# How do people perceive changes in physical bounce model for virtual racket interactions?

Sony Saint-Auret, Franck Multon, Ronan Gaugne, Ludovic Hoyet, Richard Kulpa, Valérie Gouranton

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Sony Saint-Auret, Franck Multon, Ronan Gaugne, Ludovic Hoyet, Richard Kulpa, et al.. How do people perceive changes in physical bounce model for virtual racket interactions?. SAP 2025 - ACM Symposium on Applied Perception, Aug 2025, Vancouver (BC), Canada. pp.1-10, 10.1145/3736702.3744355 . hal-05225594v2

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# How do people perceive changes in physical bounce model for virtual racket interactions?

SONY SAINT-AURET, Univ Rennes, INSA Rennes, Inria, CNRS, IRISA, France

FRANCK MULTON, RONAN GAUGNE, and LUDOVIC HOYET, Univ Rennes, Inria, CNRS, IRISA, France

RICHARD KULPA, Univ Rennes, Inria, M2S, France

VALÉRIE GOURANTON, Univ Rennes, INSA Rennes, Inria, CNRS, IRISA, France

<figure><img src="image_2.png" /><figcaption>Fig. 1. The experimental VR Setup. Left: picture of a participant performing the “Perception-Action” block. Right: generic virtual room displayed in the head mounted display.</figcaption></figure>

Nowadays, Virtual Reality is widely used in sports, to enhance physical fitness, or improve specific subskills, such as anticipation skills. However, many factors in VR can alter the experience and make it difficult to transfer the skills trained in VR to real practice. One of these factors is the physical simulation of the virtual environment, that may produce unexpected behaviours. Hence, if users are athletes in ball-based sports, the VR training simulator should compute ball trajectories that look plausible for them. In this paper, our aim is to evaluate how human perception can be influenced by variations in a ball physics’ model. We explore properties of human perception, the acceptance threshold beyond which a deviation from the reference ball trajectory is perceived more than 50% of time, and the Just-Noticeable Difference (JND) as an indicator of perceptual sensitivity. To this end, we conducted psychophysical experiments where participants were asked to either only observe, or observe and hit virtual bouncing balls simulated with varying coefficients of restitution (COR). We report the acceptance threshold and JND in different conditions. We found that participants detected variations in COR more easily when having the motor task. Additionally, their sensitivity to variations was globally higher when they first performed the perceptual task alone, before the motor task was introduced. These results

contribute to the design of credible VR environments involving bouncing objects, such as for virtual sports.

CCS Concepts: • **Software and its engineering** → *Virtual worlds training simulations*; • **Computing methodologies** → *Interactive simulation*; • **Human-centered computing** → *Virtual reality*; • **User studies**.

Additional Key Words and Phrases: Perception and cognition, Psycho-physics, Virtual reality, Racket sports, Bouncing physics

## ACM Reference Format:

Sony Saint-Auret, Franck Multon, Ronan Gaugne, Ludovic Hoyet, Richard Kulpa, and Valérie Gouranton. 2025. How do people perceive changes in physical bounce model for virtual racket interactions?. In *ACM Symposium on Applied Perception 2025 (SAP '25)*, August 9–10, 2025, Vancouver, BC, Canada. ACM, New York, NY, USA, 9 pages. https://doi.org/10.1145/3736702.3744355

## 1 Introduction

Virtual Reality is now widely used in sports, to enhance physical fitness [Sarupuri et al. 2024], or to improve specific subskills such as anticipation skills [Bideau et al. 2009]. Previous works have shown better performance enhancement when using VR compared to traditional video training [Vignais et al. 2015], with greater motivation [Makransky et al. 2019a] which suggests that VR provides a more engaging and effective learning environment.

However, for experts accustomed to perceiving and reacting to subtle stimuli, transferring skills from VR training to real practice remains a great challenge [Miles et al. 2012; Oagaz et al. 2022a]. Indeed, skills transfer is difficult to explain and assess, as it relies on numerous parameters, including display and tracking technologies [Buttussi and Chittaro 2018], multi-sensory feedback interfaces [Tsai et al. 2022], experimental design [Makransky et al. 2019b], user

Authors' Contact Information: Sony Saint-Auret, sony.saint-auret@inria.fr, Univ Rennes, INSA Rennes, Inria, CNRS, IRISA, Rennes, France; Franck Multon, franck.multon@irisa.fr; Ronan Gaugne, ronan.gaugne@irisa.fr; Ludovic Hoyet, ludovic.hoyet@inria.fr, Univ Rennes, Inria, CNRS, IRISA, Rennes, France; Richard Kulpa, richard.kulpa@univ-rennes2.fr, Univ Rennes, Inria, M2S, Rennes, France; Valérie Gouranton, valerie.gouranton@irisa.fr, Univ Rennes, INSA Rennes, Inria, CNRS, IRISA, Rennes, France.
experience [Ahn et al. 2022], and the quality of the virtual simulation [Vignais et al. 2009]. Some authors showed no effect of rendering quality [Vignais et al. 2009], while others reported performance adaptations even for little latency in VR applications [Hoyet et al. 2019]. Hence, the actual tolerance to simplifications or artifacts in VR training simulators is still an open question. In this paper, we aim to explore how the accuracy of the physical simulation is perceived in such VR sports training simulators.

