Four-Dimensional Path Planning Methodology for Collaborative Robots Application in Industry 5.0

Abstract

Industry 5.0 is a developing phase in the evolution of industrialization that aims to reshape the production process by enhancing human creativity through the utilization of automation technologies and machine intelligence. Its central pillar is the collaboration between robots and humans. Path planning is a major challenge in robotics. An offline 4D path planning algorithm is proposed to find the optimal path in an environment with static and dynamic obstacles. The time variable was embodied in an enhanced artificial fish swarm algorithm (AFSA). The proposed methodology considers changes in robot speeds as well as the times at which they occur. This is in order to realistically simulate the conditions that prevail during cooperation between robots and humans in the Industry 5.0 environment. A method for calculating time, including changes in robot speed during path formation, is presented. The safety value of dynamic obstacles, the coefficients of the importance of the terms of the agent’s distance to the ending point, and the safety value of dynamic obstacles were introduced in the objective function. The coefficients of obstacle variation and speed variation are also proposed. The proposed methodology is applied to simulated real-world challenges in Industry 5.0 using an industrial robotic arm.

1. Introduction

The continuous rapid evolution of technology and the increase in industrial demands have led to the development of a new concept called Industry 5.0. Navahandi highlighted the positive effects that Industry 4.0 has had on improving and increasing the efficiency of the production process, while also citing its impact on human and environmental costs. He also encourages the integration of advanced technological solutions through Industry 5.0 as a solution to the shortcomings of Industry 4.0 [1]. Industry 5.0 is a human-centric approach to industrial processes, especially smart manufacturing, in which humans collaborate with machines. Advanced robotics collaborates with artificial intelligence and humans to boost manufacturing productivity, problem-solving, and decision-making. This approach aims to leverage technology to support, rather than replace, human workers, thereby promoting a more flexible and creative production environment [2,3].

Several technologies can be utilized to achieve Industry 5.0’s goals, such as artificial intelligence [4], the Internet of Things [5], cloud computing [6], big data [7], blockchains [8], digital twins [9], collaborative robots [10], and 6G [11]. The integration of these technologies into the manufacturing process contributes to providing greener solutions compared to existing industrial procedures, which do not emphasize the protection of the natural environment [12]. The capabilities of Industry 5.0 will particularly serve the needs of small and medium-sized businesses, where the full automation of their production process is considered complex and expensive and there is an increased need to quickly adapt their products to the demands of their customers [13].

Industry 5.0 can be applied in several fields such as intelligent healthcare [14], supply chain management [15], cloud manufacturing [16], manufacturing [17], education [18], and disaster management [19]. Matthew et al. analyzed concerns about patient privacy and safety arising from the implementation of Industry 5.0 in the healthcare sector. They discussed issues such as security, privacy, and dependability that exist in the healthcare sector and the integration of artificial intelligence and blockchain technologies into the healthcare management system [20]. Boudouaia et al. studied the solutions that Industry 5.0 can provide to problems in supply chains, such as limited visibility in both upstream and downstream supply chains, lack of trust among the different stakeholders, transparency, and traceability [21]. Xiao combined the Internet of Things and cloud computing technologies using intelligent algorithms in manufacturing [22]. Akundi et al. evaluated the applications of Industry 5.0 and conducted an extensive literature review [23].

Collaborative robots have a prominent role in Industry 5.0. While in previous stages of industrial evolution, robots operated in isolation from humans and under regulated conditions, in Industry 5.0 they are called upon to collaborate with humans to achieve their assigned task [24]. The design of a collaborative robot may not differ from that of industrial robots. Nevertheless, it is necessary to be equipped with certain safety components that can improve the safety of a robot’s workspace [25]. Human–Robot Collaboration (HRC) systems have been developed to enable humans to work in the same environment as robots in shared workspaces. These systems are considered highly complicated and special attention needs to be paid during their design to ensure safety, injury prevention, and optimal process execution. Figure 1a shows in 2D a simplified workspace consisting of one robot and one human. As is evident, there is a common area between the robot and human workspaces that cannot be considered negligible and tasks need to be performed inside it. Figure 1b shows an expanded version of the example in Figure 1a, where multiple humans and robots collaborate in the same workspace. Although collaboration between robots and humans can bring several advantages, there are limited papers that have studied HRC systems. Research on HRC is an open scientific field and the need for more exhaustive studies is emphasized [26]. The majority of papers related to the interaction between humans and robots tend to focus primarily on safety concerns [27].

A vital issue in robotics is considered to be finding the optimal path. The path planning problem originated in the mid-1960s. The path problem can be described as the process of searching for a sequence of points from a starting to an ending position [28]. In any given environment, there may be several paths that satisfy this condition; however, the research interest here focuses on finding the optimal path. The difficulty of this challenge increases when a robot’s operating space is three-dimensional and there are several obstacles within it. Path planning methodologies can generally be distinguished into offline and online methods. In offline programming, a robot’s navigation environment is known in advance, while online, the environment is partially known or completely unknown [29]. The researchers have proposed several algorithms to find the optimal path in both 2D [30,31,32] and 3D [33,34,35].

Researchers have endeavored to incorporate the time variable into a 3D path and transform it to 4D. Liu et al. proposed an improved particle swarm optimization (PSO) algorithm for 3D path planning and a time coordination method to moderate the velocity of multiple UAVs in order to ensure that they arrive at the same destination simultaneously [36]. Tan et al. nominated a method for unmanned aircraft systems or multiple drone operations in a high-traffic urban airspace. Their method resolves the air conflicts by modifying the aircraft’s departure time [37]. Zhou et al. presented a real-time 4D trajectory planning method for aircraft in a dynamic airspace environment. Their method decomposes the multi-aircraft collision resolution problem into multiple pairwise aircraft collision resolution subproblems and uses the analytic geometry of hyperplane space to resolve collisions [38]. Ramesh et al. suggested a methodology for finding the optimal path in an environment with dynamic obstacles [39]. Rahmaniar and Rakhmania presented a genetic algorithm for finding the optimal path in a dynamic environment [40]. Papers that explore the path planning applications in a dynamic environment were included in the literature review due to their characteristics of changing over time. The research on methodologies for solving the path planning problem taking into account the four dimensions simultaneously is quite limited compared to that of 2D or 3D.

