Anne-Claire C Macquet
email: anne-claire.macquet@insep.fr

Decision-making in expert volleyball players 1 Recognition within the decision-making process: A case study from expert volleyball players

Recognition within the decision-making process: A case study from expert volleyball players

A case study from expert volleyball players

In sport, decision-making is important for athletes, who have to make the right decisions in a short time frame. Regarding sport, various studies have focused on different aspects of decision-making. They have referred to visual search strategies (e.g., [START_REF] Farrow | Do expertise and the degree of perception-action coupling affect natural anticipatory performance?[END_REF]Salvesbergh, Williams, Van Der Kamp, & Ward, 2002), knowledge bases (e.g., [START_REF] Mcpherson | Tactics, the neglected attribute of expertise[END_REF][START_REF] Williams | Declarative knowledge in sport: A by-product of experience or a characteristic of expertise[END_REF], the role of practice in decision-making (e.g., [START_REF] Kibele | Non-consciously controlled decision-making for fast motor reactions in sports-A priming approach for motor responses to non-consciously perceived movement features[END_REF][START_REF] Poolton | The influence of analogy learning on decision-making in table tennis: Evidence from behavioral data[END_REF][START_REF] Raab | T-ECHO: Model of decision-making to explain behavior in experiments and simulations under time pressure[END_REF][START_REF] Raab | Decision-making in sports: Influence of complexity on implicit and explicit learning[END_REF], and the option generation process (e.g., [START_REF] Johnson | Take the first: Option-generation and resulting choices[END_REF]. These studies, apart from that of [START_REF] Mcpherson | Tactics, the neglected attribute of expertise[END_REF], used standardized or simulated situations. These situations allowed for the reduction of the factor of uncertainty to control the studied variables. Their context was different from that of competitive situations. In competitive situations, uncertainty cannot be controlled by the researchers. Other studies concern the ecological dynamic of decision-making in sport (Araujo, Davids, & Hristovski, 2006).

In reference to the theoretical approach of the course of action [START_REF] Theureau | Le Cours d'Action: Analyse Sémio-logique[END_REF], studies have focused on the athletes' activity. This approach postulates that affective and cognitive processes cannot be separated from the situations as they develop. Activity is considered as a meaning construction process: meaning is built as activity unfolds, it does not precede it. These studies focused on: (a) the interpretation of athlete particular situation during the course of action (e.g., Arippe-Longueville, (d'), [START_REF] Arripe-Longueville (d') | Coach-athlete interaction during elite archery competitions : An application of methodological framework used in ergonomics research to sport psychology[END_REF], (b) the knowledge produced during the course of action (e.g., [START_REF] Sève | Structure of expert players' activity during competitive interaction in table tennis[END_REF], (c) the time duration in a competition during which athletes can observe, and adapt their actions to those of their opponent's (e.g., [START_REF] Sève | Table tennis scoring systems and expert players' exploration activity[END_REF], and (d) the problems Decision-making in expert volleyball players 4 encountered by athletes during their performances (Haw & Durand, 2007). These studies focused on dynamic and natural situations.

These studies and that of [START_REF] Mcpherson | Tactics, the neglected attribute of expertise[END_REF] depended on the assumption that skilled performers usually make their decisions in a conscious way and they are able to reliably verbally report them to others. This assumption has been controversial. Nisbett and Wilson (1997) considered that people cannot completely and accurately report their higher order mental processes. Ericsson andSimon (1980, 1993) considered that retrospective verbalization only allows one to verbalize the products of this activity (the reasons). In sports, [START_REF] Raab | Decision-making in sports: Influence of complexity on implicit and explicit learning[END_REF] and [START_REF] Poolton | The influence of analogy learning on decision-making in table tennis: Evidence from behavioral data[END_REF] suggested that acquiring decisionmaking requires both implicit and explicit processes and verbal reports on implicit processes lag behind what is done and known unconsciously.

The methodological framework of retrospective reports is concerned with qualitative methodologies and a first person approach (Varela & Shear, 1999). The player reenacts his/her activity in the course of action in its natural setting, without disturbing the performance. Retrospective reports take place after the competition and are concerned with the evolution of particular behaviors and psychological components such as focus, meanings and emotions in competitive situations. This method aims to study problems encountered by athletes, and the methods that are used to cope with them.

In dynamic sports, time constraints present a conflict; athletes must act quickly even though they cannot rapidly understand a given situation. These difficulties in understanding are linked to the uncertainty of the situation (e.g., locus, time, nature of the event) and to the short duration of the event. Some information is not available quickly enough, although this information seems important for the athletes to assess the situation and therefore make and carry out a decision. Athletes have to resolve a dilemma: waiting for useful information and accepting the risk of not having enough time to act, versus anticipating a decision and Decision-making in expert volleyball players 5 accepting the risk of putting into action a decision that is not adapted to the given situation. In dynamic situations, time-pressure seems to be too great to allow the player to understand all the aspects of the situation [START_REF] Amalberti | La maîtrise des situations dynamiques [Control of dynamic situations[END_REF][START_REF] Amalberti | La sécurité écologique et la maîtrise des situations : Concepts et stratégies mis en jeu par les professionnels pour assurer leur propre sécurité et celle du système [Ecological security and situation control : Concepts and strategies used by professionals to ensure their own security and those of the context[END_REF]. To cope with time-pressure, athletes have to be content with a partial and sufficient understanding of a given situation to have time to act. They are trained to reproduce efficient actions, but are sometimes overloaded with information and have difficulty accommodating their actions because of time constraints. Time to react is critical in sport. For example, in volleyball, five players can potentially attack the ball in different places and time frames [START_REF] Blain | Volley-ball: de l'apprentissage à la compétition de haut niveau [volleyball: From learning to high level competition[END_REF]. The receiver has to quickly determine where, to whom , when, and what sort of set the hitter is going to hit to predict where, when and who is going to receive the hit. The number of attack options increases with the accuracy of the initial pass. The receiver has to reduce this complexity and cope with time pressure.

Most competitive situations in sport present many similarities to dynamic situations studied using the Naturalistic Decision-Making (NDM) approach (e.g., Salas & Klein, 2001;Zsambok & Klein, 1997). These situations contain ill-structured problems and time pressure, and refer to an uncertain dynamic environment, with shifting, ill-defined or competing goals, action/feedback loops, many players, and high stakes. Like competitive sport situations, they refer to the real context of work. NDM has been used to study decision-making in many areas, such as nuclear power (e.g., [START_REF]Analysis of decision-making in nuclear plant emergencies: An investigation of aided decision-making[END_REF], aviation (e.g., [START_REF] Kaempf | Current and future applications of naturalistic decisionmaking in aviation[END_REF], military command and control (e.g., [START_REF] Pascual | Evidence of naturalistic decision-making in military command and control[END_REF], and health care (e.g., Keyser (de) & Nissen, 2001). Regarding sport and the NDM approach, we are aware of only two studies that focused on decision-making in competitive situations. [START_REF] Omodei | Using a head-mounted video camera and two-stage replay to enhance orienteering performance[END_REF] adopted a debriefing procedure to provide a method to focus on the mental aspects of athletes' performance, that can be used by coaches to enhance performance. [START_REF] Macquet | Naturalistic decision-making in expert badminton players[END_REF] studied decision-making in expert badminton players. They emphasized Decision-making in expert volleyball players 6 different types of decisions made in relation to the game and they generated information and knowledge categories associated with the types of decisions made. NDM focuses on the way people use their experience to make decisions during their work and aims at improving this process of decision-making. During the training process, athletes over-train decision-making and application of actions to rapidly make and carry out an efficient decision. During competitions, coaches expect athletes to make decisions that have been learned during the training process. These decisions are expected to be efficient and to control the situation. But how do athletes really make their decisions in competition?

In NDM, [START_REF] Klein | The Recognition-Primed Decision model: Looking back, looking forward[END_REF] developed a model of decision-making for experienced agents involved in dynamic and complex situations, who cope with personal consequences for their actions. His model is known as Recognition-Primed Decision (RPD). It aims to describe how people can use their experience to make good decisions without needing to compare the multiple alternative courses of action. This model differs from Rational Choice models (e.g., [START_REF] Simon | Rational choice and the structure of the environment[END_REF]. "Experienced agents use their experience to identify a workable course of action as the first one they consider." (Klein, 1997, p. 287). If they have to evaluate it, they mentally simulate it to see if it would work. The RPD model suggests that people can assess the given situation by recognizing its typicality. The decision maker recognizes the situation through salient features that experience has shown to be useful. Recognition has four byproducts: (a) expectancies, (b) relevant cues, (c) plausible goals, and (d) typical action.

There are three levels in experiencing the situation in a changing context: simple match, diagnose the situation, and evaluate a course of action. In level 1, the situation is rapidly perceived as typical, so the decision maker can quickly implement a course of action that corresponds to this typical situation. In level 2, the situation is not perceived as typical at first, so the decision maker has to clarify it by diagnosing it, to emphasize its typicality. Then, he/she can implement a course of action based on a typical action. In level 3, the decision Decision-making in expert volleyball players 7 maker perceives the situation as typical. Before implementing a typical action, he/she investigates several actions, evaluates them through mental simulation to perceive if they will work or not. When he/she considers that it will work, he/she implements it. However, decision makers spend more time and energy assessing what is happening rather than comparing several courses of action.

