Optimizing Path Planning for Smart Vehicles: A Comprehensive Review of Metaheuristic Algorithms
Abstract:
In the realm of smart vehicle navigation, both in known and unknown environments, the crucial aspects encompass the vehicle's localization using an array of technologies such as GPS, cameras, vision systems, laser, and ultrasonic sensors. This process is pivotal for effective motion planning within the vehicle's free configuration space, enabling it to adeptly avoid obstacles. The focal point of such navigation systems lies in devising a path from an initial to a target configuration, striving to minimize the path length and the time taken, while simultaneously circumventing obstacles. The application of metaheuristic algorithms has been pivotal in this regard. These algorithms, characterized by their ability to exploit initial solutions and explore the environment for feasible pathways, have been extensively utilized. A significant body of research in robotics and automation has focused on evaluating the efficacy of populationbased algorithms including Genetic Algorithm (GA), Ant Colony Optimization (ACO), Particle Swarm Optimization (PSO), Firefly Algorithm (FA), and Whale Optimization Algorithm (WOA). Additionally, trajectorybased methods such as Tabu Search (TS) and Simulated Annealing (SA) have been scrutinized for their proficiency in identifying short, feasible paths among the plethora of solutions. There has been a surge in the enhancement and modification of these algorithms, with a multitude of hybrid metaheuristic algorithms being proposed. This review meticulously examines various metaheuristic algorithms and their hybridizations, specifically in their application to the path planning challenges faced by smart vehicles. The exploration extends to the comparison of these algorithms, highlighting their distinct advantages and limitations. Furthermore, the review delves into potential future directions in this evolving field, emphasizing the continual refinement of these algorithms to cater to the increasingly complex demands of smart vehicle navigation.1. Introduction
The advent of technologies such as big data, cloud computing, 5G networks [1], the Internet of Things (IoT) [2], and artificial intelligence (AI) [3], [4] has been instrumental in the evolution of smart vehicles. These vehicles leverage these technologies to mitigate human error in driving, navigate traffic in selfdriving modes, assist in industrial logistics and manufacturing processes as Automated Guided Vehicles (AGVs), and operate in challenging terrains as Unmanned Ground Vehicles (UGVs). Furthermore, their applications extend into healthcare, domestic settings, and ecommerce through Autonomous Mobile Robots (AMRs).
In the domain of selfdriving vehicles, technologies like Light Detection and Ranging (LIDAR) sensors [5], cameras integrated with deep learning algorithms [6], and ultrasonic sensors [7] are employed for vehicle and object detection, traffic alerts, zebra crossing recognition, and collision avoidance with pedestrians. AGVs, utilized in warehouse logistics and material handling, navigate using lasers, magnets, vision cameras, or by following marked lines or wires. The role of AI, particularly reinforcement learning, has become prominent in route planning for AGVs [8]. AMRs have a wide array of applications including inspection [9], surveillance [10], monitoring [11], logistics, and service [12], [13], [14], [15]. They are categorized into holonomic and nonholonomic types [16], [17], with holonomic robots having controllable degrees of freedom equal to their total degrees of freedom [18], allowing movement in any direction within their configuration space. Nonholonomic mobile robots, on the other hand, have constraints on their velocities and derivatives of position [19].
For effective navigation, smart vehicles must comprehend the nature of their environment to adapt their actions for optimal goal attainment. Critical to this process are three fundamental components: mapping, localization, and path planning [20]. Mapping involves the creation or retrieval of environmental maps, providing location and orientation data for the vehicles. Localization is essential for vehicles to ascertain their position on the map, a task accomplished using cameras, GPS, and various sensors like laser, vision, and ultrasonic sensors. The location may be expressed in absolute coordinates (longitude, latitude, altitude), as a reference relative to the environment, or as topographical coordinates (e.g., in a room). Path planning is the process of determining a viable, obstaclefree route in typically congested realworld environments [21].
2. Literature Review
Path planning in the context of smart vehicles is categorized into two primary approaches: global and local. Global path planning is concerned with deriving the optimal path using extensive environmental data. This approach is most effective in static environments that are welldefined and familiar to the smart vehicle. Here, path planning algorithms generate a complete route from an origin to a destination, thereby determining the optimal trajectory for the vehicle. In contrast, local path planning is pertinent in environments that are either unfamiliar or subject to change. It involves realtime computation while the vehicle is in motion, utilizing data from onboard local sensors. This enables the smart vehicle to adaptively generate new routes in response to dynamic environmental changes. A wide array of path planning methods and algorithms have been explored in the field of robotics. Factors influencing the selection of an appropriate algorithm include the kinematics and dynamics of the environment, the computational capabilities of the smart vehicles, the type of sensors employed, and the availability of other sourced information. The decisionmaking process regarding the choice of an algorithm also involves considering the tradeoffs between algorithmic performance and complexity, which vary depending on the specific application [22]. As illustrated in Figure 1, path planning methods and algorithms can be divided into several categories, such as classical methods like cell decomposition, metaheuristic algorithms including the genetic algorithm, machine learning approaches like reinforcement learning, and sampling methods exemplified by probabilistic roadmaps [22], [23], [24], [25].
Metaheuristic algorithms represent a class of highlevel heuristic approaches that are designed to provide suitable solutions to optimization problems, particularly those characterized by incomplete information or limited computational resources. When applied to path planning, metaheuristic algorithms demonstrate proficiency in managing environments that are partially known or contain moving obstacles. This is in contrast to classical algorithms, which typically necessitate prior comprehensive knowledge of the environment [23]. Metaheuristic algorithms are categorized into two main groups: populationbased methods like Particle Swarm Optimization and trajectorybased methods such as Simulated Annealing, as depicted in Figure 2 [26]. Populationbased metaheuristics operate by generating multiple points within the search space, whereas trajectorybased methods progress through the search space by navigating a trajectory via a single point at each time step.
The GA, an optimization methodology, draws upon the principles of genetics and natural selection, first conceptualized by Bremermann [27]. Holland was the pioneer in adapting the genetic algorithm to computer science [28]. Its applications have since permeated various domains, including robot navigation and numerous scientific and technological fields. This algorithm focuses on optimizing complex problems where the objective function needs to be maximized or minimized within specified constraints. The method starts by defining a population size, where chromosomes (sets of genes) are formulated based on the given problem. Each chromosome in the population is assigned a fitness value, contingent upon the objective function. Chromosomes are then selected based on their fitness, allowing them to propagate their genes to subsequent generations through crossover processes. Mutation is employed to maintain population diversity and avert premature convergence. Table 1 elucidates the pseudocode of the genetic algorithm. The algorithm concludes its process once the population has converged [29].
The recent focus on GAbased methods, particularly in the realm of optimization problems like path planning, highlights the potential of GA in addressing these challenges [6]. This is evidenced by the success of GA in various applications, as discussed in Table 2, which outlines different studies that have employed GA for path planning. This table includes the variables considered in each study and provides insights into their findings. Moreover, the hybridization of GA with other intelligent algorithms has been an area of considerable research interest. Notable examples include the integration of GA with Fuzzy Logic [30], Intelligent Water Drop [31], and Neural Network [32], aiming to enhance the efficacy of the solutions. In the utilization of GAbased methods for path planning, distance often emerges as a common parameter [33], [34], [35], [36], [37], alongside other considerations such as path smoothness and clearance [34], [36], [37], energy evaluation [38], and factors related to robot speed. A noteworthy study by Liu et al. [39] presented an improved GA to tackle the appointment order allocation and route planning issues of Cainiao unmanned vehicles. Additionally, Wang et al. [40] proposed an optimized approach using GA to implement the MultiObjective Evolutionary Algorithm (MOEA) for planning the trajectory of a mobile robot in a known environment. This experiment involved a twowheeled mobile robot using the ArUco system within the Robotic Operating System (ROS). However, it's important to note the limitations of this method, particularly its unsuitability for rough terrains due to the omission of the mobile robot's dynamics in the planning process. Furthermore, the algorithm's deployment on a console computer, rather than within the robot's embedded system, is attributed to the limitations of the embedded system's lowend hardware.