The aforementioned approaches individually address the problems of path planning, robot speed, environmental changes, and collision avoidance. The time parameter in [36,37] is used to adjust parameters for already planned paths without examining the impact of these parameters on optimal path formation. The approach in [38] is applied to already predefined trajectories in real time and focuses on collision avoidance and application to a large number of aircraft without considering whether the resulting solution is the global optimal. In [39,40] the change in obstacles is not directly correlated with the time variable. Thus, the planned path does not accurately simulate real conditions and may not be optimal. Existing methodologies deal with 4D path planning without taking into account all four dimensions simultaneously. In industrial environments, the path of robots is directly related to their economic and operational efficiency. A short path implies a reduction in production time and cost.

Collaborative robots are an important part of Industry 5.0. The workspaces of collaborative robots, as described in Figure 1, are complex environments with several tasks executing at the same time. The sequence, execution time, and duration of each task are determined in advance during the organization of the industrial production process. Several industrial tasks need to be performed at the same time in the shared areas highlighted in Figure 1b. Planning a robot’s path within a shared area when humans or other robots are moving at the same time is challenging. In order to ensure the increase in performance offered by Industry 5.0 and to fully exploit the new capabilities of robotic systems, the generated path needs to be the global optimal. These environments are directly related to the time variable required to more accurately determine the positions of humans or other robots. 2D and 3D path planning algorithms can be used to generate trajectories; however, they cannot provide assurance that the formed paths are optimal for performing tasks in the described conditions. Incorporating the time variable into the path formation process is of primary importance for accurately representing real-world conditions and identifying optimal solutions. The accuracy of simulating the real environment in digital form is an important issue in the field of robotics. Path planning algorithms that take into account the time parameter after path formation cannot accurately simulate the real environment, and the time parameter cannot influence the shape of the formed path. On the other hand, algorithms applied to dynamic environments in the case of online path planning form locally optimal solutions that are not the global optimal solutions in all cases. Algorithms for offline path formation in dynamic environments do not take into account the time factor accurately but generally reflect the changes that occur in the real world. These methodologies can generate solutions either individually or in combination; however, to find better quality solutions, it is necessary to form paths with methodologies that take into account the time factor simultaneously with the formation of the route.

In this paper, an offline 4D path planning methodology is presented to address the challenge of robot path planning in an area where robots and humans work at the same time. The methodology is applied in cases where the trajectories of humans and robots are known in advance. The suggested methodology integrates the time variable, a robot’s velocity, environmental changes, and collision avoidance into the path planning process. The proposed methodology takes into account all four dimensions simultaneously and finds the optimal path for a single robot in environments that change over time, such as those encountered in the collaboration of robots with humans or other robots. The numerical value of the robot’s movement speed in the real world varies. These changes can significantly affect the shape of the optimal path in a dynamic environment. The proposed algorithm takes into consideration changes in a robot’s speed value and the times at which they occur in order to ensure that the formed path can be executed to accomplish real world tasks without collisions or suboptimal path choices. The proposed offline path planning method is used in an environment where changes in space through dynamic obstacle definition and robot speed values are known in advance.

The presented methodology targets solving path planning cases for a single robot where finding the optimal path in 4D path planning was achieved by incorporating the time variable into a 3D path planning algorithm. The algorithm that was modified was the enhanced fish algorithm presented in [41], due to its ability to quickly generate complete paths without any interaction with the virtual environment. By incorporating the time variable, the algorithm can identify optimal solutions even in dynamically changing environments. The enhanced AFSA uses a 3D model of 24 possible movement points to navigate the fish within the grid environment and it is integrated with a ray casting algorithm. In order to incorporate the time variable, several modifications are presented. The objective function was redesigned to be more efficient in the conditions of a dynamic environment. The modifications include the introduction of a safety value of dynamic obstacles. The coefficients w1 and w4 for the importance of the distance of the discrete point’s location to the ending point and the safety value of dynamic obstacles were introduced. Moreover, a time calculation method based on the robot’s velocity and the agent’s movement in the grid environment is presented. The coefficients of obstacle variation and speed variation were presented in order to reduce computational errors and prevent errors occurring due to inaccurate representations of the physical world in the virtual environment. Finally, a methodology was presented to more accurately calculate the time between the agent’s movements from one given position to the next, taking into account cases where intermediate changes in the robot’s speed occur. The methodology for more accurate time calculation is used in the integration of the ray casting algorithm as well as in the use of thin grid environments to ensure the safety of the robot and its surroundings. All the proposals in this paper contribute to a more realistic representation of the real operating conditions of robots and the environments encountered in Industry 5.0 during the offline path planning process.

The main contributions of this paper can be summarized as follows:

• A methodology for 4D path planning is presented in which changes in a robot’s movement speed are taken into account;
• A time calculation methodology for an agent’s movement in a grid environment is presented;
• The ray casting algorithm was modified for use in a dynamic environment;
• The safety value of dynamic obstacles to the objective function is introduced.

2. Materials and Methods

2.1. 3D Environment Creation

Simulation of the real world is achieved by using the methodology of a 3D grid. The 3D grid method transforms the physical world into many 3D elementary parallelepipeds. Each elementary parallelepiped has specific dimensions and represents a part of the physical space. In order to decrease the computational resources required, each elementary parallelepiped is represented by a point located at its center of gravity. The idea of a 3D grid environment is further expanded to achieve the 4D path planning creation. Figure 2 shows an example of the 3D grid environment and the centers of gravity of the elementary parallelepipeds are marked in red.

