Julie Boiché
email: julie.boiche@univ-montp1.fr

Mélissa Plaza

Aïna Chalabaev

Emma Guillet

Philippe Sarrazin

Social Antecedents and Consequences of Sport Gender Stereotypes during Adolescence

Keywords: adolescent attitudes, sports (attitudes toward), athletic participation, socialization, parental attitudes, sex role attitudes

Eccles' Expectancy-Value Model (Fredricks & Eccles, 2004) posits that the stereotypes endorsed by parents may influence their children's participation in leisure-time activities. This influence is presumed to occur through the mediating role of children's perceived competence and value given to the activity, predicting in turn dropout. Previous research is scarce relative to (a) the parent-child transmission of gender stereotypes and (b) the relationships among stereotypes, selfperceptions, and decision to drop out from an activity. We present three studies that examined gender stereotypes in the sport context to test these under-explored aspects of the model. Study 1 revealed significant links among perceived gender stereotypes in the social environment (i.e., general and parental beliefs), personal endorsement of stereotypes, and dropout behavior among 347 adolescents. Study 2 revealed no bound between stereotypes assessed among parents and 104 adolescent athletes. It further indicated that self-perceptions may mediate the relationship between 155 adolescents' gender stereotypes and intentions to drop out from sport. Study 3 involved 23 parent-adolescent dyads and revealed that parents' and adolescents' endorsement of gender stereotypes were not significantly related when assessed with explicit measures, but significantly correlated when assessed through an implicit test. Taken as a whole, the results of this set of studies suggest that gender sport stereotypes are conveyed from social environment to adolescents and that they can lead to dropout. The implications for parents and practitioners are discussed.

Social Antecedents and Consequences of Sport Gender Stereotypes during Adolescence

Adolescence is a key period with regard to choices involving both vocational orientations and extra-curricular activities [START_REF] Boyd | Lifespan development[END_REF]. Social psychologists advance that these decisions are partly influenced by stereotypes, that is, shared beliefs concerning the characteristics-both personality traits and behaviors-of social groups [START_REF] Leyens | Stereotypes, social cognition, and social explanation[END_REF]. More particularly, young women's and men's choices could be differentially encouraged by their parents depending on their sex. For instance, sport is conceived as a male arena in Western societies [START_REF] Jacobs | Changes in children's self-competence and values: Gender and domain differences across grades one through twelve[END_REF], and the social environment, notably parents, is thought to play a key role in the transmission of such beliefs. Indeed, parents may provide experiences and opportunities to engage in the sport context that differ between their daughters and sons. One of the most fruitful theoretical frameworks devoted to the understanding of such differentiated socialization is the Expectancy-Value Model (EVM) of Eccles and her collaborators [START_REF] Eccles | Gender role socialization in the family: A longitudinal approach[END_REF][START_REF] Fredricks | Parental influences on youth involvement in sports[END_REF]).

In the next section, we focus on the reasons why these gender-based socialization processes are detrimental to girls and women in the sport context. Then, we present the mechanisms through which parents treat boys and girls differentially from an EVM perspective, focusing specifically on parents' transmission of pervasive gender stereotypes to their children.

We review empirical support of this model to date, and we point out the tenets in need of further investigation.

Sport Participation in Youth

Gender differences in sport are often perceived as resulting from natural biological factors.

During pubescent development, physical capacities develop more among male than female adolescents, leading to a significant gap in physical abilities between them [START_REF] Eagly | The science and politics of comparing women and men[END_REF].

However, these gender differences can also be explained by social processes. Indeed, since childhood boys participate more in motor activities than girls [START_REF] Hines | Androgen, estrogen, and gender: Contributions of the early hormone environment to gender-related behavior[END_REF], and sport dropout rates are more important among adolescent girls, compared to boys [START_REF] Dumith | Physical activity change during adolescence: A systematic review and a pooled analysis[END_REF]French Minister for Youth and Sport, 2001).

This absence of involvement or increased dropout among girls, female adolescents, and women is preoccupying, considering the benefits associated with regular sport participation.

First, this kind of extra-curricular activity has been shown to foster positive youth development [START_REF] Mahoney | Organized activity participation, positive youth development, and the over-scheduling hypothesis[END_REF]. In particular, it has been associated with improved academic functioning [START_REF] Marsh | School athletic participation: Mostly gain with little pain[END_REF] and development of positive peer relationships [START_REF] Smith | Youth peer relationships in sport[END_REF]. Also, sport participation could represent an effective way to buffer the potential sexual objectification of girls-a phenomenon appearing during puberty due to physical changes, in particular breast growth. Recent theorization on embodying experiences indicates that competitive athletics may be a source of positive body image, which in turn is likely to lead to less objectification inside and outside sports [START_REF] Menzel | Embodying experiences and the promotion of positive body image: The example of competitive athletics[END_REF]. Additionally, insufficient exercise may have detrimental health consequences, such as increased risks of suffering from type-2 diabetes, certain cancers, and cardiac conditions [START_REF] Bauman | Updating the evidence that physical activity is good for health: An epidemiological review 2000-2003[END_REF]. Although these pathologies mostly concern adults, it is noteworthy that optimal physical development during childhood and adolescence is conditioned by regular energy expense and that sport participation during these periods is a significant predictor of adopting an active lifestyle during adulthood [START_REF] Perkins | Childhood and adolescent sports participation as predictors of participation in sports and physical fitness activities during young adulthood[END_REF]. Given that more boys than girls reach adequate levels of daily physical activity [START_REF] Knisel | Sport motivation and physical activity of students in three European schools[END_REF], gender-differentiated socialization in the sport context is concerning.

Expectations of Success and Value

According to the EVM, the two most proximal variables determining a young individual's sustained participation in an activity are (a) his/her expectations of success in the activity and (b) the value he/she places into it. Expectations of success are the chances of success estimated by the individual in a given situation. This variable is assumed to be directly related to perceived ability or competence, as well as the perceived difficulty of the task [START_REF] Fredricks | Parental influences on youth involvement in sports[END_REF]. Numerous studies have observed a significant link between children or adolescents' expectations of success or perceptions of competence and their physical activity or sport involvement. For example, researchers reported significant correlations between such perceptions and the time spent being physically active [START_REF] Bois | Elementary schoolchildren's perceived competence and physical activity involvement: The influence of parents' role modelling behaviours and perceptions of their child's competence[END_REF][START_REF] Eccles | Gender differences in sport involvement: Applying the Eccles' expectancy-value model[END_REF][START_REF] Kimiecik | Relationships among children's beliefs, perceptions of their parents' beliefs and their moderate to vigorous physical activity[END_REF]. Similarly, others observed that perceived competence negatively predicted dropout from the activity [START_REF] Guillet | Understanding female sport attrition in a stereotypical male sport within the framework of Eccles's expectancy-value model[END_REF][START_REF] Sarrazin | Motivation and dropout in female handballers: A 21-month prospective study[END_REF].

The second key variable, namely the value accorded to the activity, comprises four distinct components: (a) the attainment value or importance of doing well in the activity; (b) the intrinsic value or inherent interest in the activity; (c) the utility value, which refers to the potential positive outcomes of the activity; and (d) the costs or negative aspects of engaging in the activity. This last variable emerged as a strong predictor of motivation in past research [START_REF] Weiss | Attraction-and entrapment-based commitment among competitive female gymnasts[END_REF], whereas the attainment, intrinsic, and utility values were found to be positively associated with sport involvement [START_REF] Eccles | Gender differences in sport involvement: Applying the Eccles' expectancy-value model[END_REF]. Attainment and utility components are also negatively associated with dropout [START_REF] Guillet | Understanding female sport attrition in a stereotypical male sport within the framework of Eccles's expectancy-value model[END_REF]. Other studies on interest or intrinsic motivation indicated similar results (e.g., [START_REF] Fontayne | Effet du genre sur le choix et le rejet des activités physiques et sportives en éducation physique: Une approche additive et différentielle du modèle de l'androgynie. [Effect of gender on choice and rejection of sports and physical activities in physical education classes: An additive and differential approach of the androgyny model[END_REF][START_REF] Sarrazin | Motivation and dropout in female handballers: A 21-month prospective study[END_REF]. The EVM posits that a child's gender is one of the most central variables contributing to such context-specific perceptions.

Gender, Expectations of Success, and Value of Sport

Past research has consistently observed gender differences in the levels of perceived competence and value in sport across age and culture. For example, [START_REF] Eccles | Gender differences in sport involvement: Applying the Eccles' expectancy-value model[END_REF] reported that, compared to boys, sixth-grade girls rated themselves as significantly less able in sports, and girls conceived this domain as less important, useful, and enjoyable. Similar gender differences in perceived sport competence and value have been observed among American kindergarten and elementary school students [START_REF] Fredricks | Family socialization, gender, and sport motivation and involvement[END_REF] and Malaysian adolescent athletes [START_REF] Chin | Sex, age group and locality differences in adolescent athletes' beliefs, values and goal orientation in track and field[END_REF]. Finally, gender differences in perceived sport competence have been replicated among a sample of French children [START_REF] Bois | Parents' appraisals, reflected appraisals, and children's self-appraisals of sport competence: A yearlong study[END_REF].

The EVM suggests that these differences are partly due to the endorsement of sport stereotypes [START_REF] Fredricks | Parental influences on youth involvement in sports[END_REF]. Several studies brought empirical support to this hypothesis. [START_REF] Chalabaev | Stereotype endorsement and perceived ability as mediators of the girls' gender orientation-soccer performance relationship[END_REF] observed that the more female adolescents agreed with the stereotype that soccer is masculine (in accordance with stereotypes existing in France with regard to this activity), the less they felt competent in this sport. In another study, Chalabaev and Sarrazin (2009) compared adolescents' self-perceptions and motivations in a masculine (soccer) and in a feminine activity (dance). It appeared that students felt more competent and were more autonomously motivated when the stereotypes associated with the activity were favorable to their gender (i.e., dance for girls and soccer for boys). Taken together, these studies indicate that gender stereotypes in sport are likely to impact self-perceptions and behaviors in the athletic area (for a review, see [START_REF] Chalabaev | The influence of sex stereotypes and gender roles on sport participation and performance: Review and future directions[END_REF]. However, no study to our knowledge has specifically investigated the mediating role played by self-perceptions in the relationship between stereotype endorsement and actual participation (albeit in sport or in other contexts).