Genetic Algorithm  
1  Choose encode method  
2  G$\leftarrow$0  
3  G_{max }$\leftarrow$ Maximum generation  
4  Initialize population  
5  for (G < G_{max}) do  
6  for (i = 1 to maximum population) do  
7  Evaluate fitness of individual i  
8  end for  
9  Selection  
10  Crossover  
11  Mutation  
12  Move new individuals to population G + 1  
13  G $\leftarrow$ G + 1  
14  end for  
15  return best individual 
Types  Initial Population Generation Method  Population Size  Reproduction Operators  Fitness Function  Sorting and Selection Technique  Number of Generations  Type of Vehicle  Type of Obstacle  Type of Map  Software  Remarks  Ref 
Improved GA  Random  100  $F=\frac{1}{C+M P}$  Optimization guidance factor and Roulette selection  500  Multiple  Static  Topological  MATLAB R2018a  · Order allocation and route planning problem is modelled to obtain efficient picking of orders. · Provides the optimal solutions to unmanned vehicle inputs and their path planning.  [39]  
Novel GA  CBPRM  20, 25, 50  · Crossover: Ordinary · Mutation: Change, smooth and shortcut operators  $\mathrm{C}(\mathrm{k})=\mathrm{L}(\mathrm{k}) \times \mathrm{S}(\mathrm{k})$  50  Single  Static  Geometrical  CGAL 3.3.1 [41]  · Minwoski sum is used as a computational geometry based approach instead of cell based methods. · CBPRM speeds up the evolutionary process.  [42]  
hTetroGA  Random  25, 50, 100  · Crossover: Singlepoint crossover operator · Mutation: Classic GA mutation operator · Removal Operator: Translational motion command, Nontranslational motion command · Rearrangement Operator  $\begin{gathered}F= \frac{1}{1+W_{A^*}\left(P O S\left(p_{l_{p^{1}}}\right)\right)}\end{gathered}$  NSGAII technique.  50  Single (Reconfigurable tilted robot)  Static (HShaped, Spiral and 3Slit) Dynamic (Perpendicular and Parallel)  24 × 24 Grid  Robotic Operating System (ROS)  · NSGAII technique is implemented to determine best motion command sequence. · hTetroGA algorithm can be implemented for reconfigurable robots during a rescue task.  [43] 
Novel GA  Random  20  · Crossover: Singlepoint with crossover rate(pc) = 1.0 · Mutation: Bitwise flipping with mutation rate(pm) = 0.1 (1/string length)  Parent Selection: Rankbased roulette wheel Survivor selection: Elitism + crowding distance (NSGAII)  2000  Single (Two wheeled mobile robot running on STM32 microcontroller)  Static  10 × 10 Grid  Robotic Operating System (ROS) and Python 3  · Experimented mobile robot is localized by the ArUco system through a bird’s view camera. · Mobile robot can’t operate on rough environment because its dynamic behaviour is not considered in the work.  [40]  
GA  Random  · Crossover: singlepoint  $\begin{aligned} F =N\left(\alpha_1 \sum_{i=1}^n d_i\right. \left.+\alpha_2 * m\right)\end{aligned}$  Roulette  Between 100 and 500 generations  Single  Static (Special, Regular, Irregular multiple).  100 × 100 Geometrical  · Large turning angle problem in basic genetic algorithm is overcome. · Genetic algorithm implementation is separate from path smoothing process. · Experimental results are compared with Dijkstra algorithm  [44] 
The concept of ACO was first introduced by Dorigo in his Ph.D. dissertation in 1992 [45]. This algorithm is inspired by the sophisticated social behaviors exhibited by ants during their search for food. A key element of this behavior is the deposition of pheromones, which serve to guide other ants by creating a trail to the food source. The trail's pheromone concentration intensifies as more ants traverse it, thereby increasing the likelihood of it being followed by additional ants. Notably, the shortest route to the food source becomes the most popular among the ants, as it can be traversed in the least amount of time. This phenomenon was first observed in the renowned Double Bridge experiment [46], where ants consistently selected the shortest path over time when presented with multiple routes to a food source. Pheromone evaporation also plays a crucial role in this process. It serves as a mechanism to prevent the ants from getting trapped in locally optimal solutions [47]. As the pheromone evaporates, the attractiveness of a given path diminishes, reducing the likelihood of it being selected by other ants. Additionally, on the shortest path, the rate of pheromone deposition surpasses its rate of evaporation, ensuring that a high pheromone level is maintained. In the context of ACO algorithms, this concept is applied to the selection of paths between nodes. The probability of an ant, situated at node i, choosing to move to another node j in the network, is influenced by the level of pheromone deposition on the potential paths, as described in reference [47].
where, $\tau_{i j}^k$ denotes pheromone levels. Analogous to the natural tendencies of ants, paths with elevated pheromone concentrations are more likely to attract ants in the algorithm, leading to a preference for these paths over others with lower pheromone levels [48].
Ant Colony Optimization  

1  Initialize nodes and necessary parameters  
2  Initialize pheromone level of each node  
3  Define maximum iterations ITR  
4  while (ITR$>$0) do  
5  for each ant k do  
6  $\eta_j \leftarrow$ heuristic function of the search space (fitness value)  
7  Transition_probability $[j] \leftarrow p_{i j}^k(t)$  
8  Select node with the highest $p_{i j}^k(t)$  
9  Update pheromone level $\tau_{i j}(t+1)$  
10  end for  
11  ITR=ITR1  
12  end while  
13  Best solution $\leftarrow$ solution with best $\eta_j$  
14  return Best solution 
The heuristic function: $\eta_{i j}^k=\frac{1}{d_{j e}}$.
The pheromone update:
$\Delta \tau_{i j}^k(t)= \begin{cases}\frac{Q}{L_k}, & \text { if ant } k \text { passes node } i \text { and } j \\ 0, & \text { otherwise }\end{cases}$ is the quantity of pheromone deposited, where $Q$ is a constant and $L_k$ is the total length of the path that ant $k$ travels. A pseudocode of this algorithm is presented in Table 3.
Numerous scholars have conducted comprehensive research on the operational mechanics, structural design, and optimal parameterization of ACO, proposing various enhancements to address these areas, as detailed in Table 4. Additionally, a range of variables, as listed in Table 5, have been considered in these studies. Liu et al. [49] optimized crosspath nodes in the path search process using the ant colony algorithm combined with geometric optimization, which improved the algorithm’s effectiveness and path quality via pheromone updates. You et al. [50] developed a novel heuristic operator to augment the diversity and convergence of the population search. Dai et al. [51] addressed issues related to global convergence speed and path smoothing by enhancing an A* algorithmbased ACO and the maximumminimum ant system, incorporating a retraction mechanism to circumvent deadlocks. Jiao et al. [52] proposed an adaptive state transfer and pheromone update method, enhancing the significance of heuristic information and pheromone strength in the iterative process of the algorithm, thereby improving its adaptability to diverse environments and its capacity to escape local optima. Akka and Khaber [53] refined the state transfer formula to prioritize the selection of neighbor nodes with the most exits as the subsequent node. This enhanced algorithm introduces diversity to the search process and mitigates the impact of ineffective pheromones by dividing the multiheuristic function, separately rewarding and penalizing the worst path, as outlined by Yang et al. [54].
Types  m  α  β  ρ  Q  N_{max}  φ  γ  ξ  υ  δ  Selection of Next Node  Type of Vhicle  Type of Obstacles  Type of Maps  Software  Remarks  Ref. 
An improved ant colony algorithm (ACOPD)  10  1.1  12  0.5  200  0.01  $\sqrt{2}$  Single  Static  Grid  · Proposed method solves convergence speed problem in ACO. · Geometric optimization method is implemented to improve path generated by ACO.  [49]  
DLACO [PEACO and TPOA]  20 10  1 0.3  3 0.8  0.03 0.1  100 100  1  0.9  Roulette wheel selection  Single (Rikirobot)  Static  Grid  · PEACO and TPOA is combined to generate path and avoid obstacles. · Proposed method gives better results in path distance, smoothness and good solution rate when compared to APACA and MOFA. · Experimentation is done with Rikirobot powered by Raspberry Pi and Rplidar A1. · Implements piecewise BSpline to smoothen path.  [55]  
LFACO  50  1  3  0.3  100  50  1  10  5  20  Roulette wheel  Multiple robot  Static  Grid  MATLAB and Robotic Operating System (ROS)  · Proposed method aims to solve multirobot path planning. · Pheromone update in ACO incorporates pheromones of leader and follower ant. · Maximumminimum ant approach is employed for global search. · Generated path is optimized by TPOA and dynamic cutpoint method.  [54]  
APACA  120  1  5  0.9  100  200  Single (Smart wheelchairs)  Static  20 × 20 Grid (in subgraph (a) of Figure 3)  · Implementation of Direction determining Method to speed up convergence rate for global optimal search. · Proposed method shows better outcomes in number of iterations and path length when compared to IACA and GPACA.  [52]  
RMACA  50  1  5  0.5  10  54% lower than [56]  Single  Static  Grid. (Common, tunnel, trough, baffle maps)  MATLAB  · Retraction mechanism is employed to avoid local minimum. · Improved Maximumminimum ant approach performs global search. · Estimation function in A* improves search efficiency of ACO. · RMACA is better in convergence rate and bending suppression effect.  [51]  
IACA  30  1  5  2  100  Single  Static  Grid  MATLAB  · Stimulating probability is introduced to improve transition rule. · Unlimited step length principle is used as heuristic information for path search efficiency. · Dynamic change in evaporation rate increases convergence speed.  [53]  
Ant Colony Optimization and Fuzzy Control  80  1  9  0.5  1  100  Roulette wheel  Single  Static  20 × 20 Grid  MATLAB  · Fuzzy algorithm controls to adjust evaporation rate. · Proposed critical obstacle influence factor generates initial pheromone distribution.  [57]  
IACOA*  50  0, 1, 2, 3, 4, 5  0, 1, 3, 5, 7, 9  0, 0.1, 0.3, 0.5, 0.7, 0.9  1  100  Single  Static  20 × 20 Grid (in subgraph (c) of Figure 3)  MATLAB R2020b  · Proposed modelled environment is based on geometric modelling and Monte Carlo calculation. · Proposed method gives optimum results in terms of path length, cumulative radiation dose and energy consumption when compared to other algorithms.  [58]  
IAACO  50  1  7  2.5  100  1  Single  Static  20 × 20 Grid in subgraph (b) of Figure 3)  · The transition probability is induced with angle guiding factor and obstacle exclusion factor to enhance path search efficiency. · Heuristic information adaptive adjustment factor and adaptive pheromone volatilization factor are introduced into the pheromone update rule for optimum global search.  [59] 
Variables  Description 
$\mathrm{m}$  Ant's population size 
$N_{m a x}$  Maximum iteration number 
$\alpha$  Weight of Pheromone 
$\beta$  Weight of Heuristic information 
$\rho$  Pheromone evaporation ratio 
$\mathrm{Q}$  Pheromone's Intensity 
$\tau_{i j}$  Pheromone on the path between i and j 
$\eta_{i j}$  Heuristic information on j 
$\xi$  Distance factor coefficient 
$\varphi$  Distance correction parameter 
$v$  Ant’s importance on moving straight 
$\delta$  Parameter to update Pheromone 
$\gamma$  Diffusion coefficient 
PSO is inspired by the collective behavior of animals like birds, tetrapods, or fish in their pursuit of food. This approach mirrors the natural group dynamics observed in these species, where there is no singular leader guiding the group to the food source [60]. In such a group, each individual may not know the precise location of the food, but they can approximate their proximity to it. Independent efforts by each animal to reach the food source would be inefficient, leading to extended time frames and chaos. Consequently, the most effective strategy is for the members to follow those closest to the food source [29]. In PSO, each individual animal is analogous to a solution, possessing two critical pieces of information: Their fitness value, derived from the objective function; The velocities that guide the solution towards the target location.
