Fuzzy inference system-assisted human-aware navigation framework based on enhanced potential field

2.1. Definition of frames and states

To describe the motion of the AMR, the reference frames, as shown in Figure 1, are established. The inertial frame is represented by $$ \hat{\mathbf{n}}_k $$ for $$ k\ =\ 1, \ 2, \ \text{and }3 $$. The AMR's body frame is denoted by $$ \hat{\mathbf{r}}_k $$, with its origin centered on the center of mass of the AMR's body. The AMR's primary movement direction aligns with $$ \hat{\mathbf{r}}_1 $$, while $$ \hat{\mathbf{r}}_3 $$ coincides with $$ \hat{\mathbf{n}}_3 $$. Accordingly, $$ \hat{\mathbf{r}}_2 $$ is determined using the right-hand rule. Similarly, the human's body frame is defined through $$ \hat{\mathbf{h}}_k $$, with its origin situated at the center of mass of the human. Similar to the AMR's body frame, the forward direction of the human corresponds to $$ \hat{\mathbf{h}}_1 $$, and $$ \hat{\mathbf{h}}_3 $$ aligns with $$ \hat{\mathbf{n}}_3 $$. It is important to note that, given that the AMR generally lacks translational motion along $$ \hat{\mathbf{n}}_3 $$ and rotational motion along $$ \hat{\mathbf{n}}_1 $$ and $$ \hat{\mathbf{n}}_2 $$, this study adopts a 2D position vector and accounts for a rotational component about $$ \hat{\mathbf{n}}_3 $$.

Figure 1. Definition of frames representation of vectors.

The positions of the AMR, human, obstacle, and the AMR's destination in the inertial frame are denoted as $$ ^N\mathbf{p}_m\in \mathbb{R}^2 $$, $$ ^N\mathbf{p}_h\in \mathbb{R}^2 $$, $$ ^N\mathbf{p}_{o}\in \mathbb{R}^2 $$, and $$ ^N\mathbf{p}_{g}\in \mathbb{R}^2 $$, respectively. Here, the superscripts $$ N $$, $$ R $$, and $$ H $$ denote that the vector is represented in the inertial, AMR's body, and human's body frames, respectively. Note that the position vectors for the human and obstacle expressed in the AMR frame are described as

where $$ C_{RN} $$ represents the rotation matrix that maps from the inertial frame to the body frame, defined as

Here, $$ \psi $$ represents the heading angle of the AMR, while the orientation of the human is denoted as $$ \phi $$. Also, the relative angle between the AMR and the human in the AMR frame is expressed as $$ \delta $$, and the relative angle between the AMR and the obstacle within the AMR frame is defined as $$ \gamma $$, as shown in Figure 1.

2.2. Enhanced potential field

The APF serves as a widely adopted method for robot collision avoidance owing to its mathematical efficiency and its capability to generate smooth obstacle-avoiding trajectories. This approach comprises two primary components: attraction and repulsion. An AMR is steered toward its destination by an attractive PF while being simultaneously pushed away from environmental obstacles through repulsive PFs. Consequently, a net PF emerges, which is the sum of both attractive and repulsive PFs, influencing the AMR's motion. The total potential function, denoted as $$ Q_t $$ governing the PFs, is defined as^[23]

where

Here, $$ Q_a({^N}{\mathbf{p}_m}) $$ is the attractive potential function, $$ Q_r({^N}{\mathbf{p}_m}) $$ is the repulsive potential function, $$ k_a $$ and $$ k_r $$ are the attractive and repulsive coefficients, respectively, $$ n_g $$ is the potential order, $$ d(\mathbf{a}, \mathbf{b}) $$ is the distance between two generic position vectors $$ \mathbf{a} $$ and $$ \mathbf{b} $$, and $$ d_o $$ is the distance of influence. An essential improvement to note in the presented equation is the inclusion of an additional term $$ d({^N}{\mathbf{p}_m}, {^N}{\mathbf{p}}_g)^{n_g} $$, which effectively addresses a well-known issue of the conventional APF known as GNRON^[23]. Also, by taking the negative gradient of the respective terms, the total PF is derived as^[23]

