Muhammad Farhan
email: muhammad.ahmed@ec-nantes.fr

Ahmed Ls2n

Vincent Frémont
email: vincent.fremont@ec-nantes.fr

Isabelle Fantoni
email: isabelle.fantoni@ls2n.fr

Active SLAM Utility Function Exploiting Path Entropy

Keywords: Active SLAM, Mapping, Information Theory, Entropy

In this article we present a utility function for Active SLAM (A-SLAM) which utilizes map entropy along with D-Optimality criterion metrices for weighting goal frontier candidates. We propose a utility function for frontier goal selection that exploits the occupancy grid map by utilizing the path entropy and favors unknown map locations for maximum area coverage while maintaining a low localization and mapping uncertainties. We quantify the efficiency of our method using various graph connectivity matrices and map efficiency indexes for an environment exploration task. Using simulation and experimental results against similar approaches we achieve an average of 32% more coverage using publicly available data sets.

I. INTRODUCTION

Simultaneous Localization and Mapping (SLAM) encompasses a range of methods used by robots to determine their own position while simultaneously creating a map of their surroundings during navigation. Most SLAM algorithms are considered passive, wherein the robot is moving freely, and the navigation or path-planning algorithm plays no active role in directing the robot's movements or trajectory. In contrast, A-SLAM aims to address the optimal exploration of unknown environments by proposing a navigation strategy that generates future target positions and actions. These actions are designed to reduce uncertainty in the map and the robot's pose, enabling a fully autonomous SLAM system.

In A-SLAM, the robot's exploration process begins by initially identifying potential target positions within its current map estimate. Once the robot has established a map of its surroundings, it proceeds to locate positions worth exploring. One commonly used method for this task is frontierbased exploration, initially introduced by [START_REF] Yamauchi | A frontier-based approach for autonomous exploration[END_REF]. Frontier-based exploration defines the 'frontier' as the boundary separating known map locations from unknown ones, as observed by the robot's sensors. After identifying these goal frontiers, the robot computes a cost or utility function. This function relies on the potential reward associated with selecting the optimal action from a set of all possible actions. In an ideal scenario, this utility function should account for the complete joint probability distribution of both the map and the robot's poses. To quantify this uncertainty, we turn to two wellestablished domains: Information Theory (IT), and the Theory of Optimal Experimental Design (TOED), as detailed in [START_REF] Pázman | Foundations of optimum experimantal design[END_REF]. The subsequent crucial step involves executing the optimal action, guiding the robot towards its goal position using path planning techniques.

In this article, we propose a utility function for selecting the goal frontier candidate for autonomous exploration by the robot. Our function takes into account the amount of uncertainty in the map measured as path entropy and Euclidean distance to each frontier candidate. We add this utility function to the one of [START_REF] Placed | Fast Autonomous Robotic Exploration Using the Underlying Graph Structure[END_REF] which selects the frontier weighted by the Doptimality criterion as the maximum number of spanning trees (in pose-graph) towards it. Using the proposed utility function brings the advantage that it incorporates not only the SLAM uncertainty but also entropy reduction in the environment to guide the robot to promising unknown areas for the exploration task.

This article is organized as follows: Section II, provides an insight into the related work. Section III gives preliminary knowledge about the structure of modern graph SLAM, A-SLAM, and how uncertainty is measured and related to graph connectivity. In Section IV we present the approach and formulation of our proposed method. Section V presents and discusses our simulation results. Finally, we conclude in Section VII summarizing our contributions and motivating future research aspects.

II. RELATED WORK

As discussed above, A-SLAM involves frontier detection, utility computation by quantifying and minimizing the uncertainty, and finally generating the action for robot navigation.

The method proposed by [START_REF] Trivun | A-SLAM-based algorithm for autonomous exploration with Mobile Robot[END_REF] formulates a hybrid control switching exploration method. Within the occupancy grid map, each frontier is segmented, a trajectory is planned for each segment, and the trajectory with the highest map segment covariance is selected from the global cost map, which renders this method computationally expensive and limited to static obstacles. Meanwhile, the approach in [START_REF] Mammolo | A-SLAM in Crowded Environments[END_REF] deals with dynamic environments with multiple ground robots and uses frontier exploration for autonomous exploration, and a utility function based on Shannon and Renyi entropy [START_REF] Carrillo | Autonomous robotic exploration using a utility function based on Rényi's general theory of entropy[END_REF] is used for the computation of the utility of paths.

When dealing with uncertainty quantification from IT perspective, the authors of [START_REF] Vallve | Active pose slam with RRT[END_REF] address the joint entropy minimization exploration problem and propose two modified versions of Rapidly exploring Random Trees (RRT) [START_REF] Naderi | A real-time path planning algorithm based on RRT[END_REF], as dRRT* and eRRT* respectively. dRRT* uses distance, while eRRT* uses entropy change per distance traveled as the cost function. Actions are computed in terms of the joint entropy change per distance traveled. The simulation results proved that a combination of both of these approaches provides the best path-planning strategy. An interesting comparison between IT approaches is given in [START_REF] Du | A comparative study on A-SLAM and autonomous exploration with particle filters[END_REF] where frontier-based exploration is deployed to select future candidate target positions. A comparison of joint Entropy reduction between the robot path and map is done against Expected Map Mean (EMM) and Kullback-Leiber Divergence. It was concluded that most of these approaches were not able to properly address the probabilistic aspects of the problem and are most likely to fail because of high computational cost and map grid resolution dependency on performance. The authors in [START_REF] Mu | Information-based A-SLAM via topological feature graphs[END_REF] use entropy reduction only over map features and use an entropy metric based on Laplacian approximation with a unified quantification of exploration and exploitation gains.

Recently in the works of [START_REF] Khosoussi | Novel Insights into the Impact of Graph Structure on SLAM[END_REF] and [START_REF] Khosoussi | Reliable Graphs for SLAM[END_REF], the authors exploit the graph SLAM connectivity and pose it as an Estimation over Graph (EoG) problem, where each node (state vector) and vertex (measurement) connectivity is strongly related to the SLAM estimation reliability. By exploiting the spectral graph theory which deals with the Eigenvalues, Laplacian, and Degree matrix of the associated Fisher Information Matrix (FIM) and graph connectivity respectively the authors state that the graph Laplacian is related to the SLAM information matrix and the number of Weighted number of Spanning trees (WST) is directly related to the estimation accuracy in graph SLAM.