2.2. Obstacles

In the real world, within a robot’s operating space, there may be certain obstacles or areas where it is necessary to restrict its movement in order to ensure the safety of the robot and the people or objects in its surroundings. It is important for the path planning algorithm to take these restrictions into account. This is achieved by the definition of the obstacles in virtual environments. The existence of obstacles ensures that the formed path will only pass through collision-free coordinates. Obstacles encountered in the real world can be divided into two categories: static and dynamic. Static obstacles remain at the same coordinates without changing their shape or size throughout the execution of the robot’s operations. Some examples of these obstacles in the real world are walls, machines, safety zones, and robots’ movement limitations. Dynamic obstacles are obstacles that change over time in terms of their location, shape, or even existence. Some examples of such obstacles from the real world could be other robots working in the same workspace, a person, or objects that are temporarily placed in a certain position and over time move to another location.

In the 3D grid environment, the robot can move only on the center of gravity of the parallelepipeds, which are called discrete points of movement. The concepts of static and dynamic obstacles are also transferred to the grid environment. Each discrete point is accompanied by a state in which, for each examined moment of time, it is determined whether it is considered as an obstacle or not. The existence or the absence of an obstacle is represented by the numbers 0 and 1, respectively. The discrete points that represent static obstacles are considered permanent as obstacles while the state of those considered dynamic can vary between obstacle and free. Equation (1) defines for N discrete points with coordinates x, y, and z at the examined time t the cases to be static obstacles or free. Equation (2) shows the case of a discrete point to be a dynamic obstacle. t₁ and t₂ are the starting and ending times of the appearance of the dynamic obstacle, respectively.

$${\mathrm{N}}_{\mathrm{x},\mathrm{y},\mathrm{z}}(\mathrm{t})=\left\{\begin{array}{c}0,\mathrm{t}\in \left[0,\left.+\infty \right) \mathrm{static}\right.\\ 1,\mathrm{t}\in \left[0,\left.+\infty \right) \mathrm{free}\right.\end{array}\right.,$$

(1)

$${\mathrm{N}}_{\mathrm{x},\mathrm{y},\mathrm{z}}(\mathrm{t})=\left\{\begin{array}{c}0, \mathrm{t}\in \left[{\mathrm{t}}_1,{\mathrm{t}}_2\right] \mathrm{dynamic}\\ 1, \mathrm{t}\in \left[0,\left.{\mathrm{t}}_1\right)\cup \right.({\mathrm{t}}_2,\left.+\mathrm{\infty }\right)) \mathrm{free}\end{array}\right.,$$

(2)

2.3. Robot Movement

Industrial robotic arms are easily maneuverable and can move in x, y, and z directions in a 3D environment. In path planning algorithms applied in a grid environment, researchers typically use the 8-possible movement points model for agent navigation. In this paper, the 3D navigation model of 24 possible movement points presented in [41] is used. This model was chosen because of the advantages in path formation presented in [41,42]. Figure 3 shows a 2D example with the differences between the 8- and 24-possible movement points models. The agent’s location is shown in green, the purple arrows indicate their possible movement points using the 8-possible movement points model and the black arrows indicate the additional points provided by the 24-points model.

2.4. Robot’s Velocity

The velocity of the robotic arm’s end effector is directly related to robotics applications. During the execution of industrial processes, it is necessary at various times of its operating cycle to change its speed. For example, when the robot transports objects, it needs to go slower in comparison to a case where it has placed the object at the final destination and approaches the receiving point to repeat the cycle. The arithmetical value of times when the robot’s speed needs to change in a process cycle can be determined. Furthermore, the numerical value of the speed may differ from robot to robot or when performing different processes. The numerical values of the speed and their duration intervals are determined during the design of its operating cycle. The changes in the speed of the robot’s end effector occur progressively through certain accelerations and decelerations [43]. The proposed methodology integrates the numerical values of the speed and their time intervals with the path planning optimization. The acceleration values can be taken into account through these physical quantities.

2.5. Collision Detection

The characterization of district points as free, static or dynamic obstacles is incomplete for the robot to navigate in the real world without the existence of collisions. Considering only the prohibition of movement at these points, a path that in the 3D grid will appear safe when transferred to the real world can lead to robot collisions. This can happen due to the fact that each district point assigned with specific coordinates and the prohibition does not allow the robot to move only at these coordinates. To avoid collisions, it is necessary to take into account the fact that the district point represents a spatial set of coordinates and to prohibit movement on them. In this way, the prohibition of movement does not only concern the spatial coordinates of the discrete point but the entire volume it represents. In Figure 1, the discrete points of the grid environment can be distinguished along with their represented volumes. The collision detection can be achieved by using the 6 planes that form the finite elementary parallelepipeds of the obstacle. Considering a point A (x₁, y₁, z₁) where the agent is located and a point B (x₂, y₂, z₂) where the agent will be transferred, the vector AB [x₂ − x₁, y₂ − y₁, z₂ − z₁] is obtained. In this way the collision detection problem is transformed to determining if the vectors intersects one of the 6 planes of the elementary parallelepiped’s from the considered obstacles. The intersection between the vector AB and a plane defined by an equation $\mathrm{a}·\mathrm{X}+\mathrm{b}·\mathrm{Y}+\mathrm{c}·\mathrm{Z}=\mathrm{d}$ can be found by substituting x, y, z resulting from Equations (3)–(5) to the plane equation to calculate the variable m. The coordinates of the intersection point [x, y, z] are obtained by substituting m in Equations (3)–(5).

$$\mathrm{x}={\mathrm{x}1}_{}+({\mathrm{x}2}_{}-{\mathrm{x}1}_{})\times \mathrm{m},$$

(3)

$$\mathrm{y}={\mathrm{y}}_1+({\mathrm{y}}_2-{\mathrm{y}}_1)\times \mathrm{m},$$

(4)

$$\mathrm{z}={\mathrm{z}}_1+({\mathrm{z}}_2-{\mathrm{z}}_1)\times \mathrm{m},$$

(5)

If the coordinates from the intersection point are located inside the obstacle area then a collision is detected and the transition from point A to B is prohibited. The process of collision detection checking is performed on all 6 planes of the parallelepipeds. If an intersection with a plane is detected, the agent’s movement from the point where it is located to the examined point is blocked.

2.6. Artificial Fish Swarm Algorithm

Traditional AFSA uses four main behaviors for finding the optimal solution, preying, swarming, following, and random behavior of fish. The behaviors are used for the selection of the next possible fish position.