The algorithm commences with a set of solutions or particles. Each particle navigates the solution space, returning their fitness value, known as pbest, in each iteration. The best pbest value from each iteration is recorded as the global best value, gbest. Based on these two values, the algorithm updates the velocity and position of each particle. The search process concludes either upon reaching the maximum number of iterations or upon identifying the optimal solution. The formulas for updating the velocity and position of particles in PSO are outlined subsequently.
The inertia weight is denoted as $w, c_1, c_2$ are the learning factors, $r_1, r_2$ are normal distribution random numbers within the interval $[ 0,1], \eta$ represents the velocity constraint proportional factor, $v_{i d}$ represents the velocity of the $i$th particle in d dimension, and $x_{i d}$ represents the position of the $i$th particle in d dimension. The procedure to implement this algorithm is described in Table 6.
Particle Swarm Optimization  
1  Initialize particle population size S  
2  GEN $\leftarrow$0  
3  Initialize GEN_{max}  
4  for (each particle i = 1,...,S) do  
5  Initialize particle’s position x_{i}  
6  Initialize particle’s best known position to initial position: p_{i }$\leftarrow$ x_{i}  
7  Initialize particles velocity v_{i}  
8  if fitness(p_{i}) > fitness(g) then  
9  g $\leftarrow$ p_{i}  
10  end if  
11  while GEN < GEN_{max} do  
12  for each particle i = 1,...,S do  
13  for each dimension d = 1,...,n do  
14  r_{1}, r_{2} $\leftarrow$ random normal distribution in [0,1]  
15  Update particle’s velocity v_{id}  
16  Update particle’s position x_{id}  
17  end for  
18  If fitness(x_{i}) > fitness(p_{i}) then  
19  p_{i} $\leftarrow$x_{i}  
20  If fitness(p_{i}) > fitness(g) then  
21  g $\leftarrow$ p_{i}  
22  end if  
23  end if  
24  end for  
25  GEN = GEN + 1  
26  end while  
27  Best solution solution with best $\eta_j$  
28  return Best solution 
The PSO is analogous to the GA in its initiation with a randomly formed population set, subsequently evaluated based on fitness values. This methodology has been effectively applied in various navigation contexts, such as aerial robot navigation in unknown threedimensional environments [61], humanoid robot navigation [62], and industrial robot navigation [63]. The performance efficacy of PSO is contingent on the precision in adjusting, controlling, and updating its parameters.
Since its inception in 1995, numerous approaches have been suggested to refine these aspects and enhance the overall functionality of PSO. Traditional techniques for parameter adjustment and control include Fixed Inertia Weight (FIW) [64], [65], Linearly Decreasing Inertia Weight (LDIW) [64], [65], [66], [67], Time Varying Acceleration Coefficient (TVAC) [65], [68], Random Inertia Weight (RANDIW) [64], [65], [66], [68], Random Acceleration Coefficients (RANDAC) [69], and Fixed Acceleration Coefficients (FAC) [64], [65], [66], [68].
Table 7 compiles various studies that have proposed improvements and hybridizations of PSO to address path planning challenges. Dewang et al. [70] introduced an adaptive particle swarm optimization (APSO) that dynamically alters the inertia weight in each iteration, initiating the search with a high inertia weight to avoid local minima, and gradually reducing it to focus on exploitation as iterations progress. This strategy yielded superior results in comparison to standard PSO in terms of path length and planning time, as demonstrated in the environment depicted in subgraph (a) of Figure 4. Chai et al. [71] combined PSO with the Probabilistic Roadmap Method (PRM) to enhance sampling in PRM. This hybrid method leverages knowledge of sample points in obstacleladen regions to refine sampling in open spaces, particularly in narrow passages, thereby improving connectivity. Masehian and Sedighizadeh et al. [72] utilized PSO to derive the shortest and smoothest feasible paths. Particles are initialized based on points identified in free space by a robot's laser sensor, with the optimal particle's position determined by the sensor readings. PRM serves as the local planner for obstacle avoidance, and simulation results indicate a more efficient runtime compared to the basic PRM approach. Li et al. [73] presented an improved PSO that initializes particles through uniform random distribution, employs an exponentially decaying inertia weight to enhance planning efficiency, and integrates cubic spline interpolation for path smoothing. This variant was benchmarked against other PSO variants, with comparisons based on performance indices and path planning metrics, where IPSO showed promising results. Song et al. [74] developed a fractionalorder PSO variant (FOPSO) that introduces adaptive fractionalorder velocity and utilizes Bezier curves for path smoothing. Its performance was evaluated against other PSO variants using benchmark functions. Finally, Alam et al. [75] implemented PSO for random sampling along grid lines between start and goal points, with an initial spacing of points along the Euclidean path. The effectiveness of this approach was validated through simulations in various environments with static obstacles.
Types  Particle Size  Inertia Weight  Cognitive factor (c1)  Social Factor (c2)  Number of generations  Type of Vehicle  Type of Obstacles  Type of Map  Software  Remark  Ref. 
Hybrid PSO  20  10%  1.5  1.5  500  Single (Mobile robot)  Dynamic  200 × 200 Geometrical  MATLAB 2018b  · The performance of hybrid PSO and ACO on shortest path and least time constraint is measured against PSO and ACO separately. · Although it’s observed that PSO outperforms ACO, the hybrid gives a superior results.  [76] 
EDPSO  150  1  0.4  0.4  150  Single  Static and Dynamic  20 × 20 Geometrical (see in subgraph (d) of Figure 4)  · Peaks of diversity in population gives room for more exploration in the search space. · Can be applied to smart vehicles with slow movements. · Comparison was made on all the cited metaheuristics using 10 complex multimodel functions from the CEC 2019 benchmarking suite.  [77]  
PSOAWDV  200  0.9, 0.5  2.0, 1.0  1.0, 2.0  100  Single  Static  11 × 11 Geometrical (see in subgraph (c) of Figure 4)  · Quartic Bezier transition curve with three control points is implemented to smoothen planned path.  [78]  
Improved PSO  10  0.4, 0.9  2  2  3000  Single  Static  18 × 18 Geometrical  MATLAB R2016a  · Proposed mutation operation increases particle diversity. · Proposed deredundant algorithm removes needless points, thus improving generated path.  [79] 
FIMOPSO  50  0.4 to 0.9  100  Single  Static and dynamic  210 × 178 Geometrical (in subgraph (b) of Figure 4)  MATLAB  · Constraints to be minimized are path length, motor torque, travel time, robot acceleration; obstacle avoidance is maximized. · Obstacle avoidance problem is solved with Fuzzy inference system.  [80] 
The ABC algorithm, conceived by Dervis Karaboga for addressing polynomial mathematical problems [81], is inspired by the foraging behavior of honey bees. In their natural environment, honey bees use pheromones and a waggle dance to communicate information about food sources. When a bee finds a food source, it evaluates the nectar quantity, returns to the hive, and performs a waggle dance to convey information about this source. The quality of the food source is indicated by the vigor of the waggle dance. In the context of the ABC algorithm, the location of a food source represents a potential solution to an optimization problem, and the quality of the solution is analogous to the nectar content of the food source. The ABC algorithm categorizes bees into three roles: employed bees, onlooker bees, and scout bees. It is assumed that for each food source position, there is one corresponding employed bee. Employed bees share information about the location and quality of food sources with onlooker bees through the waggle dance. Onlooker bees then select food sources based on their perceived quality, meaning that higherquality sources are more likely to be chosen. If employed bees abandon a food source, they transition into scout bees, embarking on the search for new food sources. Scout bees memorize the quality of discovered food spots and compare them with known sources to identify the most promising ones. The ABC algorithm's pseudocode is illustrated in Table 8. Initially, the ABC algorithm establishes a population of food source positions (SN), where SN represents the population size. Each food source, or solution, is a Ddimensional vector, with D being the number of optimization parameters. Each food source is linked to a probability value $p_i$, influencing the decisionmaking of onlooker bees.
Artificial Bee Colony Algorithm  
1  Initialize bee population size SN = number of employed bees = number of observer bees  
2  Evaluate fitness of each bee f(sol)  
3  Set best solution, solBest $\leftarrow$ sol  
4  ITR $\leftarrow$0  
5  Initialize ITR_{max}  
6  while ITR < ITR_{max} do  
7  for each employed bee i = 1,...,SN do  
8  Select random solution and apply random neighbourhood structure  
9  Determine the probability of each solution, p_{i}  
10  end for  
11  for each employed be do  
12  sol $\leftarrow$select solution with highest probability  
13  apply random neighbourhood structure  
14  If f(sol) < f(solBest) then  
15  solBest $\leftarrow$sol  
16  end if  
17  end for  
18  ITR = ITR + 1  
19  end while  
20  return Best solution, solBest 
where, $f i t_i$ is the fitness of solution $i$, and $SN$ is number of employed bees (population size). The generation of a new food source position from an existing one is determined using the following expression:
where, $k$ and $j$ are random values in sets $\{1,2 \ldots, \mathrm{SN}\}$ and $\{1,2, \ldots, \mathrm{D}\}$ respectively. $k$ should be different from $i$. $\emptyset_{i j} \in[1,1]$ controls the production of a neighbour food source position around $x_{i j}$.