The authors in [START_REF] Placed | A General Relationship Between Optimality Criteria and Connectivity Indices for Active Graph-SLAM[END_REF] [15] extend [START_REF] Khosoussi | Reliable Graphs for SLAM[END_REF] by debating that the maximum number of WST is directly related to the Maximum Likelihood Estimate (MLE) of the underlying graph SLAM problem formulated over lie algebra. Instead of computing the D-optimality criterion over the entire SLAM sparse information matrix, a modern D-optimality criterion is computed over the weighted graph Laplacian where each vertex is weighted using edge D-Optimality. Furthermore, it is proven that the maximum number of WST of this weighted graph Laplacian is directly related to the underlying pose graph uncertainty.

III. PRELIMINARIES A. Graph SLAM

Modern SLAM approaches adopt a graphical approach (bipartite graph) where each node represents the robot or landmark pose and each edge represents a pose to pose or pose to landmark measurement. As an example for a robot with four pose states, shown in Fig. 1 x 0:3 , lm 1:3 u 1:3 , m 1:7 represent the robot pose, landmark pose, robot pose measurement, and landmark measurement respectively. The objective is to find the optimal state vector x * which minimizes the measurement error e i (x) weighted by the covariance matrix Ω i ∈ R l×l where l is the dimension of the state vector x as shown in Equation 1. We direct interested readers to [START_REF] Xuexi | SLAM Algorithm Analysis of Mobile Robot Based on Lidar[END_REF] for an introduction and comparison of SLAM methods.

x * = arg min x i e T i (x )Ω i e i (x ) (1)
In A-SLAM the robot has to navigate in an unknown environment by performing actions in the presence of noisy sensor measurements that reduce its state and map uncertainties with respect to the environment. Such a scenario is modeled as an instance of the Partially Observable Markov Decision Process (POMDP) [START_REF] Fox | Active Markov Localization for Mobile Robots[END_REF] [17] [START_REF] Placed | A survey on active simultaneous localization and mapping: State of the art and new frontiers[END_REF].

Let us consider a scenario where the robot is in state x and takes an action a to move to x ′ . The robot's goal is to choose the optimal policy χ * that maximizes the associated expected reward (E) for each state-action pair and it can be modeled as Equation 2:

χ * = argmax t ∞ t=0 EΓ t α(x t , a t ) (2)
Where x t , a t and Γ t are respectively the state, action, and discount factor evolution at time t and α(x t , a t ) is the reward function associated with the action performed to reach x. Although the POMDP formulation of A-SLAM is the most widely used approach, it is considered computationally expensive as it considers planning and decisionmaking under uncertainty. For computational convenience, it is divided into three main steps which identify the potential goal positions/waypoints, compute the cost to reach them, and then select actions based on a utility criterion that decreases map uncertainty and increases the robot's localization.

B. Uncertainty Quantification

The cost or utility is computed based on the reward value of the optimal action selected from a set of all possible actions according to Equation 2. For uncertainty quantification IT and TOED methods are used.

In IT, Shannon entropy/entropy measures the amount of uncertainty associated with a random variable or random quantity. Since robot pose and the map is estimated as a multivariate Gaussian distributions the authors in [START_REF] Stachniss | Information Gain-Based Exploration Using Rao-Blackwellized Particle Filters[END_REF] describe Shannon entropy of the map E ∈ (0, 1) as in Equation 3where the map M is represented as an occupancy grid and each cell c i,j is associated with a probabilistic estimation P (c i,j ) of its value of 1 being occupied and 0 as free. The objective is to reduce both the robot pose and map entropy. However, measuring entropy can be computationally demanding because computing probabilistic estimations of both the robot's position and the map over the entire map area is required along with its associated grid resolution.

E[p(M)] = - i,j (p(c i,j )log(p(c i,j )) + (1 -p(c i,j ))log(1 -p(c i,j )) (3)
Alternatively, if we consider task driven utility functions, the uncertainty metric is evaluated by reasoning over the propagation of uncertainty associated with the Fisher Information Matrix (FIM) ∆ = Ω -1 of graph SLAM. TOED provides many optimally criteria, which give a mapping of the covariance matrix to a scalar value. Less covariance contributes to a higher weight of the action set χ. For a covariance matrix Ω ∈ R n×n and having eigenvalues ζ n , A,D and E optimality criterion are defined which minimize the average variance, covariance ellipsoid and maximum Eigenvalue.

TOED approaches require both the robot pose and map uncertainties to be represented as covariance matrix and may be computationally expensive, especially in landmark-based SLAM where its size increases as new landmarks are discovered. Hence, IT-based approaches are preferred over TOED.

In [START_REF] Khosoussi | Novel Insights into the Impact of Graph Structure on SLAM[END_REF] and [START_REF] Khosoussi | Reliable Graphs for SLAM[END_REF] the authors debate how the graphical topology of SLAM has an impact on estimation reliability. They establish a relationship between the Weighted number of Spanning Trees (WST) and the D-Optimality criterion and show the graph Laplacian is closely related to the FIM. Three graph connectivity metrics are discussed with connection to the SLAM estimation accuracy, namely: 1) Algebraic Connectivity (A.C), defines the robustness/resilience of a graph to stay connected even after the removal of some nodes. For connected undirected graph G = (v , ϵ), having v vertices (poses) and ϵ edges (measurements), the A.C is defined as the second largest Eigen value of the weighted Laplacian λ 2 (L 0 ). 2) Average Degree ( d), indicates the number of edges incident upon its vertices. As

d = 1 v v i=1 deg(i)
increases, the number of measurements also increases which eventually improves the MLE estimate. 3) Tree Connectivity, it is shown that WST is directly related to the determinant of the weighted graph Laplacian and the MLE estimate. It is given as t w (G) = det(L w ), where L w is the weighted Laplacian.

The above graph connectivity indices provide alternate methods that are computationally less expensive to measure SLAM uncertainty as compared to TOED and IT approaches discussed previously. In Section V we will weigh the effectiveness of our proposed method using these indices.