In preying behavior, the fish selects the next possible point in its field of view randomly according to Equation (6). The indexes i and j are indicating the current and the next point of the fish, respectively, while X is the fish location and Y is the value of the objective function. After the next point, X_j, is selected, the objective function value of the selected position Y_j is compared to the value of the current position Y_i. If the value Y_j > Y_i, the artificial fish moves one step closer to the new position according to Equation (5). If the next possible position has a lower objective function value, the selection process is repeated until a position with a higher value is found or the maximum number of attempts (fishTryNum) is exceeded. If the number fishTryNum is exceeded, then random behavior is activated. The evaluation criteria for positions are defined by the objective function. In Equation (7), ||X_j − X_i|| = d_ij is the distance between the current location i and the next possible location j.

$${\mathrm{X}}_{\mathrm{j}}(\mathrm{t}+1)={\mathrm{X}}_{\mathrm{i}}(\mathrm{t})+\mathrm{Visual} \times \mathrm{Rand}(),$$

(6)

$${\mathrm{X}}_{\mathrm{i}} (\mathrm{t}+1)={\mathrm{X}}_{\mathrm{i}} (\mathrm{t})+{\displaystyle \frac{{\mathrm{X}}_{\mathrm{j}}\left(\mathrm{t}\right)-{\mathrm{X}}_{\mathrm{i}}\left(\mathrm{t}\right)}{‖{\mathrm{X}}_{\mathrm{j}}\left(\mathrm{t}\right)-{\mathrm{X}}_{\mathrm{i}}\left(\mathrm{t}\right)‖}} \times \mathrm{Step} \times \mathrm{Rand}(),$$

(7)

In swarm behavior the artificial fish selects the next point by evaluating the selections of other fish in its range field. The fish calculates the center position of the next point based on the distribution of other fish according to Equation (8). The center position is selected if its objective function value is higher than its current position and if it is not too crowded by other fish. In Equation (8), n_f is the number of fish in the examined area. Equation (9) is used to calculate the crowded degree, δ, of the investigated position, δ ∈ (0,1). If the center position meets the above criteria the swarm behavior is executed according to Equation (10). If the above conditions are not satisfied the fish performs the preying behavior.

$${\mathrm{X}}_{\mathrm{c}}={\sum }_{\mathrm{j}=1}^{{\mathrm{n}}_{\mathrm{f}}}{\displaystyle \frac{{\mathrm{X}}_{\mathrm{j}}}{{\mathrm{n}}_{\mathrm{f}}}},$$

(8)

$${\displaystyle \frac{{\mathrm{n}}_{\mathrm{f}}}{\mathrm{N}}}\le \mathsf{\delta },$$

(9)

$${\mathrm{X}}_{\mathrm{i}}(\mathrm{t}+1)={\mathrm{X}}_{\mathrm{i}}(\mathrm{t})+{\displaystyle \frac{{\mathrm{X}}_{\mathrm{C}}\left(\mathrm{t}\right)-{\mathrm{X}}_{\mathrm{i}}\left(\mathrm{t}\right)}{‖{\mathrm{X}}_{\mathrm{C}}\left(\mathrm{t}\right)-{\mathrm{X}}_{\mathrm{i}}\left(\mathrm{t}\right)‖}}\times \mathrm{S}\mathrm{t}\mathrm{e}\mathrm{p}\times \mathrm{R}\mathrm{a}\mathrm{n}\mathrm{d}(),$$

(10)

In the following behavior, the fish during the selection process of its next position takes into account the position of other fish. The objective function value of the optimal position is compared to the current position and if it is higher and not crowded the fish moves one step towards the criteria of Equation (11). If the above conditions are not satisfied the fish performs the preying behavior.

$${\mathrm{X}}_{\mathrm{i}}(\mathrm{t}+1)={\mathrm{X}}_{\mathrm{i}}(\mathrm{t})+{\displaystyle \frac{{\mathrm{X}}_{\mathrm{j}}\left(\mathrm{t}\right)-{\mathrm{X}}_{\mathrm{i}}\left(\mathrm{t}\right)}{‖{\mathrm{X}}_{\mathrm{j}}\left(\mathrm{t}\right)-{\mathrm{X}}_{\mathrm{i}}\left(\mathrm{t}\right)‖}} \times \mathrm{Step} \times \mathrm{Rand}(),$$

(11)

In the random behavior, the artificial fish moves one step towards a random location according to Equation (12). In the algorithm presented in this paper the random behavior was replaced according to the behavior proposed in [42].

$${\mathrm{X}}_{\mathrm{i}}(\mathrm{t}+1)={\mathrm{X}}_{\mathrm{i}}(\mathrm{t})+\mathrm{Step} \times \mathrm{Rand}(),$$

(12)

The objective function has a significant impact on the convergence of the algorithm and finding the optimal path. It determines the movement of fish in the virtual environment and evaluates their selections. The objective function used is similar to the one described in [41] enriched with some proposed terms as shown in Equation (13). The safety value factor in Equation (13) assists artificial fish in avoiding static obstacles and the total movement point factor in Equation (14) helps during the next possible point selection process [41]. The coefficients w1 and w4 are introduced in this paper. They represent the importance of the distance of the discrete point’s location from the ending point and the safety value of the dynamic obstacles. The coefficients w2 and w3 represent the importance of the safety value factor and the total movement point factor, respectively. The safety value and total movement point factor are calculated according to Equations (14) and (15), respectively, where k represents the total number of discrete points from the position where the fish is located, g the total number of obstacles from the same location, and w the number of discrete points eliminated due to the existence of obstacles during the process of generating the set of possible movement points. The safety value of dynamic obstacles is introduced as shown in Equation (16), where d is the total number of dynamic obstacles from the position of the fish. The safety value of dynamic obstacles assists the artificial fish to avoid the dynamic obstacles.

