Path-Planning and Navigation for Robots Considering Human–Robot–Environment Interactions in Supermarket Environments

Abstract

This study proposes a shopping assistant robot, called CartBot, to facilitate the grocery shopping experience for customers/shoppers. A grocery store environment can be complex and confusing to customers. Therefore, the main aim is to assist customers in navigating this environment efficiently while carrying their purchased items, and hence improve the overall shopping experience and reduce shopping time. To achieve this, a unified framework for implementing path planning and collision avoidance in a supermarket environment is proposed. Here, with a shopping list as the input, an efficient (or near-optimal) global path is generated to complete the shopping. Then, a real-time local planner is proposed to navigate this path while avoiding any obstacles that are encountered. Various features/strategies to facilitate navigation and obstacle interactions are also addressed in this work. Simulation studies of CartBot are carried out in a grocery store environment along with other CartBots, employees, and human shoppers to validate the performance of the proposed approach.

1. Introduction

A recent focus of robotics research has shifted to service robots for assisting humans in their daily activities [1]. For example, grocery shopping is one such activity that people carry out regularly. Supercenters and grocery stores nowadays are massive in size and have confusing layouts. This work proposes a path planning and navigation framework for autonomous robots to assist shoppers in such stores. In other words, based on the customer’s shopping list, the robot can provide the shortest path for the customer to explore and purchase such items. Such robots are helpful for new or even regular customers to find the products they are looking for quickly without getting lost or going in circles in the process. However, it is a challenging task to find optimal routes to complete shopping and navigate supermarket environments while interacting with humans. First, the problem of calculating the optimal path based on a shopping list is analogous to solving the Traveling Salesman Problem (TSP). Here, TSP is a non-deterministic polynomial-time (NP)-hard problem whose time complexity increases exponentially with the number of shopping items [2,3]. Moreover, navigating grocery stores is complex due to the fact that the robot must recognize humans/customers and other robots that are dynamically moving. From the robot’s point of view, the robot should not treat humans as obstacles while aiming for the optimum path for its mission. More importantly, customers must perceive the robot’s motion as acceptable.

Several robotic platforms have been designed to assist humans in indoor environments. Burgard et al. [4] presented a study of a mobile robot that gives guided tours to visitors in a museum. This was one of the very first robots deployed in a public space to assist humans. Thurn et al. [5] later designed a new museum tour guide robot with better human interaction and navigation performance. Vongbunyong et al. [6] presented an autonomous mobile service robot for a reception in department stores. The study showed that the robot is able to interact with customers by providing the retail stores’ locations and details and leading the way to the stores. Ever since more robots have been deployed to assist humans in indoor spaces like hotels [7], hospitals [8], homes [9], retail stores [10], and supermarkets [11]. Especially, grocery stores and supermarkets are one of the spaces most frequented by people, and thus many studies have been performed targeting such environments. Galdelli et al. [12] introduced a path planning method for a retail robot that focuses on automated inventory management by incorporating shopper behavior data. The approach utilizes heatmaps generated from customer trajectories and product interaction data to guide the robot, aiming to optimize paths in areas with significant customer activity. Su et al. [13] proposed mobile robot operational strategies for replenishment tasks. The strategies include collision-free path planning, sensor fusion, and navigation, considering both customer safety and comfort. Cleveland et al. [14] presented a robot capable of mapping and navigating a store environment with the main aim of scanning the shelves and keeping track of the inventory. Tally 3.0 [11] is a marketed robotic solution focusing on accurately capturing the inventory. There are also many solutions dedicated to guiding humans in the shopping environment. Sales et al. [15] proposed a robot to assist elderly people with grocery shopping. The robot has a basket to carry the items picked by its user while the robot actively follows them throughout their shopping. Matsuhira et al. [16] proposed an interesting idea of having two assistant robots, one to lead and interact with the human while the other follows and carries the picked items. Gross et al. [17] presented a robot that guides customers to any item location they would like to find. They proposed to make use of the map layout of the store and the merchant’s product information system to fix the site of items on the map. Similarly, this work creates an offline global map and an item database, assuming such information is available within each merchant’s system. Chen et al. [18] also presented a similar guide robot for a shopping mall setting that guides customers to the store of their choice. Even though most of the guide robot studies above plan for the shortest path, they only considered the robot to have a single goal, from the point they interact with the customer to the location the customer wishes to be guided to. In this work, however, the authors propose having the entire shopping list of a customer as an input to our assistant robot. Thompson et al. [19] presented a study where the robot takes three shopping items as input and escorts customers to their selected items. However, unlike prior studies, they used store signs to navigate instead of maps. This is not feasible in a somewhat general scenario as we aim to plan a shorter travel path to pick all the items quickly, and to create such a plan, one needs prior map information. The study by Alves et al. [20] is similar to this work as they also use a shopping list as input and propose a planning and navigation strategy for the grocery environment. However, they considered only 10 and 14 items as inputs, which is significantly lower than the typical number of shopping items for multifamily and hypermarket shoppers. Additionally, none of the previous robot-assisted studies have considered interactions with other robots and human beings in a realistic way.