The Transmission of Gender-Sport Stereotypes

The EVM posits that stereotypes are socially transmitted by the cultural milieu in which children and adolescents grow up [START_REF] Fredricks | Parental influences on youth involvement in sports[END_REF]. Several members of the social environment may contribute to the transmission of gender-sport stereotypes, such as teachers [START_REF] Chalabaev | Can sex-undifferentiated teacher expectations mask an influence of sex stereotypes? Alternative forms of sex bias in teacher expectations[END_REF] or peers [START_REF] Wachs | The boundaries of difference: Negotiating gender in recreational sport[END_REF]. However, the EVM more specifically focuses on the crucial role played by parents in this regard. Several studies examined the links between parents' and their children's perceptions in sport. For example, parents' perceptions of their child's ability were consistently found to be positively related to their child's self-perceptions of ability, either directly [START_REF] Jacobs | The impact of mothers' gender role stereotypic beliefs on mothers' and children's ability perceptions[END_REF] or indirectly through reflected appraisals [START_REF] Bois | Parents' appraisals, reflected appraisals, and children's self-appraisals of sport competence: A yearlong study[END_REF]. In the same vein, the value attached to sport participation of children, according to parents, was positively correlated with sport value reported by children [START_REF] Fredricks | Children competence and value beliefs from childhood through adolescence: Growth trajectories in two male-sex-typed domains[END_REF][START_REF] Jodl | Parents' roles in shaping early adolescents' occupational aspirations[END_REF]).

In addition, several studies reported differentiated perceptions of parents according to their child's gender, with boys being perceived as more competent than girls [START_REF] Jacobs | The impact of mothers' gender role stereotypic beliefs on mothers' and children's ability perceptions[END_REF]) and more value of sport being perceived for boys compared to girls [START_REF] Jodl | Parents' roles in shaping early adolescents' occupational aspirations[END_REF].

However, recent articles do not report such differences among parents [START_REF] Bois | Parents' appraisals, reflected appraisals, and children's self-appraisals of sport competence: A yearlong study[END_REF][START_REF] Bois | Elementary schoolchildren's perceived competence and physical activity involvement: The influence of parents' role modelling behaviours and perceptions of their child's competence[END_REF]. Several potential explanations could account for these results. First, due to the time-lag between those various studies, it is possible that gender stereotypes related to sport are less pro-masculine now than in the early 1990s. On the other hand, it can be advanced that anti-sexist norms are currently more subtle, leading certain scholars to question the validity of self-report scales to assess socially sensitive issues such as gender stereotypes [START_REF] Greenwald | Implicit social cognition: Attitudes, self-esteem, and stereotypes[END_REF].

Regarding the social transmission of stereotypes per se, [START_REF] Jacobs | The impact of mothers' gender role stereotypic beliefs on mothers' and children's ability perceptions[END_REF] observed that the more mothers tended to see sport as a masculine domain, the more they tended to display lower ability beliefs for their daughters and higher ability beliefs for their sons. Similar processes emerged in mathematics, an educational field that is socially perceived as masculine (see [START_REF] Gunderson | The role of parents and teachers in the development of gender-related math attitudes[END_REF], for a review). However, no study to our knowledge has examined whether parents' gender stereotypes are perceived by children and whether children adopt similar gender stereotypes in return, albeit in sport or other domains.

The Present Studies

Gender stereotypes represent a core variable of the EV model [START_REF] Fredricks | Parental influences on youth involvement in sports[END_REF].

However, to date, very few studies have investigated gender stereotypes per se, and the results available from past research conducted in sport failed to provide empirical support for the hypotheses of (a) parent-child transmission of those beliefs and (b) their potential consequences on self-perceptions and behavioral involvement. The aim of the present paper was thus to examine those under-explored aspects of the EVM in order to provide a comprehensive understanding of the role played by stereotypes in the gender differences observed in sport.

Gender-sport stereotype endorsement is considered as a key variable in the socialization of gender stereotypes and is measured in all three studies-among adolescents, parents, and/or both. More particularly, based on Eccles' model, we investigated two gender stereotypes: the belief that sport is more important for boys than for girls (the attainment component of value) and the belief that boys are more competent in sport than girls (competence).

The purposes of our studies are twofold. First, our aim was to examine the role played by parental stereotypes in the transmission of gender-sport stereotypes among adolescents. More particularly, we tested the mediational role of perceived parental stereotypes in the relationship between perceived social stereotypes and adolescents' stereotypes (Study 1), as well as the link between parents' and adolescents' stereotypes using both explicit (Studies 2 and 3) and implicit methods (Study 3). Next, we explored the mechanisms through which adolescents' gender-sport stereotypes may be accompanied with sport dropout. More precisely, in Study 1 we examined the links between gender-sport stereotypes and dropout behavior, whereas in Study 2 we focused on the mediational role potentially played by self-perceptions of sport value and competence in the relationship between gender-sport stereotypes and intentions to dropout. Figure 1 summarizes the constructs and relationships under the scope of the present article.

Study 1

No known study of the EVM has explored whether the level of adolescents' stereotype endorsement is associated with the stereotypes they perceive in their social environment nor whether such general beliefs are associated with sport participation or dropout. The purpose of Study 1 was thus to examine (a) the links between adolescents' perceived gender-sport stereotypes → general social stereotypes and between parents' stereotypes → stereotype endorsement and (2) the link between stereotype endorsement → sport dropout (see Figure 1).

We hypothesized that the level of stereotypes perceived in the social environment would be positively associated with the level of stereotype endorsement in adolescents, for both value and competence. Path analysis examined more particularly whether perceived parental endorsement mediated the bound between perceived stereotypes in the cultural milieu and athletes' personal endorsement of stereotypes. Regarding the second purpose, a logistic regression analysis tested whether endorsing stereotypes in favor of the other gender was significantly associated with dropout.

Methods

Participants. The sample comprised 347 adolescents (131 young women, 216 young men; M age = 15.39 years, SD = 4.19, range = 12-20 years) currently or previously involved in one of the most popular activities practiced by adolescents in France, more particularly team sports such as soccer, basketball, handball, rugby as well as individual activities such as skiing, judo, gymnastics, horse riding, cycling, climbing, tennis, and table tennis. This sample included two groups: (a) persistent participants who completed questionnaires in their sports clubs (n = 249) and (b) dropout athletes (i.e., athletes who did not enroll back in their club at the beginning of the season; n = 98). A significant t-test for independent samples indicates that participants in the dropout sample were significantly younger (M = 15.07, SD = 2.38) than those in the persistent group (M = 16.25, SD = 2.53), t(326) = -3.89, p < .001. This can be explained by the fact that the participants' postal addresses were chosen to match the age categories concerned at the time of their practice. Also, there were more female participants in the dropout sample (49%) compared to the persisting sample (33%), χ²(1) = 484.25, p < .001. This is due to the fact that the choice of dropout participants purposively over-represented female participants compared to their actual rate of participation. Given the retrospective design of the study, it cannot be concluded that age or gender are determinants of sport dropout.

Procedure and measures. Our study was conducted during the 2003/2004 season in collaboration with one local Olympic Sport Committee in France. The research team was in contact with sports committees and clubs interested in participating. The committees helped in selecting a list of dropout athletes and communicating their postal address. The clubs welcomed the first author during practice so that the study could be presented. All participants received a similar questionnaire including measures of gender stereotypes, demographic variables (date of birth and sex), and a few questions regarding sport practice (activity, years of experience, hours of weekly practice). A letter explaining the purpose of the study (understanding how certain perceptions can influence the sport experience of adolescents) was addressed to the athletes and their parents, who had the possibility not to consent to their child's participation. In that case, adolescent athletes simply did not return the anonymous questionnaire. Persistent athletes were asked to give back their sealed envelope at practice, whereas dropout athletes were given a stamped return envelope. As a whole, 600 questionnaires were distributed to persistent athletes in 22 clubs, and 600 questionnaires were mailed to dropout athletes from 48 clubs. The participation rate was higher for persistent athletes (43%) than for dropout athletes (16%).

The Personal Endorsement of Sports-related Gender Stereotypes Scale developed by [START_REF] Bonnot | Stereotype internalization and women's math performance: The role of interference in working memory[END_REF] was used to measure gender stereotypes indirectly. This scale assesses beliefs about boys and girls separately, contrary to more direct scales measuring beliefs about one gender relative to the other. The score attributed by a respondent to girls is then subtracted to the one given by that respondent to boys. Two personal stereotypes were assessed using this scale (i.e., for each stereotype, one item targeted boys and the other targeted girls): (a) a stereotype relative to sport competence ("Personally, I think that performance of boys/girls in sport is…")-with answers ranging from 1 (very poor) to 7 (very good) -and (b) a stereotype relative to the attainment component of sport value ("Personally, I think that for boys/girls doing sport is…")-with answers ranging from 1 (not important at all) to 7 (very important). 1 The same two items (targeting boys and girls separately) were used to assess athletes' perceptions of the stereotypes held (a) by the general population ("In general people think that performance of boys/girls in sport is…") and (b) by their parents ("My parents think that for boys/girls, doing sport is…").

As a whole, participants provided 2 (one item targeting boys, one item targeting girls) × 2 (stereotypes of competence and value) × 3 (personal, general, parental) or 12 answers. These 12 responses were then used to calculate six scores of stereotypes: personal endorsement of gendered stereotypes for competence and attainment value, perceived general endorsement of both, and perceived parental endorsement of both. Across all measures, a positive score indicated a pro-masculine stereotype, a negative score a pro-feminine stereotype, and a null score the absence of stereotype. For all measures, scores could range from -6 to +6, and they did span this full range in our sample.

Data Analysis

First, in order to examine the determinants of personal gender stereotype endorsement, two separate path analysis models-relevant to sport value and competence-were tested using Amos 4.0. Because they are saturated models (df = 0), the fit-indices are not relevant and the focus was only on the bounds observed between the variables. Following [START_REF] Preacher | Asymptotic and resampling strategies for assessing and comparing indirect effects in multiple mediator models[END_REF], a bootstrapping method with 5000 resamples was used, along with a 95% Corrected Bias and Accelerated Confidence Interval, in order to test the significance of indirect effects.