$${\mathrm{f}\left(\mathrm{x}, \mathrm{y}, \mathrm{z}\right)=\sqrt{{\left({\mathrm{x}}_{\mathrm{i}}-{\mathrm{x}}_{\mathrm{ending}}\right)}^2+{\left({\mathrm{y}}_{\mathrm{i}}-{\mathrm{y}}_{\mathrm{ending}}\right)}^2+{\left({\mathrm{z}}_{\mathrm{i}}-{\mathrm{z}}_{\mathrm{ending}}\right)}^2}}^{\mathrm{w}1}\times {\left({\displaystyle \frac{1}{\mathrm{S}\left({\mathrm{x}}_{\mathrm{i}},{\mathrm{y}}_{\mathrm{i}},{\mathrm{z}}_{\mathrm{i}}\right)}}\right)}^{\mathrm{w}2}\times {\left({\displaystyle \frac{1}{\mathrm{T}\left({\mathrm{x}}_{\mathrm{i}},{\mathrm{y}}_{\mathrm{i}},{\mathrm{z}}_{\mathrm{i}}\right)}}\right)}^{\mathrm{w}3}\times {\left({\displaystyle \frac{1}{\mathrm{D}\left({\mathrm{x}}_{\mathrm{i}},{\mathrm{y}}_{\mathrm{i}},{\mathrm{z}}_{\mathrm{i}}\right)}}\right)}^{\mathrm{w}4},$$

(13)

$$\mathrm{S}({\mathrm{x}}_{\mathrm{i}},{\mathrm{y}}_{\mathrm{i}},{\mathrm{z}}_{\mathrm{i}})={\displaystyle \frac{\mathrm{k}-\mathrm{g}}{\mathrm{k}}},$$

(14)

$$\mathrm{T}({\mathrm{x}}_{\mathrm{i}},{\mathrm{y}}_{\mathrm{i}},{\mathrm{z}}_{\mathrm{i}})={\displaystyle \frac{\mathrm{k}-\mathrm{w}}{\mathrm{k}}},$$

(15)

$$\mathrm{D}({\mathrm{x}}_{\mathrm{i}},{\mathrm{y}}_{\mathrm{i}},{\mathrm{z}}_{\mathrm{i}})={\displaystyle \frac{\mathrm{k}-\mathrm{d}}{\mathrm{k}}},$$

(16)

The methodologies of 3D simple and advanced elimination are also used with the 3D navigation heatmap model proposed in [41]. As well as the integration of the ray casting algorithm with the improved AFSA, it was appropriately modified to take the time variable into account. In the integration of the two algorithms the fish is equipped with an artificial laser beam that aims from the fish’s location to the ending point. If the laser beam is not intercepted by an obstacle, the fish moves to the ending point. The interception of the laser beam is detected using Equations (3)–(5) in the same way as collision detection.

In order to successfully operate the improved AFSA integrated with the above methodologies, some parameters need to be defined. The parameters used are the same as those presented in [41]. The parameters used are as follows:

• The number of maximum iterations—MaxIt;
• The maximum number of steps until a complete path formed—MaxStepsNum;
• The population size of the fish swarm—numFish;
• The fish swarm number of attempts to find the optimal location—fishTryNum;
• Crowding factor—δ;
• Heatmap parameters—obsHeatVal, heatIncrCoef, obsNearHeatVal, obsNearHeatVal2.

2.7. Time Calculation

The calculation of the current time starts from an initial value t₀ and is subsequently calculated using the magnitudes of the robot’s displacement and velocity. The calculation is performed every time the agent is transferred from one position in the grid to another. Equation (17) is used for the calculation of the current time. The indexes cu and tr are for the current time and distance traveled, respectively. Equation (18) is used to calculate the distance traveled and the indexes located, which next denote the point where the fish is located and the point where it moved, respectively. The change in time value, spatially, depends on the distance of the selected next point from the current one and the density of the grid.

$${{\mathrm{t}}_{\mathrm{cu}}=\mathrm{t}}_0+{\displaystyle \frac{{\mathrm{d}}_{\mathrm{tr}}}{{\mathrm{V}}_{\mathrm{cu}}}},$$

(17)

$${\mathrm{d}}_{\mathrm{t}\mathrm{r}}=\sqrt{{\left({\mathrm{x}}_{\mathrm{located}}-{\mathrm{x}}_{\mathrm{next}}\right)}^2+{\left({\mathrm{y}}_{\mathrm{located}}-{\mathrm{y}}_{\mathrm{next}}\right)}^2+{\left({\mathrm{z}}_{\mathrm{located}}-{\mathrm{z}}_{\mathrm{next}}\right)}^2},$$

(18)

Once it has determined the current time in the computational environment, the algorithm checks if there are any changes in the state of the dynamic obstacle or the robot’s velocity value. If there are changes the variables are updated accordingly. Due to the fact that the coordinates of the obstacle boundaries do not coincide with the coordinates of the discrete points and the changes in the current time variable occur in varying quantities and not continuously, the coefficients of obstacle variation and speed variation were introduced to simulate more accurately the real world. Equations (19) and (20) show the conditions under which the dynamic obstacles and the speed variable change, respectively. The variable t_DO is the time when the dynamic obstacle appears, t_V is the time when the change in the robot’s speed occurs, and q_DO and q_v are the coefficients of obstacle variation and speed variation, respectively. These factors represent occasions in which the amount of time required for the robot to move from one discrete point to the next in the real world is slightly longer than in the virtual environment. This may occur due to the robot’s larger size or changes in the numerical values of its accelerations.

$${\mathrm{t}}_{\mathrm{cu}}\ge {\mathrm{q}}_{\mathrm{DO}}\times {\mathrm{t}}_{\mathrm{DO}},$$

(19)

$${\mathrm{t}}_{\mathrm{c}\mathrm{u}}\ge {\mathrm{q}}_{\mathrm{V}}\times {\mathrm{t}}_{\mathrm{V}},$$

(20)

During the time change interval, changes may occur in the existence of dynamic obstacles or the robot’s speed. These changes are determined by considering a temporary variable of the current time. The temporary time variable represents how long the robot needs to travel the distance between the two points. It takes into account that different lengths of the path may be traveled at different numerical values of speed. In this way, the numerical value of the current time variable is calculated more accurately. A simplified example of the temporary time variable is shown in Figure 4.