This work not only designs a combined guidance and navigation framework for a shopping assistant robot but also focuses on the interactions of the robot with environments, other robots, and humans. The assumptions made by this study are that the robots have no localization errors and that the supermarket environment map is known. Thus, a grocery list is the only input provided to the robot. The Dijkstra’s Algorithm (DA) is used to create a node graph connecting all the feasible paths between items. Then, the Genetic Algorithm (GA) is used to solve the TSP and to find a near-optimal path within a fixed time. After planning the path, the robotic platform utilizes the Enhanced Potential Field (EPF) algorithm developed in our research group for real-time obstacle avoidance [21]. The main achievements of this work are as follows:

• Developed a unified guidance and navigation framework for ground robotic platforms operated in grocery store environments, incorporating stakeholder requirements to improve operational efficiency and enhance the customer shopping experience.
• Integrated a real-time, collision-free navigation algorithm into a global planning framework with low computational overhead, enabling robust and safe robotic operations in complex, dynamic environments.

The rest of this work is organized as follows: Section 2 introduces the algorithms used by the shopping robot, the robot model, and the environment used for simulation. The proposed methodologies and frameworks for path planning and obstacle avoidance are described in Section 3. Section 4 shows the simulation results and discusses the interactions of the shopping robot with other agents. Finally, the conclusions of this study are listed in Section 5.

2. Preliminary

This section introduces the environment, stakeholders, robot model, and the algorithms considered in this work. In the global planner, the DA is used as a search algorithm to find the shortest path between nodes (i.e., shopping items), while the GA is employed to determine a near-optimal sequence for completing the shopping task given a list of items. In the local planner, the EPF algorithm enables robots to avoid obstacles in real time while navigating the store to collect items based on the sequence provided by the global planner.

2.1. Environment

Figure 1 illustrates a two-dimensional (2D) layout of a supermarket model based on real-world measurements taken from a nearby Kroger supermarket. Here, the length of the shelves varies from 17 to 25 m, and the width of the shelves varies from 1.02 to 2 m. Moreover, the aisle distance between shelves is around 2.16 m.

Since the target environment is mainly composed of repetitive shelves, an environment with strong characteristics was selected for this study. Hence, only the colored structures in the central part of Figure 1 constitute the simulation environment. Here, the shelves and counters are indicated in blue, the entrance is at the bottom (near $(x,y)=(0,-25)$ m) between the green structures, and the checkout area is indicated in red. All the items that need to be shopped are expected to be present on the shelves. Also, a CartBot is considered to be capable of billing the items as they are placed in it. Hence, the checkout area is set as off-limits for the CartBot.

Furthermore, grocery stores universally have simpler structures (like cuboid shelves) and don’t have any overhanging structures that people might bump into. This work deems that only the 2D information of the environment will be essential to facilitate path planning and navigation. Hence, the 2D location or vertex information of all the static structures is initially saved as an environment map. This map information is then used by the CartBot to plan a global path that avoids all the static structures in the environment.

2.2. Stakeholders

All the stakeholders considered in this study are described in this section.

2.2.1. Cart-Type Robot for Shopping Assistant

The main focus of this study is to design a navigation framework for a cart-type robot for shopping assistants, called CartBot. The primary goal is to guide the shoppers in the supermarket to find the items they are looking to purchase. The CartBot provides a near-optimal route to all items on the shopping list, thus minimizing travel distance, speeding up the shopping process, saving time wasted due to searching for items, and navigating back and forth in the supermarket.

A study conducted by the Complutense University of Madrid revealed that a person typically needs 25 s to select a food item which is reduced to just 9 s for selecting meat products [22]. Therefore, this study considers that it would take anywhere from 9 to 25 s for the shopper to decide and pick that item before proceeding to the following location. The maximum speed at which the robot can travel is constrained to minimize discomfort for other humans. Experiments were conducted based on the available information about the motorized shopping cart utilized by shoppers with disabilities to fix this speed limit [23,24]. Based on these studies, the robot’s maximum linear and rotational velocities are set at 0.33 m/s and 1.18 rad/s, respectively. The size of the robot in consideration is 0.56 m in length and breadth, so essentially the robot would fit within a cylindrical radius of 0.4 m, which is denoted by $r_{CB}$.

The CartBot is capable of avoiding both static obstacles (such as shelves) and dynamic obstacles (including humans, employees, and other CartBots). It can also dynamically recalculate its path in response to changing situations. For active obstacle avoidance, this work considers the use of onboard sensors, such as LiDARs or cameras, mounted on the CartBot. The grey area in Figure 2 represents the sensing range, defined by an arc of radius $s_r$ and a central angle of 180°.