IV. METHODOLOGY

The A-SLAM methods outlined in Section II assess uncertainty either through the overall map entropy or the SLAM's covariance matrix in its entirety, resulting in high computational costs. In this article, we propose a utility function that incorporates the path entropy and distance to the frontier candidate and adds it to the modern D-Optimality criterion as defined in [START_REF] Placed | Fast Autonomous Robotic Exploration Using the Underlying Graph Structure[END_REF]. Computing path entropy is computationally efficient along with penalizing frontiers with large distances. The final utility function not only provides a reliable SLAM estimate but also maximizes the map coverage by minimizing the unknown map area. Figure 2 shows the overview of the proposed utility function. We add this utility function to that of [START_REF] Placed | Fast Autonomous Robotic Exploration Using the Underlying Graph Structure[END_REF] which uses Lidar-based Karto SLAM [START_REF] Konoli | Efficient Sparse Pose Adjustment for 2D mapping[J][END_REF] back-end, with frontier detection over an Occupancy Grid (OG) map. For each frontier centroid (frontier candidate) we apply the Bresenham's line algorithm [START_REF] Bresenham | Algorithm for computer control of a digital plotter[END_REF] to compute the number of pixels and their occupancy values in a straight line from the robot's current position. We then compute the path entropy to each frontier candidate and weigh the paths that have a higher number of unknown cells to favor the robot to explore the unknown environment resulting in better coverage. The lengthy candidate frontier paths are penalized so that the robot does not favor paths longer than a certain threshold and maintains its SLAM accuracy. Finally, the utility is computed by adding the proposed utility to that of [START_REF] Placed | Fast Autonomous Robotic Exploration Using the Underlying Graph Structure[END_REF]. Fig. 3 shows the implementation of the proposed utility function in ROS. For each candidate frontier F = {f 1 , f 2 , .....f N } ∈ R 2 , we get the occupancy values of each path as G n = {m 0 , m 1 , .....m L }, ∀n ∈ N , where m 0 ...m L are the pixel occupancy values of path length L. We assign the probability value of P unk = 0.1 for unknown pixels (with values = -1) to quantify for low entropy and high information gain (as we are more interested in unknown area of the environment). Occupancy values of obstacles and free space are mapped to probability P Of ree = 0.45, weighing high entropy since we are not interested in places already known to the robot. Equation 4 computes the path entropy E n for each frontier candidate with the above assigned probability values.

E n = E n [p(m)] m∈G n = - m∈G n (P (m i,j )log 2 (p(m i,j )) + P (1 -m i,j )log 2 (1 -p(m i,j )), ∀m i,j ∈ M (4)
Once the path entropy is computed it is normalized with the number of pixels within the path K n , as shown in Equation 5. n = {n x , n y } and R = {R x , R y } are the selected frontier and robot positions respectively.

K n = nx i=Rx ny j=Ry m i,j (5)
The path entropy computed in Equation 4 may lead to select frontiers with large distances from the robot. These large distance frontiers may decrease the localization of the robot once it moves toward them. For penalizing these frontiers to contain the SLAM uncertainly, we apply an exponential decay operator γ n as shown in Equation 6 and will use it in computing the utility function U n 2 in Equation 7. In Equation 6λ is the decay rate operator which acts as a tuning factor for removing frontiers with large distances. We fix λ = 0.6 since we assume the environment is static and the frontier path entropy remains constant when the robot moves towards the frontier. n and R are frontier and robot locations in R 2 respectively and dist( * ) measures the Euclidean distance between n and R.

γ n = exp -(λ * dist(R,n)) (6)
The utility U n 2 as shown in Equation 7is computed by weighing normalized entropy E n /K n with ρ n = 10 β , where β is a factor which depends on the number spanning trees of the weighted graph Laplacian (L n w ) computed in Equation 8. More specifically, β is the count of the number of digits before the decimal places of U n 1 and acts as a balancing factor between entropy and the number of spanning trees.

Finally, we obtain the proposed utility function U tot in Equation 9 as the maximum of the sum U n 1 and U n 2 , where Equation 8 is adopted from [START_REF] Placed | Fast Autonomous Robotic Exploration Using the Underlying Graph Structure[END_REF]. The advantage of U tot is that it not only provides a good SLAM estimate based on the modern D-Optimality criterion but also increases the coverage of the unknown map by reducing the frontier path entropy and distance.

U n 2 = (1 -E n /K n ) * ρ n + γ n (7)
U

n 1 = Spann(L n w ) (8)
U tot = max(U n 1 + U n 2 ) (9)
V. SIMULATION RESULTS

The simulations were carried out on ROS Noetic, Ubunto 20.04 on Intel Core i7 ® , with a system RAM of 32GB and NVIDIA RTX 1000 GPU. We used the approach of [START_REF] Placed | Fast Autonomous Robotic Exploration Using the Underlying Graph Structure[END_REF] and implemented the proposed approach as mentioned in IV using Open Karto as SLAM backend, Turtlebot ® equipped with Lidar, Dynamic Window Approach (DWA) [START_REF] Fox | The Dynamic Window Approach to Collision Avoidance[END_REF] and Dijkstra's algorithm as local and global planners from the ROS navigation stack. We compared the proposed approach against two different methods which use Frontier Detection based Exploration (FD) [START_REF] Jiȓí | Map-Merging for multi-robot system[END_REF] and Active Graph SLAM (AGS) of [START_REF] Placed | Fast Autonomous Robotic Exploration Using the Underlying Graph Structure[END_REF] and using two different simulation environments namely the modified Willow Garage (W.G)3 measuring 2072m 2 , having no dynamic obstacles, and modified office (Office) measuring 741m 2 with obstacles 4 . The ground truth occupancy grid maps were generated using the gazebo 2Dmap plugin 5 which uses wavefront exploration.

For qualitative comparison, we weighed our approach with performance metrics relating to graph connectivity, map efficiency, and % of the area covered. We used graph connectivity metrics like Algebraic Connectivity (A.C), average degree ( d), normalized tree connectivity (τ (G)), and evolution of uncertainty measured as edge D-optimality in the resulting posegraph. These metrics are strongly related to SLAM estimation accuracy as described in Section III. Regarding map efficiency matrices, Structural Similarity Index (SSIM) and Root Mean Square Error (RMSE) are used. SSIM∈ (0, 1) indicates the similarity between two maps by computing the covariance of their pixel values.

We conducted 15 simulations of 30 minutes each for both W.G. and office environments using FD, AGS, and our methods. The results from these simulations along with above mentioned performance matrices are elaborated in Table I and Fig. 5. The first benchmark is FD in which the robot is guided towards the nearest frontier and once it reaches the frontier, it is added to the blacklist of chosen frontiers in order to avoid detecting it again. This approach does not take into account the FIM uncertainty of the pose-graph and nor does it favor revisiting already visited frontiers for loop closing. From Table I on both environments, we can conclude that this approach provides good d because Open Karto creates many loop closure constraints between nodes but does not contribute to uncertainly reduction because the nodes are very close to one another. This method severely lacks in A.C (especially in W.G) as compared to our approach. As described in Section III A.C is directly related to the accuracy of SLAM estimation and its higher value is encouraged. Further, we observe a lower SSIM and higher RMSE, which renders this approach not suitable for area coverage tasks as compared to preceding methods. Since this method uses a greedy frontier detection search without any quantification of SLAM uncertainly or loop closure, as a result, the SLAM covariance increases, and exploration halts after some time. Eventually the resulting SLAM pose graph has high unreliability and less coverage as compared to Our method.