According to NDM, decision makers determine only a few options [START_REF] Johnson | Take the first: Option-generation and resulting choices[END_REF][START_REF] Klein | Characteristics of skilled option generation in chess[END_REF], and most often, they determine only one [START_REF] Yates | Outsider: Impressions of naturalistic decision-making[END_REF]. Implicitly, the decision-making process is based on the use of "rules"; the decision maker applies a rule in the form "if condition C holds, then take action A". The given situation is compared to a typical situation contained in his/her memory that is associated with a typical action. If the given situation is analogous to the typical situation contained in memory, then the decision maker knows what to do and can carry out the corresponding action. Observation of volleyball players' actions suggests that the players reproduce actions in a competitive context that have been over-trained during practice. The NDM approach and more specifically the RPD model might provide a theoretical and practical tool to understand decision-making in competitive situations for several reasons. NDM focuses on real settings that involve complex problems, time pressure and high stakes.

The RPD model might provide theoretical and practical insights about the athlete use of their experiences during high stakes competitive situations. Previous studies from other approaches showed the role of practice in decision-making in simulated or standardized situations (e.g., [START_REF] Raab | T-ECHO: Model of decision-making to explain behavior in experiments and simulations under time pressure[END_REF], and the use of procedural rules to make decisions in competitive situations (e.g., [START_REF] Mcpherson | Tactics, the neglected attribute of expertise[END_REF]. The RPD model could provide insights about the number of decisions made by the athletes in competitive situations. Previous studies showed that experts made several decisions before choosing one (e.g., [START_REF] Johnson | Take the first: Option-generation and resulting choices[END_REF]. NDM could extend knowledge about the information perceived in the course of action that is reported by athletes.

Decision-making in expert volleyball players 8 Previous studies showed that experts reported information that was significant for them (e.g., [START_REF] Hauw | Situated analysis of elite trampolinists' problems in competition using retrospective interviews[END_REF][START_REF] Hauw | Situated analysis of elite trampolinists' problems in competition using retrospective interviews[END_REF]salient (e.g., Salvesbergh et al., 2002).

This study aimed to test the RPD model in a high-level sports context and elicit relevant features taken into account by athletes during the decision-making process.

Method

Participants

Seven male professional level volleyball players participated voluntarily in this study.

They ranged in age from 22 to 32 years (M = 25.2 years, SD = 3.9 years). They were playing for a club team participating in the first division of the French Championship. They had all practiced volleyball for more than 10 years, apart from one player who had practiced for 9 years. Two players played at international championship level. During the following season, three other players played for the national team.

Data Collection

Two types of qualitative data were collected at different times. Data were observed and recorded during a competitive match, and data were elicited during self-confrontation interviews.

Data observed and recorded during the competitive match. The recorded match was the return match of the French Championship quarterfinal. It was a qualifying match for the European championships. Seven video cameras were used. One was placed in the stand at the back of the court, in the middle of the court width. The six others were placed on the side of the court, behind the post and the referee's chair. Each of these six cameras recorded the actions of one particular player and his immediate environment. Each cameraman followed one player during the match. When a player was substituted, the cameraman followed the new player. All action time was recorded with cameras stopped only during time outs, when the coach talked to the players, and during change of ends. The researcher took notes about other Decision-making in expert volleyball players 9 forms of data that were considered important for the study and not recorded by the camera, such as the score corresponding to a substitution.

Data collected during self-confrontation interview. Data were collected during selfconfrontation interviews with each of the seven players who had played the most during the match. They were conducted between the 2nd and the 5th day after the match. The methodology of this type of interview is similar to the first stage of the techniques of stimulated recall used by [START_REF] Omodei | Using a head-mounted video camera and two-stage replay to enhance orienteering performance[END_REF]. The interviews were based on the video recording of the athletes during the match. The athletes were invited to recall thoughts and feelings associated with what was shown on the screen. The recall process was facilitating by using some non-directive questions. In self-confrontation interviews, each player was confronted by the video of the game with one of the trained researchers. The latter began the tape during the serve and stopped it when the point was finished. The player was invited to comment on his decisions and the events that led up to the decisions. The self-confrontation interview is a method developed by Cranach (von) and Harré (1982) that aims to elicit ongoing cognitions. The athletes were presented with their activity as soon as possible after the recording. Players were asked to describe and explain their own behaviour and activity.

The duration of each interview was limited to 90 minutes. Each player was presented with two screens; one contained the video of the match and the second consisted of the interviewed player's actions and his near environment. Players could stop the video at any time to explain something that they deemed important. The interview was structured to elicit information and salient features taken into account by the players during decision-making [START_REF] Hoffman | Eliciting knowledge from experts: A methodological analysis[END_REF]. The interviewer aimed to lead the athlete to think aloud, just as he had been thinking during the course of action. The player was invited to comment on what he experienced during the course of action. Specific follow up questions were concerned with intention ("What do you want to do here?"), focussing ("What are you looking at?"), thought Decision-making in expert volleyball players 10 ("What are you thinking of?"), and decision conditions ("What's leading you to make this decision?"). Questions did not refer to the result of action. The interviewer led the player to comment on the decision made in the situation they were watching. Some signs indicated that players were reporting from the first person perspective the focus non longer on the screen, their look glazed over, and slower speech (Vermersch, 1996).

Data Processing

Data recorded during the match were watched to distinguish the situations and the players' actions. Data elicited during the self confrontation-interviews were transcribed verbatim and given to the participants to confirm the validity of transcription. Data were processed in four phases: (a) making short accounts of the situations, (b) analyzing the short accounts according to the RPD model levels, (c) generating categories of salient features taken into account by players during the decision-making process, and (d) comparing the frequencies of salient features considered during decision-making for each level of the RPD model.

The first phase consisted of describing the game context and situations from the video recorded data, noting the score, and indicating the verbal reports from the interview regarding the player's decision. The situations were concerned with the actions and the positions of the player under review, his teammates, his opponents, and the trajectory of the ball, according to each game phase. Descriptions related to the technical language of volleyball (see Table 1).

For example, a description consisted of "Paul blocked with Jim on the right of the court. He touched the ball over the net and then defended it down. The setter set."

The second phase aimed at indicating the level of each situation regarding the RPD model. Verbal reports were analyzed. For example, verbal reports might ask (a) "Could the player assess the situation first or did he need to wait for further information?" (b) "Did the situation evolve in an unexpected way or not?" (c) "Did the player evaluate the action he Decision-making in expert volleyball players 11 wanted to carry out or not?" According to the RPD model, if the player quickly assessed the situation to decide what to do, then the situation was a simple match (level 1). In this case, critical information was available, the situation seemed familiar to him, so he could rapidly assess it, choose a course of action, and implement it. For example a player said: "maybe he was afraid to hit powerfully, because his arm went down. So my arms went down too" (Table 1). If the player needed more information to assess the situation or if the situation was assessed before evolving in an unexpected way for him, then the player had to diagnose it (level 2). He had to wait for useful information to occur or he had to assess what had changed.

For example a receiver said: "I could not see if he had hit the ball well. The trajectory of the ball was powerful". So he had to wait for information regarding the return of the ball. In another example, a defender said: "that's a ball that was deflected by the block, I spent time reacting. I was expecting a powerful return of the ball and suddenly the ball was deflected".

The player had to assess the changing situation in a short time frame. If the player mentally simulated the possible result of a course of action to know if it would have worked or not, then he evaluated a course of action (level 3). Thus he chose the course of action that he thought would work. For example, a player said: "If I had hit during my run up, then I probably would have sent the ball out. If I had hit the ball in front of me, then I would have sent it out of bounds. So I changed". When the players reported a mental simulation, they often used a sentence in the form "if… then".

The third phase aimed to generate categories of salient elements noted by the players according to RPD model and specifically to the game of volleyball. We decided to generate categories within the specific domain of volleyball terminology before using the RPD model categories. This approach allowed the context of the decision during the match to be taken into account. Then we compared the generated categories to those of the RPD model. This phase consisted of two steps. The first one was concerned with an inductive analysis of the Decision-making in expert volleyball players 12 verbal reports to generate categories of the features taken into account by the players to make their decision. This analysis referred to the Grounded Theory [START_REF] Corbin | Basics of qualitative research: Grounded theory. Procedures and techniques[END_REF]. Each category was defined and their content was described to avoid the fact that one element could pertain to two categories. We analyzed the verbal reports separately. We discussed each category until we came to an agreement about them. The second step consisted of a deductive analysis; the generated categories were compared to the categories of the RPD model to test the adequacy of the RPD model in the field of volleyball.

The fourth phase referred to a comparison between the frequencies of the salient features that were reported by the players for each level of the RPD model.

Results

Making Short Accounts of the Situations

Seventy short accounts were listed in a table containing three columns: the score, the description of the game situations, and data elicited during the self confrontation interview.

As an example, three of the short accounts are reported in table 1.

Analyzing the Short Accounts According to the RPD Model Levels

According to the RPD model, the analysis of the situations reported by the players referred to three levels. Results indicated that 57 situations were recognized rapidly by the players, who could then quickly implement a course of action (level 1). Nine situations were assessed for a longer time (level 2). In four situations, the players mentally simulated the consequences of the course(s) of action to determine if it would have worked or not before implementing one (level 3).

Generating Categories of Salient Features Taken into Account during Decision-making

This phase was concerned with two steps. The first one consisted of generating the categories of salient features reported by the players and the second one referred to a comparison between generated categories and the categories of the RPD model. Regarding Decision-making in expert volleyball players 13 the first step, twelve categories were elaborated. They referred to: (a) information about their opponent(s), (b) their teammates(s), and (c) themselves, (d) the trajectory of the ball, (e) their expectation about the players' actions, (f) abilities and tendencies of their opponents, (g), their partners, (h), themselves, (i) the number of decision(s) made, (j) the rules used, (k) a preceding event, and (l) the consequences of a course of action (see Table 2).