2.2.2. Store Employees

Other stakeholders considered in this study are the employees working in the store. The employees’ role would be to go around the supermarket to help the customers with any queries, rearrange misplaced items, and restock items (keeping the item inventory in check). Therefore, it can be expected that the employees would be randomly/periodically traveling around the supermarket. Moreover, while restocking, they would carry a supply cart traveling back and forth from the storage area/inventory stockpile. Based on such behaviors, this work considers two different types of behaviors of the employees. Type I employees would continuously travel back and forth from the storage area to the shelves and restock the shelves with items. Thus, they would push around a big supply cart, stop near shelves, pick up items from the cart, and place them on the shelves. Type II employees would walk back and forth within their designated areas, periodically stopping to rearrange the items on the shelves or to guide other customers.

Every human can have a different walking pace, which varies based on their actions or thoughts. It is known that the average walking speed of humans is between 1.35 m/s to 1.6 m/s [25]. Hence, for this study, the employee model is assumed to travel at a random pace ranging from 0 m/s to 1.5 m/s. Moreover, from our observation, employees move at a reduced speed when pushing a heavy supply cart. Thus, this study fixes the maximum speed of an employee pushing a cart at 0.5 m/s, which is about one-third of the average human speed. Furthermore, for the robot to avoid an employee without using sensor information, it would be preferable to have a rough estimate of the circular area that the employee would cover. The average man has a shoulder width of 0.46 m [26]. However, in this study, the robot would assume the employee/human to be anywhere within a 0.5 m diameter from their estimated location as a safety precaution. The dimension of the supply cart considered is 0.5 m wide and 1.0 m long [27].

2.2.3. Human Shopper

A direct stakeholder in a supermarket would be the human shopper. They push shopping carts, try to find the items they need, or walk down the aisles looking for things they might need. The behavior that can be expected from them would be going around the supermarket and suddenly slowing down or stopping as they stumble on an item they want to purchase or check out.

As mentioned previously, humans have complex behavior that cannot be modeled precisely. For this reason, human behavior is randomized with acceptable reasoning. The human shopper will travel around the supermarket looking for items and stop when they find them. As discussed in Section 2.2.1, it is found that humans take between 9 to 25 s to decide on a brand, pick an item, and place that item in the cart before proceeding to the next one. Hence, human shoppers are set to wait for 9 to 25 s whenever they make a stop. Similar to the employee pushing a supply cart (Section 2.2.2), a human shopper pushing a shopping cart is also expected to move slower than average human speed. However, a human shopper’s maximum speed is fixed at two-thirds of the average human speed, as a shopping cart is lighter and easier to maneuver than a supply cart.

Furthermore, it is assumed that human shoppers would stop close enough to the item and not leave the cart to look for items. The dimension of a standard shopping cart is approximately 0.9 m long and 0.55 m wide [28]. Based on our observation, the length and width of the model of a human shopper walking with a cart are assumed to be 1.5 m and 0.55 m, respectively.

2.3. Dijkstra’s Algorithm

The DA is a well-known search algorithm used for solving the single-source shortest path problem [29]. In graph theory, this is the problem of finding the shortest path from a source node to every other node in a graph. The DA is the most widely used, easy-to-implement, and fast algorithm that works best when the node graph is undirected with non-negative edge weights [29]. The DA takes a node graph, the start node, and the node locations in the inertial reference frame as inputs. Using this information, the shortest route, called Route, from the start node to every other node in the node graph is calculated. Additionally, a travel cost, called Cost, for all these routes is also calculated as the sum of Euclidean distances.

2.4. Genetic Algorithm

The GA is a population-based metaheuristic algorithm inspired by the natural evolution process of genes. It can provide a near-optimal solution for complex optimization problems. In the conventional GA, first, the possible solutions for the given problem are randomly selected as the gene population, also referred to as the encoding step. From this population, based on fitness values evaluated, better solutions are picked, and the rest are discarded, this is the selection step. The selection step is based on the “survival of the fittest” concept from the Darwinian evolutionary theory [30,31], and the best solution is stored in this step. In the next step, random pairs of all the better solutions are mated or crossed over to form new offspring solutions. The last step of the GA is the mutation step, where the population is mutated with a user-defined probability, usually low. In this step, a random element of the selected solution, based on probability, is changed or mutated to a different random element. The new total population consists of the selected better solutions and their respective off-springs. This new population is considered the next generation, and the selection, crossover, and mutation steps are repeatedly performed to create new generations. The best-fit solution is updated at each generation selection based on the fitness, and the GA runs iteratively until the set number of generations or a stopping condition is met.

However, the cross-over and mutation steps in the conventional GA may create duplicate genes or nodes within a member of the population. This would make those members of the population invalid solutions as a solution for TSP needs to include all the genes/nodes without any repetitions. For this reason, this work utilizes a revised GA [32], which replaces the cross-over and mutation steps with flip, swap, and slide operations. Here, GA takes a cost matrix, population size, and the number of generations (or iterations) as inputs. Using this information, a sub-optimal travel sequence or item pick order (called optRoute), which achieves a minimal travel distance, is generated.