Next, in order to examine the predictors of sport dropout, a logistic regression analysis was conducted, with dropout behavior as the dependent variable, and sex, age, and personal stereotype endorsement as independent variables. Continued participation was considered as the null result, whereas dropout was coded 1. Stereotype endorsement scores were recoded so that positive scores indicated stereotypes in favor of one's gender, whereas negative scores indicated stereotypes in favor of the other gender.

Results

Table 1 presents the levels of personal as well as perceived general and parental endorsement of competence and attainment value sport stereotypes among young men and women. A MANOVA conducted on the three scores of stereotypes of sport competence (personal, general, parental) indicated a significant effect of sex, F(3, 341) = 10.48, p < .001, Wilks' λ = .92, but both the main effect of dropout, F(3, 341) = 0.42, p = .737, Wilks' λ = .99, and the interaction were not significant, F(3, 341) = 2.48, p = .061, Wilks' λ = 98. The univariate tests indicated that the multivariate effect was accounted for by gender differences in both personal, F(1, 345) = 96.43, p < .001, ηp 2 = .10, and perceived parental, F(1, 345) = 24.95, p < .001, ηp 2 = .07, stereotypes of sports competence. Specifically, male participants held stronger gender stereotypes than female athletes, and young men believed that their parents' gender stereotypes were stronger than those perceived by young women (see Table 1).

A MANOVA conducted on the three scores of stereotypes of sport attainment value (personal, general, parental) indicated a significant effect of sex, F(3, 341) = 8.86, p < .001, Wilks' λ = .93, but no significant effect of dropout, F(3, 341) = 0.60, p < .614, Wilks' λ = .99, or the interaction, F(3, 341) = 1.59, p < .192, Wilks' λ = .98. The univariate tests indicated that the multivariate effect was accounted for by gender differences in both personal, F(1, 345) = 38.18, p < .001, ηp 2 = .10, and perceived parental, F(1, 345) = 19.30, p < .001, ηp 2 = .05, stereotypes of sports value. Specifically, male participants held stronger gender stereotypes than female athletes, and young men believed that their parents' gender stereotypes were stronger than those perceived by young women (see Table 1).

Table 1 also reports the correlations among the variables of the study. There were significant correlations for each stereotype among the three levels of assessment: the more participants held strong gender stereotypes, the more they perceived strong gender stereotypes both in general and in their parents, similarly for both competence and attainment value.

Additionally, participants' answers were significantly correlated at each level of assessment: the more they held strong gender stereotypes of sport competence, the more they tended to hold strong gender stereotypes for value, and the same was true concerning perceived general and parental stereotypes endorsement.

Prediction of stereotype endorsement. Our first model was aimed at predicting sport competence gender stereotypes. As depicted in Figure 2, all the hypothesized paths were significant. More specifically, adolescent gender stereotype of sport competence was significantly predicted by perceived general stereotype (β = .25) and gender (β = .19), even after controlling for the indirect effect of perceived parents stereotype (β = .43; significant indirect effect of general stereotype [95% CI = .171 -.287] and gender [95% CI = .059 -.143]). Furthermore, the model predicting adolescent sport value gender stereotype showed similar results. Stereotype endorsement of attainment value was significantly predicted by perceived general stereotype (β = .16) and gender (β = .23), even after controlling for the indirect effect of perceived parents stereotype (β = .33; significant indirect effect of general stereotype [95% CI = .046 -.125] and gender [95% CI = .046 -.125]).

Stereotype endorsement and dropout. Table 2 summarizes the results of the logistic regression analysis carried out with sport dropout as the outcome variable. The test of the full model with the four predictors against the constant-only model was statistically significant, χ 2 (4) = 32, p < .001. The Nagelkerke R² was equal to .14. The Hosmer-Lemeshow test produced a failto-reject decision: χ 2 (8) = 12.2, p = .143, a result consistent with the assumption that the specified logistic model was correct. Independent significant predictors of behavior included age and competence stereotype, whereas importance stereotype and gender were not significant.

Given the retrospective nature of the design, the effect of age can be seen as a sampling effect rather than as an explanatory process. On the other hand, it appeared that endorsing a promasculine stereotype of competence decreased the probability to drop out from sport among boys, whereas it increased this probability among girls (OR: .843; 95% CI = .623 -1.141).

Discussion

No known previous study using the EVM examined whether the level of gender-sport stereotypes of young individuals is associated with the stereotypes they perceive in the social environment, as well as the relationship of those beliefs to sport dropout. In our study, we observed that the more adolescent athletes perceived gender-sport stereotypes in their social environment, the more they tend to adhere to such stereotypes. It is also noteworthy that adolescents' perceptions of their parents' attitudes, as proximal socialization agents, seemed to exert a particularly important role in gauging these kinds of perceptions. This finding is in line with the conception of parents as a primary source of socialization [START_REF] Fredricks | Parental influences on youth involvement in sports[END_REF].

The second purpose of our study was to examine whether gender stereotype endorsement is linked to sport dropout. Our findings showed that endorsing a sport-competence stereotype that is in favor of one's gender (a pro-masculine stereotype for boys or a pro-feminine stereotype for girls) was a significant predictor of persistent participation, but that endorsing a stereotype in favor of the other gender (a pro-masculine stereotype for girls or a pro-feminine stereotype for boys) was associated with dropping out. This result can be interpreted as consistent with the EVM, which posits that the integration of general beliefs can in turn affect self-perceptions and motivation [START_REF] Eccles | Gender role socialization in the family: A longitudinal approach[END_REF][START_REF] Fredricks | Parental influences on youth involvement in sports[END_REF]. However, given the retrospective nature of our design, another interpretation can be advanced. It is indeed possible that adolescents adjusted their gender beliefs so that they are in line with the behavior adopted-that is, dropout or persistent participation.

Study 1 adds to the literature by showing the mediational role played by perceived parental stereotypes in the relationship between perceived social stereotypes and adolescents' stereotype endorsement, which in turn may affect sport dropout. However, it does not inform us on the bounds between actual parents' gender-sport stereotypes and their child's level of endorsement. Study 2 was aimed at investigating this question.

Study 2

The purpose of Study 2 was twofold. First, no known previous research with the EVM has examined whether parents' gender-sport stereotypes are correlated to their child's level of stereotype endorsement. The first aim of Study 2 was thus to examine the link between athletes' levels of gender stereotypes relative to sport competence and value and their parents stereotypes (see Figure 1). Next, although past research indicated that gender perceptions of sport activities were associated with perceived competence and motivation among adolescents (Chalabaev et al., 2009), and Study 1 showed that gender stereotypes could be associated with sport dropout, no known study has examined the mediational role of self-perceptions in this process. The second purpose of Study 2 was thus to examine whether adolescents' gender-sport stereotype endorsement would be linked to lower self-perceptions of sport value and competence, which in turn should be associated with higher intentions of sport dropout among adolescent athletes (see Figure 1).

Method

Participants and procedure. For this study, 155 (82 female and 73 male) adolescent athletes aged between 12 and 16 years old (M age = 13.8, SD = 1.77) and their parents volunteered to participate. Adolescents were involved in one of three athletic activities: gymnastics, tennis, and judo. An envelope containing three questionnaires was given to all athletes in their sport clubs. The letter explained the purpose of the study (i.e., looking at young athletes' feelings in their sport and the opinion of their parents). One questionnaire was addressed to athletes, and As a whole, 104 datasets were complete (i.e., comprising the athlete's questionnaire and a questionnaire from each parent).

Athlete's questionnaire. Endorsements of gender-sport stereotypes for both competence and attainment value were assessed using the same scales as in Study 1. Each participant's answer to the item targeting girls was subtracted to their answer to the item targeting boys, so that a positive score indicated a pro-masculine stereotype. Regarding the gender stereotype of sport competence, the scores for our sample ranged from -5 to +3, whereas for the gender stereotype of sport value, they were between -2 and +3.

Perceived personal competence was evaluated with the four-item subscale of the Intrinsic Motivation Inventory [START_REF] Mcauley | Psychometric properties of the Intrinsic Motivation Inventory in a competitive sport setting: A confirmatory factor analysis[END_REF]. Respondents are asked to rate themselves on items such as "I feel I'm capable of learning new things in this activity" on a 6point scale ranging from 1 (strongly disagree) to 6 (strongly agree). We then calculated an average score so that higher scores indicated higher levels of perceived competence. This full four-factor scale was initially developed in the experimental setting to assess motivational states of participants regarding a task [START_REF] Plant | Intrinsic motivation and the effects of self-consciousness, self-awareness, and ego-involvement: An investigation of internally-controlling styles[END_REF] and was subsequently adapted to the sport setting [START_REF] Mcauley | Psychometric properties of the Intrinsic Motivation Inventory in a competitive sport setting: A confirmatory factor analysis[END_REF]. Factor analyses confirmed the presence of four factors, including perceived competence (α = .80), which was significantly associated with intrinsic interest toward the activity. In the current sample, the scores ranged from 1.8 to 6.0, and the internal consistency was satisfactory (α = .75).

Sport value was measured with the 4-item scale used by Fredericks and Eccles (2002).

Respondents are asked to assess the extent to which individuals consider sport as a useful and important domain along a 7-point scale ranging from 1 (not at all) to 7 (very much). The items were constructed crossing one of two anchors ("In general / Compared to most of your other activities") with one of two sentences ("how useful/important is tennis?"). An average score was calculated so that higher scores indicated higher levels of perceived value.

A larger scale was initially used by Eccles and collaborators in the longitudinal

Childhood and Beyond study, in which they confirmed the uniqueness of the value subscale (Eccles, Wigfield, & Blumenfield, 1993). This 10-item scale was used with success in past studies conducted among French adolescents by [START_REF] Guillet | Understanding female sport attrition in a stereotypical male sport within the framework of Eccles's expectancy-value model[END_REF] who reported adequate reliability (α = .81) and higher mean levels among persisting athletes compared to dropouts. [START_REF] Wigfield | Changes in children's competence beliefs and subjective task values across the elementary school years: A three-year study[END_REF] proposed to separate the importance/usefulness versus interest aspects of sport. [START_REF] Fredricks | Children competence and value beliefs from childhood through adolescence: Growth trajectories in two male-sex-typed domains[END_REF] reported acceptable consistency of such an abridged version of importance/usefulness subscale among children (four items; alphas between .61 and .92). In the current study, the mean scores ranged from 2.2 to 5.0, and the internal consistency of the scale was satisfactory (α = .71).