Information about the players consisted of the actions or positions on the court of the opponents, the teammates, the player himself. The actions could be concerned with the entire body or with a part of the body. Information about the trajectory of the ball was concerned with the direction of the ball and the possible changes of direction, the power of the hit/serve, and where the ball fell. Expectations consisted of what they thought the players could do in the current situation, they referred to an anticipation about a specific player's action or a particular trajectory of the ball. The abilities and tendencies of their opponents, teammates and themselves consisted of typical actions that were often carried out by a specific player in a typical situation, their level of expertise in specific situations, their role in the team, and their confidence in their teammates. The player made one or several decisions regarding a specific situation. Rules indicated what the player could carry out in typical game situations.

They referred to an association between a condition and an action to be carried out. They were in the form "if condition C holds then take action A". A preceding event consisted of a similar situation to the current event which had occurred during the current match or during a prior one. The players compared the current situation and event to prior ones. This comparison led them "to recognize" the situation as typical. The consequences of the course(s) of action were concerned with the possible effects of the action on the development of the situation. The player mentally simulated the possible consequences of an action to decide if it would work or not. Consequences were reported in the form "if I do this, then this will occur".

Decision-making in expert volleyball players 14 Different associations were found between decisions and some relevant features of decision-making (see Table 3). The players reported information about their own actions and positions in all decision-making. They reported most frequently: (a) one goal, (b) rules, (c) opponents' actions, and (d) teammates' action. They less frequently reported: (a) consequences of a course of action, (b) the players' abilities and tendencies (c) expectation about the players' actions. Verbal report analysis indicated that all the salient features reported by the players were linked to their role in the team. For example a hitter said: "To hit in zone 3, I have to line up the setter". His action depended on the type of set he had to hit (e.g., a short and quick set very near to the setter).

The second step of this phase was concerned with the comparison between these formerly generated salient features to the categories of the RPD model. The model suggests that recognition has four by-products: (a) relevant cues, (b) expectancies, (c) plausible goals, and (d) typical action. The results showed that the relevant cues referred to information about the actions and positions of teammates, opponents, the player himself and on the trajectory of the ball (see Table 2). Expectancies were concerned with expectations about the players' actions (i.e., anticipation of a teammate's or an opponent's action) and with the players' abilities and tendencies (i.e., specific focus on the possible actions of the player regarding his abilities and tendencies). The plausible goals were concerned with decisions that they were able to make in any given situation. Players indicated one or two possible decisions. Typical action included both the rules used by the players to make their decision and a preceding event. Rules and preceding events referred to a typical situation and thus to a decision. In a preceding event, the decisions had worked or not. If it had worked, the players reproduced it, if not, it incited them to pay more attention to the development of the situation. For example, a blocker said: "I remember, this situation seems to be like a preceding one, it was a hit from zone 6. I was expecting a hit on the end and the hit occurred in zone 6, so I was in late to Decision-making in expert volleyball players 15 block. In the current situation, I was expecting a hit to the end too, I was preparing to move and I could already see the ball in zone 6. So I jumped in front of him." In this situation, the blocker prepared to move whilst continuing to look at the trajectory of the ball.

In level 3 of the RPD model, the decision maker considers one action, evaluates it to determine if it will work or not. This assessment consisted of mentally simulating the likely consequences of the course(s) of action. In the present study, the assessment of the course(s) of action was concerned with the consequences of the player's own action or with the consequences of the other players' actions. The consequences appeared in the form "if I do this, then this will happen". The generated categories of salient features were consistent with the categories of the RPD model. The results showed that the players often utilized a typical action associated with the use of rules or with reference to a preceding event. These results suggested that decisions were partly prearranged before the decision was made. Rules were learned during the training process, and a preceding event led to a choice in the players' decision in a competition situation.

Comparing the Frequencies of the Salient Features Considered during Decision-making for each Level of the RPD Model

The RPD model consists of three levels. The comparison of the frequencies of salient features for each level of the RPD model revealed some similarities and differences between levels (Table 3). At levels 1 and 2, the players more frequently reported: (a) one goal, (b) typical action (more specifically rules), and (c) information about relevant cues. They did not report consequences of a course of action. They less frequently reported expectancies. In level 2, they did not report the players' abilities and tendencies. Whilst in level 3, they more frequently reported consequences of a course of action and relevant cues than other features.

They did not report expectation, nor opponents' and teammates' abilities and tendencies.

They reported one goal just as frequently several. In levels 1 and 2, the multiple decisions Decision-making in expert volleyball players 16 were concerned with the development of the situation; the players made a decision, then another to adapt to the changing situation. In level 1, such decisions were made early, and the player did not precisely know what would have happened. For example, a player was preparing to block at the end, while at the same time he was waiting for more information on the situation before moving. When the situation evolved, he abandoned his first decision to adapt to the changed situation. In level 2, the first decision referred to a specific situation and the second decision referred to the sudden and unexpected change in the development of the situation. For example, a player was preparing to defend a power hit in the back of the court when the block deflected the ball, so he dived to the front zone of the court. In levels 1 and 2, the multiple decisions were concerned with the development of the situation; the players made a decision, then another to adapt to the changing situation. In level 3, the multiple decisions were concerned with the outcome of the mental simulation; if they thought the course of action would not work, then they made another decision.

Discussion

The results of this study showed that the players' decision-making was based both on a process of recognition of a typical situation and on the use of associations between a typical situation and a typical action. The current situation was compared to a typical situation contained in memory. The typical situation was linked to a typical action that had to be adapted to the current situation. The situations were perceived at first as typical or not. If yes, they were simple matched, if not they were diagnosed. Sometimes, the players evaluated the course of action to know if it would have worked or not. Most often, the players reported only one decision. The results suggested that the players used their experience to carry out an action. The findings will be discussed from two perspectives: the results with the RPD model [START_REF] Klein | The Recognition-Primed Decision model: Looking back, looking forward[END_REF] and the updating of their experience.

Consistency of Results to the RPD Model

Decision-making in expert volleyball players 17

In general, the results obtained in this study reinforced the RPD model [START_REF] Klein | The Recognition-Primed Decision model: Looking back, looking forward[END_REF].

As the model predicts, the players' decision-making consisted of two steps: situation assessment and a choice of a course of action. A process of recognition of the typicality of the situation permitted the assessment of the situation. This process had four by-products: (a) relevant cues, (b) expectancies, (c) plausible goals, and (d) typical actions. The players perceived relevant cues according to their roles and their functions in the team regarding the situation. These relevant cues were concerned with the trajectory of the ball and with the positions and the specific actions of the teammates, the opponents and the player himself.

Expectancies consisted of expectation or anticipation about a specific player's action or possible players' action according to their abilities and tendencies. The players referred to a typical action contained in memory to implement an action; this typical action was linked to specific conditions (i.e. a typical situation). This typical action was concerned with the rules or a preceding event. The players investigated some plausible goals to adapt to the unexpected situation development; this development could be either sudden or progressive. Taking into account all these by-products led the players to recognize the situations as being familiar. This recognition process seemed to depend on their experience of volleyball situations. Most situations were rapidly recognized and classified as level 1.

As the model predicts, when the situations were not perceived as typical at first, the players diagnosed the situation. They had to wait for useful information. For example, a receiver said: "I could not see if he had hit the ball well. The trajectory of the ball was powerful". Moreover, the player had to be aware of the development of the situation in order to adapt. For example a defender said: "I was expecting a hit to the end and the hit occurred in zone 6 so I was in late to block". The results showed that the players pursued the situation assessment until they carried out their action. This continuing situation assessment led them to adapt their decision to the development of the situation. This continuing assessment Decision-making in expert volleyball players 18 corresponded well with the concept of situation awareness [START_REF] Endsley | Toward a theory of situation awareness in dynamic systems[END_REF][START_REF] Endsley | Designing for situation awareness[END_REF]. The environment is continuously assessed to include all new information in the player's decision. Situation assessment appeared to be dynamic and contextual. The players assessed the situations, when this assessment seemed sufficient for the players to act they made a decision while still continuing to assess the situation. Situation assessment appeared to be a continuous process, coupled with the ever-changing environment. Situation assessment could be compared to a process of natural drift; it continues until the operator involved in the situation is satisfied [START_REF] Amalberti | La sécurité écologique et la maîtrise des situations : Concepts et stratégies mis en jeu par les professionnels pour assurer leur propre sécurité et celle du système [Ecological security and situation control : Concepts and strategies used by professionals to ensure their own security and those of the context[END_REF].

As the RPD model predicts, the players assessed the consequences of their actions to confirm their choice of a particular course of action or not. This assessment aimed to know whether a particular action would work or not. If yes, they implemented it (e.g., "I told myself if I can set high, I know he would cope"); if not, they modified their action (e.g., "if I had hit the ball in front of me, I'd have sent it out of bounds. So I changed and I hit the ball toward the left side"). These results have to be considered with caution, because there were only four decisions in level 3. The results showed that the players took into account the features included in the RPD model. Nevertheless, some features were more frequently reported than others; relevant cues and typical action were more reported than expectancies and several goals. Moreover, most often, the players reported only one goal. This suggests two points.

