# Dynamic Proxemic Model for Human-Robot Interactions Using the Golden Ratio
## 1 Introduction  
The growing integration of mobile service robots into public environments [1], ranging from shopping centres [2] and airports [3] to domestic spaces [4], has created a pressing need for systems that can interact safely and comfortably with humans. Ensuring physical safety is a well-established requirement, typically addressed by industrial standards, such as ISO 13482:2014 [5] and ISO 13855:2024 [6]. However, these standards can fail to consider subjective human comfort, particularly in shared spaces where untrained individuals move unpredictably. Industrial standards are typically designed for controlled environments, assuming predictable and well-understood interactions. In contrast, real-world shared spaces are inherently unstructured: people may move erratically, individuals may be unfamiliar with the robot's function or capabilities, and human behaviour can be not only unpredictable but occasionally intentionally disruptive [7]. Moreover, various obstacles and environmental factors can influence the robot's decision-making, and even with advanced perception, the robot may misinterpret a situation [8]. This unpredictability challengesthe applicability of standard safety and comfort metrics and calls for more adaptive and context-aware approaches [9].  
Safety in human-robot interactions is fundamentally based on maintaining a safe distance, as any shared space with a robot is inherently potentially hazardous. This underpins the concept of safety zones, which dictate the robot's behaviour depending on the proximity of a person [10]. To enable this, the system must continuously monitor distance using technologies like LiDAR [11], cameras [12], or other sensors to adapt the robot's actions accordingly, whether by slowing down, stopping, or changing trajectory [13].  
Proxemics, the study of interpersonal spatial behaviour, offers a framework for designing interaction models that respect human spatial expectations [14]. The proxemic model, introduced by [PERSON] [15], defines distinct spatial zones that are culturally and situationally dependent. Although these zones provide a conceptual foundation, they do not adequately reflect the dynamic and continuous nature of human-robot encounters in real-world environments.  
In this work, we propose a dynamic proxemic model for HRI that replaces rigid zone definitions with geometrically scalable boundaries based on the golden ratio (\(\phi\approx 1.618\)). The main goal of this article is to provide description and initial validation to this novel methodology of defining comfort zones using the golden ratio. This approach could allow for more natural, adaptive transitions between zones, modulated by human position, speed, and direction relative to the robot based on the natural occurrence of the golden ratio in nature and its use in engineering and science [16; 17]. We implement this model using depth-based visual sensing and real-time computation to define hard limits (safety zone) and soft limits (comfort zones), which are then translated into robot motion constraints.  
Two experimental scenarios for HRI were created as a proof of concept of the proposed solution. The model was embedded into a real-time robotic system using NVIDIA Jetson Xavier NX and tested on two heterogeneous mobile platforms: a quadruped robot (Unitree Go1) and a wheeled robot (Agilex Scout Mini). Experiments were conducted with human participants in two interaction scenarios to evaluate perceived comfort, behavioural acceptance, and system reliability.  
This paper is structured as follows: Section 2 reviews related work on proxemics and safety in HRI. Section 3 presents details on the used materials and methods, including hardware, sensing modalities, and mathematical formulation. Section 4 presents the architecture and integration of the system. Section 5 describes the experimental design. Section 6 presents and analyses the results and discusses the implications and limitations. Section 7 concludes the study and outlines directions for future research.