Modeling Human–Robot Proxemics Based on Human Communication Theory: A Behavior–Interaction–Object-Dependent Approach

Abstract

Understanding human comfort when in the presence of robots is vital to constructing socially adaptive robotic systems. This study introduces the Human–Robot Proxemic Index (HRPI). This quantitative model estimates user comfort based on three contextual dimensions: human activity (behavior-dependent, BD), interaction type (interaction-dependent, ID), and object characteristics (object-dependent, OD). Unlike previous proxemic models that focused solely on physical distance, HRPI integrates multidimensional contextual factors and applies sigmoid-based personalization to account for individual sensitivity. A ceiling-mounted service robot and nine participants took part in experiments. Pre- and post-interaction questionnaires were used to find out how comfortable the participants felt and what distance they preferred. The collected data were normalized and incorporated into HRPI through weighted assessment, and validation with ideal dummy data in trials showed that HRPI-based control dynamically adjusted the robot’s approach distance and speed according to user preferences. These findings highlight the strengths of HRPI as a multidimensional, context-aware framework for guiding socially appropriate robot movements and suggest that its integration with topological spatial mapping could further enhance human–robot collaboration in real-world environments.

1. Introduction

Physical distance plays a crucial role in interpersonal communication by influencing social dynamics and nonverbal behavior. Edward T. Hall [1] introduced proxemics, a concept that examines how individuals utilize space to express emotions, establish social boundaries, and manage interpersonal relationships. Argyle and Dean [2] demonstrated that eye contact and spatial distance influence the perceived closeness and comfort levels between individuals. Burgoon et al. [3] emphasized that distance regulation plays a crucial role in nonverbal communication. A study by Kroczek et al. [4] indicated that distance influences physiological responses, such as heart rate and comfort perceptions, thereby supporting the idea that distance is an internalized psychological construct. The social significance of spatial proximity was underscored by the seminal study conducted by Latané et al. [5], which determined that social influence decreases exponentially as physical distance increases.

In the field of human–robot interaction (HRI), individuals have exhibited a propensity to maintain a social distance from robots, which is influenced by the context of the interaction and the robot’s appearance [6,7]. The interaction context, which includes the robot’s speed and direction of approach, has been found to affect human comfort [8]. Subsequent research has revealed that humans adapt their behavior near robots in order to better understand social cues [9,10]. Meanwhile, perceptual characteristics such as light hue and brightness influence how far apart people and robots should be [11], and a robot’s personality (whether introverted or extroverted) influences how close people are willing to approach it [12]. Leoste et al. [13] indicated that proxemic distance is shaped by individual preferences, communication objectives, and task contexts, as demonstrated by studies on telepresence robots, which require contextual distance adjustments to promote natural communication. The dimension of anthropomorphism affects proxemic comfort, consisting of two main aspects: humanness, which refers to the attribution of mental and social capabilities, and human-likeness, which pertains to the visual resemblance to humans. Both aspects influence perceptions of comfort, trust, and emotional proximity towards robots [14].

According to recent research, emotive expressions, including robotic tears, can increase the perception of human warmth and empathy, thereby reinforcing attributions of animacy and increasing the social acceptance of robots [15]. The findings collectively confirm that proxemics in human–robot interaction is multidimensional, encompassing spatial, perceptual, emotional, and anthropomorphic aspects. Although numerous studies on human–robot proxemics have been conducted in a variety of contexts, including social navigation [16], educational settings [13], and industrial collaboration [17], most models still treat interpersonal distance as a fixed variable without taking into account the social context or user perception. However, human–robot proxemic behavior is dynamic and significantly influenced by the activity, direction of interaction, and perceived risk of the objects in question. This research focuses on how these three parameters affect human comfort when engaging with a ceiling-mounted service robot, which has a vertical interaction area and upper-body actions.