2.5. Enhanced Potential Field Algorithm

The Artificial Potential Field (APF) is a widely used method that can find a path to the goal while avoiding obstacles using a combination of attractive and repulsive potential fields [33]. However, the major issue with the conventional APF is the existence of local minima, which are situations that arise when the attractive and repulsive potential fields cancel each other. The EPF [21] is an enhanced form of the APF based on the curl-free vector field concept [34]. The EPF takes the velocity of the obstacles and the relative angles from the robot’s velocity vector into consideration to decide the direction of the repulsive potential field, resulting in smoothly avoiding obstacles without local minima issues. Hence, it is an effective solution for dynamic obstacle avoidance problems.

For the EPF, the total potential field ${\mathbf{f}}_t$ in the presence of n obstacles is given as

$${\mathbf{f}}_t={\mathbf{f}}_a+\sum _{i=1}^n{\mathbf{f}}_{r_i},$$

(1)

where ${\mathbf{f}}_a$ is the attractive potential field pulling the robot towards the goal, and ${\mathbf{f}}_{r_i}$ is the repulsive potential field repelling the robot away from the $i_{th}$ obstacle. These potential fields are calculated as follows [35]:

$$\begin{array}{cc}\hfill {\mathbf{f}}_a\left(\mathbf{q}\right)& =k_a{n}_{a}d{(\mathbf{q},{\mathbf{q}}_g)}^{n_a-1}{\displaystyle \frac{\partial d(\mathbf{q},{\mathbf{q}}_g)}{\partial \mathbf{q}}},\hfill \end{array}$$

(2)

$$\begin{array}{cc}\hfill {\mathbf{f}}_r\left(\mathbf{q}\right)& =\left\{\begin{array}{cc}-k_r{n}_r{\left({\displaystyle \frac{1}{d\left(\mathbf{q},{\mathbf{q}}_o\right)}}-{\displaystyle \frac{1}{d_o}}\right)}^{n_r-1}{\displaystyle \frac{R}{d{\left(\mathbf{q},{\mathbf{q}}_o\right)}^2}}{\displaystyle \frac{\partial d(\mathbf{q},{\mathbf{q}}_o)}{\partial \mathbf{q}}}\hfill & \mathrm{if}d\left(\mathbf{q},{\mathbf{q}}_o\right)\le d_o\hfill \\ 0\hfill & \mathrm{if}d\left(\mathbf{q},{\mathbf{q}}_o\right)>d_o\hfill \end{array}\right.,\hfill \end{array}$$

(3)

where $d_o$ is the threshold distance influenced by obstacles that generate repulsive potential fields, and $\mathbf{q}$, ${\mathbf{q}}_g$, and ${\mathbf{q}}_o$ are the position vectors of the robot, goal, and obstacle, respectively. Also, ${\displaystyle \frac{\partial d(\mathbf{q},{\mathbf{q}}_g)}{\partial \mathbf{q}}}$ and ${\displaystyle \frac{\partial d(\mathbf{q},{\mathbf{q}}_o)}{\partial \mathbf{q}}}$ are the directions from the robot towards a goal and obstacle, respectively, while $d(\mathbf{q},{\mathbf{q}}_g)$ and $d(\mathbf{q},{\mathbf{q}}_o)$ are the relative distances from $\mathbf{q}$ to ${\mathbf{q}}_g$ and ${\mathbf{q}}_o$, respectively. Moreover, $k_a$ and $k_r$ are the attractive and repulsive gains, respectively, and $n_a$ and $n_r$ are the order of attractive and repulsive potential functions. Furthermore, R is the rotation matrix used to determine the direction of the repulsive field. Algorithm 1 describes how the matrix R is chosen [35]. Here, $|\dot{\mathbf{q}}|$ and $|\dot{{\mathbf{q}}_o}|$ are the speeds of the robot and obstacle, respectively, while $\psi$ and ${\psi }_o$ are the relative angles from robot to the obstacle, and from obstacle to the robot, respectively. It is important to note that the obstacle here can be static (like shelves) or dynamic (like other robots or humans). Also, $\alpha$ is a user-defined angle that can determine the magnitude of the repulsive potential field direction.

Algorithm 1: EPF direction determination

2.6. Robot Model

This work considers a mobile robot represented using the unicycle model as shown in Figure 3. This model assumes the robot to be a rigid body with a single wheel that moves by changing the linear velocity and heading angle. This model is given by

$$\begin{array}{cc}\hfill \left[\begin{array}{c}{x}_{k+1}\\ y_{k+1}\\ {\varphi }_{k+1}\end{array}\right]& = \left[\begin{array}{c}{x}_k+v_k\Delta tcos\left({\varphi }_k\right)\\ y_k+v_k\Delta tsin\left({\varphi }_k\right)\\ {\varphi }_k+{\omega }_k\Delta t\end{array}\right],\hfill \end{array}$$

(4)

where subscript k indicates a particular time step, while $\Delta t=t_{k+1}-t_k$ is the time interval. Also, x and y represent the Cartesian position of the mobile robot, with the heading angle $\varphi$, and v and $\omega$ are the linear and angular velocities of the robot, respectively. The unicycle model is simple and easy to implement compared to the actual mobile robot kinematics or dynamics. This approach’s primary advantage is that it can work with any real mobile ground robot as their dynamics equation can be modified or equated to vary based on the linear and angular velocities.

