## 3 Responsive Joint Attention System
The VAM dynamically assigns intensity values to each detected stimulus, enabling the system to prioritize stimuli based on the interaction context. The Stimuli Intensity Manager operates on a rule-based system that assigns higher intensity values to the user's gaze and arm signaling direction than to the user's face. This prioritization reflects the system's focus on attending to external cues that indicate the user's FoA shifts. Intensity values, ranging from 0 to 1, are defined empirically based on human attentional behavior and dynamically adjusted. For instance, primary JA cues such as gaze direction and arm pointing are assigned a value of 1 when the user is not facing the robot, or when they are pointing with their arms, while the user's face maintains a baseline intensity of 0.5, acting as a default attentional anchor. In specific phases involving hand gestures, the intensity of hand cues is increased to 0.8 to reflect their relevance during the activity. This rule-based assignment was inspired by how humans naturally prioritize different types of attention-related stimuli [51]. When one of these stimuli is detected, the Stimuli Ranking Module ranks the stimuli and computes the final FoA according to their intensity to direct the robot's attention to the highest-priority stimulus (e.g., the user's head orientation direction or manual pointing). After the robot attends to the detected stimulus, the bio-inspired Inhibition of Return (IOR) [52] mechanism implemented decreases the intensity of that stimulus, reprioritizing the user's face as the main focus. This process allows the robot to alternate naturally between following the user's attention and reestablishing eye contact, ensuring the natural JA closed-cycle interaction. Regarding actuation, the BBM translates the VAM's output into physical and verbal behaviors, imitating natural human attentional responses. The human vestibular system [53] inspires the motion controller, allowing the robot to perform coordinated and fluid movements of its eyes, head, and torso (see Figure 2). The robot's movements are based on standard techniques, such as inverse kinematics for joint control and a PID controller for adjusting the base during user tracking. The bio-inspired aspect lies in how the robot coordinates its eyes, head, and torso, mimicking the oculocphalic reflex and compensatory head movements. This enables human-like attention shifts through sequential activation of its degrees of freedom. For example, when a user's gaze shifts, the robot adjusts its eyes, head, and torso if necessary. In case the FoA is within the robot's field of view, it will only move its eyes toward the FoA (see Figure 2a). If the eyes reach their limits, the robot will turn its head toward the direction of the FoA while completing a counter-movement with the eyes to maintain the robot's gaze centered (see Figure 2b). Finally, the robot reorients the body towards the FoA, with the head readjusting to ensure it remains properly oriented. (see Figure 2c). If the FoA moves and leaves the robot's field of view, it will be followed by its torso camera until it reaches its maximum position or the center of the FoA. In addition to the movement, the Attention Verbal Expression Manager handles the robot's verbal responses related to JA events. It is designed to acknowledge the user's distractions and redirect their attention to the ongoing task. The expressions employed by the robot are thought to avoid open-ended questions or responses that might shift the user's focus away from the activity. Table 1 gathers the phrase list, uttered in a neutral tone, ensuring the user perceives the robot as attentive without disrupting the interaction flow. Those are randomly selected from a predefined set to enhance naturalness, avoidingrepetition to prevent predictability. This approach reinforces the robot's awareness of the user's behavior while ensuring the interaction remains centered on the primary activity to maintain user engagement.