Parallel Multiobjective Multiverse Optimizer for Path Planning of Unmanned Aerial Vehicles in a Dynamic Environment with Moving Obstacles

Abstract

Path planning with collision avoidance for unmanned aerial vehicles (UAVs) in environments with moving obstacles is a complex process of navigation, often considered a hard optimization problem. Ordinary resolution algorithms may fail to provide flyable and collision-free paths under the time-consumption constraints required by the dynamic 3D environment. In this paper, a new parallel multiobjective multiverse optimizer (PMOMVO) is proposed and successfully applied to deal with the increased computation time of the UAV path planning problem in dynamic 3D environments. Collision constraints with moving obstacles and narrow pass zones were established based on a mathematical characterization of any intersection with lines connecting two consecutive drones’ positions. For the implementation, a multicore central processing unit (CPU) architecture was proposed according to the concept of master–slave processing parallelization. Each subswarm of the entire PMOMVO population was granted to a corresponding slave, and representative solutions were selected and shared with the master core. Slaves sent their local Pareto fronts to the CPU core representing the master that merged the received set of nondominated solutions and built a global Pareto front. Demonstrative results and nonparametric ANOVA statistical analyses were carried out to show the effectiveness and superiority of the proposed PMOMVO algorithm compared to other homologous, multiobjective metaheuristics.

1. Introduction

Nowadays, unmanned aerial vehicles (UAVs) have gained great potential in several applications involving military and civilian fields due to their advantages in terms of weight, size, mobility, and price [1,2]. In robotics, path planning is the part of navigation that deals with collision avoidance of unmanned vehicles [3,4]. It is the most important issue in a vehicle’s navigation that aims to find the shortest and collision-free path from the current robot’s location to a given target. In many real-world navigation missions, a UAV’s flight environment can be dynamic with obstacles and threats that change over time. In these hard conditions, the collision-free path planning task becomes more and more difficult and expensive, leading to a great challenge in the UAV’s control and autonomous navigation framework [5,6,7,8]. In dynamic environments, a UAV drone operates regarding the location and motion speed of the moving obstacles. The drone must respect these dynamic constraints and react quickly to these types of changes. Therefore, planning algorithms used in this formalism must present high performance mainly in terms of execution time and processing speediness.

In the literature, several research works on UAV path planning problems have been proposed and continue to be developed and extended. In this framework, the aim of the path planning techniques is not only to find the shortest flyable path but also to ensure collision-free flight capability in complex environments. From this observation, the collision avoidance issue remains the main objective in any path planning task. Various methods and techniques have been investigated for collision avoidance in aviation. In [9], a dynamic collision avoidance zone modeling approach was proposed based on the emergency collision avoidance trajectory of UAVs in a 3D environment. In such a collision avoidance strategy, a virtual cylinder static protection zone was constructed for flight conflict detection and near-midair collision with intruders. In [10], the authors investigated the problem of cognitive UAV collision avoidance based on impulse differential inclusions theory. Simulation results were carried out to verify the stability of the computed collision avoidance region. In [11], a new collision and detection avoidance (CDA) algorithm was proposed to solve conflicts in flight scenarios with the simultaneous presence of aircrafts and other path constraints, such as no-fly zones, bad weather areas, hard terrain with geofencing limitations, and so on. The elaborated CDA approach leads to the detection of hazardous situations and the generation of optimal maneuvers that avoid potential collisions without causing secondary conflicts. Real-time simulations were made to show the effectiveness of the collision avoidance algorithm in challenging conflict scenarios. In [12], a modified artificial potential field (MAPF) approach was addressed for UAV collision avoidance in a dynamic 3D space. To increase the reliability of collision avoidance, a cylindrical region was assigned to each obstacle as an uncertain zone, and the planned path was checked with updated forces of MAPF while being executed by the UAV. The MAPF algorithm was designed and tested experimentally in a constraint reference frame to decouple the decomposed forces with specific physical constraints. A corrected path was then created if the forces disagreed with the physical constraints. In [13], a real-time collision avoidance strategy based on an essential visibility graph (EVG) algorithm was developed and applied for multiple unmanned aircrafts in environments with obstacles and no-fly zones. By updating the proposed EVG over the prediction time interval, a replanning procedure was carried out regarding the available multiple flying vehicles and movable obstacles. Another popular class of path planning with collision avoidance methods based on rapidly exploring random tree (RRT) techniques has been extensively used. In [14], an extended RRT algorithm was deployed for fixed-wing UAVs flying in a complex 3D environment with the avoidance of multiple obstacles. In [15], a dynamic variant of the RRT algorithm was applied to fixed-wing UAVs in a dynamic 3D environment. The RRT structure was expanded by considering the constraint equations of the vehicle’s dynamic constraints. The collision avoidance was guaranteed by considering the locations of the dynamic obstacles at the time of each step and interpolation based on the B-spline method. In [16], an improved RRT-based cooperative path planning algorithm was designed to deal with the real-time collision avoidance problem of multiple UAVs in the presence of unknown popup obstacles and teammate vehicles.

Taken as a hard optimization problem with contradictory objectives and constraints, the path planning problem with collision avoidance challenges requires many other capabilities of resolution algorithms in terms of search diversity, escape of local minima, nonpremature convergence, and computation time fastness to deal with the complexity of dynamic navigation environments. In recent years, interesting metaheuristics approaches have been proposed to solve a wide range of multicriteria path planning problems for UAVs in static and dynamic environments [17]. In [18], an algorithm based on both ant colony optimization and artificial potential field technique was proposed to solve UAV path planning problems in an environment with dynamic threats. The authors in [19] used an improved pigeon-inspired optimization algorithm for UAV path planning in a dynamic oilfield 3D environment. In [20], a novel optimization-based planner inspired by the concept of predator-prey pigeons was investigated to find the optimal trajectory for an unmanned combat aerial vehicle in a dynamic environment. In [21], an improved artificial bee colony algorithm was proposed for path planning problems in a complex dynamic environment. Although these works were developed to solve multiobjective path planning problems, they used weighted sum functions that converted the multiobjective problem into a single objective one [22]. In this case, it is difficult to determine the best weights for different conflicting goals. In order to overcome these limitations, multiobjective metaheuristics have been proposed to address multiple conflicting objectives simultaneously. In [23], a multiobjective particle swarm optimizer was applied for path planning in an uncertain environment. In [24,25,26], the authors proposed an interesting UAV collision-free path planning approach based on the multiverse multiobjective optimization concept. The comparisons carried out with other homologous algorithms have shown the effectiveness and superiority of the proposed planner in 3D environments with moving obstacles and threats. Other bioinspired metaheuristics algorithms have been used to handle the path planning problems of UAVs in dynamic environments and can be found in [27,28,29,30,31].

In a dynamic environment, the locations of moving obstacles change over time and the drone should be able to react quickly to these environmental changes. Therefore, the execution time of any path planning algorithm plays an important role in ensuring the feasibility of a real-world UAV’s navigation mission. Thus, there is a need to improve the classical optimization-based path planning methods by either modifying the search mechanism of algorithms or using advanced computational technologies to better solve these hard and large-scale planning problems. Recently, the parallelization of algorithms has been extensively developed and presents a promising and encouraging solution in this regard [32]. Architectures using graphics processing units (GPUs) and central processing units (CPUs) are the most extensively proposed parallelization techniques. With the growing development of computer technology, microprocessors currently used in almost all embedded computers have become multicore, providing the essential hardware bases for the implementation of parallel algorithms [33]. Nowadays, the computing technology of multicore CPU has enjoyed great success in the fields of scientific research and engineering practices and has become an active research area. Parallel computing in multicore CPUs is done by dividing a large computational task into several reduced subtasks that are executed simultaneously in the different cores. This parallelization technique can speed up computation time and further improve the efficiency of using hardware resources [34]. In our previous work [32], a parallel cooperative co-evolutionary grey wolf optimizer was proposed to solve the path planning problem using a multicore CPU architecture but in a static environment with single objective optimization formulation. Other works dealing with the parallelization of some classical and old metaheuristics algorithms can be found in the literature.

So far, no remarkable research work on the parallelization of recent and multiobjective metaheuristics algorithms for the dynamic path planning problems of UAVs is worth mentioning. In this paper, a new parallel multiobjective multiverse optimization (PMOMVO) algorithm is proposed and successfully applied to solve the path planning problem of UAVs flying in hard 3D environments with moving obstacles and threats. By using a multicore CPU architecture, a master–slave parallelization was implemented to lead to efficient and more suitable PMOMVO algorithms for the considered path planning problem compared with the normal MOMVO one. A modified technique for order of preference by similarity to ideal solution (TOPSIS) was used to select the best compromise solutions among all the nondominated ones in the sense of Pareto [35]. The research mechanism behind the MOMVO’s parallelization is explained and tested over a large benchmark of hard planning scenarios. To assess the usefulness of the proposed planning approach, several comparisons and statistical analyses were carried out and discussed. The main contributions of this work are summarized as follows: (1) an efficient path planning strategy is proposed to find flyable and collision-free paths in dynamic environments with moving obstacles and narrow pass zones. (2) A new parallel variant of the multiobjective multiverse optimizer based on a multicore CPU master–slave architecture is proposed and successfully applied for such a multicriteria path planning problem. (3) Nonparametric statistical analyses were carried out to show the efficiency and superiority of the proposed PMOMVO algorithm for path planning problems with moving obstacles.