3. Methodology

This section explains the path planning and collision avoidance process proposed in this work. Here, firstly, an overview of the CartBot’s unified guidance and navigation framework is illustrated. Later on, the global and local planner components are described. The global planner is responsible for generating a global path based on the shopping list and map of the environment. The local planner controls the CartBot in real time to track the global path while avoiding any potential collisions. Moreover, the local planner also performs replanning if a situation where the global path cannot be traversed occurs.

3.1. Process Overview/Framework

Figure 4 shows the overview of the CartBot’s framework. This framework consists of a Shopping List Function that creates a shopping list, which in turn acts as an input to the Path Planner. The Path Planner computes the optimal route of traversal for the CartBot with static and dynamic obstacle avoidance. The Inventory Database containing all the items and their location, and the map of the store environment, called Map, are the environment-specific static inputs for the system. In a real-world application of the system, a shopping list would be created by every customer, and the CartBot would generate navigation plans uniquely based on it. For the scope of this work, a Shopping List Function will pick N random items and their 2D locations from the Inventory Database and create a ShoppingList. This ShoppingList is the input to the Path Planner along with the Map. The Path Planner comprises a Global Planner and a Local Planner. The Global Planner, which will be explained in Section 3.2, determines the shortest path, called GlobalPath, to reach all the necessary items while avoiding all the mapped static obstacles. Next, the Local Planner aims to traverse the GlobalPath while making real-time adjustments based on the static and dynamic obstacles to avoid collisions. Here, control inputs, $v_k$ and ${\omega }_k$, are generated for the CartBot to perform collision-free navigation and will be discussed in Section 3.3.

The shopping environment has many shelves, and it is impractical to assume a simple Euclidean distance as the travel distance from one item location to the next. Therefore, this study considers using Pre-Established Waypoints (PEWs) placed at the corners of each shelf or structure. PEWs are additional intermediate waypoints created to connect the various regions of the shopping environment from where the desired items need to be picked. As shown in Figure 5, for each vertex of a shelf, points ${\mathrm{P}}_1$ and ${\mathrm{P}}_2$ are fixed with Robot Offset Distance, called ROD, along the two edges of the vertex. Then, the midpoint of these points is calculated; this point is coined as a PEW and is saved in the Map.

3.2. Global Planner

The Global Planner aims to answer the following question. Using a shopping list of items, how can the shortest possible travel route to obtain all items and then come back to the start location be determined? This resembles the classical TSP, which is a combinatorial optimization problem and is a well-known example of NP-complete problems [2]—meaning that the time required to find an optimal solution to such problems cannot be expressed as a polynomial function. Moreover, the cost of travel between items needs to be determined to solve the routing problem in a shopping environment. This study uses travel distance as a factor to determine the travel cost.

The Global Planner is an offline planner that determines the near-optimal route for the CartBot to follow based on the environment map, within a limited time. It uses the Map and the ShoppingList to plan a suitable route that the CartBot could track so that all the items are picked up with minimum cost or less travel distance, and as a result, incur less travel time. The planned path is designed to avoid all known static obstacles in the environment based on the environment map information given. There are two broad components of Global Planner, an Undirected Graph Generation function, which produces a NodeGraph, taking into consideration the ShoppingList and the Map, a Global Path Calculation function encompassing DA and GA to solve the TSP.

3.2.1. Undirected Graph Generation

The Undirected Graph function, as shown in Figure 6, utilizes the Map to generate a StaticGraph, which connects all the PEWs not separated by static obstacles of the environment. As mentioned in Section 3.1, the ShoppingList consists of the 2D location of the desired items on the shelf, which is used to determine a stopping point for the CartBot near every item on the shopping list. These points are called Robot Stopping Points (RSPs). The RSP for an item is a point at a distance of the ROD, perpendicularly away from the shelf’s edge nearest to the actual item location. Note that the ROD is set to 0.6 m, considering the size of the CartBot and a safe operating distance. Once the RSPs for all the items in the shopping list are determined through the step labeled “Determine RSPs”, they are stored in the rsp variable. Then, rsp is added to the StaticGraph, and a new set of connections are made between the RSPs and PEWs. The result of this process is a NodeGraph along with the NodeLocations.