In virtual ball sports, bounce physics typically relies on computing the post-contact Cartesian and angular velocities according to the pre-contact velocities, in combination with the physical characteristics of the ball and contact surface [Nakashima et al. 2010]. These bouncing models depend on a few parameters, such as the restitution and friction coefficients. The coefficient of restitution (COR) is the ratio of relative velocities before and after the bounce, while the coefficient of friction depends on the contact surfaces, and affects the relative tangential velocity between these two surfaces. In the event of a collision with a deformable surface, like a racket's strings, an elasticity coefficient is also included in the model. For some sports, these coefficients are well documented [International Tennis Federation 2024], taking the nature of the contact surfaces into account. However these coefficients may also vary according to the wear of the ball. Moreover, accumulated approximations in the physical simulation, may lead to inaccurate ball behaviour. This issue becomes even more pronounced with less documented sports, as with the “Jeu de Paume” (or “real tennis”), the ancestor of modern tennis, for which the appropriate coefficients are not yet standardized and should be experimentally determined [Nakashima et al. 2010]. Such elements are therefore important to account for in serious VR training systems, where users may train their perception-action-loop to incorrect ball behaviors if the simulation is inaccurate, and therefore potentially alter the transfer of skills between the VR simulation to the real practice.

For all these reasons, it is relevant to determine how users react to deviations of the ball's behaviour from expected norms. Where lies the threshold beyond which such modifications become perceptible? As perception and action are coupled in human movement, changes in perception influence action, and reversely. This perception-action coupling requires to consider not only “perception alone” but also “perception in action” user performances. To our knowledge, perceptual thresholds for the physical properties of virtual bouncing balls have not been investigated within this perception-action loop context. Therefore, we evaluate how this loop influences perception by comparing two conditions: a static “perception only” and a dynamic “perception and action” conditions. In this paper, we propose to explore this problem in the specific context of racket sports, as it relies on fast ball movements, and high timing pressure for hitting the ball with a racket, leading to using anticipatory skills. We evaluate to what extent participants can detect changes in the physical properties of a bouncing ball. After recalling the physical equations of a bouncing ball (Section 3), we introduce the experimental set-up designed in VR to address this perceptual question (Section 4).

Figure 1 depicts the VR simulator where users had to intercept virtual bouncing balls simulated with a COR that varies across trials. We measured participants’ perceptual metrics in different conditions. Each participant underwent a static “perception-only” (report if the

COR changed) and “Perception-Action” (try to intercept the ball) protocols. They were also assigned to one of two groups based on the reference COR: a Tennis group with a high COR reference (similar to tennis), or a “Jeu de Paume” group with a low COR.

# 2 Related Works

## 2.1 Perception and intuitive physics

Perception of physical properties has been widely studied, especially for applications in computer animation and physics simulation [Yeh et al. 2009]. When people observe a dynamic event, they make inferences based on the context of this event, heuristics that stem from generalizing everyday life perceptual-motor interactions with physical objects [Vicovaro 2023]. This so-called “intuitive physics” [Kubricht et al. 2017] could be observed when people estimate the relative mass of colliding objects using heuristics [Gilden and Proffitt 1989], and may also induce strong biases in how the mind links an object’s mass to its speed [Vicovaro 2014].

Hence, as perception is biased, it can induce prediction errors. Several authors proposed to quantify the users' ability to notice changes of physical properties in various tasks. Previous work [Reitsma and Pollard 2003] showed that participants observing a virtual character's jumping animation detect respectively less errors in vertical velocity and deceleration than in horizontal velocity and acceleration. When tuning the simulation of collisions between two computer-generated balls, users seem to perceive with more or less accuracy the change of post-collision angle; linear and angular velocities; and spatio-temporal gap and delay of the collision [O'Sullivan et al. 2003]. Here, participants detected less increases than decreases of these parameters. Similarly, when looking at two-character pushing animations, participants detected less over-reactions of the pushed character compared to under-reactions [Hoyet et al. 2012]. This asymmetry in perceiving changes in the simulation was also reported for overarm or underarm throw motions, where participants were less sensitive to increases in release velocity than to decreases [Vicovaro et al. 2014]. In underarm throws, participants were also less sensitive when the object was released with a delay than when it was released prematurely, while the opposite was observed for overarm throws [Yamac and O'Sullivan 2022]. These findings highlight the complexity of understanding how physical inaccuracies of a simulation are perceived.

## 2.2 Perception in the case of ball dynamics

In racket sports, the fidelity of ball dynamics is crucial for users to develop correct posture and gesture. To study the benefits of in-simulation feedback on user posture in virtual table tennis, Oagaz et al. [2022b] developed a physically accurate simulation. However, the perception of ball bounce was not examined here. This latter has been studied in psychophysics through the lens of animation [Vicovaro et al. 2023]. Various approaches exist for simulating bouncing motion: one option is to model it using physical parameters, while another involves manually animating it (either through motion capture or hand-made animation). However, determining whether an animation is simulated or manually animated is complex and often depends on the specific parameter values used. For example, participants are more likely to interpret a ball's motion as animation-based
rather than physics-based when the coefficient of restitution (COR) exceeds a value of 1, resulting in a super-elastic collision. But, while this study explored perceptual factors such as animacy and causality, it did not delve into the human sensitivity to variations in the physical parameters of bouncing motion. In particular, can they detect variations in physics when the motion subtly differs?