Athletes' behavioral intentions to pursue participation in their sport were assessed through a four-item scale derived from Ajzen and Driven (1992). Respondents used a 7-point scale from 1 (not at all) to 7 (completely) to rate items such as "I intend to play tennis again next season." An average score is then calculated so that higher scores indicate higher levels of behavioral intentions. This scale was used in past research conducted among French adolescents in which it was found to be reliable (α = .87) and significantly predicted dropout behavior 8 months later [START_REF] Guillet | Understanding female sport attrition in a stereotypical male sport within the framework of Eccles's expectancy-value model[END_REF]. In the current study, scores ranged from 2.5 to 6.0, and the internal consistency was satisfactory (α = .80).

Parents' questionnaire. The procedure described above was also used to measure parents' level of gender stereotype endorsement. The scores of gender stereotype of sport competence ranged from -1 to 1 among both fathers and mothers. The scores of gender stereotype of sport value ranged from -1 to 2 among fathers and from -1 to 1 among mothers.

Data analysis.

The first aim of Study 2 was to examine the correspondence between parents' and athletes' levels of gender-sport stereotypes. Correlations were calculated between the scores obtained by fathers and athletes, as well as between those of mothers and athletes.

Next, we used path analysis to examine the personal correlates of gender-sport stereotypes among athletes (Figure 1). In the hypothesized model, the levels of gender stereotypes of sport competence and value were considered as predictors of the levels of perceived competence and value in sport. In turn, perceived competence and value were expected to positively predict athletes' intentions to maintain their participation. Last, perceived competence was assumed to positively predict the perceived value of the activity. Indeed, these two perceptions were found to be consistently linked in previous studies [START_REF] Fredricks | Children competence and value beliefs from childhood through adolescence: Growth trajectories in two male-sex-typed domains[END_REF][START_REF] Wachs | The boundaries of difference: Negotiating gender in recreational sport[END_REF]. It has been shown in frameworks other than the EVM that when individuals face recurrent negative information relative to their competence in one domain, they can be prone to psychologically disengage themselves through a devaluation of this domain [START_REF] Major | Coping with stigma through psychological disengagement[END_REF]. Such a process has for example been observed in education when self-esteem is threatened by negative feedback [START_REF] Régner | Relationship of sociocultural factors and academic self-esteem to school grades and school disengagement in North African French adolescents[END_REF]. Given that adolescence represents a key developmental period in terms of construction of the self, it could be assumed that a similar process occurs in another achievement area such as sport and that young athletes will tend to value their activity as long as they feel successful at it. On the other hand, adolescents who have lower self-perceptions of athletic competence could be prone to see less value in their activity. In the path analysis model, stereotypes scores of girls were reversed so that positive value indicated a stereotype in favor of one's gender within the whole sample. It was tested using path analysis with Amos 4.0, with a bootstrapping technique to estimate indirect effects [START_REF] Preacher | Asymptotic and resampling strategies for assessing and comparing indirect effects in multiple mediator models[END_REF].

Results

Preliminary analyses. Descriptive statistics and correlations between the variables of the study appear in Table 3. A series of paired t-tests was run in order to compare the level of stereotype endorsement (a) among athletes and their fathers and (b) among athletes and their mothers. The level of gender stereotype related to sport value was not significantly different between athletes and fathers, t(267) = 0.83, p = .41. The level of gender stereotype related to sport competence was significantly higher in athletes compared to their fathers, t(259) = 4.63, p < .001, ηp 2 = .07. Similarly, the scores of gender stereotypes were higher in athletes compared to their mothers both for sport value, t(285) = 2.89, p < .001, ηp 2 = .03, and for sport competence, t(279) = 4.27, p < .001, ηp 2 = .06. 3 reveals that no correlation was significant between fathers' or mothers' levels of gender-sport stereotypes and athletes' level of gender-sport stereotypes, regarding neither value nor competence. This pattern does not support our hypothesized expectation that the beliefs held by parents would be transmitted to their children. Interestingly, competence and value ratings were significantly correlated for both the athletes themselves and their fathers, but not for their mothers.

An examination of Table

Mediation of self-perceptions. Our proposed model, in which athletes' self-perceptions were expected to mediate the relationship between stereotypes and intentions, globally yielded a satisfactory fit with the data: χ²(4) = 7.77, p > .10; χ² /df = 1.943; CFI = .97; IFI = .97; TLI = .92; SRMR = .04; RMSEA = .08 (see Figure 3). The results indicate that gender stereotype of sport competence was significantly associated with perceived sport competence (β = .31). In other words, endorsing a stereotype in favor of one's gender leads to higher self-perception of competence in sport. In turn, perceived sport competence was associated with significantly higher levels of sport value (β = .43). However, perceived sport competence was not significantly related to intentions to drop out (β = .13), whereas perceived sport value significantly predicted lower intentions to dropout from sport (β = .36). There were two significant indirect effects. The first one was an indirect effect of gender competence stereotype on perceived sport value, through the mediation of perceived sport competence [95% CI = .31 -.57]; the second one was an indirect effect of perceived sport competence on intentions, through the mediation of perceived sport value [95% CI = -.14 -.42].

Discussion

The first purpose of our second study was to examine the bound between each of the parents' levels of gender-sport stereotypes and young athletes' levels of gender-sport stereotypes.

No significant association was observed: neither for mothers nor for fathers, and neither for value nor for competence stereotype. A likely explanation is that parents adopted a neutral attitude because a declarative measure of stereotype, even subtle, is prone to social desirability among adults [START_REF] Riemer | Gender typing of sports: an investigation of Metheny's classification[END_REF]. Indeed, in the current sample, from 79% up to 91% of null scores were obtained among parents on the stereotypes measures, and the range for these measures was quite restricted. The use of an implicit method could represent a means to diminish such socially desirable responding.

The second goal of Study 2 was to examine the mediational role played by selfperceptions in the relationship between the level of adolescents' gender-sport stereotypes and their intention to drop out from sport. It appeared that endorsing stereotypes of competence in favor of the other gender (pro-masculine stereotype for girls, pro-feminine stereotypes for boys) was significantly related to higher intentions to dropout through a complex chain of processes involving decreased levels of personal competence and value regarding the activity. This pattern of results is thus in line with the hypothesis according to which, among adolescents, adopting gender stereotypes could lead girls to leave the athletic area in greater proportions, compared to boys [START_REF] Dumith | Physical activity change during adolescence: A systematic review and a pooled analysis[END_REF]French Minister for Youth and Sport, 2001) because girls are more prone to consider themselves as less able and to attach less value to sport participation. Although the overall patterns in our data are consistent with this reasoning, the current subsamples of male and female participants were too small to test this specific gendered hypothesis.

Study 3

Study 2 failed to provide empirical support for the hypothesis of parental transmission of stereotypes using explicit measures. The aim of Study 3 was to further examine the link between the level of parents' sport gender stereotypes and their child's level of endorsement (see Figure 1). Unlike Study 2, this relationship was examined not only through explicit measures (i.e., questionnaires), but also via a Single-Category Implicit Association Test (SC-IAT; [START_REF] Karpinski | The single category implicit association test as a measure of implicit social cognition[END_REF]. This test measures answers that participants cannot consciously control, allowing direct access to their cognitions without social desirability biases.

Method

Participants and procedure. Twenty-three adolescents (14 girls and 9 boys, M age = 15.0, SD = 1.1, range = 14-19 years) as well as one of their parents (14 mothers and 9 fathers, M age = 45.3, SD = 4.7, range = 40-58 years) volunteered to participate. Child-parent dyads were quite well distributed: seven mother-daughter dyads, seven mother-son dyads, seven father-daughter dyads, and two father-son dyads. All the adolescents were regular sport participants (M = 5.13 hours of practice per week; range = 1 -8 weeks). With the exception of two participants, all parents had practiced or continued to practice a sport on a regular basis. 2 The study took place at the participants' house. Adolescents and parents participated separately in a quiet room and did not have the possibility to communicate with each other before both had completed the tasks. All participants firstly answered to the questionnaire and then did the implicit test on a computer (HP 584037-001) equipped with Inquisit 3.0.

Measures. The explicit measure of gender stereotype, both sport competence and value, was the same as in Studies 1 and 2. The SC-IAT [START_REF] Karpinski | The single category implicit association test as a measure of implicit social cognition[END_REF] was used as our implicit measure. This computer-based test measures the strength of cognitive associations via a simple categorization task of verbal stimuli. It requires the sorting of words belonging to three conceptual categories: one target category (in our case, "Sport") and two attribute categories (i.e., "Masculine" and "Feminine"). Participants were asked to categorize the words that appeared in the middle of a computer screen as fast and as accurately as possible, using the "e" and "i" keys of the keyboard.

The 6-step sequence described by [START_REF] Karpinski | The single category implicit association test as a measure of implicit social cognition[END_REF] was followed. The different blocks presented to the participants appear in Table 4. The first part of the test is devoted to familiarization with the words belonging to each conceptual category. Then, the assessment part of the test is composed of two steps that participants carry out in a counter-balanced way. Each step consists of 23 test trials immediately followed by 72 evaluation trials.

In one of the steps, words from the "Sport" and "Masculine" categories need to be categorized in the upper-left screen, whereas words from the "Feminine" category need to be categorized in the upper-right screen. This step represents the "compatible" association. In another step, words from the "Masculine" category need to be categorized in the upper-left screen, whereas words from the "Feminine" and "Sport" categories need to be categorized in the upper-right screen. This step thus evaluates the "incompatible" association. The SC-IAT effect corresponds to the difference of performance (expressed through mean reaction time) realized for the compatible and incompatible association.