3.2.2. Global Path Calculation

It explains how the shortest path is planned given the NodeGraph and NodeLocations. Figure 7 gives an overview of all the steps in the global path calculation. This function is divided into two phases. In Phase 1, the DA is employed to find the best travel route, Route, and the travel distance of this route, Cost. The Route contains the shortest path from a given starting node, called StartNode, in the NodeGraph to all the other nodes. The Cost contains the Euclidean distance to travel the shortest path from the given starting node in the NodeGraph to all the other nodes. The DA is iterated with each RSP (obtained by the Undirected Graph step) as a StartNode, along with the NodeGraph and NodeLocations, to determine the Route and Cost to the every other node in the NodeGraph. Furthermore, all the Route and Cost from one RSP to all other RSPs is determined and saved as route matrix and cost matrix, called Rmat and Cmat, respectively. After the DA completes the run for all the RSPs, the Cmat formulated is fed into Phase 2.

The GA uses the Cmat to solve the TSP and find the best order to pick the items to minimize the travel distance. It runs an optimizer to find the order in which the items need to be picked, called optRoute, to finish the shopping in a near-optimal path.

Firstly, in the encoding step, a population is created. In each of these solutions, the first and last position, representing the checkout area, are fixed as the starting node, while the remaining positions are randomized.

Secondly, in the selection step, using Cmat, the cost of travel for all the solutions is calculated. Now, the top 25 percent of the least cost solutions are selected, and the least cost solution or fittest population member is saved as optRoute. Then, the Flip, Swap, and Slide operations are performed on each of the previously selected solutions (top 25 percent). In a flip operation, a random sequence of a few genes/items/nodes is reversed. Thirdly, in the swap operation, two nodes in random are swapped. In the slide operation, a random sequence of nodes is slid by one position. After these three operations are performed, the resulting new population/solutions along with the selected top 25 percent of the population make up the next generation of solutions. These processes of selection step and genetic operations (Flip, Swap, and Slide) are repeated iteratively. Once the GA optimizer reaches a set number of generations, the best solution is saved as the optimal item picking order, optRoute. Then, using optRoute and Rmat, GlobalPath for the CartBot is obtained.

3.3. Local Planner

Figure 8 provides an overview of the local planner. The local planner is an online/real-time process that attempts to follow the GlobalPath while making complex decisions in regard to obstacle avoidance. The local planner is implemented at each time step to determine the best path for keeping a safe distance from static and/or dynamic obstacles. The local planner primarily utilizes the EPF algorithm based on multiple input parameters. These include the Obstacle Points (OPs) of all the static obstacles obtained using the Map information, called obs, the goal information from the GlobalPath, and the locations of all stakeholders (pos) provided by the sensing information in the Sensor block. The stakeholders include human shoppers, employees, and other CartBots. Note that the Sensor block primarily focuses on providing information about dynamic obstacles, which are the stakeholders in this study, since the environment map information is known and fixed.

The EPF directs the CartBot iteratively through every node in the GlobalPath in real time, while actively avoiding all detected obstacles within the sensing range. It is important to note that the avoidance maneuver for the CartBot is activated when the relative distance between the CartBot and obstacles is less than the threshold distance $d_o$ (described in Section 2.5), considering both safety and the congested grocery environment.

The control signals, $v_k$ and ${\omega }_k$, are generated at every time step for the CartBot. In the scenario of static obstacle detection, the EPF uses the obs to generate a repulsive potential field in order to avoid them. However, for dynamic obstacles, the planning process is divided into two categories. Firstly, if the obstacle is another CartBot, the local planner considers the pos of the dynamic obstacle and plans the path accordingly. However, pos information is not always available for human shoppers or employees. It becomes available only when the human dynamic obstacle enters the sensing range.

In the scenario where a human is in motion within the CartBot’s moving human detection range, which is defined as 2 m considering the congested grocery environment and human comfort factors [36], the local planner reduces the CartBot’s velocity to zero, i.e., $v_k={\omega }_k=0$. However, if the human blocks the CartBot’s path and remains stationary for more than 10 s, the local planner considers the human a static obstacle and continues to traverse the generated path while avoiding the human. This process is repeated at each time step until all the nodes in the GlobalPath are traversed.

In addition, there may be cases where the pre-determined global path is blocked. In such cases, a re-routing process is initiated: from the CartBot’s current position, a new global path that accounts for the remaining items is recalculated and fed into the local planner to resume the trip. This is how the local planner complements the global planner to enable the CartBot to successfully complete the task.

The following sections further discuss how static and dynamic obstacles are avoided using the EPF based on the parameters listed in

Table 1. Parameters for the EPF.

Description	Variable	Value
Obstacle influence distance	$d_o$	0.25 m
Attractive gain coefficient	$k_a$	100
Degree of attractive field	$n_a$	3
Repulsive gain coefficient	$k_r$	1
Degree of repulsive field	$n_r$	3
Rotation angle for the rotation matrix	$\alpha$	60°

.