Research in the field of ABC for path planning has led to a variety of advancements and hybrid approaches, as detailed in Table 9. One notable development in the navigation of smart vehicles is the combined Artificial Bee Colony and evolutionary programming approach proposed by ContrerasCruz et al. [82]. In this methodology, the ABC algorithm serves as the local search method, while evolutionary programming is employed to enhance the potential paths obtained. This approach was initially applied in multirobot systems and later refined for use in unfamiliar environments with dynamic obstacles [83]. However, certain shortcomings were identified, such as neglecting the distance between new bee positions and obstacles. To address these issues, an improved ABCEvolutionary Programming (ABCEP) approach was proposed by Kumar and Sikander [84]. Further advancements include the Adaptive Dimension LimitArtificial Bee Colony Algorithm (ADLABC), introduced by Kamil et al. [85] for optimizing the global path of mobile robots. This algorithm demonstrated its efficacy by finding solutions with fewer iterations and reduced computational time. Another development, the Directed Artificial Bee Colony algorithm, was shown to yield better results in dense environments, such as maps with numerous static obstacles, compared to other leading algorithms [86]. In an effort to curtail computational time, particularly crucial in realtime path planning scenarios, Szczepanski and Tarczewski [87] proposed a hybrid approach combining the ABC and Dijkstra algorithms. This amalgamation aimed to balance the comprehensive search capabilities of the ABC algorithm with the efficiency of Dijkstra's algorithm, particularly in environments where quick computation is vital.
Types  Population Size  Number of Iterations  Type of Vehicle  Type of Obstacles  Type of Map  Software  Remark  Ref. 
ADLABC  100  1000  Single  Static  10×10 Geometrical  MATLAB  · Implemented dynamic control limit reduces computational time and number of iterations. · Generated path is smoothened using cubic polynomial through three via points. · Better results are observed when proposed method is compared to ABC.  [85] 
ABCEP  10  500 generations (for EP)  Single (XidooBot, mobile robot Pioneer 3AT)  Static  Various Geometrical and Grid maps. (see Figure 5)  C language  · While ABC builds a path in collision free space, EP improves the path using mutation to produce a short path. · Proposed approach was implemented in some benchmark maps. · ABCEP is deployed to an experimental platform to show its feasibility.  [82] 
Improved ABCEP  200, 400, 600, 800, 1000  100  Single  Static and Dynamic  100m×100m Geometric  MATLAB 2018b  · Best food points among randomly distributed ones which aid in finding optimum path are selected its distance to goal point and nearest obstacle. · When determining the optimum path, takes into account the distance between the new bee position (best node) and any surrounding barriers.  [84] 
Enhanced ABC  25  100  Single  Static  100×100 Grid  · Cubic Ferguson spline is introduced to smoothen path generated by ABC.  [88] 
The FA, introduced by Yang [89], is inspired by the bioluminescent communication of fireflies. The key aspect of this communication is the distinctive light patterns emitted by fireflies, where the intensity of the light is indicative of a firefly's attractiveness. In the context of the FA, this attractiveness is analogous to the fitness value of a solution. The algorithm operates on the principle that among any two fireflies, the one emitting brighter light will attract the one with dimmer light. The attractiveness of a firefly is quantified as per the following equation [90]:
where, $\beta$ denotes the attractiveness, $\beta_0$ represents the maximum attractiveness, which is typically set to $1, \gamma$ is the light absorption coefficient ranging between [0.1,10], and $r_{i j}$ is the distance between firefly $i$ and firefly $j$, calculated using the standard Euclidean distance formula:
where, $x_i$ and $x_j$ are the positions of firefly $i$ and firefly $j$ in the ddimensional space, respectively; $\alpha$ and $\phi$ are random numbers in the distribution [0,1]. The pseudocode for FA is described in Table 10.
Firefly Algorithm  
1  Objective function f(x), $\mathrm{x}=\left(x_1, x_2 \ldots, x_d\right)$  
2  Initialize population size x_{i} (i=1,2,…,n)  
3  Determine the intensity (I) of each firefly determined by f(x)  
4  Initialize GEN_{max}  
5  while (GEN < GEN_{max}) do  
6  for (i = 1 to n) do  
7  for (j = 1 to i) do  
8  if $\left(I_j>I_i\right)$ then  
9  Vary attractiveness with distance r via $\exp (\gamma r)$  
10  move firefly i towards j  
11  Evaluate new solutions and update light intensity  
12  end if  
13  end for  
14  end for  
15  Rank fireflies and find the current best  
16  GEN = GEN + 1  
17  end while  
18  return Best firefly 
The FA has been implemented in diverse research areas, including robotics [91], [92], machine learning [93], journalism [94], and cloud computing [95]. Table 11 presents a compilation of techniques proposed for FA's application in smart vehicle navigation. In an effort to enhance the convergence speed and local search accuracy of the standard FA, Chen et al. [96] introduced a modified version (PPMFA) that incorporates a Gaussian random number into the fixed step size. This addition enhances the diversity of the firefly population, helping to prevent the algorithm from stagnating in deadend zones. Additionally, a novel path center technique was employed to calculate distances between fireflies, essentially representing paths. This method involves connecting geometric centers of path segments to form new segments and repeating the process until a single segment remains, termed as the path center. The distance between two path centers is used as the measure of distance between two fireflies. The PPMFA showed superior performance in accuracy and convergence speed compared to PSO and the Standard SFA. Duan et al. [97] proposed a Developed Firefly Algorithm (DFA) to address multiobjective navigation challenges. The algorithm extends the grid map to create feasible paths and employs the Pareto dominance relationship for path comparison and segregation. Nondominant fireflies are stored in an elite record library for comparison during iterations. DFA includes an evolutionary stage for optimizing paths by adding, removing, or swapping points. When tested against NSGAII on a ZDT1 instance, the DFA demonstrated enhanced efficiency. Patle et al. [98] utilized the firefly algorithm for mobile robot navigation in dynamically obstacleladen environments. An AI mechanism navigates the robot, while a controller based on FA detects and avoids obstacles, generating paths (fireflies) and selecting the optimum one using Euclidean distance from the nearest obstacle. This approach outperformed other intelligent methods in terms of path length in various scenarios. Experiments with the KheperaII robot showed a deviation of about 5.7% from simulated results. Hassan and Fadhil [99] developed a modified firefly algorithm for path planning of mobile robots in 3D spherelike, partially dynamic environments. In this approach, each firefly is viewed as an agent navigating around obstacles, with the generated paths considered potential solutions. The optimal path is selected based on path length and completeness. This modified approach was noted for its minimal memory requirement and effective performance in spherical spaces.
Type  Population Size  Generation  $\gamma$  $\boldsymbol{\beta}_0$  $\alpha$  Type of Vehicle  Type of Obstacles  Type of Map  Software  Remark  Ref. 
FFA 




 Single  Static  10×10 Geometrical 
 · A* algorithm is implemented to obtain the shortest path. · Cubic polynomial spline is interpolated on the generated path to produce smooth trajectory using iterative random selection.  [91] 
FAMCPSO 

 1  2  0.5  Single  Static  600cm×800cm Geometrical  MATLAB 2018b  · Proposed method is a combination of MCPSO and FA. · Consideration of inverse dynamic and kinematic modelling to obtain optimum torque and velocity for wheels of AMR. · The recommended hybrid method shows good results when compared to various algorithms in different environments.  [100] 
MOFA  200  150 


 Single  Static  Grid (see subgraph (a) of Figure 6)  C/C++ language  · Path safety, the path length, and the path smoothness are considered in the design of proposed method.  [101] 
FATPM  5  100  50 – 100  0.1  1  0.11  0.11  Single (Fire Bird V robot NEX Robotics and Embedded RealTime Systems Lab, CSE IIT Bombay)  Dynamic  Grid (see in subgraph (b) of Figure 6)  Microsoft Visual C++, 2010 with OpenGL  · While TPM searches for obstacle free path, FA performs obstacle avoidance.  [102] 
Developed in 2009, the CSA is a natureinspired optimization technique designed to tackle complex problems [103]. Its conceptual framework is based on the intriguing brood parasitism behavior observed in certain cuckoo species. These birds are known for their strategy of laying eggs in the nests of other host species. Cuckoos often adapt the appearance of their eggs, mimicking the color and pattern of the host species' eggs [104], thereby reducing the likelihood of the host species detecting the foreign egg. In cases where the host species identifies and removes the cuckoo egg or abandons its nest to start anew, the cuckoo must find another host nest. Notably, cuckoo eggs typically hatch faster than those of the host species, allowing the young cuckoo to dispose of the host's eggs and monopolize the food supplied by the unwitting host [105]. The behavioral rules of the cuckoo birds, as translated into the CSA, can be summarized as follows:
Each cuckoo randomly selects a nest in which to lay its egg.
The nests that successfully retain cuckoo eggs, escaping detection and eviction by the host species, are carried forward to the next generation.
The probability of a host bird discovering an alien egg is denoted by $P \in[ 0,1]$, and the total number of available host eggs or nests within the search space remains constant.