Modern human–robot proxemics necessitates robots not only avoiding physical collisions but also maintaining an adequate social distance to ensure that interactions feel genuine. However, the majority of research persists in differentiating proxemics (distance and mobility) from trust as two distinct elements. Samarakoon et al. [18] conducted a review of various experiments and discovered that current HRI systems are deficient in adaptive mechanisms to modify distance according to real-time human feedback. Likewise, Asif et al. [19] and Neggers et al. [10] indicate that unforeseen movements may induce discomfort and diminish user interest. Simultaneously, enhanced humanization can bolster empathy and trust, but it may induce an uncanny valley effect if it fails to align with the emotional environment [14]. A consistent emotional context, exemplified by a robot shedding tears during a parting, can augment feelings of amiability and agency [15]. Consequently, a deficiency persists in the formulation of a proxemic model that amalgamates environmental elements (activity, interaction direction, object) and psychological components (trust) into a cohesive, adaptive, quantitative framework.

The main objective of this project is to develop the Human–Robot Proxemics Index (HRPI), a multidimensional framework that statistically encapsulates human comfort across three fundamental dimensions:

• Behavior-Dependent (BD): Impacted by user behaviors (e.g., working, resting, receiving objects);
• Interaction-Dependent (ID): Determined by the nature and direction of interaction (e.g., providing or receiving items);
• Object-Dependent (OD): Influenced by the attributes of objects and their associated risk levels.

The HRPI model incorporates these three features through a sigmoid-based personalization function, informed by questionnaire findings (NARS and RAS), which considers each individual’s sensitivity to allowable distance. In contrast to earlier models that relied solely on geometric distance [7,16], HRPI integrates social, perceptual, and psychological attributes into a unified index, enabling adaptive modifications of the robot’s distance and velocity. This approach underpins socially adaptable robotic behavior, fostering trust and assuring comfort in prolonged interactions.

This paper’s organization continues below. Section 2 reviews human–robot proxemics, comfort and distance modeling, and adaptive robot behavior frameworks in the related works and the state of the art. This section finishes with an overview of the research gap addressed in this work. Section 3 introduces the methodology, including the HRPI’s conceptual foundation, mathematical formulation, and parameter sensitivity analysis. Section 4 covers participant demographics, experimental setup, questionnaire design, data standardization, and weight computation. Results and simulations include subjective comfort data analysis, sample HRPI calculations, and HRPI application in robot motion control to demonstrate adaptive distance and speed regulation in Section 5. Section 6 compares HRPI to existing models, addresses limitations, and discusses real-world human–robot interaction consequences. Section 7 closes the study by summarizing major findings and suggesting future approaches, including HRPI integration with topological intelligence frameworks and learning-based adaptive spatial mapping.

2. Related Work

2.1. Human–Robot Proxemics

Preliminary studies in Human–Robot Interaction (HRI) suggested that individuals regulate their proximity to robots similarly to human-to-human proxemics, shaped by environment and form. Walters et al. proposed an empirical proxemics framework for human–robot interaction (HRI) that emphasized the impact of approach behaviors and embodiment on user comfort [6]. Mumm and Mutlu studied psychological and physical isolation, showing that users control private boundaries and social signals by navigating spatial dynamics [7]. The comfort of an individual was consistently influenced by the direction and pace of their approach, as demonstrated by Takayama and Pantofaru [8]. Frontal and rapid approaches were generally less popular. Subsequent studies highlighted human adaptation: people change their proxemic behavior to help robots comprehend social cues, indicating bidirectional accommodation [9], and differences in passing speed influence permitted margins during intimate meetings [10].

Findings specific to a particular context contribute to the overall picture: telepresence scenarios require distance regulation based on the work to maintain natural communication [13]; industrial and construction contexts exhibit navigation rules linked to safety and role expectations [17]. Along with spatial signals, anthropomorphism (humanness versus human-likeness) influences comfort and trust thresholds [14]. Expressive cues, such as robotic tears, might increase perceived warmth and empathy, hence increasing proxemic tolerance [15]. These findings confirm that human–robot proxemics is highly contextualized and socially negotiated, rather than purely geometric.

