# Dynamic Proxemic Model for Human-Robot Interactions Using the Golden Ratio
## 3 Materials and Methods
### Robot Velocity Profile  
To set the robot's velocity, the 2D Gaussian function (see Equation (3)) is used to define the maximum allowable robot speed. Inputs to this function are the person's position  
Figure 2: Distribution of safety and comfort zones based on the golden ratio in the default state. and orientation and the relative velocity between the person and robot. The 2D Gaussian function is a mathematical function providing a smooth continuous distribution across a plane or intensity in 3D space [33]. This function is suitable for subsequent use with navigation algorithms, such as the potential field algorithm [23]. Since the 2D Gaussian function is anisotropic, it makes it possible to describe the dynamically varying differences in safety and comfort zones around a person. The function can be varied in two mutually perpendicular axes according to the actual evolution of the situation, such as speed or the direction of motion. The function includes the relative position of the person from the robot, which is described as \((x_{i}^{p},y_{i}^{p})\), and the expansion parameters \(\sigma_{x}\) and \(\sigma_{y}\), which determine how the zones around the person will expand. By adding a rotation transformation with orientation angle \(\theta\) to the person coordinates \((x_{i}^{p},y_{i}^{p})\), we achieve the possibility of rotating the function. This makes it possible to rotate the function according to the direction of motion of the person relative to the robot (Figure 3). The amplitude \(A\) can be interpreted as the maximum robot speed. The output of Equation (3) is the maximum allowed robot speed. \[f_{i}^{p}(x,y)=A-A\ e^{-\frac{\left((x-x_{i}^{p})\text{max}(\theta)+(y-y_{i}^{ p}\text{min}(\theta))^{2}\right)}{2\sigma_{x}{}^{2}}+\frac{\left(-(x-x_{i}^{p}) \text{min}(\theta)+(y-y_{i}^{p})\text{max}(\theta)\right)^{2}}{2\sigma_{y}{} ^{2}}} \tag{3}\]  
where \(\theta\) and \(\sigma_{x,y}\) are calculated as follows:  
\[\theta=atan2\Big{(}\Big{(}y-\ y_{i}^{p}\Big{)},\Big{(}x-x_{i}^{p}\Big{)}\Big{)} -\theta_{i}^{p} \tag{4}\]  
\[\sigma_{x,y}=\sigma_{0}+(v_{rel}\cdot f_{v}\cdot f_{h}) \tag{5}\]  
The person-to-robot rotation angle \(\theta\) is calculated using the person's position in space and the directional angle of the person's rotation in Equation (4). The expansion parameters \(\sigma_{x}\) and \(\sigma_{y}\), calculated in Equation (5), consist of the initial expansion value \(\sigma_{0}\), the relative velocity between the person and the robot \(v_{rel}\), the velocity factor \(f_{v}\), and the directional  
Figure 3: Robot velocity profile from a person position. (**a**) Shape of the Gaussian function with initial parameters \(\sigma_{x,y}=1\), \(A=1\), \(\theta=0^{\circ}\). (**b**) Shape of the Gaussian function during motion. (**c**) Initial distribution of comfort zones at zero velocity. (**d**) Distribution of comfort zones during motion. factor \(f_{h}\). These factors take values from 0 to 1, where \(f_{v}\) is used to tune the size of the expansion zones, and \(f_{h}\) is used to adjust the expansion around the person, with the factor taking values of 1 in the direction of the person's motion and 0 in the opposite direction of the person's motion. Thus, when the person is walking forwards, the size of the zones in front of the person is the largest and behind the person is the smallest. Using the extension parameters \(\sigma_{x}\) and \(\sigma_{y}\), the factors \(f_{h}\) and \(f_{v}\), and the amplitude \(A\), it is possible to tune the way the robot slows down and thus adjust its velocity profile if necessary. The golden ratio-based proxemic model provides a flexible framework for setting incremental approach boundaries, allowing robots to better adapt to the situation and the relative speed of human movement while respecting personal comfort. Figure 3 shows the capabilities of dynamic safety and comfort zones based on the golden ratio. Figure 3a,c show the shape of the zones and velocity profile at zero speed and Figure 3b,d show the changes in the zones and velocity profile when moving.