The CSA primarily employs a random walk strategy for nest searching, with Levy flight [103] being the most commonly used method due to its efficiency in exploring the search space. In the context of path planning problems, the nests and eggs within the CSA framework can be metaphorically viewed as solutions, where host eggs in a nest represent current solutions and a cuckoo egg symbolizes a new, potentially superior solution $x^{t+1}$. The objective is to replace less effective solutions with more viable ones (represented by the cuckoo’s egg).
where, $i_{\text {i }}$ represents the $i$th particle, $t$ stands for the iteration cycle, $\alpha>0$ is the step size, and $\oplus$ denotes entrywise multiplication. Step lengths of Levy flight are distributed according to this probability.
where, $L$ represents step size length and $\gamma$ denotes the variance. $\mathrm{P}, \gamma$, and $\alpha$ are are critical to the algorithm's performance and require careful tuning to enhance solution quality. One of the advantages of the cuckoo search algorithm is its minimal need for parameter finetuning, coupled with its ability to handle multimodal objective functions effectively. The implementation of the Cuckoo Search Algorithm is further elucidated in a pseudocode format, as illustrated in Table 12.
The implementation of the CSA spans a wide array of fields, demonstrating its versatility and effectiveness. This includes applications in vehicle routing [106], [107], neural networks [108], scheduling [109], [110], medical fields [111], [112], cloud computing [113], and notably in robotics, particularly in the navigation of smart vehicles. Table 13 showcases a range of hybrid approaches combining Cuckoo Search with other methods to address path planning challenges. In the realm of multirobot collaboration and navigation within densely obstacleladen maps, Sahu et al. [114] introduced a Modified Cuckoo Search as a novel solution. This adaptation of the CSA was specifically designed to enhance collaborative strategies and navigation efficiency in complex environments. A comparative study by Ab Wahab et al. [23] assessed the performance of the cuckoo search against other metaheuristic algorithms and traditional path planning methods across various scenarios. The study highlighted CSA's strengths and potential areas for integration with other techniques. To address the dual goals of minimizing computational costs and maximizing efficiency in mobile robot path planning, Garip et al. [115] proposed a hybrid algorithm that combines the principles of cuckoo search with firefly and particle swarm optimization. This hybrid approach aimed to leverage the strengths of each method to produce a more robust and efficient path planning solution. In the context of quantum computing, Kundra et al. [92] utilized cuckoo search to prevent premature convergence in the proposed quantum firefly algorithm. This application underscores CSA's utility in enhancing the stability and performance of other advanced algorithms. Additionally, to optimize robot paths in environments with dynamic obstacles, Kumar et al. [116] implemented a Modified Cuckoo Search algorithm. This version of CSA was tailored to process obstacle distance and heading angle data from robot sensors, enabling more adaptive and responsive path planning in changing conditions.
Cuckoo Search Algorithm  
1  Objective function f(x), $x=\left(x_1, x_2, \ldots, x_d\right)$  
2  Initialize population of n host nests  
3  ITR $\leftarrow$0  
4  Initialize maximum number of generations ITR_{max}  
5  while (ITR < ITR_{max}) do  
6  i $\leftarrow$Get a cuckoo randomly by levy flight  
7  Evaluate fitness(i)  
8  j $\leftarrow$choose a nest  
9  if fitness(i) > fitness(j) then  
10  replace j by the new solution  
11  end if  
12  Abandon a fraction(P_{a}) of worst nest and build new ones  
13  Keep the best nests  
14  Rank the nests and find the current best  
15  Pass the current best solutions to the next generation  
16  ITR = ITR + 1  
17  end while  
18  return Best nest 
Type  P  $\gamma$  $\alpha$  Population Size  Number Generations  Type of Vehicle  Type of Obstacles  Type of Map  Software  Remark  Ref. 
MCSSCAPSO  0.25  30  Multiple (Epuck robot)  Static and dynamic  450 × 450 Geometrical (see subgraph (a) of Figure 7)  C language  · PSO performs local search, CSA performs global search, and sine cosine algorithm implements greedy approach.  [117]  
Improved CSA  0.25  30  Single  Dynamic  Topological  MATLAB 2014a  · Global search ability is improved by introducing mutation and crossover. · Convergence rate and optimization accuracy of algorithm are tested using unimodal and multimodal functions.  [118]  
Hybrid CSABA  20  500  Single  Static  12 × 12 Geometrical (see in subgraph (b) of Figure 7)  MATLAB  · The proposed approach is implemented in two maps with various positions of circular obstacles.  [119]  
Hybrid geneticcucking  0.25  1  400  40  Single  Static  10000 × 8000 × 5000 Geometrical  MATLAB  · Using intelligent algorithms in path planning of 3D environment is studied. · Spherical obstacles are implemented.  [120]  
CSAPSOFA  1  0.2  20  1000  Single and multiple (Kobuki mobile robot)  Static  200 × 160 and 100 × 100 Grid  MATLAB, ROS  · CSPSOFA algorithm is investigated both in simulation and experimentally.  [115]  
CSABA  0  1  30  500  Single  Static  12 × 12 Geometrical  MATLAB 2015b  · CSABA obtains a better result in path length than CSA and BA.  [121] 
The WOA, introduced by Mirjalili and Lewis [122], is an innovative algorithm that emulates the bubblenet hunting behavior of humpback whales [123]. These whales, known for their social nature, often forage in groups, preying primarily on krill and small fish located near the water's surface. A notable aspect of their hunting technique involves diving approximately 12 meters deep and then engaging in a unique strategy to encircle their prey. This involves creating a ring of bubbles as they spiral upward towards the surface, effectively trapping the prey. This hunting method, particularly the encircling maneuver and the spiral bubblenet movement, has been translated into a mathematical model for the WOA. The algorithm draws inspiration from these distinct whale behaviors to devise a search and optimization strategy. The spiral path and bubblenet formation are key elements that have been abstracted and applied in the algorithm to simulate the whale's approach to localize and encircle the prey effectively. The mathematical representation of these behaviors enables the WOA to efficiently explore and exploit the search space in various optimization problems.
(1) Encircling prey
In the WOA, the behavior of a humpback whale encircling its prey is a key mechanism. When a whale identifies the location of its prey, it begins to encircle it. In the context of WOA, this is modeled by assuming that the best current solution in the search space is akin to the whale closest to the target prey. Consequently, other search agents (whales) in the algorithm adjust their positions relative to this best agent, simulating the encircling behavior. This behavior is represented mathematically in the following equations:
where, $A$ and $C$ are coefficients, $t$ denotes current iteration, $X^*$ is the best whale position, $X$ is the current whale position, $a$ is a decreasing constant that linearly reduces from 2 to 0 over the course of iterations and is given by $a=2\frac{2 t}{M}$ (M: maximum number of iteration), $r \in[ 0,1]$ is a random value.
(2) Spiral bubblenet manoeuvre (Exploitation phase)
As the value of $a$ decreases, the radius of encircling the prey diminishes. With $A$ being a random value within the range $[a, a]$, search agents can discern the relationship between their current position and the optimal position when $A$ is reduced to the interval $[1,1]$. Additionally, during the spiral movement phase, the position of the whale relative to the prey is updated. The distance $D^{\prime}$ between the $i$th whale and the prey (which represents the best solution obtained so far) is calculated accordingly:
where, $b$ is a constant that defines the shape of the logarithmic spiral, and $l$ is again a random value in the range [1, 1]. The spiral movement is an integral part of the whale's hunting strategy, where it combines the encircling maneuver with a simultaneous inward spiral motion towards the prey. In the WOA, the whale's position is updated based on a probability of 50% to either continue encircling the prey or to engage in the spiral movement. This probabilistic approach enables the algorithm to balance between exploration and exploitation, effectively mimicking the hunting behavior of humpback whales. The decision between the two behaviors is governed by the following equation:
(3) Searching for prey (Exploration phase)
In the exploration phase of the WOA, a whale updates its position based on a randomly selected agent, rather than the current best agent. This phase is crucial for global search within the solution space. When $A<1$, the algorithm switches to exploration mode. In this case, given $X_{\text {rand }}$ as the position of a random agent, the updated position of a whale is determined using the following equations:
The pseudocode detailing the WOA is illustrated in Table 14. This algorithm has been widely applied across various research domains. It has been utilized for image segmentation [124], in the validation of welded Al/Cu bimetal sheets [125], for intelligent facial emotion recognition [126], to enhance power system stabilizers [127], and in task scheduling for microprocessor systems [128].
In robotics, WOA has been instrumental in planning the joint trajectory of robotic arms [129], enhancing robotic manufacturing processes [130], planning navigation of unmanned vehicles [131], aiding in multiple robot space exploration [132], [133], and optimizing deep neural networks [134]. Table 15 presents an overview of various studies where WOA has been implemented in smart vehicle navigation.
Whale Optimization Algorithm  
1  Initialize the whale population $X_i(i=1,2, \ldots, n)$  
2  Calculate the fitness of each whale  
3  $X_{\text {best }}$= the best search agent  
4  while (t < maximum number of iterations)  
5  for each search agent  
6  Update $a, A, C, l$ and $p$  
7  if (p < 0.5) then  
8  if ($A<1$ ) then  
9  Update current agent via Encircling Prey  
10  else  
11  Select a random agent (X_{r}and)  
12  Update current agent via Search for Prey  
13  else  
14  Update search agent via Spiral Bubblenet  
15  end for  
16  Amend the position of whales that are outside the search space  
17  Calculate the fitness of each search agent  
18  Update X_{best} if there is a better solution  
19  t = t + 1  
20  end while  
21  return $X_{\text {best }}$ 
Type  Population Size  Number of Iterations  Type of Vehicle  Type of Obstacles  Type of Map  Software  Remark  Ref. 