Initially the total length of the distance (d_total) traveled from the position where the agent is located to the next is calculated. Then, the required total time to travel the d_total distance is calculated using Equation (21) and the numerical value of the speed for the current time. In Equation (21) t_total is the temporary time, d_total the total distance and V_ct is the velocity of the current time. Then, through Equation (22), the temporary time (t_temp) is calculated, where t_ct is the current time. If during this time there is any change in velocity then the numerical value of t_temp is recalculated using Equation (23). In Equation (23), t_ch is the time that the change in velocity will happen. From Equation (24), the distance that will be traveled at the velocity of the current time is calculated by the d_temp variable. Equation (25) is used to calculate the remaining distance d_rem. The process is repeated until the agent travels to the next point using the calculated d_rem as d_total, the new arithmetic value of velocity and the new value of the current time obtained by adding t_temp to the existing t_ct. In the example in Figure 4, there are 3 different distances d₁, d₂, and d₃ that were traveled with 3 different speeds V₁, V₂, and V₃.

$${\mathrm{t}}_{\mathrm{total}}={\displaystyle \frac{{\mathrm{d}}_{\mathrm{total}}}{{\mathrm{V}}_{\mathrm{ct}}}},$$

(21)

$${\mathrm{t}}_{\mathrm{t}\mathrm{e}\mathrm{m}\mathrm{p}}={\mathrm{t}}_{\mathrm{c}\mathrm{t}}+{\mathrm{t}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}},$$

(22)

$${\mathrm{t}}_{\mathrm{t}\mathrm{e}\mathrm{m}\mathrm{p}}={\mathrm{t}}_{\mathrm{c}\mathrm{h}}-{\mathrm{t}}_{\mathrm{c}\mathrm{t}},$$

(23)

$${\mathrm{d}}_{\mathrm{t}\mathrm{e}\mathrm{m}\mathrm{p}}={\mathrm{t}}_{\mathrm{t}\mathrm{e}\mathrm{m}\mathrm{p}}\times {\mathrm{V}}_{\mathrm{c}\mathrm{t}},$$

(24)

$${\mathrm{d}}_{\mathrm{r}\mathrm{e}\mathrm{m}}={\mathrm{d}}_{\mathrm{t}\mathrm{o}\mathrm{t}\mathrm{a}\mathrm{l}}-{\mathrm{d}}_{\mathrm{t}\mathrm{e}\mathrm{m}\mathrm{p}},$$

(25)

The adoption of obstacle variation and speed variation coefficients in environments where the changes in the current time are shorter than the changes in the time intervals of the velocity and the dynamic obstacles can significantly simplify the calculations. The characteristics of these environments may be the dense grid, the dimensions of the elementary parallelepiped are short, and the robot moves at low speeds. However, when the appropriate proportion between the changes in the current time variable and the time intervals of change in speed and dynamic obstacles is not maintained, a more accurate calculation method is presented. This method can be used in environments where the grid is thin, the dimensions of elementary parallelepipeds are large, and the robot may move fast.

The most accurate calculation is achieved by more precisely examining the changes in dynamic obstacles. When the fish is located at a point and among its next possible points there is a dynamic obstacle, its characterization as free or an obstacle depends on the state it will be in when the fish approaches it from the fish’s current location. The calculation of the intersection time is achieved using the temporary time variable and intermediate changes in the robot’s speed are also taken into account.

In the case of ray casting algorithm integration, it is first examined whether the laser beam is intersected by a static obstacle. If there is not an intersection and the laser has been successfully activated, then it is examined whether it hits any dynamic obstacle. If the laser does not hit any dynamic obstacle, it can be activated and the fish can move to the ending point. However, if a dynamic obstacle interferes in its path, then the required time to travel from the point where the fish is located to the point of intersection is calculated and the state of the obstacle at the time of the collision is verified. If the laser beam hits more than one dynamic obstacle, the same process is repeated for all the contact points. Laser activation is considered successful when all dynamic obstacles hit by the laser are considered free points at the time of intersection. It is particularly important to accurately determine the time of the collision between the agent and the dynamic obstacle. Thus, regardless of the grid’s density, the temporary time variable is used to calculate more accurately the collision time.

2.8. Time Movement Policy

During the execution of the agent’s navigation process when the agent is located near a dynamic obstacle it can either stop at its current location and wait until the obstacle disappears or it can choose among the other possible solutions. The algorithm presented does not consider the possibility of immobilizing the robot at a certain position, waiting for the possibility that the disappearance of the dynamic obstacle will lead to a more optimal solution. In this study, the possibility that an optimal solution may arise from immobilizing the robot at some point is taken into account through the coefficient of obstacle variation. Changes in the time of existence of obstacles are equivalent to examining the path formation as if there had been a short wait by the agent at the point where it is located.

2.9. Flow Charts

Figure 5 presents the flow chart of the time calculation process.

Figure 6 shows the flow chart of the proposed method.

3. Results

Numerous experiments were conducted to investigate the effectiveness and performance of the proposed methodology. The proposed methodology was evaluated on dense and thin grid environments with both static and dynamic obstacles. In addition, cases with different robot velocities and time intervals for each velocity value were examined. The proposed algorithm was implemented in a real-world scenario and an industrial robotic arm with six degrees of freedom was used to execute the generated path.

The values of the parameters that were used in the conducted experiments are the following: MaxIt = 90; MaxStepsNum = 23; numFish = 140; fishTryNum = 12; δ = 0.3; obsHeatVal = 170; heatIncrCoef = 0.4; obsNearHeatVal = 90; obsNearHeatVal2 = 40; w1 = 3; w2 = 1.3; w3 = 0.7; w4= 1.4; q_DO = 0.08; and q_V = 0.03. The way the heatmap navigation model was used is the same as described in [41].

Figure 7a,b represents two different thin grid environments. The environments consist of 108 discrete points, the dimension of each 3D elementary parallelepiped is $133.3\times 200\times 133.3$ and four dynamic obstacles. There are eight and twelve static obstacles in Figure 7a and Figure 7b respectively. Green it represents the starting path point, blue the ending point, red the free points, black the static obstacle, purple the dynamic obstacles, and magenta the final path. Figure 8a,b shows the diagrams with the lengths of the paths formed during the algorithm execution for Figure 7a and Figure 7b respectively.