2.2. Comfort and Distance Models

Foundational HRI models often encode distance as fixed zones or simple functions, sometimes tuned per scenario. Torta et al. proposed a model of proxemics for natural HRI that maps interaction roles and tasks to recommended interpersonal distances, offering a rule-based foundation for social navigation [16]. Reviews by Samarakoon et al. synthesize >150 studies and conclude that most systems lack adaptive mechanisms that update spacing in response to real-time human feedback, and rarely fuse psychological with situational variables [18]. Maniscalco et al. and Leoste et al. underscore that environment type (e.g., museums, education) and communication goals shift preferred spacing, implying that any single “global distance” is insufficient [13,20].

Recent work is beginning to learn distance preferences from data: Yang et al. present learning-based proxemics models from experimental datasets, moving beyond fixed heuristics [21]. Yet these approaches typically emphasize spatial covariates and still under-represent interaction direction, object risk, and individual sensitivity as first-class variables.

2.3. Adaptive Robot Behavior Models

The use of adaptive behavior in HRI has a long history in evaluation-in-the-loop control. Kubota et al. integrated several fuzzy state-value functions with interactive learning to provide human-friendly trajectories, representing an early example of human-aware action selection [22]. The extensive research on trust indicates that the predictability of motion and suitable social cues influence acceptance [23,24,25], but surveys on collaborative tasks reveal that unforeseen movements diminish comfort and participation [19]. Research on speed profiles demonstrates direct correlations between velocity and comfortable passing distances, prompting the development of motion controls that adjust speed according to social situations [10].

Our previous research on Object Handover Position (OHP) utilized unsupervised learning to estimate handover locations from proxemic patterns, thereby validating the ability of structured social information to inform the selection of spatial actions [26]. Simultaneously, the topological intelligence agenda (e.g., multiscopic topological twin) establishes learnt spatial structure as a foundation for planning and adaptability in real-world situations [27], in accordance with developments in learning-based proxemics modeling [21].

2.4. Summary and Research Gap

In different areas, previous research has shown that (i) proxemics in HRI is affected by many things, such as embodiment, approach kinematics, job, affect, and anthropomorphism; and (ii) controllers usually store static distances or rules that are specific to the situation. A comprehensive, quantitative formulation that manages this is still lacking. Models of behavior-dependent (BD) activity, interaction-dependent (ID) direction/role, and object-dependent (OD) risk, all managed at the same time, offer customization based on user-sensitive factors (for example, questionnaire-based) and adapt continuously to motion factors (such as distance and speed).

The proposed Human–Robot Proxemics Index (HRPI) closes this gap by combining BD, ID, and OD into a single index incorporating a sigmoid-based personalization term and linking the index to the robot’s approach distance and velocity. This design incorporates insights from proxemics theory [1,2,3,4,5], context-rich human–robot interaction (HRI) studies [6,7,8,9,10,11,12,13,17,20,28], literature on trust and acceptance [23,24,25], and adaptive control and learning approaches [10,21,22,26,27], while remaining lightweight enough for deployment and future data-driven refinement. A concise comparison of these prior approaches is presented in

Table 1. Comparison of Selected Methods and the Proposed HRPI.

Approach	Context Covariates	Personalization	Output to Control	Notes
Walters’ empirical proxemics [6]	Embodiment, approach patterns	No	Indirect (design rules)	Foundational empirical framework
Takayama & Pantofaru [8]	Direction, speed	No	Indirect (design guidance)	Kinematic factors on comfort
Torta et al. model [24]	Task/role rules	Limited	Heuristic distances	Rule-based zones
Neggers et al. [10]	Speed × passing	No	Velocity tuning (implied)	Speed–distance coupling
Yang et al. (learned) [29]	Data-driven spatial cues	Limited	Model-dependent	Toward learned proxemics
HRPI (this work)	BD, ID, OD (activity, interaction direction, object risk)	Yes (sigmoid, NARS/RAS)	Direct mapping: distance and speed	Unified, adaptive, lightweight

, highlighting their limited personalization and context coverage.

