.g. using POMDP.

Fig. 1 .

 1 Fig. 1. Comparison of the outcome of maintenance strategies and visualization of the effect of maintenance parameters

Fig. 2 .

 2 Fig. 2. Example of degradation path and practical implementation of PIR and PQR policies. Thin solid lines indicate predictions (α quantile) on the different intervals

Fig. 3 .

 3 Fig. 3. Effect of prognostics information on policy output. Comparison of PQR versus BR and PIR

Fig. 5 .

 5 Fig. 5. Influence of the degradation process on the VoI obtained form a prognostic-based maintenance policy with different levels of precision and comparison with PIR policy; b and u control the trend and variability of the degradation process

Fig. 6 .

 6 Fig. 6. Influence of the inspection/monitoring cost on the VoI obtained form a prognostic-based maintenance policy with different levels of precision and comparison with BR and PIR policies

Table 1 .

 1 Parameters of the considered decision context.. Degradation rate is fixed for failure on average at time t = 200 for simplicity of visualization and analysis

	Parameter				Notation	Value
	Failure threshold			y	100
	Power law exponent		b	1
	Degradation variability		u	8
	Degradation rate			v	y/(200 b u)
	Corrective replacement cost	cc	300
	Preventive replacement cost	cp	50
	Inspection cost			c i	2
			Block Replacement
		1.4
		1.2
	Cost per unit time	0.8 1
		0.6
			0.54 @ BR:115.00
		0.4
		0.2
		50	100	150	200	250
			Block replacement time

Acknowledgement

The authors acknowledge the support of the chair System Science and the Energy Challenge, Fondation Electricité de France (EDF)