Table 1. Dynamic obstacle details for the environments of Figure 7a,b.

Dynamic Obstacle	Starting Time (s)	Ending Time (s) Figure 7a	Ending Time (s) Figure 7b
50	1	15	15
51	1	25	25
52	1	14	9
53	1	8	3

contains the required details for the definition of dynamic obstacles. While the discrete points of obstacles are the same, the duration of obstacles 52 and 53 changes. The speed values for both cases are speedValue = {90,85,120} mm/s and their changing times are speedChange = {6,15} s.

In this scenario the static obstacle can be a manufactured component and the dynamic can represent the movement of a human’s hand while working in this position for a set of time frame during the assembling process. Between the two environments there is a difference in the number and form of static obstacles. In the environment of Figure 7b, some additional static obstacles were added so that, in combination with the change in the duration of the dynamic obstacles, the behavior of the proposed methodology can be demonstrated in a different environment where the optimal path is forced to pass through the dynamic obstacles that disappear. It is noted that in the environment of Figure 7b the path that connects the starting to ending point necessarily passes only over the static obstacles, while in the environment of Figure 7a it can also pass around them. In the environment of Figure 7a the dynamic obstacles do not substantially affect the course and form of the optimal path; however, they affect the examination of alternative paths during the algorithm’s execution. During the algorithm’s execution, when the agent is located at point 35, by following the optimal path, the numerical value of the time variable is current time = 7.37 s. In environment Figure 7a, all dynamic obstacles at that time are considered as obstacles; while in environment Figure 7b, point 53 is considered as free. In the example in Figure 7a, the time interval between the current time and the time when point 53 will be considered as free is short. Thus, suboptimal path generation from the starting point to point 35 may lead to a path shorter in length because point 53 will be considered free. In this example, this possibility does not arise because the existence of dynamic obstacles does not affect the length of the optimal path.

Table 2. Results of the proposed methodology for the environments depicted in Figure 7a,b.

Environment	Max. Length (mm)	Min. Length (mm)	Average (mm)	Time (s)
Figure 7a	2215.5	1377.5	1690.9	15.70
Figure 7b	10,480.0	1527.1	3734.8	12.13

presents the results of executing the proposed methodology in the two different environments. The path of environment Figure 7a is shorter than that of the environment in Figure 7b, indicating that choosing a path where the robot moves in the xy plane in this particular environment is shorter than when the formed path also changes in the z-axis. The algorithm’s execution time in the environment in Figure 7a is longer than in Figure 7b, which implies that the 3D paths were also examined. This is also seen by comparing the diagrams in Figure 8a,b. In the case of the environment in Figure 7b, due to the extension of the static obstacles, a 3D path must necessarily be formed to connect the starting and ending points. Figure 8a shows that the lengths of these paths are greater than 1500 mm, with the values of the sub-optimal paths ranging from 1700 to 2400 mm. These path values are also found in Figure 8a. Another reason why the algorithm runs faster in environment 2 is the existence of more obstacles. Due to the fact that the 3D model of 24 possible movement points contains many choices of the next movement point selection, bio-inspired algorithms such as AFSA face the difficulty of choosing the next movement point. The presence of more obstacles reduces the number of possible movement points and facilitates the selection process [42]. It is also observed that for the case of the environment in Figure 7b, the average and maximum lengths of the formed paths are much greater compared to the environment of Figure 7a. However, as can be seen from Figure 8b, the formed paths with great length are few. This indicates that the operating environment is well explored and several cases have been examined before determining the optimal path. Moreover, the paths with great length shown in Figure 8b show that the non-optimal path formation can also cause changes in the numerical values of the robot’s speed and the current time variable increases. Therefore, if the access to the ending point is restricted by dynamic obstacles, the path length increases until the ending point is successfully reached. These cases can be limited by reducing the numerical value of the parameter MaxStepsNum. These errors do not affect the formation of the next path. This is demonstrated in Figure 8b by the numerical values of the paths formed after the paths with great length. Their numerical values are shorter and independent of the previous path. Their independence is highlighted as well as by the general attribute of the AFSA not to exchange information with the environment based on previous paths.

Figure 9a,b represents a dense grid environment with 500 discrete points, 56 static, and 21 dynamic obstacles. The dimension of each 3D elementary parallelepiped is $60\times 150\times 100$. Figure 10a,b shows the diagrams with the lengths of the paths formed during the algorithm execution for Figure 9a and Figure 9b, respectively. Figure 11 shows the necessary information to define dynamic obstacles. The dynamic and static obstacles of both environments are the same. The algorithm’s response and final results are tested in these environments by changing the robot’s speed values. In the environment shown in Figure 9a, the values of the robot’s speed are speedValue = {50,70,40} mm/s and in the environment shown in Figure 9b, the values are speedValue = {28,70,40} mm/s. The speed value changes are happening at the same time for both environments—speedChange = {26,34} s.

In this scenario, the dynamic obstacles can represent the space occupied by a robotic arm when transferring an object, the static obstacles on the environment corners represent some machines in the production area while those in the middle are some pipes. In these examples the dynamic obstacles significantly affect the course of the optimal path. If, at the time the agent approaches their location, they are considered as free, then the robot can pass through them and form a shorter path. If they are considered as obstacles, the optimal collision-free path changes considerably. This further highlights the effect of accurately calculating the time as well as the coefficients of obstacle variation and speed variation.

During the execution of the algorithm, when the agent is located at point 155 following the optimal path that connects it to the starting point, the numerical value of the time variable is current time = 10.3 s and current time = 18.4 s in the cases of Figure 9a and Figure 9b, respectively. It is also noted that due to the difference in speed values between the two cases, the time it takes the fish to overcome the dynamic obstacles when it is located at point 135 is shorter in the case of the environment in Figure 9a compared to Figure 9b.

Table 3. Results of the proposed methodology for the environments depicted in Figure 9a,b.