The first point indicates that in volleyball situations, time-pressure is so high that the players do not have enough time to consider several options to implement one of them. To cope with high time-pressure, volleyball players use a game system, which indicates that the players' actions are performed according to their position on the court and to the current game situation. For example, a hitter said: "to hit in zone 3, I've to line up the setter". The game system contains a set of associations that links typical situations to typical actions (e.g., [START_REF] Blain | Volley-ball: de l'apprentissage à la compétition de haut niveau [volleyball: From learning to high level competition[END_REF]. The players' actions appeared to be largely prearranged by the game system and their Decision-making in expert volleyball players 19 abilities regarding each phase of the game. In high time-pressure conditions, the use of a game system largely limits the relevance of considering other plausible goals. The use of a game system aims to carry out quicker, and implement efficient, decisions. The game system takes into account: (a) rules of the games, (b) typical situations of the game, (c) players' abilities, and (d) players' roles in the team. It could be compared to the Common Frame Of Reference (COFOR) developed by [START_REF] Hoc | A cognitive approach to cooperation in dynamic situations[END_REF]. This COFOR refers to shared knowledge and shared abilities.

The second point indicates that a particular decision was made with regard to the developing situation. For example, the setter said: "I saw the center blocker was preparing to jump. So I set to the end hitter". In another example, a defender said: "there was a ball that was deflected by the block and I spent time reacting. I was expecting a powerful return of the ball and suddenly the ball was deflected; I'm diving…" In these cases, the players took into account critical elements to make their decision. Such decisions included the position of the center blocker and the deflection of the trajectory of the ball. The players did not report all the information they perceived during the decision-making process. Rather they talked about critical cues regarding the situation that they were describing. These results reinforced the idea that players spend more time assessing what is happening rather than comparing several courses of action [START_REF] Klein | The Recognition-Primed Decision model: Looking back, looking forward[END_REF].

Updating of the Players' Experience

The RPD model aims to describe how agents can use their experience to make decisions without needing to compare the pay-off with the limits of their chosen course of action.

Players' experience appeared in: (a) the use of a typical action associated with rules and a preceding event, (b) the focus on critical information, and (c) the anticipation about a particular player's action according to the possible range of actions in a given game situation.

Results suggested that players experience had two sources: training and competitions. During Decision-making in expert volleyball players 20 training, players learn some rules of the game system; they learn to focus their attention on critical cues (e.g., the height of the hitter's arm) and to consider the players' abilities and tendencies. Learning about the opponents' abilities and tendencies consists of observing their opponents during video sequences or during competitions. The coach, who expects the players to pay attention to some typical game phases or to particular opponents' actions, organizes video sequences. This training aimed to develop an efficient actions register [START_REF] Macquet | La prise de décision en sport de haut niveau : Un exemple chez une joueuse experte en badminton [Decision-making in high level sport : A case study in an expert badminton player[END_REF]. It contains typical actions, chosen on a technical and a tactical register controlled by the coach. The job of the coach consists of (a) identifying typical problems encountered in the course of the matches, (b) classifying them, and (c) finding solutions that athletes can train with, to be able to use them during competitions. The efficient actions register allows for the development of the game system. The use of rules reinforced the results obtained as shown by Macquet andFleurance (2006, 2007) and [START_REF] Mcpherson | Tactics, the neglected attribute of expertise[END_REF]. The use of training experience reinforced the results obtained as shown by Kibelé (2006), [START_REF] Poolton | The influence of analogy learning on decision-making in table tennis: Evidence from behavioral data[END_REF] and [START_REF] Raab | T-ECHO: Model of decision-making to explain behavior in experiments and simulations under time pressure[END_REF][START_REF] Raab | Decision-making in sports: Influence of complexity on implicit and explicit learning[END_REF]. Results have shown that athletes acquired experience during the course of the matches also. The athletes memorized given situations to learn about other players and a given game. Players learned from the events that occurred during the match. training periods (e.g., [START_REF] Salmela | Great job coach! Ottawa[END_REF][START_REF] Weineck | Manuel d'entraînement[END_REF]. However, the present results and the results of [START_REF] Macquet | Naturalistic decision-making in expert badminton players[END_REF], [START_REF] Mcpherson | Tactics, the neglected attribute of expertise[END_REF], [START_REF] Sève | Structure of expert players' activity during competitive interaction in table tennis[END_REF], and [START_REF] Sève | Table tennis scoring systems and expert players' exploration activity[END_REF] showed that players learn during competitions. This factor is not usually taken into account by coaches although it could be. This element of our study could Decision-making in expert volleyball players 21 contribute to improved decision-making in competitive situations. These results suggest some implications for coaching. The coach could continue to develop and reinforce associations between a typical situation and a typical action to allow players to be more readily efficient in high time-pressure situations. The training process consists of overtraining some actions to reproduce them and also adapting them to the particular context. At the same time, the coach could lead the players to make some other decisions by querying players' abilities and tendencies, and events as they are perceived. A specific framework could be developed, where the players could change roles to become accomplices for the opponent and where they could investigate the game and think about what the opponent could carry out. These two implications refer to the development and the updating of the players' knowledge about the game, suggested in different perspectives (i.e., [START_REF] Macquet | Naturalistic decision-making in expert badminton players[END_REF][START_REF] Mcpherson | Tactics, the neglected attribute of expertise[END_REF][START_REF] Sève | Structure of expert players' activity during competitive interaction in table tennis[END_REF] and to the interpretation of the athlete's particular situation according to salient and significant information (e.g., [START_REF] Arripe-Longueville (d') | Coach-athlete interaction during elite archery competitions : An application of methodological framework used in ergonomics research to sport psychology[END_REF].

They also refer to a perspective concerned with a proactive attitude [START_REF] Cellier | Exigences et gestion temporelle dans les environnements dynamiques [Requirements and time management in dynamic environments[END_REF]. Taking into account both the updating of knowledge about the players and the game play also allows players to think about what could happen in a situation in which they are committed. The players expect a particular opponent's action or simulate the consequences of a course of action. Their expectations contribute to investigating a specific course of action while, at the same time, continuing to assess the way that the situation is evolving. If the situation evolves in the expected way, they carry out the action, if not, they diagnose the situation to emphasize its typicality to implement a course of action. Their mental simulation of the consequences of a course of action allows them to adapt their decision more suitably, by anticipating the outcome of their course of action on the game play. Although these simulations were not revealed in most situations, they seemed to allow the players to cope with time pressure and situation complexity in order to be efficient.

Decision-making in expert volleyball players 22

This study presents one limitation. Only seven players had participated in the study. This limits the extent of the results. Despite this limitation, this study and NDM approach provide a framework that enables new insights into athletes' decision-making. It offers new perspectives for sports psychologists to understand volleyball players' decision-making (and, more generally, athletes' decision-making) during competitive situations. It also offers the possibility for athletes and coaches to question their way of playing, to involve them in a reflexive practice (Schön, 1993). James has served, the opponents have spiked, he is trying to receive by diving (Level 2) James: "I was preparing to defend a hit in the back of the court. There was a ball that was deflected by the block and I spent time reacting. I was expecting a powerful return of the ball and suddenly the ball was deflected by the block. I had to change to adapt to the new trajectory of the ball. It was a short and a curved one. I'm diving but I don't touch the ball. It's an unusual situation. Interviewer: Do you train this situation? James: We don't really train to defend a deflected ball, sometimes we face these situations but that's all. Here it wasn't obvious, I was expected a powerful trajectory and at a moment, it was slowed down. It's a question of speed and preparation. If we are ready to receive, then we can react more rapidly." Set 4 7-4 Jim received and sent the ball between the zones 2 and 3. James was in zone 4 hitting. He moved, went along the attack line and went to the middle of the zone 2 and 3 to jump. He jumped toward the zone 2. The opponent blockers followed him and he hit between the two opponents.

(Level 3) Jim: "I was preparing to hit taking into account the trajectory of the receive. I began to move when the serve arrived in our court, and in the same time I looked at the serve trajectory. The return moved forward, it didn't come at the middle of the court, near the net. The return arrived at my right, I was in zone 4 and I had to hit between zones 3 and 2. I had a long move to do. If the serve is powerful and the receive is fast, then I've to make a diagonal in order to run up. That generates a problem: some angles are blocked. If I arrive from a diagonal to the setter, I only can hit in front of me, I can't turn myself. Interviewer: Here the receive is moved forward to the right… Jim: Yes the receive is moved forward to the right, it didn't come at the middle. I had a long move to do. That's the reason I went to the diagonal. If I have had hit during my run up then I probably would have sent the ball out. If I had hit the ball in front of me, I would have sent it out of bounds, so I changed and I hit the ball towards the left side of the court. To hit in zone 3, I have to line up the setter. The setter saw I was a little in late, so he set a short and "slow" ball and I went up with the ball. The opponents' blockers didn't jump on my hit, so it was easier for me."