The remainder of this paper is organized as follows. In Section 2, a bibliographic survey of the main methods and works on UAV path planning and collision avoidance in dynamic environments is summarized. In Section 3, the UAV path planning problem in a dynamic 3D environment with moving obstacles is formulated as a multiobjective optimization problem under operational dynamic constraints. Section 4 presents the main components of the proposed parallel processing multiobjective multiverse algorithm based on a master–slave multicore CPU architecture. In Section 5, demonstrative results, comparisons, and nonparametric statistical analyses are carried out to show the superiority and effectiveness of the proposed PMOMVO-based dynamic path planning approach. Section 6 concludes this paper.

2. Related Work

The path planning problem with collision avoidance in dynamic environments has attracted growing attention since the end of the last century. As a hard issue in UAV navigation, the main idea is to find flyable paths that can drive a vehicle safely from a start station to a target destination without colliding with moving obstacles. In the literature, many approaches using various kinds of algorithms have been proposed to tackle such a problem. Because of this broad range, many researchers have proposed different classification models to more easily identify trends and research directions. In this study, a chronological classification of the main and recent path planning methods is addressed in a horizon that ends in 2022. As summarized in

Table 1. Summary of the main UAV path planning approaches in dynamic environments.

Classes	Year	Literature	Description of the Main Topic and Used Technique
Sampling-based algorithms	2014	[36]	A Voronoï-diagram-based algorithm proposed for UAV path planning in a dynamic environment.
	2015	[37]	An Artificial Potential Field (APF) algorithm using the attractive potential field to achieve goals and the repulsive potential field to avoid both static and dynamic obstacles.
	2016	[14]	A Rapidly exploring Random Tree (RRT) algorithm employed for fixed-wing UAV path planning in a dynamic environment.
	2016	[15]	A Dynamic RRT (DRRT) technique applied to solve fixed-wing UAV path planning problems in dynamic 3D environments.
	2017	[38]	An Improved APF (IAPF)-based method proposed to solve UAV trajectory planning problems in a dynamic environment.
	2018	[16]	An Improved RRT (IRRT)-based technique investigated to generate paths for multiple UAVs in real-time scenarios with the presence of unknown pop-up obstacles.
Node-based algorithms	2014	[39]	An A-star (A*) algorithm used to obtain the shortest feasible flying path for a UAV.
	2017	[40]	A D-star (D*) algorithm proposed to find the shortest path for mobile robots in highly dynamic environments.
	2019	[41]	A variant D* Lite of D* algorithm applied to solve the 3D path planning problem for quadrotor-types of UAVs.
	2022	[42]	A Matrix Alignment Dijkstra (MAD)-based technique proposed to make drones safely move in dynamic environments.
Mathematical-based algorithms	2019	[43]	A planning strategy based on a nonlinear control system using an optimization-based avoidance strategy with account of sensor characteristics and an emergency evaluation policy.
Machine learning-based algorithms	2019	[44]	A Reinforcement Learning (RL)-based technique proposed to solve the UAV path planning problem based on global situation information.
	2021	[45]	A Deep Reinforcement Learning (DRL) approach investigated path planning problems in complex and dynamic environments using local information and relative distance.
	2021	[46]	A Multilayer Reinforcement Learning (MRL) algorithm using a neural network with two layers proposed to collect both global and local information for navigation in dynamic environments.
Metaheuristics-based algorithms	2021	[24,25,26]	A Multicriteria Multiverse Optimizer (MOMVO) associated with the TOPSIS technique proposed for path planning problems in a 3D environment with moving obstacles and threats.
	2022	[27]	An Improved Particle Swarm Optimization (IPSO) and Arithmetic Optimization Algorithm (AOA)-based hybrid planner for the path planning of multiple robots in a dynamic environment.
	2022	[28]	A Resilient UAV Path Optimization Algorithm (RUPOA) as a new planning algorithm proposed to achieve safe path planning for UAVs under uncertain dynamic environments.
	2022	[29]	An Evolutionary PSO (EPSO) algorithm used to solve the drone path planning problem in a dynamic environment.
	2022	[30]	An Improved PSO (IPSO)-based algorithm employed for a visual-SLAM-based planning strategy to optimize the paths for multi-UAVs in a dynamic environment.

, these approaches are grouped under five main classes, namely the sampling-, node-, mathematical-, machine-learning- and bioinspired metaheuristics-based algorithms. As shown from the related literature, the sampling-based algorithms are built around a required prerequisite of the knowledge of the working environment. In this case, collision-free environment information can be sampled and interpreted by a planning algorithm. The node-based techniques are used for finding flyable routes in a graph structure. In these algorithms, the predefined nature of the graph influences the applicability for unknown dynamic scenarios. For the mathematical-model-based approaches, the main idea consists of the resolution of the path planning task by a reformulation as a typical mixed-integer linear programming problem. Machine learning, i.e., reinforcement and deep learning, and bioinspired metaheuristics-based, techniques are the most extensively used approaches in recent decades. These approaches based on artificial intelligence concepts mimic the cognitive behaviors of neural networks, swarm intelligence, and biology.

3. Path Planning Problem Formulation

3.1. Dynamic Environment Modeling

In robotics, the path planning procedure consists of guiding the unmanned vehicle from a starting point $\mathit{S}:\left(x_1,y_1,z_1\right)$ to a destination point $\mathit{T}:\left(x_n,y_n,z_n\right)$ in the coordinate system XOY, as shown in Figure 1. The X-axis range that connects these two points is divided into $\left(n-1\right)\in \mathbb{N}$ equal subsegments denoted as $\left[L_i,L_{i+1}\right],i=1,2,\dots ,n-1$. To carry out this navigation mission, two key tasks must be determined, namely the 3D environment modeling (waypoints, obstacles, flyable paths, etc.) and the planning problem often formulated as a constrained multiobjective optimization problem [24,25,26,32].

In this work and without loss of generality, a moving obstacle is modeled as a sphere or ball of radius $r_j\in {ℝ}_{+}$ and center ${\mathit{C}}^j:\left(x_c^j,y_c^j,z_c^j\right)$, as shown in Figure 2. Indeed, the idea behind the use of spherical shapes of threats aims to provide a more generalized collision-free navigation for a wide range of threat forms, for example, missiles, teammate drones in the case of multi-UAV swarms, raptors attacking the flying objects, and so on. In geometry, cubes, pyramids, and even more generally polygonal forms can be approximated as spheres (or as circles in 2D environments) circumscribed to various shapes of rigid objects. They are circumscribed spheres containing the polygonal object where all its vertices are still inside of the ball. The dynamic constraints of traversing areas in spherical shapes remain virtually the same. Moreover, several literature works on the 3D path planning of UAVs continue to use spherical shapes for a wide range of 3D obstacles and threats as well as circular forms in the case of planning in 2D environments. Thus, whatever the shape of the object that is considered an obstacle, the collision avoidance constraints manipulate analytical and nonlinear constraints analogous to those later developed in this same section.

On the other hand, a UAV drone as a rigid body moving in a 3D space is modeled thanks to the well-known Newton–Euler approach that describes the motion dynamics in the body-fixed coordinate frame as follows [26,31,32]:

$$m_{uav}{\ddot{\mathit{X}}}_{uav}=m_{uav}{\dot{\mathit{V}}}_{uav}={\mathit{F}}_{uav}$$

(1)

where ${\mathit{X}}_{uav}=\left(x_{uav},y_{uav},z_{uav}\right)$ and ${\mathit{V}}_{uav}=\left(v_{uav},u_{uav},{\mathit{w}}_{uav}\right)$ denote the position and linear velocity of the drone for the translational motion, respectively, $m_{uav}$ is the mass of the vehicle, and ${\mathit{F}}_{uav}$ is the sum of external forces. These motion equations are sampled according to the adequate time step $\Delta t$.