Note that the attractive and repulsive PFs are found by

where

Note that $$ {\partial {d( {^N}{\mathbf{p}_m}, {^N}{\mathbf{p}}_g)}}/{\partial \ {^N}{\mathbf{p}}_\mathbf{m} } $$ is the direction from the AMR to the destination position, and $$ {\partial{d({^N}{\mathbf{p}_m}, {^N}{\mathbf{p}}_o)}}/{\partial \ {^N}{\mathbf{p}}_\mathbf{m} } $$ is the direction from the AMR to the obstacle. Nonetheless, a persistent challenge with the APF approach is the occurrence of local minima, particularly when one or more obstacles obstruct the AMR's path symmetrically. This issue stems from the criteria used to determine the direction of the repulsive PF. To mitigate this challenge, the following modification is introduced, replacing $$ {\mathbf{f}}_{r_o}({^N}{\mathbf{p}_r}) $$ with a new term, as outlined in^[23]:

where $$ \hat{\mathbf{q}}\in \mathbb{R}^2 $$ is a newly defined repulsive PF direction vector given by

where

Note that $$ R_{e}\in \mathbb{R}^{2x2} $$ is the rotation matrix given by

where

Here, $$ \theta $$ represents a rotation angle responsible for altering the direction of the repulsive PF.

2.3. Fuzzy inference system

A FIS is an approach that employs a rule base and fuzzy set theory^[24] to map inputs to outputs, as shown in Figure 2. When numerical data is input into the FIS, it undergoes a transformation into fuzzy inputs through fuzzification. This process involves assigning a degree of membership to each Membership Function (MF) corresponding to the numeric value. An MF precisely defines the degree to which an input value belongs to a fuzzy set. It maps input values from a precise domain onto a scale of the degree of membership, mostly ranging between 0 and 1. This scale indicates how strongly an element is associated with the set, allowing for a smooth transition between membership and non-membership states. Subsequently, the inference unit derives outputs based on user-defined rules, typically expressed in linguistic form; for example, "If Input 1 is Category 1 AND Input 2 is Category 2, then Output is Category 3". These outputs are associated with varying degrees of membership within output MFs. Finally, a process called defuzzification is employed to convert these fuzzy outputs back into numerical outputs. In this study, Mamdani-type FISs^[24] are used and are referred to as FIS. One of the key advantages of FIS-based systems is their capacity to provide explainability, which can be attributed to the linguistically defined rules and MFs^[25].

Figure 2. Schematic diagram of FIS.

3.1. Human aware navigation

HAN represents the convergence of robot motion planning and human-robot interactions^[9]. While the primary aim of robot motion planning is to guide the robot to its destination, HAN prioritizes AMR's ability to minimize stress and prevent harm to humans. Therefore, it can be defined as "navigation of the AMRs based on human-centered PFs, aiming to minimize discomfort and prevent harm"^[19].

Human discomfort stemming from AMRs can arise from various sources, including the robot's appearance, unnatural movement, production of discomforting noise, close proximity to humans, and failure to adhere to cultural norms^[9]. Kruse et al. classified discomfort in HAN into three main categories: human comfort, AMR naturalness, and AMR sociability^[9]. Human comfort revolves around ensuring maximum comfort and safety for humans in the presence of AMRs. Examples include AMRs maintaining a specified distance from humans at all times and refraining from traversing specific spaces near humans. AMR naturalness entails the robot's movement resembling that of humans, such as following smooth trajectories and having a friendly appearance, thereby enhancing acceptance. AMR sociability implies that the robot should adhere to cultural norms, such as traveling on a specific side (left/right) of humans depending on the country and overtaking humans along a certain direction. Among these categories, this work primarily focuses on human comfort concerning AMR navigation, as the robot's appearance and noise levels are not significantly related to its navigation. Safety takes precedence in this context.