Decision-making in expert volleyball players 29

Typical action Rules

A preceding event "To hit in zone 3, I have to line up the setter" "If the serve is powerful and the receive is fast, then I've to make a diagonal in order to run up" "I remember this situation. It seems be like to a preceding one, it was a hit from zone 6. I was a little similar: I was expecting a hit at the end and the hit occurred in zone 6 so I was in late to block"

Consequences of the course of action

The consequences of a course of action

"

  For example, they observed what happened in a given situation as a consequence of the implementation of a certain course of action. They observed what actions the opponents carried out most often. These results reinforced those of the Sève et al.'s study (2003) about exploration activity of table tennis players. Theories of sport training consider that in competitions, athletes have to reproduce what they have learned during

Table 1

 1 Examples of Short Accounts of Each of the three Levels

	Score Description of the game situations	Verbal reports from the self confrontation interviews
	Set 1	Paul blocked with Jim on the right of	Paul: "I moved to the end in order to block. I jumped with my
	1-2	the court. He touched the ball over	teammate. I was expecting a powered hit. During the jump,
		the net and then defended it down.	the hitter seemed to change. I saw his arms going down.
		The setter set.	Interviewer: What did you think of?
			Paul: I don't know. May be he was afraid to hit powerfully
		(Level 1)	because his arms went down. So my arms went down too. May
			be he wanted to achieve a block out. I was afraid, I didn't
			want him to succeed. So I adapted my block to his hit and my
			arms went down too.
			Interviewer: What's lead you to make this decision?
			Paul: The fact that the opponent his arms down. I've to look
			at the hitter's arm and to "follow" it in order to adapt my
			block in order to avoid a block out."
	Set 4
	14-7

Table 3

 3 RPD Model, Salient Features Categories and Verbal Reports ve called a set very near of me to the centerhitter, in order to move away the end hitters from the center hitter" "that's a ball that was deflected by the block[…] It was a short and a curve trajectory" "The receive is moved forward to the right"

	Categories of	Categories of salient features elicited	Verbal reports from the self confrontation
	the RPD model	from the verbal reports	interview
	Relevant cues	Opponents' actions	"He's hitting aiming at his left"
			"His arms went down"
		Teammates' actions	"James was well placed at the net"
			"The setter set a slow and a short ball"
		Player's actions	"I'm trying to counterbalance my arms by closing
			the angle…"
	The trajectory of the ball "I'Expectancies Expectations "I was expecting a powerful return of the ball"
			"I was expecting a hit from the end too"
		Opponents' abilities and tendencies
		Teammates' abilities and tendencies
		Player's abilities and tendencies
	Plausible goal	The number of decision(s) made	"So I set the ball to the end hitter"
			"I adapted my block to his hit and my arms went
			down"
			"I asked myself if I should open the block to open
			up a space between my teammate and me in order
			to incite him to hit between the two blockers […]
			Therefore, I decided to put pressure on my
			teammate"
			… "I wanted to block at the end […] I jumped in
			front of him (row line hitter)"

"This guy often hits to his left" "The setter aimed to set fast balls" "My teammate is a good receiver" "Our hitter trusts some hitters more than others" "I'm not the aim receiver" "The setter often sets me when he's in difficulty"

Acknowledgments

The author is grateful to the players and to the coach of the volleyball team for their participation to this study. The author acknowledges Philippe Fleurance for his participation in the data processing and the students for their participation in the video recording.

Decision-making in expert volleyball players 30


Rémi Boutteau

Xavier Savatier

Jean-Yves Ertaud

Bélahcène Mazari

A 3D Omnidirectional Sensor For Mobile Robot Applications

Introduction

In most of the missions a mobile robot has to achieve -intervention in hostile environments, preparation of military intervention, mapping, etc -two main tasks have to be completed: navigation and 3D environment perception. Therefore, vision based solutions have been widely used in autonomous robotics because they provide a large amount of information useful for detection, tracking, pattern recognition and scene understanding. Nevertheless, the main limitations of this kind of system are the limited field of view and the loss of the depth perception. A 360-degree field of view offers many advantages for navigation such as easiest motion estimation using specific properties on optical flow [START_REF] Mouaddib | Introduction à la vision panoramique catadioptrique[END_REF] and more robust feature extraction and tracking. The interest for omnidirectional vision has therefore been growing up significantly over the past few years and several methods are being explored to obtain a panoramic image: rotating cameras [START_REF] Benosman | Panoramic Stereovision Sensor[END_REF], muti-camera systems and catadioptric sensors [START_REF] Baker | A Theory of Single-Viewpoint Catadioptric Image Formation[END_REF]. Catadioptric sensors, i.e. the combination of a camera and a mirror with revolution shape, are nevertheless the only system that can provide a panoramic image instantaneously without moving parts, and are thus well-adapted for mobile robot applications. The depth perception can be retrieved using a set of images taken from at least two different viewpoints either by moving the camera or by using several cameras at different positions. The use of the camera motion to recover the geometrical structure of the scene and the camera's positions is known as Structure From Motion (SFM). Excellent results have been obtained during the last years with SFM approaches [START_REF] Pollefeys | Visual Modeling with a Hand-Held Camera[END_REF][START_REF] Nister | Automatic Dense Reconstruction from Uncalibrated Video Sequence[END_REF], but with off-line algorithms that need to process all the images simultaneous. SFM is consequently not well-adapted to the exploration of an unknown environment because the robot needs to build the map and to localize itself in this map during its world exploration. The in-line approach, known as SLAM (Simultaneous Localization and Mapping), is one of the most active research areas in robotics since it can provide a real autonomy to a mobile robot. Some interesting results have been obtained in the last few years but principally to build 2D maps of indoor environments using laser range-finders. A survey of these algorithms can be found in the tutorials of Durrant-Whyte and Bailey (Durrant-Whyte & Bailey, 2006;Bailey & Durrant-Whyte, 2006).

Vision-based SLAM algorithms are generally dedicated to monocular systems which are cheaper, less bulky, and easier to implement than stereoscopic ones. Stereoscopic systems have, however, the advantage to work in dynamic environments since they can grab simultaneously two images. Calibration of the stereoscopic sensor enables, moreover, to recover the Euclidean structure of the scene which is not always possible with only one camera. In this chapter, we propose the design of an omnidirectional stereoscopic system dedicated to mobile robot applications, and a complete scheme for localization and 3D reconstruction. This chapter is organized as follows. Section 2 describes our 3D omnidirectional sensor. Section 3 is dedicated to the modelling and the calibration of the sensor. Our main contribution, a Simultaneous Localization and Mapping algorithm for an omnidirectional stereoscopic sensor, is then presented in section 4. The results of the experimental evaluation of each step, from calibration to SLAM, are then exposed in section 5. Finally, conclusions and future works are presented section 6.

System overview

2.1 Sensor description Among all possible configurations of central catadioptric sensors described by Nayar and Baker [START_REF] Baker | A Theory of Single-Viewpoint Catadioptric Image Formation[END_REF], the combination of a hyperbolic mirror and a camera is preferable for the sake of compactness since a parabolic mirror needs a bulky telecentric lens. Although it is possible to reconstruct the environment with only one camera, a stereoscopic sensor can produce a 3D reconstruction instantaneously (without displacement) and will give better results in dynamic scenes. For these reasons, we developed a stereoscopic system dedicated to mobile robot applications using two catadioptric sensors as shown in Figure 1.

Imposing the Single-Viewpoint (SVP) Constraint

The formation of images with catadioptric sensors is based on the Single-Viewpoint theory (Baker & Nayer, 1999). The respect of the SVP constraint permits the generation of geometrically correct perspective images. In the case of a hyperbolic mirror, the optical center of the camera has to coincide with the second focus F' of the hyperbola located at a distance of 2e from the mirror focus as illustrated in Figure 2. The eccentricity e is a parameter of the mirror given by the manufacturer. Fig. 2. Image formation with a hyperbolic mirror. The camera center has to be located at 2e from the mirror focus to respect the SVP constraint.

A key step in designing a catadioptric sensor is to respect this constraint as much as possible. To achieve this, we first calibrate our camera with a standard calibration tool to determine the central point and the focal length. Knowing the parameters of both the mirror and the camera, the image of the mirror on the image plane can be easily predicted if the SVP constraint is respected as illustrated in Figure 2. The expected mirror boundaries are superposed on the image and the mirror has then to be moved manually to fit this estimation as shown in Figure 3.

Modelling of the sensor

The modelling of the sensor is a necessary step to obtain metric information about the scene from the camera. It establishes the relationship between the 3D points of the scene and their projections into the image (pixel coordinates). Although there are many calibration methods, they can be classified into two main categories: parametric and non-parametric. The first family consists in finding an appropriate model for the projection of a 3D point onto the image plane. Non-parametric methods associate one projection ray to each pixel [START_REF] Ramalingram | Towards Complete Generic Camera Calibration[END_REF] and provide a "black box model" of the sensor. They are well adapted for general purposes but they make more difficult the minimization algorithms commonly used in computer vision (gradient descent, Gauss-Newton, Levenberg-Marquardt, etc).