For each category, eight stimuli were used. Concerning sport, the list of words used by Clément-Guillotin and his collaborators (Clément-Guillotin, [START_REF] Clément-Guillotin | Is sport still a masculine domain? A psychological glance[END_REF] in their study on psychological gender was retained (namely: team, locker, stadium, competition, game, sweat, stopwatch, and running shoes). For the "Masculine" and "Feminine" categories, the lists were based on those of [START_REF] Greenwald | A unified theory of implicit attitudes, stereotypes, self-esteem, and self-concept[END_REF]. The properties (length and lexical frequency) of the French words belonging to the "Feminine" (girl, woman, mother, sister, misses, she, her, and they-female) and "Masculine" (boy, man, father, brother, mister, he, his, and they-male) categories were comparable. 3 Data analysis. Regarding our explicit measures, the procedure used to compute the two stereotypes scores was identical to the one used in Studies 1 and 2. Concerning the implicit test, following the procedure proposed by [START_REF] Karpinski | The single category implicit association test as a measure of implicit social cognition[END_REF], all the reaction times lower than 300 ms and those higher than 3000 ms were deleted as outliers. In case the participant gave a wrong answer, a penalty of 400 ms was added. On average, the participants made as many mistakes in the "compatible" block (M = 3.12, SD = 2.51) as in the "incompatible" block (M = 3.10, SD = 2.90), t(81) =.10, p < .92.

Then, the SC-IAT score was calculated using the modified algorithm proposed by [START_REF] Greenwald | Understanding and using the implicit association test: I. An improved scoring algorithm[END_REF]. The steps to calculate the D score are: (a) calculation of the "inclusive" standard deviation including every time response of every trial in steps 4 and 6, (b) calculation of the mean time response in the set of answers from steps 4 and 6, (c) calculation of the difference between the mean of "compatible" block and the mean of "incompatible" block, and (d) division of the difference by the "inclusive" standard deviation. The more important absolute value of D score reflects a strong implicit association. A positive D score reveals an automatic association between the categories "sport" and "feminine," whereas a negative D score reveals an automatic association between the categories "sport" and "masculine." The block signing order did not have a significant effect on the implicit score for compatible (M = 0.14, SD = 0.30) and incompatible (M = 0.14, SD = 0.26) pairings, t(81) = 0.02, p < .985. Thus, all participants were considered simultaneously, and the analyses were run without controlling this parameter.

Results and Discussion

The means and standard deviations across the sample are presented in Table 5, along with correlations among the variables. A series of paired t-tests was run to examine whether athletes' levels of stereotypes endorsement significantly varied from their parents' levels. Regarding explicit gender stereotype of sport competence, athletes had significantly higher scores than their parents, t(44) = 3.12, p = .003, ηp 2 = .18. Regarding explicit gender stereotype of sport value, there was no significant difference between athletes' scores and their parents' scores, t(44) = -0.76, p = .453. Last, athletes' scores on the implicit test were not significantly different from their parents' scores, t(44) = 0.38, p = .706.

Regarding the relationships between athletes' and parents' scores, consistent with Study 2, no significant relationship appeared between adolescents' and parents' scores considering explicit measures of gender-sport stereotypes for value or competence (see Table 5). In other words, no significant link between parents' and their children's answers to explicit questionnaires was observed. In contrast, a significant correlation was observed regarding the implicit D score such that the more parents reported a strong implicit association between "sport" and "masculine," the more their child did as well. Conversely, the more parents reported a strong implicit association between "sport" and "feminine," the more their child also did. Thus a significant bound between parents' and adolescents' scores emerged when considering implicit scores. Taken as a whole, this pattern of results reinforces the increasing interest observed in social cognition for implicit measures, given that they could represent a means to access cognitive constructs subject to social desirability biases.

Although not the main focus of the present study, another significant result emerged: namely, a significant relationship between!explicit adolescent competence stereotype and implicit parent stereotype (see Table 5). This result indicates that the more parents implicitly associate sport and masculine, the more their children tend to declare that boys were more competent than girls in sport. This result suggests that parents could exert a social influence on their children without being conscious of it (for example by offering more sport opportunities of reinforcing more often boys than girls), which in turn may result in a stereotype explicitly assumed by their children. This interpretation remains speculative given the cross-sectional design of our study, but future research combining implicit tests and observations of parents could shed light on this unexpected result.

General Discussion

The EVM of Eccles and her collaborators represents one of the most widely used contemporary theoretical frameworks interested in parental socialization, especially regarding the sport area [START_REF] Eccles | Gender role socialization in the family: A longitudinal approach[END_REF][START_REF] Fredricks | Parental influences on youth involvement in sports[END_REF]. This model assumes that gender stereotypes present within the cultural milieu may facilitate differentiated developmental patterns between boys and girls because they are transmitted by parents to their children through the construction of general schemas and specific self-perceptions. Even though the main hypotheses of the model have received support from empirical research in the last decades, within a sport context aspects of the model remain under-explored or studied indirectly (i.e., as part of the general social environment). The aims of the present paper were thus to examine (a) the social antecedents of adolescents' gender stereotypes relative to sport competence and value (both perceived by children and assessed in parents) and (b) the outcomes of the gender-sport stereotypes on adolescent athletes' self-perceptions, intentions, and dropout behavior. In order to reach those purposes, a series of three empirical studies involving adolescent athletes and their parents were conducted.

Social Antecedents of Gender Stereotypes

According to Eccles' model [START_REF] Eccles | Gender role socialization in the family: A longitudinal approach[END_REF][START_REF] Fredricks | Parental influences on youth involvement in sports[END_REF], a child's beliefs are likely to be influenced by several sources in his/her social environment, but a particularly important role is attributed to parents. We thus assumed that the degree of gender stereotypes displayed by adolescents would be related primarily to the beliefs they perceive in their cultural milieu, and notably in their parents. This hypothesis was supported in Study 1 for stereotypes of both competence and attainment value, with higher coefficients observed for adolescents' perceived parental stereotypes than stereotypes in general. Although this result fits the postulates of the EVM, it is not sufficient to empirically support the hypothesis of a social transmission of stereotypes. Indeed, those results do not indicate that actual beliefs shared in the society or in parents predict adolescents' personal endorsement, but only that adolescents tend to hold stronger gender stereotypes when they perceive their social environment to do so.

As a consequence, in Studies 2 and 3 we further explored the degree of endorsement of stereotypes, both among adolescents and their parents. Indeed, if certain beliefs related to sport (e.g., value) revealed significant links between parents' and their child's answers, no known study of this kind has examined stereotypes. In Study 2, no significant relationships were observed. This absence of significant relationships can be accounted for by the very low variance in parents' answers, reflecting a potential social desirability response bias more marked among parents than among adolescents [START_REF] Tobin | The intrapsychics of gender: A model of self-socialization[END_REF]. This interpretation was strengthened by the results obtained in Study 3 because, in this additional study, no significant association was observed with explicit measures, whereas the predicted bound emerged for implicit scores.

Taken together, this pattern of results across studies lends significant empirical support to the EVM hypothesis of social transmission of gender stereotypes during adolescence. It also underlines the methodological difficulties encountered in this field and, more particularly, regarding the use of explicit scales among adults. The development of implicit measures appears as an efficient strategy to capture subtle sexist opinions. Future research should as a consequence rely on both explicit and implicit measures, for instance in order to compare the behaviors observed among parents or reported by their children, depending on their pattern of answers.

(Potentially three profiles could emerge: one with neutral scores on both explicit and implicit measures, another with stereotypes on both explicit and implicit measures, and a third with neutral scores on explicit scales but stereotypes on implicit tests.)

Outcomes of Gender Stereotype Endorsement

Past research relative to gender differences and self-perceptions of value and competence is well documented, in particular regarding the sport context. However, previous studies investigating directly the role played by gender stereotypes in the motivational processes are scarce. In the current article, we thus explored how gender stereotype endorsement could impact self-perceptions of competence and value and, in turn, intentions to drop out from one's activity (which ultimately may result in actual dropout behavior). The results obtained in the present work can be seen as quite pioneering with this regard. A significant link between the adoption of a view in disfavor of one's gender category and actual sport dropout was shown in Study 1.

Although this result was only observed regarding the stereotype of sport competence, it is the first known to actually test this specific hypothesis.

Study 2 enriches the results by supporting the existence of potential mediators in this relationship. Indeed, it appeared that adopting a gender-biased view of sport could significantly predict intentions to dropout from sport, through indirect effects of self-perceptions (regarding competence and attainment value) in the sport context. As a whole, the different paths suggested by the EVM to lead from stereotype endorsement to behavior were empirically supported in our samples.

Future research may refine our results by including other variables such as sport or gender identity. Indeed, the interactions between those identity components and gender stereotype potentially account for behaviors such as maintained sport participation or dropout. In this vein, certain individuals may adopt sexist views, but those general beliefs may not affect their self-perceptions because they conceive themselves as atypical, or because gender is not a central component of their self-schema. For example, a girl who adheres to pro-masculine stereotypes relative to sport could protect her self-views, if at the same time, she considers herself as exceptional for a girl, or if being a girl does not appear as a central core of who she is [START_REF] Tobin | The intrapsychics of gender: A model of self-socialization[END_REF]. This minimized gender identification could in turn prevent her from disengaging in contexts socially seen as masculine, such as sport. Conversely, young women whose athletic identity is very important, and who also strongly identify as feminine, could be motivated to challenge the context of masculinizing sport. As a consequence, they should be likely to put a lot of effort and commitment in this context.

Limitations and Perspectives

A first limitation of the present series of studies is linked to potential stereotype accuracy. Indeed, the differentiated views according to gender in the athletic area may actually reflect real differences between both social groups. For example, [START_REF] Jacobs | The impact of mothers' gender role stereotypic beliefs on mothers' and children's ability perceptions[END_REF] have shown that mothers' beliefs in their child's athletic competence were positively predicted by their achievement on physical tests. It thus seems particularly fruitful to include such objective data in order to examine more thoroughly the role of stereotypes, conceived as an exaggeration of real inter-groups differences. In addition, the results must be taken cautiously given the respective retrospective or cross-sectional nature of the studies. A prospective design involving several points of measurement across time and an examination of persistence/dropout behavior would strengthen the empirical support to the processes hypothesized by the EVM. Moreover, whereas sport was treated as a generally pro-masculine area, it is noteworthy that certain activities carry more specific stereotypes and that some sports may be considered as strongly appropriate for one gender or carry on a more neutral social perception [START_REF] Fontayne | Effet du genre sur le choix et le rejet des activités physiques et sportives en éducation physique: Une approche additive et différentielle du modèle de l'androgynie. [Effect of gender on choice and rejection of sports and physical activities in physical education classes: An additive and differential approach of the androgyny model[END_REF]. Due to the size of our samples, it was not possible to distinguish the different activities in this way, but this seems a promising avenue for future research on sport stereotypes.