This study introduces three important terms in HAN: PR, FV, and BS, as shown in Figure 3. PR signifies that the AMR must not only maintain a predetermined relative distance from humans at all times but also avoid collisions with them. BS denotes the region behind the human, where the presence of the AMR may cause discomfort to humans when traversing. FV represents the region in front of the human where the AMR could potentially obstruct the human's path. Consequently, a successful HAN planner must ensure that the AMR avoids these defined regions.

Figure 3. Representation of HAN factors with respect to a human.

3.2. Proposed approach: HAN-EPF

3.2.1. Definitions of HAN factors

The primary focus of this study centers on the integration of factors associated with humans into the navigation of the AMR. To streamline collision-free path planning, it is assumed that the relative positions of humans with respect to the AMR are known, as this information can be obtained through optical or infrared sensors equipped with perception algorithms. Similarly, the relative position information of obstacles is obtained through lidar sensors. Note that obstacles are represented as a collection of points resembling cloud points. Data from the sensor, in the form of cloud points, is utilized to outline the boundaries of obstacles within the sensor's range. To account for HAN factors related to humans, this work introduces virtual cloud points surrounding a human. These virtual points are positioned along the perimeters of the HAN factors. To illustrate the formation of these virtual cloud points, Figure 4 depicts a parametric representation of the HAN factors in relation to a human, as introduced in the previous section.

Figure 4. Virtual cloud points for each HAN factor (left - PR, middle - BS, right - FV).

The threshold distance for PR can vary considerably, depending on factors such as human age, the type of social interaction, and cultural norms^[26]. However, in this work, PR is treated as a constant $$ \alpha_{PR} $$ around the human center, in alignment with relevant studies^[9,12,13]. Furthermore, the dimensions of the BS are defined using $$ \alpha_{BS_1} $$ and $$ \alpha_{BS_2} $$, determined based on survey results^[13]. Similarly, the angle and distance parameters for FV ($$ \alpha_{FV_1} $$ and $$ \alpha_{FV_2} $$) are chosen considering the characteristics of human vision. Beyond a certain limit, known as the monocular vision area, human vision reduces to approximately one-fifth of normal human vision, leading to limited 3D perception and difficulty in viewing^[27]. It is important to note that the generation of virtual cloud points for each HAN factor depends on the position and orientation of the human. Consequently, when the AMR detects a human within its sensing range, it generates virtual cloud points around the humans, taking into account the relevant HAN factors. These virtual cloud points for each HAN factor, as illustrated in Figure 4, are compactly formulated as

(17) $$ \begin{equation} \begin{bmatrix} {f_x} \\ {f_y} \end{bmatrix} = D_{j} \ \Omega_{j}, \end{equation} $$

where $$ f_x $$ and $$ f_y $$ are the $$ x $$ and $$ y $$ coordinates of the virtual cloud point, respectively, $$ D_j $$ encompasses the geometric formulations involving varying $$ \delta $$ angle, and $$ \Omega_j $$ contains the dimensional information for each HAN factor, denoted as $$ j $$ and covering $$ PR $$, $$ BS $$, and $$ FV $$. Here, $$ \Omega_j $$ for each $$ j $$ is defined as $$ \Omega_{PR} = \alpha_{PR} $$, $$ \Omega_{BS} = \begin{bmatrix} \alpha_{BS_1} & 0.5\alpha_{BS_2} \end{bmatrix}^\mathrm{T} $$, and $$ \Omega_{FV} = \alpha_{FV2} $$, and $$ D_j $$ is defined as

(18) $$ \begin{equation} \begin{aligned} {D_{PR}} = \begin{bmatrix} {\cos\delta} \\ {\sin\delta} \\ \end{bmatrix} \text{ for }0\le \delta < 2\pi, \end{aligned} \end{equation} $$