WOA 
 100  Single (Khepera II mobile robot)  Static  Geometrical  MATLAB  · Algorithm solves robot scheduling problem in a manufacturing environment. · Novel mathematical model is proposed and experimentally tested on 26 benchmark functions.  [135] 
Improved WOA based on GA 
 100  Single  Static  20×0 Grid 
 · Proposed method can be implemented for logistic mobile robot. · Efficiency of proposed algorithm is improved by 10.71% compared to traditional WOA.  [136] 
MWOA  100  500  Single  Static  300×500 pixels Geometrical 
 · Distance and smooth path functions are minimized. · The pareto frontoptimal solution gives the optimal solution for MWOA. · The proposed method has a lower error rate than the MultiObjective Genetic Algorithm (MOGA) method [137].  [138] 
MOWOA  50, 80, 100, 150  50, 70, 90, 110  Multiple  Static  15×15 Grid (see Figure 8)  MATLAB  · At 130 iterations and 150 waypoints, proposed algorithm outperforms compared deterministic and hybrid stochastic exploration algorithm. · Map exploration and minimum time map enhancing accuracy is the idea behind proposed algorithm.  [139] 
NWOA 
 1000  Single (Raspberry Pi (3B+))  Static and dynamic  1800×1800 Geometrical  Python  · Adaptive technology, enhanced potential field factors and virtual obstacles are introduced to optimize the convergence rate of the algorithm. · NWOA performance better in convergence rate when compared to WOA, GAWOA, and EGEWOA.  [140] 
Updated WOA 
 500  Single  Static  8×8 Geometrical 
 · Proposes a changed whale advancement calculation based Mobile robot way determination.  [141] 
Proposed by Mirjalili et al. [142], GWO is an algorithm inspired by the social hierarchy and hunting techniques of grey wolves. Grey wolves are apex predators that typically live in packs of 5 to 12 members, each adhering to a strict social structure as depicted in Figure 9.
In this hierarchy, the alphas ($\alpha$)—a male and a female—serve as the leaders. Positioned at the top, they represent the fittest solution, and their directives are followed by the rest of the pack. The beta ($\beta$) wolves, ranked second, are considered the primary candidates for alpha status. Below them are the delta ($\delta$) wolves, including scouts, sentinels, elders, hunters, and caretakers, who preside over the omega ($\omega$) wolves, often viewed as the scapegoats of the pack. Grey wolves hunt collaboratively, starting with tracking, chasing, and approaching their prey. They encircle, pursue, and harass the prey until it weakens, then launch an attack. This hunting behavior is mathematically modeled in GWO, where the alpha, beta, and delta wolves correspond to the best, secondbest, and thirdbest solutions, respectively, while the remaining solutions are considered omega wolves. The model encompasses several phases:
(1) Encircling the prey:
where, $\vec{A}$ and $\vec{C}$ represent coefficient vectors, $t$ is the present iteration, $\overrightarrow{X_P}$ is prey's position, $\vec{X}$ is the grey wolf position, $\overrightarrow{r_1}$ and $\overrightarrow{r_2}$ are random vectors within $[ 0,1]$, and $\vec{a}$ decreases linearly from 2 to 0 in the course of iterations.
(2) Hunting:
Alpha, beta, and delta wolves update their positions first, assuming better knowledge of the prey's location:
(3) Attacking the prey (Exploitation):
The mathematical representation of a prey attack in the GWO is characterized by the gradual decrease of $a$ from 2 to 0 over the iterations. The attack phase is initiated when the value of $A<1$, with $\vec{A}$ being a random value within the range of $[2 \vec{a}, 2 \vec{a}]$.
(4) Searching for prey (Exploration)
When the value of $\boldsymbol{A}>\mathbf{1}$ in the GWO, the wolves are prompted to explore for more suitable prey. This exploration phase is influenced by the parameter $\overrightarrow{\boldsymbol{C}}$, which is a random value within the range $[ 0,2]$. The role of $C$ is to introduce stochasticity into the behavior of the grey wolves, either emphasizing $(\mathrm{C}>1)$ or deemphasizing $(\mathrm{C}<1)$ their predatory attack. This mechanism allows for a balanced exploration of the search space, mimicking the adaptive hunting behavior of grey wolves in the wild. The pseudocode detailing the GWO process is provided in Table 16.
GWO has been applied to solve optimization problems in a diverse range of fields. In medicine, it has been used for various optimization tasks [143], [144]. In manufacturing, it has been employed for operation sequencing [145]. The algorithm has also been adapted for use in unmanned aerial vehicle navigation [146], multiagent systems [147], and robotics. For insights into the variety of GWO applications specifically in smart vehicle navigation, one can refer to Table 17. Gul et al. [148] proposed a hybrid algorithm combining PSO with GWO. This hybrid PSOGWO algorithm was designed to improve path length and ensure smoother trajectories for mobile robots. Furthermore, a mutation operator was introduced to refine the trajectory generated by the PSOGWO algorithm (Figure 10) [149]. To address the challenge of local minima in GWO, Dong et al. [150] suggested a modified position update mechanism specifically tailored for robot path planning. This modification aimed to enhance the algorithm's ability to navigate complex environments more effectively. In addition, Kumar et al. [151] developed a Variable Weight GWO, aimed at increasing speed and reducing the distance of planned routes for mobile robots. This variant of GWO adjusts the algorithm's parameters dynamically to optimize performance in realtime navigation tasks.
Grey Wolf Optimization  
1  Initialize the prey wolf population $X_i(i=1,2, \ldots, n)$  
2  Initialize a, A, and C  
3  Calculate the fitness of each search agent  
4  $X_\alpha$ = the best search agent  
5  $X_\beta$= the second best search agent  
6  $X_\delta$= the third best search agent  
7  while (t < maximum number of iterations)  
8  for each search agent  
9  Update the position of the current search agent  
10  end for  
11  Update a, A, and C  
12  Calculate the fitness of all search agent  
13  Update $X_\alpha, X_\beta, X_\delta$  
14  t = t + 1  
15  end while  
16  return $X_\alpha$ 
Type  Population Size  Iterations  Type of Vehicle  Type of Obstacles  Type of Map  Software  Remark  Ref. 
IGWO  30  100  Single  Static  Geometrical  MATLAB R2018b  · The algorithm is tested on 20 benchmark functions.  [150] 
VMGWO  20,25 (For map 1) 25,30 (For map2)  35,20 (For map 1) 30,40(For map 2)  Single  Static  20×10×10 Geometrical (3D map)  MATLAB 2018a  · Execution speed outperformed GWO.  [151] 
HPSOGWOEA 

 Single  Static  Geometrical
 MATLAB R2017a  · Mutation operator from evolution algorithm is introduced to solve path dafty, length and smoothness.  [148] 
HPSOGWO  100  500  Single  Static and Dynamic  100×100 Geometrical (see Figure 10)  MATLAB R2019b  · Frequencybased function is introduced to modify the search process of GWO.  [149] 
HWGO  20  100  Single 

 MATLAB R2019  · Proposed algorithm is implemented in tuning parameters of fractional order PID controller.  [152] 
The MVO is an innovative populationbased algorithm inspired by the multiverse theory in physics, which posits the existence of multiple universes interacting within a multiverse [153]. This algorithm incorporates three key cosmological concepts: white holes, black holes, and wormholes [154], [155]. In astrophysics, the Big Bang, considered the origin of the universe, is likened to a white hole [156], a concept representing regions emitting energy and matter. Conversely, black holes are known for their intense gravitational force, which attracts and absorbs matter, including light beams [157]. Wormholes are theorized as spacetime passages that link different parts of a universe or even connect separate universes. The concept of universe expansion, driven by the inflation rate or eternal inflation [158], is also integrated into this model.
In MVO, these astronomical phenomena are mathematically modeled to facilitate exploration (white holes), exploitation (wormholes), and local search (black holes) in the search space. Each 'universe' in MVO represents a potential solution, with the objects within a universe analogous to variables of that solution. The fitness value of a solution is equated to its inflation rate. Universes in MVO adhere to the following principles:
(1) High inflation rate leads to a high chance of having a white hole.
(2) Low inflation rate leads to a low chance of having a black hole.
(3) High inflation rate universes are likely to pass objects through white holes.
(4) Lower inflation rate universes have a tendency to get objects through black holes.
(5) Regardless of inflation rate, objects in all universes can move randomly towards an optimal universe via wormholes.
Assuming that $U$ is a set of universes, where $n$ is the number of possible solutions (universes) and d represents the number of parameters or variables:
then each parameter can be represented as below:
$x_i^j$ is the $j$th parameter of the $i$th universe. $N I(U i)$ indicates the normalized inflation rate of the ith universe. $U i$ is the $i$th universe. $r 1$ is a random value in $[ 0,1].\quad x_k^j$ denotes the $j$th variable of $k$th universe chosen by a roulette wheel selection mechanism. Roulette wheel, which depends on normalized inflation rate, selects a universe and determines white holes for it. Through this mechanism, exploration is done. For exploitation, each universe is assumed to have wormholes connecting it to the best universe, facilitating the exchange of objects (parameters). The update for each parameter is thus given by:
where, $X_j$ represents the $j$th parameter of the best universe formed so far, TDR (Travelling Distance Rate) and WEP (Wormhole Existence Probability) are coefficients, $l b_j$ and $u b_j$ denote the lower and upper bounds of the $j$th parameter respectively, $x_i^j$ represents the $j$th parameter of the $i$th universe and $r$2, $r$3, $r$4 are random values in $[ 0,1]$. The formulas for the coefficients are as follows:
where, $\min$ and $\max$ are the minimum and maximum respectively, $l$ tells the current iteration and $L$ is the maximum iterations.
where, $p$ denotes the exploitation accuracy over the iterations, with higher values leading to more accurate exploitation/local search. The MultiVerse Optimizer algorithm is detailed in the pseudocode shown in Table 18.