Last, sport value appears as a key variable in the dropout process among adolescents.

However, the reasons why sport is valuable for adolescents is not detailed within the EV model.

Another framework addresses this question more specifically, namely self-determination theory [START_REF] Deci | Handbook of Self-determination research[END_REF], which posits that different kinds of motives animate individuals. These motives can be distinguished as intrinsic (closest regulation from the intrinsic component of value in the EV model) or extrinsic when individuals do not act because of inherent interest but rather to reach a specific goal. Self-determination theory also posits that extrinsic motives vary in their degree of self-determination-from high, when personal goals are identified such as friendly relationships or health, to low, when external pressures such as popularity dominate.

Past research consistently indicated that self-determined motives such as intrinsic or identified regulations were associated with the maintenance of sport participation whereas low selfdetermined motives led to dropout [START_REF] Sarrazin | A self-determination theory approach to dropout in athlete[END_REF]. However, more research could explore if the combinations of such motives could account for dropout prevention in sport among adolescents.

Practice Implications

Considering the short-term stakes of sport participation in terms of positive youth development, as well as the long-term health-related outcomes of regular physical activity, gender-differentiated patterns of sport socialization appear as a potentially discriminating phenomenon and thus should be fought. The current research supports the idea that adolescents rely on their social environment to shape-or not -general beliefs relative to the appropriateness of the athletic context for young women and men. More particularly, our work indicates that adolescents tend to endorse gender beliefs related to sport competence and value that are related to the beliefs they perceive in the cultural milieu and in particular their parents and that parents could transmit such beliefs in an implicit way. In turn, it appears that endorsing stereotypes in favor of the other gender leads to lower self-perceptions in sport, lower intentions to practice, and more dropout from sports. Parents should as a consequence be careful as to how their children are exposed to gender stereotypic messages or situations, and ultimately communicate to contradict them. They should also be aware of their own responsibility in the transmission of such belief. For example, they should question themselves about the opportunities (such as the propositions of watching sports events, trying certain activities, playing sports games with them) and encouragement they deliver to their children (regarding their effort and accomplishment), depending on their child's gender. Physical education teachers and coaches should also be aware of the impact of language implying gender differences and avoid it, as well as be careful about the reinforcement they provide to female and male children and adolescents.

Conclusion

Eccles' EV Model [START_REF] Eccles | Gender role socialization in the family: A longitudinal approach[END_REF]Fredriks & Eccles, 2004) proposes that genderbased beliefs relative to achievement contexts are transmitted to young individuals by their cultural milieu and especially by their parents. The set of studies presented here bring empirical support to such a hypothesis in the sport context, which is still conceived as a male domain. The results indicate that this social transmission might operate both through explicit processes and unconsciously. They also point to the mechanisms through which adolescents' endorsement of such beliefs may in turn lead to lower self-perceptions and behavioral disengagement from organized sport, which can lead to a sedentary lifestyle and poor health consequences in the long term.

451-469.

.

  another questionnaire was addressed to their mother and father, respectively. It was indicated that the questionnaires were aimed at being filled out by the persons with whom athletes spent most of their time (i.e., parents and/or step-parents). Fully 134 mothers (M age = 42.8, SD = 5.93, range = 34-55 years) and 114 fathers (M age = 44.3, SD = 4.30, range = 35-65 years) participated.

Figure 3 .

 3 Figure 3. Path analysis of stereotype endorsement outcomes in Study 2.

Table 1

 1 Means, Standard Deviations, and Correlations among the Variables of Study 1 Note. Different subscripts for the means for young men and women (across each row) indicate significant differences. *p < .05. ** p < .01. *** p < .001.Logistic Regression of Sport Dropout on Stereotype Endorsement Regarding Sport

		Young Men	Young Women	Stereotype -Competence	Stereotype -Value
		M (SD)	M (SD)	Personal General Parental	Personal	General
	Stereotype -Competence
	Personal	1.04 a (1.72)	-0.05 b (1.21)	--
	General	1.77 a (1.89)	1.46 a (2.24)	.50***	--
	Parental	1.19 a (1.77)	0.30 b (1.29)	.62***	.54***	--
	Stereotype -Value
	Personal	0.58 a (1.28)	-0.21 b (0.91)	.48***	.19***	.26***	--
	General	0.72 a (1.34)	1.51 a (1.68)	.14**	.31***	.13*	.26***	--
	Parental	0.52 a (1.26)	-0.05 b (1.05)	.26*** .04	.31***	.42***	.26***

Table 3

 3 Descriptive Statistics and Correlations between the Variables of Study 2

				Stereotype -Competence	Stereotype -Value	Athletes' Perceptions
		M	SD	Athlete Father Mother	Athlete Father Mother	Competence	Value
	Stereotype -Competence
	Athlete	0.52	1.11	--
	Father	-0.03	0.58	-.12	--
	Mother	0.04	0.64	-.03	.17	--
	Stereotype -Value
	Athlete	0.27	0.88	.37*** -.16	.10	--
	Father	0.19	0.62	.02	.23* -.13	.07	--
	Mother	0.02	0.34	.07	.01	.04	.10	-.03	--
	Athletes' Self-Perceptions
	Perceived Sport Competence	4.66	0.75	-.07	.05	.10	-.11	-.05	-.24**	--
	Perceived Sport Value	3.88	0.65	.05	-.04	-.06	-.09	-.10	-.14	.43***	--
	Intentions to Pursue the Activity	5.54	0.80	-.03	-.12	.03	-.02	-.07	-.13	.29***	.42***

*p < .05. ** p < .01. *** p < .001.

Table 4

 4 Blocks of Presentation during the Implicit Test (Study 3)Note. N = 46; half the participants were randomly assigned to each version of the test.

				SC-IAT				Counter-balanced SC-IAT
	block tries Function	touch "E"	Touch "I"	block tries Function	touch "E"	touch "I"
	1	23	practice	sport	other	1	23	practice	sport	other
	2	23	practice	masculine	feminine	2	23	practice	feminine	masculine
	3	23	practice	masculine-sport	feminine	3	23	practice	feminine-sport	masculine
	4	72	test	masculine-sport	feminine	4	72	test	feminine-sport	masculine
	5	23	practice	masculine	feminine-sport	5	23	practice	feminine	masculine-sport
	6	72	test	masculine	feminine-sport	6	72	test	feminine	masculine-sport

Table 5

 5 Descriptive Statistics and Correlations among the Variables ofStudy 3(N = 23) Note. Different subscripts between the means for athletes and their parents (within column) signify a significant difference.

			Stereotype -Competence	Stereotype -Value	Implicit association
	M	SD	Athletes	Parents	Athletes	Parents	Athletes

*p < .05. ** p < .01.

Figure 1. Overview of the constructs and relationships related to sport gender stereotypes in adolescents (normal lines refer to Study 1, dotted lines to Study 2 and bold lines to Study 3)

Footnotes 1 In their original study, stereotype endorsement was also assessed with five items measuring the degree of acceptance of the stereotype itself (e.g., "Men are better than women at math"; α = .74). The subtle four-item measure used and the more traditional stereotype endorsement measure were correlated, r(N = 429) = .62, p < .001. This procedure has been previously used in the sport context to assess gender stereotypes relative to soccer competence, and the scores obtained were significantly related to perceived competence in a female sample [START_REF] Chalabaev | Stereotype endorsement and perceived ability as mediators of the girls' gender orientation-soccer performance relationship[END_REF]).

2 In this sample composed of parents and adolescents, we listed 15 activities. Among those activities, four were presumed feminine (dance, gymnastics, horse-riding, fitness), four were presumed masculine (cycling, soccer, basketball, judo), and seven were presumed neutral (swimming, walking, hiking, tennis, running, track and field, volleyball) with respect to French standards [START_REF] Fontayne | Effet du genre sur le choix et le rejet des activités physiques et sportives en éducation physique: Une approche additive et différentielle du modèle de l'androgynie. [Effect of gender on choice and rejection of sports and physical activities in physical education classes: An additive and differential approach of the androgyny model[END_REF]. The sample can thus be considered as rather representative of the athlete population and not specific to one category of sports with regard to gender. 3 In French, "girl" and "daughter" share the same translation; as a consequence, we decided not to use the French equivalents of "girl" and "boy," but rather to use "they," which is expressed differently depending on whether it refers to a group of female or male persons.


M Taktak-Meziou

A Chemori
email: ahmed.chemori@lirmm.fr

J Ghommam
email: jawhar.ghommam@gmail.com

N Derbel
email: n.derbel@enis.rnu.tn

RISE Feedback Control for a R/W Head Track Following in Hard Disc Drives

Keywords: Nonlinear systems, hard-disc-drives, asymptotic stability, RISE feedback, neural networks. I

In this paper, the track following problem of the Read/Write (R/W) head of a Hard-Disc-Drive (HDD) is addressed using Robust Integral of Sign Error (RISE) based Neural Network (NN) technique. The proposed control scheme is required to compensate as much as possible the nonlinear hysteresis friction behavior which degrades the HDD performance through generating important residual tracking errors. It is well shown that the RISE technique, along with the NN based feedforward control, is able to guarantee the stability of such a system. Moreover, the boundedness of the closed-loop signals is ensured.