Projection model

Using a parametric method requires the choice of a model, which is very important because it has an effect on the complexity and the precision of the calibration process. Several models are available for catadioptric sensors: complete model, polynomial approximation of the projection function and generic model. The complete model relies on the mirror equation, the camera parameters and the rigid transformation between them to calculate the projection function [START_REF] Gonzalez-Barbosa | Fast Dense Panoramic Stereovision[END_REF]. The large number of parameters to be estimated leads to an error function which is difficult to minimize because of numerous local minima [START_REF] Mei | Single View Point Omnidirectional Camera Calibration from Planar Grids[END_REF]. The polynomial approximation of the projection function was introduced by Scaramuzza [START_REF] Scaramuzza | A Flexible Technique for Accurate Omnidirectional Camera Calibration and Structure From Motion[END_REF], who proposed a calibration toolbox for his model. The generic model, also known as the unified model, was introduced by Geyer [START_REF] Geyer | A Unifying Theory for Central Panoramic Systems and Practical Implications[END_REF] and [START_REF] Barreto | A Unifying Geometric Representation for Central Projection Systems[END_REF], who proved its validity for all central catadioptric systems. This model was then modified by Mei [START_REF] Mei | Single View Point Omnidirectional Camera Calibration from Planar Grids[END_REF], who generalized the projection matrix and also took into account the distortions. We chose to work with the unified model introduced by Mei because any catadioptric system can be used and the number of parameters to be estimated is quite reasonable. As shown in Figure 4, the projection

  T v u  p of a 3D point X with coordinates   T w w w Z Y X
in the world frame can be computed using the following steps:



The coordinates of the point X are first expressed in the sensor frame by the rigid transformation W which depends on the seven parameters of the vector

  T z y x z y x w t t t q q q q  1 V
. The first four parameters are the rotation R parameterised by a quaternion and the three others are those of the translation T . The coordinates of X in the mirror frame are thus given by:

T R                       w w w Z Y X Z Y X (1)  The point   T Z Y X  X
in the mirror frame is projected from the center onto the unit sphere giving

  T S S S Z Y X  S X
. This point is then projected onto the normalized plane from a point located at a distance ξ from the center of the sphere. These transformations are combined in the function H which depends on only one parameter:

    2 V
. The projection onto the normalized plane, written

  T y x  m
is consequently obtained by:

                       S S S S Z Y Z X y x , with                                      2 2 2 2 2 2 2 2 2 Z Y X Z Z Y X Y Z Y X X Z Y X S S S
(2)



Distortions are then added to the point m using the distortion function D, which depends on 5 coefficients: 3 for radial distortions and 2 for tangential distortions.

Let

  T 5 4 3 2 1 k k k k k  3 V
be the parameter vector containing these coefficients,

                     ) 2 ( 2 ) 1 ( ) 2 ( 2 ) 1 ( 2 2 3 4 6 5 4 2 2 1 2 2 4 3 6 5 4 2 2 1 y k xy k k k k y x k xy k k k k x         d m (3)
 Final projection is a perspective projection involving the projection matrix K . This matrix contains 5 parameters: the generalized focal lengths u  and v  , the www.intechopen.com coordinates of the principal point 0 u and 0 v , and the skew  . Let K be this projection function, and

  T 0 0 v u v u     4 V
be the parameter vector. The projection

  T v u  p
of the 3D point X is given by equation ( 4).

d d m m K p              1 0 0 0 0 0 v u v u u     (4)
Let V be the global parameter vector, i.e.

  T T T T T 4 3 2 1 V V V V V  . The global projection function of a 3D point X , written ) , ( X V P
, is obtained by chain composition of the different functions presented before:

) , ( ) , ( X V X V W H D K P     (5)
These steps allow the computation of the projection onto the image plane of a 3D point knowing its coordinates in the 3D space. In a 3D reconstruction framework, it is necessary to do the inverse operation, i.e. to compute the direction of the luminous ray corresponding to a pixel. This step consists in computing the coordinates of the point S X belonging to the sphere given the coordinates   T y x of a 2D point on the normalized plane. This step of retro projection, also known as lifting, is achieved using formula (6).

                                                   1 ) )( 1 ( 1 1 ) )( 1 ( 1 1 ) )( 1 ( 1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 y x y x y y x y x x y x y x S X (6)

Calibration

Calibration consists in the estimation of the parameters of the model which will be used for 3D reconstruction algorithms. Calibration is commonly achieved by observing a planar pattern at different positions. With the tool we have designed, the pattern can be freely moved (the motion does not need to be known) and the user only needs to select the four corners of the pattern. Our calibration process is similar to that of Mei [START_REF] Mei | Single View Point Omnidirectional Camera Calibration from Planar Grids[END_REF]. It consists of a minimization over all the model parameters of an error function between the estimated projection of the pattern corners and the measured projection using the Levenberg-Marquardt algorithm [START_REF] Levenberg | A method for the solution of certain problems in least squares[END_REF][START_REF] Marquardt | An Algorithm for Least-Squares Estimation of Nonlinear Parameters[END_REF].

www.intechopen.com

If n is the number of 3D points i X , i x their projections in the images, we are looking for the parameter vector V which minimizes the cost function

) (V E :   2 1 ) , ( 2
1 ) (     n i P E i i x X V V (7)
Our calibration tool was developed in C++ using the computer vision library OpenCV and can be freely downloaded from our website (Boutteau, 2009). It does not require any commercial software and a particular attention has been given to the optimization of the computation time since a calibration with 10 images does not exceed 2 minutes.

Relative pose estimation

The estimation of the intrinsic parameters presented in the previous section allows to establish the relationship between 3D points and theirs projections for each sensor of the stereoscopic system. To obtain metric information from the scene, for example by triangulation, the relative pose of the two sensors has to be known. This step is generally performed by a pixel matching between both images followed by the estimation of the essential matrix. This matrix, originally introduced by Longuet-Higgins [START_REF] Longuet-Higgins | A computer algorithm for reconstructing a scene from two projections[END_REF], has the property to contain information on the epipolar geometry of the sensor. It is then possible to decompose this matrix into a rotation matrix and a translation vector, but the last one can only be determined up to a scale factor [START_REF] Bunschoten | Robust Scene Reconstruction from an Omnidirectional Vision System[END_REF]. The geometrical structure of the scene can consequently be recovered only up to this scale factor. Although in some applications, especially for 3D visualization, the scale factor is not needed, it is required for preparation of intervention or for navigation. To accomplish these tasks, the size of the objects and their distance from the robot must be determined. The 3D reconstruction has therefore to be Euclidean. Thus, we suggest in this section a method to estimate the relative pose of the two sensors, with a particular attention to the estimation of the scale factor. The estimation of the relative pose of two vision sensors requires a partial knowledge of the environment to determine the scale factor. For this reason, we propose a method based on the use of a calibration pattern whose dimensions are known and which must be visible simultaneously by both sensors. Let ) , ,, (

1 1 1 1 z y x C and ) , , , ( 2 2 2 2 z y x C
be the frames associated with the two sensors of the stereoscopic system, and M be a 3D point, as shown in Figure 5.

www.intechopen.com in the frame associated with the second sensor has for coordinates in the first sensor frame:

                                            1 1 0 0 0 1
                                                                                        n
1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0              (9) www.intechopen.com
The principle of the relative pose estimation consists therefore in computing the rigid transformation between the two sensors, knowing the coordinates of a set of points in the two frames. The coordinates of the points are however not known directly since the location of the pattern is not known. The pose of the pattern has to be determined in the two frames using the Levenberg-Marquardt algorithm to minimize the error between estimated projection and projections extracted from the images. The resolution of equation ( 9) then gives the relative pose   t R, of the two sensors including the scale factor.

Simultaneous Localization and Mapping

3D reconstruction of a scene using the motion of a sensor addresses two problems: localization and mapping. Localization consists in estimating the trajectory of a robot in a known map [START_REF] Thrun | Robust Monte Carlo Localization for Mobile Robots[END_REF]. The aim of mapping is the creation of a map of the environment using measurements from the sensors embedded on a robot knowing its trajectory [START_REF] Thrun | Robotic mapping: A survey[END_REF]. When neither the trajectory of the robot, nor the map of the environment are known, localization and mapping problems have to be considered simultaneously: it is the issue of SLAM (Simultaneous Localization And Mapping). The first approach to solve the SLAM problem is to assume the motion known (by odometry or by the command law) even if this one is corrupted by noise. The position of visual landmarks can consequently be predicted. Sensors embedded on the mobile robot, for example laser range-finders, provide measurements of its environment. These observations are then used to update the model containing the coordinates of the visual landmarks and the positions of the robot. These steps (prediction/observation/update) are implemented using the Kalman filter or one of its derivatives. The second approach is to optimize the geometrical structure of the scene and the positions of the sensor using the bundle adjustment method. A synthesis on bundle adjustment algorithms was published by Triggs [START_REF] Triggs | Bundle Adjustment -A modern Synthesis[END_REF]. Bundle adjustment provides more accurate results than Kalman filters [START_REF] Mouragnon | Generic and Real-Time Structure from Motion using Local Bundle Adjustment[END_REF] but needs more computing time. In most of the applications, this algorithm is used off-line to obtain a very accurate model, but it is also possible to apply it iteratively. Although bundle adjustment is commonly used with conventional cameras, there are very few works on its adaptation to omnidirectional sensors. The main works in this field are those of Lhuillier [START_REF] Lhuillier | Automatic Structure and Motion using a Catadioptric Camera[END_REF] and Mouragnon [START_REF] Mouragnon | Generic and Real-Time Structure from Motion using Local Bundle Adjustment[END_REF] who suggest to find the structure of the scene and the motion of a catadioptric sensor by a local bundle adjustment followed by a global one to obtain more accurate results. Their works highlight the difficulty of estimating the scale factor, although it is theoretically possible with a non-central sensor. The contribution of this section deals with the application of a bundle adjustment algorithm to an omnidirectional stereoscopic sensor previously calibrated to solve the ambiguity on the scale factor. Bundle adjustment relies on the non-linear minimization of a criterion, so a first estimation of the parameters as to be found to ensure the convergence. Before the presentation of the bundle adjustment algorithm, we thus expose our initialization step.