(19) $$ \begin{align} {D_{BS}} = \begin{cases} \begin{bmatrix} {0} & {0} \\ {0} & {0} \\ \end{bmatrix}, & \text{if } \delta < \frac{\pi}{2} \text{ or } \delta > \frac{3\pi}{2}, \\ \begin{bmatrix} {-\tan(\delta-\pi/2}) & {1} \\ {0} & {0} \\ \end{bmatrix}, & \text{if } \delta \ge \frac{\pi}{2} \text{ or } \delta<{{\frac{\pi}{2}+\tan^{-1}\left(\frac{\alpha_{BS_1}}{0.5\alpha_{BS_2}}\right) }}, \\ \begin{bmatrix} {\tan(\delta-\pi/2}) & {-1} \\ {0} & {0} \\ \end{bmatrix}, & \text{if } \delta > {{\frac{3\pi}{2}{-}\tan^{-1}\left(\frac{\alpha_{BS_1}}{0.5\alpha_{BS_2}}\right)}} \text{ or } \delta \le \frac{3\pi}{2}, \\ \begin{bmatrix} {0} & {0} \\ {-1} & {1/\tan(\delta-\pi/2)} \\ \end{bmatrix}, & \text{if } \delta \ge {{\frac{\pi}{2}+\tan^{-1}\left(\frac{\alpha_{BS_1}}{0.5\alpha_{BS_2}}\right) }} \text{ or } \delta \le {{\frac{3\pi}{2}{-}\tan^{-1}\left(\frac{\alpha_{BS_1}}{0.5\alpha_{BS_2}}\right) }}, \\ \end{cases} \end{align} $$

(20) $$ \begin{equation} \begin{aligned} {D_{FV}} = \begin{cases} \begin{bmatrix} {\cos{\delta}} & {\sin{\delta}} \\ {0} & {0} \\ \end{bmatrix}, & \text{if } \delta \le {\frac{\alpha_{FV_1}}{2}} \text{ or } \delta \ge {{2\pi}-\frac{\alpha_{FV_1}}{2}}, \\ \begin{bmatrix} {0} & {0} \\ {0} & {0} \\ \end{bmatrix}, & \text{if } {\frac{\alpha_{FV_1}}{2}} < \delta < {2\pi}-\frac{\alpha_{FV_1}}{2}.\\ \end{cases} \end{aligned} \end{equation} $$

To generate a specific virtual cloud point within the FV region, denoted as $$ (f_x, f_y) $$, the parameters $$ \Omega_{FV} $$ and $$ D_{FV} $$ are used. Likewise, virtual cloud points for PR and BS are obtained based on the relative angle between the human and the AMR in the AMR frame.

3.2.2. Formulation of HAN-EPF

To account for human comfort in the AMR's navigation, an additional repulsive PF term is incorporated into the total PF. Consequently, the total PF of the HAN-EPF is expressed as

(21) $$ \mathbf{f}_t({^N}{\mathbf{p}}_m) = \mathbf{f}_a({^N}{\mathbf{p}}{_r}) + \mathbf{f}_{r}({^N}{\mathbf{p}}{_m}) + \mathbf{f}_{h}({^N}{\mathbf{p}}{_m}), $$

where $$ {\mathbf{f}}_h({^N}{\mathbf{p}}{_m}) $$ is the repulsive PF associated with humans, serving as an additional component. The definition of $$ {\mathbf{f}}_h({^N}{\mathbf{p}}{_m}) $$ is as follows:

(22) $$ \textbf {f}_{h}({^N}{\mathbf{p}_m}) = \textbf {f}_{h_g}({^N}{\mathbf{p}_m}) + \textbf {f}_{h_e}({^N}{\mathbf{p}_m}), $$

where

(23) $$\mathbf{f}_{h_g}({^N}{\mathbf{p}_m}) = -\frac{1}{2}n_g k_{h}d({^N}{\mathbf{p}_m}, {^N}{\mathbf{p}}_f)^{n_g -1}\left(\frac{1}{d({^N}{\mathbf{p}_m}, {^N}{\mathbf{p}}_f)}-\frac{1}{d_o}\right)^2 \frac{\partial{d({^N}{\mathbf{p}_m}, {^N}{\mathbf{p}}_g)}}{\partial \ {^N}{\mathbf{p}}_m }{\text, } $$