To the best authors' knowledge, the suggested control solution, applied at the low frequency region of a HDD, has never been conducted before on such system. Different simulation scenarios are performed including nominal case and external disturbance rejection to demonstrate that the proposed solution is robust and efficient to achieve good tracking performances.

describing the general functioning of the considered system: track seeking and track following [START_REF] Chen | Hard Disc Drive Servo-Systems[END_REF]. The former deals with the displacement of the head from its current position to a target track with a limited control effort, and the latter aims at maintaining the head accurately around the required track while information is being read or written. The pivot bearing movement, well known as characterized by nonlinear frictions, that can deteriorate the HDD system performance. Accordingly, large settling time, significant overshoots and residual errors can arise yielding the head positioning servo system to be unable to maintain the head tip precisely on the target track. In the literature, different models of HDD frictions behavior have been proposed. In [START_REF] Liu | Analysis and measurement of torque hysteresis of pivot bearing in hard disk drive[END_REF], a detailed presentation of the hysteresis behavior is given. Researchers, such as [START_REF] Abramovitch | Disk drive pivot nonlinearity modeling part i: frequency domain[END_REF]- [START_REF] Gong | Modeling and cancellation of pivot nonlinearity in hard disk drives[END_REF], focused on the nonlinear hysteresis modeling. The LuGre friction model was proposed for the HDD as a good presentation which captures all static and dynamic features [START_REF] Wit | A new model for control of systems with friction[END_REF]. For a complete review of the friction modeling, the reader is referred to [START_REF] Armstrong-Helouvry | A survey of analysis tools and compensation methods for the control of machines with friction[END_REF]. Therefore, many researcher communities were interested in friction effects compensation through different control strategies proposed in the literature. Some of these compensation techniques are based on an accurate modeling of the friction behavior such as in [START_REF] Gong | Modeling and cancellation of pivot nonlinearity in hard disk drives[END_REF], [START_REF] Swevers | An integrated friction model structure with improved presliding behavior for accurate friction compensation[END_REF], [START_REF] Lantos | Modeling, identification, and compensation of stick-slip friction[END_REF]. However, other approaches deal with non-model-based friction estimation [START_REF] Ishikawa | Pivot friction compensation using an accelerometer and a disturbance observer for hard disk drives[END_REF] [START_REF] Ramasubramanian | Adaptive friction compensation using extended kalman-bucy filter friction estimation: a comparative study[END_REF]. Adaptive neural network techniques were developed and demonstrated as an adequate tool in eliminating the effects of nonlinearities and even external disturbances. For instance, the authors in [START_REF] San | Adaptive neural network control of hard disk drives with hysteresis friction nonlinearity[END_REF] [START_REF] Lin | Adaptive wavelet neural network control with hysteresis estimation for piezo-positionning mechanism[END_REF] have shown the effectiveness of these techniques in dealing with friction compensation and tracking performances. In this paper, the application of the recently developed control method based on Robust Integral Sign of the Error (RISE) [START_REF] Xian | A continuous asymptotic tracking control strategy for uncertain nonlinear systems[END_REF] is proposed. This technique were firstly tested on a class of uncertain and high order nonlinear systems and have shown good performances. However, since it is a high gainfeedback control, it was more attractive to blend it with a feedforward based on neural networks [START_REF] Patre | Asymptotic tracking for uncertain dynamic systems via a multilayer neural network feedforward and rise feedback control structure[END_REF]. Such combination is advantageous since it offers the possibility to reduce, under some conditions, disturbances and uncertainties affecting the system with an improved steady-state performance and minimal control effort [START_REF] Xian | A continuous asymptotic tracking control strategy for uncertain nonlinear systems[END_REF] [START_REF] Makkar | Lyapunov-based tracking control in the presence of uncertain nonlinear parameterizable friction[END_REF]. In the RISE-NN based method, the NN weights are adjusted online. Compared with previous works, this technique is able to guarantee the asymptotic stability (AS) of the closed-loop system. This paper will not only treat the compensation of inappropriate responses, but it will also perform a precise and fast desired trajectory tracking of the HDD servo-positioning system which reflects the effectiveness of the proposed control solution. The reminder of this paper is organized as follows. In section 2, the problem statement is presented. The RISE feedback NN controller is developed in Section 3. Section 4 presents a comparative study of the different numerical simulation results to highlight the effectiveness of the proposed control solution. Concluding remarks are drawn in Section 5.

II. PROBLEM FORMULATION A. HDD low-frequency modeling

At low frequencies, the Voice-Coil-Motor (VCM) actuator dynamics including the nonlinear hysteresis friction is given by [START_REF] Ren | Feedforward control based on neural networks for hard disk drives[END_REF]:

M(q) q + F(q, q) = u
(1) y = q + w out where M(q) denotes the system inertia verifying M(q) > 0. q, q and q denote the position, velocity and acceleration of the VCM-actuator head tip respectively. u is the control input, y is the actual position of the VCM-actuator in presence of the output disturbance w out . This disturbance is induced by external vibrations. F(q, q) is a nonlinear function representing the pivot bearing hysteresis friction. The behavior of F(q, q) in HDD applications was investigated in [START_REF] Liu | Analysis and measurement of torque hysteresis of pivot bearing in hard disk drive[END_REF] and the LuGre friction model, as introduced in [START_REF] Wit | A new model for control of systems with friction[END_REF], was selected to represent the static and dynamic characteristics of this hysteresis friction as follows:

F(q, q) = σ 0 z + σ 1 ż + σ 2 q (2) ż = q -α( q) | q | z (3) α( q) = σ 0 f c + ( f s -f c )e -( q qs ) 2 ( 4
)
where z is an internal state of the friction model assumed to be unmeasurable. σ 0 , σ 1 , and σ 2 are the model parameters reflecting the small displacements which are the stiffness, the micro damping, and viscous coefficient respectively. f s corresponds to the stiction force, f c is the Coulomb friction force, and the parameter q s is the Stribeck velocity [START_REF] Astrom | Revisiting the lugre model stick-slip motion and rate dependence[END_REF].

B. Control problem statement

Let q d be the desired track position. The tracking error can therefore expressed as:

e 1 = q d -q (5)
The control objective is then to ensure the displacement of the R/W head of the HDD such that it follows a given target track. The head must be kept as close as possible to the predefined track while treating data, as such the following objective is obtained:

lim t→∞ |e 1 (t)| = lim t→∞ |q d (t) -q(t)| = 0 (6)
The recent developed feedback control strategy RISE is proposed to deal with the track following problem of the HDD. This proposed solution is combined with a NN-based feedforward controller which is able to deal with the non-explicit knowledge of the friction model F(q, q) to compensate the effects external vibrations on the actuator positioning accuracy. Such combination, as shown in [START_REF] Patre | Asymptotic tracking for uncertain dynamic systems via a multilayer neural network feedforward and rise feedback control structure[END_REF], is able to ensure the asymptotic stability of the closed-loop system and enhance the steady-state performance with a reduced control effort.

III. PROPOSED CONTROL SOLUTION: A RISE

FEEDBACK WITH NN FEEDFORWARD In this section, the RISE feedback method combined with the NN feedforward control term is proposed for the trackfollowing problem of the R/W head. Fig. 2 illustrates the general structure of the proposed control methodology for the HDD servo-system. The controller is designed based on the nonlinear model and aims at achieving a good performance of the closed-loop system and to ensure a semi-global asymptotic tracking. Dynamical neural networks present an effective tool for estimation and control of nonlinear and complex systems. The universal approximation remains the feature of the NNbased controllers [START_REF] Sun | Neuro-fuzzy dynamic-inversionbased adaptive control for robotic manipulatorsldiscrete time case[END_REF]. Consider S, a compact set and f (x) a smooth function defined as f : S → R n . There exists always three-layer NN able to represent f (x) [START_REF] Patre | Asymptotic tracking for uncertain dynamic systems via a multilayer neural network feedforward and rise feedback control structure[END_REF] such that f (x) = W σ (V x) + ε(x) for given inputs x(t) ∈ R a+1 . V ∈ R (a+1)×L are bounded constant weight matrix for the first-to-second layer and W ∈ R (L+1)×1 is the ideal weight matrix for the second-to-three layer. a is the number of inputs and L is the number of neurons in the hidden layer. σ (.) ∈ R L+1 is the activation function and ε(x) ∈ R n is the functional error approximation satisfying ε(x) ≤ ε N where ε N is a known constant bound. Fig. 3 shows an illustrative description of a three-layer NN principle.

Remark 1: The activation function σ (.) can take different forms such as sigmoid, hyperbolic tangent or a radial basis function. In this paper, the considered σ (.) is a radial basis function taking the following general form:

σ (x i ) = exp x i -c i 2 σ 2 i , ∀i ∈ N
where c i is the center of the basis function and σ i is its width, which are chosen a priori and kept fixed throughout this work for simplicity. In order to calculate the NN feedforward term, some assumptions and properties have to be taken into consideration. Assumption 1: The desired position q d , as well as its first and second time derivatives exist and are all bounded, i. e., q d , qd , and qd ∈ L ∞ .

Property 1: The NN quantities are bounded such as W ≤ W m , σ ≤ σ m , where W m and σ m are known positive constants [START_REF] Dierks | Neural network control of mobile robot formations using rise feedback[END_REF].

B. Background on RISE Feedback control

In this work, the main control objective is to maintain the R/W head as close as possible to a predefined desired trajectory in order to perform an accurate track following task. Since unknown nonlinearities of the HDD dynamics are considered, a controller is developed that exploits the universal approximation property of NNs and the implicit learning of the RISE feedback for the identification of the nonlinear effects of frictions. A RISE feedback control approach with NN feedforward estimation is therefore proposed as a good solution which guarantee an asymptotic stability of the controlled HDD model described by [START_REF] Chen | Hard Disc Drive Servo-Systems[END_REF]. The control strategy is developed in this section, introducing the open-loop and closed-loop tracking error systems. Based on assumption 1, the position tracking error e 1 (t), the tracking errors denoted by e 2 (t) and r(t), are defined as follow

e 1 = q d -q (7) e 2 = ė1 + α 1 e 1 (8) r = ė2 + α 2 e 2 (9)
where α 1 and α 2 are positive tuning gains. Remark 2: The filtered tracking error r(t) is a nonmeasurable quantity since it depends on q(t).

1) Open-loop tracking error system: To develop the openloop tracking error system, the multiplication of (9) by M(q) is made. Then, using the expressions (1), [START_REF] Armstrong-Helouvry | A survey of analysis tools and compensation methods for the control of machines with friction[END_REF], and (8), the resulting system can be expressed as follow:

M(q)r = F d + S -u (10)
where F d is an auxiliary function defined by:

F d = M(q) qd + F(q d , qd ) (11)
and S is a second auxiliary function defined by: S = M(q)(α 1 ė1 + α 2 ė2 ) + F(q, q) -F(q d , qd )

Based on the NN approximation, F d can be expressed as follows:

Ḟd = W σ (V x d ) + ε(x d ) (13)
where x d = 1 q d qd qd and ε(x d ) is the bounded NN approximation error. According to assumption 1, the following inequalities hold:

ε(x d ) ≤ ε N (14) ε(x d , ẋd ) ≤ ε N ( 15
)
where ε N and ε N are known positive bounded constants.