## 2.3 Perception-Action coupling

In addition to perceptual studies, the influence of motor tasks on perceptual ones, whether enhancing or interfering, is an important topic to investigate. Human sensory systems often integrate multiple types of information to interpret environmental input and adapt accordingly. E.g., this interplay is evident in weight perception where information can be gathered from mechanoreceptors, muscle proprioceptors, or even visual cues [Lim et al. 2021]. Multimodal integration can therefore function either as a source of distraction or as a facilitator of performance in the tasks, notably depending on the task conditions [Brand-D'Abrescia and Lavie 2008; Laurienti et al. 2004]. In particular in the context of sport, for example, performing an interception task leads to higher timing pressure compared to evaluating a situation in static condition [Ripoll and Latiri 1997; Tresilian 1993]. Previous work also showed that motor task performances can be impaired by latency in VR tasks, before users consciously perceive/report the occurrence of this delay in the VR application [Hoyet et al. 2019]. These statements support the hypothesis that analyzing perception in static or more ecological conditions (including actions) may lead to different results.

## 2.4 Focus of the present study

As highlighted in this review of the literature, extensive work in psychophysics has investigated perceptual processing. However, prior studies are mostly conducted on monocular LCD screens. This setup limits the use of depth cues and lacks the ecological validity crucial in sports contexts. To address this, we leverage VR that offer stereoscopic vision and a more immersive setting, potentially enhancing participants' ability to perceive and interpret the motion more accurately. In this study, we specifically evaluate perceptual metrics related to the physics of a bouncing ball, with a focus on the COR. We examine how these metrics vary under different conditions, notably when the intensity of the stimulus, the COR, changes, and when participants are required to perform a motor task on top of the perceptual assessment.

## 3 Physical model

To provide an immersive VR environment with realistic interactions between the ball, its surroundings and the participant's racket, it's crucial to develop an accurate physical model for the ball's bounce. While Unity, the game engine we use to build the VR experiment, offers basic physics, we aim to implement a more advanced model to enhance the fidelity of the simulation. The ball's behaviour after a bounce on the ground can be described by the Cartesian and angular velocities it adopts upon impact (resp. $v'_b$ and $\omega'_b$), which depend on the Cartesian and angular velocities (resp. $v_b$ and $\omega_b$) before the bounce, and four matrices ($A_v, B_v, A_\omega, B_\omega$) containing the physical coefficients of the bounce [Nakashima et al. 2010]. This behaviour

can be represented as follows:

$$v'_b = A_v v_b + B_v \omega_b \quad (1)$$

$$\omega'_b = A_\omega v_b + B_\omega \omega_b \quad (2)$$

with

$$A_v = \begin{bmatrix} 1 - \alpha & 0 & 0 \\ 0 & 1 - \alpha & 0 \\ 0 & 0 & -e \end{bmatrix}, B_v = \begin{bmatrix} 0 & \alpha r & 0 \\ -\alpha r & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix} \quad (3)$$

$$A_w = \begin{bmatrix} 0 & -\frac{3\alpha}{2r} & 0 \\ \frac{3\alpha}{2r} & 0 & 0 \\ 0 & 0 & 0 \end{bmatrix}, B_\omega = \begin{bmatrix} 1 - \frac{3\alpha}{2r} & 0 & 0 \\ 0 & 1 - \frac{3\alpha}{2r} & 0 \\ 0 & 0 & 1 \end{bmatrix} \quad (4)$$

$$\alpha = \mu(1 + e) \frac{|v_{bz}|}{|v_{bT}|} \quad (5)$$

where $r$ is the radius of the ball, $\mu$ is the friction coefficient, $e$ is the Coefficient of Restitution (COR), $v_{bz}$ is the vertical component of the velocity (normal to the ground), and $v_{bT}$ is the vector composed of the velocity components tangent to the ground plane. Intuitively, the friction coefficient relies on the amount of horizontal speed decrease after compared to before contact. The COR ranges from 0 (no vertical speed after impact) to 1 (vertical speed after and before impact are identical, no loss of vertical velocity). These coefficients vary depending on the nature of the two surfaces, and any potential deformation, as with the racket's strings in our case.

For well known sports, $\mu$ and $e$ are documented in the literature, such as the International Tennis Federation (ITF) recommendation for the tennis' COR ($e > 0.70$) [International Tennis Federation 2024]. For less popular sports, such as Jeu de Paume, this information is missing. It can change depending on the ground surface and the ball wear, even for well-known sports. For designers of sports VR simulations, we consequently propose an experimental set-up to identify the specific COR of an application.