(24) $$ \mathbf{f}_{h_e}({^N}{\mathbf{p}_m}) = -k_{h}\frac{d({^N}{\mathbf{p}_m}, {^N}{\mathbf{p}}_g)^{n_g}}{d({^N}{\mathbf{p}_m}, {^N}{\mathbf{p}}_f)^2} \left(\frac{1}{d({^N}{\mathbf{p}_m}, {^N}{\mathbf{p}}_f)}-\frac{1}{d_o}\right) \hat{\mathbf{q}}. $$

It is important to note that, at any given time step, only the closest virtual cloud point related to human factors and the closest obstacle point are considered. For instance, in Figure 5, all the HAN factors are detected in the left figure. However, the AMR is closer to FV than PR and BS. Hence, only FV is considered when generating $$ \mathbf{f}_{h}({^N}{\mathbf{p}_m}) $$. Similarly, the obstacle point that is closer to AMR is used to compute $$ \mathbf{f}_r $$. In the right figure, only BS is considered among the HAN factors.

Figure 5. Illustrations for selecting HAN factors.

The HAN-EPF contains several unknown parameters, but the dominant factors influencing the AMR's collision avoidance behavior are the coefficients $$ k_a $$, $$ k_r $$, and $$ k_h $$. As determining proper suitable scaling factors typically involves trial and error and is a time-consuming process, this study introduced new variables, specifically the scaling factors $$ \mu_r $$ and $$ \mu_h $$, to reduce the number of parameters and analyze the AMR's behavior concerning changes in those parameters. These variables replace $$ k_r $$ and $$ k_h $$ and are defined as follows:

(25) $$ k_r = \frac{k_a}{\mu_r}, $$

(26) $$ k_h = \frac{k_a}{\mu_h}. $$

Note that $$ k_r $$ and $$ k_h $$ are only dependent on the scaling factors once $$ k_a $$ is fixed. In this work, the scaling factors $$ \mu_r $$ and $$ \mu_h $$, along with the coefficient $$ k_a $$, are determined through the FISs proposed.

3.3. Preliminary study for understanding HAN-EPF

Before designing the FISs for the scaling factors, it is crucial to gain insights into how these scaling factors influence the AMR's behavior. This foundational understanding is essential for creating the MFs and rules for the proposed FISs. To achieve this, preliminary simulation studies were conducted to investigate the AMR's behavior in terms of its path length and relative distance from obstacles during avoidance maneuvers, with a focus on changes in the scaling factors. These preliminary results serve as the basis for designing the proposed FIS and help to establish reasonable ranges for the scaling factors. The simulation parameters introduced in Section 4 were utilized for these preliminary studies. Note that the coefficient $$ k_a $$ in the attractive PF was set to 1 to specifically observe how the AMR's behavior changes with varying $$ \mu_r $$ values. The following observations were made from these preliminary studies. When the value of $$ \mu_r $$ decreases beyond a certain value, the AMR's position starts to oscillate, as illustrated in Figure 6 (left). This oscillation occurs because the AMR traverses the boundary of the obstacle's distance of influence ($$ d_o $$). On the other hand, increasing $$ \mu_r $$ beyond a certain value causes the AMR to collide with the obstacle, as shown in Figure 6 (right). A threshold distance of 0.5 m was assumed to indicate a physical collision.

Figure 6. AMR's trajectory with very small $$ \mu_r $$ (left) and relative distance between the AMR and the obstacle with very high $$ \mu_r $$ (right).

Due to the dominant attractive PF compared to the repulsive PF, the AMR is not effectively repelled from the obstacle. Consequently, when $$ \mu_r $$ is high, the AMR travels closer to the obstacle, while a low $$ \mu_r $$ results in a longer path for the AMR, as shown in Figure 7.

Figure 7. Minimum relative distance and the travel distance with respect to the scaling factor.