Initialization

Estimating the camera motion, also called ego-motion, requires to relate the images of the sequence grabbed during the motion. Relating images consists in localizing in the images the projections of a same 3D point of the scene. This step is decisive because the precision of www.intechopen.com the motion relies on the quality of this matching. The visual landmarks are detected in a majority of works by the Harris corner detector since it has good performances on luminosity change. This detector is however very sensitive to scale change and it can thus fail with large motion. To avoid this, Lowe [START_REF] Lowe | A Distinctive image features from scale-invariant keypoints[END_REF] has presented a very interesting approach for the detection and the matching of regions of interest: the Scale Invariant Feature Transform (SIFT). The SIFT principle is to detect features from images which are invariant to scale change, rotation, and small point of view change. A descriptor, which corresponds to the orientation histogram, is then associated to the features and the matching can be achieved by comparison of their Euclidean distances. Once the images are related, the epipolar geometry can be estimated between the two times 1  k and k . The epipolar geometry is interesting since it gives information on the relative pose of two vision sensors. Several works are dedicated to the estimation of the epipolar geometry for catadioptric sensors. Some of them [START_REF] Pajdla | Epipolar Geometry of Central Panoramic Catadioptrics Cameras[END_REF][START_REF] Gonzalez-Barbosa | Fast Dense Panoramic Stereovision[END_REF][START_REF] Mariottini | The Epipolar Geometry Toolbox : multiple view geometry and visual servoing for MATLAB[END_REF] give analytical solution to the estimation of the epipolar curves. Their methods need nevertheless to introduce the mirror equations and the proposed solutions are thus specific to the kind of sensor used. Other works [START_REF] Bunschoten | Robust Scene Reconstruction from an Omnidirectional Vision System[END_REF][START_REF] Negishi | Calibration of Omnidirectional Stereo for Mobile Robots[END_REF] rely on the use of panoramic images, i.e. unwrapped images, and consider the epipolar curves as the intersection of the epipolar plane with a cylinder representing the image plane. This approach is interesting because the idea is to generalize the notion of epipolar geometry to panoramic sensors. We suggest to generalize the epipolar geometry for all central sensors using the model of the equivalence sphere. With this model, the coplanarity constraint initially defined for perspective cameras (Longuet-Higgins, 1981) can be transposed to all central sensors. As shown in Figure 6, if the points S1 X and S2 X correspond to the projection of the same 3D point X onto the two spheres, then 1 C , 2 C , S1 X , S2 X and X lie in the same plane.

Fig. 6. Epipolar geometry for central sensors.

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Let t be the translation vector of the sensor and R be the rotation matrix, the coplanarity constraint can be expressed in the coordinate system ) , , , (

2 2 2 2 z y x C as: 0 ) ( T   S1 S2 X t R X (10)
The coplanarity constraint (10) can be rewritten in the matrix form as:

0 T  S1 S2 EX X ( 11
)
where

RS E 

is the essential matrix first introduced by Longuet-Higgins (Longuet-Higgins, 1981) and S is an antisymmetric matrix characterizing the translation:

              0 0 0 x y x z y z t t t t t t S (12)
The essential matrix E can be estimated from a set of matched points using the eight-point algorithm [START_REF] Longuet-Higgins | A computer algorithm for reconstructing a scene from two projections[END_REF]. Given two lifted points

  T 1 1 1 z y x  S1 X and   T 2 2 2 z y x  S2 X
corresponding to the same 3D point X , (11) becomes for each pair of matched points: and using n pairs of matched points, the set of equations ( 13) can be expressed in the matrix form as:

0 33 1 2 31 1 2 12 1 2 11 1 2      e z z e
0  Ae (14)
With more than eight points, a least squares solution can be found by singular value decomposition (SVD) of A . Because of the least squares estimation, the estimated matrix may not respect the two constraints of an essential matrix: two of its singular values have to be equal, and the third has to be zero (Hartley & Zisserman, 2004). A constraint enforcement step is therefore necessary and is achieved by the method described by Hartley (Hartley & Zisserman, 2004). Given a 3x3 matrix

T UDV E  , where ) , , ( c b a diag  D with c b a  
. The closest essential matrix to E in Frobenius norm is given by:

T ˆV D U E  (15) where ) 0 , 2 / ) ( , 2 / ) (( ˆb a b a diag    D .
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The estimation of the essential matrix may fail when there is one outlier among the set of points used in equation ( 14). The solution is therefore found by using the RANSAC algorithm. The main difference between perspective cameras and omnidirectional sensors comes from the computation of the error to determine if a point belongs to the consensus set. In the perspective case, the error is defined as the distance d between the point 2 X and the epipolar line l corresponding to the point 1 X as shown in Figure 7. In the omnidirectional case, the computation of the error is more difficult since the epipolar line becomes an epipolar curve (see Figure 8). We therefore suggest to work on the equivalence sphere using the coplanarity constraint defined by equation ( 11) to compute this error.

Given a point S1 X on the first sphere, the normal of the epipolar plane in the second sensor frame is given by

S1 2 EX N 
. The vectors S2 X and 2 N are orthogonal when the coplanarity constraint is respected. The error e can consequently be computed as the angular error given by: The essential matrix does not give directly the displacement of the sensor since it is a combination of the rotation matrix R and the antisymmetric matrix S (see equation 12).

2 S2 N X T  e ( 16
The essential matrix has therefore to be decomposed to retrieve the rotation R and the translation t . After the constraint enforcement step, the essential matrix can be written as

T ) 0 , 1 , 1 ( diag V U E 
. Let i u be the i th column of U and i v the i th column of V , there are four possible solutions for the rotation R and two solutions for the translation vector t [START_REF] Wang | A SVD decomposition of essential matrix with eight solutions for the relative positions of two perspective cameras[END_REF]:

www.intechopen.com                      
The estimation of the essential matrix, followed by its decomposition, enables to retrieve the motion of the system, but only up to a scale factor. In a large majority of works, this ambiguity is removed by the use of odometry data [START_REF] Bunschoten | Robust Scene Reconstruction from an Omnidirectional Vision System[END_REF]. This approach is interesting since it gives directly the result, but it is preferable to use only visual data to avoid any communication with the mobile robot. The stereoscopic structure of our system can be used to retrieve the scale factor since the baseline is known. The 3D coordinates 1 k X  and k X of a set of points can actually be computed at times 1  k and k and the scale factor can be retrieved by the computation of the norm of t , given by:

1 k k RX X t    (18)

Bundle Adjustment

The algorithms presented in the previous section provide initial values for the motion of the sensor and for the 3D coordinates of points. These values are however not sufficiently accurate to be used directly. An excessive error will provide, because of its accumulation, a wrong 3D model. An extra optimisation step, the bundle adjustment, is therefore necessary. This method is well known for perspective cameras (Triggs et al., 2000) but only few works are dedicated to omnidirectional sensors. In this section, we propose a bundle adjustment algorithm which takes into account the specificity of our sensor: its omnidirectional field of view and its stereoscopic structure. Bundle adjustment consists in minimizing the cost function corresponding to the error between estimated projections of 3D points and their measured projections. Let m be the number of positions of the stereoscopic system and n be the number of 3D points, the cost function can be written as:

       m j n i ij i j P F 1 1 2 ) , ( 2 1 ) , ( x X V X V (19)
where j V is the parameter vector of the th j camera, i X is the coordinates of the th i 3D point in the world frame, ij x is the projection of the th i 3D point in the image of the th j camera and ) , (

i j P X V
is the predicted projection of point i in image j .

Using a stereoscopic sensor allows to recover the scale factor, which is impossible with a monocular sensor. The projection function defined by equation ( 5) has therefore to be modified to take into account the stereoscopic structure of the sensor. Thus, the projection function provides now four elements: the coordinates low u and low v of the projection of X onto the image plane of the first sensor, and the coordinates high u and high v of its projection onto the image plane of the second sensor. The system is no longer considered as two separate sensors but as a global device. The relative pose of the two sensors ) , ( t R is consequently added to the model parameters as shown in Figure 10. z y x z y x w t t t q q q q  5 V be the new parameter vector, the rotation being parameterised by a quaternion. An extra rigid transformation, written

) ( 5 V C
, is added to the initial projection function to express coordinates either in the low sensor frame or in the high sensor frame:

) , ( ) , ( X V X V W C H D K P      (20)
where

  T T 5 T 4 T 3 T 2 T 1 V V V V V V 
The minimization of the reprojection error defined by equation ( 19) is carried out by the Levenberg-Marquardt algorithm. The key step of this algorithm lies in the resolution of the augmented normal equation:

  e J Δ I J J T T     (21)
where  is a real number varying from iteration to iteration and I is the identity matrix.

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The resolution of equation ( 21) requires the computation of the Jacobian J of the projection function. The sensor is calibrated so the parameters that have to be estimated are the poses j 1 V of the cameras and the coordinates i X of the 3D points. Thus, the partial derivatives which have to be estimated are the derivative of the projection function whith respect to the pose of the system:

j 1 V P   ( 22
)
as well as the derivative of the projection function with respect to the coordinates of the points:

i X P   ( 23
)
The form of the Jacobian matrix used in equation ( 21) is illustrated in Figure 11 for three poses of the sensors and four points. The iterative resolution of the augmented normal equation as described in [START_REF] Levenberg | A method for the solution of certain problems in least squares[END_REF][START_REF] Marquardt | An Algorithm for Least-Squares Estimation of Nonlinear Parameters[END_REF] minimizes the reprojection error and provides a good estimate of the poses of the system and of the coordinates of the 3D points.

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Experimental results

The stereoscopic omnidirectional sensor was mounted on a Pioneer robot with a Sick LD-PDS laser range-finder used to provide the ground truth. Each step, from calibration to 3D reconstruction and motion estimation, was evaluated on real images and without prior knowledge to evaluate the system in real conditions. Some results obtained with synthetic images are also presented to validate results that need a perfect knowledge of the 3D points coordinates in the sensor frame.