In this subsection, we apply this identification process to the Jeu de Paume. To this end, we used 10 synchronized and calibrated Miquis RGB Cameras (150Hz, 1280×765 pixels), positioned all around the room. A Jeu de Paume ball was released from 19 various heights and orientations into a 1m plastic tube, to measure the cartesian and angular velocities before and after bounce (Figure 2). The ball was equipped with 6 colored patches, glued on the ball. CVAT and OpenCV were used to semi-automatically label and reconstruct the 3D position of each patch, at each frame. This enabled us to estimate the average cartesian and angular velocities of the ball for the six frames before $v_{bexp}$, $\omega_{bexp}$ and after $v'_{bexp}$, $\omega'_{bexp}$ the bounce. Theoretically, these values satisfy (1) and (2). Thus, we searched $\mu$ and $e$ that minimize the following cost function:

$$J(\mu, e) = \frac{1}{N} \sum_{i=1}^{N} \left\| \frac{v'_{b}(\mu, e) - v'_{bexp}}{\sigma_{v,exp}} \right\|^2 + \left\| \frac{\omega'_{b}(\mu, e) - \omega'_{bexp}}{\sigma_{\omega,exp}} \right\|^2 \quad (6)$$

where $\sigma_{v,exp}$ and $\sigma_{\omega,exp}$ denote the standard deviations of the linear and angular velocities, respectively, and are used as weighting factors for the parameters. $v'_b(\mu, e)$ and $\omega'_b(\mu, e)$ are given by Equations (1) and (2), depending on the two unknowns ($\mu, e$) and the experimental parameters ($v_{bexp}, \omega_{bexp}$). For the Jeu de Paume, we obtained $\mu^{JdP} = 0.018$ and $e^{JdP} = 0.54$, confirming a lower COR
<figure><img src="image_3.png" /><figcaption>Fig. 2. Experimental set-up composed of a racket rigidly attached to a support, the ball with coloured patches (detailed Jeu de Paume balls on the right) going out of a tube held at different heights, orientations and positions.</figcaption></figure>

than for more flexible balls (such as tennis: $e^T > 0.70$). This result is in line with findings from Richefeu et al. [2024], who measured the COR as a ratio of the energy rather than velocities (resulting in a squared difference between their results and ours).

These values can be used to simulate the behaviour of virtual balls in VR sports simulators, and will be used in the following perceptual study.

## 4 Experimental set-up in VR

Following the optimization of the physical parameters for simulating a Jeu de Paume ball, we designed a VR experiment to evaluate how participants respond to variations in the model under different conditions. This section outlines the structure of the experiment, starting with participant demographics and the technical apparatus used to conduct the study.

### 4.1 Participants and apparatus

Thirty volunteers took part in a perceptual experiment in VR. They were separated into two groups: Jeu de Paume (3F/12M; *age* = 24.5 ± 5.3) and Tennis (5F/10M; *age* = 26.1 ± 4.0). They were all naive to the purpose of the experiment, had normal or corrected-to-normal vision, no musculoskeletal disorder, and gave written and informed consent prior to the experiment. The study was conformed to the declaration of Helsinki, and was approved by the local ethical committee.

Participants were immersed in a virtual empty racket ball court, using a Meta Quest 2 Head-Mounted Display (framerate: 72Hz; resolution: 3616×1856 pixels). Meta Quest 2's controllers were used to control the swing of the virtual racket, including vibration feedback triggered on ball contact. Input latency, i.e. the delay between controller real and simulated motion, was experimentally estimated to be less than 20 ms. Virtual eye height was also calibrated for each participant.

Prior to the experiment, participants evaluated their VR and video game experience using a 5-point Likert scale (1: Never, 2: Rarely, 3: Monthly, 4: Weekly, 5: Daily). They reported a VR experience of 1.8 ± 0.8 (close to “Rarely”) in the Jeu de Paume group, and 2.7 ± 1.2 (close to “Monthly”) in the Tennis group. Similarly, video game

experience was rated as 2.7±1.4 (close to “Monthly”) for participants in the Jeu de Paume group, and 3.8 ± 1.1 (close to “Weekly”) for those in the Tennis group.

They also evaluated their skill level in different racket sports (tennis, table tennis, badminton, Jeu de Paume, squash, padel, other) through multiple metrics: subjective rating (1-to-5 scale: “None”, “Beginner”, “Intermediate”, “Advanced”, “Professional”), years of experience, current frequency of playing, and frequency of watching sports (videos, TV, tutorials). On average, Jeu de Paume group’s participants reported an “Intermediate” skill level, 3-5 years of playing, and a monthly frequency in playing and watching. For the Tennis group, they often felt closer to beginners, with less than 1 year of playing, having rarely played recently and never watching.

### 4.2 Protocol

During the experiment, participants were immersed in a generic virtual room, in front of an automatic virtual ball launcher, as shown in Figure 1. In each trial, participants had to observe a series of 5 virtual balls launched from different randomized horizontal positions and maximum trajectory heights. The ball was launched with a velocity that ensured to always reach the same bouncing position, close to the dominant-hand side of the participant. For each trial *k*, the first 4 balls were simulated with the same COR $e_{k,1-4}$, while the last one was simulated using another COR $e_{k,5}$. $e_{k,1-4}$ or $e_{k,5}$ were randomly assigned to correspond to the reference coefficient of the corresponding group ($e_{ref}^{JdP} = 0.54$ for the Jeu de Paume group, and $e_{ref}^T = 0.75$ for the Tennis group [International Tennis Federation 2024]). For the other ball(s), the COR value was set following an interleaved-staircase design.