MultiVerse Optimizer  
1  Create random universes (U)  
2  Initialize WEP, TDR, and Best Universe  
3  SU $\leftarrow$ Sorted universes  
4  NI $\leftarrow$ Normalize inflation rate (fitness) of the universes  
5  while the end criterion is not satisfied do  
6  Evaluate the fitness of all universes  
7  for each universe indexed by i do  
8  Update WEP and TDR  
9  Black hole index $\leftarrow$ i  
10  for each object indexed by j do  
11  r_{1}_{ }$\leftarrow$ random([0,1])  
12  if r_{1 }< NI( U_{i}) then  
13  White hole index $\leftarrow$ RouletteWheelSelection(NI)  
14  U(Black hole index, j) $\leftarrow$ SU(White hole index, j)  
15  end if  
16  r_{2}_{ }$\leftarrow$ random([0,1])  
17  if r_{2}< WEP then  
18  r_{3}_{ }$\leftarrow$ random([0,1])  
19  r_{4}_{ }$\leftarrow$ random([0,1])  
20  if r_{3}_{ }< 0.5 then  
21  U(i,j) $\leftarrow$ Best Universe(j) + TDR× ((ub(j) – lb(j))×r_{4} + lb(j))  
22  else  
23  U(i,j) $\leftarrow$ = Best Universe(j)  TDR$\times$ ((ub(j) – lb(j)) $\times r_4$+ lb(j))  
24  end if  
25  end if  
26  end for  
27  end for  
28  end while  
29  return Best Universe 
The MVO has been successfully implemented in a variety of domains to address complex optimization problems. Its applications range from project scheduling [159] to enhancing kernel extreme learning machines for medical diagnosis [160], modeling solar radiation [161], and solving economic dispatch problems in power systems [162]. In the field of robotics, MVO has demonstrated its versatility and effectiveness. It has been used for path planning in threedimensional search spaces [163], tuning PID controllers [164], [165], planning the navigation of quadrotors [166], and devising routes for mobile robots [167]. However, the application of MVO in the navigation of smart vehicles is relatively nascent, with only a handful of researchers exploring its potential in this area, as detailed in Table 19.
Type  Type of Vehicle  Type of Obstacles  Type of Map  Software  Remark  Ref. 
Evolutionary MultiVerse Optimizer  Single 

 Python (3.7)  · Parameters of each solution are the weights and bias of implemented MultiLayer perceptron Network.  [167] 
MMVO  Single  Static  400×400 Geometrical (2D and 3D) 
 · 3D path planning in a modelled 3D environment is examined.  [163] 
The BA, inspired by the echolocation behavior of microbats, was developed by Yang [168]. Microbats emit short, loud bursts of sound at frequencies ranging from 25 kHz to 150 kHz and listen to the echoes bouncing back from nearby objects. This echolocation ability enables them to pinpoint the location of objects. Typically, the frequency of pulse emission and the loudness of the sound increase during prey search and decrease upon prey discovery. The BA is formulated based on idealized rules derived from this echolocation behavior:
(1) Bats estimate distance based on the echo of their sounds and can differentiate prey from other objects.
(2) Bats move randomly with a velocity $v_i$ towards a prey at position $x_i$, emitting sounds at a frequency $f_{\min }$, with wavelength $\lambda$ and loudness $A_0$.
(3) The loudness is assumed to be between a large positive value and its minimum value is defined as $A_{\min }$.
For the $\mathrm{BA}$, the frequency which is assumed to be within $\left[ 0, f_{\max }\right]$, the new velocity and position are defined below.
where, $x_*$ represents the current global best solution from all $n$ bats and $\beta$ is a vector within $[ 0,1]$. The loudness $(A)$, starts as any positive number, typically within the range $[ 1,2]$, and is then updated by a constant $\alpha \in[ 0,1]$ as shown in Eq. (36). $A=0$ when a solution is found. The rate of pulse emission $\boldsymbol{r}_i^0 \in[ 0,1]$ is controlled by a constant $\gamma$, which can be the same as $\alpha$.
For local search, new solutions for each bat are generated using a random walk strategy, once a solution is selected from the current best ones.
where, $\epsilon \in[1,1]$ denotes a random number in $[1,1]$ and $A^t$ represents average loudness of all bats at time $t$. The implementation of the Bat Algorithm is depicted in Table 20.
The implementation of the BA in the domain of mobile robot planning and navigation has yielded impressive results, as evidenced by numerous studies in the field. Table 21 provides an overview of various research efforts that have proposed enhancements to the BA, detailing the variables considered in each study. Among these advancements, Ajeil et al. [169] introduced a Modified Frequency Bat Algorithm (MFB) specifically designed to optimize the shortest path finding from a start to an end point, compared to the standard Bat Algorithm. This novel algorithm integrates obstacle detection and avoidance techniques, utilizing sensor data to dynamically plan new paths in response to moving obstacles. The algorithm's performance was evaluated through simulations conducted in a grid mat environment. The results demonstrated that the MFB outperformed the standard BA in terms of efficiency and effectiveness in pathfinding, particularly in environments with dynamic obstacles.
Bat Algorithm  
1  Objective function f(x), $\mathrm{x}=\left(x_1, \ldots . ., x_d\right)^{\mathrm{T}}$  
2  Initialize the bat population $x_i(i=1,2, \ldots, n)$ and $v_i$  
3  Define pulse frequency $f_{\mathrm{i}}$ at $x_{\mathrm{i}}$  
4  Initialize pulse rates $r_{\mathrm{i}}$ and the loudness $A_i$  
5  while (t < Max number of iterations) do  
6  Generate new solutions by adjusting frequency,  
7  and updating velocities and locations/solutions (Eqs. (2) (4))  
8  if (rand > $r_i$) then  
9  Select a solution among the best solutions  
10  Generate a local solution around by flying randomly  
11  end if  
12  Generate a new solution by flying randomly  
13  if (rand < A_{i }& f (x_{i}) < f (x_{∗})) then  
14  Accept the new solutions  
15  Increase $r_i$ and reduce A_{i}  
16  end if  
17  Rank the bats and find the current best $x_*$  
18  end while  
19  Postprocess results and visualization 
Type  Population Size  A(0)  r(0)  $\alpha$  $\gamma$  $\boldsymbol{f}_{\min }$  $f_{\max }$  Type of Vehicle  Type of Obstacles  Type of Map  Software  Remark  Ref. 
MFB  5  1  0.5  0.98  0.8  0  10  Single  Dynamic  12×12 Grid  MATLAB  · Obstacle detection and avoidance method is integrated in the algorithm.  [169] 
Type1 FLSBA  20  Single  Static  · BA modifies Type1 FLS to generate optimum trajectory. · Proposed method aims at obtaining the least mean square error in trajectory tracking.  [170] 
The TS [171] is a method of optimization that utilizes constraintbased techniques to avoid local minima in a search space. It incorporates flexible memory cycles to intensify and diversify local search patterns, thereby facilitating the discovery of optimal solutions. During the exploration process, Tabu Search meticulously tracks information about both the current solution and those previously explored [172]. The algorithm operates by employing neighborhood search methods to progress from a current solution x to a feasible solution x'within the neighborhood of x. This iterative process continues until a predetermined stopping criterion is met. One of the key features of TS is its use of memory structures to record visited solutions, preventing the algorithm from getting trapped in local minima and encouraging the exploration of unvisited areas in the search space [173]. The memory structures, also known as the tabu list, contain a set of recently visited solutions that are temporarily banned from reconsideration, typically for n iterations (where n, the tabu tenure, specifies the length of the list). The procedure for implementing this algorithm is outlined in Table 22.
TabuSearch Algorithm  
1  Generate initial solution $\left(x_0\right)$  
2  Initialize tabu list (TL $\leftarrow$ [ ])  
3  Current solution(x) $\leftarrow$ initial solution $\left(x_0\right)$  
4  Best solution(x_{best}) $\leftarrow$ current solution(x)  
5  Define maximum iteration (ITR_{max})  
6  Iteration (ITR) $\leftarrow$0  
7  while (ITR < ITR_{max}) do  
8  S_{N} $\leftarrow$ Get neighbours of $\chi_{\text {best }}$  
9  for S $\in$S_{N} do  
10  if $S \notin T L \& \&$ fitness $(S)>$ fitness $\left(x_{\text {best }}\right)$ then  
11  $x_{\text {best }}=S$  
12  end if  
13  end for  
14  add S to TL  
15  end while  
16  return $x_{\text {best }}$ 
The application of the TS in mobile robot navigation is still relatively underexplored. However, some studies, as indicated in Table 23, have developed new hybrids that enhance the basic algorithm's efficiency in path planning. Xing et al. [174] introduced a novel TS tailored for routing multiple AGVs in warehouse settings. Châari et al. [175] developed a TSbased system model for global path planning on grid maps. Kumar et al. [176] modified the TS method for navigating mobile robots in complex environments. Khaksar et al. [177] integrated TS rules into a fuzzy controller designed to address online navigation challenges. Panda et al. [178] proposed a hybrid algorithm combining TS and PSO to optimize pathfinding for AMRs.
Type  Iteration Number  Type of Vehicle  Type of Obstacles  Type of Map  Software  Remark  Ref. 