3.2. Multiobjective Optimization Problem Formulation

Considering that the actual position of the aerial vehicle is ${\mathit{w}}_i=\left(x_i,y_i,z_i\right)$ and the time $t\ge 0$ is incremented by one unit, the drone must move to the next position ${\mathit{w}}_{i+1}=\left(x_{i+1},y_{i+1},z_{i+1}\right)$ where the coordinate ${\left\{x_{i+1}\right\}}_{1\le i\le n-1}$ is selected and the waypoint coordinates $\mathit{W}={\left\{y_{i+1},z_{i+1}\right\}}_{1\le i\le n-1}$ are considered as the decision variables for the path-planning-based optimization problem. The aim of the path planning problem in a dynamic 3D environment is to calculate the vehicle’s next positions while meeting certain objectives and avoiding collision with moving obstacles. Starting from its current position and heading towards its destination, the UAV drone is placed at the time $t\ge 0$ in the spatial coordinate ${\mathit{w}}_i=\left(x_i,y_i,z_i\right)$. As the processing time is incremented $\Delta t$, the quadrotor would pass to the next position ${\mathit{w}}_{i+1}=\left(x_{i+1},y_{i+1},z_{i+1}\right)$ at the time $\left(t+\Delta t\right)$. The length of the flyable, collision-free path is therefore an essential objective. The minimization of the lines joining the consecutive waypoints {$\left(x_i,y_i,z_i\right)$,$\left(x_{i+1},y_{i+1},z_{i+1}\right)$} and {$\left(x_{i+1},y_{i+1},z_{i+1}\right)$,$\left(x_n,y_n,z_n\right)$} leads to minimizing the total length of flyable paths. Thus, the first proposed objective function is defined as follows [24]:

$$f_1\left(\mathit{W}\right)=\sqrt{{\left(x_{i+1}-x_i\right)}^2+{\left(y_{i+1}-y_i\right)}^2+{\left(z_{i+1}-z_i\right)}^2}+\sqrt{{\left(x_{i+1}-x_n\right)}^2+{\left(y_{i+1}-y_n\right)}^2+{\left(z_{i+1}-z_n\right)}^2}$$

(2)

The consideration of the dynamic characteristics of UAVs is another essential objective [24,31,32]. The straighter the UAV drone’s flyable path, the more the control system complexity and fuel cost of the flight process decrease. However, in a real-world situation, a UAV should arbitrarily change its direction around waypoints and moving obstacles. Therefore, the candidate flyable path will present curvatures at any direction change and sharp corners. To handle such dynamic constraints of UAVs, the idea to use the concepts of steering angle and the corresponding turning radius was considered [31,32]. Such turning angles define the angles between two adjacent segments along the generated sequence of flyable waypoints, as depicted in Figure 3. The steering angle is then introduced to limit the straightness of the path and constrain the motion of the vehicle. Considering this dynamic behavior around waypoints, a second objective function that models such planning dynamic constraints can be chosen as follows [32]:

$$f_2\left(\mathit{W}\right)=\left|{\phi }_{s,s+1}\right|-{\phi }_{\max }$$

(3)

where ${\phi }_{s,s+1}=\stackrel{⏜}{\left({\mathit{w}}_{i-1}{\mathit{w}}_i,{\mathit{w}}_i{\mathit{w}}_{i+1}\right)}$ is the angle between two adjacent sth and (s+1)th segments connecting three consecutive waypoints, and ${\phi }_{\max }$ denotes the constraint value of this steering angle, as shown in Figure 3.

In dynamic environments, avoiding collision with moving obstacles is more complex and harder than in the case of static ones. Indeed, the drone must be able to coordinate its movement by avoiding any possible collision with a set of obstacles evolving in the navigation environment with position and speed variables in time [24]. In a real-world navigation environment, the geometric coordinates of dynamic obstacles are difficult to define. The main characteristics of obstacles, such as the center, the radius, the initial position, and the motion speed, are assumed to be known at the start of the planning process. At each time step, the position and speed motion equations of the jth given dynamic obstacle are updated as follows:

$$v_{obs}^j\left(t+\Delta t\right)=v_{obs}^j\left(t\right)+a_{obs}^j\Delta t$$

(4)

$$x_{obs}^j\left(t+\Delta t\right)=x_{obs}^j\left(t\right)+v_{obs}^j\left(t+\Delta t\right)\Delta t+\frac{1}{2}{a}_{obs}^j\Delta t^2$$

(5)

$$y_{obs}^j\left(t+\Delta t\right)=y_{obs}^j\left(t\right)+v_{obs}^j\left(t+\Delta t\right)\Delta t+\frac{1}{2}{a}_{obs}^j\Delta t^2$$

(6)

$$z_{obs}^j\left(t+\Delta t\right)=z_{obs}^j\left(t\right)+v_{obs}^j\left(t+\Delta t\right)\Delta t+\frac{1}{2}{a}_{obs}^j\Delta t^2$$

(7)

where $v_{obs}^j$ and $a_{obs}^j$ are the motion speed and acceleration of the j^th moving obstacle, respectively, $j=1,2,\dots ,q$, $q\in \mathbb{N}$, and $\Delta t\ge 0$ denotes the incremental time step size.

In such modeling, the UAV drone moves from the actual position ${\mathit{w}}_i=\left(x_i,y_i,z_i\right)$ to the next one ${\mathit{w}}_{i+1}=\left(x_{i+1},y_{i+1},z_{i+1}\right)$ when the time is incremented by one unit. To avoid any collision, the line connecting these two waypoints should not be crossed by a moving obstacle, as shown in Figure 4. Starting from a current instantaneous position and assuming that the UAV drone should not perform backward movements, the coordinates of the next consecutive waypoint can be expressed as follows:

$$\left\{\begin{array}{l}{x}_{i+1}=x_i+t\Delta x\\ y_{i+1}=y_i+t\Delta y\\ z_{i+1}=z_i+t\Delta z\end{array}\right.$$

(8)

where $\Delta x=\left|x_{i+1}-x_i\right|$, $\Delta y=\left|y_{i+1}-y_i\right|$, and $\Delta z=\left|z_{i+1}-z_i\right|$ denote the increments on the drone’s positions according to X-, Y-, and Z-axes, respectively.

Any waypoint on the surface of the sphere has the coordinates $\left(x_i,y_i,z_i\right)$ that verify the following characteristic equation:

$${\left(x_i-x_c^j\right)}^2+{\left(y_i-y_c^j\right)}^2+{\left(z_i-z_c^j\right)}^2=r_j^2$$

(9)

where $\left(x_c^j,y_c^j,z_c^j\right)$ and $r_j$ represent the center’s coordinates and radius of the jth sphere-based modeled obstacle, respectively.

By substituting Equation (8) in Equation (9), one can obtain the following second-order polynomial equation:

$${\mathit{A}}_j{t}^2+{\mathit{B}}_{j}t+{\mathit{C}}_j=0$$

(10)

where the terms ${\mathit{A}}_j$, ${\mathit{B}}_j$, and ${\mathit{C}}_j$ are defined as follows:

$${\mathit{A}}_j=\Delta x^2+\Delta y^2+\Delta z^2$$

(11)

$${\mathit{B}}_j=2\left(\Delta x\left(x_{i-1}-x_c^j\right)+\Delta y\left(y_{i-1}-y_c^j\right)+\Delta z\left(z_{i-1}-z_c^j\right)\right)$$

(12)

$${\mathit{C}}_j={\left(x_{i-1}-x_c^j\right)}^2+{\left(y_{i-1}-y_c^j\right)}^2+{\left(z_{i-1}-z_c^j\right)}^2-r_j{}^2$$

(13)

When the discriminant of such a second-order equation is negative, there are no intersections between the line connecting two consecutive drones’ positions and the moving obstacle modeled as a sphere. Such an operational constraint is defined as follows:

$${\Delta }_{ij}\left(y_i,z_i\right)={\mathit{B}}_j^2-4{\mathit{A}}_j{\mathit{C}}_j$$

(14)

where $j=1,2,\dots ,q$ is the index of the jth moving obstacle, $i=1,2,\dots ,n-1$ is the number of waypoints, and ${\mathit{A}}_j$, ${\mathit{B}}_j$, and ${\mathit{C}}_j$ are analytical terms given by Equations (11)–(13), respectively.

In addition, another type of collision situation can occur in the case of narrow passage between two moving obstacles located at the same altitude. In this case, i.e., generation of no feasible waypoint between two obstacles’ positions, the distance separating such two circumscribed spheres is insufficient to make a safe flight of the drone regarding its geometric dimensions. In such particular planning scenarios, a virtual sphere of radius $r_v=l_{uav}/2\in {ℝ}_{+}$ and ${\mathit{C}}_v:\left(x_c^v,y_c^v,z_c^v\right)$ centered midway between two nearby obstacles is considered, as shown in Figure 5, where $l_{uav}$ denotes the vehicle’s length and ${\delta }_{narro\mathit{w}}$ is a predefined safety distance of a narrow pass.

So, novel collision-free constraints of the same type as (14) can be defined by considering the intersection avoidance of the planned path segments with such virtual spheres, i.e., circumscribed to the narrow zone between obstacles. The waypoint assumed was generated when the drone passing between two nearby obstacles should be panelized. The UAV thus changes direction along the Z-axis, i.e., can fly above these obstacles or below them while respecting the safe distance from a possible collision with the ground. Another tolerated, feasible move consists of targeting the new waypoint of the next hyperplane, i.e., changing the direction along the Y-axis with a constant altitude, as depicted in Figure 5. From the optimization point of view, such a constraint for path planning regarding narrow passes between moving obstacles can be modeled as follows:

$${\Delta }_v^{ij}\left(y_i,z_i\right)={\mathit{B}}_v^2-4{\mathit{A}}_v{\mathit{C}}_v$$

(15)

where the terms ${\mathit{A}}_v$, ${\mathit{B}}_v$, and ${\mathit{C}}_v$ can be computed using the same Formulas (11)–(13) but by replacing the radius $r_j$ with $r_v$ and the center coordinates $\left(x_c^j,y_c^j,z_c^j\right)$ with those of the defined virtual sphere $\left(x_c^v,y_c^v,z_c^v\right)$.