3. Methodology

3.1. HRPI Conceptual Framework

The Human–Robot Proxemic Index (HRPI) is intended to objectively quantify human comfort during human–robot interactions by combining behavioral, environmental, and psychological variables. Instead of setting fixed comfort zones like most distance-based models do, the HRPI changes based on the user’s actions, the direction of contact, the object’s properties, and the person’s own sensitivity. As shown in Figure 1, the HRPI framework has three contextual dimensions:

• Behavior-Dependent (BD): This describes how a person’s behavior (e.g., working, relaxing, retrieving an object) impacts their comfortable distance. Activities involving focused attention or manual operation generally require greater interpersonal space.
• Interaction-Dependent (ID): This captures the type and direction of interaction—whether the robot is giving or receiving an object—which influences how participants position themselves during the task exchange.
• Object-Dependent (OD): This measures the influence of an object’s physical characteristics and perceived risks (e.g., soft, hot, or sharp objects) on comfortable distance and movement safety.

Each dimension generates an independent comfort score from the participant questionnaire, which is then normalized to the range 0–1. The resulting scores are combined into a single proxemic index, which is adjusted using a personalization function derived from psychological measures (the Negative Attitudes Toward Robots Scale—NARS—and the Robot Anxiety Scale—RAS). This function encodes user-specific sensitivity to proximity and social engagement.

The HRPI thus bridges the gap between subjective human perception and objective robot motion parameters. The HRPI provides a human-centered basis for adaptive robot navigation and distance control, particularly in ceiling-mounted service robots operating in human-limited spaces, such as the trailer-type living laboratory used in this study.

3.2. Mathematical Formulation

To quantify proxemic comfort, the HRPI value for a given state is computed using the following expression:

$$HRPI=\left(\sum w_i{d}_i\right)\left({\displaystyle \frac{1}{1+e^{-c_u(z_u-z)}}}\right)$$

(1)

Each score was normalized to a range between 0 and 1, and then calculated using a weighted equation adjusted for individual sensitivity through a sigmoid function [29]. The sigmoid function was selected because individual responses vary nonlinearly [3]; its slope can capture differences in personal sensitivity and background revealed by the questionnaire. Moreover, the use of the sigmoid formulation personalizes the index according to individual tolerance, where $d_i$ = $\{BD\_score,ID\_score,OD\_score\}$ are the normalized comfort values obtained from questionnaires, and $w_i$ denotes the weighting factor for each dimension (BD, ID, OD). To personalize each individual’s comfort level, a sigmoid function with a multiplier factor of 1 is used as the center (no change). The parameter $c_u$ is the user-sensitivity coefficient obtained from the results of the individual NARS and RAS responses. A participant with high sensitivity will produce a steeper sigmoid curve—indicating rapid comfort loss as the robot approaches—while a participant with low sensitivity will yield a flatter curve, reflecting a higher tolerance. $z_u$ is the comfort distance from the individual questionnaire for each aspect dimension BD-ID-OD, and z is the robot’s actual normalized distance measured by the RGB-D sensor. With this approach, the resulting index value is specific to each individual, depending on their comfort level with interacting with the robot. The following are the details of the score components:

• $BD\_score$ is calculated from “$BD\_\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{v}\mathrm{i}\mathrm{t}\mathrm{y}$” + “$BD\_\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}$”. The activity function is adjusted according to ICF staging, with five activities analyzed: sitting, reading books, resting, working, and cooking, with weights of 0.1, 0.3, 0.4, 0.6, and 0.8, respectively. The $BD\_\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}$ value is added according to the actual distance in the experiment.
• $ID\_score$ is calculated from “$ID\_\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$” × “$ID\_\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}$”, where $ID\_\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$ is 1 if interaction is required (e.g., object handover) and 0 if not. The $ID\_\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}$ value is taken from the comfort level obtained from the experiment results.
• $OD\_score$ represents the type and level of risk of the object, with categories: non-hazardous (0.1), moderately hazardous (0.3), and hazardous (0.6).

