# Dynamic Proxemic Model for Human-Robot Interactions Using the Golden Ratio
## 3 Materials and Methods
### Dynamic Comfort and Safety Zone Distribution  
When determining the size and shape of the personal zones in the human-robot interaction area, ensuring the safety of the occupants must always have the highest priority. In the case of planned interaction or simply passing by a person, it is necessary to control the mutual speed and direction of movement before a possible collision. In this respect, there are useful standards to draw on. To ensure a minimum safe distance \(d_{safety}\) between humans and robots to avoid a collision in motion, it is necessary to know the relative velocity between the human and robot \(v_{rd}\), the system reaction time of the robot to register a person entering the safety zone \(t_{1}\), the reaction time of the robot to stop its movement completely \(t_{2}\), and the largest radius of the circle determining the space occupied by the human body \(r_{p}\). This radius determines the maximum distance from the centre of the body to its outermost part in a top-down view. In the ordinary case, this outermost point is the foot or part of the hand (see Figure 1). The initial minimum radius of the space occupied by a human being \(r_{p}\) is set according to [PERSON]'s personal zone at 0.45 m [15]. Based on the ISO 13482:2014 [5] and ISO 13855:2024 [6] standards and these parameters, Equation (1) was established to determine the minimum size of the safety zone around persons. \[d_{safety}=v_{rel}(t_{1}+t_{2})+r_{p} \tag{1}\]  
The distribution of comfort zones is based on the golden ratio. In contrast to the fixed zones according to [PERSON], this model proposes the determination of proxemic zones for human-robot interactions based on the golden ratio (\(\varphi\approx 1.618\)). This approach provides a more dynamic and natural flow based on aesthetic and mathematical harmony, which can be found also in humanoid walking style [32]. The golden ratio is widely known for its occurrence in nature, and its application in human--robot proxemics offers the opportunity  
Figure 1: Radius of human-occupied space. to rethink comfort zones based on a distribution according to organic principles (see Figure 2). Here, the initial zone distribution is based on the proposed proxemic model, using the golden ratio as the initial default state:  
* **Safety zone** (0-0.45 m): This zone is defined by mentioned safety distance \(d_{\textit{safety}}\), whose default value at zero speed is identical to [PERSON]'s intimacy zone. The zone represents the distance where direct physical contact occurs. If the robot application does not require the robot to approach within this distance, the robot should never violate this zone, as there is a high risk of collision with a human. * **Physical interaction zone** (0.45-0.72 m): In this range, the robot is within arm's reach of a human. Physical interactions between the human and robot should take place in this zone, with the robot moving slowly to a stop to minimise the risk of collision. * **Near social zone** (0.72-1.17 m): This zone correlates with the area where robots can transition between active physical interaction and active non-physical interaction. In this zone, feedback from the person is expected in the form of verbal or non-verbal communication, whether the person expects direct physical contact or not. * **Middle social zone** (1.17-1.9 m): Non-physical interactions between the human and robot occur in this zone. Communication and engagement in human-robot interactions occur here. During human-robot interactions, the robot should stay here in case of no physical interaction is expected. * **Distant social zone** (1.9-3.08 m): This area is for approaching people in case of a planned attempt to interact with a human. In such a case, the robot should try to draw attention to itself to indicate its presence to the human. From this closer zone, the robot approaches at an increasingly slower speed.