The second method in Table I is AGS. We can infer that this method results in good graph connectivity and map quality matrices in both environments as compared to FD. Especially in office environment the graph connectivity metrics and high. High SSIM and low RMSE indicate better SLAM estimation as compared to W.G. because this environment has more obstacles that bring structure to SLAM estimation.

Our method when compared to the preceding methods in Table I outperforms them, especially in A.C, τ (G), SSIM, RMSE, and Map size for both environments. We can observe that for Willow Garage our method has 100% more A.C, 60% more SSIM, and 6% less RMSE error than the last best value. For modified Office we get 67% more A.C, 47% more τ (G), 22% more SSIM, and 37% less RMSE. These promising values indicate the effectiveness of our proposed method. Figure 4 plots the uncertainty evolution over time (s) in the W.G and Office environment. We quantify the uncertainty as D-Optimality for each edge in the entire pose graph. The circles denote the selected goal frontiers. Each method has a different goal frontier detection frequency depending on whether the robot has reached it or not. From Fig. 4 and Table II we can deduce that initially the uncertainty is high and as the robot explores the environment, it decreases. Our approach and AGS manage to keep the uncertainty bounded to 44% and 45% of their maximum threshold respectively, while FD keeps it at 20% hence resulting in a poor SLAM estimate due to lack of loop closure and using a greedy frontier search. In Fig. 5 a comparison of the evolution of the percentage of map discovered is presented. The average values along with the standard deviation of 15 simulations, 30 min each of FD, AGS, and our are shown. Since FD uses a greedy frontier detection and exploration approach it starts with a higher slope than AGS and our method till 1000 and 500 seconds for W.G and Office respectively, after that it the SLAM covariance becomes unbounded, and the slope decreases with the final percentage of discovered map at 25.18% and 55.14% respectively. Both AGS and our method manage to keep the slope of average exploration values constant but our method eventually manages to explore 54%, 30% more area than AGS, and 20%, 25% more area than FD for both W.G and Office environments respectively. In Fig. 6 the resulting occupancy grid maps and pose graphs are overlapped on the ground truth maps to show the area covered by our approach in environments.

VI. RESULTS

The experiments were performed using four-wheel diff.

ROSBot2 robot 3 as shown in Fig. 7a and ROS (noetic) on 20.04.6 (LTS) running on Intel Xeon ® W-2235 CPU 3.80GHz x 12, 64Gb RAM and Nvidia Quadro RTX 4000 GPU.

The environment consists of a room (lab with static obstacles and two corridors measuring 81m 2 in total. We used mapping efficiency and exploration time as performance matrices for experimental results. Fig. 7b shows the computed 3 https://husarion.com/manuals/rosbot/ .

OG map, and SLAM pose graph along with robot start and end positions using our approach. The computed OG map by the robot is overlapped with that of the ground truth (blue). We can observe high similarity between the two maps, rendering the high mapping efficiency of our approach. Figure 8 shows the average percentage of four experiments (2-AGS, 2-Our) for a total duration of 350 seconds. We can observe that initially both AGS and our method approximately map 35% of the environment this is because of the high range (up to 16m) of the Lidar sensor 4 of RosBot 2 robot. With the evolution of time we can observe that using our utility function, a steep slope is obtained from 60 to 210 seconds (approx.) which helps the robot to cover the entire environment in 230 seconds as compared to 280 seconds for AGS. These results are in contrast with Section V and indicate that our approach supersedes AGS and therefore can be utilized for efficient exploration of the environment while maintaining good SLAM efficiency.

VII. CONCLUSIONS

In this article, we have presented a utility function that selects the most favorable frontier goal location within an occupancy grid map for a reliable A-SLAM with an area coverage task. The proposed utility function incorporates path entropy to select the frontier goal location which has the highest amount of unknown cells within its path thus maximizing the area coverage. Using simulation and experimental results on publicly available environment maps we have proved the efficiency of our approach as compared to similar methods. As a future prospective, we plan to incorporate our method in a multi-robot scenario utilizing efficient frontier-sharing for maximum environment exploration.

Fig. 1 :

 1 Fig. 1: Graph SLAM structure

Fig. 2 :

 2 Fig. 2: Framework and proposed utility function

Fig. 3 :

 3 Fig. 3: Proposed utility function ROS implementation, purple line = Bresenham's line, sphere = frontier candidate, Green Squares = frontier centroids.

Fig. 4 :Fig. 5 :

 45 Fig. 4: Uncertainty evolution of AGS, FD and Our in 4a Willow Garage, 4b Office Environments.

Fig. 6 :

 6 Fig.6: Obtained pose using Our approach and ground truth maps (blue).

Fig. 7 :

 7 Fig. 7: Robot used for experiments and the resulting mapped environment overlapped with ground truth map(blue).

Fig. 8 :

 8 Fig. 8: Average Coverage Percentage in four experiments.

TABLE I :

 I Average graph connectivity and map quality comparison of 15 simulations (30 minutes each for every method)

	Env.	Meth.	A.C	d	τ (G)	SSIM	RMSE
		FD	0.104	3.290	1.016	0.05	0.70
	W.G	AGS	0.426	2.907	1.139	0.05	0.64
		Our	0.845	2.925	1.205	0.08	0.60
		FD	3.061	3.179	1.229	0.09	0.83
	Office	AGS	5.740	2.742	1.312	0.07	0.80
		Our	9.617	2.612	1.941	0.11	0.77

TABLE II :

 II Uncertainty reduction (%R) comparison

	Env.	Method	D-Opti(Max&Min)	Diff	%R
		FD	3700&2900	800	20
	W.G	AGS	4800&2600	2200	45
		Our	4700&2600	2100	44
		FD	4600&3400	1200	26
	Office	AGS	4100&2700	1400	34
		Our	4900&2900	2000	40

https://github.com/arpg/Gazebo/.

https://github.com/mlherd/

https://github.com/marinaKollmitz/gazebo.

https://www.slamtec.com/en/Lidar/A2/ .

ACKNOWLEDGMENT

This work was conducted within the framework of the NExT Senior Talent Chair DeepCoSLAM, funded by the French Government through the program "Investments for the Future" administered by the National Agency for Research (ANR-16-IDEX-0007). We also extend our gratitude to the Région Pays de la Loire and Nantes Métropole for their invaluable support in facilitating this research endeavour.

*This work was carried out in the framework of the NExT Senior Talent Chair DeepCoSLAM, which was funded by the French Government, through the program Investments for the Future managed by the National Agency for Research ANR-16-IDEX-0007, and with the support of Région Pays de la Loire and Nantes Métropole.