The distance formula for $BD\_\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}$ and $ID\_\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}$ uses the same function:

$$\mathrm{z}=\left|{\displaystyle \frac{r_i-r_t}{r}}\right|$$

(2)

For $BD\_\mathrm{d}\mathrm{i}\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{e}$, the $r_t$ value is calculated from the average between the minimum distance (15 cm) and the maximum distance of the experiment (120 cm), with r as the difference between the maximum and minimum distances. Meanwhile, $ID\_\mathrm{i}\mathrm{n}\mathrm{t}\mathrm{e}\mathrm{r}\mathrm{a}\mathrm{c}\mathrm{t}\mathrm{i}\mathrm{o}\mathrm{n}$ is set at 50.56 cm, based on the average comfortable distance when interacting to give or receive objects in the experiment. Each proxemic dimension contributes to the final HRPI through its normalized sub-score $d_i,$ computed as follows:

$$d_i=\left|{\displaystyle \frac{x_i-x_{min}}{x_{max}-x_{min}}}\right|$$

(3)

where $x_i$ is the raw Likert rating (0–4), $x_{min}=0$, and $x_{max}=$ 4. To translate HRPI values into robot control parameters, two mapping equations are used:

$$D_{safe}=D_{min}+(D_{max}-D_{min})\cdot HRPI$$

(4)

$$v_{ROBOT}=v_{MAX}(1-HRPI)$$

(5)

$D_{safe}$ is the recommended stopping distance, $v_{ROBOT}$ is the robot’s approach velocity, $D_{min}$ and $D_{max}$ represent proxemic boundaries (15–120 cm in this setup), and $v_{MAX}$ is the maximum safe velocity of the ceiling robot.

To evaluate the stability of the HRPI, a parametric sensitivity analysis was conducted using representative dummy data from the experimental setup. Parameters $c_u$ (user-sensitivity coefficient), $w_i\left(dimensionweights\right),$ and $z_u$ (preferred distance) were varied within $\pm 10\%$ their assumed values to observe qualitative effects on the HRPI output. This test was not intended as a statistical validation but rather as a consistency check to ensure that the model behaves logically under small perturbations. The results are visualized in Figure 2, which shows HRPI curves for different user-sensitivity coefficients ($c_u$). As $c_u$ increases, the sigmoid slope becomes steeper, indicating that individuals with higher robot anxiety experience sharper comfort decline at shorter distances. Conversely, smaller $c_u$ values yield smoother transitions, reflecting higher proximity tolerance. The resulting curves show gradual, continuous transitions rather than abrupt changes, suggesting that the model responds smoothly to parameter variation.

Although this is only a conceptual observation using dummy data, the result indicates that the HRPI formulation behaves predictably and could be extended for further validation with real robot experiments.

4. Experimental Design

4.1. Participant and Setup

The study involved nine participants, consisting of five males and four females, with ages ranging from 22 to 35 years (M = 27.4, SD = 3.8). The participants constitute a heterogeneous user group exhibiting differing levels of familiarity with service robots. The Tokyo Metropolitan University Ethics Committee (Protocol R7-016, adopted 5 June 2025) sanctioned the procedure for acquiring informed consent.

The trials were conducted in a controlled indoor environment designed to resemble a trailer living lab [27]. A service robot was positioned on the ceiling, traversing an overhead linear rail above the participant (Figure 3). The robot was capable of vertical and horizontal movement within a range of 15 to 120 cm. An RGB-D sensor was utilized to monitor its location and ensure safety. The temperature and light conditions remained constant throughout each session.

Hall’s proxemics hypothesis identifies three proxemic zones: intimate (≤45 cm), personal (45–120 cm), and social (>120 cm) [1]. The robot can approach at three distinct speeds: slow (0.15 m/s), moderate (0.30 m/s), and fast (0.45 m/s). The system was easily controllable due to its PC-based C/C++ interface, which executed predefined approach scenarios.

