## 2 Related Works
### Measuring User Engagement in HRI  
In the context of HRI, user engagement has emerged as a key dimension in assessing the quality and sustainability of interactions between humans and robotic agents. While the previous sections have reviewed how attentional mechanisms like JA have been technically implemented, understanding their effect on the user experience requires focusing on how users commit to the interaction. Engagement acts as a bridge between reactive systems and meaningful interaction outcomes. Understood as a multifaceted construct [39], it comprises three main dimensions: affective, cognitive, and behavioral [40,41]. The affective aspect captures users' emotional responses, such as frustration or interest, toward a system. The cognitive dimension relates to the mental effort users invest, which is shaped by the interplay between task difficulty and their own skills. Finally, the behavioral component encompasses observable user actions during interaction, including clicks or queries [22]. Given its strong ties to user experience and its sensitivity to emotional and behavioral cues, engagement could be considered a valuable indicator for assessing the influence of responding joint attention (RJA) mechanisms during an experiment. Several approaches have been proposed in the literature to study user engagement in human-robot interaction (HRI). One line of research, exemplified by [PERSON] et al. [42], relied on controlled non-verbal cues--such as nodding, blinking, and gaze aversion--to examine how engaged the participants felt based on their responses to post-experiment questionnaires. In the domain of verbal interaction, [PERSON] and [PERSON] conducted extensive work. [PERSON] et al. [43] explored engagement through patterns of mutual and directed gaze, correlating these with spoken utterances. In a complementary study, [PERSON] et al. [44] applied manual labeling to differentiate between brief glances and head nods by analyzing gaze signals. Other researchers have focused specifically on gaze-based engagement analysis. For example, [PERSON] et al. [45] used eye-tracking to collect gaze data from participants interacting with conversational agents, combining these patterns to estimate engagement levels. Beyond gaze, multimodal approaches have also been explored. [PERSON] et al. [46] combined subjective data from post-interaction questionnaires with objective behavioral indicators, such as the timing and rhythm of responses to robot stimuli and the orientation of participants' gaze, extracted from RGB-D sensors. Some authors have proposed fully automatic methods. [PERSON] et al. [47] assessed engagement by analyzing body posture, specifically trunk and back inclination, in videos of human-robot interactions. Similarly, in medical settings, researchers have used both audio features and visual cues to measure user behavior [48; 49]. Finally, [PERSON] et al. [50] presented a method relying on engagement metrics obtained from accessible sensors. Their approach was based on analyzing both static and dynamic aspects of posture and gaze to evaluate human responses during social interactions with robots. While the literature on JA has primarily focused on implementing and evaluating attentional behaviors in HRI, it has not traditionally included explicit user engagement measurements. However, some works have emphasized the conceptual overlap between JA and engagement. For instance, [PERSON] and [PERSON] [31] highlighted how directed gaze and mutual facial gaze relate to RJA and JJA, as well as EJA, yet noted that most research has focused on the recognition rather than the generation or assessment of these engagement-related behaviors. In our study, we adopted a dual-approach methodology to measure user engagement. On the one hand, we used post-experimental questionnaires, following approaches such as those described by [PERSON] et al. [46], to obtain self-reported assessments of user involvement.