We ran two blocks of experiments for each participant. In the “Perception” block, participants stayed static in front of the balls and simply observed the series of 5 balls. After each trial, they were asked to answer the following question: “Did you perceive a difference between the last ball and the previous ones?” (pressing ‘Yes’ or ‘No’ buttons in the virtual environment). In the “Perception-Action” block, they were additionally instructed to attempt at hitting the ball with the virtual racket (controlled by the position and orientation of the dominant-hand VR controller). Since the ball was consistently delivered to the dominant hand’s side of each participant, they were naturally inclined to execute forehand strokes rather than backhand ones. A visual feedback was given to users when they succeeded in hitting the ball, by simulating the trajectory of the ball after the hit. Participants in this block answered the same question than in the “Perception” block.

By presenting participants with a series of 4 balls simulated using the same COR $e_{k,1-4}$, followed by a single ball simulated using another COR $e_{k,5}$, our objective was to evaluate whether participants would be able to visually identify the difference (both blocks), as well as whether adding a motor task would affect their perception (“Perception-Action” block).

Each block followed a randomly-interleaved staircase design, inspired from psychometric protocols [O’Sullivan et al. 2003; Vicovaro et al. 2013]. This design aims at gradually finding the upper $\delta_+$ and lower $\delta_-$ perceptual metrics to evaluate participants’ sensitivity to a stimulus. In our case, for a trial *k*, the stimulus observed is the
difference between a COR value and the reference COR of the group. The first metric was the acceptance threshold, which correspond to the $e_{ref} - \delta_{-}$ and $e_{ref} + \delta_{+}$ at which participants perceive the difference with $e_{ref}$ less than 50% of the time. The second metric was the Just-Noticeable Difference (JND), a value that represents the sensibility of an observer to change in the stimulus, and can be typically obtained by subtracting the 50% threshold to the 75% threshold [Vicovaro et al. 2014]. In our context, the JND reflects how sharply a participant's certainty varies in detecting a difference. A high acceptance threshold with a low JND implies detection only at larger deviations but with a clear perceptual boundary. Whereas a low threshold with a high JND suggests sensitivity to small deviations but with a more gradual certainty buildup. To find these perceptual metrics, two staircases were used. The ascending staircase started by simulations with the COR reference value, and then increased/decreased this value along the tests, up to very different one. The descending one proceeded reversely: starting with very different COR value $e_{ref} \pm 0.25$ (supposed to be easily distinguishable from the reference), and increasing/decreasing this value along the tests, up to the reference COR value. As such, we used four staircases in our experiments, that were presented in randomly interleaved order: two ascending/descending staircases for $\delta_{-}$ and two others for $\delta_{+}$. Similarly to standard perceptual studies based on psychophysics, and as suggested by Garcia-Perez [2001], we used a down/up step ratio of 0.871, and set the stopping condition to 6 reversals, in combination with a stepup = 0.02.

Prior to the experiment, we introduced a familiarization phase including 5 additional trials that were not used for the analysis.

## 4.3 Questionnaires

We used questionnaires to complete this analysis with an idiocentric study, and evaluate the impact of the changes in e on participants' VR experience. A demographic questionnaire enabled to collect the participants' age, gender, dominant hand, VR and video games experiences, skill level in racket sports (as described in Section 4.1). A Cybersickness questionnaire (CSQ-VR) [Kourtesis et al. 2023] was also filled before the experiment and after each block. This enabled us to compute a general score for cybersickness (ranging from 6 to 42) by summing the CSQ-VR's ratings [Kourtesis et al. 2023]. Similarly, participants filled a Slater-Usoh-Steed (SUS) Presence questionnaire [Steed et al. 2018] after each block. We averaged the SUS questionnaire answers to obtain a mean presence score per block for each participant. At the end of the experiment, participants were also asked experiment-specific questions to determine whether they were able to notice what varied in the trials, how they judged the difficulty of the task, and what strategies they used to identify changes in the ball's behaviour.

## 4.4 Data analysis

Through the staircase design, we obtained answers at different CORs, representing whether participants perceived a difference between the reference ($e_{ref}$) and the modified coefficients. These answers were then fitted into logistic curves for $\delta_{-}$ and $\delta_{+}$ separately for each block using psychopy [Peirce et al. 2019], based on the FitLogistic

equation:

$$y = chance + \frac{1 - chance}{1 + \exp((x_{50} - x) * JND)} \quad (7)$$

where $x_{50}$ refers to the acceptance threshold, and chance, or guess rate [Wichmann and Hill 2001], is the lower asymptote of the logistic function, i.e., the probability of a “yes” response when no difference exists. In our yes/no paradigm, we assume a guess rate of zero.