4.2. Questionnaire and Experimental Procedure

The experimental procedure consisted of three phases: a pre-test questionnaire, the experimental trials, and a post-test questionnaire. The pre-test stage collected participants’ prior perceptions and intuitions regarding interactions with robots. During the trials, the robot executed predefined approach scenarios, and participants’ responses were assessed after each interaction. Finally, the post-test questionnaire captured participants’ reflections on comfort following direct interaction with the robot. Both the Negative Attitude Toward Robots Scale (NARS) and the Robot Anxiety Scale (RAS) were utilized in the questionnaires in order to assess the level of trust and worry that individuals had toward robots. The perceived level of comfort was measured under a number of different proxemic conditions through the use of experimental trials, and post-interaction questionnaires were used to collect reflections following the exposure. On a Likert scale of five points, comfort ratings were supplied after each trial. From 0 (extremely troubled) to 4 (very comfortable), the scale ranged from 0 to 4.

4.2.1. Behavior-Dependent Scenario

The BD experiment aimed to evaluate how human activity influences perceived comfort with robot proximity. Three scenarios were designed:

• BD1: Participant engaged in focused work.
• BD2: Participant in a resting or relaxation state.
• BD3: Participant retrieving an object located near the robot.

In each scenario, the robot approached participants at a constant speed and distance. Comfort levels were measured using pre- and post-interaction questionnaires. The experimental scenarios are illustrated in Figure 4.

4.2.2. Interaction-Dependent Scenario

The ID experiment investigated how direct interaction with the robot affected proxemic preferences. Two activities were included:

• ID1: Participant handing an object to the robot.
• ID2: Participant receiving an object from the robot.

Participants evaluated their perceived comfort and preferred interpersonal distance through questionnaires after each interaction. The experimental scenarios are illustrated in Figure 5.

4.2.3. Object-Dependent Scenario

The OD experiment focused on how the type of object carried by the robot influences perceived safety and comfort. Three objects with varying levels of perceived risk were selected:

• OD1: Pillow (non-hazardous).
• OD2: Cup filled with hot water (moderately hazardous).
• OD3: Scissors (highly hazardous).

The robot approached participants while carrying each object under different speed conditions (low, medium, and high). Participants assessed both the perceived risk of the object and their comfort with the robot’s approach. Variations in speed and proximity were systematically applied to observe changes in participants’ comfort perception. The experimental scenarios are illustrated in Figure 6.

4.3. Simulation for HRPI-Based Robot Control

To investigate the applicability of the HRPI for adaptive motion planning, a numerical simulation was run to correlate HRPI values with robot control parameters such as approach distance and velocity. In this study, “simulation” refers to the use of fictitious distance data from the RGB-D camera and idealized activity settings within the trailer-type living laboratory to generate a numerical version of the HRPI model. Dummy distance samples were constructed ranging from 15 cm to 120 cm, with 15 cm increments representing the normal distance between people and robots. We used Equations (1)–(3) to determine HRPI values for each trial condition based on the Behavior-, Interaction-, and Object-Dependent parameters (BD, ID, OD).

5. Results

This section delineates the primary results from the questionnaire analysis, normalization procedure, and HRPI calculation. Three contextual variables were assessed—Behavior-Dependent (BD), Interaction-Dependent (ID), and Object-Dependent (OD)—based on the responses of nine participants utilizing a five-point Likert scale. All data were standardized to a 0–1 range for comparative analysis across factors.

5.1. Scenario Dependent on Behavior

Figure 7 demonstrates that human activity significantly influences proxemic comfort. Participants experienced the highest comfort levels during organized and goal-directed tasks, such as working or reading, but comfort diminished during relaxed or transitional stages, such as resting or retrieving objects. This pattern suggests that an individual’s behavioral involvement influences spatial expectations: when attention is directed towards a task, individuals want a broader personal space to mitigate distractions. Comparable associations between attentional burden and distance management have been documented in investigations of social communication [2,3]. Participants demonstrated a willingness to modify their distance preferences when the robot’s movements were perceived as smooth and predictable.

5.2. Scenario Dependent on Interaction

Figure 8 illustrates that both offering and receiving object interactions yielded similar comfort levels. The findings indicate that bi-directional exchanges, involving contributions from both humans and robots, improve engagement and predictability, consequently elevating perceived safety. The mean comfortable distance for handover occurrences was approximately 50.56 cm, akin to the conversational space noted in social robot scenarios [8]. The findings suggest that collaborative yet incremental robot movements enhance trust and diminish fear during physical interactions.