It is noteworthy that the appropriate range for $$ \mu_r $$ was found to be between 0.01 and 55, where neither collisions nor oscillations occurred. These preliminary simulations maintained $$ k_a $$ at 1 and the fixed distance of influence ($$ d_o $$) at 25 m, reflecting typical lidar sensor sensing ranges. In addition, these analysis results can be applied to the scaling factor $$ \mu_h $$ for humans, employing the same scaling factor range.

3.4. FIS design

This study employs three FISs to determine the coefficient $$ k_a $$ and scaling factors $$ \mu_r $$ and $$ \mu_h $$. The first step in designing these FISs involves defining the appropriate inputs and outputs for each FIS. The structure of the FISs considered in this work is illustrated in Figure 8. For FIS 1, which determines the coefficient $$ k_a $$ (the output of the FIS 1), two inputs are selected: the relative distance between the AMR and the destination position and the relative distance between the AMR and either the obstacle or the human (whichever is closer). This choice is made because the magnitude of the attractive PF is closely tied to the AMR's motion toward the goal position. FIS 2 is responsible for determining the scaling factor $$ \mu_r $$, which governs the repulsive PF for obstacle avoidance. The inputs selected for FIS 2 include the relative distance of the AMR from the obstacle and the relative angle between the AMR's heading and the direction to the obstacle. These inputs are chosen to appropriately determine $$ \mu_r $$ in relation to obstacle avoidance. FIS 3 derives the scaling factor $$ \mu_h $$, which is associated with interactions involving humans. The inputs for FIS 3 consist of the relative distance between the AMR and the human and the relative velocity of the AMR in relation to the human. These inputs are used to determine $$ \mu_h $$, which, in turn, influences the repulsive PF concerning human interactions.

Figure 8. Structure of each FIS.

For each input and output of the FISs, three MFs are considered, which are Low, Medium, and High. In particular, this work considers two sets of the MFs, as shown in Figure 9. The first set of the MFs in Figure 9 (left) is used for the inputs of all three FISs. These MFs normalize input numeric values within the range of 0 to 1. This normalization allows the FISs to be applicable across different environments and AMR velocities, provided that the maximum values are known. For FIS 1, the output uses the first set of MFs. On the other hand, the outputs of FIS 2 and FIS 3 use the second set of MFs defined in Figure 9 (right). The range of these output MFs, ranging from 0.01 to 55, is determined based on the preliminary studies described in the previous section.

Figure 9. Two sets of the MFs considered.

The next step in designing the FISs is to build the rules that define the relationship between the inputs and outputs for each FIS, aiming to achieve the desired behavior of the AMR. As stated previously, FIS 1 determines the magnitude of the attractive PF. For example, if the AMR is in close proximity to obstacles or humans while being far from the goal position, it is desirable for the AMR to generate a smaller attractive PF than the repulsive one to avoid collisions, even if the relative distance to the goal is large. Conversely, if the AMR is far from both the destination and obstacles, it should generate a large attractive PF, which corresponds to a large value of $$ k_a $$. Based on knowledge of the desired behavior of the AMR, the rules for FIS 1 are defined as shown in Table 2.

The next step in designing the FISs is to establish the rules for FIS 2, which determines $$ \mu_r $$ and, consequently, $$ k_r $$ for the obstacle-associated repulsive PF. In contrast to FIS 1, FIS 2 takes into account the relative angle between the AMR and the obstacle. When the AMR is in close proximity to the obstacle and the relative angle is small, the AMR should generate a strong repulsive PF because a small relative angle indicates that the AMR is approaching the obstacle head-on. Consequently, $$ \mu_r $$ should be set to "Low", resulting in a high value of $$ k_r $$. The rules for FIS 2 are defined in Table 3 based on the desired avoidance maneuvers of the AMR.

Similar to FIS 2, FIS 3's rule matrix is designed to govern the desired avoidance maneuver of the AMR with respect to humans. For instance, when the AMR is in close proximity to a human and the relative velocity is high, a significant repulsive PF is needed. In such a case, where the human is approaching the AMR's direction, the AMR should have the capability to maneuver without violating HAN rules. This significant repulsive PF can be generated by setting $$ \mu_h $$ to a small value. The rule matrix for FIS 3 is defined based on the AMR's motion concerning the input values, as shown in Table 4.