3.3.1. Robot–Environment Interaction (Static Obstacles)

The static obstacles considered in this study include structural components of the environment, such as shelves and counters, which are included in the Map information. These static obstacles are much larger than the CartBot and are therefore divided into point obstacles spaced at 0.025 m from each other. The local planner uses the EPF to avoid only the closest point obstacle among several static obstacles at each time step. This closest point position information (${\mathbf{q}}_o$) is used to calculate the obstacle distance $d\left(\mathbf{q},{\mathbf{q}}_o\right)$. Based on this, the EPF generates a repulsive potential field to avoid the obstacle. At the same time, an attractive potential field is generated around the goal point obtained from GlobalPath. The goal is updated when achieved by the CartBot, to the next goal. This process occurs repeatedly until all the goals provided by the GlobalPath are reached.

3.3.2. Robot–Robot or Robot–Human Interaction (Dynamic Obstacles)

Dynamic obstacles include other CartBots, human shoppers, and employees. However, unlike static obstacles where only the nearest point obstacle is avoided, all the dynamical obstacles within the obstacle threshold are actively avoided. Note that the interaction of more than two stakeholders is not considered by the local planner within the scope of this study.

First, a scenario where the CartBot encounters another CartBot as an obstacle is considered, implying that the robots are facing each other and traveling in opposite directions. This is called “robot–robot interaction”. When such a situation arises, the EPF is triggered for both robots, causing them to cooperatively avoid each other and continue on their original path, as depicted in Figure 9.

Secondly, when the CartBot encounters a dynamic human shopper within its moving human detection range, traveling either in the same or opposite direction as its planned path, it will stop moving as long as the human remains within that range. Once no human is detected within the detection range, the CartBot resumes movement to continue picking up items. An example scenario of this “robot–human interaction” is depicted in Figure 10. This process is continuously carried out until the GlobalPath is safely traversed, avoiding all dynamic obstacles within the CartBot’s sensing range.

4. Simulation Study

This section discusses the factors affecting the results and the various interactions of the CartBot. The primary performance metric in this work is the successful completion of the task with the minimal path length, i.e., navigating to every item on the shopping list via a quasi-shortest route while maintaining a safe distance from all obstacles. In the simulations, 25 shopping items are considered, based on a statistical report published by Drive Research [37], which states that U.S. customers typically purchase 20 to 25 items during a grocery store visit. To compute the global path from the global planner, this work considers 100 populations and 1000 generations for the GA, considering the number of shopping items. When the number of items increases, these parameters can be adjusted by comparing the number of shopping items currently set with the newly set number and multiplying by the same proportion.

4.1. Single Agent: Robot–Environment Interaction

Firstly, the proposed approach is tested in a single-agent environment with no dynamic obstacles. That is, all the obstacles are static, allowing us to focus on analyzing the proposed system’s performance. At the start of the simulation, the CartBot is placed in a shopping environment, as shown in Figure 11a. It can be seen that the calculated near-optimal global path, with a path length of 168.394 m, goes through all the item locations on the shopping list provided to the CartBot. Note that the CartBot is depicted as a red square, the shopping list items are shown as orange points on the shelves, and the red dotted line represents the calculated global path.

Based on the global path information, the CartBot navigates the store using the local planner, as shown in Figure 11b. It can be seen that the CartBot successfully traverses the global path, stops by all the item locations, and returns to the starting position. Here, the actual path traveled by the CartBot, based on the local planner output, is illustrated as a solid black line. Its total path length is computed as 170.640 m, which is slightly longer than the path length generated by the global planner.

The difference between the actual and pre-planned paths arises due to the CartBot’s dynamics and the activation of the EPF algorithm. As discussed in Section 3.3, the local planner generates collision avoidance maneuvers around shelf corners to maintain a safe distance from obstacles using the EPF. The EPF is triggered when the robot is too close to the shelves, mainly at corners, causing the robot to adjust its path to avoid collisions.

Figure 12 shows the relative distance history between the CartBot and the shelves. The orange dotted line represents the threshold distance within which the CartBot is influenced by an obstacle’s repulsive potential field. That is, the local planner activates the collision avoidance maneuver using the EPF approach to maintain a safe relative distance if the distance between the CartBot and an obstacle becomes smaller than $d_0$. Furthermore, the pink dotted line indicates the physical collision threshold. It can be seen that the CartBot does not collide with obstacles. Although Figure 12 displays a history spanning over 1000 s, resulting in rapidly changing relative distance values, the CartBot’s actual movement remains smooth.

4.2. Multiple Agent: Human–Robot–Environment Interaction

In the previous scenario, the CartBot successfully navigates a shopping environment with only static obstacles. In this subsection, the proposed approach is further evaluated in a more complex environment that includes both static and dynamic obstacles. The objective is to analyze the CartBot’s behavior around dynamic obstacles. The Global Planner generates a path to collect all items, as shown in Figure 13a, similar to the single-agent case. At the start of the simulation, the CartBot (number 1) is positioned at the starting location, and various stakeholders, such as human shoppers (number 5), store employees (numbers 3 and 4), and another CartBot (number 2), are dynamically interacting with the environment. Due to these interactions, the total path length is measured as 170.924 m, which is slightly longer than in the static-obstacles-only scenario.