Emeline Chauchard

Marie Dupau

Yasser Khazaal

Axelle Moreau

Rage in video gaming: Development and Validation of the Video Gaming Rage Scale (VGRS)

Keywords: Online video game, Anger, Rage, Scale validation, Emotion

Rage refers to a decrease in playing abilities, due to sudden manifestations of negative emotions (anger or frustration) and engagement in more aggressive behaviour. Rage is commonly experienced by gamers. The purpose of this article is to assess the psychometric properties of the Video Gaming Rage Scale (VGRS).

A sample of 454 players (81.7% men) completed a set of questionnaires comprising the VGRS and questionnaires assessing internet gaming disorder, anger, impulsivity, anxiety, and depression. The total sample was randomly divided into two distinct samples of a similar size, Group A and Group B, to conduct exploratory and confirmatory factor analyses.

The exploratory factor analysis identified two factors of equal weight and fitted the data from Group A well. The confirmatory factor analysis reported an excellent fit with the data for Group B. Internal reliability was good and the correlation between the two factors was moderate.

The first factor, "Emotion dysregulation", describes the emotional dysregulation of rage, particularly anger. The second factor, "cognitive dysregulation", measures cognitive dysregulation and its impact on decision-making and strategy judgement. This scale can contribute to a new area of research focusing on the trajectories of gamers and problem gamers.

Introduction

Tilt is a term commonly used by players in the field of online poker. It is defined as a "sudden and uncontrolled change in gambling behaviour (loss of control over the game) induced by frustration, a shift which, in turn, affects behavioural, emotional, and cognitive processes" (Moreau et al., 2020, p. 4). The player experiences negative emotions (anger, sadness, guilt), irrational beliefs and dissociative experiences, impacting their ability to make rational decisions and resulting in loss of money [START_REF] Barrault | Special features of poker[END_REF][START_REF] Browne | Going on tilt: Frequent poker players and control[END_REF][START_REF] Moreau | Le TILT au Poker en ligne: Un comportement pathologique transitoire?[END_REF][START_REF] Moreau | Online Poker Tilt Scale (OPTS): Creation and validation of a tilt assessment in a French population[END_REF][START_REF] Palomäki | This is just so unfair!': A qualitative analysis of loss-induced emotions and tilting in on-line poker[END_REF][START_REF] Palomäki | Losing More by Losing It: Poker Experience, Sensitivity to Losses and Tilting Severity[END_REF] In the field of video gaming, the terms "tilt" and "rage" have been adopted by gamers. Both describe a state of frustration and anger, unmanaged by the gamer, which leads to a loss of control and a decrease in the abilities needed to perform in the game [START_REF] Moreau | Rage in video gaming, characteristics of loss of control among gamers: A qualitative study[END_REF][START_REF] Przybylski | Competence-impeding electronic games and players' aggressive feelings, thoughts, and behaviors[END_REF][START_REF] Wu | Understanding Tilt in Esports: A Study on Young League of Legends Players[END_REF]. To differentiate the phenomenon when it occurs while playing poker or while playing video games, we will use the term "tilt" for poker and "rage" for video games. Rage affects players on emotional, cognitive, and behavioural levels [START_REF] Moreau | Rage in video gaming, characteristics of loss of control among gamers: A qualitative study[END_REF]. Rage-related phenomena are discussed online by many gamers, on various websites (e.g. Reddit, forums). It is an integral part of the gaming experience. For instance, one outcome could be "rage-quitting", where a gamer suddenly quits the game to manage a high-intensity, negative emotional state [START_REF] Przybylski | Competence-impeding electronic games and players' aggressive feelings, thoughts, and behaviors[END_REF].

Three qualitative studies on rage while gaming -focusing on different samples: children [START_REF] Kahila | If the game does not work, it is lagging, or you die in game, you just get furious[END_REF], adolescents engaged in e-sport [START_REF] Wu | Understanding Tilt in Esports: A Study on Young League of Legends Players[END_REF] and young adult online gamers [START_REF] Moreau | Rage in video gaming, characteristics of loss of control among gamers: A qualitative study[END_REF] -have been published recently. All three studies gave similar descriptions of rage. The triggers of rage included interactions with other players, whether team members, friends, or strangers; disappointment with one's own game performance; comparisons with other players; repeated failure and loss; and technical problems (hardware, game or Internet). Study participants also reported that the more competitive the game, the more likely they were to rage [START_REF] Kahila | If the game does not work, it is lagging, or you die in game, you just get furious[END_REF][START_REF] Moreau | Rage in video gaming, characteristics of loss of control among gamers: A qualitative study[END_REF][START_REF] Wu | Understanding Tilt in Esports: A Study on Young League of Legends Players[END_REF], and described rage as a transitory emotional reaction [START_REF] Moreau | Rage in video gaming, characteristics of loss of control among gamers: A qualitative study[END_REF]. The main emotional states associated with rage were anger and frustration, followed by feelings of annoyance, irritation, and guilt [START_REF] Moreau | Rage in video gaming, characteristics of loss of control among gamers: A qualitative study[END_REF][START_REF] Wu | Understanding Tilt in Esports: A Study on Young League of Legends Players[END_REF].

On a cognitive level, study participants described confusion, a decline in ability to concentrate, impulsive decision-making, and dissociative experiences [START_REF] Moreau | Rage in video gaming, characteristics of loss of control among gamers: A qualitative study[END_REF][START_REF] Wu | Understanding Tilt in Esports: A Study on Young League of Legends Players[END_REF]. When asked about behavioural responses, they reported a loss of control, physical (on hardware) and verbal aggression, and a difference in their usual game play [START_REF] Kahila | If the game does not work, it is lagging, or you die in game, you just get furious[END_REF][START_REF] Moreau | Rage in video gaming, characteristics of loss of control among gamers: A qualitative study[END_REF][START_REF] Wu | Understanding Tilt in Esports: A Study on Young League of Legends Players[END_REF].

Study participants reported using various regulation strategies to control their rage episodes. Some said they would quit the game, namely "raged quitting", to cut off the source of rage.

Then, they would express their emotions by crying or screaming. They might also attempt to change their emotional state by trying to calm down or changing activity; and by rationalizing their anger [START_REF] Kahila | If the game does not work, it is lagging, or you die in game, you just get furious[END_REF][START_REF] Moreau | Rage in video gaming, characteristics of loss of control among gamers: A qualitative study[END_REF][START_REF] Wu | Understanding Tilt in Esports: A Study on Young League of Legends Players[END_REF]. The possible consequences of rage included losing the game, broken equipment, loss of pleasure and interest in the game, and a spillover effect on other gamers. Few participants reported a feeling of emotional release [START_REF] Moreau | Rage in video gaming, characteristics of loss of control among gamers: A qualitative study[END_REF].