Calibration of the sensor

The calibration step was evaluated by computing the Root of Mean Squares (RMS) distances between estimated and theoretical projections of a set of 3D points. As it is very difficult to know precisely the coordinates of a 3D point in the sensor frame, we used synthetic images.

The sensor was simulated in POV-Ray, a ray-tracing software, to generate omnidirectional images containing a calibration pattern as shown in Figure 12 and these images were used to calibrate the sensor. As the projection model and the 3D coordinates of reference points are perfectly known in this case, their theoretical projection can easily be computed and compared to the projection obtained by the estimated model. Table 1 shows the mean error and the standard deviation obtained on a set of 150 points.

Mean error (pixels) 0.24 Standard deviation (pixels) 0.11 Table 1. Calibration results on synthetic images

The calibration was then evaluated on real images. Two sets of omnidirectional images containing a calibration pattern were taken. We used the first set to calibrate the sensor as described in section. The second set was used to compute the error between estimated projections of the grids points computed using estimated model and their measured projections extracted from the images. The error on real images is greater than the error on synthetic images, but both are very low since they are less than half a pixel. These results highlight the good estimation of the model www.intechopen.com parameters even with noisy data since we used a real system which is not perfect. This can be explained by the introduction of the distortions into the model which allows some misalignments between the mirror and the camera.

Relative pose estimation

The estimation of the relative pose was evaluated on synthetic images for a vertical setup of the two sensors, i.e. the same setup as the real system. Several pairs of images containing a calibration were taken as shown in Figure 13 and the relative pose was evaluated with the method presented in section 3.3. The distance between the two sensors was defined with 10cm and they are supposed perfectly aligned. The result of the relative pose estimation is presented in Table 3. The rotation matrix is almost an identity matrix since the sensors are perfectly aligned. The norm of the translation vector is 99.95mm instead of 100mm, so an error of 0.05%. These results are particularly good especially since they involve the calibration results needed for the estimation of the pose of the calibration pattern.

The estimation of the relative pose was then evaluated on real images. The sensor was mounted on a graduated rail and was moved by steps of 10cm. At each position, an omnidirectional image was grabbed with the aim of computing the displacement of the sensor according to the first position using five calibration patterns placed in the room as shown in Figure 14. The average error is greater than the one obtained with synthetic images but it is less than 0.9%. This increase can be explained by the noise in real images but also by the accumulation of the errors because the estimation of the relative pose involves the calibration values. Once the relative pose is estimated, the essential matrix can be computed and used to check the epipolar geometry properties. In Figure 14, for each pixel on the left image (red crosses), the corresponding epipolar curve (green curves) is drawn on the right image and vice versa.

Triangulation

The combination of the calibration of the two sensors and of their relative pose estimation was evaluated by triangulation. The coordinates of the 3D points of five grids disposed on three walls of a room were estimated. Images used for this step are presented in Figure 15. For each grid, the four corners were selected manually on the two images and an automatic corners detector was applied to extract all grid points. The 3D position of these points was evaluated by triangulation and is displayed in Figure 16. The position of the points is compared to the ground truth obtained by the laser range-finder. A linear regression was performed on raw laser data (in blue) to obtain the position of the walls (red lines).

www.intechopen.com For each grid, the mean error and the standard deviation of the position of its points were computed and summarized in Table 5. The 3D error is very low, around 1% at 1 meter. The good results obtained in triangulation imply a reliable estimation of the sensor parameters and of the relative pose. Due to the resolution of catadioptric sensors, this error will nevertheless increase with the distance and will be around 10% at 10 meters.

Simultaneous Localization And Mapping

Our SLAM algorithm was evaluated on a real video sequence. The mobile robot was moved along the front of a building on a distance of 30 meters. The environment used is particularly rich since there are both manmade and natural items as shown in Figure 17. At each position of the robot, a pair of omnidirectional images was grabbed and a map of the environment was acquired by the laser range-finder. These maps are then used to compute the real trajectory (ground truth) of the robot since this kind of sensor is very accurate. As shown in Figure 18, the global map of the environment obtained by the laser rangefinder is the best result since local maps are perfectly coincident. Our SLAM algorithm provides a slight deviation but the global map is far better than the map that can be obtained by odometry. The mean error on this sequence is around 12%.

Conclusion and future work

This paper has been devoted to the design of an innovative vision sensor dedicated to mobile robot application. Combining omnidirectional and stereoscopic vision offers many advantages for 3D reconstruction and navigation that are the two main tasks a robot has to achieve. In this article we have highlighted that the best stereoscopic configuration is the vertical one as it simplifies the pixel matching between images. A special care has been put on the sensor calibration to make it flexible since it only requires the use of a planar calibration pattern. Experimental results highlight a high accuracy, which foreshadows good results for the following algorithms. The fundamental issue of Simultaneous Localization And Mapping (SLAM) was then addressed. Our solution to this problem relies on a bundle adjustment between two displacements of the robot. The initialization of the motion and the coordinates of 3D points is a prerequisite since bundle adjustment is based on a non-linear minimization. This step is a tricky problem to which we answered by the generalization of the epipolar geometry for central sensors using the unified model. Our experimental results on SLAM are promising but the error is higher than the one expected further to the calibration results. Our future work will focus on the improvement of our SLAM method by adding a global bundle adjustment to avoid error accumulation. A lot of challenges in SLAM are moreover always open, for instance SLAM based only on vision systems, SLAM taking into account six degrees of freedom, or SLAM for large-scale mapping.

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Fig. 1 .

 1 Fig. 1. View of our catadioptric stereovision sensor mounted on a Pioneer robot. Baseline is around 20cm for indoor environments and can be extended for outdoor environments. The overall height of the sensor is 40cm.

Fig. 3 .

 3 Fig. 3. Adjustment of the mirror position to respect the SVP constraint. The mirror border has to fit the estimation (green circle).

Fig. 4 .

 4 Fig. 4. Unified projection model.

Fig. 5 .

 5 Fig. 5. Relative pose estimation principle. The point M with coordinates   T 2 2 2 1 z y xin the frame associated with the second

  pose of the second sensor with respect to the first one. With n control points, equation (8) yields to the following system:

)Fig. 7 .

 7 Fig. 7. Computation of the error in the perspective case.

Fig. 8 .

 8 Fig. 8. Computation of the error in the omnidirectional case.

Fig. 10 .

 10 Fig. 10. Rigid transformation between the two sensors of the stereoscopic system.Let

Fig. 11 .

 11 Fig. 11. Form of the Jacobian matrix for 3 poses and 4 points.

Fig. 12 .

 12 Fig. 12. Synthetic omnidirectional images used for the calibration

Fig. 13 .

 13 Fig. 13. Pairs of images used for the estimation of the relative pose

Fig. 14 .

 14 Fig. 14.Epipolar curves (green) corresponding to selected pixels (red crosses)

Fig. 15 .

 15 Fig. 15. Stereo pair used for the 3D reconstruction experimentation.

Fig. 16 .

 16 Fig. 16. Position of the five grids (crosses) compared to ground truth (blue and red lines).

Fig. 17 .

 17 Fig. 17. Some of the pictures of the video sequence used for the Simultaneous Localization and Mapping.

  Table 2 summarizes the results obtained on this set of real images.

	Mean error (pixels) Standard deviation (pixels) Table 2. Calibration results on real images	0.44 0.26

Table 4 .

 4 Table 4 summarizes the results. Calibration results on real images

	Displacement (mm) Estimation (mm)	100 100.13	200 200.77	300 296.48	400 393.66	500 494.70	600 594.60

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Table 5 .

 5 Triangulation results.

	Grid number Mean error (mm) Standard deviation (mm)	1 5.843 2.616	2 15.529 4.500	3 13.182 2.420	4 3.794 2.504	5 12.872 7.485

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Two solutions for the rotation matrix can be easily eliminated because they do not respect the property of a rotation matrix 1 ) ( det  R . It remains consequently four possible combinations of rotation and translation. In the perspective case, the right solution is trivial since it can be obtained after the triangulation of a point by checking the reconstructed point is in front of both cameras. This notion of "in front of the cameras" does not exist anymore in the omnidirectional case since points from the entire 3D space can have a projection onto the image plane. The right solution could be recovered using other sensors, for example odometers, but it is preferable to use only visual data to be totally independent from the mobile robot. For each possible combination, the right solution is found by triangulating the points, and then reprojecting them onto the image plane. The computation of the reprojection error allows the determination of the correct solution since the others give totally aberrant projections. The right solution is the one which gives the minimal reprojection error since the others are totally aberrant as shown in Figure 9. Mobile robots navigation includes different interrelated activities: (i) perception, as obtaining and interpreting sensory information; (ii) exploration, as the strategy that guides the robot to select the next direction to go; (iii) mapping, involving the construction of a spatial representation by using the sensory information perceived; (iv) localization, as the strategy to estimate the robot position within the spatial map; (v) path planning, as the strategy to find a path towards a goal location being optimal or not; and (vi) path execution, where motor actions are determined and adapted to environmental changes. The book addresses those activities by integrating results from the research work of several authors all over the world. Research cases are documented in 32 chapters organized within 7 categories next described.

How to reference

In order to correctly reference this scholarly work, feel free to copy and paste the following: Remi Boutteau, Xavier Savatier, Jean-Yves Ertaud and Belahcene Mazari (2010). A 3D Omnidirectional Sensor For Mobile Robot Applications, Mobile Robots Navigation, Alejandra Barrera (Ed.), ISBN: 978-953-307-076-6, InTech, Available from: http://www.intechopen.com/books/mobile-robots-navigation/a-3domnidirectional-sensor-for-mobile-robot-applications