Environment	Max. Length (mm)	Min. Length (mm)	Average (mm)	Time (s)
Figure 9a	3380.4	1419.0	1959.6	25.42
Figure 9b	3965.2	1530.1	2098.3	33.72

contains the results of executing the proposed methodologies in the environments in Figure 9a,b. While parameters such as grid density, the definition of the dynamic obstacle behavior, the positions of the starting point, the ending point, and the static and dynamic obstacles are common to both environments, the changes to the speedValue parameters can significantly affect the execution time of the algorithm. The additional time needed to execute the algorithm in the environment in Figure 9b is attributed to the increase in the repeatability of the calculations for determining the time and the slower approach to the ending point, due to the necessity of changing the directions of the path more. The path formed in the environment in Figure 9a has one deflection point, while the path in the environment in Figure 9b has two. The second explanation is also confirmed by the diagrams in Figure 10a,b where the lengths of the paths formed are longer than in the Figure 9b environment. The average path length and the maximum length of the paths formed as shown in Table 3 are also greater for the environment in Figure 9b. This is attributed to the fact that for the environment in Figure 9b, the formation of a 3D path is mandatory to approach the ending point.

A direct comparison of the proposed methodology with any existing 4D path planning methodology is unfeasible. As mentioned in the introduction, existing 4D path planning methodologies deal separately with the 3D path planning and use the time variable to adjust some parameters, such as the robot speed or the start time, to avoid collisions in existing paths [36,37]. The present methodology incorporates the time variable into the path planning process and identifies the optimal paths by taking into account the impact of time-dependent alterations. In contrast to the existing techniques, the proposed methodology embodies in the optimal path formation process additional parameters including the robot’s speed, changes in the robot’s speed, and time-dependent changes in the environment’s morphology. Finally, the differences in the results do not allow for constructive comparison between the methodologies. State of the art methodologies use the concept of time to adjust certain parameters related to the effect of time on path execution. However, the proposed methodology focuses on finding the optimal route by more effectively simulating real-world conditions through the concept of time.

An experimental set-up was utilized to implement the presented approach in real-world processes. The experimental setup includes an industrial robotic arm, a mechanism that can reciprocate a dynamic obstacle and multiple static obstacles. The reciprocating motion of the mechanism can simulate the motion that a smaller robotic arm would perform in Industry 5.0 for accomplishing a task. The reciprocating mechanism is situated within the operational environment of the industrial robot and executes a predetermined movement. The objective of the robotic arm is to transfer an object from a starting to an ending location without causing any collision along the traveled path. The experimental setting exemplifies the circumstances and requirements encountered in Industry 5.0, establishing a suitable environment for the safe investigation of the suggested approach, mitigating the risks of damaging costly equipment, inflicting injuries, or impacting facilities. The experimental set-up aims to generate environments of similar specifications to those depicted in Figure 1b. Figure 12 shows the experimental set-up with the industrial robotic arm and static and dynamic obstacles. Figure 12b–f shows the location of the robot’s end effector during the path execution. The numbers and gray points shown in Figure 12b indicate the extreme positions that the dynamic obstacle can move to. When the robot is close to the static obstacle as depicted in Figure 12c, the course of the dynamic obstacle is from position 1 to position 2. On the contrary, when the robot is in the position depicted in Figure 12d, the dynamic obstacle moves from position 2 to position 1. Figure 12g shows the 3D model of the robot’s operating environment and the resulting path. The numbers of static obstacles, dynamic obstacles, and free movement points of the grid environment are greater than the previous examples. Figure 12h contains information about dynamic obstacles and the reciprocating movement simulation. The same points are met as dynamic obstacles at different times.

4. Discussion

Path planning is a fundamental problem in robotics. Solving the problem without taking into account the time variable can lead to superficially optimal solutions. These solutions can be easily proved suboptimal when humans or robots are collaborating simultaneously in the robot’s operating space. The time variable plays a decisive role in finding the optimal path in environments that change over time. The application of 3D methods in such environments requires specially designed conditions so that the formed path can be considered optimal. In the robot’s workspace in Industry 5.0 applications, the type of dynamic obstacles analyzed in this paper are met extensively. The proposed methodology can effectively incorporate the time variable for the design of a 4D path that takes into account known dynamic and static obstacles.

The 4D path planning was accomplished by incorporating the time variable into the path creation procedure of a 3D path planning algorithm. The 3D path planning algorithm used was an improved AFSA. The improved AFSA used a 3D model of 24 possible movement points for the agent’s navigation in a grid environment and was integrated with a ray casting algorithm. The proposed modifications were related to the improved AFSA algorithm without the integration of the ray casting algorithm so that they could be adopted in other bio-inspired algorithms, as well as to the integration of the ray casting algorithm.

Changes in the robot’s speed are taken into account in the proposed methodology. The changes in robot’s speed help to more realistically simulate real-world conditions, create more accurate paths, calculate more accurately the current time and avoid collisions with dynamic obstacles. The coefficients w1 and w4 along with the safety value of dynamic obstacles lead to a more comprehensive evaluation of the possible movement points, improving the navigation of the fish and contributing to faster convergence of the algorithm.

The coefficients of obstacle variation and speed variation contribute to the reductions in the calculations without impairing the reliability of the final solutions. They prevent errors that arise from inaccuracies in the simulation of the real world in the grid environment, such as in cases where the robot takes a little more or less time to safely move from one discrete point to the next due to its larger or smaller size, or changes in the values of its accelerations or decelerations. The coefficient of obstacle variation also benefits from examining cases where a short stop at a certain point could lead to a solution with improved performance compared to the continuous transfer of artificial fish to a nearby possible movement point.

The methodology presented for more accurately calculating the time it takes the robot to move from one point to the next, taking into account changes in speed, leads to a reduction in errors and the formation of higher quality paths. This prevents the risk of collisions in the real workplace from inaccuracy errors in time calculations and the adoption of suboptimal paths.

The proposed 4D path planning methodology can successfully solve offline programming path planning problems in a known 3D environment. A potential extension of this work might be a 4D solution to path planning for multiple robots in the same environment and the study of an appropriate methodology to define dynamic obstacles based on the different sizes and movements of the multiple robots in the environment.