Based on these modeling specifications and for the generation of the ith waypoint, the UAV’s path planning process in a dynamic environment with moving obstacles can be formulated as the following constrained multiobjective optimization problem:

$$\left\{\begin{array}{l}\underset{\mathit{W}\in \mathcal{D}\subseteq {ℝ}^2}{\mathrm{Minimize}}\phi \left(\mathit{W}\right)=\left\{f_1\left(\mathit{W}\right),f_2\left(\mathit{W}\right)\right\}\\ \mathrm{s}.\mathrm{t}:\\ g_{ij}\left(\mathit{W}\right)<0\\ h_{ij}\left(\mathit{W}\right)<0\end{array}\right.$$

(16)

where $f_1\left(.\right)$ and $f_2\left(.\right)$ are the cost functions of the biobjective optimization problem given by Equations (2) and (3), $g_{ij}\left(.\right)={\Delta }_j\left(.\right)$ and $h_{ij}\left(.\right)={\Delta }_v^{ij}\left(.\right)$ are the collision-free constraints of the moving obstacles defined by Equations (14) and (15), $\mathit{W}={\left\{y_i,z_i\right\}}_{1\le i\le n-1}$ are the decision variables of the problem, and $\mathcal{D}=\left\{\mathit{W}\in {ℝ}^2|{\mathit{W}}^{\min }\le \mathit{W}\le {\mathit{W}}^{\max },g_{ij}\left(\mathit{W}\right)<0,h_{ij}\left(\mathit{W}\right)<0,i=1,2,\dots ,n;j=1,2,\dots ,q\right\}$ denotes the bounded search space.

To handle the inequality constraints of the formulated multiobjective optimization problem (16), the external, static type of penalty functions are used as follows [47]:

$${\varphi }_k\left(\mathit{W}\right)=f_k\left(\mathit{W}\right)+{\displaystyle \sum _{i=1}^n{\displaystyle \sum _{j=1}^p{\lambda }_{ij}\max {\left\{0,g_{ij}\left(\mathit{W}\right)+h_{ij}\left(\mathit{W}\right)\right\}}^2}}$$

(17)

where ${\lambda }_{ij}\in {ℝ}^{+}$ is the jth penalty parameter associated with the jth constraint, $p\le q$ is the number of obstacles in the ith waypoint’s neighborhood that denotes the number of considered inequality constraints in the optimization, and $k\in \left\{1,2\right\}$.

3.3. Proposed Path Planning Strategy

Obstacle collision avoidance is fundamental within any dynamic path planning problem with UAVs. Based on the assumptions that the number, initial positions, and range of velocity of the moving obstacles are known, the proposed UAV path planning strategy consists of determining appropriate collision-free waypoint sequences in 3D environments with moving threats. Allowing for safe navigation from a starting point to a destination station, the designed path planning technique is based on the first step of the collection of the environment’s information, i.e., start and target points, number and dimension of moving obstacles, initial positions of obstacles, etc. As shown in the flowchart of Figure 6, the modeling of the 3D flight environment regarding the workspace partition, UAV dynamic constraints, and the geometry of moving obstacles should be investigated. At each increment of the computation time, the generation of collision-free waypoints is then performed using the proposed parallel processing PMOMVO planner (see the corresponding Algorithm 2 and the flowchart of Figure 6). Such an improved PMOMVO algorithm, associated with a TOPSIS type of multicriteria decision-making approach, aims to increase the performance of the dynamic planner in terms of computation time and trapping avoidance in local minima. A geometric technique to formulate moving obstacle avoidance is proposed based on the intersection of the segments connecting the generated waypoints with any obstacle assumed to be circumscribed in a 3D sphere with known centers and radii, as illustrated in Figure 4 and Figure 5. The generated collision-free waypoints are then targeted by the UAV drone while respecting the defined dynamic constraints, i.e., steering angles, straightness limitation, and narrow passage constraints, as shown in Figure 3, Figure 4 and Figure 5.

4. Parallel Optimization Algorithm

4.1. Multiobjective Multiverse Optimizer

The standard Multiverse Optimizer (MVO) is a population-based metaheuristic inspired by physical theories of the existence of the multiverse [48]. This swarm-intelligence-based algorithm models the interaction between different universes based on the notion of black/white holes and wormholes [48,49]. The sending and receiving of objects in a universe (decision variables) through wormholes are done according to their inflation rates (fitness values) to improve the process of exploration/exploitation and to avoid entrapment in local optima. The main motion equations of such an algorithm are defined as follows:

$$W_i^j=\left\{\begin{array}{l}\left\{\begin{array}{l}{W}_j+TDR+\left(u{b}_j-l{b}_j\times ran{d}_3+l{b}_j\right)\hspace{0.33em}\hspace{0.33em}ran{d}_2<0.5\\ W_j+TDR-\left(u{b}_j-l{b}_j\times ran{d}_3+l{b}_j\right)\hspace{0.33em}\hspace{0.33em}ran{d}_2<0.5\end{array}\right.\hspace{0.33em}\hspace{0.33em}\hspace{0.33em}\mathrm{if} ran{d}_1<WEP\\ W_i^j\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\hspace{1em}\mathrm{if} ran{d}_1\ge WEP\end{array}\right.$$

(18)

where $W_i^j$ denotes the jth component in the ith solution, $W_j$ indicates the jth variable of the best universe, $l{b}_j$ and $u{b}_j$ are the lower and upper bounds, respectively, $ran{d}_{1,2,3}\in \mathcal{U}\left(0,1\right)$ are random numbers uniformly distributed in the interval [0, 1].

In Equation (18), the terms $WEP$ and $TDR$ that present the wormhole existence probability and traveling distance rate, respectively, are defined as follows:

$$WEP={\rho }_{\min }+iter\left({\rho }_{\max }-{\rho }_{\min }\right)/n_{iter}$$

(19)

$$TDR=1-ite{r}^{1/\gamma }/n_{iter}$$

(20)

where ${\rho }_{\min }$ and ${\rho }_{\max }$ are the min and max values of the wormhole existence probability, respectively, $iter=1,2,\dots ,n_{iter}\in \mathbb{N}$ denotes the current algorithm’s iteration, and $\gamma \in {ℝ}_{+}$ defines the exploitation accuracy.

In this paper, a multiobjective variant of the multiverse optimizer (MOMVO) is developed by adding the concepts of archiving to store the nondominant solutions in the sense of Pareto. The leader selection and roulette techniques are implemented to select the best solutions from the Pareto archive. Since an archive can accommodate a limited number of nondominant solutions, a probabilistic mechanism is used to remove unsatisfactory solutions [49]:

$${\delta }_i=N_i/\alpha$$

(21)

where $N_i$ defines the number of the vicinity solutions and $\alpha >1$ is a constant.

Based on the above updating equations, a pseudocode of the proposed MOMVO can be summarized as follows (Algorithm 1):

Algorithm 1: MOMVO

1. Set the control parameters of MOMVO. 2. Randomly initialize the population, i.e., positions of the universes. 3. While ($iter<n_{iter}+1$) do 4. Update $WEP$ and $TDR$ by applying Equations (19) and (20), “See Section 4.1”. 5. For each universe do Boundary checking for the universes inside the search space. Calculate the inflation rate (fitness) of universes. End For 6. Sort the fitness values. 7. Find the nondominated solutions. 8. Normalize the inflation rates of each universe. 9. Update the archive regarding the obtained nondominated solutions. 10. If the archive is full do Delete some solutions from the archive to hold the new. End if 11. Update the positions of universes according to Equation (18), “See Section 4.1”. 12. If any newly added solutions to the archive are outside boundaries do Update the boundaries to cover the new solution(s). End if 13. Increment $iter$ 14. Stop the algorithm’s execution when it reaches $n_{iter}$.

On the other hand, a decision-making approach is required to select the best compromise solution among all the nondominated Pareto ones. For the formulated multicriteria path planning problem (16), an improved technique for order of preference by similarity to ideal solution (TOPSIS) is adopted as in our previous works [35].

4.2. Parallelization of the MOMVO Algorithm

In this paper, a master–slave model is proposed for the MOMVO algorithm parallelization as shown in Figure 7. In this proposed shared memory model, one of the CPU cores is selected as the master and the other ones are defined as the slaves [32].

The master processor will be responsible for initializing the population with cardinality $n_{pop}\in \mathbb{N}$ and breaking it down into $m\in \mathbb{N}$ subpopulations denoted as ${\mathit{S}}_1,{\mathit{S}}_2,\dots ,{\mathit{S}}_m$. Each subswarm of the entire MOMVO population is granted to a corresponding slave and a representative of each of them is selected by the roulette technique and sent to the master core, as shown in Figure 7.

In this work, the parallelization mechanism is introduced to simultaneously evolve all the subpopulations in the different cores, which can minimize the execution time. Each slave is designed to evolve the allocated subpopulation by applying a MOMVO code, as described in Algorithm 1, and independently search for the set of Pareto solutions. After an evolution cycle, each slave sends to the master the best solution retained by the roulette technique [49]. The master receives the solutions from all slaves and returns the nondominated ones to each for a new cycle. After that, each slave replaces its worst solutions with the solutions received from the master which are considered as best ones. The master checks the stopping condition; if it is reached, this process stops, and the slaves send their local Pareto fronts to the core representing the master. Such a master merges the received Pareto fronts and removes the repeated solutions to make a set of nondominated solutions and, therefore, a global Pareto front. Algorithm 2 provides a pseudocode for the proposed parallel processing PMOMVO algorithm.