While most gamers experience some degree of rage when gaming, some may experience problematic or repeated episodes. Rage and its consequences should therefore be studied to better understand their possible role in gaming disorder and the negative effects of gaming (anger, sleep disturbance, emotional distress, -etc.). Moreover, the regulation of rage could be a specific focus of prevention programmes that reinforce emotional and cognitive regulation. First however, to better understand rage and its implications, we developed the Video Gaming Rage Scale (VGRS) based on the results of the qualitative study [START_REF] Kahila | If the game does not work, it is lagging, or you die in game, you just get furious[END_REF][START_REF] Moreau | Rage in video gaming, characteristics of loss of control among gamers: A qualitative study[END_REF]. This article assesses the psychometric properties of the Video Gaming Rage Scale (VGRS).

Methodology

Participants

To be included in the study, participants had to be 18 years old, fluent in French, and have played video games at least once in the last month. In total, 1054 people accessed the online questionnaire. Six hundred biased or incomplete questionnaires were excluded from our sample, which accounted for 57% of the recorded answers. The final sample consisted of 454 players, of whom 18.3% (n = 83) were women. The total sample was randomly divided into two distinct samples of similar size, named Group A and Group B. Sociodemographic and gaming characteristics are reported in Table 1. The two groups were statistically similar in these characteristics.

Materials

The online survey included several questions on sociodemographic data, namely, age, gender and level of education, as well as gaming-related information such as gaming frequency, length of gaming sessions and types of gaming activity. Additionally, participants

were invited to answer the following questionnaires:

The Video Gaming Rage Scale (VGRS)

VGRS was developed based on a qualitative study describing rage among gamers [START_REF] Moreau | Rage in video gaming, characteristics of loss of control among gamers: A qualitative study[END_REF], previous studies, and a questionnaire on online poker tilt [START_REF] Moreau | Le TILT au Poker en ligne: Un comportement pathologique transitoire?[END_REF][START_REF] Moreau | Online Poker Tilt Scale (OPTS): Creation and validation of a tilt assessment in a French population[END_REF][START_REF] Moreau | Tilt in online poker: Development of a short version of the online poker tilt scale[END_REF]. In the qualitative study [START_REF] Moreau | Rage in video gaming, characteristics of loss of control among gamers: A qualitative study[END_REF], we identified nine main topics to characterise rage: anger, impaired reasoning skills, verbal aggression, tension/restlessness, impulsivity, loss of game/decrease in level of play, loss of pleasure/interest, relationship with gaming partners/contagious aspects of rage, and rage quit. First, we created three items for each topic -27 items in total. This first version was trialled with a sample of 25 gamers. In view of comments from participants in this sample, we modified some wording and identified the "rage quit" aspect as an outcome of rage rather than a component of the phenomenon.

We decided to exclude it from the VGRS but recommend adding a subsidiary question about rage-quit frequency. The 24 items in the VGRS questionnaire are presented in Table 2. The instructions for the questionnaire were: "Please indicate how often you have encountered the following situation in the past three months -'When I play online video games, there are times when...' ". Participants answered the questionnaire on a Likert scale with five different responses: "Never" (0), "Rarely" (1), "Sometimes" (2), "Often" (3), "Almost Always" (4).

Severity of Tilting Scale

Two items from the Palomäki, Laakasuo, & Salmela study on tilt in poker [START_REF] Palomäki | Losing More by Losing It: Poker Experience, Sensitivity to Losses and Tilting Severity[END_REF] were adapted to video gaming to measure the perceived frequency of tilting episodes.

The question "According to you, how many times have you raged in the past 3 months?" was rated on a 7-point Likert scale ranging from 0 to more than 10 times and "Rage has been a problem for me in the last 3 months" was rated on a seven-point Likert scale ranging from "Completely Disagree" (1) to "Completely Agree" (7).

Online Poker Tilt Scale (OPTS-9)

The nine-item questionnaire initially used to measure the frequency of tilt episodes in online poker [START_REF] Moreau | Tilt in online poker: Development of a short version of the online poker tilt scale[END_REF] was adapted to the context of gaming. The instruction was "When I play video games online, there are times when...". Responses were reported on a 5-point Likert scale from 1 to 5 ("Never", "Rarely", "Sometimes", "Often", "Almost Always"). Scores range from 0 to 36 and the higher the score, the more frequent the episodes of rage. OPTS-9 had good reliability in our sample (ω = 0.832)

Internet Gaming Disorder Test (IGDT-10)

IGDT-10 ( [START_REF] Király | Validation of the Ten-Item Internet Gaming Disorder Test (IGDT-10) and evaluation of the nine DSM-5 Internet Gaming Disorder criteria[END_REF] was used to evaluate Internet Gaming Disorder (IGD) using the 9 criteria in DSM-5 (American Psychiatric Association, 2013). Participants answered 10 questions on a three-point Likert scale ("Never", "Sometimes", and "Often"). "Often" was scored as "Yes" (1) and "Never" and "Sometimes" were scored as "No" (0). Items 9 and 10 refer to the same criterion and are combined in the scoring. The composite score of IGDT-10 ranges from 0 to 9. A score of more than 5 indicates a potential internet gaming disorder.

IGDT-10 had low reliability in our sample (ω = 0.658).

Clinical Anger Scale (CAS)

The CAS is a measure of anger [START_REF] Snell | The clinical anger scale: Preliminary reliability and validity[END_REF]. This scale is made up of 21 items that assess anger levels in a variety of situations to determine how the individual manages anger. Each item is composed of four statements and participants must choose the one that best corresponds to their feelings or situation. Each statement is associated with a score from 0 to 3. Total scores range from 0 to 63. The CAS had good reliability in our sample (ω = 0.891).

UPPS Impulsive Behaviour Scale

The 20 items in the UPPS Impulsive Behaviour Scale-short form [START_REF] Billieux | Validation of a short French version of the UPPS-P Impulsive Behavior Scale[END_REF] assessed five aspects of impulsivity: urgency, positive urgency, lack of premeditation, lack of perseverance, and sensation seeking. Participants answered on a Likert scale ranging from 1 "I strongly agree" to 4 "I strongly disagree". Overall scores range from 20 to 80. A high score on a sub-scale and on the overall score corresponds to a high level of impulsivity. In our sample, the UPPS had good reliability (ω = 0.821).