This allowed estimation of the acceptance thresholds and JNDs, as described in Section 4.2. Logistic functions were fit to 120 condition combinations: 2 groups (Jeu de Paume, Tennis) × 15 subjects × 2 blocks (Perception and Perception-Action) × 2 threshold sides (lower, upper). A preliminary fit was performed on the data to obtain the mean acceptance threshold and JND using the logistic curves that were successfully fitted (109 out of the 120 logistic curves). These values were then used as an initial guess in the optimization process of the model (Equation 7) to fit the final logistic curves. This process ensured that only 2 out of 120 logistic curves (less than 2%) were filtered out, as psychopy was unable to reliably fit a logistic curve, likely due to an insufficient number of points in the region where the curve should plateau at $x = 0$. This issue was likely caused in these two plots due to too high random answering from the participants.

To analyze the acceptance thresholds and JNDs in the various conditions, we conducted mixed-design ANOVAs with within-subject factors Block (Perception, Perception-Action) and Threshold Side (Lower, Upper), as well as between-subject factors Group (Jeu de Paume, Tennis) and Block Order (Perception first, Perception-Action first). Shapiro-Wilk normality tests were conducted prior to statistical analyses, and we applied an Aligned Rank Transform [Elkin et al. 2021; Wobbrock et al. 2011] prior to the ANOVAs when normality was rejected ($p < 0.05$). Posthoc analyses were performed on significant effects with Bonferroni correction.

For the questionnaire analyses, when Shapiro-Wilk normality tests were satisfied, we reported the mean, otherwise, the median and interquartile range (IQR) were reported.

# 5 Results

In this section, we present the acceptance threshold and JND obtained in the various conditions and groups. We also present the results of the Presence and Cybersickness questionnaires, and the free feedback from the participants.

## 5.1 Acceptance threshold and JND

Results of the statistical analyses on acceptance thresholds, with the factors described in Section 4.4, showed a main effect of Block ($F_{1,76.188} = 4.048, p = 0.048, \eta^2 = 0.05$), where posthoc analysis revealed that the acceptance threshold for the Perception block was significantly higher than the one for the Perception-Action block. We found an interaction effect between Threshold Side and Group ($F_{1,76.186} = 5.887, p = 0.0176, \eta^2 = 0.07$), which was not confirmed by posthoc analysis. An interaction effect was also found between Block, Threshold Side and Group ($F_{1,76.185} = 7.578, p = 0.0074, \eta^2 = 0.09$), for which posthoc analysis showed that the acceptance threshold was significantly higher for Perception compared to Perception-Action in the case of the upper threshold for the Tennis
<table><caption>Table 1. Mean (± standard deviation) acceptance thresholds across participants by Block</caption><thead><tr><th>Metric</th><th>Perception</th><th>Perception-Action</th></tr></thead><tbody><tr><td>Threshold</td><td>0.093 ± 0.035</td><td>0.087 ± 0.035</td></tr></tbody></table>

<table><caption>Table 2. Mean (± standard deviation) acceptance thresholds (AT) across participants by Block, Threshold Side and Group</caption><thead><tr><th>Group</th><th>Threshold Side</th><th>Block</th><th>AT</th></tr></thead><tbody><tr><td>Jeu de Paume</td><td>Lower</td><td>Perception</td><td>0.085 ± 0.037</td></tr><tr><td>Jeu de Paume</td><td>Lower</td><td>Perception-Action</td><td>0.076 ± 0.028</td></tr><tr><td>Jeu de Paume</td><td>Upper</td><td>Perception</td><td>0.091 ± 0.043</td></tr><tr><td>Jeu de Paume</td><td>Upper</td><td>Perception-Action</td><td>0.102 ± 0.047</td></tr><tr><td>Tennis</td><td>Lower</td><td>Perception</td><td>0.096 ± 0.030</td></tr><tr><td>Tennis</td><td>Lower</td><td>Perception-Action</td><td>0.092 ± 0.028</td></tr><tr><td>Tennis</td><td>Upper</td><td>Perception</td><td>0.101 ± 0.032</td></tr><tr><td>Tennis</td><td>Upper</td><td>Perception-Action</td><td>0.080 ± 0.031</td></tr></tbody></table>

<table><caption>Table 3. Mean (± standard deviation) JNDs across participants by Block Order</caption><thead><tr><th>Metric</th><th>Perception first</th><th>Perception-Action first</th></tr></thead><tbody><tr><td>JND</td><td>0.024 ± 0.017</td><td>0.036 ± 0.027</td></tr></tbody></table>

group (being 0.101 ± 0.032 and 0.080 ± 0.031 respectively). Means and standard deviation values are reported in Table 1 and Table 2, and displayed in Figure 3 and 4.

Statistical analyses on JNDs only showed a Block Order effect ($F_{1,25,988} = 5.0622, p = 0.0331, \eta^2 = 0.16$), which revealed that the mean JND was significantly higher for participants when first presented with the Perception-Action block compared to Perception block first. Specifically, the mean JND for “Perception-Action first” was 0.036 ± 0.027 while the mean for “Perception first” was 0.024 ± 0.017. This result suggests that overall, the participants’ sensitivity to COR variations was finer when first presented with the Perception block. Means are reported in Table 3 and a visual representation of these results is provided in Figure 5.