Figure 14 shows the relative distance history of the CartBot from both static and dynamic obstacles. One can observe that the CartBot successfully avoids collisions with the shelves, human shoppers, store employees, and other CartBots. This demonstrates the effectiveness of the local planner in avoiding all types of obstacles while still following the global path.

To further validate the performance of the proposed approach, the interactions between the CartBot and other stakeholders are closely analyzed. Figure 15a a shows a sequence of robot–robot interactions between CartBot 1 and CartBot 2. Scene 1 in Figure 15a depicts the moment just before the two robots engage. Both robots are moving toward each other with non-zero velocity, and CartBot 1’s global path is directed toward CartBot 2. At this point, the local planner activates an avoidance maneuver to prevent a potential collision. Scene 2 in Figure 15a illustrates the active interaction between the two robots. They continue to move while executing collision avoidance maneuvers based on the EPF algorithm. The deviation from the global path ensures a safe traversal in the constrained aisle. In Scene 3 in Figure 15a, the robots return to their their respective global paths after completing the avoidance behavior. This highlights the capability of the proposed approach to maintain safety in a constrained space like shopping aisles, between two shelves. Figure 15b,c show the CartBot 1’s relative distance, velocity, and angular velocity. From the relative distance trends, it is evident that CartBot 1 avoids both CartBot 2 and the shelves without collisions.

Figure 16 illustrates a robot–human interaction. Scene 1 in Figure 16a shows CartBot 1 is approaching a human shopper with non-zero velocity. The local planner activates the repulsive potential field within the EPF framework to prevent a collision, similar to the robot–robot interaction scenario. In Scene 2 in Figure 16a, CartBot 1 stops as the human shopper enters its moving human detection range, illustrated in Figure 16c. This stop behavior is confirmed by CartBot 1’s velocity trend in Figure 16b. The human shopper is expected to navigate around the robot, which is seen in Scene 3 in Figure 16a. The relative distance plot in Figure 16b shows that the human maintains a safe distance of over 1 m. In Scene 4 in Figure 16a, CartBot 1 resumes following its planned path about 10 s after the human shopper exits the detection range. The complete relative distance history in Figure 16b confirms that the CartBot 1 avoids collisions, highlighting the effectiveness of the proposed approach in dynamic human–robot shared spaces.

4.3. Multiple Agent: Monte Carlo Simulation

To evaluate the robustness of the proposed framework, Monte Carlo simulations are performed with 100 trials using randomly generated item locations at each trial, along with multiple agents used in the previous section. Figure 17 depicts the overlapped navigation trajectories for the trials, and the trajectories are evenly distributed due to the randomly located items to collect. In particular, noticeable avoidance trajectories are observed around the leftmost shelf due to another CartBot’s operation in a larger space than the other spaces between the shelves. In the random simulations, no collision is observed, and the minimum relative distance between the CartBot and obstacles across the simulations is 0.0658 m, which is greater than the physical collision threshold. In addition, various evaluation metrics are summarized in

Table 2. Summary of MC simulation results.

Item	Mean	Standard Deviation
Global path length (m)	178.837	13.261
Total travel path (m)	183.229	13.476
Global path computation (s)	0.451	0.022
Local planner’s computation at each time step (s)	2.611 × ${10}^{-4}$	4.798 × ${10}^{-5}$

. Similar to the single-run results, the averaged total travel path is slightly greater than the global path. For 25 shopping items, the computation for obtaining the global path across the trials takes 0.451 s on average. That is, the global planner rapidly provides the global path that the CartBot follows. Furthermore, the computation time for collision-free navigation at each time step is observed as 2.611 × ${10}^{-4}$ s, underlining the robustness and effectiveness of the proposed framework.

5. Conclusions

This study proposes a unified guidance and navigation framework for a shopping assistant robot, CartBot, designed to assist customers during grocery shopping. The proposed framework consists of two primary parts: a global planner and a local planner. Given a shopping list provided by the user, the global planner computes a near-optimal path (or global path) that visits all required items using a combination of Dijkstra and genetic algorithms. The local planner then follows this global path while avoiding static and dynamic obstacles using the enhanced potential field approach. To evaluate the effectiveness of the proposed framework, numerical simulations were conducted in both single and multi-agent settings, where CartBot encountered static obstacles like shelves and dynamic obstacles including human shoppers, store employees, and another CartBot. The CartBot successfully navigated the planned global path to collect the listed items while maintaining a safe distance from all obstacles throughout its operation. This study serves as a benchmark for evaluating various methods and strategies aimed at developing autonomous shopping carts. In future work, hardware experiments using mobile robots in a scaled-down physical environment are planned to validate the proposed framework in real-world scenarios. Moreover, advanced techniques, such as adaptive and/or artificial intelligence-based control methods, will be considered to improve the robustness of the unified framework in the presence of uncertainties, like localization errors and sensor noise.