Algorithm 2: PMOMVO

Master process

1. Randomly initialize $n_{pop}$ agents of the population. 2. Decompose the population into $m$ subswarms denoted as ${\mathit{S}}_1,{\mathit{S}}_2,\dots ,{\mathit{S}}_m$. 3. Randomly choose a representative solution from each subswarm. 4. Send each subswarm to a corresponding slave. 5. Cycle = 0. 6. While termination criterion = false do Parallel for j = 1 to $m$ slaves Find the nondominated solutions among the representatives of the slaves. Send to the slaves the nondominant solutions. Waiting for slaves. Receive all representatives of subswarms from the slaves. End Parallel for Cycle = Cycle+1. End While 7. Merge all subpopulations’ Pareto fronts in a single one. 8. Use the multicriteria decision making TOPSIS method to find the optimal solution.

Slave [j] process

9. While true do Receive the solutions from the master process. Update the worst solutions with those received from the master. Execute the MOMVO algorithm on each subswarm ${\mathit{S}}_j$. Select and send the representative of each subswarm ${\mathit{S}}_j$ to the master process. End While

5. Simulation Results and Discussion

5.1. Software Environment for Parallel Computing

For a practical implementation of the parallel MOMVO algorithm, the hardware architecture on which the program will be executed and the software environment associated with such architecture are key elements. In this work, a multicore CPU architecture was adopted for parallelization. A computer with a Core i5 processor with 12 cores at 2.90 GHz and 8.00 GB of RAM was used. In such a parallel architecture, the multiple CPU cores of a shared memory machine can operate in parallel and share the same memory space. The “Parallel Computing Toolbox” of the MATLAB environment was investigated to provide the easiest parallel programming avenue [50]. In this paper, the simplest “Parfor” structure of the MATLAB software tool was used to illustrate this functionality. The number of iterations of the “Parfor” loop was equal to the number of workers who performed the iterations independently of each other and evolved in parallel, i.e., one per subpopulation.

5.2. Numerical Experimentations and ANOVA Tests

In order to illustrate the effectiveness of the proposed PMOMVO algorithm for the formulated path planning problem, six variants of such an algorithm were executed under various flight scenarios with an increased number of moving obstacles, as shown in

Table 2. Flight scenarios with moving obstacles for the path planning process.

	Scenarios
	1	2	3	4	5
Moving obstacles	5	9	12	15	20
Starting point [m]	[0, 0, 0]	[1, 2, 0]	[1, 2, 0]	[2, 4, 0]	[0, 0, 0]
Target point [m]	[9, 8, 0]	[10, 10, 0]	[13, 10, 0]	[16, 13, 0]	[16, 15, 0]
Initial positions [m]	[5 5 2], [3 3 2], [5 3 1], [2 1 1], [6 2 2]	[1 3 1], [3 5 1], [4 4 3], [5 5 4], [7 3 4], [8 2 1], [9 5 2], [10 8 1], [9 9 1]	[2 3 1], [2 4 1], [4 3 2], [5 3 3], [5 5 2], [6 4 1], [7 7 2], [7 3 4], [8 6 3], [10 8 2], [9 2 1], [12 9 2]	[1 3 1], [2 5 2], [2 4 3], [2 7 1], [3 2 1], [3 3 3], [4 1 2], [4 5 4], [6 7 1], [7 2 2], [8 5 2], [10 8 3], [6 12 2], [13 1 2], [15 11 3]	[2 5 1], [15 13 1], [4 2 1], [13 9 4], [10 7 2], [14 12 1], [6 2 3], [4 12 1], [14 2 4], [9 10 2], [7 12 1], [12 10 5], [8 1 2], [8 8 1], [9 14 2] [5 3 1], [10 3 2], [8 14 1], [14 2 3], [12 8 2]
Motion speeds [m/s]	[4 −2 1], [2 −2 −2], [4 2 2], [2 2 2], [−2 2 −2]	[2 1 1], [3- 3 1], [4 1 −2], [2 1 1], [−1 −2 2], [0.5 1 −1], [1 −1 2], [1 −1 1], [1 2 1]	[−1 3 −1], [−1 1 1], [2 2 1], [1 2 4], [0.2 1 3], [1 −1 1], [2 1 2], [−1 2 2], [1 2 1], [3 0.5 2], [2 −1 1], [3 1 1]	[1 2 1], [−2 1 1], [3 −1 3], [2 1 2], [−1 3 1], [1 2 −2], [2 −1 2], [4 1 2], [3 2 1], [0.5 2 −2], [1 −2 1], [1 2 3], [1 2 0.5], [0.5 −1 1], [−1 2 2]	[1 3 1], [3 −1 1], [4 1 3], [2 2 4], [−1 3 1], [2 1 1], [3 1 2], [1 3 −2], [1 1 1], [2 −0.5 2], [1 1 2], [1 2 1], [2 2 −2], [1 −2 1], [1 1 2], [1 −2 −1], [1 2 −1], [2 1 1], [1 −1 2], [−1 1 2]

. These PMOMVO variants implement parallel algorithms with different subpopulations equal to 2, 4, 6, 8, 10, and 12 according to the available cores of the used CPU-based master–slave architecture. For the rest of the paper, these variants are denoted as PMOMVO-2, PMOMVO-4, PMOMVO-6, PMOMVO-8, PMOMVO-10, and PMOMVO-12. The control parameters of the reported algorithms are set as ${\rho }_{\min }=0.2$,${\rho }_{\max }=1$, $n_{pop}=50$ and $n_{iter}=100$.

All PMOMVO variants were run 20 independent times on the formulated path planning problem (16). For all experimentations, the PMOMVO variants were compared to the normal MOMVO as described by Algorithm 1. The effects of the main design parameters, i.e., the number of times that slaves shared their best solutions $l\in \left\{0,1,3,7,9\right\}$, called the sharing rate of better solutions, and the number of slaves in the parallel CPU architecture $m\in \left\{2,4,6,8,10,12\right\}$ were analyzed and discussed. In order to evaluate the performance of the PMOMVO algorithms in terms of convergence capability and Pareto nondominated solutions diversity, the metrics hyper volume (HV) [51], maximum spread (MS) [52], and hole relative size (HRS) [53] were considered. The results of these performance measurements are summarized in

Table 3. Optimization results of problem (16) regarding the HV metric.

Algorithms	Sharing Rates of Better Pareto Solutions
		$\mathit{l}=0$	$\mathit{l}=1$	$\mathit{l}=3$	$\mathit{l}=4$	$\mathit{l}=9$
MOMVO ($m=0$)	Best	0.0915	-	-	-	-
	Mean	0.0838	-	-	-	-
	Worst	0.0654	-	-	-	-
	STD	0.0153	-	-	-	-
PMOMVO-2	Best	0.0956	0.0964	0.0974	0.0987	0.0991
	Mean	0.0803	0.0848	0.0860	0.0883	0.0912
	Worst	0.0687	0.0689	0.0715	0.0754	0.0853
	STD	0.0131	0.0135	0.0129	0.0115	0.0069
PMOMVO-4	Best	0.0964	0.0973	0.0989	0.0993	0.0995
	Mean	0.0810	0.0869	0.0879	0.0898	0.0923
	Worst	0.0691	0.0695	0.0725	0.0758	0.0864
	STD	0.0138	0.0137	0.0131	0.0119	0.0066
PMOMVO-6	Best	0.0976	0.0986	0.0991	0.0997	0.1003
	Mean	0.0816	0.0870	0.0889	0.0904	0.0950
	Worst	0.0721	0.0736	0.0754	0.0810	0.0839
	STD	0.0130	0.0124	0.0117	0.0093	0.0082
PMOMVO-8	Best	0.0981	0.0989	0.0992	0.0999	0.1014
	Mean	0.0821	0.0880	0.0897	0.0929	0.0961
	Worst	0.0732	0.0740	0.0714	0.0823	0.0845
	STD	0.0128	0.0120	0.0140	0.0088	0.0082
PMOMVO-10	Best	0.0991	0.0994	0.0997	0.1045	0.1063
	Mean	0.0933	0.0908	0.0942	0.0964	0.0989
	Worst	0.0756	0.0784	0.0792	0.0841	0.0859
	STD	0.0120	0.0103	0.0105	0.0102	0.0101
PMOMVO-12	Best	0.0995	0.0996	0.0998	0.1054	0.1084
	Mean	0.0966	0.0946	0.0961	0.0978	0.0996
	Worst	0.0755	0.0786	0.0806	0.0845	0.0862
	STD	0.0130	0.0112	0.0103	0.0105	0.0110

,

Table 4. Optimization results of problem (16) regarding the MS metric.