TSPATH  30  Single  Static  Grid (see Figure 11)  Designed a C++ simulation model  · The effectiveness of the tabu search for the global path planning problem is investigated.  [175] 
PSOTABU  31  Multiple  Static  Geometrical  Designed a C simulation environment  · In terms of solution quality and computation time, PSOTABU, outperforms the basic PSO and TABU search algorithm.  [178] 
Modified Tabu Search 
 Single (KheperaIII robot)  Static  10 × 8cm Grid (simulation) and 400 × 300cm^{2} Geometrical (experimental)  VREP simulation software  · Algorithm is verified in both simulation and experimental platform. · Deviation between the simulation and experimental results is about 4%.  [176] 
ANFIS 
 Single  Static  10×9, 10×10 and 10×14 Geometrical  MATLAB R2010b  · Heuristic rules of Tabu search is infused in fuzzy controller. · Fuzzy planner can handle online navigation task.  [177] 
Tabu Search  9  Single  Static  10×10 Grid  MATLAB  · Algorithm generates trajectories to multiple end points using the shortest possible path.  [179] 
GSTIACA 
 Multiple  Dynamic  Grid  ePuck architecture in the Webots simulation environment.  · Realtime simulation shows concurrent navigation and map building in dynamic environments.  [180] 
TS/FA  5  7  Single  Static  561×380 px^{2} and 433×430 px^{2} Grid  MATLAB 2014a and Vrep simulator  · TS/FA is an offline hybrid algorithm. · Bezier curve is used to smoothen generated path.  [181] 
An important aspect to consider when evaluating the metaheuristic algorithms discussed in this work is their performance on various benchmark functions. Benchmark tests, comprising mathematical functions, are instrumental in assessing the algorithms' ability to find solutions in a given dimension d that lead to global optima [182]. These benchmark functions, as cataloged in Table 24, can be categorized into unimodal, multimodal, or combinatory types, which blend unimodal and multimodal characteristics. Unimodal functions are designed to lead to a single optimum solution, whereas multimodal functions yield multiple optimum solutions.
The significance of metaheuristic algorithms in research, particularly in addressing complex realworld problems, is underscored by several advantages noted by Gholizadeh and Barati [183]. Their high efficiency and flexibility are key attributes that make these algorithms increasingly valuable in solving complex challenges. Additionally, the popularity and impact of a metaheuristic algorithm can often be gauged by the number of citations it receives in academic literature. Table 25 provides a ranking of the algorithms utilized in this study based on their citation count.
Selecting the appropriate algorithm for path planning in the application of smart vehicles is a critical decision. The choice of algorithm significantly depends on the specific requirements of the smart vehicle's mission. For instance, the algorithmic needs for rescue missions and urgent tasks differ markedly from those required for surveillance or logistics operations. A key consideration in this decisionmaking process is the balance between exploration and exploitation, which are inherent tradeoffs in optimization problems. Exploration entails an efficient search of the solution space, aiming to circumvent local optima in pursuit of global solutions. While this process is thorough, it often results in slower convergence speeds. On the other hand, exploitation focuses on rapidly converging to a solution, which enhances the speed but raises the risk of becoming trapped in local optima. Therefore, when choosing an algorithm for path planning in a particular context, it's crucial to weigh these factors: convergence speed and the ability to identify global optima. The chosen algorithm should align with the specific objectives of the task at hand, whether it requires rapid response times or a more comprehensive search of the solution space.
Function Name  Equation  Objective Value  Modality 
Spherical  $\sum_{i=1}^{i=d} x_i{ }^2$  0  Unimodal 
Schwefel 2.22  $\sum_{i=1}^{i=d}\leftx_i\right+\prod_{i=1}^{i=n}\leftx_i\right$  0  Multimodal 
Schwefel 2.21  $\max _{1 \leq i \leq n}\leftx_i\right$  0  Unimodal 
Rosenbrock  $\sum_{i=1}^{i=d} 100\left(x_{i+1}x_i{ }^2\right)^2+\left(1x_i\right)^2$  0  Multimodal 
Step  $\sum_{i=1}^d\leftx_i+0.5\right^2$  0  Unimodal 
Schwefel  $418.9829 d\sum_{i=1}^{i=d}x_i \sin \sqrt{\leftx_i\right}$  0  Multimodal 
Rastrigin  $10 * d+\sum_{i=1}^{i=d} x_i{ }^210 \cos \left(2 \pi x_i\right)$  0  Multimodal 
Auckley  $20 * \exp \sqrt{\frac{1}{d} \sum_{i=1}^{i=d} x_i^2}\exp \left(\frac{1}{d} \sum_{i=1}^{i=d} \cos \left(2 \pi x_i\right)\right)+e$  0  Multimodal 
Griewank  $\sum_{i=1}^d \frac{x_i^2}{4000}\prod_{i=1}^d \cos \left(\frac{x_i}{\sqrt{i}}\right)+1$  0  Multimodal 
Rank  Year  Algorithm  Number of Citations 
1  1995  Particle Swarm Optimization [60]  75041 
2  1975  Genetic Algorithm (GA) [28]  74165 
3  1992  Ant Colony Optimization [45], [184]  5685 (from 1992) 15447 (from 2006) 
4  2014  Grey Wolf Optimization [142]  9881 
5  1986  Tabu Search Algorithm [171], [185]  6320 (from 1986) 9716 (from 1989) 
6  2005  Artificial Bee Colony [186]  8060 
7  2009  Cuckoo Search Algorithm [103]  7217 
8  2016  Whale Optimization Algorithm [122]  6649 
9  2008  Firefly Algorithm [89]  6224 
10  2016  MultiVerse Optimizer [153]  1656 
In the realm of computational time and path optimization for smart vehicle applications, various metaheuristic algorithms exhibit distinct strengths. A hybrid CSA, for instance, has been shown to achieve optimal paths more rapidly than both PSO and GA [187]. PSO, on the other hand, outperforms the BA in terms of convergence when tuning omnidirectional mobile robots [188], while a hybrid PSO variant provides shorter paths in less time compared to modified BA and ABC [189].
CSA has been proven to outperform BA in finding an optimal path [121]. When it comes to covering the search space effectively, a multiobjective GWO demonstrates superior performance over multiobjective PSO and GA [190]. ACO outshines GA in obtaining optimal paths [191], and a hybrid ACOPSO algorithm yields more optimal robot paths than ACO alone [192]. Moreover, ABC is noted for achieving shorter paths than PSO, as evidenced by simulation results [88]. These findings underscore the importance of selecting appropriate algorithms for specific tasks in smart vehicle applications, as outlined in Table 26, where the bestfitted algorithms for various industrial activities such as logistics, material handling, and surveillance are detailed.
Tasks 
Vehicle scheduling 
Warehouse material handling 
Unmanned ground vehicle (military) 
Search operation 
Security and surveillance 
Cleaning and disinfection operation 
Ecommerce delivery 
The simulation of metaheuristic algorithms is carried out on various platforms, each offering unique mathematical and graphical functionalities. While some platforms provide a basic graphical representation of simulation outputs, others, like GazeboROS, offer a more immersive experience with 3D animated environments for visualizing simulated outcomes. Among the most popular choices for researchers is MATLAB, known for its comprehensive inbuilt functions that greatly facilitate the programming of metaheuristic algorithms. However, some researchers prefer custombuilt solutions, creating their own simulation platforms using fundamental programming languages such as C and C++ [175], [178]. Table 27 presents a compilation of the languages and simulation platforms commonly employed in this field of research. Additionally, a quantitative comparison of some of these simulators has been conducted by Farley et al. [193], providing insights into their respective capabilities and suitability for different types of simulations.
Platform/Programming Language  Remarks  Ref 
Python  Highlevel userfriendly programming language.  [194] 
C, C++  Highlevel programming language for generalpurpose programming.  [195] 
MATLAB  Modeling and simulation software built by Mathworks.  [196] 
CoppeliaSim (formally VREP)  Creates room for importing personally designed robots. Robotic models can be controlled using C, python, or MATLAB scripts including ROS node.  [197] 
ROS with Movelt  Movelt is the primary simulator in ROS for motion planning, 3D perception, manipulation and control.  [198] 
GazeboSim  Offers various libraries and cloud services for robot simulation.  [199] 
Webots  Offers a complete development environment to simulate robots and mechanical systems.  [200] 
MORSE  Modular Open Robot Simulator Engine based on Blender game engine. A 3D simulator that offers a set of standard sensors, actuators and robotic bases.  [201] 
USARsim  Urban Search and Rescue simulator for multirobot purposes.  [202] 
3. Conclusion
This study provides a comprehensive review of various metaheuristic algorithms and their hybrids, as developed by researchers to address path planning and navigation challenges in smart vehicles. The classification of these algorithms is primarily into two categories: populationbased (encompassing evolutionary, swarm intelligence, and natureinspired algorithms) and trajectorybased algorithms. A detailed description of each algorithm is provided, followed by reviews of recent articles spanning the last 13 years (2010  2023), with a majority of the studies concentrated between 2017 and 2023. Key parameters considered in this review include the type of vehicle (single or multiple robots), obstacle nature (static or dynamic), map type (topological, geometrical, or grid map), and the simulation platforms used for analysis.
The analysis also focuses on computational time and the efficiency of these algorithms in finding the shortest optimum path. Various tasks performed by smart vehicles are enumerated, highlighting the diverse applications. A notable observation is that navigation for smart vehicles remains an ongoing challenge, particularly in optimizing path length and reducing travel time. Researchers have made significant improvements and updates to these algorithms to address observed anomalies [118], [150]. A prominent trend is the development of hybrids between different algorithms, either to finetune parameters [136] or to combine advantages for enhanced robustness [119]. Path smoothing has been a crucial consideration in some studies, with techniques like cubic polynomials [85], [91], Bezier curves [78], [181], Bspline curves [55], and Cubic Ferguson splines [88] being used to generate smooth paths.
While most reviewed studies focused on static environments, there is a growing need to explore the navigation of smart vehicles in dynamic settings, considering moving obstacles and human interactions. One case study demonstrates the potential of using object detection sensors and refining data through neural networks with finely tuned weights for path planning in dynamic environments [167]. Additionally, combining reinforcement learning with metaheuristic algorithms could offer novel solutions for path planning challenges. For instance, incorporating reinforcement learning agents to balance exploration and exploitation in populationbased metaheuristic algorithms could significantly enhance navigation path planning.
The data used to support the research findings are available from the corresponding author upon request.
The authors declare no conflict of interest.