## 5.2 Questionnaire analyses

The results of the ANOVA test revealed no significant main or interaction effect for Presence, with an average Presence score across conditions categorized as medium (4.23 ± 1.06).

Cybersickness, as measured with the CSQ-VR scores, increased significantly between the pre-experiment and post-blocks questionnaires ($F_{2,52} = 16.204, p \approx 0, \eta^2 = 0.38$), indicating a measurable effect of the experimental setup and VR exposure on the participants’ physical comfort. However, this effect appears small as both pre-experiment and post-block scores, respectively 7.07 ± 1.68 and 9.10 ± 3.25, remained relatively low (CSQ-VR scores range from 6 to 42), suggesting a weak discomfort for participants.

<figure><img src="image_4.png" /><figcaption>Fig. 3. Top: Average acceptance thresholds per Block: Perception, Perception-Action. Error bars depict standard error. Bottom: Psychometric curves presenting, for each Block, the probability of perceiving a difference during a trial. Shaded areas represent 95% confidence intervals based on ±1.96 standard errors of threshold and JND parameters across participants. Black dashed lines represent acceptance thresholds.</figcaption></figure>

At the end of the experiment, participants were asked to evaluate the difficulty of the tasks in both blocks, on a scale of 1 to 5 (Very easy, Easy, Medium, Hard, Very Hard). Data’s normality was rejected, and no main or interaction effects was found. The median task difficulty score was 2 (Easy) ($IQR = 2 - 3$). Overall, participants generally perceived the tasks as easy to complete.

At the conclusion of the experiment, participants were also asked to reflect on perceptual differences they noticed in the behaviour of the virtual ball. A majority of them noticed differences in ball’s “height”, “bounciness” or “(post-bounce) trajectory”, aligning with expected effects of COR modifications on post-bounce linear velocity as seen in Equation (3), which would logically also alter how high the ball bounces. However, only 57% of participants explicitly mentioned noticing changes in “speed” or “acceleration” of the ball. Interestingly, only a single participant remarked on the presence of
<figure><img src="image_5.png" /><figcaption>Fig. 4. Average acceptance thresholds per Block, Threshold Side for the Jeu de Paume (left) and the Tennis (right) groups.</figcaption></figure>

post-bounce angular velocity, “rotation”, which is indeed affected
by the COR in our model, as defined by Equation (4).

# 6 Discussion

Regarding the human perceptual properties under our experimen-
tal conditions, our results show that participants start to notice
variations in a ball bounce's COR more than 50% at the lowest of
0.080 ± 0.031 (upper threshold for the perception-action block of the
tennis group), with a perceptual sensitivity varying at a lowest rate
corresponding to a JND of 0.024 ± 0.017 (when starting with the
Perception block). These findings provide initial practical insights
into the acceptable margin of error in a physical model, whether in
comparison to real-world behaviour or in the presence of variability
during real-time execution.

## 6.1 Perceptual enhancement through action and prior observation

The perception-action loop is reflected in our findings. Introducing
the motor task of hitting the ball significantly lowers the threshold
at which participants frequently detect a difference between CORs.
In this context, action appears to enhance perception, likely because
the motor task is here directly tied to the perceptual task. Rather
than acting as a distraction, it provides a complementary strategy
for detecting differences in ball physics, thereby supporting overall
task performance. This is consistent with the literature [Laurienti
et al. 2004]. Semantically congruent tasks, such as analyzing the
physics of the ball-to-ground bounce while also preparing to hit
it based on own predictions, tend to enhance task performance.
The fact that this effect is only observed in participants from the
Tennis group, who had a higher reference COR, and at the upper
threshold, may suggest that the difference is particularly amplified
for overhand movements. In fact, it was the only condition where
participants were required to move their arm above shoulder level,
due to the height at which the ball bounced. This finding also corrob-
orates the fact that most participants state that they used different

<figure><img src="image_6.png" /><figcaption>Fig. 5. Top: Average JNDs per Block Order: Perception first, Perception-Action first. Error bars depict standard error. Bottom: Psychometric curves presenting, for each Block Order, the probability of perceiving a difference during a trial. Shaded areas represent 95% confidence intervals based on ±1.96 standard errors of threshold and JND parameters across participants. Black dashed lines represent acceptance thresholds.</figcaption></figure>

strategies for each block. Indeed, some participants reported that
they compared “if the ball bounced higher/lower than their shoul-
der” in the Perception block. In the Perception-Action block, they
reported more frequently their “difficulty handling the last ball or
that they had to change their position drastically”. These results
also support the idea that fundamentally, participants seem to use
different cues to evaluate the difference between two balls simulated
with different COR, but a deeper exploration of this question would
be necessary to provide more relevant information to serious VR
training simulator designers.