Algorithms	Sharing Rates of Better Pareto Solutions
		$\mathit{l}=0$	$\mathit{l}=1$	$\mathit{l}=3$	$\mathit{l}=4$	$\mathit{l}=9$
MOMVO ($m=0$)	Best	0.9685	-	-	-	-
	Mean	0.9482	-	-	-	-
	Worst	0.9041	-	-	-	-
	STD	0.0336	-	-	-	-
PMOMVO-2	Best	0.9710	0.9796	0.9886	0.9891	0.9943
	Mean	0.9550	0.9751	0.9874	0.9883	0.9886
	Worst	0.9413	0.9614	0.9712	0.9745	0.9813
	STD	0.0148	0.0099	0.0098	0.0080	0.0065
PMOMVO-4	Best	0.9723	0.9822	0.9889	0.9894	0.9954
	Mean	0.9628	0.9765	0.9882	0.9884	0.9887
	Worst	0.9465	0.9652	0.9723	0.9765	0.9824
	STD	0.0135	0.0089	0.0094	0.0071	0.0062
PMOMVO-6	Best	0.9821	0.9836	0.9891	0.9926	0.9969
	Mean	0.9708	0.9787	0.9885	0.9894	0.9897
	Worst	0.9563	0.9663	0.9754	0.9782	0.9839
	STD	0.0130	0.0089	0.0076	0.0075	0.0064
PMOMVO-8	Best	0.9841	0.9856	0.9894	0.9952	0.9986
	Mean	0.9735	0.9793	0.9887	0.9912	0.9933
	Worst	0.9587	0.9681	0.9768	0.9863	0.9881
	STD	0.0129	0.0088	0.0072	0.0046	0.0053
PMOMVO-10	Best	0.9863	0.9874	0.9897	0.9965	1.0034
	Mean	0.9812	0.9796	0.9895	0.9931	0.9957
	Worst	0.9621	0.9685	0.9771	0.9881	0.9892
	STD	0.0127	0.0094	0.0071	0.0045	0.0067
PMOMVO-12	Best	0.9876	0.9886	0.9902	1.0005	1.0123
	Mean	0.9845	0.9804	0.9898	0.9957	0.9989
	Worst	0.9665	0.9714	0.9785	0.9898	0.9901
	STD	0.0113	0.0087	0.0067	0.0053	0.0110

and

Table 5. Optimization results of problem (16) regarding the HRS metric.

Algorithms	Sharing Rates of Better Pareto Solutions
		$\mathit{l}=0$	$\mathit{l}=1$	$\mathit{l}=3$	$\mathit{l}=4$	$\mathit{l}=9$
MOMVO ($m=0$)	Best	9.6741	-	-	-	-
	Mean	14.7315	-	-	-	-
	Worst	20.3641	-	-	-	-
	STD	5.8492	-	-	-	-
PMOMVO-2	Best	9.9874	8.9631	6.9654	6.1234	4.1231
	Mean	16.7786	12.5560	9.5141	7.2783	5.6696
	Worst	21.6521	15.8652	12.4264	11.3698	7.2541
	STD	5.8589	3.4520	2.7825	2.7552	1.5695
PMOMVO-4	Best	9.8841	8.7431	6.7412	5.8419	3.9841
	Mean	14.0749	11.3203	8.7722	7.4904	4.1613
	Worst	18.3652	15.5874	11.3210	10.2143	6.1432
	STD	4.2452	3.4582	2.2923	2.2023	1.1974
PMOMVO-6	Best	9.7436	8.5961	6.6321	5.1231	3.4561
	Mean	13.3927	10.9666	8.1799	6.1541	3.9961
	Worst	17.6354	13.8741	11.0362	9.4563	5.8741
	STD	3.9485	2.6423	2.2353	2.2614	1.2682
PMOMVO-8	Best	9.5896	8.2541	6.1536	4.8236	3.3987
	Mean	13.1597	9.8244	7.0997	5.8765	3.8388
	Worst	16.5413	12.8741	10.8413	8.1243	5.3243
	STD	3.4723	2.3482	2.4723	1.6856	1.0088
PMOMVO-10	Best	9.4132	7.1245	5.4563	4.6874	3.1716
	Mean	12.0952	8.8336	6.7727	5.2175	3.7988
	Worst	14.1452	10.2365	8.8741	7.6584	5.1413
	STD	2.3728	1.5569	1.7241	1.5823	1.0068
PMOMVO-12	Best	8.7413	6.5741	5.2416	4.2478	2.6746
	Mean	11.7824	7.6079	6.1369	5.0861	3.7444
	Worst	13.4713	9.9541	8.5542	7.2365	5.0321
	STD	2.3968	1.7319	1.7136	1.5402	1.1804

, respectively.

By observing these experimentations, one can note that for the HV and MS performance metrics, increasing the number of slaves and Pareto best solution sharing rates leads to a clear superiority of the PMOMVO algorithms in terms of obtained nondominated solution diversity. Moreover, since the HRS metric calculates the largest spacing of nondominated solutions on a Pareto front, the obtained low values of such a performance measurement for the proposed PMOMVO solvers lead to better uniformity of the compromise surfaces. As shown in Table 5, the numerical results show that for all reported PMOMVO variants, the HRS values are decreased by increasing the sharing rates of the better solutions.

Figure 8 illustrates the set of Pareto solutions obtained by the MOMVO and PMOMVO algorithms corresponding to the mean case of the optimization. Looking at these results, one can observe the large gaps in the topology distribution of Pareto fronts for the low values of the sharing rates $l=0$,$l=1$, and $l=3$. However, for the high solution sharing rate $l=9$, the distribution of the nondominated solutions along the compromise surface is more uniform. According to these demonstrative results, the number of times that slaves share information with each other clearly influences the uniformity of the obtained Pareto front. The higher the $l\in \mathbb{N}$ value, the more the distribution of Pareto solutions approximates a uniform distribution. In addition, the performance of the PMOMVO variants, especially those with the highest number of slaves, surpasses the standard MOMVO in terms of solution distribution uniformity.

Let us consider the evaluation and analysis of the generated UAV global paths. To discuss the performance of the PMOMVO algorithm in solving the dynamic path planning problem, experimentations were performed considering the commonly used performance criteria flight time (FT) and straight-line rate (SLR) [32]. The ability to avoid collisions with moving obstacles in the considered dynamic environment was also investigated. Numerical experimentations carried out for 20 independent executions led to the optimization results of

Table 6. Optimization results regarding the SLR criterion.

Algorithms	Scenarios
		1	2	3	4	5
MOMVO	Best	1.0204	1.0271	1.0163	1.0069	1.0206
	Mean	1.0321	1.0301	1.0273	1.0119	1.0378
	Worst	1.0898	1.0710	1.0354	1.0175	1.0422
	STD	0.3352	0.2982	0.1376	0.0893	0.2447
PMOMVO-2	Best	1.0094	1.0100	1.0164	1.0046	1.0205
	Mean	1.0225	1.0214	1.0230	1.0085	1.0344
	Worst	1.0344	1.0338	1.0293	1.0129	1.0376
	STD	0.1423	0.1485	0.0913	0.0673	0.1985
PMOMVO-4	Best	1.0086	1.0081	1.0105	1.0033	1.0148
	Mean	1.0180	1.0162	1.0153	1.0059	1.0205
	Worst	1.0253	1.0254	1.0232	1.0099	1.0251
	STD	0.1002	0.1042	0.0910	0.0563	0.1125
PMOMVO-6	Best	1.0079	1.0072	1.0101	1.0027	1.0146
	Mean	1.0160	1.0144	1.0162	1.0054	1.0202
	Worst	1.0236	1.0223	1.0211	1.0086	1.0241
	STD	0.0913	0.0912	0.0781	0.0496	0.1046
PMOMVO-8	Best	1.0074	1.0065	1.0084	1.0024	1.0145
	Mean	1.0144	1.0131	1.0140	1.0045	1.0203
	Worst	1.0219	1.0204	1.0175	1.0076	1.0240
	STD	0.0858	0.0825	0.0661	0.0426	0.1036
PMOMVO-10	Best	1.0071	1.0065	1.0078	1.0022	1.0143
	Mean	1.0149	1.0110	1.0128	1.0048	1.0199
	Worst	1.0205	1.0191	1.0155	1.0070	1.0236
	STD	0.0832	0.0759	0.0549	0.0414	0.1029
PMOMVO-12	Best	1.0065	1.0057	1.0067	1.0015	1.0141
	Mean	1.0137	1.0121	1.0112	1.0051	1.0203
	Worst	1.0197	1.0190	1.0141	1.0063	1.0235
	STD	0.0765	0.0783	0.0526	0.0401	0.1032

and

Table 7. Optimization results regarding the FT (sec) criterion.