2) Closed-loop tracking error system: Using the previous open-loop tracking error system [START_REF] Ishikawa | Pivot friction compensation using an accelerometer and a disturbance observer for hard disk drives[END_REF], the control input is the summation of the feedforward NN estimation term and the RISE feedback term. As detailed in [START_REF] Patre | Asymptotic tracking for systems with structured and unstructured uncertainties[END_REF], the RISE control term µ(t) is expressed as follows:

µ(t) = (k s + 1)e 2 (t) -(k s + 1)e 2 (0) (16)
+ t 0 [(k s + 1)α 2 e 2 (s) + β 1 sgn(e 2 (s))]ds
where k s , β 1 ∈ R + are positive feedback. The time derivative of ( 16) is given by:

μ(t) = (k s + 1)r(t) + β 1 sgn(e 2 (t)) (17)
Since the nonlinearities in the system's dynamics are supposed to be unknown, a new control term, denoted Fd , and generated by the NN feedforward estimation is added to approximate the uncertainties and cancel out their effects. Fd is then expressed by:

Ḟd = Ŵ σ (V x d ) (18)
where V ∈ R (a+1)×L is a bounded constant weight matrix, and Ŵ ∈ R (L+1)×1 , is the matrix of the estimates of the NN weights, which are generated on-line by:

Ẇ = K[σ (V x d )e 2 -κ Ŵ ] ( 19
)
where κ is a positive design constant parameter. K = K > 0 is a constant positive defined control gain matrix. According to property 1, the upper bound of Ẇ can be formulated as follows:

Ẇ ≤ F N σ m e 2 ( 20
)
where F N is a known bound constant. The overall control input system is then given by:

u = Fd + µ (21)
By taking the time derivative of ( 21) and substituting the expressions of μ and Ḟd given by ( 18) and ( 17) respectively, we get:

u = Ḟd + μ (22) = Ŵ σ (V x d ) + (k s + 1)r(t) + β 1 sgn(e 2 (t))
Thereby, the closed-loop tracking error system dynamics are formulated by taking the first time derivative of ( 10)

M(q)ṙ = -Ṁ(q)r + Ḟd + Ṡ -u (23) = -Ṁ(q)r + Ḟd + Ṡ -Ŵ σ (V x d ) -(k s + 1)r(t) -β 1 sgn(e 2 (t)) = - 1 2 Ṁ(q)r + W σ (V x d ) + ε(x d ) -(k s + 1)r(t) +(- 1 2 Ṁ(q)r + Ṡ + e 2 ) -β 1 sgn(e 2 (t)) -e 2
where W = W -Ŵ is the estimation error. The equation ( 23) can then be rewritten as follows: 24)

M(q)ṙ = - 1 2 Ṁ(q)r + Ñ + N B 1 + N B 2 -e 2 (
-(k s + 1)r(t) -β 1 sgn(e 2 (t))
where

Ñ = - 1 2 Ṁ(q)r + Ṡ + e 2 ( 25
)
N B 1 = ε(x d ) ( 26
)
N B 2 = W σ (V x d ) (27)
As detailed in [START_REF] Patre | Asymptotic tracking for systems with structured and unstructured uncertainties[END_REF], according to the Mean Value Theorem, Ñ is upper bounded as follows:

Ñ = - 1 2 Ṁ(q)r + Ṡ + e 2 ≤ ρ( z ) z ( 28
)
where z(t) ∈ R 3 is given by:

z(t) = e 1 e 2 r ( 29
)
and ρ( z ) is a positive nondecreasing bounding function.

In order to facilitate the stability analysis, some important inequalities are considered as given in following lemma. Lemma 1: Consider N B 1 and N B 2 as expressed respectively by ( 26) and ( 27). The following hold.

N B 1 ≤ ε N (30) ṄB 1 ≤ ε N (31) N B 2 ≤ ( W m + F N σ m e 2 )σ m ≡ ξ B 2 (32) ṄB 2 ≤ ξ 1 e 2 +ξ 2 ( 33
)
where ξ B 2 , ξ 1 , and ξ 2 are positive known constants.

Proof : Inequalities (30) and (31) can be directly determined according to equations ( 14), [START_REF] Patre | Asymptotic tracking for uncertain dynamic systems via a multilayer neural network feedforward and rise feedback control structure[END_REF], and (26). Based on Property 1 and equation ( 20) which deal with the upper bounds of the NN weights, inequality (32) can be easily justified. Then, by considering the derivative relation σm = σ m (1σ m ) together with the time derivative of N B 2 expressed as ṄB 2 = Ẇ σ m + W σm , inequality (33) is concluded.

IV. NUMERICAL SIMULATIONS In this section, numerical simulations are conducted in Matlab/ Simulink framework with a sampling time T e = 0.05ms. The 3.5-in HDD-VCM actuator is chosen to test the effectiveness of the proposed control scheme. Therefore, the full used model of the system is as described by ( 1)-( 4). The dynamic model parameters are chosen as: M(q) = 1, σ 0 = 10 5 , σ 1 = √ 10 5 , σ 2 = 0.4, f s = 1.5, f c = 1, and qs = 10 -3 . The effect of the external vibrations w out will be studied in this section. Different scenarios have been performed to show the tracking capabilities of the proposed controller. The first scenario aims at tracking both sinusoidal and constant desired trajectory q d without external disturbances [START_REF] Wit | Adaptive neural network control of hard disk drives with hysteresis friction nonlinearity[END_REF]. The sinusoidal reference is chosen as q d = A sin(πt) where A = 2µm and f = 200Hz, whereas the constant desired trajectory is chosen to be a unit step q d = 1µm. A general zero-mean Gaussian white noise w noise with a variance σ 2 = 9×10 -9 (m) 2 is considered for this scenario as well as for the other scenarios. In the second scenario, for clarity reasons, only the constant reference of 1µm is considered. Different disturbances are introduced to test if the proposed controller would be able to reject both input and output disturbances. The new system block diagram, for this scenario, is as depicted in Fig. 4. w out is an output disturbance assumed to be an impulse with an amplitude of 0.3µm applied to the system at time instance t = 4ms. The considered input disturbance w in is an unknown perturbation with |w in | ≤ 3mV . For reasons of simplicity, and in order to have a clear study of the disturbance rejection problem, w in is assumed to be persistent and maintained equal to -3mV [START_REF] Chen | Hard Disc Drive Servo-Systems[END_REF]. In simulations, all initial conditions are chosen at the origin. The saturation constraint on the control input u has been taken into consideration such as | u |≤ 3v. In each scenario, a comparative study between the combined RISE-NN and a classical PD controller is proposed.

A. Scenario 1: Tracking problem in nominal case

The performances of the proposed controllers in non disturbed conditions are illustrated in Figs. 5 and6. The different control parameters are summarized in TABLE I. It can be clearly seen that for a sinusoidal reference trajectory, the RISE-NN control method gives much more better results than the PD controller in terms of speed and accuracy. The PD controller, as shown in Fig. 5 generates large overshoots and needs a much more time to reach the desired position while the RISE-NN controller shows a rapid compensation of the tracking error to a neighborhood of zero with little overshoots. The time history of the NN weights is displayed in Fig. 6 which shows their boundedness. The closed-loop system's performances are summarized in Table II.

B. Scenario 2: Tracking problem with external disturbances

The performance of the RISE based NN controller for the disturbance rejection scenario can be seen in Fig. 7 and Fig. 8. The controller achieves a good track following despite the persistent added input disturbance w in as illustrated in Fig. 4. At time instant t = 40ms, it can be seen that the compensation of the external impulse output disturbance w out is successfully performed. The position error signal, as shown in Fig. 8(c) returns quickly to around zero. Therefore, the variation in the R/W head position can be read from the tracking error plot. Moreover, the RISE-NN control input evaluation respects the physical constraint and is kept limited within the interval [-3, 3] v. It is worth to note that the norm of the NN weights can be upper bounded by a constant as depicted in Fig. 8. Compared with the PD simulation results, the later plots have degraded performances. The 5% settling time and oscillations are much more important than those of the proposed RISE-NN approach. With the PD controller, the recovery time, needed to return to the desired track position after an external disturbance, is too large which decreases the overall system response. A summary of the closed-loop system's performances for this scenario is presented in Table II.

V. CONCLUSION AND FUTURE WORK

In this paper, the recently developed Robust Integral of the Sign of the Error (RISE) feedback controller, combined with a NN-based feedforward term, has been designed. The proposed control simulations are compared with those of a classical PD controller to highlight the effectiveness of the former to perform an accurate and fast tracking of the HDD servo-positioning system. Such control solution was proved to achieve the compensation of uncertainties and nonlinear frictions in the system with the guarantee of asymptotic stability of the closed-loop signals. Therefore, considering NN feedforward term in the RISE approach improves the tracking performance and reduces the control effort such that the NN weight estimates are kept bounded. In a future work, the optimization tools will be introduced for an extended version of the proposed RISE-NN controller.

Fig. 2 :

 2 Fig. 2: View of the control structure including the RISE feedback and the NN feedforward

Fig. 3 :

 3 Fig. 3: Schematic view of a three-layer NN

Fig. 4 :

 4 Fig. 4: Block diagram of the system considering input and output external disturbances as well as measurement noise. (Scenario 2)

Fig. 5 :Fig. 6 :Fig. 7 :Fig. 8 :

 5678 Fig. 5: Tracking of a sinusoidal reference trajectory in non disturbed case (plots with PD controller): (a) Output displacement, (b) Control input, and (c) Tracking error signal. (a) (b) (c)

TABLE I :

 I Summary of the controllers' parameters

	Reference (µm)		RISE-NN	PD
	q d	α 1	α 2	K s β 1	K p	K d
	2 sin(200Π t) 6000 1500 1850 1 1.5 × 10 7 0.02
	1	1500 1500 1850 1 9 × 10 6 0.05

TABLE II :

 II Controllers performance comparison.

	Without disturbances (Sinusoidal Reference)
		PD	RISE-NN
	Settling time	3.4 ms	1.62 ms
	Maximum overshoot	11%	0.5%
	Control input | u |	3 v	3 v
	Disturbances Rejection (Step response)
		PD	RISE-NN
	Recovery time	15 ms	2.7 ms
	Maximum overshoot	27%	16%
	Control input | u |	2.88 v	1.5 v