5.3. Scenario Dependent on Objects

Figure 9 encapsulates comfort evaluations across various object categories. Items deemed hazardous, such as scissors, significantly diminished comfort in contrast to benign objects like cushions. Moderately risky items, such as cups with hot liquids, generated moderate levels of comfort. This pattern reflects previous research indicating that perceived threat and item shape affect interpersonal distance [6,10]. Consequently, robots must adjust both approach velocity and stopping distance in accordance with the object’s risk level to uphold user confidence and safety.

5.4. Personal Variations (NARS and RAS)

The attitudinal questionnaires demonstrated significant inter-individual diversity, as seen in Figure 10. The Robot Anxiety Scale (RAS) evaluated anxiety associated with presence, interaction, and perceived threat, whereas the Negative Attitudes Toward Robots Scale (NARS) examined discomfort and safety apprehensions. Despite the absence of significant differences in mean scores before and after contact, the distribution of responses indicated varying levels of familiarity. Individuals familiar with robots typically exhibited reduced anxiety, aligning with the research of Złotowski et al. [14] and Yasuhara and Takehara [15]. The disparities warrant the incorporation of the individualized sensitivity parameter $c_u$ into the HRPI paradigm, facilitating individual adaptation according to psychological disposition.

5.5. HRPI Simulation Results

Statistical tests were conducted to examine the difference between pre- and post-interaction comfort scores across the three HRPI dimensions. The Shapiro–Wilk normality test indicated that the Behavior-Dependent (BD) data were non-normally distributed (p < 0.05), whereas the Interaction-Dependent (ID) and Object-Dependent (OD) data followed a normal distribution (p > 0.05). Accordingly, a Wilcoxon signed-rank test was applied for BD, and paired t-tests were used for ID and OD. The results showed no statistically significant differences between pre- and post-interaction scores (p > 0.05). Nevertheless, the mean values remained consistent, suggesting that the HRPI-based robot behavior maintained stable levels of human comfort. This indicates that the proposed model did not introduce additional discomfort, thereby supporting its feasibility for adaptive proxemics control.

A numerical simulation was performed to show model behavior, utilizing fictitious distance data ranging from 15 cm to 120 cm. Figure 11 illustrates a steady increase in HRPI values as the robot approached, especially under object-handover situations. The sequence of events—working → approaching → handover → receiving—illustrates a gradual increase in HRPI as proximity and interaction intensity escalate. To quantify the questionnaire results, an HRPI formula was developed by combining three main interrelated dimensions (BD, ID, OD), and the HRPI data was converted into 2D heatmaps, seen in Figure 11. The heatmap depicts how HRPI values change depending on distance and type of interaction (such as moving closer, exchanging objects, or receiving). Warmer colors indicate lower comfort levels or increased user sensitivity, whilst cooler hues indicate greater comfort and tolerance.

Figure 12 displays HRPI curves for all nine subjects under uniform settings (work activity, object transfer, moderate velocity). Each curve signifies a personalized comfort response derived from normalized questionnaire data and user sensitivity (cu). Participants with elevated anxiety levels generated steeper slopes, whereas individuals acquainted with robots displayed flatter profiles. This customized pattern enhances the efficacy of HRPI in identifying user-specific proxemic preferences that might inform adaptive motion planning. The HRPI-distance curves are used to demonstrate how comfort levels and approach control settings vary with distance and user sensitivity in Figure 12. The curve graphs depict how the HRPI changes during the approach trajectory, from the “working” to “approaching” and “object handover” phases. These patterns demonstrate that the HRPI model operates in the same manner as predicted by proxemic theory: as distance decreases and interaction risk increases, comfort declines.

6. Discussion

Human–robot proxemics is a complex process that is shaped by psychological differences as well as behavioral, interactional, and object-related factors.