The entire process is summarized in the flowchart, as shown in Figure 10. Initially, once the parameters are defined, the human positional information is utilized to generate the virtual cloud points. Concurrently, the parameters used in the calculations concerning the AMR frame are inputted into the FIS to derive the outputs: $$ k_a $$, $$ \mu_r $$, and $$ \mu_h $$. These resulting outputs, along with the virtual cloud points, are then fed into the EPF framework to calculate the total PF exerted on the AMR.

Figure 10. Overall procedure of the proposed approach.

4.1. Scenario 1

In this scenario, the AMR maneuvers in an environment designed to introduce challenges such as local minima, GNRON, and the presence of a moving human. The environment includes a circular obstacle placed to block the AMR's head-on, causing a local minima issue, and a rectangular obstacle positioned near the destination position to create a GNRON issue. The parameters for the position information of the AMR and human are listed in Table 6.

Figure 11 (left) shows the AMR's trajectory in Scenario 1, demonstrating that the AMR adaptively avoids both the obstacle and the human during its travel. Also, Figure 11 (right) provides a visual representation of the scenario, highlighting that the AMR successfully maintains a relative distance greater than the collision threshold value of 0.5 m from both obstacles and the human. The closest relative distance between the AMR and the obstacle (purple dot) is 1.6 m, while the closest distance to the virtual cloud points (red dot) is 1.07 m. Furthermore, Figure 12 shows a close-up view of the AMR's interaction with the human. At $$ 41 $$ s, the AMR starts a maneuver when it encounters the human. Notably, the closest virtual cloud point corresponds to the FV, and based on the relative angle $$ \delta $$, the AMR navigates towards the back of the human. At $$ 42 $$ s and $$ 44 $$ s, after navigating away from FV, the AMR successfully avoids PR and BS, respectively. Hence, throughout the simulation, the AMR reaches the destination position while avoiding the human and navigating safely without violating any HAN rules.

Figure 11. AMR's trajectory and relative distance from the AMR to obstacles and humans (Scenario 1).

Figure 12. AMR's trajectories near humans (Scenario 1): (left) 41 s, (middle) 42 s, and (right) 44 s.

4.2. Scenario 2

Scenario 2 presents a more complex environment compared to Scenario 1, involving obstacles and moving humans. This environment consists of two humans, one in motion and the other rotating to change gaze, along with three obstacles. The parameters related to the positions of the AMR and human are given in Table 7.

Figure 13 displays the trajectory traveled by the AMR in Scenario 2. Despite facing two humans simultaneously near a static obstacle, the AMR adeptly navigates through this complex situation. Following this maneuver, the AMR safely reaches its destination while avoiding two static obstacles. Figure 13 (right) provides a visual representation of the relative distance between the AMR, obstacles, and humans. It can be observed that the closest relative distance between the AMR and the obstacles and the virtual cloud points are $$ 1.15 $$ and $$ 1.06 $$ m, respectively, both exceeding the collision threshold distance. Consequently, no collision occurs during the maneuver.

Figure 13. AMR's trajectory and relative distance from the AMR to obstacles and humans (Scenario 2).

Figure 14 (top left) depicts the initial encounter between the AMR and human-1, who starts to rotate the AMR approaches. The AMR spends a significant portion of time navigating close to the FV of human-1, primarily because of the continuous rotation of their gaze. In response, the AMR adjusts its path, avoiding human-1 and subsequently encountering a rectangular obstacle near $$ 58 $$ s, as shown in Figure 14 (top right). When encountering the rectangular static obstacle, the robot prioritizes avoiding collision with it. Depending on the relative angle, the robot adjusts its navigation towards the right to circumvent the obstacle. Around $$ 60 $$ s, as shown in Figure 14 (bottom left), the AMR approaches a moving human and successfully navigates to the right side of the human, even though the human is moving toward the AMR. Note that the closest relative distance between the AMR and human-2 is highlighted by a red dot in Figure 13 (right), which occurs near the FV of human-2. Finally, at 70 s, as shown in Figure 14 (bottom right), the AMR bypasses human-2 and proceeds toward its destination, encountering both the rectangular obstacle and the circular obstacle along the way.