Hospital Anxiety and Depression Scale (HADS)

The Hospital Anxiety and Depression Scale (HADS) is used to evaluate depressive and anxiety symptoms [START_REF] Roberge | A psychometric evaluation of the French Canadian version of the Hospital Anxiety and Depression Scale in a large primary care population[END_REF][START_REF] Zigmond | The Hospital Anxiety and Depression Scale[END_REF]. The HADS comprises 14 items divided into two aspects -anxiety and depression. For each item, there are four response options coded from 0 to 3. The score for the two subscales varies between 0 and 21. The higher the scores, the more severe the symptomatology. HADS presented good reliability in our sample (ωdepression = 0.688; ωanxiety = 0.768).

Procedure

The questionnaire was created using LimeSurvey software and was available online from March 2019 to September 2019. Participants accessed the questionnaire via an announcement posted on social media gaming groups (primarily Facebook and Twitter). At the beginning of the questionnaire, all participants agreed to give their informed consent and participation was anonymous. This study followed the ethical guidelines set out in the Declaration of Helsinki and was approved by the ethics committee (approval number: 07122022)

Statistical Analysis

To begin, an exploratory factor analysis was performed on the 24 items in the VGRS questionnaire using a varimax rotation and taking half of the study population (Group A). As a first step, the scale, which was based on eight topics from a qualitative study, was reduced to two items per topic by deleting the item with the lowest saturation. Then, a second exploratory factor analysis was performed on these 16 items in the VGRS to identify the structure of the questionnaire. Kaiser criterion (factors with Eigen values greater than 1) and a scree-plot examination (keep all factors before the breaking point or elbow) were used to select factors. Items were assigned to a factor if they weighted greater than 0.45 on that factor [START_REF] Tabachnick | Using Multivariate Statistics[END_REF]. Finally, a confirmatory factor analysis was performed on the other half of the population (Group B). Several adjustment indices were used to evaluate the model fully [START_REF] Tabachnick | Using Multivariate Statistics[END_REF][START_REF] Thompson | Ten commandments of structural equation modeling[END_REF]:

(1) the model chi square divided by degree of freedom (χ 2 / df. With a large sample size, this value should be lower than 2.00. The lower this value, the better the fit),

(2) the Tucker-Lewis index (TLI. This value should be 0.90 or higher for a good fit; the higher this value, the better the fit) [START_REF] Bentler | Significance tests and goodness of fit in the analysis of covariance structures[END_REF][START_REF] Tucker | A reliability coefficient for maximum likelihood factor analysis[END_REF],

(3) the Comparative Fit Index (CFI. This value should be 0.90 or higher for a good fit; the higher this value, the better the fit) [START_REF] Bentler | Comparative fit indexes in structural models[END_REF], (4) the Root Mean Square Residual (RMR. This value should be 0.05 or lower; the lower this value, the better the fit), and

(5) the Root Mean Square Error of Approximation (RMSEA. This value should be 0.08 or lower; the lower this value, the better the fit) [START_REF] Snell | The clinical anger scale: Preliminary reliability and validity[END_REF].

The internal consistency of the subdimensions was determined by the ω McDonald.

To test for convergent validity, Pearson's correlations were conducted to examine relationships between scores on the VGRS and other variables (Sample A and B).

The statistics were generated using Jamovi software.

Results

Descriptive statistics

The overall population consisted of 454 participants, of whom 369 were men (81.3%) and the mean age was 25.7 years (SD = 8.5 years). Table 1 presents the sociodemographic and gaming characteristics of the overall population and for Groups A and B. When it came to gaming behaviour, 5% (n = 23) were potential problem gamers.

Item Suppression

To reduce the scale, we conducted an exploratory factor analysis (EFA) using the Group A sample and including the 24 items used to create the VGRS. The EFA proposed a two-factor solution (KMO = .897; Bartlett test, p > .001). This model explained 38.8% of the variance. The factor matrix and the items associated with each theme are presented in Table 2. At this first step, and for each of the eight topics, we deleted the item that had the lowest saturation. All items from the topic "loss of pleasure" were excluded because they weighted below 0.45. Finally, we selected 14 items for exploratory analyses.

Exploratory analyses for VGRS

Still considering the sample for Group A, a second EFA was performed using the 14 items selected in the item suppression step. Four more items had to be deleted. Item 1 was deleted because its meaning and saturation were similar to those for item 9 and its removal improved the fit of the scale (RMSEA = 0.0802). Items 2 and 20 were rejected due to their low saturation (< .45). As a result, the "relationship with partners/contagious aspects" topic is no longer represented in this scale and like rage quit, this aspect should be considered a consequence rather than a feature of rage episodes. Finally, item 15 was deleted, because it loaded similarly on both factors.

A third EFA was performed with the 10 remaining items (Table 3). This explained 53.7% of the variance, which is more than the 50% recommended for a meaningful factor solution. The Bartlett test was significant (p < .001) [START_REF] Floyd | Factor analysis in the development and refinement of clinical assessment instruments[END_REF], and KMO= .833. The factor matrix for the 10 items in the VGRS is shown in Table 3. The analysis produced a two-factor solution: the first factor, combining items oriented toward cognitive regulation, named "cognitive dysregulation"; and the second, considering items oriented toward emotional regulation, named "emotion dysregulation". These factors presented an equivalent percentage of explained variance, respectively, 28.5 and 25.2%.

Confirmatory factor analysis for VGRS on Group B

The two-factor model derived from the EFA was tested on the second half of the sample, Group B, with a confirmatory factor analysis. The correlated model fitted the data well (χ 2 / df = 2, TLI = 0.946, CFI = 0.959, RMR = 0.043, RMSEA = 0.067). All loadings were significant and the correlation between the two factors was moderate (from 0.35).

Reliability

The ω McDonald coefficient for reliability was 0.834 for emotional dysregulation and 0.838 for cognitive dysregulation, indicating good internal reliability.

Correlation

The two factors moderately correlated (r = 0.351).

Discussion

The aim of this study was to create and explore the psychometric properties of the VGRS. First, the exploratory factor analysis identified two factors of equal weight and fitted the data from the Group A sample well. Then, the confirmatory factor analysis reported an excellent fit to the data for the Group B sample. The final version of the scale included 10 items divided into two distinct factors. The first factor refers to the emotional dysregulation of rage, particularly anger. The second factor measures cognitive dysregulation and its incidence on decisionmaking and strategy judgement. These two factors are concordant with the two aspects assessed by the Online Poker Tilt Scale (OPTS) [START_REF] Moreau | Tilt in online poker: Development of a short version of the online poker tilt scale[END_REF] and are in line with the results of qualitative studies published about rage [START_REF] Kahila | If the game does not work, it is lagging, or you die in game, you just get furious[END_REF][START_REF] Moreau | Rage in video gaming, characteristics of loss of control among gamers: A qualitative study[END_REF][START_REF] Wu | Understanding Tilt in Esports: A Study on Young League of Legends Players[END_REF]. The VGRS measures the occurrence of typical characteristics of rage, without directly naming the rage. The higher the score on the scale, the more frequent the episodes of rage. No cut-off score was defined.