6.1. Analysis of HRPI Behavior

The established proxemics theory [1,2] corresponds with the ongoing increase in HRPI as the distance decreases. The model exhibits continuous and monotonic variation, where even slight modifications to the parameters result in proportional changes in the HRPI. The model demonstrates stability in numerical results and clarity in comprehension. Smooth response qualities are essential for real-time control, as abrupt changes can lead to abrupt movements that might be socially unacceptable.

6.2. Implications for Adaptive Robot Control

Integrating HRPI with robotic motion parameters, such as approach distance and speed, allows the system to adapt its functionality based on comfort observation. Low HRPI levels (0.10–0.30) indicate that caution is advised when traversing long distances. Elevated HRPI levels (0.60–1.00) facilitate quicker and closer engagement, which is advantageous for experienced users. As a bridge between psychological comfort and control engineering, this mapping (

Table 2. Mapping between HRPI range and corresponding robot control parameters.

HRPI Range	Robot Speed	Stopping Distance	Interpretation
0.10–0.30	Slow (≈0.15 m/s)	≈100–120 cm	User highly sensitive → maintain longer distance
0.30–0.60	Moderate (≈0.30 m/s)	≈60–90 cm	Comfortable engagement zone
0.60–1.00	Fast (≈0.45 m/s)	≈30–60 cm	High familiarity → closer interaction

) demonstrates that proxemic sensation can be described using quantitative concepts. Prior navigation research [10,17] has demonstrated analogous adaptive-speed relationships.

6.3. Evaluation Regarding Established Models

Previous models by Walters et al. [6], Takayama and Pantofaru [8], and Torta et al. [16] provided static zone recommendations that did not necessitate individual customization. Learning-based frameworks, such as those developed by Yang et al. [21], can illustrate spatial distributions; however, they frequently neglect the aspects of object risk and interaction direction. The HRPI integrates contextual factors (BD, ID, OD) with individual sensitivity parameters ($c_u$, $z_u$).

6.4. Potential Directions and Constraints

This investigation remains in the preliminary phase. The limited sample size and dependence on subjective questionnaires hinder the generalizability of the findings. This simulation employed theoretical numerical data rather than real-time robot evaluations. Future recommendations include (i) increasing participant diversity, (ii) incorporating objective metrics such as gaze and posture tracking, and (iii) verifying the HRPI through live robot trials with real-time feedback. The parameter weights $w_i$ were derived heuristically; however, accuracy may be improved through data-driven estimation or adaptive weighting.

6.5. Broader Implications

HRPI establishes a framework for robotic mobility that is psychologically secure and socially responsible, notwithstanding these constraints. Due to its low operational cost, it is suitable for real-time applications in ceiling-mounted or service-robot systems. HRPI enhances spatial negotiation in collaborative workspaces, healthcare settings, rehabilitation contexts, and indoor assistance scenarios. This paradigm combines spatial learning with adaptive social behaviors, thereby improving emerging topological intelligence methodologies [27].

7. Conclusions and Future Work

This study introduced the Human–Robot Proxemics Index (HRPI) as a comprehensive and personalized framework for assessing the well-being of humans during interactions with service robots. The HRPI utilizes a sigmoid-based sensitivity function and parameters that are influenced by objects, behavior, and interaction to establish a correlation between an individual’s comfort level and the robot’s movement. The results demonstrated that the robot comfort level decreased as it approached or engaged with more hazardous objects, consistent with proxemic theory [1,2]. The model generated distinct, fluid curves that can directly alter the robot’s velocity and range. This represents a nimble and adaptable substitute for black-box or fixed-zone methodologies [16,21].

Although validation was conducted with a restricted sample and an implausible situation, the HRPI exhibits the potential for real-time adaptive control in ceiling-mounted and service robots. In the near future, HRPI-based controllers will be implemented in actual robotic studies conducted within dynamic environments. This will increase the diversity of the subject pool and integrate behavioral and physiological characteristics. In summary, HRPI symbolizes a significant advancement in the advancement of socially conscious robotics that prioritize human interaction and can adjust their behavior to the user’s comfort and interactions with others.