Algorithms	Scenarios
		1	2	3	4	5
MOMVO	Best	451.6952	491.3265	736.3214	828.9541	898.2541
	Mean	462.8974	504.5230	745.9852	839.8741	915.5800
	Worst	471.2563	515.6932	752.6541	847.9852	930.6589
	STD	8.7587	11.1254	8.2263	9.5062	10.2123
PMOMVO-2	Best	404.3652	435.6523	668.3621	748.8741	794.2654
	Mean	412.6981	441.6321	674.9852	756.9612	801.5542
	Worst	420.1234	447.6521	680.6311	761.6325	810.5461
	STD	6.7824	5.9524	6.1874	6.4525	7.1256
PMOMVO-4	Best	398.8741	417.8541	642.9874	708.2145	751.2541
	Mean	406.8741	425.6952	648.7412	714.8963	757.2601
	Worst	411.6541	430.6541	654.6512	721.3241	765.2541
	STD	6.3685	5.9214	5.8326	6.5552	6.0395
PMOMVO-6	Best	391.5231	413.1782	638.6321	704.6411	748.9852
	Mean	397.8214	419.8523	643.9852	709.8521	753.9252
	Worst	404.5241	424.3251	648.1243	714.9874	760.4123
	STD	5.5208	5.0833	4.7214	5.1732	5.0820
PMOMVO-8	Best	398.8741	420.8745	643.6521	711.9881	757.1635
	Mean	406.7741	426.8741	649.8521	717.9852	763.5603
	Worst	412.8562	432.6523	656.3214	725.6521	771.2143
	STD	6.0185	5.8742	6.1524	6.8701	5.5258
PMOMVO-10	Best	403.5241	425.8541	644.3251	714.9663	763.1423
	Mean	414.6892	431.6541	651.6598	722.5871	768.6741
	Worst	421.9852	438.5241	657.9874	729.9631	776.4512
	STD	7.1979	6.3512	6.3576	7.1254	5.6589
PMOMVO-12	Best	407.6352	428.2441	649.8741	722.8810	778.5412
	Mean	418.4756	435.9874	656.9871	729.8741	784.3506
	Worst	425.7412	443.3652	662.8741	738.6954	792.1423
	STD	8.1256	7.5287	6.5586	7.5253	6.5266

. In terms of the traveled path length, the smaller the SLR metric, the better the PMOMVO-based planner efficiency. Similarly for the FT performance criterion, the lower the elapsed time, the better the optimization algorithm in terms of navigation speed and planning time management.

In order to statistically analyze these results, nonparametric ANOVA statistics in the sense of Friedman and Fisher’s LSD posthoc tests were carried out while considering the two performance criteria SLR and FT [54]. The related statistical results are summarized in

Table 8. Friedman ranking of the mean performance: SLR criterion.

	Scenarios					Ranks’ Sum.
Algorithms	1	2	3	4	5
MOMVO	7	7	7	7	7	35
PMOMVO-2	6	6	6	6	6	30
PMOMVO-4	5	5	4	5	5	24
PMOMVO-6	4	4	5	4	3	20
PMOMVO-8	2	3	3	1	4	13
PMOMVO-10	3	1	2	2	2	10
PMOMVO-12	1	2	1	3	1	8

,

Table 9. Friedman ranking of the mean performance: FT criterion.

	Scenarios					Ranks’ Sum.
Algorithms	1	2	3	4	5
MOMVO	7	7	7	7	7	35
PMOMVO-2	4	6	6	6	6	28
PMOMVO-4	3	2	2	2	2	11
PMOMVO-6	1	1	1	1	1	5
PMOMVO-8	2	3	3	3	3	14
PMOMVO-10	5	4	4	4	4	21
PMOMVO-12	6	5	5	5	5	26

,

Table 10. Paired comparison of the PMOMVO algorithms: SLR criterion.

	PMOMVO-2	PMOMVO-4	PMOMVO-6	PMOMVO-8	PMOMVO-10	PMOMVO-12
MOMVO	5	11	15	22	25	27
PMOMVO-2	-	6	10	17	20	22
PMOMVO-4	-	-	4	11	14	16
PMOMVO-6	-	-	-	7	10	12
PMOMVO-8	-	-	-	-	3	5
PMOMVO-10	-	-	-	-	-	2

and

Table 11. Paired comparison of the PMOMVO algorithms: FT criterion.

	PMOMVO-2	PMOMVO-4	PMOMVO-6	PMOMVO-8	PMOMVO-10	PMOMVO-12
MOMVO	7	24	30	21	14	9
PMOMVO-2	-	17	23	14	7	2
PMOMVO-4	-	-	6	3	10	15
PMOMVO-6	-	-	-	9	16	21
PMOMVO-8	-	-	-	-	7	12
PMOMVO-10	-	-	-	-	-	5

. The performance of the reported parallel PMOMVO algorithms was analyzed through the considered flight scenarios of Table 2. The Iman–Davenport extension of the Friedman test provides the statistics $F_{F1}=38.424$ and $F_{F2}=83.500$ for the SLR and FT criteria, respectively. For the seven algorithms ($\mu =7$) and five path planning scenarios ($\sigma =5$), the critical F-statistics value at 95% of significance and with $\mu -1$ and $\left(\mu -1\right)\left(\sigma -1\right)$ degrees of freedom is equal to $F_{6,24,0.05}=2.5082<F_{F1}<F_{F2}$. Therefore, the null hypothesis is rejected and there are significant differences between the competing PMOMVO solvers.

Fisher’s LSD posthoc test was applied to find out which PMOMVO-based planning algorithms differ from others [54]. Table 10 and Table 11 summarize the paired comparisons of all reported algorithms for the SLR and FT performance indices, respectively. The critical values computed for the absolute difference of the rank sums of the two algorithms are equal to 4.0124 for the SLR criterion and 2.7939 for the FT criterion, respectively. The bold and underlined values indicate significant differences between the performances of the competing PMOMVO algorithms. From these performed statistical tests and analyses as well as the Friedman ranking of the proposed PMOMVO-based planners, one can observe that the variants with the highest number of CPU multicores slaves, i.e., PMOMVO-10 and PMOMVO-12, outperform the standard MOMVO one as well as the other PMOMVO algorithms with the lower number of slaves. However, in terms of processing time, one can observe, according to the results of Table 7, that the PMOMVO-6 algorithm is the best variant regarding the obtained low values of the FT metrics. Such a variant of the PMOMVO algorithm can be retained for the rest of developments as the faster and more efficient solution to the UAV path planning problem in the considered dynamic 3D environment with moving obstacles.

Regarding the ability of the proposed technique to avoid collisions with moving obstacles, other experiments were conducted. The usefulness of the proposed parallel processing PMOMVO-based path planning approach was tested through three flight instances with the same starting and destination positions and moving obstacle number but with different positions and motion speeds of the dynamic threats, as shown in

Table 12. Scenarios for the moving obstacle avoidance illustration.

Instance	Starting	Destination	Moving Obstacles’ Position [m]	Moving Obstacles’ Speed [m/s]
1	[0, 0, 0]	[8, 8, 0]	[5 5 1], [3 3 2], [5 3 1], [1.5 2.5 1], [1 1 1.5]	[4 −2 1], [2 −2 −2], [4 2 2], [1 1 1.5], [1 1 1]
2	[0, 0, 0]	[8, 8, 0]	[5 6 1], [3 3 1], [6 2 1], [2 2 1], [1 2 1]	[1 2 1], [1 −1 −1], [1 2 2], [1 1 1], [1 2 1]
3	[0, 0, 0]	[8, 8, 0]	[5 2 1], [3 1 1], [7 5 1], [1.5 1.5 1], [1 3 1]	[2 −1 1], [1 2 −1], [1 −2 1], [1 1 1], [1 −1 1]

. The case of design using the best compromise of computation time and low slaves in the multicore CPU architecture, i.e., the PMOMVO-6 variant, was considered for the demonstrative results.

Figure 9, Figure 10 and Figure 11 present the simulation results of the planned paths over the different scenarios. Each situation gives the result of dynamic path planning where the UAV positions are captured relatively at different times of the navigation process. These demonstrative results show the collision-free abilities facing moving obstacles with different initial positions and motion speeds. Changing the position and motion speed of the moving obstacles affects the 3D planned path, which further improves the effectiveness and superiority of the proposed PMOMVO-based planning approach under moving obstacles.

To discuss the quality of the obtained solutions, additional simulation results were carried out, as shown in Figure 12. The same flight scenario was considered, and four runs of the PMOMVO-based planner were made. Based on this illustration, one can observe that with the same conditions of a path planning scenario, i.e., same number of obstacles, same start and destination positions, etc., the proposed PMOMVO-based planner as a metaheuristics-based design approach gives a nonreproducible result but still a good feasible solution for the addressed optimization problem. Among all these results, one can choose the best in terms of the shortness of the flyable path and collision avoidance capability regarding the moving obstacles. Through these comparable results and the high reproducibility behavior of the proposed PMOMVO algorithm, argued by the obtained small values of STD metrics, one can observe the minor and negligible deviations between all these obtained results. So, the optimality of the obtained solution can neither be affirmed nor checked theoretically, but such a solution remains quite feasible and of good quality.

5.3. Computation Tims Analysis and Comparison

The performance of the proposed PMOMVO variants is discussed in terms of computation time consumption. The algorithm’s execution time presents a key element for any efficient collision-free path planner in a real-world application of UAV navigation. First, it is important to note that the two main design parameters of the proposed parallel-processing-based algorithm are the solution sharing rate and the number of slaves. Numerical experimentations were carried out to show the effect of increasing the solution sharing rate of slaves in each variant of the PMOMVO algorithm. Figure 13 and Figure 14 show the evolution of the computation time metric over the sharing rate configuration and slave number variation, respectively. These demonstrative results highlight the superiority of the proposed PMOMVO algorithms compared to the MOMVO standard one. Based on these results, one can observe clearly the influence of the sharing rates and slave numbers on the performance of the proposed planning strategy in terms of computation times and execution fastness. As the number of slaves increases, the execution time decreases up to a certain number of slaves and then gradually increases. The reason for the increase in computation time for a high number of slaves is that the processing time related to the affected computations is not enough, and the master–slave communications require more time. In this study, one can conclude that the variant of the PMOMVO algorithm that runs with a six-slave-based multicore CPU architecture remains the best solver in terms of computing speed.