Components from three topics identified by the qualitative study [START_REF] Moreau | Rage in video gaming, characteristics of loss of control among gamers: A qualitative study[END_REF] were excluded, namely, "rage quit", "relationship with partners" / "contagious aspects", and "loss of pleasure" (Table 2). These three aspects are outcomes or consequences of rage episodes, and do not characterise it. However, we suggest the use of additional questions to the VGRS to assess the outcomes of rage on the behaviour of gamers (rage quit), their relations with gaming partners and on their own pleasure and interests in gaming (Supplementary material).

Low to medium correlations indicate that rage is an independent construct of video game addiction, anger, and impulsivity. Nevertheless, these variables are related, therefore, they deserve to be investigated in future research.

Rage is a distinct and emerging construct. The VGRS is a new scale in the field of video gaming, which can be used to measure transitory loss of control and experiences of unpleasant emotions while gaming. This scale can contribute to new areas of research that focus on the trajectories of gamers and problem gamers and which aim to understand emotional regulation and the emotional skills deployed while playing video games and esport. It could have clinical and preventative implications. The correlations found with Internet Gaming Disorder may indicate a possible contribution of such aspects to gaming disorder. Further longitudinal studies are called for. The frequency of rage episodes could help to detect problem players and could be central to prevention by encouraging gamers to develop emotional skills and regulation. Emotional skills are identified as crucial for preventing addictive behaviours and could be used and well received by video game players [START_REF] Marchica | The Role of Emotion Regulation in Video Gaming and Gambling Disorder: A Systematic Review[END_REF].

This study has several limits. As we used online data collection, our sample could not be considered representative of the video gaming population. Indeed, the sample is mainly composed of men, while women constitute 48% of the online video gaming population (Entertainment Software Association., n.d.). Finally, the psychometric properties of the scale were evaluated using a sample of French or French speaking gamers. Validation of the scale needs to be replicated among various samples of gamers, in different languages.

Conclusions

Rage is well known and identified by gamers but has been understudied in the research field.

Rage refers to a decrease in playing abilities, due to sudden manifestations of negative emotions (anger or frustration) and engagement in more aggressive behaviour. It is triggered by interactions with other players, disappointment with one's own game performance, repeated failure and loss, or technical issues [START_REF] Kahila | If the game does not work, it is lagging, or you die in game, you just get furious[END_REF][START_REF] Moreau | Rage in video gaming, characteristics of loss of control among gamers: A qualitative study[END_REF][START_REF] Wu | Understanding Tilt in Esports: A Study on Young League of Legends Players[END_REF]. The VGRS is designed to assess rage episodes. After performing exploratory and confirmatory analysis, the scale comprises ten items, divided into two factors: emotion dysregulation and cognitive dysregulation. The scale has good psychometric properties. The VGRS will be useful for developing studies in the field of problematic gaming and e-sport. The numeration of the items corresponds to the order in which they appear in the questionnaire the participants completed. Items in bold correspond to the selected items included in the second round of the exploratory analysis of the VGRS.

  The factors cognitive dysregulation (CD) and emotional dysregulation (ED) correlated with the number of episodes (rCD = 0.227; rED = 0.603); estimation of rage as a problem (rCD = 0.310; rED = 0.534); the scores for Internet Gaming Disorder (rCD = 0.171; rED = 0.287); the Clinical Anger Scale (rCD = 0.152; rED =0 .349); Positive urgency (rCD = 0.163 ; rED = 0.201); Negative urgency (rCD = 0.280; rED = 0.283); Lack of perseverance (rCD = 0.200; rED = 0.128); Lack of premeditation (rCD = 0.172; rED = 0.128); Anxiety (rCD = 0.129; rED = 0.205); and Depression(rCD = 0.107; rED = 0.211).

Table 1 .

 1 Descriptive statistics for the samples

	All Sample	Group A	Group B
	N= 454	n= 227	n= 227

Table 2 .

 2 Themes associated to each item in the VGRS and factor matrix for exploratory analysis after Varimax rotation (Sample A)

	Themes	Items from the original version	Factor 1	Factor 2
	Anger	1. I am angry [Je suis en colère]	.757
		5. Others make me angry [Les autres me mettent en colère]	.546
		9. I feel anger [Je ressens de la colère]	.733
	Alteration in	3. My decisions are no longer rational [Mes décisions ne sont plus rationnelles]
	reasoning	26. I
	abilities

play poorly [Je joue mal] 16. I make bad choices [Je fais de mauvais choix]

		.496
		.722
		.786
	Verbal	19. I
	Aggression

insult others [J'insulte les autres] 18. I criticise the game out loud [Je critique le jeu à voix haute] 17

  . I swear [Je dis des gros mots]

		.736
		.656
		.562
	Tension /	6. I hit my hardware hard (mouse, keyboard, gamepad) [Je tape fort sur mon
	restlessness	matériel (souris, clavier, manette)]
		7. I

am physically tense [Je suis tendu physiquement ] 14. I feel irritated [Je me sens énervé]

		.499
		.515
		.762
	Impulsivity	15. I play without thinking about the consequences [Je joue sans penser aux conséquences] 27. .496 .767
		<.45
	Loss of the	12. I
	game/decrease in
	level of play

I act without thinking [J'agis sans réfléchir] 10. I take more risks [Je prends plus

de risque]

no longer apply the strategy necessary to succeed [Je n'applique plus la stratégie nécessaire pour réussir]

  8. I am below my usual level [Je suis en dessous de mon niveau habituel]11.

I am not efficient [Je ne suis pas efficace]

  I felt like I was wasting my time [J'ai eu l'impression de perdre mon temps] 21. I thought the game was poorly done [J'ai trouvé que le jeu était mal fait] 13. I don't enjoy the game anymore [Je n'ai plus de plaisir à jouer]

			.700
			.608
			.720
	Loss of pleasure,	23. <.45	<.45
	lack of interest	<.45	<.45
		<.45	<.45
	Contagious	2.
	aspects/relations
	hip with partners

My gaming partners told me to calm down [Mes partenaires de jeu m'ont dit de me calmer] 20. I annoy my teammates [J'énerve mes coéquipiers] 4

  . My teammates reproach me [Mes coéquipiers me font des reproches]

	.557
	.478
	<.45
	Note:

Table 3 .

 3 Factor matrix for the second exploratory analysis performed on the 14 items selected in the first step (Group A sample)

	VGRS Items

This study followed the ethical guidelines set out in the Declaration of Helsinki and was approved by the local ethics committee (approval