Furthermore, participants showed greater sensitivity (lower JND)
when they completed the Perception block first. This may suggest
that they developed a more precise sense of the visual cues to assess
COR differences through passive observation first, which later en-
hanced their discrimination ability in the Perception-Action block.
## 6.2 Absence of Weber's Law effect and asymmetry in perception

While Weber's law posits that perceptual sensitivity scales pro-
portionally with the magnitude of the reference stimulus [Fechner
1948], no group differences were observed. It suggests that percep-
tual sensitivity to COR deviations may remain consistent across
reference values. However, this may also be due to the reference
values not differing sufficiently to have an impact, or alternatively,
because the strategies employed by the participants in this VR ex-
periment were effective in counteracting the Weber's law. Further
investigation would be needed to confirm this.

Lower and Upper thresholds did not differ significantly for each reference COR. This contrasts with prior studies [O'Sullivan et al. 2003; Vicovaro et al. 2014], which found greater sensitivity to decreasing speed compared to increasing speed. An explanation may stem from a different experimental set-up in our experiment. Indeed, we used an immersive 3D environment with an HMD, while previous studies used 2D video displays. In our experiment, the virtual balls moved towards the participants, introducing depth cues that are less accurate in 2D perceptual analyses. Unlike prior experiments where participants likely focused on isolated visual cues (e.g., post-collision speed), our first-person VR context allowed for additional self-relative cues, such as the relative position and velocity of the ball. Some participants reported strategies based on some form of proprioceptive and peripheral vision “landmarks”. Participants reported that they could locate the ball compared to their body in this immersive environment. In the “Perception-Action” block, participants reported gesture adaptations that may guide their choice. As such, these strategies may impact the asymmetry between lower and upper thresholds in perceptual tasks. This opens new avenues of research to better understand the kinematic and perceptual variables that may influence the way people evaluate the physical difference, as well as potential physical realism, of racket sports VR simulators.

## 6.3 Use of heuristics and subjective ease of task

In the bouncing ball physical model, modifying the COR directly impacts the relation between the pre and post-collision linear and angular velocities. Participants commonly reported noticing changes in the post-bounce height of the virtual ball, which is a direct consequence of post-collision velocity modifications. But, relatively few participants explicitly mentioned perceiving a change in the ball's speed. Only one participant noticed a change in angular velocity. The underlying reason may align with findings from prior research on intuitive physics [Gilden and Proffitt 1989], which suggested that individuals tend to anchor their judgments on the most visually salient cues. In this case, the most salient cues may have been bounce height, and overall trajectory. Changes in linear speed may have been less perceptible, and variations in angular velocity were likely even harder to detect.

## 6.4 Limitations

An unexpected uneven participant expertise was observed between
the two groups of subjects: the Jeu de Paume group reported greater
racket sport experience, while the Tennis group had more VR and

gaming experience. Hence, the difference of perceptual behaviour
observed between the two groups could be explained either by the
different COR value, but also by this uneven participant expertise
between groups. Further research would therefore be necessary to
clarify the effect of prior experience in such tasks.

Hardware configuration could also impact the perception cues of participants, e.g., due to different system latencies, as reported in previous works [Wang et al. 2023]. In this work, latency was estimated to be less than 20 ms. Haptic feedback was also limited to little vibrations of the controller, which cannot match the intensity of real racket-ball impacts. The weight of the controller also differed from a real racket. Further studies would help to clarify if such limitations indirectly affect the way people perceive changes in bouncing ball physics simulation.

Finally, we also computed the speed of the virtual racquet at the moment of contact with the virtual ball during our experiment, through the displacement of the Meta Quest 2 controllers. For our participants, we measured speeds ranging from 5 to 15 m.s⁻¹ at the time of contact, with a very small standard intra-individual variation. Such differences in motor action could have also influenced participants’ perceptions, which would be interesting to further investigate in future studies, e.g., by exploring the effect of hit speed and proprioceptive cues on participants’ perceptions.

## 7 Conclusion

A major challenge in serious sport VR simulators is ensuring skill transfer to real practice, which depends on multiple factors. The results reported in this paper contribute to better understand how inaccuracies in the ball physical simulation can be perceived by users, and consequently potentially affect this skill transfer. In a virtual racket sport context, sensitivity improves globally when participants can initially focus on the perception task only. Additionally, for overhand shots particularly, perceptual sensitivity benefits from active engagement, suggesting that perception-action integration enhances discrimination of ball's COR.

These findings help to establish perceptual thresholds that can support designers of serious training simulators. Hence, this can provide them with recommendations to prevent users from being surprised by unusual ball behaviour, or train in VR with visual cues that would differ in real practice. This is especially true for expert users who are used to play with specific balls and terrains (such as playing tennis on grass or clay tennis court). It would also encourage them to more carefully design the physical parameters, including carrying-out experimental protocols to estimate accurate COR, if necessary. In addition to COR, other physical parameters, such as friction or elasticity, could also be studied, to ensure an expected behaviour of the ball when hitting the racket. Such research topics would support the development of more effective VR simulators for skill transfer, especially in elite sports.

## Acknowledgments

This study was funded by the French National Research Agency within the framework of the France 2030 programme for EUR DIGIS-PORT (ANR-18-EURE-0022) and CONTINUUM (ANR-21-ESRE-0030) projects.
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