On the other hand, in order to further show the superiority of the proposed parallel PMOMVO algorithm, a comparative study of the computation times was carried out, as depicted in Figure 15. Other extensively used multiobjective optimization algorithms, namely the salp swarm algorithm (MSSA) [55], grey wolf optimizer (MOGWO) [56], nondominated sorting genetic algorithm II (NSGA-II) [57], and particle swarm optimization (MOPSO) [58] were considered for such a comparison. Retained as the most powerful variant of the PMOMVO algorithm in terms of computation time, the PMOMVO-6 optimizer largely outperformed all reported MSSA, MOGWO, MOPSO, and NSGA-II algorithms.

5.4. Parameters Sensitivity Analysis

As for any path planning approach using unconventional and nonreproducible algorithms, the influence of main control parameters should be analyzed and discussed. The algorithm sensitivity is investigated through numerical experimentations performed with randomly chosen sets of population size $n_{pop}$ and iterations $n_{iter}$. The solution sharing rate of PMOMVO slaves is set as $l=9$.

Table 13. Path length variation under iterations and population size of problem (16).

Max- Iter.	Pop. Size	Variants of PMOMVO Algorithm
		MOMVO	PMOMVO -2	PMOMVO -4	PMOMVO -6	PMOMVO -8	PMOMVO -10	PMOMVO -12
100	50	22.7612	22.6852	22.3812	22.3755	22.3712	22.3674	22.3665
	100	22.7521	22.5720	22.3805	22.3721	22.3645	22.3514	22.3465
	200	22.7132	22.5123	22.3751	22.3698	22.3612	22.3475	22.3308
200	50	22.7574	22.6584	22.3798	22.3562	22.3478	22.3452	22.3398
	100	22.7414	22.4652	22.3674	22.3462	22.3365	22.3325	22.3274
	200	22.6852	22.4036	22.3433	22.3241	22.3054	22.2974	22.2912
300	50	22.7354	22.6136	22.3569	22.3287	22.3165	22.3126	22.2975
	100	22.7058	22.4352	22.3171	22.3084	22.2987	22.2841	22.2798
	200	22.6123	22.3852	22.2893	22.2871	22.2846	22.2785	22.2672

and

Table 14. Flight time variation under iterations and population size of problem (16).

Max- Iter.	Pop. Size	Variants of PMOMVO Algorithm
		MOMVO	PMOMVO -2	PMOMVO -4	PMOMVO -6	PMOMVO -8	PMOMVO -10	PMOMVO -12
100	50	915.5800	801.5542	757.2601	753.9252	763.5603	768.6741	784.3506
	100	1178.8686	1049.3154	860.4351	836.8540	844.4646	852.6669	866.7345
	200	1772.3200	1214.5412	889.7868	856.2145	849.5713	857.1243	871.3265
200	50	1230.2829	963.9214	837.9456	808.5869	819.6385	825.8947	830.4514
	100	1849.4532	1322.0424	1024.7202	972.8741	958.6206	961.8273	977.5868
	200	3031.1316	1765.5321	1187.2541	1056.4123	1016.5471	992.2146	1031.2314
300	50	1456.9254	1134.0631	938.2590	882.8192	870.1821	881.6050	892.7623
	100	2286.0306	1572.8265	1181.4831	1098.2651	1077.8859	989.3251	998.2514
	200	3937.7016	2394.2541	1611.0361	1388.4576	1280.5775	1262.2143	1154.2651

summarize the optimization results obtained in Scenario 5 of Table 2 while considering path length (m) and flight time (s) as performance metrics. From these demonstrative results, increasing the population cardinality and iterations does not have a significant effect on planning performance in terms of obtained collision-free path lengths, unlike the flight time which is further increased.

Roughly, the superiority of the improved MOMVO algorithms over the classical MOMVO one is demonstrated with a larger population size and higher number of iterations. Obviously, with the increase in the computation time, which is due to the increase in the number of iterations and size of the population, an increase in the number of slaves is necessary to better manage the complexity of the dynamic path planning task. The effects of these two main control parameters of PMOMVO algorithms are further illustrated as shown in Figure 16 and Figure 17. Based on these experimentations and results, one can consider that the proposed parallelization-based improvements of the classical MOMVO algorithm perform better for complex path planning problems that require significant computation time.

5.5. Comparison with Other Metaheuristics Algorithms

To analyze the performance of the proposed PMOMVO algorithms, mainly the PMOMVO-6 variant, in terms of path length (SLR) and flight time (FT) performance, the multiobjective metaheuristics algorithms MSSA, MOGWO, NSGA-II, and MOPSO already retained for the comparison of computation times were reconsidered again. These optimizers run with the following control parameters:

• − MSSA [55]: without control parameters (parameters-free algorithm).
• − MOGWO [56]: grid inflation 0.1, number of grids per dimension 10, leader selection pressure 4, and extra repository member selection pressure 2.
• − NSGA-II [57]: crossover probability 0.7, mutation probability 0.4, and mutation rate 0.02.
• − MOPSO [58]: social and cognitive parameters 2, grid inflation 0.1, leader selection pressure parameter 2, and number of grids per dimension 7.

The common control parameters of the compared multiobjective algorithms were set as $n_{iter}=100$ and $n_{pop}=50$. All optimizers were run independently 20 times according to Scenario 5 of the path planning with moving obstacles given in Table 2. The experimentation results of the comparison are summarized in

Table 15. Performance comparison of PMOMVO algorithm with other metaheuristics.

Algorithms		Performance Criteria
		Path Length SLR (m)	Flight Time FT (s)
MSSA	Best	22.721	837.40
	Mean	23.391	844.29
	Worst	24.414	856.35
	STD	0.1587	6.7255
MOGWO	Best	23.654	1154.2
	Mean	24.980	1248.7
	Worst	25.547	1549.7
	STD	0.2342	7.4924
NSGA-II	Best	21.204	8150.1
	Mean	22.865	8247.3
	Worst	23.481	8892.2
	STD	0.2871	8.2456
MOPSO	Best	32.987	3194.7
	Mean	35.281	3398.5
	Worst	37.242	4009.2
	STD	0.3733	6.62
PMOMVO-6	Best	22.251	748.9852
	Mean	22.374	753.9252
	Worst	22.4602	760.4123
	STD	0.1046	5.0820

. Based on these demonstrative results, one can observe that the proposed parallel processing PMOMVO-6 variant outperformed all reported algorithms with lower values for SLR (m) and FT (sec) metrics. The STD statistics were minimal in the case of optimization with the PMOMVO-6 algorithm. The superiority of such a parallel optimizer is further improved in terms of solution quality, reproducibility capacity, and computational time fastness.

6. Conclusions

In this paper, a new parallel processing variant of the multiobjective multiverse optimizer (PMOMVO) based on a master–slave multi-core CPU model has been proposed and successfully applied to solve the UAV path planning problem in a dynamic environment with moving obstacles. To overcome the limits and drawbacks of the standard MOMVO algorithms, particularly in terms of prohibitive computation time consumption, an efficient processing parallelization based on a master–slave CPU multicore architecture was introduced and successfully implemented. The reduction in computation time of the parallel PMOMVO algorithm contributed to the effectiveness of the proposed path planning strategy in terms of avoiding collisions with moving obstacles and narrow pass zones. Such a dynamic path planning problem was reformulated based on new ideas of collision avoidance with a moving body in a 3D space. The drone became able to respect the resulting dynamic constraints of navigation and processing time consumption and, consequently, reacted quickly to such changes in the environment.

According to the available cores of a given hardware CPU architecture and the number of partitioned MOMVO subpopulations, six variants of the parallel algorithm denoted as PMOMVO-2, PMOMVO-4, PMOMVO-6, PMOMVO-8, PMOMVO-10, and PMOMVO-12 were designed. Each slave of the proposed multicore CPU architecture was implemented to evolve the allocated subpopulation by applying a MOMVO algorithm and seding the best-selected solutions after each cycle of evolution to the master. The master received the solutions and sent to each slave the Pareto nondominated ones for a new cycle. A modified TOPSIS technique was used to select solutions in the sense of Pareto. In this study, one can observe that the variant of the PMOMVO algorithm running with six slaves, i.e., PMOMVO-6, outperformed all other compared algorithms in terms of computation time performance and solution quality. Demonstrative results and nonparametric ANOVA statistical analyses based on Friedman’s and Fisher’s LSD posthoc tests show the effectiveness and superiority of the proposed parallel processing PMOMVO algorithms for UAV path planning with collision avoidance problems in dynamic 3D environments.

Future work will focus firstly on the comparison of the proposed parallel processing PMOMVO-based path planning approach with introduced enhancements in terms of computation time reduction and moving threats collision avoidance with other popular path planning techniques, such as those using the concepts of rapidly exploding random tree (RRT). Secondly, the real-world implementation of the proposed metaheuristics-based path planning algorithm will be investigated within an indoor application using the laboratory-available Parrot AR. Drone 2.0 prototype.