Figure 14. AMR's trajectories near obstacles and humans (Scenario 2): (top left) 42 s, (top right) 58 s, (bottom left) 65 s, and (bottom right) 70 s).

4.3. Scenario 3

Scenario 3 considers a more realistic environment, resembling an airport setting with various humans and infrastructure-like obstacles. Circular obstacle-1 and circular obstacle-2 can be envisioned as pillars in an airport hallway, with humans following the convention of moving along the right side. In addition, there are two rectangular obstacles that represent passenger waiting regions. The simulation includes six human agents, and their positions are listed in Table 8.

From Figure 15 (right), one can observe that throughout the simulation, the AMR never crosses the collision threshold. The minimum relative distance between the AMR and obstacles, as well as the virtual cloud points around humans, are $$ 0.89 $$ m and $$ 0.55 $$ m, respectively. Figure 16 (top left) shows the initial encounter between the AMR and human-1, where the AMR maneuvers to the right at $$ 10.5 $$ s. However, it becomes evident that human-1 is moving faster than the AMR at $$ 14.5 $$ s, as shown in Figure 16 (top right). Consequently, the AMR stops and maneuvers to the left to avoid a collision at $$ 20 $$ s, as shown in Figure 16 (middle left). Subsequently, the AMR prioritizes steering away from pillar-1 by choosing to navigate toward the left side. At around $$ 35 $$ s, the AMR faces human-2, who moves into its path and gazes in the AMR's direction, as shown in Figure 16 (middle right). The AMR successfully navigates the gap between FV of human-2 and the circular obstacle-1 during this maneuver, as depicted in Figure 16 (bottom left). Since the gap is small, the closest relative distance occurs during this maneuver of the AMR when the AMR encounters the FV of human-2. At $$ 45 $$ s, the AMR steers clear of human-2, as shown in Figure 16 (bottom left), and Figure 16 (bottom right) shows that at $$ 65 $$ s, it encounters human-3, who is moving in the same direction as the AMR but at a slower pace. Consequently, the AMR overtakes human-3 and reaches its destination. Note that human-4, located near the destination, presents a situation similar to GNRON.

Figure 15. AMR's trajectory and relative distance from the AMR to obstacles and humans (Scenario 3).

Figure 16. AMR's trajectories near obstacles and humans (Scenario 3): (top left) 10.5 s, (top right) 14.5 s, (middle left) 20 s, (middle right) 35 s, (bottom left) 45 s, and (bottom right) 65 s.

To summarize the findings of the simulation study, the closest relative distances from the AMR to humans and obstacles obtained are tabulated in Table 9. Upon comparing various scenarios, it is observed that the relative distance from the AMR to humans is smaller than the distance from the AMR to obstacles. Since humans are moving, unlike static obstacles, during the operation of the AMR, there is a high possibility that the AMR navigates around humans closely. Furthermore, it is observed that the AMR comes closest to both humans and obstacles in Scenario 3 compared to the others. These results are natural as the AMR must navigate in a complex environment, dealing with dynamically moving humans and multiple static obstacles. Despite these challenges, it is noteworthy that in all scenarios, the AMR adeptly navigates to the destination position without violating any HAN constraints.

Authors' contributions

Made substantial contributions to the conceptualization, methodology, and analysis: Sampathkumar SK, Choi D, Kim D

Contributed to approach validation, software simulation, and writing - original draft preparation: Sampathkumar SK, Choi D

Contributed to the investigation, supervision, and writing - review and preparation: Kim D

Availability of data and materials

Not applicable.

Financial support and sponsorship

None.

Conflicts of interest

All authors declared that there are no conflicts of interest.

Ethical approval and consent to participate

Not applicable.

Consent for publication

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Copyright

© The Author(s) 2024.